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AGN-3: HR-2007 p. 1

Radiative Processes

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Hot gas radiates

•Continuum emission•Thermal emission / black-body radiation (RL 1.5)• Bremsstrahlung (free-free) (KN 4.5)• Synchrotron radiation (KN 3.2-3.4, 3.5,3.6)• Thomson scattering (KN 4.1)• Compton and inverse compton scattering (KN 4.2)• Pair production/annihilation radiation

• Line emission• bound-bound; bound-free

If gas is optically thick: Absorption

Here: Continuum emission

Plus: the torus contains dust particles with a range of sizes and temperaturesthat emit (modified) black body radiation

Overview

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Black-body radiation

Q: Is the BB spectrum modified if the black box is painted blue?

Thermal emission of optically thick gas

a. Rayleigh-Jeans limit: (radio regime)

b. Wien limit:

c. Monotonicity with temperature: of two black body curve the one with thehigher temperature lies entirely above the other

d. Wien displacement law: peak frequency/wavelength shifts linearly with T

(Note: ) e. Total Power emitted (Stephan’s law): P=σ A T4 (A: area)This also defines the effective temperature as the temperature Teff that gives atotal emitted power equivalent to the total power observed.

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Bremsstrahlung or free-free emission

“Braking radiation”Potentially contributes to the production of X-ray and γ-ray continuum spectraRadiation by charge accelerating in field of other chargeDominant process: electron-ion interaction

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Thermal Bremsstrahlung (KN 4.5)• Calculate energy emitted by a single electron with a velocity v deflected by

the electric field of a charge Z• Consider hot gas with electrons having a Maxwell-Boltzman velocity

distribution• Calculate the summed radiation for the ensemble• Result: energy emitted per unit volume of the gas per unit time (RL79)

– ne, ni electron and ion density– Z electronic charge– velocity averaged Gaunt factor

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RL p.161

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• For a hydrogen plasma, integrating over all frequencies

– Where is the frequency averaged Gaunt factor and in the

• Thermal energy gas per unit volume

• Cooling time

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In the regime where (T~6 109 or 500 keV)

Similar calculation as for thermal brehmsstrahlung:

• formula for emission of a single election

• distribution of particle velocities

• summed radiation of the particle ensemble using the available Gaunt factors

Gaunt factors and resulting expressions for the emissivity (K p. 205--206) are

available for relativistic bremsstrahlung due to:

• electron-electron

• positron-positron

• electron - positron

Note that the radiation from ions can be neglected due to their high mass

Relativistic or non-thermal bremsstrahlung

AGN-3: HR-2007 p. 10

Assignment - one page max !

1. Consider a (very unrealistic) region of gas associated with a cluster ofgalaxies with a diameter of 1 Mpc, a temperature of 107 K and adensity of 0.1 cm3. What would be the cooling time and total luminositydue to Bremsstrahlung?

2. If the cooling time is then compared to the age of the universe, what isthe conclusion?

3. How does the luminosity compare to luminous quasars?

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Synchrotron radiation

Radiation from relativistic particles moving in a B-fieldRL 6; KN 3

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Angular dependence ofradiation from a acceleratedcharged particle (RL3.3)

•Polar diagram is a dipole:

with d dipole moment•Emission is polarized withErad along the projectedacceleration vector

Radiation from a single accelerated charge

R0

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Non relativistic case:Lorentz Force equation

F=-e (v x B)

Relativistic case:

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Synchrotron radiation

Relativistic particle, moving at pitch angle θ to B-field: (RL $ 6; KN $ 3; K $ 9.2)- No radiation when θ=0- Strongly forward beamed when θ≠0- Polarized- Contribution by electrons dominates

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• Gyration frequency

• When γ ∼ 1 : cyclotron emission• When γ>1 : synchrotron emission

– Radiation is beamed in the velocity direction and within in a cone of with half-angle 1/γ

– The width of the pulse is given by the time taken by the cone to sweep acrossthe line of sight, which is for the highly relativistic case:

– Ensemble of pulses is quasi continuous and peaks at critical frequency νc

– With energy emitted per unit time:with E: energy electron

And the function F defined as

the modified Bessel function of order 5/3

(KN 3.13)

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• Total emitted power by one electron

– Note that P ∼ m-2 : hence radiation by protons can be neglected• Averaging over pitch angle α:

– With σT Thomson scattering cross section– UB = B2/(8π) energy density of magnetic field

(KN-3.10)

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Radio synchrotron spectrum from an ensemble of electrons

Power emitted by the electrons as a function of the frequency of the emittedradiation is given by:

Power law distribution for the number density of electrons as a function ofenergy can be produced in a variety of ways, including acceleration throughshocks

General result

In the case of extended, transparent radio sources the observed range ofspectral indices 0.5< α < 1, leads to 2 < p < 3.

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Radiation losses

A radiating electron loses energy at a rate

Time taken by the electron to loose half its energy:

so high energy electrons cool fastest

In practical units and using the expression for the critical frequency νc

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Polarization

B⊥ is magnetic fieldprojected on plane of skyP⊥ synchrotron power emittedperpendicular to B⊥

B|| ...

Synchrotron radiation of ensembleof particles with isotropic distributionof angles is linearly polarized

General result for degree of polarization

with a dimensionless function (e.g., KN $ 3.4; K eq. 9.16)

For power-law distribution

so degree of polarization can be larger than 50%

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Optically thick emission (self-absorption)

Photon can be absorbed by electrons in B-fieldIf source is optically thick: intensity = source function

This leads to (K 9.34)

K eq. (9.29) gives A(p) for isotropic pitch angle distribution and power-lawelectron distribution; see also KN $ 3.5.1

Relation between peak, magnetic field and flux (KN 3.56)

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Radio source energetics (KN3.6)• For a radio source of volume V, the total energy in

electrons is:

• Using KN-3.10 and β =1, the total synchrotronluminosity is:

• Using definition νc (KN 3.13), and p=2α +1

with A only a function of spectral α

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• The total particle energy Up = a Ue with a>1• The total energy of the source is sum of particle

and magnetic energy

• The total energy is minimized when

• defines the equipartition field

• The total energy for the equipartition value of themagnetic field

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• Total energy mildly dependent on– Uncertainty radio source size– Cutoff energies electron distributions– Energy in the different kind of galaxies

• For radio galaxies:– Utot in the range ∼ 1057 - 1061 with Beq ∼ 10-6- 10-4 G

• Total pressure

• Minimum pressure using the minimum energy condition

(Later: how the energy input of radio AGN impacts on theformation and evolution of massive galaxies)

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Assignment - one page max !

1. Consider a (very unrealistic) region of gas associated with a cluster ofgalaxies with a diameter of 1 Mpc, a temperature of 107 K and adensity of 0.1 cm3. What would be the cooling time and total luminositydue to Bremsstrahlung?

2. If the cooling time is then compared to the age of the universe, what isthe conclusion?

3. How does the luminosity compare to luminous quasars?

4. Derive

starting from eq 3.54 in KN

5. The spectra of Giga Hertz Peaked Spectrum (GPS) radio galaxiespeak at around 1 GHz. What would be a typical size for a GPS radiogalaxies if it is at z=0.5?

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Scattering of a photon by anelectron

1. low energies hν << mc2

Thomson scattering

2. high energies hν ~ mc2

Compton scattering.

3. in scattering process photongains energy

inverse Compton scattering

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Thomson scattering• Scattering of an electromagnetic wave incident

on an electron in the case hν << mc2 = 511 keV– applicable for optical photons

• Fully elastic: no change of photon energy• Total (Thomson) cross section

• Differential cross section for unpolarized light

• Resulting degree of polarization

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• In the scattering process energy is transferred from the photon to the electron• Compton scattering becomes important for X-rays, and dominates in gamma-

regime• A result (KN 4.2)

• In the limit ε < mc2

this is Thomson scattering

• Cross-section : Klein-Nishinaformula (KN eqs 4.6 & 4.7)

– Drops to zero at large energies

Compton scattering

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The variation of the Compton scattering cross-section with energy

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Inverse Compton scatteringWhen electron energy γmec2 large → momentum transfer to photon: inverse

Compton scattering

Photon may gain factor γ2 in energy (< γmec2)

General expression for the change of energy rather complex

The resulting spectrum depends on– luminosity and spectrum incoming radiation– energy distribution of relativistic electrons– number of scattering events– Energy balance between energy gained by the photons and lost by the electrons

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• n(ε): number density of photons of energy ε in laboratoryframe S

• n’(ε’): number density of photons of energy ε‘ in frame S’of electron

• Assuming γε << mc2, then σT can be used• Number of photons in energy range (ε’,ε’+δ ε’) that are

scattered is cσT n’(ε’)dε’• The power of the scattered photons in frame S’:

• In the lab frame (RL, p. 199):

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• For an isotropic distribution of photons, we can averageover incident angles:

With Uph the total energy density of the electromagnetic radiationNet effect scattered light minus energy incident on electron per unit

time: (use γ2-1 =β2 γ2)

Recall:

So that

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Repeated scatteringIn a thermal scattering medium with T << mc2/k (6 109 K) electrons are non-relativistic

and the average energy exchange per scattering between a photon and an electronis (RL79):

When energy flows from the thermal electrons to photons

When energy flows from the photons to the thermal electrons

At the componization temperature there is equilibrium

The Comptonization parameter y indicates whether a photon will significantly change itsenergy when crossing the medium (KN $ 4.3)

y = { average fractional energy change per scattering} × { mean number of scatterings}y > 1 spectrum altered

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Compton scattering of a photon in a mildly optically thick region.The photon begins at the central dot and executes a random walkuntil it reaches the edge of the cloud and escapes. A shorter and alonger random walk are shown.

{ mean number of scatterings} = ∼ τ2 (τ >> 1) = ∼ τ (τ << 1)

(RL. PAGE 36, 210, KN 4.3)

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Optical depth effects (KN, p. 73, RL p36-39)• For a homogeneous medium, the attenuation of the intensity I is:

– I(ν) = I(ν,0) exp(-τν), τν=nσν L• With L distance traveled

– Mean free path : lν = 1 / (n σν)• Number of scattering events before photons escape a medium with size L :

– Low optical depth• Nsc =1-exp(-τν) ∼ τn

– For high optical depth• Random walk• Nsc ∼ τν2

• Taken together– Nsc ∼ τν (1+τν)

• Time spend in a medium of size L– Low optical depth: tnsc = L/c– High optical depth: tsc = Nsc lν / c

• Ratio of the two times spend in the medium

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Assignment1. Derive that the number of scatterings before escape

Nsc is given by: Nsc ∼ τν2

(See KN 4.3 or RL 1.7)

2. Derive

Starting from

Following instruction on page 79 of KN

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Assuming

After dN scatterings:

Ratio change in energy to total energy

Integrating gives the for the energy of the photon after N scatterings

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Relativistic case (KN 4.3.2)• the average energy exchange per scattering between a photon and an

electron is:

• Resulting in a componization parameter:

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Figure illustrates strong comptonisation of abremsstrahlung spectrum in an optically thick,non-relativistic medium. The bremsstrahlungspectrum dominates at low frequency andshows a characteristic self-absorption regionand a flat region. At higher frequency, photonshave been multiply scattered via the Comptonprocess so that a Wien spectrum forms.

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Synchrotron self-Compton emission

Synchrotron photons can be Compton scattered by electrons that produce them

- This boosts photons with energy ε to γ2ε,- And electrons loose energy through

(i) Synchrotron emission(ii) Compton scattering

- Compton catastrophe:

In compact high luminosity sources, as a result of multiple up-scatteringthe electron will lose their energy very rapidly to high energy photonsproduces catastrophic energy losses of electrons, so that source isquenched

- This occurs at brightness temperatures Tb∼1012 K, and indeedno higher Tb observed

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Compton catastrophy• Energy density due to synchrotron emission of a source with radius r

and luminosity Ls

• The luminosity of such a source at a distance D, subtending an θ, isrelated to the flux F:

• Using KN 4-14 & UB = B2 / 8 π

Compton scattering dominates in compact sources with high surfacebrightness, and those are the sources that are synchrotron self-absorbed. Then:

where νa cutoff frequency

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• Rewrite, using– in terms of brightness temperature TB– Self-absorption frequency in terms of energy E=kT– Condition for self absorption T ~ TB

• When T > 1012, Lc dominates, and rapid Compton cooling sets in:the Compton catastrophe.

• Sources with intrinsic (=unbeamed) brightness temperaturesexceeding 1012 K not observed

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Assignment1. Derive that the number of scatterings before escpe Nsc

is given by: Nsc ∼ τν2

(See KN 4.3 or RL 1.7)

2. Derive

Starting from

Following instruction on page 79 of KN

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Pair production and annihilation (KN 4.6.2)

If electron have enough energy to produce X-rays, they have (almost)enough energy to make electron/positron pairs e±

Krolik $ 8.4 gives cross-sections for pair production- photon + photon → pair- electron + photon → electron + pair- nucleus + photon → nucleus + pair

Inverse process: pair annihilation -> two photons (with opposite spins)

Pairs complicate computation of equilibria- Pairs may escape easily → energy loss- Calculation non-linear

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Consider a region filled with energetic photons and relativisitic particles, then thesteady state equation (KN 4.6.4):

With n(x): number density of photons of energy x = ε / (me/c2)rate of production of soft photons (in for example accretion disk)rate of production due to pair annihilationrate of production due to Compton scattering of non-thermal and thermal electronsrate of removal by Compton scattering against non-thermal electrons with optical depth τC

NΤ

+ rate removal of photons due to photon-photon interactions/pair creation with optical depth τγ γrate of escape from region

N(γ): number density of relativistic particles with energy γrate of change due to non-thermal compton scatteringrate of pair creationrate of particle injection

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Example spectra produced bypair processing

Input BB UV photons kT = 5.1 eV relativistic electrons with

γ = 7.5 × 103

Compactness parameter l ∼ Luminosity / size measure optical depth γ’s

Broken line: spectrumproduced in the absence ofpairs

Solid line: modification due topair production with the netresult of an increase of X-rayenergy at the cost of of the γrays

(From Svensson 1994)

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Literature

• Kembhavi & Narlikar §3 and §4• Robson, §4• Krolik, §8.2-8.5,8.7,9.2• Radiative Processes in Astrophysics: Rybicki G., Lightman A.P., 1979