physics of magnetically dominated plasma: dynamics, dissipation and particle acceleration maxim...
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![Page 1: Physics of magnetically dominated plasma: dynamics, dissipation and particle acceleration Maxim Lyutikov (UBC) Cracow, June 2006](https://reader030.vdocuments.us/reader030/viewer/2022032704/56649d3e5503460f94a16e06/html5/thumbnails/1.jpg)
Physics of magnetically dominated plasma:
dynamics, dissipation and particle acceleration
Maxim Lyutikov (UBC)Cracow, June 2006
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Dynamics and energy dissipation in
electromagnetically-dominated
plasmasMaxim Lyutikov
(McGill University, CITA)
Cracow, June 2003
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Relativistic outflows may be produced and collimated by
large scale B-fields AGN jets: ergosphere and Black hole itself can act as
a Faradey disk (Blandford-Znajek, Lovelace), creating B-field dominated jets
Numerical simulations begin to show this dynamically
(Koide et al. 2002),(McKinney, Gammie, Krolik, Proga)
Large scale, energetically dominant magnetic fields may be expected in the launching region of relativistic jets and may
(should?) continue into emission regions
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New plasma physics regime: magnetically dominated
plasma In plasma rest frame:
Magnetic energy U’B=B2/8π,
Plasma energy: rest-mass, ρc2
Alfven 4-velocity
Magnetization parameter(σ1/2= μ1/3)
Magnetically dominated: σ > 1
Dissipation & accelerationFluid shocks(e.g. Fermi) Magnetic dissipation
σ ~ 0
σ = ∞σ ~ 1
I. Hydrodynamics(all fluid)
II. MHDIII. Force-free
(magnetodynamics)
σ > Γ2/2
(Subsonic relativistic flow)
Non-fluid
2
2
'
'
8 c
B
U
U
p
B
2 ,2
2 critAlfven
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Dynamics
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Expansion of σ >> 1 wind supersonic (MHD), Γ2> σ,
in vacuum: flow acceleration determined by internal flow dynamics: flow passes through fast sonic points (eg ΓF~√σ), becomes causally disconnected (Michel)
subsonic, Γ2< σ, : acceleration limited by external medium, causally connected flows pressure balance at the
contact
222
2
22 ~
8~
4c
π
B
cR
Lext
r
Γ
t
1
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GRBs: σ < Γ2/2 and σ > Γ2/2 have different early dynamics
σ > Γ2/2– subsonic flow σ < Γ2/2– supersonic flow
(reached terminal Γ0)
At late times, t> > tGRB (self-similar), composition of ejecta is not important
At early times: MHD, σ < Γ2/2 Force-free σ > Γ2/2 σ > 1: weak or no
reverse shock emission
tcoord
Γ~ t-3/2
Γ=Γ0
σ < Γ2/2
20
2~ cME ISMej
Γ Γ ~ t -1/2
σ →∞
22 )(~ tcME ISMej
tobs~tGRB
(Lyutikov 2003,2006)
supersonic
subsonic
Self-similar Sedov-Blandford-McKee
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Observations? Swift results are very puzzling: flares and lighcurve breaks at t ~10 -- 105 sec (two “breaks” were expected, tGRB~100 sec and @ Γ~1/θ, 105
sec) For σ < 1 strong reverse shock emission is expected
For σ >1 no reverse shock is weak or non-existent Reverse shock in fireball: same type as internal shocks:
microphysics is fixed by prompt emission Expected optical flash m ~ 12-18. Cooling: Flux ~ t -2 ; later: cooling to radio emission.
In the Swift era absolute majority of GRBs do not show predicted RS behavior (despite UVOT and numerous robotic telescopes).
This may indicate highly magnetized ejecta, σ > 1 Other possibilities to produce some optical emission
(e.g. e± by γ)
(Nousek; Zhang)
steep
steep
shallow
t~102-103 s
t~104-5 s
flares
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Long GRBs: expansion inside a star
of a σ> 1 wind. As long as expansion is non-relativistic
there must be dissipation Energy and Bφ-flux is injected linearly with v~c for non-relativistic expansion volume is near constant
, Bφ ~ t
Energy ~ Bφ2 ~ t2 ??? (c.f. Gunn & Rees PWNs)
Need to destroy Bφ – flux: inductance break down → dissipation
Energy goes into e±-γ (~ first 3 sec): lost after photosphere
This is different from AGNs (c.f magnetic tower of Lynden-Bell), where expansion can always be relativistic, but not for GRBs
(Lyutikov & Blandford, 03)
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Dissipation in magnetically-dominated plasma
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Dissipation: σ > 1 – energy in B-field
σ > 1: shock are weak; do not exist for σ> σcrit
B-field dissipation due to current instabilities (“reconnection”)– B-fields are strongly non-linear systems: dissipation property
of the emission region, NOT of the source activity (e.g. Solar B-field generated on ~22yr time scales, flares can rise in minutes)
σ > 1 – new plasma regime – Adopt non-relativistic schemes:
• Magnetodynamical tearing mode • Relativistic reconnection
– new acceleration schemes (no hydro or non-relativistic analogues)
• Charge-starved plasma, turbulent EM cascade 3-wave processes are allowed: FFA, AAF! (in non-relativistic MHD 3-wave with ω ≠ 0 are prohibited)
k
ω
ωB
ωpCascade
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Resistive instability of relativistic force-free current
layer(unsteady reconnection) Resistivity is usually very small (τR
~ L2/η >> τ)
Current sheets are unstable – formation of small scale sub-sheets
Tearing mode τ ~ (τA τR)1/2
τA ~ L/vA ~L/c, τR ~ L2/η
Similar to hydro (waves forms shocks) resistive RFF forms dissipative current
layers Essential for RFF simulations, EM turbulent
cascade (ML,03)
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Tearing mode in σ=∞ plasma
σ= ∞ : matter inertia is not important, force-free currents ensure JxB + ρe E=0 and decay resistively
BB
E)(E-B)(BB
B)(EEj 22
x)(
j┴=( ρp - ρe )v┴j║= ρp v║,p - ρe v║,e~ E|| /η
X
j
B
X
j
B
Dissipation rate ~ ηΔB~ηB/δ2}δ
parallel currents attract
E=ηII jII
Electric field
formation of current sheet
Resistive (tearing) EM instability ARL
c
1
~
~3max - very fast
(Lyutikov 03)
New plasma physics regime, same expression for growth rate?(come from very different dynamical equations: Maxwell and MHD)
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Tearing mode in σ=∞ plasma
slow motion in σ = ∞ plasma
Non-linear stage: formation of magnetic islands
(Komissarov et al, 2006)
cc
gVt
t
t
22
2
2
B
BEV ,
8
B
08
BB
4
B)(
0)V2(
0B)BV(B
0B
very similar to incompressible MHD!Growth rates inexcellent agreementwith analytics
This may be a step towards formationof reconnection layers.Applications: magnetars (growth rate ~ msec, similar to flare rize time), AGN, GRB jets
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Applications: magnetars, AGN, GRB jets.
Giant flare SGR 1806: Time scales: observed rize
time, < 250 μsec, implies reconnection in the magnetosphere (Alfven time , t ~ RNS/c ~ 30 μsec)
Similar to Solar Coronal Mass Ejection (CME). Magnetar jets (plumes)?
Late constant velocity, sub-relativistic outflow may be just a projection effect
(Lyutikov 2006)
(Palmer et al. 2005)
θ
βapp= β ctg θ/2
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Acceleration of UHECRs
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UHECRs:
Emax ~ 3 1020 eV Isotropic, perhaps small scale clustering UHECRs must be produced locally , < 100 Mpc Perhaps dominated by protons above ~ 1018 eV Hard(ish) aceleration spectrum, p ~ 2-2.3
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Acceleration by large scale inductive E-fields: E~∫ v•E ds
Potential difference is between different flux surface (pole-equator)
In MHD plasma is moving along V=ExB/B2 – cannot cross field lines
Bring flux surfaces together –Z-pinch collapse (Trubnikov etal95)
Kinetic motion across B-fields- particle drift - (Bell, Blasi, Arons)
E
B
V
Ud
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E ┴ B : Inductive potential
To reach Φ=3 1020 eV, LEM > 1046 erg/s (for protons)This limits acceleration cites to high power AGNs (FRII, FSRQ,high power BL Lac, and GRBs)There may be few systems with enough potential within GZK sphere (internal jet power higher than emitted), the problem is acceleration scheme
I ~L ,377~4
~R
,4
c L~I ,
L 4
EM
EM
E
c
c
Lovelace 76Blandford 99
BE
Pictor A (FRII)
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Potential energy of a charge in a sheared flow
)(4
vBc
B
VE~x
Potential Φ~-x2
∆Φ~√L/c
B
VE~-x
Potential Φ~x2
Depending on sign of (scalar) quantity (B *curl v) one sign of charge is at potential maximumProtons are at maximum for negative shear (B *curl v) < 0
,)( UFvB E=-v x B
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Astrophysical location: AGN jets
There are large scale B-fields in AGN jets Jet launching and collimation (Blandford-Znajek,
Lovelace) Observational evidence of helical fields Jets may collimate to cylindrical surfaces (Heyvaerts
& Norman) Jets are sheared (fast spine, slow edge)
Er
v
x Bφ
Er
B
VE
Protons are at maximum for negative shear (B *curl v) < 0. Related to (Ω*B) on black hole
ΩBH and disk can act as Faradey disk
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Drift due to sheared Alfven wave
Electric field Er ~- vz x Bφ : particle need to move radially, but cannot do it freely (Bφ ).
Kinetic drift due to waves propagating along jet axis ω=VA kz
Bφ(z) → Ud ~ ~er
∆
BφxBφ∆
ud
B
ErF
B-fieldF
Udrfit ~ F x B
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Why this is all can be relevant? Very fast energy gain :
highest energy particles are accelerated most efficiently!!! low Z particles are accelerated most efficiently!!! (highest
rigidity are accelerated most efficiently) Acceleration efficiency does reach absolute theoretical
maximum 1/ωB
Jet needs to be ~ cylindrically collimated; for spherical expansion adiabatic losses dominate
EZeBc
ckAacc ||
1~
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Acceleration rate DOES reach absolute theoretical maximum
~ γ/ωB Final orbits (strong shear), rL ~ Rj, drift approximation is
no longer valid New acceleration mechanism For η < ηcrit < 0, ηcrit = - ½ ωB/γ, particle motion is unstable
non-relativistic:
Acceleration DOES reach theoretical maximum ~ γ/ωB
Note: becoming unconfined is GOOD for acceleration (contrary to shock acceleration)
2
11
/2
0000
B
critη=V’
1sin/1
, 1cos
tr
y
trx
BB
B
L
BBL
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Spectrum
From injection dn/dγ~γ -p → dn/dγ ~ γ -2
ddn
γ
p2
ankle
Particles below the ankle do notgain enough energy to get rL ~Rj
and do not leave the jet
Mixed composition protons
UHECRs are dominated byprotons, below the ankle: Fe
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Astrophysical viability Need powerful AGN FR I/II (weak FR I , starbursts are
excluded) UHECRs (if protons) are not accelerated by our
Galaxy, Cen A or M87 Several powerful AGN within 100 Mpc, far way → clear
GZK cut-off should be observed: Pierre Auger: powerful AGNs?
GZK cut-off few sources IGM B-field is not well known
Fluxes: LUHECR ~ 1043 erg/sec/(100 Mpc)3 –
1 AGN is enough
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(Gabuzda 03, confirmed by Taylor etal)
Faradey Rotation and gradient of linear polarization across
the jet in 3C 273 Gradient of Rotation
Measure across the jet
Possible interpretation: helical field Need lots of poloidal flux → may come from a disk, not BH
Gradient of liner polarization
B
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Conclusion
EM-dominated plasma: may be a viable model for a variety of astrophysical phenomena. Very little is done.
Macrophysical models (ideal dynamics) Microphysics (resistivity is anomalous,
η~c2/ωp, η~c2/ωB; particle acceleration) Need for simulations (both dynamics and
acceleration)EM codes with currentsPIC codes (Nordlund)
Observations seem to be coming along
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“Where have you seen plasma, especially in magnetic field?” Landau
“The magnetic field invoked is proportional to someone’s ignorance” Woltijer
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Radiative losses
Equate energy gain in E =B to radiative loss ~ UB γ2
As long as expansion is relativistic, total potential
remains nearly constant, one can wait yrs
– Myrs to accelerate
G 2-
EeV 100410 6~
2mceZcm
B
cm 3
EeV 100 1610~
3
2mcmceZ
r
3
2-
33
42
2-2-2
22
EE
EE
103
10
,L 4 0
c
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Relativistic reconnection, σ>> 1
(Sweet-Parker)
Two parameters: Lundquist S=VAL/ η »1, σ » 1
outflow is always relativistic Inflow:
σ « S – non-relativistic inflow
S « σ « S2 – relativistic inflow Relativistic reconnection can be fast, ~ light crossing
time
(Blackman & Field 1994; Lyutikov&Uzdensky2003; Lyubarsky 2004)
1σ
S
L
δ,γ
S
2Sσβ Ain
11
,1 ,2
LSS outin
1 σ)γ1(γ inout
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Particle acceleration in relativistic reconnection
Leptons
Numerical experiments are only starting (Hoshino02, Larrabee etal 02).
Spectra depend on kinetic properties and geometry
∫(v•E)dl (McClements) If escape ~ rL, then For GRB we need γ-1 (Lazatti), also TeV AGNs
(Aharonian) No calculations of acceleration at relativistic
tearing mode (should accelerate as well)
in
d
dn
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Why magnetic energy wants to dissipate
X
j
B
X
j
B
Dissipation rate ~ ηΔB~ηB/δ2
}δAnomalous resistivity η(j)
E=η j
parallel currents attract
Electric field
Next: generalize non-relativistic fluid models to new regime
formation of current sheet
What is needed for magnetic dissipation is presence of electrical current
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Wave surfing can help
Shear Alfven waves have δE~(VA/c) δB, Axial drift in δExB helps to keep particle in phase Particle also gains energy in δE
∆B
udAlfven wave with shear
Alfven wave without shear
γmax/γ0
Most of the energy gain is in sheared E-field (not E-field of the wave, c.f. wave surfing)
Er
δE
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AGN jet
In situ acceleration is required (tsynch< R/c, short time scale variability: 20 min at TeV!)
e± winds - strong losses at the sourceIon-dominated - hard to get variability, low radiation
efficiency (Celotti,Ghisellini)EM- dominated!
Currents needed for collimation; Currents are unstable
Resistive modes may not destroy the jet, but re-arrange it (eg, sawtooth in TOKAMAKs, Appl)
Relativistic FF jets stabilized by rotationHard power law may be needed for TeV
emission(Aharonian)Polarization from helical B-field (Gabuzda; ML, in
prep)
(Lesch&Birk;Lovelace;ML)
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Jets start as B-field-dominated, can σ changes on the way?
Ideal conversion: acceleration Acceleration to fast point Γfast ~ √σfast
At this point σ ~ Γ 2>> 1 : flow remains B-field dominated
Collimation σ→ 1, but it is slow ~ ln z and unlikely σ << 1
• There are some indication (Homan et al , Jorgstad et al.) moving features, (Sudou et al.) increased jet-counter jet brightness, but not conclusive (jet bending & aberration can give visible acceleration).
Dissipative: on scale > RBH Γ 2 ~ 10 17 cm (e.g relativistic reconnection βin ~ 1, Lyutikov & Uzdensky)
• blazar γ-ray emission zone (Lyutikov 2003). Variation in Γ produced locally (no large UV variations of disk are seen): (Sikora et al. 2005)
Jet can remain B-field dominated to pc scales
(Weber & Davis, Goldreich & JulianVlahakis &Konigl)
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How can the two paradigms (σ>>1 and << 1) be
distinguished?
Shocks Spectra of Fermi-
accelerated particles (kinetic property) can be derived from shock jump conditions
Electrons need to be pre-accelerated to γ~ mp/me ~ 2000
(or √mp/me ~43)
B-field “Reconnection” spectra
are not “universal”, depend on details of geometry (universal in relativistic case, p=1 ?)
No need for pre-acceleration: all particles may be accelerated
1. Acceleration scheme with predictive power (γmin, p)
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How can the two paradigms be distinguished?: very hard
spectra, p< 2 Shock typically produce
p>2, relativistic shocks have p ~ 2.2 (Ostrowki; Kirk) non-linear shocks &
drift acceleration may give p<2, e.g. p=1.5 (Jokipi, Bell & Lucek)
B-field dissipation can give p=1 (Hoshino; Larrabee et al.); such hard spectra may be needed for TeV emitting electrons (γ-γ pair production on extragalactic light Aharonyan; Schroedter).
p< 2 spectra should not be discarded as unphysical
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2. Radiation modeling: not conclusive
TeV : SSC, B-field strongly under equipartition @ 1017 cm → Γ~50-100 (Krawczynski 04). Outflows in bulk flow?
γΓbulk
Γ ~ 2 Γbulk γ
2
2
B
ncm
B
n
B
B eind
Blazar dominance by IC: Uph’ /UB’ ~ Γ4, Γ > 10;
U e± ’ < UB’ < Uph’
Equipartition (e.g in FR II hotspots, Hardcastle) U’B ~ U’ e± : Amplification: sub-equipartition Dissipation → equipartition: (μe=γ me c 2/B)
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Aberration of Π: B-field is NOT orthogonal to polarization
In plasma frame In laboratory frame
B'
n'
e'
e'┴ B', n'
3/7
1,~ max
p
pn p
)B(vnBq
q)(nq
qne
22
0v)Bn)((e)B(e
(Blandford & Konigl 79; Lyutikov,Pariev,Blandford 03)
Both B-field and velocity field are important for Π
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Π from relativistically moving cylindrical shell with helical
B-field
zz B
B
B
B
1
tan1
tan '
'
'
Γ=2, ψ’=π/4, θob=π/3
B not orthogonal to e Jet can be Bφ dominated in observer frame and Bz-
dominated in rest
B
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Π from random B-field compressed at an oblique
shockv
l
shock
upstream shock downstream
l
Π
Π not aligned with projection of l,also Cawthorn & Cobbs
v)v(l1
-v)(ln-lnq
q)(nq
qne
22
Lyutikov (in prep)
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Aberration of Π
Direction of Π depends both on B-field and velocity field
Always plot “e”, not “inferred” B-field One needs to know velocity to infer internal B-field Symmetries in the velocity field may help
e.g. if shock is conical, on average polarization along or across jets (Cawthorn & Cobbs)
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Π from cylindrical shell with helical B-field
Γ=10, p=1, different rest frame pitch angles
'
'
'tanzB
B
Π depends on p Even co-spatial
populations with different p may give different Π (eg Radio & Optical)
Π along the jet
Π across the jet
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Large scale or small scale B-fields in pc-scale AGNs jets
Bimodial distribution of PA
PA follows the jet as it bends
Sometimes a bend gives 90 change of PA
For cylindrical jet U=0, average Π along or across the axis. Only conical shocks can give the same.
For fixed ψ, Π mostly keeps its sign. Note: for plane shock there is no correlation between Π and bend direction.
Sometimes a change does occur
BL Lac 1749+701
(Aller et al)
(Gabuzda 03)
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Resolved jets
Resolved jets: center: PA ║, edges: PA ┴ Emission is generated in small range Δr <
r(shear acceleration? Ostrowski) Core is boosted away
(Gabuzda 03)
e
e
Limb-brightening Mkn 501, (Giroletti )
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Jet polarization may tell the spin of BH
Left & Right helixes look differentDifferent Π signatureDirection of BH or disk spin (if θΓ is known)
Right
Left
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Tests/unresolved issues
Firmly established flow acceleration at (sub)-pc scales: evidence of B-field conversion
More Π studies, especially CP (unidirectional B-field) with MHD codes
Spectra requiring p → 1 Acceleration rates above Very high Π > 50% in R
compressed B-fields isotropize on Aflven time scale, 3-D random B-field is dominated by small scale
turbulence will lead to isotropization Different e-acceleration mechanisms?
X-rays are displaces from O-R (e.g. Cen A) Magnetic & shock accleration? (Kirk) NB: similar in Crab pulsar
Bsacc v
c
2
2
~
Cen A (Hardcastle et al 2003)
X-ray
Radio
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Are all ultra-relativistic jet the same?
Epeak – L correlations GRBs - positive, BL Lac – negative
Internal shocks in GRBs must be highly (unreasonably?) efficient, in BL Lac – inefficient
GRS 1915: jets appear after drop of the x-ray flux,blazars: no correlation between UV flux and flares jets without BH (Cirnicus X-1)
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Prospects
Sept. 2002, Bolognia Conference: “Can one “ prove” reconnection? – Not from first principles” By analogy to some Solar phenomena Nothing else can do
May be we can…