(1) Finding the Altitude of a Planet at a Given Time; Thence the Longitude by Same
I must say that there are tables by means of which the altitude of a celestial body can be obtained, using as the known data the dead reckoning, latitude, declination, and hour angle, but not even the most careful interpolation will, as a rule, give a result accurate enough by which to set the sextant and pick up a planet in the daytime.
As Venus and Jupiter are the only planets that can be picked up with a sextant with comparative ease in the daytime. I shall limit myself to the former, which has the added advantage of being in a favorable position for daytime observation 80 per cent of the days; and consequently, if observed at about the same time as a meridian altitude of the sun, an absolute fix can be obtained at local noon.
To insure against a partly cloudy sky, find beforehand the altitudes and azimuths corresponding to the number of more or less equidistant hour angles.
In this spherical triangle (Fig. 1.), we know two sides: the P. D., the colatitude by dead reckoning, and the hour angle, which is of our own choice, it being the difference between the G. M. T. of local transit and the G. M. T. at which we wish to observe.
The azimuth must be found by Napier's first and second analogies, but it is more conveniently taken from the tables, with careful interpolation.
(2) Working an Unknown Star for Longitude
This is very useful when, on account of continued cloudy weather, the navigator is uncertain of his position.
If the horizon is reliable and the clouds show a tendency to break, be ready with sextant and hack watch, and have the pelorus ready for an azimuth. Watch sharply in the desired portion of the heavens, and as soon as a first, second, or third magnitude star shows through a break, take as good an observation as possible while someone else takes its bearing.
Correct time, altitude, and bearing.
Then, considering a spherical triangle, of which we know two sides: colatitude and zenith distance; and the included angle azimuth, the other two angles, one of which is the hour angle, can easily be found by applying Napier's first and second analogies:
Where A = hour angle, B the other angle, and C the azimuth, always remembering that the hour angle will be the greater of the two angles when the latitude is larger than the true altitude, and vice versa.
Having the hour angle, apply it to the local sidereal time found in the usual manner, the result is the star's R. A. with which the star is identified by referring to the Nautical Almanac.
The declination is not necessary for the purpose of identification, because stars of the first, second, and third magnitude are sufficiently far between in R. A.
Then proceed to work the longitude, through which draw a line of position. If the same process is followed with a star in an adjacent quarter, we have another line of position which fixes the ship beyond doubt.
COMMENT
By Commander Paul P. Blackburn, U. S. Navy
The "Sine Proportion" Formula, which Lieutenant Porta uses for computing the true altitude of Venus, is the same formula as the one on p. 148, Bowditch, paragraph 358, with the terms transposed. The ordinary cosine haversine formula of the Marcq Saint-Hilaire method is more familiar to most navigators and brings the same result as this formula of Lieutenant Porta. Solving the problem that he gives, using latitude, hour angle, and declination instead of azimuth, hour angle, and declination, gives the following:
ascension and the right ascension of Bellatrix is not far from those of the other two; Mizar and Spica, Procyon and Pollux are other pairs where the difference in right ascension is so small that a knowledge of the declination might be useful.
The method of Bowditch, p. 173, for star identification seems better than Porta's method, because it requires only 11 log functions to find both hour angle and declination, while Porta's method takes eight logs to find hour angle alone. The formulas he uses in his star identification do not seem to have been used for this purpose, but are given in Bowditch, p. 146, Article 352, for the solution of the time azimuth, to get another angle of the astronomical triangle.
In any case of star identification, hour angle and declination can be only approximated, because the observed azimuth enters into this computation and all navigators know that very refined observations of star azimuths cannot be taken at sea.