High-order solutions of invariant manifolds associated with libration point orbits in ERTBP

High-ordersolutionsofinvariantmanifoldsassociatedwithlibrationpointorbitsintheellipticrestrictedthree-bodysystem

HanlunLei·BoXu·XiyunHou·YisuiSun

Received:28October2012/Revised:9July2013/Accepted:20August2013/Publishedonline:16October2013

©SpringerScience+BusinessMediaDordrecht2013

AbstractHigh-orderanalyticalsolutionsofinvariantmanifolds,associatedwithLissajousandhaloorbitsintheellipticrestrictedthree-bodyproblem(ERTBP),areconstructedinthispaper.TheequationsofmotionofERTBPinthepulsatingsynodiccoordinatesystemhavefiveequilibriumpoints,andthethreecollinearlibrationpointsaswellastheassoci-atedcentermanifoldsareunstable.Inourcalculation,thegeneralsolutionsoftheinvari-antmanifoldsassociatedwithLissajousandhaloorbitsaroundcollinearlibrationpointsareexpressedaspowerseriesoffiveparameters:theorbitaleccentricity,twoamplitudescorrespondingtothehyperbolicmanifolds,andtwoamplitudescorrespondingtothecen-termanifolds.TheanalyticalsolutionsuptoarbitraryorderareconstructedbymeansofLindstedt–Poincarémethod,andthenthecenterandinvariantmanifolds,transitandnon-transittrajectoriesinERTBPareallparameterized.Sincethecircularrestrictedthree-bodyproblem(CRTBP)isaparticularcaseofERTBPwhentheeccentricityiszero,thegen-eralsolutionsconstructedinthispapercanbereducedtodescribethedynamicsaroundthecollinearlibrationpointsinCRTBPnaturally.Inordertocheckthevalidityoftheseriesexpansionsconstructed,thepracticalconvergenceoftheseriesexpansionsuptodifferentordersisstudied.

ElectronicsupplementarymaterialTheonlineversionofthisarticle(doi:10.1007/s10569-013-9515-6)containssupplementarymaterial,whichisavailabletoauthorizedusers.

H.Lei·B.Xu(B)·X.Hou·Y.Sun

SchoolofAstronomyandSpaceScience,NanjingUniversity,Nanjing210093,Chinae-mail:xubo@nju.edu.cnH.Lei

e-mail:hanlunlei@sina.comX.Hou

e-mail:silence@nju.edu.cnY.Sun

e-mail:sunys@nju.edu.cn

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350H.Leietal.

KeywordsEllipticrestrictedthree-bodyproblem·Invariantmanifold·Lindstedt–Poincarémethod·Lissajousorbit·Haloorbit·Collinearlibrationpoint

1Introduction

Librationpointscorrespondtothedynamicalequilibriumsolutionsoftherestrictedthree-bodyproblem,androtatearoundthebarycenterofsystemwiththesameangularveloc-ityasthatoftheprimaries.Thelibrationpointorbits,andlow-energytransfersbasedondynamicalsystemtheoryhaveplayedanimportantroleindeepspaceexplorationandattractedmuchattentionrecently,duetotheirpotentialtogeneratenewkindsofmissionoptions.

Theboundedorbits(forinstance,Lissajousandhaloorbits)aroundtheL1pointoftheSun–EarthsystemcanprovidegoodobservationsitesoftheSun,duetotheirpriv-ilegedconfigurationswithrespecttotheprimaries,andareusuallytakenasworkingorbitsofthiskindofmissions,suchasISEE–3(1978),SOHO(1995),Genesis(2001)etc.(Canaliasetal.2004).TheboundedorbitsaroundtheL2pointofSun–Earthsystemareverysuitabletoplacetheastronomytelescopes,consideringthestablethermalenvi-ronment,withoutspacedebris,gravitygradient,andmagneticfieldfromtheEarth,etc.Anumberofsuccessfulmissionswithinthelastdecadehavetakenadvantageofthespe-ciallocationandenvironmentaroundtheL2pointofSun–Earthsystem,suchasMAP(2001),PLANCK(2007),GAIA(2012)etc.(Canaliasetal.2004).Inaddition,theinfor-mationoftheMoon’soppositesideisalwaysunknowntous,andtheboundedorbitsaroundtheL2pointofEarth–Moonsystemcanprovidenominalorbitstoplacescien-tificspacecrafts(Farquhar1971).Therecentmission,ARTEMIS,isthefirstmissionthattookadvantageoftheEarth–Moonlibrationpointorbits(Foltaetal.2012;Sweetseretal.2011).

Theunstabledynamicalpropertiesofthecollinearlibrationpointorbitscanbeuti-lizedtorealizelow-energytransfersbetweendifferentthree-bodysystems(HowellandKakoi2006).TheJapaneselunarsatellite,Hiten,adoptedthelow-energytransfertra-jectorybasedonweakstabilityboundary(WSB)theory(Belbruno2004),andexploitedtheweakstabilitypropertyofSun–Earthlibrationpointregiontoreducetheexcessvelocity,sothatlessimpulsemaneuverwasneededtoinsertthespacecraftintoasta-ble,normalcircumlunarorbit.TheballisticcapturemechanismofWSBtrajectoryhasbeenunderstoodwiththeaidofdynamicalsystemtheory.Koonetal.(2001)approxi-matedtherestrictedfour-bodyproblem(includingtheSun,Earth,Moonandspacecraft)astwocircularrestrictedthree-bodyproblems,andasystematicmethodologybasedontheintersectionofinvariantmanifolds,associatedwithlibrationpointorbitsofSun–EarthandEarth–Moonsystems,hasbeendeveloped.Recently,theWSBtrajectoryhasbeenadoptedbytheGravityRecoveryandInteriorLaboratory(GRAIL)mission,whichisapartofNASA’sdiscoveryprogram,totransfertwoorbiterstothelowlunarorbit(Chung2010).TheotherkindofEarth–MoonlowenergytrajectorycorrespondingtotheinteriorcapturearoundtheMoonissystematicallyinvestigatedbyLeietal.(2013),whocon-structedthelowenergytrajectoriesinthecircularrestrictedthree-bodyproblem(CRTBP)firstly,andthensearchedthesimilarlowenergytrajectoriesintherealsystembyusingevolutionaryalgorithm,withtheinitialinformationprovidedbytheresultsobtainedinCRTBP.

Theinvariantmanifoldsassociatedwithlibrationpointorbitscanprovidetheglobalunder-standingaboutthedynamicsaroundcollinearlibrationpointsfromtheviewpointofphase

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High-ordersolutionsofinvariantmanifoldsinERTBP351

space.Thelow-thrust,low-energytransfers,incorporatingthenaturaldynamicsofthree-bodysystem,alsohavebeeninvestigatedwidely.Whentheinvariantmanifoldsoflibrationpointorbitsoftwothree-bodysystemscannotintersectinspace,theapplicationoflowthrustpropulsiontointerconnectingballistictrajectoriesoninvariantmanifoldshasbeeninvesti-gatedinPergolaetal.(2009).Dellnitzetal.(2006)constructedtheEarth–Venuslow-thrust,low-energytransfersincorporatingtheinvariantmanifoldsofthree-bodysystem.Theattain-ablesets,playingthesameroleasinvariantmanifoldsintrajectorydesign,havebeenappliedtothedesignoflow-energy,low-thrusttransferstotheMoon(Mingottietal.2009),andthedesignofinterplanetarylow-thrusttransfers(Mingottietal.2011).

Usually,thedynamicsaroundthecollinearlibrationpointsarestudiedbynumericalmeth-ods,suchasnumericalintegration,differentialcorrection,optimizationmethods,andsoon.However,thenumericalmethodisatrialanderrorprocessanddependsontheexpe-rienceofthedesigner.Analyticalsolutionscouldprovidedeepinsightsofthedynamicsaroundthelibrationpoints,andbecomemoreandmoreimportant.Therearetwokindsofmethodologiestoanalyticallydescribethedynamicsaroundequilibriumpoints,thatis,theLindstedt–Poincarémethod(L–P)andnormalformscheme.Richardson(1980)usedtheL–Pmethodtoanalyticallyconstructathird-ordersolutionforhalo-typeperiodicmotionaroundthecollinearlibrationpointsofCRTBP.InJorbaandMasdemont(1999),theL–Pmethodaswellasnormalformschemeareadoptedtosemi-analyticallyconstructthehigh-ordersolutionsaboutthedynamicsinthecentermanifoldsofthecollinearlibrationpointsinCRTBP.LeiandXu(2013a)analyticallyconstructedthehigh-orderanalyticalsolu-tionsaroundtriangularlibrationpointsinCRTBP,anddiscussedthepracticalconvergenceindetail.ConsideringthehyperbolicbehaviorstogetherwiththecenterbehaviorsaroundthecollinearlibrationpointsinCRTBP,Masdemont(2005)expandedtheinvariantmani-foldsaspowerseriesofhyperbolicandcenteramplitudes.TheseseriesexpansionscouldexplicitlydescribethegeneraldynamicsaroundcollinearlibrationpointsofCRTBP,andhavebeenusedtostudythetwo-maneuvertransferproblembetweenLEOsandLissajousorbitsintheEarth–Moonsystem(Alessietal.2010),tolookforrescuetrajectoriesthatleavethesurfaceoftheMoonbymeansofinvariantmanifoldsoflibrationpointorbits(Alessietal.2009),andtoconstructthelow-thrusttransferstothelibrationpointorbitsofSun–MarssystemfromthelowEarthorbit(LeiandXu2013b).FortheHillproblem(referringtotheequationofrelativemotion),whichcorrespondstothereducedcaseofCRTBPwhenμ=0,GómezandMarcote(2006)computedtheboundedorbitsbyusingtheL–Pmethod.TakingintoaccounttheperturbationoftheSolargravityandlunareccentricity,FarquharandKamel(1973)analyticallydevelopedthethirdordersolutionsofquasi-periodicorbitsaroundthecollinearlibrationpointL2ofEarth-Moonsystemanddiscussedtherelationshipbetweenthefrequencyandamplitudeforlargehaloorbits.ForaspacecraftmovingalongalibrationpointorbitofEarth–Moonsystem,Farquhar(1968,1971)presenteddetailedstud-iesabouttheflightmechanics,practicalapplicationsandstation-keepingproblems.IntherealEarth–Moonsystem,definedbytheJPLephemeris,thequasi-periodicmotionsaroundthetriangularandcollinearlibrationpointsareanalyticallystudiedbyHouandLiu(2010,2011b).

ComparedtoCRTBP,theellipticrestrictedthree-bodyproblem(ERTBP)couldapprox-imatetheSolarsystembetter.Duetotheexistenceofeccentricityoftheprimaries,theequationsofmotioninERTBParenon-autonomous.Fortunately,theequationsofmotionofERTBPinthepulsatingsynodicreferenceframehavethesamesymmetriesastheonesofCRTBP,meanwhile,thedynamicalpropertiesaresimilartothoseofCRTBP.Forexample,librationpointsandcorrespondingboundedorbits(Lissajousandhaloorbits)alsoexist,andareunstableinnature.However,theinvestigationaboutthedynamicsaroundthecollinear

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352H.Leietal.

librationpointsinERTBPismorecomplicatedbyanalyticalmethod.HouandLiu(2011a)constructedthehigh-orderanalyticalsolutionsofthecentermanifolds,suchasLissajousandhaloorbitsinERTBP,bymeansofsemi-analyticalmethod,anddiscussedtheapplicationsintheEarth–MoonandSun–Earth+Moonsystem.Fortheellipticequationsofrelativemotion(orellipticHillequations),correspondingtothereducedcaseofERTBPwhenμ=0,Renetal.(2012)obtainedathird-orderexpressionoftheboundedorbitsaroundthecollinearlibrationpoints,andconceptuallypresentedtheprocessofconstructinghigh-orderanalyticalsolutionsbyusingtheL–Pmethod.

ConsideringtheunstabledynamicsofthecollinearlibrationpointsandassociatedcentermanifoldsinERTBP,thegeneralsolutionsoftheequationsofmotionaroundthecollinearlibrationpointsinERTBPconsistofthehyperboliccomponent(saddlebehavior)andcen-tercomponent(centerbehavior).Therefore,inthiswork,thesolutionsofinvariantman-ifoldsassociatedwithlibrationpointorbitsinERTBPareexpandedasformalseriesoftheorbitaleccentricityandfouramplitudes,thereinto,twoamplitudescorrespondtothehyperbolicmanifolds,andtheremainingtwoamplitudescorrespondtothecentermani-folds.TheseriesexpansionsconstructedinthispapercandescribethegeneraldynamicsaroundthecollinearlibrationpointsofERTBP,andcanbeconsideredasanextensionoftheonesdiscussedinJorbaandMasdemont(1999),Masdemont(2005),andHouandLiu(2011a).Inordertocheckthevalidityoftheanalyticalsolutions,numericalsimula-tionswiththesameinitialstatesarealsoimplementedtoinvestigatethepracticalconver-gence.

Theremainderofthispaperisstructuredasfollows.InSect.2,thebasicdynamicalmodelaboutERTBPisbrieflydescribed.Sects.3and4presenttheconstructionprocessofhigh-ordersolutionsofinvariantmanifolds,associatedwithLissajousandhaloorbits,aroundthecollinearlibrationpointsinERTBP,respectively,andtheresultsarepresentedinSect.5.Atlast,theconclusionstogetherwithdiscussionaredrawninSect.6.

2Dynamicalmodel

Aninfinitesimalparticle,suchasaspacecraft,isplacedinthegravitationalfieldgeneratedbytwomassivebodies,suchastheSunandEarth,movingaroundtheircommoncenterofmassinKeplerorbits.Inthisdynamicalsystem,themassofthespacecraftismuchlessthanthatofanyprimary,thustheattractionofthespacecraftontheprimariesisneglected.Suchasystemiscalledrestrictedthree-bodyproblem(RTBP).Inparticular,iftheprimariesmovearoundeachotherincircularorbits,suchasystemisthewell-knownCRTBP,inwhichamotionintegralandfiveequilibriumpointsexist.Generally,theorbitaleccentric-itiesoftheprimariesarenotzeroandsuchasystemistheERTBP,inwhichthemotionofthespacecraftinthebarycentricsynodiccoordinatesystemisgovernedby(Szebehely1967)

⎧1󰀈󰀈󰀈⎪X−2Y=⎪⎪⎪1+ecos⎪⎪⎨1Y󰀈󰀈+2X󰀈=⎪1+ecos⎪⎪⎪⎪1⎪⎩Z󰀈󰀈+Z=

1+ecosf

with

∂Ω

,f∂X∂Ω

,f∂Y∂Ω

,∂Z(1)

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High-ordersolutionsofinvariantmanifoldsinERTBP353

Ω=

󰀌1−μ1󰀁2μ

X+Y2+Z2+μ(1−μ)++,2R1R2

(2)

whereμ=m2/(m1+m2),m1andm2arethemassesofthetwoprimaries.R1andR2

arethedistancesofthespacecraftfromthemassiveandsecondaryprimaries,respectively.ToformulatetheequationsofmotioninERTBP,dimensionlessunitsareadopted,thatis,thetotalmassoftheprimaries,theinstantaneousdistancebetweenthetwoprimaries,andtheirangularvelocityoftheprimariesarealltakenasunity,suchthatthevaluesofthegravitationalconstantandtheperiodofthesecondaryare1and2π,respectively.Equation(1)isformulatedinthebarycentricsynodicsystem,inwhichtheX-axisisdirectedfromthemassiveprimarytowardthesecondaryprimary,theZ-axisisalignedwiththemomen-tumofthesecondaryandtheY-axisisdeterminedbytheright-handcoordinatesystem.InEq.(1),thetime-likeindependentvariableisf,whichisthetrueanomalyofthesec-ondaryontheellipticorbit.Thefirstandsecondderivativesofcoordinatearedefinedasfollows:

dXX=,

df

󰀈

X=

󰀈󰀈

d2Xdf2.(3)

TheYandZcomponentshavesimilarexpressions.Sincethecomponent1+e1cosfexistsinEq.(1),theequationsofmotioninERTBParenon-autonomous.However,theequationsofmotioninERTBPhavethesamesymmetricpropertiesasthoseofCRTBP.Tobeconsistent,letS1,S2andS3denotethethreekindsofsymmetries:

S1:(f,X,Y,Z,X󰀈,Y󰀈,Z󰀈)↔(−f,X,−Y,Z,−X󰀈,Y󰀈,−Z󰀈),S2:(f,X,Y,Z,X󰀈,Y󰀈,Z󰀈)↔(−f,X,−Y,−Z,−X󰀈,Y󰀈,Z󰀈),S3:(f,X,Y,Z,X󰀈,Y󰀈,Z󰀈)↔(f,X,Y,−Z,X󰀈,Y󰀈,−Z󰀈).

(4)(5)(6)

Alltheabovethreesymmetriesareimportantfortheconstructionofhigh-ordersolutionsofinvariantmanifoldsinERTBP.

InordertoinvestigatethemotionaroundthecollinearlibrationpointL1orL2conveniently,itisnecessarytomovetheoriginofthecoordinatesystemfromthebarycenterofsystemtotheinterestedlibrationpoint,andtaketheinstantaneousdistancebetweenthelibrationpointanditsclosestprimaryaslengthunit.DenotethenewreferenceframeastheL1orL2-centeredsynodicreferenceframe,andtheaxesofthisnewcoordinatesystemarealignedwiththecorrespondingonesofthebarycentricsynodicreferenceframe.Let(x,y,z,x󰀈,y󰀈,z󰀈)representthestatevariablesinthisnewreferenceframe,andγdenotestheinstantaneousdistancebetweenthelibrationpointanditsclosestprimary.ThetransformationofcoordinatesbetweentheoriginalbarycentricsynodicframeandtheL1orL2-centeredsynodicsystemis,

X=γ(x∓1)+1−μ,Y=γy,

Z=γz,

(7)

wheretheuppersignreferstotheL1caseandtheloweronereferstotheL2case.TheequationsofmotionintheL1orL2-centeredsynodicsystemcanbeformulatedasfollows(HouandLiu2011a):

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354H.Leietal.

󰀊⎧󰀘󰀚

󰀈󰀈󰀈ii⎪x−2y−(1+2c2)x=(−e)cosf(1+2c2)x⎪⎪⎪⎪i≥1⎪⎪⎪󰀎󰀄󰀘󰀅󰀇⎪󰀘⎪⎪⎪(−e)icosifcn+1(n+1)Tn(x,y,z),+⎪⎪⎪⎪n≥2i≥0⎪⎪⎪󰀚󰀊󰀘⎪⎪󰀈󰀈󰀈ii⎪(−e)cosf(1−c2)yy+2x−(1−c2)y=⎪⎪⎪⎨i≥1

󰀄󰀘󰀅󰀇󰀘󰀎⎪⎪⎪+cn+1Rn−1(x,y,z),(−e)icosify⎪⎪⎪⎪n≥2i≥0⎪⎪󰀚󰀊⎪󰀘⎪󰀈󰀈ii⎪⎪(−e)cosf(1−c2)zz+c2z=⎪⎪⎪⎪i≥1⎪⎪󰀄󰀘󰀅󰀇⎪⎪󰀘󰀎⎪⎪⎪(−e)icosifzcn+1Rn−1(x,y,z),+⎪⎩

i≥0

n≥2

(8)

inwhichthecoefficientcn(μ)isconstantandonlydependentonthemassparameterofthe

three-bodysystem,andisgivenby

cn(μ)=

1γi3󰀏

(±1)nμ+(−1)

n+1

n(1−μ)γi

󰀂,

(9)

(1∓γi)n+1whereγi(i=1,2)istheinstantaneousdistancebetweenLianditsclosestprimary.InEq.(9),theuppersignreferstotheL1point,andthelowersignreferstotheL2point.ThecasecorrespondingtotheL3pointisnotdiscussedheresinceitisnotthefocusofthispaper.InEq.(8),TnandRnarethehomogeneouspolynomialsofdegreenandcanbecomputedbythefollowingrecurrencerelations:

Tn=

2n−1n−12

xTn−1−(x+y2+z2)Tn−2,nn

(10)

whichstartswithT0=1andT1=x,and

Rn=

2n+32n+2n+12xRn−1−Tn−(x+y2+z2)Rn−2,n+2n+2n+2

(11)

whichstartswithR0=−1andR1=−3x.

3High-ordersolutionsofinvariantmanifoldsassociatedwithLissajousorbits

inERTBP

3.1Seriesexpansionsoftheinvariantmanifolds

TheL–Pmethodisadoptedtoconstructthehigh-orderanalyticalsolutionsabouttheinvariantmanifoldsassociatedwithlibrationpointorbitsinERTBP.TheL–Pprocedureisaprocessofrecursion,andtheunknowncoefficientsofhigh-ordersolutionsarecalculatedfromthelower-ordersolutions.LinearizetheequationsofmotioninERTBPasfollows:

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High-ordersolutionsofinvariantmanifoldsinERTBP355

󰀊⎧󰀘󰀚

󰀈󰀈󰀈ii⎪x−2y−(1+2c2)x=(−e)cosf(1+2c2)x,⎪⎪⎪⎪i≥1⎪⎪󰀊⎪󰀘󰀚⎨󰀈󰀈󰀈ii

(−e)cosf(1−c2)y,y+2x−(1−c2)y=

⎪i≥1⎪⎪󰀚󰀊⎪󰀘⎪⎪󰀈󰀈ii⎪(−e)cosf(1−c2)z,⎪⎩z+c2z=

i≥1

(12)

whichisanon-autonomoussystem,andthegeneralsolutionscannotbeobtainedeasily.

However,theL–Pmethodonlyneedsastartingpoint.Wetakethesolutionwithfirst-orderamplitudeandzero-ordereccentricityasthestartingpoint,andthesolutionscorrespondingtohigh-orderamplitudeandeccentricityarecalculatedbymeansoftheL–Pmethod.Thestartingsolutionweadoptisthegeneralsolutionofthefollowingequations:

⎧󰀈󰀈

󰀈

⎪⎨x−2y−(1+2c2)x=0,y󰀈󰀈+2x󰀈+(c2−1)y=0,⎪⎩󰀈󰀈

z+c2z=0,

(13)

inwhich,themotioninthezdirectionisuncoupledfromthatinthex–yplane.Infact,Eq.(13)isthelinearizedformoftheequationsofmotioninCRTBP,whichisaparticularcaseofERTBPwhentheeccentricityiszero.TheconjugatecharacteristicrootsofEq.(13)are±λ0,±iω0and±iν0,

󰀝󰀙󰀜

󰀜c−2+9c2−8c󰀋222

2

󰀝󰀙󰀜

󰀜2−c+9c2−8c󰀋222

2

√

c2.

λ0=,ω0=,ν0=(14)

ThusthegeneralsolutionofEq.(13)consistsinaharmonicmotioninthein-planecomponent,anun-coupledoscillationintheout-of-planecomponent(linearcentermanifold)andanexponentialpart(linearhyperbolicmanifold),givenby(Masdemont2005)

⎧⎪⎨x(f)=α1exp(λ0f)+α2exp(−λ0f)+α3cos(ω0f+φ1),

¯2α2exp(−λ0f)+κ¯1α3sin(ω0f+φ1),y(f)=κ¯2α1exp(λ0f)−κ

⎪⎩

z(f)=α4cos(ν0f+φ2),

(15)

whereα1andα2refertotheunstableandstableamplitudescorrespondingtothehyperbolicmanifolds,respectively,andtheremainingamplitudesα3andα4refertothein-planeandout-of-planeamplitudescorrespondingtothecentermanifolds,respectively.κ¯1andκ¯2areconstantsandonlydependentonthemassparameterofthethree-bodysystem,

2

+2c2+1−2c2−1ω0λ2

κ¯1=−,κ¯2=0.

2ω02λ0

(16)

WhenconsideringthenonlineartermsandtheperturbationoforbitaleccentricityofEq.(8),

theinvariantmanifoldsassociatedwithLissajousorbits,aroundthecollinearlibrationpointsinERTBP,canbeexpandedaspowerseriesoftheorbitaleccentricityandfouramplitudeparameters,

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356H.Leietal.

󰀍󰀓rst⎧

󰀘cos(rf+sθ+tθ)+x12⎪pijkm⎪ijkm⎪x(f)=exp(i−j)θepα1α2α3α4,[]3⎪rst⎪⎪sin(rf+sθ+tθ)x¯12⎪pijkm⎪⎪󰀓󰀍⎪⎨󰀘yrstpijkmcos(rf+sθ1+tθ2)+ijkm

y(f)=exp[(i−j)θ3]rstα2α3α4,epα1⎪sin(rf+sθ+tθ)y¯⎪12pijkm⎪⎪󰀓rst󰀍⎪⎪⎪󰀘zpijkmcos(rf+sθ1+tθ2)+pijkm⎪⎪⎪exp[(i−j)θ3]rsteα1α2α3α4,⎩z(f)=

z¯pijkmsin(rf+sθ1+tθ2)

(17)

inwhichθ1=ωf+φ1,θ2=νf+φ2andθ3=λf,hereφ1andφ2arearbitrarily

initialphaseanglescorrespondingtothein-planeandout-of-planemotion,respectively.

rstrstxrst¯rst¯rst¯rstpijkm,ypijkmandzpijkmrefertothecoefficientsofcosines,andxpijkm,ypijkmandzpijkm

arethecoefficientsofsinescorrespondingtothecoordinateseriesx,yandztobedetermined.Duetothenonlineartermsoftheequationsofmotionandtheperturbationofeccentricity,thefrequenciesofmotionarenotconstant,andshouldalsobeexpandedasformalseries:

󰀘⎧

ijkm

⎪ω=ωpijkmepα1α2α3α4,⎪⎪⎨󰀘

ijkmα2α3α4,(18)ν=νpijkmepα1

⎪⎪⎪⎩λ=󰀘λpijkm

pijkmeα1α2α3α4.InEqs.(17)and(18),p,i,j,k,m∈N,andr,s,t∈Z.Whenα1=0andα2=0,Eq.(17)describesthestablemanifoldsassociatedwithLissajousorbitsaroundthecollinearlibrationpoints,whereas,ifα1=0andα2=0,itdescribestheunstablemanifoldsassociatedwithLissajousorbitsinERTBP.Whentimetendstoinfinity,aspacecraftapproachestheLissajousorbitexponentially,followingthestablemanifold.Whereas,thespacecraftdepartsfromtheLissajousorbitexponentially,followingtheunstablemanifold.Moreover,whenα1·α2<0,Eq.(17)candescribethetransittrajectories,followingwhichthespacecraftcouldtransferfromonesidetotheothersideofthelibrationpointfreely.Whenα1·α2>0,Eq.(17)candescribethenon-transittrajectories,alongwhichthespacecraftcanonlymoveinonesideofthelibrationpointforacertaintime,but,can’tmoveinonesideforeverduetotheArnolddiffusionphenomenoninERTBP(Xia1993).Inparticular,whenα1=α2=0,thegeneralsolutionsrepresentedbyEq.(17)willbereducedtodescribethecentermanifoldsinERTBP,suchastheLissajous,planeandverticalLyapunovorbits.ThisreducedcaseisequivalenttotheseriesexpansionsdiscussedinHouandLiu(2011a).Whentheorbitaleccentricitysatisfiese=0,itiseasytogetf=t,analyticalsolutionsrepresentedbyEq.(17)canbeusedtodescribethedynamicsaroundthecollinearlibrationpointsinCRTBP.ThisreducedcaseisequivalenttotheseriesexpansionsofthecentermanifoldsdiscussedinJorbaandMasdemont(1999),andtheinvariantmanifoldsdiscussedinMasdemont(2005).

Inourcomputation,threeordersaredefinedinthefollowingmanner.N1referstotheorderoftheorbitaleccentricity,N2representstheorderofhyperbolicmanifold,N3standsfortheorderofcentermanifold,andthetotalorderisN=N1+N2+N3.Denotetheorderoftheanalyticalsolutionsas(N1,N2,N3),andN2≤N3isrequired.Foranyp,i,j,k,m,r,sandt,itrequiresthat0≤p≤N1,0≤i+j≤N2,0≤k+m≤N3,0≤r≤p(takingintoaccountthesymmetriesofcosineandsinefunctions),−k≤s≤kand−m≤t≤m.InEq.(17),r,sandthavethesameparityofp,kandm,duetothesymmetriesofEq.(1).Asstated,Eq.(15)isthegeneralsolutionofthelinearizedequationsofmotioninERTBPwhentheeccentricityissettozero,andtheorderisdenotedas(0,n2,n3),wheren2+n3=1,whichmeansthattheorderoftheorbitaleccentricityiszero,andthetotalorderofcenterandhyperbolicmanifoldsisone.Indetail,forthecoordinateseriesxandy,

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High-ordersolutionsofinvariantmanifoldsinERTBP

000000010x01000=1,x00100=1,x00010=1,000y01000=κ¯2,

000

y00100=−¯κ2,

010

y00010=κ¯1,

357

andforthecoordinateseriesz,

001

=1.z00001

ForthelinearsolutiondescribedbyEq.(15),theorderoffrequencyis(0,0,0),thatis,

ω00000=ω0,ν00000=ν0,λ00000=λ0.

DuetothesymmetriesoftheequationsofmotioninERTBP,manycoefficientsofcoor-rstdinateandfrequencyarezero.Whenmisodd,thecoefficientsxrst¯rstpijkm,xpijkm,ypijkmand

rst¯rsty¯rstpijkmarezero.Whenmiseven,thecoefficientszpijkmandzpijkmarezero.Inaddition,the

symmetriesofthecoefficientsalsoexist,thatis,

rst

xrstpijkm=xpjikm,

rstrstrst

yrstpijkm=−ypjikm,zpijkm=zpjikm,

¯rstx¯rstpijkm=−xpjikm,y¯rst¯rst¯rstzrstpijkm=ypjikm,zpijkm=−¯pjikm.

rstItiseasytoverifythatwheni=j,onegetsx¯rst¯rstpijkm=ypijkm=zpijkm=0.Thecoefficients

correspondingtothefrequenciesarenotzeroonlyifbothkandmareeven,besidessatisfyingi=j.Thesefactscouldbeadoptedtosavecomputerstorageandhelpustocheckthevalidityoftheprocedureofconstruction.

Intheprocessofcomputation,thefirstandsecondderivativesofthecoordinatesx,yandzwithrespecttothetrueanomalyf,appearingintheleft-handsideoftheequationsofmotion,canbecomputedasfollows(takingxasanexample):

x󰀈=

and

∂x∂x∂x∂x

+ν+λ,+ω

∂f∂θ1∂θ2∂θ3

(19)

222∂2x∂2x∂2x2∂x2∂x2∂xx=+ω+ν+λ+2ω+2ν222∂f2∂f∂θ1∂f∂θ2∂θ1∂θ2∂θ3󰀈󰀈

∂2x∂2x∂2x∂2x

+2λ+2ων+2ωλ+2νλ.

∂f∂θ3∂θ1∂θ2∂θ1∂θ3∂θ2∂θ3

(20)

ThefactofL–Pmethodliesinthathigh-ordersolutionsoftheinvariantmanifoldsaround

thecollinearlibrationpointsareconstructedfromlower-ordersolutions.Ifthesolutionoftheorder(n1,n2,n3)istobesolved,theunknowncoefficientscorrespondingtocoordinates

rstrstinclude(xrst¯rst¯rst¯rstpijkm,xpijkm),(ypijkm,ypijkm)and(zpijkm,zpijkm),wherep=n1,i+j=n2and

k+m=n3,andtheunknowncoefficientscorrespondingtofrequencyincludeωpijkm,νpijkmandλpijkm,wherep+i+j+k+m=n1+n2+n3−1.Thesolutioncorrespondingtoorder(n1,n2,n3)canbeconstructedonlyifalltermsoforder(n¯1,n¯2,n¯3),n¯1≤n1,n¯2≤n2,n¯3≤n3andn¯1+n¯2+n¯3123

358H.Leietal.

seriescorrespondingtoorder(n1,n2,n3)canbedeterminedbysolvingthefollowinglinearsystemofalgebraicequations:

⎡⎤⎡rst⎤⎡c⎤⎡rst⎤

XpijkmxpijkmδxA1A2A3A4

⎢rsts⎥¯rst⎥⎢−A2A1−A4A3⎥⎢x⎥⎢δx⎢Xpijkm⎥pijkm⎥⎢⎢⎥⎢¯rst⎥(21)+=⎢⎥,crst⎣B1B2B3B4⎦⎣ypijkm⎦⎣δy⎦⎣Ypijkm⎦¯rst−B2B1−B4B3δsy¯rstYy

pijkm

pijkm

with

⎧

A1⎪⎪⎪⎨A2⎪A3⎪⎪⎩A4

2

=−(r+ω0s+ν0t)2+λ20(i−j)−(1+2c2),

=2λ0(i−j)(r+ω0s+ν0t),=−2λ0(i−j),=−2(r+ω0s+ν0t),

(22)

and

⎧

B1=2λ0(i−j),⎪⎪⎪⎪⎨B2=2(r+ω0s+ν0t),

2⎪B3=−(r+ω0s+ν0t)2+λ2⎪0(i−j)−(1−c2),⎪⎪⎩

B4=2λ0(i−j)(r+ω0s+ν0t),

(23)

and

⎧cδx⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨sδx⎪⎪⎪δc⎪y⎪⎪⎪⎪⎪⎪⎪⎩sδy

=−2(ω0+κ¯1)ωpijk−1mδr0δs1δt0δij+2(λ0−κ¯2)λpi−1jkmδr0δs0δt0δi−1j+2(λ0−κ¯2)λpij−1kmδr0δs0δt0δij−1,=0,

=2(κ¯2λ0+1)λpi−1jkmδr0δs0δt0δi−1j−2(κ¯2λ0+1)λpij−1kmδr0δs0δt0δij−1,=−2(κ¯1ω0+1)ωpijk−1mδr0δs1δt0δij.

(24)

Theunknowncoefficientsoforder(n1,n2,n3)inzaredeterminedbythesetofalgebraic

equations:

󰀄󰀅󰀄rst󰀅󰀄c󰀅󰀓rst󰀍

ZpijkmzpijkmC1δzC2

+=,(25)s¯rst−C2C1z¯rstδzZpijkmpijkmwith

󰀕

2

C1=−(r+ω0s+ν0t)2+λ20(i−j)+c2,

C2=2λ0(i−j)(r+ω0s+ν0t),

and

󰀕

c

=−2ν0νpijkm−1δr0δs0δt1δij,δzsδz=0.

(26)

(27)

rstrst¯rst¯rst¯rstInEqs.(21)and(25),Xrstpijkm,Xpijkm,Ypijkm,Ypijkm,ZpijkmandZpijkmrepresentthe

knowncomponentsoftheequationsofmotioncorrespondingtoorder(n1,n2,n3).ThesymbolδijistheKroneckerfunction,equalszerowheni=jandequalsonewheni=j.

123

High-ordersolutionsofinvariantmanifoldsinERTBP359

WehavetomentionthatthesystemofnotationadoptedinthispaperissimilartothatinMasdemont(2005).

3.2Solvingtheundeterminedcoefficients

InEqs.(21)and(25),nineunknowncoefficientsarerequiredtobecomputed,includingsixcoefficientscorrespondingtocoordinateandthreecoefficientscorrespondingtofrequency.Intheprocessofcomputation,wewillmeetthedifficultsituationthatthenumberofcoefficientstobecomputedisnotequaltothenumberofequations.Thus,somespecialproceduresshouldbecarriedout.Forclarity,wewilldiscusshowtodeterminetheunknowncoefficientsindifferentsituations.Case1r=0

Case1.1i=j

Inthissituation,theundeterminedcoefficientsonlyincludethecoefficientsofcoordinates:

rstrstxrst¯rst¯rst¯rstpijkm,xpijkm,ypijkm,ypijkm,zpijkmandzpijkm.Theseunknowncoefficientssatisfy,

⎡A1⎢−A2⎢⎣B1−B2

A2A1B2B1

A3−A4B3−B4

xrstA4pijkmrst⎥⎢x¯A3⎥⎢pijkm

B4⎦⎣yrstpijkm

rstB3y¯pijkm

⎤⎡

⎤

Xrstpijkm

⎥rst¯⎥⎢Xpijkm⎥⎥=⎢⎥,rst⎦⎢⎣Ypijkm⎦¯rstYpijkm⎤

⎡

(28)

forthexandycomponents,and

󰀄C1C2

−C2C1

󰀅󰀄zrstpijkmrstz¯pijkm

󰀅=󰀓Zrstpijkm

rst¯Zpijkm

󰀍,

(29)

forthezcomponent.TheelementsA1,A2,A3,A4,B1,B2,B3,B4,C1andC2arethesame

asthoseinSect.3.1.TheseundeterminedcoefficientscanbeobtainedbysolvingEqs.(28)and(29).

Case1.2i=j

rstAsstatedinSect.3.1,wheni=j,onegetsx¯rst¯rstpijkm=ypijkm=zpijkm=0.Inthissituation,

rstrstrsttheunknowncoefficientsonlyincludexpijkm,y¯pijkmandzpijkm,andcanbecomputedby

solvingthefollowingalgebraicequations:

󰀄A1A4−B2B3

󰀅󰀄xrstpijkmrsty¯pijkm

󰀅=󰀓Xrstpijkm

rst¯Ypijkm

󰀍,

(30)

forthex–ycomponents,and

rst

C1·zrstpijkm=Zpijkm,

(31)

forthezcomponent.

123

360H.Leietal.

Case2r=0

Case2.1s=0,t=0and|i−j|=1

Inthiscase,theunknowncoefficientssatisfy,

⎡A1⎢0⎢⎣B10

0A10B1

A30B30

⎤⎡c⎤⎡rst⎤rstXpijkmxδx0pijkm

⎢rst¯rst⎥⎢¯pijkm⎥⎢0⎥⎢XA3⎥pijkm⎥⎥⎥⎢x⎥⎢=+⎢⎥,crstrst0⎦⎣ypijkm⎦⎣δy⎦⎣Ypijkm⎦¯rstB30y¯rstYpijkmpijkm

⎤⎡

(32)

forthex–ycomponents.Inthissituation,thecoefficientmatrixissingular.However,x¯rstpijkm,

¯rst¯rsty¯rstpijkm,XpijkmandYpijkmarethecoefficientsofsin(0),thesamewayasMasdemont(2005)

¯rst¯rstisadoptedhere,thatis,takingx¯rst¯rstpijkm=0,ypijkm=0,Xpijkm=0andYpijkm=0.Forthez

¯rstcomponent,thesimilaranalysisisconsideredsothatwecansetz¯rstpijkm=0andZpijkm=0.

Thentheremainingcoefficientssatisfy,

󰀄A1A3B1B3

󰀅󰀄xrstpijkmrstypijkm

󰀅

cδx

+c

δy

󰀄󰀅=

󰀄

󰀅Xrstpijkm

,rstYpijkm

(33)

2whereA1=λ20−(1+2c2),A3=−2(i−j)λ0,B1=2(i−j)λ0,B3=λ0−(1−c2),

c=2(λ−κδx¯2)λpnnkmandδc¯2λ0+1)λpnnkm,wheren=min(i,j).There0y=2(i−j)(κ

arethreeundeterminedcoefficientsinEq.(33),whichcanbesolvedbytakingyrstpijkm=0

rst(orxpijkm=0),

󰀄

¯2)A12(λ0−κ

B12(i−j)(κ¯2λ0+1)

󰀅󰀄

xrstpijkm

λpnnkm

󰀅=

󰀄

󰀅Xrstpijkm

,rstYpijkm

(34)

rstthenxrstpijkmandλpnnkmcanbecomputed.Forthezcomponent,zpijkmcanbeobtainedby

solvingthefollowingequation:

rstrst

(λ20+c2)zpijkm=Zpijkm.

rst2Itiseasytogetzrstpijkm=Zpijkm/(λ0+c2).

(35)

Case2.2s=1,t=0andi=j

Inthissituation,theundeterminedcoefficientssatisfy

⎡A1⎢0⎢⎣0−B2

0A1B20

0−A4B30

⎤⎡rst⎤⎡c⎤⎡rst⎤

XpijkmxpijkmδxA4

¯rst⎥⎢¯rst⎥⎢0⎥⎢0⎥⎢Xpijkm⎥pijkm⎥⎥⎢x⎢⎥+⎣⎦=⎢rst⎥,rst⎦⎣⎦00ypijkm⎣Ypijkm⎦¯rstδsB3y¯rstYypijkmpijkm

(36)

2−(1+2c),A=−2ω,B=2ω,forthex−ycomponents,whereA1=−ω024020

2csB3=−ω0−(1−c2),δx=−2(ω0+κ¯1)ωpijk−1mandδy=−2(1+κ¯1ω0)ωpijk−1m.For

thezcomponent,

󰀄󰀅󰀄rst󰀅󰀓rst󰀍

ZpijkmzpijkmC10

=¯rst,(37)rst0C1z¯pijkmZpijkm

123

High-ordersolutionsofinvariantmanifoldsinERTBP361

rst2+c.Duetoi=j,onegetsxwhereC1=−ω0¯rst¯rst2pijkm=ypijkm=zpijkm=0,thenthe

remainingunknowncoefficientssatisfy

󰀄

A4A1

−B2B3

⎤󰀓󰀍xrstrstpijkmXpijkm−2(ω0+κ¯1)⎣rst

y¯pijkm⎦=¯rst,

−2(1+κ¯1ω0)Ypijkm

ωpijk−1m

󰀅⎡

(38)

forthex–ycomponents,and

rst

zrstpijkm=Zpijkm/C1,

(39)

forthezcomponent.FromEq.(38),wecancomputexrst¯rstpijkmandωpijk−1mbytakingypijkm=rst0,ory¯rstpijkmandωpijk−1mbytakingxpijkm=0(thelattercaseisadoptedinourcomputation).Case2.3s=0,t=1andi=j

Inthiscase,wecanget

⎡A1⎢0⎢⎣0−B2

0A1B20

0−A4B30

⎤⎡rst⎤rstXpijkmxA4pijkm

⎢rst¯rst⎥⎢¯pijkm⎥⎢X0⎥pijkm⎥⎥⎢x⎥=⎢rst⎥,rst⎦⎣⎦0ypijkm⎣Ypijkm⎦¯rstB3y¯rstYpijkmpijkm

⎤⎡

(40)

2−(1+2c),A=−2ν,B=2νandforthex–ycomponents,whereA1=−ν024020

2B3=−ν0−(1−c2).Forthezcomponent,

󰀄

C10

0C1

󰀅󰀄

zrstpijkmrstz¯pijkm

󰀅+

󰀄

cδz

󰀅=

󰀓

0

Zrstpijkm

rst¯Zpijkm

󰀍,

(41)

2+c=0andδc=−2νν¯rstwhereC1=−ν020pijkm−1.Duetoi=j,onegetsxzpijkm=

rstyrst¯rst¯rstpijkm=zpijkm=0,thenxpijkmandypijkmcanbeobtainedbysolving

󰀄

A4A1

−B2B3

󰀅󰀄

xrstpijkmrsty¯pijkm

󰀅=

󰀓

Xrstpijkm

rst¯Ypijkm

󰀍,

(42)

andνpijkm−1canbeobtainedbytakingzrstpijkm=0,thatis,

󰀃

νpijkm−1=−Zrstpijkm(2ν0).

Case2.4s=0,t=−1andi=j¯rstThecoefficientsxrstpijkmandypijkmcanbecomputedbysolving

󰀄A1A4−B2B3

󰀅󰀄xrstpijkmrsty¯pijkm

󰀅=󰀓Xrstpijkm

rst¯Ypijkm

󰀍,

(44)(43)

2−(1+2c),A=−2ν,B=2νandB=−ν2−(1−c).WhileinwhichA1=−ν024020320

rstrst2forthezcomponent,takezpijkm=0andz¯pijkm=0duetoC1=−ν0+c2=0inthiscase.

123

362H.Leietal.

Case2.5otherwise

Case2.5.1i=j

rstInthissituation,xrst¯rst¯rstpijkm,xpijkm,ypijkmandypijkmarecomputedbysolvingthefollowing

algebraicequations:

⎤⎡rst⎤⎡rst⎤⎡

XpijkmxpijkmA1A2A3A4

rst¯rst⎥⎥⎢⎢−A2A1−A4A3⎥⎢x⎢Xpijkm⎥pijkm⎥⎥⎢¯rst⎢(45)=⎢rst⎥,⎦⎣⎦⎣B1B2B3B4ypijkm⎣Ypijkm⎦

¯rst−B2B1−B4B3y¯rstYpijkmpijkmforthex–ycomponents,and

󰀄C1C2

−C2C1

󰀅󰀄zrstpijkmz¯rstpijkm

󰀅=󰀓Zrstpijkm

¯rstZpijkm

󰀍,

(46)

forthezcomponent.InEqs.(45)and(46),

⎧

A1⎪⎪⎪⎨A2⎪A3⎪⎪⎩A4

and

⎧

B1=2(i−j)λ0,⎪⎪⎪⎪⎨B2=2(ω0s+ν0t),

2⎪B3=−(ω0s+ν0t)2+λ2⎪0(i−j)−(1−c2),⎪⎪⎩

B4=2(i−j)λ0(ω0s+ν0t),

2

=−(ω0s+ν0t)2+λ20(i−j)−(1+2c2),

=2(i−j)λ0(ω0s+ν0t),=−2(i−j)λ0,=−2(ω0s+ν0t),

(47)

(48)

and

󰀕

2C1=−(ω0s+ν0t)2+λ20(i−j)+c2,

C2=2(i−j)λ0(ω0s+ν0t).

(49)

Case2.5.2i=j

rstAsstated,wheni=j,onegetsx¯rst¯rstpijkm=0,ypijkm=0andzpijkm=0,andtheremaining

unknowncoefficientsareobtainedbysolving

󰀅󰀄rst󰀅󰀓rst󰀍󰀄

XpijkmxpijkmA4A1

=¯rst,(50)rst−B2B3y¯pijkmYpijkmforthex–ycomponents,and

󰀃rst

zrst=ZpijkmpijkmC1,

(51)

forthezcomponent.InEqs.(50)and(51),A1,A4,B2,B3andC1havethesameexpressions

asthoseinEqs.(47–49).

123

High-ordersolutionsofinvariantmanifoldsinERTBP363

4High-ordersolutionsofinvariantmanifoldsassociatedwithhaloorbitsinERTBP4.1Seriesexpansionsoftheinvariantmanifolds

Forlargeamplitudes,consideringthenonlineartermsoftheequationsofmotioninERTBP,itispossibletogenerate‘haloorbit’(call‘haloorbit’inordertobeconsistentwiththecaseinCRTBP)whosein-planeandout-of-planefrequenciesofmotionareidentical.Itisworthytonotethat‘haloorbit’inERTBPisquasi-periodicorbitduetotheperturbationofeccentricity.SimilartothehalocaseinCRTBPdiscussedinJorbaandMasdemont(1999),modifytheequationsofmotionintheLi-centeredsynodiccoordinatesystemasfollows:󰀊⎧󰀘󰀚

󰀈󰀈󰀈ii⎪x−2y−(1+2c2)x=(−e)cosf(1+2c2)x⎪⎪⎪⎪i≥1⎪⎪⎪󰀄󰀘󰀅󰀇⎪󰀘󰀎⎪⎪⎪cn+1(n+1)Tn(x,y,z),(−e)icosif+⎪⎪⎪⎪n≥2i≥0⎪⎪⎪󰀚󰀊󰀘⎪⎪󰀈󰀈󰀈ii⎪(−e)cosf(1−c2)yy+2x−(1−c2)y=⎪⎪⎪⎨i≥1

󰀄󰀘󰀅󰀇󰀘󰀎⎪ii⎪⎪+cn+1Rn−1(x,y,z),(−e)cosfy⎪⎪⎪⎪n≥2i≥0⎪⎪󰀚󰀊⎪󰀘⎪󰀈󰀈ii⎪⎪(−e)cosf(1−c2)z+Δzz+c2z=⎪⎪⎪⎪i≥1⎪⎪󰀄󰀘󰀅󰀇⎪⎪󰀘󰀎⎪⎪ii⎪cn+1Rn−1(x,y,z),(−e)cosfz+⎪⎩

i≥0

n≥2

(52)

whichisdifferentfromtheformadoptedinHouandLiu(2011a).InEq.(52),Δrepresentsthe

residualpartofthethirdequation,andΔ=0isneededforhalocase.SimilartotheLissajouscasediscussedinabovesection,thegeneralsolutionsofinvariantmanifoldsassociatedwithhaloorbitsaroundthecollinearlibrationpointsinERTBPareexpandedas

󰀓rs󰀍⎧

󰀘cos(rf+sθ)+x1⎪pijkm⎪pijkm⎪x(f)=exp(i−j)θα1α2α3α4,e[]3⎪rs⎪⎪sin(rf+sθ)x¯1⎪pijkm⎪⎪󰀓rs󰀍⎪⎨󰀘ypijkmcos(rf+sθ1)+pijkm

y(f)=exp[(i−j)θ3]rseα1α2α3α4,⎪sin(rf+sθ)y¯⎪1pijkm⎪⎪󰀓󰀍⎪rs⎪⎪󰀘cos(rf+sθ)+z1⎪pijkmijkm⎪⎪exp[(i−j)θ3]rsα2α3α4,epα1⎩z(f)=

z¯pijkmsin(rf+sθ1)

(53)

wherep,i,j,k,m∈Nandr,s∈Z.θ1=ωf+φandθ3=λf,whereφistheinitialphasersrsangle.Thecoefficientsxrs¯rs¯rs¯rspijkm,xpijkm,ypijkm,ypijkm,zpijkmandzpijkmcorrespondingto

thecoordinatewillbecomputediterativelybyusingtheL–Pmethod.Thefirstandsecondderivativesofcoordinatewithrespecttof,appearingintheleft-handsideoftheequationsofmotion,canbecomputedinthefollowingmanner(takingxasanexample):

x󰀈=

∂x∂x∂x+ω+λ,∂f∂θ1∂θ3

(54)

123

3H.Leietal.

and

22∂2x∂2x∂2x∂2x2∂x2∂x+ω+λ+2ω+2λ+2ωλ.x=22∂f2∂f∂θ1∂f∂θ3∂θ1∂θ3∂θ1∂θ3󰀈󰀈

(55)

Thesimilarexpressionscanbederivedforyandz.

Intheprocessofconstruction,thefrequenciesωandλ,aswellasΔarenotconstant,andalsoexpandedaspowerseriesoforbitaleccentricitye,theamplitudescorrespondingtothehyperbolicmanifoldsα1(unstablecomponent)andα2(stablecomponent),andampli-tudescorrespondingtothecentermanifoldsα3(in-planecomponent)andα4(out-of-planecomponent)asfollows:

󰀘⎧

ijkm

⎪ω=ωpijkmepα1α2α3α4,⎪⎪⎨󰀘

ijkmα2α3α4,(56)λ=λpijkmepα1

⎪⎪⎪⎩Δ=󰀘dpijkm

pijkmeα1α2α3α4.SimilartotheLissajouscaseinabovesection,threeordersaredefinedinthefollowingmanner:N1representstheorderoforbitaleccentricity,N2referstotheorderofthehyperbolicmanifolds,andN3correspondstotheorderofthecentermanifolds.ThetotalorderofanalyticalsolutionisN=N1+N2+N3.Weuse(N1,N2,N3)tostandfortheorderofanalyticalsolutiontobeconstructed,andN2≤N3isrequired.Foranyp,i,j,kandm,itrequires0≤p≤N1,0≤i+j≤N2and0≤k+m≤N3.Forrands,itrequiresthat−p≤r≤pand−k−m≤s≤k+m.Moreover,randsshouldhavethesameparityofpandk+m,respectively.Consideringthesymmetriesofcosineandsinefunctions,onlythetermscorrespondingto0≤s≤k+marecomputed.ItisworthytonotethatΔ(e,α1,α2,α3,α4)=0isrequiredforhalocase,thatis,theorbitaleccentricitye,andtheamplitudesαi,i=1,...,4,arenotindependent.Forgivenparameterse,α1,α2,α3orα4,theparameterα4orα3iscomputedinordertosatisfyΔ=0.Whenα1=0andα2=0,Eq.(53)representstheunstablemanifolds,whereasifα1=0andα2=0,itrepresentsthestablemanifoldsassociatedwithhaloorbits.Inaddition,ifα1·α2<0,Eq.(53)standsforthetransittrajectories,elseifα1·α2>0,itrepresentsthenon-transittrajectoriesassociatedwithhaloorbitsinERTBP.Inparticular,whenα1=0andα2=0,Eq.(53)canbereducedtodescribethedynamicsinthecentermanifoldsaroundthecollinearlibrationpointsinERTBP,andthisreducedsituationisequivalenttothecasediscussedinHouandLiu(2011a).Furthermore,whentheorbitaleccentricitysatisfiese=0,Eq.(53)canbereducedtodescribethedynamicsintheinvariantmanifolds(includingthecenterandhyperbolicmanifolds)associatedwithhaloorbitsaroundthecollinearlibrationpointsinCRTBP.ThisreducedsituationisequivalenttothecasediscussedinJorbaandMasdemont(1999)aboutthecentermanifoldsandthecasediscussedinMasdemont(2005)abouttheinvariantmanifoldsinCRTBP.

Forhalocase,thestartingsolutionwithfirst-orderamplitudeandzero-ordereccentricityweadoptforL–Pmethodaremodifiedas

⎧⎪⎨x(f)=α1exp(λ0f)+α2exp(−λ0f)+α3cos(ω0f+φ),y(f)=κ¯2α1exp(λ0f)−κ¯2α2exp(−λ0f)+κ¯1α3sin(ω0f+φ),(57)⎪⎩

z(f)=α4cos(ω0f+φ),whereκ¯1,κ¯2,ω0andλ0havethesameformsasEqs.(14)and(16).SubstitutingthelinearsolutionintothelinearizedequationsofEq.(52)withzeroeccentricity,onegetsd00000−

123

High-ordersolutionsofinvariantmanifoldsinERTBP365

2=0,thushaloorbitsdon’texistinthelinearizedequationsofCRTBP.Accordingc2−ω0

toEq.(57),onegets

000001x01000=1,x00100=1,x00010=1,

forthexcomponent,and

00

=κ¯2,y01000

00

y00100=−¯κ2,

01

y¯00010=κ¯1,

fortheycomponent,and

01

z00001=1.

forthezcomponent.Moreover,forthecoefficientsoffrequency,onegets

2

.ω00000=ω0,λ00000=λ0,d00000=c2−ω0

Forhalocase,somecoefficientsofthegeneralsolutionarealsozeroduetothesymmetriesof

rstheequationsofmotion.Whenmisodd,thecoefficientsxrs¯rs¯rspijkm,xpijkm,ypijkmandypijkm

rsarezero.Whenmiseven,zrsandz¯arezero.Inaddition,foranyp,i,j,kandm,pijkmpijkm

thecoefficientsofcoordinatehavethefollowingsymmetries:

rs

xrspijkm=xpjikm,

rsrsrs

yrspijkm=−ypjikm,zpijkm=zpjikm,

x¯rs¯rspijkm=−xpjikm,y¯rs¯rs¯rszrspijkm=ypjikm,zpijkm=−¯pjikm.

rs¯rsItiseasytoverifythatwheni=j,onegetsx¯rspijkm=ypijkm=zpijkm=0.Forthetermsof

frequency,thecoefficientsarenotzeroonlyifbothkandmareeven,besidessatisfyingi=j.Thesepropertiescouldalsohelpustocheckthevalidityoftheprocedureofcomputationandsavecomputerstorage.

SimilartotheLissajouscase,thehigh-ordersolutionsofinvariantmanifoldsassoci-atedwithhaloorbitsinERTBPareconstructedfromlower-ordersolutionsbyutilizingtheL–Pmethod.Ifthegeneralsolutionistruncatedatorder(N1,N2,N3),thetermsoforder(n1,n2,n3),n1≤N1,n2≤N2andn3≤N3,aretobecalculated.Takingtheconstructionofanalyticalsolutioncorrespondingtoorder(n1,n2,n3)asanexample,

rsrstheunknowncoefficientsincludexrs¯rs¯rs¯rspijkm,xpijkm,ypijkm,ypijkm,zpijkmandzpijkm,where

p=n1,i+j=n2andk+m=n3,correspondingtothecoefficientsofthecoordi-nate.Thecoefficientscorrespondingtothefrequency,ωpijkm,λpijkmanddpijkm,wherep+i+j+k+m=n1+n2+n3−1,arealsounknownandwillbecomputediteratively.Thesolutioncorrespondingtotheorder(n1,n2,n3)canbecomputedonlyifthetermsoforders(n¯1,n¯2,n¯3),wheren¯1≤n1,n¯2≤n2,n¯3≤n3andn¯1+n¯2+n¯3123

366H.Leietal.

⎡

A1⎢−A2⎢⎢B3⎢⎢−B4⎢⎣00A2A1B4B300A3−A4B1−B200A4A3B2B1000000C1−C2

⎤⎡⎤⎡rs⎤⎤⎡xrscXpijkmδpijkm0

rs⎢x⎢X⎥⎢δx⎥¯rss⎥¯⎥0⎥⎢pijkm⎥⎢x⎢pijkm⎥⎥⎢yrs⎥⎢δc⎥⎢Yrs⎥

0⎥⎢⎢⎥⎥pijkmpijkmy⎢s⎥=⎢rs⎥⎢rs+⎥⎥,⎢δ⎥⎢Y¯0⎥y¯⎢⎥⎥⎢pijkm⎥⎢y⎥⎢pijkm⎥⎥⎣δc⎦⎣ZrsC2⎦⎣zrszpijkm⎦pijkm⎦sC1δz¯rsz¯rsZpijkmpijkm

(58)

rsrs¯rs¯rs¯rswhereXrspijkm,Xpijkm,Ypijkm,Ypijkm,ZpijkmandZpijkmaretheknowncomponentsofthe

equationsofmotioncorrespondingtoorder(n1,n2,n3),andtheelementsoftheconstantcoefficientmatrixinEq.(58)aregivenby

⎧A1⎪⎪⎪⎨A2⎪A3⎪⎪⎩A4

and

⎧B1⎪⎪⎪⎨B2⎪B3⎪⎪⎩B4

and

󰀕

2

=−(r+ω0s)2+λ20(i−j)−(1+2c2),

=2λ0(i−j)(r+ω0s),=−2(i−j)λ0,=−2(r+ω0s),

(59)

2=−(r+ω0s)2+λ20(i−j)−(1−c2),

=2λ0(i−j)(r+ω0s),=2(i−j)λ0,=2(r+ω0s),

(60)

2C1=−(r+ω0s)2+λ20(i−j)+c2−d0,

C2=2λ0(i−j)(r+ω0s),

and

⎧cδx⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪δs⎪⎨xδcy⎪⎪⎪⎪⎪⎪⎪⎪δs⎪y⎪⎪⎪⎪c⎪δz⎪⎪⎪⎩sδz

=−2(ω0+κ¯1)ωpijk−1mδr0δs1δij+2(λ0−κ¯2)λpi−1jkmδr0δs0δi−1j+2(λ0−κ¯2)λpij−1kmδr0δs0δij−1,=0,

=2(κ¯2λ0+1)λpi−1jkmδr0δs0δi−1j−2(κ¯2λ0+1)λpij−1kmδr0δs0δij−1,=−2(κ¯1ω0+1)ωpijk−1mδr0δs1δij,=−(dpijkm−1+2ω0ωpijkm−1)δr0δs1δij,=0.

(61)

(62)

4.2Solvingtheundeterminedcoefficients

SimilartotheLissajouscase,wecansolvetheundeterminedcoefficientsindifferentsitua-tions.

123

High-ordersolutionsofinvariantmanifoldsinERTBP367

Case1r=0

Case1.1i=j

Inthissituation,thecoefficientsoffrequencyarezero,andtheunknowncoefficientsinclud-rsrsingxrs¯rs¯rs¯rspijkm,xpijkm,ypijkm,ypijkm,zpijkmandzpijkm,satisfythefollowinglinearsystemof

equations:

⎤⎡rs⎤⎡⎤⎡xrsXpijkmpijkmA1A2A3A400

rsrs⎢⎢⎥¯x¯pijkm⎥⎢Xpijkm⎥⎢−A2A1−A4A300⎥⎥⎢⎥⎢⎢yrs⎥⎢Yrs⎥⎢B3⎥B4B1B200⎥⎢pijkm⎥⎢pijkm⎥⎢(63)⎢rs⎥=⎢¯rs⎥.⎢−B4B3−B2B100⎥y¯pijkm⎥⎢Y⎥pijkm⎢⎥⎢⎢⎥⎢rs⎥⎣0000C1C2⎦⎣zrs⎣⎦Zpijkmpijkm⎦

rsrs0000−C2C1¯z¯pijkmZpijkmThex–ycomponentsandthezcomponentareuncoupled,andcanbesolvedseparatively.Case1.2i=j

rsAsstatedinSect.4.1,wheni=j,thecoefficientsincludingx¯rs¯rspijkm,ypijkmandzpijkmare

zero,thustheunknowncoefficientssatisfy

⎤⎡⎤⎡rs⎤⎡rs

XxA1A40pijkmpijkm

⎢rsrs¯⎣−B4B10⎦⎣y⎦¯pijkm=⎣Ypijkm⎥()⎦.

rsrs00C1zpijkmZpijkmInEqs.(63)and(),A1,A2,A3,A4,B1,B2,B3,B4,C1andC2havethesameformsas

rsEqs.(59–61).Theundeterminedcoefficientsxrs¯rspijkm,ypijkmandzpijkmcanbesolvedeasily.Case2r=0

Case2.1s=0and|i−j|=1

Inthiscase,x¯rs¯rs¯rspijkm,ypijkmandzpijkmarethecoefficientsofsin(0),andcanbetakenaszero,thentheunknowncoefficientssatisfy

󰀄󰀅󰀄rs󰀅󰀄c󰀅󰀄rs󰀅

xpijkmXpijkmδxA1A3

+c=,(65)rsδyB3B1yrsYpipijkmjkm

2forthex−ycomponents,whereA1=λ20−(1+2c2),A3=−2(i−j)λ0,B1=λ0−(1−

c=2(λ−κc2),B3=2(i−j)λ0,δx¯2)λpnnkmandδc¯2λ0+1)λpnnkm,heren=0y=2(i−j)(κ

rsmin(i,j).InEq.(65),onecancomputexpijkmandλpnnkmbytakingyrspijkm=0,orcompute

rsrsypijkmandλpnnkmbytakingxpijkm=0(thefirstcaseisadoptedinourcomputation).Forthezcomponent,theunknowncoefficientssatisfy

rs

C1·zrspijkm=Zpijkm,

(66)

whereC1=λ20+c2−d0.

Case2.2s=1andi=j

rsAsanalyzedsimilarly,wheni=j,thecoefficientsincludingx¯rs¯rspijkm,ypijkmandzpijkmare

zero,thustheunknowncoefficientssatisfy

󰀍󰀄󰀅󰀄rs󰀅󰀄c󰀅󰀓rs

XxδA4A1pijkmpijkm

+x=¯rs,(67)srsδy−B4B1y¯pijkmYpijkm

2−(1+2c),A=correspondingtothex−ycomponents.InEq.(67),A1=−ω024

2cs−2ω0,B1=−ω0−(1−c2),B4=2ω0,δx=−2(ω0+κ¯1)ωpijk−1mandδy=

123

368H.Leietal.

−2(κ¯1ω0+1)ωpijk−1m.FromEq.(67),onecancomputexrspijkmandωpijk−1mbytakingrsrsrsy¯pijkm=0,orcomputey¯pijkmandωpijk−1mbytakingxpijkm=0(thelattercaseisadoptedinourcomputation).Forthezcomponent,theunknowncoefficientssatisfy

rs

C1·zrspijkm−(dpijkm−1+2ω0ωpijkm−1)=Zpijkm,

(68)

2+c−d=0.FromEq.(68),onecancomputedwhereC1=−ω020pijkm−1bytaking

rszpijkm=0.

Case2.3otherwise

Case2.3.1i=j

Inthissituation,thecoefficientsoffrequencyareallzero,andtheunknowncoefficients

rsrsincludingxrs¯rs¯rs¯rspijkm,xpijkm,ypijkm,ypijkm,zpijkmandzpijkmarecomputedbysolvingthefol-lowingalgebraicequations:

⎤⎡rs⎤⎤⎡xrs⎡XpijkmpijkmA1A2A3A400

rsrs⎢x⎥⎢X⎥¯¯⎥⎢−A2A1−A4A300⎥⎢pijkm⎥⎢pijkm⎥⎢⎥⎢Yrs⎥rs⎥⎢⎢B3yBBB00⎢⎢⎥⎥412pijkmpijkm⎥⎢rs⎢(69)⎥=⎢¯rs⎥.⎥⎢−B4B3−B2B100⎥⎢y¯pijkm⎥⎢Ypijkm⎥⎢⎢⎥⎢rs⎥⎣0000C1C2⎦⎣zrs⎣⎦Zpijkmpijkm⎦

0000−C2C1¯rsz¯rsZpijkmpijkmCase2.3.2i=j

rsThecoefficientsincludingx¯rs¯rspijkm,ypijkmandzpijkmareknownaszero,thustheundeter-minedcoefficientssatisfy

⎤⎡⎤⎡rs⎤⎡rs

XpijkmxpijkmA1A40

⎥¯rs⎣−B4B10⎦⎣y⎦=⎢¯rs(70)⎣Ypijkmpijkm⎦.

rsrs00C1zpijkmZpijkmInEqs.(69)and(70),theelementsofconstantcoefficientmatrixaregivenby

⎧222A1=−ω0s+λ2⎪0(i−j)−(1+2c2),⎪⎪⎨A=2ωλs(i−j),200⎪A3=−2λ0(i−j),⎪⎪⎩

A4=−2ω0s,and

⎧B1⎪⎪⎪⎨B2⎪B3⎪⎪⎩B4󰀕

222

=−ω0s+λ20(i−j)−(1−c2),

(71)

=2ω0λ0s(i−j),=2λ0(i−j),=2ω0s,

(72)

and

222

s+λ2C1=−ω00(i−j)+c2−d0,

C2=2ω0λ0s(i−j).

(73)

Allundeterminedcoefficientscorrespondingtobothcoordinateandfrequencycanbeeasily

computedbysolvingthecorrespondingalgebraicequations.

123

High-ordersolutionsofinvariantmanifoldsinERTBP369

Table1CPUtime(inseconds,foraPCwitha2.60GHzIntel(R)Core(TM)i5vProCPU),theaverageresidualaccelerationofseriesexpansionsdefinedinthecontext,andRAMmemory(inKilobytesforfrequency,andMegabytesforcoordinate)forthecoefficientsofanalyticalsolutionsoftheinvariantmanifoldsassociatedwithLissajousandhaloorbitsaroundtheL1pointintheSun–EarthERTBPCaseLissajous

Order(1,1,1)(2,2,2)(3,3,3)(4,4,4)(4,4,5)(4,3,6)(5,3,5)(4,2,7)

Halo

(3,3,3)(4,4,4)(5,5,5)(5,5,9)(5,3,12)

Freq.(KB)1.60E−011.861.868.388.3.315.5.311.868.388.382.09E+012.60E+01

Coord.(MB)8.09E−031.02E−016.05E−012.66E+004.79E+005.33E+004.26E+005.02E+004.70E−011.95E+006.00E+002.46E+012.48E+01

CPUtime(s)1.56E−021.72E−013.03E+003.81E+011.05E+021.17E+027.58E+011.00E+021.42E+001.39E+011.07E+021.42E+031.43E+03

Residualacceleration1.05E−043.68E−062.90E−071.55E−081.93E−091.18E−101.93E−091.98E−113.63E−041.10E−044.03E−054.80E−071.84E−08

5Results

Inourcomputation,theSun–Earthellipticrestrictedthree-bodyproblemistakenasanexample.Themassparameterofthisthree-bodysystemisμ=3.003480575402412×10−6,andtheorbitaleccentricityoftheEarthontheellipticorbitise=0.01671022.Theprocedurescorrespondingtotheconstructionofhigh-orderanalyticalsolutionsofinvariantmanifoldsassociatedwithbothLissajousandhaloorbitsinERTBParecompletedinFORTRAN95language.Itisworthytonotethat,intheprocessofconstruction,nine-andeight-dimensionalarraysforTn(andRn)appearingintheequationsofmotionareneededtodefineandcomputeforLissajousandhalocases,respectively.Unfortunately,theFortrancompilercanonlyadmitanarrayuptosevendimensions,andwetransformthemulti-dimensionalarrayintoasingle-dimensionalarraytoovercomethistechnologicaldifficulty.Duetothelargestorageofcoefficientscorrespondingtothecoordinateandfrequency,theanalyticalsolutionsofinvariantmanifoldsassociatedwithlibrationpointorbitsinERTBPcanbeconstructeduptoalimitedorder.Whenthegeneralsolutionsarereducedtodifferentcases,theorderofthesolution(N1,N2,N3)canbeadaptedasneededfordescribingthecorrespondingdynamicsaroundthecollinearlibrationpointsmoreaccurately.Forexample,theseriesexpansionstruncatedatorder(N1,0,N3)aresuitableforcomputingLissajousorhaloorbitsinERTBP,theonestruncatedatorder(0,N2,N3)aresuitableforcomputingtheinvariantmanifoldsinCRTBP,andtheonesconstructeduptoorder(0,0,N3)aresuitableforcomputingLissajousorhaloorbitsinCRTBP.

TheCPUtime,RAMmemoryrequiredtogeneratethecoefficientsoffrequencyandcoordinateseriesexpansionstruncatedatdifferentordersaresummarizedinTable1.ThelastcolumnofTable1istheaverageresidualacceleration,whichindicatestheaccuracyoftheseriesexpansionsconstructedandiscomputedinthismanner:aspacecraftisassumedtomovealonganominaltrajectoryobtainedbyseriesexpansions,atanypointofthenominal

123

370

2.521.51x 10−3H.Leietal.

x 10−42y [adim]10.50−0.5−10.98650.9870.98750.9880.98850.90.950.990.99050.991L1z [adim]0−1−20.9870.98750.9880.98850.90.950.990.9905x [adim]x 102x 10211−4−4x [adim]z [adim]z [adim]00−1L1−1−220x 10−505101520−4x 10−41000.9880.9870.990.9−2y [adim]x [adim]y [adim]Fig.1Projectionsonthecoordinateplaneanda3Drepresentationoftheanalyticalsolutions,truncatedatorder(4,4,5),correspondingtothenegativebranchoftheunstablemanifoldassociatedwithLissajousorbitaroundthecollinearlibrationpointL1intheSun–Earthellipticrestrictedthree-bodysystem.Theamplitudesofthemanifoldareα1=−1×10−5,α2=0,α3=0.02andα4=0.02.Theinitialphaseanglecorrespondingtotheout-of-planecomponentistakenasφ2=0,andtenvaluesofφ1correspondingtothein-planeinitialphaseangleinEq.(17)aretakenin[0,2π]uniformly.Thetrajectoryspecifiedby(φ1,φ2)intheunstablemanifoldsisterminatedwhentheEuclideannormofthepositiondeviationbetweentheanalyticaltrajectoryandnumericallyintegratedtrajectoryreachesthegiventolerance1×10−5

trajectory,theaccelerationofthespacecraft,computedbytheequationsofmotion,isdenotedbyaNum,andthesecondderivativesofseriesx,yandzrepresentedbyEqs.(17)and(53)aredenotedbyaAna.TheaverageresidualaccelerationmeanstheaveragevalueoftheEuclideannormofaNum−aAnaforacertainnominaltrajectoryduringπunitsofdimensionlesstime.Thedimensionlessmagnitudeofaccelerationa=1.0isabout5.93mm/s2fortheSun–Earthellipticrestrictedthree-bodysystem.ThesimilarconsiderationofresidualaccelerationaboutanalyticalsolutionscanbeseeninFarquharandKamel(1973).ForLissajouscase,thenominaltrajectoryliesinthepositivebranchoftheunstablemanifoldswithamplitudesα3=α4=0.02andisspecifiedbyφ1=φ2=0.Forhalocase,thenominaltrajectoryliesinthepositivebranchofunstablemanifoldwithα4=0.05andisspecifiedbyφ=0.

Thehigh-ordersolutionsofinvariantmanifoldsassociatedwithLissajousorbit,uptoorder(4,4,5),andhaloorbit,uptoorder(5,5,9),areconstructedfollowingtheprocedurepresentedinSects.3and4,andthecorrespondingcoefficientsoffrequenciesandcoordinatesoftheseriesexpansionsaregivenintheformofsupplementarymateriallinkedtotheelectronicversionofthispaper.Inthesupplement,thesubroutinescorrespondingtoLissajousandhalocaseswrittenusingtheFORTRANlanguageareprovidedtocomputetheanalyticalstateoftheinvariantmanifoldsinERTBPbycallingthecoefficientsprovided.Supplementarymaterial1includesadescriptionoftheonlineappendedfiles.

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High-ordersolutionsofinvariantmanifoldsinERTBP

x 105−4371

x 102−4L110y [adim]−5z [adim]0−10−15−1−200.950.990.99050.9910.99150.9920.99250.9930.9935−20.950.990.99050.9910.99150.9920.99250.9930.9935x [adim]x 102x 10211−4−4x [adim]z [adim]z [adim]00−1L1−1−200.993−2−20−15−10−50x 105−4x 10−4−10y [adim]−200.990.9910.992x [adim]y [adim]Fig.2Projectionsonthecoordinateplaneanda3Drepresentationoftheanalyticalsolutions,truncatedatorder(4,4,5),correspondingtothepositivebranchoftheunstablemanifoldassociatedwithLissajousorbitaroundthelibrationpointL1intheSun–Earthellipticrestrictedthree-bodysystem.Theamplitudesofthemanifoldsareα1=1×10−5,α2=0,α3=0.02andα4=0.02.Theout-of-planeinitialphaseangleistakenasφ2=0,andtenvaluesofin-planeinitialphaseangleφ1inEq.(17)aretakenin[0,2π]uniformly.TheterminalconditionofthesetrajectoriesinthemanifoldsisthesameasthatinFig.1

Figure1showsthenegativebranchoftheunstablemanifoldwithamplitudesα1=−1×

=0,α3=0.02andα4=0.02,andFig.2showsthepositivebranchoftheunstable

manifoldwithamplitudesα1=1×10−5,α2=0,α3=0.02andα4=0.02.ThetrajectoriesintheinvariantmanifoldspresentedinFigs.1and2areterminatedwhentheEuclideannormofthepositiondeviationbetweentheanalyticalandnumericallyintegratedorbitsreachesthegiventhresholdortolerance1×10−5.Thisterminalconditionisappliedtothecomputationoftheinvariantmanifoldsassociatedwithhaloorbitinthefollowingtests.Thesymmetriesoftheequationsofmotionindicatethatthestableandunstablemanifoldsaresymmetricwithrespecttothex–zplane,andbothbranchesofstablemanifoldscanbeobtainedeasilybyusingthissymmetricproperty.AsstatedinSect.3,Eq.(17)canalsodescribethetransit(α1·α2<0)ornon-transit(α1·α2>0)trajectoriesaroundthecollinearlibrationpoints.Figure3presentsthesetsoftransitandnon-transittrajectory.Forhalocase,theout-of-planeamplitudeofthehaloorbitistakenasα4=0.05,andthein-planeamplitudeiscomputedtobeα3=0.140151260237441inordertosatisfyΔ(e,α1,α2,α3,α4)=0.Figures4and5showthenegativeandpositivebranchesoftheunstablemanifoldsassociatedwithhaloorbitinERTBP,respectively.Thetransitandnon-transittrajectoriesassociatedwithhaloorbitinERTBPcanbeseeninFig.6.

Besidesdescribingthehyperbolicmanifolds,Eqs.(17)and(53)canbereducedtodescribethedynamicsinthecentermanifoldsaroundthecollinearlibrationpointsinERTBP.For10−5,α2

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372

x 102.5211.5−3H.Leietal.

21.5x 10−3y [adim]10.50−0.5−1−1.50.9880.90.990.9910.9920.993Ly [adim]0.50−0.5−1L11−1.5−20.98750.9880.98850.90.950.990.9905x [adim]x 102−4x [adim]2x 10−411z [adim]z [adim]00−1−1−20.9870.9880.90.990.9910.9920.993−20.98750.9880.98850.90.950.990.9905x [adim]x 102−4x [adim]x 102−411z [adim]0z [adim]−505101520−4x 100−1−1−2−2−2−1.5−1−0.500.511.5x 10−32y [adim]y [adim]x 1021−4z [adim]x 1020−215−4L1z [adim]L10−1−2210.9930.9920.991100.99500.9−50.988x 10−4y [adim]x [adim]x 10−30−10.990.9−20.988y [adim]x [adim]Fig.3Projectionsonthecoordinateplaneand3Drepresentationofthesetsoftransittrajectory(leftpanel),andnon-transittrajectory(rightpanel)associatedwithLissajousorbitaroundtheL1pointintheSun–EarthERTBP.Theinitialphasecorrespondingtotheout-of-planecomponentistakenasφ2=0,andtenvaluesofthephaseanglecorrespondingtothein-planemotionφ1inEq.(17)aretakenin[0,2π]uniformly

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High-ordersolutionsofinvariantmanifoldsinERTBP

x 106−3373

x 1054−4432y [adim]2z [adim]L110−1−2−3−40−2−40.9860.9870.9880.90.990.9910.992−50.9860.9870.9880.90.990.9910.992x [adim]6x 10−4x [adim]4x 10−4z [adim]z [adim]220−2−4L10−2−4x 10−6−6−4−20246−3−3500.9860.9880.990.9928y [adim]x 10y [adim]−50.984x [adim]Fig.4Projectionsonthecoordinateplaneand3Dviewoftheanalyticalsolutions,truncatedatorder(5,5,9),correspondingtothenegativebranchoftheunstablemanifoldassociatedwithhaloorbitaroundthecollinearlibrationpointL1intheSun–Earthellipticrestrictedthree-bodysystem.Theamplitudesoftheinvariantmanifoldareα1=−1×10−5,α2=0andα4=0.05,thenthein-planeamplitudeisα3=0.140151260237441suchthatΔ(e,α1,α2,α3,α4)=0.TenvaluesoftheinitialphaseangleφinEq.(53)aretakenin[0,2π]uniformly.TheterminalconditionofthetrajectoryinthemanifoldisthesameasthatinFig.1

example,(i)theLissajousorbit,correspondingtoα1=α2=0,α3=0andα4=0inEq.(17);(ii)theplaneLyapunovorbit,correspondingtoα1=α2=α4=0andα3=0inEq.(17);(iii)theverticalLyapunovorbit,correspondingtoα1=α2=α3=0andα4=0inEq.(17);and(iv)thehaloorbit,correspondingtoα1=α2=0,α3=0andα4=0inEq.(53).HouandLiu(2011a)directlyexpandedtheLissajousandhaloorbitsaroundthecollinearlibrationpointsinERTBPaspowerseriesoforbitaleccentricitye,in-planeamplitudeα3andout-of-planeamplitudeα4,andconstructedthehigh-ordersolutionsuptoarbitraryorderbymeansoftheL–Pmethod.Inordertocomparewiththeseriesexpansionsconstructedinthispaper,thealgebraicmanipulatorforcomputingtheanalyticalsolutionsofLissajousandhaloorbitsinERTBPdiscussedinHouandLiu(2011a)ismodifiedintermsofthefollowingaspects:(a)theinstantaneousdistancebetweenthecollinearlibrationpointanditsclosestprimaryistakenaslengthunit;(b)theequationsofmotionrepresentedbyEq.(52)aretakentoconstructthehigh-ordersolutionsofhaloorbit,whichisdifferentfromtheformadoptedbyHouandLiu(2011a);and(c)themainconstructionprocessofanalyticalsolutionsismodified.

TocomputethetrajectoriesinthecentermanifoldsinERTBP,thehigh-ordersolutionsareconstructeduptoorder(7,0,9),wheretheorderofthehyperbolicmanifoldsiszero.ThefirstthreegraphsofFigs.7and9presenttheprojectionsonthecoordinateplaneofLissajousandhaloorbits,respectively,andFig.8showsaplaneandverticalLyapunovorbit

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x 10−354321x 10−4y [adim]0−1−2−3−4−5−60.9880.90.990.9910.9920.9930.9940.995z [adim]0−1−2−3−4−50.9880.90.990.9910.9920.9930.9940.995x [adim]6x 10−4x [adim]4x 105−42z [adim]z [adim]00−2L−551−4x 10−6−6−4−2024x 106−3−300.9940.992−50.990.988y [adim]y [adim]x [adim]Fig.5Projectionsonthecoordinateplaneand3Dviewoftheanalyticalsolutions,truncatedatorder(5,5,9),correspondingtothepositivebranchoftheunstablemanifoldassociatedwithhaloorbitaroundthecollinearlibrationpointL1intheSun–Earthellipticrestrictedthree-bodysystem.Theamplitudesoftheinvariantmanifoldareα1=1×10−5,α2=0andα4=0.05.TenvaluesoftheinitialphaseangleφinEq.(53)aretakenin[0,2π]uniformly.TheterminalconditionofthetrajectoryinthemanifoldisthesameasthatinFig.1

inERTBP.TheLissajousandhaloorbitswiththesameamplitudesarealsocomputedbythemodifiedprocedurediscussedinHouandLiu(2011a),andFigs.7(bottom-rightgraph)and9(bottom-rightgraph)showtheEuclideannormofthepositiondeviationbetweentheorbitsobtainedbythetwoanalyticalapproaches,includingtheseriesexpansionsconstructedinthispaper,andtheseriesexpansionsdiscussedinHouandLiu(2011a),uptothesameorderwithin40πunitsofdimensionlesstime.ThetimehistoriesofthepositiondeviationindicatethattheseriesexpansionsrepresentedbyEqs.(17)and(53)canbeexactlyreducedtotheseriesexpansionsdiscussedinHouandLiu(2011a).

Asisknown,whentheorbitaleccentricityoftheprimaryiszero,ERTBPisreducedtothecaseofCRTBP.ThustheanalyticalsolutionsofinvariantmanifoldassociatedwithlibrationpointorbitsinERTBPcanalsodescribethedynamicsaroundthecollinearlibrationpointsinCRTBP.Thatis,whentakinge=0,Eq.(17)candescribeLissajousorbitanditsinvariantmanifolds,andEq.(53)candescribehaloorbitanditsinvariantmanifoldsinCRTBP.InJorbaandMasdemont(1999),theLissajousandhaloorbitsaroundthecollinearlibrationpointsinCRTBParedirectlyexpandedaspowerseriesofin-planeamplitudeα3andout-of-planeamplitudeα4.InMasdemont(2005),theinvariantmanifoldsassociatedwithLissajousandhaloorbitsinCRTBParedirectlyexpressedaspowerseriesofhyperbolicamplitudesα1(unstableamplitude)andα2(stableamplitude),andcenteramplitudesα3(in-planeamplitude)andα4(out-of-planeamplitude).Inordertocomparewiththeseriesexpansionsconstructedinthispaper,thealgebraicmanipulatorforcomputingtheanalyticalsolutionsofcenterand

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20.004x 10−3375

0.010.0080.006L1y [adim]−2−4−6−8−100.985L1y [adim]00.0020−0.002−0.004−0.006−0.0080.990.995−0.010.9840.9860.9880.990.9920.994x [adim]2x 10−4x [adim]x 105432−4z [adim]z [adim]0.9860.9880.990.9920.9940.9960−2−4−6−80.98410−1−2−3−4−50.9840.9850.9860.9870.9880.90.990.9910.9920.993x [adim]2x 10−4x [adim]2x 10−4z [adim]0−2−4−6−8−12z [adim]−10−8−6−4−2024x 106−30−2−4−6−8−0.015−0.01−0.00500.0050.010.015y [adim]y [adim]x 105−4x 10−4z [adim]0z [adim]20−2−4L1−550x 10−3L10.010.9950.990.0050−0.0050.9860.9880.990.992−5y [adim]−100.985x [adim]y [adim]−0.010.984x [adim]Fig.6Setsoftransittrajectories(leftpanel)andnon-transittrajectories(rightpanel)associatedwithhaloorbitaroundthecollinearlibrationpointL1intheSun–Earthellipticrestrictedthree-bodysystem,computedbyseriesexpansionstruncatedatorder(5,5,9).TenvaluesoftheinitialphaseangleφinEq.(53)aretakenin[0,2π]uniformly

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43210−1−2−3−40.980.980.990.990.99x 10−48x 10y [adim]0−2−4−6L1−80.980.980.990.990.99z [adim]20.990.99010.99010.99020.99020.990.99010.99010.99020.9902x [adim]4320.4x 10−4x [adim]10.80.6x 10−8Deviation−6−4−20246x 108−4z [adim]10−1−20.20−0.2−0.4−0.6−3−4−8−0.8−1020406080100120y [adim]Time [adim]Fig.7ProjectionsonthecoordinateplaneofaLissajousorbit(thefirstthreegraphs)aroundthecollinearlibrationpointL1intheSun–Earthellipticrestrictedthree-bodysystem,computedbyseriesexpansionsuptoorder(7,0,9),andtheEuclideannormofthepositiondeviation(bottom-rightgraph)betweentheLissajousorbitsobtainedbytheseriesexpansionsconstructedinthispaperandtheseriesexpansionsconstructeduptoorder(7,9)discussedinHouandLiu(2011a)during40πunitsofdimensionlesstime.TheamplitudesoftheLissajousorbitareα3=0.02andα4=0.03

82x 10−4α = 0.023x 104−4α4 = 0.03y [adim]z [adim]20−2−4−6L10−2−410.5x 10−6L10−0.5−80.980.980.990.990.990.990.99010.99010.99020.9902y [adim]−10.990.990.990.990.990.99x [adim]x [adim]Fig.8Plane(leftpanel)andvertical(rightpanel)LyapunovorbitsaroundthecollinearlibrationpointL1intheSun–Earthellipticrestrictedthree-bodyproblem.TheamplitudesoftheplaneLyapunovorbitareα3=0.02andα4=0,andtheonesoftheverticalLyapunovorbitareα3=0andα4=0.03

invariantmanifoldsinCRTBParedevelopedaccordingtothediscussionpresentedinJorbaandMasdemont(1999)andMasdemont(2005),respectively.

Thehigh-orderanalyticalsolutionsofinvariantmanifoldsassociatedwithLissajousandhaloorbitsinCRTBPareconstructeduptoorder(0,7,9)and(0,9,12),respectively.Thefirst

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−3377

6x 10−45432x 1042y [adim]10−1−2−3−4−50.98850.90.950.990.99050.9910.99150.992L1z [adim]0−2−4−60.98850.90.950.990.99050.9910.99150.992x [adim]6x 10−4x [adim]10.8x 10−840.60.420Deviation05z [adim]0.20−0.2−0.4−0.6−0.8−2−4−6−5−1y [adim]x 10−3024681012Time [adim]Fig.9Projectionsonthecoordinateplaneofahaloorbit(thefirstthreegraphs)withα4=0.05aroundthecollinearlibrationpointL1intheSun–EarthERTBP,computedbyseriesexpansionsuptoorder(9,0,12),andtheEuclideannormofthepositiondeviation(bottom-rightgraph)betweenthehaloorbitsobtainedbyseriesexpansionsconstructedinthispaperandseriesexpansions,uptoorder(9,12),discussedinHouandLiu(2011a),during4πunitsofdimensionlesstime

threegraphsofFigs.10and11showtheprojectionsonthecoordinateplaneofthepositivebranchesoftheunstablemanifoldsassociatedwithLissajousandhaloorbits,respectively.Atrajectoryspecifiedbyφ1=1.6πandφ2=0inthemanifoldsisrepresentedbyaredsolidline.Atthesametime,themanifoldswiththesameamplitudesarecomputedbythealgebraicmanipulatordevelopedaccordingtoMasdemont(2005).Forthetrajectoryspecifiedbyφ1=1.6πandφ2=0,theEuclideannormofthepositiondeviationbetweentheanalyticalresultsobtainedbythetwoanalyticalapproachescanbeseeninFig.10(bottom-rightgraph)correspondingtotheLissajouscaseandFig.11(bottom-rightgraph)correspondingtothehalocase.Thetimehistoriesofthedeviationindicatethatthereducedcase(correspondingtoe=0)oftheseriesexpansionsconstructedinthispaperisequivalenttotheseriesexpansionsconstructedinMasdemont(2005).

Figures12(leftpanel)and13(leftpanel)presentaLissajousorbitwithα3=0.05andα4=0.05,andahaloorbitwithα4=0.05obtainedbyseriesexpansionstruncatedatorder(0,0,15).Figures12(rightpanel)and13(rightpanel)showtheEuclideannormofpositiondeviationbetweentheorbitcomputedbyseriesexpansionsconstructedinthispaperandtheonecomputedbythealgebraicmanipulatordevelopedaccordingtoJorbaandMasdemont(1999),andindicatethattheseriesexpansionspresentedinthispapercanbefurtherreducedtotheonesdiscussedinJorbaandMasdemont(1999).

Weutilizethemethodofnumericalintegrationwiththeinitialstatesprovidedbyseriesexpansionstoinvestigatethedomainofconvergenceofthehigh-ordersolutionsconstructed

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4320x 10−40.5y [adim]−0.5z [adim]L110−1−2−3−40.95−1−1.5−20.950.990.99050.9910.99150.9920.990.99050.9910.99150.992x [adim]432x 10−4x [adim]10.80.60.4x 10−8Deviationz [adim]10−1−2−3−4−20.20−0.2−0.4−0.6−0.8−1−1.5−1−0.500.5x 101−300.511.522.533.5y [adim]Time [adim]Fig.10ProjectionsonthecoordinateplaneoftheanalyticalsolutionscorrespondingtothepositivebranchoftheunstablemanifoldassociatedwithLissajousorbitaroundL1pointintheSun–EarthCRTBP(thefirstthreegraphs),andtheEuclideannormofthepositiondeviation(bottom-rightgraph)foraparticulartrajectoryrepresentedbytheredsolidlineobtainedbytwoanalyticalapproaches,includingtheseriesexpansionsuptoorder(0,7,9)constructedinthispaperandseriesexpansionsuptoorder(7,9)discussedinMasdemont(2005),during3.8unitsofdimensionlesstime.Theamplitudesoftheinvariantmanifoldareα1=1×10−5,α2=0,α3=0.03andα4=0.03

inthispaper.ThenumericaltoolusedforintegratingtheequationsofmotionisaRunge–KuttaFehlbergseventh-orderintegratorwitheighth-orderautomaticstep-sizecontrol,calledRKF78forshort(Fehlberg1968),andthetoleranceoferroristakenas10−14.ForLissajouscase,thetrajectoryspecifiedbyφ1=0andφ2=0inthepositivebranchoftheunstablemanifoldcorrespondingtoα1=1×10−5andα2=0isselectedtotestthedomainofconvergenceofseriesexpansionstruncatedatdifferentordersintermsofthepairofin-planeandout-of-planeamplitudes(α3,α4).Inthefollowingcomputations,themaximumvaluesofα3andα4aretakenas0.1and0.3,respectively,and100×100meshpointsof(α3,α4)distributedin[0,0.1]×[0,0.3]areevaluated.Otherchoicescanbecarriedoutinasimilarway.Forgivenparameters(e,α1,α2,α3,α4,φ1,φ2),theinitialconditioncanbecomputedbytheseriesexpansionsconstructed.Then,theinitialconditionisnumericallyintegratedbyintegratorRKF78,andthenumericalintegrationisterminatedattimeπ,whichisabouttheperiodoflibrationpointorbit.Next,thestateofthenumericalorbitattimeπiscomparedagainstthestateobtainedbyseriesexpansions,andtheEuclideannormofthepositiondeviationbetweennumericalandanalyticalorbitsattimeπiscomputed.Finally,thebase10logarithmoftheEuclideannormofthepositiondeviationisconsideredastheindexofaccuracy,thatis,theindexnmeansthedeviation10n.Figure14showsthedomainofconvergenceofthehigh-orderanalyticalsolutionstruncatedatdifferentorders,suchas

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54324x 10−3379

6x 10−42L1y [adim]0−1−2−3−4−50.9880.90.99z [adim]0.9920.9930.9940.99510−2−40.991−60.9880.90.990.9910.9920.9930.9940.995x [adim]6x 10−4x [adim]10.8x 10−840.60.4Deviation2z [adim]0.20−0.2−0.4−0.6−0.80−2−4−6−5−105x 10−300.511.522.533.54y [adim]Time [adim]Fig.11Projectionsonthecoordinateplaneoftheanalyticalsolutionscorrespondingtothepositivebranchoftheunstablemanifold(thefirstthreegraphs)associatedwithhaloorbitaroundL1intheSun–Earthcircularrestrictedthree-bodyproblem,andforaparticulartrajectoryrepresentedbytheredsolidline,theEuclideannormofthepositiondeviation(bottom-rightgraph)betweentheorbitsobtainedbytheseriesexpansionstruncatedatorder(0,9,12),constructedinthispaper,andtheseriesexpansionsuptoorder(9,12)discussedinMasdemont(2005).Theamplitudesofthemanifoldareα1=1×10−5,α2=0andα4=0.05,thenα3=0.1401567247837suchthatΔ(e=0,α1,α2=0,α3,α4)=0

1−3x 10−80.8e = α1 = α2 = 0α3 = α4 = 0.050.60.4x 1010.5Deviation10−1L10.99−20.950.9910.9905z [adim]0.20−0.2−0.40−0.5−12x 10−3−0.6−0.8−1020406080100120y [adim]x [adim]Time [adim]Fig.12ALissajousorbitwithα3=0.05andα4=0.05(leftpanel)aroundthecollinearlibrationpointL1intheSun–EarthCRTBPcomputedbyseriesexpansionstruncatedatorder(0,0,15),andtheEuclideannormofthepositiondeviation(rightpanel)betweentheLissajousorbitsobtainedbyseriesexpansionsconstructedinthispaperandseriesexpansionsuptoorder15discussedinJorbaandMasdemont(1999)during40πunitsofdimensionlesstime

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x 10−80.80.60.4x 10Deviation0.90.990.9910.9920.993z [adim]5e = α = α = 012α = 0.1401568134329013α = 0.0504L1−50.20−0.2−0.4−105x 10−3−0.60−0.8−1024681012y [adim]−50.988x [adim]Time [adim]Fig.13Ahaloorbit(leftpanel)aroundL1intheSun–EarthCRTBPobtainedbyseriesexpansionstruncatedatorder(0,0,15),andtheEuclideannormofthepositiondeviation(rightpanel)betweenhaloorbitscomputedbyseriesexpansionsconstructedinthispaperandseriesexpansionstruncatedatorder15discussedinJorbaandMasdemont(1999)during4πunitsofdimensionlesstime.Theout-of-planeamplitudeistakenasα4=0.05andthein-planeamplitudeα3iscomputedaccordingtoΔ(e=0,α1=0,α2=0,α3,α4)=0

(4,4,5),(4,3,6),(5,3,5),(4,2,7).Theresultsindicatethat,withthesamethreshold,theregionofconvergenceshowninFig.14(bottom-rightgraph)isthelargestsinceitsordercorrespondingtothecentermanifoldsisthelargest.

Forhigh-ordersolutionsofinvariantmanifoldassociatedwithhaloorbit,asimilartestiscarriedout.Thetrajectoryspecifiedbyφ=0inthepositivebranchofunstablemanifoldcorrespondingtoα1=1×10−5andα2=0ischosentotestthedomainofconvergence.DuetoΔ(e,α1,α2,α3,α4)=0,amplitudesα3andα4arenolongerindependent.Inourcomputation,α4istakenasanindependentvariable,andthemaximumvalueofα4is0.25.Ahundredvaluesofα4aretakenin[0,0.25]uniformly.Theinitialconditioncorrespondingto(e,α1,α2,α3,α4,φ)canbecomputedbytheseriesexpansionsdirectly.Then,theequationsofmotionareintegrateduntilthetimereachesπunitsofdimensionlesstime.TheEuclideannormofthepositiondeviationbetweenanalyticalandnumericalorbitsattimeπcanbeobtained.Thedomainofconvergenceofseriesexpansionsuptoorder(5,5,9)and(5,3,12)correspondingtoinvariantmanifoldsassociatedwithhaloorbitinERTBPisconsidered.Figures15(leftpanel)and16(leftpanel)showtherelationshipbetweenα3andα4,andindicatethatthein-planeamplitudeα3hasaminimumnon-zerovalue.Figures15(rightpanel)and16(rightpanel)presentthedeviationforpositionasafunctionofout-of-planeamplitudeα4.ThesolidlinesontherightpanelsofFigs.15and16denotethetolerance1×10−5.Themaximumallowedvaluesofα4forseriesexpansionstruncatedatorder(5,5,9)and(5,3,12)are0.137and0.160adimintermsofthethreshold1×10−5,respectively.Thedomainofconvergencecorrespondingtootherorderscanbeexploredinasimilarmanner.

6Conclusionsanddiscussion

Thehigh-ordersolutionsofinvariantmanifolds,associatedwithLissajousandhaloorbits,aroundthecollinearlibrationpointsintheellipticrestrictedthree-bodyproblem,arecon-structedinthispaper.Thegeneralsolutionsoftheinvariantmanifoldsareexpandedaspowerseriesoffiveparameters,includingtheorbitaleccentricityofthesecondarye,twoamplitudescorrespondingtothehyperbolicmanifoldsα1(unstable)andα2(stable),andtwoamplitudescorrespondingtothecentermanifoldsα3(in-plane)andα4(out-of-plane).Forthe

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High-ordersolutionsofinvariantmanifoldsinERTBP0.20.15−3−4381−4−3−4−5−4−4−3−3−5−6−6−7−8−9α (adim)−5−50.25−6−4−5−5α4 (adim)−4−50.20.150.10.0500−8−50.1−67−−6−7−8−540.05−8−9−6−6−7−7−6−4−8−70−500.020.040.06−10−6−5−90.1−100.05−7α3 (adim)−6α (adim)3−40.20.15−3−4−5−6−7−8−9α4 (adim)α (adim)4−4−5−6−7−50.20.150.10.050−6−5−5−4−6−3−4−3−4−3−3−40.10.050−6−4−5−60Fig.14Practicalconvergenceofthehigh-orderanalyticalsolutionsofinvariantmanifolds,truncatedatdifferentorders,associatedwithLissajousorbits,aroundL1pointintheSun–Earthellipticrestrictedthree-bodysystem.Theindexshowninthecolorbarrepresentsthebase10logarithmoftheEuclideannormofthepositiondeviationbetweentheanalyticalorbitandnumericallyintegratedorbitatπunitsofdimensionlesstime.Inordertodisplay,thedeviationistakenas1×10−10whenitislessthan1×10−10,andthedeviationistakenas1×10−3whenitisgreaterthan1×10−3.Leftpanel:theanalyticalsolutionsaretruncatedatorder(4,4,5)(topgraph)and(5,3,5)(bottomgraph).Rightpanel:theanalyticalsolutionsaretruncatedatorder(4,3,6)(topgraph)and(4,2,7)(bottomgraph)

−5−50.25−4−5−6−7−5−6−7−7−6−3−8−7−7−8−90.1−10−50.020.040.06−10−7−6−5−400.05α3 (adim)α3 (adim)0.25Truncated at order (5, 5, 9)1.81.6x 10−5Truncated at order (5, 5, 9)0.21.4Error [adim]0.140.1450.150.1550.160.165α4 [adim]0.151.210.80.60.10.050.400.1350.200.050.10.150.20.25α [adim]3α [adim]4Fig.15Therelationshipbetweenα3andα4(leftpanel),andtheEuclideannormofthedeviationforpositionbetweenanalyticalorbitandcorrespondingnumericallyintegratedorbitatπunitsofdimensionlesstime(rightpanel).Theanalyticalsolutionisconstructeduptoorder(5,5,9).Theredsolidlinedenotesthethreshold1×10−5

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10.90.80.70.15x 10−4Truncated at order (5, 3, 12)0.2Error [adim]0.140.1450.150.1550.160.165α [adim]0.60.50.40.340.10.050.20.100.135000.050.10.150.20.25α3 [adim]α4 [adim]Fig.16Therelationshipbetweenα3andα4(leftpanel),andtheEuclideannormofthedeviationforpositionbetweenanalyticalorbitandcorrespondingnumericallyintegratedorbitatπunitsofdimensionlesstime(rightpanel).Theanalyticalsolutionisconstructeduptoorder(5,3,12).Theredsolidlinealsodenotesthethreshold1×10−5

analyticalsolutionsofinvariantmanifoldsassociatedwithhaloorbits,thesefiveparametersarenotindependentandshouldsatisfyΔ(e,α1,α2,α3,α4)=0.TheadvantageoftheseriesexpansionsproposedandconstructedinthispaperliesinthatitcanprovideaparametricdescriptiontothegeneraldynamicsaroundthecollinearlibrationpointsofERTBP.Thatis,thegeneralsolutionswithe=0candescribethefollowingdynamicsaroundthecollinearlibrationpointsinERTBP:(i)thestablemanifolds,correspondingtoαi=0,i=2,3,4,andα1=0;(ii)theunstablemanifolds,correspondingtoαi=0,i=1,3,4,andα2=0;(iii)thetransittrajectories,correspondingtoαi=0,i=1,...,4,andα1·α2<0;(iv)thenon-transittrajectories,correspondingtoαi=0,i=1,...,4,andα1·α2>0;(v)theLissajousandhaloorbits,correspondingtoα1=α2=0,α3=0,andα4=0;and(vi)theplaneandverticalLyapunovorbits(thespecialcaseofLissajousorbits).Furthermore,thecenterandhyperbolicmanifoldsaroundthecollinearlibrationpointsinCRTBPwhichisthespecialcaseofERTBPcanalsobedescribedbythegeneralsolutionswithe=0.Forthereducedcases,suchasthecentermanifoldsinERTBP,thehyperbolicandcentermanifoldsinCRTBP,thealgebraicmanipulatorforcomputingtheanalyticalsolutionsaredevelopedfollowingthedescriptionpresentedinHouandLiu(2011a),Masdemont(2005)andJorbaandMasdemont(1999)inordertocomparewiththeseriesexpansionsconstructedinthispaper.Theresultsindicatethattheseriesexpansionsconstructedinthispapercanbeexactlyreducedtotheseriesexpansionsdiscussedintheexistingliterature.Inordertocheckthevalidityoftheseriesexpansionsconstructed,theinitialstatesobtainedbyseriesexpansionsarenumericallyintegratedbyRKF78,andtheanalyticalresultsarecomparedagainstthenumericalresultstoinvestigatethedomainofconvergenceofseriesexpansionsuptodifferentorders.

Duetothelargestorageofthecoefficientsofcoordinateandfrequency,theseriesexpan-sionsoftheinvariantmanifoldsassociatedwithcollinearlibrationpointorbitscanbecon-structeduptoalimitedorder.TheapplicationsofanalyticalsolutionsofinvariantmanifoldsinERTBPwillbediscussedinourfuturework.

AcknowledgmentsThisworkwascarriedoutwithfinancialsupportfromtheNationalBasicResearchProgram973ofChina(2013CB834103),theNationalHighTechnologyResearchandDevelopmentProgram863ofChina(2012AA121602),theNationalNaturalScienceFoundationofChina(GrantNo.11078001)andtheResearchandInnovationProjectforCollegeGraduatesofJiangsuProvince(GrantNo.CXZZ13_0042).

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Theauthorsaremuchobligedtothereviewersfortheirmanyinsightfulcommentsthatsubstantiallyimprovedthequalityofthispaper.

References

Alessi,E.M.,Gómez,G.,Masdemont,J.J.:LeavingtheMoonbymeansofinvariantmanifoldsoflibrationpointorbits.Commun.NonlinearSci.Numer.Simul.14,4153–4167(2009)

Alessi,E.M.,Gómez,G.,Masdemont,J.J.:Two-maneuvrestransfersbetweenLEOsandLissajousorbitsintheEarth-Moonsystem.Adv.SpaceRes.45,1276–1291(2010)

Belbruno,E.:CaptureDynamicsandChaoticMotionsinCelestialMechanics:WithApplicationstotheConstructionofLowEnergyTransfers.PrincetonUniversityPress,Princeton(2004)

Canalias,E.,Gómez,G.,Marcote,M.,etal.:Assessmentofmissiondesignincludingutilizationoflibrationpointsandweakstabilityboundaries.Technicalreport18142/04/NL/MV(2004)

Chung,M.J.:Trans-LunarCruisetrajectorydesignofGRAIL(GravityRecoveryandInteriorLaboratory)mis-sion.In:AIAA/AASAstrodynamicalSpecialistConference,Toroto,OntarioCanada,2–5August(2010)Dellnitz,M.,Junge,O.,Post,M.,etal.:OntargetforVenus-setorientedcomputationofenergyefficientlowthrusttrajectories.Celest.Mech.Dyn.Astron.95,357–370(2006)

Farquhar,R.W.:Thecontrolanduseoflibrationpointsatellites.TechnicalreportTRR346,StanfordUniversityreportSUDAAR-350(1968)

Farquhar,R.W.:Theutilizationofhaloorbitsinadvancedlunaroperations.TechnicalreportNASATND-6365(1971)

Farquhar,R.W.,Kamel,A.A.:Quasi-periodicorbitsaboutthetranslunarlibrationpoint.Celest.Mech.7,458–473(1973)

Fehlberg,E.:Classicalfifth-,sixth-,seventh-,andeighth-orderRunge-Kuttaformulaswithstepsizecontrol.TechnicalreportNASATRR-287(1968)

Folta,D.C.,Woodard,M.,Howell,K.C.,etal.:Applicationsofmulti-bodydynamicalenvironments:theARTEMIStransfertrajectorydesign.ActaAstronaut.73,237–249(2012)

Gómez,G.,Marcote,M.:High-orderanalyticalsolutionsofHill’sequations.Celest.Mech.Dyn.Astron.94,197–211(2006)

Hou,X.Y.,Liu,L.:Onquasi-periodicmotionsaroundthetriangularlibrationpointsoftherealEarth-Moonsystem.Celest.Mech.Dyn.Astron.108,301–313(2010)

Hou,X.Y.,Liu,L.:Onmotionsaroundthecollinearlibrationpointsintheellipticrestrictedthree-bodyproblem.Mon.Not.R.Astron.Soc.415,3552–3560(2011a)

Hou,X.Y.,Liu,L.:Onquasi-periodicmotionsaroundthecollinearlibrationpointsintherealEarth-Moonsystem.Celest.Mech.Dyn.Astron.110,71–98(2011b)

Howell,K.C.,Kakoi,M.:TransfersbetweentheEarth-MoonandSun-Earthsystemsusingmanifoldsandtransitorbits.ActaAstronaut.59,367–380(2006)

Jorba,À.,Masdemont,J.:Dynamicsinthecentermanifoldofthecollinearpointsoftherestrictedthreebodyproblem.PhysicaD132,1–213(1999)

Koon,W.S.,Lo,M.W.,Marsden,J.E.,etal.:LowenergytransfertotheMoon.Celest.Mech.Dyn.Astron.81,63–73(2001)

Lei,H.L.,Xu,B.,Sun,Y.S.:Earth-Moonlowenergytrajectoryoptimizationintherealsystem.Adv.SpaceRes.51,917–929(2013)

Lei,H.L.,Xu,B.:High-orderanalyticalsolutionsaroundtriangularlibrationpointsincircularrestrictedthree-bodyproblem.Mon.Not.R.Astron.Soc.434,1376–1386(2013a)

Lei,H.L.,Xu,B.:LowthrusttransfertothelibrationpointorbitsofSunMarssystemfromtheEarth.J.Astronaut.34,763–772(2013b)

Masdemont,J.J.:High-orderexpansionsofinvariantmanifoldsoflibrationpointorbitswithapplicationtomissiondesign.Dyn.Syst.20,59–113(2005)

Mingotti,G.,Topputo,F.,Bernelli-Zazzera,F.:Low-energy,low-thrusttransferstotheMoon.Celest.Mech.Dyn.Astron.105,61–74(2009)

Mingotti,G.,Topputo,F.,Bernelli-Zazzera,F.:Earth-Marstransferswithballisticescapeandlow-thrustcapture.Celest.Mech.Dyn.Astron.110,169–188(2011)

Pergola,P.,Geurts,K.,Casaregola,C.,etal.:Earth-Marshalotohalolowthrustmanifoldtransfers.Celest.Mech.Dyn.Astron.105,19–32(2009)

Ren,Y.,Masdemont,J.J.,Marcote,M.,etal.:ComputationofanalyticalsolutionsoftherelativemotionaboutaKeplerianellipticorbit.ActaAstronaut.81,186–199(2012)

123

384H.Leietal.

Richardson,D.L.:Analyticconstructionofperiodicorbitsaboutthecollinearpoints.Celest.Mech.22,241–253(1980)

Sweetser,T.H.,Broschart,S.B.,Angelopoulos,V.,etal.:Artemismissiondesign.SpaceSci.Rev.165,27–57(2011)

Szebehely,V.:TheoryofOrbits.AcademicPress,NewYork(1967)

Xia,Z.H.:Arnolddiffusionintheellipticrestrictedthree-bodyproblem.J.Dyn.Differ.Equ.5,219–240(1993)

123