Example application of the model for efficient accretion. The solid lines indicate the conditions expressed in Equations (1) and (2) (for λ _B = 10⁻⁴). The green area is the most efficient region for accretion, while the red is the most inefficient one. For example, a BH located where the green star is would accrete efficiently, and conversely for the red star. The dotted lines indicate the condition in Equation (2) for two additional values of λ _B, as indicated, for reference. To calculate the active fraction we calculate the ratio of the BH mass range that allows for efficient accretion compared to the total MBH mass range. For instance, the green hatched box indicates the value of M _•,act for the density value of 10⁵ cm⁻³, while the gray hatched box indicates the whole mass range ΔM _•—see Section 2.3 for details. Figure adapted from Pacucci et al. (2017b).