Table Of ContentGraduate Texts in Physics
Elena Bannikova
Massimo Capaccioli
Foundations
of Celestial
Mechanics
Graduate Texts in Physics
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KurtH.Becker,NYUPolytechnicSchoolofEngineering,Brooklyn,NY,USA
Jean-MarcDiMeglio,MatièreetSystèmesComplexes,BâtimentCondorcet,
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·
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)
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Toourmasters
GérardHenrideVaucouleurs
VictorMoiseevichKontorovich
Aristotle
Teachingisthehighestformof
understanding.
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
