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

A cold, self-gravitating region in a molecular cloud can collapse when gravity overcomes thermal, turbulent and magnetic support. Increasing central density produces a protostar, while radiation removes part of the released gravitational energy. The cloud initially carries angular momentum; material that cannot fall radially to the centre settles into a rotating protoplanetary disk. Collisions and cooling reduce random vertical motion, while angular-momentum transport allows some material to accrete and some to move outwards. Bipolar outflows and magnetic stresses can remove additional angular momentum.

The schematic sequence is $\text{rotating cloud}\to \text{protostar plus disk}\to \text{accreting solids and gas}\to \text{planetary system plus cleared disk}$. In the cool disk, grains condense and collide; settling and concentration can produce planetesimals, whose gravitational growth leads to planetary embryos. Massive cores may capture gas before the disk disperses; sufficiently massive unstable disks can also form bound condensations. Migration and interactions modify the final orbits. Radiation, accretion and outflows eventually disperse the remaining gas. Thus planet formation uses the same rotating reservoir produced by ordinary star formation, although the efficiency and final architecture are not universal and individual growth stages remain active research topics.

Neglect disk self-gravity and take $|z|\ll r$ around a central point mass $M_{\star }$. Its vertical gravitational acceleration isVertical hydrostatic equilibrium is $dP/dz=\rho g_z$. For a vertically isothermal ideal gas, $P=\rho c_s^2$ and $c_s$ is independent of $z$, givingHere $v_K=r\Omega =\sqrt{G{M}_{\star }/r}$, $c_s^2=k_{B}T/(\mu m_p)$ is the isothermal sound speed squared, and $h$ is the Gaussian scale height. A different convention writing $e^{-z^2/h^2}$ changes the quoted height by $\sqrt{2}$ but not the physics.

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.