THEDYNAMICSOF NATURALSATELLITES OFTHEPLANETS
NIKOLAYEMELYANOV
SternbergStateAstronomicalInstitute
LomonosovMoscowStateUniversity Moscow,Russia
Elsevier
Radarweg29,POBox211,1000AEAmsterdam,Netherlands
TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates
Copyright©2021ElsevierInc.Allrightsreserved.
TheRussianversionofthisbookwaspublishedin2019:EmelyanovN.V.,TheDynamicsofNaturalSatellitesofthePlanets BasedonObservations.Vek-2,Fryazino.576pp.ISBN978-5-85099-199-9.
Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicormechanical,including photocopying,recording,oranyinformationstorageandretrievalsystem,withoutpermissioninwritingfromthepublisher. Detailsonhowtoseekpermission,furtherinformationaboutthePublisher’spermissionspoliciesandourarrangementswith organizationssuchastheCopyrightClearanceCenterandtheCopyrightLicensingAgency,canbefoundatourwebsite: www.elsevier.com/permissions
ThisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythePublisher(otherthanasmaybe notedherein).
Notices
Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperiencebroadenourunderstanding, changesinresearchmethods,professionalpractices,ormedicaltreatmentmaybecomenecessary.
Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluatingandusinganyinformation, methods,compounds,orexperimentsdescribedherein.Inusingsuchinformationormethodstheyshouldbemindfuloftheir ownsafetyandthesafetyofothers,includingpartiesforwhomtheyhaveaprofessionalresponsibility.
Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,assumeanyliabilityforanyinjury and/ordamagetopersonsorpropertyasamatterofproductsliability,negligenceorotherwise,orfromanyuseoroperationof anymethods,products,instructions,orideascontainedinthematerialherein.
LibraryofCongressCataloging-in-PublicationData
AcatalogrecordforthisbookisavailablefromtheLibraryofCongress
BritishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
ISBN:978-0-12-822704-6
ForinformationonallElsevierpublications visitourwebsiteat https://www.elsevier.com/books-and-journals
Publisher: CandiceJanco
AcquisitionsEditor: PeterLlewellyn
EditorialProjectManager: AliceGrant
ProductionProjectManager: SruthiSatheesh
Designer: VictoriaPearson
TypesetbyVTeX
3.2.5Keplerianmotionformulaswithrespecttononsingular elements(Lagrangeelements)..........................51
3.2.6ExamplesofusingLagrangeelements..................54
3.3Forcefunctionofattractionofanon-sphericalplanet.... ........55
3.3.1Forcefunctionexpansion...............................55
3.3.2Attractioninmodelsandforrealbodies .................56
3.4Anapproximateaccountoftheinfluenceofthemainsatellites onthemotionofdistantsatellitesoftheplanet .................60
3.5Variousapproachesandmethodsforconstructingmotion modelsofplanetarysatellites... ..............................61
3.6Amodelofmotionofasatelliteofanoblateplanetbasedonthe solutionofthegeneralizedproblemoftwofixedcenters........63
3.7Constructinganalyticaltheoriesofplanetarysatellitemotion usingperturbationtheorymethods... .........................66
3.7.1Generalschemeofperturbationtheory..................66
3.7.2Circumstancesinthemotionofrealcelestialbodies, allowingtheuseofperturbationtheorymethods. ........69
3.7.3Equationsforelementsoftheintermediateorbit. ........74
3.7.4Solvingequationsforintermediateorbitelements.Small parametermethod.....................................80
3.7.5Solvingequationsforintermediateorbitelements. PoissonMethod.......................................84
3.8Expansionoftheperturbingfunctionwithrespecttothe elementsoftheintermediateorbitofaplanetarysatellite........86
3.9Determinationofperturbationsofelementsoftheplanetary satelliteintermediateorbit.. ..................................91
3.10Constantperturbationofthesemi-majoraxisofthesatellite’s orbit.........................................................97
3.11Precessingellipsemodel.... .................................101
3.12Perturbedmotionatsmalleccentricitiesoftheorbits...........103
3.12.1Problemformulation..................................103
3.12.2Constructingamodelofcircularperturbedmotion......104
3.12.3TransitiontotheelementsoftheKeplerianorbit........106
3.12.4OsculatingKeplerianelementsofthesatellite’sorbitin perturbedmotionwithsmalleccentricities..............108
3.13Constructedanalyticaltheoriesofplanetarysatellitemotion ....114
3.13.1AnalyticaltheoryofthemotionofNeptune’ssatellite Triton................................................114
3.13.2PrecessingEllipsemodelsforcloseJupitersatellites
3.13.3Specialanalyticaltheoriesofthemainsatellitesofmajor planets,takingintoaccountthemutualattractionof satellites..
3.14Influenceoftidesinviscoelasticbodiesofplanetandsatelliteon thesatellite’sorbitalmotion ..................................126
3.14.1Statementoftheproblemofinfluenceoftides.
3.14.2Equationsinrectangularcoordinates..
3.14.3Solvingtheequationsforrectangularcoordinates..
3.14.4TransitiontothedifferentialequationsinKeplerian elements.............................................134
3.14.5Someimportantconclusionsabouttheinfluenceoftidal deformationsonsatellitedynamics...
4.1Theobjectiveofsolvingtheequationsofmotionofcelestial
4.2Generalpropertiesofmethodsforthenumericalintegrationof
4.3Runge–Kuttaintegrationmethodforordinarydifferential
4.4Algorithmforsolvingproblemsofmotionofacelestialbodyby numericalintegrationmethods. ..............................158
4.5Instructionsforthecomputationalprogramforthenumerical integrationofordinarydifferentialequationsbytheEverhart method.....................................................160
4.6Belikovprogramfornumericalintegrationofordinary differentialequations........................................165
4.7Testingandcomparingsomenumericalintegrationprocedures.167
4.8Approximationoftherectangularcoordinatesofplanetsand satellitesbytruncatedChebyshevseries.. ....................167
4.9Overviewofproblemsandmethodsofnumericalintegration. BookbyAvdyushev.........................................170
Chapter5Observationsofplanetarysatellites........
5.1Generalprinciplesofobservations............................173
5.2Determinationoftopocentricpositionsofplanetsandsatellites.174
5.3Planetobservations..........................................175
5.4Observationsofaplanetarysatellite.. ........................177
5.5Observationsoftwosatellitesoftheplanet ....................178
5.6Determinationofangularmeasuredvaluesduringobservations ofplanetarysatellites... .....................................180
5.7Calculationoftheangulardistancebetweensatellitesand positionangle...............................................183
5.8Determinationoftangentialcoordinatesofsatellites ...........184
5.9Determinationofthecoordinatedifferencebetweentwo satellitesoftheplanetinthecaseofphotometricobservations ofmutualeclipsesofsatellites... .............................185
5.10Conclusionregardingmeasuredvaluesduringobservationsof planetarysatellites. .........................................187
5.11Themomentofapparentapproximationofplanetarysatellites asameasurablequantityduringobservations.................188
5.12Meansandtechniquesofground-basedobservationsof planetarysatellites. .........................................190
5.13Sourcesofobservationsfromplanetarysatellites... ...........192
5.14Timescalesandcoordinatesystemsforobservationsof planetarysatellites. .........................................194 References..................................................199
Chapter6Constructionofmodelsforthemotionsofcelestialbodies basedonobservations................... ..............201
6.1Methodofdifferentialrefinementofthemotionparametersof celestialbodiesbasedonobservations.Applicationof least-squaresmethod........................................201
6.2Weakconditionalityandambiguityofsolution.... .............212
6.3Overviewoffilteringalgorithms..............................215
6.4Calculationofmeasuredvaluesandpartialderivativesofthe measuredvaluesbyrefinedparameters.......................217
6.4.1Generalorderofcalculations..........................217
6.4.2Differentialequationsforisochronousderivativesinthe three-bodyproblem.Refinementoftheinitialconditions oftheequationsofmotion............................220
6.4.3Differentialequationsforisochronousderivativesinthe three-bodyproblem.Refinementofthemassofthe perturbingbody... ...................................223
6.4.4Differentialequationsforisochronousderivativesinthe satellitemotionproblemforaoblateplanet .............224
6.4.5Constructionofconditionalequationsforangular measurementsoftopocentriccoordinates. .............227
6.5Assigningweightstoobservationsandconditionalequations..231
6.6Calculationofstatisticalcharacteristicsofresiduals............234
6.7Theproblemofrejectingroughobservations. .................236
occultationsandeclipsesofplanetarysatellites.........239
7.1Descriptionofphenomena...................................239
7.2Methodforobtainingastrometricdata........................242
7.3Asimplifiedmodelofmutualoccultationsandeclipsesof planetarysatellites... .......................................245
7.4Photometricmodelsofmutualoccultationsandeclipsesof planetarysatellites... .......................................248
7.4.1Generalphotometriccharacteristics... .................248
7.4.2Photometricmodelofthemutualoccultationofsatellites249
7.4.3Photometricmodelofsatelliteeclipse. .................251
7.5Thelawsoflightscatteringforplanetarysatellites.. ...........253
7.5.1Lommel–Seeligerlightscatteringlaw...................253
7.5.2Hapke’slightscatteringlawforasmoothsurface........254
7.5.3Hapke’slightscatteringlawforaroughsurface.. .......255
7.5.4Hapke’slawparametersfortheGalileansatellitesof Jupiter...............................................257
7.6Disk-integratedphotometriccharacteristicsofthesatellite......258
7.7Photometricmodelsofmutualoccultationsandeclipsesofthe mainsatellitesofSaturnandUranus.. ........................262
7.7.1Photometricmodelofmutualoccultationsandeclipses ofthemainsatellitesofSaturn. ........................262
7.7.2Photometricmodelofmutualoccultationsandeclipses ofthemainsatellitesofUranus ........................264
7.8Relationfortheaccuracyofastrometricresultsofobservations ofvarioustypes.............................................265
7.9Worldwidecampaignsonobservationsofsatellitesduringtheir mutualoccultationsandeclipses.............................266
7.10Obstaclestoimprovingtheaccuracyofastrometricresults.....267
7.11Periodsofthephenomenainthefuture.......................273 References..................................................274
8.1Factorsdeterminingephemerisaccuracy......................277
8.2Estimationoftheephemerisaccuracyusingobservation-errors variancebytheMonteCarlotechniques.......................279
8.3Estimationofephemerisaccuracybyvaryingthecompositionof observationsusing“bootstrap”-samples.. ....................281
8.4Estimationoftheaccuracyofephemerisbythemethodof motionparametervariation..................................284
8.5Theaccuracyoftheephemerisofthedistantsatellitesofmajor planets.....................................................286
9.3TherotationofNeptuneandtheorbitofTriton................297
9.4TheoryofrotationforPhobos..
9.5RotationoftheGalileansatellitesofJupiter,satellitesofSaturn andPluto...................................................304
9.6Chaoticrotationofplanetarysatellites.RotationofHyperion
10.1Theimpactofvariousfactorsontheevolutionoftheorbitsof planetarysatellites... .......................................315
10.2Theevolutionoftheorbitsofsatellitessubjecttothe predominantinfluenceofplanetoblateness..
10.3Evolutionoftheorbitsoftheplanetarysatellitesunderthe actionofthesolarattraction..................................319
10.3.1Averagingoftheperturbingfunction...................319
10.3.2Aspecialcase—Hillproblem...........................324
10.3.3Analysisoffamiliesofpossiblechangesinthe eccentricityof e andtheargumentofthepericenter ω for atwice-averagedperturbingfunctionintheHillcase ....325
10.3.4Orbitevolutionintimeforadouble-averagedperturbing functionintheHillcase...............................334
10.3.5Applicationsoftheconstructedtheoryoforbitevolution instudyingthedynamicsofrealplanetarysatellites .....341
10.4Refinedmodelsoftheevolutionoftheorbitsofplanetary satellites.Numericalanalyticalmethod.. ......................342
10.5Theevolutionoftheorbitsofplanetarysatellitesunderthe combinedinfluenceofvariousfactors.. ......................346
10.6ClassificationoftheorbitsofthedistantsatellitesofJupiter, Saturn,Uranus,andNeptuneaccordingtothetypesand propertiesoftheorbitevolution..............................350
10.7Theevolutionoftheorbitsandrendez-vousoccurrencesof distantsatellitesoftheplanets. ..............................352
10.7.1Modernknowledgeabouttheevolutionoftheorbitsof distantplanetarymoons.. .............................352
10.7.2Theproblemofcalculatinganddetectingthe rendez-vousoccurrencesofdistantplanetarysatellites..355
10.7.3Ananalyticaldescriptionoftheevolutionofsatellite orbits................................................356
10.7.4Determinationofminimaldistancesbetweentheorbits ofdistantplanetarysatellites.. ........................357
10.7.5ProposedInternetsourceforthestudyoftheevolutionof theorbitsandrendez-vousofdistantplanetarysatellites.358
10.7.6Examplesofcalculatingtheminimumdistancesbetween theorbitsofsatellites .................................359
10.7.7Conclusion...........................................363
10.8RefinementoftheLaplace–Lagrangesecularperturbation
Chapter11Physicalparametersofplanetarysatellites.
11.3DetectionofvolcanoesonthesatelliteofJupiterIousing groundphotometry. .........................................371
11.4Estimatesofthephysicalparametersofdistantplanetary satellites.. ..................................................372
11.4.1Featuresofdistantplanetarysatellites.. ................372
11.4.2Overviewofavailablephotometricdatafordistant planetarysatellites... .................................374
11.4.3Photometricmodelfordistantplanetarysatellites .......377
11.4.4Determinationofphotometricparametersofsatellitesby photometry. .........................................379
11.4.5Initialdataandresultsofdeterminingthephotometric parametersofsatellites.. .............................380
11.4.6Comparisonofresultsobtainedbydifferentauthors.....385
11.4.7Conclusionsontheestimatesofthephotometric parametersofdistantplanetarysatellites ...............388
12.1Variantsandchangeofversionofmotiontheoriesand
12.2Meansofprovidingaccesstodatabases,motionmodelsand
Author’spreface
Atheoryonlymakessensewhenitisnotonlyabstractconstructionsintheimaginationofafascinatedtheoretician,butwellestablishedproceduresthatproperlyservethegoalsofpractical knowledgeofnature.
Thedynamicsofplanetarysatellitesisaveryinterestingarea ofcelestialmechanics.Atfirstglance,thedynamicsofsatellites canbestudiedwithoutobservation.Theresearchercancomeup withasatellitemodelthatisclosetorealityoringeneralabstract one,trynewmethodsonit,demonstratinghishighestskill.There isanotherseductivelineofactivity:theexplanationofwhycelestialbodiesmoveexactlyastheymove.Anewexplanationof factsknowninnatureoranexplanationofpreviouslyunexplained phenomenaseemstobeasignificantscientificachievement.Of course,wethushoneourskills.However,inthesematters,the researchershouldatsomepointstopandaskhimselfthequestion:dowegetnewinformationaboutthenaturearoundus?Of course,ascientificgeneralizationoffactsatsomestagecancreate aqualitativeleapinourideasaboutnature.However,thisisprecededbyatime-consumingandsometimesexhaustingprocess ofaccumulatingfactualknowledge.Inthedynamicsofplanetary satellites,thiswayinevitablyrunsthroughthetechnicalprocessingofinformationfromobservations,throughthecompilationof prohibitivelycumbersomecomputingprogramsandtheimplementationofboringcalculations.Whichresearcherwillgothere? Eitheraresearcherwhounderstandstheharshinevitabilityofthis process,oronewhohashisownspecialscientificandtechnologicalpredilections.Tohelpjustsuchbraveresearchers,thisbook waswritten.
Asinmanyotherbranchesofastronomy,inthedynamicsof planetarysatellites,thecriterionoftruthiscompliancewithobservations.Theoristsknowthatthemoreobservations,theworse itcanbeforatheory.Wecancongratulatesuchtheorists—their theoryhasbeenreplacedbyanewone.Itispreciselyforsuch eventsthisbookisaimed.
Bothinordinarylifeandinscientificresearch,weareoftenin searchofa“helpdesk”.Nowadays,suchabureauis“theglobal virtualmind”—Internet.Instudiesofthedynamicsofplanetary satellites,asinmanyotherscientificstudies,onlysuchdataare
requiredwhichareprovidedwithinformationaboutwhoreceived thisdataandhow,andforwhichtheaccuracyandreliability areclear.Thisbookprovidesreferenceinformationonplanetary satellites,withlinkstoreliablesources.
Scientificworkisoftensuccessfulwhenitislimitedtoacertainframeworkbothintermsofobjectsandresearchmethods.In suchaharmoniousprocess,annoyingsituationssometimesoccur whenitisnecessarytogobeyondfamiliarmethodsorinformation.Insuchsituations,theproposedbookmayhelp.
Finally,thisbookhelpscounteractthethoughtthatthewords “celestialmechanics”soundold-fashionedandthatthisisnota modernfieldofastronomy.Infact,celestialmechanicsisnotlimitedtothethree-bodyproblemandthedeterminationoftheorbit fromthreeobservations.Nowadays,thisisthemostpracticaland modernfieldofastronomy.Itnotonlysatisfiesournaturalcuriosity,butalsoservestosolvetwoperennialproblemsofhumanity:expandingourhabitatandprotectingagainstthedangerous forcesofnature.
Mostofthebookisbasedonthescientificresultsandpublicationsoftheauthor.Forthosesubjectsthattheauthorhimselfwas notdirectlyinvolvedin,thebookprovidesbriefreviewsofpublicationsofotherspecialists.Anextensivebibliographyisgivenfor allsectionsofthebook.Thisisnecessaryforamoredetailedstudy ofmethodsandscientificresults.Thebibliographyitselfisareferencematerialthatisindemandforwidespreaduse.Aseparatelist isgivenforeachchapter.Somelinksmayberepeatedindifferent chapters.
Thescientificworkonthetopicofthisbookwascarriedout bytheauthorpartiallyincollaborationwithcolleaguesfromthe InstitutdeMécaniqueCélesteetdeCalculdesEphémérides(IMCCE),Paris,France.This,ontheonehand,accompaniedthework withgoodexpertise,andontheotherhand,ensuredtherelevance oftheresultsobtainedbytheauthor.
Thepossibilityofwritingthebookwasprimarilyprovided byahighlevelofeducation,whichwasgiventotheauthorby M.V.LomonosovMoscowStateUniversity.Theauthorspenthis entirescientificlifewithintheSternbergAstronomicalInstituteof MoscowStateUniversity,wherethisbookwaswritten.
TheauthorthankstheassociateprofessorG.I.Shirminforthe finaleditingandproofreadingofthebook.
Objectives,currentproblemsand generalapproachtothestudyof thedynamicsofsatellites
1.1Introduction
TheunderstandingthatthevastUniverseextendsinalldirectionshasalwaysworriedmankind.Thiscausesadoubledesire. Firstly,itwouldbenicetounderstandourplaceintheboundless spaceandtheinfinitediversityoftheworld.Peopleoftenexperienceaslightdiscomfortfromthelackofananswertosucha question.Atthesametime,adesirearisestoextractbenefitsfrom theCosmostosatisfyever-increasingneeds.Peopleareevenmore worriedwhentheydiscoverathreattotheirlivesbytheforcesof nature.Nothingscaresussomuchasanincomprehensiblephenomenon.Itissurprisinglyeasytoreassurepeoplebyexplaining terriblephenomena,evenwithnotquitefamiliarwords.Theinformationthatatleastsomeoneunderstandstheprocessesofnaturereturnsustotheusualcomfortofeverydaylife.Thatiswhy weshouldbegratefultothefewpeoplewhoworktosaveusfrom painfulquestionsaboutspaceandfate.
Sinceancienttimes,peoplehavethoughtabouttheinfluence ofcelestialbodiesonterrestriallife.Attemptstocomparecelestialphenomenawiththefateofmanweremadebybothscientists andinvestigativeindividualsnotbeingscientists.However,atall times,averyunreliableresultwasobtainedtimeandagain.Asfor thefateofthecelestialbodiesthemselves,astronomersandmathematicianshavelongcalculatedthesurprisinglystablenatureof theirmovement.Thesizesandshapesoftheorbitsoftheplanets, ortheslopesoftheaxesoftheirrotation,havenotchangedmuch evenatcosmogonictimeintervals.
NaturalscientistsandphilosophershavecometotheconclusionthatthemainreasonfortheexistenceofaCosmicMindinthe Universeisthefunctionofcognition.Ledbyreasonlifeischaracterizedbyadesiretounderstandandexplainwhatishappening asregardsthephenomenon.
AtanystageofcognitionoftheUniverse,wealreadyhavea moreorlessadequatemodelforit.New,moreaccurateobser-
TheDynamicsofNaturalSatellitesofthePlanets
https://doi.org/10.1016/B978-0-12-822704-6.00006-6
Copyright©2021ElsevierInc.Allrightsreserved.
Chapter1 Objectives,currentproblemsandgeneralapproachtothestudyofthedynamicsofsatellites vationsmayleadtoamodelmismatchwithreality.Atmost,the requiredadjustmentofthemodelisrestoredbyclarifyingthe knownparametersofthemotionorthestateofcelestialbodies. Sometimesitisnecessarytosignificantlyimprovetheories,the model-constructingtechnique,orthecalculationmethods.This processisunconsciouslyaimedatdiscoveringnew,unexplained phenomena.Atsomestage,itispossibletogetthismuch-needed “food”fortheMind,butthisisalwaysprecededbythecolossal workofscientists—observers,theorists,andcalculators.Themotionmodelsofcelestialbodiesarealsovaluableinthattheyallow ustopredicttheirlocationatanytimeinthepastorfuture.
Atheoryonlymakessensewhenitisnotonlybyabstractconstructionsintheimaginationofafascinatedtheoretician,butalso oneneedswell-establishedproceduresthatregularlyservethe purposesofpracticalknowledgeofnature.Oneofthemaintools inthisrespectispracticalcelestialmechanics.Itispracticalcelestialmechanicsthatgivesusthemostcompleteandaccurate knowledgeofthedynamicsofplanetarysatellites.
1.2Celestialmechanics—thebasisfor
studyingthedynamicsofplanetary
satellites
Celestialmechanicsisthebranchofsciencethatstudiesthe movementsofcelestialbodiesundertheactionofnaturalforces.
Thesubjectofcelestialmechanicsisthemechanicalformof themotionofmatter.
Theobjectsofresearchareallkindsofmaterialformations, fromthesmallestparticlesofcosmicdusttocolossalsystemssuch asstarclusters,galaxiesandclustersofgalaxies.
Thepurposeofcelestialmechanicsistostudythelawsofnaturethatgovernthemechanicalmovementsofcelestialbodies.
Forallnaturalsciences,celestialmechanicsplaystheroleof afoundation,withoutwhichthestudyoftheUniverseandthe explorationoftheCosmosareunthinkable.ThesignificanceofcelestialmechanicsforlifeonEarthistogainknowledgeaboutthe motionofcelestialbodiesandthenearCosmostobettermeetthe needsofmankindandtoresultinprotectionfromtheforcesof nature.ThetheoryofmotionofartificialsatellitesofEarthallows fortheuseofspacecraftforcommunicationandresearchofterrestrialresources.Thetheoryofmotionofasteroids,cometsand meteorsgivesanassessmentofthedangerofthesebodiesenteringtheatmosphereandfallingtotheEarth’ssurface.Studiesofthe motionsofthebodiesofthesolarsystemmadeitpossibletocre-
ateafundamentalreferenceframe—amodeloftheinertialsystem implementedbycelestialmechanicsandastrometryintheform ofastronomicalyearbooksandthefundamentalstarcatalogues.
Inthedevelopmentofcelestialmechanicsmanyofthemost effectivemethodsofmathematicalphysicsandcomputational mathematicsarose,tookshapeandwerefurthered.
Asanexample(andbynomeanstheonlyone!),wecanindicatemethodsforthenumericalintegrationofdifferentialequationsdescribingvariousnaturalphenomenaandman-madeprocesses.Havingarisenincelestialmechanics,theseandothernumericalmethodsarewidelyusedinscienceandtechnology.In the17–18thcenturies,withthesolutionofastronomicalproblems bythemethodsofcelestialmechanics,essentiallyalltheoretical physicsbegan.
Notonlythetheoryofsystemsofordinarydifferentialequations,asitoccurredinthelastcentury,ispredominant,but,in fact,theentiresetofmoderntoolsofappliedmathematicsisused bymoderncelestialmechanicstomodelthemovementsofspace objects.
1.3Objectivesofstudyingthedynamicsof planetarysatellites
TheprimaryobjectiveofresearchintothedynamicsofSolar Systembodiesisthedeterminationofparametersofmotionof planetsandtheirsatellites.Thisobjectiveisrelevanttotheperennialchallengeofmankind:expandingandexploringourhabitat. Satellitesofmajorplanetsarethemostsuitabletargetsforunmannedandmannedlandingmissions.Researchofthestructure anddynamicsofSolarSystembodiesisanintegralpartofdynamicalastronomy.Themethodsofcelestialmechanicsandastrometricobservationsareusedinthisresearch.Interplanetary navigation,whichattractedtheinterestofscientistsinthesecond halfofthe20thcentury,isanewproblemofthedynamicsofSolar Systembodies.
Thegeneralapproachtostudyingthedynamicsofcelestial bodiesconsistsindevelopingmodelsofmotionandephemerides ofplanets,asteroids,andplanetarysatellites.Suchmodelsare builtbasedonthegenerallawsofnature,thephysicalparameters ofcelestialbodies,and,mostimportantly,observations.Advanced mathematicalandcomputationaltechniquesareusedintheprocess.EphemeridesaretheendresultofthisresearchandincorporatetheentirebodyofknowledgeonthedynamicsofSolarSystem bodies.
Chapter1 Objectives,currentproblemsandgeneralapproachtothestudyofthedynamicsofsatellites
Ephemeridesareusedtodeterminethephysicalpropertiesof celestialbodiesandtostudytheoriginsandevolutionoftheSolarSystem.Theyarealsoneededtoprepareandlaunchspace missionstootherplanetsandhelpdiscovernewcelestialbodies.Inthemiddleofthe19thcentury,UrbainLeVerrierhadused ephemeridestopredicttheexistenceofthethenunknownplanet Neptune,andnewplanetsandsatellitesarestillbeingdiscoveredthisway.Therefore,onemayconcludethatephemeridesalso serveasaresearchtool,sincetheyincorporatealltheavailable dataonthemotionofplanetsandsatellites.
Theresultsandconclusionsofcelestialmechanicsarevisibly andinvisiblypresentinmanyotherareasofscienceandhuman practice.
1.4Basicconceptsofcelestialmechanics andastrometry
Weestablishsomebasicconceptsofpracticalcelestialmechanicsandastrometry,withwhichwewilloperateinthefollowingpresentation.
Theobjectsofourresearcharetheplanetsandsatellitesofthe SolarSystem.Thus,weoperatewithmodelsofcelestialbodies, whichinnaturedonotexist,butwhichtoacertainextentdiffer littlefromthebehaviorsofrealcelestialbodies.Examplesofsuch objectsareamaterialpointandanabsolutelysolidhomogeneous bodyboundedbythesurfaceofatriaxialellipsoid.
Lawsofmotion. Therealmanifestationofthemotionofcelestialbodiesisachangeintheirrelativeposition,whichisdeterminedbythemutualdistances.Tosetthemotionofasystem ofcelestialbodies,oneshouldsetthelawofchangeintheirmutualdistancesintime.Themathematicaldescriptionofthelaws ofmotionaretheseorotherfunctionsoftime.
Foraconvenientrepresentationofthemotionofcelestialbodies,weusetheconceptsofareferenceframe,coordinatesystem andtimescale.Theabstractconceptofacoordinatesystemis somehowconnectedwithrealcelestialbodies.Examplesinclude theGreenwichmeridianonEarthorextragalacticradiosources. Theabstractconceptofatimescaleisassociatedwithrealphysicalprocesses.ExamplesincludeEarth’srotationorelectromagneticradiationfromanatom.
Lawsofinteraction. Thebasisforstudyingthemotionofcelestialbodiesisthelawsofphysicsthatarestrictlyestablished fromobservations,whichdescribetheinteractionsofbodiesor theeffectsonthemoftheenvironmentinwhichtheymove.The
mathematicalformofthelawsofinteractionofcelestialbodies areordinarydifferentialequations,whilethemutualdistances betweencelestialbodiesortheircoordinatessatisfytheseequations.
Mechanicalmodel. Incelestialmechanics,theconceptofa mechanicalmodelisused.Themodelisdescribedbythecompositionofmovingobjectsandtheirproperties,byspecifyingthe forcesactingontheindividualcomponentsofthemodel.Mechanicalmodelsareusedeitherforanapproximatedescriptionof themotionsofcelestialbodiesorasabasisforthedevelopment ofmoreaccuratemethodsfordescribingtheirmotions.
Thetaskofpracticalcelestialmechanicsisthecreationand studyofvariousmechanicalmodels,aswellasthestudyanddescriptionofthemotionofrealcelestialbodies.
Amechanicalmodel,being,asarule,anapproximatedescriptionofthemotionsofasystemofrealcelestialbodies,canfundamentallydifferfromthem.Inparticular,thepropertiesofbodies inthemodelmaynotcorrespondtoreality,andthelawsoftheactingforcescanbespecifiedinaspecialway.Examplesincludethe motionofasystemof materialpoints inwhichcelestialbodies aredimensionless,or arestrictedthree-bodyproblem thatdoes notsatisfyNewton’sthirdlaw.
Observations.Measuredvalues. Thesourceofourknowledge ofcelestialbodiesisobservation.Inobserving,wecannotbecontentwithstatingthefactofthepresenceofacelestialbodyinthe sky.Duringastronomicalobservations,measurementsofvarious quantitiesarecarriedoutusingavarietyofinstruments.Unlike abstractcoordinates,themeasuredvalueisalwaystherealone.It isformedinthemeasuringdevice.Astronomersdealwithawide varietyofinstrumentsandmeasuredvalues.Examplesaretheanglesofrotationofthetelescopeaxisrelativetotheverticalline andthemeridianplane,thedistancebetweenimagesofcelestialbodiesonphotographicplates,thetimeintervalbetweenthe flashofthelaserrangefinderandthefixationofthelightpulsereflectedfromthecelestialbody,thebackgroundintensityfroma singlepixelofasemiconductorlightdetector,andthedifference inrecordingsofthesignalfromaspaceradiosourceattworadio telescopes.
Accuracyofobservations. Instrumentsusuallyhavemeasurementerrors.Notethatthemysteriesoftheprocessesoccurringin measuringinstrumentsleaveusonlywiththeopportunitytobuild hypothesesregardingmeasurementerrors.Themagnitudeofthe errorofanindividualmeasurementisneverknown.Oftenweassumethattheerrorsarepurelyrandom,andweconsidervarious statisticalcharacteristicsoftheerrors.Mostly,weusethecon-
Chapter1 Objectives,currentproblemsandgeneralapproachtothestudyofthedynamicsofsatellites ceptofthemostprobableroot-mean-squareerror.Thestructural propertiesofmeasuringinstrumentssometimesmakeitpossible toapproximatelyestablishtheaccuracyofmeasurements.Inthe generalcase,wearetalkingaboutthe accuracyofobservations. Time. Variationofthemeasuredvalueintimeisduetothemotionofcelestialbodies.Measurementisperformedatsomepoint intime.Thistimepointiscountedbytheclockoftheobservatory. Inpracticalcelestialmechanics,aspecifictimeofmeasurementis alwaysascribedtoameasurablequantity.
Timeisanabstractconceptandsomeinstrumentsareneeded tomeasureit.However,anydevicehasitsownmeasurementerror.First,timewasmeasuredbytheangleofrotationoftheEarth. SuchatimewascalleduniversalwasanddesignatedasUT(UniversalTime).WhendiscrepanciesbetweenthetheoryofthemotionoftheMoonandobservationswerediscovered,itbecame clearthattheEarthrotatesunevenly,andtimehasbecomethe standard,asanindependentvariableinthetheoryofmotionof theMoon.Time,measuredbyobservationsoftheMoon,was calledephemeristimeandwasdenotedET.However,theaccuracyoftheobservationsoftheMoonisstilllimited.Thesearchfor amoreaccuratetimemeterledtoanatomicclock.Thistimesensorisnowthemostaccurate.Time,averagedoverseveralofthe mostaccurateatomicclocksintheworld,iscalledinternational atomictimeandisdesignatedasIAT(InternationalAtomicTime).
Inthefuture,wewilltalkaboutobservationsofcelestialbodies, alwaysassumingthatoneoranother measuredvalue isreceivedat acertainpointintime: measurementtime.
Theaccuracyofastronomicalmeasurementshasalready reachedsuchalevelthattheinadequacyofclassicalNewtonian mechanicsfordescribingtheobservedmotionofcelestialbodieshasbecomenoticeable.Inamoreaccuratetheoryofgeneral relativity,timepassesdifferentlyatanytwopointsinspace.To connectdifferenttimescales,itisnecessarytotakeintoaccount themotionofbodiesandtheirmasses.
Motionparameters.
Whenwestudyplanetsandsatellites, starsandgalaxies,weboldlyassumethatsomeparametersinherentincelestialbodiesandtheirmotionremainconstantall thetime.Theseincludethemass,sizeandshapeofbodies,orbit parametersandmanyotherquantities.Theseparameterscannot bedirectlymeasuredusingexistinginstruments.However,their meaningsreallymanifestthemselvesintheobservedmotionof celestialbodies.Inthefuturewewillcallsuchquantities motion parameters ofcelestialbodies.
Coordinatesystems. Measuredquantitiesdonotgivevisual representationsoftheconfigurationofthesystemofcelestialbod-
iesandareevenlesssuitableforexpressinggenerallawsofmotion.Aconvenientmeansofdescribingthespatialarrangement ofbodiesanddirectionsofcelestialbodiesistheuseofcoordinate systems.Whenwetalkaboutthepositionofthestaroraboutthe orientationofthebodyinacertaincoordinatesystem,wemean theabstractcoordinateaxesinspaceandimaginarylinesinthe sky.Coordinatesystemsarechosensoastogiveaclearideaofthe lawsandpropertiesofthemotionofcelestialbodies.
Thechoiceofacoordinatesystemisduetotheconvenienceof describingandstudyingthemotionofaparticularcelestialbody. Theoriginandcoordinateaxesareassociatedeitherwiththedetailsoftheobject,forexample,theEarth’sGreenwichmeridian,or withitsdynamicproperties,forexample,withtheprincipalaxes ofinertiaofthebody,orwiththepropertiesofmotion,forexample,withtherotationaxisofthebody,orwiththepositionofthe bodyatsometimepoint,orwemaychooseacoordinatesystem inanotherparticularway.
Mostly,asystemofrectangularorCartesiancoordinatesis used,itsoriginisdenotedbytheletter O ,andtheaxesbytheletters x , y ,and z.Thesystemofsphericalcoordinatesisoftenused withthedesignationofthecentraldistancebytheletter r ,thelatitudebytheletter ϕ andthelongitudebytheletter λ.
Werefertoanycoordinatesystemswithanoriginlocatedatthe observationpointas topocentriccoordinatesystems.Inaddition, weassociatetheaxesofthetopocentricsystemwiththevertical lineandthelocalmeridian.WhentheoriginofthecoordinatesystemisplacedatthemasscenteroftheEarth,wearetalkingabout geocentriccoordinatesystems.
Thelawsofmotionofcelestialbodiesarethedependencesof thecoordinatesofbodiesontimeandmotionparameters.Dependenciescantakemanyforms.Atmost,analyticalfunctions areusedthatdescribetheexplicitdependenceofthecoordinates ontime.Insomecases,thedependenceisgiveninimplicitform, thenthecoordinatesareobtainedbycalculationswithformulas bywayofsuccessiveapproximations.Thelawofmotioncantake theformofnumericaltablesinwhichthecoordinatesofcelestialbodiesaregivenforanumberoffixedpointsintime,usually definedwithsomeconstantstep.Withsuchanumericalspecificationofthelawofmotion,thedependenceofthecoordinateson themotionparametersofthecelestialbodyislost.Inthiscase,it isdifficulttoanalyzethepropertiesofmotion,andwearelimited tothetimeintervalforwhichthecoordinateswerecalculated.
Thecoordinatesofcelestialbodiesareabstractconcepts.They cannotbemeasuredbyanyinstruments.Coordinatesystemsare
Objectives,currentproblemsandgeneralapproachtothestudyofthedynamicsofsatellites modeledusingformulasandalgorithmsandformaconstituent partofthemotionmodelofcelestialbodies.
Amodelofmotionofacelestialbody. Wedonotknowexactly howthecelestialbodiesarearrangedandbywhatexactlawsthey move.Therefore,wehavetobecontentwiththestudyofmotion models,puttingforwardtheboldhypothesisthatourmodelsdifferlittlefromreality.
Inthegeneralcase,byamodelofmotionofacelestialbody wewillmeanacertainconstructionthatallowsustodetermine thevaluesofthemeasuredquantityatanygiventimeinstantsfor knownvaluesoftheparametersofmotion.
Implementationsofthemodelofmotionofacelestialbody canhaveverydifferentforms.Thesecanbemathematicalformulas,writtenmanuallyonpaperorpublishedasprintedmaterial.Thesecanbeprintednumerictablesofcoordinatevalues. Currently,bothformulasandtablesaredisplayedincomputer memoryunits.Inthiscase,theformulasareconvertedintocalculationalgorithms,andthetablesareavailabletocomputational programsthatsolvecertainproblems.Evenintheeraofpowerfulcomputingtechnology,thecoordinatesoftheprincipalcelestialbodiescalculatedforseveralyearsinadvancearecreatedand printedintheformofastronomicalyearbooksinseveralworldresearchcenters.
Wheredoourideasofthelawsofmotionofcelestialbodies comefrom?Inancienttimes,theywereestablishedalmostempiricallyfromsimpleobservations.Now,ofcourse,thelawsof motionarefoundintheprocessofsolvingdifferentialequations ofmotionrelativetothecoordinatesofcelestialbodies.These equationsarecompiledonthebasisofstrictlyestablishedlawsof physics,whichdescribetheinteractionsofbodiesortheeffectson themoftheenvironmentinwhichtheymove.Thisisdoneaspart ofamechanicalmodel.Allfactorsaffectingthemovementofeach bodyofthesystemandincludedinthemodelunderconsiderationareclearlyfixed.Thesetofconstructsofthelawsofmotionof celestialbodies,aswellasitsresult,thelawsofmotionthemselves, arecalledthetheoryofmotion.Thisiswhatcelestialmechanics addresses.
Inthevastmajorityofproblemsofcelestialmechanics,itis impossibletoobtainanexactsolutionoftheequationsofmotion.Onehastobecontentwitheitheranapproximatesolution oftheexactequations,oranexactsolutionoftheapproximate equations.Bothanalyticalandnumericalmethodsforsolvingdifferentialequationsareused.Inbothcases,thesolutionhasan error.Thiserrorcanbemoreorlessreliablyestimatedusingthe theoryitself.
Theaccuracyofthemotionmodelofacelestialbody. Theinitialdataforthemodelofmotionofacelestialbodyaremotion parameters,which,inturn,areknownwithsomeerror.Thiserror willalsoaffecttheaccuracyofthepre-calculationofthecoordinatesofthecelestialbodyandtheaccuracyofthepre-calculation ofthemeasuredvalue.Furthermore,wewilltalkaboutthe model accuracy,implyinganerrorinthecalculationofthemeasured value.Inthiscase,weseparatetwosourcesofthiserror:theproximityoftheobtainedsolutionofthemotionequationsandthe inaccuracyofthemotionparameters.Theerrorofthesolutionof themotionequationswillalsobecalledtheerrorofthecalculationsortheerrorofthemethod.Whenwetalkabouttheaccuracy ofthetheoryofmotionofacelestialbody,itisalwaysnecessary toclarifywhethertheinaccuracyofthemotionparametersisincludedintheerrorofthetheoryorisintheaccuracyofthetheory undertheassumptionofabsolutelyaccurateparameters.
Researchmethods. Fromotherastronomicaldisciplinescelestialmechanicsdiffersonlyinresearchmethods,amongwhichare analytical,numericalandqualitative approaches.
Analyticalmethods makeitpossibletoobtainasetofanalytical relationshipsthatallowustocalculatetheapproximatepositions andvelocitiesofcelestialbodiesatgiventimepoints,omitting itsvaluesatanyintermediatetimepoints.Afeatureofanalyticalmethodsisthegreatcomplexityandgrowingbulkinessofthe calculations.Inaddition,analyticalmethodsmakeitimpossible toassessthepropertiesofthestudiedmotionsatverylargetime intervals.Anotherdrawbackisthatanalyticalmethodsarenotapplicabletoallobjects.
Thelimitationsinherentinanalyticalmethodsdonotapplyto numericalmethods,whicharesuitableforcalculatingthemotions ofanycelestialbodiesandtheirsystemswithapredeterminedaccuracy.Withtheuseofpowerfulcomputersinscientificresearch, thepreviouslyconsideredexcessivelaboriousnessofnumerical methodshasceasedtobeanobstacletotheirapplication.But theyhavetheirown“Achilles’heel”—thisisthesteadyaccumulationoferrorwithincreaseintheintegrationinterval,whilerigorousestimatesofthegrowthofthiserrorareimpossible.Another drawbackofthesemethodsisthenumericalformofpresenting theresultsandtheinevitabilityofcalculatingtheintermediate stages,althoughoftenthegoalofthestudyisthefinalconfigurationafterintegration.
Qualitativemethods ofcelestialmechanicsmakeitpossibleto judgethepropertiesofthemovementsofcelestialbodieswithout fullintegrationof(analyticalornumerical)differentialequations.
Chapter1 Objectives,currentproblemsandgeneralapproachtothestudyofthedynamicsofsatellites
Analytical,numericalandqualitativemethodscontinuetobe appliedinmodernpracticalcelestialmechanics,andthebeauty andhighefficiencyofanalyticalmethodsaresuccessfullycombinedwiththesimplicityanduniversalityofnumericalmethods, andallthisiscomplementedbythecosmogonicimportanceof theconclusionsobtainedbyqualitativeresearchmethods.
1.5Generalapproachtostudyingthe dynamicsofplanetsandsatellitesbased onobservations
Ageneralapproachtostudyingthedynamicsofplanetsand satellitesistheconstructionofamodelofmotionbasedonobservations.Itisthemodelofmotionthatisneededforthepractical knowledgeofnature.
Fig. 1.1 showsaschemeforstudyingthedynamicsofSolar Systembodiesbasedonobservations.Atanystageofresearch, wefixthecompositionofthestudiedsystemofcelestialbodies. Thelawsoftheinteractionofbodies(gravitationalattraction,resistanceofthemedium),currentlyestablished,allowforwriting downthedifferentialequationsofmotion.Usinganalyticalmethods,onecanfindageneralsolutionoftheequationsofmotion. Aftersubstitutingthevaluesofarbitraryconstants(motionparameters)intothisgeneralsolution,weobtaintherequiredmodel ofmotionofthesystemofcelestialbodies.Whenwesolveequationsofmotionbymethodsofnumericalintegrationunderknown initialconditions(motionparameters),wealsoobtainamodelof motionofasystemofcelestialbodies.Somepreliminaryvalues ofmotionparametersareusuallyknownfrompreviousstudies. Toconstructamodelofmotion,thevaluesofthephysicalparametersenteringtheequationsofmotionthroughthelawsof interaction(forexample,themassofbodies)willalsoberequired.
Themainprocedureforstudyingthedynamicsofcelestial bodiesistorefinethemodelbasedonobservations.Observations giveusthevaluesofthemeasuredquantities.Callthemmeasured values.Ontheotherhand,wehaveamotionmodelthatserves topre-calculatemeasuredvalues.Wecancalculatethemeasured valuespreciselyatthetimesofobservation.Resultsarecalledcalculatedmeasuredvalues.Valuesdifferentinoriginofthesame entitywilldifferfromeachother.Wedenotethisdifferenceofvaluesin Fig. 1.1 symbolicallyby“O-C”(Oforobservatum,Cfor calculatum).Thedifferenceisanaturalresult,sinceitcontains anerrorofobservationandanerrorinthemodelofmotionofa celestialbody.However,insomecases,thedifferences“O-C”will
Schemeofstudyingthedynamicsofcelestialbodies.
exceedthemodelerrorandtheobservationerror.New,moreaccurateobservationsrevealamodelmismatchwithreality.Inthese cases,themismatchisattributedtothesimplestandmostprobablecause—theinaccuracyoftheacceptedvaluesofthemotion parametersofthecelestialbody.Aprocesscalledrefinementof motionparametersfromobservationsisincludedinthiscase(see “Parameterrefinementmethods”in Fig. 1.1).Mostly,therequired agreementbetweenthetheoryandobservationsisachievedbyrefiningtheparameters,andthedifferences“O-C”againfallwithin theerrorsofthemodelandobservations.
Insomerarecases,thetheorycannotbereconciledwith observations—thedifferences“O-C”remainsignificant.Thenwe havetoimprovemethodsforsolvingtheequationsofmotionand calculationmethods.Thisisthemostlaboriouspartofcelestial mechanics.Thefactorsaffectingthemotionofeachcelestialbody arebeingreconsidered.New,moreaccurateformulasofthetheoryarederived.Asaresult,theformulasbecomelongerandmore complex.Inaddition,moreaccuratecalculationmethodsarebeingdevelopedandapplied.Asaresult,therequiredcomputing timeissignificantlyincreased.
Inevenrarercases,themismatchofthetheorywithobservationsremainssignificant,nomatterhowhardtheresearcherstry torefinethemotionparametersandimprovethemotionmodel. Asaresultofthegeneralizationoffacts,testingofnewhypotheses andhighertensionofintelligence,adiscoveryismade.Previously unknowncelestialbodiesornewlawsoftheinteractionofthe knownbodiesmaybediscovered.Insuchasituation,ourgeneral ideasabouttheworldaroundusexpandsignificantly.Ageneralizationofthebasiclawsofnatureismade.
Theschemepresentedhere,likeanyscheme,ismeagreand limited,itonlyingeneraltermsreflectsthemixtureofscientific researchandtheaccumulationoffacts,fantasiesanderrors.
Figure1.1.
