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Orbital perturbations due to massive rings

We analytically work out the long-term orbital perturbations induced by a homogeneous circular ring of radius Rr and mass mr on the motion of a test particle in the cases (I): r > R_r and (II): r < R_r. In order to extend the validity of our analysis to the orbital configurations of, e.g., some proposed spacecraftbased mission for fundamental physics like LISA and ASTROD, of possible annuli around the supermassive black hole in Sgr A* coming from tidal disruptions of incoming gas clouds, and to the effect of artificial space debris belts around the Earth, we do not restrict ourselves to the case in which the ring and the orbit of the perturbed particle lie just in the same plane. From the corrections to the standard secular perihelion precessions, recently determined by a team of astronomers for some planets of the Solar System, we infer upper bounds on mr for various putative and known annular matter distributions of natural origin (close circumsolar ring with R_r = 0.02-0.13 au, dust ring with R_r = 1 au, minor asteroids, Trans-Neptunian Objects). We find m_r <= 1.4 10^-4 m_E (circumsolar ring with R_r = 0.02 au), m_r <= 2.6 10^-6 m_E (circumsolar ring with R_r = 0.13 au), m_r <= 8.8 10^-7 m_E (ring with R_r = 1 au), m_r <= 7.3 10^-12 M_S (asteroidal ring with R_r = 2.80 au), m_r <= 1.1 <= 10^-11 M_S (asteroidal ring with R_r = 3.14 au), m_r <= 2.0 10^-8 M_S (TNOs ring with R_r = 43 au). In principle, our analysis is valid both for baryonic and non-baryonic Dark Matter distributions.