Cassini's laws

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Cassini's laws provide a compact description of the motion of the Moon. They were established in 1693 by Giovanni Domenico Cassini, a prominent scientist of his time.[1]

Refinements of these laws to include physical librations have been made,[1] and they have been generalized to treat other satellites and planets.[2][3][4]

Cassini's laws

Orbital inclination and rotation. When the Moon is 5.14° north of the ecliptic, its north pole is tilted 6.68° away from the Earth. The orientation of the plane containing the vectors normal to the orbits and the Moon's rotational axis rotates 360° with a period of about 18.6 years, whereas the Earth's axis precesses with a period of around 26,000 years, so the line-up of this illustration (a major lunar standstill) occurs only once every 18.6 years.
  1. The Moon has a 1:1 spin–orbit resonance. This means that the rotation–orbit ratio of the Moon is such that the same side of it always faces the Earth.
  2. The Moon's rotational axis maintains a constant angle of
    ecliptic plane
    . The Moon's rotational axis precesses so as to trace out a cone that intersects the ecliptic plane as a circle.
  3. A plane formed from a
    orbital plane
    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
. 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
.

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

obliquity remains constant.[2][3][5] The Laplace plane is defined as the plane about which a planet or satellite orbit precesses with constant inclination.[5] 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
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.[6] 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

retrograde (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 Casssini state.[7]

See also

References and notes

  1. ^
    ISBN 3-7643-5866-1
    .
  2. ^
    ISSN 0004-6256
    .
  3. ^
    S2CID 8795467
    .
  4. ISBN 3-7643-5866-1
    .
  5. ^
    ISBN 978-0-387-48339-9
    .
  6. ^ J. N. Winn and M. J. Holman (2005),"Obliquity Tides on Hot Jupiters", The Astrophysical Journal, Volume 628, Issue 2, pp. L159-L162.
  7. S2CID 12049556
    . Based on work by G. Colombo in 1966.

Further reading
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