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Trojan (celestial body)

December 18, 2023

In astronomy, a trojan is a small celestial body (mostly asteroids) that shares the orbit of a larger body, remaining in a stable orbit approximately 60° ahead of or behind the main body near one of its Lagrangian points L4 and L5. Trojans can share the orbits of planets or of large moons.

The trojan points are located on the L4 and L5 Lagrange points, on the orbital path of the secondary object (blue), around the primary object (yellow). All of the Lagrange points are highlighted in red.

Trojans are one type of co-orbital object. In this arrangement, a star and a planet orbit about their common barycenter, which is close to the center of the star because it is usually much more massive than the orbiting planet. In turn, a much smaller mass than both the star and the planet, located at one of the Lagrangian points of the star–planet system, is subject to a combined gravitational force that acts through this barycenter. Hence the smallest object orbits around the barycenter with the same orbital period as the planet, and the arrangement can remain stable over time.[1]

In the Solar System, most known trojans share the orbit of Jupiter. They are divided into the Greek camp at L4 (ahead of Jupiter) and the Trojan camp at L5 (trailing Jupiter). More than a million Jupiter trojans larger than one kilometer are thought to exist,[2] of which more than 7,000 are currently catalogued. In other planetary orbits only nine Mars trojans, 28 Neptune trojans, two Uranus trojans, and two Earth trojans, have been found to date. A temporary Venus trojan is also known. Numerical orbital dynamics stability simulations indicate that Saturn probably does not have any primordial trojans.[3]

The same arrangement can appear when the primary object is a planet and the secondary is one of its moons, whereby much smaller trojan moons can share its orbit. All known trojan moons are part of the Saturn system. Telesto and Calypso are trojans of Tethys, and Helene and Polydeuces of Dione.

Trojan minor planets edit

 
The Jupiter trojans are seen in this graphic as Greek camp at L4 ahead of Jupiter and as Trojan camp at L5 trailing Jupiter along its orbital path. It also shows the asteroid belt between Mars and Jupiter and the Hilda asteroids.
  Jupiter trojans   Asteroid belt   Hilda asteroids

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.[4] 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 point of Jupiter are named for the characters from the Greek side of the war, whereas those orbiting near the L5 of Jupiter are from the Trojan side. There are two exceptions, which were named before the convention was put in place, the Greek 624 Hektor and the Trojan 617 Patroclus.[5]

Astronomers estimate that the Jovian trojans are about as numerous as the asteroids of the asteroid belt.[6]

Later on, objects were found orbiting near the Lagrangian points of Neptune, Mars, Earth,[7] Uranus, and Venus. Minor planets at the Lagrangian points of planets other than Jupiter may be called Lagrangian minor planets.[8]

Trojans by planet edit

Planet Number in L4 Number in L5 List (L4) List (L5)
Mercury 0 0 — —
Venus 1 0 2013 ND15 —
Earth 2 0 2010 TK7, 2020 XL5 —
Mars 1 13 (121514) 1999 UJ7 many
Jupiter 7508 4044 many many
Saturn 0 0 — —
Uranus 2 0 2011 QF99, 2014 YX49 —
Neptune 24 4 many many

Stability edit

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 m1/m2 of 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 m1/m2 = 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.

See also edit

 
Look up Trojan, Trojan asteroid, Trojan moon, or Trojan planet in Wiktionary, the free dictionary.

References edit

  1. ^ Robutel, Philippe; Souchay, Jean (2010), "An introduction to the dynamics of trojan asteroids", in Dvorak, Rudolf; Souchay, Jean (eds.), Dynamics of Small Solar System Bodies and Exoplanets, Lecture Notes in Physics, vol. 790, Springer, p. 197, ISBN 978-3-642-04457-1
  2. ^ Yoshida, F.; Nakamura, T. (December 2005). "Size Distribution of Faint Jovian L4 Trojan Asteroids". The Astronomical Journal. 130 (6): 2900–2911. Bibcode:2005AJ....130.2900Y. doi:10.1086/497571.
  3. ^ Sheppard, Scott S.; Trujillo, Chadwick A. (June 2006). "A Thick Cloud of Neptune Trojans and their Colors". Science. 313 (5786): 511–514. Bibcode:2006Sci...313..511S. doi:10.1126/science.1127173. PMID 16778021. S2CID 35721399.
  4. ^ Lagrange, Joseph-Louis (1772). [Essay on the Three-Body Problem] (PDF) (in French). Archived from the original (PDF) on 22 December 2017. {{cite journal}}: Cite journal requires |journal= (help)
  5. ^ Wright, Alison (1 August 2011). "Planetary science: The Trojan is out there". Nature Physics. 7 (8): 592. Bibcode:2011NatPh...7..592W. doi:10.1038/nphys2061.
  6. ^ Yoshida, Fumi; Nakamura, Tsuko (2005). "Size distribution of faint L4 Trojan asteroids". The Astronomical Journal. 130 (6): 2900–11. Bibcode:2005AJ....130.2900Y. doi:10.1086/497571.
  7. ^ Connors, Martin; Wiegert, Paul; Veillet, Christian (27 July 2011). "Earth's Trojan asteroid". Nature. 475 (7357): 481–483. Bibcode:2011Natur.475..481C. doi:10.1038/nature10233. PMID 21796207. S2CID 205225571.
  8. ^ Whiteley, Robert J.; Tholen, David J. (November 1998). "A CCD Search for Lagrangian Asteroids of the Earth–Sun System". Icarus. 136 (1): 154–167. Bibcode:1998Icar..136..154W. doi:10.1006/icar.1998.5995.
  9. ^ "List of Martian Trojans". Minor Planet Center. Retrieved 3 July 2015.
  10. ^ de la Fuente Marcos, C.; de la Fuente Marcos, R. (15 May 2013). "Three new stable L5 Mars Trojans". Letters. Monthly Notices of the Royal Astronomical Society. 432 (1): 31–35. arXiv:1303.0124. Bibcode:2013MNRAS.432L..31D. doi:10.1093/mnrasl/slt028.
  11. ^ "List of Neptune Trojans". Minor Planet Center. 28 October 2018. Retrieved 28 December 2018.
  12. ^ Chiang, Eugene I.; Lithwick, Yoram (20 July 2005). "Neptune Trojans as a Testbed for Planet Formation". The Astrophysical Journal. 628 (1): 520–532. arXiv:astro-ph/0502276. Bibcode:2005ApJ...628..520C. doi:10.1086/430825. S2CID 18509704.
  13. ^ Powell, David (30 January 2007). "Neptune May Have Thousands of Escorts". Space.com.
  14. ^ Choi, Charles Q. (27 July 2011). "First Asteroid Companion of Earth Discovered at Last". Space.com. Retrieved 27 July 2011.
  15. ^ Man-To Hui; et al. (November 2021). "The Second Earth Trojan 2020 XL5". Astrophysical Journal Letters. 922 (2): L25. arXiv:2111.05058. Bibcode:2021ApJ...922L..25H. doi:10.3847/2041-8213/ac37bf. ISSN 2041-8205. S2CID 243860678.
  16. ^ Leah Crane (22 November 2021). "Trojan asteroid: Another object found that shares Earth's orbit". New Scientist.
  17. ^ de la Fuente Marcos, Carlos; de la Fuente Marcos, Raúl (21 May 2017). "Asteroid 2014 YX49: a large transient Trojan of Uranus". Monthly Notices of the Royal Astronomical Society. 467 (2): 1561–1568. arXiv:1701.05541. Bibcode:2017MNRAS.467.1561D. doi:10.1093/mnras/stx197.
  18. ^ Christou, Apostolos A.; Wiegert, Paul (January 2012). "A population of main belt asteroids co-orbiting with Ceres and Vesta". Icarus. 217 (1): 27–42. arXiv:1110.4810. Bibcode:2012Icar..217...27C. doi:10.1016/j.icarus.2011.10.016. S2CID 59474402.

December 18, 2023
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In astronomy a trojan is a small celestial body mostly asteroids that shares the orbit of a larger body remaining in a stable orbit approximately 60 ahead of or behind the main body near one of its Lagrangian points L4 and L5 Trojans can share the orbits of planets or of large moons The trojan points are located on the L4 and L5 Lagrange points on the orbital path of the secondary object blue around the primary object yellow All of the Lagrange points are highlighted in red Trojans are one type of co orbital object In this arrangement a star and a planet orbit about their common barycenter which is close to the center of the star because it is usually much more massive than the orbiting planet In turn a much smaller mass than both the star and the planet located at one of the Lagrangian points of the star planet system is subject to a combined gravitational force that acts through this barycenter Hence the smallest object orbits around the barycenter with the same orbital period as the planet and the arrangement can remain stable over time 1 In the Solar System most known trojans share the orbit of Jupiter They are divided into the Greek camp at L4 ahead of Jupiter and the Trojan camp at L5 trailing Jupiter More than a million Jupiter trojans larger than one kilometer are thought to exist 2 of which more than 7 000 are currently catalogued In other planetary orbits only nine Mars trojans 28 Neptune trojans two Uranus trojans and two Earth trojans have been found to date A temporary Venus trojan is also known Numerical orbital dynamics stability simulations indicate that Saturn probably does not have any primordial trojans 3 The same arrangement can appear when the primary object is a planet and the secondary is one of its moons whereby much smaller trojan moons can share its orbit All known trojan moons are part of the Saturn system Telesto and Calypso are trojans of Tethys and Helene and Polydeuces of Dione Contents 1 Trojan minor planets 1 1 Trojans by planet 2 Stability 3 See also 4 ReferencesTrojan minor planets edit nbsp The Jupiter trojans are seen in this graphic as Greek camp at L4 ahead of Jupiter and as Trojan camp at L5 trailing Jupiter along its orbital path It also shows the asteroid belt between Mars and Jupiter and the Hilda asteroids Jupiter trojans Asteroid belt Hilda asteroidsIn 1772 the Italian French mathematician and astronomer Joseph Louis Lagrange obtained two constant pattern solutions collinear and equilateral of the general three body problem 4 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 point of Jupiter are named for the characters from the Greek side of the war whereas those orbiting near the L5 of Jupiter are from the Trojan side There are two exceptions which were named before the convention was put in place the Greek 624 Hektor and the Trojan 617 Patroclus 5 Astronomers estimate that the Jovian trojans are about as numerous as the asteroids of the asteroid belt 6 Later on objects were found orbiting near the Lagrangian points of Neptune Mars Earth 7 Uranus and Venus Minor planets at the Lagrangian points of planets other than Jupiter may be called Lagrangian minor planets 8 Four Martian trojans are known 5261 Eureka 101429 1998 VF31 311999 2007 NS2 and 121514 1999 UJ7 the only Trojan body in the leading cloud at L4 9 10 There seem to be also 2001 DH47 2011 SC191 and 2011 UN63 but these have not yet been accepted by the Minor Planet Center There are 28 known Neptunian trojans 11 but the large Neptunian trojans are expected to outnumber the large Jovian trojans by an order of magnitude 12 13 2010 TK7 was confirmed to be the first known Earth trojan in 2011 It is located in the L4 Lagrangian point which lies ahead of the Earth 14 2020 XL5 was found to be another Earth trojan in 2021 It is also at L4 15 16 2011 QF99 was identified as the first Uranus trojan in 2013 It is located at the L4 Lagrangian point A second one 2014 YX49 was announced in 2017 17 2013 ND15 is a temporary Venusian trojan the first one to be identified The large asteroids Ceres and Vesta have temporary trojans 18 Trojans by planet edit Planet Number in L4 Number in L5 List L4 List L5 Mercury 0 0 Venus 1 0 2013 ND15 Earth 2 0 2010 TK7 2020 XL5 Mars 1 13 121514 1999 UJ7 manyJupiter 7508 4044 many manySaturn 0 0 Uranus 2 0 2011 QF99 2014 YX49 Neptune 24 4 many manyStability editWhether 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 gt 100m2 gt 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 lt m1 m2 m3 2 So the trojan being a mote of dust m3 0 imposes a lower bound on m1 m2 of 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 m1 m2 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 See also edit nbsp Look up Trojan Trojan asteroid Trojan moon or Trojan planet in Wiktionary the free dictionary Earth trojan Jupiter trojan Lissajous orbit List of objects at Lagrange points Tadpole orbitReferences edit Robutel Philippe Souchay Jean 2010 An introduction to the dynamics of trojan asteroids in Dvorak Rudolf Souchay Jean eds Dynamics of Small Solar System Bodies and Exoplanets Lecture Notes in Physics vol 790 Springer p 197 ISBN 978 3 642 04457 1 Yoshida F Nakamura T December 2005 Size Distribution of Faint Jovian L4 Trojan Asteroids The Astronomical Journal 130 6 2900 2911 Bibcode 2005AJ 130 2900Y doi 10 1086 497571 Sheppard Scott S Trujillo Chadwick A June 2006 A Thick Cloud of Neptune Trojans and their Colors Science 313 5786 511 514 Bibcode 2006Sci 313 511S doi 10 1126 science 1127173 PMID 16778021 S2CID 35721399 Lagrange Joseph Louis 1772 Essai sur le Probleme des Trois Corps Essay on the Three Body Problem PDF in French Archived from the original PDF on 22 December 2017 a href Template Cite journal html title Template Cite journal cite journal a Cite journal requires journal help Wright Alison 1 August 2011 Planetary science The Trojan is out there Nature Physics 7 8 592 Bibcode 2011NatPh 7 592W doi 10 1038 nphys2061 Yoshida Fumi Nakamura Tsuko 2005 Size distribution of faint L4 Trojan asteroids The Astronomical Journal 130 6 2900 11 Bibcode 2005AJ 130 2900Y doi 10 1086 497571 Connors Martin Wiegert Paul Veillet Christian 27 July 2011 Earth s Trojan asteroid Nature 475 7357 481 483 Bibcode 2011Natur 475 481C doi 10 1038 nature10233 PMID 21796207 S2CID 205225571 Whiteley Robert J Tholen David J November 1998 A CCD Search for Lagrangian Asteroids of the Earth Sun System Icarus 136 1 154 167 Bibcode 1998Icar 136 154W doi 10 1006 icar 1998 5995 List of Martian Trojans Minor Planet Center Retrieved 3 July 2015 de la Fuente Marcos C de la Fuente Marcos R 15 May 2013 Three new stable L5 Mars Trojans Letters Monthly Notices of the Royal Astronomical Society 432 1 31 35 arXiv 1303 0124 Bibcode 2013MNRAS 432L 31D doi 10 1093 mnrasl slt028 List of Neptune Trojans Minor Planet Center 28 October 2018 Retrieved 28 December 2018 Chiang Eugene I Lithwick Yoram 20 July 2005 Neptune Trojans as a Testbed for Planet Formation The Astrophysical Journal 628 1 520 532 arXiv astro ph 0502276 Bibcode 2005ApJ 628 520C doi 10 1086 430825 S2CID 18509704 Powell David 30 January 2007 Neptune May Have Thousands of Escorts Space com Choi Charles Q 27 July 2011 First Asteroid Companion of Earth Discovered at Last Space com Retrieved 27 July 2011 Man To Hui et al November 2021 The Second Earth Trojan 2020 XL5 Astrophysical Journal Letters 922 2 L25 arXiv 2111 05058 Bibcode 2021ApJ 922L 25H doi 10 3847 2041 8213 ac37bf ISSN 2041 8205 S2CID 243860678 Leah Crane 22 November 2021 Trojan asteroid Another object found that shares Earth s orbit New Scientist de la Fuente Marcos Carlos de la Fuente Marcos Raul 21 May 2017 Asteroid 2014 YX49 a large transient Trojan of Uranus Monthly Notices of the Royal Astronomical Society 467 2 1561 1568 arXiv 1701 05541 Bibcode 2017MNRAS 467 1561D doi 10 1093 mnras stx197 Christou Apostolos A Wiegert Paul January 2012 A population of main belt asteroids co orbiting with Ceres and Vesta Icarus 217 1 27 42 arXiv 1110 4810 Bibcode 2012Icar 217 27C doi 10 1016 j icarus 2011 10 016 S2CID 59474402 Portals nbsp Stars nbsp Spaceflight nbsp Outer space nbsp Science Retrieved from https en wikipedia org w index php title Trojan celestial body amp oldid 1173446255, wikipedia, wiki, book, books, library,

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