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Cosmic Rays and Galactic Field 3 March 2003 Astronomy G9001 - Spring 2003 Prof. Mordecai-Mark Mac Low v1 2 B0 B0 cs v1 2 v1 0 t 2 40 40 B0 introduce the Alfven velocity v A , and choose 4 0 plane waves v1 v1 exp ik x i t . v1 (c v ) k v1 k 2 2 s 2 A v A k v A k v1 v A v1 k k v1 v A 0 if k v A then last term vanishes, leaving magnetosonic v A v1 0 waves with v c v , while if k v A : 2 s 2 A transverse Alfven cs2 2 waves k vA v1 v2 1 k v A v1 v A 0 2 2 2 A MHD waves Robert McPherron, UCLA Galactic Magnetic Field • Scale height • Concentration by spiral arms 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 B0 v1 b1 V0 B 0 2 B 0 t turbulent EMF = <B> mean values (correlated fluctuations) 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) Stretch-Twist-Fold Dynamo Cary Forest • Zeldovich & Vainshtein (1972) • Field amplification from stretch (b) • Flux increase from twist (c), fold (d) • Requires reconnection after (d) Galactic Dynamo • Explosions lifting field • Coriolis force twisting it • Rotation folding it. Parker Instability • If field lines supporting gas in gravitational field g bend, gas flows into valleys, while field rises buoyantly • Instability occurs for wavelengths vA cs2 2 1 2 2cs g 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 The all-particle CR spectrum Galactic: Supernovae Galactic?, Neutron stars, superbubbles, reaccelerated heavy nuclei --> protons ? Extragalactic?; source?, composition? Cronin, Gaisser, Swordy 1997 Wilkes 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 Garcia-Munoz et Cosmic Ray Pathlengths al. 1987 • 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 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? 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 change of frame 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?! • 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 Fermi acceleration • Ball charged particle • Racket “magnetic mirrors” B B V B • Magnetic “inhomogeneities” or plasma waves 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) Second Order Fermi Acceleration • Direction randomized by scattering on the magnetic fields tied to the cloud E 2 , p2 E1 g E1 1 cosq1 q1 q2 E2 g E2 1 cosq2 V E 1 , p1 E 1 cosq1 cosq 2 2 cosq1 cosq 2 1 E 1 2 On average: Exit angle: < cos q2 > = 0 Entering angle: probability relative velocity (v - V cos q < cos q1 > = - / 3 Finally... E 1 2 / 3 4 2 1 E 1 2 3 second order in V/c Mean rate of energy increase Mean free path between clouds along 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 Energy spectrum • Diffusion-loss equation N N b( E ) N ( E ) Q ( E ) D 2 N t E t esc Injection rate diffusion term Flux in energy space Escape • Steady-state solution (no source, no diffusion) power-law N ( E ) constant E -x x = 1 + tacc/tesc Problems of Fermi‟s model • Inefficient – L ~ 1 pc tcoll ~ a few years ~ 10-4 2 ~ 10-8 tacc > 108 yr !!! (tCR ~ 107 yr) • Power-law index smaller scales – x = 1 + tacc/ tesc • Why do we see x ~ 2.7 everywhere ? Add one player to the game... • “Converging flow”... Marcelo Rios Guillermo Vilas V V Diffusive shock acceleration • Shock wave (e.g. supernova explosion) Shocked medium Interstellar medium 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) Every one a winner! Shocked medium Interstellar medium Vshock/ D Vshock • At each crossing, the particle sees a „magnetic wall‟ at V = (1-1/D) Vshock • only overtaking collisions. First order acceleration E 1 cosq1 cosq 2 2 cosq1 cosq 2 1 E 1 2 On average: Up- to downstream: < cos q1 > = -2/3 Down- to upstream: < cos q2 > = 2/3 Finally... E 4 4 ( D 1) Vshock E 3 3 D c first order in V/c 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 N ( E )dE E dE x = 1 + y = 1- ln Q/ln k Universal power-law index • We have seen: x ln(1 Pesc ) N ( E )dE E dE with x 1 ln(1 E / E ) • For a non-relativistic shock – Pesc << 1 Pesc D2 x 1 E/E << 1 E / E D 1 • … where D = g+1/g-1 for strong shocks is the shock compression ratio • For a monoatomic or fully ionised gas, g5/3 x = 2, compatible with observations 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 ~ 4 Vs/c2 (k1/ u1 + k2/ u2) kB E2/3qB E – acceleration rate is inversely proportional to E… • A supernova shock lives for ~ 105 years – Emax ~ 1014 eV Galactic CRs up to the knee... 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) Constrained Transport • The biggest problem with simulating magnetic fields is maintaining div B = 0 • Solve the induction equation in conservative form: Stone & Norman 1992b B v B t S v B dl t C vB Centering of Variables 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 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 x t max v, c v 2 s 2 A Lorentz Forces 1 1 1 B B B B B 2 4 4 8 • 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 Added Routines Stone & Norman 1992b • Drop shot V v v - 2V Particle deceleration Wave-particle interaction • Magnetic inhomogeneities ≈ perturbed field lines Adjustement of the first adiabatic invariant: rg << p2 / B ~ cst Nothing special... rg >> Pitch-angle scattering: a ~ B1/B0 Guiding centre drift: rg ~ r ~ rg a • Resonant scattering with Alfven (vA2 = B2/m0) and magnetosonic waves: - 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... Acceleration rate u2 u1 downstream upstream k2/u2 k1/u1 • 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 ~ E 2/3qB – Proton at 10 GeV: k ~ 1022 cm2/s – tcycle ~ 104 seconds ! • Finally, tacc ~ tcycle Vs/c ~ 1 month !

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