Gravitational potential

 ( Energy ) 

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. The reference point, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any distance. Their similarity is correlated with both associated fields having conservative forces.

Mathematically, 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.

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Potential energy The gravitational potential ( V) at a location is the gravitational potential energy ( U) at that location per unit mass:

$$V\; =\; \backslash frac\{U\}\{m\},$$

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: $$\backslash Delta\; U\; \backslash approx\; mg\; \backslash Delta\; h.$$

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Mathematical form The gravitational Scalar potential V at a distance x from a point particle 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:

(1995). 9780030973024, Harcourt Brace & Company. . ISBN 9780030973024

(2025). 9780080470696, Academic Press. . ISBN 9780080470696

(2025). 9781107697690, Cambridge University Press. . ISBN 9781107697690

(1993). 9780748715848, Nelson Thornes. . ISBN 9780748715848

$$V(\backslash mathbf\{x\})\; =\; \backslash frac\{W\}\{m\}\; =\; \backslash frac\{1\}\{m\}\; \backslash int\_\{\backslash infty\}^\{x\}\; \backslash mathbf\{F\}\; \backslash cdot\; d\backslash mathbf\{x\}\; =\; \backslash frac\{1\}\{m\}\; \backslash int\_\{\backslash infty\}^\{x\}\; \backslash frac\{G\; m\; M\}\{x^2\}\; dx\; =\; -\backslash frac\{G\; M\}\{x\},$$ 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 $$\backslash mathbf\{a\}\; =\; -\backslash frac\{GM\}\{x^3\}\; \backslash mathbf\{x\}\; =\; -\backslash frac\{GM\}\{x^2\}\; \backslash hat\{\backslash mathbf\{x\}\},$$ where x is a vector of length x pointing from the point mass toward the small body and $\backslash hat\{\backslash mathbf\{x\}\}$ is a unit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an inverse square law: $$\backslash |\backslash mathbf\{a\}\backslash |\; =\; \backslash frac\{GM\}\{x^2\}.$$

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, ..., x _n and have masses m₁, ..., m _n, then the potential of the distribution at the point x is $$V(\backslash mathbf\{x\})\; =\; \backslash sum\_\{i=1\}^n\; -\backslash frac\{Gm\_i\}\{\backslash |\backslash mathbf\{x\}\; -\; \backslash mathbf\{x\}\_i\backslash |\}.$$

If the mass distribution is given as a mass Borel measure dm on three-dimensional Euclidean space R³, then the potential is the convolution of with dm. In good cases this equals the integral $$V(\backslash mathbf\{x\})\; =\; -\backslash int\_\{\backslash R^3\}\; \backslash frac\{G\}\{\backslash |\backslash mathbf\{x\}\; -\; \backslash mathbf\{r\}\backslash |\}\backslash ,dm(\backslash mathbf\{r\}),$$ where 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 , where dv( r) is the Euclidean volume element, then the gravitational potential is the volume integral $$V(\backslash mathbf\{x\})\; =\; -\backslash int\_\{\backslash R^3\}\; \backslash frac\{G\}\{\backslash |\backslash mathbf\{x\}-\backslash mathbf\{r\}\backslash |\}\backslash ,\backslash rho(\backslash mathbf\{r\})dv(\backslash mathbf\{r\}).$$

If V is a potential function coming from a continuous mass distribution ρ( r), then ρ can be recovered using the Laplace operator, : $$\backslash rho(\backslash mathbf\{x\})\; =\; \backslash frac\{1\}\{4\backslash pi\; G\}\backslash Delta\; V(\backslash mathbf\{x\}).$$ 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. 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.

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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 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 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

Extract of page 68

(2025). 9781420067354, CRC Press. . ISBN 9781420067354

Extract of page 19 $$V(r)\; =\; \backslash frac\; \{2\}\{3\}\; \backslash pi\; G\; \backslash rho\; \backslash leftr^2\; =\; \backslash frac\{Gm\}\{2R^3\}\; \backslash leftr^2,\; \backslash qquad\; r\; \backslash leq\; R,$$ which differentiably connects to the potential function for the outside of the sphere (see the figure at the top).

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General relativity 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.

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Multipole expansion The potential at a point is given by $$V(\backslash mathbf\{x\})\; =\; -\; \backslash int\_\{\backslash R^3\}\; \backslash frac\{G\}$$\ dm(\mathbf{r}).

The potential can be expanded in a series of Legendre polynomials. Represent the points x and r as relative to the center of mass. The denominator in the integral is expressed as the square root of the square to give

\int_{\R^3} \frac{G} \sqrt{1 -2 \frac{r} \cos \theta + \left( \frac{r} \right)^2}\,dm(\mathbf{r}) \end{align} where, in the last integral, and is the angle between x and r.

(See "mathematical form".) The integrand can be expanded as a Taylor series in , by explicit calculation of the coefficients. A less laborious way of achieving the same result is by using the generalized binomial theorem. The resulting series is the generating function for the Legendre polynomials: $$\backslash left(1-\; 2\; X\; Z\; +\; Z^2\; \backslash right)\; ^\{-\; \backslash frac\{1\}\{2\}\}\; \backslash \; =\; \backslash sum\_\{n=0\}^\backslash infty\; Z^n\; P\_n(X)$$ valid for and . The coefficients P _n are the Legendre polynomials of degree n. Therefore, the Taylor coefficients of the integrand are given by the Legendre polynomials in . So the potential can be expanded in a series that is convergent for positions x such that for all mass elements of the system (i.e., outside a sphere, centered at the center of mass, that encloses the system):

\int \sum_{n=0}^\infty \left(\frac{r} \right)^n P_n(\cos \theta) \, dm(\mathbf{r})\\ &= - \frac{G} \int \left(1 + \left(\frac{r}\right) \cos \theta + \left(\frac{r}\right)^2\frac {3 \cos^2 \theta - 1}{2} + \cdots\right)\,dm(\mathbf{r}) \end{align} The integral $\backslash int\; r\; \backslash cos(\backslash theta)\; \backslash ,\; dm$ is the component of the center of mass in the direction; this vanishes because the vector x emanates from the center of mass. So, bringing the integral under the sign of the summation gives $$V(\backslash mathbf\{x\})\; =\; -\; \backslash frac\{GM\}$$ - \frac{G} \int \left(\frac{r}\right)^2 \frac {3 \cos^2 \theta - 1}{2} dm(\mathbf{r}) + \cdots

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.)

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Units and numerical values The SI unit of gravitational potential is square metre per square seconds (m²/s²) or, equivalently, joules per kilogram (J/kg). The absolute value of gravitational potential at a number of locations with regards to the mass of 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.

≥ 130 GJ/kg

≥ 130 GJ/kg

≥ 130 GJ/kg

≥ 130 GJ/kg

Compare the gravity at these locations.

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See also

• Applications of Legendre polynomials in physics
• Standard gravitational parameter ( GM)
• Geoid
• Geopotential
• Geopotential model

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Notes

• .
• Peter Rastall (1991). 9789810207786, World Scientific. ISBN 9789810207786
• .

Categories: Energy, Gravity, Potentials, Scalar Physical Quantities

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