1994AJ 108. . 71 IW THE ASTRONOMICALJOURNAL tion oftheSunandMooncauseEarth’sequatortopre- 507yr, roughly1/3ofitduetotheSunand2/3from cess andnutate.Theprecessionisretrogradeitsrate planes. obliquity anglebetweentheEarth’sequatorandecliptic distances, theorbitaleccentricitiesandinclinations, Moon. Theratedependsonthelunarandsolarmasses fines theeclipticplane.Thelunarorbitisinclined5°to curacies oftechniquesmeasuringpositionsartificialsatel- obliquity change,andnutation. examines severaltheoreticalcontributionstoprecession, motion oftheEarth’sequatorinspaceareneeded.Thispaper lites, theMoon,andradiosources.Accuratetheoriesfor period. Butseveralinfluencescauseaslighttiltofthemean sion ofthelunarorbitplanealongeclipticwithan18.6yr ecliptic planeandthestrongsolartorquesdrivepreces- plane oforbitalprecessionwith respecttotheecliptic.The these twotorquesisalunarorbit precessionalongaplane to precessthelunarorbitalong the equator.Thenetresultof Earth’s oblatenesscontributesa smalltorquewhichattempts the eclipticatdynamicalequinox, theintersectionof tilted 8"withrespecttotheecliptic andthisplaneintersects 711 Astron.J.108 (2),August1994 0004-6256/94/108(2)/711/14/$0.90 ©1994Am.Astron. Soc.711 © American Astronomical Society • Provided by the NASA Astrophysics Data System Torques ontheoblateEarthduetogravitationalattrac- The orbitoftheEarth-MoonsystemaboutSunde- Recent decadeshaveseenimpressiveadvancesintheac- 2 The precessionandnutationoftheEarth’sequatorarisefromsolar,lunar,planetarytorqueson perturbations onthelunarorbitresultintorquesoblateEarthwhichcontributetoprecession, oblate Earth.Themeanlunarorbitplaneisnearlycoincidentwiththeeclipticplane.Asmalltiltoutof bulge. Thetotalcorrectiontotheobliquityrateis—0.0247century,itanobservablemotioninspace(the ecliptic iscausedbyplanetaryperturbationsandtheEarth’sgravitationalharmonic7-These the contributionsto18.6yrnutationare—0.03mas(milliarcseconds)forin-phaseAt//plus generally beenallowedforinpastnutationtheoriesandsomeprecessiontheories.Fortheplanetaryeffect, much largerconventionalobliquityrateiswhollyfromthemotionofecliptic,notequator),andit obliquity rate,andnutationwhiletheJperturbationscontributetoprecessionnutation.Small is notpresentintheIAU-adoptedexpressionsfororientationofEarth’sequator.The7effectshave additional contributionstothesecularratesarisefromtidaleffectsandplanetarytorquesonEarth’s out-of-phase contributionscanarisebymeansotherthandissipation.Thesumoftheto out-of-phase contributionsof0.14masinAif/and—0.03Ae.Thelattertermsdemonstratethat precession alsoarisesfromtidesandchangingJ.Contributionstheimprovedtheory,masses,ecliptic precession rateisconsideredandtheinferredvalueofmomentinertiacombination(C—A)/C,which motion, andmeasuredvaluesoftheprecessionrateobliquityarecombinedtogiveexpressions obliquity ratesarenotconstantforlongtimescausingaccelerationsinbothquantities.Acceleration rate, therigidbody(C-A)/C=0.0032737634which,incombinationwithasatellite-derivedJ,gives is usedtoscalethecoefficientsinnutationseries,evaluated.Usinganupdatedvalueforprecession normalized polarmomentofinertiaClMR=0.3307007.Theplanetarycontributionstotheprecessionand (polynomials intime)forprecession,obliquity,andGreenwichMeanSiderealTime. 2 2 2 2 2 CONTRIBUTIONS TOTHEEARTH’SOBLIQUITYRATE,PRECESSION,ANDNUTATION 1. INTRODUCTION Jet PropulsionLaboratory,CaliforniaInstituteofTechnology,Pasadena,91109 Electronic mail:LOGOS::JGW,[email protected] Received 1993December21;revised1994April13 VOLUME 108,NUMBER2 James G.Williams ABSTRACT with respecttotheeclipticplane.Asaconsequenceof change intheprecessionofEarth’sequator. Earth’s oblatenessonthelunarorbitinturncausesasmall ecliptic andequatorplanes.Thissmallinfluenceofthe planetary attractions,theeclipticplanemoves.TheMoon’s plane oforbitalprecessiontotheecliptic.Therearealso closely, butnotperfectly.Thismotioncausesa1.4"tiltofthe mean planeoforbitalprecessionfollowsthemovingecliptic with respecttothedynamicalequinox.Consequently,they tation. Inaddition,theplanetsdirectlytorqueEarth.The smaller displacement.Thesetwoinfluencesonthelunarorbit direct planetarytorquesonthelunarorbitwhichcontributea IAU-adopted theoryofprecessionandobliquitychange dynamics. Theseplanetaryinfluencesarenotincludedinthe quantity, theobliquityrateisnotafreeparameterof obliquity rate.Whiletheprecessionratemustbeameasured contribute toboththeprecessionofequatorand torques fromthesethreeplanetaryinfluencesarenotaligned result intorquesontheoblateEarthwhichmodifyitsorien- nutation theories. the planetarytiltsonlunar orbit beenincludedinrecent cause accelerations.Acceleration correctionsalsoarisefrom tidal effectsandtheEarth’schanging J- (Lieske etal.1977).Neitherhavealloftheconsequences 2 The orbitplanesoftheplanetshavesmallinclinations The abovesourcesofprecession andobliquityratealso The aboveoutlinedcorrections tothemotionof AUGUST 1994 1994AJ 108. . 71 IW 2 by Kinoshita&Souchay(1990)duetotheEarth’sJand these correctionsareaddedprecessiondeveloped Earth’s equatoraredevelopedinthefollowingsections.To Also ignoredaresecond-ordereffectsduetothechangeof without theinfluenceofaliquidcore.Smalldifferencesin The Earthwillbetreatedasarigidbodywithoutoceansand to smalleffects,itisreasonableintroducesimplifications. lating themotionofEarth’sequator(orpole)inspace.As malized polarmoment,C/MR. Earth’s fractionalmomentofinertia,(C—A)/C,andthenor- motion oftheEarth’sequatorandrevisedvalues theory aredevelopednewpolynomialexpressionsforthe second-order correctionsduetonutation.Fromtherevised 712 J.G.WILLIAMS:EARTH’SOBLIQUITYRATE,PRECESSION,ANDNUTATION motion ofthefigureaxisrigidEarthwhichisdesired. for theangularmomentumaxis,butstrictlyspeakingitis for arigidbody)areignored.Theequationswillbewritten the directionsofaxesangularmomentum,instanta- the computationsofsubsequenttwosectionsarelimited The attractingbodyhasageocentricdistancerandproduct the Earth’sorientation,e.g.,precessionandnutationmodify- neous spin,andfigure(equivalenttocelestialephemerispole where thedeclinationofattractingbodyisSandright gravity fieldoftheoblateEarthis to themaximummomentC,andxaxispointstoward of thegravitationalconstantandmassGm.TheEarthhas ing thecomputationofprecessionandnutation. equinox. Thepotentialenergyoftheexternalbodyin intersection oftheeclipticandequatorplanes,dynamical axis isalignedwiththeEarth’sprincipalcorresponding moments ofinertiaA,CwithArotation aboutthexaxisby 23 © American Astronomical Society • Provided by the NASA Astrophysics Data System This sectionsetsupthefundamentalequationsforcalcu- The oblate,rigidEarthistorquedbyanexternalbody. T=-rXVV, (2) The analyticaltheoriesfortheSun, planets,andMoonare V=Gm[M/r-(C-A)(3 sin5-l)/2r],(1) T= 3Gm(C-A)sin Scosô 3Gm(C —A) 2. FUNDAMENTALS — cosa sin a 0 (3) 3 2 -6 components become In thetorquevectorproductsofequatorialcoordinate be writtenintermsofthegeocentricdistancerandgeo- The eclipticcoordinates(X,Y,Z)oftheattractingbodycan fort. Averagedoveranintegralnumberofrevolutionsthe torque maybecomputedwithgoodaccuracylittleef- centric eclipticlongitude\andlatitudeß where aisthesemimajoraxis,eorbitaleccentricity,and mated byanellipticalorbitintheeclipticplane,solar average xcomponentoftorqueis dif//dt =T/C(x)sine,wherea)isthemajorcomponentof retrograde precessionalongtheeclipticwithrate because thelunarorbitisstronglyperturbedbySun. the Earth’sangularvelocityandCa)approximatestotal The lunarorbitishighlyperturbedandtheequivalentequa- These difficultiesinthemajorprecessionandnutationeffects odic nutationterms. torque componentshavezeroaverage,butofcoursethefirst Kepler’s thirdlaw.TheanalogousprecessionfromtheMoon divided bythesumofmasses(Sun+Earth+Moon)using G/a maybereplacedwiththesquareofmeanmotion angular momentumoftheEarth’sspin. tion factor,islessprecise.Thecomputationofmanysmall tion forthelunar-inducedprecession,includinginclina- computation forthesolarprecessionisonlylargerby have beendealtwithbyKinoshita&Souchay(1990).Their ecliptic, butitisacoarserapproximationforthelunareffect solar-induced precessionoftheEarth’sequatoralong two componentshavetimevariationswhichcontributeperi- includes aninclinationfactorof1-1.5sini.Theothertwo corrections inthispapercanusetheellipticalapproximation. 2X10 soEq.(7)isaverygoodapproximationfortheSun. e theobliquity.Thexcomponentoftorquegivesrisetoa causes an8"tiltandplanetary effects causea1.5"tilt.As slightly withrespecttotheecliptic plane.TheEarth’sJ consequence ofthesesmallsizes, expansionswillbeused. xz z 2 32/ 32/ Because theEarth’spathaboutSuniswellapproxi- dt('/dt=3Gm(C-A)cos e/2a(l-e)Cw.(7) The ellipticalapproximationaboveworkswellforthe T =3Gm(C-A)únecose/2a(l—e), The lunarorbitprecessesalong aplanewhichistilted \z) \sinß! I X\jCOSyö z x Y =rcosy0sinX.(6) 3. EFFECTSDUETOTHETILTEDLUNAR MEANPLANE Z cose+Ysine Y cos€—Zsine 2 (l/2)(7-Z)sin 2e+VZcos2e —XZ cose—XYsin6 0 712 (4) (5) 1994AJ 108. . 71 IW o where isthelunarnode(Í1=L—F).Theimportant where Fisthemeanargumentoflatitude.Similarlylunar planetary-induced termsare nutation areselectedfromChapront-Touzé&Chapront mean anomaly/,eccentricitye,andsmallercontributions longitude \maybewrittenintermsofitsmeanL, orbit totheecliptici,andsmallerperturbationsAßsothat The lunarlatitudearisesfromthe5.15°inclinationof phase 3 © American Astronomical Society • Provided by the NASA Astrophysics Data System The perturbingtermsmostimportantforprecessionand sin yß^siniF+Aß,(8) The eclipticplaneisrotatingaboutalinewhichdis- Aß=-8.045" sinL+0.326"sin(L-2F), AL=E sin(il+cos2L Out-of-phase termsinthenutation theorywillarisefrom 3Gm(C—A)sin e 3Gm{C—A)cos e 3Gm(C—A)cos e — 2E'sin9]Í1}. — 2E'cos9]cosíl+[siniBsin — 22?')cos2íl+(siniEsin9)sin2Í2]. — Bsincp2L] 3 3 3 4aCct)fi 8aC(x)Q 2 4aCoL>L z z 3Gm(C —A)sinecos 3Gm(C-A)cos 2e 3Gm(C —A)cos2e + [5siniB—2F'9]cosÍ1} — Bsincpcos2L], + 2E'cos96siniF'jsinÍ1 — 2F')sin2íl+(siniEsinip)cos2Í1] 3 3 8aCcuil 4aC(oL 2 z 3 4aCo)il z (14) (16) (15) 713 1994AJ 108. . 71 IW J Tilt Table 1.NutationtermsduetoJandplanetarytilteffectsoflunarorbit. Argument Lunar meanlongitudeisL=il+F. from thephaseshiftsinplanetaryeffectswhichturn 714 J.G.WILLIAMS:EARTH’SOBLIQUITYRATE,PRECESSION,ANDNUTATION not given. cal equinox.Therearestillsmallercorrectionswithargu- have nospecialalignmentwiththeeclipticplaneordynami- Dehant 1988,1990).Theout-of-phasetermsinTable1arise et al1990)andinterioroftheEarth(Wahr&Bergen1986; Planetary ments of2L+Í1,-il,2m—íl,and3Í1whichare arise becausetheorbitplanesofplanetsotherthanEarth footnote, butmatchesthevalueinTable1ofthispaper), lunar orbit.Theout-of-phase18.6yrtermfornutationin tation theorycouldarisefromplanetaryperturbationsonthe obliquity. Kinoshita(1975,1977)consideredtheobliquity the obliquitytermwastoosmallforhiscutofflimit.Woolard longitude occursinhisTable24(itismarkedwitha?and 2 of motion.ThisassertionwillbediscussedfurtherinSec.6. rate contributiontobedueanerrorinWoolard’sequations terms contributetoprecessionandtheaccelerationof also calculatedtheobliquityratecontribution,showing planetary effectscomputedinthispaper.InKinoshita& Kinoshita’s Mjcorrectiontoprecessionis-2.68mas/yrand (p. 127)healsocommentsthattheplanetary-inducedlunar but the—0.056mas/yrplanetaryeffectismissing.Presum- to precessionreplacestheearliercorrection.Itcontainsa Souchay (1990)amoreelaborate“second-order”correction it appearstocorrespondthesumofJandin-phase 2 from J,butnottheplanetarytilt. -0.256 mas/yrinhisTable24attheyear1900.Intext ably, theirnutationscontainthecorrespondingcontribution The inclinationoftheplanetaryorbitstoeclipticwillalso torques fromtheplanetswhichcontributetoprecession. sion oftheEarth’sequator.Therearesmalladditional below. Notethattheeffectofthese directplanetarytorques contribution wasgivenbyKinoshita &Souchay(1990),but cause asmallobliquityrate.A calculation oftheprecession on theEarthisdistinctfrom tilteffectduetothedirect not theobliquityrate.Abriefderivation ofbothratesisgiven -2.60 mas/yrcorrectiontoprecessionduetheJeffects, planetary perturbationsonthelunar orbit. 2 2 2 Í1 20 n 2L 20 2L 4. RATESDUETODIRECTTORQUESOFPLANETSONEARTH © American Astronomical Society • Provided by the NASA Astrophysics Data System Woolard (1953)wasawarethatout-of-phasetermsinnu- The torquesfromtheSunandMoondominatepreces- -0.0005 -0.0301 -1.4782 0.0049 0.0003 0.0151 mas sin -0.0028 0.0000 0.0000 0.0060 0.1366 0.0000 mas cos -0.0015 -0.0277 0.0000 0.0000 0.0032 0.0000 mas sm Ae -0.0026 -0.0081 -0.0002 0.1557 0.0003 0.0029 mas cos 2 5r2 for theplanet’svariables,noprimesEarth.Theeffects tracting planet,itisnecessarytodifferencetheheliocentric where aand'denotethesemimajoraxes,iinclination will onlybecarriedtofirstdegree(cosThentheplan- bits willbetakenascirclesandtheplanetaryinclinations two approximationswillbeintroduced.Theheliocentricor- coordinates oftheplanetandEarth.Primeswillbeused are small;sotokeepthederivationfrombecomingunwieldy, et’s geocentriceclipticcoordinatesare given by the Earthandattractingplanet.Thegeocentricdistanceris arguments oflatitudemeasuredfromthatsamenodeforboth to theecliptic,Í1'nodeonandu«' variables ofintegrationto«—«'and+'makesitclear Also, sinceonly«—'appearsinthedenominator,changing numerators Y-Z,YZ,XZ,orXYwhichreversessignun- vice isuseful.Thetransformations(sinw,sin«')—►(—sin«, There resultexpressionsinvolvingproductsofsinesandco- der anyofthethreetransformationswillaveragetozero. taken togetherleaveunchangedthedistancerwhichappears disappear duringthedoubleintegrationamathematicalde- evaluated from0to277.Towinnowouttermswhichwill ((YZ/r) =fi(YZ/rduI4ttwithbothintegrals is used.Denotingtheaveragewith(),anexample Eq. (5)areformedforsubstitutioninto(3)thetorque. dinates followingEq.(4)andtheproductsofcoordinates in thedenominatorsofintegrals.Anycomponent the periodictermsGauss’methodofaveragingover«and' sines ofuand«'.Inordertoisolatethesecularratesfrom that additionalcomponentsaveragetozero.Finallyonegets terms ofcompleteellipticintegrals ofthefirstandsecond are onlyfunctionsof«—«'and theymaybeevaluatedin -sin«'), (cosm,cos«')—►(—cos«,-cos«'),andboth kind, K(k)andE(k),respectively, 2r 253 5 5 5 In ordertocomputethegeocentriccoordinatesofat- The eclipticcoordinatesarerotatedintoequatorialcoor- r =a+a'-2'cos(u—u).(18) The threedifferentaverageson the right-handsidesabove ((Y —Z)/r=(l/2)(l/r, (XY/r) =0. (XZ/r)= -(«72)sin/'sinil'(«'(l/r (YZ/r) =(a'/2)sini'cosil'(«'(l/r f f a cos(w'+n')—acos(MfT) 5 5 a sin(w'Tí!')—asin(H+iy) -«(cos(w-w')/r), — forces oftheplanetschangingheliocentricorbit.Thetilt ing discussion. the indirecteffect.Thetwocomponentshavebeencombined orbit, coupledwiththemotionofeclipticplanedueto in Sec.3.Theindirectcontributionwilldominatethefollow- indirect termsarisefromtheforceofSunonlunar terms inthelunarorbitarisetwoways.Thedirect arise fromtheforcesofplanetsonlunarorbit.The considered furtherinthissection.Theplanetary-inducedtilt not giverisetoanobliquityrateanddoneedbe been stated(Kinoshita1975,1977)thatthesetermsdonot 0 0 0o long asLandÎÎarereferredtothemovingequinox.Theydo phase willgiverisetohigherderivativesoftheprecession give risetoanobliquityrateandtheresolutionofdiffer- 0 hinges ontheoriginofphase.(2)Timevariations ence betweenthatclaim,Woolard(1953),andthispaper subject ofthephasetilttermsusedinSec.3.(1)Ithas and obliquity. 0 0 717 J.G.WILLIAMS:EARTH’SOBLIQUITYRATE,PRECESSION,ANDNUTATION v , f nuce © American Astronomical Society • Provided by the NASA Astrophysics Data System P= +P'-Q'/Qo, Qv=Q +QP/Ú.(25) dQy/dt=-Ù(Py-). (24) dPy/dt =Ù(Q-'), We wishtoexpressthesecularmotionoflunarorbit The /2~idtilttermsinSec.3havezerophaseso There aretworeasonstoconsidertheseeminglyprosaic v q 0 0v 6. THEPHASEOFTILTTERMS -1 L. Towriteitusingrequiresintroducingafunctionof in theappendixofKinoshita(1977) wasunawarethatA/3 cal coefficient[inEq.(11)itis in thephase].Thediscussion of thispaper],butiftakenliterally itisfunctionallymislead- ing becausetheequinoxdependence ishiddeninthenumeri- notation wasnumericallysuitable fortheepoch[likeEq.(11) shows thathewasawareofthe higherpowersoftime).His terms ofcos[L—IliTjO)]orintheformEq.(26)using sion showthattheA/Sexpressioncouldhavebeenwrittenin liquity rate.Fortheindirectcontribution(thereisaparallel ard wasonlycomputingthelineartimeterm(hisdiscussion IliTjO), andII(r,0)dependsonthemovingequinox.Wool- argument forthedirectcontribution),Eq.(26)anditsdiscus- son Kinoshita(1975,1977)arguedagainstWoolard’sob- constant, thentherewouldbenoobliquityrate.Forthisrea- give risetoasecularobliquityrate.Iftheargumentwas conventionally areinlunartheory,andifthecoefficientwas measured withrespecttothemovingequinox,asÍ1andL as 1.53"cos(F+fl)]wasdisplayedbecausethesinedoesnot ponents ofF+il=L,onlythecosinecomponent[hegaveit is equivalenttoEq.(11).Brokenintosineandcosinecom- When explainingit(pp.127-128)heusedthe(directplus tribution fromtheplanetary-inducedtiltinlunarorbit. indirect) A/3contributionfromBrown’slunartheorywhich phase ofthecombineddirectandindirecttermsisslightly larger than95.13°andtheamplitudeisslightly 0 moving equinoxandtheplanetarynodes,soinEq.(11) which atJ2000is95.13°or270°-n.Atthisphase its ownphasewhichdependsonthedifferencebetween precession rateminus17.36or32.93"/yr(thisiswrong 7yr, twicetherateofIImeasuredalongfixedJ2000 ventional lunartheoryaremeasuredfromthemoving(mean 1.39". in Brown’slunartheory).Thesmallerdirectcontributionhas ecliptic. ToputA/3intheformofEq.(11)withsin(L rate measuredalongthemovingeclipticplaneis-17.36 sin(L4-phase) wherethephaseisgivenbytan(ß7P')> equation canbeputintheformofanamplitudetimes of date)equinoxalongthemoving(mean)ecliptic.TheA/3 with respecttoinertialspace.Thesubscriptzerohasbeen this phaseis210°—I[{T,0).Theratethenthegeneral moving ecliptic.InthenotationofIAUprecessionpaper phase mustbemeasuredfromthemovingequinox,along used todistinguishLandflfromilwhichincon- moving ecliptic,tothenodeofrotationeclipticon equinox alongthemovingecliptic,thenforcompatibility equinox attheinitialtime(J2000)anditsretrograderateis lows theecliptic,nodeflisaquantityreferencedto Qy—Q' areusefulforseeingthatthelunarorbitnearlyfol- where themeanlongitudeLisgivenintermsof inertial coordinatesystem.WhilethedifferencesP-'and differential Eqs.(24)andtheirsolution(25)arewritteninan argument oflatitudeandnode(L=F-fiî).Notethatthe be expressedasaperturbationinthelunarlatitude +phase), withLmeasuredfromthemoving(meanofdate) 0 0 0 0 v 0 Woolard (1953)earliercomputedtheobliquityratecon- The tiltofthelunarorbitplanetomovingeclipticcan Aß=(ß7ii)cos L+(P7iio)sin,(26) 0 717 1994AJ 108. . 71 IW s 62 -62 -62 2 -62 P 'andQthetwodirectplanetaryeffects(directtilt P' andQ'derivativesinEq.(26)canbeusedwithEqs.(12) variations andtheratewillshowbothsigns.Bycontrast, value wasreasonable. zero. Itisconcludedherethattheobliquityratefrom finite partialderivativeofÀ/3withrespecttotheequinox,but had afunctionaldependenceontheequinoxwhichwasun- The computationofthenonlinearcontributionsissimilarfor variable. Theseriesfortheindirectcontributionthenbe- power seriesintime.ForuseSec.8itisconvenientto positive. time scales(>10000yr)theobliquityexhibitsquasiperiodic contributions totheobliquityratearenotconstant.Overlong parison bySouchay&Kinoshita(1991)oftheirtheorywith indirect (anddirect)tiltisrealandthatWoolard’snumerical the equinoxdependencegivenabovecausespartialtobe displayed inWoolard.Kinoshita’sdiscussiondependedona 718 J.G.WILLIAMS:EARTH’SOBLIQUITYRATE,PRECESSION,ANDNUTATION plicit andpowerseriesfortheP’sg’s(Laskar1986; the directplanetarytorquesonEarthandplan- comes —0.00392+0.000703Tforprecessionrate(units" Lieske etal(1977)orSimon(1994)differentiateP' indirect tiltwillvarywithtime.Thetimedependenceofthe small obliquityrateduetotidaldissipation(Sec.7)isalways etary orbits,itshouldbeunderstoodthatallthreeofthese direct torqueontheEarth)dependtime-varyingplan- terms whicharisefromthesamesource. obliquity ratetermnortheout-of-phase18.6yrnutation a numericalintegrationshowedasdiscrepanciesneitherthe the directtorqueeffectsonplanetaryP’sandg’isex- etary tiltonthelunarorbit.InEqs.(22)dependenceof ecliptic andequinox.Inthetwo-time(7V)powerseriesof express theseratesinacoordinatesystemmovingwiththe and (13)tocomputetheprecessionobliquityratesasa to beabout40%largerthanforthedirecttorqueeffect.Be- due tothedirecttilttermsismoredifficult,butestimated sion and2X10~"/centuriesforobliquity.Theacceleration eration andtheresultis-17X10"/centuriesforpreces- and centuries)—0.0233+3X10Tforobliquityrate. and Q'withrespecttot,seti=0,useTasthetime to thesefiguresmustbeaddedthelargeraccelerationswhich cause thepolynomialsforplanetsusedafixedequinox, Simon etal.1994)canbeused.Venusdominatestheaccel- for precessioninEq.(22)doesnotdependonplanetaryPs "/centuries forobliquity.Thetotaloftheprecedingdirect 411X10 "/centuriesforprecessionand-48X10 result fromtransformingafixedtomovingequinox: the entryforluni-solarprecession inthetable. Note thatpart(0.03269"/century)ofthedirecttorqueeffect listed under“planetarytiltanddirecttorque”inTable4. and Qs,contributesnoaccelerations,isincludedwith and indirecttilttermsthedirecttorquesonEarthare tions are(-0.0301+0.0050L) sind tothelongitudenuta- those ofthelatitudecoefficient, thenthein-phasecontribu- secular changesintheALcoefficients scaleinproportionto (Table 1)willalsohavesecular changes.Assumingthatthe © American Astronomical Society • Provided by the NASA Astrophysics Data System There isananomalythatIdonotunderstand.Thecom- Both theprecessionandobliquityratesarisingfrom Since theindirecttilttermsdependontime-varying The coefficientsoftheplanetary-tilt-induced nutation _3 -62 _6 -62 Table 4.Timeandobliquitydependenceofprecessionrates phase coefficientsisabout10/centuryandignorable. tion and(0.0029-0.0005L)cosÍ1tothelatitudenutation obliquity withtime. tions andhigherderivativeswhichwillbeutilizedwhile marized inTable4.Thattablealsogivestheedependence et al.(1977),theunitsofthatpaperarenowadopted(arc- nomial expressionsfortheprecessionquantitiesasinLieske from Eq.(7)anditslunarcounterpart.Movingtowardpoly- in thelunarorbitandEarthrotation,possiblechanges have beenconsideredinprevioustheories,plustidaleffects gral ofprecessionrate),andanothercontributiontoobliquity ("/century) whichareneededtocalculatetheevolutionofprecessionand Using theeccentricitypolynomialsinLaskar(1986)or section. solving thedifferentialequationsfororientationinnext since thechangeofobliquitycontributesadditionalaccelera- seconds andcenturies).Theresultsofthissectionaresum- (C-A)fC. Manyoftheresultsthissectioncanbederived the orbitofEarth-MoonsystemaboutSunwhich rate. Thereareeffectsduetothechangeineccentricityof (units masandcenturies).Therelativechangesoftheout-of- rivative of-3362X10"/centuries.Theeccentricitythe causes thesolar-inducedprecessionratetohaveafirstde- in theaccumulatedprecessionIAUtheorydatestode and higherderivativesintheaccumulatedprecession(inte- by theSunandthosewhichdepend ontheheliocentricec- heliocentric orbitalsoentersinto thegeodesicprecessionEq. Simon etal.(1994),¿/e/¿/í=-42.0Xl0/centuries.This Sitter &Brouwer(1938).Itisre-evaluatedhereatJ2000. induced acceleration(firstderivativeoftheprecessionrate) the precessionrate.Theevaluationofeccentricity- orbit entersintothesolar-inducedprecessionofequator and changesintheeccentricitywillaffectderivativesof 2.7XlO "/centuries.Thelunar orbitincludesperturbations the lunarlatitudeanddistance weaklyinfluencethelunar- centricity contributeaccelerations. Theseperiodictermsin (23). Thederivativeofthatrate (inretrogradesense)is As seenfromEq.(7),theeccentricityofheliocentric This sectionconsiderseffectswhichcauseaccelerations Luni-solar, directplanetarytorque Geodesic precession Second order(M3) Tides J tilt J4 precession Second order Planetary tiltanddirecttorque Planetary tiltanddirecttorque 1 idesonlunarorbit ' 2rate Fides onspinandmoments 2 7. TIDALANDNONLINEAREFFECTS -6 62 62 -0.03310 -1.919362 +2.7x1o1 -0.0140 t -0.2630 -0.01368 P cos£0-0.003395téxlO“ -0.0268 -0.0000441+3x10“1cos£ -0.000133 t -0.0001021 -0.00643 +0.0010741 +0.00260 +0.0024 sinEcose 0 Rate in"/century Precession Obliquity 2 2 2 2 3 COS £ cos £ cos 2e/sin£ cos 2e/sin£ cos £(4-7sine) 3 cos£-1 6 cose-1 COS £ cose 1 e Dependence 718 1994AJ 108. . 71 IW 2 -8 -62 2 -6 -6 2 2 -632 83 2 2 -6 -62 2 -8 -62 -4 :=:_8 6 2 -62 P parameteroftheIAUtheorydividebycos6toget "/centuries exclusiveofthecontributiongeodesicpre- terms and-5X10bytheselatitudeterms,yielding centric eccentricitychangesare-3395X10"/centuries precession rate(withoutgeodesicprecession)duetohelio- due tothesecondderivativeofe.Thetandtermsin cession. Thereisalsoasmall(higherderivative)contribution eration. Theluni-solaraccelerationis—3395X10 fluence ontheprecessionrateis1.97X10bytheseradial their energydissipationcausestheMoontorecedeand small tterm. "/centuries usedintheIAUtheoryisexcellent,aidedby Í-6X10 "/centuriest.Toconvertthecoefficientto induced precessionthroughtheirsquares.Thefractionalin- Earth’s rotationtoslow.Lunarlaseranalysesindicateasecu- da/di/a =1.00X10~/centuries.Thel/= —cotedeldt, z yz z 2z Xcos //(M+m), 21/ — tanidi/dt] X(1 —e)cosi[da/dt/2a-ede/dt/(l-e (28) 4 2 -62 4 2 2 rate—_9 -32 9 J rates.Whiletherateisclearlyvisibleinsatellitetrack- obliquity ratebyKaula(1964),whenadjustedforrecent the EarthbyMooncauseschangesinbothlunar-and tion, givesanobliquityrateof17X10~"/centuries. "/centuries. Arelated,butnotidentical,calculationofthe is (-102-1.21XHO)X10"/centuries=-23510~ is 24X10“"/centuriesandthetidalprecessionchange culations forthesolarcontribution:tidalobliquityrate the Earth.Thesolartorqueismuchlesswellknownthan secular accelerationmeasurementsandthesolarcontribu- evidence forasmallertorqueratio(Brosche&Wunsch ratio oftideheights(0.46),thoughthereissome of solartolunartorquesisproportionalthesquare lunar. Itisacommonapproximationtoassumethattheratio solar-induced precession.Thesolartidesalsoacttodespin tween theliquidcore,solidmantlepluscrust,oceans,and spin rateoftheEarthbyexchangingangularmomentumbe- which limitknowledgeofthelong-timeaverage(Watkins& Though seemingtovaryonthousandyeartimescales,the This isabout0.7%ofthe—2"/centuriesclassicalaccelera- of theseprocessesdoaffecttheprecessionthroughchanges 1990). Herethefactor1.21isusedtoamendlunarcal- been adopted;thisyields—0.014"/centuriesinprecession. from the7rateandthereappeartobeirregularities ing datatakensince1976,thatrateisimperfectlyseparated rison 1984,1985)appearstobeinaccordwiththereported nontidal accelerationoftheEarth’sspin(Stephenson&Mor- of magnitudelargerthantidallyinducedaccelerations. tion inducedbyeclipticmotion(nextsection)andtwoorders tional toC—Aandexhibitsasmallseculardecreasewhich atmosphere, buttheseleaveCa)unaffected.However,some long-time changesinJandtheappropriatecontributionto uncertainty isthelargestrecognizedinaccel- This choicewillgiveaprecessionaccelerationvalidsince Eanes 1993).ForTable4aJ°f3XlO/centurieshas change intherangeof(—11.6to—16.8)X10"/centuries. has beendetectedfromtheanalysesofrangestoLageos in C—A.TheEarth’sgravitationalharmonicJispropor- changes withartificialsatellites. the precessionwilldependuponfurthermonitoringofJ eration ofprecession.Theprecessionisonlysensitiveto questions abouttheseparationofA8.6yrtidalsignatures these studieslieintherangeof(-2.5to sion rateshouldalsoexhibitadecrease.TheJratesfrom Eanes 1993;Neremetal1993).Consequentlythepreces- and Starlettesatellites(Yoderetal1983;Rubincam1984; Cheng etal.1989;Gegout&Cazenave1991;Watkins 2 polynomial updatesbyBretagnon &Chapront(1981), Equivalent matrixformulations havealsobeenpublished adopted generalprecessionrate, obliquity,andmasses. nomials intimefororientingthe EarthbasedontheIAU- 1976, butfutureextrapolationislesscertain.TheJrate (Lieske 1979;Fabri1980).Improved eclipticmotionledto —3.6)XlO~/centuries; theyinduceasizableprecessionrate 4 z 2 2 2 2 2 2 There areahostofnontidalprocesseswhichchangethe The IAUprecessionpaper(Lieske etal1977)givespoly- 8. POLYNOMIALS 719 1994AJ 108. . 71 IW R andthetotalcontributiontoprecessionratemultiplied well. Inthispapertheoreticalcontributionstoprecessionand which ismovingwiththeequinox.Thetotalcontributionof plus updatedvaluesforprecessionrate,obliquity,masses, by sineisdenotedThese two componentsoftheequa- uity andprecessionratesofTable4useacoordinateframe rate contributionswithrespecttoinertialspace.Theobliq- gument forusewiththeJ2000epoch.Thefixedeclipticand polynomial expressionswillbederivedforasingletimear- lated (integrated)luni-solarandgeneralprecessionrates,re- except thatthetildehasbeendropped.ThesubscriptA(for mial expressions. and computed.Inthissectionthetheoreticalimprovements obliquity ratesandhigherderivativeshavebeenidentified Laskar (1986),andSimonetal.(1994).Thelatterpaperin- the fixedeclipticandmovingequator. Two ofthedifferentialequations arejusttheprojectionsof tor’s rotationvectorareinthe plane ofthemovingequator. the obliquityrate(noeclipticrate)fromTable4isdenoted paper buttheyrequireadditionaltermsduetotheobliquity Fig. 1forthedefinitionofvariables. of dateconstitutethebasicgeometry.SeeIAUpaperand spectively. Rateswillbeindicatedwithderivatives.The accumulated) denotesanangle.Thusijjandpareaccumu- and eclipticmotionwillbeusedtogeneraterevisedpolyno- cludes improvedprecessionrate,obliquity,andmassesas figure. is =FQo-¿+.ThesubscriptAnotusedwithsymbolsinthe the movingequinoxtonodeofeclipticonfixed of J2000andthemoving(meandate)equatorecliptic.Thearcfrom sion”) betweenthemovingequinox andtheintersectionof these tworatesthroughtheangle (“planetarypreces- equator planesofJ2000andthemovingecliptic Fig. 1.Relationbetweenthefixedequator(meanequator)andecliptic 720 J.G.WILLIAMS:EARTH’SOBLIQUITYRATE,PRECESSION,ANDNUTATION € A A The notationoftheIAUpaperisusedinthissection The basicdifferentialequationsweregivenintheIAU © American Astronomical Society • Provided by the NASA Astrophysics Data System an P 'andQwereusedthere. paper withadeviationofnomore than1inthatpaper’slast precision onamicrocomputer. the IAUpapernumberof digits givenforfl^exceeds planetary precessionxaprojectedontheeclipticplane.The precession ratesinthetable.AtJ2000ofgeneral digit exceptforII.Asalsoexperienced byFabri(1980),in the inputvaluesusedinIAUtheory,programwas when thezerocoefficientoftina)isfinite.Theconstant for generatingthepolynomialsÇandz.Thepolyno- merical polynomialfitswerealsoused.Twopointsarenoted derived geometricallyfromtheabovesetofvariables.Nu- iterative solutionforthepolynomialswasdoneinextended from theinitialobliquityandprecessionrateother those givenforP'andQ’apparently additionaldigitsin able tosuccessfullyreproduce the polynomialsinIAU the firstlineandaconstantgeodesicprecessionrate.With two linesforprecessionrateinTable4:thePandttermson mial foro)iscarriedtoonehigherdegreethanthoseÇ and luni-solarprecessionarelinkedthroughtherateof the form,butdifferentnumericalvalues,ofthoseonfirst d(o/dt/sin €di//evaluatedatJ2000andhasopposite obliquity (e)andgeneralprecessionrateplusthepolynomi- sions. FortheIAUtheory,threetermsareusedwhichhave signs forandz. and z.ItisnecessarytoincludeaconstantinÇ als forP'andQ'.TheconstantinTable4isdetermined The threedifferentialequationsandthetwogeometrical tions arefunctionsofXadPa•Twogeometricalequations The differentialequationfortheobliquityratewithrespectto dure isiteratedtoconvergence.InputquantitiesaretheJ2000 intervals, thepolynomialcoefficientsarefit,andproce- are neededtolinktheselattertwovariableswiththeothers ecliptic motion,theright-handsidesofdifferentialequa- tion, respectively,ofthemovingeclipticonfixedecliptic. the eclipticpolewithUandirbeingnodeinclina- where P'=sinttsinUandg'=sinircosdescribe the movingeclipticinvolvesbothmotionof als oftime.Thefiveequationsareevaluatedatequaltime cal technique.Thevariablesarerepresentedwithpolynomi- equations mustbesolvedsimultaneouslyfor(x),if/e and R Xa ,andp. A A A 0 A A0 0 A A 0 A aA A e A r R = R Parameters suchasthecommonlyusedÇ,0andzare The computerprogramwastestedagainsttheIAUexpres- The simultaneoussolutionwasperformedwithanumeri- d€/dt —cospQ'—sinP da)/dt =cosY/^e+sinXaij,, sin Y^ttsin(II4+/?)/sinoj, In additiontoP'andQ',whichareinputfunctionsforthe sin a)di///t=cosXa€- COS(V\+p) =COS^COS(n>ll A A A aiy4 A A + (1—cos7r)cos(Ilpd7r/tR,(30) A€ + sinxasin(IIi/^cos• A (31) (29) 720 1994AJ 108. . 71 IW with dueconsiderationfortheunits.Thenonlinearpartsof been evaluatednumerically.Thefundamentalparameteris Williams &Melbourne(1982)andZhuMueller(1983), Table 5.PolynomialexpressionsfororientationoftheEarth’sequator(arcsec).TimetinJuliancenturiesfromJ2000(JD2451545.0).Greenwichmean from observations.Consequently,anadditionalequationhas GMST expressionwouldalterthedeterminationofUT1 changing theprecessionexpressionswithout mial expressionforGMSTgivenbyAokietai(1982)is of UT1(theconstantcoefficientmatchesAokietal),itsrate, the conditionthatatJ2000therewouldnotbeadiscontinuity about itssymmetryaxis the rotationrateofarigidEarthwithrespecttoinertialspace specific totheIAUprecessiontheory.Aspointedoutby dynamical equationsforrotation.TheIAU-adoptedpolyno- the conventionthatGMSTexpressionisasolutionof of dateandtheGreenwichmeridian.Thispaperwillfollow sidereal time(s)at0hUT1.TimeinUT1centuriesfrom12UT1,JD2451545. UT1 determinedfromoptical astrometric measurementsof or therotationrateofEarthinspace(therearesmall cients oftheconstantandlineartermsweresetbyimposing dividing arcsecondsby15toconvertseconds.Thecoeffi- the GMSTexpressionatzerohourUT,GMST=GMST0 721 J.G.WILLIAMS:EARTH’SOBLIQUITYRATE,PRECESSION,ANDNUTATION constant andlinearcoefficients were pickedforcontinuityof ambiguities attheleveloftruncateddigits).InAokietal catalogue equinoxdrift.Itisconventional toderivethesmall the determinationofUT1sothere isnocounterparttothe ertial frame.Inertiallyreferenced techniquesnowdominate catalogue starsratherthancontinuity withrespecttoanin- (GMST) relatesUT1totheanglebetweenmeanequinox -bUTl, comefrom © American Astronomical Society • Provided by the NASA Astrophysics Data System ¿/(GMST+^)/dil//dt—XA(33) A ru Angle GMS TO £â Xa Ça e £a Ça Pa Va k C0 n Q’ z P' A a A A A 629543 24110 84381 84381 84381 -2 2 Constant 0 0 0 0 0 0 0 0 0 .54841 8640184 409000 000000 511180 511180 409000 409000 000000 000000 000000 000000 000000 967373 000000 000000 000000 2004. 2306. 2306. 5038. 5038. 5028. -867. -46. -46. -46. 46. 10. 10. -0. 4, 7928613 456501 557700 182023 456501 770000 809560 065079 071060 833960 557700 024400 919986 997570 809560 199610 -0.429466 -0.000174 -2.381366 -1.078977 -0.033506 0.0927695 0.051142 0.493164 1.092516 0.299027 0.051268 1.105407 1.558353 0.153382 0.051043 0.193971 values matchthoseusedtogenerateTable3(Sec.5).The precession rateandobliquityplus long-periodic,oratleast for longertimes.Thepolynomialsareequivalenttoexpan- future improvements:sometheoretical,certainlyinthemea- liquity, andthetheoreticalcomputationsofKinoshita& ecliptic duetotheoreticaladvancesandimprovedplanetary been improvementsinthecomputationofmotion sions ofexpressionsappropriateforlongertimes:anaverage times extendingouttoafewmillennia,butarenotsuitable fully inthepredictiveknowledgeofJrate. Table 4havebeenused.Theresultingexpressionsaregiven planetary masscorrections.Thetheoreticalcontributionsof changes, revisedexpressionsarepresentedhere.Theinput masses, bettermeasuredvaluesforprecessionrateandob- et al1992).NontidalJchange mustbetransient.Mostof cession andobliquitybehavior forancienttimes(Berger small tidalaccelerationisinexorable andmodifiesthepre- quasiperiodic, termswithperiods exceeding10000yr surement oftheprecessionconstantandobliquity,hope- est accuracynow,itshouldbeanticipatedthattherewill While theseexpressionscanservethosewhoneedthehigh- from JD2451545.0UT1=12h.onJanuary1,2000. measured fromJ2000[i=(JD-2451545.0)/36525],exceptfor ecliptic motionistakenfromSimonetal(1994)including Souchay (1990)andthispaper.Toillustratetheresulting sion usingaUT1timescale. nonlinear termsofGMST0usingalineartimescaleforthe GMST whichusessecondsandcenturiesofUT1measured in Table5.TheunitsarearcsecondsandJuliancenturies independent time,buttoevaluatetheentireGMST0expres- (Berger 1976;Laskaretal1993). Formillionsofyearsthe 2 2 t2 The polynomialexpressionsinTable5canbeusedfor Since theIAUtheoryforprecessionappeared,therehave -0.0000003 -0.000186 -0.000309 -0.041822 -0.000223 0.018017 0.000531 0.018265 0.000522 .002000 .007727 ,001208 .001141 .000076 .000026 .000124 -0.0000020 -0.000027 -0.000003 -0.000007 -0.000029 -0.000005 -0.000001 -0.000024 -0.000004 -0.000001 -0.000001 0.000000 0.000000 0.000170 0.000133 0.000000 t* 721 1994AJ 108. . 71 IW will dampdown. of theactiveEarth’sJfromspin-controlledequilibrium the Earth’soblatenessiscausedbyitsspinandfluctuations 722 J.G.WILLIAMS:EARTH’SOBLIQUITYRATE,PRECESSION,ANDNUTATION When combiningtherotationsforprecessionandnutation, the numberofrotationscanbeminimized.Finally,anexpe- able forthevarioussetsofprecessionparametersinTable5. J2000 tothemeanequatorandequinoxofdateusing modifications. liquity withoutundertakingthemoreextensiveandcomplete most important(linear)correctionstoprecessionandob- dient procedureisgivenwhichsuitableforintroducingthe cedure istoprecessfromthemeanequatorandequinoxof orient theprecessingandnutatingEarth.Thestandardpro- angles £4,,andzthentonutatethetrueequator An alternativeistoprecessbymovingalongthefixedeclip- the rotationmatrixaroundaxisÏ)is ing nutationinobliquity.Thesequenceofsixrotations(Æ,is of date,andthentorotatethetrueequatorialplaneinclud- date, applyingnutationinlongitudetoreachthetrueequinox and equinoxofdatebyrotatingintothemeanecliptic 2 then tonutateasbefore.Thesevenrotationsare rotate alongthatequatortothemeanequinoxofdate,and tic totheintersectionwithmeanequatorofdate,then fixed ecliptictotheintersectionwithmeanof date, rotatebackalongthemeanecliptic,andthennutate to nutateaswellprecess,itispossiblebypassthemean A the meanequinoxofdate)andnutationinlongitude, the movingecliptic(angle£4),rotateinto one canmovealongthefixedequatortointersectionwith for combiningprecessionandnutation.Withfourrotations along themovingecliptic.Fiverotationsisnotminimum equator ofdateandcombinetheprecessionnutation rotate tothetrueequator (64), combinetherotationalongeclipticofdate(r}to ventional precessionexpressions inthepast.Theyareillus- The angles£4,e,andr]have notbeengivenwithcon- trated inFig.1andtheexpressions aregiveninTable5. polynomials not duetoeclipticmotion,only thelasttwo A A © American Astronomical Society • Provided by the NASA Astrophysics Data System Considered inthissectionaresequencesofrotationssuit- Consider thesequenceofrotationswhichcanbeusedto Æ !(-Ae)i?3(1(Xa"a) A secondalternativeistoprecessbymovingalongthe flli-eA-AiO^-nA-pA-A^l^A) The thirdprocedurerequiresonlyfiverotations.Ifoneis R{-e^)(7ili)R {e')R^. (37) Note thatforthechangesin precession andobliquity lAi< lA3 0 X/?^)/^-^). (34) X7?3(-,AA)«i(eo)- (35) xR(Ha)Ri(€q). (36) 3 9. ROTATIONS obliquity changesaredistributedovermultiplerotations.The changed; forthefirsttwosequencesprecessionand last twosequencesmakeitclearthatachangetothepreces- ing nutationparametersasanalternativewaytointroduce sion andobliquityratescouldbeaddedintothecorrespond- (left) rotationsinthelasttwosequencesabovewouldbe them (asimilarconclusionwasreachedbyFolkneretal ear termsintheir“nutation”corrections).Adding the precessionandGMSTexpressions.Forthisexpedient rate themostimportantcorrectionswithoutreprogramming retical contributions,suchasthenonlinearcorrections,could obliquity needchangingfortheexpedientapproachsince not requirerevision.NeitherdoestheJ20Ö0valueof cations changingUT1,sothattheGMSTOpolynomialdoes approach theequationofequinoxesassociatedwithA^will dure doesnotworkforgeometricalrevisionstoeclipticmo- difference tendstocancelbetweentherotations.Othertheo- automatically satisfytheconcernsofWilliams&Melbourne (t incenturiesfromJ2000)isanexpedientwaytoincorpo- 1994 andVLBIfitstoobservationshaveoftenincludedlin- parameters) sincetheyappearinmultiplerotationsandtend tion (purelyP’,g',IÏ4,and774butalsopartsofother also beaddedasnutationcorrections.Theexpedientproce- (1982) andZhu&Mueller(1983)aboutprecessionmodifi- precession andGMSTO. rections toprecessionandobliquitywhilemorethoroughgo- expedient. Itisbestsuitedtoeasilyinsertingthelinearcor- expedient procedurebecomestoocomplicatedthenitisnot quences ofthoserevisions.However,itisobservedthatifan to cancel,butitcouldbeappliedthedynamicalconse- ing revisionscanuseTable5’spolynomialexpressionsfor -0.02447centuries ttoAeand-0.32197centuriesA'F combination oftherigid-bodyseriesKinoshita(1977)and the elasticandstructured-EarthcorrectionsduetoWahr paper therehavebeentworevisionsofrigid-bodynutation based ontherigid-bodytheoriesanditiswelltocompare tation theoriesallowingfortheEarth’selasticityandcoreare (1979, 1981).Sincethe1980IAUnutationworkinggroup Earth’s J.Ithasservedasthebasisfornonrigidbodytreat- and understandthoserigid-bodytheories. (Zhu &Groten1989;KinoshitaSouchay1990).Thenu- Herring (1990),Mathewsetal.(1991),and ments byZhuetai(1990),andtheseveralZMOAseriesof planetary termsinvolvingarguments.Inaddition adding bothsecond-ordertermsandcorrectionsforthe vised theexpressionfor(C-A)/C scalingofthenuta- they addedsmallsolartermsdue totheoffsetofEarth smaller terms,andaddedsecond-orderJeffects, (1977) extendingthenutationseriestosmallerterms,and from thecenterofmass Earth-Moon systemandre- (1991). Kinoshita&Souchayalsoextendedtheseriesto tion seriesfromtheprecession constant. InKinoshita(1977) and Kinoshita&Souchaythe J tilteffectsonthescaling and 18.6yrtermsarepresent.The in-phase18.6yrnutation 3 3 2 10. NUTATIONCORRECTIONS,SCALING,ANDCOMPARISONS The 1980IAUnutationseries(Seidelmann1982)wasa Zhu &GrotenutilizedtheearlierworkofKinoshita 722 1994AJ 108. . 71 IW work. Kinoshita&Souchayalsoadd—0.0147centuries planetary tilteffectispresentintheformer,butnotlatter present intheearlierworks.Thustherearesmalldifferences revisions tothefirst-ordercontributionswhicharenot ferent signsanddifferby0.12mas.Inaddition,the6164day poor agreement;the3231dayobliquitycoefficientshavedif- this paperwhichareaddressedbelow.OnlyKinoshita& in thescalingofKinoshita&Souchay,ZhuGroten,and second-order contributionstotheprecessionandmakesmall coefficients andthe—0.00567centuriesprecessiondueto for furthercomparisons. coefficients disagreebyafactoroftwo.SeeSouchay(1993) three ofwhichhaveamplitudes0.02masinlongitude. this paper.Thiswouldcausethe18.6yrcoefficientto tation theorywouldhavetobemultipliedby0.99993782 Souchay havethesmallcenter-of-massoffsetcorrections, planetary tilteffect,the18.6yrtermsofZhu&Grotenand gives —17.28075"inA^and9.22792"At.Thusafter change by—0.574mas.Thesecorrectionsareinadditionto increase by1.075masandthe18.6yrAtcoefficientto Comparison oftheJcontributionsintwopapersshows 723 J.G.WILLIAMS:EARTH’SOBLIQUITYRATE,PRECESSION,ANDNUTATION phase planetarytiltcontributionsofTable1needtobe correction toacommon(C—A)ICandcompensationforthe the solarterms.Forin-phase18.6yrcoefficientsthis plied by0.9999308forthelunartermsand9297 to the18.6yrcoefficientsare1.045masinA(/f(giving those ofTable1,andtakentogetherthein-phasecorrections Kinoshita &Souchaydifferbyonly0.01masinlongitude Zhu &Groten’srigid-bodynutationseriesneedtobemulti- (Sec. 5)tomatchtheprecessionrateandotherchangesof added. ToallnutationseriessinceWoolard,a—0.15mas and 0.08masinobliquity. of Kinoshita&Souchay,shouldalsobeincluded.Whileit racy, nutationtermswithplanetaryarguments,suchasthose sion needstobeaddedthenutationinlongitude(Voinov annual termfromtheyearlyvariationofgeodesicpreces- planetary arguments. to thetime-varyingheliocentricorbit.Itshouldbecompat- period termsarebeingcarriedwhentheseargumentsfit the valuesofVandL'dependverymuchuponwhichlong- can beimproveduponbyusingthevaluesofSimonetal causes minorchangesintheresultingnutationseriesevalu- ible withtheformulationusedtogeneratenutations ation, theargumentsof1980IAUseriesandother -17.280 76")and-0.571masinA*(9.22800"). the Earth’sequatorplaneand wishtousetheseobserva- (1994). Theannualargument(/')differsby5"atJ2000,but 1988; Gilletal1989;Fukushima1991).Forhighestaccu- the theoriesofprecession,obliquity rate,andnutationdesir- tions toinfertheEarth’sproperties makeimprovementsin able. Theratetermscomputedin thispapercomefromlunar 3 © American Astronomical Society • Provided by the NASA Astrophysics Data System The coefficientsofKinoshita&Souchay’srigid-bodynu- To allnutationseriessinceWoolard(1953)theout-of- To matchtheconstantsofthispapercoefficients Improvements intheaccuracy of theobservedmotion 11. SUMMARY 2 2 orbit perturbationsduetotheplanetsandEarth’sJplus direct planetarytorquesontheEarthandtidaleffects. etary torquesontheEarth(seeSec.4andTable2), The largestcontributiontotheobliquityrateinspacewas conventional —0.468"/yrobliquityrateisduesolelytoeclip- tidal influences(Sec.7,Table4).Togetherthesecorrections due toplanetaryperturbationsonthelunarorbit(Sec.3),and tic motion,nottochangesintheEarth’sorientation.The are —0.244mas/yr.Thiscorrectionisamotioninspace;the for thisdiscordantinterpretationandconcludesthattherate tioned byKinoshita(1977).Section6discussesthereason earlier computedbyWoolard(1953),butitsrealitywasques- LAU-adopted theoryofprecessionandobliquitychangesre- //Mi?=0.329 9789,withRtheequatorialradius. by theverylongbaselineinterferometryandlunarlaserrang- quires correctionforthiscontributiontotheobliquityrate. ing techniques. is real.Theobliquitymotioninspaceshouldbeobservable The sumofthevariouscontributionstoobliquityandpreces- small contributionstotheprecessionrateduedirectplan- both linearandnonlinear(intime)contributions.Thelargest sion ratesisgiveninTable3(Sec.5).Basedonrecentmea- etary torquesandlunarorbiteffects(Secs.34,Table2). surements ageneralprecessionrateof50.2877"/yr theory fororientingtheEarth(precession,obliquitychanges, nonlinear correctionarisesfromtheEarth’sJrate.The other sourcesarecomputedinSec.7.Table4summarizes constant withtimeandthehigherderivativesfromthese C/MR=0.330 7007andanormalizedmeanmoment this givesanormalizedpolarmomentofinertia to themoment-of-inertiacombination(C—A)/C at J2000wasadopted.ForarigidEarththiscorresponds 2 Table 1).Thelargestcontributionsaretothe18.6yrnuta- planets, alsogiverisetonutationcontributions(Sec.3and rate, masses,andeclipticmotion. expressions useimprovedvaluesoftheobliquity,precession In additiontothetheoreticalcorrectionsofthispaper,these and GreenwichMeanSiderealTime)isconsideredinSec.8 nutation. rotations isgivenwhichincorporatesbothprecessionand not optimizedforthenumberofrotations.Asequencefour and revisedpolynomialexpressionsarepresented(Table5). phase nutationsareconventionally consideredtoarisefrom corrections arisebecausetheperturbingplanets’nodeson tion: -0.030sinÍ1+0.137cosÍ1toA*/'(inmas)and are consideredinSec.9.Theconventionalrotationschemeis ecliptic arenotalignedwiththedynamicalequinox.Out-of- on {C—A)!Csothatanaccurate determinationofthepre- exceptions. energy dissipationintheEarth andoceans,buttheseare cession ratesetsthescaleof nutationseries.Thisscaling =0.003 2737634.Combinedwithasatellite-determinedJ -0.028 sinÍ1+0.003cosÍ1toAe.Thesmallout-of-phase 2 2 The correctionstotheobliquityrateareduedirectplan- In additiontotheobliquityrateamendments,thereare The contributionstoobliquityandprecessionratesarenot The torques,duetolunarorbitperturbationsfromthe Matrix rotationswhichcombineprecessionandnutation The torqueswhichcauseprecession andnutationdepend 723 1994AJ 108. . 71 IW Laskar, J.,Joutel,R,&Boudin,F.1993, A&A,270,522 Laskar, J.1986,A&A,157,59 Kinoshita, H.1977,Celest.Mech.15,277 Kinoshita, H.1975,SmithsonianAstrophysicalObservatorySpecialReport, Kaula, W.M.1964,ReviewsofGeophys.2,661 Herring, T.A.,Buffet,B.Mathews,RM.,&Shapiro,I.1991,J. de Sitter,W.,&Brouwer,D.1938,Bull.Astron.Inst.Neth.,8,213 Dehant, V.1990,Geophys.J.RoyalAstron.Soc.,100,477 Dehant, V.1988,inTheEarth’sRotationandReferenceFramesforGeodesy Aoki, S.,Guinot,B.,Kaplan,G.H.,Kinoshita,McCarthy,D.D.,& Kinoshita, H.,&Souchay,J.1990,Celest. Mech.andDyn.Astron.48,187 Herring, T.A.1990,inProceedingsofthe127thColloquiumInter- Gill, E.,Soffel,M.,Ruder,H.,&Schneider,M.1989,inEarth’sRotation Fukushima, T.1991,A&A,244,Lll Folkner, W.M.,Chariot,R,Finger,M.H.,Newhall,X.X,Williams,J.G., Fabri, E.1980,A&A,82,123 Gegout, R,&Cazenave,A.1991,Geophys.Res.Lett.18,1739 Cheng, M.,Eanes,R.,Shum,C,Schutz,B.,&Tapley,B.1989,Geophys. 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