< > Explanatory Supplement. to the Astronomical Almanac. P. Kenneth Seidelmann. Sean E. Urban. Foreword by Roger Sinnott THIRD EDITION.

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1 Explanatory Supplement to the Astronomical Almanac THIRD EDITION Edited by Sean E. Urban US Naval Observatory P. Kenneth Seidelmann University of Virginia Foreword by Roger Sinnott < > UNIVERSITY SCIENCE BOOKS Mill Valley, California

2 Contents Foreword Preface Contributing Authors XXVII XXXI XXXIII 1 Introduction to Positional Astronomy Introduction 1 /././ Purpose I Introduction to The Astronomical Almanac / The Astronomical Almanac and The Explanatory Supplement Conceptual Changes since the Last Edition Implementation ofthe Changes Deletions from This Edition am/the Astronomical Almanac Timescales and Calendars Atomic Timescales Dynamical Time Timescales Based on Earth's Rotation Coordinated Universal Time (UTC) Enumeration of Dates Celestial and Terrestrial Coordinates Coordinate Systems and Frames Celestial Coordinate Systems Terrestrial Coordinate Systems The Rotation of the Earth Connections between Terrestrial and Celestial Coordinates Effects of the Position and Motion of the Object and Observer Orbital Motion Motion in Two-Body Systems Ephemerides Perturbations by and on Extended Bodies Solar System Sun Planets Satellites Other Bodies Planetary Coordinates 38

3 viii Contents 1.6 Astronomical Phenomena Rising, Setting, mid Twilight Meridian Transit Conjunction, Opposition, and Elongation Eclipses, Occupations, and Transits Satellite Phenomena Physical Observations of the Sun, Moon, and Planets 42 References 43 2 Relativity for Astrometry, Celestial Mechanics and Metrology Introduction Newtonian Formalism Special Relativity and the Metric Tensor Einstein's Theory of Gravity The Problem of Observables The Ranging Observable The Spectroscopic Observable Astromelric Observables The Post-Newtonian Framework Coordinate Timescales and Proper Time The Barycentric Celestial Reference System BCRS versus ICRS The Geocentric Celestial Reference System Coordinate Transformations Relativistic Potential Coefficients Applications / Other Versions of the Geocentric Celestial Reference System Rotating Reference Systems The Einstein-lnfeld-Hoffmann Equations of Motion Equations of Motion in the GCRS Astrometry VLB! Observations Relativistic Timescales, their Definitions and Units Practical Transformations of Timescales Scaled Coordinate Timescales Units oftimescales 70 References 71 3 Time Introduction Absolute (or Newtonian) Time Proper Time Coordinate Time Time and the Earth's Rotation Sidereal Time UT1 80

4 Contents ix 3.3 Dynamical Time Ephemelis Time TDT/TDB Atomic Time International Atomic Time, TA Terrestrial Time, 77' Civil Time Coordinated Universal Time, UTC Time Zones Barycentric and Geocentric Coordinate Time Barycentric Coordinate Time, TCB and Barycentric Ephemeris Time, 7^, Geocentric Coordinate Time, TCG Julian Date Summary of Useful Relationships Timekeeping Devices International Agencies Time Dissemination High-Frequency Radio Time Signals Low-Frequency Radio Time Signals LORAN-C Global Positioning System (GPS) GLONASS GALILEO Two-Way Satellite Time Transfer (TWSTT) Time Transfer by Internet Relativistic Effects Clock Transport from a Rotating Reference Frame Non-rotating Local Inertia! Reference Frame Electromagnetic Signal Transferfrom a Rotating Reference Frame Electromagnetic Signal Transferfrom a Non-rotating, Local Inertial Frame 101 References The Fundamental Celestial Reference System Summary The ICRS, the ICRF, and the HCRF Background: Reference Systems and Reference Frames The Effect of Catalog Errors on Reference Frames Late 20th Century Developments ICRS Implementation The Defining Extragalactic Frame The Frame at Optical Wavelengths Standard Algorithms 114

5 X Contents Relationship to Other Systems Data in the ICRS Formulae 117 References Terrestrial Coordinates and the Rotation of the Earth Introduction Terrestrial Reference Systems Reference Systems and Reference Frame Concepts The Figure of the Earth and Geodetic Reference Surfaces Geodetic Datums Geocentric. Geodetic, and Astronomical Coordinates Local Coordinate Systems Gravity, Tides, and Motions of the Crust Modeling the Earth 's Gravity Field Representation of the Earth's Gravity Field Solid Earth Tides Ocean'Tide Model Site Displacement Due to Ocean and Atmospheric Loading Plate Motions Tidal Effects in UT Tidal Effects in Polar Motion Monitoring the Orientation of the Earth Very Long Baseline Interferometry Global Navigation Satellite System Laser Ranging Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS) Historical Methods Alternative Techniques International Services 184 References Precession, Nutation, Polar Motion, and Earth Rotation Introduction The Celestial Reference System (CRS) and the Terrestrial Reference System (TRS) Reference Systems in The Astronomical Almanac The Conventional International Origin and the International Terrestrial Reference Frame fl'trf) The Celestial Intermediate Reference System and the Celestial Intermediate Pole (C1P) The Equinoxes The Equation of the Equinoxes, E(,(T) 207

6 Contents Xi 6.4 The Equation of the Origins, E (T) The J to ICRS Frame Bias Matrix Precession and Nutation The IAU 2000 Precession-Nutation Theory andiau 2006/2000A Precession Nutation The Celestial Intermediate Origin (CIO) Time and Earth Orientation Apparent Solar Time, Mean Solar Time, and the Equation of Time Greenwich Mean Time Universal Time (UT) Earth Rotation Angle (ERA or 8) Sidereal Time Conversions between Universal Time and Sidereal Time Conversion of Local Apparent Sidereal Time to UT/ Local Time and Hour Angle The Terrestrial Intermediate Origin (TIO) The Ephemeris Meridian Polar Motion 240 References Positions 249 Introduction Summary of Systems, Places, and Coordinates Reference Systems Places Coordinates Summary of Notations Components Used in the Reduction of Positions Space Motion Parallax Aberration Gravitational Light-Deflection Celestial "Of Date" Positions Terrestrial "Of Date" Positions Refraction Reduction of Positions from the BCRS to the GCRS Proper Place Local Place Astrometric Place Differential Astrometry Algorithms for the Reductions of Positions Apparent and Intermediate Place Algorithmfor Planets Apparent and Intermediate Place Algorithmfor Stars 290

7 Contents Topocentric Place Algorithm Summary of the Algorithms Approximate Algorithms 298 References 301 Further Reading Orbital Ephemerides of the Sun, Moon, and Planets 8.1 Fundamental Ephemerides Previous Ephemerides Used in The Astronomical Almanac Ephemerides Prior to! Ephemeridesfrom 1984 through Ephemerides Sinning in Equations of Motion Point-Mass Interactions Solar System Barycenter Figure Effects Lunar Gravity Coefficients Lunar Physical Lihration: Coordinates Physical Lihration Differential Equations The Moment of Inertia Tensor Time Derivative of the Inertia Tensor Principal Moments Figure-Point Mass Torques upon the Moon Figure-Figure Torque upon the Moon Acceleration of the Moon from Earth Tides The Numerical Integration of DE405/LE Estimated Integration Error Adopted Constants Observational Data Fit by the Planetary and Lunar Ephemerides Optical Data Meridian Transits Photographic and CCD Astromelry Occullation Timings Astrolabe Radiometric Emission Measurements Ranging Data Orbiter Range Points Lander Range Data VLHIData Lunar Laser Range Data The Orientation of DE405/LE Adjustment ofde405 onto the 1CRF Reference Frame Various Formulae Used in the Reduction of the Observational Data Phase Corrections Corrections to Precession and Equinox Drift Computation ofranges 322

8 e Contents xiii Antenna Location Time Delay for Relativity Time Delay for the Solar Corona Time Delay for the Troposphere Modeling the Surface of Mercury Modeling the Suiface of Venus Modeling the Surface of Mars Closure Points Viking Lander Computations Initial Conditions and Constants of DE405/LE Asteroids Positional Errors of the Planetary and Lunar Ephemerides Inner Four Planets: Ephemerides from Ranging and VLBI Uncertainties from Asteroid Perturbations Outer Planets: Reliance on the Classical Optical Observations Planetary Positional Errors in the Almanacs, Planetary Position Uncertainties in Keplerian Elements for Approximate Positions of the Major Planets Formulae for Using the Keplerian Elements Solution of Kepler's Equation, M = E sin E Approximate Accuracies of the Keplerian Formulae The Availability of Ephemerides 341 References Planetary Satellites and Rings Introduction Orbital Elements and Perturbations Orbital Elements Secular Perturbations of the Orbit Perturbations due to Commensurabilities Long-Period Perturbations by Other Satellites Planetocentric Rectangular Coordinates Traditional Method The Apparent Orbit Calculating Tabulated Values Modern Method The Apparent Orbit Calculating Tabulated Values Satellite Data in The Astronomical Almanac and Notation Presented Data Notation The Satellites of Mars The Satellites of Jupiter The Galilean Satellites Jupiter's Fifth Satellite, Amalthea Jupiter's Sixth through Thirteenth Satellites 375

9 Xiv Contents 9.9 The Rings and Satellites of Saturn The Rings of Saturn The Satellites of Saturn The Rings and Satellites of Uranus The Satellites of Neptune Triton Nereid The Satellites of Pluto 396 References Physical Ephemerides of Solar System Bodies Introduction Rotational Elements and Cartographic Coordinates Pkmetocentric and Phuwlographic Coordinate Systems Coordinate Systems ofsmall Bodies Phases and Magnitudes 410 /0.3.1 Absolute and Apparent Magnitudes Surface Brightness The Apparent Disk of a Solar System Body Silly Earth Points and Sub-Solar Points The Greatest Defect of Illumination The Apparent Magnitudes of the Minor Planets Physical Ephemeris of the Sun Physical Ephemeris of the Moon Calculating Libra/ions Physical Ephemerides of the Planets Mercury Venus Mars Jupiter Saturn Uranus Neptune Physical Ephemerides of the Dwarf Planets Physical Ephemerides of the Satellites Satellites of Mars Satellites of Jupiter Satellites of Saturn Satellites of Uranus Satellites of Neptune Satellites of Pluto 447 References 447

10 Contents XV 11 Eclipses of the Sun and Moon Introduction 453 //././ Eclipse Data Available from the Nautical Almanac Offices Corrections to the Ep/wmericles The Occurrence of Lunar and Solar Eclipses Overview Geocentric Least Angular Separation Occurrence of Lunar Eclipses Occurrence of Solar Eclipses Solar Eclipses Fundamental Equations: Introduction Besselian Elements Coordinates ofthe Observer Conditional and Variational Equations Calculation of General Solar Eclipse Phenomena Local Circumstances 4H Lunar Eclipses 491 /1.4.1 Introduction Computations Eclipses Online Solar Eclipses Lunar Eclipses Transits 499 References 502 Further Reading Astronomical Phenomena General Aspects of the Night Sky Configurations of the Sun, Moon, and Planets Interesting Phenomena of the Sun, Earth, and Moon Geocentric Phenomena Heliocentric Phenomena Risings, Settings, and Twilight Sunrise, Sunset, and Twilight Moonrise and Moonsel Formulae Associated with Rising and Setting Illumination Occultations Occultations of Stars Occultations of Planets Pole Star Tables Derivation of the Pole Star Coefficients 527 References 528

11 XVi Conlents 13 Stars and Stellar Systems Introduction I.1 Data Presentation 529 J3.1.2 Sources of Data Sources of Position Stellar Data Bright Stars Double Stars Photometric Standards Radial Velocity Standards Variable Stars Exoplanets and Host Stars Pulsars Clusters and Galaxies Open Star Clusters Globular Star Clusters Bright Galaxies Quasi-Stellar Objects Wavelength Specific Sources Radio-Source Positional Calibrators Radio-Flux Calibrators X-Ray and Gamma Ray Sources 547 References Computational Techniques Introduction to Computing Techniques Interpolation and Subtabulation Introduction and Notation Interpolation Formulae Inverse Interpolation Polynomial Representations Chebyshev Polynomials Plane and Spherical Trigonometry Matrix and Vector Techniques Matrix Multiplication Rotation of Axes Using Matrices Spherical Coordinates Using Vectors Specific Coordinate Transformations Numerical Calculus Numerical Differentiation Numerical Integration Statistics Accumulation of Error The Method of Least-Squares 579 References 583

12 Contents XVii 15 Calendars Introduction Overview Uses ofcalendars Astronomical Bases of Calendars Astronomical, Observational and Arithmetic Calendars Visibility of the Crescent Moon Non-astronomical Cycles and the Week Historical Eras and Chronology The Christian Era Dates Julian Day Numbers and Julian Dates Luni-solar Calendars Accuracy of Calendars The Ancient Egyptian Calendar History ofthe Egyptian Calendars Rules ofthe Egyptian Calendar The Roman and Julian Calendars Introduction Divisions ofthe Roman Months Caesar's Reform The Julian Calendar in Medieval Europe Rules for the Julian Calendar The Gregorian Calendar History of the Gregorian Reform Rules for the. Civil Use of the Gregorian Calendar Rules for the Ecclesiastical Calendar Calculation of the Date of Easter Sunday The Jewish Calendar History of the Jewish Calendar Rules for the Modern Jewish Calendar Rules for Postponement (Dehiyyot) Determining Tishri Determining the Length of the Year Terminology of the Jewish Calendar The Islamic Calendar Introduction History of the Islamic Calendar Rules for the Arithmetic Islamic Calendar The Indian Calendars History of Indian Calendars The Traditional Indian Solar Calendar The Traditional Indian Luni-solar Calendar Rules of the Reformed Saka Calendar The Chinese Calendar History of the Chinese Calendar Sexagenary Cycle 612

13 XViii Contents Major and Minor Terms Rules for the Modern Chinese Calendar Finding the First Day of a Chinese Year Finding the First Day of Each Month in a Chinese Year The French Republican Calendar The Baha'i Calendar Calendar Conversion Algorithms Introduction Calculating the Day of the Week Interconverting Dates and Julian Day Numbers Converting Dates in the Jewish Calendar Calculating the Date of Easter Calendar Conversion Programs 621 References 622 A Bases and Constants 625 A. 1 Bases 625 A.2 Astronomical Constants 625,4.2./ CODATA 626 A.2.2 Future of the Astronomical Unit 626 References 626 B Acronyms 627 C Glossary 633 Index 661