The K2-24 planetary system revisited by CHEOPS
V. Nascimbeni L. Borsato P. Leonardi S. G. Sousa T. G. Wilson A. Fortier A. Heitzmann G. Mantovan R. Luque T. Zingales G. Piotto Y. Alibert R. Alonso T. Bárczy D. Barrado Navascues S. C. Barros W. Baumjohann T. Beck W. Benz N. Billot F. Biondi A. Brandeker C. Broeg M. -D. Busch A. Collier Cameron A. C. M. Correia Sz. Csizmadia P. E. Cubillos M. B. Davies M. Deleuil A. Deline L. Delrez O. D. S. Demangeon B. -O. Demory A. Derekas B. Edwards D. Ehrenreich A. Erikson L. Fossati M. Fridlund D. Gandolfi K. Gazeas M. Gillon M. Güdel M. N. Günther Ch. Helling K. G. Isaak F. Kerschbaum L. Kiss J. Korth K. W. F. Lam J. Laskar A. Lecavelier des Etangs M. Lendl D. Magrin P. F. L. Maxted B. Merín C. Mordasini G. Olofsson R. Ottensamer I. Pagano E. Pallé G. Peter D. Pollacco D. Queloz R. Ragazzoni N. Rando H. Rauer I. Ribas N. C. Santos G. Scandariato D. Ségransan A. E. Simon A. M. S. Smith R. Southworth M. Stalport S. Sulis M. Gy. Szabó S. Udry B. Ulmer V. Van Grootel J. Venturini E. Villaver N. A. Walton
 

   
   V. Nascimbeni et al.
   A new dynamical modeling of K2-24




   

   
INAF, Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122 Padova, Italy Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy Instituto de Astrofisica e Ciencias do Espaco, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom Weltraumforschung und Planetologie, Physikalisches Institut, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland Center for Space and Habitability, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland Observatoire astronomique de l'Université de Genève, Chemin Pegasi 51, 1290 Versoix, Switzerland Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA Dipartimento di Fisica e Astronomia "Galileo Galilei", Università degli Studi di Padova, Vicolo dell'Osservatorio 3, 35122 Padova, Italy Instituto de Astrofísica de Canarias, Vía Láctea s/n, 38200 La Laguna, Tenerife, Spain Departamento de Astrofísica, Universidad de La Laguna, Astrofísico Francisco Sanchez s/n, 38206 La Laguna, Tenerife, Spain Admatis, 5. Kandó Kálmán Street, 3534 Miskolc, Hungary Depto. de Astrofísica, Centro de Astrobiología (CSIC-INTA), ESAC campus, 28692 Villanueva de la Cañada (Madrid), Spain Departamento de Fisica e Astronomia, Faculdade de Ciencias, Universidade do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, A-8042 Graz, Austria Max Planck Institute for Extraterrestrial Physics, Gießenbachstraße, 85748 Garching, Germany Department of Astronomy, Stockholm University, AlbaNova University Center, 10691 Stockholm, Sweden Physikalisches Institut, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland Centre for Exoplanet Science, SUPA School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK CFisUC, Departamento de Física, Universidade de Coimbra, 3004-516 Coimbra, Portugal Institute of Planetary Research, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany INAF, Osservatorio Astrofisico di Torino, Via Osservatorio, 20, I-10025 Pino Torinese To, Italy Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden Aix Marseille Univ, CNRS, CNES, LAM, 38 rue Frédéric Joliot-Curie, 13388 Marseille, France Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C, B-4000 Liège, Belgium Space sciences, Technologies and Astrophysics Research (STAR) Institute, Université de Liège, Allée du 6 Août 19C, 4000 Liège, Belgium Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium ELTE Gothard Astrophysical Observatory, 9700 Szombathely, Szent Imre h. u. 112, Hungary SRON Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA Leiden, Netherlands Centre Vie dans l’Univers, Faculté des sciences, Université de Genève, Quai Ernest-Ansermet 30, 1211 Genève 4, Switzerland Leiden Observatory, University of Leiden, PO Box 9513, 2300 RA Leiden, The Netherlands Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, 439 92 Onsala, Sweden Dipartimento di Fisica, Università degli Studi di Torino, via Pietro Giuria 1, I-10125, Torino, Italy National and Kapodistrian University of Athens, Department of Physics, University Campus, Zografos GR-157 84, Athens, Greece Department of Astrophysics, University of Vienna, Türkenschanzstrasse 17, 1180 Vienna, Austria European Space Agency (ESA), European Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands Institute for Theoretical Physics and Computational Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, 1121 Budapest, Konkoly Thege Miklós út 15-17, Hungary ELTE Eötvös Loránd University, Institute of Physics, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary Lund Observatory, Division of Astrophysics, Department of Physics, Lund University, Box 118, 22100 Lund, Sweden IMCCE, UMR8028 CNRS, Observatoire de Paris, PSL Univ., Sorbonne Univ., 77 av. Denfert-Rochereau, 75014 Paris, France Institut d'astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie Curie, 98bis blvd. Arago, 75014 Paris, France Astrophysics Group, Lennard Jones Building, Keele University, Staffordshire, ST5 5BG, United Kingdom European Space Agency, ESA - European Space Astronomy Centre, Camino Bajo del Castillo s/n, 28692 Villanueva de la Cañada, Madrid, Spain INAF, Osservatorio Astrofisico di Catania, Via S. Sofia 78, 95123 Catania, Italy Institute of Optical Sensor Systems, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany ETH Zurich, Department of Physics, Wolfgang-Pauli-Strasse 2, CH-8093 Zurich, Switzerland Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK Institut fuer Geologische Wissenschaften, Freie Universitaet Berlin, Maltheserstrasse 74-100,12249 Berlin, Germany Institut de Ciencies de l'Espai (ICE, CSIC), Campus UAB, Can Magrans s/n, 08193 Bellaterra, Spain Institut d'Estudis Espacials de Catalunya (IEEC), 08860 Castelldefels (Barcelona), Spain ESOC, European Space Agency, Robert-Bosch-Str. 5, 64293 Darmstadt, Germany HUN-REN-ELTE Exoplanet Research Group, Szent Imre h. u. 112., Szombathely, H-9700, Hungary Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, United Kingdom Dipartimento di Fisica,  Università di Trento, Via Sommarive 14, 38123 Povo (TN), Italy

   

 
  K2-24 is a planetary system composed of two transiting low-density Neptunians locked in an almost perfect 2:1 resonance and showing large TTVs, i.e., an excellent laboratory to search for signatures of planetary migration. Previous studies performed with K2, Spitzer and RV data tentatively claimed a significant non-zero eccentricity for one or both planets, possibly high enough to challenge the scenario of pure disk migration through resonant capture. With 13 new CHEOPS light curves (seven of planet -b, six of planet -c), we carried out a global photometric and dynamical (RV+TTV) re-analysis by including all the available literature data as well. We got the most accurate set of planetary parameters to date for the K2-24 system, including radii and masses at 1% and 5% precision (now essentially limited by the uncertainty on stellar parameters) and non-zero eccentricities e_b=0.0498_-0.0018^+0.0011, e_c=0.0282_-0.0007^+0.0003 detected at very high significance for both planets. Such relatively large values imply the need for an additional physical mechanism of eccentricity excitation during or after the migration stage. Also, while the accuracy of the previous TTV model had drifted by up to 0.5 days at the current time, we constrained the orbital solution firmly enough to predict the forthcoming transits for the next ∼15 years, thus enabling an efficient follow-up with top-level facilities such as JWST or ESPRESSO.
  

   

   The K2-24 planetary system revisited by CHEOPSThis article uses data from  the CHEOPS program . The individual data sets are listed in Table <ref>.
    
V. Nascimbenisend offprint requests to: 1 ^https://orcid.org/0000-0001-9770-1214
    < g r a p h i c s >L. Borsato1 ^https://orcid.org/0000-0003-0066-9268
    < g r a p h i c s >P. Leonardi2,1,55 ^https://orcid.org/0000-0001-6026-9202
    < g r a p h i c s >S. G. Sousa3 ^https://orcid.org/0000-0001-9047-2965
    < g r a p h i c s >T. G. Wilson4 ^https://orcid.org/0000-0001-8749-1962
    < g r a p h i c s >A. Fortier5,6 ^https://orcid.org/0000-0001-8450-3374
    < g r a p h i c s >A. Heitzmann7 ^https://orcid.org/0000-0002-8091-7526
    < g r a p h i c s >G. Mantovan2,1R. Luque8T. Zingales2,1 ^https://orcid.org/0000-0001-6880-5356
    < g r a p h i c s >G. Piotto2,1 ^https://orcid.org/0000-0002-9937-6387
    < g r a p h i c s >Y. Alibert6,5 ^https://orcid.org/0000-0002-4644-8818
    < g r a p h i c s >R. Alonso10,11 ^https://orcid.org/0000-0001-8462-8126
    < g r a p h i c s >T. Bárczy12 ^https://orcid.org/0000-0002-7822-4413
    < g r a p h i c s >D. Barrado Navascues13 ^https://orcid.org/0000-0002-5971-9242
    < g r a p h i c s >S. C. C. Barros3,14 ^https://orcid.org/0000-0003-2434-3625
    < g r a p h i c s >W. Baumjohann15 ^https://orcid.org/0000-0001-6271-0110
    < g r a p h i c s >T. Beck5W. Benz5,6 ^https://orcid.org/0000-0001-7896-6479
    < g r a p h i c s >N. Billot7 ^https://orcid.org/0000-0003-3429-3836
    < g r a p h i c s >F. Biondi16,1A. Brandeker17 ^https://orcid.org/0000-0002-7201-7536
    < g r a p h i c s >C. Broeg5,6 ^https://orcid.org/0000-0001-5132-2614
    < g r a p h i c s >M.-D. Busch18A. Collier Cameron19 ^https://orcid.org/0000-0002-8863-7828
    < g r a p h i c s >A. C. M. Correia20Sz. Csizmadia21 ^https://orcid.org/0000-0001-6803-9698
    < g r a p h i c s >P. E. Cubillos22,15M. B. Davies23 ^https://orcid.org/0000-0001-6080-1190
    < g r a p h i c s >M. Deleuil24 ^https://orcid.org/0000-0001-6036-0225
    < g r a p h i c s >A. Deline7L. Delrez25,26,27 ^https://orcid.org/0000-0001-6108-4808
    < g r a p h i c s >O. D. S. Demangeon3,14 ^https://orcid.org/0000-0001-7918-0355
    < g r a p h i c s >B.-O. Demory6,5 ^https://orcid.org/0000-0002-9355-5165
    < g r a p h i c s >A. Derekas28B. Edwards29D. Ehrenreich7,30 ^https://orcid.org/0000-0001-9704-5405
    < g r a p h i c s >A. Erikson21L. Fossati15 ^https://orcid.org/0000-0003-4426-9530
    < g r a p h i c s >M. Fridlund31,32 ^https://orcid.org/0000-0002-0855-8426
    < g r a p h i c s >D. Gandolfi33 ^https://orcid.org/0000-0001-8627-9628
    < g r a p h i c s >K. Gazeas34 ^https://orcid.org/0000-0002-8855-3923
    < g r a p h i c s >M. Gillon25 ^https://orcid.org/0000-0003-1462-7739
    < g r a p h i c s >M. Güdel35M. N. Günther36 ^https://orcid.org/0000-0002-3164-9086
    < g r a p h i c s >Ch. Helling15,37K. G. Isaak36 ^https://orcid.org/0000-0001-8585-1717
    < g r a p h i c s >F. Kerschbaum35L. L. Kiss38,39J. Korth40 ^https://orcid.org/0000-0002-0076-6239
    < g r a p h i c s >K. W. F. Lam21 ^https://orcid.org/0000-0002-9910-6088
    < g r a p h i c s >J. Laskar41 ^https://orcid.org/0000-0003-2634-789X
    < g r a p h i c s >A. Lecavelier des Etangs42 ^https://orcid.org/0000-0002-5637-5253
    < g r a p h i c s >M. Lendl7 ^https://orcid.org/0000-0001-9699-1459
    < g r a p h i c s >D. Magrin1 ^https://orcid.org/0000-0003-0312-313X
    < g r a p h i c s >P. F. L. Maxted43 ^https://orcid.org/0000-0003-3794-1317
    < g r a p h i c s >B. Merín44 ^https://orcid.org/0000-0002-8555-3012
    < g r a p h i c s >C. Mordasini5,6G. Olofsson17 ^https://orcid.org/0000-0003-3747-7120
    < g r a p h i c s >R. Ottensamer35I. Pagano45 ^https://orcid.org/0000-0001-9573-4928
    < g r a p h i c s >E. Pallé10,11 ^https://orcid.org/0000-0003-0987-1593
    < g r a p h i c s >G. Peter46 ^https://orcid.org/0000-0001-6101-2513
    < g r a p h i c s >D. Pollacco4D. Queloz47,48 ^https://orcid.org/0000-0002-3012-0316
    < g r a p h i c s >R. Ragazzoni1,9 ^https://orcid.org/0000-0002-7697-5555
    < g r a p h i c s >N. Rando36H. Rauer21,49 ^https://orcid.org/0000-0002-6510-1828
    < g r a p h i c s >I. Ribas50,51 ^https://orcid.org/0000-0002-6689-0312
    < g r a p h i c s >N. C. Santos3,14 ^https://orcid.org/0000-0003-4422-2919
    < g r a p h i c s >G. Scandariato45 ^https://orcid.org/0000-0003-2029-0626
    < g r a p h i c s >D. Ségransan7 ^https://orcid.org/0000-0003-2355-8034
    < g r a p h i c s >A. E. Simon5,6 ^https://orcid.org/0000-0001-9773-2600
    < g r a p h i c s >A. M. S. Smith21 ^https://orcid.org/0000-0002-2386-4341
    < g r a p h i c s >R. Southworth52M. Stalport26,25S. Sulis24 ^https://orcid.org/0000-0001-8783-526X
    < g r a p h i c s >Gy. M. Szabó28,53 ^https://orcid.org/0000-0002-0606-7930
    < g r a p h i c s >S. Udry7 ^https://orcid.org/0000-0001-7576-6236
    < g r a p h i c s >B. Ulmer46V. Van Grootel26 ^https://orcid.org/0000-0003-2144-4316
    < g r a p h i c s >J. Venturini7 ^https://orcid.org/0000-0001-9527-2903
    < g r a p h i c s >E. Villaver10,11N. A. Walton54 ^https://orcid.org/0000-0003-3983-8778
    < g r a p h i c s >
          
    Submitted 23 May 2024 / Accepted 4 September 2024
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§ INTRODUCTION



The advent of space-based, high-precision photometry, inaugurated by CoRoT <cit.> and then continued by Kepler <cit.>, its second phase K2 <cit.>, and more recently by TESS <cit.> enabled not only the unexpected discovery of entirely new classes of exoplanets, but also the application of analysis techniques hitherto relegated to theory. Among these, one the most fruitful is a dynamical technique known as Transit Time Variations (TTVs; ), in which the gravitational perturbation between planets, and its time-variable effect on the measured orbital periods, is exploited to retrieve their orbital solution. In the case where two or more planets are transiting their host stars, TTVs are very effective at both confirming the planetary nature of candidates, and at measuring their masses without the need of (or in synergy with; ) radial velocity measurements (RVs).

Dynamical simulations show that the expected amplitude of TTVs in ordinary planetary systems is quite small, usually in the order of magnitude of seconds to minutes <cit.>. Close, therefore, or even below the detection limit imposed by photon noise and/or stellar activity <cit.>. A fairly interesting exception is represented by systems where planets are locked in mean-motion resonances (MMRs) or, more in general, close to commensurability[Being close to an integer ratio of orbital periods does not necessarily imply the system to be in a MMR from a dynamical point of view; see also our discussion in Section <ref>.], the orbital period ratio being close to an integer ratio. Low-order MMRs in the j+1:j form (j∈ℕ), such as 2:1 or 3:2 can boost the TTV signal by orders of magnitude, reaching hours or even days in the most favorable configurations <cit.>. A famous case, and the first one to be investigated, is the Kepler-9 system <cit.>, a pair of transiting warm Saturn-sized planets orbiting their host in about 19.2 and 38.9 days, i. e., close to a 2:1 MMR. Resonant configurations are not merely a useful playground to exploit the TTV technique. Rather, they are also extremely interesting by themselves, since they represent a unique laboratory to test planetary formation and migration theories <cit.>. In particular, how resonances can be maintained during a disk-migration phase or form/change at a later stage is currently a very active area of debate ( and references therein).

The typical time scale of the orbital period modulation induced by the TTV (sometimes called the superperiod; P_TTV) needs to be fully mapped to avoid degeneracy in the dynamical retrieval, and can reach months or even years. In the <cit.> approximation, the superperiod can be estimated as a function of the orbital periods of the inner and outer planets:

    P_ttv =  | j+1/P_out-j/P_in |^-1 .

For most orbital configurations, P_TTV can be significantly longer than the average duration of a K2 campaign (∼ 70-80 days) or a TESS sector (∼ 27 days). Even if TESS, by design, is able to revisit a given target throughout additional sectors (according mostly to its ecliptic latitude), the sampling of the transit times can be very sparse, especially in the long-period regime (P≳ 30 d). This is particularly true for those systems discovered by K2, which lie close to the ecliptic and are therefore rarely monitored by TESS, if at all. 

On the other hand, the ESA S-class mission CHEOPS <cit.>, launched in 2019, is very efficient at observing at low ecliptic latitudes and, being a single-target telescope, has the ability to gather even very long-period transits once their ephemeris is reasonably constrained. CHEOPS has been successfully exploited several times to follow-up systems discovered by K2, sometimes with a particular focus on TTV analysis. WASP-47 is one of such systems <cit.>, for which our analysis led to an improvement of the orbital/physical parameters and, in particular, of the density of planet -d. We chose K2-24 as the next system to explore for the science case mentioned above, within the CHEOPS GTO program.

K2-24, announced by <cit.> (hereafter P16), is a planetary system made by two (sub-)Saturn-sized (R_b≃ 5.6 R_⊕, R_c≃ 8 R_⊕) planets close to a 2:1 period ratio, with orbital periods of P_b≃ 21 and P_c≃ 42 days. Since the baseline of the K2 light curve was not long enough to detect TTVs, the discovery paper had to rely on the HIRES RVs alone to constrain the planetary masses, which turned out to be in the Neptunian range (M_b= 21.0± 5.4 M_⊕, M_c= 27± 7 M_⊕), hence making K2-24b and -c extremely inflated planets with unusually large H/He envelopes predicted by models. A further PSF/HARPS RV follow-up by <cit.> essentially confirmed the mass estimates at M_b= 20± 4 M_⊕ and M_c= 26± 6 M_⊕. The only follow-up transits published so far (two of -b and two of -c),  observed by Spitzer in 2015-2016, were presented by <cit.> (hereafter P18), who also merged all the existing photometric and spectroscopic data and carried out the first TTV analysis of this system, revealing even smaller masses (M_b= 19± 2 M_⊕, M_c= 15± 2 M_⊕) and tentatively detecting the presence of an outer 54±14 M_⊕ companion at ∼ 1.1 au. 

The TTV modeling by , performed through the analytic approach developed by <cit.> and based on transit data covering only ∼40% of P_TTV, did not yield a precise measurement for both eccentricities e_b and e_c. Rather, it concluded that at least one planet must have an eccentricity significantly larger than zero, adopting e_b=0.06± 0.01 and e_c<0.07 (at 90% confidence) based on dynamical stability constraints and an imposed prior derived from the distribution of ⟨ e ⟩ observed in Kepler multi-planet systems. <cit.> later presented a more detailed analysis of dynamical stability in K2-24 based on the planetary parameters by , concluding that MMR locking protects its long-term evolution, and tighter constraining the eccentricity of the outer planet to e_c<0.05.

<cit.> further investigated the dynamical architecture of K2-24 and its implications for its formation and migration history, concluding that a pure disc-induced migration is not able to reproduce the period ratio and the TTV amplitude observed, and would result in much smaller eccentricities, by a factor of ∼ 30. Rather, they proposed a two-stage scenario where the two planets are first captured in resonance at low eccentricities within the disk, then eccentricities are excited by an outer companion (such that hinted by RV observations) during the disk dispersal phase. The same authors also suggested that the actual value of e_b and e_c may be higher than the  estimate, according to their simulations.



Only for a handful of transiting planetary systems, we know accurate eccentricities for planets in or close to low-order MMRs. From the latest version of the NASA Exoplanet Archive (; v. 2023-12-28), only seven[An additional eighth system is TIC279401253 <cit.>, a 2:1 pair of giants (P_b≃ 77 d, P_c≃ 155 d) of which the outer one is not transiting and detected through RVs.] pairs of resonant[All the listed pairs lie in or close the 3:2 MMR (K2-146, Kepler-223, K2-19), or the 2:1 MMR (all the others).]  planets can be found with both eccentricities constrained at better than 3σ (Fig. <ref>). Sorted by increasing average orbital period, they are: K2-146 b/c <cit.>, Kepler-223 b/c <cit.>, K2-19 b/c <cit.>, KOI-142=Kepler-88 b/c <cit.>, Kepler-9 b/c <cit.>, TOI-2525 b/c <cit.>, and Kepler-30 b/c <cit.>. Among these, only K2-146 and Kepler-223 lie in the Neptunian mass regime, but are quite close-in at P<10 d, where tidal interactions with the host star start to become significant <cit.>. From this point of view, the K2-24 system offers us the rare opportunity to probe the “primordial” eccentricities of a pair of warm (T_eq<800 K) Neptunians, i.e., not affected by tidal effects. 



The aim of this paper is to present and analyze thirteen new CHEOPS observations of K2-24-b/c, and to merge them with all the existing literature light curves and radial velocities (RVs) to derive an updated and consistent dynamical solution,  able to 1) recover the transit ephemeris for any future follow-up, 2) improve the measurement of planetary masses, radii, and densities, with a particular focus on the implications for their inner structure, and 3) firmly detect eccentricities for both planets. We present the new observations together with the employed archival data in Section <ref>. We describe the photometric modeling of the light curves in Section <ref>, then the global TTV+RV dynamical analysis in Section <ref>. Finally, the results are compared and interpreted in Section <ref>, where prospects for the future characterization of this system are discussed.






§ OBSERVATIONS








We collected all the available photometric and spectroscopic data of K2-24 for our analysis; they are described in the following subsections.  The very long orbital periods of K2-24b and -c, together with their long duration, uncertain ephemeris (See Section <ref>) and small transit depths (2-4 mmag) make the ground-based follow-up of this system extremely difficult. Indeed, no ground-based light curves have been published so far. It is also worth mentioning that K2-24 has never been observed by TESS in its first six observing cycles (2018-2024).

All the time stamps of the photometric and spectroscopic data described below were uniformly converted to the BJD-TDB standard and referred to the mid-exposure instant, following the prescription by <cit.>.



 §.§ K2 and Spitzer photometry


K2-24 has been observed by K2 once, in Campaign 2, from 2014-08-23 to 2014-11-13. This uninterrupted, ∼75 day-long light curve contains four transits of planet -b and two transits of planet -c (plotted with green points in Fig. <ref>), and led to their discovery published by . This light curve has been corrected for systematic errors due to the spacecraft jitter and drifting by following the approach developed by <cit.>.

Four more transits were secured by Spitzer and presented by : two of planet -b on 2015-10-27 and 2016-06-13, and two of planet -c on 2015-11-12 and 2016-06-10. Both the latter time series actually cover partial transits: the scheduling was based on a simple linear ephemeris since a more sophisticated TTV model was not available yet at that time. All the Spitzer light curves have been corrected for systematics through the pixel-level decorrelation algorithm (PLD; ) as modified by <cit.>.



 §.§ HST photometry

We downloaded the publicly available HST WFC3 G141 observations of K2-24b from the MAST archive. These data cover a single transit gathered on 2016-07-04 as part of proposal GO-14455 (PI: E. Petigura, plotted with orange points in Fig. <ref>) to extract a transmission spectrum of the planetary atmosphere. These observations have been previously analyzed by <cit.>; we will get back on their results in Section <ref>.
The visit consists of a total of eight HST orbits; in our analysis we excluded the first one due to the presence of significant time-dependent systematic errors. At the beginning of each orbit, a direct image captured with the F130N filter was used for wavelength calibration. These data were collected with the GRISM256 aperture and the SPARS10 reading sequence. The total exposure time was set to 103.13 s, with 16 up-the-ramp reads for exposure. Both scanning directions were employed. 

We calibrated the raw WFC3 data and extracted the photometric information through the Iraclis dedicated pipeline (, , ). We extracted the detrended white-light curve (spectral range: 1.088 to 1.680 μm; plotted with orange points in Fig. <ref>) from the calibrated images, taking into account the tilted configuration of the WFC3/NIR detector and modeling the time-dependent systematics using the Eq. 1 of <cit.>. The HST WFC3 time series are often affected by linear long-term and exponential short-term (“orbit ramps”) trend, especially when observing bright sources. We note that both ingress and egress are missed from this visit, implying that the transit time T_0 is expected to be relatively poorly constrained.





 §.§ CHEOPS photometry


K2-24 was targeted by CHEOPS thirteen times over a span of about two years, within the GTO subprogram #25 (PID ), focused on the study of the mass-radius relation through the TTV analysis of resonant pairs of low-mass exoplanets. A complete log of the observations is reported in Table <ref>; the corresponding light curves, extracted by the CHEOPS DRP v14 pipeline <cit.>, are plotted in Fig. <ref> and labelled with matching IDs. The gaps located at regular time intervals are due to the avoidance angles of CHEOPS and to the SAA crossing events, Earth occultations and Earth stray light contamination during its 98.77-min low-earth orbit.







It is evident that, particularly for planet -c, some transits are partial ones: this is due to the transit predictions by  becoming more and more inaccurate as the time passed since the K2/Spitzer observations increased. The O-C (observed minus calculated) discrepancy is much larger than the prediction errors reported in Table 4 of , demonstrating that their dynamical solution had to be revised and improved in order to reliably predict the future transit times.



 §.§ Radial velocities


The merged data set we collected is made of 89 RV observations in total from three different instruments:


    
  * 63 RV points from HIRES, published by  (the first 32 being already analyzed by );
    
  * 16 RV points from PFS, presented by <cit.>;
    
  * 10 RV points from HARPS (PID: 095.C-0718), also presented by <cit.>. Two additional HARPS points can be found in the ESO archive (PID: 191.C-0873), but were not included in our analysis since they are affected by an RV offset introduced on 2015-06-03.


We emphasize that we are going to fit all these RV data simultaneously for the first time, as  did not include the PFS and HARPS data in their modeling. 





§ LIGHT CURVE MODELING



We performed a global modeling of our 13 CHEOPS, six K2 and one HST light curves by fitting simultaneously the signal of planet -b and -c on both data sets, with the [<https://github.com/LucaMalavolta/PyORBIT>] software version 10 <cit.>. After some preliminary tests, we decided not to incorporate the Spitzer light curves into our global fit, as their transit depths are clearly inconsistent with the K2/CHEOPS/HST data, and also with each other. This could be due to an imperfect correction of systematic errors, since the transits are partial ones and the detrending process works by extrapolation rather than interpolation. To avoid any bias on our retrieved planetary parameters, we took the transit times T_0 for our dynamical analysis (Section <ref>) from Table 1 of  instead. 


The K2/CHEOPS/HST transits were modeled with the Batman code <cit.> and parametrized as a function of the impact parameter b, the radius ratio R_p/R_⋆ and the scaled semi-major axis a/R_⋆. The transit model for K2 was super-sampled by a factor of 10 to account for the non-negligible length of the K2 exposure times (30-minute cadence). Each transit time T_0 (13 from CHEOPS, 6 from K2, 1 from HST) was treated as a free, independent parameter, so the orbital period P was fixed at its average value interpolated over our observations, i.e., P_b=20.8891 and P_c=42.3391 days. The limb darkening effect was modeled through a quadratic law, i.e., with two parameters called u_1 and u_2 for each instrument; internally, u_1 and u_2 were re-parametrized as q_1 and q_2 following the prescription by <cit.>, to minimize the correlation between the two parameters. 

The residual systematic errors present in the CHEOPS light curves f(t) were detrended as a linear combination of terms as a function of external parameters: the first and second-order derivative of the centroid offset in x and y pixel coordinates (df/dx, d^2f/dx^2, df/dy,  d^2f/dy^2), background level (df/db), photometric contamination factor  (df/dcontam), the first three harmonics of the spacecraft roll angle (in cosϕ and sinϕ) and a quadratic baseline  f_0 + df/dt + d^2f/dt^2. The roll angle term, as expected on data from a low-Earth orbit satellite, is always dominating. Since adding 15 free parameters for each CHEOPS data set would have implied 13× 15 = 195 free parameters in our global fit just for the detrending (i. e., prohibitively expensive in terms of computational time), we went through a two-stage approach as described in <cit.>. In a first pass each individual CHEOPS light curve was fitted with both a transit model and the detrending model; then the detrended light curves were fed into the final, global fitting. The latter had therefore 52 free parameters: six LD coefficients (u_1 and u_2 for CHEOPS, K2, HST), six planetary parameters (b, R_p/R_⋆, a/R_⋆ for -b and -c), 20 transit times and 20 jitter parameters (one for each light curve). 

We set uninformative priors on all our fitting parameters. The only exceptions are the six limb darkening parameters u_1, u_2 for CHEOPS, K2 and HST. We carried out two independent analyses: the first one with fully uninformative priors (“LD-free”), and a second one by centering the prior at the theoretical value computed by the  code <cit.> and increasing to 0.05 the associated Gaussian error, to accommodate for the well-known underestimation by models  (“LD-prior”). The input stellar parameters were computed by the CHEOPS Target Characterization (TS3) working group according to the procedure described by <cit.>, Section 3.2.1, and reported in Table <ref>. We set  to use  <cit.> to find a reasonable starting point in the parameter space (50 000 generations with a population size of 8× N_par, where N_par is the number of free parameters), then we initialized an MCMC optimization with  <cit.>, running for 500 000 steps and setting a thinning factor of 100. After discarding the first 50 000 steps as burn-in phase, convergence was checked by auto-correlation function analysis (ACF). 



All the final best-fit parameters of interest from the MCMC distributions are reported in Table <ref> for both the LD-free and LD-prior case; transit times are reported separately in Table <ref>. The corresponding corner plots for the transit shape parameters (i.e., excluding transit times, LD, and jitter parameters) are shown in Fig. <ref> and Fig. <ref>. The best-fit values of u_1 and u_2 are consistent between the LD-free and LD-prior case (Table <ref>), although of course LD-prior has smaller error bars; all the remaining parameters agree within 1 σ. We adopt the LD-prior solution throughout the following analysis. We plot the CHEOPS light curves of K2-24b and -c, folded on the best-fit individual T_0 and binned over 0.3-hour intervals in Fig. <ref>.






In the last column of Table <ref> we also compare our results with the literature, i. e., with  ( did not present a new set of independent planetary parameters, since it was based on priors from ). Overall there is a very good agreement. The planetary radii R_b and R_c, in particular, are consistent within 1 σ but our error bars are improved by an order of magnitude, i. e., from a relative error of ∼9% to ∼1%. The uncertainty is now limited by our current knowledge of the stellar radius σ(R_⋆)/R_⋆≃ 1% (Table <ref>).





§ DYNAMICAL MODELING



We carried out a dynamical modeling of the K2-24 system and its strong TTV signals by simultaneously fitting the three RV data sets available (see Section <ref>) and the transit times (T_0s) extracted with  (see Table <ref>) through the  code[<https://github.com/lucaborsato/trades>]
<cit.>.

We adopted a parameterisation similar to <cit.>, assuming a 3-planet[Candidate planet -d (at ≃ 1.1 au) is far enough to be dynamically decoupled from the inner pair, so in principle it should not impact the TTV signals. It has to be included into our dynamical modeling, though, because of its effect on RV,
as in .] model and fitting for the stellar mass M_⋆, planetary-to-star mass ratio M_p/M_⋆, periods P, mean longitude[The mean longitude is defined as λ=ℳ+Ω+ω, where ℳ is the mean anomaly, Ω the longitude of ascending node, ω the argument of pericenter.] λ of all planets, eccentricity e and argument of periastron passage ω in the form √(e)cosω and √(e)sinω for planet -b and -c (as specified by the indexes b and c). We also fitted a jitter term, in log_2-space, and an offset for each RV data set.

We fixed the following parameters: longitude of ascending node Ω to 180 for each planet, circular orbit of planet -d (eccentricity e_d=0 and argument of periastron ω_d=90), inclination, i, of planet b and c as in Table <ref> and to 90 for planet -d. All the parameters are defined at the reference time 
T_ref = 2 456 905 BJD_TDB. We defined parameter priors in the physical space and converted them into fitting space; all the priors used have been reported in Table <ref>.

First, we run  with  (100 different configurations for 150  000 generations) to find a suitable starting point. Then, we run  with 100 walkers for 1  000  000 steps and we apply a conservative thinning factor of 100.
As in <cit.>,  we used a combination of the differential evolution proposal <cit.> and the snooker differential evolution proposal <cit.>
as the sampler within . After checking chains convergence through Gelman-Rubin statistics <cit.>, 
Geweke criterion <cit.>, Auto-correlation Function (ACF), and visual inspections, we discarded as burn-in the first 50% of the steps.

From the posterior distributions we extracted the maximum-a-posteriori (MAP[By the term “maximum-a-posteriori” (MAP) we mean the set of parameters that maximize the log-probability of the posterior distributions. If all the priors were uninformative/uniform, then the MAP would correspond to the maximum likelihood estimation (MLE).])
as the best-fit parameters, and the uncertainties as the high density interval (HDI) at the 68.27%[
HDI at 68.27% is equivalent of the 16^th–84^th percentiles of a Gaussian distribution.].
The best-fit parameters from  and their uncertainties are reported in Table <ref>,
with a comparison with  for the parameters in common.
The TTV and RV models from the best-fit orbital solution by  are also plotted along with
the observed data points in Fig. <ref> and Fig. <ref>, respectively. 
The fit looks perfectly satisfactory with an overall reduced χ^2 (TTV+RV) of 1.33 with 94 degrees of freedom. The corresponding lnℒ and ln(probability) values are -107.242 and -107.623, respectively.









 §.§ Prediction of future transits


A useful application of our dynamical model is the prediction of future transit events for any follow-up opportunity, 
as for both planets -b and -c transits cannot be reliably scheduled according to a linear ephemeris. The combined impact of a poorly-constrained linear ephemeris with a set of orbital parameters determined over a relatively short time span has been discussed in <cit.> in the context of the scientific preparation of the Ariel mission <cit.>.

All the transits of planets -b and -c predicted by our best-fit dynamical model up to the year 2029 included
are reported along with the associated uncertainty in the Appendix, Table <ref>, <ref>, <ref>. The values are their associated uncertainties were calculated by integrating 100 orbital solutions randomly chosen from the  fit posterior, then computing the median and the 68.27% HDI interval at each transit epoch.






§ DISCUSSION AND CONCLUSIONS



In our work, we merged all the available space-based photometry of K2-24b and -c (including 13 unpublished CHEOPS light curves; Section <ref>) and derived improved stellar parameters for K2-24 (Table <ref>) to perform a global transit fit (Section <ref>), which yielded a homogeneous set of planetary parameters and transit times (Tables <ref>, <ref>). Then we fitted the latter together with all the available RVs (HIRES, PFS, HARPS) through an RV+TTV dynamical model (Section <ref>) to get a complete orbital solution for K2-24b and -c, and for candidate planet -d as well (Table <ref>). 



 §.§ Planetary parameters of K2-24b and K2-24c


All the derived parameters for planets -b and -c look statistically consistent, at least within 2 σ, with those published by , but mostly with much smaller error bars due to the increased signal-to-noise ratio of the combined data set, the improved stellar parameters, and to the much larger baseline of the observations. This is in particular true for the planetary radii (R_b/R_⊕=5.64± 0.06, R_c/R_⊕=7.93± 0.12) and masses (M_b/M_⊕=20.6_-0.3^+1.6, M_c/M_⊕=16.4_-0.2^+1.3) for which we reached a relative error of 1% and 4-5%, respectively. We confirm the unusually low density of the outer planet (ρ_c = 0.181_-0.009^+0.017 g cm^-3), implying a very large gaseous envelope, possibly larger than 50% and hence challenging a core-accretion scenario due to the onset of runaway accretion .

Several alternative scenarios has been proposed to explain the existence of such “super-puff” planets <cit.>, including light scattering from high-altitude photo-chemical hazes <cit.> or the presence of planetary rings on specific configurations <cit.>. These hypotheses, however, will require a JWST follow-up to be tested.

A particularly interesting variable to discuss is the orbital eccentricity, due to its important consequences on the planetary migration mechanisms. We measured a extremely significant non-zero eccentricity for both planets e_b=0.049_-0.002^+0.001, e_c=0.0282_-0.0007^+0.0003, confirming the findings by , who based on the <cit.> theory predicted that e_b and e_c cannot be both zero. It is worth noting that our best-fit values are perfectly compatible with their constraints, even though our analysis is based on uninformative priors only and does not adopt any assumption on the distribution of the eccentricity in the Kepler population. We also mention that, compared with the <cit.> prediction, we found the eccentricity of -c at the very limit they set (e_c< 0.05). 




 §.§ Dynamical stability



The K2-24 system hosts three planets (Table <ref>), including two Neptune-mass planets (M_b ≈ 20.6 M_⊕, M_c ≈ 16.4 M_⊕) in the vicinity of a 2:1 mean motion resonance (P_c / P_b ≈ 2.029).

Following a suggestion from the referee, we first checked the dynamical stability of our orbital solution by computing the Angular Momentum Deficit <cit.>  of the whole posterior distribution, then we explored the stability by evaluating the AMD-Hill criterion proposed in Eq. 26 of <cit.>. We found that the whole posterior is AMD-Hill stable. We also ran an N-body integration with the Mean Exponential Growth factor of Nearby Orbits <cit.> indicator through the rebound package with the  integrator <cit.>. We set a step-size equal to the 10% of the shorter period of the system and integrated for 10^5 years. We found that not only the MAP solution is stable (MEGNO = 2), but also 1000 samples, randomly selected from the posterior, are stable with MEGNO≃ 2.





 §.§ Are K2-24b and -c on a MMR configuration?



To get a wider view on the stability of the system, and also to assess whether the system is truly on a resonant configuration, we performed a dynamical analysis in a similar way as for other planetary systems <cit.>.
The system is integrated on a regular 2D mesh of initial conditions around the best fit, including planet -d (Table <ref>).
Each initial condition is integrated for 10^4 yr, using the symplectic integrator SABAC4 <cit.>, with a step size of 10^-3 yr and general relativity corrections.
Then, we perform a frequency analysis <cit.> of the mean longitude of the inner planet over two consecutive time intervals of 5000 yr, and determine the main frequency, n and n', respectively.
The stability is measured by Δ = |1-n'/n|, which estimates the chaotic diffusion of the orbits.
In Fig. <ref>, the results for planet -b (top panel) and planet -c (bottom panel) are reported in color: orange and red represent strongly chaotic unstable trajectories; yellow indicates the transition between stable and unstable regimes; green correspond to moderately chaotic trajectories, but stable on Gyr timescales; cyan and blue give extremely stable quasi-periodic orbits.
The best-fit solution obtained from our analysis (Table <ref>) is marked with a white circle.

We observe that the best-fit solution from Table <ref> is completely stable, even if we increase the eccentricities up to 0.4. However, for eccentricities up to 0.1, which include the current best fit determination (e_b ≈ 0.05, e_c ≈ 0.03), we observe that the system is outside the 2:1 mean motion resonance, which corresponds to the large stable structure above the V-shape chaotic region in the middle of the figures. 

The TTVs analysis constrains the resonant part of the architecture, as can be seen in Fig. <ref>. In that figure, we can see that the posterior of the fit lies outside the formal resonant domain (red area in the figure), unlike, for example, TOI-216, Kepler-1705 or Kepler-1972 <cit.>.
We conclude that the K2-24 three-planet orbital solution presented in Table <ref> is not in resonance, but still reliable and supple to the uncertainties in the determination of the eccentricities of the two innermost planets.

We also note that the system is on the correct side of the resonance predicted by planetary migration models <cit.>.
This feature is usually attributed to tidal interactions with the parent star <cit.>, but in this case this mechanism does not seem to be very efficient, because the orbital period of the inner Neptune-like planet is much longer than 5 days <cit.>.

Finally, from the best-fit solution we monitored the evolution over 10 000 years of some parameters of the inner pair, including the period ratio P_c/P_b, the difference between the arguments of the pericenter Δω, and the critical resonant angles ϕ_1, ϕ_2  (Fig. <ref>). Interestingly, ϕ_1 and ϕ_2 are circulating (as one would expect from a non-MMR configuration, thus confirming our previous finding), while Δω librates in an anti-aligned (180^∘) configuration. Following a more quantitative approach, we repeated the same analysis on 10 000 random samples from the posterior, to find that on 100% of them Δω is confined between ∼ 140^∘ and ∼ 220^∘ with a mean value perfectly centered on 180^∘, therefore confirming the anti-aligned scenario. 



 §.§ Candidate planet K2-24d

It is worth noting that we independently confirmed the RV signal of the planet candidate K2-24d, previously only tentatively detected by  on HIRES data alone. While its parameters look consistent, our circular fit now yields a 8-σ detection at M_d/M_⊕ = 54_-4^+9. The period ratio with respect to the inner planets is too large (and too far from a MMR) to generate a detectable TTV on the inner planets, hence all the constraint on M_d comes from RVs. At P_d≃ 470 d, the a-priori transit probability <cit.> of -d would be approximately just R_⋆/a ≃ 0.4%, yet the actual chances are much better than that since multiple planetary systems are very likely to be coplanar <cit.>. Unfortunately, the fraction of orbital phase currently mapped by K2, Spitzer and CHEOPS together (all of which are capable of detecting the transit of a ∼50 M_⊕ planet at high confidence) is only <20%, so no conclusion can be drawn about the orbital inclination of -d. We will keep on considering -d as a candidate rather than a confirmed planet since we did not run any specific validation test for it, being outside the main scope of this paper.



 §.§ Future prospects for follow-up

The K2-24 system looks like a very promising target for a follow-up with several current and future facilities. To this purpose, the list of predicted transit windows we listed in the Appendix (Tables <ref>, <ref>, <ref>) is crucial to reliably schedule the observations. The most obvious science case is a deeper study if its dynamical architecture, including the modeling of new transit timings which could unveil additional companions on orbits on external MMRs to planet -c[A planet internal to -b, i.e., at P≲ 20 d, is easily discarded by K2 photometry, if on transiting configurations. Even if we postulate a non-transiting geometry due to an unusually high mutual inclination, the currently available RVs would put an upper limit to its mass in the rocky planet regime.]. Both planets, and in particular -c, are also compelling targets for transmission spectroscopy, since their low bulk density combined with the brightness of their host star (V≃ 11.3, J≃ 9.6, K≃ 9.2) offers a unique opportunity to probe the atmospheres of a pair of warm sub-Saturns close to a MMR and to link their composition with their formation site and migration history <cit.>. If we compute the Transmission Spectroscopy Metric (TSM; ) based on our newly derived parameters on Tables <ref>, <ref> and <ref>, we get 62± 5 for -b and 177± 16 for -c (TSM scaled factor computer for >4 R_⊕ planets). We remind the reader that a value of 90 is usually considered the threshold to select the best targets amenable to atmospheric characterization with JWST <cit.>. As already mentioned (Section <ref>), HST, through WFC3/NIR, has been already exploited to search for atmospheric features on K2-24b, unfortunately with a null result. <cit.> noted, however, that the best-fit free chemistry model was preferred to a flat line at 2.5σ, suggesting the presence of NH_3, but without evidence for H_2O. 

TESS will observe K2-24, for the first time, in Sector 91 of Cycle 7, currently planned from 2025, April 9 to May 7. According to our modeling, only one event will be captured: a transit of K2-24b at 2025-04-25T21:15:40 UTC, unfortunately close to the mid-sector gap. It is difficult at this stage to predict whether or not TESS will manage to add new data to the TTV analysis.

The availability of a new, well-constrained ephemeris, on the other hand, opens an interesting opportunity for a ground-based follow-up campaign from the southern hemisphere. Both transit depths (approx. 2 000 and 4 000 ppm, respectively) are feasible with most medium-sized telescopes operating with the defocusing technique <cit.>, and even partial transits would provide us with reliable transit times and help mapping the TTV signal, as now the transit shape parameters of both planets (including duration) are constrained at high precision (Table <ref>).

As a closing note, we mention that, in the next years, both PLATO <cit.> and Ariel <cit.> could be able to follow-up K2-24. PLATO, to be launched in 2026, will unfortunately not observe this target during its LOP (long-pointing operation phase), since K2-24 lies too close to the Ecliptic to meet the engineering constraints; however, it could be monitored at a later stage for a shorter duration (2-3 months) during the so-called short-duration observing phase (SOP; ).

Ariel, on the other hand, will observe transits of K2-24b and -c in Tier 1 and 3, respectively <cit.>. A detailed study <cit.> demonstrated that the Ariel FGS light curves of K2-24 can be exploited also for accurate TTV analysis, and that ten transits would be enough to constrain the presence of an external resonant companion down to the rocky regime. 





We thank the anonymous referee for her/his valuable comments and suggestions.

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 United Kingdom. The CHEOPS Consortium would like to 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. CHEOPS data analysed in this article will be made available in the CHEOPS mission archive (<https://cheops.unige.ch/archive_browser/>). 
VNa, LBo, TZi, GPi, GMa, IPa, RRa, and GSc acknowledge support from CHEOPS ASI-INAF agreement n. 2019-29-HH.0. 
S.G.S. acknowledge support from FCT through FCT contract nr. CEECIND/00826/2018 and POPH/FSE (EC). 
The Portuguese team thanks the Portuguese Space Agency for the provision of financial support in the framework of the PRODEX Programme of the European Space Agency (ESA) under contract number 4000142255. 
TWi acknowledges support from the UKSA and the University of Warwick. 
YAl acknowledges support from the Swiss National Science Foundation (SNSF) under grant 200020_192038. 
We acknowledge financial support from the Agencia Estatal de Investigación of the Ministerio de Ciencia e Innovación MCIN/AEI/10.13039/501100011033 and the ERDF “A way of making Europe” through projects PID2019-107061GB-C61, PID2019-107061GB-C66, PID2021-125627OB-C31, and PID2021-125627OB-C32, from the Centre of Excellence “Severo Ochoa” award to the Instituto de Astrofísica de Canarias (CEX2019-000920-S), from the Centre of Excellence “María de Maeztu” award to the Institut de Ciències de l’Espai (CEX2020-001058-M), and from the Generalitat de Catalunya/CERCA programme. 
DBa, EPa, and IRi acknowledge financial support from the Agencia Estatal de Investigación of the Ministerio de Ciencia e Innovación MCIN/AEI/10.13039/501100011033 and the ERDF “A way of making Europe” through projects PID2019-107061GB-C61, PID2019-107061GB-C66, PID2021-125627OB-C31, and PID2021-125627OB-C32, from the Centre of Excellence “Severo Ochoa” award to the Instituto de Astrofísica de Canarias (CEX2019-000920-S), from the Centre of Excellence “María de Maeztu” award to the Institut de Ciències de l’Espai (CEX2020-001058-M), and from the Generalitat de Catalunya/CERCA programme. A.C.M.C. acknowledges support from the FCT, Portugal, through the CFisUC projects UIDB/04564/2020 and UIDP/04564/2020, with DOI identifiers 10.54499/UIDB/04564/2020 and 10.54499/UIDP/04564/2020, respectively. 
S.C.C.B. acknowledges support from FCT through FCT contracts nr. IF/01312/2014/CP1215/CT0004. 
0000-0003-0312-313X. 
A.C., A.D., B.E., K.G., and J.K. acknowledge their role as ESA-appointed CHEOPS Science Team Members.
ABr was supported by the SNSA. 
CBr and ASi acknowledge support from the Swiss Space Office through the ESA PRODEX program. 
ACC acknowledges support from STFC consolidated grant number ST/V000861/1, and UKSA grant number ST/X002217/1. 
P.E.C. is funded by the Austrian Science Fund (FWF) Erwin Schroedinger Fellowship, program J4595-N. 
This project was supported by the CNES. 
The Belgian participation to CHEOPS has been supported by the Belgian Federal Science Policy Office (BELSPO) in the framework of the PRODEX Program, and by the University of Liège through an ARC grant for Concerted Research Actions financed by the Wallonia-Brussels Federation. 
L.D. thanks the Belgian Federal Science Policy Office (BELSPO) for the provision of financial support in the framework of the PRODEX Programme of the European Space Agency (ESA) under contract number 4000142531. 
This work was supported by FCT - Fundação para a Ciência e a Tecnologia through national funds and by FEDER through COMPETE2020 through the research grants UIDB/04434/2020, UIDP/04434/2020, 2022.06962.PTDC. 
O.D.S.D. is supported in the form of work contract (DL 57/2016/CP1364/CT0004) funded by national funds through FCT. 
B.-O. D. acknowledges support from the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00046. 
This project has received funding from the Swiss National Science Foundation for project 200021_200726. It has also been carried out within the framework of the National Centre of Competence in Research PlanetS supported by the Swiss National Science Foundation under grant 51NF40_205606. The authors acknowledge the financial support of the SNSF. 
MF and CMP gratefully acknowledge the support of the Swedish National Space Agency (DNR 65/19, 174/18). 
DG gratefully acknowledges financial support from the CRT foundation under Grant No. 2018.2323 “Gaseous or rocky? Unveiling the nature of small worlds”. 
M.G. is an F.R.S.-FNRS Senior Research Associate. 
MNG is the ESA CHEOPS Project Scientist and Mission Representative, and as such also responsible for the Guest Observers (GO) Programme. MNG does not relay proprietary information between the GO and Guaranteed Time Observation (GTO) Programmes, and does not decide on the definition and target selection of the GTO Programme. 
CHe acknowledges support from the European Union H2020-MSCA-ITN-2019 under Grant Agreement no. 860470 (CHAMELEON). 
KGI is the ESA CHEOPS Project Scientist and is responsible for the ESA CHEOPS Guest Observers Programme. She does not participate in, or contribute to, the definition of the Guaranteed Time Programme of the CHEOPS mission through which observations described in this paper have been taken, nor to any aspect of target selection for the programme. 
K.W.F.L. was supported by Deutsche Forschungsgemeinschaft grants RA714/14-1 within the DFG Schwerpunkt SPP 1992, Exploring the Diversity of Extrasolar Planets. 
This work was granted access to the HPC resources of MesoPSL financed by the Region Ile de France and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the programme Investissements d'Avenir supervised by the Agence Nationale pour la Recherche. 
ML acknowledges support of the Swiss National Science Foundation under grant number PCEFP2_194576. 
PM acknowledges support from STFC research grant number ST/R000638/1. 
This work was also partially supported by a grant from the Simons Foundation (PI Queloz, grant number 327127). 
NCSa acknowledges funding by the European Union (ERC, FIERCE, 101052347). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them. 
GyMSz acknowledges the support of the Hungarian National Research, Development and Innovation Office (NKFIH) grant K-125015, a a PRODEX Experiment Agreement No. 4000137122, the Lendület LP2018-7/2021 grant of the Hungarian Academy of Science and the support of the city of Szombathely. 
V.V.G. is an F.R.S-FNRS Research Associate. 
JV acknowledges support from the Swiss National Science Foundation (SNSF) under grant PZ00P2_208945. 
NAW acknowledges UKSA grant ST/R004838/1.
Ple acknowledges that this publication was produced while attending the PhD program in Space Science and Technology at the University of Trento, Cycle XXXVIII, with the support of a scholarship co-financed by the Ministerial Decree no. 351 of 9th April 2022, based on the NRRP - funded by the European Union - NextGenerationEU - Mission 4 "Education and Research", Component 2 "From Research to Business", Investment 3.3 – CUP E63C22001340001.
E.V. acknowledges support from the ’DISCOBOLO’ project funded by the Spanish Ministerio de Ciencia, Innovación y Universidades under grant PID2021-127289NB-I00.



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§ ADDITIONAL PLOTS AND TABLES





















