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NASA NTRS1974

Contribution to the theory of tidal oscillations of an elastic earth. External tidal potential

The differential equations of the tidal oscillations of the earth were established under the assumption that the interior of the earth is laterally inhomogeneous. The theory was developed using vectorial and dyadic symbolism to shorten the exposition and to reduce the differential equations to a symmetric form convenient for programming and for numerical integration. The formation of tidal buldges on the surfaces of discontinuity and the changes in the internal density produce small periodic variations in the exterior geopotential which are reflected in the motion of artificial satellites. The analoques of Love elastic parameters in the expansion of exterior tidal potential reflect the asymmetric and inhomogeneous structure of the interior of the earth.

Musen, P.↗

NASA NTRS1974

A study to determine the usefulness of interval analysis in solving problems in celestial mechanics

This investigation was undertaken to determine the usefulness of interval analysis to numerical integration and matrix inversion techniques and to combine these results to determine the value of interval analysis in bounding computational errors in the two-body problem. Conclusions were that interval analysis may be worthwhile in certain small scale isolated problems, but its usefulness in any large scale problem is doubtful.

Walling, D.↗

NASA NTRS1973

Low-thrust trajectory analysis for the geosynchronous mission

Methodology employed in development of a computer program designed to analyze optimal low-thrust trajectories is described, and application of the program to a Solar Electric Propulsion Stage (SEPS) geosynchronous mission is discussed. To avoid the zero inclination and eccentricity singularities which plague many small-force perturbation techniques, a special set of state variables (equinoctial) is used. Adjoint equations are derived for the minimum time problem and are also free from the singularities. Solutions to the state and adjoint equations are obtained by both orbit averaging and precision numerical integration; an evaluation of these approaches is made.

Jasper, T. P.↗

NASA NTRS1973

Quasi-static extension of a tensile crack contained in a viscoelastic-plastic solid

Final stretch criterion of failure is applied to the problem of quasi-static extension of a crack embedded in an elastic-plastic or viscoelastic-plastic matrix. The slow growth under subcritical conditions in a rate-sensitive Tresca solid is shown to be a superposition of creep rupture and McClintock's ductile growth. This type of growth occurs at subcritical magnitude of the imposed K-factor and can be accounted for only through a recognition of inelastic properties of solids. In the subcritical range there is no unique value for K sub c independent of geometrical configuration and flaw size. Not only the produced states of stress and strain are dependent on the loading path, but also the material resistance to fracture turns out to be a function of the history of loading that precedes catastrophic failure. A nonlinear integro-differential equation of motion is derived for a crack progressing through a viscoelastic medium with some limited ability to plastic flow. Examples of numerical integration are given incorporating both monotonic and cyclic loading programs.

Wnuk, M. P.↗

NASA NTRS1973

Introduction to stellar atmospheres and interiors

Stellar atmospheres are examined, giving attention to the transfer of radiation, the physical properties of gases, model atmospheres, and questions of line absorption in stellar atmospheres. Observational data are considered together with the equations for the stellar interior and aspects of stellar evolution. The calculation of model atmospheres and interiors is also discussed, taking into account numerical integration, a model solar atmosphere, and model stellar interiors.

Novotny, E.↗

NASA NTRS1973

Dynamic analysis of a system of hinge-connected rigid bodies with nonrigid appendages

Equations of motion are derived for use in simulating a spacecraft or other complex electromechanical system amenable to idealization as a set of hinge-connected rigid bodies of tree topology, with rigid axisymmetric rotors and nonrigid appendages attached to each rigid body in the set. In conjunction with a previously published companion paper on finite-element appendage vibration equations, this paper provides a complete minimum-dimension formulation suitable for generic programming for digital computer numerical integration.

Likins, P. W.↗

NASA NTRS1973

Applications of the Kustaanheimo-Stieffel transformation of the perturbed two-body problem

The Newtonian differential equations of motion for the two-body problem can be transformed into four linear harmonic-oscillator equations by simultaneously applying the regularization step dt/ds = r and the Kustaanheimo-Stieffel (KS) transformation. The regularization step changes the independent variable from time to a new variable s, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. A derivation of a uniform, regular solution for the perturbed two-body problem in the four-dimensional space is presented. The variation-of-parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. This ten-element solution has mixed secular terms that degrade the long-term accuracy during numerical integration. Therefore, to eliminate these terms, the solution is modified by introducing two additional elements.

Bond, V. R.↗

NASA NTRS1973

Theoretical study of H2/+/ spectroscopic properties. II, III

Description of the theoretical spectroscopic properties of the 2p pi/sub u/ and 3d sigma/sub g/ excited states of the H2/+/ hydrogen molecular ion. Numerical integration of the Schrodinger equation is used to determine vibration-rotation eigenvalues. Dunham power series expansions are used to determine the equilibrium separation, potential coefficients, and spectroscopic constants. The eigenvalues are used to determine delta-G, Bv, Dv, and Hv.

Beckel, C. L.↗

NASA NTRS1974

Lunar perturbations of artificial satellites of the earth

The disturbing function for the lunar perturbations of an artificial satellite are derived, using ecliptic elements for the moon and equatorial elements for the satellite. Secular, long-period, and short-period perturbations are then computed, with the expressions kept in closed form in both inclination and eccentricity of the satellite. Alternative expressions for short-period perturbations of high satellites are also given, assuming small values of the eccentricity. The moon's position is specified by the inclination, node, argument of perigee, true (or mean) longitude, and its radius vector from the center of the earth. The results can then be applied to numerical integration by using coordinates of the moon from ephemeris tapes or to analytical representation by using results from lunar theory, with the moon's motion represented by a precessing and rotating elliptical orbit.

Giacaglia, G. E. O.↗

NASA NTRS1974

An investigation of accretion of matter onto white dwarfs as a possible X-ray mechanism

The accretion flow onto white dwarfs is investigated by numerically integrating the full nonlinear hydrodynamic equations under the assumption of spherical symmetry to determine the short-term time-dependent behavior, to determine the effects on the flow when radial modes of oscillation of the star are excited, to see if X-rays are produced, and to determine the radiation spectrum. Some applications of the results to known X-ray sources are suggestive.

Degregoria, A. J.↗

NASA NTRS1974

Nitric oxide formation by meteoroids in the upper atmosphere

The process of nitric oxide formation during atmospheric entry of meteoroids is analyzed theoretically. An ablating meteoroid is assumed to be a point source in a uniform flow with a continuum regime evolving in its wake. The amount of nitric oxide produced by high-temperature reactions of air in the continuum regime is calculated by numerical integration of chemical-rate equations. This is accomplished by assuming that flow properties are constant across the reacting region, its radius being determined from considerations of shock-wave formation and molecular diffusion. The results, when summed over the observed mass, velocity, and entry-angle distributions of meteoroids, provide annual global production rates of nitric oxide as a function of altitude. The peak production is found to occur between 90- and 100-km altitude, the total annual rate being around 40 million kg.

Menees, G. P.↗

NASA NTRS1974

Calculation of precision satellite orbits with nonsingular elements /VOP formulation/

Review of some results obtained in an effort to develop efficient, high-precision trajectory computation processes for artificial satellites by optimum selection of the form of the equations of motion of the satellite and the numerical integration method. In particular, the matching of a Gaussian variation-of-parameter (VOP) formulation is considered which is expressed in terms of equinoctial orbital elements and partially decouples the motion of the orbital frame from motion within the orbital frame. The performance of the resulting orbit generators is then compared with the popular classical Cowell/Gauss-Jackson formulation/integrator pair for two distinctly different orbit types - namely, the orbit of the ATS satellite at near-geosynchronous conditions and the near-circular orbit of the GEOS-C satellite at 1000 km.

Velez, C. E.↗

NASA NTRS1974

Geophysical subsurface probing with radio-frequency interferometry

The radio-frequency interferometry method can be used to probe interiors of celestial bodies and terrestrial areas with low conductivity. In order to interpret the interference patterns, a theoretical study is made of the electromagnetic fields due to a dipole antenna on the surface of a horizontally stratified n-layered medium. Three approaches are used to calculate the interference patterns: direct numerical integration, asymptotic evaluation by the saddle point method, and a residue series approach. The asymptotic approach leads to the geometrical-optics interpretation. The residue approach leads to modal analysis. The validity of the formulation is checked by comparisons with analog model tank experiments and actual field data obtained from glaciers.

Kong, J. A.↗

NASA NTRS1974

Stability of a dual-spin satellite with two dampers

The rotational stability of a dual-spin satellite consisting of a main body and a symmetric rotor, both spinning about a common axis, is investigated. The main body is equipped with a spring-mass damper, while a partially filled viscous ring damper is mounted on the rapidly spinning rotor. The effect of fluid motion on the rotational stability of the satellite is calculated, considering the fluid as a single particle moving in a tube with viscous damping. Time constants are obtained by solving approximate equations of motion for the nutation-synchronous and the spin-synchronous modes, and the results are found to agree well with the numerical integrations of the exact equations. A limit cycle may exist for some configurations; the nutation angle tends to increase in such cases.

Alfriend, K. T.↗

NASA NTRS1974

Time elements

Time elements are introduced for use with Sundman time transformations of the type dt = r(alpha)ds for satellite equations of motion. Two time elements are given, one providing maximum accuracy for alpha = 1, the other for alpha = 2. Time elements and time transformations reduce local truncation error and Liapunov (in track) instability, and provide analytical step size control. Numerical results show accuracy improvements of more than one order of magnitude when time elements are employed with time transformations in the numerical integration of the satellite equations, compared with using time transformations alone.

Nacozy, P. E.↗

NASA NTRS1974

A comparison of several computational techniques for solving some common aeronomic problems

Several numerical integration techniques for solving common aeronomic problems involving species rate equations are compared for speed and accuracy. A newer technique that defines families of species that are nearly conserved is found to be superior to an iterative technique when both methods are applied to simple test problems. The 'conservation' technique is also found to be more economical than the more complex Gear (1969) integration scheme for comparable accuracy.

Turco, R. P.↗

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 NTRS1974

Linear viscous stability analysis of the stratified Bickley jet

The stability of the stably stratified Bickley jet to small disturbances in a viscous, heat conducting fluid is explored by numerical integration of the governing differential equation. Curves of neutral stability are presented in the alpha-R plane and the parametric dependence of these curves upon Richardson number and Prandtl number is examined. The results show a curious bimodal behavior of the curve of neutral stability for small Richardson numbers. In addition, closed neutral curves are found for Richardson numbers in excess of the critical value for the inviscid flow. The unstable regions in the interior of these curves increase with decreasing Prandtl number. Energy balances are computed which show the relative magnitude of the Reynolds stress production, buoyancy dissipation, and viscous dissipation terms across the flow. As the thermal diffusivity becomes large compared with the kinematic viscosity, the buoyancy term is suppressed and the thermal stratification becomes irrelevant.

Gage, K. S.↗