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Finding Greenwich time while at sea using a lunar
distance. The Lunar Distance is the angle between the Moon
and a star (or the Sun). The altitudes of the two bodies are used
to make corrections and determine the time.
In celestial navigation, lunar distance is the angle
between the Moon and another celestial body. A navigator can use a lunar distance (also
called a lunar) and a nautical almanac to calculate Greenwich
time. The navigator can then determine longitude without a marine
chronometer.
The reason for
measuring lunar distances
In celestial navigation, precise
knowledge of the time at Greenwich and the positions of one or more celestial objects are combined with careful
observations to calculate latitude and longitude^{[1]}.
Reliable marine chronometers were unavailable
until the late 18th century and not affordable until the 19th
century.^{[2]}^{[3]}^{[4
]} For approximately one hundred years (from about
1767 until about 1850)^{[5]}
mariners lacking a chronometer used the method of lunar distances
to determine Greenwich time, an important step in finding their
longitude. A mariner with a chronometer could check and correct its
reading using a lunar determination of Greenwich time.^{[2]}
Method
Summary
The method relies on the relatively quick movement of the moon
across the background sky, completing a circuit of 360 degrees in
27.3 days. In an hour then, it will move about half a degree,^{[1]}
roughly its own diameter, with respect to the
background stars and the Sun. Using a sextant, the navigator precisely measures the
angle between the moon and another body.^{[1]}
That could be the Sun or one of a selected group of bright stars
lying close to the Moon's path, near the ecliptic. At that moment, anyone on the
surface of the earth who can see the same two bodies will observe
the same angle (after correcting for parallax error). The navigator then consults a
prepared table of lunar distances and the times at which they will
occur.^{[1]}^{[6
]} By comparing the corrected lunar distance with
the tabulated values, the navigator finds the Greenwich time for
that observation. Knowing Greenwich time and local time, the
navigator can work out longitude.^{[1]}
Local time can be determined from a sextant observation of the
altitude of the Sun or a star.^{[7]}^{[8]}
Then the longitude (relative to Greenwich) is readily calculated
from the difference between local time and Greenwich Time, at 15
degrees per hour.
In
Practice
Having measured the lunar distance and the heights of the two
bodies, the navigator can find Greenwich time in three steps.
 Step One – Preliminaries
 Almanac tables predict lunar distances between the centre of
the Moon and the other body (see any nautical almanac from 1767 to
c.1900). However, the observer cannot accurately find the centre of
the Moon (and Sun, which was the most frequently used second
object). Instead, lunar distances are always measured to the
sharply lit, outer edge ("limb") of the Moon and from the sharply
defined limb of the Sun. The first correction to the lunar distance
is the distance between the limb of the Moon and its center. Since
the Moon's apparent size varies with its varying distance from the
Earth, almanacs give the Moon's and Sun's
semidiameter for each day (see any nautical
almanac from the period). Additionally the observed altitudes are
cleared of dip and semidiameter.
 Step Two – Clearing
 Clearing the lunar distance means correcting
for the effects of parallax and atmospheric refraction on the
observation. The almanac gives lunar distances as they would appear
if the observer were at the center of a transparent Earth. Because
the Moon is so much closer to the Earth than the stars are, the
position of the observer on the surface of the Earth shifts the
relative position of the Moon by up to an entire degree^{[9]}^{[10]}. The
clearing correction for parallax and refraction is a relatively
simple trigonometric function of the observed lunar distance and
the altitudes of the two bodies^{[11]}.
Navigators used collections of mathematical tables to work these
calculations by any of dozens of distinct clearing methods.
 Step Three – Finding the Time
 The navigator, having cleared the lunar distance, now consults
a prepared table of lunar distances and the times at which they
will occur in order to determine the Greenwich time of the
observation.^{[1]}^{[6
]}
Having found the (absolute) Greenwich time, the navigator either
compares it with the observed local apparent time (a separate
observation) to find longitude or compares it with the Greenwich
time on a chronometer if one is available.^{[1]}
Errors
 Effect of Lunar Distance Errors on calculated Longitude
 A lunar distance changes with time at a rate of roughly half a
degree, or 30 arcminutes, in an hour.^{[1]}
Therefore, an error of half an arcminute will give rise to an
error of about 1 minute in Greenwich Time, which (owing to the
Earth rotating at 15 degrees per hour) is the same as one quarter
degree in longitude (about 15 nautical miles at
the equator).
 Almanac error
 In the early days of lunars, predictions of the Moon's position
were good to approximately half an arcminute, a source of error of
up to approximately 1 minute in Greenwich time, or one quarter
degree of longitude. By 1810, the errors in the almanac predictions
had been reduced to about onequarter of a minute of arc. By about
1860 (after lunar distance observations had mostly faded into
history), the almanac errors were finally reduced to less than the
error margin of a sextant in ideal conditions (onetenth of a
minute of arc).
 Lunar distance observation
 The best sextants at the
very beginning of the lunar distance era could indicate angle to
onesixth of a minute and later sextants (after c. 1800) measure
angles with a precision of 0.1 minutes of arc.. In practice at sea,
actual errors were somewhat larger. Experienced observers can
typically measure lunar distances to within onequarter of a minute
of arc under favourable conditions, introducing an error of up to
one quarter degree in longitude. Needless to say, if the sky is
cloudy or the Moon is "New" (hidden close to the glare of the Sun),
lunar distance observations could not be performed.
 Total Error
 The two sources of error, combined, typically amount to about
onehalf arcminute in Lunar distance, equivalent to one minute in
Greenwich time, which corresponds to an error of as much as
onequarter of a degree of Longitude, or about 15 nautical
miles (30 km) at the equator.
See also
References
 ^ ^{a}
^{b}
^{c}
^{d}
^{e}
^{f}
^{g}
^{h}
Norie, J. W. (1828). New and Complete Epitome
of Practical Navigation. London. p. 222. http://www.mysticseaport.org/library/initiative/ImPage.cfm?PageNum=3&BibId=13617&ChapterId=30. Retrieved
20070802.
 ^ ^{a}
^{b}
Norie, J. W. (1828). New and Complete Epitome
of Practical Navigation. London. p. 221. http://www.mysticseaport.org/library/initiative/ImPage.cfm?PageNum=2&BibId=13617&ChapterId=30. Retrieved
20070802.
 ^ Taylor, Janet (1851). An Epitome of Navigation
and Nautical Astronomy (Ninth ed.). p. 295f. http://books.google.com/books?id=rvg_BRfIckEC&pg=RA6PA197&vq=lunar+observation&dq=intitle:Nautical+intitle:Almanac+date:18491851#PRA6PA195. Retrieved
20070802.

^
Britten, Frederick James (1894). Former Clock &
Watchmakers and Their Work. New York: Spon &
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"Chronometers were not regularly supplied to the Royal Navy till
about 1825"
 ^
Lecky, Squire, Wrinkles in Practical Navigation
 ^ ^{
a} ^{
b} Royal
Greenwich Observatory. "DISTANCES of Moon's Center
from Sun, and from Stars EAST of her". in Garnet. The
Nautical Almanac and Astronomical Ephemeris for the year 1804.
(Second American Impression ed.). New Jersey: Blauvelt.
p. 92. http://www.mysticseaport.org/library/initiative/ImPage.cfm?PageNum=8&BibId=21382&ChapterId=9. Retrieved
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;
Wepster, Steven. "Precomputed Lunar
Distances". http://www.math.uu.nl/people/wepster/ldtab.html. Retrieved
20070802.
 ^ Norie, J. W. (1828). New and Complete Epitome
of Practical Navigation. London. p. 226. http://www.mysticseaport.org/library/initiative/Impage.cfm?PageNum=7&bibid=13617&ChapterId=30. Retrieved
20070802.
 ^ Norie, J. W. (1828). New and Complete Epitome
of Practical Navigation. London. p. 230. http://www.mysticseaport.org/library/initiative/Impage.cfm?PageNum=11&bibid=13617&ChapterId=30. Retrieved
20070802.
 ^
DuffettSmith, Peter (1988). Practical Astronomy with
Your Calculator, third edition. p. 66. http://books.google.com/books?id=DwJfCtzaVvYC&dq=practical+astronomy+with+your+calculator&pg=PP1&ots=91i8WJbtpR&sig=OXKHBpiNTnomzQu287Mjz3E_90M&hl=en&prev=http://www.google.com/search?q=practical+astronomy+with+your+calculator&ie=utf8&oe=utf8&rls=com.google:enUS:official&client=firefoxa&sa=X&oi=print&ct=title&cad=onebookwiththumbnail.
 ^
Montenbruck and Pfleger (1994). Astronomy on the Personal
Computer, second edition. pp. 45–46. http://books.google.com/books?id=WDjJIww337EC&dq=astronomy+on+the+personal+computer&pg=PP1&ots=p8o_geQnkT&sig=9B6AXajLyAyaJNfdP2s6hmNhjII&hl=en&prev=http://www.google.com/search?q=astronomy+on+the+personal+computer&ie=utf8&oe=utf8&rls=com.google:enUS:official&client=firefoxa&sa=X&oi=print&ct=title&cad=onebookwiththumbnail.
 ^
Schlyter, Paul. "The Moon's topocentric
position". http://www.stjarnhimlen.se/comp/ppcomp.html#13.
 New and complete epitome
of practical navigation containing all necessary
instruction for keeping a ship's reckoning at sea ... to which is
added a new and correct set of tables  by J. W. Norie 1828
 Andrewes, William J.H. (Ed.): The Quest for Longitude.
Cambridge, Mass. 1996
 Forbes, Eric G.: The Birth of Navigational Science.
London 1974
 Jullien, Vincent (Ed.): Le calcul des longitudes: un enjeu
pour les mathématiques, l`astronomie, la mesure du temps et la
navigation. Rennes 2002
 Howse, Derek: Greenwich Time and the Longitude. London
1997
 Howse, Derek: Nevil Maskelyne. The Seaman's
Astronomer. Cambridge 1989
 National Maritime Museum (Ed.): 4 Steps to Longitude.
London 1962
External
links