கெப்லரின் கோள் இயக்க விதிகள்

வானியலுக்கு, யொகானசு கெப்லரின் முதன்மையான பங்களிப்பு கெப்லரின் கோள் இயக்க விதிகள் எனப்படும் மூன்று விதிகளாகும். கண்பார்வைக் குறைவுள்ளவராக இருந்தும், மிகவும் திறமையுள்ளவராக விளங்கிய ஜெர்மானியக் கணிதவியலாளரான கெப்லரின் விதிகளின் உருவாக்கத்துக்கு டென்மார்க்கைச் சேர்ந்த வானியலாளரான டைக்கோ பிரா (Tycho Brahe) என்பவரது துல்லிய வானியல் குறிப்புகள் (அவதானிப்புகள்) மிகவும் துணை புரிந்தன.^[1]^[2]^[3]

ஐசாக் நியூட்டனுடைய பின்னாளைய கண்டுபிடிப்புகளான, நியூட்டனின் இயக்கவிதி மற்றும் புவியீர்ப்பு தொடர்பான விதிகள் என்பவற்றின் உருவாக்கத்துக்குக் கெப்லரின் கண்டுபிடிப்பு அடிக்களமாக (ஆதாரமாக) அமைந்தது எனலாம். தற்கால நோக்கில், கெப்லரின் விதிகள், நியூட்டனின் விதிகளின் விளைவுகளாக இருந்தாலும், வரலாற்றின்படி, கெப்லரின் விதிகளே முதலில் வெளியானவை.

கெப்லரின் முதலாவது விதி

ஒரு கோளின் சுற்றுப்பாதை, கதிரவன் ஒரு குவியத்தில் அமைந்திருக்கும் ஒரு நீள்வட்டமாகும், என்பதே கெப்லரின் முதலாவது விதியாகும்.

கெப்லரின் இரண்டாவது விதி

கெப்லரின் இரண்டாவது விதி, கோளையும், கதிரவனையும் இணைக்கும் நேர்கோடு, கோளின் சமகால இடைவெளி நகர்வில் சம பரப்பைத் தடவிச்செல்லும், என்கிறது.^[4]

கெப்லரின் மூன்றாவது விதி

கோள்களின் ஒரு முழுசுற்றுக்கால அளவின் இருபடி, அவற்றின் நீள்வட்டச் சுற்றுப்பாதையின் பெரிய அச்சின் பாதியின் (semi-major axis) முப்படிக்கு நேர் சார்புடையது (நேர் விகித சமனாகும்). மேலும் நேர்சார்புக் கெழு (மாறிலி) எல்லாக் கோளுக்கும் ஒரே மதிப்பு கொண்டதாகும். என்பது கெப்லரின் மூன்றாவது விதியாகும்.

ஒரு கோளின் முழுச்சுற்றுகாலம் T என்றும், நீள்வட்டச் சுற்றுப்பாதையின் பெரிய அச்சின் பாதியை r என்றும், கொண்டால் இம் மூன்றாம் விதியைக் கீழ்க்காணும் ஈடுகோளால் காட்டலாம்:

T^{2}={\frac {4\pi ^{2}}{GM}}r^{3}.

மேலுள்ளவற்றில் M என்பது கதிரவனின் நிறை, G என்பது நியூட்டனின் பொருளீர்ப்பு நிலையெண் (மாறிலி).

மேலுள்ள ஈடுகோளைக் கீழ்க்காணுமாறு சுருக்கி எழுதலாம்:

T^{2}=ar^{3}.\,

இதில் a என்பது ${T^{2}}\propto {r^{3}}$ என்னும் நேர்சார்புத் தொடர்பை ஈடுகோளாக்கும் கெழு (மாறிலி), அல்லது நேர்சார்புக் கெழு.

இரு வேறு கோள்களின் சுற்றுக்கால அளவுகள் T₁, T₂ என்றும், அவற்றின் சுற்றுப்பாதையின் பெரிய அச்சின் பாதியின் அளவுகள் r₁, r₂ ஆக முறையே இருந்தால், சுற்றுக்காலங்களின் விகிதமும், பெரிய அச்சின் பாதிகளின் விகிதமும் கீழ்க்காணுமாறு அமையும்.

\left({\frac {T_{1}}{T_{2}}}\right)^{2}=\left({\frac {r_{1}}{r_{2}}}\right)^{3}

மேற்கோள்கள்

↑ In his Astronomia nova, Kepler presented only a proof that Mars' orbit is elliptical. Evidence that the other known planets' orbits are elliptical was presented only in 1621.
See: Johannes Kepler, Astronomia nova … (1609), p. 285. After having rejected circular and oval orbits, Kepler concluded that Mars' orbit must be elliptical. From the top of page 285: "Ergo ellipsis est Planetæ iter; … " (Thus, an ellipse is the planet's [i.e., Mars'] path; … ) Later on the same page: " … ut sequenti capite patescet: ubi simul etiam demonstrabitur, nullam Planetæ relinqui figuram Orbitæ, præterquam perfecte ellipticam; … " ( … as will be revealed in the next chapter: where it will also then be proved that any figure of the planet's orbit must be relinquished, except a perfect ellipse; … ) And then: "Caput LIX. Demonstratio, quod orbita Martis, … , fiat perfecta ellipsis: … " (Chapter 59. Proof that Mars' orbit, … , is a perfect ellipse: … ) The geometric proof that Mars' orbit is an ellipse appears as Protheorema XI on pages 289–290.
Kepler stated that every planet travels in elliptical orbits having the Sun at one focus in: Johannes Kepler, Epitome Astronomiae Copernicanae [Summary of Copernican Astronomy] (Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 5, part 1, III. De Figura Orbitæ (III. On the figure [i.e., shape] of orbits), pages 658–665. From p. 658: "Ellipsin fieri orbitam planetæ … " (Of an ellipse is made a planet's orbit … ). From p. 659: " … Sole (Foco altero huius ellipsis) … " ( … the Sun (the other focus of this ellipse) … ).
↑ Holton, Gerald James; Brush, Stephen G. (2001). Physics, the Human Adventure: From Copernicus to Einstein and Beyond (3rd paperback ed.). Piscataway, NJ: Rutgers University Press. pp. 40–41. பன்னாட்டுத் தரப்புத்தக எண் 0-8135-2908-5. பார்க்கப்பட்ட நாள் December 27, 2009.
↑
In his Astronomia nova ... (1609), Kepler did not present his second law in its modern form. He did that only in his Epitome of 1621. Furthermore, in 1609, he presented his second law in two different forms, which scholars call the "distance law" and the "area law".
- His "distance law" is presented in: "Caput XXXII. Virtutem quam Planetam movet in circulum attenuari cum discessu a fonte." (Chapter 32. The force that moves a planet circularly weakens with distance from the source.) See: Johannes Kepler, Astronomia nova … (1609), pp. 165–167. On page 167, Kepler states: " … , quanto longior est αδ quam αε, tanto diutius moratur Planeta in certo aliquo arcui excentrici apud δ, quam in æquali arcu excentrici apud ε." ( … , as αδ is longer than αε, so much longer will a planet remain on a certain arc of the eccentric near δ than on an equal arc of the eccentric near ε.) That is, the farther a planet is from the Sun (at the point α), the slower it moves along its orbit, so a radius from the Sun to a planet passes through equal areas in equal times. However, as Kepler presented it, his argument is accurate only for circles, not ellipses.
- His "area law" is presented in: "Caput LIX. Demonstratio, quod orbita Martis, … , fiat perfecta ellipsis: … " (Chapter 59. Proof that Mars' orbit, … , is a perfect ellipse: … ), Protheorema XIV and XV, pp. 291–295. On the top p. 294, it reads: "Arcum ellipseos, cujus moras metitur area AKN, debere terminari in LK, ut sit AM." (The arc of the ellipse, of which the duration is delimited [i.e., measured] by the area AKM, should be terminated in LK, so that it [i.e., the arc] is AM.) In other words, the time that Mars requires to move along an arc AM of its elliptical orbit is measured by the area of the segment AMN of the ellipse (where N is the position of the Sun), which in turn is proportional to the section AKN of the circle that encircles the ellipse and that is tangent to it. Therefore, the area that is swept out by a radius from the Sun to Mars as Mars moves along an arc of its elliptical orbit is proportional to the time that Mars requires to move along that arc. Thus, a radius from the Sun to Mars sweeps out equal areas in equal times.
In 1621, Kepler restated his second law for any planet: Johannes Kepler, Epitome Astronomiae Copernicanae [Summary of Copernican Astronomy] (Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 5, page 668. From page 668: "Dictum quidem est in superioribus, divisa orbita in particulas minutissimas æquales: accrescete iis moras planetæ per eas, in proportione intervallorum inter eas & Solem." (It has been said above that, if the orbit of the planet is divided into the smallest equal parts, the times of the planet in them increase in the ratio of the distances between them and the sun.) That is, a planet's speed along its orbit is inversely proportional to its distance from the Sun. (The remainder of the paragraph makes clear that Kepler was referring to what is now called angular velocity.)
↑ Bryant, Jeff; Pavlyk, Oleksandr. "Kepler's Second Law", Wolfram Demonstrations Project. Retrieved December 27, 2009.

[1] In his Astronomia nova, Kepler presented only a proof that Mars' orbit is elliptical. Evidence that the other known planets' orbits are elliptical was presented only in 1621.
See: Johannes Kepler, Astronomia nova … (1609), p. 285. After having rejected circular and oval orbits, Kepler concluded that Mars' orbit must be elliptical. From the top of page 285: "Ergo ellipsis est Planetæ iter; … " (Thus, an ellipse is the planet's [i.e., Mars'] path; … ) Later on the same page: " … ut sequenti capite patescet: ubi simul etiam demonstrabitur, nullam Planetæ relinqui figuram Orbitæ, præterquam perfecte ellipticam; … " ( … as will be revealed in the next chapter: where it will also then be proved that any figure of the planet's orbit must be relinquished, except a perfect ellipse; … ) And then: "Caput LIX. Demonstratio, quod orbita Martis, … , fiat perfecta ellipsis: … " (Chapter 59. Proof that Mars' orbit, … , is a perfect ellipse: … ) The geometric proof that Mars' orbit is an ellipse appears as Protheorema XI on pages 289–290.
Kepler stated that every planet travels in elliptical orbits having the Sun at one focus in: Johannes Kepler, Epitome Astronomiae Copernicanae [Summary of Copernican Astronomy] (Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 5, part 1, III. De Figura Orbitæ (III. On the figure [i.e., shape] of orbits), pages 658–665. From p. 658: "Ellipsin fieri orbitam planetæ … " (Of an ellipse is made a planet's orbit … ). From p. 659: " … Sole (Foco altero huius ellipsis) … " ( … the Sun (the other focus of this ellipse) … ).

[Holton-2] Holton, Gerald James; Brush, Stephen G. (2001). Physics, the Human Adventure: From Copernicus to Einstein and Beyond (3rd paperback ed.). Piscataway, NJ: Rutgers University Press. pp. 40–41. பன்னாட்டுத் தரப்புத்தக எண் 0-8135-2908-5. பார்க்கப்பட்ட நாள் December 27, 2009.

[3] In his Astronomia nova ... (1609), Kepler did not present his second law in its modern form. He did that only in his Epitome of 1621. Furthermore, in 1609, he presented his second law in two different forms, which scholars call the "distance law" and the "area law".
His "distance law" is presented in: "Caput XXXII. Virtutem quam Planetam movet in circulum attenuari cum discessu a fonte." (Chapter 32. The force that moves a planet circularly weakens with distance from the source.) See: Johannes Kepler, Astronomia nova … (1609), pp. 165–167. On page 167, Kepler states: " … , quanto longior est αδ quam αε, tanto diutius moratur Planeta in certo aliquo arcui excentrici apud δ, quam in æquali arcu excentrici apud ε." ( … , as αδ is longer than αε, so much longer will a planet remain on a certain arc of the eccentric near δ than on an equal arc of the eccentric near ε.) That is, the farther a planet is from the Sun (at the point α), the slower it moves along its orbit, so a radius from the Sun to a planet passes through equal areas in equal times. However, as Kepler presented it, his argument is accurate only for circles, not ellipses.

His "area law" is presented in: "Caput LIX. Demonstratio, quod orbita Martis, … , fiat perfecta ellipsis: … " (Chapter 59. Proof that Mars' orbit, … , is a perfect ellipse: … ), Protheorema XIV and XV, pp. 291–295. On the top p. 294, it reads: "Arcum ellipseos, cujus moras metitur area AKN, debere terminari in LK, ut sit AM." (The arc of the ellipse, of which the duration is delimited [i.e., measured] by the area AKM, should be terminated in LK, so that it [i.e., the arc] is AM.) In other words, the time that Mars requires to move along an arc AM of its elliptical orbit is measured by the area of the segment AMN of the ellipse (where N is the position of the Sun), which in turn is proportional to the section AKN of the circle that encircles the ellipse and that is tangent to it. Therefore, the area that is swept out by a radius from the Sun to Mars as Mars moves along an arc of its elliptical orbit is proportional to the time that Mars requires to move along that arc. Thus, a radius from the Sun to Mars sweeps out equal areas in equal times.
In 1621, Kepler restated his second law for any planet: Johannes Kepler, Epitome Astronomiae Copernicanae [Summary of Copernican Astronomy] (Linz ("Lentiis ad Danubium"), (Austria): Johann Planck, 1622), book 5, page 668. From page 668: "Dictum quidem est in superioribus, divisa orbita in particulas minutissimas æquales: accrescete iis moras planetæ per eas, in proportione intervallorum inter eas & Solem." (It has been said above that, if the orbit of the planet is divided into the smallest equal parts, the times of the planet in them increase in the ratio of the distances between them and the sun.) That is, a planet's speed along its orbit is inversely proportional to its distance from the Sun. (The remainder of the paragraph makes clear that Kepler was referring to what is now called angular velocity.)

[4] His "distance law" is presented in: "Caput XXXII. Virtutem quam Planetam movet in circulum attenuari cum discessu a fonte." (Chapter 32. The force that moves a planet circularly weakens with distance from the source.) See: Johannes Kepler, Astronomia nova … (1609), pp. 165–167. On page 167, Kepler states: " … , quanto longior est αδ quam αε, tanto diutius moratur Planeta in certo aliquo arcui excentrici apud δ, quam in æquali arcu excentrici apud ε." ( … , as αδ is longer than αε, so much longer will a planet remain on a certain arc of the eccentric near δ than on an equal arc of the eccentric near ε.) That is, the farther a planet is from the Sun (at the point α), the slower it moves along its orbit, so a radius from the Sun to a planet passes through equal areas in equal times. However, as Kepler presented it, his argument is accurate only for circles, not ellipses.

[5] His "area law" is presented in: "Caput LIX. Demonstratio, quod orbita Martis, … , fiat perfecta ellipsis: … " (Chapter 59. Proof that Mars' orbit, … , is a perfect ellipse: … ), Protheorema XIV and XV, pp. 291–295. On the top p. 294, it reads: "Arcum ellipseos, cujus moras metitur area AKN, debere terminari in LK, ut sit AM." (The arc of the ellipse, of which the duration is delimited [i.e., measured] by the area AKM, should be terminated in LK, so that it [i.e., the arc] is AM.) In other words, the time that Mars requires to move along an arc AM of its elliptical orbit is measured by the area of the segment AMN of the ellipse (where N is the position of the Sun), which in turn is proportional to the section AKN of the circle that encircles the ellipse and that is tangent to it. Therefore, the area that is swept out by a radius from the Sun to Mars as Mars moves along an arc of its elliptical orbit is proportional to the time that Mars requires to move along that arc. Thus, a radius from the Sun to Mars sweeps out equal areas in equal times.

[Wolfram2nd-4] Bryant, Jeff; Pavlyk, Oleksandr. "Kepler's Second Law", Wolfram Demonstrations Project. Retrieved December 27, 2009.

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