  We have been studying spherically symmetric motions of gaseous stars with
physical vacuum boundary governed either by the Euler-Poisson equations in the
non-relativistic theory or by the Einstein-Euler equations in the relativistic
theory. The problems are to construct solutions whose first approximations are
small time-periodic solutions to the linearized problem at an equilibrium and
to construct solutions to the Cauchy problem near an equilibrium. These
problems can be solved when $1/(\gamma-1)$ is an integer, where $\gamma$ is the
adiabatic exponent of the gas near the vacuum, by the formulation by R.
Hamilton of the Nash-Moser theorem. We discuss on an application of the
formulation by J. T. Schwartz of the Nash-Moser theorem to the case in which
$1/(\gamma-1)$ is not an integer but sufficiently large.
