Adv. Space Res. Vol.3, No.10-12, pp.245-248, 1984 Printed in Great Britain. All rights reserved. 0273-1177/84 $0.00 + .50 Copyright © COSPAR C O M P T O N I Z A T I O N OF LOWFREQUENCY RADIATION IN A C C R E T I O N DISKS: A N G U L A R DISTRIBUTION AND POLARIZATION OF H A R D X - R A Y R A D I A T I O N R. A. Sunyaev, and L. G. Titarchuk Space Research Institute, U.S.S.R. Academy of Sciences, Moscow, U.S.S.R. ABSTRACT We present the results of computation for angular distribution and polarization of radiation for the set of the disk Thomson optical depth values. KEYWORDS X-ray sources, radiation transfer, polarization. Comptonization (photon frequency change due to multiple scatterings on hot electrons) is one of the most important mechanisms of X-ray source spectra generation. Obviously it is this process that is responsible for the get neration of the spectra observed in many binax~y X-ray sources, in the nucle~ of galaxies end quasars. Disk accretion onto neutron stars and black holes supplies energy for most of the compact X-ray sources. In the inner regions of accretion disks responsible for the X-ray radiation, electron scattering makes a predominant contribution into opacity. The hard radiation spectrum forms aS a results of Ccmptonization (Shakura, Sunyaev 1973, 1976, Shapiro et al., 1976). The shape of radiation spectra generated by Comptonization in a plasma cloud of finite optical depth was the subject of our previous paper (Sunyaev, Titarchuk, 1980, further referred to as STSO). This report deals in detail with the following questions: a)photon distribution over the time of their escape from the disk; b)sngular distribution of the hard radiation intensity of the disk; c)polarization of hard radiation! d) the distribution of the temperature of electrons over the optical depth. The main argument that forced us to the problem of Comptonization was careful measurement of the radiation spectrum of the X-ray source Cyg XS (Sunyaev, Tramper, 1979, Nolan et al., 1981, Steinle et al., 1982). Spectra agree well with the analytically derived radiation spectrum of the rediation generated due to comptonization of low-frequency photons in a cloud of isothermal plasma (S~uyaev, TrUmper, 1979, STSO). Besides, it now seems possible to determine both the plasma temperature in the cloud ( K ~ e =26.5 keV) and its optical thickness. In the case of the disk geometry the optical semithickness of the disk as respect to the electron scattering is ~o=2 (ST80). Attempts to interpret the data of X-ray observations of nuclei of active galaxies Cen A, NGC4151, NGC 1275, yield even smaller optical thickness (Pozdnyakov et al., 1983). Thus, ~nphasis should be given to the problem of the polarization and angular distribution of the radiation escaping from the disks of comparatively small optical thickness. ,a) Ge0metry of the Problem The idealized geometry is assumed throughout this paper: an infinite disk, the optical thickness respect to the electron scattering across the disk is ~ @ . Electron density may be a function of I~ (optical depth m~easured from the upper boundary of the disk). The only factor important to us is that within each infinitely thin, plane-parallel layer the electron density is coiI~aZt along the plane parallel to the central disk plane. The disk is assumed to be isothermal. Let us introduce the normal to the disk plane, @ is the angle between this normal and a given direction, ~ = ~ O . b~ Basic Results The hard rsdlation of the accretion disk forming as a result of Comptonizatio_n, underwent there in number of scatterings much higher than average ~ :@~'(see ST80 §2). Thus, it follows ~mmediately for an isothermal JASR 3:10/12-Q 245 246 R.A. Sunyaev and L.G. Titarchuk disk that 1)hard radiation spectrum does not depend on the low-frequency photon source distribution; 2) the hard-radiation spectrum is the same at any point of the disk (is independent of ¢ - coordinate); ~) the intensity and the radiation energy density within the disk are ~- dependent ; 4) the angular distribution (Figure 2) and polarization (Figure 3) of the outgoing hard radietion are the functions of ~oonly. c I Calculation Procedure C amptonization of low-frequency rediation with h ~ o < < K T m is characterized by one-one correspondence between the number of scatterings undergone by a photon in a plasma cloud and the energy it gained Hence the angular dependence of the hard radiation of the accretion disk and its polarization may be derived without solving a spectral problem. It is sufficient to estimate the angular distribution and polarization of the radiation subjected in the disk to a number of scatterings much higher than its mean value. This is what we do. C~lculations were made in the Thomson approximation: we used Thomson cross-section and Rayleigh indicatrixo The integral-differential equation of polerized radiation transfer was solved by the method of saccessive approximations over the number of scatterings (Neuman series), the k-th approximation yielded the angular distribution and polarization of photons after K scatterings in the cloud. It is striking that for K > ~ ~oa the angular distribution and polarization of photons undergone K scatterings in the cloud do not depend on K. Therefore, the angular distribution and polarization of hard X-ray radiation generated by multiple scetterings of low-frequency photons on hot electrons should not depend on energy. Summation over all scattering events was used to estimate the integral angular distribution of the radiation from the disk and its integral polarization. These radiation characteristic most drastically depend on the distribution of primary photon-sources over the disk. 1. ~NGULAR D I S T R I B U T I ~ 0i~' RADIATION The angular distribution derived by Sobolev (1949) and Chandrasekh~r (1950), for the radiation leaving the plane-parallel atmosphere with opacity dominsted by Thomson scattering, is well known. According to that solution the ratio of intensities in the directions ~=i (along the normal to the disk) and ~ = O (in its plene) is equal to 3.06 and the dependence is well approximated by a simple formula f+ ~. o 6 / ~ . Obviously that dependence ~[jw) would be different for a disk of finite optical thickness and even more, it would depend on the photon-source distribution ove~ optical depth ~- . The angular distribution of com~tonized photons Z (~,~ ~ is given for some r o in Figure I. It gives asympto%ic dependences ~ - ~ 2 for ~mm ~ which, as has already been mentioned, depend neither o n , n o r on distribution of low f r e q u ~ c ~ photon sources inside the disk. If r ~ 1 0 the asymptotic dependence _T~J,)is close to the classical Chandrasekhar-Sobolev distribution estimeted for-a semiinfinite electron atmosphere. In the case of the small optical depth the radiation field within the disk is no longer isotropic. As a result, the intensity maximum is no more at~'=~ al q~ QJ ~4 as ~ Q$ ~ ag ~.~ f"t: ~v Q't :;r////// 0.3 ! o O,i ~,Z 0,~ ~4, O,S 0,6 eta 0.1 Qg /~-J~" Fig.1 Angular distribution of hard rsdiation from the disk computed for different To. The ~ = 10 case practically coincides with the solution the classical problem°about the escape of rsdiation from the atmosphere when electron scattering is predominent. Comptonization of Low-Frequency Radiation 247 11.POL~RIZ~TION OF I~RD I ~ D ~ T i ~ FHOM ACCf~ETI~I DISKS a)Histcry of the Problem As is known from the classical monograph of Chandrasekhar (1950) and from Sobolev a paper (1949) in the plsne parallel atmosphere (the partial case of which is a geometrically thin but optically thick accretion disk) with predcminsnt electron scattering in the opacity, the radiation from the disk is characterized by a strong linear polarization. The degree of polarization depends on a) the angle between the normal to the disk plene ~nd to the observer direction and b) the true-absorption to scattering cross-section ratio. ~!ith the predominante of the electron scattering in opecity it the polerization degree gradually grows from 0 to 11.7% when ~f decreases from I to O. This result is the cornez-stone of the pioneer papers on the research of polarization of the X-ray radiation of accretion disks (Rees, 1976, Lightman, Shapiro, 1976). In the Angel's (1969) paper that employed a ~onte-Carlo method the finite character of the optical thickness (by Thomson scattering) of the emitting zone was taken into account. Lightman and Shapiro (1975) point out weaker role of Thomson scattering in the opacity with d e c r e a s i n g X-ray photon energy and the effect it has or the radiation polarization of optically thick disks. Lightman ~nd Shapiro (1976) tried to estimate also the degree of polarization of an accretion disk assuming that it is g~ometrically thin but optically thick by scattering. Their estimates relied on the celculations of Angel (1969). Connors et al (1980) took into account the effects of the g~neral relativity theory on the angular distribution end polerization of the radiation of the accretion disk around a black hole. Loskytov and Sobolev (1980) made calculations for different distributions of photon sources, ~ifferent optical thickness of the disks ( ~ o ~ 0.05 ~ I0~ ~ ) and energies of photons in the context of the standard model of a disk (Shakura, Sunyaev, 1973), They took into consideration the bremsstrahlung and absorption of photons and Thomson scattering. Ccmptonization of radiation was not accounted for. W '/0 -8 'Y 0.4 o,2 0,3 / / o,~ o.5 0.6 o.~ -i o.8 a9 J~1 Fig.2 Hard-r~diation pol~rization degree for different . When =10 it is close to the solution of the classical problem. 248 R.A. Sunyaev and L.G. Titarchuk The re sult~ of calculations P~= ~~ do not depend -f~.. d~T ) for K , ~ and describe the polarization of ~omp~onized photons. For large ~ @ ~ 10 the hard-radiation polarization coincides with the solution of the classical problem. For small ~o < ~ 10 the difference with the classical result becomes obvious (Figure 2). At ~ @ ~ 5 the polarization changes its sign. The case of Cyg X-I ( ~ @ = 2 according to SuAayaev and Tramper 1979) is characterized by the minimal polarization whose magnitude does not exceed p ~ 5.4% and which is negative over the ramge of accretion-disk inclination angles that do not contradict the data of optical and X-ray observations 37 ° ~ i ~ 55~ that is 0.57 ~ ( 0.8 (Lyutyi et al 1973, Avmi, Bahcall, 1975). Our results seem to agree well with the data of observations of linear polarization of Cyg X-I (Long et al 1980) from the 0SO-8 satellite in the energy bands h ~ 2 . 6 keV ( ~ = 2.4%~ 1.1% and 4 ~ = 5.2keV ( P = 5 . 3 % ~ 2.5%). The degree of polarization may grow with energy because the zone with ~ z 2 . 6 keV corresponds to a relatively small number of scatterings in the disk. The characteristic dependence P # ~ J sets up in a comparatively short time, but with a small number scatterings ( ~ 5 to 10), i.e. in the range ~ ~ it still depends strongly on the geometric distribution of low-frequency photon soUrces. 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