A Solution of Kepler s Equation

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1 Itratioal Joural of Astroomy ad Astrohysics, 4, 4, Publishd Oli Dcmbr 4 i SciRs. htt:// htt://dx.doi.org/.46/ijaa A Solutio of Klr s Equatio Joh N. Tokis Tchological Educatioal Istitut of Eirus, Ioaia, Grc JohTokis@ioa.ti.gr Rcivd 7 Octobr 4; rvisd 4 Novmbr 4; acctd Dcmbr 4 Acadmic Editor: Ra Zhou, Thortical High Ergy Physics Grou, Frmi Natioal Acclrator Laboratory, Chia Coyright 4 by author ad Scitific Rsarch Publishig Ic. This work is licsd udr th Crativ Commos Attributio Itratioal Lics (CC BY). htt://crativcommos.org/licss/by/4./ Abstract Th rst study dals with a traditioal hysical roblm: th solutio of th Klr s quatio for all coics (llis, hyrbola or arabola). Solutio of th uivrsal Klr s quatio i closd form is obtaid with th hl of th two-dimsioal Lalac tchiqu, xrssig th uivrsal fuctios as a fuctio of th uivrsal aomaly ad th tim. Combiig ths w xrssios of th uivrsal fuctios ad thir idtitis, w stablish o biquadratic quatio for uivrsal aomaly ( ) for all coics; solvig this w quatio, w hav a w xact solutio of th rst roblm for th uivrsal aomaly as a fuctio of th tim. Th vrifyig of th uivrsal Klr s quatio ad th traditioal forms of Klr s quatio from this w solutio ar discussd. Th lots of th llitic, hyrbolic or arabolic Klria orbits ar also giv, usig this w solutio. Kywords Klria otio, Uivrsal Klr s Equatio, Uivrsal Aomaly, Two-Dimsioal Lalac Trasforms. Itroductio I th Klria roblm, a body of mass m follows a coic orbit, for which th focus is idtifid to th ctr of attractig body (with mass m ). Th Klria motio is dscribd by th fudamtal diffrtial quatio of th hysical two-body roblm: d r r = µ () dt r whr r is th vctor ositio of th movig body rlatd to th attractio ctr ad µ th gravitatioal a- How to cit this ar: Tokis, J.N. (4) A Solutio of Klr s Equatio. Itratioal Joural of Astroomy ad Astrohysics, 4, htt://dx.doi.org/.46/ijaa.4.446

2 µ = + > (whr G th Nwtoia gravitatioal costat) (s [], Equatio (6..)). Th traditioal form of Klr s quatio, which ca b obtaid dirctly from Equatio () (s [], Sctio 6. ad []), is ormally writt as: ramtr dfid by G( m m ) For llitic orbits ( ) for hyrbolic orbits ( ) E sie = (a) E sihe = (b) whr is th cctricity, E th cctric aomaly ad th ma aomaly, which is dfid as h µ = t, < a µ = t, < ( a) ( q ) h (c) (d) µ = t, = () (s [], Equatio () ad [], Equatios (4.5) ad (4.5)). Rmark that th tim t is masurd from rictr ad a is th smimajor axis, which is ositiv for lliss, gativ for hyrbolas, ad ifiit for arabolas; also, q is th rictr distac of th orbit. Th cas of = lads to a circular orbit ad th siml solutio E = (cf., Equatio (a)), so that w will rgard > hraftr i th rst work. Johas Klr aoucd th rlvat laws of abov quatio arly i 69 ad 69 [4]. H has usd hysics as a guid i this discovry [5]. For four cturis, th Klr s roblm is to solv th oliar Klr s Equatio () for th cctric aomaly. Early aalytical solutio of Klr s quatio was cosidrd i a comrhsiv study of Tissrad [6]. Rctly, aalytical works of th solutio ad us of Klr s quatio hav b roosd by various authors (s,.g., [7]-[]). I virtually vry dcad from 65 to th rst, thr hav aard ars dvotd to th solutio of this Klr s quatio. Its xact aalytical solutio is ukow, ad thrfor, fficit rocdurs to solv it umrically hav b wll discussd i may stadard txt books of Clstial chaics ad Astrodyamics as wll as i a larg umbr of ars. Colwll [] cotais xtsiv rfrcs to th Klr roblm i his book. Durig last two dcads, studis wr carrid out by svral ivstigators of th rst roblm [] [4]-[8]. I ths studis, thy usd umrical or aroximatios mthods for solutio of th Klr s quatio. Hc, it aars that a aalytical solutio of th Klr s quatio will b of grat itrst. I th currt study, a aalytical ivstigatio of th Klr s quatio ral roots i closd form is rstd. I Sctio, w will stablish th gral form of Klr s quatio ad will clar u th usful idtitis of th uivrsal fuctios. I Sctio, usig th two-dimsioal Lalac trasform tchiqu, w will rst a aalytical solutio for th uivrsal Klr s quatio, obtaiig th uivrsal fuctios ( U, =,,,) as fuctio of th uivrsal aomaly ( ) ad th tim ( t ). I Sctio 4, i th first st w will stablish o w biquadratic quatio for uivrsal aomaly for all coics with th hl of Nwma s quatio (cf. Equatio ()) ad som idtitis of th w xrssios of uivrsal fuctios. Th, th solutio = ( t) of th rst roblm has b obtaid, solvig this biquadratic quatio for all coics. Fially, discussio of th rsults, thus obtaid, is rstd i Sctio 5; th w solutio of th roblm will rov that vrifis th traditioal form of Klr s quatios for llitic, hyrbolic or arabolic orbits. Th llitic, hyrbolic or arabolic Klria motio is asily lottd, usig this w solutio.. Gral Form of Klr s Equatio I ordr to solv th Klr s Equatio (), w us hr th gralizd form of this quatio with th uivrsal fuctios ad th uivrsal aomaly istad of th cctric aomaly (s [], Sctio 4.5). 684

3 Workig for th Klr s Equatio (), w cosidr a objct followig a ath of sam cctricity about th ctr of attractig body; th objct is at tim t i (vctor) ositio r( t ) with (vctor) vlocity v. Th tim t is masurd from th rictr assag; so, wh t = this objct was at ositio r (with r = r ) of th rictr with vlocity v (with v = υ) ad cctric aomaly E =. W mhasiz that th vctors r ad r origiat at th ctr of attractio. Th, w itroduc th uivrsal aomaly, which is dfid by Sudma trasformatio: (s [], Equatio (4.7)) ad rlatd to th classical cctric aomaly by µ dt = rd () E a, a > (4a) E a, a < (4b) D, a = (4c) whr E is th cctric aomaly agl of llitic or hyrbolic orbit ad D th arabolic cctric aomaly of arabolic orbit with dimsio L (s [9], Equatio (8)). Th α dots th rcirocal of th smimajor axis a, amly υ υ a = a r µ r µ (5a,b,c) Ddig o th sig of α or th valu of th cctricity, th ty of th orbit is dtrmid such that: α > (or < ) for llitic orbits; α < (or > ) for hyrbolic orbits ad α = (or = ) for arabolic orbits. Not that th uivrsal aomaly is a w iddt variabl with dimsio L (s [], Equatio (4.7)). From th iitial coditio E = ad th kow rlatios: cose = ra, sie = σ a for llitic orbits ad coshe = ra, sihe = σ a for hyrbolic orbits (s [], Sctios 4. ad 4.4), w hav also for th rst roblm r v q = r = a ( ), σ = (6a,b) µ whr q stads for th rictr distac of th orbit rlatd to th aramtr with th rlatio ( ) q( ) = a = + (7) Rmark that th is a o-gativ aramtr ad th rictr distac q may b ositiv or zro; both of thm hav dimsios of lgth (s [], Sctio 4.). Now, usig th uivrsal fuctios dfid by with thir followig usful rortis: U ( α) k = ( α) + ( + k) k k ; =, =,,,, (8)! (9a) U α ; = U α ; d, =,,, U( α ; ) + αu+ ( α ; ) =, =,,,, (9b)! U U ( α ; ) ( α ; ) ( α) = U ;, =,,, = au ( α ; ) (s [], Equatio (9.7)), th two forms of Klr s Equatio () ar icororatd i o uivrsal quatio (9c) (9d) 685

4 ( α ; ) ( α ; ) µ qu + U = t () which is a stadard form of th traditioal Klr s Equatios () with th och at rictr assag (s [9], Equatio () ad [], Equatio (9)). Th gral formula () is valid for all valus of α ad ; i articular, it is good for arabolic orbits whr a = a =. To fid out th xrssio of may orbital quatitis,.g. th magitud of th ositio vctor r( t ), w must solv th stadard uivrsal form () of th Klr s quatio for th uivrsal aomaly as fuctio of th tim, = t. amly. Solutio of th Uivrsal Klr s Equatio I ordr to obtai th aalytical solutio of th rst roblm, w shall solv first th uivrsal Klr s quatio Equatio (), obtaiig th uivrsal fuctios U, =,,, as a fuctio of th uivrsal aomaly ad th tim. For this uros, w will us th doubl Lalac trasformatio tchiqu, which was aalytically studid by Aghili ad Salkhordh-oghaddam [9] ad by Valkó ad Abat []. Th uivrsal fuctios U : For this cas, w itroduc a w variabl ω(,t) so that U U ( t) ω (a), U = = ω (s Equatios (9)) ad th uivrsal Klr s Equatio () bcoms (b) U = U d = ωd (c) d t (a) qω + ω = µ From th iitial coditio E = or = ad Equatio (9a), w hav th corrsodig iitial coditios to th Equatio (a) ω, t = ω, t = (b) x Th alicatio of doubl Lalac trasform (with rsct to ad t ) to th Equatios () givs th so- Ω s, ξ i trasform domai as lutio ( s, ξ ) Ω = ξ µ ( qs + ) whr s ad ξ ar th trasform variabls of ad tim t, rsctivly. For th solutio of artial diffrtial Equatio (a), w us th Adix. Now, th uivrsal fuctio U ca b obtaid by takig th ivrs trasform of Equatio () (cf., Adix). So, w gt A U = ω(, t) = µ t (4) q whr w hav abbrviatd A = L. with dimsios: [ ] Th uivrsal fuctios ( ) A q q q () si, > (5) U : For this cas, w will us a w variabl (,t) U ψ ( t) U U, ψ so that (6a) U = = U = = ψ ψ (6b) (6c) 686

5 (s Equatios (9c)) ad th uivrsal Klr s Equatio () bcoms + = t (7a) qψ ψ µ For E = or = ad Equatio (9a), w hav th iitial coditios to th Equatio (7a) ( t) ( t) ( t) ψ, = ψ, = ψ, = (7b) Similarly as i th cas of U, usig th doubl Lalac trasform (with rsct to ad t ) for th Equa- Ψ s, ξ i trasform domai as tios (7), w obtaid th solutio (cf., Adix). Ivrtig ( s, ξ ) µ Ψ ( s, ξ ) = ξ s qs ( + ) Ψ w hav th uivrsal fuctio U whr w hav dfid th o-dimsioal fuctio (8) U = ψ, t = B µ t (9) ( ) B cos q, q > () Th, substitutig th rsults of Equatio (4) ad (9) ito th rlatios U+ au = ad U + au = (cf., Equatio (9b)), rsctivly, w ca asily obtai th w uivrsal fuctios U ad U as U = B αµ t () = () U A t q αµ whr A ad B ar giv from Equatios (5) ad (), rsctivly. 4. Aalytical Solutio of th Problm I ordr to obtai a solutio for th uivrsal aomaly ( t) whr σ = r v µ uivrsal fuctios Equatio (4.8)) =, w us th xlicit xrssio for : = αµ t + σ (). This rlatio was discovrd by C.. Nwma ad its dos ot ivolv ay of th U (s [], Equatio (4.86)). Th σ ca b also giv by th kow quatio (s [], ( αqu ) σ = (4) Substitutig Equatio (4) (with U giv by ()) ito Equatio (), th followig rlatio is obtaid ( )( ) q + α q B µ t = µ t (5) To fid out two mor rlatios btw, A ad B, similar to Equatio (5), w will us th basic rlatio si ( q ) + cos ( q ) = ad th dfiitios of A ad B Equatios (5) ad (), rsctivly. Thus, w ca asily obtai th rlatio B A q + = (6) = + (s [], Equatio (4.9)); idd, with th hl of Equatios (4), () ad (), w obtai th rlatio Furthr, w ca fid o mor rlatio usig th basic idtity U U ( U ) A A ( B) αµ t t t q µ q αµ = Th thr Equatios (5), (6) ad (7) ar a systm of th thr ukows:, A ad B. Solvig this systm, w gt from two Equatios (5) ad (6) th followig rlatios (7) 687

6 q = (8a) ( B) µ t µ t ( αµ t) A µ t q µ αµ q q ( t) t = Fially, substitutig Equatios (8) ito Equatio (7), w obtai th followig biquadratic quatio for uivrsal aomaly whr w hav abbrviatd with dimsios: [ z] = L, [ b ] = L ad [ b ] L 4 (8b) z + b z + b = (9) z αµ t (a) µ t b q ( ) + q t t b µ ( ) µ 4 q q =. Th solutio of th biquadratic Equatio (9) givs th rlatio btw th uivrsal aomaly ad th tim t for all coics: llis, hyrbola or arabola. Th solutio of this quatio ca b obtaid usig th stadard formula of th solutio for biquadratic quatio. Solvig th w biquadratic Equatio (9), w gt th solutio of th rst roblm for th uivrsal aomaly at tim t as show blow whr w hav abbrviatd articularly, for llitic orbits ( ) for hyrbolic orbits ( < ) ad for arabolic orbits ( = ) (b) (c) = αµ t+ ϕ t () ϕ ( t) q ( + µ t q ) ( ) µ t q < < w hav (a) ϕ ( t) a ( )( + µ t q ) (b) ϕ ( t) ( a ) ( )( µ t q + ) (c) { } ϕ t q + µ t q (d) = 6 + >, for q >. Cosqutly, th solutio () is ral i th cass of hyrbolic ad arabolic cass sic th corrsodig ϕ ( t), giv from Equatios (c,d), ar always ral-valud. I th cas of th llitic Klria orbits, th Equatio () is ral oly i th scial cas for which ϕ ( t), giv from Equatio (b), bcoms ral, amly for th cas Rmark that th discrimiat of th biquadratic Equatio (9) is q ( µ t q ) whr µ t q 4 < or < ( ) ( ) is th ma aomaly dfid by Equatio (c). Th ur limit of this ma aomaly (,f) is d- 688

7 fid from th rlatio (f) for vry comltd tri of th orbitig body i its llitic orbit about th ctr of th attractig body; this limit is usful for dtrmiatio of ach Klria llis (cf., th alicatios i th xt sctio). I th cas of arabolic orbits whr th limitig cas a (or α = ) ad corrsods to r = q = ad σ = r v µ =, th quatio (9) ad its solutio () ar rducd to 4 + 4q 4µ t q = (a) = ϕ t = q + µ t q Th Equatio () is th solutio of th rst roblm for all coics (llis, hyrbola or arabola) ad xrsss th rlatio btw th uivrsal aomaly ad th tim t. Kowig th solutio of th uivrsal aomaly ( t), w stablish th xact xrssios of th uivrsal fuctios U, =,,, as fuctios of th tim t. Idd, usig Equatio (), th Equatio (8) ar obtaid as fuctios of th tim t : A q µ t = + µ t q q ( B) µ t µ t ϕ( t) (b) (4a) q = (4b) Th, th uivrsal fuctios (9), (4), () ad () ar xrssd as fuctios of th tim t as show blow q = (5a) U µ t ϕ t U q = + µ t q (5b) U = ϕ ( t) (5c) U = ( ) + µ t q Th magitud of th ositio vctor r of th orbitig body is ( α ; ) ( α ; ) ( α ; ) (5d) r = ru + U = q+ U (6a,b) (s [], Equatio (4.8) ad [4], Equatio (8)). Substitutig ito Equatio (6b) th w xrssios of U, giv by Equatio (5b), w obtai th tim-ddt distac r of th orbitig body from th ctr of attractio as r = q + t µ q Furthrmor, if w work i th orbital rfrc systm with th origi at th attractig ctr (or focus), w XY, la to b th la of motio with X -axis oitig toward rictr ad Y -axis i th f is 9 ; i this way th Z -axis is aralll to th agular momtum. P XY, o th coic orbit, w hav X = rcosf ad chos th dirctio for which th tru aomaly Th, if X ad Y ar th coordiats of a oit (7) r = X (8) whr is th o-gativ quatity giv by Equatio (7) (s [], Equatio (4.)). Thus, th X ad Y coordiats ca b xrssd as fuctio of th tim t, usig Equatios (8) ad th idtitis: U + au =, U = U + U ; amly, w gt 689

8 X = q U α (9a) ; ( ) ( α ; ) Y q U = + (9b) whr U ad U ar giv from Equatios (5) (s [4], Equatios (6)-(7)). 5. Discussio Usig th stadard form of th uivrsal Klr s quatio () with th och at rictr assag, w hav drivd a w biquadratic quatio (9) for uivrsal aomaly as a fuctio of th tim t. This quatio govrs th motio of a orbitig body followig a ath with ositio vctor r( t ) from th ctr of attractig body ad with vlocity vctor v ( t). Th w solutio () of th rst roblm was obtaid solvig Equatio (9) with iitial-valu coditios for th orbitig body at tim t = assags from th rictr, with miimum ositio vctor r, vlocity vctor v ad cctric aomaly E =. Th solutio () is a solutio of th rst roblm. Idd, th w xrssios of th uivrsal fuctios U, ad U giv by (5a,c) vrify uivrsal Klr s Equatio (). orovr, th solutio () is a solutio of th Equatio (9), sic it ca b asily vrifid by substitutio of () ito (9). Th solutio () vrifis also th traditioal forms of Klr s Equatio (). Particularly, i th cas of llitic orbits ( < < ) th solutio () of th roblm is rducd for th cctric aomaly (cf. Equatio (4a)) to th form whr E α ϕ t is th ma aomaly giv by Equatio (c) ad = = + (4) ( ) ϕ α ϕ Not that ϕ ( t) < ad Similarly, i th cas of hyrbolic orbits ( ) < th solutio () of th roblm is rducd, for th cctric aomaly (cf. Equatio (4b)), to whr ( t) ( t) = ( ) + ( ) < (cf. Equatio (f)) for < <. (4) ϕ h is th ma aomaly giv by Equatio (d) ad E = a = + t (4) h h ( ) ϕh( t) ( α) ϕ( t) = + ( ) + ( ) Fially, i th cas of arabolic orbits ( = ) th solutio (b) givs for th arabolic cctric aomaly (cf. Equatios (4c) ad ()): ϕ D = x = t = q +. From th othr had, th stadard ad hyrbolic trigoomtric fuctios of Equatios () ar xrssd as si sih E si U ϕ t α α (4) (44) = = = (45a) E sih U ϕ h t = α = α = whr w hav us Equatio (5) ad th rlatioshi btw th fuctio U ad th stadard ad hyrbolic trigoomtric fuctios of Equatios (): ( U = α si α ) ad U = ( α) sih ( α) for llis (45b) 69

9 ad hyrbola, rsctivly (s [], Problm 4 - ad [9], Equatio (4)). Now, w will rov that th Equatios (4) ad (4) rrst th solutios of th traditioal forms of Klr s Equatios (). Idd, it ca b show that th lft-had sids of Equatios () ar rducd to th righthad sids, amly ϕ ( t) E sie = + ϕ( t) = (46a) ( t) ϕh E sihe = h + ϕh( t) = h (46b) i accordac with th Equatios (4), (4) ad (45). It should b oitd out that our solutios for cctric aomaly (cf., Equatios (4) ad (4)) ar rady for hysical alicatios i th corrsodig Klria orbits. I additio to abov Klria orbits, th w solutio (b) of th rst roblm for th cas of arabola ( = ad α = ) vrifis th traditioal Barkr s quatio for arabolic orbits + 6q = 6µ t (47) Rmark that th arabolic Klria quatio is calld Barkr s quatio (s [], Equatio (4.4)). Idd, from th dfiitio (8) for th uivrsal fuctios w hav = 6 U ( ;) ad = U ( ;). Usig th solutio (b), ths uivrsal fuctios ar xrssd from Equatios (5a,c), so that whr ( t) = 6 t 6 q t, = t (48a,b) µ ϕ ϕ ϕ is giv from Equatio (d). Th, th lft-had sid of Equatios (47) is rducd to th righthad sid, amly + 6q = 6µ t 6qϕ t + 6qϕ t = 6µ t (49) I ordr to study th Klria orbits with th hl of th w solutios, w us also th cartsia coordiats x ad y with th origi at th ctr of th llis or hyrbola. I this cas th X ad Y coordiats of th orbital ( XY, ) la systm giv by Equatios (9) ar rlatd to th systm ( xy, ) with th rlatios X = x a ad Y = y (s [], Equatio (4.4)); so, for th cas, ths w coordiats ca b obtaid i th xlicit forms x = a ζ (5a) whr w hav dfid th o-dimsioal rlatio ( ) y = ζ a (5b) ζ ( ) + µ t q i accordac to th Equatios (9). Th, w itroduc th o-dimsioal coordiats x ζ a y ζ η = ( ) a Th w xrssios (5) vrify th followig quatios of llis ( < ) ad hyrbola ( > ), rsctivly: ζ ζ η + = η = (5) (5a) (5b) (5a) (5b) 69

10 whr ζ is dfid by Equatio (5). Rmark that th quatios (5) ar th o-dimsioal forms of th llis ad hyrbola, rsctivly, i th cartsia systm ( xy, ) with th origi at th ctr of th llis or hyrbola (s [], Equatios (A..) ad (A.4.)). I th othr had, for th cas of arabola ( = ), th coordiats X ad Y ca b also obtaid, from Equatios (9), as followig Th, w hav, from Equatios (54), ( µ ) X = q + t q ( µ ) Y = q + t q Y = 4q( q X) = ( X ) (55a,b) Th last Equatio (55) is th quatio of arabola, which asss through its rictr with coordiats (, ) (s [], Equatio (.)). Th o-dimsioal form of Equatio (55) is with th o-dimsioal coordiats (54a) (54b) η = 4ζ (56) ( t q ) ζ µ, η ζ + (57a,b) I additio to abov rsults for th o-dimsioal coordiats w hav (for hysical alicatio) th corrsodig xrssios: < < th Equatio (5) bcoms For llitic orbits ζ ( ) + for hyrbolic orbits ( < ) th Equatio (5) bcoms ζ + ( ) + for arabolic orbits ( = ) th Equatio (57a) yilds ( ) ( ) ( ) h (58a) (58b) ζ + (58c) whr, h ad ar th ma aomalis of llis, hyrbola ad arabola, rsctivly (cf., Equatios (c,d,)). Ths ma aomalis ca b varid from to k π, whr k is a itgr. Not that th odimsioal coordiats of attractiv ctr (or focus) F i th o-dimsioal cartsia systm ( ζη, ) is giv by F (,) for all Klria orbits (llis, hyrbola or arabola). I ordr to gt a hysical isight ito th w solutio of th Klr s roblm, w aly th abov rsults for th systm Earth-oo. For this systm th cctricity of th oo is =.549 ad th ur limit for th ma aomaly ca b obtaid from rlatio (f) as <.944. So, varyig th ma aomaly (with.944 ) ad usig Equatios (58a) ad (5b), th llitic Klria motio of th oo about th Earth ca b asily lottd i th o-dimsioal cartsia systm ( ζη, ) (Figur ). Rmark that th o-dimsioal coordiats of th Earth ar obtaid as (., ). Not that th us of th ur limit of th ma aomaly, giv from th rlatio < (cf. rlatio (f)), is imort for th lottig of all llitic Klria orbits. To cofirm that w giv two mor xamls: (a) W cosidr a objct followig a llitic orbit with cctricity =.5 about th ctr of attractig body; th ur limit of th ma aomaly, from rlatio (f), is <.44 ; th, varyig th ma aomaly (with.44 ) ad usig Equatios (58a) ad (5b), w lot th llitic Klria orbit i th ζη, (Figur ). (b) W cosidr aothr objct i a llitic orbit with o-dimsioal cartsia systm 69

11 Figur. Th llitic orbit of th oo ( =.549) about th Earth. Figur. Two llitic Klria orbits with cctricitis =.5 ad =.97. cctricity =.97 about th ctr of aothr attractig body; th ur limit of th w ma aomaly, from rlatio (f), is <.464 ; ad, varyig th ma aomaly (with.45 ), w lot th llitic Klria orbit i th o-dimsioal cartsia systm ( ζη, ) (Figur ). Now, varyig th ma aomaly (with.4 for th rst lot) ad usig quatios (58b) ad h h 69

12 η = ζ (cf., Equatio (5b), w lot of th hyrbolic Klria motio of a orbitig body with th cctricity =. about th attractiv ctr F (.69, ) (Figur ). Fially, varyig th ma aomaly (with for th rst lot) ad usig Equatios (58c), ad (56), w lot of th arabolic Klria motio of a orbitig body with th cctricity = about th attrac- F, (Figur 4). tiv ctr 6. Coclusios This work rsts a solutio to th wll kow Klria two body hysical roblm. From th ivstigatio Figur. Th hyrbola of a orbitig body (with =. ) about th attractiv ctr F (.69,). Figur 4. Th arabola of a orbitig body (with = ) about th attractiv ctr F (,). 694

13 for this w solutio, th mai coclusios hav b draw as followig: ) A aalytical solutio for th uivrsal Klr s quatio has b dtrmid, obtaiig th uivrsal fuctios U, =,,, as fuctio of th uivrsal aomaly ( ) ad th tim ( t ) with th hl of th two-dimsioal Lalac trasform tchiqu. ) Usig a xlicit xrssio for th uivrsal aomaly ( ) without ay of th U fuctios (cf., Equatio ()) ad som idtitis of th w obtaid uivrsal fuctios, w dvlod a biquadratic quatio for uivrsal aomaly for all coics: llis, hyrbola or arabola. ) Th solutio = ( t) of th rst roblm has b obtaid, solvig this biquadratic quatio for all coics. 4) This w aalytical solutio for th uivrsal aomaly has b discussd ad rovd that vrifis th uivrsal Klr s quatio (cf., Equatio ()), sic th tim ddd uivrsal fuctios U ad U vrify this quatio. Th, th solutios for th cctric aomaly (cf., Equatios (4) ad (4)) wr also rovd that vrify th traditioal form of Klr s quatios for llitic or hyrbolic orbits. This w solutio for th uivrsal aomaly has also rovd that vrifis th traditioal Barkr s quatio for arabolic orbits []. Th llitic, hyrbolic or arabolic Klria motio is lottd, usig this aalytical solutio. 5) To our kowldg, this work givs i closd form th actual aalytical solutio of th Klr s roblm. Th advatag of th w solutio is siml ad rady for hysical alicatios i th llitic, hyrbolic or arabolic Klria orbits. Rfrcs [] Dady, J..A. () Fudamtals of Clstial chaics. d Editio, Willma-Bll, Virgiia. [] Fukushima, T. (999) Fast Procdur Solvig Uivrsal Klr s Equatio. Clstial chaics ad Dyamical Astroomy, 75, -6. htt://dx.doi.org/./a:86884 [] Batti, R.H. (999) A Itroductio to th athmatics ad thods Astrodyamics. Rvisd Editio, AIAA Educatio Sris, Nw York. htt://dx.doi.org/.54/ [4] Volkl, J.R. () Th Comositio of Klr s Astroomia Nova. Pricto Uivrsity Prss, Nw York. htt://rss.ricto.du/titls/787.html [5] Bruc, S. (987) Klr s Physical Astroomy. Srigr-Vrlag, Nw York. [6] Tissrad, F. (894) écaiqu Célst. Vol.. Gauthir-Villars, Paris. [7] Siwrt, C.E. ad Buristo, E.E. (97) A Exact Aalytical Solutio of Klr s Equatio. Clstial chaics, 6, htt://lik.srigr.com/articl/.7/bf47#ag- [8] Tokis, J.N. (97) Effcts of Tidal Dissiativ Procsss o Stllar Rotatio. PhD Thsis, Victoria Uivrsity of achstr, achstr. [9] Frads, S.daS. () Uivrsal Closd-form of Lagragia ultilirs for Coast-Arcs of Otimum Sac Trajctoris. Joural of th Brazilia Socity of chaical Scics ad Egirig, 5, -9. htt://dx.doi.org/.59/s [] Codurach, D. ad artiusi, V. (6) Vctorial Rgularizatio ad Tmoral as i Klria otio. Joural of Noliar athmatical Physics (World Scitific),, htt://dx.doi.org/.99/jm [] Patha, A. ad Collyr, T. (6) A Solutio to a Cubic Barkr s Equatio for Parabolic Trajctoris. athmatical Gaztt, 9, [] Boubakr, K. () Aalytical Iitial-Guss-Fr Solutio to Klr s Trascdtal Equatio Usig Boubakr Polyomials Exasio Schm BPES. Airo, 7, -. htt://rdshift.vif.com/jouralfils/v7nopdf/v7nbou.df [] Colwll, P. (99) Solvig Klr s Equatio ovr Thr Cturis. Willma-Bll, Richmod. htt:// [4] Floria, L. (996) A Proof of Uivrsality of Arc Lgth as Tim Paramtr i Klr Problm. Extracta athmatica,, 5-4. htt://dml.cidoc.csic.s/df/extractaatheaticae_996 9.df [5] Fukushima, T. (998) A Fast Procdur Solvig Gauss Form of Klr s Equatio. Clstial chaics ad Dyamical Astroomy, 7, 5-. htt://lik.srigr.com/articl/.%fa%a #ag- [6] Sharaf,.A. ad Sharaf, A.A. (998) Closst Aroach i Uivrsal Variabls. Clstial chaics ad Dyamical Astroomy, 69, -46. htt://lik.srigr.com/articl/.%fa%a85#ag- [7] Sharaf,.A. ad Sharaf, A.A. () Homotoy Cotiuatio thod of Arbitrary Ordr of Covrgc for th 695

14 Two-Body Uivrsal Iitial Valu Problm. Clstial chaics ad Dyamical Astroomy, 86, 5-6. htt://lik.srigr.com/articl/./a: #ag- [8] Jia, L. () Aroximat Klr s Ellitic Orbits with th Rlativistic Effcts. Itratioal Joural of Astroomy ad Astrohysics,, 9-. htt://dx.doi.org/.46/ijaa..4 [9] Aghili, A. ad Salkhordh-oghaddam, B. (8) Lalac Trasform Pairs of N-Dimsios ad Scod Ordr Liar Partial Diffrtial Equatios with Costat Cofficits. Aals athmatica t Iformatica, 5, -. htt:// [] Valkó, P.P. ad Abat, J. (5) Numrical Ivrsio of -D Lalac Trasforms Alid to Fractioal Diffusio Equatio. Alid Numrical athmatics, 5, htt://dx.doi.org/.6/j.aum.4.. [] Ditki, V.A. ad Prudicov, A.P. (96) Oratioal Calculus i Two Variabls ad Its Alicatio. Prgamum Prss, Nw York. 696

15 Adix Solutio of artial diffrtial quatios usig two-dimsioal Lalac trasforms Th gral form of scod-ordr liar artial diffrtial quatio i two variabls is giv as followig b ar costats ad (, ) bf 5 xx + bf 4 xy + bf yy + bf x + bf y + bf = g xy,, < x, y< (A) whr g xy is sourc fuctio of x ad y or costat. W us also th abbrviatios for th iitial coditios f ( x, ) = f( x), f (, y) = f( y), (A) f x, = f x, f, y = f y, f, = f. y ad thir o-dimsioal Lalac trasformatios x 4 f4, whr s ad th trasform variabls of x ad y, rsctivly (s [9] ad [] for dtails). Th, w gt th rlatios for two-dimsioal Lalac trasforms f s, f ( ), f ( s ) ad sx y = L f xy, ;, s F s, f xy, dd xy (Aa) [ ] L f ;,, xx s = s F s sf f4 (Ab) L f ;,, xy s = sf s sf s f f (Ac) L f ;,, yy s = F s f s f s (Ad) [ ] L f ;,, x s = sf s f (A) L ;,, fy s = F s f s (Af) L ;, xy s = (Ag) s L L [ xs ;, ] [ ; s, ] = (Ah) s = (Ai) s i accordac with th two-dimsioal aalysis formula, which ca b writt as o-dimsioal aalysis i th x dirctio followd by o-dimsioal aalysis i th y dirctio: F s f xy x y sx y (, ) = (, ) d d (A4) Now, alyig doubl Lalac trasformatio to both sids of Equatio (A) ad usig Equatios (A), w obtai th solutio of Equatio (A) i th trasform domai as with th abbrviatio (, ) F s (, ) { = g( s, ) + b 5 sf + f4 ++ b 4 sf( s) + f f B s } + b f s + f s + bf + bf s (, ) 5 4. (A5a) B s = bs + bs+ b + bs+ b+ b (A5b) I ordr to ivrt this two-dimsioal Lalac trasform F( s, ), w follow th doubl ivrsio as a two- 697

16 st rocss []. I th first st w ivrt, say, o th s trasform variabl (, ) (, ) f x = L F s (A6) whr w k th scod trasform variabl as a costat. I th scod st w ivrt o th trasform variabl ad obtai, fially, (, ) = (, ) f xy L f x (A7) 698

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