  The compressible Navier-Stokes-Poisson (NSP) system is considered in $R^3$ in
the present paper and the influences of the electric field of the internal
electrostatic potential force governed by the self-consistent Poisson equation
on the qualitative behaviors of solutions is analyzed. It is observed that the
rotating effect of electric field affects the dispersion of fluids and reduces
the time decay rate of solutions. Indeed, we show that the density of the NSP
system converges to its equilibrium state at the same $L^2$-rate
$(1+t)^{-\frac34}$ or $L^\infty$-rate $(1+t)^{-3/2}$ respectively as the
compressible Navier-Stokes system, but the momentum of the NSP system decays at
the $L^2$-rate $(1+t)^{-\frac14}$ or $L^\infty$-rate $(1+t)^{-1}$ respectively,
which is slower than the $L^2$-rate $(1+t)^{-\frac34}$ or $L^\infty$-rate
$(1+t)^{-3/2}$ for the compressible Navier-Stokes system. These convergence
rates are also shown to be optimal for the compressible NSP system.
