Bremsstrahlung Rybicki & Lightman Chapter 5 Bremsstrahlung “Free-free Emission” “Braking” Radiation Radiation due to acceleration of charged particle by the Coulomb field of another charge. Relevant for (i) Collisions between unlike particles: changing dipole emission e-e-, p-p interactions have no net dipole moment (ii) e- - ions dominate: acc(e-) > acc(ions) because m(e-) << m(ions) recall P~m-2 ion-ion brems is negligible Method of Attack: (1) emission from single epick rest frame of ion calculate dipole radiation correct for quantum effects (Gaunt factor) (2) Emission from collection of e thermal bremsstrahlung or non-thermal bremsstrahlung (3) Relativistic bremsstrahlung (Virtual Quanta) A qualitative picture Emission from Single-Speed Electrons e- b v R Electron moves past ion, assumed to be stationary. b= “impact parameter” Ze ion - Suppose the deviation of the e- path is negligible small-angle scattering The dipole moment the encounter. d e Ris a function of time during - Recall that for dipole radiation where ˆ() d 2 energy dW 8 4 ˆ d ( ) 3 frequency d 3c is the Fourier Transform of d After some straight-forward algebra, (R&L pp. 156 – 157), one can derive dW(b) d in terms of impact parameter, b. Now, suppose you have a bunch of electrons, all with the same speed, v, which interact with a bunch of ions. Let ni = ion density (# ions/vol.) ne = electron density (# electrons / vol) The # of electrons incident on one ion is n # v electro area/ e # e-s /Vol d/t around one ion, in terms of b Area 2 b db So total emission/time/Vol/freq is dW dW(b) ne ni 2v bdb d dV dt d Again, evaluating the integral is discussed in detail in R&L p. 157-158. We quote the result Energy per volume per frequency per time due to bremsstrahlung for electrons, all with same velocity v. dW 16e 6 1 2 n n Z g ff (v, ) e i 3 2 ddVdt 3 3c m v ne elec trons/ vol. ni ions/vol Z#ofc harges inion gff (v,)Gaunt factor Gaunt factors are quantum mechanical corrections function of e- energy, frequency Gaunt factors are tabulated (more later) Naturally, in most situations, you never have electrons with just one velocity v. Maxwell-Boltzmann Distribution Thermal Bremsstrahlung Average the single speed expression for dW/dwdtdV over the Maxwell-Boltzmann distribution with temperature T: dW ( v , )2 m v v exp d v d dVdt 2 kT dW 2 dVdtd m v 2 d v vexp kT 2 2 The result, with d 2 d dW 2 e 2 T Z n n eg 3 dVdtd 3 mc 3 km 5 6 where 1 / 2 1 / 2 2 h / kT e i ff g velo ver Ga fa ff In cgs units, we can write the emission coefficient dW 38 2 1 / 2 h / kT 6 . 8 1 Z n 0 n T e g d Vd e i ff td ff Free-free emission coefficient ergs /s /cm3 /Hz Integrate over frequency: dW 2 e 2kT 2 Z n e n i gB 3 dVdt 3hm c 3km 5 where In cgs: 6 1/ 2 gB frequency average of t he velocity averaged Gaunt factor ff dW 27 2 1/ 2 1.4 10 Z ne niT gB dVdt Ergs sec-1 cm-3 g T , ) ff( The Gaunt factors - Analytical approximations exist to evaluate them - Tables exist you can look up - For most situations, h g ~ 1 for 10 1 ff kT 4 so just take g 1 .2 ff Handy table, from Tucker: Radiation Processes in Astrophysics ff Important Characteristics of Thermal Bremsstrahlung Emissivity (1) Usually optically thin. Then ff (2) ff is ~ constant with hν at low frequencies h kT (3) ff falls of exponentially at h~ kT Examples: Important in hot plasmas where the gas is mostly ionized, so that bound-free emission can be neglected. T (oK) Obs. of ff Solar flare 107 (~ 1keV) radio flat X-ray exponential H II region 105 radio flat Orion 104 radio-flat Sco X-1 108 optical-flat X-ray flat/exp. Coma Cluster ICM 108 X-ray flat/exp. Bremsstrahlung (free-free) absorption Brems emission photon e- eion photon e- collateral Inverse Bremss. free-free abs. Recall the emission coefficient, jν, is related to the absorption coefficient αν for a thermal gas: ff j B ( T ) is isotropic, so 4 j ff ff and thus 6 2 4e ff 2 1/ 2 3 h / kT g ff ne ni Z T 1 e 3m hc3km 1/ 2 in cgs: ff 3.7 108 ne ni Z 2T1/ 2 3 1 eh / kT g ff Important Characteristics of (1) ff h kT(e.g. X-rays) 8 2 1 / 2 3 e i ff 3 . 7 1 n n Z 0 T g ff Because of T term, is very small unless ne is very large. 1 /2 3 ff in X-rays, thermal bremsstrahlung emission can be treated as optically thin (except in stellar interiors) (2) h kT e.g. Radio: Rayleigh Jeans holds 2 3 / 2 2 e i ff 0 . 018 n n Z T g ff Absorption can be important, even for low ne in the radio regime. From Bradt’s book: BB spectrum is optically thick limit of Thermal Bremss. HII Regions, showing free-free absorption in their radio spectra: R&L Problem 5.2 Spherical source of X-rays, radius R distance L=10 kpc flux F= 10 -8 erg cm-2 s-1 (a) What is T? Assume optically thin, thermal bremsstrahlung. Turn-over in the spectrum at log hν (keV) ~ 2 E 9o T max 10 K k (b) Assume the cloud is in hydrostatic equilibrium around a central mass, M. Find M, and the density of the cloud, ρ 3 14 R ff F 2 4 L3 Vol. emission coeff. 1/r2 Vol. 1 4 R 3 27 1/ 2 2 F 1.4 10 T n n Z gB e i 2 4 L 3 - Since T=109 K, the gas is completely ionized - Assume it is pure hydrogen, so ni = ne, then n n in e 2 H ρ=mass density, g/cm3 47 2 3 . 6 10 m H 2 Z=1 since pure hydrogen gB 1.2 20 2 1 / 2 3 2 (1) F 2 . 0 10 T R L - Hydrostatic equilibrium another constraint upon ρ, R Virial Theorem: For T=109 K K.E. grav.energy 2 particle particle GMm H 3 kT R M (2) R 5 10 cm M s un 8 3 / 2 - Eqn (1) & (2) M 4 10 LF M sun 261 / 2 Substituting L=10 kpc, F=10-8 erg cm-2 s-1 3 / 2 M 4 10 g cm M sun 7 3