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Scattering models at super-luminal shocks – a numerical study
Nuclear Physics B (Proc. Suppl.) 151 (2006) 505–508 www.elsevierphysics.com
Scattering models at super-luminal shocks - a numerical study A. Melia and J. J. Quenby a b
b
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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A. Meli, J.J. Quenby / Nuclear Physics B (Proc. Suppl.) 151 (2006) 505–508
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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A. Meli, J.J. Quenby / Nuclear Physics B (Proc. Suppl.) 151 (2006) 505–508
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
2.5
Figure 3. Spectral shape for the super-luminal large
2
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
angle diffusion case, in the shock frame at the downstream side for Γ=500 and ψ=55◦ .
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Figure 4. Spectral shape for the super-luminal pitch
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A. Meli, J.J. Quenby / Nuclear Physics B (Proc. Suppl.) 151 (2006) 505–508 3
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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
log(Energy)
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-
1.
Krymsky, G.F., 1977, Dokl.Acad.Nauk. SSSR 234, 1306 2. Bell, A.R., 1978a, MNRAS 182, 147 3. Bell, A.R., 1978b, MNRAS 182, 443 4. Axford, W.I., Leer, E. and Skadron, G. 1978, 15th ICRC Plovdiv, Bulgaria, Proc. 11, 132 5. Blandford, R.D. and Ostriker, J.P., 1978, ApJLet 221, L29 6. de Hoffmann, F. and Teller, E., 1950, Pr 80, 692 7. Begelman, M.C and Kirk, J.G., 1990, ApJ 353, 66 8. Protheroe, R. J., Meli, A., and A.-C. Donea, 2002, SpScRev 143, 276 9. Ellison, D.C., Jones, F.C., Reynolds, S.P., 1990, ApJ 360, 702 10. Meli, A. and Quenby, J. J., 2003, APh 19, 637 11. Meli, A. and Quenby, J. J., 2003, APh 19, 649
Scattering models at super-luminal shocks - a numerical study A. Melia and J. J. Quenby a b
b
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-
506
A. Meli, J.J. Quenby / Nuclear Physics B (Proc. Suppl.) 151 (2006) 505–508
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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A. Meli, J.J. Quenby / Nuclear Physics B (Proc. Suppl.) 151 (2006) 505–508
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
2.5
Figure 3. Spectral shape for the super-luminal large
2
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
log(Energy)
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-
1.
Krymsky, G.F., 1977, Dokl.Acad.Nauk. SSSR 234, 1306 2. Bell, A.R., 1978a, MNRAS 182, 147 3. Bell, A.R., 1978b, MNRAS 182, 443 4. Axford, W.I., Leer, E. and Skadron, G. 1978, 15th ICRC Plovdiv, Bulgaria, Proc. 11, 132 5. Blandford, R.D. and Ostriker, J.P., 1978, ApJLet 221, L29 6. de Hoffmann, F. and Teller, E., 1950, Pr 80, 692 7. Begelman, M.C and Kirk, J.G., 1990, ApJ 353, 66 8. Protheroe, R. J., Meli, A., and A.-C. Donea, 2002, SpScRev 143, 276 9. Ellison, D.C., Jones, F.C., Reynolds, S.P., 1990, ApJ 360, 702 10. Meli, A. and Quenby, J. J., 2003, APh 19, 637 11. Meli, A. and Quenby, J. J., 2003, APh 19, 649