than it was with test case 1 with differences between the PBE and SPH results of $\simless 3$%. The separation follows the prediction of the PBE code to better than $3$% until the circumbinary disc begins to form. Once the circumbinary disc attains approximately 5% of the binary’s mass, however, the separation is always smaller than predicted by the PBE code. As described above, this is expected because the PBE code neglects the separation-decreasing effect of the interaction between the binary and the circumbinary disc. This also explains why the PBE code slightly under-estimates the binary’s mass ratio. However, it is pleasing to see that even neglecting the interaction between the binary and the circumbinary disc, the PBE code still predicts the mass of the circumbinary disc to within a factor of 2 of that given by the SPH code during the entire evolution.

The Evolution of Accreting Protobinary Systems
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As we have seen, the PBE code gives a relatively accurate description of the evolution of an accreting protobinary, but does so $\sim 10^6$ times faster than a full hydrodynamic SPH calculation. This allows us to perform many calculations to study how the evolution of a binary as it 