Plummer Sphere, Density from Potential

[SHISHKABOB]

Homework Statement

It wants me to get the density function of a Plummer sphere from its gravitational potential.

Homework 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

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

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...

 

[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).

 

[SHISHKABOB]

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).

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