10 Summary and conclusions

In this paper an analytic approach was presented which allows one to discuss the conditions for the revival of a stalled supernova shock by neutrino heating. The treatment is time-dependent in the sense that the gas flow is not assumed to be steady and the model can be used to calculate the shock radius, shock velocity, and the properties of the gain layer as functions of time.

10.1 Components of the toy model

The "atmosphere'' of the collapsed stellar core outside of the neutrinosphere is considered to consist of three distinct layers (Sect. 2). Between neutrinosphere and gain radius there is a cooling region where neutrino emission extracts energy. In a heating layer between gain radius and shock, $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ absorptions on nucleons dominate the inverse capture reactions of electrons and positrons and deposit energy. Finally, there is an infall region above the shock, where the gas of the progenitor star is accelerated to nearly the free-fall velocity. These different layers are in contact and exchange mass and energy.

The radial structure of the cooling and heating layers is assumed to be described by the conditions of hydrostatic equilibrium, which requires that the sound travel timescale is smaller than the other relevant timescales of the problem. This assumption is reasonably well fulfilled during the phase when the shock is near stagnation or just starts to gain momentum. For such conditions the layer behaves like one unit and reacts to changes in an infinitesimally short time. In combination with a simple representation of the equation of state, hydrostatic equilibrium allows one to calculate the density and pressure profiles analytically (Sect. 5). The radius and velocity of the shock then depend on integral properties of the gain layer, i.e., its total mass and energy.

Changes of the mass and energy integrals are caused by the motion of the boundaries or by active mass flow into or out of the gain layer. In Sect. 8 conservation equations for these global quantities were derived by integrating the equations of hydrodynamics, including the terms with time derivatives, over the volume of the gain layer. Following these integral quantities it was then possible to compute the shock position and shock velocity as well as other important quantities, e.g., the location of the gain radius, as functions of time. Assuming hydrostatic equilibrium thus reduces the mathematical problem to an integration of a set of ordinary differential equations with time as the independent variable. Discussing the destiny of the supernova shock therefore means solving an initial value problem. This expresses the fact that the shock evolution depends on the initial conditions, for example, on the shock stagnation radius and the initial energy in the gain layer, and is controlled by the cumulative effects of neutrino energy deposition and mass accumulation in the gain layer.

In general the gas falling through the shock will not move as a stationary flow between shock and neutrinosphere: the rate at which mass is advected through the gain radius is usually different from the mass accretion rate by the shock. Steady-state accretion or mass loss are special cases, which should be limits of the more general situation.

It is not easy to calculate the fraction of the accreted gas which stays in the gain layer. The neutron star can "swallow'' matter only at a finite rate, depending on the efficiency with which the gas gets rid of the excess energy that prevents its integration into the neutron star surface. This efficiency is a sensitive function of the conditions in the cooling layer above the neutrinosphere. Since the hot, nascent neutron star below the neutrinosphere is a source of intense neutrino radiation and these neutrinos interact still frequently in the cooling region, the temperature there should be close to the neutrinospheric value. With this assumption and with known radius and density profile it is possible to calculate the neutrino energy loss from the cooling layer. This then allows one to derive a rough estimate for the rate at which gas can be advected through the neutrinosphere into the neutron star (Sect. 7).

Up to this stage the discussion does not require a detailed solution for the velocity field of the flow. Since hydrostatic conditions are assumed to hold, in which case the kinetic energy is small compared to internal and gravitational energies, and the time evolution can be discussed by considering integral quantities, it is sufficient to know the rates at which mass enters or leaves the gain layer at both boundaries.

The model described here is based on a number of simplifications and approximations. With the analytic representation of the equation of state developed in Sect. 5.2, there is no need to monitor the radial profile and time evolution of the electron fraction. In addition to assuming hydrostatic equilibrium this, of course, limits the accuracy of the radial structure derived for the gain layer. Other shortcomings are the treatment of the neutron star as a point mass, i.e., the gravity of the atmosphere above the neutrinosphere is neglected, general relativistic effects are ignored, the energy release by nucleon recombination in the postshock medium at -2 MeV is not included, and additional neutrino heating and cooling by neutrino-electron scattering and neutrino-pair processes are not considered. Although it may be desirable to include these effects for a more detailed solution, it seems very unlikely that more refinements will change the essence of the discussion.

In calculating the neutrino energy deposition in the gain layer the neutrinospheric luminosity as well as the neutrino emission from the cooling layer, which is associated with the accretion of gas onto the neutron star, are taken into account. The most problematic and probably most serious weakness of the presented toy model, however, is the overly simplified description of the conditions in the cooling layer, which are essential for estimating both the mass accretion rate and the accretion luminosity of the neutron star. The temperature of the medium around the neutrinosphere is certainly not only determined by the interaction with the neutrino flow from the neutrinosphere, but also depends on the processes in contact with the gain layer. This is currently not included in the toy model.

Despite of these simplifications the toy model yields interesting insights into the interdependence of effects and processes which determine the post-bounce evolution of the supernova shock. Thus it may help one understanding the results of the much more complex hydrodynamic simulations.

10.2 Results and conclusions

In particular, the discussion of this paper allows one to draw the following conclusions:

1.

A criterion for shock revival could be derived in Sects. 9.1 and 9.2. It defines the conditions for which the radius $R_{\rm {s}}$ and the velocity $U_{\rm {s}} = \dot R_{\rm {s}}$ of the supernova shock can grow simultaneously. This criterion, applied to a stalled shock ( $U_{\rm {s}} \sim 0$), states that expansion ( $\dot R_{\rm {s}} > 0$) and acceleration ( $\dot U_{\rm {s}} > 0$) will occur at the same time when both the mass and the energy in the gain layer increase, i.e., when ${\rm {d}}(\Delta M_{\rm {g}})/{\rm {d}}t > 0$ and ${\rm {d}}(\Delta E_{\rm {g}})/{\rm {d}}t > 0$;

2.

For fixed neutrinospheric radius and temperature and given neutron star mass, the destiny of a shock at radius $R_{\rm {s}}$ depends on the rate at which the shock accretes matter, $\dot M$, and on the $\nu_{\rm e}$ plus $\bar\nu_{\rm e}$ luminosity $L_\nu$ coming from the neutrinosphere. In the $\dot M$-$L_\nu$ plane the two inequality relations of the shock revival criterion define two critical lines, an upper one and a lower one, which enclose the region where favorable conditions for shock expansion and acceleration occur. Neutrino heating is strong enough there to ensure ${\rm {d}}(\Delta E_{\rm {g}})/{\rm {d}}t > 0$ for a growing mass of the gain layer. The existence of a threshold value of the core neutrino luminosity (Burrows & Goshy 1993) for a given value of $\dot M$ is confirmed. Besides stronger neutrino heating in the gain layer, the main effect of a higher $L_\nu$ is reducing the neutrino emission in the cooling region and thus suppressing the rate at which mass is advected into the neutron star. If $L_\nu$ drops below the threshold value, the gain layer loses more mass than it receives by gas falling into the shock. Therefore the pressure support for the shock breaks down and the shock retreats. The same effect can be caused when the mass accretion rate $\dot M$ drops below a critical limit. This means that not only a low core luminosity but also a low mass accretion rate by the shock can prevent shock expansion;

3.

The efficiency of the neutrino cooling between neutrinosphere and gain radius is an important factor which contributes to regulating the mass advection through the gain radius and into the neutron star, and thus affects also the evolution of the gain layer and of the shock. Although the description in the toy model is greatly simplified, it demonstrates the importance of an accurate treatment of the physics, in particular of the neutrino-matter interactions, in the cooling layer. It must be suspected that excessive neutrino emission in the cooling layer, causing mass and energy loss from the gain layer, and not insufficient neutrino heating in the gain layer, may have been the main reason why spherically symmetric simulations ultimately failed to produce explosions, although the shock had expanded to larger radii, at least for some period of the post-bounce evolution (see, e.g., Bruenn 1993; Bruenn et al. 1995; Rampp & Janka 2000);

4.

The area between the two critical lines in the $\dot M$-$L_\nu$ plane grows for larger shock radii, because the conditions for neutrino heating in the gain layer improve. For parameters $(\vert\dot M\vert,L_{\nu})$ above the upper critical line, neutrino energy deposition cannot compensate for the inflow of "negative total energy'' with the gas that is falling through the shock. Since the mass in the gain layer increases, the shock nevertheless expands. But only after the shock has reached a sufficiently large radius can neutrino heating raise the total energy in the gain layer. When this radius is very far out, the total energy in the gain layer may stay negative, even for high neutrino luminosities, because only a small fraction of the gas in the gain layer experiences favorable heating conditions. In this case an explosion cannot occur. For the parameters considered in the discussed model, this happens when the (absolute value of the) mass accretion rate by the shock is larger than about 4  $M_{\odot }\,$s^-1. Also for somewhat lower accretion rates and in cases where the energy in the gain layer has become positive, explosions might not be possible. The question whether the shock is finally able to eject some part of the mantle and envelope of the progenitor star requires a global treatment of the problem. The toy model developed in this paper, however, is suitable to discuss only the phase of shock revival in dependence of the conditions around the forming neutron star. Considering the early evolution of the supernova shock is in general not sufficient to make predictions about the final outcome, in particular in cases where the postshock medium cannot acquire enough energy to gravitationally unbind the whole mass above the gain radius;

5.

The equations of the toy model were integrated for time-dependent solutions $R_{\rm {s}}(t)$ and $U_{\rm {s}}(t)$, and characteristic properties of the gain layer as functions of time. These solutions confirm the conclusions drawn from an evaluation of the shock revival criterion. In addition, they yield information about the evolution of the mass and energy in the gain layer;

6.

Accounting for the complex effects of multi-dimensional convective overturn within the simplified discussion of the toy model is not possible. Nevertheless, some consequences of convective energy transport in the gain layer can be addressed. For a suitable choice of the structural polytropic index $\gamma$ of the hydrostatic atmosphere one can describe a situation where the gain layer is essentially isentropic in contrast to a case where the energy transport by convection is less efficient and therefore negative gradients of the entropy and specific energy are present between the gain radius and the shock front. Without convection such negative gradients must develop because neutrino heating is strongest just outside of the gain radius. Despite of a slightly larger neutrino energy deposition (because of a smaller gain radius) the absence of "convection'' in the toy model has a negative effect on the shock expansion. Because convection redistributes the energy deposited by neutrinos mainly near the gain radius to regions closer to the shock, the energy loss associated with the downward advection of gas through the gain radius is reduced. Therefore more energy stays in the gain layer, in particular at larger radii, the postshock pressure is enhanced, and the shock is driven out more easily;

7.

Parametric studies with the toy model suggest that successful explosions, driven by neutrino energy deposition, cannot be very energetic. The total energy per unit mass (gravitational plus internal energy plus minor kinetic contributions) in the expanding gain layer (between gain radius and shock) was observed to saturate around 0.1 s after shock revival and was found to be always limited, even for high core luminosities $L_\nu$, by  erg$\,$ $M_{\odot}^{-1}$, (corresponding to an energy per nucleon of 5 MeV). For "typical'' mass accretion rates by the shock, $\dot M$, of a few 0.1  $M_{\odot }\,$s^-1, the integral mass in the gain layer was then between several 10^-2 $M_{\odot }$ and (1- 2) 10^-1 $M_{\odot }$, the corresponding total energy in the gain layer at most around 10⁵¹ erg. For higher accretion rates $\dot M$the mass in the gain layer was found to be larger, but the energy per nucleon at the same time lower. The maximum total energies were obtained for intermediate values of $\dot M$ (around 1  $M_{\odot }\,$s^-1) and high $\nu_{\rm e}$ plus $\bar\nu_{\rm e}$ luminosities $L_\nu$ (values up to 12 10⁵² erg$\,$s^-1 were considered), but $\Delta E_{\rm {g}}$ was less than (2- 3) 10⁵¹ erg in all cases.

10.3 Implications

The physical mechanism of powering the explosion by neutrino absorption on nucleons therefore seems to limit the explosion energy to values of at most a few 10⁵¹ erg. There is no obvious reason why neutrino-driven explosions could not be less energetic. The upper limit of the explosion energy is of the order of or a small multiple of the gravitational binding energy of the gas mass in the neutrino-heating layer around the nascent neutron star.

The reason for this energy limit is a very fundamental one, associated with the mechanism how the energy for the explosion is delivered, stored and carried outward. The energy which starts and drives the explosion is mainly transferred to the stellar gas by electron neutrino and antineutrino absorption on nucleons. As soon as the baryons have obtained a sufficiently large mean energy the expansion of the heating region sets in and the nucleons move away from the central source of the neutrino flux. The specific energy for this to happen is of the order of the gravitational binding energy of a nucleon. In fact, because of the inertia of the gain layer and the confinement by the matter falling into the shock, the nucleons can absorb more energy than that. If this were not the case, the energy of the expanding layer would be consumed by lifting the baryons in the gravitational field of the neutron star, and the kinetic energy at infinity could never be large.

The neutrinospheric luminosity $L_\nu$ as well as the mass in the gain layer behind a stalled shock decrease with time, associated with the decreasing rate $\dot M$ of mass accretion by the shock. This has negative effects on the possibility of shock revival, as discussed above on grounds of the toy model. It definitely does not improve the conditions for a strong explosion, because of the limited energy which a baryon can absorb before it starts moving outward. With little gas being exposed to neutrino heating the explosion energy will therefore stay low. For these reasons "late'' neutrino-driven explosions appear to be disfavored compared to delayed ones that develop within the post-bounce period when both $L_\nu$ and $\dot M$ are still high. This is typically the case until about half a second after core bounce.

Estimating the final explosion energy of the star requires, of course, that the energy release by nucleon recombination and possible nuclear burning in a fraction of the mass of the gain layer are added, and the gravitational binding energy of the mantle and envelope material of the progenitor star is subtracted (see, e.g., Bethe 1990, 1993; Bethe 1996a,c). Within the considered toy model these energies cannot be estimated. In case of a successful neutrino-driven explosion these terms, however, should not be the dominant ones in the total energy budget.

The total energy of the explosion should also not receive a major contribution by the energy released during the phase of the neutrino-driven wind, which succeeds the period of shock revival and early shock expansion. Different from the latter phase, the neutrino-driven wind is characterized by quasi steady-state conditions, with the mass flow rate not varying with the radius outside of a narrow region where the mass loss of the neutron star is determined. Baryons interacting with neutrinos near the surface of the neutron star cannot absorb a particle energy much larger than their gravitational binding energy before they are driven away from the neutrinosphere. Although always positive, the neutrino heating decreases rapidly when the wind accelerates outward and the distance from the source of the luminosity increases. Since the confining effect of mass infall to a shock is absent, the final net energy of a nucleon moving out with the wind will be even smaller than at earlier times. In addition, the neutrino luminosity and the neutron star radius shrink with time. Therefore the mass loss rate during the wind phase will be lower than right after shock revival. For these reasons the total mass ejected in the wind is expected to be less than a few 10^-2 $M_{\odot }$ (see Woosley & Baron 1992; Woosley 1993a; Qian & Woosley 1996).

Overcoming the stringent limit on the energy per nucleon that neutrinos can transfer to the heated matter, requires specific conditions. It could either be achieved by a sudden, luminous outburst of (energetic) neutrinos, which builds up on a timescale shorter than the expansion time of the gas around the neutron star. However, assuming a standard, hydrostatic cooling history of the nascent neutron star, there is no theoretical model to support such a scenario. Alternatively, the energy of the explosion could be absorbed and carried by non-baryonic particles, i.e., electrons and positrons and photons. In both cases the total energy is not constrained roughly by the binding energy of the gas in the gravitational potential of the neutron star. In fact, neutrino-electron scattering and neutrino-antineutrino annihilation have been suggested as important sources of energy for the explosion (Goodman et al. 1987; Colgate 1989). An accurate discussion of the physics of neutrino transport in the semi-transparent regime around the neutrinosphere (Janka 1991a,b) and a detailed evaluation of the conditions in the heating layer, however, show that both $\nu {\rm e}^\pm$ scattering (Bethe & Wilson 1985; Bethe 1990, 1993, 1995) and $\nu\bar\nu$ annihilation (Cooperstein et al. 1987; Bethe 1997) are significantly less efficient than $\nu_{\rm e}$ and $\bar\nu_{\rm e}$ absorption, and thus contribute only minor fractions to the explosion energy.

The situation may be different when the global spherical symmetry is broken, e.g., in case of a black hole that accretes gas from a thick disk formed by the collapsing matter of a rapidly rotating, massive star. The disk becomes very hot and loses energy primarily by neutrino emission. Such a scenario was suggested as source of cosmological gamma-ray bursts and possibly strange, very energetic supernova explosions (Woosley 1993b; MacFadyen & Woosley 1999; MacFadyen et al. 1999). In this case the neutrino luminosities can be higher, the region where neutrino pairs annihilate is more compact (which implies that the neutrino number densities are larger), and the geometry favors more head-on collisions between neutrinos. All these effects lead to an enhanced probability of $\nu\bar\nu$ annihilation in the close vicinity of the black hole (Popham et al. 1999).

Acknowledgements

It is a pleasure to thank M. Rampp for comments, his patience in many discussions, and a comparative evaluation of his spherically symmetric hydrodynamical simulations. The author is very grateful to an anonymous referee for thoughtful and knowledgable comments which helped to improve the manuscript significantly. The preparation of Fig. 2 by Mrs. H. Krombach is acknowledged as well as help by M. Bartelmann to tame the "L^ATEX devil''. This work was supported by the SFB-375 "Astroparticle Physics'' of the Deutsche Forschungsgemeinschaft.