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

$$V(\mathbf{r})=-f\sum _{i=1}^N\frac{m_i}{|\mathbf{r}-{\mathbf{r}}_i},\text{for}|\mathbf{r}-{\mathbf{r}}^{\prime }|\ne 0.$$

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

$$V(\mathbf{r})=-f\underset{G}{\int }\frac{\mathrm{d}m({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}=-f\underset{G}{\int }\frac{\rho ({\mathbf{r}}^{\prime }){\mathrm{d}}^3{\mathbf{r}}^{\prime }}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}$$

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.