Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 321 / 1 / b / ii

Use characteristic, vertically averaged quantities at fixed radius. Let $H$ be a disk thickness, $\rho \sim \Sigma /H$, and let the alpha disk prescription give $\mu \sim \alpha p/\Omega$, up to a convention-dependent constant. From the preceding radiation pressure and constant-opacity balance, $\mu \sim c/\kappa$ is independent of $\Sigma$; thusVertical hydrostatic equilibrium gives $p\sim \rho {\Omega }^2{H}^2\sim \Sigma {\Omega }^{2}H$, henceThe vertically averaged kinematic viscosity satisfies $\bar{\nu }\Sigma =\int \mu dz\sim \mu H$. ConsequentlyAt fixed radius the omitted factors are independent of $\Sigma$, so this is the requested inverse-density scaling of the viscous torque in an accretion disk.

This is an order-of-magnitude vertical model, as requested. Imposing both exact pointwise $\mu =\text{constant}$ and exact pointwise $\mu \propto p(z)$ would force $p$ to be constant, contrary to nontrivial vertical hydrostatic equilibrium. The characteristic-pressure prescription supplies a scaling, not an exact simultaneous vertical solution to all those idealized equations.