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

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2. Injection model

2.1. Wave-particle interaction at parallel shocks

In supernova remnants (SNRs) the relative orientation of magnetic field and shock front can be very diverse, even in one single object. For example, if a spherically symmetric shock front expands in a region of homogeneous magnetic field, the directions of the shock normal and the magnetic field change over the shock surface from parallel to perpendicular. For nearly perpendicular shocks the acceleration process can be very fast and effective due to reflections upstream of the shock (Naito & Takahara 1995). However, the velocity of the intersection point of shock and magnetic field in highly oblique shocks can be close to light velocity. This can suppress the injection efficiency of thermal particles, which are effectively tied to magnetic field lines. That is because the velocity distribution function of thermal protons, which we shall assume to be Maxwellian, drops sharply towards high energies. This purely kinematical effect has been investigated by Baring et al. (1993). Therefore, regions of SNRs where quasi-parallel shocks exist, are likely to be where the most effective injection occurs. On the other hand, effective acceleration, i.e. short acceleration time scales and hard spectra, may be realized in other parts of a SNR, where an oblique geometry of magnetic field and shock normal is found.

For quasi-parallel shocks, where the shock propagates along the mean magnetic field (x-direction), the transport properties along the mean field direction are most important. We will assume this case, with the field [FORMULA] parallel to the shock normal. The spatial diffusion of particles is produced by magneto-hydrodynamic waves, which are in turn generated by particles streaming along the magnetic field, [FORMULA]. We refer to Malkov (1998) for an extended analytical description of the particle-wave interaction for low-momentum particles, and we describe here only the results which are relevant for the implication of this model in our simulations of the time dependent acceleration at modified shocks. When particles are streaming along the magnetic field in the upstream direction, waves are generated due to the ion cyclotron instability. The resulting upstream magnetic field, which corresponds to a circularly polarized wave, can be written as


The amplitude [FORMULA] will be amplified downstream of the shocks by a factor [FORMULA]. The downstream field can be described by a parameter [FORMULA], for which, following Malkov, we assume [FORMULA], in the case of strong shocks. Note that the perpendicular component of the magnetic field leads effectively to an alternating field downstream of the shock for particles moving along the shock normal (see Fig. 1).

[FIGURE] Fig. 1. Cartoon of the injection model in the shock-frame phase-space. Plasma is moving towards [FORMULA] into the shock with velocity [FORMULA] and gets compressed, heated and decelerated to the downstream velocity [FORMULA]. Particles with positive velocity can stream back to upstream along the magnetic field [FORMULA]. These particles provide the beam, which generates the magnetic field waves. The magnetic field wave is shown schematically in configuration space. The wave amplitude, frequency, and damping length is shown only qualitatively.

2.2. Thermal leakage model

The particles with a large enough gyro radius


can have an effective velocity with respect to the wave frame, i.e. the downstream plasma would be transparent. Some of these particles that are in the appropriate part of the phase space (depending on the shock speed) would be able to cross the shock from downstream to upstream. For the protons of the plasma, the resonance condition for the cyclotron generation of the Alfvén waves gives [FORMULA], where the cyclotron frequency of protons is given by [FORMULA], and [FORMULA] is the mean downstream thermal velocity of the protons. We now have for the thermal protons [FORMULA]. This means that most of the downstream thermal protons would be confined by the wave, and only particles with higher velocity in the tail of the Maxwellian distribution are able to leak through the shock. Ions with mass-to-charge ratio higher than protons have a proportionally larger gyro radius, so that the injection efficiency of protons would yield a lower limit for the less magnetized ions. On the other hand, for thermal electrons a plasma with such proton generated waves would have a reduced transparency due to the smaller gyro radius of the electrons. However, reflection of electrons (and protons) off the shock could become efficient with increasing wave amplitude and possibly aid in their injection (e.g. Levinson 1996; McClements et al. 1997). In the following we will focus on the protons, which carry most of the energy and momentum of the plasma.

2.3. Transparency function

To find the part of the thermal distribution for which the magnetized plasma is transparent, and, which, therefore, forms the "injection pool", Malkov (1998) solves analytically the equations of motion for protons in self-generated waves. He finds a transparency function [FORMULA], which expresses the fraction of particles that are able to leak through the magnetic waves, divided by the part of the phase space for which particles would be able to cross from downstream to upstream when no waves are present. For the adiabatic wave particle interaction the transparency function is given by Malkov (1998) Eq. (33), with [FORMULA], where v is the particle velocity and [FORMULA] is the velocity of the shock in the downstream plasma frame. Here [FORMULA] is the fraction of the particles streaming back from downstream to upstream. This quantity is divided by the fraction of particles that would be able to escape upstream in the absence of waves. In order not to further increase the complexity of our numerical simulation, we use here the following approximation of the representation given in Malkov (1998):


where the particle velocity is normalized to [FORMULA] and H is the Heaviside step function. We argued above that [FORMULA] (see Malkov 1998, Eq. 42). The transparency function now solely depends on the shock velocity in the downstream flow frame, [FORMULA], the particle velocity, v, and the relative amplitude of the wave, [FORMULA].

The calculation of the transparency function and the wave-amplitude [FORMULA] uses the ergodicity of the downstream phase-space for the randomized motion of particles in the high-amplitude wave field. The upstream wave field is generated by a beam of leaking particles whose energy density is calculated from the corresponding area of the downstream phase-space. From the energy density of this beam the upstream magnetic-field wave amplitude is determined self-consistently (Malkov 1998). 2 Because of this feedback Malkov was able to constrain the quantity [FORMULA] as [FORMULA], leaving essentially no free parameter. Comparison with hybrid plasma simulations suggests [FORMULA] (Malkov & Völk 1998), consistent with their analytical results. This constraint on the wave-amplitude [FORMULA] defines the level to which the particle-wave interaction adjusts. With the estimation of this amplitude there is no free parameter describing the level of the beam strength for the injection, and, therefore, the injection efficiency. The advantage of our approach presented here is that quantities like the plasma velocity and particle momentum distribution are calculated self-consistently by solving simultaneously the hydro-dynamical equations together with the cosmic-ray transport equation (see Sect. 3.1).

The function (3) is plotted in Fig. 2 for [FORMULA] vs. the particle velocity normalized to [FORMULA]. In Fig. 2 we have also reproduced this function as given in Fig. 2 of Malkov & Völk (1998) to allow a direct comparison. The strong velocity dependence and also the asymptotic behavior is modeled reasonably well by the representation Eq. (3). In the normalization of Fig. 2, the dependence on [FORMULA] is very weak, and, therefore, not shown. To illustrate the dependence of the transparency function on small variations of the field amplitude, it is better to choose a different normalization. Therefore, the transparency function Eq. (3) is shown in Fig. 3 vs. the velocity normalized to [FORMULA] for the maximal allowed range in [FORMULA], as described above.

[FIGURE] Fig. 2. Transparency function Eq. (3) vs. the normalized particle velocity [FORMULA] for [FORMULA] (solid line). Shown as a dashed line is the transparency function given in Fig. 2 of Malkov & Völk (1998).

[FIGURE] Fig. 3. Transparency function Eq. (3) vs. the normalized particle velocity [FORMULA] for different values of [FORMULA]. We used the relation [FORMULA] (see text).

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001