## 4. Model calculations## 4.1. Background modelsTransport calculations in this paper were restricted to neutrinos
of the electron type, and were performed on a stationary matter
background denoted as "model M0", shown in Fig. 2. This model is
a tri-polytrope representative of a hot proto-neutron star in the
cooling phase following collapse and core-bounce. In Sect. 4.4 we
also briefly consider "model WW1", which is an iron core halfway in
collapse (central density
g cm
Lattimer & Swesty's (1991) equation of state was used in both models. The equation of state determines the chemical composition (mass fractions of free protons, neutrons, alpha particles and a typical nucleus) and chemical potentials, which are required to determine neutrino opacities and the equilibrium distribution. The opacities include absorption and scattering on the particles mentioned; neutrino-electron scattering and pair processes were left out (but including them would not affect the conclusions of this paper). The next two sub-sections focus on a fixed neutrino energy, MeV, roughly the average energy of the neutrinos emerging from the background model M0. In Sect. 4.4, a spectral analysis is made of the Eddington factors. From the point of view of weak and strong equivalence, we compare the TMT results with Boltzmann transport using discrete ordinate () calculations involving angular bins. A mesh of 200 (unequally spaced) radial bins was used. The code is described elsewhere (Smit 1998). ## 4.2. Weak equivalenceWe first address weak equivalence, Results for neutrino energy
MeV are shown in Fig. 3,
displaying the angular moments ,
and
versus radius. Qualitatively, all
solutions exhibit the same behaviour of the 8.1 MeV radiation
field. Below km, the radiation
follows equilibrium dictated by the matter,
, while the small flux
and the Eddington factor
, indicate that the radiation is
diffusive. At larger radii,
km,
Comparing the profiles of TMT and , good agreement is found for MEC, MCC and WMC closures which cannot be distinguished from in Fig. 3a. The largest deviation of TMT-MEC is 6% (larger) at km, for TMT-MCC it is 6% (smaller) at km, and for TMT-WMC 9% (larger) at km, all with respect to . For LPC, the differences are much larger and amount to a 30% deficit at the surface. Looking at and
in Figs. 3b-d, we see the
differences between the various solutions becoming apparent in the
semi-transparent layer, most obviously in the case of the LPC
solution. LPC reaches parallel free streaming,
and
at the surface, where the other
solutions have For and , there is again fair agreement between and TMT for all closures except LPC, although larger differences are observed than in the case of . Nevertheless, these differences are too small to stand out clearly in the plots. The flux ratio computed with MEC, is found to approximate to better than 9% at radii larger than 25 km. Below this radius, the overall differences are in the range 10-20%, but they cannot be discerned in the figure. Near the surface, and practically coincide. The MCC and WMC flux ratios, and , agree with MEC for km, but in the semi-transparent region and out to the surface, they differ from MEC and from each other. The magnitude by which they differ from is in the same range as was found for . Fig. 3d shows that just beyond km the Eddington factors begin to visibly deviate from , and, for different solutions, also deviate from each other. A special feature to note is that , and drop below at radii km. This is impossible for the one-dimensional closures MCC and LPC. The MEC and WMC solutions mimic with a precision better than 2% and 5%, respectively. However, for WMC, is imposed by construction, whereas MEC contains it as a possible solution trajectory. Finally, in Fig. 3d, in the approach to free streaming, all two-moment Eddington factors except LPC are close to the solution, with MEC providing a slightly better fit. At the surface, MEC, MCC, WMC, and LPC deviate by 1, 2, 4 and 43%, respectively. Based on this monochromatic calculation, we have no clear-cut indication to favour a particular closure, although the numbers are slightly better for TMT-MEC. On the other hand, TMT-LPC is clearly disfavoured, as was already anticipated in Sect. 3.2.2. ## 4.3. Strong equivalenceWe now turn to strong equivalence, a good correspondence between
the actual angular distribution and
a certain model distribution . While
both closures MEC and LPC were derived from model distributions, LPC
already fails to give weak equivalence. For strong equivalence we will
therefore consider only MEC. In Sect. 3.1 we noted that the
maximum entropy model distribution is a two-parameter function,
containing the two Lagrange multipliers
and to obtain and
at a particular radius. Fig. 4
shows the discrete ordinate distribution
and the model distribution
as functions of polar angle at six
radial positions of decreasing neutrino depth in model M0, at neutrino
energy MeV. Table 3 lists
the values of
Fig. 4 a does not display and , but rather their deviations from unity, and . The figure shows that at this large neutrino depth radiation is very nearly isotropic: both and deviate from unity by a minute fraction. Note that the figure displays textbook diffusion: the distribution function is linear in the cosine of the polar angle, the Eddington factor , and the diffusion approximation holds to a high degree of accuracy. The diffusive flux has negative sign here (cf. Fig. 3c). The other frames, (b)-(f), show how, moving out towards the stellar
surface, radiation becomes forward peaked. Deviation from near
isotropy is seen in frame (b) at
neutrino depth , as well
as non-linearity in the angular dependence, signaling the breakdown of
the diffusion approximation. In Table 3, The profiles in frames (c)-(e) suggest Pauli-blocking in angle space (a left-right mirrored Fermi-function) (cf. Janka 1991, Janka et al. 1992). The blocking level is, however, below one. Only if the blocking level reaches one can we be sure that angular Pauli-blocking is observed. From the set of graphs in Janka (1991, his Fig. 3.12) the blocking level cannot be inferred because the data are averaged over neutrino energy and normalised with respect to the local neutrino density. We may conclude from Figs. 4a-f that, on the whole, matches remarkably well considering that it is only a two-parameter function: it is able to reproduce the overall character of the radiation field, which changes from a simple linear dependence on polar angle to being highly forward peaked. ## 4.4. Spectral Eddington factorsEarlier sections focused on a monochromatic solution of the neutrino Boltzmann and two-moment equations. As remarked, a comparison of the two should involve the energy dependence of the radiation field, and TMT should provide an adequate approximation to the Boltzmann solution at more than one neutrino energy. It is not intended here to repeat the monochromatic analysis of the previous sections at multiple energies. However, based on a comparison of the angular moments, our calculations for several energies do support weak equivalence of the MEC solutions. An alternative multiple-energy test of TMT versus
can be made by looking at the
variable Eddington factor as a function of
For , all closures except LPC
have Eddington trajectories that agree with
equally well. But at the highest
energy, MeV, it is actually
LPC that gives the best overall fit. At this energy, the
curve in Fig. 5 coincides with
the top fat curve in Fig. 1, High energy neutrinos in the semi-transparent regions have relatively small weight due to their low abundance. Deviations of TMT with respect to at these energies may, therefore, not be so important. We check on this by considering the energy-averaged moments, and The average Eddington factor versus is shown in Fig. 6 a for and TMT with different closures. For this exercise, all transport calculations were performed with 25 energy groups in the range MeV. Agreement between TMT-MEC and is excellent: at a given average flux , average TMT-MEC and Eddington factors differ from each other by 3% at most. The energy averaged TMT-MCC Eddington trajectory, within 6% of , is just about as good,
For TMT to be useful, weak equivalence must always apply.
Background model M0 is only one snapshot in the sequel of core
collapse events. Therefore, we consider another matter background,
model WW1, already described in Sect. 4.1. Its temperature,
density and other parameters are shown (dashed) in Fig. 2. An
additional reason to consider this model is that the atmosphere of
model M0 does not extend to very large radii (an artifact of the
polytropic model). As a result, we could not explore the entire
( We compare, as before, energy averaged Eddington factors from and TMT calculations. Because model WW1 is less dense and cooler than model M0, the neutrino energy range was lowered to 85 MeV (still using 25 bins). Average Eddington factors are shown in Fig. 6b; again we find good agreement between and TMT-MEC, with differences between the two of 2% only, while TMT-MCC agrees to within 6%. The aforementioned atmospheric gap is bridged along the nearly linear track (with the expected exception of LPC). © European Southern Observatory (ESO) 2000 Online publication: April 10, 2000 |