Cosmic Rays and Galactic Field
3 March 2003
Astronomy G9001 - Spring 2003
Prof. Mordecai-Mark Mac Low




 v1
B
B
2
0
0


c



v






v




  0
s
1
1
2


t
40
4



0



2
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
waves with v  c  v , while if k v A :
2
s
2
A
v A  v1  0 
transverse Alfven
waves
2

 2
cs
2 2
2
 k vA    v1   v 2  1 k  v A  v1  v A  0
 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
t
   v1  b1     V0  B0
turbulent EMF = <B>
(correlated fluctuations)
  
mean values
2
B0
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
2

2


vA cs
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
Cosmic Ray Pathlengths
Garcia-Munoz et
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
V
B
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
E1   E1 1   cosq1 
E2   E2 1   cosq 2 
E2, p2
q1
q2
V
E1, p1
2

E 1   cosq1   cosq 2   cosq1 cosq 2

1
2
E
1 
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  
2
E
1 
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
2

b
(
E
)
N
(
E
)

Q
(
E
)


D

N


t  E
t esc
Injection rate
Flux in energy space
diffusion term
Escape
• Steady-state solution (no source, no
diffusion)
-x
 power-law
N ( E )  constant  E
x = 1 + tacc/tesc
Problems of Fermi’s model
• Inefficient
– L ~ 1 pc  tcoll ~ a few years
  ~ 10-4  2 ~ 10-8
tacc >
• Power-law index
108
7 yr)
(t
~
10
yr !!!
CR
 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
Vshock/ D
Interstellar medium
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
2
E
1 
On average:
Up- to downstream: < cos q1 > = -2/3
Down- to upstream: < cos q2 > = 2/3
Finally...
E
E
4
4 ( D  1) Vshock
 
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
N ( E )dE  E dE
with
• For a non-relativistic
shock
– Pesc << 1
 E/E << 1
ln(1  Pesc )
x 1
ln(1  E / E )
Pesc
D2
x 1

E / E D  1
• … where D = +1/-1 for strong shocks
is the shock compression ratio
• For a monoatomic or fully ionised
gas, 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
Stone & Norman 1992b
• The biggest problem with simulating
magnetic fields is maintaining div B = 0
• Solve the induction equation in conservative
form:
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 timecentering 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
t 

x
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
Stone & Norman 1992b
Added Routines
• Drop shot
V
v
v - 2V
Particle deceleration
Wave-particle interaction
• Magnetic inhomogeneities ≈ perturbed field
lines
rg << 
Adjustement of the first
adiabatic invariant:
p2 / B ~ cst
Nothing special...
rg >> 
rg ~ 
Pitch-angle scattering:
a ~ B1/B0
Guiding centre drift:
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 !