Shock acceleration of cosmic rays

Tony Bell

Imperial College, London

Reynolds, 1986

SNR suitable CR source below 1015eV

Typical max. radius of rapidly expanding SNR ~ 1017m

Radio image of SN1006 x-ray image of SN1006

Long, 2003

Shock in magnetised plasma

Sho

ck

High velocityplasma

Low velocityplasma

Upstream ISM Downstream shocked plasma

B2

B1

B2>B1

Cosmic ray wanders around shock-scattered by magnetic field

High velocityplasma

Low velocityplasma

B2

B1

CR track

Due to scattering, CR recrosses shock many times

Shock acceleration gives right spectrum

High velocityplasma

Low velocityplasma

Upstream ISM Downstream shocked plasma

B2

B1B2>B1

Simple diffusion theory:

Prob of CR crossing shock times is

Shock velocity: vs

= vs/c

m)1( m

Average fractional energy gained at each crossing is

Differential spectrum is 2)( n

Allowing for propagation matches observed spectrum )( 6.2

Cosmic ray wanders around shock-scattered by magnetic field

High velocityplasma

Low velocityplasma

B2

B1

CR track

Due to scattering, CR recrosses shock many times

‘Bohm diffusion’

rg

Mean free path cr ~ rg (proportional to 1/B)

Requires disordered magnetic field: B/B ~ 1

DBohm= crg /3

L= rg c /3vshock

CR distribution near shock

shock

downstreamupstream

Exponential distn

Want small rg (large B) for rapid acceleration to high energy

Balance between advection and ‘Bohm’ diffusion (cr = rg )

Scaleheight must be less than SNR radius

LR

shock

CR pre-cursor

RvshockB must exceed certain value

Need L<R

L=(c/3vshock)cr

(c/3vshock)cr < R

cr=rg , (proportional to 1/B)

Condition on BvR

Get original version

(Hillas, 1984)

Cosmic Ray spectrum arriving at earth

Mainly protons

Reducing the CR mean free path

Magnetic field amplification

CR/Alfven wave interaction (conventional theory)

If CR gyration length matches Alfven wavelength

• CR scattered strongly by waves

• Waves excited by CR

B

CR

Currents driving Alfven waves

BjBBpt

u

)(1

0

B

CR crj

||crj

crj dominates in conventional theory

||crj dominates when CR current is large

k in units of rg-1

in units of vS2/crg

For SNR conditions, instability strongly driven

Dispersion relation

-4

-2

0

2

4

-2 0 2 4log10(k)

log10(omega) Re()

Im()

krg=1

Growth time of fastest growing modeUncertain efficiency factor

SNR expand rapidly for ~1000 yrs

Acceleration favoured by high velocity and high density

Look to very young SNR for high energy CR

eg SN1993J in M81 (Bartel et al, 2002)

After 1 year: vs =1.5x107 ms-1 ne~106cm-3

After 9 years: vs =0.9x107 ms-1 ne~104cm-3

jcr

jthermal

jthermal = -jcr

jthermal x B

causes helix expandextends field lines

increases B

Instability mechanism

helical field line

MHD simulations show

magnetic field amplification

BjBBpt

ucr

||0

)(1

Development of previous modelling, Lucek & Bell (2000)

t=0

t=6.4 t=9.5

t=12.4 t=16.8

0.01

0.1

1

10

100

0 5 10 15

Bperp

Bparallel

Brms

Bmax

Evolution of magnetic field

Magnetic field (log) time

linear non-linear

rms field grows 30xmax. field grows 100x

Saturation magnetic field proportional to 1/2vshock3/2

k in units of rg-1

in units of vS2/crg

For SNR conditions, instability strongly driven

Dispersion relation

-4

-2

0

2

4

-2 0 2 4log10(k)

log10(omega) Re()

Im()

krg=1

CR collimate intoFilaments and Beams

Filamentation & self-focussing

proton beam jvelocity vbeam

B

MHD response to beam – mean |B| along line of sight

dyB ||

z

xt=2

t=6

t=4

t=8

Current, j

B (0.71,1.32) (0.76,1.17)

Slices of B and in z at t=2

Magnetic field Density

B (0.40,2.61) (0.54,1.59)

Slices of B and in z at t=4

Magnetic field Density

B (0.11,8.53) (0.03,4.13)

Slices of B and in z at t=6

Low density & low B in filament

Magnetic field Density

B (0.,8.59) (0.,4.51)

Slices of B and in z at t=8

Magnetic field Density

MHD response to beam – mean |B| along line of sight

dyB ||

z

xt=2

t=6

t=4

t=8

Current, j

Filamentation & self-focussing

proton beam jvelocity vbeam

E=-uxB

B

R

Energy conservation

Magnetic field growth

jEt

U turb .

t

U

jRR

E

t

B turb

1

~~

Ideal for focussing CR into beam

(focus CR, evacuates plasma)

E=0

E=0

Power carried by filament/beam

Alfven current: Beam radius = Larmor radius

00

22

c

BrI eVg

Alfven

Power in individual filament/beam

eVAlfvenAlfven IP W

=1015eV AlfvenP 1.7x1028 W = 3x10-12 Moc2yr-1

=1021eV AlfvenP 1.7x1040 W = 3 Moc2yr-1

Some questions:

future directions

Acceleration requires large BvR

magnetic fieldvelocity

size

B increases with energy density v2

Puts emphasis on v and

For >1015eV, look at high density, high velocity objects:young SNR expanding into dense mediumsupernovaeAGN

A revised perspective?

Could jets be driven by high energy CR?

Limits on shock acceleration at high density

p-p loss time: pp ~ 3x10-9 gm/cc-1 sec

Max CR energy: ~ 25 gm/cc-1 BMG (vshock/c)2 GeV

p-p Loss length: pp ~ 0.8 gm/cc-1 m

(Aharonian, 2004)

p-p loss limit

Can CR escape dense plasma?

Other (larger?) losses

a natural explanation for CR

Recent theory:

1) removes doubts about acceleration to the knee

2) acceleration beyond knee a possibility

3) directs attention to young SNR

4) filament/beaming intriguing

5) application to accretion systems/compact objects

Shock acceleration

Lucek & Bell, MNRAS 314, 65 (2000)Bell, MNRAS 353, 550 (2004)Bell, MNRAS in press (2005)

Cassiopeia A (Chandra)

jcr

jthermal

jthermal = -jcr

jthermal x B

causes helix expandextends field lines

increases B

Instability mechanism

helical field line