The mechanism of core-collapse supernovae is unknown. Despite considerable effort,
most simulations of supernovae are not successful, and it has proven difficult to
revive the stalled accretion shock, particularly for more massive stellar
progenitors. 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
heating/cooling mechanisms has never been systematically explored. 

We quantify the deep connection between the solution space of steady-state isothermal
accretion flows with bounding shocks and the neutrino mechanism. In particular, 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. We show that if c_T^2/v_esc^2 >= 3/16 = 0.1875
explosion results - the "antesonic" condition. We generalize this result to the more
complete supernova problem, where the critical condition for explosion can be written
as c_S^2/v_esc^2 = 0.19 over a broad range in accretion rate and microphysics. Other
criteria proposed in the supernova literature fail to capture the physics of L_crit. 

In addition, we explore effects of collective neutrino oscillations on L_crit.