3669:
2691:
2070:
3681:
2320:
821:
1747:
3753:
1709:
2686:{\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}}}
2065:{\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}}}
3717:
3741:
3693:
3729:
3705:
2938:
420:
1322:) 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
2942:
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
166:
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
2967:
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
2751:
1190:
254:
991:
1704:
567:
816:
2255:
1543:
1571:. 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.
1317:
The integral may be expressed in terms of known transcendental functions for all ellipsoidal shapes, including the symmetrical and degenerate ones. These include the sphere, where the three semi axes are equal; the oblate (see
1076:
1288:
891:
1598:
1349:, 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 Earth is an
666:
2933:{\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 }
2325:
1752:
1307:
469:
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
472:
713:
217:
1334:
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
415:{\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}},}
2132:
178:, 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:
606:
2736:
1379:
170:
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
158:
674:
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
1221:
167:
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.
3409:, Translated from the Russian by Audrey Littlewood. Edited by Alan Jeffrey. Pure and Applied Mathematics, vol. 3, New York: Marcel Dekker Inc.,
3668:
3568:
3363:
3333:
3303:
3261:
3230:
3199:
3135:
96:. It may also be used for solving the electrostatic and magnetostatic fields generated by uniformly charged or polarized ellipsoidal bodies.
3579:
615:
1185:{\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{\|\mathbf {x} -\mathbf {r} \|}}\,\rho (\mathbf {r} )dv(\mathbf {r} ).}
3788:
3530:
Cohl, H. S.; Tohline, J. E.; Rau, A. R. P. (2000). "Developments in determining the grativational potential using toroidal functions".
1568:
3485:
3168:
3551:
3087:
439:
181:
3603:
2121:, by explicit calculation of the coefficients. A less laborious way of achieving the same result is by using the generalized
105:
608:
is a unit vector pointing from the point mass toward the small body. The magnitude of the acceleration therefore follows an
3422:
Wang, W. X. (1988). "The potential for a homogeneous spheroid in a spheroidal coordinate system. I. At an exterior point".
3659:
2317:
for all mass elements of the system (i.e., outside a sphere, centered at the center of mass, that encloses the system):
127:
1744:
relative to the center of mass. The denominator in the integral is expressed as the square root of the square to give
1580:
986:{\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{\|\mathbf {x} -\mathbf {r} \|}}\,dm(\mathbf {r} ),}
1699:{\displaystyle V(\mathbf {x} )=-\int _{\mathbb {R} ^{3}}{\frac {G}{|\mathbf {x} -\mathbf {r} |}}\ dm(\mathbf {r} ).}
61:
transferred) per unit mass that would be needed to move an object to that point from a fixed reference point. It is
3222:
1554:
1545:
which differentiably connects to the potential function for the outside of the sphere (see the figure at the top).
1357:
1299:
171:
3773:
2955:
The absolute value of gravitational potential at a number of locations with regards to the gravitation from the
580:
427:
2696:
562:{\displaystyle \mathbf {a} =-{\frac {GM}{x^{3}}}\mathbf {x} =-{\frac {GM}{x^{2}}}{\hat {\mathbf {x} }},}
3680:
1303:
3635:
3539:
3508:
3431:
3082:
1729:
811:{\displaystyle V(\mathbf {x} )=\sum _{i=1}^{n}-{\frac {Gm_{i}}{\|\mathbf {x} -\mathbf {x} _{i}\|}}.}
465:, and thus the acceleration of a small body in the space around the massive object, is the negative
3783:
3745:
3128:
Electrostatics and magnetostatics of polarized ellipsoidal bodies: the depolarization tensor method
3070:
2126:
1584:
1558:
1319:
1311:
462:
451:
89:
42:
35:
3451:
Milon, T. (1990). "A note on the potential of a homogenous ellipsoid in ellipsoidal coordinates".
3215:
Sang, David; Jones, Graham; Chadha, Gurinder; Woodside, Richard; Stark, Will; Gill, Aidan (2014).
2748:
emanates from the center of mass. So, bringing the integral under the sign of the summation gives
820:
3733:
3721:
3685:
3587:
3107:
1564:
1008:
609:
66:
3160:
3153:
249:
that needs to be done by an external agent to bring a unit mass in from infinity to that point:
2290:. Therefore, the Taylor coefficients of the integrand are given by the Legendre polynomials in
2250:{\displaystyle \left(1-2XZ+Z^{2}\right)^{-{\frac {1}{2}}}\ =\sum _{n=0}^{\infty }Z^{n}P_{n}(X)}
3564:
3481:
3359:
3353:
3329:
3299:
3257:
3247:
3226:
3216:
3195:
3164:
3131:
671:
3340:
3310:
3185:
17:
3778:
3697:
3643:
3547:
3516:
3477:
3460:
3439:
2122:
1538:{\displaystyle V(r)={\frac {2}{3}}\pi G\rho \left={\frac {Gm}{2R^{3}}}\left,\qquad r\leq R,}
1343:
1211:
227:
115:
93:
74:
3414:
3410:
3017:
2969:
1741:
1350:
1071:
860:
3639:
3543:
3512:
3497:"Exact solutions for the gravitational potential of a family of heterogeneous spheroids"
3435:
1712:
Illustration of a mass distribution (grey) with center of mass as the origin of vectors
3757:
3673:
3253:
3191:
1067:
238:
54:
3648:
3623:
3443:
3767:
3607:
3521:
3496:
3464:
3380:
2101:
1339:
853:
3752:
3709:
3102:
1708:
50:
31:
1298:
can be recovered in the same way if the
Laplace operator is taken in the sense of
3323:
3293:
1294:
is continuous and is zero outside of a bounded set. In general, the mass measure
3387:
2300:. So the potential can be expanded in a series that is convergent for positions
1376:
from the center, giving the gravitational potential inside the sphere, which is
867:
3355:
Einstein's
General Theory of Relativity: With Modern Applications in Cosmology
3051:
1335:
1283:{\displaystyle \rho (\mathbf {x} )={\frac {1}{4\pi G}}\Delta V(\mathbf {x} ).}
78:
1720:
and the point at which the potential is being computed at the head of vector
3586:. Saint Louis University. California Institute of Technology. Archived from
3033:
2996:
2964:
62:
1356:
Within a spherically symmetric mass distribution, it is possible to solve
450:
separately. The potential has units of energy per mass, e.g., J/kg in the
466:
454:
system. By convention, it is always negative where it is defined, and as
3552:
10.1002/1521-3994(200012)321:5/6<363::AID-ASNA363>3.0.CO;2-X
82:
30:"Gravity potential" redirects here. For Earth's gravity potential, see
1342:. On the surface of the earth, the acceleration is given by so-called
77:. The reference point, where the potential is zero, is by convention
58:
3704:
1198:
is a potential function coming from a continuous mass distribution
3097:
2986:
2956:
1707:
819:
3187:
Mathematical
Methods For Physicists International Student Edition
88:
In mathematics, the gravitational potential is also known as the
81:
far away from any mass, resulting in a negative potential at any
1326:, with 𝜌 being a constant charge density) to electromagnetism.
70:
836:
contained in the distributed mass (gray) and differential mass
3624:"Prolate spheroidal harmonic expansion of gravitational field"
2991:
2960:
2100:(See "mathematical form".) The integrand can be expanded as a
3606:. Penn State Surveying Engineering Program. Archived from
1302:. As a consequence, the gravitational potential satisfies
3474:
Postprincipia: Gravitation for
Physicists and Astronomers
3218:
1308:
Green's function for the three-variable
Laplace equation
577:
pointing from the point mass toward the small body and
3358:, Springer Science & Business Media, p. 201,
3159:(4th ed.). Harcourt Brace & Company. p.
2699:
706:, then the potential of the distribution at the point
3657:
3130:(1st English ed.). Free Scientific Information.
2754:
2323:
2135:
1750:
1601:
1382:
1224:
1079:
894:
716:
661:{\displaystyle \|\mathbf {a} \|={\frac {GM}{x^{2}}}.}
618:
583:
475:
257:
184:
130:
1555:
Gravitational acceleration § General relativity
3379:
3152:
2932:
2730:
2685:
2249:
2064:
1698:
1537:
1282:
1184:
1027:) representing the density of the distribution at
985:
810:
660:
600:
561:
414:
211:
152:
1567:, the gravitational potential is replaced by the
3582:. Department of Earth and Atmospheric Sciences.
1372:inside the sphere varies linearly with distance
3559:Thornton, Stephen T.; Marion, Jerry B. (2003),
3328:(illustrated ed.). CRC Press. p. 19.
3083:Applications of Legendre polynomials in physics
2738:is the component of the center of mass in the
1728:The potential can be expanded in a series of
1559:Gravitational field § General relativity
8:
2744:direction; this vanishes because the vector
1360:. Within a uniform spherical body of radius
1141:
1125:
956:
940:
852:If the mass distribution is given as a mass
799:
776:
627:
619:
442:and is often known to higher precision than
3561:Classical Dynamics of Particles and Systems
3155:Classical Dynamics of particles and systems
1358:Poisson's equation in spherical coordinates
212:{\displaystyle \Delta U\approx mg\Delta h.}
34:. For the field of gravity potentials, see
3298:. Cambridge University Press. p. 69.
3295:A Student's Guide to Geophysical Equations
3184:Arfken, George B.; Weber, Hans J. (2005).
1070:, then the gravitational potential is the
3647:
3520:
2916:
2883:
2873:
2867:
2855:
2850:
2845:
2839:
2823:
2818:
2813:
2807:
2796:
2791:
2786:
2775:
2761:
2753:
2721:
2698:
2671:
2661:
2626:
2616:
2610:
2598:
2593:
2588:
2582:
2553:
2548:
2543:
2537:
2511:
2506:
2501:
2495:
2474:
2464:
2443:
2433:
2421:
2416:
2411:
2405:
2394:
2383:
2368:
2363:
2358:
2352:
2334:
2324:
2322:
2232:
2222:
2212:
2201:
2179:
2175:
2164:
2134:
2050:
2040:
2031:
2019:
2014:
2009:
2003:
1978:
1973:
1968:
1962:
1947:
1939:
1935:
1934:
1932:
1920:
1915:
1910:
1904:
1883:
1873:
1864:
1859:
1853:
1848:
1840:
1832:
1820:
1815:
1809:
1804:
1798:
1790:
1786:
1785:
1783:
1761:
1751:
1749:
1685:
1665:
1660:
1652:
1647:
1641:
1633:
1629:
1628:
1626:
1608:
1600:
1508:
1492:
1474:
1456:
1442:
1426:
1398:
1381:
1269:
1242:
1231:
1223:
1171:
1154:
1147:
1136:
1128:
1119:
1111:
1107:
1106:
1104:
1086:
1078:
972:
962:
951:
943:
934:
926:
922:
921:
919:
901:
893:
888:. In good cases this equals the integral
793:
788:
779:
768:
758:
749:
738:
723:
715:
647:
633:
622:
617:
587:
585:
584:
582:
545:
543:
542:
534:
520:
509:
501:
487:
476:
474:
394:
374:
357:
351:
346:
332:
324:
313:
307:
302:
288:
275:
264:
256:
183:
137:
129:
53:associating with each point in space the
3325:An Introduction to Planetary Atmospheres
2974:
434:is the gravitational force. The product
3664:
3580:"Gravity and Earth's Density Structure"
3352:Grøn, Øyvind; Hervik, Sigbjorn (2007),
3118:
2286:are the Legendre polynomials of degree
458:tends to infinity, it approaches zero.
3151:Marion, J.B.; Thornton, S.T. (1995).
601:{\displaystyle {\hat {\mathbf {x} }}}
114:) at a location is the gravitational
27:Fundamental study of potential theory
7:
2731:{\textstyle \int r\cos(\theta )\,dm}
92:and is fundamental in the study of
2395:
2213:
1260:
347:
303:
200:
185:
122:) at that location per unit mass:
25:
3602:Charles D. Ghilani (2006-11-28).
3407:Equations of mathematical physics
153:{\displaystyle V={\frac {U}{m}},}
3751:
3739:
3727:
3715:
3703:
3691:
3679:
3667:
3604:"The Gravity Field of the Earth"
3522:10.1046/j.1365-8711.2000.03524.x
3382:Advanced Engineering Mathematics
3322:Sanchez-Lavega, Agustin (2011).
3088:Standard gravitational parameter
2917:
2851:
2819:
2792:
2762:
2672:
2594:
2549:
2507:
2475:
2417:
2364:
2335:
2051:
2015:
1974:
1916:
1884:
1854:
1841:
1833:
1810:
1762:
1686:
1661:
1653:
1609:
1270:
1232:
1172:
1155:
1137:
1129:
1087:
973:
952:
944:
902:
789:
780:
724:
670:The potential associated with a
623:
588:
546:
510:
477:
440:standard gravitational parameter
325:
314:
265:
3292:Lowrie, William Lowrie (2011).
1522:
3036:(17,000 million km from Earth)
2921:
2913:
2856:
2846:
2824:
2814:
2797:
2787:
2766:
2758:
2718:
2712:
2676:
2668:
2599:
2589:
2554:
2544:
2512:
2502:
2479:
2471:
2461:
2449:
2422:
2412:
2369:
2359:
2339:
2331:
2244:
2238:
2129:for the Legendre polynomials:
2125:. The resulting series is the
2055:
2047:
2020:
2010:
1979:
1969:
1921:
1911:
1888:
1880:
1860:
1849:
1816:
1805:
1766:
1758:
1690:
1682:
1666:
1648:
1613:
1605:
1392:
1386:
1290:This holds pointwise whenever
1274:
1266:
1236:
1228:
1176:
1168:
1159:
1151:
1091:
1083:
977:
969:
906:
898:
728:
720:
592:
550:
269:
261:
106:Gravitational potential energy
1:
3563:(5th ed.), Brooks Cole,
2072:where, in the last integral,
110:The gravitational potential (
18:Gravitational potential field
866:, then the potential is the
3649:10.1088/0004-6256/147/6/152
3444:10.1088/0305-4470/21/22/026
3279:The Theory of the Potential
1581:Spherical multipole moments
1210:can be recovered using the
245:can be defined as the work
3805:
3789:Scalar physical quantities
3622:Fukushima, Toshio (2014).
3465:10.1088/0305-4470/23/4/027
3405:Vladimirov, V. S. (1971),
3386:(2nd ed.). New York:
3223:Cambridge University Press
3071:gravity at these locations
1578:
1552:
1368:, the gravitational force
172:gravitational acceleration
103:
29:
3378:Wylie, C. R. Jr. (1960).
3246:Muncaster, Roger (1993).
2980:
2977:
2947:, the opposite is true.)
1589:The potential at a point
1019:. If there is a function
3501:Mon. Not. R. Astron. Soc
3495:Conway, John T. (2000).
3277:MacMillan, W.D. (1958).
3252:(illustrated ed.).
3221:(illustrated ed.).
3126:Solivérez, C.E. (2016).
3472:Rastall, Peter (1991).
1732:. Represent the points
844:) located at the point
47:gravitational potential
3584:EAS-437 Earth Dynamics
2934:
2732:
2687:
2399:
2251:
2217:
2066:
1725:
1700:
1539:
1364:, density ρ, and mass
1284:
1186:
987:
849:
812:
754:
662:
602:
573:is a vector of length
563:
428:gravitational constant
416:
213:
154:
3453:J. Phys. A: Math. Gen
3424:J. Phys. A: Math. Gen
2935:
2733:
2688:
2379:
2252:
2197:
2089:is the angle between
2067:
1711:
1701:
1540:
1285:
1187:
988:
859:on three-dimensional
823:
813:
734:
663:
603:
564:
417:
214:
155:
3578:Zhu, Lupeia (1988).
2752:
2697:
2321:
2133:
1748:
1730:Legendre polynomials
1599:
1380:
1222:
1077:
892:
714:
616:
581:
473:
255:
182:
128:
73:playing the role of
3640:2014AJ....147..152F
3544:2000AN....321..363C
3513:2000MNRAS.316..555C
3436:1988JPhA...21.4245W
2277:. The coefficients
2127:generating function
1585:Multipole expansion
1575:Multipole expansion
1320:reference ellipsoid
1312:Newtonian potential
1066:) is the Euclidean
1011:between the points
463:gravitational field
356:
312:
90:Newtonian potential
43:classical mechanics
36:Gravitational field
3341:Extract of page 19
3311:Extract of page 68
3108:Geopotential model
2930:
2728:
2683:
2681:
2247:
2062:
2060:
1726:
1696:
1565:general relativity
1549:General relativity
1535:
1330:Spherical symmetry
1304:Poisson's equation
1280:
1182:
983:
850:
808:
658:
610:inverse square law
598:
559:
412:
342:
298:
226:The gravitational
209:
150:
67:electric potential
3570:978-0-534-40896-1
3430:(22): 4245–4250.
3365:978-0-387-69200-5
3335:978-1-4200-6735-4
3305:978-1-139-49924-8
3263:978-0-7487-1584-8
3232:978-1-107-69769-0
3201:978-0-08-047069-6
3137:978-987-28304-0-3
3067:
3066:
2905:
2861:
2829:
2802:
2648:
2604:
2559:
2517:
2427:
2374:
2193:
2187:
2038:
2037:
2025:
1984:
1926:
1871:
1870:
1675:
1671:
1481:
1406:
1258:
1145:
960:
803:
672:mass distribution
653:
595:
553:
540:
507:
407:
380:
340:
296:
283:
222:Mathematical form
145:
16:(Redirected from
3796:
3774:Energy (physics)
3756:
3755:
3744:
3743:
3742:
3732:
3731:
3730:
3720:
3719:
3718:
3708:
3707:
3696:
3695:
3694:
3684:
3683:
3672:
3671:
3663:
3653:
3651:
3618:
3616:
3615:
3598:
3596:
3595:
3573:
3555:
3538:(5/6): 363–372.
3526:
3524:
3491:
3480:. pp. 7ff.
3478:World Scientific
3468:
3447:
3417:
3392:
3391:
3385:
3375:
3369:
3368:
3349:
3343:
3339:
3319:
3313:
3309:
3289:
3283:
3282:
3274:
3268:
3267:
3243:
3237:
3236:
3212:
3206:
3205:
3190:(6th ed.).
3181:
3175:
3174:
3158:
3148:
3142:
3141:
3123:
2981:with respect to
2975:
2951:Numerical values
2943:distance to the
2939:
2937:
2936:
2931:
2920:
2906:
2901:
2888:
2887:
2874:
2872:
2871:
2866:
2862:
2860:
2859:
2854:
2849:
2840:
2830:
2828:
2827:
2822:
2817:
2808:
2803:
2801:
2800:
2795:
2790:
2784:
2776:
2765:
2743:
2737:
2735:
2734:
2729:
2692:
2690:
2689:
2684:
2682:
2675:
2660:
2656:
2649:
2644:
2631:
2630:
2617:
2615:
2614:
2609:
2605:
2603:
2602:
2597:
2592:
2583:
2564:
2560:
2558:
2557:
2552:
2547:
2538:
2518:
2516:
2515:
2510:
2505:
2496:
2485:
2478:
2448:
2447:
2438:
2437:
2432:
2428:
2426:
2425:
2420:
2415:
2406:
2398:
2393:
2375:
2373:
2372:
2367:
2362:
2353:
2338:
2316:
2314:
2299:
2276:
2274:
2266:
2264:
2256:
2254:
2253:
2248:
2237:
2236:
2227:
2226:
2216:
2211:
2191:
2190:
2189:
2188:
2180:
2174:
2170:
2169:
2168:
2123:binomial theorem
2120:
2118:
2088:
2084:
2082:
2071:
2069:
2068:
2063:
2061:
2054:
2039:
2036:
2035:
2030:
2026:
2024:
2023:
2018:
2013:
2004:
1985:
1983:
1982:
1977:
1972:
1963:
1952:
1948:
1946:
1945:
1944:
1943:
1938:
1927:
1925:
1924:
1919:
1914:
1905:
1894:
1887:
1872:
1869:
1868:
1863:
1857:
1852:
1844:
1836:
1825:
1824:
1819:
1813:
1808:
1803:
1799:
1797:
1796:
1795:
1794:
1789:
1765:
1742:position vectors
1705:
1703:
1702:
1697:
1689:
1673:
1672:
1670:
1669:
1664:
1656:
1651:
1642:
1640:
1639:
1638:
1637:
1632:
1612:
1594:
1544:
1542:
1541:
1536:
1518:
1514:
1513:
1512:
1497:
1496:
1482:
1480:
1479:
1478:
1465:
1457:
1452:
1448:
1447:
1446:
1431:
1430:
1407:
1399:
1344:standard gravity
1289:
1287:
1286:
1281:
1273:
1259:
1257:
1243:
1235:
1217:
1212:Laplace operator
1191:
1189:
1188:
1183:
1175:
1158:
1146:
1144:
1140:
1132:
1120:
1118:
1117:
1116:
1115:
1110:
1090:
1057:
1006:
1004:
992:
990:
989:
984:
976:
961:
959:
955:
947:
935:
933:
932:
931:
930:
925:
905:
883:
881:
817:
815:
814:
809:
804:
802:
798:
797:
792:
783:
774:
773:
772:
759:
753:
748:
727:
690:and have masses
667:
665:
664:
659:
654:
652:
651:
642:
634:
626:
607:
605:
604:
599:
597:
596:
591:
586:
568:
566:
565:
560:
555:
554:
549:
544:
541:
539:
538:
529:
521:
513:
508:
506:
505:
496:
488:
480:
421:
419:
418:
413:
408:
403:
395:
381:
379:
378:
369:
358:
355:
350:
341:
333:
328:
317:
311:
306:
297:
289:
284:
276:
268:
218:
216:
215:
210:
159:
157:
156:
151:
146:
138:
116:potential energy
100:Potential energy
94:potential theory
21:
3804:
3803:
3799:
3798:
3797:
3795:
3794:
3793:
3764:
3763:
3762:
3750:
3740:
3738:
3728:
3726:
3716:
3714:
3702:
3692:
3690:
3678:
3666:
3658:
3656:
3621:
3613:
3611:
3601:
3593:
3591:
3577:
3571:
3558:
3529:
3494:
3488:
3471:
3450:
3421:
3404:
3400:
3395:
3390:. p. 454 .
3377:
3376:
3372:
3366:
3351:
3350:
3346:
3336:
3321:
3320:
3316:
3306:
3291:
3290:
3286:
3276:
3275:
3271:
3264:
3256:. p. 106.
3249:A-level Physics
3245:
3244:
3240:
3233:
3225:. p. 276.
3214:
3213:
3209:
3202:
3183:
3182:
3178:
3171:
3150:
3149:
3145:
3138:
3125:
3124:
3120:
3116:
3079:
3003:Earth's surface
2970:escape velocity
2953:
2879:
2875:
2844:
2835:
2834:
2812:
2785:
2777:
2750:
2749:
2739:
2695:
2694:
2680:
2679:
2622:
2618:
2587:
2578:
2577:
2542:
2533:
2526:
2522:
2500:
2483:
2482:
2439:
2410:
2401:
2400:
2357:
2342:
2319:
2318:
2310:
2305:
2291:
2285:
2270:
2268:
2260:
2258:
2228:
2218:
2160:
2141:
2137:
2136:
2131:
2130:
2114:
2105:
2086:
2078:
2073:
2059:
2058:
2008:
1999:
1998:
1967:
1933:
1928:
1909:
1892:
1891:
1858:
1814:
1784:
1779:
1769:
1746:
1745:
1646:
1627:
1622:
1597:
1596:
1590:
1587:
1579:Main articles:
1577:
1561:
1551:
1504:
1488:
1487:
1483:
1470:
1466:
1458:
1438:
1422:
1421:
1417:
1378:
1377:
1351:oblate spheroid
1332:
1247:
1220:
1219:
1215:
1124:
1105:
1100:
1075:
1074:
1072:volume integral
1032:
996:
994:
939:
920:
915:
890:
889:
877:
871:
861:Euclidean space
787:
775:
764:
760:
712:
711:
705:
696:
689:
680:
643:
635:
614:
613:
579:
578:
530:
522:
497:
489:
471:
470:
396:
370:
359:
253:
252:
224:
180:
179:
126:
125:
108:
102:
39:
28:
23:
22:
15:
12:
11:
5:
3802:
3800:
3792:
3791:
3786:
3781:
3776:
3766:
3765:
3761:
3760:
3748:
3736:
3724:
3712:
3700:
3688:
3676:
3655:
3654:
3619:
3599:
3575:
3569:
3556:
3527:
3507:(3): 555–558.
3492:
3486:
3469:
3459:(4): 581–584.
3448:
3419:
3401:
3399:
3396:
3394:
3393:
3370:
3364:
3344:
3334:
3314:
3304:
3284:
3281:. Dover Press.
3269:
3262:
3254:Nelson Thornes
3238:
3231:
3207:
3200:
3194:. p. 72.
3192:Academic Press
3176:
3169:
3143:
3136:
3117:
3115:
3112:
3111:
3110:
3105:
3100:
3095:
3085:
3078:
3075:
3065:
3064:
3061:
3058:
3055:
3047:
3046:
3043:
3040:
3037:
3030:
3029:
3026:
3023:
3020:
3014:
3013:
3010:
3007:
3004:
3000:
2999:
2994:
2989:
2983:
2982:
2979:
2952:
2949:
2929:
2926:
2923:
2919:
2915:
2912:
2909:
2904:
2900:
2897:
2894:
2891:
2886:
2882:
2878:
2870:
2865:
2858:
2853:
2848:
2843:
2838:
2833:
2826:
2821:
2816:
2811:
2806:
2799:
2794:
2789:
2783:
2780:
2774:
2771:
2768:
2764:
2760:
2757:
2727:
2724:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2678:
2674:
2670:
2667:
2664:
2659:
2655:
2652:
2647:
2643:
2640:
2637:
2634:
2629:
2625:
2621:
2613:
2608:
2601:
2596:
2591:
2586:
2581:
2576:
2573:
2570:
2567:
2563:
2556:
2551:
2546:
2541:
2536:
2532:
2529:
2525:
2521:
2514:
2509:
2504:
2499:
2494:
2491:
2488:
2486:
2484:
2481:
2477:
2473:
2470:
2467:
2463:
2460:
2457:
2454:
2451:
2446:
2442:
2436:
2431:
2424:
2419:
2414:
2409:
2404:
2397:
2392:
2389:
2386:
2382:
2378:
2371:
2366:
2361:
2356:
2351:
2348:
2345:
2343:
2341:
2337:
2333:
2330:
2327:
2326:
2281:
2246:
2243:
2240:
2235:
2231:
2225:
2221:
2215:
2210:
2207:
2204:
2200:
2196:
2186:
2183:
2178:
2173:
2167:
2163:
2159:
2156:
2153:
2150:
2147:
2144:
2140:
2057:
2053:
2049:
2046:
2043:
2034:
2029:
2022:
2017:
2012:
2007:
2002:
1997:
1994:
1991:
1988:
1981:
1976:
1971:
1966:
1961:
1958:
1955:
1951:
1942:
1937:
1931:
1923:
1918:
1913:
1908:
1903:
1900:
1897:
1895:
1893:
1890:
1886:
1882:
1879:
1876:
1867:
1862:
1856:
1851:
1847:
1843:
1839:
1835:
1831:
1828:
1823:
1818:
1812:
1807:
1802:
1793:
1788:
1782:
1778:
1775:
1772:
1770:
1768:
1764:
1760:
1757:
1754:
1753:
1695:
1692:
1688:
1684:
1681:
1678:
1668:
1663:
1659:
1655:
1650:
1645:
1636:
1631:
1625:
1621:
1618:
1615:
1611:
1607:
1604:
1576:
1573:
1550:
1547:
1534:
1531:
1528:
1525:
1521:
1517:
1511:
1507:
1503:
1500:
1495:
1491:
1486:
1477:
1473:
1469:
1464:
1461:
1455:
1451:
1445:
1441:
1437:
1434:
1429:
1425:
1420:
1416:
1413:
1410:
1405:
1402:
1397:
1394:
1391:
1388:
1385:
1331:
1328:
1279:
1276:
1272:
1268:
1265:
1262:
1256:
1253:
1250:
1246:
1241:
1238:
1234:
1230:
1227:
1181:
1178:
1174:
1170:
1167:
1164:
1161:
1157:
1153:
1150:
1143:
1139:
1135:
1131:
1127:
1123:
1114:
1109:
1103:
1099:
1096:
1093:
1089:
1085:
1082:
1068:volume element
982:
979:
975:
971:
968:
965:
958:
954:
950:
946:
942:
938:
929:
924:
918:
914:
911:
908:
904:
900:
897:
807:
801:
796:
791:
786:
782:
778:
771:
767:
763:
757:
752:
747:
744:
741:
737:
733:
730:
726:
722:
719:
701:
694:
685:
678:
657:
650:
646:
641:
638:
632:
629:
625:
621:
594:
590:
558:
552:
548:
537:
533:
528:
525:
519:
516:
512:
504:
500:
495:
492:
486:
483:
479:
411:
406:
402:
399:
393:
390:
387:
384:
377:
373:
368:
365:
362:
354:
349:
345:
339:
336:
331:
327:
323:
320:
316:
310:
305:
301:
295:
292:
287:
282:
279:
274:
271:
267:
263:
260:
233:at a distance
223:
220:
208:
205:
202:
199:
196:
193:
190:
187:
149:
144:
141:
136:
133:
104:Main article:
101:
98:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3801:
3790:
3787:
3785:
3782:
3780:
3777:
3775:
3772:
3771:
3769:
3759:
3754:
3749:
3747:
3737:
3735:
3725:
3723:
3713:
3711:
3706:
3701:
3699:
3689:
3687:
3682:
3677:
3675:
3670:
3665:
3661:
3650:
3645:
3641:
3637:
3633:
3629:
3625:
3620:
3610:on 2011-07-18
3609:
3605:
3600:
3590:on 2011-07-26
3589:
3585:
3581:
3576:
3572:
3566:
3562:
3557:
3553:
3549:
3545:
3541:
3537:
3533:
3532:Astron. Nachr
3528:
3523:
3518:
3514:
3510:
3506:
3502:
3498:
3493:
3489:
3487:981-02-0778-6
3483:
3479:
3475:
3470:
3466:
3462:
3458:
3454:
3449:
3445:
3441:
3437:
3433:
3429:
3425:
3420:
3416:
3412:
3408:
3403:
3402:
3397:
3389:
3384:
3383:
3374:
3371:
3367:
3361:
3357:
3356:
3348:
3345:
3342:
3337:
3331:
3327:
3326:
3318:
3315:
3312:
3307:
3301:
3297:
3296:
3288:
3285:
3280:
3273:
3270:
3265:
3259:
3255:
3251:
3250:
3242:
3239:
3234:
3228:
3224:
3220:
3219:
3211:
3208:
3203:
3197:
3193:
3189:
3188:
3180:
3177:
3172:
3170:0-03-097302-3
3166:
3162:
3157:
3156:
3147:
3144:
3139:
3133:
3129:
3122:
3119:
3113:
3109:
3106:
3104:
3101:
3099:
3096:
3093:
3089:
3086:
3084:
3081:
3080:
3076:
3074:
3072:
3062:
3059:
3056:
3053:
3049:
3048:
3044:
3041:
3038:
3035:
3032:
3031:
3027:
3024:
3021:
3019:
3016:
3015:
3011:
3008:
3005:
3002:
3001:
2998:
2995:
2993:
2990:
2988:
2985:
2984:
2976:
2973:
2971:
2966:
2962:
2958:
2950:
2948:
2946:
2940:
2927:
2924:
2910:
2907:
2902:
2898:
2895:
2892:
2889:
2884:
2880:
2876:
2868:
2863:
2841:
2836:
2831:
2809:
2804:
2781:
2778:
2772:
2769:
2755:
2747:
2742:
2725:
2722:
2715:
2709:
2706:
2703:
2700:
2693:The integral
2665:
2662:
2657:
2653:
2650:
2645:
2641:
2638:
2635:
2632:
2627:
2623:
2619:
2611:
2606:
2584:
2579:
2574:
2571:
2568:
2565:
2561:
2539:
2534:
2530:
2527:
2523:
2519:
2497:
2492:
2489:
2487:
2468:
2465:
2458:
2455:
2452:
2444:
2440:
2434:
2429:
2407:
2402:
2390:
2387:
2384:
2380:
2376:
2354:
2349:
2346:
2344:
2328:
2313:
2308:
2303:
2298:
2294:
2289:
2284:
2280:
2275:| < 1
2273:
2263:
2241:
2233:
2229:
2223:
2219:
2208:
2205:
2202:
2198:
2194:
2184:
2181:
2176:
2171:
2165:
2161:
2157:
2154:
2151:
2148:
2145:
2142:
2138:
2128:
2124:
2117:
2112:
2108:
2103:
2102:Taylor series
2098:
2096:
2092:
2081:
2076:
2044:
2041:
2032:
2027:
2005:
2000:
1995:
1992:
1989:
1986:
1964:
1959:
1956:
1953:
1949:
1940:
1929:
1906:
1901:
1898:
1896:
1877:
1874:
1865:
1845:
1837:
1829:
1826:
1821:
1800:
1791:
1780:
1776:
1773:
1771:
1755:
1743:
1739:
1735:
1731:
1723:
1719:
1715:
1710:
1706:
1693:
1679:
1676:
1657:
1643:
1634:
1623:
1619:
1616:
1602:
1593:
1586:
1582:
1574:
1572:
1570:
1569:metric tensor
1566:
1560:
1556:
1548:
1546:
1532:
1529:
1526:
1523:
1519:
1515:
1509:
1505:
1501:
1498:
1493:
1489:
1484:
1475:
1471:
1467:
1462:
1459:
1453:
1449:
1443:
1439:
1435:
1432:
1427:
1423:
1418:
1414:
1411:
1408:
1403:
1400:
1395:
1389:
1383:
1375:
1371:
1367:
1363:
1359:
1354:
1352:
1348:
1345:
1341:
1340:shell theorem
1337:
1329:
1327:
1325:
1321:
1315:
1313:
1309:
1305:
1301:
1300:distributions
1297:
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1165:
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1018:
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948:
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927:
916:
912:
909:
895:
887:
880:
875:
869:
865:
862:
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847:
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839:
835:
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827:
822:
818:
805:
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769:
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755:
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745:
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739:
735:
731:
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677:
673:
668:
655:
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644:
639:
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611:
576:
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531:
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360:
352:
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280:
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272:
258:
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68:
64:
60:
56:
52:
48:
44:
37:
33:
19:
3746:Solar System
3631:
3628:Astrophys. J
3627:
3612:. Retrieved
3608:the original
3592:. Retrieved
3588:the original
3583:
3560:
3535:
3531:
3504:
3500:
3473:
3456:
3452:
3427:
3423:
3406:
3381:
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3347:
3324:
3317:
3294:
3287:
3278:
3272:
3248:
3241:
3217:
3210:
3186:
3179:
3154:
3146:
3127:
3121:
3103:Geopotential
3091:
3069:Compare the
3068:
3063:≥ 130 GJ/kg
3045:≥ 130 GJ/kg
3028:≥ 130 GJ/kg
3012:≥ 130 GJ/kg
2954:
2944:
2941:
2745:
2740:
2311:
2306:
2301:
2296:
2292:
2287:
2282:
2278:
2271:
2261:
2115:
2110:
2106:
2099:
2094:
2090:
2079:
2074:
1737:
1733:
1727:
1721:
1717:
1713:
1595:is given by
1591:
1588:
1562:
1373:
1369:
1365:
1361:
1355:
1346:
1333:
1323:
1316:
1295:
1291:
1207:
1203:
1199:
1195:
1193:
1063:
1059:
1053:
1049:
1045:
1041:
1037:
1033:
1028:
1024:
1020:
1016:
1012:
1001:
997:
885:
878:
873:
863:
856:
851:
845:
841:
837:
833:
829:
825:
707:
702:
698:
691:
686:
682:
675:
669:
574:
570:
460:
455:
447:
443:
435:
431:
423:
251:
246:
242:
234:
230:
225:
175:
169:
163:
161:
124:
119:
111:
109:
87:
51:scalar field
46:
40:
32:Geopotential
3734:Outer space
3722:Spaceflight
3686:Mathematics
3388:McGraw-Hill
2309:< |
1306:. See also
868:convolution
3784:Potentials
3768:Categories
3634:(6): 152.
3614:2009-03-25
3594:2009-03-25
3398:References
3054:from Earth
3052:light-year
2963:, and the
2304:such that
2265:| ≤ 1
2257:valid for
1553:See also:
1336:point mass
1031:, so that
239:point mass
85:distance.
79:infinitely
3698:Astronomy
3060:140 kJ/kg
3034:Voyager 1
3025:900 MJ/kg
3009:900 MJ/kg
2997:Milky Way
2978:Location
2965:Milky Way
2928:⋯
2896:−
2893:θ
2890:
2832:∫
2805:−
2773:−
2716:θ
2710:
2701:∫
2654:⋯
2639:−
2636:θ
2633:
2572:θ
2569:
2520:∫
2493:−
2459:θ
2456:
2396:∞
2381:∑
2377:∫
2350:−
2214:∞
2199:∑
2177:−
2146:−
1993:θ
1990:
1957:−
1930:∫
1902:−
1838:⋅
1827:−
1781:∫
1777:−
1658:−
1624:∫
1620:−
1527:≤
1499:−
1433:−
1415:ρ
1409:π
1338:, by the
1261:Δ
1252:π
1226:ρ
1149:ρ
1142:‖
1134:−
1126:‖
1102:∫
1098:−
957:‖
949:−
941:‖
917:∫
913:−
800:‖
785:−
777:‖
756:−
736:∑
628:‖
620:‖
593:^
551:^
518:−
485:−
392:−
348:∞
344:∫
319:⋅
304:∞
300:∫
228:potential
201:Δ
192:≈
186:Δ
63:analogous
3077:See also
3057:0.4 J/kg
3022:57 MJ/kg
3006:60 MJ/kg
2077:= |
1206:), then
1058:, where
1009:distance
467:gradient
241:of mass
3779:Gravity
3758:Science
3674:Physics
3660:Portals
3636:Bibcode
3540:Bibcode
3509:Bibcode
3432:Bibcode
3415:0268497
3042:8 MJ/kg
3039:23 J/kg
2945:surface
2113:/|
1007:is the
876:/|
854:measure
832:, with
824:Points
697:, ...,
681:, ...,
438:is the
426:is the
237:from a
65:to the
3567:
3484:
3413:
3362:
3332:
3302:
3260:
3229:
3198:
3167:
3134:
2959:, the
2315:|
2295:= cos
2269:|
2259:|
2192:
2119:|
2083:|
1674:
1557:, and
1005:|
995:|
993:where
882:|
569:where
430:, and
422:where
162:where
83:finite
75:charge
59:energy
45:, the
3710:Stars
3114:Notes
3098:Geoid
2987:Earth
2957:Earth
884:with
69:with
49:is a
3565:ISBN
3482:ISBN
3360:ISBN
3330:ISBN
3300:ISBN
3258:ISBN
3227:ISBN
3196:ISBN
3165:ISBN
3132:ISBN
3050:0.1
2267:and
2093:and
2085:and
1736:and
1716:and
1583:and
1310:and
1040:) =
1015:and
828:and
461:The
71:mass
55:work
3644:doi
3632:147
3548:doi
3536:321
3517:doi
3505:316
3461:doi
3440:doi
3161:192
3018:LEO
2992:Sun
2961:Sun
2881:cos
2707:cos
2624:cos
2566:cos
2453:cos
2104:in
1987:cos
1740:as
1563:In
1194:If
870:of
710:is
452:MKS
446:or
41:In
3770::
3642:.
3630:.
3626:.
3546:.
3534:.
3515:.
3503:.
3499:.
3476:.
3457:23
3455:.
3438:.
3428:21
3426:.
3411:MR
3163:.
3092:GM
3073:.
2972:.
2109:=
2097:.
1353:.
1314:.
1296:dm
1218::
1214:,
1060:dv
1050:dv
1048:)
1034:dm
1000:−
886:dm
857:dm
838:dm
612::
436:GM
174:,
3662::
3652:.
3646::
3638::
3617:.
3597:.
3574:.
3554:.
3550::
3542::
3525:.
3519::
3511::
3490:.
3467:.
3463::
3446:.
3442::
3434::
3418:.
3338:.
3308:.
3266:.
3235:.
3204:.
3173:.
3140:.
3094:)
3090:(
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2837:(
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2820:x
2815:|
2810:G
2798:|
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2763:x
2759:(
2756:V
2746:x
2741:x
2726:m
2723:d
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2673:r
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2590:|
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2535:(
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2476:r
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2450:(
2445:n
2441:P
2435:n
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2423:|
2418:x
2413:|
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2403:(
2391:0
2388:=
2385:n
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2365:x
2360:|
2355:G
2347:=
2340:)
2336:x
2332:(
2329:V
2312:x
2307:r
2302:x
2297:θ
2293:X
2288:n
2283:n
2279:P
2272:Z
2262:X
2245:)
2242:X
2239:(
2234:n
2230:P
2224:n
2220:Z
2209:0
2206:=
2203:n
2195:=
2185:2
2182:1
2172:)
2166:2
2162:Z
2158:+
2155:Z
2152:X
2149:2
2143:1
2139:(
2116:x
2111:r
2107:Z
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2087:θ
2080:r
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2052:r
2048:(
2045:m
2042:d
2033:2
2028:)
2021:|
2016:x
2011:|
2006:r
2001:(
1996:+
1980:|
1975:x
1970:|
1965:r
1960:2
1954:1
1950:G
1941:3
1936:R
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1917:x
1912:|
1907:1
1899:=
1889:)
1885:r
1881:(
1878:m
1875:d
1866:2
1861:|
1855:r
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1806:|
1801:G
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1774:=
1767:)
1763:x
1759:(
1756:V
1738:r
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1724:.
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1694:.
1691:)
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1617:=
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1610:x
1606:(
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1510:2
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1485:[
1476:3
1472:R
1468:2
1463:m
1460:G
1454:=
1450:]
1444:2
1440:R
1436:3
1428:2
1424:r
1419:[
1412:G
1404:3
1401:2
1396:=
1393:)
1390:r
1387:(
1384:V
1374:r
1370:g
1366:m
1362:R
1347:g
1324:G
1292:ρ
1278:.
1275:)
1271:x
1267:(
1264:V
1255:G
1249:4
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1240:=
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1233:x
1229:(
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1208:ρ
1204:r
1202:(
1200:ρ
1196:V
1180:.
1177:)
1173:r
1169:(
1166:v
1163:d
1160:)
1156:r
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1138:r
1130:x
1122:G
1113:3
1108:R
1095:=
1092:)
1088:x
1084:(
1081:V
1064:r
1062:(
1056:)
1054:r
1052:(
1046:r
1044:(
1042:ρ
1038:r
1036:(
1029:r
1025:r
1023:(
1021:ρ
1017:r
1013:x
1002:r
998:x
981:,
978:)
974:r
970:(
967:m
964:d
953:r
945:x
937:G
928:3
923:R
910:=
907:)
903:x
899:(
896:V
879:r
874:G
872:−
864:R
848:.
846:r
842:r
840:(
834:r
830:r
826:x
806:.
795:i
790:x
781:x
770:i
766:m
762:G
751:n
746:1
743:=
740:i
732:=
729:)
725:x
721:(
718:V
708:x
703:n
699:m
695:1
692:m
687:n
683:x
679:1
676:x
656:.
649:2
645:x
640:M
637:G
631:=
624:a
589:x
575:x
571:x
557:,
547:x
536:2
532:x
527:M
524:G
515:=
511:x
503:3
499:x
494:M
491:G
482:=
478:a
456:x
448:M
444:G
432:F
424:G
410:,
405:x
401:M
398:G
389:=
386:x
383:d
376:2
372:x
367:M
364:m
361:G
353:x
338:m
335:1
330:=
326:x
322:d
315:F
309:x
294:m
291:1
286:=
281:m
278:W
273:=
270:)
266:x
262:(
259:V
247:W
243:M
235:x
231:V
207:.
204:h
198:g
195:m
189:U
176:g
164:m
148:,
143:m
140:U
135:=
132:V
120:U
118:(
112:V
57:(
38:.
20:)
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