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{{Short description|Fundamental study of potential theory}}
{{redirect|Gravity potential|Earth's gravity potential|Geopotential|the field of gravity potentials|Gravitational field}}
[[File:GravityPotential.jpg|thumb|300px|Plot of a two-dimensional slice of the gravitational potential in and around a uniform spherical body. The [[inflection point]]s of the cross-section are at the surface of the body.]]

In [[classical mechanics]], the '''gravitational potential''' is a [[scalar field]] associating with each point in space the [[Work (physics)|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 (physics)|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 [[wikt:finite|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.<ref>{{cite book|title=Electrostatics and magnetostatics of polarized ellipsoidal bodies: the depolarization tensor method|first1=C.E.|last1=Solivérez|edition=1st English|year=2016|publisher=Free Scientific Information| isbn=978-987-28304-0-3}}</ref>

==Potential energy==
{{main|Gravitational potential energy}}
The gravitational potential (''V'') at a location is the gravitational [[potential energy]] (''U'') at that location per unit mass:

<math display="block">V = \frac{U}{m},</math>

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:
<math display="block">\Delta U \approx mg \Delta h.</math>

==Mathematical form==
The gravitational [[Scalar potential|potential]] ''V'' at a distance ''x'' from a [[point particle|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:<ref>{{cite book|title=Classical Dynamics of particles and systems|first1=J.B.|last1=Marion|first2=S.T.|last2=Thornton| edition=4th|page=[https://archive.org/details/classicaldynamic00mari_0/page/192 192]|year=1995|publisher=Harcourt Brace & Company |isbn=0-03-097302-3|url-access=registration|url=https://archive.org/details/classicaldynamic00mari_0/page/192}}</ref><ref>{{cite book |title=Mathematical Methods For Physicists International Student Edition |edition=6th |first1=George B. |last1=Arfken |first2=Hans J. |last2=Weber |publisher=[[Academic Press]] |year=2005 |isbn=978-0-08-047069-6 |page=72 |url=https://books.google.com/books?id=tNtijk2iBSMC&pg=PA72}}</ref><ref>{{cite book |title=Cambridge International AS and A Level Physics Coursebook |edition=illustrated |first1=David |last1=Sang |first2=Graham |last2=Jones |first3=Gurinder |last3=Chadha |first4=Richard |last4=Woodside |first5=Will |last5=Stark |first6=Aidan |last6=Gill |publisher=[[Cambridge University Press]] |year=2014 |isbn=978-1-107-69769-0 |page=276 |url=https://books.google.com/books?id=SjsDBAAAQBAJ&pg=PA276}}</ref><ref>{{cite book |title=A-level Physics |edition=illustrated |first1=Roger |last1=Muncaster |publisher=[[Nelson Thornes]] |year=1993 |isbn=978-0-7487-1584-8 |page=106 |url=https://books.google.com/books?id=Knov8XAyf2cC&pg=PA106}}</ref>

<math display="block">V(\mathbf{x}) = \frac{W}{m} = \frac{1}{m} \int_{\infty}^{x} \mathbf{F} \cdot d\mathbf{x} = \frac{1}{m} \int_{\infty}^{x} \frac{G m M}{x^2} dx = -\frac{G M}{x},</math>
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 of units|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
<math display="block">\mathbf{a} = -\frac{GM}{x^3} \mathbf{x} = -\frac{GM}{x^2} \hat{\mathbf{x}},</math>
where '''x''' is a vector of length ''x'' pointing from the point mass toward the small body and <math>\hat{\mathbf{x}}</math> is a unit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an [[inverse square law]]:
<math display="block">|\mathbf{a}| = \frac{GM}{x^2}.</math>

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 '''x'''<sub>1</sub>, ..., '''x'''<sub>''n''</sub> and have masses ''m''<sub>1</sub>, ..., ''m''<sub>''n''</sub>, then the potential of the distribution at the point '''x''' is
<math display="block">V(\mathbf{x}) = \sum_{i=1}^n -\frac{Gm_i}{|\mathbf{x} - \mathbf{x}_i|}.</math>

[[Image:Mass distribution line segment.svg|right|thumb|Points '''x''' and '''r''', with '''r''' contained in the distributed mass (gray) and differential mass ''dm''('''r''') located at the point '''r'''.]]

If the mass distribution is given as a mass [[Borel measure|measure]] ''dm'' on three-dimensional [[Euclidean space]] '''R'''<sup>3</sup>, then the potential is the [[convolution]] of {{math|−''G''/{{abs|'''r'''}}}} with ''dm''.{{CN|date=May 2023}} In good cases{{clarify|date=September 2020}} this equals the integral
<math display="block">V(\mathbf{x}) = -\int_{\R^3} \frac{G}{|\mathbf{x} - \mathbf{r}|}\,dm(\mathbf{r}),</math>
where {{math|{{abs|'''x''' − '''r'''}}}} is the [[Euclidean distance|distance]] between the points '''x''' and '''r'''. If there is a function ''ρ''('''r''') representing the density of the distribution at '''r''', so that {{math|1=''dm''('''r''') = ''ρ''('''r''') ''dv''('''r''')}}, where ''dv''('''r''') is the Euclidean [[volume element]], then the gravitational potential is the [[volume integral]]
<math display="block">V(\mathbf{x}) = -\int_{\R^3} \frac{G}{|\mathbf{x}-\mathbf{r}|}\,\rho(\mathbf{r})dv(\mathbf{r}).</math>

If ''V'' is a potential function coming from a continuous mass distribution ''ρ''('''r'''), then ''ρ'' can be recovered using the [[Laplace operator]], {{math|Δ}}:
<math display="block">\rho(\mathbf{x}) = \frac{1}{4\pi G}\Delta V(\mathbf{x}).</math>
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 [[distribution (mathematics)|distribution]]s. 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.<ref>{{cite book|title=The Theory of the Potential|first1=W.D. |last1=MacMillan|year=1958|publisher=Dover Press}}</ref> 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 gravity]] ''g'', approximately 9.8&nbsp;m/s<sup>2</sup>, 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 [[Gauss's law for gravity#Poisson's equation and gravitational potential|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<ref>{{cite book |title=A Student's Guide to Geophysical Equations |first1=William Lowrie |last1=Lowrie |publisher=Cambridge University Press |year=2011 |isbn=978-1-139-49924-8 |page=69 |url=https://books.google.com/books?id=HPE1C9vtWZ0C}} [https://books.google.com/books?id=HPE1C9vtWZ0C&pg=PA68 Extract of page 68]</ref><ref>{{cite book |title=An Introduction to Planetary Atmospheres |edition=illustrated |first1=Agustin |last1=Sanchez-Lavega |publisher=CRC Press |year=2011 |isbn=978-1-4200-6735-4 |page=19 |url=https://books.google.com/books?id=lCXYQ4phwbwC}} [https://books.google.com/books?id=lCXYQ4phwbwC&pg=PA19 Extract of page 19]</ref>
<math display="block">V(r) = \frac {2}{3} \pi G \rho \left[r^2 - 3 R^2\right] = \frac{Gm}{2R^3} \left[r^2 -3 R^2\right], \qquad r \leq R,</math>
which differentiably connects to the potential function for the outside of the sphere (see the figure at the top).

==General relativity==
{{see also|Gravitational acceleration#General relativity|Gravitational field#General relativity}}

In [[general relativity]], the gravitational potential is replaced by the [[metric tensor (general relativity)|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.<ref name="Newtonian or gravitoelectric potential">{{citation|last1=Grøn|first1=Øyvind|last2=Hervik|first2=Sigbjorn|title=Einstein's General Theory of Relativity: With Modern Applications in Cosmology|url=https://books.google.com/books?id=IyJhCHAryuUC&pg=PA201|year=2007 |publisher=Springer Science & Business Media|isbn=978-0-387-69200-5|page=201}}</ref>

==Multipole expansion==
{{main|Spherical multipole moments|Multipole expansion}}
The potential at a point {{math|'''x'''}} is given by
<math display="block">V(\mathbf{x}) = - \int_{\R^3} \frac{G}{|\mathbf{x}-\mathbf{r}|}\ dm(\mathbf{r}).</math>

[[File:Massdistribution xy.svg|right|thumb|Illustration of a mass distribution (grey) with center of mass as the origin of vectors '''x''' and '''r''' and the point at which the potential is being computed at the head of vector '''x'''.]]
The potential can be expanded in a series of [[Legendre polynomials]]. Represent the points '''x''' and '''r''' as [[position vector]]s relative to the center of mass. The denominator in the integral is expressed as the square root of the square to give
<math display="block">\begin{align}
V(\mathbf{x}) &= - \int_{\R^3} \frac{G}{ \sqrt{|\mathbf{x}|^2 -2 \mathbf{x} \cdot \mathbf{r} + |\mathbf{r}|^2}}\,dm(\mathbf{r})\\
&=- \frac{1}{|\mathbf{x}|}\int_{\R^3} \frac{G} \sqrt{1 -2 \frac{r}{|\mathbf{x}|} \cos \theta + \left( \frac{r}{|\mathbf{x}|} \right)^2}\,dm(\mathbf{r})
\end{align}</math>
where, in the last integral, {{math|1=''r'' = {{abs|'''r'''}}}} and {{mvar|θ}} is the angle between '''x''' and '''r'''.

(See "mathematical form".) The integrand can be expanded as a [[Taylor series]] in {{math|1=''Z'' = ''r''/{{abs|'''x'''}}}}, by explicit calculation of the coefficients. A less laborious way of achieving the same result is by using the generalized [[binomial theorem]].<ref name="AEM">{{cite book |first=C. R. Jr. |last=Wylie |date=1960 |title=Advanced Engineering Mathematics |url=https://archive.org/details/advancedengineer00wyli |url-access=registration |location=New York |publisher=[[McGraw-Hill]] |edition=2nd |page=454 [Theorem 2, Section 10.8] }}</ref> The resulting series is the [[generating function]] for the Legendre polynomials:
<math display="block">\left(1- 2 X Z + Z^2 \right) ^{- \frac{1}{2}} \ = \sum_{n=0}^\infty Z^n P_n(X)</math>
valid for {{math|{{abs|''X''}} ≤ 1}} and {{math|{{abs|''Z''}} < 1}}. The coefficients ''P''<sub>''n''</sub> are the Legendre polynomials of degree ''n''. Therefore, the Taylor coefficients of the integrand are given by the Legendre polynomials in {{math|1=''X'' = cos ''θ''}}. So the potential can be expanded in a series that is convergent for positions '''x''' such that {{math|''r'' < {{abs|'''x'''}}}} for all mass elements of the system (i.e., outside a sphere, centered at the center of mass, that encloses the system):
<math display="block"> \begin{align}
V(\mathbf{x}) &= - \frac{G}{|\mathbf{x}|} \int \sum_{n=0}^\infty \left(\frac{r}{|\mathbf{x}|} \right)^n P_n(\cos \theta) \, dm(\mathbf{r})\\
&= - \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^2 \theta - 1}{2} + \cdots\right)\,dm(\mathbf{r})
\end{align}</math>
The integral <math display="inline">\int r \cos(\theta) \, dm</math> is the component of the center of mass in the {{math|'''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
<math display="block"> V(\mathbf{x}) = - \frac{GM}{|\mathbf{x}|} - \frac{G}{|\mathbf{x}|} \int \left(\frac{r}{|\mathbf{x}|}\right)^2 \frac {3 \cos^2 \theta - 1}{2} dm(\mathbf{r}) + \cdots</math>

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 {{Clarify|date=May 2012}} 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]].
<!--
 theoretically, the square of the escape velocity is the potential relatively to infinity (i.e. where the inverse distance vanishes), but what means "with respect" to the central body is unclear.
 --Incnis Mrsi
Isn't the rows the location of the observer and the column the centre of the potential well? (I rephrased the sentence somewhat; though I am not sure if it is any clearer now)
 --Gunnar Larsson
-->

{| class="wikitable"
|-
! rowspan="2" | Location
! colspan="3" | with respect to
|-
! [[Earth]] !! [[Sun]] !! [[Milky Way]]
|-
| Earth's surface || 60 MJ/kg || 900 MJ/kg || ≥ 130 GJ/kg
|-
| [[Low Earth orbit|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
|-
| 0.1 [[light-year]] from Earth || 0.4 J/kg || 140 kJ/kg || ≥ 130 GJ/kg
|}

Compare the [[Micro-g environment#Absence of gravity|gravity at these locations]].<!-- BTW another original research, even worse one -->

==See also==
* [[Legendre polynomials#Applications of Legendre polynomials|Applications of Legendre polynomials in physics]]
* [[Standard gravitational parameter]] (''GM'')
* [[Geoid]]
* [[Geopotential]]
* [[Geopotential model]]

==Notes==
<references />

==References==
{{Refbegin|2}}
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* {{cite book |title=Postprincipia: Gravitation for Physicists and Astronomers | first = Peter|last=Rastall|publisher=[[World Scientific]]|date=1991|isbn=981-02-0778-6|pages=7ff}}
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* {{Citation | last1=Thornton | first1=Stephen T. | last2=Marion | first2=Jerry B. | title=Classical Dynamics of Particles and Systems | publisher=Brooks Cole | edition=5th | isbn=978-0-534-40896-1 | date=2003}}.
* {{cite web |url=http://www.eas.slu.edu/People/LZhu/teaching/eas437/gravity.ppt |title=Gravity and Earth's Density Structure |access-date=2009-03-25 |first1=Lupeia|last1=Zhu|year=1988| agency=California Institute of Technology |publisher=Saint Louis University |department=Department of Earth and Atmospheric Sciences|work=EAS-437 Earth Dynamics}}
* {{cite web|url=http://surveying.wb.psu.edu/sur351/geoid/grava.htm|title=The Gravity Field of the Earth|access-date=2009-03-25 |author=Charles D. Ghilani|publisher=Penn State Surveying Engineering Program|date=2006-11-28|url-status=dead|archive-url=https://web.archive.org/web/20110718143144/http://surveying.wb.psu.edu/sur351/geoid/grava.htm|archive-date=2011-07-18}}
* {{cite journal|first1=Toshio | last1=Fukushima | title= Prolate spheroidal harmonic expansion of gravitational field | journal= Astrophys. J. | year=2014 | volume=147 | page=152 |number=6| doi=10.1088/0004-6256/147/6/152|bibcode=2014AJ....147..152F| doi-access=free }}
{{Refend}}

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