Cassini's laws

In 1693, the astronomer Gian Domenico Cassini (1625-1712) formulated a set of empirical laws that succinctly describe the Moon's axial rotation. According to these laws:

- The Moon spins at a uniform rate that matches its mean orbital rotation rate.
- The normal to the Moon's equatorial plane subtends a fixed angle, , with the normal to the ecliptic plane.
- The normal to the Moon's equatorial plane, the normal to the Moon's orbital plane, and the normal to the ecliptic plane, are coplanar vectors that are orientated such that the latter vector lies between the other two.

Law 2 states that the angle,
, subtended between
**
**
and
**
**
is fixed. Moreover, because the angles
and
are both small (when expressed in radians), we deduce that the vectors
**
**
and
are almost parallel to
**
**
.

Law 3 states that the vectors
**
**
,
**
**
, and
all lie in the same plane, with
**
**
and
on opposite sides of
**
**
. In other words, as the normal to the Moon's orbital plane,
, regresses about the
normal to the ecliptic plane,
**
**
, the normal to the Moon's equatorial plane,
**
**
, regresses at the same
rate, such that
**
**
is always directly opposite
with respect to
**
**
.

Cassini's first law was accounted for in the previous section. The ultimate aim of this section is to account for Cassini's second and
third laws. Our approach is largely based on that of Danby (1992). In order to simplify the analysis, we shall assume that the Moon orbits around the Earth, at the
uniform angular velocity,
, in a circular orbit of major radius
. When expressed in terms of the
,
,
coordinate system,
**
**
. Furthermore, because the unit vectors
**
**
and
are
almost parallel to
**
**
, we can write

where , , , . Similarly, because the unit vector is almost parallel to the -axis, we have

where , . The position vector, , of the center of the Earth with respect to the center of the Moon is written

Finally, given Cassini's first law, and assuming that the Moon's spin axis is almost parallel to the -axis, the Moon's spin angular velocity takes the form

Here, is a unit vector such that

where , .

According to Equations (8.150), (8.151), (8.153), and (8.154),

because and . Here,

Note that , because the Moon is almost spherically symmetric. To second order in small quantities, Equations (8.196) and (8.197) yield

where use has been made of Equations (8.192)-(8.195).

The unit
vector
**
**
is stationary in an inertial frame whose coordinate axes are fixed with respect to distant stars.
Hence, in the
,
,
body frame, which rotates with respect to the
aforementioned fixed frame at the angular velocity
**
**
,
we have (see Section 6.2)

(8.202) |

It follows, from Equations (8.190), (8.194), and (8.195), that

(8.203) | ||||||

and | (8.204) |

These expressions can be combined with Equations (8.200) and (8.201) to give

It now remains to express in terms of and .

By definition, is normal to , as the vector lies in the plane of the Moon's orbit. Hence, according to Equations (8.191) and (8.192),

(8.207) |

which implies that

Let
be the ascending node of the Earth's apparent orbit about the Moon (which implies that
is the
descending node of the Moon's actual orbit about the Earth), and
let
be a unit vector parallel to
. (See Figure 8.13 and Section 4.12.) By definition,
is normal to both
and
**
**
. In fact, we can write

(8.209) |

where is the angle subtended between the vectors and

Now,

(8.211) | ||||||

and | (8.212) |

where is the angle subtended between and . (See Figure 8.13.) Thus, Equations (8.191), (8.192), and (8.210) yield

and

(8.215) |

In fact, is the longitude of the Earth relative to the ascending node of its apparent orbit around the Moon. It follows that

where is the uniform regression rate of the Earth's apparent ascending node (which is the same as the regression rate of the true ascending node of the Moon's orbit around the Earth). Here, for the sake of simplicity, we have assumed that the Earth passes through its apparent ascending node at time . Hence, Equations (8.205), (8.206), (8.208), (8.213), and (8.216) can be combined to give

The previous two equations govern the Moon's physical libration in latitude. As is the case for libration in longitude, there are both free and forced modes. The free modes satisfy

(8.219) | ||||||

and | (8.220) |

Let us search for solutions of the form

(8.221) | ||||||

and | (8.222) |

where , , are constants. It follows that

(8.223) |

Given that and are both small compared to unity, two independent free libration modes can be derived from the preceding expression. The first mode is such that and , whereas the second is such that and . In the Moon's body frame, these modes cause the normal to the ecliptic plane,

(8.224) | ||||||

and | (8.225) |

where , , , , and the constants , , , are arbitrary. The observed values of , , and are per day, , and , respectively (Konopliv et al. 1998; Dickey et al. 1994). [Actually, and are measured by fitting observations of lunar libration obtained from laser ranging to the theory described here.] Thus, it follows that per day, per day, , and . In the body frame, the first mode causes

Let us now search for forced solutions of Equations (8.217) and (8.218) of the form

(8.226) | ||||||

and | (8.227) |

where , are constants. It follows that

(8.228) | ||||||

and | (8.229) |

Hence, recalling that , , and are all small compared to unity, we obtain the following mode of forced libration:

(8.230) | ||||||

and | (8.231) |

In the Moon's body frame, this mode causes the vectors

where

(8.234) |

and use has been made of Equations (8.213), (8.214), and (8.216). Because the observed values of , , and are , , and (Konopliv et al. 1998; Dickey et al. 1994; Yoder 1995), we deduce that

(8.235) |

According to Equation (8.232),

(8.236) |

Note that this angle would be zero in the absence of any regression of the Moon's orbital ascending node (i.e., if were zero). In other words, the nonzero angle of inclination between the Moon's equatorial and orbital planes is a direct consequence of this regression, which is ultimately due to the perturbing action of the Sun. Because the regression of the Moon's orbital ascending node is also responsible for the forced nutation of the Earth's axis of rotation (see Section 8.10), it follows that this nutation is closely related to the forced inclination between the Moon's equatorial and orbital planes.