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.

💡 Research Summary

The paper presents a comprehensive analytical treatment of the long‑term gravitational perturbations exerted by a homogeneous circular ring of radius Rₙ and mass mₙ on a test particle moving either outside (r > Rₙ) or inside (r < Rₙ) the ring. Unlike many previous studies that assumed coplanarity between the ring and the orbit, the authors retain the full three‑dimensional geometry, allowing the orbital plane to be inclined by an arbitrary angle i and to have an arbitrary longitude of ascending node Ω.

Starting from the Newtonian potential of a thin circular mass distribution, they expand the potential in Legendre polynomials, separating the cases r > Rₙ and r < Rₙ. The resulting series contain terms proportional to (Rₙ/r)³, (Rₙ/r)⁵, etc., for the external case, and to (r/Rₙ)³, (r/Rₙ)⁵, etc., for the internal case. By inserting this potential into the Lagrange planetary equations and averaging over one orbital period, they derive explicit expressions for the secular rates of the argument of periapsis ω and the mean anomaly 𝓜. The leading contribution to the periapsis precession is of order (Rₙ/r)³ (or its inverse for the internal case), i.e., a high‑order correction that becomes appreciable only when the test particle’s orbit lies relatively close to the ring.

To translate these theoretical results into observational constraints, the authors exploit the most recent determinations of anomalous perihelion precessions for several Solar‑System planets. These anomalous rates have already been corrected for all known Newtonian and relativistic effects (solar J₂, Lense–Thirring, planetary perturbations, etc.), leaving a residual that can be attributed to any additional, unmodelled mass distribution. By equating the residual precession to the ring‑induced term, they solve for an upper bound on mₙ for a given Rₙ.

Four representative ring configurations are examined:

1. Inner solar “dust” ring with Rₙ = 0.02 AU (the innermost plausible location) and Rₙ = 0.13 AU (the outer edge of the hypothesised circum‑solar dust belt). Using Mercury’s and Venus’s residual precessions, they obtain mₙ ≤ 1.4 × 10⁻⁴ M⊕ and mₙ ≤ 2.6 × 10⁻⁶ M⊕, respectively.

2. A 1 AU dust ring possibly associated with the Earth–Mars region. Mars’s residual precession yields mₙ ≤ 8.8 × 10⁻⁷ M⊕.

3. Asteroidal rings centred at the main‑belt distance, with radii 2.80 AU and 3.14 AU. The corresponding limits are mₙ ≤ 7.3 × 10⁻¹² M☉ and mₙ ≤ 1.1 × 10⁻¹¹ M☉, respectively, consistent with independent estimates of the total mass of the main asteroid belt.

4. Trans‑Neptunian Object (TNO) ring at Rₙ ≈ 43 AU. Using the outer‑planet residuals, they find mₙ ≤ 2.0 × 10⁻⁸ M☉, which is compatible with current mass estimates of the Kuiper‑belt population.

All limits are conservative, reflecting the current level of orbital determination (typically a few milliarcseconds per century). The analysis is deliberately model‑independent: it applies equally to ordinary baryonic matter (dust, rocks, ice) and to hypothetical non‑baryonic dark‑matter concentrations that might form a toroidal distribution around a massive body.

The authors also discuss prospective applications to planned high‑precision space‑based experiments such as LISA and ASTROD, where the spacecraft will follow heliocentric orbits with sub‑meter ranging accuracy. In such contexts, even a ring with mass far below the limits derived here could produce a measurable effect, offering a novel probe of faint mass structures. Moreover, the formalism can be adapted to artificial debris belts encircling Earth, a growing concern for low‑Earth‑orbit operations.

In summary, the paper delivers (i) a rigorous, fully three‑dimensional analytical framework for ring‑induced orbital precessions, (ii) a set of quantitative upper bounds on the mass of several plausible natural and artificial rings based on current planetary ephemerides, and (iii) a clear pathway for future missions to exploit orbital dynamics as a sensitive detector of both baryonic and exotic mass distributions.