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Inclination to the invariable plane for the giant planets
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 momentumvector. 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,[1] 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 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 FrenchastronomerPierre 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 dR/dt, that is, its radial velocity, multiplied by the mass.
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 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.
ReferencesEdit
^ abc Heider, K.P. (3 April 2009). . Archived from the original on 3 June 2013. Retrieved 10 April 2009.
^ . 8 April 2009. Archived from the original on 3 June 2013. Retrieved 10 April 2009. (produced with Solex 10)
^ . 6 April 2009. Archived from the original on 3 June 2013. Retrieved 10 April 2009. (produced with Solex 10)
^ 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.
^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;
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.
August 25, 2023
invariable, plane, this, article, should, divided, into, sections, topic, make, more, accessible, please, help, adding, section, headings, accordance, with, wikipedia, manual, style, august, 2023, inclination, invariable, plane, giant, planets, year, jupiter, . This article should be divided into sections by topic to make it more accessible Please help by adding section headings in accordance with Wikipedia s Manual of Style August 2023 Inclination to the invariable plane for the giant planets Year Jupiter Saturn Uranus Neptune2009 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 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 1 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 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 dR dt that is its radial velocity multiplied by the mass Body Inclination toEcliptic Sun s equator Invariable plane 1 Terre strials Mercury 7 01 3 38 6 34 Venus 3 39 3 86 2 19 Earth 0 7 155 1 57 Mars 1 85 5 65 1 67 Gas amp icegiants 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 Minorplanets Pluto 17 14 11 88 15 55 Ceres 10 59 9 20 Pallas 34 83 34 21 Vesta 5 58 7 13 Description EditThe magnitude of the orbital angular momentum vector of a planet is L R 2 M 8 displaystyle L R 2 M dot theta where R displaystyle R is the orbital radius of the planet from the barycenter M displaystyle M is the mass of the planet and 8 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 Edit a b c 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 usingVitagliano Aldo Solex 10 computer program Universita degli Studi di Napoli Federico II a href Template Cite web html title Template Cite web cite web a CS1 maint url status link 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 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 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 La Place P S Marquis de 1829 1799 1825 Celestial Mechanics Translated by Bowditch Nathaniel Boston MA volume I chapter V esp page 121 a href Template Cite book html title Template Cite book cite book a CS1 maint multiple names authors list link English translation published in four volumes 1829 1839 originally published asLa Place P S Marquis de 1799 1825 Traite de mecanique celeste Treatise on Celestial Mechanics in French a href Template Cite book html title Template Cite book cite book a CS1 maint multiple names authors list link in five volumes Further reading EditSouami D Souchay J 2012 The solar system s invariable plane PDF Astronomy and Astrophysics 543 A133 Bibcode 2012A amp A 543A 133S doi 10 1051 0004 6361 201219011 Retrieved from https en wikipedia org w index php title Invariable plane amp oldid 1171741402, wikipedia, 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