According to Equations (11.131) and (11.143),
According to Equations (11.132) and (11.144),
Finally, according to Equation (11.125),
It follows from Equations (11.122)-(11.124), (11.157), (11.163), (11.164), (11.175), and (11.176), as well as the previous expressions for , , , , and , that the net perturbation of the lunar orbit due to terms in the solution of the lunar equations of motion that depend only on is
The first term on the right-hand side of Equation (11.226) is known as variation, and is clearly due to the perturbing influence of the Sun (because it depends only on the parameter , which is a measure of this influence). Variation attains its maximal amplitude around the so-called octant points, at which the Moon's disk is either one-quarter or three-quarters illuminated (i.e., when , , , or ). Conversely, the amplitude of variation is zero around the so-called quadrant points, at which the Moon's disk is either fully illuminated, half illuminated, or not illuminated at all (i.e., when , , , or ). Variation generates a perturbation in the lunar ecliptic longitude that oscillates sinusoidally with a period of half a synodic month.11.2 This oscillation period is in good agreement with observations. However, according to Equation (11.226), the amplitude of the oscillation (calculated using ) is arc seconds, which is somewhat less than the observed amplitude of arc seconds (Chapront-Touzé and Chapront 1988). This discrepancy between theory and observation is due to the fact that, for the sake of simplicity, our expression for variation only includes contributions that are fourth order, or less, in the small parameter .
Equations (11.103) and (11.225) imply that the mean radius of the lunar orbit is