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NASA NTRS1974

Spherical means of solutions of partial differential equations in a conical region

The spherical means of the solutions of a linear partial differential equation Lu = f in a conical region are studied. The conical region is bounded by a surface generated by curvilinear ti surfaces. The spherical mean is the average of u over a constant ti surface. The conditions on the linear differential operator, L, and on the orthogonal coordinates (ti, eta, zeta) are established so that the spherical mean of the solution subjected to the appropriate boundary and initial conditions can be determined directly as a problem with only space variable. Conditions are then established so that the spherical mean of the solution in one concial region will be proportional to that of a known solution in another conical region. Applications to various problems of mathematical physics and their physical interpretations are presented.

Ting, L.↗

NASA NTRS1974

Post-flight differential correction analysis using Vinti's spheroidal method for the small astronomy satellite orbit

The results of an intensive analysis of a differential orbit improvement method utilizing observational data for a 550-kilometer altitude, near-circular, near-equatorial satellite orbit are presented. Observations of the Small Astronomy Satellite (SAS-1) are in the form of direction cosines as measured at two ground interferometer tracking stations near the Equator during the first 22 orbital revolutions (approximately 37 hours) after launch of the spacecraft. Numerical results, in both tabular and graphical form, are displayed for numerous iterated fittings of various observational arcs by differential correction of the orbital elements. Parameters varied in these comparative cases include the time duration of the observational data block, the number of pairs of direction cosine data and the number of tracking station passes included in the solution, the distribution of such passes between the two available tracking stations, and the acceptance criterion for the observational residuals in the least squares fitting procedure. It was found that three observational pairs of direction cosine data, the minimum number possible for a uniquely determined solution in theory, are sufficient to promote covergence to an accurate solution, if properly selected.

Walden, H.↗

NASA NTRS1973

Differential equation of exospheric lateral transport and its application to terrestrial hydrogen

The differential equation description of exospheric lateral transport of Hodges and Johnson is reformulated to extend its utility to light gases. Accuracy of the revised equation is established by applying it to terrestrial hydrogen. The resulting global distributions for several static exobase models are shown to be essentially the same as those that have been computed by Quessette using an integral equation approach. The present theory is subsequently used to elucidate the effects of nonzero lateral flow, exobase rotation, and diurnal tidal winds on the hydrogen distribution. Finally it is shown that the differential equation of exospheric transport is analogous to a diffusion equation. Hence it is practical to consider exospheric transport as a continuation of thermospheric diffusion, a concept that alleviates the need for an artificial exobase dividing thermosphere and exosphere.

Hodges, R. R., Jr.↗

NASA NTRS1973

The lunar crust - A product of heterogeneous accretion or differentiation of a homogeneous moon

The outer portion of the moon (including the aluminum-rich crust and the source regions of mare basalts) was either accreted heterogeneously or was the product of widespread differentiation of an originally homogeneous source. Existing evidence for and against each of these two models is reviewed. It is concluded that the accretionary model presents more problems than it solves, and the model involving differentiation of an originally homogeneous moon is considered to be more plausible. A hypothesis for the formation of mare basalts is advanced.

Brett, R.↗

NASA NTRS1974

Rigid and differential rotation of the solar corona

The rotation of the solar corona has been studied using recurrence properties of the green coronal line (5303 A) for the interval from 1947 to 1970. Short-lived coronal activity is found to show the same differential rotation as short-lived photospheric magnetic field features. Long-lived recurrences show rigid rotation in the latitude interval of plus or minus 57.5 deg. It is proposed that at least part of the variability of rotational properties of the solar atmosphere may be understood as a consequence of coexistence of differential and rigid solar rotation.

Antonucci, E.↗

NASA NTRS1973

Analysis of laser differential absorption remote sensing using diffuse reflection from the earth

A computer model analysis of an infrared laser differential absorption remote sensing technique (DARS) is presented. An infrared laser source operating at two or more wavelengths and a heterodyne detection system are considered to be mounted on either an aircraft or satellite platform to monitor the differential absorption between the laser source and the earth surface. The capability of this technique for measuring gas concentrations in the lower 5 kilometers of the atmosphere is emphasized. Numerical results are presented simulating measurements of vertical burdens of CO, NO, CH4, CO2, and O3. These results indicate that measurements of expected concentrations of these gases can be made with greater than 80% accuracy using realistic laser powers and system parameters.

Seals, R. K., Jr.↗

NASA NTRS1974

Changing stepsize in the integration of differential equations using modified divided differences

Multistep methods for solving differential equations based on numerical integration formulas or numerical differentiation formulas (for stiff equations) require special provision for changing the stepsize. New algorithms are given which make the use of modified divided differences an attractive way to carry out the change in stepsize for such methods. Error estimation and some of the important factors in stepsize selection and the selection of integration order are also considered.

Krogh, F. T.↗

NASA NTRS1974

Multi-off-grid methods in multi-step integration of ordinary differential equations

Description of methods of solving first- and second-order systems of differential equations in which all derivatives are evaluated at off-grid locations in order to circumvent the Dahlquist stability limitation on the order of on-grid methods. The proposed multi-off-grid methods require off-grid state predictors for the evaluation of the n derivatives at each step. Progressing forward in time, the off-grid states are predicted using a linear combination of back on-grid state values and off-grid derivative evaluations. A comparison is made between the proposed multi-off-grid methods and the corresponding Adams and Cowell on-grid integration techniques in integrating systems of ordinary differential equations, showing a significant reduction in the error at larger step sizes in the case of the multi-off-grid integrator.

Beaudet, P. R.↗

NASA NTRS1974

Differential rotation of solar filaments

The latitudinal component of solar differential rotation and the possibility of a radial component are discussed and compared to the observed rotational velocities of solar filaments. The author's values of rotational rate versus heliographic latitude for 100 points in the solar atmosphere derived from 17 quiescent filaments are shown to be comparable to the rates found by d'Azambuja and d'Azambuja (1948). The filament rate is significantly greater than the spot rate (Newton and Nunn, 1951); the difference cannot be accounted for by the poleward migration of filaments and seems to reflect a true radial gradient of rotational velocity in the sun. It is shown that filaments in closer proximity to active regions usually exhibit no differential rotation, while those far from active regions generally show it clearly. Comparison with Mt. Wilson photospheric Doppler measurements shows that filaments rotate faster than the general photosphere and that the spot rate exceeds that for the general photosphere.

Glackin, D. L.↗

NASA NTRS1975

NASTRAN solutions of problems described by simultaneous parabolic differential equations

NASTRAN solution techniques are shown for a numerical analysis of a class of coupled vector flow processes described by simultaneous parabolic differential equations. To define one physical problem type where equations of this form arise, the differential equations describing the coupled transfers of heat and mass in mechanical equilibrium with negligible mass average velocity are presented and discussed. Also shown are the equations describing seepage when both electrokinetic and hydrodynamic forces occur. Based on a variational statement of the general problem type, the concepts of scalar transfer elements and parallel element systems are introduced. It is shown that adoptation of these concepts allows the direct use of NASTRAN's existing Laplace type elements for uncoupled flow (the heat transfer elements) for treating multicomponent coupled transfer. Sample problems are included which demonstrate the application of these techniques for both steady-state and transient problems.

Mason, J. B.↗

NASA NTRS1975

Precision Selenodesy via Differential Very-Long-Baseline Interferometry

The technique of differential very-long baseline interferometry was used to measure the relative positions of the ALSEP transmitters at the Apollo 12, 14, 15, 16, and 17 lunar landing sites with uncertainties less than 0.005 of geocentric arc. These measurements yielded improved determinations of the selenodetic coordinates of the Apollo landing sites, and of the physical libration of the moon. By means of a new device, the differential Doppler receiver (DDR), instrumental errors were reduced to less than the equivalent of 0.001. DDRs were installed in six stations of the NASA spaceflight tracking and data network and used in an extensive program of observations beginning in March 1973.

King, R. W., Jr.↗

NASA NTRS1975

Doppler measurements of the ionosphere on the occasion of the Apollo-Soyuz test project. Part 2: Inversion of differential and rotating Doppler shifts

The preparation of the analytical approach and of the related software used in the inversion of the differential and rotating Doppler data obtained from the ionospheric experiment of the Apollo-Soyuz Test Project (ASTP) is discussed. These data were collected in space-to-space paths (between the ASTP Docking Module (DM) and the Apollo Command Service Module and in space-to-ground paths (between the DM and ground). The Doppler links operated at 162 and 324 MHz and have an accuracy better than 3 MHz over 10-sec integration time. The inversion approach was tested with dummy data obtained with a computer simulation. It was found that a measurement accuracy of 1 to 10% in the value of the horizontal electron density gradient at 221-km altitude can be achieved, in space-to-space paths. For space-to-ground paths near the orbital plane, possible effects of the horizontal gradients on the received differential Doppler shifts were identified. It was possible to reduce the gradient-associated errors in the inversion that leads to the columnar electron content by approximately one-half. Accuracies of 5 to 10% in columnar electron content are achievable, with this gradient-compensation technique.

Gay, R. H.↗

NASA NTRS1975

The geometric approach to sets of ordinary differential equations and Hamiltonian dynamics

The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field V in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along V: scalar fields, forms, vectors, and integrals over subspaces. It is shown that to any field V can be associated a Hamiltonian structure of forms if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added. Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton's variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms must include the characteristics of the set.

Estabrook, F. B.↗

NASA NTRS1974

An evaluation of differential Omega for general aviation area navigation

This paper reports on a study which compared the expected cost and performance of Differential Omega with that of Loran-C and VORTAC for general aviation area navigation. Analysis is directed toward a comparison of the systems with respect to specified performance parameters and the cost-effectiveness of each system in relation to the specifications. Loran-C offers the highest performance with respect to accuracy. Differential Omega requires the least expenditure. It was found cost ineffective to attempt to obtain complete coverage by expanding the existing VORTAC system.

Hollister, W. M.↗

NASA NTRS1976

Differential cross-section for positronium formation in electron-atomic hydrogen collisions

The L=0 and 1 partial wave amplitudes obtained by a two-state coupled static approximation with correlation with the L greater than or equal to 2 Born amplitudes were combined to obtain the differential cross section for positronium formation in electron-atomic hydrogen collisions. For positron energies of 0.64 and 0.75 ryd, minima at the scattering angles of 57 deg and 51 deg are found. Total cross sections for positronium formation for low and intermediate impact energies are given. Measurement of the differential cross section for the process positron + helium yields positronium + helium ion for the detection of possible minima is suggested.

Drachman, R. J.↗

NASA NTRS1976

ALSEP: Quasar differential VLBI

A program of Apollo Lunar Surface Experiments Package (ALSEP)-Quasar Very Long Baseline Interferometry (VLBI) is being reported. These observations primarily employ a 4 antenna technique, whereby simultaneous observations with two antennas at each end of an intercontinental baseline are used to derive the differential interferometric phase between a compact extragalactic radio source (usually a quasar) and a number of ALSEP transmitters on the lunar surface. A continous ALSEP-quasar differential phase history over a few hour period leads to extremely high angular accuracy in measuring the lunar position against the quasar reference frame. This application of the 4 antenna technique has been underway for more than a year and is now producing high quality data utilizing Deep Space Network (DSN) stations in Australia, Spain, and Goldstone, California, as well as the Spaceflight Tracking and Data Network (STDN) Apollo station at Goldstone.

Slade, M. A.↗

NASA NTRS1975

Spherical means of solutions of partial differential equations in a conical region

The spherical means of the solutions of a linear partial differential equation Lu = f in a conical region are studied. The conical region is bounded by a surface generated by curvilinear xi lines and by two truncating xi surfaces. The spherical mean is the average of u over a constant xi surface. Conditions on the linear differential operator, L, and on the orthogonal coordinates xi, eta, and zeta are established so that the problem for the determination of the spherical mean of the solution subjected to the appropriate boundary and initial conditions can be reduced to a problem with only one space variable. Conditions are then established so that the spherical mean of the solution in one conical region will be proportional to that of a known solution in another conical region. Applications to various problems of mathematical physics and their physical interpretations are presented.

Ting, L.↗

NASA NTRS1977

Application of differential similarity to finding nondimensional groups important in tests of cooled engine components

The method of differential similarity is applied to the partial differential equations and boundary conditions which govern the temperature, velocity, and pressure fields in the flowing gases and the solid stationary components in air-cooled engines. This procedure yields the nondimensional groups which must have the same value in both the test rig and the engine to produce similarity between the test results and the engine performance. These results guide the experimentalist in the design and selection of test equipment that properly scales quantities to actual engine conditions. They also provide a firm fundamental foundation for substantiation of previous similarity analyses which employed heuristic, physical reasoning arguments to arrive at the nondimensional groups.

Sucec, J.↗