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Incidental Tables Sp.-V/AQuan/1999/10/27:16:16 Page 667 Chapter 27 Incidental Tables Alan D. Fiala, William F. Van Altena, Stephen T. Ridgway, and Roger W. Sinnott 27.1 The Julian Date ...................... 667 27.2 Standard Epochs ...................... 668 27.3 Reduction for Precession ................. 669 27.4 Solar Coordinates and Related Quantities ....... 670 27.5 Constellations ....................... 672 27.6 The Messier Objects .................... 674 27.7 Astrometry ......................... 677 27.8 Optical and Infrared Interferometry ........... 687 27.9 The World’s Largest Optical Telescopes ........ 689 27.1 THE JULIAN DATE by A.D. Fiala The Julian Day Number (JD) is a sequential count that begins at Noon 1 Jan. 4713 B.C. Julian Calendar. 27.1.1 Julian Dates of Specific Years Noon 1 Jan. 4713 B.C. = JD 0.0 Noon 1 Jan. 1 B.C. = Noon 1 Jan. 0 A.D. = JD 172 1058.0 Noon 1 Jan. 1 A.D. = JD 172 1424.0 A Modified Julian Day (MJD) is defined as JD − 240 0000.5. Table 27.1 gives the Julian Day of some centennial and decennial dates in the Gregorian Calendar. 667 Sp.-V/AQuan/1999/10/27:16:16 Page 668 668 / 27 INCIDENTAL TABLES Table 27.1. Julian date of selected years in the Gregorian calendar [1, 2]. Julian day at noon (UT) on 0 January, Gregorian calendar Jan. 0.5 JD Jan. 0.5 JD Jan. 0.5 JD Jan. 0.5 JD 1500 226 8923 1910 241 8672 1960 243 6934 2010 245 5197 1600 230 5447 1920 242 2324 1970 244 0587 2020 245 8849 1700 234 1972 1930 242 5977 1980 244 4239 2030 246 2502 1800 237 8496 1940 242 9629 1990 244 7892 2040 246 6154 1900 241 5020 1950 243 3282 2000 245 1544 2050 246 9807 Century years evenly divisible by 400 (e.g., 1600, 2000) are leap years. Others are not. References 1. Explanatory Supplement to the Astronomical Almanac. 1992, edited by P.K. Seidelmann (University Science Books, Mill Valley, CA), pp. 55, 56, 580, 581, 600–604 2. Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac. 1961, (Her Majesty’s Stationery Office, London), pp. 434–439 27.1.2 Conversion Algorithms Several algorithms for converting among Julian Calendar Date, Gregorian Calendar Date, Islamic Tabular Calendar Date, Indian Civil Calendar, and Julian Day Number, and computing day of the week, are given in [1]. Probably the most useful of these is the conversion from Gregorian Calendar Date to Julian Day Number, as follows [2]: Julian Day Numbers run from noon to noon. Define the following integer variables: JD = Julian Day Number, Y = calendar year, M = month, D = day of month. Given Y , M, D, compute JD: JD = (1461 × (Y + 4800 + (M − 14)/12))/4 + (367 × (M − 2 − 12 × ((M − 14)/12)))/12 − (3 × ((Y + 4900 + (M − 14)/12)/100))/4 + D − 32075. JD ≥ 0, that is, the date is after −4713 November 23. 27.2 STANDARD EPOCHS by A.D. Fiala The beginning of the Besselian (fictitious) solar year is the instant when the right ascension of the fictitious mean sun, affected by aberration and measured from the mean equinox, is 18h40m. The Julian year begins at noon on January 0. Table 27.2 gives the Julian Date of several years. Sp.-V/AQuan/1999/10/27:16:16 Page 669 27.3 REDUCTION FOR PRECESSION / 669 Table 27.2. Julian dates of Julian and Besselian years [1, 2]. Julian year Besselian year JY JD BY JD B1850.0 239 6758.203 J1900.0 241 5020.0 B1900.0 241 5020.313 J1950.0 243 3282.5 B1950.0 243 3282.423 B1975.0 244 2413.478 J2000.0 245 1545.0 B2000.0 245 1544.533 B2025.0 246 0675.588 J2050.0 246 9807.5 B2050.0 246 9806.643 J2100.0 248 8070.0 B2100.0 248 8068.753 References 1. Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac. 1961, (Her Majesty’s Stationery Office, London), pp. 434–439 2. Explanatory Supplement to the Astronomical Almanac. 1992, edited by P.K. Seidelmann (University Science Books, Mill Valley, CA), p. 8 27.3 REDUCTION FOR PRECESSION by A.D. Fiala Approximate formulas for the reduction of coordinates and orbital elements referred to the mean equinox and equator or ecliptic of date (t) are as follows, as given in the Astronomical Almanac [3], page B19, in all years since 1984: For reduction to J2000.0 For reduction from J2000.0 α0 = α − M − N sin αm tan δm α = α0 + M + N sin αm tan bm δ0 = δ − N cos αm δ = δ0 + N cos αm λ0 = λ − a + b cos(λ + c ) tan β0 λ = λ0 + a − b cos(λ0 + c) tan β β0 = β − b sin(λ + c )β= β0 + b sin(λ0 + c) 0 = − a + b cos( + c ) cot i0 = 0 + a − b sin( 0 + c) cot i i0 = i − b cos( + c ) i = i0 + b cos( 0 + c) ω0 = ω − b sin( + c ) csc i0 ω = ω0 + b sin( 0 + c) csc i where α and δ are the right ascension and declination; λ and β are the ecliptic longitude and latitude; and , i, and ω are the orbital elements (referred to the ecliptic) longitude of the node, inclination, and argument of perihelion; the subscript zero refers to epoch J2000.0; and αm and δm refer to the mean epoch. With sufficient accuracy: α = α − 1 ( + α δ), m 2 M N sin tan δ = δ − 1 α , m 2 N cos m α = α + 1 ( + α δ ), m 0 2 M N sin 0 tan 0 δ = δ + 1 α . m 0 2 N cos m Sp.-V/AQuan/1999/10/27:16:16 Page 670 670 / 27 INCIDENTAL TABLES The precessional constants M, N, etc., are given by ◦ ◦ ◦ M = 1. 281 2323T + 0. 000 3879T 2 + 0. 000 0101T 3 ◦ ◦ ◦ N = 0. 556 7530T − 0. 000 1185T 2 − 0. 000 0116T 3 ◦ ◦ a = 1. 396 971T + 0. 000 3086T 2 ◦ ◦ b = 0. 013 056T − 0. 000 0092T 2 ◦ ◦ ◦ c = 5. 123 62 + 0. 241 614T + 0. 000 1122T 2 ◦ ◦ ◦ c = 5. 123 62 − 1. 155 358T − 0. 000 1964T 2 where T = (t − 2000.0)/100 = (JD − 245 1545.0)/36 525. 27.4 SOLAR COORDINATES AND RELATED QUANTITIES by A.D. Fiala 27.4.1 The Sun’s Coordinates and the Equation of Time: Low-Precision Formulas The following formulas from any recent Astronomical Almanac [3], page C24, give the apparent coordinates of the Sun to a precision of 0.◦01 and the equation of time to a precision of 0.m1 between 1950 and 2050; on this page the time argument n is the number of days from J2000.0. n = JD − 245 1545.0 =−2557.5 + day of year (B2-B3) + fraction of day from 0h UT. The mean longitude of the Sun, corrected for aberration, is ◦ ◦ L = 280. 460 + 0. 985 647 4n. The mean anomaly is g = 357.◦528 + 0.◦985 600 3n. Put L and g in the range 0◦ to 360◦ by adding multiples of 360◦. The ecliptic longitude is λ = L + 1.◦915 sin g + 0.◦20 sin 2g. The ecliptic latitude is β = 0◦. The obliquity of ecliptic is = 23.◦439 − 0.◦000 000 4n. The right ascension (in the same quadrant as λ)isα = tan−1(cos tan λ). Alternatively, α may be calculated directly from α = λ − ftsin 2λ + ( f/2)t2 sin 4λ, where f = 180/π and t = tan2( /2). The declination is δ = sin−1(sin sin λ). The distance of the Sun from the Earth, in AU, is R = 1.000 14 − 0.016 71 cos g − 0.000 14 cos 2g. The equatorial rectangular coordinates of the Sun, in AU, is x = R cos λ, y = R cos sin λ, z = R sin sin λ. Sp.-V/AQuan/1999/10/27:16:16 Page 671 27.4 SOLAR COORDINATES AND RELATED QUANTITIES / 671 The equation of time (apparent time minus mean time) is E(in minutes of time) = 4(L − α), where L − α is in degrees. The horizontal parallax is 0.◦0024. The semidiameter is 0.◦2666/R. The light time is 0.d0058. Similar formulas for the Moon are given in any recent Astronomical Almanac [3], page D46. 27.4.2 The Sun’s Coordinates and the Equation of Time: Tables In Table 27.3 the Sun’s position and the equation of time are evaluated to low precision to represent a typical year. In addition, the UT of the transit of Aries is given and also the right ascension of Aries at 0h UT; i.e., the Universal Time of the beginning of the sidereal day, and the sidereal time of the beginning of the civil day. The quantities are calculated using the reference in the table. They are the approximate mean of values over a four-year cycle, with range on the order of two units in the last significant digit. Table 27.3. Sun’s coordinates and equation of time [1]. Equation RA of = Sun’s geocentric apparent of time midnight app − mean Transit meridian αδLong. Dist. E − 12h of Aries =∼ R Date (h m) (deg) (deg) (AU) (m) (h m) (h m) Jan. 1 18 44 −23.0 280.2 0.9833 −3.2 17 16 6 41 Jan. 16 19 50 −21.0 295.5 0.9837 −9.5 16 17 7 40 Feb. 1 20 57 −17.3 311.8 0.9853 −13.5 15 14 8 43 Feb.
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