  We are concerned with a global existence theory for finite-energy solutions
of the multidimensional Euler-Poisson equations for both compressible gaseous
stars and plasmas with large initial data of spherical symmetry. One of the
main challenges is the strengthening of waves as they move radially inward
towards the origin, especially under the self-consistent gravitational field
for gaseous stars. A fundamental unsolved problem is whether the density of the
global solution forms a delta measure ({\it i.e.}, concentration) at the
origin. To solve this problem, we develop a new approach for the construction
of approximate solutions as the solutions of an appropriately formulated free
boundary problem for the compressible Navier-Stokes-Poisson equations with a
carefully adapted class of degenerate density-dependent viscosity terms, so
that a rigorous convergence proof of the approximate solutions to the
corresponding global solution of the compressible Euler-Poisson equations with
large initial data of spherical symmetry can be obtained. Even though the
density may blow up near the origin at a certain time, it is proved that no
delta measure ({\it i.e.}, concentration) in space-time is formed in the
vanishing viscosity limit for the finite-energy solutions of the compressible
Euler-Poisson equations for both gaseous stars and plasmas in the physical
regimes under consideration.
