Spectral and polarization properties of black hole accretion disc emission: including absorption effects
Roberto Taverna, Lorenzo Marra, Stefano Bianchi, Mičhal Dovciak, René Goosmann, Frederic Marin, Giorgio Matt, Wenda Zhang
MMon. Not. R. Astron. Soc. , 1–13 (2020) Printed 14 December 2020 (MN LaTEX style file v2.2)
Spectral and polarization properties of black hole accretion discemission: including absorption effects
R. Taverna (cid:63) , L. Marra , S. Bianchi , M. Dovˇciak , R. Goosmann , F. Marin , G. Matt and W. Zhang Dipartimento di Matematica e Fisica, Universit`a degli Studi Roma Tre, via della Vasca Navale 84, I-00146 Roma, Italy Astronomical Institute, Academy of Sciences of the Czech Republic, Boˇcn´ı II 1401, CZ-14100 Prague, Czech Republic Observatoire Astronomique de Strasbourg, Universit´e de Strasbourg, CNRS, UMR, 7550, 11 rue de l’Universit´e, F-67000 Strasbourg, France
Accepted . . . . Received . . . ; in original form . . .
ABSTRACT
The study of radiation emitted from black hole accretion discs represents a crucial way tounderstand the main physical properties of these sources, and in particular the black holespin. Beside spectral analysis, polarimetry is becoming more and more important, motivatedby the development of new techniques which will soon allow to perform measurements alsoin the X- and γ -rays. Photons emitted from black hole accretion discs in the soft state areindeed expected to be polarized, with an energy dependence which can provide an estimateof the black hole spin. Calculations performed so far, however, considered scattering as theonly process to determine the polarization state of the emitted radiation, implicitly assumingthat the temperatures involved are such that material in the disc is entirely ionized. In thiswork we generalize the problem by calculating the ionization structure of a surface layer ofthe disc with the public code CLOUDY , and then by determining the polarization propertiesof the emerging radiation using the Monte Carlo code
STOKES . This allows us to account forabsorption effects alongside scattering ones. We show that including absorption can deeplymodify the polarization properties of the emerging radiation with respect to what is obtainedin the pure-scattering limit. As a general rule, we find that the polarization degree is largerwhen absorption is more important, which occurs e.g. for low accretion rates and/or spinswhen the ionization of the matter in the innermost accretion disc regions is far from complete.
Key words: stars: black holes – X-rays: binaries – accretion discs – abundances – polariza-tion.
The black hole spin in accreting stellar-mass black hole binary sys-tems is currently estimated by using spectroscopic (either the iron Kα line profile or the thermal disc continuum emission) or tim-ing (kHz QPOs) techniques (see Reynolds 2019, and referencestherein). A fourth technique, based on the energy dependence ofthe polarization degree and angle of the thermal disc emission, hasbeen proposed in the late 70s (Connors & Stark 1977; Stark & Con-nors 1977; Connors, Piran & Stark 1980), and then revisited morerecently (Dovˇciak et al. 2008; Li, Narayan & McClintock 2009;Schnittman & Krolik 2009; Taverna et al. 2020). The renewed in-terest in the polarimetric technique is due to, on one hand, thediscrepant results provided by the other techniques in a few cases(most notably GRO J1655-40, Motta et al. 2014) and, on the otherhand, to the re-opening of the X-ray polarimetric observing windowprovided by missions like IXPE (Weisskopf et al. 2013), due to be (cid:63)
E-mail: taverna@fis.uniroma3.it launched in 2021, and on a more distant future by eXTP (Zhang etal. 2019).This technique is based on the influence of strong gravity onthe polarization degree and angle of radiation. The polarization an-gle, in particular, is expected to rotate along the geodesics; the ro-tation is larger for more energetic photons, because they are mostlyemitted closer to the black hole. When convolving over the en-tire disc emission, an energy dependence of the polarization angleresults. This effect is larger the larger the spin of the black hole(because of the lower value of the Innermost Stable Circular Or-bit, ISCO), which can therefore be determined (see in particularDovˇciak et al. 2008).Schnittman & Krolik (2009) included in their analysis the con-tribution of returning radiation, i.e. photons that, following nullgeodesics in the space-time around the BH, return to the disc beforebeing reflected (in the hypothesis of albedo) towards the ob-server. They showed that spectra and polarization observables canbe deeply modified with respect to those of direct radiation onlydue to the effects of reflection at the disc surface. Taverna et al. © 2020 RAS a r X i v : . [ a s t r o - ph . H E ] D ec R. Taverna et al. (2020) investigated the effects of considering different scatteringoptical depths on the intrinsic polarization and calculated how thespectra and polarization observables of both direct and returningradiation are modified when a more realistic albedo profile is usedto characterize the disc surface.In all the previous works, however, the polarization of thethermal disc is either calculated assuming a pure scattering slabof material (Dovˇciak et al. 2008) or the Chandrasekhar’s (1960)prescription, valid for a plane-parallel, semi-infinite scattering at-mosphere. To date, a self-consistent treatment for studying spectraland polarization properties of stellar-mass BH accretion disc emis-sion accounting for absorption effects alongside scattering ones isstill lacking, although the method described in Taverna et al. (2020)to include a more realistic albedo profile for the disc surface can beconsidered as a first attempt in this direction.In the present work we still adopt the scenario of a thermal disccovered by a surface atmosphere, as in previous ones. Here, how-ever, we move a further step forward by including in the polarizedradiative transfer calculations absorption effects in the partially-ionized slab material. To this aim, we first model the ionizationstructure of an optically-thick, surface layer of the disc using thephoto-ionization code
CLOUDY (Ferland et al. 2017). Secondly, wesolve the radiative transfer for photons propagating within this sur-face layer, exploiting the Monte Carlo code
STOKES (Marin 2018,see also Goosmann & Gaskell 2007). While our physical assump-tion of an ionized slab above a black body emitting disc is certainlysimplistic, it allows us to start addressing the issue of the effect ofabsorption on the polarization properties of the emerging radiation.A self-consistent, simultaneous treatment of the disc structure andof polarized radiative transfer is beyond the scope of the present pa-per. Spectra and polarization properties are provided at the source,without considering the general relativistic corrections that affectthe photon transport to the observer; the complete description ofphoton spectra and polarization as they would be measured at in-finity will be handled in a future publication.The plan of the paper is as follows: in section 2 we recall thegeneral concepts of our theoretical model, and describe in more de-tails the numerical implementation in section 3. Results and com-parisons with previous works are introduced in section 4. Finally,we summarize our findings and present our conclusions in section5.
We recall in this section our basic assumptions. As described inprevious works (see e.g. Taverna et al. 2020; Dovˇciak et al. 2008),we consider the accretion disc as a standard disc (see Shakura &Sunayev 1973), with particles rotating around the center at the Ke-plerian velocity. The central, stellar mass BH is characterized bythe values of its mass M and dimensionless angular momentum a ,with the space-time around described by the Kerr metric (Novikov& Thorne 1973). The disc stops at the radius of the ISCO r ms (seeBardeen et al. 1972), and no-torque at the inner boundary is as-sumed.Internal viscous dissipations heat up the surface layers of thedisc. We handle the distribution of the emitted photons through alocal blackbody, with the temperature T varying with the radialdistance r from the center according to the Novikov-Thorne profile (Novikov & Thorne 1973), T ( ξ, M, ˙ M, a ) = 741 f col (cid:18) M M (cid:12) (cid:19) − / (cid:18) ˙ M M (cid:12) yr − (cid:19) / × [ f ( ξ, a )] / keV ; (1)here ξ = ( r/r g ) / (with r g = GM/c the gravitational radius), ˙ M is the BH accretion rate and f is a function of ξ and the BH spin a (see Taverna et al. 2020, for the complete expression; see alsoPage & Thorne 1974; Wang 2000). In equation (1) we account forthe energy shift of photons due to the scatterings they undergo withparticles in the deepest layers of the disc through the hardeningfactor f col (Shimura & Takahara 1995; Dovˇciak et al. 2008; Davis& El-Abd 2019).The density of the disc material is modelled according to theradial profile discussed by Comp`ere & Oliveri (2017), who providethe expressions for the total (hydrogen) density n (H) at the equa-torial plane (see Taverna et al. 2020, for more details). For ease ofreading we reported the main formulae in Appendix A. In order toobtain the corresponding values of the density at the disc surface,we assume for the sake of simplicity a Gaussian prescription forthe vertical structure, so that n (H , z ∗ ) = n (H) exp (cid:18) − z ∗ ( r ) h (cid:19) , (2)where h is the typical height of the disc at the radial distance r and z ∗ is the altitude above the disc equatorial plane at which the scat-tering optical depth calculated up to infinity is equal to 1 (Tavernaet al. 2020).We solve the ionization structure of the disc only in its surfacelayer, in which the emitted blackbody radiation is processed. Todescribe the matter that composes this layer, we adopted the typicalsolar abundance (Asplund et al. 2005), focussing in particular onthe elements with Z = 1 (hydrogen), (helium), (carbon), (nitrogen), (oxygen), (neon), (silicon), (sulfur) and (iron), neglecting the presence of dust. In order to reproduce both the spectral and polarization propertiesof the radiation emerging from the accretion disc, we use the MonteCarlo code
STOKES (Marin 2018, see also Marin et al. 2012; Goos-mann & Gaskell 2007). This code was originally developed to solvethe radiative transfer of near-IR to UV photons propagating insideregions of material where scattering represents the main source ofopacity (like e.g. in AGN). However, with the purpose of adaptingthe code to our case, we resorted to the upgraded version 2.33 of
STOKES , optimized for modeling the X-ray radiation propagatingthrough a stratified, plane-parallel atmosphere, which fully adaptsto describe the surface layer of the disc according to our model. Inorder to solve the atomic structure in this atmospheric layer, we usethe version 17.01 of
CLOUDY (last described in Ferland et al. 2017),an open-source, photo-ionization code to simulate the relevant pro-cesses that occur in astrophysical clouds. In particular, since we re-strict our investigation to the case in which collisions are the dom-inant process for ionizing matter, we used the coronal model,which allows us to calculate the ionization properties of a mate-rial slab by specifying only the (kinetic) temperature and the total(hydrogen) density of the gas.The surface atmospheric layer is, then, divided into a number N r of patches, each one at a different radial distance r from the © 2020 RAS, MNRAS , 1–13 bsorption effects on BH accretion disc emission Figure 1.
Polarization degree in the pure-scattering limit, plotted as a function of the cosine of the inclination angle θ for isotropic emission from the bottomof the disc surface slab. Positive values correspond to polarization parallel to the disc plane ( χ = 0 ◦ ), while negative ones to polarization in the direction of thedisc axis ( χ = 90 ◦ ). Different values of the Thomson optical depth τ are explored: . (cyan), . (orange), (green), (magenta), (purple), (brown)and (pink). The polarization degree angular profile as described in Chandrasekhar (1960, see Table XXIV) is also shown for comparison (red-dashed line).Stokes parameters have been summed over the – keV (left-hand panel) and the . – keV (right-hand panel) energy bands. central BH. For the sake of simplicity, we associate to each radialpatch the corresponding values of temperature and density at thedisc surface, as provided by equations (1) and (A1)–(A3), respec-tively, and we consider them as constant within the same patch.After having defined an xyz reference frame, with the z -axis cho-sen along the disc symmetry axis, the code solves the ionizationstructure inside the patch by dividing it into several slices, charac-terized by the height z with respect to the base of the surface layer.The maximum height z max that the code reaches in each run is uni-vocally determined by the input parameter N stopH , i.e. the hydrogencolumn density at which calculations are stopped. For every radialdistance r and altitude z , we finally extract, from the code output,the fractional abundance of each element in different states of ion-ization, F (i)X α = n (i)X α n totX . (3)In equation (3), n (i)X α is the number density of the ionic species X α ( i ) , with α the ionic charge and i the excitation level of theoutermost electron, while n totX denotes the total number density ofthe element X .The output of CLOUDY obtained for each single radial patch ishence processed by a specific C ++ script, in order to produce a suit-able input file for STOKES , containing all the relevant informationabout the ionization structure. The emission region, from whichphotons are injected inside the layer can be featured as well in thisinput file, choosing between different geometries of emission. Forour simulations we assumed the emitting source to be located at thebottom of the atmospheric layer ( z = 0 ); no bulk motion is con-sidered for the emission region. Following our model prescriptions(see §2), for each radial patch of the surface layer we imposed thatall the seed photons are emitted according to an isotropic blackbodyat the temperature T ( r ) (see equation 1). This is tantamount to saythat the propagation directions, along which photons are launched, are sampled by the polar angles θ e = arccos( √ r ) φ e = 2 πr (4)with respect to the z -axis and the xz -plane, respectively; in equa-tion (4) r and r are uniform deviates between and . Seed ra-diation is set as unpolarized, i.e. the photon Stokes vectors are ini-tialized to i q u v = . (5)Photons are, then, followed along their trajectory, accountingfor all the possible interactions (such as multiple scattering, free-free interactions or photoelectric absorptions) they can experiencein the surface layer. All the photons which are not absorbed insidethe layer are eventually collected in different virtual detectors, eachone identified by the inclination θ and the azimuth φ which char-acterize the corresponding viewing direction in the xyz frame. Thetotal number of virtual detectors is fixed by specifying in input thenumbers of points N θ and N φ of the ( θ, φ ) angular mesh. For eachdetector, the Stokes parameters of the photons collected along thecorresponding viewing direction are summed together, after hav-ing rotated the different Stokes parameter reference frames aroundthe detector line-of-sight, to match with the detector frame. Thefinal output of each run (corresponding to each radial distance r )consists of the Stokes parameters i , q , u and v of the emerging ra-diation as functions of the photon energy E and of the two viewingangles θ and φ . Given the axial symmetry of the adopted geometry,we actually integrated over φ . The resolution N E and the bound-aries E min – E max of the photon energy band, as well as the num-ber N phot of seed photons to be launched in each single run, canbe defined at the beginning of the STOKES input file. The linearpolarization fraction Π and the polarization angle χ are eventually © 2020 RAS, MNRAS , 1–13 R. Taverna et al.
Figure 2.
Surface temperature (blue, solid line) and density (orange, dashed line) radial profiles obtained according to equations (1) and (A1)–(A3), re-spectively, in the case of a non-rotating (left-hand panel) and a maximally-rotating (right-hand panel) BH with mass M = 10 M (cid:12) and accretion rate ˙ M = 0 . M Edd . The hardening factor f col is set to . . obtained through the usual expressions Π = (cid:112) q + u iχ = 12 arctan (cid:18) uq (cid:19) , (6)where χ = 0 corresponds to polarization vectors oriented as per-pendicular to the z axis (i.e. lying in the plane of the disc), increas-ing in the clockwise direction.Finally, it should be noted that, by construction, at each radiusthe polarization is calculated by STOKES assuming a slab with con-stant density and temperature which is indefinite in the xy -plane.As we are dealing with a geometrically thin accretion disc solution,we consider this approximation, which simplifies significantly ourcomputations, as acceptable. In this section we present the results of some significant simula-tions performed exploiting the codes illustrated in section 3. Forthe sake of comparison with previous works, we start discussingthe outputs obtained from
STOKES in the pure-scattering limit andthen we continue considering the effects of ionization in the discsurface layer. In the following, the angular dependence is resolvedin the colatitude θ , while data are summed over the azimuthal angle( N φ = 1 ). The behavior of the polarization degree, after switching-off all in-teraction processes apart Compton scattering, is shown in Figure1 as a function of the cosine of the inclination angle θ (sampledover a -point grid), for different values of the (Thomson) opticaldepth τ . For each value of τ , N phot = 10 photons are injected inthe layer according to a blackbody distribution at the temperature T = 1 keV. Positive values refer to polarization in the plane of thedisc ( χ = 0 ◦ ), while negative ones to perpendicular polarization( χ = 90 ◦ ). In the left-hand plot Stokes parameters have been inte-grated in the – keV energy band, since this is the typical range ofoperations of the new-generation X-ray polarimeters like IXPE . Inthe right-hand one, instead, the . – keV band has been consid-ered, as the best compromise between the choice of an energy rangein which all the photons are included and the need of a reasonablecomputational time.The curves are obtained following the prescriptions discussedin §3, i.e. assuming that photons are emitted isotropically froma pointlike source placed at the bottom of the atmospheric layer,which is a semi-infinite, plane-parallel slab. In almost all the casesexplored, polarization vectors turn out to be oriented as parallelto the plane of the disc ( Π (cid:38) ), with the maximum polarizationdegree attained at high inclinations and monotonically decreasingdown to ∼ at smaller ones. The only exception occurs at small op-tical depths ( τ (cid:46) . ), for which polarization may become perpen-dicular to the disc plane at small inclination angles. In particular, Π assumes negative values for θ (cid:46) ◦ when τ = 0 . , while po-larization vectors are definitely oriented in the direction of the pro-jected disc symmetry axis for practically the entire range of incli-nations for τ = 0 . . The effect can be explained as follows. Whensmall values of τ are considered, photons which are more likelyto be scattered are those emitted at large inclinations with respectto the slab normal, since they experience a larger effective opticaldepth. Because the electric vector of the scattered photons oscillatesperpendicularly to the scattering plane, these photons will emergewith orthogonal polarization. On the other hand, photons originallyemitted at small inclinations will be practically unpolarized if theyemerge at small inclinations too, while, for symmetry reasons, theirpolarization is expected to be very large and oriented in the plane ofthe disc if they emerge at large inclinations. As a result, polarizationturns out to be negative for a large interval of inclinations, becom-ing positive only close to cos θ ∼ . On the other hand, at large op-tical depths multiple scattterings can occur, which mitigate this be- © 2020 RAS, MNRAS , 1–13 bsorption effects on BH accretion disc emission Figure 3.
Emerging spectra (top row) and polarization degree (bottom row) plotted for the two cases a = 0 (left-hand column) and a = 0 . (right-handcolumn) and for θ = 12 . ◦ (cyan), . ◦ (orange), . ◦ (green), . ◦ (red) and . ◦ (purple). Here M = 10 M (cid:12) , L = 0 . L Edd , N stopH = 10 cm − and f col = 1 . . Stokes parameters, weighted for the area and the temperature of each radial patch, are summed over the radial distance r (see equations8). havior; moreover, scatterings become more and more frequent alsofor photons emitted at small inclinations, which contribute muchmore to the overall polarization pattern. The emergence of nega-tive polarization has been originally noted by Nagirner (1962) andGnedin & Silant’ev (1978), albeit in a slightly different scenario.In particular, the latter authors showed that, while only positive po-larization is present if the photons are preferentially emitted deepin an accretion disc, negative polarization arises when the sourcefunction is almost constant along the disc vertical structure. In thelatter case, as Gnedin & Silant’ev (1978) explain, many photons be-fore the last scattering travel almost parallel to the disc surface, andmoreover their scattering angle is close to 90 ◦ , a situation similarto ours in the case of small optical depths.Increasing the layer optical depth, the angular behavior of thepolarization fraction approaches, as expected, the classic solutionobtained by Chandrasekhar (1960) assuming a semi-infinite slab(red-dashed line). In this condition, photons can emerge from theupper boundary of the layer only after a large number of scatter- ings, at variance with what happens for small optical depths, whena significant number of photons can escape even without sufferingany scattering (and then remaining unpolarized). It is interesting tonote that the polarization degree attained for optical depths largerthan ∼ turns out to even slightly exceed that expected accord-ing to the Chandrasekhar’s profile, as it can be seen looking at theleft-hand panel in Figure 1. The reason is that, while the Chan-drasekhar’s solution is obtained assuming elastic (Thomson) scat-tering, in our STOKES model Compton downscattering is accountedfor . As a consequence of the energy shift toward lower energies,a certain number of photons drops out of the selected energy band,an effect of course increasing with the average number of scatter-ings (and then with τ ). Since the photons that are lost in this wayare those which have suffered more interactions, and therefore aremore isotropised and less polarized, an increase of Π with respect We point out that Compton up-scattering is not yet implemented in thecurrent version of
STOKES © 2020 RAS, MNRAS , 1–13
R. Taverna et al. to Chandrasekhar (1960) can be observed when the energy band isrestricted to the – keV range. Results closer to those given by theChandrasekhar’s solution can be obtained, in fact, by considering awider energy range, as shown in the right-hand panel of Figure 1,where Stokes parameters are integrated over the . – keV band.A similar behavior to that just discussed has been already pre-sented in Dovˇciak et al. (2008, see their Figure 1), with some sub-stantial differences: (i) the change of orientation of the polarizationvectors occurred at quite larger optical depths (for τ ∼ ); and (ii)higher polarization fractions were attained, especially for low opti-cal depths, contrary of what happens in the present case. The mainreason of these differences resides in the different layout adopted inprevious works (Dovˇciak et al. 2008, see also Taverna et al. 2020)with respect to the present paper, with the emitting region located inthe middle of the atmospheric layer (i.e. at z = z max / instead of z = 0 ). In this situation, to ensure that photons are emitted isotrop-ically in both the upper and the lower half-spaces of the layer, thedistribution of the emission angles θ e should be corrected by θ e = arccos(1 − r ) , (7)in place of that indicated in the first of equations (4). As a conse-quence, many more photons are originally emitted along the disc,rather than perpendicularly to it, and therefore the polarizationtends to be perpendicular to the disc plane even for relatively largeoptical depths (for very large depths, however, the original distri-bution of photons is of course no longer important). We then included the ionization structure of the disc surface layeras calculated by
CLOUDY . We consider the two extreme cases of anon-rotating ( a = 0 ) and a maximally-rotating ( a = 0 . ) BH,with mass M = 10 M (cid:12) . We give the results for a portion of thedisc between r = r ms and r = 30 r g , dividing the surface layerinto N r = 30 , logarithmically-spaced radial bins. In this sectionwe consider the set of parameters already used in previous works(see e.g. Taverna et al. 2020), i.e. a hardening factor f col = 1 . anda mass accretion rate ˙ M chosen in such a way that the accretion lu-minosity amounts to of the Eddington limit L Edd . The maxi-mum height z max of the surface layer at each radial patch is chosen,instead, by setting the stop column density to N stopH = 10 cm − ,which corresponds to a Thomson optical depth τ (cid:39) . . The cor-responding radial profiles of the surface temperature and densityare plotted in Figure 2. Also in this case, STOKES runs are per-formed launching N phot = 10 seed photons for each radial patch,while Stokes parameters are sampled over the – keV energyrange through a -point grid and over the – π/ inclination in-terval through a 20-point grid, which a posteriori turned out to be agood compromise between the statistical significance and the com-putational time ( ≈ day to complete one run on an Intel i7, 4-coremachine) for the chosen values of the parameters.Figure 3 shows the spectra and polarization degree obtainedin the two cases of a = 0 and a = 0 . , for five different view-ing angles θ . In order to correctly account for the different numberof photons emitted from each radial bin, in these plots the Stokesparameter fluxes are summed over the radial distance r in the fol- The calculation of the accretion efficiency is performed using the expres-sions reported in Johannsen & Psaltis (2011), see also Krawczynski (2012). lowing way, ¯ i ( E, θ ) = (cid:88) r i ( E, r, θ ) A ( r ) T ( r )¯ q ( E, θ ) = (cid:88) r q ( E, r, θ ) A ( r ) T ( r )¯ u ( E, θ ) = (cid:88) r u ( E, r, θ ) A ( r ) T ( r ) , (8)where A ( r ) = 2 πr d r is the area of the annular radial patch, withwidth d r , at the distance r from the center, while the factor T ( r ) takes into account the different surface temperatures which charac-terize each radial patch.¿From the top row of Figure 3 one can clearly see that theemerging photon flux is in general higher for a maximally rotatingBH than for the case of a = 0 . This can be easily explained bythe fact that, for a = 0 . , the disc extends up to regions muchcloser to the BH horizon; here temperatures are much higher thanfor the non-rotating case, peaking at ∼ . keV against a maxi-mum value ∼ . keV for a = 0 (see Figure 2). As a consequence,for the Schwarzschild BH the seed photon blackbody peak can beexpected to occur at ∼ keV, while starting from – keV thestrong energy dependence of the photoelectric absorption cross-section becomes more evident. On the other hand, the maximumof the injected blackbody falls at around keV in the maximally-rotating case, so that the contribution of seed photons is much morerelevant all over the selected – keV energy range. Absorptionturns out to be important as well, as shown by the occurrence ofseveral spectral features superimposed to the continuum in both the a = 0 and a = 0 . cases. These lines appear to be mostly lo-cated at low energies ( – keV), with the exception of two, quitestrong absorption features which appear at ∼ . keV and ∼ keV.The correspondent polarization degrees are plotted as func-tions of the photon energy in the bottom row of Figure 3. Lookingat the figures, it is immediately clear that, as a general rule, polar-ization degree is higher when the absorption is more relevant (seealso the comparison with the pure scattering case). This pattern canbe explained by noting that, when absorption is important, most ofthe emerging photons are those originally emitted almost verticallyand which suffer only one scattering: those photons are all polar-ized with the polarization vector parallel to the disc surface. Pho-tons originally emitted at high inclinations are instead more likelyto be absorbed; those photons, in the pure scattering limit, wouldprovide mostly perpendicular polarization, with a reduction of thenet polarization degree. Polarization of radiation emerging from thedisc surface layer decreases by decreasing the observer’s inclina-tion angle, attaining a value close to at small θ ( (cid:46) ◦ ) and atessentially all the photon energies. For a = 0 (bottom-left panel),the polarization degree is in general higher than for the maximally-rotating BH case, except for the lower energies (at around keV),where Π turns out to be very low (below ) at all the inclina-tions. This can be explained noting that, as mentioned above, pri-mary photons peak indeed at such low energies, so that photonsemerging at – keV are essentially all seed photons, which areassumed to be unpolarized (see §3). At higher energies the polar-ization fraction increases rapidly, up to a maximum value close to ∼ for the highest inclinations. This behavior corresponds tothe most important decline of the photon flux at around keV. For a = 0 . (bottom-right panel), the energy dependent behaviorof the polarization degree closely follows that just discussed forthe Schwarzschild case, notwithstanding the lower values attained,which are in general reduced by a factor of ∼ – (with a maxi-mum around at higher energies). As for the spectra, also the © 2020 RAS, MNRAS , 1–13 bsorption effects on BH accretion disc emission Figure 4.
Photon flux (top row) and polarization degree (bottom row) plotted as functions of the photon energy (sampled with a 5000-point grid) in the caseof a = 0 . , for the first radial bin of the disc surface and for five different inclinations of the viewing direction. The values of the other parameters are thesame as in Figure 3. Scattering, free-free, recombination and photo-electric absorption effects are accounted for in the plots of the left column, while resonantscattering lines are artificially turned off in those of the right one. polarization fraction plots show further peculiar features between and keV, with the occurrence of quite narrow drops which seemsto be associated, at first glance, to the analogous absorption lines inthe flux. However, contrary of one could expect, the decrease in thephoton flux would seem to correspond this time to a decrease alsoin the polarization degree.This counter-intuitive behavior can be explained by notingthat both the spectral and the polarization profiles reported here arestrongly affected by our choice to adopt a 100-point energy grid be-tween and keV. In order to better understand the nature of thesefeatures we report, for the sake of example, the results of an addi-tional run of STOKES in the case of a = 0 . and for the first ra-dial bin of the disc surface, sampling the energy range with a 5000-point mesh. Figure 4 (left column) shows the plots of the spectrumand polarization degree for five different inclination angles and fo-cussing on the energy band between and . keV. Moreover, withthe purpose of a greater clarity, we report in the right column thesame situation as in the left one, in which we artificially removed all the features which are produced by resonant scattering. This allowsone to clearly distinguish the contribution from the iron line com-plex, which in the selected energy range is essentially due to thetransitions of Fe +24 and Fe +25 in various ionization states (Sarazin1988; Fabian et al. 2000), with two emission lines at . and keV which stand out with respect to the continuum. Looking atthe related polarization fraction plot (bottom-right panel), a suddendecrease of Π occurs in correspondence with the aforementionedspectral features, as expected in the case of emission lines. Onceresonant scattering is added in the computations (left panels), sev-eral absorption features appear in the spectrum, which are muchbroader than the narrow emission lines just discussed. In particular,two important absorption lines occur at exactly the same energiesof the most important iron emission lines. A polarization degreegrowth correctly correponds to the scattering absorption lines, withthe two significant drops at . and keV due to the iron emissionlines which are still visible in the middle of the polarization frac-tion peaks. We note that the features discussed here above can be © 2020 RAS, MNRAS000
Photon flux (top row) and polarization degree (bottom row) plotted as functions of the photon energy (sampled with a 5000-point grid) in the caseof a = 0 . , for the first radial bin of the disc surface and for five different inclinations of the viewing direction. The values of the other parameters are thesame as in Figure 3. Scattering, free-free, recombination and photo-electric absorption effects are accounted for in the plots of the left column, while resonantscattering lines are artificially turned off in those of the right one. polarization fraction plots show further peculiar features between and keV, with the occurrence of quite narrow drops which seemsto be associated, at first glance, to the analogous absorption lines inthe flux. However, contrary of one could expect, the decrease in thephoton flux would seem to correspond this time to a decrease alsoin the polarization degree.This counter-intuitive behavior can be explained by notingthat both the spectral and the polarization profiles reported here arestrongly affected by our choice to adopt a 100-point energy grid be-tween and keV. In order to better understand the nature of thesefeatures we report, for the sake of example, the results of an addi-tional run of STOKES in the case of a = 0 . and for the first ra-dial bin of the disc surface, sampling the energy range with a 5000-point mesh. Figure 4 (left column) shows the plots of the spectrumand polarization degree for five different inclination angles and fo-cussing on the energy band between and . keV. Moreover, withthe purpose of a greater clarity, we report in the right column thesame situation as in the left one, in which we artificially removed all the features which are produced by resonant scattering. This allowsone to clearly distinguish the contribution from the iron line com-plex, which in the selected energy range is essentially due to thetransitions of Fe +24 and Fe +25 in various ionization states (Sarazin1988; Fabian et al. 2000), with two emission lines at . and keV which stand out with respect to the continuum. Looking atthe related polarization fraction plot (bottom-right panel), a suddendecrease of Π occurs in correspondence with the aforementionedspectral features, as expected in the case of emission lines. Onceresonant scattering is added in the computations (left panels), sev-eral absorption features appear in the spectrum, which are muchbroader than the narrow emission lines just discussed. In particular,two important absorption lines occur at exactly the same energiesof the most important iron emission lines. A polarization degreegrowth correctly correponds to the scattering absorption lines, withthe two significant drops at . and keV due to the iron emissionlines which are still visible in the middle of the polarization frac-tion peaks. We note that the features discussed here above can be © 2020 RAS, MNRAS000 , 1–13 R. Taverna et al.
Figure 5.
Polarization degree plotted as a function of the cosine of the inclination angle θ for the two cases of a = 0 (left-hand panel) and a = 0 . (right-hand panel) and the same values of parameters as in Figure 3. Stokes parameters have been summed over the radial distance from the central BH (seeequations 8) and energy integrated between and keV. Blue-solid lines mark the behavior predicted by STOKES , while the red-dashed lines that calculatedaccording the formulae by Chandrasekhar (1960, see Table XXIV). fully resolved only if the energy grid is sufficiently fine (i.e. withmore than ∼ points between and keV). However, weremark that the energy resolution we adopted throughout the paper(i.e. 100 points in the – keV range) is still far better than thatof many existing instruments, and in particular of the forthcomingphotoelectric polarimeters like IXPE . Furthermore, considering atoo precise energy grid would lead to unacceptably long compu-tational times, as well as to over-detailed spectral and polarizationbehaviors in the majority of the selected energy range, which aremostly pointless for our research. For these reasons we resort tothe coarser, but still acceptable, 100-point energy resolution in thefollowing.In order to better understand how the ionization structure ofthe disc surface layer determines the polarization pattern of theemerging radiation, Figure 5 shows the behaviors of the polariza-tion degree plotted as a function of the cosine of the inclination an-gle θ . Here the Stokes parameters, which are still summed over theradial distance as described above (see equations 8), are further in-tegrated over the photon energy in the – keV IXPE energy band.The angular distribution of the polarization degree as predicted byChandrasekhar’s (1960) formulae is also reported in the plots (red-dashed lines). The fact that Π assumes only positive values at all theinclinations considered (for both the cases of a = 0 and a = 0 . )shows that emerging radiation is mostly polarized perpendicularlyto the disc symmetry axis. This, as already discussed in §4.1, fol-lows from the original assumption of radiation emitted isotropicallyfrom the base of the disc surface layer, so that photons are essen-tially emitted upwards, with propagation direction close to the discaxis. Furthermore, in the case of a maximally-rotating BH the po-larization degree turns out to closely follow the Chandrasekhar’sprofile, contrary of what happens for a = 0 , where the polariza-tion fraction largely exceeds, at low inclinations, that predicted byChandrasekhar (1960). Since the Chandrasekhar’s profile is the ref-erence one for models that assume scattering as the only process responsible for photon polarization , one can safely conclude thatthe polarization properties in the a = 0 . case are mainly de-termined by scattering. On the other hand, absorption plays a moreimportant role in the non-rotating case. This can be further con-firmed looking again at the temperature and density distributionsreported in Figure 2. In fact, due to the higher temperature reachedclose to the BH horizon, the fraction of ionized atoms is clearlyexpected to be larger for a = 0 . , while for a = 0 absorption ef-fects become more important. In this regard, one should also noticethat the total density close to the ISCO is much larger (by a factorof ∼ ) for the non-rotating case than for the maximally rotatingone; this enhances the effects of absorption for a = 0 .To complete this analysis, Figure 6 shows the photon flux andthe polarization degree plotted as functions of the radial distance r from the center, again for a = 0 and a = 0 . ; also in thiscase Stokes parameters have been integrated over energy in the – keV band. The behavior of the photon flux (top row) turns outto naturally follow the radial profile of the disc surface tempera-ture reported in Figure 2, peaking at the distance r characterizedby the maximum temperature. On the other hand, as it can be seenin the bottom row, the polarization fraction tends to increase as thephoton flux declines, apart from photons leaving the disc surfaceat small inclination angles, which (as already discussed) are prac-tically unpolarized. Moreover, still taking as reference the plots inFigure 2, some similarities can be observed between the behav-iors of the polarization degree and the total density as functionsof the radial distance. This further confirm the previous findingsthat ascribe the rise in polarization fraction to absorption effects.In fact, in the Schwarzschild BH case Π increases dramatically go-ing towards the ISCO, where temperature is lower, since atoms areless ionized in this region and the density is quite large. Then, af-ter a minimum attained in correspondence to the maximum of the We note, however, that for the optical depth assumed in Figure 5 (i.e. τ (cid:39) . ), the curve of the polarization degree in the pure-scattering limitstands below the Chandrasekhar’s profile (see e.g. Figure 1).© 2020 RAS, MNRAS , 1–13 bsorption effects on BH accretion disc emission Figure 6.
Photon flux (top row) and polarization degree (bottom row) plotted as functions of the radial distance r in the two cases of a = 0 (left-hand column)and a = 0 . (right-hand column), for the same values of parameters as in Figure 3 and for five different inclinations of the viewing direction. Stokesparameters have been energy integrated between and keV. temperature distribution (at ∼ r g ), the polarization degree risesagain, although the density is more or less constant, as the temper-ature decreases (and the fraction of ionized atoms drops). In themaximally-rotating BH case, instead, the polarization fraction ra-dial profile is mostly monotonic, going from the minimum value(attained close the ISCO) and essentially continuing to grow up tothe outer boundary. This is due to both the temperature decreaseand the density increase, which are visible in the right panel of Fig-ure 2. The only exceptions occur very close to the ISCO, beforethe photon flux peak is reached, and farther than ∼ r g , witha slight decrease of the polarization fraction for the highest incli-nations ( (cid:38) ◦ ). While the former effect is analogous to that justdiscussed for small radii in the a = 0 case, the latter can be morelikely ascribed to the absorption features which are present at lowenergies in the behavior of Π . In fact, according to the adopted ra-dial profile, the surface temperature far from the central BH turnsout to be quite low ( ∼ . keV), so that only low-energy photonsare expected to contribute significantly at those distances. Recallingthat the plots in Figure 6 have been obtained summing the Stokes parameters in the – keV energy range, it is reasonable to con-clude that the slight decline of Π at large r and high inclinations isdue to the deep features which characterize the polarization degreeat around – keV (see the bottom-right panel of Figure 3), whichare indeed more relevant for high values of θ . After having discussed the behavior of spectral and polarization ob-servables for a given set of parameter values, we now test how theStokes parameters of the emerging radiation can change by varyingthe properties of the disc material and the accretion flow. In thisrespect, we explored two further configurations, characterized bydifferent values of mass accretion rate ˙ M and stop column density N stopH , leaving the BH mass unchanged at
10 M (cid:12) and the harden-ing factor f col = 1 . . As reported in equations (1) and (A1)–(A3),varying the accretion luminosity acts both on the temperature andthe density radial profiles. Figure 7 shows the variations of tem-perature and density profiles as a result of increasing the accretion © 2020 RAS, MNRAS , 1–13 R. Taverna et al.
Figure 7.
A comparison of the temperature (blue) and density (orange) radial profiles in the two cases of a = 0 (left) and . (right), for L = 0 . L Edd (the case also reported in Figure 2) and . L Edd . rate from (i.e. the case already described in §4.2) to ofthe Eddington limit, again for the two cases of non-rotating andmaximally rotating BHs. In particular, the temperature undergoesa general shift upwards by a factor of ∼ . , superimposed to aradial displacement of the maximum towards a slightly larger dis-tance from the center. The density, on the other hand, drops at allthe considered radial distances by a factor of ∼ .The effects of such changes on the emerging flux are shownin the top row of Figure 8, where solid lines mark the behavior for L = 0 . L Edd (i.e. the case already discussed in §4.2) and dot-ted lines that for L = 0 . L Edd ( N stopH = 10 cm − for boththese runs). As one could expect, the photon flux attains higher val-ues when a higher accretion luminosity is considered, due to theincrease of the temperature all over the disc surface. For a = 0 (top-left panel), the low-energy peak due to primary photons is sen-sibly broadened with respect to the L = 0 . L Edd case. Moreover,the spectral features ascribed to absorption are less pronounced,as a result of both the temperature increase (which also increasesthe ionization fraction in the disc material) and the lower values ofdensity. Also in the a = 0 . case (top-right panel), spectra for L = 0 . L Edd turn out to be harder than for lower luminosities.In this case no substantial differences can be observed in the ab-sorption features, since, as noted before, already at L = 0 . L Edd temperature was sufficiently high to significantly reduce absorptioneffects. Dash-dotted lines in Figure 8 mark, instead, the behavior ofthe emerging flux for N stopH = 5 × cm − , which correspondsto a Thomson optical depth τ (cid:39) . . In order to display onlythe effect of changing N stopH , for this simulation we returned to L = 0 . L Edd , so that temperature and density profiles are thosereported in Figure 2. Contrary of what happens by increasing theaccretion luminosity, in this case the number of emerging photonsturns out to be pretty lower in the entire energy range. The peaksof the spectral distributions fall at the same energy as in the orig-inal case (see §4.2), as a result of the choice to adopt the sametemperature profile. However, both in the Schwarzschild and in themaximally-rotating case absorption features are dramatically moresignificant, this effect being more evident for a = 0 . Indeed, as- suming that photons escape the layer at a z max corresponding to alarger optical depth implies that they are still involved in a conspic-uous number of scatterings, so that a lower number of emergingphotons at that altitude z max can be reasonably expected. More-over, since an increase of the optical depth translates into a de-crease of the photon mean free path inside the disc material, thisalso justifies the increase of absorption effects, despite the fact thattemperature and density are unchanged with respect to the initialcase (with N stopH = 10 cm − ).Much in the same way as the spectra, the polarization degreeof the emerging radiation is influenced as well by the changes inluminosity and optical depth. The bottom row of Figure 8 showsthe energy-dependence of Π , plotted for the same values of the pa-rameters adopted for the flux in the top row. By increasing L to . L Edd , the overall behavior turns out to be in general loweredby ∼ – with respect to the initial case, with a more importantreduction at lower energies and for high values of θ . On the otherhand, substantial differences can be observed when the value of N stopH is increased. In particular, a steep rise occurs as the inclina-tion angle increases, attaining a value ∼ for a = 0 and ∼ for a = 0 . at keV . To explore this behavior more in depth,we reported in Figure 9 the plots of the energy-integrated polar-ization fraction as a function of the viewing inclination, similarlyto those shown in Figure 5; also in this case the polarization de-gree predicted by Chandrasekhar’s (1960) prescription is displayedfor comparison. By increasing the accretion rate (dotted lines), Π turns out to be substantially reduced at high inclinations for a = 0 (down to ∼ of the value attained for L = 0 . L Edd ), but stillremaining above the level set by the Chandrasekhar’s profile. For a = 0 . , instead, Π is only slightly lower than in the initial case,similarly to what we have noted in Figure 8, with an overall re-duction not larger than ∼ . On the contrary, if the stop column We notice that the values of Π obtained in the simulations reported inFigure 8 for a = 0 can be affected by the poor statistics due to the lownumber of photons at around keV and at high energies.© 2020 RAS, MNRAS , 1–13 bsorption effects on BH accretion disc emission Figure 8.
Spectrum (top row) and polarization degree (bottom row) of the radiation emitted from the disc surface in the cases of a = 0 (left) and . (right), for N stopH = 10 cm − – L = 0 . L Edd (solid lines), N stopH = 10 cm − – L = 0 . L Edd (dotted lines) and N stopH = 5 × cm − – L = 0 . L Edd (dash-dotted lines), and for three different inclinations of the viewing direction: θ = 12 . ◦ (blue), . ◦ (orange) and . ◦ (green). Thevalues of the other parameters are chosen as in Figure 3. density is increased to × cm − (maintaining the luminosityat . L Edd ) then the polarization degree in general increases too(dash-dotted lines). Also in this case the most relevant change oc-curs in the Schwarzschild limit (attaining a maximum value close to at high inclinations), while the curve is not far from the origi-nal one (and from that given by the Chandrasekhar’s profile) in themaximally-rotating limit. All these behaviors comply with thosejust discussed for the flux. In fact, the temperature increase (withthe consequent ionization fraction rise) and the density decreasewhich occur taking L = 0 . L Edd both determine a lower influenceof absorption effects, so that the polarization fraction turns out tobe lower with respect to the case with a lower luminosity. On theother hand, considering a larger value of the stop column densitymeans that photon transfer is calculated for larger optical depths.This translates into more significant absorption effects which pro-duce an increase of the polarization degree. That being said, whatwe have concluded in the reference case (see §4.2) holds true alsofor different luminosities and column densities, i.e. radiation is ba- sically more polarized for a = 0 than for a = 0 . , confirmingthat absorption is more important for slowly rotating BHs, whilescattering dominates as the BH spin increases. In this paper we discussed a method to simulate the spectral andpolarization properties of radiation emitted from stellar-mass BHaccretion discs in the soft state. Contrary to previous works, where,following the Chandrasekhar’s (1960) prescription, polarizationwas considered to be due only to scattering of photons onto elec-trons in the disc, in our model we include a self-consistent treat-ment of absorption effects in the disc material. To this aim, wesolved the ionization structure in the surface layer of the disc us-ing the photo-ionization code
CLOUDY (Ferland et al. 2017), as-suming the Novikov & Thorne (1973) temperature profile and theComp`ere & Oliveri (2017) density profile. Seed radiation (assumed © 2020 RAS, MNRAS , 1–13 R. Taverna et al.
Figure 9.
Polarization degree plotted as a function of the cosine of the inclination angle θ in the two cases a = 0 (left-hand column) and a = 0 . (right-handcolumn), for N stopH = 10 cm − – L = 0 . L Edd (blue, solid), N stopH = 10 cm − – L = 0 . L Edd (orange, dotted) and N stopH = 5 × cm − – L = 0 . L Edd (green, dash-dotted), while the other parameters have the same values as in Figure 3. Stokes parameters have been summed over the radialdistance from the central BH (see equations 8) and energy integrated between and keV. The red-dashed lines denotes the behavior calculated according theformulae by Chandrasekhar (1960, see Table XXIV). to be unpolarized) is injected isotropically from the bottom of thislayer, according to a blackbody distribution at the local tempera-ture. The radiative transfer of photons is then calculated using theray-tracing, Monte Carlo code STOKES (Marin 2018; Goosmann &Gaskell 2007). The emerging photon Stokes parameters are deter-mined for a number of virtual detectors on the surface of the layer,displaced at different inclinations with respect to the disc axis.The main findings of our present investigation can be summa-rized in the following points. • Placing the emitting source at the base of the surface layerforces the polarization direction of most of the emerging photons tobe parallel to the plane of the disc (see Figure 1). This differs fromthe results of previous works (e.g. Dovˇciak et al. 2008, with emis-sion region placed in the middle of the slab), according to which aconsistent fraction of photons is polarized parallel to the disc sym-metry axis for τ (cid:46) . In fact, when only scattering is considered,the polarization degree at lower τ is quite small for emitting sourceon the bottom with respect to the case of emission from the mid-dle of the layer. As expected, the polarization fraction tends to theangular behavior described by Chandrasekhar’s (1960) formulae athigh optical depths ( τ (cid:38) ). • We emphasize that including absorption effects goes in gen-eral towards increasing the polarization degree (see Figure 3 and 5,left columns). This is particularly encouraging, considering that allthe theoretical models developed so far (which account for scatter-ing as the only responsible for photon polarization, see e.g. Dovˇciaket al. 2008; Schnittman & Krolik 2009; Taverna et al. 2020) predicta level of polarization of ∼ at most. • The photon spectrum and the polarization degree turn out tostrongly depend on the spin of the central BH. This can be essen-tially ascribed to the fact that, as the BH spin increases, the innerboundary of the disc extends to smaller distances from the hori-zon, where temperatures are higher and more energetic photonscan be emitted (see Figures 2 and 6). In addition, according to theprofile by Comp`ere & Oliveri (2017), the density turns out to be higher for lower values of a (see again Figure 2). As a consequence,photon polarization can be reasonably expected to be higher forslowly rotating BHs, for which absorption effects are more impor-tant, while polarization tends to approach the Chandrasekhar’s pro-file for a = 0 . (Figure 5, right panel), where scattering effectsare dominant. • Including self-consistently the absorption effects in our modelallows us to specify the non-trivial behavior of the seed photonStokes parameters as a function of photon energy, observer’s in-clination and radial distance from the center, as it can be seen inFigure 3, 4 and 6. • We could test the dependence of spectra and polarization onthe physical properties of the disc, such as the accretion luminosityand the hydrogen column density of the disc surface slab (see Fig-ures 8 and 9). In particular, we have checked that increasing the ac-cretion luminosity acts in increasing the photon flux and decreasingthe polarization degree, as a result of the correspondent temperaturerise and density drop (see Figure 7). On the other hand, placing theupper boundary of the layer at a different height modifies as wellthe polarization properties, increasing Π as a higher value of N stopH is considered.We stress again that the main goal of this paper is to explorethe effects of absorption on the polarization properties of the ther-mal disc emission, so we give spectra and polarization observablesof radiation without considering the effects of strong gravity onphoton energy and trajectory; general relativistic corrections, in-cluding returning radiation, will be properly addressed in a futurework. We also stress that our physical assumption of an ionizedslab above a standard, black body emitting disc is certainly sim-plistic. A global, completely self-consistent treatment of the discstructure including polarized radiative transfer is a very ambitiousgoal, and clearly beyond the scope of this work. Nevertheless, theresults of the present investigation show that including absorptioneffects alongside those of scattering is crucial for correctly mod-eling the polarization properties of emission from stellar-mass BH © 2020 RAS, MNRAS , 1–13 bsorption effects on BH accretion disc emission accretion discs, an important result in view of the forthcoming X-ray polarimetry missions that will be launched in the next decade. ACKNOWLEDGMENTS
We thank the anonymous referee for comments which helped usimproving the clarity of the paper. RT, GM and SB acknowledgefinancial support from the Italian Space Agency (grant 2017-12-H.0). MD acknowledges the support by the project RVO:67985815and the project LTC18058. WZ would like to thank GACR for thesupport from the project 18-00533S.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable re-quest to the corresponding author.
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APPENDIX A: DENSITY RADIAL PROFILE IN THE DISC
The equatorial density of the disc material can be expressed as fol-lows, in each of the three different regions in which the disc is di-vided, according to which component between gas and radiationdominates in the pressure and which process between scatteringand free-free dominates the opacity (Comp`ere & Oliveri 2017; Tav-erna et al. 2020): • in the inner region, where radiation pressure and scatteringopacity dominate over gas pressure and free-free opacity, respec-tively, n (H) inn = (1 . × cm − ) α − (cid:18) M (cid:12) (cid:19) × (cid:18) ˙ M g s − (cid:19) − ξ C D − R P − ; (A1) • in the middle region, where scattering is still the main sourceof opacity but gas pressure dominates over radiation pressure, n (H) mid = (4 . × cm − ) α − / (cid:18) M (cid:12) (cid:19) − / × (cid:18) ˙ M g s − (cid:19) / ξ − / C / D − / R / P / ; (A2) • in the outer region, where gas pressure and free-free opac-ity dominate over radiation pressure and scattering opacity, respec-tively, n (H) out = (3 . × cm − ) α − / (cid:18) M (cid:12) (cid:19) − / × (cid:18) ˙ M g s − (cid:19) / ξ − / C / D − / R / P / . (A3)In equations (A1)–(A3) we assumed the mean rest mass per barion m b as / the mass of an Fe atomic nucleus (Thorne & Bland-ford 2017) and α is the disc parameter (see Shakura & Sunayev1973, we take α = 0 . throughout the paper), while complete ex-pressions for the functions C , D , R and P are reported in Comp`ere& Oliveri (2017). © 2020 RAS, MNRAS000