= Negative-hydrogen-opacity alpha-disk thickness scaling
{title2=$H^{41/3}\sim\sigma\mathcal R^6/(\kappa_0\alpha\Omega^{15}\Sigma^{7/3})$}
Let $\mathcal R=k_B/(\mu_m m_p)$ for constant <mean molecular weight> $\mu_m$ in proton-mass units, and let $\sigma$ be the Stefan–Boltzmann constant in the <Stefan–Boltzmann law>. The gas <sound speed> used here satisfies $c_s^2=\mathcal R T$, and $\Omega$ is the local orbital frequency. For the local <opacity> approximation $\kappa=\kappa_0\rho^{1/3}T^{10}$, a gas-supported <optically thick> <alpha disk> has $\rho\sim\Sigma/H$ and $\mathcal R T\sim\Omega^2H^2$. Integrated viscous heating is $F_+\sim\alpha\Sigma\Omega^3H^2$. <Radiative diffusion> gives $F_-\sim(\sigma/\kappa_0)T^{-6}\Sigma^{-4/3}H^{1/3}\sim(\sigma/\kappa_0)\mathcal R^6\Omega^{-12}\Sigma^{-4/3}H^{-35/3}$. Equating the fluxes proves the thickness scaling. At fixed composition and $\alpha$, it gives $H/r\propto r^{53/82}\Sigma^{-7/41}$ and $c_s\propto r^{6/41}\Sigma^{-7/41}$ in <Keplerian rotation>. Radial powers therefore require a surface-density profile; they are not determined by <opacity> alone. The formula is local to the stated <opacity> regime and small <disk aspect ratio>.
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