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Iterative methods for the approximate solution of the nonlinear state estimation problem are investigated in which the solution is retained in the form of a finite series of Chebyshev polynomials. Algorithms are presented which allow the state to be estimated from observational data in either the batch or the sequential form. The advantages of these techniques are discussed.
Rytov's approximation, or the method of smooth perturbations, is utilized to derive the temporal frequency spectra of the amplitude and phase fluctuations of multifrequency plane and spherical waves propagating in the interplanetary medium and solar corona. It is shown that multifrequency observations of interplanetary scintillations using either compact radio stars of spacecraft radio signals are desirable because the correlation of the multifrequency waves yields additional independent measurements of the solar wind and turbulence. Measurements of phase fluctuations are also desirable because, unlike amplitude fluctuations, they provide information on the full range of scale sizes for the electron-density fluctuations. It is also shown that a coherent dual-frequency radio system is particularly useful in making such measurements. In addition to providing a means for interpreting observations of multifrequency interplanetary scintillations, the present analysis is essential for estimating the effects of solar corona turbulence on the communications and navigation of a spacecraft whose line-of-sight path passes close to the sun.
Results are presented from photoelectric scans of the interstellar 3384-A line of Ti II obtained at a resolution of 3.7 km/s for 47 stars. Although the depletion of titanium from the interstellar gas is large and highly variable, its ratio to the depletion of interstellar calcium is relatively constant; titanium is less depleted than calcium by a factor of about four. A method for determining approximate electron densities from the Ca II and Ti II column densities is applied to 51 interstellar clouds.
The two-fluid solar-wind equations have been solved by a method which is approximately 50 times faster than any previously developed, through the use of asymptotic expansions which are self-consistently iterated upon to find a solution that passes through the critical point. The energy assumptions in two-fluid solar-wind models are reexamined, and the conclusions are as follows: (1) proton thermal conduction may not be neglected, (2) the Coulomb logarithm must be calculated as a function of radius, and (3) the electron and proton temperatures at the base need not be equal, even when the time scale for energy exchange between the species is an order of magnitude smaller than the expansion time at the base. It is possible to reproduce reasonable quiet-time solar-wind parameters at 1 AU, but only if the proton temperature is approximately twice the electron temperature at 1 solar radius. This may indicate that extended proton heating is important in the outer solar corona. Winds with velocities at 1 AU of 450 km/s are generated without nonthermal energy deposition but require high proton temperatures as well as very low densities at the base. Higher-velocity solutions are not possible in a spherically symmetric geometry for reasonable particle fluxes at 1 AU, and it is suggested that these higher-velocity states probably require additional heating, acceleration mechanisms, or nonradial flow.
System analyses of cylindrical plasma centrifuges are presented, for which the velocity field and electromagnetic fields are calculated. The effects of different electrode geometrics, induced magnetic fields, Hall-effect, and secondary flows are discussed. It is shown that speeds of 10000 m/sec can be achieved in plasma centrifuges, and that an efficient separation of U238 and U235 in uranium plasmas is feasible. The external boundary-value problem for the deposition of sputtering products is reduced to a Fredholm integral equation, which is solved analytically by means of the method of successive approximations.
An unstiffened panel buckling constraint for balanced, symmetric laminated composites is included on the global design level in a mathematical programming structural optimization procedure for designing wing structures. Constraints are introduced by penalty functions, and Newton's method based on approximate second derivatives of the penalty terms is used as the search algorithm to obtain minimum-mass designs. Constraint approximations used during the optimization process contribute to the computational efficiency of the procedure. A criterion is developed that identifies the appropriate conservative form of the constraint approximations that are used with the optimization procedure. Minimum-mass design results are obtained for a multispar high-aspect-ratio wing subjected to material strength, minimum-gage, displacement, panel buckling and twist constraints. The material systems considered for the examples are all graphite-epoxy, graphite-epoxy with boron-epoxy spar caps, and all aluminum. The composite material designs are shown to have an advantage over the aluminum designs since they can often satisfy additional constraints with only small mass increases.
The surface area-to-weight ratio of the Solar Power Satellite (SPS) is of the order of sq m/kg. Therefore, solar radiation pressure will be an important perturbation on the orbit, causing large variations in eccentricity. An analytical solution is developed for the motion of eccentricity and the line of apsides. Included is a secular term due to the earth's orbital eccentricity. Several initial conditions are considered. For some cases, the eccentricity is nearly constant over a few years. General characteristics of the motion are discussed. Long-term (30 years) effects of gravity are investigated by using a numerical integration method. Eccentricities of approximately 0.08 can be expected for uncorrected SPS orbits.
Two different approaches to the problem were investigated: (1) a dynamic method that computed pass-to-pass apparent movements in station height relative to short arcs fitted to several passes of data from the same station, and (2) a quasi-geometric method that reduces the dependence on unmodelled satellite dynamics to a knowledge of only the radial position of the satellite by considering two-station simultaneous ranging at the precise time that the satellite passes through the plane defined by the two stations and the center of the mass of the earth. The dynamic method was applied to 24-hour arcs in an attempt to study the total tidal displacement at Goddard, but it was found to be unsuitable. The quasi-geometry method, on the other hand, is significantly less influenced by these effects because it is sensitive principally to the differential tidal displacements between laser stations. Using this method, a good approximation to the theoretical Love number h sub 2 was obtained from five passes over the GEOS-3 calibration area when a systematic bias of 2.16 meters was allowed in the radial distance to the satellite.
A 4.67X, plus or minus 5 deg. compound parabolic concentrator (CPC) for a large photovoltaic array in space was analyzed. The design was demonstrated to be effective in achieving a net power gain which can be varied from more than a factor of 3 down to approximately unity. A method for reducing nonuniformities in illumination to a given desired level was found. The effectiveness of this method, which involves the introduction of a degree of non-specularity in the reflector surface, was confirmed by direct measurements with prepared foil reflectors in a CPC in terrestrial sunshine as well as by computer ray tracing. Further ray tracing confirms that the CPC design is extremely tolerant to pointing and alignment errors, minor distortions, etc. A two stage non-imaging design was shown, by preliminary measurements and analysis, to provide both the desired angular tolerance and the required degree of intensity uniformity if higher concentrations (4X-10X) are necessary.
The ionization balance in diffusion dominated discharges which depends on both one and two step ionization processes is considered. The Spenke diffusion equation (D sq delta n + neutrino n + sq kn =0) describing such conditions is solved by the Rayleigh-Ritz variational method. Simple analytic approximations to the density profile, and the similarity relation between neutrino,k,D and the discharge dimensions, are derived for planar and cylindrical geometry, and compared with exact computations for certain limiting cases.
Complicated three-dimensional viscous transonic flows about bodies at high angles of attack are solved on the Illiac IV computer. It is shown that certain approximate forms of the compressible Reynolds-averaged Navier-Stokes equations can be computed about realistic three-dimensional geometries with relative ease on the Illiac IV. The ease and efficiency with which this can be done depend on the approximations made in the basic equations, the choice of the numerical algorithm used for the solution, and the data-base system that controls the data management and identifies and manipulates the vectors. A pencil data-base system is found to be particularly suitable for the approximations and numerical method chosen to produce the results presented. In addition, some comparisons are made of computer predictions with experimental results for various lows about hemisphere-cylinders in both subsonic and supersonic free streams. The same viscous model and numerical model are used, showing good qualitative agreement in the location of separation lines and pressure distributions.
The unsolved problem of how to control the attitude and shape of future very large flexible satellite structures represents a challenging problem for modern control theory. One aspect of this problem is the question of how to choose the number and locations throughout the spacecraft of the control system actuators. Starting from basic physical considerations, this paper develops a concept of the degree of controllability of a control system, and then develops numerical methods to generate approximate values of the degree of controllability for any spacecraft. These results offer the control system designer a tool which allows him to rank the effectiveness of alternative actuator distributions, and hence to choose the actuator locations on a rational basis. The degree of controllability is shown to take a particularly simple form when the satellite dynamics equations are in modal form. Examples are provided to illustrate the use of the concept on a simple flexible spacecraft.
The ionization balance in diffusion-dominated discharges may depend on both one- and two-step ionization processes. The Spenke diffusion equation describing such conditions is solved in this paper by the Rayleigh-Ritz variational method. Simple analytic approximations to the density profile, and the similarity relation between the equation parameter, and the discharge dimensions, are derived for planar and cylindrical geometry and compared with exact computations for certain limiting cases.
The far-field radiation from the end of an exhaust duct is studied using both approximate and exact methods. Experimental data of narrow-band tone noise from static tests are compared to a multimodal radiation pattern. It is pointed out that possibly the exhaust noise in the far-field is inherently more difficult to attenuate than an inlet noise using duct suppressors.
The broadening and the signal power detected of a laser beam traversing an anisotropic scattering medium were examined using the small-angle approximation to the radiative transfer equation in which photons suffering large-angle deflections are neglected. To obtain tractable answers, simple Gaussian and non-Gaussian functions for the scattering phase functions are assumed. Two other approximate approaches employed in the field to further simplify the small-angle approximation solutions are described, and the results obtained by one of them are compared with those obtained using small-angle approximation. An exact method for obtaining the contribution of each higher order scattering to the radiance field is examined but no results are presented.
A method of the approximated solution of the problem of accretion on a rapidly moving gravitational center is developed. This solution is obtained in the vicinity of the axis of symmetry in the region of the potential flow. The solution of the problem of stationary gas accretion on a moving gravitational center simulates the movement of a substance in interstellar space in the vicinity of a black hole. A detailed picture of gas accretion on a black hole is of interest in connection with the problem of observation of black holes.
A decoupling and pole-placement technique has been developed for the Annular Suspension and Pointing System (ASPS) of the Space Shuttle which uses bandwidths as performance criteria. The dynamics of the continuous-data ASPS allows the three degrees of freedom to be totally decoupled by state feedback through constant gains, so that the bandwidth of each degree of freedom can be independently specified without interaction. Although it is found that the digital ASPS cannot be completely decoupled, the bandwidth requirements are satisfied by pole placement and a trial-and-error method based on approximate decoupling.
Investigation of the origins of living things by the uniquely appropriate method of successive approximation in attempted retracement of steps in molecular evolution has yielded: a comprehensive theoretical flowsheet from archaic inanimate matter to an infrastructured, microscopic, protoreproductive, protometabolic protocell; a laboratory model of the same; and an explanatory assessment of the natural variation component of Darwinian evolution. For each of these, the significance is dependent upon awareness of the intrinsic tendency of amino acids, in mixed sets, to order themselves. Without such awareness, it is believed these vistas would have been delayed for decades. Selfordering would have in turn been difficult to recognize and support were it not for the chromatographic developments in fractionation of copolyamino acids.