## 4. Numerical results## 4.1. Luminosity and average energyIn the following, we consider the neutrino transport results after the time-dependent simulations have reached steady states. First we compare observable quantities such as the luminosity, average energy and average squared energy of the neutrino flux for models ST and GR. These quantities are calculated at the outermost spatial zone as where is again the surface radius, and the redshift corrections from the surface up to infinity are not taken into account. The results are summarized in Table 3.
The electron-type neutrino has the lowest energy while the muon-type neutrino has the highest. The reason for this is that the electron-type neutrino has the shortest mean free path due to absorptions on the abundant neutrons, and decouples in the surface-near layers where the temperature is lower. In contrast, the muon and tau neutrinos do not interact with particles of the stellar medium by charged currents and therefore their thermal decoupling occurs at a higher temperature. The luminosity for the electron-type antineutrino gets smaller as time passes, which is due to the cooling and shrinking of the protoneutron star. As can be seen in the table, the agreement of all quantities between the two methods is very good, which confirms the statistical convergence of the Monte Carlo simulations. The average energy is in general determined accurately because the energy spectrum is shaped in the region where the neutrino angular distribution is not very anisotropic. Moreover, possible effects due to the rather coarse angular resolution of the Boltzmann code essentially cancel out by taking the ratios of Eqs. (9) and (10). On the other hand, the luminosity and the number flux of the Monte Carlo computations are also well reproduced by the Boltzmann results. This is due to the fact that these quantities are also determined deep inside the star and are nearly conserved farther out. In fact, when integrating Eq. (3) over angle and energy multiplying with unity and , respectively, ignoring all time derivatives and general relativistic effects, one gets where and
are the number and energy fluxes of
neutrinos at radius As expected intuitively, the scattering kernels drop out of the
right hand side of Eq. (11) while only the isoenergetic scattering
does not contribute on the right hand side of Eq. (12). In the given
Boltzmann code Eqs. (11) and (12) are discretized in a conservative
form for the radial advection. Therefore it is clear that the
Boltzmann code can calculate the number and energy fluxes accurately
if also the number and energy exchange by the reactions are calculated
accurately in the source terms on the right hand sides of the
equations. Moreover, the radial evolutions of
and
are entirely determined by the
number and energy exchange through reactions of which the net effect
is small in an atmospheric layer which is in a stationary state and
reemits as much energy and lepton number as it absorbs. Changes of the
number fluxes and luminosities in the considered protoneutron star
atmospheres occur only in regions where the neutrino distribution is
still essentially isotropic and possible effects due to an
insufficient angular resolution in the Boltzmann S ## 4.2. Neutrino energy spectraIn the previous section we discussed only the energy and angle integrated quantities. However, we also provide information about the energy spectra of each neutrino species, because they yield more evidence about the quality of the agreement between the calculations with the different codes. Figs. 4 and 5 show energy flux spectra defined as at the protoneutron star surface for different cases. For the reasons discussed in Sect. 4.1, the spectra computed with the Boltzmann code (symbols) and the Monte Carlo code (lines) show excellent agreement. The number and the distribution of the energy bins in the Boltzmann code seem to be adequate to reproduce the highly resolved Monte Carlo spectra.
## 4.3. Flux factor and Eddington factorSo far we discussed only angle integrated quantities since they are observable. However we are also interested in the angular distributions of the neutrinos, because information about the angular distributions is important to determine the neutrino heating rate in the hot-bubble region (see Eq. (1)). Although it is an advantage of the Boltzmann solver over MGFLD that one does not have to assume an ad hoc closure relation between the angular moments of the distribution function, one should remember that the usable number of angular mesh points is limited. In the Feautrier method the computation time increases in proportion to the third power of the dimension of the blocks in the tridiagonal block matrix which has to be inverted when one chooses the radius as the outermost variable of the do-loops. The dimension of one block, in turn, is linearly proportional to the number of angular mesh points. The same dependence holds for the number of energy mesh points and the number of neutrino species. In the standard calculations we use 6 angular mesh points and 12 energy grid points, and we treat a single neutrino species at a time, which corresponds to a block matrix size of 72. On the other hand, the number of spatial grid points is about 100. The CPU time is a few seconds per inversion of the whole matrix on a single vector processor of a Fujitsu VPP500. Hence, use of more than 10 angular mesh points is almost prohibitive for calculations with three neutrino species even with a highly parallelized matrix inversion routine (Sumiyoshi & Ebisuzaki 1998). It is, therefore, important to clarify the sensitivity of the accuracy to the angular resolution. For this reason we consider the flux factor and the Eddington factor which are defined as: Here the subscript "" means that the averages are defined with the weight of energy. In MGFLD these factors are related with each other by a closure condition which can be derived from the employed flux-limiter (Janka 1991a, 1992). For simplicity we discuss here only energy integrated quantities as defined above. The fundamental features are similar for the individual energy groups. In Figs. 6-8, we show the radial evolutions of the flux factors and
the Eddington factors for all neutrino species in case of background
model W1. The upper panels give the flux factors and the lower
panels the corresponding Eddington factors. The solid lines are the
results of the Boltzmann simulations (having the finer radial
resolution) and the filled triangles are those of the Monte Carlo
simulations. As can be seen, near the inner boundary the flux factors
are almost zero while the Eddington factors are
, which implies that the neutrino
angular distribution is nearly isotropic, a consequence of the fact
that the neutrinos are in equilibrium with the surrounding matter or,
at least, scatter very frequently. As we move outward, both factors
begin to deviate from these values, reflecting the increase of the
mean free path and a more rapid diffusion. Farther out, the angular
moments increase monotonically towards unity, the value in the free
streaming limit, as the angular distribution gets more and more
forward peaked with increasing distance from the source (sphere of
last interaction). As can be seen clearly in Figs. 6-8, the Boltzmann
solver tends to
The underestimation of the angular moments by the Boltzmann solver can be explained by the limited ability to reproduce the forward peaking of the neutrino distribution in the free streaming limit with the employed Gaussian quadrature set when only 6 angular mesh points are used, because the direction cosine in the Gauss-Legendre quadrature has a maximum value that is less than unity. This problem can be diminished if a significantly larger number of angular grid points is used. Alternatively, if the number of angular mesh points is not to be increased, a quadrature set can be chosen that includes unity (e.g., Lobatto quadrature). In the latter case, however, one has to make sure that better results cannot only be obtained in the free streaming limit but also the isotropy of the neutrino distribution in the diffusion regime can still be satisfactorily described. Possible problems in this respect can a priori be avoided if a variable angular mesh is used (see Sect. 4.4). In order to test our interpretation, we repeated the transport
simulation for in model W1, and used
10 angular mesh points instead of only 6 in the Boltzmann solver.
Fig. 10 shows a comparison of the flux factor and the Eddington factor
for the two simulations. The long dashed lines represent the results
for model FA with the higher angular resolution (larger number of
angular points), and the short dashed lines depict the corresponding
results for the standard model ST2. The discrepancy between the
Boltzmann simulation and the Monte Carlo simulation is reduced when
the number of angular mesh points is increased. This confirms our
argument that the deviation stems entirely from the fact that in the
Gauss-Legendre quadrature set used for the Boltzmann simulations the
direction cosine has a maximum value less than unity. Indeed, the
observed degree of improvement is consistent with the fact that our
finite difference scheme is of first order for the angular advection,
since we always take the upwind differencing as explained above. The
changes between the cases with 6 and 10 angular bins are moderate and
even with the finer 10-zone angular mesh the deviations of both
angular moments (flux factor and Eddington factor) from the exact
values given by the Monte Carlo data are clearly visible. We find that
the convergence towards the Monte Carlo results is so slow that a
comparison of results for 6 and 8 Gauss-Legendre quadrature points can
be misleading and suggest convergence, although a direct comparison
with the exact solution (here represented by the Monte Carlo results)
shows that this is not the case. The maximum deviations,
for the flux factor and
for the Eddington factor, are
reached as one goes far out into the optically thin regime. This is
displayed in Fig. 11 where the ratios of the Boltzmann to the Monte
Carlo results are plotted for the flux factors and the Eddington
factors from the simulations with 6 and 10 angular bins, and from a
transport calculation with a variable angular mesh (Sect. 4.4). (The
relatively large discrepancies of the flux factors for smaller radii
are explained by slight differences of the treatment of the inner
boundary condition in our different calculations, see Sect. 2.3, and
by the fact that the flux factor adopts very small values in the
optically thick region.) The flux factors calculated in the recent
paper by Messer et al. (1998) show the same behavior and do not
approach unity but saturate at a nearly constant lower level (around
0.9) even far outside the neutrinosphere. This reflects the use of
for the largest
It is interesting that MGFLD has a tendency exactly opposite to that of the Boltzmann solver. While the latter underestimates the angular moments of the neutrino distribution in the semitransparent and transparent regimes when a Gauss-Legendre quadrature is employed for the angular integration, all flux limiters used so far tend to overestimate the flux factor and the Eddington factor in the optically thin region (Janka 1991a, 1992), which implies that the neutrino angular distribution approaches the free streaming limit much too rapidly (see also Messer et al. 1998). In order to confirm this statement, transport calculations with MGFLD were done for the same models with three different flux limiters, which are Bruenn's (1985) (BR), Levermore & Pomraning's (1981) (LP) and Mayle & Wilson's (MW) (Mayle 1985). We refer the reader to Janka (1992) and Suzuki (1994) and references therein for details on the flux limiters. We show in Figs. 12 and 13 the flux factors and local number densities of the electron-type neutrino and electron-type antineutrino for model W1, respectively. It is clear that all flux limiters overestimate the forward peaking of the angular distributions of the neutrinos in the optically thin region, a trend that holds for all neutrino species and is not dependent on the background model. The typical deviation of MGFLD results from the Monte Carlo results is much larger than the difference between the Boltzmann solver and the Monte Carlo method.
From the lower panels of Figs. 12 and 13 we learn that the local neutrino number density, which is given by is overestimated in case of the Boltzmann solver (by about 10%) and underestimated for MGFLD (by approximately 30%) in the optically thin region. This is understood from the fact that the number flux, which is defined as is related to the local neutrino number density by denotes the average angle cosine for the neutrino number flux and is calculated from Eq. (16) with a factor omitted under the integrals in the numerator and denominator. In the considered cases of protoneutron star atmospheres, the neutrino number flux is determined rather deep inside the neutron star, where the neutrinos are still nearly isotropic, and is conserved farther out, because the atmosphere has attained a quasi-stationary state where the emission of neutrinos is essentially balanced by their absorption and therefore the loss of lepton number and energy proceeds only on a very long timescale. For this reason, the neutrino number flux is not strongly affected by problems with the numerical representation of the angular moments of the neutrino distribution in the region around the neutrinosphere. Differences of the angular moments do indeed lead to different neutrino-matter coupling and thus to net emission or absorption of neutrinos associated with an increase or decrease of the neutrino luminosity. This, however, can only be a short-lived, transient phenomenon until the atmosphere has reached another steady state where the flux from the core is again conserved and just propagated outward. Since in our models the protoneutron star atmospheres are always very close to such a stationary state, the number and energy fluxes agree well between the Monte Carlo method and the Boltzmann solver, irrespective of the details of the angular grid and the employed quadrature set as long as the Boltzmann solver is based on conservative finite differencing in the radial direction. The good agreement of the number fluxes can be seen in Fig. 14, which depicts the radial behavior of the number flux in case of models ST1, ST2 and ST5. Even with MGFLD, where the angular moments of the neutrino distribution are approximated in a rather crude and ad hoc way, one does not obtain significantly different neutrino luminosities in cases where the neutrino flux is determined in the diffusion regime and where the atmosphere in which the neutrinos become anisotropic does not add a large fraction to the neutrino luminosity.
The situation is different during the early stages after core bounce when the collapsed stellar core still accretes a lot of matter which radiates large numbers of neutrinos. Here a sizable fraction of the total neutrino luminosity comes from layers at rather low optical depths where the angular distribution of the neutrinos is not isotropic any more. In this case the use of the Boltzmann solver as compared to MGFLD can yield large differences in the neutrino cooling rate around and slightly outside the neutrinosphere, and can lead to different results for the net heating rate in the hot-bubble region around the nascent neutron star where neutrinos are thought to deposit the energy for the supernova explosion. This was investigated in detail recently by Messer et al. (1998) who showed that MGFLD yields too low an energy transfer from the neutrinos to the stellar gas between the neutrinosphere and the supernova shock. It is clear from Eq. (20) that an underestimation (overestimation) of the flux factor leads to an overestimation (underestimation) of the neutrino number density, even when the number flux is the same. Since the neutrino heating rate is proportional to the local neutrino number density (actually: energy density)-this is why the inverse of the flux factor appears in Eq. (1)-a systematically incorrect flux factor leads to an inaccurate determination of the net exchange of energy (heating minus cooling) between the neutrinos and the stellar medium. From Figs. 12 and 13 we therefore expect an underestimation of the hot-bubble heating for all tested flux limiters. In this context one may also worry about the differences of the angular moments calculated with the Monte Carlo code and those obtained with the Boltzmann solver when the angular integration in the latter is done with Gaussian quadrature sets. Since an affordable number of angular mesh points (6-8) implies a flux factor that is less than unity even in the free streaming regime, it cannot be excluded that this might lead to an overestimation of the neutrino energy deposition in disadvantageous situations. By comparing results for 6-point and 8-point Gaussian quadrature, Messer et al. (1998) conclude that the net neutrino heating is converged by the time 6 fixed Gaussian quadrature points are used, and the differences in the net heating rate between the simulations with 6 and 8 Gaussian quadrature points are dynamically insignificant (see Fig. 7 in their paper). As one can see in Fig. 10, however, the convergence of the angular moments with an increasing number of Gaussian quadrature points is extremely slow and clear deviations from the Monte Carlo data exist even for 10-point Gaussian quadrature. Therefore it is not clear to us whether the conclusion drawn by Messer et al. (1998) can be considered as rigorous, since they could not check their results against an exact solution but their argument rests solely on a comparison of data from two computations with slightly different numbers of angular grid points. Unfortunately, the background models for protoneutron star atmospheres (and the corresponding Monte Carlo results) available to us at the moment do not allow us to make a comparison of the heating rate which could clarify this issue. Our stellar models are from rather late stages of the evolution of a collapsed stellar iron core, namely several seconds after core bounce. As described above, at that time the protoneutron star atmosphere has settled into a steady-state which is characterized by the fact that the net energy exchange between neutrinos and the stellar medium is very small. Therefore the net neutrino heating or cooling around and outside the neutrinosphere is much smaller than it is during the early postbounce phase when the energy for the supernova explosion should be deposited behind the shock by neutrinos. Since during this crucial shock revival phase dynamical effects and the corresponding (hydrodynamic and thermodynamic) evolution of the stellar background (also in response to the energy deposition by neutrinos) can play an important role, we are not even sure how far results for stationary neutrino transport on static background models can be useful to discuss the effects of different treatments of neutrino transport on the hot-bubble heating. Therefore, without actually presenting calculations for the heating rates, we want to put a word of caution here that even in case of the Boltzmann solver an accurate determination of the neutrino energy deposition between neutrinosphere and shock might require to replace the standard 6-point or 8-point Gauss-Legendre quadrature by a quadrature that ensures a better description of the radial beaming of the neutrino distribution in the free streaming limit. As already mentioned, an increase of the number of angular mesh points is not feasible. Choosing a variable angular mesh which adjusts the mesh point locations in dependence of time and spatial position might be a solution which is more flexible and more general than globally exchanging the Gaussian quadrature set by a quadrature set (e.g., Lobatto quadrature) that includes unity. This issue will be addressed in the next subsection. ## 4.4. Variable angular mesh in the Boltzmann solverHere we attempt to improve the angular resolution of the Boltzmann solver by redistributing the angular grid points in dependence of time and position so that their density is enhanced in the forward direction in the optically thin region where the neutrino angular distribution becomes strongly forward peaked and the Boltzmann solver underestimates the flux factor and the Eddington factor. This requires adding extra angular advection terms in the numerical scheme which compensate for the motions of the angular mesh points. We assume that the position of each interface of the angular grid is a function of time, baryonic mass and neutrino energy, i.e., . Integrating Eq. (3) over angular bins then leads to the following additional angular advection fluxes at each angular mesh interface I: It is easy to understand that Eqs. (21)-(23) originate from the variability of the angular mesh points because of the differentials of the with respect to time, mass and neutrino energy. Since the neutrino reaction rates are strongly energy dependent and so is the neutrino angular distribution, it would be desirable to implement the energy dependent angular mesh according to Eq. (23). In the current preliminary attempt, however, we installed only Eqs. (21) and (22) for simplicity. Incidentally, since Eq. (23) is proportional to in static background calculations, it is anyway negligible for the models considered here. We note that the motion of mesh points is not calculated implicitly, that is, the angular mesh points for the next time step are determined from the neutrino angular distribution at the current time step and are kept fixed during the implicit calculation of the transport for the next step. In Fig. 10, we show both the flux factor and the Eddington factor obtained from the computation with the variable angular mesh. The improvement is clear from a comparison with the result of model FA which employed 10 angular mesh points and is also shown in the figure. It should be emphasized that increasing the number of angular mesh points from 6 to 10 leads to an increase of CPU time by a factor of 4.5, while the additional operations for the variable angular mesh imply negligible computational load. We repeated all Boltzmann calculations with the variable angular mesh method and found that the same improvement could be achieved for all cases. Our scheme is stable at least for the static background models, although the stability for dynamical background models remains to be tested. Thus we think this method is promising in applying the Boltzmann solver to the study of neutrino heating in the hot-bubble region of supernovae, although there is room for improvement concerning the prescription of the motion of the mesh points and the implementation of the energy-dependent angular mesh. ## 4.5. Spatial and energy resolution in the Boltzmann solver and radial advectionIn this section we discuss how the numerical results change in dependence of the number of spatial and energy grid points, the boundary condition and the treatment of the radial advection in the Boltzmann solver. In Fig. 15 we show the energy spectrum of electron-type antineutrinos for model FE, in which we used 18 energy mesh points, compared with the spectrum for the corresponding standard model ST2 which has 12 energy zones. No qualitative or quantitative difference is found between the two cases. This is also true for the luminosity and the angular distribution. Thus we think that about 15 energy mesh points are sufficient for the calculation of the energy spectrum. These results are in agreement with previous findings by Mezzacappa & Bruenn (1993a, b, c).
In models CS and NI we reduced the spatial resolution, because the Monte Carlo simulations were done with only 15 radial mesh points which were used to represent the stellar background on which the reaction kernels were evaluated. Another motivation for testing the sensitivity to the radial resolution is that it is hardly possible to describe the protoneutron star atmosphere with about 100 radial grid points in the context of a full supernova simulation. In model CS we used the same 15 spatial grid points as in the Monte Carlo simulations. On the other hand, in model NI we used 105 spatial mesh points but the density, temperature and electron fraction were not interpolated between the grid points of the Monte Carlo simulations. In Fig. 16 we show the radial evolution of the average energy as defined in Eq. (9) and that of the number flux given by Eq. (19) for model CS, to be compared with the corresponding result for model ST2 in Fig. 14. It is clear that the agreement between the Monte Carlo and the Boltzmann results for both quantities is good. We note also that the angular distribution as well as the energy spectrum are hardly affected by this change of the spatial resolution. Model NI agrees with the Monte Carlo data nearly perfectly (except for the problems with the angular distribution discussed in Sect. 4.3) after averaging over spatial mesh zones in accordance to the way the Monte Carlo data represent the transport result. Since the Boltzmann results do not change with the number of radial grid points, we conclude that the quality of the numerical solutions is not degraded very much for simulations with a decreased spatial resolution.
Minor oscillations of the number flux near the inner boundary can be seen in Fig. 14. This problem results from the fact that one cannot consistently specify the distribution of neutrinos which leave the computational volume at the inner boundary. While this distribution should be determined from the transport result just above the inner boundary, the Boltzmann solver, however, requires an ad hoc specification in order to calculate the flux at the inner boundary. This leads inevitably to an inconsistency of the flux in the innermost zone and thus to the observed oscillations. In fact, when an inhomogeneous spatial mesh was used in model NU, in which the innermost grid zone was five times smaller than in the standard models, the oscillations were diminished as well. Finally, we illustrate possible errors which are associated with the treatment of the finite differencing of the spatial advection term in the Boltzmann solver. In the radial advection term a linear average of the centered difference and of the upwind difference is used with a weight factor that changes according to the ratio of the neutrino mean free path to some chosen length scale. Mezzacappa et al. (1993a, b, c) took the ratio of the mean free path to the local mesh width in order to construct the weighting. However, we found that this does not work well if the mesh width becomes of the same order as the mean free path but is much smaller than the scale height of the background. This is indeed the case in the inner optically thick region of our standard models with 105 radial zones. In a more recent version of his code, Mezzacappa (private communciation and 1998) defines the weighting factors by refering them to the neutrinospheric radius. In Fig. 17 the dashed line shows the number flux of muon-type neutrinos for model DI which used the prescription suggested by Mezzacappa et al. (1993a, b, c). The flux is slowly increasing with radius because the upwind differencing is given too large a contribution in the optically thick region where the centered differencing should actually be chosen. As demonstrated by the solid line in Fig. 17, the constancy of the flux, however, is recovered when the ratio of the mean free path to the distance up to the surface is chosen instead of the ratio of the mean free path to the local mesh width. Yet, this issue is probably not very important for realistic calculations of the entire neutron star, since the mesh width is usually not much smaller than the typical scale height of the matter distribution.
To finish, we comment briefly on a last model in which we assumed that the nucleon scattering is taken isotropic to see to what extent the result changes. No significant effect was found by modifying the angular distribution of the dominant scattering reaction. © European Southern Observatory (ESO) 1999 Online publication: March 18, 1999 |