  We revisit the flip-flop instability of two-dimensional planar accretion
using high-fidelity numerical simulations. By starting from an initially
steady-state axisymmetric solution, we are able to follow the growth of this
overstability from small amplitudes. In the small-amplitude limit, before any
transient accretion disk is formed, the oscillation period of the accretion
shock is comparable to the Keplerian period at the Hoyle-Lyttleton accretion
radius (R_a), independent of the size of the accreting object. The growth rate
of the overstability increases dramatically with decreasing size of the
accretor, but is relatively insensitive to the upstream Mach number of the
flow. We confirm that the flip-flop does not require any gradient in the
upstream flow. Indeed, a small density gradient as used in the discovery
simulations has virtually no influence on the growth rate of the overstability.
The ratio of specific heats does influence the overstability, with smaller
gamma leading to faster growth of the instability. For a relatively large
accretor (a radius of 0.037R_a) planar accretion is unstable for gamma = 4/3,
but stable for gamma > 1.6. Planar accretion is unstable even for gamma = 5/3
provided the accretor has a radius of < 0.0025R_a. We also confirm that when
the accretor is sufficiently small, the secular evolution is described by
sudden jumps between states with counter-rotating quasi-Keplerian accretion
disks.
