Search NASA⌕ Search

SEARCH · Search NASA

Results for “Approximations and 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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
NASA NTRS1973

Martian tidal pressure and wind fields obtained from the Mariner 9 infrared spectroscopy experiment

Using temperature fields derived from the Mariner 9 infrared spectroscopy experiment, the Martian atmospheric tidal pressure and wind fields are calculated. Temperature as a function of local time, latitude, and atmospheric pressure level is obtained by secular and longitudinal averaging of the data. The resulting temperature field is approximated by a spherical harmonic expansion, retaining one symmetric and one asymmetric term for wavenumber zero and wavenumber one. Vertical averaging of the linearized momentum and continuity equations results in an inhomogeneous tidal equation for surface pressure fluctuations with the driving function related to the temperature field through the geopotential function and the hydrostatic equation. Solutions of the tidal equation show a diurnal fractional pressure amplitude approximately equal to one half of the vertically averaged diurnal fractional temperature amplitude.

Pirraglia, J. A.↗
NASA NTRS1974

Martian tidal pressure and wind fields obtained from the Mariner 9 infrared spectroscopy experiment

Using temperature fields derived from the Mariner 9 infrared spectroscopy experiment, the Martian atmospheric tidal pressure and wind fields are calculated. Temperature as a function of local time, latitude, and atmospheric pressure level is obtained by secular and longitudinal averaging of the data. The resulting temperature field is approximated by a spherical harmonic expansion, retaining one symmetric and one asymmetric term each for wavenumber zero and wavenumber one. Vertical averaging of the linearized momentum and continuity equations results in an inhomogeneous tidal equation for surface pressure fluctuations with the driving function related to the temperature field through the geopotential function and the hydrostatic equation. Solutions of the tidal equation show a diurnal fractional pressure amplitude approximately equal to one-half the vertically averaged diurnal fractional temperature amplitude.

Pirraglia, J. A.↗
NASA NTRS1991

Beyond the Kirchhoff approximation. II - Electromagnetic scattering

In a paper by Rodriguez (1981), the momentum transfer expansion was introduced for scalar wave scattering. It was shown that this expansion can be used to obtain wavelength-dependent curvature corrections to the Kirchhoff approximation. This paper extends the momentum transfer perturbation expansion to electromagnetic waves. Curvature corrections to the surface current are obtained. Using these results, the specular field and the backscatter cross section are calculated.

Rodriguez, Ernesto↗
NASA NTRS1994

The heating of nova ejecta by radioactive decays of the beta-unstable nuclei

Recent nucleosynthesis and hydrodynamic calculations of the consequences of accretion onto massive ONeMg white dwarf stars show that under certain circumstances significant amounts of the beta-unstable nuclei can be produced and ejected by the resulting explosion. We use these calculations as a guide in order to obtain the conditions under which the heating of the ejected material by the nonthermal electrons and positrons produced by the decays of the beta-unstable nuclei is sufficient to overcome the cooling from adiabatic expansion and lead to the production of X-ray-emitting coronal gas. These conditions are as follows: (1) a mass fraction for Na-22 of the order of 10(exp -3) or greater, (2) an expansion velocity in the range approximately 10(exp 2) - 10(exp 3) km/s, (3) a photospheric radius of approximately 10(exp 14) cm, (4) if the density distribution in the atmosphere satisfies a power law, then the exponent must be less than 3 for heating to overcome adiabatic cooling. Both the simulations of the outburst and the model atmosphere fits to the observed energy distributions, however, imply that the exponent is greater than or = 3 during the early phases of the outburst. Nevertheless, for a value of the exponent of 2, we predict the time when hot coronal gas can form during the expansion phases of the envelope.

Pistinner, Shlomi↗
NASA NTRS1994

Rational positive real approximations for LQG optimal compensators arising in active stabilization of flexible structures

In this paper the approximation problem for a class of optimal compensators for flexible structures is considered. The particular case of a simply supported truss with an offset antenna is dealt with. The nonrational positive real optimal compensator transfer function is determined, and it is proposed that an approximation scheme based on a continued fraction expansion method be used. Comparison with the more popular modal expansion technique is performed in terms of stability margin and parameters sensitivity of the relative approximated closed loop transfer functions.

Desantis, A.↗
NASA NTRS2000

Chuar Group of the Grand Canyon: record of breakup of Rodinia, associated change in the global carbon cycle, and ecosystem expansion by 740 Ma

The Chuar Group (approximately 1600 m thick) preserves a record of extensional tectonism, ocean-chemistry fluctuations, and biological diversification during the late Neoproterozoic Era. An ash layer from the top of the section has a U-Pb zircon age of 742 +/- 6 Ma. The Chuar Group was deposited at low latitudes during extension on the north-trending Butte fault system and is inferred to record rifting during the breakup of Rodinia. Shallow-marine deposition is documented by tide- and wave-generated sedimentary structures, facies associations, and fossils. C isotopes in organic carbon show large stratigraphic variations, apparently recording incipient stages of the marked C isotopic fluctuations that characterize later Neoproterozoic time. Upper Chuar rocks preserve a rich biota that includes not only cyanobacteria and algae, but also heterotrophic protists that document increased food web complexity in Neoproterozoic ecosystems. The Chuar Group thus provides a well-dated, high-resolution record of early events in the sequence of linked tectonic, biogeochemical, environmental, and biological changes that collectively ushered in the Phanerozoic Eon.

NASA Discipline Exobiology↗
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 NTRS2018

An Overview of Methods for Deriving the Radiative Transfer Theory from the Maxwell Equations. II: Approach Based on the Dyson and Bethe-Salpeter Equations

In this paper, the vector radiative transfer equation is derived by means of the vector integral Foldy equations describing the electromagnetic scattering by a group of particles. By assuming that in a discrete random medium the positions of the particles are statistically independent and by applying the Twersky approximation to the order-of-scattering expansion of the total field, we derive the Dyson equation for the coherent field and the ladder approximated Bethe–Salpeter equation for the dyadic correlation function. Then, under the far-field assumption for sparsely distributed particles, the Dyson equation is reduced to the Foldy integral equation for the coherent field, while the iterated solution of the Bethe–Salpeter equation ultimately yields the vector radiative transfer equation.

Electromagnetic scattering↗
NASA NTRS2023

Launch Complex 39B, SWMU 009 Air Sparging System Expansion Implementation Work Plan Kennedy Space Center, Florida

This Implementation Work Plan (IWP) for expansion of the Air Sparging (AS) system at Launch Complex 39B (LC39B), presents the design and coordination specifics to expand the existing AS Interim Measure (IM) at LC39B located at Kennedy Space Center (KSC), Florida. The existing AS system at LC39B was installed in 2017 and included 279 AS wells. LC39B has been designated as Solid Waste Management Unit (SWMU) 009 under the KSC Resource Conservation and Recovery Act (RCRA) Corrective Action Program. Expansion of the AS IM will include installation of an additional five AS wells to depths of approximately 13 and 17 feet below land surface (bls) to remediate contaminated groundwater identified northwest of the launch pad, near the liquid oxygen (LOX) tank. The expansion will target concentrations of cis-1,2-dichloroethene (cDCE) and vinyl chloride (VC) greater than Florida Department of Environmental Protection (FDEP) Natural Attenuation Default Concentrations (NADCs) that were identified during direct push technology (DPT) sampling events between 2019-2022 in shallow intervals between 8 and 16 feet bls. The objective for this expansion is to remediate contaminated groundwater within the treatment zone to below NADCs to support transition to monitored natural attenuation (MNA) through a long-term monitoring (LTM) program. This expansion will treat approximately 0.07-acres of residual groundwater contamination. The IWP design for this expansion will utilize the existing AS system trailer that is already onsite and operating. The five new AS expansion wells will be integrated into the existing Zone Z4 distribution manifold to deliver compressed air to the new AS wells. To deliver airflow, three existing lateral lines will be split in the field at a lateral split junction point (using 2-way or 3-way splits). Each split lateral will supply air to AS wells at the same depth interval to minimize preferential pathways. The trenching layout for the new AS wells was developed to minimize ground disturbance due to the amount of utilities in the area. One new monitoring well will also be installed and added to LC39B’s performance monitoring program to evaluate the effectiveness of the AS system expansion. This well will be sampled on a semi-annual basis for volatile organic compounds, with results evaluated to determine if system adjustments may be required. The main components of this IWP were presented to the KSC Remediation Team (KSCRT) at the October 2022 Meeting (Meeting 2210-M05). DPT delineation data from the LOX area were also presented, which included some inferred boundaries due to physical restrictions of the LOX tank and associated concrete pad and infrastructure. Planned monitoring well sampling in this area will help determine if back-diffusion may be occurring from beneath the pad. Additionally, based on potential petroleum odors and staining identified at two DPT locations during the January 2022 DPT event in the 10-16 feet bls range, a KSCRT action item was assigned during the October 2022 meeting to collect soil samples for petroleum compounds if similar odors or staining are identified during IM implementation. Construction completion and performance monitoring will be incorporated into LC39B’s groundwater monitoring reports, which are currently submitted on an annual basis to FDEP to document operation and maintenance activities and results from the AS IM.

Deborah M. Wilson↗
NASA NTRS1979

Finite-field Calculations of Molecular Polarizabilities Using Field-Induced Polarization Functions: Second- and Third-order Perturbation Correlation Corrections to the Coupled Hartree-Fock Polarizability of H2O

Ordinary Rayleigh-Schroudinger perturbation theory with Moller-Plesset (RSMP) partitioning is used to calculate second- and third-order correlation corrections to the CHF polarizability and dipole moment of the water molecule by a finite-field procedure. Pade approximants are found to be useful in accelerating the convergence of the property perturbation expansions. Field-induced polarization functions suitable for polarizability calculations are determined. The average polarizability calculated, neglecting vibrational averaging, with Dunning's (9s5p/4s-4s2p/2s) contracted GTO basis set augmented by field-induced lslp2d/lp polarization functions is within 3 per cent of the experimental result. Correlation corrections to the dipole moment and polarizability of the water molecule calculated by the finite-field RSMP and single + double excitation CI(SDCI) methods for the same basis set are found to be in close agreement. The RSMP approach has the advantages of being size-consistent and of being capable of greater efficiency than the SCDI method. Comparative calculations carried out using Epstein-Nesbet partitioning show that through third order RSEN correlation perturbation expansions for the dipole moment and polarizability are less rapidly convergent than RSMP expansions. However, reasonable accord with RSMP results can be achieved by using Pade approximants to accelerate the convergence of RSEN energy perturbation expansions. The convergence of RSMP property correlation expansions based on the zeroth-order uncoupled-Hartree-Fock (UCHF) and coupled-Hartree-Fock (CHF) approximations are compared through third order. Whereas the CHF + RSMP expansions are for practical purposes fully converged, the UCHF + RSMP expansions are not adequately converged.

Langhoff, S. R.↗
NASA NTRS1970

Experimental investigation and analysis of two sources of nozzle-thrust misalignment

Asymmetry of nozzle's throat produces oscillatory type net side-force axial profile. Using mean values of localized static pressure and Mach number, scaling laws for flat-plate supersonic flow over protrusion are applied to nozzle expansion cone irregularities to give approximate indication of perturbed-pressure profiles and induced side forces.

Dowdy, M. W.↗
NASA NTRS1973

Digital approximation of continuous-data control systems by point-by-point state comparison

This paper presents a point-by-point state comparison method of approximating a continuous-data system by a sampled-data system. The problem is to attempt the matching of the states of the two systems at the sampling instants. A partial matching has to be conducted if the systems have more states than controls. A weighting matrix is used to regulate the partial matching and weights placed on each state. The digital approximation is affected by use of forward gain E(T) and feedback gain G(T) in the sampled-data system. It is shown that, in general, these gains can be approximated by truncated Taylor series expansions. An illustrative example is given using the one-axis dynamics of the Skylab satellite.

Kuo, B. C.↗
NASA NTRS1978

Wave propagation through a random medium - The random slab problem

The first-order smoothing approximation yields integral equations for the mean and the two-point correlation function of a wave in a random medium. A method is presented for the approximate solution of these equations that combines features of the eiconal approximation and of the Born expansion. This method is applied to the problem of reflection and transmission of a plane wave by a slab of a random medium. Both the mean wave and the covariance are calculated to determine the reflected and transmitted amplitudes and intensities.

Acquista, C.↗
NASA NTRS1980

Stability of an optically contacted etalon to cosmic radiation

An investigation has been completed to determine the effects of prolonged exposure to cosmic radiation on Zerodur spacing elements used between two dielectric reflectors on silica substrates in the plane Fabry-Perot etalon selected for flight in the Dynamics Explorer satellite. The measured radiation expansion coefficient for Zerodur is approximately -4.0 x 10 to the -12th/rad. In addition to the overall change in gap dimension, test data indicate a degradation in etalon parallelism, which is ascribed to the different doses received by the three spacers due to their differing distances from a Co-60 source. The effect is considered to be of practical use in the tuning and parallelism adjustment of fixed gap etalons. The variation is small enough not to pose a problem for the satellite instrument where expected radiation doses are less than 10,000 rads.

Killeen, T. L.↗
NASA NTRS1984

Static shape determination and control for large space structures. I - The flexible beam. II - A large space antenna

A method for determining and controlling the shape of large, continuous space structures by means of discrete or pointwise observations and control devices is presented. The general linear boundary value problem satisfied by a one-dimensional shape function is defined, and the existence of solutions is studied. The static shape control problems for one-dimensional systems with and without rigid body modes and the static shape estimation problem are presented and solved. Eigenfunction expansions are presented which provide approximations to the algorithm terms when the associated Green's function is not known. An integral operator approach is applied to the multidimensional static problem, and the results are illustrated with a finite element model of the disk of a large space antenna which assumes no rigid body modes. It is shown that the shape control algorithm must be modified for systems with rigid body modes.

Weeks, C. J.↗
NASA NTRS1986

A simple method for computing the relativistic Compton scattering kernel for radiative transfer

Correct computation of the Compton scattering kernel (CSK), defined to be the Klein-Nishina differential cross section averaged over a relativistic Maxwellian electron distribution, is reported. The CSK is analytically reduced to a single integral, which can then be rapidly evaluated using a power series expansion, asymptotic series, and rational approximation for sigma(s). The CSK calculation has application to production codes that aim at understanding certain astrophysical, laser fusion, and nuclear weapons effects phenomena.

Prasad, M. K.↗
NASA NTRS1988

Minimum weight design of rectangular and tapered helicopter rotor blades with frequency constraints

The minimum weight design of a helicopter rotor blade subject to constraints on coupled flap-lag natural frequencies has been studied. A constraint has also been imposed on the minimum value of the autorotational inertia of the blade in order to ensure that it has sufficient inertia to autorotate in the case of engine failure. The program CAMRAD is used for the blade modal analysis and CONMIN is used for the optimization. In addition, a linear approximation analysis involving Taylor series expansion has been used to reduce the analysis effort. The procedure contains a sensitivity analysis which consists of analytical derivatives of the objective function and the autorotational inertia constraint and central finite difference derivatives of the frequency constraints. Optimum designs have been obtained for both rectangular and tapered blades. Design variables include taper ratio, segment weights, and box beam dimensions. It is shown that even when starting with an acceptable baseline design, a significant amount of weight reduction is possible while satisfying all the constraints for both rectangular and tapered blades.

Chattopadhyay, Aditi↗