trojan (celestial body)

File:InnerSolarSystem-en.png are seen in this graphic as Greek camp at {{L4|nolink=yes}} ahead of Jupiter and as Trojan camp at {{L5|nolink=yes}} trailing Jupiter along its orbital path. It also shows the asteroid belt between Mars and Jupiter and the Hilda asteroids.

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In 1772, the Italian–French mathematician and astronomer Joseph-Louis Lagrange obtained two constant-pattern solutions (collinear and equilateral) of the general three-body problem.{{Cite journal |last=Lagrange |first=Joseph-Louis |author-link=Joseph-Louis Lagrange |date=1772 |title=Essai sur le Problème des Trois Corps |trans-title=Essay on the Three-Body Problem |url=http://www.ltas-vis.ulg.ac.be/cmsms/uploads/File/Lagrange_essai_3corps.pdf |journal=Œuvres complètes |language=fr |volume=6 |pages=229–331 |archive-url=https://web.archive.org/web/20171222053004/http://www.ltas-vis.ulg.ac.be/cmsms/uploads/File/Lagrange_essai_3corps.pdf |archive-date=December 22, 2017}} In the restricted three-body problem, with one mass negligible (which Lagrange did not consider), the five possible positions of that mass are now termed Lagrange points.

The term "trojan" originally referred to the "trojan asteroids" (Jovian trojans) that orbit close to the Lagrangian points of Jupiter. These have long been named for figures from the Trojan War of Greek mythology. By convention, the asteroids orbiting near the {{L4|nolink=yes}} point of Jupiter are named for the characters from the Greek side of the war, whereas those orbiting near the {{L5|nolink=yes}} of Jupiter are from the Trojan side. There are two exceptions, named before the convention was adopted: 624 Hektor in the L4 group, and 617 Patroclus in the L5 group.{{cite journal |last=Wright |first=Alison |date=1 August 2011 |title=Planetary science: The Trojan is out there |journal=Nature Physics |volume=7 |issue=8 |page=592 |bibcode=2011NatPh...7..592W |doi=10.1038/nphys2061 |doi-access=free}}

Astronomers estimate that the Jovian trojans are about as numerous as the asteroids of the asteroid belt.{{cite journal |last1=Yoshida |first1=Fumi |last2=Nakamura |first2=Tsuko |title=Size distribution of faint L4 Trojan asteroids |year=2005 |journal=The Astronomical Journal |volume=130 |issue=6 |pages=2900–11 |doi=10.1086/497571 |bibcode=2005AJ....130.2900Y|doi-access=free }}

Later on, objects were found orbiting near the Lagrangian points of Neptune, Mars, Earth,{{cite journal |last1= Connors|first1= Martin|last2= Wiegert|first2= Paul|last3= Veillet|first3= Christian|title= Earth's Trojan asteroid|date= 27 July 2011|journal= Nature|volume= 475|pages= 481–483|doi= 10.1038/nature10233|issue= 7357|bibcode= 2011Natur.475..481C|pmid= 21796207|s2cid= 205225571}} Uranus, and Venus. Minor planets at the Lagrangian points of planets other than Jupiter may be called Lagrangian minor planets.{{cite journal |last1=Whiteley |first1=Robert J. |last2=Tholen |first2=David J. |title=A CCD Search for Lagrangian Asteroids of the Earth–Sun System |journal=Icarus |volume=136 |issue=1 |date=November 1998 |pages=154–167|bibcode=1998Icar..136..154W |doi=10.1006/icar.1998.5995 }}

• Four Martian trojans are known: 5261 Eureka, {{mpl|(101429) 1998 VF|31}}, {{mpl|(311999) 2007 NS|2}}, and {{mpl|(121514) 1999 UJ|7}} – the only Trojan body in the leading "cloud" at {{L4|nolink=yes}},{{cite web |url=http://www.minorplanetcenter.org/iau/lists/MarsTrojans.html |title=List of Martian Trojans |work=Minor Planet Center |access-date=3 July 2015}}{{cite journal |last1=de la Fuente Marcos |first1=C. |last2=de la Fuente Marcos |first2=R. |title=Three new stable L5 Mars Trojans |journal=Monthly Notices of the Royal Astronomical Society |department=Letters |volume=432 |issue=1 |pages=31–35 |date=15 May 2013 |doi=10.1093/mnrasl/slt028 |doi-access=free |arxiv=1303.0124|bibcode=2013MNRAS.432L..31D }} There seem to be, also, {{mpl|(385250) 2001 DH|47}}, {{mpl|2011 SC|191}}, and {{mpl|2011 UN|63}}, but these have not yet been accepted by the Minor Planet Center.
• There are 28 known Neptunian trojans,{{cite web |title=List of Neptune Trojans |work=Minor Planet Center |date=28 October 2018 |access-date=28 December 2018 |url=http://www.minorplanetcenter.org/iau/lists/NeptuneTrojans.html}} but the large Neptunian trojans are expected to outnumber the large Jovian trojans by an order of magnitude.{{cite journal |last1=Chiang |first1=Eugene I. |last2=Lithwick |first2=Yoram |title=Neptune Trojans as a Testbed for Planet Formation |journal=The Astrophysical Journal |volume=628 |issue=1 |pages=520–532 |doi=10.1086/430825 |arxiv=astro-ph/0502276 |date=20 July 2005|bibcode=2005ApJ...628..520C |s2cid=18509704 }}{{cite news

|date=30 January 2007

|title=Neptune May Have Thousands of Escorts

|first=David |last=Powell

|work=Space.com

|url=http://www.space.com/scienceastronomy/070130_st_neptune_trojans.html

}}

• {{mpl|2010 TK|7}} was confirmed to be the first known Earth trojan in 2011. It is located in the {{L4|nolink=yes}} Lagrangian point, which lies ahead of the Earth.{{cite news |last1=Choi |first1=Charles Q. |title=First Asteroid Companion of Earth Discovered at Last |url=http://www.space.com/12443-earth-asteroid-companion-discovered-2010-tk7.html |date=27 July 2011|work=Space.com |access-date=27 July 2011}} {{mpl|(614689) 2020 XL|5}} was found to be another Earth trojan in 2021. It is also at L4.{{cite journal|display-authors=etal |last1=Man-To Hui |title=The Second Earth Trojan 2020 XL5 |journal=Astrophysical Journal Letters |date=Nov 2021 |volume=922 |issue=2 |pages=L25 |doi=10.3847/2041-8213/ac37bf |issn=2041-8205 |arxiv=2111.05058 |bibcode=2021ApJ...922L..25H |s2cid=243860678 |doi-access=free }}{{cite journal |last1=Leah Crane |title=Trojan asteroid: Another object found that shares Earth's orbit |journal=New Scientist |date=Nov 22, 2021 |url=https://www.newscientist.com/article/2298442-astronomers-have-found-a-second-trojan-asteroid-sharing-earths-orbit/}}
• {{mpl|(687170) 2011 QF|99}} was identified as the first Uranus trojan in 2013. It is located at the {{L4|nolink=yes}} Lagrangian point. A second one, {{mpl|(636872) 2014 YX|49}}, was announced in 2017.{{cite journal

|last2=de la Fuente Marcos |first2=Raúl

|last1=de la Fuente Marcos |first1=Carlos

|title=Asteroid 2014 YX₄₉: a large transient Trojan of Uranus

|journal=Monthly Notices of the Royal Astronomical Society

|date=21 May 2017

|volume=467 |issue=2

|arxiv=1701.05541

|doi=10.1093/mnras/stx197 |pages=1561–1568|doi-access=free

|bibcode=2017MNRAS.467.1561D}}

• {{mpl|2013 ND|15}} is a temporary Venusian trojan, the first one to be identified.
• The large asteroids Ceres and Vesta have temporary trojans.{{cite journal |first1=Apostolos A. |last1=Christou |first2=Paul |last2=Wiegert |title=A population of main belt asteroids co-orbiting with Ceres and Vesta |journal=Icarus |volume=217 |issue=1 |date=January 2012 |pages=27–42 |arxiv=1110.4810 |doi=10.1016/j.icarus.2011.10.016|bibcode=2012Icar..217...27C |s2cid=59474402 }}
• Saturn has 1 known trojan in the L₄ Lagrangian Point, 2019 UO₁₄.{{Cite web |author1=Robert Lea |date=2024-10-24 |title=Saturn gets its 1st confirmed Trojan asteroid — but it might be stolen |url=https://www.space.com/saturn-first-trojan-2019uo14-asteroid-stolen |access-date=2024-10-26 |website=Space.com |language=en}}

Whether or not a system of star, planet, and trojan is stable depends on how large the perturbations are to which it is subject. If, for example, the planet is the mass of Earth, and there is also a Jupiter-mass object orbiting that star, the trojan's orbit would be much less stable than if the second planet had the mass of Pluto.

As a rule of thumb, the system is likely to be long-lived if m₁ > 100m₂ > 10,000m₃ (in which m₁, m₂, and m₃ are the masses of the star, planet, and trojan).

More formally, in a three-body system with circular orbits, the stability condition is 27(m₁m₂ + m₂m₃ + m₃m₁) < (m₁ + m₂ + m₃)². So the trojan being a mote of dust, m₃→0, imposes a lower bound on {{sfrac|m₁|m₂}} of {{sfrac|25+√621|2}} ≈ 24.9599. And if the star were hyper-massive, m₁→+∞, then under Newtonian gravity, the system is stable whatever the planet and trojan masses. And if {{sfrac|m₁|m₂}} = {{sfrac|m₂|m₃}}, then both must exceed 13+√168 ≈ 25.9615. However, this all assumes a three-body system; once other bodies are introduced, even if distant and small, stability of the system requires even larger ratios.

{{Wiktionary|Trojan|Trojan asteroid|Trojan moon|Trojan planet}}

• Lissajous orbit
• List of objects at Lagrange points
• Tadpole orbit
• Trojan wave packet

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