= Solution
\b[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 have
$$
\omega^2=\kappa^2+c_s^2k^2-2\pi G\Sigma|k|,\qquad\boxed{Q_T=\frac{c_s\kappa}{\pi G\Sigma}\lesssim1,}
$$
where $\kappa$ 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 $Q_T\simeq1$.
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 $\beta=\Omega t_{\rm cool}$ below a model-dependent threshold. The often quoted $\beta\sim3$ belongs to a particular local razor-thin calculation, not every real disk.<Nonlinear Outcome of Gravitational Instability in Cooling, Gaseous Disks by Charles F. Gammie>[2001 local model: cooling, self-regulated turbulence and fragmentation]{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 \b[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.
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