2 Physical picture

Right after core bounce the hydrodynamic shock propagates outward in mass as well as in radius, being strongly damped by energy losses due to the photodisintegration of iron-group nuclei and neutrinos. The neutrino emission rises significantly when the shock breaks out into the neutrino-transparent regime. As a consequence, the pressure behind the shock is reduced and the velocities of the shock and of the fluid behind the shock, both of which were positive initially, decrease. Finally, the outward expansion of the shock stagnates, and the shock transforms into a standing accretion shock with negative gas velocity in the postshock region. The gas of the progenitor star, which continues to fall into the shock at a velocity near free fall, is decelerated abruptly within the shock. Below the shock it moves much more slowly towards the center, where it settles onto the surface of the nascent neutron star.

Figure 1: Sketch which summarizes the processes that determine the evolution of the stalled supernova shock after core bounce. Stellar matter falls into the shock at radius $R_{\rm {s}}$ with a mass accretion rate $\dot M$ and a velocity near free fall. After deceleration in the shock, the gas is much more slowly advected towards the nascent neutron star through the regions of net neutrino heating and cooling, respectively. The radius $R_{\rm {ns}}$ of the neutron star is defined by a steep decline of the density over several orders of magnitude outside the neutrinosphere at $R_{\nu }$. Heating balances cooling at the gain radius $R_{\rm {g}}$. The dominant processes of energy deposition and loss are absorption of electron neutrinos onto neutrons and electron antineutrinos onto protons as indicated in the figure. Convective overturn mixes the layer between gain radius and shock, and convection inside the neutron star helps the explosion by boosting the neutrino luminosities

Figure 1 displays the most important physical elements which determine this evolutionary stage. Around the neutrinosphere at radius $R_{\nu }$, which is close to the radius $R_{\rm {ns}}$ of the proto-neutron star (PNS), the hot and comparatively dense gas loses energy by radiating neutrinos. If this energy sink were absent, the gas that is accreted through the shock at a rate $\dot M$ would pile up in a growing, high-entropy atmosphere on top of the compact remnant (Colgate et al. 1993; Colgate & Fryer 1995; Fryer et al. 1996). But since neutrinos are emitted efficiently at the thermodynamical conditions around the neutrinosphere, the entropy of the gas is reduced so that the gas can be absorbed into the surface of the neutron star. The mass flow through the neutrinospheric region is therefore triggered by the neutrino energy loss and allows more gas to be advected inward from larger radii. In case of stationary accretion the temperature at the base of the atmosphere ensures that the emitted neutrinos carry away the gravitational binding energy of the matter which is added to the neutron star at a given accretion rate. In fact, this requirement closes the set of equations that determines the steady state of the accretion system and allows one to determine the radius $R_{\rm {s}}$ of the accretion shock (see, e.g., Chevalier 1989; Brown & Weingartner 1994; Fryer et al. 1996). At the so-called gain radius $R_{\rm {g}}$ (Bethe & Wilson 1985) between neutrinosphere $R_{\nu }$ and shock position $R_{\rm {s}}$, the temperature of the atmosphere becomes so low that the absorption of high-energy electron neutrinos and antineutrinos starts to exceed the neutrino emission. This radius therefore separates the region of net neutrino cooling below from a layer of net heating above. Since the neutrino heating is strongest just outside the gain radius and the propagation of the shock has weakened before stagnation, a negative entropy gradient is built up in the postshock region. This leads to convective overturn roughly between $R_{\rm {g}}$ and $R_{\rm {s}}$, which transports hot matter outward in rising high-entropy bubbles. At the same time cooler material is mixed inward in narrow, low-entropy downflows (Herant et al. 1994; Burrows et al. 1995; Janka & Müller 1996). Inside the nascent neutron star, below the neutrinosphere, convective motions can enhance the neutrino emission by carrying energy faster to the surface than neutrino diffusion does (Keil et al. 1996).

Figure 2: Schematic profiles of density, temperature, and mass accretion rate between neutrinosphere at radius $R_{\nu }$ and shock at $R_{\rm {s}}$ some time after core bounce. $R_{\rm {g}}$ denotes the position of the gain radius. At the shock, $\rho$ and T jump discontinuously from their preshock values $\rho _{\rm {p}}$ and $T_{\rm {p}}$ to the postshock values $\rho _{\rm {s}}$ and $T_{\rm {s}}$, respectively. For $r < R_{\rm {eos}}$ the density declines steeply because the pressure is mainly caused by the nonrelativistic Boltzmann gases of free neutrons and protons. Outside of $R_{\rm {eos}}$ the gas is radiation dominated and the density decrease much flatter. In general, some of the gas falling into the shock at rate $\dot M$ may stay in the region of neutrino heating while another part (rate $\dot M'$) is advected into the nascent neutron star. Note that $\dot M(r)$ is continuous at the shock in the rest frame of the star only in case of a stalled shock front. Between $R_{\nu }$ and $R_{\rm {eos}}$ the temperature can be considered roughly as constant, whereas its negative gradient in the radiation dominated region ensures hydrostatic equilibrium. There is net energy loss between $R_{\nu }$ and $R_{\rm {g}}$ where T(r)exceeds the temperature $T_{\rm {H=C}} \sim T_{\nu }(R_{\nu }/r)^{1/3}$, for which neutrino heating equals cooling. Net energy deposition occurs between $R_{\rm {g}}$ and $R_{\rm {s}}$

Between neutrinosphere and the supernova shock a number of approximations apply to a high degree of accuracy, which help one developing a simple analytic understanding of the effects that influence the evolution of the supernova shock. Figure 2 shows schematically the profiles of density, temperature and mass accretion rate in that region. A formal discussion follows in the subsequent sections. Outside the neutrinosphere (typically at about $10^{11}\,$g/cm³) the temperature drops slowly compared to the density decline, which is steep. When nonrelativistic nucleons dominate the pressure, the decrease of the density yields the pressure gradient which ensures hydrostatic equilibrium in the gravitational field of the neutron star. Assuming a temperature equal to the neutrinospheric temperature in this region is a reasonably good approximation for the following reasons. On the one hand, the cooling rate depends sensitively both on density and temperature, and the density drops rapidly. Therefore the total energy loss is determined in the immediate vicinity of the neutrinosphere and the details of the temperature profile do not matter very much. On the other hand, efficient neutrino heating prevents that the temperature can drop much below the neutrinospheric value. If, instead, the temperature would rise significantly above this latter value, the matter would become optically thick to the energetic neutrinos produced in the hot gas (the opacity increases roughly with the square of the neutrino energy) and the neutrinosphere would move farther out to a lower density (and thus typically a lower temperature). Below a density between $10^9\,$g/cm³ and $10^{10}\,$g/cm³, relativistic electron-positron pairs and radiation determine the pressure, provided the temperature is sufficiently high, typically around 1 MeV or more (see Woosley et al. 1986). Exterior to the corresponding radius $R_{\rm {eos}}$, where this transition from the baryon-dominated to the radiation-dominated regime takes place, the temperature must therefore decrease so that the negative temperature gradient can yield the force which balances gravity.

The gain radius $R_{\rm {g}}$ is located at the radial position where the temperature profile T(r) intersects with the curve of temperature values, $T_{\rm {H=C}}(r)$, for which heating is equal to cooling by neutrinos, roughly given by

$$T_{\rm {H=C}}(r)\,\sim\,T_{\nu}\cdot \left( {R_{\nu}\over r}\right)^{\! {1 \over 3}}$$ (1)

(Bethe & Wilson 1985). In Eq. (1) $T_{\nu}$ means the temperature at the radius $R_{\nu }$ of the neutrinosphere. The shock at $R_{\rm {s}}$is taken to be infinitesimally thin compared to the scales considered. Within the shock the density and temperature therefore jump from their preshock values $\rho _{\rm {p}}$ and $T_{\rm {p}}$, to the postshock values $\rho _{\rm {s}}$ and $T_{\rm {s}}$, respectively. A part of the gas which falls into the shock with a mass accretion rate $\dot M$ can stay in the region of neutrino heating, whereas another part is advected with rate $\dot M'$ through the cooling region to be added to the neutron star inside $R_{\nu }$.

The approach to the problem of shock revival taken in this paper is considerably different from the discussion of steady-state accretion or winds. Steady-state assumptions, for example, were also used by Burrows & Goshy (1993) in their theoretical analysis of the explosion mechanism. Having realized the fact, however, that the mass and energy in the gain layer vary because of different rates of mass flow through the boundaries and additional neutrino heating, one is forced to the following conclusions. Firstly, the discussion has to be time-dependent, which means that the time derivatives in the continuity and energy equations cannot be ignored. (Dropping the total time-derivative in the momentum equation by assuming hydrostatic equilibrium is less problematic and yields a reasonably good approximation.) Secondly, the properties of the shock and of the gain layer must be determined as solutions of an initial value problem rather than from a steady-state picture. This reflects essential physics, namely that the shock behavior is controlled by the cumulative effects of neutrino heating and mass accumulation in the gain layer. For these reasons conservation laws for the total mass and energy in the gain layer will be derived by integrating the hydrodynamic equations of continuity and energy, including the terms with time derivatives, over the volume of the gain layer. The treatment will therefore retain the time-dependence of the problem.

In this paper the discussion will be restricted to an idealized, spherically symmetric situation and possible convective mixing will be assumed to lead to efficient homogenization of the unstable layer. Certainly, this is not a good assumption for the convective overturn that takes place in the region between gain radius and shock front, where prominent, large-scale inhomogeneities develop (Herant et al. 1994; Burrows et al. 1995; Janka & Müller 1996). Bethe (1995) has made attempts to discuss the physical implications of the simultaneous presence of low-entropy downstreams and high-entropy rising bubbles. For this purpose he introduced free parameters, e.g., to quantify the fraction of neutrinos that hits the cold downflows and is effective for their heating, or to account for the part of the matter that is added to the neutron star instead of being pushed outward in the expanding bubbles. This procedure is not really satisfactory and will not be copied here. Instead, an admittedly simplified and idealized spherical situation will be considered to highlight the conditions needed for shock revival and to develop a qualitative understanding of the influence of different effects. One-dimensional analysis can help developing a better understanding of the delayed explosion mechanism, because simulations in spherical symmetry have produced successful explosions (Wilson 1985; Wilson et al. 1986; Janka & Müller 1995, 1996). Thus they have demonstrated that convection behind the shock is not an indispensable requirement for an explosion, although it may be an essential (Herant et al. 1994; Burrows et al. 1995; Janka & Müller 1996) - yet not necessarily sufficient (Janka & Müller 1996; Mezzacappa et al. 1998b; Lichtenstadt et al. 1999) - ingredient to obtain explosions, or to raise the explosion energy in cases which fail or nearly fail in spherical symmetry.