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To characterize the system, we performed and retrieved the CHaracterising ExOPlanets Satellite (CHEOPS), TESS, and ground-based photometry, the High Accuracy Radial velocity Planet Searcher (HARPS) high-resolution spectroscopy, and Gemini speckle imaging. We characterize the host star and determine $T_{\\rm eff, \\star }=4734\\pm 67\\,\\mathrm{ K}$, $R_{\\star }=0.726\\pm 0.007\\, \\mathrm{ R}_{\\odot }$, and $M_{\\star }=0.748\\pm 0.032\\, \\mathrm{ M}_{\\odot }$. We present a novel detrending method based on point spread function shape-change modelling and demonstrate its suitability to correct flux variations in CHEOPS data. We confirm the planetary nature of both bodies and find that TOI-1064\u00a0b has an orbital period of Pb\u00a0= 6.44387 \u00b1 0.00003\u00a0d, a radius of Rb\u00a0= 2.59 \u00b1 0.04\u00a0R\u2295, and a mass of $M_{\\rm b} = 13.5_{-1.8}^{+1.7}$\u00a0M\u2295, whilst TOI-1064\u00a0c has an orbital period of $P_{\\rm c} = 12.22657^{+0.00005}_{-0.00004}$\u00a0d, a radius of Rc\u00a0= 2.65 \u00b1 0.04\u00a0R\u2295, and a 3\u03c3 upper mass limit of 8.5\u00a0M\u2295. From the high-precision photometry we obtain radius uncertainties of \u223c1.6\u00a0per\u00a0cent, allowing us to conduct internal structure and atmospheric escape modelling. TOI-1064\u00a0b is one of the densest, well-characterized sub-Neptunes, with a tenuous atmosphere that can be explained by the loss of a primordial envelope following migration through the protoplanetary disc. It is likely that TOI-1064\u00a0c has an extended atmosphere due to the tentative low density, however further radial velocities are needed to confirm this scenario and the similar radii, different masses nature of this system. The high-precision data and modelling of TOI-1064\u00a0b are important for planets in this region of mass\u2013radius space, and it allow us to identify a trend in bulk density\u2013stellar metallicity for massive sub-Neptunes that may hint at the formation of this population of planets.<\/jats:p>","DOI":"10.1093\/mnras\/stab3799","type":"journal-article","created":{"date-parts":[[2022,1,5]],"date-time":"2022-01-05T20:14:26Z","timestamp":1641413666000},"page":"1043-1071","source":"Crossref","is-referenced-by-count":55,"title":["A pair of sub-Neptunes transiting the bright K-dwarf TOI-1064 characterized with <i>CHEOPS<\/i>"],"prefix":"10.1093","volume":"511","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8749-1962","authenticated-orcid":false,"given":"Thomas G","family":"Wilson","sequence":"first","affiliation":[{"name":"Centre for Exoplanet Science, SUPA School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK"}]},{"given":"Elisa","family":"Goffo","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, Universita degli Studi di Torino, via Pietro Giuria 1, I-10125 Torino, Italy"},{"name":"Th\u00fcringer Landessternwarte Tautenburg, Sternwarte 5, D-07778 Tautenburg, Germany"}]},{"given":"Yann","family":"Alibert","sequence":"additional","affiliation":[{"name":"Physikalisches Institut, University of Bern, Gesellsschaftstrasse 6, CH-3012 Bern, Switzerland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8627-9628","authenticated-orcid":false,"given":"Davide","family":"Gandolfi","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, Universita degli Studi di Torino, via Pietro Giuria 1, I-10125 Torino, Italy"}]},{"given":"Andrea","family":"Bonfanti","sequence":"additional","affiliation":[{"name":"Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, A-8042 Graz, Austria"}]},{"given":"Carina M","family":"Persson","sequence":"additional","affiliation":[{"name":"Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, SE-43992 Onsala, Sweden"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8863-7828","authenticated-orcid":false,"given":"Andrew","family":"Collier\u00a0Cameron","sequence":"additional","affiliation":[{"name":"Centre for Exoplanet Science, SUPA School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0855-8426","authenticated-orcid":false,"given":"Malcolm","family":"Fridlund","sequence":"additional","affiliation":[{"name":"Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, SE-43992 Onsala, Sweden"},{"name":"Leiden Observatory, University of Leiden, PO Box 9513, NL-2300 RA Leiden, the Netherlands"}]},{"given":"Luca","family":"Fossati","sequence":"additional","affiliation":[{"name":"Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, A-8042 Graz, Austria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0076-6239","authenticated-orcid":false,"given":"Judith","family":"Korth","sequence":"additional","affiliation":[{"name":"Department of Space, Earth and Environment, Astronomy and Plasma Physics, Chalmers University of Technology, SE-41296 Gothenburg, Sweden"}]},{"given":"Willy","family":"Benz","sequence":"additional","affiliation":[{"name":"Physikalisches Institut, University of Bern, Gesellsschaftstrasse 6, CH-3012 Bern, Switzerland"},{"name":"Center for Space and Habitability, Gesellsschaftstrasse 6, CH-3012 Bern, Switzerland"}]},{"given":"Adrien","family":"Deline","sequence":"additional","affiliation":[{"name":"Observatoire Astronomique de l\u2019Universit\u00e9 de Gen\u00e8ve, Chemin Pegasi 51, CH-1290 Versoix, 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