Azimuth Tables

As stated in The Complete On-Board Celestial Navigator the azimuth tables have been designed to produce a value of azimuth which is compatible with the sight reduction procedure described in the book. An accuracy of one or two degrees will be sufficient when the intercept is short, which will be expected when the DR position is used.

If the variables of Declination, Local Hour Angle and Latitude are used to interpolate the tables, the aforementioned accuracy is achieved. However, the process can be considerably shortened if the variables are rounded off to the nearest degree before using the tables. The accuracy of this latter process is investigated here using the same principles that were adopted to analyse the sight reduction tables which is described previously in Detailed description. This technique of assessing the accuracy of look-up tables was suggested by Dr B D Yallop, former Director of HM Nautical Almanac Office.

The azimuth tables have been investigated using a computer program that simulates a human operator selecting, combining and extracting values from the tables over the following ranges of the principle parameters:

Declination N90 to S90 : Altitude 0 to 70 : LHA 0 to 180

Each parameter takes on values at a quarter of a degree intervals and all combinations of the three parameters were used in the analysis. Theoretically there should be

(180 x 4 +1) x (70 x 4 +1) x (180 x 4 +1) = 146,075,321

possible cases to examine, but some combinations of the parameters will not form a spherical triangle and were therefore excluded. The remaining 121,677,000 cases form the sample to be tested. Such a sample size should be sufficient to assess the performance of the tables. The differences, called errors, between the azimuths calculated accurately and those derived from the solution using the tables were compiled and the results of that investigation are as follows:



It will be seen that the preponderance of acceptable errors, i.e. less than two degrees, comprises 98.8% of the sample. Very large errors i.e. greater than ten degrees occur in only 0.005% of the sample. An error in excess of ten degrees up to a maximum of less than 18 degrees is extremely rare.

The question now arises; under what circumstances do we find errors greater than two degrees? The answer to this question is not obvious. Further analysis of the previous results is given in the following table, which is in two parts and shows that (a) these errors increase in number and size with an increase in altitude and (b) at low altitudes these errors are small and occur close to the east-west line (prime vertical). The proximity to the prime vertical can be found from the value of the quantity A given at the foot of the table. For example if an observation was made at an altitude of 15 degrees the maximum error that could occur would be less than six degrees in an azimuth range of 84 96 or 264 276 degrees.

Sample Size 121,677,000

It should be noted that even though an observation may have been made at high altitude and close to the prime vertical the size of the error is also dependent on the degree of rounding off that has been made to the principal parameters before using the azimuth tables.

Ambiguity in Azimuth

An ambiguity in the value of azimuth may arise when the body is near the prime vertical.. This is because an angle near 90 or 270, when derived from its sine, will have two possible values e.g. sin.89 = sin.91 or sin.267 = sin.273. In most cases in practice, it will be obvious which of the two values to choose e.g. from prediction, compass bearing etc. If may be possible to remove this ambiguity as follows, (a) if the declination has the opposite name to the latitude, the body will lie in the southern sky in northern latitudes, and in the northern sky in south latitudes, (b) if the declination has the same name and is greater than the latitude the body will lie in the northern sky in north latitudes and in the southern sky in south latitudes. If the ambiguity has still not been resolved this will be found by comparing the values of the observed altitude with that when the body lies on the prime vertical. The tables themselves are used to determine the altitude of the body when on the prime vertical. A simple summary of the situation is given in the diagrams on page 19 of TCO-BCN.

Summary

The previous explanation and analysis of the azimuth tables may give the impression that they are difficult to use or inaccurate. From my experience, both as a practicing navigator and teacher, this has not been the case. In my opinion they are simple to use and inaccuracies will only occur in rare circumstances. These circumstances are not easy to quantify succinctly. However, as a guide, the user should interpolate the tables if rounding off has been substantial, the observed altitude is high or the body lies within about twenty degrees of the prime vertical. Alternatively use the Weir diagrams provided.

Since the publication of The Complete On-Board Celestial Navigator in 1988 I have only received one complaint and that was illustrated by a single example which used rounding off values of 29, 31 and 31 minutes of arc in the three principal parameters at an altitude of over 61 degrees and within nearly 15 degrees of the prime vertical. Such an extreme case is most unlikely to occur. Constructive, objective criticism of the table as a whole are welcome, particularly if supported by impartial analysis and examples taken from practical experience.