Astron. Astrophys. 364, 911-922 (2000)

1. Introduction

The non-thermal energy distributions of cosmic ray ions or source distributions of electrons emitting synchrotron radiation in various astrophysical objects are commonly described as produced by the first order Fermi acceleration process at shocks (for reviews see Drury 1983; Blandford & Eichler 1987; Kirk et al. 1994).

When particles diffuse ¹ off the moving scattering centers in a region divided by a velocity discontinuity (shock), these particles can be accelerated if their mean free paths exceed the shock thickness. The relative momentum gain for a cycle of two crossings of the shock is then proportional to the velocity difference across the shock, i.e. of first order with respect to the shock velocity (Bell 1978). In astrophysical collisionless plasmas an electro-magnetic field must be present to change the energy of particles. Waves or irregularities in this field provide particle scattering, which leads to diffusion. Consider a shock with velocity propagating into a plasma at rest with density and with a homogeneous magnetic field in the direction of the shock normal. The plasma is compressed to the density by the shock, and flows downstream with the velocity , where is the compression ratio. Particles with the mean downstream velocity cannot cross the shock from downstream to upstream, because . In addition the shock may not be a discontinuity for a particle at this energy, because the gyro radius of the thermal particles is of the order of the shock thickness, leading to adiabatic energy change while crossing the shock. Because the plasma is also heated downstream of the shock, some supra-thermal particles in the high energy tail of the Maxwellian velocity distribution may gain energies and have velocities that allow them to re-cross the shock. In a homogeneous magnetic field, due to lack of scattering centers these particles would escape upstream without returning to the shock. Then, the acceleration mechanism would not apply. However, the population of particles that can move upstream provide a seed particle beam, which generates Alfvén waves responsible for scattering and, therefore, diffusion, which is an essential element of the first order Fermi acceleration process.

The problem of particle acceleration from thermal energies up to relativistic particle energies is highly non-linear, as first pointed out by Eichler (1979). First, the energy transferred from the bulk of the plasma to the sub-population of accelerated particles can change the thermodynamic properties of the plasma like the temperature and density. In addition, the accelerated particles provide their own pressure in the system, which, since it differs from the thermal pressure, modifies the velocity structure of the shock transition. Second, the waves generated by particles escaping upstream determine the transport properties of the plasma, and, therefore, regulate this wave generating escape itself. The manner in which the wave-particle interactions control the fraction of plasma particles that can escape upstream to participate in the Fermi process is commonly called injection . This is a basic aspect of the plasma of collisionless shocks and is itself highly non-linear. This injection problem is fundamentally related to the question of the efficiency of particle acceleration at shocks by the Fermi process.

Different numerical methods have been used to treat the injection problem of CR modified shocks. In Monte-Carlo simulations of non-linear particle acceleration the details of a posited scattering law provide an injection parameter, but one not determined self-consistently from the particle wave interaction (e.g. Ellison et al. 1996; Baring et al. 1999). In contrast to pure kinematical effects from shock velocity, particle speed and inclination angle of magnetic field and shock, the waves responsible for particle scattering depend on plasma properties like temperature and the beam strength of the wave generating particles itself. These coupled and time dependent effects are not easy to incorporate into a Monte-Carlo approach. Time dependent Monte-Carlo simulations have been presented by Knerr et al. (1996), but still with a prescribed scattering law, as a parameterization of injection.

In the two-fluid approach the cosmic rays are treated as a diffusive gas without following their momentum distribution. The energy transfer into CRs in these models is based on a fraction of the upstream gas particles, that are instantaneously accelerated at the shock (Dorfi 1990), around the shock (Jones & Kang 1990) or at velocity gradients (Zank et al. 1993). In practical terms, because the shock is the most prominent velocity gradient in the system, these techniques are very closely related, as pointed out by Kang & Jones (1995). Essentially the same parameterization is also used in the numerical solution of the hydro-dynamical equations coupled to the momentum dependent cosmic-ray transport equation (e.g. Falle & Giddings 1987; Kang & Jones 1991; Berezhko et al. 1994).

Kang & Jones (1995) used a numerical injection model with two essentially free parameters which describe boundaries in momentum at which particles can be accelerated ( where is the peak momentum of the Maxwell distribution) and from which these contribute to the cosmic-ray pressure (). Still, these momentum boundaries are free parameters, which can be translated into a particle fraction of the upstream gas. However, models that incorporate more details of the plasma physics of the background plasma, and, therefore, the CRs at injection energies are really necessary to constrain the phase-space function and therefore determine these parameters. In this way we can incorporate self-consistent plasma physical models into numerical simulations of CR acceleration.

Such a plasma physical model based on non-linear interactions of particles with self-generated waves in a shocked plasma has been investigated numerically by solving the kinetic equations of ions in a magnetic field, and treating the electrons as a background fluid (Quest 1988). These simulations show that ions can be scattered back and forth across the shock by self-generated waves, and Quest (1988) also points out that these scattered ions can provide a seed population of cosmic rays.

Recently, the kinetic equations of ions were solved analytically for non-linear wave-fields near strong parallel shocks by Malkov & Völk (1995) and Malkov (1998). These authors were able to constrain the fraction of phase-space of the background plasma that can be injected into the acceleration process as a result of the self-regulating interaction between wave generation and particle streaming. Here we incorporate this self-consistent analytical result in numerical solutions of the hydro-dynamical equations together with the cosmic-ray transport equation. Our simulations, therefore, provide the first time dependent solution of the problem of modified shocks that includes a self-consistent plasma physical injection model. This technique enables us to determine the level of shock modification and acceleration efficiency in an evolving shock without a free parameter for the injection process (Gieseler et al. 1999).

We describe the plasma physical injection model in some more detail in Sect. 2 before we outline the coupled set of dynamical equations of the plasma and cosmic rays in Sect. 3, together with details of the numerical method we used to solve these equations. The results are presented in Sect. 4 and Sect. 5, consisting of the time dependent evolution of the plasma properties, showing especially the modification of the velocity profile and the momentum distributions of thermal and relativistic cosmic-ray particles. In addition we present our results in the form of a particle injection efficiency and an energy transfer efficiency at modified strong shocks.

© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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