courtesy of john kirk particle acceleration. basic particle motion no current
TRANSCRIPT
Courtesy of John Kirk
Particle Acceleration
Basic particle motion
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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?
)()())(,()(
)()(
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kkv
kNf
N
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)()( kk WN kkv )(g Growth and damping rate
losssourceiij
j t
f
t
f
p
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p
f
t
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)()()(
))(()(
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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
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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
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i i
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)(3
0
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ln3
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dud uuu
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d
p
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p
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so
2
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21
2 , 4
1
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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]