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