1 1. Compton Scattering and Comptonization1.1. Radiation Drag on a Charged ParticleWhen the Thomson scattering is a valid approximation, the photon transfers all its momentumto the electron but a negligible amount of energy. Hence this scattering must be accompanied bya force acting on the particle, which can also be viewed as follows. When an electromagneticwave hits a charged particle, it makes the particle oscillate and radiated.The radiation will leadto a damping force on the particle. This drag force can be obtained by averaging the force overthe period of the wave: h f i= (2/3)(q2/mc2)2hEi2 n=TU n, where U n is the ux of radiation.Note that the mean force is quadratic in the electric eld while the instantaneous force acting on acharge particle is linear in the electric eld.In the above discussion, we considered a single plane wave that scattered by a chargedparticle.A more complicated situation arises when a charged particle with velocity v is movingthrough a region of space containing an isotropic bath of radiation with energy density Urad. Thecharge particle will be constantly scattering electromagnetic waves that make up the radiationbath. Because the charge has a nonzero velocity v, this scattering will be anisotropic and hencethe momentum transfer to the charged particle will be in the direction opposite to the velocity.The reason for the drag force is easier to understand if we think of radiation as made of photons.Clearly, the charged particle will be hit by more photons in the front than in the back and thuswill experience a drag force. The radiation drag of a particle can be obtained directly as follows.Treat the radiation bath as equivalent to randomly uctuating E and B elds with hEi= hBi= 0;hE2i = hB2i = 4Urad. The mean power radiated by the charge is given by

dEdt!rad= Tc2

1 +v23c2!Urad. (1)Now this is the energy gained by the photon eld due to the scattering. We have therefore tosubtract the energy of these photons to nd the total energy gain. The rate at which energy is 2 removed from the photon eld and thus absorbed by the charge is (dE/dt)abs= cTUrad.Hencethe net energy transfer from the charge to the radiation isdEdt=43T2Uradvc2c. (2)Thus a charge particle, moving relativistically through a radiation bath , can transfer kinetic energyto the radiation. This process is called inverse Compton scattering. The above result uses theThomson scattering cross section in the rest frame of the charged particle. If the photon energy, inthe rest frame of the charge, is comparable with mc2, then its necessary to use the Klein-Nishinacross section.1.1.1. Compton ScatteringIn previous sections we have consider the scattering of a plane wave by a charged particle(which was originally at rest) and the interaction of a moving charged particle with a radiationbath. We next discuss the case of a collection of charged particles interacting with the radiationeld. It is now convenient and necessary to think of the radiation eld as made of a collection ofphotons. The interaction between the electrons and photons can cause net energy transfer to eitherparticle and we need to take into account dierent possibilities.Consider a plasma embedded in a radiation eld of temperature Trad. The scattering ofphotons by the electrons in the plasma will continuously transfer energy between the twocomponents. The high-energy photons with mev2~ mec2will transfer energy to thelow-energy electrons, but will gain energy from the high-energy electrons (with ~ mev2).Itis assumed that ~ mec2, so that the quantum electrodynamical eects, like pair production,are negligible. In thermal equilibrium, the net transfer of energy will be zero. But if the electrontemperature Te is very dierent from the photon temperature, there can be a net transfer of energy.When TeTrad, the electrons cool (on average) by transferring energy to photons. This process, 3 inverse Compton scattering, will cause the spectrum of photons to be distorted. On the other hand,if TradTe, the energy will be transferred (on the average) from the photons to the electronsand this process is called Compton scattering. In astrophysical applications, inverse Comptonscattering plays a more important role than Compton scattering, essentially because it can serve asa mechanism for generating high-energy photons.We start by considering the scattering of a photon by an electron that was originally at rest.Let the initial and nal four momenta of the photon be kai= (~i/c)[1, ni] and kaf= (~f/c)[1, nf],respectively and those of the electrons to be pai= (mc, 0) and paf= (E/cp), respectively. Theconservation of momentum and energy can be expressed by the equationpai+ kai=paf+ kaf.Squaring this equation and using the components to eliminate the nal electron momentum, wegetfi="1 +

~imec2!(1 cos )#1, (3)where cos = (ni nf). When ~ mec2, we can expand the expression in the bracket in a Taylorseries and obtainf ii=EE=

~imec2!(1 cos ). (4)To nd the mean energy transfer, this expression should be averaged over . In the rest frameof the electron, the scattering has front-back symmetry, making hcos i= 0. Hence the averageenergy lost by the photon per collision ishEi =

~imec2!~i. (5)Let us consider the average energy gained by the photon eld from the charged particle. We sawin 1.1 that the net addition of energy to the photon eld is given by (see equation [2]):dEdt=43T2Uradvc2c. (6)The mean number of photons scattered per second is Nc= Tcnrad= TcUrad/(~i), where ~i isthe average energy of the photon dened by ~i= (Urad/nrad). Hence the average energy gained 4 by the photon in one collision ishEi =PNc=432vc2~i=432vc2hEi. (7)In the relativistic limit, (E/E) (4/3)2 1, and this process can be a source of high-energyphotons (Figure 1). For example, if=103, this process can convert radio photons to UVphotons, far-infrared photons to x rays, and optical photons to gamma rays. It should, however, benoted that the scattering cross section T has to be modied when E mec2. The modied crosssection decreases for E> mec2and the inverse Compton process ceases to be eective.When v c, the energy gain by photons per collision is hE/Ei (4kBTe/mec2) asmv2 3kBTe. Combining this with [5] we nd that the mean fractional energy change of photonsper collision, is*EE+ = h~imec2+4kBTemec2=4kBTe hEimec2. (8)If 4kBTe> hEi, the net energy transfer is from electrons to photons (inverse Compton scattering),and if 4kBTe< hEi, the net energy transfer is from photons to electrons. We may say that, in atypical collision between an electron and a photon, the electron energy changes by E2/mec2andphoton energy changes by (4kBTe/mec2)E.The process described above acts as a major source of cooling for relativistic plasma as wellas a mechanism for producing high-energy photons. The time scale for Compton cooling of anindividual relativistic particle istcc mec2P 4 10312TR106K4s, (9)where TR is the radiation temperature. If electrons are nonrelativistic with temperature Te, thistimescale istcc 1nTc

mec2kBTR! = 1.3 103TR106K4s. (10)As the energy is progressively transferred from the electrons to the photons, through repeatedscattering, the mean energy will increase towards 4kBTe, when the net transfer will cease. 5 Fig. 1.Radiation spectrum emitted by mono energetic electrons inverse Comptonscatteringmonochromaticradiation. Intheultra-relativisticlimit, 1, themean scattered frequency is 420/3 and the peak of the scattered frequency is420, where 0 is the incident frequency. 6 When the electrons and photons coexist in a region of size L, the repeated scattering ofphotons by the electrons will distort the original spectrum of the photons. The mean free path ofthe photon that is due to Thomson scattering is l= (neT)1.If the size of the region L is suchthat (L/l)1, then the photon will undergo several collisions in this region; but if (L/l) 1then there will be few collisions. It is convenient to dene an optical depth e (L/l)= (neTL)so that e1 implies strong scattering.Ife 1, then the photon goes through Ns(1) collisions in traveling a distance L.from standard random-walk arguments, we have N1/2sl L so that Ns=(L/l)2=2e. Onthe other hand, if e 1, then Ns e; therefore an estimate for the number of scatterings isNs max(e, 2e).The average fractional change in the photon energy, per collision, is given by4(kBTe/mec2). Hence the condition for a signicant change of energy is1 = Ns

4kBTemec2! =

4kBTemec2!max(e, 2e). (11)Dening a parameter y (called the Compton y parameter) byy =kBTeNsmec2=

4kBTemec2!max(e, 2e), (12)the condition for signicant scattering becomes y 1/4.Because L=e/neT, the size of the region in which this process will be important isL = e/neT and the corresponding timescale is tc (L/c). Explicitlytc=Lc=(neTc)1(mec2/4kBTe) for e 1(neTc)1(mec2/4kBTe)1/2for e1(13)A more precise condition for repeated scattering to change the spectrum of the radiation eld canbe obtained as follows. The change in the energy of a typical photon after a single scattering isgiven by the factor (E0/E) = (1 + 4kBTe/mec2), with kBT mec2. After Ns scatterings, the energychange is given by the factorE0E=

1 +4kBTemec2!Ns exp

4kBTeNsmec2! = exp(4y). (14) 7 Fig. 2.The Comptonization of low frequency photons in a spherical plasma cloudhaving kBTe= 25keV. the solid curves are analytical solutions to the Kompaneetsequation; the results of Monte Carlo simulations of the Compton scattering processare shown by the histograms. 8 Suppose that the initial frequency of the radiation eld is ~i kBTe. The energy gain by thephotons (usually called Comptonization) goes on till the mean energy of the photons raises to4kBTe. The critical optical depth needed for this is determined byE0E=

4kBTe~i! = exp"4

kBTemec2!2crit#, (15)givingcrit=" mec24kBTe!ln(4kBTe~i)#1/2. (16)When the optical depth of the region is comparable with crit, the spectrum of the photons willevolve because of repeated scattering. Such an evolution is described by the Kompaneets equation.Figure 2 show how the spectrum changes as y(e) increases in which it is assumed that the inputphotons are very low energy and the electrons have a temperature kBTe= 25keV. The spectrumevolves so that for large optical depths the beginnings of the Wien peak can be observed. Atsmaller optical depths, the spectrum mimics very closely a power-law spectrum up to energies kBTe.Finally, note that when a 10 keV photons scatters a free electron, the recoil energy is 200eV, which is much larger than the typical ionization potential of valence electrons in atoms.Therefore even bound electrons (expect the inner shell electrons of high-Z atoms) behave asthough they are free in hard-x-ray scattering.