Explanatory Supplement Astronomical Almanac PDF [PDF]

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EXPLANATORY SUPPLEMENT TO THEASTRONOMICAL ALMANAC



A Revisioflto the EXPLANATORY SUPPLEMENT TOTHEASTRONOMICAL EPHEMEBIS and THEAMERICAN EPHEMERIS ANDNAUTICAL ALMANAC



Preparedby THENAUTICAL ALMANAC OFFICE,U.S.NAVALOBSERVATORY WITHCONTRIBUTIONS FROM H.M.NAUTICAL ALMANAC OFFICE,BOYALGREENWICH OBSERVATORY JET PROPULSION LABOMTORY BUREAUDESLONGITUDES, and THETIMESERVICE ANDASTROMETRY OEPARTMENTS. U.S,NAVALOBSERVATORY



Editedby P KennethSeidelmann



UNIVERSITY SCIENCEBOOKS MillValley,Califomia



AbbrEViAtEd CONTENTS University ScienceBooks 20 Edgehill Road Mill valley, CA 9494 I F a x :( 4 1 5 )3 8 3 - 3 1 6 7 Production nanager:MaryMiller Copyedibr AidanKelly andMdSy KDnD TexIdd jacketdesigner: Robenhhi TEXfomarer and illusrabr: Ed Sznyer Pr@ftqder: Jm McDemon Printerod biDder:The Maple Vail B@k Mmufacturingcrcup



LlsToF FloUBEs wii LISTOFTABLESsi FOREWOBoN PBEFACErvii



1 2 3 4 CopyrighrO 1992by Unive6iryScience B@ks Reprcducrionor r.anslarionof dy pan of rhis wo* beyond ftar pemited by Secrionl0? or IoE of lhe t9?6 UniredSrares CopyrighrAct withour lhe pemission of the copyrightowner is unlawful.Requests for pemissiono. funherinformaron shouldbeaddress€d to rhePemissions Depanment, Unive6ityScicnceBooks. Libral' of CongrcssCatalogNmbe.: 9l -65j3 I ISBN0-915702,68-7



5 6 7 8 9 10 11 12



ASTRONOMY1 INTRODUCTION TO POSITIONAL TIME 39 REFERENCE CELESTIAL SYSTEMS 95 TERRESTRIAL COORDINATES ANOTHE ROTATION OF THE EARTH .I99 ORBITALEPHEMERIDES OF THESUN,MOON,ANDPLANETS 279 ORBITAL EPHEMERIDES ANDRINGSOF SATELLITES325 PLANETS, AND PHYSICAL EPHEIUERIOES OF THESUN,I\,,IOON, SATELLITES383 ECLIPSES OFIHE SUNANDMOON 421 ASTFONOMICAL PHENOMENA475 STARSANDSTELLqRSYSTEMS 505 COMPUTATIONAL TECHNIOUES541 CALENDARS575 12.10REFERENCES 606



13 HISTORICAL INFORMATION609 14 RELATED PUBLICATIONS667 15 REFERENCE DATA 693 Prinredin theUnnedSlaresofADerica 10987654121



GLOSSAFY721 tNoEx 741



Contents



LISTOF FIGURES xvii LISTOFTABLESxxi FOREWORDxxv PBEFACE xxvii



1 / INTRODUCTIONTOPOSITIONALASTRONOMY 1 1.1



INTRODUCTION 1 1.11 Purpose 1



1,2



TIMESCALESAND CALENDARS 2 1.21 AtomicImescales 2 1.22 Dynamical'Iime 3 1.23 Rotational Tirnescales3 (UTC) 6 1.24 Coordinated UniversalTime 1.25 The Enumeration of Dales 7



1.3



CELESTIALAND TERRESTRIALCOORDINATES 8 1.31 Coordinate SystemsandFrames 8 1.32 CelestialCoordinate Systems 11 1.33 Terrestrial Coordinate Systems 13 1.34 The Rotationof the Earth 'tI 1.35 ThgConnections betweenTerrestrial andCelestial Coordinates20 1.36 Efiectsof the PositionandMotlonof ths ObjectandObserver 21



1.4



ORBITALMOTIONS 24 '1.41 Motionin Two-body Systems 24 1.42 Typesof Perturbations28 1.,+3 Perlurlrationsby and on ExtondedBodies 30



1.5



ASTRONOMICALPHENOMENA 32 '1.51 Rlslng,Setting,andTwilight 32 1.52 Meridian Transit 3it



v



v l



CONTENTS



1.53 1.54 1.55 1.56 1,6



CONTENTS



Conjunction, Opposition, and Elongation33 Eclipses, Occultations, andTransits 34 SatellitePhenomena35 PhysicalObservations of the Sun,t\roon,andplanets 96



3.22 3.23 3.24 3.25 3.26 3.27 3.28



REFERENCES 38



/ TIME 39 2,1 INTRODUCTION 39



3.3



APPABENTAND TOPOCENTRICPLACEALGORITHMS 145 for Planels 145 Algorithm 3.31 Apparent-Place for Slars 152 Algorithm 3.32 Apparent-Place Algorithms 154 of Apparent-Place 3.33 TheComputerlmplementation 3.34 Apparent-Places-Day'NumberTechnique'155 160 3.35 Topocentric-PlaceAlgorithm



3,4



ASTROMETRY 165 DIFFEBENTIAL 3.41 VirtualPlace 165 3.42 LocalPlace 165 Place 166 3.43 Astrometric



3,5



TO FK5 SYSTEMAND EPOCHJ2OOO.O167 TRANSFORIVIATION in RightAscension 167 Correction 3.51 FK4Zero-Point to the FK4Properl/otionSystemin RightAscension | 68 3.52 TheCorrection 3.53 EllipticTermsin Aberration 169 3.54 Precession 173 System 174 3.55 The Proper-Motion of Catalogsfrom81950.0to J2000.0 175 lor theTransformation 3.56 Equations Catalogs 179 ot Observational 3.57 Transformation ExamPles 180 3.58 Numerical 3.59 MatrixMethod 180



3.6



REFERENCES 187



2.2 MEASURES OF TIMEANDTHEIRRELATIONSHIPS 40 2.21 Atomic Time(TAt) 40 2.22 2.23 2.24 2.25 2.26 2.27 2.3



2.4



2.5



2.6



at



OynamicalTtme 41 SiderealIime48 UniversalTime 50 The Ephemerjs Meridjan 54 JulianDate 55 Ime Zones 56



PRACTICALDETERMINATIONS OF TIME 58 2.31 Frequency Standards andClockperformance5g 2.32 Measurement ot AtomicTime 60 2.33 EadhRotationMeasurement61 2.34 DynamicalTimeDetermjnations 69 METHODSOF TIME TRANSFER 64 2.41 BadioIme Signats 65 2.42 Portabte Ctocks 65 2.43 LORAN_C 66 2.44 Television Comparison Techniques67 2.45 Useof Satellites 68 2.46 Intercontinental CtockSynchronization by VLB| 69 2.47 Relativistlc Efiectsin lime Transfer 70 HISTORICALDEVELOPMENT OF TIIVlEKEEPING73 2.51 Inkoduction73 2.52 ApparentSolarTime,MeanSotar-Time, andthe Equation of Tjme 74 2.53 Rotationof the Earth 76 2.54 Universall"ime 77 2.55 Ephemeris l'ime 79 2.56 Historyof AtomicTjme g4 2.57 Historyof Coordinated Unive.sat Ime 85 2.58 Historyot Transmitted lime Sionats g6



4 / TERRESTRIAL COORDINATES OF THEEARTH 199 ANDTHEROTATION 199 4.1 INTRODUCTION SYSTEMS2OO COORDINATE 4,2 TERBESTRIAL 4.21 4.22 4.23 4.24 4.25



REFERENCES 88



CELESTIAL REFERENCE SYSTEMS 95 3.1 CELESTIAL BEFERENCE SYSTEMS95 3.11 FundamentalReferenceSystems 97 3.12 The DynamicalRelerenceSystem 98 3.13 The ConventionalCelestialReferenceSystem 98



3.2 BASISOF REDUCTION OF CELESTIAL COORDINATES 99 3.21 Precession 99



Nutation |09 SoaceMotion 121 Palallax 123 Aberration '127 lightdeflection 135 Gravitational PolarMotion 139 Refraction 140



4.3



Ellipsoid 200 The Figureol the Earthandthe Reference Coordinates202 andAstronomical Geodetic, Geocentric, LocalcoordinaleSystems 207 Systems 216 GeodeticDatumsandReference System 223 Terrestrial Reference The Conventional



GRAVITYTHE TIDES,AND MOTIONSOF THE CRUST 224 4.31 Modelingthe Earth'sGravityField 224 of lhe Earth'sGravilyField 227 4.32 A Represenlation 4.33 SolidEarthTides 233 4.34 OceanTideModel 240 Loading 244 Dueto OceanandAtmospheric 4.35 SateDisplacement 4.36 Platel'rotions 249 4.37 TidalEffectson UTI 250



CONTENTS



4.4 THEMONITORING OFTHEROTATION OFTHEEARTH 251 4.41 4.42 4.43 4.44 4.45 4.5



4.6



LaserFianging 254 VeryLongBaselineInterferometry255 Historical Methods 256 Alternative Techniques258 International Services 262



DETERMINATION OF PASTVARIATIONSIN LENGTHOF DAY AND THE POSITIONOF THE POLE 265 4.51 Historical Variations in UTl and Lengthof Day 265 4.52 Historical Variations in Polarlvlotion 269



CONTENTS



OFTHEMAJOR FORTHEPOSITIONS ELEMENTS 5.8 KEPLERIAN PLANETS315 317 EPHEMERIDES 5.9 BASISFORPRE-1984 317 New Constants ol 5.91 Introduction 5 . 1 0 R E F E R E N C E3S1 9



ANDRINGSOF SATELLITES325 EPHEMERIDES 6 / ORBITAL 6.1



INTBODUCTION 325 Elements325 6.11 Orbital ottheOrbit 327 Pertulbations 6.12 Secular 330 dueto Commensurabilities 6.13 Perturbations by OtherSatellites 332 Pertubations 6.14 Long-Period 333 6.15 PlanetocentricRectangularCoordinates Orbit 336 6.'t6 The APParent Values 340 Tabulated 6.17 Calculating 6.18 Notation 342



6.2



OF MABS 342 THE SATELLITES



6.3



OF JUPITER 345 THE SATELLITES Satellites345 6.31 TheGalilean Amalthea 349 6.32 The FifthSatellite, Thebe 351 Satellite' 6.33 The Fourteenth Satellites 352 6.34 TheSixththroughThirteenth



6,4



OF SATURN 354 THE RINGSAND SATELLITES 6.41 The Ringsof Saturn 354 of Saturn 356 6.42 The Satellites



6.5



OF URANUS 368 THE RINGSAND SATELLITES



6.6



OF NEPTUNE 373 THE SATELLITES 6.61 Trilon 373 6.62 Nereid 375



6,7



THE SATELLITEOF PLUTO 377



6.8



REFEBENCES 378



REFERENCES 272



ORBITAL EPHEMERIDES O F T H E S U NM , OON,AND P L A N E T S2 7 9 5.1 FUNDAMENTAL EPHEMEBIDES 279 5.11 Gravitational Model 280 5.12 The Ephemeris Reference Frame 280 5.13 TheAstronomical Constants Usedin the Ephemerides280 5.2



COI\,4PUTATION OF EPHEMERIDES 281 5.21 Mathematical Model 281 5.22 Numerical Integration286 5.23 Orientation of Ephemerides288



5.3



OBSERVATIONAL DATAFIT BY THE PLANETARY AND LUNAR EPHEIVEHIDES290 5.31 OptjcalData 290 5.32 Radar-Ranging Data 294 5.33 Spacecraft RangePoinls 297 5.34 LunarLaserRangeData 299



5.4



5.5



5.6



5.7



LEAST-SQUARES ADJUSTI\,IENT OF THE EPHEMEBIDES 3OO 5.41 TheObservational Equations 301 5.42 The SolutionParameters301 5.43 TheStandardDeviations303 NUMERICALBEPRESENTATION OF THE EPHEN/ERIDES 303 5.51 Chebyshev Polynomials305 5.52 Chebyshev Coetflcient ceneration 305 5.53 Interpolation ErrorandPolynomial Degree 306 COMPUTATION OF OBSERVATIONAL EPHEMERIDES 307 5.61 AooarentPositions 308 5.62 Astrometric Positions 308 5.63 TransitEphemerides308 OBBITAND EPHEIVIERIDES OF OTHERBODIES WITHINTHE SOLARSYSTEM 310 andComets 310 5.71 MinorPlanets



OFTHESUN,MOON,PLANETS EPHEMERIDES 7 / PHYSICAL ANDSATELLITES383 383 7.1 INTRODUCTION



7.2



Coordinates383 andCartographic Elements 7.11 Rolational andMagnitudes388 7.12 Phases PHYSICALEPHEMERISOF THE SUN 397



7,3



OF THE MOON 398 PHYSICALEPHEIV!ERIS



7.4



OF THE PLANETS 401 PHYSICALEPHEMERIDES 7.41 Metc!ry 4O1 7.42 \enus 402



CONTENTS



xll



7.43 7.44 7.45 7.46 7.47 7.48



Mars 403 Jupiter 403 Salurn 404 Uranus 405 Neptun€ 405 Pluto 406



OF THE SATELLITES 407 PHYSICALEPHEMERIDES ot Mars 407 7.5'1 Satellites of Jupiler 408 7.52 Satellites of Saturn 408 7.53 Satellites 7.54 Satellites of Uranus 411 7.55 Satellites of Neptune 413 7.56 TheSalelliteof Pluto 414 7.6



PHYSICALEPHEMERIDES OFTHE ASTEROIDS 414



7.7



REFERENCES 417



OF THESUNANDMOON 421 8 / ECLIPSES 8.1



421 INTRODUCTION fromthe NauticalAlmanac 8.11 EclipseDataAvailable Oflice 422 to the Ephemerides424 8.12 Corrections



8.2 THE OCCURRENCEOF LUNARAND SOLARECLIPSES 426 8.21 8.22 8.23 8.24



Overview 426 Geocentric LeastAngularSeparation426 Occurrence of LunarEclipses 428 Occurrence of SolarEclipses 431



8.3 SOLARECLIPSES 434 8.31 8.32 8.33 8.34 8.35 8.36



Fundamental Equations: Introduction434 Besselian Elements 435 Coordinates of theObserver 441 ConditionalandVariationalEquations 446 Calculation of GeneralSolarEclipsePhenomena450 LocalCircumstances461



4.4



LUNARECLIPSES 467 8.41 Introduction467 8.42 Computalions467



8.5



TRANSITS 471



8.6



REFERENCES 472



9 / ASTRONOMICAL PHENOMENA475 9 . 1 GENERALASPECTSOF THE NIGHTSKY 475 9.2



CONFIGURATIONS OF THE SUN, MOON,AND PLANETS 476 ol the Sun,Earth,andMoon 477 9.21 Int€resting Phenomena Phenomena478 9.22 Geocentric Phenomena481 9.23 Heliocentric



CONTENTS



x l



ANDTWILIGHT482 9.3 RISINGS. SETTINGS. 9.31 Sunrise, Sunset, andTwilight483 9.32 lvloonrise andMoonset485 9.33 Formulas Associated withRlsino andSettino486 9.34 lllumination 490 9.4 oCCULTATTONS 494 9.41 Occultations ol Stars 494 9.42 Occultations of Planets498 9.5 POLE-STARTABLES 498 9.51 De.ivationof the Pole Star Coefficients 501



9.6 BEFERENCES502



1O/STARSAND STELLAR SYSTEMS 505 10.1 SOURCES OF DATAON STARSANDSTELLAH SYSTEMS505 '10.1'lCompiled Catalogsof StellarPositions andNrotions506 10.12StandardReference Catalogs 507 10.13Observational Positional Catalogs 508 10.14OtherCatalogs andLists 509 10.15DataCentersandTheirFacilities 509



10.2 STELLAR DATAIN IHE ASIFONOMICAL ALMANAC 5O9 10.21BrightSlars 512 10.22VariableStars 513 10.23Double andMultiole Stars 515 '10.24Photometric Standards 516 10.25Badial-VelocityStandards 518 10.26SpectralClassification519 10.27Pulsars521 10.3 CLUSTERSAND GALAXIES 522 10.31OpenClusterData 522 10.32GlobularStarClusterData 526 10.33BrightGalaxies528 10.34Quasi-Stellar Obiects 530 10.4 SOURCESCATEGORIZED BY WAVELENGTHREGION 531 10.41Radio-SourcePositionalCalibrators 531 10.42Radio-Flux Calibrators532 10.43X-RaySources535 10.5 REFERENCES 534



11/ COMPUTATIONAL TECHNIQUES541 11.1 INTRODUCTION TO COMPUTING TECHNIOUES541 11.2 INTEFIPOLATION ANDSUBTABULATION 546 1l.21 Introduction andNotation546 11.22lnterpolation Formulas547 11.23 Inverse lnte.oolation 548 11.24 PolynomialRepresentations 548



CoNTENTS



xiv



coNTENTs



11.3 PLANEANDSPHERICALTRIGONOMETRY 549 1i.4 MATRIXAND vEcroR TEcHNleuEs



12.93Converting Between lstamic Tabutar catendar Date andJulianDavNumber 604



552



RotationorAxesusinsMatrjces 1141 552 Coordinates UsingVectors553 11.42Spherical Coordinate Transformations 11.43Specific 555



CALCULUS560 11.5 NUMERICAL Ditrerentiation s60 1r.s1Numericar 11.52 NumericalIntegration 562



11.6 srATlsTlcs 566 ol Error 566 11.61TheAccumulation 11.62TheMethod of LeastSquares568 '117 REFERENOES 574



13.33 3llJ3[il3i3l][::li1,:l3:]:fi*3:?::::jl',ff3:ltili:l :3: 12.10 REFERENCES606 13 / HISTORICAL INFORMATION



12.2 THE GREGORIAN CALENDAR 580 12.21 Rutesfor civit use 5g0 12.22 EcclesiasticalRules 581 12.23 Historyofthe cregoriancatendar 583



12.3rHE HEBREW oALENDAR584 12.31Rules 584 12.32History of theHebrew Calendar588 12.4 fHE ISLAMICCALENDAR 5gg 12.41Rules 589 12.42Historyof the lslamiccalendar 591 12.5 THE INDIANCALENDAR 591 12.51RulesforCivilUse 591 12.52principres oftheRerigious carendars92 12.53Historyof the IndianCatendar 594 12.6 THE CHINESECALENDAR 594 12.61Rules 595 '12.62History oftheChinese Calendar599 12.7 JULIANDAYNUMBERSAND JULIANDATE 600 12.8 THEJULIANoALENDAR 600 '12.91Rutes 601 12.82Historyof theJulianCalendar 601 '12 9 CALENDARcoNVERsloN ALGoRITHMS 603 Dayof theWeek 603 12.91Converting betweenGregorian 12.92Converting CalendarDateandJulianDayNumber 604



609



13.1 HISTORY OFTHEALMANACS609 13..11 TheAmerican Ephemeris 609 13.12TheA;erican Ephemeris andNauticalAlmanac 613 (1960to present)614 19.13TheCooperative British andAmerican Almanacs r3.2 HrsroRy oF TNTERNATToNAL coopEBATroN 616



575 12/ cALENDARS 12.1 INTRODUCTION 575 12.11 Astronomical Basesof calendars s76 12.12 NonastronomicalBases of Calendarsrthe Week 577 12.13 calendarReformand Accuracy 578 12.14 Historical Erasand chronology 579



xv



13:31 fi?ifl?lfllilliifl::""Hi"fU";".,, 13.23 OtherInternational Organizations621 13.3 HISTORICAL LIST OF AUTHORITIES 621 13 31 Introduction621 (1767-1900) 622 13 32 The NauticalAlmanac 13.33 The AmericanEphemeris(1855-1900) 631 13.34 The NauticalAlmanac,and The Ameican Ephemerls(1901-1983) 639 13.35 Systemof Constants(1968-1983) 656 134



REFERENCES



657



14 / RELATED PUBLICATIONS 667 14.1 CURRENTPUBLICATIONS667 14.1'1JointPublications andthe UnitedStates of the Royalcreenwich Observatory Navalobservatory 667 14 12 other Publications of the L'nitedstatesNavalobservatory 668 14.13OtherPublications Observatory669 oftheRoyalGreenwich 14.14Publications of OtherCountries669 142 ASTRoNoMICALPAPERSPREPAREDFoRTHE UsE oFTHEAMERIoAN EPHEIVIER|S AND NAUTICALALI\,IANAC 670 14.3 UNITEDSTATESNAVALOBSERVATORY CIRCULARS 675 14.4 PUBLICATIONS OFTHE UNITEDSTATESNAVALOBSERVATORY SECONDSERIES 677 14.5 SELECTEDNAO TECHNICALNOTESOFTHE ROYALGREENWICH OBSERVATORY684 14.6 LISTSOF APPENDICESAND SUPPLEMENTS 685 14.61TheBrtishNauticarArmanac 685 14.62TheAmericanEphemeris688 14.63JointSuDotements andAoDendices691



CONTENTS



xvl



1 5 / R E F E R E N C E D A T6A9 3



(1986Recommended Constants Values) 693 Fundamental IAU (1976) System of AstronomicalConstants 696 'Iime andStandardEpochs 698 Sun,Earth,andMoon 700 Systems 702 GeodeticRelerence Planets:MeanEl€ments 704 Data 705 Planets:Rotational Data 706 Planels:PhysicalandPhotometric OrbitalDala 708 15.9 Satellites: 't5.10Salellites: PhysicalandPhotometric Data 710 Bings 712 15.11Planetary NamesandAbbreviations713 15.12Constellation '15.'13Mathematical Constants 714 Factors 715 15.14EnergyConversion Speed, andMass 716 15.15Unitsof Length, 15.16GreekAlphabel717 15.17Internalional System of Units(Sl) 718 15.1 1s.z 15.3 15.4 15.5 15.6 15.7 '15.8



1 5 . 1 R E F E R E N C E S7 1 9 GLOSSARY721 lndex 741



LISTOF FIGURES



I.23l.l L233.1 I .3I . I I.32l.l 1.333.1 1.33'7.1 1.361.1 1.411.1 I.4l2.I



Calculationof siderealtime 5 Variationin theequationof timethroughtheyear 6 Representation of thevector/ in rectangular coordinates I 0 planes 12 Equatorialandeclipticreference Geocentric andgeodeticcoordinates 15 Relationbetweengeographic latitudeandthealtitudeof thecelestialpole l8 Parallaxofan object 22 Geometricproperties ofconic sections 25 Angularorbitalelements 27



2.25.1 2.27.1



Meridianrelationsandtime 55 Worldmapof time zones 57



3.21.1 3.21.2 3.222.1 3.251.1 3.252.1 3.26.1 3.26.2



The eclipticandequatorat epochanddate l0O Theprecession angles(r, z,r,and9e 102 The meanandtrueequators of date I 15 Light-timeaberration 128 Stellaraberration 128 Gravitational light deflection 136 Light from theplanetsandstarsdeflectedby thesun 138



4.21.1 4.22.1 4.22.2 4.22.3 4.22.4 4.231.1 4.233.1 4.242.I 4.51.1 4.51.2 4.51.3 4.51.4 4.52.1 4.52.2 4,52.3



TheEafih'ssurfaceandreference surfaces 201 Geocentric andgeodeticlatitude 202 Asrronomiclatitudeandlongitude 203 Deflectionof theverticalon a unit sphere 204 Geocentric cartesian coordinates 205 Altitudeandazimuth 208 UTM andUPSgrid zonedesignations2ll Majorgeodeticdatumlocations 219 Irngth of dayfrom I 656to 1988 267 Ar from A.D.700to 1600 268 Ar from 700B.c.to A.D.2N 268 logA, from 700B.c.to A.D.1980 269 28 to l99Ohly 2'1 2'7O Polarmotion,1980September Xand y components of polarmotionfrom theILS, 1899.9to1979.0 271 X andy components ofpola.motion,1846.0to1891.5 272



arecalculated 283 accelerations system,in whichfigure-induced 5.212.1 The{r7( coordinate 5.321.1 Diagramof geometryfor planetaryradarranging 295



xv



LtsToF FtcuREs



xviii



5.7I4.|TherclationshipbetweentheBl950.0andJ2000.0referenceframesandtheorbital plane 314 The orbital elementsusedto describethe orbital planerelativeto a referenceplane 326 Theend-onview of a greatcircleon the An equivalent form of tle orbitalelements. asa sraightline. 326 celestialsphereis reprcsented 6.12.1 The Laplacianplane 328 plane 334 6.15.1 The satelliteorbitrcfenedto an intermediate 6.15.2 Coordinates ofthe satelliterelativeto theplanet 335 Planetocentric celestialsphere 337 6.16.1 Sphericaltriangleusedto computeU,B,andP 337 6,16.2 6.16.3 Sphericaltriangleusedto computeo andp P 337 sphere 339 6.16.4 Apparentorbitof a satelliteasFojectedon thegeocentdc Reference systemfor PhobosandDeimos 344 6.2.1 Reference systemfor AmaltheaandThebe 350 6.32-l planefor Mimas,Enceladus, Tethys,andDione 357 6.421.1 R€ference systemfor Rhea,Titan,Hypedon,Iapetus,andPhoebe 361 6.422.1 Reference 6.422.2 Anglesneededto evaluatesolarperturbations on Titan 362 6.5.1 Reference systemfor the Uraniansatellites 372 6.61.1 Refercnce systemfor Triton 374 6.62.1 Reference systemfor Nereid 376 6.I 1. 1 6.ll .2



'7.Il.l 7.11.2 '7.12.1 7.12.2 1.12.3 7.12.4



Thepositionof thenorthpoleandprimemeridianof a planetin Earthequatorial coordinates 384 Planetocentric andplanetographic coordinate systems 386 The basicvectorsandtheangleof illumination 389 Thegeometricappearance of theapparent polarradius,,' 391 Thediskofa planetasseenby an obseNeron rheEanh 392 Planetocentric unit vectorsfor pointsof intereston thediskof theplanet,andtheangles betweentheEarth'sequatorofJ2000.0andtheplanet'sequatorof date 394



8.22.1 Geometricconstructfor determining whethereclipsewill occu. 426 8.231.1 Geomeficparameters ofthe Eanh'spenumbralshadow 428 8.231.2 Geometricpatameters ofthe Earth'sumbralshadow 429 8.2321.1 Limitingconditionsfor lunareclipses 430 8.242.1 Geometdcparameters for a partialsolareclipse,whenMoonis extemallytangent 432 8-2421.1 Geometricparameters for a partialsolareclipsewhenMoonis intemallytangent 434 8.321.1 Transformation oi geocentric equatorial coordinates to thefundamental plane 435 8.323.1 Components ofshadowconesin theBesselian fundamental reference system 438 8.323.2 Relationships amongangularsemidiametet apparent semidiameter, andparallax 439 8.323.3 Vertexangleofthe penumbra 440 8.323.4 Vertexangleofthe umbra 440 8.331.1 Angularquantities in thegeocentric equatori al plane 442 8.352.1 Relationship betweenrectangular andpolarreference systems 451 8.3556.1 Definitionof auxiliaryquanriry1, 458 8.3623.1 MagnitLrde is thefractionofthe lineardiameterofthe Suncoveredby theMoon 4& 8.3623.2 Obscuration is thefractionof rheareaof thesolardiskobscuredbv theMoon 465 9.31l.l



Theanalemmic curve 485



L I S TO F F I G U R E S



9.331.1 9.34.1 9,41.1 9.5.1 9.51.1



Thehorizonat risingor setring 488 Grcundilluminationfrom varioussources 491 Fundamental planeshowingpathof occultation 49't PZStrianglefor Pola.is 499 Polarisnorthpolardistance2096-2105 502



l1.l.l I1.3.1 ll.42.l



Spherical triangle 542 Arc ofcircle 551 Triangleon unit sphefe 555



XIX



LISTOFTABLES



1.342-l



Variationsin the Earth's Rateof Rotation 19



2.33.1 2.45.1 2.58.1



Techniquesfor MeasuringEanh's Rotation 62 Error Budgetsfor CommonView GPSTime Transfer 68 Time andFrequencyStepsby WWV andMC (USNO) 86



3.2ll.l 3.214.1 3.222.1 3.222,2 3-224-l 3.224.2 3-253.1 3.26.1 3.344.1 33M.2 3.58.1



Accumulated hecessionAngles 104 Precession Angles1984Januaryld0h 107 Nutation itr LongitudeandObliquity Refelredto the Mean Ecliptic of Date I l2 FundamentalAryuments I 14 Conectionsto IAU 1980NutationScries I 16 PlanetaryTermsin Nutation,CombinedDrect andIndirect Effects I l8 Coeffrcients(','tl / c) for the Major Planets 132 ApparentDeflectionAngles 138 Second-OrderTerms 159 Errors Due to NeglectingSecond-OrderTerms 160 Selected StarPositionson FK4 alld FI(5 Systems l8l



4.242.1 4,242,2 4.32.1 4.331.l



EarthEllipsoids 220 GeodeticDatums 221 (x 106) 228 GEM-TI NormalizedCoefficients Step2 Solid Tide CorrectionsWhen &z= 0. 30 in Step I (Using a Cutoff Amplitude of 9 x lO-\2 fot A^6k"H") 235 OceanTide Coefficientsfrom the SchwiderskiModel 242 DisplacementDue to OceanLoading (cm in amplitudeanddegee in phase) 245 CanesianRotationVectorfor EachPlateUsing KinematicPlateModel AM0-2 (No Net Rotation) 250 Zonal Tide Termswith PeriodsUp to 35 Days 252 Zonal Tide Termswith PeriodsGreaterthan35 Days 253



4-34.1 4.35I . I 4.36.1 4.37.1 4.37.2 5.214.1 5.31l.l 5.322.1 5-332-l 5.41.1 5.42.1 5.42.2 5.53.1



Lunar Libration AnglesandRates 285 TraNit Cide Observationsfrom the U.S. Naval Observatorythat havebeenusedin the JPL Ephemerides, DEI 18 291 Radar-RangingObservationsUsedDircctly in DEllS 297 Mariner 9 miter Normal RangePointsto MaIs 298 The ObservationalData Usedfor the AdjusEnentof DE1l8/LE62 301 ValuesandFormal StandardDeviationsofthe ConstantsUsedin DEI l8/ LEl l8 (DE2OOil-E20o) 3O2 in AU andAu/day 304 The Initial Conditionsof the Ephemeridesat JED 2,140100.5 Gnnule Length andPolynomialDegreefor the 11 EphemerisBodies 307



i I



xxi I I I



LtsToF TABLES



xxii 5.8.1 5.8.2



at theepochJ2U)0(JED2451545,O)316 ClassicalKeplerianelements Approximatemaximumerro6 ofthe Keplerianformulasovertheinterval 1800-2050316



6.13.1



AmongSatellites 332 The PrincipalCommensurabilities



7.53.2 7,54.1 7.55.1 7.56.1



fo. Mercury 402 Pammeters PhysicalEphemeris for Venus 402 Parameters PhysicalEphemeris Parameters for Mars 403 PhysicalEphemeris for Jupiter 404 PhysicalEphemeris Pammeters PhysicalEphemeris Parameters for Saturn 404 PhysicalEphemeris Parameters for Uranus 405 PhysicalEphemeris Parameterc for Neptune 406 PhysicalEphemeris Parameters for Pluto 406 RotationParameters for Mars' Satellites 407 RotationParameters for Jupiter'sSatellites 409 Standard Canographic Longitudesfor Jupiter'sSatellires 409 RotationParameters for Saturn'sSatellites 410 Standard Cartographic Longitudesfor Satum'sSarellites 4l I RotationParamete.s for Uranus'Satellites 412 RotationParameters for Neptune's Satellites 413 RotationParameters for pluto'sSatellite 414



8.1l3.l 8,422.I



U.S.N.O. SolarEctipse Circutars424 Sequences andConditionsfor ContactTimes 470



9.21I.I 9.22.I 9.22.2 9.22l.M 9.222.1 9.222.2 9,23.1 9.34.1



Timeof Commencement of theSeasons477 Geocentric Phenomena for whichf(r) = 0 479 ceocenrdcphenomena for which/,(r) = 0 479 sibilityCriteriafor Ceocentricphenomena 480 SynodicPeriodsof rheplanersandFirstAsteroids 480 long-PeriodCyclesofMercury Venus,andMars 4gl HeliocentricPhenomena482 Coefficients for CalculatingCroundllluminatjon 4gz



II I i ll.l.2 I I.1.3 I l.2l .l t ll : I l.5l.l ll.61.l



Precision of AngleandNumberof Decimalsfor Trigonometric Functions 544 The Methodof InverseUse 545 RangeofPrecisionof theInverseDetermination of an Ansle 545 Differences in Tabular Arsumenrs547 Formulasfor planeandSfhericalTriangles 550 Derivatives to anOrd".oi I0 :ot Accumulation ofEnor in Arithmetical Operarions567



7.4L| 1.42.1 7.43.1 7,44.I 7,45.1 7,46.1 7.47.l 1.48.1 7.51.1 7.52,1 7.52.2 '7.53.1



12.21-l Monthsofthe CregorianCalendar 5gl l2.3l.I Classification of yearsin the HebrewCalendar 584 12,31.2 Monthsofrhe HebrewCalendar 585 12.31.3 Terminology ofrhe HebrewCalendar 585 12.311.1 LunationConstants for DetenninirgTishri I 586



LIST OF TABLES



l2.4l.l l2.5l.l 12.52.1 12.6l.l 12.6L2 12.82.1



Monrhsof TabularIslamicCalendar 590 Monthsofthe IndianCivil Calendar 592 SolarMonthsof theIndianReligiousCalendar 593 ChineseSexagenary Cycleof DaysandYea$ 595 ChineseSolarTerms 597 RomanDatingin theJulianCalendar 602



13.348.1 AdoptedValuesfor Coefhcients of Nutation 654



15.1 15.2 15.3 15.4 15.5 15.6 15.'7 15.8 15.9 15.10 15.ll 15.12 15.13 15.14 I5.15 15.16 15.17



(1986Recommended Fundamenral Constants Values) 693 IAU (1976)Systemof AstronomicalConstants 696 TimeandStandard Epochs 698 Sun,Earth,andMoon 700 GeodeticReference Systems 702 Planets:MeanElements 7(X Planets:Rorarional Dara ?05 Planets: PhysicalandPhorometric Data 706 Satellites: OrbitalData 708 Satellites: PhysicalandPhotometric Data 710 Planetary Rings 712 Constellation NamesandAbbreviations 713 Mathematical Constants 714 EnergyConversion Factors 715 Unitsoflrngth, Speed, andMass 716 GreekAlphabet 717 Inremational System of Units(SI) 718



xxiii



Foreword



The Etplanatory Supplement to the Astronomical Ephemeris and the Americe,n Ephemeris a,nd Nautical Almanac was first published in 1961 "to provide the user of these publications with fuller explanations of these publications themselves." This supplement was reprinted with amendments in 1972, 1974, and 1977. It was allowed to go out of print because the International Astronomical Union decided to introduce new astronomical constants, a new standard epoch and equinox, new time axguments, a new astronomical reference frame, and new fundamental ephemerides, all of which required major revisions to the supplement. In addition, The Astronornical Ephemeris ar'd The American Ephem,erisand Nautical Almanoc serieswere continued from 1981 with a new tille, The Astronomical Almanac, which contains a revised content and arrangement and is printed only in the United States, The work of computation, prepa,ration, proofreading, and production of reproducible material is still shared between the United Kingdom and the United States of America. Ma,ny changes in the arrangement of the almanarc were introduced in the edition for 1981, and major changesin the basis ofthe ephemerideswere made in the edition for 1984. The changes in 1981 included: the replacement of the hourly ephemeris of the Moon by a tabulation of daily polynomial coefficientsl the introduction of a new system of rotational elements for the planets; the extension of the scope of the data on satellites; the inclusion of orbital elements and other data for minor planets of general interest; the extension of the list of bright sta.rs;the inclusion of new lists of data for other types of stars and nonstellar objectsl new explanatory material; and a glossary of terms. The changes in 1984 included: the replacement of the classical theories by the Einsteinian theories of special and general relativity; the replacement of ephemeris time by dynamical timescales; the adoption of new fundamental heliocentric ephemerides based on a numerical integration of the motion of all the planets and of the Moon; and the use of the IAU (1976) system



xxvi



FOBEWORD



of astronomical constants, the standard equinox of J2000 0' and the FKS celestial referencesystem.Anaccountofthesechangeswasgiveninthe39-pageSupplement to the Astronomi.cal Almanac fot 1984, rvhich was bound with the Almanac, and is also given here in great detail in Chapter 13' Most ofthe text in this supplement has been written for readers who are familiar with the principal concepts of spherical and dynamical astronomy but who require detailed information about the data published in The Astronomical Almanac and about how to use the data for particular purposes. Similarly, the reference data given in this supplement are presented in forms that are appropria.te to rrsers who understand the significance of the quantities whose values are given. To a la,rge extent, the chapters are independent of each other, but an introductory overview has been given in Chapter 1 and a glossary of terms has been given at the end of the volume. Referencesto textbooks and other sources of background information a.regiven at the end of each chapter. Preliminary proposals for the new edition of the Explanatory Supplement were drawn up in 1979, and more detailed outlines of the Supplement were prepared in 1986 bv the staffs of Her Majesty's Nautical Almanac Office and the U.S. Nautical Almanac Office. By 1988, it was evident that Her Majesty's Nautical Alnanac Office would not be able to participate as originally planned, and the U.S. Nautical Almanac Ofrce took over the entire project. The supplement is organized by chapters and sections such that it can be updaied in the future. It is planned that future reprints will incorporate developments and improvements. We hope that this new publication will prove to be even more useful than its predecessor.



Preface



The primary purpose of this revised.Explanatory Supplement is to provide users of The Astronomi,cal Almanac with more complete explanations of the significance, sources, methods of computation, and use of the data given in the almanac than can be included annually in the almanac itself. The secondary purpose is to provide complementary information that doesn't change annually, such as conceptual explanations, lists of constants and other data, bibliographic references,and historical information relating to the almanac. It is hoped that lhe Erplanatory Supplement will be a useful reference book for a wide range of users in the fields of astronomy, geodesy, navigation, surveying, and space sciences, and also tcachers, historians, and people interested in the field of astronomy. Many users of the almanac are not the professional astronomers for whom it is primarily designed, and so this supplement contains some explanatory material at an elementary level; it is not, however, intended for use as a basic textbook on spherical and dynamical astronomy. In some respects it does supplement such textbooks since it is concerned with new concepts or new techniques. This supplement differs in many respects from its predecessot, the Erplanatorg Supplement to The Astronomical Epherneris and The Ameri,can Ephemeris and Nauti,cal Almanac. Vector and matrix notations have been introduced and more diagrams have been provided. Simple conversion tables and tables of quantities that can be calculated directly from simple formulas have been omitted. Detailed step-by-step examples have been omitted, and approximation methods have not been given. Most of the text is new but historical material has been carried over for the convenienceof those who do not have ready accessto the previous supplement. This supplement has been prepa,red by the Nautical Almanac Office of the United States Naval Observatory. Material has been contributed by scientists from the Nautical Almanac Office of the Royal Greenwich Observatory, Jet Propulsion xxvll



)oo/iii



PREFACE



Laboratory, Bureau d.eeLongitudes, Time Service and Astrometry Departments of the U.S. Naval Observatory, alxd other scientists. The authors of each chapter have been indicated, but other individuals may have been involved in contributing, inproving, and checkiug the material. The valuable assistatrcethat has been given in many ways by other astronomereand Bcientistsis gratefully acknowledged. Suggestionsfor improvement of this supplement, a.ndof The Ashpnomical Almonac itself would be welcomed. They should be sert to the Director, Orbital Mecha.rricsDepartment, U.S. Nav'al Obeervatory, Washington, DC 20392. Jomes B. Hagan Captain, U.S. Nawy Superinteudeut, U.S. Naval Obserratory



EXPLANATORY SUPPLEMENT TO THE ASTRONOMICALALMANAC



CHAPTER 1



Introduction to Positional Astronomy by PK. Seidelmann andG.A.Wilkrns



1,1 INTRODUCTION 1.11 Puroose The Astronomical Almanac gives data on the positions and, where appropriate, orientations of the Sun, Moon, planets, satellites, and staxs as they may be seen from the surface of the Earth during the course of a year. A proper appreciation of the significance of these data requires a basic understanding of the concepts of sphericol astrononly, which explain how the varying directions of celestial objects may be represented by positions on the surface of the celestial sphere.In addition, an appreciation of why these celestial objects appear to move in the ways predicted in The Astronomical Almanac reqrires an uuderstanding of the concepts of dynamical astronorny, which provides a mathematica.l explanation of the objects' motions in space under the influence of their mutual gravitational attractions. Spherical and d1'namical astronomy together form what is referred to here as positional astronornA. This text has been written for readers familiar with the principal concepts of spherical and dynamical a€tronomy who require detailed information about the computation and use of the data published, in The Astronomical Almanac. The prima.ry purpose of this introductory chapter is to introduce the concepts, terminology, and notation that are used throughoul The Astronomical Almanac and.lhis supplement; rigorous definitions, formulas, and further explanatory information are given in the later chapters of this supplement. The glossary gives concise definitions of words pa,rticular to spherical and dynamical astronomy. The reference data are presented in forms that are appropriate to users who uaderstand the significance of the quantities whose values are given.



*



EXPLANATORYSUPPLEMENT



1.2 TIMESCALESAND CALENDARS



1.21 AtomicTimescales 1.211 International Atomic Time For scientific, practical, and legal purposes the standard unit for the measurement of intervals of time is the SI second, which is defined by the adoption of a fixed value for the frequency of a particular transition of cesium atoms. Time can be measured in this unit by the use of time standards based on processesof physics, Cesium frequency standards, hydrogen masers, ion storage devices, and other such devices are able to count seconds and subdivide them very precisely. Thus, such a device can provide a timescale whose accuracy is dependent on the precision of the measurement and the stabiliiy of the device. Such a timescale provides a measure of time for identifying the instants at which events occur; the interval of time between two events can be calculated as the differences between the times of the events. The results of the intercomparisorr of about 200 frequency standards located around the world are combined to form a standard timescale that can be used for identifying uniquely the instants of time at which events occur otr the Earth. This standard timescale is known as International Atomic Time (TAI). It is the basis for all timescales in general use. It is distributed by many difierent means, including radio time signals: navigation systems. such as the Global Positioning System, LORAN C. and O\IEGA; communication satellites: and precise time standards.



1.212 Relativistic Effects In high-precision timekeeping, and for some purposes in solar-system dvnamics and astrometry, it is necessary to take into account the efiects of special and general relativity and to recognize, lor example, that the rate of an atomic clock depends on the gravitational potential in u4rich it is placed and that the rate will appear to depend on its motion relative to another clock rvith rvhich it is compared. In particular, one should recognizethat the independentvariable, or timescale, of the equations of urotion of the bodies of the solar system (or of a subset of thern) depends upon the coordinate system to which the equations refer. The relationship between any such timescale and TAI, which is appropriate for use at sea-levelon the surface of the Earth, may be specified by an appropriate formula containing periodic terms and an arbitrary linear term. Trvo su