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## 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".

The principle of the Lunar Distance Method" was first described in 1514 by Johan Werner, but was probably known before. But it took until the middle of the 18th century before the motion of the Moon could be predicted accurately enough and instruments became precise enough for the method to be of any practical value. Nevertheless, even with a modern sextant, the obtainable accuracy is rather limited. The average motion of the Moon along the Ecliptic is about 0.5' per minute. This is about the accuracy with which sextant measurements can be performed under ideal conditions. So the best that can be expected from the Lunar Distance Method is a time accuracy of 1 minute, which is sufficient to find Longitude within about 15 minutes of arc. It is also obvious that during the period of "new Moon", when the Moon is close to the Sun, no Lunar Distance observations can be performed.

### 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 three-body gravitational problem without analytical solution) remained unsolved.
In 1753, the German astronomer Tobias Mayer published Lunar Tables of outstanding accuracy (based on a numerical solution of the three-body problem), enabling for the first time the determination of Longitude by the Lunar Distance Method with a precision that was within the limits set by the Longitude Act passed by the parliament of the United Kingdom in July 1714. From then on the Lunar Distance Method became the only competing method for John Harrison's chronometer method for determining Longitude at sea.
When Mayer died in 1762, no decision had been reached in England concerning the "longitude problem". His improved Lunar Tables on which he had been working the last 7 years before his death, were tested by Nevil Maskelyne on a journey to Barbados in 1763, on which also the H4, the latest chronometer of John Harrison was tested.
Finally, on February 9,1765, the Board of Longitude advised the British Parliament that both Mayer and Harrison should both be rewarded for their contributions to the solution of the longitude problem. But the Board signalled serious deficiencies in each of the two methods. Harrison's chronometer was not considered general enough, because it was not yet possible to produce accurate chronometers in sufficient quantities and Mayer's method was considered not practical because it entailed too much complicated calculation work.
Maskelyne who was one of the few people who had actually employed the Lunar Distance Method successfully at sea and had also been entrusted with the verification of Harrison's chronometer at Barbados, was convinced that the deficiencies signalled by the Board of Longitude were - at that time - much easier to overcome in the case of Lunar Distance Method than in the case of the chronometers. After he became Astronomer Royal at the observatory in Greenwich, he took up the plan to take as much as possible of the elaborate calculations away from the navigator at sea. This was accomplished by using pre-computed lunar distance tables that were already published (among other ephemeral data) in the first edition of the "Nautical Almanac" in 1767.
Precomputed lunar distances were published in the "Nautical Almanac" until 1905. Since the first half of the 19th century, chronometers were produced in sufficient quantities and at affordable prices such that they were used on board of an increasing number of ships. From this time on the method of Lunar Distances to determine Longitude gradually faded out, but remained useful for the required chronometer checks. By 1905, radio time signals were available as an independent time reference making the Lunar Distance Method more or less obsolete.

### Determining Greenwich Time and Longitude with Lunar Distances

Practical 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:

• correct the limb-based measurements for the semi-diameter of the involved bodies to obtain the observed angular distance between the centers of the bodies,
• correct the Altitude measurements for the effects of parallax and atmospheric refraction,
• compare the cleared Lunar Distance with the prepared Tables of Lunar Distances and determine the time at which this distance will occur in order to find the Greenwich Time of the observation.

With the Greenwich Time found from the Lunar-Distance 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 Lunar-Distance measurements.

A basic problem is that the measurements of the Lunar Distance and the two Altitudes must be preformed at the same time. Not a big issue in past days when the Navigator had some assistants at his disposal. If a single person has to do each of the three measurements there are two practical alternatives:

• the Altitudes can be measured before and after the measurement of the Lunar Distance and the corresponding values interpolated afterwards. In order to obtain reasonable results, the time span for all five (!) measurements should not exceed a few minutes.
• the Altitude values can be calculated from the estimated time and position. This method is favourable for land-bound exercising, because time and position may be known and the horizon may not be visible.

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.
Around the days of full Moon, care should be taken to use the "correct" - sharp - limb of the Moon and not the more blurred shadow line bordering the far side of the Moon.

### Pre-Clearing the Lunar Distance

A 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 semi-diameter of the Moon (and also of the Sun if that is the reference body):

```  LDcorrected = LD + SDmoon ( + SDsun)
```

#### Corrections for the Augmentation of Semi-Diameter of the Moon

With 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 semi-diameter. The correction in Minutes-of-Arc is approximated by 0.3'*sin(Hmoon) and has to be added to the geocentric semi-diameter (SDmoon). The correction values can also be taken from the following table:

 Altitude 0°- 9° 10°-29° 30°-55° 56°-90° Correction 0.0' 0.1' 0.2' 0.3'

### Clearing the Lunar Distance

The 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 semi-diameter, 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 basic situation at the moment of the Lunar Distance measurement is shown in the picture. The observer measures the Altitudes of the Moon (Mapp) and the reference celestial body (Papp) as well as the distance (angle) between these objects (ds). This yields three sides of a spherical triangle (in the Horizontal Coordinate System): T is the Zenith of the observer, the side O-Q is a segment of the local horizon defined by the position of the observer and the Azimuth Angles of the Moon (M) and the reference body (P). The segments O-Mapp and Q-Papp are the measured (apparent) Altitudes, whereas the angle ds is the measured (apparent) Lunar Distance. The Great-Circle segments O-Mtrue and Q-Ptrue are the "corrected" (true) Altitudes from which the true Lunar Distance (da) must be derived, taking into account that due to the inaccurate time information, the length of the segment O-Q along the local horizon is not exactly known.

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 (O-Q), which can be cancelled out from the set of triangular relations.

For both triangles the top angle (O-Q) can be calculated:

```  cos(O-Q) = [cos(ds) - sin(Mapp )*sin(Papp )] / [cos(Mapp )*cos(Papp )]
cos(O-Q) = [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 side-parallax and compare the time of occurrence of this calculated true Lunar Distance with the values in the pre-computed Tables.

### Precomputed Lunar Distance Tables

#### Selecting the Reference Object

The 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.

• The Sun: is selected if the difference in GHA is between 30 and 80 degrees
• The Planets: Mars, Venus and Jupiter are selected if the difference in GHA is between 30 and 60 degrees
• Bright Stars: are selected if the difference in Dec is lower than 10 degrees and the difference in GHA is between 30 and 60 degrees

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