A rotating system will fragment into self-gravitating clumps only when the growth rates of the density perturbations are faster than the collapse timescale. a, Cylindrical radial profiles of Q when considering only thermal support (red) and with thermal and turbulent support (blue), for the MMH (dashed) and the LWH (solid). The shaded region indicates where the system is unstable to fragmentation. b, The unstable regions have a characteristic growth rate, defining a growth timescale tgrow, which exhibits an increasing trend with radius for the MMH (orange) and the LWH (green). c, If the ratio of tgrow and the free-fall time tff is less than one, the region can fragment before it gravitationally collapses. In the MMH, this condition is true at radii less than 0.03 pc, indicating that small-scale fragmentation might occur but will subsequently be suppressed by a rapid monolithic collapse. The LWH exhibits this feature inside 0.1 pc but is surrounded by gas that is stable against fragmentation.