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Astron. Astrophys. 344, 533-550 (1999)

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2. Numerical methods

2.1. Boltzmann solver

2.1.1. Basic equations

Our SN code is based on a finite difference form of the general relativistic Boltzmann equation for neutrinos. We assume spherical symmetry of the star throughout this paper. For the Misner-Sharp metric (Misner & Sharp 1964):


the Boltzmann equation in the Lagrangian frame takes the following form:


In the above formula [FORMULA] and [FORMULA] are the velocity of light and the gravitational constant, respectively, t is the coordinate time and m is the baryonic mass coordinate which is related to the circumference radius r through the conservation law of the baryonic mass. In view of combination with a Lagrangian hydrodynamics code (Yamada 1997), the baryonic mass is chosen to be the independent variable instead of the radius. [FORMULA], [FORMULA] and r are the metric components which are determined by the Einstein equations. In this paper, however, these quantities are given from the background models and set to be constant with time during the neutrino transport calculations. [FORMULA] is the neutrino phase space distribution function. Under the assumption of spherical symmetry, [FORMULA] is a function of t, m, µ and [FORMULA], where µ is the cosine of the angle of the neutrino momentum with respect to the outgoing radial direction and [FORMULA] is the neutrino energy. [FORMULA] is the baryonic mass density.

The right hand side of Eq. (3) is the so-called collision term, which actually includes absorption, emission, scattering and pair creation and annihilation of neutrinos, details of which are described below. The left hand side, on the other hand, looks a little bit different from the form used, for example, in Mezzacappa et al. (1993a, b, c). This is not only because it is fully general relativistic but also because all the velocity dependent terms are expressed as time derivatives so that it can easily be coupled to the implicit general relativistic Lagrangian hydrodynamics code (Yamada 1997). In this way a fully coupled, implicit system of the radiation-hydrodynamics equations is formed, in which the time derivatives can be treated easily because they are just off-diagonal components of the matrix set up from the linearized equations. It should be noted, however, that since we assume that the matter background is static in this paper, all these time derivatives are automatically set to be zero, although these terms have been already implemented in the code.

The conserved neutrino number [FORMULA] in the absence of source terms is represented in terms of the chosen independent variables as


This suggests to cast Eq. (3) in a conservation form with respect to the neutrino number. It is also evident that the combination of [FORMULA] is more convenient to be used than [FORMULA] itself. In the following, therefore, we define the specific neutrino distribution function as in Mezzacappa et al. (1993a, b, c) by


and use this quantity as the dependent variable to be solved for.

2.1.2. Finite difference scheme of Boltzmann equations

As mentioned above the specific neutrino distribution function [FORMULA] is a function of t, m, µ and [FORMULA]. This four-dimensional phase space is discretized and Eq. (3) is written as a finite-difference equation. In the time direction we adopt a fully implicit differencing. The discretized specific distribution function [FORMULA] is defined at the mesh centers of the spatial, angular and energy grids. Here the subscripts [FORMULA] refer to the spatial, angular and energy grid points, respectively. The superscript n corresponds to the time step. The value at each cell interface is evaluated by interpolation of the distribution at two adjacent mesh centers.

Our finite difference method is essentially the same as that of Mezzacappa & Bruenn (1993a, b, c) with some modifications. For the spatial advection, the upwind difference and the centered difference are linearly averaged with the weights determined by the ratio of the mean free path to the distance to the stellar surface unlike Mezzacappa & Bruenn (1993a, b, c) who used the ratio of the mean free path to the local mesh width. In fact, in the latter case we found that the upwind distribution was given too large a weight in the optically thick region and thus the flux was overestimated. This issue will be addressed in Sect. 4.5. As pointed out to us by Mezzacappa (private communication), an updated version of the finite differencing presented in Mezzacappa & Bruenn (1993b) is given by Mezzacappa (1998) who defines the weight factors for the transition between centered and upwind differencing for the radial advection of neutrinos very similar to the prescription chosen by us.

The angular mesh is determined such that each mesh center and cell width correspond to the abscissas and weights of the Gauss-Legendre quadrature, respectively. In angular direction, the neutrino distribution at each interface is simply taken as the upwind value. The advection in the energy space is also approximated by an upwind scheme following Mezzacappa & Matzner (1989), although this does not allow to conserve both lepton number and energy in non-static situations (which are not considered here) unless a large number of energy zones is used (Mezzacappa, private communication).

In typical calculations, 105, 6 and 12 mesh points are used for the spatial, angular and energy discretizations, respectively. The dependence of the results on the numbers of mesh points will be discussed below. The finite-differenced Boltzmann equation forms a nonlinear coupled system of equations for all radial grid points which is linearized and solved iteratively by using a Newton-Raphson scheme. The linearized equations adopt a block tridiagonal matrix form, which can be efficiently solved by the Feautrier method.

2.2. Monte Carlo method

Different from the finite difference method, the Monte Carlo method constructs the statistical ensemble average by following the destinies of individual test particles and performing the average when all particles have been transported. Due to the fact that neutrinos are fermions it is impossible to propagate them independently. Instead, the full time-dependent problem has to be simulated by following a large number (typically [FORMULA]) of sample particles along their trajectories simultaneously in order to be able to construct the local phase space occupation functions and to include anisotropies as well as phase space blocking effects self-consistently into the calculation of the reaction rates and source terms. The modeling of the phase space distribution function from the local particle sample must guarantee the correct approach to chemical equilibrium. Also the Pauli exclusion principle has to be satisfied by the statistical average. We refer readers to Janka (1987) and references therein for details.

The stellar background is divided into 15 equispaced spherical shells of homogeneous composition and uniform thermodynamical conditions, the number of which was determined both from physical requirements for spatial resolution and from the requirement to have acceptably small statistical errors in the local neutrino phase space distributions constructed from the chosen number of sample particles. Although the Monte Carlo method is essentially mesh free, about 60 energy bins and approximately 35 angular bins are used only for representing the phase space distribution functions and for calculating the reaction kernels. Neutrinos are injected into the computational volume at the inner boundary in the way described in the next section, while particles passing inwardly through the inner boundary are simply forgotten. The outer boundary is treated as a free boundary, where particles escape unhindered and no neutrino is assumed to come in from outside.

2.3. Boundary conditions

In the presented calculations we are mainly interested in the neutrino transport in the region where the neutrinos decouple from the stellar background and the emitted spectra form. Therefore we calculate the neutrino transport only in the vicinity of the "neutrinosphere". For this reason we have to set an inner boundary condition as well as an outer boundary condition for each model. At the outer boundary we impose the condition that no neutrinos enter the computational volume from outside. In the Boltzmann solver, this is realized by setting


where [FORMULA] is the radius of the outermost mesh center, and [FORMULA] is the radius of the outer surface which is dislocated outward from the outermost mesh center by half a radial cell width.

On the other hand, we have to specify the distribution of the neutrinos coming into the computational volume at the inner boundary [FORMULA] which is dislocated inward from the innermost mesh center [FORMULA] by half a radial cell width. For this purpose we adopt Fermi-Dirac distribution functions, in which the temperature, chemical potential and a normalization factor are determined such that the neutrino number density at [FORMULA] (where the neutrinos are essentially isotropically distributed because the inner boundary is chosen to be located at high optical depth), the average energy [FORMULA] and the width of the energy spectrum, measured by the parameter [FORMULA] (see Janka & Hillebrandt 1989 b), are reproduced as given by Wilson's (1988) models. Thus the inner boundary condition is set as:


where A, [FORMULA] and [FORMULA] are the fitting parameters, the values of which are summarized in Table 1 for all considered models and neutrino species. Concerning the distribution of neutrinos that leave the computational volume, we impose in Eq. (7) in the Boltzmann solver the condition that it is the same as the distribution at the innermost mesh center. From the physical point of view, however, and treated correctly in the Monte Carlo simulations, it should be determined by the fraction of neutrinos which is emitted or backscattered towards the inner boundary. This is in general different from what the phase space distribution at the innermost mesh center yields because the mean free path of the neutrinos near the inner boundary is shorter than the mesh width. As a result the imposed condition for [FORMULA] in Eq. (7) leads to a minor discrepancy of the treatment of the inner boundary condition in the Monte Carlo and Boltzmann computations and sometimes causes a small oscillation of the neutrino distribution near the inner boundary in the latter computations. This issue will be revisited later.


Table 1. Fitting parameters for the Fermi-Dirac distributions which describe the spectra at the inner boundary.

2.4. Input physics

2.4.1. Neutrino reactions

The neutrino opacities of dense neutron star matter are still one of the major uncertainties of supernova simulations. Theoretical and numerical complications arise from the description and treatment of nucleon thermal motion and recoil (Schinder 1990), nuclear force effects and nucleon blocking (Prakash et al. 1997; Reddy & Prakash 1997; Reddy et al. 1998a), and nucleon correlations, spatial (Sawyer 1989; Burrows & Sawyer 1998a, b; Reddy et al. 1998b, c) as well as temporal (Raffelt & Seckel 1995; Raffelt et al. 1996). Although in particular nucleon recoil and auto-correlations might play an important role even in the sub-nuclear outer layers of the protoneutron star down to densities below [FORMULA] (Janka et al. 1996; Hannestad & Raffelt 1998) we do not concentrate on this problem here but rather employ the standard description of the neutrino opacities, according to which neutrinos interact with isolated nucleons (see, e.g., Tubbs & Schramm 1975; Bruenn 1985). Also, as in most other simulations, bremsstrahlung production of neutrino-antineutrino pairs is neglected here, although it may be important as pointed out by Suzuki (1993) and more recently by Burrows (1998) and Hannestad & Raffelt (1998). Doing so, we intend to enable comparison with other (already published) work and want to avoid the mixing of effects from a different numerical treatment of the transport with those from a non-standard description of neutrino-nucleon interactions or from the inclusion of processes typically not considered in the past.

The following neutrino reactions have been implemented in our codes.

(electron-type neutrino absorption on neutrons),

(electron-type antineutrino absorption on protons),

(neutrino scattering on nucleons),

(neutrino scattering on electrons).

In addition to the above reactions, the following processes have also been implemented in both codes, although these reactions are not used in the present paper.

(electron-type neutrino absorption on nuclei),

(neutrino coherent scattering on nuclei),

(electron-positron pair annihilation and creation),

(plasmon decay and creation).

Since the pair processes are not taken into account in this paper, we can treat each species of neutrinos separately. A test showed that in the considered protoneutron star atmospheres neutrino pair creation and annihilation as well as processes involving nuclei are unimportant to determine the fluxes and spectra.

2.4.2. EOS

In this paper we use a simplified equation of state, in which only nucleons, electrons, alpha particles and photons are included. They are all treated as ideal gases. For given density, temperature and electron fraction we derive the mass fractions and the chemical potentials of nucleons and the electron chemical potential from this equation of state. The disregard of nuclei is well justified for the densities and temperatures we are considering, where most of the nuclei are dissociated into free nucleons. This is actually confirmed by comparing our EOS with the more realistic EOS of Wolff that is based on the Skyrme-Hartree-Fock method (Hillebrandt & Wolff 1985). Only very small differences of the nucleon chemical potentials are found for the innermost region where small amounts of nuclei appear and for the outermost region where some contribution from alpha particles is mixed into the stellar medium. We also repeated some of the calculations making use of the Wolff EOS with the nuclei-related reactions [5], [6] (neutrino emission, absorption and scattering on nuclei) turned on and found qualitatively and quantitatively the same results. Hence the nuclei-related reactions are switched off in the calculations described below.

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999