Astronomy Formulas -- Page Two  © 1997 by  James Q. Jacobs PAGE ONE: PERIODICITY FORMULAS -- DEFINITIONS -- NOTATION -- TIME FORMULAS

 ILLUMINATION GEOMETRY: Apparent solar rise-set position given: L = latitude A = Azimuth d = declination dC = degrees Celsius dK = degrees Kelvin = dC + 273 h = solar angular elevation ho = Horizon angle ha = apparent height of the celestial body considered p = atmospheric pressure Apparent solar rise-set position = cos A = (sin d - sin L sin h) / (cos L cos h) Fit of the direction of a site alignment to the rising or setting of the sun.  Given a level horizon: Cos A = - sin d / cos L Horizon elevation above mathematical level changes Azimuth by the angle A as follows: A = asin { tan ho / [ ( cos d / sin L )2 - 1 ]1/2 } Horizon illumination is affected by refraction.  The variables of atmospheric pressure and temperature are included according to the following: ha = ho + ( a / b ) where: a = p ( 0.1549 + 0.0196 ho + 0.00002 ho2 ) b = ( 273 + dC ) ( 1 + 0.5050 ho + 0.0845 ho2 ) Solving refraction correction for ho when ha is known results in: ho3 + a1 ho2 + a2 ho + a3 = 0 Given the coefficients: a1= 5.9763 - ha + 0.00023669 ( p / dK ) a2= 11.8343 - 5.9763 ha + 0.2320 ( p / dK ) a3= - 11.8343 + 1.8864 ( p / dK ) Then reduced by linear transformation: ha + ( a1/ 3 ) = y results in y3 + p y - q = 0 with p = a2- ( a12 / 3 ) q = ( a1/ 3 ) [( 2 a12 / 9 ) - a2] + a3 Thus the actual horizon elevation angle can be calculated as follows: ho = [- ( q / 2 ) + D1/2 ]1/3 + [ ( q / 2 ) - D1/2 ]1/3 - ( a1/ 3 ) where D = ( q / 2 )2 / ( p / 3 )3 MISCELLANEOUS FORMULAS AND DATA: Lunar Standstills can be determined by calculation of the longitude of the ascending node of lunar orbit.  The formula is: 259.183° - 0.05295° d. + 0.002078° T2 + 0.000002° T3 given d and T are days and Julian centuries from JD 2,415,020 (Jan. 1, noon, 1900 Ephemeris Time). Anomalistic Month, the period of the moon's eccentricity of orbit, from perigee to perigee, is 27.5545465 days.  Because the moon's distance is least at perigee, parallax is then greatest. Pleasant Creek Petroglyphs, Eastern Utah. Parallax is produced by viewing the moon from a moving earth.  The direction in which the moon appears is determined by position on the earth's surface.  To adjust the angle of view on the surface to what it would be from the center of the earth the following formula is used.  The lunar parallax correction is greatest when the moon is on the horizon, and zero when the moon is at zenith. p = 0.95075° + 0.0518055555° cos l + 0.0078333333° cos M + 0.0095277777° cos ( l - M ) + 0.0028333333° cos 2l + 0.0008611111° cos ( l + M ) where l = mean anomality, mean position of the moon measured counterclockwise from perigee, and M = longitude of the Moon. Obliquity of the Ecliptic is the temporally variated angle of the axis of rotation of the earth relative to the plane of revolution around the sun. OB = 23.4392911111° - 0.0130041666...° T - 0.00000163888...° T2 + 0.0000005036111...° T3 given: T = Julian centuries (36,525 days) from 2000.0. Epoch Calc, an Excel spreadsheet calculates this formula.  You just enter the date. Annual Secular Polar Motion is the alteration in the position of the axis of rotation relative to the surface of the geoid. 0".0035 (= 0.00000972222....°) along the meridian 65° W. Centennial General Precession is the slow and gradual retrograde rotation in the direction of the axis of rotation in fixed space. longitude = (1.396291666... + 0.0006180555... T)° right ascension, m = (1.280397222... + 0.000777... T)° declination, n = (0.5569666... + 0.000236111... T)° Anomalistic Year.  The disturbing effects of the planets on the line of the apsides of the earth results in an eastward advance of about 0.032365925° per orbit, thus producing the longer anomalistic year of (365.25964134 + 3.04 x 10-6 TE) days Ephemeris Time The Ephemeris Second, the unit of measure of Ephemeris Time, is defined in terms of the tropical year 1900, the fundamental ephemeris epoch.  The tropical year 1900 = 31,556,925.9747 seconds. Velocity of Light (in a vacuum) c = 299,792,458 meters per second = 186,282.4 miles per second. Astronomical Unit = AU = 149,597,870 km Solar Parallax = 0.00244281888...° Solar Radius = 696,000 km Gaussian Gravitational Constant = k = 0.01720209895 Constant of Gravitation = G = 6.672 x 10-11 m3 kg-1 s-2 Geocentric Gravitational Constant = 3.9860310 x 10 14 m3 s2 Heliocentric Gravitational Constant = 1.32712438 x 10 20 m3 s2 Mass of the Sun = 1.9891 x 1030 kg Mass of the Earth = 5.974 x 1024 kg Mass of the Moon = 7.348031948 x 1022 kg Constant of Nutation = 0.00255625° Constant of Aberration = 0.0056932 Mean Longitude of the Sun = 279.696677778° + 36,000.768925° TE + 0.0003025° TE2 Saros Cycle is an eclipse cycle of 242 nodal months, 223 synodic periods, 239 anomalistic months and 19 eclipse years. Epoch_calc.xls includes an eclipse calculator. 242 x 27.21222 = 6585.357425 223 x 29.53059 = 6585.321321 239 x 27.55455 = 6585.536614 19 x 346.62006 = 6585.781197 Metonic Cycle is a cycle that produces the same phase of the moon on the same date of the tropical year every 19 years. 235 x 29.53059 = 6939.688388 255 x 27.21222 = 6939.116295 19 x 365.24219 = 6939.601660 The Eight Planets, Some Data: (spreadsheet version) Planet Equatorial Radius (km) Sidereal Orbital Period, days Synodic Period, days Semi-major axis of orbit in AU Eccentricity Mercury 2,439.7 87.969 115.88 0.38709893 0.20563069 Venus 6,051.8 224.701 583.92166 0.72333199 0.00677323 Earth 6,378.14 365.25636053 365.242192654 1.00000011 0.01671022 Moon 1,737.4 27.32166156 29.5305888844 .002569555 0.05400489 Mars 3,396.19 686.980 779.94 1.52366231 0.09341233 Jupiter 71,492 4,332.589 398.88 5.20336301 0.04839266 Saturn 60,268 10,759.22 378.09 9.53707032 0.05415060 Uranus 25,559 30,685.40 369.33 19.19126393 0.04716771 Neptune 24,764 60,189.0 367.49 30.06896348 0.00858587

 FOR MORE ASTRONOMIC AND GEODETIC DATA CHECK OUT THE: Astronomical Constants Index - - - Cosmographic Values Index - - - Geodesy Page Did ancient astronomers survey the Earth? - - - - Read the Archaeogeodesy Pages WORLD WIDE WEB HUBS BY THE AUTHOR: Home  |  Anthropology  |  Archaeoastronomy  |  Photo Stock  |  Web Design  |  Art ©1997 by James Q. Jacobs.  All rights reserved. Your comments, etc. are appreciated: Contact. Published Feb. 1, 1998. Cite as: http://www.jqjacobs.net/astro/astrofor_2.html. Edited Nov. 28, 2002, after Mark E. Kowitt, Ph.D., found a typo in the number of years in a Saros cycle. Thank you Mark. 