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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 there's a simpler way that's also fine.
[Moderator's note: Moved from a technical forum and thus no template.]
$$U = -\alpha \frac{GM^2}{r}$$
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.]
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