Cosmic Rays in Star-Forming Galaxies
Dissertation
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Brian Cameron Lacki
Graduate Program in Astronomy
The Ohio State University 2011
Dissertation Committee: Professor Todd Thompson, Advisor Professor John Beacom Professor Christopher Kochanek Copyright by
Brian Cameron Lacki
2011 ABSTRACT
Cosmic rays (CRs) are high energy particles that are found wherever in the Universe star formation is occurring. I investigate several problems in the propagation of CRs in star-forming galaxies. By applying analytic models and numerically solving the “leaky box” differential equation, I calculate the population of primary and secondary CR protons, electrons and positrons in model star-forming galaxies and their nonthermal emission.
Observations show that the synchrotron radio emission of star-forming galaxies grows linearly with the infrared emission from dust-obscured young stars; this is the FIR-radio correlation (FRC). To explain the correlation, I constructed one-zone models of galaxies over the dynamic range of the FRC. I found that the FRC is caused by conspiracies of several factors, including CR escape from galaxies, ultraviolet (UV) dust opacity, non-synchrotron cooling, and secondary electrons and positrons generated by CR protons. The conspiracies have great implications for the evolution of the FRC at high redshift, preserving it and allowing variations in the
FIR-radio ratio for submillimeter galaxies.
ii Recent gamma-ray observations of M82 and NGC 253 indicate that CR protons lose much of their energy to collisions in these galaxies’ dense gas, where they generate unstable pions that decay into gamma rays and secondary particles. The ratio of gamma-ray to radio luminosity indicates that secondary electrons mostly do not cool by synchrotron emission, supporting a conspiracy origin of the FRC.
I also compare the intensities of the diffuse cosmic gamma-ray background to the X-ray and radio backgrounds. From this comparison, I find that Inverse
Compton is a minority of the X-ray background, and that the radio background is probably not from starbursts.
Finally, I modeled the nonthermal X-ray emission from starburst galaxies, both synchrotron from TeV electrons and Inverse Compton from GeV electrons. The synchrotron emission is enhanced by γγ pair production in the intense infrared radiation of starbursts. Synchrotron and Inverse Compton emission make up 1 -
10% of the observed diffuse hard X-ray emission observed in starburst galaxies.
iii For science! And my parents.
iv ACKNOWLEDGMENTS
I’ll start off by thanking my advisor and mentor, Todd Thompson. He has been both energetic and approachable. Most of what I’ve learned about being a theoretical astrophysicist, I have learned from him. Todd has shared with me many insights into astronomy and physics and presentation. He has also encouraged me when I set out on my own rogue research projects.
John Beacom has served as my other mentor through my years here. Much of the rest I have learned about being a theoretical astrophysicist I learned from him, both on physics and presentation. We’ve had many interesting discussions and ideas over the years. Our collaboration has led to one published paper, on dark matter haloes around primordial black holes, and one work in preparation.
Now is my opportunity to thank my other co-authors of the papers compiled in this dissertation: Eliot Quataert, Eli Waxman, and Avi Loeb. They have provided helpful comments on presentation, encouragement, and useful insight into the propagation of cosmic rays, especially in using analytic calculations.
While working on the papers that formed the core of this dissertation, a number of additional people provided critical comments that contributed to our works:
v Rainer Beck, Boaz Katz, Matthew Kistler, Michael Micha lowski, and Diego Torres.
I would also like to acknowledge the interesting and informative discussions and exchanges I’ve had on cosmic rays with Marco Ajello, Roland Crocker, Shane Davis,
Chuck Dermer, Liu Fan, Gary Ferland, Shunsaku Horiuchi, Christopher McKee,
Norm Murray, Padelis Papadopoulos, Vasiliki Pavlidou, Troy Porter, Jack Singal,
Aristotle Socrates, Andy Strong, Meng Su, Heinrich V¨olk, Peter Williams, and Dong
Zhang. More generally, I would like to thank the events and institutions that have hosted me as a speaker: CCAPP, Berkeley, the IAS, CITA, the “Infrared Emission,
Interstellar Medium and Star Formation” conference in Heidelberg, and the “Cosmic
Ray Interactions: Bridging High and Low Energy Astrophysics” workshop in Leiden.
Not all of my work has been on galactic cosmic rays. I acknowledge Bernd
Br¨ugmann, who offered me my first research project at Penn State. Jane Charlton was my mentor during my undergraduate years, when I worked on quasar absorption lines. She gave me both experience handling real data and a taste of theory, for which
I thank her. Christopher Kochanek guided me through my first graduate project here, and has served on my candidacy and dissertation committees. I thank him for his work, his patience, and his acting as a soundboard for my Cherenkov telescope paper. I want to acknowledge Kris Stanek for his work on the difference imaging paper as well. Scott Gaudi also provided much advice, ideas, and encouragement for the Cherenkov telescope paper, for which I am grateful. I acknowledge David
Weinberg who served on my candidacy committee as well and attended my defense.
vi Finally, I would like to thank Kohta Murase, for his work on the circumstellar shell transient paper that I was a co-author on.
I have relied on the technical assistance of many in getting my code working and verifying that it worked properly. I would like to thank the GALPROP team for making their code freely available, and codifying the pionic secondary production cross sections. Igor Moskalenko in particular shared the total Galactic gamma-ray luminosity, which was an important calibration for our efforts. I also thank the
Kamae et al. (2006) team for making public their own pionic secondary production cross sections, which include neutrinos. I gratefully acknowledge recent technical discussions on the spectrum of γγ pair production e± with Felix Aharonian, Dmitry
Khangulyan, and Markus B¨ottcher. Closer to home, I relied on the assistance of the OSU Astronomy Department’s Computing Support staff. Thanks especially to David Will who maintained Condor. Rick Pogge provided valuable assistance with getting my tables working with this LaTex template. Finally, I would like to acknowledge those tools that I relied on so much: the Astrophysical Data System and arXiv. They are both free to use and indispensable.
On the administrative side, I would like to thank the staff of the Astronomy
Department, including Kristy Krehnovi, who assisted me with travel and other administrative matters during my years. I acknowledge Evan Sugarbaker, who served as the Graduate Faculty Representative during my defense.
vii In addition to my GTA and GRA positions, I acknowledge my Elizabeth Clay
Howald Presidential Fellowship, which supported my last year of work here. I would like to thank the NRAO and IAS who will be hosting me at my next position as a
Jansky Fellow. I also am grateful to CITA and the Hubble Fellowship who offered me positions.
To my fellow grad students: thanks for giving me a chance to be Sir Isaac. And on a different note, I hope you put those rainbow sprinkles to good use.
Finally, I’d like to thank my parents, who have put up with this dream of mine for a good 20 years now.
viii VITA
July 3, 1984 ...... Born – Dover, New Jersey, USA
2006 ...... B.S., Astronomy & Astrophysics and Physics The Pennsylvania State University
2006 – 2008 ...... Graduate Teaching Associate, The Ohio State University
2008 – 2011 ...... Graduate Research Associate, The Ohio State University
2010 – 2011 ...... Elizabeth Clay Howald Presidential Fellow, The Ohio State University
PUBLICATIONS
Research Publications 1. Brian C. Lacki, Christopher S. Kochanek, Krzysztof Z. Stanek, Naohisa Inada, and Masamune Oguri, “Difference Imaging of Lensed Quasar Candidates in the Sloan Digital Sky Survey Supernova Survey Region”, ApJ, 698, 428, (2009).
2. Brian C. Lacki and Jane C. Charlton, “The z = 0.0777 CIII absorber towards PHL 1811 as a case study of a low-redshift weak metal line absorber”, MNRAS, 403, 1556, (2010).
3. Brian C. Lacki, “The End of the Rainbow: What Can We Say About the Extragalactic Sub-Megahertz Radio Sky?”, MNRAS, 406, 863, (2010).
4. Brian C. Lacki, Todd A. Thompson, and Eliot Quataert, “The Physics of the FIR-Radio Correlation: I. Calorimetry, Conspiracy, and Implications”, ApJ, 717, 1, (2010).
ix 5. Brian C. Lacki and Todd A. Thompson, “The Physics of the FIR-Radio Correlation: II. Synchrotron Emission as a Star-Formation Tracer in High-Redshift Galaxies”, ApJ, 717, 196, (2010).
6. Brian C. Lacki and John F. Beacom, “Primordial Black Holes as Dark Matter: All or Nothing”, ApJL, 720, L67, (2010).
7. Brian C. Lacki, “The Gamma-Ray Background Constrains the Origins of the Radio and X-Ray Backgrounds”, ApJL, 729, L1, (2011).
8. Brian C. Lacki, Todd A. Thompson, Eliot Quataert, Abraham Loeb, and Eli Waxman, “On The GeV & TeV Detections of the Starburst Galaxies M82 & NGC 253”, ApJ, 734, 107, (2011).
FIELDS OF STUDY
Major Field: Astronomy
x Table of Contents
Abstract...... ii
Dedication...... iv
Acknowledgments...... v
Vita ...... ix
ListofTables ...... xvii
ListofFigures...... xx
Chapter 1 Introduction ...... 1
1.1 Cosmicrays...... 1
1.1.1 TheOriginofCRs ...... 4
1.1.2 Understanding CR Propagation and Emission ...... 8
1.1.3 TheEffectsofCRs ...... 13
1.2 What Did We Know on CRs in Star-Forming Galaxies? ...... 15
1.2.1 Previous observations of CRs in other galaxies ...... 15
1.2.2 PrevioustheoriesoftheFRC ...... 21
1.2.3 Previous models of CR propagation in other galaxies . . . . . 23
1.2.4 The Cosmic Backgrounds from Star-Formation Cosmic Rays . 25
1.3 TheWorkinthisDissertation ...... 27
Chapter 2 The Physics of the Far-Infrared-Radio Correlation. I. Calorimetry, Conspiracy, and Implications ...... 34
xi 2.1 Introduction...... 34
2.2 Procedure ...... 41
2.2.1 Primary CR Injection Rates ...... 43
2.2.2 Environmental Conditions ...... 45
2.2.3 Observables and Constraints ...... 53
2.3 Review of Physical Effects of Parameters ...... 59
2.3.1 Injection Parameters: ξ, η, δ, and p ...... 60
2.3.2 MagneticField ...... 62
2.3.3 EffectiveDensity ...... 64
2.3.4 The Schmidt Law and the Photon Energy Density ...... 65
2.4 Results...... 67
2.4.1 StandardModel...... 67
2.4.2 Degeneracy in the Standard Model ...... 68
2.4.3 General Features of the Particle Spectra ...... 70
2.5 Discussion...... 72
2.5.1 Is Calorimetry Correct? ...... 72
2.5.2 What Causes the FIR-Radio Correlation? ...... 76
2.5.3 The FIR-Radio Correlation at Other Frequencies ...... 81
2.5.4 The Spectral Slope α ...... 84
2.5.5 The γ-Ray (and Neutrino) Luminosities of Starbursts . . . . . 88
2.5.6 The Dynamical Importance of Cosmic Ray Pressure ...... 95
2.6 SummaryandFutureImprovements...... 97
2.7 Variants ...... 103
2.7.1 B ρa ...... 104 ∝ xii 2.7.2 Winds ...... 105
2.7.3 Other Disk Scale Heights ...... 107
2.7.4 Optically Thick Galaxies ...... 110
2.7.5 UB = Uph ...... 112
2.7.6 FastDiffusiveEscape...... 113
2.7.7 Varying Escape Times ...... 114
2.7.8 MultipleEffectsandOtherVariants...... 117
Chapter 3 The Physics of the FIR-Radio Correlation: II. Synchrotron Emission as a Star-Formation Tracer in High-Redshift Galaxies . 151
3.1 Introduction...... 151
3.2 Theory...... 153
3.3 ProcedureandAssumptions ...... 159
3.4 Results&Discussion ...... 164
3.4.1 The Evolving FIR-Radio Correlation ...... 164
3.4.2 The Radio Excess (or Deficit) of Puffy Starbursts: Submillimeter Galaxies...... 169
3.4.3 Synchrotron Radio Emission as a Star-Formation Tracer . . . 177
3.4.4 SMGs, The Radio Background, and Radio Source Counts . . . 181
3.5 SummaryandCavaets ...... 185
3.6 Derivation of Radio Suppression from CMB ...... 190
Chapter 4 On The GeV & TeV Detections of the Starburst Galaxies M82 & NGC 253 ...... 205
4.1 Introduction...... 205
4.2 γ-rayDetections...... 208
xiii 4.2.1 Fermi andTeVdetections ...... 208
4.2.2 M82 and NGC 253 Gamma-Ray Luminosities ...... 210
4.3 InterpretationasPionicEmission ...... 212
4.3.1 Motivation for Proton Calorimetry ...... 212
4.3.2 Measuring The Calorimetry Fraction ...... 214
4.3.3 Primary Uncertainties in Fcal ...... 219
4.3.4 Assessing Proton Calorimetry ...... 223
4.4 Other Sources of γ-RayEmission ...... 226
4.4.1 DiffuseLeptonicEmission ...... 226
4.4.2 Discrete γ-RaySources...... 229
4.4.3 ATeVExcess? ...... 231
4.5 The GeV-GHz Ratio: A Diagnostic of Electron Cooling and the FIR- RadioCorrelation...... 232
4.6 Implications...... 236
4.6.1 The Detectability of Other Star-Forming Galaxies ...... 236
4.6.2 The Diffuse γ-ray & Neutrino Backgrounds fromStarFormation ...... 238
4.6.3 The Dynamical Importance of Cosmic Rays in Starbursts . . . 241
4.7 Conclusion...... 243
4.8 Fermi DataAnalysisforArp220 ...... 245
Chapter 5 The γ-Ray Background Constrains the Origins of the Radio and X-Ray Backgrounds ...... 258
5.1 Introduction...... 258
5.2 Ratio of Pionic γ Rays to Secondary Emission ...... 260
5.3 The X-ray and Soft γ-rayBackgrounds ...... 263
xiv 5.4 TheRadioBackground...... 265
5.5 Conclusion...... 270
Chapter 6 Diffuse Hard X-ray Emission in Starburst Galaxies as Synchrotron from Very High Energy Electrons ...... 275
6.1 Introduction...... 275
6.2 Motivation...... 282
6.2.1 Relevant Cooling Processes ...... 282
6.2.2 PrimaryElectrons ...... 287
6.2.3 PionicSecondaries ...... 291
6.2.4 Pair Production Tertiaries ...... 294
6.3 ModellingAssumptions...... 296
6.3.1 Injection...... 297
6.3.2 Propagation...... 300
6.3.3 Constraints ...... 305
6.4 Results and Comparison With Observations ...... 307
6.4.1 The Galactic Center ...... 311
6.4.2 NGC253StarburstCore...... 319
6.4.3 M82 ...... 324
6.4.4 Arp220 ...... 330
6.5 Discussion...... 338
6.5.1 Synchrotron, the FIR-X-ray Correlation, and Submillimeter Galaxies...... 338
6.5.2 What Neutrinos and TeV γ-rays Can Tell Us ...... 343
6.6 Conclusion...... 346
6.7 Appendix: Other Pionic Cross Section and Lifetime Parameterizations 353
xv 6.8 Appendix: Other Maximum Energy Proton Cutoffs ...... 357
Chapter 7 Additional Notes on the High-Σg Conspiracy ...... 411
7.1 Radio Enhancement: Secondaries and Spectral Effects ...... 411
7.2 Radio Suppression: Balance of the cooling processes ...... 414
7.3 Summary ...... 418
Bibliography ...... 419
xvi List of Tables
1.1 Observational tools for studying CRs in other galaxies ...... 32
1.1 Observational tools for studying CRs in other galaxies ...... 33
2.1 ListofSymbolsUsed ...... 141
2.1 ListofSymbolsUsed ...... 142
2.1 ListofSymbolsUsed ...... 143
2.2 π0 γ-Ray (and π± Neutrino)Fluxes ...... 144
2.2 π0 γ-Ray (and π± Neutrino)Fluxes ...... 145
2.3 SuccessfulModels...... 146
2.3 SuccessfulModels...... 147
2.3 SuccessfulModels...... 148
2.3 SuccessfulModels...... 149
2.3 SuccessfulModels...... 150
3.1 ModelProperties ...... 204
4.1 Non-Calorimetric Galaxies: Predicted & Observed Gamma-Ray Fluxes 250
4.1 Non-Calorimetric Galaxies: Predicted & Observed Gamma-Ray Fluxes 251
4.1 Non-Calorimetric Galaxies: Predicted & Observed Gamma-Ray Fluxes 252
4.2 Possible Calorimetric Galaxies: Predicted, & Observed Gamma-Ray Fluxes ...... 253
4.2 Possible Calorimetric Galaxies: Predicted, & Observed Gamma-Ray Fluxes ...... 254
xvii 4.2 Possible Calorimetric Galaxies: Predicted, & Observed Gamma-Ray Fluxes ...... 255
4.2 Possible Calorimetric Galaxies: Predicted, & Observed Gamma-Ray Fluxes ...... 256
4.3 Fermi-LAT γ-rayfluxesofArp220 ...... 257
6.1 ModelParameters...... 377
6.1 ModelParameters...... 378
prim 6 6.2 Galactic Center X-ray luminosities when γmax = 10 ...... 379
prim 6 6.2 Galactic Center X-ray luminosities when γmax = 10 ...... 380
prim 6 6.2 Galactic Center X-ray luminosities when γmax = 10 ...... 381
prim 6 6.2 Galactic Center X-ray luminosities when γmax = 10 ...... 382
prim 9 6.3 Galactic Center X-ray luminosities when γmax = 10 ...... 383
prim 9 6.3 Galactic Center X-ray luminosities when γmax = 10 ...... 384
prim 9 6.3 Galactic Center X-ray luminosities when γmax = 10 ...... 385
prim 9 6.3 Galactic Center X-ray luminosities when γmax = 10 ...... 386
prim 6 6.4 NGC 253 X-ray luminosities when γmax = 10 ...... 387
prim 6 6.4 NGC 253 X-ray luminosities when γmax = 10 ...... 388
prim 6 6.4 NGC 253 X-ray luminosities when γmax = 10 ...... 389
prim 6 6.4 NGC 253 X-ray luminosities when γmax = 10 ...... 390
prim 9 6.5 NGC 253 X-ray luminosities when γmax = 10 ...... 391
prim 9 6.5 NGC 253 X-ray luminosities when γmax = 10 ...... 392
prim 9 6.5 NGC 253 X-ray luminosities when γmax = 10 ...... 393
prim 9 6.5 NGC 253 X-ray luminosities when γmax = 10 ...... 394
prim 6 6.6 M82 X-ray luminosities when γmax = 10 ...... 395
prim 6 6.6 M82 X-ray luminosities when γmax = 10 ...... 396
xviii prim 6 6.6 M82 X-ray luminosities when γmax = 10 ...... 397
prim 9 6.7 M82 X-ray luminosities when γmax = 10 ...... 398
prim 9 6.7 M82 X-ray luminosities when γmax = 10 ...... 399
prim 9 6.7 M82 X-ray luminosities when γmax = 10 ...... 400
prim 6 6.8 Arp 220 X-ray luminosities when γmax = 10 ...... 401
prim 6 6.8 Arp 220 X-ray luminosities when γmax = 10 ...... 402
prim 6 6.8 Arp 220 X-ray luminosities when γmax = 10 ...... 403
prim 6 6.8 Arp 220 X-ray luminosities when γmax = 10 ...... 404
prim 6 6.8 Arp 220 X-ray luminosities when γmax = 10 ...... 405
prim 9 6.9 Arp 220 X-ray luminosities when γmax = 10 ...... 406
prim 9 6.9 Arp 220 X-ray luminosities when γmax = 10 ...... 407
prim 9 6.9 Arp 220 X-ray luminosities when γmax = 10 ...... 408
prim 9 6.9 Arp 220 X-ray luminosities when γmax = 10 ...... 409
prim 9 6.9 Arp 220 X-ray luminosities when γmax = 10 ...... 410
xix List of Figures
2.1 FiducialmodelFRC ...... 119
2.2 Radio spectral slope with gas surface density ...... 120
2.3 Cosmic ray spectra in low density galaxy ...... 121
2.4 Cosmic ray spectra in high density starburst ...... 122
2.5 Cooling times for 1.4 GHz emitting electrons ...... 123
2.6 Total emissivities of CRs ...... 124
2.7 Positronfractionwithenergy ...... 125
2.8 Secondary electron fraction with energy ...... 126
2.9 Synchrotronspectrainstandardmodel ...... 127
2.10 Pionic gamma-ray spectra in standard model ...... 128
2.11 Bremsstrahlungspectrainstandardmodel ...... 129
2.12 ICspectrainstandardmodel ...... 130
2.13 Proton calorimetric fraction in several models ...... 131
2.14 Processes in the high surface density conspiracy ...... 132
2.15 FIR-radio ratio at multiple frequencies ...... 133
2.16 Radio spectral slope at different frequencies ...... 134
2.17 Predicted radio spectra for observed starbursts ...... 135
2.18 Which process contributes to the gamma-ray fluxes? ...... 136
2.19 Which radiation fields contribute to the IC fluxes? ...... 137
xx 2.20 Cosmic ray and other pressures for galaxies and starbursts ...... 138
2.21 FRCinseveralvariants...... 139
2.22 High gas surface density conspiracy in several variants ...... 140
3.1 Rest-frame FRC at high redshift in our standard model with no winds 196
3.2 Rest-frame FRC at high redshift in our standard models with winds . 197
3.3 Rest-frame FRC evolution at high redshift in our standard models . . 198
3.4 Observed k-corrected FRC evolution in our standard models . . . . . 199
3.5 Rest-frame synchrotron radio spectra of starbursts ...... 200
3.6 Radio emission as a function of star-formation rate ...... 201
3.7 Radio flux of star-forming galaxies at high redshift ...... 202
3.8 Redshift when radio flux is suppressed by CMB ...... 203
4.1 Gamma-rayspectrumofM82 ...... 247
4.2 Gamma-rayspectrumofNGC253...... 248
4.3 Ratio of gamma-ray luminosities to star-formation rates for star- forminggalaxies...... 249
5.1 Limits on cosmic IC backgrounds from pionic electrons/positrons . . 273
5.2 Limits on the cosmic radio backgrounds from pionic electrons/positrons 274
6.1 Lowenergyphotonbackgroundsofstarbursts ...... 360
6.2 Pair production optical depths of starbursts ...... 361
6.3 Predicted Galactic Center radio spectrum ...... 362
6.4 CR electron spectra in the Galactic Center ...... 363
6.5 Predicted high energy emission from the Galactic Center ...... 364
6.6 PredictedNGC253starburstradiospectrum...... 365
6.7 CR electron spectra in NGC 253’s starburst ...... 366
6.8 Predicted high energy emission from NGC 253’s starburst ...... 367
xxi 6.9 PredictedM82radiospectrum...... 368
6.10 CRelectronspectrainM82 ...... 369
6.11 Predicted high energy emission from M82 ...... 370
6.12 PredictedArp220radiospectrum...... 371
6.13 CRelectronspectrainArp220 ...... 372
6.14 Predicted high energy emission from Arp 220 ...... 373
6.15 Synchrotron and IC X-ray fraction of starburst bolometric power . . . 374
6.16 PredictedneutrinospectraofM82...... 375
6.17 Predicted TeV gamma-ray spectra of Arp 220 ...... 376
xxii Chapter 1
Introduction
1.1. Cosmic rays
Wherever there is star formation in the Universe, there are young, massive stars and the cosmic rays (CRs) they generate. CRs are high energy subatomic particles that travel at high speeds, nearly the speed of light. They can be protons, ions, electrons, or positrons. They form a relativistic gas that pervades our Galaxy, as well as galaxies across the Universe. CRs may regulate star formation in galaxies, and drive chemistry deep in molecular clouds through ionization. CRs generate broadband continuum emission that span the entire electromagnetic spectrum, from
MHz radio waves all the way to TeV gamma rays and beyond. They also emit high energy neutrinos when they crash into gas atoms.
CRs were first understood as coming from space a century ago after experiments by Victor Hess (and Domenico Pacini; De Angelis et al. 2010). Since then, CRs have been studied through direct measurements, as well as indirect inferences of CR populations through electromagnetic observations both in our Galaxy and others.
1 Direct measurements of CR composition and energies have revealed that most of the CRs in the Milky Way are protons and ions. These form a power law spectrum extending from 1 GeV 5 PeV (109 eV 5 1015 eV) of the form dN/dE E−2.7. ∼ − − × ∝ At 5 PeV, depending on what kind of nucleus, the spectrum steepens to about ∼ E−3.1, a feature known as the so-called “knee”. The spectrum steepens again to
E−3.3 at about 400 PeV (4 1017 eV; the “second knee”), before finally hardening ∼ × again to E−2.7 at 4 EeV (4 1018 eV; the “ankle”; see the reviews in, e.g., Nagano × & Watson 2000; Torres & Anchordoqui 2004; Bl¨umer et al. 2009). Finally, the spectrum appears to cutoff above 100 EeV (Abraham et al. 2008). Below a GeV,
CR protons and ions are modulated by the Solar wind (e.g., Parker 1958; Gleeson &
Axford 1968), which prevents us from directly measuring the low energy CR proton spectrum.
CR electrons and positrons (e±) are also present, though in much lower quantities; at GeV energies there are roughly 50 - 100 CR protons for every CR electron (e.g., Ginzburg & Ptuskin 1976). As with CR protons, Solar modulation distorts the CR e± spectrum below about a GeV. The CR e± spectrum above a GeV is a dN/dE E−3 power law. Near 10 GeV, the electrons outnumber the positrons ∝ by about 10 to 1 (e.g., Adriani et al. 2009). New instruments and satellites such as
PAMELA, ATIC, Fermi, and HESS are finally probing the TeV (1012 eV) CR e± spectrum. They reveal that the E−3 power law spectrum observed at GeV energies continues to about 1 TeV, contrary to expectations that it should fall off more
2 rapidly (Chang et al. 2008; Abdo et al. 2009c; Aharonian et al. 2009a). Furthermore, the positron fraction increases with energy, again contrary to expectations (Adriani et al. 2009). Above a few TeV, we have relatively weak constraints on the Galactic e± spectrum (Kistler & Y¨uksel 2009) and none on the positron fraction.
The direct observations only tell us about cosmic rays in our corner of the
Milky Way. Cosmic rays radiate, though, and from this radiation we infer that CRs are ubiquitous in star-forming galaxies. Star-forming galaxies span a wide range of physical conditions and rates at which stars are being born. Faint dwarf star-forming galaxies like the Magellanic Clouds, have star-formation rates of only a tenth of
−1 that of the Milky Way ( 0.1 M⊙ yr ). At the bright end, starburst galaxies form ∼ stars much more quickly than the Milky Way. In the nearby starburst galaxies M82 and NGC 253, the mean gas density of the interstellar medium (ISM), the radiation energy density, the magnetic field strength are all hundreds of times higher than in the Milky Way. The star-formation rate in the starburst cores of the galaxies, just a few hundred parsecs wide, is also several times that of the Milky Way. Even more extreme are the Ultra-Luminous InfraRed Galaxies (ULIRGS), with star-formation
−1 rates of > 100 M⊙ yr . The closest ULIRG, Arp 220, has two nuclei with gas ∼ densities of 104 cm−3. Finally, the submillimeter galaxies (SMGs) observed at z 2 ≈ have star-formation rates up to several thousand M⊙ per year. Yet radio emission and gamma rays have been observed from faint, Milky Way-like, and starburst galaxies, demonstrating the presence of CRs in these galaxies. The diverse physical
3 conditions in these galaxies can regulate their CR populations in different ways: it is not enough to simply assume that they work like the Milky Way.
There are three great questions in understanding cosmic rays in star-forming galaxies. First, where do the CRs come from? Second, how do they propagate through galaxies? Third, what is their effect on galaxies?
1.1.1. The Origin of CRs
Even after a century of observing CRs, we still are not certain about what accelerates them. The bulk of the CR protons at a GeV to a PeV are known to arise within the Milky Way. Old theories in which CR protons evenly pervade the entire
Universe are ruled out by the faintness of the Small Magellanic Cloud satellite galaxy in gamma rays, indicating that it has a lower CR energy density than the Milky Way
(Sreekumar et al. 1993). CR e± are likewise known to arise in the Milky Way, simply because they are cooled too quickly by the cosmic microwave background to travel through intergalactic space (Longair 1994). The ultra-high-energy CRs (UHECRs) above the “ankle” in energy (> 4 1018 eV) are generally believed to arise outside ∼ × the Galaxy (e.g., Abraham et al. 2007), although this is not universally accepted. In between the knee and the ankle, both Galactic and extragalactic origins have been proposed (e.g., Berezinsky et al. 2006; Katz et al. 2009).
4 CRs can be either primary, meaning that they were directly accelerated from low to high energies, or secondary, meaning that they were created by the collisions of already high-energy CRs with ambient particles. As inferred from propagation models and the composition of CRs (for example, by the abundance of positrons and other particles that are only created by CRs colliding with ambient nuclei), most of the CRs in the Milky Way are primaries.
The most widely accepted candidate for the source of the primary Galactic CRs are supernova remnants. When a massive star explodes in a supernova, several Solar masses of gas are expelled into the interstellar medium. The ejecta of one supernova explosion has about 1051 ergs of kinetic energy. If perhaps 10% of this energy could be converted into CRs, supernova remnants alone would have enough power to inject all of the CRs observed in the Milky Way.
A supernova origin for at least some CRs is supported by radio and X-ray observations that demonstrate that electrons are being accelerated up to TeV energies in these supernova remnants (e.g., Koyama et al. 1995; Reynolds 2008).
Gamma-ray observations also indicate CR acceleration in supernova remnants, although these gamma rays could be from either protons or electrons (e.g., Tanimori et al. 1998; Aharonian et al. 2007b; Tavani et al. 2010).
The process through which supernova remnants are thought to accelerate CRs is known as Diffusive Shock Acceleration, or first-order Fermi acceleration (e.g., Bell
5 1978; Blandford & Ostriker 1978). As the ejecta punch through the Galactic ISM at supersonic speeds, they form a shock. The supernova remnants are rarefied enough that the shock cannot be collisional, in which individual atoms or ions hit each other
(e.g., Draine & McKee 1993). Instead, the shock is collisionless, with particles being deflected by large scale electromagnetic fields within the shock. Because the shock is collisionless, there is no single temperature defining the particles in the shock and nonthermal phenomena can occur. A particle traversing the shock can be bounced back at higher energy by magnetic inhomogeneities on the other side of the shock.
On average, particles gain energy each time they bounce back and forth across the shock. Thus, particles can be accelerated to ultrarelativistic energies through
Diffusive Shock Acceleration.
In the end, the maximum energy of CRs is limited by the size of the acceleration region and the magnetic fields which confine particles within the accelerator. As particles gain energy, their Larmor radius (the radius of the circular path they travel in a magnetic field) grows larger; eventually the Larmor radius exceeds the size of the accelerator and the particles escape. This defines the Hillas criterion for UHECR acceleration: for most known astrophysical phenomena, the energy is limited to 1020 eV or less (Hillas 1984); the highest energies are thought only ∼ to be possible in extreme environments such as active galactic nuclei, gamma-ray bursts, and magnetars. Thompson & Lacki (2011) have shown that in principle, general relativity allows CRs of up to 1027 eV to be accelerated before the confining
6 magnetic field collapses into a black hole, although most realistic conditions limit the CR energy to 1023 eV.1 For supernova remnants, the Hillas criterion gives a maximum energy of 1015 eV, as observed in the knee (Gaisser 1990). ∼
There are other potential sources of CRs, including superbubbles (e.g., Higdon
& Lingenfelter 2005), stellar winds (e.g., Quataert & Loeb 2005), gamma-ray bursts
(e.g., Higdon & Lingenfelter 2005), and pulsar wind nebulae (Arons & Tavani 1994;
Bednarek & Protheroe 1997; Y¨uksel et al. 2009; Hooper et al. 2009). These all are produced by short-lived massive stars, all correlate with the star-formation rate, and thus we expect the rate at which CRs are injected into galaxies to be roughly proportional to the star-formation rate. However, without understanding what the accelerators are, much less the physics of how they work, we cannot be sure that
CR accelerators work the same way in the diverse environments of star-forming galaxies. For example, supernova remnants may evolve differently in the dense gas environments of starbursts (Chevalier & Fransson 2001).
Not all CRs are accelerated by phenomena associated with star formation.
Active galactic nuclei, powered by the supermassive black holes in the centers of galaxies, also accelerate CRs (e.g., Begelman et al. 1984; von Montigny et al. 1995).
While the central black hole is quiescent in most galaxies, including the Milky Way, weak nonthermal emission can come from these black holes (e.g., Aharonian et
1Similar limits exist for direct acceleration in an electric potential (Kardashev 1995), which applies to the case of CRs accelerated by pulsars and similar phenomena (e.g., Goldreich & Julian 1969).
7 al. 2004). Even more exotic sources of CR accelerators are possible, such as dark matter annihilation and decay. The anomalies in the CR e± spectra prompted much speculation that TeV e± are being generated by dark matter somehow. However, there are many astrophysical uncertainties, not least of which is the presence of astrophysical sources such as pulsars which could also generate these e±. In order to search for exotic signals such as dark matter, we must understand the bulk of the
CRs in our Galaxy: those associated with star formation.
1.1.2. Understanding CR Propagation and Emission
To interpret observations of CRs, it is not enough to know where cosmic rays come from. For direct measurements of CR spectra at Earth, we need to know how they travel to Earth. To understand the nonthermal emission from CRs, we need to know how they radiate in the environments of star-forming galaxies. Either way, we must understand the propagation of CRs, which involves several physical processes regulated by many different variables.
Cosmic rays can escape their accelerators, travel through their host galaxies, and then escape them as well. In the Milky Way, CR transport occurs mainly through diffusion. The magnetic field of galaxies is inhomogeneous, and these
fluctuations can scatter CRs. These inhomogeneities can even be generated by CRs themselves (Kulsrud & Pearce 1969). As a result, CRs do not simply fly out of
8 galaxies at the speed of light, but are trapped in a scattering atmosphere. In the
Milky Way, the diffusion time for CRs is known to be 30 Myr at GeV energies ∼ from the abundances of radioactive isotopes in CRs (e.g., Garcia-Munoz et al. 1977;
Connell 1998; Webber et al. 2003). However, our understanding of CR diffusion in other galaxies is indirect and ill-constrained, coming mainly from radio emission from CR e±. From these observations, we generally expect Milky Way-like galaxies to have diffusion times of the same order as the Milky Way’s (e.g., Dahlem et al.
1995; Ptuskin & Soutoul 1998). The diffusion time is energy-dependent, growing shorter at higher energy (e.g., Swordy et al. 1990).
CRs can also be advected in bulk by galactic-scale winds. These winds are observed in starbursts throughout the Universe, reaching speeds of several hundred to several thousand kilometers per second (e.g., Heckman et al. 2000; Heckman
2003; Strickland & Heckman 2009; Heesen et al. 2009). Similar winds appear to be launched in the inner few kpc of the Milky Way (Everett et al. 2008) and the Galactic
Center region (Crocker et al. 2011a). Advection is expected to be the dominant form of escape in starbursts, and the large radio haloes of nearby starbursts such as M82 and NGC 253 are interpreted as signs that CRs are being quickly advected (e.g.,
Seaquist & Odegard 1991).
Cosmic rays do not just move through galaxies, but can also cool through radiative processes. CR protons can cool mainly in two ways in star-forming galaxies.
At low energies, they ionize gas atoms, stripping off electrons and losing energy
9 in the process. The energy loss time for ionization grows essentially linearly with energy. At high energies, CR protons can also lose energy through inelastic collisions with gas nuclei, in which they create pions. Pions are unstable mesons which can be either negatively charged (π−), positively charged (π+), or neutral (π0). These eventually decay into gamma rays, neutrinos, and secondary e±:
π0 γ + γ (1.1) → π+ e+ + ν + ν +ν ¯ (1.2) → µ e µ π− e− +ν ¯ +ν ¯ + ν . (1.3) → µ e µ
Above the threshold for pion production, the energy loss time for this process is largely independent of energy (e.g., Mannheim & Schlickeiser 1994). Heavier CR ions can also lose energy through nuclear reactions in the ISM and fragment into smaller nuclei.
Cosmic ray e± can cool through many different ways. Like CR ions, they can ionize atoms in the ISM; the ionization loss time grows linearly with energy. CR e± can be deflected in the electric field of atoms, and produce bremsstrahlung radiation.
The bremsstrahlung loss time is basically independent of energy. CR e± gyrate in galactic magnetic fields, emitting synchrotron radiation as they do. Finally, CR e± can collide with low energy photons and scatter them to higher energy through the
Inverse Compton (IC) process. The loss times for both synchrotron and (IC) are inversely proportional to the e± energy (if they are relativistic; Rybicki & Lightman
10 1979). Since they are antimatter, CR positrons can also annihilate ambient electrons while still moving at relativistic energy, with an annihilation lifetime that grows with energy (e.g., Aharonian & Atoyan 1981; Beacom & Y¨uksel 2006). As a result, the CR e± spectrum can become quite complex: hardening at low energies due to ionization losses and softening at high energies from synchrotron and IC losses.
All of these emission processes generate radiation with different characteristic spectra. Because of the threshold for pion creation, pionic emission is negligible below a few tens of MeV, peaks near 70 MeV, and then has a power law similar to the CR proton spectrum (e.g., Stecker 1970). Pionic gamma-ray emission must also be accompanied by pionic neutrinos (e.g., Stecker 1979), although the difficulty of detecting neutrinos means that we have not yet observed them from CRs beyond
Earth’s atmosphere. A bremsstrahlung photon has a characteristic energy that is about half the CR e± energy (e.g., Schlickeiser 2002). Bremsstrahlung therefore traces the CR e± spectrum directly and peaks near a GeV. Synchrotron is a broad continuum at low energies, radio to X-rays (and beyond). Inverse Compton is also a broad continuum extending from X-rays to TeV gamma rays, where QED
Klein-Nishina effects cut it off. Multiwavelength observations therefore can constrain the contribution of each process and inform us on both the spectrum of CRs and the environments they propagate in.
In addition to escaping and radiating, CRs can generate secondary particles.
These must be accounted for in any calculation of CR propagation. The secondaries
11 can be knock-off electrons, which are the electrons ionized by CRs; pionic e± generated through proton collisions with the ISM; or pair-produced through interactions of gamma rays with low energy photons (γ + γ e+ + e−; e.g., Gould → & Schr´eder 1967).
To calculate all of these effects on CR populations, one uses the diffusion-loss equation. This equation includes terms for diffusion, advection, energy losses, and
finite lifetimes for radioactive particles and describes both the spatial and temporal evolution of CRs in a modeled region (e.g., Ginzburg & Ptuskin 1976; Schlickeiser
2002; Strong et al. 2007). Typically, it is solved in the steady-state approximation.
The most advanced solutions of the diffusion-loss equation in the Milky Way are done using the GalProp software (e.g., Strong & Moskalenko 1998; Moskalenko
& Strong 1998; Strong et al. 2000). This software calculates models of both the nuclear composition of CRs below a PeV and the resolved radio and gamma-ray diffuse emission from CRs in the Milky Way. The nuclear composition in particular informs us on the amount of matter CRs travel through and how long they take to get here. GalProp forms the baseline for our understanding of the CR population in the Milky Way (e.g., Strong et al. 2004; Porter et al. 2008; Strong et al. 2010).
However, in other galaxies, we have no direct measurements of CR composition and observations of nonthermal emission may be poorly-resolved, or even non- existent. Instead of modeling the exact transport of CRs in distant star-forming
12 galaxies, we typically are concerned with the bulk properties of CR populations. For this, we simplify the diffusion-loss equation by using a one-zone approach, in which the modeled region is homogeneous. This involves replacing spatially-dependent terms with escape times from the galaxy: the diffusion-loss equation then becomes the leaky box equation (e.g., Torres 2004b). Solving the leaky box equation for different galaxies is a primary goal of this dissertation.
1.1.3. The Effects of CRs
The interest in cosmic rays in star-forming galaxies is not just about their emission. In the Milky Way, cosmic rays have nearly the same pressure as the magnetic field, turbulence, radiation, and thermal pressure (e.g., Boulares & Cox
1990). This means that CRs are dynamically important in the Milky Way and can alter the large scale flows of gas. For example, CRs in the inner Milky Way appear to be launching a large-scale wind (Everett et al. 2008). Socrates et al. (2008) recently proposed that CRs build up to very large energy densities in starbursts and then drive the winds observed around these galaxies. On the one hand, starbursts have very intense star formation in very small regions, which implies high densities of CR generation. On the other, starbursts have high gas densities which might stop CR protons before they build up to high energy densities (“proton calorimetry”), and the winds themselves might remove CRs before they reach sufficiently high energy
13 densities. Only a treatment of CR propagation can decide which of these effects is more important.
Similarly, CRs can penetrate deep into molecular clouds where they drive chemistry and create a low level of residual ionization. Based on the possible high
CR energy densities of starbursts, Papadopoulos (2010) proposed that CRs in fact dominate the ionization in star-forming regions in starbursts (see also Suchkov et al.
1993). A high CR ionization rate could heat up gas and affect the conditions for star formation. Once again, however, the densities that CRs actually attain in starbursts depends on how they propagate and cool in these environments.
Finally, CRs can interact with and regulate the magnetic field in galaxies
(Parker 1966), and may be involved in galactic-scale dynamos that build up the magnetic fields (e.g., Parker 1992). It is often assumed that CRs are in equipartition with the magnetic field in other galaxies, as it is in the Milky Way, leading to the minimum energy estimate in which U U (e.g., Burbidge 1956; Longair 1994; B ≈ CR Beck & Krause 2005). However, this idea was challenged by Thompson et al. (2006), who argued that it would be easier to explain the radio emission of starbursts if the magnetic field was larger than the minimum energy estimate. A treatment of the radio emission of star-forming galaxies using the leaky box equation is necessary to address this question. The question of the magnetic field strength in star-forming galaxies is a main concern of chapter 2.
14 1.2. What Did We Know on CRs in Star-Forming
Galaxies?
1.2.1. Previous observations of CRs in other galaxies
The next few years will see the convergence of many existing and new facilities able to study CRs in star-forming galaxies: in the radio, EVLA and LOFAR (and eventually SKA) will study CR electrons and magnetic fields, while in gamma rays
Fermi and TeV telescopes like HESS and VERITAS are for the first time providing direct information about CR protons. While it has not yet detected neutrinos yet,
IceCube is developing increasingly powerful constraints on CR protons and ions at
PeV energies. Chandra and XMM-Newton continue to excel in the X-ray bands, and NuStar will extend our knowledge above 10 keV. Simultaneously, ALMA will provide detailed maps of molecular gas in normal and starburst galaxies, and with
Herschel and SOFIA will provide good data on molecular lines. I summarize the strengths and disadvantages of each method of observation in Table 1.1.
Radio waves – Until the recent gamma-ray detections of nearby starburst galaxies, our primary way of understanding CRs in other galaxies was through the radio emission of the e± (Condon 1992). Radio emission from star-forming galaxies can be detected out to z 4, and out to even higher redshift with new facilities ≈ like EVLA (e.g., Murphy 2009). It can also be resolved, allowing us to actually
15 map where CR e± are in other galaxies. The main foreground is free-free emission, thermal bremsstrahlung from ionized regions, which only dominates above 10 GHz
(Niklas et al. 1997). However, synchrotron emission gives us a constraint on the combination of the population of CR e± and the magnetic field, where the latter is not well known.
A remarkable result of the radio observations is the existence of the FIR-radio correlation (FRC), a linear, tight relation between the GHz synchrotron emission and the thermal far-infrared emission of star-forming galaxies (van der Kruit 1971, 1973; de Jong et al. 1985; Helou et al. 1985). The FRC spans many dex in luminosity and gas density, from galaxies like the Milky Way all the way to the brightest starbursts like Arp 220 (Helou et al. 1985). Except at the lowest luminosities, the dispersion in the FRC only seems to be a factor of 2 (Yun et al. 2001). The correlation also appears to work within galaxies as well on kiloparsec scales: the radio emission of nearby galaxies traces the spiral arms and their star-forming regions, implying that the CRs are not simply spread over galaxies evenly (Beck & Golla 1988; Murphy et al. 2006a,b, 2008). The FRC is so good that radio emission is used as a proxy for the FIR emission (and thus star-formation rate), and the radio emission is even assumed to map out the FIR (e.g., for SMGs, Chapman et al. 2004; Biggs & Ivison
2008). Detections of radio waves from galaxies in the distant Universe is also used to chart the cosmic star-formation rate over the history of the Universe, since unlike infrared light, radio waves can be detected from the ground.
16 Gamma rays – Gamma-ray observations allow us to constrain the protons that make up the bulk of the CR population. Unfortunately, they are much harder to detect than radio waves and gamma-ray telescopes have poor angular resolution.
Only a few star-forming galaxies have therefore been detected in gamma rays and these have little spatial detail except for the Milky Way and Magellanic Clouds.
The bulk of the γ-rays, especially above a few hundred MeV, is expected to be pionic emission from proton-proton collisions in the ISM. CR e± also contribute bremsstrahlung, Inverse Compton, and positron annihilation radiation, especially at lower energies. Finally, CR nuclei can excite or be excited by ISM nuclei, and emit nuclear de-excitation γ-rays at MeV energies (e.g., Ramaty et al. 1979), although this has never been detected from the ambient Galaxy or external galaxies (Teegarden &
Watanabe 2006).
Before Fermi, only two galaxies were detected in GeV γ-rays: the Milky Way and the Large Magellanic Cloud (Sreekumar et al. 1992). Within its first year of observation Fermi-LAT provided the first GeV detections of starbursts, M82 and
NGC 253 (Abdo et al. 2010a). In addition, Fermi has detected M31 (Abdo et al.
2010f), the Small Magellanic Cloud (Abdo et al. 2010d), and two starbursts with
Seyfert active nuclei, NGC 4945 and NGC 1068 (Abdo et al. 2010b; Lenain et al. 2010). These observations show that gamma-ray emission does correlate with star-formation rate and have led to speculation about a FIR–gamma-ray correlation
(Abdo et al. 2010f).
17 TeV gamma-rays have also been detected from the Galactic Plane (e.g., with
Milagro, Abdo et al. 2008), the Galactic Center (with HESS, Aharonian et al.
2006b), M82 (with VERITAS, Acciari et al. 2009), and NGC 253 (with HESS, Acero et al. 2009). When combined with the GeV detections, M82 and NGC 253 appear to have harder gamma-ray spectra (Γ 2.2 2.3) than the Milky Way (Γ 2.7 2.6; ≈ − ≈ − Prodanovi´cet al. 2007). The Galactic Center diffuse emission also is measured by
HESS to have a hard Γ 2.3 spectrum (Aharonian et al. 2006b). This indicates ≈ that CRs propagate differently in these systems than in the Milky Way.
X-Rays – X-ray observations can be fairly sensitive and have good angular resolution, especially with Chandra. Inverse Compton (IC) emission from GHz- emitting electrons should contribute to galaxies’ hard X-ray emission (e.g., Moran et al. 1999). The ratio of the IC and synchrotron emission then provides an independent measure of the magnetic field strength. Synchrotron emission from 10 - 100 TeV electrons is also present in the X-ray band of starbursts, providing information about the CR spectrum at TeV – PeV energies (e.g., Protheroe & Wolfendale 1980). ∼ X-rays may also be produced by the inverse bremsstrahlung process by 100 MeV ∼ CR nuclei (e.g., Dogiel et al. 2002). Unfortunately, then nonthermal X-ray emission is expected to be buried by the emission from High Mass X-ray Binaries (HMXBs;
Persic & Rephaeli 2002). While the synchrotron emission is relatively soft, the IC emission has roughly the same spectrum as the HMXBs, and so cannot be separated
18 spectrally. Thus X-rays presently only provide constraints on the CR population in galaxies, not detections of the CRs themselves.
Neutrinos – Hadronic pion production processes that make gamma rays also make neutrinos. Since starbursts are predicted to be efficient at pion production, and are known to glow in gamma rays with hard spectra, they should also make neutrinos with hard spectra. At low (GeV) energies, neutrino signals are swamped by atmospheric neutrinos made by CRs hitting the top of Earth’s atmosphere
(e.g., Gaisser & Honda 2002). However, the Antarctic cubic kilometer detector
IceCube can detect high energy PeV neutrinos. The neutrino flux should be roughly equal to the gamma-ray flux at the same energy (Stecker 1970). Unfortunately, no star-forming galaxy has been detected in neutrinos yet (Dreyer et al. 2011), and none are expected to be bright enough individually to be seen with IceCube. Starbursts throughout the Universe also should create a diffuse neutrino background (Loeb &
Waxman 2006). This diffuse background may eventually be detectable in IceCube, and would support the hadronic nature of the gamma-ray flux of starbursts (Ahrens et al. 2004). No neutrino background is detected yet either (Abbasi et al. 2011).
Molecular lines – Low energy CRs can be detected indirectly through their ionization cooling: either the level of ionization itself or the bulk heating of gas in star-forming galaxies. CR ionization is correlated directly with the abundance of
+ the H3 molecule in the interstellar medium, and in the Milky Way has been used
19 to infer the presence of a low energy component of CRs (e.g., Indriolo et al. 2009).
+ However, H3 is seen in absorption, requiring a bright background source.
The radio emission of galaxies is also correlated with molecular gas lines from molecules like CO and HCN (Israel & Rowan-Robinson 1984; Liu & Gao 2010). One proposed mechanism for these correlations is that CRs directly heat molecular gas, which cools by molecular line emission (Adler et al. 1991). However, observations of these lines do not rule out other forms of heating. High-excitation molecular lines such as high-J CO might also indicate the presence of CRs (e.g., Meijerink et al.
2006), and would be visible in Herschel and SOFIA.
Ultra-high energy CRs – As a CR’s energy increases, its Larmor radius in the magnetic field of a galaxy increases. At EeV energies, and depending on the electric charge of the CR, the Larmor radius becomes larger than the galaxy itself. Then the
CR can escape the galaxy without being confined by magnetic fields; hence, we may be able to see UHECRs from other galaxies directly.
The current AUGER experiment has revealed that the UHECRs above
4 1019 eV are not arriving isotropically. CRs at these energies are expected to only × come from galaxies within 75 Mpc due to the GZK effect, interactions between ∼ UHECR protons and the cosmic microwave background cooling protons before they can travel longer distances (Abraham et al. 2008). The UHECRs appear to be correlated with nearby large scale structure as traced by active galactic nuclei (e.g.,
20 Abraham et al. 2007), and possibly the nearby active galaxy Centaurus A specifically
(e.g., Gorbunov et al. 2008). The actual source of UHECRs is still unknown, and may be related to star formation (for example, magnetars: Ghisellini et al. 2008), or active galactic nuclei.
In this work, I concentrate mostly on radio and gamma rays, which have been observed from other galaxies, and neglect molecular lines and UHECRs. I also consider X-rays in chapters 5 and 6.
1.2.2. Previous theories of the FRC
Before the detections of gamma rays from other galaxies, the FRC has been the single most powerful constraint on the CR populations in star-forming galaxies.
At heart, the FRC is driven by star formation. Young, massive stars emit UV light which is quickly absorbed by dust in their host galaxy. The dust then radiates that absorbed power thermally in the far-infrared. Young, massive stars also undergo supernova explosions which generate the shocks presumed to accelerate CRs. The
CR electrons then radiate in the radio through synchrotron emission as they spiral in the host galaxy’s magnetic field.
In the simplest explanation for the far-infrared–radio correlation called calorimeter theory, all of the starlight from young stars in a galaxy gets absorbed by dust and re-emitted as infrared light (“UV calorimetry”). Similarly, none of the
21 CR e± escape and all of the power the supernovae pour into cosmic ray electrons is emitted as synchrotron radio (“electron calorimetry”).2 Since both supernovae and starlight mostly come from young, bright stars, both infrared and radio will trace them equally well (V¨olk 1989). The advantage of calorimetry theory is that it has no free parameters, thus explaining the small scatter in the FRC.
However, in low luminosity galaxies, UV calorimetry is known to break down
(Xu & Buat 1995; Bell 2003), which means that electron calorimetry must break down or be altered to not overproduce the observed radio emission. This has led to the creation of non-calorimetric “conspiracy” theories. For example, perhaps both
UV light and CR electrons escape in equal proportions, which can create a linear
FRC in the right conditions (e.g., Chi & Wolfendale 1990; Helou & Bicay 1993;
Lisenfeld et al. 1996a). Calorimetry also predicts steep radio spectra (spectral index
α 1 1.2), since synchrotron and IC cooling lead to soft electron spectra. However, ≈ − the actual observed values for α are more like 0.7 0.8 (Condon 1992; Niklas et al. − 1997). This has led to non-calorimeter theories where the FRC arises if the magnetic
field strength grows with density (e.g., Niklas & Beck 1997), possibly through a turbulent dynamo (Groves et al. 2003). However, point source contamination from
2Calorimetry theory also works if the magnetic field energy density grows proportionally to a much larger radiation energy density, with all of the power in CR electrons going into IC and synchrotron emission (V¨olk 1989).
22 supernova remnants (Lisenfeld & V¨olk 2000) and bremsstrahlung and ionization cooling of CR e± (Thompson et al. 2006) could ease the spectral index problem.
A final issue with all of these models is that they do not include CR secondary e±, which may be important in starbursts (Rengarajan 2005). Starbursts especially may even be in the “proton calorimeter” limit (c.f., Pohl 1994) in which the high energy CR protons lose all their energy to pionic losses (e.g., Loeb & Waxman 2006), as compared to the Milky Way where only a few percent of the proton luminosity goes into pionic losses (Strong et al. 2010). This would flood the starburst with secondary e± and potentially disrupt the FRC. A full treatment of CR propagation is necessary to treat these effects.
1.2.3. Previous models of CR propagation in other
galaxies
It was recognized early on that starbursts have higher CR energy densities than the Milky Way and that they may be bright gamma-ray sources (e.g., V¨olk et al. 1989; Akyuz et al. 1991). This led to early analytic calculations of the CR populations in other nearby galaxies to estimate whether they would be visible in EGRET (e.g., Pohl 1994; Sreekumar et al. 1994; V¨olk et al. 1996). Similarly, there was speculation that Inverse Compton X-ray and gamma-ray emission from starbursts would be bright (e.g., Schaaf et al. 1989; Soltan & Juchniewicz 1999),
23 based on low magnetic field strengths for starbursts. Analytic estimates are still sometimes used, particularly for the pionic gamma-ray emission which has relatively simple scaling at high energies (e.g., Loeb & Waxman 2006; Thompson, Quataert, &
Waxman 2007; Crocker et al. 2011a).
Paglione et al. (1996) first created a model of the nonthermal emission of
NGC 253, including both radio and gamma-rays. Since then, other galaxies have been modelled using the leaky-box or diffusion-loss equations: Arp 220 by Torres
(2004b), M82 by Persic et al. (2008) and de Cea del Pozo et al. (2009a), NGC
253 by Domingo-Santamar´ıa & Torres (2005) and Rephaeli et al. (2010), and the
Galactic Center starburst (R 112 pc) by Crocker et al. (2011b). These models ≤ have found much higher magnetic fields ( 50 100 µG for the Galactic Center, ∼ − 100 200 µG for M82 and NGC 253, milliGauss for Arp 220) than in most of the ∼ − Milky Way ( 5 10 µG), secondary e± comparable to primary electrons (except ∼ − in the Galactic Center where a powerful wind removes CR protons before they can interact with the ISM), and hard proton spectra. These models demonstrate that starburst galaxies have CR phenomena not normally seen in the Milky Way.
24 1.2.4. The Cosmic Backgrounds from Star-Formation
Cosmic Rays
Although we can consider star-forming galaxies individually, we can also consider the aggregate emission from the billions of star-forming galaxies in the
Universe. The cosmic glows from all of these galaxies are viewed as nearly isotropic backgrounds across the electromagnetic spectrum. The nonthermal emission from star-forming galaxies are thought to be significant contributors to three backgrounds: the radio background, the gamma-ray background, and the neutrino background.
The radio background from star-forming galaxies has been calculated by assuming the FRC holds out to high redshift (Protheroe & Biermann 1996; Haarsma
& Partridge 1998; Dwek & Barker 2002). Star-forming galaxies are expected to dominate the µJy mJy population of radio sources (e.g., Danese et al. 1987; − Condon 1989; Benn et al. 1993), and studies have so far shown that the FRC holds down for most star-forming galaxies as faint as 30 µJy and star-forming galaxies to z 2 at least for brighter galaxies (e.g., Appleton et al. 2004; Ibar et al. 2008; ≈ Sargent et al. 2010). However, these calculations of the radio background have recently been challenged by the enormous extragalactic radio background found by the ARCADE2 experiment (Fixsen et al. 2009; Seiffert et al. 2009). No other source than star-forming galaxies seem numerous enough to be the source of this radio background, so Singal et al. (2010) suggested that active galactic nuclei
25 activity enhance the radio emission at high z. There is also tantalizing evidence that submillimeter galaxies are radio-bright by a factor 2 5 (e.g., Kov´acs et al. ∼ − 2006; Vlahakis et al. 2007; Murphy 2009; Micha lowski et al. 2010a). However, even taking into account possible deviations from the FRC, star-forming galaxies do not seem to produce enough radio emission to account for the ARCADE observed radio background (Ponente et al. 2011).
The contribution of star-forming galaxies to the gamma-ray background is also uncertain. Blazars, a kind of active galactic nucleus where emission happens to be beamed towards the Earth, make up the majority of the resolved sources and have been argued to be the source of the unresolved background as well. Abdo et al.
(2010g) argued that there was a break in the blazar source counts at low fluxes, implying that only 20% of the unresolved gamma-ray background was blazars, ∼ although this has been disputed by Stecker & Venters (2011). The amount of room left in the gamma-ray background for star-forming galaxies is unclear. Pavlidou
& Fields (2002) argued that normal galaxies like the Milky Way must contribute about a third of the GeV gamma-ray background, based on scaling arguments on the amount of gas and star formation in galaxies. Thompson, Quataert, & Waxman
(2007) argued that starbursts could contribute another 10 20% of the gamma-ray ∼ − background, assuming proton calorimetry applies for starburst galaxies. How much of the cosmic star-formation rate is proton calorimetric remains disputed though
(Stecker 2007; Thompson et al. 2006). There have been several other calculations
26 of the star-forming galaxy contribution to the gamma-ray backgrounds, based on scaling from the Milky Way to different star-formation rates and amounts of gas,
finding that both normal galaxies and starbursts each contribute 1 50% of the ∼ − gamma-ray backgrounds (e.g., Bhattacharya & Sreekumar 2009; Fields et al. 2010;
Makiya et al. 2011; Stecker & Venters 2011; Lacki et al. 2011).
Finally, the neutrino emission from starbursts throughout the Universe may form a hard spectrum background, if much of the star formation in the Universe occurs in proton calorimetric starbursts (Loeb & Waxman 2006).
1.3. The Work in this Dissertation
My work involves using analytic and leaky box models to understand CR propagation in other galaxies. I apply these models to several problems in the field: the origin of the FRC, the population of CRs that emit gamma rays in starburst galaxies, the origin of the cosmic radio background, and the possible emission of
X-rays by CR e± in starburst galaxies.
In chapter 2, we develop a theory of the FIR-radio correlation based on leaky box models of galaxies and starbursts, and describe its implications for CR propagation in star-forming galaxies. In order to reproduce the FIR-radio correlation
(FRC) over its entire span, we had to tune several free parameters, in particular the magnetic field. We found that the radio spectra of galaxies is not simply determined
27 at all. Instead, a conspiracy of opposing factors works to preserve the linearity of the FRC for starbursts: non-synchrotron cooling reduces the radio emission, but the presence of secondary electrons and positrons restores the radio emission. Similarly, for galaxies with low star-formation rate, we found another conspiracy between
CR escape and the dust opacity, which lower the radio and IR emission together to create a roughly linear FRC. We find that normal galaxies are partial electron calorimeters, and starbursts are both proton and electron calorimeters. Chapter 2 is a copy of Lacki et al. (2010a, ApJ 717, 1) with minor modifications.
In chapter 3, we applied our theory of the FIR-radio correlation to high-z galaxies. There are two main effects to consider. The first is the increased radiation energy density of the cosmic microwave background (CMB), which will increase
Inverse Compton losses and ultimately decrease the radio luminosity of galaxies.
The effect has been predicted for almost two decades (e.g., Condon 1992; Carilli &
Yun 1999), and our theory allows us to quantify the radio-dimming. Unlike previous theories, where the radio-dimming is simply a matter of the CMB energy density exceeding the magnetic energy density in galaxies, we find that the conspiracies operating in galaxies actually preserve the radio luminosities of galaxies out to high z. Essentially, the Inverse Compton radiation off the CMB comes at the expense of bremsstrahlung and ionization, since they are the most important cooling processes, instead of synchrotron radiation. This means the FRC should hold to higher redshift than previously expected.
28 The other difference at high redshift is the different morphology of starbursts at high redshift. At low redshift, starbursts are very compact, only a few hundred parsecs wide and perhaps a hundred parsecs in height. The high redshift SMGs are very extended, though, typically several kiloparsecs wide and, crucially, perhaps a kiloparsec high (e.g., Tacconi et al. 2006). This means that at any given surface density, SMGs have low volume density ISMs, reducing the importance of bremsstrahlung and ionization cooling. We therefore predict that SMGs should be radio-bright by a factor of 2 5, because they are “puffy”. This is in line ∼ − with observations (e.g., Kov´acs et al. 2006; Vlahakis et al. 2007; Murphy 2009;
Micha lowski et al. 2010a), and is evidence that a conspiracy like the one we posit does work in starburst galaxies. Chapter 3 is a copy of Lacki & Thompson (2010a,
ApJ 717, 196) with minor modifications.
In chapter 4, we interpret the recent Fermi-LAT, VERITAS, and HESS detections of M82 and NGC 253. By comparing the observed gamma-ray luminosities with the expected supernova rates, we constrain the energetics of pion production in these starbursts. In particular, we find that the gamma-ray observations are consistent with about 20 40% of the CR proton energy being ∼ − lost to pion production if 10% of the supernova ejecta kinetic energy goes into
CRs. We list the uncertainties, particularly the supernova rate and the acceleration efficiency, in this estimate.
29 We also compare the radio and gamma-ray luminosities of M82 and NGC
253. We find that the radio luminosity of these galaxies is much smaller than the gamma-ray luminosity. If the gamma rays are primarily pionic, then this implies that
CR e± in these galaxies are losing their energy to something other than synchrotron
(or they are escaping). This is in line with what we expect in our theory. Chapter 4 is a copy of Lacki et al. (2011, ApJ 734, 107) with minor modifications.
In chapter 5, I derive an analytic constraint on the radio and X-ray backgrounds from the observed gamma-ray background. Any synchrotron or IC background from pionic secondary e± must be accompanied by a background of pionic gamma-rays.
The observed gamma-ray background places an upper limit on the cosmic background of pionic gamma rays, and therefore the pionic synchrotron and IC backgrounds.
These limits are particularly interesting in terms of the radio background reported by ARCADE, which is 10 times greater than expected from the FRC (Seiffert et ∼ al. 2009; Fixsen et al. 2009). Singal et al. (2010) concluded that an evolution of the
FRC in star-forming galaxies was responsible for this background. However, this limit suggests that starbursts, in which secondary e± are expected to be dominant, cannot be the source of the radio background. Chapter 5 is a copy of Lacki (2011,
ApJ 729, L1).
We explore in chapter 6 a way to constrain the 10 - 100 TeV e± spectra in starbursts through synchrotron X-ray emission. We found in our previous works that starbursts have higher magnetic fields than the Milky Way. This means both
30 that synchrotron cooling can be very rapid at high energies in starbursts, and that e± of a given energy emit at a higher synchrotron frequency than in the Milky Way.
Thus, synchrotron cooling is the dominant cooling mechanism for CR e± of 10 -
100 TeV in starbursts. Furthermore, starbursts should be opaque to 10 - 100 TeV
γ-rays through γ-γ pair production, because they have intense FIR radiation fields.
This converts pionic γ-rays at these energies into CR e±. We find that synchrotron emission is a few percent of M82’s diffuse 2 - 8 keV emission and 10% of Arp ∼ 220’s 2 - 10 keV X-ray emission with standard parameters. However, there are large uncertainties given the little information we have on multi-TeV e± in these starbursts. We also calculate the IC emission from GeV e±, finding that it too is a minority of the X-ray emission. Chapter 6 is a revised copy of Lacki & Thompson
(2010b, arXiv:1010.3030).
Finally I explore the physics behind the high Σg conspiracy in more detail in chapter 7. I show why the factor of 10 enhancement in radio emission from secondary e± and spectral effects is natural, as is the factor of 5 10 suppression in − radio emission from non-synchrotron cooling. The ability of the high Σg conspiracy
0.7 to work so well is shown to reduce to a requirement that B scale as Σg : then it falls naturally out of the cooling times and energetics of CRs.
31 Radio Gamma rays X-rays Neutrinos Molecular lines UHECRs
Emission synch π, brems, IC, IC, synch (& π only ioniz ··· processa e+ ann, line N brems?) Particles e± All e± (& N?) N only All (mostly N only probedb N) Particle GeV MeV – TeV IC: GeV PeV < GeV EeV energies Synch: ∼ 10 TeV – PeV N brems: 100 MeV 32 c What else B nH , Urad IC: Urad nH CR spectrum Intervening controls Synch: B shape B emission? N brems:
nH Detections Many Few None None Milky Way Cen A?d Foregrounds Small Small? Large
Radio Gamma rays X-rays Neutrinos Molecular lines UHECRs
New LOFAR, Fermi- Chandra, IceCube, EVLA, Auger, JEM- Instruments EVLA, SKA LAT, HESS, XMM- KM3NET ALMA, EUSO VERITAS, Newton, Nu- Herschel, MAGIC, CTA Star SOFIA 33
aAbbreviations are – synch: synchrotron; IC: Inverse Compton; ioniz: ionization; brems: e± bremsstrahlung; N brems: inverse (proton) bremsstrahlung; e+ ann: positron annihilation; π: proton-proton pion production; line: nuclear de-excitation lines. bAbbreviations are – e±: electrons and positrons; N: protons and nuclei
c Abbreviations are – B: magnetic field; nH : ambient gas density; Urad: ambient radiation energy density dCentaurus A is an active galaxy powered mainly by a central supermassive black hole. Chapter 2
The Physics of the Far-Infrared-Radio Correlation. I. Calorimetry, Conspiracy, and Implications
2.1. Introduction
The far-infrared (FIR) and radio luminosities of star-forming galaxies lie on a tight empirical relation, the “FIR-radio correlation” (FRC; van der Kruit 1971,
1973; de Jong et al. 1985; Helou et al. 1985; Condon 1992; Yun et al. 2001).1
The FRC spans over three decades in luminosity, remaining roughly linear across the range 109L < L < 1012.5L , from dwarf galaxies to local ultra-luminous ⊙ ∼ ∼ ⊙ infrared galaxies (ULIRGs) like Arp 220 (Yun et al. 2001). At low luminosities
(L < 109L ), the correlation shows evidence of non-linearity (Yun et al. 2001; ∼ ⊙ Bell 2003; Beswick et al. 2008). The galaxies that make up the FRC span a large dynamic range, not just in bolometric luminosity, but also in gas surface density2
−2 −2 (0.001 g cm < Σg < 10 g cm ), photon energy density, and presumably magnetic ∼ ∼ 1This chapter was published as Lacki et al. (2010a, ApJ 717, 1). 21 g cm−2 = 4800M pc−2. ⊙ 34 field strength. From the observed Schmidt law of star formation (Schmidt 1959;
Kennicutt 1998), the range in gas surface density corresponds to a range of at least
4 105 in photon energy density. Not only does the FRC hold on galactic scales, × but it exists for regions within star-forming galaxies down to a few hundred parsecs
(e.g., Beck & Golla 1988; Bicay & Helou 1990; Murphy et al. 2006a; Paladino et al.
2006; Murphy et al. 2006b, 2008).
Star formation drives the FRC. Young massive stars produce ultraviolet (UV) light, which is easily absorbed by dust grains. The dust reradiates in the FIR, producing a linear correlation between star formation rate and the FIR luminosity, if the dust is optically thick to the UV light. The non-thermal GHz radio continuum emission observed from star-forming galaxies is synchrotron radiation from cosmic ray (CR) electrons and positrons, believed to be accelerated in supernova (SN) remnants. Since SNe mainly occur in young stellar populations, this means that star formation is directly linked to normal (non-active galactic nucleus) radio emission
(reviewed in Condon 1992).
In this paper, we model the FRC, over its range in physical parameters from normal star-forming galaxies to the densest and most luminous starbursts. Our motivation is that the normalization and linearity of the FRC has strong implications for the physical properties of star-forming galaxies and the CRs they contain. For example, we can use the radio emission to estimate the energy injection rate and equilibrium energy density of both CR electrons and protons. This is important
35 because the CR pressure is known to be dynamically important in the Milky Way
(e.g., Boulares & Cox 1990), and possibly starburst galaxies (Socrates et al. 2008).
Furthermore, we can use the inferred CR proton energy density to calculate the
flux of gamma-rays from pion production in the galaxies’ host interstellar medium
(ISM; e.g., Torres 2004b; Thompson, Quataert, & Waxman 2007). Finally, the radio emission also constrains the magnetic field strength in galaxies on the FRC
(Thompson et al. 2006).
Finding the causes of the linearity and span of the FRC is the other main purpose of this paper. The FRC is affected by the density of CRs and the environment they propagate through. For the Milky Way, the propagation of
CRs has been well studied, both observationally and theoretically (e.g., Strong &
Moskalenko 1998). However, given the vast range of environments in star-forming galaxies, it is not clear that our knowledge of CR propagation in the Galaxy can be extrapolated across the entire FRC. Therefore, one aspect of our task in explaining the FRC is determining the extent to which the properties of CR injection, such as the initial spectral slope and proton-to-electron ratio, and CR propagation, such as the rate of escape by diffusion, can apply to all star-forming galaxies.
The diversity of star-forming galaxies on the FRC and the tightness of the correlation may imply a deeper, simpler principle at work. In the calorimeter theory first proposed in V¨olk (1989), the CR electrons lose all of their energy before escaping galaxies, with most of the energy radiated as synchrotron radio emission.
36 Thus, galaxies are electron calorimeters, with the energy in CR electrons being converted into an observable form. Calorimetry also requires that galaxies on the
FRC are optically thick to UV light from young stars, which is reradiated in the
FIR. These galaxies would therefore also have to be UV calorimeters. If both electron calorimetry and UV calorimetry hold, and if synchrotron is the main energy loss mechanism, then the ratio of FIR to radio emission is simply the ratio of total starlight produced to the total energy supplied to CR electrons, which is naively expected to be a constant fraction of the energy from SNe, accounting for the FRC.
Calorimeter theory has been questioned, however, both in its assumptions and its implications. For example, the assumption that all normal galaxies are optically thick to UV light is probably false: the observed UV luminosity of normal star-forming galaxies is comparable to the observed FIR luminosity at low overall luminosities (e.g., Xu & Buat 1995; Bell 2003; Buat et al. 2005; Martin et al. 2005;
Popescu et al. 2005). Nor is electron calorimetry believed to hold in the Milky Way
(and presumably similar galaxies), since the inferred diffusive escape time is shorter than the typical estimated synchrotron cooling time (see equations 2.5 and 2.12 later in this paper; or, e.g., Lisenfeld et al. 1996a).
Even in cases when calorimetry holds, the implications of standard calorimeter theory may conflict with observations. A long-standing problem with the predictions of calorimetry has been the radio spectral indices of star-forming galaxies. If electron calorimetry holds, then the synchrotron cooling timescale is much less than the
37 escape timescale. The electron population will then be strongly cooled with a steep spectrum. For an initial injection spectrum of N E−p where p 2 2.5 and a final ∝ ≈ − synchrotron-cooled steady-state spectrum N E−P , this would imply a synchrotron ∝ spectrum of F ν−α with a spectral index of α =( 1)/2= p/2 1.0 1.2. The ν ∝ P − ≈ − observed spectral indices are 0.7 0.8 for normal galaxies, suggesting that, contrary − to calorimeter theory, electrons escape before losing their energy. Lisenfeld et al.
(1996a) consider a modified calorimeter model for normal galaxies that includes escape comparable to cooling losses, and Lisenfeld & V¨olk (2000) suggest that SN remnants in the galaxies can flatten the observed radio spectrum.
More drastically, several non-calorimeter theories have been proposed (e.g.,
Helou & Bicay 1993; Niklas & Beck 1997), often involving a “conspiracy” to maintain the tightness of the FRC. A potential pitfall of non-calorimeter models stems from the enormous dynamic range in physical properties for galaxies on the FRC. For example, inverse Compton cooling alone is very quick in starbursts, implying that electrons cannot escape from these galaxies before losing most of their energy
(Condon et al. 1991a; Thompson et al. 2006).
Typical explanations of the FRC leave out two underappreciated but important effects: proton losses and non-synchrotron cooling. Models of individual starbursts, which have gas densities 103 104 times higher than the Milky Way, predict that − CR protons lose most of their energy to pion creation as they interact with the ISM.
When a CR proton collides with a proton in the ISM, it produces a pion, either
38 charged (π+ or π−), or uncharged (π0). Neutral pions decay into gamma rays, so that pion losses should act as a source of gamma-ray luminosity in starbursts. Charged pions ultimately decay into neutrinos (which may eventually be observed with neutrino telescopes), as well as secondary electrons and positrons. Therefore, dense starburst galaxies are expected to be proton calorimeters: essentially all the injected energy in CR protons ends up converted to gamma rays, neutrinos, and secondary electrons and positrons. Proton calorimetry would also imply that, unlike the Milky
Way, secondary electrons and positrons may dominate over primary electrons and positrons, depending on the ratio of injected protons to electrons (Rengarajan 2005; secondary electrons and positrons are found to be more abundant than primary electrons in starbursts by Torres 2004b, Domingo-Santamar´ıa & Torres 2005, and de Cea del Pozo et al. 2009a). Because secondary electrons and positrons radiate synchrotron, their presence poses a problem for any explanation of the FRC that requires the CR electron density to be directly proportional to the star formation rate, including both standard calorimeter theory and the theory of Niklas & Beck
(1997).
On the other hand, bremsstrahlung, ionization, and IC may all be more important in starburst galaxies. Thompson et al. (2006) point out that cooling by bremsstrahlung and ionization tends to flatten the radio spectra, thus saving calorimeter theory from the spectral index argument, at least for starbursts.
However, the energy CR electrons lose to bremsstrahlung, ionization, and IC cannot
39 go into synchrotron radio emission, an obstacle for any theory that assumes radio emission is directly proportional to the injected power of primary CR electrons.
Therefore, even electron calorimetry and UV calorimetry are not enough to guarantee a linear FRC.
We address these issues with one-zone numerical models of CRs in star-forming galaxies. These models include CR escape as well as the main cooling processes and secondary production, a combination that has not to our knowledge been done over the entire span of the FRC. CRs in individual galaxies have been studied with similar one-zone models that fit the emission across the electromagnetic spectrum
(e.g., Arp 220 in Torres 2004b; M82 in de Cea del Pozo et al. 2009a), but in this paper our focus is on the FRC itself and not any individual galaxy. Our one-zone approach allows us to efficiently parameterize unknown quantities like the magnetic
field strength, and to try a large number of scenarios. The primary independent variable in our calculations is the gas surface density Σg, which controls both the photon energy density through the observed Schmidt law and the average gas density. These simplifying parameterizations allow us to qualitatively understand the FRC over the range of star-forming galaxies, although it ignores deviations and complications that may be important for individual galaxies.
We first describe the calculations necessary to find the CR spectra and observables for each galaxy (Section 2.2). We review the effects of each parameter on the observables (Section 2.3), before presenting our results (Section 2.4).
40 We discuss the implications of our work for CR physics (Section 2.5), including whether calorimetry is correct (Section 2.5.1), what causes the FRC (Section 2.5.2), predictions for the FRC at other frequencies (Section 2.5.3), the spectral slope problem (Section 2.5.4), the gamma-ray luminosities of galaxies (Section 2.5.5), and whether CR pressure and magnetic pressure are important as feedback mechanisms in galaxies (Section 2.5.6). We finally summarize our results (Section 2.6). In
Appendix 2.7, we present results from a suite of variants on our standard model, and show that those consistent with the FRC have similar parameters to our standard model. For the reader’s convenience, we list the symbols we use in our calculations and discussions in Table 2.1.
2.2. Procedure
We construct one-zone leaky box models of galaxies across the dynamic range of the observed FRC. We treat star-forming galaxies as homogeneous disks of gas, characterized by a column density Σg, a star formation rate surface density ΣSFR, and a scale height h. We solve the steady-state diffusion-loss equation for the equilibrium CR spectra of primary and secondary electrons and positrons, as well as primary CR protons.3
3In the steady-state approximation, the ∂N/∂t term in the diffusion-loss equation is assumed to be small. For the Milky Way (and presumably other normal spirals), the CR flux is known to be constant within a factor of 2 for the last billion years from solar system studies (Arnold et ∼ 41 Under these simplifying assumptions, the diffusion-loss equation for CRs becomes
N(E) d [b(E)N(E)] Q(E) = 0, (2.1) tlife(E) − dE −
where E is the total energy, N(E) is the CR spectrum, tlife(E) is the energy- dependent lifetime to diffusive or advective escape from the system, Q(E) is the
CR source term, and b(E) = (dE/dt) is the rate of energy loss for each particle. − The equilibrium CR spectrum is a competition at every energy between injection, cooling, and escape losses. If the injected CRs initially have a spectrum of the form
Q(E) E−p and if escape is insignificant (t (E) ), the final spectrum will ∝ life → ∞ have the form N(E) E1−p/b(E). If instead cooling is insignificant compared to ∝ escape (b(E) 0), the final spectrum will have the form N(E) E−pt (E). → ∝ life
We solve the general form of equation (2.1) numerically using a Green’s function for CR protons, electrons, and positrons (see Torres 2004b). We include synchrotron, IC, bremsstrahlung, and ionization losses (e.g., Rybicki & Lightman al. 1961; Schaeffer 1975). The tightness of the FRC combined with the long timescales for galactic evolution also imply the CR population in normal galaxies is steady state. Additionally, in extreme starbursts, the IC cooling time alone for GHz-emitting CR electrons (< 104 yr) is much shorter than ∼ the characteristic timescale for the system to evolve. Therefore, we expect that the steady-state assumption is valid. However, we note that in weaker starbursts, where the cooling and escape times for CRs are several Myr, evolution may be important and the steady-state approximation may fail (see Lisenfeld et al. 1996b).
42 1979; Longair 1994). For CR protons, we also include pion losses in tlife(E) due to inelastic proton-proton collisions using the formalism of Torres (2004b).4 The publicly available GALPROP code5 (Strong & Moskalenko 1998; Strong et al. 2000;
Moskalenko et al. 2002) is used to calculate the differential cross section for electron or positron production from proton-proton collisions, as well as for calculating the spectrum of γ-rays produced by the decay of secondary π0 mesons. We also include knock-off electrons from CR proton collisions with atoms in the ISM (see Torres
2004b).
2.2.1. Primary CR Injection Rates
We assume that both the primary CR electrons and protons are injected into galaxies with power law spectra Q(E) = CE−p with 1 γ 106 (where ≤ ≤ γ = E/(mc2) is the Lorentz factor), and we consider initial spectral slopes p in the range 2.0 p 2.6. Integrating the injection spectrum times the kinetic energy per ≤ ≤ particle K gives the total power injected per unit volume for each primary species,
4Including pion losses in the cooling term b(E) instead is formally incorrect, because the losses are catastrophic instead of continuous. For p = 2.0, the resulting proton spectra are nearly identical, but for p = 2.6, including pion losses in b(E) decreases the proton spectrum N(E) by 40% when ∼ pion losses are strong. 5Specifically, we use the “PP MESON” subroutine, which calculates the cross sections for electron and photon production through pion production. GALPROP is available at http://galprop.stanford.edu.
43 ǫCR = KQ(E)dE, to set the normalization. Energetic and escape losses produce R the final, steady-state spectrum as determined by the solution to equation (2.1).
In order to normalize the CR injection spectra, we assume that a constant
51 fraction ξ and η of the kinetic energy of SN explosions (E51 = ESN/10 erg) goes into accelerating primary CR electrons and protons, respectively. The CR electron and proton emissivities ǫCR are then proportional to the emissivity in starlight photons,
6 ǫph, produced by star formation, when averaged over the star formation episode.
Following the discussion in Thompson, Quataert, & Waxman (2007), we calculate the starlight emissivity (here, in units of erg s−1 cm−3) as
2 −1 ǫph = εΣSFRc (2h) , (2.2)
where ε = 3.8 10−4 is a dimensionless initial mass function (IMF) dependent × constant that relates the luminosity in young stars to the instantaneous star formation rate, c is the speed of light, h is the CR scale height (in cm), and ΣSFR is the surface density of star formation in cgs7 units of g s−1 cm−2 (Kennicutt 1998).
Equation 2.2 essentially says that some proportion of the mass that forms stars is converted into starlight. We take the surface density of star formation ΣSFR from the observed Schmidt law (Kennicutt 1998).
6Though we assume that the SN rate is proportional to the starlight, in reality, there will be a lag between the first massive stars and the first SNe when there will be very few CRs (e.g., Roussel et al. 2003), which we do not account for. 7Note that 1 g s−1 cm−2 = 1.5 1011 M yr−1 kpc−2 × ⊙ 44 The emissivity in CR electrons can then be written as
−5 ξ ǫCR, e = 9.2 10 E51ψ17ǫph , (2.3) × 0.01!
−1 where ψ17 = (ΓSN/ε)/(17M ) and ΓSN is the SN rate per unit star formation. ⊙ Similarly, the total emissivity of the primary CR protons can be written as
ǫCR, p = δ ǫCR, e
−4 δ ξ = 9.2 10 E51ψ17ǫph , (2.4) × 10! 0.01! where δ η/ξ is the ratio of the total energy injected in CR protons to that in ≡ CR electrons per SN. Although we have normalized ξ = 0.01 and δ = 10 in the above expressions for reference, one purpose of this paper is to show explicitly that numbers in this range are in fact compatible with observations of radio emission from star-forming galaxies.
2.2.2. Environmental Conditions
Escape
The CR lifetime (tlife) for both primary electrons and protons in equation (2.1) is uncertain, and probably varies from normal spirals like our own, where losses are mainly diffusive (e.g., Longair 1994), to dense starbursts like M82 and Arp
45 220, where CRs are likely advected in a large-scale galactic wind (e.g., Seaquist &
Odegard 1991).
We use a prescription for diffusive losses motivated by observations of beryllium isotope ratios (“CR clocks”) at the Solar Circle, which suggest that the confinement timescale for CR protons with E > 3 GeV is (e.g., Garcia-Munoz et al. 1977; ∼ Engelmann et al. 1990; Connell 1998; Webber et al. 2003)
E −1/2 tdiff (E) = 26 Myr . (2.5) 3 GeV
Identifying t with a diffusion timescale on a physical scale of kpc implies that life ∼ CR protons diffuse in the ISM of the Galaxy with a scattering mean free path of order pc. Although the behavior of t at lower energies is uncertain because of ∼ life the effects of solar modulation (compare, e.g., Engelmann et al. 1990 & Webber et al. 2003), we use equation (2.5) for all fiducial models employing diffusive losses.
Variations to the diffusion constant are considered in Appendices 2.7.6-2.7.8.
There is ample evidence for large-scale mass-loaded winds in starburst galaxies
−2 with Σg > 0.05 g cm (Heckman et al. 2000; Heckman 2003). These winds can ∼ advect CRs out of their host galaxies on a short timescale with respect to equation
(2.5), thus affecting both the predicted emission and the overall equilibrium energy
46 density of CRs. For this reason, we consider in Section 2.7.2 models of starburst galaxies with
h −2 twind = 300 kyr (Σg > 0.05 g cm ), (2.6) vwind ≈
−1 where we have taken h = 100 pc, a wind speed of vwind = 300 km s , and a cutoff
−2 between starburst and non-starburst galaxies of Σg > 0.05 g cm as reference values.
−1 −1 −1 The combined escape time in equation (2.1) is then given by tlife = tdiff + twind.
Scale Height
In our one-zone models, the galaxy scale height represents the volume in which the CRs are confined for tlife (eq. 2.1). Because we specify the properties of galaxies by their surface density, h is also important in determining the average gas density seen by CRs, which, in turn, is important for bremsstrahlung, ionization, and pion losses (see Section 2.2.2).
−2 We adopt h = 1 kpc for normal galaxies (Σg < 0.05 g cm ), although we consider several other scale heights in Appendix 2.7.3. There are several relevant scale heights which are not identical, and we must choose one for our one-zone model. CRs are injected in the gas disk (h 100 pc), but we do not use this ≈ scale height since the CRs diffuse and emit synchrotron outside the gas disk. The observed beryllium isotope ratios, as interpreted by CR diffusion models imply that the CRs have a scale height of 2 - 5 kpc in normal galaxies (Lukasiak et al. 1994;
47 Webber & Soutoul 1998; Strong et al. 2000). The magnetic field scale heights of normal galaxies are also several kpc (Han & Qiao 1994; Beck 2009). Finally, radio emission in most normal galaxies come from two disks: a thin disk with h 0.3 kpc ≈ and a thick disk with h 2 kpc (e.g., Beuermann et al. 1985; Dumke & Krause ≈ 1998; Heesen et al. 2009). On average, the thin and thick disks emit the same radio power at 1.4 GHz, and one component fits of the radio emission of normal galaxies ∼ generally find a one-component scale height of 1 1.5 kpc (Dumke et al. 1995, ∼ − 2000; Krause et al. 2006).
Since we are most interested in the radio emission of normal galaxies to explain the FRC, we use the value of h = 1 kpc from the one component fits. However, this one-zone approach does not capture all the relevant physics of the CRs. In particular, the escape time in Equation (2.5) applies to the entire CR halo. Escape from the radio-emitting regions is likely quicker and is probably underestimated in our models. Conversely, a variant on our fiducial model with large h in Appendix 2.7.3 probably overestimates the synchrotron losses in normal galaxies, since it does not account for the lower magnetic field strengths far from the midplane. Note that h = 1 kpc implies a vertical diffusion constant of D 7 1027cm2 s−1(E/GeV)1/2 z ≈ × (for 1.4 GHz emitting electrons in the Milky Way, D 1.3 1028 cm2 s−1; compare z ≈ × with the values in Dahlem et al. 1995 and Ptuskin & Soutoul 1998).
48 Starburst galaxies (Σ 0.05 g cm−2) are considerably more compact, both in g ≥ terms of their star forming regions and in terms of their CR confinement zone, and for them we adopt h = 100 pc (e.g., Downes & Solomon 1998).
ISM Density
Estimates from beryllium isotopes imply that the average density experienced by CRs is about one fifth to one tenth that of the Galactic disk. This may be because the ISM is clumpy and the CRs avoid the clumps, or because the CRs spend significant time in the low-density Galactic halo.
CRs do not necessarily travel through gas with the mean ISM density. The actual average density CRs experience depends upon the injection and propagation of the CRs, which depends on the small-scale ISM structure in galaxies and starbursts.
For example, most of the volume of the ISM in galaxies is low density material, and we can imagine the CRs are injected into this low density phase and lose their energy before encountering high density clumps. Then the density experienced by
CRs is lower than the average gas density. Conversely, we can imagine that CRs are preferentially injected into high density clumps, and are confined there by magnetic
fields in the clumps, in which case, the CRs experience a higher density than the average density of the galaxy or starburst.
49 For this reason we include a parameter f, to account for these unknown propagation and injection effects, defined by
n = f n , (2.7) eff h i
that measures the effective density “seen” by CRs (neff ) with respect to the average density of the CR confinement volume, n = Σ /(2h). For f > 1 or f < 1, the CRs h i g traverse over- or under-dense material compared to n , respectively. Note that we h i are defining f with respect to the CR confinement volume and not the gas disk.
The primary importance of the parameter f is in determining the importance of bremsstrahlung and ionization losses for CR electrons and positrons, and of pion production from inelastic proton-proton collisions.
Note that even though both the star formation rate and the magnetic fields of galaxies are taken to depend on the surface density (see Section 2.2.2), they are assumed to be independent of f. This means that the radiation energy density and the magnetic field the CRs experience are assumed to be average, while the CRs are allowed to traverse through underdense or overdense material. Although we allow ourselves this freedom in the modeling, it turns out that the models with f 1 are ≈ most consistent with observations in the Milky Way. For example, our adopted gas surface density at the Solar Circle, Σ = 2.5 10−3 g cm−2 (Boulares & Cox 1990), g × and scale height h = 1 kpc imply an average number density of n = 0.24 cm−3. h i 50 Since the CRs are inferred to travel through material of density n 0.2 0.5 cm−3 eff ≈ − (e.g., Connell 1998; Schlickeiser 2002), this implies that f 1. ≈
Interstellar Radiation Field
The interstellar radiation field is important for determining the IC losses for CR electrons and positrons. The primary contributions to the interstellar radiation field are starlight and the cosmic microwave background (CMB). The latter is particularly important for low surface brightness galaxies where it dominates starlight. Both sources of radiation are included in all models.
When the galaxy is optically-thin to the re-radiated FIR emission from young stars, then the energy density in starlight, which dictates the IC cooling timescale, is simply
Uph,⋆ = F⋆/c = εΣSFRc (2.8) 1.4 −9 Σg −3 = 3 10 − erg cm , (2.9) × g cm 2 !
where the surface density of star formation ΣSFR is connected to the average gas surface density by the Schmidt law. For large gas surface densities (Σg > 0.1 1 g ∼ − 51 cm−2) galaxies become optically thick to the reradiated FIR emission and
Uph,⋆ = (τFIR + 1)F⋆/c =(τFIR + 1)εΣSFRc (2.10) 1.4 −9 Σg −3 = 3 10 (τFIR + 1) − erg cm , (2.11) × g cm 2 !
where τFIR = κFIRΣg/2 is the vertical optical depth, and κFIR is the Rosseland mean dust opacity. For parameters typical of starbursts and ULIRGs, κ 1 10 cm2 FIR ≈ − g−1 for Galactic dust-to-gas ratio and solar metallicity (Semenov et al. 2003). For our standard models (Section 2.4.1), we assume that the CRs are always in optically thin regions, so that equation (2.9) holds. However, we discuss models with τFIR > 0 in Section 2.7.4.
Magnetic Fields
A primary motivation for this work is to determine how the average magnetic energy density of galaxies scales from normal galaxies like our own to dense ULIRGs like Arp 220. Observations of Zeeman splitting in ULIRGs supports a relatively strong scaling of magnetic field strength with gas surface density (Robishaw et al.
2008). To test a suite of models for consistency with observations, we parametrize the global average magnetic field of galaxies as
a Σg B = 6 − µG, (2.12) 0.0025 g cm 2 !
52 where a is determined from comparing with the FRC, and where the normalization has been chosen to match fiducial numbers at the Solar Circle (as in Boulares &
Cox 1990; Strong et al. 2000; Beck 2001). The magnetic field energy density is then
2 a just UB = B /(8π). The Σg dependence is motivated by the Parker instability: the
2 magnetic energy density cannot exceed the gas disk midplane pressure πGΣg, or else the magnetic field will buoy up out of the disk and escape (Parker 1966). A natural scaling for B given the Parker limit would be B Σ . The Σa scaling also ∝ g g arises if the magnetic field is in equipartition with the starlight, because the Schmidt law implies that U Σ1.4; in this case a = 0.7. We consider 0.4 a 1.0. We ph ∝ g ≤ ≤ assume that the magnetic field is constant within the confinement volume of scale height h, a reasonable assumption based on the observed radio halos of galaxies and
Galactic pulsar rotation measures (Han & Qiao 1994). We also consider two other parameterizations of the magnetic field, B ρa and U = U , in Section 2.7.1 and ∝ B ph Section 2.7.5, respectively.
2.2.3. Observables and Constraints
We use two broad conditions to select successful models. We first ask if the model satisfies the FRC. However, since the FRC alone does not necessarily demand
CR protons at all, a second constraint is needed to fix the overall CR proton normalization. We consider two sets of constraints on the protons, either using
Earth-based measurements of CRs, or observations of the entire Milky Way.
53 1. Reproduce the FIR-radio correlation. The nonthermal radio luminosity is
calculated directly from our synchrotron spectrum as νǫν at ν = 1.4 GHz.
We do not include the thermal free-free contribution to the radio luminosity;
however, the thermal radio luminosity of most galaxies is typically small at
GHz frequencies. Nor do we consider the effects of free-free absorption. The
total infrared (TIR) luminosity8 is ǫ [1 (1 exp( τ ))/τ ], with the UV ph − − − UV UV
optical depth τUV through the entire disk calculated as κUVΣg. We adopt a
UV opacity of 500 cm2 g−1, which is roughly appropriate at wavelengths of
1000 A˚ (T 30, 000 K) and Galactic metallicity and dust-to-gas ratios (Li ∼ eff ≈ −2 & Draine 2001; Bell 2003 use κUV = 190 g cm , using a smaller dust-to-gas
ratio and 1500 A).˚ Then, LTIR/Lradio is simply the ratio of these luminosities,
9 and can easily be converted into qFIR, an observable quantity we calculate as
LTIR qFIR = log10 3.67 (2.13) Lradio −
8While some of the light absorbed by dust is emitted as far-infrared (40 - 120 µm), some is also emitted in near or mid infrared. For simplicity, we assume that TIR light is directly proportional to the FIR emission. For starburst galaxies, LTIR 1.75LFIR (Calzetti et al. 2000), which we apply to ≈ every galaxy for simplicity. This correction has been applied to the reported LFIR/Lradio in Yun et al. (2001) to get our quoted observed LTIR/Lradio. Bell (2003) reports a similar LTIR/LFIR 2 for ≈ galaxies on the FRC. 9 Our version of this equation divides our calculated LTIR by LTIR/LFIR 1.75 (Calzetti et al. ≈ 2000) to get to the true, observed FIR emission. See Helou et al. (1985) for the usual definition of q.
54 and defined in Helou et al. (1985). The normalization of the FRC is
L /L = 9 105 (Yun et al. 2001), which we match by adjusting ξ TIR radio × appropriately, therefore fixing the primary CR electron injection rate in
galaxies (Section 2.2.1).
Once we have the ratio LTIR/Lradio, our primary constraint is that we require
a linear FRC to exist.
We require that
max(L /L ) TIR radio 2. (2.14) min(LTIR/Lradio) ≤
2. Fix the proton normalization. We then use two sets of constraints to fix the
proton normalization, the local constraints and the integrated constraints.
Each is considered independently for each model. For simplicity, the proton
normalization is assumed to be constant across the entire range of star-forming
galaxies.
The “local” set of constraints is based on in-situ measurements of CRs at
Earth. These are:
(a) For each electron energy, we calculate the ratio of the CR positron
number density to the total number density of positrons and electrons at
GeV energies. Below GeV energies, solar modulation of CRs can affect
the observed CR spectrum. Above 1 GeV, the observed positron flux ∼ exceeds the predicted flux even in detailed models (see Moskalenko &
55 Strong 1998; Beatty et al. 2004; Adriani et al. 2009; but see Delahaye et
al. 2009). The observed value of e+/(e+ + e−) is 0.1 at GeV energies (e.g.,
Schlickeiser 2002; Adriani et al. 2009).
We require as a local constraint that 0.05 e+/(e+ + e−) 0.2 at ≤ ≤ −2 1 GeV when Σg = 0.0025 g cm .
(b) We also compute the ratio of the proton number flux and electron number
flux at GeV energies. The ratio is observed to be p/e 100 at Earth ≈ at energies of a few GeV (e.g., Ginzburg & Ptuskin 1976; Schlickeiser
2002). This value is also inferred from SN remnants, which are believed
to accelerate CRs (Warren et al. 2005).
We require as a local constraint that 50 p/e 200 at 10 GeV when ≤ ≤ −2 Σg = 0.0025 g cm .
As an alternative way to find the CR proton normalization, we considered a separate “integrated” constraint for the entire Milky Way galaxy using an average Σg inferred from the Galactic scale radius and star formation rate. Our purpose was to assess the possibility that the Earth is not in a representative location of the Galaxy; for example, it sits in the Local Bubble.
(a) We calculate the gamma-ray luminosity of the Galaxy from π0 decay.
We approximate the Milky Way as a uniform disk with R = 4 kpc,
−2 and a surface density of Σg = 0.01 g cm derived from the Schmidt
56 10 law (Section 2.2.2) and the Milky Way luminosity, L⋆ 2 10 L ≈ × ⊙ (Freudenreich 1998; similar results are obtained by using the starlight
radiation field in Strong et al. 2000, or the SN rate in Ferri`ere 2001).
Strong et al. (2000) calculate the total π0 gamma-ray luminosity to be
L 2 1039ergs s−1. π0 ≈ ×
We require as the integrated constraint that 1 1039ergs s−1 L × ≤ π0 ≤ 4 1039ergs s−1 when Σ = 0.01 g cm−2. × g
Additional checks: As an added check, we have the observed CR spectrum at
Earth. At high energies (γ 1), the observed CR electrons have (e.g., Longair ≫ 1994)
dI cN (E) E −3.3 e = e = 0.07 cm−2 s−1 sr−1 GeV−1 (2.15) dE 4π GeV and the observed CR protons have (e.g., Mori 1997; Menn et al. 2000; AMS
Collaboration et al. 2002)
dI cN (E) E −2.7 p = p 1.5 cm−2 s−1 sr−1 GeV−1. (2.16) dE 4π ≈ GeV
The predicted CR spectrum does not determine whether a model was considered formally “successful”, but it was used to select among the adequate models for the best standard set of parameters.
57 Although we did not use it directly, we also calculate the spectral slopes at
1.4 GHz from the radio synchrotron spectrum as a sanity check. These include the
4.8 instantaneous spectral slope α1.4 as well as the spectral slopes to 4.8 GHz (α1.4) and
8.4 4.8 8.4 GHz (α1.4). Unless otherwise stated, α refers to α1.4, the spectral slope from 1.4
GHz to 4.8 GHz. Typical values of α are 0.7 0.8. As a constraint, α can be very − sensitive to minor details in the model; we note that a difference of 0.2 in α results in only a 60% difference in the specific flux after one decade in frequency, and we are mainly concerned with factor of 2 accuracy in our models. Lisenfeld & V¨olk
(2000) have argued that α is decreased by 0.1 by SN remnants within galaxies, so ∼ our value of α is uncertain at that level. We also do not include free-free emission, which can flatten the spectral slope, especially in low surface density galaxies. In
ULIRGs like Arp 220, the observed α is typically 0.5 (Clemens et al. 2008), but ∼ radio emission in these galaxies may suffer free-free absorption which flattens the spectrum; the unabsorbed synchrotron spectrum α may be as high as 0.7 (Condon et al. 1991a). To some extent, a small to moderate difference in α from its observed value can be adjusted by altering p, since decreasing p by 0.1 generally decreases
α by 0.05, and p often is not well constrained in the considered range 2 2.6. − Given these uncertainties, caveats, and sensitivities in α, and given the vast range of galaxies and starbursts we are considering, and the simplified parameterizations we are using, we do not impose any direct constraint on α. Of course, models of individual galaxies should and do account for α when they model the radio emission.
58 Throughout this work, we assume that the local values of the proton normalization and propagation – in particular, δ, η, and f – are the same for both normal galaxies and starbursts. We use this assumption for simplicity, and to keep the number of free parameters reasonable. In practice, the CR acceleration efficiency and the proton-to-electron ratio may change somewhat from normal galaxies and starbursts, but we do not consider small variations necessary for a basic understanding of the FRC. More detailed models of individual systems can and do take these changes into account, and we refer readers to these models if they wish to understand starburst galaxies in detail. It is also conceivable that f changes dramatically from normal galaxies to starbursts. Again, we do not consider this possibility in this paper, although we will explore the consequences of very low f applying to only starbursts in a future paper.
2.3. Review of Physical Effects of Parameters
To search for models that satisfy the observational constraints listed in Section
2.2.3, our grid of models spanned values of a (eq. 2.12), f (eq. 2.7), ξ and δ (eqs. 2.3 and 2.4), and p (Section 2.2.1). For a listing of these parameters of the model, see
Table 2.1. As background for interpreting our results in 2.4, we briefly review the § effects of these quantities on observables.
59 2.3.1. Injection Parameters: ξ, η, δ, and p
The parameter ξ is the normalization of the injected primary CR electron spectrum, and with δ = η/ξ, the injected CR proton spectrum normalization
(eqs. 2.3 & 2.4). Changes in ξ do not affect the shape of any of the equilibrium CR spectra. For fixed δ, larger ξ linearly increases the CR luminosity and energy density within galaxies, and thus — for fixed galaxy parameters — the luminosity of CRs in all wavebands, including the radio, neutrino, and gamma-ray luminosities.
An increase in δ at fixed ξ raises the number of secondaries from protons, the
+ + − − − ratios e /(e + e ) and esec/e , and the luminosity from pion decay, Lπ.
Of note is the ratio of injected protons to electrons at relativistic energies.
−p Suppose the electrons are injected with a spectrum Qe(E)= CeE and the protons
−p are injected with a spectrum Qp(E) = CpE . Then, given our normalization
2 2 γmaxmec −p γmaxmpc −p 2 2 conditions ǫCR, e = mec CeKE dE and ǫCR, p = mpc CpKE dE, where K R R is the kinetic energy (see Section 2.2.1), it can be shown that
C m p−2 δ˜ p = δ p . (2.17) ≡ Ce me
The quantity δ˜ represents the proton to electron ratio at high energies
(m c2 E γ m c2) if there were no escape, energy losses, or secondary p ≪ ≪ max e production. Note that it is not generally equal to δ, since δ is largely dependent on the shape of the spectrum at low energies. Our injection spectra go as E−p,
60 2 where E is the total energy: the electron spectra stretch down to mec while the
2 proton spectra only extend down to mpc . For steep spectra (p> 2), the low-energy
2 particles receive most of the energy, so that electrons with E This follows from the fact that the FRC is observed at GHz frequencies, implying electron energies of order 100 MeV to 10 GeV (eq. 2.19). Thus, the actual quantity we constrain is δ˜. Note that the relationship between Cp/Ce and δ would be different for another spectrum, such as K−p or γ−p. The spectral slope p of the injected CRs in part controls the final, propagated spectral slope dlogN(E)/dlogE. The spectral slope, in turn, determines how P ≡ much the secondary particles are diluted. Protons at energy E produce secondary electrons and positrons of energy E′ < E; a steeper primary spectrum increases the number of primaries at these lower energies compared to the proton energy E. Therefore, a larger p (and thus a bigger for primary electrons) implies a smaller P + + − − − e /(e + e ) and esec/e . This dilution implies that even in the limit of full proton calorimetry, primary electrons may be more important than secondaries. Similarly, the secondary fraction is not a good measure of proton calorimetry in itself. For our 10 2 Conversely, the proton spectrum extends to a maximum energy of γmaxmpc , much greater than the maximum energy of the electrons; for shallow spectra (p < 2), the reservoir of energy in these 2 high energy protons would lower Cp/Ce at E < γmaxmec . 61 standard model, though, we find that in proton calorimeters, secondary electrons and positrons outnumber the primary electrons 4-1. ∼ 2.3.2. Magnetic Field The magnetic field strength affects the CR spectra in several ways: 1. It determines the importance of synchrotron cooling relative to other radiative and escape losses. The synchrotron cooling timescale for CR electrons and positrons emitting at frequency νGHz = ν/GHz is − − t 4.5 107 B 3/2ν 1/2 yr, (2.18) synch ≈ × 10 GHz where B10 = B/10 µG. For normal galaxies, tsynch is comparable to, but somewhat longer than, the inferred diffusive escape timescale for the CR electrons producing GHz emission in normal galaxies (eq. 2.5). For the mG (and larger) fields thought ∼ to exist in the densest starbursts, tsynch is shorter than even the advection timescale (eq. 2.6). 2. The relative importance of synchrotron also affects the propagated equilibrium spectral slope of electrons and positrons; stronger magnetic fields P imply steeper final spectra (see Section 2.3.1). In the limit that cooling dominates escape, and that synchrotron is the main form of cooling, the equilibrium spectral slope is =1+ p. P 62 3. The magnetic field strength determines the critical synchrotron frequency (νC ) for electrons and positrons: γ 2 νC 3.3 4 B10 GHz. (2.19) ≈ 10 At a fixed observed frequency (such as 1.4 GHz), a stronger magnetic field implies that we see lower energy electrons and positrons. 4. When synchrotron cooling dominates over other cooling and escape losses, a stronger magnetic field lowers the equilibrium energy density of CR electrons and positrons, because of increased losses. However, in this calorimeter limit, each electron and positron has a higher luminosity. Therefore, Lradio approaches a maximum set by ξ, and is not affected by further increases in the magnetic field strength. This effect is the essence of the original calorimeter theory. All else being equal in our models of non-calorimetric galaxies, larger magnetic fields imply that a larger share of injected CR electron power is lost to synchrotron, because of the faster synchrotron cooling time. In cases when synchrotron does not already dominate, increasing the magnetic field strength thus increases Lradio . Note that the magnetic field in our models is normalized to the local Solar Circle gas surface density (eq. 2.12), so that changing a has no effect on local Milky Way constraints discussed in Section 2.2.3. 63 2.3.3. Effective Density The ISM density encountered by CR protons controls the production rate of secondary electrons and positrons, as well as gamma rays and high-energy neutrinos, from inelastic proton-proton collisions. The proton lifetime to pion losses is −1 7 f n tπ 5 10 yr h −i , (2.20) ≈ × cm 3 ! from Mannheim & Schlickeiser (1994) (see also Torres 2004b). Higher n = f n eff h i (eq. 2.7) means more secondaries, higher Lradio, and higher gamma-ray and neutrino luminosities. The secondary electrons and positrons raise the ratios e+/(e+ + e−) − − and esec/e , and they lower the equilibrium ratio p/e. Additionally, if the ratio of primaries to secondaries changes with energy, then the combined spectral slope P for electrons and positrons can be altered, which affects the observed radio spectral slope (see Section 2.3.1). The effective ISM density also determines the efficiency of bremsstrahlung and ionization losses for CR electrons and positrons, with higher densities making these processes more efficient. The bremsstrahlung and ionization energy loss timescales are −1 7 f n tbrems 3.7 10 yr h −i , (2.21) ≈ × cm 3 ! 64 and −1 8 −1/2 1/2 f n tion 2.1 10 B10 νGHz h −i yr, (2.22) ≈ × cm 3 ! respectively, where we have again scaled the energy dependence of tion for CR electrons and positrons emitting at GHz frequencies for comparison with tsynch (eq. 2.18). Importantly, energy lost to bremsstrahlung and ionization is not radiated in the radio, so higher f implies lower Lradio from these processes, all else being equal. The energy dependence of these cooling processes also flattens the propagated equilibrium electron and positron spectra (see the discussion after eq. 2.1; Section 2.3.1). For example, when tsynch = tbrems at some energy and all other losses are negligible, then = p + 1/2 and α = p/2 1/4. Similarly, when t = t and P − synch ion there are no other losses, = p and α = p/2 1/2. P − 2.3.4. The Schmidt Law and the Photon Energy Density The energy density of photons, and thus the importance of IC losses for CR electrons and positrons in star-forming galaxies, is set by the slope and normalization of the Schmidt law. The IC cooling timescale for CR electrons and positrons emitting radio synchrotron at frequency ν is − t 1.8 108 B1/2ν 1/2U −1 yr (2.23) IC ≈ × 10 GHz ph, −12 65 −12 −3 where Uph, −12 = Uph/10 ergs cm is the photon energy density scaled to that for a typical star-forming galaxy. For optically-thin galaxies obeying the Schmidt law of Kennicutt (1998) and ignoring the CMB, F /c = U Σ Σ1.4 ⋆ ph,⋆ ∝ SFR ∝ g (eq. 2.9). Then, for fixed frequency, the IC lifetime therefore scales as t Σa/2−1.4 IC ∝ g if the magnetic field strength varies as B Σa. We do not consider variations on ∝ g the Schmidt law (e.g., Bouch´eet al. 2007), but their effects can be inferred from equation 2.23: if ΣSFR has a steeper increase with Σg, then tIC will fall more rapidly with surface density. In practice, the CMB will make IC losses more efficient in the lowest density galaxies, and any FIR opacity (Section 2.2.2) will make them more efficient in high-density starbursts. Just as bremsstrahlung and ionization losses can reduce the share of energy left for synchrotron radiation, a greater photon energy density and IC power decreases Lradio. Unlike bremsstrahlung and ionization, though, IC losses produce a steep spectrum. In the limit where they dominate other losses and escape, =1+ p, as P in the case of pure synchrotron cooling, and α = p/2. 66 2.4. Results 2.4.1. Standard Model We adopt p = 2.3, f = 1.5, a = 0.7, δ˜ = 48 (δ = 5.0), and ξ = 0.023 as our fiducial model. This model reproduces the FRC, as seen in Figure 2.1 (solid line). In this particular model, we require ξ = 0.023 to match the normalization of the FRC. The ratio of FIR to 1.4 GHz luminosities varies by only 1.7 over the entire range of Σg, and shows no obvious trend. However, the scatter appears to be concentrated at the low-Σg end of the FRC, with LTIR/Lradio varying by less than 12% in the starbursts in this model. The standard model also satisfies both local and integrated constraints on the proton normalization, as well as the observed CR spectrum. Our positron ratio at 1 GeV, e+/(e+ + e−) = 0.10, and proton-to-electron ratio at 10 GeV, p/e = 82, are good matches to the observed values. The Milky Way γ-ray luminosity in this model, 2.0 1039 ergs s−1, is also a good match to the value of Strong et al. (2000). × The predicted proton spectrum at Earth in this model is 91% 118% of its − observed value at 1, 10, and 100 GeV, implying that p is well matched to the Galactic CR spectrum. Similarly, the CR electron flux at Earth is 122% of its observed value at 10 GeV. The least satisfactory aspect of this model is the predicted spectral slope 67 α for Milky Way-type galaxies (α 0.9 1.0), which is somewhat too high. Our ≈ − results are, however, reasonable for starbursts (α 0.5 0.7; see Figure 2.2). ≈ − We emphasize that the parameters of our standard model are adequate for all star-forming galaxies on the FRC. We discuss the many competing effects that yield the FRC in Section 2.5.2. 2.4.2. Degeneracy in the Standard Model Our local set of constraints (Section 2.2.3) narrow down the allowed parameter space considerably. The models that survive have δ˜ 34 100 and a = 0.6 0.7. ≈ − − To get the correct normalization of the FRC, we must set 0.019 ξ 0.027 when ≤ ≤ p = 2.3, so that 0.097 η 0.22. Flatter injection spectra generally have lower ξ ≤ ≤ (down to 0.006 for p = 2.0) and higher η (reaching 0.28 for p = 2.1), while steeper injection spectra generally have higher ξ (up to 0.18 for p = 2.6) and lower η (as low 4.8 as 0.09, which occurs when p = 2.4). However, α1.4 is somewhat high when p > 2.2 ∼ (0.9 1 predicted compared to 0.7 0.9 observed) for normal galaxies, but is close − − to observed values for p < 2.2 (0.8 0.9 predicted). The spectral slope is sufficiently ∼ − low (α4.8 0.5 0.7 predicted) for starbursts. 1.4 ≈ − The integrated Milky Way γ-ray luminosity from π0 decay provides similar, but somewhat weaker constraints, favoring lower δ˜. At low p = 2.0, models with 10 < δ < 50 are selected by the γ-ray luminosity. Higher p models continue to ∼ ∼ 68 work so long as δ decreases, because the normalization depends on the spectrum at very low energies as discussed in Section 2.3.1. We can take the normalization into account by comparing δ˜ (eq. 2.17), and we find that the allowed δ˜ (10 < δ˜ < 100) ∼ ∼ slowly increases with p (from roughly 25 at p = 2.1 to 91 at p = 2.6 when f = 1.0) and decreases with f (from roughly 60 at p = 2.2, f = 1.0 to 50 at p = 2.2, f = 2.0). Even p = 2.6 models predict an FRC and the correct π0 luminosity of the Milky Way; we would need to take into account either the observed α in normal galaxies or the local observed CR spectral slope to further constrain p in our standard model. For example, p = 2.6 works when δ˜ 90 (δ 1), and ξ 0.17. As discussed ≈ ≈ ≈ in Section 2.3.1, a higher p dilutes secondaries and lowers the fraction of electrons that are secondaries. Therefore, the secondaries contribute a smaller fraction of the radio luminosity and are less likely to break the FRC as galaxies become proton calorimeters at high density. The FRC can then tolerate a higher secondary production rate (and ultimately higher δ˜) for high p. A higher ξ is needed, though, since more of the energy goes into unobserved low energy electrons. These high p solutions also produce steep synchrotron spectra with α 1.1 1.2 in normal ≈ − galaxies, and can be constrained by the spectral slopes. More broadly, we also consider variations on our usual parametrization, such as lifetimes including advection, different scale heights, and FIR optical depths. Many of these variations are inconsistent with the constraints in Section 2.2.3. However, 69 those that do satisfy the constraints had similar values for ξ, δ˜, p, and a as the fiducial model. We describe these variants in detail in Appendix 2.7. 2.4.3. General Features of the Particle Spectra −2 We show typical predicted CR spectra in Figure 2.3 for Σg = 0.001 g cm and 10 g cm−2, the lowest and highest surface densities we consider. In low surface density galaxies, the protons with γ 1 have a power-law ≫ spectrum with about 0.5 greater than the injected spectral index p. The increased Pp steepness comes from faster diffusive escape at higher energies (eq. 2.5; Ginzburg & Ptuskin 1976). At lower energies, the CR proton spectrum flattens due to ionization losses, which are constant with energy (Schlickeiser 2002; Torres 2004b). High surface density galaxies have harder proton spectra with = p at high energies, Pp since pion losses overwhelm escape and are roughly energy independent. In low surface density galaxies, the primary electrons behave similarly to protons. For most low energies, they have a power-law spectrum with 0.5+ p, Pe ≈ caused by diffusive escape losses (e.g., Ginzburg & Ptuskin 1976). However, synchrotron and IC losses steepen the spectrum at high energies. Bremsstrahlung and ionization flatten the spectrum at lower energies (E < 1 GeV) (compare ∼ Figure 2.6; see also Thompson et al. 2006; Condon 1992). In high surface density 70 galaxies, diffusive losses are negligible compared to synchrotron and IC losses, thus forcing 1+ p at higher energies (e.g., Ginzburg & Ptuskin 1976). Pe ≈ The spectra of secondary electrons and positrons show additional features with respect to the primary electron spectrum. The secondary (pion-produced) spectrum is flatter than the primary spectrum at low energies, because the production cross sections decreases near the pion production threshold (see Figure 2.8; Strong & Moskalenko 1998; Torres 2004b). At high energies, the pion electrons and positrons are injected with a spectrum proportional to the steady-state proton spectrum. This means that in low surface density galaxies, the pion electrons and positrons have a steeper spectrum than the primary electrons at high energies (Ginzburg & Ptuskin 1976), while in high surface density galaxies, the secondary and primary spectral slopes are the same at high energies (as seen in Figure 2.8). Furthermore, there are always more secondary pion positrons than secondary pion electrons (ultimately due to charge conservation). Knock-off electrons become increasingly important at very low energies and dominate as γ approaches one (see Torres 2004b). We also show radiation spectra in Figures 2.9 and 2.11 for synchrotron, π0 γ-rays, relativistic bremsstrahlung, and IC emission (see Section 2.5.5 for the assumptions used to estimate the IC emission). The radio, bremsstrahlung, and IC emission generally steepen with increasing frequency (compare with Figure 2.2; see also Lisenfeld et al. 1996a; Thompson et al. 2006), whereas pion γ-rays peak at 71 a few hundred MeV. Although we do not calculate it here, the overall high-energy neutrino emission is comparable to the γ-ray emission from π0 decay (Stecker 1979). 2.5. Discussion 2.5.1. Is Calorimetry Correct? We show the effects of forcing electron and UV calorimetry to hold in Figure 2.1 for our standard model (cf. Section 2.4.1). It is clear that most of the energy in −2 1.4 GHz electrons is lost radiatively in galaxies with Σg > 0.01 g cm : calorimetry ∼ holds in high- but not low-density galaxies (this behavior was first described in Chi & Wolfendale 1990 and was also predicted by Lisenfeld et al. 1996a). At lower surface densities, electron calorimetry begins to fail (decreasing the radio luminosity), but the effect of this on the FRC is largely mitigated by the decreasing optical thickness to UV photons (decreasing the FIR luminosity). This conspiracy saves the FRC, as discussed by Bell (2003), but only applies in our standard model over one decade in Σg. In our standard model, electron escape at low Σg eventually becomes the stronger effect, so that low-density galaxies would be radio dim with respect to the FRC. A high value of q is in fact observed as a nonlinearity in the FRC at low luminosities (Yun et al. 2001, though Beswick et al. 2008 find the opposite). Unfortunately, studies of the low luminosity FRC are complicated by the presence of thermal radio emission (e.g., Hughes et al. 2006, for the Large Magellanic Cloud), 72 which also correlates with FIR light and overwhelms the nonthermal synchrotron emission considered here in the lowest density galaxies. While the standard models predict that electron calorimetry holds in the inner Milky Way, there are physically motivated variants (Appendix 2.7) which predict that electron calorimetry fails for normal galaxies and the weakest starbursts. In particular, the “strong wind” variants predict a non-calorimetric inner Milky Way, because of the wind inferred by Everett et al. (2008) (see Appendix 2.7.2). Although the transition to calorimetry is model dependent, it seems unavoidable that extreme starbursts like Arp 220 are electron calorimeters. We can derive the speed vesc at which CRs would have to stream out of galaxies for electron calorimetry to fail, according to the cooling rates in our standard model. We can also compare these numbers to standard CR confinement theory, where CRs are limited to propagate at the Alfv´en speed (vA = B/√4πρ) by a streaming instability in the ionized ISM (Kulsrud & Pearce 1969). We can also invert the problem and determine the magnetic field with a high enough Alfv´en speed11 for CRs to stream 2 2 out of the galaxy in one cooling time (Besc = 4πρvesc), as well the diffusion constant 2 (Desc = h /tcool) needed to diffuse out of the system in one cooling time. The environmental conditions needed to allow CRs to escape before cooling significantly are reasonable for weak Σ 0.1 g cm−2 starbursts. We find that v = 620 km s−1, g ≤ esc 11This estimate assumes that CRs are streaming through material with the mean ISM density. In a lower density phase, vA will be larger and Besc will be smaller. 73 and winds of several hundred kilometers per second are in fact observed in starbursts. Similarly, we calculate D = 1.9 1028 cm2 s−1, and diffusion constants of order esc × 1028 cm2 s−1 are inferred for starburst galaxies (e.g., Dahlem et al. 1995). However, if the CRs stream through mean density ISM, then B 3 mG is higher than the esc ≈ equipartition magnetic field strength B = 8π2GΣ2 300 µG for Σ = 0.1gcm−2, eq g ≈ g q so that CR escape would have to be super-Alfv´enic. For higher Σg, though, −1 −2 escape would require extreme wind speeds (8000 km s when Σg =1gcm and 120, 000 km s−1 0.4c when Σ = 10 g cm−2), extremely high diffusion rates ≈ g (2.5 1029 g cm−2 for Σ =1gcm−2 and 3.6 1030 g cm−2 for Σ = 10 g cm−2), × g × g −2 or extremely strong magnetic fields (Besc = 0.11 G in the Σg =1gcm case −2 and Besc =5 G in the Σg = 10 g cm case) that are unreasonable. We therefore conclude that electron calorimetry must hold in dense starbursts. We can similarly ask whether galaxies are proton calorimeters. The low pion luminosity of the Galaxy and the secondary positron fraction at Earth imply that normal galaxies like the Milky Way are not proton calorimeters. We have estimated the proton calorimetry fraction Fcal in our models by adding the emissivity in pion products12 to the emissivity in CR protons with energy greater than 1.22 GeV, the pion production threshold energy. However, when we do this we find that even explictly proton calorimetric models with no diffusive or advective escape have 12 Since we do not calculate the neutrino spectrum, we simply assume that Qν = Qγ , which is a reasonable approximation at high energies. 74 F 0.5. This appears to be caused by an inconsistency between the pionic cal ≈ lifetime we use (eqn 2.20) from Mannheim & Schlickeiser (1994) and the GALPROP cross sections: if we add up all of the energy in all of the pionic products of a CR proton of energy GeV, the effective energy loss rate is several times smaller than ∼ implied by Mannheim & Schlickeiser (1994)13. We also note that the Mannheim & Schlickeiser (1994) pionic lifetime is twice as short at GeV energies as the pionic ∼ lifetime in Schlickeiser (2002). Finally, this approach ignores ionization losses, which do not create secondaries but will prevent lower energy protons from escaping. To account for this discrepancy in the energetics, we normalize our estimate of Fcal so that an explicitly proton calorimetric model of the same CR injection rate, Σg, p, and f has Fcal = 1. We then see in Figure 2.13 that dense starbursts with Σ 10 g cm−2 all are proton calorimeters with F 1 for several variants g ≈ cal ≈ (Appendix 2.7). As with electron calorimetry, proton calorimetry sometimes breaks −2 down for the Σg = 0.1 g cm weak starbursts, because the time to cross the 100 pc −2 starburst scale height is short. However, when Σg > 1 g cm , proton calorimetry ∼ holds in our models; a model with winds and strong diffusive losses has proton −2 calorimetry breaking down at Σg =1gcm (Fcal = 0.45). As with the electrons, we can derive the speed that the CRs would need to stream out of a starburst for proton calorimetry to fail, which is −1 −2 vesc = h/tπ = 1900 km s (Σg/g cm )f. While vesc is easily attained by winds in 13As far as we are aware, this discrepancy has not been discussed in the literature. 75 −2 starbursts with Σg = 0.1 g cm , only the fastest winds are capable of breaking −2 proton calorimetry in Σg = 1gcm starbursts. Diffusive escape limited to the mean Alfv´en speed of the starburst would require strong magnetic fields −1/2 −2 3/2 (Besc = 27 mG(h/100pc) (Σg/g cm ) f) with energy densities greater than the midplane gas pressure in starbursts to break proton calorimetry. We therefore conclude that proton calorimetry is difficult to avoid in f 1 starbursts with ≈ −2 Σg > 1 g cm . ∼ 2.5.2. What Causes the FIR-Radio Correlation? Calorimetry and the νC Effect Calorimetry provides a simple way to explain the FRC. We find that both electron and UV calorimetry hold for starbursts, and possibly the inner regions of normal galaxies, depending on the variant on our underlying model (Appendix 2.7). Calorimetry therefore serves as the foundation of our explanation for the FRC. Other effects alter the radio luminosity, both at low density and high density, but by a factor of 10, compared to the dynamic range of 104 in Σ . At the ∼ g order-of-magnitude level, calorimetry can be said to cause the FRC, and other effects are relatively moderate corrections. However, in more detail, we find that LTIR/Lradio is not in fact flat even in the simple calorimeter model, with no escape, non-synchrotron cooling, or secondaries 76 (the light dotted line in Figure 2.14). Instead, LTIR/Lradio decreases by a factor of 2.6 as Σg increases, because 1.4 GHz observations probe lower CR electron energies as the magnetic field strength increases. We call this decrease in LTIR/Lradio with Σg the “νC effect”. In general, the effect becomes more significant as p increases past 2.0, because the electron spectrum becomes steeper. It can be shown that in this simplest calorimeter limit, L /L Bp/2−1. TIR radio ∝ High-Σg Conspiracy The radio luminosity in high-density galaxies is altered from the calorimetric luminosity mainly by two mechanisms, non-synchrotron cooling and the appearance of secondary electrons and positrons. We illustrate these effects in Figure 2.14. In normal galaxies, synchrotron cooling dominates the energy losses, though bremsstrahlung and IC off the CMB can be competitive within a factor of a few or less. However, in starbursts, energy loss is mainly by bremsstrahlung and ionization. This decreases the proportion of energy lost that goes into radio. The energy diverted to bremsstrahlung and ionization therefore increases LTIR/Lradio by a factor of up to 20 in starbursts compared to normal galaxies (compare the dotted and ∼ long-dashed lines in Figure 2.14). Secondary electrons and positrons themselves radiate in the radio. In the starbursts, which are proton calorimeters, there are several times more secondaries 77 than primary electrons, while in normal galaxies, the secondary contribution is small. Secondaries increase the radio emission by a factor of 4 in starbursts compared to ∼ normal galaxies (compare the dotted and short-dashed lines in Figure 2.14). These effects each on their own alter the calorimetric radio luminosity by up to an order of magnitude. Since both are density dependent, they both become important in starbursts. However, combined with the νC effect (Section 2.5.2) in the simple calorimeter model, they largely cancel each other out to maintain a linear FRC. The exact magnitudes of these effects are model dependent, but they are always important and the direction each works in is the same in every case. It is possible that relaxing the assumptions of our approach, such as including time dependence or spatial variation, could avoid the severe non-synchrotron losses and secondary electrons and positrons giving rise to this particular high-Σg conspiracy. However, any new effects would have to be tuned to avoid the processes we already include while still reproducing the FRC, trading one conspiracy for another. There are two other effects that appear in our variants (Appendix 2.7), but not our standard model, which can change the FRC. First, if the magnetic field is assumed to depend on density instead of surface density (Section 2.7.1), the magnetic fields will be much stronger in the starbursts for the B ρa case, since ∝ the starbursts are more compact. This will make synchrotron cooling dominant again, upsetting the high-Σg conspiracy. This effect can be compensated by winds and a weak magnetic field dependence on ρ (low a). Second, if the FIR optical 78 depth is significant (Section 2.7.4), the photon energy density inside the galaxy is greater by a factor of τ than inferred from the photon flux alone. While ∼ FIR typical FIR opacities are small, the optical depth is appreciable in dense starbursts 2 −1 2 −1 (1 cm g <κFIR < 10 cm g ). This increases IC losses dramatically at the high ∼ ∼ densities, decreasing the radio luminosity. Models with large FIR optical depths have trouble reproducing the FRC (Section 2.7.4). Low-Σg Conspiracy The radio luminosity in low density normal galaxies is modified by a different pair of opposing mechanisms, the failure of electron calorimetry and the failure of UV calorimetry. This conspiracy is illustrated in Figure 2.1 for our standard model. Normal galaxies are not generally electron calorimeters – both diffusive and advective escape can operate faster than cooling. In weak starbursts −2 (Σg < 0.1 g cm ), escape can be competitive with cooling processes, but not in ∼ stronger starbursts. Escape therefore decreases the radio emission in normal galaxies compared to the calorimetric expectation. However, normal galaxies are generally not UV calorimeters either; a substantial fraction of the UV light emitted by star formation can escape without being reprocessed into FIR light (e.g., Xu & Buat 1995; Bell 2003; Buat et al. 2005; 79 Martin et al. 2005; Popescu et al. 2005). Therefore, normal galaxies also have a lower FIR luminosity compared to the calorimetric expectation. As can be seen in Figure 2.1, each of these effects alters the FRC by a factor of 4 for Σ = 0.001 g cm−2 in our standard model. Since they work in opposite ∼ g directions, the resulting LTIR/Lradio nonetheless remains the same as the calorimetric prediction (as suggested by Bell 2003). The Intermediate Case −2 The boundary between these two conspiracies occurs when Σg = 0.1 g cm . In some variants (Appendix 2.7), factors from both surface density regimes must be tuned to maintain the FRC at this surface density: escape time, secondaries, non-synchrotron cooling, and magnetic field strength all have an effect on the radio luminosity. However, these starbursts are unavoidably opaque to UV light, so the full low-Σg conspiracy cannot work for these galaxies. This becomes a problem when CR escape is quick, such as when strong winds are present (see Section 2.7.2), causing these galaxies to be radio-dim. Since the conspiracies begin to break down for the weakest starbursts, the transition from normal galaxies to starbursts may prove important in testing models of the FRC. 80 Summary The many factors described above conspire to produce the FRC, both in low-density non-calorimetric galaxies and high-density calorimetric starbursts. The traditional distinction between calorimeter and conspiracy explanations of the FRC is not clear cut in our models. We find that the FRC requires both calorimetry and conspiracy. 2.5.3. The FIR-Radio Correlation at Other Frequencies We have mainly considered the well-studied FRC at 1.4 GHz. However, the FRC is also known to exist at 150 MHz (Cox et al. 1988), 4.8 GHz (de Jong et al. 1985; Wunderlich et al. 1987), and 10.55 GHz (Niklas 1997). The correlation holds for both normal galaxies and starbursts at these frequencies, and remains even after thermal radio emission is subtracted. We show the predicted ratios of FIR to synchrotron radio fluxes in Figure 2.15. Our standard model predicts increased nonlinearity at other frequencies for a set of galaxies that span from normal galaxies to starbursts. While LTIR/Lradio varies by only 1.7 at 1.4 GHz over the full range in Σg, it varies by 2.3 at 500 MHz, and a factor of 5.4 at 100 MHz. At higher frequencies, the situation is similar, though the linear FRC is somewhat better preserved: LTIR/Lradio varies by a factor of 2.3 at 4.8 GHz, 2.8 at 8.4 GHz, and 3.6 at 22.5 GHz. As can be seen in Figure 2.15, at low 81 frequencies the FRC is predicted to tilt to the FIR with increasing Σg, while at high frequencies the correlation is predicted to tilt to the radio in starbursts. Our fiducial model with winds and B ρa (Section 2.7.2) predicts a similar increase in scatter ∝ at other frequencies (LTIR/Lradio varies by 2.1 at 500 MHz; LTIR/Lradio varies by 2.4 at 4.8 GHz). Our models also predict that the normalization of the FRC should change with the observed frequency. In general, LTIR/Lradio decreases with increasing frequency. This effect is stronger for the starburst galaxies, where the nonlinearities in the predicted FRCs appear. The radio-brightness at high frequencies is a direct consequence of the strong bremsstrahlung and ionization cooling in our models: synchrotron losses are more efficient relative to bremsstrahlung and ionization at higher energies, so that more energy goes into radio emission. Only when αν reaches 1 does the radio emission begin to decrease with frequency.14 Direct comparison between our models and observations can be difficult, because the FRC is usually considered in terms of luminosity rather than Σg and because the FRC is often fit as a nonlinear function. We can nonetheless make some qualitative comparisons between observations and our models. The observed 151 14This is also a generic prediction if there are loss processes that dominate synchrotron at low energies. For example, galaxies are radio dim at low frequencies if they have strong diffusive losses −1/2 (tdiff E ) or winds (twind constant with E). A large LTIR/Lradio also arises if there is radio ∝ absorption at low frequencies. 82 MHz correlation appears to be nonlinear, with luminous galaxies being brighter in the radio than would be predicted from the FIR (Cox et al. 1988 find L L0.87±0.04). FIR ∝ ν Our models predict the opposite effect if LTIR increases monotonically with Σg, with LTIR/Lradio increasing with Σg. Fitt et al. (1988) attribute the observed non-linearity at these frequencies to the FIR emission of old stars, and infer a linear FRC when they remove this effect. It is also worth noting that Arp 220 is radio dim at 151 MHz, though this may be due to free-free absorption (Sopp & Alexander 1989; Condon et al. 1991a). At 4.8 GHz, the FRC is known to be tight (0.2 dex dispersion) and approximately linear (de Jong et al. 1985; Wunderlich et al. 1987), though our models predict that starbursts should be radio bright compared to their FIR fluxes at these frequencies. At 10.5 GHz, most of the radio emission is thermal and not from synchrotron. Niklas (1997) estimates the contribution from synchrotron alone and finds a nonlinear dependence L L1.25±0.09, so that the FRC tilts towards ν ∝ FIR stronger radio emission at higher luminosities. Assuming that LTIR increases with Σg, we find a qualitatively similar behavior in our models. However, Niklas (1997) also finds a non-linear FRC at 1.4 GHz, with no dependence on frequency for the slope of the FRC, in contrast to Yun et al. (2001) who find a linear correlation (except at low luminosities) but only consider the FRC at 1.4 GHz. At least two effects we do not include would complicate our predictions. At low frequencies, free-free absorption may significantly lower the radio flux beyond what we predict in starbursts. Condon et al. (1991a) argue that free-free absorption 83 is important even at GHz frequencies in starbursts like Arp 220, and it becomes more effective at low frequency. This effect would make the low frequency FRC even more nonlinear than we predict. Thermal emission becomes significant at high frequencies (Niklas 1997 estimates that 30% of the radio emission is thermal at ∼ 10.5 GHz). While the thermal contribution can be estimated and subtracted off, at very high frequencies it may so overwhelm the synchrotron radiation that studying the correlation between FIR and nonthermal radio becomes impossible.15 2.5.4. The Spectral Slope α In the Milky Way, the observed spectral slope α increases (the spectrum steepens) with frequency. At low frequencies (< 100 MHz), the spectral slope is only ∼ 0.4 0.5 (e.g., Andrew 1966; Rogers & Bowman 2008) but α reaches 0.75 0.8 ≈ − ≈ − at GHz frequencies and reaches 0.8 0.9 at several GHz (Webster 1974; Platania ≈ − et al. 1998, 2003) before free-free emission flattens the spectrum (e.g., Kogut et al. 2011), though there are variations with direction and Galactic latitude (for example, Reich & Reich 1988). In fact, our models predict a steepening with frequency, though in our standard model αν is higher than observed at all frequencies for −2 Σg = 0.0025 g cm : it is 0.79 at 100 MHz, 0.91 at 1 GHz, and 1.00 at 10 GHz 15A linear correlation between thermal radio emission and radio emission is predicted and observed (e.g., Condon 1992; Niklas 1997), though it provides no information on the cosmic rays or magnetic fields in a galaxy and is beyond the scope of this paper. 84 (Figure 2.16; see also Figures 2.9 and 2.2). We note that α can be decreased by adjusting p; a value of p = 2.1 can decrease α by 0.1. We also note that we used an escape time that increased as energy decreased; if the escape time is constant or even decreasing at low energies (e.g., Engelmann et al. 1990; Webber & Higbie 2008), then our low frequency α will also decrease, and be more in line with observations. −2 The predicted αν for Σg = 0.01 g cm are somewhat better at low frequencies: 0.73 at 100 MHz, 0.93 at 1 GHz, and 1.07 at 10 GHz. Models with p 2.0 2.1 do a ≈ − better job of matching the observed spectral slopes of the Milky Way. As can be seen by the cooling and escape times in Figure 2.5, our standard model implies that escape, synchrotron, and bremsstrahlung all can shape the spectrum in normal galaxies. Escape dominates at low surface densities, while all three are comparable for the inner regions of galaxies (Σ 0.01 g cm−2). Our g ≈ greatest problem with α for normal galaxies is that it is predicted to increase slightly with Σg (Figure 2.2). This would imply that the inner regions of spirals would have steeper spectra than the outer regions, when in fact the opposite effect is observed (e.g., Murgia et al. 2005). The reason for the steepening is that escape becomes less effective as the galaxies become denser, so that cooling prevails. In normal galaxies, synchrotron dominates bremsstrahlung by a factor of a few (and ionization by an order of magnitude), and the ratio of the synchrotron to bremsstrahlung cooling times is only weakly dependent on density (compare the short-dashed synchrotron and the long-dashed 85 bremsstrahlung lines in Figure 2.5). For a constant scale height so that B n a, ∝ h i we have from equations 2.18 and 2.21 that t /t n B−3/2 n 1−3a/2, synch brems ∝ h i ∝ h i which is essentially constant for a = 0.7 and slowly changing for a = 0.5. In contrast, from equations 2.5, 2.18, and 2.19, we find that at fixed frequency t /t B−7/4 n −7a/4, roughly inversely proportional to n . This synch diff ∝ ∝ h i h i implies that synchrotron losses become much more effective than escape as density increases, but the bremsstrahlung losses remain a factor of a few less important than synchrotron losses. Therefore, as normal galaxies become calorimetric, their radio spectra will become steep in our model, since bremsstrahlung is only important enough to flatten the spectrum from its pure synchrotron-cooled limit of α = p/2 1.1 to α 0.9 0.95. ν ≈ ν ≈ − This problem remains for all of the variants (Appendix 2.7) that satisfy local or integrated constraints, except in the strong wind variant (Section 2.7.2) in which we include advective escape that would result from the wind inferred by Everett et al. (2008), and the fast diffusive escape variant (Section 2.7.6). If normal galaxies all host similar winds from their inner regions, escape prevents electrons from fully cooling, and our strong wind variant would imply that α would decrease to 0.75 0.8 at these densities. In our fast diffusive escape model, the electrons are ∼ − similarly prevented from fully cooling, and α is slightly reduced in normal galaxies to 0.85 0.90. However, the efficient escape in these models tends to break the FRC. ∼ − 86 Although calorimeter theory often is said to produce too high α, we consistently find that α is relatively low for starbursts (Figures 2.2 and 2.16). The spectral −2 slope at 1.4 GHz ranges from 0.7 for weak starbursts (Σg = 0.1 g cm ) to 0.5 for −2 extreme starbursts (Σg = 10 g cm ) in our standard model. In our models, the high densities in the starbursts (relative to the low-density radio disk of the Milky Way) cause the flat spectra. CR electrons and positrons experience severe cooling by bremsstrahlung and ionization, lowering α (cf. Thompson et al. 2006). Extreme starbursts are in fact observed to have flat spectra (Condon et al. 1991a; Clemens et al. 2008), though Condon et al. (1991a) attribute the flat spectra to free-free absorption and argues that the intrinsic α is 0.7. We also note that models that include the FIR optical depth in Uph predict steeper spectra, since IC losses are more effective: our κ = 10 cm2 g−1 model (Section 2.7.4) implies that α 0.65 in FIR ≈ starbursts. As an example of the power and limitations of our approach, we show the predicted synchrotron radio spectra of the starbursts in M82, NGC 253, and Arp 220 of our fiducial model in Figure 2.17. We calculate the radio emission using the Schmidt law, and assuming a disk geometry with the radius of the starburst from Thompson et al. (2006) and scale height h = 100 pc. Our fiducial model underpredicts the radio emission of M82 by a factor of 2 and overpredicts the radio ∼ emission of Arp 220 by a factor of 4. This is caused by scatter in the Schmidt ∼ law and the FRC, which our models do not currently account for. However, if we 87 normalize the radio spectra to the observed 1.4 GHz (dashed line in Figure 2.17), we find that our models predict the radio spectra surprisingly well. The spectra of NGC 253 and Arp 220 are slightly flatter than predicted, which is probably due to free-free absorption. A variant with winds and B ρa similarly predicts the spectral ∝ shape, although not the normalization (gray lines in Figure 2.17). While the fiducial model is no replacement for individual models of galaxies, which predict the correct normalization of the radio spectra and model the thermal emission and absorption, Figure 2.17 demonstrates that the GHz radio spectra of starbursts in general can be understood well in terms of the high-Σg conspiracy. 2.5.5. The γ-Ray (and Neutrino) Luminosities of Starbursts Starburst galaxies are predicted to be strong sources of γ-rays, observable with Fermi and Very High Energy (VHE) telescopes. Previous studies have considered NGC 253 (Domingo-Santamar´ıa & Torres 2005), M82 (Persic et al. 2008; de Cea del Pozo et al. 2009a), Arp 220 (Torres 2004b), and the diffuse γ-ray background (Thompson, Quataert, & Waxman 2007). Several starburst galaxies have already been observed in VHE γ-rays to search for the emission. Until recently, only upper limits were available on their γ-ray emission (e.g., Aharonian et al. 2005; Albert et al. 2007a). However, detections of NGC 253 and M82 have now been announced 88 with VHE telescopes (Acciari et al. 2009; Acero et al. 2009) and Fermi (Abdo et al. 2010a). Pionic γ-rays come from CR protons in the ISM of the starbursts. Since our explanation of the FRC requires that secondary electrons and positrons contribute to the radio emission, the γ-ray luminosities of starbursts are a useful test of the high-Σg conspiracy. We calculate the γ-ray flux16 from secondary π0 decay for M31, NGC 253, M82, and Arp 220 in Table 2.2 as a check on our models. We use the Schmidt law and the Σg from Kennicutt (1998) and Thompson et al. (2006) to calculate the emissivities of gamma rays for these systems, which we then multiply by volume (from the radii given in Thompson et al. 2006 and the scale heights in Section 2.2.2) to get total luminosities to be converted to fluxes. Since we are using approximate relations such as the Schmidt law, our models will be less accurate than more detailed models of individudal galaxies, and the predicted γ-ray luminosities are rough estimates only. These models are not meant to replace individual models of starburst galaxies. The main advantage of our approach is only that we consider starbursts like M82 and NGC 253 in the broad context of all star-forming galaxies spanning the range 16We do not include any optical depth to γ-rays in our calculations. However, Torres (2004b) found that Arp 220 was opaque to γ-rays only at energies above 1 TeV, and this should also be true for galaxies with a lower surface density. 89 between normal galaxies and ULIRGs; our models are necessarily more qualitative than more specific predictions. Inelastic proton-proton collisions will also create neutrinos and antineutrinos. The total neutrino (ν +ν ¯) flux is approximately equal to the π0 γ-ray flux at energies E m c2 140 MeV (Stecker 1979; Loeb & Waxman 2006). Although we do not ≫ π ≈ calculate the neutrino flux directly, we note that the values listed in Table 2.2 would also be good estimates for the neutrino fluxes, summed over all flavors and including both neutrinos and antineutrinos. Bremsstrahlung and IC emission also are expected to contribute to the gamma-ray luminosities, especially at low energies. We calculate the bremsstrahlung spectrum for M31, NGC 253, M82, and Arp 220 with our standard parameters. Both our standard model and our fiducial wind model imply that in starbursts bremsstrahlung emission equals the total pion emission at 100 MeV and decreases at higher energies (see Figure 2.18). In less dense galaxies, bremsstrahlung grows in importance, but is still a minority contributor above 100 MeV. The high energy fall-off for bremsstrahlung comes from the steepness of the electron and positron spectra relative to the proton spectra. About half of the energy in the bremsstrahlung emission is below 100 MeV, because the electron spectrum steepens above 100 MeV. The IC emission, when integrated over energy, is less than the bremsstrahlung or pion γ-ray emission (Figure 2.6). An IC gamma-ray spectrum would require an 90 incident spectrum including CMB, dust, and stellar emission. To get a feel for the IC emission, we model the background emission as three blackbodies: the CMB, a dust component (20 K in normal galaxies and 50 K in starbursts), and a direct stellar component (10000 K). The dust component and the stellar component have a total energy density of Uph,⋆ (eq. 2.9) and are scaled according the UV optical depth τUV (see Section 2.2.3). We find that IC is smaller than bremsstrahlung for energies above about 1 MeV, and smaller than pion γ-rays above about 50-80 MeV. At low energies, −2 dust emission dominates the IC emission in galaxies with Σg > 0.01 g cm (see ∼ Figure 2.19). Upscattered UV emission from stars only dominates at high energies, and the CMB dominates the low energy IC emission in low surface density galaxies. Our predicted spectra for starbursts have a precipitous fall-off in IC emission for the starbursts past 10 GeV, because there are no electrons above our 500 GeV cutoff to boost FIR photons to higher energies. In low surface density normal galaxies, the CMB continues to provide the photons for most of the IC emission, so the drop is at 1 GeV, with UV starlight providing higher energy photons (see also Figure 2.12). ∼ Most of the energy in the IC emission is at low energies, with more than half of the upscattered IC photons having less than 5 MeV. This is because the electron spectrum steepens for E > 100 MeV (γ 200) and incident photons are upscattered ≈ in energy by a factor of about γ2; an incident 10 eV photon would typically only be boosted to a few hundred keV. 91 Considering the uncertainties and approximations in our approach, our models are in loose order-of-magnitude agreement with more sophisticated models (Table 2.2). The high-energy pionic γ-ray spectra (> 1 GeV) are largely the same ∼ as previous models for M82 and Arp 220, although our M82 models are near the low range of the predictions of de Cea del Pozo et al. (2009a). We predict lower fluxes for NGC 253 than previous models by Domingo-Santamar´ıa & Torres (2005) and Rephaeli et al. (2010), by a factor of 4 13. In particular, we predict an ∼ − integrated flux of F ( 100 MeV) 4 10−9 for NGC 253. This is substantially γ ≥ ≈ × smaller than the predictions of 2.3 10−8 by Domingo-Santamar´ıa & Torres (2005) × +1.5 −8 and 1.8− 10 by Rephaeli et al. (2010). 0.8 × Interestingly, our predictions for M82 and NGC 253 are comparable to the Fermi and VHE detections (Acciari et al. 2009; Acero et al. 2009; Abdo et al. 2010a). However, this agreement is caused by a fortuitous cancellation of factors. In Table 2.2, we show that the predicted SN rates for M82 and NGC 253 using the Schmidt law are very small. If we combine the IR luminosities of Sanders et al. (2003) with equations 2.2 and 2.4, we find SN rates of 0.065 yr−1 for M82 and 0.039 yr−1 for NGC 253. Only half of the FIR luminosity of NGC 253 comes from its starburst core (Melo et al. 2002), so its starburst has a supernova rate of 0.019 yr−1. By contrast, our model using the Schmidt law implies SN rates that ∼ 92 are smaller by a factor of 3 for M82 and 2 for NGC 253’s starburst core.17 If we ∼ ∼ scale the γ-ray fluxes to these SN rates, then the γ-ray fluxes are near or somewhat above the upper ranges of previous models. Furthermore, our rescaled fluxes for M82 and NGC 253 are then about twice as high as observed. Other differences with the models arise because we also use different distances to the starbursts (we use 3.5 Mpc for NGC 253 instead of 2.5 Mpc, as Domingo- Santamar´ıa & Torres 2005 and Rephaeli et al. 2010 did), but the other models fit the observed radio emission so a greater distance would be fit with a greater luminosity in these models. We also use different low-energy energy spectra (we simply use E−p instead of K−p or q−p where q is momentum), which will tend to underestimate the low energy CR proton spectrum, although the higher energy CR proton spectrum will be largely the same. We again emphasize that our current generic models cannot replace existing models, but are meant as a demonstration of principle for the broad range of star-forming galaxies. Our predicted fluxes do not change significantly if we consider models with winds and B ρa (see Section 2.7.2), though the TeV fluxes of starbursts are higher ∝ with this variant because we use p = 2.2 instead of 2.3. The γ-ray fluxes also provide 0 good tests of the high-Σg conspiracy in our models. The π γ-ray fluxes we predict 17If we scale to the 1.4 GHz radio luminosity from Williams & Bower (2010) instead of the TIR luminosity, we find that M82 must be scaled up by 1.8 and NGC 253 by 1.3 in our fiducial ∼ ∼ model. The agreement is better still in our fiducial model with winds and B ρa. ∝ 93 for starbursts are mainly determined by proton calorimetry, the fraction of electron power lost to synchrotron, and the Milky Way CR proton normalization. Note that our fluxes are several times greater than those predicted by Thompson, Quataert, & Waxman (2007), who assumed proton calorimetry but did not take into account non-synchrotron losses. The significant bremsstrahlung, ionization, and IC losses in our model requires more (secondary) electrons and positrons to get the same radio emission, in turn requiring more protons. Data from Fermi and VHE telescopes can distinguish these scenarios. Proton calorimetry implies a hard E−2.2 γ-ray spectrum instead of the Galactic E−2.7 spectrum: proton calorimetry increases the high-energy γ-ray emission. The detections of starburst galaxies with VHE telescopes support a hard spectrum and proton calorimetry. Note that M31 is much fainter than the starbursts in VHE γ-rays, because of its steeper escape-dominated CR proton spectrum, though the flux of E < 1 GeV γ-rays from M31 is similar to that from nearby starbursts. ∼ Bremsstrahlung and IC γ-rays overwhelm π0 γ-rays at E < 100 MeV in our ∼ model, as seen in Figure 2.18: non-synchrotron losses increase the low-energy γ-ray emission. Fermi detection of this low energy emission would support the importance of non-synchrotron cooling. Finally, a different proton normalization simply changes the amount of both CR protons and secondary electrons and positrons: high proton normalization increases γ-ray emission at all energies, without changing the spectrum. 94 The current Fermi and VHE detections of M82 and NGC 253 are somewhat ambiguous, because these starbursts are relatively weak and there is no spectral information at 100 MeV yet (Abdo et al. 2010a). The implied GeV-to-TeV spectral slopes are 2.2 2.3, which is consistent with proton calorimetry (Acciari et ∼ − al. 2009; Acero et al. 2009; Abdo et al. 2010a). However, the fluxes are lower than our predicted fluxes scaled to the IR luminosities of these galaxies. This can imply that either proton calorimetry is weaker, or the high-Σg conspiracy is weaker, particularly in NGC 253. We note that several groups have estimated Σ 0.1 0.2 g cm−2 for these starbursts, so that the observed winds could be g ≈ − sufficient to break proton calorimetry. More data and more sophisticated modeling are needed to fully understand the implications of these γ-ray observations. Future detections of ULIRGs, which are more likely to be proton calorimeters, would be particularly helpful in understanding whether there is a high-Σg conspiracy. 2.5.6. The Dynamical Importance of Cosmic Ray Pressure Near the Solar Circle in the Milky Way, the CR energy density approximately equals magnetic field energy density, gas pressure, and radiation energy density. Their pressure is also comparable to the pressure needed to support the Milky Way hydrostatically, P πGΣ Σ 10πGΣ2, where Σ 10Σ is the surface hydro ≈ g tot ≈ g tot ≈ g 95 density of all matter in the Galactic disk. Extrapolating from the Milky Way (Chevalier & Fransson 1984; Everett et al. 2008), Socrates et al. (2008) hypothesize that CRs continue to provide significant pressure support and that they drive strong winds. Jubelgas et al. (2008) have also explored the dynamical importance of CRs in galaxies, and conclude that they are not important for starbursts. We show in Figure 2.20 the pressure from magnetic fields, radiation, and CRs in our standard model, compared to the hydrostatic pressure needed to support a galactic disk. Magnetic fields and radiation (including FIR light) remain comparable as Σg increases, as predicted by Thompson et al. (2006). For the inner Milky Way, the predicted CR pressure is 2.7 10−12 erg cm−3, within a factor of 2 of × the derived best-fit CR pressure in Everett et al. (2008). CR pressure remains in rough equipartition with magnetic field and radiation pressure until the weak starbursts, but then increases much more slowly. As CR pressure is mainly provided by protons, the failure of CR pressure is caused by pion losses: starbursts are proton calorimeters, converting most of the CR proton energy into gamma rays and neutrinos that escape the system. Our results are consistent with the low average CR pressure in starbursts found by Jubelgas et al. (2008), though the CR contribution may increase near the starburst edges (Appendix C of Socrates et al. 2008). 96 2.6. Summary and Future Improvements We model the FRC across the range 0.001 g cm−2 Σ 10 g cm−2, from ≤ g ≤ normal spirals to the densest starbursts. The correlation holds in several scenarios described in Appendix 2.7. We find that: We are able to reproduce a linear FRC (Figure 2.1) consistent with both local • and integrated Galactic constraints on the energy in CR protons. We find that ξ 0.021 of an SN’s energy goes into CR electrons and η 0.1 goes into CR ≈ ≈ protons when p 2.3 and using an E−p spectrum. ≈ −2 Starburst galaxies (Σg > 0.1gcm ) are UV, electron, and proton calorimeters • ∼ for most possible scenarios. In our standard model, normal galaxies with −2 Σg = 0.01 g cm are UV and electron calorimeters, but they are not proton calorimeters with only 5% 15% of CR proton energy going into pion losses. ∼ − The FRC is caused by calorimetry combined with two conspiracies operating • in different density regimes. At low Σg, decreasing electron calorimetry causes lower radio emission, but is balanced by decreasing UV opacity, which causes lower IR emission (Figure 2.1). At high Σg, bremsstrahlung, ionization, and IC losses decrease the synchrotron radio emission, while the appearance of secondaries and the effects of B on νC increase the radio emission (Figure 2.14). 97 The magnetic field strength scales as B Σ0.6−0.7, implying B 1 2 mG • ∝ g ≈ − −2 in extreme starbursts with Σg = 10 g cm . Magnetic fields are significantly below equipartition with respect to gravity in starbursts. The CR pressure remains in equipartition with radiation and magnetic field • −2 pressure for galaxies with Σg < 0.1 g cm . In starbursts, the CR pressure is ∼ significantly below equipartition, because of pion losses (long dashed line in Figure 2.20). Despite the short synchrotron and IC cooling timescales, our models reproduce • the observed flattened radio spectra of starbursts (Figure 2.2), because of the strong bremsstrahlung and ionization cooling in these galaxies. Our models predict that FRCs exist at frequencies other than 1.4 GHz, though • with increased non-linearity (Figure 2.15). Our predictions for the γ-ray emission from M82 and NGC 253 are within an • order of magnitude of the Fermi, VERITAS, and HESS detections. However, these models assume the Schmidt law holds exactly for these starbursts. If we normalize our models’ IR emission to the observed IR emission, which should scale as star formation and CR injection power, we find that our γ-ray predictions are 1.5 times higher than observations of M82 and NGC ∼ 253, possibly because of strong winds in these starbursts. Our predictions Arp 220 are roughly in line with previous theoretical models, considering the 98 approximations we make (Table 2.2). Full understanding of the γ-ray fluxes of these individual galaxies probably requires more refined models. Our models still have several unresolved issues. We have trouble matching the spectral slope α to observations of normal galaxies: we predict spectra that are too steep. A possible solution may be the presence of a wind lowering the escape time. However the addition of a wind, as observed by Everett et al. (2008), to our models of the Milky Way tends to break the FRC. It is also possible that stronger diffusive escape is present in the radio halos of normal galaxies than we used in our models, because the normal galaxy radio scale height is typically less than the CR scale height (see Section 2.7.3). Our one-zone models include CR cooling processes and escape through diffusion (winds are considered in Appendix 2.7.2), and can test a variety of parameterizations for the environment CRs travel through. Not every issue was considered in this paper, though. A natural question would be how robust the FRC is to scatter in the properties of the host galaxy environments. For example, the Schmidt law has a scatter of 0.3 dex (Kennicutt 1998), comparable to the FRC’s own scatter of ≥ about 0.26 dex (e.g., Yun et al. 2001). It is also unlikely that the magnetic field a a exactly scales as ρ or Σg, or that the overdensity f of ISM gas that CRs travel through would be exactly the same from galaxy to galaxy. 99 We focused on star formation and the CRs it produces in our models of the FRC. However both radio and FIR emission have other sources. Star formation drives thermal radio emission, which is important at high frequencies (e.g., Condon 1992). Thermal free-free emission probably also dominates the radio for very low density galaxies like the Large Magellanic Cloud, where the luminosity is low and electrons escape easily (Hughes et al. 2006). In normal galaxies, old stellar populations contribute a significant amount of FIR light without generating CRs. To account for this, we might have to distinguish between a warm component of FIR directly related to star formation and a cool FIR component that includes old stars, and make predictions for the FIR colors of galaxies and starbursts (Helou 1986). An obvious modification would be to apply our models to higher redshifts. Although the FRC has been mainly studied in the low-z universe, there have been several recent studies of high-z star-forming galaxies. Recent observations by Vlahakis et al. (2007) have found that starbursts become radio brighter at high redshift relative to the z 0 FRC (see also, e.g., Kov´acs et al. 2006; Murphy et al. ∼ 2009a; Micha lowski et al. 2010a). In general, though, calorimeter theory predicts that LTIR/Lradio should not change in starburst galaxies. Other studies have found that the FRC holds unchanged at high redshifts (e.g., Appleton et al. 2004). At high redshifts, the CMB will have a greater energy density, implying greater IC losses. However, the CMB will not be important in dense galaxies, except at the greatest redshifts (c.f. the CMB line and the starlight radiation line in Figure 2.20). 100 More important are the morphology changes. Many starbursts at high redshifts are observed to be kiloparsecs in radius instead of 100 pc (e.g., Chapman et al. 2004; ∼ Biggs & Ivison 2008; Younger et al. 2008), usually with moderate Σg but at least one with a surface density comparable to Arp 220 (Walter et al. 2009). In these starbursts, the high-Σg conspiracy can be unbalanced, altering LTIR/Lradio. We explored these effects in detail in Lacki & Thompson (2010a). While we assumed that galaxies and starbursts are homogeneous, future improvements can be made by using simple few-zone models of the ISM. In normal galaxies, the CRs are injected from a gas-rich thin disk, but can diffuse within a thicker radio disk containing much lower density ionized gas. This is reflected in the radio emission in the Milky Way, which has both a thin and a thick disk, the latter providing most of the luminosity (Mills 1959; Beuermann et al. 1985). Two-zone models can account for these density variations. Even within the gas-rich disk, the density can fluctuate wildly between the high-density molecular clouds and the low-density coronal phase. The magnetic field and density also change from spiral arms to interarm regions, as well as with distance from the centers of galaxies. Similarly in starbursts, most of the gas is believed to be in a phase with high density, while most of the volume is relatively low density (Greve et al. 2009). If the CR populations in such phases are not well mixed, bremsstrahlung and ionization losses would be weak in a low-density phase, but strong in a high-density phase. The low-density phase, with more volume, would contain most of the CRs but might 101 have a steep spectrum, while the high density phase, with more mass, would have fewer CRs but with a harder spectrum. However, including these different phases would add additional parameters to the models. The structure of the ISM phases would have to remain generic, because detailed information is only available for the Milky Way, but the FRC spans a vast range in star formation rate and gas surface density. These parameters would complicate the conspiracy even further, since there would be more parameters to tune. Ultimately, abandoning the one-zone (or few-zone) approach would be necessary for a full understanding of the FRC. Our approach only considers the global properties, but the FRC holds locally in galaxies to sub-kiloparsec scales. We could address the local properties of the FRC by making full diffusion models, similar to the GALPROP models for the Milky Way, for galaxies across the entire range of the FRC. A complete theory might have to include time evolution as well: Murphy et al. (2008) found that synchrotron emission is better correlated spatially with star formation in regions of high star formation, possibly because the CR electrons have not yet had time to diffuse. Spatial diffusion and time dependence would make modeling vastly more complicated, but including them may eventually be worthwhile with future improvements in radio and infrared observations. 102 2.7. Variants Our standard model (Sections 2.2 and 2.4.1) does not include a variety of effects that may alter the CR populations in star-forming galaxies. Our one-zone models of CR injection, cooling, and escape allow us to efficiently survey many scenarios. These include variations in essentially all of the parameterizations of Section 2.2. We search for models that are successful under the observational constraints described in Section 2.2.3. We test several combinations of these effects with sparser grids of models, spanning p = 2.0, 2.2, 2.4 and 2.6, and choose f so that fh = 1.0 kpc and 2.0 kpc. We summarize our results in Table 2.3. Table 2.3 shows the values of p, f, a, and δ˜ that satisfy local-based constraints, the integrated Milky Way pion gamma-ray luminosity, and both together (Section 2.2.3). Despite the large number of scenarios tried, we find similar successful parameters in those scenarios that worked at all. The FRC in some of these variants is shown in Figure 2.21. Overall, we did not strongly constrain p or f, with the range in allowed f being determined mainly by the observed beryllium isotope ratios at Earth. Specific variants occasionally imposed stricter constraints on p. However, our models did place strong limits on the magnetic field energy density (in the form of a; eq. 2.12) and the CR energy density (in the form of δ˜; eq. 2.17). We found a to be 0.5 - 0.8, depending on whether B was parametrized to vary with density or surface density. Table 2.3 shows the allowed 103 δ˜, because δ decreases by a factor of 100 as p increases from 2.0 to 2.6 from the normalization issue discussed in Section 2.3.1. In most cases, δ˜ is within the range 35 - 100, as expected from the local proton-to-electron ratio p/e of about 100. 2.7.1. B ρa ∝ While our standard model assumes that B Σa (eq. 2.12), the magnetic ∝ g field in galaxies may vary as ρa (Groves et al. 2003). A B ρ0.5 scaling is also ∝ observed in Galactic molecular clouds (Crutcher 1999). The difference matters most when comparing the FRC between starburst and normal galaxies, where the scale height changes by an order of magnitude in our models. At the transition between the two regimes, since h decreases by a factor of 10 for the starbursts, B in this parametrization jumps by a factor of 10a, strengthening synchrotron cooling (Section 2.3.2). This variation on our standard model typically breaks the FRC at the transition between normal galaxies and starbursts. Starbursts generally have a LTIR/Lradio that is several times smaller than normal galaxies, because of the dramatic increase in the magnetic field and corresponding synchrotron emission. No model retains the correlation and also fulfills the local-based constraints on e+/(e− + e+) and p/e, and only one model predicts the integrated Milky Way gamma-ray luminosity from pions. Models which do satisfy local constraints on the proton normalization have 104 L /L that vary by 2.4 at p = 2.6 and more at lower p ( 2.7 at p = 2.2). TIR radio ≥ However, other variants in combination with B ρa turn out to restore the FRC ∝ (Section 2.7.2). 2.7.2. Winds We test the effects of including a wind of 300 km s−1 (a relatively “weak” wind) in all starbursts with Σ 0.05 g cm−2, as discussed in Section 2.2.2 (eq. 2.6). We g ≥ find that the FRC is broken for all values of the parameters. In the best cases, L /L varies by 2.3 over the range in Σ , while our upper limit to the allowed TIR radio ∼ g variation was 2.0 (eq. 2.14), and this does not consider additional constraints on −2 the proton normalization. The difference comes from the Σg = 0.1 g cm models, because the radio emission is halved as the 1.4 GHz CR electrons escape before cooling: essentially, electron calorimetry fails for weak starbursts. While the FRC is broken in models with winds and B Σa because synchrotron cooling is not strong ∝ g enough, recall that the problem with the B ρa models (Section 2.7.1) was that ∝ synchrotron cooling was too strong in low-Σg starbursts. This suggests that models with both winds and B ρa might work. ∝ We test this conjecture by testing cases with winds and B ρa. We then do ∝ find some models that satisfy our local constraints, including preserving the FRC. Overall, the derived constraints on p, f, and δ are similar to our standard model 105 (Section 2.4.1). There is one noticeable difference in the allowed parameter space compared to our standard models: the models that do work now have a = 0.5 0.6 − instead of 0.6 0.8. However, this still implies strong magnetic fields in starbursts, − because B does not just depend on Σg but also h, where our adopted h is 10 times smaller in starbursts than in the Milky Way. If B ρ0.5, then the magnetic field ∝ −2 strengths are 0.12 mG in weak starbursts (Σg = 0.1 g cm ) and 1.2 mG in extreme −2 starbursts (Σg = 10 g cm ). These magnetic fields are comparable in strength to those that would be present if B Σ0.7. A model with p = 2.2, δ˜ = 67, a = 0.5, ∝ g f = 1.0, and ξ = 0.0119 satisfies both local-based constraints and the integrated Milky Way γ-ray luminosity. It also reproduces both the CR electron and proton fluxes at Earth above 10 GeV to within a factor of 2. The spectral slope α in ∼ starbursts is comparable to our standard models, being slightly lower because the CRs escape quicker. In this model the high-Σg conspiracy operates in starbursts largely as it does in our fiducial model (see Figure 2.22). However, its onset is more gradual because of −2 the winds in the Σg = 0.1gcm case, reducing secondary production in the weakest starbursts. Furthermore, the low a strengthens the magnetic field in the weakest starbursts, so that synchrotron can compete more effectively with bremsstrahlung and secondaries. In dense starbursts, magnetic fields are relatively weak and a strong high-Σg conspiracy sets the radio luminosity. 106 We also test stronger winds in combination with B ρa, motivated by the ∝ inference of a wind in the inner regions of the Milky Way (Everett et al. 2008), as well as the high wind speeds observed in starbursts (Heckman et al. 2000). The new −1 −2 wind speeds are v = 175 km s in the Σg = 0.01 g cm model (comparable to the Everett et al. 2008 wind), and v = 600 km s−1 in the starbursts. This variant tends −2 to break the FRC, mainly because CRs escaped too quickly in the Σg = 0.01 g cm Milky Way analog model. With a wind advecting away CRs at lower surface densities where cooling is weaker, most of these models are not sufficiently good electron calorimeters to preserve the FRC. Further variants with strong winds are discussed in Section 2.7.8. 2.7.3. Other Disk Scale Heights Small normal galaxy disk scale heights. In our standard scenario, the scale −2 −2 height jumps down from 1 kpc at Σg = 0.01 g cm to 100 pc at Σg = 0.1 g cm . The jump can cause discontinuities in the FRC if the parameters are not fine-tuned. However, the gas disk of the Milky Way is only 100 pc thick, and there is a thin radio disk associated with it. Thus, we try models where h = 100 pc for all Σg. In this run, we consider f values of 0.1 and 0.2 to continue to match the inferred ISM density that CRs propagate in the Milky Way from beryllium isotopes. Since h does not vary, Σg and ρ are directly proportional. 107 We find that models with p = 2.2 2.6, a = 0.5 0.6, f = 0.1 0.2, and − − − δ˜ = 34 100 satisfy both the integrated Galactic pion luminosity, and the local − e+/(e+ + e−) and p/e constraints. The relatively low a is preferred since f is low implying weak bremsstrahlung and ionization. Since there is no jump in bremsstrahlung and ionization losses for starbursts because they have the same scale height as normal galaxies, a strong magnetic field would make starbursts too radio bright. However, p = 2.4 2.6 produces CR spectra that are steeper than observed. − Higher p are somewhat preferred: there is a window in parameter space with a = 0.5, f = 0.1, and δ˜ = 50 100 where L /L varies over the range in Σ by 2.26 to − TIR radio g 2.75 for p = 2.0 but 1.94 to 2.21 for p = 2.4, so that higher p passes marginally. The reason for the relatively high variation in LTIR/Lradio is that f is low: starbursts are still proton calorimeters, producing secondaries that contribute to the radio; bremsstrahlung and ionization losses, which compete for the energy available for radio emission, are much weaker than in the standard model (Section 2.5.2). At high surface densities, secondaries become important and can make starbursts too radio bright; they are diluted more at higher p (Section 2.3.1), so that LTIR/Lradio varies slightly less over the range in Σg. The high-Σg conspiracy is present in starbursts for the models, but its onset is more gradual, with non-synchrotron losses growing from a factor of 5 at Σ = 0.1 g cm−2 to 10 at Σ = 10 g cm−2 (see Figure 2.22). ∼ g ∼ g The more gradual onset arises because we use low f to match the beryllium isotope constraints at Earth, weakening bremsstrahlung and ionization and reducing 108 secondaries in weak starbursts; and low a, which increases relative synchrotron strength in weak starbursts and suppresses synchrotron in dense starbursts. Furthermore, we find that the CR proton fluxes predicted at Earth at 1, 10, and 100 GeV are about 4 - 20 times higher than observed. The predicted electron fluxes are also about 10 times higher than observed. This can easily be understood: if h is shrunk 10 times, the same number of CRs are being injected into a smaller volume, so that their number density is higher. However, it is possible that the Earth resides in an atypical region of the Galaxy, in which case our integrated constraints alone allow this variant. We also consider a less extreme version of this model, where hnorm = 300 pc, about the height of the Milky Way’s thin radio disk (Beuermann et al. 1985). We use f of 0.3-0.6 to match the beryllium isotope measurements. This variant is less restrictive, allowing p 2.2 for all considered constraints. In all of the allowed ≥ models, a = 0.6 where B Σa, midway between the a = 0.5 case preferred when ∝ g hnorm = 100 pc and the a = 0.7 preferred when hnorm = 1 kpc. The CR flux predicted at Earth is still about 2 - 5 times higher than observed at Earth, for both protons and electrons in these models. Large normal galaxy disk scale heights. While the gas disks of normal galaxies are thin, the scale heights of the CRs themselves are estimated to be several kpc (see the discussion in Section 2.2.2). We therefore considered models with h = 2 kpc 109 and h = 4 kpc. When h = 2 kpc, we find that the allowed parameters are similar to our standard value. Higher a and lower fh is slightly preferred, because the jump in density between normal galaxies and starbursts is even greater than when hnorm = 1 kpc; therefore, either higher magnetic fields or lower gas densities are needed to prevent bremsstrahlung and ionization from overwhelming synchrotron losses. CR proton fluxes at Earth are somewhat small by a factor of 1.25 2 ∼ − when p = 2.2. In these models, the high-Σg conspiracy is present at an even greater magnitude than in our standard model, with non-synchrotron losses suppressing synchrotron emission by a factor of 15 instead of 10 (see Figure 2.22). ∼ ∼ When h = 4 kpc, no models preserve the FRC. Essentially, since f is constant for all Σg in our models, and since it must be large to match the beryllium isotope constraints in the Milky Way, bremsstrahlung and ionization are inevitably extremely strong in starbursts. While the density increases drastically from normal galaxies to starbursts because of the decreasing scale height, magnetic fields do a not suddenly increase if they go as Σg, and therefore synchrotron cannot properly balance bremsstrahlung and ionization for a linear FRC. 2.7.4. Optically Thick Galaxies Normally, our models assume that the CRs propagated in ISM that was optically thin to FIR light. Then, as stated in Section 2.2.2, Uph,⋆ = F⋆/c. 110 However, in a scattering atmosphere, the photon energy density may actually be greater if the environment is embedded in an optically thick region. In that case, Uph,⋆ = (1+ τFIR)F/c, where τFIR = κFIRΣg/2 acts as a midplane scattering optical depth. 2 −1 In models with κFIR = 1 cm g , we find that we are still able to recreate the FRC and match both local and integrated Galactic constraints. The parameter space allowed by this scenario is similar to our standard model (Section 2.4.1). The increased IC scattering in extreme starbursts rules out a = 0.6, so that the magnetic field energy density remains comparable to the increased photon energy density. We also typically recover the CR flux at Earth to within a factor of 2 for these models when p = 2.2 or 2.4. 2 −1 When κFIR = 10 cm g , the FRC does not survive in any of our models. The minimum variation in LTIR/Lradio is 2.03 when a = 0.8, f = 2.0, δ˜ < 25. Synchrotron ∼ losses need to keep up with IC losses in extreme starbursts, which would favor high a. However, a = 0.8 0.9 often caused too severe synchrotron losses compared to − bremsstrahlung and ionization in dense starbursts, and too weak synchrotron losses in low surface density galaxies. Those models that nearly preserve an FRC have low δ and high f, which reduces the number of secondaries and increased bremsstrahlung and ionization cooling to compensate for the increased magnetic field strength and keep LTIR/Lradio sufficiently high. 111 2.7.5. UB = Uph Radiation pressure may drive turbulence in the ISM, until the energy densities in radiation and kinetic motions are comparable. The turbulence can, in turn, generate magnetic fields. As a result, it is possible that U U U (Thompson ph ≈ turb ≈ B 2008). We test models where UB was forced to equal Uph, where the radiation energy density includes both the CMB and starlight. In optically thin models (τFIR = 0), the fiducial values for p, f, and δ˜ (given in Section 2.4.1) recreate the FRC and match both local and integrated proton constraints (see Table 2.3). Some models, generally those with p = 2.2 2.4, also correctly predict to within a factor of 2 the − CR proton flux at Earth at energies E = 1, 10, and 100 GeV, as well as the CR electron flux at Earth at E = 10 GeV. 2 −1 A few optically thick models with κFIR = 1 cm g where UB is forced to 0 Uph satisfy the integrated Galactic π luminosity, though only one satisfies local constraints. In these models, the effectiveness of synchrotron increased in high surface density galaxies, because UB equaled the quickly increasing Uph. The FRC generally fails because of a tension between intermediate densities and high densities. Bremsstrahlung and ionization are stronger at intermediate densities −2 −2 (0.01 g cm < Σg < 1 g cm ) than in the standard model (since UB is lower ∼ ∼ than our standard prescription), requiring a high secondary fraction to compensate. 112 −2 But when Σg = 10 g cm , synchrotron cooling overwhelms bremsstrahlung and ionization, making those galaxies too radio bright. The νC effect (Section 2.5.2) becomes particularly strong as τFIR becomes appreciable, since B is rapidly increasing, causing the radio emission to increase further. The model that does satisfy all constraints has low p, increasing the secondary abundance but weakening the νC effect. 2 −1 Increasing the FIR opacity to κFIR = 10 cm g in these variants completely breaks the FRC in all attempted models. Again, the bremsstrahlung and ionization −2 −2 losses are strong at intermediate densities (0.01 g cm < Σg < 1 g cm ) but small ∼ ∼ at high densities, because both the magnetic field energy density and the radiation field increase sharply at Σ 1gcm−2. Therefore starbursts would appear too radio g ≈ bright to maintain a linear FRC. 2.7.6. Fast Diffusive Escape We have used a scale height of h = 1 kpc for normal galaxies in most variants, based on the scale heights of radio disks. However, the diffusive escape time we use in Equation (2.5) applies to the entire CR scale height of the Milky Way, which is of order 2 4 kpc (see the discussion in Section 2.2.2). The escape time from the radio − disk itself may be significantly shorter. Variants in which the scale height of normal galaxies is increased (Appendix 2.7.3) still may not properly model the radio disk, 113 because they use the midplane magnetic field strength, possibly overestimating the effectiveness of synchrotron losses. We consider the effect of faster diffusive escape of by running models in which the diffusive escape time in galaxies is shortened by a factor of 4 from the nominal lifetime in equation 2.5. We find that the FRC tends is broken in almost all models, because escape reduces the radio luminosity of the lowest surface density galaxies. The minimum variation in LFIR/Lradio in any model is 1.998, barely under our criterion of 2.000; this models also satisfies the integrated Galactic pion luminosity (p = 2.0, f = 2.0, a = 0.6, δ˜ = 35, ξ = 0.018). The models with the most-preserved FRC tend to have low a of 0.6, weakening the synchrotron emission in high surface density galaxies. On the other hand, the spectral slope of normal galaxies in these models is 0.85 0.90 when p = 2.2, which is closer to the observed values than − in our standard model (see Section 2.5.4). The CR flux at Earth in models where L /L varies by 2.2 and p = 2.2 ranges from 35% to 140% of the observed FIR radio ≤ ∼ values (68% 142% when the local and integrated Galactic proton constraints hold). − 2.7.7. Varying Escape Times Constant Dz. So far, we have been simply assuming that escape time by diffusion for CRs is the same in all galaxies and starbursts. However, in our models, the scale height of starbursts is 10 times smaller than that of normal disk galaxies. 114 Another simple assumption would be that the vertical diffusion constant Dz at any given energy is constant across star-forming galaxies and starbursts. Then, since 2 tdiff = h /Dz, the escape time would be a hundred times smaller for starbursts. This could break the FRC at the transition between normal galaxies and starbursts. To see whether this variation had any effect, we modify the escape time to 2 tdiff (E) = tdiff,MW(E)(h/hMW) . We ran our grid for two cases that had worked previously: the standard model where B Σa and with no winds, and the case with ∝ g (weak) winds and B ρa (see Section 2.7.2). In standard models with constant ∝ −2 Dz, the vastly more efficient escape of CRs in weak starbursts (Σg = 0.1 g cm ) broke the FRC. Models with B ρa and winds did reproduce the FRC and were ∝ able to satisfy local and integrated constraints. The allowed values for p, f, a, and δ˜ are similar to those for our models with B ρa, winds, and constant t (c.f. ∝ diff Section 2.7.2 and Table 2.3). The increased magnetic field strength in starbursts (from B ρa, where ρ jumps up for starbursts) compensates for the decreased ∝ diffusive escape time to restore the FRC in these models. The models generally predict low CR flux at Earth when p = 2.2, with a proton flux of about 50% - 120% of its observed Earth value. The models with constant Dz weaken the high-Σg conspiracy somewhat, but it remains present (Figure 2.22). The very strong diffusive losses and the winds −2 mean that starbursts with Σg < 1 g cm are not proton calorimeters. Since B ∼ scales with ρa, synchrotron losses can compete more effectively with bremsstrahlung 115 and ionization in weak starbursts. Bremsstrahlung and ionization mainly balance the ν effect in these galaxies, which only lowers L /L by a factor of 50% C TIR radio ∼ −2 from Σg = 0.001 g cm normal galaxies. In denser starbursts, bremsstrahlung and ionization grow in importance, but are balanced by secondaries. Overall the conspiracy presented in Figure 2.22 is weaker than in the fiducial model, with non-synchrotron losses suppressing synchrotron losses by only 2 8 for starbursts, ∼ − because a is relatively high. Note that constant Dz from normal galaxies to starbursts is not necessarily correct. The speed CRs can stream out of galaxies is expected to be limited to the Alfv´en speed, v = B/√4πρ (Section 2.5.1; Kulsrud & Pearce 1969). For B ρ0.5, A ∝ this comes out to a diffusive escape time of 3.7h 106yr, where h = h/(100 pc). 100 × 100 While this is shorter than the Milky Way CR escape time (eq. 2.5), it is also 10 times longer than the constant Dz escape time (see also Figure 2.5). These considerations suggest that diffusion is weaker in starbursts than in constant Dz models, which would help preserve the FRC. D ρ−1/3. Helou & Bicay (1993) proposed that the diffusion constant scales z ∝ as ρ−1/3, which would make escape less efficient in high-density galaxies. We try models where the escape time equaled its local value (eq. 2.5) for the local surface −2 1/3 density (Σg = 0.0025 g cm ) and increasing as ρ to account for this effect. Testing this assumption against both our standard assumptions (no winds and B Σa) and B ρa with winds, we find similar results to the constant D case, ∝ g ∝ z 116 although a and f are more severely constrained. The faster escape time still breaks the FRC in otherwise-standard models. When B ρa, though, the models restore ∝ the FRC and satisfy local and integrated constraints. Again, the allowed parameters are similar to the case with B ρa, winds, and constant t (cf. Section 2.7.2 ∝ diff and Table 2.3). The CR flux at Earth is correct, within about a factor of 2 of the observed values for the energies we considered. D ρ−1. Finally, we try a rapidly scaling diffusion constant, motivated by the z ∝ parametrization in Murphy et al. (2008). As before, the escape time is normalized to −2 its local value (eq. 2.5) at the local surface density (Σg = 0.0025 g cm ), but now increasing as ρ. We once again test it against B Σa with no winds and B ρa ∝ g ∝ with winds. This variant fails in both cases to create the FRC, because escape is too −2 efficient in the lowest density galaxies (Σg = 0.001 g cm ). 2.7.8. Multiple Effects and Other Variants Combinations with weak winds. We finally consider scenarios that combine most of the previous variants. We include starburst winds, the FIR optical depth, B ρa, and constant D , D ρ−1/3, or D ρ−1 simultaneously. Our results ∝ z z ∝ z ∝ are essentially the same as the models with constant or varying Dz considered in Section 2.7.7 with B ρa and winds. For constant D , the models reproduce the ∝ z FRC and satisfy both local and integrated constraints. As before, the models predict 117 low CR proton flux at Earth. Some models with D ρ−1/3 also are consistent z ∝ with the FRC, but only only fulfills both local and integrated Galactic proton normalization constraints. The increased photon energy density suppresses radio in −2 Σg = 10 g cm starbursts, thus making it hard to maintain the FRC over the entire range in Σ . As before models with D ρ−1 fail to reproduce the FRC. g z ∝ Combinations with strong winds. The main problem with the strong wind scenario (Section 2.7.2) is the rapid escape of the CR electrons when −2 Σg = 0.01 g cm . More synchrotron is emitted before escape if the CRs have to travel a larger distance. We therefore consider models with strong winds and hnorm = 2 kpc, with f scaled to 2.0, 3.0, and 4.0 to match local isotope measurements. We use a constant diffusion rate Dz, and try models with and without FIR opacity. In both cases, we are able to satisfy both local and integrated constraints. 118 Fig. 2.1.— The non-thermal FRC, as reproduced in our standard model (p = 2.3, f = 1.5, a = 0.7, δ = 5, ξ = 0.023). While low CR escape times and low UV optical depth on their own would break the correlation at low surface densities, the two effects cancel each other out, creating a largely linear FRC. 119 Fig. 2.2.— Spectral slope as a function of the gas surface density. In this plot, α1.4 is the instantaneous spectral slope, dlog Fν/dlog ν, at 1.4 GHz. Elsewhere in the paper, 4.8 α is the observable α1.4. The parameters have their fiducial values. For our standard p = 2.3, strong cooling by synchrotron and IC alone would imply that α = 1.15. Instead the spectral indices are significantly flatter, especially in the strong cooling calorimeter limit at high Σg, as a result of ionization and bremsstrahlung losses. 120 Fig. 2.3.— The predicted kinetic energy spectra of cosmic rays in a low density −2 galaxy (Σg = 0.001 g cm ) for our standard model (p = 2.3, f = 1.5, a = 0.7, δ˜ = 48, ξ = 0.023). We mark the kinetic energy where electrons and positrons emit synchrotron radiation at ν 1.4 GHz. The cutoff in the lepton spectra at 500 GeV ≈ 6 ∼ is caused by our use of γmax = 10 ; see Section 2.4.3 for further discussion. 121 Fig. 2.4.— The predicted kinetic energy spectra of cosmic rays in a low density galaxy −2 −2 (left, Σg = 0.001 g cm ) and a high density starburst (Σg = 10 g cm ) for our standard model (p = 2.3, f = 1.5, a = 0.7, δ˜ = 48, ξ = 0.023). We mark the kinetic energy where electrons and positrons emit synchrotron radiation at ν 1.4 GHz. ≈ 6 The cutoff in the lepton spectra at 500 GeV is caused by our use of γmax = 10 ; see Section 2.4.3 for further discussion.∼ 122 Fig. 2.5.— Cooling times (for electrons and positrons) when νC = 1.4 GHz. 123 Fig. 2.6.— Emissivity (energy lost per volume per unit time) for each cooling process from protons, electrons, and positrons, integrated over energy. Pion losses include all of the energy going into secondary production as well as γ-rays and neutrinos. 124 Fig. 2.7.— The abundance of positrons as a function of energy. The saturation of the positron ratios at high Σg, and the lack of energy dependence for E > 1 GeV, is a sign of proton calorimetry (see Section 2.4.3 and Section 2.5.1). We mark∼ with filled symbols the energies where the critical synchrotron frequency νC is 1.4 GHz. 125 Fig. 2.8.— The abundance of secondary electrons compared to all electrons, as a function of energy. The saturation of the secondary electron ratios at high Σg, and the lack of energy dependence for E > 1 GeV, is a sign of proton calorimetry (see Section 2.4.3 and Section 2.5.1). We mark∼ with filled symbols the energies where the critical synchrotron frequency νC is 1.4 GHz. 126 Fig. 2.9.— The spectra of radio synchrotron predicted by our standard model (p = 2.3, f = 1.5, a = 0.7, δ˜ = 48, ξ = 0.023). Note that the synchrotron spectra steepen somewhat with frequency (also see Figures 2.2 and 2.16). 127 Fig. 2.10.— The sspectra of π0 γ-rays predicted by our standard model (p = 2.3, f = 1.5, a = 0.7, δ˜ = 48, ξ = 0.023). Proton calorimetry flattens the π0 γ-ray spectrum in the starbursts. 128 Fig. 2.11.— The spectra of bremsstrahlung γ-rays predicted by our standard model (p = 2.3, f = 1.5, a = 0.7, δ˜ = 48, ξ = 0.023). 129 Fig. 2.12.— The estimated IC spectra (right) predicted by our standard model (p = 2.3, f = 1.5, a = 0.7, δ˜ = 48, ξ = 0.023). We discuss our assumptions for the IC emission in Section 2.5.5. 130 Fig. 2.13.— The estimated proton calorimetry fraction Fcal in our models for several variants. The values are normalized so that models with no escape have Fcal = 1. Normal galaxies are not proton calorimeters, while ULIRGs with Σ 10 g cm−2 g ≈ are in all of our variants (note the convergence of all models to 1 at high Σg). In ∼ −2 all variants, proton calorimetry holds for starbursts with Σg > 1 g cm . Variants shown are our standard model (solid; p = 2.3, f = 1.5, δ˜ =∼ 48, a = 0.7); B ρa with 300 km s−1 wind in starbursts (dash; p = 2.2, f = 1.0, δ˜ = 67, a = 0.5); strong∝ −1 −2 −1 winds of 175 km s in Σg = 0.01 g cm , 600 km s for starbursts (long dash dot; a −1 p = 2.2, f = 2.0, δ˜ = 45, a = 0.5); constant Dz, B ρ , and winds of 300 km s in starbursts (long dash; p = 2.2, f = 1.5, δ˜ = 34, a =∝ 0.6); and fast diffusive escape with B Σa and no winds (dotted; p = 2.2, f = 2.0, δ˜ = 45, a = 0.6). ∝ g 131 Fig. 2.14.— The high-Σg conspiracy in our standard model (p = 2.3, f = 1.5, a = 0.7, δ˜ = 48, ξ = 0.023). The simple calorimeter model has perfect UV calorimetry and electron calorimetry, with only synchrotron cooling and no secondaries. Non- synchrotron cooling and secondaries alone each create a broken FIR-radio correlation, but conspire to make it linear at high density. 132 Fig. 2.15.— The FIR-radio correlation at other frequencies. Galaxies toward the top of the plot are radio-dim for their FIR emission, while galaxies toward the bottom of the plot are radio-bright. The shown frequencies are 100 MHz (light gray, solid), 500 MHz (light gray, long-dashed), 1.4 GHz (medium gray, long dashed/short dashed), 4.8 GHz (black, dash dot), 8.4 GHz (black, short dash), and 22.5 GHz (black, dotted). All lines are for our standard model (p = 2.3, f = 1.5, a = 0.7, δ˜ = 48, ξ = 0.023). 133 Fig. 2.16.— The instantaneous spectral slope of the (nonthermal) synchrotron emission as a function of frequency. In this plot, αν is the instantaneous spectral 4.8 slope, dlog Fν/dlog ν. Elsewhere in the paper, α is the observable α1.4. 134 Fig. 2.17.— The predicted synchrotron radio spectra of the starbursts in M82, NGC 253, and Arp 220, compared with Allen Telescope Array observations from Williams & Bower (2010). Our fiducial model is in black (solid – using Schmidt law and estimated starburst volume; dashed – scaled to 1.4 GHz flux) and our fiducial model with winds and B ρa is in gray (solid – using Schmidt law and estimated starburst volume; dotted – scaled∝ to 1.4 GHz flux). We do not include free-free absorption or thermal emission. 135 Fig. 2.18.— The fractional contributions of π0 decay (solid), bremsstrahlung (dotted), −2 −2 and IC (dashed) to the γ-ray flux for Σg = 0.001 g cm (gray) and Σg = 10 g cm (black). The drop in bremsstrahlung and IC past 100 GeV comes from the 500 GeV cutoff in our electron spectra. 136 Fig. 2.19.— The fractional contributions of the CMB, dust emission, and starlight −2 to the IC emission of galaxies. For starbursts with Σg > 0.1 g cm , upscattered emission from dust dominates at all shown energies. The drop in the contribution of the CMB near 1 GeV and dust past 10 GeV is an artifact of our 500 GeV cutoff in the electron∼ and positron spectra. ∼ 137 Fig. 2.20.— The importance of magnetic, radiation, and CR pressures compared to the hydrostatic pressure needed to support a galactic disk. The hydrostatic pressure 2 needed to support the gas alone is πGΣg. In low-density galaxies, the mass of 2 the stars implies that Phydro = 10πGΣg (see the discussion in Section 2.5.6). The cosmic ray energy density does not increase as quickly as radiation and magnetic field energy densities in starburst galaxies. None of the three components provides enough pressure to support starburst galaxies. 138 Fig. 2.21.— The FIR-radio correlation, as reproduced in our standard model and several variants. We plot the values when ξ = 0.008 in each case. Key: 1 (dash, black) – B ρa with 300 km s−1 wind in starbursts (p = 2.2, f = 1.0, δ˜ = 67, a = 0.5); 2∝ (long dash-dot, black) – B ρa and strong winds of 175 km s−1 in −2 −1 ∝ Σg = 0.01 g cm , 600 km s for starbursts (p = 2.2, f = 1.5, δ˜ = 45, a = 0.5); 3 2 −1 (dotted, black) – standard model with κFIR = 1 cm g (p = 2.2, f = 1.0, δ˜ = 45, 2 −1 a = 0.7); 4 (dotted, light gray) – model with κFIR = 10 cm g (p = 2.2, f = 2.0, δ˜ = 22, a = 0.8); 5 (solid, black) – standard model (p = 2.3, f = 1.5, δ˜ = 48, a = 0.7); 6 (long dash short dash, black) – fast diffusive escape (p = 2.2, f = 2.0, a = 0.6, a −1 δ˜ = 45); 7 (long dash, gray) – constant Dz, B ρ , and winds of 300 km s in starbursts (p = 2.2, f = 1.5, δ˜ = 34, a = 0.6); 8∝ (dash-dot, gray) – same as (7) but 2 −1 with FIR opacity of κFIR =1 cm g . 139 Fig. 2.22.— The high-Σg conspiracy in several of our variants. The line styles are the same as in Figure 2.14. At upper left is a model with B ρa and moderate winds (p = 2.2, f = 1.0, δ˜ = 67, a = 0.5; Section 2.7.2); at upper right,∝ a model with B ρa, ∝ moderate winds, and constant Dz (p = 2.2, f = 1.5, δ˜ = 34, a = 0.6; Section 2.7.7); at lower left, a model with h = 100 pc in normal galaxies (p = 2.2, f = 0.2, δ˜ = 34, a = 0.6; Section 2.7.3); and at lower right, we show a model with h = 2 kpc in normal galaxies (p = 2.2, f = 2.0, δ˜ = 67, a = 0.7; Section 2.7.3). Although the strength and the onset of the high-Σg conspiracy varies in these scenarios, it is always present in some form in the densest starbursts. 140 Symbol Section Standard Definition Value Derived parameters p 2.2.1; 2.3.1 2.3 Power law index of the injected spectrum of primary cosmic rays ξ 2.2.1; 2.3.1 0.023 Fraction of supernova kinetic energy injected into primary CR electrons η 2.2.1 0.12 Fraction of supernova kinetic energy injected into primary CR protons δ 2.2.1; 2.3.1 5 η/ξ, the ratio of energy in injected protons to injected electrons δ˜ 2.3.1 48 Proton-to-electron ratio at relativistic energy at injection; δ renormalized to remove p dependence a 2.2.2; 2.3.2 0.7 Power law scaling of galactic magnetic fields with surface or volume density 141 f 2.2.2; 2.3.3 1.5 Ratio of density through which CRs propagate to average ISM density Other input parameters Σ 2.2 Average gas column density g ··· Σ 2.2 Star formation rate per unit area SFR ··· h 2.2; 2.2.2 1 kpc 100 pc Scale height of CR disk − E 2.2 Total energy of cosmic ray ··· t (E) 2.2; 2.2.2 Escape time for a particle of energy E from the galaxy, includes both life ··· advection and diffusion Q(E) 2.2; 2.2.1 Energy spectrum of primary CRs injected into the ISM per unit volume ··· b(E) 2.2 Energy loss rate per particle (positive for energy loss) ··· C 2.2.1 Normalization of the injected energy spectrum of primary CRs ··· γ 2.2.1 Lorentz factor of cosmic ray, E/(mc2) ··· Table 2.1. List of Symbols Used (cont’d) Table 2.1—Continued Symbol Section Standard Definition Value 6 γmax 2.2.1 10 Maximum Lorentz factor of a cosmic ray at injection K 2.2.1 Kinetic energy of cosmic ray ··· ε 2.2.1 3.8 10−4 Radiative efficiency of stellar population × 51 E51 2.2.1 1 Mechanical energy per supernova, in units of 10 ergs ψ17 2.2.1 1 Conversion rate between the supernova rate per unit mass ΓSN and starlight emissivity ǫph t (E) 2.2.2 Escape time for particle of energy E from the galaxy by diffusion diff ··· tadv 2.2.2 Escape time for particle from the galaxy by advection in a wind 142 ∞ n 2.2.2 Average number density of hydrogen, Σ /(2h) h i ··· g n 2.2.2 Average hydrogen number density the CRs encounter, f n eff ··· h i U 2.2.2; 2.3.4 Energy density in starlight (UV or reprocessed FIR) ph,⋆ ··· F 2.2.2 Starlight energy flux ⋆ ··· U 2.2.2; 2.3.4 Energy density of CMB ph,CMB ··· κFIR 2.2.2; 2.3.4 0 Effective ISM opacity to far-infrared light (FIR) B 2.2.2; 2.3.2 ISM magnetic field strength ···2 −1 κUV 2.2.3 500 cm g Effective ISM opacity to ultraviolet (UV) light ν 2.3.2 Critical frequency of synchrotron radiation. C ··· Output N(E) 2.2 Final steady-state spectrum of CRs, calculated per unit volume ··· ǫ 2.2.3 Emissivity (here, power per volume), for radiation or a CR loss process ··· (cont’d) Table 2.1—Continued Symbol Section Standard Definition Value ǫ 2.2.3 Specific emissivity (emissivity per unit frequency), generally for ν ··· synchrotron radio emission L 2.2.3 Total infrared emission from young stars TIR ··· L 2.2.3 Nonthermal synchrotron radio emission radio ··· qFIR 2.2.3 Rescaled, observed logarithm of ratio LFIR/Lradio + + − ··· 143 e /(e + e ) 2.2.3 Fraction of CR positrons in CR electrons and positrons at Earth, ··· usually at 1 GeV p/e 2.2.3 Observed ratio of protons to electrons at Earth, usually at 1 GeV ··· 0 L 0 2.2.3 Gamma-ray emission from π production π ··· dI (E)/dE 2.2.3 Spectrum of CR electrons at Earth from Milky Way sources e ··· dI (E)/dE 2.2.3 Spectrum of CR protons at Earth from Milky Way sources p ··· α 2.2.3 Power law slope of the observed radio flux, dlogF /dlogν, usually ··· ν measured between 1.4 and 4.8 GHz ( ) 2.3.1 Spectral slope of the final steady-state CR spectrum, dlogN(E)/dlogE P E ··· e− /e− 2.3.1 Fraction of CR electrons that are secondaries sec ··· F 2.5.1 Fraction of CR proton luminosity going into pion losses cal ··· a a a a 0 ± b c Galaxy log10Σg R D ΓSN Integrated π (and π Neutrino) Photon Flux >E 100 MeVd 1 GeV 10 GeV 100 GeV 300 GeV 1 TeV Standard Milky -2.0 4 0.008 0.018 2.7E-4 3.4E-5 8.5E-7 1.5E-8 2.3E-9 2.8E-10 Waye (5.1E-4) M31 -3.0 20.9 0.9 0.019 3.7E-9 3.9E-10 8.1E-12 1.4E-13 2.0E-14 2.4E-15 (1.2E-8) NGC 253 -0.33 0.21 3.5 0.011 2.6E-9 5.1E-10 3.2E-11 1.6E-12 3.8E-13 7.5E-14 (4.7E-9) M82 -0.16 0.23 3.6 0.022 5.1E-9 1.0E-9 6.3E-11 3.2E-12 7.8E-12 1.6E-13 (9.1E-9) 144 Arp 220 0.78 0.12 76.6 0.12 6.4E-11 1.3E-11 8.0E-13 4.3E-14 1.1E-14 2.3E-15 (east) (1.1E-10) Arp 220 0.94 0.07 76.6 0.071 3.6E-11 7.2E-12 4.6E-13 2.4E-14 6.1E-15 1.3E-15 (west) (6.0E-11) Arp 220 1.08 0.37 76.6 3.1 1.6E-9 3.1E-10 2.0E-11 1.1E-12 2.7E-13 5.8E-14 (disk) (2.6E-9) B ρa and winds ∝ Milky -2.0 4 0.008 0.018 2.5E-4 3.3E-5 9.6E-7 2.1E-8 3.5E-9 4.8E-10 Waye (4.4E-4) M31 -3.0 20.9 0.9 0.019 3.0E-9 3.5E-10 8.8E-12 1.9E-13 3.0E-14 4.2E-15 (7.7E-9) Table 2.2. π0 γ-Ray (and π± Neutrino) Fluxes (cont’d) Table 2.2—Continued a a a a 0 ± b c Galaxy log10Σg R D ΓSN Integrated π (and π Neutrino) Photon Flux >E 100 MeVd 1 GeV 10 GeV 100 GeV 300 GeV 1 TeV NGC 253 -0.33 0.21 3.5 0.011 2.4E-9 5.2E-10 4.0E-11 2.5E-12 6.5E-13 1.4E-13 (4.1E-9) M82 -0.16 0.23 3.6 0.022 5.2E-9 1.1E-9 8.7E-11 5.5E-12 1.5E-12 3.3E-13 (8.9E-9) Arp 220 0.78 0.12 76.6 0.12 7.9E-11 1.7E-11 1.4E-12 9.2E-14 2.6E-14 6.1E-15 (east) (1.4E-10) Arp 220 0.94 0.07 76.6 0.071 4.6E-11 1.0E-11 7.9E-13 5.3E-14 1.5E-14 3.6E-15 145 (west) (7.9E-11) Arp 220 1.08 0.37 76.6 3.1 2.0E-9 4.4E-10 3.5E-11 2.3E-12 6.6E-13 1.6E-13 (disk) (3.5E-9) a −2 −1 Σg is in units of g cm , R is in units of kpc, D is in units of Mpc, and ΓSN is in units of yr . bAlthough we calculate the π0 γ-ray spectrum explicitly, we do not perform a similar calculation for neutrinos. However, the neutrino flux from π± decay at energies much higher than m c2 140 MeV is approximately equal π ≈ to the γ-ray flux, if antineutrinos and all flavors are included (Stecker 1979; Loeb & Waxman 2006). cAll fluxes are in units of cm−2 s−1. dThe flux in parentheses includes bremsstrahlung and IC γ-rays as well as pionic emission. eFor simplicity, we treat the Milky Way as a point source at the Galactic Center, and consider only its inner regions. B v a,b κ b h b t b Cutc Allowed Valuesd § wind FIR norm diff (Σg,min) p f a δ˜ 2.4.1 B Σa 0 1000 Const. t L 2.0 2.6 1.0 2.0 0.6 0.7 34 100 ∝ g ··· diff − − − − G 2.0 2.6 1.0 2.0 0.6 0.8 10 152 − − − − C 2.0 2.6 1.0 2.0 0.6 0.7 34 100 − − − − 2.7.1 B ρa 0 1000 Const. t L ∝ ··· diff ∅ ∅ ∅ ∅ G ∅ ∅ ∅ ∅ C ∅ ∅ ∅ ∅ 2.7.2 B Σa 300 (0.05) 0 1000 Const. t L ∝ g diff ∅ ∅ ∅ ∅ G ∅ ∅ ∅ ∅ C a ∅ ∅ ∅ ∅ 146 2.7.2 B ρ 300 (0.05) 0 1000 Const. t L 2.0 2.6 1.0 2.0 0.5 0.6 34 100 ∝ diff − − − − G 2.0 2.6 1.0 2.0 0.5 0.6 15 100 − − − − C 2.0 2.6 1.0 2.0 0.5 0.6 34 100 − − − − 2.7.2 B ρa 175 (0.01) 0 1000 Const. t L ∝ diff ∅ ∅ ∅ ∅ 600 (0.05) G ∅ ∅ ∅ ∅ C ∅ ∅ ∅ ∅ 2.7.3e B Σa 0 100 Const. t L 2.2 2.6 0.1 0.2 0.5 0.6 34 100 ∝ g ··· diff − − − − G 2.0 2.6 0.1 0.2 0.5 0.6 15 100 − − − − C 2.2 2.6 0.1 0.2 0.5 0.6 34 100 − − − − 2.7.3 B Σa 0 300 Const. t L 2.0 2.6 0.3 0.6 0.6 34 100 ∝ g ··· diff − − − G 2.0 2.6 0.3 0.6 0.6 0.7 15 100 − − − − C 2.0 2.6 0.3 0.6 0.6 34 100 − − − Table 2.3. Successful Models (cont’d) Table 2.3—Continued B v a,b κ b h b t b Cutc Allowed Valuesd § wind FIR norm diff (Σg,min) p f a δ˜ 2.7.3 B Σa 0 2000 Const. t L 2.0 2.4 2.0 0.7 35 67 ∝ g ··· diff − − G 2.0 2.4 2.0 0.7 35 67 − − C 2.0 2.4 2.0 0.7 35 67 − − 2.7.3 B Σa 0 4000 Const. t L ∝ g ··· diff ∅ ∅ ∅ ∅ G ∅ ∅ ∅ ∅ C ∅ ∅ ∅ ∅ 2.7.4 B Σa 1 1000 Const. t L 2.0 2.6 1.0 2.0 0.7 34 91 ∝ g ··· diff − − − 147 G 2.0 2.6 1.0 2.0 0.7 0.8 10 91 − − − − C 2.0 2.6 1.0 2.0 0.7 34 91 − − − 2.7.4 B Σa 10 1000 Const. t L ∝ g ··· diff ∅ ∅ ∅ ∅ G ∅ ∅ ∅ ∅ C ∅ ∅ ∅ ∅ 2.7.5 U = U 0 1000 Const. t L 2.0 2.6 1.0 2.0 34 100 B ph ··· diff − − ··· − G 2.0 2.6 1.0 1.5 34 100 − − ··· − C 2.0 2.6 1.0 1.5 34 100 − − ··· − 2.7.5 U = U 1 1000 Const. t L 2.0 1.5 50 B ph ··· diff ··· G 2.0 1.0 1.5 20 50 − ··· − C 2.0 1.5 50 ··· (cont’d) Table 2.3—Continued B v a,b κ b h b t b Cutc Allowed Valuesd § wind FIR norm diff (Σg,min) p f a δ˜ 2.7.5 U = U 10 1000 Const. t L B ph ··· diff ∅ ∅ ··· ∅ G ∅ ∅ ··· ∅ C ∅ ∅ ··· ∅ 2.7.6 B Σa 0 1000 Const. t L ∝ g ··· diff ∅ ∅ ∅ ∅ (1/4 nominal) G 2.0 2.0 0.6 35.0 C ∅ ∅ ∅ ∅ 2.7.7 B Σa 0 1000 Const. D L ∝ g ··· z ∅ ∅ ∅ ∅ 148 G ∅ ∅ ∅ ∅ C ∅ ∅ ∅ ∅ 2.7.7 B Σa 0 1000 D ρ−1/3 L ∝ g ··· z ∝ ∅ ∅ ∅ ∅ G ∅ ∅ ∅ ∅ C ∅ ∅ ∅ ∅ 2.7.7 B Σa 0 1000 D ρ−1 L ∝ g ··· z ∝ ∅ ∅ ∅ ∅ G ∅ ∅ ∅ ∅ C ∅ ∅ ∅ ∅ 2.7.7 B ρa 300 (0.05) 0 1000 Const. D L 2.2 2.6 1.0 2.0 0.5 0.6 34 100 ∝ z − − − − G 2.0 2.6 1.0 2.0 0.5 0.6 10 91 − − − − C 2.2 2.6 1.0 2.0 0.5 0.6 34 91 − − − − (cont’d) Table 2.3—Continued B v a,b κ b h b t b Cutc Allowed Valuesd § wind FIR norm diff (Σg,min) p f a δ˜ 2.7.7 B ρa 300 (0.05) 0 1000 D ρ−1/3 L 2.0 2.6 1.0 2.0 0.5 34 100 ∝ z ∝ − − − G 2.0 2.4 1.0 2.0 0.5 0.6 10 67 − − − − C 2.0 2.4 1.0 1.5 0.5 34 67 − − − 2.7.7 B ρa 300 (0.05) 0 1000 D ρ−1 L ∝ z ∝ ∅ ∅ ∅ ∅ G ∅ ∅ ∅ ∅ C ∅ ∅ ∅ ∅ 2.7.8 B ρa 300 (0.05) 1 1000 Const. D L 2.0 2.6 1.0 2.0 0.5 0.6 34 100 ∝ z − − − − 149 G 2.0 2.6 1.0 2.0 0.5 0.6 10 91 − − − − C 2.0 2.6 1.0 2.0 0.5 0.6 34 91 − − − − 2.7.8 B ρa 300 (0.05) 1 1000 D ρ−1/3 L 2.6 1.0 1.5 0.5 91 ∝ z ∝ − G 2.0 2.6 1.0 2.0 0.5 0.6 10 91 − − − − C 2.6 1.0 0.5 91 2.7.8 B ρa 300 (0.05) 1 1000 D ρ−1 L ∝ z ∝ ∅ ∅ ∅ ∅ G ∅ ∅ ∅ ∅ C ∅ ∅ ∅ ∅ 2.7.8 B ρa 175 (0.01) 0 2000 Const. D L 2.2 3.0 4.0 0.6 34 45 ∝ z − − 600 (0.05) G 2.0 2.2 3.0 4.0 0.6 20 45 − − − C 2.2 3.0 4.0 0.6 34 45 − − (cont’d) Table 2.3—Continued B v a,b κ b h b t b Cutc Allowed Valuesd § wind FIR norm diff (Σg,min) p f a δ˜ 2.7.8 B ρa 175 (0.01) 1 2000 Const. D L 2.2, 2.6 2.0 4.0 0.6, 0.5 34 91 ∝ z − − 600 (0.05) G 2.0 2.6 2.0 4.0 0.5 0.6 20 91 − − − − C 2.2, 2.6 2.0 4.0 0.6, 0.5 34 91 − − a −1 −2 2 −1 150 The units for vwind are km s , the units for Σg,min are g cm , the units for κFIR are cm g , and the units for hnorm are pc. bModels without a wind have a entry. Otherwise, there is a wind of speed v in models with a surface ··· wind density Σg of at least Σg,min. In the strong wind variants with more than one Σg,min listed, the wind speed is that with the greatest Σg,min less than Σg for each model. cConstraints used to select models. The FIR-radio correlation must always hold. Additional constraints on the proton normalization: L – local constraints (e+/(e− +e+) and p/e). G – integrated constraint (Milky Way gamma-ray luminosity from π0 production). C – both local and Galactic constraints. dVariants with entries could not satisfy the given constraints. Models where the magnetic field has no free ∅ parameters have a entry in the a column. ··· eSince the scale height does not vary, D is constant and B ρa for these models. z ∝ Chapter 3 The Physics of the FIR-Radio Correlation: II. Synchrotron Emission as a Star-Formation Tracer in High-Redshift Galaxies 3.1. Introduction The radio and FIR luminosities of star-forming galaxies are linearly correlated over three decades in luminosity (van der Kruit 1971, 1973; de Jong et al. 1985; Helou et al. 1985; Condon 1992; Yun et al. 2001).1 This FIR-radio correlation (FRC) is driven by star-formation. Starlight emitted by young, massive stars in the optical and UV is reprocessed by dust into far-infrared radiation. At the same time, the supernova remnants of massive stars accelerate cosmic rays (CRs), which produce synchrotron radio emission in the galaxies’ magnetic fields. The relation at z 0 is ≈ linear except in the very lowest luminosity galaxies (Condon et al. 1991b; Yun et al. 2001; Bell 2003). Since the FRC has less than a factor of 2 scatter in the local Universe, radio luminosity is often used as a tracer of star-formation at both low and high redshift (e.g., Cram 1998; Mobasher et al. 1999). 1This chapter was published as Lacki & Thompson (2010a, ApJ 717, 196). 151 Although the FRC has been mainly studied in the low-z universe, there has been recent interest in understanding how it applies to star-forming galaxies at high z. There have been several conflicting results observationally. A number of studies have found that the FRC holds unchanged or with little evolution at high redshifts (e.g., Appleton et al. 2004; Ibar et al. 2008; Rieke et al. 2009; Murphy et al. 2009a; Younger et al. 2009; Garn et al. 2009; Sargent et al. 2010; Ivison et al. 2010; Younger et al. 2010; Persic & Rephaeli 2007 suggest radio-dim ULIRGs at high z). Submillimeter galaxies, however, seem to be radio-bright, by a factor of 3 (Kov´acs et al. 2006; Vlahakis et al. 2007; Sajina et al. 2008; Murphy et al. ∼ 2009a; Seymour et al. 2009; Murphy 2009; Micha lowski et al. 2010a,b). In particular, Murphy et al. (2009a) and Micha lowski et al. (2010a) argue that their samples of submillimeter galaxies are intrinsically radio bright with respect to the local FRC, with no significant contamination from radio bright AGN. In addition to the work on individual star-forming galaxies, there have also been new investigations into both the recently-resolved FIR and the unresolved GHz radio backgrounds. The FIR background was detected by COBE and has long been attributed to star-formation (see the reviews by Hauser & Dwek 2001; Lagache et al. 2005). The star-formation origin of the FIR background was recently confirmed by BLAST (Devlin et al. 2009; Pascale et al. 2009). A detection of the extragalactic GHz radio background has also been recently reported by ARCADE2 (Fixsen et al. 2009; Seiffert et al. 2009). Predictions for the strength of the diffuse 152 radio background from star formation rely on the FRC holding out to z 1 2 ≈ − (e.g., Haarsma & Partridge 1998; Dwek & Barker 2002), where most star-formation occurs. In this paper, we describe how and why the linearity and normalization of the FRC are broken (or preserved) for different types of galaxies at high z. We describe the basic physics of our model and the elements of the high-z universe that physically allow the FRC to break down in 3.2. In 3.3, we briefly describe our § § underlying procedure. In 3.4, we discuss our results, providing predictions for when § star-forming galaxies should be more or less radio luminous than the z 0 FRC ≈ suggests. We also summarize the implications of our predictions for radio emission as a star formation tracer in 3.4.3 and estimate the effects of radio-bright SMGs § on the diffuse radio background in 3.4.4. § Throughout this paper, primed quantities denote the source’s rest-frame and unprimed quantities are for the observer-frame. 3.2. Theory Lacki et al. (2010a) (LTQ) recently presented a theory of the FRC over its entire span at low redshift, from normal galaxies like the Milky Way to the densest ULIRGs like Arp 220. 153 The linearity and normalization of the FRC provide strong constraints on the magnetic energy density in galaxies, which governs the strength of synchrotron losses. Indeed, the energy density of starlight in dense galaxies is so large that Inverse Compton losses alone shorten the CR electron and positron lifetime to 104 yr in galaxies like Arp 220 (Condon et al. 1991a). The magnetic field in Arp ∼ 220-like starbursts must exceed mG for the cosmic ray electrons and positrons to radiate a large enough fraction of their energy in synchrotron emission to account for the observed GHz radio emission. Such galaxies are likewise sufficiently dense that ionization and bremsstrahlung losses are also rapid. These considerations imply that the magnetic energy density must increase dramatically from normal spirals to dense bright starbursts. The observed Schmidt law allows us to quantify these dependences since it connects the star-formation rate surface density (ΣSFR), which is proportional to the energy density in starlight (U = F /c Σ ), to the gas surface density: ⋆ ⋆ ∝ SFR Σ Σ1.4 according to Kennicutt (1998), or Σ Σ1.7 according to Bouch´eet SFR ∝ g SFR ∝ g al. (2007). Using the Kennicutt (1998) Schmidt Law, LTQ found that the magnetic field strength B must increase as B Σ0.6−0.8 or B ρ0.5−0.6, where Σ = 2ρh, ∝ g ∝ g implying almost a thousandfold increase in B from Milky Way-like galaxies to Arp 220-like ULIRGs. This strong dependence of B on the global properties of the galaxy is largely responsible for the radio emission from starbursts and is required by the existence of the FRC. It is also crucial to the physics of how the FRC is maintained 154 or broken at high z. LTQ was unable to distinguish between the possibilities of B scaling with Σg and B scaling with ρ. LTQ found that over most of the range of the FRC, normal galaxies and starbursts are calorimetric. They are electron calorimeters, meaning CR electrons lose most of their energy before escaping. They are also UV calorimeters, meaning UV light is efficiently absorbed by dust and converted to FIR light. If CR electrons cool mainly by synchrotron losses, then the ratio of FIR emission to radio emission from primary CR electrons should be constant, since each is some fraction of the power released by star-formation. This is the essence of calorimeter theory, first proposed by V¨olk (1989). However, LTQ found that even in this strict calorimeter limit, L /L should decrease systematically by 2 from low surface brightness FIR radio ∼ galaxies to dense ULIRGs. The critical synchrotron frequency causes this “νC effect”: at fixed frequency, as the magnetic field strength increases with increasing surface density, lower energy electrons dominate the synchrotron emission because 2 the synchrotron frequency of electrons, νC , is proportional to Ee B, where Ee is the −2 electron energy. Since the injected CR spectrum is likely steeper than Ee , there is more power radiated at low energies. Thus, radio emission increases with surface density of gas and star formation, even including only synchrotron losses in the pure electron and UV calorimeter limit. On top of calorimetry, LTQ proposed that two different conspiracies combine to produce the linear FRC, each working in a different surface density regime. The “low- 155 Σg conspiracy” arises as both UV calorimetry and electron calorimetry gradually fail 2 −2 −2 −1 at relatively low surface densities (Σg < 0.01 g cm ; ΣSFR < 0.06 M kpc yr ). ∼ ⊙ Cosmic ray electrons escape diffusively without radiating all of their energy, decreasing the radio emission from the calorimetric expectation. However, some UV photons also escape without being reprocessed into FIR by absorption. Quantitatively, each effect alters the luminosity by a factor of 5 in the very lowest ∼ −3 −2 −2 −1 surface density galaxies (Σg 10 g cm ; ΣSFR 0.001 0.002 M kpc yr ). ≈ ≈ − ⊙ Therefore, both the FIR and radio emission decrease roughly equally, to maintain a constant LFIR/Lradio (see also Bell 2003). At the other extreme, high surface density compact starbursts have two completely different opposing processes that affect the radio emission. Compact starbursts are expected to be proton calorimeters: CR protons lose almost all of their energy through pion interactions with the dense ISM of starbursts, emitting γ-rays, neutrinos, and secondary electrons and positrons (Rengarajan 2005; Loeb & Waxman 2006; Thompson, Quataert, & Waxman 2007, LTQ). The secondary electrons and positrons themselves emit synchrotron radio emission, and outnumber the primary CR electrons. Proton calorimetry therefore greatly enhances the radio emission, all else being equal. Although escape losses are negligible in starbursts, synchrotron does compete with the other important cooling processes: Inverse Compton, bremsstrahlung, and ionization (Thompson et al. 2006, LTQ). These are 21 g cm−2 = 4800 M pc−2. ⊙ 156 particularly important in bright, dense starbursts, because they have very high gas densities and photon energy densities. These other losses reduce the share of CR electron and positron power going into radio by a factor of 10 20. Between ∼ − the secondaries and the νC effect increasing Lradio, and the non-synchrotron losses suppressing Lradio, a linear FRC correlation is produced. This is the “high-Σg conspiracy” of LTQ, and it operates to produce a linear FRC even though these starbursts are completely calorimetric (for protons, electrons, and positrons) and escape losses are negligible. Galaxies at high redshift can be used to test the theory of the FRC presented in LTQ. At high z, we expect that both the low- and high-Σg conspiracies are unbalanced for some galaxies. At low Σg, the energy density of the CMB (UCMB) competes with the energy density in both starlight and magnetic fields. CRs in galaxies face increasing losses from Inverse Compton since the CMB is stronger. Normal galaxies therefore should be radio dim, as Inverse Compton losses divert power from synchrotron. Starbursts, however, where CRs already face intense IC losses from starlight, should not be affected except at the highest redshifts (e.g., Condon 1992; Carilli & Yun 1999; Carilli et al. 2008). In this paper, we quantify how much the radio emission is dimmed by increased IC losses from the CMB as a function of z. On the other hand, high-z starbursts often have different morphologies than those observed of low-z ULIRGs. Local starbursts including ULIRGs are typically 157 compact, with radii of a few hundred parsecs and vertical scales heights of less than a hundred parsecs (Solomon et al. 1997; Downes & Solomon 1998). These 12 properties do not hold for submillimeter galaxies (SMGs), very bright (L > 10 L⊙) ∼ starbursts at z 2 mostly powered by star-formation and not AGNs (e.g., Pope ≈ et al. 2006; Valiante et al. 2007; Watabe et al. 2009). SMGs are usually several kpc in diameter (several times larger than low-z ULIRGs) (e.g., Chapman et al. 2004; Biggs & Ivison 2008; Younger et al. 2008; Iono et al. 2009; Younger et al. 2010), and have much lower surface densities than their low-z counterparts of the same luminosity (Tacconi et al. 2006, but see Walter et al. 2009). Submillimeter galaxies have large random velocities compared to their rotation speeds, implying scale heights of h 1 kpc (Tacconi et al. 2006; Genzel et al. 2008; Law et al. ≈ 2009). At high redshift, we therefore must consider a class of puffy starbursts3, with −2 −2 −1 Σg 0.1 g cm (ΣSFR > 2 4 M kpc yr ) but scale heights of h = 1 kpc, in ≥ ∼ − ⊙ addition to the h = 100 pc “compact starbursts” typical of local ULIRGs considered in LTQ. Importantly, puffy starbursts will have a smaller volume density for a given Σg and ΣSFR. Since bremsstrahlung, ionization, pion, and possibly synchrotron losses all depend on the volume density instead of surface density, these loss processes may all be weaker in puffy starbursts like SMGs. With these losses suppressed, the 3We use the term “puffy” instead of “extended” to emphasize that the gas and star-formation is extended vertically as well as radially. 158 high-Σg conspiracy will become unbalanced, and the FRC can be broken. These starbursts can either be radio-bright or radio-dim, depending on their magnetic field strengths. Since the scale height is so large and the volume density is relatively low for SMGs, we can determine whether B scales with Σg or ρ with these galaxies, breaking the degeneracy in LTQ. Because Σg = 2ρh, if B increases with ρ, then the magnetic field in puffy starbursts will be weak and synchrotron radio emission will be dim compared to compact starbursts with the same Σg. However, if B increases with Σg, the magnetic field will be strong and synchrotron radio emission will be bright in puffy starbursts compared to compact starbursts with the same Σg. 3.3. Procedure and Assumptions We model galaxies and starbursts as uniform disks of gas, with scale height h, 4 star formation rate surface density ΣSFR, and gas surface density Σg. We solve the diffusion-loss equation to find the steady-state equilibrium CR spectra in galaxies and starbursts. Injection, escape, and cooling losses all compete at each energy to determine the final CR spectrum (see LTQ for details). The relevant scale height is the height of the volume in which CRs are confined and produce synchrotron radiation. Normal galaxies (Σ 0.01 g cm−2 g ≤ 4In our models, the rest-frame bolometric luminosity from star-formation F ′ is directly ′ 9 −2 proportional to ΣSFR. The conversion factor is F = 5.6 10 ΣSFR 0L⊙ kpc , where ΣSFR 0 × , , is the star formation rate surface density in units of M kpc−2 yr−1. ⊙ 159 −2 −1 or ΣSFR < 0.06 M kpc yr ) have large radio halos with h 1 kpc, which we ∼ ⊙ ≈ adopt as the CR scale height, even though the gas disk is much thinner. Compact starbursts are much smaller, with h 100 pc (e.g., Solomon et al. 1997; Downes & ≈ Solomon 1998); we use this as their scale height because they are probably good enough calorimeters to prevent most of the CR electrons/positrons from escaping −2 out into the galactic haloes at high enough surface densities (Σg > 0.1 g cm or ∼ −2 −1 ΣSFR > 2 4 M kpc yr ). The CR disk scale height for our prototypical puffy ∼ − ⊙ starbursts, the SMGs, is not yet directly measured, but it must be at least the gas scale height, if the star formation is distributed throughout the gas. The gas scale height for SMGs is kinematically inferred to be h 1 kpc (e.g., Tacconi et al. 2006; ≈ Genzel et al. 2008; Law et al. 2009), which we adopt as the CR scale height of puffy starbursts. Our models span the entire observed range in Σg for galaxies and starbursts, −2 −2 from 0.001 g cm to 10 g cm . Combined with the scale height, Σg determines the rate of injection, escape, and cooling at all energies. The total rate of injection per unit volume is proportional to the volumetric star-formation rate, ΣSFR/(2h). The observed Schmidt law directly connects Σg and ΣSFR (Schmidt 1959). We consider both Σ Σ1.4 (Kennicutt 1998, hereafter K98) and Σ Σ1.7 (Bouch´eet al. SFR ∝ g SFR ∝ g 2007, hereafter B07), using the normalizations given by K98 and B07 respectively. The B07 relation was explicitly derived for high-z galaxies, including SMGs. Protons 160 and electrons are injected into our model galaxies with an energy spectrum E−p; in this paper, we use p = 2.2 in almost all cases.5 Escape, synchrotron, Inverse Compton, bremsstrahlung, and ionization losses are included for electrons and positrons, along with escape, ionization, and pion losses for protons. Diffusive escape times are normalized to the Milky Way value, with a loss time of t E−1/2. We also consider variants with advective escape, diff ∝ as in starburst super-winds. We assume that cosmic rays travel through gas with density f n , where n is the mean ISM number density and 1.0 f 2.0. The h i h i ≤ ≤ magnetic field is parametrized as B Σa or B ρa, normalized by the Milky ∝ g ∝ Way magnetic field strength. We searched for models that satisfy the local z 0 ≈ FRC for normal galaxies and compact starbursts. The CR proton spectrum is then normalized by Milky Way CR proton constraints. The parameters chosen for each variant are used to predict the FRC for puffy starbursts and at high z. In this paper, we consider several variants to get a sense of how the FRC varies with redshift: The LTQ standard model, with no winds, B Σa, and the K98 star-formation • ∝ g relation. 5The exception is the “Standard Model” from LTQ, which assumes p = 2.3. The slightly different p makes small quantitative differences, but does not change our conclusions. 161 The LTQ model with winds of 300 km s−1 in starbursts (Σ 0.1 g cm−2), • g ≥ B ρa, and the K98 star-formation relation. ∝ A variant with no winds, B Σa, and the B07 star-formation relation. • ∝ g A variant with winds of 300 km s−1 in starbursts (Σ 0.1 g cm−2), B Σa, • g ≥ ∝ g and the B07 star-formation relation. A variant with winds of 300 km s−1 in starbursts (Σ 0.1 g cm−2), B ρa, • g ≥ ∝ and the B07 star-formation relation. We show in Table 3.6 that we can reproduce an acceptably linear local FRC for each variant. We are able to adopt parameters consistent with Milky Way-derived CR proton constraints such that LTIR/Lradio varies by a factor of 1.7 - 2.2 from normal galaxies to compact starbursts. This variation is consistent with the factor of 2 ∼ scatter in the FRC6 (e.g., Yun et al. 2001). As in LTQ, the need for a linear local 0.7−0.8 FRC strongly constrains the magnetic field in galaxies to scale as either Σg or ρ0.5−0.6 (see 3.2). § 6The biggest exception is the variant with B Σa and winds, in which the winds invariably ∝ g −2 −2 −1 make electron escape too efficient when Σg = 0.1 g cm (ΣSFR 4 M kpc yr with B07), ≈ ⊙ suppressing its radio luminosity (see the detailed discussion in LTQ, Appendix A.2). We are unable ′ ′ to find any parameters where the variation in LTIR/Lradio is less than 2.0, but we consider a variation of 2.2 adequate for this paper’s purposes. 162 The dependence of the FIR emission at a given wavelength is beyond the scope of this paper, since it depends on the exact SED of the FIR emission in the galaxy. Instead, we simply use the Total Infrared (TIR) emission, assuming it is all of the ′ ′ UV light reprocessed by the dust, and use LTIR as a proxy for LFIR. Calzetti et al. (2000) estimate that L′ 1.75L′ , a correction we apply to the local value of TIR ≈ FIR L′ /L′ from Yun et al. (2001) to find the normalization of L′ /L′ at z 0 FIR radio TIR radio ≈ (see LTQ).7 We assume that the total TIR emissivity (defined here as luminosity per volume) has been measured and corrected for redshift to its rest-frame value, ǫ′ . We can also calculate the observable quantity q as log (L /L ) 3.67 TIR FIR 10 TIR radio − (Helou et al. 1985). For the radio emission, we calculate both the rest-frame radio emissivity ′ ′ ′ ′ ′ ǫradio,rest = ν ǫν(ν ) where ν = 1.4 GHz, and the rest-frame radio emissivity as estimated by an observer using a k-correction. Note that we are not actually trying to calculate the specific flux in the observer frame, but instead what an observer will infer for the rest-frame flux after using a k-correction8: the rest-frame flux is 7 ′ When calculating the qFIR observable defined in Helou et al. (1985), we then divide our calculated ′ ′ ′ LFIR by LTIR/LFIR to get the true FIR luminosity. However, we do not account for different ′ ′ LTIR/LFIR values for normal galaxies or SMGs. 8The situation is essentially the same as an observer measuring the radio flux of a z = 0 galaxy at 2 GHz and trying to infer the 1 GHz flux by using the spectral slope. The observer has an observed radio flux at 1.4 GHz, and can easily calculate the rest-frame flux at 1.4 (1 + z) GHz flux, but wants the rest-frame flux at 1.4 GHz. 163 what matters when we are considering the true, intrinsic evolution of the FRC. Hence, although the specific flux in the observer-frame will have a bandwidth compression factor, we assume the observer will take it back out to get the inferred rest-frame specific flux. To calculate the observer-inferred rest-frame emissivity at ′ ′ ′ ′ ′ rest-frame frequency ν = 1.4 GHz, we start with ν ǫν(νobs), where νobs = (1+ z)ν and ν = ν′ = 1.4 GHz. Since the synchrotron spectra are typically expected to fall off as ǫ′ ν′−0.7 from observations of local star-forming galaxies, the radio ν ∝ luminosity can be k-corrected by multiplying by (1 + z)0.7. We therefore calculate ′ ′ ′ ′ 0.7 ǫradio,inf = ν ǫν(νobs)(1+z) , which we will refer to as the “inferred” radio emissivity. ′ ′ ′ ′ The ratio of the TIR and radio luminosities LTIR/Lradio is then ǫTIR/ǫradio,rest in the ′ ′ rest frame, and would be inferred to be ǫTIR/ǫradio,inf in the rest-frame by observers. 3.4. Results & Discussion 3.4.1. The Evolving FIR-Radio Correlation We show the rest-frame FRC for our model with B Σ0.7, the B07 star- ∝ g formation law, and no winds, in Figure 3.1 as an example. The solid dark red line is the z = 0 FRC. All solid lines assume that starbursts are compact, with h = 100 pc. It is clear from Figure 3.1 that compact starbursts show little evolution in the FRC, while low surface density galaxies have lower radio luminosities at high redshift. This behavior is robust in all of the variants. The cause of 164 evolution in the FRC is Inverse Compton losses off the CMB for CR electrons and positrons. Figure 3.3 shows that the FRC should display relatively little evolution out to z 1, except for the lowest surface brightness galaxies. However, ≈ normal galaxies have suppressed synchrotron radio emission at z 2, a factor ≈ −2 −2 −1 of 2 for Σg = 0.01 g cm (ΣSFR 0.06 M kpc yr ) and of order 10 for ∼ ≈ ⊙ −2 −2 −1 Σg = 0.001 g cm (ΣSFR 0.001 0.002 M kpc yr ). The radio luminosities ≈ − ⊙ continue to fall with redshift. At z 5, IC off the CMB starts to matter even for ≈ −2 −2 −1 the weaker starbursts (Σg = 0.1 g cm ; ΣSFR 2 4 M kpc yr ). Dense ≈ − ⊙ −2 −2 −1 starbursts (Σg = 10 g cm ; ΣSFR 900 M kpc yr for the K98 law or ≈ ⊙ 9000 M kpc−2 yr−1 for the B07 law) remain on a linear FRC even at z 10. ⊙ ≈ The strong cooling from bremsstrahlung, ionization, and IC off starlight implied by the high-Σ conspiracy ( 3.2) acts as a buffer against IC losses off the CMB. g § Similarly, diffusive escape provides a similar buffering effect in low-Σg galaxies, so that the increased IC losses off the CMB does not suppress the radio emission quickly. In other words, most of the power from IC scattering of CMB photons does not come at the expense of synchrotron, but of other losses. The radio dimming cannot be described by a simple competition between magnetic field energy density and the CMB energy density; the CMB energy density must also compete with every other loss process. The buffering actually serves as an important test for both conspiracies described in 3.2. In high-Σ galaxies and starbursts, we predict that § g bremsstrahlung, ionization, and Inverse Compton of starlight already take a large 165 portion of a GHz electron’s energy budget; hence, the high-Σg conspiracy works ′ ′ to hold LTIR/Lradio constant out to quite high redshift. The buffering essentially doubles the redshift that compact starbursts remain on the FRC; the weakest −2 −2 −1 starbursts (Σg = 0.1gcm ; ΣSFR 2 4M kpc yr ) are radio-dim by a factor ≈ − ⊙ of 3 at z = 10 with buffering, instead of z = 5 without the buffering. In low-Σg galaxies, electrons and positrons can easily escape before they lose energy to Inverse Compton off the CMB at low enough z; thus, the low-Σg conspiracy also works to ′ ′ hold LTIR/Lradio constant. The suppression due to IC losses off the CMB seen in Figure 3.1 can be estimated by examining the ratios of the synchrotron cooling time to the total loss time, including both escape and cooling losses. In Appendix 3.6, we derive the ratios of loss times and we show that for a given Schmidt law, a critical redshift zcrit can be defined for each Σg and ΣSFR at which radio emission is suppressed. We define zcrit to be the redshift when the rest-frame radio luminosity at ν′ = 1.4 GHz is suppressed by a factor of 3 compared to z = 0. For our model with the B07 star-formation law, no winds, and B Σ0.7, we find the critical redshifts are approximated as ∝ g 1.4 (Normal galaxies, ΣSFR < 0.02) ∼ 0.23 5.8ΣSFR 1 (Normal galaxies, ΣSFR > 0.02) − zcrit ∼ (3.1) ≈ 5.7Σ0.23 1 (Puffy starbursts) SFR − 0.23 7.4ΣSFR 1 (Compact starbursts), − 166 −2 −1 where ΣSFR is in units of M kpc yr . We refer the reader to Appendix 3.6, ⊙ where we present similar relations for our other models of the FRC. We note that the radio suppression could, conceivably, be used to measure the temperature of the CMB at high redshift. In principle, this method could apply to any galaxy with a radio and FIR detection. However, the conspiracies would have to be accounted for, and they are affected significantly by both the gas surface density Σ and scale height h (see 3.4.2). Any measurement of the CMB g § temperature would depend on assumptions of the galaxy properties and would be model-dependent. ′ ′ As Figure 3.4 shows, the inferred rest-frame values of LTIR/Lradio show additional apparent evolution, simply because the spectral slopes of the galaxies are not exactly 0.7. Compact starbursts have flatter spectra9 with α 0.4 0.6 ≈ − at 1.4 GHz, so by adopting α = 0.7 they appear to become slightly radio brighter until z 1, after which at higher frequencies their spectra steepen, and their radio ≈ emission appears to dim again at higher z. Normal galaxies have steeper spectra with α 0.9 1.0 at 1.4 GHz, so their apparent radio luminosity sinks below their ≈ − true radio luminosity at high redshift. ′ ′ dlogFν dlogFν 9In this paper (unlike LTQ), α and α are instantaneous spectral slopes, not ≡ dlogν′ ≡ dlogν the measured spectral slopes between two observed frequencies, unless otherwise noted. 167 At higher redshifts, our models predict that galaxies have intrinsically steeper radio spectra, because of increased IC losses from the CMB. The rest-frame spectral slope α′ of normal galaxies at 1.4 GHz and asymptotes at 1.1 by z 2 3, when ∼ ≈ − losses are dominated by IC for our injection spectrum of CRs with p = 2.2. There is much less intrinsic rest-frame evolution of starburst radio spectra; even at z 5, α′ ≈ increases only by 0.1 for the weakest compact starbursts. The observable α1.4 GHz 610 MHz shows much more pronounced evolution with redshift for starbursts, increasing by 0.1 to z 1 2, and 0.2 to z 4 5. This effect arises simply because we predict the ≈ − ≈ − CR electron and positron spectra steepen with rest-frame frequency as synchrotron and IC losses become stronger (e.g., Thompson et al. 2006, LTQ): at higher redshift and fixed observing frequency, we are seeing higher energy electrons and positrons. An important effect that we do not consider is the thermal free-free radio emission. This will set a minimum total observed radio emission that is directly proportional to the star-formation luminosity. Thus, the true total radio deficit at GHz will not be as big as the synchrotron-only deficits plotted in Figures 3.1 and 3.4. Thermal free-free emission will also flatten the spectrum, especially at high frequencies. However, free-free emission is much fainter than the synchrotron luminosity at GHz frequencies, except in the faintest star-forming galaxies (Condon 1992; Hughes et al. 2006). 168 3.4.2. The Radio Excess (or Deficit) of Puffy Starbursts: Submillimeter Galaxies As shown in Figure 3.1, puffy starbursts fall on a linear FIR-radio correlation of their own (dotted lines), in line with observations of SMGs (Kov´acs et al. 2006; Murphy et al. 2009a). The radio luminosity of these galaxies is nonetheless the result of a conspiracy, between IC losses on starlight, which decrease the radio luminosity, and the enhanced radio emission from secondary electrons and positrons, and the ν effect ( 3.2). The variation in L′ /L′ for puffy starbursts alone is usually C § TIR radio less than a factor of 2 over the range 0.1 g cm−2 Σ 10 g cm−2 (see Table 3.6; ≤ g ≤ in most variants the variation is 1.6). Like compact starbursts, escape plays ∼ essentially no role in most of the models, except that winds can slightly decrease the −2 −2 −1 radio emission in relatively tenuous Σg = 0.1 g cm (ΣSFR 2 4 M kpc yr ) ≈ − ⊙ starbursts (footnote 6). B Σ0.7−0.8: Radio-Bright Puffy Starbursts ∝ g It is plain from Figure 3.1 (left panel) that the normalization of the puffy starburst FRC (dotted lines) is different than the FRC of the compact starbursts and normal galaxies (solid lines) when B Σ0.7. We show in Table 3.6 that models with ∝ g B Σ0.7−0.8 have radio-bright puffy starbursts compared to the observed local FRC, ∝ g by a factor of 2 4 at z = 0. Like the compact starbursts, puffy starbursts show − 169 ′ ′ little rest-frame evolution in LTIR/Lradio, except at relatively low surface densities −2 −2 −1 (Σg = 0.1 g cm ; ΣSFR 2 4 M kpc yr ), where they become radio dim ≈ − ⊙ at high redshifts because of IC losses on CMB photons. Therefore, we predict that puffy starbursts, which are mainly observed at high z, have intrinsically different radio properties not caused by their redshift. We propose a natural explanation of this radio excess in the framework of LTQ ( 3.2). In dense starbursts, protons are efficiently converted into secondary electrons § and positrons through inelastic proton-proton scattering, which contribute to the ′ ′ synchrotron emission. Furthermore, the νC effect increases LTIR/Lradio for starbursts which have larger B. Compact starbursts lying on the z = 0 FRC balance these effects with increased bremsstrahlung, ionization, and IC losses, which compete with the synchrotron losses and suppress the excess radio luminosity ( 3.2). In § puffy starbursts with relatively low volume densities compared to their compact cousins at fixed Σg, however, bremsstrahlung and ionization are not strong enough to compensate for these effects. Only synchrotron and IC losses remain, upsetting the conspiracy. The radio excess predicted by this picture is systematically greater when using the K98 star-formation relation, because the IC loss rate on starlight is smaller at fixed surface density by a factor of 2.5 to 11 from weak starbursts to the densest starbursts. This freedom to vary the radio-excess with B does not exist in the standard calorimeter model, where all CR electron/positron energy goes into synchrotron emission, and the radio emission saturates. The balance between 170 synchrotron and the other forms of cooling can be changed in starbursts to alter the normalization of the FRC, even though escape is negligible. There is a reason to expect our allowed B Σ0.7−0.8 specifically: radiation ∝ g pressure may drive turbulence and enhance the magnetic field until its energy density is comparable to radiation (e.g., Thompson 2008). If the K98 relation holds, then since the magnetic energy density scales as U U , B Σ0.7, and B ∝ ph ∝ g if the B08 relation holds, then B Σ0.85. This explanation is more problematic ∝ g for the B07 relation and B Σ0.7. 10 However, while we assume that there is a ∝ g parametrization for B that applied to both compact starbursts and puffy starbursts, the radio excess should arise more generally. The radio excess arises simply because synchrotron cooling time is shorter than the bremsstrahlung and ionization cooling ′ times in puffy starbursts, but longer in compact starbursts at fixed Σg: at fixed ν , t n−1 h/Σ and t B−1/2n−1 h/(B1/2Σ ). We therefore expect there brems ∝ ∝ g ion ∝ ∝ g to be a radio excess with respect to the FRC in any puffy starburst with a strong enough magnetic field, with the exact enhancement depending on magnetic field 10However, we have scaled B to the total Milky Way magnetic field strength of 6 µG near the Solar Circle. If most of the magnetic field strength in starbursts is driven by turbulence, perhaps scaling to the disordered Galactic magnetic field strength ( 2µG; Beck 2001) near the Solar Circle ∼ would make more sense, because the ordered magnetic field arises from a different process. The lower normalization for B then would partly compensate for the steeper dependence on Σg. 171 strength because of the remaining competition, after ionization and bremsstrahlung are sub-dominant, from the IC losses on starlight.11 Our results imply that a moderate radio excess at the factor of 3 level ∼ alone is not a safe indicator of the presence of a radio-loud AGN, especially at high redshifts where SMGs are observed. While radio excess with respect to the local FRC has been suggested as a selection criterion for radio-loud AGNs (e.g., Yun et 0.7−0.8 al. 2001; Yang et al. 2007; Sajina et al. 2008), our models with Σg imply that q 1.7 2.0 for puffy starbursts powered by star-formation alone. A radio excess FIR ≈ − is inexplicable in our models only when the source is an order of magnitude brighter (qFIR < 1.5) in the radio than predicted from the FIR emission. While SMGs are ∼ relatively rare and may not be a problem in small samples, we recommend that other means be used to be sure that the radio-excess is caused by an AGN, such as a flat radio spectrum, radio morphology, mid-IR colors, or the presence of strong X-ray emission (see also Murphy et al. 2009a). 11If the magnetic field strength does not go very roughly as B Σ0.7, however, there will be more ∝ g ′ ′ scatter in LTIR/Lradio for puffy starbursts at fixed h and z. SMGs would not form their own FRC if B was the same for all SMGs regardless of Σg, or if B increased very steeply with Σg, because there still must be a conspiracy with IC losses off starlight. Since SMGs do appear to form their own FRC (e.g., Murphy et al. 2009a; Micha lowski et al. 2010a), this could be evidence specifically that their magnetic field strengths increase with Σg (or ρ). 172 B ρ0.5−0.6: Radio-Dim Puffy Starbursts ∝ A magnetic field dependence of B ρ0.5 appears to hold for Galactic molecular ∝ clouds (e.g., Crutcher 1999), and LTQ found that the FIR-radio correlation was consistent with this magnetic field dependence. The existence of galaxies with different scale heights allows us to distinguish the two possibilities for magnetic field scaling. If B Σa, then the magnetic field strength will be the same for all ∝ g galaxies with the same Σ , regardless of scale height. In models with B ρ0.5−0.6, by g ∝ contrast, the magnetic field strength is weaker in puffy starbursts than in compact starbursts with the same Σg. As seen in Figure 3.2 (right panel, dotted lines), puffy starbursts again form their own FRC. In models with B ρ0.5−0.6, they are radio dim compared to the ∝ z 0 FRC. We show in Table 3.6 that the normalization of the FRC is radio-dim ≈ by a factor of 1.2 2.0. ∼ − We can explain this in the LTQ theory of the FRC as well. The magnetic field strength must increase more slowly with ρ than with Σ to reproduce the z 0 g ≈ FRC, because h Σ /ρ is 10 times smaller in compact starbursts than normal ∝ g galaxies and puffy starbursts. Compact starbursts are highly compressed with respect to normal galaxies, so they have strong magnetic fields and synchrotron radio emission is strong enough to compete with the other losses. Puffy starbursts are not compressed, so that their magnetic fields are weak and synchrotron losses 173 cannot keep up with IC losses, nor with bremsstrahlung and ionization as 1.4 GHz emission traces ever lower electron energies at higher magnetic field strengths. Puffy starbursts therefore turn out to be radio dim compared to compact starbursts on the z 0 FRC, if B ρ0.5−0.6. As before, the B07 star-formation relation predicts ≈ ∝ greater IC losses and therefore weaker radio emission. Because of the claims in the literature that SMGs are radio bright, we do not favor these models. The suggested relative radio brightness of high-z SMGs therefore provides some evidence that, in fact, B Σ0.7−0.8 rather than B ρ0.5−0.6. ∝ g ∝ However, the matter of whether high-z SMGs are in fact radio bright is not yet settled. Although LTQ concluded that B must increase dramatically from normal galaxies to dense starbursts ( 3.2), they were unable to distinguish between these § two possibilities with the z 0 FRC alone. For this reason, high-z starbursts and ≈ their qualitatively different morphologies compared to those at z 0 can distinguish ≈ theories of the FRC. Spectral slopes A prediction of all of our variants is that puffy starbursts like submillimeter galaxies should have steep non-thermal radio spectra, with α 0.8 1.0 (see ≈ − Table 3.6). The steep spectra are caused by strong synchrotron cooling in the B Σ0.7−0.8 case and the relatively stronger IC cooling off starlight in the B ρ0.5−0.6 ∝ g ∝ case. In general, puffy starbursts should have roughly the same α as normal galaxies 174 in the local universe, which tends to be somewhat higher (α 0.7 1.0) than in ≈ − compact starbursts (α < 0.7). The slope should hold even out to extremely high ∼ Σ , as long as starbursts are puffy. In contrast, we find that α 0.5 in compact g ≈ starbursts, because of efficient ionization and bremsstrahlung losses, which flatten the equilibrium CR spectrum because of their energy dependence. As we note in LTQ, our predicted spectral index for normal galaxies is somewhat too high, and this difference in α may carry over to the puffy starbursts. However, the significant difference in α between compact and puffy starbursts should remain as a general prediction of our model: compact starbursts should have flatter spectra than puffy starbursts. The high spectral slopes can be observed either with direct measurements of multifrequency data of individual submillimeter galaxies, or with single frequency observations at a variety of redshifts. There are relatively few measurements of α for submillimeter galaxies specifically; faint radio sources have α 0.5 0.7 (Huynh et ≈ − al. 2007; Bondi et al. 2007), though that sample includes both compact starbursts and AGNs. Sajina et al. (2008) do find that α1.4 GHz 0.8 for SMGs, comparable 610 MHz ≈ to our predictions. They also find that submillimeter galaxies have a radio-excess, in agreement with Figure 3.2. More recently, Ibar et al. (2010) found an average α1.4 GHz 0.75 0.06, which is somewhat flatter than our models. These spectral 610 MHz ≈ ± slopes are not different from normal star-forming galaxies, but are noticeably steeper than local ULIRGs (Clemens et al. 2008). However, we do not account for free-free 175 absorption, which probably flattens the spectra of local ULIRGs like Arp 220 at low frequency (Condon et al. 1991a), and is not well understood in SMGs. Since puffy starbursts have steeper spectra than compact starbursts, we expect ′ ′ their inferred LTIR/Lradio will increase with redshift: if the true radio spectral slopes of SMGs are greater than the assumed α by ∆α, they will appear to become radio dimmer by a factor (1 + z)∆α, or up to 40% at z = 2. ∼ In Figure 3.5, we show the expected radio synchrotron spectra of starburst galaxies, without correcting for thermal absorption or thermal emission. At a rest-frame frequency of 1 GHz, puffy starbursts (dashed) have steeper radio spectra than compact starbursts (solid). Note that at high frequencies (ν′ > 10 GHz), the ∼ ratio of the radio luminosities per unit star formation of the compact and puffy starbursts asymptotes to a value set by the ratio of UB and Uph in these starbursts. At these high frequencies, only synchrotron and IC cooling are effective, and IC cooling would be the same for puffy and compact starbursts because of the Schmidt Law ( 3.2). For B Σa, U is the same for puffy and compact starbursts, but § ∝ g B for B ρa, puffy starbursts have much smaller U . Thus, measurements of the ∝ B synchrotron radio emission of SMGs at high ν′ could determine the magnetic field strength of SMGs and determine which scenario applies. Of course, there are unlikely to be two perfectly distinct populations of compact starbursts and puffy starbursts. Instead, there may be a continuum variation in scale 176 heights from tens to thousands of parsecs. We would then expect to see a larger ′ ′ scatter, both in LTIR/Lradio and α, in a full sample of both the most compact and the most puffy starbursts. Murphy et al. (2009a) find that submillimeter galaxies do ′ have a larger scatter in qTIR than other galaxies. However, Ibar et al. (2010) find a relatively small scatter of 0.3 in SMG radio spectral index. Importantly, for larger ∼ ′ ′ ′ h, both LTIR/Lradio and α asymptote as CR electron and positron losses are entirely determined by synchrotron and IC; h 1 kpc starbursts are already near this limit. ≈ In other words, for arbitrarily large h, the radio excess with respect to the z 0 ≈ FRC asymptotes to a value of 5 10, depending on the assumed Schmidt Law, ∼ − and the radio spectral slope asymptotes to 1.1 for p = 2.2. ∼ 3.4.3. Synchrotron Radio Emission as a Star-Formation Tracer Relying on the FRC, a number of studies have used the GHz radio emission as a tracer of star-formation (e.g., Cram 1998; Mobasher et al. 1999; Haarsma et al. 2000; Carilli et al. 2008; Seymour et al. 2008; Garn et al. 2009). Radio emission has the advantage that it is unaffected by dust obscuration, making it potentially very useful in starbursts, and therefore for most of the star-formation at z > 1 (e.g., ∼ Chary & Elbaz 2001; Le Floc’h et al. 2005; Dole et al. 2006; Magnelli et al. 2009; Pascale et al. 2009). Our models let us evaluate the theoretical basis for radio as a 177 SFR indicator at all redshifts. We show the predicted radio emissivity as a function of star formation in Figure 3.6. −2 −2 −1 At low surface densities (Σg < 0.01 g cm ; ΣSFR < 0.06 M kpc yr ), ∼ ∼ ⊙ synchrotron radio has a non-linear dependence on star-formation at all redshifts. The weak radio emission is caused by electrons escaping their host galaxies, as inferred by Bell (2003) and discussed in LTQ. That is, normal galaxies are not perfect electron calorimeters, so the radio emission is not a reliable star-formation tracer at low ΣSFR. At higher redshift, synchrotron emission is diminished by Inverse Compton off the CMB. For a Milky Way-like galaxy, we find that radio is a good star-formation tracer at z < 1, but underestimates it significantly by z > 2 (eq. 3.1). ∼ ∼ Already galaxies with star formation rates similar to Galactic levels are beginning to be observed in the radio at high redshift (Garn et al. 2009), so that the IC suppression may soon be observed. However, IC losses off the CMB should not be important even in the weakest starbursts until z > 4, as is also visible by the redshift ∼ evolution of the FRC in Figure 3.3. Radio emission does grow linearly with star-formation rate between normal −2 galaxies with Σg = 0.01 g cm (at z < 2) and compact starbursts. Therefore, ∼ it serves as an acceptable star-formation indicator for these galaxies. However, if B Σ0.7−0.8, puffy starbursts like SMGs have about 2 4 times the radio emission ∝ g − at any given star-formation rate than compact starbursts. Therefore, we expect that if B Σ0.7−0.8, the usual radio emission estimate based on the z 0 FRC ∝ g ≈ 178 will overestimate their star formation rates by a factor of 2 4. The excess is ∼ − −2 −2 −1 greatest at Σg 1 g cm corresponding to ΣSFR 40 200 M kpc yr , typical ≈ ≈ − ⊙ of observed SMGs. Among the puffy starbursts themselves, the radio emission grows linearly with star-formation rate. If instead B ρ0.5−0.6, the radio emission will ∝ underestimate the star-formation rate. However, because the SMGs lie on their own FRC, there is little real redshift evolution in the radio emissivity of puffy starbursts, because of the buffering provided by IC losses off starlight (see 3.4.1, § Appendix 3.6). Assuming a spectral slope α = 0.7 will also underestimate the radio emissivity, since these galaxies can have steep radio spectra. This explains the apparent evolution with z of puffy starbursts in Figure 3.6: we have applied a k-correction using a typically employed α = 0.7 when, in fact, the true synchrotron spectra are steeper. If the radio star-formation tracer could be calibrated to the special conditions in puffy starbursts like SMGs, taking into account their different scale height, radio emissivity, and spectral slopes, we predict that radio would be a more accurate star-formation tracer for them. We can also directly calculate the observed GHz radio flux density Sν from synchrotron emission of star-forming galaxies. We show the predicted flux density per unit star-formation at observer-frame 1.4 GHz in Figure 3.7. Surveys with the Expanded Very Large Array (EVLA) will have a continuum sensitivity of 1 µJy ∼ at frequencies of 1 - 50 GHz, and it should be able to directly detect galaxies 179 −2 −1 with Σg > 0.01 g cm and star-formation rates of 1 M yr out past z 0.5. ∼ ⊙ ≈ As stated in Murphy (2009), a starburst like M82 with SFR 3 M yr−1 will ≈ ⊙ become undetectable past z 1. However, the buffering effect we emphasize in ≈ this paper preserves the radio emission of dense starbursts at high z, so that bright starbursts will be detectable further: starbursts with SFR > 100 M yr−1 will be ∼ ⊙ detectable with EVLA at 1.4 GHz out to z 4 5 and the most intense starbursts ≈ − −1 −2 (SFR > 1000 M yr and Σg > 1 g cm ) will be detectable out past z 10 ∼ ⊙ ∼ ≈ in synchrotron emission. Murphy (2009) predicts the Square Kilometer Array (SKA) will be sensitive to star-forming galaxies with flux densities of 20 nJy. If ∼ this sensitivity is attained, then Milky Way-like galaxies (Σ 0.01 g cm−2; SFR g ≈ 1 M yr−1) will be directly detectable at 1.4 GHz in synchrotron emission out to ≈ ⊙ z 2, and starbursts with SFR 1 M yr−1 will be detectable at 1 GHz beyond ≈ ≈ ⊙ z 3. Even at z 10, the synchrotron emission of dense, compact starbursts ≈ ≈ −2 −1 (Σg > 1 g cm ) with SFR greater than 10 M yr should be detectable at 1.4 ∼ ⊙ GHz with SKA (models with B ρ0.5−0.6 have radio-dim puffy starbursts, and these ∝ are detectable at z = 10 with the SKA only for SFR greater than 20 60 M yr−1). − ⊙ By contrast, Murphy (2009) found a sensitivity of 25 M yr−1, based on the ⊙ free-free emission; this limit will apply to normal galaxies and weak starbursts where the synchrotron emission is suppressed. These sensitivities assume that natural confusion, in which radio sources overlap, will not hamper the SKA; estimates for 180 the natural confusion limit vary from nJy to µJy levels (e.g., Jackson 2004; Condon 2009; Murphy 2009). SKA and EVLA will also have good spectral coverage, which may help measurements of the spectral index. If a galaxy is detected at 5σ ( µJy for EVLA ∼ and 20 nJy for SKA; Murphy 2009) at two different frequencies ν1 and ν2 with ν /ν = 5, then α2 can be constrained to 0.1 0.15 at the 1σ level. EVLA will 2 1 1 ∼ − be better at high observer-frame frequencies (1 - 50 GHz). A problem for the EVLA will be the increasing fraction of thermal emission, which is expected to dominate the emission of starbursts at ν′ 30 GHz. EVLA will therefore not easily measure ≈ the nonthermal spectral indices of galaxies at high z. On the other hand, the SKA will face free-free absorption when observing low-z starbursts; for example, Arp 220 may be optically thick even at 1 GHz (Condon et al. 1991a). At high redshift, however, the rest-frame frequencies SKA will observe will be less affected by free-free absorption. 3.4.4. SMGs, The Radio Background, and Radio Source Counts Star-formation in galaxies over cosmic time produces a diffuse radio synchrotron background. Our models indicate that submillimeter galaxies and other puffy starbursts ought to have enhanced radio emission. In principle, this means that the 181 synchrotron radio background could be up to 2 4 times higher than usually ∼ − predicted from the Cosmic Infrared Background (CIB) and a naive application of the z 0 FRC. The magnitude of this implied radio excess is interesting, because ≈ ARCADE2 recently reported an excess radio background at 3 GHz, about five times higher than expected from star formation (Fixsen et al. 2009; Seiffert et al. 2009). Singal et al. (2010) found that constraints on Inverse Compton emission require the excess to come from regions with galactic-level (>µG) magnetic fields, and suggest ∼ an evolution of the FRC as the source of the reported excess. However, while SMGs are individually very bright and contribute much to the cosmic star-formation rate at z > 2.5, they are not typical starbursts (Bavouzet et al. 2008). Instead, they seem ∼ to represent a transient phase that can survive about 100 Myr before their gas is depleted (Tacconi et al. 2006; Pope et al. 2008). We can estimate the total radio background enhancement by scaling to the contribution of SMGs to the CIB, which is largely reprocessed starlight from galaxies at z > 1 (e.g., Dole et al. 2006; Devlin et al. 2009), and adjusting by the SMG radio ∼ excess. Submillimeter galaxies do provide the majority of light at 850µm, but ∼ this is only a small fraction of the total IR background. At the peak of the CIB ( 160µm), submillimeter galaxies provide < 10% of the total power, and possibly ∼ ∼ only 2% (Chapman et al. 2005; Dye et al. 2007). Optimistically, the radio excess ∼ from SMGs would be 10% (4 1) 30%. This is significant, but not enough to × − ≈ 182 explain the very large ARCADE2 excess. More conservatively, the excess is more likely 5% 2 10%, and could be as little as a few percent. × ≈ Since the number of radio sources down to several µJy is well known, we can estimate the fraction of the expected radio background comes from bright SMGs. Dole et al. (2006) find a TIR background of about 24 nW m−2 sr−1; from the normalization of the FIR-radio correlation we expect a 1.4 GHz background of νI 2.6 10−5 nW m−2 sr−1. Chapman et al. (2005) found an average radio flux ν ≈ × density of S1.4 75 µJy for bright SMGs (S850 µm > 5 mJy). The number counts of ≈ ∼ 75 µJy sources imply that they have a density of 1500 deg−2, contributing roughly ∼ 5.1 10−6 nW m−2 sr−1 to the 1.4 GHz radio background (e.g., Gervasi et al. ∼ × 2008). Bright SMGs have an approximate density of 600 deg−2 (e.g., Wang et al. ∼ 2004; Coppin et al. 2006), so they constitute about 40% of the background from ∼ 75 µJy sources, or 8% of the expected 1.4 GHz background from star-formation. ∼ This is roughly in line with our estimate of 10%, although the uncertainties are ∼ large enough that it could be consistent with SMGs lying on the FRC. In any case, it is fairly clear that bright SMGs are not the source of the ARCADE excess. The total excess could be greater if most starbursts at z > 1 are puffy, and not ∼ just SMGs. We do not expect this simply because most current studies show that the local FRC does hold out to high redshift for most observed galaxies and starbursts (e.g., Murphy et al. 2009a; Younger et al. 2009; Garn et al. 2009). The spectral slope would also be a problem: ARCADE2 inferred α = 0.6, while we predict a spectral 183 slope α > 0.7, steeper than local compact starbursts. Resolved radio sources in the ∼ range 50 µJy (2009) find no large evolution in the FRC down to S 20 µJy. Any large radio 1.4 ≈ excess would have to come either from a new population of low luminosity galaxies or very high-z galaxies (see Figure 3.7); extrapolations of the higher flux source populations do not predict a large radio excess (e.g. Gervasi et al. 2008). Nonetheless, our work indicates that a radio excess from SMGs can be significant. Conversely, if the luminosity function of galaxies was very steep, most galaxies could be intrinsically radio-dim with respect to the FRC, because of IC losses off the CMB (see 3.4.1 and Figures 3.1 and 3.3). Then the FRC would § overestimate the strength of the radio background. However, most star-formation at high z is believed to have occurred in starbursts (Le Floc’h et al. 2005; Dole et al. 2006), so this possibility is unlikely. 184 3.5. Summary and Cavaets We have applied the theory of LTQ to predict the FRC for redshifts 0 z 10. ≤ ≤ We use one-zone models of galaxies and starbursts with CR injection, cooling, and escape to predict the equilibrium, steady-state radio spectra of galaxies and starbursts over the entire range of the FRC. Our goals were to determine how and why the low- and high-Σ conspiracies crucial to the z 0 FRC ( 3.2) affect the g ≈ § FRC at high redshift, and to provide a quantitative model for predicting the critical redshifts at which galaxies deviate from the z 0 FRC. We find the following: ≈ 1. For compact starbursts (h 100 pc), we find relatively little evolution in ≈ the FIR-radio correlation out to z 5 10 (Figures 3.1 and 3.2). This is ≈ − partly because the magnetic energy density in galaxies is strong enough to dominate the CMB even at high redshifts. However, the high-Σg conspiracy ( 3.2) also acts as a buffer against IC losses off the CMB; the increased IC § losses must compete with the already present bremsstrahlung, ionization, and IC off starlight in addition to synchrotron losses. The rest-frame radio spectral slope α′ at fixed ν′ does not change with z, but the observed α at fixed ν increases because the non-thermal synchrotron radio spectrum steepens at higher rest-frame frequency. 2. We derive in Appendix 3.6 the critical redshifts when Inverse Compton losses off the CMB suppress the radio luminosity of galaxies compared to the z 0 ≈ 185 FRC. These relations are given for our standard model in equation 3.1. The non-thermal radio luminosity is suppressed severely in Milky Way-like galaxies −2 −1 (ΣSFR 0.06 M kpc yr ) at z 2 and the weakest compact starbursts ≈ ⊙ ≈ at z > 5. The spectrum at GHz steepens to α 1 because of these enhanced ∼ ≈ IC losses. Nonetheless, the low-Σ conspiracy ( 3.2) also acts to prevent the g § radio emission from steeply falling with redshift, since Inverse Compton losses off the CMB must be more efficient than diffusive escape, not just synchrotron losses (see 3.4.1). § 3. LTQ found that the z = 0 FRC demands that B scales with ρ or Σg in galaxies lying on the Schmidt law. In models with B Σ0.7−0.8, we find that puffy ∝ g starbursts with h = 1 kpc such as SMGs are radio bright compared to the z = 0 FRC by a factor of 2 4. This follows from a breakdown of the high-Σ ∼ − g conspiracy ( 3.2): bremsstrahlung and ionization cooling are weak in puffy § starbursts relative to the compact starbursts that predominate in the z = 0 universe. In contrast, in models with B ρ0.5−0.6, we find that puffy starbursts ∝ are radio dim compared to the observed FRC, because of weak synchrotron cooling relative to the IC losses. Since several studies have reported radio excesses for SMGs, we favor the B Σ0.7−0.8 scaling; however the issue of ∝ g whether SMGs are radio-bright is still not fully resolved. In either case, puffy starbursts show little true evolution with z, though they may appear to have fainter rest-frame radio luminosities at high z because of their steep spectra. 186 Puffy starbursts inevitably have high α (> 0.7), since bremsstrahlung and ∼ ionization losses are weak with respect to synchrotron and IC. A key prediction of our scenario with B Σ0.7−0.8 is that the variations in L′ /L′ will ∝ g TIR radio be correlated with scale height at fixed ΣSFR, since the radio-excess in our models is a direct consequence of the large CR scale height and the small bremsstrahlung and ionization losses it causes. Radio-excess (low q) puffy starbursts will have steeper radio spectral slopes (bigger α), larger velocity dispersions σ compared to their rotation speeds vcirc, and possibly moderately cooler dust temperatures (smaller Tdust). 4. As previously expected, radio emission can be a poor tracer of star formation in low surface density galaxies, because of electron escape and IC losses off the CMB. For our preferred B Σ0.7−0.8 scaling, radio emission overestimates ∝ g star-formation rate by a factor of 2 4 in puffy starbursts. Star-formation rate − is underestimated by synchrotron radio emission with the B ρ0.5−0.6 scaling. ∝ 5. While SMGs may be individually radio bright compared to the local FRC, they contribute a relatively small fraction of the Cosmic Infrared Background and the total star-formation luminosity of the Universe. This means that they enhance the star-formation radio background by < 50%, and possibly around ∼ 10%, with respect to a naive application of the z = 0 FRC. ∼ 187 As in LTQ, we did not exactly match the observed radio spectral slopes of galaxies, with α 0.9 1.0 in normal galaxies and α 0.4 0.6 in compact ≈ − ≈ − starbursts. This will have a slight effect on the k-correction. An error of 0.25 in α should only affect inferred radio luminosity by 30% at z = 2 and 60% at z = 5. Nevertheless, the prediction of steeper radio spectra in puffy starbursts with respect to compact starbursts at all relevant z should be robust. ′ ′ Our explanation for the small LTIR/Lradio ratio in submillimeter galaxies as a breakdown of the high-Σg conspiracy is based purely on the steady-state spatially-averaged synchrotron emission, but the details of the FIR emission may also matter. Throughout this paper, we have simply assumed that the bolometric FIR luminosity could be correctly inferred from observations, and have assumed the same UV opacity for all galaxies at all redshifts. The total FIR emission is also likely to depend on the metallicity, and may be lower at high z for the lowest surface density galaxies. The exact far-infrared SED is important in determining the FIR emission when observations have only been made at only a few wavelengths. The presence of AGNs, a different IMF at high z, and selection biases may also affect the inferred q of SMGs. We did not include the effects of galaxy evolution on the CR spectrum in our models. We argued in LTQ that it should not matter for quiescent spirals or for extreme starbursts, because the CR lifetime is much shorter than the time dependence of stellar populations. However, galaxy evolution may play a role in 188 weaker starbursts (Lisenfeld et al. 1996b) and in post-starburst galaxies (Bressan et al. 2002). Studies of merging normal galaxies and galaxies in clusters have indeed found that they are radio bright with respect to the FRC, possibly because of compression of magnetic fields or shock acceleration (Gavazzi et al. 1991; Miller & Owen 2001; Murphy et al. 2009b). We also assumed that the magnetic field strength at a given density does not depend on redshift. It is not entirely clear how long normal galaxies take to build up their magnetic fields, or even what process is at work (see the reviews in Widrow 2002; Kulsrud & Zweibel 2008), though there are theoretical mechanisms that can rapidly generate strong magnetic fields. Studies at z 2 indicate that ≈ normal Milky Way-like galaxies had magnetic fields with similar strengths to the present (Kronberg et al. 2008; Bernet et al. 2008; Wolfe et al. 2008). At the very highest redshifts, magnetic field strengths might be weaker, because the seed fields were essentially zero compared to the present strengths. Starbursts also may build their magnetic fields up in much shorter times than normal galaxies, and through a different process than normal galaxies (Thompson et al. 2009). Finally, we have used one-zone models, which are appropriate if the CRs sample all of the gas phases in each galaxy’s ISM. However, the ISM is known to be clumpy in the Milky Way, in compact starbursts like Arp 220 (e.g., Greve et al. 2009), and even in the SMGs themselves (Tacconi et al. 2006). A full understanding of 189 the FRC will probably require models that take into account the inhomogeneity of star-forming galaxies. 3.6. Derivation of Radio Suppression from CMB At high redshift, the nonthermal synchrotron radio luminosity of galaxies is suppressed by the CMB. This is because the Inverse Compton process shortens the lifetime that CRs have to radiate synchrotron; equivalently, their energy goes into Inverse Compton photons instead of synchrotron radio. Because of the shorter times the CR electrons and positrons have to radiate synchrotron, the redshift zcrit when the radio luminosity of a galaxy is quenched by a factor is given by Q tsynch tsynch (z =0)= (z = zcrit). (3.2) Q tloss tloss The loss time includes all cooling and escape losses, so that t t t t t t t synch = 1.0+ synch + synch + synch + synch + synch + synch (3.3) tloss tIC,⋆ tIC,CMB tbrems tion tdiff tadv In some regimes, some losses may be neglected. For example, diffusive losses are unimportant in starbursts, as are IC losses off the CMB at z = 0. Advective losses may be unimportant in normal galaxies and also for the densest starbursts. 190 We can immediately see that the presence of losses besides synchrotron and IC off the CMB prevent the suppression of radio. In the case when no losses other than synchrotron and IC off the CMB exist, equation 3.2 reduces to UB 4 ( 1) + = (1+ zcrit) . (3.4) Q − UCMB(z = 0) Q The presence of other loss processes, whether escape or cooling, requires a higher z for a fixed quench factor : crit Q UB tsynch tsynch tsynch tsynch tsynch 4 ( 1) 1.0+ + + + + + = (1+zcrit) .(3.5) Q− UCMB(z = 0) tIC,⋆ tbrems tion tdiff tadv ! Q Essentially the IC losses off the CMB must compete with not only synchrotron losses, but every other cooling and escape process as well. With the high-Σg conspiracy, t /t can be of order 10 20, so the energy density of the CMB must be synch loss − 10 20 times greater to suppress the radio emission than would be naively ∼ − expected. We give the loss times for each process in LTQ. From those lifetimes, we can find the ratios of the synchrotron cooling timescale to the other loss timescales: tsynch −2 4 = 0.11B10 (1 + z) (3.6) tIC,CMB t Σ synch = 19B−2 SFR (3.7) 10 −2 −1 tIC,⋆ M kpc yr ! ⊙ tsynch ′−1/2 −3/2 −1 Σg = 98fν1.4 B10 hkpc −2 (3.8) tbrems g cm ! 191 tsynch ′−1 −1 −1 Σg = 15fν1.4 B10 hkpc −2 (3.9) tion g cm ! tsynch ′−1/4 −7/4 = 1.5ν1.4 B10 (3.10) tdiff tsynch ′−1/2 −3/2 −1 = 12ν1.4 B10 hkpcv300, (3.11) tadv where B10 = B/(10µG), hkpc = h/(1kpc), v300 is the wind speed in units of 300 −1 ′ km s , and ν1.4 is the rest-frame frequency divided by 1.4 GHz. Our models parameterize B in terms of the gas surface density Σg: −2 0.7 0.7 40(Σg/g cm ) (B Σ ) ∝ g −2 0.8 0.8 72(Σg/g cm ) (B Σg ) ∝ B10 = (3.12) 12(Σ /g cm−2)0.5h−0.5 (B ρ0.5) g kpc ∝ −2 0.6 −0.6 0.6 22(Σg/g cm ) h (B ρ ). kpc ∝ Finally, the Schmidt Law allows us to convert between gas surface density Σg and surface density of star-formation rate ΣSFR: Σ 0.71 0.078 SFR (K98) −2 −1 Σg M kpc yr ! − = 0.59 (3.13) g cm 2 ! ⊙ Σ SFR 0.048 −2 (B07). M kpc yr−1 ! ⊙ 192 We can approximately solve for the redshift z when = 3 for galaxies and crit Q starbursts in each of the scenarios in Table 3.6. For the scenario with the B07 star-formation law, B Σ0.7, and no winds, we find: ∝ g 1.4 (Normal galaxies, ΣSFR < 0.02) ∼ 0.23 5.8ΣSFR 1 (Normal galaxies, ΣSFR > 0.02) − zcrit ∼ (3.14) ≈ 5.7Σ0.23 1 (Puffy starbursts) SFR − 0.23 7.4Σ 1 (Compact starbursts), SFR − −2 −1 where ΣSFR is in units of M kpc yr . For the scenario with the B07 ⊙ star-formation law, B Σ0.8, and winds in starbursts, we find: ∝ g 1.5 (Normal galaxies, ΣSFR < 0.02) ∼ 0.24 6.6Σ 1 (Normal galaxies, ΣSFR > 0.02) SFR − ∼ 0.06 4.8ΣSFR 1 (Puffy starbursts, ΣSFR < 0.2) − zcrit ∼ (3.15) ≈ 6.5Σ0.24 1 (Puffy starbursts, Σ > 0.2) SFR − SFR ∼ 0.06 8.6ΣSFR 1 (Compact starbursts, ΣSFR < 0.8) − ∼ 9.0Σ0.24 1 (Compact starbursts, Σ < 0.8). SFR SFR − ∼ 193 For the scenario with the B07 star-formation law, B ρ0.6, and winds in starbursts, ∝ we find: 1.4 (Normal galaxies, ΣSFR < 0.02) ∼ 0.21 5.6Σ 1 (Normal galaxies, ΣSFR > 0.02) SFR − ∼ 0.04 4.5ΣSFR 1 (Puffy starbursts, ΣSFR < 0.2) − zcrit ∼ (3.16) ≈ 0.21 5.8ΣSFR 1 (Puffy starbursts, ΣSFR > 0.2) − ∼ 0.04 9.5Σ 1 (Compact starbursts, ΣSFR < 0.7) SFR − ∼ 0.21 10.0ΣSFR 1 (Compact starbursts, ΣSFR < 0.7). − ∼ For the standard model of LTQ, with the K98 star-formation law, B Σ0.7, and no ∝ g winds, we find: 1.4 (Normal galaxies, ΣSFR < 0.02) ∼ 0.25 6.6ΣSFR 1 (Normal galaxies, ΣSFR > 0.02) − zcrit ∼ (3.17) ≈ 6.8Σ0.25 1 (Puffy starbursts) SFR − 0.25 10.2ΣSFR 1 (Compact starbursts). − 194 Finally, for the model of LTQ with the K98 star-formation law, B ρ0.5, and winds ∝ in starbursts, we find: 1.4 (Normal galaxies, ΣSFR < 0.02) ∼ 0.23 5.6Σ 1 (Normal galaxies, ΣSFR > 0.02) SFR − ∼ 0.04 4.5ΣSFR 1 (Puffy starbursts, ΣSFR < 0.3) − zcrit ∼ (3.18) ≈ 0.23 5.5ΣSFR 1 (Puffy starbursts, ΣSFR > 0.3) − ∼ 0.04 9.2Σ 1 (Compact starbursts, ΣSFR < 0.7) SFR − ∼ 0.23 9.9ΣSFR 1 (Compact starbursts, ΣSFR < 0.7). − ∼ 195 Fig. 3.1.— The FIR-radio correlation at high redshift, in the rest-frame, using the 0.7 B07 star formation law. This is the case with B Σg and no winds. Solid lines have −2 ∝ −2 −1 h = 100 pc for starbursts (Σg 0.1 g cm ; ΣSFR 2 4 M kpc yr ), while dotted lines have h = 1 kpc for≥ starbursts. A linear≥ FRC− is produced⊙ at z = 0 with the correct normalization. Galaxies become radio-dim at high redshift as IC losses off the CMB increase. Note that in each case the puffy starbursts do not lie on the same FRC as compact starbursts: there is a systematic offset caused by the unbalancing of the high-Σg conspiracy ( 3.4.2). We do not include thermal radio emission, which § ′ will set a minimum radio luminosity or maximum qFIR at each Σg. 196 Fig. 3.2.— The FIR-radio correlation at high redshift, in the rest-frame, using the B07 star formation law. This is the case with B ρ0.6 and winds. Solid lines have −2 ∝ −2 −1 h = 100 pc for starbursts (Σg 0.1 g cm ; ΣSFR 2 4 M kpc yr ), while dotted lines have h = 1 kpc for≥ starbursts. A linear≥ FRC− is produced⊙ at z = 0 with the correct normalization. Galaxies become radio-dim at high redshift as IC losses off the CMB increase. Note that in each case the puffy starbursts do not lie on the same FRC as compact starbursts: there is a systematic offset caused by the unbalancing of the high-Σg conspiracy ( 3.4.2). We do not include thermal radio emission, which § ′ will set a minimum radio luminosity or maximum qFIR at each Σg. 197 Fig. 3.3.— The evolution of the FIR-radio correlation, in the rest-frame at ν′ = 1.4 GHz. Black lines have h = 100 pc for starbursts, while grey lines have h = 1 kpc 0.7 for starbursts. This is the case with B Σg , no winds, and the B07 star- formation law. Normal galaxies become radio∝ dim because of IC losses off the CMB at intermediate redshift. Both compact and puffy starbursts maintain their rest-frame radio luminosities until high redshift. All galaxies are “buffered” by non-synchrotron losses ( 3.4.1 and Appendix 3.6), so that the evolution is not as great as would be expected§ with only synchrotron and IC losses off the CMB. The inferred evolution is usually greater than the rest-frame evolution, because normal galaxies and puffy starbursts have α > 0.7. ∼ 198 Fig. 3.4.— The evolution of the FIR-radio correlation as inferred for the rest-frame from observations at ν = 1.4 GHz. Black lines have h = 100 pc for starbursts, while 0.7 grey lines have h = 1 kpc for starbursts. This is the case with B Σg , no winds, and the B07 star-formation law. Normal galaxies become radio dim∝ because of IC losses off the CMB at intermediate redshift. Both compact and puffy starbursts maintain their rest-frame radio luminosities until high redshift. All galaxies are “buffered” by non-synchrotron losses ( 3.4.1 and Appendix 3.6), so that the evolution is not as great as would be expected§ with only synchrotron and IC losses off the CMB. The inferred evolution is usually greater than the rest-frame evolution, because normal galaxies and puffy starbursts have α > 0.7. ∼ 199 −2 Fig. 3.5.— The rest-frame synchrotron radio spectra of starbursts with Σg =1gcm , using the B07 star-formation law. Puffy starbursts are dashed, while compact starbursts are solid. These spectra do not include thermal absorption (at low frequencies, < 1 GHz) or emission (at high frequencies, > 30 GHz). ∼ ∼ 200 Fig. 3.6.— The relationship between the inferred rest-frame radio emission and star- 0.7 formation, using the B07 star formation law, B ΣSFR, and no winds. Solid lines have h = 100 pc for starbursts, while dotted lines∝ have h = 1 kpc for starbursts. A flat line would indicate that radio emission is directly proportional to ΣSFR. We see that at low Σg, the radio flux underestimates the star-formation rate even at z = 0, because of CR electron escape (the low-Σ conspiracy of 3.2). The radio g § flux overestimates the star-formation rate for puffy starbursts, because the high-Σg conspiracy is unbalanced ( 3.2; 3.4.2). Finally, the radio emission is suppressed at high redshift, partly because§ of§ IC losses off the CMB, and partly because α > 0.7 for puffy starbursts and normal galaxies (α = 0.7 assumed here for the k-correction).∼ 201 Fig. 3.7.— The observed 1.4 GHz synchrotron radio flux per unit star-formation as 0.7 a function of redshift, using the B07 star formation law, B ΣSFR, and no winds. EVLA will have a sensitivity of 1 µJy and SKA will have a∝ sensitivity of 20 nJy. ∼ ∼ 202 Fig. 3.8.— The redshift zcrit when the rest-frame 1.4 GHz synchrotron luminosity is suppressed by a factor of 3 from IC losses off the CMB, in our model with the B07 0.7 star-formation law, B Σg , and no winds. The dotted line is for puffy starbursts. For starbursts, the suppression∝ is only important at very high z, typically beyond those that will be observed by EVLA and SKA. 203 SFR Winds B FRC Scatter (PS)a PS FRC Spectral slopes c Normalization (∆q′)b Norm CS PS 0.7 K98 N Σg 1.7 (1.5) 3.9 (-0.59) 0.92 - 0.96 0.44 - 0.64 0.69 - 0.92 K98 Y ρ0.5 1.7 (1.7) 0.85 (0.07) 0.88 - 0.94 0.43 - 0.61 0.78 - 0.85 0.7 B07 N Σg 1.7 (1.5) 2.2 (-0.35) 0.87 - 0.96 0.57 - 0.67 0.86 - 0.98 0.8 B07 Y Σg 2.2 (1.7) 3.4 (-0.53) 0.85 - 0.91 0.45 - 0.59 0.75 - 0.87 B07 Y ρ0.6 2.0 (2.6) 0.49 (0.31) 0.87 - 0.91 0.49 - 0.64 0.88 - 0.91 204 a ′ ′ Variation in the local LTIR/Lradio over normal galaxies and compact starbursts, as measured at z = 0 at ′ ′ ′ ν = 1.4 GHz. The value in parentheses is the variation in LTIR/Lradio for puffy starbursts alone at z = 0 with ν′ = 1.4 GHz. bAverage radio-brightness of puffy starbursts at z = 0, compared to the local normalization of the FRC for compact starbursts and normal galaxies. ∆q′ is the offset in q′ from its locally observed value for the puffy starbursts. cRange of instantaneous spectral slopes α at ν′ = 1.4 GHz at z = 0. Norm = normal galaxies; CS = compact starbursts; PS = puffy starbursts. Table 3.1. Model Properties Chapter 4 On The GeV & TeV Detections of the Starburst Galaxies M82 & NGC 253 4.1. Introduction M82 and NGC 253 are nearby (D 2.5 4.0 Mpc), prototypical starburst ≈ − galaxies, each having an intense star-forming region of about 200 pc radius in the center of a more quiescent disk galaxy.1 The starbursts are expected to have high supernova (SN) rates of about 0.03 0.3 yr−1. SN remnants are believed to − accelerate primary cosmic ray (CRs) protons and electrons. The high SN rates in starbursts imply high CR emissivities. The presence of CR electrons and positrons in these starbursts is inferred from the nonthermal synchrotron radio emission they produce (e.g., Klein et al. 1988; V¨olk et al. 1989). However, most of the CR energy is believed to be in the form of CR protons. When high energy CR protons collide with interstellar medium (ISM) nucleons, they create pions, which decay into secondary electrons and positrons, γ-rays, and 1This chapter was published as Lacki et al. (2011, ApJ 734, 107). 205 neutrinos. With their dense ISMs ( n 100 500 cm−3) and possible high CR h i ≈ − energy densities (as evinced by the bright radio emission V¨olk et al. 1989; Akyuz et al. 1991; Persic & Rephaeli 2010), M82 and NGC 253 are predicted to be bright γ-ray sources (e.g., Akyuz et al. 1991; Sreekumar et al. 1994; V¨olk et al. 1996; Paglione et al. 1996; Romero & Torres 2003; Domingo-Santamar´ıa & Torres 2005; Thompson, Quataert, & Waxman 2007 [TQW]; Persic et al. 2008; de Cea del Pozo et al. 2009a; Rephaeli et al. 2010; Lacki et al. 2010a [LTQ]). As prototypical starbursts, if M82 and NGC 253 are seen in γ-rays, starbursts in general may be sources of γ-rays (Pohl 1994; Torres et al. 2004a), with important implications for the diffuse γ-ray and neutrino backgrounds (e.g., Pavlidou & Fields 2002; Loeb & Waxman 2006 [LW06]; TQW). However, the γ-ray luminosity of starbursts depends not only on the injection rate of CRs, but also on the efficiency of converting CR proton energy into pionic γ-rays, neutrinos, and secondary electrons and positrons. In turn, this efficiency depends on the ratio of the timescale for pion production to the escape timescale. The hypothesis that CR protons in starbursts lose all of their energy to pionic collisions before escaping is called “proton calorimetry” (c.f. Pohl 1994).2 If proton calorimetry is strongly violated, then M82 and NGC 253 and, by extension, other starbursts could in fact be weak γ-ray sources. Although γ-ray emission from M82 and NGC 253 has been sought for several years with no success (at GeV, Cillis et al. 2005; and at TeV, Aharonian et al. 2Here, we consider only CR protons with kinetic energy above the threshold for pion production. 206 2005; Itoh et al. 2007), the launch of Fermi and the advent of powerful VHE γ-ray telescopes has led to recent detections of both starbursts at GeV energies (with Fermi; Abdo et al. 2010a) and in VHE γ-rays (M82 with VERITAS, Acciari et al. 2009; NGC 253 with HESS, Acero et al. 2009). These GeV and TeV detections constrain the cosmic ray (CR) population in these dense star-forming environments. In this paper, we discuss the implications of the γ-ray detections of M82 and NGC 253. The ratio of the γ-ray luminosities to the bolometric luminosities informs the question of whether or not these systems are proton calorimeters (TQW). The emission also has implications for the energy density of CRs in starbursts (e.g., Akyuz et al. 1991). Finally, combined with the radio emission, the energy losses of CR electrons and positrons are constrained (c.f., Paglione et al. 1996; Domingo-Santamar´ıa & Torres 2005; Persic et al. 2008; de Cea del Pozo et al. 2009a; Rephaeli et al. 2010). Pionic γ-rays must be accompanied by secondary positrons and electrons; the ratio of the power in these expected electrons and positrons to the observed radio emission informs us of the energy losses of the CR electrons and positrons. In particular, we derive in 4.5 the expected synchrotron luminosity from § the pionic luminosity if synchrotron cooling is the dominant loss process. In 4.2 we describe the detections of M82 and NGC 253 at GeV and TeV § energies. We then interpret the detections as γ-rays from diffuse CR protons in 4.3. § Our interpretation includes comparison of the γ-ray luminosities of M82 and NGC 253 with their CR luminosities and their IR luminosities ( 4.3.2), and a discussion § 207 of the uncertainties in these estimates ( 4.3.3). We find that a fraction 0.4 and 0.2 § of luminosity in GeV CR protons is lost to pion production in M82 and NGC ≥ 253, respectively. We discuss the implications for our estimates mean for proton calorimetry in M82 and NGC 253 are at GeV energies ( 4.3.4). Other possible § sources for the observed γ-rays are considered in 4.4. The implications of the § detections of M82 and NGC 253 for the detection of other star-forming galaxies, the starburst contribution to the diffuse extragalactic γ-ray and neutrino backgrounds, the dynamical importance of CRs in starbursts, and for the physics of the FIR-radio correlation are described in 4.6. We summarize our results in 4.7. § § 4.2. γ-ray Detections 4.2.1. Fermi and TeV detections Abdo et al. (2010a) reported the detections of M82 (6.8σ) and NGC 253 (4.8σ) with the Fermi LAT instrument. At energies above a few hundred MeV, the γ-ray spectrum of starburst galaxies is expected to be described by a power law spectrum with differential photon fluxes N(E)= N (E/E )−Γ, where Γ ( 2) is the 0 0 ∼ photon spectral index (e.g., Paglione et al. 1996). Using the GeV data point as a normalization, we adopt GeV differential fluxes of −Γ +0.5 −9 E −2 −1 −1 NM82 1.9−0.4 10 ph cm s GeV (4.1) ≈ × GeV 208 −Γ +0.4 −9 E −2 −1 −1 NNGC253 0.9−0.3 10 ph cm s GeV . (4.2) ≈ × GeV The Abdo et al. (2010a) maximum likelihood analysis of the sources find Γ of 2.2 0.2 0.05 for M82 and 1.95 0.4 0.05 for NGC 253; however, the Fermi ± ± ± ± photon statistics are insufficient for accurate measurements of Γ. Using the best-fit power laws in Abdo et al. (2010a) gives GeV normalizations that are 60% 70% of − those in equations 4.1-4.2, although these fits are influenced by the nondetections at lower and higher energies. Fermi has not detected either starburst above 20 GeV, ∼ and NGC 253 is undetected below 500 MeV. The reported spectra are shown in ∼ Figures 4.1 and 4.2. Assuming a power law spectrum from GeV to VHE energies, we can combine the Fermi detections with the HESS and VERITAS measurements to derive the spectral slope over this energy range. The integrated fluxes from M82 reported by VERITAS (at 1.3 - 3.8 TeV) correspond to GeV-VHE spectral slopes Γ of 2.19 - 2.25 (Acciari et al. 2009), in excellent agreement with the measured spectral slope from Fermi. The HESS detection of NGC 253 at 220 GeV implies Γ = 2.3 (Acero et al. 2009), steeper than the best-fit photon index from the Fermi detections, but within the quoted errors. We adopt Γ = 2.2 for M82 and Γ = 2.3 for NGC 253 throughout the rest of this paper. Our values of Γ are only appropriate if the spectrum is truly a single power law between GeV and TeV energies. A spectral bump at GeV energies will cause an 209 underlying pionic power-law spectrum to appear steeper than it really is; conversely, a spectral bump at TeV energies will cause it to appear flatter. The possibility of a “TeV excess” is notable particularly because such an excess is seen in the Milky Way (Prodanovi´cet al. 2007; Abdo et al. 2008; see 4.4.3). § Figures 4.1 and 4.2 show that the γ-ray spectra of both M82 and NGC 253 are in reasonable agreement with most of the previous detailed model predictions (Domingo-Santamar´ıa & Torres 2005; Persic et al. 2008; de Cea del Pozo et al. 2009a; Rephaeli et al. 2010). However, these models slightly overpredict both the GeV flux and the 400 MeV flux of NGC 253 by a factor of 2 3. The Paglione et al. ∼ ∼ − (1996) model greatly overestimates the GeV flux by a factor of 5 (although ≤ ∼ it used a higher CR acceleration efficiency), and predicts a much softer spectrum (Γ 2.7) than observed. Overall, the general agreement between the theory and ≈ observations of γ-rays from M82 and NGC 253 is encouraging (see also de Cea del Pozo et al. 2009b), and suggests that models of other starbursts, particularly Arp 220 (Torres 2004b, LTQ), also predict their γ-ray fluxes with fidelity ( 4.6.1). § 4.2.2. M82 and NGC 253 Gamma-Ray Luminosities −Γ For a dN/dE = N0(E/Emin) γ-ray spectrum from Emin to Emax, the total luminosity at energies greater than Emin is 39 −1 2 L ( E ) = 2.4 10 ergs s N− D β , (4.3) γ ≥ min × 9 3.5 γ 210 −9 −2 −1 −1 where N−9 = N0/(10 ph cm s GeV ), D3.5 = D/3.5 Mpc, and E 2−Γ (Γ 2)−1 1 max (Γ = 2) − " − Emin # 6 βγ = . (4.4) Emax ln (Γ = 2) E min Typically, β 2 5. γ ≈ − +0.5 The GeV flux reported by Fermi for M82 is N−9 = 1.9−0.4, and the VERITAS detections imply a Γ 2.2 spectrum extending to at least E = 3.8 TeV. This ≈ max corresponds to a > GeV luminosity of ∼ L ( GeV) 1.9+0.5 1040D2 ergs s−1. (4.5) M82 ≥ ≈ −0.4 × 3.6 If we interpret the HESS detection of NGC 253 as part of a single Γ 2.3 power law ≈ +0.4 extending at least to Emax = 220 GeV, and use the N−9 = 0.9−0.3 value from Abdo et al. (2010a), we find that the GeV luminosity of NGC 253 is L ( GeV) 5.6+2.5 1039D2 ergs s−1. (4.6) NGC253 ≥ ≈ −1.9 × 3.5 As E , the luminosity increases by only 25%. In light of the harder max → ∞ ∼ Fermi best-fit spectrum to the GeV data (Γ = 1.95 0.4 0.05), the VHE emission ± ± might be a different spectral component (perhaps pulsar wind nebulae; see 4.4.2 § and Mannheim et al. 2010) if the pionic emission falls off between 20 and 200 GeV. However, a harder spectrum with lower Emax still implies a similar γ-ray luminosity 211 ( 7 11 1039D2 ergs s−1) with those assumptions, and the Fermi data are not ∼ − × 3.5 accurate enough to conclude there is a discrepancy. 4.3. Interpretation as Pionic Emission 4.3.1. Motivation for Proton Calorimetry The hypothesis that starbursts are proton calorimeters is motivated by the short pionic energy loss time in their dense interstellar media, −1 5 neff tπ 2 10 yr −3 , (4.7) ≈ × 250 cm where neff is the average density of the ISM encountered by CR protons before escape (Mannheim & Schlickeiser 1994), and the mean gas density in the starbursts of M82 and NGC 253 is a few hundred cm−3. If tπ is less than the escape timescale, then the system is a proton calorimeter. CRs may escape by advection in galactic winds or by diffusion. The wind advection time is t h/v 2 105 yr h v−1 , (4.8) wind ≈ ≈ × 100 500 −1 for a scale height of h = 100h100pc and wind speed of v = v500km s . We expect −1/2 the diffusive escape time to go as tdiff (E) = t0(E/E0) , where t0 = 26 Myr at 212 E0 = 3 GeV in our Galaxy from radioactive isotopes in cosmic rays (cosmic ray clocks; Connell 1998; Webber et al. 2003). Little is known about the diffusion escape time in starbursts (for example, Domingo-Santamar´ıa & Torres 2005 and de Cea del Pozo et al. 2009a assume t 1 10 Myr). If we assume that CRs stream out of diff ∼ − the starbursts at the average Alfven speed vA = B/√4πρ (Kulsrud & Pearce 1969), then t 3.5 Myr h B−1 n1/2, (4.9) diff ≈ 100 200 250 −3 where B200 = B/(200 µG) and n250 = n/(250 cm ). From equations (4.7) – (4.9), pionic losses dominate advection losses if −3 −1 −3 2/3 −2/3 neff > 250 cm h100v500 and diffusive losses at a few GeV if neff > 40 cm B200 h100 , ∼ ∼ again assuming CRs stream out of the starbursts at the Alfven speed. We see that the advective and pionic loss times are roughly equal in M82 and NGC 253, implying that inelastic proton-proton collisions are an important loss process for GeV protons, and suggesting proton calorimetry (LW06, TQW). There are two ways to avoid this conclusion: more efficient escape (shorter tadv or tdiff ), or less efficient pion losses (longer tπ, smaller neff ). Both are possible. In particular, there may be a fast wind component with v 1000 2000 km s−1 from ≈ − M82 (Strickland & Heckman 2009). In addition, the pionic losses are less efficient if the CRs mainly travel through low density gas. The ISM in starbursts is clumpy, 213 with most of the volume contained in a low density phases (e.g., Lord et al. 1996; Mao et al. 2000; Rodriguez-Rico et al. 2004; Westmoquette et al. 2009). Indeed, there is γ-ray and radio evidence that CRs do not penetrate deep into molecular clouds in the Galactic Center, so that n n and proton calorimetry fails eff ≪ h i (Crocker et al. 2011a,b). If either advective losses or pionic losses dominate, the CR proton and hadronic γ-ray spectra should both be relatively hard with Γ 2.0 2.4 for standard ≈ − injection spectra. This is because the equilibrium one-zone CR proton spectrum is roughly N(E) Q(E)τ(E), where Q(E) E−p is the proton injection spectrum ≈ ∝ with 2.0 < p < 2.4 expected, and τ(E) is the CR proton lifetime including escape ∼ ∼ and (catastrophic) pion losses. Both the advective and pionic lifetimes are roughly energy-independent, so they preserve the hard injection spectrum. By contrast, in the Milky Way CR proton lifetimes are determined by diffusive escape (t E−1/2), diff ∝ so the resulting GeV to PeV proton spectra go as E−2.7 (e.g., Ginzburg & Ptuskin 1976). The harder γ-ray spectra of M82 and NGC 253 support either proton calorimetry or strong advective losses. 4.3.2. Measuring The Calorimetry Fraction We define Lπ as the power of the starburst in all pion end-products, including hadronic γ-rays, neutrinos, and secondary electrons and positrons. The ratio of Lπ 214 to the injected CR luminosity (LCR) for CR protons with kinetic energy per particle (K) above the pion-production threshold (Kth) is then the “calorimetry fraction,” which measures the degree to which starbursts are calorimeters: L t F π life . (4.10) cal ≡ L (K K ) ≈ t CR ≥ th π The lifetime tlife of CR protons with K >Kth includes all losses — pionic, ionization, diffusive, and advective. If proton calorimetry holds, then t t , life ≈ π L L (K K ), and F 1.3 In what follows, we restrict the energy π ≈ CR ≥ th cal → range over which we estimate F to be 1 GeV. In particular, we use observed cal ≥ γ-rays with energies 1 GeV to estimate L for CRs with K 1 GeV. While a ≥ π ≥ significant fraction of both the pionic γ-rays and the CRs have energies below 1 GeV, leptonic emission is expected to become increasingly important at lower energies, contaminating the estimate of the proton calorimetry fraction. The total injected CR power LCR likely scales with the star-formation rate (c.f. Abdo et al. 2010f), and concomitantly, the supernova rate, ΓSN. Assuming that with 3Note that equation (4.10) ignores the energy dependence of the losses, aside from the restriction that K > Kth; ionization losses should be subdominant for K > Kth (Torres 2004b), while both advective and pion losses are roughly independent of energy. If diffusive losses dominate in starbursts, they may be more effective at higher energy, as in the Galaxy. 215 each supernova, a fraction η′ of its kinetic energy goes to primary CR protons with K 1 GeV, ≥ L ( GeV) = 3.2 1041ergs s−1 E η′ Γ (4.11) CR ≥ × 51 0.1 SN, 0.1 ′ ′ 51 where η0.1 = η /0.1, E51 is the energy of the supernova in 10 ergs, and −1 ΓSN, 0.1 =ΓSN/0.1 yr . Even in the proton calorimetric limit (F 1), the GeV γ-ray luminosity cal → will be significantly smaller than L . First, only 1/3 of L ends up as γ-rays; CR ∼ π L 3L . Second, a fraction β of the pionic γ-rays from CR protons with K GeV π ≈ γ π ≥ 4 will have energies > GeV. We calculate βπ using the GALPROP pionic cross sections (Moskalenko & Strong 1998; Strong & Moskalenko 1998; Strong et al. 2000) based on the work of Dermer (1986a) (see also Stecker 1970; Badhwar et al. 1977; Stephens & Badhwar 1981) and a K−p spectrum from 1GeV to 1PeV, and find that it ranges from 0.9 (p = 2.0) to 0.5 (p = 2.5). Using these factors, (see eq. 4.3) 2 −1 −1 ′−1 −1 F 0.023D N− β β E η Γ . (4.12) cal ≈ 3.5 9 γ π 51 0.1 SN, 0.1 We expect that ΓSN is proportional to the luminosity from young massive stars and the star formation rate (SFR). Because most of the stellar luminosity is converted into infrared light by dust in starbursts, if proton calorimetry holds, then the γ-ray flux of M82 and NGC 253 should simply be a fraction of the 4GALPROP is available at http://galprop.stanford.edu. 216 total FIR flux. Furthermore, if η′ is constant for all starbursts, then this ratio of observed fluxes will be constant in the calorimeter limit, so that starbursts should lie on a linear “FIR-γ-ray correlation” in analogy with the FIR-radio correlation (TQW). Observations of normal and starburst galaxies do suggest some kind of correlation between SFR and Lγ, but this correlation increases faster than linearly (L ( 100 MeV) SFR1.4±0.3; Abdo et al. 2010f), as expected if escape is more γ ≥ ∝ efficient in low luminosity galaxies (c.f. Strong et al. 2010). Following TQW, we assume that SFR is related to the total FIR luminosity (L [8 1000] µm) by L = ǫSFRc2, where ǫ is an IMF-dependent constant (see, TIR − TIR e.g., Kennicutt 1998). In the calorimeter limit, cal 1 Lγ( GeV) −4 ′ ξGeV−TIR ≥ 3.1 10 E51η0.1Ψ17 (4.13) ≡ βπ LTIR ≈ × −1 where Ψ17 = (ΓSN/ǫ)/17 M⊙ depends very modestly on the star formation history of the galaxy considered; it varies from 15 to 23 for continuous star formation ∼ ∼ over timescales of 3 107 109 yr (Leitherer et al. 1999, TQW). × − We scale the TIR luminosities from Sanders et al. (2003) (see Tables 4.1 & 4.2) to the same distances as the γ-ray luminosities in equations (4.5) – (4.6). For M82 (βπ = 0.7), we find that F M82 = ξM82 /ξcal 0.4(E η′ Ψ )−1. (4.14) cal GeV−TIR GeV−TIR ≈ 51 0.1 17 217 For NGC 253, we find that Fcal is F NGC253 = ξNGC253 /ξcal 0.2(E η′ Ψ )−1 (4.15) cal GeV−TIR GeV−TIR ≈ 51 0.1 17 for Γ = 2.3 (to 220 GeV) and β 0.6. The flux uncertainties in the Fermi and π ≈ NGC253 HESS detections of NGC 253 indicate that the uncertainty in Fcal can be significantly reduced by more Fermi data. Note that we have used the entire TIR flux of these galaxies, while the starburst cores (and not the outlying disks) probably dominate the γ-ray emission (see the treatments of NGC 253 by Domingo-Santamar´ıa & Torres 2005; Rephaeli et al. 2010). Alternatively, one may estimate Fcal with a distance-independent supernova rate (e.g., from radio source counts). In this case, Fcal (eq. 4.10) retains its strong distance dependence. For M82 (D = 3.6Mpc, N−9 =2, Γ= p = 2.2, βπ = 0.7, βγ = 3.96), F M82 0.3(E η′ )−1 D2 Γ−1 , (4.16) cal ≈ 51 0.1 3.6 SN, 0.1 where we have scaled to a value of ΓSN typically quoted in the literature. Similarly, F NGC 253 0.1 (E η′ )−1 D2 Γ−1 , (4.17) cal ≈ 51 0.1 3.5 SN, 0.1 for our adopted NGC 253 spectrum (see eq. 4.15). 218 The higher numbers for Fcal in equations (4.14) & (4.15) with respect to equations (4.16) & (4.17) are easy to understand. For Ψ17 = 1, and LTIR gives us Γ 0.059 yr−1 and Γ 0.049 yr−1 for M82 and NGC 253, respectively. Thus, SN ≈ SN ≈ −1 the nominal values for ΓSN = 0.1 yr in equations (4.16) & (4.17), while well in the range of supernova rates quoted for both systems ( 4.3.3), are larger than those § inferred from the total FIR luminosity by a factor of 1.7 2. ∼ − Note that our estimates of Fcal in equations (4.15) and (4.17) for NGC 253 are still 2 5 times higher than the HESS estimate of 0.05, even though we ∼ − ∼ use similar or higher SN rates. The main reason for this is that Acero et al. (2009) assume a GeV-to-TeV spectral slope of 2.1, whereas we use a GeV-to-TeV spectral slope of 2.3, derived from the Fermi data. With this spectral slope Acero et al. (2009) effectively underestimate the GeV γ-ray luminosity of NGC 253 by a factor of 3 from 1 220 GeV. ∼ − 4.3.3. Primary Uncertainties in Fcal Other γ-ray sources – Any γ-ray source besides pionic emission from CR protons lowers our estimate for Fcal. Although it is in principle possible that other sources dominate, e.g., the TeV emission, it is likely that the GeV emission is in fact pionic. See 4.4. § 219 Other IR sources – For our main estimates of Fcal in equations (4.14) and (4.15), we have used the total infrared light of each galaxy to measure the star-formation rates of the γ-ray emitting starbursts. However, in NGC 253 only about half of the IR emission comes from the starburst (Melo et al. 2002). It is also possible that cirrus emission from old stars contributes to the observed infrared emission, although we do not expect this to be significant within the starburst itself. Excluding this additional IR light increases the estimates of Fcal. ′ ′ Acceleration efficiency – Higher η lowers our estimated Fcal. In principle, η can be as high as 1 (Ellison & Eichler 1984; Ellison et al. 2004). Efficiencies η′ > 1 are ∼ also possible if additional CR power comes from sources other than SNe. We have ′ scaled the above estimates for Fcal using η = 0.1 based on our work on FIR-radio correlation (LTQ, TQW), which constrains E η′ to be 0.1, depending on the CR 51 ∼ proton injection spectrum (see also Torres et al. 2003; Torres 2004b). We emphasize that η′ is the energy per SN explosion in CR protons with energies 1GeV and ≥ does not include low-energy CRs. Role of supernovae – Although we have assumed in equations (4.16) and (4.17) (and implicitly assumed in our definition of η′) that SNe are responsible for all of the CRs, this has not been settled (see Butt 2009). There is evidence now that SN remnants accelerate some CRs (Tavani et al. 2010), but other sources may also 220 contribute CRs. The γ-ray detections of M82 and NGC 253, combined with the γ-ray detections of quiescent star-forming galaxies, are evidence that γ-ray emission scales with star-formation rate (Abdo et al. 2010f). However, other possible sources of CRs include stellar winds (Quataert & Loeb 2005), superbubbles (Higdon & Lingenfelter 2005; Seaquist & Stankovi´c2007; Butt & Bykov 2008), pulsars (Arons & Tavani 1994; Bednarek & Protheroe 1997; Bednarek & Bartosik 2004), and gamma-ray bursts (Waxman 1995), which all presumably scale with star formation rate. It is also possible that the efficiency of some mechanisms, such as superbubble acceleration, are different in starbursts. Supernova rates – Even if SNe are responsible for CR acceleration, the SN rates in M82 and NGC 253 are highly uncertain. Estimates of ΓSN come from stellar population fitting (F¨orster Schreiber et al. 2003), line emission (Bregman et al. 2000; Alonso-Herrero et al. 2003), FIR emission (Mattila & Meikle 2001), comparison of radio sources with models of SN remnants (van Buren & Greenhouse 1994), and direct searches for SNe (Mannucci et al. 2003). Methods based on the bolometric emission are complicated by the star-formation history, potential IMF variations in starbursts, including the high-mass (> 8M⊙) slope, the shape of the IMF below ∼ 1M⊙, and the transition mass between stars that do and do not produce SNe. ∼ Each of these numbers can affect Ψ in equation (4.13), although we do not expect large variations from galaxy to galaxy (TQW; Persic & Rephaeli 2010). 221 Methods that use direct searches for SNe or their remnants are more uncertain, and complicated by biases. For example, many of the radio sources identified as SN remnants in M82 and NGC 253 may be compact HII regions (Seaquist & Stankovi´c 2007). Methods based on the expansion speed of SN remnants may be complicated by different physical conditions in starbursts (Chevalier & Fransson 2001). At present, only two confirmed SNe have been observed in M82 (SN 2004am: Singer et al. 2004; Mattila et al. 2004; SN 2008iz: Brunthaler et al. 2009b, 2010), along with three radio transients over the past 30 yr that may be radio SNe (Kronberg ∼ & Sramek 1985; Muxlow et al. 1994, 2010). In NGC 253, SN 1940e occured 53′′. (0.9D3.5 kpc) from the galaxy’s center, outside the starburst itself (Kowal & Sargent 1971). No radio SN was observed in NGC 253 over 17 years of observations, but the implied 95% confidence limits on SN rate is weak (< 2.4 yr−1; Lenc & Tingay 2006). ∼ Overall, ΓSN reported in the literature for M82 and NGC 253 span an order of magnitude, from 0.03 yr−1 to 0.3 yr−1. Early estimates were very high, with Γ 0.3 yr−1 (Rieke et al. 1980). More recent estimates have revised Γ downward SN ≈ SN to 0.1 yr−1 (Muxlow et al. 1994; Huang et al. 1994; van Buren & Greenhouse 1994; ∼ Bregman et al. 2000; F¨orster Schreiber et al. 2001; Mattila & Meikle 2001; Lenc & Tingay 2006; Fenech et al. 2008, 2010). Nonetheless, ΓSN remains uncertain at the factor of 2 3 level, and SN rates down to 0.02 yr−1 are possible for both systems ∼ − (F¨orster Schreiber et al. 2001; Colina & Perez-Olea 1992; Engelbracht et al. 1998). 222 Galaxy distances – The estimates of Fcal in equations (4.14) and (4.15) do not depend on distance, because they depend only on the ratio of TIR and γ-ray fluxes. Equations (4.16) and (4.17) instead assume ΓSN, so that the distances do matter. Models of the γ-ray emission also often assume some distance and supernova rate (e.g., Domingo-Santamar´ıa & Torres 2005, LTQ). The estimated distances to NGC 253 vary, from less than 2.3 Mpc (Davidge & Pritchet 1990) to 3.9 Mpc (Karachentsev et al. 2003).5 This range amounts to an uncertainty of a factor of 4 in the γ-ray luminosity of NGC 253. Similarly, distances typically quoted for ∼ M82 include 3.3 Mpc (Freedman & Madore 1988), 3.6 0.3 Mpc from Cepheids in ± M81 (Freedman et al. 1994), and 3.9 0.6 Mpc from the red giant branch (Sakai & ± Madore 1999), amounting to a 40% uncertainty in the luminosity of M82. ∼ 4.3.4. Assessing Proton Calorimetry The values of Fcal derived in Section 4.3.2 imply efficient proton losses in NGC 253 and M82 compared to the Milky Way. The γ-ray data imply both systems have 5Other estimates include 2.5 2.7 Mpc from de Vaucouleurs (1978), 2.6 Mpc from Puche & − Carignan (1988), 2.9 0.5 Mpc from Blecha (1986), 3.3 Mpc from Mouhcine et al. (2005), and ± 3.5 0.2 Mpc from Rekola et al. (2005). A distance of 2.5 Mpc to NGC 253 is commonly quoted (e.g., ± Mauersberger et al. 1996), and is used in the Domingo-Santamar´ıa & Torres (2005) and Rephaeli et al. (2010) models of NGC 253; the HESS analysis similarly used 2.6 Mpc (Acero et al. 2009). TQW and LTQ used 3.5 Mpc, based on the Hubble Law. 223 F 0.2 0.5 (compared with F < 0.1 in the Milky Way based on grammage cal ≈ − cal estimates and modelling; e.g., Ginzburg & Ptuskin 1976; Garcia-Munoz et al. 1987; Engelmann et al. 1990; Jones et al. 2001; Dogiel et al. 2002; Strong et al. 2010). This suggests that t t , even though both galaxies exhibit large-scale galactic π ≈ escape winds, and that these systems represent the transition to proton calorimetry. The case for calorimetry in M82 is stronger than in NGC 253, although Fcal is uncertain for NGC 253 by a factor 2. If we only use the TIR flux of the core of NGC 253 ∼ (about half the total; Melo et al. 2002), Fcal would be about twice as high, or 0.4 - 0.5 – the same as M82 (open star in Fig. 4.3). Thus, the small Fcal for NGC 253 may simply be the result of averaging the γ-rays from the calorimetric starburst with the non-calorimetric outlying disk. Figure 4.3 shows the ratio F ( GeV)/F (see Tables 4.1 & 4.2), the γ-ray γ ≥ SF flux above 1 GeV to the bolometric flux FSF produced by young stars, as a function of gas surface density for NGC 253 and M82, as well as for the LMC (Porter et al. 2009), SMC (Abdo et al. 2010d), the Galaxy (Strong et al. 2010), and M31 (Abdo et al. 2010f), together with upper limits on M33 from Abdo et al. (2010f) and Arp 220 from our own analysis of the Fermi data (see Appendix 4.8). (Note that the plotted ratio F ( GeV)/F does not include β , since we wish to plot observable γ ≥ SF π quantities.) The dashed line indicates the calorimetric expectation from equation ′ (4.13), scaled to η = 0.1 and βπ = 0.7, for p = 2.2. 224 The solid line is the prediction of the fiducial model of LTQ, derived by combining constraints from the Schmidt Law of star formation and the observed FIR-radio correlation. At low gas surface densities, CR protons easily escape, γ-ray cal emission is weak and ξGeV−TIR is small. However, as the gas surface density increases, galaxies become more proton calorimetric and in sufficiently dense starbursts cal ξGeV−TIR asymptotes (eq. 4.13). For these galaxies we expect a FIR-γ-ray correlation −2 (TQW). The discontinuity at Σg = 0.05 g cm is due to the transition in scale height from h = 1 kpc (for normal galaxies) to h = 100 pc (starbursts) in the LTQ models; in reality, the transition between normal galaxies and starbursts is smoother. The γ-ray luminosity of the LMC is consistent with the predictions of LTQ to within a factor of 2. The standard model of LTQ is tuned to reproduce the Milky Way ∼ −2 γ-ray luminosity given in Strong et al. (2000) with Σg = 0.01 g cm . A more recent estimate revises the Milky Way pionic γ-ray luminosity down by a factor of 3 ∼ (Strong et al. 2010: S10); on the other hand, the Milky Way gas surface densities compiled in Yin et al. (2009) are also 3 times lower (the star-formation rate peaks ∼ at 6 kpc, where Σ = 0.003 g cm−2), so that the Galaxy is still fairly close to the ∼ g LTQ prediction. The SMC is γ-ray dim by a factor of 4, suggesting that CRs ∼ escape much more easily than expected (Abdo et al. 2010d). Similar behavior is indicated for the CR electrons by the radio emission of irregular galaxies (Murphy et al. 2008). However, M31 is surprisingly γ-ray bright with respect to the LTQ 225 prediction. Finally, NGC 253 and M82 appear to be somewhat γ-ray faint compared to LTQ’s fiducial model, consistent with equations (4.14) and (4.15). A detection of the ULIRG Arp 220 at the level specified in Table 4.2 would improve our understanding of how Fcal evolves with Σg. With average densities in its nuclear starbursts exceeding 104 cm3 (Downes & Solomon 1998), its pionic loss time is less than 104 yr (eq. 4.7), difficult to reach with winds (eq. 4.8). According to the models of LTQ, Arp 220 should not have a significantly larger F ( GeV)/F than γ ≥ SF M82 and NGC 253. If, instead, Arp 220 is much brighter than M82 and NGC 253, that implies that either η′ is much larger in Arp 220, or escape is more efficient in M82 and NGC 253 than in the fiducial model of LTQ. 4.4. Other Sources of γ-Ray Emission Our analysis in 4.3 assumes that the γ-ray emission is primarily pionic. Here, § we consider other possibilities that could reduce Fcal. 4.4.1. Diffuse Leptonic Emission Primary and secondary CR electrons and positrons (e±) also contribute to the γ-ray emissivity of M82 and NGC 253 via bremsstrahlung and Inverse Compton (IC) of predominantly dust-reprocessed starlight. Detailed models of M82 and 226 NGC 253 indicate that these emission processes are sub-dominant for energies more than 200 MeV (Paglione et al. 1996; Domingo-Santamar´ıa & Torres 2005; Persic et al. 2008; de Cea del Pozo et al. 2009a; Rephaeli et al. 2010; de Cea del Pozo et al. 2009b). Even without detailed modelling, these processes almost certainly do not dominate L in the GeV TeV energy range on energetic grounds. First, γ − more energy goes to π0 2γ production than to the secondary e±. Second, for a → typical ratio of total energy injected in primary electrons to protons of 1/50 (e.g., Warren et al. 2005), and even if all the electron energy goes to producing γ-rays, over 90% (Fcal < 3/50) of the protons would have to escape the starburst for the ∼ proton contribution to the γ-ray luminosity to be sub-dominant. Assuming that none of the protons interact at all, the overall energy budget of the observed γ-ray emission implies that the efficiency of primary electron acceleration in M82 and NGC 253 would have to be 10 times higher than inferred in the Galaxy. Finally, the ∼ magnetic energy density (B 200 µG implying U 1000 eV cm−3) is predicted ≈ B ≈ to be at least as strong as radiation energy density (U 200 1000 eV cm−3; rad ≈ − e.g., Paglione et al. 1996; Persic et al. 2008) in many models (e.g., Condon et al. 1991a; Domingo-Santamar´ıa & Torres 2005, TQW, Persic et al. 2008; de Cea del Pozo et al. 2009a; Rephaeli et al. 2010,LTQ). Thus, synchrotron losses will increase the energetics requirements even further. However, in specific energy ranges leptonic emission can dominate the γ-ray emissivity. Bremsstrahlung and IC emission probably make up most of the γ-ray 227 emission below 100 MeV (e.g., Paglione et al. 1996; Domingo-Santamar´ıa & Torres 2005; Persic et al. 2008; de Cea del Pozo et al. 2009a; Rephaeli et al. 2010; de Cea del Pozo et al. 2009b, LTQ). Bremsstrahlung falls off steeply with energy, and is unimportant above GeV. The IC spectrum is complicated by the shape of the ∼ input photon spectrum. Given that the photon SED of starbursts is dominated by the FIR, for a CR e± injection spectrum Q(E) E−p, we expect the IC photon ∝ index to be Γ p/2+1 at TeV energies (Rybicki & Lightman 1979).6 Whether IC ≈ ∼ pionic or advective losses dominate, the pionic emission will have a photon index Γ p; thus, IC can dominate the VHE γ-ray emission, but only if p is substantially π ≈ greater than 2.0 in the calorimetric or advective limit.7 In practice, Klein-Nishina effects will suppress the IC luminosity beyond a cutoff E 18 TeV (λ/80 µm) KN ≈ for target photons of wavelength λ. For these reasons, IC is unlikely to contribute significantly to the TeV emission unless p > 2.5. ∼ At present, the Fermi detections presented in Abdo et al. (2010a) only use photons with energies greater than 200 MeV. Improving the limits on 100 MeV photons is essential to determining the leptonic contribution, which should begin to dominate at lower energies. Direct detection of bremsstrahlung and IC from electrons and positrons would have strong implications for the synchrotron radio 6Electrons at these high energies, far greater than those observed at GHz frequencies, are cooled almost entirely by IC and synchrotron. 7 For example, if p = 2.2, then Γπ = 2.2 and ΓIC = 2.1; over 1 dex in γ-ray energy, this amounts to only a 30% increase in the ratio of IC to pionic γ-rays. ∼ 228 emission, and could test the “high-Σg conspiracy” postulated by LTQ to explain the radio emission of starburst galaxies (see 4.5). § 4.4.2. Discrete γ-Ray Sources Because M82 and NGC 253 are unresolved by Fermi, VERITAS, and HESS, the γ-ray detections include diffuse emission from CRs and emission from discrete sources.8 The high energy particles in these sources responsible for γ-rays need not contribute to the general CR population. Many such sources in the Galaxy are known to be associated with star-formation, including pulsars and a large number of unidentified sources (Abdo et al. 2009a), and should be expected in abundance in starbursts. Because relatively little work has been done on the expected properties of such sources in starbursts, it is unclear if they could dominate the γ-ray emission from M82 and NGC 253. A number of star-formation phenomena are known to be TeV sources (as reviewed by, e.g., Grenier 2008; Horns 2008; Hinton & Hofmann 2009), including SN remnants, Pulsar Wind Nebulae (PWNe) (Abdo et al. 2009b), and possibly star clusters (Aharonian et al. 2007a). The observed Galactic TeV sources 8Emission from AGNs is unlikely to be important, because no variability is observed (Abdo et al. 2010a) and any AGN luminosity in NGC 253 and M82 is small compared to the star-formation luminosity (Brunthaler et al. 2009a). 229 tend to have hard spectra (Γ 2.2), similar to the observed GeV-to-TeV spectra of ≈ M82 and NGC 253. As an example, PWNe have a total energy budget set by the pulsar rotational energy, E 2 1050 ergs P −2 , where P = P/0.01 s is the pulsar spin period at rot ≈ × 0.01 0.01 birth, comparable to that injected into CR protons by the SN remnants. The typical spindown luminosity is E˙ 6 1039P 4 B2 ergs s−1, where B = B/1012 G is the rot ≈ × 0.01 12 12 pulsar magnetic field strength, corresponding to a spindown timescale of 103 yr. ∼ If all of this energy went into γ-ray emission in M82 and NGC 253, a pulsar birth rate of 0.1 yr−1 could easily power the GeV-TeV emission. However, Galactic ∼ γ-ray sources like the Crab (e.g., Albert et al. 2008), Geminga (Y¨uksel et al. 2009), and HESS J1825-137 (Aharonian et al. 2006a) have GeV-TeV luminosities several decades lower than this estimate. For example, the total γ-ray luminosity of the Crab (L 1035 ergs s−1) implies that 104 105 such objects would be needed to γ ≈ ∼ − contribute significantly to the GeV-TeV emission seen from M82 and NGC 253 (see also Mannheim et al. 2010). Given the pulsar birthrate and spindown timescale, this seems unlikely, but we cannot rule out the PWNe in starbursts are much more radiatively efficient in γ-rays than in the Galaxy, for example, through stronger IC losses. 230 4.4.3. A TeV Excess? The TeV background of the Milky Way shows a “TeV excess” above the expected pionic background (Prodanovi´cet al. 2007; Abdo et al. 2008). Whether it is caused by unresolved discrete sources or truly diffuse emission is not known; nor is it known whether it is hadronic or leptonic. The TeV excess varies with Galactic longitude, being strongest in the Galactic Center and the Cygnus regions (Abdo et al. 2008). The latitude profile of the Galactic TeV γ-ray emission supports a hadronic explanation for the TeV excess, but leptonic models are not yet excluded (Abdo et al. 2008). The TeV excess is visible in the Galaxy because the pionic spectrum is steep; effectively, the TeV excess changes Γ from 2.7 to 2.6 (Prodanovi´cet al. 2007). If M82 and NGC 253 have hard pionic γ-ray spectra at TeV energies, then the γ-rays from ambient CRs interacting with their ISM will bury any TeV excess. Furthermore, the simplest explanation for a hadronic TeV excess is that some regions of the Galaxy are denser and more proton calorimetric, and that the TeV excess is simply pionic emission from the normal CR protons. This effect is observed in molecular clouds located near CR acceleration sites in the Milky Way (Aharonian et al. 2006b; Albert et al. 2007b; Aharonian et al. 2008), but if F 1, the entire starburst cal ≈ is illuminated this way, and the total pionic γ-ray luminosity cannot be increased further. 231 4.5. The GeV-GHz Ratio: A Diagnostic of Electron Cooling and the FIR-Radio Correlation The observed γ-rays from M82 and NGC 253 have important implications for the physics of the radio emission of starburst galaxies (see LTQ and references ± therein). If pionic, Lγ necessarily implies production of secondary e s, which produce synchrotron radiation and contribute to the GHz emissivity of the starbursts. Rengarajan (2005), TQW, and LTQ have all argued that secondary e± dominate the synchrotron emission from starbursts. Detailed models of starburst regions by Paglione et al. (1996), Torres (2004b), Domingo-Santamar´ıa & Torres (2005), Persic et al. (2008), de Cea del Pozo et al. (2009a), and Rephaeli et al. (2010) find that secondary e± are the majority of GeV e±, although the number ∼ of primary electrons is still within a factor of a few of the secondaries. The γ-ray to radio ratio provides an important constraint on the cooling mechanism of GHz-emitting electrons, if the γ-rays are pionic. This ratio can be understood through a simple argument as follows. Suppose the protons (and secondary e±) have an E−2 spectrum, with equal energy in each log E bin. The protons lose energy to pions; roughly 2 times as much energy goes into γ-rays as electrons and positrons. Furthermore, since ν E1/2, the synchrotron emission C ∝ from each log bin in e± energy is spread over 2 log bins in synchrotron frequency. 232 Therefore, if the γ-ray emission is dominated by diffuse pionic emission, and if the radio emission is dominated by secondary e±, νFν(Eγ) = 4νFν(Ee)fsyn, (4.18) ± where Eγ and Ee are the energies of γ-rays and e respectively from CR protons of −1 ± the same energy, and fsyn = (tsyn/tlife) is the fraction of CR e power going into synchrotron. CR protons produce pionic γ-rays with E 0.1K and secondary e± γ ≈ p with E 0.05K E /2. Since electrons that emit GHz synchrotron radiation e ≈ p ≈ γ − have an energy of E 560 MeVB 1/2(ν/GHz)1/2, we compare GeV γ-rays and GHz ≈ 200 GHz radio emission. The GHz flux of M82 and the starburst core of NGC 253 are 9 and 3 Jy respectively (Williams & Bower 2010), while the νFν(GeV) fluxes are 3 10−12 ergs cm−2 s−1 and 2 10−12 ergs cm−2 s−1 respectively. This implies × × that f −1 is 8 for M82 and 17 for NGC 253, if all of the γ-ray flux is from syn ∼ ∼ their starburst cores. If we instead consider the total GHz radio emission of NGC 253, 6 Jy, then f −1 8. This implies strong non-synchrotron losses, consistent syn ≈ with bremsstrahlung and ionization cooling, and may be a hint that the “high-Σg conspiracy” advocated by LTQ as an explanation for the linear FRC in dense starbursts is operating in M82 and NGC 253. Note that if some radio emission were from primaries, this would require even greater non-synchrotron losses. If B is much higher than we suppose, the e± accompanying GeV γ-rays emit at higher frequencies, but since the radio spectrum only goes as νF ν0.3 (Klein et al. 1988; ν ∝ 233 Williams & Bower 2010), even an order of magnitude increase in B only changes our conclusions by a factor 2. ∼ M82 and NGC 253 appear to have steeper GeV-to-TeV spectral slopes than p = 2.0, but the basic conclusions are unchanged with a more careful analysis. Most of the e± emitting at GHz are expected to be secondaries from CR protons, injected with a E−p spectrum. The power going into e± with energy greater than K is L (K K ) = F L (K K )β (K )f , where f is GHz e e ≥ GHz cal CR,p ≥ GHz π,e GHz e e ± the fraction of pionic luminosity going to secondary e , and βπ,e is the fraction of secondary e± power from CR protons with K K that is in e± with p ≥ GHz K K . The energy going into γ-rays with energy greater than a GeV is e ≥ GHz L ( GeV) = F L (K GeV)β (GeV)f . Finally the radio luminosity is γ ≥ cal CR,p ≥ π γ νL = L ( K )(ν/GHz)1−p/2β f f , where β is a bolometric correction ν e ≥ GHz syn sec syn syn ± factor and fsec is the fraction of e that are pionic secondaries. For p = 2.2, we have from the GALPROP cross-sections β (GeV) = 0.7, β (K ) = 0.5, and β 0.1. π π,e GHz syn ≈ Since f 2f , γ ≈ e p/2−1 −1 ν Lγ(> GeV) 30νLνfsecfsyn . (4.19) ∼ ≈ GHz This implies that f −1 is 5 for M82 and 7 for NGC 253, if all of the γ-ray flux is synch ∼ ∼ from their starburst cores. It also shows that the simpler estimate in equation (4.18) is a useful approximation even when p = 2. 6 234 These estimates are consistent with the idea that most of the radio emission in M82 and NGC 253’s starburst are from secondaries undergoing strong non- synchrotron losses. However the exact values of these ratios are still fairly uncertain. The main uncertainties are the fraction of γ-rays from diffuse pionic emission, the fraction of radio emission from primary CR e±, the fraction of γ-rays and radio from the starburst cores as opposed to the outlying disk galaxies, and the uncertainties in the γ-ray fluxes. Bremmstrahlung and ionization can easily provide these extra losses. For e± radiating synchrotron at ν = νGHzGHz, the densities when the bremsstrahlung and ionization cooling timescales are comparable to the synchrotron cooling −3 1/2 3/2 timescale (tbrems/tsynch and tion/tsynch 1) are neff > 67 cm νGHzB200 and ≤ ∼ −3 neff > 54 cm νGHzB200, respectively (LTQ; Murphy 2009). These losses are also ∼ suggested by the somewhat flattened GHz synchrotron radio spectra observed in starbursts (α 0.7), whereas pure synchrotron and IC cooling would lead to steep ≈ spectra with α 1 (Thompson et al. 2006). Detailed models which do include all ≈ of these losses regularly do fit the radio spectra of M82 and NGC 253 (Paglione et al. 1996; Domingo-Santamar´ıa & Torres 2005; Persic et al. 2008; de Cea del Pozo et al. 2009a; Rephaeli et al. 2010) as well as the γ-ray spectra (de Cea del Pozo et al. 2009b). These models can improve our interpretation of the relevant loss mechanisms. Finally, with more γ-ray data from Fermi, the lower energy leptonic 235 γ-ray emission may be detected, which would directly constrain the importance of loss processes such as bremsstrahlung. 4.6. Implications A number of previous studies have found implications for γ-ray bright starbursts. We now discuss these implications in light of the current detections of M82 and NGC 253. 4.6.1. The Detectability of Other Star-Forming Galaxies Our inference of F 0.4 and 0.2 in M82 and NGC 253 implies that a number cal ≈ of local star-forming and starburst galaxies should be visible with next-generation TeV γ-ray telescopes like CTA (e.g., Kn¨odlseder 2010) and with additional Fermi data (see also Pavlidou & Fields 2001; Cillis et al. 2005, TQW). For a number of galaxies chosen from the IRAS Bright Galaxy Survey (Sanders et al. 2003), Tables 4.1 and 4.2 list the distance, total FIR luminosity and flux, estimates of the gas surface density Σ , and three determinations of the γ-ray flux: (1) F cal( GeV), the g γ ≥ purely calorimetric prediction from equation (4.13), assuming βπ = 0.7 and Ψ17 = 1, (2) F LTQ( GeV), the prediction from LTQ given Σ and scaled to F (solid line, γ ≥ g TIR Fig. 4.3), and (3) the observed flux F obs( GeV) in cases where there exists either γ ≥ 236 a detection or an upper limit. Our predictions for F cal( GeV) use equation (4.13) γ ≥ ′ with nominal values of βπ = 0.7 (p = 2.2) and η = 0.1. Tables 4.1 & 4.2 provides a useful guide to the detectability of all local galaxies to the extent that their TIR and γ-ray light is dominated by star formation. Table 4.1 provides results for normal star-forming galaxies with low Σg that are not expected to be calorimetric. Here, F cal( GeV) provides an upper limit to the γ ≥ pionic γ-ray flux, and F LTQ( GeV) provides the prediction based on the nominal γ ≥ estimate of F ( GeV)/F for Σ (solid line, Fig. 4.3) and the star-formation rate ≥ SF g listed in the table. Since emission from old stars and different UV opacities can affect the TIR emission (Bell 2003), the γ-ray predictions should be scaled to known star-formation rates when possible. Furthermore, leptonic processes may increase the γ-ray emission significantly, especially in low density galaxies. Table 4.2 gives numbers for the starbursts in the IRAS BGS. Here, F cal( GeV) F LTQ( GeV).9 γ ≥ ≈ γ ≥ Uncertainties in Table 4.2 include those in the gas surface density Σg of the starbursts and the fraction of TIR light associated with the denser starburst regions as opposed to the surrounding galaxy in the whole. Note that if they are in fact calorimetric, the starburst/AGN systems NGC 4945 and NGC 1068 in Table 4.2 should be the brightest on the sky after M82 and NGC 253 at GeV. We note that Abdo et al. (2010b) very recently announced the γ-ray detection of NGC 4945 (1FGL J1305.4-4928), with a flux within 50% of the 9Note that NGC 5128 (Cen A) has not been included in either Table. 237 calorimetric prediction and very near the LTQ prediction; Lenain et al. (2010) has also announced the detection of NGC 1068. However, it is still uncertain how much of the γ-ray flux from these starbursts is from star-formation, and how much is from the Seyfert nuclei (Lenain et al. 2010). The values in Tables 4.1 and 4.2 should also give the approximate neutrino luminosities of nearby starbursts, since charged pions decay into neutrinos (e.g., Stecker 1979). Indeed, models of M82 and NGC 253 predict them to be neutrino sources (e.g., Persic et al. 2008; de Cea del Pozo et al. 2009a; Rephaeli et al. 2010). However, NGC 253 and NGC 4945 as Southern objects are not detectable with IceCube (though they may be detectable with KM3NET; Katz 2006), and M82 is at high declination where IceCube’s sensitivity is weakest (Abbasi et al. 2009a). 4.6.2. The Diffuse γ-ray & Neutrino Backgrounds from Star Formation The detections of NGC 253 and M82 at GeV TeV energies together with − our determination of Fcal in these systems has immediate implications for the diffuse γ-ray and neutrino backgrounds from star formation, as discussed in LW06 and TQW (see also Pavlidou & Fields 2002; Bhattacharya & Sreekumar 2009). Several recent studies present calculations of the star-forming galaxy contribution to the γ-ray background, using the γ-ray brightness of nearby normal galaxies and 238 starbursts, finding contributions of order 10 50% (e.g., Fields et al. 2010; Makiya ∼ − et al. 2011; Stecker & Venters 2011). Here we present an updated version of TQW’s calculation, based on the IR background and this paper’s Fcal. The pionic emission from starburst galaxies should be concentrated above 100 MeV, with a power law spectrum above 1 GeV to PeV energies if unabsorbed. ∼ For an acceleration efficiency of η′ = 0.1 (see LTQ), for CR protons energies larger GeV, the total integrated γ-ray background above 1 GeV is, following ≥ equation (4.13) and ignoring absorption and redshift effects, F ( GeV) 1.0 10−6η′ f burstF calF TIR GeV s−1 cm−2 sr−1, (4.20) γ ≥ ≈ × 0.1 0.75 0.5 20 TIR for Γ = 2.2, where F20 is the total diffuse extragalactic TIR background in units −2 −1 burst of 20 nW m sr , f0.75 = fburst/0.75 is the fraction of the TIR extragalactic cal background produced by starburst galaxies (see TQW), and F0.5 = Fcal/0.5 is the average calorimetric fraction of these starbursts. As with individual starbursts, the observable neutrino background must be comparable to the pionic γ-ray background, and have a similar spectrum (c.f. LW06). This estimate for the γ-ray and neutrino backgrounds from CR protons implies that for Eγ > 1 GeV and Γ = 2.2, νIν(GeV) ∼ ∼ 2 10−7(E /GeV)−0.2η′ f burstF calF TIR GeV s−1 cm−2 sr−1, within a factor of × γ 0.1 0.75 0.5 20 2 of the current observations of the extragalactic γ-ray background (Abdo et al. ∼ 239 2010c; see also Keshet et al. 2004). This indicates that starburst galaxies can be a major source of the γ-ray background. More detailed modelling of redshift evolution are necessary to get the spectral dependence correct; most of the star-formation in the Universe occurs at z 1 (e.g., Hopkins & Beacom 2006), so redshift effects will ∼ be significant. Furthermore, the Universe becomes opaque to γ-rays with observed energy 100 GeV at z =1( 50 GeV at z = 2; 20 GeV at all reasonable z; ≥ ≥ ≥ e.g., Gilmore et al. 2009; Finke et al. 2010); the γ-ray background above this energy will cascade down to lower energies. The corresponding neutrino background will be affected by redshift but not by opacity. At energies below 1 GeV, the pionic emission should decline because of the decreasing pion production cross-section, as has been discussed in previous works (e.g., Prodanovi´c& Fields 2004; Stecker & Venters 2011). The primary uncertainties are still fburst and Fcal for starbursts at high z. Importantly, Daddi et al. (2010) recently presented CO luminosities of near-infrared selected BzK galaxies at z 1.5. They derive gas masses for these relatively normal ∼ star forming galaxies of 1011 M and radii of R 3 6 kpc. These numbers ∼ ⊙ ∼ − imply gas surface densities of Σ 0.3M R−2g cm−2. Comparing this with Figure g ≈ 11 5 4.3 we expect that BzK galaxies are as calorimetric as M82. These galaxies are an important contributor to the total star formation budget of the universe in the critical redshift range z 1 2, thus strengthening the case for a half-calorimetric ∼ − 240 background as in equation (4.20). However, we emphasize that this estimate for the BzK galaxies has significant uncertainties (e.g., the CO-to-H2 conversion factor). 4.6.3. The Dynamical Importance of Cosmic Rays in Starbursts CRs are dynamically important with respect to gravity in the Galaxy (Boulares & Cox 1990). They have recently been claimed to be sub-dominant with respect to gravity in starbursts because of strong pion losses (LTQ). However, CRs may be important in driving winds in such systems (Socrates et al. 2008), as in the Galaxy (Chevalier & Fransson 1984; Everett et al. 2008). The observed γ-ray emission from M82 and NGC 253 can be converted into a constraint on the product of neff (see eq. 4.7) and the energy density of the CRs, UCR, and hence inform the question of whether or not CRs are dynamically important in these systems; L (K GeV) U V f /t , (4.21) π CR ≥ ≈ CR GeV π where L (K GeV) 3β−1L ( GeV), V is the starburst volume, and f π CR ≥ ≈ π γ ≥ GeV is the fraction of the CR energy density in CRs with energy above 1 GeV. Taking values for the radius and scale height of R250 = R/250 pc and h100 = h/100 pc, −6 −1 −2 −1 2 −1 U n 6100 eV cm f R h D β β N− . (4.22) CR eff ≈ GeV 250 100 3.5 γ π 9 241 If we assume R250 = h100 = 1 and that CRs interact with an average density of 250 cm−3, then U is 300f −1 eV cm−3 for M82 (comparable with Acciari et al. ∼ CR ∼ GeV (2009)) and 100f −1 eV cm−3 for NGC 253 (extrapolating to 1.3 TeV, 12eV cm−3, ∼ GeV about twice that of Acero et al. 2009; however, they quote n 600 cm−3). For a eff ≈ K−2.3 CR spectrum stretching from 10 MeV to infinity (Torres 2004b), f 0.25, GeV ≈ mostly in low ( GeV) energy CRs. ≪ There is a degeneracy between UCR and neff : at fixed Lγ, a small neff can be accommodated by having a high UCR, and vice versa. The value of neff is not obvious since the ISM of starbursts is highly turbulent and clumpy, with most of the volume filled with gas that is underdense with respect to the mean density. The pressure required for hydrostatic balance is P πGΣ Σ , where Σ is the total hydro ≈ g tot tot 10 surface density and is approximately equal to Σg. We find that the CR pressure P = U /3 is dynamically unimportant: P /P 0.02f −1 . Alternatively, CR CR CR hydro ≈ GeV if we assume that P P , we find that n 0.02f −1 n 5 20 cm−3, CR ≈ hydro eff ≈ GeVh i ≈ − implying that t 5 Myr, approximately 25 times longer than the nominal wind π ≈ escape timescale (see eq. 4.8). This requires far more efficient CR acceleration than η′ 0.1 (see 4.3.2), which we consider unlikely. Therefore we conclude that ≈ § P P . CR ≪ hydro 10 Although the thermal pressure within M82 is an order of magnitude less than Phydro (Lord et al. 1996; Smith et al. 2006), the turbulent pressure is comparable to Phydro (Smith et al. 2006). 242 4.7. Conclusion M82 and NGC 253 have now been detected in GeV and TeV γ-rays, with fluxes roughly comparable to previous detailed predictions. We have shown that the observed γ-ray fluxes imply that a fraction F 0.2 0.4 of the energy injected into cal ≈ − high energy CR protons is lost to inelastic collisions (pion production) with protons ′ in the ISM (for η = 0.1). However, Fcal in the range of 0.1 - 1 can be accommodated with different SNe rates and acceleration efficiencies (see the uncertainties in 4.3.3 § and 4.4). We find a significantly higher F for NGC 253 than Acero et al. (2009) § cal because NGC 253 has more GeV emission than they expected. The uncertainty in Fcal will decrease significantly with more observations by Fermi, HESS, and VERITAS. A future test of proton calorimetry in M82 and NGC 253 would be a γ-ray detection of a ULIRG like Arp 220 (c.f. Torres 2004b). Arp 220 is more likely to be a proton calorimeter than M82 and NGC 253, with its extremely high average gas density. If M82 and NGC 253 are not proton calorimeters but Arp 220 is, the ratio of Arp 220’s pionic luminosity to its stellar luminosity will be greater than M82 and NGC 253 – it will be brighter in γ-rays than expected (see Figure 4.3, Tables 4.1 & 4.2). Unfortunately, Arp 220’s flux is expected to be challenging to detect with Fermi, although upper limits alone may be constraining (as in the Appendix). Stacking searches of ULIRGs may also prove useful. 243 Pionic γ-ray emission implies secondary e± production in these starbursts (c.f. Rengarajan 2005); from the GHz to GeV ratio, we found evidence of significant non-synchrotron losses. This suggests that bremsstrahlung and ionization are important energy loss mechanisms for CR electrons and positrons (c.f. Murphy 2009). This would support the idea presented in Thompson et al. (2006) that these losses flatten the GHz radio spectrum of starbursts ( 4.5). It would also § support the “high-Σg conspiracy” suggested by LTQ to explain the linearity of the FIR-radio correlation for starbursts, whereby bremsstrahlung, ionization, and IC losses suppress the synchrotron radio emission of CR electrons in starbursts, but proton calorimetry leads to secondary electrons and positrons that boost the radio emission. Whatever the underlying physics of γ-ray production in M82 and NGC 253 is, the high fluxes of these starbursts suggest that other starbursts should also be γ-ray sources. We compile our predictions in Tables 4.1 & 4.2. Considering that much of the star formation in the universe at high-z is in luminous infrared galaxies (e.g., Elbaz et al. 1999; Chary & Elbaz 2001; P´erez-Gonz´alez et al. 2005; Magnelli et al. 2009), starbursts might make up a significant fraction ( 1/2) of the entire ∼ γ-ray background (e.g., Pavlidou & Fields 2002, TQW, Bhattacharya & Sreekumar 2009; 4.6.2). If the hadronic interpretation of the γ-ray flux holds, the neutrino § background should also be large (LW06). 244 Finally, the conclusion that M82 has F 0.4 and NGC 253 has F 0.2 cal ≈ cal ≈ implies that the pion cooling timescale is nearly equal to the wind escape timescale, 2 105 yr for these systems. This, in turn, suggests that the CR protons on average ∼ × interact with ISM near the mean density. If this is correct, then the CR pressure is significantly below the pressure needed to support each starburst gravitationally, and CRs are not on average dynamically important deep within the starbursts ( 4.6.3). § 4.8. Fermi Data Analysis for Arp 220 We followed the procedure of Abdo et al. (2010a), using the publicly available Fermi data reduction software. The analysis is reviewed in the available online documentation located at http://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/ ; the unbinned likelihood tutorial, which we followed, is specifically at http://fermi.gsfc.nasa.gov/ssc/data/analysis/scitools/likelihood tutorial.html. We downloaded data from the Fermi LAT data server (available at http://fermi.gsfc.nasa.gov/ssc/data/.) for METs of 239557417 to 286813463, a total of 18 months. Data from within 20◦ of each source were downloaded. We created an exposure cube for the entire sky for this time range. We first divided the Fermi energy range into two broad bands: a low energy bin for 100 MeV E 1 GeV and a high energy bin for 1 GeV E 100 GeV. The ≤ γ ≤ ≤ γ ≤ 245 source region had a radius of 10◦. The selection was done with gtselect. We then selected high quality events with gtmktime. Using the exposure cube, we created an exposure map around each source using gtexpmap. Finally, we could perform an unbinned likelihood analysis with gtlike. We modeled all of the sources listed in the 1FGL Fermi source catalog within 15◦ of Arp 220, the extragalactic background, diffuse Galactic emission, and Arp 220 itself. In each energy band, we fit a power law to all of the point sources, including Arp 220; both differential flux and integrated flux were considered for Arp 220. We used the P6 V3 DIFFUSE response function. The Galactic background was modeled with the gll iem v02.fit background and the extragalactic background was modeled with isotropic iem v02.txt, both of which are the default models. We first found a preliminary fit using DRMNFB, and then used the results of that fit to converge to our final fit with MINUIT. We did not detect Arp 220, as expected. Our results are summarized in Table 4.8. For the high energy band, we considered both models where Γ for Arp 220 was allowed to vary, and models where it was forced to 2.2. In the former, Γ always was forced to its maximum value of 5.0, possibly because of a dearth of high energy photons. Both variants give similar results for the integrated number of photons above 1 GeV, but in the Γ 5 model, the normalization of the differential flux at 1 → GeV is much higher. 246 Fig. 4.1.— The γ-ray spectra of M82 from Fermi (solid triangles: Abdo et al. (2010a); filled squares: VERITAS). We show several models: LTQ (unscaled to SFR and using Kennicutt (1998) Σg: long-dashed; scaled to Sanders et al. (2003) LTIR and Kennicutt (1998) Σg: solid, grey; scaled to Sanders et al. (2003) LTIR and Σg in Table 4.2: solid, black), de Cea del Pozo et al. (2009a) (dotted), Persic et al. (2008) (dashed), and TQW (p = 2.0, 2.2, 2.4, dash-dotted). We plot E2 times the differential flux at each energy. At VHE energies, the main source of variation in the models is the CR injection slope p. Note that the models of LTQ are proton calorimetric, and TQW explicitly assumes proton calorimetry (see Fig. 4.3). 247 Fig. 4.2.— The γ-ray spectra of NGC 253 from Fermi (solid triangles: Abdo et al. (2010a); filled squares: HESS). We show several models: LTQ (unscaled to SFR and using Kennicutt (1998) Σg: long-dashed; scaled to Sanders et al. (2003) LTIR and Kennicutt (1998) Σg: solid, grey; scaled to Sanders et al. (2003) LTIR and Σg in Table 4.2: solid, black), Domingo-Santamar´ıa & Torres (2005) (dotted), Rephaeli et al. (2010) (dashed), Paglione et al. (1996) (long-dashed), and TQW (p = 2.0, 2.2, 2.4, dash-dotted). We plot E2 times the differential flux at each energy. At VHE energies, the main source of variation in the models is the CR injection slope p. Note that the models of LTQ are proton calorimetric, and TQW explicitly assumes proton calorimetry (see Fig. 4.3). 248 Fig. 4.3.— The ratio of the total pionic γ-ray flux at energies GeV to the total luminosity from star formation for galaxies with γ-ray detections≥ or upper limits (for Arp 220, see Appendix 4.8). See Tables 4.1 & 4.2. The dashed lines are the ′ calorimetric expectation from equation (4.13), scaled to η = 0.05 and using βπ = 0.7 (p = 2.2). The solid line is the predicted ratio for pionic γ-rays in the fiducial model of LTQ based on the Schmidt Law of star formation and the linearity of the FIR- radio correlation. The model becomes calorimetric at high surface densities where −2 the curve flattens. The kink at Σg = 0.05 g cm is a result of the scale height changing from 1 kpc for normal galaxies to 100 pc for starbursts. Shading indicates the predicted ratios in all successful models of LTQ (darker: p = 2.2 only; lighter: all p) for pionic γ-rays. The open star represents the ratio for NGC 253’s starburst core, if it has one half the TIR luminosity of the entire galaxy (Melo et al. 2002). The FSF of M31, LMC, SMC, and the Milky Way are based on their SFRs (Williams 2003; Harris & Zaritsky 2004, 2009; Yin et al. 2009) and the Kennicutt 1998 conversion factor between LFIR and SFR. The Fγ( GeV) are assumed to be pionic, except for the Milky Way, where the pionic γ-ray≥ luminosity comes from Strong et al. (2010). 249 Predicted Predicted Calorimetric LTQ Observed a b c d cale LTQf g h Name D LTIR SFR FSF Fγ Fγ Fγ Σg ( GeV) ( GeV) ( GeV) (gcm−2) ≥ ≥ ≥ LMC 0.05 8.83 0.2i 1.47E-5 3.2E-9i 6.2E-11i (5.7 1.4)E-11j 0.002k ± SMC 0.06 7.86 0.1l 5.11E-6 1.1E-9l 3.1E-11l 1E-11m 0.003n M31 (NGC 224) 0.79 9.39 1.0o 2.95E-7 6.4E-11o 6.3E-13o (2.6 0.6)E-12p 0.001q ± NGC 598 (M33) 0.84 9.07 0.5r 1.30E-7 2.8E-11 5.5E-13 < 1.5E-12s 0.002t NGC 6946u 5.32 10.16 2.6v 1.69E-8 3.7E-12 1.3E-13 ∼ 0.004t ··· NGC 5457 (M101) 6.70 10.20 1.7w 6.96E-9 1.5E-12 2.9E-14 0.002t ··· NGC 5194 (M51)x 8.63 10.42 3.6u 8.89E-9 1.9E-12 1.0E-13 <8E-11y 0.006t z ∼ t 250 NGC 3031 (M81) 3.63 9.47 0.3 4.19E-9 9.1E-13 1.3E-14 0.0015 ··· NGC 3521 6.84 9.96 0.9v 3.54E-9 7.7E-13 2.5E-14 0.0035t ··· NGC 5055 7.96 10.09 1.3v 3.77E-9 8.2E-13 2.3E-14 <8E-11y 0.003t NGC 7331 14.71 10.58 5aa 4.25E-9 9.2E-13 2.2E-14 <∼8E-11y 0.0025t ∼ aDistances in Mpc from IRAS BGS unless otherwise noted. b TIR luminosities (in log10L ) from IRAS BGS unless otherwise noted. cThe star-formation rate (in⊙ M yr−1 as inferred from the literature. The TIR flux is likely to be inaccurate as a simple SFR indicator at these⊙ luminosities (Bell 2003). dWe calculated the bolometric star-formation flux from the SFR in ergs cm−2 s−1, using F = 3.8 SF × 10−4c2SFR/(4πD2), based on the starburst IR to SFR conversion-factor in Kennicutt (1998). See cavaets in footnote c. Table 4.1. Non-Calorimetric Galaxies: Predicted & Observed Gamma-Ray Fluxes (cont’d) Table 4.1—Continued Predicted Predicted Calorimetric LTQ Observed a b c d cale LTQf g h Name D LTIR SFR FSF Fγ Fγ Fγ Σg ( GeV) ( GeV) ( GeV) (gcm−2) ≥ ≥ ≥ e −2 −1 Pionic gamma-ray flux (in ergs cm s ) predicted in the explicitly calorimetric limit: Fγ( GeV) = −4 ′ ≥ βπFSF 1.8 10 (E51η0.05Ψ17), using βπ = 0.7 as a fiducial value; see equation 4.13. f × × −2 −1 Pionic gamma-ray flux (in ergs cm s ) predicted by the fiducial model of LTQ, using FSF. See solid line in Figure 4.3. Note that leptonic emission (particularly IC) may dominate at the lowest Σg and increase the γ-ray fluxes. gMeasurement of or upper limit on integrated gamma-ray flux of energies GeV (in ergs cm−2 s−1). 251 ≥ hGas surface density. Typical uncertainty in this quantity is 0.3 dex. ∼ iHarris & Zaritsky (2009) find that a SFR of 0.2 M yr−1 in the LMC (as used in Fig. 4.3. jCalculated using Porter et al. (2009), assuming that⊙ Γ = 2.7 above 1 GeV. Integrating the total emission in Figure 8 of Abdo et al. (2010e) gives similar results. k 8 Calculated using a total gas mass of 6 10 M⊙ (Israel 1997) and R 4.9 kpc. × 25 ≈ lHarris & Zaritsky (2004) find an average SFR of 0.1M yr−1 in the SMC (as used in Fig. 4.3) over the past few Gyr, with occasional bursts of star-formation more recently.⊙ mCalculated from Abdo et al. (2010d), by integrating the total power above 1 GeV plotted in their Figure 5. n 8 Calculated using a total gas mass of 4.5 10 M⊙ (Israel 1997) and R 3.0 kpc. × 25 ≈ oWilliams (2003) find an average SFR of 1 M yr−1 in M31 (as used in Fig. 4.3). pCalculated from the Milky Way-scaled GALPROP⊙ model in Fig. 2 of Abdo et al. (2010f). See also the upper limit in Blom et al. (1999). qFrom Kennicutt (1998). The peak gas surface density compiled in Yin et al. (2009) is comparable. (cont’d) Table 4.1—Continued Predicted Predicted Calorimetric LTQ Observed a b c d cale LTQf g h Name D LTIR SFR FSF Fγ Fγ Fγ Σg ( GeV) ( GeV) ( GeV) (gcm−2) ≥ ≥ ≥ rGardan et al. (2007) and references therein find star-formation rates of 0.3 0.7 M yr−1 in M33. − sCalculated from Abdo et al. (2010f), scaling from the detection of M31 to the upper⊙ limit on M33 for the GALPROP spectral template. tFrom Kennicutt (1998). u −2 252 This system also has a central dense starburst component with Σ 0.04 g cm (Kennicutt 1998), that may be g ≈ calorimetric, and amounts to 10 % of the total star formation rate. ∼ vScaled from Leroy et al. (2008) to the distance listed here. wCalculated from Kennicutt et al. (2008) using the Kennicutt (1998) Hα luminosity to SFR conversion. xThis system also has a central dense starburst component with Σ 0.06 g cm−2 (Kennicutt 1998), that may be g ≈ calorimetric, and amounts to 25 % of the total star formation rate. ∼ yEGRET upper limits from Cillis et al. (2005). zKennicutt et al. (2008) find an Hα luminosity equivalent to 0.5 M yr−1, while Davidge (2006) find a star- formation rate from 10 to 25 Myr ago (roughly the typical lifetime of GeV⊙ CR protons in Milky Way-like galaxies) of 0.1 M yr−1. aaThilker⊙ et al. (2007) compare star-formation rates derived through several indicators and find them to be 4.4 6.3 M yr−1. − ⊙ Predicted Predicted Calorimetric LTQ Observed a b c cald LTQe f g Name D LTIR FTIR Fγ Fγ Fγ Σg −2 (Mpc) log [L⊙] ( GeV) ( GeV) ( GeV) (gcm ) 10 ≥ ≥ ≥ M82 (NGC 3034) 3.63 10.77 1.42E-7 3.08E-11 2.0E-11 1.3E-11 0.24h NGC 253 3.50i 10.54j 9.09E-8 1.97E-11 1.2E-11 (6.5 2.5)E-12 0.15k ± NGC 4945 3.92 10.48 6.23E-8 1.35E-11 8.5E-12 (9.2 3.0)E-12 l 0.19m ± NGC 1068 (M77) 13.70 11.27 3.15E-8 6.84E-12 1.0E-12 n (3.6 1.0)E-12 o 0.02n ± NGC 5236 (M83) 3.60 10.10 3.08E-8 6.68E-12 6.6E-13 < 4E 11p 0.01q,r − IC 342 4.60 10.17 2.22E-8 4.82E-12 7.2E-13 ∼ 0.02q ··· NGC 2146 16.47 11.07 1.37E-8 2.97E-12 1.8E-12 < 4E 11p 0.14q 253 − NGC 3690/IC 694 47.74 11.88 1.06E-8 2.30E-12 1.6E-12 ∼ 2.6 q ··· NGC 1808 12.61 10.71 1.02E-8 2.21E-12 1.3E-12 0.09q ··· NGC 1365 17.93 11.00 9.87E-9 2.14E-12 1.2E-12 < 8E 11p 0.08q − NGC 3256 35.35 11.56 9.21E-9 2.00E-12 1.3E-12 ∼ 0.28q ··· NGC 4631 7.73 10.22 8.81E-9 1.91E-12 2.8E-13 < 4E 11p 0.02q − Arp 220 79.90 12.21 8.06E-9 1.75E-12 1.2E-12 <∼ 8E 11p 10s − NGC 891 8.57 10.27 8.04E-9 1.74E-12 1.0E-12 ∼ 0.08q ··· NGC 3627t 10.04 10.38 7.55E-9 1.64E-12 3.6E-13 < 4E 11p 0.04q − NGC 7552 21.44 11.03 7.39E-9 1.60E-12 6.7E-13 ∼ 0.05q ··· NGC 4736 (M94) 4.83 9.73 7.30E-9 1.58E-12 3.5E-13 0.04q ··· NGC 2903 8.26 10.19 7.20E-9 1.56E-12 9.0E-13 < 8E 11p 0.08q ∼ − Table 4.2. Possible Calorimetric Galaxies: Predicted, & Observed Gamma-Ray Fluxes (cont’d) Table 4.2—Continued Predicted Predicted Calorimetric LTQ Observed a b c cald LTQe f g Name D LTIR FTIR Fγ Fγ Fγ Σg −2 (Mpc) log [L⊙] ( GeV) ( GeV) ( GeV) (gcm ) 10 ≥ ≥ ≥ ESO 173-G015 32.44 11.34 6.59E-9 1.43E-12 6.0E-13 0.05u ··· NGC 660 12.33 10.49 6.45E-9 1.40E-12 8.1E-13 < 8E 11p 0.08q − NGC 1097 16.80 10.71 5.76E-9 1.25E-12 7.4E-13 ∼ 0.1q ··· NGC 3628v 10.04 10.25 5.59E-9 1.21E-12 2.3E-13 < 4E 11p 0.03w − NGC 3079 18.19 10.73 5.15E-9 1.12E-12 7.8E-13 <∼ 8E 11p 3.7q ∼ − 254 aDistances from IRAS BGS unless otherwise noted. bTIR luminosities from IRAS BGS unless otherwise noted. c −2 −1 2 TIR flux in ergs cm s : FTIR = LTIR/(4πD ). dPionic gamma-ray flux (in ergs cm−2 s−1) predicted in the explicitly calorimetric limit: F ( GeV) = γ ≥ β F 1.8 10−4(E η′ Ψ ), using β = 0.7 as a fiducial value; see equation 4.13. π TIR × × 51 0.05 17 π ePionic gamma-ray flux (in ergs cm−2 s−1) predicted by the fiducial model of LTQ. See solid line Figure 4.3. In these models, leptonic emission is expected to be relatively minor (< 10%) when integrated above GeV energies, although it may comprise up to 25% of the differential emission at∼ 1 GeV. ∼ f Observed gamma-ray flux (in ergs cm−2 s−1) for energies GeV, or upper limit. ≥ gGas surface density. Typical uncertainty in this quantity is 0.3 dex. ∼ hWe take M = 2.3 108M (Weiß et al. 2001) and r = 250 pc for the D adopted: Σ = M /πr2. g × g g iAdopted distance different⊙ than in IRAS BGS (3.1 Mpc) for consistency with the rest of this paper. jTIR luminosity corrected for larger adopted distance. (cont’d) Table 4.2—Continued Predicted Predicted Calorimetric LTQ Observed a b c cald LTQe f g Name D LTIR FTIR Fγ Fγ Fγ Σg −2 (Mpc) log [L⊙] ( GeV) ( GeV) ( GeV) (gcm ) 10 ≥ ≥ ≥ k From Kennicutt (1998), but scaled to the CO-H2 conversion factor advocated by Mauersberger et al. (1996). lFrom the 1FGL source catalog, as announced in Abdo et al. (2010b). NGC 4945 is a Seyfert galaxy, and the AGN may contribute some γ-ray flux. mTotal gas mass within a radius of 12” ( 227 pc at D = 3.92 Mpc) is taken as M 1.7 108M (Mauersberger ∼ g ≈ × et al. 1996). ⊙ 255 nSchinnerer et al. (2000) give a gas mass of M 5.7 108M within r 1.4 kpc, implying Σ 0.02 g cm−2 g ≈ × ≈ g ≈ for NGC 1068. However, the gas mass is not uniformly distributed⊙ in this region, but is concentrated in spiral arms. If we instead use Σ 0.1 g cm−2, we find F LTQ( GeV) = 4.0 10−12 ergs cm−2 s−1. g ≈ γ ≥ × oDerived from the power law fit to NGC 1068 as found by Lenain et al. (2010). pEGRET upper limits from Cillis et al. (2005). qFrom Kennicutt (1998). rM83 has a central starburst region with a central surface density of Σ 0.07 g cm−2 and scale radius of 0.6kpc g ≈ ∼ (Lundgren et al. 2004). sFrom Downes & Solomon (1998). tOne member of the Leo Triplet (with NGC 3623 and the starburst NGC 3628). (cont’d) Table 4.2—Continued Predicted Predicted Calorimetric LTQ Observed a b c cald LTQe f g Name D LTIR FTIR Fγ Fγ Fγ Σg −2 (Mpc) log [L⊙] ( GeV) ( GeV) ( GeV) (gcm ) 10 ≥ ≥ ≥ 256 uAlso known as IRAS 13242-5713. Negishi et al. (2001) give diameter of 1.1 armin, corresponding to 9.45 kpc. ∼ −1 Using the Kennicutt (1998) relation between FIR luminosity and star formation rate, we derive 37.7M⊙ yr −1 −2 ∼ and a surface density of star formation of 0.54 M⊙ yr kpc . Assuming the Schmidt Law gives an estimate ≈ of the gas surface density of Σ 0.05 g cm−2. g ≈ vOne member of the Leo Triplet (with NGC 3623 and the starburst NGC 3627). w 8 −2 Israel (2009) gives Mg 1.5 10 M⊙ in the inner 0.6 kpc, implying Σg 0.03 g cm . On larger scales, Irwin ≈ × 9 ∼ −2 & Sofue (1996) derive M 1.7 10 M⊙ in the inner r 1.95 kpc, implying again that Σ 0.03 g cm . g ≈ × ≈ g ∼ Property Value 100 MeV - GeV a −9 −2 −1 Φ23 (1.4 0.5) 10 ph cm s b ± × Γ23 1.7 0.4 c ± √TS23 0.74 N(100 MeV)d (16 8) 10−9 ph cm−2 s−1 GeV−1 ± × GeV - 100 GeV e −9 −2 −1 Φ35 (0.30 0.05) 10 ph cm s f g± × Γ35 h ··· √TS35 2.2 N(1 GeV)i (1.2 0.7) 10−9 ph cm−2 s−1 GeV−1 ± × GeV - 100 GeV; Γ = 2.2 e −9 −2 −1 Φ35 (0.20 0.15) 10 ph cm s h ± × √TS35 1.7 N(1 GeV)i (0.24 0.17) 10−9 ph cm−2 s−1 GeV−1 ± × aBest-fit integrated flux from 100 MeV to 1 GeV. bBest-fit photon index between 100 MeV and 1 GeV. cTest statistic for power law fit between 100 MeV and 1 GeV for Arp 220; the square root is roughly the signficance of detection. dBest-fit differential flux at 100 MeV. eIntegrated flux from 1 GeV to 100 GeV. f Best-fit photon index between 1 GeV and 100 GeV. g The best-fit model for Arp 220 has a Γ35 of 5.0, the maximum allowed by our source model file, so this value is almost certainly spurious, considering the predicted faintness of Arp 220. (See Table 4.2.) hTest statistic for power law fit between 1 GeV and 100 GeV for Arp 220. iBest-fit differential flux at 1 GeV. Table 4.3. Fermi-LAT γ-ray fluxes of Arp 220 257 Chapter 5 The γ-Ray Background Constrains the Origins of the Radio and X-Ray Backgrounds 5.1. Introduction Cosmic rays (CRs) are accelerated in many environments including star-forming galaxies (SFGs; e.g., Condon 1992) and galaxy clusters (e.g., Ferrari et al. 2008; Rephaeli et al. 2008).1 The bulk of the CR energy is in protons. These collide with ambient nuclei, creating pions, which decay into γ-rays, neutrinos, and secondary electrons and positrons (e±). Whether secondary or primary, CR e± radiate synchrotron emission in magnetic fields and Inverse Compton (IC) as they scatter low energy photons. CR protons therefore contribute to the γ-ray and neutrino backgrounds, while CR e± contribute to the radio, X-ray, and γ-ray backgrounds. The origins of these backgrounds are understood to varying degrees. The γ-ray background was once attributed to blazars, but Fermi has revealed that another source may be responsible for most of the emission above 100 MeV (Abdo et al. 1This chapter was published as Lacki (2011, ApJ 729, L1). 258 2010g). SFGs are one explanation for the γ-ray background (e.g., Fields et al. 2010; Lacki et al. 2011). The neutrino background is not yet detected, although IceCube will improve sensitivity greatly (Achterberg et al. 2007). The radio background is assumed to come from CR e± in SFGs and possibly AGNs (Protheroe & Biermann 1996; Haarsma & Partridge 1998; Dwek & Barker 2002). However, the radio bolometer ARCADE detected an extragalactic radio background six times greater than expected from the radio luminosities of z 0 galaxies (Fixsen et al. 2009; ≈ Seiffert et al. 2009). Singal et al. (2010) suggested that redshift evolution of the radio properties of SFGs explains the ARCADE background. The X-ray background is the best understood, with most of it being resolved into AGNs (Gilli et al. 2007, and references therein). A powerful way of limiting one cosmic background is to compare it with another of the same origin. For example, the Waxman-Bahcall argument limits the flux of ultra-high energy neutrinos from the observed spectrum of ultra-high energy CR protons which produce the neutrinos (Waxman & Bahcall 1999; Bahcall & Waxman 2001). Simply put, the Universe must be at least as luminous in the protons that generate secondaries as in the secondaries themselves. Similarly, we can use one pionic background – either the γ-rays or neutrinos – to constrain the others: synchrotron radio or IC X-rays from secondary e±. Secondary e± may dominate over primary electrons in starburst galaxies (e.g., Thompson, Quataert, & Waxman 259 2007) and possibly galaxy clusters (e.g., Dennison 1980), so this argument applies to backgrounds from these objects. 5.2. Ratio of Pionic γ Rays to Secondary Emission Suppose a class of sources emits CR protons of energy Ep, which experience pionic losses during their propagation. The pions decay, generating γ-ray, neutrino, and leptonic backgrounds of flux intensity dJ/dE, so the power in each of the backgrounds per log bin energy is EdJ/dE, where E is the energy of the decay product. About 1/3 of the energy lost to pionic interactions goes into neutral pions, which decay into γ-rays with typical energy E 0.1E . The remaining energy is h γi ≈ p in charged pions; of this, 1/4 goes into secondary e± and the rest into neutrinos, so 1/6 of the pionic luminosity is in secondary e± while 1/2 is in neutrinos. The average energy of the neutrinos and e± is E E 0.05E E /2. Taking the ratio h ei≈h ni ≈ p ≈h γi of the background intensity in pionic γ-rays to pionic secondary e±, we have: dJe dJγ 2 Ee Eγ , (5.1) h idEe ≈h idEγ dJe dJn and similarly, 3 Ee En for neutrinos. In comparing Eγ to Ee , we h i dEe ≈ h i dEn h i h i assume the pions are relativistic; we take E 0.3E GeV as a threshold for this. γ ≥ 0.3 Far below this energy, few secondaries are expected and any emission comes from primary e±. 260 The secondary e± radiate synchrotron and IC emission, among other losses. The pitch-angle averaged rest-frame frequency of synchrotron emission 2 3 5 is νC = (3Ee eB)/(16mec ), where e is the electron’s charge and B is magnetic field strength. Since ν E2, d ln ν = 2d ln E : the synchrotron emission from C ∝ e C e one log bin in e± energy is spread over two log bins in synchrotron frequency. At most 100% of the CR e± emission can go into synchrotron, implying that νC dJe/dνC (νC )=(Ee/2)dJe/dEe, or dJe dJγ νC (synch) < (f/4)Eγ (pionic γ ray) (5.2) dνC ∼ dEγ − evaluated for ν at E = E /2, where f 1 parameterizes uncertainties in this C e γ ≈ approximation and the backgrounds (Loeb & Waxman 2006). Lower f linearly scales down the γ-ray background, either because the total background is lower than assumed here or to consider only the pionic contribution from some class of sources; similarly, higher f linearly corresponds to lower synchrotron or IC backgrounds, either from errors in the measured backgrounds, or to consider only the contribution from secondaries from some source class. This uses the δ-function approximation for the synchrotron spectrum, which is generally valid for power law spectra (e.g., Felten & Morrison 1966).2 2This approximation is accurate to 25% for an E−2 steady-state e± spectrum and is even ∼ −3 better for an E spectrum. Note that 70% of the synchrotron emission of electrons with Ee is in the 2 ln bins centered on νC . 261 Similarly, the average rest-frame energy of an IC upscattered photon of initial 2 2 4 energy ǫ0 is EIC = (4Ee ǫ0)/(3mec ) in the Thomson limit (EIC < Ee). Once again ∼ E E2, and the IC emission from one log bin in E is spread over two log bins in IC ∝ e e EIC. We have dJe dJγ νIC (IC) < (f/4)Eγ (pionic γ ray) (5.3) dνIC ∼ dEγ − evaluated for Ee = Eγ/2, again using the δ-function approximation (Felten & Morrison 1966). Note that eqs. 5.2 and 5.3 apply not just to the whole backgrounds, but to the pionic emission from each source and each population. In what follows, we conservatively assume that all of the observed γ-ray background (Abdo et al. 2010c) is pionic in origin. Removing leptonic contributions to the γ-ray background only tightens the limits. The power-law fit to the Abdo et al. (2010c) background is: −0.41 dJγ −9 Eγ Eγ (γ ray) = 2.33 10 (5.4) dEγ − × 100 MeV in cgs units of erg cm−2 sec−1 sr−1. At energies below Fermi observations, the observed total γ-ray background is bounded by eq. 5.4 (Weidenspointner et al. 2000; Strong et al. 2004; see Fig. 5.1); therefore using the observed total γ-ray background instead of eq. 5.4 gives even stronger limits on backgrounds from secondary e± than found here. At high energies, eqn. 5.4 applies only if the Universe is transparent to 262 γ-rays. This is correct below 20 GeV, our maximum Eγ for limits on z = 10 sources, and below 100 GeV out to z 1 (e.g., Gilmore et al. 2009; Finke et al. 2010). ≈ 5.3. The X-ray and Soft γ-ray Backgrounds Nonthermal emission in X-rays has been observed in galaxy clusters, and might be IC-upscattered CMB photons (see the review by Rephaeli et al. 2008). Moran et al. (1999) suggested IC upscattered ambient far-infrared (FIR) starlight in starburst galaxies contributes significantly ( 5 10%) to the X-ray background. Since pionic ∼ − γ-rays accompany pionic secondary e± production, the observed γ-ray background limits the contribution of secondary e± in these sources to the X-ray background. In the observer-frame, and assuming a typical energy of 3kTCMB(z) for CMB photons, the typical energy of upscattered CMB photons is E E2k[T (0)](1 + z)2/(m2c4). Plugging eq. 5.4 into eq. 5.3, we IC ≈ γ CMB e get: −0.205 dJe −10 EIC 0.41 νIC < 2.2 10 f (1 + z) , (5.5) dνIC ∼ × keV in cgs units. For our assumptions about pion kinematics to be valid, we impose the −1 constraint that Eγ > 0.3E0.3(1 + z) GeV: ∼ 2 EIC > 81E0.3 eV. (5.6) ∼ 263 Since the γ-ray background is only observed for E 100E GeV Abdo et al. γ ≤ 100 (2010c), we also require: 2 2 EIC < 9.0E100(1 + z) MeV, (5.7) ∼ where E 0.2 at high z because the Universe is opaque at energies above 20 100 → GeV. We proceed similarly for ambient light of temperature Tamb, finding E E2kT (1 + z)/(m2c4). Applying eq. 5.3 to eq. 5.4 gives us in cgs units: IC ≈ γ amb e −0.205 0.205 dJe −11 EIC Tamb 0.205 νIC < 9.8 10 f (1 + z) , (5.8) dνIC ∼ × MeV 50 K 2 −1 2 valid for 1.5 keVE0.3(1 + z) T50 < EIC < 165 MeVE100(1 + z)T50, where ∼ ∼ T50 = Tamb/(50 K). Figure 5.1 shows that IC-upscattered CMB light from secondary e± is only a small fraction of the X-ray background, with greater contributions possible for sources at greater z. For f = 1 and sources at z 0 (10), it makes up < 3% (< 7%) ≈ ∼ ∼ of the background below 0.5 keV, < 1% (< 4%) at 1 keV, < 0.3% (< 0.7%) at 10 ∼ ∼ ∼ ∼ keV, < 0.9% (< 2%) at 1 MeV, and < 1% (< 3%) at 10 MeV. ∼ ∼ ∼ ∼ As seen in Figure 5.1, the bounds on the contribution of upscattered FIR light from secondary e± to the X-ray and γ-ray backgrounds are relatively small. IC upscattered FIR is 4f% or less of the cosmic backgrounds from 1 keV to 1 ∼ 264 MeV, and up to 5f% of the 1 - 100 MeV background. Bounds on upscattered ∼ optical/UV light from young stars (Tamb = 10000 K) follow similarly. We find that such emission from secondary e± is < 16f% of the actual γ-ray background for ∼ z = 0 sources (< 9f% from 1 - 100 MeV), but up to 8 15f% of the 1 - 100 MeV ∼ − background and up to f/4 of the 0.1 - 10 GeV background for sources at z = 10. ∼ These results imply that IC emission from secondary e± does not contribute significantly to the X-ray or soft γ-ray backgrounds. However, they do not apply to primary electrons or to secondary e± that have been reaccelerated. 5.4. The Radio Background SFGs are expected to be a major source of the radio background. Many estimates of the cosmic radio background (such as Protheroe & Biermann 1996; Haarsma & Partridge 1998; Dwek & Barker 2002) use the FIR-radio correlation (FRC), a tight linear relation between the FIR and GHz synchrotron luminosities of SFGs (e.g., Helou et al. 1985; Condon 1992; Yun et al. 2001). Recent measurements by ARCADE suggest that the radio background is 6 times larger than expected from applying the FRC to the IR background (Fixsen et al. 2009; Seiffert et al. 2009). One way to explain this excess is if the FRC evolves with z (Singal et al. 2010). However, most bright galaxies out to z 2 seem to lie on the FRC (e.g., Appleton ≈ 265 et al. 2004; Sargent et al. 2010), or show only moderate deviations (e.g., Ivison et al. 2010). Recent work by Lacki et al. (2010a), supported by γ-ray detections of nearby starburst galaxies (Acciari et al. 2009; Acero et al. 2009; Abdo et al. 2010a), suggests that a conspiracy enforces the FRC in starburst galaxies: secondary e± dominate the primary electrons, increasing the radio emission 10 times when combined with ∼ spectral effects; while bremsstrahlung, ionization, and IC losses suppress the radio emission by a similar factor at 1 GHz (see also Lacki et al. 2011). An unbalanced conspiracy could enhance radio emission from starbursts (Lacki & Thompson 2010a), but such “extra” radio emission comes from pionic secondary e±, which are accompanied by pionic γ-rays. The pionic γ-ray background sets a hard limit on the synchrotron background from pionic e±. Based on the FRC, Loeb & Waxman (2006) and Thompson, Quataert, & Waxman (2007) calculated starbursts’ contribution to the neutrino and γ-ray backgrounds. Eq. 5.2 inverts these arguments: the γ-ray background sets upper limits on the radio background from starbursts. These limits apply to other sources of the radio background when secondary e± dominate their radio emission. ± ′ If pion production creates secondary e with source-frame energy Ee radiating synchrotron at observer-frame frequency νC , it also creates pionic γ-rays with 266 source-frame energy E′ 2E′ . The observed γ-ray background at E = E′ (1+ z)−1 γ ≈ e γ γ therefore limits the synchrotron background from secondary e± at 2 Eγ νC 3.2 B˜µGMHz, (5.9) ≈ GeV where B˜µG =(B/µG)(1 + z). The ARCADE fit to the radio background in cgs units is dJe −10 0.4 νC = 3.7 10 νGHz. (5.10) dνC × where νGHz is the observed frequency (assumed to be νC ) in GHz (Fixsen et al. 2009). The errors in the ARCADE data indicate that eq. 5.10 applies below νmax = 3.4 GHz; at higher frequencies, the errors become too large to be sure whether the background spectrum steepens. If the background is entirely from secondaries, equations 5.2, 5.4, and 5.9 limit the radio background to dJe −11 −0.21 ˜0.21 νC < 7.0 10 νGHz BµG , (5.11) dνC ∼ × as plotted in Figure 5.2. We obtain a lower limit on B by plugging the ARCADE background (eq. 5.10) into eqn. 5.11: −1 −4.9 2.95 3.4(1 + z) f νGHzmG < B. (5.12) ∼ At low frequencies, the ARCADE data is easily consistent with the γ-ray background (below the limits for all B˜µG in Figure 5.2). At higher frequencies, large B˜µG are 267 required: with higher B, lower energy e± are responsible for the emission at a given ± frequency, and eq. 5.4 allows more power at lower e energies. The limits on Je are constant in electron energy, but slowly shift in frequency (horizontally in Fig. 5.2) with different B. Our limit only applies if E′ 0.3E GeV. This combined with eq. 5.9 implies γ ≥ 0.3 1.03 2.50 −1.03 that eq. 5.12 is only valid for νGHz < 1.01(1 + z) f E0.3 and ∼ B 3.6f 2.50(1 + z)2.03E−3.0mG (5.13) lim ≈ 0.3 is the best lower limit on B that can be derived even if the ARCADE best-fit radio background extends to ν . For very high B B this means low energy → ∞ ≫ lim primary electrons must be the source of the background. The ARCADE background is marginally inconsistent with a secondary origin in most SFGs. When f = 1.0, a secondary origin for the ARCADE excess is difficult to reconcile with the γ-ray background. Eq. 5.13 rules out the intergalactic medium, clusters, and most galaxies at low redshifts. Only the densest Ultraluminous Infrared Galaxies (ULIRGs) like Arp 220 and AGNs have the milliGauss magnetic fields needed (Condon et al. 1991a; Torres 2004b; Robishaw et al. 2008). ULIRGs are among the brightest (and therefore individually detected) galaxies, and cannot make up most of the ARCADE background (Seiffert et al. 2009). For a z = 2 population, eq. 5.13 gives us B > 34f 2.5 mG (corresponding to 3.1f 2.5 GHz), but we generally ∼ 268 expect B < 20 40 mG for dynamical reasons (Thompson et al. 2006). For a source ∼ − population at z > 2, eq. 5.13 no longer is the main restriction, and the minimum ∼ allowed B decreases (eq. 5.12). Still, even at z = 10, a secondary origin for the −4.9 ARCADE background at νmax = 3.4 GHz requires B > 11f mG (eq. 5.12) in its ∼ sources. Furthermore, there is little cumulative star formation at high z (Hopkins & Beacom 2006); since eq. 5.2 applies to each individual source population, the ARCADE sources would have to be extremely efficient at accelerating CR protons and contribute most of the γ-ray background. Observations at 10 GHz can further constrain the possibility of a z 10 source: at higher frequencies, there is more ≈ power in the radio background, but at higher energies, there is less power in the γ-ray background. Eq. 5.11 demands a spectral turnover at high frequencies for the emission from secondary e±. The steep f dependence means that uncertainties in the γ-ray background and kinematics weaken the constraints on B considerably. However, greater f implies lower B, shifting the minimum allowed electron energy (0.15 E0.3 GeV) to lower frequency; this relaxes the constraint in eq. 5.13. Even for f = 2, the 3.4 GHz detections requires milliGauss magnetic fields in the sources except at the highest redshifts. Furthermore, the strong f dependence works in reverse: if even half of the γ-ray background is not pionic, or not from the sources of the ARCADE excess, then the limits on B strengthen by a factor 30. ∼ 269 Could the ARCADE excess be from primary electrons? Any radio background from primaries can be accounted for if all of the protons escape. However CR proton escape must be quite efficient; in the Milky Way, the luminosity of primary electrons is only 1 2% that of CR protons at GeV energies (Schlickeiser 2002). Previous − ∼ modeling indicates that secondaries are important in starbursts and perhaps the inner Galaxy, but unimportant for the low density outer Galaxy (Porter et al. 2008; Lacki et al. 2010a). Recent work indicates that only 20% of the Galactic GHz ∼ luminosity is from secondaries (Strong et al. 2010); primaries can greatly enhance the radio background from low density SFGs. However, galaxies have more gas at high z, making pionic losses more efficient. Another possibility is that primary CR electron (but not proton) acceleration efficiency is much higher in some SFGs, producing more synchrotron. Singal et al. (2010) suggested AGNs provided such additional primaries. IC upscattered starlight and bremsstrahlung in such galaxies would be a sign of these extra electrons. 5.5. Conclusion The observed γ-ray background limits the luminosity of pionic secondary e± in the Universe. These secondary e± may be important in galaxy clusters and starburst galaxies. We show that simple ratios can place bounds on the contribution of IC and synchrotron emission to the radio, X-ray, and γ-ray backgrounds from these 270 secondary e±. With our given assumptions, the IC upscattered optical/UV light from secondaries contributes less than f/4 of the GeV γ-ray background for sources at z = 10 and smaller fractions at lower redshift and energies; upscattered FIR and CMB from secondaries is 2f % or less of the 1 keV - 1 MeV X-ray background for ∼ sources at z = 0 and < 4% at z = 10, although with uncertainties described below. ∼ We consider the ARCADE-measured radio background in light of these bounds. Secondary e± are expected to dominate in starbursts that make up most of the star-formation at z > 1 (Dole et al. 2006; Caputi et al. 2007; Magnelli et al. ∼ 2009). The γ-ray background is marginally inconsistent with a secondary e± origin at 3 GHz, unless the sources have milliGauss magnetic fields, although with ∼ considerable uncertainty. However, we cannot rule out primary electrons in low density galaxies or other sources (where pionic losses are minimal) as the cause of the ARCADE measurement. There are multiple uncertainties in these bounds. First, we assumed the backgrounds all came from a source population with a single redshift, allowing considerable variation in the bounds as z varies. More detailed modelling of the effects of redshift evolution is needed. Second, these constraints can be tightened by measuring the hadronic contribution to the γ-ray background (e.g., Prodanovi´c& Fields 2004); using the entire γ-ray background as done here may overestimate the other hadronic backgrounds. Third, resolving out the contribution of each class of sources to the γ-ray background would tighten the limits on their contribution to 271 the other backgrounds. Note that this holds for the sources of the ARCADE excess specifically: even if most of the γ-ray background is star-formation, the sources of the ARCADE excess may contribute only a fraction of it. Finally, future pionic neutrino background measurements above 100 GeV, such as with IceCube (e.g., DeYoung et al. 2009), would help limit the IC and synchrotron backgrounds from the highest energy secondary e±. 272 Fig. 5.1.— Limits on the X-ray and γ-ray backgrounds from IC upscattering (f = 1, ± E0.3 = 1) by secondary e on CMB photons (solid), 50 K FIR photons (dotted), and 104 K UV/optical photons (dashed), based on the observed γ-ray background. Black is z = 0 while grey is z = 10. The observed backgrounds are from Gilli et al. (2007, black line and cross-hatching) and references therein, Watanabe et al. (1999, grey shading and line), Weidenspointner et al. (2000, Xs), Strong et al. (2004, triangles), and Abdo et al. (2010c, squares). 273 ± Fig. 5.2.— Limits on the radio background (f = 1, E0.3 = 1) from secondary e at a single redshift. The constraints apply for source populations at z =0(solid), z =2(long-dashed), z =5(short-dashed), and z = 10 (dotted). The ARCADE fit is the solid grey line, with uncertainties represented by shading. The predicted radio background from Dwek & Barker (2002) (for a J ν−0.7 spectrum) is the dashed ν ∝ grey line, with uncertainies represented by the striped grey area. 274 Chapter 6 Diffuse Hard X-ray Emission in Starburst Galaxies as Synchrotron from Very High Energy Electrons 6.1. Introduction Starburst galaxies are intense generators of cosmic rays (CRs), which are accelerated by supernova remnants or other star-formation processes.1 CR protons in starbursts can produce γ-rays, neutrinos, and secondary electrons and positrons. Whatever their origin, CR electrons and positrons can produce electromagnetic emission across the spectrum: bremsstrahlung losses produce γ-rays; Inverse Compton scattering of ambient photons produces a broadband spectrum extending into γ-rays; synchrotron emission is responsible for the non-thermal GHz radio emission. Starburst galaxies are observed to be luminous in hard X-rays (here 2 10 keV) as well. The total hard X-ray emission from star-formation typically ∼ − 1This chapter is adapted from Lacki & Thompson (2010b, arXiv:1010.3030). 275 has a luminosity that is 10−4 times that of the bolometric luminosity of the starburst, and is sometimes used as a star-formation indicator (Franceschini et al. 2003; Ranalli et al. 2003; Grimm et al. 2003; Persic et al. 2004). Most of these X-rays are from point sources, but there appear to be additional diffuse components. The diffuse hard X-ray emission is best studied in the nearby starburst M82 (Strickland & Heckman 2007), but a similar diffuse hard component is observed in NGC 253 (Weaver et al. 2002). The diffuse hard X-ray emission bears some superficial resemblance to the Galactic Ridge emission of the Milky Way (Strickland & Heckman 2007), although that has recently been resolved into stellar sources by Chandra (Revnivtsev et al. 2009). At the other extreme, unresolved hard X-ray emission is also observed in brighter starbursts including the Luminous Infrared Galaxy (LIRG) NGC 3256 (Moran et al. 1999; Lira et al. 2002) and the prototypical Ultraluminous Infrared Galaxy (ULIRG) Arp 220 (e.g., Clements et al. 2002; McDowell et al. 2003). Especially in these more distant galaxies, it is unclear whether these components are truly diffuse or simply faint, unresolved sources (e.g., Lira et al. 2002). The diffuse hard X-ray emission typically has a power-law continuum spectrum, with the possible addition of softer thermal emission components and spectral lines (Persic & Rephaeli 2002; Strickland & Heckman 2007; Lehmer et al. 2010). Thermal emission from hot plasma is one possible source of the hard X-rays: it is predicted by superwind theories, and thermal emission is clearly detected in soft X-rays (e.g., Ptak et al. 1997; Dahlem et al. 1998; Strickland & Stevens 2000). The detection of 276 6.7 keV iron K lines implies the existence of hot plasma that could be a source of the hard X-rays (Persic et al. 1998; Cappi et al. 1999; Iwasawa et al. 2005, 2009), although it is not clear that such emission could explain all of the hard continuum (Strickland & Heckman 2007). The hard X-ray emission is often attributed to unresolved X-ray binaries, particularly high-mass X-ray binaries (HMXBs) (e.g., Fabbiano et al. 1982; Fabbiano 1989; Griffiths & Padovani 1990; David et al. 1992; Persic & Rephaeli 2002; Grimm et al. 2003; Persic et al. 2004). HMXBs in the Milky Way and Magellanic Clouds have a power law continuum (photon spectra of dN/dE E−Γ) with a spectral slope of Γ 1.2 (White et al. 1983). Many ∝ ≈ starburst galaxies have total hard X-ray emission with Γ 1 1.5 (e.g., Rephaeli et ≈ − al. 1991, 1995), which makes HMXBs an attractive candidate for the source of most of the hard X-ray emission. However, in M82, diffuse hard X-ray emission remains even after subtracting point sources and extrapolating the luminosity function to faint luminosities, and the diffuse emission is softer (Γ − 2 3) than expected 2 8 ≈ − from HMXBs (Strickland & Heckman 2007), unless the HMXB spectrum cuts off at energies < 10 keV. ∼ Another possible explanation for this X-ray emission is Inverse Compton emission (Hargrave 1974; Schaaf et al. 1989; Moran et al. 1999; Persic & Rephaeli 2003). CR electrons and positrons (e±) are known to be present in starbursts from their synchrotron radio emission, and the intense infrared emission of starbursts provides many target photons to be upscattered to higher energies. The IC spectrum 277 is expected to extend down to the X-rays and even lower energies. However, recent estimates generally suggest that IC emission is too weak by a factor of > 10 in M82 ∼ and NGC 253 to explain the hard X-ray emission (e.g., Weaver et al. 2002; Strickland & Heckman 2007). As with HMXBs, the spectral slope of the X-ray emission in some starbursts may be difficult to explain with IC. The CR e± spectrum around 100 MeV and the resultant IC spectrum at keV energies is expected to be hard ∼ (Γ 1.0 1.5), whereas the diffuse X-ray emission is often softer, as is the case for ≈ − M82 (Strickland & Heckman 2007). Here, we consider an alternative source of hard X-rays: synchrotron emission from CR e± with energies above a TeV. Synchrotron has previously been considered as a source of Galactic diffuse X-ray emission (Protheroe & Wolfendale 1980; Porter & Protheroe 1997), but explaining the diffuse X-ray emission from the Galaxy requires CR electrons of extreme energies ( 100 TeV), and stronger magnetic fields ∼ (B > 20 µG) than in the diffuse ISM to avoid conflict with constraints on Inverse ∼ Compton emission from CASA-MIA (Aharonian & Atoyan 2000; Bi et al. 2009). Starbursts are expected not only to have large CR populations but also stronger magnetic fields than the Milky Way. The conclusion of strong magnetic fields in starbursts is supported by a number of lines of evidence: (1) minimum energy estimates applied to radio detections of starbursts imply B 50 µG (e.g., V¨olk et ≈ al. 1989; Beck & Krause 2005; Thompson et al. 2006; Persic & Rephaeli 2010); (2) detailed modeling of CR populations in starbursts with fitting of the radio and now 278 gamma-ray spectra imply B 100 µG in the Galactic Center (Crocker et al. 2011b), ≈ B 200 µG in the nearby starbursts M82 and NGC 253 (Paglione et al. 1996; ≈ Domingo-Santamar´ıa & Torres 2005; Persic et al. 2008; de Cea del Pozo et al. 2009a; Rephaeli et al. 2010; de Cea del Pozo et al. 2009b), and B > 1 mG in the ULIRG ∼ Arp 220 (Torres 2004b); (3) considerations of the linear FIR-radio correlation that applies to quiescent star-forming galaxies and starbursts, and the strong Inverse Compton, bremsstrahlung, and ionization losses expected for CR electrons in these galaxies (e.g., V¨olk 1989; Condon et al. 1991a; Thompson et al. 2006; Murphy 2009; Lacki et al. 2010a); (4) measurements of Zeeman splitting in ULIRGs, although these apply to the denser regions of the starbursts (Robishaw et al. 2008); (5) constraints on leptonic gamma-ray emission from the Galactic Center, limiting the number of CR e± (Crocker et al. 2010). Strong magnetic fields imply not only more synchrotron power per particle, but that lower energy CR electrons can produce synchrotron X-rays. The synchrotron ± 2 emission of a CR e peaks near νC = 3γ eB sin α/(4πmec) for an electron with Lorentz factor γ with a pitch angle α with respect to a magnetic field of strength B (Rybicki & Lightman 1979). For an isotropic distribution of electrons, sin α = π/4, h i and this translates to a synchrotron emission energy of Ec = hνC : E 1.0E2 B keV, (6.1) C ≈ 10TeV 200 279 where E10TeV = Ee/(10 TeV) and B200 = B/(200 µG) is the magnetic field strength in the starburst. We see that CR e± of energy in the range 10 30 TeV will emit − X-ray synchrotron emission in the 1 - 10 keV range. In this paper, we show that under the most optimistic assumptions, the synchrotron emission can explain the diffuse hard X-ray emission of starbursts, especially Arp 220. The synchrotron spectrum (νLν) rises at 1 GHz frequency and peaks near 10 - 100 GHz, but remains constant (or slowly falling) all the way to X-ray frequencies. There are three main reasons for this. (1) The synchrotron losses of GHz-emitting e± in starbursts are 1/5 1/20 of the total losses including ∼ − bremsstrahlung, ionization, and IC (Thompson et al. 2006; Murphy 2009; Lacki et al. 2010a, 2011). Thus little of the energy in these CR e± is transformed into synchrotron emission. However, while ionization and bremsstrahlung losses increase slowly with CR e± energy, synchrotron (and IC) losses become faster increasing with CR e± energy. Therefore, bremsstrahlung and ionization become completely unimportant at 10 TeV (see the loss time scales in 6.2.1). (2) Furthermore, IC § cooling will be suppressed at energies above 10 TeV. Most of the starlight generated by starbursts is efficiently absorbed by dust and re-emitted in the IR (c.f. V¨olk 1989), and the majority of this IR is blackbody emission from 40 50 K dust ∼ − 280 grains. For FIR photons of energy ǫ wavelength λ 100 µm, 20 TeV is near the IR ≈ characteristic energy scale for Klein-Nishina suppression of IC cooling: 2 4 mec EKN 21 TeV(λ/100 µm). (6.2) ≈ ǫIR ≈ (3) A more subtle but important effect of the 20 TeV Klein-Nishina cutoff is that ∼ the threshold for pair production for γ-rays on the FIR emission is near this energy. Thus the intense IR emission of starbursts converts the γ-ray emission at 10 - 100 TeV energies into 10 - 100 TeV e±. Since the power injected in pionic γ-rays is greater than the power in pionic secondary e±, these tertiary pair e± can dominate the CR spectrum at 10 - 100 TeV if the pair production optical depth is greater than unity (γγ attenuation and pair production at 10 - 100 TeV is expected to be small in galaxies like the Milky Way, see Mastichiadis et al. 1991; Moskalenko et al. 2006; Stawarz et al. 2010), providing an additional population that can radiate X-ray synchrotron. We begin in 6.2 by presenting order of magnitude estimates showing that § synchrotron may be important. Then, in later sections we construct one-zone models to evaluate these effects using standard assumptions for CR modelling and in the context of extant multiwavelength data. In 6.3, we describe our one-zone models § of the CR spectra, including the pair produced e±. In 6.4, we present the results of § these models for the Galactic Center, NGC 253, M82, and the nuclear starbursts of Arp 220. We discuss some broader implications in 6.5, including the synchrotron § 281 contribution to X-rays from submillimeter galaxies ( 6.5.1) and further constraints § on synchrotron X-rays from future neutrino and TeV experiments ( 6.5.2). Finally, § we conclude by discussing future work that can be done in 6.6. § 6.2. Motivation 6.2.1. Relevant Cooling Processes e± losses Starbursts contain dense gas and strong radiation fields, each of which can cool CR e± through bremsstrahlung, ionization, and IC emission. These cooling processes ± have associated energy loss rates for individual CR e bbrems, bion, bIC, respectively, which can be comparable to the synchrotron energy loss rate bsynch. In addition, CR e± can escape by advection or diffusion. These processes compete with synchrotron emission in the magnetic fields of starbursts for the kinetic energy of a CR e±. The synchrotron cooling time of CR e± is E/b (E) 6πm2c4/(cσ B2E) synch ≈ T (Rybicki & Lightman 1979), or − E −1 B 2 tsynch(E) 31 yr . (6.3) ≈ 10 TeV 200 µG! For energies greater than 1 TeV, and magnetic fields of at least 200 µG, this is about the light crossing time for 100 pc, roughly the scale height for a nuclear starburst. 282 Therefore, at these energies, e± are in the “calorimeter” limit, in which the CR e± cool before escaping (c.f., V¨olk 1989). ± For a gas density n, the ionization cooling time of CR e is E/bion(E), or −1 9 E n tion(E) 1.9 10 yr −3 , (6.4) ≈ × 10 TeV250 cm from e.g., Schlickeiser (2002), where we set the ln γ term to its value at 10 TeV. The bremsstrahlung cooling time is −1 5 n tbrems(E) 1.2 10 yr −3 , (6.5) ≈ × 250 cm at high energies (assuming n 0.1n ; see Strong & Moskalenko 1998). He ≈ H Synchrotron losses therefore dominate bremsstrahlung losses as long as B > 10 µG (E/TeV)−1/2(n/250 cm−3)1/2, and synchrotron losses dominates ∼ ionization losses so long as B > 0.2 µG (E/TeV)−1(n/250 cm−3)1/2. However, ∼ bremsstrahlung and ionization can dominate at the GeV energies where e± are ∼ responsible for GHz emission (Thompson et al. 2006; Murphy 2009; Lacki et al. 2010a). The Inverse Compton cooling time is more complex because Klein-Nishina effects appear at the relevant 10 100 TeV energies in the FIR-dominated ∼ − radiation fields of starbursts. Schlickeiser & Ruppel (2010) show that the IC loss 283 time in a greybody radiation field with temperature T and radiation energy density Urad can be approximated as 2 2 2 3mec γK + γ tIC 2 , (6.6) ≈ 4cσT Urad γγK where γ 0.27m c2/(k T ) = 4.0 107(T/40 K)−1. Thus, above K ≈ e B × 20 (T/40 K)−1 TeV, Klein-Nishina effects will cause the IC energy loss ∼ time to grow as γ, while the synchrotron loss time continues to fall as γ−1. If U < U , then synchrotron dominates IC when γ γ U /U 1, or: B rad ≥ K rad B − q T −1 U U 1/2 E 20 TeV rad − B . (6.7) ≈ 40 K UB The nearby, prototypical starbursts in M82 and NGC 253 have total infrared luminosities of 1010.5 L (Melo et al. 2002; Sanders et al. 2003) and radii of ∼ ⊙ 200 pc (e.g., Turner & Ho 1983; Goetz et al. 1990; Ulvestad & Antonucci 1997; ∼ Williams & Bower 2010). Plugging in these specific values for a disk geometry, the radiation field has an energy density of approximately U L/(2πR2c), or ≈ −2 −3 L R Urad 1100 eV cm . (6.8) ≈ 1010.5L ! 200 pc! ⊙ This gives us an IC loss time of −1 2 −1 Thomson L R E tIC 300 yr 10.5 (6.9) ≈ 10 L ! 200 pc! TeV ⊙ 284 in the Thomson regime, and −1 2 2 KN L R T E tIC 71 yr 10.5 (6.10) ≈ 10 L ! 200 pc! 40 K 100 TeV ⊙ in the extreme Klein-Nishina limit. For the specific values we have been using – L = 1010.5L , R = 200 pc, T = 40 K – equation 6.7 shows that synchrotron ⊙ dominates over IC above 5 TeV when B = 200 µG and 35 TeV when ∼ ∼ B = 100 µG. Therefore unless the magnetic field energy density is much lower than the radiation field energy density, synchrotron losses will be the dominant loss process at a few tens of TeV. Proton losses CR protons can modify the e± population by creating pionic secondary electrons and positrons through collisions with interstellar gas atoms. The time for CR protons to lose all of their energy through this process is given as −1 5 n tpion = 2 10 yr −3 (6.11) × 250 cm by Mannheim & Schlickeiser (1994). Different sources in the literature give different pionic loss times that can be larger by a factor 2 3; however, these decrease ∼ − slightly with proton energy as the pionic cross section increases (see the discussion 285 in Appendix 6.7), so that at TeV energies the pionic loss time is closer to equation 6.11. Since protons have much longer cooling times than TeV CR e±, it is not as obvious whether they can escape. Advection is clearly present in starbursts in the form of the observed large-scale winds (e.g., Heckman et al. 2000; Heckman 2003; Heesen et al. 2009). The wind crossing time for a starburst of scale height h is h/v, or: −1 5 h v twind = 3 10 yr −1 (6.12) × 100 pc! 300 km s In starbursts such as M82 and NGC 253, the advective and pionic lifetimes are expected to be roughly equal. The nuclear starbursts of Arp 220, with n 104cm−3 h i ≈ (e.g., Downes & Solomon 1998), have very short pionic loss times ( 5000 yr), ∼ indicating that they are “proton calorimeters”: most of the power injected into CR protons with energies above the pion-production threshold is lost through pionic interactions. However, a key uncertainty is whether CRs sample gas of the average density. The gamma-ray luminosity of the Galactic Center region (R 112 pc) relative ≤ to its star-formation rate indicates this is not the case for that region, as does the ratio of synchrotron radio from CR e± to infrared emission (Crocker et al. 2011a,b). Crocker et al. (2011b) explains these observations as being caused by a powerful 286 wind in the Galactic Center region advecting CRs out of the disk before they can enter the molecular clouds containing most of the gas. Thus CRs experience gas of much lower density than average. While the Galactic Center is underluminous in gamma-rays and radio, this is not true for the starbursts M82 and NGC 253, which fall on the FIR-radio correlation and have a larger gamma-ray to star-formation ratio than the Milky Way, consistent with CRs experiencing average gas densities in these starbursts (Lacki et al. 2011). A final uncertainty is whether diffusive escape plays any role. Diffusive escape in the Milky Way has an energy dependence t E−0.3 E−0.6 that steepens the diff ∝ − CR proton spectrum, since it is the dominant timescale (e.g., Ginzburg & Ptuskin 1976). In contrast, the relatively flat GeV-TeV spectra of observed starburst regions (M82, NGC 253, and the Galactic Center) indicate that up to TeV energies, an energy-independent process must determine the lifetimes of CR protons; however, at the still higher energies we are considering, diffusion can dominate. We consider several values of the diffusive escape time to address this uncertainty. 6.2.2. Primary Electrons Primary electrons dominate the GHz-emitting electron population in normal galaxies. However, they are generally expected to be sub-dominant in starbursts with respect to pionic secondary e±, though still a significant minority (Rengarajan 2005; 287 Thompson, Quataert, & Waxman 2007; Lacki et al. 2010a). Detailed modelling of CR populations in starburst galaxies find that primary electrons are subdominant at GeV energies in M82 (de Cea del Pozo et al. 2009a), NGC 253 (Domingo-Santamar´ıa & Torres 2005; Rephaeli et al. 2010), and Arp 220 (Torres 2004b), although models by Crocker et al. (2011b) find primary electrons dominate in the Galactic Center. At higher energies, where the protons experience stronger diffusive losses, these models indicate higher primary fractions. It is not clear where the primary electron injection spectrum ends in starbursts. As the electrons are accelerated to higher energies, they also experience more severe synchrotron and Inverse Compton cooling. At some point, the cooling losses balance the rate of acceleration, and there can be no further acceleration of primary electrons. In the standard supernova acceleration theory, equilibrium between cooling and acceleration occurs when: − E 27 TeV v B 1/2 , (6.13) e ≈ 5000 SNR,200 −1 where 5000v5000 km s is the speed of the supernova shock and 200BSNR,200 µG is the supernova remnant magnetic field (Gaisser 1990). From equation 6.1, the end of the primary synchrotron spectrum will then be at: synch 2 B Emax 7.3keV v5000 , (6.14) ≈ BSNR 288 if the magnetic field strength in the starburst B is similar to the magnetic field strength in the supernova remnant. X-ray observations of supernova remnants in the Milky Way have revealed synchrotron emission from 10 - 100 TeV electrons, confirming these energies are reached in supernova remnants (e.g., Koyama et al. 1995; Allen et al. 1997; Reynolds & Keohane 1999; Reynolds 2008; Reynolds et al. 2008). Thus, SNRs might be able to accelerate primary electrons to the energies where they will produce hard synchrotron X-rays in the diffuse ISM of starburst galaxies. However, it is not clear that SNRs are responsible for all of the primary CR electrons; other objects such as superbubbles or pulsars may contribute (e.g., Butt 2009). Pulsar Wind Nebulae (PWNe) powered by pulsar spindown after their birth supernova in particular may inject a very hard component of e± dominating at TeV energies, possibly extending to 10 - 100 TeV (Y¨uksel et al. 2009; Bamba et al. 2010). Primary CR e± from PWNe have been invoked to explain anomalies in the CR electron spectrum observed at Earth. The birthrate of pulsars are correlated with star-formation, and pulsars would be present in large numbers in starbursts Perna & Stella (c.f. 2004); Mannheim et al. (c.f. 2010). Their spin-down luminosities could provide enough power to be comparable to the main component of CRs (Lacki et al. 2011). If a hard primary CR e± spectrum extends to 10 - 100 TeV energies or beyond, then the synchrotron X-ray luminosity may be very bright. Suppose that for every 289 supernova, roughly e = E 1048 erg of CR electrons are accelerated. This could ECR 48 occur if 0.1% of a supernova remnant’s kinetic energy (1051 ergs) were converted into primary CR electrons, which is about 1% of the total energy expected to go into CRs (Strong et al. 2010, the exact ratio depends on the poorly constrained low energy CR e± spectrum). For an E−2 injection spectrum extending from 1 GeV to 1 PeV, then E2dQ/dE = e / ln(106). At very high energies, a fraction f will go ECR synch into synchrotron, where f = t /t 1/(1 + t /t ): synch life synch ≈ synch IC −1 Urad/UB fsynch 1+ 2 (6.15) ≈ " 1+(γ/γK ) # (eqs. 6.3 and 6.6), which we have argued to be near 1. Since the characteristic synchrotron emission energy EC is proportional to (Ee), the synchrotron emission q of the CR e± population in one dex of electron energy will be spread over two dex in 2 synchrotron frequency. We finally have νLν(keV) = (1/2)fsynchEe dQe/dEe (Loeb & Waxman 2006), or: 39 ΓSN −1 νLν(keV) 1.1 10 E48fsynch − ergs s , (6.16) ≈ × yr 1 ! where ΓSN is the supernova rate. For example, M82 is believed to have a supernova rate of roughly 0.1 yr−1 (although with large uncertainties), ∼ suggesting that its synchrotron X-ray luminosity from pionic e± may be 1 1038ergs s−1, about 3% of the observed luminosity of the diffuse X-ray emission × (L (2 8 keV) = (4.4 0.2) 1039 ergs s−1, or νL 3.2 1039 ergs s−1 per bin in − ± × ν ≈ × 290 ln energy; Strickland & Heckman 2007). Softer CR e± injection spectra will have still lower synchrotron X-ray luminosities. If SNRs or other discrete sources do accelerate 10 TeV primary electrons, ∼ the synchrotron X-ray emission may not be spread continuously throughout the starburst, but concentrated near the CR sources, because the synchrotron cooling time for these electrons is so short (eq. 6.3). Even if such electrons free-stream, they will not travel farther than 10 pc from their sources. If there are a small number ∼ of CR e± accelerators, then the synchrotron X-ray emission should come from a few small diffuse regions, just as patchy “cells” of high energy e± are predicted for the Milky Way (c.f., Shen 1970; Aharonian et al. 1995; Atoyan et al. 1995). If starbursts, with a high star-formation rate concentrated into a small volume, instead contain many accelerators, the primary e± confinement regions around these accelerators will overlap and the X-ray emission will arise throughout the starburst. Unlike pionic secondary e±, or pair production e± from pionic γ-rays, primary e± at these energies are unaffected by the escape because cooling is so quick. 6.2.3. Pionic Secondaries CR protons can inelastically scatter off protons in the ISM to produce pions, which decay into secondary e±, γ-rays, and neutrinos. From the lifetime calculations given in 6.2.1, starbursts are expected to convert much more of their CR § 291 proton energy into pionic products than the Milky Way and approach the “proton calorimeter” limit (Loeb & Waxman 2006; Thompson, Quataert, & Waxman 2007; Lacki et al. 2010a). From the HESS detection, Acero et al. (2009) claimed that 5% of the proton energy was converted into pionic products in NGC 253, although they assumed a hard γ-ray spectrum. From the Fermi, HESS, and VERITAS detections, Lacki et al. (2011) inferred a proton calorimetry fraction of about 1/3 for NGC 253 and M82 from the ratio of the GeV γ-ray and IR luminosities. The secondary ≥ e± are expected to dominate in starbursts at GeV energies based on physical considerations (Rengarajan 2005; Loeb & Waxman 2006; Thompson, Quataert, & Waxman 2007; Lacki et al. 2010a) and detailed models of M82, NGC 253, and Arp 220 (Torres 2004b; Domingo-Santamar´ıa & Torres 2005; de Cea del Pozo et al. 2009a; Rephaeli et al. 2010). Primaries probably dominate in the Galactic Center due to a fast wind advecting CR protons from the region before they can interact with the molecular cloud gas (Crocker et al. 2011b). In the Milky Way, primary CR protons (and nuclei) are accelerated with a power law spectrum extending to energies of at least PeV, the so-called “knee” ∼ in the CR spectrum. Above these energies, the CR spectrum steepens: this may be because Galactic CRs are not accelerated to higher energies and we are seeing the transition to a different component of CRs, or because Galactic CRs propagate differently above the knee. Essentially nothing is known about the knee in starbursts, but presumably it is also at least at a PeV in energy. 292 If most of the energy injected into CR protons is lost to pion production for CR protons of up to a PeV in energy, then the large population of secondary pionic 10 - 100 TeV e± produced by pion decay can emit bright hard X-ray emission. Suppose that roughly = 1050 erg of CR protons are accelerated per supernova (that is, ECR roughly 1/10 of the SN kinetic power goes into CRs; Strong et al. 2010). Once again we have E2dQ/dE = / ln(106) for an E−2 injection spectrum extending from 1 ECR GeV to 1 PeV. A fraction Fcal(Ep) of that power is lost to pions for a CR proton energy E ; of that, 1/6 will go into secondary e± (e.g., Steigman & Strittmatter p ∼ 2 ± 1971; Loeb & Waxman 2006) and fsynch of the secondary e power in turn goes into synchrotron emission. From a similar argument as the primaries, we finally have 2 2 νLν(keV) = (1/2)fsynchEe dQe/dEe = (Fcal/12)fsynchEp dQp/dEp (Loeb & Waxman 2006), or: 40 ΓSN −1 νLν(keV) 1.9 10 Fcal(100 TeV)fsynch − ergs s , (6.17) ≈ × yr 1 ! −1 where ΓSN is the supernova rate. For a supernova rate in M82 of 0.1 yr , we get 2 1039ergs s−1 for f = F = 1, more than half the observed luminosity of the × synch cal diffuse X-ray emission (Strickland & Heckman 2007). In practice, energy-dependent diffusive escape of the primary protons, advective escape in the starburst superwind, 2This is because the charged pions receive 2/3 of the energy, and each charged pion ultimately decays into a positron or electron and three neutrinos of roughly equal energy. Therefore, e± recieve 1/4 2/3 = 1/6 of the pion energy. Similarly, neutrinos receive 1/2 of the pion energy, and γ-rays × recieve the remaining 1/3, from the neutral pions. 293 and softer CR proton injection spectra will reduce the secondary e± luminosity further. In contrast with primary electrons, pionic e± will be generated throughout the starburst, instead of being concentrated near the source of CR protons, because CR protons can travel a longer distance during their lifetime. Therefore the synchrotron X-ray emission secondary e± will not have as patchy a structure as the primaries, but they will be generated wherever there is gas being sampled by CRs. 6.2.4. Pair Production Tertiaries Starburst galaxies are generally predicted to become opaque to γ-rays above a few TeV, because γ-rays will pair produce e± with the IR light in the starbursts (Torres 2004b; Domingo-Santamar´ıa & Torres 2005; Inoue 2011). Thus, the pionic γ-ray emission at 10 - 100 TeV will be efficiently converted into 10 - 100 TeV e±. What is less appreciated is that these e± will lose much of their energy to synchrotron cooling, emitting mostly X-rays. Thus, if multi-TeV CR protons in starbursts have efficient pionic losses, then up to half of the CR proton energy at these energies will go into synchrotron X-rays, with some fraction escaping as neutrinos. The optical depth of a starburst to multi-TeV photons is τ hn σ , where γγ ≈ IR γγ h is the height of the starburst disk, σγγ is the cross section of pair-production, and n is the number density of target IR photons. Near threshold (where E E IR γ ≈ KN 294 from eq. 6.2), σ σ /4. The average energy of IR photons in a greybody field γγ ≈ T is π4ζ(3)kT/30 2.70kT , where ζ is the Riemann zeta function. The number ≈ density of IR photons for a starburst disk of radius R (and emitting area 2πR2) is approximately n L /(2πR2c 2.70k T ), where T is the typical temperature of IR ≈ IR × B the IR photon. We have −2 −1 LFIR h R T τγγ 5.8 10.5 , (6.18) ≈ 10 L ! 100 pc! 200 pc! 40 K ⊙ demonstrating that luminous starbursts are opaque to 30 TeV photons, turning ∼ them into pair e±. In practice, since about half of the total infrared radiation is FIR (Calzetti et al. 2000), the expected τγγ will be lower by a factor of a few, but still of order unity. This is in contrast to the Milky Way, which is essentially transparent to γ-rays (Mastichiadis et al. 1991; Moskalenko et al. 2006; Stawarz et al. 2010). Pionic γ-rays are expected to dominate the VHE γ-ray luminosity of starbursts, so the calculation of the synchrotron power from tertiary pair e± is similar to that for secondary pionic e±. CR protons roughly inject 2 times more energy in pionic γ-rays than in pionic secondary e±, so the synchrotron power should likewise be twice as great for pair e± than direct secondaries (eq. 6.17): 40 ΓSN −1 νLν(keV) 3.8 10 Fcal(100 TeV)fsynch − ergs s , (6.19) ≈ × yr 1 ! for an E−2 injection spectrum and 1050 ergs per supernova in CR protons. Again comparing to M82 with a supernova rate of Γ 0.1 yr−1, we find that νL may SN ≈ ν 295 be as high as 4 1039 ergs s−1 (with F = f = 1), equal to the observed diffuse × cal synch X-ray emission. As with secondary pionic e±, diffusive and advective losses (lower Fcal) and softer injection spectra reduce the predicted synchrotron X-ray luminosity. On the other hand, non-pionic gamma-rays, such as from discrete sources like pulsars, will also produce pairs in the starburst radiation field, and this can enhance the electron population and synchrotron X-ray luminosity further. Like secondary e±, tertiary e± will not be concentrated near CR accelerators, since the pionic γ-rays that generate them are emitted everywhere in the starburst region. However, the pair e± production may be enhanced near luminous IR sources within the starbursts. A treatment of this effect requires a radiative transfer calculation, which is beyond the scope of this work. 6.3. Modelling Assumptions To understand the synchrotron and Inverse Compton X-ray emission of starburst galaxies, we create models of the steady-state CR populations of the Galactic Center, NGC 253, M82, and the nuclei of Arp 220. Our goal here is to sketch out the parameter space allowed by multiwavelength data using a few standard assumptions, and investigate the synchrotron and IC emission that arises under these assumptions. We model the starbursts as one-zone disks of radius R and midplane-to-edge scale heights h. The evolution of CR population is governed 296 by the diffusion-loss equation (e.g., Ginzburg & Ptuskin 1976; Strong et al. 2007). In steady-state one-zone models with no spatial or temporal dependence, the diffusion-loss equation reduces to the leaky box equation: N(E) d [b(E)N(E)] Q(E) = 0. (6.20) tlife(E) − dE − Here, Q(E) is the injection spectrum of CRs, b(E) is the cooling rate of CRs (including ionization, bremsstrahlung, IC, and synchrotron), and tlife(E) is the lifetime of CRs from escape (both diffusive and advective) and pionic losses for protons. The CR −1 −1 −1 −1 lifetime tlife(E) is calculated as [tlife(E)] = [tdiff (E)] + tadv + [tπ(E)] , where tdiff (E) is the diffusive escape time, tadv is the advective escape time, and tπ(E) is the pionic loss time for protons. We use the numerical code described in Lacki et al. (2010a) to find the steady-state CR spectra, employing a Green’s function given in Torres (2004b). See Table 6.1 for the parameters we used for each starburst. 6.3.1. Injection We calculate the star-formation rate by directly converting the TIR (8 1000 µm) luminosity of the starburst disk: − 2 SFR = LTIR/(εc ), (6.21) where ε = 3.8 10−4 is a dimensionless factor relating the luminosity to the × instantaneous star formation rate, and is IMF dependent (Kennicutt 1998). The 297 factor ε is derived assuming a starburst that is continuous over 10 - 100 Myr and a Salpeter IMF (Kennicutt 1998). The CR energy injection rate per unit volume is assumed to be proportional to star-formation rate: ǫ = 9.2 10−6E ψ L (ξ/0.001)/V (6.22) CR,e × 51 17 TIR ǫ = 9.2 10−4E ψ L (η/0.1)/V (6.23) CR,p × 51 17 TIR = ǫCR,eδ (6.24) (6.25) for electrons and protons respectively, where ξ is the electron acceleration efficiency, −1 δ is the ratio of proton accleration efficiency η to ξ, ψ17 = (βSN/ε)/(17 M ), ⊙ 2 V = 2πR h is the starburst volume, and βSN is the SN rate per unit star formation. 2 The total SN rate in the starbursts is ΓSN = 17LTIR/(M c ): ⊙ −1 LTIR ΓSN = 0.036 yr ψ17 . (6.26) 1010.5 L ! ⊙ We assume ψ17 = 1 throughout this work. The efficiency of primary CR electron accleration is described by the ξ parameter, and the ratio of energy going into CR protons and CR electrons is δ. We assume that CR protons and electrons are respectively injected with a momentum power law spectrum −p dQp/dq = Cpq (6.27) 298 −p dQe/dq = Ceq (6.28) per unit volume, where q is the CR proton or electron momentum. A minimum kinetic energy cutoff of Kmin = 1 MeV was used with these spectra. The CR proton p 6 spectrum is assumed to extend to γmax = 10 , corresponding to an energy of 938 p 5 7 TeV (γmax = 10 and 10 are considered in Appendix 6.8). We try different cutoffs prim 6 9 in the Lorentz factor γmax of the primary electron spectrum, both 10 and 10 . We note that we are assuming the “test particle” approach to CR acceleration here. Nonlinear effects can result in more complicated spectra, with breaks in the power law at low energy and spectral hardening at high energy (e.g., Berezhko & Ellison 1999; Ellison et al. 2000; Malkov & O’C Drury 2001; Blasi et al. 2005). We also neglect additional, harder components of e± from different CR accelerators. In general, a hardening of the injected electron spectrum at high energy will result in more synchrotron X-ray emission and a softening of the injected electron spectrum will result in less synchrotron X-ray emission. The normalizations of the CR injection spectra are then set by calculating the integral of the kinetic energy injected and equating with ǫCR: qmax,p −p 2 2 2 4 2 ǫCR,p = Cp q q c + mpc mpc dq (6.29) qmin,p − Z q qmax,e −p 2 2 2 4 2 ǫCR,e = Ce q q c + mec mec dq (6.30) qmin,e − Z q 299 2 2 prim 2 where q = (1/c) K + 2K mc and q = mc (γmax ) 1. The ratio of min min min max − q q energy in CR electrons to CR protons injected at high energies can be approximated as: − C m c2 p 2 δ˜ = p p δ. (6.31) Ce ≈ Kmin ! In the Milky Way, δ˜ 50 100, which is expected from charge conservation in ≈ − the acceleration region for a p 2.2 momentum power law injection spectrum ≈ (Schlickeiser 2002). This value of proton/electron injection ratio is also inferred from propagation studies that compare with observations of CRs in the Milky Way (e.g., Strong et al. 2010), and from the CR pressure derived from shock structure in Tycho’s supernova remnant (Warren et al. 2005). It is also the approximate ratio of the CR proton and electron energy densities at Earth (e.g., Ginzburg & Ptuskin 1976). 6.3.2. Propagation In solving the leaky box equation for CR e±, we consider energy losses from ionization, bremsstrahlung, synchrotron, and IC. CR protons experience continuous ionization losses, which are important at low energies where pionic losses are negligible. All CRs can also escape, either through energy-dependent diffusion, or energy-independent advection (winds). We assume an E−1/2 energy dependence to the diffusive escape times; the exact energy dependence is not known, although 300 our assumption is conservative in that CR protons will tend to easily escape at higher energy without producing pionic e± and γ-rays. The normalization of the diffusive escape time in starbursts is also not known, so we consider a variety of normalizations: E −1/2 tdiff = tdiff (3 GeV) , (6.32) 3 GeV with t (3 GeV) ranging from 1 Myr to (in the Milky Way, t (3 GeV) 30 Myr; diff ∞ diff ≈ Connell 1998; Webber et al. 2003). Different energy dependences would alter the CR proton population at TeV energies, and we naturally expect more secondary and pair e± (and their accompanying synchrotron X-rays) if there is a weaker energy dependence in tdiff than in equation 6.32. However, the TeV γ-ray luminosity will also be different, so in starbursts with a TeV γ-ray luminosity constraint, the allowed parameter space would change. This would weaken the effect of different energy dependence in tdiff , since there is a shorter “lever arm” between the population traced by TeV γ-rays ( 10 100 TeV protons for 1 - 10 TeV γ-rays) and synchrotron ∼ − X-rays ( 100 1000 TeV protons). For E−0.3 energy dependence (the weakest ∼ − usually expected), the diffusion time is greater than 300 kyr at PeV energies as long as tdiff (3 GeV) > 10 Myr, meaning that advection (and any pionic losses) ∼ will be more important than diffusive escape (see equation 6.12). Thus, relatively slow diffusive escape with a E−0.3 energy dependence is equivalent to no diffusive escape at all (t (3 GeV) ). In addition, in models where primary electrons diff → ∞ 301 dominate the synchrotron X-ray flux (the Galactic Center and low B models of other starbursts), diffusive escape will have little effect, since the synchrotron cooling time is so short at these energies (equation 6.3). Advective escape times are more well known; a wind with a speed of a few hundred kilometers per second will carry a CR out of the starburst in a few hundred kyr (eq 6.12). Our models of NGC 253, M82, and Arp 220 assume that −1 −1 vwind = 300 km s . We use a wind speed of 600 km s for the Galactic Center models, as Crocker et al. (2011b) infers for the Galactic Center region. To calculate the effects of pionic losses on the CR proton spectrum, we directly integrate all of the energy going into pionic secondary products: −1 E p dσ(Ep,Esec) tπ = Kp nH βCRc Esec dEsec (6.33) ± 0 dEsec e X,γ,ν Z The spectra of the pionic secondary e±, γ-rays, and neutrinos are calculated using the Kamae et al. (2006) cross sections for proton energies below 500 TeV and the Kelner et al. (2006) cross sections for proton energies above 500 TeV. This method has the advantage of being consistent between the differential cross sections and the pionic lifetime. We also consider other cross section parametrizations and pionic lifetimes in Appendix 6.7; we generally found that these produced similar results. Calculating the injection rate of pair-produced e± requires both the low-energy target photon and γ-ray spectra within the starburst. Assuming a planar geometry, 302 with γ-rays traversing vertically out of the starburst disk3, the γ-ray number density is Qγ(Eγ)h Nγ(Eγ)= [1 exp( τγγ(Eγ))], (6.34) cτγγ(Eγ) − − where Qγ is the injection rate of γ-rays per unit volume in photons per unit energy per unit volume. For a photon number density spectrum n(ǫ), we calculate the pair production opacity τγγ using τγγ(Eγ)= hn(ǫ)σγγ(ǫ, Eγ)dǫ (6.35) Z where we use the approximation in Aharonian (2004) (equation 3.23) for the differential pair-production cross section σγγ(ǫ, Eγ) given in full in Gould & Schr´eder (1967), and where we have also assumed that the radiation field is isotropic. When calculating the γ-ray spectrum observed at Earth, we replace h with ℓ⊕, the sightline to the center of the starburst disk (equal to R for perfectly edge-on disks). We use the GRASIL SEDs for the radiation fields of M82 and Arp 220, scaled to the correct luminosities (Silva et al. 1998).4 For NGC 253, we use the M82 GRASIL SED scaled to the NGC 253 starburst’s luminosity. We use the SED 3Note that γ-rays do not always travel straight out of the disk plane, but also at horizontal angles through the disk. Thus we underestimate the mean τ by a geometrical factor of order unity. 4The GRASIL SEDs are available at http://adlibitum.oat.ts.astro.it/silva/grasil/modlib/fits/fits.html. 303 of Porter et al. (2008) for the Galactic Center, adding an infrared greybody dust component from star-formation of temperature 20 K (Launhardt et al. 2002). The GRASIL luminosities are converted to energy densities as U = L/(2πR2c). We then add the CMB to the SEDs of NGC 253, M82, and Arp 220’s nuclei (the CMB is already present in the Porter et al. (2008) for the Galactic Center). We show the resultant background spectra in Figure 6.1, and the γ-ray optical depths for these SEDs in Figure 6.2. M82 and Arp 220 are both optically thick in the 10 - 100 TeV range. NGC 253’s optical depth peaks at 35 TeV with τ 0.5. However, ∼ ≈ the Galactic Center is transparent at all considered energies. Opacity from three radiation components are visible in Figure 6.2: near infrared radiation from old stars, far infrared radiation from dust grains, and the CMB. In NGC 253, M82, and Arp 220, the far infrared radiation completely dominates the γγ opacity. ± The source function Qpair(Ee) of pair production e is then calculated from the γ-ray and IR spectra (e.g., Aharonian et al. 1983; B¨ottcher & Schlickeiser 1997). We have implemented pair production in our code. We use the Aharonian et al. (1983) approximations for the source function, as given in equation 32 of B¨ottcher & Schlickeiser (1997), which are known to be accurate at high energies.5 5Note that both equations 26 and 32 in B¨ottcher & Schlickeiser (1997) are for electrons only (Aharonian & Khangulyan 2011, private communication; B¨ottcher 2011, private communication). Our attempts at calculating the pair spectrum using the B¨ottcher & Schlickeiser (1997) formula gives results that are a factor of 2 too small, both by comparing with the Aharonian et al. (1983) pair spectrum and checking energy conservation. A previous version of this chapter assumed that 304 After calculating the γγ pair e± spectrum, we then calculate the γ-ray luminosity from these pairs. We then in turn calculate the pair e± from these γ-rays, using the same procedure as we did for the γ-rays from protons and primary and secondary e±. We found that these higher-order pair e± were typically only a small fraction of the γγ pair e± population, reaching a maximum of 50% in the Arp 220 nuclei models even with B = 250 µG and under 8% for M82, NGC 253, and the Galactic Center. 6.3.3. Constraints We run grids in η, B, p, and the normalization of the diffusive escape time ± tdiff (see eq. 6.32). Hadronic models (with CR protons, secondary e , and the pair e± associated with the γ-rays from these) are run independently of leptonic models (with primary CR electrons and the pair e± from the γ-rays they generate), giving us a hadronic and leptonic template for each parameter set. The two are then added together by scaling the hadronic template with η and the leptonic template with ξ. For starbursts where there are error bars in the radio data (Galactic Center, NGC 253, and M82), we then select the free-free emission flux Stherm(GHz) at 1 GHz based on chi-square fitting of the radio data, scaling ξ for each Stherm(GHz) so that the 1.4 GHz synchrotron radio emission of the starbursts equals our model predictions. B¨ottcher & Schlickeiser (1997) equation 32 was for both e±; thus the pair injection rate was underestimated a factor of 2 in that version. 305 We use the interferometric measurements in Williams & Bower (2010) of M82 and the starburst core of NGC 253 and the GHz radio measurements in Crocker et al. ≥ (2011b) for the Galactic Center (“HESS region”) radio flux. We use the radio data compiled in Torres (2004b) for the east and west nuclei of Arp 220; since these data do not have error bars, we simply normalize ξ to match the observed 5 GHz flux. Models with negative ξ (if the secondaries alone overproduced the radio flux) were not allowed. We then require the predicted 0.3 - 10 GeV (Abdo et al. 2010g) and TeV (Acero et al. 2009; Acciari et al. 2009) emission of M82 and NGC 253 to match the observed values to within a factor 2. For the Galactic Center region, we require the predicted TeV emission to match the observed emission (Aharonian et al. 2006b) to within a factor 2, and the predicted GeV emission to be lower than the observed emission within ℓ 1◦ and b 1◦ from EGRET (Hunter et al. 1997). For Arp 220, we | | ≤ | | ≤ require both the predicted GeV and TeV to be lower than the upper limits from the Fermi-LAT one year catalog (Abdo et al. 2010g) and MAGIC (Albert et al. 2007a), respectively. We also run a purely calorimetric hadronic model for each p, B, and density, with no advective or diffusive losses. By integrating up the volumetric power generated in secondary e±, γ-rays, and neutrinos in a model, and comparing the 306 yield with the calorimetric model, we can quantify the efficiency of pionic losses as the calorimetric fraction Fcal: ∞(E dQγ + E dQe + E dQν )dE F = 0 dE dE dE . (6.36) cal ∞ dQcal cal cal R γ dQe dQν 0 (E dE + E dE + E dE )dE R We also quantify the efficiency of pionic losses for generating VHE products by calculating F ( 10 TeV), in which we change the lower bound of integration to 10 cal ≥ TeV. Usually, a range of η will be compatible with these constraints. For the sake of brevity, for a given parameter set (Σg, B, p, tdiff (3 GeV)), we consider only those models that either have (a) the minimum allowed η, (b) the maximum allowed η, (c) the η which predicts the closest match to the GeV γ-ray emission (in M82 and ∼ NGC 253) or the TeV γ-ray emission (in the Galactic Center), or (d) η fulfills (a), prim (b), or (c) for some other γmax and the model otherwise satisfies our constraints. For Arp 220’s nuclear starbursts, we simply assume that η = 0.1 since there are no γ-ray observations yet. 6.4. Results and Comparison With Observations At GeV energies, synchrotron losses must compete with extremely strong bremsstrahlung, ionization, and IC losses, which cool e± before they radiate much synchrotron. In order to account for the observed radio emission, there must be more 307 CR e± than one would naively expect for primary electrons with Milky Way-like acceleration efficiencies. In general, we find that two different kinds of models work, given the radio constraints and the γ-ray observations: 1.) In models with high B, the GeV CR e± spectrum is dominated by pionic secondaries, which enhance the radio emission. In this limit, models are not affected much by variations in sufficiently small ξ, since any small ξ will result in sub-dominant primaries. The amount of synchrotron emission from secondaries (and tertiary pair e± from pionic γ-rays) is set by the efficiency of CR proton acceleration (η 0.1, set by the γ-ray observations), the efficiency of pionic losses F and pair ≈ cal production (set by the gas density and radiation fields, not B), and the power of synchrotron with respect to other losses fsynch (which is highly dependent on B for GHz emission, but is 1 for X-ray emitting energies in high B models). Thus ∼ the hadronic component asymptotes to a constant synchrotron X-ray luminosity as B increases. However, the GHz radio synchrotron emission does increase with B, because of the dominant non-synchrotron losses; in models with too high B, the secondaries overproduce the radio emission at fixed η. This sets an upper bound on B. 2.) At low B, the synchrotron emission of secondary e± is insufficient to explain the observed GHz radio. The only way for there to be enough CR electrons is to increase the primary CR electron acceleration efficiency greatly. Thus, low B 308 models favor primary electrons dominating the CR e± spectrum at GeV energies.6 If the CR primary electron spectrum ends at low energies, the 10 - 100 TeV CR e± spectrum only consists of secondaries and tertiaries. If the primary CR electron spectrum extends to higher energies, though, the primary electrons in these models may overwhelm the secondaries and tertiaries and greatly enhance the synchrotron X-ray emission. Although we consider them, we feel that these models are contrived: in those starbursts with γ-ray constraints, these models require the CR proton acceleration efficiency to be the same or less than the in the Milky Way (η < 0.1), ∼ so that the the total hadronic and leptonic γ-ray emission not exceed observations. However, they also require that primary CR electrons be accelerated much more efficiently (ξ 0.05 0.2) in starbursts compared to ξ < 0.01 0.02 in the Milky ≈ − ∼ − Way (Lacki et al. 2010a; the numerical value depends on the shape of the CR electron spectrum at energies GeV). They can be distinguished observationally ≤ from the high B secondary-dominant models by γ-ray emission below 100 MeV: ∼ models with low B have intense bremsstrahlung and IC emission which flattens out 6 Lacki et al. (2010a) postulated a “high-Σg conspiracy” that sets the radio synchrotron luminosity of starbursts: the suppression of starbursts’ radio emission by non-synchrotron losses is compensated by the appearance of secondary e± and the dependence of the critical synchrotron frequency on B. In the low B models, the non-synchrotron losses are still present and are even more influential because B is smaller, but they are instead compensated by a much larger primary CR electron acceleration efficiency than in the Milky Way. Thus, there still is a conspiracy, but a different one involving high ξ instead of secondaries. 309 the pionic “bump” at lower energies. Indeed, the leptonic γ-ray emission sets a lower limit on B for the starbursts we model. This basic idea that different B require different electron acceleration efficiency to match the GHz radio constraints has been described before in Persic et al. (2008), de Cea del Pozo et al. (2009a), and Rephaeli et al. (2010). In Tables 6.2 through 6.9, we list the synchrotron X-ray emission in models for the Galactic Center, NGC 253, M82, and the nuclei of Arp 220, respectively. We find that the synchrotron X-ray emission is very weak in the Galactic Center and generally weak in NGC 253 and M82, especially in models where the primary e− prim 6 spectrum cuts off below a TeV (γmax = 10 ). In Arp 220, however, the synchrotron contribution to the diffuse X-ray emission is significant, even with a low energy primary cutoff. The IC emission is also a small minority of the X-ray emission. We discuss individual starbursts in 6.4.1 - 6.4.4. § Some general trends are apparent in the tables. First, the synchrotron X-ray fraction is higher if the injection spectra are harder, simply because there are more e± and protons at higher energy. Second, the amount of X-ray emission depends on the primary electron maximum energy. The cutoff dependence is especially strong in low B models where the primaries dominate the CR e± spectrum. Even in high B models, where secondaries dominate the CR e± population at GeV energies, ∼ primaries can still dominate the TeV e± population, because diffusive escape ∼ 310 becomes quicker at high energies and removes CR protons before they can interact with starburst gas. Third, a shorter diffusive escape time reduces the synchrotron X-ray contribution, particularly in high B models for M82 and NGC 253 where the secondary e± at multi-TeV energies is dominant in some models and sub-dominant in others. To simplify our presentation, we also subjectively choose a “fiducial” model for each considered starburst. To qualify as a fiducial model, δ˜ must be near its approximate Milky Way value, within the range 50 - 100, whenever this is possible among the allowed models. This criterion selects the higher B models. Because of the coarseness of our grids in B, however, we were not able to get δ˜ to match in all of our fiducial models: δ˜ is 18 in the fiducial model for Arp 220’s western nucleus, and 48 - 84 in the other starbursts. We also consider only p = 2.2 models (which also means 13 δ 25, from eqn. 6.31). Finally, we try to choose models that are close ≤ ≤ fits to any existing γ-ray observations. The parameters for these fiducial models are listed in Table 6.1. 6.4.1. The Galactic Center Introduction – The Galactic Center region in many ways resembles a mini- starburst; we refer the reader to Crocker et al. (2011b) for a complete discussion of its properties. Diffuse hard (Γ 2.3) TeV γ-rays have been observed with HESS ≈ 311 in the region with ℓ < 0.8◦ and b < 0.3◦ (R = 112 pc; h = 42 pc; Aharonian et | | | | al. 2006b; Crocker et al. 2011b), which we take as our modelled region. The source of the CRs responsible for this emission has been conjectured to be Sgr A⋆ (e.g., Ballantyne et al. 2007), supernovae in the region (e.g., B¨usching et al. 2007; Crocker et al. 2011b), and diffusive reacceleration in the intercloud medium (Wommer et al. 2008; Melia & Fatuzzo 2011). The hard CR spectrum could either be explained by either a recent non-steady-state injection of CRs (Aharonian et al. 2006b; B¨usching et al. 2007), or steady-state energy-independent transport such as advection (Crocker et al. 2011b). For the purposes of this paper, we assume like Crocker et al. (2011b) that the diffuse TeV emission is from a steady-state population of CRs accelerated by star-formation processes, and then briefly discuss the consistency of this scenario with our results. GeV emission from the nuclear starburst specifically, as opposed to the inner Galaxy, is not yet well constrained, although it probably is present (Crocker et al. 2010). Radio emission is observed both from this region and a surrounding halo-like structure that extends out to ℓ < 3◦ and b < 1◦ (R = 420 pc, h = 140 pc) | | | | which contains most of the observed radio emission. The luminosity of the dust 8 in the region is 4 10 L (Launhardt et al. 2002); taking that as LTIR, we × ⊙ derive a star-formation rate of 0.07 M yr−1 (eq. 6.21), which is compatible with ⊙ the star-formation rate over the past few million years found by Yusef-Zadeh et 312 al. (2009) (see also the extensive discussion in Crocker et al. 2011a,b).7 This IR luminosity implies that the inner starburst region observed with HESS falls off the FIR-radio correlation observed in starbursts like M82 and NGC 253, but the surrounding halo does lie on the correlation (Crocker et al. 2011a). This combined with the γ-ray faintness of the starburst region given its supernova rate implies a strong wind, with a speed we take to be 600 km s−1 following Crocker et al. (2011b). The radiation energy density from the dust luminosity is 42 eV cm−3. For the background radiation field we add this star-formation TIR radiation field to the inner Milky Way radiation field in Porter et al. (2008), which includes the contributions from stars outside the Galactic Center starburst proper. This radiation field has an energy density 15 eV cm−3, for a total radiation energy density 57 eV cm−3 (Brad = √8πUrad = 48 µG). The gas surface density that CRs traverse in the Galactic Center region is not very well constrained. Pierce-Price et al. (2000) find a gas mass of 5 107M in the × ⊙ −2 inner 200 pc, corresponding to Σg = 0.09 g cm . Ferri`ere et al. (2007) find that Σ 0.03 g cm−2 in the inner few hundred pc, although steeply falling with distance g ≈ 7Launhardt et al. (2002) find a total stellar luminosity of 2.5 109 L , mostly from the nuclear star × ⊙ cluster, but this luminosity with eq. 6.21 implies a much higher star-formation rate of 0.3 M yr−1 ⊙ if it mostly came from young stars. Since the total radio emission is mostly from primaries in our models, this would require ξ to be 6 times smaller than our derived values, or 0.001. The TeV γ- ∼ ray emission, if it is hadronic, would imply 6 times smaller η than in our models, < 0.02. Crocker ∼ ∼ et al. (2011b) however conclude that the supernova rate is consistent with the lower rate we use. 313 from the Galactic Center. Crocker et al. (2011b) find that CRs have to sample gas of less than average density to fit the multiwavelength data. We run a grid in Σg ranging from 0.003 g cm−2 to 0.1 g cm−2 to account for these uncertainties. We assume a magnetic field strength of 50 or 100 µG (Crocker et al. 2010). The Galactic Center region also has a hard X-ray and soft γ-ray excess along the Galactic ridge (e.g., Worrall et al. 1982). The central square degree has a 2 - 10 keV surface brightness of 10−9 ergs s−1 cm−2 sr−1 (Koyama et al. 1996); we scale this to the modeled region, assuming constant surface brightness, to get a luminosity of 7.4 1036 ergs s−1. Although the X-ray ridge emission was once attributed to × very hot gas, most of the emission has been resolved into faint X-ray sources, mainly associated with stars (Revnivtsev et al. 2006, 2009). At higher energies > 50 keV ∼ (particularly soft γ-rays), Inverse Compton may dominate the emission (Porter et al. 2008). Synchrotron has previously been considered as a source of the Galactic ridge X-ray emission (Protheroe & Wolfendale 1980; Porter & Protheroe 1997), but requires stronger magnetic fields than are present in much of the Milky Way to avoid overproducing TeV Inverse Compton emission (Aharonian & Atoyan 2000). Given the strong magnetic fields and relatively high star-formation rate of the Galactic Center starburst, we reconsider whether synchrotron emission may contribute to the X-rays from this region. Modelled Radio and γ-rays – The Galactic Center region has few radio data points, so it is not surprising that we are able to fit the GHz radio data. In ≥ 314 the allowed models, the radio emission is dominated by primaries (c.f. Crocker et al. 2011b), which have ξ = 0.005 0.01 for B = 100 µG and ξ = 0.03 0.06 for − − B = 50 µG. The 74 and 330 MHz data points fall well below our fits, which is probably due to the strong free-free absorption known to be present in the region (e.g., Brogan et al. 2003). −2 Models with Σg = 0.003 g cm are strongly preferred by the γ-ray constraints (see Tables 6.2 and 6.3 for the allowed models). This supports recent modeling by Crocker et al. (2011b) implying that CRs do not sample the average density in the Galactic Center region. Low gas surface densities means that Fcal is also small, 0.3 0.8% for Σ = 0.003 g cm−2 (contrast this with F 8 21% for ∼ − g cal ≈ − −2 Σg = 0.1 g cm ). Thus, in many of our models, the Galactic Center is actually less of a proton calorimeter than the Milky Way as a whole. Above 10 TeV, diffusive escape can reduce this even further, to 0.4% when t (3 GeV) = 10 Myr and ∼ diff 0.06% when t (3 GeV) = 1 Myr for Σ = 0.003 g cm−2. Secondaries are therefore ∼ diff g a minority compared to the primaries (dashed lines in Figure 6.4). Tertiary pair e± are negligible at all energies, because the Galactic Center is transparent to 10 - 100 TeV γ-rays (Figure 6.2, short-dashed line). Synchrotron X-rays – Except in a few of the most optimistic models, only a small fraction of the hard X-ray emission from the Galactic Center is synchrotron (see Tables 6.2 and 6.3 and Figure 6.5). This is perhaps not surprising, given that most of the X-ray emission has already been resolved. With CR protons and VHE 315 γ-rays escaping so readily, there are few secondary or tertiary e± at 10 - 100 TeV energies to emit synchrotron X-rays (Figures 6.4 and 6.5). Therefore the X-ray emission is determined by primary electrons. Furthermore, the relatively low B of the Galactic Center compared to the other starbursts means that a given synchrotron frequency corresponds to higher e± energies, where the cutoff in CR protons causes a steep falloff in the secondary e± spectrum. We find synchrotron fractions of 0.01% synch 35 −1 −7 to 6%, with a fiducial value of 5.9% (L − = 4.3 10 ergs s 3 10 L ). 2 10 × ≈ × TIR This is consistent with the resolution of 88 12% of the X-ray ridge emission at ± energies of 7 keV (Revnivtsev et al. 2009). For p = 2.2, we find γprim = 109 models ∼ max have synchrotron fractions of 6%; almost all allowed models with γprim = 106 have ∼ max synchrotron fractions of 0.1% or less. The synchrotron emission has a very soft spectrum (Γ − = 2.35 2.50) in 2 10 − models with γprim = 106. This is because in the weaker 50 100 µG magnetic fields max − of the Galactic Center, the 2 - 10 keV X-ray emission traces e± of very high energy (grey shading in left panel of Figure 6.4). The CR secondary e± spectrum starts to cutoff at these energies, because the CR proton spectrum ends at a PeV in our model. In reality, the “knee” of the CR proton spectrum may be at a different energy in the Galactic Center region, so the softening of the synchrotron spectrum may not be real p 7 (see Appendix 6.8, where we find harder hadronic X-ray spectra if γmax = 10 ). In prim 9 models with γ = 10 , the synchrotron spectrum has Γ − = 2.12 2.27, since the max 2 10 − primary electron spectrum remains hard at these energies (left panel of Figure 6.4). 316 Inverse Compton X-rays – The IC contribution to the observed 2 - 10 keV emission from the TeV-emitting region is also small. The main factor affecting the IC contribution is B: in models with B = 50 µG, IC makes up 0.5% of the 2 - 10 keV emission; in models with B = 100 µG, IC makes up 0.09 - 0.16% of the 2 - 10 keV emission. The IC emission has a spectral index of 1.45 1.67, due to the strong − advective losses flattening the electron spectrum. The ratio of synchrotron and IC emission in the 2 - 10 keV band has a large range (0.029 - 55; fiducial: 48) in the allowed models. However, in all of the γprim = 109 models, synchrotron is brighter than IC, with typical ratios of 3 5 max ∼ − for p = 2.4 models and 50 for p = 2.2 models. The large values of these ratios ∼ even when p = 2.4 arises because UB > Urad and winds remove electrons below 10 ∼ GeV before they can radiate much IC (eqns. 6.9 and 6.12). By contrast, in most of prim 6 the γmax = 10 models, the synchrotron to IC ratio is almost always less than unity. Conclusion – The Galactic Center in some respects is a poor target for synchrotron X-ray emission. This is because it has a large X-ray background probably from stellar X-ray sources, contains a fast wind that removes protons before they can interact with the gas and produce secondary e± and γ-rays that could cascade, and a weak radiation field compared to other starbursts that means little pairs are produced from the γ-rays. The synchrotron emission is a few percent of the observed emission only if the primary electron spectrum extends to high enough energies. IC emission in the 2 - 10 keV X-ray band is also very weak (< 1%), because 317 the Galactic Center wind removes GeV electrons before they can IC upscatter ambient radiation. However, the proximity of the Galactic Center means that many point sources can be resolved out, leaving the truly diffuse X-ray emission (e.g., Revnivtsev et al. 2009). A potential concern about our results is whether time-independence is a good approximation. Our assumed TIR luminosity of 4 108 L translates × ⊙ to a supernova rate of (2200 yr)−1, while our fiducial diffusive escape time is 6 −1/2 −1/2 tdiff = 10 yr (E/3 GeV) = 1700 yr (E/PeV) . Thus the steady-state approximation roughly holds for protons in our models until a PeV. The advective lifetime in our models, 63000 yr (c.f. eqn. 6.12) is long enough for 30 supernovae ∼ to go off in the HESS region, which would smooth out stochastic effects at energies where advection dominates. Advective escape also naturally produces the hard spectrum (Crocker et al. 2011b). We cannot rule out that diffusive escape is much faster than we suppose, though. For primary electrons, the total 10 TeV population ≥ of the entire region is not affected by diffusive escape since the synchrotron lifetime −1 −2 alone at these energies is tsynch = 120 yr (E/10 TeV) (B/100 µG) (eqn. 6.3), faster than any possible escape process. However, the level of primary CR electrons may vary over kyr timescales if the time in which CR accelerators accelerate 10 - 100 TeV electrons is shorter than 2200 years. Thus, there could be a stochastic ∼ element in the synchrotron X-ray emission from the Galactic Center region. 318 6.4.2. NGC 253 Starburst Core Introduction – NGC 253 contains one of the nearest starbursts (we adopt D = 3.5 Mpc; 1′′.= 17 pc) and one of the two detected in both GeV (by Fermi-LAT; Abdo et al. 2010a) and TeV γ-rays (by HESS; Acero et al. 2009). Previous modelling of the nonthermal emission from NGC 253 has been done by Paglione et al. (1996), Domingo-Santamar´ıa & Torres (2005), and Rephaeli et al. (2010), and the latter predictions are in good agreement with the Fermi and HESS results. Unlike the Galactic Center, the starburst core of NGC 253 lies on the FIR-radio correlation and is γ-ray bright compared to the Milky Way. NGC 253 has a large reservoir of gas in its center; the estimated mass is (2 5) 107 M from molecular lines (e.g., − × ⊙ Mauersberger et al. 1996; Harrison et al. 1999; Bradford et al. 2003; Sakamoto et al. 2011), corresponding to central gas surface densities of 0.07 0.15 g cm−2. The − starburst appears as a disk with a diameter of 20′′.- 30 ′′. (340 - 510 pc) in radio continuum (Turner & Ho 1983; Ulvestad & Antonucci 1997) and 15′′.- 30′′. (250 - 510 pc) in infrared and molecular lines (Peng et al. 1996; Ulvestad 2000; Sakamoto et al. 2011); we adopt a radius of 150 pc. The vertical extent of the radio disk is 8′′. (136 pc) (Ulvestad & Antonucci 1997), implying a maximum vertical scale height of 68 pc. On the other hand, a cylindrical geometry for NGC 253’s radio disk, given an inclination of 78◦ and a radio diameter of 340 pc, implies that 71 pc of ∼ the vertical extent is the projection of the diameter; the remainder implies a scale height of only 33 pc. Due to the uncertainties in the geometry, we adopt a scale 319 height of 50 pc (giving a gas density of 190 cm−3). Finally, the total IR luminosity of NGC 253 is 4 1010 L (Sanders et al. 2003), of which half comes from the × ⊙ nuclear starburst (Melo et al. 2002). We therefore adopt a starburst bolometric luminosity of 2 1010 L , which corresponds to a low supernova rate of 0.025 yr−1, × ⊙ and a radiation energy density 1200 eV cm−3, equal to that of a magnetic field Brad = 220 µG. NGC 253 has also been studied with Chandra and XMM-Newton (e.g., Weaver et al. 2002; Strickland et al. 2002; Bauer et al. 2008), revealing the presence of a starburst wind in soft X-rays and hard emission from the galactic disk. XMM-Newton measured the diffuse (point source subtracted) 2 - 10 keV luminosity from the optical disk as 8.5 1038 ergs s−1 (Bauer et al. 2008). We adopt this as the × X-ray luminosity to compare against, although it includes the regions outside the starburst; these outer regions host 30% of the total star-formation rate (Melo et al. 2002). Modelled Radio and γ-rays – The combination of radio, GeV, and TeV constraints still leaves a large and complex parameter space of models that work. Selected models are shown in Tables 6.4 and 6.5. We show radio emission in a typical model in Figure 6.6. The flat radio spectrum favors high GHz thermal fractions in our models, with a span of 4 28% (fiducial: ∼ − 21%) selected by our criteria. As seen in Figure 6.6, there is evidence for unaccounted 320 spectral curvature in the residuals, and the high thermal fraction overproduces the observed emission at 20 - 100 GHz. This implies that the synchrotron spectrum is actually flatter than predicted by our models at 1 GHz, and then grows steeper at higher frequency, with a lower thermal fraction. The flatter spectrum can arise from loss mechanisms with a flatter energy dependence than synchrotron or IC, such as bremsstrahlung, ionization, or advection losses. Enhancing advection losses decreases the yield of secondary electrons, but increasing the density (and bremsstrahlung and ionization losses) increases the yield. Furthermore, more primaries would be needed to account for the smaller fraction of electron power going into GHz synchrotron emission if there are additional losses, so the results of such changes are complex. Secondaries dominate the GHz synchrotron emission in models with high B, but in low to intermediate B, primaries dominate (see Figure 6.7). The high energy emission from example models is shown in Figure 6.8. The predicted GeV γ-ray emission changes from being mainly leptonic at B = 50 µG to nearly all hadronic at B = 150 µG. This implies that NGC 253’s starburst cannot have a magnetic field much lower than 50 µG, or else the leptonic γ-ray emission would exceed the observed values. Similarly, the magnetic field cannot be much higher than 150 µG, or else the radio emission from the secondaries would be overpredicted, and in fact no models with B 200 µG fit our criteria. ≥ Similar values have been derived from previous modeling of NGC 253’s starburst (Domingo-Santamar´ıa & Torres 2005; Rephaeli et al. 2010). 321 In B 100 µG models, high η models are favored, and our fiducial model has ≥ η = 0.40, three to four times higher than the other starbursts. This is because the proton calorimetry fraction is less than one half of the value for M82 in our fiducial model, even though the γ-ray to bolometric luminosity ratio is the same. The actual CR acceleration efficiency could be 0.1 and still consistent with the γ-ray data ∼ if: (1) the supernova rate in NGC 253’s core is higher than we assume (and δ is the same as in our fiducial model), since only the injection rate of CRs matter, (2) the gas density is higher than we assume, increasing Fcal, or (3) advective losses are weaker, again increasing Fcal (c.f. Lacki et al. 2011). The HESS detection also constrains the proton and electron populations at TeV energies. In low B leptonic models, the primary electron spectrum cannot prim 9 extend to γmax = 10 with p = 2.0, or else the IC emission is overproduced. Even with p = 2.2, the leptonic models are severely constrained and just barely fit the TeV constraint (see Tables 6.4 and 6.5). The high B scenarios allow a few p = 2.0 models, but only when diffusive escape time is quick and the TeV/GeV ratio is still 3 times higher than observed. ∼ In our models, NGC 253 is not a true proton calorimeter because of its strong wind, but it is much more calorimetric than the Milky Way. The calorimetry fraction obtained by integrating the pionic products over all energies is F 10 30% cal ≈ − (c.f., Lacki et al. 2011). However, energy-dependent diffusive escape plays an 322 increasing role at higher energies. Above 10 TeV, F drops to 12% when cal ∼ t (3 GeV) = 10 Myr and 2% when t (3 GeV) = 1 Myr. diff ∼ diff Synchrotron X-rays – The wide span of the parameter space allowed by the data (Tables 6.4 and 6.5) means the 10 - 100 TeV electron population is not very well constrained, even with the HESS detection. Thus, the synchrotron X-ray fraction of NGC 253’s diffuse X-ray emission ranges from 0.19% to 120%, with a fiducial synch 37 −1 −7 value of 8% (L − = 7.2 10 ergs s 9 10 L ). For p = 2.2 models, the 2 10 × ≈ × TIR prim synchrotron fraction is typically tens of percent in low B models with high γmax , a few percent in high B models, and a few percent to less than one percent in low B prim models with low γmax . Models with large synchrotron fractions (> 1/3) overproduce ∼ the observed TeV emission by a factor 2. ∼ The spectral index of synchrotron emission is 2.28 Γ − 2.42 for ≤ 2 10 ≤ prim 6 prim 9 γ = 10 and 2.00 Γ − 2.29 for γ = 10 . max ≤ 2 10 ≤ max Inverse Compton X-rays – The calculated IC X-ray emission in the 2 - 10 keV band varies because the magnetic field in NGC 253’s starburst is unknown. The models with B = 50 µG, 100 µG, and 150 µG have IC fractions of 7 19%, 2 4%, − − and 1.3 1.9%, respectively. The synchrotron-to-IC ratio varies widely from 0.013 − prim to 49 (fiducial: 2.8). Models with high γmax have higher synchrotron/IC ratios (see Tables 6.4 and 6.5). The IC emission is consistently much harder than the synchrotron emission with 1.26 Γ − 1.40. ≤ 2 10 ≤ 323 Conclusion – The GeV and TeV detections of NGC 253’s starburst inform our knowledge of the synchrotron X-ray emitting e± population, but leave room for large variations. While the synchrotron fraction of NGC 253 can be as low as 0.2% or as high as 120%, in the fiducial model it is 8%. A key uncertainty is the primary electron energy cutoff, because the e± population is dominated by primary electrons at these energies. The IC emission is also a minority of the X-ray emission, with the largest uncertainty being the unknown magnetic field in NGC 253. The higher resolution measurements of NGC 253 with Chandra can help reduce the competing emission from other sources. First, Chandra can resolve out more point sources with its higher angular resolution. Second, the emission from the outlying disk can be subtracted, leaving only the emission from the nuclear starburst proper. Chandra has revealed that there is extended hard X-ray emission in the nuclear starburst of NGC 253, but little is known about it (Weaver et al. 2002). 6.4.3. M82 Introduction – M82 (D = 3.6 Mpc; 1′′.= 17 pc) is also detected in GeV and TeV γ-rays, with Fermi-LAT and VERITAS respectively (Abdo et al. 2010a; Acciari et al. 2009). de Cea del Pozo et al. (2009a) and Persic et al. (2008) previously modelled the nonthermal emission from M82, and there is again good agreement with the observed fluxes de Cea del Pozo et al. (2009b). M82 lies on the FIR-radio correlation 324 and is γ-ray bright with respect to the Milky Way. Most of the star-formation is in the starburst in the center of M82. Weiß et al. (2001) find a gas mass of 2.0 108 M in the inner 280 pc (for D = 3.6 Mpc), corresponding to a gas × ⊙ surface density of 0.17 g cm−2, which we use here. The infrared emission of M82 is concentrated in a ring with radius 225 pc, and the gas peaks at a radius of 250 pc (see the compilation in Table 4 of Goetz et al. 1990; see also Kennicutt 1998). Williams & Bower (2010) find a radio extent for M82 of 35′′. by 10′′., corresponding to a radius of 300 pc and a scale height of 90 pc. We therefore adopt 250 pc as the radius and 100 pc as the scale height of the modeled starburst region. Other studies find radio scale heights of 50 - 200 pc (Klein et al. 1988; Seaquist et al. 1985). Values for the wind speed vary; however, Greve (2004) find that the wind accelerates over a 200 pc scale height to 400 km s−1 and Westmoquette et al. (2009) find Hα FWHM of 100 300 km s−1 in the inner few hundred parsecs. The wind speed may − asymptote to > 1000 km s−1, but in the standard theory this happens outside most ∼ of the starburst; inside the starburst we expect wind speeds near the sound speed (Chevalier & Clegg 1985), which is 200 300 km s−1 (Greve 2004). We therefore ∼ − again adopt an advection speed of 300 km s−1. The total IR luminosity of M82 is 5.9 1010 L from Sanders et al. (2003), corresponding to a supernova rate of × ⊙ 0.06 yr−1. This also implies a radiation energy density 1300 eV cm−3, equal to that of a magnetic field Brad = 220 µG. 325 The diffuse X-ray emission of M82 is well-studied. The diffuse hard emission has a luminosity of 4.4 1039 ergs s−1 and is spatially extended over a few hundred × pc, with a scale height estimated at h = 175 pc (Strickland & Heckman 2007). The spectral slope of the diffuse hard X-ray emission is Γ 2 3 and it is difficult to ≈ − explain as thermal bremsstrahlung or IC emission (Strickland & Heckman 2007). Modelled Radio and γ-rays – The GHz synchrotron radio spectrum combined with the GeV and TeV γ-ray detections leaves us with the parameter space in Tables 6.6 and 6.7. The predicted synchrotron radio emission tends to be too large below a GHz (as shown in Figure 6.9); this may be due to free-free absorption (Klein et al. 1988). Conversely, the synchrotron radio spectrum falls off steeply at high frequencies, with α 0.8 0.9 instead of the observed 0.7 (Klein et al. 5 ≈ − 1988; Williams & Bower 2010). In order to accommodate the observed spectrum, our fitting requires higher 1 GHz thermal fractions (8 - 21%; with a fiducial value of 15%) than derived in Williams & Bower (2010) (6 2%). These high thermal ± fractions overproduce the observed total radio emission at 20 - 100 GHz. The underlying synchrotron spectrum is therefore probaby flatter, which could occur with higher bremsstrahlung, ionization, or escape losses. When B 150 µG, the GeV e± ≥ spectrum is dominated by pionic secondaries (short-dashed lines in Figure 6.10), and the 100 MeV emission is almost entirely pionic γ-rays (purple line in right panel ≥ of Figure 6.11). These models have low ξ, and thus relatively little power going into primary electrons. When B 100 µG, however, the GeV e± spectrum is mostly ≤ 326 primary electrons (long-dashed line in Figure 6.10); these models usually have much higher ξ (δ˜ < 10 for p = 2.2) than the Milky Way. ∼ As with NGC 253, the GeV γ-ray emission is primarily leptonic when B = 50 µG and hadronic when B = 150 200 µG. The predicted GeV γ-ray flux is − already at least 40% higher than observed in the B = 50 µG models because of the strong bremsstrahlung and IC emission; therefore, M82’s magnetic field is probably greater than 50 µG. Indeed, most of these models only work by either cutting off the primary e± spectrum so that the IC emission doesn’t greatly overproduce the observed TeV emission, or by having very low η so there’s virtually no pionic emission and all of the TeV emission is leptonic. Thus, the TeV detection of M82 place constraints on the high energy electron spectrum when B is low. Conversely, B is unlikely to be much higher than 200 µG (if our assumptions about the gas density and radiation field are correct), or else the secondary e± that accompany the pionic γ-rays produce too much radio emission. These values for B are in accord with previous modeling for M82’s starburst (Persic et al. 2008; de Cea del Pozo et al. 2009a). The efficiency of proton acceleration can be small in the low B leptonic models, but when B 100 µG and p = 2.2, η is of order 0.1, the usual value assumed ≥ in models of star-forming galaxies. In our models, M82 is slightly more proton calorimetric than NGC 253, with F 10 50 % integrated over all energies. At the energies above 10 TeV relevant cal ≈ − 327 for synchrotron X-ray production, F is only 11% when t (3 GeV) = 10 Myr cal ∼ diff and 1% when t (3 GeV) = 1 Myr. ∼ diff Synchrotron X-rays – In Tables 6.6 and 6.7, we list the predicted X-ray synchrotron emission for a variety of model parameters. Overall, the results are similar to those for NGC 253: the diffuse synchrotron fraction varies from 0.4% to synch 37 −1 54% but has a fiducial value of 2% (L − = 9.7 10 ergs s ; assuming Γ 2.0, ∼ 2 8 × ≈ synch −7 L − 5 10 L ). For p = 2.2 models, the synchrotron fraction is typically 2 10 ≈ × TIR 15% for low B models with a high energy primary e± cutoff, 0.8 5% for high B ∼ − models, and a fraction of 1% for low B models with a low energy primary e± cutoff. Like NGC 253, most of the models at the high end of the range of synchrotron fraction have TeV luminosities that are > 1.5 times higher than observed. Interestingly enough, the spectral slope of synchrotron emission in the 2-8 keV energy range (Γ 1.98 2.34) also matches the observed diffuse hard X-ray emission ≈ − (Γ 2 3). ≈ − Inverse Compton X-rays – The IC fraction of the diffuse X-ray emission is largest (4 - 10%) in the B = 50 µG models and becomes smaller as B increases: 1.4 - 2.4% when B = 100 µG, 0.8 - 1.1% when B = 150 µG, 0.6% when B = 200 µG. This confirms the expectation in Strickland & Heckman (2007) that IC does not make up the observed diffuse hard X-ray emission in M82. Because of the poorly constrained synchrotron X-ray fraction, the synchrotron-to-IC ratio is anywhere 328 between 0.05 and 36, with a fiducial value of 2.3. We find that the IC emission has 1.23 Γ − 1.36, significantly harder than the observed spectral slope of the ≤ 2 8 ≤ diffuse hard emission in M82. Conclusion – The synchrotron fraction of M82’s diffuse hard X-ray emission is largely unknown, even with the GeV and TeV detections, but is of order 2% in our fiducial model. The unknown primary electron energy cutoff is one of the largest contributors to the uncertainty. IC is a minority (0.5 10%) of the diffuse hard − X-ray emission, with its contribution depending mainly on magnetic field strength. The synchrotron-to-IC ratio in our models is 2 in our fiducial model. ∼ Unlike NGC 253, foreground hard X-ray emission from the host galaxy has already been removed, and the extant studies with Chandra already take into account the bright point sources. However, the diffuse hard X-ray emission has a different morphology than the radio and IR disks (and therefore, the expected sources of CRs) with h 175 pc. Restricting attention to the radio disk would ≈ help studies of the nonthermal contribution. Furthermore, the IC contribution might be identified spectrally since it has a hard spectrum. Finally, the models with the largest synchrotron X-ray contributions are dominated by primaries; so large synchrotron fractions could be tested by looking for concentrations of X-ray emission near CR accelerators. 329 6.4.4. Arp 220 Introduction – Arp 220 consists of a starburst disk containing most of the gas and possibly most of the star-formation, and two intense but smaller starburst nuclei (Downes & Solomon 1998). While there is radio data for the individual starbursts, and the galaxy as a whole, there is no data on the starburst disk specifically. We therefore consider the nuclear starbursts seperately and ignore the surrounding disk. We adopt D = 79.9 Mpc from Sanders et al. (2003). The bolometric luminosity of Arp 220 is 1.6 1012 L (Sanders et al. 2003); if it is entirely due to star-formation, × ⊙ this would imply a supernova rate of 2 yr−1. Estimates of the bolometric luminosities of the nuclear starbursts vary, but we adopt 3 1011 L for each, based off × ⊙ Downes & Solomon (1998) and consistent with Sakamoto et al. (2008). These imply supernova rates of 0.34 yr−1 in each nucleus. We note that some authors argue that the bolometric luminosity of Arp 220’s western starburst is dominated by an AGN, based on its compactness, which would reduce the IR luminosity attributable to star-formation (Downes & Eckart 2007). On the other hand, the actual supernova rates may in fact be up to near 3 yr−1 in the western starburst and 1 yr−1 in the eastern starburst, based on the appearance of radio-bright supernovae (Lonsdale et al. 2006; earlier, Rovilos et al. 2005 found 0.7 yr−1). According to Lonsdale et al. (2006), the radio sources in Arp 220’s nuclear starbursts are concentrated into regions 100 pc wide, and the radio images of the nuclear starbursts indicate diameters ∼ of 80 - 120 pc. Following Sakamoto et al. (1999), we assume that the nuclear 330 starbursts are small disks, each with radius and scale height 50 pc. The radiation energy density in each nuclear starburst is then 1.6 105 eV cm−3 (B = 2.5 mG). × rad The gas mass in each disk is 109 M within a 100 pc radius of each (Downes & ∼ ⊙ Solomon 1998; Sakamoto et al. 1999), for gas surface densities of 7 g cm−2. Given −2 the uncertainties in the gas mass and distribution, we use Σg = 10 g cm . The magnetic field strength in Arp 220 is not known, but it is very large. Estimates range from the minimum energy value of 0.3 mG to 30 mG, the maximum allowed by hydrostatic balance (Thompson et al. 2006). Torres (2004b) found that B mG ≈ in the main disk, but B 5 mG in the starburst nuclei. We run a grid of models in ≈ B for values of 0.25 mG to 16 mG for each nucleus. Unlike the Galactic Center, NGC 253, and M82, Arp 220 has no γ-ray detection in either GeV or TeV bands. For a GeV upper limit, we use the 100 MeV ∼ ≥ flux limit for 5σ sources in the Fermi-LAT one year catalog (Abdo et al. 2010g). Albert et al. (2007a) provide upper limits in the VHE range for Arp 220 (specifically, an integrated photon flux from 0.36 to 1.8 TeV). To keep the parameter space manageable, the CR proton acceleration efficiency η is assumed to be a standard value of 0.1. The X-ray emission of Arp 220 has been studied by Chandra and XMM-Newton (Iwasawa et al. 2001; Clements et al. 2002; McDowell et al. 2003; Iwasawa et al. 2005). XMM-Newton has detected a Fe K line, suggesting at least some thermal emission (Iwasawa et al. 2005); but there is also a hard X-ray continuum of unknown 331 origin. There are several diffuse components, including a nuclear source roughly cospatial with the western nuclear starburst (Arp 220 X-1), a “hard halo” with a radius of a few hundred parsecs, a circumnuclear halo extended on kiloparsec scales, and vast “plumes” stretching out 10 kpc from the galactic center (Clements et al. 2002; McDowell et al. 2003). The central unresolved source Arp 220 X-1 is approximately coincident with the western nucleus, with an unabsorbed luminosity estimated at 4 1040 ergs s−1 (Clements et al. 2002). A second X-ray source, Arp × 40 −1 220 X-4 (L − 1.5 10 ergs s , after correcting for absorption), is tentatively 2 10 ≈ × identified with the eastern nucleus, although the geometry does not exactly match the radio morphology. For this paper, we assume that Arp 220 X-1 is the western starburst and Arp 220 X-4 is the eastern starburst; in Table 6.8 and 6.9, we list the (unabsorbed) 2 - 10 keV synchrotron luminosities so they can be compared with other values for the luminosity. Modelled Radio and γ-rays – We show an example radio spectrum for the western and eastern nuclei in Figure 6.12. Secondaries dominate the GHz synchrotron emission for B 4 mG, while primaries dominate for the lower magnetic field ≥ strengths. As with M82 and NGC 253, the radio data for each nucleus appears to be flatter than predicted in our models. The discrepancy at 10 20 GHz becomes ∼ − larger in low B models, since we are then probing the higher energy (more IC and synchrotron cooled) parts of the e± spectrum. However, it is possible that a thermal emission component flattens the radio spectra. The thermal dust contribution is 332 expected to be important only at 100 GHz and above (Downes & Solomon 1998; Torres 2004b). Furthermore, the 15 GHz data points appear systematically high for both the west and east nuclei, and are high even in the models of Torres (2004b). As Torres (2004b) note, the radio data use different beam sizes at different frequencies. Finally, it is possible that bremsstrahlung and ionization are even stronger than expected. In order for advective escape to be strong enough to flatten the spectrum, wind escape times would have to be < 1000 yr (compare eqns. 6.5 and 6.12), ∼ requiring speeds of 104 km s−1, which is unlikely. ≫ The non-detection of either nuclear starburst by Fermi already implies that B 250 µG in each of them, otherwise the leptonic emission alone would be ≥ observable. Note that we applied the GeV constraint to each nucleus independently; considering them together would push the lower limit on B towards 500 µG. ∼ With more data, Fermi will be able to constrain the leptonic γ-ray emission further and increase the lower limit on B. Furthermore, note that in the B = 500 µG and B = 1 mG models, we require that more power goes into CR electrons than protons. This is unlikely, considering that the reverse situation holds for the Milky Way. In all of our models, the nuclear starbursts of Arp 220 are proton calorimeters, with F = 0.84 0.98. Even at energies above 10 TeV, where diffusion is strongest, cal − F 0.6 when t (3 GeV) = 1 Myr. cal ≈ diff 333 Synchrotron X-rays – In the extremely dense gas of Arp 220, the CR protons at 10 - 100 TeV are efficiently converted into pionic products, including secondary e± and γ-rays. In turn, the extreme FIR radiation field converts 10 - 100 TeV γ-rays into pair e±. Finally, the secondary and tertiary e± emit prodigious synchrotron X-rays (see the e± spectra in Figure 6.13). Models with p 2.0 and slow diffusive escape ≈ can explain most of the observed X-ray emission from the nuclei as synchrotron. Without any significant TeV γ-ray constraints on p, we cannot rule out these models: it is possible that all of the hard X-ray emission from Arp 220 is synchrotron (though, see the later caveats about X-ray absorption). For p = 2.2, the allowed synchrotron fractions are 0.9 42% (fiducial: 15%) − for the western starburst and 2 84% (fiducial: 34%) for the eastern starburst − (Figure 6.14). In high B models ( 4 mG) with p = 2.2, the synchrotron fraction ∼ is 10% for the western starburst and 30% for the eastern starburst, consistent ∼ ∼ with previous expectations that HMXBs dominate the hard X-ray emission of starbursts. However, synchrotron is not so minor that it can be ignored, especially considering the uncertainties. Lower B models have a larger synchrotron fraction if prim 9 prim 6 γmax = 10 and a smaller fraction if γmax = 10 . At still higher p = 2.4, synchrotron X-ray emission is reduced to a few percent of the observed diffuse X-ray flux in each starburst. We find that X-ray synchrotron emission in Arp 220’s nuclear starburst is harder than in the other starbursts, with Γ − 1.84 2.23. The hardest synchrotron 2 10 ≈ − 334 spectra arise in low B models, where the Klein-Nishina bump is very prominent. With B greater than in M82 and NGC 253, lower electron energies are being probed (grey shading in Figure 6.13), where the cutoff in the proton spectra does not matter as much. To compare, the spectral slope of the diffuse hard X-ray emission in Arp 220 is not well constrained observationally, but is probably harder than synchrotron emission: Clements et al. (2002) measured Γ = 1.4 1, while Iwasawa et al. (2001) ± measured Γ = 1.7 for the total X-ray emission of Arp 220, and Iwasawa et al. (2005) +0.4 measured Γ = 1.2−0.7, though for a Galactic absorption column. Inverse Compton X-rays – The contribution of Inverse Compton emission depends on the magnetic field strength. The contributions are highest in the low B models: for B = 500 µG, IC is 1.6 - 2.8 times the unabsorbed 2 - 10 keV luminosities in the western starburst and 3.1 - 5.7 times the eastern starburst’s luminosity. However, the actual X-ray absorption is uncertain, so these cannot be said to constrain the magnetic field with any certainty. At the other extreme of B = 4 mG, IC makes up 6 7% (western) and 12 14% (eastern) of the unabsorbed 2 - 10 ∼ − − keV luminosity. Thus, IC emission in our models can be much more inefficient than usually expected (c.f. Iwasawa et al. 2001). This is because B is allowed to be much higher than the usual minimum-energy estimate; with the higher magnetic field strengths, IC is much weaker when scaling from the synchrotron radio emission. The synchrotron luminosity can be anywhere from 0.05% to 57 times the IC emission in Arp 220’s western nucleus and 0.06% to 35 times the IC emission in Arp 335 220’s eastern nucleus, an even greater range than in the other starbursts because p is not yet constrained by TeV γ-ray observations. Restricting ourselves to p = 2.2 models narrows the range (0.4 240% for the western nucleus; 0.6 280% for the − − eastern nucleus), with fiducial values for the synchrotron-to-IC ratio of 2.3 for the western nucleus and 2.7 for the eastern nucleus. Like the other starbursts, IC emission is predicted to be harder than synchrotron emission, with Γ − 1.30 1.69, which is a closer match to the observed X-ray 2 10 ≈ − emission. It is possible, then, that Arp 220 has low magnetic fields of less than a milliGauss and its X-ray emission is IC emission. However, both HMXBs and thermal emission are also expected to have the same spectral shape and dominate the X-ray spectrum (Persic & Rephaeli 2002; Iwasawa et al. 2005). Conclusion – Arp 220 is extremely efficient at converting VHE CR proton energy into synchrotron X-rays. Without the TeV γ-ray detections that exist for the Galactic Center, NGC 253, and M82, we cannot rule out hard spectra extending to very high energies. Therefore, it is entirely possible to generate or even exceed the observed absorption-corrected X-ray luminosities of Arp 220 X-1 and Arp 220 X-4 (assumed to be the western and eastern nucleus, respectively). Therefore, if the unabsorbed luminosities are correct, we can constrain p = 2.0 electron spectra extending to multi-TeV energies based on the X-rays alone. Even in p = 2.2 models, synchrotron can easily be 10% or more of the X-ray luminosities of each starburst. ∼ 336 A large uncertainty in our models of Arp 220 is the amount of hydrogen absorption. A surface density of 10 g cm−2 implies N 6 1024 cm−2, which is H ≈ × Compton thick. If this is correct, then the true X-ray luminosity of Arp 220 may be much higher. By contrast, the X-ray emission indicates hydrogen column depths of N 3 1022cm−2 (Σ = 0.05 g cm−2; Clements et al. 2002). It is possible, however, H ≈ × g that CR e± emitting synchrotron X-rays are not cospatial with the nuclear starbursts themselves, either instead diffusing out of the nuclei (though this is unlikely given the extremely rapid losses in the extreme environments; V¨olk 1989; Condon et al. 1991a) or are pair-produced from γ-rays outside the nuclei, in which case hydrogen absorption may be relatively small. It is also possible that the covering fraction of the hydrogen is less than unity, in which case, some X-ray emission will be able to escape. Hydrogen absorption will also affect other explanations of the diffuse X-ray flux in Arp 220, such as IC emission and high-mass X-ray binaries. Indeed, it might explain why Arp 220 seems underluminous in hard X-rays for its star-formation rate, as noted by Iwasawa et al. (2005) and supported by Lehmer et al. (2010). 337 6.5. Discussion 6.5.1. Synchrotron, the FIR-X-ray Correlation, and Submillimeter Galaxies Star-forming galaxies show a correlation between their FIR luminosities and −4 their 2 - 10 keV X-ray luminosities, with L − 10 L when AGNs do not 2 10 ≈ FIR contribute to the X-ray emission (David et al. 1992; Grimm et al. 2003; Persic et al. 2004; Persic & Rephaeli 2007; Lehmer et al. 2010). The correlation holds better for the most intense starbursts like ULIRGs, where there is less infrared and X-ray emission from older stellar populations (Persic & Rephaeli 2007). X-ray luminosity has often been used as a star-formation rate indicator (Ranalli et al. 2003; Grimm et al. 2003; Persic et al. 2004). The X-ray luminosity is often attributed to HMXBs, compact objects accreting matter from a massive companion. HMXBs are expected to trace the recent stellar population. Is it possible that synchrotron accounts for much of the X-ray luminosity in some galaxies? Our models show a trend, with increasing synchrotron X-ray emission with increasing gas density: synchrotron is a tiny fraction of the Galactic Center’s X-ray luminosity if primaries cut off at low energies, a small portion of M82’s diffuse hard X-ray luminosity, and possibly a substantial fraction of Arp 220’s hard X-ray luminosity. Three basic quantities affect the luminosity from secondary and pair e±: 338 (1) the amount of gas, which CR protons collide with to create pionic secondaries; (2) the amount of star-formation, which creates IR photons that pair produce off VHE γ-rays, and which is generally correlated with the amount of gas through the Schmidt Law; and (3) the physical size of the starburst. At the extreme end of synchrotron X-ray luminosity from secondary and pair e±, it is worth considering the diffuse synchrotron emission from submillimeter galaxies (SMGs), well-studied extreme starbursts at high redshift (z > 2). SMGs ∼ contain large amounts of gas, so they should be relatively efficient at converting energy in CR protons into γ-rays and secondary e±. Furthermore, they are much more extended spatially than the more compact starbursts observed at low redshift (Chapman et al. 2004; Tacconi et al. 2006; Genzel et al. 2008; Law et al. 2009; Younger et al. 2010). This means that both diffusive and advective escape will be less effective at transporting CR protons out of SMGs, simply because they have to travel farther, further supporting the case for proton calorimetry at multi-TeV energies (Lacki & Thompson 2010a). Finally, the radiation energy density in SMGs is very high, and because the SMGs are “puffy” with long sightlines through them, the optical depth to VHE γ-rays will be high. Therefore, we may expect that SMGs will be very efficient at converting energy in VHE CR protons into synchrotron synch −6 X-ray emission, like Arp 220, with LX > 10 LSF. We can evaluate how good SMGs are at creating secondaries with the estimates in 6.2.1. Tacconi et al. (2006) § find that SMGs have typical gas surface densities of Σ 0.4 g cm−2. Using a g ≈ 339 typical midplane-to-edge scale height h of 1 kpc, we find densities of n 40 cm−3, h i ≈ comparable to those found by Tacconi et al. (2006). From eqn. 6.11, the typical pionic loss time in SMGs is t 1.3 Myr. Eqn. 6.12 gives us advective loss times pion ≈ of t 3.0 Myr (h/kpc)(v /300 km s−1)−1. The ratio of these times suggests wind ≈ wind that SMGs are proton calorimeters, depending on the unknown diffusive escape time and to a lesser extent on wind speeds (though they would have to be > 700 km s−1 ∼ to reduce Fcal significantly). To study this effect, we modeled generic galaxies of several gas surface densities −2 −2 from Σg = 0.01 g cm to 30 g cm . The magnetic fields were chosen to scale −2 0.7 as B = 6 µG(Σg/0.0025 g cm ) , in line with our previous work on the radio emission of star-forming galaxies (Lacki et al. 2010a). The star-formation rate was calculated using the Schmidt law in Kennicutt (1998). The interstellar radiation field was taken to consist of the CMB, a T = 10000 K greybody field corresponding to the unobscured starlight from young stars, and a greybody corresponding to dust-obscured starlight (40 K in the compact starbursts and 30 K in the puffy starbursts). Scale heights of h = 100 pc and 1000 pc were modeled. We chose prim η = 0.1 and varied ξ from 0.003 to 0.03; several values of tdiff (3 GeV) and γmax were tried. Our models give the ratio of both synchrotron and IC emission to the bolometric power from star-formation at 2-10 keV, and these are plotted in Figure 6.15. Both −4 components are always a minority of the standard 10 LSF from the FIR-X-ray 340 correlation. Both the synchrotron and IC fraction increase with Σg. IC shows less variation in its flux, since it mostly comes from the relatively well-constrained low energy CR electron spectrum, although LIC still varies by a factor of 3 35 for 2−10 − a given Σ when h = 100 pc (5 180 for h = 1000 pc). At the highest Σ , the IC g − g emission is 2% or less of the typical 2 10 keV luminosity for h = 100 pc (6% or less − −2 when h = 1000 pc), falling to < 0.1% for Σg = 0.01 g cm (blue shading). ∼ Synchrotron X-ray emission shows a much greater variation with different parameters (see grey shading), ranging from < 0.1% (assuming p = 2.4 or that ∼ Σg is small) to 24% of the 2 - 10 keV luminosity of typical starbursts. Models prim with high γmax are much more efficient synchrotron radiators at low Σg. At high Σg, however, the synchrotron fraction starts to converge, as CR proton energy is efficiently converted to pions and γ-ray photons are efficiently converted to e± pairs. synch For our fiducial values of p = 2.2 and tdiff (3 GeV) = 10 Myr, L2−10 approaches 4% of the total 2-10 keV luminosity in the densest starbursts when h = 100 pc (3% when h = 1000 pc). Synchrotron dominates IC for our standard parameters when −2 Σg > 0.1 g cm and h = 100 pc, which are typical of compact starbursts. Thus, we ∼ do not expect synchrotron emission to usually cause large deviations in the X-ray luminosity, although it is present, and at greater levels than IC. As with the more compact starburst models, the synchrotron fraction of 2-10 keV emission of the puffy starburst models varies over many orders of magnitude with parameters. For our fiducial tdiff (3 GeV) = 10 Myr from compact starbursts, we 341 prim actually find that the synchrotron fraction is decreased in puffy starbursts when γmax is small. This is because the diffusive escape time is the same, while the density is 10 times lower at a given Σg; the lower pionic yield reduces the amount of secondary e± and the pionic γ-rays that seed pair e±. However, for a given diffusion constant, diffusion out of the larger SMGs takes longer. Setting tdiff (3 GeV) to 100 Myr restores the synchrotron fraction to near its compact starburst value. Comparing models with tdiff (3 GeV) = 100 Myr for h = 1000 pc and tdiff (3 GeV) = 10 Myr for h = 100 pc, and γprim = 106, there is a 50% enhancement in the synchrotron max ∼ emission for Σ = 0.03 0.3 g cm−2, because the γγ optical depth is greater along g − the long sightlines in puffy starbursts. Given the uncertainties in the parameters and the small contribution of synchrotron at these energies, this is not a big effect. An interesting effect that occurs in our h = 1000 pc models is that the IC fraction actually increases. Unlike the e± at multi-TeV energies, the e± responsible for 2-10 keV IC emission are relatively low energy ( 100 MeV 1 GeV, emitting ∼ − synchrotron at GHz frequencies). In compact starbursts, these e± are cooled ∼ significantly by bremsstrahlung and ionization losses, suppressing their IC emission. In puffy starbursts with the same Σg, however, the gas volume density is lower, so that bremsstrahlung and ionization losses are not as important. This means that there is more power to go into IC (and possibly synchrotron) losses at these energies, enhancing the X-ray IC (and possibly GHz synchrotron) emission in puffy starbursts relative to compact starbursts (see also Lacki & Thompson 2010a). 342 In fact, intense starbursts like SMGs, have relatively weak X-ray emission compared to other starbursts, if they lack an AGN, with L 10−4L (Alexander X ≈ SF et al. 2005). This suggests that diffuse synchrotron accounts for a few percent of the total hard X-ray emission from SMGs, and not just the diffuse hard X-ray emission. 6.5.2. What Neutrinos and TeV γ-rays Can Tell Us There are many uncertainties in our models, leading to a broad range in the synchrotron X-ray luminosities. These include the hardness of the e± spectrum, the strength of B which sets the importance of secondary e±, and the efficiency of pionic losses at multi-TeV energies. The synchrotron X-ray emission is enhanced significantly in starbursts where pionic secondary e± and pair production e± are produced at a high rate, especially if p 2.0. A necessary byproduct of pionic ≈ e± and γ-rays is pionic neutrinos. Starbursts that are bright in neutrinos should therefore be relatively bright in synchrotron X-rays. Of course, the converse is not necessarily true if the synchrotron X-ray luminosity comes largely from primary e±. TeV γ-rays are also generated by both high energy protons and electrons in the intense environments of starbursts. While 10 100 TeV γ-rays are heavily ∼ − absorbed by pair-production, there is a window at lower energies below a few TeV (see Figure 6.2). Indeed, TeV γ-rays are the most powerful constraint yet on synchrotron X-ray emission from NGC 253 and M82. 343 M82 is in the Northern hemisphere and visible to IceCube. In Figure 6.16, we see that the predicted neutrino spectrum of M82 varies widely between different models. Models with low p and high tdiff naturally predict that M82 is very bright in VHE neutrinos: the more protons at high energies, the more neutrinos are produced. The magnetic field strength B is also linked to the neutrino flux through the radio and γ-ray observations. In models with high B, the observed radio flux is mostly secondary e±, and the accompanying pionic γ-rays make up almost all of the detected GeV to TeV γ rays (as seen in Figure 6.11). The γ-ray flux therefore translates into a high neutrino flux. However, in models with low B, the observed radio flux is mostly primary e±, and leptonic emission makes up a majority of the GeV (and possibly TeV) γ-ray flux. In order not to overproduce the γ-rays, the CR protons must be less efficiently accelerated, in turn implying lower neutrino flux. The muon neutrino flux of M82 at 10 TeV is anywhere between 10−14 and 10−12 TeV cm−2 s−1 for p = 2.0 2.2. Unfortunately, M82 is at high declination − (sin δ 0.93), where IceCube’s sensitivity is relatively weak (Abbasi et al. 2009a). ≈ Even with a full year of IceCube-80, our most optimistic models predict that M82 is 5 times too faint to be observed. NGC 253 would be even fainter, and since ∼ it is located in the Southern Hemisphere sky (sin δ 0.43), the available existing ≈ − neutrino detectors are far less sensitive (Abbasi et al. 2009b; Coyle 2010; Abbasi et al. 2011). However, the diffuse neutrino background from starbursts may be detectable by IceCube, which may shed some light on the CR spectrum at high 344 energy (Loeb & Waxman 2006). Stacking searches of nearby, bright starbursts may also lead to a detectable signal (Lacki et al. 2011). Arp 220 is much fainter from Earth than M82, far beyond the reach of neutrino detectors like IceCube, but currently has no TeV γ-ray detection (the current upper limit is set by MAGIC in Albert et al. 2007a). Thus, high p is allowed by the data. Just as TeV γ-rays constrained M82, TeV γ-rays can constrain the fraction of Arp 220’s X-ray emission that comes from synchrotron. In Figure 6.17, we show the predicted VHE γ-ray flux of Arp 220’s western and eastern nuclear starbursts in several models compared to the expected 5σ sensitivity of VERITAS and Cherenkov Telescope Array (CTA) after 50 hours of integration time (Doro 2009). VERITAS will not be able to detect Arp 220 at 5 σ except in our most prim optimistic models with high γmax , low B, and p = 2.0 (Figure 6.17, lower panel). However, if B is low, the proposed CTA will be able to detect Arp 220 except in prim more pessimistic models with p = 2.2 and small γmax . If B is high, even CTA will have trouble detecting Arp 220’s nuclear starbursts (Figure 6.17, upper panel), especially if p = 2.2. We note, though, that our fiducial supernova rate for both of these starbursts (0.7 yr−1) is about one-third of that expected if the entire TIR luminosity of Arp 220 is due to star-formation (expected supernova rate of 2 yr−1). Thus, the total TeV luminosity of Arp 220 may be 3 times larger, in which case ∼ CTA will be able to detect it even in high B cases. Both the synchrotron X-ray flux and the TeV γ-ray flux are insensitive to the diffusive escape time: the protons are 345 being efficiently trapped and creating secondaries at TeV energies. The effects of pair production absorption are clearly visible in Figure 6.17: the γ-ray flux should plummet at energies higher than a TeV, an effect first predicted for Arp 220 by Torres (2004b). 6.6. Conclusion Starbursts are luminous in hard X-rays and accelerate large amounts of CRs. We have explored whether CR e± in starbursts could generate the observed diffuse hard X-ray emission through synchrotron emission. The strength of this emission depends on the very poorly constrained 10 - 100 TeV CR e± spectrum in starburst galaxies. Using one-zone models to predict this emission, we also reconsider the previously suggested contribution from IC emission of 0.1 - 1 GeV CR e±, in light of the magnetic field strengths expected in starburst galaxies. Our conclusions are as follows: We have considered the energetics for synchrotron X-ray emission in section 6.2 • with simple order of magnitude estimates. The diffuse hard X-ray emission of M82 could be synchrotron if CR escape is not important at PeV energies and the CR spectrum is hard enough (p 2.0). Much of this emission ≈ would come from the previously neglected pair-production e±, which are 346 efficiently generated from pionic γ-rays in the intense IR bright environments of starbursts. We have also constructed one-zone models of several starbursts with standard • spectra to compare with observations in radio and γ-rays. The magnetic field strength in these models is limited at the low end by constraints on leptonic γ-ray emission and at the high end by constraints on radio emission from secondary e±. For the physical conditions we assume (listed in Table 6.1), these limits are 50 µG < B < 150 µG in NGC 253, 50 < B < 200 µG in M82, ∼ ∼ ∼ ∼ and 0.5 mG < B < 4 mG for Arp 220’s nuclei. In low B models there is a ∼ ∼ large component of primary electrons, while in high B models the secondary e± dominate at GeV energies in M82 and Arp 220’s nuclei. The existing γ-ray data of the modelled starbursts rules out most p = 2.0 • models, and we find that the synchrotron emission is probably a minority of the hard X-ray emission in the Galactic Center, NGC 253, and M82 with our assumptions. Nonetheless, there is still enough variation in the parameters for synchrotron to make up from less than one to several tens of percent of the synchrotron emission. The synchrotron X-ray emission is highest in low B prim models with large γmax , lower in high B models, and lowest in low B models prim prim with small γmax . In our fiducial models (with high B and large γmax ), the fraction of unresolved hard X-ray emission contributed by synchrotron is 6% in the Galactic Center, 9% in NGC 253, 2% in M82, 15% in Arp 220’s western 347 nucleus (Arp 220 X-1), and 34% in Arp 220’s eastern nucleus (Arp 220 X-4). These results are consistent with previous predictions that X-ray binaries contribute most of the X-ray emission in starburst galaxies at a few keV (Persic & Rephaeli 2002; Persic et al. 2004). We find that the 2 - 10 keV X-ray synchrotron emission is 10−7 10−6L in these fiducial models; in models of − IR generic star-forming galaxies it peaks at 4 10−6L for p = 2.2.8 ∼ × IR Pair production is predicted to contribute significantly to the high energy • e± population in NGC 253, M82, and Arp 220, with pair e± comparable to the pionic e±. The huge densities of IR starlight photons efficiently convert 10 100 TeV γ-rays into e± pairs. These pairs then can efficiently radiate ∼ − X-ray synchrotron. However, the actual density of pair e± is directly related to γ-ray emissivity at very high energies, which in turn depends strongly on the number and escape times of CR protons at these energies. The Galactic Center region is transparent to VHE γ-rays, so that pair e± are negligible. We find that Inverse Compton emission in the 2 - 10 keV band is also a • minority of the X-ray emission. In our models, the magnetic field energy density can be much greater than the minimum energy estimate. This means the IC emission expected by scaling the GHz radio emission is lower than in previous estimates. In our fiducial models, the fraction of unresolved X-ray 8For comparison, according to the observed FIR-radio correlation, the GHz radio luminosity is −6 10 LIR (Yun et al. 2001). ∼ 348 emission contributed by IC is 0.1% in the Galactic Center, 3% in NGC 253, 1% in M82, and 10% in Arp 220’s nuclei. The IC emission at 2 - 10 keV is ∼ much harder (Γ − = 1.2 1.7) than the synchrotron emission at the same 2 10 − energies (Γ − = 1.8 2.5). 2 10 − Key uncertainties include the maximum energy of primary electrons (γprim), • max the rate of escape of CR protons at high energies (tdiff ), and the spectral slope of CRs (p). However, some of these quantities can be constrained by future neutrino telescopes, which can determine the pionic luminosities of starbursts at TeV to PeV energies. Future TeV γ-ray telescopes like CTA can constrain the electron spectrum at high energies as well, especially in Arp 220 which has relatively weak TeV upper limits. The effects of hydrogen absorption in dense starbursts also need to be explored, although this applies to IC emission and other sources of the X-ray flux, not just synchrotron. The large X-ray foregrounds from HMXBs and other sources makes detection of the synchrotron X-rays difficult. However, if the synchrotron emission can be detected, it will have important implications for our understanding of CRs in starbursts. Very little is known about the CR population at very high energies in starbursts, such as what the escape rate is or whether there is a “knee” in the CR spectrum as there is in the Milky Way. Combined with upcoming neutrino 349 measurements, detection of synchrotron X-rays will extend our understanding from the directly observed TeV energies. Even in the Milky Way, we know relatively little about the CR electron population at multi-TeV energies (Kistler & Y¨uksel 2009), and the propagation of electrons at 100 GeV TeV energies is still poorly understood (Chang et al. 2008; − Adriani et al. 2009; Abdo et al. 2009c; Aharonian et al. 2009a). It is possible that we are underestimating the primary population significantly. We assumed that the primary CR electron injection spectrum is a power law extending to TeV-PeV energies, with one source such as supernova remnants dominating the primary electron spectrum. An additional hard primary component (from PWNe, for example) on top of a softer component that explains the radio synchrotron emission would increase the synchrotron X-ray emission. However, a second component such as PWNe would have to be very hard (p < 2.0) to enhance the X-ray flux ∼ significantly; otherwise IC emission from the low energy e± may exceed the observed γ-rays. If a significant fraction of the hard X-ray emission from starbursts is synchrotron, it may be noticeably polarized. Although the radio synchrotron emission of M82’s halo is strongly polarized, the starburst core itself has very little polarization (Reuter et al. 1994). This may be because of Faraday depolarization, in which the back of the starburst is strongly Faraday rotated with respect to the front. Faraday depolarization, however, will be completely unimportant at X-ray frequencies. The 350 low polarization could also arise if the magnetic field was turbulent and isotropic on small scales, in which case the X-ray emission will also be unpolarized. However, even turbulent magnetic fields can be made anisotropic through shear and compression (e.g., Laing 1980; Sokoloff et al. 1998). Furthermore, there is observational evidence of anisotropic magnetic fields in M82 from submillimeter and infrared polarization measurements (Greaves et al. 2000; Jones 2000). Siebenmorgen & Efstathiou (2001) and Seiffert et al. (2007) found that Arp 220 has a low infrared and submillimeter polarization, though their source apertures blend the two nuclear starbursts and the outlying starburst disk together. In any case, the diffuse hard X-ray emission of M82 only has a flux of 0.1 0.2 milliCrab (Strickland & Heckman 2007; Kirsch et al. ∼ − 2005); only if it had large polarization could the Gravity and Extreme Magnetism Small Explorer (GEMS) detect its polarization (Swank et al. 2009). It is tempting to consider even higher energy synchrotron emission, in the 10 keV - MeV band, to directly probe 30 TeV - PeV e±. Presently, there are only relatively weak upper limits (and a claimed detection of NGC 253) of this emission of nearby starbursts and ULIRGs from the OSSE instrument on the Compton Gamma-Ray Observatory (Bhattacharya et al. 1994; Dermer et al. 1997), and a recent detection of M82 by Swift-BAT (Cusumano et al. 2010). With its much greater sensitivity in the 10 - 80 keV range over previous telescopes, NuSTAR can improve this situation (Harrison et al. 2005, 2010). However, above 10 keV, we ∼ generally expect the very hard IC emission to finally bury the softer synchrotron 351 emission, even if the CR electron spectrum does extend to multi-PeV energies (compare the synchrotron and IC emission in M82 in Figure 6.11). Furthermore, the angular resolution of NuSTAR is 40”, corresponding to 750 pc at the distance of M82, so point sources may contaminate the emission (Harrison et al. 2005, 2010). On the other hand, even a measurement of the IC emission itself could be useful in constraining the magnetic field strength. The synchrotron luminosity may remain within a factor of 2 of the IC until MeV energies in strong starbursts like Arp ∼ 220, if they have very strong magnetic fields (B > mG; as seen in Figure 6.14). 9 ∼ Even if the IC and point source emission was not a significant foreground, ultimately the synchrotron will be buried by bremsstrahlung and finally direct pionic γ-ray emission above 10 MeV. Since there is more power in lower energy protons than VHE protons, the synchrotron emission cannot exceed the pionic emission at higher energies. Thus, synchrotron cannot be used to probe the CR e± spectrum above about a PeV, unless an enormous new component of primaries is present. While we have considered mainly γ-rays from star-formation, γ-rays in starburst environments can come from other sources: most notably, an AGN. In particular, Sgr A⋆ in our own Galaxy is known to be a source of VHE γ-rays (Aharonian et al. 2004). It resides in the Central Cluster, a dense star cluster with a FIR energy density of 6000 eV cm−3 over a region 2 pc in diameter (Telesco et al. 1996; see ∼ ∼ 9Such MeV emission would be unaffected by hydrogen absorption as well, which would be ideal for an extremely dense starburst like Arp 220, although it is beyond the energy range of NuSTAR. 352 also (Davidson et al. 1992; Hopkins et al. 2010)). Interestingly, the VHE emission appears to have an exponential cutoff at 15 TeV (Aharonian et al. 2009b), similar to the Klein-Nishina threshold for pair production on the FIR light of a starburst. Diffuse hard X-rays are observed from the region near Sgr A⋆; these potentially could have a synchrotron contribution from pair e±. Similar considerations may apply to other central stellar clusters around VHE-emitting AGNs. 6.7. Appendix: Other Pionic Cross Section and Lifetime Parameterizations The spectrum of pionic secondary γ-rays and e± at TeV energies depends sensitively on the behavior of the pion production process at these energies (Karlsson 2008). The physics of pion production enters two ways in our models: (1) the lifetime of CR protons to pionic processes, which determines the steady-state CR spectrum and (2) the differential cross sections for production of pionic secondaries for protons of each energy. We run models where we vary the pionic lifetimes or the differential cross sections to consider the effects. We use the fiducial parameters listed in Table 6.1 for the Galactic Center, NGC 253, M82, and the two nuclei of Arp 220. We find the thermal fraction and ξ through radio spectrum chi-square fitting (or radio flux normalization for Arp 220) described in 6.3.3 for each variation on the pionic § 353 physics. We also compare the purely hadronic flux using the different assumptions of the pionic physics. Pionic lifetime variations – In most of our models, we calculate the pionic loss lifetime by integrating up the Kamae et al. (2006) cross-sections for γ-ray, neutrino, and e± production (equation 6.33). A commonly used pionic loss lifetime comes from Mannheim & Schlickeiser (1994) (MS94) and is given in equation 6.11. Finally, Schlickeiser (2002) recommends a different pionic loss lifetime, which is longer at low energies but substantially shorter at VHE energies: t = 2.2 108yr γ−0.28(n/cm−3)−1 (6.37) π × The new γ dependence comes from the multiplicity of pions produced per collision. Formally, this equation is only valid for γ < 3000, or below a few TeV. ∼ The Schlickeiser (2002) pionic lifetime predicts substantially less hadronic synchrotron X-ray emission, unless escape dominates the multi-TeV proton lifetimes. In our Arp 220 models, where pionic losses dominate even at these energies, the hadronic 2 - 10 keV synchrotron flux is only 1/4 of the predictions using the −0.28 cross-section derived tπ. This is because the γ in tπ becomes very small for 10 - 100 TeV CR protons, reducing the steady-state number of CR protons that are the source of secondary and pair e±. The secondary source functions do not correspondingly increase, so the pionic luminosity is underpredicted. The M82 354 model also gives only 64% of the hadronic synchrotron flux predicted with the cross-section integrated tπ. The TeV γ-ray flux is also reduced in these models by similar amounts. However, the synchrotron flux in the fiducial models of NGC 253 and the Galactic Center are affected by less than 10%, because diffusion sets the total number of protons in these models. Furthermore, the total synchrotron X-ray flux is barely affected in the Galactic Center, NGC 253, and M82, and only reduced to 40% of nominal in the Arp 220 nuclei models, because primary CR e± contribute heavily to the synchrotron emission. If we use the MS94 tπ, the hadronic synchrotron X-ray emission is instead slightly enhanced. In Arp 220 models, the synchrotron X-ray emission is 113% that using our standard pionic lifetime. The total synchrotron X-ray emission is significantly greater in MS94 models, because higher ξ are picked by our fitting processes: the synchrotron is 1.3 times greater in Arp 220 models using MS94 and 1.2 times greater in M82 models. There are negligible differences in the TeV γ-ray fluxes between the MS94 models and eqn. 6.33 models, but the GeV fluxes are only 80% as big in the M82 models and 70% as big in the Arp 220 models. The Schlickeiser (2002) pionic lifetime is only meant to be valid below 2.8 TeV, which is below the energy of the CR protons responsible for hard X-ray emitting e±. Furthermore, it is derived by integrating the energy of the pionic secondaries, much like we do to derive eqn. 6.33. Since we used the Kamae et al. (2006) cross sections in eqn. 6.33, which are explicitly valid up to several hundred TeV, we believe that 355 eqn. 6.33 is more likely to be accurate in deriving the X-ray synchrotron emission. Thus, the pionic lifetime would only contribute less than a factor of 2 uncertainty, ∼ prim much less than the uncertainties from γmax , p, and tdiff . Differential cross section variations – There are several possible parametrizations of the pionic cross sections. In addition to the Kamae et al. (2006) cross sections, the GALPROP cross sections (Moskalenko & Strong 1998; Strong & Moskalenko 1998; Strong et al. 2000) based on the work of Dermer (1986a) are commonly used (in turn based on Stecker 1970, Badhwar et al. 1977, and Stephens & Badhwar 1981; see also Dermer 1986b). Both the Kamae et al. (2006) and GALPROP cross sections are based on lower energy data, and Kamae et al. (2006) is only formally valid for CR protons up to 512 TeV. The pionic cross sections given in Kelner et al. (2006) are valid for CR protons at energies at 100 GeV to 100 PeV. We consider a variation where the Kamae et al. (2006) cross sections are used for CR protons at energies below 100 GeV, and the Kelner et al. (2006) cross sections are used above 100 GeV. Using the GALPROP cross sections reduces both the hadronic and total synchrotron X-ray luminosities. For the Galactic Center, the hadronic (total) GALPROP synchrotron X-ray luminosities are 67% (100%) the Kamae et al. (2006) luminosities; for NGC 253, they are 73% (83%) the Kamae et al. (2006) luminosities; for M82, the GALPROP synchrotron X-ray luminosities are 79% (75%) the Kamae 356 et al. (2006) luminosities; and for Arp 220’s nuclei, they are 105% (90%) the Kamae et al. (2006) luminosities. Smaller ξ are preferred using the GALPROP cross sections, reducing the leptonic contribution to the synchrotron X-ray emission. In the Arp 220 models, using the GALPROP cross sections leads to small increases of the hadronic γ-ray luminosities, about 15% at GeV and 20% at TeV energies. ∼ ∼ If we instead use the Kelner et al. (2006) cross sections for Ep > 100 GeV, we find even less of an effect. For the Galactic Center, the hadronic (total) Kelner et al. (2006) synchrotron X-ray luminosities are 84% (100%) the Kamae et al. (2006) luminosities; for NGC 253, they are 87% (98%) the Kamae et al. (2006) luminosities; for M82, they are 92% (97%) the Kamae et al. (2006) luminosities; and for Arp 220’s nuclei, they are 114% (110%) the Kamae et al. (2006) luminosities. The TeV γ-ray luminosity is 27% brighter in the Arp 220 models using these cross sections instead of our standard cross sections. Thus, using other parameterizations of the pionic cross sections does not seem to add even a factor of 2 uncertainty. ∼ 6.8. Appendix: Other Maximum Energy Proton Cutoffs Throughout this work, we have assumed that primary CR protons are p 6 accelerated to a maximum Lorentz factor γmax = 10 , or a maximum energy of 938 TeV. This is loosely based on the observed knee in the CR spectrum in the ∼ 357 Galaxy, but it is possible given current observational data that the knee is at a different energy in starburst regions. Since the average energy of pionic electrons is about 1/20 of the proton energy (e.g., Kelner et al. 2006), this translates to a pionic ± prim e cutoff near 50 TeV. Thus, models with low γmax or high B can be affected by the proton energy cutoff. To consider this effect, we rerun our fiducial models with p 5 6 7 γmax of 10 , 10 , and 10 . We consider both the total and the hadronic fluxes, as in Appendix 6.7. As in our fiducial models, we use the Kamae et al. (2006) cross sections for proton energies below 500 TeV and the Kelner et al. (2006) cross sections for proton energies above 500 TeV. p 5 In models where γmax = 10 , the hadronic synchrotron X-ray flux drops by almost an order of magnitude in most of the starbursts we consider. The hadronic synchrotron X-ray flux is only 9.4%, 8.5%, and 11% of its fiducial value in the Galactic Center, NGC 253, and M82 respectively. In Arp 220’s nuclei, the effect is not so severe since B is 40 times higher than in the other starbursts, and ∼ synchrotron X-rays probe e± energies that are 6 times lower (eqn. 6.1). However, ∼ the low energy proton cutoff still reduces the hadronic synchrotron X-ray flux to 43% of its fiducial value. The 2 - 10 keV hadronic synchrotron X-ray flux is also much softer in these models, because of the cutoff in the secondary e± spectrum: synch the hadronic Γ2−10 is 3.45, 3.29, 3.06, and 2.46 in the Galactic Center, NGC 253, M82, and Arp 220’s nuclei. The total synchrotron X-ray flux is affected less than prim 9 the hadronic flux, but note that we use γmax = 10 in our fiducial models so that 358 primaries dominate in most of the starbursts. Thus the total synchrotron X-ray flux is 99.7%, 90%, 64%, 58%, and 49% of the fiducial values in the Galactic Center, NGC 253, M82, Arp 220 West, and Arp 220 East, respectively. p 7 In models where γmax = 10 , the hadronic synchrotron X-ray flux in the 2 - 10 keV band is enhanced by 40%. The hadronic synchrotron X-ray flux is 141%, ∼ 141%, 136%, and 112% of its fiducial value in the Galactic Center, NGC 253, M82, and Arp 220’s nuclei respectively. The hadronic synchrotron X-ray spectrum is also synch hardened slightly: the hadronic Γ2−10 is 2.32, 2.25, 2.21, and 2.11 in the Galactic Center, NGC 253, M82, and Arp 220’s nuclei. The total synchrotron X-ray flux is affected very little, with total synchrotron fluxes 100.2%, 103%, 114%, 110%, and 112% of fiducial in the Galactic Center, NGC 253, M82, Arp 220 West, and Arp 220 East, respectively. We conclude that if the secondary and pair e± dominate at 10 - 100 TeV energies, the maximum proton energy can significantly affect the synchrotron X-ray flux and spectral slope, particularly if the cutoff is much lower than a PeV. 359 Fig. 6.1.— The photon densities of the radiation fields of the Galactic Center (short- dashed), NGC 253 (long-dashed), M82 (solid), and a starburst nucleus of Arp 220 (dotted). The FIR emission from dust dominates the photon population, except in the Galactic Center where the number of CMB photons is comparable. 360 Fig. 6.2.— The pair-production optical depths of the Galactic Center (short-dashed), NGC 253 (long-dashed), M82 (solid), and a starburst nucleus of Arp 220 (dotted), along a vertical sightline out of each starburst disk. Note that M82 and Arp 220 are opaque at tens of TeV. 361 Fig. 6.3.— Our predicted total (synchrotron + free-free) radio spectrum (red, solid) compared to the observations compiled in Crocker et al. (2011b). Open circles are fitted while Xs are not fitted. The synchrotron spectrum itself is the red long-dashed line, with components from primary and secondary e± are shown as red dotted and red short-dashed lines. The black dash-dotted line is our fit to the thermal free-free emission with these parameters. 362 Fig. 6.4.— The CR electron spectra of the Galactic Center region. Primary electrons prim 6 prim 9 are solid (γmax = 10 ) and long-dashed (γmax = 10 ), pionic secondaries are short- dashed (blue is e+, black is e−, and grey is total). Pair and knock-off electrons are insignificant. Blue and black shading denote the approximate e± energies that radiate in the 2 - 10 keV band through IC and synchrotron, respectively. The vertical line indicates the approximate energy that radiates synchrotron at 1.4 GHz. 363 Fig. 6.5.— The high energy photon spectra of the Galactic Center region. Synchrotron is red, bremsstrahlung is green, IC is blue, pionic is violet, and total prim 6 is black. Solid lines are the Earth-observed absorbed luminosities in the γmax = 10 prim 9 models. Dashed lines show γγ absorbed luminosities in γmax = 10 models. The TeV data points are from Aharonian et al. (2006b). 364 Fig. 6.6.— Our predicted total radio spectrum for the starburst core of NGC 253 (red, solid) compared to observations compiled in Williams & Bower (2010) (open circles). The line styles are the same as in Figure 6.3. The radio emission at high frequencies is overproduced by the large thermal component, suggesting that the synchrotron spectrum is intrinsically flatter than shown here and that escape, bremsstrahlung, or ionization are stronger and ξ is higher in reality. 365 Fig. 6.7.— The CR electron spectra of NGC 253’s core for models with p = 2.2, 6 prim 6 tdiff (3 GeV) = 10 yr, and η = 0.40. Primary electrons are solid (γmax = 10 ) and prim 9 + long-dashed (γmax = 10 ), pionic secondaries are short-dashed (blue is e , black is e−, and grey is total), knock-off electrons are dotted, and pair-production tertiaries prim 9 (using the γ-ray spectrum for γmax = 10 ) are dash-dotted. Note that pair production e± are comparable to pionic secondary e± at 10 TeV. Blue and black shading denote the approximate e± energies that radiate in the 2 - 10 keV band through IC and synchrotron, respectively. The vertical line indicates the approximate energy that radiates synchrotron at 1.4 GHz. 366 Fig. 6.8.— The high energy photon spectra of NGC 253’s core for models with 6 p = 2.2, tdiff (3 GeV) = 10 yr, and η = 0.40. Synchrotron is red, bremsstrahlung is green, IC is blue, pionic is violet, and black is total. The dotted lines are without prim 6 γγ absorption (γmax = 10 ), while solid are γγ absorbed according to the sightline prim 6 observed from Earth (γmax = 10 ). Dashed lines show γγ absorbed luminosities in prim 9 prim 9 γmax = 10 models. Note that the B = 50 µG model with γmax = 10 is not allowed by our constraints. Observed luminosities from from Fermi and HESS (triangles), as well as the diffuse emission from XMM-Newton (square) and total emission from BeppoSAX (star; Cappi et al. 1999), are plotted. X-ray luminosities are scaled to one ln bin in energy assuming Γ = 2.0. The effects of neutral hydrogen absorption are not shown on right. 367 Fig. 6.9.— Our predicted synchrotron radio spectrum (red, solid) compared to observations compiled in Williams & Bower (2010) (open circles) and the unfit data in Klein et al. (1988) (Xs). The residuals indicate the synchrotron spectrum is too steep at high frequencies, suggesting that escape, bremsstrahlung, or ionization are stronger and ξ is higher in reality. The line styles are the same as in Figure 6.3. 368 Fig. 6.10.— The CR electron spectra of M82 for models with p = 2.2, tdiff (GeV) = 107 yr, and η = 0.1. The line and shading styles are the same as Fig. 6.7. 369 Fig. 6.11.— The high energy photon spectra of M82 for models with p = 2.2, 7 tdiff (GeV) = 10 yr, and η = 0.1. The line and shading styles are the same as Fig. 6.8. Synchrotron emission can dominate IC emission at energies less than a few keV in high B models. Data plotted on right are Fermi and VERITAS (triangles), as well as the diffuse emission from Chandra (square) and total emission from BeppoSAX (Cappi et al. 1999), Suzaku (Miyawaki et al. 2009), and Swift (Cusumano et al. 2010) (stars). X-ray luminosities are scaled to one ln bin in energy assumingΓ=2.0. 370 Fig. 6.12.— Our predicted synchrotron radio spectrum (red, solid) compared to observations compiled in Torres (2004b). The open circles are for the western nucleus and the open squares are for the eastern nucleus. The line and shading styles are the same as in Figure 6.3. 371 Fig. 6.13.— The CR electron spectra of Arp 220 for models with p = 2.2 and 7 tdiff (3 GeV) = 10 yr. The line styles and colors are the same as in Figure 6.7. 372 Fig. 6.14.— The high energy photon spectra of Arp 220 for models with p = 2.2 7 and tdiff (3 GeV) = 10 yr. The line styles and colors are the same as in Figure 6.8. The effects of neutral hydrogen absorption are not shown. Observed absorption- corrected luminosities from Chandra are plotted, scaled to one ln bin in energy assuming Γ = 2.0. Synchrotron emission dominates IC emission at energies less than 10 keV in high B models. ∼ 373 Fig. 6.15.— Fraction of the bolometric flux from star-formation LSF in 2 - 10 keV synchrotron (black/grey) and IC (blue) emission. On the bottom, we show h = 100 pc; the top shows h = 1000 pc, as is the case for SMGs. The lines indicate p = 2.2, ξ = 0.01, and tdiff (3 GeV) = 10 Myr for compact starbursts and prim 6 tdiff (3 GeV) = 100 Myr for puffy starbursts. Solid is γmax = 10 , while dashed is prim 9 γmax = 10 . Shading indicates possible values for other values of those parameters. −4 For a typical star-formation hard X-ray luminosity L2−10 = 10 LSF, synchrotron and IC are subdominant; for optimistic choices of parameters, synchrotron can be −5 10 LSF. The dots represent the diffuse X-ray emission for the studied starbursts, while the stars represent the total (point sources included) X-ray emission that would be seen at large distances (from the Cappi et al. 1999 fluxes for M82 and NGC 253, scaled to our distances; from McDowell et al. 2003 for Arp 220). 374 Fig. 6.16.— The all-flavor neutrino (plus antineutrino) spectrum of M82 in various models with η = 0.10. Grey is p = 2.0 and black is p = 2.2; solid is tdiff (3 GeV) = 1 Myr, dashed is 10 Myr, and dotted is 100 Myr. The neutrino emission directly scales with η. 375 Fig. 6.17.— The Earth-observed pair production absorbed VHE γ-ray spectrum of prim 6 Arp 220 in various models, for γmax = 10 (black for p = 2.2 and grey for p = 2.0) prim 9 and γmax = 10 (dark blue for p = 2.2 and light blue for p = 2.0). The line styles are the same as in Fig. 6.16. 376 Parameter Units Galactic Center NGC 253 Core M82 Arp 220 West Arp 220 East Assumed Parameters B µG 50 - 100 50 - 400 50 - 400 250 - 16000 250 - 16000 −2 Σg g cm 0.003 - 0.1 0.10 0.17 10 10 h pc 42 50 100 50 50 R pc 112 150 250 50 50 a ℓ⊕ pc 112 150 250 50 50 D Mpc 0.008 3.5 3.6 79.9 79.9 −1 vwind km s 600 300 300 300 300 8 10 10 10 10 LTIR L 4 10 2 10 5.9 10 3 10 3 10 b ⊙ −1 × 36c × 38d × 39e × 40f × 40f LX (diffuse) ergs s 7.4 10 8.5 10 4.4 10 4 10 1.5 10 377 × × × × × Fiducial Parameters B µG 100 100 150 4000 4000 −2 Σg g cm 0.003 0.10 0.17 10 10 p 2.2 2.2 2.2 2.2 2.2 ··· tdiff (3 GeV) Myr 1 1 10 10 10 γprim 109 109 109 109 109 max ··· η 0.14 0.40 0.10 0.10 0.10 ··· ξ 0.0066 0.027 0.0077 0.021 0.0083 ··· S (1.0 GHz) Jy 370 0.63 1.3 therm ······ Fiducial Results L (synch)g ergs s−1 4.3 1035 7.2 1037 9.6 1037 6.0 1039 5.2 1039 X × × × × × Table 6.1. Model Parameters (cont’d) Table 6.1—Continued Parameter Units Galactic Center NGC 253 Core M82 Arp 220 West Arp 220 East Γ (synch)g 2.12 2.08 2.15 2.12 2.12 ··· L (IC)g ergs s−1 9.0 1033 2.7 1037 4.1 1037 2.6 1039 1.9 1039 X × × × × × Γ (IC)g 1.58 1.35 1.31 1.44 1.44 ··· δ˜ 84 59 51 18 48 ··· F 0.0037 0.15 0.33 0.97 0.97 cal ··· F ( 10 TeV) 6 10−4 0.016 0.11 0.93 0.93 cal ≥ ··· × a 378 Adopted sightline distance through starburst to Earth, for computing observed TeV γ-ray spectrum. bAdopted diffuse hard X-ray emission, with which synchrotron and IC are compared. cGalactic Center luminosity in 2 - 10 keV X-ray band from Koyama et al. (1996), as extrapolated to ℓ 0.8◦ | | ≤ and b 0.3◦ assuming a constant surface brightness. | | ≤ dNGC 253 disk diffuse 2 - 10 keV X-ray luminosity from Bauer et al. (2008). Note that this includes the outlying regions of the galaxy and not just the starburst core. eLuminosity of diffuse hard X-ray excess in 2 - 8 keV band from Strickland & Heckman (2007). The uncertainty is 0.2 1039 ergs s−1. × f Absorption-corrected Arp 220 X-ray luminosities from Clements et al. (2002). We assume that Arp 220 X-1 is the western nucleus and Arp 220 X-4 is the eastern nucleus. g Synchrotron and IC luminosities and photon indexes, for the hard X-ray energy bands given for LX (diffuse) (2 - 8 keV for M82, 2 - 10 keV for the other starbursts). synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 −2 Σg = 0.003 g cm 50 2.4 0.0088 ∞ ··· 0.071 ··· 0.10 0.037 1.1e33 1.5E-4 0.029 2.49 0.90 0.63 10 0.0088 ··· 0.10 ··· 1 0.0088 ··· 0.14 ··· 100 2.0 0.0088 0.0049 4.6e33 6.3E-4 0.64 2.35 0.40 0.70 379 ∞ 0.0125 0.0049 6.5e33 8.9E-4 0.91 2.35 0.41 0.99 0.0250 0.0049 1.3e34 0.0018 1.8 2.35 0.43 2.0 10 0.0125 0.0049 1.9e33 2.6E-4 0.27 2.41 0.41 0.54 0.0250 0.0049 3.9e33 5.2E-4 0.54 2.41 0.43 1.1 0.0354 0.0049 5.5e33 7.4E-4 0.76 2.41 0.45 1.5 1 0.0707 0.0050 1.5e33 2.1E-4 0.21 2.44 0.48 0.69 0.1000 0.0050 2.2e33 2.9E-4 0.30 2.44 0.53 0.98 0.2000 0.0049 4.3e33 5.9E-4 0.61 2.44 0.67 2.0 2.2 0.0088 ∞ ··· 0.025 0.0062 2.9e33 4.0E-4 0.32 2.39 0.28 0.70 0.035 0.0061 4.1e33 5.6E-4 0.46 2.39 0.31 0.98 prim 6 (cont’d) Table 6.2. Galactic Center X-ray luminosities when γmax = 10 Table 6.2—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 0.050 0.0061e 5.8e33 7.9E-4 0.65 2.39 0.35 1.4 0.071 0.0061 8.2e33 0.0011 0.92 2.39 0.42 2.0 10 0.013 ··· 0.035 0.0062 1.3e33 1.7E-4 0.14 2.45 0.31 0.57 0.071 0.0061 2.5e33 3.4E-4 0.28 2.45 0.41 1.1 0.10 0.0060 3.5e33 4.8E-4 0.40 2.45 0.50 1.6 1 0.035 ··· 0.14 0.0062 7.1e32 1.0E-4 0.079 2.49 0.56 0.56 380 0.28 0.0059 1.4e33 1.9E-4 0.16 2.49 0.92 1.1 2.4 0.10 0.011 1.6e33 2.1E-4 0.13 2.43 0.44 0.62 ∞ 0.14 0.011 2.2e33 3.0E-4 0.19 2.43 0.57 0.88 0.20 0.011 3.1e33 4.3E-4 0.27 2.43 0.76 1.2 10 0.14 0.011 7.0e32 1.0E-4 0.059 2.50 0.57 0.55 0.20 0.011 9.9e32 1.4E-4 0.085 2.50 0.75 0.77 −2 Σg = 0.01 g cm 100 2.0 10 0.0088 0.0052 4.5e33 6.1E-4 0.66 2.41 0.64 1.2 0.013 0.0052 6.4e33 8.7E-4 0.94 2.41 0.66 1.8 1 0.018 0.0053 1.3e33 1.7E-4 0.19 2.44 0.66 0.57 0.035 0.0053 2.5e33 3.5E-4 0.37 2.44 0.75 1.1 (cont’d) Table 6.2—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 0.050 0.0052 3.6e33 4.9E-4 0.53 2.44 0.82 1.6 2.2 0.0088 0.0065 3.3e33 4.5E-4 0.40 2.39 0.46 0.80 ∞ 0.013 0.0065 4.7e33 6.4E-4 0.57 2.39 0.50 1.1 0.018 0.0064 6.7e33 9.1E-4 0.81 2.39 0.55 1.6 10 0.0088 ··· 0.013 0.0065 1.5e33 2.0E-4 0.18 2.45 0.49 0.66 0.018 0.0065 2.1e33 2.8E-4 0.25 2.45 0.54 0.94 381 0.025 0.0064e 2.9e33 4.0E-4 0.35 2.45 0.61 1.3 0.035 0.0063 4.1e33 5.6E-4 0.50 2.45 0.71 1.9 1 0.0088 ··· 0.050 0.0066 8.4e32 1.1E-4 0.10 2.49 0.79 0.66 0.071 0.0064 1.2e33 1.6E-4 0.14 2.49 0.96 0.93 2.4 0.025 0.012 1.3e33 1.7E-4 0.12 2.43 0.54 0.51 ∞ 0.050 0.012 2.5e33 3.5E-4 0.24 2.43 0.80 1.0 10 0.035 ··· 0.050 0.012 8.2e32 1.1E-4 0.077 2.50 0.79 0.63 0.071 0.012 1.2e33 1.6E-4 0.11 2.50 1.0 0.90 (cont’d) Table 6.2—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 −2 Σg = 0.03 g cm 2.4 0.0088 0.014 1.3e33 1.7E-4 0.13 2.43 0.99 0.51 ∞ 382 aFraction of the observed diffuse 2 - 10 keV X-ray luminosity (scaled from Koyama et al. (1996) as described in 6.4.1) from that synchrotron emission accounts for. § bRatio of the 2 - 10 keV synchrotron and IC luminosities. cRatio of the predicted GeV γ-ray emission with the upper limit from EGRET in Hunter et al. (1997). dRatio of the predicted TeV γ-ray emission with the observed TeV flux observed from HESS in Aharonian et al. (2006b) and Crocker et al. (2011b). eThis model satisfies our constraints ( 6.3.3). While η is not the minimum or maximum allowed or the closest §prim 6 prim 9 to reproducing the TeV luminosity for γmax = 10 , this is true for γmax = 10 so we list it anyway. synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 −2 Σg = 0.003 g cm 50 2.4 0.0088 0.038 1.3e35 0.018 3.3 2.22 0.65 0.67 ∞ 0.071 0.038 1.3e35 0.018 3.4 2.22 0.84 1.1 0.10 0.038 1.3e35 0.018 3.4 2.22 0.93 1.2 10 0.0088 0.038 1.3e35 0.018 3.3 2.21 0.64 0.64 0.10 0.038 1.3e35 0.018 3.3 2.21 0.92 0.99 1 0.0088 0.040 1.3e35 0.018 3.4 2.21 0.62 0.64 0.14 0.040 1.3e35 0.018 3.4 2.21 0.98 0.76 100 2.0 0.0088 383 ∞ ··· 0.0125 ··· 0.0250 ··· 10 0.0125 ··· 0.0250 ··· 0.0354 ··· 1 0.0707 ··· 0.1000 ··· 0.2000 ··· 2.2 0.0088 0.0066 4.3e35 0.058 48 2.12 0.24 0.60 ∞ 0.025 0.0066 4.3e35 0.059 48 2.12 0.29 1.1 0.035 0.0066e 4.3e35 0.059 48 2.12 0.33 1.3 prim 9 (cont’d) Table 6.3. Galactic Center X-ray luminosities when γmax = 10 Table 6.3—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 0.050 0.0065 4.3e35 0.059 48 2.12 0.37 1.7 0.071 ··· 10 0.013 0.0066 4.3e35 0.058 48 2.12 0.25 0.56 0.035 0.0065 4.3e35 0.058 48 2.12 0.32 0.93 0.071 ··· 0.10 0.0065 4.2e35 0.058 47 2.12 0.51 2.0 1 0.035 0.0068 4.4e35 0.060 49 2.12 0.30 0.51 0.14 0.0066f 4.3e35 0.059 48 2.12 0.57 0.91 384 0.28 0.0064 4.1e35 0.056 46 2.12 0.93 1.5 2.4 0.10 0.012 4.8e34 0.0066 4.1 2.22 0.45 0.68 ∞ 0.14 0.011 4.8e34 0.0066 4.1 2.22 0.58 0.93 0.20 0.011 4.8e34 0.0065 4.1 2.22 0.76 1.3 10 0.14 0.011 4.7e34 0.0064 4.0 2.21 0.57 0.60 0.20 0.011 4.6e34 0.0063 3.7 2.21 0.75 0.82 −2 Σg = 0.01 g cm 100 2.0 10 0.0088 ··· 0.013 ··· 1 0.018 ··· 0.035 ··· (cont’d) Table 6.3—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 0.050 ··· 2.2 0.0088 0.0069 4.5e35 0.062 55 2.13 0.48 1.2 ∞ 0.013 0.0069e 4.5e35 0.062 55 2.13 0.51 1.5 0.018 0.0069 4.5e35 0.062 55 2.13 0.56 2.0 10 0.0088 0.0070 4.5e35 0.062 54 2.13 0.48 0.85 0.013 0.0070 4.5e35 0.062 54 2.13 0.51 1.0 0.018 0.0069e 4.5e35 0.061 54 2.13 0.56 1.3 385 0.025 0.0069 4.5e35 0.061 54 2.13 0.63 1.7 0.035 ··· 1 0.0088 0.0072 4.7e35 0.063 55 2.12 0.46 0.51 0.050 0.0069 4.5e35 0.061 54 2.12 0.80 1.0 0.071 0.0069 4.5e35 0.061 53 2.12 0.97 1.3 2.4 0.025 0.012 5.2e34 0.0070 4.8 2.23 0.55 0.56 ∞ 0.050 0.012 5.1e34 0.0070 4.8 2.23 0.80 1.1 10 0.035 0.012 5.1e34 0.0069 4.7 2.22 0.65 0.50 0.050 0.012 5.0e34 0.0068 4.7 2.22 0.80 0.69 0.071 ··· (cont’d) Table 6.3—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 −2 Σg = 0.03 g cm 2.4 0.0088 0.014 6.0e34 0.0082 6.4 2.27 1.0 0.58 ∞ 386 aFraction of the observed diffuse 2 - 10 keV X-ray luminosity (scaled from Koyama et al. (1996) as described in 6.4.1) from that synchrotron emission accounts for. § bRatio of the 2 - 10 keV synchrotron and IC luminosities. cRatio of the predicted GeV γ-ray emission with the upper limit from EGRET in Hunter et al. (1997). dRatio of the predicted TeV γ-ray emission with the observed TeV flux observed from HESS in Aharonian et al. (2006b) and Crocker et al. (2011b). eThis model satisfies our constraints ( 6.3.3). While η is not the minimum or maximum allowed or the closest §prim 9 prim 6 to reproducing the TeV luminosity for γmax = 10 , this is true for γmax = 10 so we list it anyway. f Chosen as our fiducial model. synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 50 2.0 0.025 0.092 2.1e37 0.024 0.37 2.30 1.2 0.64 ∞ 0.071 0.090 5.9e37 0.069 1.0 2.30 1.3 1.8 10 0.050 0.091 9.5e36 0.011 0.17 2.35 1.2 0.68 0.14 0.087 2.7e37 0.032 0.48 2.35 1.3 1.9 1 0.20 0.092 4.8e36 0.0057 0.084 2.37 1.3 0.66 0.57 0.081 1.4e37 0.016 0.24 2.37 1.6 1.8 2.2 0.050 0.15 9.0e36 0.011 0.10 2.33 1.0 0.53 ∞ 0.14 0.13 2.6e37 0.030 0.32 2.33 1.1 1.5 387 10 0.0088 ··· 0.013 ··· 0.10 0.14 4.2e36 0.0050 0.051 2.38 1.1 0.63 0.28 0.12 1.2e37 0.014 0.15 2.38 1.4 1.8 1 0.10 ··· 0.28 ··· 0.40 0.12 2.2e36 0.0026 0.027 2.40 1.4 0.67 0.57 0.11 3.1e36 0.0036 0.038 2.40 1.7 0.94 2.4 0.0088 ∞ ··· 0.050 ··· 0.20 0.34 4.8e36 0.0056 0.037 2.36 1.2 0.59 prim 6 (cont’d) Table 6.4. NGC 253 X-ray luminosities when γmax = 10 Table 6.4—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 0.57 0.23 1.3e37 0.016 0.13 2.36 1.9 1.7 10 0.0088 ··· 0.050 ··· 0.28 0.31 1.6e36 0.0019 0.013 2.42 1.3 0.54 0.57 0.24 3.2e36 0.0038 0.030 2.42 1.9 1.1 1 0.0088 ··· 0.050 ··· 0.57 388 ··· 100 2.0 10 0.14 0.028 4.3e37 0.050 2.2 2.33 0.53 1.8 1 0.14 ··· 0.20 0.029 7.8e36 0.0092 0.38 2.35 0.53 0.62 0.57 0.017 2.2e37 0.026 1.1 2.35 0.81 1.7 2.2 0.14 0.035 4.0e37 0.047 1.6 2.31 0.54 1.4 ∞ 0.20 0.030 5.7e37 0.067 2.3 2.31 0.66 2.0 10 0.14 0.036 9.8e36 0.012 0.38 2.37 0.52 0.85 0.28 0.024 2.0e37 0.023 0.81 2.37 0.79 1.7 1 0.20 ··· 0.40 0.025 3.6e36 0.0043 0.14 2.39 0.85 0.65 0.57 0.014 5.2e36 0.0061 0.21 2.39 1.1 0.92 (cont’d) Table 6.4—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 2.4 0.20 0.068 7.6e36 0.0090 0.23 2.34 0.62 0.56 ∞ 0.40 0.035 1.5e37 0.018 0.51 2.34 1.0 1.1 0.57 0.0064 2.2e37 0.025 0.80 2.34 1.4 1.6 10 0.28 0.057 2.7e36 0.0032 0.085 2.41 0.78 0.53 389 0.40 0.039 3.9e36 0.0046 0.13 2.41 1.0 0.75 0.57 0.012 5.5e36 0.0065 0.20 2.41 1.4 1.1 150 2.0 1 0.40 ··· 0.57 0.0016 2.7e37 0.032 2.5 2.33 0.63 1.7 2.2 0.20 0.0078 6.8e37 0.080 5.5 2.28 0.52 1.9 ∞ 10 0.20 0.0090 1.7e37 0.020 1.3 2.35 0.50 1.2 0.28 0.0014 2.4e37 0.029 2.1 2.35 0.66 1.7 1 0.40 9.7E-4 4.6e36 0.0054 0.35 2.38 0.72 0.64 (cont’d) Table 6.4—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 2.4 0.20 0.021 9.3e36 0.011 0.59 2.32 0.52 0.55 ∞ 0.28 0.0061 1.3e37 0.016 0.92 2.32 0.69 0.78 10 0.28 0.0083 3.5e36 0.0041 0.23 2.40 0.67 0.52 390 aFraction of the observed diffuse 2 - 10 keV X-ray luminosity (from Bauer et al. (2008)) from that synchrotron emission accounts for. bRatio of the 2 - 10 keV synchrotron and IC luminosities. cRatio of the predicted 0.3 - 10 GeV γ-ray emission with the flux detected by Fermi-LAT in Abdo et al. (2010g). dRatio of the predicted 220 GeV γ-ray emission with the observed TeV flux observed from HESS in Acero et ≥ al. (2009). synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 50 2.0 0.025 ∞ ··· 0.071 ··· 10 0.050 ··· 0.14 ··· 1 0.20 ··· 0.57 ··· 2.2 0.050 ∞ ··· 0.14 ··· 391 10 0.0088 0.18 3.6e38 0.43 3.9 2.08 1.0 2.0 0.013 0.18 3.6e38 0.43 3.9 2.08 1.0 2.0 0.10 ··· 0.28 ··· 1 0.10 0.17 3.5e38 0.41 3.8 2.03 1.1 2.0 0.28 0.14 2.9e38 0.34 3.5 2.03 1.3 2.0 0.40 ··· 0.57 ··· 2.4 0.0088 0.43 5.1e37 0.060 0.33 2.27 0.95 0.51 ∞ 0.050 0.41 5.0e37 0.059 0.34 2.27 1.0 0.61 0.20 0.34e 4.5e37 0.053 0.35 2.27 1.2 0.96 prim 9 (cont’d) Table 6.5. NGC 253 X-ray luminosities when γmax = 10 Table 6.5—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 0.57 0.23 4.1e37 0.048 0.38 2.27 1.9 1.9 10 0.0088 0.44 5.1e37 0.061 0.33 2.20 0.95 0.50 0.050 0.42 4.9e37 0.058 0.33 2.20 1.0 0.56 0.28 0.31e 3.8e37 0.045 0.31 2.20 1.3 0.88 0.57 0.24 3.1e37 0.037 0.29 2.20 1.9 1.3 1 0.0088 0.45 5.3e37 0.062 0.33 2.12 0.95 0.50 0.050 0.44 5.1e37 0.061 0.33 2.12 1.0 0.51 0.57 0.27 3.2e37 0.038 0.28 2.12 1.6 0.61 392 100 2.0 10 0.14 ··· 1 0.14 0.048 1.0e39 1.2 49 2.00 0.51 2.0 0.20 ··· 0.57 ··· 2.2 0.14 0.038 1.4e38 0.16 5.4 2.19 0.54 1.6 ∞ 0.20 ··· 10 0.14 0.039 1.1e38 0.13 4.3 2.14 0.53 1.1 0.28 0.025 8.5e37 0.10 3.5 2.14 0.80 1.9 1 0.20 0.040 1.1e38 0.12 3.9 2.08 0.56 0.59 0.40 0.027e,f 7.2e37 0.085 2.8 2.08 0.85 0.82 0.57 0.015 4.3e37 0.051 1.7 2.08 1.1 1.0 (cont’d) Table 6.5—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 2.4 0.20 0.068 1.9e37 0.022 0.56 2.29 0.62 0.61 ∞ 0.40 0.035 2.1e37 0.025 0.70 2.29 1.0 1.2 0.57 0.0064 2.3e37 0.027 0.84 2.29 1.4 1.6 10 0.28 0.058 1.2e37 0.014 0.37 2.26 0.78 0.57 393 0.40 0.039 1.0e37 0.012 0.33 2.26 1.0 0.78 0.57 0.012 7.4e36 0.0087 0.26 2.26 1.4 1.1 150 2.0 1 0.40 0.011 2.6e38 0.31 23 2.01 0.50 1.4 0.57 0.0024 8.1e37 0.095 7.4 2.01 0.63 1.7 2.2 0.20 0.0085 9.2e37 0.11 7.4 2.20 0.52 1.9 ∞ 10 0.20 0.0094 4.4e37 0.051 3.5 2.15 0.50 1.2 0.28 0.0016 2.9e37 0.034 2.4 2.15 0.66 1.7 1 0.40 0.0012 8.0e36 0.0094 0.61 2.10 0.72 0.64 (cont’d) Table 6.5—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 2.4 0.20 0.020 1.3e37 0.015 0.83 2.28 0.52 0.56 ∞ 0.28 0.0061 1.4e37 0.017 1.0 2.28 0.69 0.78 10 0.28 0.0084 5.0e36 0.0059 0.34 2.27 0.67 0.53 394 aFraction of the observed diffuse 2 - 10 keV X-ray luminosity (from Bauer et al. (2008)) from that synchrotron emission accounts for. bRatio of the 2 - 10 keV synchrotron and IC luminosities. cRatio of the predicted 0.3 - 10 GeV γ-ray emission with the flux detected by Fermi-LAT in Abdo et al. (2010g). dRatio of the predicted 220 GeV γ-ray emission with the observed TeV flux observed from HESS in Acero et ≥ al. (2009). eThis model satisfies our constraints ( 6.3.3). While η is not the minimum or maximum allowed or the closest §prim 9 prim 6 to reproducing the GeV luminosity for γmax = 10 , this is true for γmax = 10 so we list it anyway. f Chosen as our fiducial model. synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−8 f2−8 IC Γ2−8 fGeV fVHE µG (Myr) ergss−1 50 2.0 0.013 0.067 5.4e37 0.012 0.29 2.25 1.8 0.63 ∞ 0.035 0.066 1.5e38 0.035 0.83 2.25 1.9 1.8 0.035 0.066 2.5e37 0.0056 0.14 2.31 1.8 0.61 0.071 0.065 5.0e37 0.011 0.27 2.31 2.0 1.2 2.2 0.0088 ∞ ··· 0.025 ··· 0.035 0.10 3.3e37 0.0075 0.13 2.28 1.4 0.61 0.10 0.094 9.4e37 0.021 0.36 2.28 1.8 1.7 395 10 0.0088 ··· 0.071 ··· 0.10 0.096 1.6e37 0.0036 0.061 2.34 1.8 0.68 0.14 0.091 2.2e37 0.0051 0.087 2.34 2.0 0.96 1 0.0088 ··· 0.14 ··· 2.4 0.050 ∞ ··· 0.14 0.22 1.7e37 0.0040 0.047 2.31 1.9 0.56 10 0.14 ··· 100 2.0 0.0088 ∞ ··· 0.013 0.023 8.3e37 0.019 1.3 2.24 0.59 0.55 prim 6 (cont’d) Table 6.6. M82 X-ray luminosities when γmax = 10 Table 6.6—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−8 f2−8 IC Γ2−8 fGeV fVHE µG (Myr) ergss−1 0.035 0.022 2.4e38 0.054 3.7 2.24 0.67 1.6 10 0.0088 ··· 0.035 0.022 4.0e37 0.0092 0.63 2.30 0.65 0.57 0.10 0.018 1.1e38 0.026 1.8 2.30 0.84 1.6 1 0.0088 ··· 0.20 0.018 2.7e37 0.0062 0.41 2.32 0.93 0.54 0.57 0.0047 7.8e37 0.018 1.2 2.32 1.6 1.5 2.2 0.025 396 ∞ ··· 0.035 0.031 5.3e37 0.012 0.63 2.27 0.57 0.56 0.10 0.022 1.5e38 0.034 1.9 2.27 0.94 1.6 10 0.050 ··· 0.10 0.023 2.6e37 0.0060 0.32 2.34 0.89 0.65 0.28 7.5E-4 7.5e37 0.017 1.1 2.34 1.8 1.8 1 0.40 ··· 2.4 0.14 0.042 2.9e37 0.0065 0.28 2.31 1.1 0.52 ∞ 0.28 0.0081 5.7e37 0.013 0.64 2.31 2.0 1.0 150 2.0 10 0.10 0.0075 1.4e38 0.032 4.1 2.28 0.56 1.5 1 0.14 ··· 0.20 0.0067 3.4e37 0.0077 0.93 2.30 0.65 0.52 (cont’d) Table 6.6—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−8 f2−8 IC Γ2−8 fGeV fVHE µG (Myr) ergss−1 0.28 0.0036 4.8e37 0.011 1.3 2.30 0.81 0.74 2.2 0.071 0.0099 1.3e38 0.029 3.0 2.25 0.58 1.1 ∞ 0.10 0.0061 1.8e38 0.041 4.5 2.25 0.74 1.5 10 0.10 0.0071 3.3e37 0.0075 0.80 2.33 0.70 0.63 0.14 0.0020 4.6e37 0.011 1.2 2.33 0.91 0.89 397 2.4 0.14 0.0060 3.5e37 0.0079 0.74 2.29 0.97 0.51 ∞ 200 2.0 1 0.20 0.0021 3.8e37 0.0087 1.6 2.28 0.54 0.52 2.2 10 0.10 7.2E-4 3.7e37 0.0085 1.5 2.31 0.62 0.62 aFraction of the observed diffuse 2 - 8 keV X-ray luminosity (from Strickland & Heckman 2007) from that synchrotron emission accounts for. bRatio of the 2 - 8 keV synchrotron and IC luminosities. cRatio of the predicted 0.3 - 10 GeV γ-ray emission with the flux detected by Fermi-LAT in Abdo et al. (2010g). dRatio of the predicted 700 GeV γ-ray emission with the observed TeV flux observed from HESS in Acciari ≥ et al. (2009). synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−8 f2−8 IC Γ2−8 fGeV fVHE µG (Myr) ergss−1 50 2.0 0.013 ∞ ··· 0.035 ··· 0.035 ··· 0.071 ··· 2.2 0.0088 0.12 7.1e38 0.16 2.5 2.17 1.4 1.8 ∞ 0.025 0.11 6.9e38 0.16 2.5 2.17 1.5 2.0 0.035 ··· 0.10 ··· 398 10 0.0088 0.12 7.1e38 0.16 2.5 2.07 1.4 1.7 0.071 0.11 6.3e38 0.14 2.4 2.07 1.7 1.9 0.10 ··· 0.14 ··· 1 0.0088 0.13 7.5e38 0.17 2.6 2.01 1.5 1.7 0.14 0.11 6.3e38 0.14 2.3 2.01 1.8 1.6 2.4 0.050 0.28 1.0e38 0.023 0.22 2.26 1.5 0.52 ∞ 0.14 0.21 9.0e37 0.020 0.24 2.26 1.9 0.81 10 0.14 0.22 7.9e37 0.018 0.20 2.19 1.9 0.50 100 2.0 0.0088 0.037 2.3e39 0.52 35 2.12 0.62 1.9 ∞ 0.013 ··· prim 9 (cont’d) Table 6.7. M82 X-ray luminosities when γmax = 10 Table 6.7—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−8 f2−8 IC Γ2−8 fGeV fVHE µG (Myr) ergss−1 0.035 ··· 10 0.0088 0.037 2.2e39 0.51 35 2.03 0.61 1.7 0.035 0.034 2.1e39 0.48 33 2.03 0.69 2.0 0.10 ··· 1 0.0088 0.039 2.4e39 0.54 36 1.98 0.61 1.6 0.20 0.028 1.7e39 0.39 26 1.98 0.96 1.7 0.57 0.0071 5.1e38 0.12 7.6 1.98 1.6 1.8 2.2 0.025 0.034 2.9e38 0.067 3.5 2.20 0.52 0.65 399 ∞ 0.035 0.033e 3.0e38 0.068 3.6 2.20 0.58 0.80 0.10 0.024 3.3e38 0.074 4.1 2.20 0.94 1.8 10 0.050 0.032 2.5e38 0.057 3.0 2.13 0.64 0.55 0.10 0.025 2.1e38 0.049 2.6 2.13 0.90 0.83 0.28 9.4E-4 8.2e37 0.019 1.2 2.13 1.8 1.8 1 0.40 0.0064 6.1e37 0.014 0.73 2.07 1.8 0.50 2.4 0.14 0.042 4.8e37 0.011 0.47 2.28 1.1 0.55 ∞ 0.28 0.0081 6.1e37 0.014 0.68 2.28 2.0 1.1 150 2.0 10 0.10 0.012 9.0e38 0.20 26 2.05 0.57 1.8 1 0.14 0.014 9.2e38 0.21 25 2.00 0.55 0.70 0.20 0.010e 7.0e38 0.16 19 2.00 0.65 0.77 (cont’d) Table 6.7—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) η ξ L2−8 f2−8 IC Γ2−8 fGeV fVHE µG (Myr) ergss−1 0.28 0.0056 4.1e38 0.093 11 2.00 0.81 0.87 2.2 0.071 0.011 2.1e38 0.049 5.1 2.20 0.58 1.1 ∞ 0.10 0.0064 2.3e38 0.053 5.9 2.20 0.74 1.5 10 0.10 0.0077f 9.7e37 0.022 2.3 2.15 0.70 0.66 0.14 0.0023 6.5e37 0.015 1.7 2.15 0.91 0.90 2.4 0.14 0.0061 3.8e37 0.0087 0.81 2.27 0.97 0.51 ∞ 400 200 2.0 1 0.20 0.0032 2.5e38 0.057 10 2.01 0.54 0.56 2.2 10 0.10 7.0E-4 4.4e37 0.0099 1.7 2.16 0.62 0.62 aFraction of the observed diffuse 2 - 8 keV X-ray luminosity (from Strickland & Heckman 2007) from that synchrotron emission accounts for. bRatio of the 2 - 8 keV synchrotron and IC luminosities. cRatio of the predicted 0.3 - 10 GeV γ-ray emission with the flux detected by Fermi-LAT in Abdo et al. (2010g). dRatio of the predicted 700 GeV γ-ray emission with the observed TeV flux observed from HESS in Acciari ≥ et al. (2009). eThis model satisfies our constraints ( 6.3.3). While η is not the minimum or maximum allowed or the closest §prim 9 prim 6 to reproducing the GeV luminosity for γmax = 10 , this is true for γmax = 10 so we list it anyway. f Chosen as our fiducial model. synch syncha Synch b synch c d B p tdiff (3 GeV) ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 West nucleus 500 2.0 0.81 2.6e39 0.065 0.0416 1.90 0.60 0.016 ∞ 10 0.81 2.4e39 0.061 0.0388 1.90 0.60 0.016 1 0.81 1.5e39 0.039 0.0246 1.93 0.60 0.012 2.2 1.3 6.0e38 0.015 0.0072 1.94 0.49 0.0060 ∞ 10 1.3 5.6e38 0.014 0.0068 1.94 0.49 0.0058 1 1.3 3.6e38 0.0091 0.0044 1.97 0.49 0.0047 2.4 2.8 8.5e37 0.0021 8E-4 1.98 0.46 0.0015 ∞ 10 2.8 8.0e37 0.0020 7E-4 1.99 0.46 0.0014 401 1 2.8 5.3e37 0.0013 5E-4 2.02 0.46 0.0012 1000 2.0 0.25 6.5e39 0.16 0.33 1.97 0.19 0.015 ∞ 10 0.25 6.1e39 0.15 0.31 1.97 0.19 0.014 1 0.25 3.9e39 0.098 0.20 2.00 0.19 0.011 2.2 0.35 1.6e39 0.039 0.063 2.01 0.16 0.0056 ∞ 10 0.35 1.5e39 0.037 0.060 2.01 0.16 0.0054 1 0.36 9.7e38 0.024 0.040 2.04 0.16 0.0045 2.4 0.75 2.3e38 0.0058 0.0075 2.06 0.15 0.0014 ∞ 10 0.75 2.2e38 0.0054 0.0071 2.07 0.15 0.0014 1 0.76 1.5e38 0.0037 0.0048 2.10 0.15 0.0012 2000 2.0 0.077 1.3e40 0.31 1.9 2.04 0.069 0.012 ∞ prim 6 (cont’d) Table 6.8. Arp 220 X-ray luminosities when γmax = 10 Table 6.8—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 10 0.077 1.2e40 0.29 1.8 2.04 0.069 0.012 1 0.077 7.9e39 0.20 1.2 2.07 0.068 0.0096 2.2 0.097 3.2e39 0.079 0.41 2.09 0.064 0.0049 ∞ 10 0.097 3.0e39 0.075 0.39 2.09 0.064 0.0048 1 0.097 2.0e39 0.051 0.26 2.12 0.063 0.0040 2.4 0.20 4.9e38 0.012 0.054 2.15 0.061 0.0013 ∞ 10 0.20 4.7e38 0.012 0.052 2.15 0.035 0.0013 1 0.20 3.3e38 0.0082 0.036 2.19 0.061 0.0011 402 4000 2.0 0.023 1.8e40 0.46 7.8 2.07 0.030 0.010 ∞ 10 0.023 1.7e40 0.44 7.3 2.07 0.030 0.0099 1 0.024 1.2e40 0.30 5.0 2.10 0.029 0.0082 2.2 0.020 4.9e39 0.12 1.9 2.13 0.036 0.0043 ∞ 10 0.020 4.6e39 0.12 1.8 2.13 0.036 0.0042 1 0.020 3.3e39 0.082 1.3 2.16 0.035 0.0036 2.4 0.039 8.0e38 0.020 0.28 2.19 0.037 0.0012 ∞ 10 0.039 8.0e38 0.020 0.28 2.19 0.037 0.0012 1 0.040 5.5e38 0.014 0.20 2.23 0.037 0.0010 8000 2.0 0.0057 2.2e40 0.56 26 2.06 0.018 0.0092 ∞ 10 0.0057 2.1e40 0.53 24 2.06 0.018 0.0090 (cont’d) Table 6.8—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 1 0.0059 1.5e40 0.37 17 2.09 0.018 0.0075 16000 2.0 1 1.8E-4 1.6e40 0.41 57 2.06 0.015 0.0073 East nucleus 500 2.0 0.61 2.6e39 0.17 0.055 1.90 0.46 0.016 ∞ 10 0.61 2.4e39 0.16 0.051 1.90 0.46 0.016 1 0.61 1.5e39 0.10 0.032 1.93 0.46 0.012 2.2 0.95 6.0e38 0.040 0.0095 1.94 0.38 0.0060 403 ∞ 10 0.95 5.6e38 0.038 0.0089 1.94 0.38 0.0058 1 0.95 3.6e38 0.024 0.0057 1.97 0.37 0.0047 2.4 2.1 8.5e37 0.0057 0.0010 1.98 0.36 0.0015 ∞ 10 2.1 8.0e37 0.0054 9E-4 1.99 0.36 0.0014 1 2.1 5.3e37 0.0035 6E-4 2.02 0.36 0.0012 1000 2.0 0.18 6.5e39 0.43 0.43 1.97 0.15 0.015 ∞ 10 0.18 6.1e39 0.40 0.41 1.97 0.15 0.014 1 0.18 3.9e39 0.26 0.26 2.00 0.15 0.011 2.2 0.26 1.6e39 0.10 0.084 2.01 0.13 0.0056 ∞ 10 0.26 1.5e39 0.098 0.079 2.01 0.13 0.0054 (cont’d) Table 6.8—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 1 0.26 9.7e38 0.065 0.052 2.04 0.13 0.0045 2.4 0.56 2.3e38 0.015 0.0099 2.06 0.12 0.0014 ∞ 10 0.56 2.2e38 0.015 0.0093 2.07 0.12 0.0014 1 0.56 1.5e38 0.0099 0.0063 2.10 0.12 0.0012 2000 2.0 0.054 1.3e40 0.83 2.5 2.04 0.056 0.012 ∞ 10 0.054 1.2e40 0.79 2.4 2.04 0.056 0.012 1 0.055 7.9e39 0.52 1.6 2.07 0.055 0.0096 2.2 0.066 3.2e39 0.21 0.54 2.09 0.054 0.0049 404 ∞ 10 0.066 3.0e39 0.20 0.51 2.09 0.054 0.0048 1 0.066 2.0e39 0.14 0.35 2.12 0.054 0.0040 2.4 0.14 4.9e38 0.033 0.072 2.15 0.053 0.0013 ∞ 10 0.14 4.7e38 0.031 0.069 2.15 0.053 0.0013 1 0.14 3.3e38 0.022 0.048 2.19 0.053 0.0011 4000 2.0 0.014 1.8e40 1.2 10 2.07 0.026 0.010 ∞ 10 0.014 1.7e40 1.2 9.7 2.07 0.026 0.0099 1 0.014 1.2e40 0.80 6.7 2.10 0.025 0.0082 2.2 0.0077 4.9e39 0.33 2.5 2.13 0.033 0.0043 ∞ 10 0.0078 4.6e39 0.31 2.4 2.13 0.033 0.0042 1 0.0082 3.3e39 0.22 1.7 2.16 0.033 0.0036 (cont’d) Table 6.8—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 2.4 0.017 8.0e38 0.054 0.38 2.19 0.035 0.0012 ∞ 10 0.017 7.7e38 0.051 0.37 2.20 0.035 0.0012 1 0.017 5.5e38 0.037 0.26 2.23 0.034 0.0010 8000 2.0 0.00054 2.2e40 1.5 34 2.06 0.017 0.0092 ∞ 10 0.00057 2.1e40 1.4 32 2.06 0.017 0.0090 405 1 0.00083 1.5e40 0.99 23 2.09 0.017 0.0075 aFraction of the observed diffuse 2 - 10 keV X-ray luminosity (from Clements et al. (2002)) from that synchrotron emission accounts for. We assume that Arp 220 X-1 is the west nucleus, and Arp 220 X-4 is the east nucleus. bRatio of the 2 - 10 keV synchrotron and IC luminosities. cRatio of the predicted 100 MeV γ-ray emission with the Fermi-LAT one-year catalog source sensitivity at ≥ high Galactic latitude (Abdo et al. 2010g). dRatio of the predicted 0.36 - 1.8 TeV γ-ray emission with the upper limits on the flux obtained with MAGIC in Albert et al. (2007a). synch syncha Synch b synch c d B p tdiff (3 GeV) ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 West nucleus 500 2.0 1.3 1.2e41 2.9 1.9 1.85 0.67 0.23 ∞ 10 1.3 1.2e41 2.9 1.9 1.85 0.67 0.23 1 1.3 1.2e41 2.9 1.9 1.84 0.66 0.23 2.2 1.3 1.7e40 0.41 0.20 1.91 0.50 0.044 ∞ 10 1.3 1.7e40 0.41 0.20 1.91 0.50 0.044 1 1.3 1.6e40 0.41 0.20 1.92 0.50 0.043 2.4 2.8 2.3e39 0.058 0.021 1.97 0.46 0.0085 ∞ 10 2.8 2.3e39 0.058 0.021 1.98 0.46 0.0085 406 1 2.8 2.3e39 0.058 0.021 2.00 0.46 0.0082 1000 2.0 0.38 7.3e40 1.8 3.7 1.93 0.21 0.061 ∞ 10 0.38 7.3e40 1.8 3.7 1.93 0.21 0.060 1 0.38 7.1e40 1.8 3.6 1.93 0.21 0.057 2.2 0.38 1.1e40 0.28 0.45 1.99 0.16 0.014 ∞ 10 0.38 1.1e40 0.27 0.45 1.99 0.16 0.014 1 0.38 1.1e40 0.26 0.43 2.01 0.16 0.013 2.4 0.76 1.6e39 0.041 0.053 2.05 0.15 0.0029 ∞ 10 0.76 1.6e39 0.041 0.052 2.06 0.15 0.0028 1 0.76 1.6e39 0.039 0.050 2.08 0.15 0.0026 2000 2.0 0.12 4.6e40 1.1 6.9 2.01 0.073 0.020 ∞ prim 9 (cont’d) Table 6.9. Arp 220 X-ray luminosities when γmax = 10 Table 6.9—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 10 0.12 4.5e40 1.1 6.8 2.01 0.073 0.020 1 0.12 4.1e40 1.0 6.2 2.02 0.048 0.0078 2.2 0.10 7.6e39 0.19 1.0 2.07 0.064 0.0062 ∞ 10 0.10 7.5e39 0.19 0.98 2.08 0.065 0.0061 1 0.10 6.6e39 0.17 0.86 2.10 0.064 0.0053 2.4 0.20 1.2e39 0.030 0.13 2.14 0.061 0.0015 ∞ 10 0.20 1.2e39 0.029 0.13 2.15 0.061 0.0015 1 0.20 1.0e39 0.026 0.11 2.17 0.061 0.0013 407 4000 2.0 0.036 3.1e40 0.79 13 2.05 0.030 0.011 ∞ 10 0.036 3.0e40 0.76 13 2.05 0.030 0.011 1 0.036 2.5e40 0.63 11 2.07 0.030 0.0091 2.2 0.021 6.2e39 0.16 2.4 2.12 0.036 0.0044 ∞ 10 0.021e 6.0e39 0.15 2.3 2.12 0.036 0.0043 1 0.022 4.6e39 0.12 1.8 2.15 0.035 0.0037 2.4 0.039 1.0e39 0.026 0.36 2.19 0.037 0.0012 ∞ 10 0.040 9.8e38 0.025 0.35 2.19 0.037 0.0012 1 0.040 7.7e38 0.019 0.27 2.22 0.037 0.0010 8000 2.0 0.0088 2.6e40 0.65 30 2.05 0.018 0.0093 ∞ 10 0.0088 2.5e40 0.62 28 2.05 0.018 0.0091 (cont’d) Table 6.9—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 1 0.0092 1.9e40 0.46 21 2.07 0.018 0.0076 16000 2.0 1 2.7E-4 1.7e40 0.41 57 2.04 0.015 0.0073 East nucleus 500 2.0 0.95 8.9e40 5.9 1.9 1.85 0.51 0.18 ∞ 10 0.95 8.9e40 5.9 1.9 1.85 0.51 0.18 1 0.95 8.8e40 5.9 1.9 1.84 0.51 0.17 2.2 1.0 1.3e40 0.84 0.20 1.91 0.39 0.035 ∞ 408 10 1.0 1.3e40 0.84 0.20 1.91 0.39 0.035 1 1.0 1.2e40 0.83 0.20 1.92 0.39 0.034 2.4 2.1 1.8e39 0.12 0.021 1.97 0.36 0.0068 ∞ 10 2.1 1.8e39 0.12 0.021 1.98 0.36 0.0067 1 2.1 1.8e39 0.12 0.021 2.00 0.36 0.0065 1000 2.0 0.29 5.6e40 3.7 3.8 1.93 0.16 0.049 ∞ 10 0.29 5.6e40 3.7 3.8 1.93 0.16 0.049 1 0.29 5.4e40 3.6 3.6 1.93 0.16 0.046 2.2 0.28 8.6e39 0.57 0.46 1.99 0.13 0.012 ∞ 10 0.28 8.5e39 0.57 0.46 1.99 0.13 0.011 1 0.28 8.0e39 0.53 0.43 2.01 0.13 0.010 (cont’d) Table 6.9—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 2.4 0.56 1.3e39 0.085 0.054 2.05 0.12 0.0025 ∞ 10 0.56 1.3e39 0.084 0.054 2.06 0.12 0.0025 1 0.56 1.2e39 0.079 0.051 2.08 0.12 0.0023 2000 2.0 0.084 3.6e40 2.4 7.2 2.01 0.059 0.018 ∞ 10 0.084 3.5e40 2.3 7.1 2.01 0.059 0.017 1 0.085 3.1e40 2.1 6.3 2.02 0.058 0.015 2.2 0.070 6.3e39 0.42 1.1 2.07 0.055 0.0058 ∞ 10 0.070 6.1e39 0.41 1.0 2.08 0.055 0.0057 409 1 0.071 5.2e39 0.34 0.89 2.10 0.054 0.0049 2.4 0.14 9.8e38 0.065 0.14 2.14 0.053 0.0015 ∞ 10 0.14 9.5e38 0.064 0.14 2.15 0.053 0.0014 1 0.14 8.2e38 0.054 0.12 2.17 0.053 0.0013 4000 2.0 0.021 2.6e40 1.7 15 2.05 0.026 0.011 ∞ 10 0.021 2.5e40 1.7 14 2.05 0.026 0.011 1 0.022 2.0e40 1.3 11 2.07 0.026 0.0087 2.2 0.0082 5.4e39 0.36 2.8 2.12 0.033 0.0044 ∞ 10 0.0083e 5.2e39 0.34 2.7 2.12 0.033 0.0043 1 0.0087 3.8e39 0.25 2.0 2.15 0.033 0.0037 2.4 0.017 9.0e38 0.060 0.43 2.19 0.035 0.0012 ∞ (cont’d) Table 6.9—Continued synch syncha Synch b synch c d B p tdiff (3 GeV) ξ L2−10 f2−10 IC Γ2−10 fGeV fVHE µG (Myr) ergss−1 10 0.017 8.6e38 0.057 0.41 2.19 0.035 0.0012 1 0.018 6.5e38 0.043 0.31 2.22 0.034 0.0010 8000 2.0 8.4E-4 2.3e40 1.5 35 2.05 0.017 0.0092 ∞ 10 8.9E-4 2.2e40 1.4 33 2.05 0.017 0.0090 410 1 0.0013 1.5e40 1.0 23 2.07 0.017 0.0076 aFraction of the observed diffuse 2 - 10 keV X-ray luminosity (from Clements et al. (2002)) from that synchrotron emission accounts for. We assume that Arp 220 X-1 is the west nucleus, and Arp 220 X-4 is the east nucleus. bRatio of the 2 - 10 keV synchrotron and IC luminosities. cRatio of the predicted 100 MeV γ-ray emission with the Fermi-LAT one-year catalog source sensitivity at ≥ high Galactic latitude (Abdo et al. 2010g). dRatio of the predicted 0.36 - 1.8 TeV γ-ray emission with the upper limits on the flux obtained with MAGIC in Albert et al. (2007a). eChosen as our fiducial model. Chapter 7 Additional Notes on the High-Σg Conspiracy A key theme of the work in this dissertation is the complexity of cosmic ray (CR) propagation in star-forming galaxies, particularly starbursts. The different kinds of cooling, the presence of pionic secondary e± (and even γγ pair e±), and spectral effects must all be taken into account. In our models, we find that non-synchrotron losses including bremsstrahlung and ionization balance the additional radio emission from secondaries, so that the linearity of the FRC is preserved from normal galaxies to starbursts. But this requires effects which on their own adjust the radio emission by a factor 10 canceling out to good accuracy, and remaining in balance for all the ∼ starbursts. How does this cancellation of factors actually work, numerically, in our models? 7.1. Radio Enhancement: Secondaries and Spectral Effects A factor of 10 radio enhancement is naturally expected from the energetics of pion production. For simplicity, consider the case when p = 2.0, so that there 411 is equal energy per log energy bin being injected into a galaxy. At a given energy > GeV, there are δ˜ 50 100 times more primary protons than primary electrons ∼ ≈ − being injected (e.g., Ginzburg & Ptuskin 1976; Schlickeiser 2002; Warren et al. 2005; Strong et al. 2010). A fraction F 1 of that energy is converted into pionic losses; cal → of that, 1/2 goes into neutrinos, 1/3 into γ-rays, and 1/6 into secondary e± (e.g., Loeb & Waxman 2006; Thompson, Quataert, & Waxman 2007). Thus, the pionic secondary e± should enhance the radio emission over that of the primaries by 1 νL (secondary e±) F νL (protons) (7.1) ν ≈ 6 cal ν δ˜ F νL (primaries) (7.2) ≈ 6 cal ν δ˜ 12.5 FcalνLν(primaries). (7.3) ≈ 75! When p> 2, the secondaries are “diluted” with respect to the primaries, since they individually have much lower energy than their primary protons, and there are more primary electrons injected at lower energies. The secondary-to-primary ratio at a given energy is δ/˜ 6 E /E p−2, where E /E 0.05. Assuming that ×h sec pi h sec pi ≈ δ˜ 50 100, the secondaries are more prevalent than the primaries by a factor ≈ − f = 4.6 9.2 when p = 2.2, and 2.5 5.0 when p = 2.4. sec − − On the other hand, a second effect which we call the “νC effect” enhances the emission seen at a given frequency: in the higher magnetic field strengths of 412 starbursts, we see lower energy e± at a given frequency, and there are more e± at low energies than at high energies. The critical synchrotron frequency is 2 Ee B νC 0.13 GHz. (7.4) ≈ GeV 10 µG! Hence, while the secondary-to-primary ratio decreases at higher p, the observed luminosity of the primaries at each frequency becomes higher at stronger magnetic fields through the νC effect. For a power law injection spectrum dQ/dE E−p, the luminosity at an observed frequency goes as ∝ e νL (primaries) E2dQ/dE E2−p. Since ν E2B, this translates to a magnetic ν ∝ e ∝ e C ∝ e − field dependence: νL (primaries) ν(2 p)/2B(p−2)/2. In our models we found that ν ∝ C B Σ0.7 was necessary for the high-Σ conspiracy to work. Then over the entire ∝ g g 4 −2 −2 span of the FRC (a factor of 10 in Σg from 0.001 g cm to 10 g cm ), the νC effect leads to an enhancement of a factor 1.9 when p = 2.2 and a factor 3.6 ∼ ∼ when p = 2.4. At a given observing frequency, a different injection slope p decreases the secondary-to-primary ratio by E /E p−2 but increases the total primary emission h sec pi (p−2)/2 by (Bmax/Bmin) . Hence, the secondaries and the νC effect work together to keep the total enhancement of radio emission roughly constant as p varies between 2.0 and 2.4 in proton calorimetric galaxies, essentially because E /E 2 400 h sec pi ≈ is close to (B /B ) 630. Therefore the near-proton calorimetric generation max min ≈ 413 of secondaries in starbursts, combined with the νC effect consistently enhances the radio emission by a factor of 10 even when p = 2. ∼ 6 7.2. Radio Suppression: Balance of the cooling processes The ratio of synchrotron to other losses governs the efficiency of conversion of CR electron power into synchrotron emission. Normal galaxies have CR scale heights of kiloparsecs (e.g., Lukasiak et al. 1994; Han & Qiao 1994; Dumke et al. 1995; Beck 2009), whereas most of the radio emission from starburst galaxies comes from their compact h < 100 pc disks (e.g., Ulvestad & Antonucci 1997; Downes ∼ & Solomon 1998; Williams & Bower 2010). For a given gas surface density, then, bremsstrahlung and ionization cooling are increased in importance by a factor of 10 or more. However, in order to maintain the linearity of the FRC for starburst galaxies, the ratios must be constant across all starburst galaxies. The cooling times for the four main energy loss processes – ionization, bremsstrahlung, synchrotron, and IC – are given in chapter 2 (Lacki et al. 2010a). In the last section of chapter 3 (Lacki et al. 2010a), I then calculate the ratios of each energy loss process with synchrotron. Assuming the Kennicutt (1998) Schmidt Law: 1.4 ΣSFR Σg −2 = 36.3 − (7.5) M kpc yr−1 ! g cm 2 ! ⊙ 414 and our fiducial magnetic field strength dependence from chapter 2: 0.7 Σg B = 6 µG − (7.6) 0.0025 g cm 2 ! these ratios come out to be t synch = 0.44 (7.7) tIC,⋆ 0.05 −1 tsynch Σg h = 0.39 −2 (7.8) tbrems g cm ! kpc! 0.3 −1 tsynch Σg h = 0.38 −2 (7.9) tion g cm ! kpc! (7.10) at an observing frequency of 1.4 GHz, where h is the CR scale height. We see that the only one of these ratios that changes much with Σg at a given scale height is ionization. The total fraction of power going into synchrotron, fsynch = tloss/tsynch, is −1 tsynch tsynch tsynch fsynch = 1.0+ + + . (7.11) " tIC,⋆ tbrems tion # For normal star-forming galaxies, with gas surface densities 0.001 0.1 g cm−2 and − CR scale heights of 1 kpc, fsynch varies only from 0.51 to 0.57 (neglecting escape). In starburst galaxies, with gas surface densities of 0.01 10 g cm−2 and CR scale − heights of 100 pc, fsynch varies from 0.15 to 0.075, only a factor of 2. Hence the jump from 1 kpc to 100 pc scale heights reduces the radio emission by a factor of 5 10. ∼ − Just as importantly, after the jump, the loss processes remain in balance even as Σg 415 increases further, so that factor 5 10 does not change much. What causes this ∼ − extraordinary cancellation, where the vast range in physical conditions results in a small change of fsynch? ± 2 Consider a CR e with Lorentz factor γ = Ee/(mec ) in a galaxy or starburst with a magnetic field strength B, a radiation energy density Urad, and a mass density ρ. The synchrotron cooling time for that e± goes as t γ−1B−2, synch ∝ the IC cooling time goes as t γ−1U , the bremsstrahlung cooling time IC ∝ rad goes as t ρ−1, and the ionization cooling time approximately goes as brems ∝ t ρ−1γ. The ratio of synchrotron and IC loss times is therefore maintained ion ∝ when B2/(8π) = U (U + U ) – if either the magnetic field energy density B ∝ B rad is larger than the radiation energy density, or if the magnetic field energy density grows linearly with the radiation energy density. This is the essence of the original calorimeter theory by V¨olk (1989). According to the Schmidt Law, the surface areal star-formation rate scales as Σ Σ1.4 (Kennicutt 1998), and for a thin disk SFR ∝ g geometry, we expect the radiation energy density to grow as U Σ Σ1.4. rad ∝ SFR ∝ g In order to maintain the balance between IC and synchrotron, this implies that B Σ0.7. ∝ g The ratio of synchrotron to bremsstrahlung cooling times is ργ−1B−2. However, we observe the radio emission at constant frequency and not constant energy. The synchrotron-emitting frequency of an electron goes as ν γ2B, or equivalently, ∝ γ ν1/2/B1/2. Furthermore, the density can be written in terms of surface density ∝ 416 and scale height: ρ Σ /h. We then see the ratio of synchrotron to bremsstrahlung ∝ g −1/2 3/2 cooling times is Σgν B /h. At a given scale height and observing frequency, the ratio is constant if B Σ2/3 – very nearly the same condition for maintaining the ∝ g synchrotron/IC ratio. This condition, resulting from the use of constant observing frequency rather than energy, is responsible for the linearity of the FRC for starbursts in our models. In my talks, I refer to it as the “bremsstrahlung miracle”. Ionization is subdominant for most starbursts with respect to bremsstrahlung, so maintaining a constant ratio between it and synchrotron losses is less important. However, as the magnetic field strength increases, electrons at lower energies, where ionization cooling is dominant, are probed by GHz synchrotron emission. Thus, ionization may affect the FRC in the densest starbursts, like Arp 220. The ratio of synchrotron to ionization loss times are ργ−2B−2 Σ ν−1B−1/h; this ratio would ∝ g be constant only if B Σ . However, even with B Σ0.7, the restricted range of ∝ g ∝ g gas surface densities in starbursts (0.1 10 g cm−2, a factor of 100) means only a − factor 4 variation in the synchrotron to ionization ratio. Interestingly, Arp 220 is ∼ faint with respect to the FRC, and has an extremely flat spectrum (e.g., Williams & Bower 2010). Other ULIRGs also show scatter in the FRC. While typically attributed to extremely strong free-free absorption at 1 GHz (e.g., Condon et al. 1991a), these characteristics are also consistent with ionization losses affecting the timescales of CR electron losses. 417 Therefore the small scale height of starbursts enhances the effectiveness of bremsstrahlung (and to a lesser extent, ionization) by a factor 10; the B Σ0.7 ∼ ∝ g dependence assures that the ratio of synchrotron with IC and bremsstrahlung losses is very nearly constant for starbursts. Ionization may affect the densest starbursts and cause them to become radio-dim, as observed in Arp 220. Thus the total effect is that the other losses consistently reduce the radio emission by a factor 5 10. ∼ − 7.3. Summary I have shown that the high-Σg conspiracy arises from the cooling times as long as B Σ0.7 and proton calorimetry holds in starbursts. Secondaries and ∝ g spectral effects enhance the radio emission by a factor 10, while non-synchrotron ∼ losses reduce the radio emission by 5 10, so that the correlation is preserved from − normal galaxies to starbursts. The magnetic field dependence results in constant IC/synchrotron and bremsstrahlung/synchrotron cooling time ratios, and only a weakly varying ionization/synchrotron cooling time ratio. Even though this conspiracy naturally arises from the physics, there are a number of remaining questions. Why should the magnetic field strength go as B Σ0.7? How sensitive is the conspiracy to perturbations? There is a large scatter ∝ g in the Kennicutt (1998) Schmidt Law, about a dex, and there may be a large scatter in equation 7.6 for B, for all we know. Yet there is only a 0.3 dex in the FRC. 418 Might there be other, perhaps simpler, solutions to the physics that allow the FRC to arise? As we showed in chapter 4 (Lacki et al. 2011), the gamma-ray to radio flux ratio of M82 and NGC 253 indicate that the non-synchrotron losses are 10 ∼ times stronger than the synchrotron losses for GHz-emitting e±. But might some other combination of loss processes than bremsstrahlung and ionization, such as IC or winds, be responsible for this? And could that be consistent with the gamma-ray data indicating that F 0.3 in these starbursts? Is the apparent radio-brightness cal ≈ of submillimeter galaxies true, and if so, does it indicate an unbalanced conspiracy? Is it even possible to characterize the physical conditions in starbursts with single values for magnetic field strength and density? Or, since the ISM in starbursts is clumpy, might there be different populations of CRs sampling different phases of the ISM? 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