= Thermal instability from negative hydrogen ion opacity
{title2=$Q^+\propto T,\quad Q^-\propto T^{-35/6}$}
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 $\alpha\ll1$ 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.
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