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, that is, \frac{dr^2}{dt^2}\propto 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.

Position equation

Integrating the above equation twice with a change of variables yields the equation of motion:

\xi + \frac{1}{2} \sin(2\xi) = kt

where

k=\sqrt{\frac{8 \pi}{3} G \rho_0}

and

\frac{r}{r_0}=\cos^2(\xi)

Now, that equation of motion is a real hassle to deal with, so numerical solution is necessary. This, we can use mathematica for to find the roots of this equation in terms of \xi,t. Once we have \xi,t pairs, we can take the \cos^2(\xi)*r_0 to get a list of values for r for a list of discrete t.

\boxed{r(t) = r_0 \cos^2(\xi(t))}

These lists can then be plotted in mathematica:

Radius of concentric spheres over time. This plot shows the size of the outer radius of a collapsing sphere over time. The blue line shoes the outer most radius, r_0=10, the yellow is a sphere of same initial density but half the radius, and so on. From this plot, we can see that the shells do not cross over time, that the collapse speeds up over time, and that they all collapse to 0 radius in a free fall time.

Next, by using the values of r(t),t in the velocity equation, we can see study the velocity profiles of collapsing uniform spheres.

Velocity equation

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

• describes the outer velocity of a sphere that contains mass M_r as a function of r(t), where r(t) is a list of discrete radii as described above
• 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. Note that this shows that all shells continue to accelerate over time, that the furthest away shells have the highest initial acceleration, all consistent with the discussion above.

Now, instead of seeing this plot in terms of time, we can instead put in terms of radius, since r is a function of time.

Velocity of outer radius as a function of the outer radius. That is to say, "when the outer radius has fell to a new position of "r outer", the velocity of that outer radius is "v outer" How can you check this plot? You would look at the r vs. t plot, see at what time does the radius reach X, and then read off the velocity at that time. These numbers will match what you are seeing in this plot.

Now, all this beckons — what is the radial velocity profile of the collapsing sphere over time? This should be roughly vertical lines in the velocity of outer radius over time plots, and after some fancy coding in mathematica (see attached notebook if interested), one can arrive at this beauty:

Radial velocity of collapsing uniform density sphere over time At t=0, the sphere is static, and the velocity of the sphere is everywhere 0 (blue line). At the next time, we see that the radius of the sphere has shrunk, and that the velocity of the gas in the sphere is fastest at the outer edge, but goes to 0 linearly as you approach the origin (orange line). Each of these lines are evenly spaced in time, which shows the growing acceleration of the sphere. The red line is approaching the free fall time. As you can see from the velocity equation, at this point r→0, and we are getting a singularity in the velocity. This is why the line is tending toward an infinite slope.

Density

The average density of the collapsing sphere is:

\rho(t) = \frac{{M_r}}{\frac{4}{3}\pi r(t)^3}

Plotting this over time for spheres of various initial radii shows,

Density of collapsing sphere over time. Shown here are 2 overlaid initial spheres, one with radius r=r_0=10 and one with r=r_0/2=5. They are exactly the same, so you only see one line.

This plot shows that the entire interior of the sphere experiences a homologous (uniform) increase in density. The density increases everywhere exactly the same! Furthermore, the density increases by many orders of magnitude over the period of collapse. Assuming the collapse is isothermal (which is a good approximation over much of the collapse of protostars), then this implies that the Jeans mass decreases drastically over the period of collapse. This means that local regions of the collapsing sphere themselves will begin collapsing, and the whole cloud can be expected to fragment.

Profiles together of a collapsing sphere

Here is a plot showing the profiles of a collapsing uniform sphere:

Blue = acceleration, gold = velocity, green = position, red = density