Past exam of the exoplanets and Planetary Systems course of the University of Cambridge / 2015 / Paper 1 / 3 / a / Solution

A sufficiently self-gravitating disk can fragment directly, bypassing the slow assembly of a massive solid core. In the simplest thin gas-disk model the competition is between self-gravity, pressure and rotational shear. Axisymmetric perturbations havewhere $\kappa$ is the radial epicyclic frequency, not the opacity. In a nearly Keplerian disk, $\kappa \simeq \Omega$. This is Toomre's stability criterion; finite thickness and global/nonaxisymmetric modes modify the precise boundary. The fastest local wavelength is $\lambda =2c_s^2/(G\Sigma )$, of order the disk scale height times $2\pi$ when $Q_T\simeq 1$.

Instability alone does not ensure bound planets. Compression and shocks heat the gas, raising $Q_T$; a slowly cooling disk can maintain self-regulated spiral turbulence. Cooling-limited disk fragmentation requires energy loss on a sufficiently short dynamical timescale, conventionally expressed as $\beta =\Omega t_{\mathrm{cool}}$ below a model-dependent threshold. The often quoted $\beta \sim 3$ belongs to a particular local razor-thin calculation, not every real disk.^[ref] Cold massive outer disks are promising sites, but fragment contraction, tidal survival, further accretion and migration determine whether the result remains a planetary-mass body or grows into a brown dwarf.

For gravitational collapse of a settled particle layer, if “core collapse” instead denotes a solid-particle layer, its random-velocity dispersion replaces gas sound speed in an appropriate particle-disk stability condition. Settling and concentration can enable collapse into planetesimals; gas turbulence and particle stirring oppose it. This route makes solid seeds for later core accretion and is physically distinct from direct gas fragmentation. In both cases one must establish self-gravity against velocity support and tides, rather than infer collapse from high surface density alone.