Gravitational Potential
We have seen that the force experienced by a point object of mass situatated in a gravitational field can be
written
|
(1.268) |
where is the object's gravitational potential energy. It is clear from Equation (1.264)
that
. It follows that the gravitational acceleration of the object,
, can
be written
|
(1.269) |
where
is independent of . The quantity
is known as gravitational potential.
From Equation (1.264), the gravitational potential due to a point object (or a spherically symmetric object) of mass situated at the origin
is
|
(1.270) |
Consider a collection of point objects. Let the th object have mass and displacement . Given that
gravity is a superposable force, the generalization of the previous equation is clearly
|
(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
, then the previous expression generalizes to give
|
(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
|
(1.273) |
Here,
|
(1.274) |
is known as the Laplacian of
.
(See Section A.21.) Equation (1.273), which specifies the gravitational potential,
, generated by a
continuous mass distribution of mass density
, is known as Poisson's equation. Of course,
Equation (1.272) is the integral form of Poisson's equation.