Interaction between disk and extended corona in a general relativistic framework

Technical Summary: Interaction between Disk and Extended Corona in a General Relativistic Framework (Part I)

Problem Statement
Current models of two-phase accretion flows in X-ray binaries (XRBs) often struggle to explain the hard-state observations where the optically thick disk extends close to the black hole (BH), illuminated by a hard X-ray-emitting corona. Previous studies (e.g., Haardt & Maraschi 1991; Poutanen et al. 2018) established that a static slab corona above a disk has an upper limit on the Comptonized power it can produce in energy equilibrium. If the corona becomes too optically thick or hot, a runaway cascade occurs where enhanced disk illumination increases seed photons, further boosting coronal output, preventing a self-consistent equilibrium. Furthermore, existing models often assume local energy balance, neglecting radial variations and general relativistic (GR) effects, or rely on codes that do not incorporate feedback from the corona to the disk. This work addresses the gap in achieving a globally self-consistent equilibrium solution for a disk-corona system within a full GR framework, specifically testing the viability of a static slab geometry for hard states.

Methodology
The authors upgraded the Monte Carlo radiative transfer code MONK (Zhang et al. 2019) to incorporate mutual interaction between the accretion disk and an extended corona. The key methodological advancements include:

1. Iterative Global Energy Balance: The code now iteratively computes Comptonization by the corona and disk illumination until a global equilibrium is reached. The fundamental constraint is that the total observed luminosity ($L_{tot}$) must equal the total available accretion power ($L_{acc} = \eta \dot{M}c^2$).
2. Albedo and Feedback Mechanism: An albedo parameter is introduced to define the fraction of incident coronal flux reflected by the disk (modeled as a semi-infinite electron atmosphere using Chandrasekhar's formalism). The remaining fraction is absorbed, increasing the disk's effective temperature and intrinsic dissipation.
3. Viscous Dissipation Parameter ($\alpha$): Unlike previous works that assumed all accretion energy is dissipated in the corona, this model adjusts the fraction $\alpha$ of viscous power dissipated intrinsically in the disk. The code iterates $\alpha$ such that $L_{tot} = L_{acc}$. If $L_{tot} > L_{acc}$, $\alpha$ is reduced to lower the seed photon flux, and vice versa.
4. General Relativistic Framework: Photon propagation is tracked via null geodesics in Kerr geometry. The code accounts for light bending, gravitational redshift, and the four possible photon trajectories: reaching infinity, hitting the corona, self-irradiating the disk, or falling into the BH.
5. Spectral Analysis: The photon index ($\Gamma$) of the Comptonized component is derived by fitting the emergent spectrum to a power law ($N(E) \propto E^{-\Gamma}$) within an energy range defined by the overlap of multiple scattering orders.

Key Results
Using reference parameters (BH mass $10 M_\odot$, spin $a=0$, coronal temperature $T_e = 120$ keV, albedo $= 0.5$), the study yields the following findings:

• Equilibrium Solutions: Global equilibrium solutions exist for both hard and intermediate states. For a static slab corona fully covering the disk, the lowest achievable photon index is approximately 1.7–1.8 under typical conditions.
• Optical Depth Limit: There is a strict upper limit on the coronal optical depth ($\tau_{max} \approx 0.16$ for the reference case) beyond which no physical equilibrium solution exists (requiring $\alpha < 0$). As $\tau$ increases, the intrinsic disk dissipation fraction $\alpha$ must decrease to maintain energy balance, eventually reaching near-zero values where disk emission is entirely reprocessed coronal illumination.
• Parameter Dependencies:
  • BH Spin: Higher spin ($a=0.998$) leads to harder spectra ($\Delta \Gamma \sim -0.15$) due to increased light bending and disk illumination, which forces a lower $\alpha$.
  • Coronal Temperature: Higher $T_e$ results in harder spectra.
  • Albedo: Higher albedo significantly hardens the spectra. For an albedo of 1.0 (complete reflection), the system can sustain much higher optical depths ($\tau \approx 0.6$), allowing the photon index to drop to $\Gamma \sim 1.0$. This occurs because high albedo suppresses the reprocessed soft seed photons, decoupling the feedback loop.
• Spectral Shape: The Comptonized spectrum often deviates from a simple power law, exhibiting an "anisotropy break" at high energies, a feature attributed to the one-sided injection of seed photons and GR effects.
• Radial Energy Flow: Under the assumption of a spatially uniform, extended slab corona, achieving local equilibrium at each radial annulus requires energy to flow from the inner to the outer annuli, in addition to satisfying global equilibrium.

Significance and Conclusions
The authors conclude that with the upgraded MONK code, it is now possible to achieve global energy equilibrium for disk-corona systems, providing a robust framework for constraining system parameters. However, the study finds that a static slab corona is not a favorable geometry for explaining the hardest observed spectra in XRBs, even when global energy balance is enforced. The inability of the static slab to produce photon indices significantly below 1.7–1.8 (unless albedo is unity) suggests that alternative geometries are required. The authors state that future work will explore truncated disk geometries and outflowing coronae as potential solutions to these limitations. The approach also holds potential for constraining coronal geometry using polarization data in addition to flux.