Invariable plane
Encyclopedia
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. 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
). 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 and rotational planes.
This plane is sometimes called the "Laplacian" or "Laplace plane" or the "invariable plane of Laplace", though the Laplace plane
more often refers to the related concept of the plane about which orbital planes
precess. The two should not be confused, though both derive from the work of (and are at least sometimes named for) the French
astronomer
Pierre Simon Laplace. The two are equivalent only in the case where all perturbers
and resonances
are far from the precessing body. 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. 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.
, where 
is the orbital radius of the planet (from the barycenter), 
is the mass of the planet, and 
is its orbital velocity. That of Jupiter contributes the bulk of the Solar System's angular momentum, 60.3%. Then comes Saturn at 24.5%, Neptune at 7.9%, and Uranus at 5.3%. The Sun
forms a counterbalance to all of the planets, so it is near the barycenter when Jupiter is on one side and the other three jovian planets are diametrically opposite on the other side, but the Sun moves to 2.17 solar radii away from the barycenter when all jovian planets are in line on other side. The orbital angular momenta of the Sun and all non-jovian planets, moons, and small 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 an invariable plane defined on orbits alone 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 and to bodies being non-spherical. This causes a change in the magnitude of the angular momentum, as well as a change in its direction (precession) because the rotational axes are not parallel to the orbital axes. Nevertheless, these changes are exceedingly small compared to the total angular momenta of the system, and for almost all purposes the plane can be considered invariable when working in Newtonian dynamics
.
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 (the ecliptic) is inclined to the invariable plane by 1°34'59"−18"T, where T is the number of centuries since 1900. Its J2000.0 value is 1°34'43.3". The inclination of the orbit of Jupiter to the invariable plane varies over the range of 14'–28'.
Planetary system
A planetary system consists of the various non-stellar objects orbiting a star such as planets, dwarf planets , asteroids, meteoroids, comets, and cosmic dust...
, also called Laplace's invariable plane, is the plane passing through its barycenter (center of mass) perpendicular to its angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
vector. In the Solar System
Solar System
The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...
, about 98% of this effect is contributed by the orbital angular momenta of the four jovian planets (Jupiter
Jupiter
Jupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...
, Saturn
Saturn
Saturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...
, Uranus
Uranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...
, and Neptune
Neptune
Neptune is the eighth and farthest planet from the Sun in the Solar System. Named for the Roman god of the sea, it is the fourth-largest planet by diameter and the third largest by mass. Neptune is 17 times the mass of Earth and is slightly more massive than its near-twin Uranus, which is 15 times...
). 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 and rotational planes.
This plane is sometimes called the "Laplacian" or "Laplace plane" or the "invariable plane of Laplace", though the Laplace plane
Laplace plane
The Laplace plane or Laplacian plane of a planetary satellite, named after its discoverer Pierre-Simon Laplace , is a mean or reference plane about whose axis the instantaneous orbital plane of a satellite precesses....
more often refers to the related concept of the plane about which orbital planes
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...
precess. The two should not be confused, though both derive from the work of (and are at least sometimes named for) the French
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
Pierre Simon Laplace. 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. 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. 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.
Description
The magnitude of the orbital angular momentum vector of a planet is, where is the orbital radius of the planet (from the barycenter), is the mass of the planet, and is its orbital velocity. That of Jupiter contributes the bulk of the Solar System's angular momentum, 60.3%. Then comes Saturn at 24.5%, Neptune at 7.9%, and Uranus at 5.3%. The SunSun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
forms a counterbalance to all of the planets, so it is near the barycenter when Jupiter is on one side and the other three jovian planets are diametrically opposite on the other side, but the Sun moves to 2.17 solar radii away from the barycenter when all jovian planets are in line on other side. The orbital angular momenta of the Sun and all non-jovian planets, moons, and small Solar System bodies
Small solar system body
A small Solar System body is an object in the Solar System that is neither a planet nor a dwarf planet, nor a satellite of a planet or dwarf planet:...
, 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 an invariable plane defined on orbits alone 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 and to bodies being non-spherical. This causes a change in the magnitude of the angular momentum, as well as a change in its direction (precession) because the rotational axes are not parallel to the orbital axes. Nevertheless, these changes are exceedingly small compared to the total angular momenta of the system, and for almost all purposes the plane can be considered invariable when working in Newtonian dynamics
Newtonian dynamics
In physics, the Newtonian dynamics is understood as the dynamics of a particle or a small body according to Newton's laws of motion.-Mathematical generalizations:...
.
Position
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 perturbationPerturbation (astronomy)
Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....
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
Ecliptic
The ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...
, 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 (the ecliptic) is inclined to the invariable plane by 1°34'59"−18"T, where T is the number of centuries since 1900. Its J2000.0 value is 1°34'43.3". The inclination of the orbit of Jupiter to the invariable plane varies over the range of 14'–28'.