Fig. 4.
At the equinoxes, where the ecliptic and equator cross and the solstices—the vertices of the ecliptic—that is, four times a year, the true and mean sun are together, but departing from these points they do not travel with the same right ascension, remembering that right ascension is measured on the equator. Taking, for example, the earth in that quadrant of the orbit comprised between the vernal equinox and summer solstice, the apparent sun in the heavens would be by cause of obliquity alone, to the right or to the westward of the mean sun, and thus it will be seen that with the earth rotating from right to left the apparent sun will cross the meridian first; consequently between March 21st and June 21st that part of the equation of time due to the obliquity of the orbit bears a minus sign when mean time is desired from the apparent time. This correction reaches its maximum half-way or 45° from the equinox, amounting at that point to nearly 10 minutes.
Fig. 5.
Now the reason for this difference between the mean and apparent sun when each (so far as this problem is concerned) moves along its respective path—the equator and the ecliptic—at the same rate, is this: suppose the equator between the equinox and the solstice is divided into an equal number of parts and an hour circle drawn through each point of division. Beginning from the equinox (the common apex of the triangle) the arc of each hour circle between the equator and the ecliptic, forms the altitude of a right-angled triangle, while the equator and the ecliptic are base and hypothenuse respectively. Thus it will be seen that each portion of the equator (base) is shorter than the corresponding part of the ecliptic as defined the hour circle, to the extent of the ratio of the base to the hypothenuse.
This amount increases with the increasing size of the triangle, but a new element enters to counteract its effect. With the increasing divergence of the ecliptic and equator, the divergence of the hour circles becomes a factor and as the solstice is approached the divisions on the equator, are represented on the ecliptic by gradually decreasing spaces between the hour circles.
The combination of these effects produces the error due to the obliquity of the orbit. The error has the opposite effect in the next quadrant, that is, from June to September; and in opposite quadrants it is the same.
So it will be seen that error due to the eccentricity causes the apparent sun to lead the mean sun from December 31st to June 30th, reaching a maximum of 8 minutes about April 2d. This sun then falls astern until December, again attaining a maximum of -8 minutes about October 1st. The error due to the obliquity of the orbit accumulates between the equinoxes and solstices for at these points the two suns are together and there is no error, but about the 6th of May, August, November, and February, it reaches a maximum of 10 minutes; in August and February, the mean sun takes the lead.
These two errors of equation of time combined algebraically will result in the plain line of the diagram.
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