Search NASA⌕ Search

SEARCH · Search NASA

Results for “generalized spherical functions”

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.

At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS1972

Spherical balloon response to three-dimensional time-dependent flows

The concept of the Lagrangian displacement of a balloon is introduced. It is shown that the general balloon response problem is extremely complicated because the wind-forcing functions in the balloon equations of motion are functions of the wind velocity vector and its Eulerian first derivatives evaluated at the location of the balloon. The linear perturbation equations for a spherical balloon are derived by perturbing the components of velocity of the balloon about a terminal velocity state which is in equilibrium with a space-time invariant mean horizontal flow. The atmospheric flow is also perturbed such that the resulting equations can be used to analyze the responses of spherical balloons to three-dimensional time-dependent flows. The wind field is represented in terms of a four-fold Fourier integral that involves three orthogonal wave numbers and a frequency, while the balloon components of velocity are represented as Fourier integrals involving a frequency which, in turn, is a function of the wind field wave numbers and frequency and the unperturbed flow components of velocity.

Fichtl, G. H.↗
DOE OSTI2020

The Thomas theorem and the Efimov States within a generalized Bohr model

Using a generalized Bohr model and the hyper-spherical formalism for a three-body system, we derive the universal energy functions. We apply our model to 4He atom where only the Coulomb potential is dominant and find that its binding energy is well reproduced (less than 5.5% difference). Later, we focus on the equal mass three-body systems and derive the Thomas theorem assuming a simple interaction depending on the range of the potential. We discuss the conditions for which an unbound two-body system produces a bound three-body system and apply our model to 4He and triton atoms as well as to the triton nucleus. Using their scattering lengths and effective ranges, we are able to reproduce the two-body or the three-body binding energies (less than 5% difference) with only one parameter fitted. Prediction for excited (Efimov) levels are also given and in particular we demonstrate that for some hyper-angles two equal minima appear which indicate a phase (shape) transition similar to the Landau's theory of phase transition. We suggest that the observed excited levels in two different experiments for the triton nucleus are indeed Efimov levels and there may be more surprises.

74 ATOMIC AND MOLECULAR PHYSICS↗
NASA NTRS1979

Particle dynamics associated with the space environment

A major factor in dust dynamics in a microgravity environment is the absence of settling; hence, Spacelab air will contain larger particles than a comparable laboratory on earth. In addition, the presence of low level acceleration fluctuations (thruster firings, crew motions, etc.) could inhibit dust removal by surface scavenging. Because of the presence of larger particles, aerosol dynamics at larger Reynolds numbers must be considered. An accurate drag coefficient for spherical particles was developed for the higher Reynolds number phenomena. A general graphic correlation for the drag coefficient as a function of Reynolds and Knudsen numbers was developed based on currently available drag data for spheres. The general equations that govern dust dynamics were reviewed.

Sandborn, V. A.↗
NASA NTRS1989

Droplet model for autocorrelation functions in an Ising ferromagnet

The autocorrelation function of Ising spins in an ordered phase is studied via a droplet model. Only noninteracting spherical droplets are considered. The Langevin equation which describes fluctuations in the radius of a single droplet is studied in detail. A general description of the transformation to a Fokker-Planck equations and the ways in which a spectral analysis of that equation can be used to compute the autocorrelation function is given. It is shown that the eigenvalues of the Fokker-Planck operator form (1) a continuous spectrum of relaxation rates starting from zero for d = 2, (2) a continuous spectrum with a finite gap for d = 3, and (3) a discrete spectrum for d greater than 4, where d is the spatial dimensionality. Detailed solutions for various cases are presented.

Tang, Chao↗
DOE OSTI2021

Some aspects of the five-dimensional Lovelock black hole spacetime: Strong homotopy retract, perihelion precession and quasistationary levels

Highlights: • Some topological aspects are explored. • The perihelion precession is calculated. • The general Heun functions are applied to obtain analytical solutions. • The quasibound states and the radial wave eigenfunctions are obtained. In this work we explore some mathematical physics aspects of the spherically symmetric Lovelock black hole in high dimensions. Intended for this aim, we thoroughly consider the metric corresponding to the five-dimensional Lovelock black hole spacetime. We construct the strong retractions by the geodesic equations on the background under consideration. As a result, from the topological point of view, we construct the theory of strong homotopy retract, which will allow us, in principle, to better understand some of its suitable applications on astrophysics and cosmology, in particular, in the analysis of the spacetime singularities. We find the solutions of the equation of motion for both radial and angular coordinates, and then we describe the outer (“exterior”) and lower (“interior”) apparent horizons. Indeed, the outer apparent horizon is the last surface from which the light waves could still escape from the black hole. Thus, it is meaningful to analyze some physical phenomena related to quantum particles propagating outside the exterior apparent horizon, in particular, we discuss the quasistationary levels of scalar fields and their radial wave functions, which are given in terms of the general Heun functions. We also calculate the perihelion precession in this background.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Hyperheavy spherical and toroidal nuclei: The role of shell structure

The properties of toroidal hyperheavy even-even nuclei and the role of toroidal shell structure are extensively studied within covariant density functional theory. The general trends in the evolution of toroidal shapes in the Z ≈ 130–180 region of the nuclear chart are established for the first time. These nuclei are stable with respect to breathing deformations. The most compact fat toroidal nuclei are located in the Z ≈ 136, N ≈ 206 region of the nuclear chart, but thin toroidal nuclei become dominant with increasing proton number and toward proton and neutron drip lines. The roles of toroidal shell structure, its regularity, supershell structure, and shell gaps as well as the role of different groups of the pairs of the orbitals in its formation are investigated in detail. The lowest in energy solutions at axial symmetry are characterized either by large shell gaps or low density of the single-particle states in the vicinity of the Fermi level in at least one of the subsystems (proton or neutron). Related quantum shell effects are expected to act against the instabilities in breathing and sausage deformations for these subsystems. The investigation with a large set of covariant energy density functionals reveals that substantial proton Z = 154 and 186 and neutron N = 228, 308, and 406 spherical shell gaps exist in all functionals. The nuclei in the vicinity of the combination of these particle numbers form the islands of stability of spherical hyperheavy nuclei. Furthermore, the study suggests that the N = 210 toroidal shell gap plays a substantial role in the stabilization of fat toroidal nuclei.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS1992

Design and analysis of soft X-ray imaging microscopes

The spherical Schwarzschild microscope for soft X-ray applications in microscopy and projection lithography consists of two concentric spherical mirrors configured such that the third-order spherical aberration and coma are zero. Since multilayers are used on the mirror substrates for X-ray applications, it is desirable to have only two reflecting surfaces in a microscope. To reduce microscope aberrations and increase the field of view, generalized mirror surface profiles are here considered. Based on incoherent and sine wave modulation transfer function calculations, the object plane resolution of a microscope has been analyzed as a function of the object height and numerical aperture (NA) of the primary for several spherical Schwarzschild, conic, and aspherical Head reflecting two-mirror microscope configurations. The Head microscope with a NA of 0.4 achieves diffraction limited performance for objects with a diameter of 40 microns. Thus, it seems possible to record images with a feature size less than 100 A with a 40x microscope when using 40 A radiation.

Shealy, David L.↗
NASA NTRS2011

Implicit Priors in Galaxy Cluster Mass and Scaling Relation Determinations

Deriving the total masses of galaxy clusters from observations of the intracluster medium (ICM) generally requires some prior information, in addition to the assumptions of hydrostatic equilibrium and spherical symmetry. Often, this information takes the form of particular parametrized functions used to describe the cluster gas density and temperature profiles. In this paper, we investigate the implicit priors on hydrostatic masses that result from this fully parametric approach, and the implications of such priors for scaling relations formed from those masses. We show that the application of such fully parametric models of the ICM naturally imposes a prior on the slopes of the derived scaling relations, favoring the self-similar model, and argue that this prior may be influential in practice. In contrast, this bias does not exist for techniques which adopt an explicit prior on the form of the mass profile but describe the ICM non-parametrically. Constraints on the slope of the cluster mass-temperature relation in the literature show a separation based the approach employed, with the results from fully parametric ICM modeling clustering nearer the self-similar value. Given that a primary goal of scaling relation analyses is to test the self-similar model, the application of methods subject to strong, implicit priors should be avoided. Alternative methods and best practices are discussed.

Mantz, A.↗
DOE OSTI2021

Electric and magnetic black holes in a new nonlinear electrodynamics model

Highlights: • We introduce a new NED model which is comparable with the BI model in the weak-field limit. • We find an electric black hole solution in the context of Einstein’s gravity minimally coupled with the new NED model. • We study the thermal stability of the electric black hole solution. • Modified Smarr’s formula consistent with the first law of black hole thermodynamics is obtained. A new nonlinear electrodynamics (NED) model is introduced in the form of a nonpolynomial Lagrangian which admits static spherical and also plane wave solutions in a flat space. Upon coupling with gravity the electric field is finite and comparable with the Born–Infeld counterpart. The electric and magnetic black hole solutions in the Einstein’s gravity coupled with this NED model are presented. The solutions give both asymptotically and in the weak field limit Reissner–Nordström (RN) black hole and unlike the other known models our electric solution is expressed in terms of elementary functions in a closed form. We study the first law and derive the modified Smarr’s formula for the electric type extension of our model. Considerable rich structure, especially thermodynamic ones, ranging from first to the second order phase transitions are added to the RN black hole of linear electrodynamics with this NED model. Having the exact solution for the metric function at our disposal we investigate the stability of the electric black hole from both the thermodynamical and causal points of view.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1986

Hollow cathode plasma coupling study, 1986

The electron collection and emission characteristics of a simple hollow cathode contactor, an extended anode hollow cathode contactor supplied by JSC, and a ring cusp magnetic field contactor are presented and the effects of discharge power and argon or xenon expellant flowrate on these characteristics are examined. All of the contactors are shown to exhibit good electron emission performance over a wide range of discharge power and expellant type and flowrate. Good electron performance is shown to be more difficult to achieve. Results suggest that the extended anode and ring cusp contactors should perform satisfactorily to electron emission currents beyond 1000 mA and electron collection currents beyond 500 mA. All contactors performed better on xenon than argon. A general theory of plasma contactor operation in both the electron collection and electron emission modes, which describes the current-limiting effects of space-charge phenomena is given. This current-limiting and collecting phenomenon is shown to be a function of driving potential differences and emitting and collecting surface radius ratio for the case of a spherical geometry. Discharge power did not appear to influence the electron collection current substantially in the experiments so it is suggested in light of the model that the contactors are generally not limited by their ion production capabilities under conditions at which they were tested.

Wilbur, Paul J.↗
NASA NTRS1991

X ray microscope assembly and alignment support and advanced x ray microscope design and analysis

Considerable efforts have been devoted recently to the design, analysis, fabrication, and testing of spherical Schwarzschild microscopes for soft x ray application in microscopy and projection lithography. The spherical Schwarzschild microscope consists of two concentric spherical mirrors configured such that the third order spherical aberration and coma are zero. Since multilayers are used on the mirror substrates for x ray applications, it is desirable to have only two reflecting surfaces in a microscope. In order to reduce microscope aberrations and increase the field of view, generalized mirror surface profiles have been considered in this investigation. Based on incoherent and sine wave modulation transfer function (MTF) calculations, the object plane resolution of a microscope has been analyzed as a function of the object height and numerical aperture (NA) of the primary for several spherical Schwarzschild, conic, and aspherical head reflecting two mirror microscope configurations.

Shealy, David L.↗
NASA NTRS1989

Modelling asteroid brightness variations. I - Numerical methods

A method for generating lightcurves of asteroid models is presented. The effects of the shape of the asteroid and the scattering law of a surface element are distinctly separable, being described by chosen functions that can easily be changed. The shape is specified by means of two functions that yield the length of the radius vector and the normal vector of the surface at a given point. The general shape must be convex, but spherical concavities producing macroscopic shadowing can also be modeled.

Karttunen, H.↗
NASA NTRS1993

Two-integral distribution functions for axisymmetric galaxies

The new method presented for finding distribution functions, which depend only on the classical integrals of energy and angular momentum for stellar systems with known axisymmetric densities, is the analog for the axisymmetric case of Eddington's classical solution for the isotropic distribution function, depending only on energy, of a known spherical density. It is required that density be expressed as a function of the potential and of a radial coordinate. Our solution is also an integral which is derived directly from the density, and hence can be used with complicated densities. A numerical quadrature is generally required to evaluate this solution, but contour integrals can be computed accurately by numerical quadrature; this is preferable to an explicit evaluation if the latter is an infinite series, such as is obtained using Fricke's method. We give several examples, including some for which our distribution functions are new. Our method can be extended simply to the related problems of finding anisotropic distribution functions for spherical or disk systems.

Hunter, C.↗
DOE OSTI2022

Full-potential KKR within the removed-sphere method: A practical and accurate solution to the Poisson equation

Here an efficient and accurate generalization of the removed-sphere method (RSM) to solve the Poisson equation for total charge density in a solid with space-filling convex Voronoi polyhedra (VPs) and any symmetry is presented. The generalized RSM avoids the use of multipoles and VP shape functions for cellular integrals, which have associated ill-convergent large, double-internal L sums in spherical-harmonic expansions, so that fast convergence in single-L sums is reached. Our RSM adopts full Ewald formulation to work for all configurations or when symmetry breaking occurs, such as for atomic displacements or elastic constant calculations. The structure-dependent coefficients A L that define RSM can be calculated once for a fixed structure and speed up the whole self-consistent-field procedure. The accuracy and rapid convergence properties are confirmed using two analytic models, including the Coulomb potential and energy. We then implement the full-potential RSM using the Green's function Korringa-Kohn-Rostoker (KKR) method for real applications and compare the results with other first-principle methods and experimental data, showing that they are equally as accurate.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
NASA NTRS2013

Global Propagation of Gravity Waves Generated with the Whole Atmosphere Transfer Function Model

A brief review is presented of the Transfer Function Model (TFM) [e.g., Mayr et al., Space Science Reviews, 1990], which describes acoustic gravity waves (AGW) that propagate across the globe in a dissipative and static (no winds) background atmosphere with globally uniform temperature and density variations extending from the ground to 700 km. Unique among existing models, the TFM can be placed between the analytical approach on one end, and the rigorous numerical approach of general circulation models (GCM). The time consuming numerical integration of the conservation equations is restricted to compute the transfer function (TF) for a broad range of frequencies and spherical harmonics. Given TF, the atmospheric response for a chosen source configuration is then obtained in short order. Computationally efficient, the model is well suited to serve as experimental and educational tool for simulating propagating wave patterns across the globe. By design, the TFM is also semi-analytical and therefore well suited to explore the different wave modes that can be generated under different dynamical conditions.

Transfer Function Model↗
DOE OSTI2022

Reciprocal swimming at intermediate Reynolds number

In Stokes flow, Purcell's scallop theorem forbids objects with time-reversible (reciprocal) swimming strokes from moving. In the presence of inertia, this restriction is eased and reciprocally deforming bodies can swim. A number of recent works have investigated dimer models that swim reciprocally at intermediate Reynolds numbers Re≈1–1000. These show interesting results (e.g. switches of the swim direction as a function of inertia) but the results vary and seem to be case specific. Here, in this work, we introduce a general model and investigate the behaviour of an asymmetric spherical dimer of oscillating length for small-amplitude motion at intermediate Re. In our analysis we make the important distinction between particle and fluid inertia, both of which need to be considered separately. We asymptotically expand the Navier–Stokes equations in the small-amplitude limit to obtain a system of linear partial differential equations. Using a combination of numerical (finite element) and analytical (reciprocal theorem, method of reflections) methods we solve the system to obtain the dimer's swim speed and show that there are two mechanisms that give rise to motion: boundary conditions (an effective slip velocity) and Reynolds stresses. Each mechanism is driven by two classes of sphere–sphere interactions, between one sphere's motion and (1) the oscillating background flow induced by the other's motion, and (2) a geometric asymmetry induced by the other's presence. We can thus unify and explain behaviours observed in other works. Our results show how sensitive, counterintuitive and rich motility is in the parameter space of finite inertia of particles and fluid.

42 ENGINEERING↗
NASA NTRS1981

General formulation of optical paths for large zenith angles in the earth's curved atmosphere

Formulas that can be used to determine the optical path between two points along an atmospheric ray path are derived for the case when the local zenith angle of the ray path is larger than 70 deg. For angles less than 70 deg, these formulas reduce to the airmass function; viz., the secant of the zenith angle. The formulation presented in this paper is general enough to be applicable to a wide variety of atmospheric conditions, such as spherical and nonspherical atmospheres, and vertically and horizontally homogeneous as well as inhomogeneous atmospheres. A formulation for the case when atmospheric refraction is important also is presented here.

Wang, P.-H.↗
NASA NTRS1976

Rigid rotators

The two-particle, steady-state Schroedinger equation is transformed to center of mass and internuclear distance vector coordinates, leading to the free particle wave equation for the kinetic energy motion of the molecule and a decoupled wave equation for a single particle of reduced mass moving in a spherical potential field. The latter describes the vibrational and rotational energy modes of the diatomic molecule. For fixed internuclear distance, this becomes the equation of rigid rotator motion. The classical partition function for the rotator is derived and compared with the quantum expression. Molecular symmetry effects are developed from the generalized Pauli principle that the steady-state wave function of any system of fundamental particles must be antisymmetric. Nuclear spin and spin quantum functions are introduced and ortho- and para-states of rotators, along with their degeneracies, are defined. Effects of nuclear spin on entropy are deduced. Next, rigid polyatomic rotators are considered and the partition function for this case is derived. The patterns of rotational energy levels for nonlinear molecules are discussed for the spherical symmetric top, for the prolate symmetric top, for the oblate symmetric top, and for the asymmetric top. Finally, the equilibrium energy and specific heat of rigid rotators are derived.

Source record↗