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) 197.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.2 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 momentumexchange between the photon and the electron unlike Thomson scat-tering, is discussed. In astrophysical applications it is the inverseCompton scattering that plays an important role than the Comptonscattering itself. The change in the spectrum of the incident radiationcaused by the multiple Compton scattering with an thermal distribu-tion of electrons, called Comptonization is discussed. The Kompaneetsequation that describes the comptonization is derived and solution fora particularly simple case is discussed as an illustration. The wellknow Sunyaev - Zeldovich Effect (CMB Comptonization) and its ap-plications are briefly discussed.

3

θ

P , E

ν

ν

Figure 1: Figure shows the scattering of photon by electron at rest.

1 Introduction

Compton scattering is an scattering event that causes momentum exchangebetween a photon and a charged scatterer such as an electron. For lowphoton energies, hν mec

2 (where me is the rest mass of the electron),the scattering of radiation from free charges reduces to the classical case ofThomson scattering. Quantum effects appears through the kinematics of thescattering process. The kinematic effects occur because a photon possessesa momentum hν

cas 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 = ( ε

c) (1, ni)

and Pγf = ( ε′

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 theelectron be Pei = (mec, 0) and Pef = (E

c, P).

Conservation of momentum and energy is expressed by,

Pei + Pγi = Pef + Pγf . (1)

Rearranging terms and squaring gives,

|Pef |2 = |Pei + Pγi − Pγf |

2. (2)

4

Solving the above, we finally obtain

ε′ =ε

1 + εmec2

(1 − cosθ). (3)

In terms of wavelength, the above equation can be written as,

λ′ − λ = λc (1 − cosθ), (4)

where the Compton wavelength is defined to be λc = hmec

= 0.02426 A forelectron. Observe that there is a wavelength change of the order of λc uponscattering. For long wavelengths λ λc (i.e., hν mec

2) the scattering isclosely elastic. When this condition is satisfied, we can assume that there isno change in photon energy in the rest frame of the electron.

From the Eq. (3) we can calculate mean energy transfer as follows:

ω′

ω=

[

1 +hω

mec2(1 − cosθ)

]

−1

. (5)

For hω mec2, we can expand the expression in the bracket of the Eq. (5)

binomially and obtain,

∆Eγ

Eγ

= −hω

mec2(1 − cosθ). (6)

In order to find the mean energy transfer, one has to average the Eq. (6) overθ. In the rest frame of the electron, the scattering has front-back symmetry,making 〈 cosθ 〉 = 0. Hence the average energy lost by the photon percollision is,

〈∆Eγ 〉 = −〈 hω 〉

mec2〈Eγ 〉 = −

〈Eγ 〉2

mec2. (7)

The differential cross-section for Compton scattering is given by Klein-Nishina formula,

dσ

dΩ=

r02

2

ε′2

ε2

(

ε

ε′+ε′

ε− sin2θ

)

, (8)

5

where r0 is classical electron radius. Note that for ε′ = ε, the above equationreduces to the classical expression, given by

dσ

dΩ=

1

2r0

2(

1 + cos2θ)

. (9)

The total cross-section is given by,

σ = σT3

4

1 + x

x3

[

2x (1 + x)

1 + 2x− ln(1 + 2x)

]

(10)

+ σT3

4

1

2xln(1 + 2x) −

1 + 3x

(1 + 2x)2

,

where x = hνmec2

. From the above expression it is clear that, the principaleffect is to reduce the cross-section from its classical value as the photonenergy becomes large. In the non-relativistic regime (i.e., for x 1), theEq. (10) takes the form,

σ ≈ σT

(

1 − 2x +26x2

5+ · · ·

)

, (11)

whereas for the extreme relativistic regime (i.e., for x 1), the Eq. (10)takes the form,

σ ≈3

8σT x

−1(

ln2x +1

2

)

. (12)

For low x, the cross-section approaches σT and the change in energy of thephoton 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. Thescattering of photons by the electrons in the plasma will continuously trans-fer the energy between the two components. The high energy photons withmev

2 hω mec2 will transfer the energy to the low energy electrons,

but will gain energy from the high energy electrons (with hω mev2). In

thermal equilibrium, the net transfer of the energy will be zero. But if the

6

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

Cro

ss−

se

ctio

n (

in b

arn

s)

Klein−Nishina

Thomson

Figure 2: Figure shows the plot of total cross-section for Compton scatteringas 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 calledinverse Compton scattering. On the other hand, if Trad Te, the energywill be transfered (on the average) from the photons to the electrons, due toCompton scattering. In astrophysical applications, inverse Compton scatter-ing plays a more important role than Compton scattering and it can serveas 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 orradiation incident on the plasma will be distorted. This change in the spec-trum of radiation due to multiple scattering of photons by thermal electronsis called Comptonization.

Before dealing with comptonization, let us consider the energy transferdue to single scattering.

7

ε

ε

θθ1

K − Frame

1

ε

ε

θ1

K − Frame

1

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 K ′ 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 ε′ in K ′ frame is given by,

ε1′ =

ε′

1 + ε′

mec2( 1 − cosθ1

′ ). (13)

In the lab frame energy of the scattered photon, by Doppler formula is givenby,

ε1 = γ ε1′ ( 1 + β cosθ1

′ ), (14)

where γ is the Lorentz factor. For a special case of θ1′ = π

2the Eq. (14) gives,

ε1 ≈ γ ε1′. (15)

Now let us assume that in the rest frame of the electron, ε′ mec2, so that

Thomson limit is applicable and ε′ ≈ ε1′. Thus the Eq. (15) is given by,

ε1 ≈ γ ε′. (16)

But from the Doppler shift formula we have ,

ε′ = ε γ ( 1 − β cosθ ), (17)

8

and for θ = π2, the Eq. (17) gives,

ε′ ≈ γ ε. (18)

Substituting the Eq. (18) in the Eq. (16) we obtain,

ε1 ≈ γ2 ε. (19)

From the above relation it is clear that, for relativistic electron with ( γ2 −1 ) hν

mec2, initially low energy photons gain energy by a factor of γ2 in the

lab frame at the expense of the kinetic energy of the electron. This processtherefore can convert a radio photons to a UV photons, far-IR photons toX-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 asingle photon off a single electron. Now let us derive the energy transfer forthe case of a given isotropic distribution of photons scattering off a givenisotropic distribution of electrons.

Let dn = n(ε)dε be the density of photons having energy in the rangedε. But dn, in terms of the differential number of particles dN (a Lorentzinvariant), and the three dimensional volume element dV can be written as,

dn =dN

dV. (20)

We know that the four dimensional volume element dX = dx0 dx1 dx2 dx3 =dx0 dV is Lorentz invariant. Therefore dn = dN

dXdx0 transforms like the time

component (x0) of the photon position four-vector. Further, since photonfour-momentum pµ and position xµ are parallel four-vectors (i.e., their spatialcomponents are related to their time components in the same way), the ratiodx0

p0is invariant. Thus dn

εis Lorentz invariant, since dN

dXis invariant. In other

words,dn

ε=

dn′

ε′or

n(ε)dε

ε=

n(ε′)dε′

ε′. (21)

9

Now the total power emitted (i.e., scattered) in the electron’s rest framecan be obtained from,

dE1′

dt′= c σT

∫

ε1′ n(ε′) dε′, (22)

where ε1′ 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 ofelectron is negligible compared to the energy change in the lab frame, i.e.,(γ2 −1) ε

mec2, so that ε1

′ = ε′. Further dE1′

dt′is invariant since it is the ratio

of the same components of two parallel four-vectors. Thus,

dE1

dt=

dE1′

dt′. (23)

Substituting the Eq. (22) in the Eq. (23), we obtain,

dE1

dt= c σT

∫

ε′ n(ε′) dε′, (24)

= c σT

∫

ε′2 n(ε) dε

ε.

Here we have used the Eq. (21), and the assumption that γε mec2, so that

Thompson cross-section is applicable. Substituting ε′ = εγ ( 1 − β cosθ ), inthe Eq. (24), one obtains,

dE1

dt= c σT γ

2∫

( 1 − β cosθ )2 ε n(ε) dε, (25)

which refers solely to quantities in the lab frame. For an isotropic distributionof photons we have 〈 ( 1 − β cosθ )2 〉 = 1 + 1

3β2, since 〈 cosθ 〉 = 0 and

〈 cos2θ 〉 = 13. Thus the Eq. (25), takes the form,

dE1

dt= c σT γ

2(

1 +1

3β2)

Urad, (26)

where Urad =∫

ε n(ε) dε is the initial photon or radiation energy density.To get the net power gain of photon field, we need to subtract the powerirradiated onto the electron. Therefore the rate of decrease of the total initialphoton energy is,

dE

dt= −c σT

∫

ε n(ε) dε = −c σT Urad. (27)

10

Thus the net power lost by the electron, and thereby converted into increasedradiation is,

dErad

dt= c σT Urad

[

γ2(

1 +1

3β2)

− 1]

. (28)

But we know that γ2 − 1 = γ2β2, and therefore the Eq. (28) takes the form,

Pcompt =dErad

dt=

4

3c σT γ

2 β2 Urad. (29)

From the Eq. (29), one can compute the total Compton power per unitvolume, from a medium of relativistic electrons. Let N(γ)dγ be the numberof electrons per unit volume with γ in the range γ to γ + dγ. Then totalCompton power is,

Ptot(erg s−1 cm−3) =∫

PcomptN(γ) dγ. (30)

The total Compton power can be calculated, provided the distribution of theelectrons is known (see Rybicki and Lightman, 1979, for such calculations).

Now we can calculate the average power gained by the photon field fromthe electron, as follows: The mean number of photons scattered per secondis,

Nc = c σT nrad =c σT Urad

h ω, (31)

where 〈Eγ 〉 = h ω is average energy of the photon defined by 〈Eγ 〉 = Urad

nrad

.Hence the average energy gained by the photon in one collision is,

〈∆Eγ 〉 =Pcompt

Nc. (32)

Substituting the Eq. (31), in the Eq. (32), we obtain,⟨

∆Eγ

Eγ

⟩

=4

3γ2(

v

c

)2

. (33)

When v c, γ = 1 and for a thermal distribution of non-relativistic elec-trons, mev

2 = 3kBTe, the Eq. (33) can be written as,⟨

∆Eγ

Eγ

⟩

=4kBTe

mec2. (34)

11

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

⟨

∆Eγ

Eγ

⟩

=4kBTe − 〈Eγ 〉

mec2. (35)

If 4kBTe > 〈Eγ 〉, the net energy transfer is from electrons to photons (in-verse Compton scattering), and if 4kBTe < 〈Eγ 〉, the net energy transferis from photons to electrons. In other words, we may say that, in a typicalcollision between an electron and a photon, the electron energy changes by(

4kBTe

mec2

)

〈Eγ 〉.

The above process acts as a major source of cooling for relativistic plasmaas well as a mechanism for producing high-energy photons. The time scalefor Compton cooling of an individual relativistic electron is given by,

tcc 'γ me c

2

Pcompt. (36)

Substituting the Eq. (29) in the Eq. (36), and noting that Urad = σ Trad4, we

have,

tcc '3mec

4σTγσ β2 T 4rad

, (37)

where Trad is the radiation temperature. If electrons are non-relativistic (i.e.,γ = 1) with temperature Te, this time scale is given by,

tcc 'kB Te

Pcompt. (38)

The energy gain by photons i.e., comptonization continues till the meanenergy 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 energyof photon due to repeated scattering. When electrons and photons co-existin a region of size l, the repeated scattering of photons by the electrons willdistort the original spectrum of the photons (i.e., Comptonization). The

12

mean free path of the photon due to Thomson scattering is λγ = (neσT )−1.If the size of the region l is such that ( l

λγ) 1, then the photon will undergo

several collisions in this region. But if ( lλγ

) 1, then there will be few

collisions. Therefore let us define optical depth as τe ≡ lλγ

= neσT l, so that

τe 1 implies strong scattering.

If τe 1, then the photon undergoes Ns ( 1) collisions in traveling adistance l. From standard random-walk arguments, we have Ns = τe

2. Onthe other hand, if τe ≤ 1, then Ns ' τe. Therefore an estimate for thenumber of scattering is Ns ' max(τe, τe

2). The average fractional changein the photon energy per collision is given by 4kETe

mec2. Hence the condition for

a significant change of energy is

1 ' Ns

(

4kBTe

mec2

)

=

(

4kBTe

mec2

)

max( τe, τe2 ). (39)

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

y =kBTeNs

mec2. (40)

Now the condition for significant scattering is y ' 14.

A more precise condition for repeated scattering to change the spectrumof the radiation field can be obtained as follows: The change in the energyof a typical photon after a single scattering is given by the factor (ε′/ε) =(1 + 4kBTe/mec

2), with KBTe mec2. After Ns scatterings, the energy

change is by the factor

ε′

ε=

(

1 +4kBTe

mec2

)Ns

' exp

(

4kBTeNs

mec2

)

= exp(4y), (41)

where we have used the Eq. (40). Suppose that the initial mean frequencyof the radiation is ωi with hωi kBTe. The energy gain by the photons(i.e., Comptonization) goes on till the mean energy of the photons raises to4kBTe. The critical optical depth needed for this is determined by

ε′

ε=

(

4kBTe

hωi

)

= exp

[

4

(

kBTe

mec2

)

τ 2crit

]

, (42)

13

giving

τcrit =

[(

mec2

4kBTe

)

ln

(

4kBTe

hωi

)]1/2

. (43)

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

6 Kompaneets Equation

The spectrum of the photons will evolve because of the repeated scatter-ing. Such an evolution (i.e., comptonization) is described by an equationcalled Kompaneets equation. In this section we will derive this Kompaneetsequation.

Let us assume that the medium is reasonably homogeneous over thelength scales of interest and that the changes in the number n(ω) of pho-tons of frequency ω occur only because of scattering. Then the evolutionequation for photon number density is (see Padmanabhan, vol. 1, 2000, forthe derivation of the following equation)

∂n(ω)

∂t=

∫

d3p∫

dΩ

(

dσ

dΩ

)

c n(ω′) [ 1 + n(ω) ]N(E ′) (44)

−n(ω) [ 1 + n(ω′) ]N(E) ,

where we consider the scattering,

E + ω E ′ + ω′

In the Eq. (44), dσdΩ

is the electron-photon scattering cross-section, n(ω) isthe photon distribution function, and N(E) is the electron distribution func-tion. The rate of scattering of photons from frequency ω ′ to frequency ω byelectrons of energy E ′ is described by the term,

∫

d3p∫

c

(

dσ

dΩ

)

dΩn(ω′) [ 1 + n(ω) ].

14

The proportionality to n(ω′) and N(E ′) is obvious, the [ 1 + n(ω) ] term takesinto account the stimulated emission effects i.e., the probability of scatteringfrom frequency ω′ to ω is increased by the factor 1 + n(ω), because photonsobey Bose-Einstein statistics and tend toward mutual occupation of the samequantum state. The quality dσ

dΩis the differential scattering cross-section of

the Eq. (9), for Thomson scattering, and the integration over d3p takes into

account all the electrons with energy E = p2

2me. similarly, the scattering of

photons from ω to ω′ is described by the term,

∫

d3pN(E)∫

c

(

dσ

dΩ

)

dΩn(ω) [ 1 + n(ω′) ].

Apart from these quantum mechanical correction factors, the Eq. (44),is a standard form in kinetic theory. In general, the Eq. (44) can be solvedonly for special cases or with approximations.

A detailed analysis of the evolution of the spectrum in the presence ofrepeated scatterings off relativistic electrons is difficult because the energytransfer per scattering is large and one must solve the integro-differentialequation (i.e., the Eq. (44)). However, when the electrons are non-relativistic,the fractional energy transfer h∆, or frequency change ∆ per scattering issmall. In particular, the Eq. (44) may be expanded to second order in thissmall quantity ∆ yielding an approximation called the Fokker-Planck equa-tion. For photons scattering off a non-relativistic, thermal distribution byA.S.Kompaneets (1957) and is known as the Kompaneets equation.

For a thermal distribution of non-relativistic electrons, N(E) is given bythe Boltzmann distribution,

N(E) ∝ exp(

−E

kBTe

)

. (45)

Define frequency change as,

∆ = ω′ − ω. (46)

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

15

retaining up to quadratic order,

n(ω′) = n(ω) + ∆∂n

∂ω+

1

2∆2 ∂

2n

∂ω2+ · · · , (47)

N(E ′) = N(E) + h∆∂N

∂E+

(h∆)2

2

∂2N

∂E2+ · · · . (48)

Put x = hωkBTe

and use the Eq. (45) in the Eq. (47) and the Eq. (48) respec-tively, then we obtain,

n(ω′) = n(ω) +h∆

kBTe

∂n

∂x+

1

2

(

h∆

kBTe

)2∂2n

∂x2+ · · · , (49)

N(E ′) = N(E) +h∆

kBTeN(E) +

1

2

(

h∆

kBTe

)2

N(E) + · · · . (50)

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

∂n

∂t=

[

∂n

∂x+ n (n + 1 )

]

I1 +1

2

(

h

kBTe

)2

(51)

[

∂2n

∂x2+ 2 ( 1 + n)

∂n

∂x+ n (n + 1)

]

I2,

where,

I1 =h

kBTe

∫

d3p∫

dΩdσ

dΩcN(E) ∆, (52)

and I2 =∫

d3p∫

dΩdσ

dΩcN(E) ∆2. (53)

To proceed further, we need an estimate for ∆ in the individual scattering.The conservation of energy and momentum in the electron-photon scatteringcan be expressed respectively by,

h ω +p2

2me= h ω′ +

p′2

2me, (54)

andhω

cn + p =

hω′

cn′ + p′. (55)

16

Solving for p′ in the Eq. (55), squaring and substituting in the Eq. (54) andsimplifying, we obtain,

h∆ = −hωcp · ( n − n′ ) + h2ω2 ( 1 − n · n′ )

mec2 + hω ( 1 − n · n′ ) − c n′ · p. (56)

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 denominatorof the Eq. (56) are small correction to mec

2. Hence, to lowest order, theEq. (56) can be written as,

h∆ = h (ω′ − ω ) ' −hω

mecp · ( n − n′ ). (57)

Now substituting for ∆ from the Eq. (57) into the Eq. (53), we have,

I2 =(

ω

mec

)2

c∫

d3p∫

dΩ

(

dσ

dΩ

)

N(E) [p · ( n − n′ ) ]2. (58)

Let ψ be the angle between the vector p and the vector (n− n′), so that theEq. (58), can be written as,

I2 =(

ω

mec2

)2

c∫

d3p∫

dΩ

(

dσ

dΩ

)

N(p) p2 cos2ψ | n − n′ |2. (59)

Now since dσdΩ

does not depend on p, to lowest order in vc, 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,

N(p) = ne (2πmekBTe)3

2 exp

(

−p2

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,

I2 =ω2nekBTe

mec

∫

dΩdσ

dΩ| n − n′ |. (60)

17

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

I2 =2ω2nekBTe σT

mec, (61)

where we have used σT = 8π3r0

2. Now writing the above equation in termsof x, we have,

I2 =2neσT (kBTe)

2 x2

h2mec. (62)

Let us next consider I1. In the lowest order, ∆ ∝ p · ( n − n′ ) and atthis order I1 will be zero, when integrated over all p. Thus, to obtain thenon-zero contribution to I1 we need to expand the Eq. (56) to a higher orderin v

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

I1 =∆E

kBTe

σTnec, (63)

as the rate of collision is σTnec. Substituting the Eq. (35), in the Eq. (63),we obtain,

I1 =σTnec

kBTe

hω

mec2( 4kBTe − hω ). (64)

Writing the above equation in terms of x,

I1 =kBTe

mec2σTnec x( 4 − x ). (65)

Substituting the Eq. (62) and the Eq. (65), in the Eq. (51) and simplifyingone obtains,

mec2

kBTe

1

σT nec

∂n

∂t=

1

x2

∂

∂x

[

x4

(

∂n

∂x+ n + n2

)]

. (66)

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

∂y=

1

x2

∂

∂x

[

x4

(

∂n

∂x+ n + n2

)]

. (67)

18

The Eq. (67), is referred to as Kompaneets equation. The solution tothis equation describes the evolution of the photon spectrum that is dueto repeated scattering with a non-relativistic thermal bath of electrons. TheEq. (67) can be solved only for special cases and simple geometries. In generalcases, it must be solved by numerical integration.

As an illustration, let us consider the steady state solution to this Kom-paneets equation. For steady state situation, ∂n

∂y= 0, so that the Eq. (67)

can now be written as,∂n

∂x= −n (n + 1 ). (68)

Integrating the above equation we have,

n = [exp( x − x0 ) − 1 ]−1, (69)

which is the Bose-Einstein distribution with non-zero chemical potential µ =β−1 x0. We know that in the case of comptonization, the scattering betweenthe electrons and photons cannot change the total number of the photons,but can change the mean energy. Therefore the final configuration cannotbe a Planck spectrum, because of the constraints on both the number andthe energy. Hence the final distribution of the photons undergoing repeatedscattering with the electrons, will be a Bose-Einstein distribution. The βand µ of the distribution will be determined by the total number and energyof the photons. When x 1 we will have n 1 and n(x) ∝ exp−x. Thisspectrum 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, thatfills the Universe, when passes through these regions, they will be scatteredby the electrons ( which are at a much high temperature) and gain energy.(Note that Cosmic microwave background radiation is a black body radia-tion corresponding to a temperature of about 2.76K). This will distort thecosmic microwave background radiation spectrum in the vicinity of a clus-ter of galaxies. This comptonization of the cosmic microwave background

19

radiation by hot gas in the cluster of galaxies is usually referred to as theSunyaev-Zeldovich effect. It has been observed that, the highest level ofspectral deviation, around 1mK, are caused by Compton scattering of theradiation by hot gas in the clusters of galaxies. Spectral diminutions in theRayleigh-Jeans region and closer to the Planckian peak have been measuredin about a dozen clusters.

The Sunyaev-Zeldovich effect has its origin in the early work on spectraldistortions of the cosmic microwave background. A pure blackbody Planckspectrum is obviously not expected at all frequencies and in all directionsacross the sky, even after accounting for galactic effects (such as absorption,thermal emission from the interstellar gas and dust). To cause a spectraldistortion, a radiative process must occur sufficiently late in the cosmologi-cal evolution (i.e., at red-shifts z ≤ 3 × 106) to prevent the radiation fromthermalizing and regaining a pure Planck spectrum due to its weak cou-pling with the matter. In particular, the effect of Compton scattering of thecosmic microwave background radiation by the hot intergalactic medium wascalculated extensively by Sunyaev and Zeldovich (1972), and hence the nameSunyaev-Zeldovich effect.

7.1 Theory

The nature of the thermal Sunyaev-Zeldovich effect can be easily realized. Asphotons of the isotropic cosmic microwave background traverses through theintra-cluster medium, some are Compton scattered by hot intra-cluster elec-trons. Scattering off the moving electrons causes Doppler frequency shifts,and as the electron gas is very hot, the radiation gains energy. Conservationof photon number in the scattering implies that there is a systematic shift ofphotons from the Rayleigh-jeans region of the spectrum to the Wein side ofthe spectrum. The basic goal is the calculation of the spectral distributionof the scattered radiation field.

The time rate of change of the photon occupation number n, of anisotropic radiation field due to Compton scattering by isotropic, non-relativisticMaxwellian electron gas is given by the Kompaneets equation. This equa-tion - a non-relativistic Fokker-Planck approximation to the exact kinetic

20

equation, is given by

∂n

∂y=

1

x2

∂

∂x

[

x4

(

∂n

∂x+ n + n2

)]

, (70)

where x = hνkBTe

, y = kBTeσT nectmec2

, and Compton scattering here is taken in

the Thomson limit, hν mec2, 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 latterterms greatly simplifies the Eq. (70) to the following,

∂n

∂y=

1

x2

∂

∂x

(

x4 ∂n

∂x

)

. (71)

Here n is related to the radiation intensity Iν by,

n =Iνc

2

8πhν3. (72)

If the incident radiation is only weakly scattered, then an approximatesolution to the Eq. (71), can be obtained by substituting on the right handside the expression for the occupation number of a purely Planckian radiationfield, given by

n(x) =1

exp(x) − 1. (73)

Therefore the Eq. (71) takes the form,

∂n

∂y=

x exp(x)

(exp(x) − 1)2

[

x ( exp(x) + 1 )

exp(x) − 1− 4

]

. (74)

Now integrating along the path length through the cluster, we have,

∆n =x exp(x)

(exp(x) − 1)2

[

x ( exp(x) + 1 )

exp(x) − 1− 4

]

y. (75)

But from the Eq. (72),

∆Inr =8πhν3

c2∆n. (76)

21

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

∆Inr =8πh

c2

(

kBTex

h

)3

∆n. (77)

Therefore the change in the spectral intensity along the line of sight is givenby,

∆Inr = a y g(x), (78)

where a = 8π (kBTe)3

(hc)2and the subscript nr denotes the fact that the expression

was obtained in the non-relativistic limit. The spectral form of this thermalSunyaev-Zeldovich effect is expressed in the function,

g(x) =x exp(x)

(exp(x) − 1)2

[

x ( exp(x) + 1 )

exp(x) − 1− 4

]

, (79)

and is as shown in the Fig. (4). The spatial dependence of the effect iscontained in the Kompaneets parameter given by

y =∫

(

kBTe

mec2

)

ne σT dl, (80)

where the integral is over a line of sight through the cluster.

The radiation field gains energy through scattering by the much hotterintra-cluster gas, but with no change in the photon number, photons aretransferred from the Rayleigh-Jeans to the Wein side of the spectrum, asshown in the Fig. (4) by the spectral part of the intensity change, the func-tion 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 temper-ature. The Eq. (78) for the intensity change is valid approximation, in thenon-relativistic limit, only if these changes are small. Specifically, these ex-pressions are presumed to be accurate to first order in y, which is indeeda small parameter, typically of the order of the 10( − 4) in rich clusters.However, relativistic corrections are important, especially at high frequen-cies (for details see Rephaeli, 1995). Note that the above derivation holdsfor low optical thickness to Compton scattering in clusters. Note also that inthe above derivation it was implicitly assumed that neither the cluster grav-itational potential not intra-cluster gas properties change during the passageof the radiation through the cluster.

22

1 2 3 4 5 6 7 8 9 10−6

−4

−2

0

2

4

6

8

x

g(x

)

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

hνkBTe

.

7.2 Cluster Comptonization as a Probe

Comptonization of the cosmic microwave background radiation by hot gas inclusters imprints unique spectral signatures that can be used as importantastrophysical and cosmological probes. The full significance of the thermalSunyaev-Zeldovich effect as a cosmological probe has been appreciated withina decade following the work of the Sunyaev and Zeldovich (1972). Radio andmillimeter-band telescopes on the ground and sub-millimeter telescopes onthe mountain, in balloons, or in the orbit could be used to search for hotintergalactic gas in the clusters, as well as large-scale irregularities in the dis-tribution of such gas. Also one might find remote clusters and protoclustersof 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 back-ground radiation. The effect does not depend on the redshift of the cluster,a very desirable and uncommon feature that makes this effect an extremelyvaluable diagnostic cosmological and cluster evolutionary tool. The highmeasurement sensitivity now achievable at frequencies near and on the Weinside enables detection of the thermal effect at a high degree of confidence

23

based on its characteristic spectral profile there. The potential benefits frommapping the spectral imprints of Comptonization in clusters are particu-larly great when supplemented with detailed X-ray measurements of the gasdensity, temperature, and their spectral profiles. We note that because theproperties of a background radiation field are affected by the scattering bythe intra-cluster electrons, a positive intensity change across a cluster maypractically be viewed as a net emission from the cluster, whereas a negativechange makes the cluster an absorber of the cosmic microwave backgroundradiation. Thus, clusters are - in effect - powerful sinks and sources of emis-sion in the radio and submillimetric regions, respectively, whose spectralbrightness is independent of redshift.

A measurement of the thermal effect directly determines the integratedintra-cluster gas pressure, whereas that of the kinematic effect (to be dis-cussed later) yields the gas column density. Because of the linear dependenceof the Sunyaev-Zeldovich effect on gas density (ne), it is, in principle, easier tomap gas properties in the outer regions of clusters by Sunyaev-Zeldovich mea-surements than it is to map from measurements of X-ray emissivity (∝ ne

2).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-Zeldovicheffect and X-ray measurements of thermal emission from the intra-clustergas. Thus the value of the Hubble constant can be deduced from these mea-surements of a nearby cluster. Also effect is important as a source of cosmicmicrowave background anisotropy on arc-minute scales, and thereby servesas 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 rolein the growth of the large-scale structure, and it is obviously very importantto determine their velocities in order to fully study this role. The kinematicSunyaev-Zeldovich effect offers a relatively simple way to measure directly theradial component of the peculiar velocity of the cluster. The spectral changederived for thermal effect is caused by the random thermal motions of theelectrons 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 effectcan be obtained by a simple relativistic transformation, if it is assumed -based on the smallness of both the thermal and kinematic effects - that theeffects are separable. If so, the additional kinematic intensity change is

∆Ik = −a h(x)Vr

cτ, (81)

where

h(x) =x4 exp(x)

(exp(x) − 1)2, (82)

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

τ = σT

∫

ne dl, (83)

is the cluster optical depth to Compton scattering.

The observational feasibility of this method rests heavily on the relativemagnitudes of the kinematic and thermal effects. The velocity dependenceof the ratio ∆Ik/∆Inr is (Vr/c)/(ve/c)

2 (where ve is the rms thermal electronvelocity), while the spectral part of this dependence is given in the rationh(x)/g(x) (in the non-relativistic limit). Therefore, to avoid masking thekinematic effect completely, measurements have to be made at cross overfrequency (in the Fig. (4)), where ∆Inr = 0 and h(x) has a broad maximumcentered at cross over frequency. Scattering of the radiation in clusters alsoaffects its polarization state. There are several possible ways by which netpolarization can be induced; these include single and multiple scattering offintra-cluster electrons in a cluster moving perpendicular to the line-of-sight.Motion transverse to the line-of-sight induces a quadrupole component in thespatial distribution of the radiation; thus the radiation will appear linearlypolarized. thus the tangential component of the velocity of the cluster canalso be determined.

So far, we have considered only the effects of the interaction of cosmicmicrowave background photons with intra-cluster electrons. Compton scat-tering also effects the radiation at other parts of the electromagnetic back-ground. Measurements of the Compton signature may thus prove valuable

25

in the study of some of the other cosmological background radiations. Forexample, the effect can be used to distinguish between a truly cosmologicalcomponent 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. SpacePhys., 4:173.

[6] Yoel Rephaeli: 1995, Comptonization of the Cosmic Microwave Back-

ground: The Sunyaev-Zeldovich Effect, Annu. Rev. Astron. Astrophys.,33:541-79.

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