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. In other words, the furthest shells have the highest acceleration. They speed up from an initial velocity of 0 the fastest. This has to be the case, as

• all shells reach the origin at the same time (so outer shells have furthest to go in the same amount of time, i.e. they must travel faster)
• shells do not cross over the collapse
• which means that the mass contained within each concentric sphere remains a constant over the collapse

This last point is useful for solving integrating the equation. Since the mass is constant in any shell (i.e. shell by the way here is used to mean concentric sphere), we can replace M_r in the equation by M_r = \frac{4}{3} \pi r_0^3 \rho_0. 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. We will look at these equations next.

Velocity equation

\frac{dr}{dt} = -\sqrt{\frac{8\pi}{3} G \rho_0 r_0^2(\frac{r_0}{r}-1)}

• describes the outer velocity of a sphere that contains mass M_r as a function of r = r(t)
• at radius r = r_0, \frac{dr} {dt} = 0 by construction
• if r were decreasing linearly over time, then plotting v over time on a linear scale would show that v decreases as the square root. However, a look at r vs. t shows that r does not decrease linearly, so v does not strictly go at the negative square root.

Velocity of outer radius over time. This plot shows that for initial concentric spheres of the same density, by varying outer radius, spheres with larger radii have steeper velocity profiles over time. Note this velocity is the velocity out the outer edge of the sphere. The free-fall time here, as can be seen where all of the curves begin to asymptote, is about t_{ff} \approx .24. The smallest sphere plotted has an initial radius of \frac{1}{1000}r_0 (this is the green line), whereas the velocity of the outer most sphere within initial radius r_0 is the blue line.