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by Lunar Distance
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The lunar
distance is the angle between the Moon and the Sun
or a celestial body (planet, star). By this method
the Greenwich time can by calculated, using an
almanach with the tabulated values. Then the
longitude can be determined without a chronometer. Lunar Distances were used to determine longitude at sea in the period from about 1770 (Lunar Tables by Tobias Mayer) until 1850, but rarely after that date. This was a time when chronometers were not yet widely available at a reasonable price and were considered unreliable on long voyages. But the lunars tables were not dropped from the US "American Ephemeris & Nautical Almanac" until 1912, from the German "Nautisches Jahrbuch  Ephemeriden und Tafeln" as late as 1924. Captain Joshua Slocum (18441909), the first man to sail singlehandedly around the world, wrote in Sailing Alone Around the World (1900 first edition) Chapter XI: >>On the fortythird day from land,  a long time to be at sea alone  the sky being beautifully clear and the moon being "in distance" with the sun, I threw up my sextant for sights. I found from the result of three observations, after long wrestling with lunar tables, that her longitude by observation agreed within five miles of that by deadreckoning. This was wonderful; both, however, might be in error, but somehow I felt confident that both were nearly true, and that in a few hours more I should see land; and so it happened, for then I made the island of Nukahiva, the southernmost of the Marquesas group, clearcut and lofty. The verified longitude when abreast was somewhere between the two reckonings; this was extraordinary. All navigators will tell you that from one day to another a ship may lose or gain more than five miles in her sailingaccount, and again, in the matter of lunars, even expert lunarians are considered as doing clever work when they average within eight miles of the truth.<< Using a sextant, the navigator precisely measures the angle D' between the limb of the Moon and the Sun. First correction: The first correction to the observed lunar distance d is the distance between the limb of the Moon (r_{M}) and its center and the limb of the Sun and its center (r_{S}): D' = d + r_{M} + r_{S} Second correction: The almanac gives geocentric lunar distances (as they would appear from the center of the Earth). Therefore the lunar distance has to be corrected for the effects of atmospheric refraction (R) and parallax (P). The topocentric apparent altitude of the Moon H3=H' has to be converted to the geocentric altitude H4=H taking into account refraction (R) and parallax (P). The equatoreal horizontal parallax is HP = arcsin(r_{E}/d_{ME}), and the parallax P = arcsin[sin(HP)*cos(H' ). H = H'  R + P = H'  R +
arcsin[sin(HP)*cos(H' )]
The geocentric altitude h of the Sun is obtained from the topocentric apparent altitude h': h = h'  r + p = h'  r +
arcsin[sin(r_{E}/d_{SE})*cos(h' )]
Clearing the lunar distance: By Dunthorne's formula the geocentric lunar distance D can be computed from D: The
unknown time T_{D} corresponding with D
is found by (linear) interpolation from the
tabulated lunar distances D_{1} and D_{2}
at the times T_{1}
and T_{2} (D between D_{1} and D_{2}):
T_{D} = T_{1} + (T_{2}T_{1})*(DD_{1})/(D_{2}D_{1})
Accuracy: An error of only 1' in the measurement of D' is an error of approximately 2 minutes in time or 0.5° in longitude or 40 km in position at mid latitudes. Simulated
Example:
The geocentric lunar distance D at the time T is computed from the geocentric ecliptical longitudes (L_{1}, L_{2}) and latitudes (B_{1}, B_{2}) by the Planets applet (accuracy 2'', comparision with the precise results of MICA, Multiyear Interactive Computer Almanac, U.S. Naval Observatory): Berlin, 52.51°N, 13.41°E, on 2010 Feb 24 at 14:00 UT: The apparent altitudes of the Sun and the Moon (h', H') are also computed by the Planets applet, as well as the geocentric altitudes (h, H) taking into accout refraction and parallax and the apparent lunar distance D' (using Dunthorne's formula resolved for D'). Of course the time computed from D, D_{1} and D_{2} will agree with T. Assuming an error of only 1' for the apparent lunar distance D' the time error will be about 2 minutes, and the error in geographic longitude about 0.5° or 40 km in position at mid latitudes. Additional errors of a few arcminutes in the altitudes h, h', H, H' don't increase very much the errors of longitude. More corrections due to refraction (temperature, pressure, flatening of the lunar/solar disk) and to the ellipsoidal flattening of the earth have to be taken into account for precise results. 
Tests:  Total solar eclipse of 1999 Aug 11 in Karsruhe (Germany) 49.005° N, 8.386° E: The apparent lunar distance of the Sun is computed 0° 00' 00'' from 10:29 until 10:38 UT, semidiameter Sun 15' 47'', Moon 16' 01''.  Total solar eclipse of 2006 Mar 29 in Side (Turkey) 36.767° N, 31.367° E: The minimum apparent lunar distance of the Sun is computed 0° 02' 32'' at 10:56 UT, semidiameter Sun 16' 01'', Moon 16' 35''. The numerical algorithms of the applet are using the Ephemeris Time scale (ET, now continued by the Terrestrial Time TT). Thefore the difference ∆T = ET  UT is needed using UT. The values of ∆T are estimated after 2010. 
Updated: 2010, Sep 15
© 2010 J. Giesen