THE PHYSICS OF PARTICLE ACCELERATION BY COLLISIONLESS SHOCKS

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1 THE PHYSICS OF PARTICLE ACCELERATION BY COLLISIONLESS SHOCKS Joe Giacalone Lunary & Planetary Laboratory, University of Arizona, Tucson, AZ, 8572, USA ABSTRACT Using analytic theory, test-particle simulations, and self-consistent hybrid simulations, we show that quasi-perpendicular shocks those which propagate nearly perpendicular to the upstream magnetic field accelerate particles directly out of the incident thermal population to energies much higher than the upstream ram energy of the plasma. It has already been established that quasi-parallel shocks those which propagate nearly in the same direction as the upstream magnetic field efficiently accelerate particles directly out of the incident thermal population; however this has not yet been established for quasi-perpendicular shocks. Our results can be understood within the framework of the diffusive shock acceleration theory. We find that the accelerated-particle spectrum obtained from a more-general self-consistent hybrid plasma simulation are quantitatively consistent with a less-sophisticated test-particle simulation. The implications of this are discussed. INTRODUCTION The acceleration of charged particles by collisionless shocks has been studied extensively (see the reviews by Jones and Ellison, 99; and Drury 983) and yet there is still no widely-accepted theory that explains which, and how many particles of a thermal population incident on a collisionless shock will be accelerated to high energies. Diffusive shock acceleration theory a well-known and elegant theory of the acceleration of particles at shocks naturally explains the commonly observed power-law dependence of the distribution of cosmic rays on energy. It has proven useful in understanding many important phenomena related to high-energy particles in astrophysical plasmas. However, the key assumption in this theory is that the particle distribution must be isotropic to first order. This is not observed, nor is it expected from analytic theory, to be the case for particles whose energy is near that of the bulk of the plasma particles (i.e. the thermal peak). Consequently, the diffusive theory is not adequate to describe how the particles are injected. The injection of particles at shocks depends upon, among other things, the angle between the shock normal direction and the upstream magnetic field, θ Bn. For shocks with θ Bn < 30 quasi-parallel shocks it has been established that particles are efficiently accelerated from thermal energies to energies well beyond the incident ram energy of the plasma (Ellison, 98; Scholer, 990; Giacalone et al., 992; Kucharek and Scholer, 995). Conversely, for perpendicular shocks those for which θ Bn > 85 it has not yet been established that thermal particles can be accelerated to high energies (although it has been demonstrated that pickup ions may be accelerated by nearly perpendicular shocks (Ellison et al., 999; Giacalone and Ellison, 2000). It is the purpose of this paper to demonstrate that these shocks are also capable of accelerating thermal particles to high energies, but, perhaps, at a lower acceleration efficiency. ANALYTIC CONSIDERATIONS: THE LIMIT OF DIFFUSIVE SHOCK ACCELERATION

2 For purpose of illustration, we first discuss the diffusive approximation, although the numerical computations discussed below are more general. It is well known that the energy spectrum of chargedparticles which can be described by the diffusive transport equation, downstream of a collisionless shock, is a power law in the absence of losses. Since this theory does not address injection, we consider the limits under which the theory is valid. It is assumed that the pitch-angle distribution is quasiisotropic; thus, we require that the diffusive-streaming anisotropy in the plasma frame of reference to be small. The general form of this condition is given by Giacalone and Ellison (2000) δ = 3U [ + (κ A/κ ) 2 sin 2 θ Bn + ( κ /κ ) 2 sin 2 θ Bn cos 2 ] θ /2 Bn () v ((κ /κ ) sin 2 θ Bn + cos 2 θ Bn ) 2 where U is the flow speed upstream of the shock, v is the plasma-frame particle velocity, κ A is the antisymmetric component of the diffusion tensor, and κ, are the components perpendicular and parallel to the average magnetic field. f (w) w > w inj w < w inj Diffusive Non-Diffusive Super-thermal tail Maxwellian Power-law w inj w Figure : A distribution function showing a thermal peak and suprathermal tail. The distribution is power-law at high energies (w > w inj ). This figure is adapted from Giacalone (200). Shown in Figure is a cartoon illustration of a distribution function with a thermal peak and highenergy tail. w inj is a characteristic speed at which the process is diffusive (i.e. w inj is the value of v in Equation () for which δ = ). Our interpretation of the injection problem is that the physics which describes the suprathermal tail, i.e. its velocity dependence and density is not known. There are a number of processes described in the literature which address this physics, but there is currently no consensus theory. Examples of some important factors contributing to the formation of suprathermal tails are the microphysics of the shock layer (Zank et al., 996; Lee et al., 996; Giacalone et al., 99) and also statistical mechanisms such as transit-time damping (Schwadron et al., 996). We note that for a parallel shock, θ Bn 0 and () reduces to δ = 3U /v. Thus, parallel shocks are efficient accelerators of particles whose speed is on the order of the plasma flow speed. Except for low-temperature plasmas, there are usually a significant number of particles within the thermal pool which can easily be accelerated by a parallel shock. Self-consistent hybrid simulations confirm this (Scholer, 989; Giacalone et al., 992). We also note that this result is independent of the diffusion coefficient. Hence, parallel shocks are capable of accelerating low-energy particles regardless of their mean-free path. 2

3 For a perpendicular shock we take the limit θ Bn 90 in () yielding δ = 3U [ + v ( κa κ ) 2 ] /2 (2) In the case of classical scattering, κ and κ A can be written in terms of the ratio of the scattering mean-free path, λ to the particle gyroradius, r g : κ = (/3)(r g /λ )vr g, κ A = (/3)vr g. For this case, Equation (2) becomes, δ = 3U [ ( ) λ 2 ] /2 + (3) v Equation (3) implies that the stronger the scattering (and hence, the smaller the mean-free path) the more efficient the particle injection. This is in agreement with the Monte-Carlo simulations by Baring et al. (994). On the other hand, there is no reason to believe that classical scattering theory gives a valid description of the cross-field diffusion coefficient. In fact, Giacalone and Jokipii (999) showed that for certain forms of the magnetic turbulence spectra, κ does not agree with the theory of classical-scattering. We may consider other cases which seem reasonable, such as, taking κ /κ to be a constant. For this case, we have δ = 3U [ ( ) κ 2 ( + v κ r g λ /r g + (λ /r g ) 2 where we use κ A = (/3)vr g as we did before (this is a reasonable representation). ) 2 ] /2 (4) Equation 4 shows that, contrary to the case of classical scattering (Equation 3), δ decreases with the mean-free path. Thus, for the case of a constant κ /κ, weaker scattering (longer mean-free path) is required in order to have more efficient injection. A constant ratio of κ /κ may be reasonable for charged particles moving in the large-scale (much larger than the particle gyroradius) meandering of the ambient magnetic field lines which intersect a non-planar shock front. The larger diffusion coefficient implies that particles are in the diffusive limit at lower energies than expected from classical scattering theory which assumes small-scale fluctuations of the magnetic field (i.e. nearly straight magnetic field lines) and a planar shock front. TEST-PARTICLE NUMERICAL SIMULATIONS In this section, we use a more-general numerical model to illustrate the importance of the diffusivestreaming anisotropy in the role of injection and acceleration of particles at quasi-perpendicular shocks. Our model is similar to that of Baring et al. (994) except that here we treat the particles as test particles. The model described here is less sophisticated than the self-consistent plasma simulation described in the next section. We consider a planar shock in which plasma flows in one direction opposite to the shock-normal direction and abruptly decreases across the shock. Embedded in this flow is a magnetic field perpendicular to the flow direction. The field abruptly jumps at the shock to a value of 3.7 times its upstream value. This value comes from the shock-jump (or Rankine-Hugoniot) relations using an upstream Alfvén Mach number of 0 and an upstream sonic Mach number of 0 (a total plasma beta of 6/5). Test particles are released just behind the shock front (in the downstream region) with a speed, measured in the downstream plasma frame, equal to the upstream flow speed. The distribution is initially isotropic on a shell in velocity space in the downstream plasma frame of reference. The trajectories of the particles are integrated by solving the Lorentz force acting on each particle. The fields which are 3

4 used in this calculation are determined at the particle position. We ignore electromagnetic structures inside the shock layer and consider only the motional electric field (which is a constant). The particles are scattered phenomenologically by determining a scattering time based on an exponential distribution with a mean scattering time, τ. The particles are scattered isotropically in the local plasma frame of reference. An upstream free-escape boundary exists at a position 5000r 0 (where r 0 is the initial gyroradius of the particles) upstream of the shock. Upon crossing this boundary, the particle is lost. Downstream of the shock, there exists another boundary. When particles cross this boundary, which is located 0 local gyroradii downstream of the shock, the particle is tested for whether it will escape the system (based on the probability-of-escape formula described in Jones and Ellison, 99). If it is determined that it will escape, its orbit integration is complete and diagnostics are tabulated. This method produces a steady-state particle distribution function. Figure 2a shows the results of three test-particle simulations using different values of the mean scattering time τ. Clearly the scattering time has a profound influence on the resulting energy spectra. This can be understood in terms of the discussion of the previous section (see Equation 3). Note that below a particular energy, each spectra is steeper than it is at higher energies. For speeds above the critical speed, the slope of the spectra agree with the prediction of the standard diffusive theory (. for the compression ratio used in this study). The critical speed increases with increasing scattering time (which is related to the particle mean-free path), which is consistent with Equation 3. Flux, J (arbitrary units) Downstream Energy Spectra a Energy, E p E - Ωτ = 20 E -4 Ωτ = 50 Ωτ = 00 Flux, J (arbitrary units) Ωτ = 20 Ωτ = 50 Ωτ = 00 b Anisotropy, δ Figure 2: (a) Simulated energy spectra downstream of the shock for the test particle simulations. Plotted is the differential intensity (p 2 f, where p is momentum and f is the phase-space distribution function) as a function of energy. The energy is normalized to the plasma ram energy, E p. Three different mean scattering times are shown. (b) The same as (a), except that the flux is plotted against the diffusive-streaming anisotropy. We also note that the the spectra roll over at the maximum energies due to the presence of the free- 4

5 escape boundary. The spectra roll over at a different energy for different scattering times due to the fact that the quantity determining this critical energy is the distance to the free-escape boundary in units of the diffusive skin depth, d = κ /U (U is the upstream plasma flow speed). For the case of Ωτ=00, is very small and thus the distance to the free-escape boundary in units of is much larger than is the case for Ωτ=20, and particles can reach higher energies. In Figure 2b, the same spectra shown in Figure 2a are replotted against the diffusive-streaming anisotropy δ. Plotted in this way, each spectra are steeper than the diffusive-theory prediction for δ > 3, and in agreement with the theory for δ < 3. To this point, we have assumed a crude model for the shock to illustrate the importance of the diffusive-streaming anisotropy with regards to the injection process. We have not included additional physical processes such as large-scale magnetic fluctuations and the electrodynamics within the shock transition layer. Below we present a more-general treatment using a self-consistent hybrid simulation. SELF-CONSISTENT PLASMA SIMULATIONS In order to study the injection and acceleration of thermal particles incident upon a more-realistic quasi-perpendicular shock, we utilize the well-known hybrid simulation. This method is useful for studying the physics of plasmas in which the ion kinetic effects are important. The ions are treated as individual particles and the electrons are treated as a charge-neutralizing, massless fluid. The electric field is determined from the electron momentum equation. The magnetic field is determined from Faraday s law using the ion density and flow speed determined from moments collected after advancing the ions. Ampere s law and the assumption of quasi-neutrality are also used. The model used in this study is a two-dimensional version of that used in a previous study by Giacalone and Ellison (2000). We use a two-dimensional simulation because we wish to include the affects of the large-scale meandering of magnetic field lines upstream of the shock as well as those arising from a non-planar, rippled, shock front. Computational constraints restrict our model to a two dimensional one. In our model, plasma flows in the positive x direction and is continuously injected at x = 0. The plasma reflects off of a perfectly conducting wall at x = x max. The plasma piles up near the wall and a density wave propagates back into the system which forms into a shock propagating against the flow. The system is periodic in the z dimension. We use an inflow Mach number of 4; the resulting shock-frame Mach number M A 5.4. For the simulation described here, we use a box of dimensions x max = 250c/ω i and z max = 250c/ω i, where c/ω i is the ion inertial length. Other important parameters of the simulations include the total plasma beta (electron plus ion) and shocknormal angle and are given by β =, θ Bn = 90. Additional parameters, listed for completeness, are: x = z = 0.5c/ω i (grid cell size), t = 0.02Ω i (time step), T e /T i = (electron to ion temperature ratio), and η = πωi (anomalous resistivity), where Ω i is the ion cyclotron frequency, and ω i is the ion plasma frequency. The initial spatially-uniform ion distribution was generated using 20 particles per cell. Using this rather limited number of particles per cell we have found that energy is conserved to better than 8%. We also incorporate a particle-splitting algorithm to improve the statistics of the high-energy tail of the distribution. We use the same method as in previous models (Giacalone et al., 992; Giacalone et al., 994). The initial magnetic field lies in the x z plane. We superimpose on this a set of forward and reverse propagating, right- and left-hand circularly polarized Alfvén waves at the start of the simulation. These waves are also continuously injected into the system through the duration of the simulation. A 5

6 description of this method can be found in the papers by Giacalone and Ellison (2000) and Giacalone et al. (992). It has been established that for a two-dimensional magnetic field which does not point in the ignorable direction, particles are unable to move off of the magnetic line of force on which they first start (Jokipii et al., 993; Giacalone 994). Thus, cross-field diffusion is artificially suppressed. To overcome this limitation, we introduce ad hoc scattering as we did previously in a one-dimensional model (Giacalone et al., 994). For the simulation discussed here, we use Ω i τ = 50. Figure 3 shows the gray-scale intensity of B z at t = 80Ω i obtained from the simulation. The largescale rippling of the shock front is due to the shock moving into a non-uniform medium containing the injected magnetic waves that we have superimposed on the system. We note in passing that the rippling of the shock front also occurs in a simulation in which no waves are injected into the system. In this case, the rippling occurs as a result of the non-uniform excitation of waves created by reflected ions. Particles incident at different points along the non-planar shock front will see a different local shock-normal angle. Thus, if the local shock-normal angle is more parallel, then particles may be accelerated more efficiently B z t = 80 Ω ī z, c/ω i B x, c/ω i B Figure 3: Gray-scale plot of the z-component of the simulated magnetic field obtained from the hybrid simulation as a function of position at t = 80Ω i. The entire simulation domain is shown. Shown in Figure 4 are the energy spectra computed at different times (as shown) downstream of the shock. Particles are accelerated to well over 000 times the plasma ram energy E p. The acceleration is very rapid as particles are accelerated to 000E p in about 00 gyroperiods. For particles accelerated near the quasi-perpendicular region of the Earth s bow shock, this would correspond to particles being accelerated to over an MeV in about three minutes. 6

7 E - Flux, J (arbitrary units) Downstream Energy Spectra t = 40 Ω ī t = 80 Ω ī t = 20Ω ī E Energy, E p Figure 4: Differential flux as a function of energy for particles downstream of the shock obtained from the hybrid simulation. Three different times are shown. The spectra shown in Figure 4 are quite steep indicating a very inefficient acceleration process. Moreover, the spectra are quantitatively consistent with those shown in Figure 2 for the cruder test-particle model. Note that from E = E p to E = 000E p, the spectra drops 2 orders of magnitude. This is nearly the same as the case of Ωτ = 50 shown in the Figure 2a. This suggests that the perpendicular diffusion coefficient for the particles is not enhanced by the large-scale field-line random walk in this model as we anticipated. However, we chose a rather large scattering time of 50 gyroperiods which is just less than /2 of the overall simulation time. Thus, it is conceivable that a longer simulation is necessary to see the effects of the field line random walk. Another implication of the fact that the spectra obtained in the hybrid and test-particle models are consistent is that the waves which are self generated by the accelerated particles have very little energy. If they had more energy, the scattering mean-free path would be reduced and the spectra would not be consistent with the results of the test-particle simulation for Ωτ = 50. Figure 5 shows the trajectory of a typical accelerated particle in the hybrid simulations. All of the particle s energy gain comes at the shock which is indicated by a gray band in the figure. The acceleration of this particle is very rapid. It gained some 40E p is about 0 gyroperiods. A similar solar-wind ion encountering the Earth s bow shock would have been accelerated to 50keV in 20 seconds. CONCLUSIONS We have discussed the injection problem arising from the inability of analytic theory to explain the 7

8 60 shock Energy, E p Energy, E p x, c/ω i z, c/ω i x, c/ω i t, Ω ī Figure 5: A typical accelerated proton in the hybrid simulations. (left) Kinetic energy, in units of the plasma ram energy, as a function of position with the gray band indicating the shock location (which moves about a mean location). (upper right) Energy as a function of time. (middle right) Particle s z location as a function of time. (lower right) Particle s x location as a function of time. entire distribution function, from energies typical of the thermal peak to high energies, of charged particles accelerated by collisionless shocks. Using self-consistent plasma simulations, simple testparticle simulations, and analytic theory, we have shown that quasi-perpendicular shocks are capable of accelerating thermal particles to energies much higher than the plasma ram energy. Diffusive shock acceleration, which assumes that the particle distribution function is isotropic to first order, naturally explains high-energy tails (e.g. power law, exponential etc.) in the energy distribution. We have considered the validity of this theory for the special cases of parallel and perpendicular shocks. On the one hand, we have shown that the diffusive approximation is more easily met at parallel shocks than perpendicular shocks. On the other hand, we have also shown that, in general, perpendicular shocks can be effective at accelerating low-energy particles. Charged particles moving along the largescale meandering of magnetic field lines can enhance the diffusion normal to the shock front. This enables particles to remain near the shock and be accelerated. The enhanced diffusion coefficient reduces the diffusive-streaming anisotropy which means that particles are in the diffusive limit at lower energies than expected from the theory of classical scattering. ACKNOWLEDGEMENTS This work was supported by NASA under grants NAG5-225 and NAG REFERENCES Baring, M. G., D. C. Ellison, and F. C. Jones, Monte-Carlo simulations of particle-acceleration at oblique shocks, Astrophys. J. Suppl. Ser. 90, 547,994. Drury, L., An introduction to the theory of diffusive shock acceleration of energetic particles in tenuous plasma, Rep. Prog. Phys., 46, 973, 983. Ellison, D. C., Monte Carlo simulation of charged particles upstream of the Earth s bow shock, Geophys. Res. Lett., 8, 99, 98. 8

9 Ellison, D. C., F. C. Jones, and M. G. Baring, Direct acceleration of pickup ions at the solar wind termination shock: The production of anomalous cosmic rays, Astrophys. J., 52, 403, 999. Giacalone, J., The injection problem, The Outer Heliosphere: The Next Frontiers, COSPAR Colloquium series, vol., 528, Scherer et al. eds, p. 377, 200. Giacalone J., and J. R. Jokipii, The transport of cosmic rays across a turbulent magnetic field, Astrophys. J. 520, 204, 999. Giacalone, J., On the cross-field diffusion of ions in one- and two-dimensional hybrid simulations of collisionless shocks, Geophys. Res. Lett., 2, 244, 994. Giacalone, J., and D. C. Ellison, Three-dimensional hybrid simulations of particle injection and acceleration at quasi-perpendicular shocks, J. Geophys. Res. 05, 254, Giacalone, J., D. Burgess, S. J. Schwartz, and D.C. Ellison, Hybrid simulations of protons strongly accelerated by a parallel collisionless shock, Geophys. Res. Lett 9, 433, 992. Giacalone, J., J. R. Jokipii, and J. Kóta, Ion injection and acceleration at quasi-perpendicular shocks, J. Geophys. Res. 99, 935, 994. Giacalone, J., T. P. Armstrong, and R. B. Decker, J. Geophys. Res. 96, 362, 99. Jokipii, J. R., J. Kóta, and J. Giacalone, Perpendicular transport in - and 2-dimensional shock simulations, Geophys. Res. Lett., 20, 759, 993. Jones, F. C., and D. C. Ellison, The plasma physics of shock acceleration, Space Sci. Rev., 58, 259, 99. Kucharek, H., and M. Scholer, Injection and acceleration of interstellar pickup ions at the heliospheric termination shock J. Geophys. Res. 00, 745, 995. Lee, M. A., V. D. Shapiro, and R. Z. Sagdeev, J. Geophys. Res. 0, 4777, 996. Scholer, M., Diffuse ions at a quasi-parallel collisionless shock: Simulations, Geophys. Res. Lett., 7, 82, 990. Schwadron, N. A., L. A. Fisk, and G. Gloeckler, Geophys. Res. Lett. 2, 287, 996. Zank, G. P., H. L. Pauls, I. H. Cairns, and G. M. Webb, J. Geophys. Res., 0, 457,

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