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1898:). The slight changes that result from the terms in the equations – which themselves may have been simplified yet again – are used as corrections to the original solution. Because simplifications are made at every step, the corrections are never perfect, but even one cycle of corrections often provides a remarkably better approximate solution to the real problem.
1901:
There is no requirement to stop at only one cycle of corrections. A partially corrected solution can be re-used as the new starting point for yet another cycle of perturbations and corrections. In principle, for most problems the recycling and refining of prior solutions to obtain a new generation of
1920:
This general procedure – starting with a simplified problem and gradually adding corrections that make the starting point of the corrected problem closer to the real situation – is a widely used mathematical tool in advanced sciences and engineering. It is the natural extension of the "guess, check,
1980:
of the two larger celestial bodies. Other reference frames for n-body simulations include those that place the origin to follow the center of mass of a body, such as the heliocentric and the geocentric reference frames. The choice of reference frame gives rise to many phenomena, including the
2007:
2581:
Undergraduate level course by
Richard Fitzpatrick. This includes Lagrangian and Hamiltonian Dynamics and applications to celestial mechanics, gravitational potential theory, the 3-body problem and Lunar motion (an example of the 3-body problem with the Sun, Moon, and the
2224:
is an object that orbits another object (known as its primary). The term is often used to describe an artificial satellite (as opposed to natural satellites, or moons). The common noun ‘moon’ (not capitalized) is used to mean any
2384:
Guerra, André G C; Carvalho, Paulo Simeão (1 August 2016). "Orbital motions of astronomical bodies and their centre of mass from different reference frames: a conceptual step between the geocentric and heliocentric models".
1905:
The common difficulty with the method is that the corrections usually progressively make the new solutions very much more complicated, so each cycle is much more difficult to manage than the previous cycle of corrections.
1587:, France, in May 1886, the international consensus was that all ephemerides should be based on Newcomb's calculations. A further conference as late as 1950 confirmed Newcomb's constants as the international standard.
1583:, he recalculated all the major astronomical constants. After 1884 he conceived, with A.M.W. Downing, a plan to resolve much international confusion on the subject. By the time he attended a standardisation conference in
2211:
is orbital motion in a system, such as a planet and its satellites, that is contrary to the direction of rotation of the central body, or more generally contrary in direction to the net angular momentum of the entire
193:
2235:
is the combination of out-of-balance forces and accelerations of (mostly) solid bodies that raises tides in bodies of liquid (oceans), atmospheres, and strains planets' and satellites' crusts.
2101:
is a more exact theory than Newton's laws for calculating orbits, and it is sometimes necessary to use it for greater accuracy or in high-gravity situations (e.g. orbits near the Sun).
217:
1545:
to use a single polar coordinate equation to describe any orbit, even those that are parabolic and hyperbolic. This is useful for calculating the behaviour of planets and
1668:
4-body problem: spaceflight to Mars (for parts of the flight the influence of one or two bodies is very small, so that there we have a 2- or 3-body problem; see also the
1748:
1833:
comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly. (It is closely related to methods used in
2242:
are versions one mathematical theory for the orbits and positions of the major planets, which seeks to provide accurate positions over an extended period of time.
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2128:
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start with a simplified form of the original problem, which is carefully chosen to be exactly solvable. In celestial mechanics, this is usually a
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is a compilation of positions of naturally occurring astronomical objects as well as artificial satellites in the sky at a given time or times.
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is more recent than that. Newton wrote that the field should be called "rational mechanics". The term "dynamics" came in a little later with
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is a branch of mathematics, pioneered by celestial mechanicians, for calculating approximate numerical answers (such as the position of a
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472:
1922:
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is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity.
1471:(25 December 1642 – 31 March 1727) is credited with introducing the idea that the motion of objects in the heavens, such as
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2046:
1500:
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is a part of astronomy that deals with measuring the positions of stars and other celestial bodies, their distances and movements.
100:
1813:
A spacecraft orbiting Earth, a moon, or a planet (in the latter cases the approximation only applies after arrival at that orbit)
1306:
2205:
is the temporary
Keplerian orbit about a central body that an object would continue on, if other perturbations were not present.
1883:), or a circular orbit, which is only correct in special cases of two-body motion, but is often close enough for practical use.
72:
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closer to the values from the real problem, such as including the gravitational attraction of a third, more distant body (the
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is the periodic, apparently backwards motion of planetary bodies when viewed from the Earth (an accelerated reference frame).
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is a position fixing technique that was the first system devised to help sailors locate themselves on a featureless ocean.
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was the original goal of celestial mechanics, and has only been imperfectly achieved. It continues to motivate research.
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A further simplification is based on the "standard assumptions in astrodynamics", which include that one body, the
1396:. Prior to Kepler there was little connection between exact, quantitative prediction of planetary positions, using
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1426:(1571–1630) was the first to closely integrate the predictive geometrical astronomy, which had been dominant from
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1982:
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1452:. Kepler's elliptical model greatly improved the accuracy of predictions of planetary motion, years before
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changes, and interplanetary transfers, and is used by mission planners to predict the results of
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Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of
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2808:
2729:
2309:
Combot, Thierry (2015-09-01). "Integrability and non integrability of some n body problems".
2135:(JPL DE) is a widely used model of the solar system, which combines celestial mechanics with
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Problems in celestial mechanics are often posed in simplifying reference frames, such as the
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concerns pre-Newtonian explanations of the causes of the motions of the stars and planets.
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of masses are mutually interacting via the gravitational force. Although analytically not
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better solutions could continue indefinitely, to any desired finite degree of accuracy.
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was to deal with the otherwise unsolvable mathematical problems of celestial mechanics:
1575:(12 March 1835 – 11 July 1909) was a Canadian-American astronomer who revised
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techniques, and contemporary discussions of the physical causes of the planets' motion.
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Celestial
Mechanics is a Planetarium Artwork created by D. S. Hessels and G. Dunne
2357:
Great
Physicists: The life and times of leading physicists from Galileo to Hawking
1533:, analyzed the stability of planetary orbits, and discovered the existence of the
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1750:. In this case, the system is fully integrable and exact solutions can be found.
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Marshall
Hampton's research page: Central configurations in the n-body problem
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in the sky) which are too difficult to solve down to a general, exact formula.
2011:
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For detailed treatments of how his laws of planetary motion can be used, see
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and celestial mechanics to the practical problems concerning the motion of
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in the general case, the integration can be well approximated numerically.
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Doggett, LeRoy E. (1997), "Celestial
Mechanics", in Lankford, John (ed.),
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are the parameters needed to specify a
Newtonian two-body orbit uniquely.
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1758:
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1937:
1875:, which is correct when there are only two gravitating bodies (say, the
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for their explanation, but whose evolution proves the existence of
1529:(25 January 1736 – 10 April 1813) attempted to solve the
522:
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380:
2509:
Annals of the New York
Academy of Sciences, Vol. 1065, pp. 346-374
1598:(14 March 1879 – 18 April 1955) explained the anomalous
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2006:
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balls and falling apples, could be described by the same set of
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1614:– the first in 1974 – whose orbits not only require the use of
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for elliptical orbits by deriving them from the gravitational
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2624:
Professor Tatum's course notes at the
University of Victoria
2332:"Two-Body Problem -- from Eric Weisstein's World of Physics"
1911:
1880:
1850:
1480:
1444:, which he developed using his physical principles and the
1343:
395:
358:
3032:
2575:, high-school level educational web site by David P. Stern
1362:
For early theories of the causes of planetary motion, see
1334:. Historically, celestial mechanics applies principles of
2636:
Italian
Celestial Mechanics and Astrodynamics Association
2041:. The motion of these objects is usually calculated from
1895:
1862:
1557:). More recently, it has also become useful to calculate
1476:
188:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
1910:
is reported to have said, regarding the problem of the
1892:
time-rate-of-change equations for the object's position
1541:, emphasizing energy more than force, and developing a
1610:
did not provide the highest accuracy. Observations of
1437:
New Astronomy, Based upon Causes, or Celestial Physics
2997:
1730:
1700:
152:
2469:, New York: Taylor & Francis, pp. 131–140,
1634:
Celestial motion, without additional forces such as
1853:, which moves noticeably differently from a simple
60:. Unsourced material may be challenged and removed.
1742:
1712:
1604:The Foundation of the General Theory of Relativity
1579:'s table of lunar positions. In 1877, assisted by
187:
2133:Jet Propulsion Laboratory Developmental Ephemeris
1369:Modern analytic celestial mechanics started with
3081:
2167:attempts to account for the motions of the Moon.
2049:. Orbital mechanics is a core discipline within
1483:, and the motion of objects on the ground, like
1537:. Lagrange also reformulated the principles of
2383:
1549:and such (parabolic and hyperbolic orbits are
2675:
2661:
2517:Springer-Praxis 2010, XVI, 264 p., Hardcover
2491:Celestial Mechanics: The Waltz of the Planets
1724:) the configuration is much simpler than for
1300:
877:
1857:because of the competing gravitation of the
2515:Stability and Chaos in Celestial Mechanics.
1886:The solved, but simplified problem is then
2668:
2654:
2080:. Orbital mechanics focuses on spacecraft
1790:. This is also often approximately valid.
1307:
1293:
884:
870:
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2314:
2285:
2260:Trenti, Michele; Hut, Piet (2008-05-20).
2259:
1606:. This led astronomers to recognize that
120:Learn how and when to remove this message
2573:Astronomy of the Earth's Motion in Space
2060:, including both spacecraft and natural
2005:
1989:while on a geocentric reference frame.
1520:
2551:
2462:
2354:
2025:or astrodynamics is the application of
1629:
1622:, was a discovery that led to the 1993
1511:, which Newton included in his epochal
14:
3082:
2866:Atomic, molecular, and optical physics
2558:, University of Denver, archived from
2308:
1976:, where the origin coincides with the
1819:
1786:, is much smaller than the other, the
1434:, with physical concepts to produce a
2649:
2489:Alessandra Celletti, Ettore Perozzi,
2466:History of Astronomy: An Encyclopedia
2329:
2139:and astronomical and spacecraft data.
566:Newton's law of universal gravitation
2262:"N-body simulations (gravitational)"
1992:
1932:
58:adding citations to reliable sources
29:
2506:Gauge Freedom in Orbital Mechanics.
2484:Fundamentals of Celestial Mechanics
2448:John E. Prussing, Bruce A. Conway,
2436:Introduction to Celestial Mechanics
2182:numerical model of the solar system
1388:, and over a century after Newton,
1342:) to astronomical objects, such as
547:Mechanics of planar particle motion
155:
24:
2529:
1928:
1600:precession of Mercury's perihelion
1590:
1407:
25:
3116:
2545:
2143:Dynamics of the celestial spheres
2010:A satellite orbiting Earth has a
1849:'s solution for the orbit of the
1414:Kepler's laws of planetary motion
1364:Dynamics of the celestial spheres
1079:Kepler's laws of planetary motion
3067:
3055:
3043:
3031:
3019:
3007:
2552:Calvert, James B. (2003-03-28),
2536:Encyclopedia:Celestial mechanics
1998:This section is an excerpt from
1936:
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909:
851:
850:
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170:
34:
2987:Timeline of physics discoveries
1463:
1442:modern laws of planetary orbits
45:needs additional citations for
2377:
2348:
2323:
2302:
2253:
1841:.) The earliest use of modern
1683:Spaceflight to, and stay at a
13:
1:
2427:
2417:10.1088/0031-9120/51/5/055012
1916:"It causeth my head to ache."
1440:in 1609. His work led to the
473:Koopman–von Neumann mechanics
3100:Astronomical sub-disciplines
2355:Cropper, William H. (2004),
2047:law of universal gravitation
541:Non-inertial reference frame
7:
2951:Quantum information science
2105:
1923:used anciently with numbers
1801:orbiting the center of the
1670:patched conic approximation
1491:. In this sense he unified
1430:in the 2nd century to
1233:Tsiolkovsky rocket equation
468:Appell's equation of motion
338:Inertial frame of reference
10:
3121:
2782:Classical electromagnetism
2452:, 1993, Oxford Univ. Press
2216:Apparent retrograde motion
1997:
1823:
1411:
1361:
1357:
1202:Engineering and efficiency
1021:Bi-elliptic transfer orbit
2959:
2896:
2824:
2740:
2712:
2684:
2504:Michael Efroimsky. 2005.
2493:, 2007, Springer-Praxis,
2287:10.4249/scholarpedia.3930
1807:A planet orbiting the Sun
2888:Condensed matter physics
2336:scienceworld.wolfram.com
2246:
1810:A moon orbiting a planet
1228:Propellant mass fraction
1127:Gravitational influences
631:Rotating reference frame
463:Hamilton–Jacobi equation
2361:Oxford University Press
2129:Developmental Ephemeris
2043:Newton's laws of motion
1970:synodic reference frame
1776:(approx. the same mass)
1765:(approx. the same mass)
1620:gravitational radiation
1553:extensions of Kepler's
1099:Specific orbital energy
572:Newton's laws of motion
432:Newton's laws of motion
2972:Nobel Prize in Physics
2834:Relativistic mechanics
2238:Two solutions, called
2019:
1744:
1743:{\displaystyle n>2}
1714:
1332:objects in outer space
1016:Hohmann transfer orbit
599:Simple harmonic motion
512:Euler's laws of motion
306:D'Alembert's principle
189:
2977:Philosophy of physics
2513:Alessandra Celletti,
2486:, 1992, Willmann-Bell
2229:of the other planets.
2009:
1745:
1715:
1521:Joseph-Louis Lagrange
1448:observations made by
1212:Preflight engineering
944:Argument of periapsis
453:Hamiltonian mechanics
271:Statistical mechanics
190:
69:"Celestial mechanics"
2936:Mathematical physics
2149:Dynamical time scale
2123:Celestial navigation
2094:propulsive maneuvers
2053:design and control.
1869:Perturbation methods
1728:
1698:
1630:Examples of problems
1577:Peter Andreas Hansen
1392:introduced the term
1390:Pierre-Simon Laplace
1326:that deals with the
1268:Propulsive maneuvers
676:Angular acceleration
668:Rotational frequency
448:Lagrangian mechanics
441:Analytical mechanics
197:Second law of motion
150:
54:improve this article
3095:Classical mechanics
3090:Celestial mechanics
2911:Atmospheric physics
2750:Classical mechanics
2678:branches of physics
2555:Celestial Mechanics
2459:, 1961, John Wiley.
2457:Celestial Mechanics
2434:Forest R. Moulton,
2409:2016PhyEd..51e5012G
2330:Weisstein, Eric W.
2278:2008SchpJ...3.3930T
2062:astronomical bodies
2012:tangential velocity
1843:perturbation theory
1831:Perturbation theory
1826:Perturbation theory
1820:Perturbation theory
1713:{\displaystyle n=2}
1624:Nobel Physics Prize
1608:Newtonian mechanics
1581:George William Hill
1539:classical mechanics
1503:, Newton confirmed
1394:celestial mechanics
1382:celestial mechanics
1340:classical mechanics
1320:Celestial mechanics
1245:Efficiency measures
1148:Sphere of influence
1117:Celestial mechanics
899:Part of a series on
528:Harmonic oscillator
506:Equations of motion
141:Classical mechanics
135:Part of a series on
27:Branch of astronomy
18:Dynamical astronomy
2967:History of physics
2598:2002-10-01 at the
2579:Newtonian Dynamics
2455:William M. Smart,
2171:Numerical analysis
2137:numerical analysis
2099:General relativity
2020:
1974:three-body problem
1948:. You can help by
1835:numerical analysis
1740:
1710:
1616:General Relativity
1602:in his 1916 paper
1531:three-body problem
1501:his law of gravity
1458:law of gravitation
1064:Dynamical friction
844:Physics portal
458:Routhian mechanics
333:Frame of reference
185:
2995:
2994:
2982:Physics education
2931:Materials science
2898:Interdisciplinary
2856:Quantum mechanics
2523:978-3-540-85145-5
2450:Orbital Mechanics
2387:Physics Education
2370:978-0-19-517324-6
2240:VSOP82 and VSOP87
2227:natural satellite
2209:Retrograde motion
2086:orbital maneuvers
2023:Orbital mechanics
2000:Orbital mechanics
1993:Orbital mechanics
1983:retrograde motion
1966:
1965:
1873:Keplerian ellipse
1855:Keplerian ellipse
1653:, where a number
1555:elliptical orbits
1535:Lagrangian points
1386:Gottfried Leibniz
1322:is the branch of
1317:
1316:
1167:Lagrangian points
1104:Vis-viva equation
1074:Kepler's equation
921:Orbital mechanics
894:
893:
641:Centrifugal force
636:Centripetal force
592:Euler's equations
577:Relative velocity
353:Moment of inertia
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2203:Osculating orbit
2197:Orbital elements
1987:superior planets
1961:
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1921:and fix" method
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1722:two-body problem
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1685:Lagrangian point
1675:3-body problem:
1509:two-body problem
1499:dynamics. Using
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1302:
1295:
1274:Orbital maneuver
1223:Payload fraction
1203:
1184:Lissajous orbits
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1089:Orbital velocity
1036:Hyperbolic orbit
932:Orbital elements
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742:Daniel Bernoulli
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3119:
3115:
3114:
3113:
3111:
3110:
3109:
3080:
3079:
3078:
3068:
3066:
3056:
3054:
3044:
3042:
3030:
3020:
3018:
3006:
2998:
2996:
2991:
2955:
2941:Medical physics
2892:
2851:Nuclear physics
2820:
2814:Non-equilibrium
2736:
2708:
2680:
2674:
2600:Wayback Machine
2565:
2563:
2548:
2541:Expert articles
2532:
2530:Further reading
2527:
2477:
2438:, 1984, Dover,
2430:
2425:
2424:
2382:
2378:
2371:
2353:
2349:
2340:
2338:
2328:
2324:
2307:
2303:
2258:
2254:
2249:
2108:
2103:
2102:
2003:
1995:
1972:applied to the
1962:
1956:
1953:
1946:needs expansion
1931:
1929:Reference frame
1828:
1822:
1770:binary asteroid
1729:
1726:
1725:
1699:
1696:
1695:
1679:Quasi-satellite
1632:
1596:Albert Einstein
1593:
1591:Albert Einstein
1570:
1523:
1466:
1424:Johannes Kepler
1421:
1410:
1408:Johannes Kepler
1367:
1360:
1313:
1284:
1283:
1279:Orbit insertion
1269:
1261:
1260:
1246:
1238:
1237:
1213:
1205:
1201:
1194:
1193:
1189:Lyapunov orbits
1180:
1179:
1163:
1153:
1152:
1128:
1120:
1116:
1109:
1108:
1094:Surface gravity
1069:Escape velocity
1059:
1051:
1050:
1031:Parabolic orbit
1027:
1026:
993:
991:
988:two-body orbits
979:
978:
969:Semi-major axis
934:
924:
920:
890:
849:
836:
835:
828:
827:
826:
701:
693:
692:
672:
626:Circular motion
620:
610:
609:
608:
565:
535:
532:
511:
490:
482:
481:
478:
477:
435:
425:
417:
416:
415:
374:
370:Mechanical work
363:
347:
285:
277:
276:
275:
230:
222:
199:
175:
169:
165:
163:
154:
153:
151:
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126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
3118:
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3077:
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2847:
2846:
2841:
2830:
2828:
2822:
2821:
2819:
2818:
2817:
2816:
2811:
2804:Thermodynamics
2801:
2800:
2799:
2794:
2784:
2779:
2774:
2773:
2772:
2767:
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2757:
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2627:
2626:
2615:
2614:
2603:
2602:
2584:
2583:
2576:
2570:
2547:
2546:External links
2544:
2543:
2542:
2531:
2528:
2526:
2525:
2511:
2502:
2487:
2482:J.M.A. Danby,
2480:
2475:
2460:
2453:
2446:
2431:
2429:
2426:
2423:
2422:
2376:
2369:
2363:, p. 34,
2347:
2322:
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2250:
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2178:
2168:
2162:
2157:
2151:
2146:
2140:
2126:
2120:
2115:
2107:
2104:
2014:and an inward
2004:
1996:
1994:
1991:
1964:
1963:
1957:September 2023
1943:
1941:
1930:
1927:
1824:Main article:
1821:
1818:
1817:
1816:
1815:
1814:
1811:
1808:
1805:
1780:
1779:
1778:
1777:
1766:
1763:Alpha Centauri
1739:
1736:
1733:
1709:
1706:
1703:
1692:
1691:
1690:
1689:
1688:
1687:
1681:
1673:
1631:
1628:
1612:binary pulsars
1592:
1589:
1569:
1566:
1525:After Newton,
1522:
1519:
1465:
1462:
1456:developed his
1418:Kepler problem
1409:
1406:
1359:
1356:
1315:
1314:
1312:
1311:
1304:
1297:
1289:
1286:
1285:
1282:
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1276:
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1251:Gravity assist
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1084:Orbital period
1081:
1076:
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1066:
1060:
1057:
1056:
1053:
1052:
1049:
1048:
1046:Decaying orbit
1043:
1038:
1033:
1025:
1024:
1018:
1011:
1009:Transfer orbit
1007:
1006:
1005:
1003:Elliptic orbit
1000:
998:Circular orbit
994:
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984:
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671:
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665:
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653:Coriolis force
650:
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2:
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2975:
2973:
2970:
2968:
2965:
2964:
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2958:
2952:
2949:
2947:
2946:Ocean physics
2944:
2942:
2939:
2937:
2934:
2932:
2929:
2927:
2924:
2922:
2919:
2917:
2914:
2912:
2909:
2907:
2904:
2903:
2901:
2899:
2895:
2889:
2886:
2882:
2881:Modern optics
2879:
2877:
2874:
2872:
2869:
2868:
2867:
2864:
2862:
2859:
2857:
2854:
2852:
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2761:
2758:
2756:
2753:
2752:
2751:
2748:
2747:
2745:
2743:
2739:
2731:
2730:Computational
2728:
2727:
2726:
2723:
2721:
2718:
2717:
2715:
2711:
2703:
2700:
2699:
2698:
2695:
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2689:
2687:
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2679:
2671:
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2664:
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2657:
2652:
2651:
2648:
2644:
2643:
2637:
2634:
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2620:
2619:
2613:
2610:
2609:
2608:
2607:
2601:
2597:
2594:
2591:
2590:
2589:
2588:
2580:
2577:
2574:
2571:
2562:on 2006-09-07
2561:
2557:
2556:
2550:
2549:
2540:
2537:
2534:
2533:
2524:
2520:
2516:
2512:
2510:
2507:
2503:
2500:
2499:0-387-30777-X
2496:
2492:
2488:
2485:
2481:
2478:
2476:9780815303220
2472:
2468:
2467:
2461:
2458:
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2447:
2445:
2444:0-486-64687-4
2441:
2437:
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2141:
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2134:
2130:
2127:
2124:
2121:
2119:
2116:
2113:
2110:
2109:
2100:
2097:
2095:
2091:
2090:orbital plane
2087:
2083:
2079:
2075:
2071:
2067:
2063:
2059:
2054:
2052:
2051:space-mission
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2017:
2013:
2008:
2001:
1990:
1988:
1984:
1979:
1975:
1971:
1960:
1951:
1947:
1944:This section
1942:
1939:
1935:
1934:
1926:
1924:
1918:
1917:
1913:
1909:
1903:
1899:
1897:
1893:
1889:
1884:
1882:
1878:
1874:
1870:
1866:
1864:
1860:
1856:
1852:
1848:
1844:
1840:
1836:
1832:
1827:
1812:
1809:
1806:
1804:
1800:
1796:
1795:
1793:
1792:
1791:
1789:
1785:
1784:orbiting body
1775:
1771:
1767:
1764:
1760:
1756:
1755:
1753:
1752:
1751:
1737:
1734:
1731:
1723:
1707:
1704:
1701:
1686:
1682:
1680:
1677:
1676:
1674:
1671:
1667:
1666:
1664:
1663:
1662:
1660:
1656:
1652:
1651:-body problem
1650:
1645:
1641:
1637:
1627:
1625:
1621:
1617:
1613:
1609:
1605:
1601:
1597:
1588:
1586:
1582:
1578:
1574:
1573:Simon Newcomb
1568:Simon Newcomb
1565:
1563:
1560:
1556:
1552:
1551:conic section
1548:
1544:
1540:
1536:
1532:
1528:
1518:
1516:
1515:
1510:
1506:
1505:Kepler's Laws
1502:
1498:
1494:
1490:
1489:physical laws
1486:
1482:
1478:
1474:
1470:
1461:
1459:
1455:
1451:
1447:
1443:
1439:
1438:
1433:
1429:
1425:
1419:
1415:
1405:
1403:
1399:
1395:
1391:
1387:
1383:
1379:
1377:
1372:
1365:
1355:
1353:
1350:, to produce
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1310:
1305:
1303:
1298:
1296:
1291:
1290:
1288:
1287:
1280:
1277:
1275:
1272:
1271:
1265:
1264:
1257:
1256:Oberth effect
1254:
1252:
1249:
1248:
1242:
1241:
1234:
1231:
1229:
1226:
1224:
1221:
1219:
1216:
1215:
1209:
1208:
1204:
1198:
1197:
1190:
1187:
1185:
1182:
1181:
1175:
1171:
1170:
1168:
1162:
1161:N-body orbits
1157:
1156:
1149:
1146:
1144:
1143:Perturbations
1141:
1139:
1136:
1134:
1131:
1130:
1124:
1123:
1119:
1113:
1112:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1087:
1085:
1082:
1080:
1077:
1075:
1072:
1070:
1067:
1065:
1062:
1061:
1055:
1054:
1047:
1044:
1042:
1039:
1037:
1034:
1032:
1029:
1028:
1022:
1019:
1017:
1013:
1012:
1010:
1004:
1001:
999:
996:
995:
989:
983:
982:
975:
972:
970:
967:
965:
964:Orbital nodes
962:
960:
957:
955:
952:
950:
947:
945:
942:
940:
937:
936:
933:
928:
927:
923:
917:
916:
912:
908:
907:
904:Astrodynamics
903:
902:
898:
897:
887:
882:
880:
875:
873:
868:
867:
865:
864:
858:
848:
845:
840:
834:
833:
832:
831:
823:
820:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
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773:
770:
768:
765:
763:
760:
758:
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753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
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708:
705:
704:
697:
696:
689:
685:
681:
677:
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673:
669:
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661:
659:
656:
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651:
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619:
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613:
605:
602:
600:
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583:
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560:
556:
553:
552:
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542:
538:
534:
533:
529:
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524:
521:
519:
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502:
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497:
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493:
486:
485:
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471:
469:
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461:
459:
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454:
451:
449:
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445:
443:
442:
437:
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433:
428:
427:
421:
420:
412:
409:
407:
404:
402:
399:
397:
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239:
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233:
226:
225:
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166:
160:
146:
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124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
3074:Solar System
2906:Astrophysics
2764:
2720:Experimental
2641:
2640:
2630:Associations
2629:
2628:
2618:Course notes
2617:
2616:
2605:
2604:
2586:
2585:
2564:, retrieved
2560:the original
2554:
2539:Scholarpedia
2514:
2505:
2490:
2483:
2465:
2456:
2449:
2435:
2390:
2386:
2379:
2356:
2350:
2339:. Retrieved
2335:
2325:
2304:
2269:
2266:Scholarpedia
2265:
2255:
2188:
2165:Lunar theory
2118:Astrophysics
2084:, including
2082:trajectories
2066:star systems
2055:
2037:, and other
2021:
2016:acceleration
1969:
1967:
1954:
1950:adding to it
1945:
1919:
1915:
1904:
1900:
1890:to make its
1887:
1885:
1867:
1829:
1799:Solar System
1788:central body
1781:
1693:
1654:
1648:
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1603:
1594:
1571:
1562:trajectories
1524:
1513:
1496:
1492:
1469:Isaac Newton
1467:
1464:Isaac Newton
1454:Isaac Newton
1436:
1422:
1393:
1381:
1375:
1371:Isaac Newton
1368:
1319:
1318:
1115:
1041:Radial orbit
992:eccentricity
974:True anomaly
959:Mean anomaly
949:Eccentricity
686: /
682: /
680:displacement
678: /
539: /
501:Displacement
439:
430:
424:Formulations
411:Virtual work
351: /
291:Acceleration
284:Fundamentals
240:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
3062:Outer space
3050:Spaceflight
2809:Statistical
2725:Theoretical
2702:Engineering
2642:Simulations
2272:(5): 3930.
2233:Tidal force
2180:Creating a
2160:Gravitation
1888:"perturbed"
1839:are ancient
1759:binary star
1636:drag forces
1497:terrestrial
1450:Tycho Brahe
1398:geometrical
1380:. The name
1174:Halo orbits
1138:Hill sphere
954:Inclination
822:von Neumann
489:Core topics
3105:Astrometry
3084:Categories
2926:Geophysics
2916:Biophysics
2760:Analytical
2713:Approaches
2566:2006-08-21
2428:References
2400:1605.01339
2341:2020-08-28
2316:1509.08233
2112:Astrometry
2039:spacecraft
2035:satellites
2027:ballistics
1978:barycenter
1794:Examples:
1774:90 Antiope
1754:Examples:
1665:Examples:
1659:integrable
1559:spacecraft
1479:, and the
1432:Copernicus
1218:Mass ratio
1133:Barycenter
757:d'Alembert
737:Maupertuis
700:Scientists
582:Rigid body
256:Kinematics
80:newspapers
3026:Astronomy
2876:Molecular
2777:Acoustics
2770:Continuum
2765:Celestial
2755:Newtonian
2742:Classical
2685:Divisions
2296:1941-6016
2222:Satellite
2154:Ephemeris
1914:'s orbit
1803:Milky Way
1514:Principia
1493:celestial
1460:in 1686.
1446:planetary
1402:numerical
1376:Principia
1352:ephemeris
1324:astronomy
1058:Equations
986:Types of
802:Liouville
684:frequency
604:Vibration
321:potential
246:Continuum
241:Celestial
218:Textbooks
110:June 2013
2596:Archived
2587:Research
2106:See also
2064:such as
2045:and the
1879:and the
1861:and the
1837:, which
1772:, e.g.,
1761:, e.g.,
1527:Lagrange
857:Category
782:Hamilton
767:Lagrange
762:Clairaut
727:Horrocks
688:velocity
658:Pendulum
646:reactive
618:Rotation
587:dynamics
537:Inertial
523:Friction
406:Velocity
381:Momentum
261:Kinetics
251:Dynamics
229:Branches
213:Timeline
3014:Physics
3000:Portals
2960:Related
2844:General
2839:Special
2697:Applied
2606:Artwork
2582:Earth).
2405:Bibcode
2274:Bibcode
2212:system.
2131:or the
2070:planets
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2031:rockets
1694:In the
1638:or the
1473:planets
1428:Ptolemy
1358:History
1348:planets
1336:physics
1328:motions
817:Koopman
777:Poisson
772:Laplace
717:Huygens
712:Galileo
557: (
496:Damping
349:Inertia
343:Impulse
316:kinetic
266:Statics
236:Applied
208:History
94:scholar
2871:Atomic
2826:Modern
2676:Major
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2473:
2442:
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2175:planet
2078:comets
2076:, and
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1644:rocket
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1547:comets
1543:method
1485:cannon
1475:, the
1378:(1687)
1354:data.
855:
807:Appell
792:Cauchy
787:Jacobi
732:Halley
722:Newton
707:Kepler
559:linear
555:Motion
401:Torque
376:Moment
311:Energy
301:Couple
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3038:Stars
2395:arXiv
2393:(5).
2311:arXiv
2247:Notes
2190:orbit
2074:moons
1877:Earth
1859:Earth
1642:of a
1585:Paris
1344:stars
939:Apsis
812:Gibbs
797:Routh
752:Euler
391:Speed
386:Space
328:Force
101:JSTOR
87:books
2797:Wave
2692:Pure
2519:ISBN
2495:ISBN
2471:ISBN
2440:ISBN
2365:ISBN
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1851:Moon
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1481:Moon
1416:and
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396:Time
359:Mass
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2792:Ray
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2282:doi
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