BE Update

Plots of rho and {Grad(P)/rho - Grad(Phi)}:

Xi = 3, rho_weight = 1

Xi = 3, rho_weight = 10

Xi = 4, rho_weight = 1

Xi = 4, weight = 10

Xi = 5, weight = 10

Xi = 5.15, weight = 10

Xi = 5.2, weight = 10  --- UNSTABLE!

In these plots, the presence of a greater force inward outside of the sphere makes sense. Considering Gauss's law for gravity, we know that Fg ~  Menc/r^2, where Menc ~ r^3. Hence, Fq ~ r. This is like collapse of a uniform sphere, all the material in the sphere arrives at the center in the same time, i.e. material further out falls faster..