Compton Scattering in Astrophysics

Home , Thomson scattering

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 photon and a charged particle such as an electron that causes momentum exchange between the photon and the electron unlike Thomson scattering, 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 distribution 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 applications 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 momentum 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 |Pef | = |Pei + Pγi − Pγf | . (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:

− ω0 hω¯ 1 = 1 + 2 (1 − cosθ) . (5) ω " mec #

2 For hω¯ mec , we can expand the expression in the bracket of the Eq. (5) binomially and obtain,

∆Eγ hω¯ = − 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 hω¯ 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 transfer the energy between the two components. The high energy photons with 2 2 mev h¯ω mec will transfer the energy to the low energy electrons, 2 but will gain energy from the high energy electrons (with hω¯ 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 scattering 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 scattering. Due to multiple Compton scattering, the spectrum of the photons or radiation incident on the plasma will be distorted. This change in the spectrum 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

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 scattering

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. We now assume that the change in energy of the photon in the rest frame of electron is negligible compared to the energy change in the lab frame, i.e., 0 2 0 0 dE1 (γ − 1) 2 , so that = . Further 0 is invariant since it is the ratio mec 1 dt of the same components of two parallel four-vectors. Thus, 0 dE dE 1 = 1 . (23) dt dt0 Substituting the Eq. (22) in the Eq. (23), we obtain,

dE1 0 0 0 = c σT n( ) d , (24) dt Z 02 n() d = c σT . Z 2 Here we have used the Eq. (21), and the assumption that γ mec , so that Thompson cross-section is applicable. Substituting 0 = γ ( 1 − β cosθ ), in the Eq. (24), one obtains,

dE1 2 2 = c σT γ ( 1 − β cosθ ) n() d, (25) dt Z which refers solely to quantities in the lab frame. For an isotropic distribution 2 1 2 of photons we have h ( 1 − β cosθ ) i = 1 + 3 β , since h cosθ i = 0 and 2 1 h cos θ i = 3 . Thus the Eq. (25), takes the form,

dE1 2 1 2 = c σT γ 1 + β Urad, (26) dt 3 where Urad = n() d is the initial photon or radiation energy density. To get the net Rpower gain of photon ﬁeld, we need to subtract the power irradiated onto the electron. Therefore the rate of decrease of the total initial photon energy is, dE = −c σT n() d = −c σT Urad. (27) dt Z 10 Thus the net power lost by the electron, and thereby converted into increased radiation is,

dErad 2 1 2 = c σT Urad γ 1 + β − 1 . (28) dt 3 But we know that γ2 − 1 = γ2β2, and therefore the Eq. (28) takes the form, dE 4 P = rad = c σ γ2 β2 U . (29) compt dt 3 T rad From the Eq. (29), one can compute the total Compton power per unit volume, from a medium of relativistic electrons. Let N(γ)dγ be the number of electrons per unit volume with γ in the range γ to γ + dγ. Then total Compton power is,

−1 −3 Ptot(erg s cm ) = Pcompt N(γ) dγ. (30) Z The total Compton power can be calculated, provided the distribution of the electrons is known (see Rybicki and Lightman, 1979, for such calculations). Now we can calculate the average power gained by the photon ﬁeld from the electron, as follows: The mean number of photons scattered per second is, c σ U N = c σ n = T rad , (31) c T rad h¯ ω h E i h ω h E i Urad where γ = ¯ is average energy of the photon deﬁned by γ = nrad . Hence the average energy gained by the photon in one collision is,

Pcompt h ∆Eγ i = . (32) Nc Substituting the Eq. (31), in the Eq. (32), we obtain, ∆E 4 v 2 γ = γ2 . (33) * Eγ + 3 c When v c, γ = 1 and for a thermal distribution of non-relativistic elec- 2 trons, mev = 3kBTe, the Eq. (33) can be written as,

∆Eγ 4kBTe = 2 . (34) * Eγ + mec

11 Clubbing the Eq. (7) and the Eq. (34), we ﬁnd that the mean fractional energy change of photon per collision is,

∆Eγ 4kBTe − h Eγ i = 2 . (35) * Eγ + mec

If 4kBTe > h Eγ i, the net energy transfer is from electrons to photons (inverse Compton scattering), and if 4kBTe < h Eγ i, the net energy transfer is from photons to electrons. In other words, we may say that, in a typical collision between an electron and a photon, the electron energy changes by 4kB Te 2 h E i. mec γ Theabove process acts as a major source of cooling for relativistic plasma as well as a mechanism for producing high-energy photons. The time scale for Compton cooling of an individual relativistic electron is given by,

2 γ me c tcc ' . (36) Pcompt

4 Substituting the Eq. (29) in the Eq. (36), and noting that Urad = σ Trad , we have, 3mec tcc ' 2 4 , (37) 4σT γσ β Trad where Trad is the radiation temperature. If electrons are non-relativistic (i.e., γ = 1) with temperature Te, this time scale is given by,

kB Te tcc ' . (38) Pcompt The energy gain by photons i.e., comptonization continues till the mean energy of photons raises to 4kBTe and after that the net transfer will cease.

5 Compton y-parameter

Compton y-parameter gives the condition for a signiﬁcant change of energy of photon due to repeated scattering. When electrons and photons co-exist in a region of size l, the repeated scattering of photons by the electrons will distort the original spectrum of the photons (i.e., Comptonization). The

12 −1 mean free path of the photon due to Thomson scattering is λγ = (neσT ) . If the size of the region l is such that ( l ) 1, then the photon will undergo λγ several collisions in this region. But if ( l ) 1, then there will be few λγ collisions. Therefore let us deﬁne optical depth as τ ≡ l = n σ l, so that e λγ e T τe 1 implies strong scattering.

If τe 1, then the photon undergoes Ns ( 1) collisions in traveling a 2 distance l. From standard random-walk arguments, we have Ns = τe . On the other hand, if τe ≤ 1, then Ns ' τe. Therefore an estimate for the 2 number of scattering is Ns ' max(τe, τe ). The average fractional change 4kETe in the photon energy per collision is given by 2 . Hence the condition for mec a signiﬁcant change of energy is

4kBTe 4kBTe 2 1 ' Ns 2 = 2 max( τe, τe ). (39) mec ! mec !

Deﬁning a parameter y called the Compton y-parameter by,

kBTeNs y = 2 . (40) mec

1 Now the condition for signiﬁcant scattering is y ' 4 . A more precise condition for repeated scattering to change the spectrum of the radiation ﬁeld can be obtained as follows: The change in the energy of a typical photon after a single scattering is given by the factor (0/) = 2 2 (1 + 4kBTe/mec ), with KBTe mec . After Ns scatterings, the energy change is by the factor

0 Ns 4kBTe 4kBTeNs = 1 + 2 ' exp 2 = exp(4y), (41) mec ! mec ! where we have used the Eq. (40). Suppose that the initial mean frequency of the radiation is ωi with hω¯ i kBTe. The energy gain by the photons (i.e., Comptonization) goes on till the mean energy of the photons raises to 4kBTe. The critical optical depth needed for this is determined by

0 4kBTe kBTe 2 = = exp 4 2 τ crit , (42) hω¯ i ! " mec ! #

13 giving 2 1/2 mec 4kBTe τcrit = ln . (43) " 4kBTe ! hω¯ i ! #

When the optical depth of the region is comparable with τcrit, the spectrum of the photons will evolve, because of the repeated scattering. Such an evolution is described by an equation called the Kompaneets equation, which will be discussed in the next section.

6 Kompaneets Equation

The spectrum of the photons will evolve because of the repeated scattering. Such an evolution (i.e., comptonization) is described by an equation called Kompaneets equation. In this section we will derive this Kompaneets equation. Let us assume that the medium is reasonably homogeneous over the length scales of interest and that the changes in the number n(ω) of photons of frequency ω occur only because of scattering. Then the evolution equation for photon number density is (see Padmanabhan, vol. 1, 2000, for the derivation of the following equation)

∂n(ω) dσ = d3p dΩ c { n(ω0) [ 1 + n(ω) ] N(E0) (44) ∂t Z Z dΩ! − n(ω) [ 1 + n(ω0) ] N(E) }, where we consider the scattering,

0 0 E + ω )* E + ω

dσ In the Eq. (44), dΩ is the electron-photon scattering cross-section, n(ω) is the photon distribution function, and N(E) is the electron distribution function. The rate of scattering of photons from frequency ω0 to frequency ω by electrons of energy E0 is described by the term,

dσ d3p c dΩ n(ω0) [ 1 + n(ω) ]. Z Z dΩ!

14 The proportionality to n(ω0) and N(E0) is obvious, the [ 1 + n(ω) ] term takes into account the stimulated emission eﬀects i.e., the probability of scattering from frequency ω0 to ω is increased by the factor 1 + n(ω), because photons obey Bose-Einstein statistics and tend toward mutual occupation of the same dσ quantum state. The quality dΩ is the diﬀerential scattering cross-section of the Eq. (9), for Thomson scattering, and the integration over d3p takes into 2 account all the electrons with energy E = p . similarly, the scattering of 2me photons from ω to ω0 is described by the term,

dσ 0 d3p N(E) c dΩ n(ω) [ 1 + n(ω ) ]. Z Z dΩ!

Apart from these quantum mechanical correction factors, the Eq. (44), is a standard form in kinetic theory. In general, the Eq. (44) can be solved only for special cases or with approximations. A detailed analysis of the evolution of the spectrum in the presence of repeated scatterings off relativistic electrons is difficult because the energy transfer per scattering is large and one must solve the integro-differential equation (i.e., the Eq. (44)). However, when the electrons are non-relativistic, the fractional energy transfer h¯∆, or frequency change ∆ per scattering is small. In particular, the Eq. (44) may be expanded to second order in this small quantity ∆ yielding an approximation called the Fokker-Planck equation. For photons scattering off a non-relativistic, thermal distribution by A.S.Kompaneets (1957) and is known as the Kompaneets equation. For a thermal distribution of non-relativistic electrons, N(E) is given by the Boltzmann distribution, E N(E) ∝ exp − . (45) kBTe Define frequency change as,

∆ = ω0 − ω. (46)

We now consider situation in which the energy transfer is small, ∆ 1, and expand n(ω0) = n(ω + ∆) and N(E0) = N(E + h¯∆) in a Taylor series in ∆,

15 retaining up to quadratic order,

∂n 1 ∂2n n(ω0) = n(ω) + ∆ + ∆2 + · · · , (47) ∂ω 2 ∂ω2 ∂N (¯h∆)2 ∂2N N(E0) = N(E) + h¯ ∆ + + · · · . (48) ∂E 2 ∂E2 Put x = h¯ω and use the Eq. (45) in the Eq. (47) and the Eq. (48) respec- kB Te tively, then we obtain,

2 2 0 h¯∆ ∂n 1 h¯∆ ∂ n n(ω ) = n(ω) + + 2 + · · · , (49) kBTe ∂x 2 kBTe ! ∂x h¯∆ 1 h¯∆ 2 N(E0) = N(E) + N(E) + N(E) + · · · . (50) kBTe 2 kBTe !

Substituting the Eq. (49) and the Eq. (50) in the Eq. (44), and simplifying we ﬁnally obtain,

∂n ∂n 1 h¯ 2 = + n (n + 1 ) I1 + (51) ∂t " ∂x # 2 kBTe ! ∂2n ∂n + 2 ( 1 + n) + n ( n + 1) I2, " ∂x2 ∂x # where,

h¯ 3 dσ I1 = d p dΩ c N(E) ∆, (52) kBTe Z Z dΩ 3 dσ 2 and I2 = d p dΩ c N(E) ∆ . (53) Z Z dΩ To proceed further, we need an estimate for ∆ in the individual scattering. The conservation of energy and momentum in the electron-photon scattering can be expressed respectively by,

p2 p02 h¯ ω + = h¯ ω0 + , (54) 2me 2me 0 hω¯ hω¯ 0 and nˆ + p = nˆ0 + p . (55) c c

16 Solving for p0 in the Eq. (55), squaring and substituting in the Eq. (54) and simplifying, we obtain,

hω¯ c p · ( nˆ − nˆ0 ) + h¯2ω2 ( 1 − nˆ · nˆ0 ) h¯ ∆ = − . (56) 2 0 0 mec + hω¯ ( 1 − nˆ · nˆ ) − c nˆ · p

Note that while simplifying we have neglected the terms containing ∆2, as ∆ ω 1. The second term in the numerator of the Eq. (56) is a small correction to the ﬁrst term. Similarly the second and the third terms in the denominator 2 of the Eq. (56) are small correction to mec . Hence, to lowest order, the Eq. (56) can be written as,

hω¯ h¯ ∆ = h¯ ( ω0 − ω ) ' − p · ( nˆ − nˆ0 ). (57) mec Now substituting for ∆ from the Eq. (57) into the Eq. (53), we have,

2 ω 3 dσ 0 2 I2 = c d p dΩ N(E) [ p · ( nˆ − nˆ ) ] . (58) mec Z Z dΩ! Let ψ be the angle between the vector p and the vector (nˆ − nˆ0), so that the Eq. (58), can be written as,

2 ω 3 dσ 2 2 ˆ0 2 I2 = 2 c d p dΩ N(p) p cos ψ | nˆ − n | . (59) mec Z Z dΩ!

dσ v Now since dΩ does not depend on p, to lowest order in c , the integral over p may be done independently of the integral over the photon direction. Next, substitute in the Eq. (59), for the Maxwellian electron distribution,

2 3 p 2 N(p) = ne (2πmekBTe) exp − , 2mekBTe ! and let ψ be the polar angle for the d3p integration i.e., d3p = p2 dp d(cosψ) dφ. Now the Eq. (59), takes the form,

2 ω nekBTe dσ 0 I2 = dΩ | nˆ − nˆ |. (60) mec Z dΩ

17 If θ is the angle of scattering, then substituting the Eq. (9), in the Eq. (60), and carrying out the integration, we obtain,

2 2ω nekBTe σT I2 = , (61) mec 8π 2 where we have used σT = 3 r0 . Now writing the above equation in terms of x, we have, 2 2 2neσT (kBTe) x I2 = 2 . (62) h¯ mec

0 Let us next consider I1. In the lowest order, ∆ ∝ p · ( nˆ − nˆ ) and at this order I1 will be zero, when integrated over all p. Thus, to obtain the non-zero contribution to I1 we need to expand the Eq. (56) to a higher order v in c . However, we can determine I1 by an indirect procedure. Note that by deﬁnition, I1 gives the ratio of the energy transfer rate and the mean energy (kBTe) of the electrons. If ∆E is the mean energy transfer per collision, then ∆E I1 = σT nec, (63) kBTe as the rate of collision is σT nec. Substituting the Eq. (35), in the Eq. (63), we obtain, σT nec hω¯ I1 = 2 ( 4kBTe − hω¯ ). (64) kBTe mec Writing the above equation in terms of x,

kBTe I1 = 2 σT nec x( 4 − x ). (65) mec Substituting the Eq. (62) and the Eq. (65), in the Eq. (51) and simplifying one obtains,

2 mec 1 ∂n 1 ∂ 4 ∂n 2 = 2 x + n + n . (66) kBTe σT nec ∂t x ∂x " ∂x ! #

But σT nect = τe - optical depth and using the expression for Compton y- parameter from the Eq. (40), in the Eq. (66), we have, ∂n 1 ∂ ∂n = x4 + n + n2 . (67) ∂y x2 ∂x " ∂x ! #

18 The Eq. (67), is referred to as Kompaneets equation. The solution to this equation describes the evolution of the photon spectrum that is due to repeated scattering with a non-relativistic thermal bath of electrons. The Eq. (67) can be solved only for special cases and simple geometries. In general cases, it must be solved by numerical integration. As an illustration, let us consider the steady state solution to this Kom- ∂n paneets equation. For steady state situation, ∂y = 0, so that the Eq. (67) can now be written as, ∂n = − n ( n + 1 ). (68) ∂x Integrating the above equation we have,

−1 n = [exp( x − x0 ) − 1 ] , (69) which is the Bose-Einstein distribution with non-zero chemical potential µ = −1 β x0. We know that in the case of comptonization, the scattering between the electrons and photons cannot change the total number of the photons, but can change the mean energy. Therefore the final configuration cannot be a Planck spectrum, because of the constraints on both the number and the energy. Hence the final distribution of the photons undergoing repeated scattering with the electrons, will be a Bose-Einstein distribution. The β and µ of the distribution will be determined by the total number and energy of the photons. When x 1 we will have n 1 and n(x) ∝ exp−x. This spectrum is the same as the Wein’s spectrum.

7 Sunyaev - Zeldovich Eﬀect (CMB Comp- tonization)

There are regions in the Universe - like the cluster of galaxies - that con- tain hot, ionized gas. Cosmic microwave background radiation photons, that ﬁlls the Universe, when passes through these regions, they will be scattered by the electrons ( which are at a much high temperature) and gain energy. (Note that Cosmic microwave background radiation is a black body radiation corresponding to a temperature of about 2.76◦K). This will distort the cosmic microwave background radiation spectrum in the vicinity of a cluster of galaxies. This comptonization of the cosmic microwave background

19 radiation by hot gas in the cluster of galaxies is usually referred to as the Sunyaev-Zeldovich effect. It has been observed that, the highest level of spectral deviation, around 1mK, are caused by Compton scattering of the radiation by hot gas in the clusters of galaxies. Spectral diminutions in the Rayleigh-Jeans region and closer to the Planckian peak have been measured in about a dozen clusters. The Sunyaev-Zeldovich effect has its origin in the early work on spectral distortions of the cosmic microwave background. A pure blackbody Planck spectrum is obviously not expected at all frequencies and in all directions across the sky, even after accounting for galactic effects (such as absorption, thermal emission from the interstellar gas and dust). To cause a spectral distortion, a radiative process must occur sufficiently late in the cosmological evolution (i.e., at red-shifts z ≤ 3 × 106) to prevent the radiation from thermalizing and regaining a pure Planck spectrum due to its weak coupling with the matter. In particular, the effect of Compton scattering of the cosmic microwave background radiation by the hot intergalactic medium was calculated extensively by Sunyaev and Zeldovich (1972), and hence the name Sunyaev-Zeldovich effect.

7.1 Theory The nature of the thermal Sunyaev-Zeldovich effect can be easily realized. As photons of the isotropic cosmic microwave background traverses through the intra-cluster medium, some are Compton scattered by hot intra-cluster electrons. Scattering off the moving electrons causes Doppler frequency shifts, and as the electron gas is very hot, the radiation gains energy. Conservation of photon number in the scattering implies that there is a systematic shift of photons from the Rayleigh-jeans region of the spectrum to the Wein side of the spectrum. The basic goal is the calculation of the spectral distribution of the scattered radiation field. The time rate of change of the photon occupation number n, of an isotropic radiation field due to Compton scattering by isotropic, non-relativistic Maxwellian electron gas is given by the Kompaneets equation. This equation - a non-relativistic Fokker-Planck approximation to the exact kinetic

20 equation, is given by

∂n 1 ∂ ∂n = x4 + n + n2 , (70) ∂y x2 ∂x " ∂x ! #

hν kB TeσT nect where x = , y = 2 , and Compton scattering here is taken in kBTe mec 2 the Thomson limit, hν mec , so that the cross-section for scattering is given by Thomson cross-section. In the right hand side of the Eq. (70), the ﬁrst term in the parentheses is much larger than the other two terms, because Te T , where T is radiation temperature. Ignoring these latter terms greatly simpliﬁes the Eq. (70) to the following,

∂n 1 ∂ ∂n = x4 . (71) ∂y x2 ∂x ∂x !

Here n is related to the radiation intensity Iν by, I c2 n = ν . (72) 8πhν3

If the incident radiation is only weakly scattered, then an approximate solution to the Eq. (71), can be obtained by substituting on the right hand side the expression for the occupation number of a purely Planckian radiation ﬁeld, given by 1 n(x) = . (73) exp(x) − 1 Therefore the Eq. (71) takes the form,

∂n x exp(x) x ( exp(x) + 1 ) = − 4 . (74) ∂y (exp(x) − 1)2 " exp(x) − 1 # Now integrating along the path length through the cluster, we have,

x exp(x) x ( exp(x) + 1 ) ∆n = − 4 y. (75) (exp(x) − 1)2 " exp(x) − 1 #

But from the Eq. (72), 8πhν3 ∆I = ∆n. (76) nr c2

21 The above equation in terms of x can be written as,

3 8πh kBTex ∆Inr = ∆n. (77) c2 h ! Therefore the change in the spectral intensity along the line of sight is given by, ∆Inr = a y g(x), (78) 3 (kB Te) where a = 8π (hc)2 and the subscript nr denotes the fact that the expression was obtained in the non-relativistic limit. The spectral form of this thermal Sunyaev-Zeldovich eﬀect is expressed in the function, x exp(x) x ( exp(x) + 1 ) g(x) = − 4 , (79) (exp(x) − 1)2 " exp(x) − 1 # and is as shown in the Fig. (4). The spatial dependence of the eﬀect is contained in the Kompaneets parameter given by

kBTe y = 2 ne σT dl, (80) Z mec ! where the integral is over a line of sight through the cluster. The radiation field gains energy through scattering by the much hotter intra-cluster gas, but with no change in the photon number, photons are transferred from the Rayleigh-Jeans to the Wein side of the spectrum, as shown in the Fig. (4) by the spectral part of the intensity change, the function g(x). The relative change of the energy density of the radiation, exp(4y), is obviously determined by the degree of coupling to the gas and its temperature. The Eq. (78) for the intensity change is valid approximation, in the non-relativistic limit, only if these changes are small. Specifically, these expressions are presumed to be accurate to first order in y, which is indeed a small parameter, typically of the order of the 10( − 4) in rich clusters. However, relativistic corrections are important, especially at high frequencies (for details see Rephaeli, 1995). Note that the above derivation holds for low optical thickness to Compton scattering in clusters. Note also that in the above derivation it was implicitly assumed that neither the cluster grav- itational potential not intra-cluster gas properties change during the passage of the radiation through the cluster.

22 8

2 g(x)

−2

−4

−6 1 2 3 4 5 6 7 8 9 10 x

Figure 4: The spectral dependence of the non-relativistic thermal intensity change (∆Inr), g(x) as a function of the non-dimensional frequency x = hν . kBTe

7.2 Cluster Comptonization as a Probe Comptonization of the cosmic microwave background radiation by hot gas in clusters imprints unique spectral signatures that can be used as important astrophysical and cosmological probes. The full significance of the thermal Sunyaev-Zeldovich effect as a cosmological probe has been appreciated within a decade following the work of the Sunyaev and Zeldovich (1972). Radio and millimeter-band telescopes on the ground and sub-millimeter telescopes on the mountain, in balloons, or in the orbit could be used to search for hot intergalactic gas in the clusters, as well as large-scale irregularities in the distribution of such gas. Also one might find remote clusters and protoclusters of galaxies located at cosmological distances. Mere detection of the thermal Sunyaev-Zeldovich effect constitutes a di- rect observational proof of the universality of the cosmic microwave background radiation. The effect does not depend on the redshift of the cluster, a very desirable and uncommon feature that makes this effect an extremely valuable diagnostic cosmological and cluster evolutionary tool. The high measurement sensitivity now achievable at frequencies near and on the Wein side enables detection of the thermal effect at a high degree of confidence

23 based on its characteristic spectral profile there. The potential benefits from mapping the spectral imprints of Comptonization in clusters are particularly great when supplemented with detailed X-ray measurements of the gas density, temperature, and their spectral profiles. We note that because the properties of a background radiation field are affected by the scattering by the intra-cluster electrons, a positive intensity change across a cluster may practically be viewed as a net emission from the cluster, whereas a negative change makes the cluster an absorber of the cosmic microwave background radiation. Thus, clusters are - in effect - powerful sinks and sources of emission in the radio and submillimetric regions, respectively, whose spectral brightness is independent of redshift. A measurement of the thermal effect directly determines the integrated intra-cluster gas pressure, whereas that of the kinematic effect (to be discussed later) yields the gas column density. Because of the linear dependence of the Sunyaev-Zeldovich effect on gas density (ne), it is, in principle, easier to map gas properties in the outer regions of clusters by Sunyaev-Zeldovich mea- 2 surements than it is to map from measurements of X-ray emissivity (∝ ne ). By combining the two sets of measurements a substantial part of the indeter- minacy inherent in X-ray analysis of intra-cluster gas properties is removed. Doing so will enable a meaningful investigation of the degree of inhomogene- ity of the gas. The angular diameter distance to a cluster can be observa- tionally determined from measurements of the thermal Sunyaev-Zeldovich effect and X-ray measurements of thermal emission from the intra-cluster gas. Thus the value of the Hubble constant can be deduced from these measurements of a nearby cluster. Also effect is important as a source of cosmic microwave background anisotropy on arc-minute scales, and thereby serves as a probe to cluster evolution. Peculiar velocities of galaxies and clusters are basic features of the large- scale structure of the Universe. Clusters of galaxies play a fundamental role in the growth of the large-scale structure, and it is obviously very important to determine their velocities in order to fully study this role. The kinematic Sunyaev-Zeldovich effect offers a relatively simple way to measure directly the radial component of the peculiar velocity of the cluster. The spectral change derived for thermal effect is caused by the random thermal motions of the electrons whose distribution is assumed to be isotropic (in the cluster frame). Clearly, when the cluster as a whole has a finite peculiar velocity in the cosmic

24 microwave background frame, there will be an additional kinematic (Doppler) effect. The expressions for the intensity change due to the kinematic effect can be obtained by a simple relativistic transformation, if it is assumed - based on the smallness of both the thermal and kinematic effects - that the effects are separable. If so, the additional kinematic intensity change is V ∆I = −a h(x) r τ, (81) k c where x4 exp(x) h(x) = , (82) (exp(x) − 1)2

Vr, is the line-of-sight velocity of the cluster, which is positive (negative) for a receding (approaching) cluster, and

τ = σT ne dl, (83) Z is the cluster optical depth to Compton scattering. The observational feasibility of this method rests heavily on the relative magnitudes of the kinematic and thermal effects. The velocity dependence 2 of the ratio ∆Ik/∆Inr is (Vr/c)/(ve/c) (where ve is the rms thermal electron velocity), while the spectral part of this dependence is given in the ration h(x)/g(x) (in the non-relativistic limit). Therefore, to avoid masking the kinematic effect completely, measurements have to be made at cross over frequency (in the Fig. (4)), where ∆Inr = 0 and h(x) has a broad maximum centered at cross over frequency. Scattering of the radiation in clusters also affects its polarization state. There are several possible ways by which net polarization can be induced; these include single and multiple scattering off intra-cluster electrons in a cluster moving perpendicular to the line-of-sight. Motion transverse to the line-of-sight induces a quadrupole component in the spatial distribution of the radiation; thus the radiation will appear linearly polarized. thus the tangential component of the velocity of the cluster can also be determined. So far, we have considered only the effects of the interaction of cosmic microwave background photons with intra-cluster electrons. Compton scattering also effects the radiation at other parts of the electromagnetic background. Measurements of the Compton signature may thus prove valuable

25 in the study of some of the other cosmological background radiations. For example, the eﬀect can be used to distinguish between a truly cosmological component from the cluster one.

26 References

[1] Kompaneets, A.S.: 1957, Sov. Phys. JETP, 4, 730.

[2] Lightman, A.P., and Rybicki, G.B.: 1979, Radiative Processes in Astro- physics, Wiley-Interscience Publications.

[3] Padmanabhan, T.: 2000, Theoretical Astrophysics, Vol.1, Cambridge Uni- versity Press.

[4] Padmanabhan, T.: 2000, Theoretical Astrophysics, Vol.3, Cambridge Uni- versity Press.

[5] Sunyaev, R.A., and Zeldovich, Y.B.: 1972, Comments. Astrophys. Space Phys., 4:173.

[6] Yoel Rephaeli: 1995, Comptonization of the Cosmic Microwave Back- ground: The Sunyaev-Zeldovich Eﬀect, Annu. Rev. Astron. Astrophys., 33:541-79.