In astronomy, Kepler's laws of planetary motion give good approximations for the orbits of planets around the Sun. They were published by Johannes Kepler from 1609 to 1621 in three works Astronomia nova, Harmonice Mundi and Epitome Astronomiae Copernicanae. The laws were based on Kepler's concept of solar fibrils adapted to the accurate astronomical data of Tycho Brahe. These laws replaced the circular orbits and epicycles of Copernicus's heliostatic model of the planets with a heliocentric model that described elliptical orbits with planetary velocities that vary accordingly. The three laws state that: The orbit of a planet is an ellipse with the Sun at one of the two 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 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.
Comparison to Copernicus
Johannes Kepler's laws improved the model of Copernicus. According to Copernicus: 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: The planetary orbit is not a circle with epicycles, but an ellipse. The Sun is not at the center but at a focal point of the elliptical orbit. Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the area speed (closely linked historically with the concept of angular momentum) is constant.
The 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
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.
History Kepler's laws were developed based on a physical theory of planetary motion in which the Sun emitted magnetic fibrils which pulled the planets into orbits. The fibrils were somewhat elastic allowing non-circular motion driven by the inertia of the planets.
In Astronomia nova (1609), Kepler articulated his first law, showing that Mars' orbit is elliptical, 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. having found them by analyzing the astronomical observations of Tycho Brahe. Kepler had believed in the 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 eccentricity of all planets except Mercury. His first law reflected this discovery.
In his Astronomia nova (1609), Kepler did not present his second law in its modern form. He did that only in his Epitome Astronomiae Copernicanae of 1621. Kepler had two versions of the second law, related in a qualitative sense: the first "distance law" and later the "area law". The distance form was only correct for orbits that were almost circular, but the area form was correct for all elliptical orbit. The "area law" is what became the second law in the set of three. This law had little impact on astronomy because calculations of planetary positions using the law were approximate and time consuming. 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. 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]) As the century proceeded it became more widely accepted.
Kepler's third law was published in 1619 in his Harmonice Mundi.Johannes Kepler, Harmonices Mundi [The Harmony of the World] (Linz, (Austria): Johann Planck, 1619), book 5, chapter 3, p. 189. An English translation of Kepler's Harmonices Mundi is available as: Johannes Kepler with E. J. Aiton, A. M. Duncan, and J. V. Field, trans., The Harmony of the World (Philadelphia, Pennsylvania: American Philosophical Society, 1997); see especially p. 411. In 1621, Kepler noted that his third law applies to the four brightest moons of Jupiter.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, 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.) Godefroy Wendelin, the first well-known astronomer to adopt Kepler's laws, gave a detailed account of the third law in 1652.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, 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]).)
Kepler's work had little initial impact. His work was a strong defense of Copernicanism which had fallen out of fashion in part because of opposition by Tycho Brahe. In 1627 Kepler published the Rudolphine Tables containing many accurate astronomical observations accumulated by Brahe. The breadth and accuracy of the tables allowed astronomers to compare Kepler's formula to good quality data. At first these difficult calculations were off putting, but once undertaken more astronomers became convinced of Kepler's approach.
The reception in Germany changed noticeably between 1688, shortly after Newton's Principia was published and was taken to be basically Copernican, and 1690, by which time work of Gottfried Leibniz on Kepler had been published. Newton understood 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.
As three laws
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".Voltaire, Eléments de la philosophie de Newton [Elements of Newton's Philosophy] (London: 1738). See, for example:
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.)