5.4.1: Field of a Point Mass

Equation 5.3.1, together with the definition of field strength as the force experienced by unit mass, means that the field at a distance $r$ from a point mass $M$ is

$$g = \frac{GM}{r^2} \quad \text{N kg}^{-1} \text{ or m s}^{-2} \label{5.4.1} \tag{5.4.1}$$

In vector form, this can be written as

$$\textbf{g} = -\frac{GM}{r^2} \hat{\textbf{r}} \quad \text{N kg}^{-1} \text{ or m s}^{-2} \label{5.4.2} \tag{5.4.2}$$

Here $\hat{\textbf{r}}$ is a dimensionless unit vector in the radial direction.

It can also be written as

$$\textbf{g} = -\frac{GM}{r^3} \textbf{r} \quad \text{N kg}^{-1} \text{ or m s}^{-2} \label{5.4.3} \tag{5.4.3}$$

Here $\textbf{r}$ is a vector of magnitude $r$ − hence the $r^3$ in the denominator.