Neutrino-driven supernovae: An accretion instability in a nuclear physics controlled environment
ABSTRACT New simulations demonstrate that low-mode, nonradial hydrodynamic instabilities of the accretion shock help starting hot-bubble convection in supernovae and thus support explosions by the neutrino-heating mechanism. The prevailing conditions depend on the high-density equation of state which governs stellar core collapse, core bounce, and neutron star formation. Tests of this sensitivity to nuclear physics variations are shown for spherically symmetric models. Implications of current explosion models for r-process nucleosynthesis are addressed.
arXiv:astro-ph/0411347v1 12 Nov 2004
An accretion instability in a nuclear physics controlled environment
H.-T. Jankaa∗, R. Burasa, F.S. Kitaura Joyanesa, A. Mareka, M. Ramppa, and L. Schecka
aMax-Planck-Institut f¨ ur Astrophysik,
Karl-Schwarzschild-Str. 1, D-85741 Garching, Germany
New simulations demonstrate that low-mode, nonradial hydrodynamic instabilities of
the accretion shock help starting hot-bubble convection in supernovae and thus support
explosions by the neutrino-heating mechanism.
the high-density equation of state which governs stellar core collapse, core bounce, and
neutron star formation. Tests of this sensitivity to nuclear physics variations are shown
for spherically symmetric models. Implications of current explosion models for r-process
nucleosynthesis are addressed.
The prevailing conditions depend on
Supernova (SN) models in spherical symmetry (1D) — even the latest ones with “Boltz-
mann neutrino (ν) transport” [ 1] — do not explode, neither by the prompt bounce-shock
mechanism, nor by the delayed neutrino-heating mechanism. Suggestions that neutron-
finger convection in the neutron star (NS) could enhance the ν luminosity and thus the
heating behind the shock [ 2] could not be confirmed by detailed analysis [ 3]. Also Ledoux
convection in the nascent NS seems to be ineffective in this respect [ 4, 5]. A reduction of
the ν opacity by several 10% at densities below ∼1013gcm−3would also allow ν’s to leave
the accretion layer more easily and could produce explosions [ 6], but a physical effect of
corresponding size is not known.
Multi-dimensional simulations of core-collapse SNe revealed that hydrodynamic over-
turn behind the stalled accretion shock develops on a relevant timescale and increases the
efficiency of the ν-heating mechanism, enabling explosions even when spherical models
fail [ 7]. Initially there was hope that this new twist in the theory of SNe was the long-
sought guarantee for “robust” explosions. But the first generation of multi-dimensional
models employed a simplified treatment of the ν transport — at best by grey flux-limited
diffusion schemes — which fell much behind the sophistication of the transport in non-
exploding 1D models. It was therefore suspected that the transport approximations might
have contributed to the success of the multi-dimensional simulations [ 8].
Recently, two-dimensional (2D) models have become available with a significant im-
provement of the ν transport by solving the energy-dependent equations of ν number,
∗This work was supported by the Sonderforschungsbereich (SFB) 375 “Astro-Particle Physics” and by
the SFB-Transregio 7 “Gravitational Wave Astronomy” of the Deutsche Forschungsgemeinschaft.
energy, and momentum, making use of closure relations obtained from a model Boltz-
mann equation [ 4, 9]. The description of ν-matter interactions has thus reached a new
level of accuracy and refinement, although the transport is still treated to be essentially
radial (lateral flux components are ignored in the moments equations but terms associated
with lateral velocities and lateral gradients of ν pressure are included). Simulations with
thus improved treatment of the ν physics could not find explosions despite of convective
activity in the ν-heating layer behind the SN shock [ 5], confirming the suspicion that
radical transport approximations might have favored the rapid and powerful explosions
seen in other 2D and 3D models [ 7, 10].
So what is missing in the currently most advanced SN models which fail to produce
explosions? In the following, we shall first briefly summarize the status of core-collapse
modeling by the Garching group. In particular, we shall discuss that any limitation of
the angular wedge of 2D simulations to less than 180 degrees (e.g. to 90 degrees as in
many previous simulations) imposes artificial constraints to the fluid flow and suppresses
large-scale modes that can play an important role for the growth of convection. It is
further demonstrated that even a “modest” amount of rotation in the SN core, in the
ballpark of predictions of current stellar evolution models [ 11], may have an important
impact on the postbounce evolution. Finally, we shall elaborate on uncertain aspects of
the nuclear physics which the SN shows sensitivity to.
2. EXPLOSION MODELS
Simulations leading to explosions can be reported for stars near the low-mass end of
SN progenitors, namely for stars in the mass range of ∼8–10M⊙with O-Ne-Mg cores [
12], and for an 11.2M⊙progenitor [ 13], both characterized by small cores of less than
<∼1.3M⊙(O-Ne-Mg or Fe, respectively) and a steep density decrease (entropy increase)
2.1. Stars in the 8–10M⊙range with O-Ne-Mg cores
The main improvement of our new simulations of O-Ne-Mg core collapse — which
we did so far only in spherical symmetry — compared to previous approaches is the
more accurate, spectral treatment of ν transport and ν-matter interactions. Using the
nuclear equation of state (EoS) of Lattimer & Swesty [ 14] and more recently also that of
Hillebrandt & Wolff [ 15], we could not confirm the prompt explosions found in calculations
with simpler ν treatment [ 15]. The shock is created much deeper inside the collapsing
core than in the “old” models (cf. also Ref. [ 16]), typically at a mass coordinate around
∼0.45M⊙[ 17] (defined by the moment when the postshock entropy first exceeds 3kBper
nucleon), and it stalls (defined by the time when the postshock velocity becomes negative)
only 1.2ms later at ∼0.8M⊙, still well inside the neutrinosphere and before energy losses
by the prompt νeburst could have contributed to its damping [ 17]. We also do not find
the powerful shock revival by ν heating as seen in Ref. [ 18] and for some choice of the
nuclear EoS by Fryer et al. [ 19]. Instead, the shock continuously expands due to the
monotonically decreasing preshock density and the steep density decline at the interface
between C-O shell and He shell (Fig. 1). At the end of our simulation the mass accretion
rate by the shock has correspondingly dropped to less than 0.03M⊙s−1. Although our
simulation is not yet finally conclusive in this point, we see indications that a ν-driven wind
Figure 1. Left: Composition in the highly degenerate 1.38M⊙core of an 8–10M⊙progenitor [
12] with O, Ne and Mg in the central region enclosed by a C-O shell. ?X? denotes the mass
fraction of a representative neutron-rich heavy nucleus that appears in nuclear statistical equi-
librium, and Yeis the electron fraction. Right, top: Mass trajectories, shock position (thick solid
line that rises to the upper right corner of the plot, where it reaches the surface of the C-O shell),
neutrinospheres (νe: solid; ¯ νe: dash-dotted; νµ, ¯ νµ, ντ, ¯ ντ: dashed), and gain radius (dotted) for
the collapsing O-Ne-Mg core. The mass trajectories are equidistantly spaced with intervals of
0.02M⊙. The outermost line corresponds to an enclosed mass of 1.36M⊙near the outer edge
of the carbon shell. The (blue) hatched region is characterized by a dominant mass fraction
of alpha particles. Right, bottom: Luminosities of νe(solid), ¯ νe(dashed) and heavy-lepton ν’s
(individually; dotted) as functions of time. The sawtooth pattern during the collapse phase until
about 50ms is a numerical artifact. (The plots were taken from Ref. [ 17].)
begins to fill the volume between SN surface and shock and will lead to mass ejection with
a rather low explosion energy (a few 1050erg due to the wind power and nuclear burning).
Little nickel production (∼0.01M⊙), can be expected, corresponding to the wind mass
loss rate that can be estimated for an initial νe+¯ νeluminosity of ∼ 8×1052ergs−1(Fig. 1)
using the equations in Refs. [ 20]. The baryonic mass of the NS will be very close to or
only some 0.01M⊙less than the C+O core mass (which is 1.38M⊙). These findings are
very similar to the outcome of simulations of accretion induced white dwarf collapse to
NSs (AICs) [ 21].
A weak explosion of an 8–10M⊙ star has been suggested as an explanation of the
observed properties of the Crab SN remnant [ 22]. The wind-driven explosion seen in our
models, however, does not provide the conditions for the low-entropy, low-Yer-process
Figure 2. Three stages (at postbounce times of 141.1ms, 175.2ms, and 225.7ms from left
to right) during the evolution of a (non-rotating) 11.2M⊙ progenitor model from Ref. [ 13],
visualized in terms of the entropy. The scale is in km. The dense NS is visible as low-entropy
circle at the center. The computation was performed in spherical coordinates, assuming axial
symmetry, and employing the “ray-by-ray plus” variable Eddington factor technique [ 24] for
treating ν transport in multi-dimensional SN simulations. Equatorial symmetry is broken on
large scales soon after bounce, and low-mode convection begins to dominate the flow between
the NS and the strongly deformed SN shock. The model continues to develop a probably weak
explosion, the energy of which was not determined before the simulation had to be stopped
because of CPU time limitations.
nucleosynthesis discussed for prompt explosions of collapsing O-Ne-Mg cores in previous
work [ 23]. R-processing in the high-entropy environment of the ν-driven baryonic wind
can also not be expected to take place, because sufficiently high entropies, short expansion
timescales, and low proton-to-baryon ratios require the NS to be very massive (∼2M⊙)
and compact (<∼10km) (e.g., Refs. [ 20]). It is therefore unclear how O-Ne-Mg core
collapse events could contribute to the production of high-mass r-process nuclei.
2.2. Massive stars with iron core
Progenitor stars in the mass range between 11M⊙and 25M⊙were found to neither
explode by the prompt bounce-shock mechanism nor by the delayed ν-heating mechanism
in spherically symmetric simulations [ 27], in agreement with results of other groups [ 1].
This implies that state-of-the-art SN models do not support suggestions that r-process
nucleosynthesis might occur in prompt explosions of low-mass (∼11M⊙) progenitors [ 28].
We have also started to perform core-collapse simulations for the mentioned stellar
mass range in 2D, using a polar coordinate grid and assuming azimuthal symmetry [ 5, 4].
Because of the energy-dependent treatment of ν transport and ν-matter interactions the
requirements of CPU time are substantial, and we were so far able to perform only a
handful of such 2D calculations. To limit the need of computer resources we initially
constrained the computational volume to a ∼90owedge (between roughly +45oand −45o
around the equatorial plane and periodic boundary conditions for nonrotating models
and between 0oand 90owith reflecting boundaries for rotating ones), using an angular
resolution of ∼1.4o. This choice was also motivated by 2D models of the first generation
which were able to obtain explosions with a similar setup due to the help of hot-bubble
of the shock (giving a rough measure of
deformation) vs. time for 2D simulations
with a polar coordinate grid (models “2D”),
compared to 1D results with the same
physics, but no convection (models “1D”).
11.2M⊙ star [ 13], simulated with
a full 180ogrid (Model 2D(128)) and with
a 90oangular wedge around the equatorial
plane (Model 2D(64)). Bottom: 15M⊙star
(Model s15s7b2 of Ref. [ 25]) without (Model
2D(32)) and with rotation (Model 2D(rot);
see also [ 5, 26]). In the former 2D simulation
a 90owedge around the equator was used, in
the latter a 90ogrid from pole to equator.
Maximum and minimum radii
convection [ 7].
First simulations of stars of 11.2, 15, and 20M⊙with a 90owedge, however, did not
produce explosions [ 5]. This suggests that the success of the previous calculations with
simplified transport was most probably connected with the use of transport approxima-
tions. One of our new models, a 15M⊙ star (Model s15s7b2 of Ref. [ 25]), was also
collapsed with a modest rotation: The pre-collapse core was assumed to rotate (essen-
tially rigidly) with a period of 12s, i.e. Ω ≈ 0.5rads−1(see Fig. 1 in [ 26]), which is
significantly faster — but not orders of magnitude faster — than predicted by the latest
stellar evolution models [ 11]. Even this “modest” amount of rotation turned out to make
a big difference. As visible in the lower panel of Fig. 3, the shock expands to a much
larger radius than in the 2D simulation without rotation (although both simulations did
not lead to explosions within ∼270ms of postbounce evolution), mostly because of the
influence of centrifugal forces and the more violent convection in a more extended region
of ν heating [ 5].
2.3. Nonradial shock instabilities and low-mode postshock convection
While the 11.2M⊙simulation with 90owedge did not explode (see Fig. 3, upper panel),
the same model with a full 180ogrid developed a presumably weak explosion (Figs. 2 and
3). Equatorial symmetry is broken on large scales some 10ms after convective activity
in the postshock region has started (at ∼50ms p.b.), and low modes (l = 1,2) begin to
dominate the flow pattern between NS and shock after about 140ms p.b., a phenomenon
which might be linked to the observed large recoil velocities of young pulsars [ 29]. The
convection becomes significantly more violent than in the 90osimulation where obviously
important degrees of freedom were suppressed. This latter fact was emphasized in an
interesting paper by Blondin et al. [ 30], who found a similar development of low-mode
asymmetries in the postshock flow and shock oscillations as a consequence of the instability
of standing accretion shocks (“SASI”) against non-radial perturbations that are amplified
in a “vortical-acoustic feedback cycle” [ 31].
Blondin et al. considered idealized conditions in their numerical studies, assuming
steady-state mass accretion with a constant rate, a fixed inner boundary radius, a simple
ideal-gas EoS, and ignoring ν heating and cooling. We decided to test the effects of ν’s
and of the size of the angular wedge (and thus of the possible modes) in a separate set
of simulations with a realistic EoS but an approximative treatment of the ν transport (as
described in Refs. [ 29]), which allowed us to save computer time and thus to perform
faster calculations with higher resolution, and to run more models. The mass accretion
rate was given by the collapse of a 15M⊙progenitor star and the NS was replaced by a
gravitating point mass inside a contracting inner boundary that followed the behavior of
the shrinking NS in our full-scale SN simulations with detailed transport physics.
With this setup we confirmed that 2D simulations with a 180ogrid can yield explosions
even if models with 90owedge do not explode. In these studies, with ν effects switched
on or off, we found that ν losses promote the action of the vortical-acoustic cycle by
allowing matter to settle on the proto-NS surface. Our results show that corresponding
shock deformation produces growing perturbations in the postshock flow which can accel-
erate the onset of convective overturn even in cases where the infall timescale is initially
shorter than the growth timescale of Ledoux convection. We also see that dipolar shock
oscillations become more violent in case of a rapid contraction of our inner boundary
(mimicing a NS that becomes rapidly more compact due to a soft nuclear EoS), thus re-
leasing gravitational binding energy which partly is converted to turbulent kinetic energy
of the postshock flow. In this case l = 1,2 modes begin to dominate faster and the SASI
conspires with convective instability to establish favorable conditions for a high efficiency
of ν heating behind the shock. Both nonradial instabilities differ characteristically in the
way how anisotropies emerge: Convective overturn is driven by the negative entropy gra-
dient in the ν-heated layer, which becomes Rayleigh-Taylor (RT) unstable first on small
scales, while the SASI starts from vorticity producing sound wave interactions with the
shock and induces dipolar oscillations of the postshock layer before convective activity is
initiated and the typical RT mushrooms become visible.
2.4. Nuclear physics sets the stage
Nuclear physics therefore governs not only the collapse phase, where electron captures
on nuclei were recently included in a much improved way and were found to determine
the position of shock formation and the structure of the layers the shock expands into [
32] (also W.R. Hix and G. Mart´ ınez-Pinedo, this conference). Nuclear physics and in
particular the properties of the nuclear EoS also determine the contraction and size of the
nascent NS and thus may influence the growth of nonradial instabilities and anisotropies
in the postshock accretion flow during the ν-heating phase.
In fact, differences in the nuclear EoS have interesting consequences for core collapse,
bounce conditions, shock formation, and postshock evolution in 1D simulations [ 33] (for
a brief summary, see [ 27]). Figure 4 shows selected results for three different EoSs
(Lattimer & Swesty [ 14], Shen et al. [ 34], and Wolff & Hillebrandt [ 15]). The softest
of them (L&S) leads to the highest densities at bounce and the smallest enclosed mass of
the shock formation position. Lateron the nascent NS contracts most rapidly, forcing the
shock to retreat much more quickly than in case of, in particular, the stiff Wolff EoS (Fig. 4,
upper middle panel). These differences affect the ν luminosities during the νeburst and
the postbounce accretion phase (Fig. 4, upper right panel). While for the Wolff EoS the νe
s = 1
s = 1
0 50 100
050 100 150 200
Figure 4. Left: Pressure (top) and adiabatic index Γ ≡ (dlnP/dlnρ)s(bottom) vs. mass density
for an entropy s = 1kBper nucleon (and Ye= 0.4) for the EoSs of Ref. [ 15] (“Wolff”, thin lines),
Ref. [ 34] (“Shen”, medium) and Ref. [ 14] (“L&S”, thick). Middle: Shock positions (solid lines)
and neutrinospheric radii of νe(dashed) as functions of time for collapse simulations of a 15M⊙
progenitor (Model s15a28 [ 13]) with the three nuclear EoSs (top) and with two different EoSs
at densities ρ < 1011gcm−3(L&S compared to an ideal gas EoS of e−, e+, γ’s, and Boltzmann
gases of n, p, α and a representative heavy nucleus in NSE; bottom). Right: Prompt νeburst
(left panel) and postbounce luminosities of νe(solid lines), ¯ νe(dotted) and heavy-lepton ν’s and
¯ ν’s (dashed) for the simulations of the 15M⊙star with the three different nuclear EoSs (top)
and the two different low-density EoSs (bottom). (The plots were taken from Ref. [ 33].)
release in the burst is highest, a more compact NS and thus hotter neutrinosphere causes
higher post-burst ν luminosities and mean energies. In the lower middle and right panels
we show the result of a test which we performed with a four-nuclei NSE-EoS replacing the
low-density (ρ < 1011gcm−3) part of the L&S EoS (where an error in the treatment of
α-particles has recently been discovered; C. Fryer, J. Lattimer, personal communication).
The conditions in the postshock layer were hardly affected because the gas is disintegrated
into free nucleons at the high entropies encountered in simulations with relativistic gravity.
Minor differences in the evolution of the shock radius (Fig. 4, lower middle panel) and ν
luminosities (Fig. 4, lower right panel) were caused by differences in the mass accretion
rate associated with the EoS treatment in the infall region ahead of the shock.
Non-radial instability of the accretion shock can amplify vorticity in the postshock
flow and thus can support the growth of convection in the ν-heated layer. It may be an
important ingredient for eventually robust explosions by the ν-driven mechanism. Low
modes in the flow can develop when the explosion sets in slowly. The relative importance
of the different instabilities seems to depend on ν cooling and heating on the one hand
and the high-density EoS, which controls the contraction of the nascent NS, on the other.
Simulations require the use of a full 180ogrid and ultimately may have to be done in 3D.
Acknowledgements. We are grateful to K. Nomoto, A. Heger, S. Woosley, and M. Limongi
for providing us with their progenitor data. Supercomputer time at the John von Neumann
Institute for Computing in J¨ ulich and the Rechenzentrum Garching is acknowledged.
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