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Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast Centre for Plasma Physics DIAS Summer School 2018 [email protected]
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Page 1: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

Particle Acceleration - Alternatives to Diffusive Shock Acceleration

Brian RevilleQueen’s University Belfast Centre for Plasma Physics

[email protected]

Page 2: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

Outline

• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration

• Magnetic Reconnection

• Particle transport in magnetised plasmas

Page 3: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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?

Page 4: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

Outline

• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration

• Magnetic Reconnection

• Particle transport in magnetised plasmas

Page 5: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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.

Page 6: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 7: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

Outline

• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration

• Magnetic Reconnection

• Particle transport in magnetised plasmas

Page 8: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

Fermi Acceleration (1949)High velocity cloud in ISM ⟨Ucloud⟩ ≈ 30 km/s

Ucloud

Magnetic field lines

Page 9: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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 |−

Page 10: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

ν± =v ± ⟨Ucloud⟩

⟨ dpdt ⟩ = ν+ |Δp |+ − ν− |Δp |− = 4

⟨Ucloud⟩2

v2

vℓ

p (Note the scaling with Ucloud and )ℓ

⟨ dpdt ⟩ = αpIn ultrarelativistic limit v~c,

Page 11: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 12: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

(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.

Page 13: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 14: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 15: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 16: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

Outline

• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration

• Magnetic Reconnection

• Particle transport in magnetised plasmas

Page 17: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 18: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 19: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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)

Page 20: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 21: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 22: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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.

Page 23: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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.

Page 24: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

Outline

• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration

• Magnetic Reconnection

• Particle transport in magnetised plasmas

Page 25: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 26: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 27: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 28: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

Outline

• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration

• Magnetic Reconnection

• Particle transport in magnetised plasmas

Page 29: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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?

Page 30: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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!!

Page 31: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 32: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 33: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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↵

Page 34: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

Outline

• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration

• Magnetic Reconnection

• Particle transport in magnetised plasmas

Page 35: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 36: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 37: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 38: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 39: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

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)

Page 41: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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

Page 46: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

Dominant Acceleration events

Nalewajko et al 15

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Power-laws?

Sironi & Spitkovsky 14

Werner et al 16

Page 48: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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??)

Page 49: Particle Acceleration - Alternatives to Diffusive Shock …€¦ · Particle Acceleration - Alternatives to Diffusive Shock Acceleration Brian Reville Queen’s University Belfast

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:


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