  We analyze the linear stability of a stalled accretion shock in a perfect gas
with a parametrized cooling function L ~ rho^{beta-alpha} P^alpha. The
instability is dominated by the l=1 mode if the shock radius exceeds 2-3 times
the accretor radius, depending on the parameters of the cooling function. The
growth rate and oscillation period are comparable to those observed in the
numerical simulations of Blondin & Mezzacappa (2006). The instability mechanism
is analyzed by separately measuring the efficiencies of the purely acoustic
cycle and the advective-acoustic cycle. These efficiencies are estimated
directly from the eigenspectrum, and also through a WKB analysis in the high
frequency limit. Both methods prove that the advective-acoustic cycle is
unstable, and that the purely acoustic cycle is stable. Extrapolating these
results to low frequency leads us to interpret the dominant mode as an
advective-acoustic instability, different from the purely acoustic
interpretation of Blondin & Mezzacappa (2006). A simplified characterization of
the instability is proposed, based on an advective-acoustic cycle between the
shock and the radius r_nabla where the velocity gradients of the stationary
flow are strongest. The importance of the coupling region in this mechanism
calls for a better understanding of the conditions for an efficient
advective-acoustic coupling in a decelerated, nonadiabatic flow, in order to
extend these results to core-collapse supernovae.
