- #1
SHISHKABOB
- 541
- 1
Homework Statement
It wants me to get the density function of a Plummer sphere from its gravitational potential.
Homework Equations
Plummer sphere potential:
[itex]\Phi (r) = -\frac{GM}{\sqrt{r^{2}+a^{2}}}[/itex]
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
[itex]\rho = \frac{3a^{2}}{4\pi}\frac{M}{(r^{2} + a^{2})^{5/2}}[/itex]
The Attempt at a Solution
So according to Poisson's equation
[itex]\nabla ^{2} \Phi = 4\pi G \rho (x)[/itex]
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
[itex]2rGM(r^{2} + a^{2})^{-1/2}[/itex]
and then the second derivative got me
[itex]\frac{GM}{2}\left[2(r^{2} + a^{2})^{-3/2} - (r^{2} + a^{2})^{-1/2}\right][/itex]
then after a bit of rearranging I have ended up with
[itex]\frac{M}{8\pi}(r^{2} + a^{2})^{-1/2}(2 - \frac{1}{r^{2} + a^{2}})[/itex]
I'm not really sure if I'm on the right track...