Post on 18-Dec-2015
Drift acceleration of UHECRs in sheared AGN jets
Maxim Lyutikov (UBC)in collaboration withRashid Ouyed (UofC)
Acceleration of UHECRs: there is more than Fermi
Conventionally – Fermi I & II at relativistic shocks: – maximal efficiency (τacc ~ γ/ωB)
is usually either assumed or put “by hand” by using (super)-Bohm cross-field diffusion
– so far NO self-consistent simulations produced required level of turbulence
– Hardening above the “ankle” ~ 1018 eV may indicate new acceleration mechanism
shock
v1 > v2
We propose new, non-stochastic acceleration mechanism that turns on above the ankle, E> 1018 eV
General constraints on acceleration cite:
Constraints on UHECRs are so severe, “~“ estimates are useful
Maximal acceleration E-field < B-field: E=β0 B, β0 ≤ 1
Total potential Φ = E R= β0 B R
Maximal energy E~ Z e Φ = β0 Z e B R
Maximal Larmor radius rL ~ β0 R
– For β0 < 1 system can confines particles with energy large than is can accelerate to (NB: Hillas condition rL ~ R is condition on confinement, not acceleration)
Two possibilities:– E || B (or E>B) – DC field– E ┴ B – inductive E-field
} Two paradimes for UHECR acceleration
DC (linear) acceleration for UHECR
do not work Full DC accelerations schemes (with E-field || to B-
field or E>B) cannot work in principle for UHECRs1. leptons will shut off E|| by making pairs (typically
after ΔΦ << 1020 eV)2. Double layer is very inefficient way of accelerating: E-
field will generate current, current will create B-field, there will be large amount of energy associated with B-field. One can relate potential drop with total energy:
– Relativistic double layer:
– Maximal energy
– Total energy: E~ B2 R3~ I2 R c2
cZemI p maxE
4/14/1
6015 10
10eV10
R
kpc
erg
Etot
4/1~ R E ji
ie
Φ,R4/12
max
R
EZecm tot
pE
BE
E ┴ B : Inductive potential
BR0ER
1 ~ for ,~14
4 2
tottotEM LLc
BERL
cBRLB EM2
00 )(~ ,~E
I ~L ,377~4
~R EM Ec
To reach Φ=3 1020 eV, L > 1046 erg/s (for protons)This limits acceleration cites to high power AGNs (FRII, FSRQ,high power BL Lac, and GRBs)There are a few systems with enough potential, the problem is acceleration scheme
cL0
2
~
,L 4 0
c
4
c L~I
2~
c
IRB
E ┴ B: Poynting flux in the system: relate Φ to luminocity
Lovelace 76Blandford 99
Electric potential
BE
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
What is the mechanism of acceleration? Before, it was only noted that there is large potential (Lovelace, Blandford, Blasi), but no mechanism (Bell)
Kinetic motion across B-fields- particle drift
EB
V
Potential energy of a charge in a sheared flow)(
4vB
c
B
VE~x
Potential Φ~-x2
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 This derivation is outside of applicability of non-relativisitc MHD
2
4 : xv
2xB
c For linear velocity profile
Astrophysical location: AGN jets
There are large scale B-fields in AGN jets Jet launching and collimation (Blandford-Znajek,
Lovelace, Blandford-Payne; Hawley) Observational evidence in favor of helical fields in pc-
scale jets (talk by Gabuzda ,posters, Lyutikov et al 2004, )
Jets may collimate to cylindrical surfaces (Heyvaerts & Norman)
At largest scales Bφ is dominant Jets are sheared (talk by Laing)
E
v
x Bφ
E
Drift due to sheared Alfven wave
Electric field E ~ vz x Bφ ~ er For negative shear, (B *curl v) < 0, proton is at
potential maximum, but it cannot move freely along it: need kinetic drift along radial direction
Inertial Alfven wave propagating along jet axis ω=VA kz
Bφ(z) → Ud ~ ~er
∆
ud
∆BφxBφ B
ErF
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)
Jet needs to be ~ cylindrically collimated; for spherical expansion adiabatic losses dominate
EZeBc
ckck A
B
Aacc ||
1
||
11~
dt Eu)eZ eZ( vEEEnergy gain:
ZeB
tkAt 4
22EE
For linear velocity profile, V=η x, E= η x B/c, x = ud t,
cZeB
ku Ad 2
E →
Wave surfing can help
Shear Alfven waves have δE~(VA/c) δB, particle also gains energy in δE
Axial drift in δExB helps to keep particle in phase
∆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
Final orbits (strong shear), rL ~ Rj When rL becomes ~ jet radius, drift approximation is no
longer valid New acceleration mechanism Larmor radius of the order of the shear scale, η=V’
~ωB/γ (Ganguli 85)
Non-relativistic, linear shear: Vy=η x
unstable motion for η< - ωB
1sin/1
, 1cos
t
rytrx
BB
B
L
BBL
B
V
V
V
η=0η=-0.5η=-0.9η=-1.01
η=V’/ωB
Final orbits: relativistic
Relativistic
For η < ηcrit < 0 particle motion is unstable
When shear scale is ½ of Larmor radius motion is unstable
Acceleration DOES reach theoretical maximum
Note: becoming unconfined is GOOD for acceleration
2
11
/2
0000
B
crit
η=V’
Spectrum
From injection dn/dγ~γ -p → dn/dγ ~ γ -2
d
dn2
γ
2 p0
ankle
Particles below the ankle do notgain enough energy to get rL ~Rjand do not leave the jet
Mixed composition protons This is what is seen
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
Astrophysical viability Need powerful AGN FR I/II (weak FR I , starbursts are
excluded)– UHECRs (if protons) are not accelerated by Cen A or M87
Several powerful AGN within 100 Mpc, far way → clear GZK cut-off should be observed
For far-away sources hard acceleration spectrum, p~ 2 , is needed
Only every other AGN accelerates UHECRs Clustering is expected but IGM B-field is not well known
– μGauss field of 1Mpc creates extra image of a source (Sigl)– Isotropy & clustering: need ~ 10 sources (Blasi & Di Marco)
Fluxes: LUHECR ~ 1043 erg/sec/(100 Mpc)3 – 1 AGN is enough Pre-acceleration can be done outside of the jet and pulled-in Shock acceleration in galaxy cluster shock stops @ 1018 eV Matching fluxes of GCR and EGCR….
Main properties of the mechanism:
Protons are at maximum for negative shear (B *curl v) < 0
Acceleration rate increases with energy
At highest energies acceleration rate does reach absolute theoretical maximum τacc ~ γ/ωB
At a given energy, particles with smallest Z (smallest rigidity) are accelerated most efficiently (UHECRs above the ankle are protons)
produces flat spectrum Pierre Auger: powerful AGNs?
– GZK cut-off– few sources
May see ν & γ fluxes toward source (HESS, IceCube)
GRBs: L ~ 1050 erg/s
GRBs: I~ 1020 A, Emax=3 1022 eV Max. acceleration E~B (on τ~ γ/ωB), shorter than
expansion time scale c Γ/R Radiative losses (e.g. synchrotron). For ECR~ 3 1020 eV
Always fighting adiabatic losses: need to get all the available potential on less than expansion time scale
If there is GZK cut-off, and LGRB~LCR then GRBs are
viable source
G510 3~2mceZ
cmB
cm 14310~ 3
2mcmceZ
r
3
2-
33
42
2-2-2
22
1003
100
E
E
E ┴ B : Inductive potential
L ~ E B R2c ~ (BR)2 βc ~E 2 c
I ~L
377~4
~R
4
c L I
E 4or ,
L 4
EM E
E
c
Rc
Voltage Current
Sun 108 V 109 A
Pulsar 1016 V 3 1012 A
SGR 3 1014 V 1012 A
AGN 3 1020 V 1018 A
GRB 3 1022 V 1020 A
SLAC 1010 V 1 AR
Ee
4~maxE
Potential energy of a charge in a sheared flow
Consider sheared fluid motion of plasma in magnetic field
Depending on sign of (scalar) quantity (B *curl v) one sign of charge is at potential maximum
Protons are at maximum for negative shear (B *curl v) < 0
This derivation is outside of applicability of non-relativisitc MHD
,
4UF
c
)()(4
)(4
BvvBc
Bvc
Bc
vE
)(4
vBc
2
4 : x vprofile,ocity linear velFor
2xB
c z
in stationary, current-free case