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.