Homologous Collapse of a uniform density sphere

Ignoring the pressure forces in the gas (which is valid when the gas is Jeans unstable, i.e. E_{grav}>>E_{kin}), 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 condition, the initial velocity \frac{dr}{dt}=0 at the initial radius of the sphere r=r_0, leads to an equation that describes the radius of the outer sphere over time (i.e. over the course of collapse), and the velocity at this radius over time.