Emission mechanisms. I Emission mechanisms. I Giorgio Giorgio Matt Matt (Dipartimento di Fisica, Università à Roma Tre, Roma Tre, Italy Italy) ) (Dipartimento di Fisica, Universit Reference: Rybicki Rybicki & & Lightman Lightman, , “ “Radiative Radiative processes in astrophysics processes in astrophysics” ”, Wiley , Wiley Reference:
Outline of the lecture Outline of the lecture Basics (emission, absorption, Basics (emission, absorption, radiative radiative transfer) transfer) Bremsstrahlung Bremsstrahlung Synchrotron emission Synchrotron emission Compton Scattering (Inverse Compton) Compton Scattering (Inverse Compton)
Any charged particle in accelerated motion emits e.m. radiation. The intensity of the radiation is governed by Larmor ’ s formulae: where q is the electric charge, v dW 1 dv 2 2 2 q sin the particle = Θ 3 dtd dt 4 c Ω π velocity, _ the angle between the acceleration vector and the 2 dW 2 dv F direction of 2 2 P q = = ∝ emission 3 dt dt m 3 c ( Averaged over emission angles) The power is in general inversely proportional to the square of the mass of the emitting particle !! (dv/dt = F/m) Electrons emit much more than protons !
The previous formulae are valid in the non relativistic case. If the velocity of the emitting particle is relativistic, then the formula for the angle-averaged emission is: 2 2 2 dv dv 2 4 2 P q = γ + γ perp . parallel 3 dt dt 3 c where _ is the Lorenzt factor ( _=(1-v 2 /c 2 ) -1/2 ) of the emitting particle, and the acceleration vector is decomposed in the components parallel and perpendicular to the velocity. Of course, for _~1 the non relativistic formula is recovered.
The equation of radiative dE I Intensity ≡ transfer is: ν dAdtd d ν Ω dI F I cos d Flux I j ν = ϑ Ω ∫ = − α + ν ν ν ν ν ds dE ( ) j emissivity dI j ds or , with d ds ≡ = τ = α ν ν ν ν ν dVdtd d ν Ω ( optical depth ) τ ≡ dI 1 ν absorption coefficien t ν α = − v dI I ds I S ν ν = − + ν ν d τ ν j S ν ≡ ν α ν If matter is in local thermodynamic equilibrium, S _ is a universal 3 2 h ν function of temperature: S _ = B _ (T) B ( T ) = ν h ν (Kirchoff ’ s law). 2 c e 1 kT − B _ (T) is the Planck function:
Polarization Polarization The polarization vector (which is a pseudovector , i.e. modulus π ) rotates forming an ellipse. Polarization is described by the Stokes parameters : 2 2 I A B = + 2 2 Q ( A B ) cos 2 = − θ 2 2 U ( A B ) sin 2 = − θ V 2 AB = ± 2 2 2 ( I Q U V ) = + + If V=0, radiation is linearly polarized If Q=U=0, radiation is circularly polarized
Polarization Polarization Summing up the contributions of all photons, I increases while this is not necessarily so for the other Stokes parameters. Therefore: 2 2 2 I Q U V ≥ + + T T T T The net polarization degree and angle are given by: 2 2 2 Q U V + + T T T Π = I T U 1 T arctan χ = 2 Q T
Black Body emission Black Body emission If perfect thermal equilibrium between radiation and matter is reached throughout the material, I _ is independent of _. In this case the matter emits as a Black Body : 4 I B ( F T ) = = σ ν ν 2 2 kT ν hv kT I Rayleigh Jeans << = − v 2 c h 3 2 h ν ν − h kT I e Wien kT ν >> = v 2 c h 2 . 82 kT ν = max
Black body emission occurs when __ ∞ , so in practice there are always deviations from a pure Black Body spectrum due to finite opacities and surface layers effects. The only perfect Black Body is the Cosmic Microwave Background radiation.
Thomson scattering Thomson scattering It is the interaction between a photon and an electron (at rest), with h_«mc 2 . It is an elastic process. The cross section is: 2 8 e π 25 2 6 . 65 10 cm − σ = = × T 2 4 3 m c 4 d e The differential σ T = 2 sin θ cross section is: 2 4 d m c Ω The scattered radiation is polarized. E.g., the polarization degree of a parallel beam of unpolarized radiation is: 2 1 cos − θ P = 2 1 cos + θ
Pair production and annihilation Pair production and annihilation A e + -e - pair may annihilate producing two _-rays (to conserve momentum). If the electrons are not relativistic, the two photons have E=511 keV. Conversely, two _-rays (or a _-ray with the help of a nucleus) may produce a e + -e - pair Pair production is of course a threshold process. E _1 E _2 > 2m 2 c 4 The pair production cross sections are: e + e − γγ → σ ≈ σ T γγ 1 p e + e − p γ → σ ≈ ασ α ≈ p T γ 137 _-rays interacts with IR photons. The maximum distance at which an extragalactic _-ray source is observed provides an estimate of the (poorly known) cosmic IR background
Bremsstrahlung ( (“ “braking radiation braking radiation” ”) ) Bremsstrahlung a.k.a. free-free emission a.k.a. free-free emission It is produced by the deflection of a charged particle (usually an electron in astrophysical situations) in the Coulombian field of another charged particle (usually an atomic nucleus). Also referred to as free-free emission because the electron is free both before and after the deflection.
b is called the impact parameter 2 6 The interaction occurs on a dW 16 Z e timescale ≈ 3 2 2 2 d 3 c m v b ν _t ≈ 2b/v ( ) b v / << ω A Fourier analysis leads to the emitted energy per unit frequency in dW a single collision, which is inversely 0 ≈ proportional to the square of: the d ν mass, velocity of the deflected ( ) b v / >> ω particle (electron) and impact parameter.
Integrating over the impact parameter, we obtain: b max and b min must be 2 2 6 dW 32 Z e π evaluated taking into n n g = account quantum e i ff d dtdV 3 2 3 3 c m v ν mechanics. They are calculated where numerically. b 3 g ff is the so called max g ln = Gaunt factor . It is of ff b π order unity for large min intervals of the parameters. To get the final emissivity, we have to integrate over the velocity distribution of the electrons.
Thermal Bremsstrahlung Bremsstrahlung Thermal If electrons are in thermal equilibrium, their velocity distribution is Maxwellian. The bremsstrahlung emission thus becomes: 1 h 2 2 6 ν dW 64 Z e 2 π π − − T n n e g 2 kT = e i ff 3 2 d dtdV 3 c m 3 km ν Integrated over 1 2 2 6 dW 64 Z e 2 k π π T n n g 2 = frequencies e i ff 3 2 dtdV 3 hc m 3 m dW f ( T ) n n dV = ∫ Emission measure e i dt The above formulae are valid in the optically thin case. If _ >> 1, we of course have the Black Body emission
Free-free absorption Free-free absorption A photon can be absorbed by a free electron in the Coulombian field of an atom: it is the free-free absorption, which is the aborption mechanism corresponding to bremsstrahlung. Thus, for thermal electrons: dW ff ff j B ( T ) = = α ν ν ν 4 d dtdV π ν 1 h 2 6 8 Z e 2 ν π π − − ff 3 T n n − ( 1 e ) g 2 kT α = ν − e i ff ν 3 2 3 hc m 3 km At low frequencies matter in thermal equilibrium is optically thick to free-free aborption, becoming thin at high frequencies.
Polarization Polarization Bremsstrahlung photons are polarized with the electric vector perpendicular to the plane of interaction. In most astrophysical situations, and certainly in case of thermal bremsstrahlung, the planes of interaction are randomly distributed, resulting in null net polarization. For an anisotropic distribution of electrons, however, bremsstrahlung emission can be polarized.
Cooling time Cooling time For any emission mechanism, the cooling time is defined as: where E is the energy of the emitting E t cool particle and dE/dt the energy lost by = dE / dt radiation. For thermal bremsstrahlung: 1 3 6 x 10 t T yr 2 ≈ cool 2 n Z g e ff Radio image of the Orion Nebula X-ray emission of the Coma Cluster The cooling time is of order one thousand years for a HII regions, and of a few times 10 10 years (i.e. more than the age of the Universe) for a Cluster of galaxies
Synchrotron emission Synchrotron emission It is produced by the acceleration of a moving charged particle in a magnetic field due to the Lorentz force: r r r q F ( v B ) = × c The force is always perpendicular to the mcv sin particle velocity, so it does not do work. γ α r = Therefore, the particle moves in a helical B qB path with constant |v| (if energy losses by radiation are neglected). The radius v sin qB α of gyration and the frequency of the orbit ω = = B r mc are γ B (_ is the angle between v and B):
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