# Deriving the gravitational binding energy of the cluster

Gold Member
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 theres a simpler way thats also fine.

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## Answers and Replies

Gold Member
MathematicalPhysicist
Gold Member
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.

Gold Member
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
Gold Member
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.

Gold Member
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 thats also okay.