gravitational field

The gravitational field at any point \[P\] in space is defined as the gravitational force felt by a tiny unit mass (or test particle) placed at \[P\]. Mathematically, it can be defined using Newton's law of universal gravitation, i.e. \[\mathbf{g}=\frac{\mathbf{F}}{m}=-\frac{GM}{\left\lVert \mathbf{r} \right\rVert^{2}}\hat{\mathbf{r}}\] where \[M\] is the mass of the object whose field we are concerned with.

We say "a tiny unit mass" because we don't want the gravitational field from the test mass itself to disturb the system.

Since gravitational force satisfies the superposition principle (most of the time, if we're not considering extreme cases like black holes), the field around multiple particles is simply the vector sum of the fields around each individual particle.

Let \[M_{i}\] be the mass of particle \[i\] and \[\mathbf{R}_{i}=\mathbf{r}-\mathbf{r}_{i}\] (i.e. the displacement vector pointing from particle \[i\] to a test particle at position \[\mathbf{r}\]),

\begin{align*} \mathbf{g}&=\sum_{i}\mathbf{g}_{i}\\ &=\sum_{i}\frac{\mathbf{F}_{i}}{m}\\ &=\sum_{i}\left( -G \frac{M_{i}}{\left\lVert \mathbf{R}_{i} \right\rVert^{2}}\hat{\mathbf{R}}_{i} \right)\\ &=-G\sum_{i}\frac{M_{i}}{\left\lVert \mathbf{R}_{i} \right\rVert^{2}}\hat{\mathbf{R}}_{i} \end{align*}