Nonlinear dynamics in an extended neighbourhood of the translunar equilibrium point

NONLINEAR DYNAMICS IN AN EXTENDED NEIGHBOURHOOD OF THE TRANSLUNAR EQUILIBRIUM POINT  ANGEL JORBA AND JOSEP MASDEMONT Departament de Matematica Aplicada I (ETSEIB) Universitat Politecnica de Catalunya Diagonal 647, 08028 Barcelona (Spain) 1. Intro duction We are intere sted in themotionofasmall particle in some regions of the Earth-Mo on system. As a rst mo del, wewill usethe spatial Re str icte d Three Bo dy Problem (RTBP, s ee 5] for denition andmain prop ertie s). It i s well known that, in syno dic co ordinates, thi s mo del has ve equilibr ium p oints. We will fo cus on theonethatis behindthe Mo on, usually calle d the L 2 point. Our purp os e i s tode scr ib e the dynamics in an extended ne ighb ourhood of that p oint. Thi s information i s very us eful tokeep an spacecraft there, b ecaus e we can take advantageof thenatural dynamics of the problem. Themain to ols us e d are an eective computation of the central manifold of the p oint (to give a qualitativede scr iption of the dynamics) and a Lindstedt-Poincaremethod (to computeinvar ianttor i ins idethe central manifold). Thi s metho dology has already b een us e d in s imilar problems (s ee 1]). 2. Eectivereduction tothe central manifold Thi s s ection i s devoted tothe computation of the central manifold asso ciated to L 2 .The pro ce dure i s quite s imilar to a normal form computation. 2.1. THE HAMILTONIAN The reference system i s the following: the or igin i s taken at L 2 ,the x axi s i s directed opposite tothe Mo on, the z axi s i s given bythevector of angular motion of EarthandMoonand the y axis is selecte d in order to obtain a orthogonalpositive-or iente d system of reference. Then, the Hamiltonian takes the form H = 1 2 ( p 2 x + p 2 y + p 2 z )+ yp x ; xp y ; 1 ;  r E ;  r M  where r E and r M are thedistance s f rom ( x y  z )to EarthandMoonrespectively.To computethepower expans ion of H aroundthe or igin, it i s worth notingthat 1 ;  r E +  r M = X n  2 c n  n P n  x    2 where  2 = x 2 + y 2 + z 2 , P n is the Legendre p olynomial of degree n and c n = ( ; 1) n  3 "  +(1 ;  )   1+   n +1 #  (1) being  thedistance b etween L 2 and Mo on divided bytheEarth-Mo on di stance. A go o d way of implementingthi s expans ion on a computer i s totake advantageof the recurrence of the Legendre p olynomials (s ee, for instance, 2]). 2.2. LINEAR BEHAVIOUR AND EXPANSIONS The linear ization of the Hamiltonian around L 2 shows thatthelocal behaviour near thi s p ointisofthetyp e saddle  centre  centre. So, us ing a real, linear and symplectic change of var iable s the Hamiltonian takes the form H = x 1 y 1 + ! 1 2 ( x 2 2 + y 2 2 )+ ! 2 2 ( x 2 3 + y 2 3 )+   where  , ! 1 and ! 2 are real numb ers. For the following computations it i s very convenient to \diagonalize" the s econdorder terms. Thi s i s donebyusingthe complex change x i = ( q i + p ; 1 p i ) = p 2, y i =( p ; 1 q i + p i ) = p 2, i =2  3. So, the Hamiltonian lo oks like H = q 1 p 1 + p ; 1 ! 1 q 2 p 2 + p ; 1 ! 2 q 3 p 3 + X m  3 X j k j = m h k q k 1 1 p k 2 1 q k 3 2 p k 4 2 q k 5 3 p k 6 3 : (2) In f act, a go o d way of p erformingthese changesofvar iable s on the Hamiltonian i s towrite them as a unique linear transformation andtoapply the recurrence mentioned abovetothi s change. Thi s pro duce s the expans ion of the Hamiltonian in thenal var iable s. 2.3. COMPUTATION OF THE CENTRAL MANIFOLD Thenext step i s to p erform canonical transformations to (2) to obtain H = q 1 p 1 + p ; 1 ! 1 q 2 p 2 + p ; 1 ! 2 q 3 p 3 + X m  3 X j k j = m k 1 = k 2 h k q k 1 1 p k 2 1 q k 3 2 p k 4 2 q k 5 3 p k 6 3 : (3) Thechange s are doneusingPoi sson brackets (s ee, for s imilar example s, 3], 4] or 2]). The purp os e i s to kill all the monomials with k 1 6 = k 2 . Of cours e, thi s i s doneup to a nite order N .The divi sors obtained in the generatingfunction of thechange of var iable s are  ( k 2 ; k 1 )+ p ; 1 ! 1 ( k 4 ; k 3 )+ p ; 1 ! 2 ( k 6 ; k 5 ) : As weonly wantto eliminate monomials with k 1 6 = k 2 ,the corre sp onding divi sors can b e bounde d f rom b elow. Thi s shows theabs ence of small denominators in the pro ce ss. So, although thi s pro ce dure i s not convergent, it pro duce s verygoodapproximations tothe dynamics. Thelaststep i s to call I = q 1 p 1 ,andthenal Hamiltonian lo oks like H = H N ( I q 2 p 2 q 3 p 3 )+ R N +1 ( q p ) : Now, if we skip R N +1 (itisvery small near L 2 ), we s ee that I i s a rst integral, andsetting I =0 we skip thehyp erb olic part andwereduce the Hamiltonian toitscentral manifold. Finally,the re sulting Hamiltonian i s realie d. 3 3. Dynamics ins idethe central manifold Thephas e space in the central manifold i s four dimens ional. Todescribe thedynamics, we x thePoincare s ection q 3 =0(thi s will corre sp ond, at rst order, to z = 0 in the initial co ordinates) andwe us e, as a parameter,theenergy level h of the Hamiltonian. Thi s implie s that it i s enough toplotthe s e 2-D s ections for s everal levels of h toobtain a qualitative de scr iption of thephas e space. 4. Computation of invar ianttor i near L 2 Here is better toworkwiththe equations of motion instead of the Hamiltonian. They can b e wr itten in thefrom  x ; 2_ y ; (1 + 2 c 2 ) x = @ @x X n  3 c n  n P n  x     y +2_ x +( c 2 ; 1) y = @ @y X n  3 c n  n P n  x     z + c 2 z = @ @z X n  3 c n  n P n  x    where the reference system i s theoneintro duce d in s ection 2.1 and c n has b een dene d in (1). Notethat, in the s e equations, the left-handside contains thelinear part andthe r ight-hand sidethe nonlinear part. For the computation of invar ianttor i, we will us e a LindstedtPoincaremetho d (s ee 1]). Thi s i s a recurrent pro ce dure that pro duce s approximations to thesolutions startingfromthe solutions of the linear part. 4.1. THE 2-D INVARIANT TORI We lo ok for formal expans ions, in p owers of the amplitudes  and  ,ofthetyp e x = 1 X ij =1 0 @ X k j i j m j j j a ij k m cos( k 1 + m 2 ) 1 A  i  j  y = 1 X ij =1 0 @ X k j i j m j j j b ij k m s in( k 1 + m 2 ) 1 A  i  j  z = 1 X ij =1 0 @ X k j i j m j j j c ij k m cos( k 1 + m 2 ) 1 A  i  j  where  1 = !t + 1 and  2 = t + 2 .The f requencie s ! and  are expanded as ! = P 1 ij =0 ! ij  i  j and  = P 1 ij =0  ij  i  j . All the co ecients are compute d recurrently up to a nite order i + j . 4.2. HALO ORBITS Nowwe lo ok for (p er io dic) Halo orbits. In thi s cas e weonlyhaveone f requency andone amplitude. Neverthele ss, wewill keep thetwo amplitudes  and  adding a constrain 4 between them. So, the expans ions havethe form x = 1 X ij =1 0 @ X k  i + j a ij k cos( k! t ) 1 A  i  j  y = 1 X ij =1 0 @ X k  i + j b ij k s in( k! t ) 1 A  i  j  z = 1 X ij =1 0 @ X k  i + j c ij k cos( k! t ) 1 A  i  j  where ! = P 1 ij =0 ! ij  i  j andthe relation b etween b oth amplitudes can be written in the form  = P 1 ij =0  ij  i  j =0. Wewanttostressthat Halo orbits are not pre s entin thelinear ize d system andthey are only foundwhen nonlinear terms are cons idere d, s ince they are pro duce d bya1to1 re sonance b etween thetwo f requencie s of the central part. 5. Re sults All the algor ithms have b een implemented bytheauthors in C andFORTRAN, taking advantageof the symmetries and particular itie s of the problem. The us e of commercial algebraic manipulators do e s not allowto reach high orders in the expans ions to get accurate re sults(the central manifold has b een compute d up toorder 32, the Lindstedt-Poincarehas b een applie d up toorder 35 for theinvar ianttor i and45forthe Halo orbits). Wehavesummar ize d the re sults in gure 1. The rst plot i s oneofthePoincare s ections ( h =1)ofthe central manifold. The elliptic p ointin the centre corre sp onds toavertical Lyapunov orbit, thelateral elliptic p oints corre sp ondto Halo orbitsandtheboundary of the s ection i s a planar Lyapunov orbit. The s econd plot corre sp onds tooneoftheinvar ianttor i in the central manifold. It has b een plotte d as s een f rom the Earth, jointly withthe Mo on di sk. Notethe\cut" b etween the initial andthenal part of the tra jectory, just over the Moon disk. This pointhas the same p os ition in b oth cas e s but with dierentvelo citie s. Thi s implie s that, if wewantto avoid thelunar di sk, we can re start the orbit with only a mano euvre. Thetimebetween twoof these mano euvre s i s abouttwoweeks. 6. Applications to spacecraft dynamics Themostintere sting tra jector ie s are theones that, as s een f rom the Earth, avoid theMoon di sk. They can b e us eful, for instance, tokeep constantcommunication with a mi ss ion to thedarksideofthe Mo on. Halo orbits are suitable for that purp os e. Themain drawback is thatthe orbit i s very widein thehor izontal direction, so the spacecraft must carry an or ientable antenna, always p ointingtothe Earth. Thi s makes the mi ss ion more exp ens ive. Another p oss ibilityisto us e an invar ianttor i in the central manifold of L 2 .Notethat the orbit can b e s electe d more \square d", so theantennadoes not nee d to b e or ientable. On theother hand, and f rom timetotime, the spacecraft will b e eclips e d bythe Mo on. Thi s can b e avoided by doingamano euvre every twoweeks. Full details of the s e computations, as wellasthestability prop ertie s of the orbitsand the corre sp ondingcontrol strategie s will app ear in a future pap er. 5 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 Figure1. Left: Poincare s ection of the central manifold. Right: Quas ip er io dic tra jectory in the central manifold, as s een f rom the Earth(inthe centre wehave drawn the Mo on di sk). 7. Extens ions Thenext step i s totakeinto accountthemain p erturbations of the problem, namely the noncircular motion of theMoonandthe eect of theSun. Thi s will allowtogive precise details aboutthe dynamics of the real problem. Thi s workis still in progre ss. Acknowle dgements A. Jorba has b een partially supported bythe grants DGICYT PB94{0215, CIRIT GRQ93{ 1135, the Europ ean Network ERBCHRXCT 940460 andthe UPC fund PR9409. J. Masdemonthas b een supp orted bythe grant DGICYT PB94{0215 andthe UPC fund PR9409. Reference s 1. Gomez, G., Jorba, A., Masdemont, J. and Simo, C. (1991) Study renement of s emianalytical halo orbit theory, ESOC Contract 8625/89/D/MD(SC), Final Rep ort. 2. Jorba, A. and Simo, C. (1994) Eectivestability for p er io dically p erturb e d Hamiltoni an systems, in Hamiltoni an Mechanics, IntegrabilityandChaotic Behavior, J. Se imeni s (Ed.), Plenum Pre ss, New York, pp. 245{252. 3. Giorgilli , A., Delshams, A., Fontich, E., Galgani, L. and Simo, C. (1989) Eectivestability for a Hamiltoni an system near an elliptic equilibr i um p oint, withanapplication tothe Re str icte d Three Bo dy Problem, J. of Di. Equations , 77:1 ,pp. 167{198. 4. Simo, C. (1989) Estabilitatde sisteme s Hamiltonian s, Mem. de la Real Acad. de Ciencias y Artes de Barcelona , Vol. XLVI I I, no. 7 ,pp. 303{348. 5. Szebehely, V. (1967) Theory of orbits, Academic Pre ss.