Deriving the gravitational binding energy of the cluster

[Arman777]

I am trying to derive the gravitational binding energy of the cluster. Its given as

$$U = -\alpha \frac{GM^2}{r}$$

Now for the derivation I started from
$$dU = -\frac{GM(r)dm}{r}$$

I I tried to write $dm = \rho(r)4 \pi r^2dr$ and do it from there but I could not do much. Any ideas how can I proceed ?
$$dU = -\int_0^R \frac{GM(r)}{r}\rho(r)4\pi r^2dr$$

If there's a simpler way that's also fine.

[Moderator's note: Moved from a technical forum and thus no template.]

 

[MathematicalPhysicist]

There's a similar to Poisson equation for Gravity, perhaps this will help you solve this question:
https://en.wikipedia.org/wiki/Poisson's_equation

 

[Arman777]

There's a similar to Poisson equation for Gravity, perhaps this will help you solve this question:
https://en.wikipedia.org/wiki/Poisson's_equation

Hmm I don't see how that can help ..

 

[MathematicalPhysicist]

To tell you the truth I don't understand what is exactly your problem?
I mean you want to get $U=-\alpha GM^2/r$ from $dU = -\int_0^R \frac{GM(r)}{r}\rho(r)4\pi r^2dr$, but you didn't state what are the explicit dependencies of $M(r)$ and $\rho(r)$, so how can you calculate this integral?!
I mean if they are constants with respect to r then it's a simple matter to calculate this integral, but there is insufficient data to compute this integral as I see it.

 

[Arman777]

To tell you the truth I don't understand what is exactly your problem?
I mean you want to get $U=-\alpha GM^2/r$ from $dU = -\int_0^R \frac{GM(r)}{r}\rho(r)4\pi r^2dr$, but you didn't state what are the explicit dependencies of $M(r)$ and $\rho(r)$, so how can you calculate this integral?!
I mean if they are constants with respect to r then it's a simple matter to calculate this integral, but there is insufficient data to compute this integral as I see it.

If its spherically symmetric ?

 

[MathematicalPhysicist]

You mean the density $\rho(r)$ is proportional to $r^{-3}$? or is it $M(r)$?
Anyway, if you state your problem with all the assumptions as clear as possible then I believe a solution is possible.

 

[Arman777]

You mean the density $\rho(r)$ is proportional to $r^{-3}$? or is it $M(r)$?
Anyway, if you state your problem with all the assumptions as clear as possible then I believe a solution is possible.

I seen some solution here but it does not look like my thing...

https://physics.stackexchange.com/q...otential-energy-of-any-spherical-distribution

Thats the problem. I mean I thought I mentioned the spherically symmetry thing. Which I noticed I did not so I said it. I just wondered is it derivable from a unknown matter density. If its not that's also okay.