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