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.
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.