The earth and the moon travel along curved paths through space, but their separation distance remains the same on average. The center of gravity of the earth-moon system follows an approximately elliptical orbit about the sun. Superimposed on this gross translation, the earth and moon translate in approximately circular orbits about their common center of gravity. The force giving rise to the centipetal acceleration associated with this relative motion is the mutural gravitational attraction between the earth and the moon.
Thus if we let represent the total mass of the mmoon and that of the earth and take d as the mean distance between their centers then
where and are the centripetal accelerations of the centers of the moon and earth, respectively, relative to their common center of gravity. If we write
we might regard these as an expression of balance of forces where we interpret the terms and as centrifugal forces. These forces are fictitious in the sense that they merely replace the effect of and are consequently opposite in direction to the actual accelerations.
Let represent the angular speed of a straight line joining the centers of the earth and moon and take and as the distances, respectively, of the centers of the earth and moon from the common point of rotation of this line (see Fig. 2.04-1). The centripetal accelerations are, accordingly
and, from Eqs. (27), it is evidence therefore that
Inserting the values of , and d from Table 2.02-II gives
which is actually within the earth itself. The angular velocity of the earth/moon system which is necessary to maintain the balance of centrifugal and attractive forces is, from Eqs. (29a) and (27a),
Inserting the values of G, , d and gives
which indicates an orbital period of
This is consistent with the observed sidereal period of the moon in its orbit relative to the earth (see Table 2.02-II).