Transport of energetic particles in cosmic magnetic fields

1. Transport of energetic particles in cosmic magnetic fields Vladimir Ptuskin

2. particle diffusion in random magnetic field • compound diffusion,anomalous diffusion • diffusion + convection • - cosmic ray transport in intergalactic space

3. diffusion approximation intergalactic space solar wind SNR interstellar medium

4. microscopic theory p θ Larmor radius average field B0 B1 L random field (Alfvenic turbulence) effect of random field: rg ~ 1/k resonance condition

5. diffusion mean free path particle scattering by weak magnetic field perturbation, rg1 >> L: δθ = L/rg1 p δθ B1 L diffusion on pitch-angle: Dθ ~ (δθ)2/δt ~ (v/rg)×(B1,res/B0)2 with estimates L ~ rg, δt ~ rg/v at kres ~ 1/rg B1 B0 scattering back ∆θ ~ 1 at τ ~ 1/Dθ ~ (rg/v)×(B0/B1,res)2 mean free path l ~ vτ ~ rg(B0/B1,res)2 parallel diffusion Dll ~ vrg(B0/B1,res)2 coefficient

6. scattering, ∆θ~1 perpendicular diffusion D┴ ~ rg2/τ ~ vrg(B1,res/B0)2 rg B0 flux Hall diffusion grad Ncr DA ~ vrg B0

7. diffusion tensor jA– Hall component j┴ jll B0 grad Ncr Ares = B1,res/B0 diffusion equation: energy density of random field: W(k)dk ~ k-2+adk, Dll ~ vrga a = 1/3 Kolmogorov spectrum a = 1/2 Kraichnan spectrum a = 0 shocks and discontinuities a = 1 Bohm diffusion scaling, DB = vrg/3 W(k) 1/k2-a kmin~1/Lmax k

8. Armstrong et al. 1995 turbulence in the interstellar medium B1 ~ B0 at L = 100 pc a = 0.3 to 0.5 Dll≈ 5×1028PGV0.3 cm2/s

9. anomalous perpendicular diffusion - cosmic ray particles are strongly magnetized: rg/l ~ 10-6 at 1 GeV; - average Galactic magnetic field is almost pure azimuthal (in the disk): B0φ : B0r : B0z = 1 : 0.2 : 0.003 - large-scale random field is large: δB ~ B0 at L ≈ 100 pc z what is efficient perpendicular diffusion coefficient in static random field if locally D┴ = DA = 0 ? “natural” answer: Def┴ = <A2>Dll is wrong ! the right answer is Def┴ =0 !! direction of average magnetic field B0 A local total field B inclined, A < 1, sinA ≈ A

10. random component average • diffusion equation • for cosmic ray density - parallel diffusion field line Dll local diffusion B0 stochastic nonlinearity no perpendicular diffusion on average !

11. compaund diffuison Dll≠0, D┴ = DA = 0, isotropic random large-scale magnetic field B random walk of magnetic field line: r2 ~ Ls particle diffusion along the line: s2 ~ Dllt displacement of particle in r space: r2 ~ L(Dllt)1/2 hence Def = r2/t →0 at t →∞ s L r Compound diffusion works only for degenerate static diffusion tensor. Finite D┴, DAor fluctuations in time destroy compound diffusion. Particle loses correlation with initial magnetic field line and “forget” information on previous trajectory

12. spreading of magnetic field lines and cosmic ray diffusion weak static large-scale random field A << 1 B0 independent random walk, r0 >> L ∆r2 ~ A2Ls - separation of two lines r0 ∆r L r0 << L, two close lines, remain correlated up to s ~ sc sc ~ L×(ln(L/r0))/A2 - decorrelation length B0 r0 ∆rc L for flat spectrum of random magnetic field W(k) ~ k-2+a, a > 0: sc ~ L/A2(with no r0!) Particle diffusion in this fieldDef┴ ~ (∆rc)2/∆tc displacement ∆rc ~ (A2Lsc)1/2 Def┴ ~ A4 Dll - anomalous perpendicular diffuson “memory” time ∆tc ~ sc2/Dll (perturbation on A2(Dll/D┴)1/2)

13. some results for strong static random field 1D case: Def = (<D-1>)-1 2D case: isotropic random field <B> = 0 Def = (D┴Dll)1/2 ~ vrg/3 divB=0 coinsides with Bohm diffusion coefficient Remark: in the case of random Alfven waves, the approximation of static field works if Va < A2Dll/L

14. 3D case: (B1 – large-scale random field, B1 < B0) ~ B0 percolation of field lines: Defll ~ Dll, Def┴ ~ (B1/B0)4Dll(Def~Dll at B1~B0) no percolation of field lines: Defll ~ (B0/B1)2vrg, Def┴ ~ (B1/B0)2vrg (Bohm dif. Def~vrg at B1~B0) thus, two general types of scaling: Def~Dll and Def~DB

15. diffusion + convection number cosmic-ray density intensity transport equation for distribution function F(p,r,t) (4π∫dpF = Ncr, F(p) = J(E)) diffusion convection transport adiabatic change of momentum source term . explanation of the 4th term ( p = -p·div(u)/3): continuity equation for background gas: hence p ~ 1/L for isotropic expansion or compression

16. cosmic rays in expanding Universe Hubble law: u(r,t) = H(t)·r, div(u(r,t)) = 3H(t) – constant on r; ∂rIcr = 0 for uniform source distribution q = q(E,t) Hubble constant H0 = 100·h km/(s·Mpc), h ≈ 0.7 Hubble scale RMg = c/H0 = 3×103h-1 Mpc redshift dz = - (1+z)·H(z)dt, dr = cdt small z << 1 asymptotics: r(z) ≈ RMg·z·(1 - (0.5+0.75·Ωm)·z + …) H(z) = H0·((1+z)3·Ωm + ΩΛ)1/2 for a flat universe, i.e. at Ωm + ΩΛ = 1 matter density Ωm≈0.3 vacuum energy density ΩΛ≈0.7

17. interaction with cosmic microwave background • high-energy protons lose • energy as result of: • cosmological redshift • e+e- pairs production • pion production • simple approximation • for energy loss rate: • dE/dt = - E/τ0(E) • at z = 0; • τ(E,z) = (1+z)-3τ0((1+z)E) • since T ~ (1+z). Nagano & Watson 2000

18. equation for cosmic ray intensity source evolution, m = 0 - no evolution source power at z = 0 where interaction with mw photons expansion solution by method of characteristics: