Particle acceleration at a perpendicular shock
SHINE 2006
Zermatt, Utah
August 3rd
Gang Li, G. P. Zank and Olga Verkhoglyadova
Institute of Geophysics and Planetary Physics, University of California, Riverside, CA 92521, USA
Outline
• perpendicular diffusion coefficient, NLGC theory
• Can the injection requirement (isotropy) be relaxed?
• Perpendicular and parallel shocks from observation.
• Acceleration at a perpendicular shock, maximum and injection energy
Difference between parallel and perp. shock
Perpendicular shock Quasi-perp shock
Particle acceleration at a perpendicular shock
Alfven wave intensity goes to zero at a perp. shock, and _parallel ~ 1/ == > no time to reach ~ GeV.
but, _perp is smaller, so maybe a perpendicular shock acceleration?
= _parallel /(1 + (_parall/ rl)2)
Need a good theory of _perp
Simple QLT:
Non-linear-Guiding-center:
acc xx
p
dp
dt
r
u r FHG
IKJ
1 3
1
1
2
b g
NONLINEAR GUIDING CENTER THEORY
Matthaeus, Qin, Bieber, Zank [2003] derived a nonlinear theory for the perpendicular diffusion coefficient, which corresponds to a solution of the integral equation
32 2
2 2 20 03
xxxx
xx z zz
S da v
B v k k
k k
2D slabxx xx z xxS S k k S k k
Superposition model: 2D plus slab
Solve the integral equation approximately (Zank, Li, Florinski, et al, 2004):
2/3
2/3 1/322 22/3 22 2/3 1/3
2 2/3 1/32/32 2/3 1/32 20 2 0
min , 33 1 4.33 3 3.091 3
3
slabD slab
xx D slab slabslabD
a Cb ba C H H
B b B
modeled according to QLT.
Diffusion tensor: 2 2sin cosxx bn bn
Since , the anisotropy is defined by
1/ 22 2 2 2 2
2 2 2
( cos )sin31
3 ( sin cos )d bn bn
bn bn
u q
v
For a nearly perpendicular shock sin 1bn
1/ 22 2 2
2 2 2
cos3 1
( cos )1
g bn
bn
ru
v r
1
Anisotropy and the injection threshold
To apply diffusive shock acceleration
Anisotropy as a function of energy(r = 3)
Injection threshold as a function of angle for 1
Remarks: 1)Anisotropy very sensitive to2) Injection more efficient for quasi-parallel and strictly perpendicular shocks
90bn
Anisotropy and the injection threshold
Particle acceleration at Perp. shock: recipes
• STEP 1: Evaluate K_perp at shock using NLGC theory instead of wave growth expression.
2/32
2/3 22 2/3 1/322
0
2/31/32 2
2/3 1/32/3 2/3 1/32 22 0
33
min , 31 4.33 3 3.091 3
3
D
xx D
slabslab
slab slabslabD
bva C
B
a C bH H
b B
2 2 22
2 3
20% :80%slab Db b b
b R
Parallel mfp evaluated on basis of QLT (Zank et al. 1998.
• STEP 2: Evaluate injection momentum p_min by requiring the particle anisotropy to be small.
1/ 22 2 2
2 2 2
cos13
( cos )1
g bninj
bn
rv u
r
Particle acceleration at Perp. shock: recipes
• STEP 3: Determine maximum energy by equating dynamical timescale and acceleration timescale.
R t
R t
q t
up d p
p
pafaf
af a f a fb g lnmax
z12
inj
3/ 4
2 1/32 1/3
max min2
14
3 2.243slabsh slabbmV r eB R
p pr B c R
Remarks: 2
12
slabbR
B
22 3shV Rt
R
Like quasi-parallel case, p_max decreases with increasing heliocentric distance.
Particle acceleration at Perp. shock: recipes
Maximum and injection energies
Difference between parallel and perp. shock
Backup
acc xx
p
dp
dt
r
u r FHG
IKJ
1 3
1
1
2
b g
|| Bohmgvr
3
||
1 2cHard sphere scattering: c gr 3 v
Particle scattering strength
Weak scattering: 1gr 2 1gr
Strong scattering: 1gr
2 2B B
Shock acceleration time scale
