When the core of a massive star collapses to a neutron star, a huge amount of gravitational binding energy is released mainly in the form of neutrinos which are abundantly produced by particle reactions in the hot and dense plasma. In fact, the emission of neutrinos determines the sequence of dynamical events which precede the death of the star in a supernova explosion. Electron neutrinos from electron captures on protons and nuclei reduce the electron fraction and the pressure and thus accelerate the contraction of the stellar iron core to a catastrophic implosion. The position of the formation of the supernova shock at the moment of core bounce depends on the degree of deleptonization during the collapse phase. The outward propagation of the prompt shock is severely damped by energy losses due to the photodisintegration of iron nuclei, and finally stopped only a few milliseconds later when additional energy is lost in a luminous outburst of electron neutrinos. These neutrinos are created in the shock-heated matter and leave the star as soon as their diffusion is faster than the expansion of the shock. "Prompt'' supernova explosions generated by a direct propagation of the hydrodynamical shock have been obtained in simulations only when the stellar iron core is very small (Baron & Cooperstein 1990) and/or the equation of state of neutron-rich nuclear matter is extraordinarily soft (Baron et al. 1985; Baron et al. 1987).
At a later stage (roughly a hundred milliseconds after
the shock formation) the situation has changed. The
temperature behind the stalled shock has dropped such that
increasingly energetic neutrinos diffusing out from
deeper layers start to transfer energy to the stellar gas
around the nascent neutron star. If this energy deposition is
sufficiently strong, the stalled shock can be revived and a
"delayed''
explosion can be triggered (Wilson 1985; Bethe & Wilson 1985; the idea that neutrinos provide the energy of the supernova
explosion was originally brought up by Colgate & White 1966).
A small fraction of less than one per cent of the
energy released in neutrinos can account for the kinetic energy
of a typical type II supernova. This explanation is currently
favored for the explosion of massive stars in the mass range
between about 10
and roughly 25
.
It is supported by numerical simulations and analytic
considerations. A finally convincing hydrodynamical simulation,
however, is still missing (a summary of our current
understanding of the explosion mechanism can be found
in Janka 2001). The measurement of a neutrino
signal from a Galactic supernova would offer the most direct
observational test for our theoretical perception of the onset
of the explosion. Due to the central role of neutrinos in
supernovae, the neutrino transport deserves particular
attention in numerical models.
Current hydrodynamic models of neutrino-driven supernova explosions leave an ambiguous impression and have caused confusion about the status of the field outside the small community of supernova modelers. Simulations by different groups seem to be contradictory because some models show successful explosions by the neutrino-driven mechanism whereas others have found the mechanism to fail.
Wilson and collaborators have performed successful simulations for more than 15 years now (e.g., Wilson 1985; Wilson et al. 1986; Wilson & Mayle 1988; Wilson & Mayle 1993; Mayle et al. 1993; Totani et al. 1998). Their models were calculated in spherical symmetry, but shock revival was obtained by making the assumption that the neutrino luminosity is boosted by neutron-finger instabilities in the nascent neutron star (Wilson & Mayle 1988, 1993). In fact, in their calculations explosion energies comparable to typically observed values (of the order of 1051 erg) require pions to be abundant in the nuclear matter already at moderately high densities (Mayle et al. 1993). The corresponding equation of state (EoS) with pions leads to higher core temperatures and the emission of more energetic neutrinos. Both the convectively boosted luminosities and the harder spectra enhance the neutrino heating behind the stalled shock. The relevance of neutron-finger instabilities for efficient energy transport on large scales, however, has not been demonstrated by direct simulations. The existence of neutron-finger instabilities (in the sense of the definition introduced by Wilson & Mayle 1993) is indeed questioned by other investigations (Bruenn & Dineva 1996) and might be a consequence of specific properties of the high-density EoS or the treatment of the neutrino transport by Wilson and collaborators. Also the implementation of a pionic component in their EoS is at least controversial and not in agreement with other, more conventional descriptions of nuclear matter (Pethick & Pandharipande, personal communication).
Rather than finding neutron-finger instabilities, two-dimensional hydrodynamic simulations have shown that regions inside the nascent neutron star can exist where Ledoux or quasi-Ledoux convection (as defined by Wilson & Mayle 1993) develops on a time scale of tens of milliseconds after core bounce (Keil et al. 1996; Keil 1997; also Janka et al. 2001). The models were constrained to a simulation of the neutrino cooling of the nascent neutron star, but it must be expected that the convective enhancement of the neutrino luminosity, which becomes appreciable after about 200 milliseconds (see also Janka et al. 2001), can have important consequences for the revival of the stalled supernova shock (Burrows 1987). Ledoux unstable regions were also detected in spherical cooling models of newly formed neutron stars (Burrows 1987; Burrows & Lattimer 1988; Pons et al. 1999; Miralles et al. 2000). Nevertheless, their significance is controversial, because Bruenn et al. (1995) in spherically symmetric models and Mezzacappa et al. (1998a) in two-dimensional simulations observed convection inside the neutrinosphere only as a transient phenomenon during a relatively short period after core bounce.
The question is therefore undecided whether convection plays an important role in newly formed neutron stars and if so, what its implications for the supernova explosion are. Unfortunately, the various calculations were performed with different treatments of the neutrino transport, one-dimensional vs. two-dimensional hydrodynamics, general relativistic gravitational potential vs. Newtonian gravity, or even an inner boundary condition to replace the central, dense part of the neutron star core (Mezzacappa et al. 1998a). It is this inner region, however, where Keil et al. (1996) found convection to develop roughly 100 milliseconds after bounce. Convection at larger radii and thus closer below the neutrinosphere had indeed died out within only 20-30 milliseconds after shock formation in agreement with the results of Bruenn et al. (1995). Future studies with a better and more consistent handling of the different aspects of the physics are definitely needed to clarify the situation.
A second hydrodynamically unstable region develops exterior to the nascent neutron star in the neutrino-heated layer behind the stalled supernova shock. Convective overturn in this region is helpful and can lead to explosions in cases which otherwise fail (Herant et al. 1992; Herant et al. 1994; Janka & Müller 1995; Janka & Müller 1996; Burrows et al. 1995). In two-dimensional supernova models computed recently by Fryer (1999); Fryer & Heger (2000), and Fryer et al. (2002) these hydrodynamical instabilities in the postshock region are crucial for the success of the neutrino-driven mechanism, because they help transporting hot gas from the neutrino-heating region directly to the shock, while downflows simultaneously carry cold, accreted matter to the layer of strongest neutrino heating where a part of this gas readily absorbs more energy from the neutrinos. The existence of this multi-dimensional phenomenon seems to be generic for the situation which builds up in the stellar core some time after shock formation. It is therefore no matter of dispute.
All simulations showing explosions as a consequence of post-shock convection, however, have so far been performed with a strongly simplified treatment of the crucial neutrino physics, e.g., with grey, flux-limited diffusion which is matched to a "light bulb'' description at some "low'' value of the optical depth (Herant et al. 1994; Burrows et al. 1995). Alternatively, an inner boundary near the neutrinosphere had been used where the spectra and luminosities of neutrinos were prescribed to parameterize our ignorance and the potential uncertainties of the exact properties of the neutrino emission from the newly formed neutron star (Janka & Müller 1995, 1996). It is therefore not clear whether the instabilities and their associated effects in fully self-consistent and more accurate simulations will be strong enough to cause successful explosions. Doubts in that respect were raised by Mezzacappa et al. (1998b), whose two-dimensional models showed convective overturn in the neutrino-heating region but still no explosion. Mezzacappa et al. (1998b) combined two-dimensional hydrodynamics with neutrino transport results obtained by multi-group flux-limited diffusion in spherical symmetry. Although not self-consistent, their approach is nevertheless an improvement compared to previous treatments of the neutrino physics by other groups.
All computed models bear some pieces of truth. Essentially they demonstrate the remarkable sensitivity of the supernova dynamics to the different physical aspects of the problem, in particular the treatment of neutrino transport and neutrino-matter interactions, the properties of the nuclear EoS, multi-dimensional hydrodynamical processes, and general relativity. Considering the huge energy reservoir carried away by neutrinos, the neutrino-driven mechanism appears rather inefficient (the often quoted value of 1% efficiency, however, misjudges the true situation, because neutrinos can transfer between 5% and 10% of their energy to the stellar gas during the critical period of shock revival; see Janka 2001). Nevertheless, neutrino-driven explosions are "marginal'' in the sense that the energy of a standard supernova explosion is of the same order as the gravitational binding energy of the ejected progenitor mass. The final success of the supernova shock is the result of different physical processes which compete against each other. On the one hand, neutrino heating tries to drive the shock expansion, on the other hand energy losses, e.g., by neutrinos that are reemitted from the inward flow of neutrino-heated matter which enters the neutrino-cooling zone below the heating layer, extract energy and thus damp the shock revival. It is therefore not astonishing that different approximations or descriptions for one or more physical components of the problem can decide between an explosion or failure of a simulation.
None of the current supernova models includes all relevant aspects to a satisfactory level of accuracy, but all of these models are deficient in one or more respects. Wilson and collaborators obtain explosions, but their input physics is unique and cannot be considered as generally accepted. Two-dimensional (Herant et al. 1994; Burrows et al. 1995; 1999; Fryer & Heger 2000; Fryer et al. 2002) and three-dimensional (Fryer, personal communication) models show explosions due to strong postshock convection, but the neutrino transport and neutrino-matter interactions are handled at a level of accuracy which falls back behind the most elaborate treatments that have been applied in spherical symmetry. The sensitivity of the outcome of numerical calculations to details of the neutrino transport demands improvements. Self-consistent multi-dimensional simulations with a sophisticated and quantitatively reliable treatment of the neutrino physics have yet to be performed.
Most recently, spherically symmetric Newtonian (Rampp & Janka 2000; Mezzacappa et al. 2001), and general relativistic (Liebendörfer et al. 2001b) hydrodynamical simulations of stellar core-collapse and post-bounce evolution including a Boltzmann solver for the neutrino transport have become possible. Although the models do not yield explosions, they must be considered as a major achievement for the modeling of supernovae. Before these calculations only the collapse phase of the stellar core had been investigated with solving the Boltzmann equation (Mezzacappa & Bruenn 1993c). Boltzmann transport has now superseded multi-group flux-limited diffusion (Arnett 1977; Bowers & Wilson 1982; Bruenn 1985; Myra et al. 1987; Baron et al. 1989; Bruenn 1989a,b, 1993; Cooperstein & Baron 1992) as the most elaborate treatment of neutrinos in supernova models.
The differences between both methods in dynamical calculations have still to be figured out in detail, but the possibility of solving the Boltzmann equation removes imponderabilities and inaccuracies associated with the use of flux-limiters in particular in the region of semi-transparency, where neutrinos decouple from the stellar background and also deposit the energy for an explosion (Janka 1991; Janka 1992; Messer et al. 1998; Yamada et al. 1999). For the first time the transport can now be handled at a level of sophistication where the technical treatment is more accurate than our standard description of neutrino-matter interactions, which includes various approximations and simplifications (for an overview of the status of the handling of neutrino-nucleon interactions, see Horowitz 2002 and the references therein).
In this paper we describe our new numerical code for solving
the energy and time dependent
Boltzmann transport equation for neutrinos coupled to the
hydrodynamics of the stellar medium. First results
from supernova calculations with this code were published
before (Rampp & Janka 2000), but a detailed documentation of
the method will be given here. It is based on a variable
Eddington factor technique where the coupled set of Boltzmann
equation and neutrino energy and momentum equations is
iterated to convergence. Variable Eddington moments of
the neutrino phase space distribution are used for closing
the moment equations, and the integro-differential character
of the Boltzmann equation is tamed by using the zeroth and
first order angular moments (neutrino density and flux) in the
source terms on the right hand side of the Boltzmann equation.
This numerical approach is fundamentally different from the
so-called 
methods (Carlson 1967; Yueh & Buchler 1977; Mezzacappa & Bruenn 1993a; Yamada et al. 1999) which employ a
direct discretization of the Boltzmann equation in all variables
including the dependence on the angular direction of the
radiation propagation.
Some basic characteristics of our code are similar to elements described by Burrows et al. (2000). Different from the latter paper we shall discuss the details of the numerical implementation of the transport scheme and its coupling to a hydrodynamics code (Rampp 2000), which in our case is the PROMETHEUS code with the potential of performing multi-dimensional simulations. The variable Eddington factor technique was our method of choice for the neutrino transport because of its modularity and flexibility, which offer significant advantages for a generalization to multi-dimensional problems. We shall suggest and motivate corresponding approximations which we consider as reasonable in the supernova case, at least as a first step towards multi-dimensional hydrodynamics with a Boltzmann treatment of the neutrino transport.
This paper is arranged as follows: in Sect. 2 the equations of radiation hydrodynamics are introduced. Section 3 provides a general overview of the numerical methods used to solve these equations and contains details of their practical implementation. Results for a number of idealized test problems as well as applications of the new method to the supernova problem are presented in Sect. 4. A summary will be given in Sect. 5. In the Appendix the numerical implementation of neutrino opacities and the equation of state used for core-collapse and supernova simulations is described.
Copyright ESO 2002