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Cosmic Ray Transport

Prof. Yuri Litvinenko
University of Waikoto, New Zealand

and

Dr. Horst Fichtner
Astrophysics

Theoretical models of cosmic-ray acceleration and transport

The generation and transport of energetic cosmic-ray particles (electrons, protons, ions) is among the most important problems in modern astrophysics. When the particles propagate in the cosmos, they often interact with turbulent magnetic fields. The resulting complicated evolution of the cosmic-ray distribution can be mathematically described as a simpler diffusive process. The diffusion approximation and its generalisations have been employed for studying the generation and propagation of energetic particles in many astrophysical situations: in the atmosphere of the Sun, interplanetary space, and the interstellar medium.

The VIP grant will support a systematic investigation of unresolved issues of particle transport in astrophysics, with a focus on analytical and numerical modelling of diffusive particle transport. The areas of particular interest are as follows. (1) Rigorous derivation of the diffusion and telegraph approximations for cosmic-ray transport modelling, which quantifies their accuracy. (2) Development of new analytical and numerical methods for solving the equations governing the transport of solar and galactic cosmic rays. (3) Critical examination of possible mechanisms of fractional particle diffusion in the cosmos.

Figure: Particle density profiles from solutions of the Fokker-Planck equation (black curve), the telegraph equation (green curve), the hyperdiffusion equation (blue curve), and the diffusion equation (red curve) (Litvinenko and Noble, 2016).

December 2015 - February 2016

During his first three-month visit, Dr Litvinenko gave a talk on modelling sunspot and starspot decay by turbulent erosion, and he lectured on cosmic-ray transport equations. The lectures, open to the students and members of the institute, covered the following topics: the kinetic equation for the cosmic-ray distribution function, the Fokker-Planck-Kolmogorov equation, the diffusion approximation and its limitations, the Parker transport equation, the telegraph equation and other higher-order models.

Together with Dr Horst Fichtner (RUB) and Dr Frederic Effenberger (Stanford University), Yuri Litvinenko began developing a model for the observed anomalous transport of solar cosmic rays. Observations of solar energetic particles indicate that their transport in interplanetary space in the presence of turbulent scattering can be superdiffusive, that is the mean square displacement of the particles can increase with time faster than predicted by the familiar linear dependence of standard diffusion. A popular interpretation of such anomalous transport invokes solutions of cosmic-ray transport equations containing fractional derivatives. Yet the physical basis of fractional diffusion models of cosmic-ray transport remains uncertain. We suggest that a nonlinear diffusion model can provide a more physical explanation of the anomalous cosmic-ray transport. Our preliminary results will be presented at the 41st COSPAR Scientific Assembly (Istanbul, Turkey, August 2016).

December 2016 - February 2017

During his second three-month visit to the RUB Theoretische Physik IV Institute, Dr Litvinenko gave a talk on the telegraph and hyperdiffusion approximations in cosmic-ray transport to the students and members of the institute. He also actively participated in the weekly Space- and Astrophysics research seminar, which included Steffen Krakau's trial talk before his successful PhD examination.

Drs Litvinenko and Fichtner, together with a RUB postgraduate student Dominik Walter, made further progress in their theoretical investigation of cosmic-ray transport. Anomalous, superdiffusive transport appears to play a significant role in the evolution of energetic particles, accelerated at interplanetary shocks. The goal of the investigation was to explore a nonlinear diffusion model that could provide a physical explanation of the superdiffusive cosmic-ray transport. The idea is that the particle distribution function is governed by a nonlinear diffusion equation in which the diffusion coefficient is determined by the resonant interaction of the particles and the waves that they generate. As a concrete illustration, wave generation by the streaming particles was assumed to be balanced by wave dissipation by a turbulent cascade. The study has led to new analytical and numerical solutions for an evolving cosmic-ray distribution, characterised by superdiffusive scalings. In the presence of convection, the nonlinear model yields a power-law dependence of the particle density on the distance upstream of the shock. Preliminary results formed the basis of Mr Walter's Bachelorarbeit, and a joint paper is expected to be completed soon. Mr Walter's interest in the project was driven by Dr Litvinenko's presentation during his first visit.

Drs Fichtner, Litvinenko, and Effenberger also co-authored two conference papers, based on their collaborative work: "Anomalous transport of cosmic rays: A case of nonlinear diffusion?" and "Stochastic processes and fractional differential equations for anomalous diffusion problems". Both papers were to be presented during the 41st COSPAR Scientific Assembly in Istanbul, Turkey (30 July - 7 August, 2016). Sadly, the Assembly was cancelled because of the army coup against the Turkish government on 15 July.

Publications

December 2017 - February 2018

During his third three-month stay, supported by the VIP program, Dr Litvinenko regularly attended the research seminar in the RUB Theoretische Physik IV Institute, where he also presented his recent research results. Drs Litvinenko and Fichtner and Mr Dominik Walter continued a systematic theoretical investigation of a nonlinear model of cosmic-ray transport, which follows from a self-consistent treatment of the resonantly interacting cosmic-ray particles and their self-generated turbulence.

Previously we demonstrated that the nonlinear transport model can yield the observed anomalous diffusion scalings of the energetic particles in space. We obtained analytical and numerical solutions for two initial value problems and explored the roles of advection and a constant particle source. The results formed the basis of Mr Walter's Bachelorarbeit and were published in a refereed paper in The Astrophysical Journal. During this stay we explored the effect of a time-dependent particle source in the transport model by obtaining a self-similar solution and investigating its properties. A follow-up paper is in preparation. In a separate project, we explored the propagation of nonlinear acoustic waves in the solar atmosphere.

Since the purpose of the VIP Fellowship is to contribute to graduate training at RUB, we are particularly pleased to note that further development of the model will form the basis of Mr Walter's PhD thesis. He is expected to enroll in a PhD program before the end of this year. Dr Litvinenko's short final visit is planned for June 2018, at which time we intend to complete a proposal to the German Research Foundation (DFG). We will request funding to ensure continuation of our active research collaboration, primarily through Mr Walter's Doctoral Scholarship.

Publications