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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0031-9007=07=98(26)=261101(4) 261101-1 © 2007 The American Physical Society

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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261101-2

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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