Inclassical mechanics, thegravitational potential is ascalar potential associating with eachpoint in space thework (energy transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the conservativegravitational field. It isanalogous to theelectric potential withmass playing the role ofcharge. The reference point, where the potential is zero, is by conventioninfinitely far away from any mass, resulting in a negative potential at anyfinite distance. Their similarity is correlated with both associatedfields havingconservative forces.
Mathematically, the gravitational potential is also known as theNewtonian potential and is fundamental in the study ofpotential 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 gravitationalpotential energy (U) at that location per unit mass:
wherem 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, thegravitational 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:
The gravitationalpotentialV at a distancex from apoint mass of massM can be defined as the workW that needs to be done by an external agent to bring a unit mass in from infinity to that point:[2][3][4][5]
whereG is thegravitational constant, andF is the gravitational force. The productGM is thestandard gravitational parameter and is often known to higher precision thanG orM separately. The potential hasunits of energy per mass, e.g., J/kg in theMKS system. By convention, it is always negative where it is defined, and asx tends to infinity, it approaches zero.
Thegravitational field, and thus the acceleration of a small body in the space around the massive object, is the negativegradient 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 iswherex is a vector of lengthx pointing from the point mass toward the small body and is aunit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows aninverse square law:
The potential associated with amass 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 pointsx1, ...,xn and have massesm1, ...,mn, then the potential of the distribution at the pointx is
If the mass distribution is given as a massmeasuredm on three-dimensionalEuclidean spaceR3, then the potential is theconvolution of−G/|r| withdm.[citation needed] In good cases[clarification needed] this equals the integralwhere|x −r| is thedistance between the pointsx andr. If there is a functionρ(r) representing the density of the distribution atr, so thatdm(r) =ρ(r)dv(r), wheredv(r) is the Euclideanvolume element, then the gravitational potential is thevolume integral
IfV is a potential function coming from a continuous mass distributionρ(r), thenρ can be recovered using theLaplace operator,Δ:This holds pointwise wheneverρ is continuous and is zero outside of a bounded set. In general, the mass measuredm can be recovered in the same way if the Laplace operator is taken in the sense ofdistributions. As a consequence, the gravitational potential satisfiesPoisson's equation. See alsoGreen's function for the three-variable Laplace equation andNewtonian potential.
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 (seereference 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 constantG, with 𝜌 being a constantcharge density) to electromagnetism.
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 apoint mass, by theshell theorem. On the surface of the earth, the acceleration is given by so-calledstandard 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 anoblate spheroid.
Within a spherically symmetric mass distribution, it is possible to solvePoisson's equation in spherical coordinates. Within a uniform spherical body of radiusR, density ρ, and massm, the gravitational forceg inside the sphere varies linearly with distancer 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).
Ingeneral relativity, the gravitational potential is replaced by themetric 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 at a pointx is given by
The potential can be expanded in a series ofLegendre polynomials. Represent the pointsx andr asposition vectors relative to thecenter of mass. The denominator in the integral is expressed as the square root of the square to givewhere, in the last integral,r = |r| andθ is the angle betweenx andr.
(See§ Mathematical form.) The integrand can be expanded as aTaylor series inZ =r/|x|, by explicit calculation of the coefficients. A less laborious way of achieving the same result is by using the generalizedbinomial theorem.[10] The resulting series is thegenerating function for the Legendre polynomials:valid for|X| ≤ 1 and|Z| < 1. The coefficientsPn are the Legendre polynomials of degreen. Therefore, the Taylor coefficients of the integrand are given by the Legendre polynomials inX = cos θ. So the potential can be expanded in a series that is convergent for positionsx such thatr < |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 thex direction; this vanishes because the vectorx 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 thesurface, the opposite is true.)
TheSI unit of gravitational potential is thesquare metre per square second (m2/s2) or, equivalently, the joule per kilogram (J/kg).Theabsolute value of gravitational potential at a number of locations with regards to the mass of theEarth, theSun, and theMilky 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 theescape velocity.
| Location | with respect to | ||
|---|---|---|---|
| Earth | Sun | Milky Way | |
| Earth's surface | 60 MJ/kg | 900 MJ/kg | ≥ 130 GJ/kg |
| LEO | 57 MJ/kg | 900 MJ/kg | ≥ 130 GJ/kg |
| Voyager 1 (17,000 million km from Earth) | 23 J/kg | 8 MJ/kg | ≥ 130 GJ/kg |
| 0.1light-year from Earth | 0.4 J/kg | 140 kJ/kg | ≥ 130 GJ/kg |
Compare thegravity at these locations.
