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NASA NTRS1975

On the alleged collisional origin of the Kirkwood Gaps

This paper examines two proposed mechanisms whereby asteroidal collisions and close approaches may have given rise to the Kirkwood Gaps. The first hypothesis is that asteroids in near-resonant orbits have markedly increased collision probabilities and so are preferentially destroyed, or suffer decay in population density, within the resonance zones. A simple order-of-magnitude analysis shows that this hypothesis is untenable since it leads to conclusions which are either unrealistic or not in accord with present understanding of asteroidal physics. The second hypothesis is the Brouwer-Jefferys theory that collisions would smooth an asteroidal distribution function, as a function of Jacobi constant, thus forming resonance gaps. This hypothesis is examined by direct numerical integration of 50 asteroid orbits near the 2:1 resonance, with collisions simulated by random variables. No tendency to form a gap was observed.

Heppenheimer, T. A.↗

NASA NTRS1975

Open-loop response of a burning liquid monopropellant

A direct comparison was obtained between the theoretical and experimental response of a burning liquid monopropellant surface to imposed pressure oscillations. The comparison extended beyond the quasi-steady frequency regime into the range where the combustion process interacts with the thermal wave in the liquid phase. The theory considered both gas and liquid-phase transient effects, involving a numerically integrated, first-order perturbation analysis. The experiment employed hydrazine as the test fuel, burning as a strand. With kinetic parameters fixed by steady burning rate measurements, the model gave good predictions of liquid-phase temperature distributions during steady combustion as well as the amplitude and phase of the oscillatory component of the burning rate in the presence of imposed pressure oscillations. The combustion response tends to peak in two frequency ranges; one corresponding to interaction with liquid-phase transient effects, which could be verified both theoretically and experimentally; the second corresponding to gas-phase transient effects, which could only be examined theoretically. Both ranges provide conditions where the amplifying power is sufficient to drive combustion instability.

Allison, C. B.↗

NASA NTRS1976

Dynamics of spin stabilized spacecraft during deployment of telescoping appendages

The dynamics of a spin stabilized spacecraft with telescoping appendages are treated both analytically and numerically. Two types of telescoping appendages are considered: (a) where an end mass is mounted at the end of an (assumed) massless boom; and (b) where the appendage is assumed to consist of a uniformly distributed homogeneous mass throughout its length. For the telescoping system Eulerian equations of motion are developed. During all deployment sequences it is assumed that the transverse component of angular momentum is much smaller than the component along the major spin axis. Closed form analytical solutions for the time response of the transverse components of angular velocities are obtained when the spacecraft hub has a nearly spherical mass distribution. For the more general case, a series solution is obtained and this solution is limited by its radius of convergence. The comparison of the different approximate analytical methods with numerical integration results are studied and it is observed that the oscillatory nature of the responses of the transverse angular velocity components reduces rapidly with faster extension rates.

Sellappan, R.↗

NASA NTRS1976

Thermospheric winds and exospheric temperatures from incoherent scatter radar measurements in four seasons

The time-dependent equations of neutral air motion are solved subject to three constraints: two of them are the usual upper and lower boundary conditions and the third is the value of the wind-induced ion drift at any given height. Using incoherent radar data, this procedure leads to a fast, direct numerical integration of the two coupled differential equations describing the horizontal wind components and yields time dependent wind profiles and meridional exospheric neutral temperature gradients. The diurnal behavior of the neutral wind system and of the exospheric temperature is presented for two solstice and two equinox days. The data used were obtained by the St. Santin and the Millstone Hill incoherent scatter radars. The derived geographic distributions of the exospheric temperatures are compared with those predicted by the OGO-6 empirical thermospheric model.

Antoniadis, D. A.↗

NASA NTRS1976

Short-period and long-period perturbations of a spherical satellite due to direct solar radiation

On the basis of expressions derived by Kozai (1961) and those developed in this paper, a detailed semianalytic algorithm is presented for calculating radiation-pressure perturbations in the Keplerian elements. Through some simple modifications, the algorithm is also made to hold when e = 0, i = 0, or both. The perturbations are obtained by summing over the sunlit segment of the satellite's orbit during each revolution or part thereof. The end points of this segment are evaluated numerically once per revolution. The effect of the inherent uncertainties in the boundaries of the earth's shadow is discussed. The algorithm is tested by means of numerical integration of the equations of motion and through comparisons with observations of the balloon satellite 1963 30D during a 200-day interval.

Aksnes, K.↗

NASA NTRS1976

Stabilization by modification of the Lagrangian

In order to reduce the error growth during a numerical integration, a method for stabilizing the differential equations of Keplerian motion is offered. It is characterized by the use of the eccentric anomaly as independent variable in such a way that the time transformation is given by a generalized Lagrange formalism. The control terms in the equations of motion obtained by this modified Lagrangian give immediately a completely Lyapunov-stable set of differential equations. The equation of time integration is modified by a control term which leads to an integral which defined the time element for the perturbed Keplerian motion.

Baumgarte, J. W.↗

NASA NTRS1976

Second-order closure analysis of turbulent reacting flows

A reacting shear layer program for a complete second-order closure of turbulent reacting flows is developed. The program uses a typical eddy box model and many of the fluid mechanical turbulence models developed in earlier studies. The computer program actually solves a total of 30 equations, where the numerical integration of the equations is performed by a forward-time-centered space quasi-implicit upwind finite-difference scheme. Results for nonreacting and simple isothermal reacting flows are presented along with preliminary results for propane-air flames. It is shown that the program predictions for simple nonreacting flows are in good agreement with available data. The calculations for isothermal reacting flows demonstrate the importance of including a mixedness correlation in finite rate chemistry calculations, where significant errors in the flowfield properties can result from neglect of the interaction between turbulence and the chemistry. Results for propane-air diffusion flames further support this conclusion.

Varma, A. K.↗

NASA NTRS1976

Numerical analysis of a two-dimensional nonsteady detonations

In the present work a system of two-dimensional nonsteady hydrodynamic and chemical kinetic equations was numerically integrated for an exothermic system. Assumed two-step reaction model simulates practically an oxyhydrogen mixture. The calculation starts from a plane Chapman-Jouguet detonation as an initial condition. Two-dimensional disturbances are generated by artificially placing nonuniformities ahead of the detonation front. Regardless of the difference of the given initial disturbances, a fixed number of triple shock waves were produced for a fixed combination of mixture model and geometry when the transition period was over. This shows that for a given detonation tube geometry any exothermic system has its own characteristic multidimensional structure. The obtained number of triple shock waves contained in the detonation front was in agreement with existing experimental observations under the same condition.

Taki, S.↗

NASA NTRS1976

Estimates of nitric oxide production for lifting spacecraft reentry

An approximate analysis to estimate the quantity of nitric oxide that could be formed in the wake of a reentering lifting spacecraft is reported. Three different approaches are undertaken: two simplified analytical models utilizing the sudden-freezing concept named the 'trailing-edge-freezing' and 'wake-freezing' approximations, and a computer calculation involving numerical integration of chemical rate equations. The three methods predict a maximum nitric oxide production equivalent to 6, 9.5 and 8% of the mass of the spacecraft, respectively. Thus, the amount of nitric oxide expected to be produced by future space activities is negligibly small compared with that produced by the natural processes.

Park, C.↗

NASA NTRS1976

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. 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 of nitric oxide is found to occur at altitudes between 90 and 100 km, the total annual rate being about 40 million kg. The present results suggest that the large concentration of nitric oxide observed below 95 km could be attributed to meteoroids instead of photodissociation of nitrogen.

Menees, G. P.↗

NASA NTRS1976

Technique for atmospheric rate chemistry calculations

The possibility that predictions of atmospheric photochemistry/transport models are sensitive to uncertainties in reaction rates and other inputs stresses the need for rapid numerical integration schemes in rate photochemistry problems. Reducing the computational burden has a major merit in facilitating sensitivity studies to assess the effect of uncertainties on predicted ozone diminutions from NOx (NO + NO2) in the exhaust plume of SST engines. The paper discusses the validity of an algorithmic approach to integration of rate chemistry problems in combustion, developed by Rubel and Baronti for an approximate calculation of the production rate of the i-th chemical species involved. An analysis of two projected SST engines confirms the validity of the proposed algorithm. Because of the relative arithmetical simplicity, it may be easier to treat diffusion rate chemistry calculations using the Rubel and Baronti approximation than would be possible by other approaches.

Matloff, G. L.↗

NASA NTRS1976

Stabilizing influence of earth perturbations on polar lunar orbiters

A class of highly inclined lunar orbits is discussed for which earth perturbations have significant influence on the orbit evolution and the usefulness of the orbit for scientific exploration. The theory of the long-term motion of particles under third-body and zonal harmonic perturbations is taken as a starting point for investigations of orbits with relatively long lifetimes and evolutionary patterns that enhance the effectiveness of the satellite as a lunar farside relay. An equilibrium solution in the doubly-averaged three-body problem with oblateness is identified and the results of some numerical integrations are presented.

Uphoff, C.↗

NASA NTRS1976

Laser beam scintillation beyond the turbulent atmosphere A numerical computation

The extended Huygens-Fresnel formulation for propagation through turbulence is used to examine scintillation of a finite laser beam. The method is demonstrated analytically for propagation beyond a weak Gaussian phase screen. A numerical integration technique is used to extend the results to a more realistic turbulence model. Results are compared with existing Gaussian beam propagation theory.

Bufton, J. L.↗

NASA NTRS1976

A space-time tensor formulation for continuum mechanics in general curvilinear, moving, and deforming coordinate systems

Tensor methods are used to express the continuum equations of motion in general curvilinear, moving, and deforming coordinate systems. The space-time tensor formulation is applicable to situations in which, for example, the boundaries move and deform. Placing a coordinate surface on such a boundary simplifies the boundary condition treatment. The space-time tensor formulation is also applicable to coordinate systems with coordinate surfaces defined as surfaces of constant pressure, density, temperature, or any other scalar continuum field function. The vanishing of the function gradient components along the coordinate surfaces may simplify the set of governing equations. In numerical integration of the equations of motion, the freedom of motion of the coordinate surfaces provides a potential for enhanced resolution of the continuum field function. An example problem of an incompressible, inviscid fluid with a top free surface is considered, where the surfaces of constant pressure (including the top free surface) are coordinate surfaces.

Avis, L. M.↗

NASA NTRS1977

Electron dynamics in a plasma focus

Results are presented of a numerical integration of the three-dimensional relativistic equations of motion of electrons subject to given electric and magnetic fields deduced from experiments. Fields due to two different models are investigated. For the first model, the fields are those due to a circular distribution of axial current filaments. As the current filaments collapse toward the axis, large azimuthal magnetic and axial electric fields are induced. These fields effectively heat the electrons to a temperature of approximately 8 keV and accelerate electrons within the radius of the filaments to high axial velocities. Similar results are obtained for the current-reduction phase of focus formation. For the second model, the fields are those due to a uniform current distribution. Both the current-reduction and the compression phases were studied. These is little heating or acceleration of electrons during the compression phase because the electrons are tied to the magnetic field. However, during the current-reduction phase, electrons near the axis are accelerated toward the center electrode and reach energies of 100 keV. A criterion is obtained which limits the runaway electron current to about 400 A.

Hohl, F.↗

NASA NTRS1977

Determination of constant-volume balloon capabilities for aeronautical research

The proper application of constant-volume balloons (CVB) for measurement of atmospheric phenomena was determined. And with the proper interpretation of the resulting data. A literature survey covering 176 references is included. the governing equations describing the three-dimensional motion of a CVB immersed in a flow field are developed. The flowfield model is periodic, three-dimensional, and nonhomogeneous, with mean translational motion. The balloon motion and flow field equations are cast into dimensionless form for greater generality, and certain significant dimensionless groups are identified. An alternate treatment of the balloon motion, based on first-order perturbation analysis, is also presented. A description of the digital computer program, BALLOON, used for numerically integrating the governing equations is provided.

Tatom, F. B.↗

NASA NTRS1977

Analysis of separation of the space shuttle orbiter from a large transport airplane

The feasibility of safely separating the space shuttle orbiter (140A/B) from the top of a large carrier vehicle (the C-5 airplane) at subsonic speeds was investigated. The longitudinal equations of motion for both vehicles were numerically integrated using a digital computer program which incorporates experimentally derived interference aerodynamic data to analyze the separation maneuver for various initial conditions. Separation of the space shuttle orbiter from a carrier vehicle was feasible for a range of dynamic-pressure and flight-path-angle conditions. By using an autopilot, the vehicle attitudes were held constant which ensured separation. Carrier-vehicle engine thrust, landing gear, and spoilers provide some flexibility in the separation maneuver.

Wilhite, A. W.↗

NASA NTRS1977

Mutual admittance between circular waveguide-fed apertures for large spacing

A closed-form asymptotic expression for the mutual admittance between two circular waveguide-fed apertures is developed for large element spacings. A comparison with calculations obtained by a numerical integration method indicated that the present analysis gives good results for spacings greater than two wavelengths.

Bailey, M. C.↗