# Derivation of the potential of a sphere

1. May 13, 2005

### Jayse_83

Hi,

I've been told in a lecture course that the potential used in the virial theorem (for our application) is 3/5 GM/r. This describes the potential for a sphere of radius r, mass M created by bringing infinitly thin shells from infinity to form the next 'layer' of the sphere. I am having difficulty deriving this for myself, anybody wana give it a try ??

2. May 13, 2005

### SpaceTiger

Staff Emeritus
For a constant density sphere, that's:

$$E=-\frac{3}{5}\frac{GM^2}{R}$$

$$E=-\int_0^R \frac{GM_r}{r}dm$$

$$M_r=\frac{4}{3}\pi r^3\rho$$

$$dm=4\pi r^2\rho dr$$

$$E=-\frac{16\pi^2G\rho^2}{3}\int_0^R r^4dr=-\frac{16\pi^2G\rho^2}{15}R^5$$

$$M=\frac{4}{3}\pi R^3\rho$$

$$E=-\frac{3}{5}\frac{GM^2}{R}$$

Last edited: May 13, 2005
3. May 13, 2005

### whozum

Spacetiger: Where do the first two lines follow from?

4. May 13, 2005

### SpaceTiger

Staff Emeritus
The first one is just correcting the result. I'm pretty sure he had mis-typed it. The second is summing over the potentials of spherical shells at a radius r and with a width of dr (brought in from infinity and assuming the potential is zero at infinity).

Last edited: May 13, 2005
5. May 13, 2005

### Jayse_83

yep i missed the squared out ... thanks for your help :)