= Solution
Use a gas-pressure-supported, <optically thick> <alpha disk>, with $\mathcal R=k/(\mu_m m_p)$ and positive parameters. Here $\sigma$ is the Stefan–Boltzmann constant in the <Stefan–Boltzmann law>, and $F_+$ and $F_-$ below are heating and cooling flux estimates per disk face. <Hydrostatic equilibrium> and the <ideal gas> relation give $\mathcal R T\sim\Omega^2H^2$, while $\rho\sim\Sigma/H$. Integrating viscous heating over one semi-thickness and estimating <radiative diffusion> give
$$
F_+\sim\mu\Omega^2H\sim\alpha\Sigma\mathcal R T\Omega\sim\alpha\Sigma\Omega^3H^2,\qquad F_-\sim\frac{\sigma T^4}{\kappa\rho H}\sim\frac{\sigma}{\kappa_0}T^{-6}\Sigma^{-4/3}H^{1/3}.
$$
Here the stipulated <negative hydrogen ion opacity> exponent is $\kappa=\kappa_0\rho^{1/3}T^{10}$, which is a local approximation distinct from other empirical <opacity> fits. Substituting $T\sim\Omega^2H^2/\mathcal R$ and equating heating and cooling gives
$$
\boxed{H^{41/3}\sim\frac{\sigma}{\kappa_0}\mathcal R^6\Omega^{-15}\alpha^{-1}\Sigma^{-7/3}.}
$$
Dropping numerical constants of order one is essential here. This is the <negative-hydrogen-opacity alpha-disk thickness scaling>. Its angular thickness estimate is
$$
\boxed{\frac Hr\sim\left(\frac{\sigma}{\alpha\kappa_0}\right)^{3/41}\mathcal R^{18/41}\Sigma^{-7/41}\frac{\Omega^{-45/41}}r.}
$$
For a central mass $M$, <Keplerian rotation> gives $\Omega=(GM/r^3)^{1/2}$, so $H/r\propto r^{53/82}\Sigma(r)^{-7/41}$. The isothermal <sound speed> $c_s^2=\mathcal R T$ is
$$
\boxed{c_s\sim\Omega H\propto\Omega^{-4/41}\Sigma^{-7/41}\propto r^{6/41}\Sigma(r)^{-7/41}.}
$$
An adiabatic <sound speed> differs by the constant factor $\sqrt\gamma$ if the adiabatic index is fixed. For $\Sigma\propto r^{-p}$, constant $\alpha$, and fixed composition, the <disk aspect ratio> grows as $r^{(53+14p)/82}$, so there is <disk flaring> when $p>-53/14$; $c_s\propto r^{(6+7p)/41}$. The problem supplies no numerical parameters or specified $\Sigma(r)$, so no unique numerical $H/r$ or unconditional radial power can be inferred.
If the additional standard closure of a constant inward <accretion rate> well outside a zero-torque edge is intended, $\bar\nu\sim\alpha\Omega H^2$ and $\bar\nu\Sigma\simeq\dot M/(3\pi)$ imply $\Sigma\propto(\alpha\Omega H^2)^{-1}$. Eliminating it gives $H^9\propto\Omega^{-38/3}$ at fixed $\dot M,\alpha$, hence $H/r\propto r^{10/9}$ and $c_s\propto r^{11/18}$. These stronger powers use that extra steady-flow assumption. All these local scalings cease to apply if the <thin disk> or the stated <opacity> regime fails.
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