Drift acceleration of UHECRs in sheared AGN jets Maxim Lyutikov (UBC) in collaboration with Rashid...

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