## 3. Closures## 3.1. Maximum entropy closureThe use of a maximum entropy principle to find a closure dates back to Minerbo (1978), who applied the procedure to photon transport. Cernohorsky et al. (1989) first applied the principle to fermionic radiation. By maximising the spectral entropy functional under the constraints that the moments Here and in the following, In general, the functional form, Eq. (15), of the model distribution does not allow analytic inversion, and . For this reason, the maximum entropy neutrino distribution was considered originally (Cernohorsky et al. 1989; Cernohorsky & van den Horn 1990) in a Pade approximation that led to the Levermore-Pomraning closure LPC (Sect. 3.2.2). However, in the case of fermionic radiation, the assumptions involved in this approximation may lead to violation of constraints imposed by the Pauli principle. Therefore, Janka et al. (1992) explored the nature of the full maximum entropy closure by performing the inversion and numerically. While these investigations revealed that the neutrino angular distribution is well represented by the two-parameter Fermi-Dirac form of Eq. (15), it was noted that the numerical inversion was too time consuming for MEC to be of practical use in neutrino transport calculations. However, the inversion became redundant when Cernohorsky & Bludman (1994) found a closed form for the variable Eddington factor in which is the inverse of the Langevin function . The lowest-order polynomial approximation to , having the correct behaviour in the free-streaming and diffusive limits and no free parameters, is accurate to at least 2%. Using this approximation in is equivalent to interpolate between the limits for , and for in the inverse Langevin function. (This has a maximum error of at .) With the polynomial approximation, maximum entropy closure becomes a feasible option in two-moment transport, as the actual inversion of the Langevin function is bypassed. Fig. 1 shows as a function
of flux ratio
## 3.1.1. Maximum packingA limiting case of the maximum entropy distribution is obtained for Fu (1987) calls this angular degeneracy, in analogy with the zero-temperature limit of the Fermi-function in energy space: for , angular states above are filled. Janka et al. (1992) also refer to it as "maximum (or tightest) packing": all radiation is packed in a cone with the minimal possible opening angle . The maximum packing distribution yields, with (9) respectively, the moments , , . A maximum packing closure relation is readily derived: This maximum packing closure marks one boundary of maximum entropy closure in () space: in Fig. 1, is the lower fat curve above which all maximum entropy trajectories lie. ## 3.1.2. Minerbo closureThe other boundary of MEC is set by the limit , for which the distribution becomes . This is the low density or Maxwell-Boltzmann limit of MEC. The moment integrals (9) can be performed analytically and lead to Minerbo's (1978) closure This closure is shown as the upper fat curve in Fig. 1. Together with the maximum packing curve it marks the domain of MEC in () space. The closures and both satisfy the causality requirement (13). Therefore, this approach to radial free streaming is followed by all intermediate MEC trajectories as well. Fig. 1 also shows a number of other closures that we proceed to discuss in relation to MEC. The closures are summarized in Tables 1 and 2.
## 3.2. Other closures## 3.2.1. Wilson's closureand equivalent "minimal" flux limiter originally presented (Wilson et al. 1975) for use in flux-limited neutrino diffusion are still widely used in numerical simulations of gravitational collapse (e.g., Bowers & Wilson 1982; Wilson et al. 1975; Wilson 1984; Bruenn 1975, 1985; Mezzacappa & Bruenn 1993a, 1993b; Messer et al. 1998). Physically, the prescription amounts to an interpolation between the diffusive and free streaming fluxes by harmonically averaging the two. This guarantees the correct diffusive and free streaming limits, but leaves the intermediate behavior imprecise. Wilson's closure, with , does not
satisfy the causality requirement (13), and has a minimum at
(see Fig. 1). In
one-dimensional closures, such a minimum is not expected as
is a measure of the anisotropy in
the direction of ## 3.2.2. Levermore-Pomraning closureAnother closure that has been widely adopted in both photon and neutrino radiative transfer is the Levermore-Pomraning closure (LPC), corresponding to the flux limiter of Levermore & Pomraning (1981). This closure can be parametrised by The closure corresponds to an approximate angular distribution
which is assumed to be slowly varying in space and time in the
intermediate transport regime. (The Knudsen parameter The closure LPC was shown to be consistent with maximum entropy
considerations (Pomraning 1981). However, the closure stands out as
the anomalous one in Fig. 1, where it is seen to lie outside the
domain of (fermionic) MEC. The underlying distribution (see
Table 2) can be derived from the maximum entropy distribution
(15) by assuming (Cernohorsky et
al. 1989), but this assumption no longer holds away from
isotropy. This by itself is not problematic, but it may cause the
distribution, , to exceed unity. In
the case of fermions, this represents an internal inconsistency (Janka
et al. 1992). To prevent it, one must impose
, but so, at a given With given by Eq. (26), a
parametric constraint limits
It was already pointed out by Janka (1991, 1992) and Körner & Janka (1992) that LPC pushes too rapidly in regions where the opacity drops to low values. This is related to the fact that does not contain a critical point (See also Smit et al. (1997)). This behavior can be quantified by checking the approach to free streaming. For LPC one finds which is a factor of two below the value required by (13), and explains why in Fig. 1, LPC lies well above the other closures. ## 3.2.3. Janka's Monte Carlo closureJanka (1991, 1992) performed extensive Monte Carlo calculations of neutrino transport in typical hot neutron star environments. From the results he constructed several analytic fits to energy averaged transport data. The fits were parametrised as and different sets were provided. If we insist on the free streaming behaviour (13), the fit parameters should be related by This constraint is not satisfied by the parameters listed in Janka (1991) which show deviations up to 20%. The closure corresponding to the set (, , ), pertains to electron-type neutrinos in a background model resembling the model M0 which we use in Sect. 4; it is denoted as MCC and is shown in Fig. 1. As noted by Janka et al. (1992), the MCC closures were in general not well represented by conventional one-dimensional closures, but could be reproduced by two-dimensional maximum entropy solutions. © European Southern Observatory (ESO) 2000 Online publication: April 10, 2000 |