1.
Parameter
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A parameter, generally, is any characteristic that can help in defining or classifying a particular system. That is, a parameter is an element of a system that is useful, or critical, parameter has more specific meanings within various disciplines, including mathematics, computing and computer programming, engineering, statistics, logic and linguistics. Mathematical functions have one or more arguments that are designated in the definition by variables, a function definition can also contain parameters, but unlike variables, parameters are not listed among the arguments that the function takes. When parameters are present, the definition actually defines a family of functions. A parameter could be incorporated into the name to indicate its dependence on the parameter. For instance, one may define the base b of a logarithm by log b = log log where b is a parameter that indicates which logarithmic function is being used. It is not an argument of the function, and will, for instance, in some informal situations it is a matter of convention whether some or all of the symbols in a function definition are called parameters. However, changing the status of symbols between parameter and variable changes the function as a mathematical object, for instance, the notation for the falling factorial power n k _ = n ⋯, defines a polynomial function of n, but is not a polynomial function of k. Indeed, in the case, it is only defined for non-negative integer arguments. Sometimes it is useful to all functions with certain parameters as parametric family. Examples from probability theory are given further below, a variable is one of the many things a parameter is not. The dependent variable, the speed of the car, depends on the independent variable, change the lever arms of the linkage. Will still depend on the pedal position and you have changed a parameter A parametric equaliser is an audio filter that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another. These settings, the level of the peak or trough, are two of the parameters of a frequency response curve, and in a two-control equaliser they completely describe the curve. More elaborate parametric equalisers may allow other parameters to be varied and these parameters each describe some aspect of the response curve seen as a whole, over all frequencies. A graphic equaliser provides individual level controls for various frequency bands, if asked to imagine the graph of the relationship y = ax2, one typically visualizes a range of values of x, but only one value of a. Of course a different value of a can be used, generating a different relation between x and y, thus a is a parameter, it is less variable than the variable x or y, but it is not an explicit constant like the exponent 2. More precisely, changing the parameter a gives a different problem, in calculating income based on wage and hours worked, it is typically assumed that the number of hours worked is easily changed, but the wage is more static
2.
Kepler orbit
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In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. It is thus said to be a solution of a case of the two-body problem. As a theory in classical mechanics, it also does not take account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways, in most applications, there is a large central body, the center of mass of which is assumed to be the center of mass of the entire system. By decomposition, the orbits of two objects of mass can be described as Kepler orbits around their common center of mass. Variations in the motions of the planets were explained by smaller circular paths overlaid on the larger path, as measurements of the planets became increasingly accurate, revisions to the theory were proposed. In 1543, Nicolaus Copernicus published a model of the solar system. In 1601, Johannes Kepler acquired the extensive, meticulous observations of the planets made by Tycho Brahe, Kepler would spend the next five years trying to fit the observations of the planet Mars to various curves. In 1609, Kepler published the first two of his three laws of planetary motion, the first law states, The orbit of every planet is an ellipse with the sun at a focus. More generally, the path of an object undergoing Keplerian motion may also follow a parabola or a hyperbola, alternately, the equation can be expressed as, r = p 1 + e cos Where p is called the semi-latus rectum of the curve. This form of the equation is useful when dealing with parabolic trajectories. Despite developing these laws from observations, Kepler was never able to develop a theory to explain these motions, between 1665 and 1666, Isaac Newton developed several concepts related to motion, gravitation and differential calculus. However, these concepts were not published until 1687 in the Principia, in which he outlined his laws of motion, newtons law of gravitation states, Every point mass attracts every other point mass by a force pointing along the line intersecting both points. The laws of Kepler and Newton formed the basis of modern celestial mechanics until Albert Einstein introduced the concepts of special and general relativity in the early 20th century. For most applications, Keplerian motion approximates the motions of planets and satellites to relatively high degrees of accuracy and is used extensively in astronomy and astrodynamics. See also Orbit Analysis To solve for the motion of an object in a two body system, two simplifying assumptions can be made,1, the bodies are spherically symmetric and can be treated as point masses. There are no external or internal forces acting upon the other than their mutual gravitation. The shapes of large bodies are close to spheres
3.
Apsis
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An apsis is an extreme point in an objects orbit. The word comes via Latin from Greek and is cognate with apse, for elliptic orbits about a larger body, there are two apsides, named with the prefixes peri- and ap-, or apo- added to a reference to the thing being orbited. For a body orbiting the Sun, the point of least distance is the perihelion, the terms become periastron and apastron when discussing orbits around other stars. For any satellite of Earth including the Moon the point of least distance is the perigee, for objects in Lunar orbit, the point of least distance is the pericynthion and the greatest distance the apocynthion. For any orbits around a center of mass, there are the terms pericenter and apocenter, periapsis and apoapsis are equivalent alternatives. A straight line connecting the pericenter and apocenter is the line of apsides and this is the major axis of the ellipse, its greatest diameter. For a two-body system the center of mass of the lies on this line at one of the two foci of the ellipse. When one body is larger than the other it may be taken to be at this focus. Historically, in systems, apsides were measured from the center of the Earth. In orbital mechanics, the apsis technically refers to the distance measured between the centers of mass of the central and orbiting body. However, in the case of spacecraft, the family of terms are used to refer to the orbital altitude of the spacecraft from the surface of the central body. The arithmetic mean of the two limiting distances is the length of the axis a. The geometric mean of the two distances is the length of the semi-minor axis b, the geometric mean of the two limiting speeds is −2 ε = μ a which is the speed of a body in a circular orbit whose radius is a. The words pericenter and apocenter are often seen, although periapsis/apoapsis are preferred in technical usage, various related terms are used for other celestial objects. The -gee, -helion and -astron and -galacticon forms are used in the astronomical literature when referring to the Earth, Sun, stars. The suffix -jove is occasionally used for Jupiter, while -saturnium has very rarely used in the last 50 years for Saturn. The -gee form is used as a generic closest approach to planet term instead of specifically applying to the Earth. During the Apollo program, the terms pericynthion and apocynthion were used when referring to the Moon, regarding black holes, the term peri/apomelasma was used by physicist Geoffrey A. Landis in 1998 before peri/aponigricon appeared in the scientific literature in 2002
4.
Ellipse
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In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a type of an ellipse having both focal points at the same location. The shape of an ellipse is represented by its eccentricity, which for an ellipse can be any number from 0 to arbitrarily close to, ellipses are the closed type of conic section, a plane curve resulting from the intersection of a cone by a plane. Ellipses have many similarities with the two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder and this ratio is called the eccentricity of the ellipse. Ellipses are common in physics, astronomy and engineering, for example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies, the shapes of planets and stars are often well described by ellipsoids. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency, a similar effect leads to elliptical polarization of light in optics. The name, ἔλλειψις, was given by Apollonius of Perga in his Conics, in order to omit the special case of a line segment, one presumes 2 a > | F1 F2 |, E =. The midpoint C of the segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis and it contains the vertices V1, V2, which have distance a to the center. The distance c of the foci to the center is called the distance or linear eccentricity. The quotient c a is the eccentricity e, the case F1 = F2 yields a circle and is included. C2 is called the circle of the ellipse. This property should not be confused with the definition of an ellipse with help of a directrix below, for an arbitrary point the distance to the focus is 2 + y 2 and to the second focus 2 + y 2. Hence the point is on the ellipse if the condition is fulfilled 2 + y 2 +2 + y 2 =2 a. The shape parameters a, b are called the major axis. The points V3 =, V4 = are the co-vertices and it follows from the equation that the ellipse is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin
5.
Greek alphabet
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It is the ancestor of the Latin and Cyrillic scripts. In its classical and modern forms, the alphabet has 24 letters, Modern and Ancient Greek use different diacritics. In standard Modern Greek spelling, orthography has been simplified to the monotonic system, examples In both Ancient and Modern Greek, the letters of the Greek alphabet have fairly stable and consistent symbol-to-sound mappings, making pronunciation of words largely predictable. Ancient Greek spelling was generally near-phonemic, among consonant letters, all letters that denoted voiced plosive consonants and aspirated plosives in Ancient Greek stand for corresponding fricative sounds in Modern Greek. This leads to groups of vowel letters denoting identical sounds today. Modern Greek orthography remains true to the spellings in most of these cases. The following vowel letters and digraphs are involved in the mergers, Modern Greek speakers typically use the same, modern, in other countries, students of Ancient Greek may use a variety of conventional approximations of the historical sound system in pronouncing Ancient Greek. Several letter combinations have special conventional sound values different from those of their single components, among them are several digraphs of vowel letters that formerly represented diphthongs but are now monophthongized. In addition to the three mentioned above, there is also ⟨ου⟩, pronounced /u/, the Ancient Greek diphthongs ⟨αυ⟩, ⟨ευ⟩ and ⟨ηυ⟩ are pronounced, and respectively in voicing environments in Modern Greek. The Modern Greek consonant combinations ⟨μπ⟩ and ⟨ντ⟩ stand for and respectively, ⟨τζ⟩ stands for, in addition, both in Ancient and Modern Greek, the letter ⟨γ⟩, before another velar consonant, stands for the velar nasal, thus ⟨γγ⟩ and ⟨γκ⟩ are pronounced like English ⟨ng⟩. There are also the combinations ⟨γχ⟩ and ⟨γξ⟩ and these signs were originally designed to mark different forms of the phonological pitch accent in Ancient Greek. The letter rho, although not a vowel, also carries a rough breathing in word-initial position, if a rho was geminated within a word, the first ρ always had the smooth breathing and the second the rough breathing leading to the transiliteration rrh. The vowel letters ⟨α, η, ω⟩ carry an additional diacritic in certain words, the iota subscript. This iota represents the former offglide of what were originally long diphthongs, ⟨ᾱι, ηι, ωι⟩, another diacritic used in Greek is the diaeresis, indicating a hiatus. In 1982, a new, simplified orthography, known as monotonic, was adopted for use in Modern Greek by the Greek state. Although it is not a diacritic, the comma has a function as a silent letter in a handful of Greek words, principally distinguishing ό. There are many different methods of rendering Greek text or Greek names in the Latin script, the form in which classical Greek names are conventionally rendered in English goes back to the way Greek loanwords were incorporated into Latin in antiquity. In this system, ⟨κ⟩ is replaced with ⟨c⟩, the diphthongs ⟨αι⟩ and ⟨οι⟩ are rendered as ⟨ae⟩ and ⟨oe⟩ respectively, and ⟨ει⟩ and ⟨ου⟩ are simplified to ⟨i⟩ and ⟨u⟩ respectively
6.
Latin script
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Latin script is used as the standard method of writing in most Western and Central European languages, as well as in many languages in other parts of the world. Latin script is the basis for the largest number of alphabets of any writing system and is the most widely adopted writing system in the world, Latin script is also the basis of the International Phonetic Alphabet. The 26 most widespread letters are the contained in the ISO basic Latin alphabet. The script is either called Roman script or Latin script, in reference to its origin in ancient Rome, in the context of transliteration, the term romanization or romanisation is often found. Unicode uses the term Latin as does the International Organization for Standardization, the numeral system is called the Roman numeral system, and the collection of the elements, Roman numerals. The numbers 1,2,3. are Latin/Roman script numbers for the Hindu–Arabic numeral system, the Latin alphabet spread, along with Latin, from the Italian Peninsula to the lands surrounding the Mediterranean Sea with the expansion of the Roman Empire. The Latin script also came into use for writing the West Slavic languages and several South Slavic languages, the speakers of East Slavic languages generally adopted Cyrillic along with Orthodox Christianity. The Serbian language uses both scripts, with Cyrillic predominating in official communication and Latin elsewhere, as determined by the Law on Official Use of the Language and Alphabet. As late as 1500, the Latin script was limited primarily to the languages spoken in Western, Northern, the Orthodox Christian Slavs of Eastern and Southeastern Europe mostly used Cyrillic, and the Greek alphabet was in use by Greek-speakers around the eastern Mediterranean. The Arabic script was widespread within Islam, both among Arabs and non-Arab nations like the Iranians, Indonesians, Malays, and Turkic peoples, most of the rest of Asia used a variety of Brahmic alphabets or the Chinese script. It is used for many Austronesian languages, including the languages of the Philippines, Latin letters served as the basis for the forms of the Cherokee syllabary developed by Sequoyah, however, the sound values are completely different. In the late 19th century, the Romanians returned to the Latin alphabet, under French rule and Portuguese missionary influence, a Latin alphabet was devised for the Vietnamese language, which had previously used Chinese characters. In 1928, as part of Mustafa Kemal Atatürks reforms, the new Republic of Turkey adopted a Latin alphabet for the Turkish language, kazakhstan, Kyrgyzstan, Iranian-speaking Tajikistan, and the breakaway region of Transnistria kept the Cyrillic alphabet, chiefly due to their close ties with Russia. In the 1930s and 1940s, the majority of Kurds replaced the Arabic script with two Latin alphabets, although the only official Kurdish government uses an Arabic alphabet for public documents, the Latin Kurdish alphabet remains widely used throughout the region by the majority of Kurdish-speakers. In 2015, the Kazakh government announced that the Latin alphabet would replace Cyrillic as the system for the Kazakh language by 2025. In the course of its use, the Latin alphabet was adapted for use in new languages, sometimes representing phonemes not found in languages that were written with the Roman characters. These new forms are given a place in the alphabet by defining an alphabetical order or collation sequence, a digraph is a pair of letters used to write one sound or a combination of sounds that does not correspond to the written letters in sequence. Examples are ⟨ch⟩, ⟨ng⟩, ⟨rh⟩, ⟨sh⟩ in English, a trigraph is made up of three letters, like the German ⟨sch⟩, the Breton ⟨c’h⟩ or the Milanese ⟨oeu⟩
7.
Eccentric anomaly
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In orbital mechanics, eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit. The eccentric anomaly is one of three parameters that define a position along an orbit, the other two being the true anomaly and the mean anomaly. Consider the ellipse with equation given by, x 2 a 2 + y 2 b 2 =1, where a is the semi-major axis and b is the semi-minor axis. For a point on the ellipse, P = P, representing the position of a body in an elliptical orbit. The eccentric anomaly, E, is observed by drawing a right triangle with one vertex at the center of the ellipse, having hypotenuse a, the eccentric anomaly is measured in the same direction as the true anomaly, shown in the figure as f. The equation sin E = −y/b is immediately able to be ruled out since it traverses the ellipse in the wrong direction. It can also be noted that the equation can be viewed as being the similar triangle with adjacent side through P and the minor auxiliary circle, hypotenuse b. The eccentricity e is defined as, e =1 −2, with this result the eccentric anomaly can be determined from the true anomaly as shown next. The true anomaly is the angle labeled f in the figure, located at the focus of the ellipse, the true anomaly and the eccentric anomaly are related as follows. Hence, tan E = sin E cos E =1 − e 2 sin θ e + cos θ. Angle E is therefore the adjacent angle of a triangle with hypotenuse 1 + e cos θ, adjacent side e + cos θ. The eccentric anomaly E is related to the mean anomaly M by Keplers equation and it is usually solved by numerical methods, e. g. the Newton–Raphson method. Murray, Carl D. & Dermott, Stanley F, solar System Dynamics, Cambridge University Press, Cambridge, GB Plummer, Henry C. K. An Introductory Treatise on Dynamical Astronomy, Dover Publications, New York, NY Eccentricity vector Orbital eccentricity
8.
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
9.
Orbital state vectors
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State vectors are defined with respect to some frame of reference, usually but not always an inertial reference frame. The position vector r describes the position of the body in the frame of reference. Together, these two vectors and the time at which they are valid uniquely describe the bodys trajectory. The body does not actually have to be in orbit for its state vector to determine its trajectory, it only has to move ballistically, i. e. solely under the effects of its own inertia and gravity. For example, it could be a spacecraft or missile in a suborbital trajectory, if other forces such as drag or thrust are significant, they must be added vectorially to those of gravity when performing the integration to determine future position and velocity. For any object moving through space, the velocity vector is tangent to the trajectory, the state vectors can be easily used to compute the angular momentum vector as h = r × v. Because even satellites in low Earth orbit experience significant perturbations, the Keplerian elements computed from the vector at any moment are only valid at that time. Such element sets are known as osculating elements because they coincide with the actual orbit only at that moment