A cosmic-ray current driven instability in parallel shocks Brian - - PowerPoint PPT Presentation

A cosmic-ray current driven instability in parallel shocks

Brian Reville

Max-Planck-Institut für Kernphysik, Heidelberg

KINETIC MODELING OF ASTROPHYSICAL PLASMAS Krakow, 5-9 October 2008

Cosmic-ray current driven instability

◮ We consider the region upstream of a quasi-pll shock ◮ 3-component plasma ◮ relativistic beam of protons (Γb) along zeroth order field ◮ thermal electron/proton distribution kbT/mc2 = Θ ≪ Γb ◮ linear dispersion relation for circularly polarised waves ◮ plasma susceptibility:

ω2χ ≈ ω′

pb 2ω′

ǫωc − ω′

pb 2ω′

ǫωc + ω′ + c2ω2 v2

A

+ ω2

pω

ǫω3

c

• c2k2 − ω2

u2

⊥

• p

ωΘj, ckΘj ≪ |ωcj|

Cosmic-ray current driven instability

◮ Neglecting thermal effects, and provided

Γbβ2(nb/np)(ωpp/ωcp)2 ≫ 1 maximum growth rate of Im(ω) = 1 2 nb np βbωpp Same result as Bell 2004

◮ Including thermal effects, if Θp ≫ vA/c

Im(ω) = √ 3 2 nb np 2

3 vA

c 2

3 ωpp

ωc 2

3

• β2

b

• u2

⊥

• 1

3

ωc

◮ May provide saturation mechanism in relativistic plasmas

Cosmic-ray current driven instability

e.g., vA = 2 × 10−5, Γb = 10, nb/np = 1/10, ǫ = −1, ǫ = +1 Θ = 1/1000 Θ = 1/10 log ˜ k log ˜ k ˜ ω ˜ ω

• cf. BR, Kirk & Duffy 2006, PPCF

Cosmic-ray current driven instability

◮ saturation when currents associated with waves:

|k × B| ≈ 4πncreβ

◮ k ∼ 1/rg - saturated field energy

B2

w

8π = 1 2ncrΓbmpc2

◮ Entire energy of the beam goes into magnetic field

production

◮ How do we include this in acceleration models?

Cosmic-ray modified shocks

◮ Particles escape beyond some boundary ◮ escaping particles drive magnetic field growth

Steady-state solutions

◮ coupled hydrodynamic - kinetic equations

∂f ∂t + ∂ ∂x

• uf − κ(x, p) ∂f

∂x

• = ∂

∂p3

• p3f ∂u

∂x

• Lesc = κ(p∗)

u0 , e.g. p∗ = 105

R=1.1 R=2.2 R=9.5

p4f p/mc

0.001 0.01 0.1 1 10 0.1 1 10 100 1000 10000 100000 1e+06

Injection efficiency ν − R diagram

0.01

R M=10 50 100 500

ν

1 10 100 1 0.1

u0 = 5000km s−1, n0 = 1 cm−3, p∗ = 103 ν = 4π 3 mc2 ρ0u2 p4

0f0(p0),

Maximum momentum

◮ what is the location of escape boundary - transition zone

from weak to strong turbulence

◮ Lesc = κ(p∗)/u0 determines maximum confined energy ◮ diffusive current at escape boundary drives nonresonant

instability jcr(−Lesc) = −4πe ∞

p0

κ ∂f ∂x p2dp

◮ How do we determine Lesc in a self-consistant manner? ◮ Calculate CR flux (and p∗) from numerical SS solution ◮ Transition zone determined from condition

Ladv ≡ ush/Γmax ∼ Lesc

Maximum momentum

adv

L Lesc ν

0.5 1 1.5 2 2.5 3 3.5 4 0.01 0.1

105 104 103 102 BR, Kirk & Duffy, ApJ submitted

◮ Maximum energy can be calculated as a function of

injection parameter

Summary

◮ Efficient acceleration of particles at collisionless shocks

connected with magnetic field amplification

◮ nonresonant mode (Bell 2004) appears to be of greatest

importance in efficient cosmic-ray accelerating shocks

◮ reduced diffusion coefficients leads to more rapid

acceleration

◮ field amplification incorporated using free escape

boundary Lesc

◮ maximum energy determined by boundary position -

calculated self-consistently

◮ system is self-organising - maximum energy function of

injection