Cosmic ray acceleration at oblique shocks

Mon. Not. R. Astron. Soc. 418, 1208–1216 (2011) doi:10.1111/j.1365-2966.2011.19571.x Cosmic ray acceleration at oblique shocks A. R. Bell, K. M. Schure and B. Reville Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU Accepted 2011 August 2. Received 2011 August 2; in original form 2011 July 17 ABSTRACT We show that the diffusion approximation breaks down for particle acceleration at oblique shocks with velocities typical of young supernova remnants. Higher order anisotropies flatten the spectral index at quasi-parallel shocks and steepen the spectral index at quasi-perpendicular shocks. We compare the theory with observed spectral indices. 1 I N T RO D U C T I O N The well-established theory of cosmic ray (CR) acceleration by strong non-relativistic shocks predicts a differential energy powerlaw spectrum n(E) dE ∝ E−s dE with a spectral index s = 2 (Axford, Leer & Skadron 1977; Krymskii 1977; Bell 1978; Blandford & Ostriker 1978). However, a variety of spectral indices is observed. Galactic CRs have a steeper spectrum even when allowance is made for energy-dependent propagation losses (Hillas 2005). Steeper spectra are also observed in young supernova remnants (SNRs) as discussed below. Deviations from a s = 2 spectrum are well known for relativistic shocks (e.g. Kirk & Heavens 1989; Ostrowski 1991; Baring, Ellison & Jones 1993; Gieseler et al. 1999; Ruffalo 1999; Achterberg et al. 2001; Kobayakawa, Honda & Samura 2002). Here we show that corrections to theory produce a spectral index that deviates significantly from s = 2 even at shock velocities as low as 10 000 km s−1 typical of young SNRs. The magnitude of the effect depends on the ratio of the velocity of the accelerating particle to the shock velocity. Hence, it may also apply to the acceleration of non-relativistic particles at heliospheric shocks. The spectrum can either steepen or flatten depending on the angle between the shock normal and the large-scale upstream magnetic field. Quasi-perpendicular shocks (Jokipii 1982, 1987) show particularly strong steepening. The effect is small for parallel shocks where changes in the spectral index are second order in the ratio of the particle velocity to the shock velocity. Non-linear hydrodynamic feedback of the CR pressure on to the shock structure can also change the spectral index (Dury & Völk 1981; Drury 1983; Ellison, Baring & Jones 1996), but we do not take account of this in the present paper. We assume that the fluid velocity is uniform both upstream and downstream of the shock. A further assumption lying at the heart of our method is that the magnetic field can be divided into (i) a large-scale field that is uniform apart from a change of direction and magnitude at the shock and (ii) a small-scale field that fluctuates on scales smaller than the CR Larmor radius and deflects  E-mail: the CR trajectories through small angles to cumulatively scatter the CR. This ignores magnetic field line wandering which can also change the CR spectral index (Kirk, Duffy & Gallant 1996). In our calculations, we further assume that the fluid is always compressed by a factor of 4 at the shock, which neglects the increase in compression at the shock due to the different equation of state of the relativistic component (ratio of specific heats = 4/3) and the effects of finite Mach number. With these assumptions, we isolate and concentrate on the breakdown of the diffusive approximation and the effect of high-order anisotropies in the CR distribution near the shock. 2 E Q UAT I O N S CRs are thought to be accelerated to high energy by crossing back and forth across a shock in a first-order Fermi process, with a fractional energy gain of order us /c at each crossing where us is the shock velocity. A small proportion of the CR injected at low energy proceeds to cross the shock many times before escaping the shock environment. Statistically, this results in a power law stretching up to a few PeV in the Galaxy and probably approaching ZeV extragalactically. Here we consider CR protons of mass mp , but the analysis applies equally well to heavier ions or to electrons. The shape and extent of the CR spectrum is described by the CR distribution function f (x, p, t) in momentum p. We locate a planar shock at x = 0. Upstream of the shock (x < 0), the ambient plasma flows towards the shock at the shock velocity us . For a strong nonrelativistic shock, the plasma velocity is reduced to us /4 in the downstream plasma. The equation for the CR distribution function is ∂f u ∂u ∂f ∂f ∂u ∂f − +(vx + u) − px p ∂t ∂x ∂x ∂px c ∂x ∂px ∂f + ev ∧ B · = C(f ), ∂p (1) where f is defined in the local fluid rest frame moving at velocity u(x) in the x direction and relativistic effects due to γ u = (1 − u2 /c2 )−1/2 = 1 have been neglected. vx is the x-component of the CR velocity  C 2011 The Authors C 2011 RAS Monthly Notices of the Royal Astronomical Society  Key words: acceleration of particles – shock waves – cosmic rays – ISM: supernova remnants. Cosmic ray acceleration at oblique shocks v, B is the local large-scale magnetic field and C(f ) represents CR scattering by small-scale fluctuations in the magnetic field. In the local fluid rest frame, C(f ) does not change the CR energy, but merely deflects its trajectory by diffusion in angle. ∂u/∂x is zero everywhere except at the shock where it produces CR acceleration. Building on our experience of modelling laser plasmas, we solve the Vlasov–Fokker–Planck (VFP) equation by expressing the CR distribution function as a sum of spherical harmonics (Bell et al. 2006): f (x, p, t) =  flm (x, p, t)Pl|m| (cos θ)eimφ , l,m l = 0, ∞ m = −l, l  ∗ fl−m = flm ,  m m  l − m ∂fl−1 ∂flm ∂f m l + m + 1 ∂fl+1 +u l +c + ∂t ∂x 2l − 1 ∂x 2l + 3 ∂x ceBx m ceBz m f + β p l 2p l   m fm ∂u (l − m)(l − m − 1) ∂fl−2 − (l − 2) l−2 −p ∂x (2l − 3)(2l − 1) ∂p p  m m f (l − m)(l + m) ∂fl + (l + 1) l + (2l − 1)(2l + 1) ∂p p  m (l − m + 1)(l + m + 1) ∂fl fm + −l l (2l + 1)(2l + 3) ∂p p  m fm (l + m + 1)(l + m + 2) ∂fl+2 + (l + 3) l+2 + (2l + 3)(2l + 5) ∂p p   m fm u ∂u l − m ∂fl−1 − (l − 1) l−1 −p c ∂x 2l − 1 ∂p p −im l+m+1 + 2l + 3 =−   C 2011 The Authors, MNRAS 418, 1208–1216 C 2011 RAS Monthly Notices of the Royal Astronomical Society  where ωz = ceBz /p and ωx = ceBx /p. Supernova (SN) ejecta initially expand at velocities up to ∼c/5, and the SNR shock velocity decreases to ∼ c/100 during the first 1000 years. In the limit of small u/c, the l = 0 and l = 1 equations dominate. In the Chapman– Enskog expansion, valid for scalelengths much larger than the CR scattering length, the magnitude of fl m decreases with increasing l, such that fl m ∼ (u/c)l f 00 . With this expansion, fl m can be neglected for l > 1, in which case the above three equations (l = 0 and l = 1) reduce to ∂f 0 c ∂f10 + u 0 = 0, 3 ∂x ∂x ∂f00 c + 2ωz (f11 ) = −νf10 , ∂x ωz 0 f = −νf11 −iωx f11 − (4) 2 1 in the upstream plasma, where f 00 and f 01 a (...truncated)