  The nonlinear asymptotic stability of Lane-Emden solutions is proved in this
paper for spherically symmetric motions of viscous gaseous stars with the
density dependent shear and bulk viscosities which vanish at the vacuum, when
the adiabatic exponent $\gamma$ lies in the stability regime $(4/3, 2)$, by
establishing the global-in-time regularity uniformly up to the vacuum boundary
for the vacuum free boundary problem of the compressible Navier-Stokes-Poisson
systems with spherical symmetry, which ensures the global existence of strong
solutions capturing the precise physical behavior that the sound speed is
$C^{{1}/{2}}$-H$\ddot{\rm o}$lder continuous across the vacuum boundary, the
large time asymptotic uniform convergence of the evolving vacuum boundary,
density and velocity to those of Lane-Emden solutions with detailed convergence
rates, and the detailed large time behavior of solutions near the vacuum
boundary. The results obtained in this paper extend those in \cite{LXZ} of the
authors for the constant viscosities to the case of density dependent
viscosities which are degenerate at vacuum states.
