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In celestial mechanics, the gravitational potential belonging to an individual massive object, such as the earth or the sun, is a scalar field generated by that object, see also gravitational well. It is in every way analogous to the electric potential with mass playing the role of charge; the gravitational field is analogous to the electric field, and so are their mathematics and properties.

In mathematics the gravitational potential is also known as the Newtonian potential and is fundamental in the study of potential theory.

Mathematical form

The potential V at a distance r from a point mass of mass M is

${\displaystyle V=-{\frac {GM}{r}},}$

where G is the gravitational constant. V has units of energy per unit mass. V is always negative where it is defined, and as r 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. Because the potential has no angular components, its gradient is:

${\displaystyle \mathbf {a} =-{\frac {GM}{r^{3}}}\mathbf {r} =-{\frac {GM}{r^{2}}}{\hat {\mathbf {r} }},}$

where r is a vector of length r pointing from the point mass towards the small body and ${\displaystyle {\hat {\mathbf {r} }}}$ is a unit vector pointing from the point mass towards the small body. The magnitude of the acceleration therefore follows an inverse square law:

${\displaystyle |\mathbf {a} |={\frac {GM}{r^{2}}}.}$

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:

${\displaystyle V(\mathbf {x} )=\sum _{i=1}^{n}-{\frac {gm_{i}}{|\mathbf {x} -\mathbf {x_{i}} |}}.}$

If the mass distribution is given as a mass measure dm on three-dimensional Euclidean space R, then the potential is the convolution of −G/|y| with dm. In good cases this equals the integral

${\displaystyle V(\mathbf {x} )=-\int _{\mathbf {R} ^{3}}{\frac {G}{|\mathbf {x} -\mathbf {y} |}}\,dm(\mathbf {y} ).}$

If there is a continuous function ρ(x) representing the density of the distribution at x, so that dm = ρ(x)dx, where dx is the Euclidean volume element, then the gravitational potential is

${\displaystyle V(\mathbf {x} )=-\int _{\mathbf {R} ^{3}}{\frac {G}{|\mathbf {x} -\mathbf {y} |}}\,\rho (\mathbf {y} )\,d^{3}\mathbf {y} .}$

If V is a potential function coming from a continuous mass distribution ρ(x), then ρ can be recovered using the Laplace operator Δ using the formula:

${\displaystyle \rho (\mathbf {x} )={\frac {1}{4\pi G}}\Delta V(\mathbf {x} ),}$

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. Consequently the gravitational potential satisfies Poisson's equation. See also Green's function for the three-variable Laplace equation and Newtonian potential.

Gravitational potential expanded in series of Legendre polynomials

The potential at a point x is given by

${\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{|\mathbf {x} -\mathbf {y} |}}\ dm(\mathbf {y} ).}$

The potential can be expanded in a series of Legendre polynomials. For simplicity, assume that an origin is chosen to coincide with the center of mass of the system. Expanding the denominator in terms of the dot product gives

${\displaystyle {\begin{aligned}V&=-\int _{\mathbb {R} }^{3}{\frac {G}{\sqrt {|\mathbf {x} |^{2}-2\mathbf {x} \cdot \mathbf {y} +|\mathbf {y} |^{2}}}}\,dm(\mathbf {y} )\\{}&=-{\frac {1}{|\mathbf {x} |}}\int _{\mathbb {R} ^{3}}G\,\left/\,{\sqrt {1-2\cos(\theta ){\frac {r}{|\mathbf {x} |}}+\left({\frac {r}{|\mathbf {x} |}}\right)^{2}}}\right.\,dm\end{aligned}}}$

where in the last integral we have changed to spherical coordinates (r,θ,φ) in which r is the distance to the origin and θ is the axial angle with respect to the axis in the direction of x.

The integrand is precisely the generating function for the Legendre polynomials:

${\displaystyle \left(1-2XZ+Z^{2}\right)^{-{\frac {1}{2}}}\ =\sum _{n=0}^{\infty }Z^{n}P_{n}(X)}$

where the P_n are the Legendre polynomials of degree n. So the integral can be expanded in a series of the form

${\displaystyle {\begin{aligned}V(\mathbf {x} )&=-{\frac {G}{|\mathbf {x} |}}\int \sum _{n=0}^{\infty }\left({\frac {r}{|\mathbf {x} |}}\right)^{n}P_{n}(\cos(\theta ))\,dm\\{}&=-{\frac {G}{|\mathbf {x} |}}\int \left(1+\left({\frac {r}{|\mathbf {x} |}}\right)\cos(\theta )+\left({\frac {r}{|\mathbf {x} |}}\right)^{2}{\frac {3\cos(\theta )^{2}-1}{2}}+\cdots \right)\,dm\end{aligned}}}$

The second term in the expansion ${\displaystyle \int r\cos(\theta )dm}$ is the first moment of the mass distribution about the plane through the origin that is perpendicular to the axis x; this vanishes because the origin was chosen to coincide with the center of mass. So, bringing the integral under the sign of the summation gives

${\displaystyle V(\mathbf {x} )=-{\frac {GM}{|\mathbf {x} |}}-{\frac {G}{|\mathbf {x} |}}\int \left({\frac {r}{|\mathbf {x} |}}\right)^{2}{\frac {3\cos(\theta )^{2}-1}{2}}dm+\cdots }$

Spherical symmetry

A spherically symmetric mass distribution behaves to an observer completely outside the distribution as though all of the mass were 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/s, 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 the 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 and density σ, the gravitational force g inside the sphere varies linearly with distance r from the center, giving the gravitational potential inside the sphere, which is

${\displaystyle V(r)={\frac {2}{3}}\pi G\sigma (r^{2}-3R^{2}),\qquad r\leq R,}$

which smoothly connects to the potential function for the outside of the sphere (see the figure at the top).

Potential energy

The gravitational potential (V) is the potential energy (U) per unit mass:

${\displaystyle U=mV}$

where m is the mass of the object. The potential energy is the negative of the work done by the gravitational field moving the body to its given position in space from infinity. If the body has a mass of 1 unit, 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 which is nearly independent of position. For instance, in daily life, in the region close to the surface of the Earth, the gravitational acceleration 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:

${\displaystyle \Delta U=mg\Delta h}$

General relativity

In general relativity, the gravitational potential is replaced by the metric tensor.

See also

• Applications of Legendre polynomials in physics

Notes

1. Vladimirov 1984, §7.8 harvnb error: no target: CITEREFVladimirov1984 (help)
2. C. R. Wylie, Jr. 1960,Advanced Engineering Mathematics (McGraw-Hill Book Company)
3. Marion & Thornton 2003, §5.2 harvnb error: no target: CITEREFMarionThornton2003 (help)

References

• Peter Dunsby (1996-06-15). "Mass in Newtonian theory". Tensors and Relativity: Chapter 5 Conceptual Basis of General Relativity. Department of Mathematics and Applied Mathematics University of Cape Town. Retrieved 2009-03-25.
• Lupei Zhu Associate Professor, Ph.D. (California Institute of Technology, 1998). "Gravity and Earth's Density Structure". EAS-437 Earth Dynamics. Saint Louis University (Department of Earth and Atmospheric Sciences). Retrieved 2009-03-25.{{cite web}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
• Charles D. Ghilani (2006-11-28). "The Gravity Field of the Earth". The Physics Fact Book. Penn State Surveying Engineering Program. Retrieved 2009-03-25.
• Thornton, Stephen T.; Marion, Jerry B. (2003), Classical Dynamics of Particles and Systems (5th ed.), Brooks Cole, ISBN 978-0-534-40896-1.
• Rastall, Peter (1991). Postprincipia: Gravitation for Physicists and Astronomers. World Scientific. pp. 7ff. ISBN 9810207786.
• 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.

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