- #1

Arman777

Gold Member

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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.

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

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.

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