Neutrinos dominate the energetics of core-collapse supernovae.
Only about one percent or 
erg of the gravitational binding
energy released in the formation process of the compact remnant,
usually a neutron star, end up as kinetic energy of the expanding ejecta,
whereas 99% of this energy are radiated away in neutrinos. Electron
captures on protons and nuclei trigger the gravitational instability
of the iron core of an evolved massive star, because the electron number and
thus the pressure are reduced by the escape of electron neutrinos (see, e.g., Bruenn 1986a). Later the loss of energy by the diffusion of neutrinos and antineutrinos of all flavors drives the evolution of the nascent neutron star from a hot, inflated configuration to
the compact and very dense final state (Burrows & Lattimer 1986).
Colgate & White (1966) were the first to suggest that neutrinos may also play a crucial role for the explosion by taking up the gravitational binding energy of the collapsing core and depositing it in the rest of the star. Subsequent improvements and more realistic treatments of the microphysics, like equation of state (EoS) and neutrino transport, have changed our modern picture of stellar core collapse dramatically compared to the pioneering simulations by Colgate & White (1966). Because of the discovery of weak neutral currents and the corresponding importance of neutrino scattering off nucleons and nuclei, the forming neutron star was recognized to be highly opaque to neutrinos. Therefore the neutrino luminosities turned out to be too low, and the energy transfer rate by neutrinos not large enough to invert the infall of the surrounding gas into an explosion. For many years, hopes and efforts therefore concentrated on the prompt bounce-shock mechanism: the energy given to the hydrodynamical shock wave in the moment of core bounce was thought to lead directly to the ejection of the stellar mantle and envelope. Detailed models, however, showed that the shock experiences such severe energy losses by photodisintegration of iron nuclei and additional neutrino emission, that its outward propagation stops still well inside the iron core (e.g., Bruenn 1985, 1989a,b, 1993; Baron & Cooperstein 1990; Hillebrandt 1987; Myra et al. 1987, 1989).
Wilson (1985), however, discovered that neutrinos can indeed cause an explosion on a timescale much longer than previously thought. More than 100 milliseconds after core bounce the conditions for neutrino energy deposition have significantly improved (Bethe & Wilson 1985), and the mass infall rate and thus the ram pressure of the shock have decreased, making an explosion at later times easier than right after bounce (Burrows & Goshy 1993; Bethe 1995). Although Wilson et al. (1986) obtained such "delayed'' explosions via the neutrino-heating mechanism, their simulations gave rather low explosion energies, and their successes could not be confirmed by independent models with supposedly superior treatment of the neutrino physics and EoS (Bruenn 1986b, 1989a,b). Later simulations by Wilson & Mayle (1988, 1993) and Mayle & Wilson (1988) included neutron-finger convection in the nascent neutron star, which boosts the neutrino luminosities and thus increases the neutrino heating and the explosion energy. But whether neutron-finger convection actually occurs in the hot neutron star, or Ledoux-type convection (Burrows 1987; Keil et al. 1996; Pons et al. 1999), or none (Bruenn et al. 1995; Mezzacappa et al. 1998a) seems to depend on the properties of the nuclear EoS and possibly also on the treatment of the neutrino physics.
More recently, multi-dimensional simulations showed that convective overturn in the region of net neutrino heating between shock and gain radius (that is the position outside the neutrinosphere where neutrino cooling is balanced by neutrino heating; Bethe & Wilson 1985) can aid the explosion (Herant et al. 1994; Janka & Müller 1995, 1996; Burrows et al. 1995) and can produce successes even when spherically symmetric models fail. This "convective engine'' (Herant et al. 1994) or "boiling'' (Burrows et al. 1995) transports cool gas into the region of strongest heating while at the same time hot gas rises towards the shock. Both effects increase the efficiency of neutrino energy transfer, reduce the energy loss by the reemission of neutrinos from the heated gas, and raise the postshock pressure, thus leading to more favorable conditions for shock expansion. While the existence and importance of postshock convection is not questioned, simulations with the most advanced treatment of the neutrino transport applied to multi-dimensional supernova calculations so far (Mezzacappa et al. 1998b; Lichtenstadt et al. 1999) nourished doubts whether the effects of convection are sufficiently strong to cause explosions.
Therefore scepticism about the viability of the delayed explosion
mechanism by neutrino heating still remains (Thompson 2000), and seems justified
even more because of recent observations which indicate a possible connection
between gamma-ray bursts and at least some supernovae (e.g., Galama et al. 1998;
Bloom et al. 1999). If confirmed, this discovery would require to consider large energies and/or asphericities of the explosions (Iwamoto et al. 1998;
Woosley et al. 1999; Höflich et al. 1999)
which might be hard to explain by the neutrino-driven mechanism. Therefore, despite
the fact that the observations are still far from being conclusive, theorists feel
tempted to speculate about alternative ways to power stellar explosions, e.g.,
by invoking magnetically driven jets (Wang & Wheeler 1998; Khokhlov et al. 1999). However, while we know about the crucial role of neutrinos,
we have no observational evidence or convincing theoretical argument in support
of a dynamically important strength of magnetic fields in combination with a
significant degree of rotation in the iron cores of all massive stars.
Rather than in ordinary core-collapse supernovae, jets and a magnetohydrodynamic
mechanism may be at work in cases where the neutrino-driven mechanism
definitely fails, e.g., for progenitor main sequence masses above about
(Fryer 1999) and when a black hole forms at the center of a
rapidly spinning massive star (MacFadyen & Woosley 1999; MacFadyen et al. 1999).
When judging about the viability of the neutrino-driven mechanism, one must, however, keep in mind the enormous complexity of the problem. Because of this complexity a number of approximations and simplifications had to be made in even the currently most refined hydrodynamical calculations. Some of these deficiencies have probably disadvantageous consequences for the efficiency of neutrino energy deposition in the postshock layers. Until very recently, all published hydrodynamical models employed, for example, a still unsatisfactory treatment of the neutrino transport. Instead of solving the Boltzmann transport equation, they used flux-limited diffusion schemes, a fact which underestimates the neutrino heating above the gain radius and overestimates the energy loss by neutrino emission below it (Janka 1991a, 1992; Messer et al. 1998; Yamada et al. 1999). Moreover, multidimensional supernova simulations have so far not been able to resolve the convective processes inside the nascent neutron star, although cooling models of neutron stars show their potential importance (Burrows 1987; Keil et al. 1996; Pons et al. 1999). Even more, recent investigations (e.g., Raffelt & Seckel 1995; Janka et al. 1996; Burrows & Sawyer 1998, 1999; Reddy et al. 1998, 1999; Yamada 2000; Yamada & Toki 2000, and references therein) suggest that neutrino interaction rates in hot nuclear matter are suppressed compared to the standard description used in the numerical codes. Both the latter effects imply that the neutrino luminosities from the post-collapse core are most likely underestimated in current supernova models.
The neutrino-driven mechanism is by its nature sensitive to the neutrino-matter coupling in the heating region, which depends on the properties, i.e., spectra and luminosities, of the neutrino emission from the neutrinosphere and on the angular distribution of the neutrinos exterior to the neutrinosphere (Messer et al. 1998; Yamada et al. 1999; Burrows et al. 2000). These issues require not only the best possible technical treatment of the neutrino transport (cf. Mezzacappa et al. 2000; Liebendörfer et al. 2000; Rampp & Janka 2000) and of the description of the neutrino opacities, but they can vary with the structure of the progenitor star, with general relativity, and with the nuclear EoS and therefore the compactness of the nascent neutron star. Differences of the simulations by different groups may be associated with one or more of these issues. Unfortunately, a detailed analysis and direct comparison is essentially impossible because of largely different numerical approaches and a complicated interdependence of effects.
In this unclear and extremely unsatisfactory situation a better fundamental understanding of the conditions and requirements for shock revival by neutrino heating is highly desirable. Several attempts were made for a discussion by analytic means (Bruenn 1993; Bethe 1993, 1995, 1997; Shigeyama 1995; Thompson 2000) or on grounds of simplified numerical analysis (Burrows & Goshy 1993). While each of them contains interesting aspects and can shed light on certain results of simulations, they have led to contradictory conclusions, and none is general enough to be finally convincing. For example, assuming steady-state conditions (Burrows & Goshy 1993) cannot explain how accretion is reversed into expansion, and why an accretion shock should contract again after moving outward for some while, a possibility which was in fact observed in many hydrodynamical simulations. The beginning of the reexpansion of the stalled shock and the phase when most of the explosion energy is deposited can also not be described by a stationary neutrino-driven baryonic wind (Qian & Woosley 1996). Bethe (1990, 1993, 1995, 1997) gave a very useful and detailed discussion of the physics of neutrino heating, the structure and composition of the heating region, and the shock energetics and nucleosynthesis, using observational constraints from Supernova 1987A and numerical results provided mainly by Jim Wilson. Although addressing the question of the start of the shock, his analysis does not really reveal the requirements for a successful shock revival. Moreover, aspects were disregarded which have been recognized to be important for the outcome of simulations, for example the fact that rapid neutrino losses in the cooling region can weaken or even prevent an explosion (Woosley & Weaver 1994; Janka & Müller 1996; Messer et al. 1998). Bethe arrived at the conclusion that the explosion energy is delivered by neutrinos, whereas Bruenn (1993) and Thompson (2000) argued that neutrino heating is insufficient to cause an explosion because the advection timescale of the gas between shock and gain radius is too short for large energy deposition. Shigeyama (1995), on the other hand, performed a quasi-stationary analysis by expanding the physical variables in a power series of a small parameter, but his approach obscures the essential physics of shock revival rather than illuminating them.
The work presented here is a new approach for an analytic discussion of the conditions which can lead to the reexpansion of the supernova shock. The analysis is based on a simplified model for the post-bounce structure of the collapsed stellar core and generalizes the treatment of neutron star accretion by Chevalier (1989; see also Brown & Weingartner 1994; Fryer et al. 1996). It is not meant to yield quantitative results or to be able to compete with detailed hydrodynamical simulations, but it should allow one to reproduce the basic features of the shock stagnation, accretion, and shock revival phases. It is therefore a supplementary tool which helps one getting a qualitative understanding of the processes that determine the post-bounce evolution of the collapsed stellar core. In particular, the relative strength of competing effects that play a role in the neutrino-heating mechanism and their influence on the behavior of the supernova shock, i.e., its radial position and velocity as a function of time, can be estimated. This should help explaining why some models fail to produce explosions while others succeed.
The paper is organized in the following way. In Sect. 2 the physics of the post-bounce accretion phase will be described, in Sect. 3 the basic equations and corresponding assumptions used in the simplified analytic model will be introduced, in Sect. 4 the characteristic radii of the problem and their properties will be formally defined, in Sect. 5 the structure of the collapsed stellar core behind the stalled supernova shock will be discussed, in Sect. 6 expressions for the neutrino heating and cooling will be derived, in Sect. 7 the mass accretion rate of the nascent neutron star will be estimated, and in Sect. 8 the equations of mass and energy conservation will be applied to the neutrino heating layer, which leads to a criterion for the revival of a stalled supernova shock in Sect. 9. The equations derived in this paper will then be combined to an analytic toy model which allows one to integrate the shock position, shock radius, and properties of the gain layer as functions of time by solving an initial value problem. A summary and conclusions will follow in Sect. 10.
Copyright ESO 2001