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Astron. Astrophys. 356, 559-569 (2000)

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3. Closures

3.1. Maximum entropy closure

The 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

[EQUATION]

under the constraints that the moments e and [FORMULA] be given, one obtains a Fermi-Dirac type angular dependence of the radiative distribution function

[EQUATION]

Here and in the following, µ denotes the cosine of the polar angle of the momentum vector of the radiation quanta with respect to the preferential (radial) direction. Taking moments of the maximum entropy distribution [FORMULA], one obtains e, f, and p as functions of the two Lagrange multipliers [FORMULA] and a. The closure is formally obtained by inversion of [FORMULA] and [FORMULA] to express the Lagrange multipliers in terms of e and f. These latter relations may be used to write the closure in the form [FORMULA]. MEC is thus inherently a two-dimensional closure, depending explicity on the energy density e, as well as the flux f.

In general, the functional form, Eq. (15), of the model distribution does not allow analytic inversion, [FORMULA] and [FORMULA]. 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 [FORMULA] and [FORMULA] 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

[EQUATION]

The function [FORMULA] is defined as

[EQUATION]

in which [FORMULA] is the inverse of the Langevin function [FORMULA].

The lowest-order polynomial approximation to [FORMULA], having the correct behaviour in the free-streaming and diffusive limits and no free parameters,

[EQUATION]

is accurate to at least 2%. Using this approximation in [FORMULA] is equivalent to interpolate between the limits [FORMULA] for [FORMULA], and [FORMULA] for [FORMULA] in the inverse Langevin function. (This has a maximum error of [FORMULA] at [FORMULA].) 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 [FORMULA] as a function of flux ratio f at several fixed e-values. Note that an actual solution [FORMULA] of TMT will not follow any of these curves, because [FORMULA] varies with radius.

[FIGURE] Fig. 1. Closures. Solid curves denote maximum entropy closure Eddington factors [FORMULA] versus flux ratio f at fixed e-values. The two fat curves mark the boundaries of maximum entropy closure. The upper fat curve is the low density limit [FORMULA], the lower one is the maximum packing curve. In between lie, with solid gray lines, from top to bottom, [FORMULA] to [FORMULA] in steps of [FORMULA]. The dashed-dotted curve is Janka's Monte Carlo closure MCC, the dash-triple-dotted line is Wilsons minimal closure WMC, and dotted is the Levermore-Pomraning closure, LPC.

3.1.1. Maximum packing

A limiting case of the maximum entropy distribution is obtained for

[EQUATION]

Fu (1987) calls this angular degeneracy, in analogy with the zero-temperature limit of the Fermi-function in energy space: for [FORMULA], angular states above [FORMULA] 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 [FORMULA]. The maximum packing distribution [FORMULA] yields, with (9) respectively, the moments [FORMULA], [FORMULA], [FORMULA]. A maximum packing closure relation is readily derived:

[EQUATION]

This maximum packing closure marks one boundary of maximum entropy closure in ([FORMULA]) space: in Fig. 1, [FORMULA] is the lower fat curve above which all maximum entropy trajectories [FORMULA] lie.

3.1.2. Minerbo closure

The other boundary of MEC is set by the limit [FORMULA], for which the distribution becomes [FORMULA]. 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

[EQUATION]

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 ([FORMULA]) space. The closures [FORMULA] and [FORMULA] both satisfy the causality requirement (13). Therefore, this approach to radial free streaming is followed by all intermediate MEC trajectories [FORMULA] 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.


[TABLE]

Table 1. Eddington factors for two ad hoc and three statistical one-dimensional closures. The function [FORMULA] is the inverse of the Langevin function [FORMULA].



[TABLE]

Table 2. Statistics and Angular Distribution.


3.2. Other closures

3.2.1. Wilson's closure

Wilson's closure (WMC)

[EQUATION]

and equivalent "minimal" flux limiter

[EQUATION]

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 [FORMULA], does not satisfy the causality requirement (13), and has a minimum at [FORMULA] (see Fig. 1). In one-dimensional closures, such a minimum is not expected as [FORMULA] is a measure of the anisotropy in the direction of f ; indeed it does not occur in other conventional one-dimensional closures. In two-dimensional closures, on the other hand, [FORMULA] need not be monotonic increasing as a function of f (See also Janka et al. 1992).

3.2.2. Levermore-Pomraning closure

Another 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

[EQUATION]

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 R in this case is a slight generalization of (3).)

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 [FORMULA] (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, [FORMULA], to exceed unity. In the case of fermions, this represents an internal inconsistency (Janka et al. 1992). To prevent it, one must impose [FORMULA], but a is not controllable. From a given TMT solution we may work backwards to find [FORMULA] by inverting [FORMULA] and [FORMULA], and check if [FORMULA] or [FORMULA], but a priori measures cannot be undertaken. Calculating the energy density with [FORMULA] and inverting, one finds

[EQUATION]

so, at a given R, the parameter a exceeds unity when [FORMULA], where

[EQUATION]

With [FORMULA] given by Eq. (26), a parametric constraint [FORMULA] limits e at a given f value. For a neutrino transport solution this means that [FORMULA] must drop sufficiently rapidly in the outer regions where [FORMULA] increases; else the solution is inconsistent with the fermionic nature of the radiation.

It was already pointed out by Janka (1991, 1992) and Körner & Janka (1992) that LPC pushes [FORMULA] too rapidly in regions where the opacity drops to low values. This is related to the fact that [FORMULA] 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

[EQUATION]

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 closure

Janka (1991, 1992) performed extensive Monte Carlo calculations of neutrino transport in typical hot neutron star environments. From the results he constructed several analytic fits [FORMULA] to energy averaged transport data. The fits were parametrised as

[EQUATION]

and different sets [FORMULA] were provided. If we insist on the free streaming behaviour (13), the fit parameters should be related by

[EQUATION]

This constraint is not satisfied by the parameters listed in Janka (1991) which show deviations up to 20%. The closure corresponding to the set ([FORMULA], [FORMULA], [FORMULA]), 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.

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Online publication: April 10, 2000
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