"Gravity potential" redirects here. For Earth's gravity potential, see
Geopotential. For the field of gravity potentials, see
Gravitational field.
In
classical mechanics, the gravitational potential is a
scalar field 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. It is
analogous to the
electric potential with
mass playing the role of
charge. The reference point, where the potential is zero, is by convention
infinitely far away from any mass, resulting in a negative potential at any
finite distance.
In mathematics, the gravitational potential is also known as the
Newtonian potential and is fundamental in the study of
potential theory. It may also be used for solving the electrostatic and magnetostatic fields generated by uniformly charged or polarized ellipsoidal bodies.[1]
The gravitational potential (V) at a location is the gravitational
potential energy (U) at that location per unit mass:
where m is the mass of the object. 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. If the body has a mass of 1 kilogram, then the potential energy to be assigned to that body is equal to the gravitational potential. 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. In that case, the difference in potential energy from one height to another is, to a good approximation, linearly related to the difference in height:
Mathematical form
The gravitational
potentialV at a distance x from a
point mass of mass M can be defined as the work W that needs to be done by an external agent to bring a unit mass in from infinity to that point:[2][3][4][5]
where G is the
gravitational constant, and F is the gravitational force. 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. By convention, it is always negative where it is defined, and as x tends to infinity, it approaches zero.
The
gravitational field, and thus the acceleration of a small body in the space around the massive object, is the negative
gradient of the gravitational potential. Thus the negative of a negative gradient yields positive acceleration toward a massive object. Because the potential has no angular components, its gradient is
where x is a vector of length x pointing from the point mass toward the small body and is a unit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an
inverse square law:
The potential associated with a
mass distribution is the superposition of the potentials of point masses. If the mass distribution is a finite collection of point masses, and if the point masses are located at the points x1, ..., xn and have masses m1, ..., mn, then the potential of the distribution at the point x is
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
If V is a potential function coming from a continuous mass distribution ρ(r), then ρ can be recovered using the
Laplace operator, Δ:
The integral may be expressed in terms of known transcendental functions for all ellipsoidal shapes, including the symmetrical and degenerate ones.[6] These include the sphere, where the three semi axes are equal; the oblate (see
reference ellipsoid) and prolate spheroids, where two semi axes are equal; the degenerate ones where one semi axes is infinite (the elliptical and circular cylinder) and the unbounded sheet where two semi axes are infinite. All these shapes are widely used in the applications of the gravitational potential integral (apart from the constant G, with 𝜌 being a constant charge density) to electromagnetism.
Spherical symmetry
A spherically symmetric mass distribution behaves to an observer completely outside the distribution as though all of the mass was concentrated at the center, and thus effectively as a
point mass, by the
shell theorem. On the surface of the earth, the acceleration is given by so-called
standard gravityg, 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 spherically symmetric mass distribution, it is possible to solve
Poisson's equation in spherical coordinates. 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.[9]
The potential can be expanded in a series of
Legendre polynomials. Represent the points x and r as
position vectors relative to the center of mass. The denominator in the integral is expressed as the square root of the square to give
where, in the last integral, r = |r| and θ is the angle between x and r.
(See "mathematical form".) 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:
valid for |X| ≤ 1 and |Z| < 1. The coefficients Pn are the Legendre polynomials of degree n. Therefore, the Taylor coefficients of the integrand are given by the Legendre polynomials in X = cos θ. 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):
The integral is the component of the center of mass in the x direction; this vanishes because the vector x emanates from the center of mass. So, bringing the integral under the sign of the summation gives
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.)
Numerical values
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. The potential is half the square of the
escape velocity.
Vladimirov, V. S. (1971), Equations of mathematical physics, Translated from the Russian by Audrey Littlewood. Edited by Alan Jeffrey. Pure and Applied Mathematics, vol. 3, New York: Marcel Dekker Inc.,
MR0268497.
Wang, W. X. (1988). "The potential for a homogeneous spheroid in a spheroidal coordinate system. I. At an exterior point". J. Phys. A: Math. Gen. 21 (22): 4245–4250.
Bibcode:
1988JPhA...21.4245W.
doi:
10.1088/0305-4470/21/22/026.
Milon, T. (1990). "A note on the potential of a homogenous ellipsoid in ellipsoidal coordinates". J. Phys. A: Math. Gen. 23 (4): 581–584.
doi:
10.1088/0305-4470/23/4/027.
Thornton, Stephen T.; Marion, Jerry B. (2003), Classical Dynamics of Particles and Systems (5th ed.), Brooks Cole,
ISBN978-0-534-40896-1.
Zhu, Lupeia (1988).
"Gravity and Earth's Density Structure". Department of Earth and Atmospheric Sciences. EAS-437 Earth Dynamics. Saint Louis University. California Institute of Technology. Retrieved 2009-03-25.