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})]