An Analytic Model For Magnetically-Dominated Accretion Disks

TL;DR

The paper develops a self-similar analytic model for flux-frozen, magnetically dominated accretion disks around supermassive black holes, motivated by cosmological MHD simulations showing hyper-magnetized, rapidly cooling disks at super-Eddington rates. By anchoring the solution to outer boundary conditions at the free-fall radius r_ff with a trans-Alfvénic turbulent state (ψ_A ≈ 1), the model self-consistently predicts the disk’s density, magnetic field, scale height, and inflow, and demonstrates that β ≪ 1 and Q ≫ 1 persist from the outer disk down to near the BH radius of influence, ensuring gravitational stability at hyper-Eddington accretion rates. Thermal structure is explored to confirm β ≪ 1 and rapid cooling, with the outer disk remaining radiation-pressure dominated only weakly and at small radii; radiation transport and vertical stratification are argued to be mild in the outer regions, while inner regions require careful treatment. The model’s scalings are robust to variations in several assumptions and align well with simulations, distinguishing these flux-frozen disks from SS73, MAD, or magnetically elevated models, and suggesting such hyper-magnetized disks may be common in quasars, with important implications for spectral properties and the physics of accretion at high rates.

Abstract

Recent numerical cosmological radiation-magnetohydrodynamic-thermochemical-star formation simulations have resolved the formation of quasar accretion disks with Eddington or super-Eddington accretion rates onto supermassive black holes (SMBHs) down to a few hundred gravitational radii. These 'flux-frozen' and hyper-magnetized disks appear to be qualitatively distinct from classical $α$ disks and magnetically-arrested disks: the midplane pressure is dominated by toroidal magnetic fields with plasma $β\ll 1$ powered by advection of magnetic flux from the interstellar medium (ISM), and they are super-sonically and trans-Alfvenically turbulent with cooling times short compared to dynamical times yet remain gravitationally stable owing to magnetic support. In this paper, we present a simple analytic similarity model for such disks. For reasonable assumptions, the model is entirely specified by the boundary conditions (inflow rate at the BH radius of influence [BHROI]). We show that the scalings from this model are robust to various detailed assumptions, agree remarkably well with the simulations (given their simplicity), and demonstrate the self-consistency and gravitational stability of such disks even in the outer accretion disk (approaching the BHROI) at hyper-Eddington accretion rates.

Figures (1)

• Figure 1: Comparison of the model predictions here ( thick dashed lines, § \ref{['sec:analytic']}) to the numerical simulations of quasar accretion disks forming from cosmological initial conditions in Paper II ( solid lines show mean values at each $R$ at one moment/snapshot, shaded range shows the $90\%$ inclusion interval for all gas at that $R$, averaged over all simulation snapshots). We compare inflow rate $\dot{M}(r)$ (here net inflow less outflow); magnetic field strength (with the simulation example showing the toroidal/poloidal/radial decomposition); gas density; scale height $H/R$; turbulent (1D direction-averaged rms $\delta v$), thermal sound ($c_{s}$) and Alfvén ($v_{A}$) speeds; mean inflow velocity $-\langle v_{R}\rangle$; and effective Toomre $Q$ parameter (including thermal, magnetic, and turbulent support). Simulation quantities are volume-averaged in the midplane ($|z|<H$), and plotted versus cylindrical radius from the SMBH $R$, from $\sim 80\,$au (the inner simulation accretion boundary) to $\sim 1\,$pc (exterior to which the physics become more ISM-like, with significant star formation in a GMC-like complex which is being tidally disrupted by the SMBH to form the accretion disk, biasing the statistics; see Paper I). For the models, we take the parameters $M_{\rm bh}=1.3\times10^{7}\,M_{\odot}$, $\dot{M}_{\rm bh} \approx 25\,{\rm M_{\odot}\,yr^{-1}}$ (plotted; $\dot{m}\approx 100$), and $r_{\rm ff} \approx R_{\rm BHROI} \approx 5\,{\rm pc}$ directly from the simulation, and assume trans-Alfvénic turbulence ($\psi_{A}=1$, $\zeta=1/3$). We contrast a shakurasunyaev73 model with $\alpha=0.1$ ( thin dotted). The analytic model reproduces the key simulation properties reasonably well, especially compared to an SS73 model which predicts quantities like $Q$ differing by factors as large as $\sim 10^{8}$.