Geometric correction for wind accretion in binary systems

The Bondi-Hoyle-Lyttleton (BHL) accretion model is widely used to describe how a compact object accretes material from a companion's stellar wind in binary systems. However, its standard implementation becomes inaccurate when the wind velocity ($v_\mathrm{w}$) is comparable to or less than the orbital velocity ($v_\mathrm{o}$), predicting non-physical accretion efficiencies above unity. This limits its applicability to systems with low wind-to-orbital velocity ratios ($w= v_\mathrm{w} / v_\mathrm{o} \leq 1$), such as symbiotic systems. We revisit the implementation of the BHL model and introduce a geometric correction factor that accounts for the varying orientation of the accretion cylinder relative to the wind direction. This correction ensures physically plausible accretion efficiencies ($\eta \leq 1$) for all $w$ in circular orbits. Our new implementation naturally predicts the flattening of the accretion efficiency observed in numerical simulations for $w < 1$, without the need for ad hoc adjustments. We also peer into the implications of our prescription for the less-explored case of eccentric orbits, highlighting the key role of the geometric correction factor in shaping the accretion process. We compare our predictions with numerical simulations, finding good agreement for a wide range of parameters. Applications to the symbiotic star R~Aqr and the X-ray binary LS 5039 are presented. This improved implementation offers a more accurate description of wind accretion in binary systems, with implications for stellar evolution, population synthesis, and observational data interpretation.