The non-thermal radio and high-energy radiation spectra from extragalactic and galactic objects such as X-ray binaries, micro-quasars, active galactic nuclei, jets and gamma-ray bursts require the presence of non-thermal particle distributions. One of the prime mechanisms for producing such distributions is diffusive acceleration at a shock front. This theory assumes the particle distributions are isotropised by efficient scattering on wave turbulence on both sides of the shock and gain energy by a first order Fermi process upon crossing it. The equation governing the particle distribution is of the Fokker-Planck type, containing both advection (dynamical friction) and diffusion terms in the spatial variables and an advection term in the energy variable (or in the magnitude of the particle momentum p), which is proportional to the divergence of the fluid flow, and remains valid in an integral sense even across shock fronts. Many generalisations of this equation have been discussed, but for the application to non-thermal radiation spectra, the most relevant are the inclusion of synchrotron losses (Webb et al. 1984) and the extension to systems containing multiple shock fronts (Blandford & Ostriker 1980, Spruit 1988, Achterberg 1990, Schneider 1993, Melrose 1996, Melrose & Crouch 1997). However, because of the difficulty of solving the Fokker-Planck equation when the coefficients are complicated (and possibly discontinuous) functions of position, these papers either adopted an idealised situation, or developed approximation schemes valid in only part of the parameter space. A particularly interesting alternative approach is to use the equivalence of the Fokker-Planck equation to a system of stochastic differential equations (SDE's). Numerical integration of these is then akin to a Monte-Carlo simulation of the problem, which is relatively simple to implement, applies to complex flow patterns and places no restriction on the number of phase-space dimensions (e.g., Gardiner 1983). Several conventional Monte-Carlo simulations of particle acceleration exist (for a review see Jones & Ellison 1991), usually assuming a prescribed form of the mean free path of the particle as a function of the particle rigidity and plasma density. They have the advantage of being able to describe the evolution of both the thermal and suprathermal populations, as well as the non-linear back reaction of the non-thermal component on the shock profile. Relativistic shocks and large angle scattering are also easy to include in such simulations (see Ellison et al. 1990). However, this approach always makes additional assumptions concerning the particle trajectory (e.g., that it is unperturbed between isotropising `collisions') which go beyond the diffusion approximation. In comparison, the SDE approach adopts the diffusion approximation for test particle and is restricted to the transport of suprathermal particles.
It is only recently that SDE systems have been applied to astrophysical problems: in solar physics with the investigation of acceleration of fast electrons in the solar corona (MacKinnon & Craig 1991, Conway et al. 1998) and in space physics with ion acceleration at the solar termination shock (Chalov et al. 1995). In two papers Achterberg & Krülls (1992) and Krülls & Achterberg (1994 - henceforth KA94), have applied the SDE approach to the problem of particle acceleration at astrophysical shocks, including the possibility of second-order Fermi acceleration (Schlickeiser 1989). These authors drew several important conclusions. In particular, they showed that for Kolmogorov turbulence the second-order acceleration effect is restricted to a small momentum range close to that of injection, because of the increase of the diffusion time with momentum. At high momentum, in contrast, the spectrum is formed by the competing effects of synchrotron losses and the first-order Fermi process.
The numerical scheme used in KA94 is explicit , that is, the first-order Fermi acceleration term in the SDE is integrated forwards in time using the value of the fluid velocity gradient at the beginning of each step. This method requires a step short enough to resolve sharp features in the flow, such as a shock transition, and thus severely limits the ability to simulate acceleration in complex flow patterns containing structure on a large range of spatial scales. Our purpose in this paper is to propose and test an implicit method of integrating the SDE's. By implicit we mean that the coefficient of the first-order Fermi acceleration term is computed by linear interpolation between the end points of each time step. Such a scheme allows one to treat discontinuous shock structures using a finite time step and thus opens up the prospect of finding approximate numerical solutions of the advection-diffusion equation in complex flow patterns where the gradients of the fluid flow may become large over a distance much shorter than the shock thickness. This is our longer term goal; for the present, we limit ourselves to the problem of a 1D single shock or to systems of multiple shocks.
The organisation of the paper is as follows: in Sect. 2 we present the SDE system equivalent to the advection-diffusion equation and describe the explicit and implicit integration methods. We then test the implicit scheme in Sect. 3 by applying it to the (well-known) problem of acceleration at a single isolated shock front, both with and without synchrotron losses. We also compare the performances of implicit and explicit schemes. Sect. 4 focuses on the application to acceleration in a system of multiple shocks. Here, as well as testing the code against known approximate solutions, we present new solutions valid in regions of parameter space inaccessible to the analytic methods. We use the results to calculate the optically thin synchrotron radiation produced by the stationary particle distribution within a periodic pattern of shocks. In Sect. 5 we summarise our results and discuss extensions of the method to more complex problems.
© European Southern Observatory (ESO) 1999
Online publication: June 18, 1999