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Mon. Not. R. Astron. Soc. 000, 1–17 (1999) Printed 13 December 2018 (MN LaTEX style file v1.4)

Thermal synchrotron radiation and its Comptonization

in compact X-ray sources

Grzegorz Wardzinski∗ and Andrzej A. Zdziarski⋆

N. Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland

Received 1999 June 14

ABSTRACT

We investigate the process of synchrotron radiation from thermal electrons at semi-relativistic and relativistic temperatures. We find an analytic expression for the emis-sion coefficient in the case of random magnetic fields, and show that it is significantlymore accurate that those derived previously. We also present analytic approximationsto the synchrotron turnover frequency. Then, we treat Comptonization of the syn-chrotron radiation, and give simple expressions for the spectral shape and the emittedpower. We also consider modifications of the above results by bremsstrahlung.

We then study the importance of Comptonization of synchrotron radiation incompact X-ray sources. We first consider emission from hot accretion flows and fromactive coronae above optically-thick accretion discs in black-hole binaries and AGNs.We find that for plausible values of the magnetic field strength, Comptonization ofthermal synchrotron radiation is, in general, negligible in luminous sources, except forthose with hardest X-ray spectra and stellar masses. Increasing the black-hole massresults in a strong reduction of the maximum Eddington ratio possible from to thisprocess. X-ray spectra of intermediate-luminosity sources, e.g., low-luminosity AGNs,can be explained by synchrotron Comptonization only in the case of hot accretionflows. Then, bremsstrahlung emission always dominates X-ray spectra of very weaksources. Finally, we consider sources around weakly-magnetized neutron stars. Wefind that synchrotron Comptonization can account for the power-law X-ray spectraobserved in their low states.

Key words: accretion, accretion discs – gamma-rays: theory – radiation mechanisms:thermal– X-rays: galaxies – X-rays: stars.

1 INTRODUCTION

The theory of cyclotron and synchrotron radiation is a wellestabilished part of physics. However, there still remain un-certainties about the accuracy and the range of applicabil-ity of some analytic formulae describing the emission. Oneimportant example of such uncertainties concerns the spec-tra of synchrotron emission from mildly-relativistic and rel-ativistic thermal plasma, in which case numerous studiesdevoted to this field, e.g., Jones & Hardee (1979), Petrosian(1981, hereafter P81), Takahara & Tsuruta (1982), Robinson& Melrose (1984), Mahadevan, Narayan & Yi (1996, here-after MNY96), yielded results not entirely consistent witheach other.

Precise determination of spectra from the thermal syn-chrotron process is of key significance for studies of emissionfrom accretion flows onto black holes and neutron stars.

⋆ E-mail: gwar@camk.edu.pl, aaz@camk.edu.pl

Although direct, optically-thin, thermal synchrotron emis-sion is rarely observable, accurate optically-thin spectra arenecessary to determine the turnover frequency, below whichthe plasma becomes optically-thick. The resulting partially-absorbed spectrum is then observable in some cases. Fur-thermore, photons from that spectrum provide a supply ofseed photons for Comptonization in the plasma, which pro-cess gives rise to observable power-law spectra with high-energy cutoffs. On the other hand, blackbody radiation emit-ted by optically-thick accretion discs and other forms ofoptically-thick matter may constitute a competing supplyof seed photons. In addition, bremsstrahlung radiation isalways emitted by a hot plasma.

In this work, we first consider formulae for the syn-chrotron emission coefficient of a thermal plasma (Section2). We clarify the accuracy and the range of applicabilityof previously derived formulae as well as propose our ownexpressions. We concentrate on mildly relativistic and rel-ativistic plasmas, as exhaustive studies of non-relativistic

c© 1999 RAS

2 G. Wardzinski and A. A. Zdziarski

thermal cyclotron emission exist (Chanmugam et al. 1989and references therein). We then calculate the turnover fre-quency and consider effects of bremsstrahlung emission andself-absorption. Second, we investigate Comptonization ofthe synchrotron radiation (hereafter abbreviated as the CSprocess) and present convenient formulae for the resultingspectra and luminosities (Section 3). In Section 4, we applyour results to two main geometries of accretion flows onto ablack hole: a hot, two-temperature, optically thin disc, andactive regions above a cold disc (i.e., a patchy corona). Thosetwo models, for various plausible prescriptions for the mag-netic field strength, are then compared with spectral datafrom black-hole binaries and AGNs. Finally, in Section 5,we study whether the CS proces can account for power-lawspectra of weakly-magnetized, accreting, neutron stars.

2 SYNCHROTRON RADIATION FROM

THERMAL PLASMAS

2.1 Emission of a single electron

Let us consider an electron (with charge e) moving in a uni-form magnetic field, B, at a velocity, β ≡ v/c. Let ξ be theangle between v and B, and ϑ be the angle between B andthe direction towards the observer. Then, the synchrotronpower per unit frequency and unit solid angle in the ob-server frame and in cgs units is given by (e.g. Bekefi 1966;Pacholczyk 1970)

ην ≡dW

dν dΩdt(ϑ, ξ, γ) =

2πe2ν2

c× (1)

∞∑

n=1

δ(yn)

[

(

cos ϑ− β cos ξ

sin ϑ

)2

J2n(z) + β2 sin2 ξJ ′2

n (z)

]

,

where

yn ≡nνcγ

− ν(1− β cos ξ cos ϑ), z ≡νγβ sinϑ sin ξ

νc, (2)

νc ≡ eB/2πmec is the cyclotron frequency, γ = (1−β2)−1/2

is the Lorentz factor, Jn is a Bessel function of order n, andme is the electron mass.

Note that an additional factor of (1 − βµ cosϑ)−1 dueto the Doppler effect should have appeared in the formalderivation of equation (1). However, this term is a resultof the Fourier transform performed over infinite time, andit disappears in the case of an electron moving chaotically,as, e.g., in a thermal plasma. For detailed discussion, seeScheuer (1968), Rybicki & Lightman (1979, section 6.7), Pa-cholczyk (1970, section 3) and references in these works.

2.2 Synchrotron emission coefficient in a thermal

plasma

The emission coefficient, jν(ϑ), of a thermal plasma at atemperature, T , and with a uniform magnetic field, B, canbe obtained by integrating the rate of equation (1) over arelativistic Maxwellian electron distribution,

ne(γ) =ne

Θ

γ(γ2 − 1)1/2

K2(1/Θ)exp

(

−γ

Θ

)

, (3)

where Θ ≡ kT/mec2 is the dimensionless plasma tempera-

ture, and ne is the electron density. K2 is a modified Bessel

function, which can be approximated by

K2

(

1

Θ

)

≈ (4)

(

πΘ2

)1/2(

1 + 15Θ8

+ 105Θ2

128− 0.203Θ3

)

e−1/Θ, Θ 6 0.65;

2Θ2 − 12+ ln(2Θ)+3/4−γE

8Θ2 + ln(2Θ)+0.95

96Θ4 , Θ > 0.65,

where γE ≈ 0.5772 is Euler’s constant and the last coeffi-cients in the series have been adjusted to achieve a relativeerror 6 0.0008.

The integral for the emission coefficient is then,

jν(ϑ) =

∫ ∞

1

dγ1

2ne(γ)

∫ 1

−1

dµ ην(ϑ, µ, γ), (5)

where µ = cos ξ. This integration is relatively difficult tocarry out due to the complicated form of the integrand andthe presence of Jn in ην . In particular, standard numericalmethods of computing Jn become very inefficient for n ≫1. P81 obtained an approximate solution by replacing thesummation over n by integration over ν in equation (1) andthen using the principal term of an asymptotic expansionof Jn(z), where 0 6 z 6 nβ < n [see equation (9.3.7) ofAbramowitz & Stegun (1970, hereafter AS70)]. The region ofvalidity of that approximation of Jn is given by γ2 ≪ ν/νc.The integration over µ and γ by the method of the steepestdescent with some additional approximations yields

jν(ϑ) =21/2πe2neν

3cK2(1/Θ)exp

[

−(

9ν

2νcΘ2 sinϑ

)1/3]

. (6)

We see that the emission coefficient becomes very small bothat large frequencies, and at viewing angles, ϑ, significantlydifferent from π/2.

We note that the corresponding formula of equation(26) in P81 is too small by a factor of 2 due to the adoptednormalization of the electron distribution [equation (23) inP81] being also twice too small. On the other hand, Taka-hara & Tsuruta (1982) obtain a formula [equation (2.9)in their paper] for the absorption coefficient for the per-pendicular polarization of sychrotron radiation, α⊥

ν , whichagrees with our expression (6) for the emission coefficient atϑ = π/2. (They also compute the coefficient for the parallel

polarization, α‖ν , which, however, is negligibly small com-

pared to the dominant coefficient of the perpendicular po-larization.) We note that the relation between the absorptionand emission coefficients is given by Kirchhoff’s law, which,in the case of polarized radiation, contains the source func-tion for each polarization separately, i.e., B

‖ν = B⊥

ν = Bν/2(see, e.g. Chanmugam et al. 1989). The coefficients for polar-ized radiation are then related to the coefficients includingboth polarizations by jν = j

‖ν + j⊥ν , αν = (α

‖ν + α⊥

ν )/2. Wealso find that the emission coefficient computed by Jones &Hardee (1979) for ultrarelativistic plasmas, i.e., with Θ ≫ 1,appears incorrectly larger than the corresponding limit ofequation (6) by a factor of 21/2.

In our numerical calculations, we used an approxima-tion of Jn(z) in terms of the Airy function [equation (9.3.6)in AS70], which maximum relative error is ∼ 0.08 at n = 1,z = 0, but it rapidly decreases for larger values of n, z. Ifthe argument of the Airy function is < 2.25, we approximateit by its power series [equation (10.4.3) in AS70] up to the13th power. Otherwise, we use the approximation to Jn ofP81 described above, in which case the maximum error is

c© 1999 RAS, MNRAS 000, 1–17

Thermal synchrotron radiation and its Comptonization 3

∼ 0.02. Typical accuracy of the resulting approximation toJn is then < 0.01. The accuracy can be further increased byadding the first-order correction to the principal term of theasymptotic expansion of Jn, see equation (9.3.7) in AS70.The derivative, J ′

n, is calculated with the second expressionin equation (9.1.30) of AS70. Note that the above methodis more accurate, but also more complicated, than a relatedmethod given by Wind & Hill (1971).

We have checked the accuracy of the formula (6) com-pared to the results of numerical integration of equation (5)at ϑ = π/2 for 50 6 ν/νc 6 1000 and 0.05 6 Θ 6 3,which is the range of parameters most relevant for compactsources (e.g. Takahara & Tsuruta 1982; Zdziarski 1986, here-after Z86; Narayan & Yi 1995, hereafter NY95). (Below, wealso use the same ranges while testing other synchrotronrates.) We note first that our numerical results are in a verygood agreement with tabulated numerical results given forΘ <∼ 0.1 by Chanmugam et al. (1989). At small Θ, equation(6) strongly overestimates the actual value of jν and cannotbe used at all for Θ < 0.1, while for 0.1 <∼ Θ <∼ 0.2, it is cor-rect within a factor of ∼ 3 or better. For 0.2 < Θ <∼ 1, theerror generally does not exceed 30 per cent. The errors alsodiminish with increasing frequency; for instance, at Θ = 0.2,the approximate expression (6) overestimates the correct re-sult by 60 per cent at ν/νc = 100 but only by 15 per centat ν/νc = 1000. For Θ >∼ 1, the higher the temperature, thehigher frequencies are required to maintain a low error offormula (6), or, equivalently, the error increases with tem-perature at a given frequency.

Thus, the accuracy of the emission coefficient (6) is notsatisfactory for detailed modelling of astrophysical plasmas,as also found by MNY96. We propose a significantly moreaccurate approximation,

jν(ϑ) =πe2

c

(

ννcΘ

6γ0

)1/2

ne(γ0)

(

1 + 2cot2 ϑ

γ20

)

×(

1− β20 cos

2 ϑ)1/4

Z(γ0), β0 ≡ (1− γ−20 )1/2 (7)

[which uses eqs. (11) and (25) of P81]. Here Z is given by

Z(γ) ≡

[

t exp[

(1 + t2)−1/2]

1 + (1 + t2)1/2

]2n

, (8)

t ≡ (γ2 − 1)1/2 sinϑ, n ≡ν(1 + t2)

νcγ, (9)

and γ0 is the Lorentz factor corresponding to the saddlepoint of the integral (5) over γ,

γ0 =(

4νΘ

3νc sinϑ

)1/3

, (10)

which represents the energy of electrons contributing mostto the emission at ν.

For Θ = 0.05 and 50 6 ν/νc 6 1000, equation (7) givesan accuracy within a factor of ∼ 3 or better, with the ac-curacy increasing with frequency. For 0.2 <∼ Θ <∼ 0.5, therelative error does not exceed 30 per cent. For higher tem-peratures, 0.5 < Θ <∼ 1, the errors of the approximate inte-gration over γ start to increase and a 30 per cent accuracycan be achieved only for ν/νc > 100. For Θ > 1 the accuracyof formula (7) becomes comparable to that of equation (6).

The accuracy of formula (7) for lower temperatures canbe further improved by calculating the saddle point, γ0, nu-merically as the maximum of the function,

g(γ) ≡ne(γ)

γZ(γ) (11)

[instead of using γ0 of equation (10)]. With this modifica-tion, for 0.05 <∼ Θ <∼ 0.2, the accuracy of equation (7) isalways better than 20 per cent. We also find that when γ0is calculated numerically, equation (7) has the correct low-temperature and low-frequency limits. Specifically, it can beused down to Θ ∼ 10−3 with 30 per cent accuracy, providedΘν/νc > 1. It also gives results correct within a factor of ∼ 3for Θν/νc > 0.05. At Θ = 10−3, this formula matches wellthe nonrelativistic emission coefficient of Trubnikov (1958).

Our expression (7) can be compared with that of Robin-son & Melrose (1984), who have also used a method based onthat of P81 with accuracy improved with respect to that pa-per. We have found that their expression, in the form givenby Dulk (1985) but with the non-relativistic Maxwellianreplaced by the relativistic one, possesses the proper low-temperature limit as well as it is generally more accuratethan that of equation (7). Specifically, the accuracy of theirformula in the range 50 6 ν/νc 6 1000 and 0.01 6 Θ 6 1becomes typically better than 15 per cent. For Θ > 1, itsaccuracy is similar to that of expression (6). However, theirexpression is much more complicated than that of equation(7). Finally, we note that a relativistic generalization of theresults of Robinson & Melrose (1984) by Hartmann, Woosley& Arons (1988) appears incorrect. Namely, their expression(B1) should be multiplied by a factor 2π(γ2

0 − 1)−1/2 (intheir notation).

2.3 Angle-averaged emission coefficient

The emission coefficients derived above are appropriate fora plasma in a uniform magnetic field. However, in typical as-trophysical conditions, we expect either to deal with emis-sion from regions where magnetic fields are chaotic or toobserve radiation being a sum of contributions from a num-ber of regions with different orientations of magnetic field.Therefore, we would like to find the emission coefficient av-eraged over magnetic field directions (or, equivalently, overthe viewing angle), jν = ǫν/4π, where ǫν is the emittedpower per unit volume and frequency, and

jν =1

4π

∫

jν(ϑ)dΩ =1

2

∫

jν(ϑ) sinϑ dϑ. (12)

To obtain an expression for jν , we integrate equation (7)over ϑ with the method of the steepest descent. We note thatalthough integration of the formulae of Robinson & Melrose(1984) would yield results somewhat more accurate, the re-sulting formulae would be extremely complicated. Here, wetreat Z as the fast varying (with ϑ) part of the integrand(7) and take into account the dependence of γ0 from equa-tion (10) on ϑ. The saddle point is at ϑ = π/2. We havefound that this method yields a correct spectral shape, butthe resulting normalization is too large by a factor of about1.5 (around Θ ∼ 0.2 and ν/νc ∼ 102). With this correction,the resulting expression is

c© 1999 RAS, MNRAS 000, 1–17

4 G. Wardzinski and A. A. Zdziarski

jν =π3/2e2ne(γ0)Z(γ0)

(1.5)31/22c

(

ννcΘ

γ0

) 1

2

∣

∣

∣

∣

d2 lnZ(γ0)

dϑ2

∣

∣

∣

∣

− 1

2

ϑ= π

2

. (13)

We have checked that in the ranges of 0.1 6 Θ 6 2 and50 6 ν/νc 6 1000, the additional error introduced by thisapproximate integration is always < 0.1 [compared to the re-sults of numerical integration of equation (7)]. This error di-minishes with increasing frequency and, e.g., for ν/νc ∼ 1000and Θ in the range above, it is < 0.02. Note that if the steep-est decent method were applied ignoring the dependence ofγ0 on ϑ, the above renormalization would have been un-necessary, but the spectral shape given by the final formulawould have been less accurate.

On the other hand, the asymptotic emission coefficient(6) can be integrated over ϑ, as done by MNY96. This yields,

jν =21/6π3/2e2neν

35/6cK2(1/Θ)v1/6a(Θ, v) exp

[

−(

9v

2

)1/3]

, (14)

where v ≡ ν/νcΘ2 and the correction factor, a, represents

the ratio of the exact emission coefficient to that obtainedby integration of equation (6). This factor can be calculatedfrom the ratio of equations (33) to (31) in MNY96 withfitting coefficients of their Table 1.

We have tested the accuracy of the formulae (13) and(14), the latter both with and without the tabulated correc-tions, again in the range 0.05 6 Θ 6 3 and 50 6 ν/νc 6 1000and investigated their low- and high-temperature limits. Fig-ure 1a shows the relative error of equation (13). We see itis <∼ 15 per cent for 0.1 <∼ Θ <∼ 1. For 0.05 6 Θ <∼ 0.1 and1 < Θ 6 3, such accuracy is achieved only for ν/νc >∼ 200,while for lower frequencies this formula still returns an es-timate correct within a factor of ∼ 3, except at the lowesttemperature of our range, Θ ≃ 0.05, where ν/νc >∼ 100 is re-quired. Expression (13) does not apply at very low Θ due tothe breakdown of the approximation (10); however, replac-ing the latter with γ0 calculated numerically improves theoverall accuracy and yields results correct within a factor of∼ 3 down to Θ ∼ 10−3. At Θ ∼ 10, the approximation (13)yields results correct within a factor of ∼ 3 or better onlyfor ν/νc >∼ 1000.

The expression (14) with the tabulated corrections ofMNY96 matches very well the numerical results (the dif-ference is less than a few per cent), with the exception ofthe case of Θ = 0.084, where the corrections in Table 1 ofMNY96 appear misprinted as the obtained values are a fewtimes too small and the error increases with frequency.

The accuracy of formula (14) with a ≡ 1 is shown in Fig-ure 1b. We see that it provides an estimate correct within afactor of ∼ 3 or better for Θ >∼ 0.1, while it strongly overes-timates the correct result for lower values of Θ. For Θ >∼ 0.3,its error is <∼ 30 per cent. For Θ ∼ 10, it yields estimatescorrect within a factor of ∼ 3, and for ν/νc >∼ 1000 its ac-curacy is better than 30 per cent. The general behaviour ofthe above average emission coefficients is similar to thosederived for uniform magnetic field, i.e. their accuracy in-creases with ν and, for Θ >∼ 1, decreases with increasingtemperature at a given frequency. Still, equation (14) witha ≡ 1 gives a reasonable overall approximation to the emis-sion coefficient at relativistic temperatures. In particular,the frequency-integrated emissivity obtained from that for-

Figure 1. (a) Contour plots of the ratio of the approximatedemission coefficient (13) to the correct numerical value. (b) Thesame for equation (14) with a ≡ 1.

mula is only slightly larger, by a factor of 1.181, than theactual emissivity,

ǫ = 16Θ2neσTcB2

8π, Θ ≫ 1, (15)

where σT is the Thomson cross section.

2.4 Synchrotron self-absorption

The synchrotron radiation is self-absorbed by electrons up tothe turnover frequency, νt, above which the plasma becomesoptically thin to the synchrotron radiation, i.e.,

τ = ανtR = 1, (16)

where R is the characteristic size of the plasma. Below νt,the observed spectrum has the blackbody form, typically inthe Rayleigh-Jeans limit. In this limit and averaging overangles, Kirchhoff’s law implies,

jνt2ν2

tmeΘR = 1. (17)

This can be solved numerically. On the other hand, Zdziarskiet al. (1998) gives the solution using equation (14),

νtΘ2νc

=343

36ln3 C

ln C

ln C

...

, C =3

7Θ

[

πτTa exp(1/Θ)

3αfxc

] 2

7

, (18)

where xc ≡ hνc/mec2 is a dimensionless cyclotron frequency,

τT = neσTR is the Thomson optical depth of the plasma,

c© 1999 RAS, MNRAS 000, 1–17

Thermal synchrotron radiation and its Comptonization 5

Figure 2. The ratio of the turnover frequency calculated fromequation (18) with a ≡ 1 to the accurate numerical result as afunction of temperature.

and αf is the fine-structure constant. Note that νt dependson ne and R only through their product, or equivalently,τT. Typically, the correction factor, a, is a slowly-varyingfunction of frequency (MNY96) and then equation (18) forνt is explicit. We also see that the dependence of νt on ais only logarithmic, and thus the accuracy of determiningthe synchrotron emission coefficient only weakly affects theturnover frequency.

Mahadevan (1997) used a slightly different method ofdetermining νt. Namely, he equated the total flux of theRayleigh-Jeans emission to that in the optically-thin syn-chrotron one. This corresponds to setting the right-hand sideof equation (17) to 3/4 instead of 1, which changes negligiblythe resulting value of νt as compared to equation (18).

We have compared the approximated formula (18) witha ≡ 1 with results of numerical calculations. It turns outthat the error of equation (18) is almost independent of thevalues of the parameters other then Θ. Figure 2 comparesthe results of the two methods in the temperature range0.03 6 Θ 6 1 for a source with τT = 1, R = 107 cm and B =106 G. (In this parameter range, the effect of bremsstrahlungcan be neglected, see below.) We see that for 0.1 <∼ Θ <∼1 the discrepancy does not exceed 20 per cent but growsrapidly for lower temperatures. We have also found a power-law approximation for xt ≡ hνt/mec

2 as

xt ≈ 2.6× 10−6(

Θ

0.2

)0.95

τ 0.05T

(

B

106 G

)0.91

, (19)

which is accurate to <∼ 30 per cent for 0.05 <∼ Θ <∼ 0.4,0.3 <∼ τT <∼ 3, 10G <∼ B <∼ 108 G. Typical values of xt/xc

are in the range of 10–103.

2.5 Effect of bremsstrahlung on the turnover

frequency

We note that the above formulae for the turnover fre-quency do not take into account bremsstrahlung emissionand self-absorption. However, for some combinations of Θ,ne and B, bremsstrahlung emission can be comparable toor stronger than the synchrotron one and then the turnoverfrequency determined neglecting bremsstrahlung will be in-correct. Then, the absorption coefficient, ανt , should includea contribution from bremsstrahlung, αff

ν . Note that sinceαffν ∝ n2

eR, νt is no more solely a function of τT wheneverbremsstrahlung is important.

The effect of bremsstrahlung on the turnover frequencycan be checked by computing νt without taking into ac-count bremsstrahlung, and then computing the emission co-efficient, jν

t, including both processes. As long as jν

t≫ jffν

t,

equations in Section 2.4 can be used. We hereafter use formu-lae for the free-free emission coefficient of Svensson (1984).

We have calculated the turnover frequency as a functionof Θ and B for the remaining parameters fixed, see Figures3a, b, respectively. As expected, νt is determined mostlyby bremsstrahlung at low temperatures and weak magneticfields. However, though bremsstrahlung can have a negligi-ble effect on the value of νt, the total emitted power can stillbe dominated by it, see Section 4 below, and, e.g. Figure 3in Z86. On the other hand, if bremsstrahlung self-absorptiondominates over the synchrotron one, bremsstrahlung emis-sion will also dominate over the CS emission, except in thecase of spectra of the latter emission being harder than thebremsstrahlung spectrum.

3 COMPTONIZATION OF SYNCHROTRON

PHOTONS

Synchrotron photons produced in a plasma will, in general,be Compton scattered by the plasma electrons. In the caseof a thermal plasma, the synchrotron spectrum is usuallyself-absorbed up to a high value of ν/νc (Section 2.4). Inthat case, the self-absorbed synchrotron spectrum is rathernarrow, consisting of a hard Rayleigh-Jeans spectrum atν <∼ νt (in which regime scattering can be neglected, τ ≫ τT,τ ≫ 1), and a fast-declining tail of the optically-thin syn-chrotron spectrum at ν >∼ νt ≫ νc (where absorption can beneglected, τ ≪ τT, τ ≪ 1). Photons in that spectrum thenserve as seed photons for Comptonization. For that process,it is usually sufficient to treat the self-absorbed synchrotronphotons as monochromatic at νt (see, e.g., Monte Carlo sim-ulations in Z86). Also, for magnetic fields, B <∼ 109 G, andelectron temperatures, kT >∼ 10 keV, expected in compactsources, hνt ≪ kT , and photons at ν ∼ νt gain energy inthe scattering process.

To treat thermal Comptonization, we use the method ofZdziarski (1985, hereafter Z85), as applied to thermal syn-chrotron sources in Z86. This method gives an approximateform of Comptonized spectra, reproducing relatively accu-rately the energy spectral index, α, and the total power inthe scattered spectrum (see comparison with Monte Carloresults in Z86). On the other hand, it gives a relatively inac-curate shape of the high-energy cutoff of the scattered spec-trum (see, e.g., Poutanen & Svensson 1996), which, however,is of negligible importance for our applications.

c© 1999 RAS, MNRAS 000, 1–17

6 G. Wardzinski and A. A. Zdziarski

Figure 3. Example dependences of the turnover frequency on(a) temperature and (b) magnetic field strength. The dot-dashedand dashed curves give νt calculated with the analytic approx-imation of equation (18), and numerically using equation (17),respectively, with bremsstrahlung neglected in both cases. Thesolid curves give the numerical results with bremsstrahlung takeninto account.

We consider a homogeneous and isotropic source char-acterized by τT, Θ and B. The flux in scattered photons canbe approximated as a sum of a cut-off power law and a Wiencomponent,

FC(x) = NP

(

x

Θ

)−α

e−x/Θ +NW

(

x

Θ

)3

e−x/Θ, (20)

where x ≡ hν/mec2 is a dimensionless photon energy, and

NP, NW are normalizations of the power-law and Wien com-ponents, respectively.

For τT <∼ 2, α can be approximately given by,

α = −lnPsc

lnA, A ≡ 1 +

⟨

∆ν

ν

⟩

≈ 1 + 4Θ + 16Θ2, (21)

where A is the average photon energy amplification per scat-tering and Psc is the scattering probability averaged over thesource volume. Note that α > 0 in general. In spherical ge-ometry (Osterbrock 1974),

Psc = 1−3

8τ 3T

[

2τ 2T − 1 + e−2τT (2τT + 1)

]

→

3τT4

, τT ≪ 1;1− 3

4τT, τT ≫ 1,

(22)

where τT is the Thomson optical depth along the radius. Wehave found a power-law approximation to α(τT,Θ) valid at0.5 <∼ τT <∼ 2 and Θ <∼ 0.4,

α ≈3

19

1

τ4/5T Θ

. (23)

See Zdziarski et al. (1994) for Psc in the slab geometry.Note that at τT ≪ 1, scattering profiles corresponding

to subsequent scatterings are visible in the Compton spec-trum, and a power law is no longer a good approximation tothe shape of the spectrum below the cutoff (e.g., Fig. 2b inZ86). Still, the power-law description presented here givesthe shape averaged over the scattering orders, and an inte-gral over that approximate form gives a fair approximationto the total luminosity (see, e.g., Monte Carlo simulations inZ86). On the other hand, at τT >∼ 2–3, Comptonization canbe described by means of a kinetic, Fokker-Planck, equation(Sunyaev & Titarchuk 1980; see e.g. Lightman & Zdziarski1987 for relativistic and low-τT corrections). The two solu-tions can be matched at τT ∼ 2 (Z85). The remaining resultspresented in this Section can still be used at τT >∼ 2–3, ex-cept for a different prescription for α. In particular, the ratioof NW/NP given by Z85,

NW

NP=

Γ(α)

Γ(2α+ 3)Psc, (24)

where Γ is Euler’s gamma function, and which uses a resultof Sunyev & Titarchuk (1980), is a good approximation inboth the optically-thin and optically-thick regimes (see, e.g.,comparison with Monte Carlo results in Z86).

We need then to normalize the flux in the Compton-scattered spectrum, FC(x), with respect to that in theRayleigh-Jeans spectrum, FRJ(x). In general, the self-absorbed synchrotron spectrum at its peak around xt (≡hνt/mec

2) is somewhat above an extrapolation of the power-law component of the Compton spectrum, FC(x) (which fol-lows from photon conservation, Z85). Z86 finds the followingphenomenological relation providing a good approximationto the relative normalization,

FC(xt) = ϕFRJ(xt), ϕ(Θ) ≈1 + (2Θ)2

1 + 10(2Θ)2. (25)

This is valid for xt ≪ Θ, which condition we assume here-after. The flux of the Rayleigh-Jeans spectrum is given by,

FRJ(x) = πIRJ(x), IRJ (x) =2mec

3Θ

λ3C

x2, (26)

where IRJ is the specific intensity and λC ≃ 2.426 × 10−10

cm is the electron Compton wavelength. Then,

c© 1999 RAS, MNRAS 000, 1–17

Thermal synchrotron radiation and its Comptonization 7

NP = πϕIRJ(xt)(xt/Θ)α. (27)

The luminosity due to the CS emission is then given by,

LCS = A

[∫ xt

0

dx FRJ(x) +

∫ ∞

xt

dx FC(x)

]

. (28)

where A is the source area, and which can be integrated to,

LCS = A2πmec

3Θx3t

3λ3C

1 + 3ϕ(

xt

Θ

)α−1

×

[

Γ(

1− α,xt

Θ

)

+6Γ(α)Psc

Γ(2α+ 3)

]

, (29)

where the incomplete gamma function is well approximatedfor xt ≪ Θ by,

Γ(

1− α,xt

Θ

)

≃

ln Θxt

− γE, |α− 1| < 10−3;Θxt

− ln Θxt

− 1 + γE, |α− 2| < 10−3;1

α−1

(

Θxt

)α−1, α > 2.99;

Γ(1− α)− (xt/Θ)1−α

1−α+ (xt/Θ)2−α

2−α, otherwise.

(30)

At xt/Θ = 0.01, the maximum relative error of this ap-proximation of ∼ 0.02 occurs around α ≃ 3. The relativeerror declines rapidly at lower values of xt/Θ and α; e.g.,it is < 0.001 at xt/Θ = 0.01 and α 6 2.9, and < 0.002 atxt/Θ = 10−3 at any α.

In an astrophysically important case of α ∼ 0.4–0.9 (e.g.Gierlinski et al. 1997; Zdziarski, Lubinski & Smith 1999), themain contribution to the luminosity comes from the high-energy end of the spectrum, and the Wien component isrelatively unimportant. Then, we obtain an approximationvalid within a factor of <∼ 2 at xt/Θ <∼ 10−5,

LCS ≈8π2mec

3R2

(1− α)λ3C

x2+αt ϕΘ2−α, (31)

where we assumed a spherical geometry. On the other hand,for soft spectra, with α >∼ 1.1, we get an approximation validwithin ∼ 30 per cent,

LCS ≈8π2mec

3R2

3λ3C

(

1 +3ϕ

α− 1

)

x3tΘ. (32)

For order-of-magnitude estimates, we can then substitute xt

of equation (19) in equations (31)-(32). This yields LCS ∝R2Θ3.78−0.11αB1.82+0.91α and ∝ R2Θ3.66B2.73 for 0.4 <∼ α <∼0.9 and α >∼ 1.1, respectively.

We stress that the above formulae for the Comptoniza-tion spectral shape and the corresponding expressions forthe luminosity are mutually consistent. This is much prefer-able to computing Comptonization luminosity independently

of its corresponding spectrum by multiplying the power ina self-absorbed synchrotron spectrum by an amplificationfactor of Comptonization (e.g. of Dermer, Liang & Canfield1991), as often done in studies of advective flows.

4 APPLICATIONS TO ACCRETING BLACK

HOLES

Figure 4 shows an example of the CS spectrum for parame-ters typical to low-luminosity AGNs. The spectrum has theRayleigh-Jeans shape below the turnover frequency, it is a

Figure 4. An example of the spectrum of a magnetized plasma,for R = 9 × 1013 cm, B = 2.2 × 103 G, τT = 0.47 and Θ = 0.4.The dotted, short-dashed and long-dashed curves represent spec-tra due to synchrotron emission, its Comptonization and Comp-tonized bremsstrahlung, respectively. The solid curve representsthe sum of those components. This spectrum also represents amodel of a hot accretion flow fitted to the X-ray data for NGC4258 (bow-tie, Makishima et al. 1994), see Section 4.1.

power law due to Comptonization above that frequency, andthen it has a thermal high-energy cutoff. We also show thecontribution due to Comptonized bremsstrahlung, which ismoderately important for the chosen parameters. We havecomputed the latter spectrum by treating the spectrum ofequation (20) as Green’s function for Comptonization andthen integrating over the seed spectrum of optically-thinbremsstrahlung. The spectrum is computed assuming spher-ical geometry. More examples of spectra from magnetizedplasmas are given, e.g. in Z86.

For given Θ, the Rayleigh-Jeans part of the spec-trum is independent of B, and its spectral luminosity isLRJ(ν) ∝ R2. On the other hand, the CS spectrum obeysLCS(ν) ∝ R2ν2+α

t [equation (27)], i.e., its normalization in-creases quickly with increasing B via the dependence of theturnover frequency on B. When self-absorption is dominatedby the synchrotron process, νt is roughly ∝ B with no ex-plicit dependence onR, see equation (19). The total luminos-ity, LCS, follows the same dependence if α <∼ 1, see equation(31), and LCS ∝ R2ν3

t for α >∼ 1, see equation (32). Theshape of the Comptonized bremsstrahlung spectrum is al-most independent of B (except for a very weak dependenceof Comptonization on νt).

The bremsstrahlung luminosity can be constrained frombelow as

LCB

LE> 1.2× 10−6Θ1/2τ 2

Tr ≈ 1.2× 10−8α−5/2Θ−2r, (33)

where r ≡ R/Rg, Rg ≡ GM/c2 is the gravitational ra-dius, LE ≡ 4πµeGMmpc/σT ≈ 1.5 × 1038m erg s−1 is theEddington luminosity, µe = 2/(1 + X) is the mean elec-tron molecular weight, and X (≈ 0.7) is the H mass frac-tion, and m ≡ M/M⊙. The right-hand-side expressions ne-glect Comptonization and relativistic corrections, which ef-fects will increase the actual LCB. The last expression usesthe approximation of equation (23) to α(τT,Θ). Note thatLCB represents a strict lower limit to the luminosity of asource with given τT, Θ and size. Thus, the luminosity ofweak sources is dominated by LCB, with LCS being negli-gible. Note, however, that usually the plasma parameters

c© 1999 RAS, MNRAS 000, 1–17

8 G. Wardzinski and A. A. Zdziarski

Figure 5. Example dependences of the plasma luminosity on Bfor (a) R = 4 × 107 cm and (b) R = 4 × 1014 cm for threevalues of temperature. The flat parts correspond to dominantbremsstrahlung, and dashed curves give the luminosity in the CScomponent alone.

depend on luminosity, which dependence should be takeninto account when determining the luminosity below whichbremsstrahlung dominates.

Figure 5 shows some example dependences of the totalL on B at Θ = 0.05–0.4 for R = 4×107 cm and 4×1014 cm(corresponding to ∼ 30Rg for 10M⊙ and 108M⊙ black-holemass, respectively). The flat parts at low values of B cor-respond to the dominant bremsstrahlung, with its relativeimportance increasing with decreasing Θ.

Hereafter, we use the symbols L and η for the totalluminosity of a hot plasma in a source, including all ra-diative processes, and for the corresponding Eddington ra-tio, L/LE, respectively. This L then corresponds to the oneobserved (excluding components not originating in the hotplasma, e.g., a disc blackbody). We then compare observed

values of L with the CS luminosity assuming some specificprescriptions for the magnetic field strength in an accre-tion flow. Typically, the maximum possible magnetic fieldstrength corresponds to equipartition between the pressureor energy density of the field and of the gas and radiationin the source (Galeev, Rosner & Vaiana 1979).

4.1 Two-temperature accretion flows

Here, we consider the CS emission from optically thin, two-temperature accretion flow. In inner parts of hot accretionflows, the ion temperature, Ti, is typically much higher thanthe electron temperature (e.g. Shapiro, Lightman & Eardley1976; NY95), and the ion energy density is much higher thanthat of both electrons and radiation. Then, energy densityequipartition corresponds to

B2

8π=

3

2ne

µe

µikTi, (34)

where µi = 4/(1 + 3X) is the mean ion molecular weight.We note that pressure equipartition would result is a slightlydifferent condition, and that the magnetic-field pressure,B2/8π, is sometimes assumed to equal B2/24π (e.g., Ma-hadevan 1997).

Maximum possible ion temperatures are sub-virial, anddetailed accretion flow models show kTi ≈ δmpc

2/r, whereδ ≪ 1 (Shapiro et al. 1976; NY95). In particular, δ constantwith r and dependent on the flow parameters is obtainedin the self-similar advection-dominated solution of NY95.Close to the maximum possible accretion rate of the hotflow, δ ∼ 0.2 is a typical value [see equation (2.16) in NY95],which we adopt in numerical calculations below. However,the self-similar solution breaks down in the inner flow, whichresults in values of Ti much lower than that given by the self-similar solution, e.g., by a factor of ∼ 10 at r = 6 (Chen,Abramowicz & Lasota 1997). Thus, we can obtain an up-per limit on the equipartition magnetic field strength byassuming that it follows the self-similar solution above someradius, r0 ∼ 20, while it remains constant at r < r0,

B ≃ 7.3× 108 G(

δτTm

)1/2

×

1/r0, r 6 r0;1/r, r > r0.

(35)

This typically gives B <∼ 107 G and <∼ 104 G in inner regionsof binary sources and AGNs, respectively.

Then, the CS luminosity can be constrained from aboveby a sum of the contribution from the inner region (assumedhere to be spherical) with radius r0, and a contribution fromthe outer region r > r0, obtained here by radial integrationall r0 to infinity (for which we assumed a planar geome-try). We use here the upper limit on B of equation (35) andassume α and Θ constant through the flow. The latter as-sumption maximizes LCS of given α and Θ since both τTand Θ will, in fact, decrease with r in an accretion flow.(We also note that the maximum of dissipation per ln rin the Schwarzschild metric occurs at r ∼ r0 ∼ 20.) For0.4 <∼ α <∼ 0.9, the resulting upper limit is

LCS

LE≈

43× 10−2.83αδ0.91+0.46αΘ2.64−0.68αϕ

(mr20)0.46α−0.09(1− α)α0.63α+1.26

×

1 for inner region;(0.91α − 0.18)−1 for outer region,

(36)

c© 1999 RAS, MNRAS 000, 1–17

Thermal synchrotron radiation and its Comptonization 9

where we used equations (23), (31), (35) and ϕ ∼ 0.5. Thetotal LCS corresponds to the sum of the contributions fromthe two regions. In the case of α >∼ 1.1, we have an upperlimit on the luminosity of

LCS

LE≈

0.021δ1.37Θ1.96

m0.37r0.730 α1.89

(

1 +3ϕ

α− 1

)

×

1 for inner region;1.37 for outer region,

(37)

which is based on equation (32).We see from equations (36)-(37) that the predicted Ed-

dington ratio decreases with the mass. Then, if the CS pro-cess dominates, the predicted X-ray spectra would hardenwith increasing M at a given LCS/LE. In fact, the opposite

trend is observed; black-hole binaries have X-ray spectraharder on average than those of AGNs (e.g. Zdziarski et al.1999), implying that either the CS process does not domi-nate the X-ray spectra of those objects or (less likely) theEddington ratio is much lower on average in AGNs than inblack-hole binaries.

We note that this conclusion differs from a statementin NY95 that hot accretion flows with CS cooling are effec-tively mass-invariant. In particular, we have been unable toexplain their Fig. 4, in which flows onto black holes have ra-dial dependences of both the temperature at the critical (i.e.,maximum possible) accretion rate and that rate itself (in Ed-dington units) virtually identical for m = 10 and m = 108.This would then imply very similar X-ray spectra in bothcases. On the other hand, we have found a good agreementof our results with the dependences on m in the results ofMahadevan (1997).

We find that, under conditions typical to compactsources, the Comptonized bremsstrahlung luminosity, LCB,is often comparable to or higher than LCS. Figure 6 showsthe results of numerical calculations of the Eddington ra-tio for both CS and bremsstrahlung radiation as a functionof the spectral index of the CS radiation, α, for m = 10and 108. To enable direct comparison of those two com-ponents, we show the luminosity from the inner region(with constant B) only, assuming r0 = 20 and δ = 0.2.Thick and thin curves give LCS and LCB [see equation(33) with r = r0], respectively. The relative importanceof bremsstrahlung increases with increasing mass and size,e.g. LCB/LCS ∝ m0.46α−0.09r0.91α+0.82

0 and ∝ m0.37r1.730 forα <∼ 0.9 and >∼ 1.1, respectively. Note that LCB decreaseswith increasing Θ at a constant α, which is an effect of astrong dependence of LCB on τT (increasing with decreasingΘ) and a weak dependence of LCB on temperature. Also, animportant conclusion from Figure 6 is that Eddington ratios>∼ 0.01 can be obtained from this process only for very hardspectra and high electron temperatures, with this constraintbeing significantly stronger in the case of AGNs.

Figure 7a compares the upper limits on LCS/LE withvalues of L/LE inferred from observations for a numberof objects and with the corresponding luminosity frombremsstrahlung. The shown values of LCS include the con-tributions from both the inner and the outer region of theflow [equation (35)] for r0 = 20 and δ = 0.2. On the otherhand, the shown values of LCB are for emission from withinr0 only, and the expected actual LCB will be by a factorof ∼ 2 larger. For Cyg X-1 in the hard state, we used thebrightest spectrum of Gierlinski et al. (1997), with α = 0.6,

Figure 6. The Eddington ratio as a function of the spectral in-dex of CS radiation and for 3 values of electron temperature formasses (a) m = 10, and (b) m = 108. The heavy and thin curvesshow the luminosity in the CS radiation and in Comptonizedbremsstrahlung, respectively. See text for other assumptions. Thecurves are shown only for values of α corresponding to τT 6 3, asfor larger τT our formulae for α (Section 3) break down.

Θ = 0.2, and L ≈ 3 × 1037 erg s−1 (assuming the distanceof D = 2 kpc, Massey, Johnson & DeGioia-Eastwood 1995;Malysheva 1997) and m = 10. For GX 339–4 in the hardstate, we used α = 0.75, Θ = 0.1, L ≈ 3 × 1037 erg s−1,D = 4 kpc and m = 3 (Zdziarski et al. 1998). For NGC4151, we used the brightest spectra observed by Ginga andCGRO/OSSE (Zdziarski, Johnson & Magdziarz 1996), forwhich α = 0.85, Θ = 0.1, and L ≈ 9 × 1043 erg s−1, as-suming D = 16.5 Mpc (corresponding to H0 = 75 km s−1

Mpc−1), and m = 4×107 (Clavel et al. 1987). For NGC 4258we adopted L = 2×1041 erg s−1 from an extrapolation of the2–10 keV luminosity of L2−10keV = 3.1 × 1040 erg s−1 withα = 0.78 (Makishima et al. 1994) up to 200 keV, D = 6.4

c© 1999 RAS, MNRAS 000, 1–17

10 G. Wardzinski and A. A. Zdziarski

Figure 7. Comparison of Eddington ratios inferred from obser-vations (asterisks, except for Sgr A∗, where an upper limit is

shown) with the model upper limits from the CS emission (heavycurves) for a range of black-hole masses. Vertical arrows connectobserved values with the corresponding model upper limits, ex-cept for NGC 4258 and Sgr A∗, for which no sufficient spectraldata exist. On the other hand, the model curves for Θ = 0.2,α = 0.9, corresponding to the average spectrum of Seyfert 1s,have no corresponding observational point. Thin horizontal linesshow the Comptonized bremsstrahlung emission. The magneticfield energy density is in equipartition with (a) ions in a hot ac-cretion flow, and (b) with radiation energy density times c/vA ina patchy corona, see Sections 4.1 and 4.2.2, respectively.

Mpc and m = 3.6 × 107 (Miyoshi et al. 1995). Finally, forSagitarius A∗ we assume an upper limit of L < 1037 erg s−1

(Narayan et al. 1998), and m = 2.5× 106 (Eckart & Genzel1996). Finally, we show LCS for the average parameters ofSeyfert-1 spectra, α = 0.9, Θ = 0.2 (e.g. Zdziarski 1999).

We see that only in the case of Cyg X-1, with its rela-tively hard spectrum, the CS emission can contribute sub-

stantially, at <∼ 30 per cent, to the total luminosity. Givenuncertainties of our model, e.g., in the value of δ, it is in prin-ciple possible that the CS process can account for most of theemission of Cyg X-1. On the other hand, we have adoptedhere assumptions maximizing synchrotron emission and weconsider it more likely that the CS process is negligible inCyg X-1. This appears to be supported by the similaritybetween X-ray spectra and the patterns of time variabilitybetween Cyg X-1 and the other black-hole binary consid-ered by us, GX 339–4 (Miyamoto et al. 1992; Zdziarski etal. 1998). In the latter object, LCS clearly provides a negli-gible contribution to L, as shown by Zdziarski et al. (1998).Furthermore, a remarkable similarity exists between X-rayspectra of black-hole binaries and Seyfert 1s (e.g. Zdziarskiet al. 1999). For the latter, LCS/LE ∼ 10−5–10−6 are found,whereas the typical Eddington ratios of those objects arelikely to be ∼ 0.01 (e.g., Peterson 1997), i.e., 3–4 orders ofmagnitude more. Concluding, CS emission in the assumedhot-disc geometry is unlikely to be responsible for the ob-served X-ray spectra of luminous black-hole sources. To ex-plain those spectra, an additional source of soft seed photonsis required. Such seed photons are naturally provided byblackbody emission of some cold medium, e.g an optically-thick accretion disk or cold blobs, co-existing with the hotflow.

On the other hand, the CS process can clearly be impor-tant in weaker sources, e.g., NGC 4258. We have comparedpredictions of our model with the 2–10 keV spectral indexand luminosity (see above) of this object. We have obtaineda good fit to the data, as shown for δ = 0.2, Θ = 0.4 inFigure 4, taking into account emission within r0 = 17. Thisyields LCS = 1.8 × 1041 erg s−1 and LCB = 4 × 1040 ergs−1. A similar spectrum from an advection disc model wasobtained by Lasota et al. (1996).

For Θ ≈ 0.2 (typical for luminous sources), we could re-produce the L2−10keV of NGC 4258, but the 2–10 keV spec-trum was dominated by bremsstrahlung, i.e., much harderthan the observed one. The relatively high temperature,Θ ≃ 0.4, required in our model, is, in fact, consistent withpredictions of hot accretion flow models (e.g. NY95), inwhich Θ increases with decreasing luminosity as a result ofdiminishing electron cooling rate. Unfortunately, we have asyet no observational constraints on the plasma temperaturein low-luminosity sources.

As stated above, Comptonized bremsstrahlung, withLCB/LE independent of M , provides the minimum possi-ble luminosity of a plasma with given τT, Θ and r. This isshown by horizontal lines in Figure 7. Thus, sources withlower luminosities cannot have plasma parameters charac-teristic of luminous sources, which is indeed consistent withhot accretion flowmodels, in which τT quickly decreases withdecreasing L (e.g. NY95). On the other hand, CS emissioncan still be important in a certain energy range above theturnover energy even if LCB > LCS. This may be the case,e.g., in Sgr A∗ (Narayan et al. 1998).

4.2 Active coronae

4.2.1 Equipartition with radiation energy density

Di Matteo, Celotti & Fabian (1997b, hereafter DCF97) haveconsidered a model in which energy is released via mag-

c© 1999 RAS, MNRAS 000, 1–17

Thermal synchrotron radiation and its Comptonization 11

netic field reconnection in localized active regions forming apatchy corona above an optically-thick accretion disc. Theyargue that since the energy is transferred via the magneticfield, its energy density will be in equipartition with the localradiation energy density,

B2

8π≈

9

16π

L

N(rbRg)2c, (38)

where rbRg is the radius of an active blob and N is the av-erage number of blobs. The right-hand-side of this equationcorresponds to the average photon density in an opticallythin, uniformly-radiating, sphere. This yields values of Bsomewhat larger than those of DCF97, whose used a numer-ical factor of 1/4π in their expression for B. Consequently,our estimates of LCS will be higher than those obtained us-ing the formalism of DCF97. In the case of an optically andgeometrically thin reconnection region, the numerical factorabove should be 9/12π. Equation (38) can be rewritten as

B ≈ 1091

rb

(

η

Nm

)1/2

G. (39)

The number of regions active at a given time can be esti-mated to be N ∼ 10, as implied by amplitude of variabil-ity on shortest time scales (Haardt, Maraschi & Ghisellini1994). The characteristic size of a reconnection region hasbeen estimated by Galeev et al. (1979) as

rb ∼ hdα−1/3v , (40)

where hdRg is the scale height of an underlying optically-thick disc, and αv is the standard viscosity parameter. Thistypically yields a range of 0.01 <∼ rb <∼ 5, depending on thedisc parameters and increasing with accretion rate (see, e.g.Svensson & Zdziarski 1994). For the sake of simplicity, wehereafter use a typical value of rb = 2 in numerical examplesbelow, which is the same value as that adopted by DCF97.

With the above value of B, we can derive the ratio ofthe CS luminosity from the corona to the total coronal lu-minosity,

LCS

L≈

650ϕ(η/Nmr2b)0.46α−0.09Θ3.78−0.11α

102.24αα0.13+0.06α(1− α)(41)

for 0.4 <∼ α <∼ 0.9. The analogous formula for α >∼ 1.1 is

LCS

L≈ 1.25

(

1 +3ϕ

α− 1

)

(

η

Nmr2b

)0.37Θ3.66

α0.19. (42)

From equations (41)-(42), we find that, in most cases,the coronal model yields much lower values of LCS than themodel of a hot accretion flow of Section 4.1. For the hardstate of Cyg X-1 with η = 0.02 and m = 10 (Section 4.1),and forN = 10, rb = 2, we find LCS/L ≈ 1.7×10−2, which is∼ 20 times less than LCS/L obtained in the hot flow model.Only an extremely low and inconsistent with equation (40)value of rb ∼ 10−5 would lead to LCS = L in this case. Forobjects with softer spectra, even lower values of LCS/L areobtained. For objects with low luminosities, Comptonizedbremsstrahlung would dominate, see equation (33), whichapplies to all coronal models discussed here with r = Nrb.

Thus, thermal synchrotron radiation cannot be a sub-stantial source of seed photons for Comptonization in activecoronal regions with equipartition between magnetic fieldand radiation, even for weak sources. This finding can beexplained by considering consequences of this equipartition

in the presence of strong synchrotron self-absorption. Withneither self-absorption nor additional sources of seed pho-tons, equipartition between the energy densities in the fieldand in photons leads to the net Comptonization luminos-ity, LC, roughly equal to the synchrotron luminosity, LS,which then implies α ∼ 1. (An additional assumption hereis the validity of the Thomson limit, e.g. Rybicki & Lightman1979.) However, strong self-absorption in a thermal plasmawith parameters relevant to compact sources dramaticallyreduces LS, i.e., LS ≪ LC. Then, a very hard Comptoniza-tion spectrum, with α ∼ 0, is required to upscatter the fewsynchrotron photons into a spectrum with luminosity of LC.On the other hand, if α ∼ 1, as typical for compact cosmicsources, CS photons would give only a tiny contribution tothe actual coronal luminosity, L, and other processes, e.g.Comptonization of blackbody photons from the underlyingdisc would have to dominate.

This conclusion differs from findings in DCF97 and DiMatteo, Celotti & Fabian (1999) (who studied the relativeimportance of Comptonization of synchrotron and disc pho-tons as a function of rb and the height of the active regionabove the disc) that this process is often important underconditions typical to compact objects, in particular in GX339–4. This discrepancy appears to stem mostly from theirassumption of η = 1 (i.e., L = LE) for calculating the valueof B in those papers (T. Di Matteo, private communication).In the specific cases they consider, LCS ≪ LE, i.e., only asmall fraction of the dissipated power is radiated away, andthey postulate that the remaining power is stored in mag-netic fields (see Section 2.2 in DCF97). However, they do notspecify the final fate of the power supplied to the magneticfields. We note that in steady state, the supplied power hasto be either eventually radiated away or transported awayfrom the disc. In the former case, we would recover our re-sult of LCS ≪ L. In the latter, the power in magnetic fieldscould be either converted to kinetic power of a strong out-flow or advected to the black hole. Those possibilities requirestudies that are beyond the scope of this paper. We onlynote that a physical realization of a strong but effectivelynon-radiating outflow in the vicinity of a luminous binaryappears difficult. Similarly, the advection time scale of anoptically-thick disc is probably much longer than the timescale for dissipation of coronal magnetic fields. Finally, wenote that η = 1 would often require accretion rates muchhigher than those estimated from the physical parametersof X-ray binaries (see, e.g., Zdziarski et al. 1998 for the caseof GX 339–4).

In addition, the values of Θ assumed in DCF97 andDi Matteo, Celotti & Fabian (1999) for luminous states ofblack-hole sources are higher than those used in the exam-ples shown in this work (which were derived from the bestcurrently-available data). Due to the strong dependence ofLCS on Θ [see eqs. (41)-(42)], this also results in higher val-ues of LCS.

We note that equation (41) can yield LCS ∼ L forΘ >∼ 1. This is partly explained by most of energy densityin Comptonized photons being then in the Klein-Nishinalimit. Then, the energy density of photons in the Thomsonlimit is much less than the total energy density, and thusmore comparable with energy density of self-absorbed syn-chrotron photons. Thus, the CS process is expected to playan important role in sources with Θ >∼ 1. We point out that,

c© 1999 RAS, MNRAS 000, 1–17

12 G. Wardzinski and A. A. Zdziarski

however, equations (41)-(42) breaks then down as the ap-proximations of equations (19) and (23) become invalid andnumerical calculations should be employed in that case.

4.2.2 Dissipation of magnetic field

As discussed by, e.g., by Di Matteo, Blackman & Fabian(1997a) and Di Matteo (1998), the strength of the coronalmagnetic field can be higher than the equipartition value ofequation (38). If an active region is powered by dissipationof magnetic field and the dissipation velocity is gvA, whereg 6 1 and vA = B/(4πnempµe/µi)

1/2 is the Alfven speed,the energy density stored in the field will be (Di Matteo etal. 1997a) c/gvA times that in radiation (which escapes thesource with the velocity c), i.e.,

B2

8π

gvAc

≈9

16π

L

N(rbRg)2c. (43)

Using the approximation of equation (23) yields,

B ≈6.2× 108(η/gN)1/3

r5/6b m1/2α5/24Θ5/24

G. (44)

The luminosity in the CS emission is then approxi-mately given by

LCS

L≈

260Θ3.4−0.3αr0.48−0.76αb (N/η)0.39−0.3αϕ

102.44αg0.61+0.3αm0.46α−0.09α0.5+0.25α(1− α), (45)

for 0.4 <∼ α <∼ 0.9, and

LCS

L≈ 0.32

(

1 +3ϕ

α− 1

)

Θ3.09(N/η)0.09

g0.89m0.37r0.28b α0.76, (46)

for α >∼ 1.1.The factor g is related to the geometry of the dissipa-

tion region and it is expected that g ∼ hb/rb, where hbRg isthe scale height of the blob, see e.g. Di Matteo et al. (1997a).Those authors adopted g = hb/rb = 0.1, whereas Di Mat-teo (1998) assumed g = 1 and hb/rb = 0.1. Taking intoaccount theoretical uncertainties related to the reconnec-tion process, we set g = hb/rb = 1 in our examples below.Setting g = hb/rb = 0.1 and appropriately correcting theradiation density and the surface area of an active regionwould increase the predicted LCS by a factor of ∼ 5.

Although LCS in this model is higher than in the previ-ous case, it is still relatively low. This is explained by typicalvalues of vA ∼ 0.1c, which lead to only a moderate increaseof B with respect to that of equation (38). For Cyg X-1 forthe same parameters as above (η = 0.02, N = 10, rb = 2),we find LCS/L ≈ 0.2, i.e., still < 1 (virtually independent ofrb), although g = 0.1 would increase this ratio up to unity.However, for objects with softer spectra, the obtained val-ues of LCS/L are much lower, as shown in Figure 7b, whichshows comparison with the same data as Figure 7a. For themodel curve corresponding to the average Seyfert-1 spec-trum, we assumed η = 0.01. We see that predictions of thismodel for luminous sources, with η > 0.02, are similar tothose of the hot-flow model (Section 4.1).

However, this model predicts values of LCS for weakobjects much lower than those in the hot disc model. Thiscan be seen by comparing the dependence on η in bothcases. In the case of a hot flow, LCS does not depend onL, so formally LCS/L ∝ η−1 (not including dependences

Figure 8. The fraction of the total coronal luminosity producedby the CS process (heavy and medium curves for m = 10 andm = 108, respectively) and by Comptonized bremsstrahlung (thincurves applying to both values of m) as a function of the spectralindex of the CS radiation, for η = 10−2 (solid curves) and η =10−4 (dashed curves) in the case of magnetic field of equation(43). See Section 4.2.2 for other parameters.

of the plasma parameters on L). On the other hand, equa-tions (45)-(46) yield roughly LCS/L ∝ η−0.1 in the presentcase (which difference is mainly caused by the source areamuch smaller in the patchy corona model than in the hotflow model). Thus, decreasing L leads to only a marginalincrease of LCS/L. In contrast, the luminosity from Comp-tonized bremsstrahlung is about the same in the two models.Consequently, LCB ≫ LCS is typical of weak sources in thecoronal models.

This is indeed the case for NGC 4258, where we havefound it impossible to reproduce its 2–10 keV power-lawindex and luminosity even with rb ≪ 1, except for Θ >1. Although such a high temperature cannot be ruled outobservationally at present, the corresponding τT ≪ 1, whichthen would lead to a curvy spectrum reflecting individualscattering profiles. Then, the similarity of the X-ray spectralindex of that object to the average index of Seyfert 1s wouldhave to be accidental, which we consider unlikely.

Figure 8 shows a comparison of CS and Comptonizedbremsstrahlung emission as functions of α for two values ofm and two values of η, and for g = 1, N = 10, rb = 2,Θ = 0.2. As expected, bremsstrahlung dominates at lowvalues of η and high values of α, and its Eddington ratio isindependent of m.

4.2.3 Magnetic field in the disc

The magnetic field strength in a corona is limited by equa-tion (43) provided most of the energy of the field is dissipatedthere, which seems to be a likely assumption. However, thatfield depends on the factor g, whose value appears uncer-tain, and which could, in principle, be ≪ 1. Furthermore,we cannot rule out the presence of a stronger, more per-

c© 1999 RAS, MNRAS 000, 1–17

Thermal synchrotron radiation and its Comptonization 13

manent field, which would dissipate only partially. In anycase, the coronal magnetic field strength will be lower thanthat inside the disc, Bd. The latter is in turn limited byequipartition with the disc pressure, as upward buoyancyforces will rapidly remove any excess magnetic flux tubesfrom the disc (Galeev et al. 1979). Such an equipartitionfield was adopted by Di Matteo (1998) in calculations of therate at which energy was supplied to a corona. Numericalsimulations of Hawley, Gammie & Balbus (1995) also sug-gest that the magnetic field in the disc does not exceed theequipartition value.

Here, we utilize formulae for the vertically-averaged discstructure of Svensson & Zdziarski (1994), which take into ac-count that a fraction, f < 1, of the energy dissipated in thedisc is transported to the corona. However, we recalculatethe disc structure taking into the additional pressure pro-vided by the equipartition magnetic field. For consistencywith the rest of this work, we assume equipartition with en-ergy density rather than with pressure, which assumptionhas only a minor effect on our results. Expressions for Bd

below are computed at r = 11, which is approximately theradius at which a radiation-dominated region first appearswith increasing accretion rate (Svensson & Zdziarski 1994).For lower radii, somewhat higher values of Bd are obtained,but the contribution of those radii to the total dissipatedpower is small. In a disc region dominated by the gas pres-sure, we have then,

Bd ≈1.1× 108m2/5

(αvm)9/20(1− f)1/20G, (47)

where m is the dimensionless accretion rate, m ≡ Mc2/LE.In a disc region dominated by radiation pressure, we have,

Bd ≈ 5.2× 107 [αvm(1− f)]−1/2 G. (48)

Note that the radiation-pressure dominated region appearsonly for accretion rates higher than certain value mcrit. Here-after, we will assume αv = 0.1 and that half of the accretedenergy is dissipated in the corona, i.e., f = 1/2. We as-sume the efficiency of cooling-dominated accretion in theSchwarzschild metric, which gives the power dissipated inthe corona of L = 0.057mfLE.

We first compare the maximum strength of coronalmagnetic field of equation (43) with the disc field of equa-tions (47)-(48). For parameters relevant to compact objects,the disc field is always larger than the maximum coronalmagnetic field, up to two orders of magnitude, except forextremely low accretion rates, m <∼ 10−10. This shows theself-consistency of the coronal models of Sections 4.2.1-4.2.2.

As discussed above, given the uncertainties about themechanism of magnetic field reconnection in active regionsabove accretion discs, we cannot rule out the presence ofan average coronal field strength, 〈B〉, stronger than that ofequation (43), and limited by equations (47)-(48),

〈B〉 = εBd, (49)

where ε < 1 accounts for an inevitable decay of the fieldduring a flare event. The dissipation rate of the magneticfield in a reconnection event is ∝ B2 (see e.g. section 12.4in Longair 1992) so the field decays exponentially. Since theabove average is weighted by the luminosity of an activeregion, which quickly decreases with decreasing B, ε can be,

Figure 9. The ratio of the CS and bremsstrahlung luminosities(heavy and thin lines, respectively; the latter ratio is indepen-dent of m) to the power dissipated in a patchy corona as func-tions of the accretion rate in the case of the coronal magneticfield strength equal to 0.5 of that inside the disc. The solutionscorresponding to dominance of the gas pressure and radiationpressure are shown for m below and above ∼ mcrit, respectively.The chosen values of m, Θ, α are the same as in Figure 7. SeeSection 4.2.3 for details. The vertical bars mark our estimates ofthe accretion rates in 3 cases.

in principle, of the order of unity. Thus, we adopt ε = 0.5 innumerical examples below.

Assuming the magnetic field strength given by equa-tion (49), we obtain for disc regions dominated by the gaspressure,

LCS

L≈

200 × 10−3.1α

α0.13+0.06α(1− α)×

ε1.82+0.91αNr2bΘ3.78−0.11αm0.36α−0.27ϕ

α0.82+0.41αv f(1− f)0.09+0.05αm0.41α−0.18

(50)

for 0.4 <∼ α <∼ 0.9 and

LCS

L≈ 0.052

(

1 +3ϕ

α− 1

)

ε2.73Nr2bΘ3.66m0.09

α1.23v f(1− f)0.14m0.23α0.19

(51)

for α >∼ 1.1. Similar expressions can be derived for regionsdominated by the radiation pressure.

Figure 9 shows the above ratios for the same values ofα, Θ and m (except that m = 108 was assumed for theaverage Seyfert 1 spectrum) as in Figure 7 as functions of mfor ε = 0.5, rb = 2, and N = 10. We see that the obtainedluminosity is comparable to the one dissipated in the coronaonly in the case of Cyg X-1. In other cases, the CS processgives a small contribution, comparable to or lower than thecontribution of bremsstrahlung.

In the case of NGC 4258, we can reproduce the 2–10keV power-law index and luminosity with Θ = 0.43 and rb =2. We note, however, that the CS luminosity is a sensitivefunction of the size of active regions in this model, LCS ∝r2b (which dependence is much stronger than that in theprevious coronal models). Based on equation (40), we expectrb decreasing with the decreasing L. If this is indeed the case

c© 1999 RAS, MNRAS 000, 1–17

14 G. Wardzinski and A. A. Zdziarski

for NGC 4258, the model of coronal CS emission could bethen ruled out for this object (unless Θ > 1, as in the casediscussed in Section 4.2.2 above).

We therefore again conclude that the CS radiation canbe important only in the case of stellar-mass sources withhard spectra. This radiative process is negligible in thecase of objects with soft spectra, probably including low-luminosity sources.

5 APPLICATIONS TO ACCRETING

NEUTRON STARS

Power-law X-ray spectra are, in fact, observed not only fromblack-hole sources. Similar power laws are detected, e.g.,from binary systems with weakly-magnetized neutron starsin the so-called low spectral state. Unambiguous classifica-tion of some of those sources as neutron stars comes fromdetections of type-1 X-ray bursts from them. Those spectraare often modelled by thermal Comptonization (e.g. Barretet al. 1999, hereafter B99), in which case the question arisesof the origin of seed photons.

It is also of interest that accreting weakly-magnetizedneutron stars show a number of similarities to accretingblack holes (see e.g. a discussion in B99). First, this con-cerns timing properties in X-rays, namely time lags betweensoft and hard X-rays, which suggests similar geometries ofthe sources and/or similar variability mechanisms (e.g. Fordet al. 1999). Second, spectra of black holes in their hardstates fit remarkably well a correlation between the 1–20and 20–200 keV luminosities seen in neutron-star binaries(Barret, McClintock & Grindlay 1996) which may indicatethat similar radiation mechanisms are operating in the twoclasses of objects. On the other hand, the presence of theneutron star results in more sources of soft photons than inthe case of a black hole, namely thermal radiation from thesurface of the star and synchrotron radiation in the stellarmagnetic field.

Cyclotron radiation as a source of soft photons Comp-tonized in weakly magnetized accreting neutron stars hasrecently been considered by Psaltis (1998). In his model, self-absorbed cyclotron photons are produced in a layer abovethe stellar surface with kT ∼ 20 keV and a magnetic field ofB <∼ 1010 G, and then subsequently Comptonized in a hotspherical corona of kT ∼ 10 keV. Here, we consider a moregeneric geometry in which synchrotron photons are pro-duced and Comptonized in the same medium, the tempera-ture of which is determined from observations. This leads usto investigating plasmas of temperatures higher than thoseconsidered by Psaltis (1998). As in Section 4, we comparethe luminosities in our model spectra to those observed.

We analyze here spectra of four X-ray bursters. Thefirst one is 4U 0614+091, which low-state spectrum was fit-ted with the Comptonization model of Poutanen & Svensson(1996) by Piraino et al. (1999). They found the seed black-body photons have kTBB <∼ 30 eV, and the plasma param-eters are given by Θ ≃ 0.5 and α = 1.44. The luminositywas L1−200keV ≃ 3.7 × 1036 erg s−1 assuming D = 3 kpc(Brandt et al. 1992). It is interesting that the above elec-tron temperature is much higher than that seen from otherX-ray bursters so far. The parameters of the second object inour sample, 1E 1724–3045, modeled by B99 with a Comp-

Figure 10. The relation between the size of the emission regionand its magnetic field strength required for the CS process toreproduce the observed X-ray spectra and luminosities of fourX-ray bursters (heavy lines). The thin horizontal lines show theupper limits on the field strength resulting from fitted constraintson the energy of seed photons.

tonization model of Titarchuk (1994) are kTBB ≃ 1 keV,Θ ≃ 0.057, α = 0.95. (Since B99 did not give the spectralindex of their fit, we have obtained it from the Comptoniza-tion model they used.) The luminosity, given D = 6.6 kpc(Barbuy, Bica & Ortolani 1998), is L1−200keV ≃ 1.3 × 1036

erg s−1. The third source is GS 1826–238, for which B99found α = 0.73, and Θ ≃ 0.08 from the high-energy tailwith the model of Poutanen & Svensson (1996). At D = 7kpc (B99), L1−200keV ≃ 1.5 × 1036 erg s−1. Observations ofthe fourth object, XB 1916–053, were reported by Churchet al. (1998). The X-ray spectrum was fitted as a power-lawof index α = 0.61 and the temperature was estimated fromthe cut-off in the spectrum to be Θ ≃ 0.06. At D = 9 kpc,L0.5−200keV ≃ 1.1× 1036 erg s−1 (Church et al. 1998).

We consider then a generic model with a spherical, uni-form, plasma cloud, and investigate the relation between thesize of the cloud and the magnetic field strength required toreproduce the observed luminosities at their spectral indicesand temperatures. From equations (19) and (32), we expectthat this relation will be roughly of the form of B ∝ R−0.7.We note that since a CS spectrum is normalized by its self-absorbed, optically-thick, part, the luminosity is roughlyproportional to the source area, with the source geometrybeing only of secondary importance. Thus, our results mayalso approximate those for emission from a thin layer abovethe stellar surface. A second constraint on the model canbe obtained from the fitted temperature of the seed pho-tons, which is known for 4U 0614+091 and 1E 1724–3045.In those cases, we can identify the seed-photon temperaturewith the maximum possible turnover energy, hνt <∼ kTBB,which then yields an upper limit on B. This limit is B <∼ 108

G and B <∼ 6× 109 G, respectively. In the case of GS 1826–238, the seed-photon temperature was not fitted, but we canestimate, from the shape of the spectrum presented by B99,that kTBB <∼ 1 keV, which yields B <∼ 5 × 109 G. We haveno corresponding limit for XB 1916-053.

The size–magnetic field relation obtained for the fourobjects is shown in Figure 10. Those results suggest thatComptonization may take place in a corona of the size com-parable to the stellar radius (∼ 106 cm). The required mag-netic field is then B <∼ 109 G. This value is consistent both

c© 1999 RAS, MNRAS 000, 1–17

Thermal synchrotron radiation and its Comptonization 15

with the upper limits resulting from fitting the energy ofseed photons and with the maximum field strength possiblein X-ray bursters, B <∼ 1010–1011 G, see Lewin, van Paradijs& Taam (1995). (Their constraint results from the require-ment that the magnetic field does not funnel matter towardsthe magnetic poles, and more stringent constraints can beobtained for a specific structure of the accretion flow.)

We note that if Comptonization takes place close tothe stellar surface, reflection albedo close to unity would berequired. Otherwise reprocessing in the surface layers wouldresult in a strong blackbody component, which is not seen.

We also find that it is unlikely that the size of the regionwhere synchrotron photons are produced and Comptonizedis much larger than the neutron star radius. Such a configu-ration would be then similar to that of an optically thin discaround a black hole (Section 4.1) or an advection-dominatedextended corona above a cold disc (Narayan, Barret & Mc-Clintock 1997). Then the required magnetic field, B ∼ 108

G at R ∼ 107 cm, would be too strong to be sustained bythe disc [see equation (35), where we estimate the magneticfield in hot, optically thin, discs].

6 DISCUSSION

A general feature of models of accretion flows around blackholes considered above is the dependence of B ∝ m−1/2 [orvery close to it, equation (47)]. This then implies decreas-ing values of LCS/LE with m, and the relative importanceof the CS process much weaker in AGNs than in black-holebinaries. The predicted values of LCS strongly increase withincreasing spectral hardness and temperature of the plasma.Compared with observations, we find that both the hot flowand coronal models can, in principle, explain emission ofthe hardest (α ∼ 0.5) among luminous (Eddington ratios of∼ 0.01), stellar-mass, sources in terms of the CS process.However, this process provides a negligible contribution tothe luminosity, L, of luminous stellar-mass sources with softspectra and of luminous AGNs regardless of α. Taking intoaccount the overall similarity between properties of luminoussources with either hard or soft spectra and containing ei-ther stellar-mass or supermassive black holes (e.g., Zdziarski1999), it is likely that the CS process is in general energeti-cally negligible in luminous sources.

However, the hot flow and coronal models differ in theirpredictions for the relative importance of the CS processwith decreasing L/LE. In the hot flow model, there is clearlya range of L/LE ≪ 0.01, in which the CS process can yielda dominant contribution to L even in the case of AGNs. Onthe other hand, the coronal models have rather weak depen-dences of LCS/L on L/LE. However, at some low value ofL/LE, bremsstrahlung emission becomes dominant. There-fore, we have found that, in the coronal models, there is norange of L/LCS (except for hard, stellar-mass, sources, seeabove) in which the CS process could dominate energeti-cally. In both models, bremsstrahlung is expected to givethe main contribution to L in very weak sources.

There are three complications in the picture above.First, the plasma parameters are expected, in general, to de-pend on L/LE, which will modify theoretical dependencesof LCS/L on the Eddington ratio. On the other hand, theplasma parameters can often be determined by spectral fit-

ting, in which cases our results can be used directly. Second,we have considered here accretion onto non-rotating blackholes. Black-hole rotation increases, in general, the efficiencyof accretion, and the CS process may yield then luminosi-ties higher than those found here. This is, in fact, confirmedby a study of Kurpiewski & Jaroszynski (1999), who haveconsidered advection-dominated flows and have found thatan increase of the black-hole angular momentum leads toincrease of the CS emission. This happens due to both thesynchrotron emission itself as well its Comptonization be-coming much more effective in innermost regions of the disc.Analysis of those issues is, however, beyond the scope of thiswork. Third, even when other processes dominate energeti-cally, there can be a range of photon energies in which theCS process gives the dominant contribution.

Still, the strong conclusion of Section 4 is that the ther-mal CS process does not dominate observed X-ray spec-tra of luminous black-hole sources. Rather, the dominantX-ray producing process in those sources appears to beComptonization of blackbody photons emitted by someoptically-thick medium, e.g., an optically-thick accretiondisc or clumps of cold matter. Observationally, the presenceof such component is indicated by the wide-spread pres-ence of Compton reflection from a cold medium in lumi-nous black-hole sources (e.g. Zdziarski et al. 1999) as wellas observations of blackbody-like soft X-ray components inspectra of black-hole binaries (e.g. Ebisawa et al. 1994). Fur-thermore, a very strong correlation between the reflectionstrength and the X-ray spectral index has been discoveredin those sources (Zdziarski et al. 1999). The presence of suchcorrelation is naturally explained in models with plasmacooling due to blackbody emission of cold matter, whereasit cannot be explained if the CS process dominates. On theother hand, we find in Section 5 that the thermal CS pro-cess can easily account for power law emission of weaklymagnetized neutron stars in their low states.

We point out that a small non-thermal tail in the elec-tron distribution, possibly resulting, e.g. from stochastic ac-celeration or acceleration in reconnection events could signif-icantly increase the turnover frequency (since electrons withγ ≫ 1 are usually responsible for emission at the turnoverfrequency), and thus increase the CS luminosity from accret-ing black holes and neutron stars. This issue is the subjectof our work in progress.

Finally, we notice thermal Comptonization may be(at most) a secondary process in spectral formation insome classes of compact X-ray sources. One important classof such sources are black-hole binaries in the soft state,which X-ray and soft γ-ray spectra appear non-thermal (e.g.Gierlinski et al. 1999). Those spectra contain very strongblackbody components, which probably dominate over syn-chrotron photons as seeds for non-thermal Comptonization.

7 CONCLUSIONS

The main results of this work can be outlined as follows.We have derived and tested analytic approximations for

the synchrotron emission coefficient in thermal plasmas, ap-plicable especially to the semi-relativistic range of temper-atures, T ∼ 109 K. We have also obtained analytic approx-

c© 1999 RAS, MNRAS 000, 1–17

16 G. Wardzinski and A. A. Zdziarski

imations to the turnover frequency (at which the plasmabecomes optically-thick to absorption).

Then, we have presented a method to treat thermalComptonization of synchrotron radiation. Our approximateanalytic expressions allow for self-consistent calculations ofthe spectrum and the luminosity of such a source and canbe easily applied to models of accretion discs.

We have also investigated the role of the thermal CSprocess in accretion flows around stellar-mass and super-massive black holes. We have considered two main scenar-ios: a hot, two-temperature, optically-thin flow and activeregions above a cold, optically-thick disc. We have foundthat this process is only marginally important in luminousX-ray sources containing accreting black holes, and it canpossibly dominate only in stellar-mass sources with hardestspectra. The dominant radiative process in those sourcesappears to be Comptonization of blackbody radiation emit-ted by cold matter in the vicinity of the hot plasma. Onthe other hand, the CS process can explain X-ray spectraof weaker sources, e.g., low-luminosity AGNs, but only inthe hot-flow model. Finally, below certain low luminosity,bremsstrahlung becomes the dominant process.

Finally, we considered the case of weakly-magnetizedaccreting neutron stars. We have found that their power-lawX-ray spectra in the low state can be accounted for by theCS process taking place in a corona of the size comparableto the stellar radius.

ACKNOWLEDGMENTS

This research has been supported in part by a grant fromthe Foundation for Polish Science and the KBN grants2P03D00624 and 2P03D01716. It is a pleasure to acknowl-edge valuable discussions with Chris Done, Marek Gierlinski,Krzysztof Jahn and Marek Sikora. We are grateful toTiziana Di Mateo, the referee, whose thoughtful commentshave led to significant improvements of this study.

REFERENCES

Abramowicz M. A., Chen X., Kato S., Lasota J.-P., Regev O.,1995, ApJ, 438, L37

Abramowitz M., Stegun I. A., 1970, Handbook of MathematicalFunctions. Wiley, New York (AS70)

Barbuy B., Bica E., Ortolani S., 1998, A&A, 333, 117

Barret D., McClintock J. E., Grindlay J. E., 1996, ApJ, 303, 526Barret D., Olive J. F., Boirin L., Done C., Skinner G. K., Grindlay

J. E., 1999, ApJ, in press (B99)

Bekefi G., 1966, Radiation Processes in Plasmas. Wiley, New YorkBrandt S., Castro-Tirado A. J., Lund N., Dremin V., Lapshov I.,

Sunyaev R., 1992, A&A, 262, L15Chanmugam G, Barrett P. E., Wu K., Courtney M. W., 1989,

ApJS, 71, 323

Chen X., Abramowicz M. A., Lasota J. P., 1997, ApJ, 476, 61Church M. J., Parmar A. N., Ba lucinska-Church M., Oosterbrock

T., Dal Fiume D., Orlandini M., 1998, A&A, 338 556

Clavel J., et al., 1987, ApJ, 321, 251

Dermer C. D., Liang E. P, Canfield E., 1991, ApJ, 369, 410Di Matteo T., 1998, MNRAS, 299, L15

Di Matteo T., Blackman E. G., Fabian A. C., 1997a, MNRAS,291, L23

Di Matteo T., Celotti A., Fabian A. C., 1997b, MNRAS, 291, 805(DCF97)

Di Matteo T., Celotti A., Fabian A. C., 1999, MNRAS, 304, 809

Dulk G. A., 1985, ARA&A, 23, 169

Ebisawa K., Ueda Y., Inoue H., Tanaka Y., White N. E., 1996,ApJ, 467, 419

Eckart A., Genzel R., 1996, Nat, 383, 415

Ford E. C., van der Klis M., Mendez M., van Paradijs J., KaaretP., 1999, ApJ, 512, L31

Galeev A. A., Rosner R., Vaiana G. S., 1979, ApJ, 229, 318

Gierlinski M., Zdziarski A. A, Done C., Johnson W. N., EbisawaK., Ueda Y., Haardt F., Phlips B. F., 1997, MNRAS, 288, 958

Gierlinski M., Zdziarski A. A, Poutanen J., Coppi P. S., EbisawaK., Johnson W. N., 1999, MNRAS, 309, 496

Haardt F., Maraschi L., Ghisellini G., 1994, ApJ, 432, L95

Hartmann D., Woosley S. E., Arons J., 1988, ApJ, 332, 777

Hawley J. F., Gammie C. F., Balbus S. A., 1995, ApJ, 440, 742

Jones, T. W., Hardee, P. E., 1979, ApJ, 228, 268

Kurpiewski A., Jaroszynski M., 1999, A&A, 346, 713

Lasota J.-P., Abramowicz M. A., Chen X., Krolik J., Narayan R.,Yi I., 1996, ApJ, 462, 142

Lewin W. H. G., van Paradijs J., Taam R. E., 1995, in Lewin W.H. G., van den Heuvel E. P. J., van Paradijs J., eds., X-rayBinaries, Cambridge Univ. Press, Cambridge, p. 175

Lightman A. P., Zdziarski A. A., 1987, ApJ, 319, 643

Longair M. S., 1992, High Energy Astrophysics. Cambridge Uni-versity Press, Cambridge

Mahadevan R., 1997, ApJ, 477, 585

Mahadevan R., Narayan R., Yi I., 1996, ApJ, 465, 327 (MNY96)

Makishima K. at al., 1994, PASJ, 46, L77

Malysheva L. K., 1997, Astron. Lett., 23, 585 [in Russian: AZhPisma, 23, 667]

Massey P., Johnson K. E., DeGioia-Eastwood K., 1995, ApJ, 454,171

Miyamoto S., Kitamoto S., Sayuri I., Negoro H., Terada K., 1992,ApJ, 391, L21

Miyoshi M. et al., 1995, Nat, 373, 127

Narayan R., Yi I., 1995, ApJ, 452, 710 (NY95)

Narayan R., Barret D., McClintock J. E., 1997, ApJ, 482, 448

Narayan R., Mahadevan R., Grindlay J. E., Popham R. G., Gam-mie C., 1998, ApJ, 492, 554

Osterbrock D. E., 1974, Astrophysics of Gaseous Nebulae. Free-man, San Francisco

Pacholczyk A. G., 1970, Radio Astrophysics. Freeman, San Fran-cisco

Peterson B. M., 1997, Active Galactic Nuclei. Cambridge Univer-sity Press, Cambridge

Petrosian V., 1981, ApJ, 251, 727 (P81)

Piraino S., Santangelo A., Ford E. C., Kaaret P., 1999, A&A, 349,77

Poutanen J., Svensson R., 1996, ApJ, 470, 249

Psaltis D., 1998, PhD Thesis, University of Illinois

Robinson P. A., Melrose D. B., 1984, Australian J. Phys., 37, 675

Rybicki G. R., Lightman A. P., 1979, Radiative Processes in As-trophysics. Wiley-Interscience, New York

Scheuer P. A. G., 1968, ApJ, 151, L139

Shapiro S. L., Lightman A. P., Eardley D. M., 1976, ApJ, 204,187

Sunyaev R. A., Titarchuk L. G., 1980, A&A, 86, 121

Svensson R., 1984, MNRAS, 209, 175

Svensson R., Zdziarski A. A., 1994, ApJ, 436, 599

Takahara F., Tsuruta S., 1982, Prog. Theor. Phys., 67, 485

Titarchuk L., 1994, ApJ, 434, 313

Trubnikov B. A., 1958, Doklady Akad. Nauk. SSSR, 118, 913

Wind J. P., Hill E. R., 1971, Australian J. Phys., 24, 43

Zdziarski A. A., 1985, ApJ, 289, 514 (Z85)

Zdziarski A. A., 1986, ApJ, 303, 94 (Z86)

c© 1999 RAS, MNRAS 000, 1–17

Thermal synchrotron radiation and its Comptonization 17

Zdziarski A. A., 1999, in Poutanen J., Svensson R., eds., HighEnergy Processes in Accreting Black Holes, ASP Conf. Series161, p. 16

Zdziarski A. A., Fabian A. C., Nandra K., Celotti A., Rees M. J.,Done C., Coppi P. S., Madejski G. M., 1994, MNRAS, 269,L55

Zdziarski A. A., Johnson W. N., Magdziarz P., 1996, MNRAS,283, 193

Zdziarski A. A., Poutanen J., Miko lajewska J., Gierlinski M., Ebi-sawa K., Johnson W. N., 1998, MNRAS, 301, 435

Zdziarski A. A., Lubinski P., Smith D. A., 1999, MNRAS, 303,L11

This paper has been produced using the Royal AstronomicalSociety/Blackwell Science LaTEX style file.

c© 1999 RAS, MNRAS 000, 1–17