Study of null and time-like geodesics in the exterior of a Schwarzschild black hole with quintessence and cloud of strings

Abstract

Recently, an analytical study of radial and circular orbits for null and time-like geodesics that propagate in the spacetime produced by a Schwarzschild black hole associated with cloud of strings, in a universe filled by quintessence, has been done in Ref. \cite{Mustafa:2021}. In this paper, we complete the aforementioned study by investigating possible analytical solutions to the equations of motion for other types of bound orbits, besides taking into account the cases of unbound orbits. This requires an extensive study of the corresponding effective potentials that categorize the test particle motion. We follow the standard Lagrangian dynamics to parametrize the radial and angular geodesics and the resultant (hyper-)elliptic integrals of motion are treated accordingly. We also simulate the orbits which correspond to different levels of energy in the effective potentials.