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

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3. Model

3.1. Dynamical equations

The standard hydro-dynamical equations of mass, momentum and energy conservation for a gas with velocity [FORMULA], and density [FORMULA], corrected for CR pressure effects are given by

[EQUATION]

where [FORMULA] and [FORMULA] are the gas and the CR pressure, respectively, and [FORMULA] is the total energy density of the gas per unit mass. Here [FORMULA] is the total Lagrangian time derivative. We assume [FORMULA] for the thermal gas adiabatic index throughout this work. The injection energy loss term [FORMULA] accounts for the energy transferred to high energy particles and will be discussed later. Eqs. (4)-(6) are solved using a Total Variation Diminishing (TVD) code based on the scheme of Harten (1983).

We assume that the shock Mach number [FORMULA] (with [FORMULA] the upstream sound speed) exceeds the Alfvén Mach number [FORMULA] (with [FORMULA] the upstream Alfvén speed). Then the diffusion-convection equation, which describes the time evolution of the phase-space density [FORMULA] of the high energy CRs (e.g. Skilling 1975), takes the form:

[EQUATION]

The diffusion coefficient [FORMULA] is assumed to be a scalar. Transforming to the variables [FORMULA] and [FORMULA] (cf. Falle & Giddings 1987; Kang & Jones 1991), Eq. (7) can be written as

[EQUATION]

This equation is solved using an implicit Crank-Nicholson scheme, which is second order in space and time (see e.g. Falle & Giddings 1987).

The high energy particles provide an additional pressure to the system that has to be included in the set of hydro-dynamical equations (with p normalized to the proton momentum [FORMULA]):

[EQUATION]

This definition of the CR pressure [FORMULA] includes the difference of the phase-space density from the Maxwellian distribution [FORMULA] and defines the sub-population which we identify as CRs. The CR energy density is defined as

[EQUATION]

3.2. Injection scheme

We do not include an additional injection term in Eq. (8), because in our model injection is described self-consistently from the thermal distribution. Therefore, the lower boundary of the momentum distribution of the CR population must match the upper boundary of the momentum distribution of the gas. The distinction between these populations is, of course, only technical, and defined by the validity of the relevant dynamical equations. We use a Maxwell distribution according to the actual density and temperature of the plasma. Instead of a fixed momentum boundary we use here the transparency function [FORMULA] to define where the lower boundary of the CR momentum distribution matches the momentum distribution of the bulk plasma. The injection into the high energy part of the phase-space distribution (i.e. the CRs) is then directly provided by the bulk of the plasma.

The initial Maxwellian phase-space density [FORMULA] is given by:

[EQUATION]

where [FORMULA] is the particle number density, and the temperature is defined by the local gas pressure [FORMULA] and density [FORMULA] according to [FORMULA]. Here µ is the mean molecular weight which is assumed to be one, and [FORMULA] is the Boltzmann constant. The details of how the momentum distribution is calculated in a time step from [FORMULA] to t are as follows. First we define the CR part of the momentum distribution by [FORMULA]. Now the CR diffusion-convection equation (Eq. 8) is solved for the entire momentum space, including the thermal Maxwell distribution, to find the updated distribution function [FORMULA]. For momenta below the critical momentum of [FORMULA] any particle acceleration must be suppressed, and therefore the result of Eq. (8) is rejected by restoring the Maxwellian distribution given in Eq. (11). For momenta above the critical momentum and upstream of the shock, we use the transparency function as a filter for [FORMULA] as described below, since [FORMULA] corresponds to the fraction of the phase-space density at a given momentum that can cross the shock from downstream to upstream. The final distribution at time t is then given by [FORMULA] in the following way:

[EQUATION]

So effectively at the lower momentum limit where [FORMULA] the Eq. (8) has no effect at all, while at the higher momentum limit where [FORMULA] the result of Eq. (8) is used without further modification. Only in the intermediate momentum regime where [FORMULA], the transparency function represents the injection process (i.e. thermal leakage). Thus the transparency function defines self-consistently the momentum boundary above which the particle acceleration mechanism can work, and also defines the transition region between thermal plasma and accelerated particles.

The particle injection rate into the CR population can be estimated from the adiabatic change of the momentum due to the velocity gradient of the flow:

[EQUATION]

Then the energy loss rate of the gas can be written as

[EQUATION]

Note here the condition [FORMULA] in fact defines the "injection pool" where the thermal leakage takes place. Due to the steep dependence of both the Maxwell distribution and the transparency function on the particle momentum, the momentum range of the injection pool is well restricted. Either below or above this momentum range [FORMULA] constant, so [FORMULA]. If the transparency function is given by a step function [FORMULA], it becomes the injection scheme adopted by Kang & Jones (1995, 1997) in which the injection takes place at a single injection momentum rather than an extended momentum range.

The transparency function [FORMULA] given by Eq. (3) depends on the downstream plasma velocity, which is averaged over the diffusion length of the particles with momentum at the injection threshold. This dependence is also important for the injection efficiency, and leads to a regulation mechanism similar to the above beam wave interaction. If the initial injection is so strong that a significant amount of energy is transferred from the gas to high energy particles, the downstream plasma cools, and, in addition, the downstream bulk velocity decreases in the shock frame due to the shock modification of the cosmic-ray population. Because the injection pool is in the high energy tail of the Maxwellian distribution of the gas, the cooling decreases significantly the injection rate. However, the deceleration, in turn, allows for a modest increase of the phase-space of particles that can be injected. This is expressed by the [FORMULA] dependence of Eq. (3). This velocity dependence balances partly the reduction of injection due to the cooling of the plasma. Remarkably, these two effects lead to a very weak dependence of the injection efficiency on [FORMULA] in the vicinity of [FORMULA].

3.3. Diffusion model

Since the injection process is included self-consistently, the diffusion coefficient is the only remaining free parameter in our model. We assume the particle diffusion is based on the scattering off the self-generated waves which have a field component perpendicular to the plasma flow. The compression of the plasma leads to an amplification of these waves, which is described by scaling the diffusion coefficient as [FORMULA]. In our one dimensional model we have to describe diffusion along the mean magnetic field. The lower limit for the diffusion coefficient is the Bohm diffusion coefficient [FORMULA] where [FORMULA] is the magnetic field strength in units of micro-gauss. For the present calculations we assume the diffusion coefficient is simply related to Bohm diffusion as

[EQUATION]

We have introduced the factor [FORMULA] to account for the higher diffusion in the direction of the mean magnetic field, because this direction is parallel to the shock normal, and, therefore, relevant for the acceleration process at quasi-parallel shocks. Although the time scale for the cosmic-ray acceleration does depend on the diffusion coefficient, the basic self regulation process for the injection problem which we investigate here is not dependent on the choice of [FORMULA]. Therefore, since our study intends to focus on the general time dependent behavior of this injection model, we do not include a completely self-consistent scattering model, where the diffusion coefficient is coupled to the spectrum of the Alfvén waves.

3.4. Initial and boundary conditions

We assume there is no pre-existing CR population, so the initial particle distribution is purely Maxwellian with the local plasma temperature and density. We use open boundary conditions for the description of the thermal plasma in our simulations. In momentum space, the lower boundary is provided by the Maxwell distribution as discussed above. At the highest momentum and also at the upstream boundary in configuration-space we use a `free escape' boundary. Downstream we use a `no diffusive flux' boundary, where the cosmic-ray density is always kept constant across the boundary. However, the grid size was chosen so large in the simulations we present here, that the cosmic-ray pressure is at all times essentially zero at both boundaries.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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