tokens: given that the resulting cooling rate peaks in a temperature regime where the gas is expected to be fully ionized. To calculate an approximate cooling function, we solve equation ( C1 ) for a densities in the range 10 −2 –10 6 cm −3 and in the temperature range of 1–10 8 K. For each species, cooling rates are calculated at collisional excitation equilibrium. The net cooling is the sum of these terms: where the first sum is over species, n i ,s is the number density of an excited state for a given species, and ν ij is the frequency of the photon emitted by a transition from level i to level j . The two-dimensional table of cooling rates is averaged along the density axis to get an average cooling rate as a function of temperature. The average cooling function is show in Fig. C1 alongside the piecewise fit used in our models (equation 32 ). Figure C1. Average volumetric cooling rate (solid) and piecewise approximation (dashed) for collisionally excited line radiation. The shaded region indicates the 1σ error region, and the hatched region highlights the range of values averaged over to obtain the approximate cooling function used. APPENDIX D: DISC FINDING ALGORITHM We have developed an efficient parallelized disc finding routine that allows us to track the formation and evolution of formed discs in our models irrespective of scale or orientation. We implement this routine directly into athena for two reasons. First, this tool provides the advantage of ‘on-the-fly’ analysis which yields much greater time accuracy than is reasonably managed through post-processing. Secondly, the method is designed to take advantage of the parallelized structure of athena , which dramatically reduces the amount of time required to execute the necessary operations. The algorithm takes the following steps: (1) On each processor, or local grid, the total mass and angular momentum with respect to the origin are calculated within a search radius, R search . For simulations with refinement, overlapped cells are excluded from this total. The net angular momentum ( L net ) and mass ( M net ) are then calculated across all processors. (2) For each cell, the deviation angle is calculated as where L cell is the angular momentum vector of the cell. (3) The total mass for cells with θ dev θ thresh is then calculated. For our simulations, we found that setting θ thresh = 30° was a sufficiently strict condition for capturing the formation of a disc. (4) A mass fraction is then calculated as For our models, we consider f M > 0.75 to be indicative of a potential disc. (5) For disc candidates, we continue to a three-dimensional rotation of the mesh from the simulation reference frame into a reference frame in which the axis is parallel to L net , which we call the disc frame. We construct a new mesh in the disc frame with identical dimensions and hierarchical structure to the simulation frame of reference. We loop over all cells on the disc frame mesh and calculate the
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