  We consider stability of rotating gaseous stars modeled by the Euler-Poisson
system with general equation of states. When the angular velocity of the star
is Rayleigh stable, we proved a sharp stability criterion for axi-symmetric
perturbations. We also obtained estimates for the number of unstable modes and
exponential trichotomy for the linearized Euler-Poisson system. By using this
stability criterion, we proved that for a family of slowly rotating stars
parameterized by the center density with fixed angular velocity profile, the
turning point principle is not true. That is, unlike the case of non-rotating
stars, the change of stability of the rotating stars does not occur at extrema
points of the total mass. By contrast, we proved that the turning point
principle is true for the family of slowly rotating stars with fixed angular
momentum distribution. When the angular velocity is Rayleigh unstable, we
proved linear instability of rotating stars. Moreover, we gave a complete
description of the spectra and sharp growth estimates for the linearized
Euler-Poisson equation.
