Draft version August 28, 2017 Typeset using LATEX twocolumn style in AASTeX61 WEIGHING URANUS' MOON CRESSIDA WITH THE η RING Robert O. Chancia,1 Matthew M. Hedman,1 and Richard G. French2 1Department of Physics, University of Idaho, Moscow ID 83844-0903 2Astronomy Department, Wellesley College, Wellesley MA 02481 ABSTRACT The η ring is one of the narrow rings of Uranus, consisting of a dense core that is 1-2 km wide and a diffuse outer sheet spanning about 40 km. Its dense core lies just exterior to the 3:2 Inner Lindblad Resonance of the small moon Cressida. We fit the η ring radius residuals and longitudes from a complete set of both ground-based and Voyager stellar and radio occultations of the Uranian rings spanning 1977-2002. We find variations in the radial position of the η ring that are likely generated by this resonance, and take the form of a 3-lobed structure rotating at an angular rate equal to the mean motion of the moon Cressida. The amplitude of these radial oscillations is 0:667±0:113 km, which is consistent with the expected shape due to the perturbations from Cressida. The magnitude of these variations provides the first measurement of the mass and density of the moon Cressida (m = 2:5 ± 0:4 × 1017 kg and ρ = 0:86 ± 0:16 g/cm3) or, indeed, any of Uranus' small inner moons. A better grasp of inner Uranian satellite masses will provide another clue to the composition, dynamical stability, and history of Uranus' tightly packed system of small moons. Keywords: planets and satellites: individual (Uranus, Cressida) | planets and satellites: rings arXiv:1708.07566v1 [astro-ph.EP] 24 Aug 2017 Corresponding author: Robert O. Chancia [email protected] 2 1. INTRODUCTION mean motion of the moon Cressida. We argue that this In March of 1977, Elliot et al.(1977), Millis et al. structure is a result of the η ring's close proximity to (1977), and Bhattacharyya & Kuppuswamy(1977) dis- Cressida's 3:2 inner Lindblad resonance (ILR). One of covered nine narrow rings around the planet Uranus by the maxima in the ring's radius aligns with Cressida, as measuring the light blocked by each ring before and af- expected for the stable ring structure located exterior to ter Uranus occulted the star SAO158687. Since then, the resonant radius. The measured radial amplitude of the Uranian rings have been studied extensively with this ring structure and its distance from the resonance ground based stellar occultations (Millis & Wasserman allow us to estimate Cressida's mass, and thus obtain 1978; Nicholson et al. 1978; Elliot et al. 1981b,a; Nichol- the first gravity-based mass measurement of any inner son et al. 1981; French et al. 1982; Sicardy et al. 1982; Uranian moon. Elliot et al. 1983, 1984; French et al. 1986b,a; Elliot et al. We have only been able to find three previous men- 1987; French et al. 1988, 1996). Occultations provide tions of the Cressida 3:2 ILR and its association with the very precise radial locations of the rings at different lon- η ring. Porco & Goldreich(1987) identified the most gitudes in their orbits around Uranus. French et al. relevant resonances in the Uranian ring-moon system (1988) found that the main rings of Uranus consist of and made a case for Cordelia and Ophelia shepherding six measurably eccentric rings (6, 5, 4, α, β, and ) and the outermost ring through torques generated by the three nearly circular rings (η, γ, and δ). In the past, Lindblad resonances located appropriately on the ring's measurements of the η ring's radius have not shown the inner and outer edges (Goldreich & Porco 1987). They ring to be anything but circular. The η ring also features also note single resonances that could be perturbing the a broad low optical depth sheet extending approximately γ and δ rings. Finally, they state: \The only isolated 40 km exterior to its narrow core (Elliot et al. 1983). first-order satellite resonances which fall near any of the During the Voyager 2 flyby of Uranus Smith et al. remaining rings are located interior to the η ring." Porco (1986) discovered ten new small inner moons, but no one & Goldreich(1987) list both the Cressida 3:2 and the has ever measured their masses or densities. Nine of the Cordelia 13:12 resonances, located at a = 47171:6 ± 0:3 moons orbit within a radial range of 20,000 km, mak- km and a = 47173:0 ± 0:3 km respectively. These reso- ing the group one of the most tightly packed systems nances fall 3−5 km interior to the η ring. They calculate of interacting satellites in our solar system. Lissauer the widths of both resonances to be ∼ 1 km and dismiss (1995) estimated the masses of the inner moons assum- the possibility that either resonances is perturbing the ing densities equal to that of the larger moon Miranda η ring. Murray & Thompson(1988) later marked the (Jacobson et al. 1992) and shapes estimated with pho- location of the Cressida 3:2 ILR in their figure display- tometry (Thomas et al. 1989), but stated that at least ing a radial scan of a high phase image of the Uranian some of Uranus' small inner moons are significantly less rings acquired by Voyager 2. Subsequently, Murray & massive than these estimates. The lifetime of this sys- Thompson(1990) noted that this resonance needs to be tem is highly sensitive to the masses of the individual re-examined using updated satellite parameters. At the satellites (French et al. 2015). In fact, prior to the knowl- time, with a smaller data set, there was no detection of edge of the even less stable moon Cupid (Showalter & either an m = 3 or an m = 13 mode in the η ring, nor Lissauer 2003; French & Showalter 2012), Duncan & Lis- any other modes due to resonances with known satel- sauer(1997) showed that Desdemona could collide with lites having observed effects on any of the other pre- either Cressida or Juliet within the next 4 − 100 million viously noted rings' edges (French et al. 1988). Thus, years, depending on the masses of the satellites involved. it was only sensible to dismiss these resonances, and it The discovery of the dusty ν and µ rings (Showalter & is reasonable that they have not been of interest since. Lissauer 2006), near the orbits of Portia/Rosalind and We are only able to make this discovery now because Mab respectively, hints at the possibility of an evolving we have a larger set of occultation data extending from inner ring-moon system dominated by accretion (Tis- 1977 through 2002. careno et al. 2013). Kumar et al.(2015) also argue that We present the data used in this analysis in Section anomalies in Mab's orbital motion may be explained by 2, and describe our ring particle streamline model and a ring-moon system that is undergoing re-accretion after our mode detection methods in Section 3. In Section 4, a recent catastrophic disruption. we report the parameters of our fit to the η ring and Here we investigate a complete set of Uranian η calculate the mass and density of Cressida. Finally in ring occultation observations spanning their discovery Section 5, we discuss potential implications for the dy- in 1977 to 2002. We find that the η ring's radii ex- namical stability of the tightly packed system of inner hibit a 3-lobed structure rotating around Uranus at the 3 Uranian moons and the possible composition of Cres- yond the ring plane. Stations on Earth detected the sida. diffracted signal and relative phase change, to later be reconstructed into high-resolution radial optical depth 2. OBSERVATIONAL DATA profiles after the removal of diffraction effects (Tyler et al. 1986; Gresh et al. 1989). Presently, ground based The observational data used for this analysis consist occultation opportunities are rare because Uranus has of 49 individual occultation observations of the η ring. passed out of the dense Milky Way background, dras- In the appendix, Table3 contains each occultation's ring tically reducing the density of appropriate background intercept time, inertial longitude, and mid-radius deter- stars. The rings are also no longer as open to our view mined using a simple square-well model for profile fit- from Earth as they were in the 1980s because the ap- ting, developed by Elliot et al.(1984) and used in later parent aspect of the ring plane as viewed from Earth orbit determinations of the Uranian rings (French et al. changes over time. 1986b,a, 1988, 1991). Of these 49 observations, 46 are Earth-based stellar occultations, two are Voyager 2 Ra- 3. RING PARTICLE STREAMLINE MODEL AND dio Science Subsystem (RSS) radio occultations, and one FITTING METHOD is a Voyager 2 Photopolarimeter Subsystem (PPS) stel- lar occultation. Several of the observations are ingress The procedure used here follows that of French et al. and egress pairs from the same occultation of Uranus (1986b, 1988, 1991) for the Uranian rings, more recently and its rings. employed by Hedman et al.(2010), Nicholson et al. For each Earth-based occultation, an instrument (2014a,b), and French et al.(2016b) for analyses of Sat- recorded the brightness of the background star as a urn's non-circular narrow rings, gaps, and edges. After function of time. As the Earth moves relative to Uranus taking account any inclination relative to the equatorial the rings can block the star's light, leaving each ring's plane, the majority of narrow rings are well-fit by simple mark as a sharp decrease in the recorded brightness of precessing Keplarian ellipses whose radii are described the star for some amount of time related to the width by: a(1 − e2) of the ring.