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Laplace Limit
In mathematics, the Laplace limit is the maximum value of the eccentricity for which a solution to Kepler's equation, in terms of a power series in the eccentricity, converges. It is approximately : 0.66274 34193 49181 58097 47420 97109 25290. Kepler's equation ''M'' = ''E'' − ε sin ''E'' relates the mean anomaly ''M'' with the eccentric anomaly ''E'' for a body moving in an ellipse with eccentricity ε. This equation cannot be solved for ''E'' in terms of elementary functions, but the Lagrange reversion theorem gives the solution as a power series in ε: : E = M + \sin(M) \, \varepsilon + \tfrac12 \sin(2M) \, \varepsilon^2 + \left( \tfrac38 \sin(3M) - \tfrac18 \sin(M) \right) \, \varepsilon^3 + \cdots or in general : E = M \;+\; \sum_^ \frac \sum_^ (-1)^k\,\binom\,(n-2k)^\,\sin((n-2k)\,M) Laplace realized that this series converges for small values of the eccentricity, but diverges for any value of ''M'' other than a multiple of π ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eccentricity (mathematics)
In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. More formally two conic sections are similar if and only if they have the same eccentricity. One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular: * The eccentricity of a circle is zero. * The eccentricity of an ellipse which is not a circle is greater than zero but less than 1. * The eccentricity of a parabola is 1. * The eccentricity of a hyperbola is greater than 1. * The eccentricity of a pair of lines is \infty Definitions Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the eccentricity, commonly denoted as . The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is oriented ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kepler's Equation
In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was first derived by Johannes Kepler in 1609 in Chapter 60 of his ''Astronomia nova'', and in book V of his '' Epitome of Copernican Astronomy'' (1621) Kepler proposed an iterative solution to the equation. The equation has played an important role in the history of both physics and mathematics, particularly classical celestial mechanics. Equation Kepler's equation is where M is the mean anomaly, E is the eccentric anomaly, and e is the eccentricity. The 'eccentric anomaly' E is useful to compute the position of a point moving in a Keplerian orbit. As for instance, if the body passes the periastron at coordinates x = a(1 - e), y = 0, at time t = t_0, then to find out the position of the body at any time, you first calculate the mean anomaly M from the time and the mean motion n by the formula M = n(t - t_0), then solve the Kepler equation above t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Anomaly
In celestial mechanics, the mean anomaly is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classical two-body problem. It is the angular distance from the pericenter which a fictitious body would have if it moved in a circular orbit, with constant speed, in the same orbital period as the actual body in its elliptical orbit. Definition Define as the time required for a particular body to complete one orbit. In time , the radius vector sweeps out 2 radians, or 360°. The average rate of sweep, , is then :n = \frac = \frac~, which is called the '' mean angular motion'' of the body, with dimensions of radians per unit time or degrees per unit time. Define as the time at which the body is at the pericenter. From the above definitions, a new quantity, , the ''mean anomaly'' can be defined :M = n\,(t - \tau) ~, which gives an angular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eccentric Anomaly
In orbital mechanics, the 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 angular parameters ("anomalies") that define a position along an orbit, the other two being the true anomaly and the mean anomaly. Graphical representation Consider the ellipse with equation given by: :\frac + \frac = 1, where ''a'' is the ''semi-major'' axis and ''b'' is the ''semi-minor'' axis. For a point on the ellipse, ''P'' = ''P''(''x'', ''y''), representing the position of an orbiting body in an elliptical orbit, the eccentric anomaly is the angle ''E'' in the figure. The eccentric anomaly ''E'' is one of the angles of a right triangle with one vertex at the center of the ellipse, its adjacent side lying on the ''major'' axis, having hypotenuse ''a'' (equal to the ''semi-major'' axis of the ellipse), and opposite side (perpendicular to the ''major'' axis and touching th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity (mathematics), eccentricity e, a number ranging from e = 0 (the Limiting case (mathematics), limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola). An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution. Analytic geometry, Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is: : \frac+\frac = 1 . Assuming a \ge b, the foci are (\pm c, 0) for c = \sqrt. The standard parametric e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions (e.g., arcsin, log, or ''x''1/''n''). All elementary functions are continuous on their domains. Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841. An algebraic treatment of elementary functions was started by Joseph Fels Ritt in the 1930s. Examples Basic examples Elementary functions of a single variable include: * Constant functions: 2,\ \pi,\ e, etc. * Rational powers of : x,\ x^2,\ \sqrt\ (x^\frac),\ x^\frac, etc. * more general algebraic functions: f(x) satisfying f(x)^5+f(x)+x=0, which is not expressible through n-th roots or rational powers of alone * Exponential functions: e^x, \ a^x * Logarithms: \ln x, \ \log_a x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Reversion Theorem
In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions. Let ''v'' be a function of ''x'' and ''y'' in terms of another function ''f'' such that :v=x+yf(v) Then for any function ''g'', for small enough ''y'': :g(v)=g(x)+\sum_^\infty\frac\left(\frac\partial\right)^\left(f(x)^kg'(x)\right). If ''g'' is the identity, this becomes :v=x+\sum_^\infty\frac\left(\frac\partial\right)^\left(f(x)^k\right) In which case the equation can be derived using perturbation theory. In 1770, Joseph Louis Lagrange (1736–1813) published his power series solution of the implicit equation for ''v'' mentioned above. However, his solution used cumbersome series expansions of logarithms. In 1780, Pierre-Simon Laplace (1749–1827) published a simpler proof of the theorem, which was based on relations between partial derivatives with respect to the variable x and the parameter y. Charl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, ''c'' (the ''center'' of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized and extended the work of his predecessors in his five-volume ''Mécanique céleste'' (''Celestial Mechanics'') (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace. Laplace formulated Laplace's equation, and pioneered the Laplace transform which appears in many branches of mathematical physics, a field that he took a leading role in forming. The Laplacian differential operator, widely used in mathematics, is also named after him. He restated and developed the nebular hypothesis of the origin of the Solar System and was one of the first scientists to sugges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radius Of Convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges. In case of multiple singularities of a function (singularities are those values of the argument for which the function is not defined), the radius of convergence is the shortest or minimum of all the respective distances (which are all non-negative numbers) calculated from the center of the disk of convergence to the respective singularities of the function. Definition For a power series ''f'' defined as: :f(z) = \sum_^\infty c_n (z-a)^n, where *''a'' is a complex constant, the center of the disk of convergence, *''c''''n'' is the ''n''-th comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendental Equation
In applied mathematics, a transcendental equation is an equation over the real number, real (or complex number, complex) numbers that is not algebraic equation, algebraic, that is, if at least one of its sides describes a transcendental function. Examples include: :\begin x &= e^ \\ x &= \cos x \\ 2^x &= x^2 \end A transcendental equation need not be an equation between elementary functions, although most published examples are. In some cases, a transcendental equation can be solved by transforming it into an equivalent algebraic equation. Some such transformations are sketched #Transformation into an algebraic equation, below; computer algebra systems may provide more elaborated transformations. In general, however, only approximate solutions can be found. Transformation into an algebraic equation Ad hoc methods exist for some classes of transcendental equations in one variable to transform them into algebraic equations which then might be solved. Exponential equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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