We investigate ways of accurately simulating the propagation of energetic
charged particles over small times where the standard Monte Carlo approximation
to diffusive transport breaks down. We find that a small-angle scattering
procedure with appropriately chosen step-lengths and scattering angles gives
accurate results, and we apply this to the simulation of propagation upstream
in relativistic shock acceleration.Comment: 4 pages, 2 figures, proceedings of World Space Environment Forum
  (WSEF2002) to appear in Space Science Reviews, accepte

Donea, A. -C

Meli, A.

Protheroe, R. J.

Space Science Reviews

English

arXiv

Copyright © 2003. Space Science Reviews. All rights reserved. Printed in U.S.A. Submitted to Cornell University’s online archive www.arXiv.org in 2003 by Ray Protheroe. Post-print sourced from www.arxiv.org.We investigate ways of accurately simulating the propagation of energetic charged particles over small times where the standard Monte Carlo approximation to diffusive transport breaks down. We find that a small-angle scattering procedure with appropriately chosen step-lengths and scattering angles gives accurate results, and we apply this to the simulation of propagation upstream in relativistic shock acceleration.R.J. Protheroe, A. Meli and A.-C. Done

Protheroe, R.

Donea, A.

Adelaide Research & Scholarship

arXiv:astro-ph/0210011v1  1 Oct 2002Small-Angle Scattering and Diffusion: Application toRelativistic Shock AccelerationR.J. Protheroe, A. Meli∗ and A.-C. DoneaDepartment of Physics & Mathematical Physics, The University of Adelaide,Adelaide, SA 5005, Australia19 June 2002Abstract. We investigate ways of accurately simulating the propagation of ener-getic charged particles over small times where the standard Monte Carlo approx-imation to diffusive transport breaks down. We find that a small-angle scatteringprocedure with appropriately chosen step-lengths and scattering angles gives ac-curate results, and we apply this to the simulation of propagation upstream inrelativistic shock acceleration.Keywords: cosmic rays, diffusion theory, relativistic shock acceleration1. IntroductionRelativistic charged particle transport in magnetized astro-physicalplasma is strongly affected by magnetic irregularities, and may beapproximated by diffusion. Diffusive transport of particles having speedv can be simulated by a three-dimensional random walk with steps sam-pled from an exponential distribution with mean free path λ = 3D/v,where D (cm2 s−1) is the spatial diffusion coefficient, followed by large-angle (isotropic) scattering after each step (e.g. Chandrasekhar 1943),and this gives good results for distances much larger than λ.In diffusive shock acceleration at relativistic shocks problems arisewhen simulating particle motion upstream of the shock because theparticle speeds, v, and the shock speed vshock = c(1 − 1/γ2shock)1/2 areboth close to c, and so very small deflections are sufficient to causea particle to re-cross the shock. Clearly, Monte Carlo simulation bya random walk with mean free path λ and large-angle scattering isinappropriate here, and in Monte Carlo simulations of relativistic shockacceleration at parallel shocks Achterberg et al. (2001) consider insteadthe diffusion of a particle’s direction for a given angular diffusion coeffi-cient Dθ (rad2 s−1). Similarly, for a given spatial diffusion coefficient D,Protheroe (2001) and Meli & Quenby (2001) adopt a random walk witha smaller mean free path, ℓ¯ ≪ λ, followed by scattering at each stepby a small angle with mean deflection , θ¯ < 1/γshock. See Bednarz &Ostrowski (2001) for a recent review of relativistic shock acceleration.∗ visiting from Imperial College, Londonc© 2008 Kluwer Academic Publishers. Printed in the Netherlands.Protheroe.tex; 5/02/2008; 6:09; p.12 R.J. Protheroe, A. Meli & A.-C. Donea2. Small-angle scattering and diffusionWe consider propagation by small steps sampled from an exponentialdistribution with mean ℓ¯ ≪ λ, followed at each step by scatteringby a small angle sampled from an exponential distribution with meanθ¯ ≪ π. The change in direction (θ1, θ2) may then be described as two-dimensional diffusion with angular diffusion coefficient Dθ = θ¯vθ/2(rad2 s−1) where vθ = θ¯/t¯, and t¯ = ℓ¯/v such that Dθ = θ¯2v/(2ℓ¯). Thetime tiso, which gives rise to a deflection equivalent to a large-angle(isotropic) scattering, is determined by (σ2θ1 + σ2θ2)1/2 =√4Dθtiso ∼π/2, giving λ ∼ vtiso ∝ ℓ¯/θ¯2 and a spatial diffusion coefficient D ∝ℓ¯v/θ¯2 ∝ v2/Dθ. By using a Monte Carlo method it is straightforwardto test this, determine the constant of proportionality, and therebymake the connection between diffusion and small angle scattering.The solution of the diffusion equation for a delta-function sourcein position and time q(~r, t) = δ(~r)δ(t) and an infinite diffusive mediumis a three-dimensional gaussian with standard deviation σ =√2Dt(Chandrasekhar 1943). The results from several Monte Carlo randomwalk simulations are shown in Fig. 1, from which we find that theexpected dependence occurs for θ¯ < 5◦ at times t > 105 ℓ¯/v. For thiscase, we see from Fig. 1 that σ2 → 2tvℓ¯/3θ¯2, and so we obtain theconnection between small-angle scattering and diffusion theory, namelyD ≈ ℓ¯v/(3θ¯2) ≈ v2/(6Dθ).3. Application to relativistic shock accelerationAs viewed in the frame of reference of the upstream plasma, ultra-relativistic particles are only able to cross the shock from downstreamto upstream if the angle θ between their direction and the shock nor-mal pointing upstream is θ < sin−1(1/γshock), where γshock = (1 −β2shock)−1/2 and βshock = vshock/c. For highly relativistic shocks theseparticles cross the shock from downstream to upstream travelling al-most parallel to the shock normal. Similarly, having crossed the shock,only a very slight angular deflection, by ∼ 1/γshock is sufficient to returnthem downstream of the shock. This change in particle direction givesrise to a change in particle energy E′ and momentum p′, measured inthe downstream plasma frame (primed coordinates), ofE′n+1E′n≈ p′n+1p′n=1− β12 cos θn+11− β12 cos θn(1)in an acceleration cycle (downstream → upstream → downstream),where β12 is the speed of the upstream plasma as viewed from thedownstream frame.Protheroe.tex; 5/02/2008; 6:09; p.2Small-Angle Scattering and Diffusion: Application to Relativistic Shock Acceleration 3Figure 1. σ2 vs. time for a 3D random walk with isotropic injection at the originat t = 0. Step-lengths ℓ were sampled from an exponential distribution with meanℓ¯ followed by small-angle scattering with scattering angle θ sampled from an expo-nential distribution with mean θ¯ (the numbers attached to the curves). Dashed lineis σ2 = 2tvℓ¯/3θ¯2. Curves for 5◦–20◦ result from 104 simulations; 4◦ curve resultsfrom 8× 104 simulations (width shows statistical error).In “parallel shocks” the magnetic field is parallel to the shocknormal, and so the the pitch angle ψ is the angle to the shock normaland v cosψ gives the component of velocity parallel to the shock. Thusthe small-angle scattering method described in the previous section isused here to simulate particle motion upstream of a parallel relativisticshock, including the effects of pitch-angle scattering, for a given diffu-sion coefficient. We inject ultra-relativistic particles at the shock withdownstream-frame energy E′0 travelling upstream parallel to the shocknormal, i.e. θ0 = 0. We follow a particle’s trajectory until the shockcatches up with it, and it crosses from upstream to downstream withan upstream-frame angle θ1 to the shock normal and a downstream-frame energy E′1. The simulation was performed for γshock = 10, andfive different mean scattering angles θ¯, to determine the maximum θ¯-value that can safely be used for accurate simulation. The resultingdistributions of cos θ and log(E′1/E′0) are shown in Fig. 2, and showthat in this application one requires θ¯ < 0.1/γshock. Our results arequite consistent with those of Achterberg et al. (2001), who used adiffusive angular step ∆θst ≤ 0.1/γshock.Protheroe.tex; 5/02/2008; 6:09; p.34 R.J. Protheroe, A. Meli & A.-C. Doneaa. cos θ1 distribution b. log(E′1/E′0) distributionFigure 2. Small-angle scattering simulation of excursion upstream in diffusive shockacceleration at a parallel relativistic shock with γshock = 10. Results are shownfor 105 injected particles and θ¯ = 10−2/γshock (top histogram), 3 × 10−2/γshock,10−1/γshock, 0.3/γshock, 1/γshock and 3/γshock (bottom histogram). Note that thetop three histograms are almost indistinguishable.4. ConclusionThe standard Monte Carlo random walk approach to simulation ofenergetic charged particle propagation for a given spatial diffusion co-efficient D can be extended to apply accurately to times much less thanλ/v = 3D/v2 by using a small-angle scattering procedure with stepssampled from an exponential distribution with mean free path ℓ¯ = θ¯2λfollowed at each step by scattering with angular steps sampled from anexponential distribution with mean scattering angle θ¯ < 0.09 rad (5◦).The spatial and angular diffusion coefficients are then D ≈ ℓ¯v/(3θ¯2)and Dθ = θ¯2v/(2ℓ¯), and are related by D ≈ v2/(6Dθ). In simulation ofupstream propagation in relativistic shock acceleration one must useθ¯ < 0.1/γshock to obtain accurate results.ReferencesAchterberg, A., Gallant, Y.A., Kirk, J.G., and Guthmann, A.W.: 2001, MNRAS328, 393.Bednarz, J., and Ostrowski, M.: 2001, MNRAS 310, L13.Chandrasekhar, S.: 1943, Rev. Mod. Phys. 15, 1.Meli, A., and Quenby, J.: 2001, 27th Int. Cosmic Ray Conf. , Ed. M. Simon et al.vol. 7, p. 2742.Protheroe, R.J.: 2001, 27th Int. Cosmic Ray Conf. , Ed. M. Simon et al. vol. 6,p. 2006, and p. 2014.Protheroe.tex; 5/02/2008; 6:09; p.4

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Small-angle scattering and diffusion: Application to relativistic shock acceleration

Small-Angle Scattering and Diffusion: Application to Relativistic Shock
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Small-Angle Scattering and Diffusion: Application to Relativistic Shock Acceleration

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