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.
Heidelberg is a university town in Baden-Württemberg situated on the river Neckar in south-west Germany. In the 2016 census, its population was 159,914, with a quarter of its population being students. Located about 78 km south of Frankfurt, Heidelberg is the fifth-largest city in the German state of Baden-Württemberg. Heidelberg is part of the densely populated Rhine-Neckar Metropolitan Region. Founded in 1386, Heidelberg University is Germany's oldest and one of Europe's most reputable universities. A scientific hub in Germany, the city of Heidelberg is home to several internationally renowned research facilities adjacent to its university, including four Max Planck Institutes. A former residence of the Electorate of the Palatinate, Heidelberg is a popular tourist destination due to its romantic cityscape, including Heidelberg Castle, the Philosophers' Walk, the baroque style Old Town. Heidelberg is in the Rhine Rift Valley, on the left bank of the lower part of the Neckar in a steep valley in the Odenwald.
It is bordered by the Gaisberg mountains. The Neckar here flows in an east-west direction. On the right bank of the river, the Heiligenberg mountain rises to a height of 445 meters; the Neckar flows into the Rhine 22 kilometres north-west in Mannheim. Villages incorporated during the 20th century stretch from the Neckar Valley along the Bergstraße, a road running along the Odenwald hills. Heidelberg is on European walking route E1. Since Heidelberg is among the warmest regions of Germany, plants atypical of the central-European climate flourish there, including almond and fig trees. Alongside the Philosophenweg on the opposite side of the Old Town, winegrowing was restarted in 2000. There is a wild population of African rose-ringed parakeets, a wild population of Siberian swan geese, which can be seen on the islands in the Neckar near the district of Bergheim. Heidelberg is a unitary authority within the Regierungsbezirk Karlsruhe; the Rhein-Neckar-Kreis rural district surrounds it and has its seat in the town, although the town is not a part of the district.
Heidelberg is a part of the Rhine-Neckar Metropolitan Region referred to as the Rhein-Neckar Triangle. This region consists of the southern part of the State of Hessen, the southern part of the State of Rhineland-Palatinate, the administrative districts of Mannheim and Heidelberg, the southern municipalities of the Rhein-Neckar-Kreis; the Rhein-Neckar Triangle became a European metropolitan area in 2005. Heidelberg consists of 15 districts distributed in six sectors of the town. In the central area are Altstadt and Weststadt; the new district will have 5,000–6,000 residents and employment for 7,000. Further new residential space for 10,000-15,000 residents was made available in Patrick Henry Village following the departure of the US Armed Forces; the following towns and communes border the city of Heidelberg, beginning in the west and in a clockwise direction: Edingen-Neckarhausen, Schriesheim, Schönau, Neckargemünd, Gaiberg, Sandhausen, Plankstadt and Mannheim. Heidelberg has an oceanic climate, defined by the protected valley between the Pfälzerwald and the Odenwald.
Year-round, the mild temperatures are determined by maritime air masses coming from the west. In contrast to the nearby Upper Rhine Plain, Heidelberg's position in the valley leads to more frequent easterly winds than average; the hillsides of the Odenwald favour precipitation. The warmest month is July, the coldest is January. Temperatures rise beyond 30 °C in midsummer. According to the German Meteorological Service, Heidelberg was the warmest place in Germany in 2009. Between 600,000 and 200,000 years ago, "Heidelberg Man" died at nearby Mauer, his jaw bone was discovered in 1907. Scientific dating determined his remains as the earliest evidence of human life in Europe. In the 5th century BC, a Celtic fortress of refuge and place of worship were built on the Heiligenberg, or "Mountain of Saints". Both places can still be identified. In 40 AD, a fort occupied by the 24th Roman cohort and the 2nd Cyrenaican cohort; the early Byzantine/late Roman Emperor Valentinian I, in 369 AD, built new and maintained older castra and a signal tower on the bank of the Neckar.
They built a wooden bridge based on stone pillars across it. The camp protected the first civilian settlements; the Romans remained until 260 AD. The local administrative center in Roman times was the nearby city of Lopodunum. Modern Heidelberg can trace its beginnings to the fifth century; the village Bergheim is first mentioned for that period in documents dated to 769 AD. Bergheim now lies in the middle of modern Heidelberg; the people converted to Christianity. In 863 AD, the monastery of St. Michael was founded on the Heiligenberg inside the double rampart of the Celtic fortress. Around 1130, the Neuburg Monastery was founded in the Neckar valley. At the same time, the bishopric of Worms extended its influence into the valley, founding Schönau Abbey in 1142. Modern He
Orders of magnitude (length)
The following are examples of orders of magnitude for different lengths. To help compare different orders of magnitude, the following list describes various lengths between 1.6 × 10 − 35 metres and 10 10 10 122 metres. To help compare different orders of magnitude, this section lists lengths shorter than 10−23 m. 1.6 × 10−11 yoctometres – the Planck length. 1 ym – 1 yoctometre, the smallest named subdivision of the metre in the SI base unit of length, one septillionth of a metre 1 ym – length of a neutrino. 2 ym – the effective cross-section radius of 1 MeV neutrinos as measured by Clyde Cowan and Frederick Reines To help compare different orders of magnitude, this section lists lengths between 10−23 metres and 10−22 metres. To help compare different orders of magnitude, this section lists lengths between 10−22 m and 10−21 m. 100 ym – length of a top quark, one of the smallest known quarks To help compare different orders of magnitude, this section lists lengths between 10−21 m and 10−20 m. 2 zm – length of a preon, hypothetical particles proposed as subcomponents of quarks and leptons.
2 zm – radius of effective cross section for a 20 GeV neutrino scattering off a nucleon 7 zm – radius of effective cross section for a 250 GeV neutrino scattering off a nucleon To help compare different orders of magnitude, this section lists lengths between 10−20 m and 10−19 m. 15 zm – length of a high energy neutrino 30 zm – length of a bottom quark To help compare different orders of magnitude, this section lists lengths between 10−19 m and 10−18 m. 177 zm – de Broglie wavelength of protons at the Large Hadron Collider To help compare different orders of magnitude, this section lists lengths between 10−18 m and 10−17 m. 1 am – sensitivity of the LIGO detector for gravitational waves 1 am – upper limit for the size of quarks and electrons 1 am – upper bound of the typical size range for "fundamental strings" 1 am – length of an electron 1 am – length of an up quark 1 am – length of a down quark To help compare different orders of magnitude, this section lists lengths between 10−17 m and 10−16 m. 10 am – range of the weak force To help compare different orders of magnitude, this section lists lengths between 10−16 m and 10−15 m. 100 am – all lengths shorter than this distance are not confirmed in terms of size 850 am – approximate proton radius The femtometre is a unit of length in the metric system, equal to 10−15 metres.
In particle physics, this unit is more called a fermi with abbreviation "fm". To help compare different orders of magnitude, this section lists lengths between 10−15 metres and 10−14 metres. 1 fm – length of a neutron 1.5 fm – diameter of the scattering cross section of an 11 MeV proton with a target proton 1.75 fm – the effective charge diameter of a proton 2.81794 fm – classical electron radius 7 fm – the radius of the effective scattering cross section for a gold nucleus scattering a 6 MeV alpha particle over 140 degrees To help compare different orders of magnitude, this section lists lengths between 10−14 m and 10−13 m. 1.75 to 15 fm – Diameter range of the atomic nucleus To help compare different orders of magnitude, this section lists lengths between 10−13 m and 10−12 m. 570 fm – typical distance from the atomic nucleus of the two innermost electrons in the uranium atom, the heaviest naturally-occurring atom To help compare different orders of magnitude this section lists lengths between 10−12 and 10−11 m. 1 pm – distance between atomic nuclei in a white dwarf 2.4 pm – The Compton wavelength of the electron 5 pm – shorter X-ray wavelengths To help compare different orders of magnitude this section lists lengths between 10−11 and 10−10 m. 25 pm – approximate radius of a helium atom, the smallest neutral atom 50 pm – radius of a hydrogen atom 50 pm – bohr radius: approximate radius of a hydrogen atom ~50 pm – best resolution of a high-resolution transmission electron microscope 60 pm – radius of a carbon atom 93 pm – length of a diatomic carbon molecule To help compare different orders of magnitude this section lists lengths between 10−10 and 10−9 m. 100 pm – 1 ångström 100 pm – covalent radius of sulfur atom 120 pm – van der Waals radius of a neutral hydrogen atom 120 pm – radius of a gold atom 126 pm – covalent radius of ruthenium atom 135 pm – covalent radius of technetium atom 150 pm – Length of a typical covalent bond 153 pm – covalent radius of silver atom 155 pm – covalent radius of zirconium atom 175 pm – covalent radius of thulium atom 200 pm – highest resolution of a typical electron microscope 225 pm – covalent radius of caesium atom 280 pm – Average size of the water molecule 298 pm – radius of a caesium atom, calculated to be the largest atomic radius 340 pm – thickness of single layer graphene 356.68 pm – width of diamond unit cell 403 pm – width of lithium fluoride unit cell 500 pm – Width of protein α helix 543 pm – silicon lattice spacing 560 pm – width of sodium chloride unit cell 700 pm – width of glucose molecule 780 pm – mean width of quartz unit cell 820 pm – mean width of ice unit cell 900 pm – mean width of coesite unit cell To help compare different orders
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
Jet Propulsion Laboratory
The Jet Propulsion Laboratory is a federally funded research and development center and NASA field center in La Cañada Flintridge, United States, though it is referred to as residing in Pasadena, because it has a Pasadena ZIP Code. Founded in the 1930s, the JPL is owned by NASA and managed by the nearby California Institute of Technology for NASA; the laboratory's primary function is the construction and operation of planetary robotic spacecraft, though it conducts Earth-orbit and astronomy missions. It is responsible for operating NASA's Deep Space Network. Among the laboratory's major active projects are the Mars Science Laboratory mission, the Mars Reconnaissance Orbiter, the Juno spacecraft orbiting Jupiter, the NuSTAR X-ray telescope, the SMAP satellite for earth surface soil moisture monitoring, the Spitzer Space Telescope, it is responsible for managing the JPL Small-Body Database, provides physical data and lists of publications for all known small Solar System bodies. The JPL's Space Flight Operations Facility and Twenty-Five-Foot Space Simulator are designated National Historic Landmarks.
JPL traces its beginnings to 1936 in the Guggenheim Aeronautical Laboratory at the California Institute of Technology when the first set of rocket experiments were carried out in the Arroyo Seco. Caltech graduate students Frank Malina, Qian Xuesen, Weld Arnold, Apollo M. O. Smith, along with Jack Parsons and Edward S. Forman, tested a small, alcohol-fueled motor to gather data for Malina's graduate thesis. Malina's thesis advisor was engineer/aerodynamicist Theodore von Kármán, who arranged for U. S. Army financial support for this "GALCIT Rocket Project" in 1939. In 1941, Parsons, Martin Summerfield, pilot Homer Bushey demonstrated the first jet-assisted takeoff rockets to the Army. In 1943, von Kármán, Malina and Forman established the Aerojet Corporation to manufacture JATO rockets; the project took on the name Jet Propulsion Laboratory in November 1943, formally becoming an Army facility operated under contract by the university. During JPL's Army years, the laboratory developed two deployed weapon systems, the MGM-5 Corporal and MGM-29 Sergeant intermediate-range ballistic missiles.
These missiles were the first US ballistic missiles developed at JPL. It developed a number of other weapons system prototypes, such as the Loki anti-aircraft missile system, the forerunner of the Aerobee sounding rocket. At various times, it carried out rocket testing at the White Sands Proving Ground, Edwards Air Force Base, Goldstone, California. In 1954, JPL teamed up with Wernher von Braun's engineers at the Army Ballistic Missile Agency's Redstone Arsenal in Huntsville, Alabama, to propose orbiting a satellite during the International Geophysical Year; the team lost that proposal to Project Vanguard, instead embarked on a classified project to demonstrate ablative re-entry technology using a Jupiter-C rocket. They carried out three successful sub-orbital flights in 1956 and 1957. Using a spare Juno I, the two organizations launched the United States' first satellite, Explorer 1, on January 31, 1958. JPL was transferred to NASA in December 1958, becoming the agency's primary planetary spacecraft center.
JPL engineers designed and operated Ranger and Surveyor missions to the Moon that prepared the way for Apollo. JPL led the way in interplanetary exploration with the Mariner missions to Venus and Mercury. In 1998, JPL opened the Near-Earth Object Program Office for NASA; as of 2013, it has found 95% of asteroids that are a kilometer or more in diameter that cross Earth's orbit. JPL was early to employ female mathematicians. In the 1940s and 1950s, using mechanical calculators, women in an all-female computations group performed trajectory calculations. In 1961, JPL hired Dana Ulery as the first female engineer to work alongside male engineers as part of the Ranger and Mariner mission tracking teams. JPL has been recognized four times by the Space Foundation: with the Douglas S. Morrow Public Outreach Award, given annually to an individual or organization that has made significant contributions to public awareness of space programs, in 1998; when it was founded, JPL's site was west of a rocky flood-plain – the Arroyo Seco riverbed – above the Devil's Gate dam in the northwestern panhandle of the city of Pasadena.
While the first few buildings were constructed in land bought from the city of Pasadena, subsequent buildings were constructed in neighboring unincorporated land that became part of La Cañada Flintridge. Nowadays, most of the 177 acres of the U. S. federal government-owned NASA property that makes up the JPL campus is located in La Cañada Flintridge. Despite this, JPL still uses a Pasadena address as its official mailing address; the city of La Cañada Flintridge was incorporated in 1976, well after JPL attained international recognition as a Pasadena institution. There has been occasional rivalry between the two cities over the issue of which one should be mentioned in the media as the home of the laboratory. There are 6,000 full-time Caltech employees, a few thousand additional contractors working on any given day. NASA has a resident office at the facility staffed by federal managers who oversee JPL's activities and work for NASA. There are some Caltech graduate students, college student interns and co-op students.
The JPL Education Office serves educators and students by providi
The Sun is the star at the center of the Solar System. It is a nearly perfect sphere of hot plasma, with internal convective motion that generates a magnetic field via a dynamo process, it is by far the most important source of energy for life on Earth. Its diameter is about 1.39 million kilometers, or 109 times that of Earth, its mass is about 330,000 times that of Earth. It accounts for about 99.86% of the total mass of the Solar System. Three quarters of the Sun's mass consists of hydrogen; the Sun is a G-type main-sequence star based on its spectral class. As such, it is informally and not accurately referred to as a yellow dwarf, it formed 4.6 billion years ago from the gravitational collapse of matter within a region of a large molecular cloud. Most of this matter gathered in the center, whereas the rest flattened into an orbiting disk that became the Solar System; the central mass became so hot and dense that it initiated nuclear fusion in its core. It is thought that all stars form by this process.
The Sun is middle-aged. It fuses about 600 million tons of hydrogen into helium every second, converting 4 million tons of matter into energy every second as a result; this energy, which can take between 10,000 and 170,000 years to escape from its core, is the source of the Sun's light and heat. In about 5 billion years, when hydrogen fusion in its core has diminished to the point at which the Sun is no longer in hydrostatic equilibrium, its core will undergo a marked increase in density and temperature while its outer layers expand to become a red giant, it is calculated that the Sun will become sufficiently large to engulf the current orbits of Mercury and Venus, render Earth uninhabitable. After this, it will shed its outer layers and become a dense type of cooling star known as a white dwarf, no longer produce energy by fusion, but still glow and give off heat from its previous fusion; the enormous effect of the Sun on Earth has been recognized since prehistoric times, the Sun has been regarded by some cultures as a deity.
The synodic rotation of Earth and its orbit around the Sun are the basis of solar calendars, one of, the predominant calendar in use today. The English proper name Sun may be related to south. Cognates to English sun appear in other Germanic languages, including Old Frisian sunne, Old Saxon sunna, Middle Dutch sonne, modern Dutch zon, Old High German sunna, modern German Sonne, Old Norse sunna, Gothic sunnō. All Germanic terms for the Sun stem from Proto-Germanic *sunnōn; the Latin name for the Sun, Sol, is not used in everyday English. Sol is used by planetary astronomers to refer to the duration of a solar day on another planet, such as Mars; the related word solar is the usual adjectival term used for the Sun, in terms such as solar day, solar eclipse, Solar System. A mean Earth solar day is 24 hours, whereas a mean Martian'sol' is 24 hours, 39 minutes, 35.244 seconds. The English weekday name Sunday stems from Old English and is a result of a Germanic interpretation of Latin dies solis, itself a translation of the Greek ἡμέρα ἡλίου.
The Sun is a G-type main-sequence star. The Sun has an absolute magnitude of +4.83, estimated to be brighter than about 85% of the stars in the Milky Way, most of which are red dwarfs. The Sun is heavy-element-rich, star; the formation of the Sun may have been triggered by shockwaves from more nearby supernovae. This is suggested by a high abundance of heavy elements in the Solar System, such as gold and uranium, relative to the abundances of these elements in so-called Population II, heavy-element-poor, stars; the heavy elements could most plausibly have been produced by endothermic nuclear reactions during a supernova, or by transmutation through neutron absorption within a massive second-generation star. The Sun is by far the brightest object in the Earth's sky, with an apparent magnitude of −26.74. This is about 13 billion times brighter than the next brightest star, which has an apparent magnitude of −1.46. The mean distance of the Sun's center to Earth's center is 1 astronomical unit, though the distance varies as Earth moves from perihelion in January to aphelion in July.
At this average distance, light travels from the Sun's horizon to Earth's horizon in about 8 minutes and 19 seconds, while light from the closest points of the Sun and Earth takes about two seconds less. The energy of this sunlight supports all life on Earth by photosynthesis, drives Earth's climate and weather; the Sun does not have a definite boundary, but its density decreases exponentially with increasing height above the photosphere. For the purpose of measurement, the Sun's radius is considered to be the distance from its center to the edge of the photosphere, the apparent visible surface of the Sun. By this measure, the Sun is a near-perfect sphere with an oblateness estimated at about 9 millionths, which means that its polar diameter differs from its equatorial diameter by only 10 kilometres; the tidal effect of the planets is weak and does not affect the shape of the Sun. The Sun rotates faster at its equator than at its poles; this differential rotation is caused by convective motion
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