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

On Spherically Symmetric Motions of the Atmosphere Surrounding a Planet Governed by the Compressible Euler Equations