Download Radiative Processes Overview

Radiative Processes
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AGN-3: HR-2007
Overview
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 temperatures
that emit (modified) black body radiation
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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 the
higher 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 a
total 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 spectra
Radiation by charge accelerating in field of other charge
Dominant 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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Relativistic or non-thermal bremsstrahlung
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
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Assignment - one page max !
1. Consider a (very unrealistic) region of gas associated with a cluster of
galaxies with a diameter of 1 Mpc, a temperature of 107 K and a
density of 0.1 cm3. What would be the cooling time and total luminosity
due to Bremsstrahlung?
2. If the cooling time is then compared to the age of the universe, what is
the conclusion?
3. How does the luminosity compare to luminous quasars?
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Synchrotron radiation
Radiation from relativistic particles moving in a B-field
RL 6; KN 3
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Radiation from a single accelerated charge
R0
Angular dependence of
radiation from a accelerated
charged particle (RL3.3)
•Polar diagram is a dipole:
with d dipole moment
•Emission is polarized with
Erad along the projected
acceleration vector
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Non relativistic case:
Lorentz Force equation
Relativistic case:
F=-e (v x B)
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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 halfangle 1/γ
– The width of the pulse is given by the time taken by the cone to sweep across
the line of sight, which is for the highly relativistic case:
– Ensemble of pulses is quasi continuous and peaks at critical frequency νc
(KN 3.13)
– With energy emitted per unit time:
with E: energy electron
And the function F defined as
the modified Bessel function of order 5/3
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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:
(KN-3.10)
– With σT Thomson scattering cross section
– UB = B2/(8π) energy density of magnetic field
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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 emitted
radiation is given by:
Power law distribution for the number density of electrons as a function of
energy can be produced in a variety of ways, including acceleration through
shocks
General result
In the case of extended, transparent radio sources the observed range of
spectral 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 field
projected on plane of sky
P⊥ synchrotron power emitted
perpendicular to B⊥
B|| ...
Synchrotron radiation of ensemble
of particles with isotropic distribution
of 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-field
If 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-law
electron 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 synchrotron
luminosity 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 the
magnetic 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:
-6
-4
– Utot in the range ∼ 1057 - 1061 with Beq ∼ 10 - 10 G
• Total pressure
• Minimum pressure using the minimum energy condition
(Later: how the energy input of radio AGN impacts on the
formation 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 of
galaxies with a diameter of 1 Mpc, a temperature of 107 K and a
density of 0.1 cm3. What would be the cooling time and total luminosity
due to Bremsstrahlung?
2. If the cooling time is then compared to the age of the universe, what is
the 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 galaxies
peak at around 1 GHz. What would be a typical size for a GPS radio
galaxies if it is at z=0.5?
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Scattering of a photon by an
electron
1. low energies hν << mc2
Thomson scattering
2. high energies hν ~ mc2
Compton scattering.
3. in scattering process photon
gains 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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Compton scattering
•
•
•
In the scattering process energy is transferred from the photon to the electron
Compton scattering becomes important for X-rays, and dominates in gammaregime
A result (KN 4.2)
•
In the limit ε < mc2
this is Thomson scattering
•
Cross-section : Klein-Nishina
formula (KN eqs 4.6 & 4.7)
– Drops to zero at large energies
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The variation of the Compton scattering cross-section with energy
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Inverse Compton scattering
When 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 laboratory
frame 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 average
over incident angles:
With Uph the total energy density of the electromagnetic radiation
Net effect scattered light minus energy incident on electron per unit
time: (use γ2-1 =β2 γ2)
Recall:
So that
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Repeated scattering
In 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 electron
is (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 its
energy 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 walk
until it reaches the edge of the cloud and escapes. A shorter and a
longer 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
•
Time spend in a medium of size L
– Nsc ∼ τν (1+τν)
•
– 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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Assignment
1. 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 a
bremsstrahlung spectrum in an optically thick,
non-relativistic medium. The bremsstrahlung
spectrum dominates at low frequency and
shows a characteristic self-absorption region
and a flat region. At higher frequency, photons
have been multiply scattered via the Compton
process 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-scattering
the electron will lose their energy very rapidly to high energy photons
produces catastrophic energy losses of electrons, so that source is
quenched
- This occurs at brightness temperatures Tb∼1012 K, and indeed
no 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, is
related to the flux F:
•
Using KN 4-14 & UB = B2 / 8 π
Compton scattering dominates in compact sources with high surface
brightness, and those are the sources that are synchrotron selfabsorbed. 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 temperatures
exceeding 1012 K not observed
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Assignment
1. 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 the
steady state equation (KN 4.6.4):
With n(x):
N(γ):
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 annihilation
rate of production due to Compton scattering of nonthermal and thermal electrons
rate of removal by Compton scattering against nonthermal electrons with optical depth τCNΤ
+ rate removal of photons due to photon-photon
interactions/pair creation with optical depth τγ γ
rate of escape from region
number density of relativistic particles with energy γ
rate of change due to non-thermal compton scattering
rate of pair creation
rate of particle injection
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Example spectra produced by
pair 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: spectrum
produced in the absence of
pairs
Solid line: modification due to
pair production with the net
result of an increase of X-ray
energy at the cost of of the γ
rays
(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
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