The Neutrino Mechanism of Core-Collapse Supernovae
Dissertation
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Ondˇrej Pejcha
Graduate Program in Astronomy
The Ohio State University 2013
Dissertation Committee: Professor Todd Thompson, Advisor Professor Christopher Kochanek Professor Marc Pinsonneault Professor Krzysztof Stanek Copyright by
Ondˇrej Pejcha
2013 ABSTRACT
The core of a massive star at the end of its life collapses and launches an outgoing shockwave. Simulations show that the shock wave evolves into a quasi-static accretion shock, but it proves difficult to revive its outward propagation. The stalled accretion shock turns into explosion when the neutrino luminosity from the collapsed
crit core exceeds a critical value (Lν, core) (the “neutrino mechanism”).
crit I study the physics of Lν, core and its parameter dependence. I quantify the connection between the steady-state isothermal accretion flows with bounding shocks and the neutrino mechanism. I show that there is a maximum, critical sound speed
crit cT above which it is impossible to maintain accretion with a standoff shock. I derive this “antesonic” condition, which characterizes the transition to explosion over a broad range in accretion rate, proto-neutron star properties and microphysics.
crit Additionally, I characterize the effects of collective neutrino oscillations on Lν, core,
crit which can decrease Lν, core by a sizeable amount. However, I find that collective neutrino oscillations are generally suppressed by matter effects.
The physics of the explosion mechanism and the progenitor structure are imprinted in the observed properties of the explosion and the compact remnants. I
ii present preliminary study combining the formalism of critical neutrino luminosity with time evolution of a range of progenitor stars. I construct a semi-analytic model of supernovae that ties together the mass function of the compact remnants with the rate of successful explosions. Both of these quantities can be constrained by observations.
I use Bayesian analysis to compare the double neutron star mass distribution with supernova explosion models. I infer the properties of the progenitor binary population and fallback during the explosion. I constrain the mass coordinate where the explosion develops, which has direct implications for the mechanism of supernova explosions.
iii To my family
iv ACKNOWLEDGMENTS
I would like to express my deepest gratitude and thanks to my advisor Todd
Thompson. Not only has he been the source of great ideas for projects, but his enthusiasm for astronomy and science in general has been very addictive. He has provided a tremendous amount of support for my own scientific ideas, which often sidelined and delayed our joint work, but made me much happier and scientifically richer. I thank Todd for his compassion and understanding that gave me hope that it is possible not only to survive but to succeed as well.
I learned a lot about data analysis and many other areas of astrophysics from
Chris Kochanek. I admire his broad knowledge and attention to details while never losing sight of the big picture. I am thankful for astonishingly fast editing and comments on most of my papers.
I am grateful to Kris Stanek not only for many interesting ideas and projects that we worked on, but also for being available whenever I needed to talk about science or life. I learned a lot about science and astronomy from our discussions and
I thank Kris for being always very kind to me.
v I thank Marc Pinsonneault for serving on my graduation committee and teaching a great class on stars. I thank Basu Dasgupta for co-authoring the paper on collective neutrino oscillations. I admire Andy Gould for his effort to uncover the truth in science and I thank him for very useful advice and help. I thank Subo Dong for many discussions, selfless help and for arranging my talk at IAS. I am grateful to
Jose Prieto for discussions and help with my visit at Princeton. I am thankful to all professors, postdocs, and students for making astro coffee one of the things I have looked forward to every day. I thank the administrative and technical staff, including
Kristy Krehnovi, for making everything run smoothly. I acknowledge David Will for discussions and help in many areas. I am grateful to Bob Wing for great time we spent together. I thank my friends and officemates Calen, Ana, Kate, Sun, Gisella,
Jon, Katie, Claudia, Subo, Molly, and others for great company throughout the years.
I thank David Heyrovsk´yfor advising my bachelor and master theses at Charles
University and helping me with applications to the graduate school. I am grateful to all my teachers, mentors, and everybody else who helped me in my pursuit of astronomy.
I thank my family and especially my mother for support during my studies in the Czech Republic so that I could focus solely on science. Finally, I am grateful to my wife Eva for believing in me and all the love and support she has been giving me.
vi VITA
March 8, 1984 ...... Born – Brno, Czech Republic
2006 ...... Bc. Physics, Charles University, Czech Republic
2008 ...... Mgr. Theoretical Physics, Charles University, Czech Republic
2008 – 2009 ...... Distinguished University Fellow, The Ohio State University
2009 – 2012 ...... Graduate Research and Teaching Associate, The Ohio State University
2012 – 2013 ...... Distinguished University Fellow, The Ohio State University
PUBLICATIONS
Research Publications
1. C. J. Grier, B. M. Peterson, Keith Horne, M. C. Bentz, R. W. Pogge, K. D. Denney, G. De Rosa, Paul Martini, C. S. Kochanek, Y. Zu, B. Shappee, R. Siverd, T. G. Beatty, S. G. Sergeev, S. Kaspi, C. Araya Salvo, J. C. Bird, D. J. Bord, G. A. Borman, X. Che, C. Chen, S. A. Cohen, M. Dietrich, V. T. Doroshenko, Yu. S. Efimov, N. Free, I. Ginsburg, C. B. Henderson, A. L. King, K. Mogren, M. Molina, A. M. Mosquera, S. V. Nazarov, D. N. Okhmat, O. Pejcha, S. Rafter, J. C. Shields, J. Skowron, D. M. Szczygiel, M. Valluri, and J. L. van Saders, “The Structure of the Broad-line Region in Active Galactic Nuclei. I. Reconstructed Velocity-delay Maps”, ApJ, 764, 47, (2013).
vii 2. O. Pejcha, B. Dasgupta, and T. A. Thompson, “Effect of Collective Neu- trino Oscillations on the Neutrino Mechanism of Core-Collapse Supernovae”, MNRAS, 425, 1083, (2012).
3. O. Pejcha, T. A. Thompson, and C. S. Kochanek, “The observed neutron star mass distribution as a probe of the supernova explosion mechanism”, MNRAS, 424, 1570 (2012).
4. C. J. Grier, B. M. Peterson, R. W. Pogge, K. D. Denney, M. C. Bentz, Paul Martini, S. G. Sergeev, S. Kaspi, T. Minezaki, Y. Zu, C. S. Kochanek, R. Siverd, B. Shappee, K. Z. Stanek, C. Araya Salvo, T. G. Beatty, J. C. Bird, D. J. Bord, G. A. Borman, X. Che, C. Chen, S. A. Cohen, M. Dietrich, V. T. Doroshenko, T. Drake, Yu. S. Efimov, N. Free, I. Ginsburg, C. B. Henderson, A. L. King, S. Koshida, K. Mogren, M. Molina, A. M. Mosquera, S. V. Nazarov, D. N. Okhmat, O. Pejcha, S. Rafter, J. C. Shields, J. Skowron, D. M. Szczygiel, M. Valluri, and J. L. van Saders, “Reverberation Mapping Results for Five Seyfert 1 Galaxies”, ApJ, 755, 60, (2012).
5. P. Cagaˇs,and O. Pejcha, “Discovery of a double eclipsing binary with pe- riods near a 3:2 ratio”, A&A, 744, L3, (2012).
6. O. Pejcha, and C. S. Kochanek, “A Global Physical Model for Cepheids”, ApJ, 748, 107, (2012).
7. O. Pejcha, and T. A. Thompson, “The Physics of the Neutrino Mecha- nism of Core-collapse Supernovae”, ApJ, 746, 106, (2012).
8. C. J. Grier, B. M. Peterson, R. W. Pogge, K. D. Denney, M. C. Bentz, Paul Martini, S. G. Sergeev, S. Kaspi, Y. Zu, C. S. Kochanek, B. J. Shappee, K. Z. Stanek, C. Araya Salvo, T. G. Beatty, J. C. Bird, D. J. Bord, G. A. Borman, X. Che, C. Chen, S. A. Cohen, M. Dietrich, V. T. Doroshenko, Yu. S. Efimov, N. Free, I. Ginsburg, C. B. Henderson, Keith Horne, A. L. King, K. Mogren, M. Molina, A. M. Mosquera, S. V. Nazarov, D. N. Okhmat, O. Pejcha, S. Rafter, J. C. Shields, J. Skowron, D. M. Szczygiel, M. Valluri, and J. L. van Saders, “A Reverberation Lag for the High-ionization Component of the Broad-line Region in the Narrow-line Seyfert 1 Mrk 335”, ApJ, 744, L4, (2012).
9. C. B. Henderson, K. Z. Stanek, O. Pejcha, and J. L. Prieto, “An R- and I-band Photometric Variability Survey of the Cygnus OB2 Association”, ApJS, 194, 27, (2011).
10. S. Poddan´y, L. Br´at, and O. Pejcha, “Exoplanet Transit Database. Re-
viii duction and processing of the photometric data of exoplanet transits”, New Astronomy, 15, 297, (2010).
11. T. Sumi, D. P. Bennett, I. A. Bond, A. Udalski V. Batista, M. Do- minik, P. Fouqu´e, D. Kubas, A. Gould, B. Macintosh, K. Cook, S. Dong, L. Skuljan, A. Cassan, F. Abe, C. S. Botzler, A. Fukui, K. Furusawa, J. B. Hearnshaw, Y. Itow, K. Kamiya, P. M. Kilmartin, A. Korpela, W. Lin, C. H. Ling, K. Masuda, Y. Matsubara, N. Miyake, Y. Muraki, M. Nagaya, T. Nagayama, K. Ohnishi, T. Okumura, Y. C. Perrott, N. Rattenbury, To. Saito, T. Sako, D. J. Sullivan, W. L. Sweatman, P. J. Tristram, P. C. M. Yock, J. P. Beaulieu, A. Cole, Ch. Coutures, M. F. Duran, J. Greenhill, F. Jablonski, U. Marboeuf, E. Martioli, E. Pedretti, O. Pejcha, P. Rojo, M. D. Albrow, S. Brillant, M. Bode, D. M. Bramich, M. J. Burgdorf, J. A. R. Caldwell, H. Calitz, E. Corrales, S. Dieters, D. Dominis Prester, J. Donatowicz, K. Hill, M. Hoffman, K. Horne, U. G. Jørgensen, N. Kains, S. Kane, J. B. Marquette, R. Martin, P. Meintjes, J. Menzies, K. R. Pollard, K. C. Sahu, C. Snodgrass, I. Steele, R. Street, Y. Tsapras, J. Wambsganss, A. Williams, M. Zub, M. K. Szyma´nski, M. Kubiak, G. Pietrzy´nski, I. Soszy´nski, O. Szewczyk, L. Wyrzykowski, K. Ulaczyk, W. Allen, G. W. Christie, D. L. DePoy, B. S. Gaudi, C. Han, J. Janczak, C.-U. Lee, J. McCormick, F. Mallia, B. Monard, T. Natusch, B.-G. Park, R. W. Pogge and R. Santallo, “A Cold Neptune-Mass Planet OGLE-2007-BLG-368Lb: Cold Neptunes Are Common”, ApJ, 710, 1641, (2010).
12. O. Pejcha, and K. Z. Stanek, “The Structure of the Large Magellanic Cloud Stellar Halo Derived using OGLE-III RR Lyr Stars”, ApJ, 704, 1730, (2009).
13. O. Pejcha, “Time-Dependent Rebrightenings in Classical Nova Outbursts: A Late-Time Episodic Fuel Burning?”, ApJ, 701, L119, (2009).
14. O. Pejcha, and D. Heyrovsk´y, “Extended-Source Effect and Chromaticity in Two-Point-Mass Microlensing”, ApJ, 690, 1772, (2009).
15. M. Uemura, T. Kato, R. Ishioka, S. Yoshida, K. Kadota, N. Ohkura, A. Henden, O. Pejcha, K. Kinugasa, M. Fujii, M. Simonsen, J. Greaves, P. A. Dubovsky, G. Poyner, D. West, R. J. Stine, D. Taylor, M. Poxon, E. Muyllaert, J. Ripero, M. Reszelski, and C. P. Jones, “Deep Fading of the New Herbig Be Star MisV1147”, PASJ, 56, 183, (2004).
FIELDS OF STUDY
Major Field: Astronomy
ix Table of Contents
Abstract ...... ii
Dedication ...... iv
Acknowledgments ...... v
Vita ...... vii
List of Tables ...... xiv
List of Figures ...... xv
Chapter 1 Introduction ...... 1
1.1 Overview...... 1
1.2 The neutrino mechanism of core-collapse supernovae ...... 5
crit 1.3 Physical effects on Lν, core: collective neutrino oscillations ...... 10 1.4 Constraining the supernova mechanism from observations: remnant massdistribution ...... 12
Chapter 2 The Physics of the Neutrino Mechanism ...... 19
2.1 Introduction...... 19
2.2 Isothermalaccretionboundedbyashock ...... 21
2.2.1 Topology of solutions ...... 22
x 2.2.2 Thecriticalsoundspeed ...... 26
2.3 Steady-state accretion shock in core-collapse supernovae ...... 30
2.3.1 Numericalsetup...... 31
2.3.2 Properties of the fiducial calculation ...... 35
2.3.3 Correspondence with isothermal accretion ...... 40
2.3.4 Properties of the critical solutions ...... 47
2.3.5 Effect of nuclear binding energy ...... 51
crit acc 2.3.6 Interpretation of Lν, core and the importance of Lν ...... 52 2.4 Explosionconditions ...... 55
2.4.1 Positive acceleration condition of Bethe & Wilson (1985) . . . 56
2.4.2 Explosion condition of Janka (2001) ...... 56
2.4.3 Advection time vs. heating time ...... 58
2.4.4 Antesoniccondition...... 60
2.5 Analytictoymodel ...... 62
2.5.1 Constructionofthetoymodel ...... 63
crit acc 2.5.2 Relation of Lν, core to Lν ...... 72
crit 2.5.3 Lν, core asafunctionofdimension ...... 73 2.6 Discussionandsummary ...... 76
Chapter 3 Effect of Collective Neutrino Oscillations ...... 102
3.1 Introduction...... 102
3.2 Method ...... 105
3.2.1 Hydrodynamic Equations and Boundary Conditions ...... 106
3.2.2 Treatment of CνO ...... 109
3.3 Results...... 115
xi crit 3.3.1 Effect of CνO on Lν, core ...... 116
3.3.2 ComparisontoOtherEffects...... 120
3.3.3 Role of Multi-Angle Matter Effects ...... 122
3.4 Discussion & Conclusions ...... 123
Chapter 4 Observational Signatures of the Explosion Mechanism . 129
4.1 Introduction...... 129
4.2 Elementsofthesemi-analytictheory ...... 133
4.3 Results...... 138
4.4 Discussion...... 140
Chapter 5 The observed neutron star mass distribution as a probe of the supernova explosion mechanism ...... 150
5.1 Introduction...... 150
5.2 Statisticalmodel ...... 151
5.2.1 General considerations ...... 151
5.2.2 NSsasmembersofabinary ...... 153
5.2.3 Data, underlying models and implementation ...... 157
5.3 Results...... 160
5.3.1 Propertiesofthebinarymodel...... 160
5.3.2 Comparison of the individual models ...... 162
5.3.3 Extensions and limitations ...... 167
5.3.4 The NS mass distribution of Ugliano et al. (2012) ...... 170
5.4 Discussions&Conclusions ...... 172
Chapter 6 Future work ...... 187
xii 6.1 TheStabilityofSteady-StateShocks ...... 188
6.2 Constraining the Explosion Mechanism with Supernova Observables . 190
Bibliography ...... 193
xiii List of Tables
2.1 Valuesofparametersusedinthetoymodel...... 81
5.1 Summary of the remnant mass distribution models from Zhang et al. (2008)...... 178
5.2 Summaryofdoubleneutronstarsystems...... 179
xiv List of Figures
2.1 Isothermalaccretion ...... 82
2.2 Maximumisothermalsoundspeed...... 83
2.3 Antesonic factor for polytropic equation of state ...... 84
2.4 Physical variables as a function of radius ...... 85
2.5 Properties of the two branches of solutions ...... 86
2.6 Overviewoftheflowstructureinsupernovae ...... 87
2.7 Detailed structure of the flows in supernovae ...... 88
2.8 Effects of different physics on the critical curve ...... 89
2.9 Properties of critical neutrino luminosity ...... 90
2.10Criticalshockradii ...... 91
2.11 Comparison of our calculations with 1D, 2D, and 3D results ..... 92
2.12 Effect of radiation transport and accretion luminosity on the critical curve...... 93
2.13 Critical conditions of Janka (2001) ...... 94
2.14 Advection and heating times as defined by Murphy & Burrows (2008) 95
2.15 Advection and heating times defined by Thompson et al. (2005) . . . 96
2.16 Theantesoniccondition ...... 97
2.17 Components of Bernoulli integral and their radial derivatives . . . . . 98
2.18 Shock radii as a function of Lν, core forthetoymodel...... 99
2.19 Comparison of toy model to the numerical solution ...... 100
xv 2.20 Illustration of effects of modified heating and cooling on shock radii . 101
3.1 Critical core neutrino luminosity and shock radii ...... 126
3.2 Relative reduction of critical core neutrino luminosity ...... 127
3.3 The relative importance of the matter suppression ...... 128
4.1 Density profiles of supernova progenitors ...... 142
4.2 Time evolution of mass accretion rate and PNS mass ...... 143
4.3 Time evolution as a function of equation of state ...... 144
4.4 Time evolution for different progenitors ...... 145
4.5 Ratioofshockradii...... 146
4.6 Ratio of core luminosity to critical luminosity ...... 147
4.7 Remnantmassfunction...... 148
4.8 Different calibrations of the neutrino mechanism ...... 149
5.1 Probability contours of DNS masses for best model ...... 180
5.2 Probability contours of DNS masses for model with fallback . . . . . 181
5.3 Probability that the pulsar in the DNS came from the initially more massiveprogenitor ...... 182
5.4 Relative probability of the different models for the origin of the DNS massdistribution ...... 183
5.5 Cumulative probability distributions ...... 184
5.6 Relative probability of the DNS mass models as a function of entropy masscut...... 185
5.7 Probability contours of DNS masses for Ugliano et al. (2012) model . 186
xvi Chapter 1
Introduction
1.1. Overview
In 1572, Tycho Brahe and other astronomers observed an appearance of a new star in the constellation of Cassiopeia. With peak brightness rivaling that of the planet Venus, the “Tycho’s supernova”, as it is called nowadays, challenged the contemporary concepts dating back to Aristotle that the sky beyond the Moon and planets is immutable initiating the revolution that led to modern astronomy.
Over the years, more supernovae have been discovered in the Milky Way and other galaxies showing that a single supernova can outshine the rest of its host galaxy’s hundred billion stars for more than a month. As the amount of observational data accumulated, the theoretical understanding of processes that lead to the brilliant explosion has grown as well. In particular, Baade & Zwicky (1934) proposed that supernovae loudly announce the transition from a normal star at least
10 times more massive and 10 to 1000 times bigger than our Sun to a neutron ∼ 1 star1, which typically has a mass of about 1.4 times the Sun but radius of only
10 km. Given the right conditions, a supernova explosion may also produce a black hole, an object so massive and small that not even light can escape.
The matter density inside a neutron star is fantastic – if a common ant was made of the same material as the neutron star it would weigh about 2 million metric tons. Clearly, extreme physics is required to explain the formation and structure of neutron stars. The first computer simulations in the 1960s such as those carried out by Colgate & White (1966) showed that chargeless weakly-interacting particles called neutrinos likely have tremendous importance in the supernova explosions. In fact, most of the supernova energy (essentially all of the 1053 ergs of the gravitational ∼ binding energy of the remnant neutron star) is carried away by 1058 neutrinos ∼ created in a burst lasting only several seconds. The electromagnetic energy that we observe as a supernova explosion is a mere 0.01% of the total released energy. ∼
The basic picture of neutron star birth during a supernova explosion was verified by the detection of 20 neutrinos from supernova 1987A in the nearby galaxy the
Large Magellanic Cloud (e.g. Bionta et al. 1987; Hirata et al. 1987; Loredo & Lamb
1There exists another type of supernova, usually called “type Ia”, which are manifestations of a thermonuclear explosion of a carbon-oxygen white dwarf. The historical Tycho and Kepler supernovae were probably of type Ia. Although supernovae Ia are important for measuring distances in the Universe, I will discuss in this Dissertation only supernovae associated with a core collapse of a massive star.
2 2002), so far the only supernova neutrinos directly detected. However, the small number of detected neutrinos was not enough to solve the basic problem, which has plagued supernova theory since the first numerical simulations in 1960s: supernova simulations do not robustly explode. Furthermore, the very mechanism by which supernovae explode is poorely known and very hard to constrain observationally.
Even nowadays, there is no theory that links the structure of the exploding star and the extreme physics of the “supernova engine” with the observed properties of supernova explosions and the properties of the neutron stars and black holes that remain after the explosion.
This gap in knowledge has significant repercussions for the rest of the astrophysics. Supernovae affect the evolution and formation of entire galaxies, injecting matter and energy in the interstellar medium and fostering the birth of the next generation of stars. Supernovae are also an important source of chemical enrichment of the Universe, producing carbon, oxygen and iron among other elements.
In this Dissertation, I study analytic and semi-analytic models of physical processes in the crucial phase of supernova explosions and I illustrate how they connect to observed properties. In the rest of this Chapter, I review in more detail the processes and physics that govern the supernova explosion and the masses of the remnants: neutron stars and black holes.
3 In Chapter 2, I study the steady-state structure of a flow between the proto-neutron star and the accretion shock. I analyze the concept of critical neutrino luminosity, which has been used to interpret supernova simulations, and I develop a simple isothermal model, which captures its basic physics. I also derive a new
“antesonic” condition that is equivalent to the critical neutrino luminosity. I discuss implications of these findings for supernova simulations. This Chapter is based on the work published by Pejcha & Thompson (2012).
Various physical effects can increase or decrease the critical neutrino luminosity.
In Chapter 3, I incorporate the microphysical effect of collective neutrino oscillations into the formalism for the critical neutrino luminosity with the hope that it would substantially lower the critical neutrino luminosity in the right part of the parameter space to produce explosions. However, I show that collective neutrino oscillations are not effective in lowering the critical neutrino luminosity for a wide range of supernova progenitors. This work was published by Pejcha et al. (2012a).
In Chapter 4, I discuss the ongoing work of combining the physics of the neutrino mechanism with the structure of the progenitor star. I show how the structure of the progenitor star is encoded in the boundary conditions of the
flow (proto-neutron star properties and mass accretion rate) and how this yields predictions that can be observationally verified.
4 In Chapter 5, I show what constraints can be put on the mechanism of supernovae by analyzing the mass function of neutron stars. I construct a Bayesian model of the double neutron star mass distribution and compare it to theoretical supernova explosion models. This work chronologically preceeds the work in Chapter
4 and is thus based on less physical explosion models published previously by other authors. This work was published by Pejcha et al. (2012b).
In Chapter 6, I discuss perspectives on the theory of supernovae and outline possible avenues for future work on this subject.
1.2. The neutrino mechanism of core-collapse
supernovae
At the end of the life of a massive star with initial mass > 8 M⊙, the degenerate ∼ core made of heavy elements grows by episodes of nuclear shell burning until the
Chandrasekhar instability initiates collapse. When the central matter reaches nuclear densities, it stiffens dramatically as a result of the hard core repulsion of the strong force, and drives a shockwave into the super-sonically collapsing outer mantle. The shock starts at an approximately constant mass coordinate of about
0.5 to 0.7 M⊙, which depends mainly on weak-interaction physics and nuclear interactions, and little on the progenitor structure (e.g. Goldreich & Weber 1980;
Yahil 1983; Bethe 1990). The outward progress of the shockwave is halted by
5 a combination of energy losses from the dissociation of nuclei across the shock, neutrino emission as the shockwave moves from the optically-thick inner core to the optically-thin surrounding envelope, and the ram pressure of the overlying infalling outer iron core. In other words, the mass of the iron core is too big to allow for a prompt explosion (e.g. Bethe 1990). A quasi-static accretion phase then ensues, which lingers for many dynamical times before eventual explosion. It consists of a hot, optically-thick, accreting proto-neutron star (PNS) with neutrinosphere radius
30 to 60 km, radiating its gravitational binding energy in neutrinos of all flavors, ∼ surrounded by a standoff accretion shock with radius of 200 km. This is a robust ∼ feature of supernova simulations (see Janka 2012; Janka et al. 2007, 2012; Burrows et al. 2007c; Burrows 2013, for a review).
As the supernova explosion develops within the inner core of the star, the overlying layers of stellar material prevent direct observations of the region aside from neutrinos or gravity waves emitted by very nearby supernovae (e.g. Ott et al.
2004; Y¨uksel & Beacom 2007; Ott et al. 2012). Thus, our understanding of the relevant processes is largely based on numerical simulations. Despite decades of modelling effort, the mechanism responsible for reviving the shockwave to positive velocities has not been conclusively identified. The “delayed neutrino mechanism” relies on neutrino heating in the semi-transparent subsonic accretion flow to drive explosion. A fraction of the neutrinos diffusing out of the collapsed PNS deposit their energy in the accretion flow, and for a sufficiently large heating rate explosion results.
6 This mechanism was originally discussed by Colgate & White (1966), and then developed by Bethe & Wilson (1985). However, the most detailed one-dimensional models generically fail to explode, except for the lowest mass progenitors (Rampp &
Janka 2000; Bruenn et al. 2001; Liebend¨orfer et al. 2001; Mezzacappa et al. 2001;
Thompson et al. 2003; Kitaura et al. 2006; Janka et al. 2008). These models clearly neglect multi-dimensional effects such as convection, both in the PNS interior (Keil et al. 1996; Mezzacappa et al. 1998; Dessart et al. 2006), and in the heating region between the neutrinosphere and the shock (Herant et al. 1992, 1994; Burrows et al.
1995; Janka & M¨uller 1996; Fryer & Heger 2000; Fryer & Warren 2002, 2004; Buras et al. 2006a), the standing accretion shock instability (SASI) and/or vortical-acoustic instability (Foglizzo 2002; Blondin et al. 2003; Blondin & Mezzacappa 2006; Ohnishi et al. 2006; Iwakami et al. 2008; Murphy & Burrows 2008; Marek & Janka 2009;
Nordhaus et al. 2010; Fern´andez 2010), and the potential for other multi-dimensional non-linear phenomena such as the “acoustic” mechanism of Burrows et al. (2006,
2007a), which was critically evaluated by Weinberg & Quataert (2008). Indeed, many of these works have claimed that multi-dimensional effects are crucial for explosion because convection between the PNS and the shock allows the matter to stay in the region of net energy deposition longer, and an increase in the average shock radius as a result of the SASI and convection makes the heating region larger, while putting more matter higher in the gravitational potential well. Even with all the multi-dimensional effects, the simulations do not produce explosions for
7 progenitors more massive than about 15 M⊙ (e.g. Rampp & Janka 2000; Bruenn et al. 2001; Liebend¨orfer et al. 2001; Mezzacappa et al. 2001; Kitaura et al. 2006;
Buras et al. 2006b; Marek & Janka 2009; Fischer et al. 2010; Suwa et al. 2010;
Takiwaki et al. 2012; M¨uller et al. 2012a). As these simulations address primarily non-rotating progenitors, it is worth mentioning that the combination of sufficiently rapid rotation and strong magnetic fields can potentially explode a wider range of progenitor stellar masses (e.g. LeBlanc & Wilson 1970; Symbalisty 1984; Akiyama et al. 2003; Thompson et al. 2005; Burrows et al. 2007b; Dessart et al. 2008; Suwa et al.
2010; Dessart et al. 2012), but it is not likely that the magneto-rotational explosions are responsible for the majority of observed supernovae.
An approach complimentary to these multi-dimensional modelling efforts was followed by Burrows & Goshy (1993) who calculated the steady-state structure of the region between the PNS and the accretion shock, during the quasi-time-steady epoch following collapse, bounce, and shock formation. For a fixed mass accretion rate, they increased the PNS core neutrino luminosity by hand until no steady-state
crit solution was possible. The critical core neutrino luminosity (Lν, core) that bounds steady-state accretion solutions from above has been identified with the initiation of the supernova explosion. With this knowledge in hand, the task of getting a supernova simulation to explode via the neutrino mechanism can be understood as either an effort to increase the neutrino flux (or effective heating rate) up to the
crit critical value, or to decrease Lν, core so that it can be reached with the available
8 luminosities. Examples of the former include the works by Wilson & Mayle (1988),
Bruenn & Dineva (1996), and Keil et al. (1996), which appealed to salt-finger
(doubly-diffusive) convection within the PNS to increase the core luminosity at early times after collapse, or Thompson et al. (2005), who found that viscous heating in models of rotating collapse could enable explosion. Examples of the latter include
Yamasaki & Yamada (2006), who showed that convection in the heating region
crit generically decreases Lν, core, or Murphy & Burrows (2008) and Nordhaus et al.
(2010), who found the same in 2D and 3D simulations (see also Hanke et al. 2012;
Couch 2012).
However, it is to a great extent unknown why there is a critical neutrino luminosity at all, what determines its existence, and how it scales with the
crit parameters of the problem. Indeed, the question of the physics of Lν, core is particularly paradoxical because during the stalled accretion shock phase, the size of the PNS, its neutrino luminosity, and the mass accretion rate through the shock change significantly, yet the simulations show that the position of the shock moves only very slowly. This implies there are many possible near-equilibrium configurations that yield similar shock radii for very different global parameters. For example, in low mass progenitors, the incoming mass accretion rate can decrease by a factor of 10 in the first second after bounce, but the neutrino luminosities vary by less than a factor of 2, and yet the shock radius is essentially constant. Given this early-time behavior, if the neutrino luminosity were to increase very slowly at
9 a later time in the evolution, why would one expect a catastrophic change from steady accretion to dynamical outward expansion? One might instead expect that an increase in the net energy deposition would simply result in the shock just getting pushed to some different equilibrium position with larger shock radius. Contrary to
crit this expectation, the existence of Lν, core implies instead that if the energy deposition below the shock reaches a critical value, the shock can no longer exist together with steady-state accretion, the shockwave expands dynamically, and the star explodes as a supernova (e.g., Burrows & Goshy 1993; Herant et al. 1994; Burrows et al. 1995;
Janka & M¨uller 1996). My results on the physics of the critical neutrino luminosity are discussed in Chapter 2.
crit 1.3. Physical effects on Lν, core: collective neutrino
oscillations
The critical neutrino luminosity is a useful framework for evaluating the sensitivity of the neutrino mechanism to various physical effects. It is difficult to include all known physics in supernova simulations and it is thus important to identify the one or several most important pieces. Historically, most emphasis has concentrated on multi-dimensional effects such as convection (Keil et al. 1996;
Mezzacappa et al. 1998; Dessart et al. 2006; Herant et al. 1992, 1994; Burrows et al.
1995; Janka & M¨uller 1996; Fryer & Heger 2000; Fryer & Warren 2002, 2004; Buras
10 et al. 2006a; Yamasaki & Yamada 2006), the SASI (Foglizzo 2002; Blondin et al.
2003; Blondin & Mezzacappa 2006; Ohnishi et al. 2006; Iwakami et al. 2008; Murphy
& Burrows 2008; Marek & Janka 2009; Nordhaus et al. 2010; Fern´andez 2010), and their relative importance (Burrows et al. 2012; M¨uller et al. 2012b; Dolence et al. 2013; Hanke et al. 2013). Recently, some 3D simulations were claimed to explode more easily than otherwise identical 2D simulations (Nordhaus et al. 2010), but this result has not been independently confirmed and it seems that critical luminosities are similar in 2D and 3D (Hanke et al. 2012; Couch 2012; Takiwaki et al. 2012; Dolence et al. 2013). The microphysics for densities and temperatures inside supernovae are not completely understood and this could have effect on the explosion as was shown by testing different forms of equation of state (e.g. Baron et al. 1985a,b; Sumiyoshi et al. 2005; Yamasaki & Yamada 2006; Hempel et al. 2012;
Couch 2013; Suwa et al. 2013). Other recently investigated physical effects include nuclear burning (e.g. Fern´andez & Thompson 2009a,b; Nakamura et al. 2012) and general relativity (e.g. Kuroda et al. 2012; M¨uller et al. 2012a; Ott et al. 2013).
Nonetheless, none of these physical effects seem to be the one primarily responsible for supernova explosions.
Since most of the energy of the supernova is released in neutrinos, it is important to understand what level of fidelity in calculating their propagation and feedback on the supernova is necessary to capture the explosion. In Chapter 2, I show that including a simple gray radiation transport between the proto-neutron
11 star and the accretion shock lowers the critical luminosity by about 8–23%. It is difficult to separate the effect of more advanced radiation transport from other assumptions in the small number of simulations that have been ran.
Recently, it has been shown that due to the high density of neutrinos in supernovae, self-interaction between neutrinos becomes important and can lead to a range of phenomena called “collective neutrino oscillations” (e.g. Pantaleone
1992; Duan et al. 2006, 2010). In particular, there is a possibility of an instability
(Dasgupta et al. 2009) that exchanges part of the µ and τ neutrino spectra with electron neutrino and antineutrino spectra. If the luminosities and energies of neutrinos are right, this can produce significantly more heating below the shock than what is obtained in calculations neglecting neutrino oscillations, i.e., effective
crit Lν, core is decreased by the collective neutrino oscillations. In Chapter 3, I present an investigation of collective neutrino oscillations in the framework of critical neutrino luminosity and show that this effect is not important for supernova dynamics in the typical investigated progenitors.
1.4. Constraining the supernova mechanism from
observations: remnant mass distribution
A part of understanding the supernova explosion mechanism is to provide predictions of properties of supernova explosions (energies, velocities, and amounts
12 of synthesized heavy nuclei) and the remaining compact objects (neutron star and black hole mass function, distribution of kick velocities, and rotation rates).
These predictions can be compared with observations, which will in turn better constrain the explosion mechanism. However, observational constraints on the causal chain that links massive stars, the supernova mechanism and their remnants are difficult to obtain and have many open problems. For example, examinations of pre-explosion images of type IIp supernovae2 suggest that the progenitors have initial masses lower than 16.5 1.5 M (Smartt et al. 2009) and, more generally, ± ⊙ there is a dearth of high-mass progenitor stars (Kochanek et al. 2008). The upper limit on the IIp progenitor mass is surprising, because red supergiants with masses of up to 25 M⊙ are observed (Levesque et al. 2005) and are thought to explode and produce NSs (Heger et al. 2003). While the possibility that these stars do not explode at all is intriguing (Kochanek et al. 2008), the statistical significance of this
“red supergiant problem” is only 2.4σ (Smartt et al. 2009). Other explanations involve red supergiants exploding as other types of supernovae due to differences in mass-loss rates (Smith et al. 2009; Yoon & Cantiello 2010; Moriya et al. 2011;
Georgy 2012), or binary evolution (Eldridge et al. 2011; Smith et al. 2011).
Calculating the explosion properties makes use of not only the explosion mechanism, but requires understanding of the propagation of the shock through the
2The plateau in the light curve indicates a presence of thick hydrogen envelope and a red supergiant progenitor (e.g. Chevalier 1976; Arnett 1980).
13 progenitor and the accompanying nuclear reactions. The properties of the compact remnants are to a large extent set early in the explosion and are thus much more tightly linked to the explosion mechanism. This is especially true for the mass function of the compact objects. As described in Section 1.2, at the moment of explosion, the core-collapse remnant mass is set by the mass at core bounce and the mass accreted during the steady-state accretion phase. The mass enclosed by the shock at the moment of the core bounce is approximately constant (about 0.5 to 0.7 M⊙) is too low when compared with typical observed masses of neutron stars of about 1.4 M⊙ and thus a significant amount of mass has to be accreted during the initial shock propagation and during the ensuing quasistatic evolution, which is terminated by the eventual explosion. If the explosion fails and does not occur, the remnant will continue accreting and will evolve into a black hole (e.g. Burrows 1986;
Liebend¨orfer et al. 2001; Heger et al. 2003; Kochanek et al. 2008; O’Connor & Ott
2011). Alternatively, a black hole can be formed if mass that fell back during the explosion increases the remnant mass above the maximum allowed NS mass (e.g.
Woosley & Weaver 1995; Zhang et al. 2008) or if a phase transition occurs in the cooling NS (e.g. Brown & Bethe 1994; Keil & Janka 1995). The fate of the compact object and its properties are thus to a large extent set by the duration and properties of the quasistatic accretion phase. In Chapter 4, I combine the formalism of the neutrino mechanism with the inner structure of the progenitor stars to predict the
14 compact object mass function and discuss how this depends on various assumptions made in the process.
In addition to the mass at core bounce and the mass accreted through the shock before explosion, the final mass of the core-collapse remnant is determined by the amount of fallback during the explosion3. Because self-consistent supernova simulations do not generically explode, the usual approach for studying fallback, supernova nucleosynthesis, and their observational consequences is to initiate an artificial explosion by depositing enough momentum or energy at a specific mass coordinate of the progenitor to match the final supernova energy or nickel yield
(Woosley & Weaver 1995; Thielemann et al. 1996; Timmes et al. 1996; Zhang et al. 2008; Dessart et al. 2011). The boundary conditions are adjusted for each progenitor star to give an explosion with the desired properties. The amount of fallback in these models depends on the boundary conditions (MacFadyen et al.
2001) and the structure of the progenitor. Broadly speaking, these simplified models predict that low-metallicity stars explode as supernovae when they are still hot and blue. Their compact envelopes thus generate stronger reverse shocks, more fallback, and higher mass remnants (Chevalier 1989; Zhang et al. 2008). Due to mass loss, solar-metallicity stars have less massive final envelopes, which generically lead to less fallback, and the strongest reverse shocks are produced at the helium/hydrogen
3The fraction of the PNS mass lost due to neutrino-driven wind in the explosion is negligible (e.g.
Thompson et al. 2001).
15 interface (Woosley & Weaver 1995). Stars without hydrogen envelopes, such as
Wolf-Rayet stars, experience little fallback and thus always produce NSs if the supernova mechanism is successful. In any model, more energetic explosions produce less fallback and smaller remnant masses (Zhang et al. 2008). In addition to increasing the mass of the remnant, variations in fallback could introduce an element of stochasticity into the NS or BH mass function (Ozel¨ et al. 2012).
As the remnant mass function encodes information about the structure of the progenitors and the supernova explosion mechanism, considerable attention has been devoted to measuring NS and BH masses (Finn 1994; Thorsett & Chakrabarty
1999; Schwab et al. 2010; Kiziltan et al. 2010; Valentim et al. 2011; Zhang et al.
2011; Ozel¨ et al. 2012). The most precise mass measurements come from binary systems where one component is a pulsar and there are accurate measurements of at least two post-Keplerian parameters. A special group of such systems are double
NS binaries (DNS), where the first-born NS is observed as a recycled pulsar with a rotation period of 20 ms < P < 200 ms, and the companion NS is the result of a ∼ ∼ supernova from the initially less massive secondary star (e.g. Bhattacharya & van den Heuvel 1991; Portegies Zwart & Yungelson 1998; Burgay et al. 2003; Lyne et al.
2004). Six such systems are currently known, giving 12 precise mass measurements.
The masses of these NSs cluster at 1.35 M with a dispersion of only 0.06 M ∼ ⊙ ∼ ⊙ (Kiziltan et al. 2010; Ozel¨ et al. 2012), and the mean masses of the pulsars and their companions differ by only 0.03 M⊙ (Ozel¨ et al. 2012). Present theories argue for very
16 little mass transfer in these systems once the first NS is formed, so both NS masses basically represent the birth masses of the NSs. The mean observed masses are significantly higher than the Chandrasekhar mass of the pre-collapse core, which may indicate growth by fallback during the supernova (Kiziltan et al. 2010). The tight mass distribution, however, argues against significant fallback, and the properties of the distribution have also been attributed to the particular evolution history that leads to their formation (Ozel¨ et al. 2012). Schwab et al. (2010) proposed that a third of the DNSs formed as a result of electron-capture supernovae as evidenced by their lower masses, with the remaining higher mass systems forming as a result of a Fe-core collapse. Kiziltan et al. (2010) and Ozel¨ et al. (2012) have carefully fit simple analytic models to the DNS mass distributions, however, a direct comparison with more physical supernova explosion models has been lacking. In Chapter 5,
I use a Bayesian formalism to directly constrain supernova explosion models with the observed mass distribution of double neutron star binaries. Since this Chapter chronologically preceeds the work on Chapter 4, I use piston-based models of Zhang et al. (2008) and make a short note about recent calculations of Ugliano et al. (2012).
Constraining the supernova explosion mechanism with observations can be fruitful. However, more work is necessary on two fronts. First, a more detailed understanding of the supernova explosion mechanism is required to solidify the predictions of the compact remnant mass distribution. One potential avenue is obtaining a deeper understanding of the accretion shock instabilities that are
17 observed in numerical simulations. Second, a model of shock propagation through the progenitor star that includes nuclear burning and is based on a realistic explosion mechanism is necessary to predict observed properties of supernovae such as explosion energies, velocities and heavy element yields. More perspective on these potential areas of research is given in Chapter 6.
18 Chapter 2
The Physics of the Neutrino Mechanism
2.1. Introduction
Within the framework of time-steady spherically symmetric models of supernovae, this Chapter provides an answer to two questions fundamental to supernova theory, and a possible explanation for a third. First, what is the physics
crit of Lν, core? In Section 2.2, we investigate isothermal and polytropic accretion flows with a bounding shockwave and we show that even in such a simplistic setting there exists a critical value of the controlling parameter — the isothermal sound speed
— that separates steady-state accretion solutions from wind solutions, which are identified with explosions (Burrows 1987; Burrows & Goshy 1993; Burrows et al.
1995; Yamasaki & Yamada 2005). This critical value is determined by conservation of momentum and energy across the standoff accretion shock as was found by
Yamasaki & Yamada (2005, 2006); above the critical value it is not possible to satisfy these conditions with a steady-state accretion solution, and instead the
flow must rearrange itself into an outgoing wind. In Section 2.3, we calculate the steady-state structure of the accretion flow between the neutrinosphere and the shock
19 in the supernova problem. We explicitly show that the critical neutrino luminosity discovered by Burrows & Goshy (1993) is equivalent to the critical condition from the isothermal model. That is, the neutrino mechanism of supernovae as formulated by
Burrows & Goshy (1993) is identical to the statement that the shock jump conditions cannot be satisfied together with the Euler equations for steady-state accretion when the sound speed of the flow exceeds the critical value. Furthermore, we discuss the effects of radiation transport and the accretion luminosity on the absolute value of
crit Lν, core, and we show how the latter depends on the mass and radius of the PNS and the mass accretion rate.
crit The second question this Chapter answers is “What criterion does Lν, core correspond to in terms of variables within the accretion flow itself?” For example, it has been claimed that when the advection time in the heating region becomes longer than the heating timescale, explosion must result (Janka 2001; Thompson & Murray
2001; Thompson et al. 2005; Buras et al. 2006b; Scheck et al. 2008; Murphy &
Burrows 2008). Is this heuristic criterion identical to the critical neutrino luminosity?
crit The answer is no. In fact, we show in Section 2.4 that Lν, core corresponds to a virtually constant ratio of the sound speed to the escape velocity in the accretion
1 2 2 flow — the “antesonic condition:” c 0.19v , where vesc is the escape speed — S ≃ esc 1We call this condition “antesonic” because the required sound speed is significantly below the local escape velocity, and the critical radius at which this point occurs is smaller than the radius of sonic point in Bondi accretion. This condition is reached in supernovae at the time of explosion, before the flow arranges itself into a super-sonic neutrino-driven wind (see Section 2.2 for details).
20 and that this condition is a consequence of the physics of the critical luminosity itself.
Third, an intriguing observation was made by Ohnishi et al. (2006), Murphy
& Burrows (2008) and Nordhaus et al. (2010) who found that the mere fact of
crit increasing the dimension of the simulation lowers Lν, core. While going from 1D to
2D allows for the effects of convection and the SASI, it is not clear what is gained by
crit going from 2D to 3D, or why Lν, core should decrease monotonically with dimension.
crit In Section 2.5, we provide evidence that the reduction of Lν, core observed in 2D and
3D simulations occurs because the flow can organize itself to cool less efficiently and yet still maintain hydrostatic equilibrium, giving an overall lower cooling efficiency, lower critical luminosities, larger entropy, and larger shock radii. Thus, the reduction
crit of Lν, core in 2D and 3D may be due to less efficient cooling, rather than more efficient heating.
2.2. Isothermal accretion bounded by a shock
Here we review the basic physics of spherically-symmetric isothermal accretion with a shock. We find that this idealized model problem is a key to understanding the mechanism of supernovae. We also discuss here the analogous problem of polytropic accretion flows with a constant adiabatic index Γ before solving the more complete supernova problem in Section 3.
21 2.2.1. Topology of solutions
The velocity structure of spherical steady-state isothermal flows is described by the equation
1 dM 2 1 M = , (2.1) − M dx x − 2x2 where M = v/cT is the Mach number, v is the fluid velocity, cT is the isothermal
2 sound speed, x = rcT /(2GM) is the rescaled radial coordinate r, and M is the mass of the central object. It is possible for a standing shock wave in the flow to exist at a point that satisfies the two Rankine-Hugoniot shock jump conditions
ρ−M − = ρ+M +, (2.2)
ρ−(M −)2 + ρ− = ρ+(M +)2 + ρ+, (2.3) which express conservation of mass and momentum, respectively. Here, ρ is the mass density and the + and superscripts correspond to the quantities evaluated − just upstream and downstream of the shock, respectively. The physically relevant solution to equations (2.2–2.3) is M +M − = 1.
In Figure 2.1, we show with black solid lines Mach-number profiles of flows with a single constant mass accretion rate M˙ = 1 M s−1 obtained by solving − ⊙ equation (2.1). The integration starts at a fixed radius rν corresponding to the
˙ 2 surface of the star, and with a fixed velocity vν = M/(4πrνρν), where ρν is the
2 prescribed mass density at rν. Different lines correspond to different values of cT .
22 The dashed line shows the decelerating transonic solution, which goes through
2 the sonic point located at xsonic = 0.25 and Msonic = 1. It has high velocity − at large radii and M 0 as x 0, unlike Bondi accretion flow, which starts → → with zero velocity at infinity, goes through the sonic point and has M as → −∞ x 0 (shown with dash-dotted line). The decelerating transonic solution separates → accretion “breezes”, which are always subsonic ( 1 < M 0) and denoted as B, − ≤ from solutions going through M = 1 and denoted as A. Although the A-type − solutions lying below the decelerating transonic solution are considered unphysical in standard Bondi accretion theory, the parts of these solutions that span from rν to the shock are viable in this setting (McCrea 1956).
The dotted lines in Figure 2.1 show the velocities just downstream of the shock,
M −, corresponding to three different assumed profiles of the upstream velocities
M +: (1) Bondi flow (blue), (2) free fall with M + = x−1/2 (green), and (3) − pressure-less free fall (red), that is a good approximation in more realistic supernova calculations (Colgate & White 1966) and that requires modification of equation (2.3) by assuming that M + = x−1/2 1, and − ≪ −
ρ−(M −)2 + ρ− = ρ+x−1. (2.4)
2In this paper we assign the name “sonic point” to the position of the flow where the numerator and denominator of the fluid momentum equation simultaneously vanish. Although this point is generally called the “critical point”, we avoid this name to prevent confusion with the critical neutrino luminosity discussed later in the paper.
23 The physically relevant solution to equations (2.2) and (2.4) is M − =
(√x−1 4 x−1/2)/2. For the sake of clarity, we do not show the upstream − − profiles in Figure 2.1 except for Bondi flow (dash-dotted line).
The velocity profile of the shocked accretion flow for a given cT is constructed by following one of the black lines from rν until it crosses a dotted line that corresponds to the appropriate shock jump conditions. At this intersection, the solution jumps to whatever velocity profile M + was assumed to be present upstream.
The essential point of Figure 2.1 is that for a fixed M˙ the presence of a shock is not guaranteed for all values of cT . For example, consider the blue dotted line, which corresponds to Bondi accretion flow upstream of the shock. For the smallest values of cT , the required shock radius — the intersection of the black solid line with blue dotted line — would be below the radius of the star, an unphysical situation.
As we step to higher cT , the shock appears at rν and moves outward. At a critical
crit value cT corresponding to the decelerating transonic accretion flow (dashed line),
crit the shock can only coincide with the sonic point at x = xsonic = 0.25. Here there is no jump in the velocity, and the shock degenerates into a jump in the derivative of the velocity (Velli 1994; Del Zanna et al. 1998). A shock is not possible for
crit cT >cT , because the blue dotted line lies below the decelerating transonic solution
(dashed line) in the range of radii of interest and hence B-type solutions are not viable solutions inside the shock – that is, solutions of type B do not intersect the blue dotted line.
24 The situation is somewhat different when the flow upstream of the shock is in free fall, as shown with the red and green dotted lines in Figure 2.1. In these cases, the shock jump conditions allow part of the B-type solutions as viable solutions downstream of the shock. Furthermore, for values of cT higher than that corresponding to the decelerating transonic solution (dashed line) two shock radii are possible for a single value of cT , because the curvature of B-type solutions and shock jump conditions yields two intersections. When stepping to higher values of cT , the two shock radii come closer together and finally coincide at a critical value
crit cT . Contrary to the case of Bondi flow above the shock, there is no relation between
crit crit the critical sound speed cT and the sonic point. The shock radius at cT also does not coincide with the velocity minimum of any of the B-type solutions. Above the
crit critical sound speed cT , a shock in the flow is not possible, qualitatively similar to the case of Bondi flow (blue dotted line) described above.
Finally, we briefly discuss the “upper” solutions that appear when cT is higher than the value corresponding to the decelerating transonic solution (dashed line
crit in Fig. 2.1), but still cT < cT . We see that the maximum shock radius for these solutions is identical to the sonic point radius. The shock radius decreases with
crit increasing cT until it merges with the normal solution branch at cT . These upper solutions were found by Yamasaki & Yamada (2005) also in the full problem, but these authors show that the upper solutions are unstable to perturbations and thus
25 most likely do not occur in a real physical system. Because of this, we do not discuss the upper solutions further except in Sections 2.3.2 and 2.3.3.
2.2.2. The critical sound speed
crit In Figure 2.2 we show the critical values cT as a function of mass accretion rate
M˙ for the three sets of shock jump conditions shown in Figure 2.1. The three curves are similar over a significant range of M˙ despite the relatively stark differences in the shock jump conditions. Specifically, the top and the middle curves closely resemble the bottom blue line, which is, in fact, the relation between the sound speed and the mass-accretion rate for isothermal transonic accretion (Lamers & Cassinelli 1999, p. 68). The curvature of this relation in the log-log plane of Figure 2.2 arises from exponential near-hydrostatic density profile of the atmosphere, which is a good approximation to the decelerating transonic accretion flow below the sonic point.
From Figure 2.1 it is obvious that for pressure-less free fall upstream of the
crit shock the solution described by cT does not correspond to any special radial point
crit crit in the flow. However, we also see that the shock position x at cT is constant when expressed in the dimensionless coordinates since this is the point where the dotted line just meets the flow profile. Changes in M˙ , ρν or rν only move the starting
crit ˙ inner point of the flow, but one can always find a value of cT = cT (M,ρν, rν) to get the critical solution at xcrit. By plugging the solution of M − from the shock
26 jump condition in equation (2.4) into equation (2.1)3, we obtain exactly xcrit = 3/16 implying that the critical condition for existence of steady-state accretion shock in isothermal accretion is
crit 2 (cT ) 3 2 = = 0.1875, (2.5) vesc 16
where vesc = 2GM/r is the escape velocity. This result is valid when the flow p upstream of the shock is in free fall and has negligible thermal pressure. We
crit note that equation (2.5) is valid only at cT . There are indeed shock solutions
crit with xshock > x that correspond to the upper solution branch as discussed in
Sections 2.2.1 and 2.3.2 and by Yamasaki & Yamada (2005), but these solutions still
crit crit ˙ have cT < cT . The actual value of cT for given M, M, rν, and ρν is calculated numerically as is demonstrated in Figure 2.2.
As the ratio in equation (2.5) is lower than for the sonic point, we call this condition “antesonic” (see footnote 1). For the case of free fall with non- negligible pressure shown with green line in Figure 2.1, the numerical factor is
(5 √21)/2 0.2087. Furthermore, we emphasize that equation (2.5) does not − ≃
3We note that this procedure is valid only for determining the position of the critical point in isothermal accretion. As seen from Figure 2.1, the critical point occurs when the line of all possible shock positions (solution to eqs. [2.2] and [2.4], dotted red line) just touches the flow profiles (solutions to eq. [2.1], solid black lines). Thus, at this point only, the tangents and values are the same, which we then use to calculate the position of the critical point, equation (2.5).
27 imply any connection to the Parker point4 or sonic point. Thus, one cannot assume that the existence of a shock is equivalent to the existence of a sonic point in the context of supernova stalled accretion shocks (as in Shigeyama 1995). However, if the flow upstream of the shock is a Bondi accretion flow with the same sound speed, then the critical condition is exactly the same as the expression for the position of
crit 2 the sonic point: (cT /vesc) = 0.25.
Our calculation of the critical condition in equation (2.5) also explains why the critical curves for different shock jump conditions in Figure 2.2 are so similar: their values can be estimated in the same way as for the position of the sonic point
(Lamers & Cassinelli 1999, p. 68), except that for the shock jump conditions relevant in the supernova context the value of xcrit is lower than 0.25 (eq. [2.5]).
For the sake of completeness, we repeated the isothermal calculation with pressure-less free fall above the shock for a polytropic equation of state P = kρΓ, where k is the normalization factor and Γ is the adiabatic index, which we vary from
1 to 1.7. We do not assume energy conservation through the shock so that k is left as a free parameter, which we can vary to obtain its critical values. The results are qualitatively similar to those for isothermal accretion. For fixed Γ, there exists a critical value of the constant k above which there are no steady-state solutions with a bounding shock. As in the isothermal case, this critical value of k corresponds
4The Parker point occurs when the right-hand side of equation (2.1) vanishes, and in the case of isothermal flow the Parker point coincides with the sonic point (Lamers & Cassinelli 1999, p. 63).
28 to a critical condition that can be written as max (c2 /v2 ) 0.19Γ (compare with S esc ≈ eq. [4]), where cS is the adiabatic sound speed and the “antesonic factor” is linearly proportional to the adiabatic index of the flow. We illustrate the dependence of
2 2 max(cS/vesc) on Γ in Figure 2.3. where we plot the antesonic ratio as a function of k for a range of M˙ and two values of Γ. We note that although we are able to write down a single local criterion for the antesonic condition in the adiabatic case, at the
2 2 radial location where max (cS/vesc) occurs, this condition should be interpreted as a global condition on the flow, and not a local one. As in the isothermal case, the physics of the critical condition is determined by the ability of the flow to satisfy the jump conditions while simultaneously satisfying the Euler equations for steady-state accretion.
The physics of collapse, core bounce and shock stall dictate that there is a phase of steady-state accretion flow with a standing shock with time-changing M˙ and cT . If cT slowly increases at fixed M˙ , the shock radius advances until the critical
crit crit value cT is reached. Physically, cT exists because as cT is increased the shock jump conditions simply cannot be satisfied given the properties of the flow upstream and the necessity of maintaining conservation of mass and momentum. What happens
crit at cT depends on the properties of the flow upstream of the shock. In the case of
Bondi flow, the profile can continuously transition to a shock-less B-type solution
(accretion breeze) as shown by Korevaar (1989), Velli (1994, 2001) and Del Zanna
crit et al. (1998), because there is no jump in velocity at cT . However, in the case
29 relevant for supernovae, where the flow above the shock is in pressure-less free
crit fall, the shock does not degenerate at cT and a catastrophic dynamical transition must occur, most likely to a supersonic wind, which can be identified with the supernova explosion (Burrows 1987; Burrows & Goshy 1993; Burrows et al. 1995).
A similar conclusion was reached by Yamasaki & Yamada (2005). It is likely that time-dependent hydrodynamical instabilities will modify the exact realization of this scenario. Nevertheless, our results from the isothermal accretion model are an important step towards understanding the mechanism of supernova explosions, and specifically the neutrino mechanism, as we detail below. Our isothermal model also provides a robust explosion criterion of broad applicability (the “antesonic” condition, Section 2.4.4).
2.3. Steady-state accretion shock in core-collapse
supernovae
While isothermal flow calculations provide insight into the physics of steady- state accretion with shocks, we perform more involved calculations to assess the effects of individual parameters of the problem for supernova explosions. In this
Section we first describe our equations and boundary conditions and then we proceed by discussing how the solution changes when approaching the critical neutrino luminosity. We then prove the correspondence between the critical neutrino
30 luminosity and the critical sound speed from Section 2.2, and discuss properties of the critical solutions.
2.3.1. Numerical setup
We solve the time-independent Euler equations
1 dρ 1 dv 2 + + = 0, (2.6) ρ dr v dr r dv 1 dP GM v + = , (2.7) dr ρ dr − r2 dε P dρ q˙ = , (2.8) dr − ρ2 dr v
where P is the gas pressure, M is the mass within the radius of the neutrinosphere r , ε is the internal specific energy of the gas, andq ˙ = is the net heating ν H − C rate, a difference of heating and cooling . These hydrodynamic equations couple H C through the equation of state (EOS), P (ρ,T,Ye) and ε(ρ,T,Ye), where T is the gas temperature, to the equation for electron fraction Ye
dYe v = l + l + (l + l + + l¯ + l − )Ye, (2.9) dr νen e n − νen e n νep e p
+ − where lνen, le n, lν¯ep, and le p are reaction rates involving neutrons and protons.
31 Using thermodynamic expansion of ε and P we express the radial derivatives of v, ρ and T by inverting equations (2.6–2.8). The radial derivatives are
2 2 dρ ρ 4v vesc q˙ K W dYe = 2 − 2 + 2 2 + 2 2 , (2.10) dr −2r v cS v v cS v cS dr 2 − 2 − − dv v v 4c q˙ K v W dYe = esc − S , (2.11) dr −2r v2 c2 − ρ v2 c2 − ρ v2 c2 dr − S − S − S dT ρZ 4v2 v2 q˙ v2 c2 = − esc + − T dr −2rC v2 c2 vC v2 c2 − V − S V − S 2 2 1 dYe v cT ∂ε Z ∂P 2 − 2 2 2 . (2.12) −C dr v c ∂Ye − v c ∂Ye V " − S ρ,T − S ρ,T #
Combinations of thermodynamic variables in these equations are defined as
∂ε C = , (2.13) V ∂T ρ,Ye ∂P c2 = , (2.14) T ∂ρ T,Ye 1 ∂P K = , (2.15) CV ∂T ρ,Ye ∂P ∂ε W = K , (2.16) ∂Y − ∂Y e ρ,T e ρ,T P ∂ε Z = , (2.17) ρ2 − ∂ρ T,Ye 2 2 cS = cT + KZ. (2.18)
This form of thermodynamic equations is the same as the Newtonian version of equations of Thompson et al. (2001), except that we include dependency on Ye and we do keep Z and K separate, because they are related only for a special case of equation of state. The Equations (2.10–2.12) are useful for describing the connection
crit between the critical luminosity and cT in Section 2.2.
32 Our equation of state includes relativistic electrons and positrons with chemical potential and nonrelativistic free protons and neutrons (Qian & Woosley 1996). In our neutrino physics we use prescriptions of heating, cooling, opacity and reaction rates for charged-current processes5 with neutrons and protons from Scheck et al.
(2006). We explicitly calculate the degeneracy parameter of electrons and positrons, but we assume that degeneracy parameter of (anti)neutrinos is zero. We also assume that the energies of neutrinos and antineutrinos do not change between rν and rS.
We also employ, for the first time in the context of calculations of the critical luminosity, gray neutrino radiation transport using the expression
dL ν = 4πr2ρq,˙ (2.19) dr − where Lν is the combined luminosity of electron neutrinos and antineutrinos, which are assumed to be equal everywhere. We do not consider neutrinos of other
flavors. We assume that neutrinos have root-mean-square energy of ǫνe = 13 MeV
and antineutrinos ǫν¯e = 15.5 MeV (Thompson et al. 2003). Even this very simple approximation to radiation transport enables us to assess the relative importance of accretion and core neutrino luminosity for the supernova explosion.
Five equations (2.6–2.19) or alternatively (2.10–2.12) are solved on an interval of radii between rν and rS, where rν is the radius of the neutrinosphere and is fixed, and rS is the radius of the shock, which is left to vary. The matter inside of rν is the 5We also modified the heating and cooling functions to include electron-positron annihilation and
crit creation, which increased Lν, core in the fiducial calculation by about 2 to 5%.
33 PNS core and is characterized only by its mass M and luminosity Lν, core and energy of neutrinos and antineutrinos that it isotropically emits.
We thus need six boundary conditions to uniquely determine five functions: ρ, v, T , Ye and Lν, and shock radius rS. We first demand that the flow has a fixed mass accretion rate M˙ = 4πr2ρv by applying this constraint at the inner boundary. We assume that the neutrino luminosity at the inner boundary is equal to the luminosity of the PNS core, Lν(rν)= Lν, core. At the outer boundary we apply the shock jump conditions6
2 + 2 ρv + P = ρ vff , (2.20) 1 P 1 v2 + ε + = v2 , (2.21) 2 ρ 2 ff
where vff = √Υvesc is the free fall velocity, and in accordance with analytically estimated accretion data of Woosley et al. (2002) supernova progenitor models available on-line7 we choose Υ = 0.25. The quantity ρ+ is the density just upstream of the shock and can be calculated from conservation of mass. We also assume that the matter entering the shock is composed of iron and hence Ye = 26/56 at the outer boundary. The last boundary condition comes from requirement that rν is the
6We do not include nuclear binding energy in our calculations. However, the pertinent effects are qualitatively assessed in Section 2.3.5 and in Yamasaki & Yamada (2006). 7http://homepages.spa.umn.edu/ alex/stellarevolution/data.shtml ∼ 34 neutrinosphere for electron neutrinos, which gives a constraint on the optical depth
rS 2 τ = κ e ρ dr = , (2.22) ν ν 3 Zrν where κνe is the opacity to electron neutrinos. This condition is implemented by adding an extra variable corresponding to the optical depth and requiring that it is zero at the inner boundary and 2/3 at the outer boundary.
The equations and boundary conditions above are solved using a relaxation algorithm (Press et al. 1992, p. 753) with logarithmic spacing of grid points (see
Thompson et al. 2001, and references therein). The number of grid points was usually > 1000. We refer to the problem outlined by equations (2.6–2.22) and ∼ the microphysics as the “fiducial calculation”. As this problem is complex, we occasionally resort to various simplifications to make some of our points more clear.
We describe what exactly we modified with respect to the fiducial calculation at a various places in the text.
2.3.2. Properties of the fiducial calculation
We include neutrino radiation transport and hence our solutions are parameterized by the core neutrino luminosity Lν, core and yield neutrino luminosity as a function of radius Lν(r). We confirm the existence of an upper limit to the core
crit ˙ neutrino luminosity Lν, core at fixed M (Burrows & Goshy 1993; Yamasaki & Yamada
crit 2005, 2006). We determine Lν, core by starting the relaxation code at a low value
35 of Lν, core with an initial guess of power-law profiles of variables, and then increase
Lν, core in small steps while using the converged solution from the previous step as an
crit initial guess for the next step. We use bisection to trace Lν, core to a specified relative precision.
Figure 2.4 shows profiles of thermodynamic quantities for Lν, core stepping from
crit 0 to Lν, core. Similar plots except for Lν(r) were published previously by Yamasaki
& Yamada (2005, 2006), but we reproduce them here for the sake of completeness.
Our results fall within the paradigm of quasi-steady-state supernova structure (e.g.
Bruenn 1985; Mayle & Wilson 1988; Janka 2001). In particular, we find a steep decrease in the density outside rν that flattens when the dominant pressure support changes from an ideal gas of nucleons to a relativistic gas of electrons and positrons.
The temperature profile is flat at small r, but then drops as the relativistic particles become more dominant. Still, the temperature changes only from approximately
3MeV to 0.5 MeV, much less than the corresponding change by several orders of magnitude in density. This implies that our isothermal model is a legitimate rough approximation of the problem. We discuss in more detail the profiles of the adiabatic
8 sound speed cS, total specific energy in the form of Bernoulli integral
v2 P GM B = + ε + , (2.23) 2 ρ − r and neutrino luminosity Lν that are to our knowledge not previously discussed in
crit the context of the calculation of Lν, core.
8The importance of individual terms contributing to B is assessed in Section 2.5.
36 The total specific energy B is an increasing function of Lν, core in the whole region of interest. The maximum of B occurs at rgain, which is defined asq ˙(rgain) = 0.
B crit Both rgain and (rgain) increase with increasing Lν, core. We note that even at Lν, core the energy is significantly negative in the entire region of interest. It is certainly not
B crit the case that positive corresponds to explosion (Lν, core = Lν, core; Burrows et al.
1995).
The adiabatic sound speed cS was calculated using equation (2.18) and is plotted in the bottom left of Figure 2.4 as a ratio to the local escape velocity vesc(r).
Similar to B, we see that an increase of Lν, core results in a net increase of cS between rν and rS. The maximum of cS occurs at gradually larger radii for increasing Lν, core,
2 2 similar to the gain radius rgain. However, the maximum of cS/vesc does not exactly coincide with rgain; in fact, rgain is always about 1 to 2% larger for the particular calculation shown in Figure 2.4.
The middle center panel of Figure 2.4 shows profiles of Lν(r). As expected, Lν grows in the cooling layer as a result of net neutrino and antineutrino production and decreases in the gain layer as a result of heating. How much does the accreting material cooling via neutrino emission contribute to the total neutrino flux emanating
acc from below the shock? We define the accretion neutrino luminosity Lν as the difference between the neutrino luminosity at the shock and the luminosity of the
37 core,
acc L = L (rS) L core. (2.24) ν ν − ν,
The relative contribution of the accretion luminosity to the total neutrino luminosity decreases with growing Lν, core. For Lν, core = 0, the accretion luminosity is the sole contributor to the total luminosity of about 1.4 1052 ergs s−1, but for the critical × core luminosity of 6.41 1052 ergs s−1 the accretion luminosity contributes 28% to × the total neutrino output. Clearly, the accretion luminosity is a sub-dominant, but important part of the total neutrino output. In Section 2.3.4 we investigate how including the neutrino radiation transport and hence accretion luminosity affects
crit ˙ Lν, core for a range of M.
We now turn our attention to another aspect of steady-state accretion shocks: the existence of two solutions with different shock radii for some values of Lν, core, discovered by Yamasaki & Yamada (2005). These authors also performed linear stability analysis and found that the upper solution branch with higher rS is unstable, the lower branch is stable, and both branches meet at a marginally stable
crit point at Lν, core. We confirm the presence of the upper solution branch with our code.
We switch to the upper solution branch by first calculating a sequence of models on
crit crit the lower branch up to Lν, core. We take a solution close to Lν, core, manually scale up the radial coordinates by about 50% and let the relaxation algorithm converge to a solution on the upper branch.
38 We find that the profiles of thermodynamic variables for the upper solution branch are not fundamentally different from those of the lower branch presented in Figure 2.4 and hence we do not present them here. We plot in Figure 2.5 the shock radii and total specific energies at several radii of interest for both branches, and as a function of Lν, core for several values of M˙ . We see that the upper branch of solutions extends to larger shock radii, but ends at a finite radius and non-zero
Lν, core (left panel, Figure 2.5). We specifically checked that this is not an artifact in our calculation. We note however, that for different input physics the upper branch can end at much larger radii, consistent with infinity in our code. We also find that the specific energy of the upper solution is always higher than that of the lower solution for the same Lν, core at all radii. In analogy with other physical systems that have two states with different energy, we can argue that the higher energy upper solution branch is unstable to transition, in agreement with Yamasaki & Yamada
(2005) who find that the upper branch is indeed unstable to radial perturbations.
We do not discuss the properties of the upper branch in greater detail, because it is unlikely that these solutions would be realized in a real physical system. However, the existence of the upper solution branch is analoguous to the upper solutions in
Figure 2.1.
39 2.3.3. Correspondence with isothermal accretion
It has been clearly established in 1D calculations that there exists an upper
crit limit to the core neutrino luminosity Lν, core that allows for a steady-state accretion shock (Burrows & Goshy 1993; Yamasaki & Yamada 2005, 2006), and this limit has been confirmed to exist in multi-dimensional dynamical supernova simulations
(Murphy & Burrows 2008; Nordhaus et al. 2010), and in simulations of the stalled accretion shock (Ohnishi et al. 2006; Iwakami et al. 2008). To illustrate how the upper limit on the core neutrino luminosity is related to the limit on the sound speed in isothermal accretion discussed in Section 2.2, we consider the total specific energy of the system B, which has the property that its radial derivative is proportional to the net deposited heating and cooling:
dB 4πr2ρ = q.˙ (2.25) dr M˙
It is well established (e.g. Janka 2001) and we confirm this in Section 2.5, that the contribution of the kinetic term v2/2 to the total specific energy budget in
2 equation (2.23) is negligible. At a fixed radius r in the gain layer whereq ˙ L core/r , ≈ ν, the increase of Lν, core will result in an increase of enthalpy h = ε + P/ρ, and consequently, of the adiabatic sound speed, as h c2 . We showed in Section 2.3.2 ∼ S that cS as well as B at a given radius are increasing functions of Lν, core everywhere, not only in the gain layer. This correspondence between Lν, core and cS suggests that
40 crit crit Lν, core is a realization of cT (Section 2.2) in the context of the neutrino heating mechanism.
crit As shown in Section 2.2, cT occurs when the flow cannot satisfy the shock jump conditions at any point in the accretion flow (see also Yamasaki & Yamada
2005). We now prove that the same mechanism is responsible for the existence of
crit Lν, core in the supernova context. Essentially, we are trying to construct an equivalent of Figure 2.1 for non-isothermal flows and more realistic physics. In order to make this feasible we consider a simplification of the problem by setting dYe/dr = 0 and dLν/dr = 0 (eqs. [2.9] and [2.19]) and setting the heating and cooling terms
−18 2 −1 2 18 6 −1 −1 to = 1.2 10 cm g L core/r and = 2 10 (T/MeV) ergs s g , H × ν, C × respectively. We also modify the boundary conditions by requiring ρ(r ) = 3 1010 g ν × −3 cm instead of fixing τν. With these changes, the problem simplifies considerably, because ρ(rν) and v(rν) are now fixed and the only variables left are T (rν) and rS, which are determined by the two shock jump conditions. Consequently, if we know the value of T (rν), the flow profile is now uniquely determined as an initial value problem.
We want to show how the solution for the flow structure with a shock relates to all possible accretion solutions. For a given Lν, core and T (rν) we calculate the A-type solutions (those reaching M = 1 in Figure 2.1) by setting v2 = c2 at the outer − S boundary, which gives a continuum of solutions for a range of T (rν). Solutions going through the sonic point are obtained by setting the numerator and denominator of
41 the momentum equation (2.11) simultaneously to zero at the outer boundary. The accretion breezes (B-type solutions in Figure 2.1) are obtained from the A-type solutions by setting v to a specific fraction of cS at a fixed radius corresponding to v = cS in an A-type solution. By varying this fraction, we get a continuum of flows with different T (rν). We extend the breezes to larger radii with an initial-value integrator of stiff equations based on algorithms from Press et al. (1992) and using the results from our relaxation algorithm as a starting point. We verified that for
B-type solutions (subsonic accretion breezes) the ratio v/c is always less than | S| unity.
In the left panel of Figure 2.6 we show the position of the shock radius rS (solid) and the sonic point rsonic (dashed) as a function Lν, core. In the right panel, we show the values of T (rν) for the same solutions as in the left panel. We see that rsonic exhibits qualitatively similar behavior to rS, except that for low values of Lν, core there is no sonic point with a finite rsonic. Furthermore, the position of the shock is well separated from the sonic point, except for the tip of the upper solution branch, where these two appear to merge, similar to what happens in isothermal accretion in
Figure 2.1. The same trend is seen in T (rν) (the only free parameter of the problem for fixed Lν, core), which suggests that the upper branches of shock and sonic point indeed coincide at some Lν, core.
In order to explore the situation in more detail, we selected four values of Lν, core marked by vertical dot-dashed lines to illustrate the four configurations visible in
42 Figure 2.6: (1) low Lν, core with one solution with finite shock radius and no finite sonic radius, (2) intermediate Lν, core where two solutions are present for both the
crit shock and the sonic point, (3) Lν, core > Lν, core but with two sonic point solutions still present, and (4) high Lν, core with no solution for either the shock or the sonic point. In Figure 2.7 we plot flow solutions for each of these luminosities (black solid lines) with shock jump conditions overplotted (green solid lines). The Figure should be interpreted in a way similar to Figure 2.1. We found that plotting the
Mach number of the flows only obfuscates the issue because the lines cross in a complicated manner; instead we plot momentum ρv2 + P and specific energy B.
In this parameter space the different flow lines generally do not intersect and the shock jump conditions are a function of only the radius as can be seen from the equations (2.20) and (2.21).
The upper left part of Figure 2.7 shows flow profiles for low Lν, core, where there is only one solution for the shock radius (far left vertical dash-dotted line in
Figure 2.6). Each solid line corresponds to a different value of T (rν). For low T (rν),
2 2 we find that the flows are of type A. That is, they reach a point where v = cS
(not a sonic point; shown with black dots in Figure 2.7). For increasing T (rν), the intersection of the flow with the shock jump conditions moves to larger radii.
A-type solutions extend to T (r ) 3.32 MeV, where the point where v2 = c2 moves ν ≃ S to infinite radius. For yet larger T (rν) we are able to find only B-type solutions
— subsonic accretion breezes — with v/c < 1 everywhere. For a single value of | S|
43 T (rν) the profiles of momentum and energy of the flow intersect with the shock jump conditions at the same radius, which means that only at this T (rν) is a solution with a shock possible (red line). The vertical dotted lines mark radii of intersection of momentum of the flow with the momentum shock jump condition. We then plot these radii in the energy panel and we see that for flows lying below the red line the intersection of B with the appropriate shock jump condition occurs at systematically smaller radii than what is required by the momentum profiles. Conversely, for curves above the red line the intersection occurs at larger radii. Thus, the momentum and energy at the shock can only be conserved simultaneously for a single T (rν) at
51 −1 L core = 5 10 ergs s , which then defines not only T (r ), but also rS. ν, × ν
The upper right panel of Figure 2.7 displays flow profiles for intermediate
Lν, core, which allows for two solutions for the shock and the sonic point (vertical dash-dotted line labelled (2) in Figure 2.6). The topology of the solutions for this
Lν, core differs from the previous case. The A-type solutions exist only for a small range of temperatures (3.517 MeV < T (rν) < 3.675 MeV). There are two transonic solutions going through two distinct sonic points at finite but vastly different radii, which separate A-type solutions from the B-type solutions, (which occur for
T (rν) < 3.517 MeV and T (rν) > 3.675 MeV). Due to severe stiffness of the equations, we were not able to follow the B-type solutions to very large radii. In contrast to the upper left panel of Figure 2.7 at lower Lν, core, the flow profiles intersect the shock jump conditions at just two inner boundary temperatures shown by the two solid
44 red lines. The upper solutions are analoguous to those obtained in the isothermal model in Figure 2.1.
crit The lower left part of Figure 2.7 explores the region with Lν, core >Lν, core where two solutions for the sonic point are still present. The topology of the solutions is
52 −1 similar to the previously discussed case with L core = 5 10 ergs s , except that ν, × the A-type solutions exist for an even smaller range of inner boundary temperatures
(3.852 MeV < T (rν) < 3.875 MeV). However, in this case, there are no simultaneous intersections of the momentum and energy profiles of the flows with the shock jump conditions. This is explicitly shown by the vertical dotted lines, which mark several radii of intersection of the momentum profiles of the flows with the momentum shock jump condition. Clearly, the intersections of the energy profiles with the energy shock jump conditions occur at systematically smaller radii. This means that at this
Lν, core no flow profile can connect to the free-falling matter upstream, because it is not possible to simultaneously satisfy conservation of mass, momentum and energy at the shock. Although the mismatch in radii decreases as we go to the higher T (rν) of B-type solutions, they eventually cease to cross the energy shock jump condition
52 −1 altogether. Overall, as implied by Figure 2.6, for L core = 9.5 10 ergs s it ν, × is impossible to find a flow solution that simultaneously conserves momentum and energy at the shock.
For the sake of completeness we show in the lower right panel of Figure 2.7 the flow profiles for high Lν, core, where there is no possible solution with a shock
45 or a sonic point. All flows displayed are B-type solutions with v/c < 1. Despite | S| the fact that we were not able to extend some of the solutions to large radii due to stiffness of the equations, v/cS of the solutions systematically converged to zero at large radii.
With the Lν, core–T (rν) space systematically explored we can now label regions of A and B solutions in Figure 2.6 in analogy with Figure 2.1. We note that the more physically realistic simulation displayed in Figure 2.5 has exactly the same behavior except that sonic points exist at finite radii all the way to Lν, core = 0.
The upper shock solution branch, however, ends as indicated in Figure 2.5. As can
crit be seen from Figure 2.6, the position of the shock when Lν, core = Lν, core does not align with the position of the sonic point and hence a smooth transition to a B-type solution is not possible, unlike the results of Korevaar (1989), Velli (1994, 2001) and
Del Zanna et al. (1998). With the same logic as in Section 2.2 we argue that at
crit crit Lν, core the structure of the accretion flow catastrophically changes: if Lν, core >Lν, core the shock moves outwards dynamically and a neutrino-driven wind with a sonic point below the shock is established. However, multi-dimensional time-dependent hydrodynamical instabilities might modify the realization of this process.
46 2.3.4. Properties of the critical solutions
crit In Figure 2.8 we present the critical core neutrino luminosity Lν, core versus the mass accretion rate, the critical curve, as calculated with the fiducial physical model as described in Section 2.3.1. Before we dive into properties of the critical curve, we investigate whether it is a robust phenomenon from the point of view of input physics. To this end, we simplify the problem by setting the electron chemical potential to zero, shutting off neutrino radiation transport (dLν/dr = 0), electron fraction evolution (dYe/dr = 0), and choosing simplified heating and cooling
−18 2 −43 6 prescription with s = 1.2 10 L core/r and s = 8.19 10 T , respectively. H × ν, C × The critical curve for this modification is shown in Figure 2.8 with a dashed black line. With respect to the simplified version of the calculation we make additional individual changes to see how the critical curve changes: we modify cooling by multiplying it by a factor of 5, making it proportional to T 3 instead of T 6, or shutting it off altogether. We modified heating by making it proportional to r−3, constant at all radii or adding to s heating motivated by Alfv´en wave energy deposition H (Metzger et al. 2007). We also changed the EOS to be that of a pure ideal gas.
We also tried an EOS of pure relativistic particles, but we were able to converge to solution only by relaxing the condition on optical depth and replacing it with fixed density at rν. We thus do not show the critical curve here.
47 Figure 2.8 shows that even drastic changes to the functional form of the heating and cooling functions and EOS consistently yield critical values of Lν, core.
crit ˙ Naturally, the exact value of Lν, core at a given M changes considerably. However, it is interesting that we do not see wildly discrepant slopes of individual lines in
Figure 2.8. Specifically, changing the form of the heating affects the slope only very slightly, while the form of cooling has a much more pronounced effect, except when simply multiplied by a constant factor. This can be explained by the fact that cooling couples directly to the temperature, and the temperature gradient is directly responsible for the hydrostatic equilibrium of the region when relativistic gas dominates.
crit In order to obtain dependencies of Lν, core on the parameters for the fiducial problem (solid line in Figure 2.8), we determine critical core luminosities on a grid of parameter values chosen as M 1.2, 1.4, 1.6, 1.8, 2.0 M , r 20, 40, 60 km, ∈ { } ⊙ ν ∈ { } τ 1/6, 1/3, 2/3 , and 30 values of M˙ logarithmically spaced between ν ∈ { } 0.01 M s−1 and 2 M s−1. Although in principle the neutrinosphere radius r − ⊙ − ⊙ ν and core luminosity Lν, core are connected by physics of neutrino diffusion out of the
PNS, we consider them separately here. The critical curves for sample values of parameters are shown in the left panel of Figure 2.9. In addition to the well-known
crit ˙ crit dependence of Lν, core on M, we see that Lν, core increases with increasing M and decreasing rν and τν. Lower values of τν also increase the curvature of the lines in
˙ crit the M–Lν, core space. In order to quantify these observations we fit to all our results
48 a combination of power laws to get
0.723 1.84 ˙ −1.61 crit 53 −1 −0.206 M M rν Lν, core = 8.18 10 ergs s τν −1 (2.26) × M⊙ M⊙ s ! 10 km
crit The relative differences between calculated and fitted values of Lν, core are plotted in the right panel of Figure 2.9. On the first sight, the multiple power-law
fit of equation (2.26) is not perfect. However, for most of the data the power-law
crit fit is good within 30% of Lν, core. There are systematic trends in the residuals as a function of M˙ in the sense that all lines are curved upward: at low and high M˙ the
crit value of Lν, core is slightly higher than what would correspond to the mean power law
fitted to the data. We can understand this on the basis of our isothermal accretion model and its critical curve presented in Figure 2.2, which also exhibits an upward curvature of the critical curves, which is a manifestation of the nearly-hydrostatic exponential density structure of the flow (see Section 2.2).
In the fit of equation (2.26) we did not include the dependence on the remaining parameters of our calculation, including the incoming electron fraction Ye(rS), and
energy of the neutrinos ǫνe and antineutrinos ǫν¯e . We performed a simple check to
crit see how changes in these parameters affect Lν, core. We find that changing Ye(rS) from 0.45 to 0.5 shifts the critical luminosity by at most 1%. As suggested by the
crit −2 physics of the neutrino heating, Lν, core scales as ǫνe . We confirm this scaling to
within 2% by varying ǫνe from 8 MeV to 14 MeV while keeping the ratio ǫνe /ǫν¯e fixed at 13/15.5 0.839. ≃ 49 One of the uncertainties in core-collapse supernova simulations and neutron star physics in general is the EOS of dense matter. For example, a softer high-density nuclear EOS generically leads to smaller PNS radii at earlier times after collapse and bounce, but also leads to higher average neutrino energies, which might be favorable for explosion (e.g. Baron et al. 1985a,b; Sumiyoshi et al. 2005; Janka et al. 2005;
Marek et al. 2009). Can we put some constraints on the EOS using our results on
crit Lν, core? In more realistic simulations, Lν, core, rν and the temperature of emitted
neutrinos T ǫ e are connected by the physics of diffusion. In the simplest case, we ν ∼ ν can assume that these quantities follow the black-body law
2 4Lν, core rν = 4 , (2.27) 7πσTν where σ is the Stefan-Boltzmann constant. In addition to equation (2.26) we found
crit −2 that L ǫ to a very good precision. Eliminating the dependence on ǫ e using ν, core ∼ νe ν equation (2.27), we find
Lcrit τ −0.14M 1.23M˙ 0.482r−0.41. (2.28) ν, core ∝ ν ν
crit Therefore, larger neutrinosphere radii suggest lower Lν, core, when all other quantities are equal. Thus, stiff equations of state that yield higher rν are potentially preferred.
crit We note that this result is enabled by superlinear dependence of Lν, core on rν in equation (2.26). On the other hand, soft EOS leads to a faster contraction of the
PNS, higher core luminosities and neutrino energies, which can be favorable for the explosion too (Scheck et al. 2006, 2008; Marek & Janka 2009; Marek et al. 2009).
50 crit crit Finally, we briefly mention the behavior of shock radii at Lν, core, rS . In
crit crit Figure 2.10, we plot rS /rν as a function of Lν, core for data from Figure 2.9. We see that the points lie close to a line described approximately as rcrit/r (Lcrit )−0.26. S ν ∝ ν, core crit The weak dependence on Lν, core can probably be attributed to changes in the
crit ˙ efficiency of cooling as a function Lν, core and hence M (see Section 2.5.3 for a related discussion). There are no noticeable residuals as a function of rν, M, and τν.
2.3.5. Effect of nuclear binding energy
While we do not include nuclear binding energy in our energy shock jump
crit condition (eq. [2.21]), we can assess its effects on rS and Lν, core with the help of
Figure 2.7. Nuclear binding energy would appear as a positive term on the left-hand side of equation (2.21), thus the total energy B just downstream of the shock would be smaller at a given radius. This corresponds to shifting the green line down in the energy panels of Figure 2.7. This would lead to smaller shock radii, because the common intersection for the momentum and energy jump conditions would occur at
crit smaller radius. Similarly, Lν, core would increase, because the common intersection could occur at higher Lν, core. For example, consider the energy panel in lower left part of Figure 2.7. If we shift the energy shock jump condition (green line) down by a small amount, it will intersect the flow profiles (black lines) exactly at the position of the similar intersection in the momentum panel (vertical dotted lines).
crit Thus, a solution with a shock is possible and Lν, core increases. By moving the green
51 line down even more, the intersection would move to lower radii, thus decreasing rS. We verified that these qualitative assessments are correct by a sample numerical calculation. Finally, we point out that nuclear binding energy was included in similar calculations of Yamasaki & Yamada (2006).
crit acc 2.3.6. Interpretation of Lν, core and the importance of Lν
As we have demonstrated in Section 2.3.4, the critical luminosity depends on
M˙ , M and rν. A natural combination of these variables to have the dimension of luminosity is GMM/r˙ ν, an expression for power released by accretion of material.
Although the powers of individual quantities are somewhat different than those in equation (2.26), one might wonder whether these expressions for the critical
crit luminosity are intimately related in the sense that Lν, core is directly proportional
acc to Lν . This interpretation is especially tempting, because dynamical supernova models seem to be on the verge of explosion over a wide range of M˙ , which would suggest that a significant fraction of the neutrino luminosity is supplied by cooling
˙ acc of the accreting gas. Any change in M yields a proportional change of Lν , which again somehow proportionally changes the critical luminosity while keeping it higher
acc than Lν . This picture is, however, incorrect.
crit To illustrate why, we plot in Figure 2.12 Lν, core along with the neutrino
crit acc luminosity leaving the shock Lν(rS) at Lν, core, and their difference, Lν . We also plot
52 ˙ acc GMM/rν, corresponding to the maximum achievable Lν for a cold accretion flow.
acc crit We see that Lν is always a small fraction of Lν, core, and that even the maximum
acc possible value of Lν falls short of the necessary luminosity by a factor of several as found also by Burrows & Goshy (1993). Clearly, the major source of neutrinos powering the explosion is the core of the PNS. However, the accretion luminosity
crit is not entirely unimportant. We plot in Figure 2.12 Lν, core for a calculation with the same parameters but with neutrino radiation transport shut off. Including the radiation transport in the calculation has a positive effect of lowering the critical core luminosity by about 8% at M˙ = 0.01 M s−1 and by about 23% at M˙ = 2 M − ⊙ − ⊙ s−1, respectively.
Instead of regarding equation (2.26) as a manifestation of accretion luminosity, we can understand it in the terms of the critical condition on the sound speed in isothermal accretion given in equation (2.5). The dependencies of equation (2.26) can be then explained in simple terms. Increasing the mass of the PNS core makes the escape velocity higher and velocity profile steeper, and more heating is then necessary to get a mild enough velocity profile that it can no longer connect to the shock jump conditions (Section 2.2). Increasing rν shifts the region of interest to weaker gravitational potential with lower escape velocity and hence lower neutrino luminosity is required to get a mild velocity profile that again does not connect to the shock jump conditions. Decreasing τν with all other parameters fixed means less neutrinos absorbed and clearly a higher critical neutrino luminosity is required.
53 To illustrate this in greater detail, we can writeq/v ˙ dc2 /dr c2 /r and ∼ S ∼ S ν 2 hence c h qr˙ /v ρ r L core/M˙ , where h is the enthalpy and ρ is the density S ∼ ∼ ν ∼ ν ν ν, ν crit at the neutrinosphere. Plugging this expression as cT in equation (2.5), we obtain
Lcrit GMM/˙ (ρ r2). The product ρ r is proportional to the optical depth τ , ν, core ∼ ν ν ν ν ν because most of opacity to the neutrinos is concentrated close to the neutrinosphere.
crit Thus, by relating cT of the isothermal accretion model through Euler equations to
crit Lν, core we arrive through a simple estimate to essentially the same expression for
crit Lν, core as in the dimensional analysis in the beginning of this Section. The power-law dependence of these estimates is of course different than in the empirical fit in equation (2.26) due to nonlinear nature of the problem.
Several authors have produced critical curves based on either steady-state calculations like our own, or hydrodynamic simulations. Through steady-state calculations, Burrows & Goshy (1993) obtained a critical curve that is best fit by a power law, Lcrit M˙ 0.43. This power-law slope is lower by about 0.3 than what ν, core ∼ we measure. This difference can be reconciled by realizing that Burrows & Goshy
(1993) assumed that the neutrinosphere radius rν is determined by Lν, core and neutrino temperature Tν through a black-body law. If we substitute equation (2.27) to equation (2.26) to eliminate rν, we get
Lcrit τ −0.114M 1.02M˙ 0.401ǫ0.68. (2.29) ν, core ∝ ν νe
The power-law slope in M˙ is about 0.40, quite close to Burrows & Goshy (1993).
54 In Figure 2.11 we compare our results to the critical curves of Nordhaus et al. (2010) based on hydrodynamic simulations. In order to make our calculations compatible with theirs, we couple rν to Lν, core through equation (2.27) with
Tν = 4.5 MeV. We keep M fixed at 1.4 M⊙, which makes the slope of our critical curves slightly milder than that of Nordhaus et al. (2010): for a given progenitor model lower, but fixed, Lν, core means that the explosion occurred later when M
crit is higher, because the PNS had a longer time to gain mass, and hence Lν, core is higher. We see that our results are in good agreement with Nordhaus et al. (2010)
crit and that including heating from accretion luminosity reduces Lν, core by an amount comparable to going from 1D to 2D or from 2D to 3D.
2.4. Explosion conditions
Our goal in this section is to find out how well some of the conditions proposed in the supernova literature for reviving the stalled accretion shock work in diagnosing
crit Lν, core within the context of steady-state calculation. Similar work was performed by Murphy & Burrows (2008) with dynamical simulations. Our code does not allow for time evolution, but we can directly and exactly compare how a given explosion
crit ˙ condition relates to the derived steady-state value of Lν, core at various values of M.
55 2.4.1. Positive acceleration condition of Bethe &
Wilson (1985)
Bethe & Wilson (1985) were the first to analyze the delayed neutrino mechanism. They argued that shock recession creates a steeper pressure profile that in turn overcomes gravitation and gives an outward acceleration,r ¨ > 0. In the language of our steady-state model,r ¨ = v(dv/dr) > 0 is the statement of their critical explosion condition. From Figure 2.4 we clearly see that below the shock the flow decelerates everywhere, except maybe for a small range of radii, where v is basically constant. We also find through an investigation of the velocity profiles that having dv/dr = 0 somewhere in the flow does not correlate with approaching the critical curve at different M˙ . In the discussion of our toy model in Section 2.5 we show that velocity and acceleration terms are subdominant in the momentum
crit and energy equation of the problem and that Lν, core does not depend on these terms.
From this evidence we conclude that the vanishing acceleration criterion of Bethe &
Wilson (1985) does not coincide with the critical neutrino luminosity.
2.4.2. Explosion condition of Janka (2001)
Janka (2001) presented a quasi-time-dependent analytic analysis of the conditions necessary for revival of a stalled accretion shock. He found that the shock experiences both expansion and outward acceleration when both the mass and
56 energy in the gain region grow with time. These conditions define two critical lines in the Lν, core–M˙ plane that enclose a region favorable for explosion. As the approach of Janka (2001) is completely different from that of Burrows & Goshy (1993) and also ours, his critical condition is also very different. As can be seen in Figures 4 and
5 of Janka (2001), the critical luminosities are decreasing function of increasing M˙ , | | unlike the plots in Burrows & Goshy (1993), Yamasaki & Yamada (2005, 2006), and our Figure 2.12, where the critical luminosities increase with increasing M˙ . | |
In Figure 2.13, we plot as a function of Lν, core and M˙ the mass in the gain
rS 2 layer, Mgain = 4πr ρ dr, total mass between the neutrinosphere and the shock, rgain
R rS 2 Mtot, energy in the gain layer, Egain = 4πr ρ(ε GM/r)dr, and the total rgain − R energy between the neutrinosphere and the shock Etot. We see that for a model
crit with fixed mass accretion rate the masses grow with increasing Lν, core as Lν, core is approached. While the total energy at any specific radius in the gain layer increases with growing Lν, core (see Figure 2.4 for a plot of the related Bernoulli integral), the total energy within that region decreases, because the gain layer gets physically larger with increasing Lν, core.
The critical conditions of Janka (2001) demand that the mass and energy in the gain region increase with time. As there is no time dependence in our steady-state models and thus this condition is not directly testable, we might emulate the time dependence with a sequence of steady-state models. If we fix M˙ and increase Lν, core, following either of the lines in Figure 2.13, Mgain grows and Egain decreases. On
57 51 −1 the other hand, if we fix L core to, say, 5 10 ergs s and decrease M˙ , which ν, × | | corresponds to moving vertically from red through blue to black lines in Figure 2.13,
Mgain first drops and then increases. Similarly, Egain first increases and then just before reaching the critical curve it decreases. Thus, from the point of view of our steady-state models, the conditions of Janka (2001) are not equivalent to the critical neutrino luminosity.
2.4.3. Advection time vs. heating time
The ratio of advection to heating time tadv/theat has been extensively used as a metric of hydrodynamic simulations (e.g. Thompson 2000; Thompson & Murray
2001; Thompson et al. 2005; Buras et al. 2006b; Scheck et al. 2008). Murphy &
Burrows (2008) found that this condition is only a rough diagnostic of the critical condition for explosion.
Here, we specifically test two definitions of advection and heating times. The
first is an integral condition on the gain layer due to Murphy & Burrows (2008)
rS dr t′ = , (2.30) adv v Zrgain | | rS 2 r 4πr ρε dr t′ = gain . (2.31) heat rS 4πr2ρ q˙ dr Rrgain R ′ ′ We plot in Figure 2.14 values of the ratio tadv/theat for a range of mass accretion rates. To a factor of 2 the condition t′ /t′ 1 at Lcrit . However, the specific ∼ adv heat ∼ ν, core 58 value of the ratio changes from about 0.7 to almost 1.2 over the range of mass accretion rates – more than a 50% increase. Clearly, this definition does not exactly correspond to the critical luminosity.
The second definition of advection and heating times is due to Thompson et al.
(2005), who defined the times locally as
r t′′ = P , (2.32) adv v
P/ρ t′′ = , (2.33) heat q˙
−1 where rP = (dln P/dr) is the pressure scale-height at a given radius. In Figure 2.15
′′ ′′ ˙ we plot the ratio tadv/theat as a function of radius for two values M. For the smaller
M˙ , the ratio spikes to high values below the gain radius, while it stays consistently below unity above it, even though the critical luminosity was reached. Contrary to
˙ ′′ ′′ this, the higher M calculation tolerates tadv/theat > 1 in the gain layer for a number of sub-critical luminosities.
Neither definition of advection and heating times yields the desired exact
crit correspondence with Lν, core. We tried to adjust the definitions in many ways, but failed to produce a condition that would be valid over a wide range of parameters
(e.g., M˙ , M, and rν).
59 2.4.4. Antesonic condition
crit 2 2 In Section 2.2 we showed that the antesonic condition (cT ) /vesc = 0.1875 is equivalent to the critical condition for isothermal accretion with pressure-less free fall at the shock, and it can be generalized to max (c2 /v2 ) 0.19Γ for a polytropic S esc ≈ EOS. Does a similar condition hold also in the more realistic calculations? In order to test this, we studied the maximum value of the ratio of the adiabatic sound speed
2 2 cS to the escape velocity, max (cS/vesc), similarly to the polytropic case. We plot in
Figure 2.16 how this value changes with core luminosity and mass accretion rate. We
find that the value of this parameter is surprisingly constant at max(c2 /v2 ) 0.20 S esc ≈ ˙ crit over almost three orders of magnitude in M at Lν, core = Lν, core. Furthermore, we took all 1350 critical luminosities used to construct Figure 2.9 and calculated a
2 2 crit histogram of values of max (cS/vesc) at Lν, core. We find that the histogram can be very well fit with a Gaussian with a maximum at 0.193 and with a width of 0.009, that is only 5% of its value! While from Figure 2.16 we observe in critical values a slight trend with M˙ , we also see from the same Figure that the numerical noise contributes significantly to the total scatter9.
9Although critical luminosities can be determined very precisely with our code, other parameters like critical shock radii are determined less precisely. This is because ∂rS/∂Lν, core = at the | | ∞ crit critical luminosity, as can be inferred from Figure 2.5. Hence, a small uncertainty in Lν, core is dramatically amplified in rS and related quantities, as can be noted in Figures 2.14 and 2.16.
60 2 2 How does the value of max (cS/vesc) depend on input physics? We determined
2 2 max(cS/vesc) for data of Figure 2.8 and found that there is some variability. For constant heating (solid red line) the ratio is about 0.15, while for no cooling (solid blue line) and heating proportional to r−3 (red dashed line) the mean value is about 0.26. For all other cases of different physics, the mean value ranges from
0.19 to 0.22, quite close to the result from our fiducial calculation. Furthermore, we also find that increasing Υ, the fraction of free-fall velocity of the incoming matter (eq. [2.21]), from 0.25 to unity increases the value of the ratio from 0.19 to about 0.22. We found in Section 2.2 that for polytropic EOS the antesonic ratio is linearly proportional to adiabatic index Γ. With this information at hand, the close
2 2 agreement between max (cS/vesc) measured in our fiducial calculation and the value in isothermal accretion can be regarded as a coincidence. The fiducial calculation is not exactly isothermal (Γ > 1), which compensates for lowering Υ to 0.25. Even so,
crit ˙ despite the fact that Lν, core(M) changes dramatically with the large changes to the input physics considered in Figure 2.8, max (c2 /v2 ) is virtually constant 0.2. S esc ∼
2 2 Another question is how does the radius at which cS/vesc peaks relate to other significant radii of the problem like the gain and shock radii? In the isothermal calculation this critical radius coincides with the shock radius, because at this position the escape velocity is smallest. In the full problem this is clearly not the case as shown in the lower left panel of Figure 2.4. Instead we find that radius of
2 2 maximum cS/vesc nearly coincides with the gain radius, but that they are offset by
61 as much as 2% in both directions. For the case of no cooling, the maximum occurs
2 2 at the neutrinosphere, and there is no gain radius. As the radius of maximum cS/vesc closely tracks the gain radius in the fiducial calculation, their difference might not be noticeable at all in simulations.
Burrows et al. (1995) proposed a similar “coronal” condition for the explosion, where T of the substantial amount of matter needs to be higher than the local escape temperature. However, we have demonstrated that the transition from accretion to wind does not require sound speeds above the escape speeds. Instead, the sound speed vs. escape speed condition is only a manifestation of the inability to satisfy the shock jump conditions. Indeed, as Figure 2.4 shows, the Bernoulli integral is negative at explosion as noted also by Burrows et al. (1995).
2.5. Analytic toy model
In this Section we discuss an analytical toy model that retains the important properties of the full numerical steady-state solution (as characterized by Burrows
crit & Goshy 1993 and Yamasaki & Yamada 2005, 2006), namely the existence of Lν, core and two branches of solutions with different shock radii. We are able to derive
crit explicit analytic expressions for Lν, core and rS (eqs. 2.40 & 2.41).
Our toy model is based on the assumption of hydrostatic equilibrium,
crit conservation of energy, and the fact that Lν, core occurs when the shock jump
62 conditions cannot be simultaneously matched to any solution of the Euler equations
crit (Section 2.2). We find that we can realistically model Lν, core using the solution to simple algebraic equations. Specifically, we obtain a quadratic equation for rS,
crit which gives Lν, core when the discriminant vanishes. The cost of this simplicity is that we must introduce a number of approximations, albeit reasonable, and are forced to make our model somewhat internally inconsistent. Therefore, our toy model produces only qualitative agreement with the numerical solution. Even this approach, however, enables us to grasp the basic elements of the physics involved and to extend them to more complicated and more complete models.
2.5.1. Construction of the toy model
It is well known (e.g. Janka 2001) that the region between the accretion shock and the neutrinosphere is very close to hydrostatic equilibrium. Figure 2.17 shows individual components of the Bernoulli integral (equation [2.23]) along with their radial derivatives. We see that contributions of both the kinetic energy term v2/2 to the total energy budget and of the v(dv/dr) term to the momentum equilibrium are negligible. Thus, we can formulate the steady-state problem without the v(dv/dr) term in the momentum equilibrium equation and neglect the v2 terms on the left-hand side of the shock jump conditions. In this setting, the flow is hydrostatic
2 and velocity can be formally defined as v = M/˙ (4πr ρ). We also set dYe/dr = 0 and dL /dr = 0, which implies τ 1. With these simplifications the equations (2.6–2.8) ν ν ≪
63 can be reformulated as
r1 ′ GMρ(r , rS) ′ P (r1) P (r2) = dr (2.34) − − r′2 Zr2 r1 ′2 ′ GM GM 4πr ρ(r , rS) ′ ′ h(r1) h(r2) = + q˙(r ,Lν, core)dr , (2.35) − r1 − r2 M˙ Zr2 where h = ε + P/ρ is the specific enthalpy of the flow and we explicitly state that the density profile ρ(r, rS) depends on the shock radius rS.
A primary simplification of the model is that
As boundary conditions we choose the two shock jump conditions assuming that the matter up the shock is in free fall, specifically h(rS) = GM/rS and
˙ 2 P (rS) = vescM/(4πrS), and we fix the inner boundary density to ρ(rν) = ρν. To make use of the boundary conditions, we set r1 = rν and r2 = rS in equations (2.34) and (2.35). We choose the equation of state of relativistic particles which satisfies
ε = 3P/ρ and hence h = 4P/ρ. We chooseq ˙ = with heating and cooling H − C functions depending only on radius r:
aL = ν, core , (2.36) H r2 b = . (2.37) C r4
While is a good approximation, lacks the important self-regulatory feature of H C realistic cooling, which is proportional to T 6, that an excess in the internal energy can be quickly radiated away. In other words, when no heating is present, the
64 material cannot radiate away via neutrino cooling more energy than its total internal energy content. To quantify this constraint on the cooling function (r), we assume C that a cold parcel of matter falls from large distance through the shock to the neutrinosphere at rν. When no heating is present, the parcel will radiate a fraction of its gravitational potential energy
∞ 4πr2ρ GM (r)dr = η , (2.38) ν M˙ C rν Zr where η is the cooling efficiency. If the sole source of neutrino flux is the accretion
flow itself (ie. there is no L core), then η 1. In the presence of heating we still ν, ≤ consider equation (2.38) to be valid, but with a different (and potentially higher) value of η. A condition similar to equation (2.38) was employed by Fern´andez &
Thompson (2009a) to limit excessive cooling due to discreteness effects.
We verified with our relaxation code that our choices of equation of state and heating and cooling along with an assumption of hydrostatic equilibrium do not fundamentally change the phenomenology of the problem. Namely, even in this highly simplified version of the problem the critical neutrino luminosity and the two branches of solutions are still present.
We construct our toy model by evaluating the momentum balance
(equation [2.34]) and enthalpy conservation (equation [2.35]) between the neutrinosphere and the shock and then solving for rS. In order to have an
−3 analytically tractable result we fix the density profile to be ρ(r) = ρν(r/rν) .
65 However, as the problem is fully determined and the density profile is an integral part of the solution, an ad hoc prescription of the density profile violates one of the boundary conditions. We choose to keep the boundary conditions on enthalpy
˙ 2 and pressure, h(rS) = GM/rS and P (rS) = Mvesc/(πrS), and to violate the related condition on density ρ(rS) = Mv˙ esc/(πGMrS). This internal inconsistency means that our toy model will produce only qualitative agreement with the corresponding numerical solution. However, our point is to illustrate basic ingredients and physical principles of steady-state accretion, not to provide any numeric values.
Integrating equations (2.34–2.35) and applying the boundary conditions we get a polynomial equation for the shock radius
2πr3ρ aL b r 4 Mv˙ πρ r3 aL r 2 ν ν ν, core S + esc 2 ν ν ν, core S ˙ r2 − 2r4 r πρ r2 − ˙ r2 r − M ν ν ν ν ν M ν ! ν GM πρ b + ν = 0, (2.39) − rν rνM˙
where we neglect the dependence of the free-fall velocity vesc on the shock radius rS to get an analytic solution. We will discuss how this approximation changes the toy model results further below. With this assumption, equation (2.39) is solved for the shock radius yielding
66 r 2 2aL b −1 aL v M˙ 2 S = ν, core ν, core esc r r2 − r4 × r2 − 2π2ρ2r5 ± ν ν ν " ν ν ν 2GMM˙ aL b v M˙ 2 aL v2 M˙ 4 q˙(r )2 + ν, core esc ν, core + esc . (2.40) ± ν πρ r4 r2 − 2r4 − π2ρ2r5 r2 4π4ρ4r10 s ν ν ν ν ν ν ν ν ν While it is not possible to readily provide obvious physical interpretation to the individual terms in equation (2.40), we can say that they are combination of heating and cooling terms that include coefficients a and b, which measure the strength of heating and cooling, the gravitational potential energy term proportional to GM/rν, and the ram pressure term that is proportional to vescM˙ .
Our solution for the square of the shock radius has two critical points. The
2 first critical point occurs when the denominator vanishes, Lν, core = b/2arν. In this case one of the solutions stays finite and the other goes from negative (and hence unphysical) to infinite. This corresponds to the minimal neutrino luminosity that allows two solutions for the shock radius10. Note that this happens for a finite and positive value of Lν, core.
The second critical point occurs when the discriminant vanishes. At this point the two solution branches merge and there is no solution for parameter combinations 10 ∞ 2 ˙ One might be tempted to interpret rν 4πr ρq/˙ Mdr = 0 as an equivalent and general condition for the minimum luminosity that givesR two solutions for the shock radius. However, this is a mere coincidence arising from our choice of density profile and equation of state that cancels several terms in equations (2.34) and (2.35) when rS . → ∞ 67 that give a negative discriminant. Thus, the neutrino luminosity corresponding
crit to this critical point can be readily identified as the critical luminosity Lν, core.
The equation for the critical curve can be obtained by setting the discriminant in equation (2.40) to zero and solving the quadratic equation for the neutrino luminosity
˙ ˙ 2 crit b GM M vescM aLν, core = 2 + | 2 | + 2 2 3 rν πρνrν 2π ρνrν ±
GM M˙ πρνb vescb vescM˙ | 2 | 1 + 2 , (2.41) ± πρνrν s − GMM˙ rν(GM) − πρνrνGM where the physically relevant critical luminosity is the smaller of the two solutions.
To explain this result, let us keep for the moment only the first two terms on the right-hand side. Merging the terms with a and b toq ˙crit and realizing that
M˙ = 4πr2ρ v(r ), we can writeq ˙critr /v(r ) GM/r , and c2 qr˙ /v (similar ν ν ν ν ν ∼ ν S ∼ ν to the discussion in Section 2.3.4). Combining these expressions shows that
(ccrit)2 GM/r , which is basically equivalent to the critical condition for isothermal S ∼ ν accretion as given by equation (2.5). The additional terms on the right-hand side arise because within the framework of the toy model we were able to calculate the critical condition exactly.
We note at this point that the analytic and relatively short form of equations (2.39–2.41) is a result of a particular choice of the density profile and the heating and cooling functions. It is indeed possible to repeat the derivations with different power-law indices for heating and cooling and even with a density profile
68 prescription that satisfies the outer boundary condition on density, which would make the toy model internally consistent. The final polynomial equation would be of higher order and generally without analytic solution at all. Even without an analytic solution it is possible to determine the two critical points using Descartes’ rule of signs (appearance of the second solution branch for the shock radius) and setting the polynomial discriminant to vanish (maximum neutrino luminosity that gives steady-state solution). However, we decide not to discuss these more complicated toy models as the lengthy equations are not more illuminating than the simplest version we present here.
We set b = ηGM M˙ /πρ as discussed in Section 2.5 and rewrite equation (2.41) | | ν as
˙ crit GM M 1 Lν, core = | | rν πrνρνa × v M˙ v M˙ 1+ η esc (1 + η) 1 esc , (2.42) v × − 2πrνρνGM ± u − πrνρνGM ! u t where the physically relevant solution is the one that gives lower critical luminosity.
With values of other parameters of the problem given in Table 2.1 we can test how our toy model compares with our numerical solution with the same input physics. Figure 2.18 shows the shock radius rS as a function of Lν, core for
M˙ = 0.5 M s−1. We compare the analytic solution given by equation (2.40) (solid − ⊙ line) to the numerical solution of the simplified problem (dashed line). With the
69 dotted line we also plot the numerical solution to equation (2.39), which differs from the analytic solution by accounting for the fact that vesc is a function of the shock radius rS. Both analytic and numerical solution to equation (2.39) agree very well within each other and hence we prove the assumption of constant vesc is justified.
The agreement between the numerical solution without fixed density profile and the analytic solution is, however, only qualitative. From the inset plot, which shows the density profiles of the numerical solution, we see that the density profiles are to first order power laws with the slope dependent on the shock radius. This explains why there is a mismatch of 8 1051 ergs s−1 in the critical luminosity between the ∼ × numerical and analytic solution. There is also a slight curvature to the numerically obtained density profiles that results in these shock radii always being slightly higher than the analytic ones. Despite the obvious inadequacies we are able to very well reproduce the numerical shock radii with our analytic model.
Figure 2.19 shows the comparison of the analytic critical curve given by equation (2.42) with the numerical solution to the simplified problem. We see a very
−1 good agreement both in the absolute value and in the slope for M˙ < 1 M⊙ s . At | | ∼ higher M˙ , we see an upturn in both the analytic and numerical critical curves. In | | fact, the analytic critical curve ends when the term inside the square root vanishes.
We were not able to get a smoothly extending numerical critical curve above
2.5 M s−1, which suggests that it is absent or at least substantially changed at ∼ ⊙ very high mass-accretion rates. However, we did not investigate this issue further as
70 we do not see anything as nearly dramatic in the realistic critical curves presented in
Section 2.3.4 and it is thus an artifact of our simplistic assumptions in the toy model.
Furthermore, the mass-accretion rates where the critical curve ends are fairly high and steady-state approximation ceases to be valid in this area of parameter space.
To make a direct comparison with the numerical results of the full fiducial model presented in Section 2.3.4 we expand the square-root term in equation (2.42)
−1 in the limit of low mass-accretion rate ( M˙ < 1 M⊙ s ) and low cooling efficiency | | ∼ (η < 1) to get ∼
2 ˙ ˙ crit GM M 1 η 1 vescM Lν, core η | | 1+ 1+ , (2.43) ≈ rν 2πrνρνa 4 η πrνρνGM ! which is correct up to second order. The term in the square brackets is always positive and larger than 1. The denominator can be interpreted as an upper limit on the optical depth τ , because τ ∞ 4πρadr = 2πr ρ a for our fixed density ν ν ≈ rν ν ν R profile.
We conclude with an explanation of derivation of equation (2.45). The
crit discriminant in equation (2.40) vanishes at Lν, core, which allows us to plug in the
crit expression for Lν, core from equation (2.43). Under the same assumptions that were
crit used to derive equation (2.43), we can calculate that the critical shock radius rS
crit −1/2 does not depend on the particular Lν, core, but it is proportional to η .
71 crit acc 2.5.2. Relation of Lν, core to Lν
crit acc Is it possible to reach Lν, core with just Lν ? Within the framework of
crit the toy model, the answer to the question of whether Lν, core is always higher than ηGM M˙ /r depends on the ratio of the term in the square brackets in | | ν equation (2.43) (which is always positive and larger than unity) to the denominator,
2πrνρνa, an upper limit of the optical depth τν for our fixed density profile. For small η we can thus write (see eq. 2.41)
˙ crit η GM M Lν, core | |, (2.44) ≈ τν rν where the exact factor in front of the right-hand side is comparable to, but always . higher than unity. For parameters given in Table 2.1 we get τ 2πr ρ a = 0.36 ν ≈ ν ν and Lcrit > ηGM M˙ /r . We confirmed this result for all 0 <η 1 by inspecting ν, core | | ν ≤ graphs similar to Figure 2.19.
From equation (2.44) we see that if τ 1, then even if the whole region ν ∼ between the neutrinosphere and the accretion shock is immersed in a neutrino flux corresponding to the maximum integrated neutrino cooling ηGMM/r˙ ν, the critical luminosity cannot be attained. Furthermore, if η > τν then the critical luminosity ∼ is even higher than the maximum gravitational energy release rate. In reality, the accretion luminosity available for absorption will be substantially lower than in the ideal scenario described, and hence sufficient core neutrino luminosity independent of the accretion flow is required to reach an explosion. One could rightfully argue that
72 the optical depth is not constrained within our toy model and that at high τν some effect might come to play that lowers the critical luminosity below the accretion luminosity. However, as demonstrated by our numerical solution to the more detailed
crit version of the problem described in Section 2.3.4 and depicted in Figure 2.12, Lν, core is in fact much higher than the maximum possible accretion luminosity.
crit 2.5.3. Lν, core as a function of dimension
Ohnishi et al. (2006) and Iwakami et al. (2008) investigated the shock radii and stability of 2D and 3D accretion flows with heating and cooling based on steady-state models of Yamasaki & Yamada (2005). They found that going from
1D to 2D increases the shock radii even when the region is convectively stable, and that this effect is more pronounced for higher Lν, core. They also noticed that
52 −1 their L core = 6.0 10 ergs s simulation exploded even though it was stable in ν, × spherical symmetry and steady-state calculations. The common opinion shared by
Ohnishi et al. (2006) and Iwakami et al. (2008) is that the SASI is responsible for
crit the shock revival, essentially lowering Lν, core. However, we have shown in our toy model (Figure 2.18 and in more involved calculations, Figure 2.8) that decreasing
crit the cooling efficiency η lowers Lν, core, as displayed in equation (2.44). In order to illustrate this point more clearly, we plot in Figure 2.18 shock radii for somewhat
crit lower cooling efficiency η. We see that not only is Lν, core decreased, but also that the shock radii consistently increase over the whole range of allowed core luminosities
73 and that at fixed luminosity the increase of rS is both relatively and absolutely more
crit prominent. For lower η the shock radius at Lν, core increases only modestly, as can be seen from Figure 2.18.
To put this on more quantitative grounds, Ohnishi et al. (2006) found non-
52 exploding dynamical 1D models for neutrino luminosities up to L core = 6.5 10 ergs ν, × −1 52 −1 s , while the models in 2D did not explode up to L core = 5.5 10 ergs s . These ν, × crit numbers can be identified as Lν, core in 1D and 2D, respectively. Interpreted using
crit equation (2.44), these values of Lν, core suggest that η decreased by about 15% in going from 1D to 2D. However, from the physics of the toy model and our derivation of equations (2.44) and (2.40), we find that changes in η are always accompanied by
crit crit changes in rS at Lν, core, such that the ratio of two critical shock radii rS for two
crit cooling efficiencies η1 and η2 at their respective Lν, core is
rcrit(η ) η S 1 = 2 . (2.45) rcrit(η ) η S 2 r 1
This equation implies that a decrease in η of 15% should correspond to an increase
crit in rS at L by 8 9%, in good agreement with the larger shock radii found by ν, core − Ohnishi et al. (2006) in their 2D simulations (see their Fig. 4).
crit Similarly, Nordhaus et al. (2010) found that in 3D Lν, core drops to about 60% of the 1D value (see our Figure 2.11). What modification of the physics would be necessary to explain such a decrease? As shown in Figure 2.20, we find within our fiducial model that reducing cooling by a factor of two yields the desired drop
74 crit of Lν, core. Simultaneously, the entropy in the gain layer at fixed Lν, core increases
crit by about 30% and rS at Lν, core increases by about 5 to 7%. We obtain similar
crit changes in Lν, core and entropy if we increase the net heatingq ˙ by a factor of two in the gain layer, thus simulating in a crude way increased energy deposition as
crit might be provided by convection. However, in this case rS decreases by about 8 to
10%, although at fixed Lν, core rS is larger. Yamasaki & Yamada (2006) performed a calculation similar to ours, which included convection in a phenomenological way,
crit that essentially flattens the entropy gradient, and they found that while Lν, core decreased substantially, the shock radii also decreased.
We note here, that η is the only free parameter within our toy model and therefore the effects of all unknown physics get projected into η. Even so, the decrease in the critical luminosities and the increase of shock radii as a function of dimension observed in simulations (e.g. Ohnishi et al. 2006; Iwakami et al.
2008; Murphy & Burrows 2008; Nordhaus et al. 2010) may occur because higher dimensional flows have less efficient cooling, and not because of SASI or any other instability. This is supported by Nordhaus et al. (2010) who did not see in their 3D
crit simulations as vigorous SASI activity as in their 2D simulations, yet Lν, core was lower than in 1D. Furthermore, while both decreased cooling and increased heating give
crit lower Lν, core, higher entropy, and higher rS at fixed Lν, core, the latter gives lower rS at
crit Lν, core, which we consider incompatible with results of hydrodynamical simulations.
Finally, while it is somewhat difficult to disentangle the effects of modified heating
75 crit and cooling, we think that the cooling is responsible for the reduction in Lν, core because the temperature dependence of cooling offers greater prospects for modifying
crit the structure of the flow, providing lower Lν, core, higher rS, and higher entropy.
crit Looking at Figure 2.8, we see that with no cooling Lν, core can be more than order of magnitude smaller at low M˙ . As the usefulness of our 1D calculations and the | | toy model are quite limited for discussing multi-dimensional effects, our work can provide predictions and scaling relations of various quantities that will be useful in understanding the results of future simulations.
2.6. Discussion and summary
We studied spherically-symmetric accretion flows in order to understand the
crit physics of the critical neutrino luminosity, Lν, core, which separates steady-state accretion from explosion — a formulation of the neutrino mechanism of supernovae.
Our results can be summarized as follows.
1. We investigated isothermal accretion flows with fixed mass accretion rate M˙ and variable sound speed cT . In Figure 2.1 we showed that there is a maximum, critical
crit sound speed cT which allows for a steady-state shock standoff accretion shock. For
crit cT >cT the flow cannot conserve mass and momentum simultaneously at the shock and it must move outward on a dynamical timescale to establish a wind solution.
76 For the shock jump conditions relevant for the supernova problem we showed that
2 2 there is no smooth transition to an outflow solution. We studied the ratio cT /vesc
crit and we find that it is constant at cT and stays always below the value for the sonic point. Thus, for the first time, we derive the “antesonic” condition for explosion: if c2 exceeds 0.19v2 (eq. [2.5]), there is no solution for a steady-state accretion T ≃ esc flow with a standoff accretion shock, and explosion necessarily results. The physical realization of this transition is likely modified by time-dependent hydrodynamical instabilities.
crit 2. The mechanism of cT in the isothermal model does not depend on any particular heating mechanism, and we explicitly showed in Section 2.3.3 that it is directly
crit analogous to Lν, core observed in more complete steady-state calculations. Specifically, in Figure 2.7 we investigated and classified the structure of accretion flows in the supernova problem as a function of the core neutrino luminosity Lν, core with respect to the position of the sonic point and the constraints imposed by the shock jump conditions. We found that the critical solution does not correspond to the sonic point, but it is always marginally close (Figure 2.6). We showed that the maximum of the ratio c2 /v2 at Lcrit is close to 0.19 (Figure 2.16) and stays constant S esc ν, core ≃ to within 5% over a wide range of M˙ , and masses and radii of the PNS. Thus,
crit our “antesonic” condition corresponds directly to Lν, core and is thus superior in
crit monitoring the approach to Lν, core when compared to other heuristic conditions
77 proposed in the literature (Section 2.4). For example, the ratio of the advection to the heating time in the accretion flow is not constant along the critical curve
crit ˙ (Figures 2.14 & 2.15). Instead, the physics of Lν, core(M), and thus the neutrino mechanism of supernovae itself, is identical to the physics of the antesonic condition.
3. Extending the previous works of Burrows & Goshy (1993) and Yamasaki &
Yamada (2005, 2006) we studied the thermodynamical profiles of the flows for Lν, core
crit ranging from 0 to Lν, core (Figure 2.4). We confirm that there are two solutions with different shock radii and different energies for a given Lν, core (Figure 2.5), which
crit merge at Lν, core. We also found that the critical curve robustly exists for a wide range of microphysics, although the exact normalization and slope vary (Figure 2.8).
crit ˙ Finally, we quantified the dependence of Lν, core over a wide range of M, and mass and radius of the PNS, providing a useful power-law fit as a function of the key parameters of the problem (Figure 2.9 and eq. [2.26]). Our results imply that larger
crit PNS radius decreases Lν, core, potentially favoring stiff high-density nuclear equations of state (see Section 3.4; eq. [18]). However, faster contractions of the PNS in soft
EOS models can be favorable for explosion by increasing neutrino luminosities and energies.
4. We include in our calculations a simple approximation to gray neutrino radiation transport (Figure 2.4). We found that the neutrino cooling of the flow above the
78 acc crit neutrinosphere, the accretion neutrino luminosity, Lν , lowers Lν, core by a small but significant amount (Figure 2.12). Thus, it is the core, not the accretion flow, which is responsible for the transition to the explosion. Specifically, we show by analysis
crit of our numerical results and by calculations within our toy model that Lν, core is always higher than the maximum available accretion power GMM/r˙ ν (Figure 2.19
crit acc and eq. [2.44]). Even though Lν, core is always >Lν , we found in Figure 2.11 that
crit the inclusion of heating by accretion luminosity decreases Lν, core by an amount comparable to the reduction caused by going from 1D to 2D, or from 2D to 3D.
5. Our numerical calculations and toy model imply (but do not prove) that the
crit reduction of Lν, core seen in recent 2D and 3D simulations likely arises from an overall reduction in the cooling efficiency allowed by the accretion flows in multi-dimensions
(Section 2.5.3 and Figure 2.18), and not from from additional heating caused by convection or the presence of shock oscillations. In particular, while both a reduction in cooling and an increase in heating increase the entropy in the gain
crit layer and decrease Lν, core, a reduction in cooling always occurs with an increase in
crit crit rS(Lν, core), while an increase in heating results in a decrease in rS(Lν, core). Because
crit the simulations imply that rS(Lν, core) is somewhat larger in 2D and 3D with respect to 1D, our equation (2.45), derived from the toy model, implies that a decrease in
crit the cooling efficiency of the flow is the cause of the observed decrease in Lν, core.
79 However, more work on this issue is clearly warranted.
80 Parameter Value Description
M 1.2 M⊙ neutron star mass
rν 40 km neutrinosphere radius ρ 3 1010 g cm−3 density at the neutrinosphere ν × 9 −1 vesc 7.31 10 cm s velocity just outside the shock − × η 0.4 cooling efficiency a 6.45 10−19 cm2 g−1 heating function coefficient × GM|M˙ | b η πρν cooling function coefficient
Table 2.1. Values of parameters used in the toy model.
81 Fig. 2.1.— Isothermal accretion plotted in the space of Mach number M and rescaled 2 radial coordinated x = rcT /(2GM). Solid black lines show solutions to eq. (2.1) with −1 M˙ = 1 M⊙ s and M = 1.4 M⊙ starting from rν = 30 km (grey dashed line), and − ˙ 2 10 3 2 with fixed velocity vν = M/(4πrνρν), where ρν = 3 10 g cm . The value of cT increases from 4 1018 (black solid line starting at lowest× x) to 1.68 1019 cm2 s−2 (highest line) in× the steps of 6.4 1017 cm2 s−2. The dashed line is the× decelerating × transonic solution going through the sonic point Msonic = 1 at xsonic = 0.25. The dash-dot line is Bondi accretion flow. Dotted lines are veloci−ties just downstream of a shock positioned at any x, assuming that the upstream flow is either Bondi accretion flow (blue bottom line), in free fall (green top line), or in pressure-less free fall (red middle line). A viable accretion solution with a shock starts at rν and follows any of the black lines until it crosses a dotted line, where it jumps to the assumed upstream crit velocity profile. Above a certain cT there is no accreting solution with a steady- state shock. The critical value xcrit, where this happens, is shown with a vertical red dash-dot-dot line for the pressure-less free fall upstream of the shock.
82 crit Fig. 2.2.— The maximum isothermal sound speed cT that allows for a shock in the flow at a given M˙ . The PNS parameters — M, rν, ρν — and color coding — green, red, blue — are the same as in Figure 2.1.
83 Fig. 2.3.— Antesonic factor for polytropic equation of state. For each Γ, we plot 2 2 ˙ −1 max(cS/vesc) as a function of k for M ranging from 0.01 M⊙ s (black lines) to −1 − 1.0 M⊙ s (red lines). For fixed Γ, the lines overlap. The antesonic factors at each −critical k are marked with filled circles with color corresponding to M˙ . The antesonic factor at critical k is linearly proportional to Γ. The horizontal dotted line marks the value for the isothermal case, 3/16 = 0.1875.
84 Fig. 2.4.— Variables of interest as the core neutrino luminosity Lν, core approaches the critical value Lcrit for M˙ = 0.5 M s−1, M = 1.4 M and r = 50 km. Profiles ν, core − ⊙ ⊙ ν of density ρ, velocity v, temperature T , electron fraction Ye, neutrino luminosity Lν, net heatingq ˙, adiabatic sound speed cS, internal specific energy ε, and total specific B crit 52 −1 energy for 0 Lν, core Lν, core = 6.41 10 ergs s are shown. The profiles at crit ≤ ≤ × Lν, core are shown with blue thick lines.
85 Fig. 2.5.— Properties of the two branches of solutions for M = 1.4 M⊙ and rν = 50 km −1 for M˙ = 0.1 (blue), 0.5 (red) and 2.5 M⊙ s (green). Left: Shock radius rS as − −crit − 52 52 53 a function of Lν, core. Lν, core occurs at 2.17 10 , 6.42 10 , and 2.28 10 ergs −1 × × × s for the respective mass accretion rates. The black dotted line denotes rν. The solid and dashed parts of each line denote the lower (stable) and upper (unstable) solution branches, respectively. Right: Total specific energy B at rν (bottom lines), rgain (top lines) and rS (middle lines). Color coding is the same as in the left panel.
86 Fig. 2.6.— Overview of the flow structure in the simplified supernova setup with Lcrit = 9.07 1052 ergs s−1. Vertical dash-dotted lines denoted by numbers mark ν, core × values of Lν, core selected for detailed study in Figure 2.7. Left: Shock radius rS (red solid line) and sonic radius rsonic (black dashed line) as a function of Lν, core. Right: Temperature at the neutrinosphere T (rν) for the solutions with a shock (red solid line) and with a sonic point (black dashed line). Regions of different solution types are marked with letters A and B, as in Figure 2.1. The dotted line is the separatrix between A- and B-type solutions where the sonic point lies at infinity.
87 Fig. 2.7.— Detailed study of the structure of flows in the simplified supernova setup summarized in Figure 2.6. The four panels show flow structure at different values of Lν, core corresponding to different situations: single shock radius and no sonic point (upper left), two shock radii and two sonic-point solutions (upper right), no shock solution and two sonic-point solutions (lower left), and neither shock or sonic-point solution (lower right). In each panel the flows that satisfy the shock jump conditions are shown with red solid lines, flows going through the sonic point (filled triangle) are shown with black dashed lines, and all remaining flows are shown with solid black 2 2 lines. A-type solutions going through v = cS end with filled circles. Regions of different solution types are marked with letters A and B, as in Figure 2.6. Green solid lines mark the shock jump conditions. Additionally, the dotted vertical lines in the upper left and lower left panels mark radii of intersection of the flows with the momentum shock jump condition.
88 Fig. 2.8.— Effects of different physics on the critical curve for M = 1.4 M⊙, rν = 50 km. The solid black line shows the fiducial model, while the black dashed line denotes the model with simplified physics (see text for details), which was then subject to additional changes. We consider no cooling, = 0 (solid blue), cooling increased C 3 −11 3 by a factor of 5, = 5 s (blue dashed), cooling proportional to T , = 8.19 10 T C C −3 C × (blue dotted), heating proportional to r , = s (rν/r) (red dashed), constant −18 2 H H × heating rate = 1.2 10 Lν, core/rν (solid red), neutrino and Alfv´en waves heating, H21 −×1 −1 2 = s + 10 ergs s g (r /r) exp[(1 r/r )/5] (dash-dotted red; Metzger et al. H H ν − ν 2007), and EOS in the form of pure ideal gas (solid green).
89 Fig. 2.9.— Left: Curves of critical core neutrino luminosity as a function of M˙ . The black solid line is for M = 1.6 M⊙, rν = 40 km and τν = 2/3. Red dotted lines are for the same parameters except M = 2.0 M⊙ (upper line) or 1.2 M⊙ (lower line). Green dashed lines are for rν = 20 km (upper line) or 60 km (lower line). Blue dash-dotted lines are for τν = 1/3 (lower line) or 1/6 (upper line). Right: Relative residuals from the fitting of equation (2.26) to the ensemble of calculated critical luminosities. Each −1 line corresponds to a sequence of M˙ from 0.01 to 2 M⊙ s with other parameters fixed. − −
90 crit Fig. 2.10.— Critical shock radii relative to rν as a function of Lν, core for calculations presented in Figure 2.9. Different colors correspond to different values of M.
91 Fig. 2.11.— Comparison of our calculations with the results of Nordhaus et al. (2010) (red, green and blue lines with points). The black line is our fiducial model with M = 1.4 M⊙ and rν calculated from equation (2.27), while the dashed line is for our calculation with dLν/dr = 0 (eq. [2.19]). Heating from the accretion luminosity crit reduces Lν, core by an amount comparable to going from 1D to 2D or from 2D to 3D.
92 Fig. 2.12.— Effect of radiation transport and accretion luminosity on the critical crit curve. The black solid line shows critical core neutrino luminosity Lν, core for M = 1.6 M⊙ and rν = 40 km, the black dashed line is the neutrino luminosity at the shock Lν(rS), and the black dotted line is their difference, the accretion luminosity acc ˙ Lν . The blue dash-and-dotted line is the maximum accretion luminosity, GMM/rν. The red solid line is a critical curve for the same parameters except that neutrino radiation transport was shut off.
93 Fig. 2.13.— Top: Mass in the gain layer Mgain (lower set of lines) and the total mass between the neutrinosphere and the shock Mtot (upper set of lines) as a function crit of Lν, core. Filled circles at the end of the lines denote position of Lν, core. Colors −1 denote variations in the mass-accretion rate going from M˙ = 0.01 M⊙ s (black) to M˙ = 0.58 M s−1 (yellow) in uniform logarithmic steps.| | Bottom: Energy ´ala | | ⊙ Janka (2001) in the gain layer Egain (lower set of lines) and the energy between the neutrinosphere and the shock Etot (upper set of lines). The meaning of symbols and colors is the same as in the upper panel.
94 Fig. 2.14.— Ratio of advection and heating times as defined by Murphy & Burrows (2008). Each line shows the ratio as a function of core luminosity Lν, core and fixed M˙ . The color coding corresponds to different values of M˙ going lef to to right from −1 −1 | | 0.01 M⊙ s (black) to 1.95 M⊙ s (red). The filled circles at the end of the lines crit mark the value at Lν, core.
95 Fig. 2.15.— Ratio of advection and heating times as defined by Thompson et al. (2005) for two values of M˙ . Each line corresponds to a radial profile of the ratio for a given luminosity going from 1 1051 ergs s−1 (black) to the critical value (red). Filled circles at the end of the lines mark× the value just inside of the shock.
96 Fig. 2.16.— The antesonic condition. Maximum value of the ratio of the adiabatic sound speed cS to the escape velocity vesc as a function of Lν, core (individual lines). Colors distinguish between different mass accretion rates and have the same meaning crit as in Figure 2.14. Filled circles mark Lν, core. As discussed in Section 2.2 in the case of polytropic accretion flows, although the critical condition plotted here appears to be a local condition on the sound speed in the accretion flow, our analysis in Section 2.2 2 2 and in Figures 2.6 and 2.7 shows that the quantity max (cS/vesc) is a merely a scalar metric for solution space of Euler equations and thus it is a global condition.
97 Fig. 2.17.— Left: Absolute values of individual components of the Bernoulli integral for models from Figure 2.4: kinetic energy v2/2 (solid black), internal energy ε (dashed red), pressure P/ρ (dotted green) and gravitational potential energy GM/r (dash- dotted blue). Right: absolute values of the radial derivative of the profiles in the top panel: v(dv/dr) (solid black), dε/dr (dashed red), (dP/dr)/ρ (long-dashed green), P/ρ2(dρ/dr) (dotted orange), and GM/r2 (dash-dotted blue).
98 Fig. 2.18.— Shock radii rS as a function of Lν, core for the toy model described in Section 2.5. The solid blue line shows the analytic solution given by equation (2.40), the dotted green line is a numerical solution to equation (2.39) where the dependence of vesc on rS was included, and the dashed black line is the numerical solution without fixing the density profile. All have η = 0.4 (eq. [2.38]). The dash-dotted blue line shows the solution to equation (2.40) with η = 0.28. Critical points are marked with filled circles, and the upper and lower solution branches are shown with thin and thick lines, respectively. The inset plot shows density profiles from the numerical solution for a range of Lν, core. In order of increasing shock radii, the solid lines are profiles 52 −1 crit separated by 10 ergs s up to Lν, core (dashed purple line). Solid lines for higher 51 −1 shock radii (upper solution branch) are for Lν, core decreased in steps of 10 ergs s down to the minimum for the upper solution branch. The dotted orange line shows −3 the density profile assumed in the analytic toy model, ρ/ρν =(r/rν) .
99 crit Fig. 2.19.— Comparison of Lν, core from the numerical solution to eqs. (2.34) and (2.35) (dashed black line) to the analytic solution for the toy model provided in crit eq. (2.41) (solid blue line; only the solution with lower Lν, core is shown). The dash- dot red line shows ηGM M˙ /r (η = 0.4; compare with Fig. 2.18). | | ν
100 Fig. 2.20.— Illustration of effects of modified heating and cooling on rS (solid lines) crit and Lν, core (filled circles). Decreasing cooling (blue) and increasing heating (red) both crit act to decrease Lν, core, and to increase rS at fixed Lν, core with respect to the fiducial crit calculation (black). However, lower cooling efficiency at Lν, core increases rS, whereas crit higher heating decreases rS at Lν, core.
101 Chapter 3
Effect of Collective Neutrino Oscillations
3.1. Introduction
The hot proto-neutron star (PNS) cools by emission of neutrinos of all flavors.
Because νµ,ν ¯µ, ντ , andν ¯τ (hereafter collectively denoted as νx) do not interact with the PNS matter through the charged-current interactions, they decouple from matter at smaller PNS radii and higher temperatures, and thus can have higher average energy than νe andν ¯e. A fraction of the νe andν ¯e are absorbed below the accretion shock and the associated energy deposition rate per unit mass is approximately
2 2 proportional to Lνe, core(ενe + εν¯e ), where Lνe, core is the electron neutrino energy
luminosity of the PNS, and ενe and εν¯e are the electron neutrino root-mean-square energies. This energy deposition plays a significant role in the dynamics of the supernova and perhaps in the revival of the shockwave (e.g. Colgate & White 1966;
Bethe & Wilson 1985). Specifically, the “neutrino mechanism”, as formulated by Burrows & Goshy (1993), states that the steady-state accretion through the
crit shock turns into an explosion when Lνe, core exceeds a critical value, Lν, core. In
crit Pejcha & Thompson (2012) we showed using steady-state calculations that Lν, core
102 is equivalent to reaching max (c2 /v2 ) 0.19 in the accretion flow, where c is the S esc ≃ S sound speed and vesc is the local escape velocity. This “antesonic” condition is a manifestation of the inability of the flow to satisfy both the shock jump conditions and the Euler equations for the accretion flow simultaneously (Yamasaki & Yamada
crit 2005; Fern´andez 2012). We also determined the dependence of Lν, core on the key parameters of the problem, including the energies of the neutrinos over a wide range of parameter values. Specifically, and most importantly for this paper, we found that
crit Lν, core is proportional to the inverse square of the νe andν ¯e energies, as expected from the heating rate. There are a number of time-dependent multi-dimensional effects that might modify the transition from accretion to explosion. For example, accretion luminosity from cooling of the accretion flow is an important contribution
crit to Lν, core (Pejcha & Thompson 2012) and accretion simultaneously powering an asymmetric explosion is possible only in 2D and 3D (e.g. Burrows et al. 2006; Marek
& Janka 2009; Suwa et al. 2010). Furthermore, close to the critical condition for explosion the shock surface often exhibits oscillations that feed back on the neutrino emission (e.g. Murphy & Burrows 2008; Marek & Janka 2009; Nordhaus et al. 2010;
crit Hanke et al. 2012) potentially modifying Lν, core. At least in 1D, these oscillations
crit seem to occur only very close to the steady-state value of Lν, core (Fern´andez 2012).
crit The steady-state calculation is thus useful way to estimate Lν, core and to examine effects of modified physics on the critical condition for supernova explosion.
103 Most of the heating below the shock occurs due to absorption of νe andν ¯e on neutrons and protons, while the νx escape without much interaction. However, due to the high density of neutrinos in this region, self-interaction between neutrinos becomes important and can lead to a range of phenomena called “collective neutrino oscillations” (e.g. Pantaleone 1992; Duan et al. 2006, 2010). In particular, there is a possibility of an instability (Dasgupta et al. 2009) that exchanges part of the
νx spectra with the νe andν ¯e spectra. If the luminosities and energies of νe,ν ¯e, and νx are right, this can produce significantly more heating below the shock than calculations neglecting neutrino oscillations, i.e., effective Lν, core is increased by
CνO. Specifically, a strong effect on heating can be expected if luminosities are similar and νx have significantly higher energies than νe andν ¯e. The exact values of these quantities and their mutual ratios are model-dependent (e.g. Thompson et al.
2003; Marek & Janka 2009; Fischer et al. 2010, 2012; H¨udepohl et al. 2010).
Chakraborty et al. (2011a,b), Dasgupta et al. (2012), Suwa et al. (2011), and Sarikas et al. (2011) have investigated the role of CνO in the core-collapse simulations of several progenitor models. They found that there are a number of multi-angle effects, especially the effect of matter suppression, that can reduce or entirely eliminate CνO. We address the issue of increased neutrino heating due to
CνO and matter suppression without reference to detailed supernova models and
crit we evaluate them at Lν, core, which separates accretion from explosion. We treat the oscillation physics in a schematic way and the supernova physics using a steady
104 state model developed in Chapter 2 and Pejcha & Thompson (2012). Although this approach is less detailed than some recent papers, e.g., (e.g. Chakraborty et al.
2011a,b; Dasgupta et al. 2012; Sarikas et al. 2011), it allows for a parametric study to determine the potential role of CνO in shock reheating, and its dependence on the progenitor mass, radius, and accretion rate for a very broad range of parameters and without being tied to any particular progenitor model or simulation setup.
This Chapter is organized as follows. In Section 3.2, we describe our steady state model for the accretion flow based on Chapter 2 and Pejcha & Thompson
(2012), and a scheme for collective neutrino oscillations based on Dasgupta et al.
crit (2012). We present our results in Section 3.3. We quantify the changes to Lν, core and the shock radii, and compare the magnitude of the effect of CνO to other known pieces of physics. We also estimate the importance of multi-angle effects showing that they suppress CνO in the region of parameter space where they might otherwise be strong. In Section 3.4, we conclude with a discussion and review of our results.
3.2. Method
In this Section we first describe the hydrodynamic equations that we shall solve, their boundary conditions, and the input neutrino physics (Section 3.2.1). We describe our scheme of coupling the CνO effects to the hydrodynamical equations in
Section 3.2.2. Our combination of steady-state approach and simple treatment of
105 crit CνO allows us to calculate Lν, core, quantify the maximum possible effect of CνO on
crit Lν, core as a function of boundary conditions, and set limits on the parameter space, which can then be probed with more realistic methods.
3.2.1. Hydrodynamic Equations and Boundary Conditions
We use the code developed in Chapter 2 and Pejcha & Thompson (2012) to calculate the structure of the steady-state accretion flow between the neutrinosphere at radius rν and the standoff accretion shock at rS assuming spherical symmetry by solving the time-independent Euler equations
1 dρ 1 dv 2 + + = 0, (3.1) ρ dr v dr r dv 1 dP GM v + = , (3.2) dr ρ dr − r2 d P dρ q˙ E = , (3.3) dr − ρ2 dr v where P is the gas pressure, M is the mass within the radius of the neutrinosphere r , is the internal specific energy of the gas, andq ˙ is the net heating rate, a ν E difference of heating and cooling. We solve for the electron fraction Ye using
dYe v = l + l + (l + l + + l¯ + l − )Ye, (3.4) dr νen e n − νen e n νep e p
+ − where lνen, le n, lν¯ep, and le p are reaction rates involving neutrons and protons.
These equations are coupled together through the equation of state (EOS),
P (ρ,T,Ye) and (ρ,T,Ye), where T is the gas temperature. Our EOS contains E 106 relativistic electrons and positrons, including chemical potentials, and nonrelativistic free protons and neutrons (Qian & Woosley 1996). We use prescriptions of heating, cooling, opacity and reaction rates for charged-current processes with neutrons and protons from Scheck et al. (2006). We do not include the accretion luminosity, i.e. the changes in neutrino luminosities as a function of radius due to emission or absorption of neutrinos, but our neutrino luminosities change as a function of radius due to the CνO. Here, we also assume that νx do not directly interact with matter, but instead they convert to νe andν ¯e through CνO as described in Section 3.2.2.
While we explicitly calculate the degeneracy parameter of electrons and positrons, we assume that the degeneracy parameter of (anti)neutrinos is zero.
We need five boundary conditions to uniquely determine four functions: ρ, v,
T and Ye, and shock radius rS. We first demand that the flow has a fixed mass accretion rate M˙ = 4πr2ρv by applying this constraint at the inner boundary. At the outer boundary we apply the shock jump conditions
2 + 2 ρv + P = ρ vff , (3.5) 1 P 1 v2 + + = v2 , (3.6) 2 E ρ 2 ff
where vff = √Υvesc is the free fall velocity, and we choose Υ = 0.25 in agreement with the analytically estimated mass accretion rates of Woosley et al. (2002) supernova progenitors. The quantity ρ+ is the density just upstream of the shock and can be calculated from conservation of mass. We also assume that the matter entering the
107 shock is composed of iron and hence Ye = 26/56 at the outer boundary. The last boundary condition comes from requiring that rν is the neutrinosphere for electron neutrinos, which gives a constraint on the optical depth
rS 2 τ = κ ρ dr = , (3.7) νe νe 3 Zrν
where κνe is the opacity to electron neutrinos, which is proportional to the mean
2 square neutrino energy ενe .
The key parameters of the problem are the mass accretion rate M˙ through the shock, the PNS mass M, its radius rν, and the core neutrino luminosity of
each species Lνe, core. The shock radius rS is the eigenvalue of the problem and
1 crit is determined self-consistently in the calculation . The critical luminosity Lν, core
is determined by increasing Lνe, core with the remaining parameters fixed until no
crit steady-state solution is possible. For Lνe, core > Lν, core, the solution most likely transitions to a neutrino-driven wind, which is identified with the supernova explosion (Burrows 1987; Burrows & Goshy 1993; Yamasaki & Yamada 2005;
Fern´andez 2012). The highest value of Lνe, core that yields a steady-state solution is
1The only difference in the numerical setup in this Chapter with respect to Chapter 2 and Pejcha
& Thompson (2012) is that here we assume that luminosities in each neutrino species are constant throughout the region of interest. In Chapter 2 we implemented a simple gray neutrino transport, which allows for dLν /dr = 0. There is also a difference in notation in the sense that Lν,core in 6 Chapter 2 refers to the combined electron neutrino and antineutrino luminosity, while here we
parameterize the luminosities separately, Lν,core = 2Lνe, core.
108 crit crit identified with Lν, core. The value of Lν, core is polished to the desired level of precision by using a bisection method.
crit ˙ In Chapter 2 we calculated Lν, core and rS(Lνe, core) for a wide range of M, M,
crit 2 ˙ and rν. We found that Lν, core scales approximately as a power law in M, M, and rν,
Lcrit M˙ 0.723M 1.84r−1.61, (3.8) ν, core ∝ ν for 0.01 M˙ 2 M s−1, 1.2 M 2.0 M , and 20 r 60 km. We also found ≤ ≤ ⊙ ≤ ≤ ⊙ ≤ ν ≤ that for the physical solutions, rS increases with increasing Lνe, core (Yamasaki &
crit crit Yamada 2005) and that the values of rS at Lν, core, rS , depend only on the actual
crit value of Lν, core. Specifically, we found that
crit rS crit −0.26 (Lν, core) . (3.9) rν ∝
3.2.2. Treatment of CνO
In this Section, we describe our implementation of CνO effects in our
crit calculation. In order to evaluate the maximum potential effect on Lν, core, we employ a simple treatment of CνO that is relatively insensitive to details of, or approximations to, otherwise complicated calculations of CνO. Essentially, we build on the results of Dasgupta et al. (2012) who showed that this simple approach is a 2 crit ˙ We argued in Chapter 2 that the small upward curvature of Lν, core(M) is caused most likely by an exponential correction factor that arises due to nearly-hydrostatic structure of the accretion
flow.
109 good approximation to more complicated calculations, and we extend their work to
˙ crit a larger parameter space M, M, and rν, and evaluate CνO at Lν, core, the boundary between accretion and explosion.
To calculate the effective change of neutrino energies and luminosities as a function of radius we assume that a maximum possible flavor conversion occurs and we model this as a smooth non-oscillatory transition of the flavors. More specifically, in CνO the neutrinos convert to other flavors through interactions schematically written as
νeν¯e ⇆ νxν¯x. (3.10)
f With maximum possible flavor conversion, the “final” neutrino number fluxes φνe ,
f f i i φν¯e , and φνx can be expressed using the “initial” neutrino number fluxes φνe , φν¯e ,
i i i and φνx at the neutrino sphere. For example if φνe > φν¯e (which is always true for the scenarios considered in this paper), then from Equation (3.10) follows that the
i maximum number flux of νe that can be converted to νx is equal to φν¯e . Similarly,
i one quarter of the total flux in νx, φνx , is converted to νe and one quarter toν ¯e; the remaining half stays in νx. After the conversion is fully done, the νe number flux is a sum of φi φi with the original energy ε , which could not have been converted νe − ν¯e νe i due to a lack ofν ¯e, plus the number flux φνx of converted νx neutrinos with energy
110 ενx . Similar logic gives the final number fluxes in all flavors:
φf = φi φi + φi , (3.11) νe νe − ν¯e νx
f i φν¯e = φνx , (3.12)
f i i i 4φνx = 2φνx + φν¯e + φν¯e , (3.13)
f i where in the equation for φνx one instance of φν¯e represents converted νe with the appropriate energy and the other one convertedν ¯e with their energy. The total
i f neutrino number flux φν,ν¯ = φνe + φν¯e + 4φνx is conserved, specifically φν,ν¯ = φν,ν¯.
f The rms neutrino energies in each flavor after the conversion is finished, ενe ,
f f εν¯e , and ενx , are calculated as rms of the initial energies weighted by the fluxes as given in Equations (3.11)–(3.13), specifically for the electron-flavor neutrinos
(εf )2 = (φi φi )(εi )2 + φi (εi )2 /φf and εf = εi . νe νe − ν¯e νe νx νx νe ν¯e νx i For νe, the number flux at the neutrinosphere φνe is related to the core energy
luminosity Lνe, core as
L 4πr2φi = νe, core , (3.14) ν νe εi h νe i
i where ε is the mean energy of νe at the neutrinosphere. Expressions similar to h νe i Equation (3.14) are valid for φi and φi with initial mean energies εi and εi . In ν¯e νx h ν¯e i h νx i order to calculate the mean energies from the rms energies, that we use throughout the paper, we assume that the neutrino energy spectrum is Fermi-Dirac, which gives the necessary conversion factor. The differences in the number fluxes when compared
111 to using the rms energies are 13%, but as we show below the dependence of ∼ physical scale of conversion on the number flux is rather weak. The neutrinosphere radius rν of νe neutrinos is used to calculate the number fluxes at rν for all neutrino
flavors, because at rν theν ¯e and νx are already essentially free-streaming.
Motivated by more complete studies of the physical extent of the flavor conversion (e.g. Dasgupta et al. 2012), and as a numerical expedient, we assume that the neutrino number fluxes and energies for each flavor vary smoothly as a function of radius between the initial and final states. For νe,
φ (r) = [1 P (r)] φi + P (r)φf , (3.15) νe − νe νe ε (r) = [1 P (r)] εi + P (r)εf , (3.16) νe − νe νe
and similar equations are valid forν ¯e and νx. Neutrino luminosity Lνe (r) as a
function of radius is calculated as a product of φνe (r) and ενe (r), which guarantees
that Lνe (rν) = Lνe, core. The “survival probability” P (r) is modelled after the full solution to the CνO problem of Dasgupta et al. (2012) as
1 1 2r (rsync + rend) P (r)= + tanh − , (3.17) 2 2 σ(rend rsync) − where we choose σ 0.679 to have P (rsync) = 0.05 and P (rend) = 0.95. We shall use ≃ simple estimates for rsync and rend of flavor conversion derived from the analysis in
Hannestad et al. (2006), and summarized in Dasgupta et al. (2012). This approach is the most optimistic possibility for oscillations. Our aim in this paper is to ascertain
112 the maximum effect that CνO may have on the supernova shock reheating, and therefore our constraints shall be conservative. Any new physics effects that reduce the effect of CνO will only strengthen the constraints we derive here.
The two parameters rsync and rend in Equation (3.17) define the range of radii where CνO operates in a simplified treatment of nonlinear effects of neutrino-neutrino interactions (Hannestad et al. 2006). The flavor conversion occurs above a synchronization radius rsync, which is defined as
r2 µ(r ) = 4Ω 1+ ν . (3.18) sync 4r2 sync
The flavor conversion is more or less complete at radius rend defined as
2 − rν µ(rend) = ΩF 1+ 2 . (3.19) + 4r F end Here, Ω depends on the neutrino oscillation frequency and neutrino energy spectra and as in Dasgupta et al. (2012), we choose a typical value Ω = 50 km−1. The quantity / + is the ratio of the net lepton asymmetry in the system to the F− F F− neutrino flux available for oscillations +, which are defined as F
i i φνe φν¯e − = − , (3.20) F φν,ν¯ i i i φνe + φν¯e 2φνx + = − . (3.21) F φν,ν¯
The collective potential µ is defined as (Esteban-Pretel et al. 2007; Dasgupta et al.
2012)
r 2 r2 √ ν ν µ(r)= 2GFφν,ν¯ 2 2 , (3.22) r 2r rν − 113 where GF is the Fermi coupling constant. Equations (3.18) and (3.19) are solved for rsync and rend given rν and the neutrino number fluxes. We note here that the dependence of Equation (3.14) on rν introduces an absolute scale to Equations (3.18) and (3.19), and therefore rsync and rend do not scale linearly with rν. Instead, in the limit of rsync/r 1, the scaling is ν ≫
rsync 1/4 −1/2 Lνe, corerν (3.23) rν ∝ for fixed neutrino energies. The same scaling holds for rend.
The collective neutrino oscillations can be suppressed due to a variety of effects. Out of them, the most relevant here is the matter suppression of the CνO
(Esteban-Pretel et al. 2008). This occurs when the Mikheyev-Smirnov-Wolfenstein
(MSW) potential λ becomes much greater than a critical value λMA. The MSW potential is
Yeρ λ = √2GF (ne− ne+ )= √2GF , (3.24) − mn
where ne− and ne+ are the number densities of electrons and positrons, respectively, and we assume that the matter is composed only of protons, neutrons, electrons, and positrons. The critical value λMA is defined as
2 rν λMA = 2√2GFφ ¯ , (3.25) ν,ν r2 F− and it is essentially the collective potential weighted by the lepton asymmetry factor
(Dasgupta et al. 2012).
114 We emphasize that in the calculation presented here, the neutrino energies and luminosities as a function of radius self-consistently enter not only in the heating, but also in the reaction rates for the calculation of Ye (Eq. [3.4]) and in the boundary condition on optical depth (Eq. [3.7]). However, in our approximation we do not include accretion luminosity and we cannot resolve any of the multi-dimensional effects, although we attempt to effectively capture them by modifying the cooling rates as will be discussed in Section 3.3.1.
3.3. Results
In this Section, we illustrate the effect of CνO on the critical luminosity required
i i i for shock revival. First, we show our results with (ενe ,εν¯e ,ενx ) = (13, 15.5, 20) MeV
(Thompson et al. 2003), and Lνe, core = Lν¯e,core = Lνx,core, as an optimistic scenario for CνO (Section 3.3.1). Then we study a more realistic scenario with
i i i (ενe ,εν¯e ,ενx ) = (11, 13, 18) MeV, and Lνe, core = Lν¯e,core = 2Lνx,core, based on recent detailed simulations of the accretion phase (e.g. Marek & Janka 2009; Fischer et al. 2010, 2012). We discuss our results primarily for the former case, but we find that the latter case differs only in the maximum strength of CνO and not in the parameter range. We assess CνO for different sets of parameters in Section 3.3.2. We also initially ignore the matter suppression throughout the discussion to illustrate
115 the unattenuated magnitude of the effect, but we return to the matter suppression in Section 3.3.3 and show how it further constrains the parameter space for CνO.
crit 3.3.1. Effect of CνO on Lν, core
crit i i i In the left panel of Figure 3.1, we show Lν, core for (ενe ,εν¯e ,ενx ) = ˙ (13, 15.5, 20) MeV and Lνe, core = Lν¯e,core = Lνx,core including CνO as a function of M for M = 1.2 M⊙ and rν = 60 km (red dashed line) along with the fiducial calculation without the effect of CνO (black solid line). The critical curve in the fiducial calculation is approximately a power law (Chapter 2 Pejcha & Thompson 2012). We see that CνO lowers Lcrit to 0.65 times the fiducial value for M˙ < 0.01 M s−1. ν, core ∼ ⊙ −1 For higher M˙ , the critical curve turns upward and for M˙ > 0.32 M⊙ s , it essentially ∼ coincides with the fiducial calculation meaning that CνO have little effect.
The behavior seen in the left panel of Figure 3.1 is non-trivial even in our simple setup, because rsync and rend are a function of the boundary conditions. It can be understood by analyzing the position of rS relative to rsync and rend. We
crit expect that CνO will have an effect on Lν, core only if the conversion starts below the shock radius and the full effect will be obtained if the conversion is completed inside the standing shock. For the parameters we consider here, rsync is always
above the neutrinosphere. Because rS increases with Lνe, core (grey solid lines in
crit crit Fig. 3.1, right panel) and reaches a maximum rS at Lν, core, the effect of CνO is
116 crit most prominent for Lνe, core close to Lν, core. The right panel of Figure 3.1 shows
crit the effect of CνO on the shock radii. We see that rS in the fiducial calculation
(thick black solid line) closely follows the results from Chapter 2 (Eq. [3.9], dots, triangles and stars in Figure 3.1) except that the calculations presented here have
crit larger rS , because we set dLν/dr = 0. The calculation with CνO (red dashed line)
˙ crit crit closely follows the fiducial results for high M and Lν, core, where rsync > rS . Here,
crit the CνO effect is negligible and Lν, core is very close to the fiducial value. When
crit crit 52 −1 −1 r rsync, which occurs for L 1.5 10 ergs s (M˙ 0.32 M s ), the S ≃ ν, core ≃ × ≃ ⊙ crit CνO effect starts to become important, and both Lν, core and rS decrease relative
crit 51 −1 to the fiducial calculation. For Lν, core < 2.5 10 ergs s , which corresponds to ∼ × ˙ −1 crit M < 0.035 M⊙ s , rend < rS and CνO affect the structure of the flow significantly ∼ and essentially saturate. Lcrit with CνO is reduced to 0.65 of the fiducial value ν, core ∼ for low M˙ .
How does the strength of the CνO effect scale with M and rν? In Chapter 2
crit crit we showed that rS depends predominantly on Lν, core (Eq. [3.9]) and that the dependencies on other parameters like M and rν are much weaker. We plot values
crit of rS from Chapter 2 as dots, triangles and stars in Figure 3.1, right panel, for many different M and rν. We also showed in Chapter 2 that the critical luminosity scales as a power law of M˙ , M, and rν, which we reproduce in Equation (3.8). Thus,
˙ crit crit increasing M or M increases Lν, core, which in turn decreases rS and increases rsync and rend (eq. [3.23]) and the effect of CνO will be weaker. Similarly, lower rν
117 crit crit yields higher Lν, core and lower rS /rν. At the same time, rsync/rν and rend/rν will increase (eq. [3.23]) and the CνO effect will become important at smaller M˙ (smaller
crit Lν, core). Therefore, the collective oscillations will be most prominent for the sets of
crit ˙ parameters that minimize Lν, core: small M, M, and large rν.
crit crit We plot in Figure 3.2 fred—the ratio of Lν, core including CνO to Lν, core of a
crit reference calculation—essentially the reduction factor of Lν, core due to CνO. We note that in order to evaluate only the effect of CνO, we choose the reference calculation to have the same M, rν, and microphysics. Red dashed lines show fred for the neutrino parameters discussed so far. Figure 3.2 shows that to get a > 10% reduction ∼ crit ˙ −1 in Lν, core for rν = 60 km and M = 1.2 M⊙ due to CνO, we require M < 0.1 M⊙ s . ∼ crit To get the same reduction in Lν, core for M = 1.4 M⊙ and rν = 40 km, we require
M˙ 0.01 M s−1. For r = 20 km and M = 1.6 M , there is no noticeable reduction ≪ ⊙ ν ⊙ crit crit ˙ in Lν, core, because rS < rsync for the whole considered range of M.
These results can be put into context by coupling to progenitor models.
Looking at the solar-metallicity supernova progenitors of Woosley et al. (2002)3,
M˙ = 0.1 M s−1 is reached 0.65 s after the initiation of the collapse for an 11.2 M ⊙ ∼ ⊙ progenitor, but at 4 s for a 15 M progenitor. At these times, the accreted baryonic ∼ ⊙ masses are M = 1.35 and 2.2 M⊙ for the 11.2 and 15.0 M⊙ progenitors, respectively.
The lower limit of our calculations M˙ = 0.01 M s−1 is reached only after 15 s ⊙ ∼ for the 11.2 M⊙ progenitor, when the PNS has almost fully cooled (e.g. Pons et al.
3http://www.stellarevolution.org/data.shtml
118 1999; H¨udepohl et al. 2010; Fischer et al. 2010, 2012). From this investigation we
crit conclude that the decrease of Lν, core due to CνO is noticeable only for very low mass progenitors, which reach low M˙ at early times, when rν is still potentially large.
This would be possible for a stiff equation of state of dense nuclear matter, which
4 would keep rν high .
crit The reduction of Lν, core depends on the assumed neutrino energies and luminosities. So far, we have discussed the case favorable for CνO, specifically
i i i (ενe ,εν¯e ,ενx ) = (13, 15.5, 20) MeV and Lνe, core = Lν¯e,core = Lνx,core. Now we turn to neutrino energies and luminosities that more closely approximate the results of the recent sophisticated calculations (e.g. Marek & Janka 2009; Fischer et al. 2010,
i i 2012), namely (ενe ,εν¯e ,ενx ) = (11, 13, 18) MeV and Lνe, core = Lν¯e,core = 2Lνx,core, shown with green dotted lines in Figure 3.2. We see that for these parameters, the
CνO become apparent at approximately the same values of M˙ , but the effect is much smaller. For M = 1.2 M⊙ and rν = 60 km the maximum possible reduction of
Lcrit is only about 10%, much smaller than 40% for the equal luminosities and ν, core ∼ higher neutrino energies. Interestingly, unequal luminosities allow for a possibility of
having fred > 1. One case when this can happen is when φνx is low enough, so that
νe andν ¯e oscillate to νx, but there is very little νx to oscillate back. As a result, the
4 crit Note however, that to get an explosion Lν, core has to be reached by the actual core luminosity
Lνe, core, which depends on the equation of state in a more complicated way. Indeed, a softer equation
of state generally leads to higher Lνe, core at early times after bounce and may thus be favorable for explosion via the neutrino mechanism (e.g. Marek & Janka 2009).
119 luminosity in νe andν ¯e decreases and CνO can thus be detrimental for the explosion.
We do not see fred > 1 for any considered parameter combination.
3.3.2. Comparison to Other Effects
Now we compare the CνO effect to other physical effects that have been
crit shown to decrease Lν, core. We have seen that the relative positions of rS, rsync and rend determine the effect of CνO. It is known that multi-dimensional effects like convection and SASI consistently increase shock radii (e.g. Burrows et al. 1995;
Ohnishi et al. 2006; Iwakami et al. 2008; Murphy & Burrows 2008; Marek & Janka
crit 2009; Nordhaus et al. 2010) and decrease Lν, core over the corresponding 1D value
(Murphy & Burrows 2008; Nordhaus et al. 2010; Suwa et al. 2010; Hanke et al. 2012), as illustrated by the grey lines with dots in Figure 3.2. The common explanation is that multi-dimensional effects make the energy deposition of neutrinos more efficient by increasing the dwell time of the matter in the gain region (Murphy & Burrows
2008; Nordhaus et al. 2010; Takiwaki et al. 2012). In Chapter 2, we attempted to address this issue by adjusting the heating or cooling within the framework of our steady-state calculations. We found that both a decrease of cooling and an increase
crit crit of heating make Lν, core smaller and increase rS for a fixed Lνe, core. However, rS increases only for the case of reduced cooling. For this reason, based on inspection
crit of simulation results, we suggested (but did not prove) that the decrease of Lν, core seen in multi-dimensional simulations is the result of less efficient neutrino cooling
120 instead of the commonly assumed increase in heating efficiency. This decreases
crit Lν, core by about 30% compared to the fiducial case, similar to the difference in
crit Lν, core observed by Nordhaus et al. (2010), as evidenced by the lower thick grey line in Figure 3.2. The blue dash-dotted line in Figure 3.2 (and in the left panel of
crit Figure 3.1) then shows Lν, core with CνO and reduced cooling. As expected, because
crit of lower Lν, core and higher rS at fixed Lνe, core, the effect of CνO starts to be apparent
−1 −1 for M˙ < 0.5 M⊙ s and reaches full strength for M˙ < 0.04 M⊙ s . At low M˙ , ∼ ∼ the effect of CνO becomes comparable to that of increasing the dimension of the simulation from 1D to 3D, but only for fairly large rν and small M and for the less realistic energies and luminosities (red dashed lines).
In Chapter 2 we investigated the effect of a simple gray neutrino radiation transport on Lcrit (i.e. dL /dr = 0). We found that including the neutrinos ν, core ν 6 generated by the cooling of the accretion flow itself (the accretion luminosity)
crit −1 lowers Lν, core by 8% to 23% for the mass accretion rates between 0.01 and 2 M⊙ s .
crit However, the accretion luminosity was always a small fraction of Lν, core and the PNS neutrino emission has to play the major role in reviving the stalled accretion shock.
In Figure 3.2 we plot with a thick grey line fred that was obtained by including the accretion luminosity and we see that it is somewhat smaller than the maximum effect from CνO. The effect of accretion luminosity is most prominent at high M˙ and has similar importance at small M˙ and small rν. The effect of the accretion
121 luminosity is comparable to going from 1D calculations to 2D in the calculations of
Nordhaus et al. (2010).
3.3.3. Role of Multi-Angle Matter Effects
Throughout this Section we have neglected the matter-suppression effects on
CνO in order to obtain the maximum possible effect of CνO over a broad range of parameters. We found that CνO can operate only in a very restricted range in M˙ ,
M, and rν. At this point we evaluate the importance of matter suppression on our results.
In Figure 3.3 we plot the ratio λ/λMA, which estimates the relative importance of the MSW matter potential to the collective potential (Eqs. [3.24]–[3.25]). This
crit ratio is evaluated for solutions at Lν, core just inside of the shock, where it attains the smallest value. If λ/λMA 1, the collective oscillations are suppressed. We ≫ see that for large radii and small masses (M = 1.2 M⊙ and rν = 60km), the CνO are suppressed by the matter effects. However, this is also the parameter space where CνO occur for M˙ attainable by low-mass progenitors together with a stiff high-density EOS (Figure 3.2). Figure 3.3 shows that matter suppression effects are moderate for the other parameter combinations, but these combinations do not
crit yield any decrease in Lν, core due to CνO for any realistic mass accretion rates. Thus, for the region of parameter space where CνO are maximal, the matter suppression
122 effects are largest, whereas where the matter suppresion is small, the effect of CνO is negligible.
3.4. Discussion & Conclusions
We investigate the effect of collective neutrino oscillations on the neutrino mechanism of core-collapse supernovae as parameterized by the critical neutrino
crit luminosity Lν, core. We assume that neutrino energies and luminosities vary smoothly between an initial state at synchronization radius rsync and a final state end radius rend, as summarized by Dasgupta et al. (2012). The final states are dictated by the neutrino number conservation. Without matter-suppression, we found that collective
crit crit crit crit oscillations affect Lν, core if rsync < rS , where rS is the shock radius at Lν, core, and
crit the full magnitude of the effect occurs if rend < rS .
crit The reduction of Lν, core depends on the assumed energy difference between
νe,ν ¯e and νx, and on the individual luminosities. We find that neutrino energies of Thompson et al. (2003) and equal luminosities in each flavor are favorable for
crit CνO, giving reduction of L by a factor of 1.5 (fred = 0.65, Fig. 3.2). The ν, core ∼ more recent calculations of Marek & Janka (2009) and Fischer et al. (2010, 2012) predict slightly lower energies and approximately half the νx luminosity compared
crit to νe andν ¯e. For these parameter values we find that the Lν, core reduction reaches only about 10%, but the parameter space in terms of M˙ , M, and rν is essentially the
123 crit same. We do not find any increase of Lν, core for the range of parameters considered.
i Conversely, if the energy ενx was increased to 25 MeV or 35 MeV while keeping
i i crit ˙ ενe = 13 MeV, εν¯e = 15.5 MeV, and luminosities of all flavors equal, Lν, core at low M would be reduced by a factor of 2 and 3.5 (fred 0.5 and 0.3), respectively. ≃
We find that CνO can be important only for low M˙ , small M, and large rν (see
Fig. 3.1, left panel, red lines). This is best achieved in the lowest mass progenitors, in which M˙ decreases very rapidly due to their steep density structure. However, for times < 0.65 s after the collapse is initiated, M˙ is likely still too high to cause a ∼
CνO-driven explosion in an 11.2 M⊙ progenitor (see also Chakraborty et al. 2011a,b;
Dasgupta et al. 2012). At late times, M˙ decreases rapidly, but the effect of CνO
crit on Lν, core is likely offset by the simultaneous drop of rν as the PNS cools and the concomitant increase in M. Figure 3.2 shows that for smaller rν and larger M, CνO does not have any effect. Thus, only if the decrease in rν is slow enough, perhaps due
˙ crit to a stiff equation of state, M might drop enough so that the decrease in Lν, core due to CνO can potentially influence the system before black hole formation or explosion via the ordinary neutrino mechanism. However, as is indicated in Figure 3.3, these parameter combination are the ones most affected by matter suppression and CνO are unlikely.
Finally, we note that even in our implementation of CνO the parameter space where CνO could operate is small and reduced even more due to matter suppression effects. Realistically, the potential of CνO flavor conversion may be reduced even
124 more. In particular, refinements in the treatment of the CνO typically make the effect of CνO still smaller, because the physical scale of conversion is moved to larger radii due to various multi-angle effects (Esteban-Pretel et al. 2008; Chakraborty et al. 2011a,b; Dasgupta et al. 2012; Sarikas et al. 2011). We find that the multi-angle effects are important especially for small M and large rν (Figure 3.3), where CνO have the greatest potential of influencing the supernova explosion. If the physical
crit ˙ scale of conversion is pushed to a radius larger than rS for a given M, M, rν, the
crit CνO will not have any effect on Lν, core. We thus conclude that CνO are unlikely to play a role in neutrino-driven explosions by evaluating these effects at the critical luminosity.
125 crit ˙ Fig. 3.1.— Left: critical core neutrino luminosity Lν, core as a function of M for M = 1.2 M⊙ and rν = 60 km. The solid black line shows the fiducial calculation (no CνO), while the red dashed line is with CνO included. The vertical red dotted lines mark different regimes of the effect (see text and the right panel). The blue dash-dotted line includes CνO and has the neutrino cooling reduced by a factor of 2 to approximate multi-dimensional effects. Right: effect of CνO on shock radii. Dots crit crit mark rS for calculations from Chapter 2, illustrating that rS depends primarily crit on Lν, core. Blue dots, orange triangles, and green stars are for rν = 20, 40, and crit 60 km, respectively. The black solid line shows rS for the fiducial calculation from this paper for the same M and rν as in the left panel. The grey lines show rS as ˙ a function of Lνe, core for different M with CνO included. The grey lines terminate crit at rS , which are connected with a dashed red line (compare with red dashed line, crit left panel). The black dotted lines show rsync and rend. For rS < rsync, CνO has no crit effect; for rS > rend, the effect of the CνO is maximized.
126 crit Fig. 3.2.— Relative reduction of Lν, core with respect to the fiducial 1D calculation (fred) as a function of M˙ including various physical effects. Red dashed and green dotted lines show fred for CνO for two sets of neutrino energies and luminosities. The legend gives (ενe ,εν¯e ,ενx ). Lines are labeled with M and rν of the PNS. Blue dash- dotted line shows the effect of CνO and reduced cooling rate by a factor of 2 relative to a calculation with reduced cooling only. The upper thick grey line shows fred when neutrinos from cooling of the accretion flow are taken into account and the lower thick grey line illustrates fred for cooling rate reduced by a factor 2 (Chapter 2 Pejcha & Thompson 2012). The grey solid lines with points show fred for multi-dimensional effects when the dimension of the simulation is increased from 1D to 2D or from 1D to 3D (from Nordhaus et al. 2010).
127 Fig. 3.3.— The relative importance of the matter suppression of CνO parameterized by λ/λMA (Eqs. [3.24]–[3.25]). The lines are for the calculations with different M and rν presented with dashed red and dotted green lines in Figure 3.2. The ratio crit λ/λMA was evaluated just inside of the shock and at Lν, core, where the ratio λ/λMA is smallest. If λ/λMA 1, the CνO are suppressed. The upward curvature of the ≫ −1 M = 1.2 M⊙ and rν = 60 km lines for M˙ < 0.6 M⊙ s is caused by the decrease of crit crit Lν, core and rS due to CνO when matter suppression∼ effects are neglected.
128 Chapter 4
Observational Signatures of the Explosion Mechanism
4.1. Introduction
At the moment of core collapse, the interior structure of stars with different initial mass, metallicity and rotation rate is different. Figure 4.1 shows the density profile as a function of enclosed mass for Woosley et al. (2002) progenitors for three different metallicities. The vertical dashed lines mark baryonic masses of the lowest measured neutron star in a double neutron star binary (Mgrav = 1.18 M⊙; Faulkner et al. 2005) and the highest well-measured mass of a recycled pulsar (Mgrav = 1.97 M⊙;
Demorest et al. 2010). The density profiles of the progenitors in this mass range differ by many orders of magnitude for all three metallicities considered. Since the masses of neutron stars are determined by accretion of the progenitor on the proto-neutron star through the accretion shock, there is a direct correspondence between the structure of the progenitor and the outcome of the supernova – for some progenitors the accretion might continue indefinitely until a black hole is formed whereas for others an explosion stops the growth of the central neutron star.
129 During the quasistatic accretion phase, the region inside of the accretion shock is sonically disconnected from the rest of the progenitor upstream of the shock. As a result, the progenitor structure is a time-dependent boundary condition for the region inside the accretion shock. If all physical timescales inside the accretion shock were much shorter than the typical timescale over which the boundary conditions change, the full characterization of the system at any given time would require specifying only the instanteneous values of mass, momentum, and energy flow through the shock and the total enclosed mass. However, the proto-neutron star in the center is so dense that the neutrino diffusion timescale approaches 1 second and is therefore ∼ comparable to the difference of infall timescales of sufficiently different layers of the progenitor. In other words, material that is being accreted will affect the system at much later times. In this way, the structure of the part of progenitor that has been accreted up to some point is encoded in the thermodynamical properties of the region inside of the accretion shock.
In the context of the steady-state model of the region between the neutrinosphere and the accretion shock, the progenitor properties influence both the instantenous values of the boundary conditions at the shock such as the mass accretion rate M˙ , velocity upstream of the shock, and electron fraction Ye, and the global properties that depend on the past accretion history such as the total proto-neutron star mass
M, radius rν, and neutrino luminosities and energies. Some of these parameters are more important than others, but the time evolution of all of them determines
130 at what moment the system explodes, if it explodes at all, and what the resulting explosion properties will be.
Since supernova simulations are numerically expensive, it would be useful to have a criterion that would capture the differences between the progenitors and indicate which are more prone to an a neutrino-driven explosion. O’Connor & Ott
(2011) and O’Connor & Ott (2013) defined a progenitor “compactness” parameter
M /M⊙ ζM = , (4.1) r(Mbary = M )/1000 km which measures the radius r of a progenitor mass coordinate M at the moment of the core bounce. For neutron star formation, one would assume M 1.75 M , since ≈ ⊙ this is the typical mass of neutron stars. Higher ζM means that densities are higher so that r(M ) is small and the mass accretion rates are high. Progenitors with high
ζM are more prone to black hole formation (O’Connor & Ott 2011) and produce higher neutrino luminosities (O’Connor & Ott 2013). However, Ugliano et al. (2012) performed 1D simulations of a set of progenitors and found that even progenitor with low ζM fail to explode and also produce black holes.
Naively, high ζM implies high M˙ and therefore higher critical neutrino luminosity, but also high accretion luminosity. Furthermore, as argued above, it is the full mass accretion history and not the instantenous M˙ or the radial coordinate of a particular mass shell that determines the state of the system at any given time. For example, consider two progenitors with identical ζM , but with one having
131 smooth density profile and the other one with a large drop in density inside of r(M ). The first progenitor does not explode, but the second one very well might since when the density drop passes through the shock, M˙ dramatically decreases,
crit which correspondingly lowers Lν, core. However, the proto-neutron star is luminous and puffy due to the previously high value of M˙ and it is relatively easier to get
crit Lν, core >Lν, core and hence explosion for the progenitor with the density jump.
In the rest of the Chapter, we construct a semi-analytic theory of the neutrino mechanism of core-collapse supernovae and apply it to a range of progenitors with different metallicities using several equations of state. There is a significant advantage to pursuing a semi-analytic theory instead of a fully numerical approach: it is much faster and thus allows a much wider exploration of physical assumptions and calibrations. For example, the set of 1D calculations of Ugliano et al. (2012) was performed only for solar metallicity progenitors, one equation of state, and one calibration of neutrino luminosities based on SN 1987A. Although the works of O’Connor & Ott (2011, 2013) were for multiple equations of state, progenitors, and calibrations (heating factors), even such a set of 1D calculations is hard to significantly extend. In our semi-analytic theory, we run a single 1D calculation for many progenitors and equations of state, while all remaining changes and calibrations require essentially no computational time. For example, when a full knowledge of multi-dimensional instabilities becomes available, we can easily incorporate it in the
132 semi-analytic model producing results effectively based on multi-dimensional theory with a small fraction of the cost.
4.2. Elements of the semi-analytic theory
Our semi-analytic theory of the neutrino mechanism is based on comparing the critical neutrino luminosity to an actual core neutrino luminosity as a function of time for each progenitor. The critical neutrino luminosity is calculated using the steady-state code described in Chapters 2 and 3 with several changes. First, we use the Helmholtz equation of state (Timmes & Swesty 2000)1, which significantly improves convergence in some more extreme part of the parameter space. Second, we solve for the radial dependence of electron neutrino and antineutrino luminosities separately to accomodate time-dependent changes in the luminosity ratio.
The time evolution is assumed to be quasistatic and is driven by changes in
˙ the boundary conditions and global parameters: M(t), M(t), rνe (t), rν¯e (t), Lνe (t),
Lν¯e (t), ενe (t), and εν¯e (t). Given the progenitor density profile ρ(r) and mass profile
r 2 M∗(r) = 0 4πs ρ(s)ds, the mass accretion rate as a function of time can be well R estimated as
dM (r) dt −1 M˙ (t) ∗ ff , (4.2) ≈ dr dr
1http://cococubed.asu.edu/code pages/eos.shtml
133 where tff is the free-fall time
πr3/2 tff = , (4.3) 2 2GΥM∗(r) where Υ = 0p.25 in accordance with analytically estimated accretion data of Woosley et al. (2002) supernova progenitor models available on-line2. The mass of the
t ˙ ′ ′ proto-neutron star is then M(t)= 0 M(t )dt . R Unfortunately, the remaining parameters of neutrinosphere radii, neutrino luminosities and energies are set by the physics of neutrino diffusion from the proto-neutron star and it is not easy to obtain estimates with precision good enough to make the semi-analytic model work. A more involved calculation is necessary. We use the open-source code GR1D (O’Connor & Ott 2010, 2011) to calculate the time evolution of the neutrinosphere radii, neutrino luminosities and energies. Although the estimates of M˙ and M from Equation (4.2) are relatively accurate, we use the values from GR1D to prevent any mismatch in time zero point between the two codes. We use the default parameters for the collapse problem and the neutrino leakage scheme. We run the simulations for all available non-rotating progenitors of Woosley et al. (2002). The progenitors were calculated for three metallicities:
−4 primordial, Z = 10 Z⊙, and solar. For all progenitors, we use the Lattimer &
Swesty (1991) equation of state with nuclear incompressibility of K = 220 MeV
(LS220). Additionally, for the solar metallicity progenitors we run the simulations for equations of state of Lattimer & Swesty (1991) with incompressibilities of 180
2http://2sn.org/stellarevolution/data.shtml
134 (LS220) and 375 MeV (LS375), and the equation of state of Shen et al. (2011)
(Shen).
In Figure 4.2, we plot the time evolution of the mass accretion rate M˙ and proto-neutron star mass M for solar metallicity progenitors evolved with LS220. We see that the evolution of higher-mass progenitors ends earlier than for low mass. The end of the simulations is in most cases set by the moment when GR1D crashes due to thermodynamic parameters getting out of the bounds of the given equation of state. In all simulations except for the lowest mass progenitors, we reach the typical neutron star mass of 1.4 M⊙. Reaching this value in the lowest mass progenitors would require prohibitively long simulation run times. In a small number of cases, the simulation ends when a threshold time limit is reached or when the proto-neutron star forms a black hole. Since this is a preliminary work, we proceed forward even though we do not have simulations for sufficiently long time for some progenitors. In the final version, we will have time evolution for at least several seconds.
In Figure 4.3, we show the time evolution of neutrinosphere radii, and neutrino luminosities and energies for a 13 M⊙ solar metallicity progenitor for the four equations of state considered. We can see that there are basically no differences between LS180 and LS220 equations of state in any of the considered quantities. On
the other hand, L375 and Shen are significantly “stiffer” in the sense that rνe and rν¯e decrease substantially more slowly. Consequently, the proto-neutron star remains bigger and colder and the luminosities and energies are lower. Naively, it seems that
135 soft equations of state are beneficial for supernova explosions since the luminosities and neutrino energies are higher, which leads to more heating. However, as we show in Chapter 2, the critical neutrino luminosity is quite sensitive to the neutrinosphere
crit radius with larger radii lowering Lν, core. As a result, the evaluation of equation of state dependence requires putting all the pieces of the semi-analytic model together.
Figure 4.4 shows the differences between the individual solar metallicity progenitors. We see that generally progenitors with the highest iron core mass (19 to 25 M⊙) exhibit the largest neutrinosphere radii as well as the highest neutrino luminosities. This is due to high mass accretion rates, which cause matter piling up just inside the neutrinosphere, thus increasing its radius and neutrino luminosity.
Interestingly, the neutrino energies follow an almost monotonic relation with progenitor mass. The behavior of the electron neutrino and antineutrino radii, luminosities, and energies is very similar and we thus plot interquartile ranges in the right column to get a better sense of the overall behavior. We see that at about 0.2 to 0.3 seconds after the core bounce, there is a drop in the upper envelope of neutrino luminosities. This is associated with accretion of a composition interface and drop in M˙ (cf. Fig. 4.2). In low-mass progenitors, this happens about 0.1 s earlier, which is the reason for the decrease in the median neutrino luminosity. The drop of the upper envelope of luminosity and energy at 0.8 s happens as the simulations of ∼ high-mass progenitors end.
136 We use the output from GR1D to calculate the critical neutrino luminosity
crit Lν, core(t) with the steady-state code. Many pieces of physics in our steady-state code differ from GR1D and it is necessary to check whether the two approaches give comparable results. The best quantity to examine is the shock radius, which is calculated independently by both codes. In Figure 4.5, we show the relative ratio of the shock radii calculated using the two approaches as a function of M˙ and the time after core bounce. We see that for t > 0.1 s, the two methods agree very well within several tens of percent. Note that the shock radius can change by a large
crit amount when Lν, core is very close to Lν, core (Chapter 2). When plotted as a function of M˙ , we see a systematic trend in the sense that the steady-state calculation gives lower shock radii for small M˙ . The steady-state calculation gives unrealistically high shock radii for the 11.2 and 11.4 M⊙ progenitors, but this happens less than 0.1 s after the bounce, when the steady-state approximation is not yet valid. Due to the systematic trend in shock radii with M˙ , it is preferred to construct the semi-analytic model with M˙ as the independent variable, because in this way it is easier to take out the systematic trend. To summarize, apart from a small systematic trend the
−1 steady-state calculation gives credible results for M˙ < 1 M⊙ s and t> 0.1 s over a ∼ very wide range in progenitor mass.
137 4.3. Results
crit We use the steady-state code to calculate Lν, core and we compare it with
GR1D crit the actual Lν, core from . In Figure 4.6, we plot the ratio Lν, core/Lν, core as a function of M˙ for the solar metallicity progenitors. Except for the 11.2 and 11.4 M⊙ progenitors (where the steady-state approximation does not apply), the ratio stays below unity. This is equivalent to the statement that 1D supernova simulations do
crit not explode. Furthermore, there is a general trend that Lν, core/Lν, core decreases with decreasing M˙ . Part of this comes from the systematic mismatch between the steady-state and GR1D calculations, which manifests itself in shock radii (Fig. 4.5).
The important feature of the semi-analytic theory of supernovae is that any additional effects can be incorporated provided that they depend only on the boundary conditions and are not dynamical. An example of such a systematic effect would be the difference between a more sophisticated and realistic neutrino transport scheme and the simple leakage in GR1D, which probably significantly underestimates the neutrinosphere radius when compared to a two-moment scheme (Ott 2013,
′ private communication). Consider a physical effect that decreases Lcrit L crit ν, core → ν, core for small M˙
1 crit pM˙ q, if M
Figure 4.6. Such a physical effect opens possibilities of a relatively early explosion at
M˙ 0.6 M s−1, when the composition interface and an associated drop in density ≈ ⊙ −1 pass through the shock, or at late times when M˙ < 0.1 M⊙ s . ∼
These two possibilities have observational consequences: early explosions are generally assumed to have higher energy and produce neutron stars with lower mass. In Figure 4.7, we show the remnant mass of the solar metallicity progenitors evolved with LS220 assuming a calibration of the neutrino mechanism according to Equation (4.4). We see that there is no clear boundary between successful explosions, which produce neutron stars, and failed explosions, which produce black holes. In other words, black hole formation can occur for progenitors with relatively low initial mass < 20 M⊙. ∼
Now we investigate various calibrations of the neutrino mechanism, which is the most important feature of the semi-analytic model. With the time evolution of boundary conditions available, such calculation is a matter of several seconds. In
Figure 4.8, we plot the fraction of successful supernova explosions, and the mean and width of the neutron star mass distribution as a function of parameters p and q of
Equation (4.4). We see that the semi-analytic model can be immediately constrained by observations. The mean graviational mass of double neutron stars is about
1.33 M⊙, which translates to about 1.46 M⊙ baryonic, and the width of the mass distribution is about 0.05 M⊙ (Ozel¨ et al. 2012; Pejcha et al. 2012b). Comparing this
139 numbers to the contours in middle and bottom panels of Figure 4.8, we infer that
1.0 < q < 1.5 and p > 1.5. This implies that the fraction of successful explosions ∼ ∼ ∼ should be < 0.7. ∼
4.4. Discussion
The only similar study of many different progenitors so far has been the work of
Ugliano et al. (2012), who performed 1D calculations from core bounce to advanced phases of the explosion that were calibrated by properties of SN 1987A. There are similarities as well as differences between ours and Ugliano et al. (2012) remnant mass distributions. First, both calculations show a similar pattern of successful and unsuccessful explosions interweaved between each other. Second, the results agree on unsuccessful explosions for initial progenitor masses of 15 to 16 M⊙, and for some progenitors between 18 and 20 M⊙. However, there is a disagreement for progenitor masses 23–25 M⊙, where we obtain successful explosions unlike Ugliano et al. (2012).
For progenitor masses 31–38 M⊙, our core luminosities do not get above critical while Ugliano et al. (2012) obtain successful explosions.
Part of these discrepancies might be related to differences in the treatment of proto-neutron star evolution. Ugliano et al. (2012) excise the inner 1.1 M⊙ of the proto-neutron star and evolve the excision radius manually according to a prescription, which is same for all progenitors. Above the excision radius, they use
140 grey flux-limited diffusion for neutrino transport (Scheck et al. 2006). On the other hand, our calculations with GR1D self-consistently evolve the whole proto-neutron star. GR1D uses neutrino leakage while our steady-state calculations assume optically-thin gray neutrino transport. However, at this point of our preliminary investigation, we are more severely limited by the inability to evolve the system for longer times with GR1D (see Fig. 4.4). If we had much longer time-series of proto-neutron star parameters, it might be possible to select p and q to get results similar to Ugliano et al. (2012) by making the 23–25 M⊙ progenitors form a black
−1 hole, while having 31–38 M⊙ progenitors explode when M˙ < 0.1 M⊙ s . ∼
To summarize, the preliminary results of the semi-analytic supernova model are promising, but require further work to be competitive with earlier results.
Most importantly, it is necessary to be able to evolve the accreting and cooling proto-neutron star for at least couple seconds as was done by O’Connor & Ott (2011).
It is unclear whether it is necessary to implement a more sophisticated neutrino transport scheme. Althouth the two-moment scheme produces larger neutrinosphere radii (Ott 2013, private communication), it is likely that the resulting neutrino energies are smaller so the net effect on the critical neutrino luminosity is unclear. In the future, the semi-analytic model can be extended to provide additional supernova observables such as amounts of nickel produced in the explosion and supernova explosion energies.
141 1010 Neutron stars Black holes
11.0 26.0 8 10 12.0 27.0 Z = primordial 13.0 28.0 14.0 29.0 15.0 30.0 6 16.0 31.0
] 10
-3 17.0 32.0 18.0 33.0 19.0 34.0 [g cm ρ 20.0 35.0 4 10 21.0 36.0 22.0 37.0 23.0 38.0 24.0 39.0 102 25.0 40.0
100 1010
-4 8 10 11.0 18.0 25.0 32.0 39.0 46.0 53.0 Z = 10 ZO • 11.2 18.2 25.2 32.2 39.2 46.2 53.2 11.4 18.4 25.4 32.4 39.4 46.4 53.4 11.6 18.6 25.6 32.6 39.6 46.6 53.6 11.8 18.8 25.8 32.8 39.8 46.8 53.8 12.0 19.0 26.0 33.0 40.0 47.0 54.0 12.2 19.2 26.2 33.2 40.2 47.2 54.2 12.4 19.4 26.4 33.4 40.4 47.4 54.4 12.6 19.6 26.6 33.6 40.6 47.6 54.6 6 12.8 19.8 26.8 33.8 40.8 47.8 54.8 13.0 20.0 27.0 34.0 41.0 48.0 55.0 ] 10 13.2 20.2 27.2 34.2 41.2 48.2 55.2 -3 13.4 20.4 27.4 34.4 41.4 48.4 55.4 13.6 20.6 27.6 34.6 41.6 48.6 55.6 13.8 20.8 27.8 34.8 41.8 48.8 55.8 14.0 21.0 28.0 35.0 42.0 49.0 56.0 14.2 21.2 28.2 35.2 42.2 49.2 56.2 [g cm 14.4 21.4 28.4 35.4 42.4 49.4 56.4
ρ 14.6 21.6 28.6 35.6 42.6 49.6 56.6 14.8 21.8 28.8 35.8 42.8 49.8 56.8 4 15.0 22.0 29.0 36.0 43.0 50.0 57.0 10 15.2 22.2 29.2 36.2 43.2 50.2 57.2 15.4 22.4 29.4 36.4 43.4 50.4 57.4 15.6 22.6 29.6 36.6 43.6 50.6 57.6 15.8 22.8 29.8 36.8 43.8 50.8 57.8 16.0 23.0 30.0 37.0 44.0 51.0 58.0 16.2 23.2 30.2 37.2 44.2 51.2 58.2 16.4 23.4 30.4 37.4 44.4 51.4 58.4 16.6 23.6 30.6 37.6 44.6 51.6 58.6 16.8 23.8 30.8 37.8 44.8 51.8 58.8 2 17.0 24.0 31.0 38.0 45.0 52.0 59.0 10 17.2 24.2 31.2 38.2 45.2 52.2 59.2 17.4 24.4 31.4 38.4 45.4 52.4 59.4 17.6 24.6 31.6 38.6 45.6 52.6 59.6 17.8 24.8 31.8 38.8 45.8 52.8 59.8 60.0 75.0
100 10100.0
8 10 10.8 15.8 20.8 25.8 Z = ZO • 11.0 16.0 21.0 26.0 11.2 16.2 21.2 26.2 11.4 16.4 21.4 26.4 11.6 16.6 21.6 26.6 11.8 16.8 21.8 26.8 12.0 17.0 22.0 27.0 6 12.2 17.2 22.2 27.2
] 10 12.4 17.4 22.4 27.4
-3 12.6 17.6 22.6 27.6 12.8 17.8 22.8 27.8 13.0 18.0 23.0 28.0 13.2 18.2 23.2 28.2 [g cm 13.4 18.4 23.4 29.0 ρ 13.6 18.6 23.6 30.0 4 13.8 18.8 23.8 31.0 10 14.0 19.0 24.0 32.0 14.2 19.2 24.2 33.0 14.4 19.4 24.4 34.0 14.6 19.6 24.6 35.0 14.8 19.8 24.8 36.0 15.0 20.0 25.0 37.0 15.2 20.2 25.2 38.0 102 15.4 20.4 25.4 39.0 15.6 20.6 25.6 40.0
100 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
M [MO •]
Fig. 4.1.— Density profiles as a function of enclosed mass of publicly-available supernova progenitors of Woosley et al. (2002). The three panels are for different metallicities indicated in each panel. Different initial progenitor masses are shown with different colors with the progenitor–color assignment given in each panel. The two vertical grey dashed lines mark approximate baryonic masses of the lowest and highest precisely measured neutron star masses.
142 10.8 14.0 17.2 20.4 23.6 26.8 11.0 14.2 17.4 20.6 23.8 27.0 11.2 14.4 17.6 20.8 24.0 27.2 11.4 14.6 17.8 21.0 24.2 27.4 11.6 14.8 18.0 21.2 24.4 27.6 11.8 15.0 18.2 21.4 24.6 27.8 12.0 15.2 18.4 21.6 24.8 28.0 12.2 15.4 18.6 21.8 25.0 28.2 12.4 15.6 18.8 22.0 25.2 29.0 12.6 15.8 19.0 22.2 25.4 30.0 12.8 16.0 19.2 22.4 25.6 31.0 13.0 16.2 19.4 22.6 25.8 32.0 ] 13.2 16.4 19.6 22.8 26.0 33.0 13.4 16.6 19.8 23.0 26.2 34.0 -1 1.0 13.6 16.8 20.0 23.2 26.4 35.0 13.8 17.0 20.2 23.4 26.6 36.0 s 37.0
• O 38.0 39.0 40.0
M 75.0
. [ M 0.1
2.2
2.0 ]
• O 1.8 M [
M 1.6
1.4
1.2 0.1 1.0 time after bounce [s]
Fig. 4.2.— Time evolution of mass accretion rate M˙ at the shock (top panel) and the proto-neutron star mass M (lower panel) for solar metallicity progenitors evolved with LS220. The colors corresponding to individual progenitors are given in the top panel. The maximum gravitational mass of a cold neutron star in L220 is about 2 M⊙ (O’Connor & Ott 2011).
143 ν ν e e 80 LS220 LS180 60 LS375 Shen
40
20 neutrinosphere radius [km]
0 70
] 60 -1
50 ergs s 51 40
30
20
neutrino luminosity [10 10
0 20
18
16
14 rms neutrino energy [MeV]
12 0.0 0.5 1.0 1.5 0.5 1.0 1.5 time after bounce [s] time after bounce [s]
Fig. 4.3.— Time evolution of neutrinosphere radius (top row), and neutrino luminosities (middle row) and rms energies (bottom row) for solar-metallicity 13 M⊙ progenitor calculated with GR1D. Left column is for electron neutrinos, while the right one for electron antineutrinos. Each panel shows four different lines for Lattimer & Swesty (1991) equations of state with incompressibilities of 180 (dotted red), 220 (solid black), and 375 MeV (dashed blue), and for Shen et al. (2011) equation of state (dash-dotted green).
144 ν ν e e 100 10.8 15.8 20.8 25.8 11.0 16.0 21.0 26.0 11.2 16.2 21.2 26.2 11.4 16.4 21.4 26.4 11.6 16.6 21.6 26.6 11.8 16.8 21.8 26.8 80 12.0 17.0 22.0 27.0 12.2 17.2 22.2 27.2 12.4 17.4 22.4 27.4 12.6 17.6 22.6 27.6 12.8 17.8 22.8 27.8 13.0 18.0 23.0 28.0 13.2 18.2 23.2 28.2 60 13.4 18.4 23.4 29.0 13.6 18.6 23.6 30.0 13.8 18.8 23.8 31.0 14.0 19.0 24.0 32.0 14.2 19.2 24.2 33.0 14.4 19.4 24.4 34.0 14.6 19.6 24.6 35.0 40 14.8 19.8 24.8 36.0 15.0 20.0 25.0 37.0 15.2 20.2 25.2 38.0 15.4 20.4 25.4 39.0 neutrinosphere radius [km] 15.6 20.6 25.6 40.0 75.0 20
100 ] -1 80
ergs s 60 51 40
20 neutrino luminosity [10
10 24
22
20
18
16 rms neutrino energy [MeV] 14
12 0.1 1.0 0.1 1.0 time after bounce [s] time after bounce [s]
Fig. 4.4.— Time evolution of neutrinosphere radius (top row), and neutrino luminosities (middle row) and rms energies (bottom row) for solar-metallicity progenitors evolved with LS220. Left column is for electron neutrinos, where all individual progenitors are shown with colors given in the top left panel. Right column is for electron antineutrinos, but here the shaded regions mark the interquartile ranges containing 68%, 95%, and 99.7% of all progenitors. The thick black line is the median. All progenitors had equal weight for calculation of interquartile ranges.
145 2.0
1.5
1.0
0.5 ratio of shock radii
0.0 0.1 1.0 . -1 M [MO • s ] 2.0
1.5
1.0
0.5 ratio of shock radii
0.0 0.1 1.0 time after bounce [s]
Fig. 4.5.— Ratio of shock radii calculated using the steady-state code with respect to shock radii from GR1D as a function of M˙ (top panel) and time after bounce (lower panel). Calculation was done for solar-metallicity progenitors with LS220. The color coding of progenitors is the same as in Figure 4.4.
146 1.5
1.0 crit core L / core L 0.5
0.0 0.1 1.0 . -1 M [MO • s ]
Fig. 4.6.— Ratio of core neutrino luminosity to the critical neutrino luminosity as a function of M˙ for solar-metallicity progenitors evolved with LS220. The color coding of the lines is the same as in Figure 4.4. The thick dashed lines shows two possible calibrations of the neutrino mechanism according to Eq. (4.4). The gray area approximates the region where steady state assumption is invalid.
147 2.0
] 1.8 • O M 1.6
1.4
Remnant mass [ 1.2
1.0 10 15 20 25 30 35 40
Progenitor mass [MO •]
Fig. 4.7.— Baryonic masses of the remnants as a function of the initial progenitor ′crit mass. Red bars indicate successful explosions when Lν, core/Lν, core > 1 at some point, while gray bars mark failed explosions when the core luminosity never raises above the critical value. The remnant masses were not corrected for mass carried out by neutrino emission after the start of the explosion. Calibration of the neutrino mechanism using the lower thick dashed line in Figure 4.6 was assumed.
148 2.5 Successful explosion fraction 0.9 0.7
2.0 0.5
0.9
1.5
q 0.7 0.3
0.5 1.0 0.3 0.1
0.5 0.1
2.5 Mean NS mass 1.42
2.0
1.5 q
1.46 1.0 1.42 1.42 1.38
0.5 1.38
2.5 NS mass standard deviation
2.0
0.03 1.5 q
0.05
1.0
0.01 0.090.07 0.010.030.05 0.07 0.09
0.5 0.05 0.01
0.03
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 p
Fig. 4.8.— Fraction of successful explosions (top), and mean (middle) and standard deviation (bottom) of the neutron star mass distribution for different calibrations of the neutrino mechanim parameterized by p and q (Eq. [4.4]). We assumed a Salpeter (1955) initial mass function, dN/dM M −2.35. ∝
149 Chapter 5
The observed neutron star mass distribution as a probe of the supernova explosion mechanism
5.1. Introduction
We use the mass distribution of neutron stars (NSs) to probe the physics of supernova explosions. We replace the parametric models of observed NS masses used by Kiziltan et al. (2010) and Ozel¨ et al. (2012) with predictions of NS masses based on the actual physics of progenitors and supernova explosions so that we can directly constrain, compare and assess the validity of physical models for the explosion. We present a Bayesian formalism that quantitatively compares different predictions for the remnant mass function to the DNS data. We compare different DNS production models using the artificial supernova explosion models of Zhang et al. (2008) with different explosion energies and different progenitor metallicities. In Section 5.2, we outline our Bayesian framework for comparing the NS production models with the observed data. In Section 5.3, we present our results and outline extensions of our model that may further constrain the underlying physics. In Section 5.4, we discuss our results and their implications for the supernova explosion mechanism.
150 5.2. Statistical model
In this Section we outline a general Bayesian statistical model to quantitatively evaluate different hypotheses about the origin of the DNS mass distribution. We choose a Bayesian framework because it easily allows for a simultaneous comparison of multiple models with different numbers of parameters, yields best-fit parameter estimates, and naturally incorporates prior knowledge. We start by outlining a procedure for calculating the posterior probability distributions and then we formulate several hypotheses to be evaluated. We also describe the data and the underlying physical model. A major limitation for these models is the very limited availability of supernova explosion models – even those employing a simple piston at a fixed composition jump or mass cut – as a function of mass and metallicity.
5.2.1. General considerations
According to Bayes theorem, the posterior probability of hypothesis H with internal parameters θ given data D, P (θH D), is equal to the prior probability | P (θH) multiplied by the marginal likelihood P (D θH) that D arose from hypothesis | H,
P (θH D) P (θH)P (D θH). (5.1) | ∝ |
If the data D are composed of N individual measurements and the i-th measurement is characterized by a probability density in an observed pair of masses M (M1, M2), ≡ 151 P (D M), then the marginal likelihood of hypothesis H is i | N P (D θH)= P (D M)P (M θH)dM, (5.2) | i | i | i Y Z where P (M θH) is the probability that the given value of M occurs for the i | parameters θ of H for the i-th measurement. For a given hypothesis H, different values of the parameters θ yield different probabilities of the data P (D θH) such | that we can determine the “best-fit” parameters θ and their confidence intervals based on the posterior probability distribution P (θH D). |
Suppose that we have two hypotheses H1 and H2 parameterized by their individual parameter sets θ1 and θ2. Which of the two hypotheses better describes the data? Within the framework of Bayesian analysis, the relative “probability” of the two hypotheses is given by the Bayes factor
B1 P (D θ1H1)P (θ1H1)dθ1 B12 = = | . (5.3) B2 P (D θ2H2)P (θ2H2)dθ2 R | R Note that only the ratio of B1 and B2, B12, has any meaning and that it can be extended to an arbitrary number of hypotheses. Proper calculation of B12 also requires that the individual probabilities in Equation (5.2) are properly normalized with P (D M)dM 1, P (M θH)dM 1, and P (θH)dθ 1 over the i | ≡ | ≡ ≡ R R R relevant ranges of M and θ. Jeffreys (1983) groups values of B12 in several
1/2 categories: B12 > 10 implies that hypothesis H1 is “substantially” better than
2 H2. If B12 > 10 , then the evidence against H2 and in favor of H1 is decisive.
Jeffreys (1983) also gives tables to approximately relate B12 as a function of
152 number of the parameters in θ to a more commonly used χ2 difference, specifically
2 B12 exp( ∆χ /2). ∝ −
The Bayesian statistical model we present here is similar to the one developed by Ozel¨ et al. (2010, 2012) with a key difference: instead of using a phenomenological description based on a parametric function (in their case, a Gaussian), we will tie the observed NS masses directly to physical calculations of remnant masses based on supernova physics and the progenitor structure. This allows us to quantitatively compare different scenarios for the origin of the NS mass distribution.
5.2.2. NSs as members of a binary
The masses of the two NSs in a binary system are not independent and reflect the binary initial mass distribution, any mass transfer processes that occurred during the system evolution, and the supernova physics. We assume that one observation yields a pair of NS masses of the binary, M (M1, M2), where M1 is the mass of the ≡ recycled pulsar, and M2 is the mass of the companion. In Equation (5.2), P (D M) i | is the probability of observing the i-th pair of masses
P (D M)= (M1, M 1 , σ1 ) (M2, M 2 , σ2 ), (5.4) i | N ,i ,i N ,i ,i and (M 1,i, M 2,i), (σ1,i, σ2,i) are the measured masses and their uncertainties for the i-th DNS system. Here, are Gaussians defined as N 1 (x µ)2 (x, µ, σ)= exp − . (5.5) N √ 2 − 2σ2 2πσ 153 Equation (5.4) assumes that there are no correlations between the two NS mass measurements in the binary, although these could be included. More complicated models of the probability densities of the observed masses can be included as well.
An important issue for the calculation of P (M θH) is the assignment of i | the DNS components to the original primary and secondary stars in the binary.
There are two mutually incompatible possibilities, specifically, either the recycled pulsar came from the primary star and the companion from the secondary, or the reverse. Following Press (1997), we account for these two probabilities by expressing
P (M Mmax) as a combination of these two mutually incompatible hypotheses i |
P (M θH)= P (p )[p P(M1, M2 θH)+(1 p )P(M2, M1 θH)]dp , (5.6) i | i i | − i | i Z
where pi is the probability that the recycled pulsar in system i came from the primary star and P (p ) is the prior on p . P(M , M θH) is the probability density i i A B| distribution of the pair of remnant masses (MA, MB) where MA corresponds to the remnant mass of the primary star and MB to the that of the secondary star. For an uniform prior P (pi), the final probability is a simple average of the two options.
However, the binary evolution models generally require that the recycled pulsar originated from the primary star and hence the prior P (pi) is strongly peaked at p = 1. Based on the binary evolution models, we choose P (p ) = δ(p 1) for all i i i − systems. In Section 5.3.1, we investigate the appropriate choice of P (pi) for the individual systems.
154 We consider two forms of the probability P(M , M θH). First, as a A B| counterpoint to the more complicated models, we consider an “independent” star model where each star is independently drawn from the IMF. This simple model is conceptually similar to the parametric models that assume no correlation between the stars in a binary used by Kiziltan et al. (2010) and Ozel¨ et al. (2012). In this model the primary and secondary probability distributions are independent,
P(M , M θH) = P(M θH)P(M θH), and P(M θH) is the probability A B| A| B| | distribution of remnant masses of a single star
Mmax ′ P(M θH) P (M ) [M, M (M ), σtheo]dM . (5.7) | ∝ N MZmin
Here, we assume that NSs are produced by stars with initial masses1 M between
Mmin and Mmax, where Mmin is fixed and Mmax is a parameter (Mmax θ). P (M ) ∈ is the probability of progenitor mass M , which we assume to be a power law,
P (M ) M −α, with α = 2.35 to match Salpeter (1955). The function M ′(M ) ∝ provides the remnant mass M ′ for the given progenitor mass M (see Section 5.2.3).
The independent model is symmetric and thus Equation (5.6) will give the same marginal likelihood with no dependence on P (pi).
Second, we consider a genuine binary distribution that is defined as
Mmax MA P (q) ′ ′ P(MA, MB θH) dMA dMBP (MA) [MA, M (MA), σtheo] [MB, M (MB), σtheo], | ∝ MA N N MZmin MZmin (5.8)
1Throughout this paper we denote progenitor masses as M and remnant masses as M.
155 where we assume that the primaries are drawn from a Salpeter IMF,
P (M ) M −2.35, and the secondaries are drawn from a distribution P (q) A ∝ A of mass ratios q = M /M . We assume either uniform P (q) for 0.02 q 1.0, B A ≤ ≤ or a population of “twin” binaries with half of binaries distributed uniformly in the interval 0.9 q 1 and the other half uniformly distributed for 0.02 q < 0.9 ≤ ≤ ≤ (Pinsonneault & Stanek 2006; Kobulnicky & Fryer 2007; Kochanek 2009). We
1 normalize the mass ratio distribution as 0.02 P (q)dq = 1 and we drop systems with R secondaries with MB < Mmin that would produce NS-WD binaries. Systems with more massive primaries thus produce a higher relative fraction of DNSs. We neglect all binary evolution processes that could modify the relation between the initial and remnant masses M ′(M ), because the mass transfer in a DNS progenitor binary system occurs after the main sequence evolution, which fixes the size of the helium core of the primary (e.g. Bhattacharya & van den Heuvel 1991; Portegies Zwart
& Yungelson 1998). We thus assume that M ′(M ) is the same for primaries and secondaries. Again, Mmax is a free parameter (Mmax θ). ∈
We chose the form of Equations (5.7)–(5.8) for several reasons. First, the function M ′(M ) is usually tabulated only for a discrete set of M and we need
P(M θH) and P(M , M θH) to be continuous and smooth. This is because | A B| the likelihoods of many of the NS mass measurements are sharply peaked and we do not want our results to be sensitive to the exact position of the NS mass with respect to the discrete mass models of the theoretical studies. Second, the width
156 of the kernel, σtheo, can be interpreted as the uncertainty in the theoretical NS masses either due to progenitor structure, NS growth during the accretion phase, or stochasticity in the amount of fallback. In principle, one can have σtheo = σtheo(M ) and make the NS mass uncertainty depend on the progenitor mass. If σtheo is too small, P(M θH) and P(M , M θH) will have many individual peaks, while if it | A B| is too large, the structure in the NS distribution will be smeared out. We varied
0.01 M σtheo 0.05 M and found that for higher values of σtheo the relative ⊙ ≤ ≤ ⊙ probabilities of the models were smaller. However, the ordering of the models did not change. We choose σtheo = 0.025 M⊙ as a rather arbitrary compromise between the two extremes. This width is several times smaller than the typical width of the
DNS mass distributions of Kiziltan et al. (2010) and Ozel¨ et al. (2012).
The last quantity we need to evaluate Equation (5.1) is the prior on Mmax,
P (Mmax). Since Mmax attains only positive values and we do not have any physical constraints, we set the prior to be uniform in ln Mmax for 10 M Mmax 100 M , ⊙ ≤ ≤ ⊙ where the upper limit corresponds to the approximate maximum mass of a star.
5.2.3. Data, underlying models and implementation
In order to calculate P (θH D) for the independent and binary models, we | need the mapping between the initial progenitor mass and the final remnant mass
M ′(M ). We use the results of Zhang et al. (2008) summarized in Table 5.1, who
157 obtained NS and BH mass distributions for primordial (Z = 0) and solar metallicity
(Z = Z ) progenitors by positioning a piston at a particular mass coordinate and ⊙ injecting enough momentum to obtain an explosion with the desired ejecta kinetic energy E at infinity. The pistons were positioned either at the point where the entropy S/NA = 4 kB, which corresponds approximately to the base of the oxygen burning shell, or at the edge of the deleptonized core (“Ye core”), which is located deeper in the star where the electron fraction Ye decreases due to electron captures on protons. This radius roughly corresponds to the iron core. We also consider remnant masses that correspond to the Ye core and S/NA = 4 kB masses with no fallback. Looking at Table 5.1, we see that the model calculations do not extend all the way to the minimum mass for supernova explosion Mmin. It is expected that fallback is negligible for these low mass stars and that the remnant mass is equal to the core mass. Following Zhang et al. (2008), we extend the properties of the 10 M⊙ stars down to Mmin for primordial composition stars. For solar metallicity and the piston at S/NA = 4 kB, we set M = 1.37 M for 11 M 12 M and M = 1.35 M ⊙ ≤ ≤ ⊙ ⊙ for 9.1 M < 11 M . For solar metallicity and the piston at the Ye core, we set ≤ ⊙ M = 1.32 M for 9.1 M 12 M . All remnant masses were corrected for the loss ⊙ ≤ ≤ ⊙ of binding energy Ebind due to neutrino emission during the supernova event using the approximation
M 2 E = 0.075 M grav , (5.9) bind ⊙ M ⊙ 158 where Mgrav is the gravitational mass of the remnant after the correction for Ebind
(Timmes et al. 1996). The commonly assumed value of 0.084 M⊙ for the leading factor from Lattimer & Yahil (1989) differs slightly from Equation (5.9). We adopt the sample of DNS from Ozel¨ et al. (2012), which is reproduced in Table 5.2 for convenience. The sample consists of 6 NS binaries, that yield 12 precise NS mass measurements.
Because we used Gaussians for P (D M) and the kernels appearing in | Equations (5.7) and (5.8), we can evaluate the integrals over M in Equation (5.2) analytically by swapping the order of integration. This greatly speeds up the calculation, especially for the binary models. Integrals over M in Equations (5.7) and (5.8) were evaluated using the midpoint rule centered on the given progenitor
M and with dM equal to half the distance in M to the nearest progenitor models.
We did not use a more sophisticated integration method because the NS mass distribution is not intrinsically smooth and because there are significant jumps in M ′ between progenitors of similar mass. We use 5 to 10 points for the mass ranges where no directly calculated progenitors are available (9.5 M 10 M for Z = 0 and ≤ ≤ ⊙
9.1 M 12 M⊙ for Z = Z ). We assume that the maximum gravitational NS ≤ ≤ ⊙ mass is 2.0 M⊙ (Demorest et al. 2010), and we thus do not include in the calculation of P(M θH) or P(M , M θH) any progenitor producing M ′(M ) > 2.0 M . | A B| ⊙ These progenitors are assumed to yield BHs.
159 5.3. Results
We first discuss the DNS mass distribution and the ambiguities in associating a NS in a binary with a progenitor (Section 5.3.1). Then we examine a range of hypotheses on the origin of the NS mass distribution (Section 5.3.2). We also present several extensions and limitations to our analysis (Section 5.3.3).
5.3.1. Properties of the binary model
In Figures 5.1 and 5.2, we show examples of the probability distribution of
NS masses originating from a binary P(M , M Mmax) with a uniform P (q) along A B| with the distributions marginalized over MA or MB. The model in Figure 5.1 has the highest relative probability of all models considered in Section 5.3.2. For comparison, Figure 5.2 shows a model with fallback that has relative probability
2.8 B12 (Eq. [5.3]) lower by a factor of 10 . Examining Equation (5.8), we see ∼ that the primary and secondary mass ranges producing NS are quite different. For uniform P (q), the probability distribution of primary masses MA is proportional
−α to M (1 Mmin/M ), and primaries with masses close to Mmin do not A − A contribute to the distribution of NS masses MA because these systems mostly produce NS-WD binaries. The primary distribution producing NSs peaks at
(α + 1)Mmin/α 1.43Mmin for a Salpeter IMF. The distribution of secondaries ≈ producing NSs is proportional to M −α M −α for a flat P (q). Here, the cutoff is at B − max
160 high masses, while the distribution of the lowest mass progenitors is almost Salpeter.
These analytic estimates immediately show that NS masses originating from the primary and secondary stars of a binary represent different progenitor mass ranges.
For example, for Mmin = 9.1 M⊙ and Mmax = 25 M⊙, the mean progenitor masses of the primary and secondary components are 16.5 and 12.8 M⊙. This explains why the marginal distribution of MA in Figure 5.1 is not peaked at low MA and has a stronger secondary peak at M 1.6 M , when compared to the distribution of M . A ∼ ⊙ B The secondary peak is much higher for models that include fallback (Fig. 5.2). There are no observed DNSs with masses in this second peak, which leads to a preference for models with no fallback (Section 5.3.2). The secondaries MB have progenitor distributions close to the IMF. Thus, the highest peak for secondaries is at 1.22 M⊙, which is the assumed gravitational NS mass for stars with 9.1 M 12 M . ≤ ≤ ⊙
We see from Figures 5.1 and 5.2 that most of the probability is in the region where the primary produces a more massive NS. However, since M ′(M ) is not monotonic (Zhang et al. 2008), there is a small probability that the more massive
NS originated in fact from the less massive progenitor. Generally, the farther the
NS mass pair is from the diagonal (MA = MB), the smaller the probability that the more massive NS came from the less massive progenitor. In our formalism we can estimate whether the mass difference between the two NSs is enough to distinguish between an NS originating from the primary or the secondary, or essentially whether the prior P (p ) = δ(p 1) in Equation (5.6) is appropriate. i i −
161 Figure 5.3 shows the ratio of probabilities that the millisecond pulsar originated from the more massive progenitor, P(M1, M2 Mmax), as compared to the reverse |
P(M2, M1 Mmax). In Figure 5.3 we have marginalized over explosion energies and |
Mmax. We see that if M1 M2 < 2σtheo, our model cannot distinguish between | − | ∼ the primary/secondary origin of the millisecond pulsar based on the masses alone.
The case of J1906+0746 is peculiar (because the pulsar is significantly less massive than the companion) and our results show that it is unlikely to have originated from the more massive progenitor. In agreement with Lorimer et al. (2006) who give a very small characteristic pulsar age (see also our Table 5.2), we propose that the observed pulsar in J1906+0746 comes from the less massive secondary star and we set the prior on pi in Equation (5.6) to be P (pi) = δ(pi) for this system. We show this alternative assignment as an open circle and the dashed lines in Figure 5.1. This alternative assignment increases the relative probability of the binary models by a factor of 3 to 300. ∼ ∼
5.3.2. Comparison of the individual models
Next we evaluate the relative probabilities of individual models. We specifically discuss the differences between the independent and binary models, the explosion energy, and the position of the piston. We find that there is little difference in relative probability between the uniform and twin mass ratio distributions so we only discuss the uniform P (q) model, which has slightly higher probability for solar metallicity. By
162 comparing the relative probabilities of models with free Mmax to models that include all progenitors (equivalent to setting P (Mmax)= δ(Mmax 100 M ) in Eq. [5.3]), we − ⊙ also find that models with Mmax as a free parameter are not significantly preferred.
The inferred values of Mmax range from 14 M⊙ to 35 M⊙ depending on the method used to infer the “best-fit” value, but with confidence intervals covering most of the allowed range for Mmax. Here, we show models marginalized over Mmax, although models simply fixing Mmax = 100 M⊙ give essentially the same results. Finally, for the purposes of this Section, we do the calculations with the pulsar in J1906+0746 attributed to the secondary star, consistent with Lorimer et al. (2006) and our discussion in Section 5.3.1.
Figure 5.4 shows the relative probabilities (Bayes factors) of our models as a function of explosion energy. Focusing first on the primordial composition models
(left panel of Fig. 5.4), we see a general trend of increasing model probability for explosion energies up to about E 2 1051 ergs, after which the relative probability ≈ × is essentially constant. The explanation is that higher E explosions produce less fallback and hence reduce the number of high-mass NSs that expand the overall NS mass range. This is confirmed by the models that include only the mass of the core with no fallback (horizontal lines in Figure 5.4) that match the probabilities of the high-E models. For solar composition (right panel of Fig. 5.4), the effect of E is not clear. Since Zhang et al. (2008) investigated only two values of E, the probability ratios are not large (factor of 3). ∼
163 In all cases, the relative probabilities for binary versus independent mass models is between 5 and 200, which suggests a strong preference for binary models unless we assume that the pulsar in J1906+0746 came from the initially more massive star. In this case, the relative probability of the binary models decreases to a factor of 3 for Z = 0 and for Z = Z some of the models even disfavor the ∼ ⊙ binary models. This relative change was expected based on Figure 5.3. If we assume uniform P (pi) in Equation (5.6), the relative probability of binary models is again only a factor of 3 higher than for the independent mass model. Binary models are ∼ significantly punished if there is a system with MA significantly lighter than MB if that is not allowed by the underlying remnant mass model. Correct treatment of the primary/secondary assignment of the millisecond pulsar and companion is crucial for properly evaluating the relative probabilities of the independent and binary models.
Figure 5.4 also shows clear differences in the relative probabilities of the different piston positions. For Z = 0, the pistons at the Ye core are strongly disfavored, because for low-mass progenitors the masses of the Ye cores are too low.
For solar composition, the situation is reversed. Models with the piston at the Ye core are significantly more likely than those putting it at S/NA = 4 kB, because they reduce the number of NSs with M > 1.5 M⊙. Furthermore, we see that the highest probability models essentially correspond to those with no fallback. There are small changes in the relative probabilities between the two binary mass ratio distributions
(P (q)) and the primary/secondary assignment for J1906+0746. But the best overall
164 model is the one with no fallback, uniform P (q), and remnant masses equal to the
Ye core mass. The two piston positions used by Zhang et al. (2008) are somewhat arbitrary, so in Section 5.4 we address the question of whether some other piston position would produce better agreement with the observations.
While in the rest of this paper we examine the relative probability of different hypotheses for the DNS mass distribution in the Bayesian sense, at this point we make a “frequentist” diversion and compare the observations to the models in an absolute sense. We have argued that the individual NS masses in a DNS system reflect the binarity of the original system in the sense that the primary and secondary progenitor and NS mass distributions are different. Even though one of the components is observed as a pulsar, the ambiguity of which component originated from the primary or secondary star of the binary remains present for some systems unless more information is added (J1906+0746). In order to compare the models globally, we construct in Figure 5.5 cumulative distributions of the DNS total mass (M1 + M2) and the mass difference ( M1 M2 ), along with the predictions | − | from Zhang et al. (2008) models coupled to the scenarios described in Section 5.2 with Mmax = 100 M⊙.
The distribution of mass differences in the left column of Figure 5.5 shows that getting the observed mass differences is entirely within the range of the models, although most of them predict a much broader distribution of mass differences. The twin models (blue and green lines) generally give smaller mass differences than their
165 uniform P (q) counterparts. Note that the no fallback, Ye core, solar metallicity model with the highest Bayesian relative probability (thick yellow dashed line in the lower left panel) shows almost perfect agreement with the observations. We see in the cumulative distribution of total masses in the right column of Figure 5.5 that the observed distribution is much narrower than all theoretical predictions. The narrowest cumulative distributions are again produced by solar metallicity models with no fallback and remnant masses given by the Ye core masses (thick dashed lines). Implementing a cutoff for progenitor mass Mmax makes the distributions narrower. However, this is only weakly favored by the observations. Additionally, the observed minimum total mass of 2.6 M is markedly higher than any of the ∼ ⊙ minimum total masses predicted by the models (2.2 to 2.4 M⊙). The shift is 0.2 to
0.3 M⊙, which is much higher than the mass necessary to recycle the pulsar to the observed spin periods.
At this point in a “frequentist” analysis, we would compare the two cumulative distributions using a Kolmogorov-Smirnov test to ascertain whether the observations are compatible with the models in an absolute sense. However, mass differences and total masses are only particular aspects of the full 2D distributions. If we examine Figure 5.1, which shows P(M1, M2 Mmax = 100 M ) for the model with | ⊙ the highest Bayesian relative probability, we see that all 6 binaries lie within the probability contour containing 75% of the probability (if we assume that the pulsar in J1906+0746 came from the initially less massive progenitor). The chance of
166 having no system outside this contour is 0.756 = 0.178 and we would typically expect
4.5 1.1 systems within this contour given 6 systems. This suggests that the highest ± Bayesian probability model represents the data quite well – the fact that there is no DNS system with M > 1.5 M⊙ is likely only a statistical fluctuation. However, none of the three additional DNS systems with accurate total masses (Fig. 5.5, right panels) has a total mass of about 2.4 M⊙, which suggests that the finer features of the DNS mass distribution might not be entirely reflected in the theoretical models.
More DNS systems with accurate masses of both components are necessary to address this question.
5.3.3. Extensions and limitations
There are numerous possible extensions to the analyses presented in this paper. For example, if Mmin is slightly lower, as suggested by studies of Type IIp supernova progenitors (Smartt 2009), stars in this mass range will dominate the total probability due to the steepness of the Salpeter IMF. Core masses of the progenitors that were not explicitly calculated (M < 10 M⊙ for Z = 0 and M < 12 M⊙ for
Z = Z ) can also be different – Nomoto (1984) calculated presupernova structure ⊙ of a 8.8 M⊙ star, which has enclosed mass at both S/NA = 4 kB and Ye = 0.499 of
2 1.49 M⊙ (baryonic) , which is significantly higher than that assumed for low-mass
2 Note that in this model S/NA < 4 kB for all zones below 1.49 M⊙ except a single zone at
1.19 M⊙, where it reaches S/NA = 4.02 kB.
167 progenitors in the Zhang et al. (2008) models. If we vary Mmin 5 M and the ≥ ⊙ baryonic remnant mass of 1.2 Mec 1.55 M for stars where the progenitor ≤ ≤ ⊙ structure was not explicitly calculated, use uniform priors in ln Mmin and ln Mec, and hold Mmax = 100 M⊙ fixed, we find very little change in the relative probabilities of the models. There is no significant change of Mec with respect to Zhang et al. (2008).
M +1.4 The results on the minimum progenitor mass are typically min = 8.9−1.5 M⊙, which is compatible with both Zhang et al. (2008) and Smartt (2009). We also find
1.32 < Mec < 1.39 M⊙ for Z = Z , which is compatible with Zhang et al. (2008). ∼ ∼ ⊙
The current DNS sample has only six systems with precise masses. A precise measurement of a NS birth mass greater than about 1.5 M⊙ would greatly constrain the NS formation models. Until such system is found3, it might be possible to obtain additional constraints by including NS systems with less precise mass measurements.
The Bayesian formalism naturally accounts for non-Gaussian marginal likelihoods
P (D M) such as those presented by Ozel¨ et al. (2012) for some systems. Adding i | DNS systems with only a precise total mass measurement is unlikely to change our results, as their total masses are compatible with our sample (Fig. 5.5). We experimented with adding the eclipsing X-ray pulsars, which should also have masses
3 NS masses greater than 1.5 M⊙ have been measured. For example, the system J1614 2230 has −
M = 1.97 0.04 M⊙ (Demorest et al. 2010). However, these NSs are likely significantly recycled ± and do not represent NS birth masses, although Lin et al. (2011) and Tauris et al. (2011) argue that the birth mass of J1614 2230 was higher than 1.5 M⊙. − 168 near the birth mass (Rawls et al. 2011; Ozel¨ et al. 2012), and found that these measurements have uncertainties that are too large to improve the constraints.
The models can also be extended to include other types of binaries with degenerate components (e.g. BH-NS, BH-BH, NS-WD) since we can calculate the full remnant mass function for the binaries (Eq. [5.8]). That there are no known
BH-NS binaries must strongly constrain Mmax through the relative probabilities of
NS-NS, NS-BH and BH-BH for different values of Mmax. Unfortunately, this also requires estimates for the relative detection efficiencies of the individual channels.
Finally, there are also a number of limitations to this work. The current sample of DNSs with precise masses has only six systems. The best available remnant mass function of Zhang et al. (2008) is based on 1D models that artificially explode non-rotating progenitors produced by a single stellar evolution code (Woosley et al.
2002). In addition, specific conditions have to be met to produce a DNS system.
Belczynski et al. (2002) gives a comprehensive list of possible channels for DNS formation. Most of them involve mass transfer and a theoretically uncertain phase of common envelope evolution, which exposes the NS to potentially hypercritical accretion (e.g. Chevalier 1993; Brown 1995; Dewi et al. 2006; Lombardi et al. 2011).
Furthermore, Belczynski et al. (2010) investigated the relative numbers of DNS systems and isolated recycled pulsars and found a disagreement with theoretical predictions that point to a lack of understanding of massive binary star evolution or supernova explosions. Another significant effect is the disruption of binaries due
169 to mass ejection and kicks during the two supernovae. We implemented the binary survival probability after the first supernova explosion by modifying Equation (5.8) using the results of Kalogera (1996). We used the final progenitor mass of Zhang et al. (2008) for the mass of the primary star before the explosion and various combinations of the initial and final mass for the secondary at the moment of the primary explosion. We also tried a number of relative kick velocities. Adding this to the calculation had no significant consequence. We note, however, that the second supernova is more important for the survival of the system (e.g. Dewi & van den
Heuvel 2004; Willems & Kalogera 2004; Willems et al. 2004; Stairs et al. 2006; Wang et al. 2006; Wong et al. 2010). Finally, while it is believed the NS masses are little affected by mass transfer, this may not hold for the progenitor stars. However, we think that results of detailed binary evolution and population synthesis models can affect only the details of the DNS mass distribution and not the overall conclusions.
5.3.4. The NS mass distribution of Ugliano et al. (2012)
After most of the work on this Chapter has been done, Ugliano et al. (2012) published a NS mass distribution that is based on 1D simulations of neutrino-driven explosions followed from the onset of the collapse until the end of the fallback.
The simulations are normalized by comparison with the observed parameters of SN
1987A. The distribution is based on over 100 progenitors with 10 M M 40 M , ⊙ ≤ ≤ ⊙ which we supplement with remnant masses of 1.35 M for 9.1 M M < 10 M , ⊙ ⊙ ≤ ⊙
170 similarly to the models of Zhang et al. (2008) discussed in the main paper. We also corrected the remnant masses for the binding energy according to Equation (5.9). In
Figure 5.7, we show the resulting probability distribution of the DNS masses. We see that the peaks of the distribution are shifted by 0.1 M to higher masses with the ∼ ⊙ respect to the DNS data. Even though the fallback on the remnants is included, the probability distribution has little power for NS masses higher than about 1.6 M⊙, unlike the large amounts of fallback in many of the Zhang et al. (2008) models
(Fig. 5.2). Indeed, Ugliano et al. (2012) find little fallback for high-mass progenitors.
In order to compare the relative probability of the Ugliano et al. (2012) distribution to the models in Figure 5.6, we marginalize over Mmin and the remnant mass Mec for progenitors with Mmin M < 10.8 M in the same way as we did ≤ ⊙ for the models of Woosley et al. (2002). We find that the relative probability is only 103.2 on the scale of Figure 5.6, which is likely caused by the slight mismatch in the probability peaks with respect to the DNS data. If we include remnant masses without fallback, which might be more appropriate for DNS progenitors with stripped hydrogen envelopes, the relative probability raises to about 105.4. However, this is about a factor of 10 worse than our best models in Figures 5.4 and 5.6. Thus, the reduction of fallback in the Ugliano et al. (2012) NS mass distribution provides a better match to the observed masses of DNS binaries, but not as good match as the Zhang et al. (2008) Ye core models with no fallback.
171 5.4. Discussions & Conclusions
In this work, we present a Bayesian framework that directly compares the observed distribution of NS masses to the theoretical models of NS formation in supernovae. We illustrate this method on a sample of double neutron stars from
Ozel¨ et al. (2012) which did not experience significant mass accretion and should reflect the distribution of NS birth masses. We use the relation between the initial progenitor mass and the final remnant mass of Zhang et al. (2008), tabulated for a range of explosion energies, primordial and solar compositions and for a momentum piston that drives the explosion positioned either at the base of the oxygen burning shell (the S/NA = 4 kB models) or at the outer edge of the deleptonized core (the Ye core models). We also investigate models with no fallback. We assume that the NS progenitors are independently drawn from a Salpeter IMF (independent mass model) or from a binary model where the primaries (progenitors of millisecond pulsars) are drawn from a Salpeter distribution and secondaries (progenitors of companions) are drawn from a distribution of mass ratios P (q) that is either flat or strongly peaked at q = 1.
We find a strong preference for binary models over models independently drawn from a Salpeter distribution (Fig. 5.4). However, this is true only if we can correctly assign primary and secondary labels to the pulsars and their companions (especially to J1906+0746; Fig. 5.3). Otherwise, the preference for binary models is weak.
172 The mass distributions of primaries and secondaries are asymptotically consistent with the IMF, but the mass distribution of the primary progenitors is cut off at the minimum mass for NS production (Mmin), while for secondaries it is cut off at the maximum mass (Mmax), hence each DNS component probes different mean progenitor masses. We do not find any clear preference for either of the binary mass ratio distributions we considered. We do not find any preference for models with
Mmax. We find that for a primordial composition locating the piston at S/NA = 4 kB is strongly favored, while for solar composition models placing the piston at the
Ye core is more probable. Models with no fallback (or with strong explosions that produce little fallback) are always preferred because they minimize the width of the
NS mass distributions (Figs. 5.1 and 5.2). Globally, the highest probability model has a uniform distribution for the binary mass ratios, solar composition, sets the remnant mass to that of the Ye core, and has no fallback. This model also appears to be consistent with the data in an absolute “frequentist” sense (Figs. 5.1 and 5.5).
The preference for no or very little fallback could either be a feature of the supernova mechanism or a consequence of mass loss. Stripping the progenitor significantly reduces fallback, as can be seen in the high (M > 40 M⊙) mass solar ∼ metallicity Zhang et al. (2008) models. For example, their model with an initial mass of 60 M⊙ has pre-explosion mass of only 7.29 M⊙ and mass inside S/NA = 4 kB of 1.60 M , with a fallback mass of only 0.04 M (0.00 M ) for E = 1.2 1051 ergs ⊙ ⊙ ⊙ × (2.4 1051 ergs). While the mass can be transferred between the stars in the binary ×
173 and change the amount fallback when compared to a single star, the point is that our results prefer no fallback in either star of the binary. This is in agreement with the models of Portegies Zwart & Yungelson (1998) where both of the two supernovae in a binary occur in a stripped progenitor. Alternatively, the DNS mass distribution may represent a surprisingly clear fingerprint of the supernova explosion mechanism.
Specifically, with no fallback, the mass of the NS corresponds to the mass coordinate of the progenitor where the explosion was initiated. The bulk of the stellar mass growth and consequently also the bulk of the DNS population growth occurred between redshifts 1 and 2 (e.g. Juneau et al. 2005), when the metallicity was approximately solar. Our Z = Z results clearly prefer the edge of the deleptonized ⊙ core (S/NA 2.8 kB) as the mass coordinate of the explosion rather than at the base ≈ of the oxygen shell (S/NA = 4 kB).
The two piston positions used by Zhang et al. (2008) are somewhat arbitrary so we explored whether other piston position would yield better results. We took the solar metallicity progenitor models of Woosley et al. (2002)4 that cover the mass range 10.8 M 40 M and determined the mass coordinates that correspond to ≤ ≤ ⊙ a range of values of the entropy, 2.0 kB S/NA 4.0 kB. As we have no means of ≤ ≤ calculating the amount of fallback and models without fallback are preferred anyway, we assume that the remnant masses are set by the mass coordinate at the given entropy. The relative probabilities of these models were calculated with Mmin and
4http://www.stellarevolution.org/data.shtml
174 Mec as free parameters and Mmax fixed (Sec. 5.3.3) and are shown as a function of the entropy in Figure 5.6. We see that there is a clear maximum at S/NA 2.8 kB. ≈ The relative probability at maximum is almost equal to the highest probability model from Figure 5.4, which is not surprising because for many progenitors this is the entropy at the edge of the Ye core. If correct, this is an important result, because it shows that supernova explosions are initiated approximately at the edge of the iron core. Since the elapsed time from the core bounce until the accretion of all material out to the edge of the deleptonized core is 0.1to0.3 s for majority of the progenitors, this also means that the supernovae explode by a delayed mechanism. That it is not longer challenges some of the delayed neutrino-driven models that explode at
> 0.5 s after the bounce (Marek & Janka 2009; M¨uller et al. 2012a) and constrains ∼ the time available for the unstable modes that potentially drive the explosion (Fryer et al. 2012). It is important to note that the composition and density of the layers that accrete through the shock do not change smoothly with time; the composition interfaces such as the one at the edge of the iron core lead to sudden decreases in the mass accretion rate, which cause expansion of the shock. This expansion might be temporary and can be followed by recession of the shock (Marek & Janka 2009), or it might trigger an explosion such as those observed after the advection of the
Si/SiO shell interface (Buras et al. 2006a; M¨uller et al. 2012a).
Independent of any concern about binaries, the greatest limitation of this and similar studies in the near future is our poor understanding of the physics of
175 supernova explosions that link the initial progenitor mass to the final remnant mass.
While the DNS mass distribution strongly suggests that the explosion develops at the edge of the deleptonized core, supernova theory does not provide any a priori reason why this should be so and whether this is true for all progenitor masses, metallicities, rotation rates and other parameters.
To summarize, ideal theoretical calculations of the massive star remnant mass function require an understanding of the supernova explosion mechanism.
Until a self-consistent understanding is available, artificially induced supernova explosions intended to study the remnant population should increase their realism by manually inducing the explosions by either increasing the neutrino luminosity or absorption cross section in order to properly model the transition from accretion to neutrino-driven wind and the explosion5, as has been done for some studies of explosion physics (e.g. Scheck et al. 2006; Nordhaus et al. 2010; Fujimoto et al.
2011; Kotake et al. 2011; Hanke et al. 2012; Nordhaus et al. 2012). Although the
DNS distribution suggests that fallback cannot be significant for the DNS systems, it would be interesting to see if this is also a feature of “semi-physical” explosion models. Of particular importance is whether fallback is a strong or stochastic function of progenitor structure, if it occurs. The range of progenitor models and
5 After the submission of this paper, Ugliano et al. (2012) published a NS mass distribution that is based on neutrino-driven explosions that are followed until the end of the fallback. We present the resulting probability distribution and comparison with other models in the Appendix.
176 model supernovae also needs to be expanded to better cover the progenitor mass range in order to fully sample the rapidly changing core properties with mass.
Models are most needed near solar metallicities, since such stars overwhelmingly dominate any observable population of supernovae or NS binaries, and at the low masses that dominate the progenitor population due to the initial mass function.
Observations mainly constrain the outcomes of low-mass solar metallicity stars - the models least examined in theoretical studies.
177 51 Z Piston at Mmin Progenitor mass range E [10 ergs]
0 S/NA = 4 kB 9.5 M⊙ (10, 100) M⊙ 0.3, 0.6, 0.9, 1.2, 1.5, 1.8, 2.4, 3.0, 5.0, 10.0
0 S/NA = 4 kB 9.5 M⊙ (10, 100) M⊙ S/NA = 4 kB core mass, no fallback
0 Ye core 9.5 M⊙ (10, 100) M⊙ 1.2, 10.0
178 0 Ye core 9.5 M⊙ (10, 100) M⊙ Ye core mass, no fallback
Z S/NA = 4 kB 9.1 M⊙ (12, 100) M⊙ 1.2, 2.4 ⊙ Z S/NA = 4 kB 9.1 M⊙ (12, 100) M⊙ S/NA = 4 kB core mass, no fallback ⊙ Z Ye core 9.1 M⊙ (12, 100) M⊙ 1.2, 2.4 ⊙ Z Ye core 9.1 M⊙ (12, 100) M⊙ Ye core mass, no fallback ⊙
Table 5.1. Summary of the remnant mass distribution models from Zhang et al. (2008). Name M 1 σ1 P1 [ms] P˙1 M 2 σ2 Refererence
J0737-3039a 1.3381 0.0007 22.7 1.8 10−18 1.2489 0.0007 Kramer et al. (2006) × B1534+12 1.3332 0.0010 37.9 2.4 10−18 1.3452 0.0010 Stairs et al. (2002) × J1756-2251 1.40 0.02 28.5 1.0 10−18 1.18 0.02 Faulkner et al. (2005) × −14 179 J1906+0746 1.248 0.018 144.1 2.0 10 1.365 0.018 Kasian (2008), Lorimer et al. (2006) × B1913+16 1.4398 0.002 59.0 8.6 10−18 1.3886 0.002 Weisberg et al. (2010) × B2127+11C 1.358 0.010 30.5 5.0 10−18 1.354 0.010 Jacoby et al. (2006) ×
a −16 The companion is also a pulsar with P2 = 2.8s and P˙2 = 8.9 10 . ×
Table 5.2. Summary of double neutron star systems. Fig. 5.1.— Contours of probability P(M , M Mmax) for having a DNS with A B| masses MA and MB that originated from the more massive primary and less massive secondary stars of the original binary, respectively, for the most probable model discussed in Section 5.3.2. The model has uniform distribution of q and the remaining parameters are given in the upper right corner of the plot. Contour labels indicate the total probability that lies outside the contour (i.e. the contour labeled as “0.1%” encloses 99.9% of the total probability). Circles indicate pairs of DNS masses. For the red systems, the pulsar is always assumed to arise from the primary. The green circle is for J1906+0746 where we show both possible assignments (see text). The upper panel shows the distribution of remnant masses MA from the primary, with the lines marking the observed values of M 1. For J1906+0746, we show both cases, where the pulsar came from the primary (solid) or the secondary (dashed). The right panel shows the distribution in MB.
180 Fig. 5.2.— Same as in Figure 5.1, but for a model with significant mass fallback. The model has uniform P (q) and the remaining parameters are given in the upper right 2.8 corner of the plot. The relative probability B12 of this model is a factor of 10 ∼ lower than for the model in Figure 5.1 (see Section 5.3.2 for more details).
181 Fig. 5.3.— Probability that the pulsar in the DNS came from the initially more massive progenitor with respect to the reverse. DNS system names are given at the bottom while the individual NS masses are given at the top with the pulsar masses above the companion masses. The symbols indicate the median, while the error bars shows 1 and 2σ contours. The results are shown for the two piston positions, two metallicities and the two binary models (symbols are explained in the plot), and are marginalized over all values of E and Mmax, weighted by P (Mmax D). Filled squares and circles mark the median, while the error bars show 1 and 2σ |quantiles.
182 Fig. 5.4.— Relative probability of the different models for the origin of the DNS mass distribution as a function of the explosion energy and marginalized over Mmax. We show primordial composition (left panel) and solar metallicity (right panel) models for the piston positioned at S/NA = 4 kB (solid lines with filled symbols) and at the Ye core (dashed lines with open symbols). For the sake of clarity, we show only the independent mass model and the uniform P (q) binary model. The horizontal lines indicate results for the models with NS masses equal to the mass at the piston position and no fallback.
183 Fig. 5.5.— Cumulative probability distributions of the mass differences ( M M , | A − B| left panels) and the total masses (MA + MB, right panels) of the DNS systems. The results of the Zhang et al. (2008) models with Mmax = 100 M⊙ are shown with thin lines for Z = 0 (top panels) and Z = Z (middle panels). The bottom panels show solar metallicity models with no fallback.⊙ The observed cumulative distributions of M 1 M 2 and M 1 +M 2 are shown with the thick black line. The distribution of total masses| − in| the right panel includes additionally DNS systems J1518+4904 (Janssen et al. 2008), J1811 1736 (Corongiu et al. 2007), and J1829+2456 (Champion et al. 2005) that have accurate− total (but not individual) masses.
184 Fig. 5.6.— Relative probability of the DNS mass models as a function of the position of the mass cut in entropy (S/NAkB) that separates the remnant and the supernova ejecta (no fallback). The solid line shows the results for the supernova progenitors of Woosley et al. (2002), while the dashed horizontal line is for the best model based on Zhang et al. (2008) that places the piston at the Ye core and assumes no fallback (right panel of Fig. 5.4).
185 Fig. 5.7.— Same as Figure 5.1, but for a model based on the solar metallicity NS mass distribution of Ugliano et al. (2012). The distribution of P (q) is uniform and the calculation includes all progenitors of Ugliano et al. (2012).
186 Chapter 6
Future work
In the recent years, the field of supernova theory has focused on the transition from 2D to 3D simulations. Most of the 3D results up to date have been with simplified or no transport of neutrinos, but more advanced calculations are not very far. Although the supernovae in the Nature are 3D, it is not clear what will be gained by the 3D simulations with respect to 2D. Most of the recent results done in the light-bulb approximation do not show any substantial differences in critical neutrino luminosities between otherwise identical 2D and 3D simulations. Is it possible that such a difference will appear when more sophisticated neutrino transport is added to the simulations? In Chapter 2, we showed that adding a simple gray neutrino transport lowers the critical luminosity by up to 20%, but it is unclear why would ∼ the radiation transport make any significant difference between 2D and 3D.
Eventually, all supernova models have to be compared to and verified by observations. So far, the state-of-the-art self-consistent 2D simulation do not produce explosions with energies comparable to what is observed (e.g. M¨uller et al.
2012a). Instead of pushing further in the time-dependent simulations with uncertain
187 prospects for success, I propose two projects that will increase and systematize the current understanding of the supernova mechanism and connect this knowledge to observations.
6.1. The Stability of Steady-State Shocks
Radial and non-radial instabilities accompany the transition from collapse to explosion (e.g. Burrows et al. 1995; Blondin et al. 2003; Marek & Janka 2009). The different character of these instabilities in 1D, 2D, and 3D simulations is thought to explain their different outcomes. Specifically, simulations in 2D explode at lower
Lν, core than in 1D (Murphy & Burrows 2008). Explosions in 3D do not seem to be easier than in 2D (Nordhaus et al. 2010; Hanke et al. 2012; Couch 2012; Dolence et al. 2013).
Despite considerable efforts, the nature of the instabilities in the stalled shock is not completely understood (e.g. Blondin & Mezzacappa 2006; Fern´andez 2012;
Guilet & Foglizzo 2012). One of the important aspects of the problem that has so far received only very little attention is the dependence of the instability growth rates on the parameters of the problem and the microphysics (e.g. neutrino interactions and nuclear physics). For fixed microphysics, the most important parameters are the mass accretion rate through the shock M˙ , the proto-neutron star mass M, and its radius rν, all of which evolve in time as the star collapses. The values of these parameters at a particular moment depend on the structure of the progenitor.
188 High-mass progenitors maintain higher M˙ for a longer time so that the PNS mass M increases faster, and thus they occupy a different part of parameter space from the low-mass progenitors. The high computational cost and complexity of the 2D/3D time-dependent simulations prohibits any systematic study of the physical effects on the instabilities, and the steady-state calculations to date have not addressed all relevant effects (Yamasaki & Yamada 2005, 2007). In particular, radiation transport
acc and the consistent incorporation of the accretion neutrino luminosity Lν due to
crit cooling of the flow is missing even though it significantly reduces Lν, core as we showed in Chapter 2.
In order to address these important issues, I will systematically study the radial and non-radial linear stability of steady-state accretion shocks. I will gradually increase the complexity of the problem to isolate and characterize the most important effects. I will start with purely isothermal flows with gas in free fall upstream of the shock, which, as I showed in Chapter 2, captures the basic physics of the problem even if idealized. No similar stability analysis has yet been performed with boundary conditions appropriate to core-collapse supernovae. On the opposite side of the complexity spectrum will be a calculation with a realistic equation of state and a progressively more complex treatment of radiation transport to diagnose the stability
acc properties of Lν . There is an exciting possibility of a yet undiscovered instability
acc as a result of Lν , because small changes in the mass accretion rate in the region where net neutrino cooling dominates influences the matter upstream in the flow
189 acc ˙ through Lν . I will determine the dependence of the growth rates on M, M, and rν by solving the two-point boundary value problem of a system of ordinary differential equations using relaxation methods, where the eigenvalue is the instability growth rate. This will have immediate implications for which stars explode via the neutrino mechanism by showing which moments of the collapse are susceptible to instabilities and how does this change for different progenitors.
6.2. Constraining the Explosion Mechanism with
Supernova Observables
The only direct probe of the entrails of a supernova explosion has been the detection of a handful of neutrinos from SN1987A. While the general picture of core collapse was confirmed, the data produced a very limited insight into the properties of the explosion dynamics. Until a core-collapse event offers a detection of more neutrinos or gravitational waves, constraints on the mechanism have to be inferred indirectly by linking the explosion mechanism to the observables through supernova shock propagation models. These models use detailed nuclear reaction networks to follow the shock wave from 200 km, where the explosion initiates, to the surface of ∼ the progenitor star.
As supernova simulations do not explode on their own, the models have to be exploded “by hand”. This is usually achieved by the action of an artificial piston
190 (Timmes et al. 1995; Woosley & Weaver 1995). Such models are widely used for studying nucleosynthetic yields (e.g. Woosley & Weaver 1995), neutron star and black hole mass functions (e.g. Timmes et al. 1996; Zhang et al. 2008) and supernova light curves (e.g. Dessart et al. 2010, 2011). However, the recent simulations of
Ugliano et al. (2012) and my results on the neutron star mass distribution (Chapters
4 and 5) suggest that the piston-driven models give unrealistic results. Since other important observables such as nucleosynthetic yields are closely related to any problems with the remnant mass distribution, it is necessary to revisit the findings of the piston-driven models with more realistic models exploded by neutrino heating.
I will calculate a suite of dynamical models of shock propagation through the progenitor star initiated by neutrino heating below the stand-off accretion shock using the hydrodynamical code FLASH (Fryxell et al. 2000), which is equipped with nuclear reaction networks. I will vary the neutrino heating parameters informed by my detailed exploration of the explosion conditions as a function of progenitor mass. I will take into account the properties of the shock instability as a function of progenitor structure from the project described in Section 2. I will calibrate the neutrino heating based on the observations of SN1987A (Ugliano et al. 2012) or through measured IIp progenitor masses (Smartt 2009). The initial radial profiles of thermodynamic variables will be based on publicly available models of massive stars
(Heger et al. 2000; Woosley et al. 2002). Additionally, I will investigate progenitors with their surface layers stripped due to binary processes to better approximate
191 the progenitors of the double neutron star binaries, which provide the most precise birth masses of neutron stars. Since no such progenitors are publicly available, I will either modify the publicly available progenitors, or alternatively, I will compute the progenitors with the publicly available MESA code (Paxton et al. 2011, 2013).
My models will provide the nucleosynthetic yields of elements up to the iron peak, the mass function of the remnants, the dynamics of the ejecta and their mutual connections. These calculations have a value on their own as an update to previous piston-based models, but more importantly, they will allow me to constrain the supernova explosion mechanism by comparing my new neutrino-driven calculations to the observations of supernovae, their remnants and progenitors. For example, varying the driving neutrino luminosity results in different durations of the quasi-static accretion phase and hence different neutron star masses. The ensuing explosion starts from different conditions: the escape velocity is higher due to higher proto-neutron star mass, the expelled neutrino-heated matter has lower mass, and the shock does not propagate through the more dense material that was previously accreted. As a result, there will be changes in the synthesized nickel masses and expansion velocities, which are routinely determined observationally (e.g. Hamuy
2003; Smartt 2009) together with changes in the distribution of neutron star masses.
I will statistically analyze the sensitivity of the observations to model explosion parameters and determine their value. A crucial part of this analysis is to account for potential ambiguities, selection effects and differences between the datasets using the
192 Bayesian approach I took in my work on the neutron star mass distribution. I will also use my extensive experience with data modeling from my work on Cepheids.
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