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
10.
Circular orbit
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A circular orbit is the orbit at a fixed distance around any point by an object rotating around a fixed axis. Below we consider a circular orbit in astrodynamics or celestial mechanics under standard assumptions, here the centripetal force is the gravitational force, and the axis mentioned above is the line through the center of the central mass perpendicular to the plane of motion. In this case, not only the distance, but also the speed, angular speed, potential, there is no periapsis or apoapsis. This orbit has no radial version, transverse acceleration causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have a = v 2 r = ω2 r where, v is velocity of orbiting body. The formula is dimensionless, describing a ratio true for all units of measure applied uniformly across the formula. If the numerical value of a is measured in meters per second per second, then the values for v will be in meters per second, r in meters. μ = G M is the standard gravitational parameter, the orbit equation in polar coordinates, which in general gives r in terms of θ, reduces to, r = h 2 μ where, h = r v is specific angular momentum of the orbiting body. Maneuvering into a circular orbit, e. g. It is also a matter of maneuvering into the orbit, for the sake of convenience, the derivation will be written in units in which c = G =1. The four-velocity of a body on an orbit is given by. The dot above a variable denotes derivation with respect to proper time τ
11.
Sine
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In mathematics, the sine is a trigonometric function of an angle. More generally, the definition of sine can be extended to any value in terms of the length of a certain line segment in a unit circle. The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy, via translation from Sanskrit to Arabic and then from Arabic to Latin. The word sine comes from a Latin mistranslation of the Arabic jiba, to define the trigonometric functions for an acute angle α, start with any right triangle that contains an angle of measure α, in the accompanying figure, angle A in triangle ABC has measure α. The three sides of the triangle are named as follows, The opposite side is the side opposite to the angle of interest, the hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle, the adjacent side is the remaining side, in this case side b. It forms a side of both the angle of interest and the right angle, once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse. As stated, the value sin appears to depend on the choice of right triangle containing an angle of measure α, however, this is not the case, all such triangles are similar, and so the ratio is the same for each of them. The trigonometric functions can be defined in terms of the rise, run, when the length of the line segment is 1, sine takes an angle and tells the rise. Sine takes an angle and tells the rise per unit length of the line segment, rise is equal to sin θ multiplied by the length of the line segment. In contrast, cosine is used for telling the run from the angle, arctan is used for telling the angle from the slope. The line segment is the equivalent of the hypotenuse in the right-triangle, in trigonometry, a unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system. Let a line through the origin, making an angle of θ with the half of the x-axis. The x- and y-coordinates of this point of intersection are equal to cos θ and sin, the points distance from the origin is always 1. Unlike the definitions with the triangle or slope, the angle can be extended to the full set of real arguments by using the unit circle. This can also be achieved by requiring certain symmetries and that sine be a periodic function. Exact identities, These apply for all values of θ. sin = cos =1 csc The reciprocal of sine is cosecant, i. e. the reciprocal of sin is csc, or cosec. Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side, the inverse function of sine is arcsine or inverse sine
12.
Tangent
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In geometry, the tangent line to a plane curve at a given point is the straight line that just touches the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve, a similar definition applies to space curves and curves in n-dimensional Euclidean space. Similarly, the tangent plane to a surface at a point is the plane that just touches the surface at that point. The concept of a tangent is one of the most fundamental notions in geometry and has been extensively generalized. The word tangent comes from the Latin tangere, to touch, euclid makes several references to the tangent to a circle in book III of the Elements. In Apollonius work Conics he defines a tangent as being a line such that no other straight line could fall between it and the curve, archimedes found the tangent to an Archimedean spiral by considering the path of a point moving along the curve. Independently Descartes used his method of normals based on the observation that the radius of a circle is always normal to the circle itself and these methods led to the development of differential calculus in the 17th century. Many people contributed, Roberval discovered a method of drawing tangents. René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents, further developments included those of John Wallis and Isaac Barrow, leading to the theory of Isaac Newton and Gottfried Leibniz. An 1828 definition of a tangent was a line which touches a curve. This old definition prevents inflection points from having any tangent and it has been dismissed and the modern definitions are equivalent to those of Leibniz who defined the tangent line as the line through a pair of infinitely close points on the curve. The tangent at A is the limit when point B approximates or tends to A, the existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as differentiability. At most points, the tangent touches the curve without crossing it, a point where the tangent crosses the curve is called an inflection point. Circles, parabolas, hyperbolas and ellipses do not have any point, but more complicated curves do have, like the graph of a cubic function. Conversely, it may happen that the curve lies entirely on one side of a line passing through a point on it. This is the case, for example, for a passing through the vertex of a triangle. In convex geometry, such lines are called supporting lines, the geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century, suppose that a curve is given as the graph of a function, y = f
13.
Atan2
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In a variety of computer languages, the function atan2 is the arctangent function with two arguments. For any real number x and y not both equal to zero, atan2 is the angle in radians between the positive x-axis of a plane and the point given by the coordinates on it. The angle is positive for angles, and negative for clockwise angles. The purpose of using two arguments instead of one, i. e and it also avoids the problem of division by zero, as atan2 will return a valid answer as long as y is non-zero. The atan2 function was first introduced in computer programming languages, but now it is common in other fields of science. It dates back at least as far as the FORTRAN programming language and is found in many modern programming languages. Among these languages are, Cs math. h standard library, the Java Math library. NETs System. Math, the Python math module, the Ruby Math module, in addition, many scripting languages, such as Perl, include the C-style atan2 function. The one-argument arctangent function cannot distinguish between diametrically opposite directions, for example, the anticlockwise angle from the x-axis to the vector, calculated in the usual way as arctan, is π/4, or 45°. However, the angle between the x-axis and the vector appears, by the method, to be arctan, again π/4, even though the answer clearly should be −3π/4. In addition, an attempt to find the angle between the x-axis and the vector requires evaluation of arctan, which fails on division by zero, the atan2 function calculates the arc tangent of the two variables y and x. It is similar to calculating the arc tangent of y/x, except that the signs of both arguments are used to determine the quadrant of the result. Thus, the function takes into account the signs of both vector components, and places the angle in the correct quadrant, e. g. atan2 = π/4. In addition, atan2 can produce an angle of ±π/2 while the ordinary arctangent method breaks down, the atan2 function is useful in many applications involving vectors in Euclidean space, such as finding the direction from one point to another. A principal use is in computer graphics rotations, for converting rotation matrix representations into Euler angles, the function atan2 computes the principal value of the argument function applied to the complex number x+iy. That is, atan2 = Pr arg = Arg, the argument can be changed by 2π without making any difference to the angle, but to define atan2 uniquely one uses the principal value in the range (−π, π]. On implementations without signed zero, or when given positive zero arguments and it will always return a value in the range rather than raising an error or returning a NaN. In Common Lisp, where optional arguments exist, the atan function allows one to supply the x coordinate. In Mathematica, the form ArcTan is used where the one parameter form supplies the normal arctangent, Mathematica classifies ArcTan as an indeterminate expression
14.
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
15.
Kepler's laws of planetary motion
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In astronomy, Keplers laws of planetary motion are three scientific laws describing the motion of planets around the Sun. The orbit of a planet is an ellipse with the Sun at one of the two foci, a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. The square of the period of a planet is proportional to the cube of the semi-major axis of its orbit. Most planetary orbits are circular, and careful observation and calculation are required in order to establish that they are not perfectly circular. Calculations of the orbit of Mars, whose published values are somewhat suspect, from this, Johannes Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. Keplers work improved the theory of Nicolaus Copernicus, explaining how the planets speeds varied. Isaac Newton showed in 1687 that relationships like Keplers would apply in the Solar System to a approximation, as a consequence of his own laws of motion. Keplers laws are part of the foundation of modern astronomy and physics, Keplers laws improve the model of Copernicus. Keplers corrections are not at all obvious, The planetary orbit is not a circle, the Sun is not at the center but at a focal point of the elliptical orbit. Neither the linear speed nor the speed of the planet in the orbit is constant, but the area speed is constant.015. The calculation is correct when perihelion, the date the Earth is closest to the Sun, the current perihelion, near January 4, is fairly close to the solstice of December 21 or 22. It took nearly two centuries for the current formulation of Keplers work to take on its settled form, voltaires Eléments de la philosophie de Newton of 1738 was the first publication to use the terminology of laws. The Biographical Encyclopedia of Astronomers in its article on Kepler states that the terminology of laws for these discoveries was current at least from the time of Joseph de Lalande. It was the exposition of Robert Small, in An account of the discoveries of Kepler that made up the set of three laws, by adding in the third. Small also claimed, against the history, that these were empirical laws, further, the current usage of Keplers Second Law is something of a misnomer. Kepler had two versions, related in a sense, the distance law and the area law. The area law is what became the Second Law in the set of three, but Kepler did himself not privilege it in that way, Johannes Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe. Keplers third law was published in 1619 and his first law reflected this discovery
16.
Hyperbola
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In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other, the hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. If the plane intersects both halves of the double cone but does not pass through the apex of the cones, each branch of the hyperbola has two arms which become straighter further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the point about which each branch reflects to form the other branch. In the case of the curve f =1 / x the asymptotes are the two coordinate axes, hyperbolas share many of the ellipses analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term, many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids, hyperboloids, hyperbolic geometry, hyperbolic functions, and gyrovector spaces. The word hyperbola derives from the Greek ὑπερβολή, meaning over-thrown or excessive, hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones. The term hyperbola is believed to have coined by Apollonius of Perga in his definitive work on the conic sections. The rectangle could be applied to the segment, be shorter than the segment or exceed the segment, the midpoint M of the line segment joining the foci is called the center of the hyperbola. 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, C2 is called the director circle of the hyperbola. In order to get the branch of the hyperbola, one has to use the director circle related to F1. This property should not be confused with the definition of a hyperbola 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 hyperbola if the condition is fulfilled 2 + y 2 −2 + y 2 = ±2 a. Remove the square roots by suitable squarings and use the relation b 2 = c 2 − a 2 to obtain the equation of the hyperbola, the shape parameters a, b are called the semi major axis and semi minor axis or conjugate axis. As opposed to an ellipse, a hyperbola has two vertices
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International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
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OCLC
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The Online Computer Library Center is a US-based nonprofit cooperative organization dedicated to the public purposes of furthering access to the worlds information and reducing information costs. It was founded in 1967 as the Ohio College Library Center, OCLC and its member libraries cooperatively produce and maintain WorldCat, the largest online public access catalog in the world. OCLC is funded mainly by the fees that libraries have to pay for its services, the group first met on July 5,1967 on the campus of the Ohio State University to sign the articles of incorporation for the nonprofit organization. The group hired Frederick G. Kilgour, a former Yale University medical school librarian, Kilgour wished to merge the latest information storage and retrieval system of the time, the computer, with the oldest, the library. The goal of network and database was to bring libraries together to cooperatively keep track of the worlds information in order to best serve researchers and scholars. The first library to do online cataloging through OCLC was the Alden Library at Ohio University on August 26,1971 and this was the first occurrence of online cataloging by any library worldwide. Membership in OCLC is based on use of services and contribution of data, between 1967 and 1977, OCLC membership was limited to institutions in Ohio, but in 1978, a new governance structure was established that allowed institutions from other states to join. In 2002, the structure was again modified to accommodate participation from outside the United States. As OCLC expanded services in the United States outside of Ohio, it relied on establishing strategic partnerships with networks, organizations that provided training, support, by 2008, there were 15 independent United States regional service providers. OCLC networks played a key role in OCLC governance, with networks electing delegates to serve on OCLC Members Council, in early 2009, OCLC negotiated new contracts with the former networks and opened a centralized support center. OCLC provides bibliographic, abstract and full-text information to anyone, OCLC and its member libraries cooperatively produce and maintain WorldCat—the OCLC Online Union Catalog, the largest online public access catalog in the world. WorldCat has holding records from public and private libraries worldwide. org, in October 2005, the OCLC technical staff began a wiki project, WikiD, allowing readers to add commentary and structured-field information associated with any WorldCat record. The Online Computer Library Center acquired the trademark and copyrights associated with the Dewey Decimal Classification System when it bought Forest Press in 1988, a browser for books with their Dewey Decimal Classifications was available until July 2013, it was replaced by the Classify Service. S. The reference management service QuestionPoint provides libraries with tools to communicate with users and this around-the-clock reference service is provided by a cooperative of participating global libraries. OCLC has produced cards for members since 1971 with its shared online catalog. OCLC commercially sells software, e. g. CONTENTdm for managing digital collections, OCLC has been conducting research for the library community for more than 30 years. In accordance with its mission, OCLC makes its research outcomes known through various publications and these publications, including journal articles, reports, newsletters, and presentations, are available through the organizations website. The most recent publications are displayed first, and all archived resources, membership Reports – A number of significant reports on topics ranging from virtual reference in libraries to perceptions about library funding
19.
Orbit
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In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet about a star or a natural satellite around a planet. Normally, orbit refers to a regularly repeating path around a body, to a close approximation, planets and satellites follow elliptical orbits, with the central mass being orbited at a focal point of the ellipse, as described by Keplers laws of planetary motion. For ease of calculation, in most situations orbital motion is adequately approximated by Newtonian Mechanics, historically, the apparent motions of the planets were described by European and Arabic philosophers using the idea of celestial spheres. This model posited the existence of perfect moving spheres or rings to which the stars and it assumed the heavens were fixed apart from the motion of the spheres, and was developed without any understanding of gravity. After the planets motions were accurately measured, theoretical mechanisms such as deferent. Originally geocentric it was modified by Copernicus to place the sun at the centre to help simplify the model, the model was further challenged during the 16th century, as comets were observed traversing the spheres. The basis for the understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. Second, he found that the speed of each planet is not constant, as had previously been thought. Third, Kepler found a relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are respectively about 5.2 and 0.723 AU distant from the Sun, their orbital periods respectively about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter,5. 23/11.862, is equal to that for Venus,0. 7233/0.6152. Idealised orbits meeting these rules are known as Kepler orbits, isaac Newton demonstrated that Keplers laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections. Newton showed that, for a pair of bodies, the sizes are in inverse proportion to their masses. Where one body is more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body. Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force, in a dramatic vindication of classical mechanics, in 1846 le Verrier was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits, in relativity theory, orbits follow geodesic trajectories which are usually approximated very well by the Newtonian predictions but the differences are measurable. Essentially all the evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy
20.
Box orbit
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In stellar dynamics a box orbit refers to a particular type of orbit that can be seen in triaxial systems, i. e. systems that do not possess a symmetry around any of its axes. They contrast with the orbits that are observed in spherically symmetric or axisymmetric systems. In a box orbit, the star oscillates independently along the three different axes as it moves through the system, as a result of this motion, it fills in a box-shaped region of space. Unlike loop orbits, the stars on box orbits can come close to the center of the system. As a special case, if the frequencies of oscillation in different directions are commensurate, such orbits are sometimes called boxlets. Horseshoe orbit Lissajous curve List of orbits
21.
Parabolic trajectory
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In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit and it is also sometimes referred to as a C3 =0 orbit. Parabolic trajectories are escape trajectories, separating positive-energy hyperbolic trajectories from negative-energy elliptic orbits. At any position the body has the escape velocity for that position. This is entirely equivalent to the energy being 0, C3 =0 Barkers equation relates the time of flight to the true anomaly of a parabolic trajectory. There are two cases, the move away from each other or towards each other. At any time the speed from t =0 is 1.5 times the current speed
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Elliptic orbit
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In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity less than 1, this includes the special case of a circular orbit, with eccentricity equal to zero. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0, in a wider sense it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1, in a gravitational two-body problem with negative energy both bodies follow similar elliptic orbits with the same orbital period around their common barycenter. Also the relative position of one body with respect to the other follows an elliptic orbit, examples of elliptic orbits include, Hohmann transfer orbit, Molniya orbit and tundra orbit. A is the length of the semi-major axis, the velocity equation for a hyperbolic trajectory has either +1 a, or it is the same with the convention that in that case a is negative. Conclusions, For a given semi-major axis the orbital energy is independent of the eccentricity. ν is the true anomaly. The angular momentum is related to the cross product of position and velocity. Here ϕ is defined as the angle which differs by 90 degrees from this and this set of six variables, together with time, are called the orbital state vectors. Given the masses of the two bodies they determine the full orbit, the two most general cases with these 6 degrees of freedom are the elliptic and the hyperbolic orbit. Special cases with fewer degrees of freedom are the circular and parabolic orbit, another set of six parameters that are commonly used are the orbital elements. In the Solar System, planets, asteroids, most comets, the following chart of the perihelion and aphelion of the planets, dwarf planets and Halleys Comet demonstrates the variation of the eccentricity of their elliptical orbits. For similar distances from the sun, wider bars denote greater eccentricity, note the almost-zero eccentricity of Earth and Venus compared to the enormous eccentricity of Halleys Comet and Eris. A radial trajectory can be a line segment, which is a degenerate ellipse with semi-minor axis =0. Although the eccentricity is 1, this is not a parabolic orbit, most properties and formulas of elliptic orbits apply. However, the orbit cannot be closed and it is an open orbit corresponding to the part of the degenerate ellipse from the moment the bodies touch each other and move away from each other until they touch each other again. In the case of point masses one full orbit is possible, the velocities at the start and end are infinite in opposite directions and the potential energy is equal to minus infinity. The radial elliptic trajectory is the solution of a problem with at some instant zero speed
23.
Horseshoe orbit
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A horseshoe orbit is a type of co-orbital motion of a small orbiting body relative to a larger orbiting body. The orbital period of the body is very nearly the same as for the larger body. The loop is not closed but will drift forward or backward slightly each time, when the object approaches the larger body closely at either end of its trajectory, its apparent direction changes. Over an entire cycle the center traces the outline of a horseshoe, asteroids in horseshoe orbits with respect to Earth include 54509 YORP,2002 AA29,2010 SO16,2015 SO2 and possibly 2001 GO2. A broader definition includes 3753 Cruithne, which can be said to be in a compound and/or transition orbit, saturns moons Epimetheus and Janus occupy horseshoe orbits with respect to each other. The following explanation relates to an asteroid which is in such an orbit around the Sun, the asteroid is in almost the same solar orbit as Earth. Both take approximately one year to orbit the Sun and it is also necessary to grasp two rules of orbit dynamics, A body closer to the Sun completes an orbit more quickly than a body further away. If a body accelerates along its orbit, its orbit moves outwards from the Sun, if it decelerates, the orbital radius decreases. The horseshoe orbit arises because the attraction of the Earth changes the shape of the elliptical orbit of the asteroid. The shape changes are small but result in significant changes relative to the Earth. The horseshoe becomes apparent only when mapping the movement of the relative to both the Sun and the Earth. The asteroid always orbits the Sun in the same direction, however, it goes through a cycle of catching up with the Earth and falling behind, so that its movement relative to both the Sun and the Earth traces a shape like the outline of a horseshoe. Starting at point A, on the ring between L5 and Earth, the satellite is orbiting faster than the Earth and is on its way toward passing between the Earth and the Sun. But Earths gravity exerts an outward accelerating force, pulling the satellite into an orbit which decreases its angular speed. When the satellite gets to point B, it is traveling at the speed as Earth. Earths gravity is still accelerating the satellite along the orbital path, eventually, at Point C, the satellite reaches a high and slow enough orbit such that it starts to lag behind Earth. It then spends the next century or more appearing to drift backwards around the orbit when viewed relative to the Earth and its orbit around the Sun still takes only slightly more than one Earth year. Given enough time, the Earth and the satellite will be on opposite sides of the Sun, eventually the satellite comes around to point D where Earths gravity is now reducing the satellites orbital velocity
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Hyperbolic trajectory
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In astrodynamics or celestial mechanics, a hyperbolic trajectory is the trajectory of any object around a central body with more than enough speed to escape the central objects gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola, in more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one. Under standard assumptions a body traveling along this trajectory will coast to infinity, similarly to parabolic trajectory all hyperbolic trajectories are also escape trajectories. The specific energy of a hyperbolic trajectory orbit is positive, planetary flybys, used for gravitational slingshots, can be described within the planets sphere of influence using hyperbolic trajectories. Like an elliptical orbit, a trajectory for a given system can be defined by its semi major axis. However, with a hyperbolic orbit other parameters may more useful in understanding a bodys motion, the following table lists the main parameters describing the path of body following a hyperbolic trajectory around another under standard assumptions and the formula connecting them. The semi major axis is not immediately visible with an hyperbolic trajectory, usually, by convention, it is negative, to keep various equations are consistent with elliptical orbits. With a hyperbolic trajectory the orbital eccentricity is greater than 1, the eccentricity is directly related to the angle between the asymptotes. With eccentricity just over 1 the hyperbola is a v shape. At e =2 the asymptotes are at right angles, with e >2 the asymptotes are more that 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion approaches a straight line, with bodies experiencing gravitational forces and following hyperbolic trajectories it is equal to the semi-minor axis of the hyperbola. In the situation of a spacecraft or comet approaching a planet, if the central body is known the trajectory can now be found, including how close the approaching body will be at periapsis. If this is less than planets radius an impact should be expected, a body approaching Jupiter from the outer solar system with a speed of 5.5 km/h, will need the impact parameter to be at least 770, 000km or 11 times Jupiter radius to avoid collision. As, typically, all variables can be determined accurately. μ = b v ∞2 tan δ /2 where δ =2 θ ∞ − π is the angle the body is deflected from a straight line in its course. Where μ is a parameter w and a is the semi-major axis of the orbit. The flight path angle is the angle between the direction of velocity and the perpendicular to the direction, so it is zero at periapsis. For example, at a place where escape speed is 11.2 km/s,11.62 −11.22 =3.02 This is an example of the Oberth effect
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Parking orbit
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A parking orbit is a temporary orbit used during the launch of a satellite or other space probe. A launch vehicle boosts into the orbit, then coasts for a while. The alternative to an orbit is direct injection, where the rocket fires continuously until its fuel is exhausted. There are several reasons why a parking orbit may be used, geostationary spacecraft require an orbit in the plane of the equator. Getting there requires a transfer orbit with an apogee directly above the equator. Unless the launch site itself is close to the equator. Instead, the craft is placed with a stage in an inclined parking orbit. When the craft crosses the equator, the stage is fired to raise the spacecrafts apogee to geostationary altitude. Finally, a burn is required to raise the perigee to the same altitude. In order to reach the Moon or a planet at a desired time, using a preliminary parking orbit before final injection can widen this window from seconds or minutes, to several hours. For the Apollo programs manned lunar missions, a parking orbit allowed time for spacecraft checkout while still close to home, use of a parking orbit requires a rocket upper stage to perform the injection burn while under zero g conditions. Often, the upper stage which performs the parking orbit injection is used for the final injection burn. During the parking orbit coast, the propellants will drift away from the bottom of the tank and this must be dealt with through the use of tank diaphragms, or ullage rockets to settle the propellant back to the bottom of the tank. A reaction control system is needed to orient the stage properly for the final burn, cryogenic propellants must be stored in well-insulated tanks, to prevent excessive boiloff during coast. Battery life and other consumables must be sufficient for the duration of the parking coast, the Centaur and Agena families of upper stages were designed for such restarts and have often been used in this manner. The last Agena flew in 1987, but Centaur is still in production, the Briz-M stage often performs the same role for Russian rockets. The Apollo program used parking orbits, for all the mentioned above except those that pertain to geostationary orbits. When the Space Shuttle orbiter launched interplanetary probes such as Galileo, the Ariane 5 does not use parking orbits
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Hohmann transfer orbit
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In orbital mechanics, the Hohmann transfer orbit /ˈhoʊ. mʌn/ is an elliptical orbit used to transfer between two circular orbits of different radii in the same plane. The orbital maneuver to perform the Hohmann transfer uses two engine impulses, one to move a spacecraft onto the orbit and a second to move off it. This maneuver was named after Walter Hohmann, the German scientist who published a description of it in his 1925 book Die Erreichbarkeit der Himmelskörper, Hohmann was influenced in part by the German science fiction author Kurd Lasswitz and his 1897 book Two Planets. The diagram shows a Hohmann transfer orbit to bring a spacecraft from a circular orbit into a higher one. It is one half of an orbit that touches both the lower circular orbit the spacecraft wishes to leave and the higher circular orbit that it wishes to reach. The transfer is initiated by firing the engine in order to accelerate it so that it will follow the elliptical orbit. When the spacecraft has reached its orbit, its orbital speed must be increased again in order to change the elliptic orbit to the larger circular one. The engine is fired again at the lower distance to slow the spacecraft into the lower circular orbit. The Hohmann transfer orbit is based on two instantaneous velocity changes, extra fuel is required to compensate for the fact that the bursts take time, this is minimized by using high thrust engines to minimize the duration of the bursts. Low thrust engines can perform an approximation of a Hohmann transfer orbit and this requires a change in velocity that is up to 141% greater than the two impulse transfer orbit, and takes longer to complete. Typically μ is given in units of m3/s2, as such be sure to use meters not kilometers for r 1 and r 2, the total Δ v is then, Δ v t o t a l = Δ v 1 + Δ v 2. In application to traveling from one body to another it is crucial to start maneuver at the time when the two bodies are properly aligned. In the smaller circular orbit the speed is 7.73 km/s, in the elliptical orbit in between the speed varies from 10.15 km/s at the perigee to 1.61 km/s at the apogee. The Δv for the two burns are thus 10.15 −7.73 =2.42 and 3.07 −1.61 =1.46 km/s, together 3.88 km/s. It is interesting to note that this is greater than the Δv required for an orbit,10.93 −7.73 =3.20 km/s. Applying a Δv at the LEO of only 0.78 km/s more would give the rocket the escape speed, as the example above demonstrates, the Δv required to perform a Hohmann transfer between two circular orbits is not the greatest when the destination radius is infinite. The Δv required is greatest when the radius of the orbit is 15.58 times that of the smaller orbit. This number is the root of x3 − 15x2 − 9x −1 =0
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Geocentric orbit
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A geocentric orbit or Earth orbit involves any object orbiting the Earth, such as the Moon or artificial satellites. In 1997 NASA estimated there were approximately 2,465 artificial satellite orbiting the Earth and 6,216 pieces of space debris as tracked by the Goddard Space Flight Center. Over 16,291 previously launched objects have decayed into the Earths atmosphere, altitude as used here, the height of an object above the average surface of the Earths oceans. Analemma a term in astronomy used to describe the plot of the positions of the Sun on the celestial sphere throughout one year, apogee is the farthest point that a satellite or celestial body can go from Earth, at which the orbital velocity will be at its minimum. Eccentricity a measure of how much an orbit deviates from a perfect circle, eccentricity is strictly defined for all circular and elliptical orbits, and parabolic and hyperbolic trajectories. Equatorial plane as used here, an imaginary plane extending from the equator on the Earth to the celestial sphere, escape velocity as used here, the minimum velocity an object without propulsion needs to have to move away indefinitely from the Earth. An object at this velocity will enter a parabolic trajectory, above this velocity it will enter a hyperbolic trajectory, impulse the integral of a force over the time during which it acts. Inclination the angle between a plane and another plane or axis. In the sense discussed here the reference plane is the Earths equatorial plane, orbital characteristics the six parameters of the Keplerian elements needed to specify that orbit uniquely. Orbital period as defined here, time it takes a satellite to make one orbit around the Earth. Perigee is the nearest approach point of a satellite or celestial body from Earth, sidereal day the time it takes for a celestial object to rotate 360°. For the Earth this is,23 hours,56 minutes,4.091 seconds, solar time as used here, the local time as measured by a sundial. Velocity an objects speed in a particular direction, since velocity is defined as a vector, both speed and direction are required to define it. The following is a list of different geocentric orbit classifications, Low Earth orbit - Geocentric orbits ranging in altitude from 160 kilometers to 2,000 kilometres above mean sea level. At 160 km, one revolution takes approximately 90 minutes, medium Earth orbit - Geocentric orbits with altitudes at apogee ranging between 2,000 kilometres and that of the geosynchronous orbit at 35,786 kilometres. Geosynchronous orbit - Geocentric circular orbit with an altitude of 35,786 kilometres, the period of the orbit equals one sidereal day, coinciding with the rotation period of the Earth. The speed is approximately 3,000 metres per second, high Earth orbit - Geocentric orbits with altitudes at apogee higher than that of the geosynchronous orbit. A special case of high Earth orbit is the elliptical orbit
28.
Geosynchronous orbit
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A geosynchronous orbit is an orbit about the Earth of a satellite with an orbital period that matches the rotation of the Earth on its axis of approximately 23 hours 56 minutes and 4 seconds. Over the course of a day, the position in the sky traces out a path, typically in a figure-8 form, whose precise characteristics depend on the orbits inclination. Satellites are typically launched in an eastward direction, a special case of geosynchronous orbit is the geostationary orbit, which is a circular geosynchronous orbit at zero inclination. A satellite in a geostationary orbit appears stationary, always at the point in the sky. Popularly or loosely, the term geosynchronous may be used to mean geostationary, specifically, geosynchronous Earth orbit may be a synonym for geosynchronous equatorial orbit, or geostationary Earth orbit. A semi-synchronous orbit has a period of ½ sidereal day. Relative to the Earths surface it has twice this period, examples include the Molniya orbit and the orbits of the satellites in the Global Positioning System. Circular Earth geosynchronous orbits have a radius of 42,164 km, all Earth geosynchronous orbits, whether circular or elliptical, have the same semi-major axis.4418 km3/s2. In the special case of an orbit, the ground track of a satellite is a single point on the equator. A geostationary equatorial orbit is a geosynchronous orbit in the plane of the Earths equator with a radius of approximately 42,164 km. A satellite in such an orbit is at an altitude of approximately 35,786 km above sea level. It maintains the position relative to the Earths surface. The theoretical basis for this phenomenon of the sky goes back to Newtons theory of motion. In that theory, the existence of a satellite is made possible because the Earth rotates. Such orbits are useful for telecommunications satellites, a perfectly stable geostationary orbit is an ideal that can only be approximated. Elliptical geosynchronous orbits can be and are designed for satellites in order to keep the satellite within view of its assigned ground stations or receivers. A satellite in a geosynchronous orbit appears to oscillate in the sky from the viewpoint of a ground station. Satellites in highly elliptical orbits must be tracked by ground stations
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Geostationary orbit
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A geostationary orbit, geostationary Earth orbit or geosynchronous equatorial orbit is a circular orbit 35,786 kilometres above the Earths equator and following the direction of the Earths rotation. An object in such an orbit has a period equal to the Earths rotational period and thus appears motionless, at a fixed position in the sky. Using this characteristic, ocean color satellites with visible and near-infrared light sensors can also be operated in geostationary orbit in order to monitor sensitive changes of ocean environments, the notion of a geostationary space station equipped with radio communication was published in 1928 by Herman Potočnik. The first appearance of an orbit in popular literature was in the first Venus Equilateral story by George O. Smith. Clarke acknowledged the connection in his introduction to The Complete Venus Equilateral, the orbit, which Clarke first described as useful for broadcast and relay communications satellites, is sometimes called the Clarke Orbit. Similarly, the Clarke Belt is the part of space about 35,786 km above sea level, in the plane of the equator, the Clarke Orbit is about 265,000 km in circumference. Most commercial communications satellites, broadcast satellites and SBAS satellites operate in geostationary orbits, a geostationary transfer orbit is used to move a satellite from low Earth orbit into a geostationary orbit. The first satellite placed into an orbit was the Syncom-3. A worldwide network of operational geostationary meteorological satellites is used to provide visible and infrared images of Earths surface, a geostationary orbit can only be achieved at an altitude very close to 35,786 km and directly above the equator. This equates to a velocity of 3.07 km/s and an orbital period of 1,436 minutes. This ensures that the satellite will match the Earths rotational period and has a footprint on the ground. All geostationary satellites have to be located on this ring. 85° per year, to correct for this orbital perturbation, regular orbital stationkeeping manoeuvres are necessary, amounting to a delta-v of approximately 50 m/s per year. A second effect to be taken into account is the longitude drift, there are two stable and two unstable equilibrium points. Any geostationary object placed between the points would be slowly accelerated towards the stable equilibrium position, causing a periodic longitude variation. The correction of this effect requires station-keeping maneuvers with a maximal delta-v of about 2 m/s per year, solar wind and radiation pressure also exert small forces on satellites, over time, these cause them to slowly drift away from their prescribed orbits. In the absence of servicing missions from the Earth or a renewable propulsion method, hall-effect thrusters, which are currently in use, have the potential to prolong the service life of a satellite by providing high-efficiency electric propulsion. This delay presents problems for latency-sensitive applications such as voice communication, geostationary satellites are directly overhead at the equator and become lower in the sky the further north or south one travels. At latitudes above about 81°, geostationary satellites are below the horizon, because of this, some Russian communication satellites have used elliptical Molniya and Tundra orbits, which have excellent visibility at high latitudes
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Sun-synchronous orbit
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A Sun-synchronous orbit is a geocentric orbit that combines altitude and inclination in such a way that the satellite passes over any given point of the planets surface at the same local solar time. Such an orbit can place a satellite in constant sunlight and is useful for imaging, spy, more technically, it is an orbit arranged in such a way that it precesses once a year. The surface illumination angle will be nearly the same time that the satellite is overhead. This consistent lighting is a characteristic for satellites that image the Earths surface in visible or infrared wavelengths. For example, a satellite in sun-synchronous orbit might ascend across the twelve times a day each time at approximately 15,00 mean local time. This is achieved by having the orbital plane precess approximately one degree each day with respect to the celestial sphere, eastward. Typical sun-synchronous orbits are about 600–800 km in altitude, with periods in the 96–100 minute range, riding the terminator is useful for active radar satellites as the satellites solar panels can always see the Sun, without being shadowed by the Earth. The dawn/dusk orbit has been used for solar observing scientific satellites such as Yohkoh, TRACE, Hinode and PROBA2, Sun-synchronous orbits can happen around other oblate planets, such as Mars. A satellite around the almost spherical Venus, for example, will need an outside push to be in a sun-synchronous orbit.696 deg. Note that according to this approximation cos i equals −1 when the semi-major axis equals 12352 km, the period can be in the range from 88 minutes for a very low orbit to 3.8 hours. If one wants a satellite to fly over some given spot on Earth every day at the same hour, it can do between 7 and 16 orbits per day, as shown in the following table. When one says that a Sun-synchronous orbit goes over a spot on the earth at the local time each time. The Sun will not be in exactly the same position in the sky during the course of the year, very often a frozen orbit is therefore selected that is slightly higher over the Southern hemisphere than over the Northern hemisphere. ERS-1, ERS-2 and Envisat of European Space Agency as well as the MetOp spacecraft of EUMETSAT are all operated in Sun-synchronous, orbital perturbation analysis Analemma Geosynchronous orbit Geostationary orbit List of orbits Polar orbit World Geodetic System Sandwell, David T. The Gravity Field of the Earth - Part 1 Sun-Synchronous Orbit dictionary entry, centennial of Flight Commission NASA Q&A Boain, Ronald J. The A-B-Cs of Sun Synchronous Orbit Design, List of satellites in Sun-synchronous orbit
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Low Earth orbit
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A low Earth orbit is an orbit around Earth with an altitude between 160 kilometers, and 2,000 kilometers. Objects below approximately 160 kilometers will experience very rapid orbital decay, with the exception of the 24 human beings who flew lunar flights in the Apollo program during the four-year period spanning 1968 through 1972, all human spaceflights have taken place in LEO or below. The International Space Station conducts operations in LEO, the altitude record for a human spaceflight in LEO was Gemini 11 with an apogee of 1,374.1 kilometers. All crewed space stations to date, as well as the majority of satellites, have been in LEO, objects in LEO encounter atmospheric drag from gases in the thermosphere or exosphere, depending on orbit height. Due to atmospheric drag, satellites do not usually orbit below 300 km, objects in LEO orbit Earth between the denser part of the atmosphere and below the inner Van Allen radiation belt. The mean orbital velocity needed to maintain a stable low Earth orbit is about 7.8 km/s, calculated for circular orbit of 200 km it is 7.79 km/s and for 1500 km it is 7.12 km/s. The delta-v needed to achieve low Earth orbit starts around 9.4 km/s, atmospheric and gravity drag associated with launch typically adds 1. 3–1.8 km/s to the launch vehicle delta-v required to reach normal LEO orbital velocity of around 7.8 km/s. Equatorial low Earth orbits are a subset of LEO and these orbits, with low inclination to the Equator, allow rapid revisit times and have the lowest delta-v requirement of any orbit. Orbits with an inclination angle to the equator are usually called polar orbits. Higher orbits include medium Earth orbit, sometimes called intermediate circular orbit, orbits higher than low orbit can lead to early failure of electronic components due to intense radiation and charge accumulation. Although the Earths pull due to gravity in LEO is not much less than on the surface of the Earth, people, a low Earth orbit is simplest and cheapest for satellite placement. It provides high bandwidth and low communication time lag, but satellites in LEO will not be visible from any point on the Earth at all times. Earth observation satellites and spy satellites use LEO as they are able to see the surface of the Earth more clearly as they are not so far away and they are also able to traverse the surface of the Earth. A majority of satellites are placed in LEO, making one complete revolution around the Earth in about 90 minutes. The International Space Station is in a LEO about 400 km above the Earths surface, since it requires less energy to place a satellite into a LEO and the LEO satellite needs less powerful amplifiers for successful transmission, LEO is used for many communication applications. Because these LEO orbits are not geostationary, a network of satellites is required to provide continuous coverage, lower orbits also aid remote sensing satellites because of the added detail that can be gained. Remote sensing satellites can also take advantage of sun-synchronous LEO orbits at an altitude of about 800 km, envisat is one example of an Earth observation satellite that makes use of this particular type of LEO. The LEO environment is becoming congested with space debris due to the frequency of object launches and this has caused growing concern in recent years, since collisions at orbital velocities can easily be dangerous, and even deadly
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Medium Earth orbit
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Medium Earth orbit, sometimes called intermediate circular orbit, is the region of space around the Earth above low Earth orbit and below geostationary orbit. The most common use for satellites in this region is for navigation, communication, the most common altitude is approximately 20,200 kilometres ), which yields an orbital period of 12 hours, as used, for example, by the Global Positioning System. Other satellites in medium Earth orbit include Glonass and Galileo constellations, communications satellites that cover the North and South Pole are also put in MEO. The orbital periods of MEO satellites range from about 2 to nearly 24 hours, telstar 1, an experimental satellite launched in 1962, orbits in MEO. The orbit is home to a number of artificial satellites
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Molniya orbit
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A Molniya orbit is a type of highly elliptical orbit with an inclination of 63.4 degrees, an argument of perigee of −90 degrees and an orbital period of one half of a sidereal day. Molniya orbits are named after a series of Soviet/Russian Molniya communications satellites which have been using this type of orbit since the mid-1960s.4 degrees north, to get a continuous high elevation coverage of the Northern Hemisphere, at least three Molniya spacecraft are needed. The reason that the inclination should have the value 63. 4° is that then the argument of perigee is not perturbed by the J2 term of the field of the Earth. Much of the area of the former Soviet Union, and Russia in particular, is located at high latitudes, to broadcast to these latitudes from a geostationary orbit would require considerable power due to the low elevation angles. A satellite in a Molniya orbit is better suited to communications in these regions because it looks directly down on them, an additional advantage is that considerably less launch energy is needed to place a spacecraft into a Molniya orbit than into a geostationary orbit. It is necessary to have at least three spacecraft if permanent high elevation coverage is needed for an area like the whole of Russia where some parts are as far south as 45° N. If three spacecraft are used, each spacecraft is active for periods of eight hours per orbit centered at apogee as illustrated in figure 9. The Earth completes half a rotation in 12 hours, so the apogees of successive Molniya orbits will alternate between one half of the hemisphere and the other half. For example if the apogee longitudes are 90° E and 90° W, said next spacecraft has the visibility displayed in figure 3 and the switch-over can take place. Note that the two spacecraft at the time of switch-over are separated about 1500 km, so that the stations only have to move the antennas a few degrees to acquire the new spacecraft. To avoid this expenditure of fuel, the Molniya orbit uses an inclination of 63. 4° and that this is the case follows from equation of the article Orbital perturbation analysis as the factor then is zero. The reason why the orbital period shall be half a day is that the geometry relative to the ground stations should repeat every 24 hours. In fact, the precise ideal orbital period resulting in a ground track repeating every 24 hours is not precisely half a sidereal day, but rather half a synodic day. For a Molniya orbit, the inclination is selected such that Δ ω as given by the formula above is zero but Δ Ω, as given by the other equation, will be −0. 0742° per orbit. The rotational period of the Earth relative to the node will therefore be only 86,129 seconds,35 seconds less than the day which is 86,164 seconds. The primary use of the Molniya orbit was for the satellite series of the same name. After two launch failures in 1964, the first successful satellite to use this orbit was Molniya 1-01 launched on April 23,1965. The early Molniya-1 satellites were used for military communications starting in 1968
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Orbit of the Moon
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Not to be confused with Lunar orbit. The Moon orbits Earth in the direction and completes one revolution relative to the stars in approximately 27.323 days. Earth and the Moon orbit about their barycentre, which lies about 4,600 km from Earths center, on average, the Moon is at a distance of about 385,000 km from Earths centre, which corresponds to about 60 Earth radii. With a mean velocity of 1.022 km/s, the Moon appears to move relative to the stars each hour by an amount roughly equal to its angular diameter. The Moon differs from most satellites of planets in that its orbit is close to the plane of the ecliptic. The plane of the orbit is inclined to the ecliptic by about 5°. The properties of the orbit described in this section are approximations, the Moons orbit around Earth has many irregularities, whose study has a long history. The orbit of the Moon is distinctly elliptical, with an eccentricity of 0.0549. The non-circular form of the lunar orbit causes variations in the Moons angular speed and apparent size as it moves towards, the mean angular movement relative to an imaginary observer at the barycentre is 13. 176° per day to the east. The Moons elongation is its angular distance east of the Sun at any time, at new moon, it is zero and the Moon is said to be in conjunction. At full moon, the elongation is 180° and it is said to be in opposition, in both cases, the Moon is in syzygy, that is, the Sun, Moon and Earth are nearly aligned. When elongation is either 90° or 270°, the Moon is said to be in quadrature, the orientation of the orbit is not fixed in space, but rotates over time. This orbital precession is also called apsidal precession and is the rotation of the Moons orbit within the orbital plane, the Moons apsidal precession is distinct from, and should not be confused with its axial precession. The mean inclination of the orbit to the ecliptic plane is 5. 145°. The rotational axis of the Moon is also not perpendicular to its plane, so the lunar equator is not in the plane of its orbit. Therefore, the angle between the ecliptic and the equator is always 1. 543°, even though the rotational axis of the Moon is not fixed with respect to the stars. The period from moonrise to moonrise at the poles is quite close to the sidereal period, when the sun is the furthest below the horizon, the moon will be full when it is at its highest point. The nodes are points at which the Moons orbit crosses the ecliptic, the Moon crosses the same node every 27.2122 days, an interval called the draconic or draconitic month
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Polar orbit
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A polar orbit is one in which a satellite passes above or nearly above both poles of the body being orbited on each revolution. It therefore has an inclination of 90 degrees to the equator, a satellite in a polar orbit will pass over the equator at a different longitude on each of its orbits. Polar orbits are used for earth-mapping, earth observation, capturing the earth as time passes from one point, reconnaissance satellites. The Iridium satellite constellation also uses a polar orbit to provide telecommunications services, the disadvantage to this orbit is that no one spot on the Earths surface can be sensed continuously from a satellite in a polar orbit. Near-polar orbiting satellites commonly choose a Sun-synchronous orbit, meaning each successive orbital pass occurs at the same local time of day. To keep the local time on a given pass, the time period of the orbit must be kept as short as possible. However, very low orbits of a few hundred kilometers rapidly decay due to drag from the atmosphere, commonly used altitudes are between 700 km and 800 km, producing an orbital period of about 100 minutes. The half-orbit on the Sun side then takes only 50 minutes, list of orbits Molniya orbit Vandenberg Air Force Base, a major United States launch location for polar orbits Orbital Mechanics
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Tundra orbit
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A Tundra orbit is a highly elliptical geosynchronous orbit with a high inclination and an orbital period of one sidereal day. A satellite placed in this orbit spends most of its time over an area of the Earth. The ground track of a satellite in an orbit is a closed figure eight. These orbits are similar to Molniya orbits, which have the same inclination. The only current known user of Tundra orbits is the EKS satellite, until 2016, Sirius Satellite Radio, now part of Sirius XM Holdings operated a constellation of three satellites in Tundra orbits for satellite radio. The RAAN and mean anomaly of each satellite was offset by 120° so that when one satellite moved out of position, the three satellites were launched in 2000 and moved into circular disposal orbits in 2016, Sirius XM now broadcasts only from geostationary satellites. Tundra and Molniya orbits are used to high latitude users with higher elevation angles than a geostationary orbit. An argument of perigee of 270° places apogee at the northernmost point of the orbit, an argument of perigee of 90° would likewise serve the high southern latitudes. An argument of perigee of 0° or 180° would cause the satellite to dwell over the equator, the Tundra and Molniya orbits use a sin−1 √4/5 ≈63. 4° inclination to null the secular perturbation of the argument of perigee caused by the Earths equatorial bulge. With any inclination other than 63. 4° or its supplement,116. 6°, the argument of perigee would change steadily over time, and apogee would occur either before or after the highest latitude is reached