Keplerovi zakoni

U astronomiji, Keplerovi zakoni o kretanju planeta, koje je objavio Johan Kepler između 1609. i 1619. godine, opisuju orbite planeta oko Sunca. Zakoni su modifikovali heliocentričnu teoriju Nikole Kopernika, zamenjujući njene kružne orbite i epicikle eliptičnim putanjama i objašnjavajući kako se planetarne brzine razlikuju. Tri zakona navode da:^[1]^[2]

Sve planete se kreću po elipsama kojima je jedno od žarišta Sunce.
Radijus vektor (provodnica) Sunce-planeta (dužina koja spaja središte Sunca i trenutni položaj planete), prelazi u jednakim vremenskim razmacima jednake površine.
Kvadrati ophodnih vremena planeta proporcionalni su kubovima njihovih srednjih udaljenosti od Sunca.

Eliptične orbite planeta su naznačene proračunima orbite Marsa. Iz ovoga je Kepler zaključio da i druga tela u Sunčevom sistemu, uključujući ona udaljenija od Sunca, takođe imaju eliptične orbite. Drugi zakon pomaže da se utvrdi da kada je planeta bliža Suncu, ona putuje brže. Treći zakon izražava da što je planeta udaljenija od Sunca, to je njena orbitalna brzina sporija i obrnuto.

Isak Njutn je 1687. pokazao da će se odnosi poput Keplerovih primeniti u Sunčevom sistemu kao posledica njegovih sopstvenih zakona kretanja i zakona univerzalne gravitacije.

Precizniji istorijski pristup nalazi se u Astronomia nova i Epitome Astronomiae Copernicanae.

Poređenje sa Kopernikom

Zakoni Johanesa Keplera su poboljšali Kopernikov model. Prema Koperniku:^[3]^[4]

Planetarna orbita je krug sa epiciklima.
Sunce je približno u centru orbite.
Brzina planete u glavnoj orbiti je konstantna.

Iako je bio u pravu kada je rekao da se planete okreću oko Sunca, Kopernik nije adekvatno je definisao njihove orbite. Kepler je tačno definisao orbitu planeta na sledeći način:^[2]^[1]^[5]

Planetarna orbita nije krug sa epiciklima, već elipsa.
Sunce nije u centru, već u fokusnoj tački eliptične orbite.
Ni linearna brzina ni ugaona brzina planete u orbiti nisu konstantne, ali je površinska brzina (istorijski blisko povezana sa konceptom ugaonog momenta) konstantna.

Ekscentricitet Zemljine orbite čini vreme od martovske do septembarske ravnodnevice, oko 186 dana, nejednakim vremenu od septembarske do martovske ravnodnevice, oko 179 dana. Prečnik bi presekao orbitu na jednake delove, ali ravan kroz Sunce paralelna sa ekvatorom Zemlje preseca orbitu na dva dela sa površinama u odnosu 186 prema 179, tako da je ekscentricitet Zemljine orbite približno

e\approx {\frac {\pi }{4}}{\frac {186-179}{186+179}}\approx 0.015,

što je blizu tačnoj vrednosti (0,016710218). Tačnost ovog proračuna zahteva da dva izabrana datuma budu duž male ose eliptične orbite i da sredine svake polovine budu duž glavne ose. Pošto su dva datuma izabrana ovde ravnodnevnice, ovo će biti tačno kada perihel, datum kada je Zemlja najbliža Suncu, padne na solsticij. Trenutni perihel, blizu 4. januara, prilično je blizu solsticiju 21. ili 22. decembra.

Nomenklatura

Bilo je potrebno skoro dva veka da sadašnja formulacija Keplerovog dela poprimi svoj ustaljeni oblik. Volterovi Eléments de la philosophie de Newton (Elementi Njutnove filozofije) iz 1738. godine bila je prva publikacija koja je koristila terminologiju „zakona“.^[6]^[7] Biografska enciklopedija astronoma u svom članku o Kepleru (str. 620) navodi da je terminologija naučnih zakona za ova otkrića bila aktuelna barem od vremena Žozefa de Lalanda.^[8] Upravo je izlaganje Roberta Smola, u Izveštaju o astronomskim otkrićima Keplera (1814) koje je sačinilo skup od tri zakona, dodavanjem trećeg.^[9] Smol je takođe tvrdio, protiv istorije, da su to bili empirijski zakoni, zasnovani na induktivnom zaključivanju.^[7]^[10]

Dalje, trenutna upotreba „Keplerovog drugog zakona”" je donekle pogrešna. Kepler je imao dve verzije, povezane u kvalitativnom smislu: „zakon udaljenosti“ i „zakon oblasti“. „Zakon oblasti“ je ono što je postalo Drugi zakon u skupu od tri; ali sam Kepler to nije privilegovao na taj način.^[11]

Prvi Keplerov zakon

Sunce kao fokus u eliptičkoj orbiti

Planete se kreću po eliptičkim putanjama u kojem se u jednom od fokusa nalazi centar mase, sunce.

$r={\frac {p}{1+\varepsilon \cos \phi }}$

Dokaz

Napiše se Lagranževa funkcija za Sunčev sistem (neku planetu): $L=T-V={\frac {1}{2}}mv^{2}-{\Bigl (}-{\frac {\gamma Mm}{r}}{\Bigr )}={\frac {1}{2}}m{\Bigl (}{dr \over dt}{\Bigr )}^{2}+{\frac {1}{2}}mr^{2}{\Bigl (}{d\phi \over dt}{\Bigr )}^{2}+{\frac {\gamma Mm}{r}}$ ,

gde je m — masa planete, M — masa Sunca, γ — univerzalna gravitaciona konstanta, $\gamma =6\,67\cdot 10^{-11}{\frac {[N][m]^{2}}{[kg]^{2}}}$ , r — rastojanje između Sunca i planete, ${dr \over dt}$ -radijalna brzina i $r{d\phi \over dt}$ — azimutalna brzina

Sada se može uočiti jedna konstanta kretanja koristeći Lagranžove jednačine:

Elipsa sa značajnim tačkama

$-{d \over dt}{\Bigl (}{\frac {\partial L}{\partial {\dot {\phi }}}}{\Bigr )}+{\frac {\partial L}{\partial \phi }}=0$

Kako je ${\frac {\partial L}{\partial \phi }}=0$ to je ${\frac {\partial L}{\partial {\dot {\phi }}}}=const$ , odnosno $mr^{2}{\dot {\phi }}=p_{\phi }$ je konstanta kretanja ili moment impulsa planete u odnosu na Sunce. Sada napišimo Hamiltonovu funkciju sistema, koja u slučaju konzervativnih sila predstavlja i ukupnu energiju sistema: $H=T+V={\frac {1}{2}}mv^{2}-{\frac {\gamma Mm}{r}}=E$ ili ${\frac {1}{2}}m{\dot {r}}^{2}+{\frac {1}{2}}{\frac {p_{\phi }^{2}}{mr^{2}}}-{\frac {\gamma Mm}{r}}=E$ ; odavde je:

${\dot {r}}={dr \over dt}={\sqrt {{\frac {2E}{m}}-{\frac {{\dot {p_{\phi }}}^{2}}{m^{2}r^{2}}}+2{\frac {\gamma M}{r}}}}$ . Eliminiše se promenljiva t smenom: ${\dot {r}}={\frac {d\phi }{d\phi }}{\frac {dr}{dt}}={\frac {d\phi }{dt}}{\frac {dr}{d\phi }}={\frac {p_{\phi }}{mr^{2}}}{\frac {dr}{d\phi }}={\sqrt {{\frac {2E}{m}}-{\frac {{p_{\phi }}^{2}}{m^{2}r^{2}}}+2{\frac {\gamma M}{r}}}}$

Uvede se smena: ${\frac {p_{\phi }}{mr}}=u$ ; $du=-{\frac {p_{\phi }dr}{mr^{2}}}$ , pa jednačina iznad prelazi u:

$-{\frac {du}{\sqrt {{\frac {2E}{m}}-u^{2}+2{\frac {\gamma Mmu}{p_{\phi }}}}}}=d\phi$ <=> $-{\frac {du}{\sqrt {{\frac {2E}{m}}+{\Bigl (}{\frac {\gamma Mm}{p_{\phi }}}{\Bigr )}^{2}-{\Bigl (}u-{\frac {\gamma Mm}{p_{\phi }}}{\Bigr )}^{2}}}}=d\phi$

Smenom: $z={\frac {u-{\frac {\gamma Mm}{p_{\phi }}}}{\sqrt {{\frac {2E}{m}}+{\Bigl (}{\frac {\gamma Mm}{p_{\phi }}}{\Bigr )}^{2}}}}$ => ${\frac {-dz}{\sqrt {1-z^{2}}}}=d\phi$ => $z=\cos \phi$ => ${\frac {p_{\phi }}{mr}}-{\frac {\gamma Mm}{p_{\phi }}}={\sqrt {{\frac {2E}{m}}+{\Bigl (}{\frac {\gamma Mm}{p_{\phi }}}{\Bigr )}^{2}}}\cos \phi$ => $r={\frac {\frac {p_{\phi }}{m}}{{\frac {\gamma Mm}{p_{\phi }}}+{\sqrt {{\frac {2E}{m}}+{\Bigl (}{\frac {\gamma Mm}{p_{\phi }}}{\Bigr )}^{2}}}\cos \phi }}$

$r={\frac {\frac {p_{\phi }^{2}}{\gamma Mm^{2}}}{1+{\sqrt {1+{\frac {2Ep_{\phi }^{2}}{\gamma ^{2}M^{2}m^{3}}}}}\cos \phi }}$ = $r={\frac {p}{1+\varepsilon \cos \phi }}$

Drugi Keplerov zakon

Ilustracija za Keplerov drugi zakon

Radijus vektor Sunce—planeta u jednakim vremenskim intervalima opisuje jednake površine.

Ovaj zakon se matematički može predstaviti izrazom:

$r^{2}{\frac {d\theta }{dt}}=const$

Dokaz sledi iz: $mr^{2}{\dot {\phi }}=p_{\phi }$

Treći Keplerov zakon

Kvadrati perioda obilaska planeta oko sunca (T) srazmerni su kubovima velikih poluosa (a) njihovih putanja.

Matematički, ovaj zakon se može napisati izrazom: ${\frac {a_{1}^{3}}{T_{1}^{2}}}={\frac {a_{2}^{3}}{T_{2}^{2}}}=const$

Dokaz

Pođe se od izraza: $r={\frac {p}{1+\varepsilon \cos \phi }}$ i primeti da je $r={\frac {p}{1-\epsilon }}=a+e$ ; $\phi =\pi$ i:

$r={\frac {p}{1+\epsilon }}=a-e$ ; $\phi =0$ , a — velika poluosa elipse; e — ekscentritet elipse ili rastojanje fokusa od centra elipse; b — mala poluosa, $b={\sqrt {a^{2}-e^{2}}}$

$a={\frac {p}{1-\epsilon ^{2}}}$ , $e=a\epsilon$ ; $b=a{\sqrt {1-\epsilon ^{2}}}$

U ovom slučaju: $p={\frac {p_{\phi }^{2}}{\gamma Mm^{2}}}$ ; $1-\epsilon ^{2}=-{\frac {2Ep_{\phi }^{2}}{\gamma ^{2}M^{2}m^{3}}}$ ; $a=-{\frac {\gamma Mm}{2E}}$

Kako je površina elipse: $S=\pi ab=\int _{0}^{T}{r^{2}{\dot {\phi }}dt}$ = $\int _{0}^{T}{{\frac {p_{\phi }}{m}}dt}={\frac {p_{\phi }}{m}}T$

$\pi ab=\pi a^{2}{\sqrt {1-\epsilon ^{2}}}={\frac {p_{\phi }}{m}}T$ <=> $\pi ^{2}a^{4}{\Bigl (}1-\epsilon ^{2}{\Bigr )}={\frac {p_{\phi }^{2}}{m^{2}}}T^{2}$

${\frac {p_{\phi }^{2}}{m^{2}}}T^{2}=\pi ^{2}{\frac {\gamma ^{4}M^{4}m^{4}}{16E^{4}}}{\frac {2Ep_{\phi }^{2}}{\gamma ^{2}M^{2}m^{3}}}$ <=> $T^{2}={\frac {\pi ^{2}\gamma ^{2}M^{2}m^{3}}{8E^{3}}}=\pi ^{2}a^{3}$

Vidi još

Reference

^ ^a ^b „Orbits and Kepler's Laws”. NASA Solar System Exploration. Pristupljeno 13. 12. 2022.
^ ^a ^b „Kepler's Laws”. hyperphysics.phy-astr.gsu.edu. Pristupljeno 13. 12. 2022.
^ „Planetary Motion: The History of an Idea That Launched the Scientific Revolution”. earthobservatory.nasa.gov (na jeziku: engleski). 07. 07. 2009. Pristupljeno 13. 12. 2022.
^ „Nicolaus Copernicus”. HISTORY (na jeziku: engleski). Pristupljeno 13. 12. 2022.
^ Hurley, Steve (20. 01. 2018). „Johannes Kepler”. Explaining Science (na jeziku: engleski). Pristupljeno 13. 12. 2022.
^
Voltaire, Eléments de la philosophie de Newton [Elements of Newton's Philosophy] (London, England: 1738). See, for example:
- From p. 162: "Par une des grandes loix de Kepler, toute Planete décrit des aires égales en temp égaux : par une autre loi non-moins sûre, chaque Planete fait sa révolution autour du Soleil en telle sort, que si, sa moyenne distance au Soleil est 10. prenez le cube de ce nombre, ce qui sera 1000., & le tems de la révolution de cette Planete autour du Soleil sera proportionné à la racine quarrée de ce nombre 1000." (By one of the great laws of Kepler, each planet describes equal areas in equal times ; by another law no less certain, each planet makes its revolution around the sun in such a way that if its mean distance from the sun is 10, take the cube of that number, which will be 1000, and the time of the revolution of that planet around the sun will be proportional to the square root of that number 1000.)
- From p. 205: "Il est donc prouvé par la loi de Kepler & par celle de Neuton, que chaque Planete gravite vers le Soleil, ... " (It is thus proved by the law of Kepler and by that of Newton, that each planet revolves around the sun ... )
^ ^a ^b Wilson, Curtis (maj 1994). „Kepler's Laws, So-Called” (PDF). HAD News (31): 1—2. Pristupljeno 27. 12. 2016. CS1 održavanje: Format datuma (veza)
^
De la Lande, Astronomie, vol. 1 (Paris, France: Desaint & Saillant, 1764). See, for example:
- From page 390: " ... mais suivant la fameuse loi de Kepler, qui sera expliquée dans le Livre suivant (892), le rapport des temps périodiques est toujours plus grand que celui des distances, une planete cinq fois plus éloignée du soleil, emploie à faire sa révolution douze fois plus de temps ou environ; ... " ( ... but according to the famous law of Kepler, which will be explained in the following book [i.e., chapter] (paragraph 892), the ratio of the periods is always greater than that of the distances [so that, for example,] a planet five times farther from the sun, requires about twelve times or so more time to make its revolution [around the sun] ... )
- From page 429: "Les Quarrés des Temps périodiques sont comme les Cubes des Distances. 892. La plus fameuse loi du mouvement des planetes découverte par Kepler, est celle du repport qu'il y a entre les grandeurs de leurs orbites, & le temps qu'elles emploient à les parcourir; ... " (The squares of the periods are as the cubes of the distances. 892. The most famous law of the movement of the planets discovered by Kepler is that of the relation between the sizes of their orbits and the times that the [planets] require to traverse them; ... )
- From page 430: "Les Aires sont proportionnelles au Temps. 895. Cette loi générale du mouvement des planetes devenue si importante dans l'Astronomie, sçavior, que les aires sont proportionnelles au temps, est encore une des découvertes de Kepler; ... " (Areas are proportional to times. 895. This general law of the movement of the planets [which has] become so important in astronomy, namely, that areas are proportional to times, is one of Kepler's discoveries; ... )
- From page 435: "On a appellé cette loi des aires proportionnelles aux temps, Loi de Kepler, aussi bien que celle de l'article 892, du nome de ce célebre Inventeur; ... " (One called this law of areas proportional to times (the law of Kepler) as well as that of paragraph 892, by the name of that celebrated inventor; ... )
^ Robert Small, (1804). An account of the astronomical discoveries of Kepler. London: England: J Mawman. str. 298—299.
^ Robert Small, An account of the astronomical discoveries of Kepler (London, England: J. Mawman, 1804).
^ Bruce Stephenson (1994). Kepler's Physical Astronomy. Princeton University Press. str. 170. ISBN 978-0-691-03652-6.

Literatura

Kepler's life is summarized on pages 523–627 and Book Five of his magnum opus, Harmonice Mundi (harmonies of the world), is reprinted on pages 635–732 of On the Shoulders of Giants: The Great Works of Physics and Astronomy (works by Copernicus, Kepler, Galileo, Newton, and Einstein). Stephen Hawking, ed. 2002. ISBN 978-0-7624-1348-5.
A derivation of Kepler's third law of planetary motion is a standard topic in engineering mechanics classes. See, for example, pages 161–164 of Meriam, J. L. (1971) [1966]. „Dynamics, 2nd ed”. New York: John Wiley. ISBN 978-0-471-59601-1. .
Murray and Dermott (1999). Solar System Dynamics. Cambridge University Press. ISBN 978-0-521-57597-3.
V.I. Arnold (1989). Mathematical Methods of Classical Mechanics. Springer. ISBN 978-0-387-96890-2.

Spoljašnje veze

B.Surendranath Reddy; animation of Kepler's laws: applet
Crowell, Benjamin, Conservation Laws, „http://www.lightandmatter.com/area1book2.html”. Arhivirano iz originala 01. 06. 2020. g. Spoljašnja veza u |title= (pomoć), an online book that gives a proof of the first law without the use of calculus. (see section 5.2, pp. 112)
David McNamara and Gianfranco Vidali, Kepler's Second Law - Java Interactive Tutorial, https://web.archive.org/web/20060910225253/http://www.phy.syr.edu/courses/java/mc_html/kepler.html, an interactive Java applet that aids in the understanding of Kepler's Second Law.
University of Tennessee's Dept. Physics & Astronomy: Astronomy 161 page on Johannes Kepler: The Laws of Planetary Motion [1]
„Equant compared to Kepler: interactive model”. Arhivirano iz originala 26. 12. 2008. g.
Kepler's Third Law:interactive model [2]Arhivirano na sajtu Wayback Machine (26. decembar 2008)
Solar System Simulator (Interactive Applet)
"Derivation of Kepler's Laws" (from Newton's laws) at Physics Stack Exchange.
Crowell, Benjamin, Light and Matter, an online book that gives a proof of the first law without the use of calculus (see section 15.7)
Audio – Cain/Gay (2010) Astronomy Cast Johannes Kepler and His Laws of Planetary Motion
Solar System Simulator (Interactive Applet)
Kepler and His Laws, educational web pages by David P. Stern

[:1-1] „Orbits and Kepler's Laws”. NASA Solar System Exploration. Pristupljeno 13. 12. 2022.

[:0-2] „Kepler's Laws”. hyperphysics.phy-astr.gsu.edu. Pristupljeno 13. 12. 2022.

[3] „Planetary Motion: The History of an Idea That Launched the Scientific Revolution”. earthobservatory.nasa.gov (na jeziku: engleski). 07. 07. 2009. Pristupljeno 13. 12. 2022.

[4] „Nicolaus Copernicus”. HISTORY (na jeziku: engleski). Pristupljeno 13. 12. 2022.

[5] Hurley, Steve (20. 01. 2018). „Johannes Kepler”. Explaining Science (na jeziku: engleski). Pristupljeno 13. 12. 2022.

[6] Voltaire, Eléments de la philosophie de Newton [Elements of Newton's Philosophy] (London, England: 1738). See, for example:
From p. 162: "Par une des grandes loix de Kepler, toute Planete décrit des aires égales en temp égaux : par une autre loi non-moins sûre, chaque Planete fait sa révolution autour du Soleil en telle sort, que si, sa moyenne distance au Soleil est 10. prenez le cube de ce nombre, ce qui sera 1000., & le tems de la révolution de cette Planete autour du Soleil sera proportionné à la racine quarrée de ce nombre 1000." (By one of the great laws of Kepler, each planet describes equal areas in equal times ; by another law no less certain, each planet makes its revolution around the sun in such a way that if its mean distance from the sun is 10, take the cube of that number, which will be 1000, and the time of the revolution of that planet around the sun will be proportional to the square root of that number 1000.)

From p. 205: "Il est donc prouvé par la loi de Kepler & par celle de Neuton, que chaque Planete gravite vers le Soleil, ... " (It is thus proved by the law of Kepler and by that of Newton, that each planet revolves around the sun ... )

[7] From p. 162: "Par une des grandes loix de Kepler, toute Planete décrit des aires égales en temp égaux : par une autre loi non-moins sûre, chaque Planete fait sa révolution autour du Soleil en telle sort, que si, sa moyenne distance au Soleil est 10. prenez le cube de ce nombre, ce qui sera 1000., & le tems de la révolution de cette Planete autour du Soleil sera proportionné à la racine quarrée de ce nombre 1000." (By one of the great laws of Kepler, each planet describes equal areas in equal times ; by another law no less certain, each planet makes its revolution around the sun in such a way that if its mean distance from the sun is 10, take the cube of that number, which will be 1000, and the time of the revolution of that planet around the sun will be proportional to the square root of that number 1000.)

[8] From p. 205: "Il est donc prouvé par la loi de Kepler & par celle de Neuton, que chaque Planete gravite vers le Soleil, ... " (It is thus proved by the law of Kepler and by that of Newton, that each planet revolves around the sun ... )

[Wilson_1994-7] Wilson, Curtis (maj 1994). „Kepler's Laws, So-Called” (PDF). HAD News (31): 1—2. Pristupljeno 27. 12. 2016. CS1 održavanje: Format datuma (veza)

[8] De la Lande, Astronomie, vol. 1 (Paris, France: Desaint & Saillant, 1764). See, for example:
From page 390: " ... mais suivant la fameuse loi de Kepler, qui sera expliquée dans le Livre suivant (892), le rapport des temps périodiques est toujours plus grand que celui des distances, une planete cinq fois plus éloignée du soleil, emploie à faire sa révolution douze fois plus de temps ou environ; ... " ( ... but according to the famous law of Kepler, which will be explained in the following book [i.e., chapter] (paragraph 892), the ratio of the periods is always greater than that of the distances [so that, for example,] a planet five times farther from the sun, requires about twelve times or so more time to make its revolution [around the sun] ... )

From page 429: "Les Quarrés des Temps périodiques sont comme les Cubes des Distances. 892. La plus fameuse loi du mouvement des planetes découverte par Kepler, est celle du repport qu'il y a entre les grandeurs de leurs orbites, & le temps qu'elles emploient à les parcourir; ... " (The squares of the periods are as the cubes of the distances. 892. The most famous law of the movement of the planets discovered by Kepler is that of the relation between the sizes of their orbits and the times that the [planets] require to traverse them; ... )

From page 430: "Les Aires sont proportionnelles au Temps. 895. Cette loi générale du mouvement des planetes devenue si importante dans l'Astronomie, sçavior, que les aires sont proportionnelles au temps, est encore une des découvertes de Kepler; ... " (Areas are proportional to times. 895. This general law of the movement of the planets [which has] become so important in astronomy, namely, that areas are proportional to times, is one of Kepler's discoveries; ... )

From page 435: "On a appellé cette loi des aires proportionnelles aux temps, Loi de Kepler, aussi bien que celle de l'article 892, du nome de ce célebre Inventeur; ... " (One called this law of areas proportional to times (the law of Kepler) as well as that of paragraph 892, by the name of that celebrated inventor; ... )

[11] From page 390: " ... mais suivant la fameuse loi de Kepler, qui sera expliquée dans le Livre suivant (892), le rapport des temps périodiques est toujours plus grand que celui des distances, une planete cinq fois plus éloignée du soleil, emploie à faire sa révolution douze fois plus de temps ou environ; ... " ( ... but according to the famous law of Kepler, which will be explained in the following book [i.e., chapter] (paragraph 892), the ratio of the periods is always greater than that of the distances [so that, for example,] a planet five times farther from the sun, requires about twelve times or so more time to make its revolution [around the sun] ... )

[12] From page 429: "Les Quarrés des Temps périodiques sont comme les Cubes des Distances. 892. La plus fameuse loi du mouvement des planetes découverte par Kepler, est celle du repport qu'il y a entre les grandeurs de leurs orbites, & le temps qu'elles emploient à les parcourir; ... " (The squares of the periods are as the cubes of the distances. 892. The most famous law of the movement of the planets discovered by Kepler is that of the relation between the sizes of their orbits and the times that the [planets] require to traverse them; ... )

[13] From page 430: "Les Aires sont proportionnelles au Temps. 895. Cette loi générale du mouvement des planetes devenue si importante dans l'Astronomie, sçavior, que les aires sont proportionnelles au temps, est encore une des découvertes de Kepler; ... " (Areas are proportional to times. 895. This general law of the movement of the planets [which has] become so important in astronomy, namely, that areas are proportional to times, is one of Kepler's discoveries; ... )

[14] From page 435: "On a appellé cette loi des aires proportionnelles aux temps, Loi de Kepler, aussi bien que celle de l'article 892, du nome de ce célebre Inventeur; ... " (One called this law of areas proportional to times (the law of Kepler) as well as that of paragraph 892, by the name of that celebrated inventor; ... )

[9] Robert Small, (1804). An account of the astronomical discoveries of Kepler. London: England: J Mawman. str. 298—299.

[10] Robert Small, An account of the astronomical discoveries of Kepler (London, England: J. Mawman, 1804).

[Stephenson1994-11] Bruce Stephenson (1994). Kepler's Physical Astronomy. Princeton University Press. str. 170. ISBN 978-0-691-03652-6.

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