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Thermal Bremsstrahlung Thermal Bremsstrahlung ©©Radiation due to the acceleration of a charge in the Coulomb field of another charge is called bremsstrahlung or free-free emission A full understanding of the process requires a quantum treatment, since photons of energies comparable to that of the emitting particle can be produced However, a classical treatment is justified in some regimes, and the formulas so obtained have the correct functional dependence for most of the physical parameters. Therefore, we first give a classical treatment and then state the quantum results as corrections (Gaunt factors) to the classical formulas©© George B. Rybicky & Alan P. Lightman in ©©Radiative processes in astrophysics©© p. 155 Processi di radiazione e MHD ± THERMAL BRESSTRAHLUNG ± Daniele Dallacasa Radiative processes: foreword Continuum from radiative processes from free-free transitions c. physics holds when λ √ smaller that the typical scale d dB= hmkT of the interaction and quantum mechanics (∆x∆p ≥ h) can be avoided h/mv << d h = Planck©s constant m = mass of the (lightest) particle v = velocity of the (lightest) particle d = characteristic scale (of the interaction or λ of the electromagnetic wave if it is involved) λ dB= de Broglie wavelength implications: particles like points --- ν Luis (7th duke of) de Broglie 1892 -1987 h << Eparticle (Nobel 1929) Bremsstrahlung Foreword Free electrons are accelerated (decelerated) in the Coulomb field of atomic (ionized) nuclei (free-free) and radiate energy Electrons and nuclei are in thermal equilibrium at the temperature T Electrons move at velocity v wrt the ions and during ©©collisions©© are deflected. The electrostatic interaction decelerates the electron which emits a photon bringing away part of its kinetic energy N.B. e and p have also homologous interactions (e with e; p with p) but they do not emit radiation since the electric dipole is zero. Astrophysical examples: 1. HII regions (104 oK) 2. Galactic hot-coronae (107 oK) 3. Intergalactic gas in clusters/groups (107-8 oK) 4. Accretion disks (around evolved stars & BH) Bremsstrahlung is the main cooling process for high T (> 107 K) plasma b Bremsstrahlung single event Radiated power (single particle) is given by the Larmor©s formula: v 2 dE 2 e 2 _ P = − = a e 3 b dt 3 c x acceleration is from Coulomb©s law Ze+ Ze 2 Ze 2 F = m a ≃ − a = x 2 m x 2 where x= actual distance between e- and Ze+ Remarks: − − 1. a ~ m 1 P ~ m 2 − − 2. a ~ x 2 P ~ x 4 Radiated power is relevant 1. for electrons (positrons) only i.e. light particles 2. when the two charges are at the closest distance (x ~ b) The interaction lasts very shortly: ∆t ~ 2b/v where v = velocity Bremsstrahlung Pulses and spectrum The total emitted energy in a single ªcollisionº (consider x ~ b) 2 2 e 2 Z e 2 2b 4 Z 2 e 6 1 P t = = x ≃b P(t) 3 c 3 m x 2 v 3 c 3m2 b 3 v Radiation comes in pulses ∆ t long: the profile is slightly asymmetric due to the decrease of the e± velocity t ν The Fourier analysis of the pulse provides the spectral P( ) distribution of the radiated energy: it is flat up to a frequency 1 v ≃ ≈ max 2 t 4b ν ν max related to the kinetic energy of the electrons (exponential decay beyond ν ) max Bremsstrahlung radiated energy P(ν) The radiated energy per unit frequency (assuming flat spectrum) from a single event can be written as ν ν max P t P t 16 Z 2 e 6 1 1 ≈ = 2P t 2 ≃ ≈ 3 2 2 2 2 2 max 3 c me b v b v Bremsstrahlung real plasma The real case: a cloud with ion number density nZ and free electron number density ne In a unit time, each electron experiences a number of collisions with an impact parameter between b and b + db is 2n Z v b db and the total number of collisions is 2ne n Z v b db Total emissivity from a cloud emitting via bremsstrahlung (single velocity!) b 2 6 dW max 16 Z e 1 J v , = = 2n n v b db = br e Z ∫b 3 2 2 2 d dV dt min 3 c me b v 6 6 b 32e 2 max db 32 e 1 2 b max = 3 2 ne n Z Z ∫ = 3 2 n e nZ Z ln bmin 3c me v b 3c me v b min Bremsstrahlung impact parameter Evaluation of : bmax at a given v, we must consider only those interactions where electrons have impact parameters corresponding to v > v and then max v b ≤ is a function of frequency max 4 Evaluation of : bmin a) classical physics requires that Dv ≤ v Ze2 2 b 2 Ze2 v ≃ a t = 2 ≤ v b min−c ≥ 2 me b v me v b) quantum physics and the Heisemberg principle DpDx ³ h/2p h h h p=me v ≈ me v ≥ ≈ b min−q ≥ 2 x 2bmin 2me v we must consider which has to be used bmin Bremsstrahlung classical/quantum limit which [classical/quantum] limit prevails: b h /2m v min−q e h v 137 v ≥ ≈ 2 2 = 2 c ≈ 1 when v 0.01c bmin−c 2 Z e /me v 4Z e c Z c but b min−q 137 3k 1/ 2 v = 3 k T / m e then ≈ T b Zc m min−c e Examples: HII region (T~104) Hot IGM cluster gas (T~108) Bremsstrahlung Gaunt factor The Gaunt Factor ranges from about 1 to about 10 in the radio domain Values around 10-15 in the radio domain, slightly higher than 1 at higher energies Thermal bremsstrahlung electron velocities / total emissivity A set of particles at thermal equilibrium follow Maxwell-Boltzmann distribution: 3/ 2 m −m v 2 /2 kT f v dv = 4 e e e v 2 dv 2k T Then is the number density of electrons whose velocity ne(v) = ne f(v) dv is between and and replaces in v v+dv ne(v) ne Jbr(n) When the plasma is at equilibrium, the process is termed thermal bremsstrahlung Total emissivity: must integrate over velocity (energy) distribution (0 £ v ?) dW ∞ J br , T = = ∫v J br ,v f v dv = dt dV d min −38 −1/ 2 −h /kT 2 −1 −3 −1 = 6.8⋅10 T e ne n Z Z g ff ,T erg s cm Hz T ~ particle energy high energy cut-off (T) bremsstrahlung ± interlude: photon frequency .vs. temperature Consider an electron moving at a speed of v =1000 km s-1. In case it would radiate all its kinetic energy in a single interaction 1 2 − 6 − 2 − E = h = m v = 0.5 ×9.1×10 31 kg ×10 m s 1 = 4.55 ×10 19 J 2 −19 2 −2 E 4.55 ×10 kg m s 14 the frequency of the emitted radiation is = = − − = 6.87 ×10 Hz h 6.626×10 34 kg m2 s 1 in case this electron belongs to a population of particles at thermal equilibrium for which − v = 1000 km s 1 is the typical speed, determine which is the temperature of the plasma −19 2 −2 E 4.55 ×10 kg m s 4 T = = − 2 − − = 3.30×10 K k 1.38×10 23 kg m s 2 K 1 k = 1.3806503 × 10 -23 m2 kg s -2 K -1 h = 6.626068 × 10 -34 m2 kg s -1 -31 me = 9.10938188 × 10 kg bremsstrahlung ± interlude (2): the cut-off frequency The cut-off frequency in the thermal bremsstrahlung spectrum depends on the temperature only, and is conventionally set when the exponential equals 1/e. −23 2 −2 −1 h k 1.38 ×10 kg m s K 10 = 1 = T = − 2 − × T = 2.08×10 ×T Hz kT h 6.626×10 34 kg m s 1 It is interesting to determine the cut-off frequency for a warm plasma (HII region, T ~ 10 4 K) and for a hot plasma (IGM in clusters of galaxies, T ~ 10 8 K). ⇨ In observations, the measure of the cut-off frequency is a way to determine the plasma temperature Thermal bremsstrahlung total emissivity(2) Total emissivity does not depend on frequency (except the cut ± off) and the spectrum is flat Total emitted energy per unit volume: must integrate over velocity (energy) distribution (0 £ v ?) at a given frequency v the velocity of the particle v must be as such 1 2h m v 2 ≥ h v ≥ 2 min m e and integrating over the whole spectrum: dW −27 1/2 2 − − J T = ≃ 2.4⋅10 T n n Z g T erg s 1 cm 3 br dt dV e Z ff g f f T average gaunt factor Thermal bremsstrahlung cooling time Let©s consider the total thermal energy of a plasma and the radiative losses: the cooling time is tot E th 3/2n en pkT t b r = = we can assume ne = n p J br T J br T 11 3 3ne kT 1.8⋅10 / 6⋅10 / ≃ = T 1 2 sec = T 1 2 yr −27 1/2 2 n g n g 2.4⋅10 T ne gff e ff e ff N.B. The ªaverageº Gaunt Factor is about a factor of a few Astrophysical examples: 2 3 −3 4 HII regions: ne ~ 10 −10 cm ; T ~ 10 K −3 −3 7 8 Galaxy clusters ne ~ 10 cm ; T ~ 10 −10 K (thermal) bremsstrahlung is the main cooling process at temperatures above T~ 107 K N.B.-2- all galaxy clusters have bremsstrahlung emission bremsstrahlung practical applications(1) A galactic HII region: M42 (Orion nebula) distance : 1,6 kly ~ 0.5 kpc angular size ~ 66 © ~ 1.1° compute: −16 −1 −11 h cutoff ~ kT = 1.38⋅10 erg K ×6.24⋅10 erg /eV T courtesy of NRAO/AUI − = 8.61⋅10 5 T eV 3 6⋅10 1/2 t br = T yr ne gff −27 1/2 2 −1 −3 j br T = 2.4⋅10 T n e gff erg s cm −1 Lbr = j br⋅V erg s bremsstrahlung practical applications(2) A cluster of galaxies: Coma (z=0.0232), size ~ 1 Mpc Typical values of IC hot gas have radiative cooling time exceeding 10 Gyr all galaxy clusters are X- ray emitters bremsstrahlung practical applications(3) Cluster of galaxies: a laboratory for plasma physics bremsstrahlung self-absorption (1) Thermal free-free absorption: energy gained by a free moving electron.
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