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invariable

Invariable plane

The invariable plane of a planetary system is the plane passing through its barycenter (center of mass) which is perpendicular to its angular momentum vector. In the Solar system, about 98% of this effect is contributed by the orbital angular momenta of the four jovian planets (Jupiter, Saturn, Uranus, and Neptune). It is also called the Laplacian plane (not to be confused with the similarly-named Laplace plane) after the French astronomer who first recognized it, Pierre Simon Laplace. The invariable plane is within 0.5° of the orbital plane of Jupiter, and may be regarded as the weighted average of all planetary orbital planes.

The magnitude of the orbital angular momentum vector of a planet is L = RMV, where R is the orbital radius of the planet (from the barycenter), M is the mass of the planet, and V is its orbital velocity. That of Jupiter contributes the bulk of the solar system's angular momentum, 60.3%. Then come Saturn at 24.5%, Neptune at 7.9%, and Uranus at 5.3%. The Sun forms a counterbalance to all of the planets, so the barycenter moves from the center of the Sun when Jupiter is on one side and the other three jovian planets are diametrically opposite on the other side of the Sun, to 2.17 solar radii away from the center of the Sun when all are in line on one side (admittedly unlikely). The orbital angular momenta of the Sun and all non-jovian planets, moons, and minor solar system bodies, as well as the axial rotation momenta of all bodies, including the Sun, total only about 2%.

If all solar system bodies were point masses, or were rigid bodies having spherically symmetric mass distributions, then the invariable plane would be truly invariable and would constitute an inertial frame of reference. But almost all are not, allowing the transfer of a very small amount of momenta from axial rotations to orbital revolutions due to tidal friction. Although this causes no change in the magnitude of the angular momentum, it causes a slight change in its direction because the rotational axes are not parallel to the orbital axes. Precession of the various spin axes also cause a slight change in its direction. Nevertheless, these changes are infinitesimal relative to the total angular momenta of the system, so for all intents and purposes the plane is invariable.

All planetary orbital planes wobble around the invariable plane, meaning that they rotate around its axis while their inclinations to it vary, both of which are caused by the gravitational influence of the other planets. That of Earth rotates with a quasi-period of 100,000 years and an inclination which varies from 0.1° to 3°. If long term calculations are performed relative to the present ecliptic, which is inclined to the invariable plane by about 1.5°, it appears to rotate with a period of 70,000 years and an inclination that varies between 0° and 4°. Specifically, Earth's orbit is inclined to the invariable plane by 1°34'59"−18"T, where T is the number of centuries since 1900. The inclination of the orbit of Jupiter to the invariable plane varies over the range of 14'–28'.

Laplace called the invariable plane the plane of maximum areas, where the area is the product of the radius and its differential time change dR/dt, that is, its velocity, multiplied by the mass.

References

  • La Place, Marquis de (Pierre Simon Laplace). Mécanique Céleste, translated by Nathaniel Bowditch. Boston: 1829, in four volumes (1829–1839). See volume I, chapter V, especially page 121. Originally published as Traite de mécanique céleste (Treatise on Celestial Mechanics) in five volumes, 1799–1825.
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