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

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5. Summary

In this paper an extensive comparison was made between a newly developed Boltzmann neutrino transport code based on the discrete ordinate (SN) method as described by Mezzacappa & Bruenn (1993a, b, c), and a Monte Carlo transport treatment (Janka 1987, 1991a) by performing time-dependent calculations of neutrino transport through realistic, static profiles of protoneutron star atmospheres under the assumption of spherical symmetry. In particular, the sensitivity of the results of the Boltzmann solver to the employed numbers of radial, angular and energy grid points and to the treatment of the radial advection terms was investigated. The flux factor and Eddington factor, which contain information about the angular distribution of the neutrinos in the neutrino-decoupling region, were also compared with the approximate treatment of this regime by a multi-energy-group flux-limited diffusion code (MGFLD).

The Boltzmann and Monte Carlo results showed excellent agreement for observables such as the luminosity and the flux spectra which are determined in those regions of the star where the neutrino-matter interactions are still very frequent and thus the neutrino distributions are still nearly isotropic. Since the luminosity and the spectra are essentially conserved farther out, the spatial evolutions as well as the surface values exhibit this agreement, as long as the finite differencing of the Boltzmann solver is done in a conservative way.

Because the requirements of computer time increase steeply with a better resolution of the neutrino distribution function, the number of angular grid points that can be used in the Boltzmann solver is rather limited. Typically computations with only 6-10 angular bins between zero and 180 degrees are feasible. For a Gauss-Legendre quadrature set, the direction cosines have a maximum value [FORMULA] less than unity. It turns out that in this case the Boltzmann code cannot describe the strong forward peaking of the neutrino distribution in the semitransparent and transparent regimes very accurately, and it is therefore not possible to reproduce the exact limits for the flux factor, [FORMULA], and for the Eddington factor, [FORMULA], at large distances away from the average "neutrinosphere". Instead, the Boltzmann solver systematically tends to underestimate the anisotropy and thus the values of the angular moments of the neutrino distribution in the optically thin regime. This trend in also visible in recent results published by Messer et al. (1998). Interestingly, it is exactly opposite (but significantly smaller) than the deficiencies of MGFLD which tends to overestimate the angular beaming outside the neutrinosphere because all flux limiters enforce the free-streaming limit when the optical depth of the stellar background becomes very low (Janka 1991a, 1992).

Now, the energy transfer from neutrinos to the stellar plasma through the dominant processes of electron neutrino and antineutrino absorption scales with the neutrino energy density (or number density). Since for given luminosity the latter is inversely proportional to the flux factor (Eq. 20 and Figs. 12 and 13), the energy deposition rate by neutrinos in the hot-bubble region of the supernova core is larger for smaller values of the flux factor (see Eq. 1 and also Messer et al. 1998). Therefore one has to worry that an overestimation (underestimation) of the angular moments of the neutrino distribution will lead to an underestimation (overestimation) of the neutrino heating rate. In case of MGFLD this problem was investigated and analysed in very much detail recently by Messer et al. (1998). Comparing calculations with their Boltzmann code and with MGFLD, Messer et al. (1998) demonstrated that MGFLD has serious deficiencies to describe the neutrino transport and neutrino-matter coupling in collapsed stellar cores at early postbounce times. Convergence of their Boltzmann results was checked by Messer et al. (1998) by comparing simulations with 6-point and 8-point Gaussian quadrature. Their tests suggest that convergence of the neutrino net heating rate beneath the shock is reached by the time 6 fixed Gaussian quadrature points are used, and the differences in the net heating rate between 6-point and 8-point Gaussian quadrature are dynamically insignificant. According to Messer et al. (1998) this suggests that a fairly small number of fixed angular grid points may be sufficient for an accurate computation of the neutrino net heating of the shocked core material.

However, Messer et al. (1998) did not have the possibility to check their numerical results against exact solutions. Our computations show that convergence of the Boltzmann results for the angular moments of the neutrino distribution is extremely slow when the number of angular points for the Gaussian quadrature is increased. Deviations from the exact Monte Carlo results are still visible (although not dramatically large) even when 10 angular grid points are used (Figs. 10 and 11). Unfortunately, the models of static protoneutron star atmospheres which we used as background profiles in this work are for rather late times (several seconds) after core bounce where the neutrino energy deposition rates are at least one to two orders of magnitude smaller than during the crucial phase of shock revival. Therefore we did not attempt to investigate the effects on the neutrino heating rate that are associated with the remaining differences between the results from the Boltzmann solver and from the Monte Carlo code. Of course, this question should be studied by future work, also with respect to neutrino pair processes like neutrino-antineutrino annihilation into electron-positron pairs which are sensitive to both the flux factor [FORMULA] and the Eddington factor [FORMULA] (see Janka 1991b).

The forward peaking of the neutrino angular distribution in the optically thin regime can be better reproduced by the Boltzmann solver and thus the agreement with the highly accurate Monte Carlo data can be improved by taking a significantly larger number of fixed angular grid points for the Gaussian quadrature. Here, however, strict limits are set by the strong increase of the computation time. Alternatively, one might choose a quadrature set, e.g., a Lobatto quadrature set, that includes unity, in which case an increase of the total number of angular grid points may be unnecessary. However, making this replacement globally might cause problems in the optically thick limit where the isotropy of the neutrino distribution has to be described. Certainly the most elegant and flexible way to improve the treatment of the neutrino angular distribution with a small number of angular mesh points is the implementation of a variable angular mesh in the Boltzmann solver. The positions of the angular grid points must be moved at each time step and in every spatial zone such that they are clustering in the forward direction in the optically thin regime.

We found that the energy spectra can be well calculated with about 15 energy mesh points. The fact that a reduction of the number of spatial grid points from more than 100 to only 15 in the neutron star atmosphere did not change the quality of the transport results means that the Boltzmann code can be reliably applied to realistic simulations which involve the whole supernova core. Moreover, it was demonstrated that the details of the interpolation between centered differencing and upwind differencing in the spatial advection term can affect the accuracy of the transport results.

The excellent overall agreement of the results obtained with the Boltzmann code and the Monte Carlo method confirms the reliability of both of them. Good performance of the SN method for solving the Boltzmann equation of neutrino transport has been found before by Mezzacappa & Bruenn (1993a, b, c) and Messer et al. (1998) even for realistic dynamic situations. We hope that the work described here also helps to reveal possible deficiencies and weaknesses and thus will serve for further improvements of the numerical treatment of neutrino transport in supernovae and protoneutron stars.

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999