![FIG. 4 (color). Spectra of the characteristic GW strain hchar of all models and the LIGO (optimal) rms noise curves [32].](/figures/fig-4-color-spectra-of-the-characteristic-gw-strain-hchar-of-ell2831w.webp)
3D Collapse of Rotating Stellar Iron Cores in General Relativity Including Deleptonization
and a Nuclear Equation of State
C. D. Ott,
1,5,
*
H. Dimmelmeier,
2
A. Marek,
2
H.-T. Janka,
2
I. Hawke,
3
B. Zink,
2,4
and E. Schnetter
4
1
Max-Planck-Institut fu
¨
r Gravitationsphysik, Albert-Einstein-Institut, Am Mu
¨
hlenberg 1, D-14476 Potsdam, Germany
2
Max-Planck-Institut fu
¨
r Astrophysik, Karl-Schwarzschild-Straße 1, D-85741 Garching, Germany
3
School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK
4
Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA
5
Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA
(Received 29 September 2006; published 29 June 2007)
We present 2D and 3D simulations of the collapse of rotating stellar iron cores in general relativity
employing a nuclear equation of state and an approximate treatment of deleptonization. We compare fully
general relativistic and conformally flat evolutions and find that the latter treatment is sufficiently accurate
for the core-collapse supernova problem. We focus on gravitational wave (GW) emission from rotating
collapse, bounce, and early postbounce phases. Our results indicate that the GW signature of these phases
is much more generic than previously estimated. We also track the growth of a nonaxisymmetric
instability in one model, leading to strong narrow-band GW emission.
DOI: 10.1103/PhysRevLett.98.261101 PACS numbers: 04.25.Dm, 04.30.Db, 95.30.Sf, 97.60.Bw
Introduction.—For more than two decades, astrophysi-
cists have struggled to compute the gravitational wave
(GW) signal produced by rotating stellar core collapse
and the subsequent supernova evolution. Besides the co-
alescence of black hole and neutron star binaries, core-
collapse events are considered to be among the most
promising sources of detectable GWs. Theoretical predic-
tions are still hampered by three major problems: (i) the
rotational configuration prior to gravitational collapse is
still uncertain since multi-D evolutionary calculations
of rotating massive stars have not yet been performed;
(ii) reliable waveform estimates require a general relativ-
istic (GR) treatment, since both high densities and veloc-
ities in combination with strong gravitational fields are
encountered in this problem; and (iii) an adequate treat-
ment of the nuclear equation of state (EOS) and the neu-
trino microphysics and radiative transfer is crucial for
obtaining realistic collapse, bounce, and postbounce dy-
namics and waveforms. GW emission from core-collapse
supernovae may arise from rotating collapse and bounce,
postbounce neutrino-driven convection, anisotropic neu-
trino emission, nonaxisymmetric rotational instabilities
of the protoneutron star (PNS), or from the recently pro-
posed PNS core g-mode oscillations. Previous estimates of
the GW signature of core-collapse supernovae have either
relied on Newtonian simulations [1–6] (to some extent
approximating GR effects [7]) or GR simulations with
simplified analytic (so-called hybrid) EOSs and no neu-
trino treatment [8–11]. Depending on rotation, softness of
the EOS at subnuclear densities, and inclusion of GR
effects, the collapse dynamics and accordingly the GW
signature can differ significantly.
Here, we present new results from GR simulations,
focusing on the rotating collapse, bounce, and early post-
bounce phases. These are the first-ever multi-D GR simu-
lations with presupernova models from stellar evolution
calculations, a microphysical nuclear EOS, and a simple
but effective treatment of electron capture and neutrino
radiation effects during collapse. In this way, we obtain the
most accurate estimates of the GW signature of rotating
stellar core collapse in full GR to date.
Method and initial models.—We perform all 3D simu-
lations in 3 1 GR using
CACTUS [ 12], Cartesian coordi-
nates, and mesh refinement provided by
CARPET [13].
Spacetime is evolved using the BSSN formulation (see,
e.g., [14]) with 1 log slicing and a hyperbolic shift [15].
We use the finite-volume GR hydrodynamics code
WHISKY
[16]. Typical simulation grids extend to 3000 km and use 9
refinement levels. The central resolution is r 350 m.In
addition, we perform axisymmetric (2D) simulations for
all models using the
COCONUT code [8,17], approximating
GR by the conformal flatness condition (CFC).
COCONUT
utilizes spherical coordinates with 250 logarithmically-
spaced radial zones with r 250 m and 45 equidistant
angular zones. Resolution tests with both codes do not
yield significant qualitative or quantitative differences.
We extract GWs using a variant of the Newtonian quadru-
pole formula [9].
We employ the microphysical nuclear EOS of Shen et al.
[18] as implemented in [19]. Deleptonization is approxi-
mated as proposed by [20]: The electron fraction Y
e
is
parameterized as a function of density based on data from
spherically symmetric radiation-hydrodynamics calcula-
tions with standard electron capture rates [21]. After core
bounce, Y
e
is passively advected, and further lepton loss is
neglected, but neutrino pressure continues to be taken into
account above trapping density [20].
In this Letter, we focus on the collapse of massive
presupernova iron cores with at most moderate differential
rotation, and rotation rates that may be too fast to match
PRL 98, 261101 (2007)
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garden-variety pulsar birth spin estimates [22,23], but
could be relevant in the collapsar-type gamma-ray burst
context. As initial data, we use the nonrotating 20M
presupernova model s20 of Woosley et al. [24], which
we force to rotate according to the rotation law discussed
in [5,8]. We parameterize our models in terms of the
differential rotation parameter A and the initial ratio of
rotational kinetic to gravitational energy
i
T=jWj.In
addition, we perform a calculation with the 20M
model
E20A of [25], which includes an approximate 1D treatment
of rotation. In Table I, we list the model parameters.
Results.—In Fig. 1, we compare GW signals computed
with
COCONUT in 2D-CFC and those computed with our
3D-full-GR approach. Model s20A2B2 (red lines) is a
moderate rotator with a
i
0:50%, rotating rigidly in
its central region. It stays axisymmetric throughout its
numerical evolution. The agreement of 2D-CFC with 3D-
full-GR is excellent for this model: Both waveforms match
almost perfectly at bounce and during the very early post-
bounce phase. A few ms after bounce, when convection in
the region behind the stalling shock sets in due to a
negative entropy gradient, the signals begin to differ quan-
titatively while remaining in phase. We attribute this small
mismatch to the choice of coordinate grids and to differ-
ences in the growth and scale of vortical postbounce mo-
tions between 2D and 3D. Model s20A1B5 rotates with
constant in the entire core. Despite its very large
i
4%, it remains essentially axisymmetric during the time
covered by our simulation, since most of its angular mo-
mentum is attached to material at large radii that falls
inward and spins up only slowly. The waveforms in CFC
and full GR agree very well. Again, both waveforms match
best for the strong burst related to core bounce during
which more than 90% of the total GW energy are emitted
in an axisymmetric model. The overall excellent agree-
ment of CFC with full GR confirms results of [9,11] and
proves that CFC is a very good approximation to full GR in
the core-collapse scenario.
In Fig. 2, we present waveforms of models with varying
initial degree of differential rotation A and rotation rate
i
.
We find that the inclusion of a microphysical EOS and of
electron capture yields results that differ considerably from
those obtained in previous, less sophisticated studies.
Figure 2 exemplifies that largely independent of the initial
rotational configuration, the GW signal of rotating collapse
has a generic shape: a slow rise in the prebounce phase, a
large negative amplitude at bounce when the motion of the
inner core is reversed, followed by a ring-down. This so-
called ‘‘Type I’’ signature corresponds to a baryonic pres-
sure dominated bounce [1,2,5,8]. Thus, all our models
undergo core bounce dominated by the stiffening of the
EOS at nuclear density.
TABLE I. Summary.
b
is the density at bounce, the maximum characteristic GW strain
h
char;max
is at a distance of 10 kpc, and E
gw
is the energy emitted in GWs. Values for E20A
pb
include GW emission from late-time 3D dynamics.
Model A [10
8
cm]
i
[%]
b
[%]
b
10
14
g
cm
3
h
char;max
[10
21
] E
gw
[10
9
M
c
2
]
s20A1B1 50.0 0.25 0.90 3.29 1.46 0.6
s20A1B5 50.0 4.00 10.52 2.90 9.68 26.9
s20A2B2 1.0 0.50 6.72 3.07 8.77 22.0
s20A2B4 1.0 1.80 16.33 2.35 4.28 9.4
s20A3B3 0.5 0.90 16.57 2.33 4.58 12.4
E20A 0.37 11.31 2.79 12.18 36.9
E20A
pb
24.23 75.4
t-t
bounce
(ms)
h
+
(10
−21
at 10 kpc)
-5 0 5 10
-10
-8
-6
-4
-2
0
2
4
6
s20A2B2 3D full GR
s20A2B2 2D CFC
s20A1B5 3D full GR
s20A1B5 2D CFC
FIG. 1 (color). GW strain h
along the equator for models
s20A2B2 and s20A1B5. We compare 2D-CFC and 3D-full-GR
results.
-5 0 5 10 15
-12
-10
-8
-6
-4
-2
0
2
4
6
t-t
bounce
(ms)
h
+
(10
−21
at 10 kpc)
s20A2B1
s20A2B2
s20A2B4
s20A3B3
E20A
FIG. 2 (color). Equatorial GW strain h
for a representative
subset of the models listed in Table I. Note the generic shape of
the core bounce GW signal.
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This is in stark contrast to the studies using the simple
hybrid EOS [2,8–10], where initial models with rotation
rates in the range investigated here develop sufficient
centrifugal support during contraction to stop collapse at
subnuclear densities, resulting in several consecutive cen-
trifugal bounces separated by phases of coherent reexpan-
sion of the inner core. While in GR, models exhibiting such
multiple centrifugal bounce and the corresponding ‘‘Type
II’’ GW signals are only rarer compared to Newtonian
gravity [8], we do not observe any such model in this study.
An evident example is model s20A2B4: In previous studies
without detailed microphysics, the corresponding model
with identical initial rotation parameters (A2B4G1)
showed clear ‘‘Type II’’ behavior in both Newtonian and
GR calculations [2,8].
The suppression of the multiple centrifugal bounce sce-
nario is due to two physical effects: (i) In contrast to the
simple hybrid EOS, in our case the mass and dynamics of
the inner core (which is most important for the GW emis-
sion) is not merely determined by the adiabatic index
d lnP=d ln (at constant entropy) of the EOS, but also by
deleptonization during collapse. This leads to considerably
smaller inner cores with less angular momentum and
weaker pressure support. (ii) Since multiple centrifugal
bounce was observed for a model with moderately fast
rotation in a previous Newtonian study [1] employing
both a microphysical EOS and deleptonization, the ab-
sence of this collapse type in our study is not only caused
by microphysical effects, but also by the effectively
stronger gravity in GR. This is in accordance with simu-
lations using the simple hybrid EOS [8]. For a more de-
tailed discussion of these effects, see [26].
Model E20A possesses the largest GW amplitude of all
our models. In addition, it reaches a high
b
at core bounce
(see Table I) and settles at a postbounce
f
of 9%.A
previous Newtonian study [27] has found a low-T=jWj
nonaxisymmetric instability for a PNS with similar
f
.
In order to verify their findings, we trace the evolution of
model E20A to 70 ms after bounce and perform an analysis
of azimuthal density modes / e
im’
by computing complex
Fourier amplitudes C
m
1
2
R
2
0
$; ’; z 0e
im’
d’
on rings of constant coordinate radius with respect to the
coordinate center of mass. The latter stays within the
innermost zone at all times. In the top panel of Fig. 3,we
display the normalized mode amplitudes extracted at
15 km radius. Without adding artificial seed perturbations
to model E20A, discretization errors trigger m f1; 2; 3g
modes, which rise to a level of 10
5
during the 220 ms
collapse phase. After bounce, the m 1 mode exhibits the
fastest growth. This growth on a dynamical time scale,
lasting over tens of ms until saturation, is closely followed
by a growth of m f2; 3g daughter modes [27,28]. Note
that after core bounce, model E20A remains dynamically
stable to the m 4 grid mode. In the lower panel of Fig. 3,
we plot the GW strains h
and h
as seen along the polar
axis. The rotational symmetry of E20A before and shortly
after bounce is reflected in the fact that h
and h
are
practically zero until E20A develops considerable nonax-
isymmetry when the m 1 mode becomes dominant and
its m 2, GW-emitting harmonic reaches a sizable ampli-
tude. In agreement with expectations for a spinning bar, h
and h
oscillate at the same frequency ( 930 Hz) and are
phase-shifted by a quarter cycle.
Discussion.—Our results indicate that the GW signature
of the collapse, bounce, and early postbounce phases of the
core-collapse supernova evolution is much more generic
than previously thought. We find that the dynamics of core
bounce are dominated by mainly gravity and microphysics,
reducing the importance of centrifugal support for the
rotation rates considered here. Importantly, for our model
set, we do not observe rotationally induced multiple core
bounce at subnuclear density as proposed by previous
studies that did not include a microphysical EOS and
electron capture treatment in combination with GR.
t-t
bounce
(ms)
h
+/×,pole
(10
−21
at 10 kpc)
-10 0 10 20 30 40 50 60 70
-3
-2
-1
0
1
2
h
+ ,pole
h
×,pole
A
m
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
m= 1
m= 2
m= 3
m= 4
FIG. 3 (color). Normalized mode amplitudes A
m
jC
m
j=C
0
at postbounce times (upper panel) and GW strains h
and h
along the poles (lower panel) for model E20A.
f (Hz)
h
char
at 10 kpc
LIGO I
advanced
LIGO
10
2
10
3
10
−22
10
−21
10
−20
E20A
s20A2B1
s20A2B2
s20A2B4
s20A1B5
s20A3B3
FIG. 4 (color). Spectra of the characteristic GW strain h
char
of
all models and the LIGO (optimal) rms noise curves [32].
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Thus, we predict that the core-bounce waveform of models
in a large parameter space of initial rotation rate and degree
of differential rotation will likely both qualitatively and
quantitatively resemble those presented in Fig. 2.
Model E20A, which we evolve to later postbounce
times, exhibits the dynamical growth of a nonaxisymmetric
low-T=jWj corotation-type m 1 instability [27–29]. We
also find m f2; 3g contributions and significant GW
emission from the quadrupole components. We emphasize
that we observe this instability not only in E20A, but also
in other models with comparable values of
f
. Our results,
which remove the limitations of previous studies
[10,27,30,31], demonstrate that the development of non-
axisymmetric structures is neither limited to Newtonian
gravity, simple matter models, equilibrium configurations,
nor high values of , but may rather be a phenomenon
occurring generically in differentially rotating compact
stars.
For the GW signals from the axisymmetric collapse and
core-bounce phase, we obtain peak amplitudes of up to h
10
20
at 10 kpc, while the nonaxisymmetric structures in
model E20A developing later emit GWs with h only a
factor 5 smaller. However, since the latter emission
process operates over several tens of ms, the total energy
E
gw
emitted in GWs is larger than that from the
core-bounce signal. This is evident in Fig. 4, where we
display the characteristic GW strain spectra h
char
R
1
2
2
dE
gw
=df
q
[5] for all models, evolving E20A
for 70 ms after bounce (see also Table I). Considering
only the core-bounce waveforms, h
char
has its maximum
between 300 and 800 Hz, while for model E20A it peaks at
930 Hz, the frequency of the GW-emitting component of
its nonaxisymmetric structures. We conclude that the core-
bounce GW signals of all models investigated here may be
detectable by current and future LIGO-class detectors from
anywhere in the Milky Way. Models that develop non-
axisymmetric instabilities may be detectable out to much
larger distances if the instability persists for a sufficiently
long time.
We point out that due to the nature of the approximation
used for the neutrino effects, we can only accurately model
the GW emission in the collapse, bounce, and early post-
bounce epoch of the core-collapse supernova scenario. In
future work, we plan to improve upon this and carry out
longer-term postbounce evolutions, where other GW emis-
sion mechanisms may be relevant [6,7].
We thank A. Burrows, E. Mu
¨
ller, S. Ou, L. Rezzolla,
E. Seidel, D. Shoemaker, N. Stergioulas, and J. Tohline for
help and stimulating discussions. This research was par-
tially supported by the DFG (Nos. SFB/TR7, SFB 375) and
by the NCSA (Grant No. AST050022N).
*cott@as.arizona.edu
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