Breaking the Radio – Gamma-Ray Connection in Arp 220

MNRAS 000, 000–000 (0000) Preprint 16 January 2019 Compiled using MNRAS LATEX style ﬁle v3.0

Tova M. Yoast-Hull? and Norman Murray† Canadian Institute for Theoretical Astrophysics, University of Toronto, ON, M5S 3H8, Canada

16 January 2019

ABSTRACT Recent analyses of the γ-ray spectrum from the ultra-luminous infrared galaxy Arp 220 have revealed a discrepancy in the cosmic ray energy injection rates derived from the γ-rays versus the radio emission. While the observed radio emission is consistent with the star formation rate inferred from infrared observations, a significantly higher cosmic ray population is necessary to accurately model the measured γ-ray flux. To resolve this discrepancy between the radio and γ-ray observations, we find that we must increase the cosmic ray energy injection rate and account for an infrared optical depth greater than unity. Raising the energy injection rate naturally raises the total γ-ray flux but also raises the radio flux unless there is also an increase in the energy loss rate for cosmic ray leptons. A optically thick medium results in an increase in energy losses via inverse Compton for cosmic ray leptons and preserves agreement with submillimeter, millimeter, and infrared wavelength observations. Key words: cosmic rays – galaxies: individual (Arp 220) – galaxies: starburst – gamma rays: galaxies – radiative transfer – radio continuum: galaxies

1 INTRODUCTION collide with the ISM to produce neutral pions that subse- quently decay into γ-rays. Over the past decade, eight star-forming galaxies (mostly gi- In our current understanding, the correlation between ant spirals) have been detected in γ-rays by Fermi, including radio and FIR emission seen in star-forming galaxies requires M31 (NGC 0224), NGC 0253, NGC 1068, NGC 2146, M82 two additional conditions to hold. First, either star-forming (NGC 3034), NGC 4945, Arp 220, and Circinus (Abdo et al. galaxies must be cosmic ray electron calorimeters, meaning 2010b,a; Lenain et al. 2010; Hayashida et al. 2013; Tang that cosmic ray electrons are confined to the galaxy and radi- et al. 2014; Peng et al. 2016; Griffin et al. 2016). Observa- ate the majority of their energy within their injection volume tions of these galaxies show a tentative correlation between (Voelk 1989), or both cosmic ray electrons and ultraviolet their γ-ray and far-infrared (FIR) luminosities (Ackermann radiation escape from star-forming galaxies in proportional et al. 2012; Rojas-Bravo & Araya 2016), indicating a physi- amounts (Lacki et al. 2010). Secondly, in most star forming cal relationship between these two quantities in star-forming galaxies, energy losses via synchrotron must be larger than galaxies as predicted by Thompson et al.(2007). Recent sim- losses via inverse Compton for cosmic ray electrons, implying ulations show further evidence of a link between the FIR that the energy density in magnetic fields must be slightly emission and γ-rays from galaxies via their star formation larger than the energy density in radiation for star-forming rate (SFR; e.g., Martin 2014; Pfrommer et al. 2017). galaxies (U U ; Lisenfeld et al. 1996; Thompson et al. The essential physical processes underlying this corre- B & rad 2006). arXiv:1901.04564v1 [astro-ph.HE] 14 Jan 2019 lation are tied to massive stars, which dominate the dif- While star-forming galaxies are generally believed to fuse emission in star-forming galaxies. Massive stars radi- be electron calorimeters, most normal star-forming galax- ate predominately in the ultraviolet; much of this emission ies are not proton calorimeters (e.g., Pfrommer et al. 2017). is absorbed by the surrounding interstellar medium (ISM) In quiescent systems, the diffusion time-scale for cosmic ray and re-radiated in the infrared. At the ends of their lives, protons is significantly shorter than their energy loss time- massive stars explode as supernovae, and the resulting rem- scale, and thus, cosmic ray protons preferentially escape into nants accelerate electrons and protons to relativistic ener- the halo. Additionally, these are systems in which primary gies (e.g., de Jong et al. 1985; Helou et al. 1985). Cosmic ray cosmic ray electrons are likely dominant over secondary cos- electrons interacting with galactic magnetic fields produce mic ray electrons and positrons produced in interactions of non-thermal synchrotron emission, while cosmic ray protons cosmic ray protons with the ISM. In contrast, intensely star-forming systems (such as Arp ? E-mail: [email protected] 220) are at least partial proton calorimeters (e.g., Lacki et al. † Canadian Research Chair in Astrophysics 2011; Yoast-Hull et al. 2013; Eichmann & Becker Tjus 2016)

c 0000 The Authors 2 T. M. Yoast-Hull & N. Murray

9 3.25 Star-Forming 10− Fermi Active Galaxy Veritas Upper Limits ] 3.00 1 − s

Circinus 2 −

2.75 m c

) Arp 220 z V H e G 4 G .

[ 10 1 2.50 10− L

NGC 2146 x

\ u

NGC 0253 l γ F L

(

2.25 y g NGC 4945 a o l R

M82 NGC 1068 a

2.00 m

m 51 1 a εCR = 3.2 10 erg yr−

G × 51 1 ε = 1.6 10 erg yr− 1.75 M31 CR × 50 1 11 ε = 8.0 10 erg yr− 10− CR × 50 1 εCR = 4.0 10 erg yr− 1.50 × 10 1 100 101 102 103 104 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 − Energy (E ) [GeV] log(Radio Luminosity) [L ] γ ¯

Figure 1. The logarithm of the ratio of γ-ray to radio luminosity, as a function of logarithmic radio luminosity, is shown for nearby East Data star-forming galaxies. The mean ratio is 2.266 with a standard East Model deviation of 0.335, depicted here as solid and dashed lines respec- West Data West Model tively. Local galaxy M31 and the Seyfert galaxy Circinus are clear outliers, while Arp 220 is a marginal outlier. Galaxies with γ-ray detections by Fermi are shown with filled black symbols and γ- 10 1 ray flux upper limits are denoted with open gray symbols. See − AppendixA for details on the data used. Radio Flux [Jy] and secondary cosmic rays dominate over primary cosmic ray electrons (e.g., Lacki & Thompson 2013; Peretti et al. 51 1 ε = 3.2 10 erg yr− CR × 2018). As a result, in these systems, we would expect both 51 1 εCR = 1.6 10 erg yr− 2 × the non-thermal radio and the γ-ray emission to be pro- 50 1 10− ε = 8.0 10 erg yr− CR × 50 1 portional to the population of cosmic ray protons. Thus, for ε = 4.0 10 erg yr− CR × starburst systems in which secondary cosmic rays dominate, 100 101 102 there may be a constant ratio between the γ-ray and radio Frequency (ν) [GHz] fluxes, regardless of whether the galaxy is a complete proton calorimeter. Observations clearly show that strong correlations exist Figure 2. γ-ray and radio spectra are shown for varying cosmic ray energy injection rates. For the γ-rays, energy injection rates between the FIR, radio, and γ-ray emission in star-forming larger than 1051 erg yr−1 are necessary; however, energy injection galaxies (e.g., Ackermann et al. 2012; Schöneberg et al. 2013; rates lower than 1051 erg yr−1, are necessary to fit the radio data. Rojas-Bravo & Araya 2016), see Fig. A1 in AppendixA. Data for the γ-rays are from Peng et al.(2016); Fleischhack & Examining the ratio of γ-ray flux to radio fluxes in star- VERITAS Collaboration(2015), and data for the radio are from forming galaxies detected with Fermi, we find tentative ev- Downes & Solomon(1998); Rodr´ıguez-Ricoet al.(2005); Williams idence for a constant γ-ray to radio flux ratio in starbursts & Bower(2010); Barcos-Muñozet al.(2015). galaxies, see Fig.1. The ratio for normal star-forming galaxies is lower than that for starbursts, as expected due to the dominance of primary cosmic ray electrons and the lack of the inferred energy injection rates depends on assumptions proton calorimetry in quiescent galaxies. In contrast, the ra- about the cosmic ray populations in the nuclei of Arp 220 tio of γ-ray to radio flux for the Seyfert galaxy Circinus is (Yoast-Hull et al. 2017). significantly higher than the starburst ratio, likely due to To examine these assumptions in detail, we begin by the dominance of the central active galactic nucleus (AGN; defining the injection spectrum for cosmic ray protons such Wojaczyńskiet al. 2015). that −Γp Fig.1 also shows Arp 220, the closest ultra-luminous Γp − 2 ηεCR Ep infrared galaxy (ULIRG), as slightly more than a standard qp(Ep) = −Γ × × , (1) E p V 1 GeV deviation away from the mean γ-ray to radio luminosity ra- min tio for starbursts. In investigating Arp 220 in more detail, where Γp is the cosmic ray (injection) spectral index, εCR modeling of the γ-ray and radio spectra reveal a larger cos- is the cosmic ray energy injection rate, and η = 0.1 is the mic ray energy injection rate is necessary to reproduce the acceleration efficiency of cosmic ray protons. The critical observed γ-ray flux than is necessary for the radio flux, see factor here, besides the energy injection rate (see Fig.3), is Fig.2. Of course, the magnitude of this discrepancy between the cosmic ray proton spectral index: as the spectral index

MNRAS 000, 000–000 (0000) Optical Depths in Arp 220 3

Radio Flux

Eq. 2

Sy Sy j ν (ν)=∫ dEe N e (E e)×Pν (Ee) Lower Synch Power

Eq. 4 Lower Magnetic Field 1/ 3 Sy 2 2 −ν/ ν Lower Energy Spectrum ( )∝ β γ ν c Pν E e U B e e 4 / 3 e Obs. Constraint νc Eq. 5 B≃1−5mG

N e (E e)=qe (Ee)×t( Ee ) Lower Injection Spectrum

Eq. 6

tot prim sec Lower Confinement Time qe (Ee)=qe (E e)+qe (E e) Lower Secondaries

Eq. 9 Eq. 8

−1 −1 −1 −1 sec t e (Ee) =tloss( Ee ) +t diff (Ee) +t adv Lower Primaries qe (E e)∝∫dE p N p(E p)σ p− p(E p)

Eq. 7

−Γ Γ −2 ξ ηε E e Lower Energy Loss Time q prim(E )= e × CR ×( e ) e e −Γe Lower Protons Emin V 1 GeV Eq. 10 Eq. 1

−Γ Ee p Γ p−2 ηεCR E p t loss(Ee)≃ ≪t adv ,t diff (E e) q p(E p)= −Γ × ×( ) −dE e /dt Lower e/p Ratio p Emin V 1 GeV Std. Assumption ξ=0.01 Lower Energy Loss Rate Lower Injection Rate Eq. 11 – 15 Obs. Constraint Lower Spectral Index from F ( E ) Lower B, Raise n ,U γ γ ISM rad Obs. Constraint Obs. Constraints from Γ γ from CO, FIR

Figure 3. Flow chart depicting the possible paths for lowering the radio flux. steepens, the energy injection rate required to supply a set models for starburst galaxies (Yoast-Hull et al. 2013, 2014, Γp γ-ray flux increases (because the factor 1/Emin decreases). 2015), we explore how changes in energy injection rate, Thus, for flatter spectral indices (Γp ∼ 2.1 − 2.2), there infrared optical depth, and magnetic field strength affect exist models where the radio and γ-ray fluxes imply cosmic the cosmic ray electron populations and the resulting non- ray energy injection rates which are consistent with each thermal emission, see Fig.3. We find that accounting for other (e.g., Torres 2004; Lacki et al. 2011; Lacki & Thomp- an infrared optical depth greater than unity in the mod- son 2013), and for steeper spectral indices (Γp ∼ 2.3 − 2.5), els, as required by submillimeter and millimeter wavelength the energy injection rates derived from the γ-ray and radio observations, easily resolves the tension between the energy fluxes are inconsistent with each other (e.g., Yoast-Hull et al. injection rates derived from the radio and γ-ray spectra, re- 2017). As the observed γ-ray spectrum is best fit by spectral spectively. indices of Γγ = 2.35 ± 0.16 (Peng et al. 2016), both sets of This paper is structured as follows: In Section2, we models are consistent with the observations within the sta- lay out the equations relating to the cosmic ray spectra and tistical uncertainty on the γ-ray spectral index1. Through- radio emission. In Section3, we discuss observational con- out the rest of this paper, we will be concentrating on solu- straints on the physical quantities that are input into the tions to the tension between the radio and γ-ray observations models. In Section4, we describe how our models are con- for steeper spectral indices. structed and tested, and we report our results. In Section5, In this paper, we examine possible mechanisms which we compare our results with observed and other quantities. can resolve the discrepancy between the observed radio and Our conclusions are summarized in Section6. γ-ray fluxes from Arp 220. Using our previously developed

2 THEORETICAL FRAMEWORK 1 It should be noted here that Γp ∼ Γγ , as the cross section for neutral pion production depends on energy such that σp−p(Ep) ∝ To resolve the diﬀerence between the cosmic ray energy in- lnEp (e.g., Kelner et al. 2006). jection rates inferred from γ-ray and radio spectra, we must

MNRAS 000, 000–000 (0000) 4 T. M. Yoast-Hull & N. Murray lower the radio ﬂux while maintaining or raising the γ-ray

flux for a given cosmic ray energy injection rate. Unfortu- B = Advection 0.0 5 mG nately, we cannot simply attribute the radio and γ-ray fluxes B = Diffusion 6 0. 10 5 m G Losses to different cosmic ray populations. B = 5 mG The primary hadronic γ-ray emission mechanism is neu- B = 5 0 m ] G tral pion decay. In the process of producing neutral pions, r

y 5 1 [ 10 v = 200 km s−

cosmic ray protons also create charged pions which subse- ) p 1 E v = 500 km s− quently decay into secondary electrons and positrons, along ( p

t 1 v = 1000 km s− with various neutrinos. Hence, the initial interactions lead- , e

l 4 1 v = 2000 km s− ing to hadronic γ-rays also lead to radiation from secondary a 10 c

s 3 - n = 10,000 cm−

leptons and thus radio emission. Therefore, whether the γ- e 3

m n = 20,000 cm− rays are hadronic or leptonic in origin, there should also be i T 3 3 a substantial amount of corresponding radio emission. 10 n = 50,000 cm− 3 To explore all the variables affecting radio emission, we n = 100,000 cm− break down the various equations governing the radio synchrotron emissivity in Fig3. Inputs affecting the total syn- 102 chrotron flux include: the magnetic field strength (B), the 1 0 1 2 3 10− 10 10 10 10 average ISM gas density (nISM), the radiation field energy Energy (Ep) [GeV] density (Urad), the electron-to-proton (number) ratio (ξ), the cosmic ray injection spectral index (Γe), and the cosmic Figure 4. Confinement time-scales for cosmic ray protons are ray energy injection rate (εCR). Both the radio and γ-ray spectra provide constraints on shown, including the diffusion time-scale (dashed lines), advection time-scale (dotted lines), and the energy-loss time-scale (solid the cosmic ray spectral index, and the γ-ray flux, in particu- lines). The energy-loss time-scale for protons is dominated by the lar, provides a lower limit on the the cosmic ray energy injec- pion production time-scale, which is dependent on the average tion rate as discussed in Section1, limiting our avenues of al- ISM gas density. For the ISM densities found in Arp 220, the tering the radio flux. Additionally, lowering the electron-to- energy-loss time-scale is significantly lower than either the diffu- proton (number) ratio has limited effect due to the presence sion or advection time-scales. of secondary electrons and positrons. Further observational constraints on the magnetic field strength, ISM density, and The cosmic ray injection spectrum, q(E), is assumed to be radiation field energy density are discussed in Section3. a power-law, proportional to the energy injection rate and inversely proportional to the volume of the injection region, 2.1 Cosmic Ray Leptons as seen in the previous section. The cosmic ray lepton spectrum includes both primary Non-thermal emission dominates the radio spectrum at GHz electrons, accelerated alongside the primary protons, and frequencies, and so, we concentrate on the synchrotron emis- secondary electrons and positrons, the decay products of sivity, charged pions produced during collision of cosmic ray pro- Z ∞ tons with the surrounding ISM. Thus, the total cosmic ray jSy = dE N (E )P Sy(γ ), (2) ν e e e ν e lepton injection spectrum is given by Emin Sy qtot(E ) = qprim(E ) + qsec(E ), (6) where Ne(Ee) is the total leptonic energy spectrum. Pν , the e e e e e e full synchrotron radiative power for a single electron, is given where the injection spectrum of the primary electrons is by (Ginzburg 1969; Blumenthal & Gould 1970; Rybicki & given by Lightman 1979; Schlickeiser 2002; Dermer & Menon 2009; −Γe Boettcher et al. 2012; Ghisellini 2013) prim Γe − 2 ξηεCR Ee √ qe = −Γ × × . (7) 3 E e V 1 GeV Sy 3qe B min Pν (γe, ψ) = 2 sin(ψ)F (x). (3) mec Here, ξ is the electron-to-proton (number) ratio, which is ∞ R usually taken to be ξ = 0.01 for Ee = Ep = 1 GeV. For Here, ψ is the pitch angle, and F (x) = x x dξK5/3(ξ). K5/3(ξ) is a modified Bessel function of the second kind, simplicity, we will assume that the spectral indices for the 2 and x = ν/νψ where νψ = 3qeB/4πmec × γe sin(ψ). Assum- injection spectra for the cosmic ray protons and electrons ing an isotropic pitch angle distribution and adopting the are the same, Γp = Γe. Then, the injection spectrum for asymptotic approximation by Boettcher et al.(2012), the secondary cosmic rays depends directly on the cosmic ray synchrotron power is given by proton energy spectrum such that (e.g., Schlickeiser 2002; 2 2 1/3 Kelner et al. 2006) Sy 32πc qe 2 2 ν −ν/νc Pν (γe) = 2 UB βe γe 4/3 e , (4) Z 9Γ(4/3) mec sec νc qe (Ee) ∝ dEpNp(Ep)σp−p(Ep). (8) where Γ(4/3) is the Gamma function and the critical fre- 2 quency νc = 3qeγe B/4πmec. The cosmic ray confinement time is comprised of the The energy spectrum for cosmic ray leptons is given by energy-loss time-scale (tloss(Ee)), the diffusion time-scale the product of the cosmic ray injection spectrum, q(E), and (tdiff (Ee)), and the energy-independent advective time-scale the cosmic ray confinement time-scale, t(E), see Fig.4. (tadv) such that −1 −1 −1 −1 N(E) = q(E) × t(E). (5) te(Ee) = tloss(Ee) + tdiff (Ee) + tadv. (9)

MNRAS 000, 000–000 (0000) Optical Depths in Arp 220 5

lepton energy spectrum (N (E )), see Fig.3. From equa- 103 e e tion4, we can see that the only means by which we can Ionization 3 3 3 3 − − − − m m m m Bremsstrahlung c c c c 2 V V V V decrease the synchrotron power is by decreasing the mag- 10 e e e e 0 0 0 0 ] IC / Synch 0 0 0 0

1 0 0 0 2 0 0 5 6 netic ﬁeld strength.

− 2 0 2 6 1 = = s = = U 1 U , U U , G The situation in regard to the energy spectrum for cos- V 10 , , G m G G m 3

e 5 m m 0 . − [ 0 0 . 0 cm . . 1 5 2 = 00 mic ray leptons is more complicated. To decrease the lep- ) 0 = 0 t = = B 2 B = 3 d 0 B B n m− tonic energy spectrum, we could decrease the energy injec- / 10 c e 00 50

E = n 3 tion spectrum. However, the observed γ-ray spectrum pro- d − ( cm 1 00 10 e 10− = vides stringent constraints on the cosmic ray spectral index

t n 3

a − cm R 00 (Γp) and a lower limit on the cosmic ray energy injection 2 = s 2 n s 10− rate (εCR). Thus, lowering the population of secondary elec- o 3

L n = 20000 cm−

y trons and positrons is not possible within the conﬁnes of the

g 3 3

r 10− n = 5000 cm−

e observed γ-ray ﬂux. Hence, the primary avenue for lowering n 3 E n = 1000 cm− the leptonic energy spectrum is via the conﬁnement time- 4 10− 3 scale, see Fig.3. As the energy-loss time-scale dominates the n = 200 cm−

5 total conﬁnement time-scale, see Fig.4, we will concentrate 10− 1 0 1 2 3 10− 10 10 10 10 our eﬀorts on the energy loss rate for cosmic ray electrons. Energy (Ee) [GeV]

2.2 Energy Loss Rates 106 T = 25 K As in previous papers, we include energy losses via ioniza- T = 50 K tion, bremsstrahlung, inverse Compton, and synchrotron in

] 5 T = 100 K 1 10 our models. The energy loss rates for these mechanisms are − T = 250 K s directly proportional to the gas density (n), the radiation V T = 500 K e [ 4 T = 1000 K ﬁeld energy density (Urad), and the magnetic ﬁeld strength ) 10 t

d (B) such that (Ginzburg 1969; Schlickeiser 2002; Dermer & / e

E Menon 2009; Schlickeiser & Ruppel 2010) d (

e 10 t

a dEe 9 3

R − = m c σ n [6.85 + ln(γ )] , (11) e T mol e

s dt 4 s Ion

o 2

L 10 ion

y dEe 3α

g − = cσT nionZ(Z + 1)Ee r

e dt Br 2π n 1 E 10 1 × ln(2γ ) − , (12) e 3 0 10 dE mol 3.9α 101 102 103 104 105 e s−s − = cσT φH i nmolEe, (13) Energy (Ee) [GeV] dt Br 8π 2 2 dEe 4 γKγe − = cσT Urad 2 2 , (14) Figure 5. Top: Energy loss rates for cosmic ray leptons dt IC 3 γK + γe are shown, including losses due to ionization (dotted lines), 2 dEe 4 B 2 bremsstrahlung (dashed lines), and synchrotron / inverse Comp- − = cσT γe , (15) ton (solid lines). Bottom: Energy loss rates from inverse Compton dt Sy 3 8π are shown for varying temperature. As the temperature increases, s−s the energy at which the energy loss rate becomes energy indepen- where φH i ≈ 45 is the scattering function for strongly dent decreases. shielded gas. The loss rates for ionization and bremsstrahlung both scale with the gas density (note that nmol nion) and be- Because of the extreme environment of Arp 220, the energy- cause of the relative energy dependencies, bremsstrahlung loss time-scale is significantly shorter than either the diffu- losses dominate over ionization losses starting at Ee & sion or advective time-scales, see Fig.4. Further, an advec- 0.5 GeV, see Fig.5. Similarly, the loss rates for inverse tive time-scale competitive with the energy-loss time-scale Compton and synchrotron have the same energy depen- would only result in the need for a larger cosmic ray energy dence, and thus, inverse Compton losses dominate for Urad > injection rate for the observed γ-ray spectrum, increasing UB and synchrotron losses dominate for Urad < UB , where 2 the discrepancy between the injection rate derived from the UB = B /8π. γ-ray spectrum and the observed supernova rate. Thus, to For ‘low’ densities (n ∼ 200 cm−3) and ‘high’ magnetic obtain a conservative lower limit on the cosmic ray injection or radiation fields (U > 105 eV cm−3), synchrotron or in- rate, we will assume that the total lifetime for the cosmic verse Compton losses dominate for all relevant energies (see ray electrons is simply Fig.5). However, for ‘moderate’ densities ( n ∼ 1000 cm−3) and more ‘moderate’ magnetic or radiation fields (U > te(Ee) = t (Ee) ' Ee/(−dE/dt). (10) loss 1000 eV cm−3), bremsstrahlung losses are dominate at low Based on the above equations, lowering the radio flux energies (Ee < 1.0 GeV) and synchrotron or inverse Comp- Sy requires lowering either the synchrotron power (Pν ) or the ton losses dominate at higher energies, see Fig.5.

MNRAS 000, 000–000 (0000) 6 T. M. Yoast-Hull & N. Murray

In equation (14), γK is the critical√ Klein-Nishina Table 1. Fiducial Inputs 2 Lorentz factor which is given by γK = 3 5mec /8πkTrad (Schlickeiser & Ruppel 2010). For γ γK, the inverse Parameters East West CND Ref. Compton cross section is well approximated by the Thomp- Radius son cross section, while for γ ≥ γK the full Klein-Nishina 85 65 15 1,2 cross section must be used and the energy loss rate from R [pc] inverse Compton becomes energy independent. As shown in Scale Height 30 30 10 1 Fig.5, this change in the cross section becomes important h [pc] for either high energies (E 10 TeV), high dust tempera- e & Column Density 1025 1025 1026 1,3 tures (Trad & 250 K), or high optical depths. −2 NH2 [cm ] Molecular Gas Massa 8 10 5.9 5.9 4 MH2 [10 M ] 3 OBSERVATIONAL CONSTRAINTS Average ISM Densitya 1.1 1.2 65 n [104 cm−3] As in previous work (Yoast-Hull et al. 2015, 2017), we model H2 the eastern nucleus with a one-zone model and the western FIR Luminosity 11 3.0 3.0 6.0 1,3 nucleus as two zones: an inner circumnuclear disc (CND) LIR [10 L ] and an outer torus. Here, we will discuss the relevant prop- FIR Energy Densitya erties for both nuclei, and we have listed these properties in 4 −3 2.8 4.2 150 UIR [10 eV cm ] Table1. Dust Temperaturea Based on observations of the molecular media and high- 50 55 135 resolution radio continuum measurements, we assume radii TIR [K] of R = 85 pc, R = 65 pc, and R = 15 pc for the eastern Notes. aDerived from above parameters; References: nucleus, the outer western nucleus, and the CND, respec- (1) Scoville et al.(2017); (2) Barcos-Mu˜nozet al.(2015); tively (Barcos-Mu˜nozet al. 2015; Scoville et al. 2017). We (3) Wilson et al.(2014); (4) Bolatto et al.(2013) assume scale-heights of h = 30 pc in the eastern and outer western nuclei and h = 10 pc for the CND (Scoville et al.

2017; Sakamoto et al. 2017). 4 From the Stefan-Boltzmann law (UIR = aTIR), we can also calculate an effective dust temperature from the infrared luminosity such that 3.1 Molecular Gas Masses 1/4 LIR Millimeter wavelength observations provide probes of the H2 TIR = , (18) 25 −2 (2πR2 + 2πRH) × ca column density (NH2 ) ranging from 10 cm to greater 26 −2 than 10 cm (Wilson et al. 2014; Scoville et al. 2017), as- where a is the radiation constant a = 4σ/c. suming a Galactic CO-to-H2 conversion factor of XCO,20 = For the CND, estimates from millimeter wavelength ob- 2. Modeling and observations suggest that the conversion 11 servations put the infrared luminosity at LIR ' 6 × 10 L factor should be lower in ULIRGs, with potential values (Wilson et al. 2014; Scoville et al. 2017). This corresponds ranging from XCO,20 ' 0.2 − 1.0 (Bolatto et al. 2013, and to an effective dust temperature of 135 K with an energy references therein). We require that the molecular gas mass density of 1.5 × 106 eV cm−3. Adopting slightly less intense be less than the dynamical mass. So, for a disc of radius luminosities for the eastern nucleus and the outer western R and total height H, we calculate the molecular gas mass nucleus gives us effective dust temperatures of ∼ 50 K and such that energy densities of ∼ 104 eV cm−3, see Table1. ULIRG XCO MH2 = πRH × mH2 NH2 MW . (16) XCO 3.3 Optical Depths For the eastern and outer western nuclei, we adopt moderate column densities of 1025 cm−2, and for the CND, we To lower the total radio emission while simultaneously pre- assume a density of 1026 cm−2 based on Scoville et al.(2017). serving a high energy injection rate, one could increase the ULIRG By choosing a moderate conversion factor of XCO,20 = 0.6, radiation field energy density (or the infrared optical depth), 9 we settle on molecular gas masses of ∼ 10 M for each increase the average ISM density, or decrease the magnetic of the nuclei, see Table1, which is consistent with other field strength. From Zeeman splitting in OH masers in Arp estimates of the gas mass (e.g., Barcos-Muñozet al. 2015). 220, observations confirm that the magnetic field strengths in the Arp 220 nuclei are of mG strength (McBride et al. 2015). Magnetic fields of similar strengths are found even in 3.2 Radiation Fields extremely dense molecular gas clouds in our own Milky Way (e.g., Crutcher 2012, and references therein). Thus, we can- Measurements of the infrared luminosity give us a means to not decrease the magnetic field strength significantly from calculate a lower limit on the radiation field energy density previous models (Yoast-Hull et al. 2016). (UIR) such that Similarly, estimates of the dynamical masses for the nu- 9 L clei are ∼ 2×10 M which provides a rough upper limit on U = IR . (17) IR 4πR2 × c the molecular gas mass (Scoville et al. 2017). Therefore, we

MNRAS 000, 000–000 (0000) Optical Depths in Arp 220 7 cannot increase the average ISM density significantly above Table 2. Cosmic Ray Energy Injection Rates previous models (Yoast-Hull et al. 2017). Based on these observational constraints, we will be fo- East West CND cusing on the optical depth and thus the radiation field en- Scenario A (Protons & Electrons) ergy density Urad. For an optically thick disc, the radiation field energy density in the disc is higher than the emergent Range 0.5 − 10 0.5 − 10 1.0 − 20 radiation field energy density, inferred from observations us- Step Size 0.5 0.5 1.0 ing equation (17). The energy density and dust temperature in the disc are related to the emergent flux and radiation Scenario B (Protons Only) temperature such that Range 1.0 1.0 2.0 − 40 Step Size 0 0 2.0 Urad = τIRUIR (19) 1/4 Scenario C (Electrons Only) Trad = τIR TIR. (20) Range 1.0 1.0 101 − 105 The effective optical depth (τIR) used here is the opti- Step Size 0 0 100.2 cal depth of the integrated infrared spectrum, which peaks near 60 µm (González-Alfonsoet al. 2004). We will discuss Notes: In all cases, the units for the injection rates 50 −1 the equivalent optical depth for millimeter wavelengths and are 10 erg yr . compare with observational estimates in Section 5.1. Using these relations, we find that for the optically thick inject only primary protons, including secondary electrons case (τIR ≥ 1) and positrons but not primary electrons. In Scenario C, we similarly fix the injection rates in the eastern and western dE − ∝ U τ , (21) nuclei and test a range of rates from 1051 − 1055 erg yr−1 in dt IR IR IC steps of 100.2 yr−1 in the CND, see Table2. For Scenario C, 1/2 UIRτIR we only include primary electrons and no primary protons qIC(Eγ ) ∝ 2 . (22) (kTIR) or secondary electrons and positrons. In each scenario, we test the models against both the Additionally, from equations (B10)&(B12), we know that γ-ray and radio spectra while varying the injection spectral the inverse Compton emissivity (qIC) depends on equa- index (2.3, 2.4, 2.5), magnetic field strength (0.1−100 mG), 1/4 4 tion (B11), I(u × τIR kTIR,Eγ ), which depends on the elec- and optical depth (1 − 10 ). For the radio spectra, we also tron loss time-scale and thus the electron energy loss rates, vary the absorption fraction (0.2 − 1.0) and the ionized gas see AppendixB for details. density (50 − 800 cm−3), both of which affect thermal radio emission. For our χ2 tests against the γ-ray data, we assume values for the spectral index and optical depth are 4 RESULTS equal across each of the regions (East, West, CND) in order to reduce the computation time, and we assume that the 4.1 Model Setup magnetic field strength in the CND is ∼ 7 times larger than in the East and the West based on the difference in ISM The goal of this study is to find a solution or set of solu- densities. tions in parameter space via χ2 tests that simultaneously Regarding radio data, there exist separate spectra for fit the γ-ray and radio spectra. For the γ-ray spectra, the the eastern and western nuclei. Thus, in Scenario A, we as- most critical parameters are the cosmic ray spectral index sume separate values for spectral index, optical depth, mag- and the cosmic ray energy injection rate. Because the γ-ray netic field, ionized gas density, and absorption fraction in observations provide no information about regarding spatial the eastern nucleus versus the western nucleus, and we also distribution, we test three different scenarios for where and assume that the spectral index, optical depth, and absorp- how energy is injected into cosmic rays. tion fraction are constant across the outer western nucleus Previously, we began with injection rates of 0.7 × and the inner CND. We also assume that the ionized gas 1050 erg yr−1, 0.7 × 1050 erg yr−1, and 1.3 × 1051 erg yr−1 density increases by a factor of ∼ 50 from the outer western for the eastern nucleus, western nucleus, and CND, respec- nucleus to the inner CND. tively (Yoast-Hull et al. 2015; Varenius et al. 2016). Thus, In Scenarios B & C, we also assume that the bulk of in Scenario A, we begin by assuming similar initial injec- the γ-ray emission and the minority of the radio emission tion rates of 0.5 × 1050 erg yr−1, 0.5 × 1050 erg yr−1, and comes from the CND. Thus, for the CND, we require that 1050 erg yr−1 and increase the injection rates in equal pro- the radio emission be less than 1% of the total emission from portion up to a total injection rate of 4 × 1051 erg yr−1, see the western nucleus. As such, we can test separate spectral Table2. This model includes primary protons along with indices, magnetic field strengths, and optical depths for the primary and secondary electrons, with the assumption that West versus the CND. the primary electron-to-proton (number) ratio is ξ = 0.02 (Yoast-Hull et al. 2013). In Scenario B, we fix the injection rates in the eastern 4.2 Gamma-Ray Spectra and outer western nuclei to 1050 erg yr−1 in each region to be consistent with the observed supernova rates, see Vare- The best-fitting results from the χ2 tests, performed against nius et al.(2016). In the CND, we test a range of injection the γ-ray data from Peng et al.(2016), require energy injec- rates from 2 × 1050 to 4 × 1051 erg yr−1, see Table2, and tion rates of 2×1051 erg yr−1 and 1051 erg yr−1 for Scenarios

MNRAS 000, 000–000 (0000) 8 T. M. Yoast-Hull & N. Murray

Gamma-Rays (Scenario A) Gamma-Rays (Scenario C, B = 2.5 mG) 104 101 104 101 Γp = 2.3 Γp = 2.3

103 103

102 102 ) ) R R I

I 1

1 τ

τ 10 10 ( (

h h t t p p

e 0 0 e 100 10 10 2 2 D χ D χ Γp = 2.4 Γp = 2.5

Γp = 2.4 Γp = 2.5 l l a a

0 c i c 10

i 3 3 t t 10

10 p p O O 2 102 10

1 101 10

0 1 100 10 10− 101 102 103 104 101 102 103 104 105 20 40 20 40 50 1 50 1 Injection Rate (εCR) [10 erg yr− ] Injection Rate (εCR) [10 erg yr− ]

Gamma-Rays (Scenario C, B = 10 mG) Gamma-Rays (Scenario B) 4 1 4 1 10 10 10 10 Γp = 2.3 Γp = 2.3 103 103

102 102 ) R ) I 1 τ R

I 10 1 (

τ 10 ( h

t h p t

e 0 0 p 10 10 2 D e 0 χ 10 2 Γp = 2.4 Γp = 2.5 l

D χ

Γp = 2.4 Γp = 2.5 a l c i a 0 3 t c 10 10 i 3 p t 10 O p O 102 102

101 101

0 1 10 10− 100 101 102 103 104 101 102 103 104 105 20 40 20 40 50 1 Injection Rate (εCR) [10 erg yr− ] 50 1 Injection Rate (εCR) [10 erg yr− ]

Figure 6. Chi-squared contours for Scenarios A and B for γ-ray Gamma-Rays (Scenario C, B = 40 mG) 104 101 data, injection rate (εCR) versus optical depth (τIR) The γ-ray Γp = 2.3 emission in Scenarios A and B is dominated by hadrons. Thus, 103 the results are not dependent on magnetic ﬁeld strength or optical depth. 102 ) R I 1 τ 10 (

h t

A and B, respectively, see Fig.6. These results for Scenar- p

e 0 0 10 10 2

D χ ios A and B are independent of optical depth (see Fig.6), Γp = 2.4 Γp = 2.5 l a c as hadronic γ-rays dominate the emission in these models. i 3 t 10 p

Additionally, these models are best fit with spectral indices O of Γp ' 2.4, closely matching the spectral index (Γγ = 2.35) 102 derived from fitting a pure power-law to the γ-ray spectrum (Peng et al. 2016). 101 For Scenario C, we assume injection of cosmic ray lep- 0 1 10 10− tons, and thus the energy injection rate required to fit the 101 102 103 104 101 102 103 104 105 50 1 γ-ray data changes with both optical depth and magnetic Injection Rate (εCR) [10 erg yr− ] field strength, see Fig.7. Overall, preferred models have energy injection rates of ≤ 1052 erg yr−1, and the best-fitting Figure 7. Chi-squared contours for Scenario C for γ-ray data, models require both high optical depths and high magnetic injection rate (εCR) versus optical depth (τIR). The γ-ray emis- field strengths. sion in Scenario C is dominated by leptons. Thus, as magnetic Examining Fig.7, we can see that at low magnetic field field strength increases, the dependency of the energy injection strengths, the energy injection rate is nearly independent of rate on optical depth increases. optical depth. However, as magnetic field strength increases,

MNRAS 000, 000–000 (0000) Optical Depths in Arp 220 9 the relationship between optical depth and injection rate nears a pure power-law. Further, these models show a clear 1.6 λ(ν) requirement for an optical depth significantly greater than τIR ≈ τν . (23) 60 µm unity, see Fig.7. In Wilson et al.(2014), the authors calculated optical depths of 1.7 and 5.3 for the eastern and western nuclei, 4.3 Radio Spectra respectively. The optical depths were derived from a com- bination of submillimeter observations with the Atacama For the radio spectra, we have collected data from Downes Large Millimeter/submillimeter Array (ALMA) at 691 GHz & Solomon(1998); Rodr´ıguez-Rico et al.(2005); Barcos- (434 µm) and at 345 GHz (869 µm). Converting these op- Muñozet al.(2015), covering a frequency range from 1.7 to tical depths to optical depths in the infrared range, we get 43 GHz. Here, we limit our analysis to models with only the τ60 = 40.3 and τ60 = 125.7 for the eastern and western nu- best-fitting energy injection rates from the γ-ray analysis. clei, respectively. These values would suggest that Scenarios For Scenario A, our best-fitting models for both the eastern B and C are more likely than Scenario A, as optical depths and western nuclei are restricted to magnetic field strengths larger than 103 were necessary for joint solutions between greater than 10 mG and optical depths greater than 103, the radio and γ-ray data in Scenario A. see Fig.8. For Scenarios B and C, where we have assumed constant injection rates of 1050 erg yr−1 for the East and West, we find solutions with magnetic field strengths as low 5.2 Energy Densities 3 as a few mG and optical depths ranging from unity to ∼ 10 . From our results, it is clear that there is significantly higher In the case of the CND, we consider any model produc- energy density in infrared radiation in the nuclei of Arp 220 ing radio emission that is less than 1% of the total radio than in the free-streaming emergent (observed) radiation emission from the western nucleus to be acceptably negli- field. This is consistent with submillimeter and millimeter gible in regards to the total radio emission. These models wavelength estimates of the optical depth at those wave- 2 are marked as having a χ value of 1 (corresponding to dark lengths (e.g., Wilson et al. 2014; Scoville et al. 2017). To purple) in Fig.8. If we allow for magnetic field strengths evaluate the how the energy density in the trapped radia- below 1 mG, then the minimum requirement for the opti- tion field compares with the other components of the ISM, cal depth for a joint solution is simply unity. If we require we evaluate the energy densities for turbulence, magnetic that the magnetic field strength be greater than or equal to fields, radiation fields, and cosmic rays below. 1 mG, then the optical depth must be larger than 10 for The turbulent energy density is given as Uturb = Scenario B and larger than 100 for Scenario C, see Fig.8. 2 ρσturb/2, where ρ is the ISM gas density (ρH2 = mH2 nH2 ). In all cases, it is clear that as magnetic field strength Measurements by Scoville et al.(2017) show the turbulent increases, optical depth must also increase, see Fig.8. This velocities to be 120 km s−1 and 250 km s−1 in the eastern is expected because as magnetic field strength increases, the and western nuclei, respectively. Combined with gas den- total synchrotron emissivity increases. Thus, to keep the ra- sities of ∼ 104 and 6.5 × 105 cm−3, these velocities give us dio emission small, the optical depth must also increase. turbulent energy densities of 1.7×106 and 4.2×108 eV cm−3. Above, we also found that for Scenario C, as the energy For magnetic field strengths of ∼ 4 − 20 mG, the corre- injection rate increases and assuming high (≥ 10 mG) mag- 2 5 sponding energy densities (UB = B /8π) range from 4×10 netic field strengths, the optical depth must decrease, see eV cm−3 to 107 eV cm−3. Thus, the magnetic field energy Fig.7. For the γ-rays, the γ-ray flux increases as the energy density lags slightly behind the turbulent energy density in injection rate increases. Thus, to avoid over-predicting the the eastern and outer western nuclei. However, in the CND, γ-ray flux, the optical depth must decrease as the energy the magnetic field energy density clearly lags behind the tur- injection rate increases. Regardless, there is clearly a mini- bulent energy density by more than an order of magnitude. mum optical depth necessary to provide a joint solution to For the radiation field, the total energy density is given the radio and γ-ray spectra. by Urad = τIRUIR. Thus, for the East and the West, the total radiation field energy densities range from ∼ 104 − 107 eV cm−3 based on the results of our χ2 analysis in Scenarios B and C. In the CND, the total radiation field energy density 5 DISCUSSION must be greater than ∼ 107 eV cm−3 and greater than ∼ 108 eV cm−3 in Scenarios B and C, respectively. 5.1 Observed Optical Depths If we take our infrared optical depths approximated As stated above, the best-fitting optical depths derived from from the Wilson et al.(2014) measurements, the correspond- our χ2 analysis are effective optical depths over the whole ing energy densities are 1.1 × 106 eV cm−3 for the eastern of the infrared range. Thus, a full and complete comparison / outer western nucleus versus 1.9 × 108 eV cm−3 for the between our τIR and optical depth values derived at specific CND. These specific values are within the range of energy wavelengths from observational measurements is complex. densities from our acceptable models and they agree well However, a rough comparison is simple enough if we assume with the turbulent energy densities. that our effective optical depth is approximately represen- Finally, for the cosmic rays, the energy density involves tative of the optical depth at the peak of the infrared spec- an integral over the energy spectrum such that UCR = trum, ∼ 60 µm in Arp 220 (González-Alfonsoet al. 2004; R EN(E)dE. For Scenarios B and C, the cosmic ray energy Rangwala et al. 2011). Assuming optical depth depends on densities for the East and West are 1.8 × 103 and 3 × 103 frequency to the 1.6 power in ULIRGs (Casey 2012), we have eV cm−3. Because Scenarios B and C have low assumed

MNRAS 000, 000–000 (0000) 10 T. M. Yoast-Hull & N. Murray

Radio (East, Scenario A) Radio (East, Scenario B / C) 104 103 104 103 Γp = 2.3 Γp = 2.3

103 103

102 102 ) ) R R I 1 I 1 τ 10 τ 10 ( (

h h t t p p

e 0 2 e 0 2 10 10 2 10 10 2

D χ D χ Γp = 2.4 Γp = 2.5 Γp = 2.4 Γp = 2.5 l l a a c c i 3 i 3 t 10 t 10 p p O O

102 102

101 101

100 101 100 101 4 3 2 4 3 2 1 4 3 2 4 3 2 1 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− Magnetic Field Strength (B) [Gauss] Magnetic Field Strength (B) [Gauss]

Radio (West, Scenario A) Radio (West, Scenario B / C) 104 103 104 103 Γp = 2.3 Γp = 2.3

103 103

102 102 ) ) R R I 1 I 1 τ 10 τ 10 ( (

h h t t

p 2 p 2

e 0 e 0 10 10 2 10 10 2

D χ D χ Γp = 2.4 Γp = 2.5 Γp = 2.4 Γp = 2.5 l l a a c c i 3 i 3 t 10 t 10 p p O O

102 102

101 101

101 101 100 100 4 3 2 4 3 2 1 4 3 2 4 3 2 1 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− Magnetic Field Strength (B) [Gauss] Magnetic Field Strength (B) [Gauss]

Radio (CND, Scenario B) Radio (CND, Scenario C) 104 101 104 101

103 103

102 102 ) ) R R I 1 I 1 τ 10 τ 10 ( (

h h t t Γp = 2.3 Γp = 2.3 p p

e 0 e 0 10 2 10 2

D χ D χ

l l a a c c i 3 i 3 t 10 t 10 p p O O

102 102

101 101

Γp = 2.4 Γp = 2.5 Γp = 2.4 Γp = 2.5 100 100 100 100 4 3 2 4 3 2 1 4 3 2 4 3 2 1 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− 10− Magnetic Field Strength (B) [Gauss] Magnetic Field Strength (B) [Gauss]

Figure 8. Chi-squared contours for radio data, magnetic ﬁeld strength (B) versus optical depth (τIR).

MNRAS 000, 000–000 (0000) Optical Depths in Arp 220 11

105 10-37 E = 1019 eV τ = 1 p, max τIR = 1

1 ] τ = 10 4 τ = 10 1 IR − ) 10 -39 p-p interactions 2 s γ τ = 100 τ = 10 10 IR 3 − γ 3 −

τ τ = 10 ( m

3 4 c

h 10 τ = 10 1 t − p 10-41 e V e D

G l 2

10 [ a

) c i π t -43 p E 10 ( O 1 π

10 q

-45 0 10 10 7 8 9 10 11 12 101 102 103 104 105 106 10 10 10 10 10 10 Energy (Eπ) [GeV] Energy (Eγ) [GeV]

Figure 9. Optical depths for photon-photon interactions are shown for diﬀering values of the infrared optical depth. As the τIR = 1 10-60 ] infrared optical depth increases, the photon-photon optical depth 1 − s

-102

also increases and the peak shifts down in energy. However, even 3 10 −

at for very high infrared optical depth values, the photon-photon m 15

c -144 Ep, max = 10 eV 10 optical depth is not greater than unity below 100 GeV. 1 − 16 Ep, max = 10 eV V -186 e 10 17

G Ep, max = 10 eV [

) 18

π -228 10 Ep, max = 10 eV E energy injection rates, along with high energy loss rates ( 19 π Ep, max = 10 eV q -270 for cosmic rays, the corresponding energy densities are low 10 20 Ep, max = 10 eV and consistent with our previous study in Yoast-Hull et al. (2015). For the CND, in Scenarios B and C, the total cosmic 106 107 108 109 1010 1011 1012 1013 ray energy densities are ∼ 4×104 eV cm−3. Thus, even with Energy (Eπ) [GeV] the additional cosmic ray population inferred from the γ-ray flux, the cosmic ray energy density is well below the other Figure 10. The source function for photopion production is energy densities. shown for varying infrared optical depths (top) and for varying maximum cosmic ray energies (bottom). The top figure also includes the source function for pion production from proton- 5.3 Further Interactions with Infrared Photons proton interactions (solid top lines). Maximum cosmic ray energies of greater than ∼ 1018 are necessary for pion production from Having established that the infrared optical depth in the nu- photon-proton interactions to even approach the pion production clei of Arp 220 is greater than unity, we must assess how this from proton-proton interactions. increase in the radiation energy density affects other inter- action rates. Specifically, we examine how the optical depth for photon-photon interactions and the source function for extends from 0.1 − 10 GeV. Thus, while there is a signifi- photopion production increase with increasing infrared op- cant increase in the γ-ray absorption due to photon-photon tical depth. interactions above 100 GeV, the spectral region covered by the Fermi γ-ray observations is optically thin. At present, there are no γ-ray detections at energies 5.3.1 Photon-Photon Interactions above 10 GeV, only an upper limit at 100 GeV from VER- ITAS (Fleischhack & VERITAS Collaboration 2015). How- From previous studies, we know that even considering only ever, a future detection by VERITAS or the Cherenkov Tele- the emergent infrared energy density, the radiation fields in scope Array (CTA) could provide valuable additional con- Arp 220 are large enough for photon-photon (γ − γ) interac- straints on the infrared optical depth and the cosmic ray tions to create significant absorption in the TeV energy γ-ray spectral index. spectrum (Torres 2004; Lacki & Thompson 2013; Yoast-Hull et al. 2015). The optical depth for γ − γ interactions depends on the integral over the photon number density over 5.3.2 Proton-Photon Interactions the square of the photon energy and is thus directly proportional to the radiation field energy density and inversely In an environment with extremely intense radiation fields, proportional to the temperature to the third power, see Ap- pions produced in proton-photon (p − γ) interactions can be pendixC for details. Hence, τγ−γ depends on the effective the dominant source of γ-ray and neutrino production. Here, 1/4 infrared optical depth such that τγ−γ ∝ τIR . we compare the source function for pions from p − γ pro- Calculating the photon-photon optical depth for vari- cesses against that from p − p processes, see Fig. 10. Using ous infrared optical depths, we naturally find that as τIR in- the simplified models for photo-hadronic interactions from creases, the peak optical depth for γ − γ interactions shifts Hümmeret al.(2010), we plot the source function for pho- up in value and down in energy, see Fig.9. However, we find topions for a variety of effective infrared optical depths and that for even very large values for the infrared optical depth, for varying maximum cosmic ray energies, see AppendixC τγ−γ does not rise above unity for energies below 100 GeV. for details about the source function. The γ-ray energy spectrum measured in Peng et al.(2016) Examining Fig. 10, we find that even at large optical

MNRAS 000, 000–000 (0000) 12 T. M. Yoast-Hull & N. Murray depths, the pion ﬂux from p−p interactions is orders of mag- ACKNOWLEDGEMENTS nitude larger than the pion ﬂux from p − γ interactions. Ad- We thank Jay Gallagher, Todd Thompson, and Almog Ya- ditionally, the maximum cosmic ray energy must be greater linewich for useful discussions on Arp 220 and on greybodies than 1018 eV for the cosmic ray protons in order for the pho- and radiative transfer. We also thank Mubdi Rahman for topion source function to even approach the source function discussions on statistics. This work was performed in part of pions from proton-proton interactions. at the Aspen Center for Physics, which is supported by Na- tional Science Foundation grant PHY-1607611. We acknowl- edge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC). This research was undertaken, in part, thanks to funding from the Canada Research Chairs program.

6 SUMMARY

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Reimer O., eds, Astrophysics and Space Science Proceedings In Fig. A1, we show FIR, γ-ray, and radio luminosities Vol. 34, Cosmic Rays in Star-Forming Environments. p. 299 for 60 nearby galaxies. Galaxies with γ-ray detections are (arXiv:1209.1196), doi:10.1007/978-3-642-35410-6˙21 Scoville N., et al., 2017, ApJ, 836, 66 shown in filled symbols and those with only γ-ray upper Soifer B. T., Boehmer L., Neugebauer G., Sanders D. B., 1989, limits are shown in open symbols. Pure star-forming galaxies AJ, 98, 766 are denoted with circular symbols and active galaxies are Stecker F. W., 1971, NASA Special Publication, 249 denoted with triangular symbols. Tang Q.-W., Wang X.-Y., Tam P.-H. T., 2014, ApJ, 794, 26 For each analysis, we ran 10,000 samples with a burn Thompson T. A., Quataert E., Waxman E., Murray N., Martin sample of 1000. Modeling for the radio and FIR luminosi- C. L., 2006, ApJ, 645, 186 ties converged. However, there were ∼ 2000 and ∼ 200 di- Thompson T. A., Quataert E., Waxman E., 2007, ApJ, 654, 219 vergences for the FIR and γ-ray luminosities and the ra- Torres D. F., 2004, ApJ, 617, 966 dio and γ-ray luminosities, respectively. With an extremely Varenius E., et al., 2016, A&A, 593, A86 Voelk H. J., 1989, A&A, 218, 67 small sample size of 8 galaxies and probable outliers (e.g., Williams P. K. G., Bower G. C., 2010, ApJ, 710, 1462 Circinus), there is little to be done to create convergence. Wilson C. D., Rangwala N., Glenn J., Maloney P. R., Spinoglio Thus, we see a significantly larger spread in the posterior L., Pereira-Santaella M., 2014, ApJ, 789, L36 models for the bottom two figures in Fig. A1. WojaczyńskiR., Nied´zwiecki A., Xie F.-G., Szanecki M., 2015, A&A, 584, A20 Xu C., Helou G., 1996, ApJ, 456, 152 A2 Multi-Wavelength Observations Yoast-Hull T. M., Everett J. E., Gallagher III J. S., Zweibel E. G., The total far-infrared fluxes for each galaxy are calculated 2013, ApJ, 768, 53 such that Yoast-Hull T. M., Gallagher III J. S., Zweibel E. G., 2014, ApJ, −11 790, 86 FFIR = 1.26 × 10 (2.58S60 + S100), (A1) Yoast-Hull T. M., Gallagher J. S., Zweibel E. G., 2015, MNRAS, −2 −1 453, 222 where FFIR is given in units of erg cm s (Sanders Yoast-Hull T. M., Gallagher J. S., Zweibel E. G., 2016, MNRAS, & Mirabel 1996, and references therein). Infrared data 457, L29 are taken from the IRAS Revised Bright Galaxy Sample Yoast-Hull T. M., Gallagher III J. S., Aalto S., Varenius E., 2017, (Sanders et al. 2003) and radio data are taken from the MNRAS, 469, L89 NRAO VLA Sky Survey (NVSS; Condon et al. 1998). γ-ray Yun M. S., Reddy N. A., Condon J. J., 2001, ApJ, 554, 803 data in Table A1 are taken from the Third Fermi Source de Jong T., Klein U., Wielebinski R., Wunderlich E., 1985, A&A, Catalog (Acero et al. 2015) and from Lopez et al.(2018) 147, L6 and Ackermann et al.(2016) for the SMC and the LMC, respectively. Upper-limits for γ-ray fluxes in Table A2 are taken from Rojas-Bravo & Araya(2016). Exceptions: APPENDIX A: NEARBY GALAXIES M31 (NGC 0224): The infrared fluxes are taken from Xu & Helou(1996), and the radio flux is taken from Denni- A1 Regression Analysis son et al.(1975). The correlation between the diffuse FIR and radio emission NGC 1144 : The infrared fluxes are taken from the IRAS in star-forming galaxies is well-established. Recent detec- Bright Galaxy Sample (Soifer et al. 1989).

MNRAS 000, 000–000 (0000) 14 T. M. Yoast-Hull & N. Murray

NGC 3627 : The radio ﬂux used is the sum of both NVSS 7 Star-Forming components associated with the galaxy, following Condon Active Galaxy et al.(2002). NGC 4945 : The radio ﬂux is taken from Elmouttie et al. 6 (1997). ] ¯ L

[ NGC 5055 : The radio ﬂux is taken from Condon et al.

) y

t (2002) and is inconsistent with the ﬂux listed in the NVSS i s

o 5

n catalog. i m

u M51 (NGC 5194): The radio ﬂux is taken from Condon L

o i et al.(2002) and references therein. d a

R 4 Circinus: The infrared ﬂuxes are taken from the IRAS ( g o

l Point Source Catalog (Helou & Walker 1988), and the radio ﬂux is taken from Elmouttie et al.(1998). 3 NGC 7331 : The radio ﬂux used is the sum of both NVSS components associated with the galaxy, following Condon et al.(2002). 9.0 9.5 10.0 10.5 11.0 11.5 12.0 log(Far Infrared Luminosity) [L ] ¯

APPENDIX B: LEPTONIC EMISSIVITIES 10 Star-Forming B1 Bremsstrahlung Active Galaxies 9 Magellanic Clouds The emissivity for bremsstrahlung is given as ]

¯ Z ∞ L

[ dσ

) 8 qBr = cnISM dEeNe(Ee) , (B1) y t i E dEγ

s min o n i 7 p m 2

u where Emin = Eγ (2mec + Eγ ) and the diﬀerential cross L

y section is given by (Ginzburg 1969; Blumenthal & Gould a R

a 1970; Stecker 1971; Schlickeiser 2002) m m

a "" 2# #

( 5 dσ 3ασT Eγ Eγ g = 1 + 1 − φ + 1 − φ .

o 1 2 l dEγ 8πEγ Ee Ee 4 (B2) The scattering functions φ1 and φ2 are related such that 2 3 φ2 = −2/3φ1 and are equivalent to to Z φu where φu 8 9 10 11 12 log(Far Infrared Luminosity) [L ] is given by (Ginzburg 1969; Blumenthal & Gould 1970; ¯ Schlickeiser 2002) 2Ee Ee − Eγ 1 10 φu = 4 ln 2 − . (B3) Star-Forming mec Eγ 2 Active Galaxy 9 ]

¯ B2 Photon Number Density L [

) y t i 8 Throughout the paper, we assume a greybody distribution s o n

i for optical depths of unity or less. The photon number den- m u

L sity for such a distribution is given by (Rybicki & Lightman 7 y

a 1979) R

m 2 β m Cdil

a 6

G nph() = , (B4) ( 2 3 3 /kTrad g π ~ c e − 1 0 o l 5 where β = 1.6 and λ0 = 200 µm, parameters which are adopted from Casey(2012). The number density is normal- ized such that (e.g., Ginzburg 1969; Rybicki & Lightman 4 3 4 5 6 7 1979; Ghisellini 2013) log(Radio Luminosity) [L ] ¯ Z ∞ Urad = nph()d (B5) Figure A1. For nearby star-forming galaxies, we show radio ver- 0 Z ∞ 3+β sus infrared luminosity (top), gamma-ray versus infrared luminos- Cdil −β = 2 3 3 0 /kT d ity (middle), and gamma-ray versus radio luminosity (bottom). π ~ c 0 e rad − 1 Galaxies with γ-ray detections by Fermi are shown with ﬁlled C −β symbols and open gray symbols indicate galaxies with γ-ray ﬂux dil 0 4+β = 2 3 3 (kTrad) Γ(4 + β)ζ(4 + β), upper limits. Posterior predictive regression lines (100 of 10,000) π ~ c are shown in solid light blue, with the mean trend line shown in dashed red. MNRAS 000, 000–000 (0000) Optical Depths in Arp 220 15

Table A1. Data for Star-Forming Galaxies with Gamma-Ray Detections

Galaxy S60 µm σ60 µm S100 µm σ100 µm S1.4 GHz σ1.4 GHz F0.1−100 GeV σ0.1−100 GeV (Jy) (mJy) (Jy) (mJy) (mJy) (mJy) (erg cm−2 s−1) (erg cm−2 s−1)

NGC 0224 595 2.8 × 104 3073 3.16 × 105 8600 1100 5.88 × 10−12 0.863 × 10−12 NGC 0253 968 65 1290 644 2990 114 8.65 × 10−12 9.22 × 10−13 NGC 1068 196 108 257 228 4850 170.4 6.97 × 10−12 8.55 × 10−13 NGC 2146 147 63 194 240 1070 39.9 3.58 × 10−12 6.75 × 10−13 NGC 3034 1480 74 1370 694 6200 209 1.07 × 10−11 8.37 × 10−13 NGC 4945 625 133 1330 797 6450 250 1.38 × 10−11 1.24 × 10−12 CIRCINUS 249 24.9 316 31.6 1370 – 1.30 × 10−11 2.01 × 10−12 ARP 220 104 112 115 138 327 9.8 1.92 × 10−12 4.30 × 10−13 SMC 6.69 × 103 – 1.50 × 104 – – – 2.90 × 10−11 1.0 × 10−12 LMC 8.29 × 104 – 1.85 × 105 – – – 2.24 × 10−10 –

Notes: NGC 0224 = M31, NGC 3034 = M82.

where Γ(x) is the gamma function and ζ(x) is the Riemann Compton emissivity, we start by designating the second in- zeta function. Solving for the dilution factor, Cdil, we have tegral as I(, Eγ ) such that 2 3 3 β Z ∞ π ~ c 0 Urad 3 nph() qIC(Eγ ) = cσT d I(, Eγ ), (B10) Cdil = 4+β , (B6) (kTrad) Γ(4 + β)ζ(4 + β) 4 0 and substituting back into the number density gives us, where Z ∞ 2+β Ne(Ee) Urad I(, Eγ ) = dEe 2 F (q, Γ). (B11) nph() = 4+β × /kT . (B7) γ (kTrad) Γ(4 + β)ζ(4 + β) e rad − 1 Emin e In the optically thin case, β = 1.6 and thus Γ(5.6)ζ(5.6) ≈ Then, substituting our expression for the greybody distribu- 63. In the optically thick case, β = 0 and thus Γ(4)ζ(4) = tion from above into our expression for emissivity, we have 2 π /15 ≈ 6.5. Z ∞ 1+β 3cσT Urad qIC(Eγ ) = 4+β d /kT I(, Eγ ), 4K(β)(kTrad) 0 e rad − 1 B3 Inverse Compton Emissivity where K(β) = Γ(4 + β)ζ(4 + β). Changing variables from The emissivity for inverse Compton emission is given as to u = /kTrad gives Z ∞ (Blumenthal & Gould 1970; Schlickeiser 2002) 3cσT Urad qIC(Eγ ) = du × (kTrad) Z ∞ Z ∞ 4+β 3 nph() Ne(Ee) 4K(β)(kTrad) 0 qIC(Eγ ) = cσT d dEe 2 F (q, Γ), 1+β 1+β 4 γ u × (kTrad) 0 Emin e × I(u × (kT ),E ) (B8) eu − 1 rad γ with units of GeV−1 cm−3 s−1, where the minimum cosmic Z ∞ 1+β 3cσT Urad u du ray electron energy is given by qIC(Eγ ) = 2 u I(u × kTrad,Eγ ). 4K(β)(kTrad) 0 e − 1 " 2 4 1/2# (B12) 1 mec Emin = Eγ 1 + 1 + . 2 Eγ Thus, the inverse Compton emissivity decreases as the square of the dust temperature and also changes as I(, Eγ ) Here, Eγ is the energy of the resulting γ-ray and is the changes. As I(, Eγ ) depends on Ne(Ee) = qe(Ee)te(Ee), energy of the incident photon. The function F (q, Γ) is part the expression will change as the electron lifetime changes of the Klein-Nishina cross section and is given by (Ginzburg with dust temperature. 1969; Blumenthal & Gould 1970; Stecker 1971; Rybicki & Lightman 1979; Schlickeiser 2002; Dermer & Menon 2009; Boettcher et al. 2012; Ghisellini 2013) APPENDIX C: PHOTON INTERACTIONS Γ2q2(1 − q) F (q, Γ) = 2q ln(q) + (1 + q − 2q2) + , (B9) 2(1 + Γq) For photon-photon (γ − γ) interactions, the cross section is given by where 3 2 4γe Eγ σγ−γ (βcm) = σT (1 − βcm) Γ = 2 and q = 2 . 16 mec Γ(γemec − Eγ ) 4 1 + βcm 2 × (3 − βcm) ln − 2βcm(2 − βcm) , Here, Γ is a product of the photon and electron energies and 1 − βcm is not to be confused with the gamma function, Γ(x), or the (C1) cosmic ray spectral index, Γp. −2 1/2 To better understand the eﬀects of the dust tempera- where βcm = (1 − γcm ) and γcm is the center-of- ture (Trad) and to simplify our expression for the inverse momentum frame Lorentz factor of the produced electron

MNRAS 000, 000–000 (0000) 16 T. M. Yoast-Hull & N. Murray

Table A2. Data for Star-Forming Galaxies without Gamma-Ray Detections

Galaxy S60 µm σ60 µm S100 µm σ100 µm S1.4 GHz σ1.4 GHz F0.1−100 GeV (Jy) (mJy) (Jy) (mJy) (mJy) (mJy) (10−12 erg cm−2 s−1)

IC 1623 22.9 62 31.6 113 249 9.8 0.481 NGC 0520 31.5 30 47.4 146 176 5.3 0.396 NGC 0660 65.5 88 115 134 373 13.9 1.06 NGC 0695 7.59 31 13.6 167 74.8 3.1 1.36 NGC 0828 11.5 53 25.3 113 104 3.7 0.715 MRK 1027 5.28 39 8.57 108 53.2 1.6 0.521 NGC 0891 66.5 71 172 353 239 7.9 2.04 NGC 1022 19.7 35 27.3 111 46.7 1.5 1.12 NGC 1055 23.4 51 65.3 82 199 6.7 1.56 NGC 1144 5.06 38 11.5 198 155 5.4 0.328 NGC 1365 94.3 33 166 126 376 13 1.33 IC 0342 181 70 392 434 191 7.3 1.83 NGC 1530 9.88 43 25.8 128 67.3 2.7 1.77 NGC 1614 32.1 83 34.3 430 137 4.9 1.14 VII Zw 31 5.51 31 10.1 126 41.4 1.3 0.151 NGC 2276 14.3 58 29.0 133 267 8.6 0.522 ARP 055 6.07 48 10.3 109 36.8 1.2 0.583 NGC 2903 60.5 52 130 208 445 13.9 1.08 UGC 05101 11.7 34 19.9 137 170 5.8 0.383 NGC 3079 50.7 64 105 141 769 27.1 1.40 NGC 3147 8.17 38 29.6 222 89.9 3.4 0.192 ARP 148 6.38 34 10.3 106 36.4 1.2 0.797 NGC 3556 32.6 68 76.9 231 217 7.1 1.11 NGC 3627 66.3 59 137 118 453 11.2 1.01 NGC 3628 54.8 77 106 240 291 8.7 0.866 ARP 299 113 52 111 133 677 25.4 1.08 NGC 3893 15.6 54 36.8 63 139 5 0.431 NGC 4030 18.5 46 50.9 117 154 5.3 0.399 NGC 4041 14.2 19 31.7 126 102 3.8 0.875 NGC 4414 29.6 80 70.7 72 240 7.9 0.578 NGC 4631 85.4 62 160 260 446 14.1 0.635 MRK 0231 30.8 42 29.7 108 309 12.1 0.651 NGC 4826 36.7 77 81.7 99 99.9 3.7 0.755 NGC 5005 22.2 21 63.4 131 181 6.2 1.11 NGC 5055 40.0 49 140 356 350 – 0.626 ARP 193 17.0 88 24.4 120 104 3.2 0.983 NGC 5135 16.9 46 31.0 177 200 7.1 1.28 NGC 5194 97.4 193 221 329 1570 – 0.723 NGC 5236 266 64 524 302 405 15 0.609 MRK 0273 22.5 42 22.5 70 145 5.1 0.449 NGC 5678 9.67 29 25.7 85 110 3.9 0.496 NGC 5713 22.1 65 37.3 88 158 5.7 0.203 NGC 5775 23.6 46 55.6 97 280 9.1 0.680 NGC 6240 22.9 54 26.5 174 426 15 0.269 NGC 6701 10.1 28 20.1 67 88.9 3.1 0.426 NGC 6946 130 71 291 458 187 6.7 0.551 IC 5179 19.4 62 37.3 90 169 6.2 0.293 NGC 7130 16.7 44 25.9 132 190 7.5 0.227 NGC 7331 45.0 91 110 468 328 8.1 1.19 NGC 7469 27.3 40 35.2 599 181 5.4 0.847 NGC 7479 14.9 49 26.7 247 99.0 3.7 1.95 NGC 7771 19.7 135 40.1 839 141 5.2 0.905 MRK 0331 18 46 22.7 194 70.7 2.2 0.303

Notes: NGC 5194 = M51, NGC 5236 = M83.

MNRAS 000, 000–000 (0000) Optical Depths in Arp 220 17 and positron (Dermer & Menon 2009; Boettcher et al. 2012). Then, the optical depth for γ − γ interactions is dτ 2 Z ∞ n () Z s0 γ−γ ( ) = d ph dssσ (s), (C2) dx 1 2 γ−γ 1 1/1 0 where s0 = 1 is the maximum center-of-momentum energy for head-on collisions. Dermer & Menon(2009) de- fine dτγ−γ /dx as the absorption probability per unit path length, so the approximate optical depth is merely τγ−γ ' lpath × dτγ−γ /dx. For an inclined ellipse, we take the path ◦ length through the disc to be lpath = π/2 × R sec(90 − i). If we substitute into equation C2 with our expression for the greybody distribution from Section B2 and change variables from to u = /kTrad, we have dτ 2 3cσ U Z ∞ γ−γ ( ) = × T rad du × (kT ) dx 1 4K(β)(kT )4+β rad 1 rad umin β β Z s0 u × (kTrad) × u dssσγ−γ (s), (C3) e − 1 0 dτ 2 3cσ U Z ∞ uβ γ−γ ( ) = × T rad du dx 1 4K(β)(kT )3 eu − 1 1 rad umin Z s0 × dssσγ−γ (s). (C4) 0 Thus, for γ − γ interactions, the optical depth is directly 3 proportional to Urad/(kTrad) . For proton-photon (p−γ) interactions, we utilize the parameterizations presented in Hümmer et al.(2010), which is in turn based on the SOFIA code (Mücke et al. 2000). Their parameterizations cover photopion production through both direct and multi-photon production and via the ∆-resonance and higher order ∆- and N- resonances. The parameterized source functions (equation 28) take the form E m Z ∞ m yχIT QIT = N b p dyn p M ITf IT(y), b p χIT E γ E b b th/2 b (C5) where y = Ep/mp and χ, M are parameterization constants given in Tables 4-6 in Hümmer et al.(2010). The functions f(y) are further parameterizations given in their equations 31-35 and 38-39. The parameterization functions are given for each of the types of production (resonances, direct, multi- pion) and the parameterization constants are further broken down for each type of pion (π+, π−, π0).

MNRAS 000, 000–000 (0000)