Trojan (celestial body)

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.

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.

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 and Uranus probably do 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

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 Lagrangian 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

PlanetNumber in L4Number in L5List (L4)List (L5)
Mercury00
Venus102013 ND15
Earth202010 TK7, 2020 XL5
Mars113(121514) 1999 UJ7many
Jupiter75084044manymany
Saturn00
Uranus202011 QF99, 2014 YX49
Neptune244manymany

Stability

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 onm1/m2 of25+√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 ifm1/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

  • Jupiter trojan
  • Lissajous orbit
  • List of objects at Lagrange points
  • Tadpole orbit

References

  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" (PDF). Science. 313 (5786): 511–514. Bibcode:2006Sci...313..511S. doi:10.1126/science.1127173. PMID 16778021. S2CID 35721399.
  4. ^ Lagrange, Joseph-Louis (1772). "Essai sur le Problème des Trois Corps" [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.

Media files used on this page

Solar System Template Final.png
Major Solar System objects. Sizes of planets and Sun are roughly to scale, but distances are not. This is not a diagram of all known moons – small gas giants' moons and Pluto's S/2011 P 1 moon are not shown.
Wiktionary-logo-en-v2.svg
Author/Creator: Dan Polansky based on work currently attributed to Wikimedia Foundation but originally created by Smurrayinchester, Licence: CC BY-SA 4.0
A logo derived from File:WiktionaryEn.svg, a logo showing a 3 x 3 matrix of variously rotated tiles with a letter or character on each tile. The derivation consisted in removing the tiles that form the background of each of the shown characters. File:WiktionaryEn.svg is under Creative Commons Attribution-Share Alike, created by Smurrayinchester, and attributed to Wikimedia Foundation. This is the version without the wordmark.
Solar system.jpg
This is a montage of planetary images taken by spacecraft managed by the Jet Propulsion Laboratory in Pasadena, CA. Included are (from top to bottom) images of Mercury, Venus, Earth (and Moon), Mars, Jupiter, Saturn, Uranus and Neptune. The spacecraft responsible for these images are as follows:
  • the Mercury image was taken by Mariner 10,
  • the Venus image by Magellan,
  • the Earth and Moon images by Galileo,
  • the Mars image by Mars Global Surveyor,
  • the Jupiter image by Cassini, and
  • the Saturn, Uranus and Neptune images by Voyager.
  • Pluto is not shown as it is no longer a planet, and no spacecraft has yet visited it when this montage was taken. The inner planets (Mercury, Venus, Earth, Moon, and Mars) are roughly to scale to each other; the outer planets (Jupiter, Saturn, Uranus, and Neptune) are roughly to scale to each other. PIA 00545 is the same montage with Neptune shown larger in the foreground. Actual diameters are given below:
  • Sun (to photosphere) 1,392,684 km
  • Mercury 4,879.4 km
  • Venus 12,103.7 km
  • Earth 12,756.28 km
  • Moon 3,476.2 km
  • Mars 6,804.9 km
  • Jupiter 142,984 km
  • Saturn 120,536 km
  • Uranus 51,118 km
  • Neptune 49,528 km
Crab Nebula.jpg
This is a mosaic image, one of the largest ever taken by NASA's Hubble Space Telescope, of the Crab Nebula, a six-light-year-wide expanding remnant of a star's supernova explosion. Japanese and Chinese astronomers recorded this violent event in 1054 CE, as did, almost certainly, Native Americans.

The orange filaments are the tattered remains of the star and consist mostly of hydrogen. The rapidly spinning neutron star embedded in the center of the nebula is the dynamo powering the nebula's eerie interior bluish glow. The blue light comes from electrons whirling at nearly the speed of light around magnetic field lines from the neutron star. The neutron star, like a lighthouse, ejects twin beams of radiation that appear to pulse 30 times a second due to the neutron star's rotation. A neutron star is the crushed ultra-dense core of the exploded star.

The Crab Nebula derived its name from its appearance in a drawing made by Irish astronomer Lord Rosse in 1844, using a 36-inch telescope. When viewed by Hubble, as well as by large ground-based telescopes such as the European Southern Observatory's Very Large Telescope, the Crab Nebula takes on a more detailed appearance that yields clues into the spectacular demise of a star, 6,500 light-years away.

The newly composed image was assembled from 24 individual Wide Field and Planetary Camera 2 exposures taken in October 1999, January 2000, and December 2000. The colors in the image indicate the different elements that were expelled during the explosion. Blue in the filaments in the outer part of the nebula represents neutral oxygen, green is singly-ionized sulfur, and red indicates doubly-ionized oxygen.
The Earth seen from Apollo 17 with transparent background.png
"The Blue Marble" is a famous photograph of the Earth taken on December 7, 1972 by the crew of the Apollo 17 spacecraft en route to the Moon at a distance of about 29,000 kilometers (18,000 statute miles). It shows Africa, Antarctica, and the Arabian Peninsula.
He1523a.jpg
Author/Creator: ESO, European Southern Observatory, Licence: CC BY 4.0
Artist's impression of "the oldest star of our Galaxy": HE 1523-0901
  • About 13.2 billion years old
  • Approximately 7500 light years far from Earth
  • Published as part of Hamburg/ESO Survey in the May 10 2007 issue of The Astrophysical Journal
RocketSunIcon.svg
Author/Creator: Me, Licence: Copyrighted free use
SVG replacement for File:Spaceship and the Sun.jpg. A stylized illustration of a spaceship and the sun, based on the description of the emblem of the fictional Galactic Empire in Isaac Asimov's Foundation series ("The golden globe with its conventionalized rays, and the oblique cigar shape that was a space vessel"). This image could be used as a icon for science-fiction related articles.
Earth-moon.jpg
This view of the rising Earth greeted the Apollo 8 astronauts as they came from behind the Moon after the fourth nearside orbit. Earth is about five degrees above the horizon in the photo. The unnamed surface features in the foreground are near the eastern limb of the Moon as viewed from Earth. The lunar horizon is approximately 780 kilometers from the spacecraft. Width of the photographed area at the horizon is about 175 kilometers. On the Earth 240,000 miles away, the sunset terminator bisects Africa.
InnerSolarSystem-en.png
The inner Solar System, from the Sun to Jupiter. Also includes the asteroid belt (the white donut-shaped cloud), the Hildas (the orange "triangle" just inside the orbit of Jupiter), the Jupiter trojans (green), and the near-Earth asteroids. The group that leads Jupiter are called the "Greeks" and the trailing group are called the "Trojans" (Murray and Dermott, Solar System Dynamics, pg. 107)
This image is based on data found in the en:JPL DE-405 ephemeris, and the en:Minor Planet Center database of asteroids (etc) published 2006 Jul 6. The image is looking down on the en:ecliptic plane as would have been seen on 2006 August 14. It was rendered by custom software written for Wikipedia. The same image without labels is also available at File:InnerSolarSystem.png.
Lagrange very massive.svg
Diagram of Lagrange points in a system where the primary is much more massive than the secondary (e.g. SunEarth).