© The Authors 2025

1. Introduction

Black hole (BH) X-ray binaries (BHXBs) evolve from a low lunimosity with hard spectrum state (LHS) to a high luminosity with soft spectrum state (HSS). In these states, the corona, presumably, dominates the X-ray emission, particularly in the LHS and the intermediate state (IS) (Montanari et al. 2009). But the geometry of the corona is not known yet. The geometries of the corona considered up to now are: a sphere above a BH as a lamp-post (Wilkins & Fabian 2012), a Compton cloud (CC) as a flat (planer) atmosphere (Haardt & Matt 1993), and a quasi-spherical CC between an accretion disk (AD) and a central BH (Shaposhnikov & Titarchuk 2006).

We return to the problem of the X-ray spectral formation in compact objects, in particular for BHs, because of the quite exact measurements of the linear polarization (LP) for these objects (see e.g., Krawczynski et al. 2022; Steiner et al. 2024; Dovčiak et al. 2024; Marin et al. 2024). We remind the reader details of the old paper (hereafter ST85) by Sunyaev & Titarchuk (1985), who investigated analytically the X-ray spectral and LP formation, P, in the slab (plain) geometry for a wide range of the Thomson optical depth from 0.1 to more than 10. It is remarkable that, for an optical depth (half of the slab) larger than 10, they obtained a value of polarization that follows the classical Chandrasekhar distribution versus the inclination, i, which is the angle between the direction of observation and the normal to the slab (Chandrasekhar 1950). It is well known that the shape of the observed X-ray spectra agrees with the analytically derived Comptonization spectra Sunyaev & Titarchuk (1980), Titarchuk (1994), hereafter ST80 and T94, respectively. In this paper, it is important to remind the reader of the expected characteristics of the linear X-ray polarization produced in a flat CC. We interpret the X-ray data from several Galactic BHXBs and one active galactic nucleus (AGN) for which remarkable polarization measurements have been obtained.

Stellar and supermassive BHs have similar accretion processes. All these BHs show similar components in their spectra: a thermal component in the soft X-ray spectra, and a Comptonization component in the hard X-rays. The thermal component is likely generated within the AD, which is typically presented as a geometrically thin, optically thick structure, such as the Shakura-Sunyaev disk (see Shakura & Sunyaev 1973). The Comptonized component is formed in the CC, a region of hot plasma that up-scatters photons from the AD to higher energies via inverse Compton scattering, producing the hard X-ray emission observed in BHXBs and AGNs (see Titarchuk 1994).

In the HSS, the AD blackbody emission dominates in the emergent spectrum (see the X-ray spectral evolution in a BH XTE J1650–500, Montanari et al. 2009). Some authors can think that in the HSS a geometrically thin optically thick AD continues down to the innermost stable circular orbit (Saade et al. 2024), while in the LHS the AD is possibly truncated. A cartoon showing the different geometry of a BH in the HSS and the LHS is shown in Fig. 1 in ST85 and in Fig. 3 here.

Fig. 1. Percentage of LP, P, as a function of μ = cos i, where i is the angle between the normal to the CC and a given direction (see ST85 and our Fig. 3). Red and blue boxes are drawn for the measured linear polarization degree in the 2–8 keV for the soft and hard states, correspondingly, for different BH sources (source names are indicated by arrows next to the corresponding μ). The vertical width of these boxes corresponds to the error bars of the IXPE measurements.

Polarization provides a direct constraint on the geometry of the X-ray-emitting region that is not possible by timing and spectral measurements only. With this in mind, we applied this new independent way to derive relevant parameters of the accretion processes through the polarization data as demonstrated in ST85. For the first time ever, the Imaging X-ray Polarimetry Explorer (IXPE) (Weisskopf et al. 2022; Soffitta et al. 2021), a NASA-ASI mission with crucial INAF and INFN contributions that was launched on December 9, 2021, provides energy resolved X-ray polarization data in the 2–8 keV energy band for a large number of stellar BHs (Dovčiak et al. 2024) in their different states and for a few selected radio-quiet AGNs (Marin et al. 2024) .

In Sect. 2 we recall the polarization properties of the hard X-ray radiation produced within a planar CC around a BH. In Sect. 3 we use the polarization properties of BHs to derive the main parameters of these objects. In Sect. 4 we point out the importance of our results, and in Sect. 5 we draw the conclusions of our polarization study of the Galactic and extragalactic black holes (EBHs).

2. The polarization X-ray photons emerging from Compton clouds

We assume that a primary source of the soft photons is distributed over the CC slab with different configurations: in the center of the CC slab, a uniform distribution inside the cloud, or on the CC boundary. ST85 showed that the polarization properties of photons that undergo a number of scatterings, N ≫ τ₀², much larger than the average, for a given optical depth of the CC slab, 2τ₀ (see Fig. 1 in ST85), are independent of any of these distributions.

Up-scattering (Comptonization) of low-frequency (soft) photons of hν ≪ k_BT_e leads to energy gain up to photon energies, hν ≤ 3k_BT_e. A thermal, nonrelativistic (i.e., Maxwellian) electron distribution is considered (see ST80, ST85).

It is a remarkable result of ST85 that the polarization of X-ray photons should be independent of the photon energy, which is essentially confirmed by the IXPE observations. Solving the integro-differential equation of the transport of the polarized radiation, ST85 used the method of consecutive approximations. For k ≫ τ₀ the polarization of photons after these k scatterings is independent of k (see ST85). When k_BT_e < 70 keV and the Thomson approximation is valid, the result of the ST85 calculations is correct (Pozdnyakov et al. 1983).

We introduce a two-component vector, $\overline{I}=(I_l,I_r)$, denoting the radiation intensities related to the electric field oscillations in the plane defined by the disk axis and by direction of photon propagation (plane called meridional) and in the plane perpendicular, to it, respectively. Changes in I_l and I_r in the CC plane have been calculated by the following transport equations (Chandrasekhar 1950):

$$\begin{array}{c}\hfill \mu \frac{\mathrm{d}}{\mathrm{d}\tau }\begin{array}{c}{I}_l(\tau ,\mu )\\ I_r(\tau ,\mu )\end{array}=\begin{array}{c}{I}_l(\tau ,\mu )\\ I_r(\tau ,\mu )\end{array}{\displaystyle {\int }_0^1}P(\mu ,{\mu }^{\prime })\begin{array}{c}{I}_l(\tau ,{\mu }^{\prime })\\ I_r(\tau ,{\mu }^{\prime })\end{array}\mathrm{d}{\mu }^{\prime }-\begin{array}{c}{F}_l(\tau )\\ F_r(\tau ),\end{array}\end{array}$$(1)

where dτ = −σ_TN_edz, μ = cos i, and

$$\begin{array}{c}\hfill P(\mu ,{\mu }^{\prime })=\frac{3}{8}\left\{\begin{array}{cc}2(1-{\mu }^2)(1-{{\mu }^{\prime }}^2)+{\mu }^2{{\mu }^{\prime }}^2\hfill & {\mu }^2\hfill \\ {{\mu }^{\prime }}^2\hfill & 1\hfill \end{array}\right\}\end{array}$$

is the scattering matrix. The vector $\overline{F}=(F_l,F_r)$ notes the distribution of primary sources. The boundary condition of the problem is

$$\begin{array}{c}\hfill \overline{I}(\tau =0,-\mu )=\overline{I}(\tau =2{\tau }_0,\mu )=0\end{array}$$(2)

for any 0 ≤ μ ≤ 1.

The degree of polarization is defined as

$$\begin{array}{c}\hfill P=\frac{I_r-I_l}{I_r+I_l}\end{array}$$(3)

and the intensity is a sum, I(0, μ) = I_r(0, μ)+I_l(0, μ).

We should find the photon distribution that undergoes k scatterings. One should also determine the distribution of photons that are not scattered in the medium; namely,

$$\begin{array}{c}\hfill \overline{I}(\tau ,\mu )={\displaystyle {\int }_{\tau }^{2{\tau }_0}}exp[-({\tau }^{\prime }-2{\tau }_0)/\mu )]\overline{F}({\tau }^{\prime })\mathrm{d}{\tau }^{\prime }/\mu ,\end{array}$$(4)

for any initial photon distribution, $\overline{F}(\tau )$.

When the intensities of the first and next iterations are calculated one can proceed with the intensity of photons that undergo k scatterings:

$$\begin{array}{c}\hfill \mu \frac{\mathrm{d}}{\mathrm{d}\tau }\begin{array}{c}{I}_l^k(\tau ,\mu )\\ I_r^k(\tau ,\mu )\end{array}=\begin{array}{c}{I}_l^k(\tau ,\mu )\\ I_r^k(\tau ,\mu )\end{array}-{\displaystyle {\int }_0^1}P(\mu ,{\mu }^{\prime })\begin{array}{c}{I}_l^{k-1}(\tau ,{\mu }^{\prime })\\ I_r^{k-1}(\tau ,{\mu }^{\prime })\end{array}\mathrm{d}{\mu }^{\prime }.\end{array}$$(5)

The solution of the k iteration depends on its solution for (k − 1) one (see also ST85).

In Table 1 of ST85, the authors showed that the polarization degree and angle distribution of k-scattered photons are independent of the primary source distribution if k ≫ τ₀. To illustrate this statement, the authors made their calculations for τ₀ = 2 and the iteration number N ≥ 20. The calculation results for k ≫ τ₀ are independent of k and demonstrate that these results can be applied to the polarization of the Comptonized photons (see more details in ST85, Appendix A2). It is worth emphasizing that for large optical depths (τ ≫ 5) X-ray polarization converges on the classical solution by Sobolev (1949) and Chandrasekhar (1950). The change in the polarization sign for τ < 4 (see Fig. 1 and ST85) can be explained by a simple example of the optically thin CC (see Sect. 5.3 in ST85).

The appropriate radiative transfer equation, Eq. (1), can be rewritten in the operator form (see also Sect. A2 in ST85). The system of equations of the polarized radiation, Eq. (1), can be rewritten as

$$\begin{array}{c}\hfill \overline{I}=L\overline{I}+\overline{F},\end{array}$$(6)

where L is the polarization integral operator. One can check that the operator, L, meets all the conditions of the Hilbert-Schmidt theorem (see ST85). Thus, L has an eigenvalue, $\overline{p_i}$, and ortho-normalized eigen functions, φ_i and p_i → 0, while i → ∞. A vector function, $\overline{F}$, can be expanded in a generalized Fourier series:

$$\begin{array}{c}\hfill \overline{F}={\mathrm{\Sigma }}_{i=1}^{\infty }{a}_i\overline{{\phi }_i}.\end{array}$$(7)

Solving Eq. (6) by the method of successive approximations, we find that the term $\overline{I_k}$, related to the photons, underwent k scatterings:

$$\begin{array}{c}\hfill {\overline{I}}_k={\mathrm{\Sigma }}_{i=1}^{\infty }{a}_i{p}_i^k\overline{{\phi }_i}.\end{array}$$(8)

Because of the sequence of p_i decreasing with k, we find that

$$\begin{array}{c}\hfill {\overline{I}}_k\simeq a_1{p}_1^k\overline{{\phi }_1}.\end{array}$$(9)

This mathematical result is illustrated by Table 2 of ST85 for k ≫ τ₀².

3. Determination of the main parameters of the Compton cloud

Sunyaev & Titarchuk (ST80; see also ST85) solved the problem of X-ray spectral formation in the bounded medium. In particular, for the slab geometry they derived a formula for the spectral index, α (and Γ = α + 1):

$$\begin{array}{c}\hfill \alpha =-\frac{3}{2}+\sqrt{\frac{9}{4}+\gamma },\end{array}$$(10)

where γ = m_ec²β/k_BT_e, β = π²/[12(τ₀ + 2/3)²], α = Γ − 1, and τ₀ is the optical depth of one half of the slab.

Using these formulae one can find the electron temperature, k_BT_e, as a function of α and β:

$$\begin{array}{c}\hfill k_{\mathrm{B}}{T}_{\mathrm{e}}=\frac{\beta m_{\mathrm{e}}{c}^2}{{(\alpha +3/2)}^2-9/4}.\end{array}$$(11)

ST85 calculated the transfer of the polarized radiation using the iteration method. The number of iterations should be much more than the average number of scatterings in a given flat CC. As a result they made a plot of the LP of the Comptonized photons) as a function of μ = cos i, where i is the inclination of the system for different Thomson optical depths of τ₀. ST85 calculated the LP for τ₀ from 0.1 to 10, and for τ₀ > 10 they found that the LP follows the well-known Chandrasekhar distribution (see Fig. 1).

For quite a few BHs, the LP, P, was measured. For example, for Cyg X-1 using IXPE data, (Steiner et al. 2024) showed that in the soft (HSS) and hard (LHS) states of the source, respectively, P_s ∼ (2 ± 0.5)% and P_h ∼ (3.5 ± 0.5)% are almost independent of the photon energy. The measured photon indices are Γ_HSS = 2.5, Γ_LHS = 1.4, and k_BT_e = 9 keV, and 90 keV, respectively (see Fig. 2 and Table 1).

Fig. 2. IXPE spectra of Cyg X-1 for which X-ray polarization is detected during the LHS and HSS. The photon index is Γ = 1.4, Eline = 6.5 keV, and χred2 = 158/149; see left panel (ID = 01002901). For the HSS spectrum, these parameters are Γ = 2.4, Eline = 6.4 keV, and χred2 = 151/149; see right panel (ID=02008601). The best-fit model consists of const*tbabs*(comptb + Gaussian).

Fig. 3. Geometry of the CC. Using the polarimetric measurement, we assume that the planer CC is present. The CC is smaller and thicker for the soft state (dark blue) than that for the hard state (light blue).

For the HSS of 4U 1630-47, (Rodriguez Cavero et al. 2023) found that Γ ∼ 2.6 − 2.9. The inclination, i, is in the range of 60 − 75° (see Kuulkers 1998), and using the plot of the LP versus μ (see Fig. 1) we find that τ₀ is around 1.7 and k_BT_e ≈ 10 keV (see Eqs. (10) and (11) and Table 1]).

It is worth noting the trend of P, which ranges from 6−12% in the 2–8 keV energy band, as in Table 1 (Ratheesh et al. 2024). On the other hand, the photon indices, Γ, in these measurements vary from 2.6 to 4.5 (see Ratheesh et al. 2024 and also Rodriguez Cavero et al. 2023). This indicates that the polarization detections were made while the source underwent significant spectral state changes. It is important to emphasize that the quasi-constancy of P with energy can only be ensured if the P value is determined within the same spectral state.

For the BHXRB 4U 1957+115, (Marra et al. 2024, hereafter M24) obtained P_h = (1.9 ± 0.6)%, which is observed in the HSS. Knowing the inclination, i ∼ 72°, and using the LP versus μ plot, we find that τ₀ ∼ 2.5 for 4U 1957+115 (Γ is about 2, see our Table 1). This is consistent with the result of M24, who found that Γ = 1.93 ± 0.21 (Table 1 in M24).

For LMC X–1 (Podgorný et al. 2023, hereafter P23) report an IXPE observation in the HSS using NICER, NuSTAR, and IXPE spectra (see their Fig. 3). The photon index, Γ ≈ 2.6, was obtained using the XSPEC NTHCOMP model. P23 assume that i is about 36°. Thus, using the P values and μ = cos36° ∼ 0.8, we find that the optical depth, τ₀, is about 2.5.

For Swift J1727.8–1613 (Veledina et al. 2023), hereafter V23, report a significant polarization [P = (4.1 ± 0.2)%] that is almost independent of the photon energy (see Table 1 in V23). If we take into account that i is within the interval 30° −60° (Casares & Jonker 2014), then we obtain τ₀ ∼ 1.5. V23 also report that Γ ∼ 1.8 is determined for a particular observation when the LP is about 4%. In this case the optical depth, τ₀, is about 2.

For GX 339–4 (Mastroserio et al. 2025) published a significant (4σ) polarization degree, P = (1.3 ± 0.3)%, in the 3–8 keV band (where the spectrum is dominated by the corona) . We assume that the inclination in this source is 30° as a low limit (see Mastroserio et al. 2025, Table 3). Thus, we get a value of τ₀ ≈ 3.

IXPE observed a select number of radio-quiet AGNs. One of the brightest Seyfert-1 galaxies, NGC 4151, shows a significant polarization, P = (4.5 ± 0.9)%, parallel to the radio emission. We obtain τ₀ ≈ 2 using Fig. 1 and the inclination, i, within the range of 58.1° ±0.9° (Bentz et al. 2022).

4. Discussion

One can argue about the sign of the observed polarization degree (see Fig. 1). If one chooses positive values then one infers a value of τ₀ > 5 for almost any value of μ = cos i. However, the derived value of τ₀ > 5 contradicts the observed value of the index, Γ > 1.5 (α > 0.5), and the Comptonization parameter, Y ∼ (k_BT_e/m_ec²)(2τ₀)² ∝ α⁻¹ (see Table 1). For these values of τ₀ > 5, the Y parameter is higher than 2. In this case the emergent spectrum has a Wien shape (see e.g., Rybicki & Lightman 1979), which contradicts the X-ray observations.

We recall again here the definition of polarization described by ST85. Negative polarization of the observed radiation implies that the polarization vector lies within the meridional plane, which is the plane defined by the disk normal and the propagation direction of the outgoing photons. Indeed, the IXPE observations of BHXRBs and Seyfert-1 galaxies show that the polarization vector is consistently parallel to the disk axis for all of them (see, e.g., Krawczynski et al. 2022; Dovčiak et al. 2024; Marin et al. 2024). This confirms that the polarization vector resides within the meridional plane, explicitly ruling out polarization vectors oriented parallel to the disk plane.

The plane geometry is unexpected for the CCs. As we noted in the introduction, any type of CC geometry can be assumed: jets above the disk or a quasi-spherical CC around a BH. But there can only be strong LP if there is a strong asymmetry in the CC around a BH. It is natural that for the spherical CC around the central object (e.g., a NS) the polarization degree is close to zero (see e.g., Farinelli et al. 2024). If we choose negative values of polarization, P, then this leads to values of τ₀ from 0.1 to 3 (see Figs. 5 and 8 in ST85 and Fig. 1 here).

Saade et al. (2024, hereafter S24) make a comparison of the X-ray polarimetric properties of stellar and supermassive BHs. In their Fig. 1, they present the polarization degree versus their inclination, i. The range of i for the Galactic BHs is almost determined, while EBHs have a very broad distribution (within approximately the 60° interval).

The PD values of these EBHs are typical of the ones calculated in ST85 and presented here in Fig. 1. It is not by chance that S24 noted that the polarization properties of these two types of sources are similar. It is easy to see that the range of the optical depth, τ₀, in this case can be estimated within an interval of 1 − 1.5. If these EBHs are observed in the LHS then Γ < 2 and the electron temperature is k_BT_e ∼ 40 − 60 keV (see Table 1).

It is well known that the best-fit values of the optical depth, τ, and k_BT_e are around 2 and 60 keV, respectively, in the LHS and τ > 3, k_BT_e < 10 keV in the HSS (see Table 1 and e.g., ST85, T94).

5. Conclusions

A theoretical idea of the LP formation in a BH relies on the ST85 study. The linearly polarized X-ray radiation is the result of the multiple up-scattering of the initially X-ray soft photons (or UV ones in the AGN case) by the hot CC. Multiple scattering of the soft disk photons leads to the formation of the relatively hard emergent Comptonization spectrum. Furthermore, this specific spectrum is linearly polarized if the CC has a flat geometry. The polarization degree of these multiple up-scattered photons, P, should be independent of the photon energy.

ST85 made a plot of P as a function of μ = cos i, where i is inclination of the source. For different Thomson optical depths, τ₀, of the flat CC, ST85 calculated the LP, P, for τ₀ from 0.1 to 10 (see also their Figs. 5 and 8).

IXPE observed quite a few BHs and these observations have already been analyzed (see e.g., Dovčiak et al. 2024). Cyg X–1 data were studied by Krawczynski et al. (2022), Steiner et al. (2024), who established that in the soft and hard states P_s ∼ 2% and P_h ∼ 3.5%, respectively. Rodriguez Cavero et al. (2023) analyzed the data for 4U 1630–47 and found that P_h ∼ 7%. Marra et al. (2024) analyzed the data from a BH source, 4U 1957+115. They obtained P_h ∼ 2%.

The analysis of IXPE data for LMC X–1 was made by Podgorný et al. (2023) finding that P is about 1%, while for 4U 1630–47 Rodriguez Cavero et al. (2023) found P to be about 7%. In the case of Swift J1727.8–1613 (Veledina et al. 2023) reported that P ∼ 4%. For GX 339–4 an analysis of IXPE observations was made by Mastroserio et al. (2025) finding that P ∼ 1.3%. Finally for a bright Seyfert-1 galaxy (NGC 4151), Gianolli et al. (2024) found P ∼ 4.5%. For all these BHs, using the plot of P versus μ and values P for a given source, we estimated the CC optical depth, τ₀, and using the results of the spectral analysis, we obtained the photon index, Γ, and the plasma temperature, k_BT_e, without any free parameters (see Table 1).

IXPE observed a polarization vector aligned to the disk axis. This is consistent with the negative polarization inferred in this work based on a spectral shape analysis. A similar pattern is observed in radio-quiet Seyfert 1 galaxies studied by IXPE (Marin et al. 2024), highlighting a remarkable similarity across BHs that spans an enormous range of BH masses (e.g., see Saade et al. 2024).

Acknowledgments

The Imaging X-ray Polarimetry Explorer (IXPE) is a joint US and Italian mission. The US contribution is supported by the National Aeronautics and Space Administration (NASA) and led and managed by its Marshall Space Flight Center (MSFC), with industry partner Ball Aerospace (now, BAE Systems). The Italian contribution is supported by the Italian Space Agency (Agenzia Spaziale Italiana, ASI) through contract ASI-OHBI-2022-13-I.0, agreements ASI-INAF-2022-19-HH.0 and ASI-INFN-2017.13-H0, and its Space Science Data Center (SSDC) with agreements ASI-INAF-2022-14-HH.0 and ASI-INFN 2021-43-HH.0, and by the Istituto Nazionale di Astrofisica (INAF) and the Istituto Nazionale di Fisica Nucleare (INFN) in Italy. This research used data products provided by the IXPE Team (MSFC, SSDC, INAF, and INFN) and distributed with additional software tools by the High-Energy Astrophysics Science Archive Research Center (HEASARC), at NASA Goddard Space Flight Center (GSFC). In addition, we acknowledge the fruitful discussion with the referee on the content of the presented paper.

References

All Tables

All Figures

Fig. 1. Percentage of LP, P, as a function of μ = cos i, where i is the angle between the normal to the CC and a given direction (see ST85 and our Fig. 3). Red and blue boxes are drawn for the measured linear polarization degree in the 2–8 keV for the soft and hard states, correspondingly, for different BH sources (source names are indicated by arrows next to the corresponding μ). The vertical width of these boxes corresponds to the error bars of the IXPE measurements.

In the text

	Fig. 2. IXPE spectra of Cyg X-1 for which X-ray polarization is detected during the LHS and HSS. The photon index is Γ = 1.4, Eline = 6.5 keV, and χred2 = 158/149; see left panel (ID = 01002901). For the HSS spectrum, these parameters are Γ = 2.4, Eline = 6.4 keV, and χred2 = 151/149; see right panel (ID=02008601). The best-fit model consists of const*tbabs*(comptb + Gaussian).
In the text

	Fig. 3. Geometry of the CC. Using the polarimetric measurement, we assume that the planer CC is present. The CC is smaller and thicker for the soft state (dark blue) than that for the hard state (light blue).
In the text