Pitch-angle Fokker-Planck Coefficient Research Articles

Solutions of the cosmic ray velocity diffusion equation

In order to describe the propagation and acceleration of cosmic rays, one usually uses a diffusive transport equation. The most fundamental equation is the pitch-angle dependent diffusion equation which is usually called the Fokker-Planck equation. In the current paper we solve the position integrated equation numerically and analytically. For a constant pitch-angle Fokker-Planck coefficient we derive an exact solution of the corresponding transport equation and compare it with numerical solutions. We show that even if the scattering coefficient is assumed to be constant, the solution behaves well. As a second example we consider the case of a linear pitch-angle Fokker-Planck coefficient. Again we solve the corresponding transport equation numerically and analytically. In all cases considered, we find similar distribution functions. We also compute the corresponding velocity correlation functions and parallel diffusion coefficients. Our results are relevant for improving analytical theories of perpendicular diffusion, for code tests, and for different astrophysical applications where a pitch-angle dependent description of the particle motion is required.

Numerical analysis of the Fokker-Planck equation with adiabatic focusing: Realistic pitch-angle scattering

We solve the focused transport equation of cosmic rays numerically to investigate non-isotropic models of the pitch-angle scattering coefficient. In previous work, the Fokker-Planck equation was solved either analytically by using approximations, or by using a numerical approach together with simple models for the pitch-angle scattering coefficient. It is the purpose of the current article so compute particle distribution functions as well as the parallel diffusion coefficient by solving numerically the focused transport equation for a more realistic Fokker-Planck coefficient of pitch-angle scattering. Our analytical form for the scattering parameter is based on non-linear diffusion theory that takes into account realistic scattering at pitch-angles close to 90°. This general form contains the isotropic form as well as the quasi-linear limit as special cases. We show that the ratio of the diffusion coefficients with and without focusing sensitively depends on the ratio of the turbulent magnetic field and the mean field. The assumed form of the pitch-angle Fokker-Planck coefficient has an influence on the parallel diffusion coefficient. In all considered cases we found a reduction of the ratio of the diffusion coefficients if the ratio of magnetic fields is reduced.

Detailed numerical investigation of 90∘ scattering of energetic particles interacting with magnetic turbulence

In the present paper, we re-visit a well-known problem in diffusion theory, namely the 90∘ scattering problem. We use a test-particle code to compute the pitch-angle Fokker-Planck coefficient at 90∘ for different values of the turbulent magnetic field strength and the magnetic rigidity. We consider a slab model and compare our numerical findings with the analytical result provided by second-order quasilinear theory. We show that the latter theory accurately describes 90∘ scattering. We also replace the slab model by a more realistic two-component model to explore the influence of the turbulence model on 90∘ scattering.

Pitch-angle scattering in magnetostatic turbulence

Context. Spacecraft observations have motivated the need for a refined description of the phase-space distribution function. Of particular importance is the pitch-angle diffusion coefficient that occurs in the Fokker-Planck transport equation. Aims. Simulations and analytical test-particle theories are compared to verify the diffusion description of particle transport, which does not allow for non-Markovian behavior. Methods. A Monte-Carlo simulation code was used to trace the trajectories of test particles moving in turbulent magnetic fields. From the ensemble average, the pitch-angle Fokker-Planck coefficient is obtained via the mean square displacement. Results. It is shown that, while excellent agreement with analytical theories can be obtained for slab turbulence, considerable deviations are found for isotropic turbulence. In addition, all Fokker-Planck coefficients tend to zero for high time values.

Pitch-angle scattering in magnetostatic turbulence

Aims. The process of pitch-angle isotropization is important for many applications ranging from diffusive shock acceleration to large-scale cosmic-ray transport. Here, the basic analytical description is revisited on the basis of recent simulation results. Methods. Both an analytical and a numerical investigation were undertaken of the Fokker-Planck equation for pitch-angle scattering. Additional test-particle simulations obtained with the help of a Monte-Carlo code were used to verify the conclusions. Results. It is shown that the usual definition of the pitch-angle Fokker-Planck coefficient via the mean-square displacement is flawed. The reason can be traced back to the assumption of homogeneity in time which does not hold for pitch-angle scattering. Conclusions. Calculating the mean free path via the Fokker-Planck coefficient has often proven to give an accurate description. For numerical purposes, accordingly, it is the definition that has to be exchanged in favor of the pitch-angle correlation function.

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.

Analytical reduction of pitch-angle scattering in isotropic turbulence

The scattering of energetic particles in stochastic magnetic fields cannot be described using quasi-linear theories except for simple turbulence geometries. A promising candidate for the analytical description of such scattering processes is the second-order quasi-linear theory [A. Shalchi, Phys. Plasmas 12 (2005) 052905]. For isotropic turbulence, the theory has been evaluated only semi-numerically but the results are in good agreement with numerical simulations [R.C. Tautz, A. Shalchi, R. Schlickeiser, Astrophys. J. 685 (2008) L165]. Here, using well-established methods from the summation of Bessel functions, the equation for the pitch-angle Fokker–Planck coefficient is reduced to a few single and double integrals that are left to numerical evaluation. In doing so, the evaluation of transport parameters is facilitated, thus enabling one to conduct parameter studies.

Analytical description of nonlinear cosmic ray scattering: isotropic and quasilinear regimes of pitch-angle diffusion

Aims. We investigate pitch-angle scattering, which is a fundamental process in the physics of cosmic rays. Methods. By employing the second-order quasilinear theory, the pitch-angle Fokker-Planck coefficient is calculated analytically for the first time. Results. We demonstrate that for sufficiently strong turbulence the pitch-angle Fokker-Planck coefficient is isotropic. The derived results can be used to compute the parallel mean free path for all forms of the turbulence spectrum. We also consider applications, namely the transport of solar energetic particles and the propagation of cosmic rays in the Galaxy. Conclusions. The previously used assumption of isotropic pitch-angle diffusion is indeed correct for sufficiently strong turbulence. An analytical description of nonlinear particle scattering is possible.

Pitch-angle scattering in pure two-dimensional and two-component turbulence

Aims. It has been demonstrated that quasilinear theory is inappropriate for describing pitch-angle scattering and parallel spatial diffusion, if a turbulence model with a strong two-dimensional component is assumed. We aim to revisit this problem using a more systematic and reliable approach. Methods. We consider particle transport in pure two-dimensional turbulence, by using a vector-potential description of pitch-angle scattering. Results. It is demonstrated that pitch-angle scattering in pure two-dimensional turbulence behaves subdiffusively and, thus, parallel scattering is expected to behave superdiffusively. In the slab/2D composite model, the total pitch-angle Fokker-Planck coefficient reaches the slab limit when very long timescales are considered.

The 90° problem of scattering theory revisited: dynamical turbulence versus nonlinear effects

The 90° problem of cosmic-ray transport theory is revisited in this paper. By using standard forms of the wave spectrum in the solar wind, the pitch-angle Fokker–Planck coefficient and the parallel mean free path are computed for different resonance functions. A critical comparison is made of the strength of 90° scattering due to plasmawave effects, dynamical turbulence effects and nonlinear effects. It is demonstrated that, only for low-energy cosmic particles, dynamical effects are usually dominant. The novel results presented here are essential for an effective comparison of heliospheric observations for the parallel mean free path with the theoretical model results.

Second-order quasilinear theory of cosmic ray transport

The problem of pitch-angle diffusion close to 90° is well known in cosmic ray astrophysics. If the pitch-angle Fokker–Planck coefficient for pure slab geometry is calculated, the quasilinear approximation results in vanishing pitch-angle scattering. For a realistic wave spectrum with a steep dissipation range this vanishing coefficient generates an infinitely large parallel mean free path. It is well known from numerical simulations that the 90° problem is a problem of quasilinear theory and not a problem of reality. In the current paper quasilinear theory is used to calculate corrections of the unperturbed orbit. These corrections can be resubstituted into transport theory to calculate a second-order pitch-angle Fokker–Planck coefficient. The second-order quasilinear theory is an applicable theory which agrees with simulations for pitch-angle diffusion.

Spurious contribution to cosmic ray scattering calculations

The quasi-linear theory for cosmic ray propagation is a well-known and widely accepted theory. In this paper, we discuss the different contributions to the pitch-angle Fokker-Planck coefficient from large and small scales for slab geometry using the damping model of dynamical turbulence. These examinations will give us a hint on the limitation range where the quasi-linear approximation is a good approximation.

Nonlinear Parallel and Perpendicular Diffusion of Charged Cosmic Rays in Weak Turbulence

The problem of particle transport perpendicular to a magnetic background field is well known in cosmic-ray astrophysics. Whereas it is widely accepted that quasi-linear theory (QLT) of particle transport does not provide the correct results for perpendicular diffusion, it was assumed for a long time that QLT is the correct theory for parallel diffusion. In the current paper we demonstrate that QLT is in general also incorrect for parallel particle transport if we consider composite turbulence geometry. Motivated through the recent success of the so-called nonlinear guiding center theory of perpendicular diffusion, we present a new theory for parallel and perpendicular diffusion of cosmic rays. This new theory is a nonlinear extension of QLT and provides us with a coupled system of nonlinear Fokker-Planck coefficients. By solving the resulting system of integral equations we obtain new results for the pitch-angle Fokker-Planck coefficient and the Fokker-Planck coefficient of perpendicular diffusion. By integrating over pitch angle we calculate the parallel and perpendicular mean free path. To our knowledge the new theory is the first that can deal with both parallel and perpendicular diffusion in agreement with simulations.

The Parallel Mean Free Path of Heliospheric Cosmic Rays in Composite Slab/Two‐dimensional Geometry. I. The Damping Model of Dynamical Turbulence

The parallel mean free path of cosmic-ray particles in partially turbulent electromagnetic fields is calculated in the damping model of dynamical turbulence for pure two-dimensional turbulence geometry. Using the general results for the pitch-angle Fokker-Planck coefficient for pure slab geometry from Teufel & Schlickeiser, the parallel mean free path for composite slab/two-dimensional geometry is also calculated. The rigidity dependence and the absolute value of the mean free path for this general turbulence geometry are compared with observational results. We demonstrate that the composite model is able to explain observational results.

Analytic calculation of the parallel mean free path of heliospheric cosmic rays

The parallel mean free path of cosmic ray particles in partially turbulent electromagnetic fields is calculated for two particular turbulence models: slab-like dynamical and random sweeping turbulence. Using the general results for the pitch-angle Fokker-Planck coefficient from Teufel & Schlickeiser (2002) the rigidity dependence and the absolute value of the mean free path for these specific turbulence models are calculated for a turbulence power spectrum with finite wave amplitude at very small wavenumbers. We demonstrate that this modification affects especially the mean free path at very large rigidities. We also derive approximations for the mean free path for realistic Kolmogorov-type turbulence power spectra which include the steepening at high wavenumbers due to turbulence dispersion and/or dissipation.