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Astron. Astrophys. 317, 140-163 (1997)

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1. Introduction

Currently the most promising model of Type II supernova explosions, which are powered by the gravitational binding energy released in the collapse of the iron-nickel core of a massive star with a main sequence mass of [FORMULA], relies on the so-called delayed explosion mechanism by neutrino heating (Wilson 1985; Bethe & Wilson 1985; Wilson et al. 1986; Colgate 1989; Janka 1993). According to this mechanism energy is deposited in the layers between the nascent neutron star and the stalled prompt supernova shock during a period of a few 100 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-bubble region and eventually causes the supernova explosion by reviving the stalled shock wave.

After the shock wave stalls ( [FORMULA] ms after bounce) and before the delayed neutrino heating mechanism begins to become effective ( [FORMULA] ms after bounce) the outer parts of the proto-neutron star are convectively unstable because, firstly, the deleptonization occurring in the shocked matter outside the neutrinosphere produces a negative lepton gradient and because, secondly, the weakening (and the eventual stalling) of the prompt shock wave gives rise to a negative entropy gradient in the same region. According to Epstein (1979) this is a sufficient condition for convective motion to occur. This is a situation commonly encountered in most supernova simulations (Burrows 1987; Burrows & Lattimer 1988; Bruenn 1993). The existence of these instabilities was demonstrated in several multi-dimensional hydrodynamical simulations (Burrows & Fryxell 1992, 1993; Burrows et al. 1995; Janka & Müller 1993a,b; 1995a,b; Müller 1993; Müller & Janka 1994).

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

Provided sufficient energy is deposited by neutrinos in the postshock matter an energetically typical Type II supernova explosion ( [FORMULA] erg) results. Whether this explosion mechanism is successful or not depends very sensitively on the neutrino flux that is emitted from the neutrinosphere of the hot proto-neutron star and its decay with time (Janka 1993; Janka & Müller 1995a, b, 1996).

While propagating outward through the stellar envelope the shock wave will produce density and pressure stratifications, which are Rayleigh-Taylor unstable in the neighbourhood of composition interfaces, i.e. near nuclear burning shells (Bandiera 1984; Benz & Thielemann 1989). This third kind of instability (actually a set of instabilities) in Type II supernova explosions has numerically been investigated in great detail (Fryxell et al. 1991; Hachisu et al. 1990, 1991; Herant & Benz 1991, 1992; Müller et al. 1991) and will not be considered further here.

In Table 1 an overview over the different types of hydrodynamical instabilities in Type II supernova explosions is presented and some important characteristics are summarized.

The non-spherical stratification and mass flow resulting from these instabilities, or, more precisely, the time-dependent mass quadrupole moment, is a potential source of gravitational radiation (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 convective regions are located at relatively small radii (see Table 1). The instabilities in the stellar envelope occur at such large radii that no 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 estimate the gravitational wave amplitudes and energy associated with the anisotropic neutrino emission from our models (Epstein 1978, Turner 1978). In particular the convective processes around the neutrinosphere can lead to anisotropies of the neutrino luminosity which are caused by convective transport of neutrinos and by temperature and density variations in the neutrinospheric region.

In the following we present the first detailed investigation to determine the gravitational radiation from post-bounce convective motions in Type II supernova explosions. The results are based on several two-dimensional hydrodynamical simulations and on two three-dimensional ones (Janka & Müller 1995a, 1996; Müller 1993). Special attention is therefore focussed on the comparison of the characteristics of the convective overturn in the two- and three-dimensional cases and on the corresponding differences of the gravitational-wave emission. Our analysis is in some sense complementary to other investigations, in which the gravitational radiation from collapsing, rotating cores was computed (Mönchmeyer et al. 1991; Bonazzola & Marck 1993; Zwerger 1995; Müller & Zwerger 1995). In rotational core collapse the gravitational radiation results from a time-dependent quadrupole moment due to a coherent, large-scale deviation from spherical symmetry caused by the action of centrifugal forces, whereas the gravitational radiation emitted from the convectively unstable regions is produced by small-scale statistical deviations from sphericity. Another potential source of gravitational radiation from Type II supernova are asphericities which can already exist in the Si- and O-shells of the progenitor star (Bazan & Arnett 1994) and which might be amplified during core collapse (Burrows & Hayes 1996).

The paper is organized as follows. In Sect. 2 we describe the results of our two- and three-dimensional simulations with the main emphasis being put on the simulations of convection inside the proto-neutron star, because the results for the convective overturn processes in the hot-bubble region have already been presented and discussed earlier (see Janka & Müller 1995a, 1996). In Sect. 3 first the formalism is described which was used to calculate the gravitational wave signature of our models. Thereafter, the quadrupole waveforms of the gravitational radiation, the frequency-dependent spectra of the wave amplitudes, the spectral energy densities (i.e., the differential energy emitted at different frequencies), and the total, i.e. spectrally integrated, energies emitted in gravitational waves are given for all our models. Finally, we summarize our results and discuss their implications in Sect. 4.

[TABLE]

Table 1. Hydrodynamical instabilities in Type II supernova explosions

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