Periodic orbits of neutral test particles in Reissner-Nordstr\"{o}m naked singularities

Abstract

We conduct studies on Levin's taxonomy of periodic orbits for neutral test particles around a Reissner-Nordstr\"{o}m naked singularity. It was known that naked singularities could harbor two distinct regions of time-like bound orbits and thus we expect periodic orbits to appear in both regions. It is possible for a pair of periodic orbits from both regions to possess the exact same angular momentum $L$ and energy $E$ values. We chart the sets of periodic orbits in $(L,E)$-parameter space and highlight the general distribution pattern of these sets for three possible scenarios. Regions within $(L,E)$-space can be partitioned into multiple domains $\mathcal{D}_k$ based on the roots configuration of the quartic polynomial $P(u)$ where $u$ is the inverse radial coordinate. Consequently, each domain and interestingly enough, portions of certain periodic orbits sets that lie in different $\mathcal{D}_k$ require different analytical solutions to plot the resulting orbit. Furthermore, we uncover physical properties of some hypothetical circular orbits residing in the inner region from analysing the $(L,E)$-space.