ISCOs and OSCOs in the Presence of a Positive Cosmological Constant in Massive Gravity

Abstract

We study the impact of a non-vanishing (positive) cosmological constant on the innermost and outermost stable circular orbits (ISCOs and OSCOs, respectively) within massive gravity in four dimensions. The gravitational field generated by a point-like object within this theory is known, generalizing the usual Schwarzschild–de Sitter geometry of General Relativity. In the non-relativistic limit, the gravitational potential differs by the one corresponding to the Schwarzschild–de Sitter geometry by a term that is linear in the radial coordinate with some prefactor γ, which is the only free parameter. Starting from the geodesic equations for massive test particles and the corresponding effective potential, we obtain a polynomial of fifth order that allows us to compute the innermost and outermost stable circular orbits. Next, we numerically compute the real and positive roots of the polynomial for several different structures (from the hydrogen atom to stars and globular clusters to galaxies and galaxy clusters) considering three distinct values of the parameter γ, determined using physical considerations, such as galaxy rotation curves and orbital precession. Similarly to the Kottler spacetime, both ISCOs and OSCOs appear. Their astrophysical relevance as well as the comparison with the Kottler spacetime are briefly discussed.

Binary X-ray Sources in Massive Brans–Dicke Gravity

“…For the geometry considered here, the mass term (∝ r 2 ) is the key ingredient, and its inclusion shifts the curves upwards, maintaining the position of the peak. To obtain the modification with respect to the standard Schwarzschild case, we first obtain the r ISCO by computing the roots of the effective potential V effec (see, e.g., [89,90]) for different values of the parameter l (or equivalently m). Subsequently, we perform the coefficient z ≡ r ISCO /r 0 , where r 0 is the Schwarzschild radius in kilometers, adjusting the X-ray emission spectrum accordingly.…”

“…In particular, the details on how to compute r ISCO can be found, for instance, in [89,90]. Avoiding the details, we need to find the lowest (among the real and positive) root of P 4 (r), a function found to be…”

Spherical Particle Orbits around a Rotating Black Hole in Massive Gravity

“…This black hole solution incorporates a positive cosmological constant allowing for the formation of an ISCO and an OSCO, as recently studied in ref. [40]. However, as highlighted in the opening of this section, it is worth noting that stable circular orbits represent only a subset of the broader category of spherical particle orbits, which forms the primary focus of our investigation.…”