The whys and wherefores of navigation

by Gershom Bradford

Mean time is reckoned by the revolutions of a fictitious sun, called the mean sun, and the length of one of these revolutions is the average of a year of apparent days. This, owing to its uniformity, is the time used for the everyday purposes of life. The difference between apparent and mean time is called the equation of time, and, by applying it according to signs given in the Nautical Almanac, one can be converted to the other as desired.

Sidereal time is indicated by the position of the First Point of Aries relative to the meridian; it is star time. The stars make a complete revolution of the heavens in 4 minutes less time than is required by the mean sun. Therefore the sidereal day is that amount shorter than the solar day.

The point of the celestial vernal equinox or First Point of Aries is a sort of celestial “bench mark;” besides indicating sidereal time, it serves as a point from which right ascension is measured eastward. This subject has been discussed previously, but as it is intimately associated with sidereal time, perhaps it may be made clearer since the latter has been so fully explained.

Right ascension is measured on the celestial equator, precisely the same as longitude on the earth, excepting that it is always measured eastward through the full 24 sidereal hours, contrary to the diurnal movement of heavenly bodies. Moreover, the meridian passing through the vernal equinox is called the celestial prime meridian, and sometimes the Greenwich of the heavens. There is another point of distinction, however, between this prime meridian of the sky and our meridian of Greenwich, which, while it does not effect practical navigation, has to receive consideration in the long run; our longitude values on the earth remain at all times constant, but owing to the precession of the equinoxes the celestial prime meridian is slowly moving westward, thus causing the right ascensions measured from it to become very slowly in error (50´´ yearly).

The hour angle of a body is the angle formed at the pole between the meridian and the hour circle passing through the body measured westward.

With all these important facts well in mind we will go ahead under a slow bell, through a few more statements which may be found a little perplexing. However, a careful study of the Time Diagram will, no doubt, drive away the haziness so often surrounding the subject of time.

This diagram represents the plane of the equator looking down upon the North Pole. The 75° W meridian is chosen as that of the observer and local time reckoned therefrom. The arrows on the outer circumference indicate the directions of the earth’s rotation; the other smaller arrows indicate the direction in which each element is reckoned.
Fig. 3.

The hour angle of the mean sun is the local mean time, and the hour angle of the First Point of Aries is the local sidereal time. The local mean time and longitude in time accelerated by Table III Nautical Almanac plus right ascension of the mean sun is equal to the local sidereal time. The right ascension of the meridian is the same thing, exactly, as the hour angle of the First Point of Aries, and both of these are identical with the local sidereal time. The sidereal time of Greenwich mean noon is the same as the right ascension of the mean sun at that time. The hour angle of a star plus the right ascension of the same star is equal to the local sidereal time.

Difference of longitude can be represented by an interval of sidereal time or by a difference of right ascension, precisely the same as by a difference of solar time. Thus with the local sidereal time calculated from an observation of a star, and the corresponding Greenwich sidereal time taken from the Nautical Almanac, the longitude is at hand, by turning their difference into arc. The fact that the actual time interval is longer in the case of solar time than in an interval of the same number of hours of sidereal time, has no influence on the resulting difference of longitude. The number of degrees in any arc can be the same, yet vary in linear measurement, but the same number of hours of solar and sidereal time represent the same proportionate part of a circle. It was just stated that the hour angle of a star plus its right ascension is the same as the local sidereal time; now this is also true of the sun. The hour angle of the mean sun plus the right ascension of the mean sun is equal to the local sidereal time; by means of this equality we are able to find the Greenwich sidereal time on any occasion. It is necessary to have this element in order to compare it with the local sidereal time, which we find by observation of a star, to obtain difference of longitude (in time). In page 2 of the Nautical Almanac will be found the Right Ascension of the Mean Sun at Greenwich Mean Noon; this must always be taken out for the preceding noon. We now have a measure of sidereal time to be added to a measure of mean time, but it will be remembered in early arithmetic that an apple and a peach can not be added together any more than ½ can be added to ⅓. The only course to steer is to reduce the quantities to a common denominator, or like quantities. So in handling these two varieties of time, solar time must be accelerated by adding a correction to it, or sidereal time retarded by subtracting an amount necessary to make it equal to a corresponding value of solar time. The tables for the conversion of one of these varieties of time to the other are found in the American Ephemeris Tables II and III, Bowditch Tables 8 and 9, and at the foot of pages 2 and 3, Nautical Almanac.