Cooling-limited disk fragmentation

A gas disk near $Q_T\sim 1$ need not form bound fragments: gravitational compression and shocks can heat it into a self-regulated spiral state. Fast enough cooling, often parameterized by $\beta =\Omega t_{\mathrm{cool}}$, permits collapse to continue. A numerical critical beta depends on thermodynamics, dimensionality and the cooling model; it is not universal.^[ref]

A fragment formed by gravitational instability of an astrophysical disk must remain bound after it forms. Its own gravity competes with the surrounding tidal field, while heating, cooling and accretion change its size and internal support.

For a low-mass clump on a circular orbit around a dominant star, its Hill radius estimates the region over which its attraction can resist the local stellar tide. Cooling and contraction can make a fragment more compact, whereas inward migration can shrink that tidal region. A calculation demonstrating initial fragmentation therefore does not alone establish a long-lived giant planet: survival requires the subsequent thermal and orbital evolution as well.