  We consider spherically symmetric motions of inviscid compressible gas
surrounding a solid ball under the gravity of the core. Equilibria touch the
vacuum with finite radii, and the linearized equation around one of the
equilibria has time-periodic solutions. To justify the linearization, we should
construct true solutions for which this time-periodic solution plus the
equilibrium is the first approximation. But this leads us to difficulty caused
by singularities at the free boundary touching the vacuum. We solve this
problem by the Nash-Moser theorem.
