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1 PERIODICITY FORMULAS: Sidereal Orbit Tropical Year Eclipse Year Anomalistic Year Sidereal Lunar Orbit Lunar Mean Daily Sidereal Motion Lunar Synodical Period Centenial General Precession Longitude ( x 10-7 TE) days ( x 10-6 TE) days ( x 10-5 TE) days ( x 10-6 TE) days ( x 10-7 T) days ( x 10-8 T) ( x 10-7 T) days ( T) Given TE = Julian centuries from day 0.5, 1900 ET Given T = Tropical centuries from N DEFINITIONS: Sidereal Orbit Sidereal Noon Fundamental Epoch of Sidereal Time Ephemeris Time Tropical Year (YT) Eclipse Year (YE) Temporal Unit (TU) Greenwich Mean Sidereal Time (GT) Universal Time (UT) Lunar Synodic Period (S9) Precession (PR) Annual Parallax Annual Aberration Diurnal Parallax Diurnal Aberration Atmospheric Refraction is a revolution relative to a fixed celestial position. is the instant of transit of mean equinox relative to a fixed meridian position. (FE) is the instant 12 hours, 0 days, 1900 years A.D. with hours in mean sidereal time. is the actual count of solar days from a fixed meridian. is the period from equinox to equinox. is the period between the earth and lunar orbit planes node crossings. is 36,525 mean solar days since Jan. 0.5, 1900, UT. = 0.0 hours UT = 12 hours + afms has replaced Mean Solar Time due to a recognition of the non-uniform rotation rate of the earth. is the period of time from one full or new moon to another, that is the time between consecutive alignments of the sun, earth and moon on a plane perpendicular to the plane of solar revolution. is the retrograde rotation of the earth's axis relative to fixed celestial reference. is the viewpoint difference due to the change in the earth's position relative to the sun. For the nearest star the angle is about is the angular shift in apparent position resulting from motion velocity of viewing from orbiting (moving) earth. is the viewpoint difference due to the rotation of the earth. The amount varies with the latitude of the observer. is the result of observing from a spining observing position on the surface of the earth. Velocity of the observer causes apparent shift to a maximum correction of about at the equator. is the bending of light rays by the earth's atmosphere.

2 NOTATION afms Ficticious Mean Solar position DMS Day, Mean Sidereal d, h, m, s, day, hour, minute, second ES ET FE GT JC JD Ephemeris Second Ephemeris Time Fundamental Epoch Greenwich Mean Sidereal Time Julian Century Julian Day L Longitude of the Mean Sun R Period of Sidereal Rotation T TE TU UT Tropical Centuries from N Temporal Epoch Temporal Solar Based Unit Universal Time TIME FORMULAS afms d s628 TU TU 2 DMS 86,400.s DMS / P x TE DMS / P ( x TE) -1 ES Tropical Year 1900 / 31,556, FE 12h 0d 1900 A.D. (hours in mean sidereal time) FE geometric mean solar longitude : mean GT 12h + afms GT d s8138 T'U + 0.s0929 T'U 2 JC 36,525 days ephemeris time L , TE TE 2 Mean Solar Day : Mean Sidereal Time Mean Solar Day : Mean Sidereal Time x TU ( x TU) -1 TE One JC from 0.d5, 1900 (JD 2,415,020.) TU T'U 36,525 mean solar days from 12h, Jan. 0, 1900 UT. 36,525 mean solar days from 12h, Jan. 0, 2000 UT1.

3 ILLUMINATION GEOMETRY Apparent solar rise-set position given: L = latitude A = Azimuth d = declination dc = degrees Celsius dk = degrees Kelvin = dc 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 ( ho ho 2 ) b = ( dc ) ( ho ho 2 ) Solving refraction correction for ho when ha is known results in: ho 3 + a1 ho 2 + a2 ho + a3 = 0 Given the coefficients: a1= ha ( p / dk ) a2= ha ( p / dk ) a3= ( p / dk ) Then reduced by linear transformation: ha + ( a1/ 3 ) = y results in y 3 + p y - q = 0 with p = a2- ( a1 2 / 3 ) q = ( a1/ 3 ) [( 2 a1 2 / 9 ) - a2] + a3 Thus the actual horizon elevation angle can be calculated as follows: ho = [- ( q / 2 ) + D 1/2 ] 1/3 + [ ( q / 2 ) - D 1/2 ] 1/3 - ( a1/ 3 ) where D = ( q / 2 ) 2 / ( p / 3 ) 3

4 MISCELLANEOUS FORMULAS AND DATA Lunar Standstills can be determined by calculation of the longitude of the ascending node of lunar orbit. The formula is: d T T 3 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 days. Because the moon's distance is least at perigee, parallax is then greatest. 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 = cos l cos M cos ( l - M ) cos 2l 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 = T T T 3 given: T = Julian centuries (36,525 days) from Annual Secular Polar Motion is the alteration in the position of the axis of rotation relative to the surface of the geoid. 0".0035 (= ) 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 = ( T) right ascension, m = ( T) declination, n = ( 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 per orbit, thus producing the longer anomalistic year of ( 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, 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 = Solar Radius = 696,000 km Gaussian Gravitational Constant = k =

5 Constant of Gravitation = G = x m 3 kg -1 s -2 Geocentric Gravitational Constant = x m 3 s 2 Heliocentric Gravitational Constant = x m 3 s 2 Mass of the Sun = x kg Mass of the Earth = x kg Mass of the Moon = x kg Constant of Nutation = Constant of Aberration = Mean Longitude of the Sun = , TE TE 2 Saros Cycle is an eclipse cycle of 242 nodal months, 223 synodic periods, 239 anomalistic months and 17 eclipse years. 242 x = x = x = x = 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 = x = x = SOME PLANETARY DATA Planet Equatorial Radius (km) Orbital Period in days Semi-major axis of orbit in AU Eccentricity Mercury Venus Earth Mars Jupiter 71, Saturn 60,000 10, Uranus 25,400 30, Neptune 24,300 60, Pluto ,

6 COSMOGRAPHIC VALUES INDEX A SET OF COSMOGRAPHIC VALUES WITH CODE CONSTANT NUMERIC VALUE CODE Earth, Radius, Equatorial mi. 6,378,140 m ER Earth, Circumference, Equatorial Earth, Circumference, Mean Earth, Arc Degree, Mean Earth, Radius, Mean Earth, Radius, Polar 24, mi. 40,075, m 24, mi. 40,030, m mi. 111, m mi. 6,370, m mi. 6,356, m CE CRM AM RA RP Flattening, axis ratio FL Flattening inverse FI Flattening of the Earth FE Inclination, Lunar Orbit, Mean IL Wobble, Lunar Orbit Inclination ± WL Lunar Major ( OB + IL ) OL Lunar Minor ( OB - IL ) OM Moon, Distance 384,399,070 m MD Lunar Orbit Eccentricity LE Moon, Radius, Mean 1,738,000 m MR Obliquity of the Ecliptic OB Solar distance 149,597,870,000 m AU Geographical Mile 1, m MG MODULES in degrees CODE Lunar Motion per Mean Rotation R27 Mean Daily Lunar Motion C27 Solar Orbit per Day SO Solar Orbit per Sidereal Rotation SI Solar Orbit per Nodal Month S22 Solar Orbit per Sidereal Month S27 Solar Orbit per Synodic Month S29

7 ASTRONOMIC CONSTANTS INDEX A SET OF ASTRONOMIC CONSTANTS WITH CODE CONSTANT NUMERIC VALUE CODE Earth, Orbit, Days DO Earth, Orbits per Day OD Earth, Orbits per Lunar Orbit OL Earth, Orbits per Nodal Month ON Earth, Rotations, Lunar Orbit RL Earth, Rotations, Nodal Month RN Earth, Rotations, Orbit RO Earth, Rotations, Synodic Month R9 Earth, Rotations, Year RY Eclipse Year, Days YE Eclipse Year, Rotations RE Lunar Nodal Month, Days S2 Lunar Nodal Months per Day ND Lunar Nodal Months per Orbit NO Lunar Nodal Months per Rotation NR Lunar Orbit, Days S7 Lunar Orbits per Day LD Lunar Orbits per Orbit LO Lunar Orbits per Rotation LR Lunar Sidereal Month, Days S7 Lunar Synodic Month, Days S9 Nutation Cycle, Orbits NUO Nutation Cycle, Years YN Precession Cycle, Years PR Rotation, Seconds SRO Sidereal Day, Seconds SDS Solar Day, Seconds SD Year, Eclipse, Days YE Year, Rotations YR Year, Tropical, Days YT Years per Day YD Years per Lunar Orbit YL Years per Nutation YN Years per Orbit O YO

8 Solar System Planet Equatorial Radius (km) Orbital Period in days Semi-major axis of orbit in AU Mercury Venus Earth Mars Jupiter 71, Saturn 60,000 10, Uranus 25,400 30, Neptune 24,300 60, Pluto , Solar Radius 696,000 Earth and Moon Astronomical Unit = AU = 149,597,870 km Velocity of Light (in a vacuum) c = 299,792,458 meters per second = 186,282.4 miles per second. CONSTANT Earth, Radius, Equatorial Earth, Circumference, Equatorial Earth, Circumference, Mean Earth, Arc Degree, Mean Earth, Radius, Mean NUMERIC VALUE mi. 6,378,140 m 24, mi. 40,075, m 24, mi. 40,030, m mi. 111, m mi. 6,370, m Earth, Radius, Polar mi. 6,356, m Flattening, axis ratio Flattening inverse Flattening of the Earth Inclination, Lunar Orbit, Mean Wobble, Lunar Orbit Inclination ± Moon, Distance 384,399,070 m Lunar Orbit Eccentricity

9 Moon, Radius, Mean 1,738,000 m Obliquity of the Ecliptic Solar distance 149,597,870,000 m Geographical Mile 1, m Periodicity Saros Cycle is an eclipse cycle of 242 nodal months, 223 synodic periods, 239 anomalistic months and 17 eclipse years. 242 x = x = x = x = 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 = x = x = MODULES in degrees Lunar Motion per Mean Rotation Mean Daily Lunar Motion Solar Orbit per Day Solar Orbit per Sidereal Rotation Solar Orbit per Nodal Month Solar Orbit per Sidereal Month Solar Orbit per Synodic Month Periodicity Formulas Sidereal Orbit Tropical Year Eclipse Year Anomalistic Year Sidereal Lunar Orbit Lunar Mean Daily Sidereal Motion Lunar Synodical Period Centenial General Precession Longitude Given TE = Julian centuries from day 0.5, 1900 ET Given T = Tropical centuries from N Definitions ( x 10-7 TE) days ( x 10-6 TE) days ( x 10-5 TE) days ( x 10-6 TE) days ( x 10-7 T) days ( x 10-8 T) ( x 10-7 T) days ( T) Sidereal Orbit Sidereal Noon is a revolution relative to a fixed celestial position. is the instant of transit of mean equinox relative to a fixed meridian position.

10 Fundamental Epoch of Sidereal Time Ephemeris Time Tropical Year (YT) Eclipse Year (YE) Temporal Unit (TU) Greenwich Mean Sidereal Time (GT) Universal Time (UT) Lunar Synodic Period (S9) Precession (PR) Annual Parallax Annual Aberration Diurnal Parallax Diurnal Aberration Atmospheric Refraction (FE) is the instant 12 hours, 0 days, 1900 years A.D. with hours in mean sidereal time. is the actual count of solar days from a fixed meridian. is the period from equinox to equinox. is the period between the earth and lunar orbit planes node crossings. is 36,525 mean solar days since Jan. 0.5, 1900, UT. = 0.0 hours UT = 12 hours + afms has replaced Mean Solar Time due to a recognition of the non-uniform rotation rate of the earth. is the period of time from one full or new moon to another, that is the time between consecutive alignments of the sun, earth and moon on a plane perpendicular to the plane of solar revolution. is the retrograde rotation of the earth's axis relative to fixed celestial reference. is the viewpoint difference due to the change in the earth's position relative to the sun. For the nearest star the angle is about is the angular shift in apparent position resulting from motion velocity of viewing from orbiting (moving) earth. is the viewpoint difference due to the rotation of the earth. The amount varies with the latitude of the observer. is the result of observing from a spining observing position on the surface of the earth. Velocity of the observer causes apparent shift to a maximum correction of about at the equator. is the bending of light rays by the earth's atmosphere. GRAVITY Mass of the Sun = x kg Mass of the Earth = x kg Mass of the Moon = x kg

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