(11.1) |

for , where

(See Chapter 4.) Here,

(11.3) |

Furthermore,

because the barycenter of the Earth-Moon-Sun system lies at the origin.

Let

be the position vector of the Moon relative to the Earth, and let

be the position vector of the Sun relative to the Earth-Moon barycenter, . It follows that

(11.7) | ||||||

and | (11.8) |

Now, assuming that , we can write

(11.9) |

We conclude that

(11.10) | ||||||

(11.11) | ||||||

and | (11.12) |

so that

(11.15) | ||||||

(11.16) | ||||||

and | (11.17) |

which implies that

Let

It follows from Equations (11.18)-(11.20) that

Note that , where and are the major radii of the lunar orbit about the Earth, and the orbit of the Earth-Moon barycenter about the Sun, respectively (Yoder 1995). Now,

(11.25) |

for and , where the are Legendre polynomials (Abramowitz and Stegun 1965b). Thus, expanding Equations (11.23) and (11.24) to third order in the small parameters and , respectively, we obtain

(11.26) | ||||||

and | ||||||

(11.27) |

respectively. It follows from Equation (11.2) that

where use has been made of

(11.29) | ||||||

and | (11.30) |

(Abramowitz and Stegun 1965b), as well as Equation (11.21).

Let (see Chapter 4)

(11.31) | ||||||

and | (11.32) |

where per day is the mean orbital angular velocity of the Moon around the Earth, and per day is the mean orbital angular velocity of the Earth-Moon barycenter around the Sun (Yoder 1995). Here, we have neglected the combined mass of the Earth and the Moon, , with respect to the mass of the Sun, , because (Yoder 1995). It follows from Equations (11.13), (11.14), and (11.28) that the equations of motion of the Moon and the Earth-Moon barycenter can be written

respectively, where

According to Equations (11.33) and (11.35), the relative size of the lowest-order (quadrupole) solar disturbance of the lunar orbit is

On the other hand, according to Equation (11.36), the relative size of the next-order (octupole) solar disturbance of the lunar orbit is

(11.39) |

Finally, according to Equations (11.34) and (11.37), the relative size of the lowest-order (octupole) disturbance of the orbit of the Earth-Moon barycenter around the Sun, due to the combined effect of the Earth and the Moon, is

(11.40) |

Here, use has been made of the fact that the ratio of the terrestrial to the lunar mass is (Yoder 1995). Clearly, it is an excellent approximation to neglect the octupole disturbance of the orbit of the Earth-Moon barycenter around the Sun, while retaining the quadrupole and octupole disturbances of the orbit of the Moon around the Earth. Adopting this approximation, we conclude that the former orbit is an unperturbed Keplerian ellipse.

One obvious way of proceeding would be to express the right-hand side of the lunar equation of motion, Equation (11.33), as the gradient of a disturbing function (see Exercise 1), and then to use this function to determine the time evolution of the Moon's osculating orbital elements from Lagrange's planetary equations. (See Chapter 10.) Unfortunately, this approach is fraught with mathematical difficulties. (See Brouwer and Clemence 1961). It is actually more straightforward to directly solve Equation (11.33) in a Cartesian coordinate system. This method of solution is outlined below.