Gravitational radiation from convective instabilities in...

Astron. Astrophys. 317, 140–163 (1997) ASTRONOMYAND

ASTROPHYSICS

Gravitational radiation from convective instabilitiesin Type II supernova explosionsEwald Muller and H.-Thomas Janka

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

Received 5 February 1996 / Accepted 23 April 1996

Abstract. We present two- and three-dimensional simulationsof convective instabilities during the first second of a Type II su-pernova explosion. Convective overturn occurs in two distinct,spatially well separated regions: (i) inside the proto-neutron starimmediately below the neutrinosphere (r <∼ 50 km) and (ii) inthe neutrino-heated “hot-bubble” region interior to the outwardpropagating revived shock wave (100 km <∼ r <∼ 1000 km). Wehave calculated the gravitational wave signature of both con-vective instabilities including the quadrupole waveforms, theenergy spectra, and the total amount of the emitted gravitationalwave energy. Moreover, we have estimated the amplitude andenergy of gravitational waves associated with the anisotropicneutrino emission that is caused by the convective transport ofneutrinos and by aspherical perturbations of temperature anddensity in the neutrinospheric region.

For a supernova located at a distance of 10 kpc the maxi-mum dimensionless gravitational wave amplitudes due to con-vective mass motions range from |hTT| ≈ 2 × 10−22 for thethree-dimensional simulation to |hTT| ≈ 3×10−21 for the moststrongly radiating two-dimensional model. The total emittedenergy varies from 3 × 10−14 Mc2 to 5 × 10−10 Mc2. Theconvective mass motions inside the proto-neutron star producea stronger signal than convection in region (ii) with up to a fac-tor of 10 larger amplitudes and 1000 times more gravitationalwave energy. Because of smaller convective eddies and struc-tures and slower overturn velocities, the wave amplitudes ofthree-dimensional models are more than a factor of 10 smaller,and the energy emitted in gravitational waves is almost 3 ordersof magnitude less than in the corresponding two-dimensionalsituation.

In two dimensions the gravitational wave amplitude asso-ciated with the anisotropic emission of neutrinos can be larger(factor 5) than the wave amplitude due to mass motions in theproto-neutron star, although the energy in the neutrino tidal fieldis 20 times smaller. In three dimensions the neutrino gravita-tional wave amplitude is reduced by a factor of about 10 andthe gravitational wave energy by a factor of roughly 100 rela-tive to the two-dimensional results. Nevertheless, the neutrinotidal field is more than a factor of 10 larger than the gravitational

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wave amplitude from mass motions and the corresponding grav-itational wave energies can be of similar size.

Most of the gravitational radiation from convection insidethe proto-neutron star is emitted in the frequency band 100–1000 Hz, while convective motions in the hot-bubble regiongenerate waves from several 100 Hz down to a few Hz. Gravi-tational waves from the anisotropic neutrino emission have mostpower at frequencies between some 10 Hz and a few 100 Hz anda low-frequency contribution at about 1 Hz to several Hz.

Features in the gravitational-wave signal from the neutrino-heated region are well correlated with structures in the neu-trino signal, both being associated with sinking and rising lumpsof matter and with temporal variations of aspherical accretionflows towards the proto-neutron star. A simultaneous measure-ment of both signals would impose important constraints on thedynamics of Type II supernovae and theoretical models of theexplosion mechanism.

Key words: supernovae: general – stars: neutron – gravitational:waves – hydrodynamics – convection – instabilities

1. Introduction

Currently the most promising model of Type II supernova ex-plosions, which are powered by the gravitational binding energyreleased in the collapse of the iron-nickel core of a massive starwith a main sequence mass of 8 M <∼M <∼ 25 M, relies onthe so-called delayed explosion mechanism by neutrino heating(Wilson 1985; Bethe & Wilson 1985; Wilson et al. 1986; Col-gate 1989; Janka 1993). According to this mechanism energyis deposited in the layers between the nascent neutron star andthe stalled prompt supernova shock during a period of a few100 ms by absorption of a minor fraction (between 1 and 10%)of the neutrinos emitted from the collapsed core. This neutrino-energy deposition produces a radiation dominated hot-bubbleregion and eventually causes the supernova explosion by reviv-ing the stalled shock wave.

E. Muller & H.-T. Janka: Gravitational radiation from convective instabilities in Type II supernova explosions 141

After the shock wave stalls (≈ 10 ms after bounce) andbefore the delayed neutrino heating mechanism begins to be-come effective (≈ 100 ms after bounce) the outer parts of theproto-neutron star are convectively unstable because, firstly,the deleptonization occurring in the shocked matter outsidethe neutrinosphere produces a negative lepton gradient and be-cause, secondly, the weakening (and the eventual stalling) ofthe prompt shock wave gives rise to a negative entropy gra-dient in the same region. According to Epstein (1979) this isa sufficient condition for convective motion to occur. This isa situation commonly encountered in most supernova simula-tions (Burrows 1987; Burrows & Lattimer 1988; Bruenn 1993).The existence of these instabilities was demonstrated in sev-eral multi-dimensional hydrodynamical simulations (Burrows& Fryxell 1992, 1993; Burrows et al. 1995; Janka & Muller1993a,b; 1995a,b; Muller 1993; Muller & Janka 1994).

Noting that the hot bubble is convectively unstable, Bethe(1990) and Colgate (1989; see also Colgate et al. 1993) pointedout that this situation may give rise to a dynamical overturn ofhot, neutrino-heated, rising material and cold postshock matter,and can lead to large-scale deviations from spherically symmet-rical supernova explosions. This instability was indeed found innumerical simulations (Burrows et al. 1995; Herant et al. 1992,1994; Janka & Muller 1993b, 1995a, 1996; Yamada et al. 1993;Shimizu et al. 1993, 1994; Muller & Janka 1994; see, however,Miller et al. 1993).

Provided sufficient energy is deposited by neutrinos in thepostshock matter an energetically typical Type II supernova ex-plosion (Eexp ≈ 1051 erg) results. Whether this explosion mech-anism is successful or not depends very sensitively on the neu-trino flux that is emitted from the neutrinosphere of the hotproto-neutron star and its decay with time (Janka 1993; Janka& Muller 1995a,b, 1996).

While propagating outward through the stellar envelopethe shock wave will produce density and pressure stratifica-tions, which are Rayleigh-Taylor unstable in the neighbour-hood of composition interfaces, i.e. near nuclear burning shells(Bandiera 1984; Benz & Thielemann 1989). This third kindof instability (actually a set of instabilities) in Type II super-nova explosions has numerically been investigated in great de-tail (Fryxell et al. 1991; Hachisu et al. 1990, 1991; Herant &Benz 1991, 1992; Muller et al. 1991) and will not be consid-ered further here.

In Table 1 an overview over the different types of hydro-dynamical instabilities in Type II supernova explosions is pre-sented and some important characteristics are summarized.

The non-spherical stratification and mass flow resultingfrom these instabilities, or, more precisely, the time-dependentmass quadrupole moment, is a potential source of gravitationalradiation (for a recent review, see Thorne 1995). Obviously,only convection inside the proto-neutron star and inside the hot-bubble region is a promising source, because both convectiveregions are located at relatively small radii (see Table 1). Theinstabilities in the stellar envelope occur at such large radii thatno strong gravitational wave signal can be expected. Hence,we have only analysed the gravitational wave signature of the

two former instabilities. In addition, we have attempted to es-timate the gravitational wave amplitudes and energy associatedwith the anisotropic neutrino emission from our models (Ep-stein 1978, Turner 1978). In particular the convective processesaround the neutrinosphere can lead to anisotropies of the neu-trino luminosity which are caused by convective transport ofneutrinos and by temperature and density variations in the neu-trinospheric region.

In the following we present the first detailed investigation todetermine the gravitational radiation from post-bounce convec-tive motions in Type II supernova explosions. The results arebased on several two-dimensional hydrodynamical simulationsand on two three-dimensional ones (Janka & Muller 1995a,1996; Muller 1993). Special attention is therefore focussed onthe comparison of the characteristics of the convective overturnin the two- and three-dimensional cases and on the correspond-ing differences of the gravitational-wave emission. Our analy-sis is in some sense complementary to other investigations, inwhich the gravitational radiation from collapsing, rotating coreswas computed (Monchmeyer et al. 1991; Bonazzola & Marck1993; Zwerger 1995; Muller & Zwerger 1995). In rotationalcore collapse the gravitational radiation results from a time-dependent quadrupole moment due to a coherent, large-scaledeviation from spherical symmetry caused by the action of cen-trifugal forces, whereas the gravitational radiation emitted fromthe convectively unstable regions is produced by small-scale sta-tistical deviations from sphericity. Another potential source ofgravitational radiation from Type II supernova are asphericitieswhich can already exist in the Si- and O-shells of the progen-itor star (Bazan & Arnett 1994) and which might be amplifiedduring core collapse (Burrows & Hayes 1996).

The paper is organized as follows. In Sect. 2 we describethe results of our two- and three-dimensional simulations withthe main emphasis being put on the simulations of convectioninside the proto-neutron star, because the results for the convec-tive overturn processes in the hot-bubble region have alreadybeen presented and discussed earlier (see Janka & Muller 1995a,1996). In Sect. 3 first the formalism is described which wasused to calculate the gravitational wave signature of our mod-els. Thereafter, the quadrupole waveforms of the gravitationalradiation, the frequency-dependent spectra of the wave ampli-tudes, the spectral energy densities (i.e., the differential energyemitted at different frequencies), and the total, i.e. spectrallyintegrated, energies emitted in gravitational waves are given forall our models. Finally, we summarize our results and discusstheir implications in Sect. 4.

2. Simulations of convection

All simulations were performed with a modified version ofthe explicit hydrodynamical code PROMETHEUS (Fryxell etal. 1989; Muller et al. 1991), which is a direct Eulerian im-plementation of the Piecewise Parabolic Method of Colella &Woodward (1984).

142 E. Muller & H.-T. Janka: Gravitational radiation from convective instabilities in Type II supernova explosions

Table 1. Hydrodynamical instabilities in Type II supernova explosions

type of instability convection inside the convection in the Rayleigh-Taylor instabilitiesproto-neutron star hot-bubble region in the stellar envelope(unstable mass region: (unstable mass region: at composition and shell0.7 M <∼ m(r) <∼ 1.20 M 1.25 M <∼ m(r) <∼ 1.40 M interfaces after shockfor iron core masses for iron core masses passage≈ 1.3 – 1.4 M) ≈ 1.3 – 1.4 M)

cause of instability prompt deleptonization burst energy deposition by neutrinos unsteady shock propagationand failure of prompt shock causing unstable entropy through stellar envelopecausing unstable entropy gradient in layers behind causing unstable density andand/or lepton gradients stalled and revived shock wave pressure stratifications

radial location 10 (?) – 50 km ≈ 100 – 1000 km >∼ 5000 km

onset after shock ≈ 10 – 20 ms ≈ 50 – 80 ms ≈ 0.3 – 0.5 hrsformation

duration ≈ 20 ms – sec (?) ≈ 100 – 500 ms ≈ 2 – 3 hrs

convective overturn ≈ 1 – 10 ms ≈ 40 – 100 mstimescale

maximum gravitational ≈ 3 × 10−21 ≈ 5 × 10−22

wave amplitude fora source at 10 kpc

energy radiated in form of <∼ 5 × 10−10 Mc2 <∼ 1 × 10−12 Mc2

gravitational waves

typical frequency of 100 – 1000 Hz 10 – 100 Hzgravitational waveemission

2.1. Convection inside the proto-neutron star

The simulations concentrating on the convective processes in-side the proto-neutron star have been performed neglecting ef-fects due to neutrinos completely. This is justified, since themain purpose of the simulations was to demonstrate the exis-tence of the convective instability inside the proto-neutron starand to reveal the relevant length scales and timescales of theconvective overturn. We followed these processes only over aperiod of about 30–50 ms and do not expect any crucial mod-ification of the gas flow by neutrino-energy gain or loss. Inparticular, the convective velocities seen in our simulations areso high that convective mixing is probably faster than neutrinodiffusion.

Neutrino viscosity should also be negligible, because theReynolds number of neutrino shear viscosity is relatively largefor the considered densities ρ ≈ 1011–1012 g/cm3, tempera-tures T ≈ 3–10 MeV, length scales l ≈ 106 cm and velocitiesu ≈ 109 cm/s in the convective region. Estimates based on theexpressions given by van den Horn & van Weert (1981) yieldfor the dynamical shear viscosity of neutrinos in the diffusiveregime

ην ≈ 3× 1023 T 25

ρ12

[ gcm s

](1)

and for the corresponding Reynolds number

<(ν) =ρul

ην≈ 3000

(ρ12

T5

)2

u9 l6 , (2)

where T5 is the temperature in 5 MeV, ρ12 the density in1012 g/cm3, u9 the fluid velocity in 109 cm/s and l6 the typi-cal length scale in 106 cm.

All simulations were performed with an elaborate, vector-ized equation of state, which contains contributions from neu-trons, protons, alpha particles, and a representative heavy nu-cleus in nuclear statistical equilibrium. Electrons, treated as anarbitrarily degenerate and relativistic gas, positrons, and pho-tons were taken into account.

The initial model was a spherically symmetrical (non-rotating) post-bounce model of Hillebrandt (1987) which rep-resented the approximately 1.4 M iron core of a 20 M starabout 12 ms after core bounce. As this model was computedwith a general relativistic correction to the gravitational poten-tial, the model had to be relaxed before it was mapped ontothe multi-dimensional grid. This was achieved by evolving themodel with the 1D Lagrangian hydrodynamics code of Janka(unpublished) for another 17 ms. Then all waves created by theinitial force imbalance had died out.


In order to save a significant amount of computer time weused an inner boundary condition at a fixed finite radius Rin =15 km, which corresponds to an interior massMin = 0.639 M.Hence, we did not simulate the flow in the inner part of the proto-neutron star. We assumed hydrostatic equilibrium at the innerboundary and approximated the interior mass by an equivalentcentral point mass. The usage of such an inner boundary is welljustified for sufficiently short times, because initially the innerpart of the proton-neutron star is convectively stable (see Figs. 1and 2). Of course, one must take care that the computational gridextends far enough into the stable region.

Self-gravity of the matter in the computational domain wascomputed with the efficient algorithm of Muller & Steinmetz(1995), which solves Poisson’s equation in integral form byan expansion into spherical harmonics. In the 3D simulationsa spherically symmetrical potential was used, which resultsfrom the angular-averaged density distribution outside the innerboundary and the central point mass. As the overall matter distri-bution remains nearly spherically symmetrical during the evo-lution of the convective instability, this approximation shouldcause only small errors. Two-dimensional test calculations withand without a spherically symmetrical potential confirmed thesmallness of the errors.

The multi-dimensional simulations were started by mappingthe one-dimensional post-bounce model onto the 2D (or 3D)grid and by perturbing the radial velocity on the whole gridwith a random perturbation of 10−3 amplitude. When reducingthe perturbation to 10−5 the results changed only little.

2.1.1. Two-dimensional simulations

The two-dimensional simulations of convection inside theproto-neutron star were performed in spherical coordinates us-ing a computational grid of 400 (non-equidistant) radial and 90(equidistant) angular zones. Axial and equatorial symmetry wasassumed. The radius of the inner boundary was set to 15 km (seeabove), while the outer edge of the grid was located at a radiusof 2000 km.

In Figs. 1 and 2 the Ye and entropy distributions, respec-tively, are displayed at several moments of time during the evo-lution. In both figures the solid curve gives the initial distribu-tion.

The deep trough in the Ye profile is the result of neutrinolosses and deleptonization around the neutrinosphere, whereneutrinos decouple from the stellar gas and where their modeof propagation changes from diffusion to free streaming. Sinceelectron neutrinos are created when electrons and protons com-bine to neutrons, the gas close to the proto-neutron star “sur-face” quickly neutronizes and the concentration of electrons,Ye, decreases. The entropy maxima at M (r) = 0.88 M andM (r) = 1.26 M are caused by a combination of shock propa-gation and post-bounce oscillations of the collapsed iron core.

Figs. 1 and 2 clearly show that for 0.87 M <∼ M (r) <∼1.07 M the initial entropy gradient as well as the initial Yegradient are negative, i.e., the stratification is convectively un-stable. From the inner boundary at M (r) = 0.639 M out to

Fig. 1. Time evolution of the angular-averaged electron number fraction〈Ye〉 plotted versus enclosed mass M (r) for a 2D simulation: solid(t = 0), dotted (t = 12 ms), short dashed (t = 14 ms), long dashed(t = 18 ms), and dash-dotted (t = 32 ms). Note that at t = 0 ms themodel is spherically symmetrical

Fig. 2. Same as Fig. 1 but showing the angular-averaged (total) entropyper nucleon

M (r) = 0.69 M and for 0.77 M <∼ M (r) <∼ 0.87 Mthe Ye gradient is negative but the entropy gradient is positive.The mass layer in between (0.69 M <∼ M (r) <∼ 0.77 M)is stable as both gradients are positive. Outside the convec-tively unstable region the entropy gradient remains negativeup to M (r) = 1.2 M, while the initial Ye gradient becomespositive. Even further out both gradients are positive up to thelocal entropy maximum at M (r) = 1.26 M, implying a con-vectively stable stratification. In a narrow adjacent mass layer(1.26 M <∼ M (r) <∼ 1.29 M) which is located immediatelybehind the shock the entropy gradient is strongly negative whilethe lepton gradient, which is positive at the inner edge of thelayer, quickly becomes zero. According to the stability criterionthis is a convectively unstable situation. We point out that suchunstable stratifications can also be found in other core collapse


simulations, as e.g., in post-bounce models of Bruenn (1993).Note that although the different regions may be rather narrow interms of the mass coordinate M (r), they can nevertheless havea sizable radial extension of several ten kilometers.

The evolution of the angular-averaged electron number frac-tion 〈Ye〉 and of the angular-averaged entropy 〈S〉 is shown forthe two-dimensional simulation in Fig. 1 and Fig. 2, respec-tively. While the trough in the initial 〈Ye〉-profile is filled in,further inside the adjacent, initially stable plateau is more andmore eroded due to convective undershooting. The correspond-ing effect can be recognized in the evolution of the angular-averaged entropy distribution, where the trough in the profile atM (r) ≈ 1.15 M is filled in, while the entropy maximum atM (r) ≈ 0.9 M is removed. Due to convective overshootingboth profiles are also flattened out to M (r) ≈ 1.22 M whichis well beyond the outer edge of the convectively unstable masslayer at M (r) = 1.07 M. Figs. 1 and 2 clearly exhibit con-vective mixing also immediately behind the shock wave in theregion 1.26 M ≤ M (r) ≤ 1.29 M where only the entropygradient is unstable. Here some convective over- and under-shooting penetrate into the stable layers above and below andinfluence the 〈Ye〉 and 〈S〉 profiles within about 0.1 M aroundthe convecting shell.

According to Figs. 1 and 2 the convective instability needsa growth time of about 10 ms before significant modificationsof the entropy and lepton number distributions occur. It thentakes another 10 to 20 ms to completely homogenize both theentropy and the Ye distributions in the unstable layers and inthe adjacent regions influenced by convective under- and over-shooting. Note in this respect that the extent of undershootingat the inner edge of the innermost unstable region steadily in-creases with time and has almost reached the inner boundary ofthe computational grid at the end of the simulation. This may betaken as an indication that eventually the whole proto-neutronstar might be involved in this process und thus neutrino transportby convective motions could be more important than diffusivetransport in deleptonizing the proto-neutron star. However, onlysimulations which cover a longer interval of the evolution andwhich also consider the whole proto-neutron star can definitelyconfirm this possibility.

The development of the convective instability in the mantleof the proto-neutron star is further illustrated in Figs. 3 to 7. Theright upper panels in Figs. 3a and 4a show that first the innerunstable layer develops Rayleigh-Taylor fingers that penetrateinward and plumes that rise outward. This is best seen fromthe entropy plot (Fig. 4) which shows the two entropy max-ima (in orange and red) and the low-entropy zones (in blue)inside, between, and outside these maxima. The position of theshock wave is at the inner edge of the outermost blue region.The yellow-red ring immediately behind the shock marks theouter unstable layer where the convective instability requires alonger timescale (about 15–20 ms) to grow into the nonlinearregime (lower two panels in Fig. 4). The Rayleigh-Taylor fin-gers are subject to Kelvin-Helmholtz instabilities that lead tothe formation of mushroom-like caps and strong bending of themushroom stems. As a consequence the initially narrow fingers

begin to merge into successively larger blobs (upper and lowerleft panels in Figs. 3a and 4a). After another 5–10 ms the wholeunstable region is involved in a convective overturn (lower rightpanels in Figs. 3 and 4). Finally, after about 25–30 ms, the innerregion is completely mixed and homogenized.

The convective mixing releases gravitational binding energyby establishing a more compact stratification in the mantle ofthe proto-neutron star (see also Figs. 7 and 17). The liberatedbinding energy is used to temporarily push the shock out toabout 400 km which is more than twice its initial radius. Thisis illustrated in Fig. 5 which shows the evolution of the entropydistribution between 20 and 32 ms after the beginning of thesimulation. The outward propagation of the shock is clearlyseen, as well as the mushrooms rising up from the outer con-vectively unstable layer. The shock expansion comes to a haltat t ≈ 30–40 ms and subsequently a re-contraction sets in.

As the convective velocities reach and even partially exceedthe local sound speed (vconv ≈ 109 cm/s) strong pressure wavesand even weak shock waves are generated by the convectiveflow. This is illustrated in Fig. 6, where the divergence of thevelocity field is shown for the same four snapshots of time asdisplayed in Fig. 5. Besides the propagation of the shock wave,one recognizes several strong, interacting sound waves “emit-ted” from the convective region and propagating outward behindthe supernova shock front. The convective layer itself shows aturbulent wave activity which reflects the non-stationary char-acter of the convective overturn. Moreover, we see from Fig. 6that the shock wave is not perfectly spherically symmetricaland exhibits large-scale deformations caused by the interactionwith rising convective elements. The overall, moderately pro-late deformation of the shock front along the symmetry axis ofthe two-dimensional computational grid for t >∼ 25 ms is pro-duced by a large convective element that rises rapidly near thesymmetry axis (see Figs. 3, 4 and 5). We point out, however,that its outward velocity may in fact be overestimated becauseof the imposed axial symmetry which implies that convectiveelements near the symmetry axis are blob-like while those atthe equator are torus-like.

The convective activity inside the proto-neutron star causessignificant inhomogeneities in the temperature and density strat-ifications as illustrated by Fig. 7. The inhomogeneities vary intime and space implying non-radial mass motions and a time-dependent mass quadrupole moment associated with the outerlayers of the proto-neutron star. Both effects generate gravita-tional radiation with typical frequencies in the range of severalhundred Hz to about one kHz (see Sect. 3). In the transition fromthe upper panels (at t = 12 ms after the start of the simulation)to the lower panels (at t = 21 ms) in Fig. 7 the more compactstratification of the later state is clearly visible.

Although not included in the present simulation, we mayspeculate about another, possibly very important effect of con-vection, which has implications for the explosion mechanism.Convective mixing around and below the neutrinospheric re-gion near the proto-neutron star surface may be accompaniedby an increase of the neutrino luminosities during the early phaseof the supernova explosion. Since the convection is dynamical


Fig. 3. Convection inside the nascent neutron star. The four pan-els show the time evolution of the electron number fraction Ye ina region between 15 km and 155 km. The panels are arranged incounter-clockwise order, starting from the right upper side, and showsnapshots at 12 ms, 14 ms, 18 ms, and 21 ms after the start of the2D simulation. The colours correspond to Ye in the range [0.1, 0.5]with increasing values from blue, over green, yellow, and orange tored. Blue regions have Ye <∼ 0.12, yellow regions correspond to0.13 <∼ Ye <∼ 0.16, orange regions to 0.16 <∼ Ye <∼ 0.21, and redones to Ye >∼ 0.21

Fig. 4. Same as Fig. 3 but showing the time evolution of the entropydistribution. The colour levels correspond to entropies in the range[3.2, 12.3] with increasing values from blue, over dark green and yel-low to red. Blue regions have S <∼ 5, dark green regions correspondto 5 <∼ S <∼ 6, yellow regions to 6 <∼ S <∼ 8, and red regions to8 <∼ S <∼ 12

Fig. 5. Convective overturn in the proto-neutron star and transient shockexpansion. The four panels show the time evolution of the entropyin a region between 15 km and 403 km. The panels are arranged incounter-clockwise order, starting from the right upper side, and showsnapshots at 20 ms, 25 ms, 28 ms, and 32 ms after the start of the 2Dsimulation. The colour coding is the same as in Fig. 4

Fig. 6. Same as Fig. 5 but showing the evolution of the divergence ofthe flow field. The colour coding is chosen such that compression wavesare strongly enhanced. Dependent on their strength shocks are encodedfrom dark blue to light blue while sound waves appear in yellow


Fig. 7. Structural effects of convective instabilities on the proto-neutronstar. The four panels show two snapshots of the time evolution of thetemperature distribution (left two panels) and of the logarithm of thedensity distribution (right two panels) in a region between 15 km and95 km. The upper two snapshots are taken 12 ms and the two lowerones 21 ms after the start of the 2D simulation. The colours correspondto temperature values in the range 2.5× 1010 K ≤ T ≤ 1.8× 1011 Kand to density values in the range 10.5 ≤ log ρ ≤ 13.3, both increasingfrom blue, over green and yellow to red. Yellow regions have a temper-ature of about 1011 K and a density of approximately 5× 1011 g/cm3.Below and near the neutrinospheric region which is at a density ofabout 1011 g/cm3, density contrasts of about a factor of 2 develop andtemperature differences of up to ∼ 30% occur

Fig. 8. Time evolution of the minimum value of the angular-averagedelectron number fraction 〈Ye〉min in the convective region of theproto-neutron star: solid (2D simulation); dotted (3D simulation ina 60 sector); dashed (3D simulation in a 20 sector)

Fig. 9. Convection inside the nascent neutron star. The snapshot shows ameridional cut (φ = 0) of Ye 14 ms after the start of the 3D simulationwith a cone of opening angle 60. Note that the shapshot is taken atthe same time as the upper left panel in Fig. 3. The radial size of thedisplayed region and the colour coding is the same as in Fig. 3

Fig. 10. Same as Fig. 9 but showing the evolution of the entropy dis-tribution. The colour coding is the same as in Fig. 4


and violent and the gas velocities are close to the local speedof sound, neutrinos could be transported out of the dense in-terior of the newly-formed neutron star much faster than bydiffusion. The corresponding increase of the neutrino emissionmay provide an important aid to the neutrino-powered explosionmechanism. Moreover, the neutron star shrinks faster due to theenhanced cooling of its surface layers. Since, in addition, thesupernova shock is driven further out as a consequence of theconvective settling of the outer parts of the proto-neutron star,ideal conditions for efficient neutrino-heating in the postshockregion are established.

2.1.2. Three-dimensional simulations

The two three-dimensional calculations were done on a grid(r, θ, φ) of 320 × 60 × 60 and 320 × 20 × 20 zones, re-spectively. The non-equidistant radial zones covered the region15 km ≤ r ≤ 1000 km, while the equidistant angular zonescovered a cone of opening angle 60 and 20, respectively. Inboth simulations the cone was centered at θ = 90 and φ = 0

and periodic boundary conditions were imposed in the angulardirections.

The development of the convective instability inside theproto-neutron star looks qualitatively quite similar when com-paring the results of 2D and 3D simulations. The growth rates ofthe convective instability are practically identical, as can be seenfrom the time evolution of the minimum value of the angular-averaged electron number fraction 〈Ye〉min in the convective re-gion (Fig. 8). In comparison with the 2D simulations the amountof overshooting and undershooting is somewhat smaller in thetwo 3D simulations which explains the smaller final 〈Ye〉min,〈Ye〉3D

min ≈ 0.175 compared to 〈Ye〉2Dmin = 0.182, by less mix-

ing of high-Ye matter from the layers above and below into theconvective region (see also Fig. 16).

The qualitative similarity of the 2D and 3D results is alsoconfirmed by Figs. 9 and 10, which show snapshots of the distri-butions of electron number fraction (Fig. 9) and of the entropy(Fig. 10) in a meridional cut of the computational domain 14 msafter the start of the simulation. The features present in Figs. 9and 10 are qualitatively very similar to those found in the 2Dsimulations at the same epoch (see upper left panel of Fig. 3 andFig. 4, respectively). However, the finger-like or blob-like struc-tures seen in the 2D runs are actually toroidal structures (becauseof the assumed axial symmetry), whereas the inhomogeneitiesin the 3D simulations are genuine three-dimensional structureswith no geometrical restrictions imposed. Their developmentfrom small fingers or bubbles to mushroom-like features whichsubsequently merge is illustrated in Fig. 11, which shows threesnapshots of the surface Ye = 0.16 at times t = 12.2 ms,13.6 ms, and 16.6 ms, respectively.

In order to compare the 2D and 3D results in a more quan-titative way we have performed a normal mode analysis byFourier transforming the electron number fraction (which is agood tracer of the inhomogeneities) at different epochs of theevolution. The relative power of the turbulent motions at dif-ferent wave numbers and at different radial positions inside the

Fig. 11. Surface of constant electron fraction Ye = 0.160 att = 12.2 ms, 13.6 ms, and 16.6 ms (from top to bottom) for the 3Dsimulation that was performed in a wedge of opening angle 60 (be-tween θ = 60 and θ = 120)


convectively unstable layer is shown for the 3D simulation with60 cone in Fig. 12 by contour plots at three moments of time.A corresponding contour plot for the 2D simulation is given inFig. 13 at t = 13.9 ms which is very close to the time of thesecond snapshot in Fig. 12. In Figs. 12 and 13 the left ordinategives the wave number of the normal mode normalized to onequadrant k4 ≡ k/4 = (π/2)/λ, while the right ordinate givesthe corresponding wavelength (in degrees) of the mode. Thismeans that the typical angular size of rising or falling lumpsof matter is half this wavelength. Note that in the 3D case, wehave averaged the Ye distribution inφ-direction before perform-ing the Fourier transformation. If we instead average over thedistribution in θ-direction the power spectrum does not changesignificantly.

Fig. 12 shows that the evolution of the power spectrum ischaracterized by small-scale features growing into large-scaleones in the whole unstable layer, an effect which reflects the suc-cessive merging of smaller Rayleigh-Taylor fingers into largerones as well as the increasing homogenization of the convec-tive layer by the mixing process. At t = 16.6 ms (third panelin Fig. 12) all power has become concentrated into structureswhich have a wavelength larger than about 10 degrees, whileearly on in the evolution (first panel of Fig. 12) a significantamount of power is contained in features with a wavelength ofseveral degrees only. Comparing the second panel of Fig. 12with Fig. 13 one recognizes that at the epoch of maximum con-vective activity the structures in the 3D simulation are only abouthalf as large with respect to their angular extent than those in the2D simulation. In particular, in the three-dimensional case thedominant modes have (angular) wavelengths of about 20–30

(i.e., clump sizes of about 10–15), while the correspondingvalues are around 40–60 (i.e., clump sizes of approximately20–30) in the two-dimensional simulation.

The angular-integrated angular kinetic energy as a functionof time also reveals quantitative differences between the two-and three-dimensional results (see Fig. 14). The angular kineticenergy is about a factor of 3–4 smaller in the 3D simulations thanin the 2D one. Fig. 14 also shows that the angular kinetic energyin the 3D simulation performed within the 20 sector is signifi-cantly reduced compared to the 3D simulation performed withinthe 60 sector. This reduction of the angular kinetic energy iscaused by the insufficient angular extent of the computationaldomain, which hampers the development of the convective in-stability, because the dominant structures have wavelengths ofabout 20–30 which are too large to be correctly modelled ina 20 sector. Note that early on in the evolution (t <∼ 14 ms),when the dominant structures still have a small angular size, theangular kinetic energy of both 3D simulations agrees very well.

Further quantitative differences are found when comparingthe root mean squared angular velocity

vrmsang ≡

√〈v2

θ〉 + 〈v2φ〉 (3)

of the 2D and 3D simulations at a given epoch. Fig. 15 showsthat in the three-dimensional simulations vrms

ang is roughly a factorof two smaller than in the two-dimensional simulation. More-

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Fig. 12. Relative power of the turbulent motions at different angu-lar scales for a three-dimensional simulation of convection inside theproto-neutron star. The simulation was performed in a wedge of open-ing angle 60 (between θ = 60 and θ = 120). The normal modeanalysis was done with the electron fraction Ye. The three snapshotsare taken at 12.2 ms, 13.6 ms, and 16.6 ms, respectively (from top tobottom). The levels of constant power are chosen linearily in steps of10% of the maximum value. Note that the scale on the right of eachpanel is nonlinear and gives the wavelength (i.e., twice the angularsize) of a structure in degrees

over, in 2D vrmsang shows larger variations as a function of the

enclosed mass than in the 3D case. This is simply caused bythe fact that the 2D convective structures are twice as large andhave 3–4 times more specific angular kinetic energy than the 3Dones. Hence, in the two-dimensional simulation the flow quan-tities are more inhomogeneous and show larger spatial fluctu-ations. Compared to the result of the 3D simulation with the60 sector, the root mean squared angular velocity obtained inthe 3D simulation with the 20 angular sector turns out to besmaller in the unstable region where initially both the lepton


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Fig. 13. Relative power of the turbulent overturn motions at different(angular) scales for a two-dimensional simulation of convection insidethe proto-neutron star as deduced from the electron concentration. Thesnapshot is taken at 13.9 ms. The levels of constant power are chosenlinearily in steps of 10% of the maximum value. Note that the scale onthe right is nonlinear and gives the wavelength (i.e., twice the angularsize) of a structure in degrees

Fig. 14. Angular-integrated angular kinetic energy as a function oftime: solid (2D simulation); dotted (3D simulation in a 60 sector);dashed (3D simulation in a 20 sector)

and the entropy gradient are negative. As discussed before, thisis a numerical artifact of the insufficiently large angular grid incase of the simulation performed with the 20 cone.

Because the root mean squared angular velocity and the an-gular size of the structures is smaller in the three-dimensionalsimulations than in the two-dimensional models, the genuinethree-dimensional finger-like and blob-like structures have lessmomentum than their two-dimensional counterparts. This ex-plains why in the 3D simulations the extent of undershootingat the inner edge of the convectively unstable layer is reducedcompared to the undershooting found in the 2D simulation (seeFig. 16). Note that the “reference” Ye distribution, which isdisplayed as the solid curve in Fig. 16 and which all multi-dimensional simulations are compared with, refers to the resultof a one-dimensional simulation which leaves the initial Ye dis-tribution (as function of the enclosed mass) unchanged due to

Fig. 15. Root mean squared angular velocity versus enclosed mass18 ms after the start of the simulations: solid (2D simulation); dotted(3D simulation in a 60 sector); dashed (3D simulation in a 20 sector)

Fig. 16. Angular-averaged electron number fraction 〈Ye〉 plotted ver-sus enclosed mass for different simulations at 32 ms after the start ofthe computations: solid (1D); dotted (2D); short-dashed (3D with 60

sector); long-dashed (3D with 20 sector)

the disregard of neutrinos in the hydrodynamical simulation andthe lack of convective mixing in the 1D case.

In two spatial dimensions convection penetrates by meansof undershooting about 1.2 pressure scale heights into the adja-cent convectively stable layers. In three spatial dimensions theundershooting is reduced to about 0.8 pressure scale heights,i.e., the depth of the undershooting region is smaller by about50% than in the axisymmetrical models. Because of the insuffi-ciently large angular grid in the 3D simulation performed withthe 20 sector, the extent of undershooting is less strong byanother 10 to 20% in this case (Fig. 16).

The effect of convective overturn and mixing on the den-sity stratification and on the shock propagation is illustratedin Fig. 17, which shows the density and mass distributions att = 32 ms for the 1D, 2D and 3D simulations. One notices that


Fig. 17. Angular-averaged density and integral mass distribution 32 msafter the start of the simulations: solid (1D simulation); dotted (2D sim-ulation); dashed (3D simulation in a 60 sector). Note the dependenceof the shock position and of the compactness of the proto-neutron staron the spatial dimension of the simulation

the result for the 3D case lies in between the 1D and 2D re-sults and that in two spatial dimensions the structural effectsare largest. As already mentioned above, along with the con-vective overturn of the matter potential energy is liberated inthe multi-dimensional simulations, which drives a re-expansionof the stalled shock wave. Hence, the shock is located at largerradii in the 2D and 3D simulations than in the 1D model. Conse-quently, the mass distribution at small radii is more compact inthe multi-dimensional cases, i.e., more mass is located inside aradius of about 45 km than in the spherically symmetrical model(Fig. 17). Compared to the 2D model these effects are reduced inthe 3D simulation, because the undershooting is weaker. There-fore the mass region involved in the mixing is smaller and theproto-neutron star becomes less compact, which in turn leads toless potential energy release and to a less strong re-expansionof the shock wave, r1D

sh ≈ 220 km < r3Dsh ≈ 280 km < r2D

sh ≈400 km at t = 32 ms, rsh being the shock radius.

2.2. Convection in the hot-bubble region

We have performed a second set of simulations, which wereaimed at studying the convective instability in the hot-bubbleregion and which included a simple, but nevertheless reasonablywell justified treatment of neutrino effects in the stellar matter(for details, see Janka & Muller 1995a, 1996).

In these simulations the inner part of the collapsed stellarcore slightly inside the neutrinosphere was cut out and time-dependent neutrino fluxes were imposed at this inner boundary.Thereby, the neutrino emission from the central part of the proto-neutron star was mimiced. The radius of the inner boundary waseither kept fixed, or was allowed to shrink with time to take intoaccount the contraction of the cooling neutron star.

Due to neutrino heating an unstable entropy gradient buildsup between the shock and the neutrinosphere. Depending on

the imposed neutrino flux at the inner boundary, this hot-bubbleregion becomes convective after about 50–80 ms. Thus, in oursecond set of simulations two convective regions are present: (i)the lepton and entropy unstable layer inside the proto-neutronstar just below the neutrinosphere, and (ii) the entropy-unstablehot-bubble region (see Table 1). Although we located the in-ner boundary somewhat inside the neutrinosphere, our compu-tational domain did only partially encompass the convectivelyunstable layer in the proto-neutron star. Putting the inner bound-ary deeper inside the star reduces the time step appreciably andthus makes the simulation significantly more expensive, and, inparticular, would have made the long time (up to 1 s past corebounce) hot-bubble simulations discussed in Janka & Muller(1995a, 1996) impracticable.

Because only part of the inner convection zone is includedin the simulations the radial extent of the convective eddiesand of the undershooting is constrained and suppressed by theimposed inner boundary condition. Convection inside the proto-neutron star is therefore weaker in these models than we wouldexpect in simulations without an inner boundary at finite radius.Hence, the gravitational waves produced by anisotropies in thesemodels are primarily caused by non-radial motions in the hot-bubble region (see Sect. 3).

The initial model was the collapsed 1.31 M iron core ofa 15 M star at a time about 25 ms after core bounce. By thattime the prompt shock wave had transformed into a standingaccretion shock at a radius of about 115 km, enclosing a massof 1.25 M. The matter behind the shock has small negativevelocities and settles onto the forming neutron star. The initialmodel was provided by Bruenn (private communication; seealso Bruenn 1993).

The two-dimensional simulations of convection in the hot-bubble region were performed in spherical coordinates. Thecomputational grid consisted of 400 radial zones distributednon-equidistantly between the time-dependent inner boundaryand the fixed outer edge of the grid at 17000 km, with 100 radialzones out to 290 km and 42 radial zones within the innermost100 km. Initially the inner boundary was located at 31.7 km(ρ = 1.7× 1012 g/cm3) with an interior mass (constant in time)of 0.848 M. Over a period of about 0.5 s the radius of theinner boundary was reduced to a value of 15 km. Thereby, wehave simulated the contraction of the proto-neutron star due tolepton and energy losses. The gravitational potential was com-puted in the same way as in the 2D simulations of convectioninside the proto-neutron star, i.e., as the superposition of contri-butions from the spherically symmetrical point-mass potentialassociated with the mass inside the inner boundary and from the2D matter distribution in the computational domain. In angulardirection between 90 and 180 equidistant zones were used. Inall simulations axial symmetry was assumed, and in some sim-ulations, in addition, equatorial symmetry was imposed. Thelatter restriction was found, however, to influence the growth oflarge-scale (>∼ 30) angular modes negatively (for more detailssee Janka & Muller 1995a, 1996)


3. Gravitational wave signature

The non-radial motions and the resulting time-dependence andasphericities of the density stratification due to convection inthe unstable layers both inside the proto-neutron star and in theneutrino-heated hot-bubble will produce gravitational radiation.Thus, we have analysed the two- and three-dimensional mod-els discussed in the previous section and have computed theamount and the signature of gravitational radiation that can beexpected from convection in Type II supernovae. Moreover, thedensity and temperature variations in the neutrino-decouplingregion (see Fig. 7) and the fast convective overturn below theneutrinosphere lead to anisotropies of the neutrino emissionwhich will also be a source of gravitational waves (Epstein 1978,Turner 1978).

3.1. Formalism

The gravitational wave signal was calculated using a post-Newtonian approach, where numerically troublesome higher or-der time derivatives of the quadrupole moment are transformedinto much better tractable spatial derivatives. In particular, forthe gravitational quadrupole radiation field, hTT, we used an ex-pression derived independently by Nakamura & Oohara (1989),and by Blanchet et al. (1990):

hTTij (X, t) =

2Gc4R

Pijkl(N ) ×

×∫

d3x ρ

[2vkvl − xk ∂lΦ− xl ∂kΦ

], (4)

where R = |X| is the distance between the observer and thesource, Φ is the Newtonian gravitational potential, ρ is the mass-density and v is the velocity. The other quantities have theirusual meaning except for Pijkl(N ) (with N = X/R) whichdenotes the transverse-traceless (TT) projection operator ontothe plane orthogonal to the outgoing wave direction N , actingon symmetrical Cartesian tensors according to

Pijkl(N ) = (δik −NiNk) (δjl −NjNl) +

− 12

(δij −NiNj) (δkl −NkNl) . (5)

∂i represents the partial derivative with respect to the xi coordi-nate. The integrand in Eq. (4) is defined on a compact manifoldand is known to the (2nd order) accuracy level of the numer-ical algorithm of the hydro-code. Eq. (4) can be shown to beequivalent to the standard representation

hTTij (X, t) =

2Gc4R

Pijkl(N )∂2

∂t2Qkl (t− R

c) (6)

where the trace-free part of the mass-quadrupole tensor of thematter distribution is given by

Qij(t) =∫

d3x ρ(x, t)

(xixj − 1

3δijx

2

). (7)

It can easily be shown that evaluating the integral of Eq. (4) by anintegration scheme (of at least 2nd order) is by one order of accu-racy superior to twice applying numerical time-differentiationmethods to quadrupole data given at discrete points of time (seeMonchmeyer et al. 1991).

3.1.1. Evaluation in two dimensions

The gravitational radiation field gives direct information aboutthe second time derivative of the mass-quadrupole tensor (seeEq. (7)). In case of axisymmetry the quadrupole moment, Q,is the only independent component of the quadrupole tensor.Its relation to the Cartesian components Qij of the radiativemass-quadrupole tensor is

Q ≡ 34

√5πQzz . (8)

As all our simulations were performed in spherical coordi-nates, it is natural to represent the (total) radiation field hTT interms of the “pure-spin tensor harmonics” T E2,lm

ij and TB2,lmij

with amplitudes AE2lm and AB2

lm in the following way (Thorne1980):

hTTij (X, t) =

1R

∞∑l=2

+l∑m=−l

AE2lm(t− R

c) T E2,lm

ij (θ, φ) +

+ AB2lm(t− R

c) TB2,lm

ij (θ, φ)

. (9)

For the definitions ofT E2,lmij ,TB2,lm

ij ,AE2lm, andAB2

lm, see Thorne(1980). In spherical coordinates the coefficients AE2

lm and AB2lm

have especially simple integral representations over the source.By symmetry, there is only one nonvanishing quadrupole termin Eq. (9), namely AE2

20 . Higher-order terms are neglected inthe quadrupole approximation hTT of the gravitational radiationfield hTT. Transforming Eq. (4) to spherical coordinates andexpressing vi in terms of unit vectors in the r, θ and φ direction,one obtains by comparison of Eq. (4) with the lowest-order termof Eq. (9) for the quadrupole wave amplitudeAE2

20 the expression

AE220 (t) =

G

c4

16π3/2

√15

∫ 1

−1

∫ ∞

0%(r, z, t)

[vrvr(3z2 − 1) +

+ vθvθ(2− 3z2)− vφvφ − 6vrvθz√

1− z2 +

− r∂rΦ(3z2 − 1) + 3∂θΦz√

1− z2

]r2drdz , (10)

where ∂r = ∂/∂r, ∂θ = ∂/∂θ, and z = cos θ.

From the definition of T E2,2mij (Thorne 1980, Eq. (2.39e))

one derives for the components of hTT the formula

hTTθθ =

18

√15π

sin2θAE2

20 (t)R

. (11)


The only other nonzero component ishTTφφ = −hTT

θθ . The total en-ergy radiated in gravitational waves is then given by the generalexpression

E =c3

G

132π

∑l,m

∫ +∞

−∞

(dAE2

lm

dt

)2

+

(dAB2

lm

dt

)2

dt . (12)

The total energy radiated per unit frequency will be denotedby dE/dν and was calculated by interpolating the data for theamplitudes AE2

20 that are given at non-equidistant times onto anequidistant temporal grid and using the fast Fourier transformtechnique with a rectangular window function.

3.1.2. Evaluation in three dimensions

If the source is of genuine three-dimensional nature, it is com-mon to express the gravitational quadrupole radiation field, hTT,in the following tensorial form (see, e.g., Misner et al. 1973)

hTTij (X, t) =

1R

(A+e+ + A×e×) (13)

with the unit linear-polarization tensors

e+ = eθ ⊗ eθ − eφ ⊗ eφ , (14)

e× = eθ ⊗ eφ + eφ ⊗ eθ , (15)

eθ andeφ being the unit polarization vectors in θ andφ-directionof a spherical coordinate system and ⊗ the tensor product. Theamplitudes A+ and A× represent the only two independentmodes of polarization in the TT gauge, and are given by thefollowing expressions for θ = 0, φ = 0

A+ = Ixx − Iyy , (16)

A× = 2Ixy , (17)

and for θ = π/2, φ = 0 by

A+ = Izz − Iyy , (18)

A× = −2Iyz , (19)

where

Iij =G

c4

∫d3x ρ

(2vivj − xi∂jΦ− xj∂iΦ

). (20)

The total energy radiated in form of gravitational waves is thengiven by

E =c3

5G

∫ +∞

−∞

[ddt

(Aij − 1

3δijAll

)]2

dt (21)

=2c3

15G

∫ +∞

−∞

[A2xx + A2

yy + A2zz − AxxAyy − AxxAzz+

− AyyAzz + 3 (A2xy + A2

xz + A2yz)]

dt (22)

with

Aij ≡ Iij . (23)

3.1.3. Gravitational waves from neutrino emission

In order to estimate the gravitational wave signal associated withthe anisotropic emission of neutrinos, we use Eq. (16) of Ep-stein (1978) in the limit of a very distant source,R = |X| → ∞.In addition, we make use of the approximation that the gravita-tional wave signal measured by an observer at time t is causedonly by radiation emitted at time t′ = t−R/c. Hence, we taket− t′ = const = R/c, which means that only a neutrino pulse it-self is assumed to cause a gravitational wave signal but memoryeffects after the pulse has passed the observer are disregarded.With these simplifications, one gets for the dimensionless grav-itational wave amplitude

hTTij (X, t) =

4Gc4R

∫ t−R/c

−∞dt′∫

4πdΩ′

(ninj

)TT

1− cos θ· dLν(Ω′, t′)

dΩ′(24)

with θ being the angle between the direction towards the ob-server and the direction Ω′ of the radiation emission, anddLν(Ω, t)/dΩ denoting the direction dependent neutrino lumi-nosity, i.e., the energy radiated at time t per unit of time and perunit of solid angle into direction Ω. The angular integral overthe radiation source is performed over dΩ′ = −d(cosϑ′)dϕ′ andthus over all angles ϑ′ and ϕ′ which specify the (beam) direc-tion in the source coordinate frame (x′, y′, z′) that we identifywith the coordinate frame used for the hydrodynamical simu-lations. With the angles θ and φ defining the radiation direc-tion in the observer’s frame (x, y, z) (where the observer islocated at distance R along the z-axis) one has (nxnx)TT =12

(1− cos2θ

) (2 cos2φ− 1

)= 1

2

(1− cos2θ

)cos(2φ), and the

gravitational wave amplitude is given by

hTTxx = −hTT

yy = −hTT+ =

=2Gc4R

∫ t−R/c

−∞dt′∫

4πdΩ′ (1 + cos θ) cos(2φ)

dLν(Ω′, t′)dΩ′

. (25)

Replacing cos(2φ) by sin(2φ) in Eq. (25) yields hTTxy = hTT

× .In Eq. (25) θ and φ need to be expressed in terms of the an-

gles ϑ′, ϕ′, ϑ, and ϕ when ϑ and ϕ define the orientation of theobserver’s coordinate system (observer located in z-directionand y-axis lying in the x′-y′-plane) relative to the source coor-dinate frame. Choosing ϕ = 0 the y- and y′-axes coincide andthe expressions become rather simple. For ϑ = 0 the z-axis andthe z′-axis are identical, too, and the observer is situated alongthe system’s z′- (polar) axis. In that case one obtains(hTTxx

)p

=2Gc4R

∫ t−R/c

−∞dt′∫

4πdΩ′

(1 + cosϑ′

)cos(2ϕ′)

dLν(Ω′, t′)dΩ′

. (26)

When dLν/dΩ is axially symmetrical, the gravitational wavesignal for an observer on the symmetry axis vanishes and

E. Muller & H.-T. Janka: Gravitational radiation from convective instabilities in Type II supernova explosions 153(hTTxx

)p

= 0. For ϑ = π/2 the observer is positioned perpen-

dicular to the source’s z′-axis in the equatorial plane (z-axisand x′-axis coincide) and hTT

xx becomes

(hTTxx

)e

=2Gc4R

∫ t−R/c

−∞dt′∫

4πdΩ′

(1 + sin ϑ′ cosϕ′

) cos2ϑ′ − sin2ϑ′ sin2ϕ′

cos2ϑ′ + sin2ϑ′ sin2ϕ′

× dLν(Ω′, t′)dΩ′

. (27)

Equations (26) and (27) can be rewritten as

hTTxx =

2Gc4R

∫ t−R/c

−∞dt′ Lν(t′) · α(t′) (28)

with the anisotropy parameter α(t) being defined by

α(t) ≡ 1Lν(t)

·∫

4πdΩ′ Ψ(ϑ′, ϕ′) · dLν(Ω′, t)

dΩ′(29)

where Ψ(ϑ′, ϕ′) denotes the angle dependent factors appearingin the integrals of Eqs. (26) and (27), respectively, and Lν(t) isthe total neutrino luminosity,

Lν(t) =∫

4πdΩ′

dLν(Ω′, t)dΩ′

. (30)

Using the dimensionless gravitational wave amplitude ofEq. (29) and taking into account that there is only one non-zero component of the quadrupole amplitude, Eqs. (11) and (12)allow one to estimate the total energyE(t) that is associated withthe gravitational waves produced by the anisotropic emissionof neutrinos in the two-dimensional (axially symmetrical) caseuntil time t:

E(t) = β · Gc5

∫ t−R/c

−∞dt′ L2

ν(t′) · α2(t′) . (31)

β is a numerical factor of order unity, typically around 0.5. Em-ploying Eq. (13) and Eqs. (16)–(23) and assuming that all am-plitudes Aij contribute roughly equally, one finds that Eq. (31)also holds for the three-dimensional situation but with a slightlydifferent value of β.

3.2. Two-dimensional results

The results of our gravitational wave analysis for four differ-ent models are displayed in Figs. 18 to 21 which show thequadrupole wave amplitudes, the quadrupole amplitude spec-tra, the spectral energy densities, and the energies radiated inform of gravitational waves, respectively. One first notices thatthe signal forms and signal strengths as well as the spectral dis-tributions of the gravitational wave energy depend on whetherthe gravitational radiation is produced by convection inside theproto-neutron star or by convection in the hot-bubble region.Generally speaking, in case of convective overturn processes

in the proto-neutron star, the maximum gravitational wave am-plitude is significantly larger (≈ 350 cm instead of <∼ 50 cm),the spectral energy distribution is peaked at higher frequen-cies (at 500 Hz to 1000 Hz instead of at about 100 Hz), andmuch more energy is radiated in form of gravitational waves(5 × 10−10 Mc2 instead of <∼ 10−12 Mc2) as compared tomulti-dimensional processes in the hot-bubble region.

In model HB2D, which refers to the two-dimensional sim-ulation of convection inside the proto-neutron star starting withHillebrandt’s configuration of a post-bounce stellar core, thestrongly unstable lepton and entropy gradients give rise to largeconvective velocities which exceed 109 cm/s and are hence tran-sonic. As the unstable layer involves a relatively large amountof mass (≈ 0.45 − 0.5 M) the angular kinetic energy of theconvective flow is substantial (Eang

kin<∼ 3.8 × 1050 erg). More-

over, because of the compact and quite massive collapsed stel-lar core (1.4 M) in model HB2D the non-radial flow takesplace deep in a strong gravitational potential with convectiveelements undershooting the stable layer down to a radius ofabout 15 km. Consequently, this model produces a relativelystrong gravitational wave signal with a maximum amplitude of|AE2

20 | = 366 cm (Fig. 18) and a total emitted gravitational waveenergy ofEGW = 5.5×10−10 Mc2 (Fig. 21). Corresponding tothe characteristic overturn timescales of the convective eddies ofabout 2–10 ms, the frequency spectrum of the quadrupole am-plitude has a broad maximum at about 100–500 Hz (Fig. 19)and the energy spectrum shows most power being radiated be-tween 100 and roughly 1000 Hz (Fig. 20).

As discussed in Sect. 2.2, convection inside the proto-neutron star is much weaker in those of our models which weparticularly used to study convection in the hot-bubble region.There are three reasons for this. Firstly, only the outer part of theconvective zone in the proto-neutron star is included in the sim-ulations, i.e., only about 0.15 M of the roughly 0.3 M thatare unstable against convection. Secondly, the lepton and en-tropy gradients in Bruenn’s initial model are less unstable thanthose in Hillebrandt’s collapsed stellar core. Thirdly, due to thesmaller core mass of only about Mcore ≈ 1.3 M the gravita-tional potential is weaker in Bruenn’s model and there is lessmass in the unstable layer around and below the neutrinosphere.In the hot-bubble simulations the convective velocities insidethe proto-neutron star are therefore significantly smaller, onlyabout vconv ≈ 2 × 108 cm/s, and, correspondingly, the angularkinetic energies reach only about Eang

kin ≈ 4–5 × 1048 erg at atime when the convection inside the proto-neutron star is fullydeveloped (at t ≈ 30 ms, compare Fig. 9 in Janka & Muller1996). Thus, the resulting gravitational wave signal from theconvective overturn inside the proto-neutron star is very weakin these models as compared to model HB2D. In fact, in themodels where we followed the neutrino effects in the hot-bubbleregion, the gravitational wave emission is strongly dominatedby the waves produced by mass motions in the neutrino-heatedlayer between proto-neutron star and supernova shock.

We have analysed three such models, namely modelsML2D, LP2D, and EP2D (equivalent to models T2c, T3c, andT4c, respectively, in Janka & Muller 1996), whose gravitational


Fig. 18. Quadrupole amplitude AE220 [cm] of

various models versus time. The upper leftpanel shows the amplitude obtained from atwo-dimensional simulation of convection in-side the proto-neutron star (see Sect. 2.1). Theremaining three panels give the amplitudes ob-tained from two-dimensional simulations ofconvection in the hot-bubble region. The ini-tial neutrino flux imposed at the inner bound-ary increases along the model sequence ML2D(top right), LP2D (bottom left), and EP2D (bot-tom right) from L0

νe = 2.0 × 1052 erg/s toL0νe = 2.25 × 1052 erg/s (see Janka & Muller

1996), leading to increasingly faster explo-sions and higher supernova explosion energies(0.5×1051, 1.1×1051, and 1.3×1051 erg, respec-tively). The additional thin curve in the lowerright panel gives the quadrupole amplitude pro-duced by the convection inside the proto-neutronstar alone. It is very small in models ML2D,LP2D, and EP2D because the convectively un-stable region inside the proto-neutron star wasonly partially included in these simulations

wave signature we discuss in the following. We point out herethat the actual gravitational wave signal of these models willbe considerably larger if the whole convection zone inside theproto-neutron star, which yields only a minor contribution tothe gravitational wave emission in the presented models, is in-cluded in the simulations. Models ML2D, LP2D, and EP2Dare nevertheless useful to investigate especially the character-istics of the gravitational waves originating from turbulent mo-tions in the hot-bubble region. We find that the structure of theproduced waves contains detailed information about duration,strength, and pattern of the accretion and convection processesbehind the supernova shock. When these dynamical processeshave direct influence on the neutrino emission, e.g., when mat-ter accreted onto the neutron star produces additional neutrinoemission, we notice correlations of the neutrino luminosity withthe gravitational wave amplitude and luminosity. Moreover, ourmodels suggest that strength and duration of turbulent processes

around the proto-neutron star are correlated with the size andtemporal decay of the neutrino fluxes from the proto-neutron starand with the explosion energy of the supernova. For this reasonwe expect, and indeed observe, characteristic differences of thegravitational wave signature of models with different explosiondynamics and different explosion parameters.

In model ML2D the temporal modulations of the gravita-tional wave signal and the neutrino losses from the hot-bubbleregion show clear correlations (compare the upper right panel inFig. 18 with Fig. 18 in Janka & Muller 1996). Neutrino emis-sion and gravitational wave emission from the turbulent layersaround the proto-neutron star in this model are produced bydownflows of cold material from the postshock region. Thesenarrow downflows reach very high velocities of more than twicethe speed of sound and are abruptly decelerated at a radius ofabout 80 km (see Figs. 15 and 16 in Janka & Muller 1996). Thepeaks and characteristic features in the neutrino emission are


Fig. 19. Similar to Fig. 18 but showing thequadrupole amplitude spectra |AE2

20 (ν)| [kpc/Hz]as a function of the frequency of the emitted grav-itational radiation

associated with the dissipation of kinetic energy of the gas inthe surroundings of the proto-neutron star. The aspherical, dy-namical gas motions also act as source of gravitational waves.The large variations of the gravitational wave amplitude aroundt ≈ 120–140 ms (Fig. 18) are directly correlated with spikes inthe lepton number loss/gain rate and the energy loss/gain rateof the stellar gas at the same epoch (Fig. 18 in Janka & Muller1996). Similarly, the variation of the gravitational wave ampli-tude between 150 ms and 200 ms is mirrored by a correlatedactivity in the lepton and energy loss/gain rates. Actually, onecan observe a slight time-lag of the neutrino emission. This shiftcan be explained by the fact that the gravitational wave emis-sion traces the dynamical infall of the downflows, whereas theneutrino emission peaks at the moment when the infalling gasreaches highest temperatures and densities, i.e., typically at themoment when it is decelerated and strongly compressed nearthe proto-neutron star. The characteristic timescale of gas mo-

tions and convective overturn in the hot-bubble region is of theorder of several ten to about 100 ms. The frequency spectrumof the quadrupole amplitude therefore peaks at frequencies ofabout 10 to roughly 100 Hz (Fig. 19), and the energy spectrumhas a maximum between 50 and 200 Hz with significant powerin the frequency range from 10 to 400 Hz (Fig. 20).

The gravitational wave signal of model LP2D is charac-terized by the low-frequency emission from the aspherical ex-pansion of postshock material and by the superimposed high-frequency modes that are produced by small-scale convectiveprocesses in the neutrino heated hot-bubble region. The pro-late, large-scale deformation of the supernova shock leads toa time-dependent mass quadrupole moment that varies withina typical time of about 100 ms. While the effect of the expan-sion of supernova shock and ejecta becomes prominent in thegravitational wave signature at times later than about 100 msafter the start of the simulation, the overturn of neutrino-heated


Fig. 20. Similar to Fig. 18 but showing the spec-tral energy density dE(ν)/dν [ Mc2/Hz] ofthe quadrupole radiation as a function of the fre-quency of the emitted gravitational radiation

material determines the wave amplitude during the first 100 mswhere signal variations on timescales of 10–20 ms are visible.The gravitational wave amplitude reaches a maximum value of|AE2

20 | = 10 cm (Fig. 18) and has typical frequencies betweenabout 10 and 100 Hz (Fig. 19). The energy spectrum peaksat about 50–70 Hz ((Fig. 20) and the total energy radiated ingravitational waves is EGW = 5 × 10−14 Mc2 which is al-most a factor of 20 smaller than in model ML2D where it isEGW = 8 × 10−13 Mc2 within a similar time of 200–300 msafter supernova shock formation. While the major part of thegravitational wave emission of model ML2D is produced byanisotropic, dynamical downflows of matter from the postshockregion to the proto-neutron star, roughly half of the gravitationalwave energy of model LP2D results from overturn motions ofneutrino-heated gas and the other half from the large-scale ex-pansion.

In model EP2D the explosion happens faster (due to highercore neutrino fluxes) and the convective overturn in the neutrino-heated region is correspondingly shorter. The contribution of thelarge-scale deformation and expansion of the postshock regionto the gravitational wave signal therefore clearly dominates thewave amplitude after about 100 ms (Fig. 18). On a timescale ofhundreds of milliseconds the wave amplitude exhibits the slowvariation associated with the change of the mass quadrupolemoment due to the global dynamical evolution of the explo-sion. However, it shows only little substructure of higher fre-quencies because the rapid expansion of model EP2D limits theduration of the phase of convective overturn in the hot-bubbleregion to about 100 ms. This can be verified in the quadrupoleamplitude spectrum (Fig. 19) which confirms the clear dom-inance of low-frequency modes (around 1–10 Hz) and essen-tially no pronounced features at higher frequencies. The corre-sponding energy spectrum (Fig. 20) is very flat and spans the


Fig. 21. Similar to Fig. 18 but showing the en-ergy radiated in form of gravitational wavesEGW

[10−13 Mc2] as a function of time

range of frequencies between about 1 Hz and roughly 100 Hz.Even stronger than in case of model LP2D, the global, asym-metrical expansion of model EP2D on a timescale of several100 ms is reflected in the fact that significant, or even most,power is in a fundamental, low-frequency signal of a few Hz.Model EP2D emits about 3.4 × 10−14 Mc2 of gravitationalwave energy which is about 50% less than model LP2D. Inthe lower right panels of Figs. 18–21 the thin solid lines corre-spond to the gravitational wave signal that originates from theincompletely represented (see above) convective region insidethe proto-neutron star for model EP2D. A comparison of the thinand thick lines, the latter representing the total signal, shows thatmost of the emission is produced by the convection in the hot-bubble region. Only at early times after bounce (t <∼ 50 ms,which is when the convective overturn in the neutrino-heatedregion is not yet fully developed) do convective motions aroundand inside the neutrinosphere contribute significantly to the total

gravitational wave emission (see Fig. 18) at frequencies above50 Hz (Figs. 19 and 20) and with an integrated energy of about1.1 × 10−14 Mc2. At later times the signal of convection inthe proto-neutron star is minor and the results shown for mod-els ML2D, LP2D, and EP2D in Figs. 18–21 do indeed primarilyoriginate from the mass motions in the hot-bubble region andfrom the explosive expansion of the supernova.

Comparing the three models we see that with increasing neu-trino flux (imposed as an inner boundary condition) and hencewith increasing explosion energy the convective activity in thehot-bubble region changes from violent, long-lasting convectiveoverturn associated with accretion processes (model ML2D) torapid expansion and relatively slowly changing large-scale de-formation (model LP2D) which eventually dominate the overallasymmetry and the quadrupole moment of the exploding star(model EP2D). This change of the characteristics of non-radialmotions in the hot-bubble and postshock regions is directly re-


flected in the dominant frequencies of the gravitational wavesignal, which drop from about 200 Hz (model ML2D) downto less than 10 Hz (model EP2D). Thus, a measurement of thefrequency of the wave signal provides important insights intothe explosion dynamics. Moreover, since the signal producedby the convection inside the proto-neutron star is typically ofmuch higher frequency (500–1000 Hz), such a measurementwould also allow to discriminate the contributions from bothconvection zones. Unfortunately, the calculated maximum di-mensionless amplitudes hTT (Eq. (11)) are too small to be de-tected for a supernova outside our own Galaxy, because they liein the range |hTT(2D)| ≈ 1 × 10−22 ... 3 × 10−21 for a sourceat a distance of 10 kpc.

3.3. Three-dimensional results

The analysis of the three-dimensional model caused some prob-lems because our simulation volume involved only a 60 by 60degree sector of a full sphere. Simply extending the data fromthe computational volume to the whole sphere by making use ofthe periodic boundary conditions imposed in angular directionduring the simulation did not make sense because a sphericalpotential was used in the simulation and because the resultingconfiguration was highly symmetrical in angular direction. Thecomputed sector fits into the full sphere exactly three times inθ-direction and six times in φ-direction which gives rise to a 60,120, and 180 degree rotational symmetry around the z-axis.

In order to compute the gravitational radiation we insteadproceeded as follows. First we divided the simulated sector into36 subsectors of 10 by 10 degree each. Then these 36 subsectorswere randomly distributed over the full sphere. This process isjustified, because the angular distribution of the dynamical vari-ables varies on scales smaller than or, at least, not much largerthan 10 degrees (see Sect. 2.1.2 and Fig. 12). However, one hasto take into account that in φ-direction the linear extent of thesubsectors decreases like sin θ when approaching the poles atθ = 0 and θ = π. We therefore mapped one 10 by 10 degreesubsector as constructed in the simulated 60×60 degree wedgeover several pole-near subsectors such that the length scalesof the structures were approximately conserved. This mappingprocedure of the data was repeated for all radial shells and forall times while keeping the association of 10× 10 subsectorsof the computational cone with their randomly chosen mappinglocations on the sphere fixed.

The resulting angular distribution is displayed in Fig. 22showing the density on a sphere of radius 65 km (which is rightinside the convective layer in the proto-neutron star) at differentmoments of the evolution. Fig. 22 shows that the structures areindeed small enough and the mapping process is random enoughto produce statistically homogeneous angular distributions. Theresult is so perfect that if we omitted the white solid line thatmarks the boundary of the computed sector, it would be impos-sible to locate the wedge of the simulation by eye inspection.

Fig. 22 further shows that the angular scale of the fluctu-ations is time-dependent and grows steadily with time fromthe onset of the instability at about 9 ms (upper left panel in

Fig. 22). The level of the fluctuations reaches a maximum atabout 13.6 ms (lower left panel in Fig. 22), when the ratio ofmaximum to minimum density is 2.32 at 65 km. Within thenext 3 ms the ratio drops to a value of 1.72 (lower right panelin Fig. 22) which reflects the increasing homogenization of themixing layer.

From the density distribution on the full sphere as con-structed by the mapping procedure we computed the corre-sponding three-dimensional gravitational potential taking intoaccount the central point mass (see Sect. 2.1). The three-dimensional Poisson solver employed in the calculation of thepotential is an extension of the two-dimensional solver of Muller& Steinmetz (1995) and was provided by Zwerger (personalcommunication). Using the computed three-dimensional gravi-tational potential and the mapped density and velocity distribu-tions, we were able to derive the gravitational wave signatureof model HB3D from Eqs. (13–23).

According to Fig. 23 model HB3D, which is the three-dimensional analogue of model HB2D, emits significantly lessenergy in form of gravitational waves. There are several reasonsfor that. Firstly, the convective elements are smaller, only abouthalf of the typical size found in two dimensions (l3D ≈ 0.5l2D),and the mass motions are therefore less coherent and do notcause the strong large-scale deformations seen in 2D. Secondly,the rising and sinking convective elements move with smallervelocities in three spatial dimensions, v3D <∼ 5 × 108 cm/s ≈0.5v2D, which leads to reduced over- and undershooting (onlyabout 0.8 instead of 1.2 pressure scale heights, see Sect. 2.1)and is another reason for the weaker large-scale deformationof the outer layers of the proto-neutron star in the 3D modelHB3D. While model HB2D emits a gravitational wave energyof EGW = 5.5 × 10−10 Mc2, model HB3D radiates an en-ergy of only EGW = 7.5 × 10−13 Mc2 during the same timeinterval of 32 ms (compare Figs. 18 and 23). The quadrupoleamplitudes of both polarizations (A+ and A×) are also shownin Fig. 23 for an observer at the pole and at the equator, respec-tively. The maximum absolute values of the amplitudes neverexceed 4 cm which is about a factor of 100 smaller than inthe two-dimensional model HB2D (see Fig. 18). This corre-sponds to a maximum dimensionless gravitational wave ampli-tude hTT (Eq. (13)) of |hTT(3D)| ≈ 2 × 10−22 for a source atdistance 10 kpc, about a factor of 15 smaller than in HB2D. Acomparison of Figs. 19 and 20 with Fig. 23 (upper and lowerright panels, respectively) reveals that the frequency spectrumof the quadrupole amplitude and the spectral energy densityshow more relative power, respectively are more peaked, to-wards lower frequencies in case of model HB3D. In HB3D thespectral maxima are around 100–200 Hz and the spectra droprapidly towards higher frequencies, whereas the amplitude spec-trum of HB2D has a broad region of highest power between 100and 700 Hz (Fig. 18) and the spectral energy density of HB2Dpeaks between about 200 and 600 Hz. This difference of thewave frequencies is a result of the smaller convective velocitiesand the correspondingly longer overturn timescales and less vi-olent convection in the three-dimensional situation.


Fig. 22. The four panels show the time evolution of the density fluctuations on a sphere with a radius of 65 km located in the middle of theconvective layer inside the proto-neutron star. The snapshots are taken at 9.0 ms (top left), 12.2 ms (top right), 13.6 ms (bottom left), and16.6 ms (bottom right) after the start of the 3D simulation. The ratio of maximum density to minimum density is 1.11, 2.09, 2.32, and 1.72,respectively. In each panel the white frame marks the 60 by 60 degree sector in which the simulation was performed. Note that the fluctuationsare homogeneously distributed over the sphere at all times and that the size of the structures increases with time

3.4. Gravitational waves from anisotropic neutrino emission

The gravitational wave emission associated with the anisotropicradiation of neutrinos can only be estimated for the presentedmodels because no multi-dimensional neutrino transport wasused in the simulations, in fact some of the models were com-puted without including neutrino effects at all.

In order to estimate the anisotropy parameter α(t) definedin Eq. (29) we used the following procedure. Assuming thatthe neutrino flux can be approximated by black-body emissionand thus scales with the fourth power of the temperature T andwith the area r2dΩ of the emitting surface region, we employeddLν(Ω, t)/dΩ ∝ 〈T 4r2〉 to evaluate Eq. (29). Here 〈T 4r2〉 de-notes a mass-weighted radial average of T 4r2 in the neutri-


Fig. 23. The gravitational wave signal of con-vective instabilities inside the proto-neutron staraccording to the three-dimensional model HB3D.The upper left panel shows the quadrupole wave-forms of the two independent signal amplitudesA+ and A× at the pole (θ = 0, φ = 0; solidand dotted lines) and at the equator (θ = 90,φ = 0; dashed and dashed-dotted lines), respec-tively. The upper right panel shows the frequencyspectra of the polar amplitudes Ap

+ (solid curve)and Ap

× (dotted curve), the lower left panel theenergy radiated in form of gravitational waves,EGW [10−12 Mc2], as a function of time, andthe lower right panel displays the correspondingspectral energy density

nospheric layer that is considered to encompass densities of5 × 1010 g/cm3 to 1011 g/cm3. The mean 〈T 4r2〉 is evaluatedby adding up the mass-weighted contributions of all grid cellsalong a specified radial beam direction Ω. Repeating this for allangular directions and for all given time levels of a model yieldsthe needed input dLν(Ω, t)/dΩ into Eq. (29).

Evaluating the two-dimensional model HB2D we find thatthe anisotropic neutrino emission associated with the convec-tive processes inside the proto-neutron star leads to an equatorialanisotropy parameter of αe(2D) ≈ 10−2 ... 10−1 (Eq. (29) withthe angular factor of Eq. (27)) at a time when the convectiveoverturn is fully developed (t >∼ 13 ms after the start of thesimulation). We obtain numerical values that are rather close tothose of the relative quadrupole moment of r2T 4. A test inte-gration, moreover, was in agreement with the analytical resultfor the axially symmetrical case, namely that the correspond-ing polar anisotropy parameter αp (Eq. (29) with the angular

factor of Eq. (26)) is negligibly small. Using the temperaturedistribution from the three-dimensional simulation HB3D weget αe(3D) ≈ αp(3D) ≈ 5 × 10−4 ... 5 × 10−3 for t >∼ 13 ms,again fairly similar to what is obtained from the quadrupoleformula.

The anisotropy parametersα therefore turn out to be roughlya factor of 10 smaller in the 3D case than in the 2D situation.Correspondingly, when going from 2D to 3D, the gravitationalradiation field hTT (Eq. (28)) becomes about one order of mag-nitude smaller and the gravitational wave energyEGW (Eq. (31))approximately two orders of magnitude, if the neutrino lumi-nosities of both cases are similar. Assuming constant luminosityLν and anisotropy α for an emission time ∆t, one gets

RhTTν ∼ 1.6× 104 α

(Lν

1053 erg/s

)(∆t

1 s

)cm (32)


and

EGW,ν ∼ 1.5× 10−8 β α2

(Lν

1053 erg/s

)2(∆t

1 s

)Mc2 . (33)

For Lν = 1053 erg/s and a typical emission time of one sec-ond – which assumes that convective overturn and anisotropicneutrino emission continue during most of the time when thegravitational binding energy of the proto-neutron star (a few1053 erg) is lost by neutrino emission – one finds |RhTT

ν (2D)| tobe of the order of several 100 cm, i.e., |hTT

ν (2D)| <∼ few×10−20

for a source at a distance of 10 kpc. The gravitational waveenergy is EGW,ν(2D) <∼ few × 10−11 Mc2. For the three-dimensional case these numbers are |hTT

ν (3D)| <∼ few× 10−21

and EGW,ν(3D) <∼ few× 10−13 Mc2.The gravitational wave signal from anisotropic mass mo-

tions due to convection inside the proto-neutron star wasfound (Sects. 3.2 and 3.3) to be |hTT(2D)| <∼ 3 × 10−21

and EGW(2D) ≈ 5.5 × 10−10 Mc2 for model HB2D and|hTT(3D)| <∼ 2 × 10−22 and EGW(3D) ≈ 7.5 × 10−13 Mc2

for the three-dimensional model HB3D. A comparison showsthat the gravitational wave amplitude associated with the neu-trino emission is somewhat larger than the wave amplitude dueto convective motions. In two dimensions the neutrino gravita-tional wave amplitude is about 5 times larger, in three dimen-sions the factor can become even 10. The total energy radiatedin gravitational waves, however, is dominated by the contribu-tions from the mass quadrupole moment. Neutrino gravitationalwaves account only for a minor fraction (for a few per cent atmost) of the total gravitational wave energy in two-dimensionalmodels, while in 3D they contribute up to several 10% of thegravitational wave energy.

Like the gravitational waves from convective motions insidethe newly formed neutron star, the neutrino gravitational wavesare predominantly emitted in the frequency band between sev-eral 10 Hz and a few 100 Hz because of the common origin andthus similar timescales of the anisotropic processes. In case ofthe neutrino gravitational waves there is also a superimposedlow-frequency component (∼ 1 Hz to a few Hz) caused by thelong-time variation of the neutrino emission from the coolingneutron star on a timescale of about one second.

4. Summary and discussion

We have presented two- and three-dimensional simulations ofconvective instabilities during the first second of a Type II su-pernova explosion. Convective processes occur in two distinct,spatially well separated regions: (i) inside the proto-neutron starimmediately below the neutrinosphere, and (ii) in the neutrino-heated hot-bubble region interior to the outward propagatingrevived shock front. The convective overturn around and be-low the neutrinosphere (region (i)) leads to anisotropic neutrinoemission which is also a source of gravitational waves. We havecalculated the gravitational wave signals from mass motions inboth convectively unstable regions and from the aspherical neu-trino emission, including the quadrupole waveforms, the power

spectra, and the total amount of the emitted gravitational waveenergy.

For a supernova located at a distance of 10 kpc the maxi-mum dimensionless gravitational wave amplitudes |hTT| associ-ated with convective mass motions range from about 1× 10−22

to 3× 10−21 and the total amount of the emitted energy variesfrom 3×10−14 Mc2 to 5×10−10 Mc2. Convective motionsinside the proto-neutron star involve more mass and are moreviolent and therefore produce the stronger gravitational wavesignal with up to a factor of 10 larger wave amplitudes. How-ever, most of the gravitational radiation from convection insidethe proto-neutron star is emitted in the frequency band 100–1000 Hz, while convective motions in the hot-bubble regiongenerate waves from several 100 Hz down to a few Hz.

Comparing different two-dimensional models we find thatwith increasing total neutrino luminosity and hence with in-creasing explosion energy the convective activity in the hot-bubble region changes from violent convective overturn associ-ated with anisotropic accretion processes to rapid overall expan-sion and relatively slowly changing large-scale deformations ofthe expanding shells behind the outward propagating supernovashock. This change of the characteristics of non-radial motionsin the hot-bubble region is directly reflected in the dominant fre-quencies of the gravitational wave signal. While turbulent over-turn around the proto-neutron star produces gravitational waveswith most power at frequencies of 100–200 Hz, the dominantfrequencies are at only some 10 Hz when the period of con-vective activity is short and the non-sphericity of the model isdetermined by the explosive expansion. Thus, a measurementof the frequency of the wave signal would provide important in-sights into the explosion dynamics. Moreover, since the signalproduced by the convection inside the proto-neutron star is typ-ically of much higher frequency (about 1000 Hz), it would alsobe possible to discriminate the contributions from the two con-vection zones to the measured signal. Interestingly, structures inthe gravitational wave signal are well correlated with prominentfeatures in the neutrino emission if both gravitational wave andneutrino production are associated with dynamical processes inand around the nascent neutron star. Simultaneous informationfrom both neutrino and gravitational wave measurements wouldtherefore impose important constraints on theoretical models ofType II supernova explosions.

The anisotropic neutrino emission itself generates gravita-tional waves, too. We estimate the degree of anisotropy from thedensity and temperature inhomogeneities associated with con-vective and turbulent processes in the neutrinospheric region.We find that for typical post-bounce neutrino luminosities thegravitational wave amplitude can be larger than the wave am-plitude due to mass motions by a considerable factor of 5–10,although the energy in the neutrino tidal field is only a minorcontribution to the total energy radiated in gravitational waves.

Our three-dimensional simulation of convection inside theproto-neutron star gives a strongly reduced gravitational wavesignal compared to the corresponding two-dimensional model.The main reason for this is that in three spatial dimensions theconvective structures and elements are smaller (l3D ≈ 0.5l2D),


move less fast (v3D <∼ 5 × 108 cm/s ≈ 0.5v2D), and, cor-respondingly, show less strong overshooting and undershoot-ing (0.8 instead of 1.2 pressure scale heights). The maximumquadrupole amplitudes due to mass motions are reduced byabout a factor of 15, the gravitational wave amplitudes asso-ciated with the anisotropic neutrino emission by as much as afactor of 10. The total amount of energy radiated in form ofgravitational waves is 2–3 orders of magnitude smaller in 3D.A similarly strong reduction of the signal strength is to be ex-pected for the gravitational waves emitted by turbulent motionsin the hot-bubble region when the simulations will be performedin three spatial dimensions.

The gravitational wave signal from convective and turbulentprocesses inside the nascent neutron star and in the surroundingneutrino-heated region is rather weak compared with other po-tential astrophysical sources of gravitational waves. Neutron starmergers produce signals that are about 100–1000 times strongerwith dimensionless wave amplitudes |hTT|of up to (3–4)×10−18

for a source at 10 kpc distance (for recent calculations, see, e.g.,Ruffert et al. 1996 and references therein). Two-dimensionalsimulations of the gravitational collapse of rotating stellar coreswith realistic input physics yield maximum wave amplitudesof 2 × 10−20 (Monchmeyer et al. 1991). Computing a broadvariety of models with different angular momentum and differ-ent angular momentum distribution in the pre-collapse stellarcore and considering different equations of state during corecollapse, Zwerger (1995) and Muller & Zwerger (1995) foundthat the most efficient models have |hTT| <∼ 10−19. However,their most weakly emitting two-dimensional models producedwave amplitudes about one order of magnitude smaller thanthe maximum signal that can be expected from the simulationspresented in this paper.

Our results are in rough qualitative agreement with the find-ings of Burrows & Hayes (1996) based on two-dimensionalmodels, although there are quantitative differences. We empha-size that the results for the gravitational wave signal associatedwith non-spherical neutrino emission depend sensitively on theduration of the phase of anisotropic neutrino loss and on thetemporal evolution of the total luminosity in all kinds of neu-trinos. These characteristics will vary with the properties of theexploding star and thus with the parameters of the forming neu-tron star and will also be sensitive to the details of the numericalscheme and physical input used for the simulations. Moreover,calculations based on two-dimensional models tend to overes-timate the gravitational wave emission by about one order ofmagnitude.

The models that were analysed in this work are only an-other preliminary step towards the full, complex problem ofstellar core collapse and supernova explosion in three dimen-sions and towards a detailed quantitative understanding of theassociated emission of gravitational waves. We have performedhydrodynamical calculations in two and three dimensions andhave analyzed the effects of post-bounce convective activity inthe forming neutron star and in the exploding star. However,our models have a number of restrictions and approximations.These will have to be removed and their possible and probable

quantitative influence on the presented results will have to beinvestigated in future work.

Our simulations did start from progenitor star models thatwere neither evolved to the onset of core collapse in two or threedimensions, nor followed through core collapse and bouncewith a multi-dimensional code, although the multi-dimensionaldescription of these evolutionary phases could be importantto determine the structure of the initial state of our computa-tions, even in the case that rotation is absent in the star (see,e.g., Bazan & Arnett 1994; Burrows et al. 1995; Goldreich etal. 1995). Furthermore, self-consistency was violated by map-ping the initial configuration of a collapsed stellar core as givenfrom a one-dimensional, general relativistic simulation into ourmulti-dimensional code which treated gravity in the Newtonianapproximation. In addition, only in the two-dimensional runsnon-sphericities of the gravitational potential were taken intoaccount via an expansion into spherical harmonics, whereas inthe 3D simulations a spherically symmetrical potential derivedfrom the mean density distribution was used. Moreover, theneutrino physics in the current models needs to be improved oreven waits for being included in the multi-dimensional mod-elling. Finally, current progenitor star models and the presentedsupernova simulations assume zero angular momentum. Withthe combined effects of rotation, convection inside the proto-neutron star, and neutrino-driven overturn in the hot-bubble re-gion, however, the quantitative results of the gravitational waveemission could be significantly modified.

Acknowledgements. The post-collapse models provided to us byS.W. Bruenn and W. Hillebrandt to be used as initial models in oursimulations are kindly acknowledged. We would like to thank Maxi-milian Ruffert, Gerhard Schafer, and Thomas Zwerger for many en-lightening discussions and valuable comments. In particular, we aregrateful to Thomas Zwerger for providing to us his 3D Poisson solver.All hydrodynamical simulations were performed on the CRAY-YMP4/64 of the Rechenzentrum Garching. The Rechenzentrum Garchingis also acknowledged for generous financial support concerning thepublication costs.

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