Bondi radius ($r_B=GM/c_s^2$)

The thermal sphere-of-influence scale $r_B=GM/c_s^2$ compares gravitational potential per unit mass with the ambient gas sound-speed squared. A convention based on escape speed instead introduces a factor of two. The scale is not a capture cross-section or an accretion rate without a flow and dissipation model.

With planet-to-star mass ratio $q$ and disk aspect ratio $h$, $r_B/a=q/h^2$ and $r_H/a=(q/3{)}^{1/3}$. Equating the two gives $q=h^3/\sqrt{3}$. The thermally limited radius grows faster with planet mass than the tidally limited radius; an envelope outer scale is often the smaller of the two.