Astron. Astrophys. 332, 969–983 (1998) ASTRONOMYAND

ASTROPHYSICS

Simulations of non-axisymmetric rotational core collapse

M. Rampp?, E. Muller??, and M. Ruffert ???

Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str. 1, Postfach 1523, D-85740 Garching, Germany

Received 17 November 1997 / Accepted 13 January 1998

Abstract. We report on the first three-dimensional hydrody-namic simulations of secular and dynamical non-axisymmetricinstabilities in collapsing, rapidly rotating stellar cores whichextend well beyond core bounce. The resulting gravitationalradiation has been calculated using the quadrupole approxima-tion.

We find that secular instabilities do not occur during thesimulated time interval of several 10 ms. Models which becomedynamically unstable during core collapse show a strong non-linear growth of non-axisymmetric instabilities. Both randomand coherent large scale initial perturbations eventually give riseto a dominant bar-like deformation (exp(±imφ) with m = 2).In spite of the pronounced tri-axial deformation of certain partsof the core no considerable enhancement of the gravitationalradiation is found. This is due to the fact that rapidly rotatingcores re-expand after core bounce on a dynamical time scale be-fore non-axisymmetric instabilities enter the nonlinear regime.Hence, when the core becomes tri-axial, it is no longer verycompact.

Key words: gravitation – hydrodynamics – instabilities – stars:neutron – stars: rotation – supernovae: general – gravitationalwaves

1. Introduction

The rotation rate of a self-gravitating fluid body is usually quan-tified by the parameterβ := Erot/|Epot|, whereErot is therotational energy andEpot the potential energy of the body(e.g., Tassoul 1978). It is well known from linear stability anal-ysis that global non-axisymmetric instabilities can develop in arotating body whenβ is sufficiently large (e.g., Chandrasekhar1969; Tassoul 1978). Rotational instability arises from non-radial azimuthal (or toroidal) modesexp(±imφ), whereφ is theazimuthal coordinate and wherem is characterizing the modeof perturbation. The mode withm = 2 is known as the “bar”mode. Driven by hydrodynamics and gravity, a rotating fluid

Send offprint requests to: M. Rampp? e-mail: mjr@mpa-garching.mpg.de

?? e-mail: emueller@mpa-garching.mpg.de??? e-mail: mruffert@ast.cam.ac.uk

body becomes dynamically unstable to non-axisymmetric per-turbations on a time scale of approximately one rotation periodwhenβ > βdyn. Secular instabilities develop due to dissipativeprocesses such as gravitational radiation reaction (GRR; Chan-drasekhar 1970), viscosity (Roberts & Stewartson 1963), or acombination of both (Lindblom & Detweiler 1977), and growon a time scale of many rotation periods (Schutz 1989) whenβsec < β < βdyn.

The values of the critical rotation parametersβsec andβdyndepend on the density stratification and the angular momen-tum distribution of the rotating body. For MacLaurin spheroids,i.e., for incompressible, rigidly rotating equilibrium configura-tions, secular and dynamical bar instabilities set in forβsec =0.1375 andβdyn = 0.2738, respectively (e.g., Chandrasekhar1969; Tassoul 1978). In differentially rotating polytropes (p ∝%1+1/n) both the angular momentum distribution and to a lesserextent the polytropic indexn affect the value ofβsec at whichthe secular bar mode sets in (Imamura et al. 1995). Forn = 3/2polytropes andm = 2 perturbations Pickett et al. (1996) findthat the classical dynamical stability limitβdyn ≈ 0.27 holds, ifthe angular momentum distribution is similar to those of Mac-Laurin spheroids. However, considerably lower values ofβdynare found when the latter gives rise to extended Keplerian-disk-like equilibria.

It is unclear to what extent these results obtained analyti-cally for MacLaurin spheroids and numerically for compressibleconfigurations are applicable to collapsing cores, since they arederived for stationaryequilibrium models. It has been shown,however, that during axisymmetric rotational core collapse con-servation of angular momentum can lead to very rapidly rotat-ing configurations, whose rotation rates exceedβsec and evenβdyn (Tohline 1984; Eriguchi & Muller 1985; Monchmeyer etal. 1991; Zwerger & Muller 1997). But whether such super-critical rotation in collapsing cores indeed leads to the growthof non-axisymmetric instabilities has not yet been demonstratedin hydrodynamic simulations.

Besides of possibly being relevant for the collapse dy-namics of rapidly rotating stellar cores, tri-axial instabilitieshave also been envisaged to boost the gravitational wave signalfrom rotational core collapse (for a review, see e.g., Piran 1990;Thorne 1995). This would be of great importance for the fourlong-baseline laser interferometric gravitational wave detectors

970 M. Rampp et al.: Simulations of non-axisymmetric rotational core collapse

(GEO600, LIGO, TAMA, VIRGO) which will become opera-tional within the next few years. They are designed to achievesensitivities (for single bursts) down to gravitational wave am-plitudes of|h| ∼ 10−22 (see e.g., Abramovici et al. 1992). Inorder to obtain a few events per year, one has to detect allcore collapse supernovae out to the Virgo cluster of galaxies(distance' 10 Mpc). However, simulations of axisymmetricrotational core collapse predict gravitational wave amplitudesof at most|h| ∼ 10−23 at that distance (Muller 1982; Finn& Evans 1990; Monchmeyer et al. 1991; Yamada & Sato 1995;Zwerger & Muller 1997). Significantly larger gravitational wavesignals could be produced if non-axisymmetric instabilities dueto rotation act in the collapsing core. According to this idea, arapidly spinning core will experience a centrifugal hang-up andwill be transformed into a bar-like configuration that spins end-over-end like an American football, if its rotation rate exceedsthe critical rotation rate(s). One has further speculated, that thecore might even break up into two or more massive pieces,if β > βdyn (e.g., Bonnell & Pringle 1995). It has been sug-gested that the resulting gravitational radiationcouldbe almostas strong as that from coalescing neutron star binaries (Thorne1995). The actual strength of the gravitational wave signal willsensitively depend on (i) the radius at which the centrifugalhang-up occurs and (ii) what fraction of the angular momentumof the non-axisymmetric core goes into gravitational waves, andwhat fraction into hydrodynamic waves. These sound and shockwaves are produced as the bar or lumps, acting like a twirling-stick, plow through the surrounding mass layers.

Recently, both semi-analytic and numerical methods havebeen used to compute the gravitational radiation produced bynon-axisymmetric instabilities in rapidly rotating stars. Lai &Shapiro (1995) have studied secular instabilities in rapidly rotat-ing neutron stars and have computed the resulting gravitationalradiation using linearized dynamical equations and a compress-ible ellipsoid model. Houser et al. (1994), Houser & Centrella(1996) and Smith et al. (1996) have used three-dimensional hy-drodynamic codes to simulate the nonlinear growth of the dy-namical tri-axial instability in rapidly (β = 0.3) rotating poly-tropes (n = 3/2, n = 1, n = 1/2). Scaling their results to neu-tron star dimensions (i.e., a polytrope with massM ∼ 1.4 Mand radiusR ∼ 10 km) they have also calculated the gravi-tational radiation from the non-axisymmetric instability. Theyobtained a maximum dimensionless gravitational wave ampli-tude |h| ∼ 2 10−22 for a source at a distance of 10 Mpc, theenergy lost to gravitational radiation being∆E ∼ 10−3Mc2.

Concerning investigations, like those just discussed, it is im-portant to note that they can be used to predict the gravitationalradiation from rapidly rotating, stationary neutron stars, whichmight (or might not!) form as a consequence of rotational corecollapse. They are not appropriate, however, for predicting thegravitational wave signature of thecollapseof rapidly rotatingstellar cores. We stress this difference here, because it is oftenoverlooked.

Up to now three-dimensional hydrodynamic simulations ofrotational core collapse, which can follow the nonlinear growthof non-axisymmetric instabilitiesduring collapse, have only

been performed by Bonazzola & Marck (1993, 1994), and byMarck & Bonazzola (1992). For their simulations they used apseudo-spectral hydrodynamic code and a polytropic equationof state. They computed the evolution of several initial mod-els and found that the gravitational wave amplitude is withina factor of two of that of 2D simulations for the same initialdeformation of the core (Bonazzola & Marck 1994). However,their simulations were restricted to the pre-bounce phase of thecollapse, and thus are less relevant.

In the following we report on the first three-dimensionalhydrodynamic simulations of non-axisymmetric instabilities incollapsing, rapidly rotating stellar cores which extend well be-yond core bounce. In addition to the dynamics, we have alsocomputed the gravitational radiation emitted during the evolu-tion. Our study is a continuation and an extension of the recentwork of Zwerger (1995) and Zwerger & Muller (1997), whohave performed a comprehensive parameter study of axisym-metric rotational core collapse. The initial models, the equationof state and the hydrodynamic method are adopted from theirstudy. From their set of 78 models we have only considered thosewhich exceed the critical rotation rate(s) during core collapse.

The paper is organized as follows: In Sect. 2 we presentthe computational procedure used to follow the hydrodynamicevolution of the core and to compute the gravitational wavesignal. Subsequently, in Sect. 3, we first discuss the results of atwo-dimensional simulation, which serves as a reference pointfor the three-dimensional runs. We then present various aspectsof the evolution of the non-axisymmetric models and discuss thegravitational wave signature of the models. Finally, in Sect. 4,we summarize our results, and conclude with a discussion ofthe shortcomings of our approach.

2. Computational procedure

2.1. Hydrodynamic method and equation of state

The simulations were performed with the Newtonian mul-ti-dimensional finite-volume hydrodynamic code PROMET-HEUS developed by Bruce Fryxell and Ewald Muller (Fryxell etal. 1989). PROMETHEUS is a direct Eulerian implementationof the Piecewise Parabolic Method (PPM) of Colella & Wood-ward (1984). For the three-dimensional simulations we used avariant of PROMETHEUS due to Ruffert (1992), which utilizesmultiple-nested refined equidistant Cartesian grids to enhancethe spatial resolution. We do not include any general relativisticeffects for the fluid and have neglected effects due to neutrinotransport.

Matter in the core is approximated by a perfect fluid using asimplified analytic equation of state (Janka et al. 1993; see alsoZwerger & Muller 1997 for details of the implementation). Thepressure is a function of the density% and energy densityu ofthe core matter and consists of two parts:

P (%, u) = Pp + Pth . (1)

The polytropic part

Pp = K · %Γp , (2)

M. Rampp et al.: Simulations of non-axisymmetric rotational core collapse 971

which depends only on density, describes the pressure contri-bution of degenerate relativistic electrons or that due to the re-pulsive nuclear forces at high densities. The thermal partPthmimics the thermal pressure of shock-heated matter. It is com-puted from the corresponding internal energy densityuth by

Pth = (Γth − 1)uth, with Γth = 1.5 . (3)

The thermal energy densityuth in turn is given by the totalenergy densityu through the relation

u = up + uth , (4)

whereup is the energy density of the degenerate electron gas.To account for the repulsive part of the nuclear forces the“polytropic” adiabatic exponentΓp (see Eq. (2)) is increasedfrom a valueΓ1 close to4/3 (resembling degenerate, rela-tivistic electrons) to a valueΓ2 = 2.5, if the density of afluid element exceeds nuclear matter density which is set to%nuc = 2.0 1014g cm−3.

2.2. Gravitational wave emission

In the transverse-traceless (TT) gauge a metric perturbationhcan be decomposed as (e.g., Misner et al. 1973, chap. 35)

hTT = h+e+ + h×e× (5)

with the unit linear polarization tensors defined, in sphericalcoordinates(r, θ, φ), as

e+ = eθ ⊗ eθ − eφ ⊗ eφ

e× = eθ ⊗ eφ + eφ ⊗ eθ , (6)

whereeθ andeφ are the unit vectors in the corresponding co-ordinate directions and⊗ is the tensor product. The two inde-pendent polarizationsh+ andh× are calculated from the fluidvariables in Post-Newtonian approximation, keeping only themass-quadrupole in the multipole expansion of the field. Let usconsider an observer located at coordinates(r, θ, φ) in a spher-ical coordinate system whose origin coincides with the centerof mass of the core. Then the two independent polarizations aregiven by (e.g., Zhuge et al. 1994)

h+ =G

c41r

((I−xx cos2 φ+ I−yy sin2 φ+ I−xy sin 2φ) cos2 θ

+I−zz sin2 θ − (I−xz cosφ+ I−yz sinφ) sin 2θ

−I−xx sin2 φ− I−yy cos2 φ+ I−xy sin 2φ)

(7)

h× =G

c42r

( 12(I−yy − I−xx) sin 2φ cos θ + I−xy cos 2φ cos θ

+(I−xz sinφ− I−yz cosφ) sin θ). (8)

Here the quantitiesI−ij are the second time derivatives ofthe Cartesian components of the reduced quadrupole momenttensorI−. They are defined according to

I−ij :=d2

dt2

∫% (x, t− r/c) ·

(xixj − 1

3δijxkx

k

)d3x , (9)

where we have used the Einstein sum convention(i.e., summation over repeated indices) and whereδij isthe Kronecker symbol. Note that the origins of the sphericalcoordinate system (used to specify the observer’s position) andthe Cartesian coordinate system (used to describe the matterdistribution of the source) coincide. The coordinate systemsare oriented relative to each other such thatθ = 0 correspondsto the positivez-direction.

We assume that the source possesses equatorial symmetry(i.e., it is symmetric with respect to the transformationz→ −z),and henceI−xz = I−yz ≡ 0. Inspection of the remaining terms inEqs. (7) and (8) then yieldsθ ∈ 0, π/2 andθ = 0 as necessaryconditions for extrema of the waveformsh+ andh× (consideredas functions of the observer’s position), respectively. Thus, wecalculateh× along the positivez-axis andh+ in the equatorialplane (at an azimuthal positionφ = 0) to yield the maximumstresses1. We have checked for all models that we do not misspossible additional (short term) maxima ofh+ that could bepresent for an observer at some azimuthal positionφ /= 0 in theequatorial plane but could be overlooked by an observer locatedatφ = 0.

The quadrupole part of the total gravitational wave energyis given by

EGW =15G

c5

∫ ∞

−∞

...

I− ij

...

I− ij dt . (10)

Instead of numerically approximating Eq. (9) with stan-dard finite differences, it is superior (e.g., Finn & Evans 1990;Monchmeyer 1993) to use an equivalent expression derived in-dependently by Nakamura & Oohara (1989) and by Blanchet etal. (1990):

I−ij = STF

2∫% · (vivj − xi∂jΦ) d3x

, (11)

where the symmetric and trace free part of a doubly indexedquantityAij is defined by

STFAij :=12Aij +

12Aji − 1

3δijAll . (12)

The spatial derivatives∂jΦ of the gravitational potentialappearing in Eq. (11) are approximated by centered finite dif-ferences.

For two reasons no back-reaction of the gravitational waveemission on the fluid (GRR) has been implemented. Firstly,the peak luminosity of gravitational radiation that can be ex-pected from rotational core collapse is at most of the order of1050 erg s−1 (Muller 1982; Finn & Evans 1990; Monchmeyeret al. 1991; Zwerger & Muller 1997). When comparing this lu-minosity with the typical energies (kinetic, rotational, potential,internal)E ' 1052 erg of the core one finds a GRR time scaleof the order of 100 s, which is at least by a factor of104 largerthan the time interval over which we can calculate the core’sevolution. Secondly, total energy conservation is violated by

1 For all models discussed below, the value ofh+ at θ = 0 turnedout to be negligibly small.

972 M. Rampp et al.: Simulations of non-axisymmetric rotational core collapse

an amount∆E ' 1049 erg in the best resolved 2D calculationand by an amount∆E ' 1050 erg in the 3D runs. This vio-lation means an acceptable 1%-error as far as the dynamics isconcerned. However, it dominates the change of energy radi-ated due to GRR (EGW <∼ 1047 erg) in the time interval of oursimulations by about a factor of103.

2.3. Two-dimensional simulations

Closely following Zwerger & Muller (1997), the axisymmetric(reference) simulations were performed in spherical coordinatesassuming equatorial symmetry. Thus, the angular grid coveredthe rangeθ ∈ [0, π/2]. The number of equidistant angular zoneswasnθ = 18, 45 or 90, which corresponds to an angular res-olution of 5, 2 or 1, respectively. The moving radial gridconsisted of 360 zones. The grid was moved in such a way thatthe inner core (i.e., the subsonic inner part of the core) was al-ways resolved by 180 radial zones. The radial zoning variedfrom about 0.5 km in the unshocked inner core to about 50 kmin the outermost parts of the outer core (for more details seeZwerger & Muller 1997).

As in Zwerger & Muller (1997), the gravitational wave am-plitude was computed by expanding the gravitational field into“pure-spin tensor harmonics” (Thorne 1980). The only non-vanishing quadrupole contribution is then given by

AE220 =

G

c416π3/2√

15d2

dt2

∫ ∞

0

∫ 1

−1%

(32z2 − 1

2

)r4 dzdr , (13)

which is related to the (dimensionless) metric perturbations ofEq. (5) by

h+ =18

√15π

sin2 θAE2

20

r, h× ≡ 0 (14)

Using the counterpart of Eq. (11) in spherical coordinates(e.g., Zwerger & Muller 1997, Eq. 20), the second time deriva-tive as well as ther2-weighting of mass elements, both numeri-cally troublesome (e.g., Finn & Evans 1990), can be eliminated.

In the quadrupole approximation the total gravitational waveenergy is given by

EGW =c3

G

132π

∫ ∞

−∞

(dAE2

20

dt

)2

dt . (15)

Higher-order terms in the multipole expansion of the grav-itational wave field are negligible both for the wave amplitudesand the amount of radiated energy in the axisymmetric case(e.g., Monchmeyer et al. 1991, Zwerger 1995).

2.4. Three-dimensional simulations

The three-dimensional calculations have been performed onmultiple-nested refined equidistant Cartesian grids (see Ruffert(1992) for details of the method). As in the 2D calculations weassume the equatorial planez = 0 to be a plane of symmetry. Sixcentered grids were used each of which was refined by a factor oftwo. In direction of the rotation (z-) axis each grid had 32 zones,

while the perpendicular grid planes consisted of64× 64 zones.The coarsest grid had a spatial extent of 2560 km covering theentire core with cubes of edge size 40 km. The finest grid cov-ered the innermost 80 km with a linear resolution of about 1 km.Thus, the resulting spatial resolution was comparable to that ofthe 2D simulations.

For the three-dimensional simulations we considered ax-isymmetric models of Zwerger & Muller (1997), which are sta-ble against non-axisymmetric perturbations in the early stages ofcollapse (i.e.,β(t) <∼ 0.1), but later exceed the critical rotationrate(s). Test calculations showed that non-axisymmetric insta-bilities did not grow in perturbed axisymmetric models withβ <∼ 0.1 on time scales of a few 10 ms. Using trilinear inter-polation an axisymmetric model was mapped onto the three-dimensional grid when its rotation parameter reached a value of≈ 0.1. Typically, this occured a few milliseconds before corebounce.

We then imposed certain non-axisymmetric perturbationsand followed the subsequent dynamical evolution of the core inthree spatial dimensions. Due to the cubic cells of our computa-tional grid, the mapping procedure itself introduces deviationsfrom axisymmetry that are symmetric with respect to the coor-dinate transformations

φ → φ+ π/2 and φ → φ+ π . (16)

This implies that only azimuthal modesexp(imφ) with evenvalues ofm are contained in the “grid-mapping noise”. The am-plitudes are of the order of a few percent measured as relativedeviationsδψ := (ψ − 〈ψ〉φ) / 〈ψ〉φ of a scalar fieldψ(r, θ, φ)from its azimuthal mean〈ψ〉φ := (2π)−1

∫ 2π

0 ψ(r, θ, φ) dφ.This defines a lower limit for the size of the perturbation ampli-tudes, which we can impose explicitly onto the axisymmetricmodels. We point out that density perturbationsδ% ' 1% areabout two orders of magnitude larger than those typically used innumerical stability analysis (e.g., Houser et al. 1994; Aksenov1996; Smith et al. 1996).

We used explicit random perturbations of the density withan amplitude of 10%, i.e., the density distribution of the mappedmodel was modified according to

%(r, θ, φ) → %(r, θ, φ) · [1 + 0.1 f(r, θ, φ)

], (17)

wheref denotes a random function distributed uniformly in theinterval[−1, 1]. All azimuthal modes that can be resolved on agiven grid are therefore present in the initial data, them = 4 andm = 2 “grid-modes” mentioned above presumably being dom-inant. The corresponding model is refered to in the following asmodel MD1.

In a second model MD2, we imposed in addition a largescalem = 3 azimuthal perturbation with an amplitude of 5%,i.e.,

%(r, θ, φ) → %(r, θ, φ) · [1 + 0.05 sin(0.6 + 3φ)

]. (18)

Note that for convenience of notation we have given theform of the perturbation in spherical coordinates, although it isintroduced on a Cartesian grid.

M. Rampp et al.: Simulations of non-axisymmetric rotational core collapse 973

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a b

c d

Fig. 1a–dSnapshots of the density distribution (in units of [g cm−3]) and the velocity field in a meridional plane for the axisymmetric modelA4B5G5. Thez-axis coincides with the axis of rotation (and symmetry). The contours are logarithmically spaced with intervals of 0.25 dex,they are shaded with darker grey values for higher density regions and labeled with their respective values. The time of the snapshot and thevelocity scale are given in the top right corner of each panel.

3. Results

Zwerger & Muller (1997) have studied the overall dynamics ofaxisymmetric rotational core collapse and calculated the result-ing gravitational wave signal. In particular, they have investi-gated how the dynamics and the gravitational radiation dependon the (unknown) initial amount and distribution of the core’sangular momentum, and on the equation of state. Among the78 models computed by them, Zwerger & Muller (1997) foundonly one model (A4B5G5), whose rotation rate parameter con-siderably exceedsβdyn near core bounce. However,β remainslarger thanβdyn for only roughly one millisecond, because thecore rapidly re-expands after bounce and hence slows down. Inaddition, Zwerger & Muller (1997) found three more models(A3B5G4, A3B5G5 and A4B5G4) that fulfilledβ > βsec forseveral 10 ms (see Fig. 4 of Zwerger & Muller 1997).

The two “classical” dissipation mechanisms that can in prin-ciple drive secular non-axisymmetric instabilities, namely vis-

cosity of core matter (Roberts & Stewartson 1963) and gravita-tional radiation reaction (GRR; Chandrasekhar 1970), are neg-ligible during the time scales we have simulated evolution ofthe core (a few 10 ms). Nevertheless, some inner fraction of thecollapsing core could still become unstable, because of acousticcoupling with or due to advection of matter into the outer core(see e.g., Schutz 1983 and references therein). Therefore, wehave experimented with two of the three models exceedingβsec(but notβdyn) for several 10 ms by imposing non-axisymmetricperturbations on different hydrodynamic quantities (%, P , v)of different spatial character (m = 2, random) and amplitude(5%, 10%) at different epochs of the evolution. On a time scaleof several 10 ms we have found neither indications for secularinstabilities nor a significant enhancement of the gravitationalwave signal (compared with the corresponding axisymmetricmodel).

Thus, the only model remaining of the set of models ofZwerger & Muller (1997), which is a promising candidate for the

974 M. Rampp et al.: Simulations of non-axisymmetric rotational core collapse

Fig. 2. Radii of selected mass shells (for their definition see Eq. (20)in the text) as a function of time for two three dimensional modelsMD1 (dashed lines) and MD2 (dashed-dotted lines), and the axisym-metric model A4B5G5 (solid lines). The mass inside the shown massshells ranges from0.1 M to 0.9 M in intervals of0.2 M. Linescorresponding to0.1 M and0.7 M are drawn in bold.

growth of tri-axial instabilities during its gravitational collapse,is model A4B5G5 (see above). Its properties are described inthe next subsection. Some global quantities of the model havealready been published by Zwerger & Muller (1997).

3.1. The axisymmetric model A4B5G5

All pre-collapse models of Zwerger & Muller (1997) are ax-isymmetricn = 3 polytropes (p ∝ %1+1/n) in rotational equi-librium whose angular velocityΩ only depends on the distance$ from the axis of rotation according to the rotation law

Ω($) ∝(1 + (

$

A)2

)−1. (19)

Collapse is initiated by reducing the adiabatic indexΓ fromthe value 4/3, which is consistent with the structure of an = 3polytrope, to a somewhat smaller valueΓ1 (see Eq. (2)).

The initial model A4B5G52 is the most extreme one in thelarge parameter set considered by Zwerger & Muller (1997). Itis thefastest(βi = 0.04) andmost differentially(A = 107 cm;see Eq. (19)) rotating model being evolved with thesoftestequation of state(Γ1 = 1.28). According to the criterion de-rived by Ledoux (1945; Eq. 77) for rotating stars, a rotationrate ofβ = 0.04 requiresΓ1 > 1.3 for the star to be stableagainst the fundamental radial mode, i.e., collapse. The result-ing model has a mass of1.66M, a total angular momentum ofLz = 3.85 1049 erg s and an initial equatorial radius of 1280 km.

2 If not stated otherwise, quoted numbers always refer to the 2Dsimulation with the best angular resolution, i.e., the one performedwith nθ = 90 (see Sect. 2.3).

Fig. 3. Rotation rate parameterβ as a function of time for the three-dimensional models MD1 (dashed) and MD2 (dashed-dotted). Forcomparison, the evolution ofβ of the two-dimensional model is shown,too (solid). The solid horizontal lines mark the critical valuesβsec

(lower line) andβdyn (upper line), where MacLaurin spheroids be-come secularly and dynamically unstable to non-axisymmetric bar-likeperturbations.

Fig. 1a shows that model A4B5G5 has a torus-like densitystratification with an off-center density maximum, which is lo-cated in the equatorial plane 200 km away from the rotationaxis. During collapse a kind of “infall channel” forms along therotation axis, a structure which has already been found in somemodels of Monchmeyer et al. (1991). There matter falls almostat the speed of free-fall towards the center. With decreasing lat-itude (i.e., increasing polar angleθ) matter is increasingly de-celerated by centrifugal forces (Fig. 1b). Att ≈ 28 ms, whichis about 3 ms before core bounce, an oblate shock is visible(Fig. 1c). It begins already to form at the bottom of the infallchannel 7 ms before core bounce (Fig. 1b). Pressure gradientsdecreasing with increasing polar angleθ lead to an asymmetricpropagation of the shock its speed being larger in polar than inequatorial regions. The result is a “prolate shock in an oblatestar” (Finn & Evans 1990).

A second shock, which forms at the edge of the torus shortlybefore the density in the core reaches its maximum value insidethe torus, propagates behind the outer shock (Fig. 1d). Like theouter shock, the inner shock eventually becomes prolate, too.

The overall dynamics of the core can be conveniently an-alyzed in terms of the radial extent of different “mass shells”M(R) defined by

M(R) := 2π∫ R

0

∫ π

0〈%(r, θ)〉φ r

2 sin θ dθdr , (20)

where〈〉φ denotes the azimuthal mean of the density to be cal-culated in the 3D simulations.M(R) is the mass contained ina sphere of radiusR. Note that in more than one spatial dimen-

M. Rampp et al.: Simulations of non-axisymmetric rotational core collapse 975

Fig. 4.Maximum density on the grid as a function of time for the three-dimensional models MD1 (dashed) and MD2 (dashed-dotted). Forcomparison we also show the maximum density of the two-dimensionalmodel A4B5G5 (solid). The solid horizontal line marks the assumednuclear matter density%nuc = 21014g cm−3.

sion these mass shells are not identical with the well knownLagrangian mass shells.

Inverting M(R) one obtains the radiusRM of the massshell. This quantity is plotted as a function of time for differentmassesM in Fig. 2. One recognizes that maximum compressionis reached in model A4B5G5 att = 30.67 ms, which is followedby an expansion lasting for about 2 ms. Without any furthersignificant oscillations the inner core evolves towards its newequilibrium: The radii of mass shells withM <∼ 0.7M arenearly constant with time fort >∼ 33 ms (Fig. 2). The samebehaviour can be deduced from the evolution of the rotationparameter (β ∝ R−1 for a homogeneous sphere with radiusR)and of the maximum density on the grid (Figs. 3 and 4). Fig. 2also shows that the central coreM(R) < 0.7M oscillateswith a single dominant volume mode, which is in agreementwith the results obtained by Monchmeyer et al. (1991). Theyfound single volume modes in cores with coherent motion inequatorial and polar directions. This certainly occurs in modelA4B5G5, since matter in the torus which contains most of themass shows almost no motion in polar directions at all.

Matter in the torus (about0.7M) stays in sonic contact,i.e., it contracts coherently. It also remains unshocked well be-yond core bounce. Hence, the torus resembles the qualitativefeatures of a homologously contracting inner core that has beenfound analytically for the spherical collapse of polytropic cores(Goldreich & Weber 1980, Yahil 1983) and numerically inslowly rotating cores with ellipsoidal density stratification (Finn& Evans 1990; Monchmeyer et al. 1991).

The main features of the quadrupole amplitudeAE220 calcu-

lated for model A4B5G5 are: A slow monotonic increase withtime for the first 25 ms is followed by a pronounced negative

spike at the time of bounce with no subsequent short-period os-cillations (see Fig. 9). We therefore consider the gravitationalwaveform to be closest resembled by signal type II, which ischaracterized by prominent spikes arising at the bounce(s) dueto single dominant volume mode(s) (Monchmeyer et al. 1991).The non-vanishing polarization, calculated for a source at10 Mpc and for an observer located in the equatorial plane ofthe core, has a peak value ofh+ = −3.5 10−23. The energyof the quadrupole radiation isEGW = 7.8 10−8M c2. Mostof the spectral power is radiated at a frequency ofν ≈ 200 Hz.This is the largest signal strength obtained for any axisymmetriccollapse models in the set of Zwerger & Muller (1997). How-ever, the amplitude is still by at least one order of magnitude toosmall to be detectable even with the “advanced” LIGO interfer-ometer (see Abramovici et al. 1992 for the sensitivity limits), ifthe source is located in the Virgo cluster.

3.2. Evolution of non-axisymmetric models

The initial non-axisymmetric perturbations were introduced att0 = 28.13 ms (Figs. 5a and 7a). This is about 2.5 ms be-fore bounce (when the density reaches a maximum inside thetorus), and about 2 ms before the core’s rotation ratesβ >∼ 0.3,i.e., when it should become dynamically unstable. The addi-tional perturbation of 5% amplitude andm = 3 azimuthal de-pendence imposed on model MD2 can hardly be distinguishedfrom the 10% random noise perturbations of both models. Theinner torus has contracted from an initial radius of 200 km to oneof 60 km att = t0. Note that we refer to its density maximum,when we give its radial position.

The subsequent rapid contraction of the rotating core is re-flected by a steep rise of the maximum density, which peaks att = 30.68 ms in both 3D models and in the axisymmetric one(Fig. 4). At approximately the same time the contraction (Fig. 2),the rotation parameterβ (Fig. 3), the dissipation-rate of kinetic-infall energy into thermal energy and the gravitational potentialenergy reach their peak values. Thus, we considert = 30.68 msto be the time of bounce, although thecentral density doesnot reach its first peak then, which is usually considered as thebounce criterion. However, because of the torus-like densitystratification of the models, the central density has less mean-ing. During the simulations it remained always much lower thanthe maximum density inside the torus (by 1 – 3orders of mag-nitude). One also has to be careful, in particular in 3D simula-tions, when deriving implications from “local” quantities. Forexample, one might be tempted to conclude from the fact thatthe maximum density exceeds nuclear matter density in modelMD2, i.e.,%max > 2 1014g cm−3 (Fig. 4), that this model con-trary to model MD1 suffered a bounce due to the stiffening ofthe equation of state. However, nuclear matter density is ex-ceeded only in a few zones inside the torus (see Fig. 7c), whilethe corresponding azimuthal average, which is determining theoverall dynamics, is practically identical to that in model MD1.Therefore, we always compare the value of the maximum den-sity with the corresponding azimuthal mean, before drawing anyconclusions.

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Up tot ≈ 33 ms, i.e., up to about 2 ms after bounce, the non-axisymmetric initial perturbations have not grown far enoughto produce any significant changes in the overall dynamics ofthe collapsing core. This can be seen by comparing the timeevolution of the radii of mass shells defined by Eq. (20) (seeFig. 2). Also locally, non-axisymmetries are still small pertur-bations on a rapidly changing axisymmetric background. Thedensity contrast in the torus isδ% <∼ 0.15 for model MD2 att ≤ 33 ms (Figs. 7a–c). Hence, the distribution of hydrodyna-mic variables in planes perpendicular to the rotation axis is stillgiven by Fig. 1.

The subsequent evolution differs from that of the axisym-metric model and is also different between the two 3D models.

Model MD1:Due to the Cartesian geometry of the computa-tional grid, the toroidal mode withm = 4 sticks out of the noisefirst. The inner torus assumes a transient square-like structure,which exists for≈ 7 ms. This does not influence the overall dy-namics of the model significantly compared to the axisymmetric

one, although we find a somewhat less compact density strati-fication when comparing the radii of the innermost mass shellswith those of model A4B5G5 fort >∼ 33 ms (Fig. 2). When westopped the simulation att = 45.04 ms, the final configurationshows a prominent off-center density maximum in the equato-rial plane rotating with a period ofT ≈ 5 ms at a distance of$ ≈ 30 km.

Model MD2: The development of the instability isshown in Figs. 5 and 6. By the time of bounce three distinct den-sity maxima are visible the density contrast beingδ% <∼ 0.15 in-side the torus. These density maxima eventually grow into threedistinct clumps (Fig. 7c). The further evolution is characterizedby intensive hydrodynamic activity produced by the twirling-stick action of the three clumps. Most notably are trailing “spiralarms” causing mass and angular momentum to be transportedaway from the non-axisymmetric, high density regions (Figs.7d, e). This is mirrored in the evolution of the innermost(M ≤ 0.7M) mass shells, whose radii decrease during the

M. Rampp et al.: Simulations of non-axisymmetric rotational core collapse 977

28.12 ms 12.0 29.24 ms 12.0

30.52 ms 12.0 33.88 ms 12.0

36.92 ms 12.0 40.57 ms 12.0

Fig. 6.Surfaces of constant density% = 1012g cm−3 for model MD2.The edges of the cubic box have a length of 160 km. The snapshots aretaken at the same times as those in Fig. 7.

time when the spiral arms are present. The largest deviationsfrom the monotonic expansion of the mass shells of model MD1occur betweent ≈ 33 ms andt ≈ 38 ms, when the spiral armsare most prominent in model MD2. Eventually, the three spiralarms merge (Fig. 6 second last snapshot and Fig. 7e) and forma bar-like object inside theM = 0.7M mass shell (Fig. 6 lastsnapshot and Fig. 7f).

3.3. Growth region of the instability

For both 3D models we find that although perturbations havebeen imposed on the whole computational grid with the same

relative amplitude initially, they grow only in the immediatevicinity of the inner torus, i.e., inside a cube with an edgelength of ≈ 100 km (Figs. 6 and 7). This can be seen moreclearly in Fig. 8, where we have plotted the density (upper“curves”) of each grid cell in the equatorial plane versus its(cylindrical) coordinate distance$ :=

√x2 + y2 from the

origin. One notices considerable deviations from axisymme-try (which is mirrored in Fig. 8 by a large spread in den-sity at a given distance$) only for $ <∼ 40 km. That thisis not an artefact due to the usage of a multiple-nested re-fined grid can be seen as follows: The cube-shaped boundariesof the single grids are located at aCartesiancoordinate dis-tanceξbound ∈ 40 km, 80 km, 160 km, . . . from the origin(ξ ∈ x, y, z). Depending on the azimuthal angleφ this im-plies for each grid a doubling of the (linear) zone size atcylin-drical coordinate distances$bound ∈ [ξbound,

√2 · ξbound].

If the growth of the perturbations was damped notably by thecoarser numerical resolution at larger radii one would expectthe spread in density as function of$ to change significantly atthe limiting values of$bound, which is obviously not the casein Fig. 8.

In contrast, we observe a striking correlation between theMach-numberMa∗ calculated in a rotating coordinate systemwhose angular velocityΩ ≈ 1800s−1 is that of the torus att = 30 ms (whenβ = βdyn) and the domain where the densityvaries considerably with azimuthal angle: Non-axisymmetricperturbations grow only in regions where the flow is subsonic,i.e., whereMa∗ < 1 (Fig. 8).

Our qualitative interpretation of this result is as follows:Non-axisymmetric instabilities have been shown to occur inMacLaurin spheroids as well as in a large variety of (differ-entially) rotating compressible fluid bodiesin equilibrium, ifthe rate of rotation is sufficiently large. However, being equi-librium configurations, the instabilities in these models grow inthe absence of any radial motion. When viewed from a suit-ably chosen rotating coordinate system sonic contact along theazimuthal coordinate is established in these models and allowsfor a coherent growth of the instability. According to our re-sults, the latter condition seems to besufficientfor the growthof non-axisymmetric instabilities also in collapsing rotators. Ifon the other hand sonic contact is not given along the azimuthalcoordinate due to large radial velocities — like in the outer coreof our models — it is difficult to imagine how global tri-axialdeformations can develop coherently. Thus, we suppose thatsonic communication in azimuthal direction is also anecessarycondition for the growth of non-axisymmetric instabilities.

One might argue that these results depend on our particularchoice of the rotating coordinate system. However, establishingco-rotation at any distance$ (i.e., chosing a coordinate systemrotating with an angular velocity equal to that of the fluid atthat distance) outside the domain of the inner torus does not al-low to “transform away” supersonic velocities in the collapsing(outer) core, since the radial component dominates the angularcomponent of the velocity field by a large amount.

978 M. Rampp et al.: Simulations of non-axisymmetric rotational core collapse

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M. Rampp et al.: Simulations of non-axisymmetric rotational core collapse 979

30.20 ms 30.52 ms

31.16 ms 31.48 ms

Fig. 8. Scatter plots of the density (upper “curves”, logarithmic scale,in units of [g cm−3]) and the Mach numberMa∗ (lower “curves”,calculated in a coordinate system rigidly rotating with the torus) inthe equatorial planez = 0 versus distance$ :=

√x2 + y2 from

the rotation axis at different instants in time (see upper right cornerof each panel). The bold horizontal line separates supersonic (upper)from subsonic (lower) flow regions. The thin vertical lines indicate theboundariesξbound of the nested grids (see text).

3.4. Gravitational wave signal

Although models MD1 and MD2 show prominent deviationsfrom axisymmetry, the gravitational wave signal is changedonly marginally compared with the axisymmetric calculation. Inparticular, the maximum wave amplitudes are equal for the 3Dand the 2D simulations within an accuracy of 2%. The wave-forms and the energy radiated in form of gravitational wavesare displayed in Fig. 9. The peak amplitude ofh+ is reached atabout the time of bounce in both the 2D and the 3D models. Upto this point in the evolution the waveforms are identical, too.This is not surprising, since the initial perturbations of mod-els MD1 and MD2 have not grown significantly until the timeof bounce, which occurs only 0.5 ms afterβdyn is reached. Asdiscussed above, the density contrast isδ% <∼ 0.15 inside the in-ner torus before bounce. Accordingly,h×, which is of genuinenon-axisymmetric origin nearly vanishes until bounce (Fig. 9).

The maximum amplitudes ofh× are reached during the fur-ther evolution, when the spiral arms merge to form the final bar(model MD2) or when the transientm = 4 structure evolvesto some “bar-like” configuration (model MD1). These maximaare only of the order of 10% of the maximum value of|h+|.The amplitudes of the cross- and plus-polarizations finally be-come comparable, because the inner core (torus) approachesits new rotational equilibrium and thus the time derivatives of

Fig. 9. Gravitational wave amplitudesrh+(x=r, y=0, z=0), (thicklines),rh×(x=0, y=0, z=r), (thin lines) and radiated energyEGW

as functions of time for the three-dimensional models MD1 (dashed)and MD2 (dashed-dotted) in comparison with the axisymmetric calcu-lation (solid). The peak values|rh| ' 1000 cm correspond to dimen-sionless amplitudes|h| ≈ 3 10−23 for a source atr = 10 Mpc.

the quadrupole moments due to radial motion become steadilysmaller. The periodicity seen inh× is due to a nearly solid-bodytype rotation of the high density regions of the core.

Comparing the 3D results with the 2D signal, one noticesthat the 3D waveforms do not reproduce the small additionallocal minimum inh+ visible att ≈ 31 ms for the axisymmet-ric model. Also the (absolute values of the) time derivatives ofh+ are smaller for the following 2 ms. This leads to a smallervalue of the amount of energy radiated in form of gravitationalwaves (Eq. 10; for model MD1h+ is roughly proportional toI−xx and I−yy during this time interval because of the near ax-isymmetry of the core at this time). We have analyzed carefullythe unexpected fact that compared to the axisymmetric mod-els we obtain only 65% of the gravitational wave energy forthe two models MD1 and MD2, which during this epoch areapproximately, but not exactly axisymmetric. Several tests andcomparisons (e.g., with poorer resolved 2D calculations) haveshown that this isnot a possibly unnoticed 3D effect, but dueto a somewhat lower “angular” resolution of the 3D simulationcompared with the best resolved 2D run (see Rampp (1997) fordetails). The lower resolution gives rise to a larger violation oftotal energy conservation, most of which occurs during bounce.Since all dynamical quantities of model MD1 – local as well asglobal ones – agree with the 2D results within an accuracy ofa few percent, we consider the observationally relevantwave-formscalculated in the 3D simulations to be reliable within theunderlying approximations (cf. sect. 2). We also point out thefact that the squared time derivatives of the momentsI−ij en-ter the formula for the radiated energy. Small deviations in thewaveforms therefore can account for quite large differences in

980 M. Rampp et al.: Simulations of non-axisymmetric rotational core collapse

Fig. 10. Rotation parameterβ as a function of time for the three-dimensional model MD3 (dotted), which suffers a second collapse. Forcomparison the evolution ofβ of model MD1 is shown, too (dashed).

the energies. It is not surprising, though not easy to explainin detail, that the 3D waveforms differ from the axisymmet-ric waveforms at later timest >∼ 33 ms, when the dynamicalevolution of these models is changed significantly due to theirnon-axisymmetric inner cores.

Given the dynamical evolution of the core, the small mag-nitude of the amplitudesh× can easily be explained by uti-lizing the well known order-of-magnitude argument for thequadrupole waveforms

rh× ' 2G

c4I−

T 2 ' 2G

c40.1MR2

T 2 ' 100 cm. (21)

According to the results of our computations, we have in-serted the quadrupole momentI− of a homogeneous bar withmassM ∼ 0.5M, lengthR ∼ 100 km and rotation pe-riod T ∼ 5 ms. Approximating the non-axisymmetrically dis-tributed mass of∼ 0.5M as two point masses orbiting eachother at a distance of 100 km with an angular velocity equal tothe observed one, yields the same order-of-magnitude forh×.

Finally, by using the quadrupole formulae (Eqs. 7 and 8) wemay have underestimated the amount of gravitational radiationproduced by model MD2, particularly during the phase whenthem = 3 symmetry of the inner core is most prominent. Weestimate the order-of-magnitude of the mass-octupole contribu-tion to the signal by (e.g., Blanchet et al. 1990, Eq. 6.8)

rhTToct ' 2

G

c43c

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(vc

)3<∼ 500 cm, (22)

wherev ∼ 0.1 c andM ∼ 0.5M and a Schwarzschild radiusRs ∼ 1.5 105 cm was assumed. Hence, the mass-octupole ra-diation cannot account for significant enhancement of the peakamplitudes of our model MD2 compared with the axisymmetriccase (Fig. 9) or MD1, although it might change the details of thewaveforms.

3.5. Second collapse of a non-axisymmetric core

After bounce, the proto-neutron star settles into its new rota-tional equilibrium, and cools and contracts on a secular timescale, which is given by the neutrino-loss time scale (' 10 s;see e.g., Keil & Janka 1995 and references cited therein). Ac-cording to linear stability analysis tri-axial perturbations growon this time scale, provided the rotation parameterβ >∼ 0.14. Fornumerical reasons (because our hydrodynamic code is explicitand thus the time step is limited by the CFL stability condition;see e.g., LeVeque 1992) as well as for physical reasons (neglectof weak interactions and neutrino transport) our approach is in-adequate for simulating this secular evolution. However, we areable to consider asuddenreduction of the stabilizing pressurein a deformed, non-axisymmetric, rapidly rotating post-bouncecore (model MD3). This extreme case cannot provide the an-swer to the question whether secular instabilities will indeedgrow to nonlinear amplitudes and what their influence will beon the evolution of the core. Nevertheless, such a simulation canshed some light on the problem how much gravitational radia-tion can be expected, when a rapidly rotating non-axisymmetricneutron star forms.

We choose model MD1 att = 43.12 ms as the starting pointfor the simulation (Fig. 11a). At this time we (suddenly) reduceΓ1 from its original value of 1.28 to a value of 1.2. This rendersthe rotating core unstable against a second dynamic collapse. Inpassing we note that the stability criterion due to Ledoux (1945;Eq. 77) would requireΓ1 <∼ 1.18 for a rigidly rotating coreinequilibriumwith β ≈ 0.16 to collapse.

The overall contraction, bounce and re-expansion of modelMD3 is reflected in the time evolution of the rotation parameterβ. Note, that concerning the development of non-axisymmetricinstabilities the absolute value ofβ is not very relevant here,because the initial “perturbations” are already in the non-linearregime. Compared to the (first) collapse of model MD1 we find asomewhat longer time scale for the contraction and re-expansionin model MD3. The peak value ofβ ≈ 0.23 is considerablysmaller than that of model MD1 (β ≈ 0.34), because onlymass shells withM <∼ 0.5M contract to radii similar to thosereached during the collapse of model MD1. Mass shells of modelMD3 withM >∼ 0.8M remain nearly unaffected by the suddenpressure reduction att = 43.12 ms.

During the collapse of model MD3 several pronounced den-sity maxima are formed. They are all located inside the torusreaching peak values of% = 5.3 1014g cm−3. The five ini-tial clumps forming at the density maxima contain a mass of≈ 0.1M each (Fig. 11b). During the further evolution theirnumber decreases as they merge one after the other. Eventually,just two of them remain, which form a bar-like or binary-likedeformed central region (Fig. 11c and d).

The maximum density on the grid (reached in the clumps)sharply rises from2 1013g cm−3 to 5.3 1014g cm−3 within afraction of a millisecond after pressure reduction. The collapseis, however, rapidly decelerated by the action of centrifugalforces, which also eventually cause a bounce att ≈ 46 ms(Fig. 10). The pressure increase resulting from matter whose

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density exceeds nuclear matter density is dynamically unim-portant, as only a negligible amount of mass inside some of theclumps is involved (Fig. 11).

Model MD3 shows a highly non-axisymmetric, bar-likedensity stratification alreadybeforethe (second) bounce occursat t ≈ 46 ms (Fig. 11b). This is different from the behaviourof models MD1 and MD2, where large non-axisymmetries(quadrupole moments) only appear wellafterbounce, i.e., afterthe most rapid phase of the evolution. Accordingly, one expectslarger time derivatives of larger quadrupole moments and there-fore a stronger gravitational wave signal from model MD3 thaneither from model MD1 or MD2.

The actual waveforms for model MD3 are shown in Fig. 12.Compared to model MD1 one notices a considerably largervalue for the genuinely non-axisymmetric polarization ampli-tudeh×. The maximum values|rh×| ≈ 800 cm and|rh+| ≈300 cm do, however, neither exceed the peak value of|rh+| ob-tained in the axisymmetric simulation at bounce nor those ofthe three-dimensional models MD1 and MD2 (Fig. 12).

Additional (axisymmetric) simulations withΓ1 reduced to1.2 or 1.25 have shown that the peak value ofβ, the time scaleof the (second) collapse as well as the peak values of the grav-itational waveforms do not depend strongly on the exact valueof Γ1.

4. Summary and discussion

We have presentedthree-dimensionalhydrodynamic simula-tions of the collapse of rapidly rotating stellar iron cores. Thematter in the core has been described by a simple analyticalequation of state. Effects due to neutrino transport have beenneglected. Hydrodynamics and self-gravity has been treated inNewtonian approximation. The initial models for our simula-tions have been taken from the comprehensive parameter studyof axisymmetricrotational core collapse performed by Zwerger& M uller (1997).

For our study we have selected those axisymmetric mod-els which are secularly or dynamically unstable with respect

982 M. Rampp et al.: Simulations of non-axisymmetric rotational core collapse

Fig. 12. Gravitational wave amplitudesrh+(x = r, y = 0, z = 0),(thick lines) andrh×(x=0, y=0, z=r), (thin lines) as functions oftime for the three-dimensional model MD3 (dotted), wherer denotesthe distance to the source. For comparison the amplitudes of modelMD1 are shown with dashed lines.

to non-axisymmetric perturbations. Hence, in order to be se-lected the rotation rate parameterβ of a model had to exceedthe critical rotation ratesβsec ≈ 0.14 or βdyn ≈ 0.27 dur-ing its evolution. These critical rotation rates have been derivedfor incompressible MacLaurin spheroids, but also hold approx-imately for a wide range of compressibleequilibriumconfigu-rations (e.g., Tassoul 1978). After mapping the rapidly rotatingaxisymmetric cores onto the three-dimensional computationalgrid, we imposed an initial non-axisymmetric perturbation andsimulated the further evolution with a three-dimensional vari-ant of the PROMETHEUS hydrodynamic code. The simulationscover a time interval from a few milliseconds before core bounceup to several tens of milliseconds after bounce.

We have used the quadrupole formula to calculate the grav-itational wave signal and have estimated the contribution ofthe next-leading radiative-multipole order, which is negligiblysmall. Because of its smallness gravitational radiation reactionwas not taken into account.

Our 3D simulations show that in two models, which aresecularly but not dynamically unstable (0.15 ≤ β ≤ 0.2) non-axisymmetric perturbations do not grow. Consistent with thisresult, we also observe no significant enhancement of the grav-itational wave emission in these models compared to the ax-isymmetric case.

Among the models investigated by Zwerger & Muller (1997)there is only one model whereβ > βdyn during the evolution.This is their most rapidly and most differentially rotating modelevolved with the softest equation of state. This axisymmetricmodel was perturbed about 2 ms before bounce by imposinga random density distribution of 10% amplitude. Despite therandom initial perturbation,m = 2 andm = 4 toroidal modesare found to dominate the early evolution of the model, becauseof the set of cubic grids used in the simulation. Therefore, a

second simulation was performed, where an additionalm = 3toroidal perturbation of 5% amplitude was imposed initially.

The gross features of the evolution are quite similar in the ax-isymmetric and non-axisymmetric simulations. In both cases theoverall evolution is characterized by a rapid contraction, bounceand re-expansion of the inner core. In the 3D simulations we, inaddition, observe the growth of the initial perturbations. Untilseveral milliseconds (or dynamical time scales of the inner core)after bounce the density distribution resembles the symmetry ofthe initially dominating modes. Subsequently, the core evolvestowards a bar-like shaped configuration. In none of the 3D mod-els the gravitational wave amplitude exceeds the axisymmetricvalue of|h| = 3.5 10−23 (for a source at 10 Mpc), three dimen-sional effects on the waveforms being only of the order of 10%.Even when the collapsed, bar-like, rapidly rotating inner core isforced into a second collapse (by artificially reducing its adia-batic index), we do not observe gravitational wave amplitudessignificantly larger than|h| ' 10−23 (for a source at 10 Mpc).

In order to judge the implications of our results for rotationalcore collapse in general and for gravitational wave astronomyin particular, the following possible limitations of our approachshould be kept in mind:

(i) SinceGM/Rc2 <∼ 0.2 for the axisymmetric models ofZwerger & Muller (1997), general relativistic effects canbe viewed as moderate corrections to Newtonian gravity asfar as the collapse dynamics is concerned. However, sincegeneral relativity counteracts the stabilizing influence of ro-tation on radial modes (e.g., Tassoul 1978), its influence onthe stability properties of (even only moderately compact)rotating iron cores can be of considerable importance. WhenGR is taken into account, pre-collapse models with a largeramount of angular momentum (and the same EOS) than inthe Newtonian approximation can collapse. Furthermore,GR models will maintain higher densities as well as largerrotation parameters for a longer time interval after bounceas compared to Newtonian models (see the discussion inZwerger & Muller 1997).

(ii) The secular evolution(i.e., the evolution on time scales ofhundreds of milliseconds) of the collapsed core is not knowneven in the axisymmetric case. Simulating this evolutionrequires models with detailed microphysics and thermody-namics, and an adequate treatment of the neutrino transport.However, even in 2D such simulations are still prohibitivelytime consuming. Thus, there might exist additional initialmodels different from those presently available, which couldbecome unstable to tri-axial perturbations on a secular timescale (see however the next point).

(iii) Concerning thelimited set of axisymmetric initial models,we adopt the arguments given in Zwerger & Muller (1997).They claim that the large parameter space considered in theirstudy most probably comprises the whole domain whereactual pre-collapse rotating iron cores are to be found. Theyfurther argue that using a realistic equation of state insteadof a polytropic one will change the details of the evolution,but will not give rise to qualitatively different results. In

M. Rampp et al.: Simulations of non-axisymmetric rotational core collapse 983

particular, it is unlikely that there exists a much larger setof pre-collapse initial models, which fulfillβ > βdyn andwhich are sufficiently compact for an extended time intervalafter core bounce.But even if there are models which fulfill these conditionsand become triaxial, we do not expect them to produce a con-siderably stronger gravitational wave signal than the “best”axisymmetric models. Provided the initial amount of an-gular momentum is not unreasonably large, the critical ro-tation rates for non-axisymmetric instabilities can only beexceeded in collapsing cores with relatively soft equationsof state (Eriguchi & Muller 1985, Zwerger & Muller 1997).But these soft equations of state drastically reduce both themass and the radius of the inner core (Zwerger & Muller1997). However, according to our 3D calculations, it is onlyin the inner core, where non-axisymmetric perturbations cangrow significantly. Taken together these considerations sug-gest that bar-like inner cores have masses less than0.5M,radii less than100 km, and rotation periods greater than1 ms. This implies a maximum gravitational wave ampli-tude|h| <∼ 10−22 (for a source at 10 Mpc).

Given the most recent rates of core collapse supernovaevents (e.g., Cappellaro et al. 1997), which not all may in-volve a rapidly rotating core, we conclude that it is ratherunlikely to expect gravitational wave signals from dynamicnon-axisymmetric instabilities with sufficient strengthandrate(i.e., several per year) to be observable with the large interfer-ometric gravitational wave antennas presently under construc-tion.

Movies in MPEG format of the dynamical evolution of allmodels are available in the world-wide-web athttp:// www.mpa-garching.mpg.de/˜mjr/GRAV/grav3.html

Acknowledgements.The calculations were performed at the Rechen-zentrum Garching on a Cray J90. This research was supported in partby the National Science Foundation under Grant No. PHY 94-07194.

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