Precession of lunar perigee

According to Equations (11.206) and (11.284), the mean ecliptic longitude of the lunar perigee evolves in time as

$$\alpha = \alpha_0 + \alpha'\,n'\,t,$$ (11.337)

where

$$\alpha' =c\,m= \frac{3}{4}\,m+ \frac{225}{32}\,m^{\,2} + {\cal O}(m^{\,3}).$$ (11.338)

Equation (11.337) implies that the perigee precesses (i.e., its longitude increases in time) at the mean rate of $360\,\alpha'$ degrees per year. (Of course, a year corresponds to ${\mit\Delta} t=2\pi/n'$.) Furthermore, it is clear that this precession is entirely due to the perturbing influence of the Sun, because it depends only on the parameter $m$, which is a measure of this influence. Given that $m=0.07480$, we find that the perigee advances by $34.36^\circ$ degrees per year. Hence, we predict that the perigee completes a full circuit about the Earth every $1/\alpha' = 10.5$ years. In fact, the lunar perigee completes a full circuit every $8.85$ years. Our prediction is somewhat inaccurate because our previous analysis neglected ${\cal O}(m^{\,2})$, and smaller, contributions to the parameter $c$ [see Equation (11.284)], and these turn out to be significant. (See Section 11.17.)