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*Astron. Astrophys. 356, 559-569 (2000)*
## 5. Conclusions
We have computed numerical neutrino transport using two methods: a
discrete ordinate method, , 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
, *i.e.,* weak equivalence of
the radiation field . Four different
closures, MEC, WMC, LPC, and MCC were used in TMT. Of these, LPC is
not weakly equivalent to the three moments in
. 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,
, as a function of polar angle, gave
a fair enough description of the radiation field
as calculated with the
method.
Spectral solutions of showed
that the Eddington trajectories are
different at different energies. One-dimensional closures are unable
to account for this, but , the
two-dimensional closure MEC, has extra freedom in (*f*-*p*)
space. Thus, for example, MEC can follow a
trajectory. The closure of Wilson,
WMC, does have a minimum where , 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 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
solution may be closely tracked by
the Minerbo limit of MEC. On the other hand, the maximum packing limit
was never attained in the
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
was found in the energy averaged
Eddington trajectories versus
, 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
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 () 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.
© European Southern Observatory (ESO) 2000
Online publication: April 10, 2000
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