POSITION-LINE-NAVIGATION using SEXTANT and CLOCK,    - V.07
     Copyright © 2022 see below

DESCRIPTION:

General: This program calculates a position-line (Marc St. Hilaire-method) from the observation of sextant-altitude of a celestial body together with the precise time. A second position line, either from a different celestial body or from the same body some two hours or so later, and its intersection with the first position-line gives the position of the ship. If the ships position has changed between the two measurements, the first position line must be shifted by the same amount on the sea-chart.

Usage: Enter date and time of the sextant-observation of a celestial body. Then enter your assumed geographic position (=Dead Reckoning Position, DR) by latitude and longitude. Finally enter the angle above horizon measured by the sextant, and give values for index-correction, temperature, pressure and height-of-eye above sea-level. Finally enter 0 or 1 for either the lower (=0) or the upper rim (=1) when measured on the sun or moon. (This is irrelevant for planets and stars). At last hit "Calculate" to complete the calculation. Since sofar you have not specified which celestial body you have observed, the program offers a comparison of the observed with the calculated altitudes for all bodies from Sun through Saturn and Star. The difference is given in the last column denoted by "Δh [nm]" in arcminutes corresponding to nautical miles. The POSITION-LINE then must be drawn orthogonal to the Azimuth-direction of the observed body, and shifted away from DR-position by an amount ±Δh in parallel towards (+) or away from (-) the oberved celestial body (respectively its image point on the surface of the earth). -
In detail, the program calculates geocentric altitudes and azimuths of bodies for an assumed position, with corrections for atmospheric refraction and parallax, and returns the corrected observed sextant-height and deviation Δh from calculated sextant-height (sight reduction). Azimuths are measured clockwise (0°...360°) from the geographic north direction, as is common in marine navigation. See Henning Umlands web-page for the theory behind. When the difference Δh between observed and calculated altitude is larger than about 3°, a wiggly line "~~~" is displayed, otherwise the Δh-value in arc-minutes, equivalent to nautical miles. The results have been tested against various tutorial tables for astronomical navigation. Experience shows that the uncertainties of the measurement and the atmospheric conditions are much larger than the uncertainties of the geocentric altitudes calculated by this ephemeris-program.

Astronomical Almanac: The ephemeris used is identical to the computer almanac Astronomical-Almanac-106.html by the same authors. 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).

Furthermore, the following quantities are provided:

Geocentric semidiameter (SD) and equatorial horizontal parallax (HP) for Sun, Moon, and planets
Equation of time
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)
Day of the week

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.

A separate program AstroNavig-V13f.c (written in Fortran and "C", on this web-site) allows for a complete determination of position at sea using 2 position lines.
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) 

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: 29.oct.2022).