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Astron. Astrophys. 344, 533-550 (1999)

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3. Models

3.1. Stellar models

The time-dependent transport calculations presented here were performed for background profiles which are representative of protoneutron star atmospheres during the quasi-static neutrino cooling phase (Wilson 1988). At this stage, several seconds after core bounce, the typical evolution timescale of density, temperature, and electron fraction is much longer than the timescale for neutrinos to reach a stationary state. Therefore our assumption of a static and time-independent background is justified. In addition, our interest is focused on the radial evolution of the Eddington factors and on a test of the influence of the energy and angle resolution used in the SN Boltzmann solver. Both aims do not require a fully self-consistent approach which takes into account the evolution of the stellar background (in particular of the temperature and composition). In fact, the Eddington factors are normalized angular integrals of the radiation intensity and as such reflect very general characteristics of the geometrical structure of the atmosphere where neutrinos and matter decouple.

Profiles from Wilson's (1988) protoneutron star model were taken for three different times, 3.32, 5.77, and 7.81 s after core bounce. With the chosen fundamental variables density [FORMULA], temperature T, and electron fraction [FORMULA], the thermodynamical state is defined for the plasma consisting of non-relativistic free nucleons, arbitrarily relativistic and degenerate electrons and positrons, and photons in thermal equilibrium. Figs. 1-3 show the input used for the three models. In Fig. 3 also the general relativistic metric coefficients [FORMULA] and [FORMULA] are given as provided by Wilson's data and used for a comparative general relativistic calculation of the neutrino transport in model W3.

[FIGURE] Fig. 1. The profile of density, temperature and electron fraction for Wilson's (1988) post bounce core model W1 ([FORMULA]s).

[FIGURE] Fig. 2. The profile of density, temperature and electron fraction for Wilson's post bounce core model W2 ([FORMULA]s).

[FIGURE] Fig. 3. The profile of density, temperature and electron fraction as well as the metric coefficients [FORMULA] (open squares) and [FORMULA] (filled squares) for Wilson's post bounce core model W3 ([FORMULA]s).

3.2. Computed transport models

All models computed with the Boltzmann code are summarized in Table 2. Models ST are the standard models, in which 105 uniform spatial, 6 angular and 12 energy mesh points were used. The energy mesh is logarithmically uniform and covers 0.9-110 MeV. The numbers of angular grid points and energy grid points were increased in models FA and FE, respectively. In model CS we used the same radial grid as in the Monte Carlo simulations where 15 radial zones were chosen. 105 spatial mesh points were again used in model NI with no interpolations of density, temperature and electron fraction in the radial grid of the Monte Carlo simulations. Model GR took into account the general relativistic effects. We used a non-uniform spatial mesh in model NU. A different interpolation of up-wind differencing and centered differencing was tried for the radial advection term in model DI. We assumed the nucleon scattering to be isotropic in model IS. As is understood from Table 2, most of the comparisons were done for the electron-type antineutrinos, since they are most important from the observational point of view.


Table 2. Characteristics of the calculated models. [FORMULA] implies that 105 spatial mesh points, 6 angular mesh points and 12 energy mesh points are used. See the text for details.

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999