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KOLACZEK 6 LACCESS, M NUMD)RI 0 WApas) 'COD[" 'A ," (NA • CRORTMXORADNUMMER) 3 ICATWOOFty) 4 4N, Ai V14. 0 Smithsonian Astrophysical Observatory SPECIAL REPORT 286 1 Research in Space Sciei.ce SAO Special Report No. 286 r SELENOCENTRIC AND LUNAR TOPOCENTRIC COORDINATES OF DIFFERENT SPHERICAL SYSTEMS B. Kolaczek September 20, 1968 Smithsonian Institution Astrophysical Observatory Cambridge, Massachusetts 02138 PRECEDING PAGE BLANK NOT FIL WO, TABLE OF CONTENTS Section Pale ABSTRACT ............................... vii 1 INTRODUCTION ............................ 1 2 SELENOEQUATORIA& COORDINATE SYSTEM ....... 5 2.1 Definition; Transformation of Mean Geoequatorial into Mean Selenoequatorial Coordinates ....... 5 2.2 Transformation of Geo-apparent Geoequatorial into Seleno-apparent Selenoequatorial Coordinates ... 10 2.3 Calculation of the Mean Selenoequatorial Coordinates .... .. ... .... .. .. .... .. 12 2.4 Transformation of Mean into Seleno-apparent Selenoequatorial Coordinates . .. .. .... .. .. 15 3 ECLIPTIC COORDINATE SYSTEM ................ 19 3. 1 Introduction . .. .. .. .. .. 19 3. 2 Mean Ecliptic Coordinates ................ 19 3. 3 Seleno-apparent Ecliptic Coordinates ......... 24 3.4 Tranformation of the Ecliptic into the Seleno- equatorial Coordinate System .............. 27 4 GEOEQUA.TORIAL COORDINATE SYSTEM .......... 31 5 STELLAR COORDINATE SYSTEM .. ......... • .... 33 5. 1 Introduction and Definition ................ 33 5.2 Transformation of Stellar Coordinates into Other Coordinates, and Conversely .............. 34 6 MOON'S HORIZONTAL COORDINATE SYSTEM ....... 37 b. 1 Basic Formulas ....................... 37 6.2 Transformation of Ecliptic into Lunar Horizontal Coordinates ................... 39 6. 3 Apparent Motion of the Moon's Celestial Sphere .. 45 7 SELENO- RECTANGULAR COORDINATE SYSTEMS ..... 51 8 CONCLUSIONS ............................. 57 iii TABLE OF CONTENTS (Cont.) pendixAp Page A PRECESSION OF THE MOON ................... 59 B PHYSICAL LIBRATION OF THE MOON ..... ....... 63 C ABERRATION OF THE MOON'S MOTION'S ........... 67 C. 1 Lunar Daily Aberration ................. 67 C.2 Lunar Monthly Aberration ................ 67 s C. 3 Lunar Annual Aberration ................. 72 D PARALLAX OF THE RADIUS OF THE MOON, THE RADIUS OF ITS ORBIT, AND THE MOON--SUN DISTANCE ................................ 79 D.1 Introduction ......................... 79 D.2 The Lunar Daily Parallax ................ 80 D. 3 Lunar Monthly Parallax ................. 89 D. 4 Lunar Annual Parallax .................. 96 E GENERAL PARALLACTIC FORMULAS IN RECTANGULAR AND SPHERICAL COORDINATES ................ 99 F GENERAL FORMULAS FOR THE INFLUENCE OF AN ABERRATION IN SPHERICAL COORDINATES ........ 103 GTABLE OF CONSTANTS ...................... 105 G. 1 The IAU System of Astronomical Constants .... 105 G. 2 The Moon's Constants ....... .. ...... .. 108 G. 3 The Earth's Constants .................... 114 r G.4 Constants of Precession ................. 117 f ACKNOWLEDGMENT........................ 119 REFERENCES AND BIBLIOGRAPHY ............. 120 iv m LIST OF ILLUSTRATIONS Figure Page 1 Spherical triangle EP(S showing the relation of the ecliptic to the selenoequatorial coordinates of a star 7 2 Spherical triangle P( P% showing the relation of the geo- equa.torial to the selenoequatorial coordinates of a star... 9 3 The Moon's precession in selenoequatorial coordinates .. 14 • 4 The parallactic angle n of a star S in the spherical triangle SEP.................................... 21 5 a) Woolf's network in the ecliptic coordinate system for X = t 90°; b) Woolf's network in the selenoequatorial coor- dinate system for aT = t 90* .................... 29 6 Spherical triangles S2QS1 and SQS1 showing the relation of the stellar coordinates K, X to the coordinates of other systems v, 4 ............................. 35 7 The Moon's astronomical triangle ................ 38 8 Spherical triangles P(EZ( and SEZ( showing the relation of the ecliptic to the lunar horizontal coordinates ...... 40 9 Spherical polygon ........................... 43 10 The Moon's astronomical triangle (DP( Z( ........... 50 11 The relation of the rectangular coordinates of the point P(x, y, z) to its polar coordinates (pp, u, v) ........... 52 C-1 The ecliptic and selenoequatorial coordinates of the apex C of the Moon's orbital motion .................... 70 C-2 The projection of the velocities of the Moon's and of the Earth's orbital motions on the plane perpendicular to the ecliptic .................................. 72 * C-3 The projection of the velocities of the Earth's and of the Moon's orbital motion on the Moon's orbital plane ...... 74 C-4 The lunar annual aberration of a star .............. 76 D-1 The relation between lunar hour angles of different points on the Moon's celestial sphere ................... 87 E-1 Translation of the rectangular coordinate system ...... 100 F-1 Aberrational displacement of a star ............. .. 103 v LIST OF TABLES Table Page 1 Value s of a - X ............................ 28 2 Values of d0 - 0 ............................ 29 3 Values of (dzL)max i. one Earth sidereal second ...... 39 4 Lunar horizontal coordinates of the north ecliptic pole ... 41 5 Moon's horizontal coordinates and lunar hour angles for special points on the Moon's celestial sphere ... ...... 46 6 The selenocentric coordinates of the special points on the Moon's celestial sphere ....................... 47 C-1 The coordinates of the apex of the Moon's orbital motion and the notation of the true and apparent (at this motion) spherical coordinates in the different coordinate systems ....... 71 D-i The lunar daily parallaxes of the distances of the lunar artificial satellites .......................... 80 D-2 The lunar daily parallaxes of the Sun and the planets .... 81 D-3 The selenocentric coordinates of a point on the Moon's sur- face and the notation of the selenocentric and lunar topo- centric coordinates in the different coordinate systems 84 D-4 The approximate values of the lunar monthly parallaxes of the Sun and of the planets ...................... 90 D-5 The selenocentric coordinates of the Earth and the notation of the selenocentric and geocentric coordinates in different systems ................ ................ 91 D-6 Transformation of the formulas for the Earth's geocentric parallax into the formulas for the lunar monthly parallax . 94 a w vi ABSTRACT The paper contains a short outline of the Moon's spherical astronomy. The problems of the mean and of the apparent selenocentric and lunar topocentric spherical coordinates is treated deeply. The advantages and disadvantages of the different spherical coordinate systems, such as seleno- equatorial, geoequatorial, ecliptic, stellar, and Moon's horizontal, for orientation in space from the Moon are discussed. The necessary formulas are given to calculate the mean and the apparent positions of stars and other •` celestial bodies in each of the coordinate systems, regarded equally as selenocentric or lunar topocentric. The appendices contain short descrip- tions of all the phenomena related to the discussed coordinate systems: the Moon's precession and notation, lunar aberrations and lunar parallaxes; the general aberrational and parallactic formulas are also given. RESUME Le momoire contient un court sche'ma de 1 1 astronomie spheri- qua de la lune. Nous traitons a fond le probleme dos coordonnees spheriques selenoeentrfques at topocentriques lunaires moyennes at apperentes. En vue dune orientation dons 1'espace a partir de Is lune, nous discutons des aventages at desavantages de differents systemes de coordonnees spheriques, par example Is systeme sale- .: noequatorial, Is systeme gs'oequatorial, is systeme de 1'eclipti- Is qua, Is systeme stellaire, at systeme du plan horizontal de In lune. Nous donnons lea formulas necessaires pour calcuatr lea positions moyennes at apparentes des etoiles at autres corps ce- lestas dons chocun des systemes do coordonnees, consideres a is foie comma selenocentriques at topocentriques lunaires. Les appendices contiennent de courtas descriptions de toes lea phe- Is nom nos rattachis aux systemes do coordonnees discute's: pro- cession at is nutation do In luny , lea aberrations at lea pa- rallaxes luna ires; sous donnons ausai lea formulas gene rates d's- berrations at do parallaxes. vii { KOHCI W b 3TOR CTaTbe npHBOAHTCA KpaTKHA Omepy c4.*pHKecxo# aCTpOHOMNH J1!iHbl. IipotmeMbi CBA3aHHwe CO CpeAHRMH H xaMymHMHCA ceAeHOUeHTpNMOC- KHMH M AyHKhWH TOnoueHTPMmecvmmH CA.epmmecyHMH KOOpAHHaTaMH AeTaAbHO 143N MaJOTCA. 06CVzAaVTC19 npOY.MyMeCTBa Y HeAOCTSTKK Apyrxx ctpepume-