Graduate Texts in Physics Elena Bannikova Massimo Capaccioli Foundations of Celestial Mechanics Graduate Texts in Physics SeriesEditors KurtH.Becker,NYUPolytechnicSchoolofEngineering,Brooklyn,NY,USA Jean-MarcDiMeglio,MatièreetSystèmesComplexes,BâtimentCondorcet, UniversitéParisDiderot,Paris,France SadriHassani,DepartmentofPhysics,IllinoisStateUniversity,Normal,IL,USA MortenHjorth-Jensen,DepartmentofPhysics,Blindern,UniversityofOslo,Oslo, Norway BillMunro,NTTBasicResearchLaboratories,Atsugi,Japan RichardNeeds,CavendishLaboratory,UniversityofCambridge,Cambridge,UK WilliamT.Rhodes,DepartmentofComputerandElectricalEngineeringand ComputerScience,FloridaAtlanticUniversity,BocaRaton,FL,USA SusanScott,AustralianNationalUniversity,Acton,Australia H.EugeneStanley,CenterforPolymerStudies,PhysicsDepartment,Boston University,Boston,MA,USA MartinStutzmann,WalterSchottkyInstitute,TechnicalUniversityofMunich, Garching,Germany AndreasWipf,InstituteofTheoreticalPhysics,Friedrich-Schiller-UniversityJena, Jena,Germany Graduate Texts in Physics publishes core learning/teaching material for graduate- andadvanced-levelundergraduatecoursesontopicsofcurrentandemergingfields withinphysics,bothpureandapplied.ThesetextbooksservestudentsattheMS-or PhD-levelandtheirinstructorsascomprehensivesourcesofprinciples,definitions, derivations,experimentsandapplications(asrelevant)fortheirmasteryandteaching, respectively.Internationalinscopeandrelevance,thetextbookscorrespondtocourse syllabisufficientlytoserveasrequiredreading.Theirdidacticstyle,comprehensive- nessandcoverageoffundamentalmaterialalsomakethemsuitableasintroductions or references for scientists entering, or requiring timely knowledge of, a research field. · Elena Bannikova Massimo Capaccioli Foundations of Celestial Mechanics ElenaBannikova MassimoCapaccioli DepartmentofAstronomyandSpace DepartmentofPhysics“EttorePancini” Informatics UniversityofNaplesFedericoII V.N.KarazinKharkivNationalUniversity Naples,Italy Kharkiv,Ukraine ISSN 1868-4513 ISSN 1868-4521 (electronic) GraduateTextsinPhysics ISBN 978-3-031-04575-2 ISBN 978-3-031-04576-9 (eBook) https://doi.org/10.1007/978-3-031-04576-9 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsofreprinting,reuseofillustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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MustafaKemalAtatürk Agoodteacherislikeacandlethatconsumes itselfwhilelightingthewayforothers. Introduction VladimirIgorevichArnold Celestialmechanicsistheoriginofdynamicalsystems,linear algebra,topology,variationalcalculus,andsymplectic geometry. Initsmostclassicalsense,celestialmechanicsisthedisciplinethatstudiesthemotion oftheSolarSystembodiessuchasplanets,theirsatellites,asteroids,andcomets,as wellasartificialsatellitesandspaceprobes,inthecontextofclassicalmechanicsand Newton’stheoryofgravitation.1Thenatureoftheproblemsaddressedandthedegree ofprecisionrequiredfortheresultsdonotmandatetheuseofgeneralrelativitybut inspecialcases,whenEinstein’sgravitationaltheorycompensatesforafalsification ofclassicalmechanics(forexample,inthefullexplanationofMercury’sperihelion advance; see the last two sections of Chap. 4) or when the measurements are so accurate as to challenge relativity itself (for instance, in experiments designed to verifytheLense–Thirringeffect2). Theoriginsoratleastthemotivationsofcelestialmechanicsareintertwinedwith theveryrootsofastronomy,theoldestofallnaturalsciences(see,forinstance,[1]and [2]).AlreadyduringPrehistory,theclosestandbrightestcelestialbodiesandthestars visibletothenakedeyewereobservedwiththeaimofdescribingandthenpredicting theirmotions.Thereasonsforthisinterestresideintheapparentcentralityofmankind 1ThelocutioncelestialmechanicswascoinedbytheFrenchmathematicianPierreSimondeLaplace inthelateeighteenthcentury:from1798to1825,hepublishedthefivevolumesofhismonumental Traité de mécanique céleste. The term dynamics, from the Greek word for force, was instead introduced(originallyinFrench)bytheGermanpolymathGottfriedLeibniz(1646–1716)righta centuryearlier. 2Joseph Lense (1890–1985) and Hans Thirring (1888–1976) were Austrian physicists. They predictedtherelativisticcorrectiontotheprecessionofagyroscopenearalargerotatingmass. Forexample,whileclassicallyasatelliteorbitingasphericalsymmetricbodyisunaffectedbythe possiblerotationoftheprimary,relativityrequiresthattheplaneofthesatellite’sorbitsuffersa precessioninthedirectionoftheprimary’srotationaxis. vii viii Introduction withrespecttothefirmamentandinthemanifestperfectionofthestars,bothfixed (truestars)andwandering(planets,fromtheancientGreektermforvagabond).These otherwiseunchanginglightsources3appearedtobesubjecttoperiodicmovementsas wellasrecurringphenomena(Lunarphases,SolarandLunareclipses,Metoncycle4). Theyshowednosignsofbirthordeath,5 whichplacedthemoutsidethedomainof time,makingthemeternal.Bycontrast,anythingonEarthappearedcorruptibleand imperfect.Despitethisremarkablediversity,eternityandcelestialperfectionseemed tobeabletotuneinwellwithearthlyevents(suchasseasonalvariations)andhence with the rhythms of human life. his reassuring synergy generated the belief of a conspiracybetweenthetwodistinctAristotelianworlds,celestialandterrestrial,one withintheotherandwithaboundaryatthesphereoftheMoon,6andeachonewith itsowncompositionandphysics.7 Inthewakeofananthropocentricreadingofthe world, the next logical step was to postulate that the whole construction had been 3Apartfromafewnovaeandsupernovaevisibletothenakedeye,thephotometricvariabilityof starswasvirtuallyunknowntoancientastronomers,andinanycaseignored.TheMoonandthe planetsdidchangewithtimetheirluminosityandaspect(Moonphases),butthesealterationswere periodic. 4Ifaphenomenonisproducedbyacombinationoftwoindependentperiodicphenomenawith periods P1 and P2, it is also periodic with period P = mP1 = nP2, where m and n are the smallestintegerssatisfyingtherelation.Forinstance,about2700yearsagoChaldeanastronomers discovered that every 223 synodic months, equivalent to about 18.03 Julian years (each lasting 365.25days),theSunandtheMoonresumethesamerelativepositions.Hence,thechronologyof theeclipsesrepeatsalmostidenticallyevery18.03years(butSolareclipseswilloccurindifferent placesonEarth).Thisperiodicity,christenedtheSaroscyclein1686bytheEnglishastronomer EdmundHalley(1656–1742)whoderivedthenamefromaBabylonianorGreektermofuncertain meaning,istheconsequenceoftworequirementsforaneclipsetooccur:(1)theMoonmustbe in conjunction or opposition with the Sun, which happens with the period of a synodic month, every 29.53 days, and (2) the Moon, whose orbit is inclined by ∼ 5◦, must lie on the ecliptic, otherwisethealignmentisincomplete(henceineffective).ThisrequiresthattheMoonbetransiting through one node of its orbit, which happens twice in a draconic month of 27.21 days. Now, 223×29.53(cid:4)242×27.21(cid:4)6585days =18.03years.TheSaroscyclehashadgreatimportance since,owingtotheextraordinarysuggestivenessofeclipses(especiallySolar),predictingthemgave greatpowertothosewhowereabletodoso. AnotherexampleisprovidedbytheMetoncycle.InthefifthcenturyBC,theAthenianMeton discovered(orlearnedfromtheChaldeans)thatthephasesoftheMoonrepeatatapproximately thesamedatesevery19years.Thedurationofthiscyclecomesfromtheequation235×29.53(cid:4) 19×365.25days. 5Randomappearancesofacometornovastarweretreatedasmeteorological(earthly)phenomena. 6AccordingtoamodeldevisedbyPlatoandthenacceptedbytheGreek-Alexandrianculturewitha fewexceptionsandsomecomplicationsdictatedbytheneedforgreateradherencetoobservational evidence,theworldconsistedofspheresconcentrictotheEarth,eachassignedtoawanderingstar accordingtothesequence;Moon,Mercury,Venus,Sun,Mars,Jupiter,andSaturn. 7Tothefourultimateelementsthatmakeuptheterrestrialworld(fire,air,water,andearth)according toEmpedoclesofAkragas,Sicily(5thcenturyBC),Aristotle(384–322BC)addedtheetheror quintessence(LatinatenamefortheGreekether),fillingthecelestialspheres.Notethat,according totheStagirite,twoarethetypesofmotion,naturalandviolent.Causeofnaturalmotionisthereturn Introduction ix created8 for the central lodger, the human being, a creature privileged by God(s). Wandering stars were then interpreted as pencils with which the deity represented themessagesaddressedtomen,toguide,admonish,orsimplyinformthem9.This claimmadeastronomybecometheinstrumentofthemysteriousastrology,judgedas thehighestformofknowledgeandthereforedeliveredintothehandsofwisepriests. Apseudosciencethateasilyslippedintosuperstition: Thisistheexcellentfopperyoftheworld,that,whenwearesickinfortune,oftenthesurfeit ofourownbehaviour,wemakeguiltyofourdisastersthesun,themoon,andthestars;asif wewerevillainsonnecessity;foolsbyheavenlycompulsion;knaves,thieves,andtreachers bysphericalpre-dominance;drunkards,liars,andadulterersbyanenforc’dobedienceof planetary influence; and all that we are evil in, by a divine thrusting on. An admirable evasion of whore-master man, to lay his goatish disposition to the charge of a star! My fathercompoundedwithmymotherundertheDragon’sTail,andmynativitywasunder UrsaMajor,sothatitfollowsIamroughandlecherous.Fut!IshouldhavebeenthatIam, hadthemaidenlieststarinthefirmamenttwinkledonmybastardizing. W.Shakespeare,KingLear[Act1,Sc.2] Thesimpleuseofcelestialphenomenafororientation,timekeeping,orpractical life,asintheWorksandDaysoftheGreekHesiod(c.750–650BC)whichregards agriculture,whilesoimportant,haslongappearedassecondarytoastrology.10Exam- plesofsuchattemptstointerpretthemotionofthestarsaresomeNeolithicconstruc- tionsdatingbacktothatperiod,suchasStonehengeonSalisburyPlaininWiltshire, England,andthemenhirs(longstonesinCelticlanguages).Later,variouspopulations intheMiddleEast,Egypt,CentralAmerica,andChinaelaboratedthisrelationship with the sky through frequent and well-documented observations of astronomical phenomena and through the construction of huge architectural works such as the EgyptianpyramidsandtheMesopotamianziggurats. ActivitiesofthiskindarewidespreadinGreece.PythagorasofSamos(c.570–495 BC)placedtheSunandtheotherplanetsonconcentricspheresaroundacentralfire andmadethiscorrespondtotheharmonyofthemusicalscales.Plato(c.428–348BC) wasoneofthefirsttoformulateageocentrichypothesisfortheSolarSystem,stating thatthecelestialspherescarryingthewanderingstarsrotatearoundtheEarth,seen asamotionlessglobe.HeguessedthesphericityoftheSun,claimingthattheMoon ofabodytoitsnaturalplace(upforairandfire,downforwaterandearth)whereitwillremainin quiet;thevelocityisproportionaltothesizeofthebody.Thecauseofviolentmotionisanexternal motorincontactwiththebody(remoteactionsarenotallowed);thespeedisdirectlyproportional totheappliedforceandinverselytothemagnitudeofthebody.Aristotlealsopostulatedthatthe naturalmotionsarerectilinear(butnotuniform)onEarth,whiletheyarecircularanduniformin thesky. 8Theideaofagenesisiscommontomostancientcivilizations(butnottothethoughtofAristotle), andtomoderncosmologytoo(theBigBang). 9Inoneofhiswritings,JohannesKeplerstated:Thattheskydoessomethingtomanisobvious enough:butwhatitdoesspecificallyremainshidden. 10IntheHarmoniesoftheWorld(1619),wherehereportsthediscoveryofhisfamousThirdLaw, Keplerwrites:Thesoulofthenewlybornbabyismarkedforlifebythepatternofthestarsatthe momentitcomesintotheworld,unconsciouslyremembersit,andremainssensitivetothereturnof configurationsofasimilarkind. x Introduction doesnotshineonitsownbutreceivesitslightfromtheSun.Othercontributionstothe knowledgeoftheSolarSystemweremade,forexample,bytheMilesianphilosophers Thales (c. 624–548 BC) and Anaximander (c. 610–546 BC), the Pythagorean and Pre-SocraticPhilolausofCroton(470–385BC),andHeraclidesPonticus(385–322 or310BC),philosopherandastronomer,contemporarytoEudoxus. RetrogrademotionofMars.Theapparentreversalinthedirectionoftheplanet’smotioniscaused bythedifferentangularvelocityofMarsrelativetothatoftheEarth’sobserver.Credit:NASA The concept of the geocentric system, placing the Earth at the center of the universe, started to spread in the Western World. With various upgrades, it would endure for many centuries. Eudoxus of Cnidus (c. 408–355 BC), student of Plato andthePythagoreanArchytas(428–360BC),elaboratedthemodelofhomocentric spheres:auniverseasaclockworkwiththeEarthatthegeometriccenter.Eachsphere carriesaplanetinacircularanduniformmotion,differentfromthatofothers.This simple model worked reasonably well; but in order to match the observations (in particular, the so-called retrograde motion, when a planet reverses temporarily its prograde motion with respect to fixed stars), more spheres were needed for some planets,somuchsothattheoverallnumbergrewlargely. The reason why this geocentric conception managed to survive until the time ofCopernicus,andevenlongerinsomecircles,ismainlythreefold.Itwasableto account for some phenomena such as the lack of an annual parallax (mistakenly assigned to the Earth’s immobility and not to the enormous distances of the stars) ortheabsenceofanyapparentconsequence ofahighorbitalvelocityrequiredfor anEarthrevolvingabouttheSun.Itwasalsoinkeepingwithmen’sinnermostneed toconsiderthemselvesatthecenteroftheuniverseasproofofadirectconnection to God(s). The third reason has been the enormous cultural influence of Aristotle, thefirstphysicistofancientphilosophers,theMasterofthosewhoknow,asDante callshiminCantoIVoftheInferno.Hetheorizedthegeocentricmodel,discussing allofitsconsequences.AttheendoftheMiddleAges,thescholasticphilosophers, represented chiefly by the most learned Italian scholar Thomas Aquinas (1225– 1274), merged the geocentric cosmology with the Christian theology, which also required an immobile Earth to match the prayer of Joshua to the Lord reported in theOldTestament:Sun,standstilloverGibeon,andyou,moon,overtheValleyof