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.

Inclination to the invariable plane for the giant planets

Year	Jupiter	Saturn	Uranus	Neptune
2009[1]	0.32°	0.93°	1.02°	0.72°
142400[2]	0.48°	0.79°	1.04°	0.55°
168000[3]	0.23°	1.01°	1.12°	0.55°

Solar System

In the Solar System, about 98% of this effect is contributed by the orbital angular momenta of the four giant planets (Jupiter, Saturn, Uranus, and Neptune). The invariable plane is within 0.5° of the orbital plane of Jupiter,^[1] and may be regarded as the weighted average of all planetary orbital and rotational planes.

Terminology and definition

This plane is sometimes called the "Laplacian" or "Laplace plane" or the "invariable plane of Laplace", though it should not be confused with the Laplace plane, which is the plane about which the individual orbital planes of planetary satellites precess.^[4] Both derive from the work of (and are at least sometimes named for) the French astronomer Pierre-Simon Laplace.^[5] The two are equivalent only in the case where all perturbers and resonances are far from the precessing body. The invariable plane is derived from the sum of angular momenta, and is "invariable" over the entire system, while the Laplace plane for different orbiting objects within a system may be different. Laplace called the invariable plane the plane of maximum areas, where the "area" in this case is the product of the radius R and its time rate of change ⁠ d R / d t ⁠, that is, its radial velocity, multiplied by the mass.

Body		Inclination to
		Ecliptic	Sun's equator	Invariable plane[1]
Terre- strials	Mercury	7.01°	3.38°	6.34°
	Venus	3.39°	3.86°	2.19°
	Earth	0°	7.25°[6]	1.57°
	Mars	1.85°	5.65°	1.67°
Gas & ice giants	Jupiter	1.31°	6.09°	0.32°
	Saturn	2.49°	5.51°	0.93°
	Uranus	0.77°	6.48°	1.02°
	Neptune	1.77°	6.43°	0.72°
Minor planets	Pluto	17.14°	11.88°	15.55°
	Ceres	10.59°		9.20°
	Pallas	34.83°		34.21°
	Vesta	5.58°		7.13°

Description

The magnitude of the orbital angular momentum vector of a planet is ${\displaystyle L=R^{2}M{\dot {\theta }}}$, where ${\displaystyle R}$ is the orbital radius of the planet (from the barycenter), ${\displaystyle M}$ is the mass of the planet, and ${\displaystyle {\dot {\theta }}}$ is its orbital angular 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 R_☉ away from the barycenter when all jovian planets are in line on the 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, and further if there were no external effects due to the uneven gravitation of the Milky Way Galaxy, 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 orbital 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 momentum of the system, which is very nearly conserved despite these effects. For almost all purposes, the plane defined from the giant planets' orbits alone can be considered invariable when working in Newtonian dynamics, by also ignoring the even tinier amounts of angular momentum ejected in material and gravitational waves leaving the Solar System, and the extremely tiny torques exerted on the Solar System by other stars passing nearby, Milky Way galactic tides, etc.

References

1. Heider, K.P. (3 April 2009). "The mean plane (invariable plane) of the Solar System passing through the barycenter". Archived from the original on 3 June 2013. Retrieved 10 April 2009.

   produced using
   

   Vitagliano, Aldo. "Solex 10" (computer program). Università degli Studi di Napoli Federico II. Archived from the original on 2015-05-24. Retrieved 2010-11-23.
2. "MeanPlane (invariable plane) for 142400/01/01". 8 April 2009. Archived from the original on 3 June 2013. Retrieved 10 April 2009. (produced with Solex 10)
3. "MeanPlane (invariable plane) for 168000/01/01". 6 April 2009. Archived from the original on 3 June 2013. Retrieved 10 April 2009. (produced with Solex 10)
4. Tremaine, S.; Touma, J.; Namouni, F. (2009). "Satellite dynamics on the Laplace surface". The Astronomical Journal. 137 (3): 3706–3717. arXiv:0809.0237. Bibcode:2009AJ....137.3706T. doi:10.1088/0004-6256/137/3/3706. S2CID 18901505.
5. La Place, P.-S., Marquis de (1829) [1799–1825]. Celestial Mechanics. Translated by Bowditch, Nathaniel. Boston, MA. volume I, chapter V, esp. page 121.{{cite book}}: CS1 maint: multiple names: authors list (link) — English translation published in four volumes, 1829–1839;

   originally published as
   

   La Place, P.-S., Marquis de (1799–1825). Traité de mécanique céleste [Treatise on Celestial Mechanics] (in French).{{cite book}}: CS1 maint: multiple names: authors list (link) in five volumes.
6. Planetary Fact Sheets, at http://nssdc.gsfc.nasa.gov

Further reading

• Souami, D.; Souchay, J. (2012). "The solar system's invariable plane" (PDF). Astronomy and Astrophysics. 543: A133. Bibcode:2012A&A...543A.133S. doi:10.1051/0004-6361/201219011.