Magnetic field and accelerated shock acceleration

Magnetic fieldand

accelerated shock acceleration

Tony Bell

Imperial College, London

Lucek & Bell, MNRAS 314, 65 (2000)Bell & Lucek, MNRAS 321, 433 (2001)Bell, MNRAS 353, 550 (2004)Bell, MNRAS 358,181 (2005)

Reynolds, 1986

SNR suitable CR source below 1015eV

Radio image of SN1006 x-ray image of SN1006

Long, 2003

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

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

Scaleheight must be less than SNR radius

LR

shock

CR pre-cursor

Need L<R

Bohm diffusion: cr = rg L= rg c /3vshock

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

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

k in units of rg-1

in units of vS2/crg

For SNR conditions, instability strongly driven- changes nature of turbulence

-4

-2

0

2

4

-2 0 2 4log10(k)

log

10(o

meg

a) Re()

Im()

krg=1

CR interaction with short wavelength waves

CR trajectory

B

CR trajectories unaffected by B

Wave growth driven by jcr||xB

||crj

Electric currents carried by CR and thermal plasma

Density of 1015eV CR: 10-3 m-3

Current density: jcr ~ 10-17 Amp m-2

LR

shock

CR pre-cursorjcr

CR current must be balanced by current carried by thermal plasma

jthermal = - jcr

jthermalxB force acts on plasma to balance jcrxB force on CR

CRcurrent

Current carried by thermal plasma

Magnetic fieldfrozen into

thermal plasma

j x B

j x Bj

j

j x B force expands the spiralLengthens field linesIncreases magnetic fieldIncreases j x B force POSITIVE FEEDBACK (INSTABILITY)

Unstable growth of magnetic field

Time sequence: four adjacent field lines

a)

d)

b)

No reason for non-linear saturation of a single mode

c)

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

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

Shock velocity drops in Sedov phase – reduces max. CR energy

MHD simulations demonstrate

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

3

0

2

~ scrssat vU

c

vB

Estimate of saturationmagnetic field

-4

-2

0

2

4

-2 0 2 4log10(k)

log1

0(gr

owth

rat

e)

Linear growth

kmin kmax

kmin= (CR Larmor radius) -1 ~ B

kmax B = 0 jCR

B increases during non-linear growth

“kmin” increases, “kmax” decreases

Growth saturates when kmax = kmin = 1/CR Larmor radius

3

0

2

vc

v~ scr

ssat UB

Cassiopeia A (Chandra)

Indicates magnetic field amplification at shock

(Vink & Laming, 2003; Völk, Berezhko, Ksenofontov, 2005)

Structure of turbulence

Magnetic field Density

Cavities in density and magnetic field

Slices perpendicular to CR flux at t=6

Field lines – wandering spirals

Cavities and Filaments

Spiral field lines configured as a single mode

Alternative configuration

j x B j x B

Spiral expands leaving a central cavity

Expanding filament

Magnetic field(theta component)

Density

Cavity in density and magnetic field

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

Filamentation & self-focussing

proton beam jvelocity vbeam

E=-uxB

B

R

Magnetic field growtht

U

jRR

E

t

B turb

1

~~

Ideal for focussing CR into beam

Focuses CR, evacuates cavity

E=0

E=0

CR exhausts and jets

1) SN in circumstellar wind, aligned rotator

2) CR source at centre of accretion disk

Supernova inWind from star with dipole aligned with rotation axis

CR flux drives cavity along axisLow energy CR escape through cavity

Number of e-foldings ~

c CR pressure CR Larmor radiusvs vs

2 cavity radius

1/2

vs = SNR shock velocity

Accretion disk jets

Central source of CR

Disk wind carries magnetic field

CR flux produces cavity

Exhaust of low energy CR & thermal plasma

Rotating disk threaded by magnetic field

Consequences:• Magnetic field spirals clockwise• Jets on 2 sides or none

Power carried by filament/beam

Natural evolution:1) Beam radius = Larmor radius2) Beam carries Alfven current

00

22

c

BrI eVg

Alfven

Power in individual filament/beam

eVAlfvenAlfven IP W

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

=1020eV AlfvenP 1.7x1038 W = 0.03 Moc2yr-1

Black holes: characteristic parameters (Begelman, Blandford & Rees, 1984; based on Eddington luminosity LE)

T

pE

cGMmL

4

2c

GMR

GM

cn

Te

2

2

0

2

2cmn

Bpe

CR energy for which:1) Larmor radius rg = R2) Alfven current carries LE

eV103o

16

2/1

M

M

R

rg

Mass depth independent of black hole mass M2mkg25 RmnR pe

(R for p-p energy loss = 800 kg m-2)

Hillas, 2005

Conclusions

Lucek & Bell, MNRAS 314, 65 (2000)Bell & Lucek, MNRAS 321, 433 (2001)Bell, MNRAS 353, 550 (2004)Bell, MNRAS 358,181 (2005)

• Magnetic field amplification increases max CR energy

• Historical SNR produce CR up to knee

• Very young SNR may get beyond knee

• Exhaust model may connect high energy CR to jets