1.
Minor planet
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A minor planet is an astronomical object in direct orbit around the Sun that is neither a planet nor exclusively classified as a comet. Minor planets can be dwarf planets, asteroids, trojans, centaurs, Kuiper belt objects, as of 2016, the orbits of 709,706 minor planets were archived at the Minor Planet Center,469,275 of which had received permanent numbers. The first minor planet to be discovered was Ceres in 1801, the term minor planet has been used since the 19th century to describe these objects. The term planetoid has also used, especially for larger objects such as those the International Astronomical Union has called dwarf planets since 2006. Historically, the asteroid, minor planet, and planetoid have been more or less synonymous. This terminology has become complicated by the discovery of numerous minor planets beyond the orbit of Jupiter. A Minor planet seen releasing gas may be classified as a comet. Before 2006, the IAU had officially used the term minor planet, during its 2006 meeting, the IAU reclassified minor planets and comets into dwarf planets and small Solar System bodies. Objects are called dwarf planets if their self-gravity is sufficient to achieve hydrostatic equilibrium, all other minor planets and comets are called small Solar System bodies. The IAU stated that the minor planet may still be used. However, for purposes of numbering and naming, the distinction between minor planet and comet is still used. Hundreds of thousands of planets have been discovered within the Solar System. The Minor Planet Center has documented over 167 million observations and 729,626 minor planets, of these,20,570 have official names. As of March 2017, the lowest-numbered unnamed minor planet is 1974 FV1, as of March 2017, the highest-numbered named minor planet is 458063 Gustavomuler. There are various broad minor-planet populations, Asteroids, traditionally, most have been bodies in the inner Solar System. Near-Earth asteroids, those whose orbits take them inside the orbit of Mars. Further subclassification of these, based on distance, is used, Apohele asteroids orbit inside of Earths perihelion distance. Aten asteroids, those that have semi-major axes of less than Earths, Apollo asteroids are those asteroids with a semimajor axis greater than Earths, while having a perihelion distance of 1.017 AU or less. Like Aten asteroids, Apollo asteroids are Earth-crossers, amor asteroids are those near-Earth asteroids that approach the orbit of Earth from beyond, but do not cross it
2.
Centaur (minor planet)
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Centaurs are minor planets with a semi-major axis between those of the outer planets. They have unstable orbits that cross or have crossed the orbits of one or more of the giant planets, Centaurs typically behave with characteristics of both asteroids and comets. They are named after the centaurs that were a mixture of horse. It has been estimated there are around 44,000 centaurs in the Solar System with diameters larger than 1 km. The first centaur to be discovered, under the definition of the Jet Propulsion Laboratory, however, they were not recognized as a distinct population until the discovery of 2060 Chiron in 1977. The largest confirmed centaur is 10199 Chariklo, which at 260 km in diameter is as big as a mid-sized main-belt asteroid, however, the lost centaur 1995 SN55 may be somewhat larger. No centaur has been photographed up close, although there is evidence that Saturns moon Phoebe, imaged by the Cassini probe in 2004, in addition, the Hubble Space Telescope has gleaned some information about the surface features of 8405 Asbolus. As of 2008, three centaurs have been found to display comet-like comas, Chiron,60558 Echeclus, and 166P/NEAT, Chiron and Echeclus are therefore classified as both asteroids and comets. Other centaurs, such as 52872 Okyrhoe and 2012 CG, are suspected of having shown comas, any centaur that is perturbed close enough to the Sun is expected to become a comet. The generic definition of a centaur is a body that orbits the Sun between Jupiter and Neptune and crosses the orbits of one or more of the giant planets. Though nowadays the MPC often lists centaurs and scattered disc objects together as a single group, the Jet Propulsion Laboratory similarly defines centaurs as having a semi-major axis, a, between those of Jupiter and Neptune. In contrast, the Deep Ecliptic Survey defines centaurs using a classification scheme. These classifications are based on the change in behavior of the present orbit when extended over 10 million years. The DES defines centaurs as non-resonant objects whose instantaneous perihelia are less than the osculating semi-major axis of Neptune at any time during the simulation and this definition is intended to be synonymous with planet-crossing orbits and to suggest comparatively short lifetimes in the current orbit. The collection The Solar System Beyond Neptune defines objects with an axis between those of Jupiter and Neptune and a Jupiter – Tisserands parameter above 3. The JPL Small-Body Database lists 324 centaurs, there are an additional 65 trans-Neptunian objects with a perihelion closer than the orbit of Uranus. The Committee on Small Body Nomenclature of the International Astronomical Union has not formally weighed in on either side of the debate, thus far, only the binary objects Ceto and Phorcys and Typhon and Echidna have been named according to the new policy. Other objects caught between these differences in classification methods include 944 Hidalgo which was discovered in 1920 and is listed as a centaur in the JPL Small-Body Database
3.
Perihelion and aphelion
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The perihelion is the point in the orbit of a celestial body where it is nearest to its orbital focus, generally a star. It is the opposite of aphelion, which is the point in the orbit where the body is farthest from its focus. The word perihelion stems from the Ancient Greek words peri, meaning around or surrounding, aphelion derives from the preposition apo, meaning away, off, apart. According to Keplers first law of motion, all planets, comets. Hence, a body has a closest and a farthest point from its parent object, that is, a perihelion. Each extreme is known as an apsis, orbital eccentricity measures the flatness of the orbit. Because of the distance at aphelion, only 93. 55% of the solar radiation from the Sun falls on a given area of land as does at perihelion. However, this fluctuation does not account for the seasons, as it is summer in the northern hemisphere when it is winter in the southern hemisphere and vice versa. Instead, seasons result from the tilt of Earths axis, which is 23.4 degrees away from perpendicular to the plane of Earths orbit around the sun. Winter falls on the hemisphere where sunlight strikes least directly, and summer falls where sunlight strikes most directly, in the northern hemisphere, summer occurs at the same time as aphelion. Despite this, there are larger land masses in the northern hemisphere, consequently, summers are 2.3 °C warmer in the northern hemisphere than in the southern hemisphere under similar conditions. Apsis Ellipse Solstice Dates and times of Earths perihelion and aphelion, 2000–2025 from the United States Naval Observatory
4.
Astronomical unit
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The astronomical unit is a unit of length, roughly the distance from Earth to the Sun. However, that varies as Earth orbits the Sun, from a maximum to a minimum. Originally conceived as the average of Earths aphelion and perihelion, it is now defined as exactly 149597870700 metres, the astronomical unit is used primarily as a convenient yardstick for measuring distances within the Solar System or around other stars. However, it is also a component in the definition of another unit of astronomical length. A variety of symbols and abbreviations have been in use for the astronomical unit. In a 1976 resolution, the International Astronomical Union used the symbol A for the astronomical unit, in 2006, the International Bureau of Weights and Measures recommended ua as the symbol for the unit. In 2012, the IAU, noting that various symbols are presently in use for the astronomical unit, in the 2014 revision of the SI Brochure, the BIPM used the unit symbol au. In ISO 80000-3, the symbol of the unit is ua. Earths orbit around the Sun is an ellipse, the semi-major axis of this ellipse is defined to be half of the straight line segment that joins the aphelion and perihelion. The centre of the sun lies on this line segment. In addition, it mapped out exactly the largest straight-line distance that Earth traverses over the course of a year, knowing Earths shift and a stars shift enabled the stars distance to be calculated. But all measurements are subject to some degree of error or uncertainty, improvements in precision have always been a key to improving astronomical understanding. Improving measurements were continually checked and cross-checked by means of our understanding of the laws of celestial mechanics, the expected positions and distances of objects at an established time are calculated from these laws, and assembled into a collection of data called an ephemeris. NASAs Jet Propulsion Laboratory provides one of several ephemeris computation services, in 1976, in order to establish a yet more precise measure for the astronomical unit, the IAU formally adopted a new definition. Equivalently, by definition, one AU is the radius of an unperturbed circular Newtonian orbit about the sun of a particle having infinitesimal mass. As with all measurements, these rely on measuring the time taken for photons to be reflected from an object. However, for precision the calculations require adjustment for such as the motions of the probe. In addition, the measurement of the time itself must be translated to a scale that accounts for relativistic time dilation
5.
Semi-major and semi-minor axes
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In geometry, the major axis of an ellipse is its longest diameter, a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semi-major axis is one half of the axis, and thus runs from the centre, through a focus. Essentially, it is the radius of an orbit at the two most distant points. For the special case of a circle, the axis is the radius. One can think of the axis as an ellipses long radius. The semi-major axis of a hyperbola is, depending on the convention, thus it is the distance from the center to either vertex of the hyperbola. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction. Thus a and b tend to infinity, a faster than b, the semi-minor axis is a line segment associated with most conic sections that is at right angles with the semi-major axis and has one end at the center of the conic section. It is one of the axes of symmetry for the curve, in an ellipse, the one, in a hyperbola. The semi-major axis is the value of the maximum and minimum distances r max and r min of the ellipse from a focus — that is. In astronomy these extreme points are called apsis, the semi-minor axis of an ellipse is the geometric mean of these distances, b = r max r min. The eccentricity of an ellipse is defined as e =1 − b 2 a 2 so r min = a, r max = a. Now consider the equation in polar coordinates, with one focus at the origin, the mean value of r = ℓ / and r = ℓ /, for θ = π and θ =0 is a = ℓ1 − e 2. In an ellipse, the axis is the geometric mean of the distance from the center to either focus. The semi-minor axis of an ellipse runs from the center of the ellipse to the edge of the ellipse, the semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the axis that connects two points on the ellipses edge. The semi-minor axis b is related to the axis a through the eccentricity e. A parabola can be obtained as the limit of a sequence of ellipses where one focus is fixed as the other is allowed to move arbitrarily far away in one direction
6.
Orbital eccentricity
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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 an orbit, values between 0 and 1 form an elliptical orbit,1 is a parabolic escape orbit. The term derives its name from the parameters of conic sections and it is normally used for the isolated two-body problem, but extensions exist for objects following a 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 that defines its shape. 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 have zero angular momentum and hence eccentricity equal to one, keeping the energy constant and reducing the angular momentum, elliptic, parabolic, and hyperbolic orbits each tend to the corresponding type of radial trajectory while e tends to 1. For a repulsive force only the trajectory, including the radial version, is applicable. For elliptical orbits, a simple proof shows that arcsin yields the projection angle of a circle to an ellipse of eccentricity e. For example, to view the eccentricity of the planet Mercury, next, tilt any circular object by that angle and the apparent ellipse projected to your eye will be of that same eccentricity. From Medieval Latin eccentricus, derived from Greek ἔκκεντρος ekkentros out of the center, from ἐκ- ek-, eccentric first appeared in English in 1551, with the definition a circle in which the earth, sun. Five years later, in 1556, a form of the word was added. The eccentricity of an orbit can be calculated from the state vectors as the magnitude of the eccentricity vector, e = | e | where. For elliptical orbits it can also 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, rp is the radius at periapsis. For Earths annual orbit path, ra/rp ratio = longest_radius / shortest_radius ≈1.034 relative to center point of path, the eccentricity of the Earths orbit is currently about 0.0167, the Earths orbit is nearly circular. Venus and Neptune have even lower eccentricity, over hundreds of thousands of years, the eccentricity of the Earths orbit varies from nearly 0.0034 to almost 0.058 as a result of gravitational attractions among the planets. The table lists the values for all planets and dwarf planets, Mercury has the greatest orbital eccentricity of any planet in the Solar System. Such eccentricity is sufficient for Mercury to receive twice as much solar irradiation at perihelion compared to aphelion, before its demotion from planet status in 2006, Pluto was considered to be the planet with the most eccentric orbit
7.
Mean anomaly
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In celestial mechanics, the mean anomaly is an angle used in calculating the position of a body in an elliptical orbit in the classical two-body problem. Define T as the time required for a body to complete one orbit. In time T, the radius vector sweeps out 2π radians or 360°. The average rate of sweep, n, is then n =2 π T or n =360 ∘ T, define τ as the time at which the body is at the pericenter. From the above definitions, a new quantity, M, the mean anomaly can be defined M = n, because the rate of increase, n, is a constant average, the mean anomaly increases uniformly from 0 to 2π radians or 0° to 360° during each orbit. It is equal to 0 when the body is at the pericenter, π radians at the apocenter, if the mean anomaly is known at any given instant, it can be calculated at any later instant by simply adding n δt where δt represents the time difference. Mean anomaly does not measure an angle between any physical objects and it is simply a convenient uniform measure of how far around its orbit a body has progressed since pericenter. The mean anomaly is one of three parameters that define a position along an orbit, the other two being the eccentric anomaly and the true anomaly. Define l as the longitude, the angular distance of the body from the same reference direction. Thus mean anomaly is also M = l − ϖ, mean angular motion can also be expressed, n = μ a 3, where μ is a gravitational parameter which varies with the masses of the objects, and a is the semi-major axis of the orbit. Mean anomaly can then be expanded, M = μ a 3, and here mean anomaly represents uniform angular motion on a circle of radius a
8.
Degree (angle)
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A degree, usually denoted by °, is a measurement of a plane angle, defined so that a full rotation is 360 degrees. It is not an SI unit, as the SI unit of measure is the radian. Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians, the original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the path over the course of the year. Some ancient calendars, such as the Persian calendar, used 360 days for a year, the use of a calendar with 360 days may be related to the use of sexagesimal numbers. The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle, a chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree, Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically. Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes, eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts. Furthermore, it is divisible by every number from 1 to 10 except 7 and this property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24-hour day convention. Finally, it may be the case more than one of these factors has come into play. For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for geographic coordinates, degree measurements may be written using decimal degrees, with the symbol behind the decimals. Alternatively, the sexagesimal unit subdivisions can be used. One degree is divided into 60 minutes, and one minute into 60 seconds, use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the arcminute and arcsecond, are represented by a single and double prime. For example,40. 1875° = 40° 11′ 15″, or, using quotation mark characters, additional precision can be provided using decimals for the arcseconds component. The older system of thirds, fourths, etc. which continues the sexagesimal unit subdivision, was used by al-Kashi and other ancient astronomers, but is rarely used today
9.
Orbital inclination
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Orbital inclination measures the tilt of an objects orbit around a celestial body. It is expressed as the angle between a plane and the orbital plane or axis of direction of the orbiting object. For a satellite orbiting the Earth directly above the equator, the plane of the orbit is the same as the Earths equatorial plane. The general case is that the orbit is tilted, it spends half an orbit over the northern hemisphere. If the orbit swung between 20° north latitude and 20° south latitude, then its orbital inclination would be 20°, the inclination is one of the six orbital elements describing the shape and orientation of a celestial orbit. It is the angle between the plane and the plane of reference, normally stated in degrees. For a satellite orbiting a planet, the plane of reference is usually the plane containing the planets equator, for planets in the Solar System, the plane of reference is usually the ecliptic, the plane in which the Earth orbits the Sun. This reference plane is most practical for Earth-based observers, therefore, Earths inclination is, by definition, zero. Inclination could instead be measured with respect to another plane, such as the Suns equator or the invariable plane, the inclination of orbits of natural or artificial satellites is measured relative to the equatorial plane of the body they orbit, if they orbit sufficiently closely. The equatorial plane is the perpendicular to the axis of rotation of the central body. An inclination of 30° could also be described using an angle of 150°, the convention is that the normal orbit is prograde, an orbit in the same direction as the planet rotates. Inclinations greater than 90° describe retrograde orbits, thus, An inclination of 0° means the orbiting body has a prograde orbit in the planets equatorial plane. An inclination greater than 0° and less than 90° also describe prograde orbits, an inclination of 63. 4° is often called a critical inclination, when describing artificial satellites orbiting the Earth, because they have zero apogee drift. An inclination of exactly 90° is an orbit, in which the spacecraft passes over the north and south poles of the planet. An inclination greater than 90° and less than 180° is a retrograde orbit, an inclination of exactly 180° is a retrograde equatorial orbit. For gas giants, the orbits of moons tend to be aligned with the giant planets equator, the inclination of exoplanets or members of multiple stars is the angle of the plane of the orbit relative to the plane perpendicular to the line-of-sight from Earth to the object. An inclination of 0° is an orbit, meaning the plane of its orbit is parallel to the sky. An inclination of 90° is an orbit, meaning the plane of its orbit is perpendicular to the sky
10.
Longitude of the ascending node
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The longitude of the ascending node is one of the orbital elements used to specify the orbit of an object in space. It is the angle from a direction, called the origin of longitude, to the direction of the ascending node. The ascending node is the point where the orbit of the passes through the plane of reference. Commonly used reference planes and origins of longitude include, For a geocentric orbit, Earths equatorial plane as the plane. In this case, the longitude is called the right ascension of the ascending node. The angle is measured eastwards from the First Point of Aries to the node, for a heliocentric orbit, the ecliptic as the reference plane, and the First Point of Aries as the origin of longitude. The angle is measured counterclockwise from the First Point of Aries to the node, the angle is measured eastwards from north to the node. pp.40,72,137, chap. In the case of a star known only from visual observations, it is not possible to tell which node is ascending. In this case the orbital parameter which is recorded is the longitude of the node, Ω, here, n=<nx, ny, nz> is a vector pointing towards the ascending node. The reference plane is assumed to be the xy-plane, and the origin of longitude is taken to be the positive x-axis, K is the unit vector, which is the normal vector to the xy reference plane. For non-inclined orbits, Ω is undefined, for computation it is then, by convention, set equal to zero, that is, the ascending node is placed in the reference direction, which is equivalent to letting n point towards the positive x-axis. Kepler orbits Equinox Orbital node perturbation of the plane can cause revolution of the ascending node
11.
Argument of periapsis
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The argument of periapsis, symbolized as ω, is one of the orbital elements of an orbiting body. Parametrically, ω is the angle from the ascending node to its periapsis. For specific types of orbits, words such as perihelion, perigee, periastron, an argument of periapsis of 0° means that the orbiting body will be at its closest approach to the central body at the same moment that it crosses the plane of reference from South to North. An argument of periapsis of 90° means that the body will reach periapsis at its northmost distance from the plane of reference. Adding the argument of periapsis to the longitude of the ascending node gives the longitude of the periapsis, however, especially in discussions of binary stars and exoplanets, the terms longitude of periapsis or longitude of periastron are often used synonymously with argument of periapsis. In the case of equatorial orbits, the argument is strictly undefined, where, ex and ey are the x- and y-components of the eccentricity vector e. In the case of circular orbits it is assumed that the periapsis is placed at the ascending node. Kepler orbit Orbital mechanics Orbital node
12.
Apparent magnitude
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The apparent magnitude of a celestial object is a number that is a measure of its brightness as seen by an observer on Earth. The brighter an object appears, the lower its magnitude value, the Sun, at apparent magnitude of −27, is the brightest object in the sky. It is adjusted to the value it would have in the absence of the atmosphere, furthermore, the magnitude scale is logarithmic, a difference of one in magnitude corresponds to a change in brightness by a factor of 5√100, or about 2.512. The measurement of apparent magnitudes or brightnesses of celestial objects is known as photometry, apparent magnitudes are used to quantify the brightness of sources at ultraviolet, visible, and infrared wavelengths. An apparent magnitude is measured in a specific passband corresponding to some photometric system such as the UBV system. In standard astronomical notation, an apparent magnitude in the V filter band would be denoted either as mV or often simply as V, the scale used to indicate magnitude originates in the Hellenistic practice of dividing stars visible to the naked eye into six magnitudes. The brightest stars in the sky were said to be of first magnitude, whereas the faintest were of sixth magnitude. Each grade of magnitude was considered twice the brightness of the following grade and this rather crude scale for the brightness of stars was popularized by Ptolemy in his Almagest, and is generally believed to have originated with Hipparchus. This implies that a star of magnitude m is 2.512 times as bright as a star of magnitude m +1 and this figure, the fifth root of 100, became known as Pogsons Ratio. The zero point of Pogsons scale was defined by assigning Polaris a magnitude of exactly 2. However, with the advent of infrared astronomy it was revealed that Vegas radiation includes an Infrared excess presumably due to a disk consisting of dust at warm temperatures. At shorter wavelengths, there is negligible emission from dust at these temperatures, however, in order to properly extend the magnitude scale further into the infrared, this peculiarity of Vega should not affect the definition of the magnitude scale. Therefore, the scale was extrapolated to all wavelengths on the basis of the black body radiation curve for an ideal stellar surface at 11000 K uncontaminated by circumstellar radiation. On this basis the spectral irradiance for the zero magnitude point, with the modern magnitude systems, brightness over a very wide range is specified according to the logarithmic definition detailed below, using this zero reference. In practice such apparent magnitudes do not exceed 30, astronomers have developed other photometric zeropoint systems as alternatives to the Vega system. The AB magnitude zeropoint is defined such that an objects AB, the dimmer an object appears, the higher the numerical value given to its apparent magnitude, with a difference of 5 magnitudes corresponding to a brightness factor of exactly 100. Since an increase of 5 magnitudes corresponds to a decrease in brightness by a factor of exactly 100, each magnitude increase implies a decrease in brightness by the factor 5√100 ≈2.512. Inverting the above formula, a magnitude difference m1 − m2 = Δm implies a brightness factor of F2 F1 =100 Δ m 5 =100.4 Δ m ≈2.512 Δ m
13.
Numerical integration
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This article focuses on calculation of definite integrals. The term numerical quadrature is more or less a synonym for numerical integration, Some authors refer to numerical integration over more than one dimension as cubature, others take quadrature to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral ∫ a b f d x to a degree of accuracy. If f is a smooth function integrated over a number of dimensions. The term numerical integration first appears in 1915 in the publication A Course in Interpolation, Quadrature is a historical mathematical term that means calculating area. Quadrature problems have served as one of the sources of mathematical analysis. Mathematicians of Ancient Greece, according to the Pythagorean doctrine, understood calculation of area as the process of constructing geometrically a square having the same area and that is why the process was named quadrature. For example, a quadrature of the circle, Lune of Hippocrates and this construction must be performed only by means of compass and straightedge. The ancient Babylonians used the trapezoidal rule to integrate the motion of Jupiter along the ecliptic, for a quadrature of a rectangle with the sides a and b it is necessary to construct a square with the side x = a b. For this purpose it is possible to use the fact, if we draw the circle with the sum of a and b as the diameter. The similar geometrical construction solves a problem of a quadrature for a parallelogram, problems of quadrature for curvilinear figures are much more difficult. The quadrature of the circle with compass and straightedge had been proved in the 19th century to be impossible, nevertheless, for some figures a quadrature can be performed. The quadratures of a surface and a parabola segment done by Archimedes became the highest achievement of the antique analysis. The area of the surface of a sphere is equal to quadruple the area of a circle of this sphere. The area of a segment of the cut from it by a straight line is 4/3 the area of the triangle inscribed in this segment. For the proof of the results Archimedes used the Method of exhaustion of Eudoxus, in medieval Europe the quadrature meant calculation of area by any method. More often the Method of indivisibles was used, it was less rigorous, john Wallis algebrised this method, he wrote in his Arithmetica Infinitorum series that we now call the definite integral, and he calculated their values. Isaac Barrow and James Gregory made further progress, quadratures for some algebraic curves, christiaan Huygens successfully performed a quadrature of some Solids of revolution
14.
Curve fitting
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Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a smooth function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, most commonly, one fits a function of the form y=f. Starting with a first degree polynomial equation, y = a x + b and this is a line with slope a. A line will connect any two points, so a first degree polynomial equation is an exact fit through any two points with distinct x coordinates. If the order of the equation is increased to a second degree polynomial and this will exactly fit a simple curve to three points. If the order of the equation is increased to a third degree polynomial and this will exactly fit four points. A more general statement would be to say it will exactly fit four constraints, each constraint can be a point, angle, or curvature. Angle and curvature constraints are most often added to the ends of a curve, identical end conditions are frequently used to ensure a smooth transition between polynomial curves contained within a single spline. Higher-order constraints, such as the change in the rate of curvature, many other combinations of constraints are possible for these and for higher order polynomial equations. If there are more than n +1 constraints, the curve can still be run through those constraints. An exact fit to all constraints is not certain, in general, however, some method is then needed to evaluate each approximation. The least squares method is one way to compare the deviations, there are several reasons given to get an approximate fit when it is possible to simply increase the degree of the polynomial equation and get an exact match. Even if a match exists, it does not necessarily follow that it can be readily discovered. Depending on the algorithm used there may be a divergent case and this situation might require an approximate solution. The effect of averaging out questionable data points in a sample, rather than distorting the curve to fit them exactly, runges phenomenon, high order polynomials can be highly oscillatory. If a curve runs through two points A and B, it would be expected that the curve would run somewhat near the midpoint of A and B, as well
15.
Two-body problem
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In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, the two-body problem can be re-formulated as two one-body problems, a trivial one and one that involves solving for the motion of one particle in an external potential. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved, by contrast, the three-body problem cannot be solved in terms of first integrals, except in special cases. Let x1 and x2 be the positions of the two bodies, and m1 and m2 be their masses. The goal is to determine the trajectories x1 and x2 for all t, given the initial positions x1 and x2. The two dots on top of the x position vectors denote their second derivative with respect to time, adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. Adding equations and results in an equation describing the center of mass motion, by contrast, subtracting equation from equation results in an equation that describes how the vector r = x1 − x2 between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories x1 and x2. The resulting equation, R ¨ =0 shows that the velocity V = dR/dt of the center of mass is constant, hence, the position R of the center of mass can be determined at all times from the initial positions and velocities. The motion of two bodies with respect to each other always lies in a plane, introducing the assumption that the force between two particles acts along the line between their positions, it follows that r × F =0 and the angular momentum vector L is constant. We now have, μ r ¨ = F r ^, Kepler orbit Energy drift Equation of the center Eulers three-body problem Gravitational two-body problem Kepler problem n-body problem Virial theorem Two-body problem Landau LD, Lifshitz EM. Two-body problem at Eric Weissteins World of Physics
16.
Barycenter
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The barycenter is the center of mass of two or more bodies that are orbiting each other, or the point around which they both orbit. It is an important concept in such as astronomy and astrophysics. The distance from a center of mass to the barycenter can be calculated as a simple two-body problem. In cases where one of the two objects is more massive than the other, the barycenter will typically be located within the more massive object. Rather than appearing to orbit a center of mass with the smaller body. This is the case for the Earth–Moon system, where the barycenter is located on average 4,671 km from the Earths center, when the two bodies are of similar masses, the barycenter will generally be located between them and both bodies will follow an orbit around it. This is the case for Pluto and Charon, as well as for many binary asteroids and it is also the case for Jupiter and the Sun, despite the thousandfold difference in mass, due to the relatively large distance between them. In astronomy, barycentric coordinates are non-rotating coordinates with the origin at the center of mass of two or more bodies, the International Celestial Reference System is a barycentric one, based on the barycenter of the Solar System. In geometry, the barycenter is synonymous with centroid, the geometric center of a two-dimensional shape. The barycenter is one of the foci of the orbit of each body. This is an important concept in the fields of astronomy and astrophysics. If a is the distance between the centers of the two bodies, r1 is the axis of the primarys orbit around the barycenter. When the barycenter is located within the massive body, that body will appear to wobble rather than to follow a discernible orbit. The following table sets out some examples from the Solar System, figures are given rounded to three significant figures. If Jupiter had Mercurys orbit, the Sun–Jupiter barycenter would be approximately 55,000 km from the center of the Sun, but even if the Earth had Eris orbit, the Sun–Earth barycenter would still be within the Sun. To calculate the motion of the Sun, you would need to sum all the influences from all the planets, comets, asteroids. If all the planets were aligned on the side of the Sun. The calculations above are based on the distance between the bodies and yield the mean value r1
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National Optical Astronomy Observatory
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The National Optical Astronomy Observatory is the United States national observatory for ground based nighttime ultraviolet-optical-infrared astronomy. The National Science Foundation funds NOAO to provide forefront astronomical research facilities for US astronomers, however, professional astronomers from any country in the world may apply to use the telescopes operated by NOAO under the NSFs open skies policy. Astronomers submit proposals for review to gain access to the telescopes which are scheduled every night of the year for observations. The combination of open access and the merit based science proposal process makes NOAO unique in the world. The NOAO headquarters are located in Tucson, Arizona and are co-located with the headquarters of the National Solar Observatory, the NOAO is operated by the Association of Universities for Research in Astronomy, under a cooperative agreement with the NSF. NOAO operates world class research telescopes in both the northern and southern hemispheres and these telescopes are located at Kitt Peak and Cerro Tololo in the US and Chile, respectively. Complemented with similar instruments, the two sites allow US astronomers to make observations over the entire sky. Instrumentation includes optical to near infrared wavelength cameras and spectrometers, CTIO has a base and office facility in the seaside town of La Serena, Chile. The CTIO telescopes are located some 70 km inland in the foothills of the Chilean Andes, access to the observatory is made through the picturesque Elqui Valley. The Blanco 4m played the role in discovery of Dark Energy. The Blanco began hosting a new 3-degree field of view called the Dark Energy Camera, also known as DECam. This camera is being built at Fermilab in Chicago, USA, CTIO operates, and is a partner in the 4. 1m Southern Astrophysical Research Telescope. SOAR concentrates on high resolution observations and will soon deploy an adaptive optics module to help support such observations. KPNO is located near Tucson, AZ, USA, the mountain, Kitt Peak, is part of the tribal lands of the Native American people the Tohono Oodham. The mountain has been leased from the Tohono Oodham since 1958, the native name for the mountain is loligam which means manzanita. The observatory was established in 1958, and its largest telescope, a new wide field imager working at near infrared wavelengths has been deployed to advance studies of galactic star formation, cosmology, and the structure and evolution of galaxies. NOAO also manages US participation in the international Gemini Observatory, Gemini is a partnership of Argentina, Australia, Brazil, Canada, the United Kingdom, and the United States. The US holds a 50% share of the project which provides public access time on each of Geminis two 8m telescopes, one telescope is located near CTIO in Chile, and the other is located on the island of Hawaii
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Marc William Buie
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In 2008 Marc Buie moved to Boulder, Colorado to work at the Southwest Research Institute in the Space Science Department. Buie grew up in Baton Rouge, Louisiana and received his B. Sc. in physics from Louisiana State University in 1980 and he then switched fields and earned his Ph. D. in Planetary Science from the University of Arizona in 1984. Dr. Buie was a fellow at the University of Hawaii from 1985 to 1988. Dr. Buie joined the staff at Lowell Observatory in 1991, since 1983 Pluto has been a central theme of research done by Buie, who has published over 85 scientific papers and journal articles. His first result was to prove that the methane visible on Pluto was on its surface and he is also one of the co-discoverers of Plutos moons, Nix and Hydra. He has been working with the Deep Ecliptic Survey team who have been responsible for the discovery of over 1,000 of these distant objects, beyond the work of just locating these objects, he additionally seeks to develop a better picture of the structure and nature of them. A spin-off project from this endeavor is his participation in the project to locate a Kuiper belt object that is within the range of the New Horizons mission once it passes by Pluto. In an effort closer to home, he also studies near-Earth asteroids to try to more about these potentially dangerous solar system neighbors. Most of these research efforts involve the use of Lowell Observatory telescopes in addition to use of the Hubble. The inner main-belt asteroid 7553 Buie was named in the honor on 28 July 1999. He is also profiled as part of an article on Pluto in Air & Space Smithsonian magazine, from the desk of Marc W. Buie page from Lowell Portrait of Marc Buie by Dan Coogan
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Jet Propulsion Laboratory
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The Jet Propulsion Laboratory is a federally funded research and development center and NASA field center in La Cañada Flintridge, California and Pasadena, California, United States. The JPL is managed by the nearby California Institute of Technology for NASA, the laboratorys primary function is the construction and operation of planetary robotic spacecraft, though it also conducts Earth-orbit and astronomy missions. It is also responsible for operating NASAs Deep Space Network and they are also responsible for managing the JPL Small-Body Database, and provides physical data and lists of publications for all known small Solar System bodies. The JPLs 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. Malinas 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, Malina, Parsons, Forman, Martin Summerfield, in 1943, von Kármán, Malina, Parsons, and Forman established the Aerojet Corporation to manufacture JATO motors. The project took on the name Jet Propulsion Laboratory in November 1943, during JPLs 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 and it also developed a number of other weapons system prototypes, such as the Loki anti-aircraft missile system, and the forerunner of the Aerobee sounding rocket. At various times, it carried out testing at the White Sands Proving Ground, Edwards Air Force Base. A lunar lander was developed in 1938-39 which influenced design of the Apollo Lunar Module in the 1960s. The team lost that proposal to Project Vanguard, and instead embarked on a project to demonstrate ablative re-entry technology using a Jupiter-C rocket. They carried out three successful flights in 1956 and 1957. Using a spare Juno I, the two organizations then launched the United States first satellite, Explorer 1, on February 1,1958, JPL was transferred to NASA in December 1958, becoming the agencys primary planetary spacecraft center. JPL engineers designed and operated Ranger and Surveyor missions to the Moon that prepared the way for Apollo, JPL also led the way in interplanetary exploration with the Mariner missions to Venus, Mars, 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 Earths orbit. JPL was early to employ women 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 their first woman engineer to work alongside male engineers as part of the Ranger and Mariner mission tracking teams, when founded, JPLs site was a rocky flood-plain just outside the city limits of Pasadena. Almost all of the 177 acres of the U. S, the city of La Cañada Flintridge, California was incorporated in 1976, well after JPL attained international recognition with a Pasadena address
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Center of mass
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The distribution of mass is balanced around the center of mass and the average of the weighted position coordinates of the distributed mass defines its coordinates. Calculations in mechanics are simplified when formulated with respect to the center of mass. It is a point where entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the equivalent of a given object for application of Newtons laws of motion. In the case of a rigid body, the center of mass is fixed in relation to the body. The center of mass may be located outside the body, as is sometimes the case for hollow or open-shaped objects. In the case of a distribution of separate bodies, such as the planets of the Solar System, in orbital mechanics, the equations of motion of planets are formulated as point masses located at the centers of mass. The center of mass frame is a frame in which the center of mass of a system is at rest with respect to the origin of the coordinate system. The concept of center of mass in the form of the center of gravity was first introduced by the ancient Greek physicist, mathematician, and engineer Archimedes of Syracuse. He worked with simplified assumptions about gravity that amount to a uniform field, in work on floating bodies he demonstrated that the orientation of a floating object is the one that makes its center of mass as low as possible. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes, Newtons second law is reformulated with respect to the center of mass in Eulers first law. The center of mass is the point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. In analogy to statistics, the center of mass is the location of a distribution of mass in space. Solving this equation for R yields the formula R =1 M ∑ i =1 n m i r i, solve this equation for the coordinates R to obtain R =1 M ∭ Q ρ r d V, where M is the total mass in the volume. If a continuous mass distribution has density, which means ρ is constant. The center of mass is not generally the point at which a plane separates the distribution of mass into two equal halves, in analogy with statistics, the median is not the same as the mean. The coordinates R of the center of mass of a system, P1 and P2, with masses m1. The percentages of mass at each point can be viewed as projective coordinates of the point R on this line, another way of interpreting the process here is the mechanical balancing of moments about an arbitrary point
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Small Solar System body
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A Small Solar System Body is an object in the Solar System that is neither a planet, nor a dwarf planet, nor a natural satellite. The term was first defined in 2006 by the International Astronomical Union, all other objects, except satellites, orbiting the Sun shall be referred to collectively as Small Solar System Bodies. These currently include most of the Solar System asteroids, most Trans-Neptunian Objects, comets and this encompasses all comets and all minor planets other than those that are dwarf planets. Except for the largest, which are in equilibrium, natural satellites differ from small Solar System bodies not in size. The orbits of satellites are not centered on the Sun, but around other Solar System objects such as planets, dwarf planets. Some of the larger small Solar System bodies may be reclassified in future as dwarf planets, the orbits of the vast majority of small Solar System bodies are located in two distinct areas, namely the asteroid belt and the Kuiper belt. These two belts possess some internal structure related to perturbations by the planets, and have fairly loosely defined boundaries. Other areas of the Solar System also encompass small bodies in smaller concentrations and these include the near-Earth asteroids, centaurs, comets, and scattered disc objects
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Minor-planet moon
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A minor-planet moon is an astronomical object that orbits a minor planet as its natural satellite. It is thought that many asteroids and Kuiper belt objects may possess moons, 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 minor planet 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, many hours later, several observations were reported in Mexico attributed to the occultation by Hebe itself. Although not confirmed this documents the first formally documented case of a companion of an asteroid. As of October 2016, there are over 300 minor planets known to have moons, in addition to the terms satellite and moon, the term binary is sometimes used for minor planets with moons, and triple for minor planets with two moons. If one object is much bigger it can be referred to as the primary, 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, small satellites are often referred to as moonlets. As of February 2017, over 330 moons of planets have been discovered. For example, in 1978, stellar occultation observations were claimed as evidence of a satellite for the asteroid 532 Herculina, however, later more-detailed imaging by the Hubble Telescope did not reveal a satellite, and the current consensus is that Herculina does not have a significant satellite. There were other reports of asteroids having companions in the following years. 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 asteroids in the Jupiter trojans. The first trans-Neptunian binary after Pluto–Charon,1998 WW31, was resolved in 2002. Triple asteroids, or trinary asteroids, are known since 2005 and this was followed by the discovery of a second moon orbiting 45 Eugenia. Also in 2005, the Kuiper belt object Haumea was discovered to have two moons, making it the second KBO 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 trinary asteroids were discovered, more continue to be discovered at a rate of one a year. Most recently discovered was a moon orbiting the belt asteroid 130 Elektra. List of multiple planets, The data about the populations of binary objects are still patchy
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Asteroid
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Asteroids are minor planets, especially those of the inner Solar System. The larger ones have also been called planetoids and these terms have historically been applied to any astronomical object orbiting the Sun that did not show the disc of a planet and was not observed to have the characteristics of an active comet. As minor planets in the outer Solar System were discovered and found to have volatile-based surfaces that resemble those of comets, in this article, the term asteroid refers to the minor planets of the inner Solar System including those co-orbital with Jupiter. There are millions of asteroids, many thought to be the remnants of planetesimals. The large majority of known asteroids orbit in the belt between the orbits of Mars and Jupiter, or are co-orbital with Jupiter. However, other orbital families exist with significant populations, including the near-Earth objects, individual asteroids are classified by their characteristic spectra, with the majority falling into three main groups, C-type, M-type, and S-type. These were named after and are identified with carbon-rich, metallic. The size of asteroids varies greatly, some reaching as much as 1000 km across, asteroids are differentiated from comets and meteoroids. In the case of comets, the difference is one of composition, while asteroids are composed of mineral and rock, comets are composed of dust. In addition, asteroids formed closer to the sun, preventing the development of the aforementioned cometary ice, the difference between asteroids and meteoroids is mainly one of size, meteoroids have a diameter of less than one meter, whereas asteroids have a diameter of greater than one meter. Finally, meteoroids can be composed of either cometary or asteroidal materials, only one asteroid,4 Vesta, which has a relatively reflective surface, is normally visible to the naked eye, and this only in very dark skies when it is favorably positioned. Rarely, small asteroids passing close to Earth may be visible to the eye for a short time. As of March 2016, the Minor Planet Center had data on more than 1.3 million objects in the inner and outer Solar System, the United Nations declared June 30 as International Asteroid Day to educate the public about asteroids. The date of International Asteroid Day commemorates the anniversary of the Tunguska asteroid impact over Siberia, the first asteroid to be discovered, Ceres, was found in 1801 by Giuseppe Piazzi, and was originally considered to be a new planet. In the early half of the nineteenth century, the terms asteroid. Asteroid discovery methods have improved over the past two centuries. This task required that hand-drawn sky charts be prepared for all stars in the band down to an agreed-upon limit of faintness. On subsequent nights, the sky would be charted again and any moving object would, hopefully, the expected motion of the missing planet was about 30 seconds of arc per hour, readily discernible by observers
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Aten asteroid
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The Aten asteroids are a group of asteroids, whose orbit brings them into proximity with Earth. The group is named after 2062 Aten, the first of its kind, since then, more than 1,000 Atens have been discovered, of which many are classified as potentially hazardous asteroids. For a list of existing articles, see Aten asteroids and List of Aten asteroids, Aten asteroids are defined by having a semi-major axis of less than one astronomical unit, the average distance from the Earth to the Sun. They also have a greater than 0.983 AU. Asteroids orbits can be highly eccentric, an Aten orbit need not be entirely contained within Earths orbit, as nearly all known Aten asteroids have their aphelion greater than 1 AU although their semi-major axis is less than 1 AU. Observation of objects inferior to the Earths orbit is difficult and this difficulty may be the cause of some sampling bias in the apparent preponderance of eccentric Atens, Aten asteroids account for only about 6% of the known near-Earth asteroid population. Many more Apollo-class asteroids are known than Aten-class asteroids, possibly because of the sampling bias, the shortest semi-major axis for any known Aten asteroid is 2008 EY5 at 0.626 AU. A very small possibility of impact remained for 2036, but this was also eliminated, there are also sixteen known Apohele asteroids, traditionally listed as a subclass of Atens, but generally regarded a separate class of their own. Unlike Atens, Apoheles permanently stay within Earths orbit and do not cross it
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Asteroid belt
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The asteroid belt is the circumstellar disc in the Solar System located roughly between the orbits of the planets Mars and Jupiter. It is occupied by numerous irregularly shaped bodies called asteroids or minor planets, the asteroid belt is also termed the main asteroid belt or main belt to distinguish it from other asteroid populations in the Solar System such as near-Earth asteroids and trojan asteroids. About half the mass of the belt is contained in the four largest asteroids, Ceres, Vesta, Pallas, the total mass of the asteroid belt is approximately 4% that of the Moon, or 22% that of Pluto, and roughly twice that of Plutos moon Charon. Ceres, the belts only dwarf planet, is about 950 km in diameter, whereas Vesta, Pallas. The remaining bodies range down to the size of a dust particle, the asteroid material is so thinly distributed that numerous unmanned spacecraft have traversed it without incident. Nonetheless, collisions between large asteroids do occur, and these can form a family whose members have similar orbital characteristics. Individual asteroids within the belt are categorized by their spectra. The asteroid belt formed from the solar nebula as a group of planetesimals. Planetesimals are the precursors of the protoplanets. Between Mars and Jupiter, however, gravitational perturbations from Jupiter imbued the protoplanets with too much energy for them to accrete into a planet. Collisions became too violent, and instead of fusing together, the planetesimals, as a result,99. 9% of the asteroid belts original mass was lost in the first 100 million years of the Solar Systems history. Some fragments eventually found their way into the inner Solar System, Asteroid orbits continue to be appreciably perturbed whenever their period of revolution about the Sun forms an orbital resonance with Jupiter. At these orbital distances, a Kirkwood gap occurs as they are swept into other orbits. Classes of small Solar System bodies in other regions are the objects, the centaurs, the Kuiper belt objects, the scattered disc objects, the sednoids. On 22 January 2014, ESA scientists reported the detection, for the first definitive time, of water vapor on Ceres, the detection was made by using the far-infrared abilities of the Herschel Space Observatory. The finding was unexpected because comets, not asteroids, are considered to sprout jets. According to one of the scientists, The lines are becoming more and more blurred between comets and asteroids. This pattern, now known as the Titius–Bode law, predicted the semi-major axes of the six planets of the provided one allowed for a gap between the orbits of Mars and Jupiter
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Asteroid family
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An asteroid family is a population of asteroids that share similar proper orbital elements, such as semimajor axis, eccentricity, 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 can have only about ten identified members. About 33% to 35% of asteroids in the belt are family members. There are about 20 to 30 reliably recognized families, with tens of less certain groupings. 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 also several families which resulted from a large cratering event which did not disrupt the parent body. Such cratering families typically consist of a large body and a swarm of asteroids that are much smaller. Some families have complex structures which are not satisfactorily explained at the moment. Due to the method of origin, all the members have closely matching compositions for most families, notable exceptions are those families which formed from a large differentiated parent body. Asteroid families are thought to have lifetimes of the order of a billion years and this is significantly shorter than the Solar Systems age, so few if any are relics of the early Solar System. Such small asteroids then become subject to such as the Yarkovsky effect that can push them towards orbital resonances with Jupiter over time. Once there, they are relatively 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, and this is the basis of the age determinations and it is supposed that many very old families have lost all the smaller and medium-sized members, leaving only a few of the largest intact. A suggested example of old family remains are the 9 Metis and 113 Amalthea pair. Further evidence for a 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, when the orbital elements of main belt asteroids are plotted, a number of distinct concentrations are seen against the rather uniform background distribution of generic asteroids. These concentrations are the asteroid families, the proper elements are related constants of motion that remain almost constant for times of at least tens of millions of years, and perhaps longer
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Jupiter trojan
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The Jupiter trojans, commonly called Trojan asteroids or just Trojans, are a large group of asteroids that share the orbit of the planet Jupiter around the Sun. Relative to Jupiter, each Trojan librates around one of Jupiters two stable Lagrangian points, L4, lying 60° ahead of the planet in its orbit, and L5, 60° behind. Jupiter trojans are distributed in two elongated, curved regions around these Lagrangian points with an average 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 6,178 Jupiter trojans have been found as of January 2015. By convention they are named after a mythological figure from the Trojan War. The total number of Jupiter trojans larger than 1 km in diameter is believed to be about 1 million, like main-belt asteroids, Jupiter trojans form families. Jupiter trojans are bodies with reddish, featureless spectra. 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 Systems formation or slightly later, during the migration of giant planets. NASA has announced the discovery of an Earth trojan, the trapped body will librate slowly around the point of equilibrium in a tadpole or horseshoe orbit. These leading and trailing points are called the L4 and L5 Lagrange points, however, no asteroids trapped in Lagrange points were observed until more than a century after Lagranges 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 saw the recently discovered Saturnian satellite Phoebe, which was only two away in the sky at the time, or possibly an asteroid. The objects identity was not realized until its orbit was calculated in 1999, 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, by May 2003, the number had grown to 1,600. Asteroids in the L4 group are named after Greek heroes, confusingly,617 Patroclus was named before the Greece/Troy rule was devised, and a Greek name thus appears in the Trojan node. The Greek node also has one misplaced asteroid,624 Hektor, 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
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Near-Earth object
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A near-Earth object is any small Solar System body whose orbit brings it into proximity with Earth. By definition, a solar system body is a NEO if its closest approach to the Sun is less than 1.3 astronomical unit and it is now widely accepted that collisions in the past have had a significant role in shaping the geological and biological history of the Earth. NEOs have become of increased interest since the 1980s because of increased awareness of the potential danger some of the asteroids or comets pose, and mitigations are being researched. In January 2016, NASA announced the Planetary Defense Coordination Office to track NEOs larger than 30 to 50 meters in diameter and coordinate an effective threat response, NEAs have orbits that lie partly between 0.983 and 1.3 AU away from the Sun. When a NEA is detected it is submitted to the IAUs Minor Planet Center for cataloging, some NEAs orbits intersect that of Earths so they pose a collision danger. The United States, European Union, and other nations are currently scanning for NEOs in an effort called Spaceguard. In the United States and since 1998, NASA has a mandate to catalogue all NEOs that are at least 1 kilometer wide. In 2006, it was estimated that 20% of the objects had not yet been found. In 2011, largely as a result of NEOWISE, it was estimated that 93% of the NEAs larger than 1 km had been found, as of 5 February 2017, there have been 875 NEAs larger than 1 km discovered, of which 157 are potentially hazardous. The inventory is much less complete for smaller objects, which still have potential for scale, though not global. Potentially hazardous objects are defined based on parameters that measure the objects potential to make threatening close approaches to the Earth. Mostly objects with an Earth minimum orbit intersection distance of 0.05 AU or less, objects that cannot approach closer to the Earth than 0.05 AU, or are smaller than about 150 m in diameter, are not considered PHOs. This makes them a target for exploration. As of 2016, three near-Earth objects have been visited by spacecraft, more recently, a typical frame of reference for looking at NEOs has been through the scientific concept of risk. In this frame, the risk that any near-Earth object poses is typically seen through a lens that is a function of both the culture and the technology of human society, NEOs have been understood differently throughout history. Each time an NEO is observed, a different risk was posed and it is not just a matter of scientific knowledge. Such perception of risk is thus a product of religious belief, philosophic principles, scientific understanding, technological capabilities, and even economical resourcefulness.03 E −0.4 megatonnes. For instance, it gives the rate for bolides of 10 megatonnes or more as 1 per thousand years, however, the authors give a rather large uncertainty, due in part to uncertainties in determining the energies of the atmospheric impacts that they used in their determination