If neutrino transport calculations are done in the
diffusion approximation, where the distribution function is
truncated at first order
the expansion in (21) is also truncated at the same order
and, hence, there is no need to calculate high order terms of the kernels.
However, in the
semitransparent region where the diffusion approximation breaks down
and the flux factor is big,
one needs to use a closure relation which is different from
and
the terms in 
for 
must be taken
into account because, as we will demonstrate,
keeping only the first order term
would give a wrong value of the energy exchange between neutrinos
and matter.
To see this fact more clearly, let us study the influence on
the energy source of 
and 
order terms truncating
at l=3.
In this expression we have written inside brackets 
the part of the correction due to the distribution function. The value of
these factors is restricted to the interval [0,1]. Therefore, the
maximum value of each new term is 
, which
we have plotted in Figs. 1 (click here)a,b and c for l=1,2,3, respectively.
As we can see from the plots, if terms in brackets are not small,
the contribution should be included for all energies and
the order term has a significant contribution for low
energies. Only when the terms in brackets are much smaller than 1 the
expansion can be truncated at first order.
Figure 1: Ratios 
for i=1,2,3
as functions of 
and 
for an electron degeneracy parameter 
and 
-
neutrino type
The closure relation becomes, therefore, a fundamental point in
the calculation of the 
emission-absorption rate.
There are several closure relations used by different authors
and the question "which is the best one?'' has no answer yet.
For the sake of comparison, in this work we include four different
closure relations: MB (Minerbo 1978),
LP (Levermore & Pomraining 1981,
CB (Cernohorsky & Bludman 1994) and MH (Mihalas 1984).
Cernohorsky closure depends on both, the 
and 
moments of the distribution function p=p(I0,f) and
q=q(I0,f) and the rest of them
are uniparametric closures, this is, p=p(f) and
q=q(f). In the small occupation limit (
) CB closure
is equivalent to Minerbo's one and in the maximal forward angular
packing limit is equivalent to the vacuum approximation closure (see below).
The form of these closures is the following:
where 
with 
being the inverse of the Langevin
function 
. The following expression fits this function well
(Cernohorsky & Bludman 1994).
In Fig. 2 (click here) we show p(f) and q(f) for the different
closure relations, taking an occupation level for the CB case of I0=0.1.
In Fig. 3 (click here), the combinations of the different moments
that appear inside brackets in (24) are plotted as a function of f. In both
figures the solid line is for CB closure, dotted line for MH,
dashed line for MB and dashed-dotted line for LP. We also plot
with crosses the closure obtained in the vacuum approximation.
As can be seen from the figures, there are important differences between
different closures for 
. This can lead to relevant differences
in the energy exchange between neutrinos and matter for large values
of the flux factor f.
Figure 2: Plots of p(f) and q(f) for different closure
relations. The solid line is for CB closure, dotted line for MH,
dashed line for MB and dashed-dotted line for LP. We also plot
with crosses the closure obtained in the vacuum approximation
Figure 3: Terms in brackets in the and order
contributions in the Legendre expansion for different closures. The meaning
of the lines is the same as in Fig. 2 (click here)
To illustrate this feature, let us study a simple model consisting
of a sphere of radius R radiating neutrinos and antineutrinos
isotropically into vacuum, the so called vacuum approximation
(Cooperstein et al. 1986). The closure consistent
with this model (VA) is
We assume that the neutrino (and antineutrino) spectrum
at the surface of the sphere is
Fermi-Dirac with zero chemical potential and 
MeV.
Of course in a real case the
neutrinospheres of neutrinos and antineutrinos are located in
different places for different energies, but this example is
illustrative of the general behaviour of the heating rate. We
calculate the net (heating minus cooling)
heating rate of the matter per unit volume for given
distance to the center of the sphere (d)
and matter temperatures (T).
For 
cooling dominates over heating and we find
that the effect of
including new orders is negligible. In contrast, for low T the
dominant term in Eq. (19) is the term proportional to 
and there
are remarkable differences in the heating rates obtained including
the new terms. These effects are also closure dependent and we
compare the closure relations discussed previously in order to estimate
the differences between them.
Figure 4: Total energy deposited for different values of
in units of
1020 erg 
. The solid line
is the exact solution after numerical integration and the different symbols
stand for the closures LP (triangle), MH (star), MB (diamond) and
CB (crosses)
In Fig. 4 (click here) we present the result of our calculations. We fix the matter temperature at T=0.5 MeV and we study the influence of higher order terms and closures for different distances from the center of the sphere (different flux factors). For the sake of comparison, we have performed Monte Carlo integration of the complete expression for the reaction rate, shown in the figure as the solid line. We overplot the total energy deposition after including each new order, using different symbols for the different closures. As can be seen, at first order there is an underestimation of the deposition rate that is worse for high flux factors, even changing the sign (this means emission instead absorption of energy) for f>0.8. The inclusion of the second order term gives a big improvement if one uses the closure consistent with the form of the distribution function in the model. The third order term is a small correction that can be omitted in all cases unless high accuracy is required. Using different closures instead of VA gives worse results at small x, but solves the problem in the sign for high values of x. We also observe remarkable differences between the results obtained using different closures, even though, we cannot deduce which one would have better behaviour in a realistic case.
To summarize, we restate our main conclusions.
The first conclusion is that, in the semi-transparent region and
for matter temperature lower than neutrino temperature,
it is necessary to consider the expansion up to order.
This procedure gives good results when combined with an appropriate
closure relation. The order term does not lead
to a substantial improvement
in the solution
since, as we have shown, it is only a small correction.
For
cooling dominates over heating and we find that the effect of
including new orders is negligible.
The second conclusion is that, even though convergence is reached
with order corrections, the result is very sensitive to the
closure relation chosen.
Best results are obtained
when using a closure relation consistent with
the particular distribution function used in the model
and therefore, detailed study of the closure in each particular
problem is needed to obtain good estimates of interaction rates
in a multigroup flux-limited diffusion problem.
These results can be applied in all problems concerning neutrino
transport such as stellar core collapse or cooling of newly born
neutron stars.
Acknowledgements
This work has been supported by the Spanish DGCYT grant PB94-0973 and partially by the US Dept. of Energy grant DE-AC02-87ER40317. We thank A. Pérez for careful reading the manuscript and J.A.P. thanks J. Lattimer, M. Prakash and S. Reddy for useful discussions.
