Gravitational potential
In classical mechanics, the gravitational potential is a scalar potential associating with each point in space the work (energy transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the conservative gravitational field.
It is analogous to the electric potential with mass playing the role of charge.
Their similarity is correlated with both associated fields having conservative forces.
It may also be used for solving the electrostatic and magnetostatic fields generated by uniformly charged or polarized ellipsoidal bodies.
Potential energy is equal (in magnitude, but negative) to the work done by the gravitational field moving a body to its given position in space from infinity.
So the potential can be interpreted as the negative of the work done by the gravitational field moving a unit mass in from infinity.
In some situations, the equations can be simplified by assuming a field that is nearly independent of position.
For instance, in a region close to the surface of the Earth, the gravitational acceleration, g, can be considered constant.
The product GM is the standard gravitational parameter and is often known to higher precision than G or M separately.
The potential has units of energy per mass, e.g., J/kg in the MKS system.
where |x − r| is the distance between the points x and r. If there is a function ρ(r) representing the density of the distribution at r, so that dm(r) = ρ(r) dv(r), where dv(r) is the Euclidean volume element, then the gravitational potential is the volume integral
This holds pointwise whenever ρ is continuous and is zero outside of a bounded set.
In general, the mass measure dm can be recovered in the same way if the Laplace operator is taken in the sense of distributions.
As a consequence, the gravitational potential satisfies Poisson's equation.
See also Green's function for the three-variable Laplace equation and Newtonian potential.
The integral may be expressed in terms of known transcendental functions for all ellipsoidal shapes, including the symmetrical and degenerate ones.
On the surface of the earth, the acceleration is given by so-called standard gravity g, approximately 9.8 m/s2, although this value varies slightly with latitude and altitude.
The magnitude of the acceleration is a little larger at the poles than at the equator because Earth is an oblate spheroid.
Within a uniform spherical body of radius R, density ρ, and mass m, the gravitational force g inside the sphere varies linearly with distance r from the center, giving the gravitational potential inside the sphere, which is[7][8]
which differentiably connects to the potential function for the outside of the sphere (see the figure at the top).
In general relativity, the gravitational potential is replaced by the metric tensor.
When the gravitational field is weak and the sources are moving very slowly compared to light-speed, general relativity reduces to Newtonian gravity, and the metric tensor can be expanded in terms of the gravitational potential.
Represent the points x and r as position vectors relative to the center of mass.
The integrand can be expanded as a Taylor series in Z = r/|x|, by explicit calculation of the coefficients.
A less laborious way of achieving the same result is by using the generalized binomial theorem.
[10] The resulting series is the generating function for the Legendre polynomials:
So the potential can be expanded in a series that is convergent for positions x such that r < |x| for all mass elements of the system (i.e., outside a sphere, centered at the center of mass, that encloses the system):
This shows that elongation of the body causes a lower potential in the direction of elongation, and a higher potential in perpendicular directions, compared to the potential due to a spherical mass, if we compare cases with the same distance to the center of mass.
(If we compare cases with the same distance to the surface, the opposite is true.)
The absolute value of gravitational potential at a number of locations with regards to the gravitation from [clarification needed] the Earth, the Sun, and the Milky Way is given in the following table; i.e. an object at Earth's surface would need 60 MJ/kg to "leave" Earth's gravity field, another 900 MJ/kg to also leave the Sun's gravity field and more than 130 GJ/kg to leave the gravity field of the Milky Way.
