Dynamical friction and massive black hole orbits: analytical predictions and numerical solutions

We investigate the orbital decay of a massive BH embedded in a dark matter halo and a stellar bulge, using both analytical and numerical simulations with the aim of developing and validating a reliable dynamical friction (DF) correction across simulation resolutions. We develop a Python-based library to solve the equations of motion of the BH and provide an analytical framework for the numerical results. Then, we carry out simulations at different resolutions and for different softening choices using the Tree-PM code OpenGADGET3, where we implement an improved DF correction based on a kernel-weighted local density estimation. Our results demonstrate that the DF correction significantly accelerates BH sinking and ensures convergence with increasing resolution, closely matching analytical predictions. We find that in low-resolution regimes - particularly when the BH mass is smaller than that of the background particles - our DF model still effectively controls BH dynamics. Contrary to expectations, the inclusion of a stellar bulge can delay sinking due to numerical heating, an effect partially mitigated by the DF correction. We conclude that our refined DF implementation provides a robust framework for modeling BH dynamics both in controlled simulation setups of galaxies and in large-scale cosmological simulations. This will be crucial for future simulation campaigns, to enable more accurate predictions of AGN accretion and feedback, and to estimate gravitational-wave event rates.