Acceleration and radiation mechanisms in extragalactic gamma ray - - PowerPoint PPT Presentation

Acceleration and radiation mechanisms in extragalactic gamma ray sources

E.V. Derishev

Institute of Applied Physics, Nizhny Novgorod, Russia

Are jets efficient emitters?

Cyg A

AGNs:

AGN jet luminosity up to 1048 erg/s radio lobe luminosity

• f the order of

1044 erg/s

?

Prompt emission Afterglow

GRBs:

GRB (prompt) energetics up to few×1054 erg afterglow energetics up to few×1052 erg

Not too much room for inefficient sources

Timescales in a relativistic jet

R

θ

variability timescale, tv dinamical timescale, td light-crossing time, tl In the comoving frame: tv ≃ td td = R/(Γc) tl = min [R/(Γc), θR/c] For a wide jet (θ ≥ Γ−1): tv ≃ td ≃ tl

Are the jets wide?

AGNs: you can see it

M 87

GRBs: check observed correlations

Narrow jets imply a correlation between peak energy εp and energetics EGRB:

EGRB ∝ ε3

p Observed correlation:

EGRB ∝ ε2

p Amati, 2006

Radiation mechanisms

electrons

• Synchrotron radiation

undulator radiation

• Inverse Compton radiation
• Bremsstrahlung

protons

• Synchrotron radiation
• Inelastic nucleon collisions
• Coulomb losses

Lsy = 4 3γ2σTcB2 8π εsy ∼ γ2eB mec

When coupled to diffusive shock acceleration,

εsy mec2/αf ∼ 70 MeV

due to radiative losses, that limit acceleration

Radiation mechanisms

electrons

• Synchrotron radiation

undulator radiation

• Inverse Compton radiation
• Bremsstrahlung

protons

• Synchrotron radiation
• Inelastic nucleon collisions
• Coulomb losses

B

ν 2 ν 4/3 νFν

synchrotron undulator

ν

d

Toptygin & Fleishman 1987, Medvedev 2000

Lund = 4 3γ2σTcB2 8π εund ∼ γ2c d

differs from synchrotron if

d < mec2/(eB)

Radiation mechanisms

electrons

• Synchrotron radiation

undulator radiation

• Inverse Compton radiation
• Bremsstrahlung

protons

• Synchrotron radiation
• Inelastic nucleon collisions
• Coulomb losses

Thomson regime (εph ≪ mec2/γ):

LIC = 4 3γ2σTcwph εIC ∼ γ2εph

Klein-Nishina regime (εph mec2/γ):

LIC < 4 3γ2σTcwph εIC ∼ γmec2

εph – background photons’ energy wph – background radiation energy density

Interlude: Two-photon absorption

In the limit εγ ≫ mec2 we have σγγ ≈ 2 σeγ

Two-photon absorption

Optical depth for two-photon absorption τγγ(ω) ≃ σγγ Nph(ω∗) R Inverse Compton energy losses per particle ˙ ε ≃ 1 2ε σeγ Nph(ω∗) c

Under assumption of high radiation efficiency ( ˙

ε > ε/tv)

the optical depth of a source with size

R ≃ ctv

is

τγγ > 2 σγγ(ε/2) σeγ(ε) ≫ 1

Nph (ω∗) – number density of photons with frequency ∼ ω∗

2nd interlude: SSC vs ERC

• Efficient cooling means that

w′

ph >

mec

32 π 9

γ′

(

e2 mec2

)2 Γtv

• Photons’ occupation number

K ≃ w′

ph

2π2(c)3 ε4

∗

• Comptonization in the Thomson regime, i.e.

ε∗ < mec2/γ′

and

Γγ′ > ε/mec2 Hence,

ε∗ – the energy of comptonized photon in the comoving frame α – the fine-structure constant λc – the electron Compton wavelength

K > 9 π 16

(

ε mec2

)3

λc α2Γ4 ctv (for SSC) K > 9 π 16

(

ε mec2

)3

λc α2Γ2 ctv (for ERC)

SSC vs ERC

Assume: the photons’ occupation number does not exceed its magnitude at the peak of black-body spectrum, i.e. K < 0.02

• btain:

independent lower limit to the Lorentz factor

• For synchrotron self Compton

Γ > 3 α1/2

(

ε mec2

)3/4 ( λc

c tv

)1/4

≃ 2 ε3/4

12

t1/4

3

• For external radiation Compton

Γ > 9 α

(

ε mec2

)3/2 ( λc

c tv

)1/2

≃ 4 ε3/2

12

t1/2

3

Radiation mechanisms

electrons

• Synchrotron radiation

undulator radiation

• Inverse Compton radiation
• Bremsstrahlung

protons

• Synchrotron radiation
• Inelastic nucleon collisions
• Coulomb losses

emission power density:

˙ wff = 2 παf σTc n2

e

√

Tmec2 G(ne, T)

At an optical depth τ each electron on average radiates

wff ne = 2 π αf τ

√

Tmec2 G(ne, T)

inefficient unless

T α2

f mec2 ∼ 25 eV

Radiation mechanisms

electrons

• Synchrotron radiation

undulator radiation

• Inverse Compton radiation
• Bremsstrahlung

protons

• Synchrotron radiation
• Inelastic nucleon collisions
• Coulomb losses

At a given energy

L(p)

sy =

 me

mp

 

4

L(e)

sy ∼ 10−13 L(e) sy very slow mechanism, works only in TeV range

Radiation mechanisms

electrons

• Synchrotron radiation

undulator radiation

• Inverse Compton radiation
• Bremsstrahlung

protons

• Synchrotron radiation
• Inelastic nucleon collisions
• Coulomb losses

end up with energetic electrons, which radiate by either of the electron mechanisms

Acceleration in shear flows

Berezhko & Krymskii 1981 Ostrowski 1990, 1998 Rieger & Duffy 2006

Acceleration rate ˙ ε = 3D

(∇V

c

)2

ε

For Bohm diffusion

˙ ε = (∇V )2 ωB ε = (∇V )2 eBc ε2

ε – particle’s energy ωB – gyrofrequency, D = c2 3 ωB Acceleration fails against losses if

R0 < 3 × 1030 L3/2

45

Γ4 cm

R0 – size of the central engine

Diffusive shock acceleration

Acceleration rate ˙ ε ≈ 1 6 r − 1 r V 2 D ε

 η ≃ V 2

c2

 

r = u1 u2 ≤ 4 – shock compression

ratio

V

– velocity of the shock

D = p c2 3 eB

for Bohm diffusion

Spectrum of accelerated particles: dN dε ∝ ε−α α = −r + 2 r − 1

Acceleration at relativistic shocks

θ ∼ 2/Γ

The energy gain factor g = (1/2) (Γθ)2 ≃ 2 The probability of particle injection back to upstream must be ∼ 1 to get efficient

• acceleration. The actual probability

depends on the (unknown) magnetic field geometry. Favorable geometry gives, e.g., dN dε ∝ ε−22

9

(Keshet & Waxman, PRL 2005) “Realistic” geometry leads to very soft particle distributions, with energy concentrated near Γ2mc2 (Niemiec & Ostrowski, ApJ 2006; Lemoine, Pelletier & Revenu, ApJ 2006)

• cf. talk by L. Sironi

Diffusive shock acceleration

“Thermal particles ” Acceleratedparticles

f()

• Maximumacceleration

energy

mp me

()

me

B

c 2 h

2 1

• Γ – Lorentz-factor of the shock,

γ – Lorentz-factor of an electron, ωB = eB/mec – gyrofrequency, α – fine structure constant, f(γ) – injection function

Diffusive shock acceleration gives f(γ) ∝ γ−s , where s ≃ 2.2 (universal power-law)

Schematic broad-band spectrum

ν

ε sy ε IC

νFν Synchrotron peak IC peak Synchrotron cut−off

Standard assignment of spectral features –

F – log scale

ν

ν

Cut-off “Thermal” break – log scale ν

... – standard problems

• Position of the peak is too sensitive to the shock

Lorentz-factor

Photon energy at the peak in the comoving frame ε′

peak ∼

(

Γmp me

)2 eB

mec in the laboratory frame εpeak ∝ Γ4 (since B ∝ Γ)

• The spectrum well above the peak frequency is

universal and too hard Nγ ∝ γ−3.2 ⇒ νFν ∝ ν−0.1

... – standard problems

• Low-frequency asymptotics in the fast-cooling regime

is too soft

The hardest possible injection f(γ) = δ(γ − γ0) gives νFν ∝ γη for γ < γ0 ; ⇒ νFν ∝ ν1/2, if η =

synchrotron losses total radiative losses = const

more details on this point were given in the talk by E. Lefa

Another assignment of spectral features –

F – log scale

ν

ν

Cut-off “Thermal” break – log scale ν

... – other problems

• The synchrotron cut-off frequency is too high

At the maximum energy, the scattering length (gyroradius) equals to the radiation length: η

(

4 3γσT B2 8π

)−1

= γmec2 eB So that γ2

max

eB

mec ≃ η mec2 α σT – Thomson cross-section

• Low-frequency asymptotics in the fast-cooling regime

is too soft

The hardest possible injection f(γ) = δ(γ − γ0) gives νFν ∝ γη for γ < γ0 ; ⇒ νFν ∝ ν1/2, if η = const

A universal acceleration-radiation scheme for relativistic outflows?

What limits acceleration?

Particle escape from the accelerator Degradation of particles’ energy

• Sinchrotron radiation
• Inelastic collisions
• Inverse Compton losses (for electrons)
• Photomeson interactions and creation of e−e+ pairs

(for protons and nuclei) The probability of photon-induced reaction is usually small, ≪ 1

How small has to be “small” to become dynamically negligible?

For a non-relativistic shock, a probability ≪ 1 is always small For a relativistic flow, the answer is either ≪ 1 or ≪ 1/Γ2, depending on what you are talking about

If some energy leaks from downstream to upstream and mixes up with the upstream particles, we feed back to the shock Γ2 times the initial energy! Γ is the Lorentz factor of the flow

When “small” is REALLY small (1) (2) (3) 3 2 1 θ ∼ π

Full isotropization in the upstream (θ ∼ 1) gives the energy gain factor g = 1 2 (Γθ)2 ∼ Γ2 in each shock-crossing cycle Photon-induced reactions reversibly “convert” accelerated particles to neutrals ⇒ Converter acceleration mechanism Derishev, Aharonian, Kocharovsky & Kocharovsky, PRD 2003; Stern, MNRAS 2003

Conversion to neutrals for protons

The proton cycle

• Requires presence of dense photon field or dense

baryonic matter

• Operates down to mildly relativistic bulk velocities
• Accompanied by powerful neutrino emission
• If efficient, quenches the electron cycle

Conversion to neutrals for electrons/positrons

The electron cycle

• Operates in relatively week photon fields
• Usually requires bulk Lorentz factor in excess of a few
• Has two regimes: externally pumped cascade and

acceleration

Emerging particle distribution

ε

ε

log logdN

d __

logΓ2

Distribution’s envelope for monoenergetic injection: dN dε ∝ ε−α α = 1 − ln p cn ln g

— spectral index

p cn — conversion probability

Estimations of the conversion probability

LBLR – luminosity of the broad-line region EXray – energy released in the form of hard X-rays

Changes in the beam-pattern

Derishev, Aharonian & Kocharovsky, ApJ 2007

comoving frame laboratory frame

• Low-energy particles

ε ≪ εcr

• Critical-energy particles

ε ≃ εcr

• High-energy particles

ε ≫ εcr

εcr = 3 2

(

mec2)2 e3/2B1/2;

ω ∼ 70 × Γ MeV

— for synchrotron losses

Gamma-ray sources without a counterpart

lg vF

v

lg v lg vF

v

lg v

• n-axis view
• ff-axis view

Distinctive features of the converter mechanism

• Protons are accelerated, but not nuclei
• Accelerated particles reach supercritical energies,

so that the spectrum of their synchrotron emission extends to much higher frequencies (up to Γ2 times higher compared to diffusive shock acceleration mechanism)

Distinctive features of the converter mechanism

• Emerging particle distribution is not universal, but

instead depends on the source parameters Good for explaining the whole variety of spectra?

• Broadening of beam pattern (up to becoming

nearly isotropic) for high-energy emission in the sub-GeV – TeV range with possible applications to: high-latitude unidentified EGRET sources = off-axis blazars? prolonged GeV emission from Gamma-Ray Bursts = geometrically retarded off-axis emission?

Diffusive shock acceleration

Probability of injection is sensitive to the magnetic field geometry Acceleration efficiency is sensitive to the magnetic field geometry Smoothing out sharp discontinuities progressively decreases efficiency Acceleration starts from thermal ion energy Does not depend on presence of photon fields Works for both relativistic and non-relativistic outflows

Converter acceleration

Injection is equally easy at any point in the downstream Acceleration does not depend on the magnetic field geometry Efficiently works even in absence

• f any discontinuities

Acceleration is efficient only past certain energy threshold Requires intense photon fields Requires relativistic hydrodynamical velocities

A tale of large Lorentz factor

Increase Lorentz factor!

• The emitting region moves further away

R ∝ Γ2

• Comoving photon density rapidly decreases

w′

ph ≃ 1

Γ2 L 4πR2c ≃ L 4π(ctv)2Γ6c

L – the isotropic luminosity

• Two-photon absorption threshold increases

An arbitrarily large Lorentz factor ...

(1) The synchrotron peak is at ε = Γγ′ 2 eB mec (2) Radiation flux is a fraction

• f the magnetic-energy flux

w′

ph = η1

B2 8π (3) Radiation efficiency is η2 ≤ 4 9 Γtv γ′

(

e2 mc2

)2

B2 mec For a given isotropic luminosity w′

ph =

L tv 4π(Γ2ctv)3

tv – the dynamical timescale ( ≈ the observed variability timescale) ε – the observed photon energy Γ – the jet’s Lorentz factor γ′ – the electron’s Lorentz factor in the jet comoving frame

... cannot be arbitrarily large.

Derishev, Kocharovsky, Kocharovsky, A&A 372, 1071 (2001) Begelman, Fabian, Rees, MNRAS Letters 384, L19 (2008)

Substitute: γ′ from expression (1) into inequality (3) B from expression (2) into inequality (3) Obtain: the upper limit to the Lorentz factor Γ <

  [

211/38] e14 ε2 L3 η3

1η4 2 2 (mec2)10 c7 t2 v

 

1/16

≃ 55

 

1 η3

1η4 2

 

1/16 ε1/8 6

L3/16

45

t1/8

3

Self-Compton radiation

(1) The peak is at ε = Γγ′ 4 eB mec (Thomson regime!) (2) Radiation flux is a fraction

• f the magnetic-energy flux

w′

ph = η1

B2 8π (3) Radiation efficiency is η2 ≤ 4 9 Γtv γ′

(

e2 mc2

)2

B2 mec

• r

η2 ≤ 4 9 η1/2

1

Γtv γ′

(

e2 mc2

)2

B2 mec

The limiting Lorentz factor (inverse Compton radiation) Γ <

  [

223/316] e30 ε2 L7 η7

1η8 2 2 (mec2)22 c15 t6 v

 

1 36

For active galactic nuclei Γ ≤ 21

 

1 η7

1η8 2

 

1/36 ε1/18 12

L7/36

45

t1/6

3

For gamma-ray bursts Γ ≤ 1400

 

1 η7

1η8 2

 

1/36 ε1/18 6

L7/36

51

t1/6

−3

η1 =

radiated energy magnetic field energy ∝

L tv R3B2 ∝ L t2

vB2 ≈ Compton y parameter

η2 =

dynamical timescale particle cooling timescale ∝ γ B2tv ∝ ε1/2 sy B3/2tv Observed parameters:

L, tv, εsy

Assumption:

R ≈ ctv

Free parameter: Γ

Inefficient sources

Efficient sources

Living rooms for GRBs and TeV Blazars BKN = 1 Γ3

( εsy

mec2

)3 m2

ec3

e

= Γ3

 mec2

εic

 

3 m2 ec3

e

Gamma-ray Bursts

There is a window in the parameter space, where the sources are

• radiatively efficient
• opaque for the inverse Compton radiation

at the same time. This leads to the condition

B < BKN ,

equivalent to

tv > 1 α re c

 2 L re/c

mec2

 

1/2 

mec2

ε

 

3

≃ 3 × 10−5 s L1/2

51 ε−3 6 α – the fine-structure constant re – the classical electron radius

A byproduct limit for GRB sources

If a Gamma-Ray Burst is powered by a black hole, then its mass must be

M > 3 L1/2

51 ε−3 6

M⊙

TeV Blazars

A good TeV Blazar must be

• radiatively efficient
• transparent for the inverse Compton radiation

The window opens if

η1(BKN) > 1 ,

which may be treated in two ways (1)

tv < 1 Γ6 1 α re c

 2 L re/c

mec2

 

1 2 (

ε mec2

)3

≃ 1.5 × 1012 s L1/2

45 ε3 12

Γ6

(2)

Γ <

 2 L re/c

α2 mec2

 

1 12 ( re

ctv

)1

6

(

ε mec2

)1

2 ≃ 30 L1/12

45

ε1/2

12

t1/6

3 α – the fine-structure constant re – the classical electron radius

A byproduct limit for blazars

If a TeV blazar is powered by a black hole, then its mass must be

M < 1.5 × 1017 L1/2

45 ε3 12

Γ6 M⊙

Fast cooling regime

N ( ) – log scale γ

γ

Cut-off Γ mp me

( )

me

ωB

c 2 h

2 1

α

”Thermal” break – log scale N(γ) – electrons’ distribution function Continuity equation ∂N ∂t + div (˙ γN) = f(γ) gives stationary solution N(γ) = −1 ˙ γ

∫ ∞

γ

f(γ′) dγ′ The corresponding spectrum (provided ν ∝ γx) is : νFν ∝ dF d ln γ ∝ γη

∫ ∞

γ

f(γ′) dγ′ η(γ) – the fraction of electron’s energy transferred to the observed radiation

Chaotic magnetic field

Scattering length: ℓs = ℓc

(rg

ℓc

)2

=

(

γmec2)2 e2B2ℓc Acceleration limit: ℓs = η

(

4 3γσT B2 8π

)−1

Consequently, γ3

max ≃ η ℓc

re

ℓc – correlation length rg = γmec2 eB – gyroradius re – classical radius

• f the electron

Typical energy of synchrotron photons: γ2

max

eB

mec ≃

 η ℓc

rg0

 

2/3 (αB

Bcr

)1/3 mec2

α

Bcr ≃ 4.5 × 1013 G – Schwinger magnetic field rg0 = mec2 eB – ”cold” gyroradius

Decaying magnetic field (1)

• Distancetotheshockfront,

r

wph B

wph – effective energy density of photons

wph(γ) =

∫ mec2

hγ

wν dν

The electrons are advected ⇒ dγ dr = 3 c ∂γ ∂t = −4γ2σT

(

wph + B2 8π

)

For a power-law photon spectrum wν ∝ νq

(−1 < q < 0):

wph(γ) ∝ γ−1−q

Decaying magnetic field (2)

• Let

η ≪ 1 and wν ∝ νq

dγ dr ∝ −γ1−q ⇒ γ =

(

r r0

)1

q

for γ ≪ γ0 Injecting delta-function gives: N(γ) ∝ γq−1 for γ < γ0

• Let

B ∝ r−y

The synchrotron efficiency: η ≃ B2 8πwph ∝ r−2yγ1+q ∝ γ1+q−2qy Typical synchrotron frequency: ν ∝ γ2B ∝ γ2−qy

Emerging spectrum: νFν ∝ γη ∝ ν

2+q−2qy 2−qy q = −1: νFν ∝ ν

1+2y 2+y

q = 0: νFν ∝ ν η = 1: νFν ∝ ν

1−2y 2−3y