impetus: consistent SPH calculations of 3D spherical Bondi accretion on to a black hole

Abstract

We present three-dimensional calculations of spherically symmetric Bondi\naccretion onto a stationary supermassive black hole (SMBH) of mass $10^{8}$\n$M_{\\odot}$ within a radial range of $0.02-10$ pc, using a modified version of\nthe smoothed particle hydrodynamics (SPH) \\gad \\sp code, which ensures\napproximate first-order consistency (i.e., second-order accuracy) for the\nparticle approximation. First-order consistency is restored by allowing the\nnumber of neighbours, $n_{\\rm neigh}$, and the smoothing length, $h$, to vary\nwith the total number of particles, $N$, such that the asymptotic limits\n$n_{\\rm neigh}\\to\\infty$ and $h\\to 0$ hold as $N\\to\\infty$. The ability of the\nmethod to reproduce the isothermal ($\\gamma =1$) and adiabatic ($\\gamma =5/3$)\nBondi accretion is investigated with increased spatial resolution. In\nparticular, for the isothermal models the numerical radial profiles closely\nmatch the Bondi solution, except near the accretor, where the density and\nradial velocity are slightly underestimated. However, as $n_{\\rm neigh}$ is\nincreased and $h$ is decreased, the calculations approach first-order\nconsistency and the deviations from the Bondi solution decrease. The density\nand radial velocity profiles for the adiabatic models are qualitatively similar\nto those for the isothermal Bondi accretion. Steady-state Bondi accretion is\nreproduced by the highly resolved consistent models with a percent relative\nerror of $\\lesssim 1$\\% for $\\gamma =1$ and $\\sim 9$\\% for $\\gamma =5/3$, with\nthe adiabatic accretion taking longer than the isothermal case to reach steady\nflow. The performance of the method is assessed by comparing the results with\nthose obtained using the standard Gadget and the Gizmo codes.\n