Thermal stability of an accretion disk

Thermal stability of an accretion disk asks whether a small temperature disturbance decays while the local surface density is effectively fixed. Write net heating as $Q^{+}-Q^{-}$; with positive effective heat capacity, stability requires ${\partial }_T(Q^{+}-Q^{-})<0$ at equilibrium. A hotter patch must gain a restoring excess of cooling. The comparison belongs on the thermal timescale of an accretion disk, before slower radial redistribution changes the column. Its outcome depends on the stress and opacity prescriptions. The standard total-pressure alpha disk closure predicts instability in a radiation-supported, electron-scattering regime.^[ref]

Partial hydrogen ionization changes disk cooling and permits cold and hot stable thermal branches separated by unstable equilibria. A dwarf nova can cycle between a cold matter-storing disk and a hot, rapidly accreting disk. The cycle is described by the evolution of local surface density and temperature, coupled to viscous evolution of an accretion disk.

At fixed disk radius, balancing local heating and cooling gives thermal equilibria parameterized by surface density and temperature. In the hydrogen-ionization disk instability, this locus has cold and hot stable branches joined by unstable equilibria. Crossing the turning points drives the state changes of a dwarf nova. Thermal stability of an accretion disk tests the local heating and cooling response, while viscous evolution of an accretion disk changes the surface density.

For a gas-supported alpha disk with $\kappa \propto {\rho }^{1/3}{T}^{10}$, compare rates at fixed surface density of a disk, radius, $\alpha$ and composition on the thermal timescale of an accretion disk. Assume that the vertical dynamical timescale of a disk is shorter than that thermal time, and that the viscous timescale is longer; the separation is controlled for $\alpha \ll 1$ and small disk aspect ratio. Vertical hydrostatic equilibrium gives $H\propto T^{1/2}$, $\rho \propto T^{-1/2}$, so $\kappa \propto T^{59/6}$. Viscous heating satisfies $Q^{+}\propto T$ and radiative cooling satisfies $Q^{-}\propto T^4/(\kappa \Sigma )\propto T^{-35/6}$. At equilibrium a positive temperature perturbation changes the net heating by $(41/6)(Q_0/T)\delta T>0$. With positive effective heat capacity, it grows rather than relaxes. This is a local thermal statement and uses the fixed-column, rather than fixed-volume-density, comparison.

The thermal timescale compares the internal energy stored in a disc annulus with its local heating or cooling. In a gas-pressure-supported alpha disk, $t_{\mathrm{th}}\sim (\alpha {\Omega }_K{)}^{-1}$, whereas the viscous timescale is longer by approximately $(R/H{)}^2$. A local thermal perturbation is therefore studied at fixed surface density of a disk, rather than fixed steady-state mass accretion rate.

At fixed surface density, a total-pressure alpha disk supported by radiation pressure has viscous heating $Q^{+}\propto T_c^8$ but radiative cooling $Q^{-}\propto T_c^4$ when electron scattering supplies a nearly constant opacity. Heating therefore wins after a positive temperature perturbation.