It requires 24 hours for this star to complete the small circle of revolution, the same time required by every star; its movement is necessarily very slow. By computing its hour angle, we can locate its position on this circle, and hence by applying a correction to its altitude, subtracting or adding according to the position of the star above or below the pole, we will obtain the altitude of the pole.
A rough estimate of the position of the pole may be secured by noting the position of the Big Dipper, the second star in the handle, called Mizar, is approximately in line with Polaris and the pole.
We will now proceed to show the method by which the hour angle is obtained:
In the talk on Time, it was stated that the local (astronomical) mean time plus the right ascension of the mean sun is equal to the local sidereal time; and again, that the right ascension of a star plus its hour angle equals local sidereal time. With these facts as a basis, the formula for latitude by Polaris given in the Nautical Almanac will be followed in explanation.
Fig. 6.
The time of observation must be noted by chronometer and converted into local (astronomical) mean time; this must be corrected by Table III (Nautical Almanac) in order to change this solar interval into a sidereal time interval; to this converted time must be added the Greenwich sidereal time of mean noon (page 2); that is, the hour angle of the First Point of Aries, or what is the same thing, the right ascension of the mean sun; to this sum must be applied a correction for longitude, in time, taken from the foot of page 2, N. A. The sum is the local sidereal time.
The reason for the correction of longitude is this: The difference between the right ascension of the mean sun at noon on two successive days is 3 m. 56 s., the same as the accumulated difference between solar and sidereal time in 1 day. Now we take from the Nautical Almanac this element for Greenwich mean noon, yet the sun has since covered the distance equal to the longitude, and during the interval required to do this, the sidereal time has accelerated over the solar an amount which bears the same ratio to the 3 m. 56 s., that the longitude in time bears to 24 hours. The Nautical Almanac handles the terms of this proportion in tabular form at the foot of page 2. It is stated that the sun has traveled from the meridian of Greenwich to the local meridian, and it might be suggested that at the time of observation the sun has covered this amount plus the local hour angle or the local astronomical mean time. This is true but the amount of local hour angle has been previously accelerated to sidereal time by the correction to local astronomical mean time.
With the local sidereal time enter Table I (Nautical Almanac) and pick out the correction to be applied according to sign to the altitude. It is probably needless to say that the observed altitude must be corrected for index error, dip and refraction before applying this latter correction, which converts it into latitude.
This is called the Nautical Almanac method and is sufficiently accurate for navigational purposes, but should a greater refinement be desired there are tables of further corrections given in the American Ephemeris and Nautical Almanac.