Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S30 / 2 / c / Solution

At $T=100\mathrm{K}$ with $\mu =2.3$, $c_s=\sqrt{k_{B}T/(\mu m_p)}\simeq 599\mathrm{m}{\mathrm{s}}^{-1}$. A solar-mass Keplerian disk has $v_K\simeq 29.8\mathrm{km}{\mathrm{s}}^{-1}$ at $1\mathrm{AU}$, so $\boxed{h/r\simeq 0.020}$: the thin-disk approximation is consistent.

For fixed central mass, $h/r\propto T^{1/2}{r}^{1/2}$. If $T\propto r^{-q}$, thenA flared outer disk intercepts more stellar radiation than a flat or self-shadowed disk, affecting its temperature, chemistry, vertical structure and observed spectral energy distribution. Heating changes the sound speed and therefore transport and dispersal; it can also enhance irradiation-driven mass loss. The actual coupled evolution depends on optical properties and transport, so flaring alone does not fix a unique lifetime. “Mid-planet” in the source is read as mid-plane.