ASTRONOMICAL AND NAUTICAL ALMANAC FOR SUN, MOON, BRIGHTER PLANETS,
 58 NAVIGATIONAL STARS AND LUNAR DISTANCES  V1.06n

 Copyright © 2021 Henning Umland & Heiner Müller-Krumbhaar

DESCRIPTION:

This is a computer almanac developed with special attention to the needs of navigators and amateur astronomers. It is based upon the VSOP87D Theory (P. Bretagnon, G. Francou)), the 1980 IAU Nutation Theory (P. Seidelmann) including J. Laskar's formula, the LEA406b Theory for the Moon (S. Kudryavtsev) which is based on models of the NASA-Jet-Propulsion Laboratory, and formulas published in Astronomical Algorithms by J. Meeus and Textbook on Spherical Astronomy by W. M. Smart. The program calculates Greenwich hour angle (GHA), the right ascension (RA), and declination (Dec) for Sun, Moon, Venus, Mars, Jupiter, Saturn, and navigational stars (apparent geocentric positions).

Optionally, the program also calculates geocentric altitudes and azimuths of bodies for an assumed position (sight reduction). Corrections for parallax and atmospheric refraction are not included. Azimuths are measured clockwise (0°...360°) from the geographic north direction, as is common in marine navigation.

Furthermore, the following quantities are provided:

Geocentric semidiameter (SD) and equatorial horizontal parallax (HP) for Sun, Moon, and planets
Equation of time
Illuminated fraction of the apparent disks of Moon and planets
Phase of the Moon
Greenwich mean sidereal time (GMST)
Greenwich apparent sidereal time (GAST)
Apparent longitude of the Sun (equinoxes = 0°,180°; solstices = 90°,270°)
Equation of the equinoxes (= GAST−GMST)
Nutation in longitude (Δψ)
Nutation in obliquity (Δε)
Mean obliquity of the ecliptic
True obliquity of the ecliptic (= mean obliquity + Δε)
Julian date (JD)
Julian ephemeris date (JDE)
Geocentric lunar distance of Sun, planets, and selected star (center−center)
Day of the week

The phases of the Moon are indicated as follows:

New
+cre = waxing crescent
FQ = first quarter
+gib = waxing gibbous
Full
–gib = waning gibbous
LQ = last quarter
–cre = waning crescent

By definition, the phase is defined by the difference between the ecliptic longitudes of Moon and Sun. It does not exactly correlate with the illuminated fraction of the Moon's disk since the plane of the Moon's orbit is inclined to the ecliptic.

GHA and RA refer to the true equinox of date. The relation between these is: GHA = GAST - RA (each in either degrees or hours)

The apparent positions of the planets refer to their respective center. A phase correction is not included.

The apparent semidiameter of Venus includes the cloud layer of the planet and may be slightly greater than values calculated with other software (referring to the solid surface). The semidiameters of Jupiter and Saturn refer to the respective equator.

For determining longitude or time by lunar distances, bodies near the ecliptic are most suitable. Those are Sun, all planets, and stars with an ecliptic latitude of less than approx. ±10°. The recommended stars are marked by an asterisk in the drop-down menu. Since the lunar distance of each body passes through a minimum during each lunar orbit, the rate of change of the lunar distance becomes zero at such an instant. Therefore, the rate of change should be checked by calculating the lunar distance of the chosen body at two instants 1 h apart. This rate of change (absolute value) should be as high as possible (roughly 0.5°/h) for best precision. If it is too small, another body should be chosen. - A separate program Lunars-V13f.c (written in Fortran and "C", on this web-site) allows for a complete determination of time and position at sea using Lunar distances.

The almanac covers a time span of 400 years (recommended range: 1800...2200), provided the ΔT value (= TT−UT1) for the given date is known. The rotation-speed of the earth is slowing down with time (UT1 vs. TT) due to tidal friction, mostly caused by the moon. But there are some irregular changes which cannot be predicted for long times. In turn, those irregularities are coupled back to the orbit of the moon. Therefore errors in ΔT have a much greater influence on the coordinates of the Moon than on the other results. A value of ΔT accurate to approx. ±1s is sufficient for most applications. ΔT is obtained through the following formula:

ΔT = 32.184s + (TAI−UTC) − (UT1−UTC)

Current values for TAI−UTC and UT1−UTC are regularly published on the web site of the International Earth Rotation and Reference Systems Service, IERS (IERS Bulletin A, General Information). Instead of UT1−UTC, here DUT1 (= UT1−UTC rounded to a precision of 0.1s) can be used. UTC deviates from UT1 by at most 0.9s because of the annual introduction of leap seconds into UTC.

Example:   TAI−UTC = 37s,   DUT1 = −0.2s,   ΔT = 32.184 + 37 − (−0.2) = 69.4s

ΔT tends to increase in the long term (tidal friction). Since the evolution of ΔT is also influenced by various random processes, accurate long-term predictions are not possible. Here are some ΔT values of the past (Proc. Royal Society A, 472, 2016, by F.Stephenson, L.Morrison and C.Hohenkerk (="SMH"), see: https://astro.ukho.gov.uk/nao/lvm/ ):

Year= 500AD:+5590s(±40s)   600:+4650s   700:+3760s   800:+2940s   900:+2230s(±30s)  
1000:+1650s   1100:+1220s   1200:+910s   1300:+680s   1400:+480s(±20s) 
1500:+290s   1600:+109s   1700.0:+14.0s(±5s)   1800.0:+18.4s   1850.0:+9.3s   1900.0: -1.98s(±0.05s)  
1950.0:+28.93s   1960.0:+33.07s   1970.0:+39.93s   1980.0:+50.36s   1990.0:+56.99s  
2000.0:+63.81s   2010.0:+66.06s   2020.0:+69.36s  
Prediction only (from year 2020):
2030:+67s(±2s)  2040:+68s  2050:+70s  2100:+80s(±10s)  2200:+160s  2300:+330s  2400:+610s  2500:+1000s(±100s) 

Results for sun, moon and planets were compared with Interactive Computer Ephemeris 0.51 (US Naval Observatory) and in particular with JPL-HORIZONS (NASA Jet Propulsion Laboratory, models DE431and DE441). Note that NASA-JPL uses ΔT=TDB-UT1 for years before 1962, but ΔT=TDB-UTC for later years ! Here TDB is solar-systems Barycentric Dynamical Time, differing from TT only periodically by about 2 milliseconds. The difference UT1-UTC can be obtained from IERS, see above.
The positions of stars are based on the fifth fundamental catalogue FK5 for epoch J2000.0. Results for stars were compared with Astronomical Almanac 5.6 (Stephen L. Moshier) in reference also to the book "Astronomical Algorithms" by J. Meeus. (The definition of star-velocities is a non-trivial task. Because of 200 years of observations FK5 may partly give more reliable longterm data than FK6).

^*GHA and RA for the triple-star system Polaris seemingly have a lower precision which is only geometrically caused (small polar distance). The meridians converge at the poles, and the equatorial coordinate system has a singularity there in each case. For Polaris, the possible error Δ therefore cannot be taken directly as δ GHA but must be viewed in combination with the distance from the pole as Δ = δ GHA*cos(Dec), with cos(90°)=0. Again, for that combination our precision is ± 1 arcsecond.
   (last update: 8.may.2023).