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# A dense ring of the trans-Neptunian object Quaoar outside its Roche limit

## Abstract

Planetary rings are observed not only around giant planets1, but also around small bodies such as the Centaur Chariklo2 and the dwarf planet Haumea3. Up to now, all known dense rings were located close enough to their parent bodies, being inside the Roche limit, where tidal forces prevent material with reasonable densities from aggregating into a satellite. Here we report observations of an inhomogeneous ring around the trans-Neptunian body (50000) Quaoar. This trans-Neptunian object has an estimated radius4 of 555 km and possesses a roughly 80-km satellite5 (Weywot) that orbits at 24 Quaoar radii6,7. The detected ring orbits at 7.4 radii from the central body, which is well outside Quaoar’s classical Roche limit, thus indicating that this limit does not always determine where ring material can survive. Our local collisional simulations show that elastic collisions, based on laboratory experiments8, can maintain a ring far away from the body. Moreover, Quaoar’s ring orbits close to the 1/3 spin–orbit resonance9 with Quaoar, a property shared by Chariklo’s2,10,11 and Haumea’s3 rings, suggesting that this resonance plays a key role in ring confinement for small bodies.

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## Data availability

The observational data that support this paper and other findings of this study are available at the Strasbourg astronomical Data Center (CDS).

## Code availability

This research made use of SORA, a python package for stellar occultations reduction and analysis, developed with the support of ERC Lucky Star and LIneA/Brazil, within the collaboration Rio–Paris–Granada teams. Instructions for downloading and installing SORA can be found on Python Package Index (https://pypi.org/project/sora-astro/), on GitHub (https://github.com/riogroup/SORA) and on its online documentation (https://sora.readthedocs.io/). The method and examples of earlier numerical simulations results have been summarized in ref. 45, anybody interested in using this code in collaborative mode can contact H.S. Other specific codes written especially for this project are available from the corresponding author upon request.

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## Acknowledgements

We dedicate this paper to the memory of our recently deceased friend and colleague, Tom Marsh, who was instrumental in the development of HiPERCAM. This work was carried out under the Lucky Star umbrella that agglomerates the efforts of the Paris, Granada and Rio teams, which is funded by the ERC under the European Community’s H2020 (ERC grant agreement no. 669416). We thank C. D. Murray for help to calculate the expansion of Weywot’s potential to sixth order in eccentricity. Part of the results were obtained using CHEOPS data. CHEOPS is an ESA mission in partnership with Switzerland with important contributions to the payload and the ground segment from Austria, Belgium, France, Germany, Hungary, Italy, Portugal, Spain, Sweden and the UK. The CHEOPS Consortium gratefully acknowledge the support received by all the agencies, offices, universities and industries involved. Their flexibility and willingness to explore new approaches were essential to the success of this mission. The design and construction of HiPERCAM was supported by the ERC under the European Union’s Seventh Framework Programme (FP/2007-2013) under ERC-2013-ADG grant agreement no. 340040 (HiPERCAM). HiPERCAM operations and V.S.D. are funded by the Science and Technology Facilities Council (grant no. ST/V000853/1). The GTC is installed at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, on the island of La Palma. This work has made use of data from the ESA mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). This study was financed in part by the National Institute of Science and Technology of the e-Universe project (INCT do e-Universo, CNPq grant no. 465376/2014-2). This study was financed in part by CAPES - Finance Code 001. The following authors acknowledge the respective (1) CNPq grants to B.E.M. no. 150612/2020-6; F.B.-R. no. 314772/2020-0; R.V.-M. no. 307368/2021-1; M.A. nos. 427700/2018-3, 310683/2017-3 and 473002/2013-2; and J.I.B.C. nos. 308150/2016-3 and 305917/2019-6. (2) CAPES/Cofecub grant to B.E.M. no. 394/2016-05. (3) FAPERJ grant no. M.A. E-26/111.488/2013. (4) FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) grants to A.R.G.-J. no. 2018/11239-8 and R.S. no. 2016/24561-0. (5) CAPES-PrInt Program grant to G.B.-R. no. 88887.310463/2018-00, mobility number 88887.571156/2020-00. (6) DFG (the German Research Foundation) grant to R.S. no. 446102036. P.S-S. and R.D. acknowledge financial support by the Spanish grant no. AYA-RTI2018-098657-J-I00 ‘LEO-SBNAF’ (MCIU/AEI/FEDER, UE). J.L.O., P.S-S., R.D. and N.M. acknowledge financial support from the State Agency for Research of the Spanish MCIU through the ‘Center of Excellence Severo Ochoa’ award for the Instituto de Astrofísica de Andalucía (grant no. SEV-2017-0709), and they also acknowledge the financial support by the Spanish grant nos. AYA-2017-84637-R and PID2020-112789GB-I00, and the Proyectos de Excelencia de la Junta de Andalucía grant nos. 2012-FQM1776 and PY20-01309. G.B-R. and I.P. acknowledge support from CHEOPS ASI-INAF agreement no. 2019-29-HH.0. A.B. was supported by the SNSA. A.C.-C. and T.G.W. acknowledge support from STFC consolidated grant nos. ST/R000824/1 and ST/V000861/1, and UK Space Agency grant no. ST/R003203/1. U.K. and R.B. acknowledge support by The OpenSTEM Laboratories, an initiative funded by the Higher Education Funding Council for England and the Wolfson Foundation. J.W. gratefully acknowledges financial support from the Heising-Simons Foundation, C. Masson and P. A. Gilman for Artemis, the first telescope of the SPECULOOS network situated in Tenerife, Spain. The ULiege’s contribution to SPECULOOS has received funding from the ERC under the European Union’s Seventh Framework Programme (FP/2007-2013) (grant agreement no. 336480/SPECULOOS), from the Balzan Prize and Francqui Foundations, from the Belgian Scientific Research Foundation (F.R.S.-FNRS; grant no. T.0109.20), from the University of Liege and from the ARC grant for Concerted Research Actions financed by the Wallonia-Brussels Federation. TRAPPIST is a project funded by the Belgian Fonds (National) de la Recherche Scientique (F.R.S.-FNRS) under grant no. PDR T.0120.21. TRAPPIST-North is a project funded by the University of Liege, in collaboration with the Cadi Ayyad University of Marrakech (Morocco). E.J. is FNRS Senior Research Associate. This research includes data from observers.

## Author information

Authors

### Contributions

B.E.M., B.S. and H.S. wrote the paper and made the figures, with substantial contributions from F.B.-R., C.L.P., T.S., R.S. and F.V. F.B.-R., B.S., J.D., J.L.O. and P.S.-S. organized the stellar occultation observational campaigns with the collaboration of W. Beisker and J.L. B.E.M., F.B.-R., C.L.P., M.A. and G.M. analysed the data and obtained the physical parameters presented in the paper, with inputs from B.S., J.L.O., P.S.-S. and R.V.-M. H.S. designed and performed the N-body simulations. B.S., T.S. and R.S. performed dynamical studies presented in this project. F.V. computed Weywot orbital solution used in this project. B.E.M., A.R.G.-J., R.C.B., G.B.-R., F.L.R. and M.A. are the main developers of data analysis software used in this project. R.V.-M., E.F.-V., G.B.-R., B.J.H., M.K., J.I.B.C., R.D. and D. Souami collaborated with the interpretations of the results. F.J. and J.P.T. observed Quaoar occultation on 2 September 2018. F.B.-R., J.L.O, P.S.-S., V.S.D., N.M., E.J., T.R.M., S.P.L., Z.B., R.B., A. Burdanov, M.F., M.G., U.K., D. Sebastian, C.S. and J.W. participated in the observation of Quaoar occultation on 5 June 2019. W. Benz, G.B., I.P., A. Brandeker, A.C.-C., H.G.F., N.H., G.O. and T.G.W. are members of ESA CHEOPS mission that detected Quaoar occultation on 11 June 2020, and the data analysis was done with substantial contributions from A.R.G.-J. J. Broughton, D.H., J. Bradshaw, R.L., D.G., W.H., S.K. and P.N. observed Quaoar occultation on 27 August 2021, where J. Broughton, R.L. and J. Bradshaw detected the dense arc of material within the Quaoar ring. The opportunity to review the results and comment on the manuscript was given to all authors.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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## Extended data figures and tables

### Extended Data Fig. 1 Multi-band light curve observed by HiPERCAM on 05 June 2019.

The flux observed in the gs, rs, is and zs bands (black points) and the models (red line) vs. time relative to the observer’s closest approach time (03:00:31.858 utc). The blue shaded regions are enlarged in the side panels. Only one common model (the one obtained for is) is plotted as no statistically significant differences are observed between filters (Table 1). Note the increase of the light curve quality with wavelength.

### Extended Data Fig. 2 Dynamical environment of Quaoar’s ring.

The inner ellipse is Quaoar’s Roche limit, assuming particles with bulk densities of ρ = 400 kg m−3, see Methods for details. The corotation radius corresponds to the synchronous orbit, where the orbital period of particles matches Quaoar’s rotation period. The blue and green zones delimit the location of the 1/3 Quaoar spin-orbit resonance (SOR) and the 6/1 Weywot mean motion resonance (MMR), respectively (Table 1). The width of each zone represents the 1-sigma uncertainties on the resonance locations, dominated by the uncertainty on Quaoar’s mass. The two outer black ellipses outline the two possible Quaoar ring solution of Table 1.

### Extended Data Fig. 3 Examples of Quaoar’s ring time evolution.

The run shown here uses an optical depth τ = 0.25 and the Model 4 for the ϵ(vn)-relation (Extended Data Fig. 4). The curves show the evolution of radial velocity dispersion, labeled according to the bulk density ρ of the particles in kg m−3. The thick black curves indicate that the system has formed a gravitational aggregate. The inserts (160 m × 160 m in size) show snapshots from the ρ = 6, 000 kg m−3 case around the time when accretion begins. The labels indicate the time in units of orbital periods and the impact frequency f in units of impacts/particle/orbital period. Our criterion for detecting accretion is that f is larger than twice the average pre-accretion value of f, which corresponds to 189 orbital periods. For small ρ the steady state cst is practically the same as in the non-gravitating case. The drop of cst with larger ρ’s follows from the pairwise pre-impact acceleration: increased impact speeds reduce the effective ϵ and thus allow for a lower cst (ref. 42).

### Extended Data Fig. 4 Various models for the collisional restitution coefficient.

Left: Model 1: Frost-covered ice spheres at temperature T = 210 K, $${\epsilon }({v}_{{\rm{n}}})={({v}_{{\rm{n}}}/{v}_{{\rm{c}}})}^{-0.234}$$, with vc = 0.0077 cm s−1 (ref. 25); Model 2: Frost-covered ice at T = 123 K, $${\epsilon }({v}_{{\rm{n}}})=0.48{{v}_{{\rm{n}}}}^{-0.20}$$ (ref. 8); Model 3: as Model 1 but with vc = 0.077 cm s−1 (ten times the value of Model 1); Model 4: particles of radius R = 20 cm with compacted frost at T = 123 K, $${\epsilon }({v}_{{\rm{n}}})=0.90\,\exp (-0.22{v}_{{\rm{n}}})+0.01{{v}_{{\rm{n}}}}^{-0.6}$$ (ref. 8). Right: Theoretical relation for the dependence of critical coefficient of restitution ϵcr on optical depth, required for the balance between dissipation and the viscous gain of energy due to local viscosity 43. In case of velocity-dependent elasticity, the system adjusts its impact velocities (via velocity dispersion) so that the effective mean ϵ corresponds to the ϵcr(τ). In case of constant ϵ < ϵcr the system flattens to a near-monolayer state with a minimum c ≈ 2 to 3Rn ≈ 0.01 cm s−1R/1 m (where n is the mean motion) determined by the non-local viscous gain associated with the finite size of the particles. If the constant ϵ > ϵcr, no thermal balance is possible and the system disperses via exponentially increasing c.

### Extended Data Fig. 5 The influence of optical depth and particle size.

In the upper panel, we show Model 4 simulations with τ = 0.25 and τ = 1.0. In both series of simulations particle size is R = 1 m. The same conventions as in Fig. 3 are used, but now the scale for c is linear, not logarithmic. For larger τ the steady-state velocity dispersion is reduced, and thus the condition cst/vesc 2 is achieved for smaller ρ. The bottom panel compares the three assumed particle radii (0.33 m, 1 m, and 3 m) on accretion, using a common optical depth of τ = 0.25 and assuming the velocity-dependent elasticity law of Model 2. Since cst is nearly independent of R, the critical density corresponding to cst/vesc ~ 2 scales roughly as ρcr 1/R2 since $${v}_{{\rm{esc}}}\propto \sqrt{R}$$ (eq. (9)). Note that if a constant ϵ 0.5 is assumed, the particle size has no effect on the accretion limit, since in this case cst scales linearly with R in a similar fashion to vesc

### Extended Data Fig. 6 Topology of the Quaoar 1/3 Spin-Orbit Resonance (SOR).

The bottom graph shows the maximum eccentricity $${e}_{\max }$$ reached by a particle starting on an initially circular orbit of semi-major axis a perturbed by the Quaoar 1/3 SOR resonance. This resonance is driven by a mass anomaly whose amplitude is quantified by the dimensionless parameter ϵ1/3. The exact resonance radius a1/3 is marked by the dashed vertical tick mark. The top plots show the phase portraits in the eccentricity vector space (X, Y) corresponding to particular values of a, with $$X=e\cos ({\varphi }_{1/3})$$, $$Y=e\sin ({\varphi }_{1/3})$$, where e is the orbital eccentricity and $${\varphi }_{1/3}=(-{\lambda }^{{\prime} }+3\lambda -2\varpi )/2$$ is the resonant critical angle, see Methods for details. In an interval of width W = (16ϵ1/3/3)a1/3 in semi-major axis centerd on the resonance, the origin of the phase portrait is an unstable hyperbolic point. Particles are then forced to reach a maximum eccentricity $${e}_{\max }=\sqrt{4/3}\sqrt{(W/2-\Delta a)/{a}_{1/3}}$$, where Δa = a − a1/3 is the initial distance of the particle to the resonance. The value of $${e}_{\max }$$ peaks at $${e}_{{\rm{peak}},1/3}=(8/3)\sqrt{| \,{{\epsilon }}_{1/3}\,| /3}$$ for Δa = − (8/9)ϵ1/3a1/3. Outside this interval, $${e}_{\max }=0$$. Units are arbitrary in all the plots.

## Supplementary information

### Supplementary Information

Supplementary Table 1 and Fig. 1.

### Supplementary Data

Source data for Supplementary Fig. 1.

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Morgado, B.E., Sicardy, B., Braga-Ribas, F. et al. A dense ring of the trans-Neptunian object Quaoar outside its Roche limit. Nature 614, 239–243 (2023). https://doi.org/10.1038/s41586-022-05629-6

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