Dynamical friction and galaxy merging time-scales

The time-scale for galaxies within merging dark matter haloes to merge with each other is an important ingredient in galaxy formation models. Accurate estimates of merging time-scales are required for predictions of astrophysical quantities such as black hole binary merger rates, the build-up of stellar mass in central galaxies and the statistical properties of satellite galaxies within dark matter haloes. In this paper, we study the merging time-scales of extended dark matter haloes using N-body simulations. We compare these results to standard estimates based on the Chandrasekhar theory of dynamical friction. We find that these standard predictions for merging time-scales, which are often used in semi-analytic galaxy formation models, are systematically shorter than those found in simulations. The discrepancy is approximately a factor of 1.7 for M_sat/M_host ≈ 0.1 and becomes larger for more disparate satellite-to-host mass ratios, reaching a factor of ∼3.3 for M _sat/M_host ≈ 0.01. Based on our simulations, we propose a new, easily implementable fitting formula that accurately predicts the time-scale for an extended satellite to sink from the virial radius of a host halo down to the halo's centre for a wide range of M_sat/M _host and orbits. Including a central bulge in each galaxy changes the merging time-scale by ≲10 per cent. To highlight one concrete application of our results, we show that merging time-scales often used in the literature overestimate the growth of stellar mass by satellite accretion by ≈40 per cent, with the extra mass gained in low mass ratio mergers.