A spherically symmetric radiating shell in the presence of a spinless radiating central body

We construct a solution of Einstein's field equations with a possible astrophysical interest by matching a Vaidya solution with another Vaidya solution through a thin spherical shell of radially radiating matter. We study the system of equations for the motion of the shell and the radiation fields in two simple cases. In one of them we consider a dust shell of constant proper mass radiating at constant rate. In the other case we restrict the motion of the shell to a stationary configuration and assume it to be totally opaque. We find that this later solution is unstable under small perturbations in the radius of the shell but there exist stable stationary solutions if the pressure within the shell is different from zero.