Compton Scattering in Astrophysics Sampoorna. M. JAP Student 2003, Indian Institute of Science. December 12, 2003 1 Contents 1 Introduction 4 2 Comptonization 6 3 Energy transfer - scattering from electrons in motion 8 4 Inverse Compton power for single scattering 9 5 Compton y-parameter 12 6 Kompaneets Equation 14 7 Sunyaev - Zeldovich Effect (CMB Comptonization) 19 7.1 Theory . 20 7.2 Cluster Comptonization as a Probe . 23 2 Abstract Compton Scattering, a scattering phenomenon between the pho- ton and a charged particle such as an electron that causes momentum exchange between the photon and the electron unlike Thomson scat- tering, is discussed. In astrophysical applications it is the inverse Compton scattering that plays an important role than the Compton scattering itself. The change in the spectrum of the incident radiation caused by the multiple Compton scattering with an thermal distribu- tion of electrons, called Comptonization is discussed. The Kompaneets equation that describes the comptonization is derived and solution for a particularly simple case is discussed as an illustration. The well know Sunyaev - Zeldovich Effect (CMB Comptonization) and its ap- plications are briefly discussed. 3 ν ν θ P , E Figure 1: Figure shows the scattering of photon by electron at rest. 1 Introduction Compton scattering is an scattering event that causes momentum exchange between a photon and a charged scatterer such as an electron. For low 2 photon energies, hν mec (where me is the rest mass of the electron), the scattering of radiation from free charges reduces to the classical case of Thomson scattering. Quantum effects appears through the kinematics of the scattering process. The kinematic effects occur because a photon possesses hν a momentum c as well as an energy hν. The scattering will no longer be elastic because of the recoil of the charge. The energy of the scattered photon is determined by setting up the mo- mentum and energy conservation relations in the rest frame of the electron. Let the initial and final four-momentum of the photon be Pγi = ( c ) (1; ni) 0 and Pγf = ( c ) (1; nf ), where ni and nf are the initial and final directions of the photon (see Fig. (1)). And let the initial and final momenta of the E electron be Pei = (mec; 0) and Pef = ( c ; P). Conservation of momentum and energy is expressed by, Pei + Pγi = Pef + Pγf : (1) Rearranging terms and squaring gives, 2 2 jPef j = jPei + Pγi − Pγf j : (2) 4 Solving the above, we finally obtain 0 = : (3) 1 + 2 (1 − cosθ) mec In terms of wavelength, the above equation can be written as, 0 λ − λ = λc (1 − cosθ); (4) where the Compton wavelength is defined to be λ = h = 0:02426 A˚ for c mec electron. Observe that there is a wavelength change of the order of λc upon 2 scattering. For long wavelengths λ λc (i.e., hν mec ) the scattering is closely elastic. When this condition is satisfied, we can assume that there is no change in photon energy in the rest frame of the electron. From the Eq. (3) we can calculate mean energy transfer as follows: − !0 h!¯ 1 = 1 + 2 (1 − cosθ) : (5) ! " mec # 2 For h!¯ mec , we can expand the expression in the bracket of the Eq. (5) binomially and obtain, ∆Eγ h!¯ = − 2 (1 − cosθ): (6) Eγ mec In order to find the mean energy transfer, one has to average the Eq. (6) over θ. In the rest frame of the electron, the scattering has front-back symmetry, making h cosθ i = 0. Hence the average energy lost by the photon per collision is, 2 h h!¯ i h Eγ i h ∆Eγ i = − 2 h Eγ i = − 2 : (7) mec mec The differential cross-section for Compton scattering is given by Klein- Nishina formula, 2 02 0 dσ r0 2 = 0 + − sin θ ; (8) dΩ 2 2 ! 5 0 where r0 is classical electron radius. Note that for = , the above equation reduces to the classical expression, given by dσ 1 = r 2 1 + cos2θ : (9) dΩ 2 0 The total cross-section is given by, 3 1 + x 2x (1 + x) σ = σT − ln(1 + 2x) (10) 4 ( x3 " 1 + 2x # ) 3 1 1 + 3x + σT ln(1 + 2x) − 2 ; 4 ( 2x (1 + 2x) ) hν where x = 2 . From the above expression it is clear that, the principal mec effect is to reduce the cross-section from its classical value as the photon energy becomes large. In the non-relativistic regime (i.e., for x 1), the Eq. (10) takes the form, 26x2 σ ≈ σT 1 − 2x + + · · · ; (11) 5 ! whereas for the extreme relativistic regime (i.e., for x 1), the Eq. (10) takes the form, 3 −1 1 σ ≈ σT x ln2x + : (12) 8 2 For low x, the cross-section approaches σT and the change in energy of the photon is also very small. As x increases, the energy transfer becomes larger, and the cross-section drops. This fact is clearly shown in the Fig. (2). 2 Comptonization Consider a plasma embedded in a radiation field of temperature Trad. The scattering of photons by the electrons in the plasma will continuously trans- fer the energy between the two components. The high energy photons with 2 2 mev h¯! mec will transfer the energy to the low energy electrons, 2 but will gain energy from the high energy electrons (with h!¯ mev ). In thermal equilibrium, the net transfer of the energy will be zero. But if the 6 0.7 Thomson 0.6 0.5 0.4 0.3 Cross−section (in barns) Klein−Nishina 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10 x Figure 2: Figure shows the plot of total cross-section for Compton scattering as a function of x temperature of the electron Te is very different from the photon temperature, there can be a net transfer of energy. When Te Trad, the electrons cool (on the average) by transferring energy to photons. This process is called inverse Compton scattering. On the other hand, if Trad Te, the energy will be transfered (on the average) from the photons to the electrons, due to Compton scattering. In astrophysical applications, inverse Compton scatter- ing plays a more important role than Compton scattering and it can serve as a mechanism for generating high energy photons. In astrophysical situations one often encounters multiple Compton scat- tering. Due to multiple Compton scattering, the spectrum of the photons or radiation incident on the plasma will be distorted. This change in the spec- trum of radiation due to multiple scattering of photons by thermal electrons is called Comptonization. Before dealing with comptonization, let us consider the energy transfer due to single scattering. 7 ε ε 1 1 ε θ θ θ 1 1 ε K − Frame K − Frame Figure 3: Scattering geometry in lab and rest frame of the electron. 3 Energy transfer - scattering from electrons in motion Let K be the lab or observer's frame, and K0 be the rest frame of the electron. The scattering event as seen in each frame is shown in the Fig. (3). The energy of the scattered photon 0 in K0 frame is given by, 0 0 1 = 0 0 : (13) 1 + 2 ( 1 − cosθ ) mec 1 In the lab frame energy of the scattered photon, by Doppler formula is given by, 0 0 1 = γ 1 ( 1 + β cosθ1 ); (14) 0 π where γ is the Lorentz factor. For a special case of θ1 = 2 the Eq. (14) gives, 0 1 ≈ γ 1 : (15) 0 2 Now let us assume that in the rest frame of the electron, mec , so that 0 0 Thomson limit is applicable and ≈ 1 . Thus the Eq. (15) is given by, 0 1 ≈ γ : (16) But from the Doppler shift formula we have , 0 = γ ( 1 − β cosθ ); (17) 8 π and for θ = 2 ; the Eq. (17) gives, 0 ≈ γ . (18) Substituting the Eq. (18) in the Eq. (16) we obtain, 2 1 ≈ γ . (19) From the above relation it is clear that, for relativistic electron with ( γ2 − hν 2 1 ) 2 , initially low energy photons gain energy by a factor of γ in the mec lab frame at the expense of the kinetic energy of the electron. This process therefore can convert a radio photons to a UV photons, far-IR photons to X-ray photons, and optical photons to gamma-ray photons. 4 Inverse Compton power for single scatter- ing In the previous section, we derived the energy transfer for scattering of a single photon off a single electron. Now let us derive the energy transfer for the case of a given isotropic distribution of photons scattering off a given isotropic distribution of electrons. Let dn = n()d be the density of photons having energy in the range d. But dn, in terms of the differential number of particles dN (a Lorentz invariant), and the three dimensional volume element dV can be written as, dN dn = : (20) dV We know that the four dimensional volume element dX = dx0 dx1 dx2 dx3 = dN dx0 dV is Lorentz invariant. Therefore dn = dX dx0 transforms like the time component (x0) of the photon position four-vector. Further, since photon four-momentum pµ and position xµ are parallel four-vectors (i.e., their spatial components are related to their time components in the same way), the ratio dx0 dn dN p0 is invariant. Thus is Lorentz invariant, since dX is invariant. In other words, dn dn0 n()d n(0)d0 = or = : (21) 0 0 9 Now the total power emitted (i.e., scattered) in the electron's rest frame can be obtained from, 0 dE1 0 0 0 0 = c σT 1 n( ) d ; (22) dt Z 0 where 1 is the energy of the scattered photon in electron's rest frame.