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Radiative Processes in High Energy Astrophysics

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Radiative Processes in High Energy Astrophysics RADIATIVE PROCESSES IN HIGH ENERGY ASTROPHYSICS Gabriele Ghisellini INAF – Osservatorio Astronomico di Brera February 28, 2012 arXiv:1202.5949v1 [astro-ph.HE] 22 Feb 2012 2 Contents 1 Some Fundamental definitions 7 1.1 Luminosity ............................ 7 1.2 Flux................................ 8 1.3 Intensity.............................. 8 1.4 Emissivity............................. 8 1.5 Radiativeenergydensity. 8 1.6 How to go from L to u ...................... 9 1.6.1 Weareoutsidethesource . 9 1.6.2 We are inside a uniformly emitting source . 9 1.6.3 We are inside a uniformly emitting shell . 10 1.7 How to go from I to F ...................... 10 1.7.1 Flux from a thick spherical source . 11 1.8 Radiative transport: basics . 11 1.9 Einsteincoefficients. 14 1.9.1 Emissivity, absorption and Einstein coefficients . 16 1.10 Meanfreepath .......................... 17 1.10.1 Randomwalk....................... 17 1.11 Thermalandnonthermalplasmas . 19 1.11.1 Energy exchange and thermal plasmas . 20 1.11.2 Non–thermal plasmas . 20 1.12 CoulombCollisions . 21 1.12.1 Crosssection . .. .. .. .. .. .. 22 1.13 The electric field of a moving charge . 24 1.13.1 Total emitted power: Larmor formula . 26 1.13.2 Pattern .......................... 27 2 Bremsstrahlung and black body 29 2.1 Bremsstrahlung.......................... 29 2.1.1 Free–free absorption . 31 2.1.2 From bremsstrahlung to black body . 32 2.2 Blackbody ............................ 34 3 4 CONTENTS 3 Beaming 37 3.1 Rulersandclocks......................... 37 3.2 Photographs and light curves . 38 3.2.1 Themovingbar ..................... 38 3.2.2 Themovingsquare. 39 3.2.3 Rotation, not contraction . 41 3.2.4 Time............................ 42 3.2.5 Aberration ........................ 43 3.2.6 Intensity.......................... 45 3.2.7 Luminosityandflux . 45 3.2.8 Emissivity......................... 46 3.2.9 Brightness Temperature . 48 3.2.10 Moving in an homogeneous radiation field . 49 3.3 Superluminalmotion . 51 3.4 Aquestion............................. 53 4 Synchrotron emission and absorption 55 4.1 Introduction............................ 55 4.2 Totallosses ............................ 55 4.2.1 Synchrotron cooling time . 59 4.3 Spectrum emitted by the single electron . 59 4.3.1 Basics ........................... 59 4.3.2 Therealstuff ....................... 60 4.3.3 Limits of validity . 63 4.3.4 From cyclotron to synchrotron emission . 63 4.4 Emission from many electrons . 65 4.5 Synchrotronabsorption: photons . 66 4.5.1 From thick to thin . 68 4.6 Synchrotron absorption: electrons . 69 4.7 Appendix:UsefulFormulae . 71 4.7.1 Emissivity......................... 71 4.7.2 Absorption coefficient . 72 4.7.3 Specific intensity . 73 4.7.4 Self–absorption frequency . 74 4.7.5 Synchrotronpeak. 74 5 Compton scattering 75 5.1 Introduction............................ 75 5.2 TheThomsoncrosssection . 75 5.2.1 Why the peanut shape? . 76 5.3 Direct Compton scattering . 78 5.4 The Klein–Nishina cross section . 79 5.4.1 Anotherlimit....................... 81 5.4.2 Pause ........................... 82 CONTENTS 5 5.5 Inverse Compton scattering . 82 5.5.1 Thomsonregime .. .. .. .. .. .. 83 5.5.2 Typical frequencies . 83 5.5.3 Cooling time and compactness . 88 5.5.4 Single particle spectrum . 89 5.6 Emission from many electrons . 90 5.6.1 Non monochromatic seed photons . 92 5.7 ThermalComptonization . 94 5.7.1 Average number of scatterings . 94 5.7.2 Average gain per scattering . 95 5.7.3 Comptonization spectra: basics . 97 6 Synchrotron Self–Compton 101 6.1 SSCemissivity ..........................101 6.2 Diagnostic.............................103 6.3 Whyitworks ...........................104 7 Pairs 107 7.1 Thermalpairs...........................108 7.1.1 First important result . 110 7.1.2 Second important result . 110 7.2 Non–thermalpairs . 113 7.2.1 Optical depth and compactness . 115 7.2.2 Photon–photon absorption . 115 7.2.3 An illustrative case: saturated pair production . 118 7.2.4 Importance in astrophysics . 121 8 Active Galactic Nuclei 123 8.1 Introduction............................ 123 8.2 Thediscovery...........................124 8.2.1 Marteen Schmidt discovers the redshift of 3C 273 . 124 8.3 Basiccomponents. 125 8.4 Thesupermassiveblackhole. 126 8.5 Accretion disk: Luminosities and spectra . 127 8.5.1 Spectrum .........................128 8.6 Emissionlines........................... 131 8.6.1 Broad emission lines . 131 8.6.2 Narrow emission lines . 133 8.7 TheX–raycorona ........................ 133 8.7.1 The power law: Thermal Comptonization . 134 8.7.2 Compton Reflection . 136 8.7.3 TheIronLine....................... 137 8.8 The torus and the Seyfert unification scheme . 140 8.8.1 TheX–raybackground. 142 6 CONTENTS 8.9 Thejet...............................144 8.9.1 Flat and steep radio AGNs . 144 8.9.2 Blazars ..........................146 8.9.3 Thepowerofthejet . 152 8.9.4 Lobes as energy reservoirs . 154 8.10 Openissues ............................156 Chapter 1 Some Fundamental definitions 1.1 Luminosity By luminosity we mean the quantity of energy irradiated per second [erg s−1]. The luminosity is not defined per unit of solid angle. The monochromatic luminosity L(ν) is the luminosity per unit of frequency ν (i.e. per Hz). The bolometric luminosity is integrated over fequency: ∞ L = L(ν) dν (1.1) Z0 Often we can define a luminosity integrated in a given energy (or frequency) range, as, for instance, the 2–10 keV luminosity. In general we have: ν2 L[ν1−ν2] = L(ν) dν (1.2) Zν1 Examples: Sun Luminosity: L = 4 1033 erg s−1 • ⊙ × Luminosity of a typical galaxy: L 1011L • gal ∼ ⊙ Luminosity of the human body, assuming that we emit as a black– • body at a temperature of (273+36) K and that our skin has a surface of approximately S =2m2: L = SσT 4 1010 erg/s 103 W (1.3) body ∼ ∼ This is not what we loose, since we absorb from the ambient a power L = SσT 4 8.3 109 erg/s if the ambient temperature is 20 C amb ∼ × (=273+20 K). 7 8 CHAPTER 1. SOME FUNDAMENTAL DEFINITIONS 1.2 Flux The flux [erg cm−2 s−1] is the energy passing a surface of 1 cm2 in one second. If a body emits a luminosity L and is located at a distance R, the flux is L L(ν) ∞ F = ; F (ν) = ; F = F (ν)dν (1.4) 4πR2 4πR2 Z0 1.3 Intensity The intensity I is the energy per unit time passing through a unit surface located perpendicularly to the arrival direction of photons, per unit of solid angle. The solid angle appears: [erg cm−2 s−1 sterad−1]. The monochro- matic intensity I(ν) has units [erg cm−2 s−1 Hz−1 sterad−1]. It always obeys the Lorentz transformation: I(ν) I′(ν′) = = invariant (1.5) ν3 (ν′)3 where primed and unprimed quantities refer to two different reference frames. The intensity does not depend upon distance. It is the measure of the irra- diated energy along a light ray. 1.4 Emissivity The emissivity j is the quantity of energy emitted by a unit volume, in one unit of time, for a unit solid angle erg j = (1.6) dVdtdΩ If the source is transparent, there is a simple relation between j and I: I = jR (optically thin source) (1.7) 1.5 Radiative energy density We can define it as the energy per unit volume produced by a luminous source, but we have to specify if it is per unit solid angle or not. For simplicity consider the bolometric intensity I. Along the light ray, construct the volume dV = cdtdA where dA (i.e. one cm2) is the base of the little cylinder of height cdt. The energy contained in this cylinder is dE = IcdtdAdΩ (1.8) 1.6. HOW TO GO FROM L TO U 9 In that cylinder I find the light coming from a given direction, I can also say that: dE = u(Ω)cdtdAdΩ (1.9) Therefore I u(Ω) = (1.10) c If I want the total u (i.e. summing the light coming from all directions) I must integrate over the entire solid angle. 1.6 How to go from L to u There at least 3 possible ways, according if we are outside the source, at a large distance with respect to the size of the source, or if we are inside a source, that in turn can emit uniformly or only in a shell. 1.6.1 We are outside the source Assume to be at a distance D from the source. Consider the shell of surface 4πD2 and height cdt. The volume of this shell is dV = 4πD2cdt. In the time dt the source has emitted an energy dE = Ldt. This very same energy is contained in the spherical shell. Therefore the energy density (integrated over the solid angle) is: dE L u = = (1.11) dV 4πD2c 1.6.2 We are inside a uniformly emitting source Assume that the source is optically thin and homogeneous. In this case the energy density will depend upon the mean escape time from the source. For a given fixed luminosity we have that the longer the escape time, the larger the energy density (more photons accumulate inside the source before escaping). Therefore: L u = t (1.12) V h esci For a source of size R one can think that the average escape time is t h esci∼ R/c, but we should consider that this is true for a photon created at the center. If the photon is created at the border of the sphere, it will have a probability 1/2 to escape immediately, and a probability less than 1/2 to ∼ pass the entire 2R diameter of the sphere. Therefore the mean escape time will be less that R/c. Indeed, for a sphere, it is t (3/4)R/c Therefore: h esci∼ 3L 3R 9 L u = = (1.13) 4πR3 4c 4 4πR2c 10 CHAPTER 1.
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