Cassini's Laws

Cassini’s Laws are three empirical laws that describe the rotation of the Moon about its center of mass. These were formulated in 1693 by Gian Domenico Cassini.

First law: Rotation rate synchronous with mean orbital rate

The first law states that the Moon has a spin-orbit resonance of 1:1. This means that it rotates uniformly about its own axis once in the same time that it takes to revolve around the Earth. In other words, the Moon rotates about a fixed axis with constant angular velocity in a period of rotation that is equal to its mean sidereal period (sidereal period refers to the orbit time measured relative to stars). This accounts for the fact that virtually the same portion of the Moon (the “near side”) is always turned toward Earth.

Second law: Constant inclination of Moon’s equator to ecliptic

The ecliptic refers the apparent great-circle annual path of the sun in the celestial sphere (the imaginary sphere containing the celestial bodies). The second law states that the Moon’s equator (or rotational axis) has a constant angle of inclination from the ecliptic plane (or ecliptic pole). The angle of inclination of the Moon’s equator to the ecliptic plane is about 1.53 degrees of arc. This law implies that at any given time, the spin axis of the Moon is somewhere on a cone centered on a normal to the ecliptic plane.

Third law: The Moon’s spin axis, orbit normal, and ecliptic normal are coplanar

Consider a normal to the ecliptic plane and originating from the same point, a normal to the orbital plane of the Moon. These two vectors will describe a third plane that is perpendicular to the first two. The third law states that, at any given moment, the spin axis of the Moon will always be in this third plane. This means that the Moon's equator intersects the ecliptic right at the lunar nodes: the "ascending" node of the lunar equator resides at the “descending node” of the lunar orbit.

Cassini’s Laws imply that the Moon's spin axis has a precessional period (i.e. the time taken for the axis to "wobble" once in a circle) equal to the precessional period of the orbital plane of the Moon with respect to the ecliptic. This period is a little more than 18 years.

It should be noted that Cassini’s Laws are not entirely accurate. The Moon’s rotation deviates slightly from these laws; this deviation is known as “physical libration”. The physical libration is always less than a few hundredths of a degree.

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