Longitude by Chronometer is an astronomical navigation method of calculating an observer's position on earth. This 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 get a fix. 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 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 GMT. 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 such as a bearing of a point of land or crossing a depth contour on a chart.
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 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.
