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)

In addition,

• 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 integrating the equation. Since the mass is constant in any shell (by the way, 'shell' here is analogous to a 'concentric sphere'), we can replace M_r in the equation by M_r = \frac{4}{3} \pi r_0^3 \rho_0 when integrating this equation for a sphere of initial radius r0. Integrating this equation under the boundary conditions: the initial velocity is zero, \frac{dr}{dt}|_{t=0}=0, and this occurs when r=r_0, leads to equations that describe the position of the radius of the 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, and a numerical solution is necessary (i.e. a closed form solution DNE). Thus, we must find discrete solutions to this equation. For this, we can use mathematica 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 corresponding to a list of discrete t:

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

These lists can then be plotted in mathematica. For example, here is the position over time of some concentric spheres of the same initial density:

Radius of concentric spheres over time. This plot shows the radius of a collapsing sphere over time. The blue line shows the radius of a sphere with initial radius r_0=10, the yellow line is a sphere of same initial density but half the radius (and thus can be envisioned as a smaller, inner, concentric sphere), 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 discrete values of r(t) and t in the velocity equation, we can 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 velocity of a sphere at its outer most edge 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 with time, then plotting v over time (on a linear scale) would show that v decreases as the negative square root. However, a look at r vs. t shows that r does not decrease linearly, so v does not strictly go as the negative square root:

Velocity of outer radius over time. This plot shows that for concentric spheres of the same initial density, spheres with larger radii have steeper velocity profiles over time (again, this is the velocity at the outer edge of the spheres). 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 sphere with initial radius r_0 is the blue line. Note this shows all shells continue to accelerate over time and that the shells that are furthest away 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 it in terms of radius, since r is a function of time. Here that is shown for the sphere with initial radius, r=r_0:

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 from the v vs. t plot. These numbers will match what you are seeing in this plot.

Now, all this beckons — what is the radial velocity profile of a collapsing uniform density 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 (r=r_0) 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 shrank, 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 spheres, one with initial radius r=r_0=10 and one with r=r_0/2=5. They are exactly the same, so you only see one line. That is, the density grows uniformly everywhere inside a collapsing uniform density sphere, and the average density is the exact density at any radius.

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), 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.

Here is another plot of density (note, the plot range is different here from the previous plot, to show the first time states in greater detail), now as a function of radius with the different lines corresponding to different times. This plot confirms that over time the density increases uniformly throughout the sphere. (Again, time is tracked by the shrinking radius of the sphere):

Profiles together of a uniform density collapsing sphere

Here is a plot showing the key temporal profiles of a collapsing uniform sphere. They thus show the given quantity at the edge of the collapsing sphere, over time:

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

Ram pressure

With the density over time, and the velocity profiles over time, I can get a ram pressure radial profile over time of a collapsing uniform sphere. This is what I was after all along, as I want to match this to the outward ram pressure of the splashed region in the MHD colliding flows runs.

Here she is:

Ram pressure profiles over time of a collapsing uniform sphere. You can track the passage of time by the shrinking outer radius of the sphere, and the growing ram pressure. A horizontal line (from the shock conditions of the flow) will be placed on this plot, and where it intersects the ram pressure profile(s) will be a prediction of where the ring will form in the splash region of the sims (​see description here). If the horizontal line were to cross 2 profiles, we would have to check whether given the speed of the splashed material, if there has been time enough for that material to have reached the splash radius.

Note, the full plot range for the ram pressure plot is not shown here — if it were the tremendous growth of density and velocity as the sphere approached the freefall time would overwhelm everything else.