According to Equations (11.131) and (11.143),

(11.215) | ||||||

and | (11.216) |

It follows from Equations (11.192), (11.193), and (11.197) that

According to Equations (11.132) and (11.144),

(11.219) | ||||

and | (11.220) |

where use has been made of the previous expressions for and . It follows from Equations (11.192), (11.193), and (11.198) that

(11.221) | ||||

and | (11.222) |

Finally, according to Equation (11.125),

(11.223) |

where use has been made of the previous expressions for and . Equation (11.189) yields

(11.224) |

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 previous expressions are accurate to .

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

(11.228) |

In other words, is the effective major radius of the orbit. Likewise, Equations (11.97), (11.104), and (11.226) suggest that the mean orbital angular velocity of the Moon is

(11.229) |

Hence, making use of Equation (11.31), we deduce that

Here, is the mass of the Earth, and the mass of the Moon. The previous expression is accurate to . Equation (11.230) implies that, when applied to the geocentric lunar orbit, Kepler's third law of orbital motion (see Sections 4.7 and 4.16) is slightly modified by the perturbing influence of the Sun (which is parameterized by ).