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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

For the continuous case the potential, V at point b is computed as an integral over the distributed mass,

${\displaystyle V=-\int {\frac {G}{r}}\ dm=-\int {\frac {G}{|\mathbf {R} -\mathbf {x} |}}\ dm,}$

where dm equals density times differential volume, where x is a vector from the center of mass to the differential element of mass, where r is a vector from the differential element of mass to point b i.e. ${\displaystyle \mathbf {r} =\mathbf {R} -\mathbf {x} \,}$, , and where R is a vector from the center of mass to point b as shown in figure.

Expanding the denominator by taking the square root of the square, carrying out the dot product, and factoring R out of the denominator we get the more useful expression,

${\displaystyle V=-\int {\frac {G}{\sqrt {R^{2}-2\mathbf {R} \cdot \mathbf {x} +x^{2}}}}\,dm\ =}$

${\displaystyle -{\frac {1}{R}}\int G\,/\,{\sqrt {1-2cos(\beta ){\frac {x}{R}}+\left({\frac {x}{R}}\right)^{2}}}\,dm.\ }$ V can be expanded in a series of Legendre Polynomials

Consider the function,

${\displaystyle f({\frac {x}{R}})=\left(1-2cos(\beta )\left({\frac {x}{R}}\right)+\left({\frac {x}{R}}\right)^{2}\right)^{-{\frac {1}{2}}}\ .}$

Expanding ${\displaystyle f({\frac {x}{R}})}$ in a Taylor series about ${\displaystyle \left({\frac {x}{R}}\right)=0}$, we get the approximation,

${\displaystyle f({\frac {x}{R}})\approx 1+\left({\frac {x}{R}}\right)\ cos(\beta )+\left({\frac {x}{R}}\right)^{2}{\frac {3cos(\beta )^{2}-1}{2}}\ .\ }$

The coefficients are Legendre polynomials with argument ${\displaystyle cos(\beta )}$:

${\displaystyle f({\frac {x}{R}})\approx P_{0}(cos(\beta ))+\left({\frac {x}{R}}\right)\ P_{1}(cos(\beta ))+\left({\frac {x}{R}}\right)^{2}\ P_{2}(cos(\beta ))\ }$.

The appearance of the first three terms of the Taylor series expansion for ${\displaystyle f({\frac {x}{R}})}$ suggests that the infinite series might follow the same pattern. On page 454, section 10.8 of Theorem 2 which states that ${\displaystyle \left(1-2XZ+Z^{2}\right)^{-{\frac {1}{2}}}\ =\sum _{n=0}^{\infty }Z^{n}P_{n}(X)\ }$, is proven. Also see Applications of Legendre polynomials in physics. Applied to the notation used here Theorem 2 states that

${\displaystyle \left(1-2cos(\beta )\left({\frac {x}{R}}\right)+\left({\frac {x}{R}}\right)^{2}\right)^{-{\frac {1}{2}}}\ =\sum _{n=0}^{\infty }\left({\frac {x}{R}}\right)^{n}P_{n}(cos(\beta ))\ .}$

We thus get the result,

${\displaystyle V=-{\frac {G}{R}}\int \sum _{n=0}^{\infty }\left({\frac {x}{R}}\right)^{n}P_{n}(cos(\beta ))\,dm\ {\rm {,}}}$

where the ${\displaystyle P_{n}(cos(\beta ))}$ are Legendre Polynomials of degree n and argument ${\displaystyle cos(\beta )\,.}$ This is a form which is quite useful for applications as shown in section 11.7 Gravitational Potential ... of . Taking integrals of terms in the series we get:

${\displaystyle V=-{\frac {G}{R}}\int (1+\left({\frac {x}{R}}\right)\ cos(\beta )+\left({\frac {x}{R}}\right)^{2}{\frac {3cos(\beta )^{2}-1}{2}}\ dm+...}$

Note that ${\displaystyle \int x\ cos(\beta )dm=0.}$ This is true since ${\displaystyle x\ cos(\beta )}$ is the component of the location with respect to the point cm of the differential element of mass along the direction of R as can be seen from the figure. But the point cm is the location of the center of mass. Therefore we get the desired result that ${\displaystyle \int x\ cos(\beta )dm=0\ }$ and that ${\displaystyle \int \left({\frac {x}{R}}\right)\ cos(\beta )dm=0\ }$ since R is invariant over the integral. We therefore get

${\displaystyle V=-{\frac {GM}{R}}-{\frac {G}{R}}\int \left({\frac {x}{R}}\right)^{2}{\frac {3cos(\beta )^{2}-1}{2}}dm+...}$

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. Leonard Meirovitch 1970, Methods of Analytical Dynamics (McGraw-Hill Book Company)
4. 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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