Monte Carlo simulations

The aim of paper Meli et al. (2005) is to examine which is the role that different scattering models (large angle scattering or pitch angle diffusion) can play in reference to the spectral shape at very high gamma plasma flows, by considering superluminal shock configurations. It is necessary to note that the flow into and out of the shock discontinuity is not along the shock normal, but a transformation is possible into the normal shock frame to render the flows along the normal (Begelman and Kirk, 1990) and for simplicity it was assumed such transformation has already been made. For these simulation runs a Monte Carlo technique is applied by considering the motion of a particle of momentum p in a magnetic field B. As it was mentioned above, in super-luminal conditions it is not possible to transform into a frame where E = 0 (i.e., into De Hoffmann-Teller frame) a condition that is only possible for the sub-luminal case where u < ush tan^ . Thus, the frames to be used in this simulation will be the fluid frames, where still the electric field is zero E = 0, and the shock frame, which it will be used only as a 'check frame' to test whether upstream or downstream conditions apply. Initially the particles are injected at 50X, where X is the particles' mean free path, from the shock and their guiding center is followed upstream, at the upstream frame until the particle reaches the shock at xsh = 0 followed by an appropriate transformation to the shock frame. At injection the speed of the plasma upstream is highly relativistic and the values between r = 10 and 1000 are kept. For the pitch angle scattering Mali et al. (2005) follow a standard Monte Carlo random walk approach to simulate the energetic particle propagation by using a small angle scattering procedure with steps sampled from an exponential distribution with mean free path L = 89 X and keeping the angle 89 less than 0.1/r (Berezhko and Volk, 2000) while following the diffusion approximation of the scattering during and immediately before the particle reaches the shock front. Since there is no easy approximation at this juncture to determine the probability of shock crossing or reflection, Meli et al. (2005) change the model following the helical trajectory of the particle, in the fluid frames upstream (index 1) or downstream (index 2) at E = 0, respectively, where the velocity coordinates of the particle are calculated in a three-dimensional space. They assume that the tip of a particle's momentum vector undergoes randomly a small change 0 in its direction on the surface of a sphere and within a small range of polar angle (after a small increment of time). If the particle had an initial pitch angle 0o, it was calculate its new pitch angle 0 by a trigonometric formula (Ryu et al., 1993). After they follow the trajectory in time, using <\=<0 + cat, and t is the time from first detecting the shock presence at xsh, ysh, zsh and assuming that St = rg/Hc, where rg is the Larmor radius,

H > 100 . After the suitable calculations it was checked whether the particle meets the shock again by transforming to the shock frame. If the particle meets the shock then the suitable transformations to the upstream frame are made again made again and they follow the particle's trajectory as described above. If the particle never meets the shock its guiding center is followed, the same way as mentioned earlier for the upstream side after the injection and it is left to leave the system if it reaches a well defined Emax momentum boundary or a spatial boundary of 100a .

4.30.3. Main results

Main results are shown in Fig. 4.30.1

4.30.3. Main results

Main results are shown in Fig. 4.30.1

Fig. 4.30.1. Spectrum for the super-luminal pitch angle diffusion case, in the shock frame at the downstream side for r = 10 and y = 89°. As an indication a gamma of 105 corresponds to ~ 100 GeV for protons. For the right panel it is note that the steep cutoff may suggest a connection to the spectra of relativistic electrons originating from observed hot spots (super-luminal shocks) in extragalactic radio sources. From Meli et al. (2005).

Fig. 4.30.1. Spectrum for the super-luminal pitch angle diffusion case, in the shock frame at the downstream side for r = 10 and y = 89°. As an indication a gamma of 105 corresponds to ~ 100 GeV for protons. For the right panel it is note that the steep cutoff may suggest a connection to the spectra of relativistic electrons originating from observed hot spots (super-luminal shocks) in extragalactic radio sources. From Meli et al. (2005).

log (Energy) log (Energy) log (Energy)

Fig. 4.30.2. Spectral shapes for r = 500 (left panel), r = 1000 (middle panel) for y = 76°. In the right panel are shown two spectra for r = 50 and an inclination of y = 50° and y = 89°, respectively. It may be see no dependence of the spectrum with the angle of the magnetic field at the shock normal. This is tested for all gamma and different shock angles. The behavior is the same. From Meli et al. (2005).

log (Energy) log (Energy) log (Energy)

Fig. 4.30.2. Spectral shapes for r = 500 (left panel), r = 1000 (middle panel) for y = 76°. In the right panel are shown two spectra for r = 50 and an inclination of y = 50° and y = 89°, respectively. It may be see no dependence of the spectrum with the angle of the magnetic field at the shock normal. This is tested for all gamma and different shock angles. The behavior is the same. From Meli et al. (2005).

From Fig. 4.30.1-4.30.2 one may understand that for pitch angle diffusion, the spectral shape of the accelerated particles follows a rather smooth power-law shape in comparison to the large angle scattering where the spectral shape gives a steep sudden cut-off. For both cases the simulations show that most of the particles are 'swept' downstream the shock after only a cycle. This condition limits the particle's ability to gain very high energies, contrary to the simulation findings in Meli and Quenby (2003a,b) for highly relativistic sub-luminal shocks, where plateau structured spectral shapes are seen, however.

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