  Energy deposition by neutrinos can rejuvenate the stalled bounce shock and
can provide the energy for the supernova explosion of a massive star. This
neutrino-heating mechanism, however, is not finally accepted or proven as the
trigger of the explosion. Part of the problem is that the complexity of the
hydrodynamic models often hampers a clear and simple interpretation of the
results. This demands a deeper theoretical understanding of the requirements of
a successful shock revival. A toy model is presented here for discussing the
neutrino heating phase analytically by a time-dependent treatment, which allows
one to calculate the radius and velocity of the supernova shock from global
properties of the gain layer as solutions of an initial value problem. A
criterion is derived for the requirements of shock revival. It confirms the
existence of a minimum neutrino luminosity needed for shock expansion, but also
demonstrates the importance of a sufficiently large mass infall rate to the
shock. The possibility of very energetic neutrino-driven explosions seems
excluded because the total specific energy transferred to nucleons is limited
by about 1e52 ergs per solar mass (about 5 MeV per nucleon) and the total mass
in the gain layer is typically only around 0.1 solar masses. Energy transport
by convection from the region of maximum heating to radii closer behind the
shock is found to support the explosion by reducing the energy loss associated
with the inward advection of neutrino-heated matter through the gain radius.
(abridged)
