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48-567: Mathematically, 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. The gravitational potential ( V ) at a location is the gravitational potential energy ( U ) at that location per unit mass: V = U m , {\displaystyle V={\frac {U}{m}},} where m

96-478: A celestial body is the product of the gravitational constant G and the mass M of that body. For two bodies, the parameter may be expressed as G ( m 1 + m 2 ) , or as GM when one body is much larger than the other: μ = G ( M + m ) ≈ G M . {\displaystyle \mu =G(M+m)\approx GM.} For several objects in the Solar System ,

144-401: A circular orbit around a central body, where the centripetal force provided by gravity is F = mv r : μ = r v 2 = r 3 ω 2 = 4 π 2 r 3 T 2 , {\displaystyle \mu =rv^{2}=r^{3}\omega ^{2}={\frac {4\pi ^{2}r^{3}}{T^{2}}},} where r

192-649: 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 1 , ..., x n and have masses m 1 , ..., m n , then the potential of the distribution at the point x is V ( x ) = ∑ i = 1 n − G m i ‖ x − x i ‖ . {\displaystyle V(\mathbf {x} )=\sum _{i=1}^{n}-{\frac {Gm_{i}}{\|\mathbf {x} -\mathbf {x} _{i}\|}}.} If

240-461: A pendulum oscillating above the surface of a body as: μ ≈ 4 π 2 r 2 L T 2 {\displaystyle \mu \approx {\frac {4\pi ^{2}r^{2}L}{T^{2}}}} where r is the radius of the gravitating body, L is the length of the pendulum, and T is the period of the pendulum (for the reason of the approximation see Pendulum in mechanics ). G M E ,

288-713: A distance x from a 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: V ( x ) = W m = 1 m ∫ ∞ x F ⋅ d x = 1 m ∫ ∞ x G m M x 2 d x = − G M x , {\displaystyle 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 {GmM}{x^{2}}}dx=-{\frac {GM}{x}},} where G

336-440: A point x is given by V ( x ) = − ∫ R 3 G | x − r |   d m ( r ) . {\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{|\mathbf {x} -\mathbf {r} |}}\ dm(\mathbf {r} ).} The potential can be expanded in a series of Legendre polynomials . Represent

384-573: A relative uncertainty of the order of 10 . During the 1970s to 1980s, the increasing number of artificial satellites in Earth orbit further facilitated high-precision measurements, and the relative uncertainty was decreased by another three orders of magnitude, to about 2 × 10 (1 in 500 million) as of 1992. Measurement involves observations of the distances from the satellite to Earth stations at different times, which can be obtained to high accuracy using radar or laser ranging. G M ☉ ,

432-591: 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: Δ U ≈ m g Δ h . {\displaystyle \Delta U\approx mg\Delta h.} The gravitational potential V at

480-535: Is a compactly supported Radon measure . It satisfies the Poisson equation Δ w = μ {\displaystyle \Delta w=\mu } in the sense of distributions . Moreover, when the measure is positive , the Newtonian potential is subharmonic on R . If f is a compactly supported continuous function (or, more generally, a finite measure) that is rotationally invariant , then

528-463: Is a potential function coming from a continuous mass distribution ρ ( r ), then ρ can be recovered using the Laplace operator , Δ : ρ ( x ) = 1 4 π G Δ V ( x ) . {\displaystyle \rho (\mathbf {x} )={\frac {1}{4\pi G}}\Delta V(\mathbf {x} ).} This holds pointwise whenever ρ is continuous and

SECTION 10

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576-509: Is a solution of the Poisson equation Δ w = f , {\displaystyle \Delta w=f,} which is to say that the operation of taking the Newtonian potential of a function is a partial inverse to the Laplace operator. Then w will be a classical solution, that is twice differentiable, if f is bounded and locally Hölder continuous as shown by Otto Hölder . It

624-514: Is a vector of length x pointing from the point mass toward the small body and x ^ {\displaystyle {\hat {\mathbf {x} }}} is a unit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an inverse square law : ‖ a ‖ = G M x 2 . {\displaystyle \|\mathbf {a} \|={\frac {GM}{x^{2}}}.} The potential associated with

672-413: Is constant and equal to 2 μ . For elliptic and hyperbolic orbits magnitude of μ = 2 times the magnitude of a times the magnitude of ε , where a is the semi-major axis and ε is the specific orbital energy . In the more general case where the bodies need not be a large one and a small one, e.g. a binary star system, we define: Then: The standard gravitational parameter can be determined using

720-474: Is standard for planets orbiting the Sun or most moons and greatly simplifies equations. Under Newton's law of universal gravitation , if the distance between the bodies is r , the force exerted on the smaller body is: F = G M m r 2 = μ m r 2 {\displaystyle F={\frac {GMm}{r^{2}}}={\frac {\mu m}{r^{2}}}} Thus only

768-737: Is the distance 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 volume element , then the gravitational potential is the volume integral V ( x ) = − ∫ R 3 G ‖ x − r ‖ ρ ( r ) d v ( r ) . {\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{\|\mathbf {x} -\mathbf {r} \|}}\,\rho (\mathbf {r} )dv(\mathbf {r} ).} If V

816-464: 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. By convention, it is always negative where it is defined, and as x tends to infinity, it approaches zero. The gravitational field , and thus

864-815: 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 V ( x ) = − G M | x | − G | x | ∫ ( r | x | ) 2 3 cos 2 ⁡ θ − 1 2 d m ( r ) + ⋯ {\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 } This shows that elongation of

912-421: 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

960-441: Is the orbit radius , v is the orbital speed , ω is the angular speed , and T is the orbital period . This can be generalized for elliptic orbits : μ = 4 π 2 a 3 T 2 , {\displaystyle \mu ={\frac {4\pi ^{2}a^{3}}{T^{2}}},} where a is the semi-major axis , which is Kepler's third law . For parabolic trajectories rv

1008-436: Is the volume of the unit d -ball (sometimes sign conventions may vary; compare ( Evans 1998 ) and ( Gilbarg & Trudinger 1983 )). For example, for d = 3 {\displaystyle d=3} we have Γ ( x ) = − 1 / ( 4 π | x | ) . {\displaystyle \Gamma (x)=-1/(4\pi |x|).} The Newtonian potential w of f

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1056-452: 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 . 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

1104-415: The Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian , on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object of study in potential theory . In its general nature, it is a singular integral operator , defined by convolution with a function having a mathematical singularity at

1152-516: The ( d  − 1)-dimensional Hausdorff measure , then at a point y of S , the normal derivative undergoes a jump discontinuity f ( y ) when crossing the layer. Furthermore, the normal derivative of w is a well-defined continuous function on S . This makes simple layers particularly suited to the study of the Neumann problem for the Laplace equation. Standard gravitational parameter The standard gravitational parameter μ of

1200-1831: 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): V ( x ) = − G | x | ∫ ∑ n = 0 ∞ ( r | x | ) n P n ( cos ⁡ θ ) d m ( r ) = − G | x | ∫ ( 1 + ( r | x | ) cos ⁡ θ + ( r | x | ) 2 3 cos 2 ⁡ θ − 1 2 + ⋯ ) d m ( r ) {\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}}} The integral ∫ r cos ⁡ ( θ ) d m {\textstyle \int r\cos(\theta )\,dm}

1248-687: The Newtonian kernel Γ {\displaystyle \Gamma } in dimension d {\displaystyle d} is defined by Γ ( x ) = { 1 2 π log ⁡ | x | , d = 2 , 1 d ( 2 − d ) ω d | x | 2 − d , d ≠ 2. {\displaystyle \Gamma (x)={\begin{cases}{\frac {1}{2\pi }}\log {|x|},&d=2,\\{\frac {1}{d(2-d)\omega _{d}}}|x|^{2-d},&d\neq 2.\end{cases}}} Here ω d

1296-535: The Newtonian potential is instead thought of as an electrostatic potential . The Newtonian potential of a compactly supported integrable function f {\displaystyle f} is defined as the convolution u ( x ) = Γ ∗ f ( x ) = ∫ R d Γ ( x − y ) f ( y ) d y {\displaystyle u(x)=\Gamma *f(x)=\int _{\mathbb {R} ^{d}}\Gamma (x-y)f(y)\,dy} where

1344-399: The Newtonian potential of μ is referred to as a simple layer potential . Simple layer potentials are continuous and solve the Laplace equation except on S . They appear naturally in the study of electrostatics in the context of the electrostatic potential associated to a charge distribution on a closed surface. If d μ = f d H is the product of a continuous function on S with

1392-598: The Poisson equation in suitably regular domains, and for suitably well-behaved functions f : one first applies a Newtonian potential to obtain a solution, and then adjusts by adding a harmonic function to get the correct boundary data. The Newtonian potential is defined more broadly as the convolution Γ ∗ μ ( x ) = ∫ R d Γ ( x − y ) d μ ( y ) {\displaystyle \Gamma *\mu (x)=\int _{\mathbb {R} ^{d}}\Gamma (x-y)\,d\mu (y)} when μ

1440-406: 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 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

1488-600: 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 a = − G M x 3 x = − G M x 2 x ^ , {\displaystyle \mathbf {a} =-{\frac {GM}{x^{3}}}\mathbf {x} =-{\frac {GM}{x^{2}}}{\hat {\mathbf {x} }},} where x

Gravitational potential - Misplaced Pages Continue

1536-408: 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.) The absolute value of gravitational potential at a number of locations with regards to

1584-505: The constant G , with 𝜌 being a constant charge density) to electromagnetism. 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 m/s, although this value varies slightly with latitude and altitude. The magnitude of

1632-464: The convolution of f with Γ satisfies for x outside the support of f f ∗ Γ ( x ) = λ Γ ( x ) , λ = ∫ R d f ( y ) d y . {\displaystyle f*\Gamma (x)=\lambda \Gamma (x),\quad \lambda =\int _{\mathbb {R} ^{d}}f(y)\,dy.} In dimension d  = 3, this reduces to Newton's theorem that

1680-677: The gravitation from 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 . Compare the gravity at these locations . Newtonian potential In mathematics ,

1728-461: The gravitational parameter for the Earth as the central body, is called the geocentric gravitational constant . It equals (3.986 004 418 ± 0.000 000 008 ) × 10  m ⋅s . The value of this constant became important with the beginning of spaceflight in the 1950s, and great effort was expended to determine it as accurately as possible during the 1960s. Sagitov (1969) cites a range of values reported from 1960s high-precision measurements, with

1776-439: The gravitational parameter for the Sun as the central body, is called the heliocentric gravitational constant or geopotential of the Sun and equals (1.327 124 400 42 ± 0.000 000 0001 ) × 10  m ⋅s . The relative uncertainty in G M ☉ , cited at below 10 as of 2015, is smaller than the uncertainty in G M E because G M ☉ is derived from the ranging of interplanetary probes, and

1824-550: The gravitational potential inside the sphere, which is V ( r ) = 2 3 π G ρ [ r 2 − 3 R 2 ] = G m 2 R 3 [ r 2 − 3 R 2 ] , r ≤ R , {\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,} which differentiably connects to

1872-871: The last integral, r = | r | and θ is the angle between x and r . (See "mathematical form".) 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 . The resulting series is the generating function for the Legendre polynomials: ( 1 − 2 X Z + Z 2 ) − 1 2   = ∑ n = 0 ∞ Z n P n ( X ) {\displaystyle \left(1-2XZ+Z^{2}\right)^{-{\frac {1}{2}}}\ =\sum _{n=0}^{\infty }Z^{n}P_{n}(X)} valid for | X | ≤ 1 and | Z | < 1 . The coefficients P n are

1920-584: The mass distribution is given as a mass measure dm on three-dimensional Euclidean space R , then the potential is the convolution of − G /| r | with dm . In good cases this equals the integral V ( x ) = − ∫ R 3 G ‖ x − r ‖ d m ( r ) , {\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{\|\mathbf {x} -\mathbf {r} \|}}\,dm(\mathbf {r} ),} where | x − r |

1968-485: The origin, the Newtonian kernel Γ {\displaystyle \Gamma } which is the fundamental solution of the Laplace equation . It is named for Isaac Newton , who first discovered it and proved that it was a harmonic function in the special case of three variables , where it served as the fundamental gravitational potential in Newton's law of universal gravitation . In modern potential theory,

Gravitational potential - Misplaced Pages Continue

2016-1215: The points x and r as position vectors relative to the center of mass. The denominator in the integral is expressed as the square root of the square to give V ( x ) = − ∫ R 3 G | x | 2 − 2 x ⋅ r + | r | 2 d m ( r ) = − 1 | x | ∫ R 3 G 1 − 2 r | x | cos ⁡ θ + ( r | x | ) 2 d m ( r ) {\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}}} where, in

2064-403: The potential energy of a small mass outside a much larger spherically symmetric mass distribution is the same as if all of the mass of the larger object were concentrated at its center. When the measure μ is associated to a mass distribution on a sufficiently smooth hypersurface S (a Lyapunov surface of Hölder class C ) that divides R into two regions D + and D − , then

2112-420: The potential function for the outside of the sphere (see the figure at the top). In general relativity , the gravitational potential is replaced by the 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. The potential at

2160-421: The product of G and M is needed to predict the motion of the smaller body. Conversely, measurements of the smaller body's orbit only provide information on the product, μ , not G and M separately. The gravitational constant, G , is difficult to measure with high accuracy, while orbits, at least in the solar system, can be measured with great precision and used to determine μ with similar precision. For

2208-446: The symmetrical and degenerate ones. 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

2256-427: The value of μ is known to greater accuracy than either G or M . The SI unit of the standard gravitational parameter is m ⋅ s . However, the unit km ⋅ s is frequently used in the scientific literature and in spacecraft navigation. The central body in an orbital system can be defined as the one whose mass ( M ) is much larger than the mass of the orbiting body ( m ), or M ≫ m . This approximation

2304-457: Was an open question whether continuity alone is also sufficient. This was shown to be wrong by Henrik Petrini who gave an example of a continuous f for which w is not twice differentiable. The solution is not unique, since addition of any harmonic function to w will not affect the equation. This fact can be used to prove existence and uniqueness of solutions to the Dirichlet problem for

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