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Astron. Astrophys. 356, 559-569 (2000)

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5. Conclusions

We have computed numerical neutrino transport using two methods: a discrete ordinate method, [FORMULA], to obtain a direct solution of the Boltzmann equation, and two-moment transport, TMT, with a variable Eddington factor. The two were compared first by looking at the angular moments [FORMULA], i.e., weak equivalence of the radiation field [FORMULA]. Four different closures, MEC, WMC, LPC, and MCC were used in TMT. Of these, LPC is not weakly equivalent to the three moments in [FORMULA]. The remaining three closures, MEC, MCC, and WMC, give more or less the same, good accuracy in monochromatic transport, with maximum entropy closure (MEC) being slightly the better of the three. In addition to weak equivalence, MEC displayed strong equivalence at this typical energy, i.e., the maximum entropy distribution function, [FORMULA], as a function of polar angle, gave a fair enough description of the radiation field [FORMULA] as calculated with the [FORMULA] method.

Spectral solutions of [FORMULA] showed that the Eddington trajectories [FORMULA] are different at different energies. One-dimensional closures are unable to account for this, but [FORMULA], the two-dimensional closure MEC, has extra freedom in (f-p) space. Thus, for example, MEC can follow a [FORMULA] trajectory. The closure of Wilson, WMC, does have a minimum where [FORMULA], but will always invoke it in a TMT solution, even when the actual radiation field may not display this feature. The MEC trajectories may cover a domain bounded by the limiting curves representing the Minerbo and maximum packing closure relations. In their approach to free streaming, all of these trajectories obey the causality constraint (13) (as do the [FORMULA] solutions). While MCC can be constructed to also meet this requirement, the closures WMC and LPC always violate this condition. In the low density regime the [FORMULA] solution may be closely tracked by the Minerbo limit of MEC. On the other hand, the maximum packing limit was never attained in the [FORMULA] solutions. Therefore, in our experience, Minerbo's closure may lead to a good representation of non-diffusive neutrino transport, but maximum packing cannot be recommended as a closure. The closure LPC, although originally shown to be consistent with maximum entropy considerations, lies essentially outside the domain of fermionic MEC.

Very good agreement between TMT-MEC and [FORMULA] was found in the energy averaged Eddington trajectories [FORMULA] versus [FORMULA], indicating that the neutrino spectrum is on average well represented by TMT-MEC. This was also found for TMT-MCC, but with TMT-MEC again being superior. This average weak equivalence of TMT-MEC/MCC and [FORMULA] was found for two different background models, representing an early and a late stage of core collapse. We may, therefore, expect that TMT-MEC and TMT-MCC are likely to give an accurate average representation of the neutrino radiation field during the entire core collapse scenario.

In this respect let us mention that velocity dependent terms encountered in actual dynamical (or relativistic) calculations introduce the third order moment (beyond f and p), for which a convenient practical inversion scheme is lacking. While from a given angular model distribution one can calculate the moments, in the case of the maximum entropy angular distribution a practical closure on the third order moment would be available only in the Minerbo and maximum packing limits. Another example is the third moment of the LP-distribution (see Van Thor et al. 1995), which of course will fail the weak equivalence requirement. The Minerbo and maximum packing closures, on the other hand, may be as adequate as discussed.

Summarizing, two-moment transport (TMT) gave the best overall fit to the discrete ordinate ([FORMULA]) solution when using the maximum entropy (MEC) and Janka's Monte Carlo (MCC) closures. In view of its physical basis and greater p-f domain, we favour MEC over MCC as a closure in two-moment neutrino transport.

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© European Southern Observatory (ESO) 2000

Online publication: April 10, 2000