The Concept of Using a Sextant Altitude

Using the altitude of a celestial body is similar to using the altitude of an object of known height, to obtain a distance.

One object or body provides a distance but the observer can be anywhere on a circle of that radius away from the object. At least two distances are necessary for a position. (Three avoids ambiguity.)

Using a Sextant for Celestial Navigation

The main difference using a star or other celestial body is that calculations are carried out on an imaginary sphere surrounding the Earth; the Celestial Sphere.

Working on this sphere, the distance becomes [90° - Altitude.] The point on the sphere corresponding to the Observer is known as his Zenith.

Using a Nautical Almanac to find the position of the body, the body’s position could be plotted on an appropriate chart and then a circle of the correct radius drawn around it. In practice we need to be more precise than that.

Position Lines

Each circle found using a sextant altitude is of immense radius therefore the short length of interest can be considered a straight line

Comparing the observed distance to the body and the calculated distance using an estimated position provides the distance towards, or away from the body.

The observed distance is known as the True Zenith Distance (TZD.) The value based on the assumed position is the Calculated Zenith Distance (CZD.) The difference between the two is known as the Intercept.

The closest point on this circle is known as the Intercept Terminal Position (ITP) and the line representing the circle at that position is called a Position Line.

Additional sights provide additional position lines which intersect to provide a Fix

A Running Fix

A vessel is usually moving between sights therefore they are combined "on the run."

The position line from a first sight must be moved to allow it to be combined with another position line for a different time.

A double-headed arrow identifies a Transferred Position Line.

After a second sight has been calculated, its position line can be plotted and combined with the first to provide a fix.

Notes on Running Fixes

Under normal conditions, one would expect an error of +/- 0’.3 in the Position Lines. (This error is mainly due to the time recorded under practical conditions.) Land Surveyors achieve accuracy comparable to GPS using more sophisticated instruments but the same calculations/ method.

Final accuracy is obviously improved by taking more observations. Six star sights will typically provide a fix within 0’.2 of the true position.

Most people adopt some shortcuts in the interest of speed. These have a cost in terms of accuracy. The Sun's Total Correction Tables assume that the Sun's semi-diameter is either 15'.9 or 16'.2. A Sun Sight in April (SD = 16'.0) is immediately in error by 0'.2. Tables are rounded to the nearest 0'.1 which could introduce a cumulative error of 0'05 for every item.

With Star sights, the short interval between the first and last sight means that many people use a single position for all the sights and plot the results without allowing for the vessel's movement. The error is larger than above, but more than acceptable in mid-ocean.

Before Calculators and GPS

The method used until the 1980s was the Haversine Formula with Log Tables. A few navigators, mainly military, used Sight Reduction Tables but most preferred the longer method in the interests of accuracy and flexibility.

The Haversine formula is a rearrangement of the Cosine formula substituting Haversines for the Cosine terms. (Hav(Angle) = ½ x [1 – Cos(Angle) ] ). This makes a calculation using logarithms slightly easier, as the terms are always positive.

Hav(CZD) = Hav(Lat difference Dec) + Hav(LHA) x Cos(Lat) x Cos Dec)

Next Section

Sight Calculations and Obtaining a Position