  In this paper, we study a free boundary problem for compressible spherically
symmetric Navier-Stokes equations with a gravitational force and degenerate
viscosity coefficients. Under certain assumptions that imposed on the initial
data, we obtain the global existence and uniqueness of the weak solution and
give some uniform bounds (with respect to time) of the solution. Moreover, we
obtain some stabilization rate estimates in $L^\infty$-norm and weighted
$H^1$-norm of the solution. The results show that such system is stable under
the small perturbations, and could be applied to the astrophysics.
