Velocity correlation functions of charged particles derived from the Fokker–Planck equation

An important ingredient in theories for diffusion of charged particles across a mean magnetic field are velocity correlation functions along and across that field. In the current article we present an analytical study of these functions by investigating the two-dimensional Fokker–Planck equation. We show that for an isotropic pitch-angle Fokker–Planck coefficient, the parallel velocity correlation function is an exponential function in agreement with the standard model used previously. For other forms of the pitch-angle diffusion coefficient, however, we find non-exponential forms. Also a new, velocity correlation function based, approach for deriving the so-called Earl-relation is presented. This new derivation is more systematic and simpler than previous derivations. We also discuss higher-order velocity correlations and the applicability of the quasi-normal hypothesis in particle diffusion theory. Furthermore, we compute velocity correlation functions across the mean field and develop an alternative theory for perpendicular diffusion.     

Dust in magnetised plasmas: Basic theory and some applications

In this paper we review the theory of charged test particle motion in magnetic fields. This theory is then extended to charged $\text{dust}$ particles, for which gravity and charge fluctuations play an important role. It is shown that systematic drifts perpendicular to the magnetic field and stochastic transport effects may then have to be considered - none of which occur in the case of atomic particles (with the exception of charge exchange reactions). Some applications of charged dust particle transport theory to planetary rings are then briefly discussed.     

Numerical investigation of the influence of large turbulence scales on the parallel and perpendicular transport of cosmic rays

In recent analytical investigations it has been demonstrated that the turbulence behavior at large scales has a very strong influence on the perpendicular diffusion coefficient of charged particles. In the present paper we use computer simulations to investigate numerically cross field transport and particle propagation along the mean magnetic field for different turbulence models at large scales. Our results are compared with quasilinear theory and nonlinear diffusion theories. We show that for different forms of the turbulence spectrum at large scales, the perpendicular mean free paths obtained numerically are in agreement with recent predictions made by analytical theory. It is also shown that the parallel diffusion coefficient contains always a strong nonlinear contribution which is, however, independent of the assumed spectrum at large scales.     

On the effects of dynamical turbulence on the perpendicular diffusion of low-energy cosmic ray electrons

Diffusion perpendicular to the heliospheric magnetic field plays an integral role in the transport of charged particles in the heliosphere. In this study the perpendicular diffusion coefficient of low-energy cosmic ray electrons is calculated, using an equation derived from the random ballistic decorrelation interpretation of nonlinear guiding centre theory. An observationally motivated 2D turbulence power spectrum is assumed and the effects of various turbulence inputs on the resulting perpendicular diffusion coefficient are investigated. The perpendicular diffusion coefficients are first determined at 1 AU, for both magnetostatic and dynamical turbulence conditions. These solutions are also evaluated for radial distances of 0.1 AU to 10 AU to further investigate the values of the perpendicular diffusion coefficients in the very inner heliosphere. The results of this study show that the dissipation range of the turbulence power spectrum provides a negligible contribution towards the perpendicular diffusion coefficient, and that solutions derived using only the energy containing range serve as good approximations for solutions derived assuming the full 2D turbulence power spectrum. Finally, it is shown that the effects of dynamical turbulence, as considered in the present study, do not affect the perpendicular diffusion coefficients derived from the scattering theory considered here.     

Simulation study of motion of charged particles trapped in Earth’s magnetosphere

This article aims to understand the motion of the charged particles trapped in the Earth’s inner magnetosphere. The emphasis is on identifying the numerical scheme, which is appropriate to characterize the trajectories of the charged particles of different energies that enter the Earth’s magnetosphere and get trap along the magnetic field lines. These particles perform three different periodic motions, namely: gyration, bounce, and azimuthal drift. However, often, the gyration of the particle is ignored, and only the guiding center of the particle is traced to reduce the computational time. It is because the simulation of all three motions (gyro, bounce, and drift) together needed a robust numerical scheme, which has less numerical dissipation. We have developed a three-dimensional test particle simulation model in which the relativistic equation of motion is solved numerically using the fourth and sixth-order Runge-Kutta methods. The stability of the simulation model is verified by checking the conservation of total kinetic energy and adiabatic invariants linked with each type of motion. We found that the sixth-order Runge-Kutta method is suitable to trace the complete trajectories of both proton and electron of a wide energy range, 5 keV to 250 MeV for L = 2 – 6. We have estimated the bounce and drift periods from the simulations, and they are found to be in good agreement with the theory. The study implies that a simulation model with sixth-order Runge-Kutta method can be applied to the time-vary, non-analytical form of magnetic configuration in future studies to understand the dynamics of charged particles trapped in Earth’s magnetosphere.     

Comparison of the effects of two models for perpendicular diffusion on cosmic-ray latitudinal gradients

We compare the effects of two different models for perpendicular diffusion on the latitudinal gradients of galactic cosmic ray protons during solar minimum conditions. These two models correspond to the newly developed non-linear guiding center theory [Matthaeus, W.H., Qin, G., Bieber, J.W., Zank, G.P. Nonlinear collisionless perpendicular diffusion of charged particles. Astrophys. J. Lett., 590 (1), L53–L56, 2003] and the theory based on a velocity correlation function approach [Bieber, J.W., Matthaeus, W.H. Perpendicular diffusion and drift at intermediate cosmic-ray energies. Astrophys. J., 485 (2) 655–659, 1997]. In this ab initio study a steady-state two-dimensional numerical modulation model is used which incorporates a state-of-the-art turbulence model. We show that the non-linear guiding center theory predicts a mean free path that has a rigidity dependence that better accounts for the latitudinal gradients measured by Ulysses during its first fast latitude scan in 1994/1995.     

磁层高能粒子运动的Fokker Planck方程位置项扩散系数研究

利用准线性理论计算了磁层高能粒子运动的Fokker Planck方程在可观测相空间的位置项扩散系数,并与绝热不变量径向扩散系数进行对比分析.研究发现:位置项扩散系数随径向距离呈现R6的比例关系快速增大.相同径向距离条件下,由于空间位置项z分量的作用,高纬度地区的位置项扩散系数小于低纬度地区.通过与径向扩散系数对比发现,两者具有相同的量级,但两者的相对大小需要根据具体的扰动形态进行分析.此研究对使用测试粒子模拟磁层高能粒子运动,尤其是根据引导中心理论,利用蒙特卡洛方法求解磁层高能粒子运动的Fokker Planck方程,建立磁层空间高能粒子运动的精细化模型具有重要意义.      

Cosmic-ray diffusion in the inner heliosphere

For about the last 40 years, we have been trying to understand the propagation of cosmic rays and other energetic charged particles through the interplanetary medium. Identification of the basic processes affecting the propagation, namely diffusion, convection by the solar wind, adiabatic deceleration, and gradient and curvature drifts, was attained early on, but reaching detailed physical understanding, particularly of the roles of diffusion and gradient and curvature drifts, continues as an active topic of research to this day. Particularly unclear is the nature of the cross-field propagation. Many observations seem to require more efficient cross-field propagation than theoretical propagation models can easily produce. At the same time, there are other observations that seem to show strong guidance of the particles by the interplanetary magnetic field. With current measurements from spacecraft near Earth and from the Ulysses spacecraft, which samples nearly the complete range of heliographic latitudes in the inner heliosphere, critical tests of the ways in which cosmic rays and other energetic charged particles propagate through the interplanetary medium are possible. I briefly review the status of observations that are relevant to the characterization of diffusive propagation in the inner heliosphere and will present evidence for a possibly previously overlooked contribution from transport along magnetic flux tubes that deviate dramatically from the average interplanetary spiral configuration.     

A method for operative calculation of charged particle penetration into the LEO

We develop a simple method for calculating the effective vertical cutoff rigidity of charged particles, taking into account the Kp-index and the local time, on the basis of generalization of the results of extensive trajectory calculations for trial particles moving in the geomagnetic field. The vertical cutoff rigidities, calculated by the Tsyganenko-89 model, are presented as an International Geomagnetic Reference Field (IGRF) model calculated and thereafter corrected in accordance with the geomagnetic disturbance and local time conditions. The fits from the proposed method agree with the results of cutoff rigidity measurements carried out by satellites. The method is intended for applications using cutoff calculations, such as evaluating particle penetration of spatial boundaries, calculating magnetospheric transmissions for low-orbital spacecrafts flights and interpreting the results of orbital experiments.     

带电粒子在近地球区运动的计算方法

基于磁层粒子动力学理论,根据单粒子方法和偶极磁场模型,在近地球区(L＜10),详细讨论了带电粒子的运动特征以及适合不同能量的带电粒子的计算方法,定量分析了磁层中的电场和磁场对各种能量的带电粒子漂移运动的影响.结果表明,对于能量低于105 eV的电子和102 eV的质子,宜采用引导中心近似方法;对于能量介于105～108 eV之间的电子和102～108 eV之间的质子,可在部分区域内采用引导中心近似方法,若运动区在10 Re内,则只能采用变步长的轨道法;而对于引导中心方法,粒子能量高于105 eV时可以忽略电场漂移的影响,粒子能量低于103 eV时不必考虑磁场漂移的影响,从而简化了引导中心方程组,提高了数值计算效率.      

Self-similar flow of a rotating dusty gas behind the shock wave with increasing energy, conduction and radiation heat flux

A self-similar solution is obtained for one dimensional adiabatic flow behind a cylindrical shock wave propagating in a rotating dusty gas in presence of heat conduction and radiation heat flux with increasing energy. The dusty gas is assumed to be a mixture of non-ideal (or perfect) gas and small solid particles, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-condition is maintained and variable energy input is continuously supplied by the piston (or inner expanding surface). The heat conduction is expressed in terms of Fourier’s law and the radiation is considered to be of the diffusion type for an optically thick grey gas model. The thermal conductivity K and the absorption coefficient α_R are assumed to vary with temperature only. In order to obtain the similarity solutions the initial density of the ambient medium is assumed to be constant and the angular velocity of the ambient medium is assumed to be decreasing as the distance from the axis increases. The effects of the variation of the heat transfer parameters and non-idealness of the gas in the mixture are investigated. The effects of an increase in (i) the mass concentration of solid particles in the mixture and (ii) the ratio of the density of solid particles to the initial density of the gas on the flow variables are also investigated.     

三维试验粒子轨道法在磁层粒子全球输运中的应用

根据磁层粒子动力学理论, 通过偶极磁场模型验证利用三维试验粒子轨道方法模拟近地球区(r < 8R_e)带电粒子运动特征的可靠性. 在此基础上, 以太阳风和磁层相互作用的全球MHD模拟结果为背景, 利用三维试验粒子轨道方法, 对非磁暴期间南向行星际磁场背景下太阳风离子注入磁层的情形进行数值模拟, 并对北向行星际磁场背景下太阳风离子注入极尖区以及内磁层的几种不同情形进行了单粒子模拟. 模拟结果反映了南向和北向行星际磁场离子向磁层的几种典型输入过程, 揭示出行星际磁场南向时太阳风粒子在磁层内密度分布的晨昏不对称性以及其在磁鞘和磁层内的大致分布, 并得出统计规律. 模拟结果与理论预测和观测结论相一致, 且通过数值模拟发现, 行星际磁场北向时靠近极尖区附近形成的非典型磁镜结构对于能量粒子经由极尖区注入环电流区域过程有重要的影响和作用.      

On the universality of asymptotic limits in the theory of field line diffusion and perpendicular transport of energetic particles

We discuss the random walk of magnetic field lines in astrophysical plasmas. Based on the standard theory of field line diffusion we show that there are two asymptotic limits. In these limits field line wandering is universal because in both regimes the field line diffusion coefficient depends only on fundamental length scales and absolute magnetic field strengths. As examples we discuss the field line diffusion coefficient for different prominent turbulence models namely the slab model, the two-dimensional model, and the Goldreich–Sridhar model. We show that the field line diffusion coefficient for the latter model agrees with the results obtained for slab and two-dimensional turbulence in limiting cases. We also discuss the transport of energetic particles perpendicular with respect to the mean magnetic field. Based on the unified nonlinear transport theory we consider again asymptotic limits. It is shown that one can identify four different regimes in which the transport is again universal. In all four cases perpendicular transport only depends on fundamental length scales of turbulence, magnetic field values, and the parallel diffusion coefficient.     

真空和微重力下烟幕粒子动力学特性研究

为了研究红外烟幕在外层空间的适用性,基于外层空间环境高真空和微重力的特殊性,分析烟幕粒子的真空和微重力效应,采用分子运动论理论近似模拟真空和微重力下烟幕粒子的动力学特性,建立烟幕粒子间距函数模型,利用Mathematica方法计算了烟幕材料粒径、密度及粒子运动时间对粒子间距大小的影响.结果表明,随着烟幕材料粒径及密度的增大,粒子间距逐渐减小,形成烟幕的稳定性逐渐增大;粒子运动时间愈长烟幕愈不稳定.      

Parallel,adaptive, multi-object trajectory integrator for space simulation applications

Computer simulation is a very helpful approach for improving results from space born experiments. Initial-value problems (IVPs) can be applied for modeling dynamics of different objects – artificial Earth satellites, charged particles in magnetic and electric fields, charged or non-charged dust particles, space debris. An ordinary differential equations systems (ODESs) integrator based on applying different order embedded Runge–Kutta–Fehlberg methods is developed. These methods enable evaluation of the local error. Instead of step-size control based on local error evaluation, an optimal integration method is selected. Integration while meeting the required local error proceeds with constant-sized steps. This optimal scheme selection reduces the amount of calculation needed for solving the IVPs. In addition, for an implementation on a multi core processor and parallelization based on threads application, we describe how to solve multiple systems of IVPs efficiently in parallel.     

Preliminary validation of computational procedures for a new atmospheric ionizing radiation (AIR) model.

A new computational procedure to determine particle fluxes in the Earth's atmosphere is presented. The primary cosmic ray spectrum has been modeled through an analysis of simultaneous proton and helium measurements made on high altitude balloon flights and spacecraft. An improved global fit to the data was achieved through applying a unique technique utilizing the Fokker-Plank equation with a non-linear rigidity-dependent diffusion coefficient. The propagation of primary particles through the Earth's atmosphere is calculated with a three-dimensional Monte Carlo transport program called FLUKA. Primary protons and helium nuclei (alphas) are generated within the rigidity range of 0.5 GV-20 TV uniform in cos2 theta. For a given location, primaries above the effective cutoff rigidity are transported through the atmosphere. Alpha particles are initially transported with a separate package called HEAVY to simulate fragmentation. This package interfaces with FLUKA to provide interaction starting points for each nucleon originating from a helium nucleus. Results from this calculation are presented and compared to measurements.     

Charged-particle mutagenesis II. Mutagenic effects of high energy charged particles in normal human fibroblasts.

The biological effects of high LET charged particles are a subject of great concern with regard to the prediction of radiation risk in space. In this report, mutagenic effects of high LET charged particles are quantitatively measured using primary cultures of human skin fibroblasts, and the spectrum of induced mutations are analyzed. The LET of the charged particles ranged from 25 KeV/micrometer to 975 KeV/micrometer with particle energy (on the cells) between 94-603 MeV/u. The X-chromosome linked hypoxanthine guanine phosphoribosyl transferase (hprt) locus was used as the target gene. Exposure to these high LET charged particles resulted in exponential survival curves; whereas, mutation induction was fitted by a linear model. The Relative Biological Effect (RBE) for cell-killing ranged from 3.73 to 1.25, while that for mutant induction ranged from 5.74 to 0.48. Maximum RBE values were obtained at the LET of 150 keV/micrometer. The inactivation cross-section (alpha i) and the action cross-section for mutant induction (alpha m) ranged from 2.2 to 92.0 micrometer2 and 0.09 to 5.56 x 10(-3) micrometer2, respectively. The maximum values were obtained by 56Fe with an LET of 200 keV/micrometer. The mutagenicity (alpha m/alpha i) ranged from 2.05 to 7.99 x 10(-5) with the maximum value at 150 keV/micrometer. Furthermore, molecular analysis of mutants induced by charged particles indicates that higher LET beams are more likely to cause larger deletions in the hprt locus.     

Measurements of the radiation quality factor Q at aviation altitudes during solar minimum (2006–2008)

In radiation protection, the Q-factor has been defined to describe the biological effectiveness of the energy deposition or absorbed dose to humans in the mixed radiation fields at aviation altitudes. This particular radiation field is generated by the interactions of primary cosmic particles with the atoms of the constituents of the Earth’s atmosphere. Thus the intensity, characterized by the ambient dose equivalent rate H∗(10), depends on the flight altitude and the energy spectra of the particles, mainly protons and alpha particles, impinging on the atmosphere. These charged cosmic projectiles are deflected both by the interplanetary and the Earth’s magnetic field such that the corresponding energy spectra are modulated by these fields. The solar minimum is a time period of particular interest since the interplanetary magnetic field is weakest within the 11-year solar cycle and the dose rates at aviation altitudes reach their maximum due to the reduced shielding of galactic cosmic radiation. For this reason, the German Aerospace Center (DLR) performed repeated dosimetric on-board measurements in cooperation with several German airlines during the past solar minimum from March 2006 to August 2008. The Q-factors measured with a TEPC range from 1.98 at the equator to 2.60 in the polar region.