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Sousa, S. G., Adibekyan, V., Delgado-Mena, E., et al. 2021, A&A, 656, A53

Speagle, J. S. 2020, MNRAS, 493, 3132

Stassun, K. G., Oelkers, R. J., Paegert, M., et al. 2019, AJ, 158, 138

Stumpe, M. C., Smith, J. C., Van Cleve, J. E., et al. 2012, PASP, 124, 985

Stumpe, M. C., Smith, J. C., Catanzarite, J. H., et al. 2014, PASP, 126, 100

Sun, L., Ioannidis, P., Gu, S., et al. 2019, A&A, 624, A15

Sun, Q., Deliyannis, C. P., Steinhauer, A., Anthony-Twarog, B. J., & Twarog, B. A. 2023, ApJ, 952, 71

Tokovinin, A. 2018, PASP, 130, 035002

Tsantaki, M., Sousa, S. G., Adibekyan, V. Z., et al. 2013, A&A, 555, A150

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Ulmer-Moll, S., Osborn, H. P., Tucson, A., et al. 2023, A&A, 674, A43

Vanderburg, A., & Johnson, J. A. 2014, PASP, 126, 948

Venturini, J., & Helled, R. 2017, ApJ, 848, 95

Venturini, J., Guilera, O. M., Haldemann, J., Ronco, M. P., & Mordasini, C. 2020, A&A, 643, L1

Weiss, L. M., Marcy, G. W., Petigura, E. A., et al. 2018, AJ, 155, 48

Wielen, R. 1977, A&A, 60, 263

Winn, J. N. 2010, in Exoplanets, ed. S. Seager, 55

Winn, J. N., Holman, M. J., Henry, G. W., et al. 2007, AJ, 133, 1828

Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868

Zechmeister, M., & Kürster, M. 2009, A&A, 496, 577

Zeng, L., Sasselov, D. D., & Jacobsen, S. B. 2016, ApJ, 819, 127

Ziegler, C., Tokovinin, A., Briceno, C., et al. 2020, VizieR Online Data Catalog: J/AJ/159/19

19 Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università degli Studi di Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy

20 INAF Osservatorio Astronomico di Padova, Vicolo dell' Osservatorio 5, 35122 Padova, Italy

21 Department of Astronomy, Stockholm University, AlbaNova University Center, 10691 Stockholm, Sweden

22 Dipartimento di Fisica, Università degli Studi di Torino, via Pietro Giuria 1, 10125, Torino, Italy

23 Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK

24 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK

25 Département de Physique, Institut Trottier de Recherche sur les Exoplanètes, Université de Montréal, Montréal, Québec H3T 1J4, Canada

26 Instituto de Astrofísica de Canarias, Via Lactea s/n, 38200 La Laguna, Tenerife, Spain

27 Departamento de Astrofísica, Universidad de La Laguna, Astrofísico Francisco Sanchez s/n, 38206 La Laguna, Tenerife, Spain

28 Institut de Ciencies de l'Espai (ICE, CSIC), Campus UAB, Can Magrans s/n, 08193 Bellaterra, Spain

29 Institut d'Estudis Espacials de Catalunya (IEEC), Gran Capità 2-4, 08034 Barcelona, Spain

30 Admatis, 5. Kandó Kálmán Street, 3534 Miskolc, Hungary

31 Depto. de Astrofísica, Centro de Astrobiología (CSIC-INTA), ESAC campus, 28692 Villanueva de la Cañada (Madrid), Spain

32 Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France

33 Caltech/IPAC-NASA Exoplanet Science Institute, 770 S. Wilson Avenue, Pasadena, CA 91106, USA

34 Centre for Exoplanet Science, SUPA School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK

35 Université de Paris Cité, Institut de physique du globe de Paris, CNRS, 1 rue Jussieu, 75005 Paris, France

36 Institute of Planetary Research, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany

37 European Southern Observatory, Av. Alonso de Cordova 3107, Casilla 19001, Santiago de Chile, Chile

38 Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden

39 Department of Physics and McDonnell Center for the Space Sciences, Washington University, St. Louis, MO 63130, USA

40 Aix-Marseille Univ, CNRS, CNES, LAM, 38 rue Frédéric Joliot-Curie, 13388 Marseille, France

41 Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C, 4000 Liège, Belgium

42 Space sciences, Technologies and Astrophysics Research (STAR) Institute, Université de Liège, Allée du 6 Août 19C, 4000 Liège, Belgium

43 ETH Zurich, Institute for Particle Physics and Astrophysics, Wolfgang-Pauli-Strasse 27, 8093 Zurich Switzerland

44 Institut d'astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie Curie, 98bis bd. Arago, 75014 Paris, France

45 Airbus Defence and Space SAU, C/ Aviocar 2, 28906 Getafe, Madrid, Spain

46 Leiden Observatory, University of Leiden, PO Box 9513, 2300 RA Leiden, The Netherlands

47 Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, 439 92 Onsala, Sweden

48 Department of Astrophysics, University of Vienna, Türkenschanzstrasse 17, 1180 Vienna, Austria

49 European Space Agency (ESA), European Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands

50 Institute for Theoretical Physics and Computational Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria

51 Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, 1121 Budapest, Konkoly Thege Miklós út 15-17, Hungary

1 Observatoire de Genève, Université de Genève, Chemin Pegasi 51, 1290 Versoix, Switzerland
e-mail: angeliki.psaridi@unige.ch

2 Center for Space and Habitability, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland

3 Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

4 Weltraumforschung und Planetologie, Physikalisches Institut, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland

5 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal

6 Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal

7 Space Research Institute, Austrian Academy of Sciences, Schmiedlstraße 6, 8042 Graz, Austria

8 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

9 Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, Bd de l'Observatoire, CS 34229, 06304 Nice cedex 4, France

10 Institute for Computational Science, University of Zurich, Winterthurerstr. 90, 8057 Zurich, Switzerland

11 Astrophysics Group, Keele University, Staffordshire ST5 5BG, UK

12 Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA

13 NASA Ames Research Center, Moffett Field, CA 94035, USA

14 Research Institute for Advanced Computer Science, Universities Space Research Association, Washington, DC 20024, USA

15 Department of Physics and Astronomy, Union College, 807 Union St., Schenectady, NY 12308, USA

16 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK

17 Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA

18 Department of Physics, Engineering and Astronomy, Stephen F. Austin State University, 1936 North St, Nacogdoches, TX 75962, USA
Sousa, S. G., Santos, N. C., Adibekyan, V., Delgado-Mena, E., & Israelian, G. 2015, A&A, **577**, A67

Sousa, S. G., Adibekyan, V., Delgado-Mena, E., et al. 2021, A&A, **656**, A53

Speagle, J. S. 2020, MNRAS, **493**, 3132

Stassun, K. G., Oelkers, R. J., Paegert, M., et al. 2019, AJ, **158**, 138

Stumpe, M. C., Smith, J. C., Van Cleve, J. E., et al. 2012, PASP, **124**, 985

Stumpe, M. C., Smith, J. C., Catanzarite, J. H., et al. 2014, PASP, **126**, 100

Sun, L., Ioannidis, P., Gu, S., et al. 2019, A&A, **624**, A15

Sun, Q., Deliyannis, C. P., Steinhauer, A., Anthony-Twarog, B. J., & Twarog, B. A. 2023, ApJ, **952**, 71

Tokovinin, A. 2018, PASP, **130**, 035002

Tsantaki, M., Sousa, S. G., Adibekyan, V. Z., et al. 2013, A&A, **555**, A150

Tu, L., Johnstone, C. P., Güdel, M., & Lammer, H. 2015, A&A, **577**, A3

Twicken, J. D., Clarke, B. D., Bryson, S. T., et al. 2010, Proc. SPIE, **7740**, 774023

Twicken, J. D., Catanzarite, J. H., Clarke, B. D., et al. 2018, PASP, **130**, 064502

Ulmer-Moll, S., Osborn, H. P., Tucson, A., et al. 2023, A&A, **674**, A43

Vanderburg, A., & Johnson, J. A. 2014, PASP, **126**, 948

Venturini, J., & Helled, R. 2017, ApJ, **848**, 95

Venturini, J., Guilera, O. M., Haldemann, J., Ronco, M. P., & Mordasini, C. 2020, A&A, **643**, L1

Weiss, L. M., Marcy, G. W., Petigura, E. A., et al. 2018, AJ, **155**, 48

Wielen, R. 1977, A&A, **60**, 263

Winn, J. N. 2010, in Exoplanets, ed. S. Seager, 55

Winn, J. N., Holman, M. J., Henry, G. W., et al. 2007, AJ, **133**, 1828

Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, **140**, 1868

Zechmeister, M., & Kürster, M. 2009, A&A, **496**, 577

Zeng, L., Sasselov, D. D., & Jacobsen, S. B. 2016, ApJ, **819**, 127

Ziegler, C., Tokovinin, A., Briceno, C., et al. 2020, VizieR Online Data Catalog: J/AJ/159/19

19 Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università degli Studi di Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy

20 INAF Osservatorio Astronomico di Padova, Vicolo dell' Osservatorio 5, 35122 Padova, Italy

21 Department of Astronomy, Stockholm University, AlbaNova University Center, 10691 Stockholm, Sweden

22 Dipartimento di Fisica, Università degli Studi di Torino, via Pietro Giuria 1, 10125, Torino, Italy

23 Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK

24 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK

25 Département de Physique, Institut Trottier de Recherche sur les Exoplanètes, Université de Montréal, Montréal, Québec H3T 1J4, Canada

26 Instituto de Astrofísica de Canarias, Via Lactea s/n, 38200 La Laguna, Tenerife, Spain

27 Departamento de Astrofísica, Universidad de La Laguna, Astrofísico Francisco Sanchez s/n, 38206 La Laguna, Tenerife, Spain

28 Institut de Ciencies de l'Espai (ICE, CSIC), Campus UAB, Can Magrans s/n, 08193 Bellaterra, Spain

29 Institut d'Estudis Espacials de Catalunya (IEEC), Gran Capità 2-4, 08034 Barcelona, Spain

30 Admatis, 5. Kandó Kálmán Street, 3534 Miskolc, Hungary

31 Depto. de Astrofísica, Centro de Astrobiología (CSIC-INTA), ESAC campus, 28692 Villanueva de la Cañada (Madrid), Spain

32 Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France

33 Caltech/IPAC-NASA Exoplanet Science Institute, 770 S. Wilson Avenue, Pasadena, CA 91106, USA

34 Centre for Exoplanet Science, SUPA School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK

35 Université de Paris Cité, Institut de physique du globe de Paris, CNRS, 1 rue Jussieu, 75005 Paris, France

36 Institute of Planetary Research, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany

37 European Southern Observatory, Av. Alonso de Cordova 3107, Casilla 19001, Santiago de Chile, Chile

38 Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden

39 Department of Physics and McDonnell Center for the Space Sciences, Washington University, St. Louis, MO 63130, USA

40 Aix-Marseille Univ, CNRS, CNES, LAM, 38 rue Frédéric Joliot-Curie, 13388 Marseille, France

41 Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C, 4000 Liège, Belgium

42 Space sciences, Technologies and Astrophysics Research (STAR) Institute, Université de Liège, Allée du 6 Août 19C, 4000 Liège, Belgium

43 ETH Zurich, Institute for Particle Physics and Astrophysics, Wolfgang-Pauli-Strasse 27, 8093 Zurich Switzerland

44 Institut d'astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie Curie, 98bis bd. Arago, 75014 Paris, France

45 Airbus Defence and Space SAU, C/ Aviocar 2, 28906 Getafe, Madrid, Spain

46 Leiden Observatory, University of Leiden, PO Box 9513, 2300 RA Leiden, The Netherlands

47 Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, 439 92 Onsala, Sweden

48 Department of Astrophysics, University of Vienna, Türkenschanzstrasse 17, 1180 Vienna, Austria

49 European Space Agency (ESA), European Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands

50 Institute for Theoretical Physics and Computational Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria

51 Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, 1121 Budapest, Konkoly Thege Miklós út 15-17, Hungary

1 Observatoire de Genève, Université de Genève, Chemin Pegasi 51, 1290 Versoix, Switzerland
e-mail: angeliki.psaridi@unige.ch

2 Center for Space and Habitability, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland

3 Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

4 Weltraumforschung und Planetologie, Physikalisches Institut, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland

5 Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal

6 Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal

7 Space Research Institute, Austrian Academy of Sciences, Schmiedlstraße 6, 8042 Graz, Austria

8 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

9 Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, Bd de l'Observatoire, CS 34229, 06304 Nice cedex 4, France

10 Institute for Computational Science, University of Zurich, Winterthurerstr. 90, 8057 Zurich, Switzerland

11 Astrophysics Group, Keele University, Staffordshire ST5 5BG, UK

12 Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA

13 NASA Ames Research Center, Moffett Field, CA 94035, USA

14 Research Institute for Advanced Computer Science, Universities Space Research Association, Washington, DC 20024, USA

15 Department of Physics and Astronomy, Union College, 807 Union St., Schenectady, NY 12308, USA

16 Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK

17 Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA

18 Department of Physics, Engineering and Astronomy, Stephen F. Austin State University, 1936 North St, Nacogdoches, TX 75962, USA
52 ELTE Eötvös Loránd University, Institute of Physics, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary

53 IMCCE, UMR8028 CNRS, Observatoire de Paris, PSL Univ., Sorbonne Univ., 77 av. Denfert-Rochereau, 75014 Paris, France

54 Proto-Logic LLC, 1718 Euclid Street NW, Washington, DC 20009, USA

55 European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching bei München, Germany

56 INAF Osservatorio Astrofisico di Catania, via S. Sofia 78, 95123 Catania, Italy

57 Zentrum für Astronomie und Astrophysik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany

58 Institut fuer Geologische Wissenschaften, Freie Universitaet Berlin, Maltheserstrasse 74–100, 12249 Berlin, Germany

59 Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

60 Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

61 ELTE Eötvös Loránd University, Gothard Astrophysical Observatory, 9700 Szombathely, Szent Imre h. u. 112, Hungary

62 HUN-REN-ELTE Exoplanet Research Group, Szent Imre h. u. 112, Szombathely, 9700, Hungary

63 NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA

64 Centre for Exoplanets and Habitability, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK
52 ELTE Eötvös Loránd University, Institute of Physics, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary

53 IMCCE, UMR8028 CNRS, Observatoire de Paris, PSL Univ., Sorbonne Univ., 77 av. Denfert-Rochereau, 75014 Paris, France

54 Proto-Logic LLC, 1718 Euclid Street NW, Washington, DC 20009, USA

55 European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching bei München, Germany

56 INAF Osservatorio Astrofisico di Catania, via S. Sofia 78, 95123 Catania, Italy

57 Zentrum für Astronomie und Astrophysik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany

58 Institut fuer Geologische Wissenschaften, Freie Universitaet Berlin, Maltheserstrasse 74–100, 12249 Berlin, Germany

59 Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

60 Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

61 ELTE Eötvös Loránd University, Gothard Astrophysical Observatory, 9700 Szombathely, Szent Imre h. u. 112, Hungary

62 HUN-REN-ELTE Exoplanet Research Group, Szent Imre h. u. 112, Szombathely, 9700, Hungary

63 NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA

64 Centre for Exoplanets and Habitability, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK
Appendix A: TESS light curves

<figure><img src="image_15.png" /><figcaption>Fig. A.1: TESS SAP flux light curves with the best-fit *juliet* model shown with the black line (see Sect. 4 for details on the modeling). The transits of TOI-815b and TOI-815c are indicated with purple and orange, respectively.</figcaption></figure>
Appendix A: TESS light curves

<figure><img src="image_16.png" /><figcaption>Fig. A.1: TESS SAP flux light curves with the best-fit juliet model shown with the black line (see Sect. 4 for details on the modeling). The transits of TOI-815b and TOI-815c are indicated with purple and orange, respectively.</figcaption></figure>
## Appendix B: RVs with ESPRESSO

<table><caption>Table B.1: ESPRESSO RV observations for TOI-815, along with activity indicator measurements, i.e. FWHM, bisector, and $\log R'_{HK}$.</caption><thead><tr><th>Time<br>[BJD-2 450 000]</th><th>RV<br>[m s<sup>-1</sup>]</th><th>σ<sub>RV</sub><br>[m s<sup>-1</sup>]</th><th>FWHM<sup>(2)</sup><br>[m s<sup>-1</sup>]</th><th>BIS<sup>(2)</sup></th><th>logR'<sub>HK</sub><sup>(3)</sup></th><th>t<sub>exp</sub><br>[s]</th></tr></thead><tbody><tr><td>9308.60433</td><td>-2973.3950</td><td>0.6145</td><td>6474.143</td><td>-33.038</td><td>-4.7084</td><td>600</td></tr><tr><td>9310.60679</td><td>-2970.1704</td><td>0.7233</td><td>6470.941</td><td>-33.052</td><td>-4.7338</td><td>600</td></tr><tr><td>9313.73506<sup>(1)</sup></td><td>-2962.6373</td><td>0.5712</td><td>6492.966</td><td>-30.291</td><td>-4.6757</td><td>600</td></tr><tr><td>9316.52823</td><td>-2967.0427</td><td>0.7459</td><td>6494.718</td><td>-23.625</td><td>-4.7107</td><td>600</td></tr><tr><td>9318.53033</td><td>-2968.9726</td><td>0.5863</td><td>6478.690</td><td>-29.060</td><td>-4.6762</td><td>600</td></tr><tr><td>9324.54484</td><td>-2979.7547</td><td>0.8117</td><td>6513.799</td><td>-28.181</td><td>-4.6859</td><td>600</td></tr><tr><td>9327.50777<sup>(1)</sup></td><td>-2965.7154</td><td>0.4397</td><td>6482.962</td><td>-32.938</td><td>-4.6229</td><td>600</td></tr><tr><td>9329.52592<sup>(1)</sup></td><td>-2971.0105</td><td>0.9842</td><td>6543.347</td><td>-29.283</td><td>-4.7529</td><td>600</td></tr><tr><td>9332.57219</td><td>-2979.7266</td><td>0.6474</td><td>6491.382</td><td>-26.738</td><td>-4.6824</td><td>600</td></tr><tr><td>9336.49956<sup>(1)</sup></td><td>-2982.4246</td><td>1.3115</td><td>6515.936</td><td>-40.000</td><td>-4.9931</td><td>600</td></tr><tr><td>9339.48187</td><td>-2973.6498</td><td>0.5988</td><td>6468.715</td><td>-33.267</td><td>-4.7189</td><td>600</td></tr><tr><td>9341.51194<sup>(1)</sup></td><td>-2966.5835</td><td>0.9108</td><td>6513.090</td><td>-35.131</td><td>-4.7516</td><td>600</td></tr><tr><td>9346.54064</td><td>-2972.2321</td><td>0.5959</td><td>6482.420</td><td>-24.888</td><td>-4.6930</td><td>600</td></tr><tr><td>9373.49668</td><td>-2973.5189</td><td>0.5467</td><td>6462.905</td><td>-33.991</td><td>-4.6367</td><td>600</td></tr><tr><td>9375.51475</td><td>-2971.6794</td><td>0.6513</td><td>6471.909</td><td>-32.600</td><td>-4.6265</td><td>600</td></tr><tr><td>9406.50596</td><td>-2975.2108</td><td>0.7212</td><td>6492.234</td><td>-32.504</td><td>-4.6149</td><td>600</td></tr><tr><td>9410.48054</td><td>-2971.8872</td><td>0.6759</td><td>6499.208</td><td>-30.202</td><td>-4.6978</td><td>600</td></tr><tr><td>9415.50797</td><td>-2971.5355</td><td>0.8107</td><td>6491.252</td><td>-26.700</td><td>-4.6620</td><td>600</td></tr><tr><td>9673.54348</td><td>-2977.4158</td><td>0.5684</td><td>6488.390</td><td>-34.257</td><td>-4.6894</td><td>700</td></tr><tr><td>9675.51515</td><td>-2973.7947</td><td>0.9406</td><td>6487.917</td><td>-25.694</td><td>-4.6418</td><td>700</td></tr><tr><td>9677.65231</td><td>-2972.6960</td><td>0.8151</td><td>6492.405</td><td>-31.643</td><td>-4.6580</td><td>700</td></tr><tr><td>9679.55323<sup>(1)</sup></td><td>-2973.4334</td><td>0.5445</td><td>6497.504</td><td>-32.766</td><td>-4.6080</td><td>700</td></tr><tr><td>9683.53185</td><td>-2981.6090</td><td>0.4694</td><td>6497.328</td><td>-29.483</td><td>-4.6329</td><td>700</td></tr><tr><td>9685.55122</td><td>-2980.4065</td><td>0.7458</td><td>6496.394</td><td>-24.517</td><td>-4.6203</td><td>700</td></tr><tr><td>9687.52944</td><td>-2972.0070</td><td>0.4487</td><td>6495.493</td><td>-29.672</td><td>-4.6432</td><td>700</td></tr><tr><td>9695.52289</td><td>-2967.3200</td><td>0.9486</td><td>6512.740</td><td>-29.566</td><td>-4.7299</td><td>700</td></tr><tr><td>9697.58163</td><td>-2965.9727</td><td>0.5759</td><td>6511.966</td><td>-26.858</td><td>-4.6611</td><td>700</td></tr><tr><td>9784.49262</td><td>-2977.5647</td><td>0.6128</td><td>6525.410</td><td>-22.486</td><td>-4.6248</td><td>700</td></tr><tr><td>9904.83330</td><td>-2969.5919</td><td>0.4342</td><td>6487.460</td><td>-35.249</td><td>-4.6440</td><td>700</td></tr><tr><td>9905.77809</td><td>-2970.1154</td><td>0.6533</td><td>6491.720</td><td>-34.630</td><td>-4.6797</td><td>700</td></tr><tr><td>9906.80910</td><td>-2970.1976</td><td>0.4551</td><td>6492.162</td><td>-33.028</td><td>-4.6399</td><td>700</td></tr><tr><td>9907.79772</td><td>-2969.7455</td><td>1.3028</td><td>6534.337</td><td>-38.113</td><td>-4.8075</td><td>700</td></tr><tr><td>9910.75047</td><td>-2966.0686</td><td>0.7246</td><td>6503.361</td><td>-25.248</td><td>-4.6902</td><td>700</td></tr><tr><td>9917.72905</td><td>-2978.0604</td><td>0.6064</td><td>6481.170</td><td>-33.996</td><td>-4.6673</td><td>700</td></tr></tbody></table>

**Notes:**

(1) These data did not pass the ESPRESSO flux quality control and therefore were excluded from the analysis.

(2) The uncertainties of FWHM and BIS are twice the uncertainties of the RVs ($2\sigma_{RV}$).

(3) The median uncertainty of $\log R'_{HK}$ is 0.03.
## Appendix B: RVs with ESPRESSO

<table><caption>Table B.1: ESPRESSO RV observations for TOI-815, along with activity indicator measurements, i.e. FWHM, bisector, and $\log R'_{HK}$.</caption><thead><tr><th>Time<br>[BJD-2 450 000]</th><th>RV<br>[m s<sup>-1</sup>]</th><th>σ<sub>RV</sub><br>[m s<sup>-1</sup>]</th><th>FWHM<sup>(2)</sup><br>[m s<sup>-1</sup>]</th><th>BIS<sup>(2)</sup></th><th>logR'<sub>HK</sub><sup>(3)</sup></th><th>t<sub>exp</sub><br>[s]</th></tr></thead><tbody><tr><td>9308.60433</td><td>-2973.3950</td><td>0.6145</td><td>6474.143</td><td>-33.038</td><td>-4.7084</td><td>600</td></tr><tr><td>9310.60679</td><td>-2970.1704</td><td>0.7233</td><td>6470.941</td><td>-33.052</td><td>-4.7338</td><td>600</td></tr><tr><td>9313.73506<sup>(1)</sup></td><td>-2962.6373</td><td>0.5712</td><td>6492.966</td><td>-30.291</td><td>-4.6757</td><td>600</td></tr><tr><td>9316.52823</td><td>-2967.0427</td><td>0.7459</td><td>6494.718</td><td>-23.625</td><td>-4.7107</td><td>600</td></tr><tr><td>9318.53033</td><td>-2968.9726</td><td>0.5863</td><td>6478.690</td><td>-29.060</td><td>-4.6762</td><td>600</td></tr><tr><td>9324.54484</td><td>-2979.7547</td><td>0.8117</td><td>6513.799</td><td>-28.181</td><td>-4.6859</td><td>600</td></tr><tr><td>9327.50777<sup>(1)</sup></td><td>-2965.7154</td><td>0.4397</td><td>6482.962</td><td>-32.938</td><td>-4.6229</td><td>600</td></tr><tr><td>9329.52592<sup>(1)</sup></td><td>-2971.0105</td><td>0.9842</td><td>6543.347</td><td>-29.283</td><td>-4.7529</td><td>600</td></tr><tr><td>9332.57219</td><td>-2979.7266</td><td>0.6474</td><td>6491.382</td><td>-26.738</td><td>-4.6824</td><td>600</td></tr><tr><td>9336.49956<sup>(1)</sup></td><td>-2982.4246</td><td>1.3115</td><td>6515.936</td><td>-40.000</td><td>-4.9931</td><td>600</td></tr><tr><td>9339.48187</td><td>-2973.6498</td><td>0.5988</td><td>6468.715</td><td>-33.267</td><td>-4.7189</td><td>600</td></tr><tr><td>9341.51194<sup>(1)</sup></td><td>-2966.5835</td><td>0.9108</td><td>6513.090</td><td>-35.131</td><td>-4.7516</td><td>600</td></tr><tr><td>9346.54064</td><td>-2972.2321</td><td>0.5959</td><td>6482.420</td><td>-24.888</td><td>-4.6930</td><td>600</td></tr><tr><td>9373.49668</td><td>-2973.5189</td><td>0.5467</td><td>6462.905</td><td>-33.991</td><td>-4.6367</td><td>600</td></tr><tr><td>9375.51475</td><td>-2971.6794</td><td>0.6513</td><td>6471.909</td><td>-32.600</td><td>-4.6265</td><td>600</td></tr><tr><td>9406.50596</td><td>-2975.2108</td><td>0.7212</td><td>6492.234</td><td>-32.504</td><td>-4.6149</td><td>600</td></tr><tr><td>9410.48054</td><td>-2971.8872</td><td>0.6759</td><td>6499.208</td><td>-30.202</td><td>-4.6978</td><td>600</td></tr><tr><td>9415.50797</td><td>-2971.5355</td><td>0.8107</td><td>6491.252</td><td>-26.700</td><td>-4.6620</td><td>600</td></tr><tr><td>9673.54348</td><td>-2977.4158</td><td>0.5684</td><td>6488.390</td><td>-34.257</td><td>-4.6894</td><td>700</td></tr><tr><td>9675.51515</td><td>-2973.7947</td><td>0.9406</td><td>6487.917</td><td>-25.694</td><td>-4.6418</td><td>700</td></tr><tr><td>9677.65231</td><td>-2972.6960</td><td>0.8151</td><td>6492.405</td><td>-31.643</td><td>-4.6580</td><td>700</td></tr><tr><td>9679.55323<sup>(1)</sup></td><td>-2973.4334</td><td>0.5445</td><td>6497.504</td><td>-32.766</td><td>-4.6080</td><td>700</td></tr><tr><td>9683.53185</td><td>-2981.6090</td><td>0.4694</td><td>6497.328</td><td>-29.483</td><td>-4.6329</td><td>700</td></tr><tr><td>9685.55122</td><td>-2980.4065</td><td>0.7458</td><td>6496.394</td><td>-24.517</td><td>-4.6203</td><td>700</td></tr><tr><td>9687.52944</td><td>-2972.0070</td><td>0.4487</td><td>6495.493</td><td>-29.672</td><td>-4.6432</td><td>700</td></tr><tr><td>9695.52289</td><td>-2967.3200</td><td>0.9486</td><td>6512.740</td><td>-29.566</td><td>-4.7299</td><td>700</td></tr><tr><td>9697.58163</td><td>-2965.9727</td><td>0.5759</td><td>6511.966</td><td>-26.858</td><td>-4.6611</td><td>700</td></tr><tr><td>9784.49262</td><td>-2977.5647</td><td>0.6128</td><td>6525.410</td><td>-22.486</td><td>-4.6248</td><td>700</td></tr><tr><td>9904.83330</td><td>-2969.5919</td><td>0.4342</td><td>6487.460</td><td>-35.249</td><td>-4.6440</td><td>700</td></tr><tr><td>9905.77809</td><td>-2970.1154</td><td>0.6533</td><td>6491.720</td><td>-34.630</td><td>-4.6797</td><td>700</td></tr><tr><td>9906.80910</td><td>-2970.1976</td><td>0.4551</td><td>6492.162</td><td>-33.028</td><td>-4.6399</td><td>700</td></tr><tr><td>9907.79772</td><td>-2969.7455</td><td>1.3028</td><td>6534.337</td><td>-38.113</td><td>-4.8075</td><td>700</td></tr><tr><td>9910.75047</td><td>-2966.0686</td><td>0.7246</td><td>6503.361</td><td>-25.248</td><td>-4.6902</td><td>700</td></tr><tr><td>9917.72905</td><td>-2978.0604</td><td>0.6064</td><td>6481.170</td><td>-33.996</td><td>-4.6673</td><td>700</td></tr></tbody></table>

**Notes:**

(1) These data did not pass the ESPRESSO flux quality control and therefore were excluded from the analysis.

(2) The uncertainties of FWHM and BIS are twice the uncertainties of the RVs ($2\sigma_{RV}$).

(3) The median uncertainty of $\log R'_{HK}$ is 0.03.
Appendix C: Spectral synthesis

<figure><img src="image_16.png" /><figcaption>Fig. C.1: Spectral synthesis around the Li doublet. The vertical lines mark the position of the lines.</figcaption></figure>
**Appendix C: Spectral synthesis**

<figure><img src="image_17.png" /><figcaption>Fig. C.1: Spectral synthesis around the Li doublet. The vertical lines mark the position of the lines.</figcaption></figure>
Appendix D: Generalized Lomb-Scargle periodograms

<figure><img src="image_17.png" /><figcaption>Fig. D.1: GLS periodogram of the TOI-815 spectroscopic and photometric time series. More specifically: (1) RV ESPRESSO measurements, (2) RV residuals, after the subtraction of TOI-815c signal, (3) RV residuals, after the subtraction of TOI-815b and c signals, (4) FWHM, bisector, and Ca II H & K line activity indicator (logR'_{HK}), (5) TESS Sector 63 photometric data, and (6) TESS photometric data (2008, 2012, and 2023). The two black vertical lines correspond to the minimum and maximum rotation periods (P_{rot,min} ~ 14.2 days and P_{rot,max} ~ 16.5 days). The purple and orange vertical lines correspond to the orbital periods of planets b (P_b ~ 11.2 days) and c (P_c ~ 35 days), respectively. The gray horizontal lines indicating the 1% FAP.</figcaption></figure>
Appendix D: Generalized Lomb-Scargle periodograms

<figure><img src="image_18.png" /><figcaption>Fig. D.1: GLS periodogram of the TOI-815 spectroscopic and photometric time series. More specifically: (1) RV ESPRESSO measurements, (2) RV residuals, after the subtraction of TOI-815c signal, (3) RV residuals, after the subtraction of TOI-815b and c signals, (4) FWHM, bisector, and Ca II H & K line activity indicator (logR'_{HK}), (5) TESS Sector 63 photometric data, and (6) TESS photometric data (2008, 2012, and 2023). The two black vertical lines correspond to the minimum and maximum rotation periods (P_{rot,min} ~ 14.2 days and P_{rot,max} ~ 16.5 days). The purple and orange vertical lines correspond to the orbital periods of planets b (P_b ~ 11.2 days) and c (P_c ~ 35 days), respectively. The gray horizontal lines indicating the 1% FAP.</figcaption></figure>
## Appendix E: Priors

<table><caption>Table E.1: Maximum values and 68% confidence intervals of the posterior distributions from joint modeling with juliet.</caption><thead><tr><th>Parameter</th><th>Prior<sup>(a)</sup></th><th>Value<sup>(b)</sup></th></tr></thead><tbody><tr><td colspan="3"><strong>Jump stellar parameters</strong></td></tr><tr><td>Stellar density, ρ<sub>s</sub> (ρ<sub>☉</sub>)</td><td>N(2396, 139)</td><td>2431<sup>+122</sup><sub>-125</sub></td></tr><tr><td colspan="3"><strong>Jump parameters for TOI-815b</strong></td></tr><tr><td>Orbital period, P<sub>orb</sub> (days)</td><td>U(10, 12)</td><td>11.197259<sup>+0.000018</sup><sub>-0.000017</sub></td></tr><tr><td>Transit epoch, T<sub>0</sub> (BJD)</td><td>N(2459287.6, 0.1)</td><td>2459287.6028<sup>+0.0009</sup><sub>-0.0009</sub></td></tr><tr><td>Scaled planetary radius, R<sub>P</sub>/R<sub>s</sub></td><td>U(0, 1)</td><td>0.0349<sup>+0.0004</sup><sub>-0.0004</sub></td></tr><tr><td>Impact parameter, b</td><td>U(0, 1)</td><td>0.279<sup>+0.058</sup><sub>-0.073</sub></td></tr><tr><td>Eccentricity, e</td><td>F(0)</td><td>0 (adopted, 3σ &lt; 0.20)</td></tr><tr><td>Argument of periastron, ω<sub>s</sub> (deg)</td><td>F(90)</td><td>-</td></tr><tr><td>RV semi-amplitude, K (m s<sup>-1</sup>)</td><td>U(0, 100)</td><td>2.6<sup>+0.5</sup><sub>-0.5</sub></td></tr><tr><td colspan="3"><strong>Jump parameters for TOI-815c</strong></td></tr><tr><td>Orbital period, P<sub>orb</sub> (days)</td><td>U(34, 36)</td><td>34.976145<sup>+0.000099</sup><sub>-0.000097</sub></td></tr><tr><td>Transit epoch, T<sub>0</sub> (BJD)</td><td>N(2459288.6, 0.1)</td><td>2459288.6331<sup>+0.0023</sup><sub>-0.0024</sub></td></tr><tr><td>Scaled planetary radius, R<sub>P</sub>/R<sub>s</sub></td><td>U(0, 1)</td><td>0.0312<sup>+0.0011</sup><sub>-0.0010</sub></td></tr><tr><td>Impact parameter, b</td><td>U(0, 1)</td><td>0.839<sup>+0.015</sup><sub>-0.016</sub></td></tr><tr><td>Eccentricity, e</td><td>F(0)</td><td>0 (adopted, 3σ &lt; 0.22)</td></tr><tr><td>Argument of periastron, ω<sub>s</sub> (deg)</td><td>F(90)</td><td>-</td></tr><tr><td>RV semi-amplitude, K (m s<sup>-1</sup>)</td><td>U(0, 100)</td><td>5.5<sup>+0.5</sup><sub>-0.5</sub></td></tr><tr><td colspan="3"><strong>Instrumental photometric parameters</strong></td></tr><tr><td>Offset relative flux, M<sub>TESS<sub>S9</sub></sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>-8.6<sup>+8.0</sup><sub>-8.2</sub></td></tr><tr><td>Jitter, σ<sub>w,TESS<sub>S9</sub></sub> (ppm)</td><td>logU(0.01, 1000)</td><td>0.5<sup>+6.5</sup><sub>-0.5</sub></td></tr><tr><td>Offset relative flux, M<sub>TESS<sub>S36</sub></sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>-72<sup>+411</sup><sub>-417</sub></td></tr><tr><td>Jitter, σ<sub>w,TESS<sub>S36</sub></sub> (ppm)</td><td>logU(0.01, 1000)</td><td>466<sup>+8</sup><sub>-8</sub></td></tr><tr><td>Offset relative flux, M<sub>TESS<sub>S63</sub></sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>-3594<sup>+3821</sup><sub>-6137</sub></td></tr><tr><td>Jitter, σ<sub>w,TESS<sub>S63</sub></sub> (ppm)</td><td>logU(0.01, 1000)</td><td>457<sup>+11</sup><sub>-11</sub></td></tr><tr><td>Offset relative flux, M<sub>CHEOPS<sub>1</sub></sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>-197<sup>+46</sup><sub>-44</sub></td></tr><tr><td>Jitter, σ<sub>w,CHEOPS<sub>1</sub></sub> (ppm)</td><td>logU(0.01, 1000)</td><td>299<sup>+23</sup><sub>-22</sub></td></tr><tr><td>Offset relative flux, M<sub>CHEOPS<sub>2</sub></sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>-474<sup>+48</sup><sub>-48</sub></td></tr><tr><td>Jitter, σ<sub>w,CHEOPS<sub>2</sub></sub> (ppm)</td><td>logU(0.01, 1000)</td><td>404<sup>+24</sup><sub>-23</sub></td></tr><tr><td>Offset relative flux, M<sub>CHEOPS<sub>3</sub></sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>-322<sup>+34</sup><sub>-35</sub></td></tr><tr><td>Jitter, σ<sub>w,CHEOPS<sub>3</sub></sub> (ppm)</td><td>logU(0.01, 1000)</td><td>293<sup>+19</sup><sub>-19</sub></td></tr><tr><td>Offset relative flux, M<sub>LCO</sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>1089<sup>+88</sup><sub>-88</sub></td></tr><tr><td>Jitter, σ<sub>w,LCO</sub> (ppm)</td><td>logU(0.01, 5000)</td><td>986<sup>+72</sup><sub>-68</sub></td></tr><tr><td>Offset relative flux, M<sub>ASTEP</sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>761<sup>+133</sup><sub>-131</sub></td></tr><tr><td>Jitter, σ<sub>w,ASTEP</sub> (ppm)</td><td>logU(0.01, 5000)</td><td>3369<sup>+89</sup><sub>-85</sub></td></tr><tr><td colspan="3"><strong>GP parameters</strong></td></tr><tr><td>σ<sub>GP,TESS<sub>S9</sub></sub> (relative flux)</td><td>F(0)</td><td>-</td></tr><tr><td>ρ<sub>GP,TESS<sub>S9</sub></sub> (days)</td><td>F(0)</td><td>-</td></tr><tr><td>σ<sub>GP,TESS<sub>S36</sub></sub> (10<sup>-4</sup> relative flux)</td><td>logU(10<sup>-6</sup>, 100)</td><td>12<sup>+2</sup><sub>-2</sub></td></tr><tr><td>ρ<sub>GP,TESS<sub>S36</sub></sub> (days)</td><td>logU(0.001, 100)</td><td>0.94<sup>+0.13</sup><sub>-0.11</sub></td></tr><tr><td>σ<sub>GP,TESS<sub>S63</sub></sub> (10<sup>-4</sup> relative flux)</td><td>logU(10<sup>-6</sup>, 100)</td><td>57<sup>+44</sup><sub>-22</sub></td></tr><tr><td>ρ<sub>GP,TESS<sub>S63</sub></sub> (days)</td><td>logU(0.001, 100)</td><td>9.11<sup>+3.37</sup><sub>-2.77</sub></td></tr><tr><td colspan="3"><strong>Instrumental RV parameters</strong></td></tr><tr><td>Jitter, σ<sub>w,ESPRESSO</sub> (m s<sup>-1</sup>)</td><td>logU(0.001, 100)</td><td>1.88<sup>+0.29</sup><sub>-0.25</sub></td></tr><tr><td>Systemic RV, μ<sub>ESPRESSO</sub> (m s<sup>-1</sup>)</td><td>U(-3500, -1000)</td><td>-2973.61<sup>+0.36</sup><sub>-0.36</sub></td></tr><tr><td colspan="3"><strong>Limb darkening parameters</strong></td></tr><tr><td>q<sub>1,ESPRESSO</sub></td><td>N(0.408, 0.017)</td><td>0.407<sup>+0.015</sup><sub>-0.015</sub></td></tr><tr><td>q<sub>2,ESPRESSO</sub></td><td>N(0.349, 0.022)</td><td>0.345<sup>+0.020</sup><sub>-0.019</sub></td></tr><tr><td>q<sub>1,CHEOPS</sub></td><td>N(0.519, 0.012)</td><td>0.518<sup>+0.011</sup><sub>-0.011</sub></td></tr><tr><td>q<sub>2,CHEOPS</sub></td><td>N(0.408, 0.012)</td><td>0.406<sup>+0.011</sup><sub>-0.011</sub></td></tr><tr><td>q<sub>1,LCO</sub></td><td>N(0.324, 0.008)</td><td>0.324<sup>+0.007</sup><sub>-0.007</sub></td></tr><tr><td>q<sub>2,LCO</sub></td><td>N(0.314, 0.014)</td><td>0.314<sup>+0.012</sup><sub>-0.012</sub></td></tr><tr><td>q<sub>1,ASTEP</sub></td><td>N(0.492, 0.016)</td><td>0.492<sup>+0.014</sup><sub>-0.014</sub></td></tr><tr><td>q<sub>2,ASTEP</sub></td><td>N(0.376, 0.017)</td><td>0.378<sup>+0.015</sup><sub>-0.016</sub></td></tr></tbody></table>

Notes: (a) For the priors, $N(μ, σ^2)$ indicates a normal distribution with mean $μ$ and variance $σ^2$, $U(a, b)$ a uniform distribution between $a$ and $b$, $logU(a, b)$ a log-uniform distribution between $a$ and $b$ and $F(a)$ a parameter fixed to value $a$. (b) The posterior estimate indicates the median value and then error bars for the 68% credibility intervals.
## Appendix E: Priors

<table><caption>Table E.1: Maximum values and 68% confidence intervals of the posterior distributions from joint modeling with juliet.</caption><thead><tr><th>Parameter</th><th>Prior<sup>(a)</sup></th><th>Value<sup>(b)</sup></th></tr></thead><tbody><tr><td colspan="3"><strong>Jump stellar parameters</strong></td></tr><tr><td>Stellar density, ρ<sub>s</sub> (ρ<sub>☉</sub>)</td><td>N(2396, 139)</td><td>2431<sup>+122</sup><sub>-125</sub></td></tr><tr><td colspan="3"><strong>Jump parameters for TOI-815b</strong></td></tr><tr><td>Orbital period, P<sub>orb</sub> (days)</td><td>U(10, 12)</td><td>11.197259<sup>+0.000018</sup><sub>-0.000017</sub></td></tr><tr><td>Transit epoch, T<sub>0</sub> (BJD)</td><td>N(2459287.6, 0.1)</td><td>2459287.6028<sup>+0.0009</sup><sub>-0.0009</sub></td></tr><tr><td>Scaled planetary radius, R<sub>P</sub>/R<sub>s</sub></td><td>U(0, 1)</td><td>0.0349<sup>+0.0004</sup><sub>-0.0004</sub></td></tr><tr><td>Impact parameter, b</td><td>U(0, 1)</td><td>0.279<sup>+0.058</sup><sub>-0.073</sub></td></tr><tr><td>Eccentricity, e</td><td>F(0)</td><td>0 (adopted, 3σ &lt; 0.20)</td></tr><tr><td>Argument of periastron, ω<sub>s</sub> (deg)</td><td>F(90)</td><td>-</td></tr><tr><td>RV semi-amplitude, K (m s<sup>-1</sup>)</td><td>U(0, 100)</td><td>2.6<sup>+0.5</sup><sub>-0.5</sub></td></tr><tr><td colspan="3"><strong>Jump parameters for TOI-815c</strong></td></tr><tr><td>Orbital period, P<sub>orb</sub> (days)</td><td>U(34, 36)</td><td>34.976145<sup>+0.000099</sup><sub>-0.000097</sub></td></tr><tr><td>Transit epoch, T<sub>0</sub> (BJD)</td><td>N(2459288.6, 0.1)</td><td>2459288.6331<sup>+0.0023</sup><sub>-0.0024</sub></td></tr><tr><td>Scaled planetary radius, R<sub>P</sub>/R<sub>s</sub></td><td>U(0, 1)</td><td>0.0312<sup>+0.0011</sup><sub>-0.0010</sub></td></tr><tr><td>Impact parameter, b</td><td>U(0, 1)</td><td>0.839<sup>+0.015</sup><sub>-0.016</sub></td></tr><tr><td>Eccentricity, e</td><td>F(0)</td><td>0 (adopted, 3σ &lt; 0.22)</td></tr><tr><td>Argument of periastron, ω<sub>s</sub> (deg)</td><td>F(90)</td><td>-</td></tr><tr><td>RV semi-amplitude, K (m s<sup>-1</sup>)</td><td>U(0, 100)</td><td>5.5<sup>+0.5</sup><sub>-0.5</sub></td></tr><tr><td colspan="3"><strong>Instrumental photometric parameters</strong></td></tr><tr><td>Offset relative flux, M<sub>TESS<sub>S9</sub></sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>-8.6<sup>+8.0</sup><sub>-8.2</sub></td></tr><tr><td>Jitter, σ<sub>w,TESS<sub>S9</sub></sub> (ppm)</td><td>logU(0.01, 1000)</td><td>0.5<sup>+6.5</sup><sub>-0.5</sub></td></tr><tr><td>Offset relative flux, M<sub>TESS<sub>S36</sub></sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>-72<sup>+411</sup><sub>-417</sub></td></tr><tr><td>Jitter, σ<sub>w,TESS<sub>S36</sub></sub> (ppm)</td><td>logU(0.01, 1000)</td><td>466<sup>+8</sup><sub>-8</sub></td></tr><tr><td>Offset relative flux, M<sub>TESS<sub>S63</sub></sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>-3594<sup>+3821</sup><sub>-6137</sub></td></tr><tr><td>Jitter, σ<sub>w,TESS<sub>S63</sub></sub> (ppm)</td><td>logU(0.01, 1000)</td><td>457<sup>+11</sup><sub>-11</sub></td></tr><tr><td>Offset relative flux, M<sub>CHEOPS<sub>1</sub></sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>-197<sup>+46</sup><sub>-44</sub></td></tr><tr><td>Jitter, σ<sub>w,CHEOPS<sub>1</sub></sub> (ppm)</td><td>logU(0.01, 1000)</td><td>299<sup>+23</sup><sub>-22</sub></td></tr><tr><td>Offset relative flux, M<sub>CHEOPS<sub>2</sub></sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>-474<sup>+48</sup><sub>-48</sub></td></tr><tr><td>Jitter, σ<sub>w,CHEOPS<sub>2</sub></sub> (ppm)</td><td>logU(0.01, 1000)</td><td>404<sup>+24</sup><sub>-23</sub></td></tr><tr><td>Offset relative flux, M<sub>CHEOPS<sub>3</sub></sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>-322<sup>+34</sup><sub>-35</sub></td></tr><tr><td>Jitter, σ<sub>w,CHEOPS<sub>3</sub></sub> (ppm)</td><td>logU(0.01, 1000)</td><td>293<sup>+19</sup><sub>-19</sub></td></tr><tr><td>Offset relative flux, M<sub>LCO</sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>1089<sup>+88</sup><sub>-88</sub></td></tr><tr><td>Jitter, σ<sub>w,LCO</sub> (ppm)</td><td>logU(0.01, 5000)</td><td>986<sup>+72</sup><sub>-68</sub></td></tr><tr><td>Offset relative flux, M<sub>ASTEP</sub> (10<sup>-6</sup>)</td><td>N(0, 0.1)</td><td>761<sup>+133</sup><sub>-131</sub></td></tr><tr><td>Jitter, σ<sub>w,ASTEP</sub> (ppm)</td><td>logU(0.01, 5000)</td><td>3369<sup>+89</sup><sub>-85</sub></td></tr><tr><td colspan="3"><strong>GP parameters</strong></td></tr><tr><td>σ<sub>GP,TESS<sub>S9</sub></sub> (relative flux)</td><td>F(0)</td><td>-</td></tr><tr><td>ρ<sub>GP,TESS<sub>S9</sub></sub> (days)</td><td>F(0)</td><td>-</td></tr><tr><td>σ<sub>GP,TESS<sub>S36</sub></sub> (10<sup>-4</sup> relative flux)</td><td>logU(10<sup>-6</sup>, 100)</td><td>12<sup>+2</sup><sub>-2</sub></td></tr><tr><td>ρ<sub>GP,TESS<sub>S36</sub></sub> (days)</td><td>logU(0.001, 100)</td><td>0.94<sup>+0.13</sup><sub>-0.11</sub></td></tr><tr><td>σ<sub>GP,TESS<sub>S63</sub></sub> (10<sup>-4</sup> relative flux)</td><td>logU(10<sup>-6</sup>, 100)</td><td>57<sup>+44</sup><sub>-22</sub></td></tr><tr><td>ρ<sub>GP,TESS<sub>S63</sub></sub> (days)</td><td>logU(0.001, 100)</td><td>9.11<sup>+3.37</sup><sub>-2.77</sub></td></tr><tr><td colspan="3"><strong>Instrumental RV parameters</strong></td></tr><tr><td>Jitter, σ<sub>w,ESPRESSO</sub> (m s<sup>-1</sup>)</td><td>logU(0.001, 100)</td><td>1.88<sup>+0.29</sup><sub>-0.25</sub></td></tr><tr><td>Systemic RV, μ<sub>ESPRESSO</sub> (m s<sup>-1</sup>)</td><td>U(-3500, -1000)</td><td>-2973.61<sup>+0.36</sup><sub>-0.36</sub></td></tr><tr><td colspan="3"><strong>Limb darkening parameters</strong></td></tr><tr><td>q<sub>1,ESPRESSO</sub></td><td>N(0.408, 0.017)</td><td>0.407<sup>+0.015</sup><sub>-0.015</sub></td></tr><tr><td>q<sub>2,ESPRESSO</sub></td><td>N(0.349, 0.022)</td><td>0.345<sup>+0.020</sup><sub>-0.019</sub></td></tr><tr><td>q<sub>1,CHEOPS</sub></td><td>N(0.519, 0.012)</td><td>0.518<sup>+0.011</sup><sub>-0.011</sub></td></tr><tr><td>q<sub>2,CHEOPS</sub></td><td>N(0.408, 0.012)</td><td>0.406<sup>+0.011</sup><sub>-0.011</sub></td></tr><tr><td>q<sub>1,LCO</sub></td><td>N(0.324, 0.008)</td><td>0.324<sup>+0.007</sup><sub>-0.007</sub></td></tr><tr><td>q<sub>2,LCO</sub></td><td>N(0.314, 0.014)</td><td>0.314<sup>+0.012</sup><sub>-0.012</sub></td></tr><tr><td>q<sub>1,ASTEP</sub></td><td>N(0.492, 0.016)</td><td>0.492<sup>+0.014</sup><sub>-0.014</sub></td></tr><tr><td>q<sub>2,ASTEP</sub></td><td>N(0.376, 0.017)</td><td>0.378<sup>+0.015</sup><sub>-0.016</sub></td></tr></tbody></table>

Notes: (a) For the priors, $N(μ, σ^2)$ indicates a normal distribution with mean $μ$ and variance $σ^2$, $U(a, b)$ a uniform distribution between $a$ and $b$, $logU(a, b)$ a log-uniform distribution between $a$ and $b$ and $F(a)$ a parameter fixed to value $a$. (b) The posterior estimate indicates the median value and then error bars for the 68% credibility intervals.
## Appendix F: CHEOPS alias hunting

<table><caption>Table F.1: CHEOPS photometric observations of TOI-815.</caption><thead><tr><th>Visit</th><th>Expected planet</th><th>Start time<br>[UTC]</th><th>Duration<br>[h]</th><th>Eff.<br>[%]</th><th>RMS<br>ppm</th><th>Alias<br>[d]</th><th>Detection</th><th>File key</th><th>Detrending<sup>(d)</sup></th></tr></thead><tbody><tr><td>1</td><td>c</td><td>2022-03-06 00:36</td><td>8.34</td><td>73</td><td>422</td><td>22.26</td><td>no</td><td>CH_PR110048_TG018001_V0300</td><td>(e)</td></tr><tr><td>2</td><td>c</td><td>2022-03-06 16:26</td><td>8.37</td><td>60</td><td>461</td><td>20.99</td><td>no</td><td>CH_PR110048_TG018901_V0300</td><td>(e)</td></tr><tr><td>3</td><td>c</td><td>2022-03-17 04:35</td><td>9.84</td><td>61</td><td>417</td><td>22.95</td><td>no</td><td>CH_PR110048_TG018101_V0300</td><td>(e)</td></tr><tr><td>4<sup>(a)</sup></td><td>b</td><td>2022-03-18 08:36</td><td>11.64</td><td>66</td><td>514</td><td>11.20</td><td>yes</td><td>CH_PR100031_TG050801_V0300</td><td>t, x, y, y<sup>2</sup>, sin(φ), cos(φ), cos(2φ)</td></tr><tr><td>5</td><td>c</td><td>2022-04-13 01:20</td><td>8.47</td><td>60</td><td>429</td><td>17.92</td><td>no</td><td>CH_PR110048_TG025201_V0300</td><td>(e)</td></tr><tr><td>6</td><td>c</td><td>2022-04-30 06:47</td><td>8.34</td><td>51</td><td>497</td><td>29.38</td><td>no</td><td>CH_PR110048_TG027601_V0300</td><td>t (clear flux trend in data)</td></tr><tr><td>7</td><td>c</td><td>2023-02-15 11:09</td><td>8.29</td><td>69</td><td>520</td><td>31.93</td><td>no</td><td>CH_PR110048_TG036501_V0300</td><td>(e)</td></tr><tr><td>8</td><td>c</td><td>2023-02-22 01:40</td><td>9.04</td><td>68</td><td>447</td><td>25.33</td><td>no</td><td>CH_PR110048_TG036701_V0300</td><td>(e)</td></tr><tr><td>9</td><td>c</td><td>2023-02-27 14:48</td><td>8.22</td><td>64</td><td>580</td><td>19.85</td><td>no</td><td>CH_PR110048_TG036901_V0300</td><td>(e)</td></tr><tr><td>10<sup>(b)</sup></td><td>b</td><td>2023-02-28 14:16</td><td>8.39</td><td>62</td><td>460</td><td>18.83</td><td>yes</td><td>CH_PR110048_TG037001_V0300</td><td>t<sup>2</sup>, x, y, y<sup>2</sup>, bg, sin(φ), sin(2φ), cos(2φ)</td></tr><tr><td>11<sup>(c)</sup></td><td>c</td><td>2023-04-23 08:31</td><td>10.09</td><td>58</td><td>462</td><td>34.98</td><td>yes</td><td>CH_PR110048_TG042201_V0300</td><td>t, t<sup>2</sup>, y, y<sup>2</sup>, bg, cos(φ)</td></tr></tbody></table>

**Notes:**

(a) Figure 7, left.

(b) Figure 7, center.

(c) Figure 7, right.

(d) The correlation terms were determined using `pycheops`.

(e) Detrended with: x, y, bg, cos(φ), cos(2φ), cos(3φ), ΔT, sin(φ), sin(2φ), sin(3φ), bg<sup>2</sup>, x<sup>2</sup>, y<sup>2</sup>

<figure><img src="image_18.png" /></figure>

Fig. F.1: CHEOPS unsuccessful transit observations that ruled out the most likely period aliases of TOI-815c. The data have been detrended using the detrending terms presented in Table F.1.
## Appendix F: CHEOPS alias hunting

<table><caption>Table F.1: CHEOPS photometric observations of TOI-815.</caption><thead><tr><th>Visit</th><th>Expected planet</th><th>Start time [UTC]</th><th>Duration [h]</th><th>Eff. [%]</th><th>RMS ppm</th><th>Alias [d]</th><th>Detection</th><th>File key</th><th>Detrending<sup>(d)</sup></th></tr></thead><tbody><tr><td>1</td><td>c</td><td>2022-03-06 00:36</td><td>8.34</td><td>73</td><td>422</td><td>22.26</td><td>no</td><td>CH_PR110048_TG018001_V0300</td><td>(e)</td></tr><tr><td>2</td><td>c</td><td>2022-03-06 16:26</td><td>8.37</td><td>60</td><td>461</td><td>20.99</td><td>no</td><td>CH_PR110048_TG018901_V0300</td><td>(e)</td></tr><tr><td>3</td><td>c</td><td>2022-03-17 04:35</td><td>9.84</td><td>61</td><td>417</td><td>22.95</td><td>no</td><td>CH_PR110048_TG018101_V0300</td><td>(e)</td></tr><tr><td>4<sup>(a)</sup></td><td>b</td><td>2022-03-18 08:36</td><td>11.64</td><td>66</td><td>514</td><td>11.20</td><td>yes</td><td>CH_PR100031_TG050801_V0300</td><td>t, x, y, y<sup>2</sup>, sin(φ), cos(φ), cos(2φ)</td></tr><tr><td>5</td><td>c</td><td>2022-04-13 01:20</td><td>8.47</td><td>60</td><td>429</td><td>17.92</td><td>no</td><td>CH_PR110048_TG025201_V0300</td><td>(e)</td></tr><tr><td>6</td><td>c</td><td>2022-04-30 06:47</td><td>8.34</td><td>51</td><td>497</td><td>29.38</td><td>no</td><td>CH_PR110048_TG027601_V0300</td><td>t (clear flux trend in data)</td></tr><tr><td>7</td><td>c</td><td>2023-02-15 11:09</td><td>8.29</td><td>69</td><td>520</td><td>31.93</td><td>no</td><td>CH_PR110048_TG036501_V0300</td><td>(e)</td></tr><tr><td>8</td><td>c</td><td>2023-02-22 01:40</td><td>9.04</td><td>68</td><td>447</td><td>25.33</td><td>no</td><td>CH_PR110048_TG036701_V0300</td><td>(e)</td></tr><tr><td>9</td><td>c</td><td>2023-02-27 14:48</td><td>8.22</td><td>64</td><td>580</td><td>19.85</td><td>no</td><td>CH_PR110048_TG036901_V0300</td><td>(e)</td></tr><tr><td>10<sup>(b)</sup></td><td>b</td><td>2023-02-28 14:16</td><td>8.39</td><td>62</td><td>460</td><td>18.83</td><td>yes</td><td>CH_PR110048_TG037001_V0300</td><td>t<sup>2</sup>, x, y, y<sup>2</sup>, bg, sin(φ), sin(2φ), cos(2φ)</td></tr><tr><td>11<sup>(c)</sup></td><td>c</td><td>2023-04-23 08:31</td><td>10.09</td><td>58</td><td>462</td><td>34.98</td><td>yes</td><td>CH_PR110048_TG042201_V0300</td><td>t, t<sup>2</sup>, y, y<sup>2</sup>, bg, cos(φ)</td></tr></tbody></table>

**Notes:**

(a) Figure 7, left.

(b) Figure 7, center.

(c) Figure 7, right.

(d) The correlation terms were determined using `pycheops`.

(e) Detrended with: x, y, bg, cos(φ), cos(2φ), cos(3φ), ΔT, sin(φ), sin(2φ), sin(3φ), bg<sup>2</sup>, x<sup>2</sup>, y<sup>2</sup>

<figure><img src="image_19.png" /><figcaption>Fig. F.1: CHEOPS unsuccessful transit observations that ruled out the most likely period aliases of TOI-815c. The data have been detrended using the detrending terms presented in Table F.1.</figcaption></figure>