# The two rings of (50000) Quaoar (Corrigendum)

C. L. Pereira, B. Sicardy, B. E. Morgado, F. Braga-Ribas, E.

Fernández-Valenzuela, D. Souami, B. J. Holler, R. C. Boufleur, G. Margoti,

M. Assafin, et al.

► To cite this version:

C. L. Pereira, B. Sicardy, B. E. Morgado, F. Braga-Ribas, E. Fernández-Valenzuela, et al. The two rings of (50000) Quaoar (Corrigendum). Astronomy & Astrophysics - A&A, 2024, 683, 10.1051/0004-6361/202346365e. insu-04726539

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LETTER TO THE EDITOR

# The two rings of (50000) Quaoar
(Corrigendum)

C. L. Pereira<fnref n="1" />,<fnref n="2" /> Ⓓ, B. Sicardy<fnref n="3" />, B. E. Morgado<fnref n="4" />,<fnref n="1" />,<fnref n="2" /> Ⓔ, F. Braga-Ribas<fnref n="5" />,<fnref n="1" />,<fnref n="2" /> Ⓔ, E. Fernández-Valenzuela<fnref n="6" />,<fnref n="7" />, D. Souami<fnref n="3" />,<fnref n="8" />,<fnref n="9" />,<fnref n="10" />,\* Ⓔ, B. J. Holler<fnref n="11" /> Ⓔ, R. C. Boufleur<fnref n="1" />,<fnref n="2" />, G. Margoti<fnref n="5" /> Ⓔ, M. Assafin<fnref n="4" />,<fnref n="2" /> Ⓔ, J. L. Ortiz<fnref n="7" /> Ⓔ, P. Santos-Sanz<fnref n="7" /> Ⓔ, B. Epinat<fnref n="12" />,<fnref n="13" /> Ⓔ, P. Kervella<fnref n="3" /> Ⓔ, J. Desmars<fnref n="14" />,<fnref n="15" /> Ⓔ, R. Vieira-Martins<fnref n="1" />,<fnref n="2" /> Ⓔ, Y. Kilic<fnref n="16" />,<fnref n="17" /> Ⓔ, A. R. Gomes-Júnior<fnref n="18" />,<fnref n="19" />,<fnref n="2" /> Ⓔ, J. I. B. Camargo<fnref n="1" />,<fnref n="2" /> Ⓔ, M. Emilio<fnref n="20" />,<fnref n="1" />,<fnref n="5" />, M. Vara-Lubiano<fnref n="7" />, M. Kretlow<fnref n="7" />,<fnref n="21" />,<fnref n="22" />, L. Albert<fnref n="23" />,<fnref n="24" /> Ⓔ, C. Alcock<fnref n="25" /> Ⓔ, J. G. Ball<fnref n="26" />, K. Bender<fnref n="27" />, M. W. Buie<fnref n="28" />, K. Butterfield<fnref n="29" />, M. Camarca<fnref n="30" />, J. H. Castro-Chacón<fnref n="31" /> Ⓔ, R. Dunford<fnref n="29" />, R. S. Fisher³² Ⓔ, D. Gamble<fnref n="29" />,³³, J. C. Geary<fnref n="25" />, C. L. Gnilka³⁴, K. D. Green³⁵ Ⓔ, Z. D. Hartman<fnref n="26" />, C-K. Huang³⁶ Ⓔ, H. Januszewski<fnref n="13" />, J. Johnston³⁷, M. Kagitani³⁸, R. Kamin<fnref n="29" />, J. J. Kavelaars³⁹ Ⓔ, J. M. Keller³⁷, K. R. de Kleer<fnref n="30" /> Ⓔ, M. J. Lehner³⁶ Ⓔ, A. Luken³², F. Marchis⁴⁰,⁴¹, T. Marlin<fnref n="30" />, K. McGregor⁴² Ⓔ, V. Nikitin<fnref n="29" />,⁴³, R. Nolthenius<fnref n="27" />, C. Patrick<fnref n="29" />, S. Redfield⁴² Ⓔ, A. W. Rengstorf⁴⁴ Ⓔ, M. Reyes-Ruiz<fnref n="31" /> Ⓔ, T. Seccull<fnref n="26" /> Ⓔ, M. F. Skrutskie⁴⁵, A. B. Smith<fnref n="26" />, M. Sproul⁴⁶ Ⓔ, A. W. Stephens<fnref n="26" /> Ⓔ, A. Szentgyorgyi<fnref n="25" /> Ⓔ, S. Sánchez-Sanjuán<fnref n="31" /> Ⓔ, E. Tatsumi⁴⁷,⁴⁸ Ⓔ, A. Verbiser⁴⁵ Ⓔ, S-Y. Wang³⁶, F. Yoshida⁴⁹,⁵⁰ Ⓔ, R. Young⁵¹, and Z-W. Zhang³⁶

<footnote n="1">¹ Observatório Nacional/MCTI, R. General José Cristino 77, CEP 20921-400 Rio de Janeiro, RJ, Brazil
e-mail: chrystianpereira@on.br</footnote>

<footnote n="2">² Laboratório Interinstitucional de e-Astronomia - LIneA, Rio de Janeiro, RJ, Brazil</footnote>

<footnote n="3">³ LESIA, Observatoire de Paris, Université PSL, Sorbonne Université, Université de Paris, CNRS, 92190 Meudon, France</footnote>

<footnote n="4">⁴ Universidade Federal do Rio de Janeiro – Observatório do Valongo, Ladeira Pedro Antônio 43, CEP 20.080-090 Rio de Janeiro, RJ, Brazil</footnote>

<footnote n="5">⁵ Federal University of Technology – Paraná (UTFPR-Curitiba), Rua Sete de Setembro, 3165, CEP 80230-901 Curitiba, PR, Brazil</footnote>

<footnote n="6">⁶ Florida Space Institute, UCF, 12354 Research Parkway, Partnership 1 building, Room 211, Orlado, USA</footnote>

<footnote n="7">⁷ Instituto de Astrofísica de Andalucía – Consejo Superior de Investigaciones Científicas, Glorieta de la Astronomía s/n, 18008 Granada, Spain</footnote>

<footnote n="8">⁸ Departments of Astronomy, and of Earth and Planetary Science, 501, Campbell Hall, University of California, Berkeley, CA 94720, USA</footnote>

<footnote n="9">⁹ naXys, Department of Mathematics, University of Namur, Rue de Bruxelles 61, 5000 Namur, Belgium</footnote>

<footnote n="10">¹⁰ Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, Bd de l'Observatoire, CS 34229, 06304 Nice Cedex 4, France</footnote>

<footnote n="11">¹¹ Space Telescope Science Institute, Baltimore, Maryland, USA</footnote>

<footnote n="12">¹² Aix Marseille Université, CNRS, CNES, LAM (Laboratoire d'Astrophysique de Marseille) UMR 7326, 13388 Marseille, France</footnote>

<footnote n="13">¹³ Canada-France-Hawaii Telescope, 65-1238 Mamalahoa Highway, Kamuela, HI 96743, USA</footnote>

<footnote n="14">¹⁴ Institut Polytechnique des Sciences Avancées IPSA, 63 boulevard de Brandebourg, 94200 Ivry-sur-Seine, France</footnote>

<footnote n="15">¹⁵ Institut de Mécanique Céleste et de Calcul des Éphémérides, IMCCE, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, Univ. Lille, France</footnote>

<footnote n="16">¹⁶ Akdeniz University, Faculty of Sciences, Department of Space Sciences and Technologies, 07058 Antalya, Turkey</footnote>

<footnote n="17">¹⁷ TÜBİTAK National Observatory, Akdeniz University Campus, 07058 Antalya, Turkey</footnote>

<footnote n="18">¹⁸ Institute of Physics, Federal University of Uberlândia, Uberlândia-MG, Brazil</footnote>

<footnote n="19">¹⁹ UNESP – São Paulo State University, Grupo de Dinâmica Orbital e Planetologia, CEP 12516-410 Guaratinguetá, SP, Brazil</footnote>

<footnote n="20">²⁰ Universidade Estadual de Ponta Grossa, O.A. – DEGEO, Ponta Grossa (PR), Brazil</footnote>

<footnote n="21">²¹ Internationale Amateursternwarte (IAS) e. V., Mittelstr. 6, 15749 Mittenwalde, Germany</footnote>

<footnote n="22">²² International Occultation Timing Association – European Section (IOTA/ES), Am Brombeerhag 13, 30459 Hannover, Germany</footnote>

<footnote n="23">²³ Département de Physique et Observatoire du Mont-Mégantic, Université de Montréal, C.P. 6128, Succ. Centre-ville, Montréal H3C 3J7, Québec, Canada</footnote>

<footnote n="24">²⁴ Institut Trottier de Recherche sur les exoplanètes, Université de Montréal, Canada</footnote>

<footnote n="25">²⁵ Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA</footnote>

<footnote n="26">²⁶ Gemini Observatory/NSF's NOIRLab, Hilo, Hawaii, USA</footnote>

<footnote n="27">²⁷ Cabrillo College Astronomy, Aptos, CA, USA</footnote>

<footnote n="28">²⁸ Southwest Research Institute, 1050 Walnut Street, Suite 300, Boulder, CO 80302, USA</footnote>

<footnote n="29">²⁹ International Occultation Timing Association (IOTA), Topeka, KS, USA</footnote>

<footnote n="30">³⁰ California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, USA</footnote>

<footnote n="31">³¹ Universidad Nacional Autónoma de México, Instituto de Astronomía, AP 106, Ensenada 22800, BC, Mexico</footnote>

<footnote>* Fulbright Visiting Scholar (2022–2023) at University of California, Berkeley.</footnote>
32 University of Oregon, Eugene, OR, USA

33 CanCON - Canadian Collaborative Occultation Network, Canada

34 NASA Ames Research Center, Moffett Field, California, USA & NASA Exoplanet Science Institute, Caltech/IPAC, Mail Code 100-22, Pasadena, CA, USA

35 University of New Haven, Dept. of Mathematics and Physics, 300 Boston Post Road, West Haven, CT 06477, USA

36 Academia Sinica Institute of Astronomy and Astrophysics, 11F of AS/NTU Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan, ROC

37 University of Colorado, 2000 Colorado Avenue, Boulder, CO 80309, USA

38 Planetary Plasma and Atmospheric Research Center, Graduate School of Science, Tohoku University, 6-3 Aramaki Aza-Aoba, Aoba-ku, Sendai 980-8578, Japan

39 Department of Physics and Astronomy, University of Victoria, Building, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada

40 Unistellar, 5 allée Marcel Leclerc, bâtiment B, 13008 Marseille, France

41 SETI Institute, Carl Sagan Center, Suite 200, 339 Bernardo Avenue, Mountain View, CA 94043, USA

42 Astronomy Department and Van Vleck Observatory, Wesleyan University, Middletown, CT 06459, USA

43 Research and Education Collaborative Occultation Network, USA

44 Purdue University Northwest, Department of Chemistry and Physics, Hammond, IN, USA

45 University of Virginia, Department of Astronomy, PO Box 400325, Charlottesville, VA 22904, USA

46 Private Observatory, PA, USA

47 Instituto de Astrofísica de Canarias, University of La Laguna, Tenerife, Spain

48 Dept. Earth and Planetary Science, The University of Tokyo, Tokyo, Japan

49 School of Medicine Department of Basic Sciences University of Occupational and Environmental Health, 1-1 Iseigaoka, Yahata, Kitakyusyu 807-8555, Japan

50 Planetary Exploration Research Center, Chiba Institute of Technology, 2-17-1 Tsudanuma, Narashino, Chiba 275-0016, Japan

51 Naylor Observatory, Lewisberry, PA, USA

A&A, 673, L4 (2023), https://doi.org/10.1051/0004-6361/202346365

**Key words.** methods: data analysis – methods: observational – techniques: photometric – Kuiper belt objects: individual: Quaoar – planets and satellites: rings – errata, addenda

We identified an error in our opacity calculations for Quaoar's rings in the original paper.

We used the Cuzzi (1985) calculations to obtain the ring opacity, accounting for both the Fresnel diffraction caused by the ring and the Airy diffraction caused by individual particles. These calculations show that the actual ring optical depth is half the apparent optical depth observed during Earth-based stellar occultations, as detailed in the correct equations of our Appendix E. However, this correction was applied twice in our fitting code, resulting in optical depths underestimated by a factor of two in the text and Table E.1.

The corrected values of optical depths are listed in Table 1 below, with the associated corrections on equivalent widths and equivalent depths. This does not change the main conclusion of the original paper, but using the incorrect normal optical depths

presented previously would result in underestimated values of the particle filling factor and thus collision rate. Also, when ring photometry becomes available, it would underestimate the particle albedo.

Moreover, we mentioned an optical depth peaking at $\tau_N \sim 0.4$ for the dense part of ring Q1R. Previously, this value was obtained from the equivalent optical depth and the full width at half maximum (FWHM) of the fitted Lorentzian. Here we used the exact peak value to calculate the normal opacities, $p_N$, and optical depths, $\tau_N$, for the Q1R dense region.

## References

Cuzzi, J. N. 1985, Icarus, 63, 312
<table><caption>**Table 1.** Revised physical parameters of rings Q1R and Q2R.</caption><thead><tr><th>Ring</th><th>Detection</th><th>Mid-time<br>August 9th, 2022</th><th>r<br>(km)</th><th>W<sub>r</sub><br>(km)</th><th>p<sub>N</sub></th><th>τ<sub>N</sub></th><th>E<sub>p</sub><br>(km)</th><th>A<sub>τ</sub><br>(km)</th></tr></thead><tbody><tr><td rowspan="3">Q1R- ing</td><td>CFHT</td><td>06:31:15.8 (0.1)</td><td>3,994.5 (2.0)</td><td>105.6 (4.4)</td><td>0.0059 (0.0004)</td><td>0.0060 (0.0004)</td><td>0.6 (0.2)</td><td>0.6 (0.2)</td></tr><tr><td>Gemini (z')</td><td>06:31:15.72 (0.02)</td><td>3,995.5 (0.4)</td><td>76.5 (0.9)</td><td>0.0051 (0.0002)</td><td>0.0052 (0.0002)</td><td>0.4 (0.1)</td><td>0.4 (0.1)</td></tr><tr><td>Gemini (r')</td><td>06:31:15.69 (0.04)</td><td>3,995.9 (0.6)</td><td>85.5 (1.4)</td><td>0.0066 (0.0006)</td><td>0.0066 (0.0006)</td><td>0.6 (0.2)</td><td>0.6 (0.2)</td></tr><tr><td rowspan="5">Q1R- egr</td><td>CFHT<sup>(a)</sup></td><td>06:38:32.58 (0.07)</td><td>4,123.69 (0.05)</td><td>6.09 (0.11)</td><td>0.17 (0.02)</td><td>0.2 (0.1)</td><td>2.01 (0.03)</td><td>2.41 (0.03)</td></tr><tr><td>Gemini (z')<sup>(a)</sup></td><td>06:38:32.44 (0.04)</td><td>4,122.88 (0.02)</td><td>5.29 (0.06)</td><td>0.18 (0.02)</td><td>0.26 (0.06)</td><td>1.87 (0.02)</td><td>2.21 (0.02)</td></tr><tr><td>Gemini (r')<sup>(a)</sup></td><td>06:38:32.43 (0.04)</td><td>4,122.62 (0.04)</td><td>5.1 (0.1)</td><td>0.17 (0.03)</td><td>0.2 (0.1)</td><td>1.74 (0.03)</td><td>2.03 (0.03)</td></tr><tr><td>TUHO</td><td>06:38:37.4 (0.4)</td><td>4,113.6 (7.7)</td><td>78 (18)</td><td>0.06 (0.02)</td><td>0.07 (0.03)</td><td>4.6 (3.5)</td><td>5.2 (4.0)</td></tr><tr><td>TAOS II</td><td>06:34:55.7 (0.3)</td><td>4,131.4 (4.5)</td><td>35 (16)</td><td>0.09 (0.04)</td><td>0.12 (0.06)</td><td>3.3 (3.1)</td><td>4.0 (3.9)</td></tr><tr><td rowspan="3">Q2R- ing</td><td>CFHT</td><td>06:32:43.86 (0.09)</td><td>2,490.6 (1.5)</td><td>16.1 (3.3)</td><td>0.0045 (0.0008)</td><td>0.0045 (0.0008)</td><td>0.07 (0.04)</td><td>0.07 (0.04)</td></tr><tr><td>Gemini (z')</td><td>06:32:43.60 (0.04)</td><td>2,493.9 (0.6)</td><td>11.3 (1.1)</td><td>0.0065 (0.0009)</td><td>0.0065 (0.0009)</td><td>0.07 (0.04)</td><td>0.07 (0.04)</td></tr><tr><td>Gemini (r')</td><td>n.d.<sup>(b)</sup></td><td>n.d.<sup>(b)</sup></td><td>n.d.<sup>(b)</sup></td><td>n.d.<sup>(b)</sup></td><td>n.d.<sup>(b)</sup></td><td>n.d.<sup>(b)</sup></td><td>n.d.<sup>(b)</sup></td></tr><tr><td rowspan="3">Q2R- egr</td><td>CFHT</td><td>06:37:00.21 (0.02)</td><td>2,540.8 (0.3)</td><td>11.2 (0.4)</td><td>0.0090 (0.0009)</td><td>0.0092 (0.0009)</td><td>0.10 (0.04)</td><td>0.10 (0.04)</td></tr><tr><td>Gemini (z')</td><td>06:37:00.08 (0.02)</td><td>2,540.4 (0.3)</td><td>6.8 (0.8)</td><td>0.009 (0.001)</td><td>0.010 (0.001)</td><td>0.06 (0.03)</td><td>0.06 (0.03)</td></tr><tr><td>Gemini (r')</td><td>06:36:59.91 (0.08)</td><td>2,537.7 (1.3)</td><td>9.4 (2.9)</td><td>0.006 (0.002)</td><td>0.007 (0.002)</td><td>0.06 (0.05)</td><td>0.06 (0.05)</td></tr></tbody></table>

**Notes.** The normal opacity, $p_N$, and normal optical depth, $\tau_N$, were calculated from the ring opening angle, $B$, and position angle, $P$, on August 9, 2022, derived from the orientation of the body obtained from the 2018–2022 data. The other parameters are: mid-time in UT and the error bars in parentheses given in seconds; $r$, the radial distance from Quaoar’s center in kilometers; $W_r$, the radial width in kilometers; $E_p$ and $A_τ$, the equivalent width and equivalent depth in kilometers, respectively. The terms “ing” and “egr” stand for ingress and egress, respectively, and refer to the fact that the detection occurred before and after the occultation by Quaoar’s main body, respectively. The error bars in parentheses are at the $1\sigma$ level. <sup>(a)</sup>From Lorentzian fit. The width $W_r$ is defined as the FWHM of the $\tau_N$ profile. The $E_p$ and $A_τ$ values were obtained from respective integrals in the ring profile (see Sect. 3 of original file); <sup>(b)</sup>Not detected.