# Longitude by chronometer

Longitude by Chronometer, known by mariners as "Long by Chron", is an astronomical navigation method of calculating an observer's position on earth. The method gives the observer a position line on which the observer is situated. Usually the observer will take two sets of sights at an interval of approximately 3 hours and "run-on" the earlier position line to the time of the second observation to give a 'fix'.

Methodology

This method uses an assumed latitude and calculates the longitude that a position line crosses it. The position line obtained is actually part of a small circle, as opposed to great circle, where any observer can stand and the heavenly object would have the same altitude in the sky. When plotting the small segment of this circle on a chart it is drawn as a straight line, the resulting tiny errors are too small to be significant.

The assumed latitude is usually obtained from a DR or Dead Reckoning position. This is worked out by applying the distance from the last known position either by log or by the estimated speed over time with the course steered. A sight is taken; that is the distance above the horizon of a heavenly object is measured with a sextant and the exact time noted in UTC. The sextant angle obtained is corrected for dip (the error caused by the observers height above the sea) and refraction to obtain the true altitude of the object above the horizon. This is then subtracted from 90° to obtain the angular distance from the position directly above, the zenith. This is referred to as the True Zenith Distance. The true zenith distance of the object is also the distance (in arc) on the earth's surface from the observer to where that object is overhead, the geographical position of the object.

Using a nautical almanac, the declination (celestial latitude), and the Greenwich hour angle (celestial longitude) are obtained of the observed object for the time of observation. Using the haversine formula the local hour angle of the position where the position circle crosses the assumed latitude is calculated. The local hour angle is the difference in longitude from the observer's position and the geographical position of the observed object. Hour angles, unlike longitude which is measured east and west from Greenwich, are always measured west from 0° through to 360°.

The local hour angle is then added to the Greenwich hour angle to obtain the longitude where the position line passes through the assumed latitude.

To draw the position line on a chart the azimuth or bearing of the heavenly object must be known. It is usually calculated but could have been observed. A line at right angles to the azimuth is drawn through the calculated position. The observer is somewhere on this line.

To obtain a fix (a position) this line must be crossed with another position line either from another sight or from elsewhere e.g. a bearing of a point of land or crossing a depth contour such as the 200 metre depth line on a chart.

ights

Until the age of satellite navigation ships usually took sights at dawn, noon and dusk. The morning and evening sights were taken during "navigational twilight" whilst the horizon was visible and the stars, planets and/or moon were visible, at least through the telescope of a sextant. Three or more observations were required to give a position accurate to within a mile under favourable conditions.

Running Fix

The noon sight is a running fix. A sight of the sun is taken around 9 am ship's time and the position line is run up to noon. At noon a meridian altitude is taken which obtains the latitude which is crossed with the run up earlier sight and a noon position obtained. As the noon sight is a running fix it is not as accurate as the star sights taken at dawn and dusk.

Calculating a sight using the Longitude by Chronometer method can be done with nautical tables using the haversine formula.

$extrm\left\{Hav., H\right\} = \left\{ extrm\left\{\left(Hav. Z, - Hav.\left(L, +, or, -, D\right)\right), Sec.L, Sec.D, ,$

Where

H = Local Hour AngleZ = True Zenith DistanceL = LatitudeD = Declination

An experienced navigator can do a sight from start to finish in about 5 minutes using nautical tables or a scientific calculator.

Sight reduction tables are often used for star sights, as they can greatly speed up the process. Professional navigators, in general, do not use them for the sun, moon and planets as it is quicker to calculate these using tables or a scientific calculator.

The sight reduction tables in the British Nautical Almanac are designed to be used only if there are no tables, proper sight reduction tables or calculators available.

Accuracy and versatility

The Longitude by Chronometer method of calculating sights is at its most accurate when the azimuth of the object is due east or west. As the azimuth changes towards the south or north, depending which hemisphere the observer is in, the cross of the position line with the assumed latitude becomes more and more oblique and the position obtained is therefore less accurate. For this reason it is a less versatile method of calculating sights than the intercept method which can be used for all azimuths and the "Long by Chron" method has therefore fallen out of favour.

* Latitude
* Longitude
* Haversine formula
* Intercept method
* Meridian altitude

References

*"Nicholls's Concise Guide, Volume 1", by Charles H. Brown F.R.S.G.S. Extra Master
*"Norie's Nautical Tables", edited by Capt. A.G. Blance

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