Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration -...
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Particle Acceleration - Alternatives to Diffusive Shock Acceleration
Brian RevilleQueen’s University Belfast Centre for Plasma Physics
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Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
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The high-energy Universe is non thermal
Beatty & Westerhoff ‘09
Cosmic-Rays
Knee
Ankle
Not Maxwellian!
GRB 080916C
Fermi Collaboration ‘09
Yuan et al. ‘11
Crab Nebula spectrum
How, where and to what degree, different systems distribute their energy budget is a fascinating area of physics -
Are shocks the only game in town?
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Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
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Transport in Magnetised Plasma
dp
dt= q (E + v ⇥B)
E0 = �u⇥B
Particle motion determined by Lorentz force:
Astrophysical plasmas are to a reasonable approximation ideal:
Electric field vanishes in local fluid frame (u=0)
Hence, to zeroth order, particles simply gyrate about mean magnetic field (Note, energy is a constant of motion if E=0)
If we now introduce some “scattering” (fluctuating field components), particle trajectory undergoes random small angle deflections.
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Diffusion in Magnetised Plasma
Collisions are produced by small-angle deflections on fluctuating electric and magnetic fields. For now quantify these through the so-called Hall parameter , where is the gyro-frequency, and the scattering time.h = !g⌧B !g ⌧B
The resulting transport is thus diffusive with diffusion coefficients along the field: and across field Dk =
1
3hrgv D? =
h
1 + h2rgv
We will see where these come from after introducing Fokker Planck theory
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Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
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Fermi Acceleration (1949)High velocity cloud in ISM ⟨Ucloud⟩ ≈ 30 km/s
Ucloud
Magnetic field lines
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Fermi Acceleration (1949)
Energy approximately conserved in frame of cloud, but can be scattered/mirrored. On exiting, transform back to ambient frame p’’ = p’+m Ucloud = p + 2 m Ucloud , Net change:
Consider particle with initial momentum p (>> mUcloud). Transform to frame of moving cloud: p’ = p+m Ucloud
Ucloud
Magnetic field lines
High velocity cloud in ISM ⟨Ucloud⟩ ≈ 30 km/s
Δp = − 2p ⋅ Ucloud
v
Particles can lose or gain energy depending on sign of with p ⋅ Ucloud |Δp |+ = |Δp |−
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Fermi Acceleration (1949)
Ucloud
Magnetic field lines
High velocity cloud in ISM ⟨Ucloud⟩ ≈ 30 km/s
Let be the average distance between clouds. The mean rate of collisions must depend on the relative velocity
ℓ
ν± =v ± ⟨Ucloud⟩
ℓ
⟨ dpdt ⟩ = ν+ |Δp |+ − ν− |Δp |− = 4
⟨Ucloud⟩2
v2
vℓ
p (Note the scaling with Ucloud and )ℓ
⟨ dpdt ⟩ = αpIn ultrarelativistic limit v~c,
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Fokker-Planck Collisions
Following Chandrasekhar (1943) we consider a probability that in a (brief) time a particle will change its momentum from
thus
W(p, Δp)Δt
p → p + Δp
f(p, t) = ∫ f(p − Δp, t − Δt) W(p − Δp, Δp) d3(Δp)
|Δp | ≪ | p |Assume and Taylor expand (about (p,t) )
f(p, t) = ∫ {f(p, t)W(p, Δp) − ΔtW(p, Δp)∂f∂t
c
− Δp∂
∂p[ f(p, t)W(p, Δp)]
+12
ΔpΔp :∂
∂p∂
∂p[ f(p, t)W(p, Δp)] + …}d3(Δp)
Since is a probability, it must satisfy W(p, Δp) ∫ W(p, Δp) d3(Δp) = 1
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(Aside) — W(p, Δp) for Fermi II
Consider the non-relativistic case (the extension to relativistic particles is trivial)
As we already say, a particle with initial momentum p, after a head-on collision has momentum p+2m Ucloud , and the probability of such a collision is proportional to the relative velocity ~ v + Ucloud.
Now consider a particle with initial momentum p+2m Ucloud undergoing an overtaking collision. The momentum after collision is now simply p, and the probability is (v+2 Ucloud) - Ucloud = v + Ucloud.
Hence, the two processes are exactly symmetric. Or more specifically, we can conclude that
W(p − Δp, Δp) = W(p, − Δp)W(p, Δp) = W(p + Δp, − Δp) or
This is called detailed balance, and simplifies the analysis considerably. We will use this identity in the next slide.
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Fokker-Planck CollisionsA slight rearrangement and we have the Fokker-Planck collision operator
∂f∂t
c
= −∂
∂pif(p, t)⟨
Δpi
Δt ⟩ +12
∂∂pi
∂∂pj
f(p, t)⟨ΔpiΔpj
Δt ⟩
We can go one step further. If the process is symmetric a typical situation in test-particle limit (Recall our head on -vs- overtaking collisions) Taylor expanding yet again, we immediately see to the same order as before
W(p − Δp, Δp) = W(p, − Δp)
⟨Δpi
Δt ⟩ =1Δt ∫ ΔpW( p, Δp) d3(Δp) and ⟨
ΔpiΔpj
Δt ⟩ =1Δt ∫ ΔpiΔpjW( p, Δp) d3(Δp)
W(p, − Δp) = W(p − Δp, Δp) = W(p, Δp) − Δpi∂W∂pi
+12
ΔpiΔpj∂2W
∂pi∂pj+ …
1 = 1 − Δt∂
∂pi ⟨Δpi
Δt ⟩ −12
∂∂pj ⟨
ΔpiΔpj
Δt ⟩=0
on integration gives
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Fokker-Planck CollisionsSubbing in
∂f∂t
c
= −12
∂∂pi
f( p, t)∂
∂pj ⟨ΔpiΔpj
Δt ⟩ +12
∂∂pi
∂∂pj
f( p, t)⟨ΔpiΔpj
Δt ⟩ =12
∂∂pi ⟨
ΔpiΔpj
Δt ⟩ ∂f∂pj
Dij =12 ⟨
ΔpiΔpj
Δt ⟩ =1
2Δt ∫ ΔpiΔpjW(p, Δp) d3(Δp)
Using the fact that both f(p) and df/dp —> 0 as p —> ∞, multiply by energy and integrate by parts (twice)
∂f∂t
=∂
∂pi (Dij∂f∂pj )
⟨ ∂ε∂t ⟩ = ⟨ ∂
∂pj ∫∂ε∂pi
ΔpiΔpjW(p, Δp) d3(Δp)⟩ ≡ ⟨ ·ε⟩⟨…⟩ = ∫ (…) fd3p
ε
where
we find
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Stochastic AccelerationReturning to Fermi acceleration, if particle distribution remains near isotropic, using spherical momentum coordinates,
∂f∂t
=∂
∂pi (Dij∂f∂pj ) ≈
1p2
∂∂p (p2Dpp
∂f∂p )
Dpp =12 ⟨ ΔpΔp
Δt ⟩ ≈12 ⟨( p ⋅ U
v )2
⟩ vd
=13
p2U2
vℓ
·ε =∂
∂pj ∫∂ε∂pi
ΔpiΔpjW(p, Δp) d3(Δp) =∂
∂pj
∂ε∂p
pi
pDij ≈
1p2
∂∂p [vp2Dpp]
If f is a function of |p| only (i.e. isotropic)
dεdt
=43
U2
ℓp = αε α =
43
⟨Ucloud⟩2
v2
vℓ
In ultra-relativistic limit
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Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
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Acceleration in Magnetised Turbulence
θB0
B0
ω/k = vA cos θ
Fast
Slow
ω/k
ω/k
ω/k = c2s + v2
A
Zhou, Matthaeus & Dmitruk ’04
Situation likely more complex in fully developed turbulence, but basic picture (Kulsrud & Ferrari 71, Achterberg 81)
Resonance condition:
Particle scattering dominated by interaction with Alfven waves (s=1) Acceleration due to fast magneto sonic modes (s=0)
Alfven
ω − k∥v∥ − sΩ = 0
Dpp =8π
p2 ∫ dk∫ dωω
k∥v (1 −ω2
k2∥v2 )
2
I(k, ω) ≈cv ( δB
B0 )2
ωp2
MHD phase diagrams for warm plasma
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Fermi II - Maximum Energy
Fermi II has a simple thermodynamic interpretation. Consider the original Fermi 1949 picture, with random moving clouds.
Each cloud has an assumed volume >> rg3 ,
and mean velocity ⟨Ucloud⟩ ≈ 30 km/s
Fast particles will (until some loss process / size limitation) attempt to come into thermal equilibrium with these clouds
kT ≈12
MU2cloud ∼ ρISMVolcloudU2
cloud = ∞
One can more generally rely on Hillas limit to apply on scale of accelerating system
εHillas = Z 1014 ( u30 km/s ) ( B
μG ) Lkpc eV
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Particle Spectrum
Supplementing our previous equation with an escape condition,
∂f∂t
=1p2
∂∂p (p2Dpp
∂f∂p ) −
fτesc
Taking and we readily find for steady stateDpp ∝ pq f ∝ p−s
s =q + 1
2+ ( q + 1
2 )2
+τacc
τescτacc ≡
p2
Dppwhere
Power laws only recovered in peculiar limit of τacc(p)/τesc(p) = const
More generally, we can solve the full transport equation (e.g. Schlickeiser 88)
∂f∂t
=1p2
∂∂p (p2 (Dpp
∂f∂p )) −
fτesc
+Source4πp2
Can be solved for arbitrary initial conditions and power-law dependence on all variables (eg. Kardashev 62, Mertsch ’11)
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Particle Spectrum
10-1 1 10 102 103 104 105
10-8
10-6
10-4
10-2
1
x
nHx,x 0,t,t 0L
t = 0.01t = 0.03t = 0.1t = 0.3t = 1t = 3
rates const.q = 3ê2
10-1 1 10 102 103 104 105
10-8
10-6
10-4
10-2
1
x
nHx,x 0,t,t 0L
t = 0.01t = 0.03t = 0.1t = 0.3t = 1t = 3steady state
rates const.q = 3ê2
10-1 1 10 102 103 104 105
10-8
10-6
10-4
10-2
1
x
nHx,x 0,t,t 0L
t = 0.01t = 0.03t = 0.1t = 0.3t = 1t = 3
rates const.q = 5ê3
10-1 1 10 102 103 104 105
10-8
10-6
10-4
10-2
1
x
nHx,x 0,t,t 0L
t = 0.01t = 0.03t = 0.1t = 0.3t = 1t = 3steady state
rates const.q = 5ê3
10-1 1 10 102 103 104 105
10-8
10-6
10-4
10-2
1
x
nHx,x 0,t,t 0L
t = 0.01t = 0.03t = 0.1t = 0.3t = 1t = 3
rates const.q = 2
10-1 1 10 102 103 104 105
10-8
10-6
10-4
10-2
1
x
nHx,x 0,t,t 0L
t = 0.01t = 0.03t = 0.1t = 0.3t = 1t = 3steady state
rates const.q = 2
Impulsive Injection Continuous InjectionFigures from Mertsch JCAP 2011
Dpp ∝ pq
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Observations of “curved” spectraF(E) = K2E−a−b log EF(E) = K1E−s
Significantly better fit of XMM data Mrk 421 using log-parabolic fit (Tramacere et al 07)
Southern lobes of Cen A, Stawarz et al ‘13
Cowsik & Sarkar 84
Non thermal x-ray hotspots in outer lobes
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Semi-relativistic limit
Does the QLT theory hold at mildly relativistic phase velocities? vA~ 0.2 c
O’Sullivan, BR, Taylor 09
δB/B = 0.1 δB/B = 1
Integrate test particles in synthetic spectrum of Alfven waves.
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Fermi-I -vs- Fermi-IIFermi II has been largely superseded (for good reasons) by DSA. However …..
tacc,DSA =3
u1 − u2 ( κ1
u1+
κ2
u2 ) ≈ 3κ∥
u2sh
=v2
u2sh
λv
tacc,FII ≈v2
v2A
λv
κ =13
λv
Accelerating shocks are super-Alfvenic (in ISM vA ~ 10 km/s) so this effect is likely a small correction in most cases.
If shocks are highly non-linear : magnetic fields are amplified & incoming flow decelerated. As vA —> c the two processes can operate on similar timescale
tacc,FII
tacc,DSA≈ M2
A
Acceleration time (e.g. Drury 83)
But recall our result for Fermi II
i.e.
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Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
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Particle TransportEarlier we introduced the concept of parallel and cross-field diffusion. Now that we are more familiar with Fokker-Planck methods, let’s revisit.
∂f∂t
+ v ⋅∂f∂x
+ q (E +1c
v × B) ⋅∂f∂p
=∂
∂p⋅ (D
∂f∂p )
E = −1c
u × B
p′� = p − γmu
∂f∂t
+ (v′� + u) ⋅∂f∂x
+qc
(v′� × B) ⋅∂f∂p′�
= [( p′� ⋅ ∇)u] ⋅∂f∂p′�
+∂
∂p′�⋅ (D
∂f∂p′�)
f(x, p, t) = f0(x, p, t) +pp
⋅ f1 + . . .
We have derived the Vlasov-Fokker-Planck equation:
On sufficiently large length scales (MHD) , so working in the local fluid frame will prove advantageous. Consider the transformation
Next, we consider a simple expansion in angular basis functions
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Particle Transport
∂∂p
⋅ (D∂f∂p ) ≈
ν2 { ∂
∂μ [(1 − μ2)∂f∂μ ] +
11 − μ2
∂2f∂ϕ2 }
∂f0∂t
+ ∇ ⋅ (uf0) +v3
∇ ⋅ f1 =∂
∂p3 [p3(∇ ⋅ u)f0]∂f1
∂t+ (u ⋅ ∇)f1 + Ω × f1 + v∇f0 + νf1
dominant terms
=∂ub
∂xaf b1 +
∇ ⋅ u3
p2 ∂∂p ( f1
a
p ) +15
p2 ∂∂p ( f1
b
p ) σab
To lowest order, we can assume most energy fluctuations are contained in the change of reference frame, and consider only scattering in angle
Simply taking moments ( )of the mixed frame VFP equation, we find:∫ dΩ
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Particle Transport
Ω × f1 + v∇f0 + νf1 = 0 ⟹ f1 =v/ν
1 + h2 [h × ∇ − ∇ − h(h ⋅ ∇)] f0
∂f0∂t
+ ∇ ⋅ (uf0) =∂
∂p3 [p3(∇ ⋅ u)f0] + ∇ ⋅ [ DB
1 + h2 [∇ + h(h ⋅ ∇) − h × ∇] f0]
h = Ω/ν
To derive the final transport equation, we consider the dominant terms:
∂f0∂t
+ ∇ ⋅ (uf0) +v3
∇ ⋅ f1 =∂
∂p3 [p3(∇ ⋅ u)f0]∂f1
∂t+ (u ⋅ ∇)f1 + Ω × f1 + v∇f0 + νf1
dominant terms
=∂ub
∂xaf b1 +
∇ ⋅ u3
p2 ∂∂p ( f1
a
p ) +15
p2 ∂∂p ( f1
b
p ) σab
recall the Hall parameter
Substitute back into the f0 equation, we recover the “usual” transport equation
Exercise: Consider 2 extreme cases of gradient in f parallel and perpendicular to h
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Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
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Fermi Acceleration in jets
e.g. NGC315 Worrall et al 2007
~ 15-2
0 kpc
Min-Energy fields in bright regions ~
rg =E18
BµGkpc
40 µG
Jet velocity on axis ~0.9 c (Canvin et al ’05)
Could jets be sources of UHECRs? And if so, how are they accelerated?
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Shear Acceleration
Γj > 1, βj > 0
ΓA = 1, β = 0θ1
θ2
Consider discontinuous shear profile, βx = βjet y>0, ux=0 y<0
As in DSA, we consider ideal MHD scenario, energy conserved in each half-plane (electric field vanishes in local rest frame)
Hence, as per usual, there is a kinematic energy gain on each crossingu0′� = Γj(u0 − βju1) = Γjγ0(1 − βj cos θ1)
u0′�′� = Γj(u0′� + βju1′�) = Γ2j γ0(1 + βj cos θ′�2)(1 − βj cos θ1)
Ambient —> JetJet —> Ambient
Net changeΔγγ
= βjcos θ2 − cos θ1
1 − βj cos θ2
Understanding transport is crucial!!
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Acceleration in gradual shear flowsParticle scatters across a sheared flow, assume energy is conserved in local frame (classical Fermi acceleration scenario)
Gains energy if collision is head on, loses energy otherwise
Originally considered by Berezhko & Krymskii (1981) using transport equation, and kinetically by Rieger & Duffy (2006). Both finding
dp
dt=
4 + ↵
15
✓du
dy
◆2
⌧p ⌧ / p↵where scatter time
Acceleration time inversely proportional to MFP (iff !!)!g⌧ << 1
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Acceleration in gradual shear flows
dp
dt=
4 + ↵
15
✓du
dy
◆2
⌧p
!g⌧ << 1
!gtacc ⇠15
4 + ↵
1
�2
L2/r2g!g⌧
or
and since acceleration substantially sub-Bohm
Disfavours shear models of UHECR acceleration, which generally require c.f. Lemoine’s talk
ωgtacc ≳ 1
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Acceleration in gradual shear flows
Rieger & Duffy 06
p�5 p�3.5
Accelerated particle distribution for different scattering rates with synchrotron cooling
⌧sc / p↵
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Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
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Magnetic Reconnection
Originally suggested by Giovanelli (1947) and Dungey (1953).
Developed by Sweet, Parker, Furth & Petschek in 50s and 60s.
Plays major role in solar physics • magnetic flares • solar storms • coronal / solar wind heating? Important in pulsars, magnetars & possibly GRBs/AGN jets
Recent studies of High-Mach number shocks also show reconnecting fields
www.wikipedia.com
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Magnetic ReconnectionSchindler & Hornig 01 Sweet - Parker picture
v1 ⇠ E ⇥B1
B21
⇠ Ez
B1
• Particles drift into diffusion region
whereEz ⇡ ⌘jz ⇠ ⌘B1/�
• Mass conservation impliesLv1 = �v2
and energy conservation Two observations: 1. flow exits diffusion region at ~ Alfven velocity Hence:
2. This number is small
v1v2
⇠ v1vA
⇠ �
Lv1vA
⇠ ⌘
�vA⇠
r⌘
LvA= Rm�1/2
B21/8⇡ = ⇢v22/2
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Magnetic ReconnectionSchindler & Hornig 01 Petschek picture
• Same rules about particle conservation,
• and outflow is still Alfvenic However, Petschek suggests inflow can be much faster , if diffusion region small.
Lv1 = �v2
Requires standing slow mode shocks to allow plasma to cross into outflow region. Opening angle of outflow region must increase to accommodate for greater inflow rate. This sets a maximum reconnection rate
! vA
v1vA
⇡ 1
ln(Rm)Note, while this rate is more consistent with observations, slow mode shocks have never been observed in self-consistent simulations
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MHD vs Hall MHD simulations
P. Cassak Thesis
Hall MHD
Resistive MHD
So-called Hall physics is essential to achieve “fast” reconnection in electron-ion plasmas
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Collisionless reconnection
nemedue
dt= − ∇Pe − ∇ ⋅ π − nee(E +
1c
ue × B) + νeineme(ui − ue)
E = −1c
ue × B −∇Pe
nee−
∇ ⋅ πnee
− medue
dt+
νeime
nee2nee(ui − ue)
j = Zeniui − eneue ≈ ene(ui − ue)
Let’s derive a generalised Ohm’s law. Using electron momentum equation:
Rearrange
And using definition of electric current
E +1c
ui × B = η j +1
neecj × B −
∇Pe
nee−
∇ ⋅ πnee
− medue
dttextbook Ohm’s law Hall term pressure
gradients inertiapressure anisotropy
Note, in absence of collisions, ( η = 0 ) how do magnetic fields diffuse?
Magnetic reconnection forbidden in ideal MHD (magnetic flux conserved exactly).
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Hall reconnection
Huang et al, ‘11
Hall term by itself can not cause reconnection (magnetic field simply frozen into electrons), but appears to be a key aspect of opening the outflow region, increasing the reconnection rate. When is it important?
Compare:ui × B1
neej × B
∼vABc
4πneeB2
δ
=ωpiδ
c
i.e. Hall term dominates on scales < ion inertial length
j =c
4π∇ × B ∼
c4π
Bδ
where we have used
Hall overtakes from SP if δ > δSPc
ωpi> LR−1/2
m ⇒ λMFP >me
miβ1/2L
β =4πkTe
B2
Exercise:consider case of strong guide field (out of plane)
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Particle AccelerationImportant Questions:
1. How are particles accelerated? 2. Acceleration versus heating 3. Ion versus electron acceleration 4. Role of guide field 5. Non-relativistic versus relativistic 6. electron-ion versus electron-positron 7. driven versus non-driven reconnection 8. Importance of anomalous resistivity (micro-turbulence) 9. Turbulent reconnection
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How to Investigate Acceleration in reconnecting plasmas
1. Test particle:• Easy to implement, large particle statistics can be used• Can be used to test analytic models• Not self consistent (no feedback on Ohm’s law)
2. PIC• Self-consistent• Expensive• Not always easy to interpret
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Speiser Orbits
Sonnerup 71
Speiser 65
Using simple model of current sheet, the peculiar orbits were first described by Speiser and colleagues. Particle drifts play a vital role. More realistic configurations, the particle trajectories in diffusion layer very quickly become chaotic (e.g. Buechner & Zelenyi 89)
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Test particle simulations
No Guidefield
GuideField
Mignone et al. 18B = B0 ( yL
,xL
,Bz
B0 ) E = (0,0,Ez)
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Relativistic Reconnection & Plasmoid instability - PIC simulations
Nalewajko et al 15Sironi & Spitkovsky 14
Magnetic “islands” form and merge. Recall ∂B∂t
= − c∇ × E
topological field change -> E-fields
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Dominant Acceleration events
Nalewajko et al 15
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Power-laws?
Sironi & Spitkovsky 14
Werner et al 16
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electron vs ion acceleration
Dahlin, Drake & Swisdak 18
Recent work by Dahlin et al 18, present in depth study of • 2D vs 3D effect • guide field • mass ratio
As one moves towards more realistic PIC simulations, electron heating appears favoured over ion heating.
Note: no power-law (no escape??)
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Diffusive Reconnection Acceleration?
Drury ‘12
Consider a black box about the reconnection region, high energy particles move (scatter) back and forth across the reconnection region.
Flux in momentum space across shock
Φ = ∫R
4πp3
3f(p)( − ∇ ⋅ U) =
4πp3
3f(p)∫∂S
U =4πp3
3f(p)(2U1A1 − 2U2A2)
Steady state —> ∂Φ∂p
= − 2A2U24πp2 f ⟹∂ ln f∂ ln p
=−3rr − 1
≈ − 3
A key assumption in this is maintaining isotropy. i.e. Scattering in sub-Alfvenic flows!! So is Fermi II at play here too?
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Turbulence -> Reconnection
Wu et al. 14, PRL
Turbulence in Collisionless plasmas dissipates in current sheets.
Can heat both ions and electrons, energy budget depends on the “level” of turbulence.
https://www.youtube.com/watch?v=DAtLhKrF37o
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Summary• Astrophysical fluids are almost universally turbulent • On escaping the thermal pool, scattering in momentum (both in
direction and magnitude) is inevitable • Second order Fermi can play an important role in many environments • Kinetic behaviour of energetic particles in reconnecting plasmas is far
from understood, with many open questions • Acceleration at current layers may be amenable to similar methods as
shocks. Shock spectra might be influenced by turbulence. Turbulence
can drive reconnection and affect the acceleration rate. • The contribution of any of these processes can produce Galactic
cosmic rays or indeed UHECRs in the case of ExGal CRs remains controversial.
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A few other select acceleration mechanisms
Shock Drift & Shock surfing
Converter mechanisms: (Derishev, Stern & Poutanen) Particle transport involves conversion of hadron/lepton into a neutral particle (neutron/photon) that then converts (via some mechanism) back to charged product, carrying a large energy fraction of parent
Linear acceleration: Unscreened fields in vicinity of black-hole/pulsar. Direct linear acceleration.
Centrifugal Acceleration (Rieger & Mannheim)
It was not possible to cover all possible acceleration mechanisms. Below is a list of some additional methods that may apply in different conditions: