Courtesy of John Kirk

Particle Acceleration

Basic particle motion

gqt

m

tFBuE

up )(

d

d

d

d

2

//// ˆ

qB

eu

drift

driftgyro

BFu

uuu

gyroeuBq

muR uuE

//// ˆ , :0

2

:0

B

q

drift

BEu

EFE

3

2

2

2

2 :Gradient

qB

mu

B

mu

drift

BBu

BF

22

2//

2

2//

)(

:Curvature

qBR

mu

R

mu

c

cdrift

c

c

c

BRu

BBR

RF

No current

Dreicer DC electric fields (focusing on electrons)

Electric force vs. drag force

Reaching maximum at the thermal speed

ln

4

2

0 te

peD v

eE

[Dreicer, 1959, 1960]

Coulomb logarithm

E > ED: super-Dreicer

E < ED: sub-Dreicer qE vs. tvmf edrag )(

< qE above v=vc, electrons will run-away

ED for typical flares is ~ 10-4 V cm-1.

Holman [1985] work:

E ~ 10-7 V cm-1, spatial scales of L ~ 30 Mm (the size of a typical flare loop), yielding electron energies of W ~ 100 keV for an temperature of T ~ 107 K, a collision frequency of 2x103 s-1, a length scale of 10 Mm.

In principle, the sub-Dreicer DC electron field mode can explain the thermal-plus-nonthermal distributions as observed in hard X-ray spectra.

However, there are a number of open issues:

1) Require a large extent along the current sheet that is unstable.

2) Contradicts to the observed time-of-flight delays [Aschwanden 1996]

3) Electron beam current require counter-streaming return currents that can limit the acceleration efficiency severely. [Brown & Melrose 1977; Brown & Bingham 1984; LaRosa & Emslie 1989; Litvinenko & Somov 1991]

Litvinenko [1996] work:

B ~ 100 G, E ~ 10 V cm-1, d ~ 100 m the width of the current sheet, yielding electron energies of W ~ 100 keV, an acceleration length of 100 m.

Stochastic Acceleration

Is broadly defined as any process in which a particle can either gain or lose energy in a short interval of time, but where the particles systematically gain energy over longer times.

wave-particle interaction

It’s more important for particle acceleration in flares.

Gain energy: , escape rate: b, and the escape probability of a particle with moment > p: P

abppPap

pbP

dp

pdP

tbpP

pPtbppP

tapp

)()()(

d)(d

)(d1)d(

dd

How?

)()())(,()(

)()(

kkpkr

kkv

kNf

N

t

Ncollg

)()( kk WN kkv )(g Growth and damping rate

losssourceiij

j t

f

t

f

p

fND

p

f

t

f

)()()(

))(()(

)()( ppp

kr

ppv

p

Neglect the evolution of wave spectrum

In an isolated homogeneous volume

0

)(1

)(

))()(()()2(

d)(

////

////

3

3

vks

vks

,A

kkN,D

s

jiij

kpkp,

kkpk

p

Doppler resonance condition

Melrose, Plasma Astrophysics I & II, Gordon & Breach Publishers, 1980; Benz, Plasma Astrophysics (2nd edition), 2003.

Second order of 1/vi

For typical coronal conditions: peepiicoll

Consider an interaction of ions with very low-frequency waves, for example, Alfven waves

1

0 &

////

////

AA v

vv

v

vks

222AvkThe dispersion relation is

To be accelerated, an ion needs to have a threshold energy. For typical coronal Alfven speeds, 2000 km s-1, the threshold should be > ½*mpvA

2~20 keV.

A problem is how to accelerate ions from their thermal energy (~1 keV) to the threshold energy.

Resonance with a single small-amplitude wave: the gain energy oscillate with frequency of ω, the maximum energy gain is small and zero on average.

E

t

ω1

ω2

ω3

ω4

A broadband spectrum of waves is thus typically required to accelerate particles to high energies.

explain the enhanced ion abundances with the stochastic acceleration.

In the scenario of turbulent MHD cascades: long-wavelength Alfven waves cascade to shorter wavelengths, gyroresonant interactions are first enabled for the lowest Ω, such as iron, and proceed then to higher Ω.

Shock drift acceleration

1) A drift at shock front like drift

2) A convective electric field in the (opposite) direction

So particles gain energy when crossing shocks.

B

Bu

Diffusive shock acceleration [Jones & Ellison, 1991]

pv

uup du

3

2

N

i i

du

v

uupNp

10 3

21)(

One crossing

N crossesings

20

)(2

1dd)( vuvuvuvvu d

uv

xdxd

u

v

xxd

d

d

From the downstream to the upstream

From the upstream to the downstream

Probability of return (two crossings)2

/1

/1)return(

vu

vuΡ

d

d

22/

1 /1

/1)(

N

i id

id

vu

vuNΡ

upstreamdownstream

2

0

)(2

1dd)( vuvuvuvvu d

uv

xdxd

v

u

xxd

d

d

ud

In downstream frame

2/

1

532/

1 5

1

3

14

/1

/1ln2)(ln

N

i i

d

i

d

i

dN

i id

id

v

u

v

u

v

u

vu

vuNP

Assuming u << vi, only the first order of 1/vi is kept.

2/

1

14)(ln

N

i id vuNP

2/

110

1

3

4

3

21ln

)(ln

N

i idu

N

i i

du

vuu

v

uu

p

Np

)(3

0

0

)(

ln3

)(ln

dud uuu

du

d

p

ppP

p

p

uu

upP

so

2

00

0

2

00

00

)(

21

2 , 4

1

3

3)()(

f

r

r

u

ur

p

p

r

r

p

n

p

p

uu

u

p

n

p

pP

u

unpf

d

u

uuuu

du

u

d

u

dudu

spectral index > 1

Problems and limitations

v >> uu, ud => the second order and more of u/v could be neglected.

Velocity distribution should be isotropic in all relevant frames.

Shock thickness should be much smaller than mean free path of particles.

For a particle energy E = 1 MeV electron (rg ~ 108 cm , v ~ 1010 cm/s) tacc ~ 102 s proton (rg ~ 1011 cm , v ~ 109 cm/s) tacc ~ 104 s ~ 0.1 day

E = 1 GeV rg ~ 1012 cm , v ~ 1010 cm/s tacc ~ 106 s ~ 0.1 AU ~ 1 month

E = 1 PeV (= 1015 eV) rg ~ 1018 cm , v ~ 1010 cm/s tacc ~ 1012 s ~ 1 pc ~ 105 yr

E= 1 EeV (=1018 eV) rg ~ 1021 cm , v ~ 1010 cm/s tacc ~ 1015 s ~ 1 kpc ~ 108 yr

u

v

u

Er

up

tpt gacc

)(~

2

05.0005.0~//

//

With a given time, Eperp > Epar

Acceleration time scale

[Jokipii et al.1995; Giacalone and Jokipii 1999; Zank et al. 2004; Bieber et al. 2004]

[courtesy of Ho et al., 2004]

[Reams, 1999]

ESP (Energetic Storm Particle) events

[courtesy of Ho et al., 2004]