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}

This acceleration is due to all of the mass inside of the sphere, i.e. the mass contained within a radius r, M_r. Since this equation governs the local acceleration of the gas at any radius, we can consider how the acceleration of any shell within the sphere behaves over time, once we know r(t) of course. Initially (t=0), however, we do know the radius, so we can check the behavior of this equation at t=0 for various shells within the sphere.

Consider first the outer most radius. A sphere with radius r=r_0 has within it mass that goes like M_{r0} \propto r_0^3. Therefore, we see that initially, the acceleration of the gas is proportional to r_0. The density of the sphere is uniform, and this means repeating this procedure for any smaller radii shows that the acceleration of interior shells decreases as r decreases.

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.