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Fractional Parker equation for the transport of cosmic rays: steady-state solutions

机译：宇宙射线传输的分数帕克方程：稳态解

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Context. The acceleration and transport of energetic particles in astrophysical plasmas can be described by the so-called Parker equation, which is a kinetic equation comprising diffusion terms both in coordinate space and in momentum space. In the past years, it has been found that energetic particle transport in space can be anomalous, for instance, superdiffusive rather than normal diffusive. This requires a revision of the basic transport equation for such circumstances. Aims. Here, we extend the Parker equation to the case of anomalous diffusion by means of fractional derivatives that generalize the usual second-order spatial diffusion operator. Methods. We introduce the left and right Caputo fractional derivatives in space. These derivatives are one of the tools used to describe anomalous transport. We consider the case of steady-state solutions upstream and downstream of a planar shock. Results. We obtain an estimate of the particle acceleration time at shocks in the case of superdiffusion. An analytical solution of the steady-state fractional Parker equation is given by the Mittag-Leffler functions, which correspond to a power-law profile for the energetic particle intensity far upstream of the shock, in agreement with the results obtained from a probabilistic approach to superdiffusion. These functions also correspond to a stretched exponential close upstream of the shock. Conclusions. These results can help to model more precisely the measured fluxes of energetic particles that are accelerated at both interplanetary shocks and supernova remnant shocks.
机译：上下文。高能粒子在天体等离子体中的加速和传输可以通过所谓的Parker方程来描述，该方程是一个动力学方程，包括坐标空间和动量空间中的扩散项。在过去的几年中，已经发现空间中的高能粒子传输可能是异常的，例如，超扩散而不是正常扩散。在这种情况下，这需要修改基本的运输方程。目的在这里，我们通过分数微分将一般的二阶空间扩散算子进行推广，将Parker方程扩展到异常扩散的情况。方法。我们介绍了空间中的左和右Caputo分数导数。这些导数是用于描述异常传输的工具之一。我们考虑平面冲击上游和下游的稳态解的情况。结果。在超扩散的情况下，我们获得了冲击时粒子加速时间的估计。 Mittag-Leffler函数给出稳态分数阶Parker方程的解析解，该函数对应于冲击上游的高能粒子强度的幂律分布，与从概率方法获得的结果一致超扩散。这些功能还对应于电击上游的拉伸指数接近。结论。这些结果可以帮助更精确地建模在行星际激波和超新星残余激波时加速的高能粒子通量。

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