Plummer Sphere, Density from Potential

1. Feb 26, 2013

SHISHKABOB

1. The problem statement, all variables and given/known data
It wants me to get the density function of a Plummer sphere from its gravitational potential.

2. Relevant equations
Plummer sphere potential:

$\Phi (r) = -\frac{GM}{\sqrt{r^{2}+a^{2}}}$

where phi is the potential as a function of radius from the mass, M. And a is a scale factor of the model. I think I'm just supposed to take M as a constant here.

I am supposed to end up with

$\rho = \frac{3a^{2}}{4\pi}\frac{M}{(r^{2} + a^{2})^{5/2}}$

3. The attempt at a solution

So according to Poisson's equation

$\nabla ^{2} \Phi = 4\pi G \rho (x)$

So to solve for ρ I just took the derivative of Phi twice with respect to r twice and then divided by 4πG

first derivative got me

$2rGM(r^{2} + a^{2})^{-1/2}$

and then the second derivative got me

$\frac{GM}{2}\left[2(r^{2} + a^{2})^{-3/2} - (r^{2} + a^{2})^{-1/2}\right]$

then after a bit of rearranging I have ended up with

$\frac{M}{8\pi}(r^{2} + a^{2})^{-1/2}(2 - \frac{1}{r^{2} + a^{2}})$

I'm not really sure if I'm on the right track...

2. Feb 26, 2013

voko

The Laplacian is not "taking the derivative with respect to r twice". Look up its expression in spherical coordinates (or derive it, if you are not supposed to know).

3. Feb 26, 2013

SHISHKABOB

ohhh yes you are right, thank you. That explains a lot