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Scattering models at super-luminal shocks - a numerical study A. Melia and J. J. Quenby a b

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Institut f¨ ur Physik, Universit¨at Dortmund, D-44221 Dortmund, Germany Imperial College London, Astrophysics Group, Blackett Laboratory SW7, 2BW, London UK

We present Monte Carlo calculations of the cosmic ray spectral shapes and cosmic ray energy gains for diﬀusive shock acceleration at highly oblique relativistic shocks, namely super-luminal shocks, with high upstream ﬂow Lorentz gamma factors, from 10 up to 1000, which could be relevant to models of relativistic astrophysical environments like Active Galactic Nuclei jets and Gamma Ray Burst sites. Both large angle and pitch angle diﬀusion of the accelerated cosmic rays is considered, applying to the relevant astrophysical models. Our physical/numerical approach allows that the particle’s trajectory is followed along the magnetic ﬁeld lines during shock crossings where the equivalent of a guiding centre approximation is inappropriate, measuring its phase space co-ordinates in the ﬂuid frames, where E=0. We ﬁnd striking diﬀerences in the cosmic ray energy spectral shapes comparing large angle scattering versus pitch angle diﬀusion methods; showing that the cosmic ray case of diﬀusion in superluminal shocks could be the ’key’ of the problem of the universality of the diﬀusive shock acceleration (Fermi acceleration).

1. Introduction It is widely believed that shocks are a source of particle (cosmic rays) acceleration. In particular, the development of the diﬀusive shock acceleration model, whereby cosmic rays are repeatedly accelerated in multiple crossings of a shock interface due to collisions with upstream and downstream magnetic scattering centers, has been stimulated by the detection of 106 − 1020 eV nuclei, implying their presence through much of the observable universe. The work of the late 70s by a number of authors such as [1], [2],[3] [4], [5] established the basic mechanism of particle diﬀusive acceleration (in non-relativistic ﬂows). Shocks can be classiﬁed into parallel and oblique ones with respect to the orientation of the magnetic ﬁeld with the shock normal, and oblique shocks can be classiﬁed into sub-luminal and super-luminal ones. The diﬀerence between these last two categories is that in the sub-luminal case, it is possible to ﬁnd a relativistic transformation to the frame of reference (de Hoﬀmann-Teller frame [6]), in which the shock front is stationary and the electric ﬁeld is zero (E=0) in both upstream and downstream regions. On the other 0920-5632/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2005.07.090

hand, super-luminal shock fronts do not admit a transformation to such a frame, as they correspond to shock fronts in which the point of intersection of the front with a magnetic ﬁeld line moves with a speed greater than c. Furthermore the super-luminal shocks can have non-relativistic and relativistic velocities. Here we have established a numerical Monte Carlo method to study the cosmic ray acceleration in highly relativistic super-luminal shocks which still puzzles the researchers in the ﬁeld. 2. Monte Carlo simulations - Results The simulations run with both large angle and pitch angle elastic scattering which is taking place in the ﬂuid frames. An investigation of the role of diﬀerent scattering models in reference to spectral shape at very high gamma plasma ﬂows is the aim, and the plasma ﬂow velocity ranges from Γ = 10 − 103 . As we mentioned earlier, we consider super-luminal shocks and we choose the angle to range as follows:ψ ≤ 75o where ψ is the angle between the shock normal and the magnetic ﬁeld, seen in the shock frame. In general we note here that the ﬂow into and out of the shock dis-

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continuity is not along the shock normal, but a transformation is possible into the normal shock frame to render the ﬂows along the normal [7]. For simplicity we assume such a transformation has already been made. For these simulation runs a Monte Carlo technique is applied while considering the motion of a particle of momentum p in a magnetic ﬁeld B. As mentioned, in super-luminal conditions it is not possible to transform into a frame where E=0 (de Hoﬀmann-Teller frame) a condition that is only possible for the subluminal case where, V ≤ Vsh tanψ. Thus, the frames to be used in this simulation will be the ﬂuid frames (where the electric ﬁeld is still zero, E=0) and the shock frame, which will be used only as a ’check frame’ to test whether upstream or downstream conditions apply. Initially the particles are injected at 50λ 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 ﬂow of the plasma upstream is highly relativistic as we keep the gamma of the ﬂow between 10 and 1000, as mentioned earlier. For the pitch angle scattering we 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 = δθ2 λ, keeping the angle δθ less than 0.1/Γ ([8] private communication) while following the diﬀusion approximation of the scattering during and immediately before the particle reaches the shock front. There is no easy approximation at this juncture to determine the probability of shock crossing or reﬂection, so we change the model following the helical trajectory of the particle, in the ﬂuid frames upstream (’1’) or downstream (’2’)(E=0), respectively, where the velocity coordinates of the particle are calculated in a three dimensional space. We assume that the tip of a particle’s momentum vector undergoes randomly a small change θ1 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 θo , we calculate its new pitch angle θ by a trigonometric formula [9]. We follow the

trajectory in time, using φ1 = φ◦ + ωt, and t is the time from ﬁrst detecting the shock presence at xsh , ysh , zsh and assuming that δt = rg /Hc, where rg is the Larmor radius, H ≥ 100. After the suitable calculations we check to see 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 and we follow the particle’s trajectory as described above. If the particle never meets the shock, then 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 100λ. In ﬁgures 1-8 the reader may see the spectral shapes for the large angle scattering and pitch angle diﬀusion for diﬀerent super-luminal angles. The ﬁndings are that 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 of [10] and [11] for highly relativistic sub-luminal shocks, where plateau structured spectral shapes are seen.

Figure 1. Spectral shape for the super-luminal large angle scattering case, in the shock frame at the downstream side for Γ=10 and ψ=48◦ .

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angle diﬀusion case, in the shock frame at the downstream side for Γ=50 and ψ=55◦ .

Figure 2. Spectral shape for the super-luminal large angle scattering case, in the shock frame at the downstream side for Γ=30 and ψ=75◦ .

Figure 5. Spectral shape for the super-luminal pitch angle diﬀusion case, in the shock frame at the downstream side for Γ=200 and ψ=50◦ . 3

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Figure 3. Spectral shape for the super-luminal large

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angle scattering case, in the shock frame at the downstream side for Γ=40 and ψ=75◦ .

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Figure 6. Spectral shape for the super-luminal pitch 0.5

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Figure 4. Spectral shape for the super-luminal pitch

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Figure 7. Spectral shape for the super-luminal pitch angle diﬀusion case, in the shock frame at the downstream side for Γ=1000 and ψ=46◦ . 2

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tering perturbation of the particle’s helical trajectory to be followed along the magnetic ﬁeld lines during shock crossings where the equivalent of a guiding centre approximation is inappropriate while measuring its phase space co-ordinates in the ﬂuid frames, where E=0. Striking diﬀerences in the cosmic ray energy spectral shapes while using large angle scattering versus pitch angle diﬀusion show that the characteristic powerlaw spectra of the diﬀusive acceleration is not expected for both cases. The energy gain of the cosmic rays in superluminal shocks is more limited, either due to a sharp cutoﬀ or the steepness of the power law which shows negative slopes of about 2, rather than closer to 1, as seen at intermediate values of Γ between 10-100. This is also possible due to the fact that they are swept away easily by the plasma ﬂow which runs with velocities close to the speed of light and is comparable to the cosmic rays velocity. The diﬀerences in the cosmic ray spectral shapes could give further insights into the problem of the universality of the diﬀusive (Fermi) acceleration in highly oblique (super-luminal) relativistic shocks at the source; with further consequences to the subsequent wavelength dependent emission properties from such environments. Further detailed study will concern the acceleration rates for this kind of acceleration.

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REFERENCES

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Figure 8. Spectral shape for the super-luminal pitch angle diﬀusion case, in the shock frame at the downstream side for Γ=1000 and ψ=76◦ .

3. Conclusions We presented Monte Carlo calculations of the cosmic ray spectral shapes for particle shock acceleration at super-luminal shock waves with two diﬀerent scattering models (namely large angle scattering and pitch angle diﬀusion). We used high upstream ﬂow Lorentz gamma factors, ranging from 10 up to 1000, which could be relevant to models of relativistic astrophysical environments like Active Galactic Nuclei jets and Gamma Ray Bursters. Our numerical approach allows scat-

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