Cassini's Laws
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Cassini's laws provide a compact description of the motion of the
Moon The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...
. They were established in 1693 by
Giovanni Domenico Cassini Giovanni Domenico Cassini (8 June 1625 – 14 September 1712) was an Italian-French mathematician, astronomer, astrologer and engineer. Cassini was born in Perinaldo, near Imperia, at that time in the County of Nice, part of the Savoyard sta ...
, a prominent scientist of his time.For the original statement of the laws, see Refinements of these laws to include physical
libration In lunar astronomy, libration is the cyclic variation in the apparent position of the Moon that is perceived by observers on the Earth and caused by changes between the orbital and rotational planes of the moon. It causes an observer to see ...
s have been made, and they have been generalized to treat other satellites and planets.


Cassini's laws

# The Moon has a 1:1 spin–orbit resonance. This means that the
rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
ratio of the Moon is such that the same side of it always faces the
Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
. # The Moon's rotational axis maintains a constant angle of
inclination Orbital inclination measures the tilt of an object's orbit around a celestial body. It is expressed as the angle between a reference plane and the orbital plane or axis of direction of the orbiting object. For a satellite orbiting the Eart ...
from the
ecliptic plane The ecliptic or ecliptic plane is the orbital plane of Earth around the Sun. It was a central concept in a number of ancient sciences, providing the framework for key measurements in astronomy, astrology and calendar-making. From the perspec ...
. The Moon's rotational axis precesses so as to trace out a cone that intersects the ecliptic plane as a circle. # A plane formed from a normal to the ecliptic plane and a normal to the Moon's
orbital plane The orbital plane of a revolving body is the geometric plane in which its orbit lies. Three non-collinear points in space suffice to determine an orbital plane. A common example would be the positions of the centers of a massive body (host) a ...
will contain the Moon's rotational axis. In the case of the Moon, its rotational axis always points some 1.5 degrees away from the North
ecliptic pole An orbital pole is either point at the ends of the orbital normal, an imaginary line segment that runs through a focus of an orbit (of a revolving body like a planet, moon or satellite) and is perpendicular (or normal) to the orbital plane. Pr ...
. The normal to the Moon's orbital plane and its rotational axis are always on opposite sides of the normal to the ecliptic. Therefore, both the normal to the orbital plane and the Moon's rotational axis precess around the ecliptic pole with the same period. The period is about 18.6 years and the motion is
retrograde Retrograde may refer to: Film and television * Retrograde (2004 film), ''Retrograde'' (2004 film), a film by Christopher Kulikowski * Retrograde (2022 American film), ''Retrograde'' (2022 American film), a documentary film by Matthew Heineman * ...
.


Cassini state

A system obeying these laws is said to be in a Cassini state, that is: an evolved rotational state where the spin axis, orbit normal, and normal to the
Laplace plane The Laplace plane or Laplacian plane of a planetary satellite, named after its discoverer Pierre-Simon Laplace (1749–1827), is a mean or reference plane about whose axis the instantaneous orbital plane of that satellite precesses. Laplace's name ...
are coplanar while the
obliquity In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orbital ...
remains constant. The Laplace plane is defined as the plane about which a planet or satellite orbit precesses with constant inclination. The normal to the Laplace plane for a moon is between the planet's spin axis and the planet's orbit normal, being closer to the latter if the moon is distant from the planet. If a planet itself is in a Cassini state, the Laplace plane is the
invariable plane The invariable plane of a planetary system, also called Laplace's invariable plane, is the plane passing through its barycenter (center of mass) perpendicular to its angular momentum vector. Solar System In the Solar System, about 98% of this e ...
of the stellar system. Cassini state 1 is defined as the situation in which both the spin axis and the orbit normal axis are on the same side of the normal to the Laplace plane. Cassini state 2 is defined as the case in which the spin axis and the orbit normal axis are on opposite sides of the normal to the Laplace plane. Earth's Moon is in Cassini state 2. In general, the spin axis moves in the direction perpendicular to both itself and the orbit normal, due to the
tidal force The tidal force or tide-generating force is the difference in gravitational attraction between different points in a gravitational field, causing bodies to be pulled unevenly and as a result are being stretched towards the attraction. It is the ...
exerted by the object being orbited (planet or star) and other objects in the system. (In the case of the Moon, its spin axis moves mostly under the influence of the Earth, whereas the smaller tidal influence of the Sun works in the same direction at full moon and in the opposite direction at new moon and is thus negligible.) The rate of movement of the spin axis goes to zero if the spin axis coincides with the orbit normal. If the orbit normal precesses in a regular circular motion (due to tidal influences from other objects, such as the Sun in the case of the Moon), it is possible to characterize the solutions to the differential equation for the motion of the spin axis. It turns out that the spin axis traces out loops on the unit sphere that rotates at the speed of the orbital precession (so that the orbit normal and the normal to the Laplace plane are fixed points in the sphere). With certain values of the parameters, there are three areas on the sphere in each of which we have circulation around a point inside the area where the spin axis doesn't move (in this rotating frame of reference). These points are Cassini states 1 and 2 and a third Cassini state in which the rotation is
retrograde Retrograde may refer to: Film and television * Retrograde (2004 film), ''Retrograde'' (2004 film), a film by Christopher Kulikowski * Retrograde (2022 American film), ''Retrograde'' (2022 American film), a documentary film by Matthew Heineman * ...
(which would not apply to a moon like ours that is tidally locked). The three areas are separated by a separatrix that crosses itself, and the point where it crosses itself is the unstable Cassini state 4. (Under other parameter values only states 2 and 3 exist, and there is no separatrix.) If an object flexes and dissipates kinetic energy, then these solutions are not exact and the system will slowly evolve and approach a stable Cassini state. This has happened with the Moon. It has reached a state with a constant obliquity of 6.7°, at which the precession of the spin axis takes the same 18.6 years as taken by the precession of the orbit normal, and is thus in a Cassini state.See Based on work by G. Colombo in 1966.


See also

*
Orbit of the Moon The Moon orbits Earth in the retrograde and prograde motion, prograde direction and completes one orbital period, revolution relative to the March Equinox, Vernal Equinox and the fixed stars in about 27.3 days (a tropical month and sidereal mont ...


References and notes


Further reading


Cassini Laws – from Eric Weisstein's World of Physics
*{{cite journal , author=Eckhardt, Donald H. , year=1981 , title=Theory of the Libration of the Moon , journal=Earth, Moon, and Planets , volume=25 , issue=1 , pages=3–49 , publisher=
Springer Netherlands Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...
, doi=10.1007/BF00911807, bibcode = 1981M&P....25....3E , s2cid=123650603
Cassini's 3 laws
Astrophysics Eponymous laws of physics
Laws Law is a set of rules that are created and are law enforcement, enforceable by social or governmental institutions to regulate behavior, with its precise definition a matter of longstanding debate. It has been variously described as a Socia ...