Gravitational potential

Consider two point masses, $m$ and $m'$, located at position vectors ${\bf r}$ and ${\bf r}'$, respectively. According to the preceding analysis, the acceleration ${\bf g}$ of mass $m$ as a result of the gravitational force exerted on it by mass $m'$ takes the form

$${\bf g} = G\,m'\,\frac{{\bf r}'-{\bf r}}{\vert{\bf r}'-{\bf r}\vert^{\,3}}.$$ (3.3)

Now, the $x$-component of this acceleration is written

$$g_x = G\,m'\,\frac{x'-x}{[(x'-x)^2+(y'-y)^2+(z'-z)^2]^{\,3/2}},$$ (3.4)

where ${\bf r}=(x,\, y,\, z)$ and ${\bf r}' = (x',\,y',\,z')$. However, as is easily demonstrated,

$$\frac{x'-x}{[(x'-x)^2+(y'-y)^2+(z'-z)^2]^{\,3/2}}\equiv \frac{\partial}{\partial x}\!\left(\frac{1}{[(x'-x)^2+(y'-y)^2+(z'-z)^2]^{\,1/2}}\right).$$ (3.5)

Hence,

$$g_x = G\,m'\,\frac{\partial}{\partial x}\!\left(\frac{1}{\vert{\bf r}'-{\bf r}\vert}\right),$$ (3.6)

with analogous expressions for $g_y$ and $g_z$. It follows that

$${\bf g}= -\nabla{\mit\Phi},$$ (3.7)

where

$${\mit\Phi}({\bf r})= - \frac{G\,m'}{\vert{\bf r}'-{\bf r}\vert}$$ (3.8)

is termed the gravitational potential. Of course, we can write ${\bf g}$ in the form of Equation (3.7) only because gravity is a conservative force. (See Section 2.4.)

It is well known that gravity is a superposable force. In other words, the gravitational force exerted on some point mass by a collection of other point masses is simply the vector sum of the forces exerted on the former mass by each of the latter masses taken in isolation. It follows that the gravitational potential generated by a collection of point masses at a certain location in space is the sum of the potentials generated at that location by each point mass taken in isolation. Hence, using Equation (3.8), if there are $N$ point masses, $m_i$ (for $i=1, N$), located at position vectors ${\bf r}_i$, then the gravitational potential generated at position vector ${\bf r}$ is simply

$$\index{gravitational potential!of collection of point masses} {\mit\Phi}({\bf r}) = - G\sum_{i=1,N} \frac{m_i}{\vert{\bf r}_i-{\bf r}\vert}.$$ (3.9)

Suppose, finally, that, instead of having a collection of point masses, we have a continuous mass distribution. In other words, let the mass at position vector ${\bf r}'$ be $\rho({\bf r}')\,d^{\,3}{\bf r}'$, where $\rho({\bf r}')$ is the local mass density, and $d^{\,3}{\bf r}'$ a volume element. Summing over all space, and taking the limit $d^{\,3}{\bf r}'\rightarrow 0$, we find that Equation (3.9) yields

$${\mit\Phi}({\bf r}) = - G\int\frac{\rho({\bf r}')}{\vert{\bf r}'-{\bf r}\vert}\,d^{\,3}{\bf r}',$$ (3.10)

where the integral is taken over all space. This is the general expression for the gravitational potential, ${\mit\Phi}({\bf r})$, generated by a continuous mass distribution, $\rho({\bf r})$.