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 the theoretical potential of the sphere outside of the 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.