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

Aims. 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. Methods. We developed a Python-based library to solve the equations of motion characterising the BH and we provided an analytical framework for the numerical results. We carried out simulations at different resolutions and for a range of softening choices using the Tree-PM code OpenGADGET3, where we implemented an improved DF correction based on a kernel-weighted local density estimation. Results. Our results demonstrate that the DF correction significantly accelerates BH sinking and ensures convergence with increasing resolution, closely matching the 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. This effect can be partially mitigated by the DF correction. Conclusions. 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 approach will be crucial for future simulation campaigns, enabling more accurate predictions of active galactic nucleus (AGN) accretion and feedback, while allowing for the estimation of gravitational wave event rates.