Thermal equilibrium curves of accretion disks driven by magnetorotational instability

Abstract

Analogous to the HR diagram for stars, the thermal equilibrium curve encodes the thermodynamics of accretion disks by expressing the local balance between heating -- primarily via viscous dissipation -- and cooling -- typically through radiative transfer. These curves are commonly plotted as surface density versus effective temperature. When an S-shaped locus appears, local annuli become bistable, and limit-cycle oscillations arise when the external mass-transfer rate falls within an unstable band. This behavior underpins the disk instability model for recurring outbursts in cataclysmic variables. This paper reviews first-principles thermal equilibrium curves for accretion disks driven by magnetorotational instability (MRI), with emphasis on dwarf novae. Unlike the parameterized $α$-viscosity approach, the curves are obtained by solving the governing equations with radiation magnetohydrodynamics simulations, thereby reproducing S-shaped loci without prescribing $α$. The disk instability in dwarf-nova systems and the physical origin of angular-momentum transport (shear stresses) are also briefly reviewed. Notes on the stability of radiation-dominated accretion flows are included in the Appendix.

Figures (11)

• Figure 1: Thermal equilibrium curves in the $\Sigma$--$T_{\rm eff}$ plane for several disk classes (colors denote classes). Values in parentheses are $\log_{10}\Omega ~[\mathrm{s}^{-1}]$. BH disk at $30 r_\text{g}$ ($P_{\rm rad}\gg P_{\rm gas}$): Refs. Hirose_2009aHirose_2009bBlaes_2011. BH disk at $150r_\text{g}$ ($P_{\rm rad}\sim P_{\rm gas}$): Refs. Krolik_2007Blaes_2007. BH disk at $300r_\text{g}$ ($P_{\rm rad}\ll P_{\rm gas}$): Ref. Hirose_2006. dwarf-nova disk at $14r_{\rm WD}$: Ref. Hirose_2014. Protoplanetary disk at 0.04 AU (hydrogen ionization): Ref. 10.1093/mnras/stv203. Protoplanetary disk at 1 AU (MRI dead zone): Ref. Hirose_2011. Protoplanetary disk at 50 AU (self-gravity): Refs. 10.1093/mnras/stx82410.1093/mnras/stz163. Detailed parameters used to calculate the gravitational radius $r_\text{g}$ and the white dwarf radius $r_\text{WD}$ are given in the respective references. All data and scripts needed to reproduce this figure are available at Ref. RMHD_TEs.
• Figure 2: Left: Schematic S-shaped thermal equilibrium curve in the $\Sigma$--$T_{\rm eff}$ plane. Right: Rosseland-mean opacities (black) near hydrogen--ionization temperatures at a density $\rho=10^{-6}\ \mathrm{g\,cm^{-3}}$optab14.
• Figure 3: Shear stress versus total pressure ($P_{\rm gas} + P_{\rm rad}$) for different disk classes; data from the references listed in the caption of Fig. \ref{['fig:TEcurve_all']}. All data and scripts needed to reproduce this figure are available at Ref. RMHD_TEs.
• Figure 4: Thermal-balance equilibria at $\Omega = 6.4\times10^{-3}\ \mathrm{s}^{-1}$. The color indicates the total optical depth based on the Rosseland-mean opacity.
• Figure 5: $T_{\rm mid}$ versus $T_{\rm eff}$. The dotted curve indicates $T_{\rm mid} \propto T_{\rm eff}^4$. The color encodes $T_{\rm eff}$.