Structure of the Uranian Rings: II. Ring orbits and widths

Abstract

We have used the square-well model of J. L. Elliot, R. G. French, K. J. Meech, and J. H. Elias (1984, Astron. J. 89, 1587–1603) to determine the midtimes, widths, and optical depths of all available Uranus ring occultation observations from 1977 to 1983. We have fitted an improved kinematical model for the nine known rings, from which we find: (i) J 2 = (3.3461 ± 0.0030) × 10 −3, J 4 = (−3.21 ± 0.37) × 10 −5, pole of the equatorial plane (1950.0) α = 5 h06 m25.6 s ± 4.6 s, δ =+15°01′56″ ± 2′09″; typical uncertainties in ring orbital elements are as follows: σ (semimajor axis) ⋍ 5 km, σ ( eccenticity) ⋍2×10 −5, σ ( inclination) ⋍0.005° ; (ii) typical post-fit residuals of 0.2–0.06 km in the ring plane radius for rings 6, 5, 4, α, β, η, and ε are comparable to the uncertainties in the midtimes for many data sets; (iii) upper limits to shepherd satellite masses of m < 10 18 g ( Δ/100km) 2 for shepherds near these seven rings, from observed radial perturbations, and m < 4 × 10 17 g ( Δ/100 km) 2 from anomalous ring precession rates (where Δ is the distance between the ring and the shepherd satellite); (iv) the γ and δ rings have rms radial perturbations ⋍ 3 km , well above the uncertainties in the occultation profile midtimes; (v) these perturbations appear to vary slowly with true anomaly, raising the possibility that they may be associated with Lindblad resonances. From the ring width determinations, we find: (vi) all nine rings have significant width perturbations, well above the uncertainties in the width measurements; (vii) typical width perturbations are 0.5–2.0 km, much larger than the orbit radius perturbations for all but the γ and δ rings; (viii) newly determined width-longitude relations, including apsidal twists, and dynamical models yield ring masses of (4.2 ± 0.9) × 10 16, (3.8 ± 0.6) × 10 16, and (6.1 ± 0.1) × 10 18 g, surface densities of 2.0 ± 0.4, 1.5 ± 0.2, and 32.9 ± 0.6 cm −2, and dispersion velocities of 0.15 ± 0.03, 0.17 ± 0.02, and (0.17 ± 0.25) cm sec −1 for the α, β, and ε rings, respectively, and a typical ring thickness of ⋍8 m , consistent with the results of P.D. Nicholson and K. Matthews (1983, Bull. Amer. Astron. Soc. 15, 816); (ix) there are significant deviations (>1 km) from the best-fitting sinusoidal width-longitude relations for the α, β, and ε rings; (x) the narrow eccentric rings 6, 5, and 4 do not follow simple width-longitude relations; (xi) the quasi-circular rings η, γ, and δ show large width variations (greater than a factor of 2), even for ring profiles at the same true anomaly obtained at different times. From optical depth measurements of the rings, we find: (xii) accurate width-optical depth products (“equivalent depths”), based on a square-well model, for profiles of all nine rings; (xiii) the rings are not optical monolayers-there is significant particle shadowing for all nine rings; (xiv) the “equivalent depths” of the γ and δ rings are not conserved, probably indicative of unresolved radial structure within the rings.