Gravitational Potential

We have seen that the force experienced by a point object of mass $m$ situatated in a gravitational field can be written

$${\bf f} = -\nabla U,$$ (1.268)

where $U$ is the object's gravitational potential energy. It is clear from Equation (1.264) that $U\propto m$. It follows that the gravitational acceleration of the object, ${\bf g}={\bf f}/m$, can be written

$${\bf g} = -\nabla{\mit\Phi},$$ (1.269)

where ${\mit\Phi}=U/m$ is independent of $m$. The quantity ${\mit\Phi}$ is known as gravitational potential.

From Equation (1.264), the gravitational potential due to a point object (or a spherically symmetric object) of mass $M$ situated at the origin is

$${\mit\Phi}({\bf r}) = -\frac{G\,M}{r}.$$ (1.270)

Consider a collection of $N$ point objects. Let the $i$th object have mass $m_i$ and displacement ${\bf r}_i$. Given that gravity is a superposable force, the generalization of the previous equation is clearly

$${\mit\Phi}({\bf r})=- \sum_{i=1,N} \frac{G\,m_i}{\vert{\bf r}-{\bf r}_i\vert}.$$ (1.271)

Moreover, the gravitational acceleration due to the collection of objects is again given by Equation (1.269). Finally, if, instead of having a collection of point mass objects, we have a continuous mass distribution characterized by a mass density $\rho({\bf r})$, then the previous expression generalizes to give

$${\mit\Phi}({\bf r}) = -\int\frac{G\,\rho({\bf r}')}{\vert{\bf r}-{\bf r}'\vert}\,dV',$$ (1.272)

where the integral is over all space.

Equation (1.269) can be combined with the differential form of Gauss's theorem, (1.249), to give

$$\nabla^2{\mit\Phi} = 4\pi\,G\,\rho.$$ (1.273)

Here,

$$\nabla^2{\mit\Phi}\equiv \nabla\cdot\nabla{\mit\Phi} = \frac{\par... ...i}}{dx^2} +\frac{\partial^2{\mit\Phi}}{dy^2}+\frac{\partial^2{\mit\Phi}}{dz^2},$$ (1.274)

is known as the Laplacian of ${\mit\Phi}$. (See Section A.21.) Equation (1.273), which specifies the gravitational potential, ${\mit\Phi}({\bf r})$, generated by a continuous mass distribution of mass density $\rho({\bf r})$, is known as Poisson's equation. Of course, Equation (1.272) is the integral form of Poisson's equation.