Longitude by chronometer

Longitude by chronometer, is an astronomical navigation method of calculating the longitude of an observer's position on Earth. The longitude derived by this method must be combined with the latitude of the observer's position (derived through some other method) to resolve a "fix" or exact position of the observer on the Earth.

Background

Determining longitude by use of a chronometer is perhaps the easiest method of determining longitude, albeit one of less accuracy than other methods. However, while other methods of celestial navigation require extensive use of sight reduction tables and tedious, error-prone calculations, determining longitude by use of the chronometer is fairly simple and straightforward. As the name of process implies, other than a sextant (to determine local apparent noon), the primary tool required is an accurate chronometer. The only celestial body necessary is a clear shot of the Sun at one's local apparent noon.

The principle of determining longitude by chronometer is based on the regular apparent movement of the Sun across the sky each day. Ignoring the fact that the Earth revolves, the Sun’s apparent orbit around the Earth each day takes 24 hours to pass through the 360 degrees of the Earth’s circumference or 15 degrees each hour (360/24 = 15) or one degree every 4 minutes (60/15 = 4). By determining the difference between the time of one’s local apparent noon with that of local apparent noon at a known reference longitude, a navigator can determine his own longitude by multiplying the result by the speed of the Sun’s movement across the sky at 15 degrees each hour or 1 degree every 4 minutes. This concept was well known to ancient mariners but was unable to be implemented through lack of an accurate clock or chronometer that could consistently and accurately provide the time at the known reference longitude.

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 the French, proposed their own reference longitudes as a standard, the world’s navigators have generally come to accept as the standard the reference longitude adopted by the British who, incidentally, were also the first to create a working chronometer. The reference longitude adopted by the English 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).

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, noon at 15 degrees west longitude would take place at exactly 1300 hours UTC. 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 of his local apparent noon. Because the Sun reached noon at 1200 UTC on the Prime Meridian, by subtracting the UTC of local apparent noon from 1200 UTC and multiplying the result by the Sun’s movement of 15 degrees each hour 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.

Corrections to the process

Unfortunately, the Sun does not make a perfect apparent orbit around the Earth. 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) 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

Calculating longitude by time sight.

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: $cos LHA = (sin Ho - sin Dec * sin B )/(cos Dec * cos B)\,$

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

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, by Dava Sobel

External links

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