Interaction between disk and extended corona in a general relativistic framework

TL;DR

This work extends the Monte Carlo radiative transfer code MONK to model mutual energy feedback between an accretion disk and an extended, relativistic corona in Kerr spacetime, enforcing global energy balance via an iterative disk-dissipation fraction $oldsymbol{\\alpha}$. The introduction of disk albedo and a semi-infinite electron atmosphere for reflection enables self-consistent equilibria across hard and intermediate spectral states, revealing that a static slab corona cannot explain the hardest X-ray binary spectra ($oldsymbol{oldsymbol{\\Gamma_{min}} \\sim 1.7$–$1.8$ for typical parameters). The results show how higher BH spin, higher coronal temperature, and especially higher albedo harden the spectra, with the equilibrium solution constraining the relative disk/corona/reflection contributions and the maximum allowable optical depth $oldsymbol{\\tau_{max}}$ under global balance. The study also highlights the need for radial coronal structure (e.g., nonuniform Te or truncated disks) to achieve local energy balance, and discusses limitations of current reflection modeling due to neglected atomic transitions. Overall, the framework offers a path to constrain coronal geometry and polarization via energy balance and radiative feedback, with future work aimed at more realistic geometries and refined reflection physics.

Abstract

The energy equilibrium between the corona and the underlying disk in a two-phase accretion flow sets a lower limit on the achievable photon index. A slab corona may not explain the hard state observations of X-ray binaries (XRBs). We incorporate energy feedback to the accretion disk resulting from illumination by an extended corona, and vice versa. The interaction between these two components allows for the possibility of finding an energetically self-consistent equilibrium solution for a given disk-corona system. We have upgraded the existing Monte Carlo radiative transfer code, MONK, to incorporate the interaction between the disk and the extended corona within the general relativistic framework. We introduce an albedo parameter to specify the fraction of the incident flux that is reflected by the disk, while the remainder is absorbed and added to the intrinsic dissipation. Reflection is modeled assuming a semi-infinite electron atmosphere. We find global equilibrium solutions by iterating interaction between disk and extended slab corona. A higher black hole spin, higher coronal temperature, and higher albedo all lead to harder spectra. For typical coronal temperatures and disk albedo, the lowest achievable photon index with a static slab corona fully covering the disk is approximately 1.7-1.8. With the upgraded version of MONK, we are now able to achieve global energy equilibrium for a given disk-corona system. This approach holds significant potential for constraining the coronal geometry using not only the observed flux but also polarization. A static slab does not appear to be a favorable coronal geometry for the hard state of XRBs, even when global energy balance is taken into account. In future work, we will explore truncated disk geometries and outflowing coronae as potential alternatives. (shortened)

Figures (6)

• Figure 1: Benchmarking the self-irradiated reflection computed using MONK, assuming the disk to be a semi-infinite electron atmosphere, against the results of Taverna2020 for a Kerr black hole with spin parameter $a = 0.998$ and an inclination angle of 75$^\circ$. The data points from Taverna2020 were extracted using the https://automeris.io/
• Figure 2: Change in the observed spectrum at an inclination of 60$^\circ$ due to the imposition of global energy equilibrium ($L_{\rm tot} = L_{\rm acc}$). The coronal optical depth is fixed at 0.15, and all other parameters are set to their reference values as described in Section \ref{['sec_input_parameters']}. Different spectral components are shown using distinct colors. The solid lines correspond to the equilibrium solution obtained through iteration of the disk dissipation fraction $\alpha$, which converges to a value of 0.07. The dashed lines represent the case with $\alpha = 1$, where no iteration is performed and the equilibrium condition $L_{\rm tot} = L_{\rm acc}$ is not enforced. The dotted black line shows the total spectrum for the $\alpha = 1$ case, but renormalized such that $L_{\rm tot} = L_{\rm acc}$. The solid magenta line represents a power law fit to the scattered spectrum of the equilibrium solution, yielding a photon index of 1.82. The two vertical lines mark the energy range of 2.5–43.7 keV over which the power law fit is performed.
• Figure 3: Equilibrium solutions for increasing coronal optical depth, computed using the reference set of parameters. Panel (a): Variation of the disk dissipation fraction $\alpha$ with optical depth, illustrating a reduction in intrinsic disk dissipation as optical depth increases. Panel (b): Fitted photon index as a function of optical depth for different viewing inclinations. Panel (c): Fractional contributions of the disk, corona, and reflection components to the total luminosity (integrated over all energies) at an inclination of 60$^\circ$.
• Figure 4: Effect of different parameters on the spectral hardness of equilibrium solutions at an inclination of 60$^\circ$. In each panel, a single parameter is varied while keeping all others fixed at their reference values, as described in Section \ref{['sec_input_parameters']}. The reference parameter set (black hole spin $a = 0$, coronal electron temperature $T_{\rm e} = 120$ keV, and disk albedo = 0.5) is shown by black circles in all panels. Panel (a), (b), and (c) shows the effect of black hole spin, coronal electron temperature, and disk albedo respectively. In general, higher spin, higher electron temperature, and higher albedo lead to harder spectra at a given optical depth. However, among these, only increasing the albedo results in a significant reduction of the minimum photon index $\Gamma_{\rm min}$.
• Figure 5: Variation of the escaping fraction ($f_{\rm e}$) and anisotropic fraction ($f_{\rm a}$) with the coronal optical depth ($\tau$). The equilibrium solutions are computed using the same reference parameter set as in Fig. \ref{['fig_alpha_flux_fraction']}. To facilitate comparison, the theoretical expectation for $f_{\rm e}$, given by $\exp(-2\tau)$, is also shown. The factor of 2 arises because $\tau$ in MONK is defined as the vertical optical depth from the mid-plane of the slab to its surface.