Laplace plane
Encyclopedia
The Laplace plane or Laplacian plane of a planetary satellite, named after its discoverer Pierre-Simon Laplace
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...

 (1749–1827), is a mean or reference plane about whose axis the instantaneous orbital plane
Orbital plane (astronomy)
All of the planets, comets, and asteroids in the solar system are in orbit around the Sun. All of those orbits line up with each other making a semi-flat disk called the orbital plane. The orbital plane of an object orbiting another is the geometrical plane in which the orbit is embedded...

 of a satellite
Satellite
In the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavour. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....

 precesses
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...

.

This plane is sometimes called the satellite's "invariable plane", but the latter term more often refers
Invariable plane
The invariable plane of a planetary system, also called Laplace's invariable plane, is the plane passing through its barycenter 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...

 to the related concept of the sum of all angular momentum vectors in a system. The two are equivalent only in the case where all perturbers
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...

 and resonances
Orbital resonance
In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other, usually due to their orbital periods being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of...

 are far from the precessing body.

Definition

The axis of this Laplace plane is coplanar with, and between, (a) the polar axis of the parent planet's spin, and (b) the orbital axis of the parent planet's orbit around the Sun. The Laplace plane arises because the equatorial oblateness of the parent planet tends to cause the orbit of the satellite to precess around the polar axis of the parent planet's equatorial plane, while the solar perturbations tend to cause the orbit of the satellite to precess around the polar axis of the parent planet's orbital plane around the Sun. The two effects acting together result in an intermediate position for the reference axis for the satellite orbit's precession.

Explanation

In effect, this is the plane normal to the orbital precession pole of the satellite. It is a kind of "average orbital plane" of the satellite, around which the instantaneous orbital plane of the satellite precesses
Precession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...

, and to which it has a constant additional inclination.

In most cases, the Laplace plane is very close to the equatorial plane of its primary planet (if the satellite is very close to its planet) or to the plane of the primary planet's orbit around the Sun (if the satellite is far away from its planet). This is because the strength of the planet's perturbation on the satellite's orbit is much stronger for orbits close to the planet, but drops below the strength of the Sun's perturbation for orbits farther away. Examples of satellites whose Laplace plane is close to their planet's equatorial plane include the satellites of Mars and the inner satellites of the giant planets. Examples of satellites whose Laplace plane is close to their planet's orbital plane include Earth's Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

 and the outer satellites of the giant planets. Some satellites, such as Saturn's Iapetus
Iapetus (moon)
Iapetus ), occasionally Japetus , is the third-largest moon of Saturn, and eleventh in the Solar System. It was discovered by Giovanni Domenico Cassini in 1671...

, are situated in the transitional zone and have Laplace planes that are midway between their planet's equatorial plane and the plane of its solar orbit.

So the varying positions of the Laplace plane at varying distances from the primary planet can be pictured as putting together a warped or non-planar surface, which may be pictured as a series of concentric rings whose orientation in space is variable: the innermost rings are near the equatorial plane of rotation
Plane of rotation
In geometry, a plane of rotation is an abstract object used to describe or visualise rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, such as two, four or more dimensions.Mathematically such...

 and oblateness of the planet, and the outermost rings near its solar orbital plane. Also, in some cases, larger satellites of a planet (such as Neptune's Triton
Triton (moon)
Triton is the largest moon of the planet Neptune, discovered on October 10, 1846, by English astronomer William Lassell. It is the only large moon in the Solar System with a retrograde orbit, which is an orbit in the opposite direction to its planet's rotation. At 2,700 km in diameter, it is...

) can affect the Laplace planes of smaller satellites orbiting the same planet.

The work of Laplace

The Laplace or Laplacean plane, as discussed here, relates to the orbit of a planetary satellite. It is to be distinguished from another and quite different plane, also discovered by Laplace, and which is also sometimes called the "Laplacian" or "Laplace plane", but more often the invariable plane
Invariable plane
The invariable plane of a planetary system, also called Laplace's invariable plane, is the plane passing through its barycenter 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...

 (or the "invariable plane of Laplace"). The invariable plane is simply derived from the sum of angular momenta, and is "invariable" over the entire system, while the Laplace plane may be different for different orbiting objects within a system. Confusingly, a satellite's Laplace plane (as defined here) is also sometimes called its "invariable plane".

The Laplace plane is a result of perturbational effects, which were discovered by Laplace while he was investigating the orbits of Jupiter’s principal moons (the Galilean satellites of Jupiter). Laplace found that the effects of the solar perturbing force, and of the planet’s oblateness (its equatorial bulge), together gave rise to an "inclinaison propre", an "own inclination", in the plane of the satellite orbits, relative to the plane of Jupiter’s equator.
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