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Polarized X-Rays from Windy Accretion in Cygnus X-1

Polarized X-Rays from Windy Accretion in Cygnus X-1 Recent X-ray polarimetric data on the prototypical black hole X-ray binary Cyg X-1 from the Imaging X-ray Polarimetry Explorer present tight constraints on accretion geometry in the hard spectral state. Contrary to general expectations of a low, 1% polarization degree (PD), the observed average PD was found to be a factor of 4 higher. Aligned with the jet position angle on the sky, the observed polarization favors geometry of the X-ray emission region stretched normally to the jet in the accretion disk plane. The high PD is, however, difficult to reconcile with the low orbital inclination of the binary i ≈ 30°. We suggest that this puzzle can be explained if the emitting plasma is outflowing with a mildly relativistic velocity 0.4 c. Our radiative transfer simulations show that Comptonization in the outflowing medium elongated in the plane of the disk and radiates X-rays with the degree and direction of polarization consistent with observations at i ≈ 30°. Unified Astronomy Thesaurus concepts: Stellar mass black holes (1611); Starlight polarization (1571); Accretion (14); Polarimetry (1278); X-ray binary stars (1811) 1. Introduction too-soft X-ray spectra (Stern et al. 1995) because a large fraction of the coronal emission becomes reprocessed to soft Accreting black holes (BHs) in X-ray binaries display radiation by the underlying disk, and the reflection component different spectral states, “hard” and “soft,” distinguished by the becomes stronger than the observed one. It was suggested that spectral shape and the variability properties (Zdziarski & reprocessing could be reduced by the high ionization of the Gierliński 2004; Remillard & McClintock 2006). In the soft disk surface (Nayakshin & Dove 2001; Malzac et al. 2005; state, the dominant contribution to the X-ray flux comes from a Poutanen et al. 2018). thermal-looking spectrum peaking at ∼1 keV energies, which The observed spectrum may also be explained by a mildly is commonly attributed to the multi-temperature blackbody relativistic outflowing corona, which beams hard X-rays away accretion disk (Novikov & Thorne 1973; Shakura & from the accretion disk, reducing their reflection and reproces- Sunyaev 1973). An additional power-law-like tail is likely sing (Beloborodov 1999; Malzac et al. 2001). Outflows are produced by inverse Compton scattering by relativistic generally expected to accompany the heating process in the electrons in the corona (Poutanen & Coppi 1998; Gierliński magnetized corona. For instance, magnetic flares eject plasma, et al. 1999; Zdziarski et al. 2001; McConnell et al. 2002). In the resembling solar flares. Furthermore, if energy is released in hard state, the spectrum is power-law-like in the standard X-ray compact flares, the plasma can become dominated by electron– band and shows a cutoff at ∼100 keV, which is interpreted as a positron pairs (Svensson 1984; Stern et al. 1995; Poutanen & signature of thermal Comptonization in a hot electron medium. Svensson 1996), which have a tiny inertial mass. In this case, The geometry of this medium and the source of soft seed the plasma flows out with a saturated speed controlled by the photons is a matter of debate (Done et al. 2007; Poutanen & local radiation field. Coronal outflows have also been observed Veledina 2014; Bambi et al. 2021). in global simulations of accretion, which have begun to The most popular model for the hot medium is the inner hot, implement radiative effects in electron-ion plasma (Liska et al. geometrically thick, optically thin flow (e.g., Shapiro et al. 2022). Mildly relativistic “winds” are expected to surround the 1976; Ichimaru 1977; Poutanen et al. 1997; Esin et al. 1998; more relativistic “jets” seen in radio observations during the Yuan & Narayan 2014). It is supported by a number of hard state (Fender 2001; Stirling et al. 2001). arguments, including correlations between characteristic varia- In spite of a large body of data on timing, spectra, and bility frequencies of the aperiodic noise (Axelsson et al. 2005), imaging of BH X-ray binaries, there is still a large uncertainty the central frequencies of quasiperiodic oscillations (Pottsch- in the geometry of their X-ray emission region. Polarization has midt et al. 2003; Ibragimov et al. 2005), the spectral slope of long been anticipated to provide an opportunity to determine Comptonization continuum, and the amplitude of Compton the accretion geometry (Lightman & Shapiro 1976). One of the reflection (Zdziarski et al. 1999, 2003). best-studied BH X-ray binaries Cyg X-1 was observed with the A corona on top of a cold disk near the BH was also Imaging X-ray Polarimeter Explorer (IXPE; Weisskopf et al. considered a source of hard X-rays (Haardt & Maraschi 1993). 2022) on 2022 May 15–21 and June 18–20. A rather high Simple models with a slablike corona were found to produce polarization degree (PD) = 4.0% ± 0.2% in the 2–8 keV range at the polarization angle PA = −20°± 2°, consistent with the Original content from this work may be used under the terms jet position angle (Stirling et al. 2001), was detected of the Creative Commons Attribution 4.0 licence. Any further (Krawczynski et al. 2022). Various models for the X-ray distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. emitting region, such as a slab corona, a static inner hot flow, 1 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov and conical or spherical lamppost corona at the disk rotation axis above the BH have been considered. Krawczynski et al. (2022) suggest that the data can be reproduced only by models where the emission region is extended orthogonally to the jet (which is itself presumably perpendicular to the accretion disk). However, none of the considered models reproduces the high +0.8 PD at the measured inclination of i = 27.5 (Miller-Jones -0.6 et al. 2021); higher inclination, i  45°, was required in order to reproduce the data. In this paper, we show that the observed PD can be reproduced if the hot medium forms an outflow from the accretion disk. The mildly relativistic speed of the outflow affects polarization of scattered radiation because the angular distribution of seed photons is significantly aberrated in the plasma rest frame (Beloborodov 1998). This results in a higher polarization of outgoing radiation at low inclinations. In Section 2 we describe the setup of the model, and Section 3 presents the results of our radiative transfer simulations. We discuss the obtained results and conclude in Section 4. Appendices A and B present the radiative transfer equation (RTE) describing our model and a few examples of simulations. 2. Models for Polarization 2.1. Geometry Figure 1. Three alternative geometries considered in the work: panel (a) the We assume that the observed X-rays in the hard state of Cyg outflowing corona with the underlying cold disk producing seed photons X-1 are produced close to the compact object within the hot corresponding to model (A), panel (b) is for the outflowing hot flow with the medium. We consider three alternative geometries sketched in internal synchrotron seed photons corresponding to model (B), and panel (c) is for the outflowing hot flow with the seed photons from the outer truncated disk Figure 1: (model C). (A) Hot corona covering the cold accretion disk (aka slab corona) with the seed unpolarized blackbody photons of following the profile temperature kT = 0.1 keV from the underlying accre- bb tion disk. Radiation produced by multiple Compton bt () =b,1 ( ) scattering in such a slab is polarized at a level of ∼10% (e.g., Poutanen & Svensson 1996; Schnittman & where τ is the Thomson optical depth measured from the Krolik 2010). central plane and τ is the total optical depth through the (B) Hot medium situated within the truncated cold accretion disk; this model is usually referred to as a hot-flow model. corona. The flow diverges sideways at some height H, which Here the seed photons could be produced within the flow, leads to a rapid drop of density. This height then can be for example, by the synchrotron emission of nonthermal considered as the upper boundary of the outflow. electrons with the peak at around 1 eV (Malzac & We note here that the continuity equation implies that Belmont 2009; Poutanen & Vurm 2009; Veledina et al. n (z)β(z) = n β is uniform with z in a steady outflow. For the e 0 0 2013) or by cold clouds embedded in the hot flow (Celotti velocity profile given by Equation (1), the optical depth then et al. 1992; Poutanen et al. 2018; Liska et al. 2022).We can be expressed as consider the first case here. (C) Same as model (B) but with the seed unpolarized photons b t 0 0 dzts ()== n ()z dzs n dz= s n dz,2 () eT T 0 T0 coming from the outer cold truncated disk of the b() z t() z temperature kT = 0.1 keV. In this model, the incident bb angles of the seed photons are limited by the aspect ratio resulting in of the flow h = H/R (see Appendix A for details). tt ()zz== ( H),2t snH. (3) 0 0T0 For all models, the scattering geometry is approximated by a plane-parallel slab extended along the disk plane. We thus get the electron density and velocity dependence on height 2.2. Parameters of the Outflow -12 12 nz ()== n (z H),. bb ()z (z H) (4) The hot medium itself can be in a dynamical state, where in e0 0 addition to the azimuthal and radial motions, some fraction of The densest part of the outflow near the mid-plane has a small the accreting matter leaves the system in a mildly relativistic optical deptht()zz µ and makes a negligible contribution to wind. For our simulations, to account for the influence of the scattering, so the details of how the inflow becomes the outflow wind on polarization properties, we assume that the whole hot medium has the vertical velocity (aka outflowing corona) near the mid-plane are not very important. 2 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov Our model implies mass-loss rate from the disk of radius R of MR =2, pbc nmm ()5 out 00 p where μ is the mean molecular weight per electron in the outflow. On the other hand, the standard estimate for accretion rate in a disk with luminosity L and radiative efficiency ò is 4pRm c L ℓ gp M == ,6 () acc  c s where ℓ = L/L , L = 4πGMm c/σ is the Eddington Edd Edd p T luminosity and R = GM/c is the gravitational radius. This gives  R m mb t M sb n R out T0 0 00 e e == .7 () M ℓ 24Rℓh R acc g g Our fiducial model has h = 1, β = 1/2, and τ = 1. The 0 0 observed luminosity of Cyg X-1 implies ℓ ∼ 0.02, assuming the BH mass M ∼ 10M . Then, for an outflow made of electron-  proton plasma (μ = 1),we find MM ~ 6R R . This e out acc g estimate is consistent with a large fraction of the accretion flow being diverted into an outflow before reaching the BH. A significantly lower estimate for M is found if the coronal out heating occurs intermittently in space and time, in magnetic flares. In this case, the local radiation density can reach values sufficient for copious e pair production, reducing μ and M . out 2.3. Method The parameters of the corona are the electron temperature T , the Thomson optical depth τ , and the terminal velocity β , 0 0 which is varied from 0 to 0.6. In the simulations, we considered Figure 2. Angular distribution of the PD for different outflow velocities β = 0 kT = 100 keV, which is close to the observed values (black solid), 0.2 (red dotted), 0.4 (green dashed), and 0.6 (blue dotted–dashed) (Gierlinski et al. 1997). To reproduce the spectral energy in the middle of the IXPE range at 4 keV for the three models are shown in distribution of Cyg X-1 in the hard state (Gierlinski et al. 1997; Figure 1. Coronal parameters are kT = 100 keV, kT = 0.1 keV, and Γ = 1.6. e bb +0.8 The vertical blue stripe marks the i = 27.5 inclination (Miller-Jones Krawczynski et al. 2022), we iterated τ to achieve the -0.6 et al. 2021) and the horizontal beige stripe corresponds to the observed X-ray observed photon index Γ = 1.6 in the 2–10 keV energy range. PD from Cyg X-1 of 4.0% ± 0.2% (Krawczynski et al. 2022). The RTE that accounts for multiple Compton scattering of linearly polarized radiation in the moving medium is given in normal), the scattered radiation is polarized perpendicular to Appendix A. The only difference with the previously the flow normal (i.e., along the disk), resulting in negative PD. considered static models is that the effect of relativistic On the other hand, if they come from the outer truncated disk aberration has to be taken into account. We start simulations (i.e., sideways), the scattered radiation is polarized parallel to with some initial τ and solve the RTE in the plane-parallel the normal, resulting in positive PD. In the latter case, (slab) approximation assuming azimuthal symmetry. Once the polarization can reach 33% in an ideal case when photons outgoing spectrum is obtained, we find the photon index Γ in are injected in a plane and scattering is coherent (Sunyaev & the 2–10 keV range (for an observer at i = 30°) and correct τ . Titarchuk 1985; Poutanen et al. 2022). If the corona is moving Iterations continue until the desired Γ is reproduced. The upwards, external radiation will be affected by relativistic escaping polarized radiation is described by two Stokes aberration, resulting in parallel polarization for 0.1  β  0.8 parameters I and Q, with U being identically zero because of (Beloborodov 1998). When photons undergo many scatterings azimuthal symmetry. We can then define PD = Q/I, which is in an optically thin plane-parallel medium, polarization is in positive when the dominant direction of oscillations of the general parallel to the flow normal independently of the angular electric vector (aka the polarization vector) is parallel to distribution of seed photons. the slab normal. If the polarization vector is perpendicular to the flow normal, then the PD is negative. For example, the optically thick electron-scattering atmosphere produces nega- 3. Results tive PD (Chandrasekhar 1960). The basic physics that controls the polarization production Figure 2(a) shows the angular distribution of the PD for can be described as follows. Polarization of radiation scattered model (A) in the middle of the IXPE range at 4 keV. We see in the hot flow once depends strongly on the angular that for static corona, the peak in the PD is reached at ∼70°, distribution of the incoming seed photons. If they come from while for the outflowing corona, the PD is growing with the underlying cold disk (and therefore beamed along the flow inclination more or less monotonically. This figure clearly 3 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov polarization is negative (i.e., parallel to the disk) at photon energies where the first scattering dominates (see Appendix B for spectral decomposition in different scattering orders);it changes the sign at around 1 keV and continues to grow to higher energies where the photons undergo more than two scatterings. At i = 30°, the PD does not exceed 3% even at 100 keV. We note that a static slab-corona model cannot produce the hard spectra (Γ ≈ 1.6) observed in Cyg X-1 once the reflection and reprocessing of radiation are taken into account: thermalization of the hard radiation in the disk would produce too many soft photons that cool the corona producing soft spectra (Poutanen et al. 2018). Motion of the corona away from the disk can be reconciled with the spectral hardness (Beloborodov 1999) resulting also in a higher PD (see dotted, dashed, and dotted–dashed curves in Figure 3(a)). Because of the relativistic aberration, the seed photon angular distribution is less beamed along the normal in the flow-comoving frame (see Figure 4), and even single- scattered photons have polarization parallel to the disk axis (see also Beloborodov 1998). At an inclination i < 30°, the PD of 4% can be reached only for β > 0.6, which also predicts an increasing trend of the PD with energy. Let us now consider a truncated disk geometry with the inner hot flow. Figure 3(b) shows the results for model (B) corresponding to the synchrotron seed photons. In the static case, the PD is 2%–2.5% at i = 30° (see blue solid line). Because photons reaching the IXPE range undergo many scatterings, the PD is largely energy independent (see also Figure 5). At the same inclination, the PD increases with the outflow velocity, reaching ∼4% at β ∼ 0.4 and 5% at β ∼ 0.6. At a higher inclination of i = 60°, the PD is about twice as large as for the i = 30° case (see the set of red curves). Finally, we consider model (C) with the hot-flow geometry and the truncated disk providing seed photons for Comptoniza- Figure 3. Energy dependence of the PD for the three models shown in tion. The results for the case of H/R = 1 are shown in Figure 2. The blue and red lines correspond to the inclinations of 30° and 60°, Figure 3(c). Here we see that the static hot flow produces nearly respectively. Solid, dotted, dashed, and dotted–dashed lines correspond to the outflow velocities of β = 0, 0.2, 0.4, and 0.6, respectively. In the middle constant PD of ∼2.5% in the IXPE range at i = 30°. For the panel, the red curves for β = 0.4 and 0.6 coincide. outflow velocity β = 0.2, the anisotropy of the seed photons in the outflow rest frame is large enough to produce parallel shows that the static slab corona can produce a PD of 4% only polarization at a 4% level for single-scattered photons, which at inclinations exceeding 60°. The outflowing corona, on the then drops somewhat at higher scattering orders. At β = 0.4, other hand, can produce this high PD at i = 30° for β  0.6. the PD reaches 4.5%–5% and weakly depends on energy in the The angular distribution of the PD for model (B) is shown in IXPE range. At even higher β = 0.6, the PD reaches 6%. Figure 2(b). We see that the PD for the static flow of β =0is larger than in the case of underlying disk seed photons. The 4. Discussion and Summary reason is simple: the synchrotron seed photons take more scattering to get to the IXPE range, resulting in a more In our simulations, we ignored relativistic effects related to anisotropic radiation field and higher PD. The observed 4% PD the rotation of the flow resulting in a rotation of the polarization is reached only at i ≈ 40°. For β = 0.2, this PD is reached at plane due to relativistic effects. We note here that these effects inclination of ≈33°, and the observed value of 4% at i < 30° is are more important when the local emitted spectrum is produced when β  0.4. blackbody-like, when relativistic aberration and Doppler The angular distribution of the PD for model (C) is shown in boosting affect strongly the polarization vector just above the Figure 2(c). It is similar to the other models but shows a peak of the blackbody (Connors et al. 1980; Dovčiak et al. slightly higher PD. We see that the PD observed in Cyg X-1 is 2008; Loktev et al. 2022). However, for a power-law-like well reproduced for the outflow velocity of β ≈ 0.4. We also spectrum, the effect is less pronounced because of a relatively considered a model of the seed photons from the cold clouds larger contribution from large radii of the flow to the observed within the hot flow, but its results are very similar to those of spectrum in a given energy range. Depolarization effects are models (B) and (C).We find that at all considered velocities, 1% and can explain the difference between the required models (B) and (C) predict parallel polarization exceeding inclinations for the static models found in this work and in ∼10% at inclinations i = 60°–70°. Krawczynski et al. (2022), where relativistic effects are taken Let us now consider the energy dependence of PD. The into account. results are shown in Figure 3 for two selected inclinations of Our simulations of the dynamic corona accounting for 30° and 60°. For model (A) and the case of a static corona, Comptonization confirm previous results for Thomson 4 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov Figure 4. Spectro-polarimetric properties of model (A) of a slab corona of T = 100 keV and T = 0.1 keV with the outflow of terminal velocity β = 0.4. (a) e bb 0 37 −1 Spectral energy distribution of the escaping radiation at inclination i = 30° scaled to produce the angle-integrated bolometric luminosity of L = 10 erg s . Blue solid, dotted, dashed, dotted–dashed, triple-dotted–dashed, and long-dashed lines show the contribution of different scattering orders n = 0, 1, 2, 4, 7, and 10, respectively. The black solid line gives the total spectrum. (b) The PD as a function of energy at the same inclination for the total radiation (black solid) and for different scattering orders is shown. (c) Angular distribution of the intensity at optical depth τ = (3/4)τ (where β = 0.3) for the same scattering orders as above at the corresponding peaks of EI . The black and red curves correspond to the lab and comoving frame intensities, respectively. The lab-frame intensities are normalized to unity at the maximum. (d) Angular dependence of PD for the same scattering orders at the same energies as the intensity. The IXPE energy range is marked by a beige vertical stripe in panels (a) and (b). scattering that the PD strongly depends on the angular multiple scatterings are important, and the angular distribution distribution of incident photons (Beloborodov 1998; Belobor- of seed photons does not play any role. Once we combine the odov & Poutanen 1999). This effect is more important at the mildly relativistic velocity with the anisotropy of the seed energies where the first scattering dominates. For the slab photons coming from the outer truncated disk, even higher PDs corona, the static model results in a rapidly growing PD in the can be achieved. We conclude that in order to get 4% parallel IXPE range, but it stays below 2% for i = 30°. Increasing the polarization throughout the IXPE range assuming inclination of velocity of the outflow gives a higher PD, which can reach the 30°, a mildly relativistic outflow velocity β ≈ 0.4 is needed observed 4% at i = 30° only when β > 0.6. Such a high together with either anisotropic distribution of seed soft terminal velocity is required to produce the necessary photon photons coming from the outer cool disk or dominance of anisotropy in the flow rest frame due to relativistic aberration. multiple scattering, which is a natural outcome of synchrotron We note that the average velocity of the outflow in that case is a photon injection at very low energies. factor of 2 smaller and therefore is consistent with the estimate We also note that our model predicts PDs exceeding 10% at β = 0.3 by Beloborodov (1999) that is required to achieve the high inclinations i > 60°, which potentially could be observed photon spectral slope and the reflection fraction observed in in other BH X-ray binaries. Similarly high polarization is also Cyg X-1. The terminal velocity is somewhat larger than the expected from highly inclined accretion flows in Seyfert equilibrium velocity of the electron–positron pairs, which may galaxies of intermediate types (see, e.g., Gianolli et al. 2023 for indicate the presence of a significant amount of protons and the case of NGC 4151). ions in the outflow and the role of magnetic processes (rather than radiation pressure) in the flow acceleration. We thank P. Abolmasov and the referee A. A. Zdziarski for The model with the internal synchrotron seed photons can valuable suggestions. J.P. and A.V. acknowledge support from reproduce an energy-independent 4% PD at i ≈ 30° for a more the Academy of Finland grant 333112. A.M.B. is supported by modest velocity of β = 0.4–0.5 because in the IXPE range, NSF grant AST2009453, NASA grant 21-ATP21-0056, and 5 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov Figure 5. Same as in Figure 4 but for the seed synchrotron photons of model (B). Simons Foundation grant 446228. Nordita is supported in part is the Doppler factor andg=- 11 b is the Lorentz factor by NordForsk. of the flow. Comoving frame quantities are marked with sub- or Facility: IXPE. superscript c. We define the Thomson optical depth across the Software: COMPPS (Poutanen & Svensson 1996). slab as a measure of the column density: dτ = n (z)σ dz, where e T z is the vertical coordinate and n is the electron concentration Appendix A measured in the lab frame. The source function in the lab frame Radiative Transfer Equation is related to that in the comoving frame by the Lorentz The RTE describing Comptonization of polarized radiation transformation (Rybicki & Lightman 1979): in the plane-parallel atmosphere in the lab frame (marked with l3c superscript l) can be written in the form (Mihalas & SS () tm ,,xx =  (t, ,m), (A2) Mihalas 1984; Nagirner & Poutanen 1994; Beloborodov & Poutanen 1999): where the angles are related by the aberration formulae dx I() tm ,, mb + mb - m =-[(1, bt)m][-s(xx )I(t,m) m = , m = .A()3 dt c 1 - bm 1 + bm + S() tm ,, x ]. (A1) The source function in the comoving frame is Here the photon energy is measured in units of the electron rest mass x = E/m c , μ is the cosine of the angle the photon ¥dx 1 c 2 c momentum makes with the slab normal, I = (I, Q) is the ¢ S() tm ,,xx = dm c còò 0 -1 Stokes vector that fully describes linear polarization (the Stokes c c c ¢¢ ¢¢ U parameter is zero due to the azimuthal symmetry), S is the ´RI () xx ,;mm, (t,x ,m) = I , (A4) c c c c source function (also Stokes vector) in the lab frame, and σ(x ) is the dimensionless total Compton scattering cross section (in where R is the 2 × 2 azimuth-averaged redistribution matrix units of the Thomson cross section σ ) for isotropic describing Compton scattering by isotropic hot electrons (see Maxwellian electron gas as a function of the photon energy Appendix A1 in Poutanen & Svensson 1996). The Stokes in the comoving framexx = , where =- 11 [(gbm)] vector I in the comoving frame is related to that in the lab 6 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov frame as assume it to be unpolarized, isotropic, and optical-depth independent. Its spectral energy distribution is computed c -3l II () tm ,,xx =  (t, ,m). (A5) assuming a thermal electron distribution of kT = 100 keV with a weak nonthermal tail. The resulting synchrotron We solve the RTE using the iterative scattering method spectrum has a shape of a broken power law with a peak at (Poutanen & Svensson 1996) accounting exactly for polariza- ∼1eV (Wardziński & Zdziarski 2001; Veledina et al. 2013). tion by Compton scattering on isotropic (in the flow frame) The iteration procedure now involves steps 5 and 6 and then 3 thermal electron gas (Nagirner & Poutanen 1993). The only and 4, etc. difference from Poutanen & Svensson (1996) is that here we account for bulk motion. At step 1, we specify the boundary condition at τ = 0, either on the bottom of the slab corona or in Appendix B the middle of the slab, in the case of the hot-flow–truncated Spectro-polarimetric Properties of Outflowing Coronae disk geometry. We approximate the spectrum of seed photons (marked by index 0) by the (unpolarized) blackbody We show a few examples of simulations of polarized radiationescapingfromanoutflowing corona for the same l 1 I() tm== 0,xB , (T) (m-∣m∣),A(6) three models considered in the main text. We concentrate x bb 0 0 () here on the case β = 0.4, which describes well Cyg X-1 data, and the observer inclination i = 30°. Figure 4 shows where  is the Heaviside step function and μ = 1 for the slab the main results for model (A), where seed unpolarized corona, while for the truncated disk, the injection of seed blackbody photons are injected from the bottom of out- photons is limited to the zenith angles corresponding to the flowing corona. We see the spectral energy distribution of aspect ratio of the inner hot flow h = H/R, ∣mm ∣<= the total radiation as well as for selected scattering orders in hh 1 + . At step 2, we use the formal solution of the RTE panel (a).The IXPE 2–8 keV range is dominated by photons (Equation (A1); with zero source function S) to get the intensity scattered two to four times in the flow. Panel (b) shows the in all other layers of the slab: energy dependence of the PD. The single-scattered photons have small a PD, which becomes negative in the IXPE l l II () tm ,,xx=- (0, ,m) exp( t m), (A7) 0 0 range, because the angular distribution of the seed photons in the comoving frame of the flow is still rather beamed wherett=- s() x [1,b(t0)m] is the energy- and angle- x c outwards in spite of a high velocity as is seen in panel (c). dependent optical depth measured from the bottom to a The black curves there show the angular distribution of the given layer andb is the average velocity from the slab center intensity at the optical depth τ = (3/4)τ at the energy where (or bottom for slab-corona model) τ =0tothe given τ, EI of the corresponding scattering order peaks. The red defined as curves are the corresponding intensities in the comoving t frame. We see that relativistic aberration is responsible for bt() ,t¢= b(td) t anisotropy of the radiation, with the intensity of multiple t¢ tt-¢ scattered photons in the lab frame peaking at μ ≈ 0.1–0.3 bt ()+¢ bt ( ) tt+¢ (i ≈ 70°–80°). On the other hand, the peak of the intensity in = = b .A()8 22t the comoving frame corresponds to μ ≈− 0.2, i.e., it is l directed more downwards. The angular distribution of the At step 3, we transform the Stokes vector I in the lab frame for PD in both frames is shown in panel (d).Wesee that thePD photons scattered n times to the comoving frame using grows with the scattering order. It reaches maximum roughly Equation (A5). At step 4, we compute the source function in at the same angles where the intensity peaks. Because of the the comoving frame for photons scattered n + 1 times using relativistic aberration, all curves in the lab frame are shifted cc Equation (A4) as S = I . At step 5, we get the source nn +1 to higher μ, so that at a given inclination, the PD grows function in the lab frame using Equation (A2). And finally at with β. step 6, we obtain the intensity of radiation scattered n + 1 times Figure 5 shows the same quantities for model (B) with the from the formal solution of the RTE: seed synchrotron unpolarized photons, which are isotropic in dt¢ ⎧ --() tt¢s(x)[1, -b(tt¢)m] m S () tm¢- ,,xe [1 b(t¢)m] , m>0, ò n+1 ⎪ min I () tm ,, x = () A9 n+1 ⎨ dt¢ l -¢() tt- s(x)[1, -b(t¢t)m](-m) S () tm¢- ,,xe [1 b(t¢)m] ,m<0, ò n+1 () -m where t = 0 for the slab corona andtt =- for the hot min min 0 the comoving frame. The PD is increasing monotonically with flow, and b is given by Equation (A8). The iteration procedure the number of scatterings (panel (d)), but already at around continues with steps 3–6 until the desired accuracy for the total n > 5, very little variation is observed. Because about seven to ¥ l Stokes vector I = I is achieved at all optical depths, å eight scatterings are needed for seed photons to achieve the n=0 angles, and energies. IXPE range, the PD is nearly energy independent (panel (b)). For synchrotron seed photons, we start instead from step 4 The angular distributions of both the intensity and the PD and specify the source function in the comoving frame S .We (panels (c) and (d)) saturate. 7 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov Figure 6. Same as in Figure 4 but for the seed photons from the truncated disk with the coronal aspect ratio H/R = 1 of model (C). Figure 6 shows the results for model (C) with coronal aspect Dovčiak, M., Muleri, F., Goosmann, R. W., Karas, V., & Matt, G. 2008, MNRAS, 391, 32 ratio H/R = 1. In this model, we see strong anisotropy of seed Esin, A. A., Narayan, R., Cui, W., Grove, J. 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Polarized X-Rays from Windy Accretion in Cygnus X-1

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Abstract

Recent X-ray polarimetric data on the prototypical black hole X-ray binary Cyg X-1 from the Imaging X-ray Polarimetry Explorer present tight constraints on accretion geometry in the hard spectral state. Contrary to general expectations of a low, 1% polarization degree (PD), the observed average PD was found to be a factor of 4 higher. Aligned with the jet position angle on the sky, the observed polarization favors geometry of the X-ray emission region stretched normally to the jet in the accretion disk plane. The high PD is, however, difficult to reconcile with the low orbital inclination of the binary i ≈ 30°. We suggest that this puzzle can be explained if the emitting plasma is outflowing with a mildly relativistic velocity 0.4 c. Our radiative transfer simulations show that Comptonization in the outflowing medium elongated in the plane of the disk and radiates X-rays with the degree and direction of polarization consistent with observations at i ≈ 30°. Unified Astronomy Thesaurus concepts: Stellar mass black holes (1611); Starlight polarization (1571); Accretion (14); Polarimetry (1278); X-ray binary stars (1811) 1. Introduction too-soft X-ray spectra (Stern et al. 1995) because a large fraction of the coronal emission becomes reprocessed to soft Accreting black holes (BHs) in X-ray binaries display radiation by the underlying disk, and the reflection component different spectral states, “hard” and “soft,” distinguished by the becomes stronger than the observed one. It was suggested that spectral shape and the variability properties (Zdziarski & reprocessing could be reduced by the high ionization of the Gierliński 2004; Remillard & McClintock 2006). In the soft disk surface (Nayakshin & Dove 2001; Malzac et al. 2005; state, the dominant contribution to the X-ray flux comes from a Poutanen et al. 2018). thermal-looking spectrum peaking at ∼1 keV energies, which The observed spectrum may also be explained by a mildly is commonly attributed to the multi-temperature blackbody relativistic outflowing corona, which beams hard X-rays away accretion disk (Novikov & Thorne 1973; Shakura & from the accretion disk, reducing their reflection and reproces- Sunyaev 1973). An additional power-law-like tail is likely sing (Beloborodov 1999; Malzac et al. 2001). Outflows are produced by inverse Compton scattering by relativistic generally expected to accompany the heating process in the electrons in the corona (Poutanen & Coppi 1998; Gierliński magnetized corona. For instance, magnetic flares eject plasma, et al. 1999; Zdziarski et al. 2001; McConnell et al. 2002). In the resembling solar flares. Furthermore, if energy is released in hard state, the spectrum is power-law-like in the standard X-ray compact flares, the plasma can become dominated by electron– band and shows a cutoff at ∼100 keV, which is interpreted as a positron pairs (Svensson 1984; Stern et al. 1995; Poutanen & signature of thermal Comptonization in a hot electron medium. Svensson 1996), which have a tiny inertial mass. In this case, The geometry of this medium and the source of soft seed the plasma flows out with a saturated speed controlled by the photons is a matter of debate (Done et al. 2007; Poutanen & local radiation field. Coronal outflows have also been observed Veledina 2014; Bambi et al. 2021). in global simulations of accretion, which have begun to The most popular model for the hot medium is the inner hot, implement radiative effects in electron-ion plasma (Liska et al. geometrically thick, optically thin flow (e.g., Shapiro et al. 2022). Mildly relativistic “winds” are expected to surround the 1976; Ichimaru 1977; Poutanen et al. 1997; Esin et al. 1998; more relativistic “jets” seen in radio observations during the Yuan & Narayan 2014). It is supported by a number of hard state (Fender 2001; Stirling et al. 2001). arguments, including correlations between characteristic varia- In spite of a large body of data on timing, spectra, and bility frequencies of the aperiodic noise (Axelsson et al. 2005), imaging of BH X-ray binaries, there is still a large uncertainty the central frequencies of quasiperiodic oscillations (Pottsch- in the geometry of their X-ray emission region. Polarization has midt et al. 2003; Ibragimov et al. 2005), the spectral slope of long been anticipated to provide an opportunity to determine Comptonization continuum, and the amplitude of Compton the accretion geometry (Lightman & Shapiro 1976). One of the reflection (Zdziarski et al. 1999, 2003). best-studied BH X-ray binaries Cyg X-1 was observed with the A corona on top of a cold disk near the BH was also Imaging X-ray Polarimeter Explorer (IXPE; Weisskopf et al. considered a source of hard X-rays (Haardt & Maraschi 1993). 2022) on 2022 May 15–21 and June 18–20. A rather high Simple models with a slablike corona were found to produce polarization degree (PD) = 4.0% ± 0.2% in the 2–8 keV range at the polarization angle PA = −20°± 2°, consistent with the Original content from this work may be used under the terms jet position angle (Stirling et al. 2001), was detected of the Creative Commons Attribution 4.0 licence. Any further (Krawczynski et al. 2022). Various models for the X-ray distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. emitting region, such as a slab corona, a static inner hot flow, 1 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov and conical or spherical lamppost corona at the disk rotation axis above the BH have been considered. Krawczynski et al. (2022) suggest that the data can be reproduced only by models where the emission region is extended orthogonally to the jet (which is itself presumably perpendicular to the accretion disk). However, none of the considered models reproduces the high +0.8 PD at the measured inclination of i = 27.5 (Miller-Jones -0.6 et al. 2021); higher inclination, i  45°, was required in order to reproduce the data. In this paper, we show that the observed PD can be reproduced if the hot medium forms an outflow from the accretion disk. The mildly relativistic speed of the outflow affects polarization of scattered radiation because the angular distribution of seed photons is significantly aberrated in the plasma rest frame (Beloborodov 1998). This results in a higher polarization of outgoing radiation at low inclinations. In Section 2 we describe the setup of the model, and Section 3 presents the results of our radiative transfer simulations. We discuss the obtained results and conclude in Section 4. Appendices A and B present the radiative transfer equation (RTE) describing our model and a few examples of simulations. 2. Models for Polarization 2.1. Geometry Figure 1. Three alternative geometries considered in the work: panel (a) the We assume that the observed X-rays in the hard state of Cyg outflowing corona with the underlying cold disk producing seed photons X-1 are produced close to the compact object within the hot corresponding to model (A), panel (b) is for the outflowing hot flow with the medium. We consider three alternative geometries sketched in internal synchrotron seed photons corresponding to model (B), and panel (c) is for the outflowing hot flow with the seed photons from the outer truncated disk Figure 1: (model C). (A) Hot corona covering the cold accretion disk (aka slab corona) with the seed unpolarized blackbody photons of following the profile temperature kT = 0.1 keV from the underlying accre- bb tion disk. Radiation produced by multiple Compton bt () =b,1 ( ) scattering in such a slab is polarized at a level of ∼10% (e.g., Poutanen & Svensson 1996; Schnittman & where τ is the Thomson optical depth measured from the Krolik 2010). central plane and τ is the total optical depth through the (B) Hot medium situated within the truncated cold accretion disk; this model is usually referred to as a hot-flow model. corona. The flow diverges sideways at some height H, which Here the seed photons could be produced within the flow, leads to a rapid drop of density. This height then can be for example, by the synchrotron emission of nonthermal considered as the upper boundary of the outflow. electrons with the peak at around 1 eV (Malzac & We note here that the continuity equation implies that Belmont 2009; Poutanen & Vurm 2009; Veledina et al. n (z)β(z) = n β is uniform with z in a steady outflow. For the e 0 0 2013) or by cold clouds embedded in the hot flow (Celotti velocity profile given by Equation (1), the optical depth then et al. 1992; Poutanen et al. 2018; Liska et al. 2022).We can be expressed as consider the first case here. (C) Same as model (B) but with the seed unpolarized photons b t 0 0 dzts ()== n ()z dzs n dz= s n dz,2 () eT T 0 T0 coming from the outer cold truncated disk of the b() z t() z temperature kT = 0.1 keV. In this model, the incident bb angles of the seed photons are limited by the aspect ratio resulting in of the flow h = H/R (see Appendix A for details). tt ()zz== ( H),2t snH. (3) 0 0T0 For all models, the scattering geometry is approximated by a plane-parallel slab extended along the disk plane. We thus get the electron density and velocity dependence on height 2.2. Parameters of the Outflow -12 12 nz ()== n (z H),. bb ()z (z H) (4) The hot medium itself can be in a dynamical state, where in e0 0 addition to the azimuthal and radial motions, some fraction of The densest part of the outflow near the mid-plane has a small the accreting matter leaves the system in a mildly relativistic optical deptht()zz µ and makes a negligible contribution to wind. For our simulations, to account for the influence of the scattering, so the details of how the inflow becomes the outflow wind on polarization properties, we assume that the whole hot medium has the vertical velocity (aka outflowing corona) near the mid-plane are not very important. 2 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov Our model implies mass-loss rate from the disk of radius R of MR =2, pbc nmm ()5 out 00 p where μ is the mean molecular weight per electron in the outflow. On the other hand, the standard estimate for accretion rate in a disk with luminosity L and radiative efficiency ò is 4pRm c L ℓ gp M == ,6 () acc  c s where ℓ = L/L , L = 4πGMm c/σ is the Eddington Edd Edd p T luminosity and R = GM/c is the gravitational radius. This gives  R m mb t M sb n R out T0 0 00 e e == .7 () M ℓ 24Rℓh R acc g g Our fiducial model has h = 1, β = 1/2, and τ = 1. The 0 0 observed luminosity of Cyg X-1 implies ℓ ∼ 0.02, assuming the BH mass M ∼ 10M . Then, for an outflow made of electron-  proton plasma (μ = 1),we find MM ~ 6R R . This e out acc g estimate is consistent with a large fraction of the accretion flow being diverted into an outflow before reaching the BH. A significantly lower estimate for M is found if the coronal out heating occurs intermittently in space and time, in magnetic flares. In this case, the local radiation density can reach values sufficient for copious e pair production, reducing μ and M . out 2.3. Method The parameters of the corona are the electron temperature T , the Thomson optical depth τ , and the terminal velocity β , 0 0 which is varied from 0 to 0.6. In the simulations, we considered Figure 2. Angular distribution of the PD for different outflow velocities β = 0 kT = 100 keV, which is close to the observed values (black solid), 0.2 (red dotted), 0.4 (green dashed), and 0.6 (blue dotted–dashed) (Gierlinski et al. 1997). To reproduce the spectral energy in the middle of the IXPE range at 4 keV for the three models are shown in distribution of Cyg X-1 in the hard state (Gierlinski et al. 1997; Figure 1. Coronal parameters are kT = 100 keV, kT = 0.1 keV, and Γ = 1.6. e bb +0.8 The vertical blue stripe marks the i = 27.5 inclination (Miller-Jones Krawczynski et al. 2022), we iterated τ to achieve the -0.6 et al. 2021) and the horizontal beige stripe corresponds to the observed X-ray observed photon index Γ = 1.6 in the 2–10 keV energy range. PD from Cyg X-1 of 4.0% ± 0.2% (Krawczynski et al. 2022). The RTE that accounts for multiple Compton scattering of linearly polarized radiation in the moving medium is given in normal), the scattered radiation is polarized perpendicular to Appendix A. The only difference with the previously the flow normal (i.e., along the disk), resulting in negative PD. considered static models is that the effect of relativistic On the other hand, if they come from the outer truncated disk aberration has to be taken into account. We start simulations (i.e., sideways), the scattered radiation is polarized parallel to with some initial τ and solve the RTE in the plane-parallel the normal, resulting in positive PD. In the latter case, (slab) approximation assuming azimuthal symmetry. Once the polarization can reach 33% in an ideal case when photons outgoing spectrum is obtained, we find the photon index Γ in are injected in a plane and scattering is coherent (Sunyaev & the 2–10 keV range (for an observer at i = 30°) and correct τ . Titarchuk 1985; Poutanen et al. 2022). If the corona is moving Iterations continue until the desired Γ is reproduced. The upwards, external radiation will be affected by relativistic escaping polarized radiation is described by two Stokes aberration, resulting in parallel polarization for 0.1  β  0.8 parameters I and Q, with U being identically zero because of (Beloborodov 1998). When photons undergo many scatterings azimuthal symmetry. We can then define PD = Q/I, which is in an optically thin plane-parallel medium, polarization is in positive when the dominant direction of oscillations of the general parallel to the flow normal independently of the angular electric vector (aka the polarization vector) is parallel to distribution of seed photons. the slab normal. If the polarization vector is perpendicular to the flow normal, then the PD is negative. For example, the optically thick electron-scattering atmosphere produces nega- 3. Results tive PD (Chandrasekhar 1960). The basic physics that controls the polarization production Figure 2(a) shows the angular distribution of the PD for can be described as follows. Polarization of radiation scattered model (A) in the middle of the IXPE range at 4 keV. We see in the hot flow once depends strongly on the angular that for static corona, the peak in the PD is reached at ∼70°, distribution of the incoming seed photons. If they come from while for the outflowing corona, the PD is growing with the underlying cold disk (and therefore beamed along the flow inclination more or less monotonically. This figure clearly 3 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov polarization is negative (i.e., parallel to the disk) at photon energies where the first scattering dominates (see Appendix B for spectral decomposition in different scattering orders);it changes the sign at around 1 keV and continues to grow to higher energies where the photons undergo more than two scatterings. At i = 30°, the PD does not exceed 3% even at 100 keV. We note that a static slab-corona model cannot produce the hard spectra (Γ ≈ 1.6) observed in Cyg X-1 once the reflection and reprocessing of radiation are taken into account: thermalization of the hard radiation in the disk would produce too many soft photons that cool the corona producing soft spectra (Poutanen et al. 2018). Motion of the corona away from the disk can be reconciled with the spectral hardness (Beloborodov 1999) resulting also in a higher PD (see dotted, dashed, and dotted–dashed curves in Figure 3(a)). Because of the relativistic aberration, the seed photon angular distribution is less beamed along the normal in the flow-comoving frame (see Figure 4), and even single- scattered photons have polarization parallel to the disk axis (see also Beloborodov 1998). At an inclination i < 30°, the PD of 4% can be reached only for β > 0.6, which also predicts an increasing trend of the PD with energy. Let us now consider a truncated disk geometry with the inner hot flow. Figure 3(b) shows the results for model (B) corresponding to the synchrotron seed photons. In the static case, the PD is 2%–2.5% at i = 30° (see blue solid line). Because photons reaching the IXPE range undergo many scatterings, the PD is largely energy independent (see also Figure 5). At the same inclination, the PD increases with the outflow velocity, reaching ∼4% at β ∼ 0.4 and 5% at β ∼ 0.6. At a higher inclination of i = 60°, the PD is about twice as large as for the i = 30° case (see the set of red curves). Finally, we consider model (C) with the hot-flow geometry and the truncated disk providing seed photons for Comptoniza- Figure 3. Energy dependence of the PD for the three models shown in tion. The results for the case of H/R = 1 are shown in Figure 2. The blue and red lines correspond to the inclinations of 30° and 60°, Figure 3(c). Here we see that the static hot flow produces nearly respectively. Solid, dotted, dashed, and dotted–dashed lines correspond to the outflow velocities of β = 0, 0.2, 0.4, and 0.6, respectively. In the middle constant PD of ∼2.5% in the IXPE range at i = 30°. For the panel, the red curves for β = 0.4 and 0.6 coincide. outflow velocity β = 0.2, the anisotropy of the seed photons in the outflow rest frame is large enough to produce parallel shows that the static slab corona can produce a PD of 4% only polarization at a 4% level for single-scattered photons, which at inclinations exceeding 60°. The outflowing corona, on the then drops somewhat at higher scattering orders. At β = 0.4, other hand, can produce this high PD at i = 30° for β  0.6. the PD reaches 4.5%–5% and weakly depends on energy in the The angular distribution of the PD for model (B) is shown in IXPE range. At even higher β = 0.6, the PD reaches 6%. Figure 2(b). We see that the PD for the static flow of β =0is larger than in the case of underlying disk seed photons. The 4. Discussion and Summary reason is simple: the synchrotron seed photons take more scattering to get to the IXPE range, resulting in a more In our simulations, we ignored relativistic effects related to anisotropic radiation field and higher PD. The observed 4% PD the rotation of the flow resulting in a rotation of the polarization is reached only at i ≈ 40°. For β = 0.2, this PD is reached at plane due to relativistic effects. We note here that these effects inclination of ≈33°, and the observed value of 4% at i < 30° is are more important when the local emitted spectrum is produced when β  0.4. blackbody-like, when relativistic aberration and Doppler The angular distribution of the PD for model (C) is shown in boosting affect strongly the polarization vector just above the Figure 2(c). It is similar to the other models but shows a peak of the blackbody (Connors et al. 1980; Dovčiak et al. slightly higher PD. We see that the PD observed in Cyg X-1 is 2008; Loktev et al. 2022). However, for a power-law-like well reproduced for the outflow velocity of β ≈ 0.4. We also spectrum, the effect is less pronounced because of a relatively considered a model of the seed photons from the cold clouds larger contribution from large radii of the flow to the observed within the hot flow, but its results are very similar to those of spectrum in a given energy range. Depolarization effects are models (B) and (C).We find that at all considered velocities, 1% and can explain the difference between the required models (B) and (C) predict parallel polarization exceeding inclinations for the static models found in this work and in ∼10% at inclinations i = 60°–70°. Krawczynski et al. (2022), where relativistic effects are taken Let us now consider the energy dependence of PD. The into account. results are shown in Figure 3 for two selected inclinations of Our simulations of the dynamic corona accounting for 30° and 60°. For model (A) and the case of a static corona, Comptonization confirm previous results for Thomson 4 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov Figure 4. Spectro-polarimetric properties of model (A) of a slab corona of T = 100 keV and T = 0.1 keV with the outflow of terminal velocity β = 0.4. (a) e bb 0 37 −1 Spectral energy distribution of the escaping radiation at inclination i = 30° scaled to produce the angle-integrated bolometric luminosity of L = 10 erg s . Blue solid, dotted, dashed, dotted–dashed, triple-dotted–dashed, and long-dashed lines show the contribution of different scattering orders n = 0, 1, 2, 4, 7, and 10, respectively. The black solid line gives the total spectrum. (b) The PD as a function of energy at the same inclination for the total radiation (black solid) and for different scattering orders is shown. (c) Angular distribution of the intensity at optical depth τ = (3/4)τ (where β = 0.3) for the same scattering orders as above at the corresponding peaks of EI . The black and red curves correspond to the lab and comoving frame intensities, respectively. The lab-frame intensities are normalized to unity at the maximum. (d) Angular dependence of PD for the same scattering orders at the same energies as the intensity. The IXPE energy range is marked by a beige vertical stripe in panels (a) and (b). scattering that the PD strongly depends on the angular multiple scatterings are important, and the angular distribution distribution of incident photons (Beloborodov 1998; Belobor- of seed photons does not play any role. Once we combine the odov & Poutanen 1999). This effect is more important at the mildly relativistic velocity with the anisotropy of the seed energies where the first scattering dominates. For the slab photons coming from the outer truncated disk, even higher PDs corona, the static model results in a rapidly growing PD in the can be achieved. We conclude that in order to get 4% parallel IXPE range, but it stays below 2% for i = 30°. Increasing the polarization throughout the IXPE range assuming inclination of velocity of the outflow gives a higher PD, which can reach the 30°, a mildly relativistic outflow velocity β ≈ 0.4 is needed observed 4% at i = 30° only when β > 0.6. Such a high together with either anisotropic distribution of seed soft terminal velocity is required to produce the necessary photon photons coming from the outer cool disk or dominance of anisotropy in the flow rest frame due to relativistic aberration. multiple scattering, which is a natural outcome of synchrotron We note that the average velocity of the outflow in that case is a photon injection at very low energies. factor of 2 smaller and therefore is consistent with the estimate We also note that our model predicts PDs exceeding 10% at β = 0.3 by Beloborodov (1999) that is required to achieve the high inclinations i > 60°, which potentially could be observed photon spectral slope and the reflection fraction observed in in other BH X-ray binaries. Similarly high polarization is also Cyg X-1. The terminal velocity is somewhat larger than the expected from highly inclined accretion flows in Seyfert equilibrium velocity of the electron–positron pairs, which may galaxies of intermediate types (see, e.g., Gianolli et al. 2023 for indicate the presence of a significant amount of protons and the case of NGC 4151). ions in the outflow and the role of magnetic processes (rather than radiation pressure) in the flow acceleration. We thank P. Abolmasov and the referee A. A. Zdziarski for The model with the internal synchrotron seed photons can valuable suggestions. J.P. and A.V. acknowledge support from reproduce an energy-independent 4% PD at i ≈ 30° for a more the Academy of Finland grant 333112. A.M.B. is supported by modest velocity of β = 0.4–0.5 because in the IXPE range, NSF grant AST2009453, NASA grant 21-ATP21-0056, and 5 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov Figure 5. Same as in Figure 4 but for the seed synchrotron photons of model (B). Simons Foundation grant 446228. Nordita is supported in part is the Doppler factor andg=- 11 b is the Lorentz factor by NordForsk. of the flow. Comoving frame quantities are marked with sub- or Facility: IXPE. superscript c. We define the Thomson optical depth across the Software: COMPPS (Poutanen & Svensson 1996). slab as a measure of the column density: dτ = n (z)σ dz, where e T z is the vertical coordinate and n is the electron concentration Appendix A measured in the lab frame. The source function in the lab frame Radiative Transfer Equation is related to that in the comoving frame by the Lorentz The RTE describing Comptonization of polarized radiation transformation (Rybicki & Lightman 1979): in the plane-parallel atmosphere in the lab frame (marked with l3c superscript l) can be written in the form (Mihalas & SS () tm ,,xx =  (t, ,m), (A2) Mihalas 1984; Nagirner & Poutanen 1994; Beloborodov & Poutanen 1999): where the angles are related by the aberration formulae dx I() tm ,, mb + mb - m =-[(1, bt)m][-s(xx )I(t,m) m = , m = .A()3 dt c 1 - bm 1 + bm + S() tm ,, x ]. (A1) The source function in the comoving frame is Here the photon energy is measured in units of the electron rest mass x = E/m c , μ is the cosine of the angle the photon ¥dx 1 c 2 c momentum makes with the slab normal, I = (I, Q) is the ¢ S() tm ,,xx = dm c còò 0 -1 Stokes vector that fully describes linear polarization (the Stokes c c c ¢¢ ¢¢ U parameter is zero due to the azimuthal symmetry), S is the ´RI () xx ,;mm, (t,x ,m) = I , (A4) c c c c source function (also Stokes vector) in the lab frame, and σ(x ) is the dimensionless total Compton scattering cross section (in where R is the 2 × 2 azimuth-averaged redistribution matrix units of the Thomson cross section σ ) for isotropic describing Compton scattering by isotropic hot electrons (see Maxwellian electron gas as a function of the photon energy Appendix A1 in Poutanen & Svensson 1996). The Stokes in the comoving framexx = , where =- 11 [(gbm)] vector I in the comoving frame is related to that in the lab 6 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov frame as assume it to be unpolarized, isotropic, and optical-depth independent. Its spectral energy distribution is computed c -3l II () tm ,,xx =  (t, ,m). (A5) assuming a thermal electron distribution of kT = 100 keV with a weak nonthermal tail. The resulting synchrotron We solve the RTE using the iterative scattering method spectrum has a shape of a broken power law with a peak at (Poutanen & Svensson 1996) accounting exactly for polariza- ∼1eV (Wardziński & Zdziarski 2001; Veledina et al. 2013). tion by Compton scattering on isotropic (in the flow frame) The iteration procedure now involves steps 5 and 6 and then 3 thermal electron gas (Nagirner & Poutanen 1993). The only and 4, etc. difference from Poutanen & Svensson (1996) is that here we account for bulk motion. At step 1, we specify the boundary condition at τ = 0, either on the bottom of the slab corona or in Appendix B the middle of the slab, in the case of the hot-flow–truncated Spectro-polarimetric Properties of Outflowing Coronae disk geometry. We approximate the spectrum of seed photons (marked by index 0) by the (unpolarized) blackbody We show a few examples of simulations of polarized radiationescapingfromanoutflowing corona for the same l 1 I() tm== 0,xB , (T) (m-∣m∣),A(6) three models considered in the main text. We concentrate x bb 0 0 () here on the case β = 0.4, which describes well Cyg X-1 data, and the observer inclination i = 30°. Figure 4 shows where  is the Heaviside step function and μ = 1 for the slab the main results for model (A), where seed unpolarized corona, while for the truncated disk, the injection of seed blackbody photons are injected from the bottom of out- photons is limited to the zenith angles corresponding to the flowing corona. We see the spectral energy distribution of aspect ratio of the inner hot flow h = H/R, ∣mm ∣<= the total radiation as well as for selected scattering orders in hh 1 + . At step 2, we use the formal solution of the RTE panel (a).The IXPE 2–8 keV range is dominated by photons (Equation (A1); with zero source function S) to get the intensity scattered two to four times in the flow. Panel (b) shows the in all other layers of the slab: energy dependence of the PD. The single-scattered photons have small a PD, which becomes negative in the IXPE l l II () tm ,,xx=- (0, ,m) exp( t m), (A7) 0 0 range, because the angular distribution of the seed photons in the comoving frame of the flow is still rather beamed wherett=- s() x [1,b(t0)m] is the energy- and angle- x c outwards in spite of a high velocity as is seen in panel (c). dependent optical depth measured from the bottom to a The black curves there show the angular distribution of the given layer andb is the average velocity from the slab center intensity at the optical depth τ = (3/4)τ at the energy where (or bottom for slab-corona model) τ =0tothe given τ, EI of the corresponding scattering order peaks. The red defined as curves are the corresponding intensities in the comoving t frame. We see that relativistic aberration is responsible for bt() ,t¢= b(td) t anisotropy of the radiation, with the intensity of multiple t¢ tt-¢ scattered photons in the lab frame peaking at μ ≈ 0.1–0.3 bt ()+¢ bt ( ) tt+¢ (i ≈ 70°–80°). On the other hand, the peak of the intensity in = = b .A()8 22t the comoving frame corresponds to μ ≈− 0.2, i.e., it is l directed more downwards. The angular distribution of the At step 3, we transform the Stokes vector I in the lab frame for PD in both frames is shown in panel (d).Wesee that thePD photons scattered n times to the comoving frame using grows with the scattering order. It reaches maximum roughly Equation (A5). At step 4, we compute the source function in at the same angles where the intensity peaks. Because of the the comoving frame for photons scattered n + 1 times using relativistic aberration, all curves in the lab frame are shifted cc Equation (A4) as S = I . At step 5, we get the source nn +1 to higher μ, so that at a given inclination, the PD grows function in the lab frame using Equation (A2). And finally at with β. step 6, we obtain the intensity of radiation scattered n + 1 times Figure 5 shows the same quantities for model (B) with the from the formal solution of the RTE: seed synchrotron unpolarized photons, which are isotropic in dt¢ ⎧ --() tt¢s(x)[1, -b(tt¢)m] m S () tm¢- ,,xe [1 b(t¢)m] , m>0, ò n+1 ⎪ min I () tm ,, x = () A9 n+1 ⎨ dt¢ l -¢() tt- s(x)[1, -b(t¢t)m](-m) S () tm¢- ,,xe [1 b(t¢)m] ,m<0, ò n+1 () -m where t = 0 for the slab corona andtt =- for the hot min min 0 the comoving frame. The PD is increasing monotonically with flow, and b is given by Equation (A8). The iteration procedure the number of scatterings (panel (d)), but already at around continues with steps 3–6 until the desired accuracy for the total n > 5, very little variation is observed. Because about seven to ¥ l Stokes vector I = I is achieved at all optical depths, å eight scatterings are needed for seed photons to achieve the n=0 angles, and energies. IXPE range, the PD is nearly energy independent (panel (b)). For synchrotron seed photons, we start instead from step 4 The angular distributions of both the intensity and the PD and specify the source function in the comoving frame S .We (panels (c) and (d)) saturate. 7 The Astrophysical Journal Letters, 949:L10 (9pp), 2023 May 20 Poutanen, Veledina, & Beloborodov Figure 6. Same as in Figure 4 but for the seed photons from the truncated disk with the coronal aspect ratio H/R = 1 of model (C). Figure 6 shows the results for model (C) with coronal aspect Dovčiak, M., Muleri, F., Goosmann, R. W., Karas, V., & Matt, G. 2008, MNRAS, 391, 32 ratio H/R = 1. In this model, we see strong anisotropy of seed Esin, A. A., Narayan, R., Cui, W., Grove, J. 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Published: May 1, 2023

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