of $2 r_{\rm acc}$. Next, we check to see whether the smoothing length of the particle is less than the ‘accretion radius’ of the sink particle that it will become. This ensures that when the sink particle forms it can instantly accrete at least $\sim 50$ neighboring particles (we adopt 50 neighbors in these simulations). The third test is to make sure that the candidate sink particle and its neighbors are on the same integration time-step.

Once these three preliminary criteria are met, the dynamical state of the possible sink particle and its neighbors are assessed to ensure that the particles are indeed undergoing gravitational collapse and are not about to re-expand from their dense state. This takes the form of a further four tests. Firstly, we require that

$$\alpha \le \frac{1}{2}$$

where $\alpha$ is the ratio of the thermal energy to the magnitude of the gravitational energy of the particles. Secondly, we ensure that

$$\alpha + \beta \le 1$$

where $\beta$ is the ratio of rotational energy to the magnitude of the gravitational energy. The third condition is that the total energy of the particles must be negative (which actually renders the above checks redundant, but can help to 