An Analytic Model of Angular Momentum Transport by Gravitational Torques: From Galaxies to Massive Black Holes

Abstract: We present analytic calculations of angular momentum transport and gas inflow in galaxies, from scales of ~kpc to deep in the potential of a central black hole (BH). We compare these analytic calculations to numerical simulations and use them to develop a sub-grid model of BH growth that can be incorporated into semi-analytic models or cosmological simulations. Both analytic calculations and simulations argue that the strongest torque on gas arises when non-axisymmetric perturbations to the stellar gravitational potential produces orbit crossings and shocks in the gas. This is true both at large radii ~0.01-1 kpc, where bar-like modes dominate the non-axisymmetric potential, and at smaller radii <10 pc, where a lopsided/eccentric disk dominates. The traditional orbit crossing criterion is not always adequate to predict the locations of, and inflow due to, shocks in gas+stellar disks with finite sound speeds. We derive a modified criterion that predicts the presence of shocks in stellar dominated systems even absent formal orbit crossing. We then derive analytic expressions for the loss of angular momentum and the resulting gas inflow rates in the presence of such shocks. We test our analytic predictions using hydrodynamic simulations at a range of galactic scales, and show that they successfully predict the mass inflow rates and quasi-steady gas surface densities with small scatter (0.3 dex). We use our analytic results to construct a new estimate of the BH accretion rate given galaxy properties at larger radii. This captures the key scalings in the numerical simulations. Alternate estimates such as the local viscous accretion rate or the spherical Bondi rate fail systematically to reproduce the simulations.