  Linear stability of solid body rotating flows with axisymmetric density
variations is addressed analytically. Considering inviscid disturbances, a non
trivial dispersion relation is obtained and it is shown that the instability is
of Rayleigh-Taylor type in cylindrical frame. The viscous correction is
derived, in the limit of large Reynolds numbers and large azimuthal wave
numbers, allowing the determination of the most unstable mode. Theoretical
predictions are checked by comparing them to (spectrally accurate) computed
eigenvalues and direct numerical simulations.
