Kepler's laws of planetary motion

8444: 7897: 5197: 8439:{\displaystyle {\begin{aligned}\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{aligned}}} 4691: 14789: 12254:(The arc of the ellipse, of which the duration is delimited by the area AKM, should be terminated in LK, so that it 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. 11383: 11763:(No less clever 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 a very long and very learned letter. So, just as in 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 periods (which are 1.769 days for the innermost , 3.554 days for the next to the innermost , 7.164 days for the next to the outermost , and 16.756 days for the outermost ).) 5192:{\displaystyle {\begin{aligned}{\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{aligned}}} 3057: 3047: 3035: 3025: 10951: 3993: 11378:{\displaystyle {\begin{aligned}\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 }}}}\\&={\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{aligned}}} 11737:, "Marius" presents these distances, from Jupiter, of the four of Jupiter: 3, 5, 8, 13 (or 14 Galileo) ... 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 the proportion is greater than double, thus greater than of the distances 3, 5, 8, 13 or 14, although less than of the squares, which double the proportions of the distances, namely 9, 25, 64, 169 or 196, just as 3/2 is also greater than 1 but less than 2.) 1031: 181: 30: 1023: 13777: 10843: 14825: 12239:( ... , 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. 9042: 14849: 11948:"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." 14801: 12269:(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.) 2039: 14837: 4597: 11950:(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.) 14813: 10621: 6586: 4416: 10212: 3660: 6315: 5897: 2085:

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. A correct proof can be shown through this. Since the cross product of two vectors gives the area of a

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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.

6349: 12016:(... but according to the famous law of Kepler, which will be explained in the following book (para. 892), the ratio of the periods is always greater than that of the distances a planet five times farther from the sun, requires about twelve times or so more time to make its revolution ...) 856:

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

12014:"... 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; ..." 10838:{\displaystyle \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 }}.} 1821: 9929: 5670: 6761: 11746:

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,

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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.

9784: 1641: 12023:"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; ..." 8970: 6119: 2067:

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 inverse square

1512: 11729:(... just as it is clearly among the six planets around the Sun, so also it is among the four of Jupiter, because around the body of Jupiter any that can go farther from it, orbits slower, and even that is not in the same proportion, but greater ; that is, 3/2 ( 7277: 717:

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:

3497: 659:, 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. 4592:{\displaystyle {\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} }}} 12025:(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 require to traverse them; ...) 973:

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.

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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

12032:"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; ..." 2418: 6046: 10616: 8872: 5526: 6593: 3232: 6973: 851: 1929: 11582: 7332:, 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, 2321: 1674: 5433: 2071:

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

9621: 2687: 12403:(But when he noticed that it didn't agree at all with observations, he changed his thinking, and he declared that a line a planet's true motion, sweeps out equal areas of an ellipse in equal periods of time: ... ) 3005: 2810: 10944: 3977:

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.

11727:" ... 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." 4286:

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

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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:

6581:{\displaystyle 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).} 3404:

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,

12034:(Areas are proportional to times. 895. This general law of the movement of the planets become so important in astronomy, namely, that areas are proportional to times, is one of Kepler's discoveries; ...) 7189: 1154: 10956: 6112: 5956: 1553: 1416: 6871: 778:, 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 10207:{\displaystyle {\begin{aligned}|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{aligned}}} 1548: 8877: 5351: 2603: 5204: 1375: 1282: 10522: 1411: 9934: 7902: 4696: 4356: 1679: 931:

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

3199: 12401:"Sed cum id Observationibus nequaquam congruere animadverteret, mutavit sententiam, & lineam veri motus Planetæ æqualibus temporibus æquales areas Ellipticas verrere professus est: ... " 9544: 9252: 2472:, 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. 4628: 3655:{\displaystyle {\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}}} 14757: 2470: 11394: 9201: 8627: 12141:( ... 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: 12110:

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.

11649: 9614: 8777: 7647: 7429: 6310:{\displaystyle 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 ).} 5491: 2920: 10265: 8592: 7393: 12377:"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; ..." 2075:

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.

7047: 4659: 4407: 7109: 6902: 2858: 2103: 157: 9336: 4073: 3867: 12267:"Dictum quidem est in superioribus, divisa orbita in particulas minutissimas æquales: accrescete iis moras planetæ per eas, in proportione intervallorum inter eas & Solem." 7016: 7306: 5963: 14172: 10549: 10466: 7357: 9901: 9462: 8612: 6344: 5702: 5521: 3793: 2548: 1975: 6912: 6799: 9296: 9148: 7326: 7149: 4679: 3130:. 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. 12145:(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. 7598: 4245:

that he considers forces from a mathematical point of view, not a physical, thereby taking an instrumentalist view. Moreover, he does not assign a cause to gravity.

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were in its favour. As the century proceeded it became more widely accepted. The reception in Germany changed noticeably between 1688, the year in which Newton's

9414: 9389: 5360: 2501: 2327: 10544: 8797: 7413: 7129: 4376: 3771: 3751: 3727: 3707: 3683: 3463: 3443: 3423: 2735: 2612: 2521: 1313: 1177: 14107: 5892:{\displaystyle 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}}}.} 7882:{\displaystyle \mathbf {\ddot {r}} _{\text{Moon}}=Gm_{\text{Earth}}r_{{\text{Moon}},{\text{Earth}}}^{-2}{\hat {\mathbf {r} }}_{{\text{Moon}},{\text{Earth}}}} 3394:{\displaystyle 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}} 8814: 2928: 10865: 7768:{\displaystyle \mathbf {\ddot {r}} _{\text{Earth}}=Gm_{\text{Sun}}r_{{\text{Earth}},{\text{Sun}}}^{-2}{\hat {\mathbf {r} }}_{{\text{Earth}},{\text{Sun}}}} 12123:, Kepler presented only a proof that Mars' orbit is elliptical. Evidence that the other known planets' orbits are elliptical was presented only in 1621. 12041:"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; ..." 4282:

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.

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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.

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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

1854: 14737: 12237:" ... , quanto longior est αδ quam αε, tanto diutius moratur Planeta in certo aliquo arcui excentrici apud δ, quam in æquali arcu excentrici apud ε." 1816:{\displaystyle {\begin{aligned}p&=\left({\frac {r_{\max }^{-1}+r_{\min }^{-1}}{2}}\right)^{-1}\\pa&=r_{\max }r_{\min }=b^{2}\,\end{aligned}}} 12043:(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; ... ) 12840:

A derivation of Kepler's third law of planetary motion is a standard topic in engineering mechanics classes. See, for example, pp. 161–164 of

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

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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

12139:" ... ut sequenti capite patescet: ubi simul etiam demonstrabitur, nullam Planetæ relinqui figuram Orbitæ, præterquam perfecte ellipticam; ... " 6051: 5904: 6804: 12217:

of 1621. Furthermore, in 1609, he presented his second law in two different forms, which scholars call the "distance law" and the "area law".

14075: 12902: 5665:{\displaystyle {\frac {d\left(r^{2}{\dot {\theta }}\right)}{dt}}=r\left(2{\dot {r}}{\dot {\theta }}+r{\ddot {\theta }}\right)=ra_{\theta }=0.} 4210: 962: 6756:{\displaystyle 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}.} 5289: 3095:

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.

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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

12742: 10471: 7158:. The solutions to this differential equation include the Keplerian motions, as shown, but they also include motions where the orbit is a 7151:

has the same value for all the planets. So the inverse square law for planetary accelerations applies throughout the entire Solar System.

12151:(Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 5, part 1, III. De Figura Orbitæ (III. On the figure of orbits), 14164: 14002: 3491:. However, with planet masses being so much smaller than the Sun, this correction is often ignored. The full corresponding formula is: 13813: 12925: 11733:) of the proportion of each of the distances from Jupiter, which is clearly the very as is used for the six planets above. In his 923:

Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of

14287: 9779:{\displaystyle |zsp|={\frac {b}{a}}\cdot |zsx|={\frac {b}{a}}\cdot |zcy|={\frac {b}{a}}\cdot {\frac {a^{2}M}{2}}={\frac {abM}{2}},} 4272: 887: 670: 12133:

After having rejected circular and oval orbits, Kepler concluded that Mars' orbit must be elliptical. From the top of page 285:

1105: 14712: 14689: 14445: 14393: 4682: 1636:{\displaystyle {\begin{aligned}b&={\sqrt {r_{\max }r_{\min }}}\\b&={\frac {p}{\sqrt {1-\varepsilon ^{2}}}}\end{aligned}}} 576: 12936: 11761:"III. Non minus Kepleriana ingeniosa est Vendelini ... & D. 7. 164/1000. pro penextimo, & D. 16. 756/1000. pro extimo." 11601: 8965:{\displaystyle \mathbf {v} ={\frac {\sqrt {\mu a}}{r}}\left\langle -\sin {E},{\sqrt {1-\varepsilon ^{2}}}\cos {E}\right\rangle } 8729: 8468:

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

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Using Newton's law of gravitation (published 1687), this relation can be found in the case of a circular orbit by setting the

12853: 12582: 12495: 12468: 12432: 12187: 12095: 10226: 8553: 7362: 1507:{\displaystyle {\begin{aligned}a&={\frac {r_{\max }+r_{\min }}{2}}\\a&={\frac {p}{1-\varepsilon ^{2}}}\end{aligned}}} 903:(1814) that made up the set of three laws, by adding in the third. Small also claimed, against the history, that these were 2553: 1332: 1239: 14423: 14134: 4319: 4307: 4294:

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

417: 12600:"Original form of Kepler's Third Law and its misapplication in Propositions XXXII-XXXVII in Newton's Principia (Book I)" 6978: 3143: 14884: 14336: 12539: 12224:(Chapter 32. The force that moves a planet circularly weakens with distance from the source.) See: Johannes Kepler, 11695: 7272:{\displaystyle \mathbf {F} =m_{\text{planet}}\mathbf {\ddot {r}} =-m_{\text{planet}}\alpha r^{-2}{\hat {\mathbf {r} }}} 1002:

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

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absent the third law in 1609 and fully in 1619, describe the orbits of planets around the Sun. The laws replaced the

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The orbits of a planet are circular (Kepler discovered his Second Law before his First Law, which contradicts this).

17: 14722: 14547: 13912: 13709: 13589: 13008: 11957:"Il est donc prouvé par la loi de Kepler & par celle de Neuton, que chaque Planete gravite vers le Soleil, ..." 11717: 9868: 8993: 2425: 991: 684: 14788: 12954: 9155: 4241:

This implies that the Sun may be the physical cause of the acceleration of planets. However, Newton states in his

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In the special case where there are only two bodies in the Solar System, Earth and Sun, the acceleration becomes

4004:(average of aphelion and perihelion) of some Solar System orbits (crosses denoting Kepler's values) showing that 12293:

proportio qua est inter binorum quorumcunque Planetarum tempora periodica, sit præcise sesquialtera proportionis

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Kepler stated that every planet travels in elliptical orbits having the Sun at one focus in: Johannes Kepler,

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which is the acceleration of the Kepler motion. So this Earth moves around the Sun according to Kepler's laws.

644: 238: 12330: 5275:{\displaystyle {\ddot {\mathbf {r} }}=a_{r}{\hat {\boldsymbol {r}}}+a_{\theta }{\hat {\boldsymbol {\theta }}}} 2869: 14627: 13806: 2219:{\displaystyle dA={\frac {1}{2}}({\vec {r}}\times {\vec {dx}})={\frac {1}{2}}({\vec {r}}\times {\vec {v}}dt)} 11959:(It is thus proved by the law of Kepler and by that of Newton, that each planet revolves around the sun ...) 11759:. (Wendelin's clever speculation about the movement and distances of Jupiter's satellites.) From p. 492: 665:

showed in 1687 that relationships like Kepler's would apply in the Solar System as a consequence of his own

14537: 13694: 13164: 9797: 7021: 4633: 4381: 3445:, of the elliptical relative motion of one mass relative to the other, as well as replacing the large mass 7076: 6878: 2825: 14732: 14591: 14388: 13639: 12983:

by Jens Puhle (Dec 27, 2023) – a video explaining and visualizing Kepler's three laws of planetary motion

9303: 3056: 3046: 3034: 569: 502: 95: 14482: 12037: 12028: 12019: 12010: 4037: 3831: 3126:" according to precise laws, and express it in terms of musical notation. It was therefore known as the 3119:

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

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with the cube of the semi-major axis of its orbit is the same for all objects orbiting the same primary.

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The red ray rotates at a constant angular velocity and with the same orbital time period as the planet,

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of the acceleration is inversely proportional to the square of the planet's distance from the Sun (the

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It took nearly two centuries for the current formulation of Kepler's work to take on its settled form.

666: 14497: 14413: 14067: 13757: 12247: 12232: 12229: 12130: 11813: 11480:{\displaystyle (1-\varepsilon )\tan ^{2}{\frac {\theta }{2}}=(1+\varepsilon )\tan ^{2}{\frac {E}{2}}} 8713:{\displaystyle (1-\varepsilon )\tan ^{2}{\frac {\theta }{2}}=(1+\varepsilon )\tan ^{2}{\frac {E}{2}}} 3024: 2060:

Planets are pushed around the Sun by a force from the Sun. This false assumption relies on incorrect

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proportion between the periodic times of any two planets is precisely the sesquialternate proportion

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Kepler enunciated in 1619 this third law in a laborious attempt to determine what he viewed as the "

14894: 14874: 14869: 14368: 14309: 14292: 14277: 14222: 13888: 13799: 13742: 13267: 12899: 9429: 8453: 7532:{\displaystyle \mathbf {\ddot {r}} _{i}=G\sum _{j\neq i}m_{j}r_{ij}^{-2}{\hat {\mathbf {r} }}_{ij}} 4688:

Differentiate the position vector twice to obtain the velocity vector and the acceleration vector:

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is the angle to the planet's current position from its closest approach, as seen from the Sun. So (

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O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, as

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A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

14346: 14099: 13752: 13060: 12246:(Chapter 59. Proof that Mars' orbit, ... , is a perfect ellipse: ... ), Protheorema XIV and XV, 9087: 8469: 5499: 3776: 2526: 1944: 896: 368: 11862: 9045:

Geometric construction for Kepler's calculation of θ. The Sun (located at the focus) is labeled

3992: 865:. The current perihelion, near January 4, is fairly close to the solstice of December 21 or 22. 14717: 14707: 14472: 14435: 14408: 14302: 13982: 13614: 13216: 13136: 13124: 12213:... (1609), Kepler did not present his second law in its modern form. He did that only in his 10859: 7416: 7183: 3730: 1222:= 0, the orbit is a circle with the Sun at the centre (i.e. where there is zero eccentricity). 748:

Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the

562: 285: 12943: 12414: 12316: 12286: 12262: 12152: 12085: 12059: 11953: 11944: 11752: 11722: 11715:

In 1621, Johannes Kepler noted that Jupiter's moons obey (approximately) his third law in his

7778:

If the two bodies in the Solar System are Moon and Earth the acceleration of the Moon becomes

6768: 14764: 14671: 14557: 14552: 14341: 14254: 14229: 14021: 13955: 13866: 13737: 13679: 13649: 13437: 13314: 13282: 13252: 13211: 13196: 13075: 12572: 12177: 9259: 9111: 7329: 7311: 7155: 7134: 4664: 950:

also made this observation in 1643. The second law, in the "area law" form, was contested by

606: 470: 305: 213: 12485: 11698:, relatively easy derivation of Kepler's laws starting with conservation of angular momentum 7573: 2694: 14678: 14398: 14155: 13996: 13856: 13762: 13584: 13368: 13257: 13226: 13154: 13129: 13104: 13065: 13046: 13001: 12611: 12420: 11905: 11837: 11685: 10221: 9365:

It is solved in steps. Kepler considered the circle with the major axis as a diameter, and

8473: 7542: 2413:{\displaystyle {\frac {dA}{dt}}={\frac {1}{m}}{\frac {1}{2}}({\vec {r}}\times {\vec {p}}).} 1326:, the distance is maximum (by definition, aphelion is – invariably – perihelion plus 180°) 932: 763: 343: 300: 290: 218: 8975: 6041:{\displaystyle -{\frac {p{\dot {r}}}{r^{2}}}=-\varepsilon \sin(\theta )\,{\dot {\theta }}} 2034:

A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

8: 14853: 14487: 14467: 14383: 14373: 14234: 13987: 13624: 13419: 13159: 11975: 11786: 7052: 5674:

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

3468: 3207: 3070: 2061: 908: 614: 385: 223: 12615: 12424: 12222:"Caput XXXII. Virtutem quam Planetam movet in circulum attenuari cum discessu a fonte." 11909: 9396: 9371: 2483: 2078:

The area swept out is proportional to the overall time. This is also approximately true.

966:

was published and was taken to be basically Copernican, and 1690, by which time work of

14841: 14829: 14702: 14418: 14331: 14059: 13297: 13186: 13084: 12634: 12599: 12554: 12460: 10611:{\displaystyle {\frac {r}{a}}={\frac {1-\varepsilon ^{2}}{1+\varepsilon \cos \theta }}} 10529: 10299:

But note: Cartesian position coordinates with reference to the center of ellipse are (

8782: 8449: 7398: 7114: 6906: 4361: 3970:'s recent invention of logarithms and log-log graphs before he discovered the pattern. 3756: 3736: 3712: 3692: 3668: 3448: 3428: 3408: 3123: 2720: 2506: 1298: 1162: 767: 458: 333: 12511: 9786:

is by Kepler's second law proportional to time since perihelion. So the mean anomaly,

8867:{\displaystyle \mathbf {p} =r\left\langle \cos {\theta },\sin {\theta }\right\rangle } 7891:

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

14661: 14633: 14515: 14492: 14440: 14358: 14244: 13928: 13836: 13664: 13562: 13492: 13247: 13201: 13119: 12882: 12868: 12849: 12831: 12695: 12666: 12639: 12578: 12535: 12491: 12464: 12428: 12357:

Nicolai Mercatoris Hypothesis astronomica nova, et consensus eius cum observationibus

12183: 12091: 12071: 9907: 9347: 8544: 3796: 3134: 1208: 1180: 967: 951: 947: 892: 775: 373: 310: 189: 12928:", an interactive Java applet that aids in the understanding of Kepler's Second Law. 12361:

Nicolaus Mercator's new astronomical hypothesis, and its agreement with observations

12315:(Philadelphia, Pennsylvania: American Philosophical Society, 1997); see especially 12182:(3rd paperback ed.). Piscataway, NJ: Rutgers University Press. pp. 40–41. 14805: 14621: 14597: 14573: 14403: 14149: 13962: 13920: 13644: 13576: 13340: 13302: 13176: 13146: 13099: 12629: 12619: 12389: 12252:"Arcum ellipseos, cujus moras metitur area AKN, debere terminari in LK, ut sit AM." 11913: 6968:{\displaystyle \mathbf {\ddot {r}} =-{\frac {\alpha }{r^{2}}}{\hat {\mathbf {r} }}} 4288: 1058: 1015: 987: 755: 741: 630: 618: 543: 492: 257: 201: 12921:

that gives a proof of the first law without the use of calculus (see section 15.7)

12624: 12244:"Caput LIX. Demonstratio, quod orbita Martis, ... , fiat perfecta ellipsis: ... " 12143:"Caput LIX. Demonstratio, quod orbita Martis, ... , fiat perfecta ellipsis: ... " 11838:"Planetary Motion: The History of an Idea That Launched the Scientific Revolution" 846:{\displaystyle e\approx {\frac {\pi }{4}}{\frac {186-179}{186+179}}\approx 0.015,} 14899: 14609: 14378: 14272: 13936: 13904: 13851: 13822: 13684: 13277: 13181: 13171: 13070: 12994: 12958: 12906: 12827: 12823: 12815: 12811: 12798: 12308: 11721:(Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 4, part 2, 11660: 11489:

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

9549: 3807:

The following table shows the data used by Kepler to empirically derive his law:

1518: 1388: 1381: 696: 678: 598: 548: 453: 363: 338: 1924:{\displaystyle \varepsilon ={\frac {r_{\max }-r_{\min }}{r_{\max }+r_{\min }}}.} 651:

The elliptical orbits of planets were indicated by calculations of the orbit of

14793: 14652: 14477: 14460: 14430: 14217: 13975: 13896: 13780: 13732: 13724: 13719: 13604: 13599: 13530: 13510: 13501: 13094: 13080: 13056: 13051: 13026: 11680: 11665: 9070: 3112: 2606: 1525: 939: 750: 640: 602: 520: 436: 430: 353: 278: 272: 267: 12974: 12453: 11577:{\displaystyle r(1+\varepsilon \cos \theta )=a\left(1-\varepsilon ^{2}\right)} 9061:. The shaded sectors are arranged to have equal areas by positioning of point 14863: 14697: 14579: 14521: 13968: 13634: 13629: 13548: 13191: 13109: 12843: 7167: 4271:

The Sun plays an unsymmetrical part, which is unjustified. So he assumed, in

2316:{\displaystyle {\frac {dA}{dt}}={\frac {1}{2}}({\vec {r}}\times {\vec {v}}).} 1651: 904: 771: 525: 358: 315: 12964: 11757:

Vendelini ingeniosa speculatio circa motus & intervalla satellitum Jovis

1026:

Kepler's first law placing the Sun at one of the foci of an elliptical orbit

14817: 14585: 14455: 14264: 14239: 14202: 14044: 13846: 13699: 13609: 13483: 13466: 13324: 13221: 13089: 12918: 12819: 12688: 12659: 12643: 12393: 12328: 11675: 9465: 9339: 9036: 8618: 8524: 8457: 7657:, and the vector summation is over all bodies in the Solar System, besides 7171: 5428:{\displaystyle a_{\theta }=r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}.} 4313: 4215: 4205: 979: 662: 656: 243: 233: 228: 12257:

In 1621, Kepler restated his second law for any planet: Johannes Kepler,

10285:

is the inverse problem. Iterative numerical algorithms are commonly used.

2682:{\displaystyle {\frac {dA}{dt}}={\frac {r^{2}}{2}}{\frac {d\theta }{dt}}.} 885:) of 1738 was the first publication to use the terminology of "laws". The 14603: 14212: 14091: 13704: 13539: 13309: 13289: 13206: 12942:

University of Tennessee's Dept. Physics & Astronomy: Astronomy 161, "

12261:(Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 5, 9831: 8500: 4410: 3967: 2816: 2422:

Since the final expression is proportional to the total angular momentum

983: 924: 407: 12137:(Thus, an ellipse is the planet's path; ... ) Later on the same page: 10858:

A computationally more convenient form follows by substituting into the

621:

and explained how planetary velocities vary. The three laws state that:

14727: 14615: 13272: 12986: 12448: 8492: 5901:

Kepler's first law states that the orbit is described by the equation:

4685:

on top of the variable signifies differentiation with respect to time.

3686: 3204:

Then, expressing the angular velocity ω in terms of the orbital period

1230: 1030: 1014:

The orbit of every planet is an ellipse with the sun at one of the two

975: 935:

of all planets except Mercury. His first law reflected this discovery.

927:. Kepler's third law was published in 1619. Kepler had believed in the 858: 700: 487: 443: 402: 180: 12307:

is available as: Johannes Kepler with E. J. Aiton, A. M. Duncan, and

11917: 9918:

by solving Kepler's equation below, and then compute the true anomaly

3000:{\displaystyle {\frac {dA}{dt}}={\frac {abn}{2}}={\frac {\pi ab}{T}}.} 2805:{\displaystyle T\cdot {\frac {r^{2}}{2}}{\frac {d\theta }{dt}}=\pi ab} 29: 13689: 13041: 12792:

Kepler's life is summarized on pp. 523–627 and Book Five of his

12570: 12175: 10939:{\displaystyle \tan ^{2}{\frac {x}{2}}={\frac {1-\cos x}{1+\cos x}}.} 7328:

has the same value for all planets in the Solar System. According to

7159: 4630:

is the unit vector whose direction is 90 degrees counterclockwise of

590: 12743:"Memorandum № 1: Keplerian Orbit Elements → Cartesian State Vectors" 12416:

Encyclopedia of the Scientific Revolution: From Copernicus to Newton

12179:

Physics, the Human Adventure: From Copernicus to Einstein and Beyond

12007:, vol. 1 (Paris: Desaint & Saillant, 1764). See, for example: 2038: 14249: 14190: 14013: 12913: 7163: 1323: 1022: 874: 862: 13791: 12950: 11751:... (Bologna (Bononia), (Italy): Victor Benati, 1651), volume 1, 10847:

The result is a usable relationship between the eccentric anomaly

8479:

The procedure for calculating the heliocentric polar coordinates (

6904:

of any planet obeying Kepler's first and second law satisfies the

14207: 14195: 14083: 13594: 12810:: The Great Works of Physics and Astronomy (works by Copernicus, 11670: 7049:

is the unit vector pointing from the Sun towards the planet, and

943: 779: 728: 626: 12483: 4218:

of a planet moving according to Kepler's first and second laws.

89:

are equal, and are swept out in equal times by Planet 1's orbit.

14761: 9041: 12329:

National Earth Science Teachers Association (9 October 2008).

12159:(Of an ellipse is made a planet's orbit ... ). From p. 659: 1099:

Mathematically, an ellipse can be represented by the formula:

13399: 13018: 8452:

is complicated. But Keplerian approximation is the basis for

4249: 208: 12285:(Linz, (Austria): Johann Planck, 1619), book 5, chapter 3, 8874:

and the Cartesian velocity vector can then be calculated as

8448:

These accelerations are not those of Kepler orbits, and the

1149:{\displaystyle r={\frac {p}{1+\varepsilon \,\cos \theta }},} 9865:) is, however, not elementary. Kepler's solution is to use 9150:

distance between center and perihelion, the semimajor axis,

9017:. Because the uniform circular motion was considered to be 6107:{\displaystyle p{\dot {r}}=nab\,\varepsilon \sin(\theta ).} 5951:{\displaystyle {\frac {p}{r}}=1+\varepsilon \cos(\theta ).} 4253: 1935: 652: 14812: 9021:, a deviation from this motion was considered an anomaly. 6866:{\displaystyle a_{\text{r}}=-{\frac {n^{2}a^{3}}{r^{2}}}.} 12937:

Ep. 189: Johannes Kepler and His Laws of Planetary Motion

12382:

Philosophical Transactions of the Royal Society of London

12731:, p. 943 in the translation of I.B. Cohen and A. Whitman 12718:, p. 408 in the translation of I.B. Cohen and A. Whitman 11755:

In the margin beside the relevant paragraph is printed:

11500:

The fourth step is to compute the heliocentric distance

2503:

the planet sweeps out a small triangle having base line

938:

In 1621, Kepler noted that his third law applies to the

33:

Illustration of Kepler's laws with two planetary orbits.

12559:

The Metaphysical Foundations of Modern Physical Science

11891:"The great Martian catastrophe and how Kepler fixed it" 5346:{\displaystyle a_{r}={\ddot {r}}-r{\dot {\theta }}^{2}} 861:, the date the Earth is closest to the Sun, falls on a 12163:( ... the Sun (the other focus of this ellipse) ... ). 10292:, the next step is to calculate the true anomaly 9338:

the direction to the planet as seen from the Sun, the

9053:. The auxiliary circle is an aid to calculation. Line 7186:, the gravitational force that acts on the planet is: 4040: 3834: 2598:{\textstyle dA={\frac {1}{2}}\cdot r\cdot r\,d\theta } 2556: 713:

The speed of the planet in the main orbit is constant.

655:. From this, Kepler inferred that other bodies in the 98: 14777: 11604: 11514: 11397: 10954: 10868: 10624: 10552: 10532: 10474: 10403: 10229: 9932: 9871: 9849:

is computed, the goal is to compute the true anomaly

9800: 9624: 9558: 9478: 9432: 9399: 9374: 9358:) of the planet from the time since perihelion, 9306: 9262: 9211: 9158: 9114: 8978: 8880: 8817: 8785: 8732: 8630: 8600: 8556: 7900: 7784: 7670: 7614: 7576: 7545: 7432: 7401: 7365: 7338: 7314: 7287: 7192: 7137: 7117: 7079: 7055: 7024: 6981: 6915: 6881: 6807: 6771: 6596: 6352: 6325: 6122: 6054: 5966: 5907: 5710: 5683: 5529: 5502: 5448: 5363: 5292: 5207: 4694: 4667: 4636: 4607: 4419: 4384: 4364: 4322: 3779: 3759: 3739: 3715: 3695: 3671: 3500: 3471: 3451: 3431: 3411: 3235: 3226:

and then rearranging, results in Kepler's Third Law:

3210: 3146: 3073: 2931: 2872: 2828: 2746: 2723: 2697: 2615: 2529: 2509: 2486: 2428: 2330: 2243: 2106: 2064:

that an object needs to be pushed to maintain motion.

1947: 1857: 1677: 1551: 1414: 1370:{\displaystyle r_{\max }={\frac {p}{1-\varepsilon }}} 1335: 1301: 1277:{\displaystyle r_{\min }={\frac {p}{1+\varepsilon }}} 1242: 1165: 1108: 791: 12072:

An account of the astronomical discoveries of Kepler

12056:

An account of the astronomical discoveries of Kepler

10517:{\displaystyle a\cos E=a\varepsilon +r\cos \theta .} 9416:

the point on the circle such that the sector areas |

9391:

the projection of the planet to the auxiliary circle

9057:

is perpendicular to the base and through the planet

3030:

Planet orbiting the Sun in a circular orbit (e=0.0)

2475: 901:

An account of the astronomical discoveries of Kepler

710:

The Sun is approximately at the center of the orbit.

92:

The ratio of Planet 1's orbit time to Planet 2's is

12898:B.Surendranath Reddy; animation of Kepler's laws: 4351:{\displaystyle \mathbf {r} =r{\hat {\mathbf {r} }}} 4279:

All bodies in the Solar System attract one another.

12687: 12658: 12452: 12375: 11643: 11576: 11479: 11377: 10938: 10837: 10610: 10538: 10516: 10460: 10259: 10206: 9895: 9818: 9778: 9608: 9538: 9456: 9408: 9383: 9330: 9290: 9246: 9195: 9142: 8984: 8964: 8866: 8791: 8771: 8712: 8606: 8586: 8438: 7881: 7767: 7641: 7592: 7558: 7531: 7407: 7387: 7351: 7320: 7300: 7271: 7143: 7123: 7103: 7062: 7041: 7010: 6967: 6896: 6865: 6793: 6755: 6580: 6338: 6309: 6106: 6040: 5950: 5891: 5696: 5664: 5515: 5485: 5427: 5345: 5274: 5191: 4673: 4653: 4622: 4591: 4401: 4370: 4350: 4067: 3861: 3787: 3765: 3745: 3721: 3701: 3677: 3654: 3483: 3457: 3437: 3417: 3393: 3218: 3194:{\displaystyle mr\omega ^{2}=G{\frac {mM}{r^{2}}}} 3193: 3085: 2999: 2914: 2852: 2804: 2729: 2709: 2681: 2597: 2542: 2515: 2495: 2464: 2412: 2315: 2218: 1969: 1923: 1815: 1635: 1506: 1369: 1307: 1276: 1171: 1148: 982:much later identified a symmetry principle in the 845: 151: 14738:Statal Institute of Higher Education Isaac Newton 12297:(But it is absolutely certain and exact that the 8463: 994:does via rotational symmetry for the second law. 754:(closely linked historically with the concept of 643:is proportional to the cube of the length of the 14861: 12412: 12176:Holton, Gerald James; Brush, Stephen G. (2001). 12083: 9539:{\displaystyle |zsp|={\frac {b}{a}}\cdot |zsx|.} 9247:{\displaystyle b=a{\sqrt {1-\varepsilon ^{2}}},} 7131:is the period, according to Kepler's third law, 7070:is the distance between the planet and the Sun. 4316:point of view consider the vector to the planet 1910: 1897: 1885: 1872: 1790: 1780: 1726: 1705: 1583: 1573: 1450: 1437: 1341: 1248: 1195:is the distance from the Sun to the planet, and 12926:Kepler's Second Law – Java Interactive Tutorial 12532:Brush and Holton - Physics: the Human Adventure 9910:as an intermediate variable, and first compute 6590:Substituting the equation of the ellipse gives 3062:Planet orbiting the Sun in an orbit with e=0.8 3052:Planet orbiting the Sun in an orbit with e=0.2 3040:Planet orbiting the Sun in an orbit with e=0.5 3010:Orbits of planets with varying eccentricities. 1045:) for ellipse. Also shown are: semi-major axis 707:The planetary orbit is a circle with epicycles. 676:A more precise historical approach is found in 12571:Gerald James Holton, Stephen G. Brush (2001). 12534:. Princeton University Press. pp. 42–43. 12277: 12275: 12077: 8999:The important special case of circular orbit, 7177: 4623:{\displaystyle {\hat {\boldsymbol {\theta }}}} 3799:, the average distance from earth to the sun. 1218: < 1 ; in the limiting case 14029: 13807: 13002: 12944:Johannes Kepler: The Laws of Planetary Motion 12455:The Scientific Revolution in National Context 12161:" ... Sole (Foco altero huius ellipsis) ... " 11969: 11967: 10318:With reference to the Sun (with coordinates ( 7894:In the three-body case the accelerations are 7422:The acceleration of Solar System body number 2691:The area enclosed by the elliptical orbit is 2465:{\displaystyle ({\vec {r}}\times {\vec {p}})} 570: 12363:] (in Latin). London, England: Leybourn. 12291:"Sed res est certissima exactissimaque quod 9790:, is proportional to time since perihelion, 9196:{\displaystyle \varepsilon ={|cs| \over a},} 9024:The proof of this procedure is shown below. 4252:acting on a planet to be the product of its 3055: 3045: 3033: 3023: 2094:vectors, for some short piece of the orbit, 12879:Mathematical Methods of Classical Mechanics 12529: 12484:Victor Guillemin; Shlomo Sternberg (2006). 12272: 12090:. Princeton University Press. p. 170. 11884: 11882: 8811:) can now be written as a Cartesian vector 4211:Philosophiæ Naturalis Principia Mathematica 4017:For comparison, here are modern estimates: 691: 14036: 14022: 13814: 13800: 13776: 13009: 12995: 12504: 12447: 12331:"Data Table for Planets and Dwarf Planets" 11964: 11644:{\displaystyle r=a(1-\varepsilon \cos E).} 9609:{\displaystyle |zcy|={\frac {a^{2}M}{2}}.} 8772:{\displaystyle r=a(1-\varepsilon \cos E),} 7642:{\displaystyle {\hat {\mathbf {r} }}_{ij}} 4264:Every planet is attracted towards the Sun. 3367: 3363: 3304: 3300: 577: 563: 12976:"Kepler's Three Laws of Planetary Motion" 12768:"Equation of Time – Problem in Astronomy" 12633: 12623: 12171: 12169: 11888: 7059: 6211: 6168: 6082: 6025: 5486:{\displaystyle r^{2}{\dot {\theta }}=nab} 4200: 2902: 2883: 2588: 2533: 1963: 1808: 1130: 13016: 12577:. Rutgers University Press. p. 45. 12490:. American Mathematical Soc. p. 5. 12373: 12354: 11879: 9040: 8487:) of a planet as a function of the time 6875:This means that the acceleration vector 3991: 3973:Upon finding this pattern Kepler wrote: 2915:{\displaystyle r^{2}\,d\theta =abn\,dt.} 2037: 1029: 1021: 888:Biographical Encyclopedia of Astronomers 28: 12924:David McNamara and Gianfranco Vidali, " 12806:), is reprinted on pp. 635–732 of 12786: 12564: 12530:Holton, Gerald; Brush, Stephen (2001). 10260:{\displaystyle M=E-\varepsilon \sin E.} 9837: 8587:{\displaystyle M=E-\varepsilon \sin E,} 7388:{\displaystyle \alpha =Gm_{\text{Sun}}} 5263: 5238: 5173: 5029: 4997: 4967: 4823: 4611: 4543: 4515: 4493: 4304:Polar coordinate § Vector calculus 4226:of the acceleration is towards the Sun. 3773:is the elliptical semi-major axis, and 3111:The ratio of the square of an object's 14: 14862: 13655:Transposition, docking, and extraction 12848:(2nd ed.). New York: John Wiley. 12841: 12765: 12685: 12656: 12597: 12459:. Cambridge University Press. p. 12166: 12135:"Ergo ellipsis est Planetæ iter; ... " 11973: 10546:and inserting from Kepler's first law 10288:Having computed the eccentric anomaly 4297: 14288:Newton's law of universal gravitation 14017: 13795: 12990: 12220:His "distance law" is presented in: 12157:"Ellipsin fieri orbitam planetæ ... " 7042:{\displaystyle {\hat {\mathbf {r} }}} 5960:Differentiating with respect to time 5437: 4654:{\displaystyle {\hat {\mathbf {r} }}} 4402:{\displaystyle {\hat {\mathbf {r} }}} 4273:Newton's law of universal gravitation 2051: 954:in a book from 1664, but by 1670 his 14446:Newton's theorem of revolving orbits 14043: 11939:Eléments de la philosophie de Newton 11808: 11806: 11781: 11779: 11703: 11594:the final equation for the distance 9298:the distance between Sun and planet. 7104:{\displaystyle n={\frac {2\pi }{T}}} 6897:{\displaystyle \mathbf {\ddot {r}} } 2853:{\displaystyle n={\frac {2\pi }{T}}} 879:Eléments de la philosophie de Newton 152:{\textstyle ({a_{1}}/{a_{2}})^{3/2}} 14394:Leibniz–Newton calculus controversy 14135:standing on the shoulders of giants 13821: 12971:by David P. Stern (10 October 2016) 12867:, Cambridge University Press 1999, 12740: 12303:An English translation of Kepler's 11941:(London: 1738). See, for example: 9331:{\displaystyle \theta =\angle zsp,} 4308:Mechanics of planar particle motion 4068:{\textstyle {\frac {a^{3}}{T^{2}}}} 3862:{\textstyle {\frac {R^{3}}{T^{2}}}} 37:The orbits are ellipses, with foci 24: 12510:Bryant, Jeff; Pavlyk, Oleksandr. " 11696:Specific relative angular momentum 10389: 9878: 9439: 9313: 9080:the Sun (at one focus of ellipse); 9027: 8722:Compute the heliocentric distance 7011:{\displaystyle \alpha =n^{2}a^{3}} 4378:is the distance to the planet and 3137:equal to the gravitational force: 25: 14911: 13842:Kepler's laws of planetary motion 13715:Kepler's laws of planetary motion 12892: 12242:His "area law" is presented in: 11803: 11776: 11586:Using the relation above between 9618:The area swept since perihelion, 9069:The Keplerian problem assumes an 7301:{\displaystyle m_{\text{planet}}} 2476:In terms of elliptical parameters 2030:Kepler's second law states that: 595:Kepler's laws of planetary motion 349:Kepler's laws of planetary motion 14847: 14835: 14823: 14811: 14799: 14787: 14723:Isaac Newton Group of Telescopes 13913:Epitome Astronomiae Copernicanae 13775: 13710:Interplanetary Transport Network 13590:Collision avoidance (spacecraft) 12694:. New York: Dover. p. 286. 12665:. New York: Dover. p. 304. 12374:Mercator, Nic. (25 March 1670). 12259:Epitome Astronomiae Copernicanae 12149:Epitome Astronomiae Copernicanae 11718:Epitome Astronomiae Copernicanae 9472:The sector areas are related by 8994:standard gravitational parameter 8882: 8819: 8803:The position polar coordinates ( 8406: 8331: 8266: 8263: 8229: 8154: 8089: 8086: 8052: 7977: 7912: 7909: 7853: 7792: 7789: 7739: 7678: 7675: 7620: 7510: 7440: 7437: 7426:is, according to Newton's laws: 7259: 7217: 7214: 7194: 7029: 6955: 6922: 6919: 6888: 6885: 5212: 5102: 4912: 4881: 4843: 4792: 4758: 4736: 4703: 4681:is the polar angle, and where a 4641: 4579: 4460: 4430: 4389: 4338: 4324: 3107:Kepler's third law states that: 1980:The special case of a circle is 1295:= 270° the distance is equal to 1214:For an ellipse 0 < 1057:; center of ellipse and its two 1034:Heliocentric coordinate system ( 1010:Kepler's first law states that: 992:conservation of angular momentum 726:a circle with epicycles, but an 685:Epitome Astronomiae Copernicanae 179: 14743:Newton International Fellowship 14424:generalized Gauss–Newton method 14337:Newton's method in optimization 13675:Astronomical coordinate systems 13429:Longitude of the ascending node 12759: 12734: 12721: 12708: 12679: 12650: 12591: 12548: 12523: 12487:Variations on a Theme by Kepler 12477: 12441: 12406: 12367: 12348: 12322: 12301:of their mean distances, ... ") 12250:On the top p. 294, it reads: 12203: 12113: 12104: 12064: 11740: 11709: 8495:, is the following five steps: 4505: 4012:² is constant (green line) 2226:for a small piece of the orbit 883:Elements of Newton's Philosophy 868: 629:with the Sun at one of the two 13748:Retrograde and prograde motion 12520:. Retrieved December 27, 2009. 12517:Wolfram Demonstrations Project 12048: 11997: 11931: 11855: 11830: 11635: 11614: 11539: 11518: 11492: 11451: 11439: 11410: 11398: 11240: 11222: 11216: 11195: 11190: 11172: 11166: 11145: 10724: 10703: 10461:{\displaystyle |cd|=|cs|+|sd|} 10454: 10443: 10435: 10424: 10416: 10405: 10102: 10091: 10083: 10072: 10057: 10043: 10022: 10008: 10000: 9986: 9978: 9964: 9952: 9938: 9710: 9696: 9675: 9661: 9640: 9626: 9574: 9560: 9529: 9515: 9494: 9480: 9346:The problem is to compute the 9281: 9270: 9180: 9169: 9133: 9122: 8763: 8742: 8684: 8672: 8643: 8631: 8550:by solving Kepler's equation: 8464:Position as a function of time 8410: 8335: 8233: 8158: 8056: 7981: 7857: 7743: 7624: 7514: 7352:{\displaystyle m_{\text{Sun}}} 7308:is the mass of the planet and 7263: 7033: 6959: 6801:gives the simple final result 6554: 6548: 6430: 6424: 6301: 6295: 6208: 6202: 6165: 6159: 6098: 6092: 6022: 6016: 5942: 5936: 5442:Kepler's second law says that 5266: 5241: 5176: 5106: 5032: 5000: 4970: 4916: 4885: 4826: 4796: 4762: 4740: 4645: 4614: 4583: 4546: 4518: 4496: 4464: 4434: 4393: 4342: 3546: 3534: 3364: 3301: 2459: 2453: 2438: 2429: 2404: 2398: 2383: 2374: 2307: 2301: 2286: 2277: 2213: 2201: 2186: 2177: 2161: 2155: 2135: 2126: 970:on Kepler had been published. 699:'s laws improved the model of 132: 99: 13: 1: 12625:10.1016/j.heliyon.2019.e01274 11818:NASA Solar System Exploration 11791:hyperphysics.phy-astr.gsu.edu 11769: 11387:Multiplying by 1 + 10353:True anomaly would be arctan( 9906:as seen from the centre, the 9896:{\displaystyle E=\angle zcx,} 9457:{\displaystyle M=\angle zcy,} 8540:is the time since perihelion. 7649:is the unit vector from body 7600:is the distance between body 5496:The transversal acceleration 4413:pointing towards the planet. 2819:of the planet around the Sun 2025: 14364:Newton's theorem about ovals 13695:Equatorial coordinate system 12969:From Stargazers to Starships 12574:Physics, the Human Adventure 12295:mediarum distantiarum, ... " 12289:From the bottom of p. 189: 8607:{\displaystyle \varepsilon } 7154:The inverse square law is a 6339:{\displaystyle a_{\text{r}}} 5697:{\displaystyle a_{\text{r}}} 3425:, with the semi-major axis, 3102: 1005: 997: 671:law of universal gravitation 625:The orbit of a planet is an 65:for Planet 2. The Sun is at 7: 14733:Sir Isaac Newton Sixth Form 14389:Corpuscular theory of light 14315:Schrödinger–Newton equation 12355:Mercator, Nicolaus (1664). 12087:Kepler's Physical Astronomy 12074:(London: J. Mawman, 1804). 12058:(London: J Mawman, 1804), 11974:Wilson, Curtis (May 1994). 11653: 9922:from the eccentric anomaly 7178:Newton's law of gravitation 5516:{\displaystyle a_{\theta }} 3811:Data used by Kepler (1618) 3788:{\displaystyle {\text{AU}}} 3709:is the mass of the planet, 2543:{\displaystyle r\,d\theta } 1970:{\displaystyle A=\pi ab\,.} 895:. It was the exposition of 703:. According to Copernicus: 503:Tsiolkovsky rocket equation 10: 14916: 14142:Notes on the Jewish Temple 13447:Longitude of the periapsis 12931:Cain, Gay (May 10, 2010), 12808:On the Shoulders of Giants 12419:. Routledge. p. 603. 11976:"Kepler's Laws, So-Called" 11814:"Orbits and Kepler's Laws" 10851:and the true anomaly 10397:Note from the figure that 9034: 6116:Differentiating once more 4301: 4256:and the acceleration (see 3753:is the orbital period and 1233:, the distance is minimum 1061:marked by large dots. For 957:Philosophical Transactions 918: 472:Engineering and efficiency 291:Bi-elliptic transfer orbit 14885:Eponymous laws of physics 14751: 14688: 14643: 14566: 14508: 14263: 14183: 14118: 14051: 13947: 13880: 13829: 13771: 13758:Specific angular momentum 13663: 13575: 13519: 13455: 13408: 13348: 13339: 13235: 13145: 13034: 13025: 12770:. Acta Physica Polonica A 12413:Wilbur Applebaum (2000). 12084:Bruce Stephenson (1994). 11842:earthobservatory.nasa.gov 11691:Laplace–Runge–Lenz vector 9095:the center of the ellipse 8624:by solving the equation: 3066: 986:of planetary motion (the 639:The square of a planet's 14293:post-Newtonian expansion 14173:Corruptions of Scripture 14165:Ancient Kingdoms Amended 13889:Mysterium Cosmographicum 12949:Solar System Simulator ( 12881:, Ch. 2. Springer 1989, 12313:The Harmony of the World 11889:Gingerich, Owen (2011). 9926:. Here are the details. 6794:{\displaystyle b^{2}=pa} 6319:The radial acceleration 5677:The radial acceleration 5355:transversal acceleration 3802: 1832:coefficient of variation 770:makes the time from the 745:of the elliptical orbit. 692:Comparison to Copernicus 498:Propellant mass fraction 397:Gravitational influences 14483:Absolute space and time 14347:truncated Newton method 14320:Newton's laws of motion 14283:Newton's law of cooling 13753:Specific orbital energy 12335:Windows to the Universe 12125:See: Johannes Kepler, 11508:by Kepler's first law: 9291:{\displaystyle r=|sp|,} 9143:{\displaystyle a=|cz|,} 8470:transcendental equation 7321:{\displaystyle \alpha } 7144:{\displaystyle \alpha } 4674:{\displaystyle \theta } 4258:Newton's laws of motion 3996:Log-log plot of period 3966:Kepler became aware of 739:at the center but at a 722:The planetary orbit is 369:Specific orbital energy 14890:Equations of astronomy 14718:Isaac Newton Telescope 14708:Isaac Newton Institute 14478:Newton–Puiseux theorem 14473:Parallelogram of force 14461:kissing number problem 14451:Newton–Euler equations 14354:Gauss–Newton algorithm 14303:gravitational constant 13983:Kepler space telescope 13165:Geostationary transfer 12842:Meriam, J.L. (1971) . 12804:harmonies of the world 12598:Vijaya, G. K. (2019). 12394:10.1098/rstl.1670.0018 11725:. From pp. 554–555: 11645: 11578: 11504:from the true anomaly 11481: 11379: 10940: 10860:trigonometric identity 10839: 10612: 10540: 10518: 10462: 10261: 10208: 9897: 9845:When the mean anomaly 9820: 9780: 9610: 9540: 9458: 9410: 9385: 9332: 9292: 9248: 9197: 9144: 9066: 9003: = 0, gives 8986: 8966: 8868: 8799:is the semimajor axis. 8793: 8773: 8714: 8608: 8588: 8440: 7883: 7769: 7643: 7594: 7593:{\displaystyle r_{ij}} 7560: 7533: 7417:gravitational constant 7409: 7389: 7353: 7322: 7302: 7273: 7145: 7125: 7105: 7064: 7043: 7012: 6969: 6898: 6867: 6795: 6757: 6582: 6340: 6311: 6108: 6042: 5952: 5893: 5698: 5666: 5517: 5487: 5429: 5347: 5276: 5193: 4675: 4655: 4624: 4593: 4403: 4372: 4352: 4201:Planetary acceleration 4069: 4013: 3990: 3985:Harmonies of the World 3863: 3789: 3767: 3747: 3731:gravitational constant 3723: 3703: 3679: 3656: 3485: 3459: 3439: 3419: 3395: 3220: 3195: 3117: 3087: 3060: 3050: 3038: 3028: 3001: 2916: 2854: 2806: 2731: 2711: 2710:{\displaystyle \pi ab} 2683: 2599: 2544: 2517: 2497: 2466: 2414: 2317: 2220: 2044: 2036: 1971: 1925: 1817: 1646:The semi-latus rectum 1637: 1508: 1371: 1309: 1278: 1173: 1150: 1096: 1053:and semi-latus rectum 1027: 1020: 847: 286:Hohmann transfer orbit 163: 153: 14880:Copernican Revolution 14672:Isaac Newton Gargoyle 14582: (nephew-in-law) 14558:Copernican Revolution 14553:Scientific Revolution 14414:Newton–Cotes formulas 14278:Newton's inequalities 14255:Structural coloration 13956:Die Harmonie der Welt 13738:Orbital state vectors 13680:Characteristic energy 13650:Trans-lunar injection 13438:Argument of periapsis 13115:Prograde / Retrograde 13076:Hyperbolic trajectory 12865:Solar System Dynamics 11863:"Nicolaus Copernicus" 11753:page 492 Scholia III. 11646: 11579: 11482: 11380: 10941: 10840: 10613: 10541: 10519: 10463: 10262: 10209: 9898: 9821: 9819:{\displaystyle M=nt,} 9781: 9611: 9541: 9459: 9411: 9386: 9333: 9293: 9249: 9198: 9145: 9073:and the four points: 9044: 8987: 8967: 8869: 8794: 8774: 8715: 8609: 8589: 8441: 7884: 7770: 7644: 7595: 7561: 7559:{\displaystyle m_{j}} 7534: 7410: 7390: 7354: 7323: 7303: 7274: 7156:differential equation 7146: 7126: 7106: 7065: 7044: 7013: 6970: 6899: 6868: 6796: 6758: 6583: 6341: 6312: 6109: 6043: 5953: 5894: 5699: 5667: 5518: 5488: 5430: 5348: 5277: 5194: 4676: 4656: 4625: 4594: 4404: 4373: 4353: 4070: 4029:Semi-major axis (AU) 3995: 3975: 3864: 3790: 3768: 3748: 3724: 3704: 3680: 3657: 3486: 3460: 3440: 3420: 3396: 3221: 3196: 3109: 3088: 3059: 3049: 3037: 3027: 3002: 2917: 2855: 2807: 2732: 2712: 2684: 2600: 2545: 2518: 2498: 2467: 2415: 2318: 2230:and time to cover it 2221: 2041: 2032: 1972: 1926: 1818: 1638: 1509: 1372: 1310: 1279: 1174: 1151: 1033: 1025: 1012: 848: 482:Preflight engineering 214:Argument of periapsis 154: 32: 14679:Astronomers Monument 14369:Newton–Pepys problem 14342:Apollonius's problem 14310:Newton–Cartan theory 14223:Newton–Okounkov body 14156:hypotheses non fingo 14145: (c. 1680) 13997:Astronomers Monument 13585:Bi-elliptic transfer 13105:Parabolic trajectory 12863:Murray and Dermott, 12787:General bibliography 12686:Caspar, Max (1993). 12657:Caspar, Max (1993). 11735:The World of Jupiter 11602: 11512: 11395: 10952: 10866: 10622: 10550: 10530: 10472: 10401: 10269:This equation gives 10227: 9930: 9869: 9798: 9622: 9556: 9476: 9430: 9397: 9372: 9304: 9260: 9209: 9156: 9112: 8985:{\displaystyle \mu } 8976: 8878: 8815: 8783: 8730: 8628: 8614:is the eccentricity. 8598: 8554: 7898: 7782: 7668: 7612: 7574: 7566:is the mass of body 7543: 7430: 7399: 7363: 7336: 7312: 7285: 7190: 7135: 7115: 7077: 7053: 7022: 6979: 6913: 6879: 6805: 6769: 6594: 6350: 6323: 6120: 6052: 5964: 5905: 5708: 5681: 5527: 5500: 5446: 5361: 5290: 5205: 4692: 4665: 4634: 4605: 4417: 4382: 4362: 4320: 4038: 3832: 3777: 3757: 3737: 3713: 3693: 3669: 3498: 3469: 3449: 3429: 3409: 3233: 3208: 3144: 3124:music of the spheres 3071: 2929: 2870: 2826: 2744: 2721: 2695: 2613: 2554: 2527: 2507: 2484: 2426: 2328: 2241: 2104: 2062:Aristotelian physics 1945: 1855: 1675: 1549: 1412: 1333: 1299: 1240: 1163: 1106: 940:four brightest moons 789: 538:Propulsive maneuvers 96: 14488:Luminiferous aether 14436:Newton's identities 14409:Newton's cannonball 14384:Classical mechanics 14374:Newtonian potential 14235:Newtonian telescope 13872:Keplerian telescope 13625:Low-energy transfer 12965:Kepler and His Laws 12911:Crowell, Benjamin, 12616:2019Heliy...501274V 12512:Kepler's Second Law 12425:2000esrc.book.....A 12265:. From page 668: 11910:2011PhT....64i..50G 9838:Eccentric anomaly, 9254:the semiminor axis, 8456:calculations. (See 8400: 8325: 8223: 8148: 8046: 7971: 7847: 7733: 7504: 7184:Newton's second law 7063:{\displaystyle r\,} 7018:is a constant, and 5284:radial acceleration 4298:Acceleration vector 4248:Newton defined the 4022: 4000:vs semi-major axis 3812: 3484:{\displaystyle M+m} 3219:{\displaystyle {T}} 3086:{\displaystyle T=1} 3011: 1738: 1717: 909:inductive reasoning 615:Nicolaus Copernicus 611:heliocentric theory 515:Efficiency measures 418:Sphere of influence 387:Celestial mechanics 169:Part of a series on 14713:Isaac Newton Medal 14518: (birthplace) 14332:Newtonian dynamics 14230:Newton's reflector 13867:Kepler's Supernova 13620:Inclination change 13268:Distant retrograde 12957:2018-12-13 at the 12951:Interactive Applet 12905:2013-10-06 at the 12766:Müller, M (1995). 12235:, Kepler states: 11641: 11574: 11477: 11375: 11373: 10936: 10835: 10608: 10536: 10514: 10458: 10257: 10204: 10202: 9893: 9816: 9776: 9606: 9536: 9454: 9409:{\displaystyle y,} 9406: 9384:{\displaystyle x,} 9381: 9328: 9288: 9244: 9193: 9140: 9067: 8982: 8962: 8864: 8789: 8769: 8710: 8604: 8584: 8450:three-body problem 8436: 8434: 8373: 8298: 8196: 8121: 8019: 7944: 7879: 7820: 7765: 7706: 7639: 7590: 7556: 7529: 7484: 7473: 7405: 7385: 7349: 7330:Newton's third law 7318: 7298: 7269: 7141: 7121: 7101: 7073:Since mean motion 7060: 7039: 7008: 6965: 6907:inverse square law 6894: 6863: 6791: 6753: 6578: 6336: 6307: 6104: 6038: 5948: 5889: 5694: 5662: 5513: 5483: 5438:Inverse square law 5425: 5343: 5272: 5189: 5187: 4671: 4651: 4620: 4589: 4399: 4368: 4348: 4235:inverse square law 4065: 4020: 4014: 3859: 3810: 3785: 3763: 3743: 3719: 3699: 3675: 3652: 3481: 3455: 3435: 3415: 3391: 3216: 3191: 3083: 3061: 3051: 3039: 3029: 3009: 2997: 2912: 2850: 2802: 2727: 2707: 2679: 2605:, so the constant 2595: 2540: 2513: 2496:{\displaystyle dt} 2493: 2462: 2410: 2313: 2216: 2052:History and proofs 2045: 1984:= 0, resulting in 1967: 1938:of the ellipse is 1921: 1813: 1811: 1721: 1700: 1633: 1631: 1504: 1502: 1367: 1305: 1274: 1169: 1146: 1097: 1049:, semi-minor axis 1028: 843: 768:orbit of the Earth 334:Dynamical friction 164: 149: 51:for Planet 1, and 18:Kepler's laws 14775: 14774: 14667: (sculpture) 14634:Abraham de Moivre 14588: (professor) 14516:Woolsthorpe Manor 14468:Newton's quotient 14441:Newton polynomial 14399:Newton's notation 14130: (1661–1665) 14011: 14010: 14003:List of namesakes 13929:Rudolphine Tables 13857:Kepler's equation 13837:Kepler conjecture 13830:Scientific career 13789: 13788: 13763:Two-line elements 13571: 13570: 13493:Eccentric anomaly 13335: 13334: 13202:Orbit of the Moon 13061:Highly elliptical 12855:978-0-471-59601-1 12584:978-0-8135-2908-0 12497:978-0-8218-4184-6 12470:978-0-521-39699-8 12434:978-1-135-58255-5 12388:(57): 1168–1175. 12281:Johannes Kepler, 12189:978-0-8135-2908-0 12097:978-0-691-03652-6 11918:10.1063/PT.3.1259 11704:Explanatory notes 11686:Kepler's equation 11475: 11434: 11391:gives the result 11366: 11343: 11314: 11273: 11244: 11130: 11127: 11078: 11025: 10980: 10931: 10890: 10830: 10786: 10683: 10606: 10561: 10539:{\displaystyle a} 10273:as a function of 10222:Kepler's equation 10198: 10165: 10136: 10110: 9914:as a function of 9908:eccentric anomaly 9771: 9750: 9725: 9690: 9655: 9601: 9509: 9348:polar coordinates 9239: 9203:the eccentricity, 9188: 8944: 8902: 8898: 8792:{\displaystyle a} 8708: 8667: 8545:eccentric anomaly 8474:Kepler's equation 8428: 8420: 8413: 8389: 8381: 8370: 8353: 8345: 8338: 8314: 8306: 8295: 8275: 8269: 8251: 8243: 8236: 8212: 8204: 8193: 8176: 8168: 8161: 8137: 8129: 8118: 8098: 8092: 8074: 8066: 8059: 8035: 8027: 8016: 7999: 7991: 7984: 7960: 7952: 7941: 7921: 7915: 7875: 7867: 7860: 7836: 7828: 7817: 7801: 7795: 7761: 7753: 7746: 7722: 7714: 7703: 7687: 7681: 7627: 7517: 7458: 7443: 7408:{\displaystyle G} 7382: 7346: 7295: 7266: 7236: 7220: 7208: 7124:{\displaystyle T} 7099: 7036: 6962: 6949: 6925: 6891: 6858: 6815: 6738: 6691: 6672: 6657: 6607: 6568: 6532: 6483: 6413: 6363: 6333: 6284: 6235: 6178: 6135: 6067: 6035: 5999: 5985: 5916: 5884: 5834: 5810: 5773: 5752: 5733: 5718: 5691: 5632: 5614: 5602: 5577: 5560: 5468: 5419: 5407: 5389: 5334: 5315: 5269: 5244: 5219: 5179: 5162: 5150: 5132: 5109: 5084: 5065: 5040: 5035: 5021: 5003: 4991: 4973: 4961: 4949: 4924: 4919: 4903: 4888: 4874: 4850: 4829: 4817: 4799: 4785: 4770: 4765: 4743: 4729: 4710: 4648: 4617: 4586: 4572: 4554: 4549: 4533: 4521: 4499: 4487: 4472: 4467: 4449: 4437: 4396: 4371:{\displaystyle r} 4345: 4198: 4197: 4063: 3964: 3963: 3857: 3797:astronomical unit 3783: 3766:{\displaystyle a} 3746:{\displaystyle T} 3722:{\displaystyle G} 3702:{\displaystyle m} 3678:{\displaystyle M} 3650: 3649: is constant 3645: 3637: 3625: 3595: 3565: 3523: 3458:{\displaystyle M} 3438:{\displaystyle a} 3418:{\displaystyle r} 3347: 3298: 3260: 3189: 3135:centripetal force 3100: 3099: 2992: 2971: 2950: 2848: 2788: 2768: 2730:{\displaystyle T} 2674: 2654: 2634: 2574: 2516:{\displaystyle r} 2456: 2441: 2401: 2386: 2372: 2362: 2349: 2304: 2289: 2275: 2262: 2204: 2189: 2175: 2158: 2138: 2124: 1916: 1826:The eccentricity 1743: 1627: 1626: 1588: 1498: 1459: 1365: 1308:{\displaystyle p} 1272: 1209:polar coordinates 1181:semi-latus rectum 1172:{\displaystyle p} 1141: 968:Gottfried Leibniz 952:Nicolaus Mercator 948:Godefroy Wendelin 893:Joseph de Lalande 832: 806: 776:September equinox 619:elliptical orbits 587: 586: 437:Lagrangian points 374:Vis-viva equation 344:Kepler's equation 191:Orbital mechanics 75:The shaded areas 16:(Redirected from 14907: 14852: 14851: 14850: 14840: 14839: 14838: 14828: 14827: 14826: 14816: 14815: 14804: 14803: 14802: 14792: 14791: 14783: 14763: 14658: (monotype) 14622:William Stukeley 14618: (disciple) 14598:Benjamin Pulleyn 14574:Catherine Barton 14493:Newtonian series 14404:Rotating spheres 14150:General Scholium 14045:Sir Isaac Newton 14038: 14031: 14024: 14015: 14014: 13963:Katharina Kepler 13921:Harmonices Mundi 13862:Kepler polyhedra 13816: 13809: 13802: 13793: 13792: 13779: 13778: 13720:Lagrangian point 13615:Hohmann transfer 13560: 13546: 13537: 13528: 13508: 13499: 13490: 13481: 13477: 13473: 13464: 13444: 13435: 13426: 13417: 13397: 13393: 13384: 13375: 13366: 13346: 13345: 13315:Heliosynchronous 13264:Lagrange points 13217:Transatmospheric 13032: 13031: 13011: 13004: 12997: 12988: 12987: 12977: 12914:Light and Matter 12859: 12780: 12779: 12777: 12775: 12763: 12757: 12756: 12754: 12752: 12747: 12738: 12732: 12725: 12719: 12712: 12706: 12705: 12693: 12683: 12677: 12676: 12664: 12654: 12648: 12647: 12637: 12627: 12595: 12589: 12588: 12568: 12562: 12552: 12546: 12545: 12527: 12521: 12508: 12502: 12501: 12481: 12475: 12474: 12458: 12445: 12439: 12438: 12410: 12404: 12397: 12379: 12371: 12365: 12364: 12352: 12346: 12345: 12343: 12341: 12326: 12320: 12305:Harmonices Mundi 12283:Harmonices Mundi 12279: 12270: 12207: 12201: 12200: 12198: 12196: 12173: 12164: 12117: 12111: 12108: 12102: 12101: 12081: 12075: 12068: 12062: 12052: 12046: 12001: 11995: 11994: 11992: 11990: 11980: 11971: 11962: 11935: 11929: 11928: 11926: 11924: 11895: 11886: 11877: 11876: 11874: 11873: 11859: 11853: 11852: 11850: 11849: 11834: 11828: 11827: 11825: 11824: 11810: 11801: 11800: 11798: 11797: 11783: 11764: 11749:Almagestum novum 11744: 11738: 11713: 11650: 11648: 11647: 11642: 11583: 11581: 11580: 11575: 11573: 11569: 11568: 11567: 11486: 11484: 11483: 11478: 11476: 11468: 11463: 11462: 11435: 11427: 11422: 11421: 11384: 11382: 11381: 11376: 11374: 11367: 11359: 11354: 11353: 11344: 11342: 11331: 11320: 11315: 11313: 11296: 11279: 11274: 11272: 11261: 11250: 11245: 11243: 11193: 11143: 11135: 11131: 11129: 11128: 11126: 11106: 11089: 11080: 11079: 11077: 11057: 11040: 11031: 11026: 11024: 11007: 10990: 10981: 10973: 10968: 10967: 10945: 10943: 10942: 10937: 10932: 10930: 10913: 10896: 10891: 10883: 10878: 10877: 10844: 10842: 10841: 10836: 10831: 10829: 10809: 10792: 10787: 10785: 10765: 10755: 10751: 10750: 10749: 10698: 10684: 10682: 10662: 10661: 10660: 10644: 10617: 10615: 10614: 10609: 10607: 10605: 10585: 10584: 10583: 10567: 10562: 10554: 10545: 10543: 10542: 10537: 10523: 10521: 10520: 10515: 10467: 10465: 10464: 10459: 10457: 10446: 10438: 10427: 10419: 10408: 10385: 10384: 10369:), magnitude of 10266: 10264: 10263: 10258: 10213: 10211: 10210: 10205: 10203: 10199: 10194: 10171: 10166: 10161: 10157: 10156: 10146: 10137: 10132: 10128: 10127: 10117: 10111: 10106: 10105: 10094: 10086: 10075: 10069: 10060: 10046: 10025: 10011: 10003: 9989: 9981: 9967: 9955: 9941: 9902: 9900: 9899: 9894: 9825: 9823: 9822: 9817: 9785: 9783: 9782: 9777: 9772: 9767: 9756: 9751: 9746: 9742: 9741: 9731: 9726: 9718: 9713: 9699: 9691: 9683: 9678: 9664: 9656: 9648: 9643: 9629: 9615: 9613: 9612: 9607: 9602: 9597: 9593: 9592: 9582: 9577: 9563: 9545: 9543: 9542: 9537: 9532: 9518: 9510: 9502: 9497: 9483: 9463: 9461: 9460: 9455: 9415: 9413: 9412: 9407: 9390: 9388: 9387: 9382: 9337: 9335: 9334: 9329: 9297: 9295: 9294: 9289: 9284: 9273: 9253: 9251: 9250: 9245: 9240: 9238: 9237: 9222: 9202: 9200: 9199: 9194: 9189: 9184: 9183: 9172: 9166: 9149: 9147: 9146: 9141: 9136: 9125: 9071:elliptical orbit 9016: 8991: 8989: 8988: 8983: 8971: 8969: 8968: 8963: 8961: 8957: 8956: 8945: 8943: 8942: 8927: 8922: 8903: 8891: 8890: 8885: 8873: 8871: 8870: 8865: 8863: 8859: 8858: 8844: 8822: 8798: 8796: 8795: 8790: 8778: 8776: 8775: 8770: 8719: 8717: 8716: 8711: 8709: 8701: 8696: 8695: 8668: 8660: 8655: 8654: 8613: 8611: 8610: 8605: 8593: 8591: 8590: 8585: 8535: 8515: 8445: 8443: 8442: 8437: 8435: 8431: 8430: 8429: 8426: 8421: 8418: 8415: 8414: 8409: 8404: 8399: 8391: 8390: 8387: 8382: 8379: 8372: 8371: 8368: 8356: 8355: 8354: 8351: 8346: 8343: 8340: 8339: 8334: 8329: 8324: 8316: 8315: 8312: 8307: 8304: 8297: 8296: 8293: 8277: 8276: 8273: 8271: 8270: 8262: 8254: 8253: 8252: 8249: 8244: 8241: 8238: 8237: 8232: 8227: 8222: 8214: 8213: 8210: 8205: 8202: 8195: 8194: 8191: 8179: 8178: 8177: 8174: 8169: 8166: 8163: 8162: 8157: 8152: 8147: 8139: 8138: 8135: 8130: 8127: 8120: 8119: 8116: 8100: 8099: 8096: 8094: 8093: 8085: 8077: 8076: 8075: 8072: 8067: 8064: 8061: 8060: 8055: 8050: 8045: 8037: 8036: 8033: 8028: 8025: 8018: 8017: 8014: 8002: 8001: 8000: 7997: 7992: 7989: 7986: 7985: 7980: 7975: 7970: 7962: 7961: 7958: 7953: 7950: 7943: 7942: 7939: 7923: 7922: 7919: 7917: 7916: 7908: 7888: 7886: 7885: 7880: 7878: 7877: 7876: 7873: 7868: 7865: 7862: 7861: 7856: 7851: 7846: 7838: 7837: 7834: 7829: 7826: 7819: 7818: 7815: 7803: 7802: 7799: 7797: 7796: 7788: 7774: 7772: 7771: 7766: 7764: 7763: 7762: 7759: 7754: 7751: 7748: 7747: 7742: 7737: 7732: 7724: 7723: 7720: 7715: 7712: 7705: 7704: 7701: 7689: 7688: 7685: 7683: 7682: 7674: 7648: 7646: 7645: 7640: 7638: 7637: 7629: 7628: 7623: 7618: 7599: 7597: 7596: 7591: 7589: 7588: 7565: 7563: 7562: 7557: 7555: 7554: 7538: 7536: 7535: 7530: 7528: 7527: 7519: 7518: 7513: 7508: 7503: 7495: 7483: 7482: 7472: 7451: 7450: 7445: 7444: 7436: 7414: 7412: 7411: 7406: 7394: 7392: 7391: 7386: 7384: 7383: 7380: 7358: 7356: 7355: 7350: 7348: 7347: 7344: 7327: 7325: 7324: 7319: 7307: 7305: 7304: 7299: 7297: 7296: 7293: 7278: 7276: 7275: 7270: 7268: 7267: 7262: 7257: 7254: 7253: 7238: 7237: 7234: 7222: 7221: 7213: 7210: 7209: 7206: 7197: 7150: 7148: 7147: 7142: 7130: 7128: 7127: 7122: 7110: 7108: 7107: 7102: 7100: 7095: 7087: 7069: 7067: 7066: 7061: 7048: 7046: 7045: 7040: 7038: 7037: 7032: 7027: 7017: 7015: 7014: 7009: 7007: 7006: 6997: 6996: 6974: 6972: 6971: 6966: 6964: 6963: 6958: 6953: 6950: 6948: 6947: 6935: 6927: 6926: 6918: 6903: 6901: 6900: 6895: 6893: 6892: 6884: 6872: 6870: 6869: 6864: 6859: 6857: 6856: 6847: 6846: 6845: 6836: 6835: 6825: 6817: 6816: 6813: 6800: 6798: 6797: 6792: 6781: 6780: 6762: 6760: 6759: 6754: 6749: 6748: 6739: 6737: 6736: 6727: 6726: 6725: 6716: 6715: 6705: 6697: 6693: 6692: 6684: 6673: 6665: 6658: 6656: 6655: 6646: 6645: 6644: 6635: 6634: 6625: 6624: 6614: 6609: 6608: 6605: 6587: 6585: 6584: 6579: 6574: 6570: 6569: 6561: 6533: 6531: 6530: 6521: 6520: 6519: 6510: 6509: 6500: 6499: 6489: 6484: 6482: 6481: 6472: 6471: 6470: 6461: 6460: 6451: 6450: 6440: 6414: 6412: 6411: 6402: 6401: 6400: 6391: 6390: 6381: 6380: 6370: 6365: 6364: 6361: 6345: 6343: 6342: 6337: 6335: 6334: 6331: 6316: 6314: 6313: 6308: 6285: 6283: 6282: 6273: 6272: 6271: 6262: 6261: 6252: 6251: 6241: 6236: 6234: 6233: 6224: 6213: 6180: 6179: 6171: 6137: 6136: 6128: 6113: 6111: 6110: 6105: 6069: 6068: 6060: 6047: 6045: 6044: 6039: 6037: 6036: 6028: 6000: 5998: 5997: 5988: 5987: 5986: 5978: 5971: 5957: 5955: 5954: 5949: 5917: 5909: 5898: 5896: 5895: 5890: 5885: 5883: 5882: 5873: 5872: 5871: 5862: 5861: 5852: 5851: 5841: 5836: 5835: 5827: 5821: 5820: 5815: 5811: 5809: 5808: 5799: 5788: 5775: 5774: 5766: 5760: 5759: 5754: 5753: 5745: 5735: 5734: 5726: 5720: 5719: 5716: 5703: 5701: 5700: 5695: 5693: 5692: 5689: 5671: 5669: 5668: 5663: 5655: 5654: 5639: 5635: 5634: 5633: 5625: 5616: 5615: 5607: 5604: 5603: 5595: 5578: 5576: 5568: 5567: 5563: 5562: 5561: 5553: 5550: 5549: 5531: 5522: 5520: 5519: 5514: 5512: 5511: 5492: 5490: 5489: 5484: 5470: 5469: 5461: 5458: 5457: 5434: 5432: 5431: 5426: 5421: 5420: 5412: 5409: 5408: 5400: 5391: 5390: 5382: 5373: 5372: 5352: 5350: 5349: 5344: 5342: 5341: 5336: 5335: 5327: 5317: 5316: 5308: 5302: 5301: 5281: 5279: 5278: 5273: 5271: 5270: 5262: 5259: 5258: 5246: 5245: 5237: 5234: 5233: 5221: 5220: 5215: 5210: 5198: 5196: 5195: 5190: 5188: 5181: 5180: 5172: 5169: 5165: 5164: 5163: 5155: 5152: 5151: 5143: 5134: 5133: 5125: 5111: 5110: 5105: 5100: 5097: 5093: 5092: 5091: 5086: 5085: 5077: 5067: 5066: 5058: 5047: 5043: 5042: 5041: 5036: 5028: 5026: 5023: 5022: 5014: 5005: 5004: 4996: 4993: 4992: 4984: 4975: 4974: 4966: 4963: 4962: 4954: 4951: 4950: 4942: 4931: 4927: 4926: 4925: 4920: 4915: 4910: 4908: 4905: 4904: 4896: 4890: 4889: 4884: 4879: 4876: 4875: 4867: 4852: 4851: 4846: 4841: 4831: 4830: 4822: 4819: 4818: 4810: 4801: 4800: 4795: 4790: 4787: 4786: 4778: 4772: 4771: 4766: 4761: 4756: 4754: 4745: 4744: 4739: 4734: 4731: 4730: 4722: 4712: 4711: 4706: 4701: 4680: 4678: 4677: 4672: 4660: 4658: 4657: 4652: 4650: 4649: 4644: 4639: 4629: 4627: 4626: 4621: 4619: 4618: 4610: 4598: 4596: 4595: 4590: 4588: 4587: 4582: 4577: 4574: 4573: 4565: 4556: 4555: 4550: 4542: 4540: 4534: 4532: 4524: 4523: 4522: 4514: 4507: 4501: 4500: 4492: 4489: 4488: 4480: 4474: 4473: 4468: 4463: 4458: 4456: 4450: 4448: 4440: 4439: 4438: 4433: 4428: 4421: 4408: 4406: 4405: 4400: 4398: 4397: 4392: 4387: 4377: 4375: 4374: 4369: 4357: 4355: 4354: 4349: 4347: 4346: 4341: 4336: 4327: 4289:two-body problem 4208:computed in his 4081: 4077: 4074: 4072: 4071: 4066: 4064: 4062: 4061: 4052: 4051: 4042: 4023: 4019: 3988: 3987:by Kepler (1619) 3983:translated from 3875: 3871: 3868: 3866: 3865: 3860: 3858: 3856: 3855: 3846: 3845: 3836: 3813: 3809: 3794: 3792: 3791: 3786: 3784: 3781: 3772: 3770: 3769: 3764: 3752: 3750: 3749: 3744: 3728: 3726: 3725: 3720: 3708: 3706: 3705: 3700: 3684: 3682: 3681: 3676: 3661: 3659: 3658: 3653: 3651: 3648: 3646: 3644: 3643: 3638: 3635: 3632: 3631: 3626: 3623: 3620: 3618: 3617: 3596: 3594: 3593: 3592: 3579: 3571: 3566: 3564: 3563: 3562: 3549: 3529: 3524: 3522: 3521: 3512: 3511: 3502: 3490: 3488: 3487: 3482: 3464: 3462: 3461: 3456: 3444: 3442: 3441: 3436: 3424: 3422: 3421: 3416: 3400: 3398: 3397: 3392: 3390: 3389: 3377: 3376: 3362: 3361: 3352: 3348: 3346: 3338: 3337: 3336: 3323: 3314: 3313: 3299: 3297: 3296: 3287: 3279: 3271: 3270: 3265: 3261: 3256: 3248: 3225: 3223: 3222: 3217: 3215: 3200: 3198: 3197: 3192: 3190: 3188: 3187: 3178: 3170: 3162: 3161: 3092: 3090: 3089: 3084: 3012: 3008: 3006: 3004: 3003: 2998: 2993: 2988: 2977: 2972: 2967: 2956: 2951: 2949: 2941: 2933: 2921: 2919: 2918: 2913: 2882: 2881: 2859: 2857: 2856: 2851: 2849: 2844: 2836: 2811: 2809: 2808: 2803: 2789: 2787: 2779: 2771: 2769: 2764: 2763: 2754: 2736: 2734: 2733: 2728: 2717:. So the period 2716: 2714: 2713: 2708: 2688: 2686: 2685: 2680: 2675: 2673: 2665: 2657: 2655: 2650: 2649: 2640: 2635: 2633: 2625: 2617: 2604: 2602: 2601: 2596: 2575: 2567: 2549: 2547: 2546: 2541: 2522: 2520: 2519: 2514: 2502: 2500: 2499: 2494: 2480:In a small time 2471: 2469: 2468: 2463: 2458: 2457: 2449: 2443: 2442: 2434: 2419: 2417: 2416: 2411: 2403: 2402: 2394: 2388: 2387: 2379: 2373: 2365: 2363: 2355: 2350: 2348: 2340: 2332: 2322: 2320: 2319: 2314: 2306: 2305: 2297: 2291: 2290: 2282: 2276: 2268: 2263: 2261: 2253: 2245: 2225: 2223: 2222: 2217: 2206: 2205: 2197: 2191: 2190: 2182: 2176: 2168: 2160: 2159: 2154: 2146: 2140: 2139: 2131: 2125: 2117: 1976: 1974: 1973: 1968: 1930: 1928: 1927: 1922: 1917: 1915: 1914: 1913: 1901: 1900: 1890: 1889: 1888: 1876: 1875: 1865: 1822: 1820: 1819: 1814: 1812: 1807: 1806: 1794: 1793: 1784: 1783: 1757: 1756: 1748: 1744: 1739: 1737: 1729: 1716: 1708: 1698: 1642: 1640: 1639: 1634: 1632: 1628: 1625: 1624: 1609: 1605: 1589: 1587: 1586: 1577: 1576: 1567: 1513: 1511: 1510: 1505: 1503: 1499: 1497: 1496: 1495: 1476: 1460: 1455: 1454: 1453: 1441: 1440: 1430: 1376: 1374: 1373: 1368: 1366: 1364: 1350: 1345: 1344: 1314: 1312: 1311: 1306: 1283: 1281: 1280: 1275: 1273: 1271: 1257: 1252: 1251: 1191:of the ellipse, 1178: 1176: 1175: 1170: 1155: 1153: 1152: 1147: 1142: 1140: 1116: 1094: 1081: 1077: 1064: 1041: 988:orthogonal group 929:Copernican model 852: 850: 849: 844: 833: 831: 820: 809: 807: 799: 756:angular momentum 579: 572: 565: 544:Orbital maneuver 493:Payload fraction 473: 454:Lissajous orbits 388: 359:Orbital velocity 306:Hyperbolic orbit 202:Orbital elements 192: 183: 166: 165: 158: 156: 155: 150: 148: 147: 143: 130: 129: 128: 118: 113: 112: 111: 21: 14915: 14914: 14910: 14909: 14908: 14906: 14905: 14904: 14895:Johannes Kepler 14875:1619 in science 14870:1609 in science 14860: 14859: 14858: 14848: 14846: 14836: 14834: 14824: 14822: 14810: 14800: 14798: 14786: 14778: 14776: 14771: 14770: 14769: 14768: 14767: 14760: 14747: 14703:Newton's cradle 14684: 14639: 14612: (student) 14610:William Whiston 14606: (student) 14562: 14543:Religious views 14504: 14419:Newton's method 14379:Newtonian fluid 14273:Bucket argument 14259: 14179: 14114: 14047: 14042: 14012: 14007: 13989:Johannes Kepler 13943: 13916:(1618, 1620-21) 13905:Astronomia nova 13876: 13852:Kepler triangle 13825: 13823:Johannes Kepler 13820: 13790: 13785: 13767: 13685:Escape velocity 13666: 13659: 13640:Rocket equation 13567: 13559: 13553: 13544: 13535: 13526: 13515: 13506: 13497: 13488: 13479: 13475: 13471: 13462: 13451: 13442: 13433: 13424: 13415: 13404: 13395: 13391: 13387:Semi-minor axis 13382: 13378:Semi-major axis 13373: 13364: 13358: 13331: 13253:Areosynchronous 13237: 13231: 13212:Sun-synchronous 13197:Near-equatorial 13141: 13021: 13015: 12975: 12959:Wayback Machine 12907:Wayback Machine 12895: 12856: 12828:Stephen Hawking 12799:Harmonice Mundi 12789: 12784: 12783: 12773: 12771: 12764: 12760: 12750: 12748: 12745: 12741:Schwarz, René. 12739: 12735: 12726: 12722: 12713: 12709: 12702: 12684: 12680: 12673: 12655: 12651: 12596: 12592: 12585: 12569: 12565: 12553: 12549: 12542: 12528: 12524: 12509: 12505: 12498: 12482: 12478: 12471: 12446: 12442: 12435: 12411: 12407: 12372: 12368: 12353: 12349: 12339: 12337: 12327: 12323: 12302: 12280: 12273: 12226:Astronomia nova 12211:Astronomia nova 12208: 12204: 12194: 12192: 12190: 12174: 12167: 12146: 12127:Astronomia nova 12124: 12121:Astronomia nova 12118: 12114: 12109: 12105: 12098: 12082: 12078: 12069: 12065: 12053: 12049: 12002: 11998: 11988: 11986: 11978: 11972: 11965: 11936: 11932: 11922: 11920: 11893: 11887: 11880: 11871: 11869: 11861: 11860: 11856: 11847: 11845: 11836: 11835: 11831: 11822: 11820: 11812: 11811: 11804: 11795: 11793: 11787:"Kepler's Laws" 11785: 11784: 11777: 11772: 11767: 11745: 11741: 11714: 11710: 11706: 11701: 11661:Circular motion 11656: 11603: 11600: 11599: 11563: 11559: 11552: 11548: 11513: 11510: 11509: 11498: 11467: 11458: 11454: 11426: 11417: 11413: 11396: 11393: 11392: 11372: 11371: 11358: 11349: 11345: 11332: 11321: 11319: 11297: 11280: 11278: 11262: 11251: 11249: 11194: 11144: 11142: 11133: 11132: 11107: 11090: 11088: 11081: 11058: 11041: 11039: 11032: 11030: 11008: 10991: 10989: 10982: 10972: 10963: 10959: 10955: 10953: 10950: 10949: 10914: 10897: 10895: 10882: 10873: 10869: 10867: 10864: 10863: 10810: 10793: 10791: 10766: 10745: 10741: 10734: 10730: 10699: 10697: 10663: 10656: 10652: 10645: 10643: 10623: 10620: 10619: 10586: 10579: 10575: 10568: 10566: 10553: 10551: 10548: 10547: 10531: 10528: 10527: 10473: 10470: 10469: 10453: 10442: 10434: 10423: 10415: 10404: 10402: 10399: 10398: 10395: 10376: 10374: 10365: 10346: sin 10334: cos 10311: sin 10303: cos 10228: 10225: 10224: 10201: 10200: 10172: 10170: 10152: 10148: 10147: 10145: 10138: 10123: 10119: 10118: 10116: 10113: 10112: 10101: 10090: 10082: 10071: 10070: 10068: 10061: 10056: 10042: 10027: 10026: 10021: 10007: 9999: 9985: 9977: 9963: 9956: 9951: 9937: 9933: 9931: 9928: 9927: 9870: 9867: 9866: 9853:. The function 9843: 9799: 9796: 9795: 9757: 9755: 9737: 9733: 9732: 9730: 9717: 9709: 9695: 9682: 9674: 9660: 9647: 9639: 9625: 9623: 9620: 9619: 9588: 9584: 9583: 9581: 9573: 9559: 9557: 9554: 9553: 9550:circular sector 9528: 9514: 9501: 9493: 9479: 9477: 9474: 9473: 9431: 9428: 9427: 9398: 9395: 9394: 9373: 9370: 9369: 9305: 9302: 9301: 9280: 9269: 9261: 9258: 9257: 9233: 9229: 9221: 9210: 9207: 9206: 9179: 9168: 9167: 9165: 9157: 9154: 9153: 9132: 9121: 9113: 9110: 9109: 9049:and the planet 9039: 9033: 9004: 8977: 8974: 8973: 8952: 8938: 8934: 8926: 8918: 8908: 8904: 8889: 8881: 8879: 8876: 8875: 8854: 8840: 8833: 8829: 8818: 8816: 8813: 8812: 8784: 8781: 8780: 8731: 8728: 8727: 8700: 8691: 8687: 8659: 8650: 8646: 8629: 8626: 8625: 8599: 8596: 8595: 8555: 8552: 8551: 8527: 8503: 8466: 8433: 8432: 8425: 8417: 8416: 8405: 8403: 8402: 8401: 8392: 8386: 8378: 8377: 8367: 8363: 8350: 8342: 8341: 8330: 8328: 8327: 8326: 8317: 8311: 8303: 8302: 8292: 8288: 8278: 8272: 8261: 8260: 8259: 8256: 8255: 8248: 8240: 8239: 8228: 8226: 8225: 8224: 8215: 8209: 8201: 8200: 8190: 8186: 8173: 8165: 8164: 8153: 8151: 8150: 8149: 8140: 8134: 8126: 8125: 8115: 8111: 8101: 8095: 8084: 8083: 8082: 8079: 8078: 8071: 8063: 8062: 8051: 8049: 8048: 8047: 8038: 8032: 8024: 8023: 8013: 8009: 7996: 7988: 7987: 7976: 7974: 7973: 7972: 7963: 7957: 7949: 7948: 7938: 7934: 7924: 7918: 7907: 7906: 7905: 7901: 7899: 7896: 7895: 7872: 7864: 7863: 7852: 7850: 7849: 7848: 7839: 7833: 7825: 7824: 7814: 7810: 7798: 7787: 7786: 7785: 7783: 7780: 7779: 7758: 7750: 7749: 7738: 7736: 7735: 7734: 7725: 7719: 7711: 7710: 7700: 7696: 7684: 7673: 7672: 7671: 7669: 7666: 7665: 7630: 7619: 7617: 7616: 7615: 7613: 7610: 7609: 7581: 7577: 7575: 7572: 7571: 7550: 7546: 7544: 7541: 7540: 7520: 7509: 7507: 7506: 7505: 7496: 7488: 7478: 7474: 7462: 7446: 7435: 7434: 7433: 7431: 7428: 7427: 7400: 7397: 7396: 7379: 7375: 7364: 7361: 7360: 7343: 7339: 7337: 7334: 7333: 7313: 7310: 7309: 7292: 7288: 7286: 7283: 7282: 7258: 7256: 7255: 7246: 7242: 7233: 7229: 7212: 7211: 7205: 7201: 7193: 7191: 7188: 7187: 7180: 7136: 7133: 7132: 7116: 7113: 7112: 7088: 7086: 7078: 7075: 7074: 7054: 7051: 7050: 7028: 7026: 7025: 7023: 7020: 7019: 7002: 6998: 6992: 6988: 6980: 6977: 6976: 6954: 6952: 6951: 6943: 6939: 6934: 6917: 6916: 6914: 6911: 6910: 6883: 6882: 6880: 6877: 6876: 6852: 6848: 6841: 6837: 6831: 6827: 6826: 6824: 6812: 6808: 6806: 6803: 6802: 6776: 6772: 6770: 6767: 6766: 6744: 6740: 6732: 6728: 6721: 6717: 6711: 6707: 6706: 6704: 6683: 6664: 6663: 6659: 6651: 6647: 6640: 6636: 6630: 6626: 6620: 6616: 6615: 6613: 6604: 6600: 6595: 6592: 6591: 6560: 6538: 6534: 6526: 6522: 6515: 6511: 6505: 6501: 6495: 6491: 6490: 6488: 6477: 6473: 6466: 6462: 6456: 6452: 6446: 6442: 6441: 6439: 6407: 6403: 6396: 6392: 6386: 6382: 6376: 6372: 6371: 6369: 6360: 6356: 6351: 6348: 6347: 6330: 6326: 6324: 6321: 6320: 6278: 6274: 6267: 6263: 6257: 6253: 6247: 6243: 6242: 6240: 6229: 6225: 6214: 6212: 6170: 6169: 6127: 6126: 6121: 6118: 6117: 6059: 6058: 6053: 6050: 6049: 6027: 6026: 5993: 5989: 5977: 5976: 5972: 5970: 5965: 5962: 5961: 5908: 5906: 5903: 5902: 5878: 5874: 5867: 5863: 5857: 5853: 5847: 5843: 5842: 5840: 5826: 5825: 5816: 5804: 5800: 5789: 5787: 5783: 5782: 5765: 5764: 5755: 5744: 5743: 5742: 5725: 5724: 5715: 5711: 5709: 5706: 5705: 5688: 5684: 5682: 5679: 5678: 5650: 5646: 5624: 5623: 5606: 5605: 5594: 5593: 5589: 5585: 5569: 5552: 5551: 5545: 5541: 5540: 5536: 5532: 5530: 5528: 5525: 5524: 5507: 5503: 5501: 5498: 5497: 5460: 5459: 5453: 5449: 5447: 5444: 5443: 5440: 5411: 5410: 5399: 5398: 5381: 5380: 5368: 5364: 5362: 5359: 5358: 5337: 5326: 5325: 5324: 5307: 5306: 5297: 5293: 5291: 5288: 5287: 5261: 5260: 5254: 5250: 5236: 5235: 5229: 5225: 5211: 5209: 5208: 5206: 5203: 5202: 5186: 5185: 5171: 5170: 5154: 5153: 5142: 5141: 5124: 5123: 5119: 5115: 5101: 5099: 5098: 5087: 5076: 5075: 5074: 5057: 5056: 5055: 5051: 5027: 5025: 5024: 5013: 5012: 4995: 4994: 4983: 4982: 4965: 4964: 4953: 4952: 4941: 4940: 4939: 4935: 4911: 4909: 4907: 4906: 4895: 4894: 4880: 4878: 4877: 4866: 4865: 4864: 4860: 4853: 4842: 4840: 4839: 4836: 4835: 4821: 4820: 4809: 4808: 4791: 4789: 4788: 4777: 4776: 4757: 4755: 4753: 4752: 4735: 4733: 4732: 4721: 4720: 4713: 4702: 4700: 4699: 4695: 4693: 4690: 4689: 4666: 4663: 4662: 4640: 4638: 4637: 4635: 4632: 4631: 4609: 4608: 4606: 4603: 4602: 4578: 4576: 4575: 4564: 4563: 4541: 4539: 4538: 4525: 4513: 4512: 4508: 4506: 4491: 4490: 4479: 4478: 4459: 4457: 4455: 4454: 4441: 4429: 4427: 4426: 4422: 4420: 4418: 4415: 4414: 4388: 4386: 4385: 4383: 4380: 4379: 4363: 4360: 4359: 4337: 4335: 4334: 4323: 4321: 4318: 4317: 4310: 4300: 4203: 4079: 4075: 4057: 4053: 4047: 4043: 4041: 4039: 4036: 4035: 4016: 3989: 3982: 3873: 3869: 3851: 3847: 3841: 3837: 3835: 3833: 3830: 3829: 3825: 3820: 3805: 3780: 3778: 3775: 3774: 3758: 3755: 3754: 3738: 3735: 3734: 3714: 3711: 3710: 3694: 3691: 3690: 3687:mass of the Sun 3670: 3667: 3666: 3647: 3639: 3634: 3633: 3627: 3622: 3621: 3619: 3610: 3606: 3588: 3584: 3580: 3572: 3570: 3558: 3554: 3550: 3530: 3528: 3517: 3513: 3507: 3503: 3501: 3499: 3496: 3495: 3470: 3467: 3466: 3450: 3447: 3446: 3430: 3427: 3426: 3410: 3407: 3406: 3385: 3381: 3372: 3368: 3357: 3353: 3339: 3332: 3328: 3324: 3322: 3318: 3309: 3305: 3292: 3288: 3280: 3278: 3266: 3249: 3247: 3243: 3242: 3234: 3231: 3230: 3211: 3209: 3206: 3205: 3183: 3179: 3171: 3169: 3157: 3153: 3145: 3142: 3141: 3105: 3072: 3069: 3068: 2978: 2976: 2957: 2955: 2942: 2934: 2932: 2930: 2927: 2926: 2877: 2873: 2871: 2868: 2867: 2837: 2835: 2827: 2824: 2823: 2780: 2772: 2770: 2759: 2755: 2753: 2745: 2742: 2741: 2722: 2719: 2718: 2696: 2693: 2692: 2666: 2658: 2656: 2645: 2641: 2639: 2626: 2618: 2616: 2614: 2611: 2610: 2566: 2555: 2552: 2551: 2528: 2525: 2524: 2508: 2505: 2504: 2485: 2482: 2481: 2478: 2448: 2447: 2433: 2432: 2427: 2424: 2423: 2393: 2392: 2378: 2377: 2364: 2354: 2341: 2333: 2331: 2329: 2326: 2325: 2296: 2295: 2281: 2280: 2267: 2254: 2246: 2244: 2242: 2239: 2238: 2196: 2195: 2181: 2180: 2167: 2147: 2145: 2144: 2130: 2129: 2116: 2105: 2102: 2101: 2054: 2028: 2005: 1998: 1946: 1943: 1942: 1909: 1905: 1896: 1892: 1891: 1884: 1880: 1871: 1867: 1866: 1864: 1856: 1853: 1852: 1847: 1840: 1810: 1809: 1802: 1798: 1789: 1785: 1779: 1775: 1768: 1759: 1758: 1749: 1730: 1725: 1709: 1704: 1699: 1697: 1693: 1692: 1685: 1678: 1676: 1673: 1672: 1667: 1660: 1630: 1629: 1620: 1616: 1604: 1597: 1591: 1590: 1582: 1578: 1572: 1568: 1566: 1559: 1552: 1550: 1547: 1546: 1541: 1534: 1519:semi-minor axis 1501: 1500: 1491: 1487: 1480: 1475: 1468: 1462: 1461: 1449: 1445: 1436: 1432: 1431: 1429: 1422: 1415: 1413: 1410: 1409: 1404: 1397: 1389:arithmetic mean 1382:semi-major axis 1354: 1349: 1340: 1336: 1334: 1331: 1330: 1300: 1297: 1296: 1261: 1256: 1247: 1243: 1241: 1238: 1237: 1164: 1161: 1160: 1120: 1115: 1107: 1104: 1103: 1093: 1083: 1079: 1076: 1066: 1062: 1039: 1008: 1000: 921: 871: 821: 810: 808: 798: 790: 787: 786: 697:Johannes Kepler 694: 679:Astronomia nova 645:semi-major axis 603:circular orbits 599:Johannes Kepler 597:, published by 583: 554: 553: 549:Orbit insertion 539: 531: 530: 516: 508: 507: 483: 475: 471: 464: 463: 459:Lyapunov orbits 450: 449: 433: 423: 422: 398: 390: 386: 379: 378: 364:Surface gravity 339:Escape velocity 329: 321: 320: 301:Parabolic orbit 297: 296: 263: 261: 258:two-body orbits 249: 248: 239:Semi-major axis 204: 194: 190: 162: 139: 135: 131: 124: 120: 119: 114: 107: 103: 102: 97: 94: 93: 88: 81: 71: 64: 57: 50: 43: 23: 22: 15: 12: 11: 5: 14913: 14903: 14902: 14897: 14892: 14887: 14882: 14877: 14872: 14857: 14856: 14844: 14832: 14820: 14808: 14796: 14773: 14772: 14759: 14758: 14756: 14755: 14753: 14749: 14748: 14746: 14745: 14740: 14735: 14730: 14725: 14720: 14715: 14710: 14705: 14700: 14694: 14692: 14686: 14685: 14683: 14682: 14675: 14668: 14659: 14649: 14647: 14641: 14640: 14638: 14637: 14636: (friend) 14631: 14630: (friend) 14625: 14624: (friend) 14619: 14613: 14607: 14601: 14595: 14594: (mentor) 14592:William Clarke 14589: 14583: 14577: 14570: 14568: 14564: 14563: 14561: 14560: 14555: 14550: 14548:Occult studies 14545: 14540: 14535: 14530: 14525: 14519: 14512: 14510: 14506: 14505: 14503: 14502: 14501: 14500: 14490: 14485: 14480: 14475: 14470: 14465: 14464: 14463: 14453: 14448: 14443: 14438: 14433: 14431:Newton fractal 14428: 14427: 14426: 14416: 14411: 14406: 14401: 14396: 14391: 14386: 14381: 14376: 14371: 14366: 14361: 14359:Newton's rings 14356: 14351: 14350: 14349: 14344: 14334: 14329: 14328: 14327: 14317: 14312: 14307: 14306: 14305: 14300: 14295: 14285: 14280: 14275: 14269: 14267: 14261: 14260: 14258: 14257: 14252: 14247: 14245:Newton's metal 14242: 14237: 14232: 14227: 14226: 14225: 14218:Newton polygon 14215: 14210: 14205: 14200: 14199: 14198: 14187: 14185: 14181: 14180: 14178: 14177: 14169: 14161: 14152:" (1713; 14146: 14138: 14131: 14122: 14120: 14119:Other writings 14116: 14115: 14113: 14112: 14104: 14096: 14088: 14080: 14072: 14064: 14055: 14053: 14049: 14048: 14041: 14040: 14033: 14026: 14018: 14009: 14008: 14006: 14005: 14000: 13993: 13985: 13980: 13972: 13966: 13960: 13951: 13949: 13945: 13944: 13942: 13941: 13933: 13925: 13917: 13909: 13901: 13897:De Stella Nova 13893: 13884: 13882: 13878: 13877: 13875: 13874: 13869: 13864: 13859: 13854: 13849: 13844: 13839: 13833: 13831: 13827: 13826: 13819: 13818: 13811: 13804: 13796: 13787: 13786: 13784: 13783: 13781:List of orbits 13772: 13769: 13768: 13766: 13765: 13760: 13755: 13750: 13745: 13740: 13735: 13733:Orbit equation 13730: 13722: 13717: 13712: 13707: 13702: 13697: 13692: 13687: 13682: 13677: 13671: 13669: 13661: 13660: 13658: 13657: 13652: 13647: 13642: 13637: 13632: 13627: 13622: 13617: 13612: 13607: 13605:Gravity assist 13602: 13600:Delta-v budget 13597: 13592: 13587: 13581: 13579: 13573: 13572: 13569: 13568: 13566: 13565: 13557: 13551: 13542: 13533: 13531:Orbital period 13523: 13521: 13517: 13516: 13514: 13513: 13511:True longitude 13504: 13502:Mean longitude 13495: 13486: 13469: 13459: 13457: 13453: 13452: 13450: 13449: 13440: 13431: 13422: 13412: 13410: 13406: 13405: 13403: 13402: 13389: 13380: 13371: 13361: 13359: 13357: 13356: 13353: 13349: 13343: 13337: 13336: 13333: 13332: 13330: 13329: 13328: 13327: 13319: 13318: 13317: 13312: 13307: 13306: 13305: 13292: 13287: 13286: 13285: 13280: 13275: 13270: 13262: 13261: 13260: 13258:Areostationary 13255: 13250: 13241: 13239: 13233: 13232: 13230: 13229: 13227:Very low Earth 13224: 13219: 13214: 13209: 13204: 13199: 13194: 13189: 13184: 13179: 13174: 13169: 13168: 13167: 13162: 13155:Geosynchronous 13151: 13149: 13143: 13142: 13140: 13139: 13137:Transfer orbit 13134: 13133: 13132: 13127: 13117: 13112: 13107: 13102: 13097: 13095:Lagrange point 13092: 13087: 13078: 13073: 13068: 13063: 13054: 13049: 13044: 13038: 13036: 13029: 13023: 13022: 13017:Gravitational 13014: 13013: 13006: 12999: 12991: 12985: 12984: 12972: 12961: 12947: 12940: 12933:Astronomy Cast 12929: 12922: 12909: 12894: 12893:External links 12891: 12890: 12889: 12877:V. I. Arnold, 12875: 12861: 12854: 12838: 12788: 12785: 12782: 12781: 12758: 12733: 12720: 12707: 12700: 12678: 12671: 12649: 12590: 12583: 12563: 12547: 12541:978-0813529080 12540: 12522: 12503: 12496: 12476: 12469: 12440: 12433: 12405: 12366: 12347: 12321: 12271: 12256: 12255: 12240: 12202: 12188: 12165: 12155:From p. 658: 12153:pages 658–665. 12112: 12103: 12096: 12076: 12070:Robert Small, 12063: 12054:Robert Small, 12047: 12045: 12044: 12035: 12026: 12017: 11996: 11963: 11961: 11960: 11951: 11930: 11878: 11854: 11829: 11802: 11774: 11773: 11771: 11768: 11766: 11765: 11739: 11707: 11705: 11702: 11700: 11699: 11693: 11688: 11683: 11681:Kepler problem 11678: 11673: 11668: 11666:Free-fall time 11663: 11657: 11655: 11652: 11640: 11637: 11634: 11631: 11628: 11625: 11622: 11619: 11616: 11613: 11610: 11607: 11572: 11566: 11562: 11558: 11555: 11551: 11547: 11544: 11541: 11538: 11535: 11532: 11529: 11526: 11523: 11520: 11517: 11497: 11491: 11474: 11471: 11466: 11461: 11457: 11453: 11450: 11447: 11444: 11441: 11438: 11433: 11430: 11425: 11420: 11416: 11412: 11409: 11406: 11403: 11400: 11370: 11365: 11362: 11357: 11352: 11348: 11341: 11338: 11335: 11330: 11327: 11324: 11318: 11312: 11309: 11306: 11303: 11300: 11295: 11292: 11289: 11286: 11283: 11277: 11271: 11268: 11265: 11260: 11257: 11254: 11248: 11242: 11239: 11236: 11233: 11230: 11227: 11224: 11221: 11218: 11215: 11212: 11209: 11206: 11203: 11200: 11197: 11192: 11189: 11186: 11183: 11180: 11177: 11174: 11171: 11168: 11165: 11162: 11159: 11156: 11153: 11150: 11147: 11141: 11138: 11136: 11134: 11125: 11122: 11119: 11116: 11113: 11110: 11105: 11102: 11099: 11096: 11093: 11087: 11084: 11076: 11073: 11070: 11067: 11064: 11061: 11056: 11053: 11050: 11047: 11044: 11038: 11035: 11029: 11023: 11020: 11017: 11014: 11011: 11006: 11003: 11000: 10997: 10994: 10988: 10985: 10983: 10979: 10976: 10971: 10966: 10962: 10958: 10957: 10935: 10929: 10926: 10923: 10920: 10917: 10912: 10909: 10906: 10903: 10900: 10894: 10889: 10886: 10881: 10876: 10872: 10834: 10828: 10825: 10822: 10819: 10816: 10813: 10808: 10805: 10802: 10799: 10796: 10790: 10784: 10781: 10778: 10775: 10772: 10769: 10764: 10761: 10758: 10754: 10748: 10744: 10740: 10737: 10733: 10729: 10726: 10723: 10720: 10717: 10714: 10711: 10708: 10705: 10702: 10696: 10693: 10690: 10687: 10681: 10678: 10675: 10672: 10669: 10666: 10659: 10655: 10651: 10648: 10642: 10639: 10636: 10633: 10630: 10627: 10604: 10601: 10598: 10595: 10592: 10589: 10582: 10578: 10574: 10571: 10565: 10560: 10557: 10535: 10513: 10510: 10507: 10504: 10501: 10498: 10495: 10492: 10489: 10486: 10483: 10480: 10477: 10456: 10452: 10449: 10445: 10441: 10437: 10433: 10430: 10426: 10422: 10418: 10414: 10411: 10407: 10394: 10390:True anomaly, 10388: 10357: 10277:. Determining 10256: 10253: 10250: 10247: 10244: 10241: 10238: 10235: 10232: 10197: 10193: 10190: 10187: 10184: 10181: 10178: 10175: 10169: 10164: 10160: 10155: 10151: 10144: 10141: 10139: 10135: 10131: 10126: 10122: 10115: 10114: 10109: 10104: 10100: 10097: 10093: 10089: 10085: 10081: 10078: 10074: 10067: 10064: 10062: 10059: 10055: 10052: 10049: 10045: 10041: 10038: 10035: 10032: 10029: 10028: 10024: 10020: 10017: 10014: 10010: 10006: 10002: 9998: 9995: 9992: 9988: 9984: 9980: 9976: 9973: 9970: 9966: 9962: 9959: 9957: 9954: 9950: 9947: 9944: 9940: 9936: 9935: 9892: 9889: 9886: 9883: 9880: 9877: 9874: 9842: 9836: 9815: 9812: 9809: 9806: 9803: 9775: 9770: 9766: 9763: 9760: 9754: 9749: 9745: 9740: 9736: 9729: 9724: 9721: 9716: 9712: 9708: 9705: 9702: 9698: 9694: 9689: 9686: 9681: 9677: 9673: 9670: 9667: 9663: 9659: 9654: 9651: 9646: 9642: 9638: 9635: 9632: 9628: 9605: 9600: 9596: 9591: 9587: 9580: 9576: 9572: 9569: 9566: 9562: 9535: 9531: 9527: 9524: 9521: 9517: 9513: 9508: 9505: 9500: 9496: 9492: 9489: 9486: 9482: 9470: 9469: 9453: 9450: 9447: 9444: 9441: 9438: 9435: 9425: 9405: 9402: 9392: 9380: 9377: 9344: 9343: 9327: 9324: 9321: 9318: 9315: 9312: 9309: 9299: 9287: 9283: 9279: 9276: 9272: 9268: 9265: 9255: 9243: 9236: 9232: 9228: 9225: 9220: 9217: 9214: 9204: 9192: 9187: 9182: 9178: 9175: 9171: 9164: 9161: 9151: 9139: 9135: 9131: 9128: 9124: 9120: 9117: 9103: 9102: 9096: 9090: 9081: 9035:Main article: 9032: 9028:Mean anomaly, 9026: 8981: 8960: 8955: 8951: 8948: 8941: 8937: 8933: 8930: 8925: 8921: 8917: 8914: 8911: 8907: 8901: 8897: 8894: 8888: 8884: 8862: 8857: 8853: 8850: 8847: 8843: 8839: 8836: 8832: 8828: 8825: 8821: 8801: 8800: 8788: 8768: 8765: 8762: 8759: 8756: 8753: 8750: 8747: 8744: 8741: 8738: 8735: 8720: 8707: 8704: 8699: 8694: 8690: 8686: 8683: 8680: 8677: 8674: 8671: 8666: 8663: 8658: 8653: 8649: 8645: 8642: 8639: 8636: 8633: 8615: 8603: 8583: 8580: 8577: 8574: 8571: 8568: 8565: 8562: 8559: 8541: 8521: 8520:is the period. 8465: 8462: 8424: 8412: 8408: 8398: 8395: 8385: 8376: 8366: 8362: 8359: 8349: 8337: 8333: 8323: 8320: 8310: 8301: 8291: 8287: 8284: 8281: 8279: 8268: 8265: 8258: 8257: 8247: 8235: 8231: 8221: 8218: 8208: 8199: 8189: 8185: 8182: 8172: 8160: 8156: 8146: 8143: 8133: 8124: 8114: 8110: 8107: 8104: 8102: 8091: 8088: 8081: 8080: 8070: 8058: 8054: 8044: 8041: 8031: 8022: 8012: 8008: 8005: 7995: 7983: 7979: 7969: 7966: 7956: 7947: 7937: 7933: 7930: 7927: 7925: 7914: 7911: 7904: 7903: 7871: 7859: 7855: 7845: 7842: 7832: 7823: 7813: 7809: 7806: 7794: 7791: 7757: 7745: 7741: 7731: 7728: 7718: 7709: 7699: 7695: 7692: 7680: 7677: 7636: 7633: 7626: 7622: 7587: 7584: 7580: 7553: 7549: 7526: 7523: 7516: 7512: 7502: 7499: 7494: 7491: 7487: 7481: 7477: 7471: 7468: 7465: 7461: 7457: 7454: 7449: 7442: 7439: 7404: 7378: 7374: 7371: 7368: 7342: 7317: 7291: 7265: 7261: 7252: 7249: 7245: 7241: 7232: 7228: 7225: 7219: 7216: 7204: 7200: 7196: 7179: 7176: 7140: 7120: 7098: 7094: 7091: 7085: 7082: 7058: 7035: 7031: 7005: 7001: 6995: 6991: 6987: 6984: 6961: 6957: 6946: 6942: 6938: 6933: 6930: 6924: 6921: 6890: 6887: 6862: 6855: 6851: 6844: 6840: 6834: 6830: 6823: 6820: 6811: 6790: 6787: 6784: 6779: 6775: 6752: 6747: 6743: 6735: 6731: 6724: 6720: 6714: 6710: 6703: 6700: 6696: 6690: 6687: 6682: 6679: 6676: 6671: 6668: 6662: 6654: 6650: 6643: 6639: 6633: 6629: 6623: 6619: 6612: 6603: 6599: 6577: 6573: 6567: 6564: 6559: 6556: 6553: 6550: 6547: 6544: 6541: 6537: 6529: 6525: 6518: 6514: 6508: 6504: 6498: 6494: 6487: 6480: 6476: 6469: 6465: 6459: 6455: 6449: 6445: 6438: 6435: 6432: 6429: 6426: 6423: 6420: 6417: 6410: 6406: 6399: 6395: 6389: 6385: 6379: 6375: 6368: 6359: 6355: 6329: 6306: 6303: 6300: 6297: 6294: 6291: 6288: 6281: 6277: 6270: 6266: 6260: 6256: 6250: 6246: 6239: 6232: 6228: 6223: 6220: 6217: 6210: 6207: 6204: 6201: 6198: 6195: 6192: 6189: 6186: 6183: 6177: 6174: 6167: 6164: 6161: 6158: 6155: 6152: 6149: 6146: 6143: 6140: 6134: 6131: 6125: 6103: 6100: 6097: 6094: 6091: 6088: 6085: 6081: 6078: 6075: 6072: 6066: 6063: 6057: 6034: 6031: 6024: 6021: 6018: 6015: 6012: 6009: 6006: 6003: 5996: 5992: 5984: 5981: 5975: 5969: 5947: 5944: 5941: 5938: 5935: 5932: 5929: 5926: 5923: 5920: 5915: 5912: 5888: 5881: 5877: 5870: 5866: 5860: 5856: 5850: 5846: 5839: 5833: 5830: 5824: 5819: 5814: 5807: 5803: 5798: 5795: 5792: 5786: 5781: 5778: 5772: 5769: 5763: 5758: 5751: 5748: 5741: 5738: 5732: 5729: 5723: 5714: 5687: 5661: 5658: 5653: 5649: 5645: 5642: 5638: 5631: 5628: 5622: 5619: 5613: 5610: 5601: 5598: 5592: 5588: 5584: 5581: 5575: 5572: 5566: 5559: 5556: 5548: 5544: 5539: 5535: 5510: 5506: 5482: 5479: 5476: 5473: 5467: 5464: 5456: 5452: 5439: 5436: 5424: 5418: 5415: 5406: 5403: 5397: 5394: 5388: 5385: 5379: 5376: 5371: 5367: 5340: 5333: 5330: 5323: 5320: 5314: 5311: 5305: 5300: 5296: 5268: 5265: 5257: 5253: 5249: 5243: 5240: 5232: 5228: 5224: 5218: 5214: 5184: 5178: 5175: 5168: 5161: 5158: 5149: 5146: 5140: 5137: 5131: 5128: 5122: 5118: 5114: 5108: 5104: 5096: 5090: 5083: 5080: 5073: 5070: 5064: 5061: 5054: 5050: 5046: 5039: 5034: 5031: 5020: 5017: 5011: 5008: 5002: 4999: 4990: 4987: 4981: 4978: 4972: 4969: 4960: 4957: 4948: 4945: 4938: 4934: 4930: 4923: 4918: 4914: 4902: 4899: 4893: 4887: 4883: 4873: 4870: 4863: 4859: 4856: 4854: 4849: 4845: 4838: 4837: 4834: 4828: 4825: 4816: 4813: 4807: 4804: 4798: 4794: 4784: 4781: 4775: 4769: 4764: 4760: 4751: 4748: 4742: 4738: 4728: 4725: 4719: 4716: 4714: 4709: 4705: 4698: 4697: 4670: 4647: 4643: 4616: 4613: 4585: 4581: 4571: 4568: 4562: 4559: 4553: 4548: 4545: 4537: 4531: 4528: 4520: 4517: 4511: 4504: 4498: 4495: 4486: 4483: 4477: 4471: 4466: 4462: 4453: 4447: 4444: 4436: 4432: 4425: 4395: 4391: 4367: 4344: 4340: 4333: 4330: 4326: 4299: 4296: 4284: 4283: 4280: 4269: 4268: 4265: 4239: 4238: 4227: 4202: 4199: 4196: 4195: 4192: 4189: 4186: 4182: 4181: 4178: 4175: 4172: 4168: 4167: 4164: 4161: 4158: 4154: 4153: 4150: 4147: 4144: 4140: 4139: 4136: 4133: 4130: 4126: 4125: 4122: 4119: 4116: 4112: 4111: 4108: 4105: 4102: 4098: 4097: 4094: 4091: 4088: 4084: 4083: 4060: 4056: 4050: 4046: 4033: 4032:Period (days) 4030: 4027: 3980: 3962: 3961: 3958: 3955: 3952: 3948: 3947: 3944: 3941: 3938: 3934: 3933: 3930: 3927: 3924: 3920: 3919: 3916: 3913: 3910: 3906: 3905: 3902: 3899: 3896: 3892: 3891: 3888: 3885: 3882: 3878: 3877: 3854: 3850: 3844: 3840: 3827: 3822: 3819:Mean distance 3817: 3804: 3801: 3762: 3742: 3718: 3698: 3674: 3663: 3662: 3642: 3630: 3616: 3613: 3609: 3605: 3602: 3599: 3591: 3587: 3583: 3578: 3575: 3569: 3561: 3557: 3553: 3548: 3545: 3542: 3539: 3536: 3533: 3527: 3520: 3516: 3510: 3506: 3480: 3477: 3474: 3454: 3434: 3414: 3402: 3401: 3388: 3384: 3380: 3375: 3371: 3366: 3360: 3356: 3351: 3345: 3342: 3335: 3331: 3327: 3321: 3317: 3312: 3308: 3303: 3295: 3291: 3286: 3283: 3277: 3274: 3269: 3264: 3259: 3255: 3252: 3246: 3241: 3238: 3214: 3202: 3201: 3186: 3182: 3177: 3174: 3168: 3165: 3160: 3156: 3152: 3149: 3113:orbital period 3104: 3101: 3098: 3097: 3082: 3079: 3076: 3064: 3063: 3053: 3042: 3041: 3031: 3020: 3019: 3016: 2996: 2991: 2987: 2984: 2981: 2975: 2970: 2966: 2963: 2960: 2954: 2948: 2945: 2940: 2937: 2923: 2922: 2911: 2908: 2905: 2901: 2898: 2895: 2892: 2889: 2886: 2880: 2876: 2861: 2860: 2847: 2843: 2840: 2834: 2831: 2813: 2812: 2801: 2798: 2795: 2792: 2786: 2783: 2778: 2775: 2767: 2762: 2758: 2752: 2749: 2726: 2706: 2703: 2700: 2678: 2672: 2669: 2664: 2661: 2653: 2648: 2644: 2638: 2632: 2629: 2624: 2621: 2607:areal velocity 2594: 2591: 2587: 2584: 2581: 2578: 2573: 2570: 2565: 2562: 2559: 2539: 2536: 2532: 2512: 2492: 2489: 2477: 2474: 2461: 2455: 2452: 2446: 2440: 2437: 2431: 2409: 2406: 2400: 2397: 2391: 2385: 2382: 2376: 2371: 2368: 2361: 2358: 2353: 2347: 2344: 2339: 2336: 2312: 2309: 2303: 2300: 2294: 2288: 2285: 2279: 2274: 2271: 2266: 2260: 2257: 2252: 2249: 2215: 2212: 2209: 2203: 2200: 2194: 2188: 2185: 2179: 2174: 2171: 2166: 2163: 2157: 2153: 2150: 2143: 2137: 2134: 2128: 2123: 2120: 2115: 2112: 2109: 2083: 2082: 2079: 2076: 2073: 2069: 2065: 2053: 2050: 2027: 2024: 2003: 1996: 1978: 1977: 1966: 1962: 1959: 1956: 1953: 1950: 1932: 1931: 1920: 1912: 1908: 1904: 1899: 1895: 1887: 1883: 1879: 1874: 1870: 1863: 1860: 1845: 1838: 1824: 1823: 1805: 1801: 1797: 1792: 1788: 1782: 1778: 1774: 1771: 1769: 1767: 1764: 1761: 1760: 1755: 1752: 1747: 1742: 1736: 1733: 1728: 1724: 1720: 1715: 1712: 1707: 1703: 1696: 1691: 1688: 1686: 1684: 1681: 1680: 1665: 1658: 1644: 1643: 1623: 1619: 1615: 1612: 1608: 1603: 1600: 1598: 1596: 1593: 1592: 1585: 1581: 1575: 1571: 1565: 1562: 1560: 1558: 1555: 1554: 1539: 1532: 1526:geometric mean 1515: 1514: 1494: 1490: 1486: 1483: 1479: 1474: 1471: 1469: 1467: 1464: 1463: 1458: 1452: 1448: 1444: 1439: 1435: 1428: 1425: 1423: 1421: 1418: 1417: 1402: 1395: 1378: 1377: 1363: 1360: 1357: 1353: 1348: 1343: 1339: 1304: 1285: 1284: 1270: 1267: 1264: 1260: 1255: 1250: 1246: 1168: 1157: 1156: 1145: 1139: 1136: 1133: 1129: 1126: 1123: 1119: 1114: 1111: 1091: 1074: 1007: 1004: 999: 996: 920: 917: 905:empirical laws 870: 867: 854: 853: 842: 839: 836: 830: 827: 824: 819: 816: 813: 805: 802: 797: 794: 760: 759: 758:) is constant. 746: 733: 715: 714: 711: 708: 693: 690: 667:laws of motion 649: 648: 641:orbital period 637: 634: 585: 584: 582: 581: 574: 567: 559: 556: 555: 552: 551: 546: 540: 537: 536: 533: 532: 529: 528: 523: 521:Gravity assist 517: 514: 513: 510: 509: 506: 505: 500: 495: 490: 484: 481: 480: 477: 476: 469: 466: 465: 462: 461: 456: 448: 447: 439: 435: 434: 429: 428: 425: 424: 421: 420: 415: 410: 405: 399: 396: 395: 392: 391: 384: 381: 380: 377: 376: 371: 366: 361: 356: 354:Orbital period 351: 346: 341: 336: 330: 327: 326: 323: 322: 319: 318: 316:Decaying orbit 313: 308: 303: 295: 294: 288: 281: 279:Transfer orbit 277: 276: 275: 273:Elliptic orbit 270: 268:Circular orbit 264: 255: 254: 251: 250: 247: 246: 241: 236: 231: 226: 221: 216: 211: 205: 200: 199: 196: 195: 188: 185: 184: 176: 175: 171: 170: 161: 160: 146: 142: 138: 134: 127: 123: 117: 110: 106: 101: 90: 86: 79: 73: 69: 62: 55: 48: 41: 34: 9: 6: 4: 3: 2: 14912: 14901: 14898: 14896: 14893: 14891: 14888: 14886: 14883: 14881: 14878: 14876: 14873: 14871: 14868: 14867: 14865: 14855: 14845: 14843: 14833: 14831: 14821: 14819: 14814: 14809: 14807: 14797: 14795: 14790: 14785: 14784: 14781: 14766: 14762: 14754: 14750: 14744: 14741: 14739: 14736: 14734: 14731: 14729: 14726: 14724: 14721: 14719: 14716: 14714: 14711: 14709: 14706: 14704: 14701: 14699: 14698:Newton (unit) 14696: 14695: 14693: 14691: 14687: 14681: 14680: 14676: 14674: 14673: 14669: 14666: 14664: 14660: 14657: 14655: 14651: 14650: 14648: 14646: 14642: 14635: 14632: 14629: 14628:William Jones 14626: 14623: 14620: 14617: 14614: 14611: 14608: 14605: 14602: 14600: (tutor) 14599: 14596: 14593: 14590: 14587: 14584: 14581: 14580:John Conduitt 14578: 14576: (niece) 14575: 14572: 14571: 14569: 14565: 14559: 14556: 14554: 14551: 14549: 14546: 14544: 14541: 14539: 14536: 14534: 14531: 14529: 14526: 14523: 14522:Cranbury Park 14520: 14517: 14514: 14513: 14511: 14509:Personal life 14507: 14499: 14496: 14495: 14494: 14491: 14489: 14486: 14484: 14481: 14479: 14476: 14474: 14471: 14469: 14466: 14462: 14459: 14458: 14457: 14456:Newton number 14454: 14452: 14449: 14447: 14444: 14442: 14439: 14437: 14434: 14432: 14429: 14425: 14422: 14421: 14420: 14417: 14415: 14412: 14410: 14407: 14405: 14402: 14400: 14397: 14395: 14392: 14390: 14387: 14385: 14382: 14380: 14377: 14375: 14372: 14370: 14367: 14365: 14362: 14360: 14357: 14355: 14352: 14348: 14345: 14343: 14340: 14339: 14338: 14335: 14333: 14330: 14326: 14325:Kepler's laws 14323: 14322: 14321: 14318: 14316: 14313: 14311: 14308: 14304: 14301: 14299: 14298:parameterized 14296: 14294: 14291: 14290: 14289: 14286: 14284: 14281: 14279: 14276: 14274: 14271: 14270: 14268: 14266: 14262: 14256: 14253: 14251: 14248: 14246: 14243: 14241: 14238: 14236: 14233: 14231: 14228: 14224: 14221: 14220: 14219: 14216: 14214: 14211: 14209: 14206: 14204: 14201: 14197: 14194: 14193: 14192: 14189: 14188: 14186: 14184:Contributions 14182: 14175: 14174: 14170: 14167: 14166: 14162: 14159: 14157: 14151: 14147: 14144: 14143: 14139: 14137:" (1675) 14136: 14132: 14129: 14128: 14124: 14123: 14121: 14117: 14110: 14109: 14105: 14102: 14101: 14097: 14094: 14093: 14089: 14086: 14085: 14081: 14078: 14077: 14073: 14070: 14069: 14065: 14062: 14061: 14057: 14056: 14054: 14050: 14046: 14039: 14034: 14032: 14027: 14025: 14020: 14019: 14016: 14004: 14001: 13999: 13998: 13994: 13992: 13990: 13986: 13984: 13981: 13978: 13977: 13973: 13970: 13969:Jakob Bartsch 13967: 13964: 13961: 13958: 13957: 13953: 13952: 13950: 13946: 13939: 13938: 13934: 13931: 13930: 13926: 13923: 13922: 13918: 13915: 13914: 13910: 13907: 13906: 13902: 13899: 13898: 13894: 13891: 13890: 13886: 13885: 13883: 13879: 13873: 13870: 13868: 13865: 13863: 13860: 13858: 13855: 13853: 13850: 13848: 13845: 13843: 13840: 13838: 13835: 13834: 13832: 13828: 13824: 13817: 13812: 13810: 13805: 13803: 13798: 13797: 13794: 13782: 13774: 13773: 13770: 13764: 13761: 13759: 13756: 13754: 13751: 13749: 13746: 13744: 13741: 13739: 13736: 13734: 13731: 13729: 13728:-body problem 13727: 13723: 13721: 13718: 13716: 13713: 13711: 13708: 13706: 13703: 13701: 13698: 13696: 13693: 13691: 13688: 13686: 13683: 13681: 13678: 13676: 13673: 13672: 13670: 13668: 13662: 13656: 13653: 13651: 13648: 13646: 13643: 13641: 13638: 13636: 13633: 13631: 13630:Oberth effect 13628: 13626: 13623: 13621: 13618: 13616: 13613: 13611: 13608: 13606: 13603: 13601: 13598: 13596: 13593: 13591: 13588: 13586: 13583: 13582: 13580: 13578: 13574: 13564: 13556: 13552: 13550: 13549:Orbital speed 13543: 13541: 13534: 13532: 13525: 13524: 13522: 13518: 13512: 13505: 13503: 13496: 13494: 13487: 13485: 13470: 13468: 13461: 13460: 13458: 13454: 13448: 13441: 13439: 13432: 13430: 13423: 13421: 13414: 13413: 13411: 13407: 13401: 13390: 13388: 13381: 13379: 13372: 13370: 13363: 13362: 13360: 13354: 13351: 13350: 13347: 13344: 13342: 13338: 13326: 13323: 13322: 13320: 13316: 13313: 13311: 13308: 13304: 13303:Earth's orbit 13301: 13300: 13299: 13296: 13295: 13293: 13291: 13288: 13284: 13281: 13279: 13276: 13274: 13271: 13269: 13266: 13265: 13263: 13259: 13256: 13254: 13251: 13249: 13246: 13245: 13243: 13242: 13240: 13234: 13228: 13225: 13223: 13220: 13218: 13215: 13213: 13210: 13208: 13205: 13203: 13200: 13198: 13195: 13193: 13190: 13188: 13185: 13183: 13180: 13178: 13175: 13173: 13170: 13166: 13163: 13161: 13160:Geostationary 13158: 13157: 13156: 13153: 13152: 13150: 13148: 13144: 13138: 13135: 13131: 13128: 13126: 13123: 13122: 13121: 13118: 13116: 13113: 13111: 13108: 13106: 13103: 13101: 13098: 13096: 13093: 13091: 13088: 13086: 13082: 13079: 13077: 13074: 13072: 13069: 13067: 13064: 13062: 13058: 13055: 13053: 13050: 13048: 13045: 13043: 13040: 13039: 13037: 13033: 13030: 13028: 13024: 13020: 13012: 13007: 13005: 13000: 12998: 12993: 12992: 12989: 12982: 12978: 12973: 12970: 12966: 12962: 12960: 12956: 12952: 12948: 12945: 12941: 12938: 12934: 12930: 12927: 12923: 12920: 12916: 12915: 12910: 12908: 12904: 12901: 12897: 12896: 12888: 12887:0-387-96890-3 12884: 12880: 12876: 12874: 12873:0-521-57597-4 12870: 12866: 12862: 12857: 12851: 12847: 12846: 12839: 12837: 12836:0-7624-1348-4 12833: 12829: 12825: 12821: 12817: 12813: 12809: 12805: 12801: 12800: 12795: 12791: 12790: 12769: 12762: 12744: 12737: 12730: 12724: 12717: 12711: 12703: 12701:9780486676050 12697: 12692: 12691: 12682: 12674: 12672:9780486676050 12668: 12663: 12662: 12653: 12645: 12641: 12636: 12631: 12626: 12621: 12617: 12613: 12610:(2): e01274. 12609: 12605: 12601: 12594: 12586: 12580: 12576: 12575: 12567: 12560: 12556: 12551: 12543: 12537: 12533: 12526: 12519: 12518: 12513: 12507: 12499: 12493: 12489: 12488: 12480: 12472: 12466: 12462: 12457: 12456: 12450: 12444: 12436: 12430: 12426: 12422: 12418: 12417: 12409: 12402: 12395: 12391: 12387: 12383: 12378: 12370: 12362: 12358: 12351: 12336: 12332: 12325: 12318: 12314: 12310: 12306: 12300: 12296: 12292: 12288: 12284: 12278: 12276: 12268: 12264: 12260: 12253: 12249: 12245: 12241: 12238: 12234: 12231: 12227: 12223: 12219: 12218: 12216: 12212: 12206: 12191: 12185: 12181: 12180: 12172: 12170: 12162: 12158: 12154: 12150: 12144: 12140: 12136: 12132: 12128: 12122: 12116: 12107: 12099: 12093: 12089: 12088: 12080: 12073: 12067: 12061: 12057: 12051: 12042: 12039: 12036: 12033: 12030: 12027: 12024: 12021: 12018: 12015: 12012: 12009: 12008: 12006: 12003:De la Lande, 12000: 11984: 11977: 11970: 11968: 11958: 11955: 11952: 11949: 11946: 11943: 11942: 11940: 11934: 11919: 11915: 11911: 11907: 11903: 11899: 11898:Physics Today 11892: 11885: 11883: 11868: 11864: 11858: 11843: 11839: 11833: 11819: 11815: 11809: 11807: 11792: 11788: 11782: 11780: 11775: 11762: 11758: 11754: 11750: 11743: 11736: 11732: 11728: 11724: 11723:pages 554–555 11720: 11719: 11712: 11708: 11697: 11694: 11692: 11689: 11687: 11684: 11682: 11679: 11677: 11674: 11672: 11669: 11667: 11664: 11662: 11659: 11658: 11651: 11638: 11632: 11629: 11626: 11623: 11620: 11617: 11611: 11608: 11605: 11597: 11593: 11589: 11584: 11570: 11564: 11560: 11556: 11553: 11549: 11545: 11542: 11536: 11533: 11530: 11527: 11524: 11521: 11515: 11507: 11503: 11496: 11490: 11487: 11472: 11469: 11464: 11459: 11455: 11448: 11445: 11442: 11436: 11431: 11428: 11423: 11418: 11414: 11407: 11404: 11401: 11390: 11385: 11368: 11363: 11360: 11355: 11350: 11346: 11339: 11336: 11333: 11328: 11325: 11322: 11316: 11310: 11307: 11304: 11301: 11298: 11293: 11290: 11287: 11284: 11281: 11275: 11269: 11266: 11263: 11258: 11255: 11252: 11246: 11237: 11234: 11231: 11228: 11225: 11219: 11213: 11210: 11207: 11204: 11201: 11198: 11187: 11184: 11181: 11178: 11175: 11169: 11163: 11160: 11157: 11154: 11151: 11148: 11139: 11137: 11123: 11120: 11117: 11114: 11111: 11108: 11103: 11100: 11097: 11094: 11091: 11085: 11082: 11074: 11071: 11068: 11065: 11062: 11059: 11054: 11051: 11048: 11045: 11042: 11036: 11033: 11027: 11021: 11018: 11015: 11012: 11009: 11004: 11001: 10998: 10995: 10992: 10986: 10984: 10977: 10974: 10969: 10964: 10960: 10946: 10933: 10927: 10924: 10921: 10918: 10915: 10910: 10907: 10904: 10901: 10898: 10892: 10887: 10884: 10879: 10874: 10870: 10861: 10856: 10854: 10850: 10845: 10832: 10826: 10823: 10820: 10817: 10814: 10811: 10806: 10803: 10800: 10797: 10794: 10788: 10782: 10779: 10776: 10773: 10770: 10767: 10762: 10759: 10756: 10752: 10746: 10742: 10738: 10735: 10731: 10727: 10721: 10718: 10715: 10712: 10709: 10706: 10700: 10694: 10691: 10688: 10685: 10679: 10676: 10673: 10670: 10667: 10664: 10657: 10653: 10649: 10646: 10640: 10637: 10634: 10631: 10628: 10625: 10602: 10599: 10596: 10593: 10590: 10587: 10580: 10576: 10572: 10569: 10563: 10558: 10555: 10533: 10524: 10511: 10508: 10505: 10502: 10499: 10496: 10493: 10490: 10487: 10484: 10481: 10478: 10475: 10450: 10447: 10439: 10431: 10428: 10420: 10412: 10409: 10393: 10387: 10383: 10380: · 10379: 10372: 10368: 10364: 10360: 10356: 10351: 10349: 10345: 10341: 10337: 10333: 10329: 10325: 10321: 10316: 10314: 10310: 10306: 10302: 10297: 10295: 10291: 10286: 10284: 10280: 10276: 10272: 10267: 10254: 10251: 10248: 10245: 10242: 10239: 10236: 10233: 10230: 10223: 10219: 10214: 10195: 10191: 10188: 10185: 10182: 10179: 10176: 10173: 10167: 10162: 10158: 10153: 10149: 10142: 10140: 10133: 10129: 10124: 10120: 10107: 10098: 10095: 10087: 10079: 10076: 10065: 10063: 10053: 10050: 10047: 10039: 10036: 10033: 10030: 10018: 10015: 10012: 10004: 9996: 9993: 9990: 9982: 9974: 9971: 9968: 9960: 9958: 9948: 9945: 9942: 9925: 9921: 9917: 9913: 9909: 9905: 9890: 9887: 9884: 9881: 9875: 9872: 9864: 9860: 9857: = 9856: 9852: 9848: 9841: 9835: 9833: 9829: 9813: 9810: 9807: 9804: 9801: 9793: 9789: 9773: 9768: 9764: 9761: 9758: 9752: 9747: 9743: 9738: 9734: 9727: 9722: 9719: 9714: 9706: 9703: 9700: 9692: 9687: 9684: 9679: 9671: 9668: 9665: 9657: 9652: 9649: 9644: 9636: 9633: 9630: 9616: 9603: 9598: 9594: 9589: 9585: 9578: 9570: 9567: 9564: 9551: 9546: 9533: 9525: 9522: 9519: 9511: 9506: 9503: 9498: 9490: 9487: 9484: 9467: 9451: 9448: 9445: 9442: 9436: 9433: 9426: 9423: 9419: 9403: 9400: 9393: 9378: 9375: 9368: 9367: 9366: 9363: 9361: 9357: 9353: 9349: 9341: 9325: 9322: 9319: 9316: 9310: 9307: 9300: 9285: 9277: 9274: 9266: 9263: 9256: 9241: 9234: 9230: 9226: 9223: 9218: 9215: 9212: 9205: 9190: 9185: 9176: 9173: 9162: 9159: 9152: 9137: 9129: 9126: 9118: 9115: 9108: 9107: 9106: 9100: 9097: 9094: 9091: 9089: 9085: 9082: 9079: 9076: 9075: 9074: 9072: 9064: 9060: 9056: 9052: 9048: 9043: 9038: 9031: 9025: 9022: 9020: 9015: 9011: 9007: 9002: 8997: 8995: 8979: 8958: 8953: 8949: 8946: 8939: 8935: 8931: 8928: 8923: 8919: 8915: 8912: 8909: 8905: 8899: 8895: 8892: 8886: 8860: 8855: 8851: 8848: 8845: 8841: 8837: 8834: 8830: 8826: 8823: 8810: 8806: 8786: 8766: 8760: 8757: 8754: 8751: 8748: 8745: 8739: 8736: 8733: 8725: 8721: 8705: 8702: 8697: 8692: 8688: 8681: 8678: 8675: 8669: 8664: 8661: 8656: 8651: 8647: 8640: 8637: 8634: 8623: 8620: 8616: 8601: 8581: 8578: 8575: 8572: 8569: 8566: 8563: 8560: 8557: 8549: 8546: 8542: 8539: 8534: 8530: 8526: 8522: 8519: 8514: 8510: 8506: 8502: 8498: 8497: 8496: 8494: 8490: 8486: 8482: 8477: 8475: 8471: 8461: 8459: 8455: 8451: 8446: 8422: 8396: 8393: 8383: 8374: 8364: 8360: 8357: 8347: 8321: 8318: 8308: 8299: 8289: 8285: 8282: 8280: 8245: 8219: 8216: 8206: 8197: 8187: 8183: 8180: 8170: 8144: 8141: 8131: 8122: 8112: 8108: 8105: 8103: 8068: 8042: 8039: 8029: 8020: 8010: 8006: 8003: 7993: 7967: 7964: 7954: 7945: 7935: 7931: 7928: 7926: 7892: 7889: 7869: 7843: 7840: 7830: 7821: 7811: 7807: 7804: 7776: 7755: 7729: 7726: 7716: 7707: 7697: 7693: 7690: 7662: 7660: 7656: 7653:towards body 7652: 7634: 7631: 7607: 7603: 7585: 7582: 7578: 7569: 7551: 7547: 7524: 7521: 7500: 7497: 7492: 7489: 7485: 7479: 7475: 7469: 7466: 7463: 7459: 7455: 7452: 7447: 7425: 7420: 7418: 7402: 7376: 7372: 7369: 7366: 7340: 7331: 7315: 7289: 7279: 7250: 7247: 7243: 7239: 7230: 7226: 7223: 7202: 7198: 7185: 7175: 7173: 7169: 7168:straight line 7165: 7161: 7157: 7152: 7138: 7118: 7096: 7092: 7089: 7083: 7080: 7071: 7056: 7003: 6999: 6993: 6989: 6985: 6982: 6944: 6940: 6936: 6931: 6928: 6909: 6908: 6873: 6860: 6853: 6849: 6842: 6838: 6832: 6828: 6821: 6818: 6809: 6788: 6785: 6782: 6777: 6773: 6765:The relation 6763: 6750: 6745: 6741: 6733: 6729: 6722: 6718: 6712: 6708: 6701: 6698: 6694: 6688: 6685: 6680: 6677: 6674: 6669: 6666: 6660: 6652: 6648: 6641: 6637: 6631: 6627: 6621: 6617: 6610: 6601: 6597: 6588: 6575: 6571: 6565: 6562: 6557: 6551: 6545: 6542: 6539: 6535: 6527: 6523: 6516: 6512: 6506: 6502: 6496: 6492: 6485: 6478: 6474: 6467: 6463: 6457: 6453: 6447: 6443: 6436: 6433: 6427: 6421: 6418: 6415: 6408: 6404: 6397: 6393: 6387: 6383: 6377: 6373: 6366: 6357: 6353: 6327: 6317: 6304: 6298: 6292: 6289: 6286: 6279: 6275: 6268: 6264: 6258: 6254: 6248: 6244: 6237: 6230: 6226: 6221: 6218: 6215: 6205: 6199: 6196: 6193: 6190: 6187: 6184: 6181: 6175: 6172: 6162: 6156: 6153: 6150: 6147: 6144: 6141: 6138: 6132: 6129: 6123: 6114: 6101: 6095: 6089: 6086: 6083: 6079: 6076: 6073: 6070: 6064: 6061: 6055: 6032: 6029: 6019: 6013: 6010: 6007: 6004: 6001: 5994: 5990: 5982: 5979: 5973: 5967: 5958: 5945: 5939: 5933: 5930: 5927: 5924: 5921: 5918: 5913: 5910: 5899: 5886: 5879: 5875: 5868: 5864: 5858: 5854: 5848: 5844: 5837: 5831: 5828: 5822: 5817: 5812: 5805: 5801: 5796: 5793: 5790: 5784: 5779: 5776: 5770: 5767: 5761: 5756: 5749: 5746: 5739: 5736: 5730: 5727: 5721: 5712: 5685: 5675: 5672: 5659: 5656: 5651: 5647: 5643: 5640: 5636: 5629: 5626: 5620: 5617: 5611: 5608: 5599: 5596: 5590: 5586: 5582: 5579: 5573: 5570: 5564: 5557: 5554: 5546: 5542: 5537: 5533: 5508: 5504: 5494: 5493:is constant. 5480: 5477: 5474: 5471: 5465: 5462: 5454: 5450: 5435: 5422: 5416: 5413: 5404: 5401: 5395: 5392: 5386: 5383: 5377: 5374: 5369: 5365: 5356: 5338: 5331: 5328: 5321: 5318: 5312: 5309: 5303: 5298: 5294: 5285: 5255: 5251: 5247: 5230: 5226: 5222: 5216: 5199: 5182: 5166: 5159: 5156: 5147: 5144: 5138: 5135: 5129: 5126: 5120: 5116: 5112: 5094: 5088: 5081: 5078: 5071: 5068: 5062: 5059: 5052: 5048: 5044: 5037: 5018: 5015: 5009: 5006: 4988: 4985: 4979: 4976: 4958: 4955: 4946: 4943: 4936: 4932: 4928: 4921: 4900: 4897: 4891: 4871: 4868: 4861: 4857: 4855: 4847: 4832: 4814: 4811: 4805: 4802: 4782: 4779: 4773: 4767: 4749: 4746: 4726: 4723: 4717: 4715: 4707: 4686: 4684: 4668: 4599: 4569: 4566: 4560: 4557: 4551: 4535: 4529: 4526: 4509: 4502: 4484: 4481: 4475: 4469: 4451: 4445: 4442: 4423: 4412: 4365: 4331: 4328: 4315: 4309: 4305: 4295: 4292: 4290: 4281: 4278: 4277: 4276: 4274: 4266: 4263: 4262: 4261: 4259: 4255: 4251: 4246: 4244: 4236: 4232: 4228: 4225: 4221: 4220: 4219: 4217: 4213: 4212: 4207: 4193: 4190: 4187: 4184: 4183: 4179: 4176: 4173: 4170: 4169: 4165: 4162: 4159: 4156: 4155: 4151: 4148: 4145: 4142: 4141: 4137: 4134: 4131: 4128: 4127: 4123: 4120: 4117: 4114: 4113: 4109: 4106: 4103: 4100: 4099: 4095: 4092: 4089: 4086: 4085: 4058: 4054: 4048: 4044: 4034: 4031: 4028: 4025: 4024: 4018: 4011: 4007: 4003: 3999: 3994: 3986: 3979: 3974: 3971: 3969: 3959: 3956: 3953: 3950: 3949: 3945: 3942: 3939: 3936: 3935: 3931: 3928: 3925: 3922: 3921: 3917: 3914: 3911: 3908: 3907: 3903: 3900: 3897: 3894: 3893: 3889: 3886: 3883: 3880: 3879: 3852: 3848: 3842: 3838: 3828: 3823: 3818: 3815: 3814: 3808: 3800: 3798: 3760: 3740: 3732: 3716: 3696: 3688: 3672: 3640: 3628: 3614: 3611: 3607: 3603: 3600: 3597: 3589: 3585: 3581: 3576: 3573: 3567: 3559: 3555: 3551: 3543: 3540: 3537: 3531: 3525: 3518: 3514: 3508: 3504: 3494: 3493: 3492: 3478: 3475: 3472: 3452: 3432: 3412: 3386: 3382: 3378: 3373: 3369: 3358: 3354: 3349: 3343: 3340: 3333: 3329: 3325: 3319: 3315: 3310: 3306: 3293: 3289: 3284: 3281: 3275: 3272: 3267: 3262: 3257: 3253: 3250: 3244: 3239: 3236: 3229: 3228: 3227: 3212: 3184: 3180: 3175: 3172: 3166: 3163: 3158: 3154: 3150: 3147: 3140: 3139: 3138: 3136: 3131: 3129: 3125: 3120: 3116: 3114: 3108: 3096: 3080: 3077: 3074: 3065: 3058: 3054: 3048: 3044: 3043: 3036: 3032: 3026: 3022: 3021: 3017: 3014: 3013: 3007: 2994: 2989: 2985: 2982: 2979: 2973: 2968: 2964: 2961: 2958: 2952: 2946: 2943: 2938: 2935: 2909: 2906: 2903: 2899: 2896: 2893: 2890: 2887: 2884: 2878: 2874: 2866: 2865: 2864: 2845: 2841: 2838: 2832: 2829: 2822: 2821: 2820: 2818: 2799: 2796: 2793: 2790: 2784: 2781: 2776: 2773: 2765: 2760: 2756: 2750: 2747: 2740: 2739: 2738: 2724: 2704: 2701: 2698: 2689: 2676: 2670: 2667: 2662: 2659: 2651: 2646: 2642: 2636: 2630: 2627: 2622: 2619: 2608: 2592: 2589: 2585: 2582: 2579: 2576: 2571: 2568: 2563: 2560: 2557: 2537: 2534: 2530: 2510: 2490: 2487: 2473: 2450: 2444: 2435: 2420: 2407: 2395: 2389: 2380: 2369: 2366: 2359: 2356: 2351: 2345: 2342: 2337: 2334: 2323: 2310: 2298: 2292: 2283: 2272: 2269: 2264: 2258: 2255: 2250: 2247: 2235: 2233: 2229: 2210: 2207: 2198: 2192: 2183: 2172: 2169: 2164: 2151: 2148: 2141: 2132: 2121: 2118: 2113: 2110: 2107: 2099: 2097: 2093: 2089: 2080: 2077: 2074: 2070: 2066: 2063: 2059: 2058: 2057: 2049: 2040: 2035: 2031: 2023: 2021: 2017: 2013: 2009: 2002: 1995: 1991: 1987: 1983: 1964: 1960: 1957: 1954: 1951: 1948: 1941: 1940: 1939: 1937: 1918: 1906: 1902: 1893: 1881: 1877: 1868: 1861: 1858: 1851: 1850: 1849: 1844: 1837: 1833: 1829: 1803: 1799: 1795: 1786: 1776: 1772: 1770: 1765: 1762: 1753: 1750: 1745: 1740: 1734: 1731: 1722: 1718: 1713: 1710: 1701: 1694: 1689: 1687: 1682: 1671: 1670: 1669: 1664: 1657: 1653: 1652:harmonic mean 1649: 1621: 1617: 1613: 1610: 1606: 1601: 1599: 1594: 1579: 1569: 1563: 1561: 1556: 1545: 1544: 1543: 1538: 1531: 1527: 1523: 1520: 1492: 1488: 1484: 1481: 1477: 1472: 1470: 1465: 1456: 1446: 1442: 1433: 1426: 1424: 1419: 1408: 1407: 1406: 1401: 1394: 1390: 1386: 1383: 1361: 1358: 1355: 1351: 1346: 1337: 1329: 1328: 1327: 1325: 1321: 1316: 1302: 1294: 1291:= 90° and at 1290: 1268: 1265: 1262: 1258: 1253: 1244: 1236: 1235: 1234: 1232: 1228: 1223: 1221: 1217: 1212: 1210: 1206: 1202: 1198: 1194: 1190: 1186: 1182: 1166: 1143: 1137: 1134: 1131: 1127: 1124: 1121: 1117: 1112: 1109: 1102: 1101: 1100: 1090: 1086: 1073: 1069: 1060: 1056: 1052: 1048: 1044: 1037: 1032: 1024: 1019: 1017: 1011: 1003: 995: 993: 989: 985: 981: 977: 971: 969: 965: 964: 959: 958: 953: 949: 945: 941: 936: 934: 930: 926: 916: 912: 910: 906: 902: 898: 894: 890: 889: 884: 880: 876: 866: 864: 860: 840: 837: 834: 828: 825: 822: 817: 814: 811: 803: 800: 795: 792: 785: 784: 783: 781: 777: 773: 772:March equinox 769: 765: 757: 753: 752: 747: 744: 743: 738: 734: 731: 730: 725: 721: 720: 719: 712: 709: 706: 705: 704: 702: 698: 689: 687: 686: 681: 680: 674: 672: 668: 664: 660: 658: 654: 647:of its orbit. 646: 642: 638: 635: 632: 628: 624: 623: 622: 620: 616: 612: 608: 604: 600: 596: 592: 580: 575: 573: 568: 566: 561: 560: 558: 557: 550: 547: 545: 542: 541: 535: 534: 527: 526:Oberth effect 524: 522: 519: 518: 512: 511: 504: 501: 499: 496: 494: 491: 489: 486: 485: 479: 478: 474: 468: 467: 460: 457: 455: 452: 451: 445: 441: 440: 438: 432: 431:N-body orbits 427: 426: 419: 416: 414: 413:Perturbations 411: 409: 406: 404: 401: 400: 394: 393: 389: 383: 382: 375: 372: 370: 367: 365: 362: 360: 357: 355: 352: 350: 347: 345: 342: 340: 337: 335: 332: 331: 325: 324: 317: 314: 312: 309: 307: 304: 302: 299: 298: 292: 289: 287: 283: 282: 280: 274: 271: 269: 266: 265: 259: 253: 252: 245: 242: 240: 237: 235: 234:Orbital nodes 232: 230: 227: 225: 222: 220: 217: 215: 212: 210: 207: 206: 203: 198: 197: 193: 187: 186: 182: 178: 177: 174:Astrodynamics 173: 172: 168: 167: 144: 140: 136: 125: 121: 115: 108: 104: 91: 85: 78: 74: 68: 61: 54: 47: 40: 36: 35: 31: 27: 19: 14854:Solar System 14765:Isaac Newton 14677: 14670: 14662: 14653: 14586:Isaac Barrow 14524: (home) 14324: 14265:Newtonianism 14240:Newton scale 14203:Impact depth 14176: (1754) 14171: 14168: (1728) 14163: 14153: 14140: 14125: 14111: (1711) 14106: 14103: (1707) 14098: 14095: (1704) 14090: 14087: (1704) 14082: 14079: (1687) 14074: 14071: (1684) 14066: 14063: (1671) 14058: 14052:Publications 13995: 13988: 13974: 13971:(son-in-law) 13954: 13935: 13927: 13919: 13911: 13903: 13895: 13887: 13847:Kepler orbit 13841: 13743:Perturbation 13725: 13714: 13700:Ground track 13610:Gravity turn 13561: 13554: 13547: 13538: 13529: 13509: 13500: 13491: 13484:True anomaly 13482: 13467:Mean anomaly 13465: 13445: 13436: 13427: 13418: 13398: 13385: 13376: 13369:Eccentricity 13367: 13325:Lunar cycler 13298:Heliocentric 13238:other points 13187:Medium Earth 13085:Non-inclined 12968: 12932: 12912: 12878: 12864: 12844: 12807: 12803: 12797: 12793: 12772:. Retrieved 12761: 12749:. Retrieved 12736: 12728: 12723: 12715: 12710: 12689: 12681: 12660: 12652: 12607: 12603: 12593: 12573: 12566: 12558: 12555:Burtt, Edwin 12550: 12531: 12525: 12515: 12506: 12486: 12479: 12454: 12443: 12415: 12408: 12400: 12385: 12384:(in Latin). 12381: 12369: 12360: 12356: 12350: 12338:. Retrieved 12334: 12324: 12312: 12304: 12298: 12294: 12290: 12282: 12266: 12258: 12251: 12248:pp. 291–295. 12243: 12236: 12230:pp. 165–167. 12228:... (1609), 12225: 12221: 12214: 12210: 12205: 12195:December 27, 12193:. Retrieved 12178: 12160: 12156: 12148: 12142: 12138: 12134: 12129:... (1609), 12126: 12120: 12115: 12106: 12086: 12079: 12066: 12060:pp. 298–299. 12055: 12050: 12040: 12038:From p. 435: 12031: 12029:From p. 430: 12022: 12020:From p. 429: 12013: 12011:From p. 390: 12004: 11999: 11989:December 27, 11987:. Retrieved 11982: 11956: 11954:From p. 205: 11947: 11945:From p. 162: 11938: 11933: 11921:. Retrieved 11904:(9): 50–54. 11901: 11897: 11870:. Retrieved 11866: 11857: 11846:. Retrieved 11844:. 2009-07-07 11841: 11832: 11821:. Retrieved 11817: 11794:. Retrieved 11790: 11760: 11756: 11748: 11742: 11734: 11730: 11726: 11716: 11711: 11676:Kepler orbit 11595: 11591: 11587: 11585: 11505: 11501: 11499: 11494: 11488: 11388: 11386: 10947: 10857: 10852: 10848: 10846: 10526:Dividing by 10525: 10396: 10391: 10381: 10377: 10370: 10366: 10362: 10358: 10354: 10352: 10347: 10343: 10339: 10335: 10331: 10327: 10323: 10319: 10317: 10312: 10308: 10304: 10300: 10298: 10293: 10289: 10287: 10282: 10281:for a given 10278: 10274: 10270: 10268: 10217: 10216:Division by 10215: 9923: 9919: 9915: 9911: 9903: 9862: 9858: 9854: 9850: 9846: 9844: 9839: 9827: 9791: 9787: 9617: 9547: 9471: 9466:mean anomaly 9424:| are equal, 9421: 9417: 9364: 9359: 9355: 9351: 9345: 9340:true anomaly 9104: 9098: 9092: 9083: 9077: 9068: 9062: 9058: 9054: 9050: 9046: 9037:Mean anomaly 9029: 9023: 9018: 9013: 9009: 9005: 9000: 8998: 8808: 8804: 8802: 8723: 8621: 8619:true anomaly 8617:Compute the 8547: 8543:Compute the 8537: 8532: 8528: 8525:mean anomaly 8523:Compute the 8517: 8512: 8508: 8504: 8499:Compute the 8488: 8484: 8480: 8478: 8467: 8458:Lunar theory 8454:perturbation 8447: 7893: 7890: 7777: 7663: 7658: 7654: 7650: 7605: 7601: 7567: 7423: 7421: 7280: 7181: 7172:Kepler orbit 7153: 7072: 6905: 6874: 6764: 6589: 6318: 6115: 5959: 5900: 5676: 5673: 5495: 5441: 5354: 5283: 5200: 4687: 4600: 4314:heliocentric 4311: 4293: 4285: 4270: 4247: 4242: 4240: 4234: 4230: 4223: 4216:acceleration 4209: 4206:Isaac Newton 4204: 4021:Modern data 4015: 4009: 4005: 4001: 3997: 3984: 3976: 3972: 3965: 3821:to sun (AU) 3806: 3664: 3403: 3203: 3132: 3128:harmonic law 3127: 3121: 3118: 3110: 3106: 3094: 2924: 2862: 2814: 2690: 2479: 2421: 2324: 2236: 2231: 2227: 2100: 2095: 2091: 2087: 2084: 2055: 2046: 2033: 2029: 2019: 2015: 2011: 2007: 2000: 1993: 1989: 1985: 1981: 1979: 1933: 1842: 1835: 1827: 1825: 1662: 1655: 1647: 1645: 1536: 1529: 1521: 1516: 1399: 1392: 1384: 1379: 1319: 1317: 1292: 1288: 1286: 1226: 1224: 1219: 1215: 1213: 1204: 1200: 1196: 1192: 1189:eccentricity 1184: 1158: 1098: 1088: 1084: 1071: 1067: 1054: 1050: 1046: 1042: 1035: 1013: 1009: 1001: 980:Wilhelm Lenz 972: 961: 955: 937: 933:eccentricity 922: 913: 900: 897:Robert Small 886: 882: 878: 872: 869:Nomenclature 855: 764:eccentricity 761: 749: 740: 736: 727: 723: 716: 695: 683: 677: 675: 663:Isaac Newton 661: 657:Solar System 650: 594: 588: 348: 311:Radial orbit 262:eccentricity 244:True anomaly 229:Mean anomaly 219:Eccentricity 83: 76: 66: 59: 52: 45: 38: 26: 14842:Outer space 14830:Spaceflight 14665:by Paolozzi 14604:Roger Cotes 14213:Newton disc 14127:Quaestiones 14100:Arithmetica 13705:Hill sphere 13540:Mean motion 13420:Inclination 13409:Orientation 13310:Mars cycler 13248:Areocentric 13120:Synchronous 12919:online book 12830:, ed. 2002 12794:magnum opus 12774:23 February 12727:I. Newton, 12714:I. Newton, 12309:J. V. Field 12233:On page 167 11867:history.com 9832:mean motion 8501:mean motion 4411:unit vector 3968:John Napier 2817:mean motion 2523:and height 984:phase space 925:Tycho Brahe 907:, based on 742:focal point 735:The Sun is 444:Halo orbits 408:Hill sphere 224:Inclination 14864:Categories 14752:Categories 14728:XMM-Newton 14645:Depictions 14616:John Keill 14538:Apple tree 14533:Later life 14528:Early life 14108:De Analysi 13645:Rendezvous 13341:Parameters 13177:High Earth 13147:Geocentric 13100:Osculating 13057:Elliptical 12449:Roy Porter 12311:, trans., 12005:Astronomie 11937:Voltaire, 11872:2022-12-13 11848:2022-12-13 11823:2022-12-13 11796:2022-12-13 11770:References 11493:Distance, 9101:the planet 9088:perihelion 8493:perihelion 6346:satisfies 5282:where the 4302:See also: 4177:30687.153 4163:10775.599 4149:4332.8201 2863:satisfies 2737:satisfies 2026:Second law 1231:perihelion 976:Carl Runge 859:perihelion 751:area speed 701:Copernicus 488:Mass ratio 403:Barycenter 14806:Astronomy 14567:Relations 14076:Principia 13690:Ephemeris 13667:mechanics 13577:Maneuvers 13520:Variation 13283:Libration 13278:Lissajous 13182:Low Earth 13172:Graveyard 13071:Horseshoe 12729:Principia 12716:Principia 11985:(31): 1–2 11630:⁡ 11624:ε 11621:− 11561:ε 11557:− 11537:θ 11534:⁡ 11528:ε 11465:⁡ 11449:ε 11429:θ 11424:⁡ 11408:ε 11405:− 11361:θ 11356:⁡ 11340:ε 11329:ε 11326:− 11311:θ 11308:⁡ 11294:θ 11291:⁡ 11285:− 11276:⋅ 11270:ε 11259:ε 11256:− 11238:θ 11235:⁡ 11226:ε 11214:θ 11211:⁡ 11205:ε 11188:θ 11185:⁡ 11176:ε 11170:− 11164:θ 11161:⁡ 11155:ε 11124:θ 11121:⁡ 11115:ε 11104:θ 11101:⁡ 11092:ε 11075:θ 11072:⁡ 11066:ε 11055:θ 11052:⁡ 11043:ε 11037:− 11019:⁡ 11002:⁡ 10996:− 10970:⁡ 10925:⁡ 10908:⁡ 10902:− 10880:⁡ 10827:θ 10824:⁡ 10818:ε 10807:θ 10804:⁡ 10795:ε 10783:θ 10780:⁡ 10774:ε 10763:θ 10760:⁡ 10743:ε 10739:− 10722:θ 10719:⁡ 10713:ε 10701:ε 10692:θ 10689:⁡ 10680:θ 10677:⁡ 10671:ε 10654:ε 10650:− 10638:ε 10629:⁡ 10603:θ 10600:⁡ 10594:ε 10577:ε 10573:− 10509:θ 10506:⁡ 10494:ε 10482:⁡ 10373:would be 10249:⁡ 10243:ε 10240:− 10220:/2 gives 10189:⁡ 10180:⋅ 10177:ε 10168:− 10005:− 9879:∠ 9728:⋅ 9693:⋅ 9658:⋅ 9512:⋅ 9440:∠ 9314:∠ 9308:θ 9231:ε 9227:− 9160:ε 8980:μ 8950:⁡ 8936:ε 8932:− 8916:⁡ 8910:− 8893:μ 8856:θ 8852:⁡ 8842:θ 8838:⁡ 8758:⁡ 8752:ε 8749:− 8698:⁡ 8682:ε 8662:θ 8657:⁡ 8641:ε 8638:− 8602:ε 8576:⁡ 8570:ε 8567:− 8411:^ 8394:− 8336:^ 8319:− 8267:¨ 8234:^ 8217:− 8159:^ 8142:− 8090:¨ 8057:^ 8040:− 7982:^ 7965:− 7913:¨ 7858:^ 7841:− 7793:¨ 7744:^ 7727:− 7679:¨ 7625:^ 7604:and body 7515:^ 7498:− 7467:≠ 7460:∑ 7441:¨ 7367:α 7316:α 7264:^ 7248:− 7240:α 7227:− 7218:¨ 7160:hyperbola 7139:α 7093:π 7034:^ 6983:α 6960:^ 6937:α 6932:− 6923:¨ 6889:¨ 6822:− 6702:− 6681:− 6675:− 6558:− 6552:θ 6546:⁡ 6540:ε 6434:− 6428:θ 6422:⁡ 6416:ε 6299:θ 6293:⁡ 6287:ε 6206:θ 6200:⁡ 6194:ε 6176:˙ 6173:θ 6163:θ 6157:⁡ 6151:ε 6133:¨ 6096:θ 6090:⁡ 6084:ε 6065:˙ 6033:˙ 6030:θ 6020:θ 6014:⁡ 6008:ε 6005:− 5983:˙ 5968:− 5940:θ 5934:⁡ 5928:ε 5838:− 5832:¨ 5777:− 5771:¨ 5750:˙ 5747:θ 5737:− 5731:¨ 5652:θ 5630:¨ 5627:θ 5612:˙ 5609:θ 5600:˙ 5558:˙ 5555:θ 5523:is zero: 5509:θ 5466:˙ 5463:θ 5417:˙ 5414:θ 5405:˙ 5387:¨ 5384:θ 5370:θ 5332:˙ 5329:θ 5319:− 5313:¨ 5267:^ 5264:θ 5256:θ 5242:^ 5217:¨ 5177:^ 5174:θ 5160:˙ 5157:θ 5148:˙ 5130:¨ 5127:θ 5107:^ 5082:˙ 5079:θ 5069:− 5063:¨ 5038:˙ 5033:^ 5030:θ 5019:˙ 5016:θ 5001:^ 4998:θ 4989:¨ 4986:θ 4971:^ 4968:θ 4959:˙ 4956:θ 4947:˙ 4922:˙ 4917:^ 4901:˙ 4886:^ 4872:¨ 4848:¨ 4827:^ 4824:θ 4815:˙ 4812:θ 4797:^ 4783:˙ 4768:˙ 4763:^ 4741:^ 4727:˙ 4708:˙ 4669:θ 4646:^ 4615:^ 4612:θ 4584:^ 4570:˙ 4567:θ 4561:− 4552:˙ 4547:^ 4544:θ 4519:^ 4516:θ 4497:^ 4494:θ 4485:˙ 4482:θ 4470:˙ 4465:^ 4435:^ 4394:^ 4343:^ 4312:From the 4243:Principia 4231:magnitude 4224:direction 4191:60190.03 4135:686.9796 4121:365.2564 4107:224.7008 3612:− 3604:× 3598:≈ 3586:π 3568:≈ 3556:π 3379:∝ 3365:⟹ 3330:π 3302:⟹ 3254:π 3155:ω 3103:Third law 2980:π 2888:θ 2842:π 2794:π 2777:θ 2751:⋅ 2699:π 2663:θ 2593:θ 2583:⋅ 2577:⋅ 2550:and area 2538:θ 2454:→ 2445:× 2439:→ 2399:→ 2390:× 2384:→ 2302:→ 2293:× 2287:→ 2202:→ 2193:× 2187:→ 2156:→ 2142:× 2136:→ 2043:velocity. 1955:π 1878:− 1859:ε 1751:− 1732:− 1711:− 1618:ε 1614:− 1489:ε 1485:− 1362:ε 1359:− 1269:ε 1138:θ 1135:⁡ 1128:ε 1006:First law 998:Formulary 963:Principia 835:≈ 815:− 801:π 796:≈ 607:epicycles 591:astronomy 328:Equations 256:Types of 14690:Namesake 14656:by Blake 14250:Spectrum 14191:Calculus 14160: ) 14060:Fluxions 13965:(mother) 13456:Position 13081:Inclined 13052:Circular 12955:Archived 12903:Archived 12845:Dynamics 12824:Einstein 12644:30886926 12561:. p. 52. 12451:(1992). 12340:2 August 12263:page 668 11983:HAD News 11731:sescupla 11654:See also 10468:so that 8972:, where 8959:⟩ 8906:⟨ 8861:⟩ 8831:⟨ 8536:, where 8516:, where 7661:itself. 7164:parabola 5353:and the 4188:30.0690 4185:Neptune 4174:19.1913 4160:9.53707 4146:5.20336 4143:Jupiter 4132:1.52366 4104:0.72333 4093:87.9693 4090:0.38710 4087:Mercury 4082:AU/day) 3981:— 3957:10759.2 3943:4332.62 3937:Jupiter 3881:Mercury 3876:AU/day) 2925:And so, 2815:and the 2072:physics. 1834:between 1654:between 1528:between 1391:between 1324:aphelion 1322:= 180°, 1080:θ = 180° 1078:and for 875:Voltaire 863:solstice 14794:Physics 14780:Portals 14208:Inertia 14196:fluxion 14092:Queries 14084:Opticks 14068:De Motu 13979:(opera) 13959:(opera) 13948:Related 13937:Somnium 13665:Orbital 13635:Phasing 13595:Delta-v 13400:Apsides 13394:, 13192:Molniya 13110:Parking 13047:Capture 13035:General 12981:YouTube 12816:Galileo 12635:6395789 12612:Bibcode 12604:Heliyon 12421:Bibcode 12287:p. 189. 12215:Epitome 12209:In his 12131:p. 285. 12119:In his 11923:27 July 11906:Bibcode 11671:Gravity 10618:to get 10375:√ 10326:,0) ), 10322:,0) = ( 10307:, 9830:is the 9420:| and | 8992:is the 8472:called 7415:is the 7170:. (See 4260:). So: 4171:Uranus 4157:Saturn 4026:Planet 4008:³/ 3951:Saturn 3929:686.95 3915:365.25 3901:224.70 3826:(days) 3824:Period 3816:Planet 3795:is the 3729:is the 3685:is the 1830:is the 1650:is the 1524:is the 1387:is the 1203:, 1187:is the 1179:is the 1159:where 944:Jupiter 919:History 780:equator 774:to the 766:of the 729:ellipse 627:ellipse 609:in the 14900:Orbits 14663:Newton 14654:Newton 13976:Kepler 13940:(1634) 13932:(1627) 13924:(1619) 13908:(1609) 13900:(1606) 13892:(1596) 13321:Other 13222:Tundra 13090:Kepler 13066:Escape 13019:orbits 12900:applet 12885: 12871: 12852: 12834: 12822:, and 12820:Newton 12812:Kepler 12698: 12690:Kepler 12669: 12661:Kepler 12642: 12632: 12581: 12538: 12494: 12467: 12431: 12317:p. 411 12186: 12094: 9826:where 9019:normal 8779:where 8594:where 8491:since 7539:where 7395:where 7294:planet 7281:where 7235:planet 7207:planet 7111:where 6975:where 4661:, and 4601:where 4358:where 4306:, and 4194:7.504 4180:7.506 4166:7.498 4152:7.504 4138:7.495 4124:7.496 4115:Earth 4110:7.496 4101:Venus 4096:7.496 4080: 4076: 3954:9.510 3926:1.524 3909:Earth 3898:0.724 3895:Venus 3887:87.77 3884:0.389 3874: 3870: 3665:where 1229:= 0°, 1207:) are 1063:θ = 0° 1040: 14818:Stars 14498:table 13881:Works 13563:Epoch 13352:Shape 13290:Lunar 13244:Mars 13236:About 13207:Polar 13027:Types 12967:" in 12917:, an 12751:4 May 12746:(PDF) 12359:[ 11979:(PDF) 11894:(PDF) 9552:area 8511:rad)/ 8427:Earth 8388:Earth 8369:Earth 8242:Earth 8203:Earth 8167:Earth 8128:Earth 8097:Earth 7998:Earth 7959:Earth 7940:Earth 7874:Earth 7835:Earth 7816:Earth 7752:Earth 7713:Earth 7686:Earth 7359:. So 7166:or a 4409:is a 4250:force 4129:Mars 3960:7.43 3946:7.49 3940:5.20 3932:7.50 3923:Mars 3918:7.50 3904:7.52 3890:7.64 3803:Table 3601:7.496 3465:with 3018:High 2237:Thus 899:, in 838:0.015 617:with 209:Apsis 13355:Size 13294:Sun 13273:Halo 13125:semi 12883:ISBN 12869:ISBN 12850:ISBN 12832:ISBN 12776:2013 12753:2018 12696:ISBN 12667:ISBN 12640:PMID 12579:ISBN 12536:ISBN 12492:ISBN 12465:ISBN 12429:ISBN 12342:2018 12197:2009 12184:ISBN 12092:ISBN 11991:2016 11925:2023 11598:is: 11590:and 10948:Get 9548:The 9464:the 9105:and 9086:the 8507:= (2 8419:Moon 8380:Moon 8344:Moon 8305:Moon 8274:Moon 8250:Moon 8211:Moon 8192:Moon 8073:Moon 8034:Moon 8015:Moon 7866:Moon 7827:Moon 7800:Moon 4254:mass 4229:The 4222:The 4214:the 3636:days 2090:and 2068:law. 2014:and 1936:area 1934:The 1841:and 1661:and 1535:and 1517:The 1398:and 1380:The 1059:foci 1016:foci 978:and 762:The 682:and 669:and 653:Mars 631:foci 605:and 82:and 58:and 44:and 13991:ATV 13130:sub 13042:Box 12979:on 12935:, " 12826:). 12630:PMC 12620:doi 12514:", 12461:102 12390:doi 11914:doi 11627:cos 11531:cos 11456:tan 11415:tan 11347:tan 11305:cos 11288:cos 11232:cos 11208:cos 11182:cos 11158:cos 11118:cos 11098:cos 11069:cos 11049:cos 11016:cos 10999:cos 10961:tan 10922:cos 10905:cos 10871:tan 10821:cos 10801:cos 10777:cos 10757:cos 10716:cos 10686:cos 10674:cos 10626:cos 10597:cos 10503:cos 10479:cos 10330:= ( 10246:sin 10186:sin 9422:zsx 9418:zcy 8947:cos 8913:sin 8849:sin 8835:cos 8755:cos 8689:tan 8648:tan 8573:sin 8460:.) 8352:Sun 8313:Sun 8294:Sun 8175:Sun 8136:Sun 8117:Sun 8065:Sun 8026:Sun 7990:Sun 7951:Sun 7920:Sun 7760:Sun 7721:Sun 7702:Sun 7381:Sun 7345:Sun 7182:By 7174:.) 7162:or 6543:cos 6419:cos 6290:cos 6197:cos 6154:cos 6087:sin 6048:or 6011:sin 5931:cos 5704:is 5357:is 5286:is 5201:So 4683:dot 4291:.) 4078:(10 3872:(10 3015:Low 2609:is 2004:max 1997:min 1992:= 1911:min 1898:max 1886:min 1873:max 1846:max 1839:min 1791:min 1781:max 1727:min 1706:max 1666:max 1659:min 1584:min 1574:max 1540:max 1533:min 1451:min 1438:max 1403:max 1396:min 1342:max 1318:At 1287:At 1249:min 1225:At 1132:cos 1092:max 1075:min 942:of 877:'s 829:179 823:186 818:179 812:186 737:not 724:not 613:of 589:In 260:by 14866:: 13478:, 13474:, 13083:/ 13059:/ 12953:) 12818:, 12814:, 12796:, 12638:. 12628:. 12618:. 12606:. 12602:. 12557:. 12463:. 12427:. 12380:. 12333:. 12274:^ 12168:^ 11981:. 11966:^ 11912:. 11902:64 11900:. 11896:. 11881:^ 11865:. 11840:. 11816:. 11805:^ 11789:. 11778:^ 10862:: 10855:. 10386:. 10350:) 10342:, 10340:ae 10338:– 10324:ae 10315:) 10296:. 9834:. 9794:. 9362:. 9055:xd 9012:= 9008:= 8996:. 8726:: 8533:nt 8531:= 8476:. 7608:, 7570:, 7419:. 5660:0. 4275:: 4237:). 4118:1 3912:1 3782:AU 3733:, 3689:, 3624:AU 3608:10 3093:. 2234:. 2232:dt 2228:dx 2098:. 2096:dx 2092:dx 2022:. 2020:πr 2018:= 2010:= 2006:= 1999:= 1988:= 1848:: 1668:: 1542:: 1405:: 1315:. 1211:. 1183:, 1087:= 1082:, 1070:= 1065:, 946:. 911:. 688:. 673:. 593:, 14782:: 14158:" 14154:" 14148:" 14133:" 14037:e 14030:t 14023:v 13815:e 13808:t 13801:v 13726:n 13558:0 13555:t 13545:v 13536:n 13527:T 13507:l 13498:L 13489:E 13480:f 13476:θ 13472:ν 13463:M 13443:ϖ 13434:ω 13425:Ω 13416:i 13396:q 13392:Q 13383:b 13374:a 13365:e 13010:e 13003:t 12996:v 12963:" 12946:" 12939:" 12860:. 12858:. 12802:( 12778:. 12755:. 12704:. 12675:. 12646:. 12622:: 12614:: 12608:5 12587:. 12544:. 12500:. 12473:. 12437:. 12423:: 12396:. 12392:: 12386:5 12344:. 12319:. 12199:. 12100:. 11993:. 11927:. 11916:: 11908:: 11875:. 11851:. 11826:. 11799:. 11639:. 11636:) 11633:E 11618:1 11615:( 11612:a 11609:= 11606:r 11596:r 11592:E 11588:θ 11571:) 11565:2 11554:1 11550:( 11546:a 11543:= 11540:) 11525:+ 11522:1 11519:( 11516:r 11506:θ 11502:r 11495:r 11473:2 11470:E 11460:2 11452:) 11446:+ 11443:1 11440:( 11437:= 11432:2 11419:2 11411:) 11402:1 11399:( 11389:ε 11369:. 11364:2 11351:2 11337:+ 11334:1 11323:1 11317:= 11302:+ 11299:1 11282:1 11267:+ 11264:1 11253:1 11247:= 11241:) 11229:+ 11223:( 11220:+ 11217:) 11202:+ 11199:1 11196:( 11191:) 11179:+ 11173:( 11167:) 11152:+ 11149:1 11146:( 11140:= 11112:+ 11109:1 11095:+ 11086:+ 11083:1 11063:+ 11060:1 11046:+ 11034:1 11028:= 11022:E 11013:+ 11010:1 11005:E 10993:1 10987:= 10978:2 10975:E 10965:2 10934:. 10928:x 10919:+ 10916:1 10911:x 10899:1 10893:= 10888:2 10885:x 10875:2 10853:θ 10849:E 10833:. 10815:+ 10812:1 10798:+ 10789:= 10771:+ 10768:1 10753:) 10747:2 10736:1 10732:( 10728:+ 10725:) 10710:+ 10707:1 10704:( 10695:= 10668:+ 10665:1 10658:2 10647:1 10641:+ 10635:= 10632:E 10591:+ 10588:1 10581:2 10570:1 10564:= 10559:a 10556:r 10534:a 10512:. 10500:r 10497:+ 10491:a 10488:= 10485:E 10476:a 10455:| 10451:d 10448:s 10444:| 10440:+ 10436:| 10432:s 10429:c 10425:| 10421:= 10417:| 10413:d 10410:c 10406:| 10392:θ 10382:r 10378:r 10371:r 10367:x 10363:r 10361:/ 10359:y 10355:r 10348:E 10344:b 10336:E 10332:a 10328:r 10320:c 10313:E 10309:b 10305:E 10301:a 10294:θ 10290:E 10283:M 10279:E 10275:E 10271:M 10255:. 10252:E 10237:E 10234:= 10231:M 10218:a 10196:2 10192:E 10183:a 10174:a 10163:2 10159:E 10154:2 10150:a 10143:= 10134:2 10130:M 10125:2 10121:a 10108:2 10103:| 10099:x 10096:d 10092:| 10088:. 10084:| 10080:s 10077:c 10073:| 10066:= 10058:| 10054:x 10051:c 10048:s 10044:| 10040:h 10037:t 10034:i 10031:w 10023:| 10019:x 10016:c 10013:s 10009:| 10001:| 9997:x 9994:c 9991:z 9987:| 9983:= 9979:| 9975:x 9972:s 9969:z 9965:| 9961:= 9953:| 9949:y 9946:c 9943:z 9939:| 9924:E 9920:θ 9916:M 9912:E 9904:x 9891:, 9888:x 9885:c 9882:z 9876:= 9873:E 9863:M 9861:( 9859:f 9855:θ 9851:θ 9847:M 9840:E 9828:n 9814:, 9811:t 9808:n 9805:= 9802:M 9792:t 9788:M 9774:, 9769:2 9765:M 9762:b 9759:a 9753:= 9748:2 9744:M 9739:2 9735:a 9723:a 9720:b 9715:= 9711:| 9707:y 9704:c 9701:z 9697:| 9688:a 9685:b 9680:= 9676:| 9672:x 9669:s 9666:z 9662:| 9653:a 9650:b 9645:= 9641:| 9637:p 9634:s 9631:z 9627:| 9604:. 9599:2 9595:M 9590:2 9586:a 9579:= 9575:| 9571:y 9568:c 9565:z 9561:| 9534:. 9530:| 9526:x 9523:s 9520:z 9516:| 9507:a 9504:b 9499:= 9495:| 9491:p 9488:s 9485:z 9481:| 9468:. 9452:, 9449:y 9446:c 9443:z 9437:= 9434:M 9404:, 9401:y 9379:, 9376:x 9360:t 9356:θ 9354:, 9352:r 9350:( 9342:. 9326:, 9323:p 9320:s 9317:z 9311:= 9286:, 9282:| 9278:p 9275:s 9271:| 9267:= 9264:r 9242:, 9235:2 9224:1 9219:a 9216:= 9213:b 9191:, 9186:a 9181:| 9177:s 9174:c 9170:| 9163:= 9138:, 9134:| 9130:z 9127:c 9123:| 9119:= 9116:a 9099:p 9093:c 9084:z 9078:s 9065:. 9063:y 9059:P 9051:P 9047:S 9030:M 9014:M 9010:E 9006:θ 9001:ε 8954:E 8940:2 8929:1 8924:, 8920:E 8900:r 8896:a 8887:= 8883:v 8846:, 8827:r 8824:= 8820:p 8809:θ 8807:, 8805:r 8787:a 8767:, 8764:) 8761:E 8746:1 8743:( 8740:a 8737:= 8734:r 8724:r 8706:2 8703:E 8693:2 8685:) 8679:+ 8676:1 8673:( 8670:= 8665:2 8652:2 8644:) 8635:1 8632:( 8622:θ 8582:, 8579:E 8564:E 8561:= 8558:M 8548:E 8538:t 8529:M 8518:P 8513:P 8509:π 8505:n 8489:t 8485:θ 8483:, 8481:r 8423:, 8407:r 8397:2 8384:, 8375:r 8365:m 8361:G 8358:+ 8348:, 8332:r 8322:2 8309:, 8300:r 8290:m 8286:G 8283:= 8264:r 8246:, 8230:r 8220:2 8207:, 8198:r 8188:m 8184:G 8181:+ 8171:, 8155:r 8145:2 8132:, 8123:r 8113:m 8109:G 8106:= 8087:r 8069:, 8053:r 8043:2 8030:, 8021:r 8011:m 8007:G 8004:+ 7994:, 7978:r 7968:2 7955:, 7946:r 7936:m 7932:G 7929:= 7910:r 7870:, 7854:r 7844:2 7831:, 7822:r 7812:m 7808:G 7805:= 7790:r 7756:, 7740:r 7730:2 7717:, 7708:r 7698:m 7694:G 7691:= 7676:r 7659:i 7655:j 7651:i 7635:j 7632:i 7621:r 7606:j 7602:i 7586:j 7583:i 7579:r 7568:j 7552:j 7548:m 7525:j 7522:i 7511:r 7501:2 7493:j 7490:i 7486:r 7480:j 7476:m 7470:i 7464:j 7456:G 7453:= 7448:i 7438:r 7424:i 7403:G 7377:m 7373:G 7370:= 7341:m 7290:m 7260:r 7251:2 7244:r 7231:m 7224:= 7215:r 7203:m 7199:= 7195:F 7119:T 7097:T 7090:2 7084:= 7081:n 7057:r 7030:r 7004:3 7000:a 6994:2 6990:n 6986:= 6956:r 6945:2 6941:r 6929:= 6920:r 6886:r 6861:. 6854:2 6850:r 6843:3 6839:a 6833:2 6829:n 6819:= 6814:r 6810:a 6789:a 6786:p 6783:= 6778:2 6774:b 6751:. 6746:2 6742:b 6734:2 6730:r 6723:2 6719:a 6713:2 6709:n 6699:= 6695:) 6689:r 6686:p 6678:1 6670:r 6667:p 6661:( 6653:2 6649:r 6642:2 6638:b 6632:2 6628:a 6622:2 6618:n 6611:= 6606:r 6602:a 6598:p 6576:. 6572:) 6566:r 6563:p 6555:) 6549:( 6536:( 6528:2 6524:r 6517:2 6513:b 6507:2 6503:a 6497:2 6493:n 6486:= 6479:3 6475:r 6468:2 6464:b 6458:2 6454:a 6448:2 6444:n 6437:p 6431:) 6425:( 6409:2 6405:r 6398:2 6394:b 6388:2 6384:a 6378:2 6374:n 6367:= 6362:r 6358:a 6354:p 6332:r 6328:a 6305:. 6302:) 6296:( 6280:2 6276:r 6269:2 6265:b 6259:2 6255:a 6249:2 6245:n 6238:= 6231:2 6227:r 6222:b 6219:a 6216:n 6209:) 6203:( 6191:b 6188:a 6185:n 6182:= 6166:) 6160:( 6148:b 6145:a 6142:n 6139:= 6130:r 6124:p 6102:. 6099:) 6093:( 6080:b 6077:a 6074:n 6071:= 6062:r 6056:p 6023:) 6017:( 6002:= 5995:2 5991:r 5980:r 5974:p 5946:. 5943:) 5937:( 5925:+ 5922:1 5919:= 5914:r 5911:p 5887:. 5880:3 5876:r 5869:2 5865:b 5859:2 5855:a 5849:2 5845:n 5829:r 5823:= 5818:2 5813:) 5806:2 5802:r 5797:b 5794:a 5791:n 5785:( 5780:r 5768:r 5762:= 5757:2 5740:r 5728:r 5722:= 5717:r 5713:a 5690:r 5686:a 5657:= 5648:a 5644:r 5641:= 5637:) 5621:r 5618:+ 5597:r 5591:2 5587:( 5583:r 5580:= 5574:t 5571:d 5565:) 5547:2 5543:r 5538:( 5534:d 5505:a 5481:b 5478:a 5475:n 5472:= 5455:2 5451:r 5423:. 5402:r 5396:2 5393:+ 5378:r 5375:= 5366:a 5339:2 5322:r 5310:r 5304:= 5299:r 5295:a 5252:a 5248:+ 5239:r 5231:r 5227:a 5223:= 5213:r 5183:. 5167:) 5145:r 5139:2 5136:+ 5121:r 5117:( 5113:+ 5103:r 5095:) 5089:2 5072:r 5060:r 5053:( 5049:= 5045:) 5010:r 5007:+ 4980:r 4977:+ 4944:r 4937:( 4933:+ 4929:) 4913:r 4898:r 4892:+ 4882:r 4869:r 4862:( 4858:= 4844:r 4833:, 4806:r 4803:+ 4793:r 4780:r 4774:= 4759:r 4750:r 4747:+ 4737:r 4724:r 4718:= 4704:r 4642:r 4580:r 4558:= 4536:= 4530:t 4527:d 4510:d 4503:, 4476:= 4461:r 4452:= 4446:t 4443:d 4431:r 4424:d 4390:r 4366:r 4339:r 4332:r 4329:= 4325:r 4059:2 4055:T 4049:3 4045:a 4010:T 4006:a 4002:a 3998:T 3853:2 3849:T 3843:3 3839:R 3761:a 3741:T 3717:G 3697:m 3673:M 3641:2 3629:3 3615:6 3590:2 3582:4 3577:M 3574:G 3560:2 3552:4 3547:) 3544:m 3541:+ 3538:M 3535:( 3532:G 3526:= 3519:2 3515:T 3509:3 3505:a 3479:m 3476:+ 3473:M 3453:M 3433:a 3413:r 3387:3 3383:r 3374:2 3370:T 3359:3 3355:r 3350:) 3344:M 3341:G 3334:2 3326:4 3320:( 3316:= 3311:2 3307:T 3294:2 3290:r 3285:M 3282:m 3276:G 3273:= 3268:2 3263:) 3258:T 3251:2 3245:( 3240:r 3237:m 3213:T 3185:2 3181:r 3176:M 3173:m 3167:G 3164:= 3159:2 3151:r 3148:m 3081:1 3078:= 3075:T 2995:. 2990:T 2986:b 2983:a 2974:= 2969:2 2965:n 2962:b 2959:a 2953:= 2947:t 2944:d 2939:A 2936:d 2910:. 2907:t 2904:d 2900:n 2897:b 2894:a 2891:= 2885:d 2879:2 2875:r 2846:T 2839:2 2833:= 2830:n 2800:b 2797:a 2791:= 2785:t 2782:d 2774:d 2766:2 2761:2 2757:r 2748:T 2725:T 2705:b 2702:a 2677:. 2671:t 2668:d 2660:d 2652:2 2647:2 2643:r 2637:= 2631:t 2628:d 2623:A 2620:d 2590:d 2586:r 2580:r 2572:2 2569:1 2564:= 2561:A 2558:d 2535:d 2531:r 2511:r 2491:t 2488:d 2460:) 2451:p 2436:r 2430:( 2408:. 2405:) 2396:p 2381:r 2375:( 2370:2 2367:1 2360:m 2357:1 2352:= 2346:t 2343:d 2338:A 2335:d 2311:. 2308:) 2299:v 2284:r 2278:( 2273:2 2270:1 2265:= 2259:t 2256:d 2251:A 2248:d 2214:) 2211:t 2208:d 2199:v 2184:r 2178:( 2173:2 2170:1 2165:= 2162:) 2152:x 2149:d 2133:r 2127:( 2122:2 2119:1 2114:= 2111:A 2108:d 2088:r 2016:A 2012:b 2008:a 2001:r 1994:r 1990:p 1986:r 1982:ε 1965:. 1961:b 1958:a 1952:= 1949:A 1919:. 1907:r 1903:+ 1894:r 1882:r 1869:r 1862:= 1843:r 1836:r 1828:ε 1804:2 1800:b 1796:= 1787:r 1777:r 1773:= 1766:a 1763:p 1754:1 1746:) 1741:2 1735:1 1723:r 1719:+ 1714:1 1702:r 1695:( 1690:= 1683:p 1663:r 1656:r 1648:p 1622:2 1611:1 1607:p 1602:= 1595:b 1580:r 1570:r 1564:= 1557:b 1537:r 1530:r 1522:b 1493:2 1482:1 1478:p 1473:= 1466:a 1457:2 1447:r 1443:+ 1434:r 1427:= 1420:a 1400:r 1393:r 1385:a 1356:1 1352:p 1347:= 1338:r 1320:θ 1303:p 1293:θ 1289:θ 1266:+ 1263:1 1259:p 1254:= 1245:r 1227:θ 1220:ε 1216:ε 1205:θ 1201:r 1197:θ 1193:r 1185:ε 1167:p 1144:, 1125:+ 1122:1 1118:p 1113:= 1110:r 1095:. 1089:r 1085:r 1072:r 1068:r 1055:p 1051:b 1047:a 1043:θ 1038:, 1036:r 1018:. 881:( 841:, 826:+ 804:4 793:e 732:. 633:. 578:e 571:t 564:v 446:) 442:( 293:) 284:( 159:. 145:2 141:/ 137:3 133:) 126:2 122:a 116:/ 109:1 105:a 100:( 87:2 84:A 80:1 77:A 72:. 70:1 67:F 63:3 60:F 56:1 53:F 49:2 46:F 42:1 39:F 20:)

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Knowledge

Kepler's laws of planetary motion

Index