- Demonstrate that the lunar equation of motion, Equation (11.33), can be written in the canonical form
- Consider the equation of motion of the Earth-Moon barycenter, (11.34).
Approximating the orbit of the barycenter around the Sun as a circle of major radius
, and that of the
Moon and the Earth about the barycenter as a circle of major radius
, and then averaging over the motions of the Moon and the Earth,
show that Equation (11.34) reduces to
- Because it is the Earth-Moon barycenter, rather than the Earth, that describes a Keplerian orbit about the Sun, it follows that, when observed from the Earth, the Sun will
not apparently move in a Keplerian orbit. Let
and
be the associated displacements of the Sun, in ecliptic longitude and latitude, respectively, with
respect to a Keplerian orbit. Demonstrate that

where is the mass of the Moon, the mass of the Earth, the major radius of the lunar orbit, the major radius of the barycenter's orbit, the inclination of the lunar orbit to the ecliptic plane, the mean elongation of the Moon from the Earth, and the lunar mean argument of latitude. Show that and . - Demonstrate that the lowest-order evection term,
- Suppose that the major radius of the lunar orbit were reduced by a multiplicative factor
; that is,
, where
. Assuming that the masses of the Earth and Sun, and the major radius of the terrestrial
orbit, remain constant, demonstrate that the parameter
, which measures the lowest-order perturbing influence of the Sun on
the lunar orbit, would be reduced by a factor
; in other words,
. Given that
for
the true lunar orbit, how small would
have to be before the (theoretical) precession rate of the lunar
perigee became equal to the regression rate of the ascending node to within
percent? What is the corresponding major radius
of the lunar orbit in units of mean Earth radii? (The true major radius of the lunar orbit is
mean Earth radii.)
- An artificial satellite orbits the Moon in a low-eccentricity orbit whose major radius is twice the lunar radius.
The plane of the satellite orbit is slightly inclined to the plane of the Moon's orbit about the Earth. Given that the mass
of the Earth is
times that of the Moon, and the major radius of the lunar orbit is
times the lunar radius,
estimate the precession period of the satellite orbit's perilune (i.e., its point of closest approach to the Moon) in months due to
the perturbing influence of the Earth. Likewise, estimate the regression rate of the satellite orbit's ascending node (with respect to the
plane of the lunar orbit) in months. (Assume that the Moon is a perfect sphere.)
- The mean ecliptic longitudes (measured with respect to the vernal equinox at a fixed epoch) of the Moon and the Sun increase at the rates
per day and
per day, respectively. However, the vernal equinox regresses in such a manner that, on average, it completes a full
circuit every
years. Furthermore, the lunar perigee precesses in such a manner that, on average, it completes a full circuit every
years, whereas the lunar ascending mode regresses in such a manner that, on average, it completes a full circuit every
years (Yoder 1995). A
*sidereal month*is the mean period of the Moon's orbit with respect to the fixed stars, a*tropical month*is the mean time required for the Moon's ecliptic longitude (with respect to the true vernal equinox) to increase by , a*synodic month*is the mean period between successive new moons, an*anomalistic month*is the mean period between successive passages of the Moon through its perigee, and a*draconic month*is the mean period between successive passages of the Moon through its ascending node. Use the preceding information to demonstrate than the lengths of a sidereal, tropical, synodic, anomalistic, and draconic month are , , , , and days, respectively. - To first order in the Moon's orbital eccentricity and inclination, the geocentric ecliptic longitude and latitude of the Moon, relative
to the Sun, are written
and

respectively, where and radians. Here, and are the mean geocentric orbital angular velocities of the Moon and Sun, respectively, is the mean orbital angular velocity of the lunar perigee, and is the mean orbital angular velocity of the lunar ascending node. Note that , , and , where , , and are the lengths of a synodic, anomalistic, and draconic month, respectively. At , we have . In other words, at , the Moon and Sun have exactly the same geocentric ecliptic longitudes and latitudes, which implies that a solar eclipse occurs at this time. Suppose we can find some time period that satisfies , where , , are positive integers. Demonstrate that at . Thus, if the period , which is known as the*saros*, existed then solar (and lunar) eclipses would occur in infinite sequences spaced synodic months apart (Roy 2005). Show that for , the closest approximation to the saros is obtained when , , and . Demonstrate that if at (i.e., if there is a solar eclipse at ) then, exactly synodic months later, and . It turns out that these values of and are sufficiently small that the eclipse reoccurs. In fact, because synodic months almost satisfies the saros condition, solar (and lunar) eclipses occur in series of about 70 eclipses spaced synodic months, or 18 years and 11 days, apart. - Let the
-,
-, and
-axes be the lunar principal axes of rotation passing through the lunar
center of mass. Because the Moon is not quite spherically symmetric, its
principal moments of inertia are not exactly equal to one another. Let us label the principal axes such that
.
To a first approximation, the Moon is spinning about the
-axis, which is orientated normal to its orbital plane. Moreover,
the Moon spins in such a manner that the
-axis always points approximately in the direction of the Earth. Let
be the (small) angle subtended between the
-axis and the line joining the centers of the Moon and the Earth.
A slight generalization of the analysis in Section 8.11 reveals that
We can write