View source for Orbit

{{Short description|Curved path of an object around a point}}
{{About|orbits in celestial mechanics, due to gravity}}
{{pp-semi-vandalism|small=yes}}
{{Use dmy dates|date=September 2019}}
[[File:Animation of Orbital eccentricity.gif|right|thumb|Variation of orbital eccentricity<br>{{legend inline|OrangeRed|size=80%|0.0}}&nbsp;{{legend inline|Lime|size=80%|0.2}}&nbsp;{{legend inline|Cyan|size=80%|0.4}}&nbsp;{{legend inline|Gold|size=80%|0.6}}&nbsp;{{legend inline|hotpink|size=80%|0.8}}]]

In [[celestial mechanics]], an '''orbit''' is the curved [[trajectory]] of an [[physical body|object]]<ref>{{Cite encyclopedia | url=https://www.britannica.com/EBchecked/topic/431123/orbit | title=orbit (astronomy) | encyclopedia=Encyclopædia Britannica | edition=Online | access-date=28 July 2008 | archive-date=5 May 2015 | archive-url=https://web.archive.org/web/20150505012919/https://www.britannica.com/EBchecked/topic/431123/orbit | url-status=live }}</ref> under the influence of an attracting force. Known as an '''orbital revolution''', examples include the trajectory of a [[planet]] around a star, a [[natural satellite]] around a planet, or an [[satellite|artificial satellite]] around an object or position in space such as a planet, moon, asteroid, or [[Lagrange point]]. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow [[elliptic orbit]]s, with the [[barycenter|center of mass]] being orbited at a focal point of the ellipse,<ref>{{Cite web |url=http://spaceplace.nasa.gov/barycenter/ |title=The Space Place :: What's a Barycenter |access-date=26 November 2012 |archive-date=8 January 2013 |archive-url=https://web.archive.org/web/20130108073405/http://spaceplace.nasa.gov/barycenter/ |publisher=NASA |url-status=live }}</ref> as described by [[Kepler's laws of planetary motion]].

For most situations, orbital motion is adequately approximated by [[Newtonian mechanics]], which explains [[Newton's law of universal gravitation|gravity]] as a force obeying an [[inverse-square law]].<ref>{{cite book | first=Thomas S. | last=Kuhn | title=The Copernican Revolution | pages=238, 246–252 | publisher=Harvard University Press | orig-year=1957 |date=1985 | isbn=978-0-674-17103-9 | url=https://books.google.com/books?id=sWScX_aduGMC&pg=PA246 }}</ref> However, [[Albert Einstein]]'s [[general theory of relativity]], which accounts for gravity as due to curvature of [[spacetime]], with orbits following [[geodesic]]s, provides a more accurate calculation and understanding of the exact mechanics of orbital motion.

==History==
[[File:Planisphaerium Ptolemaicum siue machina orbium mundi ex hypothesi Ptolemaica in plano disposita (2709983277).jpg|thumb|alt=Andreas Cellarius, a Dutch mathematician and geographer in the 17th century, compiled a celestial atlas with theories from astronomers like Ptolemy and Copernicus. This illustration shows the Earth at the center, with the Moon and planets orbiting around it, based on Ptolemy's geocentric model before Copernicus' heliocentric model.|The Earth-centered universe according to Ptolemy; illustration by [[Andreas Cellarius]] from ''[[Harmonia Macrocosmica]]'', 1660]]

Historically, the apparent motions of the planets were described by European and Arabic philosophers using the idea of [[celestial spheres]]. This model posited the existence of perfect moving spheres or rings to which the stars and planets were attached. It assumed the heavens were fixed apart from the motion of the spheres and was developed without any understanding of gravity. This concept originated with [[Hellenistic astronomy]], particularly [[Eudoxus of Cnidus|Eudoxus]] and [[Aristotle]]. After the planets' motions were more accurately measured, theoretical mechanisms such as [[deferent and epicycle]]s were added by [[Ptolemy]].<ref>{{cite journal | last=Americo | first=M. | year=2017 | title=A brief history of premodern astronomical models | journal=The Classical Outlook | volume=92 | issue=3 | pages=94–101 | jstor=26431167 }}</ref> Although the model was capable of reasonably accurately predicting the planets' positions in the sky, more and more epicycles were required as the measurements became more accurate, hence the model became increasingly unwieldy.<ref>{{cite book | title=The Ellipse: A Historical and Mathematical Journey | first=Arthur | last=Mazer | publisher=John Wiley & Sons | year=2011 | page=12 | isbn=978-1-118-21143-4 | url=https://books.google.com/books?id=twWkDe1Y9YQC&pg=PA12 }}</ref> Originally [[geocentric model|geocentric]], it was modified by [[Copernicus]] to place the Sun at the centre to help simplify the model. The model was further challenged during the 16th century, as comets were observed traversing the spheres.<ref>{{cite book | title=Experiment and Natural Philosophy in Seventeenth-Century Tuscany: The History of the Accademia del Cimento | series=Studies in History and Philosophy of Science | first=Luciano | last=Boschiero | publisher=Springer Science & Business Media | year=2007 | isbn=978-1-4020-6246-9 | pages=216–217 | url=https://books.google.com/books?id=gfeBxXt9VxEC&pg=PA217 }}</ref><ref>{{cite book | first=M. | last=Caspar | translator-first=C. Doris | translator-last=Hellmann | title=Kepler | orig-year=1959 | year=2012 | publisher=Courier Corporation | series=Dover Books on Astronomy | pages=131–140 | isbn=978-0-486-15175-5 | url=https://books.google.com/books?id=cYQUAwAAQBAJ&pg=PA131 }}</ref>

[[File:Solar system orbital period vs semimajor axis.svg|right|thumb|Distance from Sun vs. orbital period for Solar System bodies. Each object lies along the same line because the Sun has a much higher mass.]]
The basis for the modern description of orbits was first formulated by [[Johannes Kepler]] whose results are summarised in his three laws of planetary motion. First, he found that the orbits of the planets in the [[Solar System]] are elliptical, not [[circle|circular]] (or [[epicycle|epicyclic]]), as had previously been believed, and that the Sun is not located at the center of the orbits, but rather at one [[focus (geometry)|focus]]. Second, he found that the orbital speed of each planet is not constant, as had previously been thought, but rather that the speed depends on the planet's distance from the Sun. Third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods.<ref name="Kepler's Laws of Planetary Motion">{{cite journal | title=The Mathematical Relationship between Kepler's Laws and Newton's Laws | first=Andrew T. | last=Hyman | journal=The American Mathematical Monthly | publisher=Taylor & Francis, Ltd. | volume=100 | issue=10 | date=December 1993 | pages=932–936 | doi=10.2307/2324215 | jstor=2324215 }}</ref>

Jupiter and Venus, for example, are respectively about 5.2 and 0.723 [[astronomical unit|AU]] distant from the Sun, their orbital periods respectively about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter:<ref>{{cite web | url=https://nssdc.gsfc.nasa.gov/planetary/factsheet/jupiterfact.html | title=Jupiter Fact Sheet | last=Williams | first=David R. | date=October 2, 2024 | publisher=NASA Goddard Space Flight Center | access-date=2025-08-13 }}</ref>
: <math display=inline>\tfrac{5.204^3}{11.862^2} \approxeq 1.002</math>
is practically equal to that for Venus,<ref>{{cite web | title=Venus Fact Sheet | last=Williams | first=David R. | date=November 25, 2020 | publisher=NASA Goddard Space Flight Center | url=https://nssdc.gsfc.nasa.gov/planetary/factsheet/venusfact.html | access-date=2021-04-15 }}</ref>
: <math display=inline>\tfrac{0.723^3}{0.615^2} \approxeq 0.999</math>
in accord with the relationship. Idealised orbits meeting these rules are known as [[Kepler orbits]].

[[Isaac Newton]] demonstrated that Kepler's laws were derivable from his theory of [[gravitation]],<ref>{{cite book | title=Newton's Gravity: An Introductory Guide to the Mechanics of the Universe | series=Undergraduate Lecture Notes in Physics | first=Douglas W. | last=MacDougal | publisher=Springer Science & Business Media | year=2012 | isbn=978-1-4614-5444-1 | pages=127–133 | url=https://books.google.com/books?id=t77juT326MYC&pg=PA127 }}</ref> and that, in general, the orbits of bodies subject to gravity were [[conic section]]s,<ref>{{cite journal | title=Newton's Principia and Inverse-Square Orbits | last=Nauenberg | first=M. | year=1994 | journal=The College Mathematics Journal | volume=25 | issue=3 | pages=212–222 | doi=10.1080/07468342.1994.11973610 }}</ref> under his assumption that the force of gravity propagates instantaneously.<ref name=Kembhavi_Khare_2020>{{cite book | title=Gravitational Waves: A New Window to the Universe | series=Physics and Astronomy | first1=Ajit | last1=Kembhavi | first2=Pushpa | last2=Khare | publisher=Springer Nature | year=2020 | isbn=978-981-15-5709-5 | pages=38–42 | url=https://books.google.com/books?id=ugn2DwAAQBAJ&pg=PA41 }}</ref> To satisfy Kepler's third law, Newton showed that, for a pair of bodies, the orbit size (''a''), [[orbital period]] (''T''), and their combined masses (''M'') are related to each other by:<ref>{{cite book | title=Newton's Principia For The Modern Student | first=Joseph | last=Gallant | publisher=World Scientific | year=2025 | page=22 | isbn=978-981-12-7653-8 | url=https://books.google.com/books?id=xV1PEQAAQBAJ&pg=PA22 }}</ref>
: <math>T^2 \propto \frac{a^3}{M}</math>
and that those bodies orbit their common [[center of mass]].<ref>{{cite book | title=Isaac Newton's Scientific Method: Turning Data into Evidence about Gravity and Cosmology | first=William L. | last=Harper | publisher=Oxford University Press | year=2011 | isbn=978-0-19-161790-4 | pages=305–314 | url=https://books.google.com/books?id=hgFREAAAQBAJ&pg=PA305 }}</ref> Where one body is much more massive than the other (as is the case of an artificial satellite orbiting a planet), it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.

Advances in Newtonian mechanics were then used to explore variations from the simple assumptions behind Kepler orbits, such as the perturbations due to other bodies, or the impact of spheroidal rather than spherical bodies. [[Joseph-Louis Lagrange]] developed a [[Lagrangian mechanics|new approach]] to [[Newtonian mechanics]] emphasizing energy more than force,<ref>{{cite book | title=Mathematical Methods for Physical and Analytical Chemistry | first=David Z. | last=Goodson | publisher=John Wiley & Sons | year=2011 | isbn=978-1-118-13517-4 | page=208 | url=https://books.google.com/books?id=kd_6EKIsQYgC&pg=PA208 }}</ref> and made progress on the [[three-body problem]], discovering the [[Lagrangian points]] with [[Leonhard Euler|Euler]].<ref>{{cite book | chapter=Periodic solutions near the Lagrange equilibrium points in the restricted three-body broblem, for mass ratios near Routh's critical value | first=Gianfausto | last=Dell'antonio | title=Mechanics, Analysis and Geometry: 200 Years after Lagrange | series=North-Holland Delta Series | editor-first=M. | editor-last=Francaviglia | publisher=Elsevier | year=2012 | page=19 | isbn=978-0-444-59737-3 | chapter-url=https://books.google.com/books?id=0PrVtercCVQC&pg=PA19 }}</ref> In a dramatic vindication of classical mechanics, in 1846 [[Urbain Le Verrier]] was able to predict the position of [[Neptune]] based on unexplained perturbations in the orbit of [[Uranus]].<ref>{{cite journal | title=The contrivance of Neptune | last=Krajnović | first=Davor | journal=Astronomy & Geophysics | volume=57 | issue=5 | pages=5.28–5.34 | date=October 2016 | doi=10.1093/astrogeo/atw183 | arxiv=1610.06424 | bibcode=2016A&G....57e5.28K }}</ref>

[[Albert Einstein]] in his 1916 paper ''The Foundation of the General Theory of Relativity'' explained that gravity was due to curvature of [[space-time]] and removed Newton's assumption that changes in gravity propagate instantaneously. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits. In [[relativity theory]], orbits follow geodesic trajectories which are usually approximated very well by the Newtonian predictions (except where there are very strong gravity fields and very high speeds) but the differences are measurable. Essentially all the experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy.<ref name=Kembhavi_Khare_2020/> The original vindication of general relativity is that it was able to account for the remaining unexplained amount in [[Tests of general relativity#Perihelion precession of Mercury|precession of Mercury's perihelion]] first noted by Le Verrier.<ref name=O'Donnell_2014/> However, Newton's solution is still used for most short term purposes since it is significantly easier to use and sufficiently accurate.<ref name=Kembhavi_Khare_2020/>

==Planetary orbits==
{{Multiple image
| total_width       = 260
| direction         = vertical
| image1            = SunAroundBarycenterXY.svg
| alt1              = Looping curve passing into and out of the circular Sun, marked with yearly intervals
| caption1          = Motion of the barycenter of the Solar System relative to the Sun, 2000–2050
| image2            = Perihelion-Aphelion.svg
| alt2              = An ellipse with the apsides marked at the extreme ends. The Sun at the ellipse focus closest to the perihelion
| caption2          = The apsides of an object in an elliptical orbit with the Sun
}}
Within a [[planetary system]], various non-stellar objects follow [[elliptical orbit]]s around the system's [[Barycentric coordinates (astronomy)|barycenter]]. These objects include planets, [[dwarf planet]]s, [[asteroid]]s and other [[minor planet]]s, [[comet]]s, [[meteoroid]]s, and even [[space debris]].<ref>{{cite book | title=Planetary Geoscience | first=Harry Y. | last=McSween | publisher=Cambridge University Press | year=2019 | isbn=978-1-107-14538-2 | pages=3–15 | url=https://books.google.com/books?id=oTCdDwAAQBAJ&pg=PA15 }}</ref> A comet in a [[Parabolic trajectory|parabolic]] or [[Hyperbolic trajectory|hyperbolic]] orbit about a barycenter is not gravitationally bound to the star and therefore is not considered part of the star's planetary system.<ref>{{cite encyclopedia | last=Raymond | first=S. N. | year=2023 | title=Ejection, Hyperbolic | editor-last=Gargaud | editor-first=M. | display-editors=etal | encyclopedia=Encyclopedia of Astrobiology | page=881 | publisher=Springer | location=Berlin, Heidelberg | doi=10.1007/978-3-662-65093-6_489 | isbn=978-3-662-65092-9 }}</ref> Bodies that are gravitationally bound to one of the planets in a planetary system, including [[natural satellite]]s, [[satellite|artificial satellites]], and the objects within [[ring system]]s, follow orbits about a barycenter near or within that planet.<ref name=Hahn_2020>{{cite book | title=Basic Calculus of Planetary Orbits and Interplanetary Flight: The Missions of the Voyagers, Cassini, and Juno | series=Mathematics and Statistics | first=Alexander J. | last=Hahn | publisher=Springer Nature | year=2020 | isbn=978-3-030-24868-0 | pages=120–121 | url=https://books.google.com/books?id=ILHWDwAAQBAJ&pg=PA120 }}</ref>

Owing to mutual [[Perturbation (astronomy)|gravitational perturbations]], the [[eccentricity (orbit)|eccentricities]] and [[Orbital inclination|inclination]]s of the planetary orbits vary over time.<ref>{{cite journal | first1=Bonan | last1=Pu | first2=Dong | last2=Lai | title=Eccentricities and inclinations of multiplanet systems with external perturbers | journal=Monthly Notices of the Royal Astronomical Society | volume=478 | issue=1 | date=July 2018 | pages=197–217 | doi=10.1093/mnras/sty1098 | doi-access=free | arxiv=1801.06220 }}</ref> [[Mercury (planet)|Mercury]], the smallest planet in the Solar System, has the most eccentric orbit. At the present [[Epoch (astronomy)|epoch]], [[Mars]] has the next largest eccentricity while the smallest orbital eccentricities are seen with [[Venus]] and [[Neptune]].<ref>{{cite book | title=Solar System Evolution: A New Perspective | first=Stuart Ross | last=Taylor | publisher=Cambridge University Press | year=2001 | isbn=978-0-521-64130-2 | url=https://books.google.com/books?id=R3QKq4K0jIEC&pg=PA164-IA2 }}</ref>

As two objects orbit each other, the [[periapsis]] is that point at which the two objects are closest to each other. Less properly, "perifocus"{{cn|date=October 2025}} or "pericentron" are used.<ref name=Wells_1965/> The [[apoapsis]] is that point at which they are the farthest, or sometimes apifocus{{cn|date=October 2025}} or apocentron.<ref name=Wells_1965>{{cite book | title=Analytic lifetime studies of a close-lunar satellite | first=William R. | last=Wells | series=NASA Technical Note | volume=D-2805 | publisher=National Aeronautics and Space Administration | year=1965 | page=20 | url=https://books.google.com/books?id=QqabGkoXnP8C&pg=RA5-PA20 }}</ref> A line drawn from periapsis to apoapsis is the ''[[line of apsides|line-of-apsides]]''. This is the major axis of the ellipse, the line through its longest part.<ref>{{cite encyclopedia | last=Fairbridge | first=R. W. | year=1997 | title=Apsis, apsides | encyclopedia=Encyclopedia of Planetary Science | series=Encyclopedia of Earth Science. | page=26 | publisher=Springer | location=Dordrecht | doi=10.1007/1-4020-4520-4_17 | isbn=0-412-06951-2 }}</ref>

More specific terms are used for specific bodies. For example, ''perigee'' and ''apogee'' are the lowest and highest parts of an orbit around Earth, while ''perihelion'' and ''aphelion'' are the closest and farthest points of an orbit around the Sun.<ref name=González–López_2025/> Things orbiting the [[Moon]] have a [[perilune]] and [[apolune]] (or [[periselene]] and [[aposelene]] respectively).<ref>{{cite book | title=Interplanetary Astrodynamics | first1=David B. | last1=Spencer | first2=Davide | last2=Conte | publisher=CRC Press | year=2023 | page=362 | isbn=978-1-000-85974-4 | url=https://books.google.com/books?id=6J63EAAAQBAJ&pg=PA362 }}</ref> An orbit around any [[star]], not just the Sun, has a [[periastron]] and an [[apastron]].<ref name=González–López_2025>{{cite book | title=Classical Mechanics | first=Artemio | last=González–López | publisher=CRC Press | year=2025 | isbn=978-1-040-38463-3 | page=95 | url=https://books.google.com/books?id=rq15EQAAQBAJ&pg=PA95 }}</ref>

In the case of planets orbiting a star, the mass of the star and all its satellites are calculated to be at a single point called the barycenter. The individual satellites of that star follow their own elliptical orbits with the barycenter at one focal point of that ellipse.<ref name=Hahn_2020/> At any point along its orbit, any satellite will have a certain value of kinetic and potential energy with respect to the barycenter, and the sum of those two energies is a constant value at every point along its orbit. As a result, as a planet approaches periapsis, the planet will increase in speed as its potential energy decreases; as a planet approaches apoapsis, its velocity will decrease as its potential energy increases.<ref>{{cite book | title=The Story of Physics | first1=Lloyd | last1=Motz | first2=Jefferson Hane | last2=Weaver | publisher=Springer | year=2013 | page=99 | isbn=978-1-4899-6305-5 | url=https://books.google.com/books?id=EcYGCAAAQBAJ&pg=PA99 }}</ref>

==Principles==
An orbit can be explained by combining [[Newton's laws of motion]] with his [[Newton's law of universal gravitation|law of universal gravitation]]. The laws of motion are as follows:<ref name=Newton_Oxford>{{cite web | title=Newton's laws of motion | work=Oxford Reference | publisher=Oxford University Press | url=https://www.oxfordreference.com/display/10.1093/oi/authority.20110803100232420 | access-date=2025-10-29 }}</ref>

* A body continues in a state of uniform rest or motion unless acted upon by an external force.
* The acceleration produced when a force acts is directly proportional to the force and takes place in the direction in which the force acts.
* To every action there is an equal and opposite reaction.

By the first law of motion, in the absence of gravity, a physical object will continue to move in a straight line due to [[inertia]]. According to the second law, a force, such as gravity, pulls the moving object toward the body that is the source of the force and thus causes the object to follow a curved [[trajectory]]. If the object has enough [[tangential velocity]], it will not fall into the gravitating body but can instead continue to follow the curved trajectory caused by the force indefinitely. The object is then said to be orbiting the body. According to the third law, each body applies an equal force on the other, which means the two bodies orbit around their center of mass, or barycenter.<ref name=Arfken_2012>{{cite book | title=University Physics | first=George | last=Arfken | publisher=Academic Press | year=2012 | isbn=978-0-323-14202-1 | pages=107–108 | url=https://books.google.com/books?id=PfadZy35Wh0C&pg=PA107 }}</ref>

[[File:Gravity assist still Jupiter.svg|right|thumb|A [[gravity assist]] fly-by uses a hyperbolic orbit to change a spacecraft's velocity and heading<ref>{{cite conference | title=The celestial mechanics of gravity assist | first=R. | last=Broucke | conference=Astrodynamics Conference, 15 August 1988 - 17 August 1988, Minneapolis, MN, U.S.A. | doi=10.2514/6.1988-4220 }}</ref>]]
Because of the law of universal gravitation, the strength of the gravitational force depends on the masses of the two bodies and their separation. As the gravity varies over the course of the orbit, it reproduces Kepler's laws of planetary motion.<ref name=Arfken_2012/> Depending on the evolving energy state of the system, the velocity relationship of two moving objects with mass can be considered in four practical classes, with subtypes:
; No orbit
; [[Sub-orbital spaceflight|Suborbital trajectories]]: a range of interrupted elliptical paths
; Orbital trajectories (or simply, orbits): {{unordered list
 | Range of elliptical paths with closest point opposite firing point
 | Circular path
 | Range of elliptical paths with closest point at firing point
 }}
; [[escape orbit|Open (or escape) trajectories]]: {{unordered list
 | Parabolic paths
 | Hyperbolic paths
 }}

To achieve orbit, conventional rockets are launched vertically at first to lift the rocket above the dense lower atmosphere (which causes frictional drag), and gradually pitch over and finish firing the rocket engine parallel to the atmosphere to achieve [[orbital injection]].<ref>{{cite book | title=Space Vehicle Dynamics and Control | series=AIAA education series | first=Bong | last=Wie | publisher=AIAA | year=1998 | isbn=978-1-56347-261-9 | pages=263–270 | url=https://books.google.com/books?id=n97tEQvNyVgC&pg=PA263 }}</ref> Once in orbit, their speed keeps them above the atmosphere. If an elliptical orbit dips into dense air, the object will lose speed and re-enter, falling to the ground. Occasionally a space craft will intentionally intercept the atmosphere, in an act commonly referred to as an [[aerobraking]] maneuver.<ref>{{cite web | title=Technologies for aerobraking | first1=David M. | last1=Cooper | first2=James O. | last2=Arnold  | work=NASA Technical Reports Server | date=March 1991 | url=https://ntrs.nasa.gov/citations/19910029132 | access-date=2025-10-29 }}</ref>

===Illustration===
{{main|Newton's cannonball}}
[[File:Newton Cannon.svg|thumb|300px|[[Newton's cannonball]], an illustration of how objects can "fall" in a curve]]

As an illustration of an orbit around a planet, the [[Newton's cannonball]] model may prove useful (see image). This is a '[[thought experiment]]', in which a cannon on top of a tall mountain is able to fire a cannonball horizontally at any chosen muzzle speed. The effects of air friction on the cannonball are ignored (or perhaps the mountain is high enough that the cannon is above the Earth's atmosphere, which is the same thing).<ref>See {{cite book | url=https://books.google.com/books?id=rEYUAAAAQAAJ&pg=PA6 | pages=6 to 8  | first=Isaac | last=Newton | title=Treatise of the System of the World | archive-url=https://web.archive.org/web/20161230132051/https://books.google.com/books?id=rEYUAAAAQAAJ&pg=PA6 | access-date=2016-12-30 | archive-date=2016-12-30 | date=1685 }} Translated into English 1728, see [[Philosophiæ Naturalis Principia Mathematica#Preliminary version|Newton's 'Principia' – A preliminary version]]), for the original version of this 'cannonball' thought-experiment.</ref>

If the cannon fires its ball with a low initial speed, the trajectory of the ball curves downward and hits the ground (A). As the firing speed is increased, the cannonball hits the ground farther (B) away from the cannon, because while the ball is still falling towards the ground, the ground is increasingly curving away from it (see first point, above). All these motions are actually "orbits" in a technical sense—they are describing a portion of an elliptical path around the center of gravity—but the orbits are interrupted by striking the Earth.

If the cannonball is fired with sufficient speed, the ground curves away from the ball at least as much as the ball falls—so the ball never strikes the ground. It is now in what could be called a non-interrupted or circumnavigating, orbit. For any specific combination of height above the center of gravity and mass of the planet, there is one specific firing speed (unaffected by the mass of the ball, which is assumed to be very small relative to the Earth's mass) that produces a [[circular orbit]], as shown in (C).

As the firing speed is increased beyond this, non-interrupted elliptic orbits are produced; one is shown in (D). If the initial firing is above the surface of the Earth as shown, there will also be non-interrupted elliptical orbits at slower firing speed; these will come closest to the Earth at the point half an orbit beyond, and directly opposite the firing point, below the circular orbit.

At a specific horizontal firing speed called [[escape velocity]], dependent on the mass of the planet and the distance of the object from the barycenter, an open orbit (E) is achieved that has a [[parabolic trajectory|parabolic path]]. At even greater speeds the object will follow a range of [[hyperbolic trajectory|hyperbolic trajectories]]. In a practical sense, both of these trajectory types mean the object is "breaking free" of the planet's gravity, and "going off into space" potentially never to return. However, the object remains under the influence of the Sun's gravity.<ref>{{cite book | title=Analysis of Weapons | volume=2 | series=Weapons systems fundamentals | publisher=U.S. Government Printing Office | year=1963 | page=157 | url=https://books.google.com/books?id=nBNIAQAAIAAJ&pg=PA157 }}</ref>

==Newton's laws==
===Gravity and motion===
In most real-world situations, [[Newton's laws]] provide a reasonably accurate description of motion of objects in a gravitational field. The adjustments needed to accommodate the [[theory of relativity]] become appreciable in cases where the object is in the proximity of a significant gravitational source such as a star,<ref>{{cite book | title=Differential Equations, Mechanics, and Computation | series=IAS/Park city mathematical subseries, student mathematical library | volume=51 | first1=Richard S. | last1=Palais | first2=Robert A. | last2=Palais | publisher=American Mathematical Society | year=2009 | isbn=978-0-8218-2138-1 | page=95 | url=https://books.google.com/books?id=ncqFAwAAQBAJ&pg=PA95 }}</ref> or a high level of accuracy is needed.

[[File:Gravity turn in tangential axes.svg|right|thumb|A rocket experiences a gravitational force ''g'' and acceleration from propulsion ''a<sub>e</sub>'', resulting in a net acceleration ''a''.]]
The acceleration of a body is equal to the combination of the forces acting on it, divided by its mass. The gravitational force acting on a body is proportional to the product of the masses of the two attracting bodies and decreases inversely with the square of the distance between them.<ref name=Arfken_2012/> For a [[two-body problem]], defined as an isolated system of two spherical bodies with known masses and sufficient separation, this Newtonian approximation of their gravitational interaction can provide a reasonably accurate calculation of their trajectories.<ref>{{cite book | title=Celestial Mechanics: The Waltz of the Planets | series=Springer Praxis Books | first1=Alessandra | last1=Celletti | first2=Ettore | last2=Perozzi | publisher=Springer Science & Business Media | year=2007 | isbn=978-0-387-68577-9 | page=12 | url=https://books.google.com/books?id=y7yar3lIj-AC&pg=PA12 }}</ref>

If the heavier body is much more massive than the smaller, as in the case of a satellite or small moon orbiting a planet or for the Earth orbiting the Sun, it is accurate enough and convenient to describe the motion in terms of a [[coordinate system]] that is centered on the heavier body, and we say that the lighter body is in orbit around the heavier. For the case where the masses of two bodies are comparable, an exact Newtonian solution is still sufficient and can be had by placing the coordinate system at the center of the mass of the system.<ref>{{cite book | title=An Introduction to Celestial Mechanics | first=Richard | last=Fitzpatrick | publisher=Cambridge University Press | year=2012 | isbn=978-1-107-02381-9 | pages=2–5 | url=https://books.google.com/books?id=K5wdRGI3w8kC&pg=PA5 }}</ref>

===Energy and conic sections===
Energy is associated with [[gravitational field]]s. A stationary body far from another can do external work if it is pulled towards it, and therefore has gravitational ''[[potential energy]]''. Since work is required to separate two bodies against the pull of gravity, their gravitational potential energy increases as they are separated, and decreases as they approach one another. For point masses, the gravitational energy decreases to zero as they approach zero separation. It is convenient and conventional to assign the potential energy as having zero value when they are an infinite distance apart, and hence it has a negative value (since it decreases from zero) for smaller finite distances.<ref>{{cite book | title=Dynamics of Stellar Systems | first=K. F. | last=Ogorodnikov | publisher=Elsevier | year=2016 | isbn=978-1-4831-3745-2 | page=178 | url=https://books.google.com/books?id=EdU_DQAAQBAJ&pg=PA178 }}</ref>

When only two gravitational bodies interact, their orbits follow a [[conic section]]. The orbit can be open (implying the object never returns) or closed (returning). Which it is depends on the total [[energy]] ([[kinetic energy|kinetic]] + [[potential energy]]) of the system. In the case of an open orbit, the speed at any position of the orbit is at least the [[escape velocity]] for that position, in the case of a closed orbit, the speed is always less than the escape velocity. Since the kinetic energy is never negative if the common convention is adopted of taking the potential energy as zero at infinite separation, the bound orbits will have negative total energy, the parabolic trajectories zero total energy, and hyperbolic orbits positive total energy.<ref name=Curtis_2023>{{cite book |
title=Orbital Mechanics for Engineering Students | series=Aerospace Engineering | first=Howard D. | last=Curtis | edition=3 | publisher=Butterworth-Heinemann | year=2013 | isbn=9780-08-097748-5 | pages=80–104 | url=https://books.google.com/books?id=2U9Z8k0TlTYC&pg=PA104 }}</ref><ref name=Spencer_Conte_2023>{{cite book | title=Interplanetary Astrodynamics | first1=David B. | last1=Spencer | first2=Davide | last2=Conte | publisher=CRC Press | year=2023 | isbn=978-1-000-85976-8 | pages=39–43 | url=https://books.google.com/books?id=h-i2EAAAQBAJ&pg=PR39 }}</ref>

[[File:Conic Sections.svg|right|thumb|Conic sections for different orbit types]]
An open orbit will have a parabolic shape if it has the velocity of exactly the escape velocity at that point in its trajectory, and it will have the shape of a [[hyperbola]] when its velocity is greater than the escape velocity.<ref name=Curtis_2023/><ref name=Spencer_Conte_2023/> When two bodies approach each other with escape velocity or greater (relative to each other), they will briefly curve around each other at the time of their closest approach, and then separate and fly apart.

All closed orbits have the shape of an [[ellipse]]. A circular orbit is a special case, wherein the foci of the ellipse coincide.<ref name=Curtis_2023/><ref name=Spencer_Conte_2023/>

===Kepler's laws===
Bodies following closed orbits repeat their paths with a certain time called the period. This motion is described by the empirical laws of Kepler, which can be mathematically derived from Newton's laws. These can be formulated as follows:<ref name=NASA_Kepler_2024>{{cite web | title=Orbits and Kepler's Laws | publisher=NASA/JPL | date=May 2, 2024 | url=https://science.nasa.gov/solar-system/orbits-and-keplers-laws/ | access-date=2025-10-30 }}</ref>

# The orbit of a planet around the [[Sun]] is an ellipse, with the Sun in one of the focal points of that ellipse. [This focal point is actually the [[barycenter]] of the [[Solar System|Sun-planet system]]; for simplicity, this explanation assumes the Sun's mass is infinitely larger than that planet's.] The planet's orbit lies in a plane, called the '''[[Orbital plane (astronomy)|orbital plane]]'''.<ref name=NASA_Kepler_2024/>
# As the planet moves in its orbit, the line from the Sun to the planet sweeps a constant area of the [[Orbital plane (astronomy)|orbital plane]] for a given period of time, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its [[perihelion]] than near its [[aphelion]], because at the smaller distance it needs to trace a greater arc to cover the same area.<ref name=NASA_Kepler_2024/> This law is usually stated as "equal areas in equal time."
# For a given orbit, the ratio of the cube of its [[semi-major axis]] to the square of its period is constant.<ref name=NASA_Kepler_2024/>

===Limitations of classical mechanics===
Ideally, the bound orbits of a point mass or a spherical body with a [[Newtonian gravitational field]] form closed [[ellipse]]s, which repeat the same path exactly and indefinitely. However, any non-spherical or non-Newtonian effects will cause the orbit's shape to depart from the ellipse. Such effects can be caused by a slight [[Flattening|oblateness]] of the body,<ref>{{cite book | last1=Burnett | first1=E. R. | last2=Schaub | first2=H. | chapter=Approximating orbits in a rotating gravity field with oblateness and ellipticity perturbations | title=Celestial Mechanics and Dynamical Astronomy | volume=134 | article-number=5 | year=2022 | doi=10.1007/s10569-022-10061-z | arxiv=2108.09607 }}</ref> [[Mass concentration (astronomy)|mass anomalies]],<ref>{{cite book | title=Exploring the Moon: The Apollo Expeditions | series=Springer Praxis Books | first=David M. | last=Harland | edition=2nd | publisher=Springer Science & Business Media | year=2008 | isbn=978-0-387-74641-8 | url=https://books.google.com/books?id=wcVsFg9Rff0C&pg=PA23 }}</ref> [[tide|tidal]] deformations,<ref>{{cite book | chapter=Tidal Friction Parameters from Satellite Observations | first=A. | last=Cazenave | title=Tidal Friction and the Earth's Rotation II: Proceedings of a Workshop Held at the Centre for Interdisciplinary Research (ZiF) of the University of Bielefeld, September 28–October 3, 1981 | editor1-first=P. | editor1-last=Brosche | editor2-first=Jürgen | editor2-last=Sündermann | publisher=Springer Science & Business Media | year=2012 | isbn=978-3-642-68836-2 | chapter-url=https://books.google.com/books?id=QbB9CAAAQBAJ&pg=PP19 }}</ref> or [[Theory of relativity|relativistic effects]],<ref name=O'Donnell_2014>{{cite book | title=Essential Dynamics and Relativity | first=Peter J. | last=O'Donnell | publisher=CRC Press | year=2014 | isbn=978-1-4665-8841-7 | pages=127–129 | url=https://books.google.com/books?id=7ZDUDwAAQBAJ&pg=PA127 }}</ref> thereby changing the gravitational field's behavior with distance.

The two-body solutions were published by Newton in [[Philosophiae Naturalis Principia Mathematica|Principia]] in 1687.<ref name=Newton_Oxford/> In 1912, [[Karl Fritiof Sundman]] developed a converging infinite series that solves the general [[three-body problem]]; however, it converges too slowly to be of much use. The restricted three-body problem, in which the third body is assumed to have negligible mass, has been extensively studied. The solutions to this case include the [[Lagrangian point]]s.<ref name=Karttunen_et_al_2007>{{cite book | title=Fundamental Astronomy | display-editors=1 | editor1-first=Hannu | editor1-last=Karttunen | editor2-first=Pekka | editor2-last=Kröger | editor3-first=Heikki | editor3-last=Oja | editor4-first=Markku | editor4-last=Poutanen | editor5-first=Karl Johan | editor5-last=Donner | edition=5th | publisher=Springer Science & Business Media | year=2007 | isbn=978-3-540-34143-7 | pages=120–121 | url=https://books.google.com/books?id=RacoITLoz_oC&pg=PA120 }}</ref> In the case of [[lunar theory]], the 19th century work of [[Charles-Eugène Delaunay]] allowed the motions of the Moon to be predicted to within its own diameter over a 20-year period.<ref>{{cite book | title=Astronomical Observations: Astronomy and the Study of Deep Space | series=An Explorer's Guide to the Universe | editor-last=Gregersen | editor-first=Erik | publisher=Britannica Educational Publishing | year=2009 | page=86 | isbn=978-1-61530-054-9 | url=https://books.google.com/books?id=ib2bAAAAQBAJ&pg=PA86 }}</ref> No universally valid method is known to solve the equations of motion for a system with four or more bodies.

==Formulation==
{{main|Orbit modeling}}

=== Newtonian analysis of orbital motion ===
{{further|Kepler orbit|orbit equation|Kepler's first law}}

The following derivation applies to an elliptical orbit. The assumption is that the central body is massive enough that it can be considered to be stationary and so the more subtle effects of [[general relativity]] can be ignored.

==== Force and acceleration ====
[[File:NewtonsLawOfUniversalGravitation.svg|right|thumb|Gravitational force for masses ''m''<sub>1</sub> and ''m''<sub>2</sub> with separation ''r'']]
The [[classical mechanics|Newtonian]] law of gravitation states that the gravitational acceleration of the second mass towards the central body is related to the inverse of the square of the distance between them, namely:<ref name=Fitzpatrick_2023/>

: <math> F_2 = -\frac {G m_1 m_2}{r^2} </math>

where ''F''<sub>2</sub> is the force acting on the mass ''m''<sub>2</sub> caused by the gravitational attraction mass ''m''<sub>1</sub> has for ''m''<sub>2</sub>, ''G'' is the universal [[gravitational constant]], and ''r'' is the distance between the two masses centers.

From Newton's second law, the summation of the forces acting on ''m''<sub>2</sub> related to that body's acceleration:

: <math>F_2 = m_2 A_2</math>

where ''A''<sub>2</sub> is the acceleration of ''m''<sub>2</sub> caused by the force of gravitational attraction ''F''<sub>2</sub> of ''m''<sub>1</sub> acting on ''m''<sub>2</sub>.

Combining Eq. 1 and 2:

: <math> -\frac {G  m_1  m_2}{r^2} = m_2  A_2 </math>

Solving for the acceleration, ''A''<sub>2</sub>:

: <math> A_2 = \frac{F_2}{m_2} = - \frac{1}{m_2} \frac{G m_1 m_2}{r^2} =  -\frac{\mu}{r^2} </math>

where <math> \mu\, </math> is the [[standard gravitational parameter]], in this case <math>G m_1</math>.<ref>{{cite book | title=Using SI Units in Astronomy | first=Richard | last=Dodd | publisher=Cambridge University Press | year=2011 | isbn=978-1-139-50440-9 | page=96 | url=https://books.google.com/books?id=UC_1_804BXgC&pg=PA96 }}</ref> It is understood that the system being described is ''m''<sub>2</sub>, hence the subscripts can be dropped.

====Polar coordinates====
[[File:Polar coordinates with unit vectors.svg|right|thumb|Unit vectors in polar coordinates]]
The position of the orbiting object at the current time <math> t </math> is located in the [[orbital plane]] using [[vector calculus]] in [[polar coordinates]], both with the standard Euclidean [[Basis (linear algebra)|basis]] and with the polar basis with the origin coinciding with the center of force. Let <math> r </math> be the distance between the object and the center and <math> \theta </math> be the angle it has rotated. Let <math> \hat{\mathbf{x}} </math> and <math> \hat{\mathbf{y}} </math> be the standard [[Euclidean space|Euclidean]] bases and let:<ref name=Fitzpatrick_2023/>

: <math> \hat{\mathbf{r}} = \cos(\theta)\hat{\mathbf{x}} + \sin(\theta)\hat{\mathbf{y}} </math>
: <math> \hat{\boldsymbol \theta} = - \sin(\theta)\hat{\mathbf{x}} + \cos(\theta)\hat{\mathbf{y}} </math>

be the radial and transverse [[Polar coordinate system#Vector calculus|polar]] basis. The first is the [[unit vector]] pointing from the central body to the current location of the orbiting object and the second is the orthogonal unit vector pointing in the direction that the orbiting object would travel if orbiting in a counter clockwise circle. Then the vector to the orbiting object is:

: <math> {\mathbf{O}} = r \cos(\theta)\hat{\mathbf{x}} + r \sin(\theta)\hat{\mathbf{y}} = r \hat{\mathbf{r}} </math>

[[Notation for differentiation#Newton's notation|Newton's notation]] <math> \dot r </math> and <math> \dot \theta </math> denotes the standard [[derivative]]s of how this distance and angle change over time.<ref>{{cite book | title=Newton's Principia For The Modern Student | first=Joseph | last=Gallant | publisher=World Scientific | year=2025 | isbn=978-981-12-7653-8 | pages=9–11 | url=https://books.google.com/books?id=xV1PEQAAQBAJ&pg=PA9 }}</ref> Taking the derivative of a vector to see how it changes over a tiny increment of time, <math> \delta t </math>, subtracts its location at time <math> t + \delta t </math> from that at time <math> t </math> and divides by <math>\delta t </math>. The result remains a vector.

====Kepler's second law====
Because the basis vector <math> \hat{\mathbf{r}} </math> moves as the object orbits, the first step is to differentiate it to determine the radial rate of change with time. From time <math> t </math> to <math> t + \delta t </math>, the vector <math> \hat{\mathbf{r}} </math> keeps its beginning at the origin and rotates from angle <math> \theta </math> to <math> \theta  + \dot \theta\ \delta t </math>, which moves its head a distance <math> \dot \theta\ \delta t </math> in the perpendicular direction <math> \hat{\boldsymbol \theta} </math>, giving a derivative of <math> \dot \theta \hat{\boldsymbol \theta} </math>.

: <math>\begin{align}
  \hat{\mathbf{r}}
    &= \cos(\theta)\hat{\mathbf{x}} + \sin(\theta)\hat{\mathbf{y}} \\
  \frac{\delta \hat{\mathbf{r}}}{\delta t} = \dot{\mathbf r}
    &= -\sin(\theta)\dot \theta \hat{\mathbf{x}} + \cos(\theta)\dot \theta \hat{\mathbf{y}} = \dot \theta \hat{\boldsymbol \theta} \\
  \hat{\boldsymbol \theta}
    &= -\sin(\theta)\hat{\mathbf{x}} + \cos(\theta)\hat{\mathbf{y}} \\
  \frac{\delta \hat{\boldsymbol \theta}}{\delta t} = \dot{\boldsymbol \theta}
    &= -\cos(\theta)\dot \theta \hat{\mathbf{x}} - \sin(\theta) \dot \theta \hat{\mathbf{y}} = -\dot \theta \hat{\mathbf r}
\end{align}</math>

The velocity and acceleration of the orbiting object can now be determined.<ref name=Fitzpatrick_2023/>
: <math>\begin{align}
   {\mathbf{O}} &= r \hat{\mathbf{r}} \\
   \dot{\mathbf{O}} &= \frac{\delta r}{\delta t} \hat{\mathbf{r}} + r \frac{\delta \hat{\mathbf{r}}}{\delta t}
                     = \dot r \hat {\mathbf r} + r \left[ \dot \theta \hat {\boldsymbol \theta} \right] \\
  \ddot{\mathbf{O}} &= \left[\ddot r \hat {\mathbf r} + \dot r \dot \theta \hat {\boldsymbol \theta}\right] +
                       \left[\dot r \dot \theta \hat {\boldsymbol \theta} +
                             r \ddot \theta \hat {\boldsymbol \theta} -
                             r \dot \theta^2 \hat {\mathbf r}
                       \right] \\
                    &= \left[\ddot r - r\dot\theta^2\right]\hat{\mathbf{r}} +
                       \left[r \ddot\theta + 2 \dot r \dot\theta\right] \hat{\boldsymbol \theta}
\end{align}</math>

In the last line, the coefficients of <math> \hat{\mathbf{r}} </math> and <math> \hat{\boldsymbol \theta} </math> give the accelerations in the radial and transverse directions. As said, Newton gives the first due to gravity as <math> -\mu/r^2 </math> and the second is zero per Newton's first law. Thus:<ref name=Fitzpatrick_2023/>

{{NumBlk|:|<math> \ddot r - r\dot\theta^2 = - \frac{\mu}{r^2} </math>|1}}
{{NumBlk|:|<math> r \ddot\theta + 2 \dot r \dot\theta = 0 </math>|2}}

[[File:Second law of Kepler.svg|right|thumb|For a time interval ''t'', the rate of angular change <math display=inline>\dot \theta</math> varies to maintain a constant area ''A'']]
Equation (2) can be rearranged using [[integration by parts]].

: <math> r \ddot\theta + 2 \dot r \dot\theta = \frac{1}{r}\frac{d}{dt}\left( r^2 \dot \theta \right) = 0 </math>

Both sides can now be multiplied through by <math> r </math> because it is not zero unless the orbiting object crashes. Having the derivative be zero indicates that the function is a constant.

{{NumBlk|:|<math>r^2 \dot \theta = h </math>|3}}

which is actually the theoretical proof of [[Kepler's second law]] (A line joining a planet and the Sun sweeps out equal areas during equal intervals of time).<ref name=Fitzpatrick_2023>{{cite web | title=Introduction to Celestial Mechanics | edition=2nd | date=September 16, 2023 | first=Richard | last=Fitzpatrick | publisher=The University of Texas at Austin | url=https://farside.ph.utexas.edu/teaching/celestial/Celestial/node22.html | access-date=2025-11-01 }}</ref> The [[constant of integration]], ''h'', is the [[specific relative angular momentum|angular momentum per unit mass]].<ref>{{cite book | title=Fundamentals of Astrodynamics and Applications | volume=12 | series=Space Technology Library | first=D. A. | last=Vallado | edition=2nd | publisher=Springer Science & Business Media | year=2001 | isbn=978-0-7923-6903-5 | pages=23–25 | url=https://books.google.com/books?id=PJLlWzMBKjkC&pg=PA23 }}</ref>

====Kepler's first law====
In order to get an equation for the orbit from equation (1), the time variable needs to be eliminated. (See also [[Binet equation]].)
In polar coordinates, this would express the distance <math> r </math> of the orbiting object from the center as a function of its angle <math> \theta </math>. However, it is easier to introduce the auxiliary variable <math> u = 1/r  </math> and to express <math> u </math> as a function of <math> \theta </math>. Derivatives of <math>r</math> with respect to time may be rewritten as derivatives of <math>u</math> with respect to angle.<ref name=Fitzpatrick_2023/>

: <math>u = { 1 \over r }</math>
: <math>\dot\theta = \frac{h}{r^2} = hu^2</math> (reworking (3))
: <math>\begin{align}
      \frac{\delta u}{\delta \theta} &= \frac{\delta}{\delta t}\left(\frac{1}{r}\right)\frac{\delta t}{\delta \theta } = -\frac{\dot{r}}{r^2\dot{\theta }} = -\frac{\dot{r}}{h} \\
  \frac{\delta^2 u}{\delta \theta^2} &= -\frac{1}{h}\frac{\delta \dot{r}}{\delta t}\frac{\delta t}{\delta \theta } = -\frac{\ddot{r}}{h\dot{\theta}} = -\frac{\ddot{r}}{h^2 u^2}
  \ \ \ \text{ or } \ \ \  \ddot r = - h^2 u^2 \frac{\delta^2 u}{\delta \theta^2}
\end{align}</math>

Plugging these into (1) gives

: <math>\begin{align}
                                                          \ddot r - r\dot\theta^2 &= -\frac{\mu}{r^2} \\
   -h^2 u^2 \frac{\delta^2 u}{\delta \theta^2} - \frac{1}{u} \left(h u^2\right)^2 &= -\mu u^2
\end{align}</math>

Hence:<ref name=Fitzpatrick_2023/>

{{NumBlk|:|<math> \frac{\delta^2 u}{\delta \theta^2} + u = \frac{\mu}{h^2}</math>|4}}

So for the gravitational force – or, more generally, for ''any'' inverse square force law – the right hand side of the equation becomes a constant and the equation is seen to be the [[harmonic oscillator|harmonic equation]] (up to a shift of origin of the dependent variable). The solution is:

:<math> u(\theta) = \frac\mu {h^2} + A \cos(\theta - \theta_0) </math>

[[File:Ellipse Polar.svg|thumb|right|Polar coordinates centered at focus. The semi-major axis ''a'' is the distance from the center (C) to an apsis (A or B).]]
where ''A'' and ''θ''<sub>0</sub> are arbitrary constants. This resulting equation of the orbit of the object is that of an [[ellipse#Polar form relative to focus|ellipse in Polar form]] relative to one of the focal points. This is put into a more standard form by letting <math> e \equiv h^2 A/\mu </math> be the [[eccentricity (orbit)|eccentricity]], which when rearranged we see:
:<math> u(\theta) = \frac\mu {h^2} (1+ e\cos(\theta - \theta_0))</math>

Note that by letting <math> a \equiv h^2/\mu\left(1 - e^2\right) </math> be the semi-major axis and letting <math> \theta_0 \equiv 0 </math> so the long axis of the ellipse is along the positive ''x'' coordinate yields:<ref name=Fitzpatrick_2023/>
:<math>r(\theta) = \frac{a \left(1 - e^2\right)}{1 + e\cos\theta}</math>

When ''e'' is zero, the result is a circular orbit with ''r'' equal to ''a''.

====Kepler's third law====
Incorporating Newton's laws, the constant in Kepler's third law can be shown to be:<ref>{{cite book | title=Elements of Newtonian Mechanics | first1=Jens M. | last1=Knudsen | first2=Poul G. | last2=Hjorth | publisher=Springer Science & Business Media | year=2012 | isbn=978-3-642-97599-8 | pages=352–355 | url=https://books.google.com/books?id=rkP1CAAAQBAJ&pg=PA352 }}</ref>

: <math>\frac{a^3}{T^2} = \frac{G(M + m)}{4\pi^2} \approx \frac{GM}{4\pi^2} \approx 7.496 \times 10^{-6} \frac{\text{AU}^3}{\text{days}^2}</math>

where <math>M</math> is the [[Solar mass|mass of the Sun]], ''G'' is the gravitational constant, <math>m</math> is the mass of the planet, <math>T</math> is the orbital period and <math>a</math> is the elliptical semi-major axis, and <math>\text{AU} </math> is the [[astronomical unit]], the average distance from earth to the sun. From this, the orbital period can be derived from the semi-major axis.

====Applying torque====
A torque to a satellite can result, for example, due to perturbation from a non-sperical mass.<ref>{{cite book | title=Earth System Geophysics | series=AGU Advanced Textbooks | first=Steven R. | last=Dickman | publisher=John Wiley & Sons | year=2024 | isbn=978-1-119-62797-5 | page=360 | url=https://books.google.com/books?id=OzsxEQAAQBAJ&pg=PA360 }}</ref> When the two-body system is under the influence of torque, the angular momentum ''h'' is not a constant. After the following calculation:
:<math>\begin{align}
                   \frac{\delta r}{\delta \theta} &= -\frac{1}{u^2} \frac{\delta u}{\delta \theta} = -\frac{h}{m} \frac{\delta u}{\delta \theta} \\
               \frac{\delta^2 r}{\delta \theta^2} &= -\frac{h^2u^2}{m^2} \frac{\delta^2 u}{\delta \theta^2} - \frac{hu^2}{m^2} \frac{\delta h}{\delta \theta} \frac{\delta u}{\delta \theta} \\
  \left(\frac{\delta \theta}{\delta t}\right)^2 r &= \frac{h^2 u^3}{m^2}
\end{align}</math>

The result is the [[Sturm-Liouville equation]] of two-body system.<ref>{{cite journal |last1=Luo |first1=Siwei |title=The Sturm-Liouville problem of two-body system |journal=Journal of Physics Communications |date=22 June 2020 |volume=4 |issue=6 |page=061001 |doi=10.1088/2399-6528/ab9c30|bibcode=2020JPhCo...4f1001L |doi-access=free }}</ref> 
{{NumBlk|:|<math> \frac{\delta}{\delta \theta} \left(h \frac{\delta u}{\delta \theta}\right) + hu = \frac{\mu}{h}</math>|5}}

===Relativistic orbital motion===
The classical ([[classical mechanics|Newtonian]]) analysis of [[orbital mechanics]] assumes that the more subtle effects of [[general relativity]], such as [[frame dragging]] and [[gravitational time dilation]] are negligible. Relativistic effects cease to be negligible when near massive bodies (as with the [[Kepler problem in general relativity|precession of Mercury's orbit]] about the Sun),<ref name=O'Donnell_2014/> or when extreme precision is needed (as with calculations of the [[orbital elements]] and time signal references for [[Global Positioning System#Relativity|GPS]] satellites.<ref>{{cite web | last=Pogge | first=Richard W. | url=http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html | title=Real-World Relativity: The GPS Navigation System | publisher=Ohio State University | date=March 11, 2017 | archive-url=https://web.archive.org/web/20151114135709/http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html | archive-date=2015-11-14 | access-date=2008-01-25 }}</ref>)

Because of general relativity, there exists a smallest possible radius for which a particle can stably orbit a [[black hole]]. Any inward perturbation to this orbit will lead to the particle spiraling into the black hole. The size of this [[innermost stable circular orbit]] depends on the spin of the black hole and the [[spin (physics)|spin]] of the particle itself,<ref>{{Cite journal | display-authors=1 | last1=Jefremov | first1=Paul I. | last2=Tsupko | first2=Oleg Yu. | last3=Bisnovatyi-Kogan | first3=Gennady S. | title=Innermost stable circular orbits of spinning test particles in Schwarzschild and Kerr space-times | journal=Physical Review D | year=2015 | volume=91 | issue=12 | article-number=124030 | doi=10.1103/PhysRevD.91.124030 | arxiv=1503.07060 | bibcode=2015PhRvD..91l4030J }}</ref> but with no rotation the theoretical orbital radius is just three times the radius of the [[event horizon]].<ref>{{cite journal | last1=Bardeen | first1=James M. | last2=Press | first2=William H. | last3=Teukolsky | first3=Saul A. | date=December 1972 | title=Rotating Black Holes: Locally Nonrotating Frames, Energy Extraction, and Scalar Synchrotron Radiation | journal=The Astrophysical Journal | volume=178 | pages=347–370 | doi=10.1086/151796 | bibcode=1972ApJ...178..347B }}</ref>

==Specification==
{{Main|Ephemeris}}
{{See also|Keplerian elements}}
[[File:Orbit1.svg|right|thumb|Elements of an elliptical orbit, showing the orbital plane alignment is given by the inclination, longitude of the ascending node, and argument of periapsis]]
Six parameters are required to specify a [[Keplerian orbit]] about a body. For example, the three numbers that specify the body's initial position, and the three values that specify its velocity will define a unique orbit that can be calculated forwards (or backwards) in time.<ref>{{cite book | title=Quo Vadis: Evolution of Modern Navigation: The Rise of Quantum Techniques | first=F. G. | last=Major | publisher=Springer Science & Business Media | year=2013 | isbn=978-1-4614-8672-5 | page=318 | url=https://books.google.com/books?id=3d69BAAAQBAJ&pg=PA318 }}</ref>

An [[perturbation theory|unperturbed]] orbit is two-dimensional in a plane fixed in space, known as the [[orbital plane]]. In three dimensions, the orientation of this plane relative to a [[Orbital plane of reference|plane of reference]], such as the [[plane of the sky]], can be determined by three angles. Extending the analysis to three dimensions requires simply rotating the two-dimensional plane to the required angles relative to the poles of the planetary body involved.

By tradition, the standard set of orbital elements is called the set of [[Orbital elements|Keplerian elements]], after Johannes Kepler and his laws. These six Keplerian elements are as follows:<ref>{{cite book | title=Space Flight Dynamics | series=Aerospace Series | first=Craig A. | last=Kluever | edition=2nd, reprint | publisher=John Wiley & Sons | year=2018 | isbn=978-1-119-15790-8 | pages=57–60 | url=https://books.google.com/books?id=Cp1PDwAAQBAJ&pg=PA57 }}</ref>

* [[Inclination]] (''i'')
* [[Longitude of the ascending node]] (Ω)
* [[Argument of periapsis]] (ω)
* [[orbital eccentricity|Eccentricity]] (''e'')
* [[Semimajor axis]] (''a'')
* [[True anomaly]] at [[Epoch (astronomy)|epoch]] (''θ''<sub>0</sub>).

The [[orbital period]] is simply how long an orbiting body takes to complete one orbit, which can be derived from the semimajor axis and the combined masses. In principle, once the orbital elements are known for a body, its position can be calculated forward and backward indefinitely in time. However, in practice, orbits are affected or [[Perturbation (astronomy)|perturbed]], by other forces than simple gravity from an assumed point source, and thus the orbital elements change over time.

Note that, unless the eccentricity is zero, ''a'' is not the average orbital radius. The time-averaged orbital distance is given by:<ref>{{cite book | title=Comets II | first1=M. | last1=Festou | first2=H. Uwe | last2=Keller | first3=Harold A. | last3=Weaver | year=2004 | page=157 | issue=2 | isbn=978-0-8165-2450-1 | publisher=University of Arizona Press | url=https://books.google.com/books?id=ehA8EAAAQBAJ&pg=PA157 }}</ref>

:<math>\bar{r} = a \left(1 + \frac{e^2}{2} \right)</math>

which only equals ''a'' when ''e'' is zero, for a circular orbit.

==Perturbations==
{{main|Perturbation (astronomy)}}
{{further|Osculating orbit#Perturbations|Orbit modeling#Perturbations}}

An orbital perturbation is when a force or impulse causes an acceleration that changes the parameters of the orbit over time. This perturbation is much smaller than the overall force or average impulse of the main gravitating body. Potential sources of perturbation include departure from sphericity, third body contributions, [[radiation pressure]], [[atmospheric drag]], and [[tidal acceleration]].<ref>{{cite book | title=Orbital Mechanics | edition=3rd | series=AIAA education series | editor-first=Vladimir A. | editor-last=Chobotov | publisher=AIAA | year=2002 | isbn=978-1-60086-097-3 | pages=182–195 | url=https://books.google.com/books?id=SuPQmbqyrFAC&pg=PA185 }}</ref>

===Radial, transverse, and normal perturbations===
[[File:Hohmann transfer orbit.svg|right|thumb|A [[Hohmann transfer orbit]] is a [[orbital maneuver|maneuver]] to change the altitude of an orbit (from 1 to 3) with two tangential [[Impulse (physics)|impulses]] (Δv and Δv')<ref>{{cite book | title=Celestial Mechanics and Astrodynamics: Theory and Practice | volume=436 | series=Astrophysics and Space Science Library | first1=Pini | last1=Gurfil | first2=P. Kenneth | last2=Seidelmann | publisher=Springer | year=2016 | isbn=978-3-662-50370-6 | pages=411–412 | url=https://books.google.com/books?id=VNfBDAAAQBAJ&pg=PA411 }}</ref>]]
For a body in orbit, a perturbing force can be divided into three [[Orthogonality|orthogonal]] components: radial, transverse, and normal. The first two are in the orbital plane (in the direction of the gravitating body and along the path of a circular orbit, respectively) and the third is away from the orbital plane.<ref>{{cite book | title=Physics and Chemistry of the Solar System | first=John S. | last=Lewis | edition=Revised | publisher=Academic Press | year=1997 | isbn=978-0-12-446742-2 | pages=52–53 | url=https://books.google.com/books?id=ApJCRO-_mSUC&pg=PA52 }}</ref> A small radial impulse given to a body in orbit changes the [[Eccentricity (mathematics)|eccentricity]], but not the [[orbital period]] (to first order). A [[Direct motion|prograde]] or [[Retrograde motion|retrograde]] transverse impulse (i.e. an impulse applied along the orbital motion) changes both the eccentricity and the [[orbital period]]. Notably, a prograde impulse at periapsis raises the altitude at apoapsis, and vice versa and a retrograde impulse does the opposite. A normal impulse (out of the orbital plane) causes rotation of the [[Orbital plane (astronomy)|orbital plane]] without changing the [[Orbit (dynamics)|period]] or eccentricity. In all instances, a closed orbit will still intersect the perturbation point.

===Orbital decay===
{{Main|Orbital decay}}
For an object in a sufficiently close orbit about a planetary body with a significant atmosphere, the orbit can decay because of [[drag (physics)|drag]].<ref name=Vukovich_Kim_2019>{{cite journal | title=Satellite orbit decay due to atmospheric drag | first1=G. | last1=Vukovich | first2=Y. | last2=Kim | journal=International Journal of Space Science and Engineering | volume=5 | issue=2 | date=January 23, 2019 | pages=159–180 | doi=10.1504/IJSPACESE.2019.097438 | bibcode=2019IJSSE...5..159V }}</ref> Particularly at each periapsis for an orbital with appreciable eccentricity, the object experiences atmospheric drag, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum.<ref>{{cite book | chapter=A general analytical method for artificial-satellite lifetime determination | first=Frank | last=Garcia, Jr. | title=NASA Technical Note D-4281 | date=December 1967 | location=Washington, D.C. | publisher=National Aeronautics and Space Administration | chapter-url=https://books.google.com/books?id=0a9F1oRmJP4C&pg=RA1-PA1 }}</ref> This is similar to the effect of slowing a pendulum at its lowest point; the highest point of the pendulum's swing becomes lower. Eventually, the effect becomes so great that the maximum kinetic energy is not enough to return the orbit above the limits of the atmospheric drag effect. When this happens the body will rapidly spiral down and intersect the central body.

The region for experiencing atmospheric drag varies by planet; a re-entry vehicle needs to draw much closer to Mars than to Earth,<ref>{{cite book | title=Algorithms for Satellite Orbital Dynamics | series=Springer Series in Astrophysics and Cosmology | first=Lin | last=Liu | translator-first=Shengpan | translator-last=Zhang | publisher=Springer Nature | year=2023 | isbn=978-981-19-4839-8 | pages=373–374 | url=https://books.google.com/books?id=u33GEAAAQBAJ&pg=PA373 }}</ref> for example, and the drag is negligible for Mercury. The bounds of an atmosphere vary significantly due to [[solar forcing]] and [[space weather]].<ref>{{cite journal | title=Effects of plasma drag on low Earth orbiting satellites due to solar forcing induced perturbations and heating | display-authors=1 | first1=Victor U. J. | last1=Nwankwo | first2=Sandip K. | last2=Chakrabarti | first3=Robert S. | last3=Weigel | journal=Advances in Space Research | volume=56 | issue=1 | date=July 2015 | pages=47–56 | publisher=Elsevier | doi=10.1016/j.asr.2015.03.044 }}</ref> During a [[solar maximum]], the Earth's atmosphere causes drag up to a hundred kilometres higher than during a solar minimum.

Orbits can be artificially influenced through the use of rocket engines, which change the kinetic energy of the body at some point in its path. In this way, changes in the orbit shape or orientation can be facilitated. [[Solar sail]]s or [[magnetic sail]]s are forms of propulsion that require no propellant or energy input other than that of the Sun, and so can be used indefinitely for station keeping.<ref>{{cite journal | title=Simple Strategy for Geostationary Stationkeeping Maneuvers Using Solar Sail | first=Christian | last=Circi | journal=Journal of Guidance, Control, and Dynamics | volume=28 | issue=2 | date=March 2005 | pages=249–253 | doi=10.2514/1.6797 }}</ref><ref>{{cite journal | title=Applications of magnetic sails | first1=S. G. | last1=Love | first2=D. G. | last2=Andrews | journal=Acta Astronautica | volume=26 | issue=8–10 | date=August–October 1992 | pages=643–651 | publisher=Elsevier | doi=10.1016/0094-5765(92)90154-B }}</ref> (See [[statite]] for one such proposed use.) Satellites with long conductive tethers can experience orbital decay because of electromagnetic drag from the [[Earth's magnetic field]].<ref>{{cite journal | title=Analysis of Bare-Tether Systems for Deorbiting Low-Earth-Orbit Satellites | first1=E. | last1=Ahedo | first2=J. R. | last2=Sanmartin | journal=Journal of Spacecraft and Rockets | volume=39 | issue=2 | date=March 2002 | pages=198–205 | doi=10.2514/2.3820 }}</ref> As the wire cuts the magnetic field it acts as a generator, moving electrons from one end to the other. The orbital energy is converted to heat in the wire.

For objects below the [[synchronous orbit]] for the body they're orbiting, orbital decay can occur due to [[tidal force]]s.<ref>{{cite journal | title=Tidal decay of close planetary orbits | display-authors=1 | first1=F. A. | last1=Rasio | first2=C. A. | last2=Tout | first3=S. H. | last3=Lubow | first4=M. | last4=Livio | date=May 1996 | journal=Astrophysical Journal | volume=470 | page=1187 | arxiv=astro-ph/9605059 | doi=10.1086/177941 }}</ref> The gravity of the orbiting object raises [[tidal bulge]]s in the primary, and since it is below the synchronous orbit, the orbiting object is moving faster than the body's surface so the bulges lag a short angle behind it. The gravity of the bulges is slightly off of the primary-satellite axis and thus has a component along the direction of the satellite's motion. The near bulge slows the object more than the far bulge speeds it up, and as a result, the orbit decays.

Conversely, the gravity of the satellite on the bulges applies [[torque]] on the primary and speeds up its rotation. Artificial satellites are too small to have an appreciable tidal effect on the planets they orbit, but several moons in the Solar System are undergoing orbital decay by this mechanism.<ref>{{cite book | title=Planetary Geology: An Introduction | first1=Dominic | last1=Fortes | first2=Claudio | last2=Vita-Finzi | edition=3rd | publisher=Liverpool University Press | year=2025 | isbn=978-1-78046-111-3 | url=https://books.google.com/books?id=ZVxHEQAAQBAJ&pg=PT41 }}</ref> Mars' innermost moon [[Phobos (moon)|Phobos]] is a prime example and is expected to either impact Mars' surface or break up into a ring in 20 to 40 million years.<ref>{{cite journal | title=The demise of Phobos and development of a Martian ring system | first1=Benjamin A. | last1=Black | first2=Tushar | last2=Mittal | journal=Nature Geoscience | volume=8 | pages=913–917 | year=2015 | issue=12 | doi=10.1038/ngeo2583 | bibcode=2015NatGe...8..913B }}</ref>

Orbits can decay via the emission of [[gravitational wave]]s. This mechanism is extremely weak for most stellar objects, only becoming significant in cases where there is a combination of extreme mass and extreme acceleration, such as [[compact object]]s that are orbiting each other closely.<ref>{{cite journal | title=Gravitational Waves from Merging Compact Binaries | last=Hughes | first=Scott A. | journal=Annual Review of Astronomy & Astrophysics | volume=47 | issue=1 | pages=107–157 | date=September 2009 | doi=10.1146/annurev-astro-082708-101711  | arxiv=0903.4877 | bibcode=2009ARA&A..47..107H }}</ref>

===Oblateness===
The standard analysis of orbiting bodies assumes that all bodies consist of uniform spheres, or more generally, concentric shells each of uniform density. Mathematically, such bodies are gravitationally equivalent to point sources per the [[shell theorem]].<ref>{{cite journal | title=A note on Newton's shell-point equivalency theorem | first=B. Cameron | last=Reed | journal=American Journal of Physics | volume=90 | pages=394–396 | year=2022 | issue=5 | doi=10.1119/5.0072584 | bibcode=2022AmJPh..90..394R }}</ref> However, in the real world, many bodies rotate, and this introduces [[oblateness]]; known as an [[equatorial bulge]]. This adds a [[Quadropole#Gravitational quadrupole|quadrupole moment]] to the gravitational field, which is significant at distances comparable to the radius of the body.<ref>{{cite journal | last=Iorio | first=L. | date=2011 | title=Perturbed stellar motions around the rotating black hole in Sgr A* for a generic orientation of its spin axis | journal=[[Physical Review D]] | volume=84 | issue=12 | article-number=124001 | bibcode=2011PhRvD..84l4001I | doi=10.1103/PhysRevD.84.124001 | arxiv=1107.2916 | s2cid=118305813 }}</ref><ref>{{cite book | title=Dynamics of Planetary Systems | series=Princeton Series in Astrophysics | first=Scott | last=Tremaine | publisher=Princeton University Press | year=2023 | isbn=978-0-691-24422-8 | pages=355–356 | url=https://books.google.com/books?id=FNGAEAAAQBAJ&pg=PA355 }}</ref> In the general case, the gravitational potential of a rotating body such as a planet can be [[Multipole expansion|expanded in multipoles]] to account for the departure from spherical symmetry.<ref>{{cite journal | last1=Chao | first1=B. F. | last2=Shih | first2=S. A. | title=Multipole Expansion: Unifying Formalism for Earth and Planetary Gravitational Dynamics | journal=Surveys in Geophysics | volume=42 | pages=803–838 | year=2021 | issue=4 | doi=10.1007/s10712-021-09650-8 | bibcode=2021SGeo...42..803C }}</ref>

From the point of view of satellite dynamics, of particular relevance are the so-called even zonal harmonic coefficients, or even zonals, since they induce secular orbital perturbations which are cumulative over time spans longer than the orbital period.<ref>{{cite journal | last=Renzetti | first=G. | date=2013 | title=Satellite Orbital Precessions Caused by the Octupolar Mass Moment of a Non-Spherical Body Arbitrarily Oriented in Space | journal=[[Journal of Astrophysics and Astronomy]] | volume=34 | issue=4 | pages=341–348 | bibcode=2013JApA...34..341R | doi=10.1007/s12036-013-9186-4 | s2cid=120030309 }}</ref><ref>{{cite journal | last=Renzetti | first=G. | date=2014 | title=Satellite orbital precessions caused by the first odd zonal J3 multipole of a non-spherical body arbitrarily oriented in space | journal=[[Astrophysics and Space Science]] | volume=352 | issue=2 | pages=493–496 | bibcode=2014Ap&SS.352..493R | doi=10.1007/s10509-014-1915-x | s2cid=119537102 }}</ref> They do depend on the orientation of the body's symmetry axis in the space, affecting, in general, the whole orbit, with the exception of the semimajor axis.

===Tidal locking===
{{Main|Tidal locking}}
Tidal locking between a pair of co-orbiting astronomical bodies occurs when one of the objects reaches a state where there is no longer any net transfer of [[angular momentum]] over the course of a complete orbit.<ref>{{cite book | title=Formation and Evolution of Exoplanets | editor1-first=Rory | editor1-last=Barnes | publisher=John Wiley & Sons | year=2010 | isbn=978-3-527-40896-2 | page=248 | url=https://books.google.com/books?id=-7KimFtJnIAC&pg=PA248 | access-date=2016-08-16 | archive-date=2023-08-06 | archive-url=https://web.archive.org/web/20230806163538/https://books.google.com/books?id=-7KimFtJnIAC&pg=PA248 | url-status=live }}</ref> Their gravitational interaction forces steady changes to their orbits and rotation rates as a result of [[energy transfer|energy exchange]] and heat [[dissipation]] until the locked state is formed. The object tends to stay in this state because leaving it would require adding energy back into the system. An example is the planet Mercury, which is locked in a state of completing three rotations about its axis for every two orbits.<ref>{{citation | title=Rotational Period of the Planet Mercury | last=Colombo | first=G. | journal=Nature | volume=208 | issue=5010 | page=575 | date=November 1965 | doi=10.1038/208575a0 | bibcode=1965Natur.208..575C | s2cid=4213296 | doi-access=free }}</ref>

In the case where a tidally locked body possesses synchronous rotation, the object takes just as long to rotate around its own axis as it does to revolve around its partner. In this case, one side of the celestial body is permanently facing its host object. This is the case for the Earth's Moon and for both members of the Pluto-Charon system.<ref>{{citation | title=On the Existence of Regular and Irregular Outer Moons Orbiting the Pluto–Charon System | display-authors=1 | last1=Michaely | first1=Erez | last2=Perets | first2=Hagai B. | last3=Grishin | first3=Evgeni | journal=The Astrophysical Journal | volume=836 | issue=1 | id=27 | pages=7 | date=February 2017 | doi=10.3847/1538-4357/aa52b2 | bibcode=2017ApJ...836...27M | arxiv=1506.08818 | s2cid=118068933 | doi-access=free }}</ref>

===Multiple gravitating bodies===
{{Main|n-body problem}}
[[File:Moon apsidal precession.png|thumb|alt=Apsidal precession refers to the rotation of the Moon's elliptical orbit over time, with the major axis completing one revolution every 8.85 years.|Apsidal precession of the Moon's elliptical orbit (not to scale and eccentricity exaggerated)]]
The effects of other gravitating bodies can be significant. For example, the [[orbit of the Moon]] cannot be accurately described without allowing for the action of the Sun's gravity as well as the Earth's.<ref>{{cite book | title=The Discovery of Dynamics: A Study from a Machian Point of View of the Discovery and the Structure of Dynamical Theories | first=Julian B. | last=Barbour | publisher=Oxford University Press | year=2001 | page=157 | isbn=978-0-19-513202-1 | url=https://books.google.com/books?id=pXc8DwAAQBAJ&pg=PA157 }} See also: [[evection]].</ref> One approximate result is that bodies will usually have reasonably stable orbits around a heavier planet or moon, in spite of these perturbations, provided they are orbiting well within the heavier body's [[Hill sphere]].<ref>{{cite journal | first1=A. M. | last1=Kisare | first2=D. C. | last2=Fabrycky | title=Tidal dissipation in satellites prevents Hill sphere escape | journal=Monthly Notices of the Royal Astronomical Society | volume=527 | issue=3 | date=January 2024 | pages=4371–4377 | doi=10.1093/mnras/stad3543 | doi-access=free | arxiv=2309.11609 | bibcode=2024MNRAS.527.4371K }}</ref>

A long-term impact of multi-body interactions can be [[apsidal precession]], which is a gradual rotation of the [[Apse line|line between the apsides]]. For an elliptical system, the result is a [[rosetta orbit]]. The ancient Greek astronomer [[Hipparchus]] noted just such an apsidal precession of the Moon's orbit, as the revolution of the Moon's apogee with a period of approximately 8.85 years.<ref>{{cite journal | first=Alexander | last=Jones, A. | title=The Adaptation of Babylonian Methods in Greek Numerical Astronomy | journal=Isis | volume=82 | number=3 | pages=440–453 | date=September 1991 | doi=10.1086/355836 | bibcode=1991Isis...82..441J | s2cid=92988054 }}</ref>  Apsidal precession can result from tidal perturbation, rotational perturbation, general relativity,<ref name=O'Donnell_2014/> or a combination of these effects. The detection of apsidal precession in a distant [[binary star system]] can be an indicator of the purturbative effect of an unseen third stellar companion.<ref>{{cite conference | title=Third-body perturbed apsidal motion in eclipsing binaries | display-authors=1 | first1=T. | last1=Borkovits | first2=E. | last2=Forgács-Dajka | first3=S. A. | last3=Rappaport | series=EAS Publications Series | volume=82 | date=June 2019 | conference=Astro Fluid: An International Conference in Memory of Professor Jean-Paul Zahn's Great Scientific Achievements. Tides in Stars and Planets section | pages=99–106 | doi=10.1051/eas/1982010 }}</ref>

When there are more than two gravitating bodies it is referred to as an [[n-body problem]]. Most n-body problems have no [[Closed-form expression|closed form solution]], although some special cases have been formulated.

===Approaches to many-body problems===
{{main|Orbit modeling}}

Rather than an exact closed form solution, orbits with many bodies can be approximated with arbitrarily high accuracy. One method is to take the pure elliptic motion as a basis and add [[perturbation (astronomy)|perturbation]] terms to account for the gravitational influence of multiple bodies.<ref>{{cite book | title=Theory of Orbit Determination | first1=Andrea | last1=Milani | first2=Giovanni | last2=Gronchi | publisher=Cambridge University Press | year=2010 | isbn=978-0-521-87389-5 | pages=54–55 | url=https://books.google.com/books?id=5oLK7QtX6j4C&pg=PA54 }}</ref> This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moons, planets, and other bodies are known with great accuracy, and are used to generate [[ephemeris|tables]] for [[celestial navigation]].<ref>{{cite book | title=New accuracy levels for solar system ephemerides | first1=E. M. | last1=Standish | first2=X. X. | last2=Newhall | series=Symposium - International Astronomical Union | year=1996 | volume=172 | pages=29–36 | doi=10.1017/S0074180900127081 }}</ref> Still, there are [[secular phenomena]] that have to be dealt with by [[Parameterized post-Newtonian formalism|post-Newtonian]] methods.

An incremental approach uses [[differential equation]]s for scientific or mission-planning purposes.<ref>For example: {{cite journal | display-authors=1 | last1=Peláez | first1=J. | last2=Hedo | first2=J. M. | last3=de Andrés | first3=P. R. | title=A special perturbation method in orbital dynamics | journal=Celestial Mechanics and Dynamical Astronomy | volume=97 | pages=131–150 | year=2007 | issue=2 | doi=10.1007/s10569-006-9056-3 | url=http://oa.upm.es/9819/ }}</ref> According to Newton's laws, each of the gravitational forces acting on a body will depend on the separation from the sources. Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial values of position and velocity corresponds to solving an [[initial value problem]]. Numerical methods calculate the positions and velocities of the objects a short time in the future, then repeat the calculation ad nauseam. However, tiny arithmetic errors from the limited accuracy of a computer's math are cumulative, which limits the long-term accuracy of this approach.

Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large assemblages of objects have been simulated.<ref>{{Cite journal | display-authors=1 | last1=Carleton | first1=Timothy | last2=Guo | first2=Yicheng | last3=Munshi | first3=Ferah | last4=Tremmel | first4=Michael | last5=Wright | first5=Anna | title=An excess of globular clusters in Ultra-Diffuse Galaxies formed through tidal heating | journal=Monthly Notices of the Royal Astronomical Society | year=2021 | volume=502 | pages=398–406 | doi=10.1093/mnras/stab031 | doi-access=free | arxiv=2008.11205 }}</ref>

===Radiation and magnetic fields===
For smaller bodies particularly, light<ref name=Vukovich_Kim_2019/> and [[stellar wind]] can cause significant perturbations to the [[attitude (geometry)|attitude]] and direction of motion of the body, and over time can be significant. Objects with residual magnetic fields can interact with a planetary [[magnetosphere]], perturbing their orbit.<ref name=Vukovich_Kim_2019/> Of the planetary bodies, the motion of [[asteroid]]s is particularly affected over large periods by the [[Yarkovsky effect]] when the asteroids are rotating relative to the Sun.<ref>{{cite journal | display-authors=1 | last1=Bottke, Jr. | first1=William F. | last2=Vokrouhlický | first2=David | last3=Rubincam | first3=David P. | last4=Nesvorný | first4=David | title=The Yarkovsky and YORP Effects: Implications for Asteroid Dynamics | journal=Annual Review of Earth and Planetary Sciences | volume=34 | pages=157–191 | year=2006 | doi=10.1146/annurev.earth.34.031405.125154 | bibcode=2006AREPS..34..157B | s2cid=11115100 | url=https://astro.troja.mff.cuni.cz/davok/papers/ann_rev_06.pdf | access-date=2021-08-12 | archive-date=2021-08-12 | archive-url=https://web.archive.org/web/20210812022037/https://astro.troja.mff.cuni.cz/davok/papers/ann_rev_06.pdf | url-status=live }}</ref>

==Strange orbits==
[[File:Klemperer rosette.svg|right|thumb|A simple hexagonal [[Klemperer rosette]] with two kinds of body,<ref>{{cite journal | last=Klemperer | first=W. B. | author-link=Wolfgang Klemperer | date=April 1962 | title=Some properties of rosette configurations of gravitating bodies in homographic equilibrium | journal=Astronomical Journal | volume=67 | number=3  | pages=162–167 | doi=10.1086/108686  | bibcode=1962AJ.....67..162K | doi-access=free}}
</ref> which [[Wolfgang Klemperer|Klemperer]] notes is the nearest to being stable.]]
Mathematicians have discovered that it is possible in principle to have multiple bodies in non-elliptical orbits that repeat periodically, although most such orbits are not stable regarding small perturbations in mass, position, or velocity. However, some special stable cases have been identified, including a planar figure-eight orbit occupied by [[Three-body problem|three moving bodies]].<ref>{{Cite journal | last1=Chenciner | first1=Alain | last2=Montgomery | first2=Richard | date=October 31, 2000 | title=Tidal Decay of Close Planetary Orbits | journal=The Astrophysical Journal | volume=470 | page=1187 | doi=10.1086/177941 | arxiv=math/0011268 }}</ref> Further studies have discovered that nonplanar orbits are also possible, including one involving 12 masses moving in 4 roughly circular, interlocking orbits [[topology|topologically]] equivalent to the edges of a [[cuboctahedron]].<ref name="Peterson">{{cite web | last=Peterson | first=Ivars | title=Strange Orbits | url=https://www.sciencenews.org/article/strange-orbits-1 | website=Science News | language=en | date=September 23, 2013 | access-date=21 July 2017 | archive-date=22 November 2015 | archive-url=https://web.archive.org/web/20151122231340/https://www.sciencenews.org/article/strange-orbits-1 | url-status=live }}</ref>

Finding such orbits naturally occurring in the universe is thought to be extremely unlikely, because of the improbability of the required conditions occurring by chance.<ref name="Peterson" />

==Astrodynamics==
{{Main|Orbital mechanics}}
Orbital mechanics or astrodynamics is the application of [[ballistics]] and [[celestial mechanics]] to the practical problems concerning the motion of [[rocket]]s and other [[spacecraft]].<ref>{{cite book | title=Fundamental Spacecraft Dynamics and Control | first=Weiduo | last=Hu | publisher=John Wiley & Sons | year=2015 | isbn=978-1-118-75429-0 | pages=3–5 | url=https://books.google.com/books?id=EjuCCgAAQBAJ&pg=PA3 }}</ref> The motion of these objects is usually calculated from [[Newton's laws of motion]] and [[Newton's law of universal gravitation]]. It is a core discipline within space mission design and control. Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of [[gravity]], including spacecraft and natural astronomical bodies such as star systems, [[planet]]s, [[Natural satellite|moon]]s, and [[comet]]s. Orbital mechanics focuses on spacecraft [[trajectory|trajectories]], including [[orbital maneuver]]s, orbit plane changes, and interplanetary transfers,<ref>{{cite book | title=Orbital Mechanics for Engineering Students | series=Aerospace Engineering | first=Howard D. | last=Curtis | edition=4th, revised | publisher=Butterworth-Heinemann | year=2020 | isbn=978-0-323-85345-3 | page=287 | url=https://books.google.com/books?id=k3oBEAAAQBAJ&pg=PA287 }}</ref> and is used by mission planners to predict the results of [[Spacecraft propulsion|propulsive maneuvers]]. [[General relativity]] is a more exact theory than Newton's laws for calculating orbits, and is sometimes necessary for greater accuracy or in high-gravity situations (such as orbits close to the Sun or planets).<ref>{{cite journal | last1=Sośnica | first1=K. | last2=Gałdyn | first2=F. | title=Orbital relativistic correction resulting from the Earth's oblateness term | journal=Journal of Geodesy | volume=99 | article-number=52 | year=2025 | issue=7 | doi=10.1007/s00190-025-01973-3 | doi-access=free }}</ref>

==Earth orbits==
{{Main|List of orbits}}
* [[Low Earth orbit]] (LEO): [[Geocentric orbit]]s with altitudes up to 2,000 [[km]] (0–1,240 [[mile]]s).<ref>{{cite web|url=http://www.orbitaldebris.jsc.nasa.gov/library/NSS1740_14/nss1740_14-1995.pdf |title=NASA Safety Standard 1740.14, Guidelines and Assessment Procedures for Limiting Orbital Debris |publisher=Office of Safety and Mission Assurance |date=1 August 1995 |url-status=dead |archive-url=https://web.archive.org/web/20130215143933/http://orbitaldebris.jsc.nasa.gov/library/NSS1740_14/nss1740_14-1995.pdf |archive-date=15 February 2013 }}, pp. 37–38 (6-1, 6-2); figure 6-1.</ref>
* [[Medium Earth orbit]] (MEO): [[Geocentric orbit]]s ranging in altitude from 2,000 [[km]] (1,240 [[mile]]s) to just below [[geosynchronous orbit]] at {{convert|35786|km|mi|sp=us}}. Also known as an [[intermediate circular orbit]]. These are "most commonly at {{convert|20200|km|mi|sp=us}}, or {{convert|20650|km|mi|sp=us}}, with an orbital period of 12 hours."<ref name=nasa_orbit_definition/>
* Both [[geosynchronous orbit]] (GSO) and [[geostationary orbit]] (GEO) are orbits around Earth matching Earth's [[sidereal rotation]] period. All geosynchronous and geostationary orbits have a [[semi-major axis]] of {{convert|42164|km|mi|0|abbr=on}}. All geostationary orbits are also geosynchronous, but not all geosynchronous orbits are geostationary. A geostationary orbit stays exactly above the equator, whereas a geosynchronous orbit may swing north and south to cover more of the Earth's surface. Both complete one full orbit of Earth per sidereal day (relative to the stars, not the Sun).<ref>{{cite book | title=Satellite Communications Systems Engineering: Atmospheric Effects, Satellite Link Design and System Performance | first=Louis J. | last=Ippolito, Jr. | edition=2nd | publisher=John Wiley & Sons | year=2017 | isbn=978-1-119-25941-1 | pages=23–24 | url=https://books.google.com/books?id=81I_DgAAQBAJ&pg=PA23 }}</ref>
* [[High Earth orbit]]: [[Geocentric orbit]]s above the altitude of [[geosynchronous orbit]] 35,786 [[km]] (22,240 [[mile]]s).<ref name=nasa_orbit_definition>
{{cite web |title=Orbit: Definition |url=http://gcmd.nasa.gov/add/ancillaryguide/platforms/orbit.html |work=Ancillary Description Writer's Guide, 2013 |publisher=National Aeronautics and Space Administration (NASA) Global Change Master Directory |access-date=29 April 2013 |url-status=dead |archive-url=https://web.archive.org/web/20130511114407/http://gcmd.nasa.gov/add/ancillaryguide/platforms/orbit.html |archive-date=11 May 2013  }}</ref>

==Scaling in gravity==
{{Importance section|date=November 2025}}
The [[gravitational constant]] ''G'' has been calculated as:{{physconst|G|round=4|after=.}}
* (6.6742 ± 0.001) × 10<sup>−11</sup> (kg/m<sup>3</sup>)<sup>−1</sup>s<sup>−2</sup>.

Thus the constant has dimension density<sup>−1</sup> time<sup>−2</sup>. This corresponds to the following properties.

[[Scale factor|Scaling]] of distances (including sizes of bodies, while keeping the densities the same) gives [[Similarity (geometry)|similar]] orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence velocities are halved and orbital periods and other travel times related to gravity remain the same. For example, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the Earth.

Scaling of distances while keeping the masses the same (in the case of point masses, or by adjusting the densities) gives similar orbits; if distances are multiplied by 4, gravitational forces and accelerations are divided by 16, velocities are halved and orbital periods are multiplied by 8.

When all densities are multiplied by 4, orbits are the same; gravitational forces are multiplied by 16 and accelerations by 4, velocities are doubled and orbital periods are halved.

When all densities are multiplied by 4, and all sizes are halved, orbits are similar; masses are divided by 2, gravitational forces are the same, gravitational accelerations are doubled. Hence velocities are the same and orbital periods are halved.

In all these cases of scaling. if densities are multiplied by 4, times are halved; if velocities are doubled, forces are multiplied by 16.

These properties are illustrated in the formula (derived from the [[Orbital period#Small body orbiting a central body|formula for the orbital period]])

:<math> GT^2 \rho = 3\pi \left( \frac{a}{r} \right)^3, </math>

for an elliptical orbit with [[semi-major axis]] ''a'', of a small body around a spherical body with radius ''r'' and average density ''ρ'', where ''T'' is the orbital period. See also [[Kepler’s third law]].

==See also==
{{Div col|colwidth=21em}}
* [[Free drift]]
* [[Jacobi coordinates]]
* [[List of orbits]]
* [[Molniya orbit]]
* [[Orbit determination]]
* [[Orbital spaceflight]]
* [[Perifocal coordinate system]]
* [[Polar orbit]]
* [[Radial trajectory]]
* [[VSOP model]]
{{Div col end}}

==References==
{{Reflist|30em}}

==Further reading==
* {{cite book|last1=Abell |first1=George O. |last2=Morrison |first2=David |last3=Wolff |first3=Sidney C. |name-list-style=amp|title=Exploration of the Universe |url=https://archive.org/details/explorationofuni0005abel|url-access=registration|edition=Fifth|date=1987|publisher=Saunders College Publishing|isbn=978-0-03-005143-2}}
* {{cite book|last=Linton|first=Christopher|title=From Eudoxus to Einstein: A History of Mathematical Astronomy|url=https://books.google.com/books?id=aJuwFLGWKF8C|year=2004|publisher=Cambridge University Press|isbn=978-1-139-45379-0}}

==External links==
{{Wiktionary}}
{{Commons|Orbits}}
* [http://www.calctool.org/CALC/phys/astronomy/planet_orbit CalcTool: Orbital period of a planet calculator]. Has wide choice of units. Requires JavaScript.
* [http://www.phy.hk/wiki/englishhtm/Motion.htm Java simulation on orbital motion]. Requires Java.
* [http://www.ncdc.noaa.gov/paleo/forcing.html NOAA page on Climate Forcing Data] includes (calculated) data on Earth orbit variations over the last 50 million years and for the coming 20 million years
* [https://web.archive.org/web/20100201160808/http://www.bridgewater.edu/~rbowman/ISAW/PlanetOrbit.html On-line orbit plotter]. Requires JavaScript.
* [https://web.archive.org/web/20120204054322/http://www.braeunig.us/space/orbmech.htm Orbital Mechanics] (Rocket and Space Technology)
* [https://web.archive.org/web/20060210012245/http://www.astrobiology.ucla.edu/OTHER/SSO/ Orbital simulations] by Varadi, [[Michael Ghil|Ghil]] and Runnegar (2003) provide another, slightly different series for Earth orbit eccentricity, and also a series for orbital inclination. Orbits for the other planets were also calculated, by {{cite journal | author=F. Varadi | author2=B. Runnegar | author3=M. Ghil | title=Successive Refinements in Long-Term Integrations of Planetary Orbits | journal=The Astrophysical Journal | volume=592 | date=2003 | issue=1 | pages=620–630|doi=10.1086/375560 | bibcode=2003ApJ...592..620V| doi-access=free }}, but only the [https://web.archive.org/web/20041031055131/http://www.astrobiology.ucla.edu/OTHER/SSO/Misc/ eccentricity data for Earth and Mercury] are available online.
* [http://www.lri.fr/~dragice/gravity/ Understand orbits using direct manipulation] {{Webarchive|url=https://web.archive.org/web/20171108140606/http://www.lri.fr/~dragice/gravity/ |date=8 November 2017 }}. Requires JavaScript and Macromedia
* {{cite web|last=Merrifield|first=Michael|title=Orbits (including the first manned orbit)|url=http://www.sixtysymbols.com/videos/orbit.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]}}

{{Orbits}}
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[[Category:Periodic phenomena]]
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