The whys and wherefores of navigation

by Gershom Bradford

The shape of the triangle is determined by the declination of the body, its altitude and the latitude of the vessel, and the polar or hour angle; and it stands to reason that a formula will not produce the same accuracy in the hour angle with every shape of the triangle. For instance, in high latitudes or when the body has a declination approaching 90°, the accuracy of the time sight formula becomes effected.

Another very important point to bear in mind when observing a body with the view of computing its hour angle, is its azimuth. When the bearing is nearly east or west, on the prime vertical, the body is rising or falling faster than at any other time, and an error in altitude or latitude will produce the least error in the resulting longitude. The necessity for close attention to this point is increased with the latitude. Observations for time taken when the body has an azimuth of less than 45° or over 135° are wholly unreliable.

The sun does not always cross the prime vertical in his daily track across the heavens, for under certain conditions, say during the northern winter, he will rise southward of east and set southward of west. Under these adverse conditions, the calculation of longitude is not dependable, and the best a navigator can do when using the sun is to observe as soon as he has an altitude sufficient to clear the excess refraction existing near the horizon.

It is under such circumstances that star sights are of incalculable value, for a star can always be found in a suitable position with but little waiting, or we may employ the New Navigation method, where the azimuth of the sun is as good one place as another.

In order that a body will cross the prime vertical, the latitude must be of the same name and greater than the declination. In conditions cited above the declination of the sun is south and the latitude is north, hence the body will never be on the prime vertical. If the latitude is less than the declination, the sun’s diurnal track is tilted toward the zenith, instead of away from it as when the latitude is greater, and the result is that the sun, while never on the prime vertical, approaches it for a time after rising, then recedes again. It should be observed when at its nearest point to the bearing of east of west.

The bearing of various bodies can be readily found by an inspection of Hydrographic Office Azimuth Tables Nos. 71 and 120, the declination and latitude being used as arguments.

There is a method of finding the longitude known as the equal altitude method, but it is not valuable. The conditions are exacting where accurate results are required and when these conditions exist the ordinary time sight is available and at its best advantage, so longitude by equal altitudes is not popular. To secure good results, the body must have an altitude above 70° and near the prime vertical; and, furthermore, the ship must be kept on an east or west course or remain stationary. The theory of the problem is simplicity itself, and for this reason is very alluring, but the best use that equal altitudes can be put to is the determination of chronometer error ashore, and in these days of radio time signals even this use is almost obsolete. The rule is as follows: Observe the sun’s altitude, simultaneously noticing the time by chronometer and clamp the sextant to prevent any chance of the altitude becoming disturbed. When the sun has fallen to the same altitude as of the forenoon sight, note the time again by the chronometer. The mean of the two times, corrected for chronometer error, equation of time, and the equation of equal altitudes due to change in declination, in case of the sun, is the Greenwich apparent time corresponding to our local noon or our longitude in time, which should then be converted into arc.

The stars and planets are available as well as the sun for the finding of longitude and when there is a distinct horizon, stellar sights have many advantages. The problem depends upon the solution of the astronomical triangle by the same formula as with the sun.

There are a few points of difference between a time sight of the sun and one of a star or planet needing explanation. In the case of the former body, we naturally compare the solar time of Greenwich with the solar time of the local meridian, but in stellar work we employ for this comparison stellar time, or, as it is more popularly called, sidereal time. So it becomes necessary to turn the Greenwich mean time of the chronometer into Greenwich sidereal time and compare it with local sidereal time. The difference, as in mean time, is the longitude in time, which is converted into arc in precisely the same way.

The Greenwich mean time is turned into sidereal time by adding to it the right ascension of the mean sun, taken from the Nautical Almanac and the acceleration for the Greenwich mean time (Table 9, Bowditch). The local sidereal time is the result of an addition of the star’s right ascension and the star’s hour angle, the right ascension is taken from the Nautical Almanac without correction if a fixed star is being considered and the computation of a time sight gives the star’s hour angle. The local sidereal time being the right ascension of the meridian, it follows that the angle from the vernal equinox to the star plus the angle from the star to the meridian is what we desire; hence the above rule for obtaining the local sidereal time.