Ignoring the pressure of the ambient, the local acceleration of a gas parcel at a distance r away from the origin for a uniform density sphere is given by:

\frac{d^2r}{dt^2} = -G\frac{M_r}{r^2}

Integrating this equation under the boundary conditions, the initial velocity \frac{dr}{dt}=0 and the initial radius is the radius of the sphere r=r_0, leads to an equation that describes the radius of the outer sphere over time,