Longitude by chronometer

Longitude by chronometer is a method, in navigation, of determining longitude using a marine chronometer, which was developed by John Harrison during the first half of the eighteenth century. It is an astronomical method of calculating the longitude at which a position line, drawn from a sight by sextant of any celestial body, crosses the observer's assumed latitude. In order to calculate the position line, the time of the sight must be known so that the celestial position i.e. the Greenwich Hour Angle (Celestial Longitude - measured in a westerly direction from Greenwich) and Declination (Celestial Latitude - measured north or south of the equational or celestial equator), of the observed celestial body is known. All that can be derived from a single sight is a single position line, which can be achieved at any time during daylight, when both the sea horizon and the sun are visible. To achieve a fix, more than one celestial body and the sea horizon must be visible. This is usually only possible at dawn and dusk.

The angle between the sea horizon and the celestial body is measured with a sextant and the time noted. The Sextant reading is known as the 'Sextant Altitude'. This is corrected by use of tables to a 'True Altitude' . The actual declination and hour angle of the celestial body are found from astronomical tables for the time of the measurement and together with the 'True Altitude' are put into a formula with the assumed latitude. This formula calculates the 'True Hour Angle' which is compared to the assumed longitude providing a correction to the assumed longitude. This correction is applied to the assumed position so that a position line can be drawn through the assumed latitude at the corrected longitude at 90° to the azimuth (bearing) on the celestial body. The observer's position is somewhere along the position line, not necessarily at the found longitude at the assumed latitude. If two or more sights or measurements are taken within a few minutes of each other a 'fix' can be obtained and the observer's position determined as the point where the position lines cross.

The azimuth (bearing) of the celestial body is also determined by use of astronomical tables and for which the time must also be known.

From this it can be seen that a navigator will need to know the time very accurately so that the position of the observed celestial body is known just as accurately. The position of the sun is given in degrees and minutes north or south of the equational or celestial equator and east or west of Greenwich, established by the English as the Prime Meridian.

The desperate need for an accurate chronometer was finally met in the mid 18th century when an Englishman, John Harrison, produced a series of chronometers that culminated in his celebrated model H-4 that satisfied the requirements for a ship-board standard time-keeper.

Other nations, notably France, proposed its own reference longitudes as a standard, although the world’s navigators have generally come to accept the reference longitudes tabulated by the British. The reference longitude adopted by the British became known as the Prime Meridian and is now accepted by most nations as the starting point for all longitude measurements. The Prime Meridian of zero degrees longitude runs along the meridian passing through the Royal Observatory at Greenwich, England. Longitude is measured east and west from the Prime Meridian. To determine "longitude by chronometer", a navigator requires a chronometer set to the local time at the Prime Meridian. Local time at the Prime Meridian has historically been called Greenwich Mean Time (GMT), but now, due to international sensitivities, has been renamed as Coordinated Universal Time (UTC), and is known colloquially as "zulu time".

Noon sight for Longitude

Noon on the Prime Meridian occurs at 1200 hours UTC. The Sun moves west from that point at a rate of 15 degrees each hour. Therefore, solar noon at 15 degrees west longitude would take place at exactly 1300 hours UTC. Solar noon at 30 degrees west longitude would take place at 1400 hours UTC. A navigator uses his sextant to track the rise of the Sun in the sky to determine the exact moment that it reaches its highest point in the sky—local apparent noon. The navigator then notes the UTC (on his chronometer) at this local apparent noon. By subtracting from the UTC of local apparent noon, 1200 UTC and multiplying the result by the Sun’s movement of 15 degrees for each hour's difference, a navigator can calculate the number of degrees of longitude the Sun has crossed from the Prime Meridian to his current meridian of longitude. For example, if the navigator reads 1704 hours UTC on his chronometer at his local apparent noon, he can subtract 1200 hours UTC to arrive at 5 hours and 4 minutes of travel time for the Sun at a rate of 15 degrees per hour or one degree in 4 minutes. Multiplication results in a calculation of 75 degrees west longitude plus one additional degree of west longitude to account for the :04 minutes of time past 1700 hours for a total of 76 degrees west longitude. In the time lapse from local apparent noon at the Prime Meridian to the local apparent noon at the navigator's position, the Sun has travelled 76 degrees west. Incidentally, with the same sextant sight values, the UTC of local apparent noon and the Nautical Almanac, the navigator can also determine his latitude thereby achieving a positional fix with a single noon shot of the Sun. The significance of the noon sight of the Sun has made it an integral component of nautical lore.

Longitude cannot accurately be measured at noon, when it is very easy to determine the observer's latitude without knowing the exact time. At noon the sun's change of altitude is very slow so determining the exact time that the sun is at its zenith is impossible to measure to the degree of accuracy necessary to give an accurate longitude.

Corrections to the process

Unfortunately, the Earth does not make a perfect circular orbit around the Sun. Due to the elliptical nature of the Earth’s true orbit around the Sun, the speed of the Sun’s apparent orbit around the Earth varies throughout the year and that causes it to appear to speed up and slow down very slightly. Consequently, noon at the Prime Meridian is rarely if ever exactly at 1200 UTC, but rather it occurs some minutes and seconds before or after that time each day. This slight daily variation has been calculated and is listed for each day of the year in the Nautical Almanac under the title of “Equation of Time”. This variation must be added to or subtracted from the UTC of local apparent noon to improve the accuracy of the calculation. Even with that, other factors, including the difficulty of determining the exact moment of local apparent noon due to the flattening of the Sun’s arc across the sky at its highest point, diminish the accuracy of determining longitude by chronometer as a method of celestial navigation. Accuracies of less than 10 nautical miles (19 km) error in position are difficult to achieve using the "longitude by chronometer" method. Other celestial navigation methods involving more extensive use of both the Nautical Almanac and sight reduction tables are used by navigators to achieve accuracies of one nautical mile (1.9 km) or less.

Time sight

File:TimeSight.png

Calculating longitude by time sight.

This only calculates a longitude at the assumed latitude though a position line can be drawn. The observer is somewhere along the position line.

Time sight is a general method for determining longitude by celestial observations using a chronometer; these observations are reduced by solving the navigational triangle for meridian angle and require known values for altitude, latitude, and declination; the meridian angle is converted to local hour angle and compared with Greenwich hour angle.

If Dec is the declination of the observed celestial body and Ho is its observed altitude, the local hour angle, LHA, is obtained for a known latitude B by:

$\cos(LHA) = \frac{\sin(Ho) - \sin(Dec) \cdot \sin(B)}{\cos(Dec) \cdot cos(B)}\,$

The time sight was a complement to the noon sight or latitude by Polaris in order to obtain a fix.

See also

• Celestial navigation
• Navigation
• Latitude
• Longitude
• Haversine formula
• Intercept method
• Meridian altitude
• Lunar distance
• Navigational Algorithms
• Sight reduction

References

• Sailing Alone Around the World, by Joshua Slocum
• Sextant Instructions on Use, by Davis Instruments Corp, Hayward, Ca.
• The Nautical Almanac 2009, published by Government Printing Office, Washington D.C.
• Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time, by Dava Sobel

External links

• Navigational Algorithms http://sites.google.com/site/navigationalalgorithms/
• Navigation Spreadsheets: Noon shots