Interaction between disk and extended corona in a general relativistic framework

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)

💡 Research Summary

The authors present a substantial upgrade to the Monte‑Carlo radiative‑transfer code MONK, enabling a self‑consistent treatment of the feedback between an accretion disc and an extended slab corona in full Kerr spacetime. In the original MONK framework, the disc supplied seed photons for Comptonisation, while the corona reprocessed them, but the reciprocal illumination of the disc by the corona was not modelled, breaking the expected rapid light‑travel equilibrium. To close this loop, the authors introduce an albedo parameter that determines what fraction of the coronal illumination is reflected (using Chandrasekhar’s semi‑infinite electron‑atmosphere formalism) and what fraction is absorbed and added to the intrinsic viscous dissipation of the disc.

The iterative scheme works in two nested loops. In the inner loop, for a given coronal electron temperature (kT_e) and optical depth (τ), MONK computes photon trajectories from each disc annulus, tracks their possible fates (escape to infinity, capture by the black hole, return to the disc, or entry into the corona), and simulates Compton scattering in the corona. Reflected photons are treated as new seed photons and may undergo a second round of scattering. The total emergent luminosity L_tot (including direct disc, coronal, reflected, and self‑irradiated components) is then compared with the accretion power L_acc = η Ṁ c^2, where η depends on the black‑hole spin.

If L_tot exceeds L_acc, the excess energy must be supplied by the corona; if it is lower, the disc must provide more intrinsic power. This is handled by a global scaling factor α (0 ≤ α ≤ 1) that partitions the viscous power: α L_acc is radiated directly by the disc, while (1‑α) L_acc heats the corona. After each inner‑loop convergence (ΔL_tot/L_tot < 1 %), α is updated via α_new = (L_acc / L_tot) α_old, and the whole process repeats until L_tot = L_acc within tolerance. This double‑iteration guarantees a true global energy balance, unlike previous local‑balance models.

Parameter studies reveal three key drivers of spectral hardness: black‑hole spin, coronal temperature, and disc albedo. Higher spin raises η, allowing more power to be channeled into the corona without violating energy conservation, thus hardening the spectrum. Raising kT_e increases the Compton y‑parameter, boosting the coronal output and the illumination of the disc; a higher albedo returns more photons to the corona for a second scattering, further hardening the slope. Despite these effects, the authors find a robust lower limit to the photon index for a static, fully covering slab corona: Γ ≈ 1.7–1.8 for typical coronal temperatures (≈50–100 keV) and albedos (≈0.2–0.3). This limit is significantly softer than the hardest observed hard‑state spectra of X‑ray binaries (Γ ≈ 1.4–1.6).

The study also highlights that imposing both local and global energy balance forces an inward‑to‑outward energy transport within the disc‑corona system, a requirement that is difficult to reconcile with a static slab geometry. Consequently, the authors conclude that a static slab corona is unlikely to be the geometry responsible for the hard state of X‑ray binaries, even when full relativistic effects and disc‑corona feedback are accounted for. They propose future investigations of truncated‑disc configurations and outflowing coronae, which can be readily implemented in the upgraded MONK code. Such models will allow simultaneous fitting of spectral shape, luminosity, and X‑ray polarization, offering a powerful diagnostic for the true geometry of accretion flows around black holes.