Cosmic Rays and Galactic Field

1. Cosmic Rays and Galactic Field 3 March 2003 Astronomy G9001 - Spring 2003 Prof. Mordecai-Mark Mac Low

3. MHD waves Robert McPherron, UCLA

4. Galactic Magnetic Field • Scale height • Concentration by spiral arms

5. Dynamo Generation of Fields • Seed field must be present • advected from elsewhere • or generated by “battery” (eg thermoelectric) • No axisymmetric dynamos (Cowling) • Average resistive induction equation to get mean field dynamo equation turbulent EMF = <B> (correlated fluctuations) mean values

6. Dynamo Quenching • α-dynamo purely kinematic • growing mean field does not react back on flow • Strong enough field prevents turbulence • effectively reduces α • Open boundaries may be necessary for efficient field generation (Blackman & Field)

7. Stretch-Twist-Fold Dynamo • Zeldovich & Vainshtein (1972) • Field amplification from stretch (b) • Flux increase from twist (c), fold (d) • Requires reconnection after (d) Cary Forest

8. Galactic Dynamo • Explosions lifting field • Coriolis force twisting it • Rotation folding it.

9. Parker Instability • If field lines supporting gas in gravitational field g bend, gas flows into valleys, while field rises buoyantly • Instability occurs for wavelengths

10. Relativistic Particles • ISM component that can be directly measured (dust, local ISM also) • Low mass fraction, but energy close to equipartition with field, turbulence • Composition includes H+, e-, and heavy ions • Elemental distribution allows measurement of spallation since acceleration: pathlength

11. The all-particle CR spectrum Galactic: Supernovae Galactic?, Neutron stars, superbubbles, reacceleratedheavy nuclei --> protons ? Extragalactic?; source?, composition? Cronin, Gaisser, Swordy 1997

12. Wilkes

13. Solar Modulation • Solar wind carries B field outward, modifying CR energy spectrum below few GeV • diffusion across field lines • convection by wind • adiabatic deceleration • Energy loss depends on radius in heliosphere, incoming energy of particle

14. Garcia-Munoz et al. 1987 Cosmic Ray Pathlengths • Spallation • relative abundances of Li, B, Be to C,N,O much greater than solar; sub-Fe to Fe also. • primarily from collisions between heavier elements & H leading to fission • equivalent to about 6 g cm-2 total material • Diffusion out of Galaxy • Models of path-length distribution suggest exponential, not delta-function • Produced by leaky-box model • total pathlength decreases with increasing energy

15. Leaky Box / Galactic Wind • Peak in pathlengths at 1 GeV can be fit by galactic wind driven by CRs from disk • High energy CRs diffuse out of disk • Pressure of CRs in disk drives flow outwards, convecting CRs, gas, B field • If convection dominates diffusion in wind, low energy CRs removed most effectively by wind • Typical wind velocities only of order 20 km/s • Could galactic fountain produce same effect?

16. Slides adapted from Parizot (IPN Orsay) Magnetic fields and acceleration • How is it possible? • B fields do NOT work (F B) • In a different frame, pure B is seen as E • E' = v  B (for v/c << 1) • In principle, one can always identify the effective E field which does the work • but description in terms of B fields is often simpler  acceleration by changeofframe

17. Trivial analogy... • Tennis ball bouncing off a wall • No energy gain or loss v v rebound = unchanged velocity v v same for a steady racket... How can one accelerate a ball and play tennis at all?!

18. Moving racket • No energy gain or loss... in the frame of the racket! V v Guillermo Vilas v + 2V unchanged velocity with respect to the racket  change-of-frame acceleration

19. Fermi acceleration • Ball  charged particle • Racket  “magnetic mirrors” B B V B • Magnetic “inhomogeneities” or plasma waves

20. Fermi stochastic acceleration • When a particle is reflected off a magnetic mirror coming towards it in a head-on collision, it gains energy • When a particle is reflected off a magnetic mirror going away from it, in an overtaking collision, it loses energy • Head-on collisions are more frequent than overtaking collisions  net energy gain, on average (stochastic process)

21. E2, p2 q1 q2 V E1, p1 Second Order Fermi Acceleration • Direction randomized by scattering on the magnetic fields tied to the cloud

22. On average: • Exit angle: < cos q2 > = 0 • Entering angle: • probability  relative velocity (v - V cos q)  < cos q1 > = - b / 3 Finally... second order in V/c

23. Mean rate of energy increase Mean free path between cloudsalong a field line: L Mean time between collisions L/(c cos f) = 2L/c Acceleration rate dE/dt = 2/3 (V2/cL)E  E/tacc Energy drift function b(E)  dE/dt = E/tacc

24. Energy spectrum • Diffusion-loss equation Injection rate diffusion term Flux in energy space Escape • Steady-state solution (no source, no diffusion)  power-law x = 1 + tacc/tesc

25. Problems of Fermi’s model • Inefficient • L ~ 1 pc  tcoll ~ a few years • b ~ 10-4  b2 ~ 10-8 (tCR ~ 107 yr) tacc > 108 yr !!! • Power-law index • x = 1 + tacc/ tesc • Why do we see x ~ 2.7 everywhere ?  smaller scales

26. Add one player to the game... • “Converging flow”... Marcelo Rios Guillermo Vilas V V

27. Diffusive shock acceleration Shocked medium Interstellar medium • Shock wave (e.g. supernova explosion) Vshock • Magnetic wave production • Downstream: by the shock (compression, turbulence, hydro and MHD-instabilities, shear flows, etc.) • Upstream: by the cosmic rays themselves •  ‘isotropization’ of the distribution (in local rest frame)

28. Every one a winner! Shocked medium Interstellar medium Vshock Vshock/ D • At each crossing, the particle sees a ‘magnetic wall’ at V = (1-1/D) Vshock •  only overtaking collisions.

29. First order acceleration On average: • Up- to downstream: < cos q1 > = -2/3 • Down- to upstream: < cos q2 > = 2/3 Finally...  first order in V/c

30. Energy spectrum • At each cycle (two shock crossings): • Energy gain proportional to E: En+1 = kEn • Probability to escape downstream: P = 4Vs/rv • Probability to cross the shock again: Q = 1 - P • After n cycles: • E = knE0 • N = N0Qn • Eliminating n: • ln(N/N0) = -y ln(E/E0), where y = - ln(Q)/ln(k) • N = N0 (E/E0)-y x = 1 + y = 1- ln Q/ln k

31. Universal power-law index with • We have seen: • For a non-relativistic shock • Pesc << 1 • DE/E << 1 • … where D = g+1/g-1 for strong shocks is the shock compression ratio • For a monoatomic or fully ionised gas, g=5/3 x = 2, compatible with observations

32. The standard model for GCRs • Both analytic work, simulations and observations show that diffusive shock acceleration works! • Supernovae and GCRs • Estimated efficiency of shock acceleration: 10-50% • SN power in the Galaxy: 1042 erg/s • Power supply for CRs: eCR Vconf/ tconf ~ 1041 erg/s ! • Maximum energy: • tacc ~ 4Vs/c2  (k1/ u1 + k2/ u2) • kB = Eb2/3qB E •  acceleration rate is inversely proportional to E… • A supernova shock lives for ~ 105 years • Emax ~ 1014 eV  Galactic CRs up to the knee...

33. Assignments • MHD Exercise • get as far as you can this week. Turn in what you’ve done at the next class. If need be, we’ll extend this long exercise to a second week. • You will need to have completed the previous exercises (changing the code, blast waves) to tackle this one effectively. • Read NCSA documentation (see Exercise) • Read Heiles (2001, ApJ, 551, L105)

34. Constrained Transport • The biggest problem with simulating magnetic fields is maintaining div B = 0 • Solve the induction equation in conservative form: Stone & Norman 1992b

35. Centering of Variables

36. Method of Characteristics Stone & Norman 1992b • Need to guarantee that information flows along paths of all MHD waves • Requires time-centering of EMFs before computation of induction equation, Lorentz forces

37. MHD Courant Condition • Similarly, the time step must include the fastest signal speed in the problem: either the flow velocity v or the fast magnetosonic speed vf2 = cs2 + vA2

38. Lorentz Forces • Update pressure term during source step • Tension term drives Alfvén waves • Must be updated at same time as induction equation to ensure correct propagation speeds • operator splitting of two terms

39. Added Routines Stone & Norman 1992b

40. Drop shot V v v - 2V Particle deceleration

41. Wave-particle interaction • Magnetic inhomogeneities ≈ perturbed field lines Adjustement of the first adiabatic invariant: p2 / B ~ cst rg<<l Nothing special... rg>>l Pitch-angle scattering:Da ~ B1/B0Guiding centre drift:r ~ rgDa rg ~ l

42. Resonant scattering with Alfven (vA2 = B2/m0r) and magnetosonic waves: w - kv = nW (W = qB/gm = v/rg : cyclotron frequency) • Magnetosonic waves: • n = 0 (Landau/Cerenkov resonance) • Wave frequency doppler-shifted to zero •  static field, interaction of particle’s magnetic moment with wave’s field gradient • Alfven waves: • n = ±1 • Particle rotates in phase with wave’s perturbating field •  coherent momentum transfer over several revolutions...

43. u1 u2 k2/u2 k1/u1 Acceleration rate downstream upstream • Time to complete one cycle: • Confinement distance: k/u • Average time spent upstream: t1 ≈ 4k / cu1 • Average time spent upstream: t2 ≈ 4k / cu2 • Bohm limit: k = rgv/3 ~ Eb2/3qB • Proton at 10 GeV: k ~ 1022 cm2/s •  tcycle ~ 104 seconds ! • Finally, tacc ~ tcycle Vs/c ~ 1 month !