Showing papers on "Legendre polynomials published in 1975"

Theory of Energy-Balance Climate Models.

Abstract: A class of mean annual, zonally averaged energy-balance climate models of the Budyko-Sellers type are studied by a spectral (expansion in Legendre polynomials) method. Models with constant thermal diffusion coefficient can be solved exactly, The solution is approached by a rapidly converging sequence with each succeeding approximant taking into account information from ever smaller space and time scales. The first two modes represent a good approximation to the exact solution as well as to the present climate. The two-mode approximation to a number of more general models are shown to be either formally or approximately equivalent to the same truncation in the constant diffusion case. In particular, the transport parameterization used by Budyko is precisely equivalent to the two-mode truncation of thermal diffusion. Details of the dynamics do not influence the first two modes which fortunately seem adequate for the study of global climate change. Estimated ice age temperatures and ice line latitud...

A three parameter analytic phase function for multiple scattering calculations

Abstract: A simple procedure was developed to fit the first three moments of an actual phase function with a three parameter analytic phase function. The exact Legendre Polynomial decomposition of this function is, suitable for multiple scattering calculations. The use of this function is expected to yield excellent flux values at all depths within a medium. Since it is capable of reproducing the glory, it can be used in synthetic spectra computations from planetary atmospheres. Accurate asymptotic radiance values can also be achieved as long as the single scattering albedo omega sub 0 is greater than or equal to 0.9.

Statistical theory of angular momentum polarization in chemical reactions

Abstract: A general statistical treatment applicable to any vector property of reactive scattering is derived from angular correlation theory. This pertains to the usual experimental situation in which two or three vector directions are observed but numerous other vectors are random or unobserved, particularly various angular momentum vectors. The dependence of the cross section on the angles relating the observed vectors is expanded as a Legendre polynomial series, with coefficients which represent averages of angular momentum functions over the unobserved vectors. An algorithm for calculating these angular correlation coefficients is provided by the statistical theory. All non-vanishing terms involve only even-order Legendre polynomials. In many experiments, one or two terms are predominant. Classical and quantal versions give the same algorithm in the correspondence principle limit, which often holds for chemical reactions. The angular correlations involving the initial and final relative velocity vector directi...

Critical Points of Smooth Functions

Abstract: / = ± x\ ± '•• ± x\ + const using suitable local coordinates. Every degenerate critical point bifurcates into some nondegenerate points after an arbitrarily small deformation ("morsification"). So generically, functions have no degenerate critical points. Degenerate critical points appear naturally when the function depends upon parameters. For example, the function f(x) = x — tx has a degenerate critical point for the value t = 0 of the parameter. Every family of functions close enough to this one-parameter family has a similar degenerate critical point for some small value of the parameter, When the parameters are few, only the simplest degeneracies appear generically, and one can explicitly list them, giving normal forms for functions and families. When the number of parameters increases, more complicated degeneracies appear, and their classification seems hopeless. In recent years it has been found, however, that at least the initial part of the hierarchy of singularities is remarkably simple, as is described below, Families of functions appear in all branches of analysis and mathematical physics. In this report three applications will be discussed: Lagrange singularities (or caustics), Legendre singularities (or wavefronts), and oscillating integrals (or stationary phase method).

Qualitative behavior of heavy-ion elastic scattering angular distributions

Abstract: To study the qualitative features of elastic scattering in the presence of strong absorption, the scattering amplitude is decomposed into what the semiclassical approach calls positive- and negative-deflection-angle contributions. For an amplitude obtained from partial-wave summation this is done without approximation by considering the two amplitudes corresponding to the decomposition of each Legendre polynomial into its two traveling-wave components. It is necessary to separately consider the amplitude arising from the infinite-range Coulomb interaction which does not admit a partial-wave expansion. To decompose this amplitude we follow an approach which leads to increased understanding of the "diverging lens" effect of the Coulomb field and the related Fraunhofer and Fresnel diffraction structure of angular distributions. In addition, the absence or presence of oscillation at back angles is shown to be related to the dominance of reflective or "encirclement" scattering. By examining the phase of the positive-deflection-angle contribution to the scattering amplitude, we are able to conclude, in agreement with Frahn, that the smooth fall in tandem heavy-ion elastic angular distributions arises from a diffractive shadow and not a refractive or Coulomb-rainbow shadow.NUCLEAR REACTIONS HI elastic angular distributions interpreted.

Theoretical study of inelastic scattering of H2 by Li+ on SCF and CI potential energy surfaces

Abstract: Integral and differential cross sections for pure rotational and simultaneous rotational−vibrational excitation of H2 by Li+ impact have been computed following the coupled−channel formalism using two different SCF potential energy hypersurfaces and a CI hypersurface at 0.6 and 1.2 eV. Sensitivity of integral cross sections to (a) choice of ab initio potential energy surface and (b) expansion length of a Legendre polynomial representation of one of the energy surfaces is examined. It is seen that preparation of H2 in the v = 0, j = 2 state leads to four− and fivefold increases in excitation cross sections to the v′ = 1, j′ = i, i = 0,2,4 states relative to excitation of ground state (v = 0, j = 0) H2. Differential cross sections are reported at 1.2 eV for up to five quantum rotational and for vibrational transitions on one of the energy hypersurfaces. All angular distributions required for determining ratios (inelastic : elastic) of differential cross sections needed for comparison with recent time−of−fli...

Intermolecular potentials and isotope effects for molecular hydrogen–inert gas complexes

Abstract: An exact power series expansion is derived which interrelates the coefficients of the Legendre expansions for an atom–diatomic molecule potential function expanded about the centers of mass of two different isotopes of the diatomic. Its application to the anisotropic potential recently obtained for H2– and D2–Ar yields a potential for HD–Ar, which is then used to predict the transition frequencies in the infrared absorption spectrum of the latter. The relatively large discrepancies between these predictions and the recent measurements of McKellar [J. Chem. Phys. 61, 4636 (1974)] are attributed to the unusual sensitivity of such predictions to the shape of the isotropic potential in the region between its minimum at Re(0) and its zero energy turning point, σ(0).

Step by step Tau method—Part I. Piecewise polynomial approximations

Abstract: Lanczos remarked that approximations obtained with the Tau method using a Legendre polynomial perturbation term defined in a finite interval J, give accurate estimations at the end point of J. This fact, coupled with a recursive technique for the generation of Tau approximations described by the author elsewhere (Ortiz, 1969, 1974), is used to construct a step by step formulation of the Tau method in which the error is minimized at the matching point of successive steps. This formulation is applied to the construction of accurate piecewise polynomial approximations with an almost equioscillant error and various degrees of smoothness at the breaking points. A technique based on the mapping of a master element Tau approximation defined over a finite interval of variable length is used in order to simplify the computational process. Numerical examples and an estimation of the step size in relation to the size of the error in the equation are also discussed.

Multiple scattering processes: Inverse and direct

Abstract: The purpose of the work is to formulate inverse problems in radiative transfer, to introduce the functions b and h as parameters of internal intensity in homogeneous slabs, and to derive initial value problems to replace the more traditional boundary value problems and integral equations of multiple scattering with high computational efficiency. The discussion covers multiple scattering processes in a one-dimensional medium; isotropic scattering in homogeneous slabs illuminated by parallel rays of radiation; the theory of functions b and h in homogeneous slabs illuminated by isotropic sources of radiation either at the top or at the bottom; inverse and direct problems of multiple scattering in slabs including internal sources; multiple scattering in inhomogeneous media, with particular reference to inverse problems for estimation of layers and total thickness of inhomogeneous slabs and to multiple scattering problems with Lambert's law and specular reflectors underlying slabs; and anisotropic scattering with reduction of the number of relevant arguments through axially symmetric fields and expansion in Legendre functions. Gaussian quadrature data for a seven point formula, a FORTRAN program for computing the functions b and h, and tables of these functions supplement the text.

High zonal harmonics of rapidly rotating planets

Abstract: 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.

On the formulation of the gravitational potential in terms of equinoctial variables

Abstract: Analytical averaging techniques are used to expand the disturbing potential in the equinoctial coordinate frame by considering third body harmonics and zonal functions harmonics. General results are developed through applications of Legendre and associated Legendre polynomials and the Q sub nm functions for the gravitational potential.

Effect of electron correlation on the H2CO‐He interaction potential

Abstract: A previously reported Hartree‐Fock (HF) interaction potential between H2CO(1A1) and He(1s) is modified through a series of configuration interaction (CI) calculations. The CI contribution is described by a three‐term (l=0,1,2) Legendre polynomial expansion in the angle τ formed by the direction of incidence of He and the CO bond of formaldehyde. No significant azimuthal angle dependence is obtained. Correlation is found to have little effect in the strongly anisotropic repulsive region of the interaction potential but dominates the well and long‐range regions. The maximum well depth is attained for in‐plane approaches of He and lies in the range 35–40 °K for arbitrary τ at center of mass separation of 7.5 a.u. The CI contribution in the region of the minimum is beleived accurate to ∼20%.

Formulas of the dirichlet-mehler type

Abstract: A fractional integral formula of Erdelyi is used to show that the fractional calculus leads in a natural way to useful formulas of the Dirichlet-Mehler type for the Jacobi polynomials and the generalized Legendre functions. Some important applications are pointed out, and a discrete analogue of Erdelyi's formula is derived and used to obtain a discrete Dirichlet-Mehler type formula for the Hahn polynomials (a discrete analogue of the Jacobi polynomials).

On the magnetization of the triangular lattice Ising model with triplet interactions

Abstract: The analytic properties of the spontaneous magnetization M for the triangular lattice Ising model with pure triplet interactions are investigated. In particular, it is shown that M is a solution of Legendre’s elliptic modular equation of degree 3. The following explicit hypergeometric formula for M is also derived:

High-Order Response Matrix Equations in Two-Dimensional Geometry

Abstract: Response matrix equations in two-dimensional geometry have been derived in the form of a set of coupled integral equations of the Fredholm type that have been solved by the moments method. The set of Legendre polynomials defined at the material interfaces has been chosen as the base for representing the partial interface currents and the response matrices.The method has been applied to the solution of the one-group diffusion equation and its convergence has been investigated in a series of numerical experiments, involving expansions of up to order 14. It turned out that the P1 approximation should be adequate for the majority of the two-dimensional problems occurring in power reactor design. Furthermore, the response method has a substantially higher computer efficiency than the finite difference method, both in processor time and in storage locations. As a by-product, the nature of the singularities around edges and corners of material interfaces has been analyzed by numerical experimentation.

Correlation between eigenvalues of random matrices

Abstract: The so-called «Legendre» ensemble of random matrices is studied for the nontrivial cases β=1 and 4. As for the more trivial case β=2 the level density has a behaviour in qualitative agreement with nuclear and atomic densities,i.e. it is concave upward and rises rapidly. The nearest-neighbour spacing distributions are obtained for β=1,4; they are the same as for the Gaussian and circular ensembles.

Solution to the Boltzmann equation for a model polyvalent metal and resistivity calculations

Abstract: An approximate solution to the coupled electron and phonon Boltzmann equations for an idealized model of Al, In, and other metals whose Fermi surface intersects the Brillouin-zone boundary has been found. Using an expansion in Legendre polynomials for the nonequilibrium distribution function and a spherical Fermi surface, the umklapp part of the scattering term is expanded in powers of $\frac{T}{{\ensuremath{\Theta}}_{D}}$ (where ${\ensuremath{\Theta}}_{D}$ is the Debye temperature). The solution to this equation is found which exhibits an unusual reduction in the nonequilibrium distribution function at the region of intersection of the Fermi surface with the zone boundary, as first anticipated by Klemens and Jackson. The resistivity calculated using this distribution function shows good agreement with experimental results for Al except in the very-low-temperature low-impurity region. A subsequent modification to the solution which attempts to roughly approximate the distortion of the Fermi surface near the Brillouin-zone boundary results in good agreement with experimental results even in this region.

Relativistic Compton scattering from moving electrons and angular moments

Abstract: Using the invariant, four-dimensional representation of the photon-electron interaction derived from quantum electrodynamics, we obtain analytical expressions for the Legendre moments of the scattering kernel, given arbitrary distributions of initial electrons and specified energy ranges. Monoenergetic and Maxwellian electrons distributions are briefly discussed and the partial wave expansion of the static Klein-Nishina formula included.

Some properties of polynomials orthogonal over the set 〈1, 2, ...N〉

Abstract: Using identities being discrete counterparts of those which are statisfied by the Legendre polynomials, the author proves that if the polynomials 〈Ψm(N)(t)〉 (m=0, 1, ..., N−1) form an orthogonal set over the set 〈1, 2, ..., N〉 with equal weight attached to its elements, then $$\left| {\Psi _m^{(N)} (t)} \right|< \left| {\Psi _m^{(N)} (1)} \right| = \left| {\Psi _m^{(N)} (N)} \right| (t = 2,3,...,N - 1)$$ when m(m+1)⩽N−1. This result is then extended to a wide class of Hahn polynomials.

Finite elements on the basis of continuum mechanics

Abstract: The formulation of finite element models on the basis of different variational principles is reviewed. The degrees-of-freedom of the elements are defined in an abstract way without the help of nodal points. In this manner it is possible to describe elements of arbitrary shape and accuracy. The formulation is confined to linear elasto-statics. For two-dimensional structures two hybrid element models are developed using Legendre polynomials on the element boundaries. Examples of plane stress problems are used to test the generation of convergence by increasing the accuracy of the elements vs. by increasing the number of elements.

Fifth-approximation system of equations for the theory of figure

Abstract: A system of equations for the theory of figure is obtained in the fifth approximation. The equations are expressed in terms of a generalized radius and the mean radius. (AIP)