Astronomy Formulas  Page Two © 1997 by James Q. Jacobs ILLUMINATION GEOMETRY  MISCELLANEOUS FORMULAS  PLANETARY DATA 
PAGE ONE: PERIODICITY FORMULAS  DEFINITIONS  NOTATION  TIME FORMULAS 

Apparent solar riseset position
given:
L = latitudeApparent solar riseset 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:
Horizon elevation above mathematical level changes Azimuth by the angle A as follows:
Horizon illumination is affected by refraction. The variables of atmospheric pressure and temperature are included according to the following: ha = ho + ( a / b ) where: b = ( 273 + dC ) ( 1 + 0.5050 ho + 0.0845 ho^{2} ) Solving refraction correction for ho when ha is known results in:
Given the coefficients: a_{2}= 11.8343  5.9763 ha + 0.2320 ( p / dK ) a_{3}=  11.8343 + 1.8864 ( p / dK ) Then reduced by linear transformation:
p = a_{2} ( a_{1}^{2} / 3 ) q = ( a_{1}/ 3 ) [( 2 a_{1}^{2} / 9 )  a_{2}] + a_{3} Thus the actual horizon elevation angle can be calculated as follows:
where D = ( q / 2 )^{2} / ( p / 3 )^{3} 
PAGE ONE  ILLUMINATION GEOMETRY  PLANETARY DATA 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° T^{2} + 0.000002° T^{3} 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 )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.where l = mean anomality, mean position of the moon measured counterclockwise from perigee, and OB = 23.4392911111°  0.0130041666...° T  0.00000163888...° T^{2} + 0.0000005036111...° T^{3}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)°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) daysEphemeris 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} m^{3} kg^{1} s^{2} Geocentric Gravitational Constant = 3.9860310 x 10 ^{14} m^{3} s^{2} Heliocentric Gravitational Constant = 1.32712438 x 10 ^{20} m^{3} s^{2} Mass of the Sun = 1.9891 x 10^{30} kg Mass of the Earth = 5.974 x 10^{24} kg Mass of the Moon = 7.348031948 x 10^{22} kg Constant of Nutation = 0.00255625° Constant of Aberration = 0.0056932 Mean Longitude of the Sun = 279.696677778° + 36,000.768925° TE + 0.0003025° TE^{2} 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 = 6585.357425Metonic 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 





Synodic Period, days






115.88






583.92166






365.242192654



Moon

1,737.4

27.32166156

29.5305888844

.002569555

0.05400489




779.94






398.88






378.09






369.33






367.49



More data here: NASA Planets and
Pluto: Physical Characteristics 
Astronomical
Constants Index    Cosmographic Values
Index    Geodesy Page ©1997 by James Q.
Jacobs. All rights reserved. 