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### Kepler's laws of planetary motion

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Kepler's laws of planetary motion
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{{short description|Scientific laws describing motion of planets around the Sun}}{{hatnote|For a more precise historical approach, see in particular the articles Astronomia nova and Epitome Astronomiae Copernicanae.}}{{Astrodynamics}}In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun, published by Johannes Kepler between 1609 and 1619. These improved the heliocentric theory of Nicolaus Copernicus, replacing its circular orbits with epicycles with elliptical trajectories, and explaining how planetary velocities vary. The laws state that:(File:Kepler laws diagram.svg|thumb|Figure 1: Illustration of Kepler's three 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 focal points F1 and F2 for the first planet and F1 and F3 for the second planet. The Sun is placed in focal point F1.| The two shaded sectors A1 and A2 have the same surface area and the time for planet 1 to cover segment A1 is equal to the time to cover segment A2.| The total orbit times for planet 1 and planet 2 have a ratio left(frac{a_1}{a_2}right)^frac{3}{2}.}}){{Astrodynamics |Equations}}
1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.Bryant, Jeff; Pavlyk, Oleksandr. "Kepler's Second Law", Wolfram Demonstrations Project. Retrieved December 27, 2009.
3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
The elliptical orbits of planets were indicated by calculations of the orbit of Mars.NEWS,weblink After 400 Years, a Challenge to Kepler:He Fabricated His Data, Scholar Says, The New York Times, 1990-01-23, Broad, William J., 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 helps to establish that when a planet is closer to the Sun, it travels faster. The third law expresses that the further a planet is from the Sun, the longer its orbit, and vice versa.Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System to a good approximation, as a consequence of his own laws of motion and law of universal gravitation.

## Comparison to Copernicus

Kepler's laws improved the model of Copernicus. If the eccentricities of the planetary orbits are taken as zero, then Kepler basically agreed with Copernicus:
1. The planetary orbit is a circle
2. The Sun is at the center of the orbit
3. The speed of the planet in the orbit is constant
The eccentricities of the orbits of those planets known to Copernicus and Kepler are small, so the foregoing rules give fair approximations of planetary motion, but Kepler's laws fit the observations better than does the model proposed by Copernicus.Kepler's corrections are not at all obvious:
1. The planetary orbit is not a circle, but an ellipse.
2. The Sun is not at the center but at a focal point of the elliptical orbit.
3. 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
e approx frac{pi}{4} frac{186 - 179}{186 + 179} approx 0.015,
which is close to the correct value (0.016710219) (see Earth's orbit).The calculation is correct when perihelion, the date the Earth is closest to the Sun, falls on a solstice. The current perihelion, near January 3, is fairly close to the solstice of December 21 or 22.

## Nomenclature

It took nearly two centuries for current formulation of Kepler's work to take on its settled form. Voltaire's ElÃ©ments de la philosophie de Newton (Elements of Newton's Philosophy) of 1738 was the first publication to use the terminology of "laws".Voltaire, ElÃ©ments de la philosophie de Newton [Elements of Newton's Philosophy] (London, England: 1738). See, for example:
• From p. 162: "Par une des grandes loix de Kepler, toute Planete dÃ©crit des aires Ã©gales en temp Ã©gaux : par une autre loi non-moins sÃ»re, chaque Planete fait sa rÃ©volution 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 rÃ©volution de cette Planete autour du Soleil sera proportionnÃ© Ã  la racine quarrÃ©e 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.)
• From p. 205: "Il est donc prouvÃ© 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 â€¦ )"MEMBERWIDE">
FIRST=CURTIS TITLE=KEPLER'S LAWS, SO-CALLED ISSUE=31 URL=HTTPS://HAD.AAS.ORG/SITES/HAD.AAS.ORG/FILES/HADN31.PDF, December 27, 2016, The Biographical Encyclopedia of Astronomers in its article on Kepler (p. 620) states that the terminology of scientific laws for these discoveries was current at least from the time of Joseph de Lalande.De la Lande, Astronomie, vol. 1 (Paris, France: Desaint & Saillant, 1764). See, for example:
• From page 390: " â€¦ mais suivant la fameuse loi de Kepler, qui sera expliquÃ©e dans le Livre suivant (892), le rapport des temps pÃ©riodiques est toujours plus grand que celui des distances, une planete cinq fois plus Ã©loignÃ©e du soleil, emploie Ã  faire sa rÃ©volution douze fois plus de temps ou environ; â€¦ " ( â€¦ but according to the famous law of Kepler, which will be explained in the following book [i.e., chapter] (paragraph 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]; â€¦ )
• From page 429: "Les QuarrÃ©s des Temps pÃ©riodiques sont comme les Cubes des Distances. 892. La plus fameuse loi du mouvement des planetes dÃ©couverte par Kepler, est celle du repport qu'il y a entre les grandeurs de leurs orbites, & le temps qu'elles emploient Ã  les parcourir; â€¦ " (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; â€¦ )
• From page 430: "Les Aires sont proportionnelles au Temps. 895. Cette loi gÃ©nÃ©rale du mouvement des planetes devenue si importante dans l'Astronomie, sÃ§avior, que les aires sont proportionnelles au temps, est encore une des dÃ©couvertes de Kepler; â€¦ " (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; â€¦ )
• From page 435: "On a appellÃ© cette loi des aires proportionnelles aux temps, Loi de Kepler, aussi bien que celle de l'article 892, du nome de ce cÃ©lebre Inventeur; â€¦ " (One called this law of areas proportional to times (the law of Kepler) as well as that of paragraph 892, by the name of that celebrated inventor; â€¦ ) It was the exposition of 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.Robert Small, An account of the astronomical discoveries of Kepler (London, England: J Mawman, 1804), pp. 298â€“299. Small also claimed, against the history, that these were empirical laws, based on inductive reasoning.Robert Small, An account of the astronomical discoveries of Kepler (London, England: J. Mawman, 1804).
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.BOOK, Bruce Stephenson, Kepler's Physical Astronomy,weblink 1994, Princeton University Press, 978-0-691-03652-6, 170,

## History

Johannes Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe.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.See: Johannes Kepler, Astronomia nova â€¦ (1609), 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 PlanetÃ¦ iter; â€¦ " (Thus, an ellipse is the planet's [i.e., Mars'] path; â€¦ ) Later on the same page: " â€¦ ut sequenti capite patescet: ubi simul etiam demonstrabitur, nullam PlanetÃ¦ relinqui figuram OrbitÃ¦, prÃ¦terquam perfecte ellipticam; â€¦ " ( â€¦ 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; â€¦ ) And then: "Caput LIX. Demonstratio, quod orbita Martis, â€¦ , fiat perfecta ellipsis: â€¦ " (Chapter 59. Proof that Mars' orbit, â€¦ , is a perfect ellipse: â€¦ ) The geometric proof that Mars' orbit is an ellipse appears as Protheorema XI on pages 289â€“290.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 OrbitÃ¦ (III. On the figure [i.e., shape] of orbits), pages 658â€“665. From p. 658: "Ellipsin fieri orbitam planetÃ¦ â€¦ " (Of an ellipse is made a planet's orbit â€¦ ). From p. 659: " â€¦ Sole (Foco altero huius ellipsis) â€¦ " ( â€¦ the Sun (the other focus of this ellipse) â€¦ ).BOOK, Physics, the Human Adventure: From Copernicus to Einstein and Beyondauthor2=Brush, Stephen G., 40â€“41,weblinkisbn=978-0-8135-2908-0, Rutgers University Pressaccessdate=December 27, 2009, 2001, 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 â€¦ (1609), pp. 165â€“167. On page 167, Kepler states: " â€¦ , quanto longior est Î±Î´ quam Î±Îµ, tanto diutius moratur Planeta in certo aliquo arcui excentrici apud Î´, quam in Ã¦quali arcu excentrici apud Îµ." ( â€¦ , as Î±Î´ is longer than Î±Îµ, so much longer will a planet remain on a certain arc of the eccentric near Î´ than on an equal arc of the eccentric near Îµ.) That is, the farther a planet is from the Sun (at the point Î±), 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, â€¦ , fiat perfecta ellipsis: â€¦ " (Chapter 59. Proof that Mars' orbit, â€¦ , is a perfect ellipse: â€¦ ), Protheorema XIV and XV, pp. 291â€“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.

## Formulary

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

### First law of Kepler (law of orbits)

{{bq|The orbit of every planet is an ellipse with the Sun at one of the two foci.}}(File:kepler-first-law.svg|thumb|Figure 2: Kepler's first law placing the Sun at the focus of an elliptical orbit)(File:Ellipse latus rectum.svg|thumb|Figure 4: Heliocentric coordinate system (r,{{nbsp}}Î¸) for ellipse. Also shown are: semi-major axis a, semi-minor axis b and semi-latus rectum p; center of ellipse and its two foci marked by large dots. For {{nowrap|Î¸ {{=}} 0Â°}}, {{nowrap|r {{=}} r{{sub|min}}}} and for {{nowrap|Î¸ {{=}} 180Â°}}, {{nowrap|r {{=}} r{{sub|max}}}}.)Mathematically, an ellipse can be represented by the formula:
r = frac{p}{1 + varepsilon, costheta},
where p is the semi-latus rectum, Îµ is the eccentricity of the ellipse, r is the distance from the Sun to the planet, and Î¸ is the angle to the planet's current position from its closest approach, as seen from the Sun. So (rÎ¸) are polar coordinates.For an ellipse 0

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