R_{\rm Hill} \simeq a_{\rm p} \, (1-ecc) \, ( M_{\rm p} / (3 (M_{\rm p} + M_{\star}))  )^{1/3}
\label{eq: R Hill}
\end{equation}

As in Eq. <ref>, we did not take into account planet's eccentricity.
Compared to , $R_{\rm He}/R_{\rm Hill}$ takes into account the Hei absorption peak instead of the EW of the signal, and the relation between masses of the system and the $a_{\rm p}$ instead of the star radius.
Figure <ref> presents $R_{\rm He}/R_{\rm Hill}$ as function of the . As it is shown, Hei detections are aligned in the diagram, hinting that the dimensionless parameter $R_{\rm He}/R_{\rm Hill}$ is a good indicator to investigate Hei correlations. $R_{\rm He}/R_{\rm Hill}$– have a Pearson correlation r = 0.9034, and we obtained really similar correlation with – (r = 0.8985).
For a more quantitative analysis, we fitted the trend that seems to be between $R_{\rm He}/R_{\rm Hill}$ and . Using the same code as in Sect. <ref>, we fitted the equation $R_{\rm He}/R_{\rm Hill}\,=\, A \, \log_{10}($) + $\log_{10}(B)$ and we get A = 0.213$\pm$0.027 and B = 2.57$\pm$0.15.

We find that gas giant planets are able to fill a larger fraction of their Hill radius (or Hill sphere) than mini-Neptunes. However, Hei detections on mini-Neptunes have comparable $R_{\rm He}/R_{\rm Hill}$ values to those from larger planets like HD 189733 b or TOI-1268 b. This result could point to an observational bias for small planet observations where we are only detecting those exoplanets that their He signal is $R_{\rm He}/R_{\rm Hill}$ $\gtrsim$0.2.
Moreover, compared to Fig. <ref>, some planets move from one side to the other of the line defined by the Hei detections, finding planets coded as non-detections above the line or upper limits under the line. TOI-1807 b is a clear example of a non-detection above the line. In particular, the planets with low  value seems to be consistently above the line, which could indicate another bias as well. But, this bias could be more related with the current capabilities of getting more precise measurements. Since they are likely low irradiated by  from the host star, the amount of atoms of He detectable via the NIR triplet might be also very low and really challenging thus for the current telescopes and instruments. The recent Hei detection on TOI-2134 b (Zhang et al., 2023) is consistently in the lower corner on Figures <ref> and <ref>, and might indicate the current limit in terms of signal detectability.

§.§ Hei detections across stellar types

Histogram of Hei triplet observations as function of spectral types (M, K, G, F, and O, B and A). The number of planets with detections, non-detections, and non-conclusive observations are marked by the blue, red, and orange lines, respectively. We indicate the total number of inspected planets in black on top of each column. We indicate the percentage of detections for each spectral type in blue.

Oklopčić, 2019 indicated that late-type stars are the most favorable host stars to detect the Hei triplet on their close-in transiting exoplanets, specially the K-type stars. We explored this theoretical prediction computing the histograms of detections, non-detections and non-conclusive Hei triplet observations as function of the different spectral types. Figure <ref> shows that K-type exoplanet hosting stars are the most targeted ones (29 planets), but also the spectral type with a larger detection percentage (45%). In fact, the two detections presented in this work, TOI-1268 b and TOI-2018 b, are both around K stars. For M-type stars, GJ 3470 b is the only clear Hei triplet detection (Palle et al., 2020, Ninan et al., 2020), along with the tentative detection on GJ 1214 b reported by Orell-Miquel et al., 2022.
The three detections around F-type stars (HD 209458 b, Alonso-Floriano et al., 2019; HAT-P-67 b, Gully-Santiago et al., 2023; and HAT-P-32 b, Czesla et al., 2022) are interesting because these planets are hot Jupiters, and HAT-P-67 b and HAT-P-32 b possess very extended He structures (Gully-Santiago et al., 2023, Zhang et al., 2023). To date, there are no Hei detections of exoplanets orbiting G, A, B, or O stars.
To further understand the dependence on spectral type, we also computed the histogram of Hei detections and non-detections as function of the stellar mass (Fig. <ref>). The number of detections increases from 0.47 $M_{\odot}$ (GJ 3470) until peaking at 0.9 $M_{\odot}$. Above 0.9 $M_{\odot}$, non-detections are dominant over detections. Therefore, we can estimate a limit in host stellar mass at $\sim$0.9 $M_{\odot}$ for the Hei detections, which deserves further investigation.

We note that all the Hei detections from Table <ref> are found in planets orbiting stars with $T_{\rm eff}$ $\lesssim$ 6250 K, which is the temperature at which the Kraft Break happens (Kraft, 1967). Exoplanet spin-orbit alignment seems to be related to the Kraft Break (e.g., Winn et al., 2010, Brown et al., 2017, Attia et al., 2023). To date, the hottest host star with Hei detection is HAT-P-67 with $T_{\rm eff}$ $\sim$ 6400 K. The observations of other ten planets orbiting $\gtrsim$6250 K stars resulted in upper limits only. The Kraft Break marks also the transition between stars with outer convection zones and those without. These convection zones are predicted to generate magnetic fields able to heat the chromosphere and the corona, increasing the XUV radiation (Wright et al., 2011). Therefore, stars without this heating mechanism are less likely to significantly populate the He metastable level of their exoplanets' atmospheres. The link between the stellar $T_{\rm eff}$, the flux in the XUV range and the Hei triplet detections deserves further investigation.

Equilibrium temperature ($T_{\rm eq}$) vs planet density ($\rho_{\rm p}$) diagram for the Hei detections (blue circles) and non-detections (red crosses). The black line (drawn `by-hand') indicates the hinted upper boundary for the Hei detections. Coloured dashed vertical lines indicate the Earth's density ($\rho_{\oplus}$; orange line at $\sim$5.5 g cm$^{-3}$), and a representative density where the puffy planet population starts ($\rho_{\rm puffy}$; blue line at $\sim$1.3 g cm$^{-3}$; Luque & Pallé, 2022). All known planets with $T_{\rm eq}$ and $\rho_{\rm p}$ determined with a precision better than 50% are marked with grey dots (data from NASA Exoplanet Archive).

§.§ Hei dependence on planetary parameters

One of the aims of the MOPYS project was also to find a relation between planetary and stellar properties that could explain, and predict, the Hei detections and non-detections, but without the need to involve the strength of the signal itself. From the many explored relations and diagrams, only the equilibrium temperature ($T_{\rm eq}$) and the planet density ($\rho_{\rm p}$) showed some hints of such relationship. Figure <ref> presents the $T_{\rm eq}$ vs $\rho_{\rm p}$ diagram where the information about the Hei observations is encoded in the colour markers (blue circles are detections, red crosses are non-detections, and non-conclusive observations are not shown). Although both are planet properties, $T_{\rm eq}$ encapsulates information about the star via the $T_{\rm eff}$ and the $a_{\rm p}/R_{\star}$ parameters.

From Fig. <ref>, one clear result is that there are no Hei detections on exoplanets with $T_{\rm eq}$ $\gtrsim$ 2000 K. The hottest detection at $T_{\rm eq}$ $\sim$ 2000 K comes from the puffy planet HAT-P-67 b. In fact, the data in Figure <ref> hint to a density/$T_{\rm eq}$ upper boundary for the detections. We draw `by-hand' this observational limit for a better visualisation, which follows $T_{\rm eq}$ $\simeq$ $-$500 log$_{10}$($\rho_{\rm p}$) trend, approximately. If we assume that detections and non-detections are distributed randomly across the $T_{\rm eq}$–$\rho_{\rm p}$ parameter space, the probability of extracting 14 non-detections from a pool of 17 detections and 36 non-detections is only 0.16 %. Although it is an approximate calculation, the very low probability supports the robustness of the boundary. For planets falling above the line it is unlikely that one can detect Hei, as all the inspected planets resulted in non-detections. Below that line, Hei detections and non-detections are equally found.

At face value our results indicate that for high-density planets (from Earth-size to sub-Neptunes) Hei absorptions are present only in cool atmospheres. A possible interpretation is that these type of planets quickly lose any H/He atmosphere when subjected to strong stellar instellation (assuming that $T_{\rm eq}$ is directly related at the population level to stellar flux). For low-density planets (gas giants) Hei detection are found up to $\sim$ 2000 K.

Further Hei detections will confirm or disprove the observational boundary proposed here, and define its exact dependence on different planet parameters.

§ CONCLUSIONS

This work shows the first results of the MOPYS survey, a project that aims to test the predictions of atmospheric evolution theories by confronting them with observations of evaporation proxies. We present CARMENES and GIARPS high-resolution transmission spectroscopy observations of 20 exoplanets. We analysed their atmospheres searching for Hei triplet absorption in the NIR, and H$\alpha$ in the VIS as well. We report two new detections of the Hei triplet for TOI-1268 b and TOI-2018 b, and a new H$\alpha$ detection for TOI-1136 d. We also found hints of Hei on HD 63433 b, and of H$\alpha$ on 73 b and c, which need further confirmation.

We complemented our target list with other planets from the literature with Hei triplet and H$\alpha$ (and Ly$\alpha$ for some particular planets as well) observations, for a total of 70 planets. We considered those lines as proxy of evaporation to test mass-loss theories. The main findings of our study are:

* Our age distribution of evaporation detections does not favor either photo-evaporation or core-powered mass-loss as planet formation mechanisms.

* We find no trend in atmospheric evaporation with stellar age. Young ($<$1-Gyr-old) planets do not exhibit more Hei or H$\alpha$ detections than older planets.

* Evaporation (Hei or H$\alpha$) signals are more frequent or easier to detect for planets around stellar hosts within $\sim$1–3 Gyr ages.

* The fraction of planets that show Hei detections is much larger for K-type stars (45%) than for any other spectral type.

* We find no evaporation detections of young and old planets below the radius gap, confirming rocky planets are not under extreme evaporation, at least after the first $\sim$300 Myr.

* We find hints that evaporation of puffy sub-Neptunes ($R_{\rm p}$ $>$ 3 $R_{\oplus}$) happens at  ages older than 1 Gyr, although the number of planets is small, and a larger sample is needed to confirm this finding.

* We provide new constrains to the cosmic shoreline, by using the evaporation detections as evidences for the existence of planetary atmospheres. Our cosmic shoreline move to higher irradiation flux, reducing the parameter space of bare rocky planets.

* We present the He-related dimensionless parameter $R_{\rm He}/R_{\rm Hill}$, as a new valid parameter to study the Hei detections and upper limits.

* All the Hei detections are found in planets orbiting stars with $T_{\rm eff}$ $\lesssim$ 6250 K, which is the temperature at which the Kraft Break happens.

* We determine a statistically significant observational upper boundary for Hei detections in the $T_{\rm eq}$ vs $\rho_{\rm p}$ parameter space. The line decays at a rate of $T_{\rm eq}$ $\simeq$ $-$500 log$_{10}$($\rho_{\rm p}$), approximately. Planets falling above that boundary are unlikely to show Hei absorption signals.

We encourage further evaporation observations (Ly$\alpha$, H$\alpha$, and Hei triplet) to increase and complete the sample presented in this work.
Although this work represents the biggest evaporation survey to date (specially focused on $<$1Gyr-old planets), some key questions of planet formation and atmospheric evaporation remained unanswered. In particular, the detection and atmospheric characterisation of really young planets ($<$100 Myr) is key for discriminating between different mass-loss processes in planetary atmospheres. It is precisely because of their age that they are the most difficult planets to analyse their atmospheres.

CARMENES is an instrument at the Centro Astronómico Hispano en Andalucía (CAHA) at Calar Alto (Almería, Spain), operated jointly by the Junta de Andalucía and the Instituto de Astrofísica de Andalucía (CSIC).

The authors wish to express their sincere thanks to all members of the Calar Alto staff for their expert support of the instrument and telescope operation.

CARMENES was funded by the Max-Planck-Gesellschaft (MPG),
the Consejo Superior de Investigaciones Científicas (CSIC),
the Ministerio de Economía y Competitividad (MINECO) and the European Regional Development Fund (ERDF) through projects FICTS-2011-02, ICTS-2017-07-CAHA-4, and CAHA16-CE-3978,
and the members of the CARMENES Consortium
(Max-Planck-Institut für Astronomie,
Instituto de Astrofísica de Andalucía,
Landessternwarte Königstuhl,
Institut de Ciències de l'Espai,
Institut für Astrophysik Göttingen,
Universidad Complutense de Madrid,
Thüringer Landessternwarte Tautenburg,
Instituto de Astrofísica de Canarias,
Hamburger Sternwarte,
Centro de Astrobiología and
Centro Astronómico Hispano-Alemán),
with additional contributions by the MINECO,
the Deutsche Forschungsgemeinschaft (DFG) through the Major Research Instrumentation Programme and Research Unit FOR2544 “Blue Planets around Red Stars”,
the Klaus Tschira Stiftung,
the states of Baden-Württemberg and Niedersachsen,
and by the Junta de Andalucía.

Based on observations collected at the Centro Astronómico Hispano-Alemán (CAHA) at Calar Alto, proposals 21B-3.5-004, 22A-3.5-007, 22B-3.5-009, 23A-3.5-009, and 23B-3.5-002, operated jointly by Junta de Andalucía and Consejo Superior de Investigaciones Científicas (IAA-CSIC). We used data from the CARMENES data archive at CAB (CSIC-INTA).

We acknowledge the use of the ExoAtmospheres database during the preparation of this work

This work is partly supported by JSPS KAKENHI Grant Number JPJP24H00017 and JSPS Bilateral Program Number JPJSBP120249910.
This paper is based on observations made with the MuSCAT2 instrument, developed by ABC, at Telescopio Carlos Sánchez operated on the island of Tenerife by the IAC in the Spanish Observatorio del Teide.

We acknowledge financial support from the Agencia Estatal de Investigación (AEI/10.13039/501100011033) of the Ministerio de Ciencia e Innovación and the ERDF “A way of making Europe” through projects
Ariel Postdoctoral Fellowship program of the Swedish National Space Agency (SNSA),
and the Centre of Excellence “Severo Ochoa” and “María de Maeztu” awards to the Instituto de Astrofísica de Canarias (CEX2019-000920-S), Instituto de Astrofísica de Andalucía (CEX2021-001131-S) and Institut de Ciències de l'Espai (CEX2020-001058-M).

J.O-M agraeix el recolzament, suport i ànims que sempre ha rebut per part de Padrina Conxa, Padrina Mercè, Jeroni, Mercè i més familiars i amics.
J.O.M. gratefully acknowledge the inspiring discussions with Maite Mateu, Alejandro Almodóvar, and Joan Perelló, and the support from Guillem, Benet, and Montse.
This research has made use of resources from VallAlbaida-Mallorca collaboration. J.O.M. acknowledges the contributions and patience of Jorge Terol Calvo, i molt especialment a tu, Yess.

L.N. and F.L. acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project number 314665159.
E.K. and S.H.A. acknowledge the support from the Danish Council for Independent Research through grant No.2032-00230B.
E.N. acknowledges the support by the DFG Research Unit FOR2544 “Blue Planets around Red Stars”.
S.C. acknowledges the support of the DFG priority program SPP 1992 “Exploring the Diversity of Extrasolar Planets” (CZ 222/5-1).

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§ ADDITIONAL FIGURES

Same as Fig. <ref> for MASCARA-2 b observations with CARMENES.

Same as Fig. <ref> for V1298 Tau c observations with CARMENES.

Same as Fig. <ref> for TOI-1431 b observations with CARMENES.

Same as Fig. <ref> for TOI-2048 b observations with CARMENES.

Same as Fig. <ref> for HD 63433 b observations with CARMENES.

Same as Fig. <ref> for HD 63433 c observations with CARMENES.

Same as Fig. <ref> for HD 73583 b observations with HARPS-N.

Same as Fig. <ref> for HD 73583 c observations with GIARPS.

Same as Fig. <ref> for HD 235088 b observations with CARMENES.

Same as Fig. <ref> for K2-77 b observations with CARMENES.

Same as Fig. <ref> for TOI-2046 b observations with GIARPS.

Same as Fig. <ref> for TOI-1807 b observations with CARMENES.

Same as Fig. <ref> for TOI-1136 d H$\alpha$ observations with HARPS-N (14-May-2021) and CARMENES VIS (30-Jan-2023).

Same as Fig. <ref> for TOI-2076 b observations with CARMENES.

Same as Fig. <ref> for TOI-1683 b observations with GIANO-B.

Same as Fig. <ref> for WASP-189 b observations with GIANO-B.

Same as Fig. <ref> for HAT-P-57 b observations with CARMENES.

Same as Fig. <ref> for TOI-2018 b individual observations of H$\alpha$ with HARPS-N.

Same as Fig. <ref> for TOI-2018 b idividual observations of HeI triplet with GIANO-B.

Time evolution of the activity indicators for K2-100 b (top left), V1298 Tau c (top right), TOI-2048 b (mid left), HD 63433 b (1 November 2021, mid right; and 27 November 2022, low left), and HD 63433 c (low right). The vertical dashed and dotted lines represent the different contacts during the transit.

Same as Fig. <ref> for HD 73583 b (top left), HD 73583 b (top right), K2-77 b (mid left), TOI-2046 b (mid right), and TOI-1807 b (16 December 2021, low left; and 23 December 2022, low right).

Same as Fig. <ref> for TOI-1136 d (14 May 2021, top left; and 30 January 2023, top right), TOI-1268 b (mid left), and TOI-2076 b (mid right), and TOI-2018 b (9 April 2022, low left; and 15 June 2022, low right).

Histogram of Hei triplet observations (detections: blue, non-detections: red) as function of stellar mass.

§ V1298 TAU EXTRA MATERIAL

Corner plot for the nested sampling posterior distribution of V1298 Tau c H$\alpha$ (left) and Hei (right) feature.

Prior and posterior distributions from the nested sampling fitting for V1298 Tau c H$\alpha$ feature (see Fig. <ref>). Prior label $\mathcal{U}$ represents uniform distribution.

Parameter     Prior     Posterior

Absorption [%]     $\mathcal{U}(-2, 0)$     $-$1.10$\pm$0.13

$\lambda_0$ [Å]     $\mathcal{U}(6560,6570)$     6564.90$^{+0.11}_{-0.10}$

$\sigma$ [Å]      $\mathcal{U}(0,1)$     0.55$^{+0.10}_{-0.08}$

$\Delta$v [km s$^{-1}$]     –     12$^{+5}_{-4}$

FWHM [Å]     –      1.30$^{+0.24}_{-0.17}$

EW [mÅ]      –     14.8$^{+1.7}_{-1.6}$

Prior and posterior distributions from the nested sampling fitting for V1298 Tau c HeI feature (see Fig. <ref>). Prior label $\mathcal{U}$ represents uniform distribution.

Parameter     Prior     Posterior

Absorption [%]     $\mathcal{U}(-4.5,4.5)$      $-$3.75$\pm$0.12

$\lambda_0$ [Å]     $\mathcal{U}(10830,10835)$     10832.18$\pm$0.04

$\sigma$ [Å]      $\mathcal{U}(0.0,2)$     1.02$\pm$0.04

$\Delta$v [km s$^{-1}$]     –      $-$28.7$\pm$1.1

FWHM [Å]     –      2.40$\pm$0.10

EW [mÅ]      –     95.9$^{+3.2}_{3.0}$

TS around the H$\alpha$ (left) and HeI triplet (right) lines for the transit phases from top to bottom (consecutively offset): start (blue), centre (red), end (green), and post-transit (orange). Master-Out spectrum is overplotted in black along with the individual TS for comparison. Dotted vertical lines indicate the lines positions. The wavelengths in this figure are given in vacuum.

§ TOI-2048 EXTRA MATERIAL

We analysed the TESS data from Sectors 16, 23, 24, 50, and 51 with the procedure described in Sect. <ref>. We used the  GP quasi-periodic kernel to account for the young star variability and we adopted the stellar parameters from Newton et al., 2022 for a proper comparison.

The fitted parameters with their prior and posterior values, and the derived parameters for TOI-2048 b are shown in Table <ref>. The TESS data along with the best transiting and GP models is shown in Fig. <ref> and TOI-2048 b phase folded transit is shown in Fig. <ref>. We obtained similar results using the  GP exponential kernel.

To derive an estimation of TOI-2048 b's semi-amplitude $K_{\star}$, first we forecasted its mass using the probabilistic mass-radius relationship for sub-neptune-sized planets (${R}_{\mathrm{p}}$ < 4 ${R}_\oplus$) of Wolfgang et al., 2016. We predicted a planetary mass of $\sim$9 $\pm$ 2 ${M}_\oplus$, which is translated into $K_{\star}$ $\simeq$ 2.8 $\pm$ 0.6 m s$^{-1}$ using the equation

\begin{equation}
K_{\star} = 28.4\,{\rm m\,s^{-1}} ~ ( P_{\mathrm{pl}}/ {\rm year} )^{-1/3} ~  ( M_{\mathrm{pl}} / {\rm M_{\mathrm{Jup}}} ) ~ ( M_{\star}/{\rm M_{\odot}} )^{-2/3}
\label{eq: Semi-amplitude}
\end{equation}

TOI-2048 two-min cadence SAP TESS photometry from Sectors 16, 23, 24, 50, and 51 along with the transit plus GP model. Upward-pointing red triangles mark the transits for TOI-2048 b.

Prior and posterior distributions from the  fitting for TOI-2048 b. Prior labels $\mathcal{U}$, $\mathcal{N}$, $\mathcal{F}$, and $\mathcal{J}$ represents uniform, normal, fixed, and Jeffrey's distribution, respectively.

Parameter     Prior     Posterior

$P$ [d]     $\mathcal{N}(13.7905,0.0001)$     13.790546 (55)

$t_0$ $^{(a)}$     $\mathcal{N}(1739.11,0.01)$     1739.1123 (27)

ecc     $\mathcal{F }( 0)$     –

$\omega$ (deg)     $\mathcal{F }( 90)$     –

$r_{1}$     $\mathcal{U }(0,1 )$     0.54$\pm$0.12

$r_{2}$     $\mathcal{ U}(0,1 )$     0.0306$\pm$0.0018

$\rho_{\star}$ [kg m$^{-3}$]      $\mathcal{ N }(2367.0,500.0 )$     2552$^{+350}_{-420}$

$\mu_{\textit{TESS}}$ (ppm)     $\mathcal{N }(0.0,0.1)$     $-$100$^{+200}_{-180}$

$\sigma_{\textit{TESS}}$ (ppm)     $\mathcal{J }( 10^{-6}, 10^{6})$     5600$^{+14000}_{-5600 }$

$q_{1,\textit{TESS}}$     $\mathcal{U }(0,1 )$     0.51$\pm$0.28

$q_{2,\textit{TESS}}$     $\mathcal{ U}(0,1 )$     0.46$\pm$0.30

GP$_\mathrm{B}$ (ppm)     $\mathcal{J }(10^{-6}, 10^{6})$     33$^{+16}_{-10}$

GP$_\mathrm{L}$ [d]      $\mathcal{J }(10^{-3}, 10^{3})$     43.5$^{+20}_{-12}$

GP$_\mathrm{C}$ (ppm)     $\mathcal{J }(10^{-6}, 10^{6})$     200$^{+ 40000}_{-180}$

GP$_\mathrm{P_{\mathrm{rot}}}$ [d]     $\mathcal{N }(8,2)$     7.48$\pm$0.12

3c Derived planetary parameters

$p = {R}_{\mathrm{p}}/{R}_{\star}$          0.0306$\pm$0.0018

$b =(a_{\rm p}/{R}_{\star}) \cos{ i_{\mathrm{p}} }$           0.30$^{+0.16}_{-0.18}$

$i_{\mathrm{p}}$ (deg)          89.41$\pm$0.35

$T_{14}$ [h]           3.5$\pm$0.1

$T_{12}$ [min]           6.9$^{+1.2}_{-0.7}$

$R_{\mathrm{p}}$ [${R}_\oplus$]          2.60$\pm$0.20

$a_{\mathrm{p}}$ [AU]           0.1078$\pm$0.0080

$T_{\mathrm{eq}}$ [K] $^{(b)}$           675$^{+22}_{-16}$

$^{(a)}$ Central time of transit ($t_0$) units are BJD $-$ 2 457 000.
$^{(b)}$ Equilibrium temperatures were calculated assuming zero Bond albedo.

TOI-2048 TESS photometry (blue dots with error bars) phase-folded to the period $P$ and central time of transit $t_0$ (shown above the panel, $t_0$ units are BJD $-$ 2 457 000) derived from the  fit. The black line is the best transit model for TOI-2048 b. Orange points show binned photometry for visualisation.

§ HD 63433 SYSTEM EXTRA MATERIAL

Contribution of the RM and CLV effects to HD 63433 b's transmission spectra around the H$\alpha$ (top) and HeI NIR triplet (bottom) lines. Vertical cyan lines mark the position of the lines of interest.

Prior and posterior distributions from the nested sampling fitting for HD 63433 b H$\alpha$ feature (see Fig. <ref>). Prior label $\mathcal{U}$ represents uniform distribution.

Parameter     Prior     Posterior

Absorption [%]     $\mathcal{U}(-3, 3)$     $-$1.21$^{+0.22}_{-0.24}$

$\lambda_0$ [Å]     $\mathcal{U}(6560.0,6570.0)$     6564.60$\pm$0.03

$\sigma$ [Å]      $\mathcal{U}(0.01,0.5)$     0.15$^{+0.04}_{-0.03}$

$\Delta$v [km s$^{-1}$]     –     $-$2.0$\pm$1.5

FWHM [Å]     –      0.35$^{+0.09}_{-0.07}$

EW [mÅ]      –     4.6$\pm$0.8

Corner plot for the nested sampling posterior distribution of HD 63433 b H$\alpha$ feature from partial transit.

§ HD 73583 SYSTEM EXTRA MATERIAL

We analaysed 73 TESS data from Sectors 8, 34, and 61. Because our purpose for this system is not to study the young stellar activity but derive precise ephemeris, we used the  GP exponential kernel to account for the stellar variability and rotation. We adopted the stellar parameters used in Barragán et al., 2022.

The fitted parameters with their prior and posterior values, and the derived parameters for 73 b and c are shown in Table <ref>. The TESS data along with the best transiting and GP models is shown in Fig. <ref> and 73 b and c phase folded transits are shown in Fig. <ref>.

Prior and posterior distributions from the  fitting for 73 b, and c. Prior labels $\mathcal{U}$, $\mathcal{N}$, $\mathcal{F}$, and $\mathcal{J}$ represents uniform, normal, fixed, and Jeffrey's distribution, respectively.

Parameter     Prior     Posterior

$P_b$ [d]     $\mathcal{N}(6.398,0.01)$     6.3980580 (26)

$t_{0,b}$ $^{(a)}$     $\mathcal{N}(2592.55,0.1)$     2592.56287$^{+0.00025}_{-0.00022}$

ecc$_b$     $\mathcal{F }(0)$     –

$\omega_b$ (deg)     $\mathcal{F }( 90)$     –

$r_{1,b}$     $\mathcal{U }(0,1 )$     0.718$^{+0.009}_{-0.012}$

$r_{2,b}$     $\mathcal{ U}(0,1 )$     0.0387$\pm$0.0004

$P_c$ [d]     $\mathcal{N}(18.8797,0.01)$     18.879300 (48)

$t_{0,c}$ $^{(a)}$     $\mathcal{N}(2949.6,0.1)$     2949.58243$^{+0.00077}_{-0.00094}$

ecc$_c$     $\mathcal{F }(0)$     –

$\omega_c$ (deg)     $\mathcal{F }( 90)$     –

$r_{1,c}$     $\mathcal{U }(0,1 )$     0.353$^{+0.015}_{-0.012}$

$r_{2,c}$     $\mathcal{ U}(0,1 )$     0.03301$\pm$0.00077

$\rho_{\star}$ [kg m$^{-3}$]      $\mathcal{ N }(3500.0,500.0)$     3750$^{+160}_{-110}$

$\mu_{\textit{TESS}}$ (ppm)     $\mathcal{N }(0.0,0.1)$     2.0$^{+1.6}_{-1.4}$ $\times$10$^{3}$

$\sigma_{\textit{TESS}}$ (ppm)     $\mathcal{J }( 10^{-6}, 10^{6})$     436.3$\pm$2.2

$q_{1,\textit{TESS}}$     $\mathcal{U }(0,1 )$     0.12$^{+0.06}_{-0.04}$

$q_{2,\textit{TESS}}$     $\mathcal{ U}(0,1 )$     0.54$^{+0.24}_{-0.27}$

GP$_\mathrm{\sigma}$ (ppm)     $\mathcal{J}(10^{-6}, 10^{6})$     8.5$^{+1.2}_{-0.9}$ $\times$10$^{3}$

GP$_\mathrm{\rho}$ [d]      $\mathcal{J}(10^{-3}, 10^{3})$     2.04$^{+0.20}_{-0.17}$

3c Derived planetary parameters for 73 b

$p_b = {R}_{\mathrm{p}}/{R}_{\star}$          0.0387$\pm$0.0004

$b_b =(a_{\rm p}/{R}_{\star}) \cos{ i_{\mathrm{p}} }$           0.577$^{+0.013}_{-0.018}$

$i_{\mathrm{p}, b}$ (deg)          88.35$^{+0.07}_{-0.05}$

$T_{14, b}$ [h]           2.100$^{+0.015}_{-0.013}$

$T_{12, b}$ [min]           6.93$^{+0.17}_{-0.22}$

$R_{\mathrm{p}, b}$ [${R}_\oplus$]          2.78$\pm$0.09

$a_{\mathrm{p}, b}$ [AU]           0.0618$\pm$0.0020

$T_{\mathrm{eq}, b}$ [K] $^{(b)}$           710$\pm$18

3c Derived planetary parameters for 73 c

$p_c = {R}_{\mathrm{p}}/{R}_{\star}$          0.0330$\pm$0.0008

$b_c =(a_{\rm p}/{R}_{\star}) \cos{ i_{\mathrm{p}} }$           0.030$^{+0.022}_{-0.017}$

$i_{\mathrm{p}, c}$ (deg)          89.96$\pm$0.03

$T_{14, c}$ [h]           3.60$^{+0.03}_{-0.05}$

$T_{12, c}$ [min]           6.90$\pm$0.20

$R_{\mathrm{p}, c}$ [${R}_\oplus$]          2.38$\pm$0.09

$a_{\mathrm{p}, c}$ [AU]           0.1270$\pm$0.0040

$T_{\mathrm{eq}, c}$ [K] $^{(b)}$           495$\pm$12

$^{(a)}$ Central time of transit ($t_0$) units are BJD $-$ 2 457 000.
$^{(b)}$ Equilibrium temperatures were calculated assuming zero Bond albedo.

73 TESS photometry (blue dots with error bars) phase-folded to the period $P$ and central time of transit $t_0$ (shown above each panel, $t_0$ units are BJD $-$ 2 457 000) derived for planet b (left), and c (right) from the  fit. The black line is the best transit model for each planet. Orange points show binned photometry for visualisation.

73 two-min cadence TESS photometry from Sectors 8, 34, and 61 along with the transit plus GP model. Upward-pointing red and green triangles mark the transit times for 73 b and c, respectively.

Corner plot for the nested sampling posterior distribution of 73 b (left) and 73 c (right) H$\alpha$ features.

Prior and posterior distributions from the nested sampling fitting for 73 b H$\alpha$ feature (see Fig. <ref>). Prior label $\mathcal{U}$ represents uniform distribution.

Parameter     Prior     Posterior

Absorption [%]     $\mathcal{U}(-3, 3)$     $-$0.46$\pm$0.16

$\lambda_0$ [Å]     $\mathcal{U}(6562,6566)$     6564.51$^{+0.07}_{-0.06}$

$\sigma$ [Å]      $\mathcal{U}(0,1)$     0.18$^{+0.11}_{-0.06}$

$\Delta$v [km s$^{-1}$]     –     $-$6.1$^{+3.2}_{-2.5}$

FWHM [Å]     –      0.43$^{+0.26}_{-0.14}$

EW [mÅ]      –     2.1$^{+0.7}_{-0.6}$

Prior and posterior distributions from the nested sampling fitting for 73 c H$\alpha$ feature (see Fig. <ref>). Prior label $\mathcal{U}$ represents uniform distribution.

Parameter     Prior     Posterior

Absorption [%]     $\mathcal{U}(-3, 3)$     $-$0.54$^{+0.13}_{-0.14}$

$\lambda_0$ [Å]     $\mathcal{U}(6562,6566)$     6564.65$^{+0.07}_{-0.08}$

$\sigma$ [Å]      $\mathcal{U}(0.12,1)$     0.30$^{+0.10}_{-0.09}$

$\Delta$v [km s$^{-1}$]     –     0.6$\pm$3.4

FWHM [Å]     –      0.68$^{+0.22}_{-0.21}$

EW [mÅ]      –     3.8$^{+1.0}_{-0.9}$

§ K2-77 B PLANETARY PARAMETERS

We analaysed K2-77 b TESS' data from Sectors 5, 42, 43, and 44. Because we did not see a clear rotation modulation in the light curves, we used the  GP exponential kernel to account for the stellar variability. We adopted the stellar parameters used in Gaidos et al., 2017.

The fitted parameters with their prior and posterior values, and the derived parameters for K2-77 b are shown in Table <ref>. The TESS data along with the best transiting and GP models is shown in Fig. <ref> and K2-77 b phase folded transit is shown in Fig. <ref>.

As we did in App. <ref>, we forecasted K2-77 b mass and we computed the estimated $K_{\star}$ using Eq. <ref>. We predicted a planetary mass of $\sim$9 $\pm$ 2 ${M}_\oplus$, and a semi-amplitude $K_{\star}$ of $\sim$3.4 $\pm$ 0.8 m s$^{-1}$. [Gaidos et al., 2017] only reported an upper limit to K2-77 b's mass of 1.9 ${M}_{J}$.

Prior and posterior distributions from the  fitting for K2-77 b. Prior labels $\mathcal{U}$, $\mathcal{N}$, $\mathcal{F}$, and $\mathcal{J}$ represents uniform, normal, fixed, and Jeffrey's distribution, respectively.

Parameter     Prior     Posterior

$P$ [d]     $\mathcal{N}(8.2,0.01)$     8.200139 (60)

$t_0$ $^{(a)}$     $\mathcal{N}(2522.6,0.1)$     2522.6338$^{+0.0037}_{-0.0033}$

ecc     $\mathcal{F }(0)$     –

$\omega$ (deg)     $\mathcal{F }( 90)$     –

$r_{1}$     $\mathcal{U }(0,1 )$     0.66$^{+0.09}_{-0.17}$

$r_{2}$     $\mathcal{ U}(0,1 )$     0.0309$^{+0.0022}_{-0.0025}$

$\rho_{\star}$ [kg m$^{-3}$]      $\mathcal{ N }(2577.0,500.0)$     2700$^{+450}_{-460}$

$\mu_{TESS}$ (ppm)     $\mathcal{N }(0.0,0.1)$     300$^{+810}_{-770}$

$\sigma_{TESS}$ (ppm)     $\mathcal{J }( 10^{-6}, 10^{6})$     4.5$^{+1700}_{-4.5}$ $\times$10$^{3}$

$q_{1,TESS}$     $\mathcal{U }(0,1 )$     0.55$^{+0.30}_{-0.34}$

$q_{2,TESS}$     $\mathcal{ U}(0,1 )$     0.52$^{+0.31}_{-0.33}$

GP$_\mathrm{\sigma}$ (ppm)     $\mathcal{J }(10^{-6}, 10^{6})$     4.2$^{+0.5}_{-0.4}$ $\times$10$^{3}$

GP$_\mathrm{\rho}$ [d]      $\mathcal{J }(10^{-3}, 10^{3})$     1.65$^{+0.17}_{-0.15}$

3c Derived planetary parameters

$p = {R}_{\mathrm{p}}/{R}_{\star}$          0.0309$^{+0.0022}_{-0.0025}$

$b =(a_{\rm p}/{R}_{\star}) \cos{ i_{\mathrm{p}} }$           0.50$^{+0.13}_{-0.25}$

$i_{\mathrm{p}}$ (deg)          88.7$^{+0.7}_{-0.4}$

$T_{14}$ [h]           2.68$\pm$0.22

$T_{12}$ [min]           6.3$^{+1.5}_{-1.1}$

$R_{\mathrm{p}}$ [${R}_\oplus$]          2.55$\pm$0.20

$a_{\mathrm{p}}$ [AU]           0.0751$^{+0.0045}_{-0.0050}$

$T_{\mathrm{eq}}$ [K] $^{(b)}$           760$^{+25}_{-21}$

$^{(a)}$ Central time of transit ($t_0$) units are BJD $-$ 2 457 000.
$^{(b)}$ Equilibrium temperatures were calculated assuming zero Bond albedo.

K2-77 TESS photometry (blue dots with error bars) phase-folded to the period $P$ and central time of transit $t_0$ (shown above the panel, $t_0$ units are BJD $-$ 2 457 000) derived from the  fit. The black line is the best transit model for K2-77 b. Orange points show binned photometry for visualisation.

K2-77 two-min cadence TESS photometry from Sectors 5, 42, 43, and 44 along with the transit plus GP model. Upward-pointing red triangles mark the transits for K2-77 b.

§ TOI-1807 B EXTRA MATERIAL

We analaysed TOI-1807 TESS' data from Sectors 22, 23, and 49. Because our purpose is not to model the young stellar activity but derive precise ephemerides, we used the unspecific  GP exponential kernel to account for the stellar variability and rotation. We adopted the stellar parameters used in Nardiello et al., 2022 for a proper comparison.

The fitted parameters with their prior and posterior values, and the derived parameters for TOI-1807 b are shown in Table <ref>. The TESS data along with the best transiting and GP models is shown in Fig. <ref> and TOI-1807 b phase folded transit is shown in Fig. <ref>.

Prior and posterior distributions from the  fitting for TOI-1807 b. Prior labels $\mathcal{U}$, $\mathcal{N}$, $\mathcal{F}$, and $\mathcal{J}$ represents uniform, normal, fixed, and Jeffrey's distribution, respectively.

Parameter     Prior     Posterior

$P$ [d]     $\mathcal{N}(0.54929,0.001)$     0.54937084 (65)

$t_0$ $^{(a)}$     $\mathcal{N}(2664.07,0.1)$     2664.06930$^{+0.00073}_{-0.00075}$

ecc     $\mathcal{F }(0)$     –

$\omega$ (deg)     $\mathcal{F }( 90)$     –

$r_{1}$     $\mathcal{U }(0,1 )$     0.70$^{+0.05}_{-0.06}$

$r_{2}$     $\mathcal{ U}(0,1 )$     0.01776$^{+0.00045}_{-0.00052}$

$\rho_{\star}$ [kg m$^{-3}$]      $\mathcal{ N }(3300.0,600.0)$     3430$^{+530}_{-550}$

$\mu_{\textit{TESS}}$ (ppm)     $\mathcal{N }(0.0,0.1)$     $-$0.6$^{+2.0}_{-2.0}$ $\times$10$^{3}$

$\sigma_{\textit{TESS}}$ (ppm)     $\mathcal{J }( 10^{-6}, 10^{6})$     500$^{+4.3}_{-4.1}$

$q_{1,\textit{TESS}}$     $\mathcal{U }(0,1 )$     0.15$^{+0.24}_{-0.11}$

$q_{2,\textit{TESS}}$     $\mathcal{ U}(0,1 )$     0.26$^{+0.36}_{-0.20}$

GP$_\mathrm{\sigma}$ (ppm)     $\mathcal{J}(10^{-6}, 10^{6})$     8.7$^{+1.5}_{-1.1}$ $\times$10$^{3}$

GP$_\mathrm{\rho}$ [d]      $\mathcal{J}(10^{-3}, 10^{3})$     2.30$^{+0.28}_{-0.25}$

3c Derived planetary parameters

$p = {R}_{\mathrm{p}}/{R}_{\star}$          0.01776$^{+0.00045}_{-0.00052}$

$b =(a_{\rm p}/{R}_{\star}) \cos{ i_{\mathrm{p}} }$           0.550$^{+0.077}_{-0.100}$

$i_{\mathrm{p}}$ (deg)          81.7$\pm$0.18

$T_{14}$ [h]           0.970$\pm$0.022

$T_{12}$ [min]           1.45$^{+0.25}_{-0.18}$

$R_{\mathrm{p}}$ [${R}_\oplus$]          1.33$\pm$0.08

$a_{\mathrm{p}}$ [AU]           0.0121$\pm$0.0009

$T_{\mathrm{eq}}$ [K] $^{(b)}$           1720$^{+55}_{-50}$

$^{(a)}$ Central time of transit ($t_0$) units are BJD $-$ 2 457 000.
$^{(b)}$ Equilibrium temperatures were calculated assuming zero Bond albedo.

TOI-1807 TESS photometry (blue dots with error bars) phase-folded to the period $P$ and central time of transit $t_0$ (shown above the panel, $t_0$ units are BJD $-$ 2 457 000) derived from the  fit. The black line is the best transit model for TOI-1807 b. Orange points show binned photometry for visualisation.

TOI-1807 two-min cadence TESS photometry from Sectors 22, 23, and 49 along with the transit plus GP model. Upward-pointing red triangles mark the transit times for the ultra-short period TOI-1807 b.

§ TOI-1136 D EXTRA MATERIAL

Contribution of the RM and CLV effects to TOI-1136 d's transmission spectra around the H$\alpha$ (top) and HeI NIR triplet (bottom) lines. Vertical cyan lines mark the position of the lines of interest.

Prior and posterior distributions from the nested sampling fitting for TOI-1136 d H$\alpha$ feature (see Fig. <ref>). Prior label $\mathcal{U}$ represents uniform distribution.

Parameter     Prior     Posterior

Absorption [%]     $\mathcal{U}(-3, 3)$     $-$1.12$^{+0.12}_{-0.13}$

$\lambda_0$ [Å]     $\mathcal{U}(6562,6566)$     6564.464$^{+0.024}_{-0.022}$

$\sigma$ [Å]      $\mathcal{U}(0,1)$     0.186$^{+0.030}_{-0.025}$

$\Delta$v [km s$^{-1}$]     –     $-$8.0$\pm$1.0

FWHM [Å]     –      0.45$^{+0.07}_{-0.06}$

EW [mÅ]      –     5.20$^{+0.57}_{-0.53}$

Corner plot for the nested sampling posterior distribution of TOI-1136 d H$\alpha$ feature.

§ TOI-1268 B EXTRA MATERIAL

MuSCAT2 ground-based transit observation of TOI-1268 b. Raw light curve (left panel), detrended light curve (centre panel), and residuals (right panel) for the Sloan $g$ (blue), $r$ (green), $i$ (yellow), $z_s$ (red) filters. The points show the individual observations, the circles represent binned data points, and the best-fit model is shown by the black line.

Prior and posterior distributions from the nested sampling fitting for TOI-1268 b HeI signal (see Fig. <ref>). Prior label $\mathcal{U}$ represents uniform distribution.

Parameter     Prior     Posterior

Absorption [%]     $\mathcal{U}(-3, 3)$     $-$2.00$^{+0.15}_{-0.16}$

$\lambda_0$ [Å]     $\mathcal{U}(10831,10834)$     10832.760$\pm$0.050

$\sigma$ [Å]      $\mathcal{U}(0,1.5)$     0.39$^{+0.036}_{-0.034}$

$\Delta$v [km s$^{-1}$]     –     $-$12.8$\pm$1.3

FWHM [Å]     –      0.91$^{+0.9}_{-0.08}$

EW [mÅ]      –     19.1$^{+1.8}_{-1.9}$

Corner plot for the nested sampling posterior distribution of TOI-1268 b Hei signal.

§ TOI-2076 B EXTRA MATERIAL

Prior and posterior distributions from the  fitting for TOI-2076 b data from LCO (see Fig. <ref>). Prior labels $\mathcal{U}$, $\mathcal{N}$, $\mathcal{F}$, and $\mathcal{J}$ represents uniform, normal, fixed, and Jeffrey's distribution, respectively.

Parameter     Prior     Posterior

$P$ [d]     $\mathcal{N}(10.355183,0.000065)$     10.355184 (58)

$t_0$ $^{(a)}$     $\mathcal{U}(3079.47,3079.62)$     3079.5495$^{+0.0061}_{-0.0045}$

ecc     $\mathcal{F}(0)$     –

$\omega$ (deg)     $\mathcal{F }( 90)$     –

$b =(a_{\rm p}/{R}_{\star}) \cos{ i_{\mathrm{p}} }$     $\mathcal{N}(0.149,0.089)$     0.158$^{+0.071}_{-0.074}$

$p = {R}_{\mathrm{p}}/{R}_{\star}$     $\mathcal{N}(0.02998,0.00035)$     0.03003 (30)

$\rho_{\star}$ [kg m$^{-3}$]      $\mathcal{ N }(2544,120)$     2535$\pm$100

$\mu_{\textit{LCO}}$ (ppm)     $\mathcal{N }(0.0,0.1)$     $-$0.6$^{+11}_{-25}$ $\times$10$^{3}$

$\sigma_{\textit{LCO}}$ (ppm)     $\mathcal{J }( 10^{-6}, 10^{6})$     1175$^{+76}_{-72}$

$q_{1,\textit{LCO}}$     $\mathcal{U }(0,1 )$     0.57$^{+0.30}_{-0.35}$

GP$_\mathrm{\sigma}$ (ppm)     $\mathcal{J}(10^{-6}, 10^{6})$     13$^{+80}_{-12}$ $\times$10$^{3}$

GP$_\mathrm{\rho}$ [d]      $\mathcal{J}(10^{-3}, 10^{3})$     2.5$^{+25}_{-2.3}$

3c Derived planetary parameters

$i_{\mathrm{p}}$ (deg)          89.63$\pm$0.18

$T_{14}$ [h]           3.31$\pm$0.06

$R_{\mathrm{p}}$ [${R}_\oplus$]          2.523$\pm$0.032

$^{(a)}$ Central time of transit ($t_0$) units are BJD $-$ 2 457 000.
$^{(b)}$ Equilibrium temperatures were calculated assuming zero Bond albedo.

TOI-2076 LCO photometric data (blue dots with error bars). The period $P$ and central time of transit $t_0$ derived from the  fit are shown above the panel. The black line is the best transit model for TOI-2076 b. Orange points show binned photometry for visualisation. GP contribution was removed from the data.

§ TOI-1683 B EXTRA MATERIAL

Prior and posterior distributions from the  fitting for TOI-1683 b. Prior labels $\mathcal{U}$, $\mathcal{N}$, $\mathcal{F}$, and $\mathcal{J}$ represents uniform, normal, fixed, and Jeffrey's distribution, respectively.

Parameter     Prior     Posterior

$P$ [d]     $\mathcal{N}(3.057,0.001)$     3.057541$^{+0.000014}_{-0.000010}$

$t_0$ $^{(a)}$     $\mathcal{N}(2522.7,0.1)$     2522.7001$^{+0.0012}_{-0.0010}$

ecc     $\mathcal{F }( 0)$     –

$\omega$ (deg)     $\mathcal{F }( 90)$     –

$r_{1}$     $\mathcal{U }(0,1 )$     0.799$^{+0.030}_{-0.040}$

$r_{2}$     $\mathcal{ U}(0,1 )$     0.0319$\pm$0.0014

$\rho_{\star}$ [kg m$^{-3}$]      $\mathcal{ N }(3800.0,600.0 )$     3900$^{+570}_{-580}$

$\mu_{\textit{TESS}}$ (ppm)     $\mathcal{N }(0.0,0.1)$     510$\pm$3900

$\sigma_{\textit{TESS}}$ (ppm)     $\mathcal{J }( 10^{-6}, 10^{6})$     334$^{+11}_{-12}$

$q_{1,\textit{TESS}}$     $\mathcal{U }(0,1 )$     0.62$\pm$0.25

$q_{2,\textit{TESS}}$     $\mathcal{ U}(0,1 )$     0.56$^{+28}_{-34}$

GP$_\mathrm{B}$ (ppm)     $\mathcal{J }(10^{-6}, 10^{6})$     62$^{+60}_{-25}$

GP$_\mathrm{L}$ [d]      $\mathcal{J }(10^{-3}, 10^{3})$     100$^{+100}_{-42}$

GP$_\mathrm{C}$ (ppm)     $\mathcal{J }(10^{-6}, 10^{6})$     500$^{+ 33000}_{-500}$

GP$_\mathrm{P_{\mathrm{rot}}}$ [d]     $\mathcal{U }(1,50)$     20.0$\pm$1.0

3c Derived planetary parameters

$p = {R}_{\mathrm{p}}/{R}_{\star}$          0.0319$\pm$0.0014

$b =(a_{\rm p}/{R}_{\star}) \cos{ i_{\mathrm{p}} }$           0.70$^{+0.05}_{-0.06}$

$i_{\mathrm{p}}$ (deg)          86.80$\pm$0.38

$T_{14}$ [h]           1.43$^{+0.07}_{-0.05}$

$T_{12}$ [min]           5.0$^{+0.8}_{-0.6}$

$R_{\mathrm{p}}$ [${R}_\oplus$]          2.21$\pm$0.13

$a_{\mathrm{p}}$ [AU]           0.0368$\pm$0.0023

$T_{\mathrm{eq}}$ [K] $^{(b)}$           910$\pm$30

$^{(a)}$ Central time of transit ($t_0$) units are BJD $-$ 2 457 000.
$^{(b)}$ Equilibrium temperatures were calculated assuming zero Bond albedo.

TOI-1683 TESS photometry (blue dots with error bars) phase-folded to the period $P$ and central time of transit $t_0$ (shown above the panel, $t_0$ units are BJD $-$ 2 457 000) derived from the  fit. The black line is the best transit model for TOI-1683 b. Orange points show binned photometry for visualisation.

TOI-1683 two-min cadence TESS photometry from Sectors 19, 43, and 44 along with the transit plus GP model. Upward-pointing red triangles mark the transit times for the TOI-1683 b.

Rotation period distribution as a function of colour $G$–$J$ for the Pleiades ($\sim$125 Myr; Rebull et al., 2016), Praesepe ($\sim$590 Myr; Douglas et al., 2017), Hyades ($\sim$650 Myr; Douglas et al., 2019), NGC 6811 ($\sim$1000 Myr; Curtis et al., 2019), and NGC 6774 ($\sim$2500 Myr; Gruner & Barnes, 2020) clusters. The gold star represents TOI-1683.

Equivalent width distribution of Lii as a function of the effective temperature for the Pleiades ($\sim$125 Myr; Bouvier et al., 2018), Praesepe ($\sim$590 Myr), and Hyades ($\sim$650 Myr; Cummings et al., 2017). The gold triangle represents TOI-1683.

$UVW$ velocity diagram for TOI-1683 (gold star). The members of the Castor moving group, Hyades supercluster (Hs), IC 2391 supercluster, Local Association (LA), and Ursa Major group (UMa) from [Montes et al., 2001] are included. The ellipses represent the 3$\sigma$ values of the $UVW$ for each Young Moving Group.

We analaysed TOI-1683 TESS data from Sectors 19, 43, and 44. Because our purpose is to model the stellar rotation, we used the  GP quasi-periodic kernel. We adopted the stellar parameters used in Zhang et al., 2022 for a proper comparison. The fitted parameters with their prior and posterior values, and the derived parameters for the planet candidate TOI-1683 b are shown in Table <ref>. The TESS data along with the best transiting and GP models is shown in Fig. <ref> and TOI-1683 b phase folded transit is shown in Fig. <ref>.

We computed the GLS periodogram <cit.> in the light curve of the three available TESS Sectors. We derived a stellar rotation period of 22.5 $\pm$ 5.2 days, which is consistent with the GP$_{\rm P_{rot}}$ value from the photometric fit (GP$_{\rm P_{rot}}$ 20 $\pm$ 1 d).
We estimated TOI-1683's age using the gyrochronology methods from <cit.>, <cit.>, and <cit.>, and the derived ages are 1450$^{+700}_{-550}$ Myr, 2300$^{+1300}_{-900}$ Myr, and 2600 $\pm$ 1200 Myr, respectively. All the three methods are consistent with an age older than 1 Gyr.
Figure <ref> shows the distribution of rotation periods as a function of the $G-J$ colour for various young clusters. From this qualitative analysis, it is evident that TOI-1683 appears to be older than the NGC 6811 cluster (which has an age $\sim$1 Gyr), and it is positioned above the sequence of NGC 6774 ($\sim$2.5 Gyr).

On the other hand, the presence of atmospheric absorption lines, such as LiI at 6709.61 Å, can serve as a valuable age indicator for the stars. We searched for the presence of LiI in the co-added spectrum generated by the  from the CARMENES spectra. However, no clear LiI feature was detected in the spectrum. Therefore, we set an upper limit for the LiI equivalent width of 1 m$\AA$ at 3$\sigma$. As Figure <ref> shows, this upper limit suggests that the star is older than the Hyades or Praesepe clusters, which have ages between 590–650 Myr.  Lastly, to investigate the kinematic properties of TOI-1683 and to determine if it shares any characteristics with known clusters, moving groups, or associations, we calculated the $UVW$ galactocentric space velocities using the astrometry and systemic velocity data from $Gaia$. Figure <ref> places TOI-1683 outside of any shown young moving group or cluster. Additionally, we conducted a search in the literature of cluster catalogs, and TOI-1683 was not found to be associated with any of these known stellar groups. These results suggest that it is not part of any well-established stellar association or cluster. Based on our analysis, we adopt an age older than 1 Gyr for TOI-1683 and, therefore, it is not a young object in terms of planet formation and evolution.

The age resulting of combining the three gyrochronology methods is 2000$^{+1300}_{-900}$ Myr, which is the age used in this work.
However, Zhang et al., 2022 claim that TOI-1683 is a young star ($<$1 Gyr) with an age of 500 $\pm$ 150 Myr based on its gyrochronology analysis. The main difference with our analyses is that they derived a rotation period of 11.3 $\pm$ 1.5 days, which is consistent with the half of the rotation period that we derived. Looking at a single TESS sector it is possible to confuse the stellar rotation period with one of its harmonics when it is of the order of the duration of the TESS sector ($\sim$28 days). This could be the case of TOI-1683 Sector 19. However, the consecutive Sectors 43 and 44 make it clear that the stellar rotation period of TOI-1683 is greater than 11.3 days.

§ TOI-2018 B EXTRA MATERIAL

MuSCAT2 ground-based transit observation of TOI-2018 b. Raw light curve (left panel), detrended light curve (centre panel), and residuals (right panel) for the Sloan $g$ (blue), $r$ (green), $i$ (yellow), $z_s$ (red) filters. The points show the individual observations, the circles represent binned data points, and the best-fit model is shown by the black line.

Prior and posterior distributions from the nested sampling fitting for TOI-2018 b HeI signal from the first night. Prior label $\mathcal{U}$ represents uniform distribution.

Parameter     Prior     Posterior

Absorption [%]     $\mathcal{U}(-3, 3)$     $-$1.10$^{+0.26}_{-0.30}$

$\lambda_0$ [Å]     $\mathcal{U}(10831,10834)$     10833.61$\pm$0.07

$\sigma$ [Å]      $\mathcal{U}(0,1.5)$     0.24$^{+0.07}_{-0.06}$

$\Delta$v [km s$^{-1}$]     –     8.4$\pm$1.9

FWHM [Å]     –      0.57$^{+0.17}_{-0.15}$

EW [mÅ]      –     6.6$^{+1.6}_{-1.5}$

Prior and posterior distributions from the nested sampling fitting for TOI-2018 b HeI signal from the second night. Prior label $\mathcal{U}$ represents uniform distribution.

Parameter     Prior     Posterior

Absorption [%]     $\mathcal{U}(-3, 3)$     $-$1.15$^{+0.33}_{-0.50}$

$\lambda_0$ [Å]     $\mathcal{U}(10831,10834)$     10833.21$^{+0.14}_{-0.20}$

$\sigma$ [Å]      $\mathcal{U}(0,1.5)$     0.50$^{+0.30}_{-0.20}$

$\Delta$v [km s$^{-1}$]     –     $-$0.2$^{+4.0}_{-5.5}$

FWHM [Å]     –      1.20$^{+0.65}_{-0.45}$

EW [mÅ]      –     14.7$^{+4.9}_{-4.5}$

Corner plot for the nested sampling posterior distribution of TOI-2018 b Hei signals for first (top) and second (bottom) nights.

Corner plot for the nested sampling posterior distribution of TOI-2018 b Hei signals from the combined nights.

Prior and posterior distributions from the nested sampling fitting for TOI-2018 b HeI signal from the combined nights. Prior label $\mathcal{U}$ represents uniform distribution.

Parameter     Prior     Posterior

Absorption [%]     $\mathcal{U}(-3, 3)$     $-$1.02$^{+0.19}_{-0.22}$

$\lambda_0$ [Å]     $\mathcal{U}(10831,10834)$     10833.55$^{+0.08}_{-0.07}$

$\sigma$ [Å]      $\mathcal{U}(0,1.5)$     0.30$^{+0.07}_{-0.06}$

$\Delta$v [km s$^{-1}$]     –     6.8$^{+2.1}_{-1.9}$

FWHM [Å]     –      0.71$^{+0.17}_{-0.14}$

EW [mÅ]      –     7.8$\pm$1.5

§ HEI DATABASE INFORMATION

As Zhang et al., 2023 remarked, the compilation of consistent parameters and r tudy the Hei triplet is a challenging work. We thank the effort done by Zhang et al., 2023 to construct Table 3 therein. We based our Hei database (Table <ref>) on the compilations by Kirk et al., 2022, Czesla et al., 2022, Fossati et al., 2022, Fossati et al., 2023, Guilluy et al., 2023, Zhang et al., 2023, and J. Sanz-Forcada, priv. comm., which we further complemented.

Because the Hei results are given in terms of the equivalent width (EW) or absorption peak of the line, we firstly needed a relation between both magnitudes to convert one to the other.
The EW for a spectral line is the width of a rectangle, with a height of the continuum level, such that the area of the spectral line is equal to the area of the rectangle. Usually, the planetary spectral lines are fitted using a Gaussian profile as G($\lambda$) = $A \exp{(-(\lambda - \lambda_0)^2/(2\sigma^2))}$ where $A$ is the amplitude of the signal, $\lambda_0$ is the central position of the line, and $\sigma$ is the standard deviation. Thus, the area under the curve described by G($\lambda$) can be computed as

\begin{equation}
\int_{-\infty}^{+\infty} G(\lambda) ~ d\lambda = A \sqrt{2 \pi \sigma^2}
\label{eq: AREA GAUSSIAN}
\end{equation}

If the spectra are normalised to 1, we can calculate EW = $\mid$ $A$ $\mid$ $\sqrt{2 \pi \sigma^2}$, where $A$ is in units of normalised flux. EW is in the units of $\sigma$.
Then, we calculated from the papers where EW and absorption values are reported (e.g., Guilluy et al., 2020, Casasayas-Barris et al., 2021, Zhang et al., 2022, Zhang et al., 2022, Orell-Miquel et al., 2022, Orell-Miquel et al., 2023) a master-$\sigma$ as the mean value. We obtained master-$\sigma$ = 0.43 m$\AA$.


or may not have this effect.)

#### Herding / voting privacy:

There is anecdotal evidence suggesting that social pressure causes
herding—specifically that voters align themselves with whales or voting blocs
[72]. We may view the effect as a shift in utility function. As we show in
Section 3.5, this shift has a centralizing effect.

Herding arises because votes are publicly visible. Voting privacy in principle
alleviates such pressure and therefore has a decentralizing effect.

Snapshot, a popular platform for DAO voting, has recently implemented a form
of privacy called shielded voting [73].This form of privacy, however, is only
ephemeral: Votes are private when submitted, but revealed at the end of the
vote-casting period. So it is unclear that it can fully address the
centralizing effects of herding.

End-to-end verifiable voting systems have been proposed in the literature for
decades that achieve both voting integrity and confidentiality [12]. How to
implemented them with token-based weighting is, to the best of our knowledge,
though, an open problem.

#### Voter bribery:

Our work shows a relationship between centralization and bribery. In general,
bribery causes an increase in centralization, as it has the effect of aligning
the utility functions of other players with those of the briber, as we show in
Section 3.7.

We also show that as decentralization increases, bribery cost increases.
Roughly speaking, increasing diversity in utility functions means increasing
cost to align them.

DAOs today are largely centralized [72, 39, 25, 26, 76]. Bribery may not be
especially useful, as whales generally exert strong control and require
relatively little coordination to align utility functions into a favorable
voting bloc. Voter bribery, however, is a problem in many settings, both in
political voting [60] and in corporate governance (see, e.g., [71]).

One implication of our results is that as DAO decentralization increases, in
order for bribery to succeed, it will need to be systemic. DAO designers
should therefore recognize large-scale bribery as a future risk.

#### Dark DAO risks:

We hypothesize that the most technically feasible way to implement large-scale
bribery is through a Dark DAO.

We have presented in Section 6 the first fully functional private Dark DAO
capable of subverting votes on Ethereum. Our architecture leverages the
privacy assurances of TEEs in Oasis, but bridges to Ethereum, where most DAOs
operate. Our results show that Dark DAOs are technically feasible (and incur
low transaction costs) and thus represent a viable future threat.

Dark DAOs pose not just a technical threat, but also a psychological one. The
mere existence of a Dark DAO may create a perception of vote-manipulation even
if the Dark DAO has minimal impact. Moreover, Dark DAOs can be used not just
for direct bribery but also for more subtle attacks. They can, for instance,
subvert quadratic voting schemes even when such schemes rely on well-
functioning decentralized identity systems.

One possible countermeasure DAO designers may ultimately wish to consider is
requiring voting participants to execute complete-knowledge (CK) proofs on
their keys [51].

#### Voting slates / bundling proposals:

A common trick for passing pieces of legislation that are unpopular or have a
narrow base of support is to bundle them together in large, bills. Earmarks
are a prime example [14].

This practice may be regarded as a form of utility-function “smoothing”: The
utility function of the bill as a whole (for the legislators voting on it)
differs from that of its components.

As the practice of bundling proposals / measures has the goal of aligning
utility functions, from the standpoint of VBE, it generally has a centralizing
effect, as we show in Section 3.6. DAOs may therefore wish to consider
limiting the practice and instead explore way to unbundle multi-component
proposals.

#### Data collection:

There is no practical way to compute VBE directly—since players’ typically do
not express their utility functions. As we discuss in Section 4, however,
there are ways to estimate it for a DAO based on voting history. We have found
it challenging to collect full voting histories for even popular DAOs. A
recommendation for the community is to establish and adhere to standards for
archival preservation of DAO voting data. A few

##  9 Related Work

#### DAOs:

Research literature on DAOs has been limited to date, but fairly broad. It has
included measurement studies [39, 72], retrospectives on the failure of The
DAO (e.g., [36]) and ways of addressing related technical flaws in smart
contracts such as dangerous reentrancy (e.g., [55, 28]), DAO mechanism design
(e.g., [16]), and exploration of DAOs from the standpoint of legal theory
(e.g., [46, 79]) and economics and governance (e.g., [18]).

Works exploring measurement of DAOs’ degree of decentralization most notably
include Feichtinger et al. [39],who explore Gini and Nakamoto indices, as well
as participation rates and the monetary cost of governance, Sharma et al.
[72], who consider various notions of entropy, as well as participation rates
and graph-based measures of decentralization, and [83], which taxonomizes DAOs
by comparison with other autonomous systems. Sun et al. use clustering to
identify voting blocs in a study of MakerDAO [75]. Also of note is the
informal notion of “credible neutrality,” a community standard articulated in,
e.g., [23, 21].

#### Social choice and voting theory:

A long line of work on social choice and voting theory investigates how best
to aggregate preferences of individual voters—the same functionality that DAOs
seek to provide in the decentralized setting. There are some major differences
in the DAO setting however, which may reduce how effective existing techniques
will be. For instance, the permissionless nature of DAOs allows for the
presence of Sybils which is not typically accounted for in existing voting
theory literature. Further, while the threat of large-scale voter bribery is
typically safe to ignore in classical voting, both due to the high likelihood
of detecting such an attack, as well as the the challenge in coordinating the
attack itself, as shown in our paper, Dark DAOs invalidate these prior
assumptions in the DAO setting.

Still, we believe that DAOs can provide an excellent practical battleground
for experimenting with different social choice and voting techniques in the
real world.

#### Vote-buying / coercion:

There is a considerable literature on the notion of coercion-resistance in
end-to-end verifiable voting [49, 34, 54]. Broadly speaking, coercion-
resistance means that a voter cannot convince a would-be briber or coercer of
how she voted. Influential proposed coercion-resistant voting systems include
notably Civitas [30] and, more recently, MACI [20]. None of these definitions
or system designs contemplate the risk of key encumbrance. Dark DAOs
effectively break all of them.

#### Unauthorized credential delegation:

Daian et al. put forth the notion of a Dark DAO—a DAO that aims to subvert
voting in other DAOs—in [33]. In related work, Matetic et al. propose use of
TEEs as a tool for secure credential delegation—which may be unauthorized
[59], and Puddu et al. explore malicious uses, including subversion of
e-voting [70].

##  10 Conclusion and Open Research Questions

We have proposed Voting-Bloc Entropy (VBE) as a new metric for DAO
decentralization. VBE measures the entropy of voting blocs. It is in fact a
framework into which it is possible to plug any desired method of clustering
to identify blocs and any notion of entropy.

Evaluating VBE—instantiated with $\epsilon$-threshold ordinal clustering and
min-entropy—we have proven a number of results that held shed light how a
number of practices may impact DAO decentralization. We have also shown both
in theory and through implementation of a practical system how Dark DAOs pose
a potential long-term threat.

Our work gives rise to a number of open research questions. A few deserve
particular mention:

  * •

Privacy: Our results suggest the potential decentralizing effects of ballot
secrecy, i.e., private voting. Existing verifiable end-to-end voting systems
implement a one-vote-per-person policy [12]. One open research question is
whether token-weighted variants are possible. Additionally, we emphasize in
our work the importance of collecting voting data to facilitate VBE estimates.
How to harmonize these opposing goals represents a second research challenge.

  * •

Forking and escape hatches: DAOs may suffer catastrophic failures, as was
famously the case with The DAO [40]. Proposed remedies including forking /
splitting, in which a new, quasi-independent or independent DAO is created and
escape hatches, which are committee-controlled shutdowns [38]. How their
existence and use impact decentralization are unclear and deserves study.

  * •

VBE impact: VBE is designed to formalize a view of decentralization in DAOs
reflected in the literature and in the views of practitioners. A natural
question is what impact high VBE has on decision making in DAOs. Does it
correlate with community growth, participation, and financial outcomes in
DAOs? How does it relate to notions of democratic participation in non-
blockchain settings?

## Acknowledgements

This work was funded by NSF grant CNS-2112751 and by generous support from IC3
industry partners. Andrés Fábrega is funded in part by a Uniswap Foundation
TLDR Fellowship.

Thanks to Oasis Labs for answering technical questions, Phil Daian for
extensive discussions about Dark DAOs, and Sylvain Bellemare for suggesting
the pronunciation “vibe” for VBE.

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