Particle Acceleration by Shocks
Tony Bell
with
Brian Reville, Klara Schure,
Gwenael Giacinti
University of Oxford
http://hubblesite.org/newscenter/archive/releases/2006/30/image/a/format/xlarge_web/
Cassiopeia A
Radio
(VLA)
Infrared
(Spitzer)
chandra.harvard.edu/photo/
0237/0237_radio.jpg
NASA/ESA/
Hubble Heritage
(STScI/AURA))
NASA/JPL
NASA/CXC/MIT/UMass
Amherst/
M.D.Stage et al.
NASA/JPL-Caltech/
O Krause(Steward Obs)
Optical
(Hubble)
X-ray
(Chandra)
Historical shell supernova remnants
Chandra observations
Tycho 1572AD
Kepler 1604AD
HESS observation
SNR RX J1713.7-3946
Aharonian et al
Nature (2004)
SN1006
NASA/CXC/Rutgers/
J.Hughes et al.
NASA/CXC/NCSU/
S.Reynolds et al.
Cas A 1680AD
NASA/CXC/Rutgers/
J.Warren & J.Hughes et al.
NASA/CXC/MIT/UMass Amherst/
M.D.Stage et al.
Cosmic Ray (CR) acceleration
This talk:
• How do CR escape SNR?
• Can SNR accelerate CR to 1 PeV – and when?
• Importance of magnetic field amplification for the above
Observations: TeV emission outside SNR
For related discussion :
• Drury (2011) MNRAS 415 1807
• Malkov, talk on Weds
• Reville, talk on Weds
Cosmic ray acceleration
B1
B2
CR track
High velocity
plasma
Low velocity
plasma
Due to scattering, CR recrosses shock many times
Gains energy at each crossing
CR acceleration time
shock
u
ncr
upstream
L=D/ushock
Time needed for acceleration (Lagage & Cesarsky)

4Dupstream
2
ushock
D increases with CR energy
4Ddownstream 8Dupstream


2
2
(ushock / 4)
ushock
L~R/8
Max CR energy set by  = R/ushock
R
Shock moves distance R = 8L during CR acceleration time 
SNR
If so, CR never escape upstream
Theory is simplistic
CR
precursor
shock
Maximum CR energy
Magnitude of the problem: CR Larmor radius:

4Dupstream
2
ushock

rg 
 PeV
BG
parsec
4Ddownstream 8Dupstream

2
2
(ushock / 4)
ushock
Max CR energy set by  = R/ushock
Bohm is minimum diffusion coefficient:
Maximum CR energy:
DBohm 
crg
3
 eV
3BTesla
  83 ushock BR
Young SNR: age=300yrs, B=3G, ushock=5000 km s-1
Conclusion:

Max CR energy = 1013eV
Need amplified magnetic field,
D varies with time, space, CR energy
Tycho
L~R/8
CR
precursor
Streaming CR excite instabilities
R
SNR
shock
Shock
CR streaming ahead of shock
Excite instabilities
Amplify magnetic field
upstream
Amplify magnetic field
Lucek & Bell (2000)
downstream
Conditions for PeV acceleration
  83 ushock BR
Equipartition magnetic field
B2
0
2
 ushock
Maximum CR energy: 20PeV
Theoretical saturation, matches observation (Vink 2006,2008)
B2
0

ushock
2
ushock
c
Maximum CR energy: 0.5 PeV
Within error bars, but tough!
Are Tycho, Kepler already too old and too slow?
  0.03
(young SNR)
 = CR efficiency factor
Time for magnetic field amplification?
Growth rate of fastest growing mode:
 max  12 j
0

CR electric current density:
Upstream energy fluxes:
nCR evdrift j  u
3
shock
 j
Shortest growth time:
1
 max

3
ushock
j
j
Energy of CR carrying current
50 PeV
years
3
0.03u7 ncm
Density in cm-3
CR efficiency/0.03
ushock in 10,000 km s-1
Cannot assume instability reaches saturation
The scalelength issue
CR Larmor radius:
rg  3 1016
Wavelength of fastest growing mode:
 PeV
BG
m
2 / kmax  2 1014 BG m
for ushock=10,000 km s-1 and n =1 cm-3
Fortunately: instability grows non-linearly by spatial expansion
Routes to large-scale structure with CR response included:
1) Filamentation (Brian Reville)
2) Include scattering (Klara Schure)
Numerical simulation of interacting physics
Coupled questions:
• Does the instability have time to grow?
• Does the instability saturate?
• How large is the magnetic field?
• What is the maximum CR energy?
• Do CR escape upstream of the shock?
Simulation code:
• MHD background plasma coupled to kinetic CR treatment through jxB
• Include shock, precursor & escape
• Self-consistent magnetic field generation
• CR respond to magnetic field (not diffusion model)
• 2D or 3D with momentum-dependent beyond-diffusion CR treatment
• Time-dependent
CR model:


f 
1  p3 f
f
f
 .( fu)  .u 2
 v.  ev  B.  0
t r
3p
p
r
p
f  f0 ( p, r, t )  fi ( p, r, t ) pi  fij ( p, r, t ) pi p j
isotropic
drift
off-diagonal part of stress tensor
i j
CR distribution defined in local fluid rest frame
See Schure & Bell (2011) for instability analysis with stress tensor
Flow into reflecting wall (2D simulation)
Parallel magnetic field
CR free
expansion
Thermal pressure
CR energy density
Magnetic energy density
Perpendicular magnetic field
wall
Flow at 0.1c
7.7rg
(64 cells)
shock
370rg (3104 cells)
Section near shock
shock
Thermal pressure
CR energy density
Magnetic energy density
7.7rg
61rg
Momentum dependence
Two populations at low CR energy
p  pinject
• Confined by magnetic field
• Freely escaping, excite instability
High energy CR escape freely:
Large mean free path
Generated once low energy CR confined
p  10 pinject
Escape and confinement (t=2t0/3)
3D simulation
shock
240rg
7.7rg
Thermal pressure
escaping CR
CR energy density
Perpendicular magnetic field
Perpendicular field
Perpendicular slices
Confined
CR
Instability growth
CR energy density
Perpendicular magnetic field
Stationary box in upstream plasma
Max growth rate
 max  12 j
Number of e-foldings:
  maxdt  12
0

0
jdt
 
Number of CR passed through box (times charge)
CR only confined if enough CR escaped upstream
How many e-foldings
(Fixed current simulations 2004)
1
 max
 0.8
1
5 max
Condition for CR confinement:

max
dt  5  10
Instability growth
CR energy density
Perpendicular magnetic field
Condition for CR confinement:
Upstream energy fluxes:
0
jdt  10


nCR evdrift j  u
Mean energy of escaping CR:
3
shock
j 
Max CR energy a few times larger:
 0
10
(matches simulation)
3
ushock
j
j
3
1/ 2 3
ushock
t  ncm
u7 t300 PeV
in 300 yrs
 max   j
Make a guess:  = 3
in 10,000 km s-1
in cm-3
Compare with saturation limit on CR energy
Instability growth time (depends on CR escaping upstream)
  maxdt  5
1/ 2 3
 max  33 ncm
u7 t300 PeV
max = j  = 3
Instability saturation + acceleration time
B2
0

ushock
2
ushock
c
Suggests:
1/ 2 7 / 2
 max  5 ncm
u7 t300 PeV
in 300 yrs
in cm-3
in 10,000 km s-1
• PeV acceleration lies on limit for both growth times and saturation
• High energy CR escape upstream (with efficiency ~ almost by definition)
Growth time limit
3
 max  0.3 0 ushock
t
Saturation limit
 max  0.4 0 
ushock 3
ushock t
c
Evolution of max CR energy as limited by growth times
1/ 2 3
 max   ncm
u7 t300 PeV
assume  = 3
1987A after 6 years
u7  3.5
1/ 2
 max  3 ncm
PeV
Cas A
u7  0.6, t300  1, ncm  1
 max  0.6 PeV
During Sedov phase
0.1 0.8
4/ 3
 max  0.2 E440.6 ncm
t300 PeV ( R2  ushock
)
Blast wave energy in 1044J
Conclusions
• Instability growth/saturation limits acceleration
• Some CR must escape/get ahead of main precursor to excite magnetic field
• Energy

of escaping CR determined by
j   1
• Pre-Sedov SNR reach PeV, but only just
• Max CR energy drops during Sedov
• Young high velocity SNR into high density might exceed PeV