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

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4. Summary and discussion

We have presented two- and three-dimensional simulations of convective instabilities during the first second of a Type II supernova explosion. Convective processes occur in two distinct, spatially well separated regions: (i) inside the proto-neutron star immediately below the neutrinosphere, and (ii) in the neutrino-heated hot-bubble region interior to the outward propagating revived shock front. The convective overturn around and below the neutrinosphere (region (i)) leads to anisotropic neutrino emission which is also a source of gravitational waves. We have calculated the gravitational wave signals from mass motions in both convectively unstable regions and from the aspherical neutrino emission, including the quadrupole waveforms, the power spectra, and the total amount of the emitted gravitational wave energy.

For a supernova located at a distance of 10 kpc the maximum dimensionless gravitational wave amplitudes [FORMULA] associated with convective mass motions range from about [FORMULA] to [FORMULA] and the total amount of the emitted energy varies from [FORMULA] to [FORMULA]. Convective motions inside the proto-neutron star involve more mass and are more violent and therefore produce the stronger gravitational wave signal with up to a factor of 10 larger wave amplitudes. However, most of the gravitational radiation from convection inside the proto-neutron star is emitted in the frequency band 100-1000 Hz, while convective motions in the hot-bubble region generate waves from several 100 Hz down to a few Hz.

Comparing different two-dimensional models we find that with increasing total neutrino luminosity and hence with increasing explosion energy the convective activity in the hot-bubble region changes from violent convective overturn associated with anisotropic accretion processes to rapid overall expansion and relatively slowly changing large-scale deformations of the expanding shells behind the outward propagating supernova shock. This change of the characteristics of non-radial motions in the hot-bubble region is directly reflected in the dominant frequencies of the gravitational wave signal. While turbulent overturn around the proto-neutron star produces gravitational waves with most power at frequencies of 100-200 Hz, the dominant frequencies are at only some 10 Hz when the period of convective activity is short and the non-sphericity of the model is determined by the explosive expansion. Thus, a measurement of the frequency of the wave signal would provide important insights into the explosion dynamics. Moreover, since the signal produced by the convection inside the proto-neutron star is typically of much higher frequency (about 1000 Hz), it would also be possible to discriminate the contributions from the two convection zones to the measured signal. Interestingly, structures in the gravitational wave signal are well correlated with prominent features in the neutrino emission if both gravitational wave and neutrino production are associated with dynamical processes in and around the nascent neutron star. Simultaneous information from both neutrino and gravitational wave measurements would therefore impose important constraints on theoretical models of Type II supernova explosions.

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

Our three-dimensional simulation of convection inside the proto-neutron star gives a strongly reduced gravitational wave signal compared to the corresponding two-dimensional model. The main reason for this is that in three spatial dimensions the convective structures and elements are smaller ( [FORMULA] ), move less fast ( [FORMULA] ), and, correspondingly, show less strong overshooting and undershooting (0.8 instead of 1.2 pressure scale heights). The maximum quadrupole amplitudes due to mass motions are reduced by about a factor of 15, the gravitational wave amplitudes associated with the anisotropic neutrino emission by as much as a factor of 10. The total amount of energy radiated in form of gravitational waves is 2-3 orders of magnitude smaller in 3D. A similarly strong reduction of the signal strength is to be expected for the gravitational waves emitted by turbulent motions in the hot-bubble region when the simulations will be performed in three spatial dimensions.

The gravitational wave signal from convective and turbulent processes inside the nascent neutron star and in the surrounding neutrino-heated region is rather weak compared with other potential astrophysical sources of gravitational waves. Neutron star mergers produce signals that are about 100-1000 times stronger with dimensionless wave amplitudes [FORMULA] of up to (3- [FORMULA] for a source at 10 kpc distance (for recent calculations, see, e.g., Ruffert et al. 1996 and references therein). Two-dimensional simulations of the gravitational collapse of rotating stellar cores with realistic input physics yield maximum wave amplitudes of [FORMULA] (Mönchmeyer et al. 1991). Computing a broad variety of models with different angular momentum and different angular momentum distribution in the pre-collapse stellar core and considering different equations of state during core collapse, Zwerger (1995) and Müller & Zwerger (1995) found that the most efficient models have [FORMULA]. However, their most weakly emitting two-dimensional models produced wave amplitudes about one order of magnitude smaller than the maximum signal that can be expected from the simulations presented in this paper.

Our results are in rough qualitative agreement with the findings of Burrows & Hayes (1996) based on two-dimensional models, although there are quantitative differences. We emphasize that the results for the gravitational wave signal associated with non-spherical neutrino emission depend sensitively on the duration of the phase of anisotropic neutrino loss and on the temporal evolution of the total luminosity in all kinds of neutrinos. These characteristics will vary with the properties of the exploding star and thus with the parameters of the forming neutron star and will also be sensitive to the details of the numerical scheme and physical input used for the simulations. Moreover, calculations based on two-dimensional models tend to overestimate the gravitational wave emission by about one order of magnitude.

The models that were analysed in this work are only another preliminary step towards the full, complex problem of stellar core collapse and supernova explosion in three dimensions and towards a detailed quantitative understanding of the associated emission of gravitational waves. We have performed hydrodynamical calculations in two and three dimensions and have analyzed the effects of post-bounce convective activity in the 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 be investigated in future work.

Our simulations did start from progenitor star models that were neither evolved to the onset of core collapse in two or three dimensions, nor followed through core collapse and bounce with a multi-dimensional code, although the multi-dimensional description of these evolutionary phases could be important to determine the structure of the initial state of our computations, even in the case that rotation is absent in the star (see, e.g., Bazan & Arnett 1994; Burrows et al. 1995; Goldreich et al. 1995). Furthermore, self-consistency was violated by mapping the initial configuration of a collapsed stellar core as given from a one-dimensional, general relativistic simulation into our multi-dimensional code which treated gravity in the Newtonian approximation. In addition, only in the two-dimensional runs non-sphericities of the gravitational potential were taken into account via an expansion into spherical harmonics, whereas in the 3D simulations a spherically symmetrical potential derived from the mean density distribution was used. Moreover, the neutrino physics in the current models needs to be improved or even waits for being included in the multi-dimensional modelling. Finally, current progenitor star models and the presented supernova simulations assume zero angular momentum. With the combined effects of rotation, convection inside the proto-neutron star, and neutrino-driven overturn in the hot-bubble region, however, the quantitative results of the gravitational wave emission could be significantly modified.

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