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

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cosmic rays and galactic field

Cosmic Rays and Galactic Field

3 March 2003

Astronomy G9001 - Spring 2003

Prof. Mordecai-Mark Mac Low

mhd waves
MHD waves

Robert McPherron, UCLA

galactic magnetic field
Galactic Magnetic Field
  • Scale height
  • Concentration by spiral arms
dynamo generation of fields
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

dynamo quenching
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
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

galactic dynamo
Galactic Dynamo
  • Explosions lifting field
  • Coriolis force twisting it
  • Rotation folding it.
parker instability
Parker Instability
  • If field lines supporting gas in gravitational field g bend, gas flows into valleys, while field rises buoyantly
  • Instability occurs for wavelengths
relativistic particles
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
The all-particle CR spectrum

Galactic: Supernovae

Galactic?, Neutron stars, superbubbles, reacceleratedheavy nuclei --> protons ?

Extragalactic?; source?, composition?

Cronin, Gaisser, Swordy 1997

solar modulation
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
cosmic ray pathlengths

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
leaky box galactic wind
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?
magnetic fields and acceleration

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

trivial analogy
Trivial analogy...
  • Tennis ball bouncing off a wall
    • No energy gain or loss



rebound = unchanged velocity



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!



Guillermo Vilas

v + 2V

unchanged velocity with respect to the racket

 change-of-frame acceleration

fermi acceleration
Fermi acceleration
  • Ball  charged particle
  • Racket  “magnetic mirrors”





  • Magnetic “inhomogeneities” or plasma waves
fermi stochastic acceleration
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

E2, p2




E1, p1

Second Order Fermi Acceleration
  • Direction randomized by scattering on the magnetic fields tied to the cloud

On average:

  • Exit angle: < cos q2 > = 0
  • Entering angle:
  • probability  relative velocity (v - V cos q)  < cos q1 > = - b / 3


second order in V/c

mean rate of energy increase
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

energy spectrum
Energy spectrum
  • Diffusion-loss equation

Injection rate

diffusion term

Flux in energy space


  • Steady-state solution (no source, no diffusion)

 power-law

x = 1 + tacc/tesc

problems of fermi s model
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

add one player to the game
Add one player to the game...
  • “Converging flow”...

Marcelo Rios

Guillermo Vilas



diffusive shock acceleration
Diffusive shock acceleration

Shocked medium

Interstellar medium

  • Shock wave (e.g. supernova explosion)


  • 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
Every one a winner!

Shocked medium

Interstellar medium


Vshock/ D

  • At each crossing, the particle sees a ‘magnetic wall’ at V = (1-1/D) Vshock
  •  only overtaking collisions.

First order acceleration

On average:

  • Up- to downstream: < cos q1 > = -2/3
  • Down- to upstream: < cos q2 > = 2/3


 first order in V/c

energy spectrum30
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

universal power law index
Universal power-law index


  • 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

the standard model for gcrs
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...

  • 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
Constrained Transport
  • The biggest problem with simulating magnetic fields is maintaining div B = 0
  • Solve the induction equation in conservative form:

Stone & Norman 1992b

method of characteristics
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
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
lorentz forces
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
added routines
Added Routines

Stone & Norman 1992b

Drop shot



v - 2V

Particle deceleration

wave particle interaction
Wave-particle interaction
  • Magnetic inhomogeneities ≈ perturbed field lines

Adjustement of the first adiabatic invariant: p2 / B ~ cst


Nothing special...


Pitch-angle scattering:Da ~ B1/B0Guiding centre drift:r ~ rgDa

rg ~ l

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...
acceleration rate





Acceleration rate



  • 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 !