Shell dynamics in the presence of a central body

We construct an idealized spherically symmetric relativistic model of an exploding object in the framework of the theory of surface layers in general relativity and match a Vaidya solution for a radially radiating star to another Vaidya solution through a thin spherical shell. We reduce the equations of motion and the radiation density of the Vaidya solution given by the matching conditions to a first-order system and analyze the general characteristics of the motion. We use a post-Newtonian approximation to find the equation of motion of a spherically symmetric radiating shell moving in a central gravitational potential.