= Solution
Assume the <vertical dynamical timescale of a disk> is shorter than the <thermal timescale of an accretion disk>, which is itself shorter than the <viscous timescale>. This ordering is controlled in a <thin disk> with $\alpha\ll1$: $t_{\rm dyn}\sim\Omega^{-1}$, $t_{\rm th}\sim(\alpha\Omega)^{-1}$ and $t_\nu\sim(r/H)^2t_{\rm th}$. Therefore compare heating and cooling at fixed <surface density of a disk>, radius, $\alpha$ and composition. Vertical <hydrostatic equilibrium> adjusts on the <vertical dynamical timescale of a disk>, giving $H\propto T^{1/2}$ and $\rho\propto T^{-1/2}$. Consequently the stipulated <negative hydrogen ion opacity> varies as $\kappa\propto T^{10-1/6}=T^{59/6}$. The rates per disk face then scale as
$$
\boxed{Q^+\sim\alpha\Sigma\mathcal R\Omega T\propto T,\qquad Q^-\sim\frac{\sigma T^4}{\kappa\Sigma}\propto T^{-35/6}.}
$$
Thus heating increases while cooling decreases after a positive <temperature> perturbation. At equilibrium $Q^+=Q^-=Q_0$, a small change gives $\delta(Q^+-Q^-)=(41/6)(Q_0/T)\delta T$, which drives the perturbation further from equilibrium for positive effective <heat capacity>. A negative <temperature> perturbation likewise increases the net cooling. This proves <thermal instability from negative hydrogen ion opacity> within the local closure; it is the temperature-slope test of <thermal stability of an accretion disk>. Holding volume density rather than <surface density of a disk> fixed would give a different cooling exponent, but still the same qualitative instability; the fixed-column comparison is the relevant thin-disk thermal test.
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