4  Mass distribution

The gravitational potential of a spatial mass distribution follows the principle of linear superposition.

Figure 4.1

The potential can be composed as the sum of the individual potentials of point masses

\[ V(\vb r) = -f \sum\limits_{i=1}^N \frac{m_i}{|\vb r - \vb r_i}, \qfor |\vb r - \vb r'| \ne 0. \]

For a continuous mass distribution with density \(\rho(\vb r)\), a mass \(\dd m(\vb r) = \rho(\vb r) \dd{^3 {\vb r}}\) must be assigned to each volume element. The summation gets replaced by an integration

\[ V(\vb r) = -f \int\limits_G \frac{\dd m (\vb r')}{| \vb r - \vb r'|} = -f \int\limits_G \frac{ \rho(\vb r') \dd{^3 {\vb r'}}}{| \vb r - \vb r'|} \]

This is Newton’s volume potential.

In practical applications the difficulty is to integrate over non-trivial, complicated geometries of the domain \(G\).

In the next section we demonstrate the integration over spherical shells and spheres.