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Gravitational potential

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</style><div role="note" class="hatnote navigation-not-searchable">"Gravity potential" redirects here. For Earth's gravity potential, see <a href="Geopotential" title="Geopotential">Geopotential</a>. For the field of gravity potentials, see <a href="Gravitational_field" title="Gravitational field">Gravitational field</a>.</div>

In <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>, the gravitational potential is a <a href="Scalar_potential" title="Scalar potential">scalar potential</a> associating with each point in space the <a href="Work_(physics)" title="Work (physics)">work</a> (<a href="Energy" title="Energy">energy</a> transferred) per unit mass that would be needed to move an object to that point from a fixed reference point in the conservative <a href="Gravitational_field" title="Gravitational field">gravitational field</a>. It is <a href="Analogous" class="mw-redirect" title="Analogous">analogous</a> to the <a href="Electric_potential" title="Electric potential">electric potential</a> with <a href="Mass" title="Mass">mass</a> playing the role of <a href="Charge_(physics)" title="Charge (physics)">charge</a>. The reference point, where the potential is zero, is by convention <a href="Infinitely" class="mw-redirect" title="Infinitely">infinitely</a> far away from any mass, resulting in a negative potential at any <a href="https://en.wiktionary.org/wiki/finite" class="extiw external" title="wikt:finite">finite</a> distance. Their similarity is correlated with both associated <a href="Conservative_field" class="mw-redirect" title="Conservative field">fields</a> having <a href="Conservative_force" title="Conservative force">conservative forces</a>.

Mathematically, the gravitational potential is also known as the <a href="Newtonian_potential" title="Newtonian potential">Newtonian potential</a> and is fundamental in the study of <a href="Potential_theory" title="Potential theory">potential theory</a>. It may also be used for solving the electrostatic and magnetostatic fields generated by uniformly charged or polarized ellipsoidal bodies.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Potential_energy">Potential energy</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Gravitational_potential_energy" class="mw-redirect" title="Gravitational potential energy">Gravitational potential energy</a></div>

The gravitational potential (V) at a location is the gravitational <a href="Potential_energy" title="Potential energy">potential energy</a> (U) at that location per unit mass:

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<annotation encoding="application/x-tex">{\displaystyle V={\frac {U}{m}},}</annotation>

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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 <a href="Gravitational_acceleration" title="Gravitational acceleration">gravitational acceleration</a>, 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:

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<annotation encoding="application/x-tex">{\displaystyle \Delta U\approx mg\Delta h.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Mathematical_form">Mathematical form</h2></div>

The gravitational <a href="Scalar_potential" title="Scalar potential">potential</a> V at a distance x from a <a href="Point_particle" title="Point particle">point mass</a> 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:<a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a><a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle V(\mathbf {x} )={\frac {W}{m}}={\frac {1}{m}}\int _{\infty }^{x}\mathbf {F} \left(\mathbf {x} '\right)\cdot d\mathbf {x} '={\frac {1}{m}}\int _{\infty }^{x}{\frac {GmM}{x'^{2}}}dx'=-{\frac {GM}{x}},}</annotation>

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where G is the <a href="Gravitational_constant" title="Gravitational constant">gravitational constant</a>, and F is the gravitational force. The product GM is the <a href="Standard_gravitational_parameter" title="Standard gravitational parameter">standard gravitational parameter</a> and is often known to higher precision than G or M separately. The potential has <a href="Units_of_energy" title="Units of energy">units of energy</a> per mass, e.g., J/kg in the <a href="MKS_system_of_units" class="mw-redirect" title="MKS system of units">MKS</a> system. By convention, it is always negative where it is defined, and as x tends to infinity, it approaches zero.

The <a href="Gravitational_field" title="Gravitational field">gravitational field</a>, and thus the acceleration of a small body in the space around the massive object, is the negative <a href="Gradient" title="Gradient">gradient</a> 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

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =-{\frac {GM}{x^{3}}}\mathbf {x} =-{\frac {GM}{x^{2}}}{\hat {\mathbf {x} }},}</annotation>

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where x is a vector of length x pointing from the point mass toward the small body and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {x} }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {x} }}}</annotation>

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</math><img src="./8a5451354f837d5d89774e1de386b44a903d929d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {x} }}}" loading="lazy"> is a <a href="Unit_vector" title="Unit vector">unit vector</a> pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an <a href="Inverse_square_law" class="mw-redirect" title="Inverse square law">inverse square law</a>:

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<annotation encoding="application/x-tex">{\displaystyle \|\mathbf {a} \|={\frac {GM}{x^{2}}}.}</annotation>

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The potential associated with a <a href="Mass_distribution" title="Mass distribution">mass distribution</a> 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, ..., xn and have masses m1, ..., mn, then the potential of the distribution at the point x is

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<annotation encoding="application/x-tex">{\displaystyle V(\mathbf {x} )=\sum _{i=1}^{n}-{\frac {Gm_{i}}{\|\mathbf {x} -\mathbf {x} _{i}\|}}.}</annotation>

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If the mass distribution is given as a mass <a href="Borel_measure" title="Borel measure">measure</a> dm on three-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> R3, then the potential is the <a href="Convolution" title="Convolution">convolution</a> of −G/|r| with dm. In good cases this equals the integral

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<annotation encoding="application/x-tex">{\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{\|\mathbf {x} -\mathbf {r} \|}}\,dm(\mathbf {r} ),}</annotation>

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where |x − r| is the <a href="Euclidean_distance" title="Euclidean distance">distance</a> between the points x and r. If there is a function ρ(r) representing the density of the distribution at r, so that dm(r) = ρ(r) dv(r), where dv(r) is the Euclidean <a href="Volume_element" title="Volume element">volume element</a>, then the gravitational potential is the <a href="Volume_integral" title="Volume integral">volume integral</a>

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<annotation encoding="application/x-tex">{\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{\|\mathbf {x} -\mathbf {r} \|}}\,\rho (\mathbf {r} )dv(\mathbf {r} ).}</annotation>

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If V is a potential function coming from a continuous mass distribution ρ(r), then ρ can be recovered using the <a href="Laplace_operator" title="Laplace operator">Laplace operator</a>, Δ:

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<annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {x} )={\frac {1}{4\pi G}}\Delta V(\mathbf {x} ).}</annotation>

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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 <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>. As a consequence, the gravitational potential satisfies <a href="Poisson's_equation" title="Poisson's equation">Poisson's equation</a>. See also <a href="Green's_function_for_the_three-variable_Laplace_equation" title="Green's function for the three-variable Laplace equation">Green's function for the three-variable Laplace equation</a> and <a href="Newtonian_potential" title="Newtonian potential">Newtonian potential</a>.

The integral may be expressed in terms of known transcendental functions for all ellipsoidal shapes, including the symmetrical and degenerate ones.<a href="#cite_note-6">[6]</a> These include the sphere, where the three semi axes are equal; the oblate (see <a href="Reference_ellipsoid" class="mw-redirect" title="Reference ellipsoid">reference ellipsoid</a>) 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 <a href="Charge_density" title="Charge density">charge density</a>) to electromagnetism.

<div class="mw-heading mw-heading2"><h2 id="Spherical_symmetry">Spherical symmetry</h2></div>

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 <a href="Point_mass" class="mw-redirect" title="Point mass">point mass</a>, by the <a href="Shell_theorem" title="Shell theorem">shell theorem</a>. On the surface of the earth, the acceleration is given by so-called <a href="Standard_gravity" title="Standard gravity">standard gravity</a> 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 <a href="Oblate_spheroid" class="mw-redirect" title="Oblate spheroid">oblate spheroid</a>.

Within a spherically symmetric mass distribution, it is possible to solve <a href="Gauss's_law_for_gravity#Poisson's_equation_and_gravitational_potential" title="Gauss's law for gravity">Poisson's equation in spherical coordinates</a>. 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<a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a>

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<annotation encoding="application/x-tex">{\displaystyle V(r)={\frac {2}{3}}\pi G\rho \left[r^{2}-3R^{2}\right]={\frac {Gm}{2R^{3}}}\left[r^{2}-3R^{2}\right],\qquad r\leq R,}</annotation>

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

<div class="mw-heading mw-heading2"><h2 id="General_relativity">General relativity</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Gravitational_acceleration#General_relativity" title="Gravitational acceleration">Gravitational acceleration § General relativity</a>, and <a href="Gravitational_field#General_relativity" title="Gravitational field">Gravitational field § General relativity</a></div>

In <a href="General_relativity" title="General relativity">general relativity</a>, the gravitational potential is replaced by the <a href="Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">metric tensor</a>. 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.<a href="#cite_note-Newtonian_or_gravitoelectric_potential-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="Multipole_expansion">Multipole expansion</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Spherical_multipole_moments" title="Spherical multipole moments">Spherical multipole moments</a> and <a href="Multipole_expansion" title="Multipole expansion">Multipole expansion</a></div>

The potential at a point x is given by

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<annotation encoding="application/x-tex">{\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{|\mathbf {x} -\mathbf {r} |}}\ dm(\mathbf {r} ).}</annotation>

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The potential can be expanded in a series of <a href="Legendre_polynomials" title="Legendre polynomials">Legendre polynomials</a>. Represent the points x and r as <a href="Position_vector" class="mw-redirect" title="Position vector">position vectors</a> relative to the <a href="Center_of_mass" title="Center of mass">center of mass</a>. The denominator in the integral is expressed as the square root of the square to give

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V(\mathbf {x} )&=-\int _{\mathbb {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 _{\mathbb {R} ^{3}}{\frac {G}{\sqrt {1-2{\frac {r}{|\mathbf {x} |}}\cos \theta +\left({\frac {r}{|\mathbf {x} |}}\right)^{2}}}}\,dm(\mathbf {r} )\end{aligned}}}</annotation>

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where, in the last integral, r = |r| and θ is the angle between x and r.

(See <a href="#Mathematical_form">§ Mathematical form</a>.) The integrand can be expanded as a <a href="Taylor_series" title="Taylor series">Taylor series</a> in Z = r/|x|, by explicit calculation of the coefficients. A less laborious way of achieving the same result is by using the generalized <a href="Binomial_theorem" title="Binomial theorem">binomial theorem</a>.<a href="#cite_note-AEM-10">[10]</a> The resulting series is the <a href="Generating_function" title="Generating function">generating function</a> for the Legendre polynomials:

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<annotation encoding="application/x-tex">{\displaystyle \left(1-2XZ+Z^{2}\right)^{-{\frac {1}{2}}}\ =\sum _{n=0}^{\infty }Z^{n}P_{n}(X)}</annotation>

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valid for |X| ≤ 1 and |Z| < 1. The coefficients Pn are the Legendre polynomials of degree n. Therefore, the Taylor coefficients of the integrand are given by the Legendre polynomials in X = cos θ. So the potential can be expanded in a series that is convergent for positions x such that r < |x| for all mass elements of the system (i.e., outside a sphere, centered at the center of mass, that encloses the system):

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<annotation encoding="application/x-tex">{\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(\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{aligned}}}</annotation>

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The integral <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int r\cos(\theta )\,dm}">

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<annotation encoding="application/x-tex">{\textstyle \int r\cos(\theta )\,dm}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle 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 }</annotation>

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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.)

<div class="mw-heading mw-heading2"><h2 id="Unit_and_numerical_values">Unit and numerical values </h2></div>

The <a href="SI_unit" class="mw-redirect" title="SI unit">SI unit</a> of gravitational potential is the <a href="Square_metre" title="Square metre">square metre</a> per square second (m2/s2) or, equivalently, the joule per kilogram (J/kg).

The <a href="Absolute_value" title="Absolute value">absolute value</a> of gravitational potential at a number of locations with regards to the mass of the <a href="Earth" title="Earth">Earth</a>, the <a href="Sun" title="Sun">Sun</a>, and the <a href="Milky_Way" title="Milky Way">Milky Way</a> is given in the following table; i.e. an object at Earth's surface would need 60 MJ/kg to "leave" <a href="Gravity_of_Earth" title="Gravity of Earth">Earth's gravity</a> 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 <a href="Escape_velocity" title="Escape velocity">escape velocity</a>.

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<th colspan="3">with respect to

</th></tr>

<tr>

<th><a href="Earth" title="Earth">Earth</a></th>

<th><a href="Milky_Way" title="Milky Way">Milky Way</a>

</th></tr>

<tr>

<td>Earth's surface</td>

<td>≥ 130 GJ/kg

</td></tr>

<tr>

<td>≥ 130 GJ/kg

</td></tr>

<tr>

<td><a href="Voyager_1" title="Voyager 1">Voyager 1</a> (17,000 million km from Earth)</td>

<td>≥ 130 GJ/kg

</td></tr>

<tr>

<td>0.1 <a href="Light-year" title="Light-year">light-year</a> from Earth</td>

<td>≥ 130 GJ/kg

</td></tr></tbody></table>

Compare the <a href="Micro-g_environment" class="mw-redirect" title="Micro-g environment">gravity at these locations</a>.

<ul><li><a href="Legendre_polynomials#Applications_of_Legendre_polynomials" title="Legendre polynomials">Applications of Legendre polynomials in physics</a></li>

<li><a href="Standard_gravitational_parameter" title="Standard gravitational parameter">Standard gravitational parameter</a> (GM)</li>

<li><a href="Geoid" title="Geoid">Geoid</a></li>

<li><a href="Geopotential" title="Geopotential">Geopotential</a></li>

<li><a href="Geopotential_model" class="mw-redirect" title="Geopotential model">Geopotential model</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFSolivérez2016" class="citation book cs1">Solivérez, C.E. (2016). Electrostatics and magnetostatics of polarized ellipsoidal bodies: the depolarization tensor method (1st English ed.). Free Scientific Information. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-987-28304-0-3</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFMarionThornton1995" class="citation book cs1">Marion, J.B.; Thornton, S.T. (1995). <a rel="nofollow" class="external text" href="https://archive.org/details/classicaldynamic00mari_0/page/192">Classical Dynamics of particles and systems</a> (4th ed.). Harcourt Brace & Company. p. <a rel="nofollow" class="external text" href="https://archive.org/details/classicaldynamic00mari_0/page/192">192</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-03-097302-3</bdi>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFArfkenWeber2005" class="citation book cs1">Arfken, George B.; Weber, Hans J. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tNtijk2iBSMC&pg=PA72">Mathematical Methods For Physicists International Student Edition</a> (6th ed.). <a href="Academic_Press" title="Academic Press">Academic Press</a>. p. 72. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-08-047069-6</bdi>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFSangJonesChadhaWoodside2014" class="citation book cs1">Sang, David; Jones, Graham; Chadha, Gurinder; Woodside, Richard; Stark, Will; Gill, Aidan (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SjsDBAAAQBAJ&pg=PA276">Cambridge International AS and A Level Physics Coursebook</a> (illustrated ed.). <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 276. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-107-69769-0</bdi>.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFMuncaster1993" class="citation book cs1">Muncaster, Roger (1993). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Knov8XAyf2cC&pg=PA106">A-level Physics</a> (illustrated ed.). <a href="Nelson_Thornes" title="Nelson Thornes">Nelson Thornes</a>. p. 106. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7487-1584-8</bdi>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFMacMillan1958" class="citation book cs1">MacMillan, W.D. (1958). The Theory of the Potential. Dover Press.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFLowrie2011" class="citation book cs1">Lowrie, William Lowrie (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HPE1C9vtWZ0C">A Student's Guide to Geophysical Equations</a>. Cambridge University Press. p. 69. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-139-49924-8</bdi>.</cite> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HPE1C9vtWZ0C&pg=PA68">Extract of page 68</a>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFSanchez-Lavega2011" class="citation book cs1">Sanchez-Lavega, Agustin (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lCXYQ4phwbwC">An Introduction to Planetary Atmospheres</a> (illustrated ed.). CRC Press. p. 19. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4200-6735-4</bdi>.</cite> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lCXYQ4phwbwC&pg=PA19">Extract of page 19</a>

</li>

<li id="cite_note-Newtonian_or_gravitoelectric_potential-9"><a href="#cite_ref-Newtonian_or_gravitoelectric_potential_9-0">^</a> <cite id="CITEREFGrønHervik2007" class="citation cs2">Grøn, Øyvind; Hervik, Sigbjorn (2007), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IyJhCHAryuUC&pg=PA201">Einstein's General Theory of Relativity: With Modern Applications in Cosmology</a>, Springer Science & Business Media, p. 201, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-69200-5</bdi></cite>

</li>

<li id="cite_note-AEM-10"><a href="#cite_ref-AEM_10-0">^</a> <cite id="CITEREFWylie1960" class="citation book cs1">Wylie, C. R. Jr. (1960). <a rel="nofollow" class="external text" href="https://archive.org/details/advancedengineer00wyli">Advanced Engineering Mathematics</a> (2nd ed.). New York: <a href="McGraw-Hill" class="mw-redirect" title="McGraw-Hill">McGraw-Hill</a>. p. 454 [Theorem 2, Section 10.8].</cite>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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<ul><li><cite id="CITEREFVladimirov1971" class="citation cs2">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., <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0268497">0268497</a></cite>.</li>

<li><cite id="CITEREFWang1988" class="citation journal cs1">Wang, W. X. (1988). "The potential for a homogeneous spheroid in a spheroidal coordinate system. I. At an exterior point". J. Phys. A: Math. Gen. 21 (22): 4245–4250. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988JPhA...21.4245W">1988JPhA...21.4245W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0305-4470%2F21%2F22%2F026">10.1088/0305-4470/21/22/026</a>.</cite></li>

<li><cite id="CITEREFMilon1990" class="citation journal cs1">Milon, T. (1990). "A note on the potential of a homogenous ellipsoid in ellipsoidal coordinates". J. Phys. A: Math. Gen. 23 (4): 581–584. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0305-4470%2F23%2F4%2F027">10.1088/0305-4470/23/4/027</a>.</cite></li>

<li><cite id="CITEREFRastall1991" class="citation book cs1">Rastall, Peter (1991). Postprincipia: Gravitation for Physicists and Astronomers. <a href="World_Scientific" title="World Scientific">World Scientific</a>. pp. 7ff. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>981-02-0778-6</bdi>.</cite></li>

<li><cite id="CITEREFConway2000" class="citation journal cs1">Conway, John T. (2000). <a rel="nofollow" class="external text" href="https://doi.org/10.1046%2Fj.1365-8711.2000.03524.x">"Exact solutions for the gravitational potential of a family of heterogeneous spheroids"</a>. Mon. Not. R. Astron. Soc. 316 (3): 555–558. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000MNRAS.316..555C">2000MNRAS.316..555C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1046%2Fj.1365-8711.2000.03524.x">10.1046/j.1365-8711.2000.03524.x</a>.</cite></li>

<li><cite id="CITEREFCohlTohlineRau2000" class="citation journal cs1">Cohl, H. S.; Tohline, J. E.; Rau, A. R. P. (2000). "Developments in determining the grativational potential using toroidal functions". Astron. Nachr. 321 (5/6): 363–372. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000AN....321..363C">2000AN....321..363C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F1521-3994%28200012%29321%3A5%2F6%3C363%3A%3AAID-ASNA363%3E3.0.CO%3B2-X">10.1002/1521-3994(200012)321:5/6<363::AID-ASNA363>3.0.CO;2-X</a>.</cite></li>

<li><cite id="CITEREFThorntonMarion2003" class="citation cs2">Thornton, Stephen T.; Marion, Jerry B. (2003), Classical Dynamics of Particles and Systems (5th ed.), Brooks Cole, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-534-40896-1</bdi></cite>.</li>

<li><cite id="CITEREFZhu1988" class="citation web cs1">Zhu, Lupeia (1988). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110726164033/http://www.eas.slu.edu/People/LZhu/teaching/eas437/gravity.ppt">"Gravity and Earth's Density Structure"</a>. Department of Earth and Atmospheric Sciences. EAS-437 Earth Dynamics. Saint Louis University. California Institute of Technology. Archived from <a rel="nofollow" class="external text" href="http://www.eas.slu.edu/People/LZhu/teaching/eas437/gravity.ppt">the original</a> on 2011-07-26. Retrieved 2009-03-25.</cite></li>

<li><cite id="CITEREFCharles_D._Ghilani2006" class="citation web cs1">Charles D. Ghilani (2006-11-28). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110718143144/http://surveying.wb.psu.edu/sur351/geoid/grava.htm">"The Gravity Field of the Earth"</a>. Penn State Surveying Engineering Program. Archived from <a rel="nofollow" class="external text" href="http://surveying.wb.psu.edu/sur351/geoid/grava.htm">the original</a> on 2011-07-18. Retrieved 2009-03-25.</cite></li>

<li><cite id="CITEREFFukushima2014" class="citation journal cs1">Fukushima, Toshio (2014). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0004-6256%2F147%2F6%2F152">"Prolate spheroidal harmonic expansion of gravitational field"</a>. Astrophys. J. 147 (6): 152. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014AJ....147..152F">2014AJ....147..152F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0004-6256%2F147%2F6%2F152">10.1088/0004-6256/147/6/152</a>.</cite></li></ul>

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