Table Of ContentPRECESSION, NUTATION, AND WOBBLE
OF THE EARTH
Earth rotation has long been used as a measure of time, together with using the stars as
referencepoints,todeterminetravellers’whereaboutsontheglobe.However,therotation
oftheEarthisnotuniformanditsorientationwithrespecttotheskygoesonchanging.
Coveringbothastronomicalandgeophysicalperspectives,thisbookdescribeschanges
intheEarth’sorientation,specificallyprecessionandnutation,andhowtheyareobserved
and computed in terms of tidal forcing and models of the Earth’s interior. Following an
introductiontokeyconceptsandelementarygeodetictheory,thebookdescribeshowprecise
measurementsoftheEarth’sorientationaremadeusingobservationsofextragalacticradio
sources by very long baseline interferometry (VLBI) techniques. It demonstrates how
models are used to accurately pinpoint the location and orientation of the Earth with
referencetothestarsandhowtodeterminevariationsinitsrotationspeed(length-of-day
variations).Atheoreticalframeworkisalsopresentedthatdescribestheroleplayedbythe
structureandpropertiesoftheEarth’sdeepinterior.
Incorporatingsuggestionsforfuturedevelopmentsinnutationtheoryforthenextgen-
erationmodels,thisbookisidealforadvanced-levelstudentsandresearchersinthefields
ofsolidEarthgeophysics,planetaryscience,andastronomy.
professor v. dehantisSeniorResearchScientistattheRoyalObservatoryofBelgium,
responsiblefortheOperationalDirectionReferenceSystemsandPlanetology,andisapart-
time professor at the Universite´ Catholique de Louvain. Her research is focused on the
interior,rotation,andhabitabilityoftheterrestrialobjectsofourSolarSystem,mostlyin
relationtotheevolutionoftheirinteriorsandatmospheres.ProfessorDehantisalsoPrincipal
Investigator and Co-Investigator of several instruments in ESA and NASA missions, and
has received several awards including the Descartes Prize of the European Union. She
is involved in several advisory committees such as the ESA Space Science Advisory
Committee(SSAC)andisamemberoftheRoyalAcademyofBelgium(ScienceClass).
Sheistheauthorofmorethan400publications.
professor p. m. (sonny) mathewsservedasSeniorProfessorofTheoreticalPhysics
at the University of Madras in India, and is now retired. His main research interests are
in the Earth’s nutation and precession and other concomitant phenomena, as well as the
properties of, and processes in, the Earth’s deep interior. Professor Mathews is the lead
originatoroftheMathews–Herring–Buffett(MHB)theoryofnutationadoptedbytheIAU
andIUGG.InrecognitionofhisworkontheMHBmodelofnutation,hewaselevatedto
theFellowshipoftheAmericanGeophysicalUnion(2004).HewaselectedaFellowofthe
IndianAcademyofSciencesin1975andoftheIndianNationalScienceAcademyin1983.
Hehaspublished120papers.
PRECESSION, NUTATION, AND
WOBBLE OF THE EARTH
V. DEHANT
RoyalObservatoryofBelgium
P. M. (SONNY) MATHEWS
FormerlyoftheUniversityofMadras
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
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©V.DehantandP.M.Mathews2015
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Firstpublished2015
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Contents
Preface pagexi
Listofabbreviations xii
1 Introduction–Fundamentaldefinitions–Motivation 1
1.1 RotationandglobalshapeoftheEarth 1
1.2 OrbitoftheEarth 1
1.3 Earthorientation–precessionandnutation 2
1.4 Primarycauseofprecessionandnutation 3
1.5 Nutationofanon-rigidEarth 4
1.6 ModelsoftheEarth’sinterior 6
1.7 Precession,nutation,andgeodynamics 8
1.8 TheEarth’snormalmodes 8
1.9 Motivationforthebook 9
1.10 Organizationofthebook 10
2 Conceptsandelementarytheory 12
2.1 Gravitationalpotential 12
2.2 AxesassociatedwithEarthrotation 12
2.3 Celestialsphere,celestialpoles,andequatorsofaxes 13
2.4 Terminologyfornutation 15
2.5 Normalmodes 15
2.6 Wobbleandsway 15
2.7 Length-of-day(LOD)variation 16
2.8 SolidEarthandoceantides 17
2.9 Inertiatensor;principalaxesandmomentsofinertia 18
2.10 Lowandhighfrequencynutations 19
2.11 Interplayofnutation,tides,ocean,andatmosphere 20
2.12 Remainingconceptssuchasfundamentalreferenceplanes 21
v
vi Contents
2.13 Signconventions 23
2.14 Referenceframes:basicaspects 25
2.15 Precessionandnutationofdifferentaxes 28
2.16 Precessionandnutationvariables 29
2.17 Definitionofpolarmotion 32
2.18 Ellipticalmotions 34
2.19 Decompositionintocircularmotions 36
2.20 Freerotationalmodes 40
2.21 Causesoftheforcedmotions 42
2.22 Equationsofrotationalmotion 43
2.23 GravitationalactionofacelestialbodyontheEarth 47
2.24 Solutionsforasimplemodel 56
2.25 PolarmotionofthepoleoftheCRS:relationtonutation 62
2.26 Kinematicalrelationsbetweenwobbleandnutation 64
2.27 Transferfunction 68
2.28 Oppolzerterms 69
2.29 Notationusedinthischapter 71
3 Referencesystemsandframes 77
3.1 Celestialandterrestrialreferencesystemsandframes 77
3.2 Transformationsbetweenreferenceframes 82
3.3 Notationusedinthischapter 92
4 Observationaltechniques–ephemerides 96
4.1 VLBI 96
4.2 ConstructionoftheICRF 104
4.3 GPSobservations 108
4.4 Orbitalmotionsandfundamentalarguments 110
4.5 Existingephemerides 114
4.6 EphemeridesusedfortherigidEarthnutationtheory 117
4.7 Otherkindsofobservationsneededfornutationstudies 120
5 RigidEarthprecessionandnutation 121
5.1 ApproachestothedeterminationofrigidEarthnutations 122
5.2 Hamiltonianapproach 124
5.3 Earth’sgravitationalpotentialonacelestialbody 152
5.4 TorqueonanaxiallysymmetricEarth 165
5.5 Tidegeneratingpotential(TGP) 168
5.6 Wobblesandnutationsexcitedbyarbitrarypotentials 184
5.7 TorqueintheCRFandnutationarguments 187
5.8 Nutationofthefigureaxis 191
Contents vii
5.9 Numericalintegration 198
5.10 Argumentsofthenutations:additionalremarks 198
5.11 Orbitalmotionsofthesolarsystembodies 200
5.12 Additionalsmalleffects 203
5.13 Precessionconstant 205
6 DeformableEarth–Lovenumbers 210
6.1 HydrostaticequilibriumEarthmodels 210
6.2 Perturbingpotentials 212
6.3 DeformationsofasphericalEarth 218
6.4 Lovenumbers 219
6.5 LoadLovenumbers 220
6.6 Equationsfordisplacementandgravitationalfields 222
6.7 Sphericalharmonicexpansions;radialfunctions 225
6.8 Theinertiatensor:contributionfromdeformations 242
7 Nutationsofanon-rigidEarth 257
7.1 Formulationsofthetheoryforanon-rigidEarth 257
7.2 Idealizedtwo-layerEarth:Poincare´’stheory 259
7.3 AnalyticaltreatmentofmoregeneralEarthmodels 269
7.4 Earthwithfluidoutercoreandsolidinnercore 289
7.5 Displacementfieldapproach 297
8 Anelasticity 330
8.1 ComplexrheologicalparametersfortheanelasticEarth 331
8.2 Anelasticitycontributionstonutation 336
9 Oceanandatmosphericcorrections 338
9.1 EffectsofthefluidlayersonEarthrotation 339
9.2 Linksbetweentorqueandangularmomentum 357
9.3 EffectofafluidactingonthesolidEarth 361
9.4 Non-tidaloceaneffectsontheEarthrotation 367
10 Refinementsofnon-rigidnutation 370
10.1 Effectofathermalconductivelayeratthetopofthecore 371
10.2 CMB/ICBcouplingeffectsonnutation 372
10.3 Electromagneticcoupling 374
10.4 Viscousandelectromagneticcoupling 392
10.5 Topographiccoupling 398
10.6 Viscosityoftheinnercore 400
10.7 Effectofstratificationinthecore 401
10.8 Discussiononprecessionconstant 402
viii Contents
10.9 Triaxiality 404
10.10 Second-ordereffects 405
11 Comparisonobservation-theory 415
11.1 Empiricalmodelsfornutations(suchasIERS96) 415
11.2 EstimationofbasicEarthparametersfromVLBI 415
11.3 Considerationoftheoceantideeffectsonnutation 418
11.4 Inclusionofanelasticityeffects 420
11.5 Nutationusingestimatedparameters(MHB2000) 421
11.6 Freecorenutationmodel 422
11.7 Present-dayprecision 423
11.8 Futuredecimalresearch 423
12 Conventions 425
12.1 Definitionofequator 425
12.2 Definitionofequinox 426
12.3 DefinitionofCIPrelatedtoprecessionandnutations 427
12.4 Classicalrepresentationsoftheprecession–nutationmatrix 430
12.5 DefinitionofCIO(non-rotatingorigin) 432
12.6 Stellarangle 439
12.7 LinkbetweenICRFandtheITRFinvolvingtheNRO 443
12.8 AlternativederivationoftheEarthrotationangle 447
12.9 LinkbetweenICRFandtheITRFinvolvingCIPandNRO 450
13 Marsnutations 454
13.1 RotationandobliquityofMars 454
13.2 Marsnormalmodes 456
13.3 PrecessionandnutationofMars 458
13.4 Length-of-dayvariationsandpolarmotionofMars 461
13.5 RotationmeasurementsofMars 462
AppendixA Rotationrepresentation 465
A.1 Rotationandinfinitesimalrotationsofavector 465
A.2 Rotationvectorandrotationmatrix 466
A.3 Rotationfromchangeofpoleposition 467
A.4 Changeinpoleposition 469
AppendixB Clairauttheory 471
B.1 ExpressionoftheEarth’sgravitationalpotential 471
B.2 Equipotentialsurfaceatthefirstorder–Clairaut’sellipsoid 474
B.3 GravityonthesurfaceofClairaut’sellipsoid 477
Contents ix
AppendixC Definitionsofequinoxes 479
C.1 Definitionofeclipticandorbitalplane 479
C.2 Definitionof“inertial”and“rotating”equinox 479
C.3 Expressionininertialframe 480
Bibliography 482
Index 534
