  The onset of convection in a rotating cylindrical annulus with parallel ends
filled with a compressible fluid is studied in the anelastic approximation.
Thermal Rossby waves propagating in the azimuthal direction are found as
solutions. The analogy to the case of Boussinesq convection in the presence of
conical end surfaces of the annular region is emphasized. As in the latter case
the results can be applied as an approximation for the description of the onset
of anelastic convection in rotating spherical fluid shells. Reasonable
agreement with three-dimensional numerical results published by Jones et al.
(J. Fluid Mech., vol. 634, 2009, pp. 291-319) for the latter problem is found.
As in those results the location of the onset of convection shifts outward from
the tangent cylinder with increasing number $N_\rho$ of density scale heights
until it reaches the equatorial boundary. A new result is that at a much higher
number $N_\rho$ the onset location returns to the interior of the fluid shell.
