Application of Sturm's theorem to marginal stable circular orbits of a test body in spherically symmetric and static spacetimes

Marginally stable circular orbits in the Schwarzschild black hole surrounded by quintessence matter

“…All other cases can be mapped to this case by an appropriate coordinate transformation which is not the main topic of discussion here. The necessary condition for the existences of MSCOs is given in terms of a second-order differential equation involving both h(r) and f (r) [22,26] d dr…”

“…If in a spacetime, an MSCO exist then the solution of (2) gives us the radius of such an orbit. The constants of motion E and L are given by [26]…”

“…Otherwise the root r is unphysical. The existence of such orbits for both Schwarzschild and Kottler black holes was proved in [26] with the application of Strum's theorem. They also applied the analysis on spherically symmetric spacetimes in the Weyl conformal gravity.…”

“…It may be helpful in understanding gravitational field around black holes. In this regard several black hole spacetimes have been investigated in the literature [1]- [26]. In the theory of general relativity the radius of circular orbits of particles in the vicinity of black holes has a lower bond which are known as inner most stable circular orbits (ISCOs).…”

“…These two types of orbits form a boundary between the two regions, i.e., a stable region and an unstable region respectively. In the literature this boundary is known as the MSCO [26]. If in a spacetime geometry the number of MSCOs is only two then the smaller is called the ISCO and the larger is known as the OSCO.…”