SEARCH · Engineering Papers

Results for “spherical harmonics”

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.

SourceResearch connectionFrom yearThrough year

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

At least 271 records · Page 15

All sources NASA NTRS DOE OSTI

NASA NTRS1973

An analytic study on bounds for the associated Legendre functions

The bounds for the normalized associated Legendre functions P sub nm were studied to provide a rational basis for the truncation of the geopotential series in spherical harmonics in various orbital analyses. The conjecture is made that the largest maximum of the normalized associated Legendre function lies in the interval which indicates the greatest integer function. A procedure is developed for verifying this conjecture. An on-line algebraic manipulator, IAM, is used to implement the procedure and the verification is carried out for all n equal to or less than 2m, for m = 1 through 6. A rigorous proof of the conjecture is not available.

Payne, M. H.↗

NASA NTRS1972

A simplified model for the gravitational potential of the atmosphere and its effect on the geoid

The earth's atmosphere is considered as made up of oblate spheroidal layers of variable density lying over an oblate spheroidal earth. The gravitational attraction of the atmosphere at exterior points is computed and its contribution to the usual spherical harmonic gravitational expansion is assessed. The potential is also found for points at the bottom of the model atmosphere. This latter result is of interest for determination of the potential at the surface of the geoid. The atmospheric correction to the geoid determination from satellite coefficients is given.

Madden, S. J., Jr.↗

NASA NTRS1974

Error enhancement in geomagnetic models derived from scalar data

Models of the main geomagnetic field are generally represented by a scalar potential gamma expanded in a finite number of spherical harmonics. Very accurate observations of F were used, but indications exist that the accuracy of models derived from them is considerably lower. One problem is that F does not always characterize gamma uniquely. It is not clear whether such ambiguity can be encountered in deriving gamma from F in geomagnetic surveys, but there exists a connection, due to the fact that the counterexamples of Backus are related to the dipole field, while the geomagnetic field is dominated by its dipole component. If the models are recovered with a finite error (i.e. they cannot completely fit the data and consequently have a small spurious component), this connection allows the error in certain sequences of harmonic terms in gamma to be enhanced without unduly large effects on the fit of F to the model.

Stern, D. P.↗

NASA NTRS1973

Mars gravity derived from the long-period motion of Mariner 9

The spherical harmonic coefficients in the Mars gravity field are determined using a two-stage filtering process. In the first stage, a weighted least-squares orbit determination processor is applied to Mariner 9 radio data to estimate the mean Kepler elements of each orbit. A finite difference method is then used to generate orbital element rates. In the second stage, the gravity coefficients are determined using another weighted least-squares processor which fits the long period Lagrange perturbation equations to the estimated rates. The resulting gravity potential and associated equipotential surface is presented and error sources discussed. Gravity field results are applied to various physical properties of Mars, such as moments of inertia.-

Ferrari, A. J.↗

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 NTRS1974

Lunar gravity analysis results from Explorer-49

An analysis of the lunar gravity field utilizing 234 days of Explorer-49 mean Keplerian elements is presented. A third-degree and third-order gravity model developed at the Goddard Space Flight Center (GSFC) for mission operations is compared with published models derived from Lunar Orbiter data. Improved predictions of e and Omega using the GSFC model are observed. The results suggest that use of Explorer-49 orbital data in a multisatellite solution would lead to improved global estimates of low-order spherical harmonic coefficients of the lunar gravitational potential.

Bryant, W. C., Jr.↗

NASA NTRS1974

Analysis of acoustic radiation in a jet flow environment

The analysis of sound fields from arbitrary source distributions in terms of Legendre and spherical Hankel functions is well known. The purpose of this paper is to extend this classical method of analysis to environments such as jet flows where flow and flow gradients are inherently present. The wave-equation governing the radiation of sound in such an environment is derived. The steady state flow and flow gradients in the axial and transverse directions appear as coefficients in the terms of the wave-equation. A semi-numerical method is used to solve the wave-equation in terms of modified spherical harmonics yielding the phase velocities and the directivities of an infinite set of modes. The directivity of each mode is obtained in terms of modified Legendre functions by numerical integration. Some results of these directivity and phase-velocity calculations are presented for a limited number of frequency and flow parameters. Both convective and shear refraction are shown to be important.

Mungur, P.↗

NASA NTRS1975

The complex magnetic field of Jupiter

An analysis of the characteristics of the magnetic field of the planet Jupiter is presented. The data were obtained during the flight of Pioneer 11 space probe, using a high field triaxial fluxgate magnetometer. The data are analyzed in terms of traditional Schmitt normalized spherical harmonic expansion fitted to the observations in a least squares sense. Tables of data and graphs are provided to summarize the findings.

Acuna, M. H.↗

NASA NTRS1975

Downward continuation of gravity information from satellite to satellite tracking or satellite gradiometry in local areas

Integral formulas in the parameter domain are used instead of a representation by spherical harmonics. The neglected regions will cause a truncation error. The application of the discrete form of the integral equations connecting the satellite observations with surface gravity anomalies is discussed in comparison with the least squares prediction method. One critical point of downward continuation is the proper choice of the boundary surface. Practical feasibilities are in conflict with theoretical considerations. The properties of different approaches for this question are analyzed.

Rummel, R.↗

NASA NTRS1975

Revision of geodetic parameters

Laser data from nine satellites and 12 stations are combined with surface-gravity data to obtain spherical harmonics representing the geopotential complete through degree and order 18. This laser-data-only solution provides a reasonable improvement to the gravity field.

Gaposchkin, E. M.↗

NASA NTRS1975

Analyses of the solid earth and ocean tidal perturbations on the orbits of the GEOS-1 and GEOS-2 satellites

The luni-solar tidal perturbations in the inclination of the GEOS-I and GEOS-II satellite orbits were analyzed for the solid Earth and ocean tide conditions. Precision reduced camera and TRANET Doppler observations spanning periods of over 600 days for each satellite were used to derive mean orbital elements. Perturbations due to the earth's gravity field, solar radiation pressure, and atmospheric drag were modelled, and the resulting inclination residuals were analyzed for tidal effects. The amplitudes of the observed total tidal effects were about 1.2 arc seconds (36 meters) in the inclination of GEOS-I and 4.5 arc seconds (135 meters) for GEOS-II. The solid earth tides were then modelled using earth tide measurements, earth rotation observations, and seismic data. The resulting inclination residuals were analyzed for ocean tide parameters. The derived parameters consist of one second degree coefficient and an accompanying phase angle in a spherical harmonic expansion of the ocean tidal potential for each tidal constituent. The results are presented.

Felsentreger, T. L.↗

NASA NTRS1974

Detailed gravimetric geoid for the GEOS-C altimeter calibration area

The GEOS-C spacecraft scheduled for launch in late 1974 will carry a radar altimeter for the purpose of measuring sea surface topography. In order to calibrate and evaluate the performance of the altimeter system, ground truth data are required. In this respect a detailed gravimetric geoid has been computed for the GEOS-C altimeter calibration area in the Atlantic Ocean off the East Coast of the U.S. This geoid is based upon a combination of mean free air surface gravity anomalies and the Goddard Space Flight Center GEM-6 satellite-derived spherical harmonic coefficients. Surface gravity anomalies have been used to provide information on the short wave length undulations of the geoid while the satellite-derived coefficients have provided information on the long wave length components. As part of these analyses, GSFC, SAO and OSU satellite-derived gravity models were used in the computations. Although geoid heights based upon the various satellite models differed by as much as 30 meters in the Southern Hemisphere, the differences in this Atlantic Ocean area were less than 4 meters.

Marsh, J. G.↗

NASA NTRS1975

The gravity field of Mars

A complete sixth-degree spherical-harmonic model of the gravitational potential of Mars has been produced from an analysis of Doppler data obtained from the radio tracking of the Mariner-9 spacecraft for over 200 orbital revolutions. The equipotential surface defined by this model is estimated to be accurate to within 100 m below 60 deg north latitude and to within 300 m above. The dominant feature of this surface is a Tharsis-related bulge which rises nearly two kilometers above the surrounding areas. Gravitational 'lows' are found to correspond to the Hellas depression and the Valles Marineris.

Reasenberg, R. D.↗

NASA NTRS1974

A general circulation model of stratospheric ozone

The paper reports on a general circulation model being developed with the ultimate objective of assessing the effect of SST operations on the atmospheric ozone distribution. The model variables are represented in the spectral domain using 79 spherical harmonics and 26 vertical levels between the ground and 70 km. Initial calculations have attempted to simulate the seasonal variations of the unperturbed global ozone distribution. The model shows the presence of a westerly stratospheric jet in the winter hemisphere with amplitudes similar to those observed in the polar night jet. Easterlies dominate in the summer hemisphere. The model also shows ozone to be transported polewards and downwards. Furthermore the annual cycle in ozone concentrations at mid-latitudes has been simulated. Preliminary comparisons are also made with other features of the dynamics and the ozone distribution in the stratosphere.

Cunnold, D. M.↗

NASA NTRS1975

Mariner 9 - An instrument of dynamical science

We review and evaluate the contributions of Mariner 9 in improving our knowledge of the dynamical characteristics of Mars and its two satellites, Phobos and Deimos. Primary results include the discovery of the large gravitational and topographical bulge in the Tharsis region, the development of a detailed gravity model representable as coefficients in a spherical harmonic expansion, the development of a topographic model exhibiting a three kilometer displacement of the center of figure from the center of mass, and the determination of the size, shape and motion of Phobos and Deimos.

Jordan, J. F.↗

NASA NTRS1975

A determination of the lunar moment of inertia

An estimate of the second zonal coefficient of the spherical-harmonic representation of the lunar gravitational field has been obtained from an analysis of particular orbital-element variations of the Explorer 35 and Explorer 49 spacecraft. Data from these spacecraft were used because the orbital configurations resulted in variations of the longitude of periapse and node which were, to first order, dependent only on the even zonal harmonics. The data time span for each satellite was extremely long: 2138 days for Explorer 35 and 230 days for Explorer 49. The value of the harmonic coefficient is determined and used to obtain a value of the lunar moment of inertia.

Gapcynski, J. P.↗

NASA NTRS1975

Winter bulge and diurnal variations in hydrogen inferred from AE-C composition measurements

In situ mass spectrometric measurements of ion and neutral particle thermospheric compositions have been used to infer the latitudinal and diurnal variations of thermospheric atomic hydrogen for solstice conditions. Local time-dependent and local time-independent components of the observed hydrogen distribution were separated on the basis of a model generated by expanding the log of neutral hydrogen concentration in terms of spherical harmonics. Results are compared with analogous data on N2 concentration; the comparison reveals an anticorrelation between gas temperature and hydrogen concentration. The slope of this anticorrelation line represents the 'zero flux condition' from the exosphere theory of Hodges (1973), thus further bearing out the conclusion that exospheric flow is the dominant process governing the global distribution of hydrogen.

Brinton, H. C.↗

NASA NTRS1974

Geodesic synchrotron radiation in the Kerr geometry by the method of asymptotically factorized Green's functions

The scalar, electromagnetic, and gravitational geodesic-synchrotron-radiation (GSR) spectra are determined for the case of a test particle moving on a highly relativistic circular orbit about a rotating (Kerr) black hole. It is found that the spectral shape depends only weakly on the value of the angular-momentum parameter (a/M) of the black hole, but the total radiated power drops unexpectedly for a value of at least 0.95 and vanishes as the value approaches unity. A spin-dependent factor (involving the inner product of the polarization of a radiated quantum with the source) is isolated to explain the dependence of the spectral shape on the spin of the radiated field. Although the scalar wave equation is solved by separation of variables, this procedure is avoided for the vector and tensor cases by postulating a sum-over-states expansion for the Green's function similar to that found to hold in the scalar case. The terms in this sum, significant for GSR, can then be evaluated in the geometric-optics approximation without requiring the use of vector or tensor spherical harmonics.

Chrzanowski, P. L.↗