Analytical solutions for weak black hole kicks

TL;DR

This paper develops analytical solutions for the motion of black holes subjected to weak kicks, incorporating effects like accretion and dynamical friction, and validates these solutions against numerical simulations.

Contribution

It introduces a general analytical framework for modeling black hole responses to weak kicks, including additional physical effects, with validation against numerical results.

Findings

Analytical solutions agree with numerical simulations within valid parameter ranges.

Amplitude of black hole oscillations depends linearly on initial kick velocity.

Proportionality constant closely matches simulation results.

Abstract

The black hole is modeled by a combined gravitational potential of the bulge, disk and halo and is subjected to an initial weak kick. The resulting differential equations are set up, and shown to possess analytical solutions. The effects of black hole accretion and dynamical friction are also incorporated into this analytical framework. The resultant frequencies and amplitudes are computed, and are compared with the ones obtained from numerical simulations. Within the valid range of parameters of the analytical model, the two sets of results are shown to be in reasonable agreement. It is shown that this model reproduces the linear dependence of the amplitude on the initial kick velocity, and the constant of proportionality is close to that obtained from the simulations. The analytical treatment presented is quite general, and its applications to other areas are also indicated.