## 2. Injection model## 2.1. Wave-particle interaction at parallel shocksIn 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 ( The amplitude will be amplified downstream of the shocks by a factor . The downstream field can be described by a parameter , for which, following Malkov, we assume , 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).
## 2.2. Thermal leakage modelThe 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 , where the cyclotron frequency
of protons is given by , and
is the mean downstream thermal
velocity of the protons. We now have for the thermal protons
. ## 2.3. Transparency functionTo 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
where the particle velocity is normalized to
and The calculation of the transparency function and the wave-amplitude
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). The function (3) is plotted in Fig. 2 for vs. the particle velocity normalized to . 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 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 for the maximal allowed range in , as described above.
© European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |