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NASA NTRS1978

Third-order solution of an artificial-satellite theory

A third-order solution is developed for the motions of artificial satellites moving in the gravitational field of the earth, whose potential includes the second-, third-, and fourth-order zonal harmonics. Third-order periodic perturbations with fourth-order secular perturbations are derived by Hori's perturbations method. All quantities are expanded into power series of the eccentricity, but the solution is obtained so as to be closed with respect to the inclination. A comparison with the results of numerical integration of the equations of motion indicates that the solution can predict the position of a close-earth satellite with a small eccentricity with an accuracy of better than 1 cm over 1 month.

Kinoshita, H.↗

NASA NTRS1978

Error probability performance of unbalanced QPSK receivers

A simple technique for calculating the error probability performance and associated noisy reference loss of practical unbalanced QPSK receivers is presented. The approach is based on expanding the error probability conditioned on the loop phase error in a power series in the loop phase error and then, keeping only the first few terms of this series, averaging this conditional error probability over the probability density function of the loop phase error. Doing so results in an expression for the average error probability which is in the form of a leading term representing the ideal (perfect synchronization references) performance plus a term proportional to the mean-squared crosstalk. Thus, the additional error probability due to noisy synchronization references occurs as an additive term proportional to the mean-squared phase jitter directly associated with the receiver's tracking loop. Similar arguments are advanced to give closed-form results for the noisy reference loss itself.

Simon, M. K.↗

NASA NTRS1978

Advanced panel-type influence coefficient methods applied to unsteady three dimensional potential flows

A panel method for solving unsteady, subsonic wind-body-tail flow problems is formulated and partially verified. The method is applicable to general aircraft configurations consisting of arbitrary arrangements of wings, bodies, tails, and nacelles. The wake may be located arbitrarily and the unsteady, transverse component of vorticity in the wake may be assigned any covection velocity. The wake in the unsteady flow problem, therefore, can be given the location and convection velocity of the wake produced by a steady flow which is the mean flow of the unsteady flow problem. The panel method has been used as a basis for expanding the unsteady kernel function in a power series to obtain panel influence coefficients which can be integrated in closed form.

Dusto, A. R.↗

NASA NTRS1978

On isostatic geoid anomalies

In regions of slowly varying lateral density changes, the gravity and geoid anomalies may be expressed as power series expansions in topography. Geoid anomalies in isostatically compensated regions can be directly related to the local dipole moment of the density-depth distribution. This relationship is used to obtain theoretical geoid anomalies for different models of isostatic compensation. The classical Pratt and Airy models give geoid height-elevation relationships differing in functional form but predicting geoid anomalies of comparable magnitude. The thermal cooling model explaining ocean floor subsidence away from mid-ocean ridges predicts a linear age-geoid height relationship of 0.16 m/m.y. Geos 3 altimetry profiles were examined to test these theoretical relationships. A profile over the mid-Atlantic ridge is closely matched by the geoid curve derived from the thermal cooling model. The observed geoid anomaly over the Atlantic margin of North America can be explained by Airy compensation. The relation between geoid anomaly and bathymetry across the Bermuda Swell is consistent with Pratt compensation with a 100-km depth of compensation.

Haxby, W. F.↗

NASA NTRS1978

Five micron limb-darkening and the structure of the Jovian atmosphere

A combined radiative, dynamical, and chemical model of the Jovian atmosphere is presented which reproduces the observed IR limb-darkening curves for several regions on the disk of Jupiter. Two approaches are used to perform the inversion required for determination of the source function, and a power-series-expansion model is examined, along with a convective-equilibrium model, cloud and intermediate-zone models, and a thin-shell model. The statistical methods used in the analysis are outlined, and model fits to the observations are discussed. The chemistry of phase transitions in the hot spots of Jupiter's equatorial belt is briefly investigated.

Newman, W. I.↗

NASA NTRS1979

Total ozone determination by spectroradiometry in the middle ultraviolet

A method has been developed to determine total ozone from multispectral measurements of the direct solar irradiance. The total ozone is determined by a least squares fit to the spectrum between 290 nm and 380 nm. The aerosol extinction is accounted for by expanding it in a power series in wavelength; use of the linear term proved adequate. A mobile laboratory incorporating a sky scanner has been developed and used to obtain data to verify the method. Sun tracking, wavelength setting of the double monochromator, and data acquisition are under control of a minicomputer. Results obtained at Wallops Island, Virginia, and Palestine, Texas, agree well with simultaneous Dobson and Canterbury spectrometer and balloon ECC ozonesonde values. The wavelength calibration of the monochromator and the values for the normalized ozone absorption are the most important factors in an accurate determination of total ozone.

Garrison, L. M.↗

NASA NTRS1979

Correction of instrumental distortion by analytical deconvolution of data

A general analytical theorem developed by van de Hulst (1946) for inverting the convolution integral is reviewed and illustrated both with synthetic data and with experimental data from time-of-flight measurements. If the undesired influence of an instrument used in an experimental measurement can be represented by the convolution integral, the original undistorted or true distribution may sometimes be recovered in postprocessing the data by means of deconvolution. Analytical deconvolution is achieved by using the coefficients from a power series representation of the distorted output distribution and a set of 'solving polynomials' which may be readily derived from the response function of the instrument.

Morton, D. C.↗

NASA NTRS1979

Study of periodic motions of a satellite with a magnetic damper

The motion of a satellite with a magnetic damper in the plane of a circular polar orbit is studied. The asymptotics of periodic solutions are constructed for a satellite close to axisymmetric and the radius of convergence is evaluated for the power series obtained. In a broad range of values of parameters, a periodic solution is obtained by numerical integration of equations of motion of the satellite. The asymptotics of a bifurcated curve obtained (the curve on which origin of a pair of periodic solutions occurs) in the space of the parameters agrees well with the results of numerical computation with all physical values of these parameters. A breakdown is made of the space of the initial data of phase variables in the field of effect of different types of periodic motion.

Sadov, Y. A.↗

NASA NTRS1979

An asymptotic expansion approach to the inverse radiative transfer problem

An iterative technique which recovers density profiles in a nonhomogeneous absorbing atmosphere is derived. The technique is based on the concept of factoring a function of the density profile into the product of a known term and a term which is not known, but whose power series expansion can be found. This series converges rapidly under a wide range of conditions. A demonstration example of simulated data from a high resolution infrared heterodyne instrument is inverted. For the examples studied, the technique is shown to be capable of extracting features of ozone profiles in the troposphere and to be particularly stable.

Gomberg, R. I.↗

NASA NTRS1980

Grid generation using differential systems techniques

The errors in approximating the derivatives of a function by traditional central differences at grid points of a curvilinear coordinate system were examined. The implications concerning the accuracy of the numerical solution of a partial differential equation are explained by considering several numerical examples. Although this study only considers the two dimensional case, the techniques and implications are equally valid for three dimensional grids. An interesting feature of the error analysis is its simplicity. Most of the results follow by merely working with the truncation terms of some power series expansion. These series expansions also give rise to higher order difference approximations which can significantly reduce error when the grid spacing changes rapidly, as might be the case in problems with shock waves or thin boundary layers.

Thompson, J. F.↗

NASA NTRS1981

Verification of reflectance models in turbid waters

Inherent optical parameters of very turbid waters were used to evaluate existing water reflectance models. Measured upwelling radiance spectra and Monte Carlo simulations of the radiative transfer equations were compared with results from models based upon two flow, quasi-single scattering, augmented isotropic scattering, and power series approximation. Each model was evaluated for three separate components of upwelling radiance: (1) direct sunlight; (2) diffuse skylight; and (3) internally reflected light. Limitations of existing water reflectance models as applied to turbid waters and possible applications to the extraction of water constituent information are discussed.

Tanis, F. J.↗

NASA NTRS1982

Applicaton of radiative transfer theory to microwave transmission medium calibrations

Precise determinations of the transmission medium loss and noise temperature contribution which are important to the performance characterization of low noise microwave receiving systems and thermal noise standards are discussed. Tropospheric loss is frequently inferred from microwave radiometer noise temperature measurements. Interpretation of these measurements requires an inversion of the radiative transfer integral equation. This is inconvenient even with computer techniques. Solutions of a rapidly convergent power series of the radiative transfer equations are presented. This solution is applicable to a low loss medium with either uniform or nonuniform loss distributions. A four layer atmosphere model is investigated to demonstrate the accuracy of the solution relative to the model. Applications include thermal noise standards and single- and dual-frequency water radiometers.

Stelzried, C. T.↗

NASA NTRS1982

Primary proton and helium spectra in the energy range 10 to the 12th to 10 to the 14th eV

Measurements of proton and helium spectra have been made in the energy range 10 to the 12th to 10 to the 14th eV. Large area thin emulsion calorimeters were used in the Japanese American Cooperative Emulsion Experiment balloon flight series. Power indices of the integral spectra for both nuclei are consistent with published data at lower energies. Absolute intensities are also consistent for helium and proton fluxes with extrapolations of previous data. No steepening of the proton spectrum is indicated.

Gregory, J. C.↗

NASA NTRS1982

Approximate solution to the Hopf Phi equation for isotropic homogeneous fluid turbulence

Consistent with the observed t to the -n decay laws for isotropic homogeneous turbulence and the form of the longitudinal correlation function f(r, t) for small r, the Hopf Phi equation is shown to be satisfied approximately by an asymptotic power series in t to the -n. This solution features a self-similar universal equilibrium functional which manifests Kolmogoroff-type scaling.

Rosen, G.↗

NASA NTRS1981

Measurement of very large flow angles with non-nulling seven-hole probes

This paper describes a method for measuring local direction and total and static pressures of a flow by means of a fixed probe, provided that the local air flow does not make an angle of more than 80 degrees with the axis of the probe. The probe is easily manufactured from standard-sized tubing materials. The power series calibration method used with the probe results in explicit polynomial expressions for the desired aerodynamic properties. The calibration method is easily programmed on a data acquisition system. This paper includes an example of a complete incompressible calibration.

Gallington, R. W.↗

NASA NTRS1982

The Deep Space Network: Noise temperature concepts, measurements, and performance

The use of higher operational frequencies is being investigated for improved performance of the Deep Space Network. Noise temperature and noise figure concepts are used to describe the noise performance of these receiving systems. The ultimate sensitivity of a linear receiving system is limited by the thermal noise of the source and the quantum noise of the receiver amplifier. The atmosphere, antenna and receiver amplifier of an Earth station receiving system are analyzed separately and as a system. Performance evaluation and error analysis techniques are investigated. System noise temperature and antenna gain parameters are combined to give an overall system figure of merit G/T. Radiometers are used to perform radio ""star'' antenna and system sensitivity calibrations. These are analyzed and the performance of several types compared to an idealized total power radiometer. The theory of radiative transfer is applicable to the analysis of transmission medium loss. A power series solution in terms of the transmission medium loss is given for the solution of the noise temperature contribution.

Stelzried, C. T.↗

NASA NTRS1982

Numerical computation of exponential matrices using the Cayley-Hamilton theorem

A method for computing exponential matrices, which often arise naturally in the solution of systems of linear differential equations, is developed. An exponential matrix is generated as a linear combination of a finite number (equal to the matrix order) of matrices, the coefficients of which are scalar infinite sums. The method can be generalized to apply to any formal power series of matrices. Attention is focused upon the exponential function, and the matrix exponent is assumed tri-diagonal in form. In such cases, the terms in the coefficient infinite sums can be extracted, as recursion relations, from the characteristic polynomial of the matrix exponent. Two numerical examples are presented in some detail: (1) the three dimensional infinitesimal rotation rate matrix, which is skew symmetric, and (2) an N-dimensional tri-diagonal and symmetric finite difference matrix which arises in the numerical solution of the heat conduction partial differential equation. In the second example, the known eigenvalues and eigenvectors of the finite difference matrix permit an analytical solution for the exponential matrix, through the theory of diagonalization and similarity transformations, which is used for independent verification. The convergence properties of the scalar infinite summations are investigated for finite difference matrices of various orders up to ten, and it is found that the number of terms required for convergence increases slowly with the order of the matrix.

Walden, H.↗

NASA NTRS1983

Analysis of oscillatory motion of a light airplane at high values of lift coefficient

A modified stepwise regression is applied to flight data from a light research air-plane operating at high angles at attack. The well-known phenomenon referred to as buckling or porpoising is analyzed and modeled using both power series and spline expansions of the aerodynamic force and moment coefficients associated with the longitudinal equations of motion.

Batterson, J. G.↗