Lunar distance (navigation)

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.
Illustration by Clive Sutherland

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 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. But reliable marine chronometers were unavailable or unaffordable until well into the 19th century. For approximately one hundred years (from about 1767 until about 1850), 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.

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, 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. 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 errors). The navigator then consults a prepared table of lunar distances and the times at which they will occur. 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. Local time can be determined from a sextant observation of the altitude of the Sun or a star. 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. 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. 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.

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.

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. 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 an insignificant level (less than 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 corrsponds to an error of as much as one-quarter of degree of Longitude, or about 15 nautical miles (30 km) at the equator.

History

Main article: History of longitude

Motivated by a number of maritime disasters, at least partly attributable to errors in reckoning longitude at sea, including the disastrous shipwreck of Sir Cloudesley Shovell's fleet on the Isles of Scilly in 1707, the British Parliament offered the Longitude Prize in 1714. Prizes were to be awarded for any practical method for determining the longitude of a ship at sea. A Board of Longitude was established to judge suggested methods for determining longitude and award prize money as compensation. These prizes, worth millions of dollars in today's currency, motivated astronomers and clockmakers, as well as countless other inventors, to apply themselves to the longitude problem. The lunar distance method was widely regarded as the obvious astronomical solution, and it had been recognized as such, at least in principle, for over two centuries.

The first published proposal of a method of lunar distances for determining longitude was Johannes Werner's In hoc opere haec continentur Nova translatio primi libri geographiae Cl. Ptolomaei, published at Nürnberg in 1514. The method was discussed in detail by Petrus Apianus in his Cosmographicus liber (Landshut 1524).

The method of lunar distances required an extraordinarily accurate predictive model of the Moon's motion. Isaac Newton's theory of Universal Gravitation represented the first step forward in the analysis of the Moon's motion in centuries, yet even Newton famously described the theory of the Moon's motion as a problem that made his head ache.

After decades of observation and analysis, by the latter half of the 18th century, German astronomer Tobias Mayer developed suitably accurate tables of the Moon's motion. Mayer's analysis was based in large part on the theoretical models of Leonard Euler Mayer submitted his tables along with practical, systematic suggestions for observing lunar distances to the Board of Longitude for evaluation and consideration for the longitude prize. After remarkably successful empirical tests at sea in 1762, Nevil Maskelyne proposed annual publication of lunar distance predictions in an official nautical almanac. Maskelyne asserted that Mayer's tables could fulfill the original goal of the Longitude Prize, determining longitude at sea to within half a degree. Mayer did not live to see his tables applied to practical navigation, but his widow received a large award from the Board of Longitude.

The first nautical almanac with accurate lunar distances was published with data for the year 1767, it included daily tables of the positions of the Sun, Moon, and planets and other astronomical data as well as tables of lunar distances giving the distance of the Moon from the Sun and nine stars suitable for lunar observations (ten stars for the first few years). This almanac has been published annually ever since.

Though the British Parliament rewarded John Harrison for his marine chronometer in 1773, chronometers remained very expensive and the lunar distance method continued to be used for decades. Lunar distances were widely used at sea during the period from 1767 to about 1850 and in land exploration until the beginning of the 20th century. Lunar distance tables last appeared in the USNO Nautical Almanac for 1912 and an appendix explaining how to generate single values of lunar distances was published as late as the early 1930s. Similarly, the British Nautical Almanac published the tables until 1906 and the instructions until 1924. The presence of these tables does not imply common usage. Expert navigators learned lunars as late as 1905 since they were a requirement for certain licenses. However, by this late date, the vast majority of navigators had ceased using the method of lunar distances because affordable, reliable marine chronometers had been available for decades. It was less expensive to buy three chronometers, which could serve as checks on each other, than it was to acquire a high-quality sextant which was essential for lunar distance navigation.

See also

• Royal Greenwich Observatory 1675
• Nautical Almanac
• Nevil Maskelyne
• John Harrison
• Board of Longitude 1714
• Longitude prize 1714
• Henry Raper
• American Practical Navigator
• Carry On, Mr. Bowditch

References

• 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