When a star or planet is observed at sea for obtaining the position of a ship, means of identification are generally furnished by its own characteristics of magnitude and color and by the relative location of other bodies—a star chart being employed to assist the observer if he is not familiar with the heavens. Cloudy weather, moonlight, or daylight may, however, prevent identification by this means.
If in doubt as to the name of the body at the time of taking the sight, it should be made an invariable rule to observe its bearing by compass, whence the true azimuth may be approximately deduced by applying the compass error. If the observer plots the position of his zenith on a star chart, using the local sidereal time for right ascension and the latitude for declination, the co-altitude (zenith distance) and azimuth may be laid off approximately and, with a knowledge of the star's magnitude, lead to the identification; or an estimate may be made of hour angle and declination, (the hour angle applied to local sidereal time giving the right ascension), and the star or planet thus recognized from an inspection of the tabulated values in the Nautical Almanac. These rough methods generally suffice where the body is the only one of its magnitude within an extensive region of the heavens; but cases often arise where a much closer approximation is necessary. The method to be described may then be employed.
The quantities given are the corrected altitude and azimuth, and the dead reckoning latitude; those to be found are the declination and hour angle.
In the astronomical triangle, PMZ, shown in the figure, let Z =azimuth, t = hour angle, p= polar distance, d=- declination and h= altitude. Then,
sin Z = sin t; or, sin Z cos h = sin t cos d
sin p cos h
The value of sin Z cos h (calculated from the given azimuth and altitude) must therefore equal sin t cos d, whatever the values of t and d may prove to be.
From a given latitude, azimuth and declination, the hour angle may be found either by azimuth tables of an azimuth diagram; or from a given latitude, azimuth and hour angle, the declination may be found by the same means. If, therefore, some probable value of the declination be assumed, using the known latitude and azimuth, we may ascertain the corresponding hour angle; or if the hour angle be assumed the corresponding declination is obtained; then the product of sin t cos d may be calculated, and if it agrees substantially with sin Z cos h, the trial values of the hour angle and declination are the correct ones; if not, other trials may be made until the correct ones are found. It may be remembered that absolutely exact results are not sought.
In practice, the operation may be made very short; the values of the quantities may be taken in even degrees and the logarithms need not be carried beyond the third place; the sum of the logarithms will suffice and the corresponding numbers do not have to be taken out. The possibility that the observed body may have been a planet must always be kept in mind in looking it up in the star table or chart.
As an example, suppose that in latitude 5° N., longitude 2h som W., by D. R., a star is observed whose corrected altitude is 38°, and true azimuth N. 107° E. The Greenwich sidereal time (as computed for use in the regular working of the sight) is 12h. 53m. Let it be required to identify the body.
First find the logarithm of sin Z cos h.
Z, 107°, sin 9.981
h, 38°, cos 9.897
sin Z cos h, log 9.878
Now suppose the observer estimates from the position of the body that its declination is 3° S. Look in the azimuth table on the page of latitude 5° (declination contrary name to latitude) and note the hour angle (P. M.) corresponding to Dec. 3rd and Az. 107°; this is found to be about 1h. 40m. with d= 3°, t= 1h 40m., find sin t cos d. (Sin t may be found either by converting time into arc and taking from the table in the usual way, or by multiplying by 2 and finding it from the column headed "Hour P. M." Thus in the present case, find the sine of 25° 00' or of 3h. 20M. In using the time column, be careful to take the name from the foot of the page when the double angle exceeds 6h.)
t, 1h.40m., sin 9.626
d, 3°, cos 9.999
log 9.625
As this logarithm should equal 9.878, it is seen that the assumption is incorrect. Try a value of the declination 5° farther south, that is, 8° S. The corresponding hour angle is 2h. 50m.
t, 2h. 50m., sin 9.830
d, 8°, cos 9.996
log 9.826
The logarithm is not yet quite large enough; assume declination 10° S.; the hour angle is therefore 3h. 20M.
t, 3h. 20M., sin 9.884
d, 10°, cos 9.993
log 9.877
This is practically identical with the logarithm of sin Z cos h, and the correct values are, therefore, t= 3h. 20m., d=10° S.
We have:
G. S. T. 12h. 53m.
Long. 2 50W.
L. S. T. 10 03
H. A. 3 20E.
R. A. 13 23
By star chart or table, it is found that the right ascension of
Spica is 13h. 20M. and the declination 100 39' S.; this is therefore
the body observed.
Taking another example, suppose that on January 1, 1902, in latitude 26° S., longitude 5h. 42m. E., by D. R., the altitude of a body is observed as 41°, and its azimuth as S. 84° W., the Greenwich sidereal time being 17h. 59m.; to find the name of the body.
Z, 84°, sin 9.998
h, 4.1° , cos 9.878
sin Z cos h, log 9.876
Assume first an hour angle of 3h. 00m.; the corresponding declination is 23° (same name as latitude).
t, 3h. 00m., sin 9.849
d, 23°, cos 9.964
log 9.813
Next assume an hour angle of 3h. 30m.; the declination is then 21° S.
t, 3h. 3om., sin 9.899
d, 210, cos 9.970
log 9.869
Assume hour angle 3h. 35m.; declination is still nearest to 21° S.
t, 3h. 35m., sin 9.907
d, 21°, cos 9.970
log 9.877
The last assumption is therefore correct.
We then have:
G. S. T. 17h. 59m.
Long. 5 42 E.
L. S. T. 23 41
H. A. 3 35W.
R. A. 20 06
As there is no fixed star corresponding to these coordinates the tables for the planets should be consulted. On January 1, 1902, the right ascension of Mars is 20h. 08m., and the declination, 21’ 20' S.; this is therefore the body that was observed.
It is to be remarked that the exactness with which the comparison of logarithms is carried out will depend upon the possibility of errors of identification in the region of the heavens involved. It will not usually be necessary to find the correspondence as closely as has been done in the examples given, and the cases will be rare when, with a fair estimate of hour angle or declination at beginning, a sufficiently accurate knowledge of the values can not be arrived at after the second approximation; and frequently the first will suffice for identification.
Azimuth tables intended for the sun are not available for use with bodies of greater declination than 23°. Azimuth diagrams give all values of the declination; the U. S. Hydrographic Office has in press azimuth tables which include declinations up to 70°.
The following is a summary of the method employed:
1. Reduce time of observation to Greenwich sidereal time and find the true altitude to the nearest degree. (These operations must be performed before any sight can be worked; they are, therefore, not strictly a part of the process of identification.)
2. Correct the observed azimuth for deviation and variation.
3. Find the logarithm of sin Z cos h to the third place.
4. Assume a declination and find the corresponding hour angle that will produce the given azimuth at the given latitude; or assume an hour angle and find the corresponding declination. Use an azimuth table or diagram for the purpose.
5. Find the logarithm of sin t cos d to the third place.
6. Observe whether this agrees with the logarithm of sin Z cos h, and if it does not, repeat trials until an agreement is found.
7. Having found the hour angle and declination, convert the Greenwich sidereal time into local sidereal time and subtract the hour angle if west, or add it if east; the result is the right ascension of the observed body, by which, with the declination and magnitude, the identification is accomplished.