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Results for “Legendre expansions”

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.

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All sourcesNASA NTRSDOE OSTI
NASA NTRS1973

Note on the modified two-stream approximation of Sagan and Pollack.

The modified two-stream approximation is derived from the radiative transfer equation using a two-point gaussian quadrature and neglecting terms of order l equal to or greater than 3 in the Legendre expansion of the phase function. The results are compared with the Eddington approximation and with exact results for the special cases of perfect absorption, perfect scattering, and a semi-infinite layer with isotropic scattering.

Lyzenga, D. R.↗
NASA NTRS1990

A spherical harmonics method for asteroid pole determination

A new and general method is presented for determining pole orientations of unresolved solar system bodies that exhibit brightness variation with rotation. The accuracy of the method is limited only by the number of available apparitions, which imposes the degree where the Legendre expansion must be truncated. It is shown that power spectra of lightcurves provide enough information to extract pole coordinates from the data, independent of body shape and albedo variegation. The method has been tested using several computer-generated synthetic light-curves, and the accuracy is found to be about 5 deg in both the longitude and latitude of the pole, which is considerably better than that obtained from the standard triaxial ellipsoid model. Finally, the new spherical harmonics method has been applied to six well-observed asteroids: 16 Psyche, 22 Kalliope, 39 Laetitia, 44 Nysa, 45 Eugenia, and 349 Dembowska.

Lumme, K.↗
NASA NTRS1995

An efficient algorithm for choosing scattering directions in Monte Carlo work with arbitrary phase functions

This paper describes an efficient Monte Carlo algorithm for choosing a new direction of a photon after a scattering interaction. The algorithm chooses a scattering angle by linear interpolation in a table of the inverse cumulative scattering probability. A Legendre expansion of the phase function makes it easy to apply Clenshaw's algorithm to build the interpolation table. The points in the table are close enough together that linear interpolation is accurate. With a table of 100,000 entries, we can keep the absolute and relative errors in matching the probability distribution below 10(exp -5).

Barkstrom, Bruce R.↗
NASA NTRS2018

New Treatment of Strongly Anisotropic Scattering Phase Functions: The Delta-M+ Method

The treatment of strongly anisotropic scattering phase functions is still a challenge for accurate radiance computations. The new delta-M+ method resolves this problem by introducing a reliable, fast, accurate, and easy-to-use Legendre expansion of the scattering phase function with modified moments. Delta-M+ is an upgrade of the widely used delta-M method that truncates the forward scattering peak with a Dirac delta function, where the ‘‘+’’ symbol indicates that it essentially matches moments beyond the first M terms. Compared with the original delta-M method, delta-M+ has the same computational efficiency, but for radiance computations, the accuracy and stability have been increased dramatically.

Lin, Zhenyi↗
NASA NTRS1987

Simultaneous stochastic inversion for geomagnetic main field and secular variation. I - A large-scale inverse problem

The method of stochastic inversion is extended to the simultaneous inversion of both main field and secular variation. In the present method, the time dependency is represented by an expansion in Legendre polynomials, resulting in a simple diagonal form for the a priori covariance matrix. The efficient preconditioned Broyden-Fletcher-Goldfarb-Shanno algorithm is used to solve the large system of equations resulting from expansion of the field spatially to spherical harmonic degree 14 and temporally to degree 8. Application of the method to observatory data spanning the 1900-1980 period results in a data fit of better than 30 nT, while providing temporally and spatially smoothly varying models of the magnetic field at the core-mantle boundary.

Bloxham, Jeremy↗
NASA NTRS1972

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.↗
NASA NTRS1979

Analysis of the vibratory excitation of gear systems. II - Tooth error representations, approximations, and application

The second part of a theory for predicting the vibratory excitation of gear systems from fundamental descriptions of gear tooth elastic properties and deviations of tooth faces from perfect involute surfaces is presented. The first part of the theory provides expressions for the Fourier-series coefficients of the vibratory excitation, and this paper gives expressions for these Fourier-series coefficients in terms of easily interpreted gear tooth metrics that are readily evaluated from tooth-face measurements. Results are given for rectangular tooth-face contact regions using two-dimensional Legendre polynomial expansions of local tooth-pair stiffnesses and stiffness-weighted deviations of tooth faces from perfect involute surfaces. A rigorous transfer function approach is developed that permits separation of the effects of gear tooth errors and gear design parameters; the theory is applicable to helical and spur gears and is illustrated with measurements of tooth-spacing errors and tooth profiles obtained from a pair of spur gears.

Mark, W. D.↗
NASA NTRS1981

N-stream approximations to radiative transfer

Schuster's two-stream approximation (1905) is first derived from Chandrasekhar's radiative transfer equation (1950), and then extended to an arbitrary number of streams. The resulting technique for solving the transfer function similar to the discrete ordinate and spherical harmonic methods, is useful for modeling atmospheres with complicated phase functions and moderate optical depths. The resulting n coupled linear differential equations are simple and consume less computer time than other approximations, yet have the same required accuracy. The approximation is also flexible with respect to the choice of patch functions, and no approximations are made on the form of the phase function, other than its expansion into Legendre polynomials. A four-stream approximation is evaluated for a Henyey-Greenstein phase function with an asymmetry factor equal to 0.5.

Acquista, C.↗
NASA NTRS1982

Line shape of the non-thermal 6300 A O/1D/ emission

The two-population model of Schmitt, Abreu and Hays (1981) is used to calculate the line shape of the atomic oxygen metastable state, nonthermal O(1D) 6300 A emission, in order to simulate observations made from a space platform at different zenith angles and altitudes. The Addition theorem, for spherical harmonics of a Legendre polynomial expansion of the nonthermal population distribution function, is used to obtain nonthermal line shapes observed at zenith angles other than the local vertical one.

Schmitt, G. A.↗
NASA NTRS1986

Exact formulas for multipole moments using Slater-type molecular orbitals

A triple infinite sum of formulas expressed as an expansion in Legendre polynomials is generated by use of computer algebra to represent the potential from the midpoint of two Slater-type orbitals; the charge density that determines the potential is given as the product of the two orbitals. An example using 1s orbitals shows that only a few terms are needed to obtain four-figure accuracy. Exact formulas are obtained for multipole moments by means of a careful study of expanded formulas, allowing an 'extrapolation to infinity'. This Loewdin alpha-function approach augmented by using a C matrix to characterize Slater-type orbitals can be readily generalized to all cases.

Jones, H. W.↗
NASA NTRS1988

Helioseismology from the South Pole: Comparison of 1987 and 1981 Results

Full disk images with 10 arc sec pixels and filtered to a 7 A pass band centered on the Ca II K line were obtained from the geographic South Pole in 1981 and 1987. In 1981, 50hr of essentially uninterrupted data were obtained. In 1987, three such runs were obtained over a period of 325 hours for a duty cycle of 47 percent. The 1987 observations are characterized by a much lower level of solar activity than 1981, a much improved CCD camera, considerably better image stability and a varying amount of instrumental scatter. The 1987 data have a substantially better signal-to-noise ratio than the 1981 data so that oscillations with degrees from 0 to 150 and frequencies from 2 to 7 mHz are well observed. The observations were reduced to spectra in l, m, and v. A comparison of p-mode frequencies measured in 1981 and 1987, and coefficients of Legendre polynomial expansions of frequency shifts caused by solar rotation are presented. The time behavior of systematic frequency shifts which depend upon m but which do not arise from rotation is described.

Jefferies, S. M.↗
NASA NTRS1993

Solar activity and oscillation frequency splittings

Solar p-mode frequency splittings, parameterized by the coefficients through order N = 12 of a Legendre polynomial expansion of the mode frequencies as a function of m/L, were obtained from an analysis of helioseismology data taken at Big Bear Solar Observatory during the 4 years 1986 and 1988-1990 (approximately solar minimum to maximum). Inversion of the even-index splitting coefficients confirms that there is a significant contribution to the frequency splittings originating near the solar poles. The strength of the polar contribution is anti correlated with the overall level or solar activity in the active latitudes, suggesting a relation to polar faculae. From an analysis of the odd-index splitting coefficients we infer an uppor limit to changes in the solar equatorial near-surface rotatinal velocity of less than 1.9 m/s (3 sigma limit) between solar minimum and maximum.

Woodard, M. F.↗
NASA NTRS1975

High zonal harmonics of rapidly rotating planets

A new perturbation expansion is derived for the structure of rotating bodies in hydrostatic equilibrium. The method uses an expansion of the density on Legendre polynomial functions of angle, and can be developed analytically in a manner analogous to the standard level-surface perturbation theory. The new theory proceeds from a prescribed pressure-density relation rather than from a prescribed density distribution, and is both simpler and more physically transparent than the level-surface approach. High zonal harmonics are shown to arise via a transfer function involving derivatives of the interior sound velocity, and via mixing of multipole density components in the outer shell of the planet. Sample calculations for polytropic sequences are presented, as well as standard gravity models for Jupiter and Saturn. Mathematical subleties of the theory are discussed in an appendix.

Hubbard, W. B.↗
NASA NTRS1977

A note of expansions of functions of velocity in the two-body problem

Fourier expansions based on both the true anomaly and the mean anomaly are obtained for the functions of velocity in the two-body problem; the series of coefficients is written from classical formulae involving associated Legendre polynomials, Gegenbauer polynomials, or Bessel functions. The Fourier expansions are compared with the expansions in powers of eccentricity developed by Broucke (1974) through use of computerized Poisson series manipulation.

Kinoshita, H.↗