  (Abridged) Neutrino heating may drive core-collapse supernova explosions.
Although it is known that the stalled accretion shock turns into explosion when
the neutrino luminosity from the collapsed core exceeds a critical value
(L_crit) (the "neutrino mechanism"), the physics of L_crit, as well as its
dependence on the properties of the proto-neutron star (PNS) and changes to the
microphysics has never been systematically explored. We solve the
one-dimensional steady-state accretion problem between the PNS surface and the
accretion shock. We quantify the deep connection between the solution space of
steady-state accretion flows with bounding shocks and the neutrino mechanism.
We show that there is a maximum, critical sound speed above which it is
impossible to maintain accretion with a standoff shock, because the shock jump
conditions cannot be satisfied. The physics of this critical sound speed is
general and does not depend on a specific heating mechanism. For the simple
model of pressure-less free-fall onto a shock bounding an isothermal accretion
flow with sound speed c_T, we show that if c_T^2/v_escape^2 > 3/16 explosion
results. We generalize this result to the more complete supernova problem,
showing explicitly that the same physics determines L_crit. We find that the
critical condition for explosion can be written as c_S^2/v_escape^2 = 0.19,
where c_S is the adiabatic sound speed. This "antesonic" condition describes
L_crit over a broad range in accretion rate and microphysics. We show that the
addition of the accretion luminosity (L_acc) reduces L_crit non-trivially. As
in previous work, we find that L_crit is always significantly higher than the
maximum possible value of L_acc. Finally, we provide evidence that the
reduction in L_crit seen in recent multi-dimensional simulations results from a
reduction in the efficiency of cooling, rather than an increase in the heating
rate.

High Energy Astrophysical Phenomena;Solar and Stellar Astrophysics