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At least 613 records · Page 34

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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.↗

NASA NTRS1977

Boundary-fitted curvilinear coordinate systems for solution of partial differential equations on fields containing any number of arbitrary two-dimensional bodies

A method is presented for automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multi-connected two-dimensional region containing any number of arbitrarily shaped bodies. No restrictions are placed on the shape of the boundaries, which may even be time-dependent, and the approach is not restricted in principle to two dimensions. With this procedure the numerical solution of a partial differential system may be done on a fixed rectangular field with a square mesh with no interpolation required regardless of the shape of the physical boundaries, regardless of the spacing of the curvilinear coordinate lines in the physical field, and regardless of the movement of the coordinate system in the physical plane. A number of examples of coordinate systems and application thereof to the solution of partial differential equations are given. The FORTRAN computer program and instructions for use are included.

Thompson, J. F.↗

NASA NTRS1977

MACSYMA's symbolic ordinary differential equation solver

The MACSYMA's symbolic ordinary differential equation solver ODE2 is described. The code for this routine is delineated, which is of interest because it is written in top-level MACSYMA language, and may serve as a good example of programming in that language. Other symbolic ordinary differential equation solvers are mentioned.

Golden, J. P.↗

NASA NTRS1964

Analytical Solutions for Non-Linear Differential Equations with the Help of a Digital Computer

A technique was developed with the help of a digital computer for analytic (algebraic) solutions of autonomous and nonautonomous equations. Two operational transform techniques have been programmed for the solution of these equations. Only relatively simple nonlinear differential equations have been considered. In the cases considered it has been possible to assimilate the secular terms into the solutions. For cases where f(t) is not a bounded function, a direct series solution is developed which can be shown to be an analytic function. All solutions have been checked against results obtained by numerical integration for given initial conditions and constants. It is evident that certain nonlinear differential equations can be solved with the help of a digital computer.

Cromwell, P. C.↗

NASA NTRS1977

The effect of temperature and pressure on the distribution of iron group elements between metal and olivine phases in the process of differentiation of protoplanetary material

The distribution patterns of Ni, Co, Mn, and Cr were studied in olivines of various origins: from meteorites (chondrites, achondrites, pallasites), which are likely analogs of the protoplanetary material, to peridotite inclusions in kimberlite pipes, which are analogs of mantle material. According to X-ray microanalysis data, nickel is concentrated in peridotite olivines, while manganese is concentrated in meteoritic olivines. The maximum chromium content was found in ureilites, which were formed under reducing conditions. Experiments at pressures of 20 to 70 kbar and temperatures of 1100 to 2000 C have shown that in a mixture of olivine and Ni metal or NiO, nickel enters the silicate phase, displacing Fe into the metallic phase. Equilibrium temperatures were estimated from the Fe, Ni distribution coefficients between the metal and olivine: 1500 K for pallasites, 1600 K for olivine-bronzite H6 chondrites, 1200 K for olivine-hypersthene L6, 900 K for LL6, and 1900 K for ureilites (at P = 1 atm). The equilibrium conditions of peridotites are close to T = 1800 K and P over 100 kbar. It is concluded that there is a sharp difference between the conditions of differentiation of the protoplanetary material at the time meteorites were formed and the conditions of differentiation of the planets into concentric layers.

Vinogradov, A. P.↗

NASA NTRS1977

Mechanical processess affecting differentiation of protolunar material

Mechanisms prior to lunar formation are sought to account for the loss of volatiles, the depletion of iron, and the enrichment of plagioclase. Some of the same mechanisms are necessary to account for achondritic, stony-iron, and iron meteorites. Collisions seem marginally capable of providing the heat to accomplish the differentiation into iron, magnesian silicates, and plagioclase. Once this differentiation is accomplished, the subsequent mechanical history should have been sufficient to sort material according to composition in the protolunar circumterrestrial cloud. Effects operating include the correlation of body size with mechanical strength; the lesser ability of the cloud to trap the larger, denser infalling bodies; the more rapid drawing into the Earth of the larger moonlets; and the higher energy orbits for dominantly plagioclase smaller pieces broken off by collision.

Kaula, W. M.↗

NASA NTRS1978

Differential optoacoustic absorption detector

A differential optoacoustic absorption detector employed two tapered cells in tandem or in parallel. When operated in tandem, two mirrors were used at one end remote from the source of the beam of light directed into one cell back through the other, and a lens to focus the light beam into the one cell at a principal focus half way between the reflecting mirror. Each cell was tapered to conform to the shape of the beam so that the volume of one was the same as for the other, and the volume of each received maximum illumination. The axes of the cells were placed as close to each other as possible in order to connect a differential pressure detector to the cells with connecting passages of minimum length. An alternative arrangement employed a beam splitter and two lenses to operate the cells in parallel.

Shumate, M. S.↗

NASA NTRS1977

Coplanar tail-chase aerial combat as a differential game

A reduced-order version of the one-on-one aerial combat problem is studied as a pursuit-evasion differential game. The coplanar motion takes place at given speeds and given maximum available turn rates, and is described by three state equations which are equivalent to the range, bearing, and heading of one aircraft relative to the other. The purpose of the study is to determine those relative geometries from which either aircraft can be guaranteed a win, regardless of the maneuver strategies of the other. Termination is specified by the tail-chase geometry, at which time the roles of pursuer and evader are known. The roles are found in general, together with the associated optimal turn maneuvers, by solution of the differential game of kind. For the numerical parameters chosen, neither aircraft can win from the majority of possible initial conditions if the other turns optimally in certain critical geometries.

Merz, A. W.↗

NASA NTRS1977

Boolean differentiation and integration using Karnaugh maps

Algorithms are presented for differentiation and integration of Boolean functions by means of Karnaugh maps. The algorithms are considered simple when the number of variables is six or less; in this case Boolean differentiation and integration is said to be as easy as the Karnaugh map method of simplifying switching functions. It is suggested that the algorithms would be useful in the analysis of faults in combinational systems and in the synthesis of asynchronous sequential systems which utilize edge-sensitive flip-flops.

Tucker, J. H.↗

NASA NTRS1977

ALSEP-quasar differential VLBI

A program of ALSEP-quasar very-long-baseline interferometry (VLBI) is being carried out. These observations employ primarily a 'four-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 continuous ALSEP-quasar differential phase history over a period of a few hours will lead to milliarcsecond angular accuracy in measuring the lunar position against the quasar reference frame if suitable calibration measurements are obtained. Development of this application of the four-antenna technique has been underway for more than a year and is producing high-quality data utilizing Deep Space Network stations. These high-accuracy observations are of value to tie the lunar ephemeris to a nearly inertial extragalactic reference frame, to test gravitational theories, and to measure the earth-moon tidal friction interaction.

Slade, M. A.↗

NASA NTRS1977

Compositional variation in late-stage differentiates in mare lavas, as indicated by silicate melt inclusions

Questions regarding an interpretation of data from silicate melt inclusions are considered, taking into account the fragmental nature of the record, the possible effects of cooling rate on late-stage melt composition, and the crystallization of the trapped melt. New data related to 484 new analyses are provided in a separate table. Textural and mineralogical features and bulk chemical composition for an overall characterization of late-stage differentiates are discussed. A description is presented of a grouping and recognition of rock types on the basis of late-stage inclusion analyses. Attention is given to the bimodality of potassium values and the delineation of differentiation trends. It is found that many of the problems in interpretation and much of the scatter found in the composition of silicate melt inclusions in lunar samples can be attributed to the incomplete nature of the record and to crystallization after trapping.

Roedder, E.↗

NASA NTRS1975

Rb-Sr study of a lunar dunite and evidence for early lunar differentiates

An Rb-Sr study of Boulder 3 from Station 2 sampled during the Apollo 17 mission is reported. The investigated boulder is a metaclastic rock, inferred to have come from the South Massif, and contains a variety of lithic clasts. Since the rock type may be critical in understanding lunar differentiation processes, an extensive investigation was carried out of the dunite clast and other clasts as well as of the fine-grained matrix. The investigation had the purpose to define the age of crystallization of the dunite and to determine the extent to which later metamorphism of the host rock may have effected the isotopic systematics of this clast. The isochron determined yielded an age of 4.55 plus or minus 0.10 AE for the dunite. This result demonstrates that differentiation began close to the time of formation of the moon and that relics of this process are preserved.

Papanastassiou, D. A.↗