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| [[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.]]
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| In [[classical mechanics]], the '''gravitational potential''' at a location is equal to the [[Work (physics)|work]] ([[energy]] transferred) per unit mass that is done by the force of [[gravity]] to move an object to a fixed reference location. It is analogous to the [[electric potential]] with [[mass]] playing the role of [[charge (physics)|charge]]. The reference location, where the potential is zero, is by convention infinitely far away from any mass, resulting in a negative potential at any finite distance.
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| In mathematics the gravitational potential is also known as the [[Newtonian potential]] and is fundamental in the study of [[potential theory]].
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| ==Potential energy==
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| The gravitational potential (''V'') is the [[potential energy]] (''U'') per unit mass:
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| :<math>U = m V,</math>
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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 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.
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| In some situations, the equations can be simplified by assuming a field that 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:
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| :<math>\Delta U = mg \Delta h.</math>
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| ==Mathematical form==
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| The [[Scalar potential|potential]] ''V'' at a distance ''x'' from a [[point particle|point mass]] of mass ''M'' can be defined as the work done by the gravitational field bringing a unit mass in from infinity to that point:
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| :<math>V(x) = \frac{W}{m} = \frac{1}{m} \int\limits_{\infty}^{x} F \ dx = \frac{1}{m} \int\limits_{\infty}^{x} \frac{G m M}{x^2} dx = -\frac{G M}{x},</math>
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| where ''G'' is the [[gravitational constant]]. The potential has units of energy per unit 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.
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| 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
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| :<math>\mathbf{a} = -\frac{GM}{x^3} \mathbf{x} = -\frac{GM}{x^2} \hat{\mathbf{x}},</math>
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| 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]]:
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| :<math>|\mathbf{a}| = \frac{GM}{x^2}.</math>
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| 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>V(\mathbf{x}) = \sum_{i=1}^n -\frac{Gm_i}{|\mathbf{x} - \mathbf{x_i}|}.</math>
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| [[File:Massdistribution xy 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'''.]]
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| 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 −G/|'''r'''| with ''dm''.<ref>{{harvnb|Vladimirov|1984|loc=§7.8}}</ref> In good cases this equals the integral
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| :<math>V(\mathbf{x}) = -\int_{\mathbf{R}^3} \frac{G}{|\mathbf{x} - \mathbf{r}|}\,dm(\mathbf{r}),</math>
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| where |'''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 ''{{nowrap|dm''('''r'''){{=}} ''ρ''('''r''')''dv''('''r''')}}, where ''dv''('''r''') is the Euclidean [[volume element]], then the gravitational potential is the [[volume integral]]
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| :<math>V(\mathbf{x}) = -\int_{\mathbf{R}^3} \frac{G}{|\mathbf{x}-\mathbf{r}|}\,\rho(\mathbf{r})dv(\mathbf{r}).</math>
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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 [[Laplace operator]], Δ:
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| :<math>\rho(\mathbf{x}) = \frac{1}{4\pi G}\Delta V(\mathbf{x}).</math>
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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 [[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]].
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| ==Spherical symmetry==
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| 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<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]].
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| Within a spherically symmetric mass distribution, it is possible to solve [[Gauss'_law_for_gravity#Relation to gravitational potential and Poisson's equation|Poisson's equation in spherical coordinates]]. Within a uniform spherical body of radius ''R'' and density ρ 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>{{harvnb|Marion|Thornton|2003|loc=§5.2}}</ref>
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| :<math>V(r) = \frac {2}{3} \pi G \rho (r^2-3R^2),\qquad r\leq R,</math>
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| 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==
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| 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">Grøn, Øyvind; Hervik, Sigbjørn [http://books.google.com/books?id=IyJhCHAryuUC&pg=PA201&dq=%22%CF%86+is+the+Newtonian+or+gravitoelectric+potential%22&hl=en&ei=jzzmTaS-HYPTsgaZ4rmfCA&sa=X&oi=book_result&ct=result&resnum=1 Einstein's general theory of relativity: with modern applications in cosmology] Springer, 2007, p. 201</ref>
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| ==Multipole expansion==
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| {{main|spherical multipole moments|Multipole expansion}}
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| The potential at a point '''x''' is given by
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| :<math>V(\mathbf{x}) = - \int_{\mathbb{R}^3} \frac{G}{|\mathbf{x}-\mathbf{r}|}\ dm(\mathbf{r}).</math>
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| [[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 tail of vector '''x'''.]]
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| 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
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| :<math>\begin{align}
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| 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})\\
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| {}&=- \frac{1}{|\mathbf{x}|}\int_{\mathbb{R}^3} G \, \left/ \, \sqrt{1 -2 \frac{r}{|\mathbf{x}|} \cos \theta + \left( \frac{r}{|\mathbf{x}|} \right)^2}\right.\,dm(\mathbf{r})
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| \end{align}</math>
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| where in the last integral, r = |'''r'''| and θ is the angle between '''x''' and '''r'''.
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| The integrand can be expanded as a [[Taylor series]] in ''Z'' = ''r''/|'''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 |year=1960 |title=Advanced Engineering Mathematics |location=New York |publisher=McGraw-Hill |edition=2nd |page=454 [Theorem 2, Section 10.8] |isbn= }}</ref> The resulting series is the [[generating function]] for the Legendre polynomials:
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| :<math>\left(1- 2 X Z + Z^2 \right) ^{- \frac{1}{2}} \ = \sum_{n=0}^\infty Z^n P_n(X)</math>
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| valid for |''X''| ≤ 1 and |''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 ''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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| :<math> \begin{align}
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| 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})\\
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| {}&= - \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})
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| \end{align}</math>
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| The integral <math>\int r\cos \theta dm</math> is the component of the center of mass in the '''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
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| :<math> 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>
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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.)
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| ==Numerical values==
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| 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]].
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| <!--
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| 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.
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| --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)
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| --Gunnar Larsson
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| -->
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| {| class="wikitable"
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| |-
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| ! Location !! W.r.t. [[Earth]] !! W.r.t. [[Sun]] !! W.r.t. [[Milky Way]]
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| |-
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| | Earth's surface || 60 MJ/kg || 900 MJ/kg || ≥ 130 GJ/kg
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| |-
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| | [[Low Earth orbit|LEO]] || 57 MJ/kg || 900 MJ/kg || ≥ 130 GJ/kg
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| |-
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| | [[Voyager 1]] (17,000 million km from Earth) || 23 J/kg || 8 MJ/kg || ≥ 130 GJ/kg
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| |-
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| | 0.1 [[light-year]] from Earth || 0.4 J/kg || 140 kJ/kg || ≥ 130 GJ/kg
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| |}
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| Compare the [[Micro-g_environment#Absence_of_gravity|gravity at these locations]].<!-- BTW another original research, even worse one -->
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| ==See also==
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| *[[Legendre_polynomials#Applications_of_Legendre_polynomials_in_physics|Applications of Legendre polynomials in physics]]
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| * [[Standard gravitational parameter]] ''(GM)''
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| ==Notes==
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| <references/>
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| ==References==
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| {{Refbegin}}
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| * {{cite web |url=http://www.mth.uct.ac.za/omei/gr/chap5/node4.html |title=Mass in Newtonian theory |accessdate=2009-03-25 |author=Peter Dunsby |work=Tensors and Relativity: Chapter 5 Conceptual Basis of General Relativity |publisher=Department of Mathematics and Applied Mathematics University of Cape Town |date=1996-06-15 }}
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| * {{cite web |url=http://www.eas.slu.edu/People/LZhu/teaching/eas437/gravity.ppt |title=Gravity and Earth's Density Structure |accessdate=2009-03-25 |author=Lupei Zhu Associate Professor, Ph.D. (California Institute of Technology, 1998) |publisher=Saint Louis University (Department of Earth and Atmospheric Sciences)|work=EAS-437 Earth Dynamics}}
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| * {{cite web |url=http://surveying.wb.psu.edu/sur351/geoid/grava.htm |title=The Gravity Field of the Earth|accessdate=2009-03-25 |author=Charles D. Ghilani|work=The Physics Fact Book|publisher=Penn State Surveying Engineering Program|date=2006-11-28}}
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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 | year=2003}}.
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| *{{cite book |title=Postprincipia: Gravitation for Physicists and Astronomers | first = Peter|last=Rastall|publisher=World Scientific|year=1991|isbn=981-02-0778-6|pages=7ff.}}
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| *{{Citation | last1=Vladimirov | first1=V. S. | title=Equations of mathematical physics | publisher=Marcel Dekker Inc. | location=New York | series=Translated from the Russian by Audrey Littlewood. Edited by Alan Jeffrey. Pure and Applied Mathematics | mr=0268497 | year=1971 | volume=3}}.
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| {{Refend}}
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| [[Category:Energy (physics)]]
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| [[Category:Gravitation]]
| |
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