The whys and wherefores of navigation

by Gershom Bradford

At the time of the autumnal equinox, an area similar to that of the vernal equinox is covered. As the earth approaches perihelion, the radius is gradually shortened by the eccentricity of the sun in this part of the orbit and the increased attraction causes the earth to speed up correspondingly. At the increased speed, the shorter radius sweeps the same area in a week as at other parts of the orbit and Kepler’s law still holds good.

As the apparent motion of the real sun corresponds exactly with the real motion of the earth, it is evident from the above that the real sun apparently moves at different rates of speed along the ecliptic, faster in winter and slower in summer than the mean sun.

The value of a mean solar day is the average of a year of apparent days, or in other words, there is the same number of mean solar days in a year as there are apparent days.

In considering the effect of the variable motion of the earth in its orbit, we will recall the conditions used when defining sidereal and solar days. The former comprises the interval between two successive passages of a certain star across the meridian, or perhaps better, between two successive passages of the meridian over a star. This is the true length of the earth’s rotation and is the standard to which we may refer the length of the mean or apparent solar days.

Now it requires about 3 minutes 56 seconds longer for the meridian to sweep around from sun to sun than from star to star, owing to the fact that the mean sun moves uniformly eastward that amount daily, thereby requiring the meridian, after reaching its position of yesterday noon, to overhaul the mean sun this 3 minutes 56 seconds of eastward movement. The mean sun maintains this uniform difference between its days and the length of the sidereal day. Without this daily easting of the sun, the sidereal and solar day would be the same.

But, in considering the apparent sun, we find the length of its days continually varies from both that of the sidereal and mean day. This is explained by the fact that the eastward movement of the apparent sun is due to the movement of the earth in its orbit, and as this movement becomes faster or slower the eastward movement of the sun becomes correspondingly faster or slower. Thus, we readily see that with the apparent sun moving eastward faster or slower at times, the length of the apparent day must vary accordingly and we cannot establish a uniform difference between it and the sidereal day, as in the case of the latter and the mean day. The apparent days exceed the mean days in length, between September and March, while the earth is traveling fastest in its orbit. Beginning at the autumnal equinox with the apparent sun eight minutes behind the mean sun, the former gains slowly at first but with increasing rapidity. About the end of December, at perihelion, it overhauls the mean sun and they are coincident as regards this correction only. Leaving perihelion, the apparent sun rapidly takes the lead but with a gradually decreasing amount until at the equinox in March, reaches its maximum lead of 8 minutes. Entering that portion of the year March to September, we find the earth traveling slower and the mean sun gaining on the apparent sun; between the vernal equinox and aphelion, the mean sun gains until both are together at the summer solstice and then forging ahead the mean sun attains a lead of 8 minutes in September.

It must be borne in mind that this error is caused only by the eccentricity of the orbit and is but a component part of the whole correction of the equation of time. The other portion is due to the obliquity of the orbit, or its inclination to the equator.

This error is introduced through the fact that the apparent sun moves in the ecliptic and the mean sun is assumed to proceed along the celestial equator. In considering this phase of the question, we will ignore entirely, for the time being, the error of eccentricity described above.

The error of equation of time due to the obliquity of the orbit is a simple one to see, but like many simple things it is easier to show it by a diagram than to explain in words, so the reader is referred to the accompanying figure, that a study of it may be made before proceeding.