Transient behavior of a phase-locked loop in the presence of noise
Transient behavior of first order phase locked loop in presence of noise, solving Fokker-Planck equation for loop dynamics by numerical integration
Dominiak, K. E.↗
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Electron loss from the Van Allen zones due to pitch angle scattering by electromagnetic disturbances
Fokker-Planck equation used to describe process of electron loss from Van Allen zones due to pitch angle scattering by electromagnetic disturbances
Roberts, C. S.↗
Nonlinear analysis of an absolute value type of an early-late gate bit synchronizer
Absolute value type early-late gate bit synchronizer steady state phase noise performance evaluation by Fokker-Planck method
Simon, M. K.↗
Approximations in the theory of solar-cycle modulation
Force field model for galactic cosmic rays modulation by solar cycles, noting agreement with Fokker-Planck equation analytic and numerical solutions
Axford, W. I.↗
Convection-dominated transport of solar cosmic rays
Fokker-Planck equation for convection dominated transport of solar cosmic rays solved for exponential decay phase of solar particle events
Forman, M. A.↗
Transient analysis of nonlinear tracking systems
Fokker-Planck boundary value solutions to transient phase error response of nonlinear phase locked tracking systems
La Frieda, J. R.↗
Particle flux associated with stochastic processes
Stochastic scattering process charged particle flux by Lagrange expansion based on Fokker- Planck equation
Newman, C. E.↗
Solar flare particle propagation: Comparision of a new analytic solution with spacecraft measurements
An analytic solution was obtained to the complete Fokker-Planck equation for solar flare particle propagation including the effects of convection, energy-change, corotation, and diffusion. It is assumed that the particles are injected impulsively at a single point in space, and that a boundary exists beyond which the particles are free to escape. Several solar flare particle events were observed with solar and galactic cosmic ray experiment aboard OGO 6. Detailed comparisons of the predictions of the solution with observations of 1 to 70 MeV protons show that the model adequately describes both the rise and decay times. The solution also yields a time evolution for the vector anisotropy which agrees well with reported observations.
Lupton, J. E.↗
Transport of solar flare protons: Comparison of a new analytic model with spacecraft measurements
An analytic solution was obtained to the complete Fokker-Planck equation including the effects of convection, interplanetary deceleration and acceleration, corotation, and anisotropic diffusion. The solution yields a time to maximum for the particle flux of about 10 h and an exponential decay time of about 5 h. Several solar flare particle events were observed with the solar and galactic cosmic ray experiment on OGO-6. Comparisons of the calculated time dependence of the fluxes with these observations of 1 to 70 MeV protons show that the model adequately describes both the rise and decay times.
Lupton, J. E.↗
Modulation of low energy cosmic rays
The power spectrum of the interplanetary magnetic field were tested, as well as the power spectrum-cosmic diffusion ray coefficient relation at low energies. A first order perturbation solution of the Fokker-Planck equation governing the diffusion, convection, and adiabatic deceleration of galactic cosmic rays in the solar medium was found to relate intensity fluctuations of low energy cosmic rays to local changes in the propagation parameters. Diffusion coefficients and their day to day variations were calculated from interplanetary magnetic field data obtained by Pioneer 6 in 1965/1966. These are compared to simultaneous observations by IMP 3 of the proton flux in three energy channels (20 to 40, 40 to 60, 60 to 80 MeV).
Sari, J. W.↗
Kinetic theory of a two-dimensional magnetized plasma.
Several features of the equilibrium and nonequilibrium statistical mechanics of a two-dimensional plasma in a uniform dc magnetic field are investigated. The charges are assumed to interact only through electrostatic potentials. The problem is considered both with and without the guiding-center approximation. With the guiding-center approximation, an appropriate Liouville equation and BBGKY hierarchy predict no approach to thermal equilibrium for the spatially uniform case. For the spatially nonuniform situation, a guiding-center Vlasov equation is discussed and solved in special cases. For the nonequilibrium, nonguiding-center case, a Boltzmann equation, and a Fokker-Planck equation are derived in the appropriate limits. The latter is more tractable than the former, and can be shown to obey conservation laws and an H-theorem, but contains a divergent integral which must be cut off on physical grounds. Several unsolved problems are posed.
Vahala, G.↗
Brownian motion from Boltzmann's equation.
Two apparently disparate lines of inquiry in kinetic theory are shown to be equivalent: (1) Brownian motion as treated by the (stochastic) Langevin equation and Fokker-Planck equation; and (2) Boltzmann's equation. The method is to derive the kinetic equation for Brownian motion from the Boltzmann equation for a two-component neutral gas by a simultaneous expansion in the density and mass ratios.
Montgomery, D.↗
Shock tube technique for measuring opacities at high pressures for gaseous-core nuclear rockets.
A new method of opacity determination is described that uses an improved shock tube design and is applicable to measurements of Planck and Rosseland mean opacities in hydrogen or seeded hydrogen at hydrogen partial pressures of 500 atm, at temperatures from 2300 to 8000 K. The selection of seed materials to be added to the gases to make them semiopaque is discussed. Currently favored seeds are silicon and carbon or graphite for reactive seeds, and depleted uranium and tungsten 184 for nonreactive seeds.
Patch, R. W.↗
Stratospheric thickness determined directly from satellite radiance measurements.
Discussion of the use of satellite radiance data for determining the thickness of deep stratospheric layers. Empirical regression equations are shown to provide better estimates of stratospheric thickness than do mean weighted temperatures obtained from the Planck equation. The best regression equations were found for thick layers emitting a substantial portion of the CO2-band infrared radiation measured by satellites. By adding the layer thickness to the observed height field for the lower boundary, it is possible to construct constant-pressure maps at very high altitudes.
Quiroz, R. S.↗
Random deflections of a string on an elastic foundation.
The paper is concerned with the problem of a taut string on a random elastic foundation subjected to random loads. The boundary value problem is transformed into an initial value problem by the method of invariant imbedding. Fokker-Planck equations for the random initial value problem are formulated and solved in some special cases. The analysis leads to a complete characterization of the random deflection function.
Sanders, J. L., Jr.↗
Determining optical depth in different parts of the Fraunhofer line contour
The method of approximate determination of the optical depth tau in the profile of the Fraunhofer line is considered. In the method the observed profile and allocation of tau to the temperature in the photosphere must be known. If the source function B prime (tau) in the line seems not to be equal to the Planck function, the dependence B prime (tau) on tau is needed. The method for the determination of B prime (tau) using the center-to-limb observations of the central intensities of the line is suggested.
Gurtovenko, E. A.↗
An improved separability approximation for line radiative transport in nonhomogeneous media.
A simple modification to the constant half-width approximation of Wilson and Greif, which is an extension of the well-known Curtis-Godson method to the treatment of temperature variations along the integration path, is introduced which permits a more accurate evaluation of line radiative transport in nonhomogeneous gases. To demonstrate the method's accuracy, comparisons are made with Wilson and Greif and numerical frequency integrated results for the line equivalent width and radiative flux in a planar slab with prescribed Planck function and line half-width spatial variations. These variations are chosen to represent typical shock layer conditions. It is found that the modified procedure reduces the inaccuracies inherent in Wilson and Greif's approximation by factors ranging from 5 to 10, while retaining the latter method's ease of application.
Boughner, R. E.↗
A thick-thin approximation for radiative transfer.
First- and second-order formulations of a thick-thin approximate technique for application to the solution of energy transport when nongray radiation is important are presented. The virtue of the technique is that it eliminates the cumbersome integral terms which usually appear. It is shown that the temperature distributions calculated with the technique agree well with the results of more rigorous calculations if a proper value of the critical optical depth, which separates the thick and thin regions of the spectrum, is used. The proper value of the critical optical depth is less than one and may be less than 0.5. It is also shown that the jump boundary conditions which are frequently employed with the Rosseland, or diffusion, approximation should incorporate the Planck mean absorption coefficient for a nongray gas. This is in contradiction to suggestions appearing in the literature.
Olstad, W. B.↗