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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 arc-minutes, in an hour.[1] Therefore, an error of half an arc-minute 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 arc-minute, 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 one-quarter 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 (one-tenth of a minute of arc).
Lunar distance observation
The best sextants at the very beginning of the lunar distance era could indicate angle to one-sixth 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 one-quarter 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 one-half arc-minute in Lunar distance, equivalent to one minute in Greenwich time, which corresponds to an error of as much as one-quarter of a degree of Longitude, or about 15 nautical miles (30 km) at the equator.

## References

1. ^ a b c d e f g h Norie, J. W. (1828). New and Complete Epitome of Practical Navigation. London. p. 222. Retrieved 2007-08-02.
2. ^ a b Norie, J. W. (1828). New and Complete Epitome of Practical Navigation. London. p. 221. Retrieved 2007-08-02.
3. ^ Taylor, Janet (1851). An Epitome of Navigation and Nautical Astronomy (Ninth ed.). p. 295f. Retrieved 2007-08-02.
4. ^ Britten, Frederick James (1894). Former Clock & Watchmakers and Their Work. New York: Spon & Chamberlain. p. 230. Retrieved 2007-08-08. "Chronometers were not regularly supplied to the Royal Navy till about 1825"
5. ^ Lecky, Squire, Wrinkles in Practical Navigation
6. ^ 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. Retrieved 2007-08-02.  ;
Wepster, Steven. "Precomputed Lunar Distances". Retrieved 2007-08-02.
7. ^ Norie, J. W. (1828). New and Complete Epitome of Practical Navigation. London. p. 226. Retrieved 2007-08-02.
8. ^ Norie, J. W. (1828). New and Complete Epitome of Practical Navigation. London. p. 230. Retrieved 2007-08-02.
9. ^
10. ^
11. ^
• 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