is done [*before*]{} any new candidate sinks are considered.

In addition to the sink particles, we have also implemented an external pressure term [e.g. @benz90], that enables us to model a constant pressure boundary, as opposed to the vacuum or periodic boundary conditions that are the only choices available in the standard version of Gadget. We modify the standard gas pressure contribution to the Gadget 2 momentum equation,

$$\frac{d v_{i}}{d t} = - \sum_{j} m_{j} \left[
f_{i}\frac{P_{i}}{\rho_{i}^{2}} \nabla_{i} W_{ij}(h_{i})
+ f_{j}\frac{P_{j}}{\rho_{j}^{2}}\nabla_{i} W_{ij}(h_{j}) \right] ,$$

by replacing $P_{i}$ and $P_{j}$ with $P_{i} -  P_{\rm ext}$ and $P_{j} -  P_{\rm ext}$ respectively, where $P_{\rm ext}$ is the external pressure, and all quantities have the usual meaning, consistent with those used by @springel05. The pair-wise nature of the force summation over the SPH neighbors ensures that $P_{\rm ext}$ cancels for particles that are surrounded by other particles. At the edge, where the term does not disappear, it mimics the pressure contribution from a surrounding medium. The values of $P_{\rm ext}$ used in this study are $3 \times 10^{7} k_{\rm B} \: {\rm K \: cm^{-3}}$ for the Pop III.1 clouds, and $7.5 \times 10^{6} k_{\rm B} \: {\rm K \: cm^{-3}}$ for the 