Consider the Earth–Moon–Sun system. See Figure 11.1. Let the , for , denote the masses of the Earth, the
Moon, and the Sun, respectively. Furthermore, let the , for , denote the position vectors of the Earth, the Moon, and the
Sun, respectively, in a nonrotating reference frame in which the Earth–Moon–Sun barycenter, , is at rest at the origin. The equations of motion of the three bodies that
make up the Earth–Moon–Sun system can be written

(11.1) 
for , where

(11.2) 
(See Chapter 4.) Here,

(11.3) 
Furthermore,

(11.4) 
because the barycenter of the Earth–Moon–Sun system lies at the origin.
Figure: 11.1
The Earth–Moon–Sun system. Here, denotes the Earth, the Moon, and the Sun. Furthermore, is the
barycenter of the Earth–Moon–Sun system, whereas is the barycenter of the Earth–Moon system.

Let

(11.5) 
be the position vector of the Moon relative to the Earth, and let

(11.6) 
be the position vector of the Sun relative to the Earth–Moon barycenter, . It follows that
Now, assuming that
,
we can write

(11.9) 
We conclude that
so that
Equations (11.4)–(11.6) yield
which implies that
Let

(11.21) 
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 EarthMoon 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
respectively.
It follows from Equation (11.2) that
where use has been made of
(Abramowitz and Stegun 1965b),
as well as Equation (11.21).
Let (see Chapter 4)
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 lowestorder (quadrupole) solar disturbance of the lunar orbit is

(11.38) 
On the other hand, according to Equation (11.36), the relative size of the nextorder (octupole) solar disturbance of the lunar
orbit is

(11.39) 
Finally, according to Equations (11.34) and (11.37), the relative size of the lowestorder (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 righthand side of the lunar equation of motion, Equation (11.33), as the gradient
of a disturbing function (see Section 11.18, 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.