Diffusive shock acceleration is the prime candidate for efficient
acceleration of cosmic rays. Galactic cosmic rays are believed to originate
predominantly from this process in supernova remnant shock waves. Confinement
of the cosmic rays in the shock region is key in making the mechanism
effective. It has been known that on small scales (smaller than the typical
gyroradius) high-amplitude non-resonant instabilities arise due to cosmic ray
streaming ahead of the shock. For the efficiency of scattering of the highest
energy cosmic rays it is of interest to determine the type of instabilities
that act on longer length scales, i.e. larger than the cosmic ray gyroradius.
We will present the results of our analysis of an instability that acts in this
regime and will discuss its driving mechanism and typical growth times.Comment: 4 pages, 1 figure. Proceedings for the conference on Cosmic Rays and
  the Interstellar Medium (CRISM) in June 2011, Montpellier, France. To appear
  in MSA

Bell, A. R.

Schure, K. M.

English

arXiv

Diffusive shock acceleration is the prime candidate for efficient acceleration of cosmic rays. Galactic cosmic rays are believed to originate predominantly from this process in supernova remnant shock waves. Confinement of the cosmic rays in the shock region is key in making the mechanism effective. It has been known that on small scales (smaller than the typical gyroradius) high-amplitude non-resonant instabilities arise due to cosmic ray streaming ahead of the shock. For the efficiency of scattering of the highest energy cosmic rays it is of interest to determine the type of instabilities that act on longer length scales, i.e. larger than the cosmic ray gyroradius. We will present the results of our analysis of an instability that acts in this regime and will discuss its driving mechanism and typical growth times

Schure, KM

Bell, Anthony

Oxford University Research Archive

Confining the high-energy cosmic rays

Schure, Km

ORA - Oxford University Research Archive

arXiv:1110.1096v1  [astro-ph.HE]  5 Oct 2011Mem. S.A.It. Vol. 75, 282c© SAIt 2008 Memorie dellaConfining the high-energy cosmic raysK.M. Schure1 and A.R. Bell1Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, OxfordOX1 3PU, United Kingdom e-mail:k.schure1@physics.ox.ac.ukAbstract. Diffusive shock acceleration is the prime candidate for effici nt acceleration ofcosmic rays. Galactic cosmic rays are believed to originatepredominantly from this processin supernova remnant shock waves. Confinement of the cosmic rays in the shock region iskey in making the mechanism eff ctive. It has been known that on small scales (smallerthan the typical gyroradius) high-amplitude non-resonantinstabilities arise due to cosmicray streaming ahead of the shock. For the efficiency of scattering of the highest energycosmic rays it is of interest to determine the type of instabilities that act on longer lengthscales, i.e. larger than the cosmic ray gyroradius. We will present the results of our analysisof an instability that acts in this regime and will discuss its driving mechanism and typicalgrowth times.Key words. cosmic rays — acceleration of particles1. IntroductionGalactic cosmic rays are widely believed tobe accelerated, be it predominantly or not, insupernova remnants (SNRs). If solely SNRsare responsible for the acceleration, a signifi-cant fraction of kinetic energy has to be trans-ferred onto cosmic rays. Additionally, the ac-celeration needs to be efficient enough to ar-rive at energies typical of the knee in the spec-trum, being around 1015 eV. In order to getthose acceleration rates, Bohm diffusion is nor-mally assumed, since this yields the quickestisotropisation needed for the confinement ofcosmic rays around the shock region. In real-ity, the diffusion is dependent on the power inmagnetic fluctuations on the scale of the gy-roradius. Scattering of the highest-energy cos-mic rays requires a significant component ofmagnetic field fluctuations on the order of par-sec scales. A cosmic-ray streaming instabilitythat initialises growth on those scales is de-rived from the Boltzmann equation for the cos-mic rays, coupled to the MHD equations of thebackground plasma. Typical growth times arepresented.2. Cosmic ray streaming instability inthe long-wavelength limitIt has long been known that cosmic rays can actto trigger magnetic fluctuations. The first cal-culations on this were in the resonant regime,where energy is transferred from the cosmicrays to magnetic fluctuations that have thesame scale as the gyroradius of the cosmicrays (Lerche, 1967; Kulsrud & Pearce, 1969;Wentzel, 1974; Skilling, 1975). When the per-turbed magnetic field grows to values similarto the background field, the resonance condi-tion is lost, and any magnetic field amplifica-tion beyondδB/B ≈ 1 must be non-resonant.K.M. Schure and A.R. Bell: Confining the high-energy cosmic rays 283With the development of diffusive shockacceleration, the need for efficient scatteringaround the shock region arised in order to get tocosmic ray energies of 1015 eV. Additionally,observational evidence of narrow confinementregions around the shock for relativistic elec-trons required field strengths of the order of100 µG, rather than just tens that could beexpected based on compression of interstellarmagnetic fields alone (e.g. Vink & Laming,2003; Völk et al., 2005). The non-resonant in-stability developed by Bell (2004) resolvedthis. However, care should be taken by trans-lating a magnetic field strength inferred fromobservations of TeV electrons to the magneticfield strength confining the PeV cosmic rays.On scales smaller than the gyroradius ef-fective growh of magnetic fluctuations is gen-erated because of a plasma current. The cur-rent is a reaction to the cosmic-ray current,which requires balancing in the local plasma.Magnetic field fluctuations perpendicular tothe current result in aj × B force that, fora given polarisation, focusses the cosmic rayswithin the growing magnetic loops, and pushesthe plasma outwards. With a frozen-in mag-netic field, the stretching of the field linesresults in growth. In the approximation thatthe larmor radius of the cosmic rays is muchlarger than the fluctuations, the cosmic raycurrent is effectively undisturbed and this canresult in significant magnetic field amplifica-tion. Simulations of this instability in the non-linear regime indicate that this indeed is thecase (Bell, 2004). An estimate for the satura-tion value of the magnetic field results in thisvalue being proportional to the shock veloc-ity to the power 3/2. Therefore, especially foryoung remnants where the shock velocities arehigh, magnetic field growth to orders of mGseems to be feasible.The scales of the fluctuations triggeredby this non-resonant instability are smallerthan the gyro-radii of the driving cosmic rays.Scattering, being most efficient for cosmic rayshaving a larmor radius of the same order asthe magnetic fluctuations, can become veryefficient. Cosmic rays with low energies cantherefore be confined near the shock region.However, in order to confine the high-energyend of the cosmic rays, magnetic fluctuationsneed to grow on a scale larger than the gyrora-dius of the dominant cosmic ray component.In a recent work, we describe the growth ofmagnetic fields on scales longer than the gyro-radius, due to the influence of the stress tensorarising from gradients in the perturbed cosmicray current (Schure & Bell, 2011). Small-scaleturbulence, resulting from the non-resonant in-stability on small scales, induces an effectivediffusivity that couples the right- and the left-hand polarisation through its action on theplasma.From the dispersion relation inSchure & Bell (2011), the influence ofcontributions from the stress tensor and col-lisions can be evaluated. The full dispersionrelation, taking into account the stress tensorand collisions, looks like:ω2 = ∓Ω2(iω −k2c25(3ν ∓ iωg))(1)/(ν − iω ∓ iωg +k2c25(3ν ∓ iωg)).The upper signs correspond to the left-handedpolarisation (similar to the polarisation of agyrating proton) and the lower signs to theright-hand polarisation (the one that is unsta-ble to the small-scale nonresonant Bell insta-bility). The magnetic fluctuations grow whenthe imaginary part of the square root of theabove equation is positive.ω is the complexfrequency,k the wavenumber,c speed of light,ν the effective scattering frequency,ωg the gy-rofrequency, andΩ the growth rate of the non-sonant Bell instability that contains informa-tion on the driving current:Ω =√k j0B/(ρc),with B the unperturbed magnetic field strengthparallel tok, j0 the return current, andρ theplasma mass density.The terms including factors ofk2 resultfrom the inclusion of the stress tensor. Effectsfrom the small-scale non-resonant instabilityare included throughν, the effective scatter-ing frequency. Scattering on small scales is ex-pected to arise earlier than on long scales, sincethe growth rate increases rapidly forkrg ≫ 1.284 K.M. Schure and A.R. Bell: Confining the high-energy cosmic raysBohm diffusion is the regime where the eff c-tive scattering frequency is of the same orderas the gyrofrequency, i.e.ν ≈ ωg.The method in Schure & Bell (2011) re-covers the 2004 growth rate for the right po-larisation in the regime wherekc ≫ ωg. Theresonant instability is not directly captured inthe mono-energetic approach. When the col-lisionality is zero,ν = 0, the only growingmode on small scales is the right-hand one,whereas on long scales only the left-hand modeis unstable. Whenν , 0 both modes are un-stable in the long-wavelength regime, althoughthe left-hand mode continues to dominate untilν/ωg > 1/√3.Bykov et al. (2011) also investigated an in-stability on the large scales that takes into ac-count the effect of mean field dynamo the-ory. Averaging of the magnetic fluctuations onsmall scales, arising from the 2004 Bell in-stability, generates a ponderomotive force thatacts to amplify the field on large scales. Theyfound rapid growth, and their analysis also in-dicates that the left-hand mode is the dominantgrowing mode in the long-wavelength regime.A third option of growth on larger scales is thatof growing small scale fluctuations. The fila-ments resulting from the small-scale nonreso-nant instability can merge and grow in scales(Reville & Bell, 2011).3. Typical growth timesThe driving current is a result of the stream-ing of cosmic rays, being proportional to theshock velocity:j0 = ncrqvs, with ncr the cos-mic ray number density,q the unit charge,and vs the shock velocity. Typical values forgrowth rates are summarized in Table 1 andplotted in Figure 1. A current density ofj0 =10−8 stA cm−2 is assumed at the shock, cor-responding toncr ≈ 10−5 cm−3 and vs =109 cm s−1.If ncr is constant, the current scales with(ETeV)−1 ( vs109 cm s−1), given a powerlaw spectrumwith a slope ofp = 2, and a specific scat-tering efficiency. The particle energy is thatof the driving cosmic rays, which is increas-ing with distance upstream of the shock whenFig. 1. Growth time in years as a function ofwavenumber. The solid (dashed) lines repre-sent the right- (left-) hand polarisation. Valuesare given for different efficiencies of the small-scale scattering rate. On long scales the left-hand mode has the quickest growth, whereason scales smaller than resonance, the right-hand mode dominantes.diffusion propagates proportional to the par-ticle energy. Bohm diffusion can be consid-ered to be the minimal diffusion rate. Dividingthis by the shock velocity gives a length-scale to which particles can diffuse upstream,given by L = D/vs = (rgc)/(3vs). The cor-responding time scale can be found by di-viding again over the shock velocity, to de-termine when the shock has caught up withthe location where the instability is triggered,giving a lower limit of τ = crg/(3v2s) =0.35(ETeV) (B3µG)−1 ( vs109 cm s−1)−2yr. The growthtime scales with(ETeV) (B3µG)−1 ( vs109 cm s−1)−1/2,for ncr = 10−5. The number of e-foldings forgrowth, γτ, can be seen to scale withv−3/2s ,leaving only the shock velocity as a free pa-rameter to increase the linear growth effec-tively.Alternatively, it can be considered thatncr,and thereforej0, scales with the kinetic en-ergy of the shock, such that a fixed propor-tion of ρv2s is converted into cosmic rays. Fora mono-energetic approach, this translates tocrEcr = η 12ρv2s such that the current den-K.M. Schure and A.R. Bell: Confining the high-energy cosmic rays 285Table 1. Linear growth times for the right (and left between brackets) hand modes, givenfor a particle energy of 1 TeV, magnetic field strengthB = 3µG, shock velocityvs =109cm s−1, and current j0 = 10−8 stA cm−2. It can be derived for other values by us-ing the fact that it scales with(ETeV) (B3µG)−1 ( vs109 cm s−1)−1/2if ncr is constant in time, or with(ETeV) (B3µG)−1 ( vs109 cm s−1)−3/2 ( η0.014)−1/2if ncr is a fraction of the shock kinetic energy.ν/ωg γ−1 (yr) γ−1 (yr) γ−1 (yr) γ−1 (yr) γ−1 (yr) γ−1 (yr)(krg = 0.1) (krg = 0.5) (krg = 0.8) (krg = 1) (krg = 5) (krg = 10)1/10 600 (119) 50.2 (10.5) 22.1 (5.00) 4.47 (1.10) 0.651 (13.1)0.498 (47.4)1/3 264 (164) 23.0 (14.7) 10.8 (7.31) 2.34 (1.67) 0.700 (4.62) 0.503 (14.6)1/√3 245 (246) 21.7 (22.2) 10.6 (11.2) 2.35 (2.58) 0.776 (3.53) 0.513 (8.92)1 283 (462) 25.3 (41.6) 12.5 (20.9) 8.95 (15.2) 0.949 (3.58) 0.542 (6.01)sity becomesj0 = ncrqvs = ηq 12ρv3s/Ecrand thus scales with(ETeV)−1 ( vs109 cm s−1)3. Thegrowth time in that case scales asγ−1 ∝(ETeV) (B3µG)−1 ( vs109 cm s−1)−3/2 ( η0.014)−1/2and thenumber of e-foldings is proportional toγτ ∝ v−1/2s . Of course in these equations(ETeV) (B3µG)−1can be replaced by(rg1.11×1015cm),with rg the gyroradius. The numbers in Table 1correspond to a value ofη = 0.014. It is ex-pected that the fraction of kinetic energy go-ing into cosmic rays can be significantly higherthan this, resulting in shorter growth times bya factor of the square root of this efficiency.For an efficiency of 10% this would result inreduced growth times compared to those inTable 1 by a factor 2.7.4. Discussion and conclusionAcceleration of cosmic rays in supernova rem-nants requires the growth of magnetic fluctu-ations in order to confine cosmic rays to theshock region. For the highest-energy cosmicrays the non-resonant regime on scales largerthan the gyroradius of the dominant cosmic raycomponent is of importance in confining cos-mic rays as they reach higher energies. A non-resonant instability is discussed that is drivenby the return current in the plasma, result-ing from a reaction to the cosmic ray cur-rent. The stress-tensor is crucial in generatingan instability in the long-wavelength regime.Scattering on small scales aids in mediating theinstability between the two polarisations. Thenumber of e-folding times for the instability ata given scattering efficiency is shown to dependonly on the shock velocity, and typical growthtimes are given.Acknowledgements. This research was supportedby the UK Science Technology and FacilitiesCouncil grant ST/H001948/1.ReferencesBell, A. R. 2004, MNRAS, 353, 550Bykov, A. M., Osipov, S. M., & Ellison, D. C.2011, MNRAS, 410, 39Kulsrud, R. & Pearce, W. P. 1969, ApJ, 156,445Lerche, I. 1967, ApJ, 147, 689Reville, B. & Bell, A. R. 2011, ArXiv e-printsSchure, K. M. & Bell, A. R. 2011, MNRAS,1451Skilling, J. 1975, MNRAS, 173, 245Vink, J. & Laming, J. M. 2003, ApJ, 584, 758Völk, H. J., Berezhko, E. G., & Ksenofontov,L. T. 2005, A&A, 433, 229Wentzel, D. G. 1974, ARA&A, 12, 71

 Diffusive shock acceleration is the prime candidate for efficient acceleration of cosmic rays. Galactic cosmic rays are believed to originate predominantly from this process in supernova remnant shock waves. Confinement of the cosmic rays in the shock region is key in making the mechanism effective. It has been known that on small scales (smaller than the typical gyroradius) high-amplitude non-resonant instabilities arise due to cosmic ray streaming ahead of the shock. For the efficiency of scattering of the highest energy cosmic rays it is of interest to determine the type of instabilities that act on longer length scales, i.e. larger than the cosmic ray gyroradius. We will present the results of our analysis of an instability that acts in this regime and will discuss its driving mechanism and typical growth times. </p

Bell, A

Oxford University Research Archive (ORA)

Diffusive shock acceleration is the prime candidate for efficient
acceleration of cosmic rays. Galactic cosmic rays are believed to originate
predominantly from this process in supernova remnant shock waves. Confinement
of the cosmic rays in the shock region is key in making the mechanism
effective. It has been known that on small scales (smaller than the typical
gyroradius) high-amplitude non-resonant instabilities arise due to cosmic ray
streaming ahead of the shock. For the efficiency of scattering of the highest
energy cosmic rays it is of interest to determine the type of instabilities
that act on longer length scales, i.e. larger than the cosmic ray gyroradius.
We will present the results of our analysis of an instability that acts in this
regime and will discuss its driving mechanism and typical growth times

Confining the high-energy cosmic rays

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