Lunar Distances
The "Lunar Distance Method" is a way of finding "absolute time" and through this the Longitude
of a position on the Earth without the need for an accurate clock.
Reliable marine chronometers were unavailable until the late 18th century and unaffordable
until the 19th century. During this period of about one century (from about 1750 until 1850)
mariners  such as James Cook  used the Method of Lunar Distances to determine
Greenwich time needed to work out their Longitude.
The Lunar Distance Method is based on measuring the position of the Moon relative to the Sun,
the Planets or stars on the Ecliptic. Since the Moon is close to the Earth compared to the
other celestial objects, it's position chances relatively fast against the celestial background.
The Moon orbits the Earth in 27.3 days relative to the stars and it's position as seen from
the Earth, changes by about 13° a day or 33' per hour.
So, although the Moon rotates from East to West as does the rest of the Celestial Sphere,
the Moon is lagging this motion a little bit, making it "creep" slowly West to East compared
to the background stars. In this sense, the position of the Moon relative to
the celestial background, can be thought of as similar to the position of the hands of a clock
relative to the dial and the current position might be used to determine time similar as
"reading" a chronometer.
The stars along the Ecliptic would simply be the clock dial in the case of this "moon clock".
There are however some problems: the motion of the Moon with 0.5°/hr is very slow
(compare this with the hour hand of a chronometer, which moves 30°/hr) and the time scale
of the "moon clock" has no direct relationship with the chronometer time on the Earth.
The first problem was tackled by instrument makers who constructed a precision optical instrument
for measuring angles (the Sextant), whereas the second problem had to be tackled by astronomers
and mathematicians who had to develop methods for precise prediction of the motion of the Moon.
With the resulting Lunar Distance tables, the "moon time" can be translated into "earth time".
History
One of the earliest tabulations of the positions of celestial bodies was "Ephemerides",
compiled and published by the German astronomer Regiomontanus in 1474.
In the early years of 1600, while studying the celestial observations of Tycho Brahe,
Johannes Keppler was able to formulate his laws of planetary motion, which enabled
significant improvements on the celestial model of Copernicus and better understanding of
the motion of the planets.
However, the accurate prediction of the motion of the Moon (a threebody gravitational problem
without analytical solution) remained unsolved. Determining Greenwich Time and Longitude with Lunar DistancesPractical measurements of Lunar Distances with the sextant will be performed by  seen through the sextant telescope  bringing the limb of the Moon in contact with the Limb of the Sun or in contact with a planet or an appropriate star (close to the Ecliptic). Having measured the Lunar Distance and the Altitude of the two bodies involved, the Greenwich Time is obtained in three basic steps:
With the Greenwich Time found from the LunarDistance observation, the Longitude
can be derived by comparing the Greenwich Time with the observed local time.
The derived Greenwich Time can also be used to check the deviation of the chronometer used
for the LunarDistance measurements.
Measuring angular distanced on the celestial sphere is different
from measuring altitudes. Instead of holding the sextant in the vertical plane, it must me held inclined,
so that the instrument plane includes the center of the Earth and the two celestial objects (only under these
conditions the "shortest"  great circle distance between the objects is measured).
A practical way to obtain such a measurement is to "rock" the sextant, to find the inclination that gives the
lowest possible angle reading. This is challenging and requires some exercising. PreClearing the Lunar DistanceA Lunar distance measurement is always an indirect measurement in the sense that the distance from the Moon limb to the reference body is measured. The measured distance has to be corrected for at least the semidiameter of the Moon (and also of the Sun if that is the reference body): LDcorrected = LD + SDmoon ( + SDsun) Corrections for the Augmentation of SemiDiameter of the MoonWith increasing Altitude of the Moon(Hmoon), there is a slight increase in the Moon's size due to the fact that the distance between observer and Moon decreases if the Moon is high overhead. This is called augmentation of the Moon's semidiameter. The correction in MinutesofArc is approximated by 0.3'*sin(Hmoon) and has to be added to the geocentric semidiameter (SDmoon). The correction values can also be taken from the following table:
Clearing the Lunar DistanceThe Lunar Distance measured with a sextant, is the apparent angle between the Moon and as reference body as seen by the observer at an Earth bound position. As with the standard Altitude observations, this distance (angle) must be corrected for semidiameter, atmospheric disturbances (refraction) and them "reduced" to the situation of a geocentric observer taking into account the effects of dip and parallax. This procedure is called "Clearing the Lunar Distance". Mathematical Solution
The solution of the above triangular problem, is based on the fact that the two spherical triangles
(Mapp  T  Papp) and
(Mtrue  T  Ptrue) share the same (unknown) top angle
in the Zenith (OQ), which can be cancelled out from the set of triangular relations. cos(OQ) = [cos(ds)  sin(Mapp )*sin(Papp )] / [cos(Mapp )*cos(Papp )] cos(OQ) = [cos(da)  sin(Mtrue)*sin(Ptrue)] / [cos(Mtrue)*cos(Ptrue)]The result of the equation solving can be formulated in the following awful form (different other forms are also in use): cos(da) = [{cos(Mapp)*cos(Papp)} / {cos(Mtrue)*cos(Ptrue)} * {cos(ds) + cos(Mtrue + Ptrue)}]  [cos(Mapp + Papp)]
Through this calculation, the result for the true Lunar Distance at the moment of observation is obtained.
The next steps are to take into account the effect of the sideparallax and compare the time of occurrence
of this calculated true Lunar Distance with the values in the precomputed Tables. The Method of Jean Borda for Clearing the Lunar Distance (1787)The Approximation of John Merrifield for Clearing the Lunar Distance (1884)Precomputed Lunar Distance TablesSelecting the Reference ObjectThe celestial reference bodies recorded in the daily pages of the tables are selected from the following objects: the Sun, the Planets and the brightest stars along the path of the Moon on the celestial sphere. This is the order of preference. Ideally, the reference object should have the same Declination as the Moon, but since this is usually not the case a certain amount of Declination mismatch must be allowed. From a rough analytical analysis, it can be concluded, that a Declination difference can be tolerated if the GHA distance is significantly larger than the Declination difference.

Cover << Sail Away << Celestial Navigation << Special Techniques << .  last updated: 17Nov2016 