REVISTA MEXICANA DE FíSICA 47 (6) problem in l1ewcoordinates, bidimensional case

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1 INVESTlGACIÓN REVISTA MEXICANA DE FíSICA 47 6) DICIEMBRE 2001 The Sun-Earth-Moon problem in l1ewcoordinates, bidimensional case A. Escalona Bucndía l and E. Piña Garza:.! Departamelllo de FÚ'ica, Uniwrsidad Autónoma AlctHJw!itana./z.tapa!apa Apartado postal 55534, Mh-ico, [)..~,MOLleo e-ii/j i1: Recibido e16 de marzo de 200J: ilceplado el 14 de ag.o",lodc 2001 \Ve pre:-.elll a lin approach 10 the study of the bidimensional ea:-.e for the Slln-E<H"lh.~loon problem using!lew eoordinatcs intnxluccd by Pilla ano J;lIléncz-i.ara. By adding the hypothesis that the Eartb-Moon sy:-.tellldescribe", a circular orhit amulld the Sun, we can eliminare 1m: 01' lhe coordinalcs. rhus rhe system is redlleeo into a problem wilh IWOdegree!'>of freedom Kl'Yll'Imis: Three bojy problem; mooll theory; celestial mechanics Pn:SClll:llllíl:-'una primera aproximación al estujio del caso bidimcnsional del pruhlema Sol-Tierra-Luna ulilizando las nuevas coordenadas pre:-.ellladas por Piña y Jiméncz Lara, Al agregar la hipótesis de que el sislema Ticrra-Luna ue:-.cihe una órhita circular alrededor del Sol, polkltloseliminar una de las coordenadas, de esta mancra el sistema qucda reducido a UIl prohlema Je dos grados de libertad. /)e,w.,-ili)f"'s: Problema de tres cuerpos; teoría lunar. lllcdnica celesle!'acs: 40.01l.-j: h: %20.-n 1.!ntroduction I.el IJ' I/I,:.!,In: l he the mas ses of Sun, Earth and!\1001l and JI. 1. and,. be the distances Earth-Moon, SUll-Moon, and SlIn-Earth. respectivcly, The new set uf coardinates is delined in ; noll-incrtial rcference syslem, parallcl to the directions of Ihe principal nenia axes of the triangle described by lhe three bodies. \Vith the origin at lhe barycenler [1,2). The lransfonnalioll of this frame inlo :'1 inertial system is made by" rolalion matrix. r = GS i i = 1,2,3), Becallse the problcm is restricted lo the bidimensional case, we need jllsl lllle ElIler angle so :"'/) - sin';; 0) G = Si' ;") CO;~1/1 ~. 1) This coordinate 4} \ViIIenable lis to sludy the motion of Ihe hree parlicles as a whole body sec Fig. 1). In Ihe rotaling frame, the conditioll that lhe orie:in is at lile baryccnter can be \\.Titten ~ L:.mj~i= O. 2) Tlle panicular selection of the reference systeill makcs Ihe inerlia Illtltrix lo be diagonal, this can be wrilten in the fol~ lowin,g equilliolls: 3) Ikc:lu... e tllree particles conform ti plane, just lwo 01'the lilolllcnts of inenia are independent: [:~ = l[ + ["l' Pii1a :lild Jiménel-Lara define t\vocoordinates with dimensions of k'ngth Jl l, n:.! relatcd ta these lllalllents of inertia [2] 4) o) J, = d ;, 6) FIjURE where l. L' Ang.h:. S,I" In this way, Eq. 2) becomes /f/'1/1'2"),:~ 1 [ + 1,'2 + 'ln J ' = '';l;f'''''2.,:.'j:~j')' S,~ '';1'.';"21/' '';:')' In. Sr = 1 S IJ = O. where III = In l' li/:.!, //1.:\). And Eqs. 3 )-5) become./ S,.MS" = O. S MST J ',/1'1 = [= I/,Ri, 8) Therc is anothcr coordinatc. an angle a, related to {he conflguratioll of lile syslcm. Thcre is not a simple geometrical interprclalían for this coordinate; the definition is given in Eqs. 1CJ) :lnd 20), and we \ViIIlIlake a redefinition i; Sec, 4. Equations 2)-5) can be sllllllllarized d~fining t\vo veclors 9) 10) 1 1) 12) 13) S MS T','o,l. = 1'1 = phi, 14 )

2 THE SUN-EARHI.MOON PROBLEM IN NEW COORDINATES, BIDI \lensional CASE \\'here '/1, M= O Lel a ami h bc'two vectors. in a spac~ of mi.lsscs. orthogollal to JII ami orthogonal to each oth~r Ihe last condilion is arhitrary hltt simplitics the calculalions). a. 1 = h. JlI 1) f/i'2 J J) J. 1I1;l = él h = J, 15 ) Sinl"L ooth Sr and S!I are orthogonal to lll, lhey musl be linear colllhinalions of a and b, Assuming wherc matrix II depends on lhe masses of the paj1icles 0- Iy.,c. 2. EIllIali)lIs )f JII)li)lI.,., nl"' /j.,., I/I.-:'l~.,,1 The lagrangian of lhe systclli is given by [3] -filial!ji) -fii:!fl.,l!j.2. 22) -I: l rt. I!J: l a~iar = 1 ', 16) 23) hmh T = Il. al\ih T = J. 17) where w is thc angular velm:ity. I is the malrix of inertia, and 18) Piil<l and Jilll ncz-lara define a sllch that [2] S, = af:!cosa + hu:!silla, S'I = -ar t silla + hui cosa, Inlhis way, Eqs. )- 14) are satislicd. Si tlct' 1" = Is, - s:l'.,i' = Is" - s,i'. r' = Is, - s,i'. /") ni si"',, + I!j cos'" ) /I.!. = 13 Ri 'os:! rt + nj sill:! rt, rol 2Rj-Ui)sillrTCOSa 19) 20) lile rebtion tor lile new coordinates and the distances IJ,. ;lild 1" L';1llht: \vritlen as 21 ) RestriclL'd lo Ihe bidillll'j1sional case w by E'Is. 12)- 14). 19) ""d 20) we gel L = ~[ili + ii; + Jli + ;),,' + J,') - 'ir, n,,,j,j- F. 24) Thc corrcsponding canonicall1lol1lents are /''2 = I,R'2' So \\-'c gcl thl' hal1liltonian 'ff = I,f?"f t trj);,- - '21/[1nl~';I. 25) 26) 27) 28) H = -1 [ Pi " + 1', + 2/,. 29) AmI Ihe t'quatiolls ni -= N., -=,. of l1lolioll 1', t' ):! /', JI')l' "1' n n RI + ~ v + - l ll',:!fj] 1'1Ii - un'.10). ).12) 1', JI' Da ' -/1, /)-") + p~ :W,.._-- + lii l' " "1" n').l I I - ll~ a, :,>Ri + R~ +')--=.fj 1) - - /' " "Ui - un.l 35) 36).'3) /', = l, ByEq.21) _->!!.J.. ) J' - ", /' :lrj + F!f Ui - nj)' JI' ijr'2 37) ijl' 0.\ ui sill'" + 1,', <'OS'" ) n.,.., ~.,,. u-.-',u:; :-illl- 1 + Ni l'os- f, ').\ " '2 /{0. - H ) SlIl J "os a.18) Rn'. "'-In. 1-'1.\. -l7 6) 200l) 525-5~1

3 A. ESCALONA ljuenoia ANO E. P :':A I"I{Zt\ 527 wlll'n: X can be any ol' r, ni' or n:!. p~.'., is a constant ol' 1tion. in ract it is the anglll:ir lllolllcntulll of the syslem. 3. SUII-Earth.MoolI case \Ve lieed an explicit form 1'01'a and h in order to calculate th\? lllalrix B. Since f\la is onhogonal lo h it must he a lineal l"\lihinatloll of a and 1 Ma = ; :<l+ Jllll. FIGUJ{E") Sllll-Earth.fvlo0 syslern. Solvjn~ rol"a, orlhogonality wilh 1leads lo which is,1 qlladratic equalion 1'01':r. Vector b obeys the same I:qllation. so, taking the higgcr solutioll, :1:1' for a ami the ",1;i1In,.1'", for h..ti" and.1, can be calculated using Eqs. 16) dllll 17l.1I:spcctively. :nr Ihe SlIn.Earth-,\loon case If/ I» 1H:.!, the matrix U can h\.' appra:\imated by I/I~.: 1It~) +--'- 1IJ:.! -.1' f/l.:f - ~r =. In Ihis fonn, Piña ami Jiméllcz-Lara gd the fojlowing rcslllts. :2}: f2 f2 Il/.:.! ':1 B = f21 :1 f2 III:.! -1,,) 1 I kes its lllaxillllllll allt!minilllum vallles at full t\"iooil,lild ne\\' l\1001l respeclively, oscillating aroul1d ;: constanl valllc +0_1 ). TII I h) J' lakes this valllc at tlrst ami last quarter wherc, /:::/'» JI, thus This allows to Illtlkc an estimation: J:! > :J550R 1. lile tri;lll~1e conformed by ti1\.' thrcc bodies is very nar- 1'0\.. '. and ] ys 0the.r axis in the rotating frame, with the Sun ar Ihe.1: > ti.. ide and the Earth-l\loon couple at the oppositc side see Hg. 2). 4. Redefillill~ a "2..., III:.! + 1fI:~ :.! /' = ----R,. J'I/.:.!III,:~ J::: 1" ,.,..., Jrn 2 l:j :! = ----R.). In'l + t 3 - Considering tllat the vallle 01' 1 has small oscillations aroul1d!ti \Ve redefine.12) This 17 1 \';lriahle describes slllal] oscillations arallnd zera, r!lcn \Ve \.'an rewrite El]. 21) as so \ve gel Ihe disl;lnces in terms 01' 17l' Detlning 1" = ;''101 { " -' = G.:.12SGo. ItI:.!, IS- 1" e ::: S1Il 2 al) 2 SIn a n US 1"., ",1- J'1l., "os- 1" n' = Be. lhe transfol'lllalioll llltltrix depends 0J'q only: -.~"'iil 17" 'C)San ) SII 17" C'IS J'",.. ) "os. 1" - SII ) \Hiting it in terj1ls oi" the lliasses 1, 1." - J1II.,III,,) e -= /1:1 1"2 Jm:!1II::. 46) /. 2 1-//1:: :!~ -'2JII/:!TlI:1 1/1."1-/:1 1mder lo recove,- Eqs. 1<))aJHi 20) in tcrms 01' ]l' \'ccfors a alld h are modifted too a' = a "lis 1" + h sin J'", 47) This causes a'. h' i- J. our Ihis condition is not relevanl as long as Eqs. 16)- IH) rcmain satislicd. Hel'. Me.\. Fú. -l7 «l) 2001)

4 5.28 TItE SlJN.EARTlI-MOON PROBLEM IN NEW COOR[>INATES. BIDI.\1ENSIO,"JAL CASE 13'= Thl"n:fore. for Ihe Sun.Eanll-/'vloon case \Ve llave I;wm l!lis POilll '''''e call this malrix. B and we call r7 lo rr l' lhe only dlange in lile hamiltonian ami the equalions 01' 10- ioll is lhe L'.\plicil fol"ln of lhe Illiltrix n. j\'lultiplying Eq. 21) al lhe right side by the inverse matri.\ 13-'= 1H'2+'1,: )'2 JITI'2'I1:l..//1/'21T1: lll'.! + 'I':l JrII'21:~ JIII'.!'Il.;j JIII"].'II:1 II mi +0_1) 49)., III~ J'''"}.III':f ~ r 1//"].+I:1 1//"2 1 :l )/1/"21.: III'.! + 1:1 fl/ ':1 +0 _1_), 50) lit 1 In f rst ami lasl juaner. where a ~ ti and l' ~ rccovt'r tall '2a = )/"2"':1"., --~-V Ift:\ = /li. 1., + 1. '" ',') -,., 1"- = n-, , ~ - Rj q, we 54) 55) Dividing Eq. 53) by lht' ditterellce 01' Eqs. 5 and 52) '.. e gel 56) \Ve can Illa~e the estim<llion: In full 01" new Moon. JI:! = q'.! +,.".2-2jl". f Ul '1.a == :t.tj,ooo,j.jgg. therefore we have JlI'.! - 3 I/I'.!+I/I:~ -1.,. 1.,'J Hís1I1 a+r1.cos-a, 51) 1"1 < 1I.III1I:l. 5. Rcco\'cry 01' a two hody problcl The differencc in three ordel"s of magnitudc between R I <lne! H:! allows I /1; - j)' 10 Illake :-'Olllt:'approximations: 1 [ ' n')] -1 1+'2--i+) --+. /1,, [1, 57),, l/l.) -/1/..\., rr - 1" p- = I'"2 + 1//:\ '.') '2 U} - Rj siu a {'osa. 52) 53) lile lerll\ R1/Uj is )Itl- lo ). so Ihe hamillonian can be ap R)XimaICd by 1 [" 1'; +. 1';,,p"p" _/1,] \"" «8) = - 1',- + P;+ --,--+ ", +._, '2,1 - /} R"}. 1?1 The appro.\imated equations of motion for n 1 and ti) are 1, - :! - f',; + P,~, +h iljl/llll/..)l/i..~ , ,- l/ni,p1p t " ni, [/'1 + 1J/.:~ sill:! a {2;1:: 1 {2;H"!. 1) "os'.! a l/ni H"!. - _. )'/II"}.ll1-: l ' l r' /.".2ti /:\ r l I"].+ IIt:\ {2; "., + _- _ + _ ) In:~ {j"j \ l/i'.!,..1 III'.! + 1/1:: r,.:~ /1/1 + "':1 1 2'" 1),ir,'" 1 1 ) 2 sil!" nls."]. " = ~ 1'. 1 + '2--!!...-' P ). til" P", -, 60) hom Ihe available astronomical dala H,5] \I,.'e hnd that l1t/ P, l < O,lIt) L thus,\'e can ~ISSllIllC thal ; '" I 'l" I. J = --:;:;-. 61) I,[lj \Vilh Ihe S;UIlt: appro.\illlatioll in Eq. 59), whcn 1 =: ) lid ') ~ j,ve ohlain., :' />:, t ' I1 V'''"}..''':I ~.I.,I/I.'I "1.)_ + /,)., IIHJ - r ni )1 1 ' 3 in this,,,:ay. when H'.!» NI lhl" equations for R'1 and l/j can be I"educcd 10 Ihose rol" lw) hody problclll. Therefore, we can expecl that lhe behavior dcscribed for U l alllll/j consisl'i in slllall oscillations.lrolllhi a kcplerian orbie 62)

5 ,\ ESCALONA BUENOíA ANO 1:'.. PIÑA GARZA m,. 529 FH;UKE 4. Surface 01'seclion 1:. lllotion are invariant under the transformation 6. Eliminating ~ Ld H be lhe Jacobi vector, \',:hich points from the barycenter ni"lhe Eanh-t\loon systclll lo lhe Sun {ti] sec f-ig. 3). From lhe fcolllctry of the system, \"C can writc a relation of t!lis \ t:.''lorand vcclors p. q. and r; in lhe inertial refercnce syslcm P = J':'1 - q = 1", - l':!, r~= R - np, Ih}) 64) 1"= 1", -I" = -H.-lip. 1:;) \\'hcre l = 1I1,:!./In:! + m:~), j = 1fI:J1fI.2 + 1It:: ). As,1 consequcncc,.' = R' + 13'1>" + 21Jp. n. Suhstitution in El). 5)) leads ro uf sin:! a + Ri eos"!. J = 1/1"2 + '/:1 n:!.. JlI/ 1 1/l:1 Thl'refore we can wrile R 1 in lerls 01' U. /l aud a: h6) 67) hs) As lhe cccclllricity of Earth's orbit around Ihe Sun is...mal!. abaut J.Oli. in arder 10 make a tirst approximation lo rhe prohh:m. d., " = III,+II JR' -~--~ - JI"') I SlIl a --,-o 1 - JII1!,'H 3 "os a we add the hypothesis fl = 'llistilll!.. 69) Thc 1Il0tiOll 01"the Earth-Moon systcm around the Sun is rcduced 10 a llniforin circular motion, meanwhile the connguralion oftlle lriangle confonned by the three bodies is described by U [ nd J. \Vith this idea, the smne used by Hill [6.7], lhe ~y~telll is reduced inlo a problem wilh Iwo degrees of freedomo Thcl"e are t\\'o inval'iant planes linder this ransformation: R.f'./'" = O) and U, = O.a.f'"). Weusethelatteras.1 surface of scction, callcd ~~ sce Fig. 4). On this surface the hamiltonian is reduced to I, =;- I' -+ 2' t fmm which, \Ve can \\TiIC p:! + p~ n <- ni - f'} + 2 LH - F). 70) >j -+f'; + I'J,). [{j + \, 71 ) 72) By Ihis procedure, giving Il and P1.' and uslllg Eq. 69), 1'1 is determined for cach point on except for a signo \Ve define a Poincaré Illapping 73) as the integralion of the equatiolls af mol ion starting from initial condition~ on ~. \\.'ith P J > O, unlil the solutions intersect Ihe sallle surface, with PI > O. Transformation lo is an il1volulioll. that is 16 = l. Thus fhc rc\"crsibilify propcrty can be \vrinen [8J as Dcflning file involulions: 1) = 1\1/ lo we get a transfornlj~ tiolls group 0 74) M}-' = 1I 75) 1 " lj+~ = M}I,., 76) 1j _~_= I}M'. 77) A syllllllclry tille is the set 01' points on the surface of scclion invarianl lindel':ln involulion 7. S)'nllnctr~' Hnes r} = {S E LV: = lis}. 78) 'I'hi."i is a reversible system, so wc can apply the technique of th!..' SYIlllllctl'Y lines. The hamiltonian and the equalions 01" The propcrties 01' these syllllllctry lines are given by Ihe transfortl1<.llions group. Firsl of all, Ihey are related with cach R{'\,. AleJ:. Fú. 47 «1) 2001) 525-5~ 1

6 530 THE SUN-EARTH-MOON PROOLEM IN NEw COORDINATES, OIDIMENSIONAL CASE r. FIjllRE 5. Pc:riodil.: orhib. R AU) 000' r, üooj) í 0000, r. \ "." \ ~ \ \' FH UR.I: 7. V"IUI.?<'llf a. "1 " ~~, ID.y) \~.OOl~ -"IXJJJ f;. O,uUo '. olher by 79) 1,[', = ['2)_'" 80) rhen. \\'L' l';lil gl't he whole family 01'symmetry ines. :-.tarting: trolll ro dlhl I'J' T\\'o reslllts lllakc tbis furmalislli ver)' llseful: /) :\ poillt in Ihe intcrscctioll oj 1WO s)'lll1j1ctry lilles S El', n 1', is a periodic poinl s=;\ lr'ls ;l1ld ih pt'riod is Dile of lhe divisors of l.i - 1.:1. XII FI UR.I; X. SYlllllldry linc:-.. p. u..u o h) An l'\'l'1l pl'riod orbit has 1\\'0 poinls 0r" or {\Vn points )1l \'1. \nd ;In odd periad orbit has une point 0 )' and olle poill! 0 l' 1 sec f"ig. 5). This redut't's lhe scarching rol" periadic orbirs into scarching ror intl'rsl..'ctiolls 01' lhe symlllclry lines with ro and r 1" In Ibis I.'ase, in ~. lhe horizontal axis F rr = O) is in- \arian! lkr1\: Iransformation ). so this is Ihe s)'llliliell"y I ne 1'0'.- X. Thl' Jcriodic orhits From tl1e da1<l of october of 1999 [.IJ. whcll Ihe mean Earth- Sun di:-.l;llltt "'<1:-' U.D970 ::i: O.))'2i AU, ami Ihc hypolhe- "i" n == 1 AU. \vc calculate succcssive vallles fol' U I ami r. In Fi~",. ) ;lnd 7 v.c can sec Ihe extreme values of 1 and Ihc zcro.. '..ni' Ji 1 ; full ~ 100n anl! ne\v r..iooil, Ihe 9th and '2.~,lhd; y:-. rc"pcclivl..'ly. The phasc locking. betwcell R1 and 1 is evidenl l Eq.I~7l...' conditions R 1 ::::. l, 1'., ;:::O imply ir ::::.J. so lhe /L'rol':-' n ni match ", lh Ihc t:xtrcmal vallle:-. 01"1 in ;lll) perio\ic orhil OH 1'' I;rolll aslr<hl\lmical dala 1,1,;)] we make an estimalioll or Ihe I..'nc r ;y ami an,8ular 1I10mcntum orthe syslcm thal \VC need F\ ttr.e 9. POllearé lll:lppill,8. foi"lhe Poincaré mapping 1'v1. Figure 8 shows lhe sylllmetl'y lines l' 1 10 r l' it is ca..)' lo :-.ec lhal alt of lhclll inlerscci ro in Iwn POilllS, indicaling lhe presence o.t\\..o period.! points. hgurc l).hows a PoilKaré mapping conllrming lhe exis- SUlllilL"IlCL" 01" 1\\'0 st; hk per;odic nrbits. The pcriod of the llrhit inllll' T > l..ilil' i" ".li',if;.-, days. very clo...e ~othe sidereal Illol1ll1. :li'.:~i:ll day... ThL" pcriod uf lhe moit in lhe 1 < \l side is 17 A 78 days. I~lis would be lhe OI"bil of a 1\100 w;th n:trogradc 1Il0vt'1Ilcnl. 1</'1'. \In. F/\..t7 6) COOI)

7 A. ESCALONA BUENDIA A!'\D E. PISA larza E. Piii,l. Ce/e,\"{. i\kch. 7'" t 1991.J) 1)J. 2. E. Pil-laamI L. Jiméncz Lara. C'it'st. M/'ch. o he puhlishcd). :~. E. Pilla, /Jil/ÍlIliclI lit> Ro/acirmn UniVl:fSidad Autónoma \klrnpolit,lllil-i ztapalapa. Mé:":.ico. 196). 1. \I/I/I/rio lji.)j tlellnstitl/lo tll' ;\'\'ti"{jl/0i 1 dt'!ti UNA"" ljniwrsldad N,u:íollal Autónoma de i\1é:":.íl'o,,1\1éxio, )1)<)). S. \mll/rio 2))) dd Inslit/0 dl' A\/rorlOlfll'a dl' la UNAA1 Universidad Nacional Autónoma de :\1éxico_ Méx.ico, 200). j. D. Brouwcr,.1 J. Hori, 1'lIy.ü'.\' 1l1/l1,h/lDl101IlY oj lhe Moo/l cditl'd hy ZdL'IlCCKopill. I\CiKlclllic Prcss. Ncw York, 192), 7. M. C1utz\\'ilkr. Rn Atod. I'''n, R) SS9. s E. Pub anj L. JillléllCl Lara. I'hniw /J 2) ll)~o) 31), R'\'. I\kr. :t~."'7 6) 2)OI) )25-:')31

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