Cosmic Rays and Galactic Field

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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
                          40                  40  
                                                     
                                                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 q2 > = 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 q2 > = 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   D2
                                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, g5/3       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  E2/3qB 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

         vB
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 <<                                      p2 / 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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