## Ptolemys Model For The Moon

The fourth book of the Almagest is devoted to a detailed study of the motion of the Moon. This subject presented to Ptolemy one of the most challenging problems of his whole astronomical system because the Moon's motion is subject to several irregularities that are not present in the relatively simple case of the Sun. In addition to the inherent complications of the lunar motion, Ptolemy's observations followed a very particular pattern: the primary set of observations were made when the Moon was at syzygies, that is to say, at new Moon and full Moon, when the Sun, Moon, and Earth lie in a straight line. He developed a model for these observations, noticed that it was in conflict with observations of the Moon at first and third quarters, and modified the model to account for these differences. He noticed that the resulting model was slightly at odds with the measurement of the Moon at octants and modified the model once more to account for this fact.

It is worthwhile to consider what we know today about the motion of the Moon. The Moon is a member of the three-body system of the Moon, Sun, and Earth. Its primary motion takes place in an ellipse of small eccentricity, with the Earth at one focus. This motion is disturbed by the action of the Sun, a disturbance that results in several changes to simple elliptical motion. The most important change that occurs is a rotation in the direct sense of the line of apsides of the Moonâ€”the apogee, or position of minimum lunar velocity, moves in a direct direction along the ecliptic by an amount equal to about three degrees per month. Separate from the rotation of the apogee, there are several other corrections that need to be made to simple elliptical motion. The largest of these changes is called the evection, a periodic correction that reaches its maximum when the Moon is 90 degrees from the Sun and a minimum when the Moon is aligned with the Sun and Earth. Next to the evection, there is a correction called the variation; it is also periodic and determined by the elongation of the Moon from the Sun.

The modern account depends on an analysis using the theory of perturbations, in which the mean, or average, motion is supplemented by a series of terms resulting from the mutual gravitational pull of the Earth and Sun on the Moon. Of course, Ptolemy knew none of this, but he did notice two observational facts:

the motion of the Moon along the ecliptic takes place with variable velocity, and the point of minimum velocity itself is not fixed but moves in a direct sense along the ecliptic; and the Moon's motion is subject to small changes that are connected to the relative position of the Moon and the Sun. To account for the variable velocity, Ptolemy used a deferent-epicycle model, in which the Moon lies on an epicycle whose center lies on a deferent with Earth at the center. The epicycle rotates with a period of one month, and the center of the epicycle revolves on the deferent around the Earth in about the same period. However, to account for the direct movement of the lunar apogee, one lets the period of the epicycle be slightly smaller than the period of its center on the deferent. The resulting model nicely accounts for the first observational fact mentioned above. Ptolemy also identified the evection and modified the model to account for it. He placed the center of the deferent on a small circle with center at the Earth. The center of the deferent rotates on this circle, in the process alternately drawing the epicycle closer to the Earth and extending it farther away in a reciprocating motion. One of the particular features of Ptolemy's lunar model is that the distance from the Moon to the Earth varies by a very considerable degree, with the maximum distance being almost twice the minimum distance. (This fact conflicts with the lack of any observed variation in the angular size of the Moon, a problem we discuss below.) Ptolemy did some further minor tampering with the model, but we will not follow him in this.

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