Studying the gravitational potential of the BE sphere/ambient system

What should the global gravitational potential, phi, of the BE sphere/ambient system look like in a) a "light" ambient medium and b) a "matched" ambient medium?

To answer this question, first some preliminaries:

1) How massive is the critical BE sphere in my simulations compared to the ambient medium?

• A) The BE sphere is 150 solar masses. The mass of the ambient medium then is given by the equation:

4\over3} \pi \rho ({r^3-Rbe^3})

Where rho is the uniform ambient density, and we are approximating the box as a sphere. Note, when these quantities are in computational units, one converts to mass in cgs, as defined in the BE problem module, by multiplying this equation by Mscale found in scales.data.

By this method, I found the mass in the light ambient medium to be:

= {0.0007 * {4 \over 3} \pi ({15.4^3 -1 })} = 10.7 * Mscale = 4.77 * 10^36 g = 2,248 Solar Masses

That is, Mamb~14Mbe.

Since in the matched case, the density becomes 0.07, Mamb=100*Mamb_light~1,400 Mbe. Or, in astronomical units, Mamb_matched=224,800 Solar Mass!

2. What is a theoretical approximation to the potential of the sphere outside of Rbe?

• A) We can look at this problem in 2 ways that give the same answer. First we can add the potential of the BE sphere as though it were a point charge with M=Mbe to the potential of the ambient, approximated as a uniform sphere of r=l/2, where l is box length:

phi = -{G*Mbe \over r } - {G*Mr \over r} \space (for \space r > Rbe)

Here we say

r = Mass \space enclosed = {4 \over 3} * \pi * \rho * {(r^3 - Rbe^3)}

⇒

phi = -{G \over r} [Mbe + {4 \over 3} * \pi *\rho ({r^3-Rbe^3})]

From this we can that when rho is small in ambient, phi is dominated by the first term. That is phi can be approximated as due to the BE sphere as a point gravity source. For non-negligible rho, however, we can expect the r² term to dominate at large r. This would make phi more and more negative.

We can arrive at this same equation by taking a slightly different conceptual route. We can consider phi in ambient medium to be due to the superposition of a uniform sphere on top of a point mass object. In pictures, this is like:

In equations, this is like:

Point:

= Mbe - {4 \over 3} * \pi * Rbe^3 * \rho

Total Phi:

phi = - \space{ G {(Mbe - {4 \over 3} * \pi* \rho * Rbe^3)} \over r} + {-G * {4 \over 3} * \pi * r^3 * \rho \over r}

Some manipulation shows this to be the equation as above.

Comparing the function for phi with simulation data

Now, to see how this equation compares in the different cases let's look at lineouts of pseudo-color plots of rho in the different cases.

Light Case

Here is the pseudocolor plot of phi for the light ambient case. We would expect that outside of Rbe, the potential should resemble point gravity phi especially for small r.

Ah, that looks about right.

Here is the function of phi as found above:

.

I see that for small r, there behavior is qualitatively similar, and on the same order. The theoretical curve, considering both phi from a point source and phi from a uniform sphere seems to fail at large r. This indicates that the ambient is not sufficiently important at this density to effect the point gravity potential beyond Rbe. That is, phi is well approximated by simply a point source object of mass on the order of Mbe for all r within simulation box.

Matched Case

From my sim:

Here is the function of phi as found above:

.

Now, this is where things get hard for me to understand. First, I see the range of phi, and magnitude of phi is drastically different for this matched ambient case. This may make sense, based on more mass in the ambient meaning stronger gravitational potential due to the ambient. However, I would expect that at very small radii, just above Rbe, the approximation of the ambient as a uniform sphere should be very good. This is because there is SOOOO much more mass in the ambient than in the sphere (see above calculation). Thus, very early on, I would expect the potential to be growing more and more negative as more and more mass is enclosed in this uniform sphere. Indeed, we see this behavior from the function derived above, but NOT in the simulation. Can someone please explain this discrepancy?