Trajectories of a BH placed on a circular orbit at 20 kpc from the centre of a DM halo for three different halo resolutions. From left to right, the halo is sampled with 10⁴, 10⁵ and 10⁶ particles. The dashed black lines indicate the Plummer-equivalent softening length of each simulation. Orange lines represent the BH trajectory when the DF correction is applied according to Eq. (33), while the blue lines correspond to the original DF correction in Eq. (6) of D24, the red line shows the analytical prediction from OTIS (Sect. 2.4). The ratio of the BH mass to the surrounding DM particle mass is indicated at the top of each panel.