The whys and wherefores of navigation

by Gershom Bradford

Long. in arc	40°	35´	45´´
	2 hrs.	40 m.	20 s.
		2	3
Long. in time	2 hrs.	42 m.	23 s.

This may appear complicated at first but is much the quickest way of conversion. However, Table No. 7, Bowditch is always available if desired.

In getting an understanding of any time problem, that is such as changing mean time into sidereal time; obtaining the hour angle of a star or planet; in seeking the local time from the chronometer, or any time values that are found perplexing, always draw a diagram. Make the circle on the plane of the equator, with the pole as the center, and meridians radiating from it towards the circumference of the circle. Now imagine for the moment that you are at the north pole, and the date is the 21st of September. The sun is traveling in the horizon, and if the direction of the Greenwich meridian is known, this body serves as a time piece, for the angle between this meridian (direction) and the sun is the Greenwich time. This angle corresponds with twice the angle between XII hours on the watch and the hour hand; or would (disregarding equation of time) coincide with it if the watch’s face was divided into 24 hours. Likewise when the vernal equinox or First Point of Aries lies in the direction of Greenwich it is Greenwich sidereal noon, and the subsequent angle that appears through the rotation of the earth, shows the Greenwich sidereal time.

Equation of Time

It is necessary in considering this subject to reiterate some of the statements made in the preceding talk on time, but, as they are very important, no time is wasted by further impressing them on the mind. Let it be understood that the apparent orbit of the sun is actually due to the earth’s revolution around him, yet for simpler explanation it is considered to be the sun’s own revolution.

The apparent movement of the real sun is not of uniform speed and, in consequence, it has become necessary to devise a fictitious sun whose assumed revolutions around the earth are at all times regular in their rate.

The equation of time is the difference between these two suns and, as they are at times in conjunction and at other times attain a distance from each other of 16 minutes 20 seconds, and, moreover, as the real sun is sometimes ahead and again in the wake of the mean sun, it becomes evident that the equation of time is an ever varying quantity.

The irregularity of the sun’s apparent movement as compared with the uniformity of the mean sun, is subject to two causes: First, the earth travels in an ellipse, and, as the length of a degree varies in the different parts of the circumference, the motion would appear to be irregular, that is, if the sun actually traveled at a uniform rate, it would, from the above fact, appear to us to be variable in its motion; furthermore, the laws of forces only allow a body traveling in a circle the privilege of a uniform speed so the earth, owing to its varying distance from the sun, experiences a corresponding change in the amount of attraction exerted upon it by the sun and its velocity, actually becomes variable. Thus, during the winter, December and January, when they are nearest each other, the attraction is strongest and the earth increases its speed in revolution; while in June and July the earth is at its greatest distance from the sun and the attraction is less, resulting in a slowing down in the rate of the onward movement. As the sun appears to us to take on movements corresponding to those of the earth, these variable movements of the latter are seen in the apparent motion of the sun. Second, the plane of the earth’s orbit is inclined at an angle to that of the equator, which makes the sun appear to be traveling at a variable speed along the ecliptic.

With these two errors combined influencing the apparent sun, he becomes unreliable for regulating timepieces. The mean sun, which was originated to obviate these irregularities, is assumed to travel in a circle with the earth located in the centre, which disposes of the first reason for an apparent variable motion; and again, the mean sun revolves in the plane of the equator, thus eliminating the second obstacle in the way of uniform time.

Now we will continue a little farther into the explanation of the reasons for the irregular movement of the real sun. A law discovered by Kepler, and named for him, provides that a radius from the sun to the earth covers sectors of equal areas in equal times; a sector equal in area to any sector covered in the same time. That is, when the earth is in that part of the orbit near the vernal equinox, the radius of the orbit will in a given time, say a week, sweep over a certain area; the earth proceeds toward aphelion and when in the vicinity of that point, the radius becomes greatly increased in length. Now in a week with this longer radius, a far greater area would be covered if the earth maintained the same rate of speed as at the equinox, but Kepler’s law says, “equal areas in equal times,” so in order to conform with the law, the earth’s speed of revolution must be reduced. The earth does not slow down just for the sake of obeying Kepler, but at this part of the orbit it is at its greatest distance from the sun and hence the reduced attraction causes the earth to lag a little.