## 1. IntroductionNeutrino energy transfer to the matter adjacent to the nascent neutron star is supposed to trigger the explosion of a massive star () as a type II supernova. Since the energy released in neutrinos by the collapsed stellar iron core is more than 100 times larger than the kinetic energy of the explosion, only a small fraction of the neutrino energy is sufficient to expel the mantle and envelope of the progenitor star. Numerical simulations have demonstrated the viability of this neutrino-driven explosion mechanism (Wilson 1982; Bethe & Wilson 1985; Wilson et al. 1986) but the explosions turned out to be sensitive to the size of the neutrino luminosities and the neutrino spectra (Janka 1993; Janka & Müller 1993; Burrows & Goshy 1993) both of which determine the power of the neutrino energy transfer to the matter outside the average neutrinosphere. The rate of energy deposition per nucleon via the dominant processes of electron neutrino absorption on neutrons and electron antineutrino absorption on protons is given by: Here and are the number fractions of free neutrons and protons, respectively; the normalization with the baryon density indicates that the rate per baryon is calculated in Eq. (1). denotes the luminosity of either or in units of and is the radial position in . The average of the squared neutrino energy, , is measured in units of and enters through the energy dependence of the neutrino and antineutrino absorption cross sections. is the angular dilution factor of the neutrino radiation field (the "flux factor", which is equal to the mean value of the cosine of the angle of neutrino propagation relative to the radial direction) which varies between values much less than unity deep inside the protoneutron star atmosphere, about 0.25 around the neutrinosphere, and 1 for radially streaming neutrinos very far out. The factor determines the local neutrino energy density according to the relation and thus enters the heating rate of Eq. (1). Only far away from the neutrino emitting star, , and dilutes like . For the precise definitions of the moments , and of the neutrino phase space distribution function, which have to be used for an accurate determination of the neutrino heating rate, see Eqs. (4)-(6) in Messer et al. (1998). Although it was found in two-dimensional simulations that convective instabilities in the neutrino-heating region can help the explosion (Herant et al. 1994; Janka & Müller 1993, 1996; Burrows et al. 1995; Miller et al. 1993; Shimizu et al. 1994) by the exchange of hot gas from the heating layer with cold gas from the postshock region, the strength of this convective overturn and its importance for the explosion is still a matter of debate (Janka & Müller 1996, Mezzacappa et al. 1998, Lichtenstadt et al. 1998). In addition, it turns out that the development of an explosion remains sensitive to the neutrino luminosities and the mean spectral energies even if convective overturn lowers the required threshold values. This is the case because convective instabilities can develop sufficiently quickly only when the heating is fast and an unstable stratification builds up more quickly than the heated matter is advected from the postshock region through the gain radius (which is the radius separating neutrino cooling inside from neutrino heating outside) down onto the neutron star surface (Janka & Müller 1996, Janka & Keil 1998). "Robust" neutrino-driven explosions might therefore require larger accretion luminosities (to be precise: a larger value of the product in Eq. (1)) during the early postbounce phase, or might call for enhanced neutrino emission from the core. The latter could be caused, for example, by convective neutrino transport within the nascent neutron star (Burrows 1987; Mayle & Wilson 1988; Wilson & Mayle 1988, 1993; Keil et al. 1996) or, alternatively, by a suppression of the neutrino opacities at nuclear densities through nucleon correlations (Sawyer 1989; Horowitz & Wehrberger 1991; Raffelt & Seckel 1995; Raffelt et al. 1996; Keil et al. 1995; Janka et al. 1996; Burrows & Sawyer 1998a, b; Reddy et al. 1998b, c), nucleon recoil and blocking (Schinder 1990) and/or nuclear interaction effects in the neutrino-nucleon interactions (Prakash et al. 1997; Reddy & Prakash 1997; Reddy et al. 1998a), all of which have to date not been taken into account fully self-consistently in supernova simulations. The diffusive propagation of neutrinos out from the very opaque inner core is determined by the value of the diffusion constant and thus sensitive to these effects. Most of the current numerical treatments of neutrino transport, however, are deficient not only concerning their description of the extremely complex neutrino interactions in the dense nuclear plasma but also concerning their handling of the transition from diffusion to free streaming. While the core flux is fixed in the diffusive regime, the accretion luminosity as well as the spectra of the emitted neutrinos depend on the transport in the semitransparent layers around the sphere of last scattering. Since neutrino-matter interactions are strongly dependent on the neutrino energy, neutrinos with different energies interact with largely different rates and decouple in layers with different densities and temperatures. The spectral shape of the emergent neutrino flux is therefore different from the thermal spectrum at any particular point in the atmosphere. Even more, through the factor in the denominator of Eq. (1) the energy deposition rate depends on the angular distribution of the neutrinos in the heating region. A quantitatively reliable description of these aspects requires the use of sophisticated transport algorithms which solve the Boltzmann equation instead of approximate methods like flux-limited diffusion techniques (Janka 1991a, 1992; Mezzacappa & Bruenn 1993a, b, c; Messer et al. 1998). The detection of electron antineutrinos from SN 1987A in the Kamiokande II (Hirata et al. 1987) and IMB laboratories (Bionta et al. 1987) and the construction of new, even larger neutrino experiments for future supernova neutrino measurements have raised additional interest in accurate predictions of the detectable neutrino signals from type II supernovae. Neutrino transport in core collapse supernovae is a very complex problem and difficult to treat accurately even in the spherically symmetric case. Some of the major difficulties arise from the strong energy dependence of the neutrino interactions, the non-conservative and anisotropic nature of the scattering processes such as neutrino-electron scattering, the non-linearity of the reaction kernels through neutrino Fermi blocking, and the need to couple neutrino and antineutrino transport for the neutrino-pair reactions. Therefore various simplifications and approximations have been employed in numerical simulations of supernova explosions and neutron star formation. The so far most widely used approximation with a high degree of sophistication is the (multi-energy-group) flux-limited diffusion (MGFLD) (Bowers & Wilson 1982, Bruenn 1985, Myra et al. 1987, Suzuki 1990, Lichtenstadt et al. 1998) where a flux-limiting parameter is employed in the formulation of the neutrino flux to ensure a smooth interpolation between the diffusion regime (where the neutrinos are essentially isotropic) and the free streaming regime (where the neutrinos move radially outward). Although these limits are accurately reproduced, there is no guarantee that the intermediate, semitransparent regime is properly treated. In a situation where the flux and the mean energy of the emitted neutrinos are determined in the diffusion regime and are constant farther out, e.g., in case of a cooling protoneutron star, little change of the luminosity and mean neutrino energy is found when the flux-limiter is varied (Suzuki 1990) or the transport equation is directly solved, e.g., by Monte Carlo calculations (Janka 1991a). This, however, is not true for situations where a significant fraction of the neutrino luminosity originates from the semitransparent regions, which is the case early after core bounce when the nascent neutron star still accretes a lot of matter (compare Messer et al. 1998). Differences also have to be expected for the spectra of the emitted neutrinos, because the spectral form is shaped in the semitransparent surface-near layers. Moreover, the use of a flux-limiter is problematic when accurate information about the angular distribution of the neutrinos is needed in the region between the diffusion and free streaming limits. Due to the factor appearing in Eq. (1) the hot-bubble heating is such a problem, neutrino-antineutrino pair annihilation outside the neutrinosphere is another problem of this kind. In fact, Monte Carlo simulations (Janka & Hillebrandt 1989a, b; Janka 1991a; Janka et al. 1992) have shown that all flux-limiters overestimate the anisotropy of the radiation field above the neutrinosphere, i.e., is enforced too rapidly (see also Cernohorsky & Bludman 1994). This leads to an underestimation of the neutrino heating in the hot-bubble region between neutrinosphere and supernova shock (Eq. (1)), and sensitivity of the supernova dynamics to the employed flux-limiting scheme must be expected (Messer et al. 1998, Lichtenstadt et al. 1998). Modifications of flux-limited diffusion have been suggested (Janka 1991a, 1992; Dgani & Janka 1992, Cernohorsky & Bludman 1994) by which considerable improvement can be achieved for spherically symmetric, static and time-independent backgrounds (Smit et al. 1997), but satisfactory performance for the general time-dependent and non-stationary case has not been demonstrated yet. Therefore the interest turns towards direct solutions of the Boltzmann equation for neutrino transport, also because the need to check the applicability of any approximation with more elaborate methods remains. Moreover, the rapid increase of the computer power and the wish to become independent of ad hoc constraints on generality or accuracy yield a motivation for the efforts of several groups (in particular Mezzacappa & Bruenn 1993a, b, c and Messer et al. 1998; more recently also Burrows 1998) to employ such Boltzmann solvers in neutrino-hydrodynamics calculations of supernova explosions. There are different possibilities to solve the Boltzmann equation
numerically, one of which is by straightforward discretization of
spatial, angular, energy, and time variables and conversion of the
differential equation into a finite difference equation which can then
be solved for the values of the neutrino phase space distribution
function at the discrete mesh points. Dependent on the number Another, completely different approach to solve the Boltzmann equation is the Monte Carlo (MC) method by which the probabilistic history of a large number of sample neutrinos is followed to simulate the neutrino transport statistically (Tubbs 1978; Janka 1987, 1991a; Janka & Hillebrandt 1989a). In principle, the accuracy of the results is only limited by the statistical fluctuations associated with the finite number of sample particles. Since the MC transport essentially does not require the use of angle and energy grids, it allows one to cope with highly anisotropic angular distributions and to treat with high accuracy neutrino reactions with an arbitrary degree of energy exchange between neutrinos and matter. However, the MC method is also computationally very time consuming, in particular if high accuracy on a fine spatial grid or at high optical depths is needed. Therefore it is not the transport scheme of one's choice for coupling it with a hydrodynamics code. In the present work, we make use of the advantages of the MC method
in order to test the accuracy and reliability of a newly developed
neutrino transport code that follows the lines of the
S The paper is organized as follows. The details of the Boltzmann
solver and essential information for the MC method are given in
Sect. 2. In Sect. 3 we describe the background models. Sect. 4
presents the results of our comparative calculations, i.e., neutrino
spectra, luminosities, and Eddington factors. Some of our calculations
are also compared against results obtained with a MGFLD code developed
by Suzuki (1994). The dependence of the results from the
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