View source for Celestial mechanics

{{short description|Branch of astronomy}}
{{Classical mechanics|cTopic=Branches}}
{{Astrodynamics}}
'''Celestial mechanics''' is the [[History of astronomy|branch of]] [[astronomy]] that deals with the [[motion (physics)|motion]]s and [[gravitational interaction]]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. The computation of the motion of the bodies through [[orbital mechanics]] can be simplified by using an appropriate [[inertial frame of reference]]. This leads to the use of various different coordinate systems, such as the [[Heliocentric coordinate system|Heliocentric (Sun-centered) coordinate system]].

In a binary system of objects interacting through gravity, [[Newtonian mechanics]] can used to produce a set of [[orbital elements]] that will predict with reasonable accuracy the future position of the two bodies. This method demonstrates the correctness of [[Kepler's laws of planetary motion]]. Where one of the bodies is sufficiently massive, [[general relativity]] must be included to predict [[apsidal precession]]. The problem becomes more complicated when another body is added, creating a [[three-body problem]] that can not be solved exactly. [[Perturbation theory]] is used to find an approximate solution to this problem.

==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".<ref>{{cite book | title=A History of Astronomy | series=Dover Books | first=Anton | last=Pannekoek | publisher=Courier Corporation | year=1989 | isbn=978-0-486-65994-7 | pages=264–265 | url=https://books.google.com/books?id=I1LGdDe0NYcC&pg=PA265 }}</ref> The term "dynamics" came in a little later with [[Gottfried Leibniz]],<ref>{{cite book | title=Leibniz: Body, Substance, Monad | first=Daniel | last=Garber | publisher=OUP Oxford | year=2009 | isbn=978-0-19-157062-9 | pages=129–133 | url=https://books.google.com/books?id=s_1QEAAAQBAJ&pg=PA130 }}</ref> and over a century after Newton, [[Pierre-Simon Laplace]] introduced the term ''celestial mechanics''.<ref>{{cite book | title=Calculus Reordered: A History of the Big Ideas | first=David M. | last=Bressoud | publisher=Princeton University Press | year=2021 | isbn=978-0-691-21878-6 | page=103 | url=https://books.google.com/books?id=GkgHEAAAQBAJ&pg=PA103 }}</ref> 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]] was 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.<ref>{{cite book | chapter=The New Astronomy | title=Johannes Kepler and the New Astronomy | series=Oxford Portraits in Science | first=James R. | last=Voelkel | publisher=Oxford University Press | year=2001 | isbn=978-0-19-028774-0 | chapter-url=https://books.google.com/books?id=0-7hBwAAQBAJ&pg=PT80 }}</ref>

===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.<ref>{{cite book | title=Basic Calculus of Planetary Orbits and Interplanetary Flight: The Missions of the Voyagers, Cassini, and Juno | first=Alexander J. | last=Hahn | publisher=Springer Nature | year=2020 | isbn=978-3-030-24868-0 | pages=9–20 | url=https://books.google.com/books?id=ILHWDwAAQBAJ&pg=PA9 }}</ref>

===Three-body problem===
{{main article | Three-body problem}}
[[File:Lagrange 2 mass.gif|left|thumb|The five Lagrange points for a circular orbit]]
Following Newton, mathematicians attempted to solve the more complex problem of predicting the future motion of three bodies interacting through gravity: the [[three-body problem]]. The first to provide a periodic solution was the Swiss mathematician [[Leonhard Euler]], who in 1762 demonstrated three equilibrium points lie along a straight line passing through the two primary masses. If a body of infinitesimal mass occupied one of these points, it would remain there in a stable orbit. French mathematician [[Joseph-Louis Lagrange]] attempted to solve this restricted three-body problem in 1772, and discovered two more stable orbits at the vertices of [[equilateral triangle]]s with the two primary masses. Collectively, these solutions became known as the [[Lagrange point]]s.<ref>{{cite book | title=Periodic Orbits: F. R. Moulton's Quest for a New Lunar Theory | volume=45 | series=History of Mathematics | first=Craig A. | last=Stephenson | publisher=American Mathematical Society | year=2021 | isbn=978-1-4704-5671-9 | pages=7–11 | url=https://books.google.com/books?id=m30wEAAAQBAJ&pg=PA7 }}</ref>

Lagrange reformulated the principles of [[classical mechanics]], emphasizing energy more than force,<ref>{{cite book | title=Classical Mechanics | first=Tai L. | last=Chow | edition=3rd | publisher=CRC Press | year=2024 | isbn=978-1-040-13480-1 | page=83 | url=https://books.google.com/books?id=Rz0pEQAAQBAJ&pg=PA83 }}</ref> 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).<ref>{{cite book | title=Astronomical Observations: Astronomy and the Study of Deep Space | editor-last=Gregersen | editor-first=Erik | publisher=Britannica Educational Publishing | year=2009 | isbn=978-1-61530-054-9 | pages=83–85 | url=https://books.google.com/books?id=ib2bAAAAQBAJ&pg=PA84 }}</ref><ref>{{cite book | title=Orbital Motion | first=A. E. | last=Roy | edition=4th | publisher=CRC Press | year=2020 | isbn=978-1-4200-5688-4 | pages=223–226 | url=https://books.google.com/books?id=Hzv7k2vH6PgC&pg=PA223 }}</ref> More recently, it has also become useful to calculate [[spacecraft]] [[trajectory|trajectories]].<ref>For example: {{cite journal | title=Lagrange's planetary equations for the motion of electrostatically charged spacecraft | display-authors=1 | last1=Gangestad | first1=Joseph W. | last2=Pollock | first2=George E. | last3=Longuski | first3=James M. | journal=Celestial Mechanics and Dynamical Astronomy | volume=108 | issue=2 | pages=125–145 | date=October 2010 | doi=10.1007/s10569-010-9297-z | bibcode=2010CeMDA.108..125G }}</ref>

[[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's ''Principia''.<ref>{{cite book | first=J. | last=Stillwell | title=Mathematics and its history | date=2 August 2010 | page=254 | publisher=Springer Science & Business Media | isbn=978-1-4419-6052-8 | url=https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA254 }}</ref><ref name=James_2002>{{cite book | title=Remarkable Mathematicians: From Euler to Von Neumann | series=Spectrum series of the Mathematical Association of America | first=Ioan | last=James | publisher=Cambridge University Press | year=2002 | isbn=978-0-521-52094-2 | pages=237–245 | url=https://books.google.com/books?id=fptj8AbgUDgC&pg=PA243 }}</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.<ref>{{cite book | title=Investigation of Corrections to Hansen's Tables of the Moon: With Tables for Their Application | first=Simon | last=Newcomb | publisher=U.S. Government Printing Office | year=1876 | url=https://books.google.com/books?id=AXf05ylfRUgC&pg=PP1 }}</ref> In 1877, assisted by [[George William Hill]], he recalculated all the major astronomical constants. After 1884 he conceived, with [[Arthur Matthew Weld Downing|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.<ref>{{cite book | chapter=Simon Newcomb and his work | title=Sky and Ocean Joined: The US Naval Observatory 1830-2000 | first=Steven J. | last=Dick | publisher=Cambridge University Press | year=2003 | isbn=978-0-521-81599-4 | pages=274–291 | chapter-url=https://books.google.com/books?id=DNwfG5hQ7-YC&pg=PA275 }}</ref>

===Anomalous precession of Mercury===
{{main|Apsidal precession}}
[[File:Perihelion precession.svg|right|thumb|Apsidal precession of Mercury's orbit around the Sun (not to scale)]]
In 1849, [[Urbain Le Verrier]] reported that Mercury's closest approach the Sun, its [[perihelion]], was observed to advance at the rate of {{Val|43|ul=arcsecond}} per century. This [[Tests of general relativity#Perihelion precession of Mercury|precession of Mercury's perihelion]] could not be accounted for by known gravitational perturbations using Newton's law. Instead, Le Verrier later attributed the effect to a proposed planet orbiting inside the orbit of Mercury. Dubbed [[Vulcan (hypothetical planet)|Vulcan]], subsequent searches failed to locate any such body. The cause remained a mystery until [[Albert Einstein]] explained the [[apsidal precession]] 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 in proximity to massive bodies.<ref>{{cite book | title=The Cambridge Guide to the Solar System | first=Kenneth R. | last=Lang | edition=2nd | publisher=Cambridge University Press | year=2011 | isbn=978-1-139-49417-5 | pages=217–219 | url=https://books.google.com/books?id=S4xDhVCxAQIC&pg=PA217 }}</ref> This led to attempts to solve the [[two-body problem in general relativity]] and the discovery of [[gravitational radiation]].<ref>{{cite book | title=Relativistic Celestial Mechanics of the Solar System | display-authors=1 | first1=Sergei | last1=Kopeikin | first2=Michael | last2=Efroimsky | first3=George | last3=Kaplan | publisher=John Wiley & Sons | year=2011 | isbn=978-3-527-40856-6 | page=501 | url=https://books.google.com/books?id=RfR2GawB-xcC&pg=PA501 }}</ref><ref>{{cite book | title=Traveling at the Speed of Thought: Einstein and the Quest for Gravitational Waves | first=Daniel | last=Kennefick | publisher=Princeton University Press | year=2016 | isbn=978-1-4008-8274-8 | pages=41–42 | url=https://books.google.com/books?id=T0uUCwAAQBAJ&pg=PA42 }}</ref>

==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 | display-authors=1 | last1=Trenti | first1=Michele | last2=Hut | first2=Piet | date=May 20, 2008 | 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=September 2015 | 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.<ref>{{cite book | title=Introduction to Satellite Remote Sensing: Atmosphere, Ocean, Land and Cryosphere Applications | first1=William | last1=Emery | first2=Adriano | last2=Camps | publisher=Elsevier | year=2017 | isbn=978-0-12-809259-0 | url=https://books.google.com/books?id=sZLUDQAAQBAJ&pg=PA567 }}</ref>

: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 the ''"guess, check, and adjust"'' method used in [[numerical analysis]], which [[Methods of computing square roots#Heron's method|is 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]].<ref>{{cite book | title=The Three-body Problem from Pythagoras to Hawking | first1=Mauri | last1=Valtonen | first2=Joanna | last2=Anosova | first3=Konstantin | last3=Kholshevnikov | first4=Aleksandr | last4=Mylläri | first5=Victor | last5=Orlov | first6=Kiyotaka | last6=Tanikawa | display-authors=1 | publisher=Springer | year=2016 | isbn=978-3-319-22726-9 | page=101 | url=https://books.google.com/books?id=4-wgDAAAQBAJ&pg=PA101 }}</ref> Additional sources of orbital perturbation include [[atmospheric drag]], [[solar radiation pressure]], and non-uniform gravitational fields.<ref>{{cite book | title=Introduction to Fundamental Astronomy | first=Naveen | last=Basu | publisher=Educohack Press | year=2025 | isbn=978-93-6152-114-0 | url=https://books.google.com/books?id=EZlIEQAAQBAJ&pg=PT37 }}</ref>

[[Perturbation theory|Perturbation methods]] start with a simplified form of the original problem, which is 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, 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. The changes that result from the terms in the equations are used as corrections to the original solution.<ref>{{cite book | chapter=The use of CAS Pirhana for the Construction of Motion Equations of the Planetary System Model | first1=A. S. | last1=Perminov | first2=E. D. | last2=Kuznetsov | title=Applications of Computer Algebra: Kalamata, Greece, July 20–23 2015 | volume=198 | series=Springer Proceedings in Mathematics & Statistics | editor1-first=Ilias S. | editor1-last=Kotsireas | editor2-first=Edgar | editor2-last=Martínez-Moro | publisher=Springer | year=2017 | isbn=978-3-319-56932-1 | page=386 | chapter-url=https://books.google.com/books?id=tW0uDwAAQBAJ&pg=PA386 }}</ref> Because simplifications are made at every step, the corrections are never perfect, but even one cycle of corrections often provides a better approximation.

A partially corrected solution can be re-used as the new starting point for another cycle of perturbations and corrections. In principle, the recycling of prior solutions to obtain a better solution could continue indefinitely. The  difficulty is that the corrections usually progressively make the new solutions more complicated. [[Isaac Newton|Newton]] is reported to have said, regarding the problem of the [[Moon]]'s orbit ''"It causeth my head to ache."''.<ref>{{cite book | last=Cropper | first=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 | year=2004 | page=34 }}</ref>

==Reference frame<span class="anchor" id="Reference frame"></span>==
{{main|Frame of reference}}
[[File:Inertial frames.svg|right|thumb|An [[inertial frame of reference]] (brown) mapped from a baseline coordinate system (blue)]]
A reference frame is an arbitrary defined [[coordinate system]], whose origin, orientation, and scale are specified in [[physical space]]. The frame is aligned via a set of reference points, such as distant galaxies.<ref>{{cite book | last1=Kovalevsky | first1=J. | author1-link=Jean Kovalevsky | last2=Mueller | first2=Ivan I. | author-link2=Ivan I. Mueller | title=Reference Frames | chapter=Introduction | series=Astrophysics and Space Science Library | publisher=Springer Netherlands | publication-place=Dordrecht | year=1989 | volume=154 | issn=0067-0057 | doi=10.1007/978-94-009-0933-5_1 | pages=1–12 | isbn=978-94-010-6909-0 }}</ref> Problems in celestial mechanics are often posed in simplifying reference frames, such as the '''synodic reference frame'''<!-- redirect points 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=August 1, 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 phenomena such as the [[Retrograde and prograde motion|retrograde motion]] of [[Inferior and superior planets|superior planets]] in a geocentric reference frame.<ref>{{cite book | title=Essentials of Physics | volume=1 | first1=Sanjay Moreshwar | last1=Wagh | first2=Dilip Abasaheb | last2=Deshpande | publisher=PHI Learning Pvt. Ltd. | year=2012 | isbn=978-81-203-4642-0 | page=41 | url=https://books.google.com/books?id=-DmfVjBUPksC&pg=PA41 }}</ref>

An [[Inertial frame of reference]] is employed for bodies with mass. Thus a Lunar Reference System defines an Earth Inertial frame with Earth as the origin, the Lunar Inertial frame having an origin of the Moon, and an Earth-Moon Barycentric Rotating frame anchored to the rotating [[Earth-Moon barycenter]].<ref>{{cite encyclopedia | title=Inertia, Inertial frame | first=Caitlin | last=Ahrens | encyclopedia=Encyclopedia of Lunar Science | editor-first=Brian | editor-last=Cudnik | publisher=Springer Nature | year=2023 | isbn=978-3-319-14541-9 | pages=330–331 | url=https://books.google.com/books?id=E72-EAAAQBAJ&pg=PA330 }}</ref> [[Positioning system]]s such as [[GPS]] or [[GLONASS]] use a reference frame based on the Earth. However, these are unsuitable for navigation in space.<ref>{{cite journal | title=A null frame for spacetime positioning by means of pulsating sources | display-authors=1 | last1=Tartaglia | first1=Angelo | last2=Ruggiero | first2=Matteo Luca | last3=Capolongo | first3=Emiliano | journal=Advances in Space Research | volume=47 | issue=4 | pages=645–653 | date=February 2011 | doi=10.1016/j.asr.2010.10.023 | arxiv=1001.1068 | bibcode=2011AdSpR..47..645T }}</ref> For interplanetary trajectories, a heliocentric (Sun-centered) coordinate system is used, with the XY plane aligned with the [[ecliptic]] as defined for a particular [[epoch]].<ref>{{cite book | title=Interplanetary Astrodynamics | first1=David B. | last1=Spencer | first2=Davide | last2=Conte | publisher=CRC Press | year=2023 | isbn=978-1-000-85974-4 | page=163 | url=https://books.google.com/books?id=6J63EAAAQBAJ&pg=PA163 }}</ref>

The [[local standard of rest]] (LSR) is a reference frame based on the mean motion of stellar objects in the neighborhood of the Sun. The [[peculiar velocity]] of the Sun relative to this framework is 13.4&nbsp;km/s in the direction of the [[solar apex]].<ref>{{cite book | title=Galaxies: Structure and Evolution | first=Roger John | last=Tayler | publisher=Cambridge University Press | year=1993 | isbn=978-0-521-36710-3 | url=https://books.google.com/books?id=mxb1IxSyu7wC&pg=PA27 }}</ref> There are two possible definitions for the LSR: the first is based on the kinetic motion of nearby stars, and the second is a dynamical standard that follows the Sun in its orbit around the galaxy. These two drift apart with the passage of time as the stars follow the gravitational potential around the galaxy. Perturbations in a star's galactic orbit result in [[epicycle]] motions.<ref>{{cite book | title=Fundamentals of Galaxy Dynamics, Formation and Evolution | series=JSTOR Open Access monographs | first=Ignacio | last=Ferreras | publisher=UCL Press | year=2019 | isbn=978-1-911307-61-7 | pages=74–82 | url=https://books.google.com/books?id=N7ykDwAAQBAJ&pg=PA74 }}</ref>

The [[cosmic microwave background]] has its own co-moving frame of reference. The relative motion of the Sun at {{Val|365|u=km/s}} results in a [[dipole anisotropy]] due to [[redshift]] of the isotropic radiation from this source.<ref>{{cite book | title=An Introduction to Galaxies and Cosmology | first=David John | last=Adams | editor1-first=Mark H. | editor1-last=Jones | editor2-first=Robert J. | editor2-last=Lambourne | publisher=Cambridge University Press | year=2004 | isbn=978-0-521-54623-2 | url=https://books.google.com/books?id=36K1PfetZegC&pg=PA298 }}</ref>

==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.
* [[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.
* [[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.
* [[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).
* [[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.

==References==
{{Reflist}}

==Further reading==
* 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. 
* {{cite book | 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 = 978-0-8153-0322-0 | 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}}
*[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]

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