Core-collapse supernovae and gravitational wavesC.Y. Cardallab∗

aPhysics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6354bDepartment of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996-1200

Core-collapse supernovae are dramatic events with a rich phenomenology, including gravitational radiation.Simulations of these events in multiple spatial dimensions with energy- and angle-dependent neutrino transportare still in their infancy. Core collapse and bounce in a supernova in our galaxy may well be visible by first-generation LIGO, but detailed understanding waits on improvements in modeling both stellar progenitors andthe supernova process.

1. CORE-COLLAPSE SUPERNOVAE

Core-collapse supernovae—those of Type Ib,Ic, and II—result from the catastrophic collapseof the core of a massive star. Depending on theproperties of the progenitor star, the collapse mayresult in a black hole; alternatively, the remnantcan be a neutron star with GM/R ∼ 0.1c2, whereG is the gravitational constant, M and R are theneutron star mass and radius, and c is the speedof light. Such a system is manifestly relativis-tic; and, if the high densities and infall velocitiesimplied by the above relation are combined withsufficient asphericity, the violence of core collapseand its aftermath may be expected to producesignificant gravitational radiation. (Type Ia su-pernovae, which are caused by the thermonucleardetonation or deflagration of a white dwarf star,are not expected to be interesting sources of grav-itational radiation.)

I begin by outlining our current understanding(and lack of understanding) of the core-collapsesupernova process.

For most of their existence stars burn hydrogeninto helium. In stars at least eight times as mas-sive as the Sun, temperatures and densities be-come sufficiently high to burn to carbon, oxygen,

∗This work was supportedby ScientificDiscovery ThroughAdvanced Computing (SciDAC), a program of the Officeof Science of the U.S. Department of Energy (DoE); and byOak Ridge National Laboratory, managed by UT-Battelle,LLC, for the DoE under contract DE-AC05-00OR22725.

neon, magnesium, and silicon and iron group ele-ments. The iron group nuclei are the most tightlybound, and here burning in the core ceases.

The iron core—supported by electron degen-eracy pressure—eventually becomes unstable. Itsinner portion undergoes homologous collapse (ve-locity proportional to radius), and the outer por-tion collapses supersonically. Electron capture onnuclei is one instability leading to collapse, andthis process continues throughout collapse, pro-ducing neutrinos. These neutrinos escape freelyuntil densities and temperatures in the collaps-ing core become so high that even neutrinos aretrapped.

Collapse is halted soon after the matter exceedsnuclear density. At this point (called “bounce”),a shock wave forms at the boundary betweenthe homologous and supersonically collapsing re-gions. The shock begins to move out, but afterthe shock passes some distance beyond the sur-face of the newly born neutron star, it stalls asenergy is lost to neutrino emission and dissocia-tion of heavy nuclei falling through the shock.

The details of how the stalled shock is re-vived sufficiently to continue plowing through theouter layers of the progenitor star are unclear.Some combination of neutrino heating of mate-rial behind the shock, convection, instability ofthe spherical accretion shock, rotation, and mag-netic fields launches the explosion.

It is natural to consider neutrino heating as a

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mechanism for shock revival, because neutrinosdominate the energetics of the post-bounce evo-lution. Initially, the nascent neutron star is ahot thermal bath of dense nuclear matter, elec-tron/positron pairs, photons, and neutrinos, con-taining most of the gravitational potential energyreleased during core collapse. Neutrinos, havingthe weakest interactions, are the most efficientmeans of cooling. They diffuse outward on atime scale of seconds, and eventually escape withabout 99% of the released gravitational energy.

Because neutrinos dominate the energetics ofthe system, a detailed understanding of theirevolution will be integral to any detailed anddefinitive account of the supernova process. Ifwe want to understand the explosion—which ac-counts for only about 1% of the energy budget ofthe system—we should carefully account for theneutrinos’ much larger contribution to the energybudget.

What sort of computation is needed to followthe neutrinos’ evolution? Deep inside the newly-born neutron star, the neutrinos and the fluidare tightly coupled (nearly in equilibrium), butas the neutrinos are transported from inside theneutron star, they go from a nearly isotropic dif-fusive regime to a strongly forward-peaked free-streaming region. Heating of material behindthe shock occurs precisely in this transition re-gion, and modeling this process requires accu-rately tracking both the energy and angle depen-dence of the neutrino distribution functions at ev-ery point in space.

While a full treatment of this six-dimensionalneutrino radiation hydrodynamics problem re-mains too costly for currently available compu-tational resources, there is much that has beenlearned over the years through detailed modeling.

2. SIMULATING THE EXPLOSION

Supernovae have a rich phenomenology—observations of many types that modelers wouldlike to reproduce and explain. Chief among theseis the explosion itself, which has not yet been pro-duced robustly and convincingly in simulations.Other observables of interest include neutrino sig-natures; neutron star spins, kick velocities, and

magnetic fields; synthesized element abundances;all kinds of measurements across the electromag-netic spectrum; and of course the subject of thisconference session, gravitational waves.

Simulations of core collapse, bounce, and itsimmediate aftermath have mostly aimed at thefirst few of these observables: the explosion mech-anism, neutrino signatures, remnant pulsar prop-erties, and gravitational waves. I will now de-scribe some of the progress in this work in thepast decade or so, focusing in particular on theexplosion mechanism.

Throughout the 1990s, several groups per-formed simulations in two spatial dimensions.Even in two spatial dimensions, computationallimitations required approximations that simpli-fied the neutrino transport.

One simplification allowed for neutrino trans-port in two spatial dimensions, but with neutrinoenergy and angle dependence integrated out—effectively reducing a five dimensional problem toa two dimensional one (see for example [1,2]).

These simulations exhibited explosions, sug-gesting that the enhancements in neutrino heat-ing behind the shock resulting from convectionprovided a robust explosion mechanism. Morerecent simulations in three spatial dimensionswith this same approximate treatment of neutrinotransport showed similar outcomes [3].

A different simplification of neutrino trans-port employed in the 1990s was the impositionof energy-dependent neutrino distributions fromspherically symmetric simulations onto fluid dy-namics computations in two spatial dimensions[4]. Unlike the multidimensional simulations dis-cussed above, these did not exhibit explosions,casting doubt upon claims that convection-aidedneutrino heating constituted a robust explosionmechanism.

The nagging qualitative difference betweenmultidimensional simulations with different neu-trino transport approximations renewed the mo-tivation for simulations in which both the energyand angle dependence of the neutrino distribu-tions were retained. Of necessity, the first suchsimulations were performed in spherical symme-try (actually a three-dimensional problem, de-pending on one space and two momentum space

C.Y. Cardall / Nuclear Physics B (Proc. Suppl.) 138 (2005) 436–438 437

variables). Results from three different groups arein accord. Spherically symmetric models do notexplode, even with solid neutrino transport [5–7].

Recently, one of these groups performed simu-lations in two spatial dimensions, in which theirenergy- and angle-dependent neutrino transportwas made partially dependent on spatial polarangle as well as radius [8,9]. Explosions werenot seen in any of these simulations, except forone in which certain terms in the neutrino trans-port equation corresponding to Doppler shiftsand angular aberration due to fluid motion weredropped. This was a surprising qualitative dif-ference induced by terms contributing what aretypically thought of as small corrections. Thecontinuing lesson is that getting the details of theneutrino transport right makes a difference.

Where, then, do simulations aiming at the ex-plosion mechanism stand? The above history sug-gests that elucidation of the mechanism will re-quire simulations that feature truly spatially mul-tidimensional neutrino transport. In addition, in-clusion of magnetic field dynamics seems increas-ingly strongly motivated as a possible driver ofthe explosion, because simulations with “better”neutrino transport have failed to explode—evenin multiple spatial dimensions.

3. GRAVITATIONAL RADIATION

Core bounce and associated phenomena (haltof the collapse, shock formation, and “ring-down”) would be the strongest source of gravita-tional radiation from a core-collapse supernova.Should such an event occur in our galaxy (thishappens only once every few-several decades), itprobably would be detectable by first-generationgravitational wave interferometers. This conclu-sion is reached in several studies; a recent exampleis the work of Ott et al. [10].

These authors also make some points that high-light the need for more complete simulations.Most often, the results of spherically symmetricstellar evolution computations are artificially putinto rotation for use as progenitors in studies ofcore collapse, but in simulations with the mostrecent progenitors that include rotation and mag-netic fields, the gravitational radiation can be an

order of magnitude weaker. In addition, somefeatures of the waveform (such as coherent post-bounce oscillations) probably will change with theinclusion of realistic neutrino transport because ofthe associated stalling of the shock.

In addition to the violence of core bounce andits immediate aftermath, additional phenomenain the subsequent evolution may produce grav-itational radiation at lower amplitudes. Theseinclude triaxial instabilities in the neutron star,anisotropic neutrino emission, convection, and(on longer time scales than the supernova pro-cess itself) R-mode instabilities [11]. Another lesswell-known possible source of gravitational wavesis the standing accretion shock instability [12].

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