Acceleration of Low-Energy Ions at Parallel Shocks With a Focused Transport Model

Abstract

We present a test particle simulation on the injection and acceleration of low-energy suprathermal particles by parallel shocks with a focused transport model. The focused transport equation contains all necessary physics of shock acceleration, but avoids the limitation of diffusive shock acceleration (DSA) that requires a small pitch angle anisotropy. This simulation verifies that the particles with speeds of a fraction of to a few times the shock speed can indeed be directly injected and accelerated into the DSA regime by parallel shocks. At higher energies starting from a few times the shock speed, the energy spectrum of accelerated particles is a power law with the same spectral index as the solution of standard DSA theory, although the particles are highly anisotropic in the upstream region. The intensity, however, is different from that predicted by DSA theory, indicating a different level of injection efficiency. It is found that the shock strength, the injection speed, and the intensity of an electric cross-shock potential (CSP) jump can affect the injection efficiency of the low-energy particles. A stronger shock has a higher injection efficiency. In addition, if the speed of injected particles is above a few times the shock speed, the produced power-law spectrum is consistent with the prediction of standard DSA theory in both its intensity and spectrum index with an injection efficiency of 1. CSP can increase the injection efficiency through direct particle reflection back upstream, but it has little effect on the energetic particle acceleration once the speed of injected particles is beyond a few times the shock speed. This test particle simulation proves that the focused transport theory is an extension of DSA theory with the capability of predicting the efficiency of particle injection.

Simulation of Energetic Particle Transport and Acceleration at Shock Waves in a Focused Transport Model: Implications for Mixed Solar Particle Events

“…Further simulations with different values for the compression ratio and the energy range of the protons showed results which are qualitatively similar to the ones presented above. We find that for a one-dimensional planar shock our method reproduces the results of previous simulations based on pitch-angle dependent transport by le Roux et al (2007) and Zuo et al (2013), and the predictions of DSA theory with regard to the spatial dependence and the energy spectrum of the protons for the case that scattering mean free paths are very small (below 1% ) compared with the dimensions of the system, and boundary conditions remain constant until a steady state is reached. It appears that in the above case effects of the finite speed of the particles and a moderate anisotropy in the vicinity of he shock-which are not considered in the spatial diffusion treatment-nonetheless do not lead to substantial deviations from the predictions of DSA theory.…”

“…As compared to previous investigation of the above process by le Roux et al (2007), le Roux & Webb (2012), and Zuo et al (2011Zuo et al ( , 2013, which considered in detail the effects of preheating mechanisms, of the obliquity of the shock, and resulting efficiencies in comparison with DSA theory, we have restricted ourselves to parallel shocks and particles with high enough speeds to be directly injected into the first-order Fermi process. Where the subjects of the investigation overlap, i.e., for a stationary parallel shock and values of the mean free paths which are small compared to the scale on which the geometry of the shock is constant, our model similarly reproduces the basic features of standard DSA theory: the spatial variation of the intensity in the up-and downstream regions, and a powerlaw energy distribution with a spectral index which is determined by the compression ratio at the shock, and this for distribution functions which are not close to being isotropic.…”

“…We have not made an attempt here toinvestigate the acceleration efficiency as a function of the compression ratio and other parameters as in Zuo et al (2013), and the absolute intensities in both panels of Figure 1 are scaled arbitrarily. We find that the shapes of the pitch-angle averaged spatial distribution for protons in the energy range 0.5-0.63 MeV (left panel of the figure) as well as in the energy range 7.94-10 MeV (right panel of the figure) are in good agreement with the predictions of DSA, indicating that a steady state has been reached.…”

“…With a similar technique, le Roux & Webb (2012) considered acceleration at a fast traveling shock and in particular investigated the effects of various pre-heating mechanisms for the injection of thermal particles into firstorder Fermi acceleration. Zuo et al (2011Zuo et al ( , 2013 treated particle acceleration at parallel and oblique shocks by solving the set of corresponding stochastic differential equations (SDEs), including the one which explicitly describes the energy changes, and examined the efficiency of the acceleration compared to DSA theory as a function of the shock's compression ratio and the injection speed of the particles. Similar to thediffusion-convection DSA theory, acceleration emerges as particles are being scattered back andforth through the shock, and the acceleration efficiency and the energy spectrum is, in the case of a planar parallel shock, solely determined by the shock's compression ratio and the scattering mean free paths up-and downstream of the shock, respectively.…”

Scatttering of High-Energy Particles at a Collisionless Shock Front: Dependence on the Shock Angle

“…Particle transport is often considered in such parallel shock geometries (see, e.g., Zuo et al 2013), although the number of observed parallel (or quasi-parallel) shocks is not large. Figure 1 shows the distribution of shocks observed by the STEREO mission as a function of the angle between the shock normal and the upstream magnetic field (the list of 2007-2014 STEREO shocks by Lan Jian at http://www-ssc.igpp.ucla.edu/ forms/stereo/stereo_level_3.html).…”