View source for Celestial mechanics

{{short description|Branch of astronomy}}
{{More citations needed|date=June 2013}}
{{Classical mechanics|cTopic=Branches}}
{{Astrodynamics}}
'''Celestial mechanics''' is the [[History of astronomy|branch of]] [[astronomy]] that deals with the [[motion (physics)|motion]]s of [[celestial object|objects in outer space]]. Historically, celestial mechanics applies principles of [[physics]] ([[classical mechanics]]) to astronomical objects, such as [[star]]s and [[planet]]s, to produce [[ephemeris]] data.

==History==
{{For|early theories of the causes of planetary motion|Dynamics of the celestial spheres}}

Modern analytic celestial mechanics started with [[Isaac Newton]]'s [[Philosophiæ Naturalis Principia Mathematica|''Principia'' (1687)]]. The name '''celestial mechanics''' is more recent than that. Newton wrote that the field should be called "rational mechanics". The term "dynamics" came in a little later with [[Gottfried Leibniz]], and over a century after Newton, [[Pierre-Simon Laplace]] introduced the term ''celestial mechanics''. Prior to [[Johannes Kepler|Kepler]], there was little connection between exact, quantitative prediction of planetary positions, using [[Greek astronomy#Eudoxan astronomy|geometrical]] or [[Babylonian astronomy#Neo-Babylonian astronomy|numerical]] techniques, and contemporary discussions of the physical causes of the planets' motion.

===Laws of planetary motion===
{{For|detailed treatments of how his laws of planetary motion can be used|Kepler's laws of planetary motion|Kepler problem}}

[[Johannes Kepler]] as the first to closely integrate the predictive geometrical astronomy, which had been dominant from [[Ptolemy]] in the 2nd&nbsp;century to [[Copernicus]], with physical concepts to produce a [[Astronomia nova|''New Astronomy, Based upon Causes, or Celestial Physics'']] in 1609. His work led to the [[Kepler's laws of planetary motion|laws of planetary orbits]], which he developed using his physical principles and the [[planet]]ary observations made by [[Tycho Brahe]]. Kepler's elliptical model greatly improved the accuracy of predictions of planetary motion, years before Newton developed his [[Newton's law of universal gravitation|law of gravitation]] in 1686.

===Newtonian mechanics and universal gravitation===
[[Isaac Newton]] is credited with introducing the idea that the motion of objects in the heavens, such as [[planet]]s, the [[Sun]], and the [[Moon]], and the motion of objects on the ground, like [[cannon]] balls and falling apples, could be described by the same set of [[physical law]]s. In this sense he unified ''celestial'' and ''terrestrial'' dynamics. Using [[Newton's law of universal gravitation|his law of gravity]], Newton confirmed [[Kepler's laws of planetary motion|Kepler's laws]] for elliptical orbits by deriving them from the gravitational [[two-body problem]], which Newton included in his epochal ''[[Philosophiæ Naturalis Principia Mathematica]]'' in 1687.

===Three-body problem===
{{main article | Three-body problem}}
After Newton, [[Joseph-Louis Lagrange]] attempted to solve the three-body problem in 1772, analyzed the stability of planetary orbits, and discovered the existence of the [[Lagrange point]]s. Lagrange also reformulated the principles of [[classical mechanics]], emphasizing energy more than force, and developing a [[Lagrangian mechanics|method]] 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 [[comet]]s and such (parabolic and hyperbolic orbits are [[conic section]] extensions of Kepler's [[elliptical orbit]]s). More recently, it has also become useful to calculate [[spacecraft]] [[trajectory|trajectories]].

[[Henri Poincaré]] published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). Poincaré showed that the three-body problem is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms of [[algebra]]ic and [[transcendental functions]] through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since Isaac Newton.<ref>J. Stillwell, Mathematics and its history, [https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA254 page 254]</ref>

These monographs include an idea of Poincaré, which later became the basis for mathematical "[[chaos theory]]" (see, in particular, the [[Poincaré recurrence theorem]]) and the general theory of [[dynamical system]]s. He introduced the important concept of [[Bifurcation theory|bifurcation points]] and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability. For this discovery, Poincaré received the Gold Medal of the Royal Astronomical Society (1900).<ref name="Gold Medal to Poincaré">{{cite journal |date=1900 |title=Address Delivered by the President, Professor G. H. Darwin, on presenting the Gold Medal of the Society to M. H. Poincaré |journal=Monthly Notices of the Royal Astronomical Society |volume=60 |issue=5 |pages=406–416 |doi=10.1093/mnras/60.5.406 |issn=0035-8711 |doi-access=free |author-last=Darwin |author-first=G.H.}}</ref>

===Standardisation of astronomical tables===
[[Simon Newcomb]] was a Canadian-American astronomer who revised [[Peter Andreas Hansen]]'s table of lunar positions. In 1877, assisted by [[George William Hill]], 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 [[Paris]], France, in May&nbsp;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.
===Anomalous precession of Mercury===
[[Albert Einstein]] explained the anomalous [[Tests of general relativity#Perihelion precession of Mercury|precession of Mercury's perihelion]] in his 1916 paper ''The Foundation of the General Theory of Relativity''. [[General relativity]] led astronomers to recognize that [[Newtonian mechanics]] did not provide the highest accuracy.

==Examples of problems==
Celestial motion, without additional forces such as [[drag force]]s or the [[thrust]] of a [[rocket]], is governed by the reciprocal gravitational acceleration between masses. A generalization is the [[n-body problem|''n''-body problem]],<ref>{{Cite journal|last1=Trenti|first1=Michele|last2=Hut|first2=Piet|date=2008-05-20|title=N-body simulations (gravitational)|journal=Scholarpedia|language=en|volume=3|issue=5|pages=3930|doi=10.4249/scholarpedia.3930|bibcode=2008SchpJ...3.3930T|issn=1941-6016|doi-access=free}}</ref> where a number ''n'' of masses are mutually interacting via the gravitational force. Although analytically not [[integrable]] in the general case,<ref>{{cite arXiv|last=Combot|first=Thierry|date=2015-09-01|title=Integrability and non integrability of some n body problems|class=math.DS|eprint=1509.08233}}</ref> the integration can be well approximated numerically.

:Examples:
:*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 [[patched conic approximation]])
:*3-body problem:
:**[[Quasi-satellite]]
:**Spaceflight to, and stay at a [[Lagrangian point]]

In the <math>n=2</math> case ([[two-body problem]]) the configuration is much simpler than for <math>n>2</math>. In this case, the system is fully integrable and exact solutions can be found.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Two-Body Problem -- from Eric Weisstein's World of Physics|url=https://scienceworld.wolfram.com/physics/Two-BodyProblem.html|access-date=2020-08-28|website=scienceworld.wolfram.com|language=en}}</ref>

:Examples:
:*A [[binary star]], e.g., [[Alpha Centauri]] (approx. the same mass)
:*A [[binary asteroid]], e.g., [[90 Antiope]] (approx. the same mass)

A further simplification is based on the "standard assumptions in astrodynamics", which include that one body, the [[orbiting body]], is much smaller than the other, the [[central body]]. This is also often approximately valid.

:Examples:
:*The [[Solar System]] orbiting the center of the [[Milky Way]]
:*A planet orbiting the Sun
:*A moon orbiting a planet
:*A spacecraft orbiting Earth, a moon, or a planet (in the latter cases the approximation only applies after arrival at that orbit)

==Perturbation theory==
{{main|Perturbation theory}}
[[Perturbation theory]] 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 [[numerical analysis]], which [[Methods of computing square roots#Heron's method|are ancient]].) The earliest use of modern [[perturbation theory]] was to deal with the otherwise unsolvable mathematical problems of celestial mechanics: [[Isaac Newton|Newton]]'s solution for the orbit of the [[Moon]], which moves noticeably differently from a simple [[Kepler's laws of planetary motion|Keplerian ellipse]] because of the competing gravitation of the [[Earth]] and the [[Sun]].

[[Perturbation theory|Perturbation methods]] start with a simplified form of the original problem, which is carefully chosen to be exactly solvable. In celestial mechanics, this is usually a [[Kepler's laws of planetary motion|Keplerian ellipse]], which is correct when there are only two gravitating bodies (say, the [[Earth]] and the [[Moon]]), or a circular orbit, which is only correct in special cases of two-body motion, but is often close enough for practical use.

The solved, but simplified problem is then ''"perturbed"'' to make its [[differential equation|time-rate-of-change equations for the object's position]] closer to the values from the real problem, such as including the gravitational attraction of a third, more distant body (the [[Sun]]). 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.

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 better solutions could continue indefinitely, to any desired finite degree of accuracy.

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. [[Isaac Newton|Newton]] is reported to have said, regarding the problem of the [[Moon]]'s orbit ''"It causeth my head to ache."''<ref>{{Citation |last1=Cropper |first1=William H. |title=Great Physicists: The life and times of leading physicists from Galileo to Hawking |publisher=[[Oxford University Press]] |isbn=978-0-19-517324-6 |date=2004 |page=34}}.</ref>

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, and fix" method [[Methods of computing square roots#Heron's method|used anciently with numbers]].

==Reference frame==
{{expand-section|date=September 2023}}
Problems in celestial mechanics are often posed in simplifying reference frames, such as the '''synodic reference frame'''<!-- redirects here --> applied to the [[three-body problem]], where the origin coincides with the [[barycenter]] 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.<ref>{{Cite journal |last1=Guerra |first1=André G C |last2=Carvalho |first2=Paulo Simeão |date=1 August 2016 |title=Orbital motions of astronomical bodies and their centre of mass from different reference frames: a conceptual step between the geocentric and heliocentric models |journal=Physics Education |volume=51 |issue=5|doi=10.1088/0031-9120/51/5/055012 |arxiv=1605.01339 |bibcode=2016PhyEd..51e5012G }}</ref> The choice of reference frame gives rise to many phenomena, including the [[Retrograde and prograde motion|retrograde motion]] of [[Inferior and superior planets|superior planets]] while on a geocentric reference frame. 

==Orbital mechanics==
{{excerpt|Orbital mechanics|template=-Astrodynamics}}

==See also==
* [[Astrometry]] is a part of astronomy that deals with measuring the positions of stars and other celestial bodies, their distances and movements.
* [[Astrophysics]]
* [[Celestial navigation]] is a position fixing technique that was the first system devised to help sailors locate themselves on a featureless ocean.
* [[Jet Propulsion Laboratory Developmental Ephemeris|Developmental Ephemeris]] or the [[Jet Propulsion Laboratory Developmental Ephemeris]] (JPL DE) is a widely used model of the solar system, which combines celestial mechanics with [[numerical analysis]] and astronomical and spacecraft data.
* [[Dynamics of the celestial spheres]] concerns pre-Newtonian explanations of the causes of the motions of the stars and planets.
* [[Dynamical time scale]]
* [[Ephemeris]] is a compilation of positions of naturally occurring astronomical objects as well as artificial satellites in the sky at a given time or times.
* [[Gravitation]]
* [[Lunar theory]] attempts to account for the motions of the Moon.
* [[Numerical analysis]] is a branch of mathematics, pioneered by celestial mechanicians, for calculating approximate numerical answers (such as the position of a [[planet]] in the sky) which are too difficult to solve down to a general, exact formula.
* Creating a [[Numerical model of solar system|numerical model of the solar system]] was the original goal of celestial mechanics, and has only been imperfectly achieved. It continues to motivate research.
* An ''[[orbit]]'' is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity.
* [[Orbital elements]] are the parameters needed to specify a Newtonian two-body orbit uniquely.
* [[Osculating orbit]] is the temporary Keplerian orbit about a central body that an object would continue on, if other perturbations were not present.
* [[Retrograde motion]] 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 system.
* [[Apparent retrograde motion]] is the periodic, apparently backwards motion of planetary bodies when viewed from the Earth (an accelerated reference frame).
* [[Satellite]] 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 [[natural satellite]] of the other planets.
* [[Tidal force]] 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.
* Two solutions, called [[VSOP (planets)|VSOP82 and VSOP87]] 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.

==Notes==
{{Reflist}}

==References==
{{refbegin}}
* Forest R. Moulton, ''Introduction to Celestial Mechanics'', 1984, Dover, {{ISBN|0-486-64687-4}}
* John E. Prussing, Bruce A. Conway, ''Orbital Mechanics'', 1993, Oxford Univ. Press
* William M. Smart, ''Celestial Mechanics'', 1961, John Wiley. 
* {{ Citation | last = Doggett | first = LeRoy E. | editor-last = Lankford | editor-first = John | date = 1997 | title = History of Astronomy: An Encyclopedia | chapter = Celestial Mechanics | publisher = Taylor & Francis | place = New York | pages = 131–140 | isbn = 9780815303220 | url = https://books.google.com/books?id=fIzMMe3VczkC }}
* J.M.A. Danby, ''Fundamentals of Celestial Mechanics'', 1992, Willmann-Bell
* Alessandra Celletti, Ettore Perozzi, ''Celestial Mechanics: The Waltz of the Planets'', 2007, Springer-Praxis, {{ISBN|0-387-30777-X}}.
* Michael Efroimsky. 2005. ''Gauge Freedom in Orbital Mechanics.'' [https://archive.today/20130105074903/http://www3.interscience.wiley.com/journal/118692589/abstract?CRETRY=1&SRETRY=0 Annals of the New York Academy of Sciences, Vol. 1065, pp. 346-374]
* Alessandra Celletti, ''Stability and Chaos in Celestial Mechanics.'' Springer-Praxis 2010, XVI, 264 p., Hardcover {{ISBN|978-3-540-85145-5}}

{{refend}}

==Further reading==
*[http://www.scholarpedia.org/article/Encyclopedia:Celestial_mechanics Encyclopedia:Celestial mechanics] [[Scholarpedia]] Expert articles
* {{cite book |last=Poincaré |first=H. |date=1967 |title=New Methods of Celestial Mechanics |url=https://archive.org/details/newmethodsofcele0000poin |url-access=registration |edition=3 vol. English translated |publisher=American Institute of Physics |isbn=978-1-56396-117-5 }}

==External links==
{{Commons category}}
*{{Citation | last = Calvert | first = James B. | date = 2003-03-28 | url = http://www.du.edu/~jcalvert/phys/orbits.htm | title = Celestial Mechanics | publisher = University of Denver | access-date = 2006-08-21 | archive-url = https://web.archive.org/web/20060907120741/http://www.du.edu/~jcalvert/phys/orbits.htm | archive-date = 2006-09-07 | url-status = dead }}
*[http://www.phy6.org/stargaze/Sastron.htm Astronomy of the Earth's Motion in Space], high-school level educational web site by David P. Stern
*[http://farside.ph.utexas.edu/teaching/336k/Newton.pdf Newtonian Dynamics] 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 Earth).

'''Research'''
* [http://www.math.washington.edu/~hampton/research.html Marshall Hampton's research page: Central configurations in the n-body problem] {{Webarchive|url=https://web.archive.org/web/20021001170939/http://www.math.washington.edu/~hampton/research.html |date=2002-10-01 }}

'''Artwork'''
* [https://web.archive.org/web/20190428072152/http://www.cmlab.com/ Celestial Mechanics is a Planetarium Artwork created by D. S. Hessels and G. Dunne]

'''Course notes'''
* [http://orca.phys.uvic.ca/~tatum/celmechs.html Professor Tatum's course notes at the University of Victoria]

'''Associations'''
* [http://www.mat.uniroma2.it/simca/english.html Italian Celestial Mechanics and Astrodynamics Association]

'''Simulations'''

{{Physics-footer}}
{{Portal bar|Physics|Astronomy|Stars|Spaceflight|Outer space|Solar System}}
{{Authority control}}

{{DEFAULTSORT:Celestial Mechanics}}
[[Category:Celestial mechanics| ]]
[[Category:Classical mechanics]]
[[Category:Astronomical sub-disciplines]]
[[Category:Astrometry]]