The Jupiter trojans called Trojan asteroids or Trojans, are a large group of asteroids that share the planet Jupiter's orbit around the Sun. Relative to Jupiter, each Trojan librates around one of Jupiter's two stable Lagrange points: L4, lying 60° ahead of the planet in its orbit, L5, 60° behind. Jupiter trojans are distributed in two elongated, curved regions around these Lagrangian points with an average semi-major axis of about 5.2 AU. The first Jupiter trojan discovered, 588 Achilles, was spotted in 1906 by German astronomer Max Wolf. A total of 7,040 Jupiter trojans have been found as of October 2018. By convention, they are each named from Greek mythology after a figure of the Trojan War, hence the name "Trojan"; the total number of Jupiter trojans larger than 1 km in diameter is believed to be about 1 million equal to the number of asteroids larger than 1 km in the asteroid belt. Like main-belt asteroids, Jupiter trojans form families. Jupiter trojans are dark bodies with featureless spectra.
No firm evidence of the presence of water, or any other specific compound on their surface has been obtained, but it is thought that they are coated in tholins, organic polymers formed by the Sun's radiation. The Jupiter trojans' densities vary from 0.8 to 2.5 g·cm−3. Jupiter trojans are thought to have been captured into their orbits during the early stages of the Solar System's formation or later, during the migration of giant planets; the term "Trojan Asteroid" refers to the asteroids co-orbital with Jupiter, but the general term "trojan" is sometimes more applied to other small Solar System bodies with similar relationships to larger bodies: for example, there are both Mars trojans and Neptune trojans, as well as a recently-discovered Earth trojan. The term "Trojan asteroid" is understood to mean the Jupiter trojans because the first Trojans were discovered near Jupiter's orbit and Jupiter has by far the most known Trojans. In 1772, Italian-born mathematician Joseph-Louis Lagrange, in studying the restricted three-body problem, predicted that a small body sharing an orbit with a planet but lying 60° ahead or behind it will be trapped near these points.
The trapped body will librate around the point of equilibrium in a tadpole or horseshoe orbit. These leading and trailing points are called the L5 Lagrange points; the first asteroids trapped in Lagrange points were observed more than a century after Lagrange's hypothesis. Those associated with Jupiter were the first to be discovered. E. E. Barnard made the first recorded observation of a trojan, 1999 RM11, in 1904, but neither he nor others appreciated its significance at the time. Barnard believed he had seen the discovered Saturnian satellite Phoebe, only two arc-minutes away in the sky at the time, or an asteroid; the object's identity was not understood until its orbit was calculated in 1999. The first accepted discovery of a trojan occurred in February 1906, when astronomer Max Wolf of Heidelberg-Königstuhl State Observatory discovered an asteroid at the L4 Lagrangian point of the Sun–Jupiter system named 588 Achilles. In 1906–1907 two more Jupiter trojans were found by fellow German astronomer August Kopff.
Hektor, like Achilles, belonged to the L4 swarm, whereas Patroclus was the first asteroid known to reside at the L5 Lagrangian point. By 1938, 11 Jupiter trojans had been detected; this number increased to 14 only in 1961. As instruments improved, the rate of discovery grew rapidly: by January 2000, a total of 257 had been discovered; as of October 2018 there are 4,601 known Jupiter trojans at L4 and 2,439 at L5. The custom of naming all asteroids in Jupiter's L4 and L5 points after famous heroes of the Trojan War was suggested by Johann Palisa of Vienna, the first to calculate their orbits. Asteroids in the leading orbit are named after Greek heroes, those at the trailing orbit are named after the heroes of Troy; the asteroids 617 Patroclus and 624 Hektor were named before the Greece/Troy rule was devised, resulting in a Greek spy in the Trojan node and a Trojan spy in the Greek node. Estimates of the total number of Jupiter trojans are based on deep surveys of limited areas of the sky; the L4 swarm is believed to hold between 160–240,000 asteroids with diameters larger than 2 km and about 600,000 with diameters larger than 1 km.
If the L5 swarm contains a comparable number of objects, there are more than 1 million Jupiter trojans 1 km in size or larger. For the objects brighter than absolute magnitude 9.0 the population is complete. These numbers are similar to that of comparable asteroids in the asteroid belt; the total mass of the Jupiter trojans is estimated at 0.0001 of the mass of Earth or one-fifth of the mass of the asteroid belt. Two more recent studies indicate that the above numbers may overestimate the number of Jupiter trojans by several-fold; this overestimate is caused by the assumption that all Jupiter trojans have a low albedo of about 0.04, whereas small bodies may have an average albedo as high as 0.12. According to the new estimates, the total number of Jupiter trojans with a diameter larger than 2 km is 6,300 ± 1,000 and 3,400 ± 500 in the L4 and L5 swarms, respectively; these numbers would be reduced by a factor of 2 if small Jupiter trojans are more reflective than large ones. The number of Jupiter trojans observed in the L4
An asteroid family is a population of asteroids that share similar proper orbital elements, such as semimajor axis and orbital inclination. The members of the families are thought to be fragments of past asteroid collisions. An asteroid family is a more specific term than asteroid group whose members, while sharing some broad orbital characteristics, may be otherwise unrelated to each other. Large prominent families contain several hundred recognized asteroids. Small, compact families may have only about ten identified members. About 33% to 35% of asteroids in the main belt are family members. There are about 20 to 30 reliably recognized families, with several tens of less certain groupings. Most asteroid families are found in the main asteroid belt, although several family-like groups such as the Pallas family, Hungaria family, the Phocaea family lie at smaller semi-major axis or larger inclination than the main belt. One family has been identified associated with the dwarf planet Haumea; some studies have tried to find evidence of collisional families among the trojan asteroids, but at present the evidence is inconclusive.
The families are thought to form as a result of collisions between asteroids. In many or most cases the parent body was shattered, but there are several families which resulted from a large cratering event which did not disrupt the parent body; such cratering families consist of a single large body and a swarm of asteroids that are much smaller. Some families have complex internal structures which are not satisfactorily explained at the moment, but may be due to several collisions in the same region at different times. Due to the method of origin, all the members have matching compositions for most families. Notable exceptions are those families. Asteroid families are thought to have lifetimes of the order of a billion years, depending on various factors; this is shorter than the Solar System's age, so few if any are relics of the early Solar System. Decay of families occurs both because of slow dissipation of the orbits due to perturbations from Jupiter or other large bodies, because of collisions between asteroids which grind them down to small bodies.
Such small asteroids become subject to perturbations such as the Yarkovsky effect that can push them towards orbital resonances with Jupiter over time. Once there, they are rapidly ejected from the asteroid belt. Tentative age estimates have been obtained for some families, ranging from hundreds of millions of years to less than several million years as for the compact Karin family. Old families are thought to contain few small members, this is the basis of the age determinations, it is supposed that many old families have lost all the smaller and medium-sized members, leaving only a few of the largest intact. A suggested example of such old family remains are 113 Amalthea pair. Further evidence for a large number of past families comes from analysis of chemical ratios in iron meteorites; these show that there must have once been at least 50 to 100 parent bodies large enough to be differentiated, that have since been shattered to expose their cores and produce the actual meteorites. When the orbital elements of main belt asteroids are plotted, a number of distinct concentrations are seen against the rather uniform distribution of non-family background asteroids.
These concentrations are the asteroid families. Interlopers are asteroids classified as family members based on their so-called proper orbital elements but having spectroscopic properties distinct from the bulk of the family, suggesting that they, contrary to the true family members, did not originate from the same parent body that once fragmented upon a collisional impact. Speaking and their membership are identified by analysing the proper orbital elements rather than the current osculating orbital elements, which fluctuate on timescales of tens of thousands of years; the proper elements are related constants of motion that remain constant for times of at least tens of millions of years, longer. The Japanese astronomer Kiyotsugu Hirayama pioneered the estimation of proper elements for asteroids, first identified several of the most prominent families in 1918. In his honor, asteroid families are sometimes called Hirayama families; this applies to the five prominent groupings discovered by him.
Present day computer-assisted searches have identified more than a hundred asteroid families. The most prominent algorithms have been the hierarchical clustering method, which looks for groupings with small nearest-neighbour distances in orbital element space, wavelet analysis, which builds a density-of-asteroids map in orbital element space, looks for density peaks; the boundaries of the families are somewhat vague because at the edges they blend into the background density of asteroids in the main belt. For this reason the number of members among discovered asteroids is only known and membership is uncertain for asteroids near the edges. Additionally, some interlopers from the heterogeneous background asteroid population are expected in the central regions of a family. Since the true family members caused by the collision are expected to have similar compositions, most such interlopers can in principle be recognised by spectral properties which do not matc
Vesta is one of the largest objects in the asteroid belt, with a mean diameter of 525 kilometres. It was discovered by the German astronomer Heinrich Wilhelm Olbers on 29 March 1807 and is named after Vesta, the virgin goddess of home and hearth from Roman mythology. Vesta is the second-most-massive and second-largest body in the asteroid belt, after the dwarf planet Ceres, it contributes an estimated 9% of the mass of the asteroid belt, it is larger than Pallas, though more massive. Vesta is the only known remaining rocky protoplanet of the kind. Numerous fragments of Vesta were ejected by collisions one and two billion years ago that left two enormous craters occupying much of Vesta's southern hemisphere. Debris from these events has fallen to Earth as howardite–eucrite–diogenite meteorites, which have been a rich source of information about Vesta. Vesta is the brightest asteroid visible from Earth, its maximum distance from the Sun is greater than the minimum distance of Ceres from the Sun, though its orbit lies within that of Ceres.
NASA's Dawn spacecraft entered orbit around Vesta on 16 July 2011 for a one-year exploration and left orbit on 5 September 2012 en route to its final destination, Ceres. Researchers continue to examine data collected by Dawn for additional insights into the formation and history of Vesta. Heinrich Olbers discovered Pallas in the year after the discovery of Ceres, he proposed. He sent a letter with his proposal to the English astronomer William Herschel, suggesting that a search near the locations where the orbits of Ceres and Pallas intersected might reveal more fragments; these orbital intersections were located in the constellations of Virgo. Olbers commenced his search in 1802, on 29 March 1807 he discovered Vesta in the constellation Virgo—a coincidence, because Ceres and Vesta are not fragments of a larger body; because the asteroid Juno had been discovered in 1804, this made Vesta the fourth object to be identified in the region, now known as the asteroid belt. The discovery was announced in a letter addressed to German astronomer Johann H. Schröter dated 31 March.
Because Olbers had credit for discovering a planet, he gave the honor of naming his new discovery to German mathematician Carl Friedrich Gauss, whose orbital calculations had enabled astronomers to confirm the existence of Ceres, the first asteroid, who had computed the orbit of the new planet in the remarkably short time of 10 hours. Gauss decided on the Roman virgin goddess of Vesta. Vesta was the fourth asteroid to be discovered, hence the number 4 in its formal designation; the name Vesta, or national variants thereof, is in international use with two exceptions: Greece and China. In Greek, the name adopted was the Hellenic equivalent of Hestia. In Chinese, Vesta is called the'hearth-god star', 灶神星 zàoshénxīng, naming the asteroid for Vesta's role rather than transliterating her name into Chinese, as is done with other bodies discovered in modern times, including Uranus and Pluto. Upon its discovery, Vesta was, like Ceres and Juno before it, classified as a planet and given a planetary symbol.
The symbol representing the altar of Vesta with its sacred fire and was designed by Gauss. In Gauss's conception, this was drawn. After the discovery of Vesta, no further objects were discovered for 38 years, the Solar System was thought to have eleven planets. However, in 1845, new asteroids started being discovered at a rapid pace, by 1851 there were fifteen, each with its own symbol, in addition to the eight major planets, it soon became clear that it would be impractical to continue inventing new planetary symbols indefinitely, some of the existing ones proved difficult to draw quickly. That year, the problem was addressed by Benjamin Apthorp Gould, who suggested numbering asteroids in their order of discovery, placing this number in a disk as the generic symbol of an asteroid. Thus, the fourth asteroid, acquired the generic symbol ④; this was soon coupled with the name into an official number–name designation, ④ Vesta, as the number of minor planets increased. By 1858, the circle had been simplified to parentheses, which were easier to typeset.
Other punctuation, such as 4) Vesta and 4, was used, but had more or less died out by 1949. Today, either Vesta, or, more 4 Vesta, is used. Photometric observations of Vesta were made at the Harvard College Observatory in 1880–1882 and at the Observatoire de Toulouse in 1909; these and other observations allowed the rotation rate of Vesta to be determined by the 1950s. However, the early estimates of the rotation rate came into question because the light curve included variations in both shape and albedo. Early estimates of the diameter of Vesta ranged from 383 to 444 km. E. C. Pickering produced an estimated diameter of 513±17 km in 1879, close to the modern value for the mean diameter, but the subsequent estimates ranged from a low of 390 km up to a high of 602 km during the next century; the measured estimates were based on photometry. In 1989, speckle interferometry was used to measure a dimension that varied between 498 and 548 km during the rotational period. In 1991, an occultation of the star SAO 93228 by Vesta was observed from multiple locations
A minor-planet moon is an astronomical object that orbits a minor planet as its natural satellite. As of February 2019, there are 352 minor planets suspected to have moons. Discoveries of minor-planet moons are important because the determination of their orbits provides estimates on the mass and density of the primary, allowing insights of their physical properties, not otherwise possible; the first modern era mention of the possibility of an asteroid satellite was in connection with an occultation of the bright star Gamma Ceti by the asteroid 6 Hebe in 1977. The observer, amateur astronomer Paul D. Maley, detected an unmistakable 0.5 second disappearance of this naked eye star from a site near Victoria, Texas. Many hours several observations were reported in Mexico attributed to the occultation by 6 Hebe itself. Although not confirmed, this documents the first formally documented case of a suspected companion of an asteroid. In addition to the terms satellite and moon, the term "binary" is sometimes used for minor planets with moons, "triple" for minor planets with two moons.
If one object is much bigger it can be referred to as the primary and its companion as secondary. The term double asteroid is sometimes used for systems in which the asteroid and its moon are the same size, while binary tends to be used independently from the relative sizes of the components; when binary minor planets are similar in size, the Minor Planet Center refers to them as "binary companions" instead of referring to the smaller body as a satellite. A good example of a true binary is the 90 Antiope system, identified in August 2000. Small satellites are referred to as moonlets. Prior to the era of the Hubble Space Telescope and space probes reaching the outer Solar System, attempts to detect satellites around asteroids were limited to optical observations from Earth. For example, in 1978, stellar occultation observations were claimed as evidence of a satellite for the asteroid 532 Herculina; however more-detailed imaging by the Hubble Telescope did not reveal a satellite, the current consensus is that Herculina does not have a significant satellite.
There were other similar reports of asteroids having companions in the following years. A letter in Sky & Telescope magazine at this time pointed to simultaneous impact craters on Earth, suggesting that these craters were caused by pairs of gravitationally bound objects. In 1993, the first asteroid moon was confirmed when the Galileo probe discovered the small Dactyl orbiting 243 Ida in the asteroid belt; the second was discovered around 45 Eugenia in 1998. In 2001, 617 Patroclus and its same-sized companion Menoetius became the first known binary asteroids in the Jupiter trojans; the first trans-Neptunian binary after Pluto–Charon, 1998 WW31, was optically resolved in 2002. Triple or trinary minor planets, are known since 2005, when the asteroid 87 Sylvia was discovered to have two satellites, making it the first known triple system; this was followed by the discovery of a second moon orbiting 45 Eugenia. In 2005, the dwarf planet Haumea was discovered to have two moons, making it the second trans-Neptunian object after Pluto known to have more than one moon.
Additionally, 216 Kleopatra and 93 Minerva were discovered to be trinary asteroids in 2008 and 2009 respectively. Since the first few triple minor planets were discovered, more continue to be discovered at a rate of about one a year. Most discovered were two moons orbiting large near-earth asteroid 3122 Florence, bringing the number of known trinary systems in the Solar System up to 14; the following table lists all satellites of triple systems chronologically by their discovery date, starting with Charon, discovered in 1978. The data about the populations of binary objects are still patchy. In addition to the inevitable observational bias the frequency appears to be different among different categories of objects. Among asteroids, an estimated 2% would have satellites. Among trans-Neptunian objects, an estimated 11% are thought to be binary or multiple objects, the majority of the large TNOs have at least one satellite, including all four IAU-listed dwarf planets. More than 50 binaries are known in each of the main groupings: near-Earth asteroids, belt asteroids, trans-Neptunian objects, not including numerous claims based on light-curve variation.
Two binaries have been found so far among centaurs with semi-major axes smaller than Neptune. Both are double ring systems around 2060 Chiron and 10199 Chariklo, discovered in 1994–2011 and 2013 respectively; the origin of minor-planet moons is not known with certainty, a variety of theories exist. A accepted theory is that minor-planet moons are formed from debris knocked off of the primary by an impact. Other pairings may be formed. Formation by collision is constrained by the angular momentum of the components, i.e. by the masses and their separation. Close binaries fit this model. Distant binaries however, with components of comparable size, are unlikely to have followed this scenario, unless considerable mass has been lost in the event; the distances of the components for the known binaries vary from a few hundreds of kilometres to more than 3000 km for the asteroids. Among TNOs, the known separations vary from 3,000 to 50,000 km. What is "typical" for a binary system tends to depend on its location in the Solar System (presumably because of different modes
The orbital eccentricity of an astronomical object is a parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit, greater than 1 is a hyperbola; the term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section. It is used for the isolated two-body problem, but extensions exist for objects following a Klemperer rosette orbit through the galaxy. In a two-body problem with inverse-square-law force, every orbit is a Kepler orbit; the eccentricity of this Kepler orbit is a non-negative number. The eccentricity may take the following values: circular orbit: e = 0 elliptic orbit: 0 < e < 1 parabolic trajectory: e = 1 hyperbolic trajectory: e > 1 The eccentricity e is given by e = 1 + 2 E L 2 m red α 2 where E is the total orbital energy, L is the angular momentum, mred is the reduced mass, α the coefficient of the inverse-square law central force such as gravity or electrostatics in classical physics: F = α r 2 or in the case of a gravitational force: e = 1 + 2 ε h 2 μ 2 where ε is the specific orbital energy, μ the standard gravitational parameter based on the total mass, h the specific relative angular momentum.
For values of e from 0 to 1 the orbit's shape is an elongated ellipse. The limit case between an ellipse and a hyperbola, when e equals 1, is parabola. Radial trajectories are classified as elliptic, parabolic, or hyperbolic based on the energy of the orbit, not the eccentricity. Radial orbits hence eccentricity equal to one. Keeping the energy constant and reducing the angular momentum, elliptic and hyperbolic orbits each tend to the corresponding type of radial trajectory while e tends to 1. For a repulsive force only the hyperbolic trajectory, including the radial version, is applicable. For elliptical orbits, a simple proof shows that arcsin yields the projection angle of a perfect circle to an ellipse of eccentricity e. For example, to view the eccentricity of the planet Mercury, one must calculate the inverse sine to find the projection angle of 11.86 degrees. Next, tilt any circular object by that angle and the apparent ellipse projected to your eye will be of that same eccentricity; the word "eccentricity" comes from Medieval Latin eccentricus, derived from Greek ἔκκεντρος ekkentros "out of the center", from ἐκ- ek-, "out of" + κέντρον kentron "center".
"Eccentric" first appeared in English in 1551, with the definition "a circle in which the earth, sun. Etc. deviates from its center". By five years in 1556, an adjectival form of the word had developed; the eccentricity of an orbit can be calculated from the orbital state vectors as the magnitude of the eccentricity vector: e = | e | where: e is the eccentricity vector. For elliptical orbits it can be calculated from the periapsis and apoapsis since rp = a and ra = a, where a is the semimajor axis. E = r a − r p r a + r p = 1 − 2 r a r p + 1 where: ra is the radius at apoapsis. Rp is the radius at periapsis; the eccentricity of an elliptical orbit can be used to obtain the ratio of the periapsis to the apoapsis: r p r a = 1 − e 1 + e For Earth, orbital eccentricity ≈ 0.0167, apoapsis= aphelion and periapsis= perihelion relative to sun. For Earth's annual orbit path, ra/rp ratio = longest_radius / shortest_radius ≈ 1.034 relative to center point of path. The eccentricity of the Earth's orbit is about 0.0167.
A trans-Neptunian object written transneptunian object, is any minor planet in the Solar System that orbits the Sun at a greater average distance than Neptune, which has a semi-major axis of 30.1 astronomical units. TNOs are further divided into the classical and resonant objects of the Kuiper belt, the scattered disc and detached objects with the sednoids being the most distant ones; as of October 2018, the catalog of minor planets contains 528 numbered and more than 2,000 unnumbered TNOs. The first trans-Neptunian object to be discovered was Pluto in 1930, it took until 1992 to discover a second trans-Neptunian object orbiting the Sun directly, 15760 Albion. The most massive TNO known is Eris, followed by Pluto, 2007 Makemake and Haumea. More than 80 satellites have been discovered in orbit of trans-Neptunian objects. TNOs vary in color and are either grey-blue or red, they are thought to be composed of mixtures of rock, amorphous carbon and volatile ices such as water and methane, coated with tholins and other organic compounds.
Twelve minor planets with a semi-major axis greater than 150 AU and perihelion greater than 30 AU are known, which are called extreme trans-Neptunian objects. The orbit of each of the planets is affected by the gravitational influences of the other planets. Discrepancies in the early 1900s between the observed and expected orbits of Uranus and Neptune suggested that there were one or more additional planets beyond Neptune; the search for these led to the discovery of Pluto in February 1930, too small to explain the discrepancies. Revised estimates of Neptune's mass from the Voyager 2 flyby in 1989 showed that the problem was spurious. Pluto was easiest to find because it has the highest apparent magnitude of all known trans-Neptunian objects, it has a lower inclination to the ecliptic than most other large TNOs. After Pluto's discovery, American astronomer Clyde Tombaugh continued searching for some years for similar objects, but found none. For a long time, no one searched for other TNOs as it was believed that Pluto, which up to August 2006 was classified a planet, was the only major object beyond Neptune.
Only after the 1992 discovery of a second TNO, 15760 Albion, did systematic searches for further such objects begin. A broad strip of the sky around the ecliptic was photographed and digitally evaluated for moving objects. Hundreds of TNOs were found, with diameters in the range of 50 to 2,500 kilometers. Eris, the most massive TNO, was discovered in 2005, revisiting a long-running dispute within the scientific community over the classification of large TNOs, whether objects like Pluto can be considered planets. Pluto and Eris were classified as dwarf planets by the International Astronomical Union. On Monday, December 17, 2018 the discovery of 2018 VG18, nicknamed “Farout”, was announced. Farout is the most distant solar system object so-far observed and is about 120 AU away from the sun taking more than 1,000 years to complete one orbit. According to their distance from the Sun and their orbital parameters, TNOs are classified in two large groups: the Kuiper belt objects and the scattered disc objects.
The diagram to the right illustrates the distribution of known trans-Neptunian objects in relation to the orbits of the planets and the centaurs for reference. Different classes are represented in different colours. Resonant objects are plotted in classical Kuiper belt objects in blue; the scattered disc extends to the right, far beyond the diagram, with known objects at mean distances beyond 500 AU and aphelia beyond 1000 AU. The Edgeworth-Kuiper belt contains objects with an average distance to the Sun of 30 to about 55 AU having close-to-circular orbits with a small inclination from the ecliptic. Edgeworth-Kuiper belt objects are further classified into the resonant trans-Neptunian object, that are locked in an orbital resonance with Neptune, the classical Kuiper belt objects called "cubewanos", that have no such resonance, moving on circular orbits, unperturbed by Neptune. There are a large number of resonant subgroups, the largest being the twotinos and the plutinos, named after their most prominent member, Pluto.
Members of the classical Edgeworth-Kuiper belt include 50000 Quaoar and Makemake. The scattered disc contains objects farther from the Sun, with eccentric and inclined orbits; these orbits are non-planetary-orbit-crossing. A typical example is the most massive known Eris. Based on the Tisserand parameter relative to Neptune, the objects in the scattered disc can be further divided into the "typical" scattered disc objects with a TN of less than 3, into the detached objects with a TN greater than 3. In addition, detached objects have a time-averaged eccentricity greater than 0.2 The Sednoids are a further extreme sub-grouping of the detached objects with perihelia so distant that it is confirmed that their orbits cannot be explained by perturbations from the giant planets, nor by interaction with the galactic tides. Given the apparent magnitude of all but the biggest trans-Neptunian objects, the physical studies are limited to the following: thermal emissions for the largest objects colour indices, i.e. comparisons of the apparent magnitudes using different filters analysis of spectra and infraredStudying colours and spectra provides insight into the objects' origin and a potential correlation with other classes of objects, namely centaurs and some satellites of giant planets, suspected to originate in the Kuiper belt.
Proper orbital elements
The proper orbital elements of an orbit are constants of motion of an object in space that remain unchanged over an astronomically long timescale. The term is used to describe the three quantities: proper semimajor axis, proper eccentricity, proper inclination; the proper elements can be contrasted with the osculating Keplerian orbital elements observed at a particular time or epoch, such as the semi-major axis and inclination. Those osculating elements change in a quasi-periodic and predictable manner due to such effects as perturbations from planets or other bodies, precession. In the Solar System, such changes occur on timescales of thousands of years, while proper elements are meant to be constant over at least tens of millions of years. For most bodies, the osculating elements are close to the proper elements because precession and perturbation effects are small. For over 99% of asteroids in the asteroid belt, the differences are less than 0.02 AU, 0.1, 2°. This difference is non-negligible for any purposes where precision is of importance.
As an example, the asteroid Ceres has osculating orbital elements while its proper orbital elements are A notable exception to this small-difference rule are asteroids lying in the Kirkwood gaps, which are in strong orbital resonance with Jupiter. To obtain proper elements for an object, one conducts a detailed simulation of its motion over timespans of several millions of years; such a simulation must take into account many details of celestial mechanics including perturbations by the planets. Subsequently, one extracts quantities from the simulation which remain unchanged over this long timespan; these are the proper orbital elements. Various approximate analytic calculations were made, starting with those of Kiyotsugu Hirayama in the early 20th century. Analytic methods included thousands of perturbing corrections for each particular object. Presently, the method of choice is to use a computer to numerically integrate the equations of celestial dynamics, extract constants of motion directly from a numerical analysis of the predicted positions.
At present the most prominent use of proper orbital elements is in the study of asteroid families, following in the footsteps of the pioneering work of Hirayama. A Mars-crosser asteroid 132 Aethra is the lowest numbered asteroid to not have any proper orbital elements. Hirayama family Perturbation Z. Knežević et al; the Determination of Asteroid Proper Elements, p. 603-612 in Asteroids III, University of Arizona Press. Z. Knežević: COMPUTATION OF ASTEROID PROPER ELEMENTS: RECENT ADVANCES, Serbian Astronomical Journal, vol. 195, pp. 1-8. Latest calculations of proper elements for numbered minor planets at astDys. Asteroid proper orbital elements dataset at Asteroid Families Portal