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Radiative Processes p. 1 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 AGN-3: HR-2007 p. 2 1 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. AGN-3: HR-2007 p. 3 AGN-3: HR-2007 p. 4 2 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 AGN-3: HR-2007 p. 5 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 AGN-3: HR-2007 p. 6 3 AGN-3: HR-2007 RL p. 161 p. 7 • 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 AGN-3: HR-2007 p. 8 4 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 AGN-3: HR-2007 p. 9 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? AGN-3: HR-2007 p. 10 5 Synchrotron radiation Radiation from relativistic particles moving in a B-field RL 6; KN 3 p. 11 AGN-3: HR-2007 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 AGN-3: HR-2007 p. 12 6 Non relativistic case: Lorentz Force equation Relativistic case: F=-e (v x B) AGN-3: HR-2007 p. 13 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 AGN-3: HR-2007 p. 14 7 • 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 AGN-3: HR-2007 p. 15 AGN-3: HR-2007 p. 16 8 • 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 AGN-3: HR-2007 p. 17 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. AGN-3: HR-2007 p. 18 9 p. 19 AGN-3: HR-2007 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 AGN-3: HR-2007 p. 20 10 p. 21 AGN-3: HR-2007 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% 11 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) AGN-3: HR-2007 p. 23 AGN-3: HR-2007 p. 24 12 AGN-3: HR-2007 p. 25 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 AGN-3: HR-2007 p. 26 13 • 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 p. 27 AGN-3: HR-2007 • 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) AGN-3: HR-2007 p. 28 14 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? p. 29 AGN-3: HR-2007 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 AGN-3: HR-2007 p. 30 15 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 p. 31 AGN-3: HR-2007 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 AGN-3: HR-2007 p. 32 16 The variation of the Compton scattering cross-section with energy AGN-3: HR-2007 p. 33 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 AGN-3: HR-2007 p. 34 17 • 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): AGN-3: HR-2007 p. 35 • 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 AGN-3: HR-2007 p. 36 18 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 p. 37 AGN-3: HR-2007 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) AGN-3: HR-2007 p. 38 19 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 p. 39 AGN-3: HR-2007 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 AGN-3: HR-2007 p. 40 20 Assuming After dN scatterings: Ratio change in energy to total energy Integrating gives the for the energy of the photon after N scatterings AGN-3: HR-2007 p. 41 Relativistic case (KN 4.3.2) • the average energy exchange per scattering between a photon and an electron is: • Resulting in a componization parameter: AGN-3: HR-2007 p. 42 21 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. AGN-3: HR-2007 p. 43 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 AGN-3: HR-2007 p. 44 22 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 AGN-3: HR-2007 • p. 45 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 AGN-3: HR-2007 p. 46 23 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 AGN-3: HR-2007 p. 47 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 AGN-3: HR-2007 p. 48 24 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 p. 49 AGN-3: HR-2007 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) AGN-3: HR-2007 p. 50 25 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 AGN-3: HR-2007 p. 51 26