ASTRONOMICAL AND NAUTICAL ALMANAC FOR SUN, MOON, BRIGHTER PLANETS,
58 NAVIGATIONAL STARS AND LUNAR DISTANCES V1.06f
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)
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,
see: https://astro.ukho.gov.uk/nao/lvm/ ):
Year= 500AD:+5590s(±40s)
600:+4650s
700:+3760s
800:+2940s
900:+2230s
1000:+1650s
1100:+1220s
1200:+910s
1300:+680s
1400:+480s
1500:+290s
1600:+109s
1700.0:+14.0s
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)
GHA and Dec of Sun, Moon, planets, and stars: | ±1" |
RA of Sun, Moon, planets, and stars: | ±0.07s |
GHA of Polaris*: | ±20" |
RA of Polaris*: | ±1.33s |
Dec of Polaris*: | ±1" |
HP and SD: | ±0.1" |
Equation of Time: | ±0.1s |
GAST, GMST, and Equation of Equinoxes: | ±0.001s |
Nutation (Δψ and Δε): | ±0.001" |
Mean Obliquity of the Ecliptic: | ±0.001" |
Lunar distance: | ±1" |
Altitude: | ±1" |
Azimuth: | ±1" |
Results were compared with Interactive Computer Ephemeris 0.51 (US Naval Observatory),
JPL-HORIZONS (NASA Jet Propulsion Laboratory, models DE431and DE441),
and Astronomical Almanac 5.6 (Stephen L. Moshier).
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.
*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: 20.sept.2022).