Negative hydrogen ion opacity ($\kappa \propto {\rho }^{1/2}{T}^9$)

The negative hydrogen ion supplies important continuum opacity in cool hydrogen-rich stellar atmospheres. A useful local approximate law is $\kappa ={\kappa }_0{\rho }^{1/2}{T}^9$; it is not valid over arbitrary temperatures and compositions. In an ideal gas power-law opacity radiative envelope matched to ${\nabla }_{\mathrm{phot}}=1/8$, it gives ${\nabla }_{\mathrm{rad}}=-1/3+(11/24)(T/T_{\mathrm{eff}}{)}^{9/2}$, rising rapidly toward convective instability.