3. Test particle calculations
In order to solve the equation of motion (Eq. 1) with a magnetic and electric field given by Eqs. 2-4 we used a fourth-order Runge-Kutta method for ordinary differential equations (Press et al. 1992). Starting with a single proton reflected at the shock transition zone and moving back into the upstream region (cf. Fig. 2) there are two possibilities as depicted in Fig. 3. Either the proton is able to go through the approaching SLAMS (denoted by 'escape' in Fig. 3a) or the proton is reflected and consequently returns to the shock transition (denoted by 'return' in Fig. 3b). The two protons in Fig. 3 started in a shock-SLAMS-system given by and . The SLAMS approach the shock transition with and the particle starting velocities are and in Fig. 3a and 3b, respectively. Thus, the only difference between Fig. 3a and b is the magnitude of the starting velocity.
In general, the proton trajectories are influenced by six parameters (cf. Fig. 2): the magnetic field compression within the SLAMS, the initial proton velocity , the initial position of the SLAMS , and the angles , and . In order to study under which conditions the proton initially reflected at the shock front returns towards the shock after its interaction with the SLAMS we analysed situations with three fixed parameters. In a first step we analysed proton trajectories for fixed magnetic field compression , starting velocity and starting angle in a varying shock-SLAMS geometry, i.e., varying the angles and (see Fig. 4). In each panel of Fig. 4 representing different magnetic field compressions we find regions, in which all protons starting at the transition zone with a fixed angle are either reflected ('return') by the incoming SLAMS or go through them ('escape'). In the grey shaded regions the reflection behaviour depends on the starting position of the SLAMS. The location of the different kinds of regions is -for fixed - determined by . The width of the grey shaded area separating these regions is caused by the magnitude of the starting velocity, which causes different gyroradii , with as particle velocity perpendicular to the local magnetic field b. As long as the particle's gyroradius is small in comparison with the length scale on which the magnetic field changes, i.e., , the so-called adiabatic theory is applicable (e.g. Northrop 1963). According to this theory the reflection process at magnetic mirrors is purely determined by the particle's pitch angle (, particle velocity parallel to the local magnetic field). In each panel of Fig. 4 we show the theoretical computated dividing lines according to adiabatic theory for two different starting conditions and . The dividing line for a quasi-parallel shock without incoming SLAMS () is shown in the right half as a straight line. The parallel dotted line at shows the region of shock-reencounter (Schwartz et al. 1983). The second dotted line () takes into account, that SLAMS exist within an angular range .
The reflection condition in adiabatic theory can be obtained as follows (e.g. Krall & Trivelpiece 1986). For particles moving in smoothly varying magnetic fields and vanishing electric fields there are two constants of motion: the kinetic energy of the particle and the so-called "magnetic moment" defined by . From this it follows that the particle is reflected if the pitch angle of the particle is larger than the so-called loss-coss angle , with as maximum value of the total magnetic field strength. In order to use this simple relationship one has to look for a frame of reference in which the electric field vanishes, the so-called de Hoffmann-Teller frame (de Hoffmann & Teller 1950). This frame can be found by setting ( is the particle velocity in the shock rest frame), calculating the transformed electric field and requiring that the transformed electric field is equal to zero. Thus, using the Lorentz transformation formula for the electric field in first order of the transformed electric field is given by . Taking the magnetic field from Eq. 2 (with ) and the electric field according to Eq. 3 (with ) we obtain for the components of the de Hoffmann-Teller velocity
Furthermore, the reflection condition in the de Hoffmann-Teller frame can be written as
Since the velocities perpendicular and parallel to the magnetic field in the de Hoffmann-Teller frame contained in depend on trigonometrical functions of and (cf. Eq. 5), we obtain a transcendental equation for the reflection condition. The theoretical curves in Fig. 4 (the lines in the grey shaded areas) were computed numerically from Eq. 6 for and . The difference between the theoretical computed lines and the results from the test particle calculations are due to the fact that adiabatic theory is not valid for large gyroradii. The gyroradii for the proton trajectories depicted in Fig. 3 are about 2 and 4 times the ion inertial length for and , respectively, while the scale length of the SLAMS are in the order of 10 ion inertial lengths. In this velocity ranges the proton behaviour depends not only on the geometrical starting conditions of the particle, i.e., , but also on the specific conditions while entering the SLAMS.
The difference between the predictions of adiabatic theory and the effects of a finite gyroradius is also visible in Fig. 5, where we determined the magnetic field compression that is necessary for a reflection of the incoming protons. Fig. 5a shows the computated field compression corresponding to Eq. 6 for and and the results of our test particle caculations for protons starting with a velocity of for different values of . Fig. 5b shows the adiabatic behaviour for and and the test particle analysis for . Thus, Fig. 5 shows two different effects influencing the proton reflection process: Firstly, the geometric effect causing the location of the dotted lines. For protons starting nearly parallel, i.e., a small , to the propagation direction of the SLAMS (Fig. 5b) we need a large magnetic field compression to reflect them. This becomes clear from Eq. 6, which shows that we need a large to diminish . Secondly, the effect of a finite Larmor radius causes the different widths of the reflection zones in Fig. 5a and b. Furthermore, our test particle calculations showed, that the width for large starting velocities () increases in such a way that the simple picture of two divided areas for reflection and transmission breaks down. On the other hand we find for the plasmas under consideration here, i.e., the plasma in the interplanetary medium, a normalized thermal proton velocity in the order of 1, so that the velocity of slightly superthermal particles is smaller than 10.
In case of adiabatic particle behaviour the energy gained during the reflection process can be calculated using the aforementioned de Hoffmann-Teller frame. In the de Hoffmann-Teller frame the reflection process is described by and (see e.g. Krall & Trivelpiece 1986). Thus, the transformation back into the shock rest frame leads to
The difference between the starting velocity and the velocity of the reflected proton parallel to the undisturbed magnetic field according to this equation obtained for , , and is . On the other hand we found for the difference between the starting velocity and the mean velocity of the reflected protons parallel to the shock normal according to our test particle calculation a mean value of in good agreement with the theoretical expectations. Thus, Eq. 7 shows that the velocity gain (in the shock rest frame) is determined by the shock velocity and geometry on one hand (second term of the right side of Eq. 7) and by the velocity (third term) and the geometry (second and third term) of the approaching SLAMS.
© European Southern Observatory (ESO) 1998
Online publication: January 8, 1998