Comptonization RL 7.4, 7.5, 7.6, 7.7
Photons scatter off (relativistic) electrons and gain energy hence electrons cool (Inverse Compton cooling, or ICC) We discussed spectra for: Inverse Compton recap / / 0 Single electron energy, single photon energy : 2 1 2 [ 0 4 2, 4 2 ], < 1! I /, > 1! I / 0 0 0 0 Power law electron energies c ( ) / U r 0 / p (p 1)/2 with U r 0, single photon energy: the photon energy density Power law electron energies, arbitrary input spectrum: same scaling, except when the integration limits of U r depend on the electron distribution too
Comptonization Comptonization is the process of taking an input ( seed ) spectrum and (inverse)-compton-scattering it into a new spectrum The importance of this process is measured by the Compton parameter (in finite media):! mean number average fractional energy y = of scatterings change per scattering! y If y>1 then Comptonization generally changes the spectrum Let s think about the first part first; consider the optical depth for electron scattering (Thomson optical depth) for a constant density electron cloud with size : Mean free path: R T = n e l =(n e T ) 1 T R
Number of scatterings T 1 Say the medium is optically thin, value of the number of scatterings is: N T ; then the expectation If the medium is optically thick, the photon takes a random walk, and so scatters around until it reaches the region s edge: d 2 p 1/2 p N = Nl = = p N R n e T T d 2 1/2 = R ) N = 2 T y! mean number So we can write the first part of of scatterings as: = max( T, 2 T)
Fractional energy change Now for the second part, the average fractional energy change We already saw its value for a relativistic electron: Now let s consider this more generally; assume a thermal distribution of electrons 4 3 2 1 4 3 2 Non-relativistic means Maxwell-Boltzmann: f(v) = Relativistic means Maxwell-Jüttner: f( )= me 2 k B T 2 K 2 (1/ ) exp 3/2 4 v 2 exp me v 2 with 2k B T k BT m e c 2 So: f NR / 2 exp( 1 2 2 / ) and f R / 2 exp( / )
Fractional energy change relativistic Now that we have a distribution function, we can calculate an 2 expectation value for : R 2 1 R 2 1 f = 1 R ( )d 4 exp( / )d 1 = R 1 1 f R ( )d R 1 1 2 exp( / )d 12 2 So the average fractional energy change is: x x 0 = 4 3 2 1 16 kb T m e c 2 2 (remember: x h /(m e c 2 ))
Fractional energy change non-relativistic You may recall the average kinetic energy of a thermal electron: R 1 1 0 2 m ev 2 f NR (v)dv he k i = R 1 0 f NR (v)dv Some electrons will have a lower energy than the incoming photons and downscatter them, while those with a higher energy scatter them up thermalized electron-photon equilibrium = 3 2 k BT Some unknown parameter determines the balance (fraction of the electron kinetic energy taken by the photon): x x 0 = he ki m e c 2 x 0 = 3 2 x 0 We can find by considering the equilibrium conditions
Fractional energy change non-relativistic When thermal electrons only interact with photons through scattering, in equilibrium the photons will follow: f(x) / x 2 exp( Also in equilibrium, we should have x/ ) h xi =0, so: h xi = 3 2 hxi x 2 =0 ) 3 x 2 2 = We calculate the expectation values using f(x) hxi =3 x 2 = 12 2 ) ) 3 2 =4 (Wien distribution, same scaling as relativistic electrons!) hxi just like before: So: x =4 x 0 = 4k BT h 0 x 0 m e c 2
We can now write down an expression for the Compton :! mean number y = of scatterings change per scattering y NR = 4k BT 0 ) y = hn sc i Assuming Compton y parameter 0 4k B T x x 0 ) 8 >< >: Per scattering the energy gain is y! average fractional energy m e c 2 max( T, T) 2 2 kb T y R = 16 m e c 2 max( T, T) 2 (so no downscattering) we can write: y [16 2 +4 ]( T + 2 T) [16 2 +4 ] 0 d dn = [16 2 +4 ] 0 ) = 0 e [16 = 0 e y, so differentially:
Thermal Compton low optical depth For a power-law electron distribution we got a power-law spectrum; what about for a thermal cloud of electrons? k k A Let s call the photon energy after scatterings and the amplification after one scattering; if the optical depth is low, then: A 1 0 y T +1 Assuming that the starting photon energy is low enough compared to the electrons (i.e. 0 2 1/2 me c 2 ), we can write: k = A k 0 e T 1 T The cloud is optically thin, so a fraction escapes without scattering; a fraction scatters only once, etc so: T p( k ) k T (k>0)
Thermal Compton low optical depth T 2 T 3 T 4 T 5 T 6 T 7 T 8 T power law!
Thermal Compton low optical depth For a power-law electron distribution we got a power-law spectrum; what about for a thermal cloud of electrons? Answer: also a power law, through repeated scatterings! I( k ) / I( 0 ) k T Every scattering moves the peak down with another factor, and to the right with another factor A, so: ln T a = ln T ln A = 0 ln T a ln y Note that the bumpiness of the spectrum, caused by the scatterings being discrete, depends on the values of and T A
Thermal Compton low optical depth A# A"
Thermal Compton low optical depth T # T "
Medium optical depth T & 1 For a (slightly) opaque medium,, the photons are diffusing through the electron cloud, and we have to consider more complicated thermodynamics (and quantum effects) The result is the Kompaneets equation (assuming NR electrons): kb T 1 @n @t K = m e c 2 x 2 @ @x apple x 4 @n @x + n + n2 n t K =(n e T c)t x h /k B T is the phase-space density of the photons, is time in units of mean time between scatterings, and For mild optical depths, this can still result in a power law spectrum, now with: a = 3 2 ± r 9 4 + 4 y
High optical depth T 1 For a high opacities,, things become a bit easier again: the photons and electrons reach an equilibrium, and the photon spectrum becomes a Wien spectrum (see earlier): I( ) / 3 e h /k BT (To be more precise, the photons distribution becomes Bose- Einstein with a non-zero chemical potential) Note that this spectrum is similar to a blackbody, but harder (i.e. steeper at low frequencies) Since the electrons and photons reach an equilibrium, we call this saturated Comptonization
Quasi-saturation Interesting things happen when we consider a scenario in which the seed photons are produced throughout an opaque cloud Almost all near the center will scatter their way to equilibrium, but some near the edge will exit the cloud before that point 1/ T 1/ T 1/ T A fraction will escape without scattering; another fraction escapes after one scattering event; after two, etc. Most photons take a very long time to escape, but after they reach equilibrium they no longer change their spectrum so most exit with the Wien spectrum at some fixed peak energy! Quasi-saturated spectrum: significant part of the spectrum is unsaturated, but for high T most of the energy will be in the saturated part
Quasi-saturation / 3 exp[ /(k B T )] / 0 / 2
Quasi-saturation T # T "
Imagine a magnetized region with relativistic electrons: these will produce synchrotron radiation, but on their way out the photons may interact with the electrons and (inverse) Compton scatter The electrons both produce and scatter the radiation, so we call this synchrotron self-compton (SSC) From last week: for a power-law electron distribution, the synchrotron and IC scattered spectra have the same slope Specifically, if Synchrotron self-compton j syn ( 0 )=j syn,0 0 j ssc ( ) = (4/3) 1 2 then: / n e TKRj syn ( )ln( max / min ) T 1 T ln j syn ( ) =(p 1)/2
Synchrotron self-compton T ln
Synchrotron self-compton Crab Nebula Next week: atomic lines