  We determine the asymptotic conditions under which the Boussinesq
approximation is valid for oscillatory convection in a rapidly rotating fluid.
In the astrophysically relevant parameter regime of small Prandtl number, we
show that the Boussinesq prediction for the onset of convection is valid only
under much more restrictive conditions than those that are usually assumed. In
the case of an ideal gas, we recover the Boussinesq results only if the ratio
of the domain height to a typical scale height is much smaller than the Prandtl
number. This requires an extremely shallow domain in the astrophysical
parameter regime. Other commonly-used "sound-proof" approximations generally
perform no better than the Boussinesq approximation. The exception is a
particular implementation of the pseudo-incompressible approximation, which
predicts the correct instability threshold beyond the range of validity of the
Boussinesq approximation.
