Shape and ionization of equatorial matter near compact objects from X-ray polarization reflection signatures

TL;DR

This work develops a flexible X-ray spectro-polarimetric model for equatorial, optically thick reflectors around accreting compact objects, exploring how ionization and 3D geometry imprint polarization signatures in the 0.1–100 keV band. Using a central point-like X-ray source and three inner-wall geometries (cone, elliptical torus, bowl) together with non-LTE plane-parallel reflection tables resolved by a Monte Carlo code, the study demonstrates that the ionization profile ξ and the viewing geometry can drive polarization up to tens of percent, often with distinct energy dependencies and polarization angles. The results show that ionization can rival geometry in shaping p, while the reflector shape can yield differences up to ∼30% in certain regimes, especially at small Θ and high i; they also examine the transition to optically thin winds, finding energy-dependent polarization trends bridging thick and thin limits. These findings have practical implications for IXPE and future X-ray polarimeters, and the authors propose an XSPEC model to fit real data, while noting limitations such as the neglect of relativistic effects and the assumption of a central compact source. Overall, the paper lays a foundation for interpreting polarization measurements as probes of 3D accretion structures and winds, guiding future simulations and observational campaigns.

Abstract

Motivated by the success of the IXPE mission, we elucidate what can be inferred about 3D matter structures forming about the equatorial plane of accreting compact objects from 0.1-100 keV linear polarization induced by non-relativistic large-scale reflection. We construct a model of an optically thick elevated axially symmetric reflecting medium with arbitrary ionization profile, representing the known diverse scattering environments: from thick winds and super-Eddington funnel structures formed around black holes and neutron stars, to Compton-thick dusty tori of active galactic nuclei and their broad line regions. We assume a central X-ray power-law source with an isotropic, cosine, and slab-corona emission distribution, including intrinsic polarization. The reprocessing is based on constant-density local reflection tables produced with a Monte Carlo method combined with detailed non-LTE radiative transfer, although we also show examples with classical (semi-)analytical reflection prescriptions. We conclude that varying ionization has a similarly strong impact on observed polarization as the observer's inclination and the skew and opening angle of the reflector's inner walls, altogether producing up to tens of % of reflected polarization both parallelly or perpendicularly to the projected axis, depending on the parameter values combination. After testing 3 different ad-hoc shapes of the reflector: a cone, an elliptical torus, and a bowl, we conclude that while in some configurations, their altered curvature produces more than 30% absolute difference in observed total polarization, in others, the adopted shape has a marginal impact. Lastly, we discuss the change of the observed polarization due to relaxing the optically thick assumption on equatorial winds and accreted matter, providing a continuous range of energy-dependent examples between the optically thick and thin scenarios.

Figures (10)

• Figure 1: The reflecting surface in 3D Cartesian global coordinate system, as viewed by an inclined observer. From left to right we display for the same $\Theta$ and $\rho/\rho_\mathrm{in}$ the cone-shaped, torus-shaped, and bowl-shaped geometries, respectively, that are examined in this work. The yellow star represents the central source of emission.
• Figure 2: The parametrization of the cone-shaped (green), torus-shaped (purple), and bowl-shaped (blue) reflectors. The sketch is in the meridional plane, and the reflecting structure is further rotated around the main axis and remains axially symmetric. We highlight in solid lines only the upper half of the reflecting surface above the equatorial plane in each geometry, although the model allows for taking into account reflection from the bottom half-space of a mirror-symmetric surface to the equator. In the dashed purple line, we show the part of the inner walls of the torus, which may self-obscure some other reflecting part of the torus for a highly inclined observer, but is not directly illuminated by the source due to self-shadowing. We show in the dotted purple line the ellipse defining the torus-shaped reflector in both half-spaces from the equator. Similarly, we show in the dotted green line the corresponding straight edge of an off-centered geometrical cone in the meridional plane that defines the cone-shaped reflector in the upper half-space. Similarly, we also show in the dotted blue line the ellipse defining the bowl-shaped reflector in the upper half-space. The observer is inclined at an inclination $i$. Each reflecting geometry is fully described by its half-opening angle $\Theta$, the inner radius $\rho_\mathrm{in}$, and the maximal illuminated radius $\rho$. When $i > \Theta$, the primary source is obscured for the observer.
• Figure 3: Top row: the spectra for a distant observer at $D = 1$ kpc and $\rho_\mathrm{in} = 10^9$ cm. We show the logarithm of $EF_\mathrm{E}/\xi_0$ in $\textrm{cm}^{-3}$ to compensate for the slope given by the power-law index $\Gamma = 2$ and to level the amplitude, given to a large extent by the ionization parameter $\xi_0$. In the color code we show the reflected-only spectra for $\xi_0 = 10 \, \, \textrm{erg} \cdot \textrm{cm} \cdot \textrm{s}^{-1}$ (blue), $\xi_0 = 500 \, \, \textrm{erg} \cdot \textrm{cm} \cdot \textrm{s}^{-1}$ (green), $\xi_0 = 10\,000 \, \, \textrm{erg} \cdot \textrm{cm} \cdot \textrm{s}^{-1}$ (red), and for the Chandrasekhar's formulae for diffuse reflection with 100% albedo (magenta). The primary radiation is shown in gray. The observer is inclined at $i = 51^\circ$ and the half-opening angles are $\Theta = 40^\circ$ (solid lines) and $\Theta = 70^\circ$ (dashed lines). We show the reflection from a cone (left), torus (center), and bowl (right), all for $\rho = \rho_\mathrm{c}$, $B = 1$, and unpolarized isotropic irradiation with $\beta = 2$. Specifically for the cone geometry and $\Theta = 40^\circ$, we show in black solid line the corresponding spectra in the same configuration, but integrating the fully neutral reflection tables obtained with the STOKES code. Middle and bottom row: the corresponding reflected-only polarization degree, $p$, versus energy. For clarity, we show the results for different opening angles $\Theta = 40^\circ$ (solid lines) and $\Theta = 70^\circ$ (dashed lines) in separate rows of panels. In the color-code, we provide in addition the results for Thomson single-scattering approximation (yellow). For the cone geometry and $\Theta = 70^\circ$ (bottom left), we show in addition the total polarization degree for $\xi_0 = 10 \, \, \textrm{erg} \cdot \textrm{cm} \cdot \textrm{s}^{-1}$ (blue dotted lines) and $\xi_0 = 10\,000 \, \, \textrm{erg} \cdot \textrm{cm} \cdot \textrm{s}^{-1}$ (red dotted lines). For the torus (center) and bowl (right) geometries, we show the polarization degree difference $\Delta p$ in %, which is the polarization degree $p$ of the reflecting cone subtracted from the polarization degree $p$ of the reflecting torus or bowl, respectively, in identical configurations. This is to display examples of the impact of changing curvature of the inner walls of the reflector on the resulting polarization.
• Figure 4: Images of the reflecting walls of a cone (left), a torus (center), and a bowl (right) with $\xi_0 = 500 \, \, \textrm{erg} \cdot \textrm{cm} \cdot \textrm{s}^{-1}$ and half-opening angles $\Theta = 40^\circ$ (top) and $\Theta = 70^\circ$ (bottom). Each image contains 25x25 pixels with spectro-polarimetric information integrated in the 3.5--6 keV band, if subtending the observable reflecting area. We use the color code for the background of each reflecting pixel to emphasize the polarized flux, $pF/F_*$, where $F_*$ is the flux of the primary source in observer's direction, for the cone; and the polarized flux difference, $\Delta pF/F_*$, for the torus and bowl, which is the polarized flux, $pF/F_*$, of the reflecting cone subtracted from the polarized flux, $pF/F_*$, of the reflecting torus or bowl in the same pixel for identical configurations. For the cone, each reflecting pixel contains a polarization bar, whose length is proportional to the observed polarization degree from that pixel and whose tilt from the vertical direction is corresponding to its polarization angle. For the torus and bowl, each reflecting pixel contains a number, which is the observed polarization degree in % from that pixel, and a separately color-coded arc, whose length, direction and color are altogether highlighting the polarization degree difference $\Delta p$ in %, which is the polarization degree $p$ of the reflecting cone subtracted from the polarization degree $p$ of the reflecting torus or bowl, respectively, in the same pixel for identical configurations. All other parameter values are the same as in Fig. \ref{['fig:energy_dep']}.
• Figure 5: The 3.5--6 keV averaged polarization degree versus the equatorial ionization parameter, $\xi_0$, for two different surface ionization profiles with $\beta = -2$ (top) and $\beta = 2$ (bottom). For the cone geometry (left), we show the reflected-only polarization degree, $p$ in %. For the torus (center) and bowl (right) geometries, we show the polarization degree difference, $\Delta p$ in %, which is the polarization degree $p$ of the reflecting cone subtracted from the polarization degree $p$ of the reflecting torus or bowl, respectively, in identical configurations. The results are plotted for $\Theta = 40^\circ$ (brown) and $\Theta = 70^\circ$ (turquoise), and for two different inclinations: $i = 79^\circ$, where reflected radiation is the only observable component (dashed lines), and $i = 28^\circ$, where we show the reflected-only polarization (dotted lines) and the total observed polarization (solid lines). We show the case of $\rho = \rho_\mathrm{c}$, $B = 1$, and unpolarized isotropic irradiation. For the torus, the configuration with $\Theta = 40^\circ$ and $i = 79^\circ$ (brown dashed lines) is already under full eclipse, hence it is not displayed.