198 9MNRAS.2 38.152 9H h h The EquationofTimeisthedifferencebetweensolartimeandmeantime.Solaratany uniform rate.BothorbittheEarthinexactlyonetropicalyear.Someantimeisindicatedby the Sun;andmeanSun(afictitiousobject)thatmovesalongcelestialequatorata two Suns,theactualSun(knownusuallyasapparentSun)whichmovesalongcelestial instant isthehourangleofSunatthatinstant.Thedifferencecalculatedbyconsidering by sundials. mean Sun,andapparentsolartimeisindicatedbytheactualSunis,forexample,measured ecliptic atavaryingrate,thisvariationbeingduetotheeccentricityofEarth’sorbitaround E =GHA(apparentSun)-(mean ( 1) 1 Introduction where GHAistheGreenwichhourangle.Analternative formulaoftenquotedintextbooksis where ST°istheGreenwich apparentsiderealtimeindegrees,whichisafunction ofUT,a°is Rewriting equation(1)intermsthatarecalculablegives In oldertextbooksthetermsonright-handside ofequation(2)areusuallyreversed. the apparentrightascension oftheSunindegreesandUTisUniversalTime inhours. E =rightascensionofthefictitiousmeanSun- ascension oftheapparent(actual)Sun.(2) (UT )andEphemerisTime (ET).Theprecisemeaningofthesetermsisimportant astheyboth E° =(ST°-a°)-(15UT180°) (3) The precisedefinitionoftheEquationTimeis Since 1960mosttextbooks havetakenintoaccountthedifferencebetweenUniversal Time © Royal Astronomical Society • Provided by theNASA Astrophysics Data System The EquationofTime Mon. Not.R.astr.Soc.(1989)238,1529-1535 David W.HughesDepartmentofPhysics,UniversitySheffield, function ofUniversalTime.Thisenablesittobecalculatedforanyepoch Accepted 1989January16.Received13;inoriginalform1987August21 Time, andsomaybecomparedwithexpressionsgiveninoldtextbooks. ignores thedistinctionbetweentime-scaleofEphemerisandUniversal within 30centuriesofthepresentday,toaprecisionabout3stime.We B. D.YallopandC.Y,HohenkerkRoyalGreenwichObservatory, Summary. AnequationisdevelopedwhichgivestheEquationofTimeasa Sheffield, S37RH also giveseveralexpressionsfortheEquationofEphemerisTimewhich Herstmonceux Castle,Hailsham,EastSussex,BN271RP 198 9MNRAS.2 38.152 9H formally definedbyamathematicalformulaasfunctionofsiderealtime.UTisdetermined from observationsofthediurnalmotionsstars.Thetime-scalethatisdetermined occur asargumentsintheEquationofTime.Definitionsthesetime-scalestakenfrom by polarmotion,thetime-scaleUT1isobtained.Wheneverusedinthispaper directly fromsuchobservationsisdesignatedUTOandslightlydependentontheplaceof Glossary ofTheAstronomicalAlmanacarerepeatedinthissectionforclarification. implied. Thetime-scaleavailablefrombroadcastsignalsiscalledCoordinatedUniversalTime observation. WhenUTOiscorrectedfortheshiftinlongitudeofobservingstationcaused all civiltimekeeping.SincetheEarthisslowingdownitnotauniformtime-scale.UT maintained within±0^9ofUT1. (UTC). ItisbasedontheInternationalAtomicTimeScaleTAI(whichuniform)and Time (DT).Thetime-scaleofephemeridesforobservationsfromthesurfaceEarthis variable ingravitationaltheoriesofthesolarsystem.In1984,ETwasreplacebyDynamical 1530 D.W.Hughes,B.YallopandC.Y.Hohenkerk it isreferredtothebarycentreofSolarSystem.IncontextEquationTime called TerrestialDynamicalTime(TDT);itisBarycentric(TDB)when ference betweenUTandEThasbeenignored.OntheotherhandtermEquationof gravitational potentialaroundtheEarth’sorbit,doesnotexceed0?002. not necessarytodistinguishbetweenthembecausethedifference,duevariationsin being thepositionofperihelionEarth’sorbitwithrespecttoFirstPointAries, Time isusedwhenitexpressedasafunctionofUT,whichmoreusefulinpracticalcases. Equation ofEphemerisTimewasmoreappropriateinsuchcases.Inthispaperwehave Woolard &Clemence(1966)suggestedthatreplacingthetermEquationofTimeby variation oftheEquationEphemerisTimeasafunctiontimeoverlongperiodsis the obliquityofeclipticandeccentricityEarth’sorbit.Themainreasonfor longer timeperiodsduetochangeswiththethreeparametersthatitisdependentupon,these the Equationvariesslightlythroughoutleapyearcycle.Secondly,over adopted thissuggestionsothatthetermEquationofEphemerisTimeisusedwhendif- whereas nowitoccursaroundJanuary2. the precessionoflineapsidesEarth’sorbit.Inessencelongitudeperigeeis Time inSection3allowsustolookattheformof functionthroughitsparameters.Section increasing byabout1?72percentury.Soaroundtheyear4000BCattimeofbuilding versions inoldtextbooks. over 60centuries,toanaccuracyof3stime.The formulation oftheEquationEphemeris of theGreatPyramidinEgypt,perihelionpassageoccurrednearautumnalequinox, This algorithmcalculates the EquationofTimeforanyspecifiedcalendardate andUniversal Time accurateto3softime overaperiodofwithin30centuriesthepresent day. 4 givesseveralexpressionsfortheEquationofEphemeris Timewhichcanbecomparedwith published inTheAstronomical Almanac,whiletheexpressionforsidereal time intermsof 1921) didnotdistinguishbetweenUTandETwhendiscussingtheEquationofTime. Universal Timeisbasedon the1976IAUsystemofastronomicalconstants, the1980LAU 2 AlgorithmforcalculatingtheEquationofTime Universal Time(UT)conformscloselywiththemeanmotionofSunandisbasis Ephemeris Time(ET)wastheuniformtime-scaleusedbefore1984asindependent The useoftheEquationTimeiscomplicatedbytwootherfactors.First,exactform Older textbooks(e.g.Ferguson1803;Hymers1890;Godfray1906;Ball1908;Hosmer In Section2ofthispaperwegiveanalgorithmfor calculating theEquationofTime,valid The argumentsofthesolar orbitarebasedontheepochofJ2000.0and theformulae © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 198 9MNRAS.2 38.152 9H 0=z2 83 h o=2 h 23 h 2 theory ofnutationandtheequinoxFK5catalogue.Anapproximationmadeinthis date Y-M—DwhereYistheyear,Mmonthanddayofasfollows: mean quantities.TherearenewimprovedtheoriesforthemotionofEarthbasedon Bretagnon (1982)whichshouldbeusedifgreateraccuracyweredesired. algorithm istoignoretheeffectofnutation,whichsmall,andreplaceapparentquantitiesby /= [365.25(y+4712)][30.6m0.5]59D-0.5 Step A where [x]means‘taketheintegerpartofx\ Calculate theGreenwichmeansiderealtimefrom For theJuliancalendar,G=0. Thus JD =/+G. mean anomaly(G),theobliquityofecliptic (e),andtheequationofcentre(C). Calculate therightascensionofapparentSunusing theDynamicalTimeintervalT. StepC where AT(¿)=0betweenAD1650and1900 in DynamicalTime: ST 100.4606-36000.77005t+0.00038813x 10“1 This expressionforATisbasedontheworkbyMuller(1975).Morerecentestimatesare and time(UT)requiredinJuliancenturiesof36525d,Tthecorrespondinginterval Calculate thetimearguments,intervalfrom2000January1at12UTtodate[JD) Step D given byStephenson&Morrison(1984). Step B and L=280?46607 +36000?76980T+0?0003025T /=(//) +f/T/24-2451545.0)/36525 G =38-[3[49+y/100]/4]. e° =23?4393-0?01300T- 0?0000002T+0005 The JuliandateJDat0UT(i.e.midnight)iscalculated(seeHatcher1984)fromthecalendar n n C° =(1?9146-0?00484T- 0?000014T)sinG+(0?01999-0?000082G. G° =357?528+35999Î0503T r=/-HAT(/), n (2) Calculate (1) IfM>2sety=Yandm=M-3,otherwiseY--1m*=M+9. (3) IfthecalendardateisGregorian,set (4) CalculatetheJuliandatefrom (1) Calculatethesolararguments;geometricmean eclipticlongitudeofdate(L),the © Royal Astronomical Society • Provided by theNASA Astrophysics Data System =:28 A'F(¿)[”3.6 +1.35(¿2.33)]x10“elsewhere. The EquationofTime1531 198 9MNRAS.2 38.152 9H 2 2 0h 2 3 h 2468 2 1532 D.W.Hughes,B.YallopandCY.Hohenkerk where y=tane/2and/=180/Jr. L% =L°+C°-0?0057. aberration andthecorrectiontocentre.Thisgives As statedinequation(3)theEquationofTime,E,degreesisgivenby This lastlineensuresthatthediscontinuitiesat360°aretakenintoaccount. If £>10°thenE=E-360° Step E a° =Lo~y/sin2L+0.5y/sin4L, from thepresentepoch.Thesetermsappearasfirsttwoofequation(9)where formula makestheapproximationthatST°=L°—180andDynamicalTimeUT.The form In thepastmanytextbooksquoteEquationofEphemerisTimeformulainsimplified E ={ST°-a)—(\5UT-180°) where ListhelongitudeofmeanSunandarightascensionapparentSun.This the EquationofEphemerisTime. 4 TheEquationofEphemerisTime correct formulaforsiderealtimehasbeenusedtoformalong-termexpressioncalculating approximation thatsiderealtimeequalstheSun’smeanlongitudeshiftedby180°ignoressmall E =L-a, As eisapproximately0.017,termsbeyondthecube canbeignoredhere.Notethatequations longitude ofperigeetheSun’sorbitaroundEarth andeistheeccentricityofthisorbit. The first(markedEinFig. 1)isduetotheellipticityofEarth’sorbit.By substituting into expressing thetrueanomalyofSuninterms of themeananomaly,co=L—Gis and fistheobliquityofecliptic,Llongitude oftheapparentSunandisobtainedby ( 5)and(6forLaareinradians. Lq =L+Î2c-^j-|sin(L-ft>)^esin{2L-2œ) csin(3L-3et>)+... where expansion is 0o At perihelion(January2, 0500 in1986)theapparentSunandmean arecoincident. The apparentSun,however, hasitsgreatestvelocityatthistimeandwillshoot aheadofthe minutes oftime.TheEcurve crossesthezeroabscissaatperihelionandaphelion passagetime. equation (6)itcanbeseen thattheamplitudeof‘componentE’curveinFig. 1islessthan8 mean Sun,the intervalbetweenthemcontinuing toincreaseaslong astheangularvelocity of a =E(?)tan-sin2L+-sin4L—sin6L+sin8L+..., T andtermswhichneverthelessintroducessignificanterrorsafteraboutathousandyears 0 0 0 (2) Calculatetheeclipticlongitudeofdate,(L),frommeanbyapplying (3) Calculatetherightascensionfrom 0 An approximateexpressionforaisgiveninStepD(3)Section2;moreprecise Using thisformulationitcanbeseenthattheEquation ofTimehastwomajorcomponents. © Royal Astronomical Society • Provided by theNASA Astrophysics Data System u 234 £ IfIf (4) (6) (5) 198 9MNRAS.2 38.152 9H marked ‘O’isduetotheapparentSunmovingalongeclipticwhereasmeanmovescelestial Figure 1.ThepresentdayEquationofTimeisplottedasafunctiontimethroughouttheyear.When period ofoneyearandcrossestheabscissaatperihelionpassagetimeapheliontime.The Equation ofTimehastobeaddeditsreadinggivemean-time.Theismadeuptwo respectively. equator. Thiscrossestheabscissaatequinoxesandsolstices.Thesearenotseparatedby90°oflongitude.At components bothshownasdashedcurves.Theonemarked‘E’isduetotheellipticityofSun’sorbit.Ithasa curve isabovetheabscissaapparentsolartimelessthanmeantime,i.e.sundialslowand the presenttime,forexample,winter,spring,summerandautumnoccupy87?71,91?42,92?3288?55 noon andthecomponentofEquationTimeduetothiscauseisadditive.Similar meridian overtakesthemeanSunbeforeapparentSun.Sonoonoccurstrue the apparentSunexceedsitsmeanvalue.Justbeyondasolarlongitudeof90°fromperihelion, the twoangularvelocitiesbecomeequalandafterthattimeintervalbetween reasoning showsthattheaphelionandperihelionmovementsresultsinanegativequantity. diminishes untilitbecomeszeroataphelion.Betweenperihelionandaphelionaspecificearth A muchsimplifiedexpression fortheEquationofTimecanbeobtained bysubstituting (for example,atthepresenttime,summersolsticetoautumnalequinoxis~93.6dwhereas amplitude ofabout9.9minutestimeandcrossesthezeroabscissaatequinoxes celestial equatorwhereastheapparentSunmovesalongecliptic.Thiscurvehasan the Earth’sorbitisonlyanellipsetoafirstorderapproximation. Thegravitationalinfluenceof solstices. Thecurveisonlyquasi-sinusoidalbecausethelengthsofseasonsarenotconstant equations (5) and (6)intoequation(4) thenneglectingtheminor terms.Thus the MoonandotherplanetsgivesEarthasomewhat irregularpath. autumnal equinoxestowintersolsticeis^89.8d,seeHughes1989). 4 Approximateexpressions fortheEquationofEphemerisTime given tothenearestday.Itshouldalsobenotedthat thetimebetweenperihelionandaphelion to thenearesthour(seeforexample,Roberts&Boksenberg, 1985).Priorto1975itwasonly Earth-Sun distance.Asthisisaveryslowlyvarying function, thisminimaisusuallyonlygiven An alternativeapproachtotheproblemisproduce along-termmeanorbitfortheEarth/ passage isonlyapproximatelyhalfayearandeasilycan varybyadayortwoaroundthisvalue. simplest approachistolookeachyearfortheminima indrjdt,thetimedifferentialof Moon barycentreandobtain theperihelionpassagetimeforthisorbit. The secondcomponent(markedOinFig.1)isduetothemeanSunmovingalong The journeybeyondthecomplexitiesofequations(5) and(6)isnothelpedbythefactthat The calculationof,forexample,theperihelionpassage timeisnotstraightforward.The © Royal Astronomical Society • Provided by theNASA Astrophysics Data System The EquationofTime1533 198 9MNRAS.2 38.152 9H 2 2 where Eisexpressedinradians,anddoesnotdeviate,forthepresentepoch,bymorethan18s 1534 (1971 ),preciseto1arcmin.HisexpressionforEindegreesis of timefromthoseobtainedbyusingthemorecompleteapproach. E =-2esin(L-co)+tan-sin2L £ =-(0.388+0.0593T-0.00006T)sinL(1.8020.01550.00086cos where TismeasuredinJuliancenturiesfrom1900January0.5ET(i.e.Day the twocalculatedvaluesneverexceeds13softime. a preciseformulationneverexceeds4softime(1arcmin).Forepoch0thedifferencebetween 2415020.0). ForepochAD2000,thedifferencebetweenYallop’svalueandonederivedfrom has beencalculatedforninedifferent epochs.AnnoDomini4000(a),3000(b),2000(c),1000(d), 0(e),-1000(f), Figure 2.TheEquationofEphemerisTimeisplottedasafunction oftimethroughouttheyear.Thisequation plotted forthefollowing400d in eachcase.Datesafter,andincluding,AD2000areaccording totheGregorian noon, oftheyearinquestion. These zerosareequivalenttoJulianDays3182030.5,2816788.0, 2451545.0, calendar, datesbeforeAD2000 areJulian. 2086308.0, 1721058.0,1355808.0, 990558.0,625308.0and260058.0,respectively.The Equation hasbeen - 2000(g),3000(h)and4000(i). Ineachcasethezeropointonabscissacorresponds toJanuary0,at Yallop (1978)quotedaformulafortheEquationofEphemerisTimebasedononebySmart © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 2 2 + (2.487-0.0034T-0.00004T)sin2L(0.0060.0012cos2L(8) + (0.0160.0025T)sin3L(0.081-0.0009T-0.00004cos3L - (0.053~0.0001T)sin4L, 0 100200 300 400 D. W.Hughes,B.YallopandC.Y.Hohenkerk Time, indays (7) 198 9MNRAS.2 38.152 9H 2 -62 adequate foruseoveratimeperiodof60centuriesmaybederivedfromthemethodinSection where e=0.016708-0.000423T-0.00000013T,istheeccentricityofEarth’sorbit,and the equationsforL,Gand£aregiveninStepD(l).Thetwomostdominantperiodicterms 2 bysettingAT=0,i.e.T=t.Thus produced byusingthisformulaarewithin±0!8orabout3.2s. this expressionarethosegiveninequation(7)above.Overarangeof60centuriestheerrors £ =4.47x10T+1.49xT-2esmG-¡m2G+ysm2L-t>ym4L Ferguson, J.,1803.AstronomyexplainedonSirIsaacNewton'sprinciples...,11thedn,London. Ball, SirRobertS.,1908.ATreatiseonSphericalAstronomy,CambridgeUniversityPress. References thousand yearsbetween4000BCandAD4000. Woolard, A.W.&Clemence,G.M.,1966.SphericalAstronomy,AcademicPress,NewYorkandLondon. Hatcher, D.W.,1984.Q.Jl.R.astr.Soc.,25,53. Godfray, Hugh,1906.ATreatiseonAstronomyfortheuseofCollegesandSchools,6thedn,Macmillan&Co., Bretagnon, P.,1982.Astr.Astrophys.,114,278. Yallop, B.D.,1978.TechnicalNote46,HMNauticalAlmanacOffice,RoyalGreenwichObservatory. Roberts, K.C.&Boksenberg,A.,1985.TheAstronomicalAlmanacfortheyear1986,USGovernmentPrinting Muller, P.M.,1975.PhDthesis,UniversityofNewcastle. Hymers, J.,1840.TheElementsofAstronomy,CambridgeUniversityPress. Hughes, D.W.,1989.J.Br.astr.Ass.,inpress. Hosmer, GeorgeL.,1921.Text-bookonPracticalAstronomy,JohnWiley,NewYork;Chapman&Hall, Stephenson, F.R.&Morrison,L.V.1984.Phil.Trans.Soc.Lond.A,313,47. Smart, W.M.,1971.Text-BookonSphericalAstronomy,5thedn,CambridgeUniversityPress. A short,mediumprecisionexpressionfortheEquationofEphemerisTimeinradians, Fig. 2showsthevariationofEquationEphemerisTime,usingthisexpression,every © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 2 + 4eysinGcos2L|eysin2Gcos2L-4eyinGcos4L-8einGin2L,(9) London. Office, WashingtonandHerMajesty’sStationaryLondon,p.A9(DiaryofPhenomena). London. The EquationofTime1535