  The linear stability of isothermal Bondi accretion with a shock is studied
analytically in the asymptotic limit of high incident Mach number M_1. The flow
is unstable with respect to radial perturbations as expected by Nakayama
(1993), due to post-shock acceleration. Its growth time scales like the
advection time from the shock r_sh to the sonic point r_son. The growth rate of
non-radial perturbations l=1 is higher by a factor M_1^{2/3}, and is therefore
intermediate between the advection and acoustic frequencies. Besides these
instabilities based on post-shock acceleration, our study revealed another
generic mechanism based on the cycle of acoustic and vortical perturbations
between the shock and the sonic radius, independently of the sign of post-shock
acceleration. The vortical-acoustic instability is fundamentally non-radial. It
is fed by the efficient excitation of vorticity waves by the isothermal shock
perturbed by acoustic waves. The growth rate exceeds the advection rate by a
factor log M_1. Unstable modes cover a wide range of frequencies from the
fundamental acoustic frequency ~c/r_sh up to a cut-off ~c/r_son associated with
the sonic radius. The highest growth rate is reached for l=1 modes near the
cut-off. The additional cycle of acoustic waves between the shock and the sonic
radius is responsible for variations of the growth rate by a factor up to 3
depending on its phase relative to the vortical-acoustic cycle. The instability
also exists, with a similar growth rate, below the fundamental acoustic
frequency down to the advection frequency, as vorticity waves are efficiently
coupled to the region of pseudosound. These results open new perspectives to
address the stability of shocked accretion flows.
