  We consider stability of non-rotating viscous gaseous stars modeled by the
Navier-Stokes-Poisson system. Under general assumptions on the equations of
states, we proved that the number of unstable modes for the linearized
Navier-Stokes-Poisson system equals that of the linearized Euler-Poisson system
modeling inviscid gaseous stars. In particular, the turning point principle
holds true for non-rotating stars with or without viscosity. That is, the
transition of stability only occurs at the extrema of the total mass and the
number of unstable modes is determined by the mass-radius curve. For the proof,
we establish an infinite dimensional Kelvin-Tait-Chetaev theorem for a class of
linear second order PDEs with dissipation. Moreover, we prove that linear
stability implies nonlinear asymptotic stability and linear instability implies
nonlinear instability for Navier-Stokes-Poisson system.
