Simulations of the anisotropic kinetic and magnetic alpha effects

Using simulations of isotropically forced helical turbulence the contributions to kinetic and magnetic alpha effects are computed. It is shown that for the parameter regimes considered in an earlier publication (Brandenburg & Subramanian 2005), the expressions for isotropic and anisotropic alpha effects give quantitatively similar results. Both kinetic and magnetic alpha effects are proportional to a relaxation time whose value, in units of the turnover time, is shown to be approximately unity and independent of the magnetic Reynolds number. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Monoenergetic-source solutions of the steady-state cosmic-ray equation of transport

A non spherically-symmetric monoenergetic-point-source solution of the steady-state equation of transport for cosmic-rays in the interplanetary region, in which monoenergetic particles are released isotropically and continuously from a fixed heliocentric position is derived by a Laplace transform method. The solution is for a spherically-symmetric model of the propagating region incorporating anisotropic diffusion, with a diffusion tensor symmetric about the radial direction, and the solar wind velocity is radial and of constant speedV. The spherically-symmetric monoenergeticsource solution of Webb and Gleeson (1973) and of Toptygin (1973) is regained from the spherically-symmetric component of the point-source solution.     

Application of the three-dimensional telegraph equation to cosmic-ray transport

An analytical solution to the three-dimensional telegraph equation is presented.This equation has recently received some attention but so far the treatment has been one-dimensional.By using the structural similarity to the Klein-Gordon equation,the telegraph equation can be solved in closed form.Illustrative examples are used to discuss the qualitative differences from the diffusion solution.A comparison with a numerical test-particle simulation reveals that some features of an intensity profile can be better explained using the telegraph approach.     

The unusual anisotropic solar particle event of November 18, 1968

The ground level event of November 18, 1968, was unusual in that pronounced anisotropy persisted essentially until the cessation of the arrival at the earth of relativistic protons. Real and significant differences were observed in the intensity profiles at different neutron monitor stations. By a method of successive approximations, the geographical coordinates of the axis of symmetry were determined as: latitude = - 25° ± 10° and longitude = -35° ± 10°. The exponent in a power law spectral representation over the narrow range covered by the nucleonic intensity measurements was 5.The onset times of the GLE were related to the angular differences between the mean directions of viewing and the axis of symmetry. The event differed markedly from earlier isotropic events that were well described by diffusion theory, but was compatible with the predictions of a model envisaging diffusion only in the vicinity of the Sun (radius R of solar envelope 34 solar radii). The diffusion coefficient D 3 × 10²¹ cm²/s is consistent with that deduced for earlier isotropic events, but the values of other parameters are smaller than those derived for the only previously observed event to which the model of diffusion in an inhomogeneous medium was applicable (May 4, 1960).It is concluded that the observed one hundred percent anisotropy was a direct consequence of the propagation mechanism rather than other postulated causes. Furthermore, the observed large pitchangle distribution was produced by scattering from small scale field irregularities.This research was sponsored by the National Science Foundation and Air Force Cambridge Research Laboratories, Office of Aerospace Research, under Contract No. F19628-70-C-0190, but the report does not necessarily reflect endorsement by the sponsor.     

Time-dependent Green's functions of the cosmic ray equation of transport

Two spherically symmetric time-dependent Green's functions of the equation of transport for cosmic rays in the interplanetary region are derived by transform techniques. The solar wind velocity is assumed radial and of constant speedV. In the first model the radial diffusion coefficient =₀ r (₀ constant), and in the second solution =₀= constant. The solutions are for monoenergetic, impulsive release of particles from a fixed heliocentric radius. Integration of the solutions over timet, fromt=0 tot=, gives the steady-state Green's functions obtained previously.     

Solutions of the spherically-symmetric cosmic-ray transport equation in interplanetary space

Numerical solutions of the well-known spherically-symmetric transport equation for interstellar cosmic rays in the interplanetary medium are presented. It is shown: (a) why nuclei and electron modulations differ, (b) that it is easy to read more in the model than it contains, (c) that the only significant parameter that determines the level of modulation is the product of the solar wind speed and the number of diffusion mean free paths between observer and boundary, and (d) that observations do exist which cannot be explained in terms of this simple model.     

On the particle number conservation inherent in the diffusion approach of the Fokker-Planck equation

Starting from the Fokker-Planck equation we show that, in a strict sense, the usual diffusive transport of charged particles in weak magnetic field fluctuations is not valid until the space-integrated number of particles per velocity interval has reached its final overall constant value and is conserved. Large anisotropies are thus not compatible with diffusion. Diffusion becomes exactly valid after an infinite time, but has reached a good approximation to the real transport before that. The particle-number conservation holds for quite general mean magnetic field configurations, and is not limited to the case of a constant mean field.     

On the equation of transport for cosmic-ray particles in the interplanetary region

In this paper we provide two new alternative derivations of the equation of transport for cosmic-ray particles in the interplanetary region. Both derivations are carried out by using particle positionr and timet in a frame of reference fixed in the solar system, and the particle momentump is specified relative to a local frame of reference moving with the solar wind. The first derivation is carried out by writing down a continuity equation for the cosmic rays, taking into account particle streaming and energy changes, and subsequently deriving the streaming and energy change terms in this equation. The momentum change term in the continuity equation, previously considered to be due to the adiabatic deceleration of particles in the expanding magnetic fields carried by the solar wind, appears in the present analysis as a dynamic effect in which the Lorentz force on the particle does not appear explicitly. An alternative derivation based on the ensemble averaged Liouville equation for charged particles in the stochastic interplanetary magnetic field using (r, p,t) as independent coordinates is also given. The latter derivation confirms the momentum change interpretation of the first derivation. We also provide a new derivation of the adiabatic rate as a combination of inverse-Fermi and betatron deceleration processes.     

Exact solution of transport equation in finite multiplying media

One speed neutron transport equation in a finite multiplying medium is exactly and uniquely solved following the first author's new technique (Das Gupta, 1978b) based on Laplace transformation and Wiener-Hopf technique, to obtain the angular neutron density at any depth. Criticality condition is derived.     

Separation of the steady-state cosmic ray equation of transport: A generalization

The one-dimensional, steady-state equation of transport for cosmic rays including convection, diffusion and adiabatic deceleration is separated for a spatial diffusion coefficient with an arbitrary momentum dependence and for an arbitrary spatial dependence of the convection velocityV, and applies for planar, cylindrical and spherical geometries. As an application, the previously obtained spherically symmetric steady-state Green's functions, describing the propagation of cosmic rays in interplanetary space, are generalized to the case where the convection velocity is a function of position.     

Green's theorem and Green's functions for the steady-state cosmic-ray equation of transport

Green's Theorem is developed for the spherically-symmetric steady-state cosmic-ray equation of transport in interplanetary space. By means of it the momentum distribution functionF _o(r,p), (r=heliocentric distance,p=momentum) can be determined in a regionr _arr_bwhen a source is specified throughout the region and the momentum spectrum is specified on the boundaries atr _a andr _b . Evaluation requires a knowledge of the Green's function which corresponds to the solution for monoenergetic particles released at heliocentric radiusr _o , Examples of Green's functions are given for the caser _a =0,r _b = and derived for the cases of finiter _a andr _b . The diffusion coefficient is assumed of the form = _o(p)r ^b . The treatment systematizes the development of all analytic solutions for steady-state solar and galactic cosmic-ray propagation and previous solutions form a subset of the present solutions.     

On the role of injection in kinetic approaches to non-linear particle acceleration at non-relativistic shock waves

The dynamical reaction of the particles accelerated at a shock front by the first-order Fermi process can be determined within kinetic models that account for both the hydrodynamics of the shocked fluid and the transport of the accelerated particles. These models predict the appearance of multiple solutions, all physically allowed. We discuss here the role of injection in selecting the real solution, in the framework of a simple phenomenological recipe, which is a variation of what is sometimes referred to as thermal leakage. In this context we show that multiple solutions basically disappear and when they are present they are limited to rather peculiar values of the parameters. We also provide a quantitative calculation of the efficiency of particle acceleration at cosmic ray modified shocks and we identify the fraction of energy which is advected downstream and that of particles escaping the system from upstream infinity at the maximum momentum. The consequences of efficient particle acceleration for shock heating are also discussed.     

Neutron star models based on an improved equation of state

Using an equation of state for cold degenerate matter which takes nuclear forces and nuclear clustering into account, neutron star models are constructed. Stable models were obtained in the mass range above 0.065M and density range 10^14.08 to 10^15.4 gm/cm³. All of these models were found to be bound. The outer crystalline layer of the star was found to have a thickness of 200 m or more depending on the mass of the model.     

On the energy spectrum of galactic primary cosmic rays: Results from the transport equation

The spectrum of galactic primary cosmic rays at relativistic monenta is calculated. The primaries are assumed to be accelerated continuously from the thermal galactic background medium by first- and second-order Fermi acceleration. We show that the observed spectrum is readily obtained from the transport equation conventionally invoked to discuss propagation and loss of cosmic rays in our Galaxy from a distribution of sources. We have previously (Lerche and Schlickeiser, 1985) shown that the observed secondary to primary ratio is satisfactorily explained by a similar use of the transport equation, allowing for secondary production from the primaries. Accordingly, when the results of this paper are added to those concerning the secondary/primary ratio behaviour, it would seem that continuous Fermi acceleration accounts, in a quantitative fashion, for the spectral behaviours observed at Earth.     

基于CPLD的高精度可调脉冲信号发生器研制

为满足精密时间间隔测量设备的测试需要,研制了一种时间间隔可调的高精度脉冲信号发生器。利用计算机串口控制的方式,结合复杂可编程逻辑器件（CPLD）集成度高、可靠性好及工作速度快的优点,采用A1tera公司的设计软件QuartusII进行设计仿真及实现。仿真与实测实验表明,该脉冲信号发生器不仅可以产生单路可调脉冲信号,而且能产生多路可调脉冲信号,产生的单路秒脉冲信号的1s取样Allan方差为1．84×10^-11;产生的时间间隔为100ns的多路脉冲信号的1s取样Allan方差为2．36×10^-11,2路信号之间的时间间隔数据系列的峰一峰值为101ps,可以满足多通道时间间隔测量设备测试要求的稳定度与准确度。     

Exact solution of the transport equation for imperfect rayleigh scattering in a semi-infinite atmosphere

By performing the one-sided Laplace transform on the matrix integro-differential equation for a semi-infinite plane parallel imperfect Rayleigh scattering atmosphere we derive an integral equation for the emergent intensity matrix. Application of the Wiener-Hopf technique to this integral equation will give the emergent intensity matrix in terms of singularH-matrix and an unknown matrix. The unknown matrix has been determined considering the boundary condition at infinity to be identical with the asymptotic solution for the intensity matrix.     

Radiation transport in a cylinder. I. Resolvent of the fundamental integral equation

A new method is presented for calculating spectral line radiation transport in a homogeneous circular cylinder with infinitely long axis for isotropic scattering with complete frequency redistribution or no change in frequency. In addition to the cylinder, scattering in a layer symmetric about its midpoint and in a sphere are also studied. Equations are derived which are satisfied by the resolvent of the fundamental integral equation describing these cases of scattering. By applying a Hankel transform over a finite interval, the resolvent is expressed in terms of an auxiliary function. An equation is derived for this auxiliary function which is easily soluble by iteration, with the speed of its convergence increasing with growth in the radius of the scattering region.Translated from Astrofizika, Vol. 37, No. 1, pp. 111–127, January–March, 1994.This study was carried out with financial support from the Russian Fund for Fundamental Studies (project 93-02-2957).