Transport 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-3). It has been evolved from an initial iron core at the center of a red giant of Woosley & Weaver (1995), which was kindly provided to us by S. Woosley. The evolution from the initial model with g cm-3 to WW1 was calculated with Newtonian hydrodynamics coupled to two-moment neutrino transport using the maximum entropy closure.
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).
We first address weak equivalence, i.e., the agreement between the lowest three angular moments obtained by approximate and exact transport calculations.
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, e is no longer equal to b, p differs from 1/3, and on a linear scale, f begins to deviate from zero noticeably. From an eye-on inspection of Figs. 3 a-d it is hard to judge which of the TMT solutions is in better agreement with , except that the LPC solution is clearly worse as the surface is approached. We will proceed with a more quantitative comparison.
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 f and p still well below these limiting values. The tendency of LPC to push towards a purely radial flow () too rapidly was already referred to in Sect. 3.2.2. An additional point to note is that the TMT-LPC solution obtained here does not satisfy the fermion-constraint discussed in that section anywhere in the iron core, i.e., the occupation density exceeds the limiting value given by Eq. (28) at all radii.
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.
We 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 a used in the maximalization procedure, see Eq. (15). This function, , can be calculated a posteriori from a TMT-MEC calculation of , by (numerical) inversion of the set of equations
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 a and at these positions, and the angular moments from both solutions.
Table 3. For six positions in the star shown in Fig. 4, this table lists e and f as obtained with the method (second and third column) and the TMT method (fourth and fifth). The last three columns list the Lagrange multipliers a and corresponding to a given TMT-MEC set , and the angle .
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, a increases with decreasing depth, and changes from a large negative to a large positive value. The MEC distribution is point-symmetric around (maximum packing), with angular states above more populated than below. In frames (c)-(e), is in the range , and can be associated with a real angle . The angle decreases outwards, in agreement with peaking of the radiation getting stronger.
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.
Earlier 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 f, like a one dimensional closure. The nature of the radiation field is for a large part contained in how behaves as a function of f alone. This is clear, because if the relationship were known for the true radiation field, it could be used in the two-moment equations to find the exact solutions and . In Fig. 5, trajectories are plotted for four neutrino energies. While the one-dimensional closures MCC, WMC en LPC remain the same in all four plots, it is clear that the Eddington trajectory is different at each neutrino energy. The Eddington trajectories of TMT-MEC, , are also different, and, due to the additional freedom of MEC in () space, are able to follow more closely on average. Notice in particular that TMT-MEC, in accordance with , has a minimum in the Eddington trajectories of the lower two energies, while at the higher two energies, TMT-MEC does not display this minimum, again in agreement with .
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, i.e., the Minerbo closure . Because the density of these high energy neutrinos is very low in the atmosphere, MEC tunes to the Minerbo closure which is the low density limit . For MeV, the curve lies outside the dynamic range of MEC in (f-p) space, i.e., in the semi-transparent regime the radiation field is peaked more strongly than MEC can account for. Cernohorsky & Bludman (1994) claim that fermionic radiation should be confined to the fermionic MEC (f-p) domain. Here we see that this need not be so: there is no reason why a particular neutrino radiation field need comply with a statistical maximum entropy principle. In fact, lies within the bosonic maximum entropy domain which is bounded by the Minerbo curve and the Levermore-Pomraning curve.
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,
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 (f-p) space: calculations reached to , , leaving a gap towards , the radial streaming limit.
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