Core-Collapse Supernova Explosion Theory
CCore-Collapse Supernova Explosion Theory
A. Burrows (cid:63) & D. Vartanyan Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA Department of Astronomy, University of California, Berkeley, CA 94720-3411 (cid:63) e-mail: [email protected]
Most supernova explosions accompany the death of a massive star. These explosions givebirth to neutron stars and black holes and eject solar masses of heavy elements. However,determining the mechanism of explosion has been a half-century journey of great complexity.In this paper, we present our perspective of the status of this theoretical quest and the physicsand astrophysics upon which its resolution seems to depend. The delayed neutrino-heatingmechanism is emerging as a robust solution, but there remain many issues to address, notthe least of which involves the chaos of the dynamics, before victory can unambiguously bedeclared. It is impossible to review in detail all aspects of this multi-faceted, more-than-half-century-long theoretical quest. Rather, we here map out the major ingredients of explosionand the emerging systematics of the observables with progenitor mass, as we currently seethem. Our discussion will of necessity be speculative in parts, and many of the ideas may notsurvive future scrutiny. Some statements may be viewed as informed predictions concerningthe numerous observables that rightly exercise astronomers witnessing and diagnosing thesupernova Universe. Importantly, the same explosion in the inside, by the same mechanism,can look very different in photons, depending upon the mass and radius of the star upon ex-plosion. A 10 -erg (one “Bethe”) explosion of a red supergiant with a massive hydrogen-richenvelope, a diminished hydrogen envelope, no hydrogen envelope, and, perhaps, no hydrogenenvelope or helium shell all look very different, yet might have the same core and explosionevolution.1 Core-Collapse Supernova Explosions Stars are born, they live, and they die. Many terminate their thermonuclear lives after billions ofyears of cooking light elements into heavier elements by ejecting their outer hydrogen-rich en-velopes over perhaps hundreds of years. In the process, they give birth to compact white dwarfstars, half as massive as the Sun, but a hundred times smaller. Such dense remnants cool off overbillions of years like dying embers plucked from a fire. A subset of these white dwarfs in binarystellar systems will later (perhaps hundreds of millions of years) ignite in spectacular thermonu-clear explosions, many of these the so-called Type Ia supernovae used, due to their brightness fromacross the Universe, to take its measure.However, some stars, those more massive than ∼ (cid:12) , die violently in supernova explosionsthat inject freshly synthesized elements, generation after generation progressively enriching theinterstellar medium with these products of existence. They too leave behind remnants, but neutron1 a r X i v : . [ a s t r o - ph . S R ] O c t tars and black holes. The former could become radio pulsars, are only the size of a city, and haveon average masses 50% again as massive as the Sun. The latter are perhaps a few to ten times moremassive than a neutron star, but even more compact and more exotic.The supernova explosions of these massive stars, the so-called core-collapse supernovae(CCSNe), have been theoretically studied for more than half a century and observationally studiedeven longer. Yet, the mechanism of their explosion has only recently come into sharp focus. Awhite dwarf is birthed in these stars as well, but before their outer envelope can be ejected this whitedwarf achieves the “Chandrasekhar mass” near ∼ (cid:12) . This mass is gravitationally unstable toimplosion. After a life of perhaps ∼ − a supernova needs to be launched most of the time to be consistentwith observed rates and statistics.What has emerged recently in the modern era of CCSN theory is that the structure of theprogenitor star, turbulence and symmetry-breaking in the core after bounce, and the details of theneutrino-matter interaction are all key and determinative of the outcome of collapse. Spherical sim-ulations seldom lead to explosion. Multi-dimensional turbulent convection in the core necessitatescomplicated multi-D radiation(neutrino)/hydrodynamic simulation codes, and these are expensiveand resource intensive. It is this complexity and the chaos in the core dynamics after implosion thathas retarded progress on this multi-physics, multi-dimensional astrophysical problem, until now.In the first era of CCSN simulations, the state-of-the-art was good spherical codes that handled theradiation acceptably. These models rarely, if ever, exploded. Multi-D codes were not yet useful.Then, two-dimensional (axisymmetric) codes arrived, captured some aspects of the overturningconvection about which 1D models are mute, but were slow − only a few runs could be accom-plished per year. This era was followed by the advent of some 3D capability, and at the same timemany 2D runs could be performed to map out some of parameter space and gain intuition concern-ing the essential physics and behavior. We are now in the era of multiple 3D simulations per year,wherein we can explore core dynamics and explosion in the full 3D of Nature without the fear thata mistake in a single expensive run which could take a year on a supercomputer would set us back.This progress has been enabled by the parallel expansion of computer power over the decades. Itis the pivotal role of multi-dimensional turbulence and the breaking of spherical symmetry in themechanism of explosion itself, coupled with the driving role of neutrino heating, that necessitatedthe decades-long numerical and scientific quest for the mechanism of core-collapse supernovae.What Nature does effortlessly in a trice has taken humans rather longer to unravel.However, there are now strong recent indications that the dominant explosion mechanismand rough systematics of the outcomes with progenitor star are indeed yielding to ongoing multi-2ronged international theoretical efforts. Moreover, code comparisons are starting to show generalconcordance . Many recent multi-dimensional simulations employing sophisticated physics andalgorithms are exploding naturally and without artifice. These include those from our group ,using the state-of-the-art code F ORNAX , and those from others . Neutrino heating in theso-called gain region behind a stalled shock, aided by the effects of neutrino-driven turbulenceand spherical symmetry breaking, together seem, in broad outline, to be the agents of explosionfor the major channel of CCSNe. Other subdominant channels might be thermonuclear (what dothe terminal cores of ∼ (cid:12) stars actually do? , but see ) or magnetically-driven (so-called“hypernovae”, ∼
26, 27 ; long-soft gamma-ray bursts, < . %). And indeed, there remain for theneutrino mechanism numerous interesting complications concerning nuclear and neutrino physics,the progenitor structures, and numerical challenges to be resolved before this central problem canbe retired. It is generally agreed that the stall of the roughly spherical bounce shock wave sets up a quasi-hydrostatic structure interior to it that accretes the matter falling through the shock from the outercore that is still imploding . The rate of accretion ( ˙ M ) through the stalled shock and onto theinner core is an important evolving quantity that depends essentially upon the density structure ofthe progenitor’s core just prior to Chandrasekhar instability (see Figure 1) and determines much ofwhat follows. The core is so dense and the neutrino particle energies are so high (10’s to 100’s ofMeV, million electron volts) that this structure interior to ∼ g cm − is opaque to neutrinos ofall species - the structure is a “neutrino star” with “neutrinosphere” radii that depend upon the neu-trino species and particle energy, but are initially ∼ ν e ) is generated. It is thisburst that saps energy from the shock and leads to its stalling into accretion. A secondary causeof its stalling is the shock dissociation of the infalling nuclei into nucleons. This effect lowers theeffective “ γ ” of the gas that connects internal thermal energy with pressure by diverting energy intonuclear breakup, thereby channeling less efficiently the gravitational energy otherwise available toprovide pressure support for the shock. The stalled shock radius initially hovers around 100-200km. Just interior to the shock is the semi-(neutrino)transparent “gain region” where the “optical”depth to neutrinos is ∼ ν e and ¯ ν e neutrino absorption on free neutrons andprotons in the gain region would easily power a dynamical outflow. This is similar to a thermalwind. However, the accretion ram and neutrino heating are competing to determine instabilityto explosion, with the added complication that accretion is also powering a changing fraction ofthe driving neutrino luminosities. The explosion is akin to a bifurcation between quasi-stationaryaccretion and explosion solutions, with control parameters related to the accretion rate and theneutrino luminosities (for two), but a simple analytic explosion condition in the context of realisticsimulations has proven elusive. Hence, detailed simulations are required.What has emerged is that only those progenitor models with very steep outer density profilesthat translate into rapidly decreasing post-bounce accretion rates can explode in spherical symme-try (1D) via the neutrino mechanism. Among the representative progenitor models shown in Figure1, only the 9 M (cid:12) star comes close to fitting that description. However, not even it explodes in our1D simulations. Multi-D seems required. Classically, the 8.8 M (cid:12) model of Nomoto explodesspherically, as do a few others with similar very steep outer density profiles
3, 33 . However, due tonew ideas concerning the character of thermonuclear burning and electron capture in such compactcores, this lowest-mass progenitor region is undergoing a modern reappraisal . Note that thereare strong arguments in tension with this alternate perspective . The current explosion paradigmfor most massive stars is gravitational-energy sourced, neutrino-driven, and turbulence-aided, andwe now summarize some of what we have learned concerning the roles of various specific physicaleffects. Efficiency
Since a hot and lepton-rich PNS radiates ∼ × ergs in neutrinos as it transitionsinto a tightly bound, cold neutron star and supernova explosion energies are “typically” one Bethe,it is often stated that the neutrino mechanism of core-collapse explosion is one of less than 1%tolerances. This is not true. During the 100s of milliseconds to few-second timescales after bounceover which the neutrino heating mechanism operates, the efficiency of energy deposition in thegain region, the fraction of the emitted energy absorbed there, is ∼ − , after the phase duringwhich we think the explosion energy is fully determined. Turbulent Convection
Turbulence is fundamentally a multi-dimensional phenomenon and can’tbe manifest in spherical (1D) symmetry (and, therefore, in 1D simulations). The turbulence inthe gain region interior to the stalled shock is driven predominantly by the neutrino heating itself,which produces a negative entropy gradient unstable to overturn. This is similar to boiling water ona stove, via absorptive heating from below . Figure 2 depicts the inner turbulent convective regionearly after bounce before explosion, showing accreted matter tracers swirling randomly about thePNS core. A larger neutrino heating rate will increase both the vigor of the turbulence and theentropy of this mantle material. The matter that accretes through the shock on its way inward tothe PNS during the pre-explosion phase contains perturbations that arise during pre-collapse4tellar evolution which will seed the convective instability. The larger and more prevalent theseseeds the quicker the turbulence grows to saturation and in vigor. One feature of turbulence isturbulent pressure. The addition of this stress to the gas pressure helps push the shock to a largerstalled shock radius. This places matter in more shallow reaches of the gravitational potential wellout of which it must climb and helps to overcome the subsequently smaller ram pressure due toinfalling matter from the outer core still raining in. The turbulence also forces the accreted matter toexecute non-radial trajectories as it settles, increasing the time during which it can absorb neutrinoenergy before settling on the PNS and, hence, the average entropy that can be achieved in the gainregion . Therefore, through the combined agency of both neutrino heating and neutrino-driventurbulence, the quasi-stationary structure that is the PNS, plus mantle, plus stalled shock wave ismore likely to reach a critical condition wherein the steady infalling solution bifurcates into anexplosive one. The huge binding energy accumulated in the PNS does not need to be overcome − only its mantle (and with it the rest of the star) needs to be ejected.Moreover, the turbulent hydrodynamic stress is anisotropic, with its largest component alongthe radial direction . Turbulent magnetic stress might also be a factor . Importantly, turbulenceis more effective at using energy to generate stress/pressure than a gas of nucleons, electrons, andphotons. As much as ∼ −
40% of the stress behind the stalled shock when the turbulence is fullydeveloped can be in turbulent stress. Hence, partially channelling gravitational energy of infall intoturbulence instead of into thermal energy helps support and drive the shock more efficiently . Neutrino-Matter Interactions
The predominant processes by which energy is transferred fromthe radiated neutrinos to the matter behind the shock in the gain region are electron neutrinoabsorption on neutrons via ν e + n → e − + p , anti-electron neutrino absorption on protons via ¯ ν e + p → e + + n , and inelastic scattering of neutrinos of all species off of both electrons and nucle-ons. The two super-allowed charged-current absorption reactions dominate and provide a powerapproximately equal to the product of the neutrino luminosity and the neutrino optical depth inthe gain region. The latter can be ∼ ρ ) above 10 − g cm − , nucleon-nucleon interactions in-troduce correlations in density and spin. Such non-Poissonian correlations modify the scatteringand absorption neutrino-matter interaction rates (generally suppressing them), hence, they affectthe emergent neutrino luminosities . This is relevant to the instantaneous power deposition inthe gain region, and, hence, the neutrino-driving mechanism itself. These many-body effects in-crease with density and some of the associated correction factors have been estimated
48, 49 to beof order 10-20% at 10 g cm − , near and just interior to the neutrinospheres. However, such5orrections depend upon a detailed and self-consistent treatment of the opacities along with thenuclear equation of state (EOS), and this goal has yet to be achieved. Nevertheless, using scat-tering suppression factors researchers
4, 50 have shown that these effects can facilitate explosion.They do this by decreasing the opacities, thereby increasing the neutrino loss rates. This leads to amore rapid shrinking of the PNS, which due to consequent compression heats the neutrinosphereregions. This increases the mean energy of the emitted neutrinos. Since the neutrino absorptionrates via the charged-current reactions quoted aobve increase approximately as the square of theneutrino energy and the luminosities themselves are elevated, the neutrino power deposition inthe gain region is augmented, thereby facilitating explosion. The effect is not large, but when anexplosion is marginal it can be determinative.During the early collapse phase, increasing densities lead to increasing electron Fermi ener-gies and higher electron capture rates on both free protons and nuclei. Electron capture decreasesthe electron fraction (Y e , the ratio of the electron density to the proton plus neutron density) of theinfalling gas, and this decreases the electron pressure. A decrease in the electron pressure slightlyaccelerates the infall and the mass accretion rate ( ˙ M ) versus time. As already stated, ˙ M after innercore bounce is a key parameter determining, among other things, the accretion ram pressure exter-nal to the shock and the accretion component of the neutrino luminosities. Therefore, the rate ofcapture on infall can affect explosion timing and, perhaps, its viability. The effect is not large, butwhen things are marginal altering the evolution of ˙ M can be important. However, the capture rateon the mix of nuclei in the imploding core is not known to better than perhaps a factor of five .Hence, clarifying this important issue remains of interest to modelers.Finally, the energy transfer to matter via inelastic scattering off electrons and nucleons pro-vides a subdominant component of the driving heating power behind the shock wave. The effectmay be only 10%-15%, but, again, when the core teeters on the edge of explosion such effectsmatter. Neutrino scattering off electrons (akin to Compton scattering, but for neutrinos) results in alarge energy transfer, but has a small rate. Energy transfer to the heavier nucleons is small, but thescattering rate is large. The net effect results in comparable matter heating rates for both effects,with a slight advantage to neutrino-nucleon scattering . However, calculating such spectral energyredistribution is numerically difficult, and represents one of the major computational challenges inthe field
45, 54, 55 . Explosion
The stalled shock radius can be decomposed into spherical harmonics in solid angle.The onset of the explosion of a stalled accretion shock is a monopolar instability in the quasi-spherical shock. However, approximately when the monopole becomes unstable, the dipole oftenseems to as well
6, 56 . Therefore, the explosion picks an axis, seemingly at random for a non-rotatingprogenitor, and the blast has a dipolar structure with a degree of asymmetry that seems to be lowfor quickly exploding models and larger for those whose explosion is more delayed. Figure 3 de-picts an example blast structure manifesting such a dipole . Generally, but not completely reliably,the lower-mass progenitors (such as a 9-M (cid:12) progenitor
7, 9 ) explode earlier and before turbulence isvigorous and the more massive progenitors seem to explode later and after turbulence has achieved6ome vigor. Hence, the latter generally, though not every time, explode more asymmetrically, witha larger dipolar component . The chaos of the turbulence makes the outcome stochastic, so thatthe direction of explosion is not easily predicted. Importantly, the chaos of the turbulent flow willresult in distribution functions of explosion times, directions, explosion energies, explosion mor-phologies, residual neutron star masses, Ni yields, general nucleosynthesis, and kick velocities,etc., even for the same star. It is not even known whether those functions are broad or narrow for agiven star.In addition, exploding more along an axis, as depicted in Figure 3, allows the flow externalto the shock to wrap around the prevailing axis and accrete along a pinched waist in an equatorialstructure. This breaking of symmetry, impossible in spherical symmetry, allows simultaneousaccretion and explosion. Whereas a 1D explosion by its nature turns accretion off in all directions,and thereby throttles back the accretion component of the driving neutrino luminosity, in multi-D the accretion component of the luminosity can be maintained. Hence, the breaking of sphericalsymmetry supports the driving luminosity and facilitates explosion, just as it is getting started. Thissymmetry breaking is an important aspect of viable explosion models and is impossible in sphericalmodels. Nature unchained to manifest overturning instability leading to turbulence employs thisfreedom to facilitate explosions that might be thwarted in 1D. Both the turbulent stress and theoption of simultaneous accretion in one direction while exploding in another are important featuresof the CCSN explosion mechanism.Convection in the progenitor star upon collapse will create perturbations in velocity, den-sity, and entropy that seed overturn and turbulence in the post-shock matter exterior to the innerPNS and the neutrinospheres. The magnitude of these perturbations generally increases with pro-genitor mass, but their true character is only now being explored in detail. Recently, a numberof groups have embarked upon 3D stellar evolution studies during the terminal stages of massivestars
17, 22, 24, 58–60 . The potential role of aspherical perturbations in the progenitor models in inau-gurating and maintaining turbulent convection behind the stalled shock wave is an active area ofresearch
4, 5, 17, 37, 60 and these studies might soon reveal the true nature of accreted asphericities andtheir spatial distribution. It might also be that low-order modes in the progenitors would naturallyresult in angular asymmetries in the mass accretion through the shock and provide a path of leastresistance that would (however randomly) set the explosion dipole and direction, whatever its mag-nitude. Such low accretion-rate paths might actually facilitate explosion in circumstances when itwould otherwise be problematic.Another convective phenomenon that can help achieve the critical condition for explosion isproto-neutron-star convection. This is not the neutrino-heating-driven convection in and near thegain region just behind the stalled shock, but overturn driven by electron lepton loss from beneaththe neutrinospheres. As electron neutrinos are liberated from the inner PNS mantle around a radiusof ∼
20 km, the resulting negative Y e gradient is convectively unstable. This is akin to instabil-ities in stars due to composition gradients. All proto-neutron stars show this instability, whichlasts for the entire duration of PNS evolution and likely continues long after (many seconds to one7inute) the explosion is launched (if it is). PNS convection
10, 61, 62 accelerates energy loss (partic-ularly via ν µ , ν τ , ¯ ν µ , and ¯ ν τ neutrinos) and electron lepton loss in the PNS, thereby acceleratingcore shrinkage. In a manner similar to the many-body effect, such core shrinkage leads to higherneutrinosphere temperatures and a stronger absorptive coupling to the outer gain region.We end this section by emphasizing that the most important determinant of explosion, allelse being equal, is the mass density structure of the unstable Chandrasekhar core. The densityprofile translates directly into the mass accretion rate after bounce and this determines both theaccretion tamp and the accretion component of the driving neutrino luminosity. Figure 1 providesan example set of density profiles ( ρ ( r ) ) from 9 M (cid:12) to 27 M (cid:12) . This set spans most (but notall) massive stars that give birth to CCSNe. There are a few trends in ρ ( r ) worth noting. First,the lowest-mass massive stars generally have slightly higher central densities and steeper outerprofiles and the higher-mass massive stars have lower central densities and significantly shallowerouter density profiles. However, the trend in the slope of the outer density profiles is not strictlymonotonic with progenitor mass, with some “chaos” in the structures. Ambiguities in the handlingof convection, overshoot, doubly-diffusive instabilities, and nuclear rates have led to variationsfrom modeler to modeler in progenitor stellar models up to collapse that have yet to converge.Furthermore, the effects of fully 3D stellar evolution and rotation have not yet been fully assessed.Therefore, the summary behavior depicted in Figure 1 is provisional.Given these caveats, important general insights are emerging. The first is that the sili-con/oxygen shell interface in progenitors (seen for many models in Figure 1) constitutes an im-portant density jump, which if large enough can kickstart a model into explosion. In many of ourmodels, the shock is “revived” upon encountering this interface
9, 64 . The associated abrupt drop inaccretion rate and inhibiting ram pressure at the shock upon the accretion of this interface is notimmediately followed by a corresponding drop in the driving accretion luminosity. This is due tothe time delay between accretion to the shock at 100 −
200 km and accretion to the inner core wherethe gravitational energy is converted into useful accretion luminosity. This time delay pushes thestructure closer to the critical point for explosion. However, sometimes the density jump is notsufficiently large and its magnitude in theoretical stellar models has not definitively been pinneddown.Second, the very steep density profiles seen for the lower-mass massive stars lead to ear-lier explosions. However, the associated steeply decreasing rates of accretion also result in lessmass in the gain region and a lower optical depth to the emerging neutrino fluxes. The lower ab-sorption depths in the mantle times the lower neutrino luminosities lead to lower driving powersin the exploding mantle. This results in lower explosion energies generically under the neutrinoheating paradigm for those stars with steep outer density profiles. Conversely, those stars withshallow density profiles, more often the more massive CCSN progenitors, generally explode later.However, their shallow mass profiles result in more mass in the gain region with a greater opticaldepth. The larger depth times the larger accretion luminosities lead to greater driving neutrinopower deposition. The net effect is often higher asymptotic supernova explosion energies. Hence,8ith exceptions, state-of-the-art models suggest that the explosion energy is an increasing functionof progenitor mass and the shallowness of the outer density profile of the initial core. In addition,“explodability” does not seem to be a function of “compactness”
4, 64, 65 (a measure of the ratio ofprogenitor interior mass to radius), with both high and low compactness models exploding. It hadbeen suggested that only low-compactness structures exploded. Not only does this not seem to betrue, but it seems that only the higher compactness models can result in explosion energies near thecanonical one Bethe. It may be, however, that very high compactness structures have outer mantlebinding energies for which the neutrino mechanism can’t provide sufficient driving power. Theseobjects may lead either to weak explosions or fizzles, with many of these leading to black holes(and not neutron stars). In fact, the gravitational binding energy of the mantle of the Chandrasekharcore may set the scale of the explosion energy, and if too high might thwart explosion altogether.This topic deserves much more attention.
Two-dimensional (axisymmetric) and three-dimensional simulations do not behave the same. Theaxial constraint and artificial turbulent cascade of the former compromise the interpretation of theresults. However, 2D simulations do allow the breaking of important symmetries and overturningmotions and are less expensive to perform. Importantly, due to their much lower cost, 2D numeri-cal runs can easily be carried out to many seconds after bounce, something that we and others havefound is required for many stars to asymptote to final blast kinetic energies in the context of theneutrino mechanism . So, in order to get a bird’s-eye view of some of the systematic behaviorwith progenitor mass, we have conducted for this paper a suite of longer-term 2D models using thestellar models of Sukhbold et al. as starting points. For this collection, we have found usuallythat when a 2D model explodes, its more realistic 3D counterpart does as well, and when it doesn’tneither does the 3D simulation. In our experience, this is usually, but not always, the case, thoughthere is some disagreement on this in the literature
15, 18, 56, 67–71 .In our recent set of models, it is onlythe 12-M (cid:12) and 15-M (cid:12) models that do not explode. The 2D models generally seem to explode a bitearlier than the 3D models. For instance, the 20-M (cid:12) and 25-M (cid:12) stars explode ∼
100 ms and ∼ ∼
150 and ∼
400 ms after bounce for all these 2D explod-ing models. This timescale depends, no doubt, upon simulation details (microphysics, resolution,algorithms, etc.), as well as the character of the seed perturbations. For these simulations, we didnot impose extra perturbations and left the inauguration of the initial overturning instabilities tonumerical noise.Figure 4 portrays the development of the mean shock radius for all the models of this study.The left-hand panel shows the launch phase, while the right-hand panel provides a later, larger-scale glimpse. The mean shock speeds settle between 10,000 km s − and 15,000 km s − . Table1 lists the explosion energy, baryonic and gravitational masses, and post-bounce run time. Theenergies have asymptoted to within a few tens of percent for all models of their final supernova9nergies and range from 0.09 to 2.3 Bethes. The 24-M (cid:12) model is being further scrutinized and isnot included here. The higher energies are statistically, but not monotonically, associated with moremassive progenitors. The growth of the blast energy is depicted in Figure 5. Those models thatasymptote early do so at lower energies. Those models that eventually achieve higher explosionenergies not only do so later, but experience deeper negative energies for a longer time beforeemerging into positive territory. As described in §2, this is what is expected for models withmassive (shallow) density mantles, if they explode, and these are generally, though not exclusively,for the most massive progenitors ( > (cid:12) ?).The energies shown in Table 1 and Figure 5 include the gravitational, thermal, and kineticenergies, as well as the nuclear reassociation energies, of the ejecta. They also include the outermantle binding energies of the as yet unshocked material. In this way, all the components ofthe blast energy are accounted for, except the thermonuclear term. The latter could be as muchas ∼
10% of the total, and will slightly increase our numbers. However, 0.1 M (cid:12) of oxygen pro-vides only ∼ ∼ Ni yieldswould be higher for the more densely mantled stars, so that the thermonuclear energy contributionand Ni yield would be correlated with one another and with progenitor mass
72, 73 . Curiously, ifthe speculations concerning the possible thermonuclear character of the lowest-mass progeni-tors bear out (though see ), this correlation might be preserved, though for the other end of themassive-star mass distribution (“mass function”). Note that the mass function is weighted towardsthe lower masses.Figure 6 superposes the theoretical explosion energies of Table 1 onto a plot of the obser-vationally inferred Type IIp (plateau) supernova energies versus inferred ejecta masses. For ourtheory numbers, we shift the initial progenitor mass by 1.6 M (cid:12) to account for an average residualneutron star. In so doing, we do not account for the pre-explosion mass loss of the star, which couldbe significant. However, the general trend of the inferred energy with a measure of stellar massis reproduced by the theoretical (black) dots. There is scatter in both the theory and observations,the latter due to systematic uncertainties in the models employed and observational limitations,and the former due to numerical and astrophysical uncertainties. However, natural chaos in thedynamics would naturally lead to a spread in energies (§1, §3), to a degree as yet unknown, evenfor the same initial stellar structure. We note that there seems to be a larger observational spreadin the inferred energies at lower masses. This could reflect natural chaos in the turbulent neutrinomechanism, measurement uncertainties, the effects of unknown rotation, or the possibility that thelowest-mass progenitors explode thermonuclearly just after the onset of a collapse that does notachieve nuclear densities. However, it is too soon to draw any definitive conclusions on this score.Be that as it may, the observed very roughly monotonically increasing trend of explosion energywith mass and the ability of the neutrino mechanism to reproduce the observed range of explosionenergies are both encouraging. 10inally, the infalling accretion matter plumes that hit the PNS core generate sound waves thatare launched outward. Much of the energy of these sound waves is absorbed behind the shock waveand can modestly contribute to the explosion energy. Such a component is automatically includein our bookkeeping. Though difficult to estimate separately, we don’t envision that acoustic powercan contribute more than ∼ −
10% to the total.
Figure 7 depicts the evolution of the residual baryon mass of the PNS core for the suite of 2Dmodels investigated here. Such masses flatten early, since the mass accretion rates drop quicklyafter the explosion commences. The final baryon masses at the last timesteps are given in Table 1,as are the corresponding gravitational masses. The latter include the gravitational binding energy(negative) of the core. These masses range from a low near ∼ (cid:12) to a high near ∼ (cid:12) ,nicely spanning the observed range . The neutron star masses we find are closely, but not perfectly,monotonic with progenitor mass and the shallowness of the Chandrasekhar mantle, except for thosemodels that don’t explode. Presumably, these models will eventually collapse to black holes, buton timescales longer than we have simulated. The issue of the ejecta elemental composition is fundamental to supernova theory. The shallownessof the outer mantle density profile and the associated mass of the inner ejecta are roughly corre-lated with the yields of oxygen and intermediate-mass (e.g., Ar, Si, Ca) elements. As suggestedin §3, such a structure is also likely to explode (if via the neutrino mechanism) with higher ener-gies. Therefore, more of this inner ejecta will be able to achieve the higher temperatures that cantransform oxygen and silicon into iron-peak species as well. This includes Ni. Therefore, oneexpects that in the context of the neutrino mechanism of explosion Ni yields are roughly increas-ing functions of progenitor mass, with the exceptions to strict monotonicity alluded to previously.Specifically, if a 9-M (cid:12) star explodes by the neutrino mechanism it can not have much Ni in itsejecta and if a ∼ −
25 M (cid:12) star explodes by the same mechanism the Ni yield should be moresignificant.All the inner ejecta from the region interior to the stalled shock wave, before and just afterexplosion, are very neutron-rich ( Y e ∼ . − . ). As they expand outward, absorption by ν e and ¯ ν e neutrinos on balance tends to push the ejecta Y e upward. If the expansion is fast, then some ofthe ejecta can freeze out slightly neutron-rich below Y e = 0 . . However, if the expansion is slow,there is plenty of time for some of the debris to become proton-rich (Y e > . ). However, generallyY e = 0.5 seems to predominate in the bulk. Therefore, those models that explode early and fastshould provide some neutron-rich ejecta, though more of their ejecta could still be proton-rich,while those models that explode later and more slowly (generally, the more massive progenitors)11ill be the most proton-rich. This is what we see, where Y e s from ∼ ∼ Se, Kr, and Sr) . However, these numbers should be viewed as preliminary, depending asthey do on detailed neutrino transport and the complicated trajectory histories of the ejecta parcels.We note that observations of Ni in SN1987A, inferred to be a ∼
18 M (cid:12) progenitor, require thatno material with Y e s lower than 0.497 could have been ejected . Also, none of the ejecta seen inmodern simulations can be the site of all the r-process, though the first peak is not excluded. Thetimescales and Y e s are not at all conducive.Furthermore, as stated, inner supernova matter explodes quite aspherically, with bubble,botryoidal, and fractured structures predominating. However, the spatial distribution of Y e in theejecta can have a roughly dipolar component, with one hemisphere more proton-rich than its coun-terpart. Figure 8 depicts a snapshot of a simulation of a 19-M (cid:12) model. The bluish veil is the shock,while the fractured surface is an isoentropy surface painted by Y e . As seen, there is an orange-purple dichotomy which reflects the fact that the ejecta have a dipole in Y e that persists. Even aninitially uniform ejecta Y e distribution may be unstable to the establishment of such a dipole. Ifnear and exterior to the ν e neutrinosphere at the “surface” of the PNS a perturbation in Y e arisesin a given angular patch of the inner ejecta, that perturbation can grow due the concommitant ef-fect on the absorptive opacity at those angles, which in turn will either suppress or enhance the ν e emissions to push the Y e evolution of that matter parcel in the same direction. The progressivediminution of this absorptive Y e shift effect with distance can freeze the Y e perturbation. Theupshot is then a crudely dipolar distribution in Y e that tracks a crudely dipolar angular distributionin the ν e and ¯ ν e luminosities and the so-called LESA (“Lepton Emission Sustained Asymmetry”)phenomenon
18, 57, 79, 80 . Whether this dipolar asymmetry in Y e in the ejecta is a generic outcomeremains to be seen. The neutron stars born as proto-neutron stars in the supernova cauldron are the source of the radiopulsars known to be darting throughout the galaxy with speeds that average ∼
350 km s −
81, 82 andcan range up to ∼ − . The most natural explanation for these galactic motions is recoilsduring the supernova explosion directly related to asymmetric matter ejection and/or asymmet-ric neutrino emission. Hence, momentum conservation in the context of at times very asphericalejection can easily yield the observed speeds. Moreover, it is known that neutrino emissions canhave a dipolar component and that the associated net momentum can be large. Neutrinos travel atvery, very close to the speed of light and constitute in sum as much as 0.15 M (cid:12) c of mass-energy.Therefore, a mere 1% asymmetry in angle can translate into a kick of ∼
300 km s − . However, itis not known how the ejecta and neutrino momentum vectors sum, in particular whether they addor subtract and what the integrated magnitude of the latter is.Nevertheless, one can speculate about the trends with progenitor star of the magnitude of12he kicks experienced . We have seen that the lowest-mass massive stars tend to explode a bitmore spherically, eject less core mass, and emit less energy in neutrinos. The radiated bindingenergies of the PNS are lower, given the lower accretion rates and lower PNS mass. Hence, weexpect the kicks to be smaller for the lower mass progenitors. Conversely, the more massiveprogenitors tend to explode a bit more aspherically, ejecting more core mass and emitting moremass-energy in neutrinos. Hence, we posit that they produce neutron stars with the greatest kickspeeds. There is likely to be some noise in these suggestions, but on average these trends withprogenitor mass (actually progenitor structure; see Figure 1) are compelling in the context of theneutrino mechanism of CCSN explosions. Moreover, one would predict that stellar-mass blackholes born in the context of core collapse would have low kick speeds, since they are generallyexpected to have much more inertia/mass than neutron stars and the momentum in any matterejecta their birth may entail should be smaller. However, the neutrino kicks may be as significantas for neutron-star birth; therefore, the momentum in any such black hole birth kick might becomparable. Stars have angular momentum and spin. As they evolve, the angular momentum is redistributedinternally (likely by magnetic torques) and lost in winds. It is not known what the internal birth spindistribution of massive stars is, but crude theoretical calculations suggest that angular momentumis gradually transported out of their cores as they evolve , much of it lost to stellar winds. Instars for which the internal spin rates can be measured due to observed rotational splitting ofsurface pulsational modes, models of the interior spin evolution leave their interiors rotating tentimes too fast . Therefore, the theory of angular momentum redistribution is incomplete. Inaddition, the cores of massive stars shrink and spin up. So, the spin of a Chandrasekhar core justbefore collapse is a product of the initial angular momentum distribution, wind angular momentumloss, redistribution torquing during evolution, and progressive evolutionary compression of thecore. Furthermore, the spin of the collapsing core can be affected by the stochastic shedding ofhydrodynamic waves generated during oxygen and silicon core convection just before collapse .Without many observational constraints, the final core spins before core collapse are unknown.However, radio pulsars have average surface dipole magnetic fields of ∼ gauss and areobserved on average to be rotating slowly, with average periods of ∼
500 milliseconds . Aneutron star needs to be spinning with a period of ∼ ∼ ergs, so periods of ∼
500 ms imply rotational kinetic energies that are four ordersof magnitude below supernova energies and these are not dynamically important. Nevertheless,the birth spin of a neutron star is an important observable and predicting this number should be agoal of theory.Even given the spin rate of the Chandrasekhar white dwarf that collapses, one can’t easilypredict the birth spin of the neutron star that eventually emerges. Assuming angular momentum13onservation, collapsing from this initial configuration to a neutron star spins the residue up byapproximately a factor of one thousand. So, a ∼ ∼ . Very large fields in the magnetar range (10 gauss) can spin down the nascent PNS on timescales of seconds.Furthermore, after bounce the accreting PNS can be spun up by accreting matter plumesstochastically, with a jumble of streams with both positive and negative angular momenta thatdon’t necessarily cancel. This can lead after the mass cut between the nascent PNS and the ejectato a spinning neutron star, even if the initial star was non-rotating. Figure 9 depicts emergentrotation at later times. The possibility that a previously non-rotating core could be left rotating hasbeen vigorously studied , with a range of spin periods predicted by this process alone from asecond or two to tens of milliseconds.Be that as it may, one would like to determine whether the final neutron star spin is pre-dictable. To date, we don’t know. However, if the total stellar mass and angular momentum lossare determining factors, one would expect neutron stars born in low-metallicity (low abundance ofnon-H/He elements) massive stars to be faster rotators, all else being equal. Moreover, if angulartransfer from the massive star core to its mantle is a continuous process, since the more massivestars evolve more quickly they are likely to leave cores with more angular momentum, again allelse being equal. So, lower-metallicity, more-massive massive stars would birth neutron stars withfaster spins, but, as implied here, there are still too much uncertainties and too many effectors todraw reliable conclusions. The prejudice of many researchers is that the frozen-in magnetic flux of the unstable Chandrasekharcore determines the neutron-star fields. This can’t be correct. As stated in §7, turbulence behind theshock and in the PNS itself are natural venues for dynamo growth. At the very least, such violentconvective motions will advect and tangle any initial seed fields post-bounce and the multipolaritystructure will be radically altered. Even without exponential dynamos, rotation will wind up aninitial field and the toroidal and poloidal components will evolve significantly. For large fields near ∼ gauss, the field can act back on the rotational profile significantly. Therefore, it is not at allclear what the origin of radio pulsar and magnetar B-fields is, nor what the systematic dependenceof these fields might be on progenitor characteristics; clearly, this is a rich and important topic forfuture research and there have been numerous recent papers attacking aspects of it
10, 41, 103–105 .A small subset ( ∼
1% ?) of core-collapse supernova are so-called “hypernovae” that seem to14e missing links with long-soft gamma-ray bursts (GRBs) . Their inferred explosion energiesare near ∼ ergs ( ∼
10 Bethes) and seem to be too energetic to be powered mostly by theneutrino mechanism. The best explanation is that these are powered by MHD jets that tap the largespin kinetic energy of fast-spinning (few-millisecond) proto-neutron stars. Such fast rotators mayexperience strong dynamo action that can achieve magnetar fields. Hence, the natural consequenceof rapid rotation may also be large B-fields, that together naturally lead to strong jets
26, 27 that candrive quasi-dipolar outflows. As very tentatively suggested in §7, the progenitors for hypernovaemay therefore be the more massive massive stars with low-metallicity, and a subset of them mayyield gamma-ray bursts. If the latter is the case, a transition within seconds to a black hole asthe maximum mass of a neutron star is reached is indicated , since relativistic jets are the bestexplanation for such GRBs and only black holes seem able to produce them. Therefore, this class ofGRBs would have a non-relativistic jet precursor lasting a few seconds that could be more energeticthan the relativistic jet that follows and would eventually overtake the former as it blasted out of theprogenitor star. This non-relativistic/energetic to relativistic/less-energetic phasing of jets has yetto be observed in either the hypernova or the GRB context, but is suggested by emerging theory.In any case, even if the neutrino mechanism described in this paper predominates amongCCSNe, the residual neutron star will likely be rotating, however fast, and will have a magneticfield with a dipole contribution. After the neutrino mechanism has cleared out the inner cavityaround the nascent spinning/magnetic neutron star, this object will be able to transfer power viaweak jets or pulsar action to the inner debris. This may constitute a mere 0.01% to 1% of thetotal supernova energy. Eventually, its effects will be manifest in the blast remnant, but perhapsas a sub-dominant, under-energetic phenomenon. So, for the canonical case, a standard neutrino-driven explosion could be followed by a weaker magnetically-driven secondary effect most ofthe time
26, 73 . Signatures of this sort need to be sought, but the supernova remnant Cas A, withits sub-dominant jet/counter-jet structure , may be such a structure. When the initial core israpidly rotating (we think in a small subset of cases), this magnetically-driven component mightovertake in energy the neutrino component. There may also be intermediate cases for which thetwo mechanisms compete. Such a continuum from weak to strong effects of magnetic fields is anintriguing possibility.
If and when the PNS mass exceeds the maximum gravitational mass of a neutron star (with suitablesmall thermal and compositional corrections), it will collapse to a black hole and continue toaccrete. This maximum mass is in the range of 2.1 to 2.4 M (cid:12) gravitational, about 2.4 to 2.7 M (cid:12) baryonic, and depends upon the only modestly-constrained nuclear equation of state. How muchmass is subsequently accreted depends on how much of the progenitor star is ejected. If none ofthe star is ejected, and 1) most of the progenitors of such “stellar-mass” black holes are the moremassive massive stars with high envelope binding energies (§2) and 2) these have experiencedsignificant pre-collapse wind and/or episodic mass loss, then one would expect the canonical mass15f the product black hole to be ∼ (cid:12) . This is the helium core mass of those model starsthat have very high envelope binding energies exterior to the Chandrasekhar core. However, we had earlier witnessed that stars with initial masses in the 13 −
15 M (cid:12) range didn’t explode. Thisresult could easily be model-dependent and is not the final word on what such stars do. But it ispossible that the black-hole outcome is peppered about the massive-star mass function
64, 72, 110–112 .However, the consensus is that most massive stars with initial masses less than ∼
20 M (cid:12) will leadto neutron stars and most stars with initial masses greater than ∼
30 M (cid:12) should lead to black holes.If most stellar-mass black holes birthed via collapse have masses in the range of ∼
10 M (cid:12) and neutron stars have a maximum mass a bit above ∼ (cid:12) , then there would be a “mass gap”between them. Such a gap is suggested by the data, but is not proven . It may be that a shockis relaunched, but has insufficient energy to eject enough of the inner mass and that will then fallback, still launching an explosion wave that unbinds the rest of the stellar envelope. Where thismass cut occurs would determine the birth mass of such a black hole. A “fallback” black hole isa distinct possibility , but may be a small subset of the massive-star mass function. It is likelythat most collapses lead to neutron stars, but what the neutron-star/black-hole birth ratio is for thepopulation of massive stars is a subject of much current research.Finally, there are a few points of principle that need to be articulated. The first is that inthe context of the collapse of a Chandrasekhar core, it is impossible to collapse directly to a blackhole − there must always be a proto-neutron-star intermediary. This is because the bouncing innercore is out of sonic contact with the outer infalling core. At bounce, the object does not knowthat it will eventually exceed the maximum mass. This means that even when a black hole is thefinal outcome, the PNS core will always have a significant neutrino and gravitational-wave signature. A signature of the subsequent dynamical collapse to a black hole will be the abruptcessation of both signals . Second, given that neutrino energy losses in the range ∼ (cid:12) c are “inevitable,” the outer stellar envelope will experience a decrease in the gravitational potentialit feels. This will lead to its readjustment on dynamical timescales and the likely the ejectionof matter to infinity . Hence, there should always be some sort of explosion, even when ablack hole forms. Whether it is such a “potential-shift” explosion, one with significant fallback,or one via a disk jet after the black hole and accretion disk form, it is difficult to imagine a purelyquiescent black hole birth.
10 Final Thoughts
As should now be clear, from the vantage of theory, a multitude of effects are of importance indetermining the viability, character, and strength of a CCSN explosion. The roles of the initial pro-genitor structure; multi-dimensional neutrino radiation transport; general relativity; instabilities,turbulence, and chaos; the nuclear interaction and equation of state; neutrino-matter processes andmany-body effects; resolution and numerical technique; rotation; and B-fields must all be assessedon the road to a resolution of this complex problem. It is this complexity that has paced progress16n this multi-physics, multi-dimensional, and multi-decade puzzle. However, modern theory hasgrappled with all these issues and inputs, with the result that state-of-the-art simulations from manygroups evince explosions via the neutrino mechanism with roughly the correct general characterand properties. Not all researchers agree on the details. nor do they obtain precisely the same re-sults. Nevertheless, to zeroth-order, the neutrino mechanism seems to work, and one is, therefore,tempted to declare that the overall problem of the mechanism of supernova explosions is solved,the rest being details. However, these details include the credible mapping of progenitor mass andproperties to important observables, such as explosion energy, neutron-star mass, nucleosynthesis,morphology, pulsar kicks and spins, and B-field magnitudes and multipolarities. Chaos will com-plicate all this, as will remaining uncertainties in microphysics and numerics. Yet, despite this, weare confident that core-collapse supernova theory in the year 2020 has reached a milestone, fromwhich it need never look back. 17able 1:
Explosion Energies and Neutron-Star Masses.
The 9-, 10-, and 11-M (cid:12) progenitors arefrom the Sukhbold et al. (2016) suite and were evolved on spherical grids with radial extentsof 30,000, 50,000, and 80,000 kilometers (kms), respectively. Progenitors from 12 to 26.99-M (cid:12) were inherited from the Sukhbold et al. (2018) suite. The 12-, 13-, and 14-M (cid:12) progenitors wereevolved on a spherical grid spanning 80,000 km in radius. All other progenitors were evolved onspherical grids spanning 100,000 km in radius. All models were evolved in 2D axisymmetry with1024 radial cells and 128 ( θ ) angular cells. Thus, there are some small differences in resolutionfor the lower-mass progenitors, where the progenitor grid is truncated at smaller radii so that thetemperature would remain within our equation-of-state table. All models except the 12- and 15-M (cid:12) progenitors explode. The Run Time quoted is the time after bounce at nuclear densities. Model Explosion Energy Run Time Baryonic Mass Gravitational Mass [M (cid:12) ] [B] [s] [M (cid:12) ] [M (cid:12) ] -0.03 2.75 1.82 1.62 -0.17 1.04 1.93 1.71 .0 0.5 1.0 1.5 2.0 2.5 M [M fl ] ρ [ g c m − ] fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl Figure 1:
Progenitor mass density profiles.
Plotted are the mass density (in g cm − ) versusinterior mass (in M (cid:12) ) profiles of the cores of the progenitor massive stars used as initial conditionsfor the supernova simulations we highlight in this paper. The associated spherical stellar evolutionmodels were calculated by Sukhbold et al.
63, 72 up to the point of core collapse, at which pointthey were mapped into our supernova code F
ORNAX .19igure 2: Inner matter trajectories as the explosion is about to launch.
Shown are the interiorsof an explosion only ∼
150 milliseconds after core bounce (vertical physical scale ∼
350 kilome-ters). At this time the shock wave is at ∼
150 kilometers, just before explosion. The inner ball isthe newly-birthed proto-neutron star (PNS) (rendered as an isodensity surface at 10 g cm − , col-ored by Y e ), surrounded by swirling, turbulent matter, most of which will settle onto the PNS. Thetrajectories depict the recent 5 milliseconds in the positions of individual accreted matter elements.They are colored by local entropy. The turbulence of this inner region is manifest.20igure 3: Early 3D explosion of the core of a 16-M (cid:12) star using F
ORNAX . Portrayed is a stillnear 500 milliseconds after core bounce at nuclear densities. The red is a volume rendering of thehigh-entropy of the ejecta in the neutrino-heated bubbles that constitute the bulk of the volume ofthe exploding material. The green surface is an isoentropy surface near the leading edge of theblast, the supernova shock wave. Note the asymmetric, though roughly dipolar, character of theexplosion and the pinched “wasp-waist” structure of the flow between the lobes. The dot at thecenter in the newly born neutron star. In this model, as in many others, there is clearly simultaneousaccretion at the waist, while there is ejection in the wide-angle lobes. Simultaneous accretion in onesector during concommitant explosion elsewhere maintains the driving neutrino luminosity and isa signature of the useful breaking of spherical symmetry possible in multi-dimensional flow. Thiscontrasts sharply with the artificially enforced situation in 1D/spherical simulations. Simulationperformed by the Princeton supernova group . 21 .0 0.1 0.2 0.3 0.4 0.5Time after bounce [s]0200400600800100012001400 M e a n S h o c k R a d i u s [ k m ] fl
10 M fl
11 M fl
12 M fl
13 M fl
14 M fl
15 M fl
16 M fl
17 M fl
18 M fl
19 M fl
20 M fl
21 M fl
22 M fl
23 M fl
24 M fl
25 M fl
26 M fl fl M e a n S h o c k R a d i u s [ k m ] fl
10 M fl
11 M fl
12 M fl
13 M fl
14 M fl
15 M fl
16 M fl
17 M fl
18 M fl
19 M fl
20 M fl
21 M fl
22 M fl
23 M fl
24 M fl
25 M fl
26 M fl fl Figure 4:
Mean shock radii of 2D models.
Depicted are the angle-averaged shock radii (inkilometers) of the 2D model suite calculated for this paper versus time (in seconds) after bounce.Most of the models explode, while the 12- and 15-M (cid:12) progenitor structures do not. The top panelshows the behavior during the first half second after bounce and in the inner 1500 kilometers,with models exploding (when they do) between t = 0 . and t = 0 . seconds. The bottom panelportrays the shock motion on a larger physical scale (15000 kilometers) and to latter times. Manyof the models were, in fact, carried to ∼ − for most of the simulation. Themodels were conducted on grids from 30,000 to 100,000 kilometers, with the smaller values forthe smaller-mass progenitors. See the caption to Table 1 for specifics.22 E x p l o s i o n E n e r g y [ B ] fl
10 M fl
11 M fl
12 M fl
13 M fl
14 M fl
15 M fl
16 M fl
17 M fl
18 M fl
19 M fl
20 M fl
21 M fl
22 M fl
23 M fl
25 M fl
26 M fl fl Figure 5:
The evolution of the total explosion energy (in Bethes) with time (in seconds afterbounce).
As the figure indicates, many models start bound (negative energies), even though theirshocks have been launched. It can take more than one second for some to achieve positive energies,the true signature of an explosion. Moreover, as shown on this figure, it can take ∼ − (cid:12) and 15-M (cid:12) stars in this investigation do not explode.23
10 15 20 25 300123
Figure 6:
Comparison of theoretical and empirical of explosion energy versus ejecta mass.
Plotted are the empirically inferred explosion energies versus the inferred ejecta masses, with errorbars, for a collection of observed Type IIp (plateau) supernovae. Our theoretical numbers, takenfrom Table 1, are superposed as black dots. It must be recalled that these are 2D models, and thatthere are quantitative differences between 2D and 3D simulations. We assume for conveniencethat the theoretical ejecta masses are the progenitor masses, minus the baryon mass of a putativeresidual neutron star of 1.6 M (cid:12) . This ignores any mass loss prior to explosion, surely an incorrectassumption by ∼ − (cid:12) . Nevertheless, the rough correspondence between theory and measure-ment is quite encouraging. Note that the error bars on the measurements are not firm, and do notinclude any systematic errors in the light-curve modeling procedures. In any case, the general av-erage trend from low to high explosion energy from lower to higher massive-star progenitor massreflected in the observations is reproduced rather well by the theory, both quantitatively and quali-tatively. Note also that at a given mass there is an inferred measured spread in supernova energies.This may represent a real variation in explosion energy at a given progenitor mass due in part to thenatural chaos in turbulent flow. Indeed, it is theoretically expected that Nature would map a givenstar’s properties to distribution functions in the outcomes and products of its supernova death. Theempirical estimates were taken from Morozova et al. , Martinez & Bersten , Pumo et al. ,and Utrobin & Chugai . 24 P N S M a ss [ M fl ] fl
10 M fl
11 M fl
12 M fl
13 M fl
14 M fl
15 M fl
16 M fl
17 M fl
18 M fl
19 M fl
20 M fl
21 M fl
22 M fl
23 M fl
24 M fl
25 M fl
26 M fl fl Figure 7:
Theoretical baryon mass (in M (cid:12) ) of the residual neutron star versus time afterbounce (in seconds) for the 2D models of this study.
The evolution of the residual neutron starmass is generally rather quick, with the final mass determined to within ∼
5% generally (thoughnot universally) within ∼ one second of bounce. The range of residual masses ranges from ∼ (cid:12) to 2.2 M (cid:12) for this model set. This is equivalent to a range of neutron-star gravitational massesbetween ∼ (cid:12) and ∼ (cid:12) , roughly what is empirically seen. Generally, the lower-massprogenitors give birth to lower-mass neutron stars, though this is not rigorously monotonic. Notethat the 12-M (cid:12) and 15-M (cid:12) models that don’t explode are still gradually increasing their residualmasses by the end of those simulations (see Table 1).25igure 8:
3D Explosion structure of a representative massive star progenitor model.
Theassociated simulation was performed with a 678 (radius) ×
256 ( θ ) ×
512 ( φ ) grid to render slightlyfiner details. The snapshot is taken ∼
800 milliseconds after core bounce, about 500 millisecondsinto the explosion. The blue-gray veil is the shock wave. The colored isosurface is of constantentropy, colored with the electron fraction, Y e . We note that there is a large region of purple (higherY e , more proton-rich) matter on one side and a largish region of orange-yellow (relatively lower Y e ,less proton-rich) matter on the other. This global Y e asymmetry is created by persistent angularasymmetries in the electron-neutrino and anti-electron-neutrino emission from the core duringexplosion, which by absorption in the ejecta create this asymmetry in the electron fraction of theejecta. The latter translates into an asymmetry in the nucleosynthetic element angular distribution.Results derived from a simulation done by the Princeton supernova group .26igure 9: Similar to Figure 2, but highlighting the generic swirling motions just exterior to thePNS a few hundred milliseconds after explosion.
The physical scale top to bottom is ∼
200 km.Upon accretion into this inner region, the matter blobs can stream to one side or another of the corebefore finally settling onto it. This stochastic, almost random, accretion of angular momentumcan sum over time to leave a net angular momentum and spin, despite the fact that the originalprogenitor model was non-rotating . 27. Chandrasekhar, S.
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The authors thank Joe Insley and Silvio Rizzi of the Argonne National Laboratoryand the Argonne Leadership Computing Facility (ALCF) for their considerable support with the 3D graph-ics. The authors acknowledge ongoing collaborations with Hiroki Nagakura, Davide Radice, Josh Dolence,Aaron Skinner, and Matt Coleman. They also acknowledge Evan O’Connor regarding the equation of state,Gabriel Mart´ınez-Pinedo concerning electron capture on heavy nuclei, Tug Sukhbold and Stan Woosley forproviding details concerning the initial models, and Todd Thompson and Tianshu Wang regarding inelasticscattering. Funding was provided by the U.S. Department of Energy Office of Science and the Office ofAdvanced Scientific Computing Research via the Scientific Discovery through Advanced Computing (Sci-DAC4) program and Grant DE-SC0018297 (subaward 00009650) and by the U.S. NSF under Grants AST-1714267 and PHY-1804048 (the latter via the Max-Planck/Princeton Center (MPPC) for Plasma Physics).Awards of computer time were provided by the INCITE program using resources of the ALCF, which is aDOE Office of Science User Facility supported under Contract DE-AC02-06CH11357, under a Blue Waterssustained-petascale computing project, supported by the National Science Foundation (awards OCI-0725070and ACI-1238993) and the state of Illinois, under a PRAC allocation from the National Science Foundation(
Author contributions
A.B. organized the paper and wrote most of it. D.V. conducted the 2D calculations.Otherwise, the authors contributed equally to the document.
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