The lunar and solar perturbations in the motion of an artificial satellite due to the fourth degree Legendre polynomial
Lunar and solar perturbations in motion of artificial satellite due to fourth degree Legendre polynomial
SOLAR EFFECT↗
SEARCH · Search NASA
Results for “LEGENDRE POLYNOMIALS”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Computer program ETC improves computation of elastic transfer matrices of Legendre polynomials P/0/ and P/1/
Computer program ETC improves computation of elastic transfer matrices of Legendre polynomials P/0/ and P/1/. Rather than carrying out a double integration numerically, one of the integrations is accomplished analytically and the numerical integration need only be carried out over one variable.
Gibson, G.↗
Principal Component Analysis of Up-The-Ramp Sampled Infrared Array Data
We describe the results of principal component analysis (PCA) of up-the-ramp sampled infrared (IR) array data from the Hubble Space Telescope wide field camera 3 (WFC3 IR), James Webb Space Telescope NIRSpec, and prototype Wide Field Infrared Survey Telescope’s wide field instrument detectors. These systems use, respectively, Teledyne H1R, H2RG, and H4RG-10 near-IR detector arrays with a variety of IR array controllers. The PCA shows that the Legendre polynomials approximate the principal components of these systems (i.e., they roughly diagonalize the covariance matrix). In contrast to the monomial basis that is widely used for polynomial fitting and linearization today, the Legendre polynomials are an orthonormal basis. They provide a quantifiable, compact, and (nearly) linearly uncorrelated representation of the information content of the data. By fitting a few Legendre polynomials, nearly all of the meaningful information in representative WFC3 astronomical datacubes can be condensed from 15 up-the-ramp samples down to 6 compressible Legendre coefficients per pixel. The higher order coefficients contain time domain information that is lost when one projects up-the-ramp sampled datacubes onto two-dimensional images by fitting a straight line, even if the data are linearized before fitting the line. Going forward, we believe that this time domain information is potentially important for disentangling the various nonlinearities that can affect IR array observations, i.e., inherent pixel nonlinearity, persistence, burn in, brighter-fatter effect, (potentially) nonlinear interpixel capacitance, and perhaps others.
Rauscher, Bernard J.↗
Direct solution for thermal stresses in a nose cap under an arbitrary axisymmetric temperature distribution
The design of a nose cap for a hypersonic vehicle is an iterative process requiring a rapid, easy to use and accurate stress analysis. The objective of this paper is to develop such a stress analysis technique from a direct solution of the thermal stress equations for a spherical shell. The nose cap structure is treated as a thin spherical shell with an axisymmetric temperature distribution. The governing differential equations are solved by expressing the stress solution to the thermoelastic equations in terms of a series of derivatives of the Legendre polynomials. The process of finding the coefficients for the series solution in terms of the temperature distribution is generalized by expressing the temperature along the shell and through the thickness as a polynomial in the spherical angle coordinate. Under this generalization the orthogonality property of the Legendre polynomials leads to a sequence of integrals involving powers of the spherical shell coordinate times the derivative of the Legendre polynomials. The coefficients of the temperature polynomial appear outside of these integrals. Thus, the integrals are evaluated only once and their values tabulated for use with any arbitrary polynomial temperature distribution.
Davis, Randall C.↗
Meshless Local Petrov-Galerkin (MLPG) Method with Orthogonal Polynomials for Euler-Bernoulli Beam Problems
In this paper, the feasibility of orthogonal polynomials in the meshless local Petrov Galerkin method (MLPG) method is studied. The orthogonal polynomials, Chebyshev and Legendre polynomials, are used in this MLPG method as trial functions. The test functions used were power functions with smooth derivatives at their ends. The performance of these methods is studied by applying these methods to Euler-Bernoulli beam problems. The MLPG-Galerkin and Legendre methods passed all the patch tests for simple beam problems. Next the formulations are tested on complex beam problems such as beams with partial loadings and continuous beam problems. Problems with load discontinuities and additional supports require special attention. Near discontinuities, judicious choice of number of nodes and nodal placements are needed to obtain accurate deflections, slopes, moments and shear forces. As polynomial functions are used, the large number of nodes can create a transformation matrix that is ill-conditioned, resulting in problems with the inversion of the matrix. The conditioning worsens as the number of nodes are increased beyond 20. Quadruple precision was needed for models to obtain accurate solutions. Even with quadruple precision the accuracy of the method suffers as the number of nodes is increased beyond 20. This appears to be a drawback of the MLPG-Chebyshev and MLPG-Legendre methods.
Raju, Ivatury S.↗
Normalization and Implementation of Three Gravitational Acceleration Models
Unlike the uniform density spherical shell approximations of Newton, the consequence of spaceflight in the real universe is that gravitational fields are sensitive to the asphericity of their generating central bodies. The gravitational potential of an aspherical central body is typically resolved using spherical harmonic approximations. However, attempting to directly calculate the spherical harmonic approximations results in at least two singularities that must be removed to generalize the method and solve for any possible orbit, including polar orbits. Samuel Pines, Bill Lear, and Robert Gottlieb developed three unique algorithms to eliminate these singularities. This paper documents the methodical normalization of two of the three known formulations for singularity-free gravitational acceleration (namely, the Lear and Gottlieb algorithms) and formulates a general method for defining normalization parameters used to generate normalized Legendre polynomials and Associated Legendre Functions (ALFs) for any algorithm. A treatment of the conventional formulation of the gravitational potential and acceleration is also provided, in addition to a brief overview of the philosophical differences between the three known singularity-free algorithms.
Eckman, Randy A.↗
Numerical construction of the Hill functions.
As an aid in the numerical construction of Hill functions and their derivatives, an algorithm using local coordinates and an expansion in Legendre polynomials is proposed. The algorithm is shown to possess sufficient stability, and the orthogonality of the Legendre polynomials simplifies the computation when the Ritz-Galerkin technique is used.
Segethova, J.↗
On the Gibbs phenomenon 3: Recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise analytic function
The investigation of overcoming Gibbs phenomenon was continued, i.e., obtaining exponential accuracy at all points including at the discontinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. It was shown that if we are given the first N expansion coefficients of an L(sub 2) function f(x) in terms of either the trigonometrical polynomials or the Chebyshev or Legendre polynomials, an exponentially convergent approximation to the point values of f(x) in any sub-interval in which it is analytic can be constructed.
Gottlieb, David↗
Data compression preserving statistical independence
The purpose of this study was to determine the optimum points of evaluation of data compressed by means of polynomial smoothing. It is shown that a set y of m statistically independent observations Y(t sub 1), Y(t sub 2), ... Y(t sub m) of a quantity X(t), which can be described by a (n-1)th degree polynomial in time, may be represented by a set Z of n statistically independent compressed observations Z (tau sub 1), Z (tau sub 2),...Z (tau sub n), such that The compressed set Z has the same information content as the observed set Y. the times tau sub 1, tau sub 2,.. tau sub n are the zeros of an nth degree polynomial P sub n, to whose definition and properties the bulk of this report is devoted. The polynomials P sub n are defined as functions of the observation times t sub 1, t sub 2,.. t sub n, and it is interesting to note that if the observation times are continuously distributed the polynomials P sub n degenerate to legendre polynomials. The proposed data compression scheme is a little more complex than those usually employed, but has the advantage of preserving all the information content of the original observations.
Morduch, G. E.↗
A formulation consideration for orthogonal collocation procedures
Numerical instabilities often arise in the use of high-ordered collocation approximations for numerically solving parabolic partial differential equations. These problems may be reduced by formulations involving evaluation of collocation polynomials rather than combination of the polynomials into a power series. As an illustration, two formulations using shifted Legendre polynomials of order 26 and less are compared.
Lashmet, P. K.↗
Oceanic Tidal Mixing As a Contributor to Milankovitch-scale Climate Change
We propose that changes in the magnitude of oceanic tidal mixing on long time scales is an important, but previously unrecognized, contributor to global climate change. it is well known that Earth's orbital and rotational state changes significantly on 10(exp 4)-10(exp 5) year time scales, and that this influences the spatial and temporal pattern of incident radiation. It is widely supposed that climatic variations on these same time scales are, in large part, a response of the ocean-atmosphere-cryosphere system to this radiative forcing. Our proposal is that variations in the luni-solar tidal potential, induced by these same orbital and rotational variations, influences oceanic mixing and thus modulates meridional heat transport, by amounts which are competitive with the radiative forcing. There are some obvious differences between tidal potential and insolation. First is that the Sun and Moon both contribute to tides, whereas the radiation is entirely of solar origin. Second is that the Earth is transparent to gravity but opaque to radiation. Clipping associated with this opacity makes the radiation pattern temporal spectrum rather more complex than the tidal spectrum. A third point is that solar radiation directly delivers energy to Earth's surface whereas tidal mixing will only expedite lateral transport of heat in association with oceanic thermohaline circulation. The diurnal average insolation pattern is best parameterized via a Fourier series in time of year and Legendre polynomials in sine of latitude. Our present focus will be on the annual average terms. The Legendre degree n=0 term describes the global average insolation, and is nearly constant. The degree n=l term describes differences between northern and southern hemispheres, and the annual mean is zero. The degree n=2 term is the main contributor to the equator to pole variations, and varies with obliquity and orbital eccentricity, with the obliquity variation dominating. The lowest order decomposition of the tidal potential recognizes 3 constituents: semi-diurnal, diurnal, and long period, with solar and lunar contributions to each. Our present focus will be on long term variations in the mean square amplitude of the semi-diurnal constituent, with averaging over all the short period variations. For the solar tide that includes the day and year. For the lunar tide it includes the day, month, year, and the apsidal (8.85 year) and nodal (18.6 year) periods. We present calculations of the variations in radiative and tidal forcing for the past 3 million years. The two signals are quite similar. Both vary by approximately 1% of their respective mean values, are dominated by obliquity variations, and exhibit only secondary influence from orbital eccentricity.
Munk, Walter↗
On the coefficients of differentiated expansions of ultraspherical polynomials
A formula expressing the coefficients of an expression of ultraspherical polynomials which has been differentiated an arbitrary number of times in terms of the coefficients of the original expansion is proved. The particular examples of Chebyshev and Legendre polynomials are considered.
Karageorghis, Andreas↗
A conforming spectral collocation strategy for Stokes flow through a channel contraction
A formula expressing the coefficients of an expansion of ultraspherical polynomials which has been differentiated an arbitrary number of times in terms of the coefficients of the original expansion is proved. The particular examples of Chebyshev and Legendre polynomials are considered.
Phillips, Timothy N.↗
Survey and comparative analysis of current geophysical models
Legendre polynomials for gravitational field and orbit perturbation
Von Puttkamer, J.↗
The optimum addition of points to quadrature formulae.
Optimum addition of abscissas in quadrature formulas by expansion of abscissas equation in Legendre polynomials
Patterson, T. N. L.↗
Nonlinear axisymmetric flexural vibration of spherical shells
Axisymmetric responses are presented of a nonshallow thin-walled spherical shell on the basis of nonlinear bending theory. An ordinary differential equation with nonlinearity of quadratic as well as cubic terms associated with variable time is derived. The derivation is based on the assumption that the deflection mode is the sum of four Legendre polynomials, and the Galerkin procedure is applied. The equation is solved by asymptotic expansion, and a first approximate solution is adopted. Unstable regions of this solution are discussed.
Kunieda, H.↗
Nearly spherical constant-power detonation waves as driven by focused radiation.
Analysis is made of the flow within a 3-D spark created in a gas absorbing energy from a steady conical beam of radiation (typically obtained from an array of lasers) with nearly spherical symmetry. First-order deviations are represented by series of Legendre polynomials of the polar angle. Self-similarity is demonstrated. For harmonics 1 to 5 radial profiles of velocity components, density and pressure are presented. The linear perturbation scheme exhibits a singularity at the focus. This reflects the existence of a 'forbidden zone', as is shown by a study of the full non-linear flow equations.
George, Y. H.↗
Nearly spherical constant-power detonation waves as driven by focused radiation
An analysis is made of the flow within a three-dimensional explosion, or spark, created in a gas absorbing energy from a steady conical beam of radiation with nearly spherical symmetry. The radiation, typically from an array of lasers with a common focus, is assumed to be very intense and absorbed immediately behind an outwardly advancing strong shock. Departures of the laser power from spherical uniformity, which would result from practical problems of arrangement, are conveniently represented by an ascending series of Legendre polynomials in the polar angle. For nonuniformities of small amplitude, first-order perturbations of the flow field are analyzed in detail. Self-similarity is shown to be retained for zero counter-pressure and power constant with time. For the first five harmonics in power distortion, the resulting fourth-order system of equations is solved numerically for profiles of velocity components, density, pressure, and shock shape. Results are presented graphically.
George, Y. H.↗