View source for Kepler's laws of planetary motion

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[[File:Kepler laws diagram.svg|thumb|Illustration of Kepler's laws with two planetary orbits.{{ordered list
| list_style = margin-left:0; list-style-position:inside;
| item_style = margin-top:0.3em;|The orbits are ellipses, with foci ''F''<sub>1</sub> and ''F''<sub>2</sub> for Planet 1, and ''F''<sub>1</sub> and ''F''<sub>3</sub> for Planet 2. The Sun is at ''F''<sub>1</sub>.|The shaded areas ''A''<sub>1</sub> and ''A''<sub>2</sub> are equal, and are swept out in equal times by Planet 1's orbit.|The ratio of Planet 1's orbit time to Planet 2's is <math display="inline">({a_1}/{a_2})^{3/2}</math>.
}}]]
{{Astrodynamics}}

In [[astronomy]], '''Kepler's laws of planetary motion''', published by [[Johannes Kepler]] in 1609 (except the third law, which was fully published in 1619), describe the orbits of planets around the Sun. These laws replaced [[circular orbit]]s and [[deferent and epicycle|epicycles]] in the [[Copernican heliocentrism|heliocentric theory]] of [[Nicolaus Copernicus]] with [[elliptical orbit]]s and explained how planetary velocities vary. The three laws state that:<ref name=":0">{{Cite web |title=Kepler's Laws |url=http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html |access-date=2022-12-13 |website=hyperphysics.phy-astr.gsu.edu}}</ref><ref name=":1">{{Cite web |title=Orbits and Kepler's Laws |url=https://solarsystem.nasa.gov/resources/310/orbits-and-keplers-laws |access-date=2022-12-13 |website=NASA Solar System Exploration|date=26 June 2008 }}</ref>
# The orbit of a planet is an [[ellipse]] with the Sun at one of the two [[focus (geometry)|foci]].
# A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
# The square of a planet's [[orbital period]] is proportional to the cube of the length of the [[semi-major and semi-minor axes|semi-major axis]] of its orbit.

The elliptical orbits of planets were indicated by calculations of the orbit of [[Mars]]. From this, Kepler inferred that other bodies in the [[Solar System]], including those farther away from the Sun, also have elliptical orbits. The second law establishes that when a planet is closer to the Sun, it travels faster. The third law expresses that the farther a planet is from the Sun, the longer its orbital period.

[[Isaac Newton]] showed in 1687 that relationships like Kepler's would apply in the Solar System as a consequence of his own [[Newton's laws of motion|laws of motion]] and [[Newton's law of universal gravitation|law of universal gravitation]].

A more precise historical approach is found in ''[[Astronomia nova]]'' and ''[[Epitome Astronomiae Copernicanae]]''.

==Comparison to Copernicus==

[[Johannes Kepler]]'s laws improved the model of [[Copernicus]]. According to Copernicus:<ref>{{Cite web |date=2009-07-07 |title=Planetary Motion: The History of an Idea That Launched the Scientific Revolution |url=https://earthobservatory.nasa.gov/features/OrbitsHistory |access-date=2022-12-13 |website=earthobservatory.nasa.gov |language=en}}</ref><ref>{{Cite web |title=Nicolaus Copernicus |url=https://www.history.com/topics/inventions/nicolaus-copernicus |access-date=2022-12-13 |website=history.com |date=9 November 2009 |language=en}}</ref>

# The planetary orbit is a circle with epicycles.
# The Sun is approximately at the center of the orbit.
# The speed of the planet in the main orbit is constant.

Despite being correct in saying that the planets revolved around the Sun, Copernicus was incorrect in defining their orbits. Introducing physical explanations for movement in space beyond just geometry, Kepler correctly defined the orbit of planets as follows:<ref name=":0" /><ref name=":1" /><ref name=Gingerich />{{rp|53–54}}

# The planetary orbit is ''not'' a circle with epicycles, but an ''[[ellipse]]''.
# The Sun is ''not'' at the center but at a ''[[Focus (geometry)|focal point]]'' of the elliptical orbit.
# Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the ''[[areal velocity|area speed]]'' (closely linked historically with the concept of [[angular momentum]]) is constant.

The [[Orbital eccentricity|eccentricity]] of the [[orbit of the Earth]] makes the time from the [[March equinox]] to the [[September equinox]], around 186 days, unequal to the time from the September equinox to the March equinox, around 179 days. A diameter would cut the orbit into equal parts, but the plane through the Sun parallel to the [[equator]] of the Earth cuts the orbit into two parts with areas in a 186 to 179 ratio, so the eccentricity of the orbit of the Earth is approximately
:<math>e \approx \frac{\pi}{4} \frac{186 - 179}{186 + 179} \approx 0.015,</math>

which is close to the correct value (0.016710218). The accuracy of this calculation requires that the two dates chosen be along the elliptical orbit's minor axis and that the midpoints of each half be along the major axis. As the two dates chosen here are equinoxes, this will be correct when [[perihelion]], the date the Earth is closest to the Sun, falls on a [[solstice]]. The current perihelion, near January 4, is fairly close to the solstice of December 21 or 22.

==Nomenclature==

It took nearly two centuries for the current formulation of Kepler's work to take on its settled form. [[Voltaire]]'s ''Eléments de la philosophie de Newton'' (''Elements of Newton's Philosophy'') of 1738 was the first publication to use the terminology of "laws".<ref>Voltaire, ''Eléments de la philosophie de Newton'' [Elements of Newton's Philosophy] (London: 1738).  See, for example:

* [https://books.google.com/books?id=t3UiO3NFQigC&pg=PA162 From p. 162:]  ''"Par une des grandes loix de Kepler, toute Planete décrit des aires égales en temp égaux : par une autre loi non-moins sûre, chaque Planete fait sa révolution autour du Soleil en telle sort, que si, sa moyenne distance au Soleil est 10. prenez le cube de ce nombre, ce qui sera 1000., & le tems de la révolution de cette Planete autour du Soleil sera proportionné à la racine quarrée de ce nombre 1000."''  (By one of the great laws of Kepler, each planet describes equal areas in equal times; by another law no less certain, each planet makes its revolution around the sun in such a way that if its mean distance from the sun is 10, take the cube of that number, which will be 1000, and the time of the revolution of that planet around the sun will be proportional to the square root of that number 1000.)
* [https://books.google.com/books?id=t3UiO3NFQigC&pg=PA205 From p. 205:]  ''"Il est donc prouvé par la loi de Kepler & par celle de Neuton, que chaque Planete gravite vers le Soleil, ..."''  (It is thus proved by the law of Kepler and by that of Newton, that each planet revolves around the sun ...)</ref><ref name="Wilson 1994">{{cite journal
|last=Wilson |first=Curtis |date=May 1994 |title=Kepler's Laws, So-Called |journal=HAD News |issue=31 |pages=1–2 |url=https://had.aas.org/sites/had.aas.org/files/HADN31.pdf
|access-date=December 27, 2016 }}</ref> The ''[[Biographical Encyclopedia of Astronomers]]'' in its article on Kepler (p.&nbsp;620) states that the terminology of scientific laws for these discoveries was current at least from the time of [[Joseph de Lalande]].<ref>De la Lande, ''Astronomie'', vol. 1 (Paris:  Desaint & Saillant, 1764).  See, for example:
*  [https://books.google.com/books?hl=en&id=Sg8OAAAAQAAJ&pg=PA390 From p. 390:]  ''"... mais suivant la fameuse loi de Kepler, qui sera expliquée dans le Livre suivant (892), le rapport des temps périodiques est toujours plus grand que celui des distances, une planete cinq fois plus éloignée du soleil, emploie à faire sa révolution douze fois plus de temps ou environ; ..."'' (... but according to the famous law of Kepler, which will be explained in the following book [i.e., chapter] (para. 892), the ratio of the periods is always greater than that of the distances [so that, for example,] a planet five times farther from the sun, requires about twelve times or so more time to make its revolution [around the sun] ...)
*  [https://books.google.com/books?hl=en&id=Sg8OAAAAQAAJ&pg=PA429 From p. 429:]  ''"Les Quarrés des Temps périodiques sont comme les Cubes des Distances.  892.  La plus fameuse loi du mouvement des planetes découverte par Kepler, est celle du repport qu'il y a entre les grandeurs de leurs orbites, & le temps qu'elles emploient à les parcourir; ..."'' (The squares of the periods are as the cubes of the distances.  892.  The most famous law of the movement of the planets discovered by Kepler is that of the relation between the sizes of their orbits and the times that the [planets] require to traverse them; ...)
*  [https://books.google.com/books?hl=en&id=Sg8OAAAAQAAJ&pg=PA430 From p. 430:]  ''"Les Aires sont proportionnelles au Temps.  895.  Cette loi générale du mouvement des planetes devenue si importante dans l'Astronomie, sçavior, que les aires sont proportionnelles au temps, est encore une des découvertes de Kepler; ..."''  (Areas are proportional to times.  895.  This general law of the movement of the planets [which has] become so important in astronomy, namely, that areas are proportional to times, is one of Kepler's discoveries; ...)
*  [https://books.google.com/books?hl=en&id=Sg8OAAAAQAAJ&pg=PA435 From p. 435:]  ''"On a appellé cette loi des aires proportionnelles aux temps, Loi de Kepler, aussi bien que celle de l'article 892, du nome de ce célebre Inventeur; ..."''  (One called this law of areas proportional to times (the law of Kepler) as well as that of para. 892, by the name of that celebrated inventor; ... )</ref> It was the exposition of [[Robert Small (minister)|Robert Small]], in ''An account of the astronomical discoveries of Kepler'' (1814) that made up the set of three laws, by adding in the third.<ref>Robert Small, ''An account of the astronomical discoveries of Kepler'' (London:  J Mawman, 1804), [https://archive.org/details/accountofastrono00unse/page/298 pp. 298–299.]</ref> Small also claimed, against the history, that these were [[empirical law]]s, based on [[inductive reasoning]].<ref name="Wilson 1994"/><ref>Robert Small, [https://archive.org/details/accountofastrono00unse An account of the astronomical discoveries of Kepler] (London:  J. Mawman, 1804).</ref>

Further, the current usage of "Kepler's second law" is something of a misnomer. Kepler had two versions, related in a qualitative sense: the "distance law" and the "area law". The "area law" is what became the second law in the set of three; but Kepler did himself not privilege it in that way.<ref name="Stephenson1994">{{cite book|author=Bruce Stephenson|title=Kepler's Physical Astronomy|url=https://books.google.com/books?id=pxCYAeOqJg8C&pg=PA170|date=1994|publisher=Princeton University Press|isbn=978-0-691-03652-6|page=170}}</ref>

==History==
Kepler published his first two laws about planetary motion in 1609,<ref>Astronomia nova Aitiologitis, seu Physica Coelestis tradita Commentariis de Motibus stellae Martis ex observationibus G.V. Tychnonis.Prague 1609; Engl. tr. W.H. Donahue, Cambridge 1992.</ref> having found them by analyzing the astronomical observations of [[Tycho Brahe]].<ref>In his ''Astronomia nova'', Kepler presented only a proof that Mars' orbit is elliptical.  Evidence that the other known planets' orbits are elliptical was presented only in 1621.<br>
See:  Johannes Kepler, ''Astronomia nova'' ... (1609), [https://archive.org/stream/Astronomianovaa00Kepl#page/284/mode/2up p. 285.] After having rejected circular and oval orbits, Kepler concluded that Mars' orbit must be elliptical.  From the top of page 285:  ''"Ergo ellipsis est Planetæ iter; ... "'' (Thus, an ellipse is the planet's [i.e., Mars'] path; ... ) Later on the same page:  ''" ... ut sequenti capite patescet:  ubi simul etiam demonstrabitur, nullam Planetæ relinqui figuram Orbitæ, præterquam perfecte ellipticam; ... "'' ( ... as will be revealed in the next chapter:  where it will also then be proved that any figure of the planet's orbit must be relinquished, except a perfect ellipse; ... )  And then:  ''"Caput LIX.  Demonstratio, quod orbita Martis, ... , fiat perfecta ellipsis: ... "'' (Chapter 59.  Proof that Mars' orbit, ... is a perfect ellipse: ... )  The geometric proof that Mars' orbit is an ellipse appears as Protheorema XI on pages 289–290.<br>
Kepler stated that every planet travels in elliptical orbits having the Sun at one focus in:  Johannes Kepler,  ''Epitome Astronomiae Copernicanae'' [Summary of Copernican Astronomy] (Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 5, part 1, III.  De Figura Orbitæ (III.  On the figure [i.e., shape] of orbits), [https://books.google.com/books?id=wa2SE_6ZL7YC&pg=PA658 pages 658–665.] From p. 658:  ''"Ellipsin fieri orbitam planetæ ... "'' (Of an ellipse is made a planet's orbit ... ).  From p. 659:  ''" ... Sole (Foco altero huius ellipsis) ... "'' ( ... the Sun (the other focus of this ellipse) ... ).</ref><ref name=Holton>{{cite book
|title=Physics, the Human Adventure: From Copernicus to Einstein and Beyond
|author=Holton, Gerald James |author2=Brush, Stephen G.  |pages=40–41
|url=https://books.google.com/books?id=czaGZzR0XOUC&pg=PA40
|edition=3rd paperback |isbn=978-0-8135-2908-0 |publisher=Rutgers University Press
|location=Piscataway, NJ |access-date=December 27, 2009 |date=2001}}</ref><ref>In his ''Astronomia nova'' ... (1609), Kepler did not present his second law in its modern form.  He did that only in his ''Epitome'' of 1621.  Furthermore, in 1609, he presented his second law in two different forms, which scholars call the "distance law" and the "area law".
*  His "distance law" is presented in:  ''"Caput XXXII.  Virtutem quam Planetam movet in circulum attenuari cum discessu a fonte."'' (Chapter 32.  The force that moves a planet circularly weakens with distance from the source.)  See:  Johannes Kepler, ''Astronomia nova'' ... (1609), [https://archive.org/stream/Astronomianovaa00Kepl#page/164/mode/2up pp. 165–167.] [https://archive.org/stream/Astronomianovaa00Kepl#page/166/mode/2up On page 167], Kepler states:  ''" ... , quanto longior est αδ quam αε, tanto diutius moratur Planeta in certo aliquo arcui excentrici apud δ, quam in æquali arcu excentrici apud ε."'' ( ... , as αδ is longer than αε, so much longer will a planet remain on a certain arc of the eccentric near δ than on an equal arc of the eccentric near ε.)  That is, the farther a planet is from the Sun (at the point α), the slower it moves along its orbit, so a radius from the Sun to a planet passes through equal areas in equal times.  However, as Kepler presented it, his argument is accurate only for circles, not ellipses.
*  His "area law" is presented in:  ''"Caput LIX.  Demonstratio, quod orbita Martis, ... , fiat perfecta ellipsis: ... "'' (Chapter 59.  Proof that Mars' orbit, ... , is a perfect ellipse: ... ), Protheorema XIV and XV, [https://archive.org/stream/Astronomianovaa00Kepl#page/284/mode/2up pp. 291–295.] On the top p. 294, it reads:   ''"Arcum ellipseos, cujus moras metitur area AKN, debere terminari in LK, ut sit AM."''  (The arc of the ellipse, of which the duration is delimited [i.e., measured] by the area AKM, should be terminated in LK, so that it [i.e., the arc] is AM.) In other words, the time that Mars requires to move along an arc AM of its elliptical orbit is measured by the area of the segment AMN of the ellipse (where N is the position of the Sun), which in turn is proportional to the section AKN of the circle that encircles the ellipse and that is tangent to it.  Therefore, the area that is swept out by a radius from the Sun to Mars as Mars moves along an arc of its elliptical orbit is proportional to the time that Mars requires to move along that arc.  Thus, a radius from the Sun to Mars sweeps out equal areas in equal times.

In 1621, Kepler restated his second law for any planet:  Johannes Kepler, ''Epitome Astronomiae Copernicanae'' [Summary of Copernican Astronomy] (Linz ("Lentiis ad Danubium"), (Austria):  Johann Planck, 1622), book 5, [https://books.google.com/books?id=wa2SE_6ZL7YC&pg=PA668 page 668].  From page 668:  ''"Dictum quidem est in superioribus, divisa orbita in particulas minutissimas æquales:  accrescete iis moras planetæ per eas, in proportione intervallorum inter eas & Solem."''  (It has been said above that, if the orbit of the planet is divided into the smallest equal parts, the times of the planet in them increase in the ratio of the distances between them and the sun.)  That is, a planet's speed along its orbit is inversely proportional to its distance from the Sun.  (The remainder of the paragraph makes clear that Kepler was referring to what is now called angular velocity.)</ref><ref name=Gingerich>{{cite journal|url=https://pubs.aip.org/physicstoday/article-pdf/64/9/50/9881314/50_1_online.pdf|title= The great Martian catastrophe and how Kepler fixed it |journal=Physics Today|date= 2011 |volume=64|issue=9|pages=50–54|doi=10.1063/PT.3.1259|access-date=27 July 2023 |last1= Gingerich |first1= Owen |bibcode= 2011PhT....64i..50G }}</ref>{{rp|53}} Kepler's third law was published in 1619.<ref name = "Kepler 1619">Johannes Kepler, ''Harmonices Mundi'' [The Harmony of the World] (Linz, (Austria):  Johann Planck, 1619), book 5, chapter 3, [https://archive.org/details/ioanniskepplerih00kepl/page/189 p. 189.] From the bottom of p. 189:  ''"Sed res est certissima exactissimaque quod ''proportio qua est inter binorum quorumcunque Planetarum tempora periodica, sit præcise sesquialtera proportionis'' mediarum distantiarum, ... "''  (But it is absolutely certain and exact that the ''proportion between the periodic times of any two planets is precisely the sesquialternate proportion'' [i.e., the ratio of 3:2] of their mean distances, ... ")<br>
An English translation of Kepler's ''Harmonices Mundi'' is available as:  Johannes Kepler with E. J. Aiton, A. M. Duncan, and [[Judith V. Field|J. V. Field]], trans., ''The Harmony of the World'' (Philadelphia, Pennsylvania:  American Philosophical Society, 1997); see especially [https://books.google.com/books?id=rEkLAAAAIAAJ&pg=PA411 p. 411].</ref><ref name=Holton /> Kepler had believed in the [[Copernican heliocentrism|Copernican model]] of the Solar System, which called for circular orbits, but he could not reconcile Brahe's highly precise observations with a circular fit to Mars' orbit – Mars coincidentally having the highest [[Orbital eccentricity|eccentricity]] of all planets except Mercury.<ref>{{cite web|url=https://www.windows2universe.org/?page=/our_solar_system/planets_table.html
|title=Data Table for Planets and Dwarf Planets |website=Windows to the Universe
|date=9 October 2008
|author=National Earth Science Teachers Association
|access-date=2 August 2018
}}</ref> His first law reflected this discovery.

In 1621, Kepler noted that his third law applies to the [[Galilean moons|four brightest moons]] of [[Jupiter]].<ref group="Nb">In 1621, Johannes Kepler noted that Jupiter's moons obey (approximately) his third law in his ''[[Epitome Astronomiae Copernicanae]]'' [Epitome of Copernican Astronomy] (Linz ("Lentiis ad Danubium"), (Austria):  Johann Planck, 1622), book 4, part 2, [https://books.google.com/books?id=wa2SE_6ZL7YC&pg=PA554 pages 554–555].  From pp. 554–555:  ''" ... plane ut est cum sex planet circa Solem, ... prodit Marius in suo mundo Ioviali ista 3.5.8.13 (vel 14. Galilæo) ... Periodica vero tempora prodit idem Marius ... sunt maiora simplis, minora vero duplis."'' (... just as it is clearly [true] among the six planets around the Sun, so also it is among the four [moons] of Jupiter, because around the body of Jupiter any [satellite] that can go farther from it, orbits slower, and even that [orbit's period] is not in the same proportion, but greater [than the distance from Jupiter]; that is, 3/2 (''sescupla'') of the proportion of each of the distances from Jupiter, which is clearly the very [proportion] as is used for the six planets above. In his [book] ''The World of Jupiter'' [''Mundus Jovialis'', 1614], [Simon Mayr or] "Marius"  [1573–1624] presents these distances, from Jupiter, of the four [moons] of Jupiter: 3, 5, 8, 13 (or 14 [according to] Galileo) [Note: The distances of Jupiter's moons from Jupiter are expressed as multiples of Jupiter's diameter.] ... Mayr presents their time periods: 1 day 18 1/2 hours, 3 days 13 1/3 hours, 7 days 2 hours, 16 days 18 hours: for all [of these data] the proportion is greater than double, thus greater than [the proportion] of the distances 3, 5, 8, 13 or 14, although less than [the proportion] of the squares, which double the proportions of the distances, namely 9, 25, 64, 169 or 196, just as [a power of] 3/2 is also greater than 1 but less than 2.)</ref> [[Godefroy Wendelin]] also made this observation in 1643.<ref group="Nb">Godefroy Wendelin wrote a letter to Giovanni Battista Riccioli about the relationship between the distances of the Jovian moons from Jupiter and the periods of their orbits, showing that the periods and distances conformed to Kepler's third law.  See:  Joanne Baptista Riccioli, ''Almagestum novum'' ... (Bologna (Bononia), (Italy):  Victor Benati, 1651), volume 1, [https://books.google.com/books?id=_mJDAAAAcAAJ&pg=PA492 page 492 Scholia III.] In the margin beside the relevant paragraph is printed: ''Vendelini ingeniosa speculatio circa motus & intervalla satellitum Jovis''.  (Wendelin's clever speculation about the movement and distances of Jupiter's satellites.)  From p. 492:  ''"III.  Non minus Kepleriana ingeniosa est Vendelini ... & D. 7. 164/1000. pro penextimo, & D. 16. 756/1000. pro extimo."''  (No less clever [than] Kepler's is the most keen astronomer Wendelin's investigation of the proportion of the periods and distances of Jupiter's satellites, which he had communicated to me with great generosity [in] a very long and very learned letter.  So, just as in [the case of] the larger planets, the planets' mean distances from the Sun are respectively in the 3/2 ratio of their periods; so the distances of these minor planets of Jupiter from Jupiter (which are 3, 5, 8, and 14) are respectively in the 3/2 ratio of [their] periods (which are 1.769 days for the innermost [Io], 3.554 days for the next to the innermost [Europa], 7.164 days for the next to the outermost [Ganymede], and 16.756 days for the outermost [Callisto]).)</ref> The second law, in the "area law" form, was contested by [[Nicolaus Mercator]] in a book from 1664, but by 1670 his ''[[Philosophical Transactions]]'' were in its favour.<ref>{{cite book |last1=Mercator |first1=Nicolaus |title=Nicolai Mercatoris Hypothesis astronomica nova, et consensus eius cum observationibus |trans-title=Nicolaus Mercator's new astronomical hypothesis, and its agreement with observations |date=1664 |publisher=Leybourn |location=London, England |language=Latin}}</ref><ref>{{cite journal |last1=Mercator |first1=Nic. |title=Some considerations of Mr. Nic. Mercator, concerning the geometrick and direct method of signior Cassini for finding the apogees, excentricities, and anomalies of the planets; ... |journal=Philosophical Transactions of the Royal Society of London |date=25 March 1670 |volume=5 |issue=57 |pages=1168–1175 |doi=10.1098/rstl.1670.0018 |url=https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1670.0018 |language=Latin|url-access=subscription }}  Mercator criticized Cassini's method of finding, from three observations, an orbit's line of apsides.  Cassini had assumed (wrongly) that planets move uniformly along their elliptical orbits.  From p. 1174:  ''"Sed cum id Observationibus nequaquam congruere animadverteret, mutavit sententiam, & lineam veri motus Planetæ æqualibus temporibus æquales areas Ellipticas verrere professus est: ...  "''  (But when he noticed that it didn't agree at all with observations, he changed his thinking, and he declared that a line [from the Sun to a planet, denoting] a planet's true motion, sweeps out equal areas of an ellipse in equal periods of time: ... [which is the "area" form of Kepler's second law])</ref> As the century proceeded it became more widely accepted.<ref>{{cite book|author=Wilbur Applebaum|title=Encyclopedia of the Scientific Revolution: From Copernicus to Newton|url=https://books.google.com/books?id=k43Q9RHuGXgC&pg=PT603|date=2000|publisher=Routledge|isbn=978-1-135-58255-5|page=603|bibcode=2000esrc.book.....A}}</ref> The reception in Germany changed noticeably between 1688, the year in which Newton's ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' was published and was taken to be basically Copernican, and 1690, by which time work of [[Gottfried Leibniz]] on Kepler had been published.<ref>{{cite book|author=Roy Porter|author-link=Roy Porter|title=The Scientific Revolution in National Context|url=https://archive.org/details/isbn_0521396999|url-access=registration|date=1992|publisher=Cambridge University Press|isbn=978-0-521-39699-8|page=[https://archive.org/details/isbn_0521396999/page/102 102]}}</ref>

Newton was credited with understanding that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law, whereas the other laws do depend on the inverse square form of the attraction. [[Carl Runge]] and [[Wilhelm Lenz]] much later identified a symmetry principle in the [[phase space]] of planetary motion (the [[orthogonal group]] O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, as [[conservation of angular momentum]] does via rotational symmetry for the second law.<ref>{{cite book|author1=Victor Guillemin|author2=Shlomo Sternberg|title=Variations on a Theme by Kepler|url=https://books.google.com/books?id=3NXFth0gDQgC&pg=PR5|date=2006|publisher=American Mathematical Soc.|isbn=978-0-8218-4184-6|page=5}}</ref>

==Formulary==
The mathematical model of the kinematics of a planet subject to the laws allows a large range of further calculations.

===First law ===
Kepler's first law states that:

<blockquote>The orbit of every planet is an ellipse with the sun at one of the two [[Focus (geometry)|foci]].</blockquote>

[[File:kepler-first-law.svg|thumb|Kepler's first law placing the Sun at one of the foci of an elliptical orbit]]

[[File:Ellipse latus rectum.svg|thumb|Heliocentric coordinate system (''r'',{{nbsp}}''θ'') for ellipse. Also shown are: semi-major axis ''a'', semi-minor axis ''b'' and semi-latus rectum ''p''; center of ellipse and its two [[Focus (geometry)|foci]] marked by large dots. For {{nowrap|θ {{=}} 0°}}, {{nowrap|''r'' {{=}} ''r''{{sub|min}}}} and for {{nowrap|θ {{=}} 180°}}, {{nowrap|''r'' {{=}} ''r''{{sub|max}}}}.]]

Mathematically, an ellipse can be represented by the formula:
:<math>r = \frac{p}{1 + \varepsilon\, \cos\theta},</math>

where  <math>p</math> is the [[semi-latus rectum]], ''ε'' is the [[Eccentricity (mathematics)|eccentricity]] of the ellipse, ''r'' is the distance from the Sun to the planet, and ''θ'' is the angle to the planet's current position from its closest approach, as seen from the Sun. So (''r'',&nbsp;''θ'') are [[polar coordinates]].

For an ellipse 0&nbsp;<&nbsp;''ε''&nbsp;<&nbsp;1 ; in the limiting case ''ε'' = 0, the orbit is a circle with the Sun at the centre (i.e. where there is zero eccentricity).

At ''θ'' = 0°, [[perihelion]], the distance is minimum
:<math>r_\min = \frac{p}{1 + \varepsilon}</math>

At ''θ'' = 90° and at ''θ'' = 270° the distance is equal to <math>p</math>.

At ''θ'' = 180°, [[aphelion]], the distance is maximum (by definition, aphelion is – invariably – perihelion plus 180°)
:<math>r_\max = \frac{p}{1 - \varepsilon}</math>

The [[semi-major axis]] ''a'' is the [[arithmetic mean]] between ''r''<sub>min</sub> and ''r''<sub>max</sub>:
:<math>\begin{align}
  a &= \frac{r_\max + r_\min}{2} \\[3pt]
           a &= \frac{p}{1 - \varepsilon^2}
\end{align}</math>

The [[semi-minor axis]] ''b'' is the [[geometric mean]] between ''r''<sub>min</sub> and ''r''<sub>max</sub>:
:<math>\begin{align}
  b &= \sqrt{r_\max r_\min} \\[3pt]
                 b &= \frac{p}{\sqrt{1 - \varepsilon^2}}
\end{align}</math>

The semi-latus rectum ''p'' is the [[harmonic mean]] between ''r''<sub>min</sub> and ''r''<sub>max</sub>:
:<math>\begin{align}
 p &= \left(\frac{r_\max^{-1} + r_\min^{-1}}{2}\right)^{-1} \\
                             pa &= r_\max r_\min = b^2\,
\end{align}</math>

The eccentricity ''ε'' is the [[coefficient of variation]] between ''r''<sub>min</sub> and ''r''<sub>max</sub>:
:<math>\varepsilon = \frac{r_\max - r_\min}{r_\max + r_\min}.</math>

The [[area]] of the ellipse is
:<math>A = \pi a b\,.</math>

The special case of a circle is ''ε'' = 0, resulting in ''r'' = ''p'' =  ''r''<sub>min</sub> = ''r''<sub>max</sub> = ''a'' = ''b'' and ''A'' = ''πr''<sup>2</sup>.

===Second law===
Kepler's second law states that:

<blockquote>A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.<ref name="Wolfram2nd">Bryant, Jeff; Pavlyk, Oleksandr. "[http://demonstrations.wolfram.com/KeplersSecondLaw/ Kepler's Second Law]", ''[[Wolfram Demonstrations Project]]''. Retrieved December 27, 2009.</ref></blockquote>

[[File:kepler-second-law.gif|thumb|The same (blue) area is swept out in a fixed time period. The green arrow is velocity. The purple arrow directed towards the Sun is the acceleration. The other two purple arrows are acceleration components parallel and perpendicular to the velocity.]]

The orbital radius and angular velocity of the planet in the elliptical orbit will vary. This is shown in the animation: the planet travels faster when closer to the Sun, then slower when farther from the Sun. Kepler's second law states that the blue sector has constant area.

====History and proofs====
 
Kepler notably arrived at this law through assumptions that were either only approximately true or outright false and can be outlined as follows:

# Planets are pushed around the Sun by a force from the Sun. This false assumption relies on incorrect [[Physics (Aristotle)|Aristotelian physics]] that an object needs to be pushed to maintain motion.
# The propelling force from the Sun is inversely proportional to the distance from the Sun.  Kepler reasoned this, believing that gravity spreading in three dimensions would be a waste, since the planets inhabited a plane.  Thus, an inverse instead of the [correct] inverse square law.
# Because Kepler believed that force would be proportional to velocity, it followed from statements #1 and #2 that velocity would be inverse to the distance from the sun.  This is also an incorrect tenet of Aristotelian physics.
# Since velocity is inverse to time, the distance from the sun would be proportional to the time to cover a small piece of the orbit. This is approximately true for elliptical orbits.
# The area swept out is proportional to the overall time. This is also approximately true.
# The orbits of a planet are circular (Kepler discovered his second law before his first law, which contradicts this).

Nevertheless, the result of the second law is exactly true, as it is logically equivalent to the conservation of angular momentum, which is true for any body experiencing a radially symmetric force.<ref>{{cite book |last1=Holton |first1= Gerald|last2=Brush|first2=Stephen |date=2001 |title=Brush and Holton - Physics: the Human Adventure |location= |publisher=Princeton University Press |pages=42–43 |isbn=978-0813529080}}</ref>  A correct proof can be shown through this. Since the cross product of two vectors gives the area of a parallelogram possessing sides of those vectors, the triangular area dA swept out in a short period of time is given by half the cross product of the ''r'' and ''dx'' vectors, for some short piece of the orbit, ''dx''.

<math>dA = \frac{1}{2} (\vec{r} \times \vec{dx})
= \frac{1}{2} (\vec{r} \times \vec{v} dt)
</math>
for a small piece of the orbit ''dx'' and time to cover it ''dt''.

Thus <math>\frac{dA}{dt} = \frac{1}{2} (\vec{r} \times \vec{v}).</math>

<math>\frac{dA}{dt} = \frac{1}{m} \frac{1}{2} (\vec{r} \times \vec{p}).</math>

Since the final expression is proportional to the total angular momentum <math>(\vec{r} \times \vec{p})</math>, Kepler's equal area law will hold for any system that conserves angular momentum.  Since any radial force will produce no torque on the planet's motion, angular momentum will be conserved.

====In terms of elliptical parameters====

In a small time <math>dt</math> the planet sweeps out a small triangle having base line <math>r</math> and height <math>r \, d\theta</math> and area <math display="inline">dA = \frac{1}{2} \cdot r \cdot r \, d\theta</math>, so the constant [[areal velocity]] is <math display="block">\frac{dA}{dt} = \frac{r^2}{2} \frac{d\theta}{dt}.</math>

The area enclosed by the elliptical orbit is <math>\pi ab</math>. So the period <math>T</math> satisfies
:<math>T \cdot \frac{r^2}{2} \frac{d\theta}{dt} = \pi ab</math>

and the [[mean motion]] of the planet around the Sun
:<math>n = \frac{2\pi}{T}</math>

satisfies
:<math>r^2\,d\theta = abn\,dt.</math>

And so, <math display="block">\frac{dA}{dt} = \frac{a b n}{2} = \frac{\pi a b}{T}.</math>

{| class="wikitable" style=width:500px
|+ Orbits of planets with varying eccentricities. 
|-
! Low !! High
|-
| [[File:Circular orbit of planet with (eccentricty of 0.0).gif|frameless|upright=1.2|center]] Planet orbiting the Sun in a circular orbit (e=0.0)
 || [[File:Ellipitical orbit of planet with an eccentricty of 0.5.gif|frameless|upright=1.2|center]] Planet orbiting the Sun in an orbit with e=0.5
|-
| [[File:Ellipitical orbit of planet with an eccentricty of 0.2.gif|frameless|upright=1.2|center]] Planet orbiting the Sun in an orbit with e=0.2
| [[File:Ellipitical orbit of planet with an eccentricty of 0.8.gif|frameless|upright=1.2|center]] Planet orbiting the Sun in an orbit with e=0.8
|-
| colspan=2 |  The red ray rotates at a constant angular velocity and with the same orbital time period as the planet, <math>T=1</math>.  
S: Sun at the primary focus, 
C: Centre of ellipse, 
S': The secondary focus.  
In each case, the area of all sectors depicted is identical.
|}

===Third law===
Kepler's third law states that:

<!--The article Gravitation links here. Please do not change the title of this subsection without creating an appropriate anchor.-->
<blockquote>The ratio of the square of an object's [[orbital period]] with the cube of the semi-major axis of its orbit is the same for all objects orbiting the same primary.</blockquote>

This captures the relationship between the distance of planets from the Sun, and their orbital periods.

Kepler enunciated in 1619<ref name = "Kepler 1619"/> this third law in a laborious attempt to determine what he viewed as the "[[Musica universalis|music of the spheres]]" according to precise laws, and express it in terms of musical notation.<ref>[[Edwin Arthur Burtt|Burtt, Edwin]]. ''The Metaphysical Foundations of Modern Physical Science''. p. 52.</ref> It was therefore known as the ''harmonic law''.<ref name=Holton3>{{cite book |title=Physics, the Human Adventure |author=Gerald James Holton, Stephen G. Brush |page=45 |url=https://books.google.com/books?id=czaGZzR0XOUC&pg=PA45|isbn=978-0-8135-2908-0 |publisher=Rutgers University Press |date=2001}}</ref> The original form of this law (referring to not the semi-major axis, but rather a "mean distance") holds true only for planets with small eccentricities near zero.<ref>{{cite journal | pmc=6395789 | date=2019 | last1=Vijaya | first1=G. K. | title=Original form of Kepler's Third Law and its misapplication in Propositions XXXII-XXXVII in Newton's Principia (Book I) | journal=Heliyon | volume=5 | issue=2 | pages=e01274 | doi=10.1016/j.heliyon.2019.e01274 | doi-access=free | pmid=30886926 | bibcode=2019Heliy...501274V }}</ref>

Using Newton's law of gravitation (published 1687), this relation can be found in the case of a circular orbit by setting the [[centripetal force]] equal to the gravitational force:
: <math>mr\omega^2 = G\frac{mM}{r^2}</math>

Then, expressing the angular velocity ω in terms of the orbital period <math>{T}</math> and then rearranging, results in Kepler's third law:
: <math>mr\left(\frac{2\pi}{T}\right)^2 = G\frac{mM}{r^2} \implies T^2 = \left(\frac{4\pi^2}{GM} \right)r^3 \implies T^2 \propto r^3</math>

A more detailed derivation can be done with general elliptical orbits, instead of circles, as well as orbiting the center of mass, instead of just the large mass. This results in replacing a circular radius, <math>r</math>, with the semi-major axis, <math>a</math>, of the elliptical relative motion of one mass relative to the other, as well as replacing the large mass <math>M</math> with <math>M + m</math>. However, with planet masses being so much smaller than the Sun, this correction is often ignored. The full corresponding formula is:
:<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} \text{ is constant}</math>

where <math>M</math> is the [[Solar mass|mass of the Sun]], <math>m</math> is the mass of the planet, <math>G</math> is the [[gravitational constant]], <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.

==== Table ====
The following table shows the data used by Kepler to empirically derive his law:
{| class="wikitable"
|+ Data used by Kepler (1618)
! Planet
! Mean distance <br />to sun (AU)
! Period <br />(days)
! <math display="inline">\frac{R^3}{T^2}</math>{{nbsp}}(10{{sup|−6}}{{nbsp}}AU{{sup|3}}/day{{sup|2}})
|-
|Mercury
|0.389
|87.77
|7.64
|-
|Venus
|0.724
|224.70
|7.52
|-
|Earth
|1
|365.25
|7.50
|-
|Mars
|1.524
|686.95
|7.50
|-
|Jupiter
|5.20
|4332.62
|7.49
|-
|Saturn
|9.510
|10759.2
|7.43
|}

Kepler became aware of [[John Napier]]'s recent invention of logarithms and log-log graphs before he discovered the pattern.<ref>{{Cite book| title=Kepler| url=https://archive.org/details/kepler00casp| url-access=registration| last=Caspar|first=Max| page=304| publisher=Dover| year=1993| location=New York| isbn=9780486676050}}</ref>

Upon finding this pattern Kepler wrote:<ref>{{Cite book| title=Kepler| url=https://archive.org/details/kepler00casp| url-access=registration| last=Caspar|first=Max| page=286| publisher=Dover| year=1993| location=New York| isbn=9780486676050}}</ref>

{{Blockquote|I first believed I was dreaming... But it is absolutely certain and exact that the ratio which exists between the period times of any two planets is precisely the ratio of the 3/2th power of the mean distance.
| source = translated from ''Harmonies of the World'' by Kepler (1619)}}

{{solar_system_orbital_period_vs_semimajor_axis.svg|280px}}

For comparison, here are modern estimates:{{citation needed|date=March 2024}}
{| class="wikitable"
|+ Modern data
! Planet
! Semi-major axis (AU)
! Period (days)
! <math display="inline">\frac{a^3}{T^2}</math>{{nbsp}}(10{{sup|−6}}{{nbsp}}AU{{sup|3}}/day{{sup|2}})
|-
|Mercury
|0.38710
|87.9693
|7.496
|-
|Venus
|0.72333
|224.7008
|7.496
|-
|Earth
|1
|365.2564
|7.496
|-
|Mars
|1.52366
|686.9796
|7.495
|-
|Jupiter
|5.20336
|4332.8201
|7.504
|-
|Saturn
|9.53707
|10775.599
|7.498
|-
|Uranus
|19.1913
|30687.153
|7.506
|-
|Neptune
|30.0690
|60190.03
|7.504
|}

==Planetary acceleration==
[[Isaac Newton]] computed in his ''[[Philosophiæ Naturalis Principia Mathematica]]'' the [[acceleration]] of a planet moving according to Kepler's first and second laws.
# The ''direction'' of the acceleration is towards the Sun.
# The ''magnitude'' of the acceleration is inversely proportional to the square of the planet's distance from the Sun (the ''inverse square law'').

This implies that the Sun may be the physical cause of the acceleration of planets. However, Newton states in his ''Principia'' that he considers forces from a mathematical point of view, not a physical, thereby taking an instrumentalist view.<ref>I. Newton, ''Principia'', p. 408 in the translation of I.B. Cohen and A. Whitman</ref> Moreover, he does not assign a cause to gravity.<ref>I. Newton, ''Principia'', p. 943 in the translation of I.B. Cohen and A. Whitman</ref>

Newton defined the [[force]] acting on a planet to be the product of its [[mass]] and the acceleration (see [[Newton's laws of motion]]). So:
# Every planet is attracted towards the Sun.
# The force acting on a planet is directly proportional to the mass of the planet and is inversely proportional to the square of its distance from the Sun.

The Sun plays an unsymmetrical part, which is unjustified. So he assumed, in [[Newton's law of universal gravitation]]:
# All bodies in the Solar System attract one another.
# The force between two bodies is in direct proportion to the product of their masses and in inverse proportion to the square of the distance between them.

As the planets have small masses compared to that of the Sun, the orbits conform approximately to Kepler's laws. Newton's model improves upon Kepler's model, and fits actual observations more accurately. (See [[two-body problem]].)

Below comes the detailed calculation of the acceleration of a planet moving according to Kepler's first and second laws.

===Acceleration vector===
{{See also|Polar coordinate#Vector calculus}}
From the [[heliocentric]] point of view consider the vector to the planet <math>\mathbf{r} = r\hat{\mathbf{r}} </math> where <math> r</math> is the distance to the planet and <math>\hat{\mathbf{r}}</math> is a [[unit vector]] pointing towards the planet.
<math display="block">
  \frac{d\hat{\mathbf{r}}}{dt} = \dot{\hat{\mathbf{r}}} = \dot{\theta}\hat{\boldsymbol\theta},\qquad
  \frac{d\hat{\boldsymbol\theta}}{dt} = \dot{\hat{\boldsymbol\theta}} = -\dot{\theta}\hat{\mathbf{r}}</math>

where <math>\hat{\boldsymbol\theta}</math> is the unit vector whose direction is 90 degrees counterclockwise of <math>\hat{\mathbf{r}}</math>, and <math>\theta</math> is the polar angle, and where a [[Notation for differentiation#Newton's notation|dot]] on top of the variable signifies differentiation with respect to time.

Differentiate the position vector twice to obtain the velocity vector and the acceleration vector:
<math display="block">\begin{align}
   \dot{\mathbf{r}} &= \dot{r}\hat{\mathbf{r}} + r\dot{\hat{\mathbf{r}}}
                     = \dot{r}\hat{\mathbf{r}} + r\dot{\theta}\hat{\boldsymbol{\theta}}, \\
  \ddot{\mathbf{r}} &= \left(\ddot{r}\hat{\mathbf{r}}
                             + \dot{r}\dot{\hat{\mathbf{r}}}
                       \right)
                       + \left(\dot{r}\dot{\theta} \hat{\boldsymbol{\theta}}
                               + r\ddot{\theta}\hat{\boldsymbol{\theta}}
                               + r\dot{\theta}\dot{\hat{\boldsymbol{\theta}}}
                         \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>

So
<math display="block">\ddot{\mathbf{r}} = a_r \hat{\boldsymbol{r}}+a_\theta\hat{\boldsymbol{\theta}}</math>
where the '''radial acceleration''' is
<math display="block">a_r = \ddot{r} - r\dot{\theta}^2</math>
and the '''transversal acceleration''' is
<math display="block">a_\theta = r\ddot{\theta} + 2\dot{r}\dot{\theta}.</math>

===Inverse square law===
Kepler's second law says that <math display="block">r^2\dot{\theta} = nab</math> is constant.

The transversal acceleration <math>a_\theta</math> is zero:
<math display="block">\frac{d\left(r^2\dot{\theta}\right)}{dt}
  = r\left(2\dot{r}\dot{\theta} + r{\ddot{\theta}}\right)
  = ra_\theta = 0.
</math>

So the acceleration of a planet obeying Kepler's second law is directed towards the Sun.

The radial acceleration <math>a_\text{r}</math> is
<math display="block">a_\text{r}
  = \ddot{r} - r\dot{\theta}^2
  = \ddot{r} - r\left(\frac{nab}{r^2}\right)^2
  = \ddot{r} - \frac{n^2 a^2 b^2}{r^3}.
</math>

Kepler's first law states that the orbit is described by the equation:
<math display="block">\frac{p}{r} = 1 + \varepsilon\cos(\theta).</math>

Differentiating with respect to time
<math display="block">-\frac{p\dot{r}}{r^2} = -\varepsilon\sin(\theta)\,\dot{\theta}</math>
or <math display="block">p\dot{r} = nab\,\varepsilon\sin(\theta).</math>

Differentiating once more
<math display="block">p\ddot{r}
  = nab\varepsilon\cos(\theta)\, \dot{\theta}
  = nab\varepsilon\cos(\theta)\, \frac{nab}{r^2}
  = \frac{n^2 a^2 b^2}{r^2}\varepsilon\cos(\theta).
</math>

The radial acceleration <math>a_\text{r}</math> satisfies
<math display="block">pa_\text{r}
  = \frac{n^2 a^2 b^2}{r^2}\varepsilon\cos(\theta) - p\frac{n^2 a^2 b^2}{r^3}
  = \frac{n^2 a^2 b^2}{r^2}\left(\varepsilon\cos(\theta) - \frac{p}{r}\right).
</math>

Substituting the equation of the ellipse gives
<math display="block">pa_\text{r} = \frac{n^2 a^2 b^2}{r^2}\left(\frac{p}{r} - 1 - \frac{p}{r}\right) = -\frac{n^2 a^2}{r^2}b^2.</math>

The relation <math>b^2 = pa</math> gives the simple final result
<math display="block">a_\text{r} = -\frac{n^2 a^3}{r^2}.</math>

This means that the acceleration vector <math>\mathbf{\ddot{r}}</math> of any planet obeying Kepler's first and second law satisfies the '''[[Inverse-square law|inverse square law]]'''
<math display="block">\mathbf{\ddot{r}} = -\frac{\alpha}{r^2}\hat{\mathbf{r}}</math>
where
<math display="block">\alpha = n^2 a^3</math>
is a constant, and <math>\hat{\mathbf{r}}</math> is the unit vector pointing from the Sun towards the planet, and <math>r\,</math> is the distance between the planet and the Sun.

Since mean motion <math>n=\frac{2\pi}{T}</math> where <math>T</math> is the period, according to Kepler's third law, <math>\alpha</math> has the same value for all the planets. So the inverse square law for planetary accelerations applies throughout the entire Solar System.

The inverse square law is a [[differential equation]]. The solutions to this differential equation include the Keplerian motions, as shown, but they also include motions where the orbit is a [[hyperbola]] or [[parabola]] or a [[straight line]]. (See [[Kepler orbit]].)

===Newton's law of gravitation===
By [[Newton's second law]], the gravitational force that acts on the planet is:
<math display="block">\mathbf{F} = m_\text{planet} \mathbf{\ddot{r}} = - m_\text{planet} \alpha r^{-2} \hat{\mathbf{r}}</math>

where <math>m_\text{planet}</math> is the mass of the planet and <math>\alpha</math> has the same value for all planets in the Solar System. According to [[Newton's third law]], the Sun is attracted to the planet by a force of the same magnitude. Since the force is proportional to the mass of the planet, under the symmetric consideration, it should also be proportional to the mass of the Sun, <math>m_\text{Sun}</math>. So
<math display="block">\alpha = Gm_\text{Sun}</math>
where <math>G</math> is the [[gravitational constant]].

The acceleration of Solar System body number ''i'' is, according to Newton's laws:
<math display="block">\mathbf{\ddot{r}}_i = G\sum_{j \ne i} m_j r_{ij}^{-2} \hat{\mathbf{r}}_{ij} </math>
where <math>m_j </math> is the mass of body ''j'', <math>r_{ij} </math> is the distance between body ''i'' and body ''j'', <math>\hat{\mathbf{r}}_{ij} </math> is the unit vector from body ''i'' towards body ''j'', and the vector summation is over all bodies in the Solar System, besides ''i'' itself.

In the special case where there are only two bodies in the Solar System, Earth and Sun, the acceleration becomes
<math display="block">\mathbf{\ddot{r}}_\text{Earth} = Gm_\text{Sun} r_{\text{Earth},\text{Sun}}^{-2} \hat{\mathbf{r}}_{\text{Earth},\text{Sun}}</math>
which is the acceleration of the Kepler motion. So this Earth moves around the Sun according to Kepler's laws.

If the two bodies in the Solar System are Moon and Earth the acceleration of the Moon becomes
<math display="block">\mathbf{\ddot{r}}_\text{Moon} = Gm_\text{Earth} r_{\text{Moon},\text{Earth}}^{-2} \hat{\mathbf{r}}_{\text{Moon},\text{Earth}}</math>

So in this approximation, the Moon moves around the Earth according to Kepler's laws.

In the three-body case the accelerations are
<math display="block">\begin{align}
    \mathbf{\ddot{r}}_\text{Sun} &=
      Gm_\text{Earth} r_{\text{Sun},\text{Earth}}^{-2} \hat{\mathbf{r}}_{\text{Sun},\text{Earth}} +
      Gm_\text{Moon} r_{\text{Sun},\text{Moon}}^{-2} \hat{\mathbf{r}}_{\text{Sun},\text{Moon}} \\
  \mathbf{\ddot{r}}_\text{Earth} &=
      Gm_\text{Sun} r_{\text{Earth},\text{Sun}}^{-2} \hat{\mathbf{r}}_{\text{Earth},\text{Sun}} +
      Gm_\text{Moon} r_{\text{Earth},\text{Moon}}^{-2} \hat{\mathbf{r}}_{\text{Earth},\text{Moon}} \\
   \mathbf{\ddot{r}}_\text{Moon} &=
      Gm_\text{Sun} r_{\text{Moon},\text{Sun}}^{-2} \hat{\mathbf{r}}_{\text{Moon},\text{Sun}} +
      Gm_\text{Earth} r_{\text{Moon},\text{Earth}}^{-2} \hat{\mathbf{r}}_{\text{Moon},\text{Earth}}
\end{align}</math>

These accelerations are not those of Kepler orbits, and the [[three-body problem]] is complicated. But Keplerian approximation is the basis for [[perturbation (astronomy)|perturbation]] calculations. (See [[Lunar theory]].)

==Position as a function of time{{anchor|position_function_time}}==
Kepler used his two first laws to compute the position of a planet as a function of time. His method involves the solution of a [[Transcendental function|transcendental equation]] called [[Kepler's equation]].

The procedure for calculating the heliocentric polar coordinates (''r'',''θ'') of a planet as a function of the time ''t'' since [[perihelion]], is the following five steps:

# Compute the [[mean motion]] {{math|1=''n'' = (2''π'' rad)/''P''}}, where ''P'' is the period.
# Compute the [[mean anomaly]] {{math|1=''M'' = ''nt''}}, where ''t'' is the time since perihelion.
# Compute the [[eccentric anomaly]] ''E'' by solving Kepler's equation: <math display="block">M = E - \varepsilon\sin E ,</math> where <math>\varepsilon</math> is the eccentricity.
# Compute the [[true anomaly]] ''θ'' by solving the equation: <math display="block">(1 - \varepsilon) \tan^2 \frac{\theta}{2} = (1 + \varepsilon)\tan^2\frac{E}{2}</math>
# Compute the heliocentric distance ''r'': <math display="block">r = a(1 - \varepsilon\cos E) ,</math> where <math>a</math> is the semimajor axis.
The position polar coordinates (''r'',''θ'') can now be written as a Cartesian vector <math>\mathbf{p} = r \left\langle \cos{\theta}, \sin{\theta}\right\rangle</math> and the Cartesian velocity vector can then be calculated as <math>\mathbf{v} = \frac{\sqrt{\mu a}}{r} \left\langle -\sin{E}, \sqrt{1-\varepsilon^2} \cos{E}\right\rangle</math>, where <math>\mu</math> is the [[standard gravitational parameter]].<ref>{{cite web|last1=Schwarz|first1=René|title=Memorandum № 1: Keplerian Orbit Elements → Cartesian State Vectors |url=https://downloads.rene-schwarz.com/download/M001-Keplerian_Orbit_Elements_to_Cartesian_State_Vectors.pdf|access-date=4 May 2018}}</ref>

The important special case of circular orbit, ''ε''&nbsp;=&nbsp;0, gives {{nowrap|1=''θ'' = ''E'' = ''M''}}. Because the uniform circular motion was considered to be ''normal'', a deviation from this motion was considered an anomaly.

The proof of this procedure is shown below.

===Mean anomaly, ''M''===
{{Main|Mean anomaly}}
[[File:Mean Anomaly.svg|thumb|upright|Geometric construction for Kepler's calculation of θ. The Sun (located at the focus) is labeled ''S'' and the planet ''P''. The auxiliary circle is an aid to calculation. Line ''xd'' is perpendicular to the base and through the planet ''P''. The shaded sectors are arranged to have equal areas by positioning of point ''y''.]]
The Keplerian problem assumes an [[Elliptic orbit|elliptical orbit]] and the four points:
* ''s'' the Sun (at one focus of ellipse);
* ''z'' the [[Perihelion and aphelion|perihelion]]
* ''c'' the center of the ellipse
* ''p'' the planet
and
* <math>a = |cz|,</math> distance between center and perihelion, the semimajor axis,
* <math>\varepsilon = {|cs| \over a},</math> the eccentricity,
* <math>b = a\sqrt{1 - \varepsilon^2},</math> the semiminor axis,
* <math>r = |sp| ,</math> the distance between Sun and planet.
* <math>\theta = \angle zsp,</math> the direction to the planet as seen from the Sun, the [[true anomaly]].

The problem is to compute the [[polar coordinates]] (''r'',''θ'') of the planet from the time since perihelion,&nbsp;''t''.

It is solved in steps. Kepler considered the circle with the major axis as a diameter, and
*<math>x,</math> the projection of the planet to the auxiliary circle
*<math>y,</math> the point on the circle such that the sector areas |''zcy''| and |''zsx''| are equal,
*<math>M = \angle zcy,</math> the [[mean anomaly]].

The sector areas are related by <math>|zsp| = \frac{b}{a} \cdot |zsx|.</math>

The [[circular sector]] area <math>|zcy| = \frac{a^2 M}2.</math>

The area swept since perihelion,
<math display="block">|zsp| = \frac{b}{a} \cdot|zsx| = \frac{b}{a} \cdot |zcy| = \frac{b}{a} \cdot \frac{a^2 M}{2} = \frac{abM}{2},</math>
is by Kepler's second law proportional to time since perihelion. So the mean anomaly, ''M'', is proportional to time since perihelion, ''t''.
<math display="block">M = nt,</math>
where ''n'' is the [[mean motion]].

===Eccentric anomaly, ''E''===
When the mean anomaly ''M'' is computed, the goal is to compute the true anomaly ''θ''. The function ''θ''&nbsp;=&nbsp;''f''(''M'') is, however, not elementary.<ref>{{cite web| url=http://info.ifpan.edu.pl/firststep/aw-works/fsII/mul/mueller.html| title=Equation of Time – Problem in Astronomy| last=Müller| first=M| date=1995| publisher=Acta Physica Polonica A | access-date=23 February 2013}}</ref> Kepler's solution is to use
<math display="block">E = \angle zcx,</math> ''x'' as seen from the centre, the [[eccentric anomaly]]
as an intermediate variable, and first compute ''E'' as a function of ''M'' by solving Kepler's equation below, and then compute the true anomaly ''θ'' from the eccentric anomaly ''E''. Here are the details.
<math display="block">\begin{align}
            |zcy| &= |zsx| = |zcx| - |scx| \\
with  |scx| &= \frac{|cs|  . |dx|}{2} \\
  \frac{a^2 M}{2} &= \frac{a^2 E}2 - \frac{a\varepsilon \cdot a\sin E}{2}
\end{align}
</math>

Division by ''a''<sup>2</sup>/2 gives [[Kepler's equation]]
<math display="block">M = E - \varepsilon\sin E.</math>

This equation gives ''M'' as a function of ''E''. Determining ''E'' for a given ''M'' is the inverse problem. Iterative numerical algorithms are commonly used.

Having computed the eccentric anomaly ''E'', the next step is to calculate the true anomaly&nbsp;''θ''.

But note:  Cartesian position coordinates with reference to the center of ellipse are (''a''&nbsp;cos&nbsp;''E'',&nbsp;''b''&nbsp;sin&nbsp;''E'')

With reference to the Sun (with coordinates (''c'',0) = (''ae'',0) ), ''r'' = (''a''&nbsp;cos&nbsp;''E'' – ''ae'', ''b''&nbsp;sin&nbsp;''E'')

True anomaly would be arctan(''r''<sub>''y''/''r''</sub>''x''), magnitude of ''r'' would be {{radic|''r''&nbsp;·&nbsp;''r''}}.

===True anomaly, ''θ''===
Note from the figure that
<math display="block">|cd| = |cs| + |sd|</math>
so that
<math display="block">a\cos E = a \varepsilon + r\cos\theta.</math>

Dividing by <math>a</math> and inserting from Kepler's first law
<math display="block">\frac{r}{a} = \frac{1 - \varepsilon^2}{1 + \varepsilon\cos\theta}</math>
to get
<math display="block">
  \cos E = \varepsilon + \frac{1 - \varepsilon^2}{1 + \varepsilon\cos\theta} \cos\theta
         = \frac{\varepsilon (1 + \varepsilon\cos\theta) + \left(1 - \varepsilon^2\right)\cos\theta}{1 + \varepsilon\cos\theta}
         = \frac{\varepsilon + \cos\theta}{1 + \varepsilon\cos\theta}.
</math>

The result is a usable relationship between the eccentric anomaly ''E'' and the true anomaly&nbsp;''θ''.

A computationally more convenient form follows by substituting into the [[trigonometric identity]]:
<math display="block">\tan^2\frac{x}{2} = \frac{1 - \cos x}{1 + \cos x}.</math>

Get
<math display="block">\begin{align}
  \tan^2\frac{E}{2}
    &= \frac{1 - \cos E}{1 + \cos E}
     = \frac{1 - \frac{\varepsilon + \cos\theta}{1 + \varepsilon\cos\theta}}{1 + \frac{\varepsilon + \cos \theta}{1 + \varepsilon\cos\theta}} \\[8pt]
    &= \frac{(1 + \varepsilon\cos\theta) - (\varepsilon + \cos\theta)}{(1 + \varepsilon\cos\theta) + (\varepsilon + \cos\theta)}
    = \frac{1 - \varepsilon}{1 + \varepsilon} \cdot \frac{1 - \cos\theta}{1 + \cos\theta}
    = \frac{1 - \varepsilon}{1 + \varepsilon} \tan^2\frac{\theta}{2}.
\end{align}</math>

Multiplying by 1&nbsp;+&nbsp;''ε'' gives the result
<math display="block">(1 - \varepsilon)\tan^2\frac{\theta}{2} = (1 + \varepsilon)\tan^2\frac{E}{2}</math>

This is the third step in the connection between time and position in the orbit.

=== Distance, ''r'' ===
The fourth step is to compute the heliocentric distance ''r'' from the true anomaly ''θ'' by Kepler's first law:
<math display="block">r(1 + \varepsilon\cos\theta) = a\left(1 - \varepsilon^2\right)</math>

Using the relation above between ''θ'' and ''E'' the final equation for the distance ''r'' is:
<math display="block">r = a(1 - \varepsilon\cos E).</math>

==See also==
{{cols}}
* [[Circular motion]]
* [[Free-fall time]]
* [[Gravity]]
* [[Kepler orbit]]
* [[Kepler problem]]
* [[Kepler's equation]]
* [[Laplace–Runge–Lenz vector]]
* [[Specific relative angular momentum]], relatively easy derivation of Kepler's laws starting with conservation of angular momentum
{{colend}}

== Explanatory notes ==
<references group="Nb"/>

== References ==
{{Reflist}}

== General bibliography ==
* Kepler's life is summarized on pp.&nbsp;523–627 and Book Five of his ''magnum opus'', ''[[Harmonice Mundi]]'' (''harmonies of the world''), is reprinted on: {{Cite book |title=On the shoulders of giants: the great works of physics and astronomy |title-link=On the Shoulders of Giants (book) |date=2002 |publisher=[[Running Press]] |isbn=978-0-7624-1348-5 |editor-last=Hawking |editor-first=Stephen |editor-link=Stephen Hawking |location=Philadelphia |pages=635–732}}
* A derivation of Kepler's third law of planetary motion is a standard topic in engineering mechanics classes. See, for example: {{Cite book |last=Meriam |first=J. L. |url=https://archive.org/details/dynamics00meri |title=Dynamics |date=1971 |publisher=[[Wiley (publisher)|Wiley]] |isbn=978-0-471-59601-1 |edition=2nd |location=New York |pages=161–164 |orig-year=1966}}.
* {{Cite book |last1=Murray |first1=Carl D. |author-link1=Carl D. Murray |url=https://books.google.com/books?id=NY9iQgAACAAJ |title=Solar system dynamics |last2=Dermott |first2=S. F. |author-link2=Stanley Dermott |date=1999 |publisher=[[Cambridge University Press]] |isbn=978-0-521-57295-8 |location=Cambridge; New York}}
* {{Cite book |last=Arnolʹd |first=V. I. |author-link=Vladimir Arnold |title=Mathematical methods of classical mechanics |title-link=Mathematical Methods of Classical Mechanics |date=1997 |publisher=[[Springer Publishing]] |isbn=978-0-387-96890-2 |edition=2nd |series=Graduate texts in mathematics |location=New York |chapter=Chapter 2: Investigation of the Equations of Motion}}

==External links==
{{commons category|Kepler motions}}
* [https://feynmanlectures.caltech.edu/I_07.html#Ch7-S2 The Feynman Lectures on Physics - Kepler's laws]
* Crowell, Benjamin, ''[http://www.lightandmatter.com/lm Light and Matter]'', an [[On-line book|online book]] that gives a proof of the first law without the use of calculus (see section 15.7)
* David McNamara and Gianfranco Vidali, "[https://web.archive.org/web/20060910225253/http://www.phy.syr.edu/courses/java/mc_html/kepler.html Kepler's Second Law – Java Interactive Tutorial]", an interactive Java applet that aids in the understanding of Kepler's second law.
* Cain, Gay (May 10, 2010), ''Astronomy Cast'', "[http://www.astronomycast.com/history/ep-189-johannes-kepler-and-his-laws-of-planetary-motion/ Ep. 189: Johannes Kepler and His Laws of Planetary Motion]"
* University of Tennessee's Dept. Physics & Astronomy: Astronomy 161, "[http://csep10.phys.utk.edu/astr161/lect/history/kepler.html Johannes Kepler: The Laws of Planetary Motion]"
* Solar System Simulator ([http://physinf.com/NPM/NPM.html Interactive Applet]) {{Webarchive|url=https://web.archive.org/web/20181213115626/https://www.physinf.com/NPM/NPM.html |date=2018-12-13 }}
* "[http://www.phy6.org/stargaze/Skeplaws.htm Kepler and His Laws]" in ''From Stargazers to Starships'' by David P. Stern (10 October 2016)
*{{YouTube|dG9J0o7AwVs|"Kepler's Three Laws of Planetary Motion"}} by Jens Puhle (Dec 27, 2023) – a video explaining and visualizing Kepler's three laws of planetary motion

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