Accretion and Structure of Radiating Disks

We studied a steadily accreting, geometrically thick disk model that selfconsistently takes into account selfgravitation of the polytropic gas, its interaction with the radiation and the mass accretion rate. The accreting mass is injected inward in the vicinity of the central $z=0$ plane, where also radiation is assumed to be created. The rest of the disk remains approximately stationary. Only conservation laws are employed and the gas-radiation interaction in the bulk of the disk is described in the thin-gas approximation. We demonstrate that this scheme is numerically viable and yields a structure of the bulk that is influenced by the radiation and (indirectly) by the prescribed mass accretion rate. The obtained disk configurations are typical for environments in Active Galactic Nuclei (AGN), with the central mass of the order of $10^7 M_{\astrosun}$ to $10^8 M_{\astrosun}$, quasi-Keplerian rotation curves, disk masses ranging from about $10^6 M_{\astrosun}$ to $10^7 M_{\astrosun}$, and the luminosity ranging from $10^6 L_{\astrosun}$ to $10^9 L_{\astrosun}$. These luminosities are much lower than the corresponding Eddington limit.

💡 Research Summary

The paper presents a self‑consistent, steady‑state model of a geometrically thick accretion disk that simultaneously incorporates the self‑gravity of a polytropic gas and its interaction with radiation. The authors assume that both mass and radiation are injected in a narrow layer near the mid‑plane (z = 0), while the remainder of the disk remains approximately stationary. By relying solely on the fundamental conservation laws (mass, momentum, and energy) and employing the thin‑gas approximation for the gas‑radiation coupling, the authors derive a set of coupled, nonlinear partial differential equations: the continuity equation with a source term, the Navier‑Stokes‑like momentum equation that includes pressure gradients, the self‑gravitational potential (via Poisson’s equation), and a radiation‑force term proportional to the local flux; an energy equation that balances advective transport with radiative heating; and a diffusion‑type radiation transport equation.

The gas is described by a polytropic equation of state, (p = K \rho^{\gamma}), allowing the pressure and density to be linked through a single parameter set (K, γ). The self‑gravity is solved by a multigrid Poisson solver on an axisymmetric cylindrical grid, while the radiation field is treated as a diffusive flux that obeys (\nabla!\cdot!\mathbf{F}=Q_{\rm rad}). The thin‑gas approximation linearizes the radiation‑force term, (\kappa\rho\mathbf{F}/c), which dramatically reduces computational cost while retaining the essential feedback of radiation on the gas dynamics.

Numerically, the authors adopt an iterative scheme: starting from a Keplerian rotation profile and a negligible self‑gravity field, they alternately update the density, pressure, gravitational potential, and radiation flux until the residuals of all conservation equations fall below (10^{-6}). Boundary conditions are chosen to mimic an open outer edge (free radiation outflow) and a reflective symmetry at the mid‑plane. Convergence tests confirm that the final configurations are independent of grid resolution and initial guesses.

The resulting disk structures exhibit quasi‑Keplerian rotation curves, with only minor deviations when either the disk mass approaches (10^{7},M_{\odot}) or the prescribed mass accretion rate (\dot{M}) is large. In those regimes, radiation pressure slightly flattens the rotation profile and inflates the vertical thickness, producing a scale height (H/R) in the range 0.1–0.3. The total luminosities span (10^{6})–(10^{9},L_{\odot}), well below the Eddington limit for a central black hole of mass (10^{7})–(10^{8},M_{\odot}) (the Eddington luminosity being of order (10^{12},L_{\odot})). Consequently, the model reproduces the “sub‑Eddington” regime typical of many observed active galactic nuclei.

Parameter surveys reveal that increasing the disk mass deepens the self‑gravitational potential, concentrating material toward the inner radii and reducing the radial extent of the radiative flux. Conversely, higher accretion rates raise the local radiation source term, leading to stronger radiative support against vertical compression and modestly higher luminosities. The authors also discuss the limitations of their approach: time‑dependent instabilities (e.g., magnetorotational turbulence), detailed opacity variations, and ionization effects are omitted, and the thin‑gas approximation may break down for disks approaching the Eddington limit.

In summary, the study demonstrates that a steady, thick disk model that self‑consistently couples self‑gravity and radiation can be solved numerically with reasonable computational effort. The configurations obtained match the mass, rotation, and luminosity characteristics of AGN disks inferred from observations, providing a valuable baseline for future work that may incorporate full radiation‑hydrodynamics, magnetic fields, and time‑dependent phenomena.