Scaling Laws in High-Energy Inverse Compton Scattering

0.34 MB·English

by  Satoshi Nozawa

Scaling Laws in High-Energy Inverse Compton Scattering
Satoshi Nozawa∗
Josai Junior College, 1-1 Keyakidai, Sakado-shi, Saitama, 350-0295, Japan
Yasuharu Kohyama and Naoki Itoh
Department of Physics, Sophia University, 7-1 Kioi-cho, Chiyoda-ku, Tokyo, 102-8554, Japan
(Dated: January 22, 2010)
Basedupontherateequationsforthephotondistributionfunctionobtainedinthepreviouspaper,
we study the inverse Compton scattering process for high-energy nonthermal electrons. Assuming
the power-law electron distribution, we find a scaling law in the probability distribution function
0
P1(s), where the peak height and peak position depend only on the power index parameter. We
1
solved the rate equation analytically. It is found that the spectral intensity function also has the
0
scalinglaw,wherethepeakheightandpeakpositiondependonlyonthepowerindexparameter. The
2
present study will be particularly important to the analysis of the X-ray and gamma-ray emission

n models from various astrophysical objects such as radio galaxies and supernovaremnants.
a
J PACSnumbers: 95.30.Cq,95.30.Jx,98.65.Cw,98.70.Vc

2 Keywords: cosmology: cosmicmicrowavebackground—cosmology: theory—galaxies: clusters: general—
2 radiationmechanisms: nonthermal—relativity

]
O I. INTRODUCTION Inthepresentpaper,weextendtheformalismobtained
C by NK to the case of high-energy electrons. This exten-
sion will be particularly interesting for the analysis of
.
h TheinverseComptonscatteringisoneofthemostfun-
X-ray and gamma-ray emissions, for example, from ra-
p damental reactions which have variety of applications to
dio galaxies[2] and supernova remnants[4, 5], where the
- astrophysics and cosmology. They are, for example, the
o inverseComptonscatteringoftheCMBphotonsoffnon-
Sunyaev-Zeldovich (SZ) effects[1] for clusters of galax-
r thermal high-energy electrons plays an essential role. In
t ies (CG), cosmic-rayemission from radio galaxies[2] and
s the present approach, we push analytic techniques as
a clusters of galaxies[3], and radio to gamma-ray emission
muchaspossibleinordertoobtainanalyticsolutions. In
[ from supernova remnants[4, 5]. Therefore, theoretical
contrast to the direct numerical calculation, the present
studies on the inverse Compton scattering have been
2 approachwill haveanadvantagethatone may reveales-
v done quite extensively for the last forty years, start- sential physics properties behind the numerical results.
2 ing from the works by Jones[6], and Blumenthal and
In the present paper, under a specific condition for the
2 Gould[7]tothe recentworks,forexample,by Fargion[8],
electron distribution which is typically realized, we will
2 Colafrancesco[9, 10], and Petruk[11].
2 showthata universalscalinglawis valid forthe spectral
. In particular, remarkable progress has been made in intensity function.
2
theoretical studies for the SZ effects for CG. Wright[12]
1
and Rephaeli[13] calculated the photon frequency redis-
9
0 tribution function in the electron rest frame, which is
: called as the radiative transfer method. On the other
v
hand,ChallinorandLasenby[14]andItoh,Kohyama,and
i
X Nozawa[15] solved the relativistically covariant Boltz-
r mann collisional equation for the photon distribution The present paper is organized as follows: Starting
a function, which is called the covariant formalism. Al- from the rate equations derived in the NK paper, we
though the two are very different approaches, the ob- derive in Sec. II the analytic expressions for the redis-
tained results for the SZ effect agreed extremely well. tribution function P(s,γ) and probability distribution
This has been a longstanding puzzle in the field of the function P1(s). Assuming the power-law electron dis-
relativistic study of the SZ effect for the last ten years. tribution, we show that P1(s) has a scaling law, where
Very recently, however, Nozawa and Kohyama[16] (de- the peak height and peak position depend only on the
notedNKhereafter)showedthatthetwoformalismswere powerindex parameter. We calculatedthe rate equation
indeed mathematically equivalent in the approximation and obtained the analytic expression for the spectral in-
oftheThomsonlimit. Thisexplainedthereasonwhythe tensityfunctiondI(X)/dτ. We showthatdI(X)/dτ also
twodifferentapproachesproducedsameresultsfortheSZ has the scaling law, where the peak height and peak po-
effect even in the relativistic energies for electrons. sition depend only on the power index parameter. In
Sec. III we apply the scaling law to the observation of
the spectral intensity in the X-ray and gamma-ray en-
ergy regions. Finally, concluding remarks are given in
∗Electronicaddress: [email protected] Sec. IV.
2
II. HIGH-ENERGY INVERSE COMPTON follows:
SCATTERING
β =(1 e−|s|)/(1+e−|s|), (9)
min
−
A. Rate Equations in Thomson Approximation µ′0 =[1−es(1−βµ0)]/β, (10)
1 for s 0
In the NK paper, it was shown that the covariant µ1(s)= −[1 e−s(1+β)]/β for s>≤0 , (11)
(cid:26) −
formalism[15] and radiative transfer method[12] were [1 e−s(1 β)]/β for s<0
mathematically equivalent in the following (Thomson) µ2(s)= 1 − − for s 0 . (12)
approximation: (cid:26) ≥
The total probabilities for P(s,β) and P (s) are given
ω 1
γ 1, (1) by
m ≪
γ = 1 , (2) +λβ
1 β2 dsP(s,β)=1, (13)
− Z−λβ
p ∞
whereωisthephotonenergy,γ istheLorentzfactor,and dsP (s)=1, (14)
1
β and m are the velocity and rest mass of the electron, Z−∞
respectively. Throughout this paper, we use the natu-
ral unit ~ = c = 1, unless otherwise stated explicitly. where
For the cosmic microwave background (CMB) photons 1+β
(skaBtisTfiCeMd.BT=hen2.3ω4/8m×<101−4e1V0)−,8,ωw<hic5h×im1p0l−ie3seVγ is 1w0e8l.l λβ =ln(cid:18)1−β(cid:19) . (15)
× ≪
Therefore as far as the CMB photons are concerned, It should be noted that the following useful relations
Eq. (1) is fully valid from non-relativistic electrons to
extreme-relativistic electrons of the order of TeV region. P(s,β)e−3s =P( s,β), (16)
−
Therateequationsforthephotondistributionfunction P (s)e−3s =P ( s) (17)
1 1
n(x) andspectralintensity functionI(x) werederivedin −
the NK paper under the assumption of Eq. (1). Here, are valid.
x = ω/k T is the photon energy in units of the
B CMB
thermal energy of the CMB. We recall the results here
tomakethepresentpapermoreself-contained. Theyare B. P(s,β) for Extreme-Relativistic Electrons
given as follows[16, 17]:
In this section, we derive the analytic expression of
∂n(x) = ∞ dsP (s)[n(esx) n(x)] , (3) thefrequencyredistributionfunctionP(s,β)forextreme-
1
∂τ − relativistic electrons. In Eq. (7), the integral of µ can
Z−∞ 0
∂I(x) ∞ be done analytically. One obtains as follows: for s<0,
= dsP (s) I(e−sx) I(x) , (4)
1
∂τ −
Z−∞τ =neσ(cid:2)Tt, (cid:3) (5) P(s,β)= 32β32γ4 −C1(β)−C2(β)es+C3(β)e2s
+C (β)((cid:2)λ +s)(es+e2s)+C (β)e3s , (18)
where I(x) = I x3n(x), I = (k T )3/2π2, n is the 4 β 1
0 0 B CMB e
electron number density, σT is the Thomson scattering and for s 0, (cid:3)
crosssection. InEqs.(3)and(4),P (s)istheprobability ≥
1
distribution function for the photon of a frequency shift 3
P(s,β)= C (β)+C (β)es C (β)e2s
s, which is defined by es =x′/x, 32β2γ4 1 3 − 2
+C (β)((cid:2)λ s)(es+e2s) C (β)e3s , (19)
4 β 1
1 − −
P1(s)= dββ2γ5pe(E)P(s,β), (6) where the coefficients are (cid:3)
Zβmin
es µ2(s) 1 1
P(s,β)= 2βγ4 dµ0(1 βµ )2f(µ0,µ′0) , (7) C1(β)= β4γ2 , (20)
Zµ1(s) − 0
1
f(µ0,µ′0)= 38 1+µ20µ′02+ 21(1−µ20)(1−µ′02) , (8) C2(β)= β4(1+β) 4β4−β3−13β2−3β+9 , (21)
(cid:20) (cid:21) 1 (cid:0) (cid:1)
C (β)= 4β4+β3 13β2+3β+9 , (22)
wherep (E)istheelectrondistributionfunctionofamo- 3 β4(1 β) −
mentumepwhichisnormalizedby ∞dpp2p (E)/m3 =1. 2 − (cid:0) (cid:1)
0 e C (β)= β2 3 . (23)
Variables appearing in Eqs. (6) – (8) are summarized as 4 β4 −
R
(cid:0) (cid:1)
3
Note that Eqs. (18) and (19) satisfy the relation of Letusnowcomparethepresentresultswiththelitera-
Eq. (16). It should be also noted that Eqs. (18) and ture. ItisstraightforwardtoshowthatEqs.(30)and(31)
(19) agree with Eqs. (23a)and (23b) of Fargionet al.[8], are equivalent to Eqs. (38) and (40) of Jones[6], respec-
respectively. tively. We showthe equivalencebetweenthe presentfor-
Now let us consider the case for electrons of extreme- malism and Jones’s formalism in Appendix A. It should
relativistic energies E (= γmc2) mc2. Thus, γ 1 be also mentioned that Eqs. (24a) and (24b) of Fargion
≫ ≫
andβ 1areassumed. Inthisapproximation,Eqs.(18) etal.[8]differfromourEqs.(30)and(31). Thedifference
and (1≈9) are written as follows: for s<0, comes from O(1/γ2) terms as mentioned in their paper.
Before closing this subsection, it should be also noted
3 1
P(s,γ)= +2es+8γ2e2s the following: In the present formalism, the cases s 0
32γ4 −γ2 and s<0 correspondto the Compton scattering and≥in-
(cid:20)
1 verse Compton scattering, respectively. This is because
−4(λγ +s) es+e2s + γ2e3s , (24) of the definition x = e−sx′, where x′ and x are the en-
(cid:21) ergies (in units of k T ) of initial and final photons,
(cid:0) (cid:1) B CMB
and for s 0, respectively. Equations (27) and (28) suggest that prob-
≥
ability distribution for the CMB photon scattering by
3 1 high-energyelectronsisdominatedbytheComptonscat-
P(s,γ)= +8γ2es+2e2s
32γ4 γ2 tering process instead of the inverse Compton scattering
(cid:20)
process.
1
4(λ s) es+e2s e3s , (25)
− γ − − γ2
(cid:21)
λ =2ln(2γ), (cid:0) (cid:1) (26)
γ
C. Scaling Law of P1(s) for Nonthermal Electrons
wheretheexpressionP(s,γ)wasusedinsteadofP(s,β).
Equations (24) and (25) can be integrated analytically.
In order to proceed calculation for practical applica-
One obtains as follows: for s<0,
tions, let us specify the electron distribution function.
0 1 High-energyelectronsinthesupernovaremnantsandac-
dsP(s,γ)=O , (27) tivegalacticnuclei,forexample,aremostlikelynonther-
γ2
Z−λγ (cid:18) (cid:19) mal. It is standard to describe the nonthermal distribu-
tion in terms of the power-law distribution function of
and for s 0,
≥ three parameters:
λγ 1
Z0 dsP(s,γ)=1+O(cid:18)γ2(cid:19) . (28) pe(γ)= 0N,γγ−σ, γelmseinwh≤erγe≤γmax , (32)
(cid:26)
In Eq. (28), the terms contributing to O 1/γ2 are the
higher-orderterms. Thereforeone caneliminate the cor- whereγ istheLorentzfactorandN isthenormalization
(cid:0) (cid:1) γ
respondingterms fromEq.(25), whichgivesthe new ex- constant. In Eq. (32), σ is the power index parameter,
pression for s 0. Then the new expression for s < 0 γ and γ are parameters of minimum and maxi-
≥ min max
is obtained with the relation of Eq. (16). Therefore, the mum values for γ, respectively. Then, Eq. (6) can be
total probability reexpressed as follows: for s<0,
+λγ 1 γmax
dsP(s,γ)=1+O (29)
P (s)= dγp (γ)P(s,γ), (33)
γ2 1 e
Z−λγ (cid:18) (cid:19) Zmax(γmin,e−s/2/2)
is satisfied for P(s,γ). The explicit forms are as follows:
where P(s,γ) is given by Eq. (30), and for s 0,
for s<0, ≥
3 1 γmax
P(s,γ)= +2es+8γ2e2s P (s)= dγp (γ)P(s,γ), (34)
32γ4 −γ2 1 e
h Zmax(γmin,es/2/2)
4(λ +s)es , (30)
γ
−
where P(s,γ) is given by Eq. (31). In deriving Eqs. (33)
i
and for s 0, and(34), β 1wasassumed,andthephasespacefactor
≥ γ2 was abso≈rbed, for simplicity, by the power index σ in
3 p (γ).
P(s,γ)= 8γ2es+2e2s e
32γ4 In the case of the power-law distribution of Eq. (32),
h
1 equations (33) and (34) can be integrated analytically.
4(λ s)e2s e3s . (31)
− γ − − γ2 Theexplicitformsaregivenasfollows: for 2ln2γmax <
−
i
4
s< 2ln2γ , the Compton scattering and Inverse Compton scatter-
min
−
ing, respectively. First, we define P (s,R) as follows:
C
3 1 1 for 2ln2γ <s<0,
P1(s)= 32Nγ −σ+5 2σ+5e(σ+5)s/2− γσ+5 − min
(cid:26) (cid:18) max(cid:19) σ 1 2
+σ+2 3(cid:20)σσ+−312σ+3e(σ+3)s/2 PC(s,R)=31−−R1σ−1σ(cid:26)−1σ+5(cid:0)1−Rσ+5(cid:1)e3s
1 σ 1 + − +2s
−γσ+3 σ+−3 −2s−4ln2γmax es σ+3(cid:20)σ+3
max (cid:18) (cid:19)(cid:21) σ 1
8 1 Rσ+3 − +2s+4lnR e2s
+ 2σ+1e(σ+1)s/2 e2s , (35) − σ+3
σ+1 − γσ+1 (cid:18) (cid:19)(cid:21)
(cid:18) max(cid:19) (cid:27) 1
+ 1 Rσ+1 es , (40)
σ+1 −
for 2ln2γmin <s<0, (cid:27)
− (cid:0) (cid:1)
and for 0<s<2ln(γ /γ ),
max min
3 1 1 1
P (s)= N
1 32 γ(cid:26)−σ+5(cid:18)γmσ+in5 − γmσ+ax5(cid:19) PC(s,R)=31 σ−Rσ1−1 −σ+2 5 e−(σ+5)s/2−Rσ+5 e3s
2 1 σ 1 − (cid:26) (cid:16) (cid:17)
+σ+3 γσ+3 σ−+3 −2s−4ln2γmin + 1 σ−1e−(σ+3)s/2
(cid:20) min (cid:18) (cid:19) σ+3 σ+3
1 σ 1 (cid:20)
−γmσ+ax3 (cid:18)σ+−3 −2s−4ln2γmax(cid:19)(cid:21)es −Rσ+3 σσ+−31 +2s+4lnR e2s
8 1 1 (cid:18) (cid:19)(cid:21)
+ e2s , (36) 1
σ+1 γσ+1 − γσ+1 + e−(σ+1)s/2 Rσ+1 es . (41)
(cid:18) min max(cid:19) (cid:27) σ+1 −
(cid:27)
(cid:16) (cid:17)
for 0<s<2ln2γmin, Similarly, PI′C(s,R) is for −2ln(γmax/γmin)<s<0,
3 1 1 1 σ 1 2
P (s)= N e3s P′ (s,R)=3 − e(σ+5)s/2 Rσ+5
1 32 γ −σ+5 γσ+5 − γσ+5 IC 1 Rσ−1 −σ+5 −
(cid:26) (cid:18) min max(cid:19) − (cid:26) (cid:16) (cid:17)
+ 2 1 σ−1 +2s 4ln2γ + 1 σ−1e(σ+3)s/2
σ+3 γσ+3 σ+3 − min σ+3 σ+3
(cid:20) min (cid:18) (cid:19) (cid:20)
1 σ−1 +2s 4ln2γ e2s Rσ+3 σ−1 2s+4lnR es
−γmσ+ax3 (cid:18)σ+3 − max(cid:19)(cid:21) − (cid:18)σ+3 − (cid:19)(cid:21)
+ 8 1 1 es , (37) + 1 e(σ+1)s/2 Rσ+1 e2s , (42)
σ+1(cid:18)γmσ+in1 − γmσ+ax1(cid:19) (cid:27) σ+1(cid:16) − (cid:17) (cid:27)
and for 0<s<2ln2γ ,
min
and for 2ln2γ <s<2ln2γ ,
min max
σ 1 2
P′ (s,R)=3 − 1 Rσ+5
3 1 1 IC 1 Rσ−1 −σ+5 −
P1(s)= 32Nγ −σ+5 2σ+5e−(σ+5)s/2− γσ+5 e3s −1 σ(cid:26) 1 (cid:0) (cid:1)
(cid:26) (cid:18) max(cid:19) + − 2s
2 σ 1 σ+3 σ+3 −
+ − 2σ+3e−(σ+3)s/2 (cid:20)
σ+3 σ+3 σ 1
(cid:20) Rσ+3 − 2s+4lnR es
1 σ 1 − σ+3 −
− +2s 4ln2γ e2s (cid:18) (cid:19)(cid:21)
−γσ+3 σ+3 − max 1
max (cid:18) (cid:19)(cid:21) + 1 Rσ+1 e2s . (43)
8 1 σ+1 −
+ 2σ+1e−(σ+1)s/2 es . (38) (cid:27)
σ+1(cid:18) − γmσ+ax1(cid:19) (cid:27) It is straightforwardto s(cid:0)how that (cid:1)
It should be noted that the normalization constant is P (s,R)e−3s =P′ ( s,R) (44)
C IC −
given by
is satisfied by Eqs. (40)–(43).
N =(σ 1)γσ−1 (39) Comparing Eqs. (35)–(38) with Eqs. (40)–(43), the
γ − min probabilitydistributionfunctionP (s)isdescribedasfol-
1
lows:
for the case γ .
max
→∞
Let us now introduce new functions P (s,R) and P (s+2ln2γ ,R) fors<0
C IC min
P′ (s,R) in order to express Eqs. (35)–(38) in unified P (s)= , (45)
IC 1 
forms, where R = γ /γ . Here, C and IC denote P (s 2ln2γ ,R) fors 0
min max  C − min ≥

5
where
1
P (s,R) P′ (s,R). (46)
IC ≡ 64γ6 IC
min
LetusnowconsiderthecaseR γ /γ 1. We
min max
fix γ = 108 throughout the p≡aper. In Fig.≪1(a), we
max
plot P (s) as a function of s for a typical value σ = 2.5.
1
The solid curve, dash-dotted curve, dashed curve, and
dotted curve correspond to γ = 10, 102, 103, and
min
104, respectively. It can be seen that the height of P (s)
1
is independent of γ . In Fig. 1(b), we plot the same
min
curves as a function of new variable s which is defined
C
by
s =s 2ln2γ . (47)
C min
−
In Fig.1(b)the four curvesare totally indistinguishable,
whichexhibitsascalinglawforP (s). Thereasonforthis
1
scaling lawis as below. For largeγ 1, as shownby
min
≫
Figs. 1(a), 1(b), and Eqs. (45) and (46), the probability
distribution function P (s) is dominated by P (s ,0),
1 C C
i.e. by the Compton scattering process.
Before closing this subsection, we study the σ-
dependences on the peak position s and peak height
peak
P (s ). As shown in Figs. 1(a) and 1(b), the γ -
1 peak min
dependence of P (s) is described by Eq. (47), namely, s
1
=s +2ln2γ . Therefore,wedefinethepeakposition
C min
by
s =s(σ)+2ln2γ , (48)
peak min
where s(σ) depends only on σ. The peak position is
calculated by solving the equation
FIG.1: PlottingofP1(s)andP1(sC)forσ=2.5. Figures1(a)
∂P1(s) and1(b)areP1(s)andP1(sC),respectively. Thesolid curve,
=0. (49)
dash-dottedcurve,dashedcurve,anddottedcurvecorrespond
∂s
(cid:12)(cid:12)speak to γmin = 10, 102,103, and 104, respectively.
(cid:12)
The analytic expressions(cid:12)for s(σ) in the first-order and
third-order approximations are given as follows:
plot s(σ) and P (s ), respectively. The dashed curve,
1 peak
(σ 1)(σ2+4σ+11) dash-dottedcurve,andsolidcurvecorrespondtos (σ),
1st
s (σ)= − , (50)
1st −5σ3+23σ2+51σ+17 s3rd(σ) andsnum(σ), respectively. InFig.2(b), the solid
curveanddash-dottedcurveareindistinguishable. Itcan
1
s (σ)= 3σ2+14σ+19 (51) be seenfromFigs.2(a)and2(b)thatthe third-orderap-
3rd −2(4σ2+21σ+29)" proximation is sufficiently accurate for the present pur-
poses.
1/3
(σ+1)2A3+B2+B
+
σ+1
p !
D. Scaling Law for Spectral Intensity Function
1/3
(σ+1)2A3+B2 B
− , (52)
− p σ+1 !  Let us now solve the rate equations of Eqs. (3) and
(4) with the result of Eq. (45) for P (s). We consider
1
 the CMB photons for the initial distribution. For the
A=7σ4+64σ3+254σ2+520σ+451, (53) inverseCompton scattering by high-energyelectrons, we
are interested in high-energy spectrum such as X-rays
B =3σ7 21σ6 582σ5 4378σ4 18589σ3
− − − − ( keV) and gamma-rays ( MeV). Therefore, one can
48333σ2 70688σ 44036. (54) s∼afely assume ∼
− − −
We also solved Eq. (49) numerically and obtained the ω
x 1 (55)
numerical solution snum(σ). In Figs. 2(a) and 2(b), we ≡ kBTCMB ≫
6
lowing results:
dI(X) ∞ dt 1
=I 3(σ 1)X3
dτ 0 − t et 1
(cid:20) ZX −
2 1 σ 1 t t 1 t2
+ − 2ln +
× −σ+5 σ+3 σ+3 − X X σ+1X2
(cid:26) (cid:18) (cid:19) (cid:27)
6(σ 1)(σ2+4σ+11) 1 X t(σ+3)/2
+ − dt ,(57)
(σ+1)(σ+3)2(σ+5) X(σ−1)/2 Z0 et−1 #
dn(X) 1 1 dI(X)
= , (58)
dτ 64γ6 I X3 dτ
min 0
x
X = , (59)
4γ2
min
where I = (k T )3/2π2. It should be empha-
0 B CMB
sized that the function dI(X)/dτ depends on γ only
min
throughX. Therefore,dI(X)/dτ hasthescalinglaw. On
theotherhand,thefunctiondn(X)/dτ doesnothavethe
scaling law because of the factor 1/64γ6 in Eq. (58).
min
For X 1, Eq. (57) is further simplified as follows:
≫
dI(X) 6(σ 1)(σ2+4σ+11)
=I −
dτ 0(σ+1)(σ+3)2(σ+5)
σ+5 σ+5
Γ ζ X−(σ−1)/2, (60)
× 2 2
(cid:18) (cid:19) (cid:18) (cid:19)
1 ∞ tz−1
ζ(z)= dt , (61)
Γ(z) et 1
Z0 −
where ζ(z) is the Riemann’s zeta function. We call
Eq. (60) the power-law approximation.
In Fig. 3, we plot dI(X)/dτ of Eq. (57) as a function
FIG.2: Plottingofs(σ)andP1(speak). Figures2(a)and2(b)
of X for typical σ-values for illustrative purposes. The
ares(σ)andP1(speak),respectively. Thedashedcurve,dash-
dashed curve, dash-dotted curve, and solid curve corre-
dotted curve, and solid curve correspond to the first-order
approximation,third-orderapproximation,andnumericalso- spond to σ=2.5, 3.5,and 4.5,respectively. The peak po-
lution, respectively. sition and peak height depend only on the power-index
parameter. It should be noted that dI(X)/dτ X for
X 1, and dI(X)/dτ X−(σ−1)/2 for X 1.∝There-
≪ ∝ ≫
fore, the slope of the downward curves in Fig. 3 will de-
termine the σ-value.
for scattered photons. For the γ-parameters, we assume In Figs. 4(a) and 4(b), we plot the peak position and
the same condition used in the scaling law for P (s), peak height of the spectral intensity function as a func-
1
namely, tionofσ,respectively. Thesolidcurvescorrespondtothe
numericalvalues. The dash-dottedcurvesarethe results
of analytical fitting functions. They are given by
1
1≪γmin ≪γmax. (56) Xpeak =1+ Σ a0+a1Σ1/4+a2Σ1/2 , (62)
dI(X ) 3 (cid:16)Σ 4+6Σ+Σ2 (cid:17)
peak
= , (63)
dτI 4(b +b Σ+b Σ2+Σ3)
0 0 (cid:0)1 2 (cid:1)
Undertheseassumptions,Eqs.(3)and(4)aremuchsim- Σ σ 1. (64)
plified, and can be solved analytically. The derivation is ≡ −
straightforward, however, it is lengthy. Therefore, we The fitting parameters are a = 2.18351, a =5.37131
0 1
−
give the derivation in Appendix B in detail, and quote anda = 2.02638forthepeakposition,andb =2.60331,
2 0
−
the final results here. b =6.6352 and b =5.6526 for the peak height. The er-
1 2
rors of the fitting functions in the region 2 σ 10 are
≤ ≤
According to Eqs. (B38) and (B40), one has the fol- less than 0.15% and 0.10% for X and dI(X )/dτ,
peak peak
7
FIG.3: PlottingofdI(X)/dτ. Thedashedcurve,dash-dotted
curve, and solid curve correspond to σ = 2.5, 3.5, and 4.5,
respectively.
respectively. In Fig. 4, two curves are totally indistin-
guishable.
Before closing this section, let us compare the present
resultwiththeliterature. InthetextbookbyRybickiand
FIG. 4: Plotting of the peak position and peak height of the
Lightman[18], the expression for the power-law approxi-
spectralintensityfunctionasafunctionofσ. Thesolidcurves
mation is given by Eq. (7.31). The scaling law is hidden
correspond to the numerical values. The dash-dotted curves
in the expression of Eq. (7.31). Inserting the explicit are theresults of theanalytical fitting functions.
form of the normalization constant C = n (σ 1)γσ−1
e − min
in Eq. (7.31), one finally obtains as follows:
X as follows:
dE
I G(σ)X−(σ−1)/2, (65) 1
0
dVdτdǫ ∝ 1 X , (68)
1
≪ ≪ R2
6(σ 1)(σ2+4σ+11)
G(σ)= −
(σ+1)(σ+3)2(σ+5) which again agrees with the condition of the present pa-
per. It is needless to mention that the full expression of
σ+5 σ+5
Γ ζ , (66) Eq.(57) hasto be usedforX O(1)as showninFig.3.
× 2 2 ≤
(cid:18) (cid:19) (cid:18) (cid:19)
which agrees with Eq. (60). The applicable range of the
III. ASTROPHYSICAL APPLICATIONS OF
power-law approximation of Eq. (7.31) is given[18] by
SCALING LAWS
ǫ
4γm2in ≪ ǫ¯1 ≪4γm2ax, (67) A. X-ray region
where ¯ǫ is a typical energy of initial photon distribution. In the present section, we show an application of the
In the case of the CMB photondistribution, one can use scaling law. Recently, observational studies on the in-
ǫ¯= k T . Therefore, one obtains the condition for verse Compton scattering have been done quite exten-
B CMB
8
Eq. (63). This will serve as an independent check for
the σ-value. On the other hand, the γ -value is deter-
min
mined by measuring the peak position ω in Fig. 5.
peak
Using the relation of ω with X , namely,
peak peak
1 ω
peak
X = , (70)
peak 4γ2 k T
min B CMB
the γ -value is determined by
min
1/2
1 ω
peak
γ = , (71)
min
4X k T
(cid:20) peak B CMB(cid:21)
where X is calculated by the RHS of Eq. (62) with
peak
the measured σ-value. It can be seen from Fig. 5 that
the X-ray observations have sensitivities to γ =500
min
∼
3 103 region.
×
Before closing this subsection, let us study the appli-
cability of the power-law approximation used in the lit-
erature. The condition for the power-law approximation
X 1 reads
≫
ω 4γ2 k T . (72)
≫ min B CMB
Inthecaseofγ =1 103,forexample,onehasω 1
min
× ≫
keV. As shown in Fig. 5(b), the error of the power-law
approximationis quite large in ω O(1) keV region.
∼
B. gamma-ray region
With the scaling law for the spectral intensity func-
tion,onecanextendthepresentformalismtothegamma-
FIG. 5: Plotting of dI(X)/dτ as a function of the photon
energy ω in X-ray energy regions for a typical value σ=2.5. ray region. In Fig. 6, we plot the same figure as Fig. 5
Figures 5(a), 5(b), 5(c) and 5(d) correspond to γmin=500, for the gamma-ray region. Because of the scaling law,
1×103, 2×103 and 3×103, respectively. The solid curve cor- the factor √1000( 31.6)should be simply multiplied to
≈
responds to thefull calculation of Eq. (57). The dash-dotted the γ -values of the keV region in order to obtain the
min
curveis thepower-law approximation of Eq. (60). spectralintensityfunctionintheMeVregion. Therefore,
observations in this energy region will have sensitivities
to γ =16 103 95 103 region. Similarly, the factor
min
sively,for example,X-rayobservationsfrom radiogalax- 1000 should×be m∼ultipl×ied to the γ -values of the keV
min
ies with Chandra[19, 20]. regioninordertoobtaintheparametervaluesintheGeV
In Fig. 5, we plot dI(X)/dτ as a function of the pho- region.
ton energy ω in X-ray energy region for a typical value
σ=2.5. Figures 5(a), 5(b), 5(c) and 5(d) correspond to
γmin=500, 1 103, 2 103 and 3 103, respectively. The IV. CONCLUDING REMARKS
× × ×
solidcurvecorrespondstothefullcalculationofEq.(57).
Thedash-dottedcurveisthepower-lawapproximationof
IntheNKpaper[16],wederivedthefrequencyredistri-
Eq.(60). InFig.5,thepeakheightisindependentofthe
bution function P(s,β) for a frequency shift s and elec-
γ -values as pointed in the last section. On the other
min tron velocity β. The form was derived in the Thomson
hand,thepeakpositionisshiftingtowardtohigh-energy
approximation, however, it was mathematically equiva-
side as the γ -value increases.
min lenttothatinthecovariantformalism[15]. Thereforethe
By measuring the slope of the downward curve in
frequency redistribution function can be applicable from
Fig. 5, one can determine the σ-value, because
nonrelativisticelectronstoextreme-relativisticelectrons.
In the present paper, we have extended the formal-
dI(ω)
ω−(σ−1)/2 (69) ism to extreme-relativistic electrons. First, we derived
dτ ∝
the analytic expression for P(s,γ) in the approximation
is valid. One can also determine the σ-value by mea- γ 1. It has been found that the present formalism is
≫
suring the peak height in Fig. 5 with the expression of equivalent to Jones’s formalism[6].
9
tion of the spectral intensity function in the X-ray and
gamma-ray energy regions. It has been found that the
sensitivities of the observationin the X-ray and gamma-
ray regionsare γ =500 3 103 and γ =16 103
min min
∼ × × ∼
95 103, respectively.
×
Finally,wehavestudiedtheapplicabilityofthepower-
law approximation used in the literature. In the case of
γ = 1 103, for example, the error of the power-law
min
×
approximationis quite large in ω O(1) keV region.
∼
Acknowledgments
This work is financially supported in part by the
Grant-in-Aid of Japanese Ministry of Education, Cul-
ture,Sports,Science,andTechnologyunderthecontract
#21540277. We wouldlike to thank our referee for valu-
able suggestions.
Appendix A: Comparison with Jones’s Formalism
The double differential cross sections for extreme-
relativistic electrons are given by Eqs. (38) and (40) of
Jones’s paper[6] as follows:
d2N 2πr2c
= 0 2qlnq+(1+2q)(1 q) , (A1)
dtdα α1γ2 " − #
d2N πr2c 2
= 0 (q′ 1) 1+ 2lnq′ . (A2)
dtdα 2α γ4 − q′ −
1 (cid:20) (cid:18) (cid:19) (cid:21)
FIG. 6: Plotting of dI(X)/dτ as a function of the photon
energy ω in gamma-ray energy regions for a typical value The variables in Eqs. (A1) and (A2) are related to the
σ=2.5. Figures 6(a), 6(b), 6(c) and 6(d) correspond to variables of the present paper as follows: σT = 8πr02/3,
γmin=15.8×103, 31.6×103, 63.2×103 and 94.9×103, respec- α/α1 = e−s, q = e−s/4γ2, q′ = 4γ2e−s, α = θCMBx,
tively. The solid curve corresponds to the full calculation of where θ = k T /mc2. With these variables,
CMB B CMB
Eq.(57). Thedash-dottedcurveisthepower-lawapproxima- Eqs. (A1) and (A2) are rewritten as follows:
tion of Eq. (60).
d2N 3σ c1 1
= T e−3s +2es+8γ2e2s
dtdx 32γ4 x −γ2
ByaveragingP(s,γ)overthenonthermalelectrondis- h
tributionfunction,wehavecalculatedtheprobabilitydis- 4(λγ +s)es , (A3)
−
tribution function P1(s). As for the nonthermal distri- d2N 3σ c1 i
bution function, we have adopted a standard power-law = T e−3s 8γ2es+2e2s
dtdx 32γ4 x
distribution function of three parameters: the power in-
h
1
dexσ,minimumvalueγmin,andmaximumvalueγmaxof 4(λ s)e2s e3s . (A4)
the distribution range. For the case γ 1, we have − γ − − γ2
min
≫ i
found a scaling law in P (s), where the peak position
1
Let us denote the photon distribution function in
depends on s 2ln2γ , and the peak height depends
min
− Jones’s formalism as n (α ). Then one has
only on the power index parameter σ. J 1
Wehavecalculatedthespectralintensityfunction. For
m3 α2
the case ofhigh-energyphotons ofx 1,we havefound n (α )dα = e 1 dα
ascalinglawindI(x)/dτ, wherethef≫unctiondependson J 1 1 π2(~c)3eα1/θCMB 1 1
−
a new variable X = x/(4γm2in). The peak position and = (kBTCMB)3x3e3sn(esx)ds, (A5)
peak height depend only on the power index parameter π2(~c)3
σ. The γ -dependence of dI(X)/dτ is included in the
min
variable X. where n(esx) = 1/(eesx 1). Averaging Eqs. (A3) and
−
We haveappliedthe presentformalismtothe observa- (A4) over the photon momentum with the distribution
10
function, one finally obtains where R γ /γ . In Eqs. (B2) and (B3), the func-
min max
≡
tions n (x), ..., n (x) are expressed as follows:
1 4
d2N (k T )3
Z dtdxnJ(α1)dα1 = Bπ2(C~Mc)B3 σTcx2 n1(x)= 0 dsPC(s,0)n(esY) , (B4)
dsP (s,γ)n(esx), (A6) Z−2ln2γmin
× J ∞
Z n (x)= dsP (s,0)n(esY) , (B5)
2 C
Z0
where the redistribution function in Jones’s formalism is 0
given by n (x)= dsP (s,0)n(esX) , (B6)
3 IC
Z−∞
3 1 2ln2γmin
P (s,γ)= +2es+8γ2e2s n (x)= dsP (s,0)n(esX) . (B7)
J 32γ4 −γ2 4 IC
h Z0
4(λ +s)es , (A7)
− γ In deriving Eqs. (B4)–(B7), we put R=0 and used new
3 i variables:
P (s,γ)= 8γ2es+2e2s
J 32γ4
x
h 1 X , (B8)
4(λ s)e2s e3s . (A8) ≡ 4γ2
− γ − − γ2 min
i Y ≡4γm2inx. (B9)
Comparing Eqs. (A7) and (A8) with Eqs. (30) and (31),
Introducingt=esY intoEqs.(B4)and(B5)andt=esX
respectively, one finds
into Eqs.(B6) and(B7), andinsertingthe explicit forms
of Eqs. (40)–(43), one obtains as follows:
P(s,γ)=P (s,γ), (A9)
J
1 Y 2 t2
which shows the equivalence of the two formalisms for n (x)=3(σ 1) dt
1 − Y −σ+5Y2
extreme-relativistic electrons. Zx n
1 σ 1 t t 1
+ − +2ln + n(t), (B10)
σ+3 σ+3 Y Y σ+1
(cid:18) (cid:19) o
6(σ 1)(σ2+4σ+11)
Appendix B: Derivation of Equations (57) and (58) n (x)= − Y(σ−1)/2
2 (σ+1)(σ+3)2(σ+5)
∞
We assume x 1 for the scattered photons, and 1 dtt−(σ+1)/2n(t), (B11)
γmin γmax fo≫r the γ-parameters. Let us first sol≪ve ×ZY
≪
Eq. (3). It can be rewritten as follows:
6(σ 1)(σ2+4σ+11) 1 1
∂n(x)
n (x)= −
= 3 (σ+1)(σ+3)2(σ+5) x3X(σ−1)/2
∂τ
2lnγmax/γmin γ X
dsP s, min n es4γ2 x dtt(σ+3)/2n(t), (B12)
C γ min ×
Z−2ln2γmin (cid:18) max(cid:19) Z0
+Z−22llnn2γγmmaixn/γmindsPIC(cid:18)s,γγmmainx(cid:19)n(cid:0)(cid:18)es4γxm2in(cid:1)(cid:19) n4(x1)=3(σσ−11)Xx33 ZXxt dttnt−σ+215 t2
n(x). (B1) + − 2ln + n(t).(B13)
− σ+3 σ+3 − X X σ+1X2
(cid:18) (cid:19)
o
Then we introduce the following new functions:
NowletusconsidertheCMBphotondistributionfunc-
tion
2ln1/R
dsP (s,R)n es4γ2 x
C min 1
Z−2lnn2γm(xin)+n (x), (cid:0) (cid:1) (B2) n0(t)= et 1 (B14)
≡ 1 2 −
for the initial distribution. Inserting Eq. (B14) into
Eqs. (B10)–(B13), one has for x 1
2ln2γmin x ≫
dsP (s,R)n es
IC 4γ2 n (x)=0, (B15)
Z−2ln1/R (cid:18) min(cid:19) 1
n (x)+n (x), (B3) n (x)=0, (B16)
3 4 2
≡