Anisotropic diffusion and the cosmic ray anisotropy
aa r X i v : . [ a s t r o - ph . H E ] N ov Anisotropic diffusion and the cosmic ray anisotropy
M. Kachelrieß
Institutt for fysikk, NTNU, Trondheim, Norway
Abstract.
We argue that the diffusion of cosmic rays in the Galactic magnetic field has to bestrongly anisotropic. As a result, the number of CR sources contributing to the local CR fluxis reduced by a factor ∼
1. Introduction
The observed distribution of cosmic ray (CR) arrival directions is highly isotropic. Since GalacticCR sources are strongly concentrated in the Galactic disc, an efficient mechanism for theisotropisation of the CR momenta exists. Agent of this isotropisation are turbulent magneticfields, since charged CRs scatter efficiently with resonant field modes which wavelength matchestheir Larmor radius. As a result, CRs perform on scales larger than the coherence length of theturbulent field a random walk, and the memory of the initial source location is mostly erased.Residual anisotropies are connected to the structure of the local magnetic field and, e.g., to aremaining net flux of CRs.Since large wavelengths of the turbulent field modes are less abundant, CRs with higherenergy are scattered less efficiently. Therefore, the diffusion picture predicts that the CRanisotropy should increase monotonically with energy. More precisely, if the turbulent fieldfollows a Kolmogorov power law as suggested by the observed B/C ratio, the dipole anisotropy δ should increase with energy as δ ∝ E / . Both the energy-dependence and the absolutevalue of the dipole anisotropy predicted in simple isotropic diffusion models do not agree withobservations. This discrepancy was dubbed the “CR anisotropy problem” by Hillas [1].In this short review based on the results of Refs. [2, 3], we will first argue that the diffusionof CRs in the Galactic magnetic field (GMF) has to be strongly anisotropic. As a result, thenumber of CR sources contributing to the local CR flux is strongly reduced. Therefore, the CRdensity is less smooth, and the contribution of individual sources to the CR dipole anisotropybecomes more prominent than in the standard picture. Then we argue that the observed plateauin the CR dipole anisotropy around 2–20 TeV is connected to a 2–3 Myr old CR source whichdominates in this energy range the local CR flux. Finally, we comment on the alternative thata young source like Vela is responsible for the observed plateau in the dipole anisotropy.
2. Galactic magnetic field and anisotropic diffusion
In the diffusion approach to CR propagation one considers typically the CR density in thestationary limit. The measured ratios of CR isotopes like Be /Be and of secondary/primaryratios like B/C indicate a residence time of CRs with rigidity R of order τ esc ≃ few × E E D [ c m / s ] E [eV]B/C5/64 µ G5/8 µ G5 µ G40 µ G2.e-11 G
Figure 1.
CR diffusion coefficient D ( E ) in pure isotropic Kolmogorov turbulence with L max =25 pc for four values of the strength B rms of the turbulent field. The green band shows the rangeof magnetic field strengths for which the diffusion coefficient satisfies D = (3 − × cm /sat E = 10 GeV; from Ref. [3].10 yr ( R / − β with β ≃ /
3. Then the flux from some 10 sources accumulates at lowrigidities, forming a “sea” of Galactic CRs, if one assumes that the main CR sources aresupernovae (SN) injecting ≃ erg every ≃
30 yr in the form of CRs. Since many sourcescontribute, the discrete nature of the CR sources can be neglected. Assuming additionallythat the turbulent magnetic field dominates relative to the regular field, one often replaces thediffusion tensor D ij by a scalar diffusion coefficient D .The diffusion approach based on the approximations described above has been sufficient todescribe the bulk of experimental data obtained until ≃ D ij = lim t →∞ N t N X a =1 ( x ( a ) i − x i, )( x ( a ) j − x j, ) (1)calculated numerically following the trajectories x ( a ) i ( t ) of N CRs injected into a pure randomfield with a Kolmogorov power spectrum with L max = 25 pc for various field strengths. Thetransition at R L ( E cr ) = L max between the asymptotic low-energy ( D ∝ E / , large-anglescattering) and high-energy ( D ∝ E , small-angle scattering) behaviour is clearly visible.However, for all used field strengths the diffusion coefficients are much smaller than thoseextracted using e.g. Galprop [4] or DRAGON [5]. Therefore CR propagation cannot be isotropic,because otherwise CRs overproduce secondary nuclei like boron for any reasonable values of thestrength and the coherence scale of the turbulent field, cf. with Fig. 2. Such an anisotropy -5 -4 -3 L c oh / p c B rms / µ G KraichnanKolmogorov
Figure 2.
Allowed ranges of B rms and L coh compatible with D = (3 − × cm /s at E = 10 GeV for Kolmogorov and Kraichnan turbulence. These ranges should be comparedwith the typical order-of-magnitude values that are relevant for the Galactic magnetic field: B rms ∼ (1 − µ G and L coh < ∼ a few tens of pc, from Ref. [3].may appear if the turbulent field at the considered scale does not dominate over the orderedcomponent, or if the turbulent field itself is anisotropic.One can estimate the level of anisotropy required considering the following toy model: Letus adopt a thin matter disc with density ρ/m p ≃ and height h = 150 pc around theGalactic plane, while CRs propagate inside a larger halo of height H = 5 kpc. We assume thatthe regular magnetic field inside this disc and halo has a tilt angle ϑ with the Galactic plane, sothat the component of the diffusion tensor relevant for CR escape is given by D z = D ⊥ cos ϑ + D k sin ϑ . (2)Applying a simple leaky-box approach, the grammage follows as X = cρhH/D z . Using now asallowed region for the grammage 5 ≤ X ≤
15 g/cm , the permitted region in the ϑ – η plane shownin the left panel of Fig. 3 follows, where η ≡ B rms /B describes the turbulence level. For not toolarge values of the tilt angle, ϑ < ∼ ◦ , the regular field should strongly dominate, η < ∼ .
35. Thisresults in a strongly anisotropic propagation of CRs, where the diffusion coefficient perpendicularto the ordered field can be between two and three orders of magnitude smaller than the parallelone, D ⊥ ≪ D || . As a result, the z component of the regular magnetic field can drive CRsefficiently out of the Galactic disk. For instance, the “X-field” in the Jansson-Farrar model [6]for the GMF leads to the correct CR escape time, if one chooses η ≃ .
25 [7].For this choice, the diffusion coefficients satisfy D k ≃ D iso and D ⊥ ≃ D iso / D iso denotes the isotropic diffusion coefficient D iso satisfying the B/C constraints. In the regime,where the CRs emitted by a single source fill a Gaussian with volume V ( t ) = π / D ⊥ D / k t / ,the CR density is increased by a factor 500 / √ ≃
200 compared to the case of isotropic diffusion.The smaller volume occupied by CRs from each single source leads to a smaller number of sourcescontributing substantially to the local flux, with only ∼ sources at R ∼
10 GV and about ∼
10 most recent SNe in the TeV range. This reduction of the effective number of sources mayinvalidate the assumption of a continuous CR injection and a stationary CR flux. .10.20.40.61 0 10 20 30 40 50 60 70 80 90 η ϑ /degree5g/cm Figure 3. Left: Grammage X crossed by CRs in a “disc and halo” model as a function of thetilt angle ϑ between the regular magnetic field and the Galactic plane, and of the turbulencelevel η . Right: Fit of the diffusion coefficients D k and D ⊥ at E = 10 eV as a function of η ,from Ref. [3]. 3. A local source and the cosmic ray anisotropy In the diffusion approximation, Fick’s law is valid and the net CR current j ( E ) is determinedby the gradient of the CR number density n ( E ) = d N/ (d E d V ) and the diffusion tensor D ab ( E )as j a = − D ab ∇ b n . The dipole vector δ of the CR intensity I = c/ (4 π ) n follows then as δ a = 3 c j a n = − D ab c ∇ b nn . (3)In the case of a strong ordered magnetic field B , the tensor structure of the diffusion tensorsimplifies to D ab ∝ B a B b . This corresponds to a projection of the CR gradient onto the magneticfield direction [8]. Hence, anisotropic diffusion predicts that the dipole anisotropy should alignwith the local ordered magnetic field instead of pointing to the source [8, 2]. Note that theordered magnetic field corresponds to the sum of the regular magnetic field and the sum ofturbulent field modes with wavelengths larger than the Larmor radius at the corresponding CRenergy.In the case of an (anisotropic) three-dimensional Gaussian CR density n , the formula (3) canbe evaluated analytically. The result δ = 3 R/ (2 cT ) for a single source with age T and distance R is independent of the regular and turbulent magnetic field. In Ref. [2], it was shown that theCR density of a single source is quasi-Gaussian, if CRs propagate over length scales l ≫ L coh .Numerically, the dipole anisotropy δ of a source contributing the fraction f i to the total observedCR flux is thus δ i = f i R cT ≃ . × − f i (cid:18) R 200 pc (cid:19) (cid:18) T (cid:19) − . (4)In Fig. 4, we show experimental data for the dipole anisotropy from Refs. [9, 10] as a greenband: The anisotropy grows as function of energy until E ≃ ∼ ◦ . Such a flip is naturally explained by the projectioneffect on the magnetic field line, if above 20 TeV another source, which is located in the oppositehemisphere, dominates the CR dipole anisotropy.More specifically, it was suggested in Ref. [2] that a 2–3 Myr old source at the distance200–300 pc dominates the dipole anisotropy in the range 2–20 TeV. Previously, it was shown in -5 -4 -3 -2 100 TeV 1 PeV δ E [eV] all dataIceCube 2013fit 1 SNtotalGalactic Figure 4. Lower and upper limit (green band) on the dipole anisotropy and data from IceCube(blue error-bars) compared to the contribution from the local source (red, for two values of E max ) and from the average CR sea (magenta) as function of energy; adapted from Ref. [2].Ref. [11] that the same source can explain the “positron excess”, as well as the breaks of thenuclei spectra and the different slope of the proton spectrum [12]. The contribution of this localsource is shown by two magenta lines for two different high-energy cutoffs: In one case, it wasassumed that the source can accelerate up 100 TeV, in the other that it is a PeVatron. In bothcases, the CR flux was calculated following the trajectories of individual CRs, as discussed inRefs. [7, 13, 11, 12]. Additionally, the total anisotropy beyond 10 eV of all Galactic SNe isshown by red error-bars which is calculated in the escape model which uses the same magneticfield configuration as the one used for the loal source [7, 13].A characteristic feature of this proposal is that a relatively old source dominates the observedCR flux. This is only possible in the case of anisotropic diffusion, and requires additionally thatthe perpendicular distance d ⊥ of the Sun to the magnetic field line connecting it to the sourceis not too large. Even for small d ⊥ , the CR flux from the single source is suppressed at low-energies, because of the slower perpendicular diffusion. In Refs. [12], the value d ⊥ ≃ 70 pc wasestimated requiring that the low-energy break in the source spectrum explains the breaks in theenergy spectra of CR nuclei. For this choice of d ⊥ , the flux of the local source is suppressedbelow ≃ f i and the transition to thestandard δ ∝ E / behavior below this energy.Another choice for the age of the source was suggested in Refs. [14, 15]. Here, Vela with theage around 11,000 yr and distance 300 pc was proposed as the single source responsible for theplateau in the dipole anisotropy in the energy range 2–20 TeV. In this case, the contribution ofVela to the dipole amplitude has to be suppressed by a factor ≃ D ab ∇ b n can reduce the dipole [16, 15]. Second, the measuredCR dipole is a projection into the equatorial plane and is thus reduced compared to the trueone. Finally, the CR flux contributed by Vela may be small. Calculating the CR fluxes fromnearby young sources using the standard isotropic diffusion coefficient and taking into accounthese effects, Ref. [15] argued that Vela leads to correct level of anisotropy. There is howevera caveat in this conclusion: While Ref. [15] calculates the CR fluxes from individual sourcesusing an isotropic diffusion coefficient, the remaining analysis is based on the assumption ofstrongly anisotropic diffusion. In the latter case, the CR flux depends however crucially on theperpendicular distance d ⊥ of the source to the magnetic field line connecting it with the Sun,and a calculation of the CR flux following the lines of Refs. [2, 11, 12] is required. Moreover,the number of sources is strongly reduced and correspondingly the flux of nearby sources withsmall perpendicular distance strongly enhanced. 4. Conclusions We have argued that the diffusion in the GMF has to be strongly anisotropic, because otherwiseCRs overproduce secondary nuclei like boron for any reasonable values of the strength and thecoherence scale of the turbulent field. Therefore the number of CRs contributing to the local CRflux is strongly reduced compared to the “standard picture”. As a result, the CR density is lesssmooth, and the contribution of individual sources to the CR dipole anisotropy becomes moreprominent. In this picture, the observed plateau in the CR dipole anisotropy around 1–20 TeVcan be explained by a 2–3 Myr old CR source which dominates the local CR flux in this energyrange. Such a source can explain also several other CR puzzles such as the “positron excess” ,the difference in the slope of the proton and nuclei spectra as well as their breaks [12]. Acknowledgments It is a pleasure to thank Gwenal Giacinti, Andrii Neronov, Volodymyr Savchenko and DimitriSemikoz for fruitful collaborations. References [1] Hillas A M 2005 J. Phys. G31 R95–R131[2] Savchenko V, Kachelrieß M and Semikoz D V 2015 Astrophys. J. L23 ( Preprint )[3] Giacinti G, Kachelrieß M and Semikoz D V 2018 JCAP 051 ( Preprint )[4] J´ohannesson G et al. Astrophys. 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