kdkarekhluxnthikhxngdawekhraahkhxngekhphephlxr xngkvs Kepler s laws of planetary motion khuxkdthangkhnitsastr 3 khxthiklawthungkarekhluxnthikhxngdawekhraahinrabbsuriya nkkhnitsastraelankdarasastrchaweyxrmnchux oyhnenis ekhphephlxr ph s 2114 2173 epnphukhnphbphaphaesdngkd 3 khxkhxngekhphephlxrthimiwngokhcrdawekhraah 2 wng 1 wngokhcrepnwngridwycudofks f1 aela f2 sahrbdawekhraahdwngaerkaela f1 aela f3 sahrbdawekhraahdwngthi 2 dwngxathityxyuthicud f1 2 swnaerenga 2 swn A1 aela A2 miphiwphunethaknaelaewlathidawekhraah 1 thbphunthi A1 ethakbewlathithbphunthi A2 3 ewlarwmkhxngwngokhcrsahrbdawekhraah 1 aeladawekhraah 2 misdswnethakb a13 2 a23 2 displaystyle a1 3 2 a2 3 2 ekhphephlxridsuksakarsngektkarnkhxngnkdarasastrphumichuxesiyngchawednmarkchuxthuxok praexx odypraman ph s 2148 ekhphephlxrphbwakarsngekttaaehnngkhxngdawekhraahkhxngbrahepniptamkdngay thangkhnitsastr kdkhxngekhphephlxrthathaydarasastrsayxrisotetilaelasaythxelmiaelakdthangfisiksinkhnann ekhphephlxryunynwaolkekhluxnthiepnwngrimakkwawngklm aelayngidphisucnwakhwamerwkarekhluxnthimikhwamphnaeprdwy sungepnkarepliynaeplngkhwamruthangdarasastraelafisiks xyangirkdi khaxthibayechingfisiksekiywkbphvtikrrmkhxngdawekhraahkidpraktchdecnidinxikekuxbstwrrstxma emuxixaesk niwtn samarthsrupkdkhxngekhphephlxridwaekhaknkbkdkarekhluxnthiaelakdkhwamonmthwngsaklkhxngniwtnexngodyichwichaaekhlkhulsthiekhakhidsrangkhun rupcalxngaebbxunthinamaichmkihphlphidphladkd 3 khxkhxngekhphephlxrwngokhcrkhxngdawekhraahthukdwngepnwngri odymidwngxathityepncudsunyklangcudhnung wngriekidcakkarmicudsunyklang 2 suny dngphaph dngnnekhphephlxrcungkhdkhankhwamechuxinaenwkhxngxrisotetil potelmiaelaokhepxrnikhsthiwawngokhcrepnwngklm inkhnathidawekhraahekhluxnipinwngokhcr esntrngthiechuxmrahwangdawekhraahkbdwngxathitykwadphunthietha kninrayaewlaethakn sunghmaykhwamwadawekhraahokhcrerwkwaemuxxyuikldwngxathityaelachalngemuxxyuhangdwngxathity dwykdkhxni ekhphephlxridlmthvsdidarasastrxrisotetilthiwadawekhraahekhluxnthidwykhwamerwkhngthi khxngkhabkarokhcrkhxngdawekhraahepnodytrngkbkhxngkungaeknexk khrunghnungkhxngkhwamyawwngri khxngwngokhcr sunghmaykhwamwa imephiyngaetwngokhcrthiihykwaethannthimirayaewlanankwa aetxtrakhwamerwkhxngdawekhraahthimiwngokhcrthiihykwannkokhcrchakwawngokhcrthielkkwaxikdwy kdkhxngekhphephlxridaesdngiwkhanglang aelaepnkdthimacakkdkhxngniwtnthiichrabbphikdechingkhwsunysuriya r 8 displaystyle r theta xyangirktam kdkhxngekhphephlxryngsamarthekhiynxyangxunidodyichphikdkharthiesiynraylaexiydthangkhnitsastrkdkhxthi 1 kdekhphephlxrkhxthi 1 kdkhxaerkklawwa wngokhcrkhxngdawekhraahthukdwngepnrupwngrithimidwngxathityepncudofkscudhnung khnitsastrkhxngwngriepndngni smkarkhux r p1 ϵ cos 8 displaystyle r frac p 1 epsilon cdot cos theta odythi p khux semi latus rectum aela e khux khwameyuxngsunyklang eccentricity sungmikhamakkwahruxethakbsuny aelanxykwahnung emux 8 0 dawekhraahcaxyuthicudikldwngxathitythisud rmin p1 ϵ displaystyle r mathrm min frac p 1 epsilon emux 8 90 r p aelaemux 8 180 dawekhrahcaxyuthicudikldwngxathitythisud rmax p1 ϵ displaystyle r mathrm max frac p 1 epsilon kungaeknexkkhxngwngri a khuxmchchimelkhkhnitkhxng rmin aela rmax a p1 ϵ2 displaystyle a frac p 1 epsilon 2 kungaeknothkhxngwngri b khuxmchchimerkhakhnitkhxng rmin aela rmax b p1 ϵ2 displaystyle b frac p sqrt 1 epsilon 2 nxkcakniyngepnmchchimerkhakhnitrahwangkungaeknexkkbkungeltserktm ab bp displaystyle frac a b frac b p kdkhxthi 2 phaphaesdngkdekhphephlxrkhxthi 2 kdkhxthi 2 esntrngthiechuxmrahwangdawekhraahkbdwngxathity kwadphunthietha kninrayaewlaethakn kdniruckinxikchuxhnungthiwakdphunthietha sungepnphlsubenuxngodytrngcakkdkarxnurksomemntmechingmum oprddukarkarxnuphththdngphaph karkhanwnmi 4 khndngni 1 khanwn mumkwadechliy mean anomaly M caksutrM 2ptP displaystyle M frac 2 pi t P dd 2 khanwn mumkwadeyuxngsunyklang eccentric anomaly E odykaraek smkarkhxngekhphephlxr M E ϵ sin E displaystyle M E epsilon cdot sin E dd 3 khanwn mumkwadcring true anomaly 8 odyichsmkar tan 82 1 ϵ1 ϵ tan E2 displaystyle tan frac theta 2 sqrt frac 1 epsilon 1 epsilon cdot tan frac E 2 dd 4 khanwn rayahangsunysuriya heliocentric distance r cakkdkhxaerk r p1 ϵ cos 8 displaystyle r frac p 1 epsilon cdot cos theta dd kdkhxthi 3 kdkhxthi 3 khxngkhabkarokhcrkhxngdawekhraahepnodytrngkbkhxngkungaeknexkkhxngwngokhcr dngnn imephiyngkhwamyawwngokhcrcaephimdwyrayathangaelw khwamerwkhxngkarokhcrcaldlngdwy karephimkhxngrayaewlakarokhcrcungepnmakkwakarepnsdswn P2 a3 displaystyle P 2 propto a 3 P displaystyle P khabkarokhcrkhxngdawekhraah a displaystyle a aeknkungexkkhxngwngokhcr dngnn P2 a 3 mikhaehmuxnknsahrbdawekhraahthukdwnginrabbsuriyarwmthngolk emuxhnwyhnungthukeluxk echn P thiwdepnpidarakhti aela a inhnwydarasastr P2 a 3 mikha 1 sahrbdawekhraahthukdwnginrabbsuriya inhnwyexsix P2a3 3 00 10 19s2m3 0 7 displaystyle frac P 2 a 3 3 00 times 10 19 frac s 2 m 3 pm 0 7 taaehnnginfngkchnkhxngewla pyhaekhphephlxrxnumankarokhcrwngriaelacud 4 cud s dwngxathity n ofkshnungkhxngwngri z cudikldwngxathitythisud c sunyklangkhxngwngri p dawekhraah aela a cz displaystyle a cz kungaeknexk rayacaksunyklangthungcudikldwngxathitythisud nnkhuxkungaeknexk e cs a displaystyle varepsilon cs over a khwameyuxngsunyklang b a1 e2 displaystyle b a sqrt 1 varepsilon 2 kungaeknoth r sp displaystyle r sp rayacakdwngxathitythungdawekhraah n zsp displaystyle nu angle zsp taaehnngdawekhraahtamthiehncakdwngxathity nnkhux mumkwadcring pyhakhuxkarkhanwnphikdechingkhw r n khxngdawekhraahcakewlanbtngaetdawekhraahphancudikldwngxathitythisud t zsx ab zsp displaystyle zsx frac a b cdot zsp zcy zsx displaystyle zcy zsx aela M zcy displaystyle M angle zcy y cakthiehncaksunyklang nnkhuxmumkwadechliy zcy a2M2 displaystyle zcy frac a 2 M 2 zsp ba zsx ba zcy ba a2M2 abM2 displaystyle zsp frac b a cdot zsx frac b a cdot zcy frac b a cdot frac a 2 M 2 frac abM 2 M 2ptT displaystyle M 2 pi t over T ody T khuxkhabkarokhcr zcy zsx zcx scx displaystyle zcy zsx zcx scx a2M2 a2E2 ae asin E2 displaystyle frac a 2 M 2 frac a 2 E 2 frac a varepsilon cdot a sin E 2 Division by a 2 gives Kepler s equation M E e sin E displaystyle M E varepsilon cdot sin E E M e 18e3 sin M 12e2sin 2M 38e3sin 3M displaystyle E approx M left varepsilon frac 1 8 varepsilon 3 right sin M frac 1 2 varepsilon 2 sin 2M frac 3 8 varepsilon 3 sin 3M cdots a cos E a e r cos n displaystyle a cdot cos E a cdot varepsilon r cdot cos nu ra 1 e21 e cos n displaystyle frac r a frac 1 varepsilon 2 1 varepsilon cdot cos nu to get cos E e 1 e21 e cos n cos n e 1 e cos n 1 e2 cos n1 e cos n e cos n1 e cos n displaystyle cos E varepsilon frac 1 varepsilon 2 1 varepsilon cdot cos nu cdot cos nu frac varepsilon cdot 1 varepsilon cdot cos nu 1 varepsilon 2 cdot cos nu 1 varepsilon cdot cos nu frac varepsilon cos nu 1 varepsilon cdot cos nu tan2 x2 1 cos x1 cos x displaystyle tan 2 frac x 2 frac 1 cos x 1 cos x caid tan2 E2 1 cos E1 cos E 1 e cos n1 e cos n1 e cos n1 e cos n 1 e cos n e cos n 1 e cos n e cos n 1 e1 e 1 cos n1 cos n 1 e1 e tan2 n2 displaystyle tan 2 frac E 2 frac 1 cos E 1 cos E frac 1 frac varepsilon cos nu 1 varepsilon cdot cos nu 1 frac varepsilon cos nu 1 varepsilon cdot cos nu frac 1 varepsilon cdot cos nu varepsilon cos nu 1 varepsilon cdot cos nu varepsilon cos nu frac 1 varepsilon 1 varepsilon cdot frac 1 cos nu 1 cos nu frac 1 varepsilon 1 varepsilon cdot tan 2 frac nu 2 khundwy 1 e 1 e aelaisrakthisxng caidphllphth tan n2 1 e1 e tan E2 displaystyle tan frac nu 2 sqrt frac 1 varepsilon 1 varepsilon cdot tan frac E 2 inkhnthisamnieracaidkhwamechuxmoyngknrahwangewlakbtaaehnnginwngokhcr khnthisikhuxkarkhanwnrayahangsunysuriya r cakmumkwadcring n dwykdkhxaerkkhxngekhphephlxr r a 1 e21 e cos n displaystyle r a cdot frac 1 varepsilon 2 1 varepsilon cdot cos nu karxnuphthth Derivation kdkhxngniwtnkarxnuphththkhxngkdekhphephlxrkhxthi 2 m r M mr2 r G displaystyle m cdot ddot mathbf r frac M cdot m r 2 cdot hat mathbf r cdot G r 8 8 displaystyle dot hat mathbf r dot theta hat boldsymbol theta where 8 displaystyle hat boldsymbol theta is the tangential unit vector and 8 8 r displaystyle dot hat boldsymbol theta dot theta hat mathbf r So the position vector r rr displaystyle mathbf r r hat mathbf r is differentiated twice to give the velocity vector and the acceleration vector r r r rr r r r8 8 displaystyle dot mathbf r dot r hat mathbf r r dot hat mathbf r dot r hat mathbf r r dot theta hat boldsymbol theta r r r r r r 8 8 r8 8 r8 8 r r8 2 r r8 2r 8 8 displaystyle ddot mathbf r ddot r hat mathbf r dot r dot hat mathbf r dot r dot theta hat boldsymbol theta r ddot theta hat boldsymbol theta r dot theta dot hat boldsymbol theta ddot r r dot theta 2 hat mathbf r r ddot theta 2 dot r dot theta hat boldsymbol theta Note that for constant distance r displaystyle r the planet is subject to the centripetal acceleration r8 2 displaystyle r dot theta 2 and for constant angular speed 8 displaystyle dot theta the planet is subject to the coriolis acceleration 2r 8 displaystyle 2 dot r dot theta Inserting the acceleration vector into Newton s laws and dividing by m gives the vector r r8 2 r r8 2r 8 8 GMr 2r displaystyle ddot r r dot theta 2 hat mathbf r r ddot theta 2 dot r dot theta hat boldsymbol theta GMr 2 hat mathbf r Equating component we get the two of motion one for the radial acceleration and one for the tangential acceleration r r8 2 GMr 2 displaystyle ddot r r dot theta 2 GMr 2 r8 2r 8 0 displaystyle r ddot theta 2 dot r dot theta 0 imsamarthaeykwiekhraahid SVG samarthepidichngan MathML phanplkxinkhxngebrawesxr kartxbsnxngthiimthuktxng Math extension cannot connect to Restbase cakesirfewxr http localhost 6011 th wikipedia org v1 displaystyle r dot theta 8 8 2r r 0 displaystyle frac ddot theta dot theta 2 frac dot r r 0 and integrate log 8 2log r log ℓ displaystyle log dot theta 2 log r log ell where log ℓ displaystyle log ell is a and exponentiate r28 ℓ displaystyle r 2 dot theta ell This says that the r28 displaystyle r 2 dot theta is a even if both the distance r displaystyle r and the 8 displaystyle dot theta vary The area swept out from time t1 to time t2 t1t212 base height dt t1t212 r r8 dt 12 ℓ t2 t1 displaystyle int t 1 t 2 frac 1 2 cdot base cdot height cdot dt int t 1 t 2 frac 1 2 cdot r cdot r dot theta cdot dt frac 1 2 cdot ell cdot t 2 t 1 depends only on the duration t2 t1 This is Kepler s second law karxnuphththkhxngkdekhphephlxrkhxthi 1 p ℓ2G 1M 1 displaystyle p ell 2 G 1 M 1 u pr 1 displaystyle u pr 1 and get GMr 2 ℓ2p 3u2 displaystyle GMr 2 ell 2 p 3 u 2 and 8 ℓr 2 ℓp 2u2 displaystyle dot theta ell r 2 ell p 2 u 2 X dXd8 8 dXd8 ℓp 2u2 displaystyle dot X frac dX d theta cdot dot theta frac dX d theta cdot ell p 2 u 2 Differentiate r pu 1 displaystyle r pu 1 twice r d pu 1 d8 ℓp 2u2 pu 2dud8 ℓp 2u2 ℓp 1dud8 displaystyle dot r frac d pu 1 d theta cdot ell p 2 u 2 pu 2 frac du d theta cdot ell p 2 u 2 ell p 1 frac du d theta r dr d8 ℓp 2u2 dd8 ℓp 1dud8 ℓp 2u2 ℓ2p 3u2d2ud82 displaystyle ddot r frac d dot r d theta cdot ell p 2 u 2 frac d d theta ell p 1 frac du d theta cdot ell p 2 u 2 ell 2 p 3 u 2 frac d 2 u d theta 2 Substitute into the radial equation of motion r r8 2 GMr 2 displaystyle ddot r r dot theta 2 GMr 2 and get ℓ2p 3u2d2ud82 pu 1 ℓp 2u2 2 ℓ2p 3u2 displaystyle ell 2 p 3 u 2 frac d 2 u d theta 2 pu 1 ell p 2 u 2 2 ell 2 p 3 u 2 Divide by ℓ2p 3u2 displaystyle ell 2 p 3 u 2 d2ud82 u 1 displaystyle frac d 2 u d theta 2 u 1 u 1 displaystyle u 1 d2ud82 u 0 displaystyle frac d 2 u d theta 2 u 0 These solutions are u ϵ cos 8 A displaystyle u epsilon cdot cos theta A where ϵ displaystyle epsilon and A displaystyle A are arbitrary constants of integration So the result is u 1 ϵ cos 8 A displaystyle u 1 epsilon cdot cos theta A Choosing the axis of the coordinate system such that A 0 displaystyle A 0 and inserting u pr 1 displaystyle u pr 1 gives pr 1 1 ϵ cos 8 displaystyle pr 1 1 epsilon cdot cos theta If ϵ lt 1 displaystyle epsilon lt 1 this is Kepler s first law kdekhphephlxrkhxthi 3 T2 4p2GM r3 displaystyle T 2 frac 4 pi 2 GM cdot r 3 where T planet s sidereal period r radius of the planet s circular orbit G the gravitational constant M mass of the sun T2 4p2G M m a3 displaystyle T 2 frac 4 pi 2 G M m cdot a 3 ody T object s sidereal period a object s semimajor axis G the gravitational constant 6 67 10 11 N m kg M mass of one object m mass of the other object 12 1 ϵ a VAdt 12 1 ϵ a VBdt displaystyle begin matrix frac 1 2 end matrix cdot 1 epsilon a cdot V A dt begin matrix frac 1 2 end matrix cdot 1 epsilon a cdot V B dt 1 ϵ VA 1 ϵ VB displaystyle 1 epsilon cdot V A 1 epsilon cdot V B VA VB 1 ϵ1 ϵ displaystyle V A V B cdot frac 1 epsilon 1 epsilon mVA22 GmM 1 ϵ a mVB22 GmM 1 ϵ a displaystyle frac mV A 2 2 frac GmM 1 epsilon a frac mV B 2 2 frac GmM 1 epsilon a VA22 VB22 GM 1 ϵ a GM 1 ϵ a displaystyle frac V A 2 2 frac V B 2 2 frac GM 1 epsilon a frac GM 1 epsilon a VA2 VB22 GMa 1 1 ϵ 1 1 ϵ displaystyle frac V A 2 V B 2 2 frac GM a cdot left frac 1 1 epsilon frac 1 1 epsilon right VB 1 ϵ1 ϵ 2 VB22 GMa 1 ϵ 1 ϵ 1 ϵ 1 ϵ displaystyle frac left V B cdot frac 1 epsilon 1 epsilon right 2 V B 2 2 frac GM a cdot left frac 1 epsilon 1 epsilon 1 epsilon 1 epsilon right VB2 1 ϵ1 ϵ 2 VB2 2GMa 2ϵ 1 ϵ 1 ϵ displaystyle V B 2 cdot left frac 1 epsilon 1 epsilon right 2 V B 2 frac 2GM a cdot left frac 2 epsilon 1 epsilon 1 epsilon right VB2 1 ϵ 2 1 ϵ 2 1 ϵ 2 4GMϵa 1 ϵ 1 ϵ displaystyle V B 2 cdot left frac 1 epsilon 2 1 epsilon 2 1 epsilon 2 right frac 4GM epsilon a cdot 1 epsilon 1 epsilon VB2 1 2ϵ ϵ2 1 2ϵ ϵ2 1 ϵ 2 4GMϵa 1 ϵ 1 ϵ displaystyle V B 2 cdot left frac 1 2 epsilon epsilon 2 1 2 epsilon epsilon 2 1 epsilon 2 right frac 4GM epsilon a cdot 1 epsilon 1 epsilon VB2 4ϵ 4GMϵ 1 ϵ 2a 1 ϵ 1 ϵ displaystyle V B 2 cdot 4 epsilon frac 4GM epsilon cdot 1 epsilon 2 a cdot 1 epsilon 1 epsilon VB GM 1 ϵ a 1 ϵ displaystyle V B sqrt frac GM cdot 1 epsilon a cdot 1 epsilon dAdt 12 1 ϵ a VBdtdt 12 1 ϵ a VB displaystyle frac dA dt frac frac 1 2 cdot 1 epsilon a cdot V B dt dt begin matrix frac 1 2 end matrix cdot 1 epsilon a cdot V B 12 1 ϵ a GM 1 ϵ a 1 ϵ 12 GMa 1 ϵ 1 ϵ displaystyle begin matrix frac 1 2 end matrix cdot 1 epsilon a cdot sqrt frac GM cdot 1 epsilon a cdot 1 epsilon begin matrix frac 1 2 end matrix cdot sqrt GMa cdot 1 epsilon 1 epsilon dd T dAdt pa 1 ϵ2 a displaystyle T cdot frac dA dt pi a sqrt 1 epsilon 2 a T 12 GMa 1 ϵ 1 ϵ p 1 ϵ2 a2 displaystyle T cdot begin matrix frac 1 2 end matrix cdot sqrt GMa cdot 1 epsilon 1 epsilon pi sqrt 1 epsilon 2 a 2 T 2p 1 ϵ2 a2GMa 1 ϵ 1 ϵ 2pa2GMa 2pGMa3 displaystyle T frac 2 pi sqrt 1 epsilon 2 a 2 sqrt GMa cdot 1 epsilon 1 epsilon frac 2 pi a 2 sqrt GMa frac 2 pi sqrt GM sqrt a 3 T2 4p2GMa3 displaystyle T 2 frac 4 pi 2 GM a 3 T2 4p2G M m a3 displaystyle T 2 frac 4 pi 2 G M m a 3 s t ph xangxingHyman Andrew A Simple Cartesian Treatment of Planetary Motion 2011 08 07 thi ewyaebkaemchchin European Journal of Physics Vol 14 pp 145 147 1993 Kepler s Second Law by Jeff Bryant with Oleksandr Pavlyk duephimkhwamonmthwngaehlngkhxmulxunCrowell Benjamin Conservation Laws http www lightandmatter com area1book2 html 2020 06 01 thi ewyaebkaemchchin an online book that gives a proof of the first law without the use of calculus see section 5 2 p 112 David McNamara and Gianfranco Vidali Kepler s Second Law JAVA Interactive Tutorial http www phy syr edu courses java mc html kepler html 2006 09 10 thi ewyaebkaemchchin an interactive JAVA applet that aids in the understanding of Kepler s Second Law University of Tennessee s Dept Physics amp Astronomy Astronomy 161 page on Johannes Kepler The Laws of Planetary Motion 1 Equant compared to Kepler interactive model 2 2008 12 26 thi ewyaebkaemchchin Kepler s Third Law interactive model 3 2008 12 26 thi ewyaebkaemchchin