Leading Relativistic Corrections to the Kompaneets Equation

0.17 MB·English

by  Lowell S. Brown

LA-UR 12-00207
Leading Relativistic Corrections to the Kompaneets Equation
Lowell S. Brown and Dean L. Preston
Los Alamos National Laboratory
Los Alamos, New Mexico 87545
Abstract
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1
0 We calculate the first relativistic corrections to the Kompaneets equation for the evolution of
2
n the photon frequency distribution brought about by Compton scattering. The Lorentz invariant
a
J Boltzmann equation for electron-photon scattering is first specialized to isotropic electron and

1
3 photondistributions,thesquaredscatteringamplitudeandtheenergy-momentumconservingdelta

] function are each expanded to order v4/c4, averages over the directions of the electron and photon
O
C momenta are then carried out, and finally an integration over the photon energy yields our Fokker-
.
h
Planckequation. TheKompaneetsequation, whichinvolves onlyfirst-andsecond-orderderivatives
p
-
o with respect to the photon energy, results from the order v2/c2 terms, while the first relativistic
r
st corrections of order v4/c4 introduce third- and fourth-order derivatives. We emphasize that our
a
[
result holds when neither the electrons nor the photons are in thermal equilibrium; two effective

2
v temperatures characterize a general, non-thermal electron distribution. When the electrons are
6
0 in thermal equilibrium our relativistic Fokker-Planck equation is in complete agreement with the
6
5 most recent published results, but we both disagree with older work.
.
1
0
2
1
:
v
i
X
r
a
1
I. INTRODUCTION
The Kompaneets [1] equation,
∂ σ n 1 ∂ ∂f(t,ω)
f(t,ω) = T e ω4 T +h¯ 1+f(t,ω) f(t,ω) , (1.1)
∂t mec ω2 ∂ω ( ∂ω )
h i
describes the scattering of unpolarized, low energy photons of frequency ω on a dilute dis-
tribution of non-relativistic electrons when all the particles — photons and electrons — are
distributed isotropically in their momenta. The non-relativistic total photon-electron cross
section is the Thomson cross section σ . The electron number density and mass are denoted
T
by n and m . The photon phase space distribution f(t,ω) is normalized such that the
e e
number n of photons per unit volume is given by
γ
(d3k)
n (t) = 2 f(t,ω), (1.2)
γ (2π)3
Z
in which the prefactor 2 counts the number of photon polarizationstates and k is the photon
wave-number vector with k c = ω. If the electrons are in thermal equilibrium described
| |
by a Maxwell-Boltzmann distribution, then T is the temperature (in energy units) T of
e
this thermal distribution. However, the Kompaneets equation (1.1) holds for any isotropic
distribution of electron momenta with T defined to be 2/3 of the average energy in this
distribution [2]. For photons with a Planck distribution,
1
f(t,ω) f(0)(ω) = . (1.3)
→ exp h¯ω/T 1
γ
{ }−
The terms in the curly braces in the Kompaneets equation (1.1) vanish when T = T. In
γ
particular, if T = T = T , there is a time-independent photon distribution in thermal
γ e
equilibrium with the electrons1.
Our purpose here is to examine the first relativistic corrections to the Kompaneets equa-
tion. These corrections have been previously computed by Challinor and Lasenby (C&L)
[3] for the case in which the electrons are in a thermal distribution. Using the method of
C&L, Itoh, Kohyama, and Nozawa [4] carried out the expansion to a much higher order
in v/c. Subsequently, Sazonov and Sunyaev [5] confirmed the previous work of Challinor
1 Since Compton scattering preserves the photon number, the collision term on the right-hand side of
Eq. (1.1) also vanishes for a general Bose-Einstein distribution of massless particles at temperature T,
f(t,ω) f(α)(ω)=[exp (h¯ω/T) α 1]−1.
→ { − }−
2
and Lasenby. Here we use a method that is quite different from that employed by C&L, a
method that does not require that the electrons be in thermal equilibrium. Although the
structure of our result is quite different from that found by C&L, we agree with C&L in
the number of higher-order derivatives with respect to the photon frequency ω which must
supplement the Kompaneets equation to correctly account for the relativistic corrections.
Such higher-order derivative terms are missing from the ad-hoc treatments of Cooper [6]
and of Prasad, Shestakov, Kershaw, and Zimmerman [7]. These authors assume (incor-
rectly) that the relativistic corrections may be accounted for by simply replacing the factor
ω4 that stands just before the curly braces in Eq. (1.1) by a function α(ω,T) which is de-
termined so as to give the rate of change of the photon energy density including the first
relativistic corrections. We compute both the rate of energy exchange between the photons
and electrons and the Sunyaev–Zel’dovich effect [9–11] which follow from the relativistically
corrected Kompaneets equation. Including the first relativistic corrections, our results en-
tail two effective temperatures T and T which are defined by energy moments of the
eff1 eff2
electron phase-space distribution. When the electron distribution is restricted to a thermal,
relativistic Maxwell-Boltzmann distribution at temperature T, T = T = T and we
eff1 eff2
find, after some algebra, that our results that have a completely different structure are, in
fact, in complete agreement with those of C&L. Moreover, the rate of energy exchange that
we compute (also written down by C&L) agrees with that found earlier by Woodward [8].
Our presentation is organized as follows: After describing the general method we use in
Sec. II, we then outline the calculation in Sec. III using the results of several Appendices.
Finally, ourresultsareshowninSec.IV:Sec. IVApresents ourgeneralresult, Sec. IVBgives
its restriction to the case in which the photons are in thermal equilibrium at temperature
T , Sec. IVC derives the rate of energy transport between photons at temperature T and
γ γ
the electrons in a general distribution, and finally, in Sec. IVD the Sunyaev-Zel’dovich effect
for non-thermal electrons with the first relativistic correction is briefly described.
3
II. RELATIVISTIC BOLTZMANN EQUATION FOR ISOTROPIC SCATTERING
We start from the Lorentz invariant form of the Boltzmann equation for electron-photon
scattering:
(d3p′) 1 (d3k′) 1 (d3p) 1
k∂f(x,k) = (2π)4δ4(p′ +k′ p k)
(2π)3 2E′ (2π)3 2ω′ (2π)3 2E − −
Z
′ ′ 2 ′ ′ ′
T(p,k ;p,k) [1+f(x,k)]2g(x,p)f(x,k ) [1+f(x,k )]2g(x,p)f(x,k) .
−
(cid:12) (cid:12) n o
(cid:12) (cid:12) (2.1)
(cid:12) (cid:12)
Here we revert to units in which h¯ = 1 = c, but we shall return to conventional units when
we write the final result. The left-hand side of the equation involves the relativistic scalar
k∂ = ω(∂/∂t) +k . We are assuming that the electrons and photons are not polarized.
·∇
Hence T 2 denotes the square of the Lorentz invariant scattering amplitude that is summed
| |
over the initial and final electron and photon spins. It is divided by the initial electron spin
weight g = 2 so as to describe the average scattering from an initially unpolarized ensemble
e
of electrons. It is divided by the square of the photon spin weight g2 = 4 because initially
γ
there is an unpolarized mixture and finally the scattering is into the scalar density f(x,k)
that describes a typical photon [with the factor g = 2 needed to provide the photon number
γ
count in Eq. (1.2)]. The function g(x,p) is the electron phase space density. We choose our
Lorentz metric to have signature ( + ++) so that t = x0 = x while for the spatial
0
− −
coordinates xk = x .
k
We now specialize to the isotropic case of interest where f(x,k) f(t,ω) and g(x,p)
→ →
g(t,E), with the electron number density given by
(d3p)
n = 2 g(t,E). (2.2)
e (2π)3
Z
′
The integration variables p and p in Eq. (2.1) are dummy variables. We shall make the
′
interchange p p in the first set of terms in Eq. (2.1) so as to have a common factor of
↔
g(t,E) for the two ‘scattering in to’ and ‘scattering out of’ terms. To keep a convenient
form, we shall also use the detailed balance relation
T(p′,k′;p,k) 2 = T(p,k;p′,k′) 2 (2.3)
| | | |
for this first term in Eq. (2.1). Finally, we note that the p′ integration is best performed
4
using
(d3p′) 1 (d3p′) 1 (d4p′)
= = δ p′2 +m2 (2.4)
(2π)3 2E′ (2π)3 2 p′2 +m2 (2π)3 e
e (cid:16) (cid:17)
q
against the four-dimensional delta function which now replaces
′ ′
p = p+k k , (2.5)
−
giving
p′2 +m2 = 2p(k k′) 2kk′. (2.6)
e − −
In this fashion, we obtain
∂ (d3k′) 1 (d3p) 1
′ ′
ω f(t,ω) = 2g(t,E)2πδ(2p(k k ) 2kk )
∂t (2π)3 2ω′ (2π)3 2E − −
Z
′ ′ 2 ′
T(p,k;p,k ) [1+f(t,ω)]f(t,ω )
n(cid:12)(cid:12) T(p(cid:12)(cid:12)′,nk′;p,k) 2[1+f(t,ω′)]f(t,ω) , (2.7)
(cid:12) (cid:12)
−
(cid:12) (cid:12) o
(cid:12) (cid:12)
′ ′ (cid:12) ′ 2 (cid:12)
in which the four-momentum p in T(p,k;p,k ) is determined by Eq. (2.5).
The angular part of the integrat(cid:12)(cid:12)ions over p a(cid:12)(cid:12)nd k′ pick out the completely rotationally
(cid:12) (cid:12)
invariant part of the integrand. Thus, with angular brackets denoting the average over all
the orientations of the vectors within it, we may make the replacement
′ ′ ′ ′ 2 ′ ′ ′ ′ 2
δ 2p(k k ) 2kk T(p,k ;p,k) δ 2p(k k ) 2kk T(p,k ;p,k)
− − → − −
(cid:16) (cid:17)(cid:12) (cid:12) (cid:28) (cid:16) (cid:17)(cid:12) (cid:12) (cid:29)
(cid:12) (cid:12) s(p;ω′,ω). (cid:12) (cid:12) (2.8)
(cid:12) (cid:12) (cid:12) (cid:12)
≡
In view of these remarks, we may write Eq. (2.7) as
∂ 1 (d3p) 1
f(t,ω) = 2g(p)F(t,ω;p), (2.9)
∂t ω2 (2π)3 2E
Z
with
ω ∞
′ ′ ′ ′ ′ ′
F(t,ω;p) = ω dω s(p;ω,ω )[1+f(t,ω)]f(t,ω ) s(p;ω ,ω)[1+f(t,ω )]f(t,ω) .
2π −
Z0 n o
(2.10)
For the evaluation of Eq. (2.10) it is convenient to separate the weight that appears there
into symmetric and antisymmetric parts:
s(p;ω′,ω) = sS(p;ω′,ω)+sA(p;ω′,ω), (2.11)
5
with
sS(p;ω,ω′) = +sS(p;ω′,ω), (2.12)
and
sA(p;ω,ω′) = sA(p;ω′,ω). (2.13)
−
With this decomposition, Eq. (2.10) becomes
ω ∞
F(t,ω;p) = ω′dω′ sS(p;ω′,ω) sA(p;ω′,ω) f(t,ω′)
2π −
Z0 ω h ∞ i
f(t,ω) ω′dω′ sS(p;ω′,ω)+sA(p;ω′,ω)
−2π
ω Z0 ∞ h i
2f(t,ω) ω′dω′sA(p;ω′,ω)f(t,ω′). (2.14)
−2π
Z0
III. EXPANSIONS AND ANGULAR AVERAGES
It proves convenient to use the anglesα , α′ between pandk , k′, andthe angleθ between
k and k′. We also use the velocity
p p
v = | | = | | < 1. (3.1)
E E( p )
| |
The delta function in Eq. (2.8) now becomes
1
′ ′ ′ ′ ′
δ 2p(k k ) 2kk = δ ω(1 vcosα) ω (1 vcosα) (ωω /E)(1 cosθ) ,
− − 2E − − − − −
(cid:16) (cid:17) (cid:16) (cid:17)
(3.2)
and the squared scattering amplitude (A8) now appears as
′
1 cosθ ωω
T(p′,k′;p,k) 2 = 6πm2σ 2+ − (1 cosθ) 2 1 v2
| | e T( (1 vcosα)(1 vcosα′) " E2 − − − #
− − (cid:16) (cid:17)
2 (1 cosθ)2
+ 1 v2 − . (3.3)
− (1 vcosα)2(1 vcosα′)2)
(cid:16) (cid:17) − −
As we shall see, the Kompaneets equation results from expanding Eqs. (3.2) and (3.3) in
powers of v, with this equation resulting from the order v2 terms. The leading relativistic
corrections that concern us require that the expansion be carried out to order v4. We note
that since in the applications that we envisage, ω, ω′ T p2/m v2E, ω/E or ω′/E
e
∼ ∼ ∼
should be counted as being of order v2.
6
The needed expansion of the delta function (3.2) in powers of v reads
′ ′ ′
2Eδ 2p(k k ) 2kk = δ ω ω
− − −
(cid:16) (cid:17) (cid:16) (cid:17)
′ ′ ′ ′
v(ωcosα ω cosα)δ ω ω
− " − − #
(cid:16) (cid:17)
1
+ v2(ωcosα ω′cosα′)2δ′′ ω ω′
"2 − −
(cid:16) (cid:17)
′
ωω
′ ′
(1 cosθ)δ ω ω
− E − − #
(cid:16) (cid:17)
1
v3(ωcosα ω′cosα′)3δ′′′ ω ω′
− "3! − −
(cid:16) (cid:17)
′
ωω
′ ′ ′′ ′
v(ωcosα ω cosα) (1 cosθ)δ ω ω
− − E − − #
(cid:16) (cid:17)
1
+ v4(ωcosα ω′cosα′)4δ′′′′ ω ω′
"4! − −
(cid:16) (cid:17)
′
1 ωω
v2(ωcosα ω′cosα′)2 (1 cosθ)δ′′′ ω ω′
−2 − E − −
(cid:16) (cid:17)
′ 2
1 ωω
+ (1 cosθ)2δ′′ ω ω′ + (v5). (3.4)
2 E ! − − # O
(cid:16) (cid:17)
The term of order v0, δ(ω ω′), makes no contribution, since for it the two parts of the
−
collision integral, the ‘scattering in to’ and the ‘scattering out of’, cancel. Hence we need
expand the squared amplitude (3.3) only to order v3 to obtain results good to order v4:
T(p′,k′;p,k) 2 6πm2σ v0 1+cos2θ
| | ≃ e T( " #
(cid:16) (cid:17)
′
2v (1 cosθ) (cosα+cosα) cosθ
− − " #
+ v2 (1 cosθ) cos2α+cos2α′ +2cosαcosα′
− "
(cid:16) (cid:17)
+ 2 3cos2α 3cos2α′ 4cosαcosα′ cosθ
− − − #
(cid:16) (cid:17)
+ v3 (1 cosθ) 2cosα 2cosα′ +2cos3α+2cos3α′
− " − −
(cid:16)
+4cos2αcosα′ +4cosαcos2α′
(cid:17)
4cosα 4cosα′ +4cos3α+4cos3α′
− − −
(cid:16)
+6cos2αcosα′ +6cosαcos2α′ cosθ . (3.5)
#)
(cid:17)
We now multiply the expressions (3.4) and (3.5) together and retain the resulting terms
up to those of order v4. We then use the result of angular averaging over the directions of p
7
detailed in Appendix B and subsequently average over the direction between k and k′, the
average over cosθ. To present the results in a compact form, we separately record the order
v2 result, the one that gives the Kompaneets equation,
s (p;ω′,ω) = T(p′,k′;p,k) 2δ 2p(k k′) 2kk′
2
h| | − − i 2
= 4π m2e σ v2 ωω(cid:16)′ δ′′ ω ω′ ωω(cid:17)′ δ(cid:12)(cid:12)(cid:12)′ ω ω′ , (3.6)
T
E ( 3 − − E − )
(cid:16) (cid:17) (cid:16) (cid:17)
and the order v4 result which gives the leading relativistic correction to the Kompaneets
equation
s (p;ω′,ω) = T(p′,k′;p,k) 2δ 2p(k k′) 2kk′
4
h| | − − i 4
= π m2e σ v4 2((cid:16)ω ω′)2ωω′+ 14(cid:17)ω2(cid:12)(cid:12)(cid:12)ω′2 δ′′′′ ω ω′
T
15 E ( (cid:20) − 5 # (cid:16) − (cid:17)
′
2 14 ωω
10v2 v2 (ω ω′)ωω′+ (ω ω′)2 + ωω′ δ′′′ ω ω′
− "25 − (cid:18) − 5 (cid:19) E # (cid:16) − (cid:17)
′ ′ 2
16 ωω ωω
+ v4ωω′ +4v2 (ω ω′) +42 δ′′ ω ω′
"− 5 − E E ! # −
(cid:16) (cid:17)
′
ωω
+ 12v2 δ′ ω ω′ . (3.7)
" E # − )
(cid:16) (cid:17)
There are no terms of odd order in v in the somewhat lengthly algebra required to obtain
these formulae. They simplify with the aid of the delta function identities presented in
Appendix C:
ωω′ (ωω′)2
sS(p;ω′,ω) = 2σ v2 δ′′ ω ω′ , (3.8)
2πm2 2 T 3E −
e (cid:16) (cid:17)
ωω′ (ωω′)2
sA(p;ω′,ω) = 2σ δ′ ω ω′ , (3.9)
2πm2 2 − T E2 −
e (cid:16) (cid:17)
and
ωω′ (ωω′)2 7
sS(p;ω′,ω) = σ v4ωω′δ′′′′(ω ω′)
2πm2 4 T 15E (5 −
e
′
58 ωω
+ v4 +21 δ′′(ω ω′) , (3.10)
" 5 E2 # − )
ωω′ (ωω′)2
sA(p;ω′,ω) = σ v2 14ωω′δ′′′ ω ω′ +28δ′(ω ω′) . (3.11)
2πm2 4 − T 15E2 ( − − )
e (cid:16) (cid:17)
8
IV. RESULTS
It is now a straightforward although tedious matter to insert the forms above into
′
Eq. (2.14), perform the ω integrals, and place the results into Eq. (2.9) to secure the
Kompaneets equation and its leading relativistic corrections.
A. General Result
∂ n d df(t,ω)
ω2 f(t,ω) = σ e ω4 T +h¯ [1+f(t,ω)] f(t,ω)
T eff1
∂t mec dω ( dω
1 5 df(t,ω) 47
+ T2 +21T2 + T h¯ [1+f(t,ω)] f(t,ω)
m c2 2 eff2 eff1 dω 2m c2 eff1
e (cid:18) (cid:19) e
7h¯ω2 df(t,ω) 2 df(t,ω) 1 d3f(t,ω)
6T h¯ +T2
− 10mec2 " eff1 dω ! − dω eff1h¯ dω3 #)
21n σ d2 df(t,ω)
e T ω5 T2 +2T2 +3T h¯[1+f(t,ω)]f(t,ω)
− 5m2ec3 dω2 ((cid:16) eff2 eff1(cid:17) dω eff1 (cid:27)
7n σ d3 df(t,ω)
+ e T ω6 T2 +T2 +2T h¯[1+f(t,ω)]f(t,ω) .
10m2c3dω3 ( eff2 eff1 dω eff1 )
e (cid:16) (cid:17)
(4.1)
Here we have reverted to conventional units and used the effective temperature definitions
(D5) and (D7) in Appendix D which, for convenience, we repeat here:
1 (d3p) p2c2
T = 2g(t,E), (4.2)
eff1 n (2π)3 3E(p)
e Z
and
4 (d3p) p2 2
T2 = 2g(t,E). (4.3)
eff2 15ne Z (2π)3 2me!
Wehave chosen toorder the terms inour result (4.1) so asto have successive partsinvolve
overall higher derivatives. Since each part starts out with at least one overall derivative, the
result conserves photon number as it must. Only the part with the single overall derivative
d/dω contributes to the rate of energy exchange between the photons and the electrons.
Similarly, the rate at which the second moment (h¯ω)2 changes with time is affected only
by the parts involving d/dω and d2/dω2 while all the parts contribute to the time rate of
change of the (h¯ω)3 moment. We have kept some photon frequency derivatives within the
sequence ofincreasingly higheroverall derivativesothatthesumofthetermsineachofthese
9
groups vanishes in thermal equilibrium. Each of these groupings in the result (4.1) vanishes,
in fact, in the more general situation in which the photon distribution is of the Planck
form (1.3) but with the electron g(E) constrained only to have T = T = T . With
eff1 eff2 γ
the electrons in thermal equilibrium, T reduces without approximation to the electron
eff1
temperature T , and T T in first approximation, an approximation that suffices since
e eff2 e
→
T only occurs in the relativistic correction terms. In this case, each of these groupings of
eff2
terms vanishes for a Planck distribution when T = T . Our expression (4.1) is in complete
γ e
agreement with the work of Challinor and Lasenby [3] in the limit in which the electrons are
in thermal equilibrium at temperature T . However, as mentioned previously, the structure
e
of our expression (4.1) for electrons in thermal equilibrium — which is equivalent to that of
C&L — differs completely from the previous (incorrect) results of Cooper [6] and of Prasad,
Shestakov, Kershaw, and Zimmerman [7].
B. Photons In Thermal Equilibrium
For photons in thermal equilibrium at a temperature T [the Planck distribution (1.2)
γ
or, more generally, a Bose-Einstein distribution of photons] the result reduces to
∂ d n 21 h¯2ω2
ω2 f(t,ω) = 4 ω3σ e 1 T T
∂t − dω T mec ( − 20 mec2Tγ! eff1 − γ
(cid:16) (cid:17)
5
+ T2 T T f (ω)
2mec2 eff2 − eff1 γ ) 0
(cid:16) (cid:17)
d2 n 7 h¯2ω2
+ ω4σ e 1 T T
dω2 T mec ( − 10 mec2Tγ! eff1 − γ
(cid:16) (cid:17)
47
+ T2 T T f (ω)
2mec2 eff2 − eff1 γ ) 0
(cid:16) (cid:17)
d3 42n
ω5σ e T2 T T f (ω)
− dω3 T 5m2c3 eff2 − eff1 γ 0
e (cid:16) (cid:17)
d4 7n
+ ω6σ e T2 T T f (ω). (4.4)
dω4 T 10m2c3 eff2 − eff1 γ 0
e (cid:16) (cid:17)
C. Energy Transport
The energy transfer per unit volume to the photons is given by
(d3k) ∂
u˙ = 2 ω f(t,ω). (4.5)
γ
(2π)3 ∂t
Z
10