  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.
