Shell theorem

In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.

Isaac Newton proved the shell theorem and stated that:

A corollary is that inside a solid sphere of constant density, the gravitational force varies linearly with distance from the centre, becoming zero by symmetry at the centre of mass.

This is easy to see: take a point within such a sphere, at a distance ${\displaystyle r}$ from the centre of the sphere. Then you can ignore all the shells of greater radius, according to the shell theorem. So, the remaining mass ${\displaystyle m}$ is proportional to ${\displaystyle r^{3}}$, and the gravitational force exerted on it is proportional to ${\displaystyle m/r^{2}}$, so to ${\displaystyle r^{3}/r^{2}=r}$, so is linear in ${\displaystyle r}$.

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