  We explore an unresolved controversy in the literature about the accuracy of
Smoothed Particle Hydrodynamics (SPH) in modeling the accretion of gas onto a
binary system, a problem with important applications to the evolution of
proto-binaries as well as accreting binary super massive black holes. It has
previously been suggested that SPH fails to model the flow of loosely bound
material from the secondary to primary Roche lobe and that its general
prediction that accretion drives mass ratios upwards is numerically flawed.
Here we show with 2D SPH that this flow from secondary to primary Roche lobe is
a sensitive function of gas temperature and that this largely explains the
conflicting claims in the literature which have hitherto been based on either
'cold' SPH simulations or 'hot' grid based calculations. We present simulations
of a specimen 'cold' and 'hot' accretion scenario which are numerically
converged and evolved into a steady state. Our analysis of the conservation of
the Jacobi integral of accreting particles indicates that our results are not
strongly compromised by numerical dissipation. We also explore the low
resolution limit and find that simulations where the ratio of SPH smoothing
length to disc scale height at the edge of the circumsecondary is less than 1
accurately capture binary accretion rates.
