Particle Acceleration - Alternatives to Diffusive Shock Acceleration
Brian RevilleQueen’s University Belfast Centre for Plasma Physics
Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
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?
Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
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.
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
Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
Fermi Acceleration (1949)High velocity cloud in ISM ⟨Ucloud⟩ ≈ 30 km/s
Ucloud
Magnetic field lines
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 |−
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,
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
(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.
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
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
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
Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
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
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
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)
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
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
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.
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.
Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
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
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
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
Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
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?
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!!
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
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
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↵
Outline
• Second order Fermi • Fermi’s clouds • MHD turbulence • Shear Acceleration
• Magnetic Reconnection
• Particle transport in magnetised plasmas
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
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
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
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
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).
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)
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
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
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)
Test particle simulations
No Guidefield
GuideField
Mignone et al. 18B = B0 ( yL
,xL
,Bz
B0 ) E = (0,0,Ez)
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
Dominant Acceleration events
Nalewajko et al 15
Power-laws?
Sironi & Spitkovsky 14
Werner et al 16
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??)
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?
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
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.
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: