Mean Sun and Apparent SunWe'll cover the following things here:
What is the mean sun?
The ecliptic of Ptolemy's sun moves uniformly around the earth, but has a single epicycle. In this diagram, found on p. 96 in the Great Books series, the earth is at D, with the mean sun at A moving uniformly around the earth, while the apparent sun is at E, at a distance of AE = AF from the point on the ecliptic A. The epicycle rotates at the same rate as the concentric, creating simple eccentric motion for the apparent sun. (see the aside from Part II on eccentric motion). Thus the mean sun A moves uniformly around the earth, while the actual apparent sun at E, does not appear to do so. The ratio between the circle ABC and the epicycle EFG is given by Ptolemy as 120:5 in The Almagest, III.4.
Why would Ptolemy use it for oppositions?
“For let ABC be the star's [Mars's] eccentric circle on which the epicycle's center is borne, and let it be about center D. And let AC be the diameter through the apogee; and point E on it the ecliptic's center; and F the center of the eccentric with respect to which the mean longitudinal position of the epicycle is considered. [F is the equant.] With the epicycle GHKL described about B, let the straight lines FLBH and GBKEM be joined.
Ptolemy does not seek to see the planet at opposition in order that it be seen in the same direction by both the sun and the earth. Rather, since the planet's epicycle and the mean sun (the center, A, of the sun's epicycle EFG in the previous diagram) both move uniformly, and at the same rate, we can tell when the epicycle lines up to show us the epicycle's center in the same direction as the planet by looking for its opposition to the mean sun. Thus, Ptolemy considers the effect of the second inequality (the epicycle) removed. Here is an animation comparing the two oppositions:
Here, you can see that it would be incorrect, in Ptolemy's system, to use apparent opposition, since at apparent opposition, the earth's view of planet Mars K does not line up with the center of its epicycle B, and thus would not remove the effect of the second inequality. At mean opposition, however, the second inequality is indeed removed. The assumption here is that the mean sun's direction from the earth is exactly parallel to the planet's direction on its epicycle from the center of the epicycle. That is, that EM is parallel to BK, an assumption that Ptolemy made: he "chose the mean motion, thinking that the difference (if any) between taking the mean sun and the apparent sun could not be perceived in the observations, but that the form of computation and of the proofs would be easier if the sunís mean motion were taken." (New Astronomy p.121)
What does Copernicus mean by the mean sun?
“For when Copernicus transformed the Ptolemaic hypothesis into his own general form, he supposed the observer to be stationed at some nearly motionless point near the sun, distant from the sun's own body by the entire eccentricity of the solar orb.” (New Astronomy, p. 142)
At first, this does not sound like Ptolemy's idea of the mean sun. Here, we have a stationary point, while Ptolemy had a moving one. But, remember the equivalence of the concentric-with-epicycle and the eccentric. Ptolemy has the mean sun move in a circle centered on the earth. Copernicus has the earth move in a circle centered on the mean sun. Ptolemy has the apparent sun move around the earth, but with an epicycle: you could say that the apparent sun moves on an eccentric with respect to the earth. Copernicus has the earth move on an eccentric with respect to the apparent sun. As far as the geometry is concerned, the two hypotheses are equivalent. And just as the earth moves uniformly on its eccentric, so too does the mean sun (the center of the earth's orbit for Copernicus) move uniformly through the zodiac during the year.
Why not just name it the "mean earth" and be honest?
Why did Copernicus use the mean sun for oppositions?Because he copied Ptolemy!
Perhaps the opening of Copernicus's commentaries on Mars will shed some light on this question:
The other two oppositions take place in the months of Pharmuthi and Epiphi. Now, what on earth is Copernicus doing, in using millenium-old, poorly made observations? Perhaps he is simply comparing them with his three observations, made in 1512, 1518, and 1523. But Ptolemy's original purpose for taking oppositions at the mean sun (to have the correct orientation of the epicycle) is vitiated by the Copernican hypothesis, which assigns the cause of the second inequality to the motion of the earth. Unlike Ptolemy, who used opposition as a timer to determine when the planet is facing the earth on its epicycle, Copernicus seeks oppositions because they afford the opportunity to know the sidereal position of the planet as observed by the center of the world. And according to Copernicus, the center of the world is the sun, right? Not really. Copernicus actually puts the center of the earth's orbit as the center of the world. This point, while being near the sun, is in no way a characteristic of it.
Had Copernicus stuck with reality, rather than following "his master Ptolemy," he would have no reason whatsoever for using the mean sun. Then, Kepler could not have written about him in Part I: his hypothesis would not have been equivalent!