Energies and angular momenta of periodic Schwarzschild geodesics
Yen-Kheng Lim, Zhi Cheng Yeo
Abstract
We consider physical parameters of Levin and Perez-Giz's ``periodic table of orbits'' around the Schwarzschild black hole, where each periodic orbit is classified according to three integers $(z,w,v)$. In particular, we chart its distribution in terms of its angular momenta $L$ and energy $E$. In the $(L,E)$-parameter space, the set of all periodic orbits can be partitioned into domains according to their whirl number $w$, where the limit of infinite $w$ approaches the branch of unstable circular orbits. Within each domain of a given whirl number $w$, the infinite zoom limit ${\mathrm{lim}}_{z\ensuremath{\rightarrow}\ensuremath{\infty}}(z,w,v)$ converges to the common boundary with the adjacent domain of whirl number $w\ensuremath{-}1$. The distribution of the periodic orbit branches can also be inferred from perturbing stable circular orbits, using the fact that every stable circular orbit is the zero-eccentricity limit of some periodic orbit, or arbitrarily close to one.