The MOPYS project: A survey of 70 planets in search of extended Hei and H atmospheres

No evidence of enhanced evaporation in young planets

The MOPYS project

J. Orell-Miquel<ref>,<ref> F. Murgas<ref>,<ref> E. Pallé<ref>,<ref> M. Mallorquín<ref>,<ref> M. López-Puertas<ref> M. Lampón<ref> J. Sanz-Forcada<ref>
L. Nortmann<ref> S. Czesla<ref>
E. Nagel<ref>
I. Ribas<ref>,<ref>
M. Stangret<ref>
J. Livingston<ref>,<ref>,<ref>
E. Knudstrup<ref>,<ref>
S. H. Albrecht<ref>
I. Carleo<ref>,<ref>
J. Caballero<ref>
F. Dai<ref>,<ref>,<ref>
E. Esparza-Borges <ref>,<ref> A. Fukui<ref>,<ref>
K. Heng<ref>
Th. Henning<ref>
T. Kagetani<ref>
F. Lesjak<ref>
J. P. de Leon<ref>
D. Montes<ref>
G. Morello<ref>,<ref>
N. Narita<ref>,<ref>,<ref>
A. Quirrenbach<ref>
P. J. Amado<ref>
A. Reiners<ref>
A. Schweitzer<ref>
J. I. Vico Linares<ref>

Instituto de Astrofísica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain

Departamento de Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain
Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la Astronomía s/n, E-18008 Granada, Spain
Centro de Astrobiología (CSIC-INTA), Camino Bajo del Castillo s/n, Villanueva de la Cañada, E-28692 Madrid, Spain
Institut für Astrophysik und Geophysik, Georg-August-Universität, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
Thüringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany
Institut de Ciències de l'Espai (ICE, CSIC), Campus UAB, Can Magrans s/n, 08193 Bellaterra, Barcelona, Spain
Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122, Padova, Italy
Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Astronomical Science Program, Graduate University for Advanced Studies, SOKENDAI, 2-21-1, Osawa, Mitaka, Tokyo, 181-8588, Japan
Department of Space, Earth and Environment, Chalmers University of Technology, 412 93, Gothenburg, Sweden
Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
NASA Sagan Fellow
Division of Geological and Planetary Sciences, 1200 E California Blvd, Pasadena, CA, 91125, USA
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan
Faculty of Physics, Ludwig Maximilian University, Scheinerstrasse 1, D-81679, Munich, Bavaria, Germany
Max-Planck-Institute für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany
Department of Multi-Disciplinary Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo
Departamento de Física de la Tierra y Astrofísica  and IPARCOS-UCM (Instituto de Física de Partículas y del Cosmos de la UCM),  Facultad de Ciencias Físicas, Universidad Complutense de Madrid, E-28040, Madrid, Spain
Landessternwarte, Zentrum für Astronomie der Universität Heidelberg, Königstuhl 12, D-69117 Heidelberg, Germany
Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany
Centro Astronómico Hispano en Andalucía, Observatorio Astronómico de Calar Alto, Sierra de los Filabres, 04550 Gérgal, Almería, Spain

During the first Gyr of their life, exoplanet atmospheres suffer from different atmospheric escape phenomena that can strongly affect the shape and morphology of the exoplanet itself. These processes can be studied with Ly$\alpha$, H$\alpha$ and/or Hei triplet observations.
We present high-resolution spectroscopy observations from CARMENES and GIARPS checking for Hei and H$\alpha$ signals in 20 exoplanetary atmospheres: V1298 Tau c, K2-100 b, HD 63433 b, HD 63433 c, HD 73583 b, HD 73583 c, K2-77 b, TOI-2076 b, TOI-2048 b, HD 235088 b, TOI-1807 b, TOI-1136 d, TOI-1268 b, TOI-1683 b, TOI-2018 b, MASCARA-2 b, WASP-189 b, TOI-2046 b, TOI-1431 b, and HAT-P-57 b. We report two new high-resolution spectroscopy Hei detections for TOI-1268 b and TOI-2018 b, and an H$\alpha$ detection for TOI-1136 d. Furthermore, we detect hints of Hei for HD 63433 b, and H$\alpha$ for HD 73583 b and c, which need to be confirmed.
The MOPYS (Measuring Out-flows in Planets orbiting Young Stars) project aims to understand the evaporating phenomena and test their predictions from the current observations.
We compiled a list of 70 exoplanets with Hei and/or H$\alpha$ observations, from this work and the literature, and we considered the Hei and H$\alpha$ results as proxy for atmospheric escape.
Our principal results are that 0.1–1Gyr-old planets do not exhibit more Hei or H$\alpha$ detections than older planets, and evaporation signals are more frequent for planets orbiting $\sim$1–3 Gyr-old stars.
We provide new constrains to the cosmic shoreline, the empirical division between rocky planets and planets with atmosphere, by using the evaporation detections and explore the capabilities of a new dimensionless parameter, $R_{\rm He}/R_{\rm Hill}$, to explain the Hei triplet detections. Furthermore, we present a statistically significant upper boundary for the Hei triplet detections in the $T_{\rm eq}$ vs $\rho_{\rm p}$ parameter space. Planets located above that boundary are unlikely to show Hei absorption signals.

§ INTRODUCTION

Along with their stars, planets evolve and change over time. During their early stages of formation, in particular, they suffer severe changes in their physical and orbital properties due to internal and external forces (Baruteau et al., 2016). Exoplanets form embedded in the proto-planetary disc, from where they can accrete H and He which are the major constituents of their gaseous envelopes (Guenther et al., 2023). Following the disc dispersion in $\sim$1–10 Myr (Haisch et al., 2001, Hernández et al., 2007, Baruteau et al., 2016), different phenomena occur favouring the mass-loss processes in the planetary atmospheres, namely:
i) the gas disk dissipates leaving the exoplanet and its H/He-rich atmosphere exposed to direct radiation of the host star (Fedele et al., 2010, Barenfeld et al., 2016, Dawson & Johnson, 2018);
ii) stars are more active and have their highest levels of X-ray and extreme ultraviolet (XUV, 1–912 Å) energy irradiation until $\sim$ 100 Myr (Jackson et al., 2012);
iii) exoplanets are undergoing contraction and cooling processes until $\sim$1 Gyr (Ginzburg et al., 2016, Ginzburg et al., 2018, Gupta & Schlichting, 2020);
iv) initial gas envelopes tend to be inflated by internal and external heat, resulting in more extended atmospheres (Sanz-Forcada et al., 2011, Owen & Wu, 2016, and references therein); and
v) the XUV radiation supports the population of metastable Hei (Sanz-Forcada & Dupree, 2008).
Therefore, at this stage of the system’s evolution, the conditions are optimal for detecting the young primordial extended atmospheres.

Photo-evaporation processes are stronger when they are driven by X-ray radiation, which is more significant during the saturated luminosity state (from $\lesssim$100 Myr until $\sim$1 Gyr for late spectral types; Jackson et al., 2012).
After this timescale, photo-evaporation is mainly driven, progressively, by the extreme ultraviolet (EUV, 100-912 Å) flux, and its effects on the exoplanet atmosphere are less significant (Sanz-Forcada et al., 2011, Owen & Jackson, 2012, Owen & Wu, 2013).
Furthermore, atmospheric escape can also be driven by the heat released from the cooling core of the planet. This mechanism, called core-powered evaporation, plays an important role, and this process can act on $\sim$Gyr time scales (Ginzburg et al., 2016, Ginzburg et al., 2018, Gupta & Schlichting, 2020).

Both escape mechanisms have some overlap in time and may act simultaneously in shaping the final atmosphere of mature exoplanets.
In this context, the study of planets at the early stages of evolution is crucial for a better comprehension of the different processes, and prove theory predictions, such as planet formation and migration, giant planets' gas accretion, or escape of the primary atmospheres of rocky-core planets with gas envelopes (e.g., Baruteau et al., 2016, Owen & Wu, 2017, Dawson & Johnson, 2018).
In particular, core-powered and photo-evaporation processes have been suggested to explain the formation of the “radius gap” in the small (1–4 R$_{\oplus}$) planets radius distribution (e.g., Fulton et al., 2017, Fulton & Petigura, 2018, Van Eylen et al., 2018, Van Eylen et al., 2021), although formation and migration mechanisms have also been proposed to explain the valley [Luque & Pallé, 2022].

Exoplanets can also experience atmospheric mass loss during their later lifetime because photo-evaporation and core-powered mechanisms can act at longer times scales than $\sim$0.1–1 Gyr, but at a reduced total mass-loss rates (Sanz-Forcada et al., 2011, Owen & Jackson, 2012).
According to stellar and planetary evolution models and assuming that high mass-loss rates are easier to detect, the number of evaporation detections should be higher in the $<$100Myr-old period, where the X-ray photo-evaporation and core-powered mechanisms act together (Sanz-Forcada et al., 2011, Owen & Wu, 2013, Ginzburg et al., 2016). After $\sim$1 Gyr, the number of detections should decrease as the core-powered atmospheric escape is weakening during the first few Gyrs (Ginzburg et al., 2016, Ginzburg et al., 2018). As a rough approximation, we should expect a plateau of atmospheric mass-loss detections after the $\sim$Gyr time-scale.
At the time photo-evaporation is mainly driven by the EUV flux (Sanz-Forcada et al., 2011), the planet properties are already fixed. For gas giants and Jupiter-mass planets, the impact of this `late-evaporation' on the planet mass is close to null (Owen & Wu, 2013), while for low-mass planets only small changes in their masses are expected (Lopez et al., 2012, Owen & Wu, 2013, Owen & Jackson, 2012). Therefore, the detection of hydrodynamic escape in old ($\gtrsim$1 Gyr) planets may not imply a significant change in their total mass and it may have no evolutionary consequences. Hydrodynamic escape by Roche lobe overflow can be considered as an exception (see e.g., Koskinen et al., 2022). The middle atmosphere of extremely close-in planets (with orbital periods of $\lesssim$1 d) extends to the Roche lobe due to the stellar gravitational tide, producing a very strong atmospheric mass-loss rate. The Roche lobe mechanism can be assumed stellar–age independent, and may not introduce a detectable change in the number of detections across stellar age.

The study of atmospheric escape in exoplanet atmospheres is mainly performed with the observations of three evaporation tracers: the H Lyman-$\alpha$ line (Ly$\alpha$) at 1216 $\AA$ in the ultraviolet (UV) wavelengths, the H Balmer-$\alpha$ line (H$\alpha$) at 6564 $\AA$ at optical wavelengths, and the Hei triplet at 10833 $\AA$ in the near-infrared (NIR) wavelengths (all the wavelengths in this work are referenced in the vacuum).

The Ly$\alpha$ was the first line used for probing evaporating atmospheres, when Vidal-Madjar et al., 2003 detected its absorption in the upper atmosphere of HD 209458 b.
However, the Ly$\alpha$ line has some limitations:
$i$) UV observations cannot be carried out from ground-based facilities, relegating its study to space observations with the Hubble Space Telescope (HST) instrument STIS (Space Telescope Imaging Spectrograph).
$ii$) Ly$\alpha$ is strongly affected by the interstellar medium extinction, limiting its observation to only the closest stars due to their large relative motions.
$iii$) the H in the upper layers of Earth's atmosphere (geocorona) have Ly$\alpha$ emission.
In practice, the core of the Ly$\alpha$ line cannot be observed and only the broad wings of the line are accessible. This has led to a very limited number of planetary atmospheres being accessible to Ly$\alpha$ observations.

The H$\alpha$ line was used by Yan & Henning, 2018 to prove the evaporation of KELT-9 b's atmosphere. The core and wings of this optical line can be observed from space but also from ground-based facilities, allowing for line profile characterisation when high-resolution spectrographs are used. However, the H$\alpha$ line has some caveats as well:
$i$) relatively low mass-loss rates can produce large Ly$\alpha$ absorption signals, while remaining non-detected in H$\alpha$ (e.g., GJ 436 b, Kulow et al., 2014, Ehrenreich et al., 2015, Cauley et al., 2017);
$ii$) stellar lines in the visible are more sensitive to stellar activity than those in the NIR (e.g., HPaschen lines and Hei triplet), making their analysis very challenging in the presence of such activity (Fuhrmeister et al., 2020, Palle et al., 2020, Howard et al., 2023).

The Hei triplet was proposed by Seager & Sasselov, 2000, and later modelled by Oklopčić & Hirata, 2018, as an alternative to search for evidence of atmospheric escape in the NIR.
The first detections came almost simultaneously from ground- and space-based observations (Spake et al., 2018, Nortmann et al., 2018, Allart et al., 2018). Moreover, ground-based high-resolution spectroscopy observations allowed to retrieve several physical parameters from line profile fitting (Lampón et al., 2020, Lampón et al., 2021, Lampón et al., 2021, Lampón et al., 2023). Since stellar lines in the NIR are less sensitive to stellar activity, but not exempted from it (e.g., Spake et al., 2018, Salz et al., 2018), than the optical lines, the Hei triplet is a good tracer to explore young exoplanet atmospheres.

Unfortunately, the Hei triplet has some disadvantages too, namely:
$i$) there are telluric absorption and emission lines surrounding the NIR triplet wavelengths that can overlap with the planet signal depending on the observing epoch.
$ii$) the Hei ionisation wavelength cut-off ($\lambda$ $\leq$ 504 $\AA$) is lower than that of Hi ($\lambda$ $\leq$ 912 $\AA$).
$iii$) the lifetime of the helium metastable state is $\sim$2.2 h [Drake, 1971], and a strong and constant XUV irradiation is needed to maintain the helium metastable state population detectable (Sanz-Forcada & Dupree, 2008).

This work presents the first results of the Measuring Out-flows in Planets orbiting Young Stars (MOPYS) survey. The MOPYS project aims to provide observational constrains to the time scales of atmospheric evolution and mass loss processes, and test the predictions of planetary evolution models by comparing them to the evaporation tracers observations. In particular, we are interested in testing the mass-loss processes in planet formation at different time scales.
Assuming that young planets undergo higher mass-loss rates than older ones, and higher mass-loss rates are, in general, easier to detect, we search for signs of evaporation using the Hei triplet and/or H$\alpha$ line observations as proxy, focusing on transmission spectroscopy observations of $\lesssim$1 Gyr-old exoplanets.
Moreover, we study the detection rate between young and old planet populations in our sample and compare the observed Hei triplet signals with predicted photo-evaporation mass-loss rates. We scheduled the observations such that telluric contamination of the Hei triplet is minimized, taking advantage of a favorable barycentric velocity of the Earth. The majority of the high-resolution spectrographs that we used allow to observe the Hei triplet and H$\alpha$ simultaneously: CARMENES (Quirrenbach et al., 2014, Quirrenbach et al., 2020), HARPS-N+GIANO-B (GIARPS; Claudi et al., 2016), and HARPS+NIRPS (Mayor et al., 2003, Wildi et al., 2022). Although H$\alpha$ might be more affected by the probable stellar activity, the simultaneous observation of the two lines increases the capability to detect ongoing evaporation. For consistency, we complemented the database of Hei triplet and H$\alpha$ observations with some Ly$\alpha$ observations from the literature when possible, although this work does not focus on this UV line.

This manuscript is organized as follows. We describe the observational datasets and the spectroscopic analysis methodology in Section <ref>. The atmospheric results for each planet are detailed in Section <ref>, while Section <ref> introduces further H and/or Hei atmospheric results from the literature. Section <ref> we detail our criteria to consider atmospheric detections. In Section <ref>, we discuss our main findings regarding atmospheric evaporation and the Hei triplet. The conclusions of this work can be read in Section <ref>.

§ OBSERVATIONS AND DATA ANALYSIS

§.§ Spectroscopic observations

Observing log of the transits analysed in this work. Columns from left to right are: Planet, name of the planet; Instrument, spectrograph used; Date, date of starting night of the observations; Start/End, starting/ending time of observations; $T_{\rm c}$, central time of the scheduled transit; $\sigma_{\rm T_c}$, 1$\sigma$ uncertainty of $T_{\rm c}$; $T_{\rm exp}$, median exposure time of the observations; $N_{\rm T_{14}}$/$N_{\rm obs}$, Number of spectra fully taken between the first ($T_1$) and fourth ($T_4$) contacts ($N_{\rm T_{14}}$) compared to the total number of spectra taken during the observations ($N_{\rm obs}$); Median S/N, median signal-to-noise ratio (S/N) value at the spectral orders of H$\alpha$ and/or HeI lines, respectively.

Planet     Instrument     Date     Start     End     $T_{\rm c}$     $\sigma_{\rm T_c}$     $T_{\rm exp}$     $N_{\rm T_{14}}$/$N_{\rm obs}$     Median

[UT]       [UT]      [UT]       [min]     [min]          S/N

MASCARA-2 b     CARMENES     2017 Aug 23     21:09     02:36     23:53     0.5     3.3     50/70     101/86

K2-100 b     CARMENES     2020 Jan 07     21:21      01:18      23:18     1     3.3     23/60     28/28

V1298 Tau c     CARMENES     2020 Jan 05     18:27      01:08      20:21     16     20     12/20      97/104

TOI-1431 b     GIANO-B     2020 May 31     23:46      05:14      01:14     1     1.6     50/116      44

TOI-2048 b     CARMENES     2021 Jun 07     21:11      03:04      23:27     8     10     18/32      32/39

HD 63433 b     CARMENES     2021 Nov 01     02:08      05:10      02:43     2     1     79/111      40/43

CARMENES     2022 Nov 27     22:48      03:10      01:00     2.5     5     26/36      111/124

HD 63433 c     CARMENES     2023 Feb 18     21:40      03:00      00:23     4.4     5     41/54      142/139

HD 73583 b     HARPS-N     2022 Jan 12     23:44      03:33      01:31     1     10     12/23      86

HD 73583 c     GIARPS     2023 Jan 04     23:41       05:56      01:59     1     10     20/37      69/72

HD 235088 b     CARMENES     2022 Aug 06     21:00      01:18      23:07     0.8     5     28/44      72/86

K2-77 b     CARMENES     2022 Sep 27     23:40      04:28      03:21     9     10     14/26       23/29

TOI-2046 b     GIARPS     2022 Sep 29     19:51      01:05      21:39     9     10     13/30      22/21

TOI-1807 b     CARMENES      2021 Dec 16     02:48      05:39      05:34     1     10     3/16      32/40

CARMENES     2022 Dec 23     02:05      05:20     03:45     2     6.6     8/25      61/77

TOI-1136 d     HARPS-N     2021 May 14     21:29       03:55      00:36     7     15     15/26      43

CARMENES     2023 Jan 30     23:41       06:15      03:17     3     15     15/24      91/88

TOI-1268 b     GIANO-B     2023 Feb 24     23:15      06:50      04:02     0.5     10     21/42     39

TOI-2076 b     CARMENES     2022 May 11     21:55      03:29      – $^{(a)}$     –     5     0/54     72/85

CARMENES     2023 Apr 13     22:01      04:35      23:37     6     10     15/32     80/90

TOI-1683 b     CARMENES     2022 Sep 11     00:33      01:15      01:40     3.7     6.6     3/6     26/33

GIANO-B     2023 Sep 19     01:05      05:55      02:09     3.8     10     8/24     39

GIANO-B     2023 Sep 22     00:57      05:43      03:32     3.8     10     8/26     41

WASP-189 b     GIANO-B     2019 May 06     21:42     04:46     01:18     0.1     1.6     95/156     67

HAT-P-57 b     CARMENES     2018 Jul 08     21:24      02:56      00:05     6     10     18/30     /47

TOI-2018 b     GIARPS     2022 Apr 09     23:34      06:00      00:12     3.3     5     19/67     29/53

GIARPS     2022 Jun 15     21:52      02:33      22:17     3     5     11/45     33/40

$^{(a)}$ No planetary transit was observed.

In this work we analysed 16 transits observed with the CARMENES (Calar Alto high-Resolution search for M dwarfs with Exoearths with Near-infrared and optical Échelle Spectrographs; Quirrenbach et al., 2014, Quirrenbach et al., 2020) spectrograph located at the Calar Alto Observatory, Almería, Spain. CARMENES has two spectral channels: the optical channel (VIS), which covers the wavelength range from 0.52–0.96 $\mu$m with a resolving power of $\mathcal{R}$ = 94 600, and the near-infrared channel (NIR), which covers 0.96–1.71 $\mu$m with a resolving power of $\mathcal{R}$ = 80 400. The targets were observed with both channels simultaneously.

Fibre A was used to observe the targeting star, while fibre B was placed on blank sky in order to monitor the sky emission lines (fibres A and B are separated by 88 arcsecs in the east-west direction). The observations were reduced using the CARMENES pipeline  [Caballero et al., 2016], and both fibres were extracted with the flat-optimised extraction algorithm (Zechmeister et al., 2014).

We also analysed 11 transits observed with HARPS-N (High Accuracy Radial velocity Planet Searcher for the Northern hemisphere; Cosentino et al., 2012) and/or GIANO-B (Oliva et al., 2012) spectrographs mounted on the 3.6m Telescopio Nazionale Galileo (TNG) at Roque de los Muchachos Observatory, La Palma, Spain. HARPS-N is an optical spectrograph which covers the wavelength range from 0.383–0.693 $\mu$m with a resolving power of $\mathcal{R}$ = 115 000. HARPS-N spectra were extracted using the standard Data Reduction Software (DRS) pipeline [Cosentino et al., 2014]. GIANO-B (Oliva et al., 2012) is a near-infrared spectrograph which covers the wavelength range from 0.95–2.45 $\mu$m with a resolving power of $\mathcal{R}$ $\simeq$ 50 000. When possible, the observations were done in GIARPS mode (Claudi et al., 2016), which allows simultaneous use of HARPS-N and GIANO-B spectrographs.

GIANO-B observations were carried out with the nodding acquisition mode, where the object is observed at two different predefined positions on the slit (A and B) following an ABAB pattern (Claudi et al., 2016). The nodding technique allows to monitor the sky with the slit position that is not on the object, and then efficiently subtract the thermal background and telluric emission lines. The GIANO-B spectra were wavelength calibrated, and extracted using the  pipeline (Rainer et al., 2018).

Table <ref> shows the observing log of the planetary transits analysed in this work, indicating the instrument used in each case. We computed the central time of transit ($T_{\rm c}$) and its uncertainty ($\sigma_{T_{\rm c}}$) with the Transit and Ephemeris Service tool from NASA Exoplanet Archive[<https://exoplanetarchive.ipac.caltech.edu/cgi-bin/TransitView/nph-visibletbls?dataset=transits>] and the parameters from Table <ref>.
The core of the MOPYS observations were conducted as part of the 21B-3.5-004, 22A-3.5-007, 22B-3.5-009, 23A-3.5-009, and 23B-3.5-002 observing programmes (PI J. Orell-Miquel) with CARMENES, and CAT21B_61, CAT22A_9, CAT22B_43, and CAT23A_100 observing programmes (PI J. Orell-Miquel) at TNG. K2-100 b, V1298 Tau c, HD 235088 b, and TOI-2048 b were observed using GTO time by the CARMENES consortium.

We processed the VIS and NIR CARMENES, and HARPS-N observations with [<https://github.com/mzechmeister/serval>] [Zechmeister et al., 2018], which derives the relative radial velocities (RVs) and several activity indicators: the chromatic radial velocity index (CRX), the differential line width (dLW), and the H$\alpha$, NaI D1 and D2 and CaII IRT line indices. Moreover, for the HARPS-N datasets we also used the YABI tool[Available at <http://ia2-harps.oats.inaf.it:8000>.], an online version of the HARPS-N DRS pipeline, to derive absolute RVs and spectral activity indicators: cross-correlation function (CCF) full width at half maximum (FWHM), CCF constrast (CTR), bisector (BIS), and Mont-Wilson S-index.
For the targets only observed with GIANO-B (because the GIARPS mode was not possible), we inspected the H Paschen lines in the NIR as stellar activity indicators. We constructed the stellar light curve of the H Paschen $\beta$ (Pa-$\beta$, 12821.6 Å), H Paschen $\gamma$ (Pa-$\gamma$, 10941.1 Å), and H Paschen $\delta$ (Pa-$\delta$, 10052.1 Å) lines.
Abrupt and/or strong variations in the time evolution of the activity indices or lines may indicate stellar activity or flares during the transit, and could compromise or challenge the detection of planetary signals, in particular absorption lines in the visible part of the spectrum [Palle et al., 2020, Orell-Miquel et al., 2023].

RV measurements during a transit can be used to look for the Rossiter-McLaughlin (RM; Rossiter, 1924, McLaughlin, 1924) effect. During the crossing of a planet in front of its host star, the planet blocks different parts of the stellar disc. That produces an RV anomaly during the transit known as the RM effect. The detection of the RM effect in RV time series taken during a planetary transit enables the confirmation of the presence of the transiting planet and also helps to determine the orbital configuration and architecture of the planetary system. This technique has been successfully applied to young planets unveiling the architecture of its planetary systems, such as AU Mic b [Palle et al., 2020] or DS Tuc A b [Benatti et al., 2021].
Because we used the CARMENES instrumental configuration where fibre B points at the sky, there were no simultaneous Fabry-Pérot calibrations during the observations. Without them, the RV time evolution is dominated by the instrument drift during the night.

§.§ Photometric observations

We refined the ephemerides and planetary properties of TOI-2048 b, HD 73583 b & c, K2-77 b, TOI-1807 b, TOI-1683 b, TOI-2076 b, TOI-2018 b, and TOI-1268 b using photometric data.

From the Transiting Exoplanet Survey Satellite (TESS; Ricker et al., 2015), we analysed the 2-minute cadence TESS simple aperture photometry (SAP; Morris et al., 2017) using [<https://juliet.readthedocs.io/en/latest/index.html>]. This  library is based on other public packages for transit light curve (, Kreidberg, 2015), and for Gaussian process (GP; , Foreman-Mackey et al., 2017) modelling and uses nested sampling algorithms (, Speagle, 2020; , Feroz et al., 2009, Buchner et al., 2014) to explore all the parameter space.
In the fitting procedure, we adopted a quadratic limb-darkening law with the ($q_1$,$q_2$) parameterisation introduced by Kipping, 2013, and we considered the uninformative sample ($r_1$,$r_2$) parameterisation introduced in Espinoza, 2018 to explore the impact parameter of the orbit ($b$) and the planet-to-star radius ratio ($p$ $=$ ${R}_{\mathrm{p}}/{R}_{\star}$) values. We modelled the photometric variability of the young host stars adding to fit the  GP exponential or  GP quasi-periodic kernels. We followed the same procedure applied to analyse the HD 235088 system in [Orell-Miquel et al., 2023].

One transit of TOI-1268 b and TOI-2018 b each were observed by the multi-colour imager MuSCAT2 (Narita et al., 2019) mounted at the 1.52-m Telescopio Carlos Sánchez (TCS) in the Teide Observatory in Tenerife, Spain. MuSCAT2 obtained simultaneous photometric data of the transits in four bands (Sloan $g$, $r$, $i$, $z_s$). The exposure times for each band were optimised for each night, CCD and target to avoid the saturation of the target and comparison stars in the field. Standard data reduction, aperture photometry, and transit modelling including systematic effects was performed by the MuSCAT2 custom pipeline (Parviainen et al., 2019).

One partial transit of TOI-2076 b was observed on the night of 14 May 2023 with the Sinistro imager mounted on one of the two 1-m telescopes (Dome B) operated by Las Cumbres Observatory (LCO; Brown et al., 2013) at the Teide Observatory in Tenerife, Spain. The observations were performed through the rp-band filter, in the full-frame mode, and with an exposure time of 10 s. To avoid detector saturation, the telescope was defocused such that the FWHM of the stellar point spread function was 7"–8". The observation started at 22:00 UT, about 1.6 hours before the expected ingress time, and halted at 01:03 UT, in the middle of the expected transit, due to a technical problem. The obtained images were processed by the BANZAI pipeline [McCully et al., 2018] for dark and flat corrections. The light curve of TOI-2076 was then extracted by aperture photometry using a custom pipeline [Fukui et al., 2011], and the transit modelling was performed with .

More details of each transit analysis can be found in their corresponding sections (TOI-1268 b: Sect. <ref>, TOI-2018 b: Sect. <ref>, and TOI-2076 b: Sect. <ref>).

§.§ Telluric correction

The main objective of these observations was the analysis of the HeI triplet at $\sim$10833 Å. However, the HeI lines are surrounded by emission and absorption telluric lines.
There is an H$_2$O absorption line at 10835 $\AA$ and there are four emission lines of hydroxyl (OH) at 10832.1 $\AA$, 10832.4 $\AA$, 10834.2 $\AA$, 10834.3 $\AA$ (Oliva et al., 2015), although the  last two are detected in the spectra as a single peak. Due to the Earth's orbital motion the relative positions of the planet and telluric spectral lines change with the epoch.
Thus, we only scheduled the observations during transiting epochs that minimise overlapping and mitigate the contamination over the HeI triplet (see Orell-Miquel et al., 2022, Spake et al., 2022 for further details).

When analysing the transits presented in this work, we encountered another source of contamination of the HeI triplet. We detected an emission line at $\sim$10833 $\AA$ in few transits where we needed to extend the observations into the astronomical twilight. This emission line only appears in the last few spectra and its strength increases towards the end of the night. We presume it is scattered sunlight from the incoming dawn.

Because each spectrograph has its own particularities, we handled with the telluric emission and absorption contamination differently for each of the instruments, see next subsections.

§.§.§ CARMENES telluric correction

We corrected the CARMENES VIS and NIR spectra from telluric absorptions following the approach described in Nagel et al., 2023 with the  package in version 1.5.9 (Smette et al., 2015, Kausch et al., 2015). Then, we corrected the Hei region of the CARMENES NIR data from telluric OH emission following the methodology described in previous HeI studies (e.g., Nortmann et al., 2018, Palle et al., 2020, Czesla et al., 2022), in particular [Orell-Miquel et al., 2022, Orell-Miquel et al., 2023].

We used fibre B to generate a synthetic emission model fitting simultaneously the three main OH peaks with three independent Gaussian profiles. The amplitude, central position, and standard deviation of the Gaussian profiles were set free, and we only introduced an initial guess for their central positions.
Prior to applying the emission model to fibre A, we accounted for the different efficiency of the two injection fibres. For each dataset, we computed the scaling factor between fibres comparing the strongest OH peak from the co-added spectra of each fibre. We assumed the scaling factor to be constant during the night. Finally, for each pair of target and sky spectra, we divided the science spectra by its particular OH emission model, multiplied by the nightly scaling factor.
When we detected the solar HeI emission line in fibre B, we added a fourth Gaussian profile in the fitting procedure.

§.§.§ HARPS-N telluric correction

We corrected the HARPS-N spectra from telluric absorptions with  (v4.2) via its implementation in the  (Spectral Lines Of Planets with python, Sicilia et al., 2022) package. We only used  for the purpose of running  easily on HARPS-N data, and correct those spectra from telluric absorptions.

Although the wavelength solution from the original HARPS-N data is based on wavelengths in air, the  correction provides a wavelength solution in the vacuum. Because CARMENES and GIANO-B wavelengths are in the vacuum, we also used the vacuum wavelengths for the HARPS-N data for consistency.

§.§.§ GIANO-B telluric correction

Due to our planning of the observations, the H$_2$O absorption line is always far from the Hei triplet. Thus, we decided to not correct the GIANO-B spectra from telluric absorption. If a particular residual map or transmission spectrum shows strong telluric residuals, we simply masked that spectral region.

The sky emission lines are corrected by the  pipeline, taking advantage of the nodding technique. This procedure is very efficient in removing emission lines that have similar strength in consecutive exposures. However, the HeI emission increases too quickly between exposures for the ABAB nodding procedure to provide an accurate correction for the emission contamination. We inspected the individual spectra and masked the affected wavelength region of particular spectra.

§.§ Transmission spectrum analysis

Transit and system parameters used to compute the transmission spectra for each planet analysed in this work.

$P$     $T_0$ $^{(a)}$     $T_{14}$ $^{(b)}$     $T_{12}$ $^{(b)}$     $a_{\rm p}$     $i_{\rm p}$     $\gamma$     $K_{\star}$     $K_{\rm p}$ $^{(c)}$

Planet     [d]     [d]     [h]     [min]     [au]     [deg]     [km s$^{-1}$]     [m s$^{-1}$]     [km s$^{-1}$]

K2-100 b      1.6739035 (4) $^{(2)}$      140.71941 (27) $^{(2)}$     1.60$\pm$0.01 $^{(1)}$     2.8$\pm$0.8 $^{(1)}$     0.0301 (14) $^{(2)}$     81.27$\pm$0.37 $^{(2)}$      $+$34.393 (3) $^{(2)}$     10.6$\pm$3.0 $^{(2)}$     193$\pm$9

MASCARA-2 b      3.4741070 (19) $^{(1)}$      909.5906 (3) $^{(2)}$      3.57$\pm$0.02 $^{(1)}$      29$\pm$1 $^{(1)}$      0.0542 (21) $^{(1)}$      86.15$^{+0.28}_{-0.27}$ $^{(1)}$      $-$21.07 (3) $^{(2)}$      322.51 $^{(3)}$      170$\pm$7 $^{(3)}$

V1298 Tau c     8.249071 (58) $^{(1)}$      1854.3479 (11) $^{(1)}$     4.66$\pm$0.12 $^{(2)}$      12 $^{(2)}$     0.0841 (13) $^{(3)}$       88.5$^{+0.9}_{-0.7}$ $^{(2)}$      $+$14.64 (14) $^{(4)}$       4$^{+5}_{-3}$ $^{(3)}$     111$\pm$2

TOI-1431 b     2.6502409 (41) $^{(1)}$      1739.17728 (11) $^{(1)}$     2.489$\pm$0.009 $^{(1)}$      0.716$\pm$0.022 $^{(1)}$      0.0465 (17) $^{(1)}$       80.30$^{+0.18}_{-0.17}$ $^{(1)}$      $-$25.154 $^{(1)}$     294.1$\pm$1.1 $^{(1)}$      188.15 $^{(2)}$

TOI-2048 b     13.790546 (55) $^{(1)}$      1739.1123 (27) $^{(1)}$     3.5$\pm$0.1 $^{(1)}$      6.9$^{+1.2}_{-0.7}$ $^{(1)}$      0.1078 (80) $^{(1)}$       89.41$\pm$0.35 $^{(1)}$      $-$7.6 (2) $^{(2)}$     $\sim$2.8 $^{(1)}$      85$\pm$6

HD 63433 b     7.107789 (10) $^{(2)}$      1916.45142 (32) $^{(2)}$      3.22$\pm$0.03 $^{(1)}$      9$\pm$1 $^{(1)}$      0.0719 (44) $^{(1)}$     89.4$^{+0.4}_{-0.6}$ $^{(1)}$     $-$15.81 (10) $^{(1)}$      1.4$^{+1.4}_{-0.9}$ $^{(3)}$      110$\pm$6

HD 63433 c     20.543888 (46) $^{(2)}$      1844.05824 (48) $^{(2)}$      4.07$\pm$0.03 $^{(1)}$      8$\pm$1 $^{(1)}$      0.1458 (62) $^{(1)}$     89.15$^{+0.07}_{-0.20}$ $^{(1)}$     $-$15.81 (10) $^{(1)}$      3.6$^{+1.1}_{-1.0}$ $^{(3)}$      77$\pm$3

HD 73583 b     6.3980580 (26) $^{(1)}$       2592.56287 (25) $^{(1)}$     2.100$^{+0.015}_{-0.013}$ $^{(1)}$       6.9$\pm$0.2 $^{(1)}$     0.0618 (20) $^{(1)}$      88.35$\pm$0.07 $^{(1)}$       +21.52 $^{(3)}$      4.37$^{+1.5}_{-1.3}$ $^{(2)}$     105$\pm$3

HD 73583 c     18.879300 (48) $^{(1)}$       2949.58243 (77) $^{(1)}$     3.60$^{+0.03}_{-0.05}$ $^{(1)}$       6.9$\pm$0.20 $^{(1)}$     0.1270 (40) $^{(1)}$      89.96$\pm$0.03 $^{(1)}$       +21.52 $^{(3)}$      2.89$^{+0.53}_{-0.51}$ $^{(2)}$     73.2$\pm$2.3

HD 235088 b      7.4341394 (60) $^{(1)}$      2798.4635 (56) $^{(1)}$     2.700$\pm$0.025 $^{(1)}$      4.4$\pm$0.5 $^{(1)}$      0.0725 (35) $^{(1)}$      88.85$\pm$0.30 $^{(1)}$      $-$27.370 (2) $^{(2)}$       $\sim$2.5 $^{(1)}$     106$\pm$5

K2-77 b     8.200139 (60) $^{(1)}$      2522.6338 (35) $^{(1)}$      2.68$\pm$0.22 $^{(1)}$      6.3$^{+1.5}_{-1.1}$ $^{(1)}$       0.0751 (45) $^{(1)}$      88.7$^{+0.7}_{-0.4}$ $^{(1)}$      +7.35 (20) $^{(2)}$      $\sim$3.4 $^{(1)}$      100$\pm$6

TOI-2046 b     1.4971842 (6) $^{(1)}$     1792.2767 (23) $^{(1)}$      2.410 (32) $^{(1)}$     21$\pm$1 $^{(2)}$      0.0267 (18) $^{(2)}$     83.6$\pm$0.9 $^{(1)}$      $−$9.480 (43) $^{(1)}$     374.7$\pm$7.8 $^{(1)}$      193$\pm$13

TOI-1807 b     0.54937084 (65) $^{(1)}$     2664.06930 (75) $^{(1)}$     0.970$\pm$0.022 $^{(1)}$     1.45$^{+0.25}_{-0.18}$ $^{(1)}$     0.0121 (9) $^{(1)}$     81.7$\pm$1.8 $^{(1)}$      $-$7.33$\pm$0.59 $^{(3)}$     2.39$^{+0.45}_{-0.46}$ $^{(2)}$      235$\pm$6

TOI-1136 d     12.51937 (41) $^{(1)}$     2349.525 (5) $^{(1)}$     4.12$\pm$0.11 $^{(1)}$      11.0$\pm$0.5  $^{(1)}$      0.1057 (46) $^{(1)}$     89.41$\pm$0.28 $^{(1)}$      $+$6.91$\pm$0.33 $^{(2)}$     $\sim$2.2$\pm$0.6 $^{(1)}$      92$\pm$4

TOI-1268 b     8.1577094 (45) $^{(1)}$     3000.66841 (14) $^{(2)}$     4.001$\pm$0.025 $^{(1)}$      21$\pm$2 $^{(1)}$      0.0711 (63) $^{(1)}$     88.63$^{+0.32}_{-0.30}$ $^{(1)}$      $+$3.79$\pm$0.14 $^{(3)}$     31.7$^{+2.5}_{-2.6}$ $^{(1)}$      105$\pm$8

TOI-2076 b     10.355183 (65) $^{(1)}$     3079.5495 (45) $^{(2)}$     3.251$\pm$0.03 $^{(3)}$      8.5$\pm$0.2 $^{(3)}$       0.0682 (13) $^{(3)}$     88.9$\pm$0.11 $^{(3)}$      $-$13.19$\pm$0.21 $^{(4)}$     $\sim$2 $^{(3)}$      71.6$\pm$1.3

TOI-1683 b     3.057541 (14) $^{(1)}$     2522.7001 (12) $^{(1)}$     1.43$^{+0.07}_{-0.05}$ $^{(1)}$      5.0$^{+0.8}_{-0.6}$ $^{(1)}$      0.0368 (23) $^{(1)}$     86.80$\pm$0.38 $^{(1)}$      $+$38.4$\pm$0.3 $^{(2)}$     $\sim$5 $^{(3)}$      130$\pm$8

WASP-189 b      2.7240330 (42) $^{(1)}$     1926.5416960 (65) $^{(1)}$     4.351$\pm$0.026 $^{(2)}$      18.6$\pm$6.2 $^{(2)}$       0.05053 (98) $^{(1)}$     84.03$\pm$0.14 $^{(1)}$      $-$24.465$\pm$0.012 $^{(1)}$     182$\pm$13 $^{(1)}$      200$\pm$4

HAT-P-57 b     2.4652950 (32) $^{(1)}$     5113.48127 (48) $^{(a , 1)}$     3.499$\pm$0.019 $^{(1)}$      19.1$\pm$0.8 $^{(1)}$      0.0406 (11) $^{(1)}$     88.26$\pm$0.85 $^{(1)}$      $-$5.99$\pm$0.35 $^{(1)}$     $<$215 $^{(1)}$      180$\pm$5

TOI-2018 b     7.435583 (22) $^{(1)}$     2746.4287 (21) $^{(2)}$     2.36$\pm$0.09 $^{(1)}$      6.5$\pm$0.5 $^{(1)}$      0.0609 (22) $^{(1)}$     88.52$\pm$0.22 $^{(1)}$      $-$25.617$\pm$0.002 $^{(3)}$     4.4$\pm$1.0 $^{(1)}$      89.1$\pm$3.3

K2-100b: $^{(1)}$ Stefansson et al., 2018, $^{(2)}$ Barragán et al., 2019.
MASCARA-2 b: $^{(1)}$ Lund et al., 2017 , $^{(2)}$ Talens et al., 2018, $^{(3)}$ Casasayas-Barris et al., 2019.
V1298 Tau c: $^{(1)}$ J. Livingston, priv. comm., $^{(2)}$ David et al., 2019, $^{(3)}$ Suárez Mascareño et al., 2021, $^{(4)}$ Gaidos et al., 2022.
TOI-1431 b: $^{(1)}$ Addison et al., 2021, $^{(2)}$ Stangret et al., 2021.
TOI-2048 b: $^{(1)}$ This work (App. <ref>), $^{(2)}$ [Newton et al., 2022]
HD 63433 b: $^{(1)}$ Mann et al., 2020, $^{(2)}$ Zhang et al., 2022, $^{(3)}$ Mallorquín et al., 2023
HD 63433 c: $^{(1)}$ Mann et al., 2020, $^{(2)}$ Zhang et al., 2022, $^{(3)}$ Mallorquín et al., 2023
HD 73583 b: $^{(1)}$ [Barragán et al., 2022], $^{(2)}$ El Mufti et al., 2021.
HD 73583 c: $^{(1)}$ [Barragán et al., 2022], $^{(2)}$ El Mufti et al., 2021.
HD 235088 b: $^{(1)}$ [Orell-Miquel et al., 2023], $^{(2)}$ Gaia DR2 (Soubiran et al., 2018).
K2-77 b: $^{(1)}$ This work (App. <ref>), $^{(2)}$ [Gaidos et al., 2017].
TOI-2046 b: $^{(1)}$ [Kabáth et al., 2022], $^{(2)}$ Computed using this table parameters.
TOI-1807 b: $^{(1)}$ This work (App. <ref>), $^{(2)}$ [Nardiello et al., 2022], $^{(3)}$ Gaia DR2 (Soubiran et al., 2018).
TOI-1136 d: $^{(1)}$ [Dai et al., 2023], $^{(2)}$ Gaia DR2 (Soubiran et al., 2018).
TOI-1268 b: $^{(1)}$ [Šubjak et al., 2022], $^{(2)}$ This work (Sect. <ref>), $^{(3)}$ Gaia DR2 (Soubiran et al., 2018).
TOI-2076 b: $^{(1)}$ [Zhang et al., 2022], $^{(2)}$ This work (Sect. <ref>), $^{(3)}$ [Osborn et al., 2022], $^{(4)}$ Gaia DR2 (Soubiran et al., 2018).
TOI-1683 b: $^{(1)}$ This work (Sect. <ref>) , $^{(2)}$ Gaia DR2 (Soubiran et al., 2018), $^{(3)}$ [Zhang et al., 2022].
WASP-189 b: $^{(1)}$ Lendl et al., 2020 , $^{(2)}$ Anderson et al., 2018.
HAT-P-57 b: $^{(1)}$ Hartman et al., 2015.
TOI-2018 b: $^{(1)}$ [Dai et al., 2023], $^{(2)}$ This work (Sect. <ref>), $^{(3)}$ Gaia DR2 (Soubiran et al., 2018).

$^{(a)}$ $T_0$ given in BJD $-$ 2 457 000, except for HAT-P-57 b that is BJD $-$ 2 450 000.
$^{(b)}$ $T_{14}$ is the total transit duration between the first ($T_1$) and fourth ($T_4$) contacts, and $T_{12}$ is the duration of the ingress/egress.
$^{(c)}$ Calculated from $K_{\rm p} = 2 \pi ~ a_{\rm p} ~ P^{-1} \sin{i_{\rm p}}$ using the parameters in this table, when there is no reference.

We analysed the spectroscopic observations via the well-established transmission spectroscopy technique (e.g., Wyttenbach et al., 2015, Casasayas-Barris et al., 2017) successfully applied in [Orell-Miquel et al., 2022, Orell-Miquel et al., 2023]. The planetary and stellar parameters needed to compute the transmission spectra for each planet are shown in Table <ref>.

The telluric corrected spectra are normalised by a first degree polynomial. We used a blue and red region near the line of interest and free of tellurics to fit the polynomial. The spectra are shifted into the stellar rest frame, accounting for the Earth's barycentric movement, the stellar systemic velocity ($\gamma$), and the stellar Doppler shift induced by the planet.
Then, we created a high signal-to-noise (S/N) stellar spectrum (Master-Out; MO) by calculating the mean of all the spectra taken completely during out-of-transit, and we divided all the spectra by the MO. With this step, the stellar spectrum is removed from the spectroscopic time series. Next, we moved the spectra to the planetary rest frame using the formula

\begin{equation}
v_{\rm p}(t) = K_{\rm p} \sin{(2 \pi \phi(t))}
\label{eq: Planet rest frame}
\end{equation}

where $K_{\rm p}$ is the radial velocity semi-amplitude that the star induces to the planet, and $\phi$ is the orbital phase.
We neglected orbital eccentricity because the values provided for the planets in the literature are usually poorly constrained and the impact for all but one case is of the same order as the uncertainties derived for the signal shifts. The exception to this is TOI-1268 b for which we used the  (Fulton et al., 2018) package to account for the eccentricity shift during the transit.

To compute the transmission spectrum (TS), we only consider those spectra taken entirely between the first ($T_1$) and fourth ($T_4$) contacts. We averaged them using the inverse of the squared propagated errors (1/$\sigma^2$) as weights. If the residual map contains strong telluric residuals and/or stellar line variability near the inspected line, we mask those affected regions to compute the TS. However, each night is different, and we explain further details of their particularies in the data analysis section (Sect. <ref>), where needed.

When the TS shows a clear absorption at the expected position, we fitted the feature with a Gaussian profile using the MultiNest algorithm (Feroz et al., 2009) via its python implementation  [Buchner et al., 2014].
Otherwise, when the residual map or the transmission spectrum do not show evidences of absorption from the planetary atmosphere, we derived conservative limits for the planetary absorption. We estimated a 3$\sigma$ upper limit absorption peak as three times the root-mean-squared (RMS) value of flat spectral region in the continuum of the spectrum close to the line(s) of interest. The exact spectral range might be different in each case to avoid including strong residuals or very scattered regions. The equivalent width (EW) of a line is a more appropriate measure because it is independent from the instrumental resolution, but it is difficult to compute when there is no line to measure. We give the upper limits in terms of absorption peak and we translate them into EWs (or from EW to absorption peak) as it is explained in App. <ref>.

To analyse the data from GIANO-B, we followed some of the recommendations from [Guilluy et al., 2020], [Fossati et al., 2022], and [Guilluy et al., 2023]. We considered the science spectra taken at positions A and B as different instruments, and we combined them after computing both TS. GIANO-B is known to be affected by a fringing pattern that affects high S/N spectra (Guilluy et al., 2020).
However, the lower S/N of our spectra, due to its maximum exposure time of 600 s and the faintness of our targets, prevented us to detect hints of the fringing pattern. The MO spectra obtained from GIANO-B did not show any clear sinusoidal pattern, as seen in <cit.>. Thus, we decided to not correct the fringing pattern, as it was done in Fossati et al., 2022. Lastly, the wavelength solution of GIANO-B may not be stable during the night. [Guilluy et al., 2020] found a typical instrument drift of half of the GIANO-B pixel element during four hours of observations. That could be an issue for the cross-correlation technique, where really good precision in the wavelength solution is needed (Brogi et al., 2018). However, the transmission spectra technique does not require such extreme precision, and hence we did not apply any correction to our GIANO-B observations, which have lower S/N than in [Guilluy et al., 2020].

Another aspects to take into account with transit observations are the Rossiter-McLaughlin (RM) and center-to-limb-variation (CLV) effects, which can interfere or mimic single line absorptions (Casasayas-Barris et al., 2021). Because these effects scale with $R_{\rm p}/R_{\star}$, they have larger impact on gas giant transmission observations. Previous Hei triplet line works proved that they have negligible contributions on the Hei results (e.g., Nortmann et al., 2018, Guilluy et al., 2023). Here, we computed the RM and CLV models for HD 63433 b ($R_{\rm p}$ = 2.1 $R_{\oplus}$) and TOI-1136 d ($R_{\rm p}$ = 4.6 $R_{\oplus}$) whose contribution to the H$\alpha$ and Hei triplet signals is negligible. These analyses and results are presented in Sect. <ref>.
The models of RM and CLV effects were created with the  [Plez, 2012], using the Kurucz ATLAS9 [Castelli & Kurucz, 2003] and VALD3 [Ryabchikova et al., 2015] line lists. The stellar models were created for 21 limb-darkening angles assuming local thermodynamical equilibrium (LTE). In the next step, taking into account the system parameters, we calculated the stellar models for orbital phases from $-$0.05 to 0.05, remembering that the planet covers different stellar regions with different limb-darkening during its transit. Next, all models were divided by the out-of-transit spectrum, creating the RM and CLV effect model for each orbital phase and wavelength.
However, it is necessary to first measure the RM in the RV time series to compute the models, and this is not detected for some of our targets.

§ TRANSMISSION SPECTROSCOPY RESULTS

Residuals maps and transmission spectra around the H$\alpha$ line (top panels) and HeI NIR triplet (bottom panels) lines for K2-100 b observations with CARMENES.
Left panels: Residual maps in the stellar rest frame. Time since mid-transit time ($T_{\rm c}$) is shown in the vertical axis, wavelength is in the horizontal axis, and relative absorption is colour-coded. Dashed and dotted white horizontal lines indicate the different contacts during the transit. Dashed cyan tilted lines indicate the predicted trace of the planetary signals. Solid green vertical lines indicate the position of the OH emission telluric lines. Solid blue line indicates the position of the H$_2$O absorption telluric line.
Right panels: Planet transmission spectra (TS) in the planet rest frame. We show the original data in light gray and the data binned by 0.2 Å in black. When an absorption signal is fitted, a red line and shaded region show the best Gaussian fit model with its $1\sigma$ uncertainties. Dotted cyan vertical lines indicate the H$\alpha$ (top) and the HeI triplet (bottom) lines positions. All the wavelengths in this figure are referenced in the vacuum.

In this section we present the results and the details of the residuals maps and TS calculations planet by planet. We present an example of a residuals map and TS figure in Figure <ref>, and the figures for the rest of transits without detections are shown in Appendix <ref>. Furthermore, the time evolution of the activity indicators derived from the analysed transits can be found in Appendix <ref> as well.

§.§ K2-100 b

The resulting residual maps and TS centred in the spectral regions of the H$\alpha$ and Hei triplet are shown in Figure <ref>. The time evolution of all the activity indices is mainly flat, and there is no evidence of strong stellar variability during the transit.

The HeI triplet TS shows a flat spectrum, while the H$\alpha$ TS is flat with some emission features from stellar variations in the line core during the transit. We found no significant absorption in either of the lines, and we placed a 3$\sigma$ upper limit to the excess absorptions of 1.4 % and 1.3 % for H$\alpha$ and HeI, respectively.

K2-100 b was already observed with the InfraRed Doppler (IRD) spectrograph on the 8.2-m Subaru telescope ($\mathcal{R}$ = 70 000; Kotani et al., 2018). Gaidos et al., 2020 derived an EW 99% confidence detection upper limit of 5.7 m$\AA$. From Fig. 8 therein, we estimated an absorption depth of $\sim$1.2%, which is consistent with the upper limit derived from the CARMENES observations.

§.§ MASCARA-2 b

[Casasayas-Barris et al., 2019] already detected H$\alpha$ in MASCARA-2 b using CARMENES and HARPS-N observations. The H$\alpha$ absorption measured with CARMENES VIS data was $-$0.85$\pm$0.03 %, and while combining 3 nights with HARPS-N was $-$0.68$\pm$0.06 %.

Here, we inspected the CARMENES NIR spectra, which were not analysed in [Casasayas-Barris et al., 2019], looking for HeI excess absorption. Figure <ref> presents the residual map and TS around the HeI triplet.
Because some telluric OH variability was still remaining in the residual map, we masked the affected regions.
Despite the structure in the final TS, due to the masked regions, we found no significant planetary absorption. We computed a $3\sigma$ upper limit to the HeI excess absorption of 0.5 $\%$.

§.§ V1298 Tau c

V1298 Tau planetary system is known to have transit time variations (TTVs) that can complicate the analysis of high-resolution spectroscopy observations (David et al., 2019). According to non-linear ephemerids for the V1298 Tau system (J. Livingston, priv. comm.), a nearly complete transit of planet c was observed the night of 5 of January 2020. Unfortunately, we missed pre-transit observations which are critical to compute a good MO and to check for stellar variability during the observations.

For this transit, the  H$\alpha$ index shows a clear decrease only from about mid-transit to egress, and an increase at the end of the observations. We did not include those last spectra in the computation of the MO spectrum, although they are plotted in the residual maps. The other  products do not show signs of strong stellar activity (see Fig. <ref>).

The H$\alpha$ residual map (Fig. <ref> left) shows a clear absorption region coincident with the  H$\alpha$ index decrease, suggesting the stellar origin of the signal. We computed the TS only with the non-affected spectra from the first half of the transit. We fitted a $-$1.10 $\pm$ 0.13 % feature at the H$\alpha$ position, which is consistent with the 3$\sigma$ upper limit derived from the nearby spectral region (1.1 %). The priors and posterior values from the fit are listed in Table <ref>, and the posterior distributions are shown in Fig. <ref>.

The first half of the transit in the HeI NIR triplet residual map (Fig. <ref> left) shows an extended absorption region, which does not coincide in time with the H$\alpha$ variability detected. In the second half of the transit, there is a vertical absorption feature in the stellar rest frame that does not follow the planetary track when we shift the spectra into the planet rest frame. This feature seems to be related to telluric residuals. We computed the TS with those unaffected spectra from the first half of the transit. The HeI TS has a deep ($-$3.75 $\pm$ 0.12 %) and broad (1.02 $\pm$ 0.04 Å) absorption feature, which is larger than the 3$\sigma$ upper limit (1.1 %). The priors and posterior values from the fit are listed in Table <ref>, and the posterior distributions are shown in Fig. <ref>.

To explore the origin of the H$\alpha$ and Hei features, we computed the TS from the unaffected spectra (start phase), the centre of the transit (centre phase), the end of the transit (end phase), and post-transit. We compare those TS with the MO spectra in Fig. <ref>. The start TS from H$\alpha$ and Hei lines show a different behaviour from centre and end. Furthermore, the comparison between TS and MO evidences the stellar line profile variation during the observations. Therefore, we cannot confidently attribute the detected features to V1298 Tau c atmosphere. Moreover, the H$\alpha$ TS in Fig. <ref> show a tiny bump at $\sim$6563 $\AA$ which can be attributed to the star and its youth.

The H$\alpha$ line of V1298 Tau c was previously explored by Feinstein et al., 2021 and Schlawin et al., 2021. Feinstein et al., 2021 studied with the Gemini/GRACES spectrograph the behaviour of the H$\alpha$ and CaII infrared triplet during a transit, finding significant variations of the H$\alpha$ during the observations. Schlawin et al., 2021 analysed one transit looking for H$\alpha$ absorption with the PEPSI spectrograph, but the observations were affected by stellar variability. Furthermore, Vissapragada et al., 2021 explored the HeI triplet of planet c with the Habitable-zone Planet Finder (HPF) spectrograph, but the in-transit spectra were affected by a stellar flare. Lastly, a transit of planet c was observed with narrow-band helium filter photometry, but Vissapragada et al., 2021 did not detect the transit during the observations.

Unfortunately, our observations join the previous unsuccessful attempts to analyse the atmosphere of V1298 Tau c. Due to the many complications that affected the observations, we adopted the absorption peaks from the H$\alpha$ and HeI features as upper limits of any possible planetary absorption. Our upper limits to H$\alpha$ and HeI excess absorption are 1.1 % and 3.7 %, respectively.

§.§ TOI-1431 b

Stangret et al., 2021 analysed two HARPS-N and one EXPRES transits, finding no signs of atomic or molecular absorption. A tentative detection of H$\alpha$ appeared only during the first HARPS-N dataset (0.33 $\pm$ 0.07 %), but it was considered as stellar variability because it was not reproduced in the other two visits. We consider that absorption feature as an upper limit for the purpose of this work.

Here, we looked for HeI excess absorption in the GIANO-B spectra from the first night of Stangret et al., 2021. The residual map and TS are shown in Fig. <ref>. The TS is mainly flat and we placed a 3$\sigma$ upper limit to the HeI excess absorption of 0.4 %.

§.§ TOI-2048 b

[Newton et al., 2022] noted that ground-based follow-up observations found a tentative transit detection $\sim$20 min later than expected. We analysed the new available TESS data to derive updated ephemerides. The details of the photometric fit are explained in Appendix <ref>. We refined TOI-2048 b ephemerides and planet properties (Table <ref>), and confirmed the detected delay from ground-based observations. Furthermore, because TOI-2048 b does not have mass measurements, we estimated its probable mass from its radius (Wolfgang et al., 2016) to predict its semi-amplitude $K_{\star}$. This parameter is required for a proper measurement of the TS. We forecasted a mass of 9.4 $\pm$ 1.0 M$_\oplus$ and a $K_{\star}$ of 2.8 $\pm$ 0.3 m s$^{-1}$. The details of all these calculations are explained in App. <ref> as well.

The time evolution of all the activity indicators is mainly flat except for the last four H$\alpha$ index points. We excluded those spectra from the MO combination, although they are plotted in the residual maps. Furthermore, we masked a region near the HeI triplet because some residuals still remained after correcting the OH emission. We obtained similar results with or without applying the OH emission correction.

H$\alpha$ shows a narrow absorption feature (see Fig. <ref>), but we consider it too narrow to have a planetary origin (it is only one binned point). Moreover, there are emission signals around the absorption signal in the residual map, and the absorption peak is of the order of the 3$\sigma$ upper limit of 1.5 %, and is difficult to distinguish from the continuum in the final TS. We found no significant absorption features around HeI triplet (Fig. <ref>). We placed a 3$\sigma$ upper limit to the HeI excess absorption of $\sim$1 %.

§.§ HD 63433 b

The night on 1 November 2021 was cloudy until $\sim$01:40 UT, and we could only cover the transit partially, without pre-transit information. Then, we observed a complete transit one year later, on 27 November 2022. The time evolution of all the activity indicators is mainly flat, without evidence of stellar activity during the observations in both nights.
Although the RM effect was detected by [Mann et al., 2020], the RM and CLV contribution to the H$\alpha$ and Hei triplet transmission spectra is well within the error bars of the TS (see Fig. <ref>), so we decided to not correct the data from these marginal effects.

The H$\alpha$ transmission spectrum from the first partial transit displays an absorption feature of $-$1.21$^{+0.22}_{-0.24}$ % (Fig. <ref>). Table <ref> shows the priors and posterior values from the fit and Fig. <ref> shows the posterior distribution. However, we did not recover that H$\alpha$ absorption in our second visit, which covered the entire transit. We consider the H$\alpha$ feature does not have a planetary origin. Due to the different quality around the H$\alpha$ from both nights, we did not combine the two datasets. The residual map and TS for the H$\alpha$ from the second transit is shown in Fig. <ref>. We placed a 3$\sigma$ upper limit to the H$\alpha$ excess absorption of 0.4 %.

The residual maps and TS for the HeI triplet from both nights separately are shown in Fig. <ref>. The HeI triplet was completely uncontaminated from tellurics, and both transmission spectra show some residuals from the stellar SiI line at 10830 $\AA$. Due to the different quality of the nights, we did not combine them and focused on the second event which is less noisy. The TS presents a $\sim$0.25 % excess absorption feature just at the expected position, but there is noise structure of similar amplitude. Finally, we adopted a conservative 3$\sigma$ upper limit of 0.4 %.

Our results on the HeI triplet are consistent with the previous upper limit of $\sim$0.5 % reported by [Zhang et al., 2022]. Their partial transit also showed a $\sim$0.2% excess absorption, but they found a correlation between the stellar HeI and H Paschen $\gamma$ lines.
We also compared the H Paschen lines with the HeI line, but we could not explain the HeI triplet feature as stellar line variability.
Further high-S/N observations of HD 63433 b are required to confirm the interesting feature of $\sim$0.2% around the HeI triplet found in this work and [Zhang et al., 2022].

Although the study of Ly$\alpha$ observations is not the focus of this work, it is worth mentioning [Zhang et al., 2022] presented HST/STIS observations of HD 63433 b around the Ly$\alpha$ line, but no absorption was found for this planet.

§.§ HD 63433 c

The time evolution of the  activity indicators is flat, except the H$\alpha$ line index that displays a narrow peak close to mid-transit, which is also visible in the H$\alpha$ residual map (Fig. <ref>). Therefore, we excluded those spectra to compute the H$\alpha$ TS, which is mainly flat. The H$\alpha$ TS displays a too narrow absorption feature that originates only from the first half of the transit. Thus, we attributed this signal to a stellar origin rather than to absorption from the planet. We put a 3$\sigma$ upper limit of 0.4 % to any possible H$\alpha$ excess absorption from the planet's atmosphere.

The HeI triplet residual map and TS do not show features from the planetary atmosphere, or from stellar variability (Fig. <ref>).
We placed a 3$\sigma$ upper limit to the HeI excess absorption of 0.4 %, consistent with the previous upper limit of $\sim$0.5% reported by [Zhang et al., 2022] with Keck/NIRSPEC.

[Zhang et al., 2022] detected an excess absorption of Ly$\alpha$ in one visit of HD 63433 c atmosphere with HST/STIS. However, we do not detect the evaporating atmosphere of HD 63433 c via H$\alpha$ in the VIS, or the HeI triplet in the NIR. Similar situation applies to, e.g., GJ 436 b where Ly$\alpha$ was detected (Kulow et al., 2014), but not H$\alpha$ or HeI triplet (Cauley et al., 2017, Nortmann et al., 2018, Villarreal D'Angelo et al., 2021).

§.§ HD 73583 b

We analysed the new TESS data to update the ephemerides on 73 planetary system (see App.<ref>) and we used them to analyse the transit of HD 73583 b observed with HARPS-N. The MO spectra was computed without the last three spectra, which show different values in the  H$\alpha$ indicator and YABI S-index. The time evolution for the other activity indicators is mainly flat.

The H$\alpha$ residual map and TS are shown in Fig. <ref>, where the last spectra clearly exhibit higher flux in the H$\alpha$ line core.
The TS shows a small absorption feature at the H$\alpha$ position, which seems to come from the second half of the transit according to the residual map. The signal is $-$0.46$\pm$0.16 % deep (significance $<$3$\sigma$), and has a net shift of $-$6.1$^{+3.2}_{-2.5}$ km s$^{-1}$. The priors and posteriors from the fit are shown in Table <ref>, and the posterior distributions are plotted in Fig. <ref>.
However, it is hard to link that absorption to the planet atmosphere for two reasons: $i$) the absorption comes mostly from the second half of transit (see the residual map on Fig. <ref>), and $ii$) the shift of the feature is in the opposite direction of the HeI detection from [Zhang et al., 2022].
We considered the $\sim$0.5% absorption feature as the upper limit for H$\alpha$ absorption.

Because GIANO-B was not on the telescope the night of the observations, we could not explore the Hei triplet. [Zhang et al., 2022] detected an excess absorption of 0.68$\pm$0.08 % from one full and one partial transits of 73 b with Keck/NIRSPEC. These observations were re-analysed in [Zhang et al., 2022] reporting a new excess absorption measurement of 0.72$\pm$0.08 %. The HeI triplet signal has a blue shift, rather than the usual red-shift reported in the other Hei detections.

§.§ HD 73583 c

As we mention in the previous section, the TESS photometric analysis for the HD 73583 planetary system is shown in App. <ref>. Using the values from [Barragán et al., 2022], the uncertainty on HD 73583 c $T_{\rm c}$ would be 50 minutes, while we now obtained a precision of $\sim$1 min.

The time evolution of the  H$\alpha$ and dLW indicators show a peak at the beginning of the transit that extends until mid-transit. Thus, to compute the TS, we only considered the spectra from the second half of the transit. The other activity indicators show a scattered but flat time evolution. The H$\alpha$ residual map and TS are shown in Fig. <ref>, where the H$\alpha$ variability is visible in the residual map.
The TS has a small feature at the expected position. We fitted an absorption of $-$0.54$^{+0.13}_{-0.14}$ %, and we present the prior and posterior values in Table <ref>, and the posterior distributions are shown in Fig. <ref>. Although the fitted absorption is significant ($\sim$3.8$\sigma$), we consider that value as an upper limit or a tentative detection. A second transit with better S/N and no stellar activity will confirm these results.

The Hei triplet residual map and TS is shown in Fig. <ref>.
There is no clear evidence of a planetary trace on the residual map or planetary absorption detected on the TS. We placed a 3$\sigma$ upper limit to the HeI excess absorption of 0.5 %.

§.§ HD 235088 b

We observed one transit of this sub-Neptune-sized planet during this project and its results were analysed in depth in [Orell-Miquel et al., 2023]. We detected a HeI blueshifted absorption signal of $-$0.91$\pm$0.11 %, confirming the previous detection from [Zhang et al., 2022] of $-$0.64$\pm$0.06 % absorption. The residual maps and transmission spectra for the H$\alpha$ and HeI triplet are shown in Fig. <ref> (adapted from Orell-Miquel et al., 2023).

In this work we adopted the results presented in [Orell-Miquel et al., 2023], where an age of 600–800 Myr was estimated for HD 235088.

§.§ K2-77 b

Because K2-77 b ephemerides are based on K2 photometry, we decided to analyse the available TESS data and improve the uncertainties on $T_{\rm c}$ for this target (see App. <ref>).

According to our results, the last spectrum from the CARMENES observations was taken during the egress, missing the post-transit coverage to compute the MO spectrum. Moreover, the pre-transit observations have low S/N due to higher airmass.

The difficulties to calculate the MO spectrum are visible in the residual maps and TS for both H$\alpha$ and HeI triplet lines (Fig. <ref>). We could only derive broad 3$\sigma$ upper limits for H$\alpha$ and HeI excess absorptions of 2.5 % and 2.7 %, respectively.

§.§ TOI-2046 b

We set GIARPS mode observations with similar exposure times than for HARPS-N and GIANO-B. Due to the faintness of the host star (J = 10.4 mag), we used GIANO-B's maximum time exposure of 600 s for both spectrographs. The spectra have relatively low S/N around the H$\alpha$ and the HeI triplet.

Also, the  pipeline could not properly remove the biggest telluric OH emission peak, and we decided to mask those telluric affected regions. The masked regions are far enough from the Hei line to ensure that they do not affect the final results.

H$\alpha$ and HeI residual maps and TS are shown in Figures <ref>. We only could derive very broad 3$\sigma$ upper limits for H$\alpha$ and HeI excess absorptions of 5 % and 2.9 %, respectively.

§.§ TOI-1807 b

We analysed the newly available TESS data to derive updated ephemerids, and refine the $T_{\rm c}$. The details of the photometric fit are explained in App. <ref>, and our results for the planet properties are shown in Table <ref>.
However, we planned the observations with older ephemerides (Nardiello et al., 2022), missing part of the transit on 16 December 2022.
We observed a second full transit on 23 December 2022, also with better S/N. The H$\alpha$ and HeI residual maps and TS for both events are shown in Figures <ref>. The second visit does not show evidences of absorption features.
We only used the data from the second transit to compute upper limits due to the different quality between both observations. We derived a 3$\sigma$ upper limit of 0.95 % for H$\alpha$ excess absorption, and an upper limit of 0.80 % for the HeI triplet.

A transit of this ultra-short period (USP) Earth-like density planet was already observed with IRD, reporting an upper limit to the HeI triplet EW of 4 m$\AA$ <cit.>. Because our upper limit is less constraining than the previous observations, we used the results from Gaidos et al., 2022 in the discussion in Section 4.

§.§ TOI-1136 d

Same as Fig. <ref> for TOI-1136 d. H$\alpha$ results (top panels) are the combination of HARPS-N and CARMENES VIS observations. Hei triplet results (bottom panels) are from CARMENES NIR observations.

Transit light curve of H$\alpha$ line from the combined observations of TOI-1136 d. The H$\alpha$ light curve has been constructed integrating the counts of the residual map in the planet rest frame around $\lambda_0$ using $\sigma$ (blue) and FWHM (orange) wavelength band passes from Table <ref>. The vertical lines represent the different contacts during the transit.

One HARPS-N transit of TOI-1136 d was observed to check for the planetary RM signature in the RV time series [Dai et al., 2023]. In this work, we inspected the dataset looking for H$\alpha$ planetary absorption signal. Furthermore, we scheduled a second transit with CARMENES to inspect the whole spectral range. We used nonlinear ephemerides to determine the $T_{\rm c}$ for the second transit (F. Dai, priv. comm.). The RM and CLV effects over the final TS of H$\alpha$ and Hei triplet lines are negligible (Fig. <ref>). Thus, we did not correct the data from these marginal effects.

The residual maps and TS of the H$\alpha$ for the individual nights are shown in Figure <ref>, while the final results are shown in Figure <ref>. HARPS-N results show a noisy TS with a small absorption feature. However, a similar feature is also detected in the CARMENES H$\alpha$ TS, which has higher S/N.
The results from both nights confirm the H$\alpha$ detection on the atmosphere of TOI-1136 d. The combined H$\alpha$ residual map and TS are shown in Fig. <ref>. We fitted the signal obtaining an absorption of $-$1.12$^{+0.12}_{-0.13}$ %. Table <ref> presents the priors and posteriors, and Fig. <ref> shows the posterior distributions.
Although the signal is a bit deeper in the first half of the transit, the H$\alpha$ transit light curve (Fig. <ref>) shows an absorption consistent with the transit duration without evidences of tail-like structure.

For the HeI triplet, because the CARMENES observations were performed close to the end of the night, the last spectra are affected by solar HeI emission. Although we corrected the emission line, some residuals still persisted. Then, we decided to mask the affected spectral range only on particular spectra. The HeI residual map and TS are shown in Fig. <ref>, and do not show clear evidences of HeI absorption. We placed a 3$\sigma$ upper limit to the HeI triplet excess absorption of $\sim$0.5 %.

§.§ TOI-1268 b

Same as Fig. <ref> for TOI-1268 b Hei triplet observations with GIANO-B.

Transit light curve of HeI triplet from TOI-1268 b detection. The HeI light curve has been constructed integrating the counts of the residual map in the planet rest frame around $\lambda_0$ using $\sigma$ (blue) and FWHM (orange) wavelength band passes from Table <ref>. The vertical lines represent the different contacts during the transit.

The transit of TOI-1268 b on the 24 of February 2023 was only followed spectroscopically by GIANO-B, and we missed the information about H$\alpha$, and the stellar variability or activity from the visible wavelength range covered by HARPS-N.
MuSCAT2 covered the same transit photometrically with the exposure times initially set to $g=5$ s, $r$=10 s, $i$=10 s, and $z_s$=15 s and later modified to  $g$=5 s, $r$=10 s, $i$=8 s, and $z_s$=12 s to avoid the saturation of the target star. Our MuSCAT2 transit fit ha a central time of $T_{\rm c}$ = 2460000.66841 $\pm$ 0.00014 BJD.

The four bands of the MuSCAT2 photometry do not show the presence of strong stellar activity or the planet crossing in front of any big starspot(s). Moreover, the Pa-$\beta$, Pa-$\gamma$, and Pa-$\delta$ lines are mainly flat with no correlation with the stellar Hei line.
Because our observations ended close to the twilight, our last spectra are affected by Hei sunlight contamination. Because this kind of emission increases quickly between exposures as the observations are approaching the twilight, it is not well corrected by the ABAB procedure. We decided to mask those regions affected by the Hei emission only in the selected spectra.

The residual map and TS are shown in Fig. <ref>, where the residual map shows an absorption region at the expected position of the planetary trace. The signal is well detected in the residual map and the TS, where we fitted a blue-shifted absorption of $-$2.00$^{+0.15}_{-0.16}$ % (EW = 19.1$\pm$1.9 mÅ). Table <ref> presents the priors and posteriors, and Fig. <ref> shows the posterior distributions. These high-resolution spectroscopy results confirm the Hei detection using narrow-band photometry from [Pérez González et al., 2023].
We also constructed the Hei triplet transit light curve, shown in Fig. <ref>. The absorption seems to extend further than the end of the white light transit hinting an He tail, that needs to be confirmed in further observations.

§.§ TOI-2076 b

In this section we present the analyses and results from one CARMENES transit of TOI-2076 b observed in 2023. Previously in May 2022, we missed a transit of TOI-2076 c due to imprecise ephemerides from [Hedges et al., 2021]. Nowadays, the periods of TOI-2076 planets are well-constrained and refined (Osborn et al., 2022, Zhang et al., 2022), and we could confirm that none of the three planets were transiting the night of 11 May 2022. However, we take advantage of those observations to study the behaviour of the star between epochs. The values derived from the  activity indicators for the two datasets are consistent, and there are no differences between epochs. The levels of stellar activity of both visits are very similar and TOI-2076 b transit seems unaffected by stellar activity o variability.
Due to bad weather conditions at the beginning of the night of 13 April 2023, we missed the pre-transit baseline and a very small part of the transit. Moreover, the median S/N of the in-transit spectra is lower ($\sim$60) than after transit ($\sim$90).

The TTVs of the TOI-2076 system (Osborn et al., 2022) make the scheduling of its transits complicated and increase the probability of missing part of them. Thus, we complemented our observations on 2023 with a photometric follow-up. We could not perform simultaneous observations on 13 April 2023, but we detected TOI-2076 b ingress one month after (14 May 2023). The details of the photometric analysis and their results are explained in App. <ref>.

The H$\alpha$ residual map and TS are shown in Fig. <ref>. The TS close to the H$\alpha$ has consistently negative values that could be considered as a planetary absorption. Although the lack of pre-transit baseline could be the origin of the signal, as for HD 63433 b transit on 1 November 2021 (Sect. <ref>). Then, a full transit coverage is needed to confirm this tentative signal. We set a 3$\sigma$ upper limit of 0.7% to H$\alpha$ exoplanet absorption.

The HeI triplet residual map and TS are shown in Fig. <ref>. We do not detect clear evidences of HeI planetary absorption. We looked for correlations of the HeI NIR line with the HeI D3 (5877.2 $\AA$), and H Paschen lines, with no conclusive results. We could only put a 3$\sigma$ upper limit of $\sim$1% for any HeI triplet planetary absorption.

TOI-2076 b atmosphere was already targeted looking for Hei. The same partial transit was observed from the same mountain with two different 8-m telescopes with their high-resolution spectrographs. In both cases, a consistent feature was detected but there are differences in the interpretation of the signal. Zhang et al., 2022 with Keck/NIRSPEC reported a Hei detection with an excess absorption of 1.01$\pm$0.05 % (EW = 10.0$\pm$0.7). On the other hand, [Gaidos et al., 2022] fitted a significant excess EW of 8.5$\pm$1.4 m$\AA$, but their analyses did not rule out the stellar origin of the signal.
Although CARMENES has the capability to detect such Hei excess absorptions (e.g., Salz et al., 2018, Alonso-Floriano et al., 2019, Orell-Miquel et al., 2023), the S/N of our transit only allows to put an upper limit at $\sim$1%, which is at the same level of the previous detected feature. Further observations in good weather conditions are needed to firmly confirm the TOI-2076 b HeI signal. For the purpose of this work, we will use the positive results from [Zhang et al., 2022] in the discussion.

We also used the dataset from May 2022 to check for H/He structures around TOI-2076, recently reported in other exoplanets (Zhang et al., 2023, Gully-Santiago et al., 2023). However, we obtained very similar results when we included those spectra in the analyses of TOI-2076 b H$\alpha$ and HeI lines.

§.§ TOI-1683 b

We scheduled three transits of TOI-1683 b for the MOPYS project, one with CARMENES and two with GIARPS. CARMENES transit was stopped due to bad weather and the few taken spectra are useless for atmospheric analyses. The two other transits were observed only with GIANO-B due to problems with the GIARPS mode, and we could not observe until ingress for the first GIANO-B visit (19 September 2023).

[Zhang et al., 2022] reported an age of 500$\pm$150 Myr, based on gyrochronology relations. Here, we derived a new age estimation of 2$^{+1.3}_{-0.9}$ Gyr based on gyrochronology, stellar kinematics, and stellar Lii analyses, indicating that TOI-1683 is not a young star. Hence, we do not consider TOI-1683 b as a young planet. The details of our stellar age analyses are explained in Appendix <ref>. Furthermore, we derived new ephemerides for TOI-1683 b using the available TESS data from Sectors 19, 43, and 44 in the MAST archive. The photometric analyses and results are shown in Appendix <ref> as well.

Zhang et al., 2022 observed one transit in bad weather conditions with Keck/NIRSPEC, and reported a HeI detection of 0.84$\pm$0.17 %. Although we also observe in poor weather and seeing conditions, we do not detect a significant Hei excess absorption in the individual or combined nights from GIANO-B. The residual maps and TS from the individual and combined nights are shown in Fig. <ref>. We placed a 3$\sigma$ upper limit from our two visits with GIANO-B to the Hei excess absorption of 0.7 %, which is $\sim$1$\sigma$ consistent with Zhang et al., 2022 detection. We use the positive detection from [Zhang et al., 2022] in the discussion.

§.§ WASP-189 b

[Stangret et al., 2022] already analysed one HARPS-N and two HARPS observations finding a H$\alpha$ absorption feature of $-$0.13$\pm$0.02 %. However, because the line was affected by the RM effect, they were cautious claiming a detection. Here, we inspected the GIANO-B data taken simultaneously to those HARPS-N observations.

The SiI and HeI stellar lines are not detectable in the spectra and the only visible spectral features are the H$_2$O telluric absorption lines, which we masked in the residual map and TS, shown in Fig. <ref>. Although the TS has some structure, there is no evidence of Hei triplet absorption. We set a 3$\sigma$ upper limit of 0.3 % to the HeI absorption on WASP-189 b.

§.§ HAT-P-57 b

HAT-P-57 b is an `adolescent' planet (1.00$^{+0.67}_{-0.51}$ Gyr; Hartman et al., 2015), where [Stangret et al., 2022] found an H$\alpha$ absorption feature ($-$0.7$\pm$0.2 %). However, the signal is narrow compared to other H$\alpha$ detections from the literature and it is surrounded by pulsations from the host star.

We analysed one archival CARMENES NIR transit to inspect the Hei triplet. Although we masked some regions of strong tellurics, the residual map and TS in Fig. <ref> do not show any significant planetary absorption signal. We set a 3$\sigma$ upper limit of 1 % to the planetary HeI absorption.

§.§ TOI-2018 b

Same as Fig. <ref> for TOI-2018 b combined observations of H$\alpha$ with HARPS-N and Hei triplet with GIANO-B.

Transit light curve of HeI triplet from the combined nights of TOI-2018 b. The HeI light curve has been constructed integrating the counts of the residual map in the planet rest frame around $\lambda_0$ using $\sigma$ (blue) and FWHM (orange) wavelength band passes from Table <ref>. The vertical lines represent the different contacts during the observation.

The planet candidate TOI-2018 b was initially alerted to the community as a `young planet', but final analyses of its age were ambiguous and it was not possible to confirm an age below 1 Gyr (2.4$^{+0.8}_{-0.2}$; Dai et al., 2023). Although we can not include TOI-2018 b in our sample of young planets, the results obtained from two partial transits observed with GIARPS deserve to be included in this work.

The GIARPS transit on the night of 15 June 2023 was photometrically followed with MuSCAT2, observing the transit in four bands with the exposure times set to $g$=15 s, $r$=15 s, $i$=10 s, and $z_s$=10 s. Our MuSCAT2 transit fit found a central time of $T_{\rm c}$ = 2459746.4287$\pm$0.0021 BJD, confirming that we missed partially the transits in both visits.

The H$\alpha$ residual map and TS from the individual nights and their combination are shown in Figures <ref> and <ref>, respectively. The TS is flat, and the 3$\sigma$ upper limit for H$\alpha$ excess absorption is set to 1.5 %.

The Hei triplet residual map and TS from the individual nights and their combination are shown in Figures <ref> and <ref>, respectively. Although the quality of the two nights is different, the results of both nights are consistent within 1$\sigma$, and we detect a consistent excess absorption of $\sim$1 % at the expected position of the planetary Hei signal. When we combine both nights, we confirm the detection of a red-shifted Hei absorption of $-$1.02$^{+0.19}_{-0.22}$ % (EW = 7.8$\pm$1.5 mÅ). All the nested sampling material for the individual and combined datasets is shown in App. <ref>.
The Hei triplet transit light curve (Figure <ref>) shows an absorption signal consistent with the transit duration, with no evidence of an extended He signal. Although the Hei signal is consistent and well detected in the two parcial transits, a full transit observation will help to confirm our detection and the study of TOI-2018 b atmospheric evaporation.

§ ADDITIONAL LITERATURE EVAPORATION TRACERS OBSERVATIONS

Compilation of H$\alpha$ and HeI observations from high-resolution spectroscopy facilities of young ($\lesssim$1 Gyr) transiting exoplanets, from the literature and this work (Sect.<ref>).
A cross ($\times$) indicates when the stellar activity prevented to measure any planetary absorption or derive upper limits.

Planet      H$\alpha$ [$\%$]     HeI [$\%$]                                               Planet      H$\alpha$ [$\%$]     HeI [$\%$]

MASCARA-2 b      0.85$\pm$0.03 $^{(1)}$     $<$0.5         TOI-2046 b      $<$5     $<$2.4

K2-100 b      $<$1.4     $<$1.3         TOI-1807 b      $<$0.95     $<$0.80

$\lesssim$1.2 $^{(2)}$                                    $\lesssim$0.4 $^{(8)}$

TOI-1431 b      $<$0.33 $^{(3)}$     $<$0.4         TOI-2048 b     $<$1.5     $<$1.0

HD 63433 b      $<$0.4     $<$0.4         HD 63433 c      $<$0.4     $<$0.4

$\lesssim$0.5 $^{(4)}$               –     $\lesssim$0.5 $^{(4)}$

HD 73583 b      $\lesssim$0.5     0.72$\pm$0.08 $^{(5,6)}$         HD 73583 c      $\lesssim$0.5     $<$0.5

HD 235088 b      $\times$ $^{(7)}$      0.91$\pm$0.11 $^{(7)}$         TOI-2076 b      $<$0.7      $\lesssim$1

0.64$\pm$0.06 $^{(6)}$                     1.01$\pm$0.05 $^{(6)}$

V1298 Tau b      –     $\lesssim$1.7 $^{(20)}$         V1298 Tau c      <1.1     <3.7

TOI-1268 b      –     2.00$^{+0.15}_{-0.16}$          TOI-1136 d       1.12$^{+0.12}_{-0.13}$     $<$0.5

WASP-189 b      0.13$\pm$0.02 $^{(9)}$     <0.3         HAT-P-57 b      <0.7$\pm$0.2$^{(9)}$     <1

K2-77 b     $<$2.5     $<$2.7          K2-25 b      –     $<$1.7 $^{(12)}$

AU Mic b      $\times$ $^{(13)}$     $<$0.34 $^{(14)}$         K2-136 c      –     $<$2.3 $^{(15)}$

DS Tuc b      $\times$ $^{(16)}$     –          WASP-52 b      0.86$\pm$0.13 $^{(17)}$     3.44$\pm$0.31 $^{(18)}$

HAT-P-70 b      1.56$\pm$0.15 $^{(19)}$     –         WASP-80 b      –     <0.85 $^{(21)}$

KELT-9 b      1.15$\pm$0.05 $^{(10)}$     $<$0.33 $^{(11)}$

TOI-2018 b $^{(\dagger)}$     $<$1.5     1.02$^{+0.19}_{-0.22}$         TOI-1683 b $^{(\dagger)}$      –     <0.7

0.84$\pm$0.17 $^{(6)}$

$^{(1)}$ Casasayas-Barris et al., 2019.
$^{(2)}$ EW at 99% confidence of 5.7 m$\AA$ [Gaidos et al., 2020].
$^{(3)}$ Stangret et al., 2021.
$^{(4)}$ Zhang et al., 2022.
$^{(5)}$ Zhang et al., 2022.
$^{(6)}$ Zhang et al., 2022.
$^{(7)}$ Orell-Miquel et al., 2023.
$^{(8)}$ EW at 99% confidence of 4 m$\AA$ [Gaidos et al., 2022].
$^{(9)}$ [Stangret et al., 2022].
$^{(10)}$ Yan & Henning, 2018.
$^{(11)}$ Nortmann et al., 2018.
$^{(12)}$ EW at 99% confidence of 17 m$\AA$ [Gaidos et al., 2020].
$^{(13)}$ Palle et al., 2020
$^{(14)}$ EW at 99% confidence of 3.7 m$\AA$ [Hirano et al., 2020].
$^{(15)}$ EW at 99% confidence of 25 m$\AA$ [Gaidos et al., 2021].
$^{(16)}$ Benatti et al., 2021.
$^{(17)}$ Chen et al., 2020.
$^{(18)}$ Kirk et al., 2022.
$^{(19)}$ Bello-Arufe et al., 2022.
$^{(20)}$ Gaidos et al., 2022.
$^{(21)}$ Fossati et al., 2022

$^{(\dagger)}$ Note that TOI-2018 b and TOI-1683 b are included in the table but they are likely not young planets.

§.§ Young planets from the literature

In the framework of the MOPYS project, we adopted the 1 Gyr stellar age as the threshold to classify exoplanets as `young' ($\lesssim$1 Gyr) or `old' ($\gtrsim$1 Gyr). We used this nomenclature in the discussion in Section <ref>. The 1 Gyr threshold is mainly based on the core-powered evaporation timescale, which is longer than the strong photo-evaporation initial stage (X-ray driven).

Young exoplanets are an interesting population that have called the attention of different research groups. To put the young planet results obtained in this work into context, we inspected the ExoAtmospheres[<http://research.iac.es/proyecto/exoatmospheres/index.php>] database looking for other young exoatmospheric analyses. Table <ref> presents a compilation of literature high-resolution spectroscopy studies of young transiting exoplanet atmospheres targeting the H$\alpha$ or the HeI triplet[We also acknowledge the Hei observations of K2-33 b (Hirano] et al., 2024), which appeared during the peer review process of the manuscript.. Here, we give a short context to those observations from the literature included in Table <ref>.

KELT-9 b: As the hottest exoplanet known to date, this planet attracted attention to probe and investigate its extreme atmosphere. [Yan & Henning, 2018] detected an evaporating exosphere of H$\alpha$. However, <cit.> could only place an upper limit to the HeI triplet absorption.

K2-25 b: The HeI triplet was observed using the IRD spectrograph, although the transit was contaminated by telluric OH emission. Gaidos et al., 2020 reported a 99% confidence upper limit to the transit-associated EW of 17 m$\AA$.

AU Mic b: Due to the host star's youth, the H$\alpha$ observations with ESPRESSO were strongly affected by stellar activity, making it impossible to set an upper limit [Palle et al., 2020]. Recently, Rockcliffe et al., 2023 reported a Ly$\alpha$ detection but only in one of the two visits with HST/STIS.
IRD and Keck/NIRSPEC spectrographs obtained a 99% confidence upper limit to the HeI EW of 4.4 and 3.7 m$\AA$, respectively [Hirano et al., 2020]. [Allart et al., 2023] observed one transit with SPIRou spectrograph finding a significant Hei absorption feature (0.37 $\pm$ 0.09 %). However, the authors finally reported a conservative 3$\sigma$ upper limit of 0.26 %, consistent with previous observations.

K2-136 c: Gaidos et al., 2021 put a 99% confidence upper limit to the HeI triplet EW of 25 m$\AA$ with one transit observed with the IRD spectrograph. The HeI lines position in between the telluric OH emission and H$_2$O absorption lines complicated the calculation of the transmission spectra.

DS Tuc b: The H$\alpha$ was analysed with ESPRESSO and HARPS spectrographs observations. However, the transits were affected by stellar activity and an H$\alpha$ upper limit could not be set [Benatti et al., 2021].

V1298 Tau system: This 20-Myr old multi-planet system has seen different attempts to study the presence of HeI in their planetary atmospheres. [Gaidos et al., 2022] used the IRD spectrograph to observe, during different nights, the star alone and a transit of planet b. An increasing HeI absorption was detected during the transit, but the authors proposed other explanations besides the planetary absorption. Using the narrowband helium filter technique, Vissapragada et al., 2021 observed the transits of planets b and d. For planet b they do not require extra absorption to explain the flux decrease,  while they found a tentative excess absorption combining two partial transits of planet d, but it requires a significant transit time variation.

WASP-52 b: The planet was observed with ESPRESSO detecting H$\alpha$, and other atomic species as well [Chen et al., 2020]. The presence of HeI in its atmosphere has been studied, first with a tentative detection using narrow-band photometry [Vissapragada et al., 2020, Vissapragada et al., 2022] and later confirmed with the Keck/NIRSPEC spectrograph [Kirk et al., 2022]. However, a recent paper by [Allart et al., 2023] did not find Hei absorption in two visits with the SPIRou spectrograph, and derived a 3$\sigma$ upper limit of 1.69 %.

HAT-P-70 b: The atmosphere of this ultra-hot Jupiter was inspected by [Bello-Arufe et al., 2022] using HARPS-N. They detected absorption coming from the H$\alpha$, H$\beta$, and H$\gamma$ lines, and many other atomic and molecular species as well.

WASP-80 b: Its atmosphere has been targeted in search of Hei absorption but without success (Fossati et al., 2022, Allart et al., 2023). Salz et al., 2015 is the only reference which provides a formerly calculated age of $<$200 Myr. However, the observed X-ray luminosity ($\log L_{\rm X}$ $\sim$ 27.8 erg s$^{-1}$ in Salz et al., 2015, down to 27.5 erg s$^{-1}$ in Sanz-Forcada et al. in prep.) yields a $\log L_{\rm X}/L_{\rm bol}$ $\sim$ $-5.0$, implying an age of $\sim$ 3 Gyr using Sanz-Forcada et al., 2011 age–$L_{\rm X}$ relations. Moreover, the TESS light curve do not suggest a rotational period of $\lesssim$ 15 d. Therefore, WASP-80 is probably older than 1 Gry according to the gyrochronology method shown in Figure <ref> ($G - J$ = 2.05).

§.§ Older planets from the literature

To put in context the young exoplanet Hei  (planets with ages $\lesssim$ 1 Gyr) findings, we complemented them with old exoplanet high-resolution Hei observations (planets with age $\gtrsim$ 1 Gyr) from the literature.
We have compiled a Hei triplet database, detailed in Table <ref>, which we constructed using the ExoAtmospheres database and literature results. For consistency, we only considered Hei triplet results from high-resolution spectrographs. That is, the narrow-band photometry detections of HAT-P-26 b (Vissapragada et al., 2022) and TOI-1420 b (Vissapragada et al., 2024) were not included.
Although we analysed the H$\alpha$ and we cited some results on the Ly$\alpha$ as well, Table <ref> only reports the H (H$\alpha$ or Ly$\alpha$) observations for the planets with Hei triplet observations, except for the young hot Jupiter HAT-P-70 b with H$\alpha$ detection (Bello-Arufe et al., 2022). The young Neptune DS Tuc b is not included in Table <ref> because no upper limit could be set from H$\alpha$ observations (Benatti et al., 2021) and no Hei triplet observations could be found in the literature.

Given the number of planets and details of each observation, we do not discuss the planets individually here, and refer the reader to Table <ref> and the appropriate references.

§.§ A note on He I variability

The planetary Hei triplet is known to show variability in its strength but also in its detectability (e.g., Palle et al., 2020, Zhang et al., 2022). In this work, we presented two upper limits for TOI-2076 b and TO1683 b, for which Zhang et al., 2022 reported Hei detections. Our upper limits are consistent at 1$\sigma$ with Zhang et al., 2022 detections. Thus, Hei triplet variability may be one reason for our absence of planetary signals. Variability was also invoked by Orell-Miquel et al., 2023 to explain the $\sim$2$\sigma$ deeper absorption found in HD 235088 b. We note that Gaidos et al., 2022 derived non-conclusive results for TOI-2076 b Hei signal, and TOI-1683 b transit from Zhang et al., 2022 was performed in poor observing conditions. We want to stress the importance of re-observing targets as a sanity check to confirm previous results (detections or upper limits), but also to study the planetary Hei triplet variability.

On the other hand, Krolikowski et al., 2023 analysed the variability of the Hei triplet of young stars. In particular, V1298 Tau, K2-100, K2-136, K2-77, HD 63433, and TOI-2048 were in the list of observed stars. They found that young stars show higher variability, with V1298 Tau being extremely variable, which could explain the non-conclusive results from the Hei observations. The stellar Hei variability decreases rapidly and keeps constant for stars older than $\sim$300 Myr (Krolikowski et al., 2023). Although the stellar Hei triplet line is variable, the short-term variability might not have a significant impact on the transit observations performed during the same night (Fuhrmeister et al., 2020, Krolikowski et al., 2023). In fact, H$\alpha$ is more sensitive to stellar variability than the Hei triplet (Fuhrmeister et al., 2020) and the analyses of those lines on AU Mic b and DS Tuc b are good examples (Palle et al., 2020, Hirano et al., 2020, Benatti et al., 2021).

§ CRITERIA FOR DETECTIONS, NON-DETECTIONS AND NON-CONCLUSIVE MEASUREMENTS

Relationship between the `observed' ($\dot{m}_{\rm obs}$) and the `theoretical' ($\dot{m}_{\rm theory}$) energy-limited mass-loss rates. We define XUV until the Hei ionisation range ($\lambda$ = 5 – 504 Å). The black line indicated the fitted relationship (shown in the legend) and the shaded area the 1$\sigma$ uncertainty.
Hei observations are coded as blue circles for detections, red crosses for non-detections, orange pointing-down triangles for non-conclusive.
We did not plot the error bars of $\dot{m}_{\rm theory}$ due to the very large uncertainties associated to the  values and its calculation.
Every planet has its name labelled, in black for detections and the rest is in grey. The two new detections presented in this work, TOI-2018 b and TOI-1268 b, are in good agreement with the predicted trend.

A problem we encountered when studying the Hei signal (and H$\alpha$ as well) was how to deal with the observations where only an upper limit value is given, which are the majority  of the cases (52 of 69 in Table <ref>). In this work, we classified the upper limit measurements in two groups: $a$) non-conclusive measurements (i.e. the upper limit value is large, and it does not really constrain the presence of the atom in the exoplanet atmosphere) and $b$) non-detection (i.e. the upper limit value is low enough to confidently assume there is no significant presence of that atom in the exoplanet atmosphere).

To assign observations to either category, for the He observations, we used the relationship that the observed and theoretical mass-loss rates ( and , respectively) seem to follow, represented in Figure  <ref>. We present  and  equations in Sect. <ref>, and we further explore and discuss the trend in that section.
For the time being, the fitted line between – seems to account for the differences between planetary characteristics, and populations (Vissapragada et al., 2022, Zhang et al., 2023, and Sect. <ref>). Therefore, we consider that relation to classify the Hei upper limits. Although other criteria might be chosen, it is used consistently for the whole sample.

We computed the  upper limits from the Hei EW upper limit values. If the  upper limit falls above the fitted line, the  upper limit value is higher than the a priori expected signal, and so it is not constrained. Thus, we refer to those measurements as non-conclusive (orange down-pointing triangles in Fig. <ref>).
On the other side, the  upper limits falling below the line can be considered as non-detections (red crosses in Fig. <ref>). We also considered as non-detections the  upper limits that fall within 1$\sigma$ of the fit (grey shaded are in Fig. <ref>).

We note that  is unknown for some of the exoplanets in our sample. For those planets with no measured , we performed a similar procedure but in a 1$/\rho_{\rm XUV}$ vs  diagram (not shown), which is a proxy for  vs  . $\rho_{\rm XUV}$ is the planet density when considering as the radius its XUV radius, which is different from the planet density $\rho_{\rm p}$ (see Sect. <ref>).

We did some exceptions when classifying some Hei observations. GJ 1214 b: we re-classified as a non-detection the tentative Hei signal reported by Orell-Miquel et al., 2022 due to other upper limits reported in the literature (Petit dit de la Roche et al., 2020, Kasper et al., 2020, Spake et al., 2022, Allart et al., 2023). V1298 Tau b: because of the unclear origin of the signal detected by Gaidos et al., 2022, we set this observation as non-conclusive. V1298 Tau c is in the same situation as planet b, and its Hei upper limit is non-conclusive. Moreover, the measurements for the V1298 Tau planets are consistent with the stellar Hei line variability range derived for their host star (Krolikowski et al., 2023). WASP-76 b: its Hei feature was presented as upper limit in Casasayas-Barris et al., 2021 but here we consider it as a non-detection due to the planet position well below the – line.

For H$\alpha$, we simply considered the upper limits as non-detections. However, if the H$\alpha$ upper limit absorption is comparable to that of Hei, we used the Hei classification for both measurements.
We only considered few Ly$\alpha$ detections for some particular exoplanets. The classification and nomenclature described in this section (detection, non-detection, and non-conclusive) is used all across the manuscript.

We note that absoprtion measurements of the three lines (Ly$\alpha$, H$\alpha$, and Hei triplet) would be the ideal case to determine whether an exoplanet is undergoing strong atmospheric escape. However, these observations are not available for all targets, and may be hard (even impossible) to obtain for some individual stars (e.g., Ly$\alpha$ interstellar medium extinction, H$\alpha$ variability in active stars, or low Hei triplet population). When possible, we consider the results from the three lines to determine atmospheric escape detections.

§ DISCUSSION

We have organised the discussion of our results in the following manner: Section <ref> gives a general view of the evaporation tracers across stellar age, planet radius, period, and mass. In Sect. <ref> we constrain the cosmic shoreline from Hei detections. In Section <ref> we explore the relation between Hei detections and the energy-limited mass-loss rates. Finally, we explore further relations between Hei detections and different planet and stellar properties in Sections <ref>, <ref>, and <ref>.

§.§ Evaporation tracers of planetary atmospheres across stellar age

Planetary radius versus stellar age diagram of evaporation (H and Hei) detections (blue circles), non-detections (red crosses), and non-conclusive (orange point-down triangles) observations. Planetary radii and stellar ages are from Table <ref>. Top panel presents the summed histogram of evaporation detections (blue line) and non-detections (red line) across stellar age. Grey points represent all known planets with radius and age determined with precisions better than 30% and 50%, respectively (data from NASA Exoplanet Archive).

Equivalent width (EW) of Hei detections (blue circles), non-detections (red crosses), and non-conclusive (orange point-down triangles) observations as function of stellar age. Top panel shows the summed histogram of Hei detections (blue line) and non-detections (red line) across stellar age bins.

Radius-period diagrams for planets with ages $\leq$ 1 Gyr (top panels) and $>$ 1 Gyr (bottom panels) with Hei (left panels), H (H$\alpha$ or Ly$\alpha$, middle panels) observations, and evaporation (combination of Hei and/or H observations, right panels). Detections and non-detections are marked with blue circles and red crosses, respectively. Radius gap [Van Eylen et al., 2018] is marked as dashed green line. Grey points represent all known planets with period and radius determined with precisions better than 25% (data from NASA Exoplanet Archive).

Radius-mass diagrams for planets with ages $\leq$ 1 Gyr (top panels) and $>$ 1 Gyr (bottom panels) with Hei (left panels), H (H$\alpha$ or Ly$\alpha$, middle panels) observations, and evaporation (combination of Hei and/or H observations, right panels). Detections and non-detections are marked with blue circles and red crosses, respectively. Solid lines are the theoretical models from Zeng et al., 2019 for Earth-like (green) and 50% H$_2$O+50% rocky (blue) compositions. Grey points represent all known planets with mass and radius determined with a precision better than 20% (data from NASA Exoplanet Archive).

Figure <ref> displays the planetary radius versus stellar age, with the planets colour-coded according to their evaporation measurements. We considered the Hei triplet and/or H$\alpha$ absorption detections (and Ly$\alpha$ in some particular cases) as proxies of evaporation signs. Thus, an evaporation detection means that either Hei or H$\alpha$ or both have been positively detected.

Figure <ref> is similar to Figure <ref> but focuses only on the Hei triplet, and presents a comparison between the Hei EW signals and the stellar age. The planets are marked and colour-coded according to their Hei results. [Allart et al., 2023] reported that a trend between Hei and stellar age was noticeable in their sample of eleven planets, however the authors refrained from further conclusions due to their small sample size and the lack of precise stellar ages.
Although in this work we used a larger sample (18 young and 35 old planets, including detections and non-detections, but less homogeneous than the Allart et al., 2023 one), we do not notice a clear correlation between Hei EW and stellar age.

The first 100 Myr are critical for the planetary atmospheric evolution, according to photo-evaporation models (see Introduction in Sect. <ref>).
Unfortunately, there are only a few planets in Figures <ref> and <ref> with ages $\lesssim$ 100 Myr. The youngest objects have ages of $\sim$20 Myr (AU Mic b and V1298 Tau c & b) and then there is a lack of HeI and H$\alpha$ observations until $\sim$100 Myr (WASP-80 b, and K2-77 b). WASP-80 b, with a formerly calculated age of $<$200 Myr [Salz et al., 2015], has no detection of HeI triplet. However, a possible older age of WASP-80 (see Section <ref>) and the calculated $\log L_{\rm XUV}$ at the planet separation (Table <ref>) could explain the lack of detection of the Hei triplet. AU Mic b non-detection is harder to reconcile, but different scenarios can be invoked (e.g., stellar activity masking possible detections or H/He ratio). Hei and H$\alpha$ observations of V1298 Tau b & c resulted in non-conclusive measurements, mainly due to the strong stellar activity levels of the young host star. However, the masses derived for planets b and e indicate that their densities are similar to older planets, suggesting they are not inflated and that they contracted faster than expected (Suárez Mascareño et al., 2021).

While the non-detections of WASP-80 b, AU Mic b, V1298 Tau b & c do not seem to fit the predictions from photo-evaporation models (Lopez et al., 2012, Owen & Jackson, 2012, Owen & Wu, 2013, Owen & Wu, 2017, Owen & Lai, 2018, Dawson & Johnson, 2018), the sample of $\lesssim$ 100 Myr-old planets is too small to draw conclusions. A larger sample of $\lesssim$150-Myr-old planets is needed to explore the photo-evaporation time scales.

From Fig. <ref> and <ref>, the youngest planet with Hei detection is TOI-1268 b (245$\pm$135 Myr), and the youngest planet with H$\alpha$ detection is MASCARA-2 b (200$^{+100}_{-50}$ Myr). The number of planets with evaporation detections increases until peaking at $\sim$2 Gyr, and then it is roughly constant, but this is related to the age distribution in our sample. The proportion between detections and non-detections remains constant (within error bars) with age, except at old ages (>5 Gyr) when non-detections dominate (see Figs. <ref>, and <ref>). This non-detections domination is more pronounced when considering only the Hei observations (Fig. <ref>), and extended over all stellar ages, except over the 1–3 Gyr range.

Our results on evaporation time scales agree with the conclusions from Loyd et al., 2020 on the radius gap, and Christiansen et al., 2023 on the hot Neptune population (see Fig. 6 in their work), and photo-evaporation mechanism is not more supported than the core-powered one. The fact that we do not see a decrease in the evaporation tracers, until very old ages ($\sim$5 Gyr, see Figs. <ref>, and <ref>), might be marginally more consistent with the core-powered time scale of Gyr (Ginzburg et al., 2016, Ginzburg et al., 2018, Gupta & Schlichting, 2020).

§.§.§ Radius-Period diagram across stellar age

Figure <ref> compares the population of young ($\leq$1 Gyr, top panels) and old ($>$1 Gyr, bottom panels) planets in a radius-period diagrams for only Hei triplet observations, only H (mainly H$\alpha$ with few Ly$\alpha$) observations, and both evaporation proxies. Figures <ref>, <ref>, and <ref> show no clear differences for the evaporation of the gas giant planets before and after 1 Gyr. The detections of evaporation are evenly spread over the stellar ages, with more preference of H$\alpha$ detection than from Hei for the young gas giants.  We note that there are no evaporation detections of young and old planets below the radius gap  or for Earth-like planets, supporting rocky planets are not under extreme atmospheric mass-loss processes, at least after $\sim$300 Myr which is the age of the youngest rocky planet in our dataset (TOI-1807 b).

Allan et al., 2023 simulated how the Hei triplet planetary signal from a highly irradiated ($a_{\rm p}$ = 0.045 AU) gas giant planet ($M_{\rm p}$ = 0.3 M$_{\rm J}$) around a K-dwarf star changes with the stellar age. They assumed a typical H/He ratio of 98/2 (Lampón et al., 2020, Lampón et al., 2021, Orell-Miquel et al., 2023). From their simulations, they derive excess absorption peaks of 4–7 % for young (16–550 Myr) Hot Jupiters, while at 5 Gyr the excess absorption would be of $\sim$1.5%.
They state that a close-in ($a_{\rm p}$ $<$ 0.1 AU) planet with a radius of 1-2 R$_{\rm J}$ transiting a $<$150-Myr-old K dwarf star would be the best target to test their evolution models.
Although the Hei triplet detection is favoured by the extreme radiation in XUV range ($\lambda$ < 504 nm; Sanz-Forcada & Dupree, 2008) and the TS simulations from Allan et al., 2023, none of the three 20-Myr-old planets analysed here present a clear detection of Hei.
WASP-80 b ($a_{\rm p}$ = 0.035 AU, $R_{\rm p}$ = 1 R$_{\rm J}$, $M_{\rm p}$ = 0.5 M$_{\rm J}$, $<$200 Myr, spectral type $\sim$K7V) is the closest planet from Table <ref> to the simulated one in Allan et al., 2023 but it has really low Hei upper limits (Fossati et al., 2022, Allart et al., 2023). WASP-52 b might be in agreement with the simulations, although the largest Hei excess absorptions to date come from older planets (e.g., HAT-P-67 b, Gully-Santiago et al., 2023; HAT-P-32 b, Czesla et al., 2022; and WASP-107 b, Kirk et al., 2020).
In Fig. <ref> WASP-52 b is the only young gas giant planet with a Hei detection. The ratio between Hei detections/non-detections seems larger for old rather than young planets, but there are fewer differences when comparing the H detections/non-detections.

§.§.§ Small planets evaporation across stellar age

Figure <ref> shows the radius-mass diagram for small planets ($R_{\rm p}$ $<$5 R$_{\oplus}$ and $M_{\rm p}$ $<$ 30 M$_{\oplus}$), also comparing evaporation proxies for young and old planets. Planets with $R_{\rm p}$ $\sim$ 1.5–3 R$_{\oplus}$ fall in a degenerated region of the mass-radius diagram, and their bulk compositions can be consistent with a large range of models, from water worlds (planets with a large water mass fraction) to  planets with rocky cores with H/He envelopes (Zeng et al., 2019). For these planets, in Figure <ref>, we found a mixture of detections and non-detections, with no difference between young and old planets.

Planets with $R_{\rm p} > $3 R$_{\oplus}$ are well above the water-rich composition line and are supposed to be gaseous with very light envelopes and low densities (Luque & Pallé, 2022). In our sample there are four young and five old puffy planets with evaporation observations (see right panels of Fig. <ref>). While only 1 in 4 young planets has an evaporation detection, 3 out of 5 old planets have a detection, hinting that atmospheric escape of puffy sub-Neptunes is stronger at ages older than $\sim$ 1 Gyr. Still, the numbers are small, and a larger sample is needed to confirm these findings.

For the planets above the radius valley, Malsky et al., 2023 predicted a He enhancement due to the diffusive separation of the atmospheric constituents where the He and metals are preferentially retained while H is evaporated. The time scale of this mechanism is comparable to the planet lifetime ($\sim$10 Gyr) and planets with $\lesssim$1 Gyr have not had time to show the effects of favoured H evaporation (Malsky et al., 2023). So, our young planet population is too young to suffer this differential escape process, and even some of our old planets could be considered `young' as well.
Then, the long time scale might explain why Figure <ref> does not show more Hei detections for the old waterworlds than for the young ones.
Malsky et al., 2023 predict that TOI-1235 b will have an He enhanced atmosphere at 10 Gyr, but [Krishnamurthy et al., 2023] put a really restricted Hei upper limit, although its age is poorly constrained (5$^{+5}_{-4.4}$ Gyr). The other planets listed in <cit.> have no H/He observations to test their predictions.

Although Malsky et al., 2023 focus only on the small-sized planets population ($\lesssim$3 $R_{\oplus}$), the diffusive separation and the subsequent preferential H evaporation could be the mechanism to explain some observational results from  Fig. <ref> and <ref> regarding large- and intermediate-sized planets.
i) There are no Hei detections in young puffy planets but there are for old ones. TOI-1136 d is the only young puffy planet with H$\alpha$ detection, and the only three old puffy planets with H observations both show H evaporation.
ii) WASP-52 b is the only young gas giant with Hei detection (and also has H$\alpha$ detection) while there are several detections on old gas giants, and H$\alpha$ is extensively detected in young and old gas giant planets.

§.§ Cosmic shoreline from Hei observations

Cosmic shoreline compared to Hei observations. Hei non-detections are denoted by red errorbars with no marker. Hei detections are circles scaled by their R$_{\rm p}$ and colour-coded by their stellar age (lateral colour bar).
Instellation relative to Earth vs escape velocity (left panel) and cumulative instellation in the XUV range relative to Earth vs escape velocity (right panel) graphics.
We marked the cosmic shoreline, as I $\propto$ $^4$, from <cit.> and constrained from the observations (dashed black line).
Planets sitting above/on the cosmic shore and with H/He detections are labelled.

The cosmic shoreline [Zahnle, 1998] is an empirical division found in the Solar System bodies that splits them between those that retain a certain amount of atmosphere and those that are purely bare rocks without atmosphere. The division was extended to the extrasolar planet population with success by Zahnle & Catling, 2017. Determining precisely the cosmic shoreline is important as it has often been used to predict the existence of an atmosphere for newly discovered planets, and to argue for atmospheric characterisation follow-up (e.g., with the James Webb Space Telescope). Observationally, extended atmospheres are relatively easy to detect via H and HeI absorptions. Here, we use these observations to better constrain the cosmic shoreline.

Zahnle & Catling, 2017 found a power law between planet bolometric instellation ($I_{\rm p}$) and planet velocity escape  that follows $I_{\rm p}$ $\propto$ $^4$. However, the radiation that shapes the exoplanet atmosphere is the X-rays and EUV stellar flux received by the planet during its early stages. Zahnle & Catling, 2017 assumed the approximation of X-ray luminosity saturation (Jackson et al., 2012) to estimate the total extreme radiation (X+EUV radiation) received by the planet. Following <cit.>, we compute the cumulative X-rays and EUV instellation ($I_{\rm X+EUV}$; considering $\lambda$ < 100 nm) as a function of the planet instellation ($I_{\rm p}$) and the stellar bolometric luminosity ($L_{\star}$):

\begin{equation}
I_{\rm X+EUV} = I_{\rm p} \, (L_{\star}/L_{\sun})^{-0.6}
\label{eq: total instellation XUV}
\end{equation}

We computed  as

\begin{equation}
v_{\rm esc} = \sqrt{ 2  \, G M_{\rm p}/ R_{\rm p} }
\label{eq: vesc}
\end{equation}

where G is the universal gravitational constant.
The empirical relation found between $I_{\rm X+EUV}$ with  also follows the same power law ($I_{\rm X+EUV}$ $\propto$ $^4$), as for $I_{\rm p}$.

Zahnle & Catling, 2017 presented the relationships between $I_{\rm p}$ and , and between $I_{\rm X+EUV}$ and  for the planets and small bodies of the Solar System along with the exoplanet population.
To test the predictions of the cosmic shoreline, we compared the proposed empirical relationships to the actual Hei detections. Figure <ref> reproduces Figures 1 and 2 from Zahnle & Catling, 2017, and includes also the Hei observations.
Because the cosmic shorelines are empirical relations, only the slope in logarithmic scale is determined. Thus, we get from <cit.> the independent terms to plot the empirical equations.

By definition, all Hei (and H$\alpha$) detections must be below the cosmic shorelines plotted in Fig. <ref>. However, we find seven gas planets with Hei (or H$\alpha$) detections in the region of supposedly bare rocky planets. Those are the ones labelled in Fig. <ref> panels.
In Fig. <ref> left panel, HAT-P-32 b and HAT-P-67 b are clearly located above the line, although their 1$\sigma$ uncertainties fall in the limit of the cosmic shoreline. TOI-1136 d, WASP-76 b, and KELT-9 b which have Hei non-detections but H$\alpha$ detections, are also over or above the cosmic shoreline.
Moreover, in Fig. <ref> right panel, HAT-P-32 b and WASP-107 b fall over the line, and GJ 3470 b and TOI-1136 d are clearly above the cosmic shoreline for energy-limited regime. With the exception of TOI-1136 d, the planets that contradict the shoreline are not young planets, with stellar ages $>$1 Gyr.

[Zahnle & Catling, 2017] stated that the $I$ $\propto$ $^4$ and $I_{\rm X+EUV}$ $\propto$ $^4$ lines are `drawn in by hand to guide the eye'. Thus, the evaporation detections above the shoreline can help to constrain the independent terms of those by-hand equations. For that purpose, we considered the planet whose uncertainties have the largest separation from the `by-hand' line to compute the limits of the cosmic shoreline. We took the coordinates of the extreme uncertainty as a point to calculate the line equation. We took HAT-P-67 b for the $I$ $\propto$ $^4$ line, and GJ 3470 b for $I_{\rm X+EUV}$ $\propto$ $^4$ although very similar results were obtained with TOI-1136 d. Then, it is trivial to get the line equation with the slope and one point. The cosmic shoreline equations constrained from the evaporation detections are:

\begin{equation}
\log(I/I_{\oplus}) = 4 \log(v_{\rm esc} \,[\rm {km\,s^{-1}}]) \, -2.04
\label{eq: I ves equation}
\end{equation}

\begin{equation}
\log(I_{\rm X+EUV}/I_{\rm X+EUV \oplus}) = 4 \log(v_{\rm esc} \,[{\rm km\,s^{-1}}]) \, -2.51
\label{eq: Ixuv ves equation}
\end{equation}

They are shown in Figure <ref> left and right panels as dashed black lines, respectively.
In both cases, the cosmic shoreline moved to higher radiation levels reducing the amount of exoplanets with no atmosphere.
Further atmospheric observations and more precise measurements of the planets close to the shoreline will allow to better constrain the observational cosmic shoreline.

§.§ Observed vs theoretical mass-loss rates

Hei transmission signal strength for the planets with detections (blue circles with error bars labelled in black) and non-detections (red pointing down triangles labelled in grey), as a function of the stellar  ($\lambda$ = 5 – 504 Å) at the planet distance.
We show the equivalent height of the HeI atmosphere, $\delta_{Rp}$, normalised by the atmospheric scale height, $H_{\mathrm{eq}}$. Data from Table <ref> and references therein.

Since the first HeI detection in planetary atmospheres, there have been many attempts to predict the presence of this line using other observable planetary parameters. Nortmann et al., 2018 performed one of the first attempts by computing $\delta_{Rp}/H_{\rm eq}$. That is: the equivalent height of the Hei absorbing atmosphere $\delta_{Rp}$ = $( R_{\rm p}^{2} + D_{\rm He} \, R_{\star}^{2} )^{1/2} - R_{\rm p}$, where $D_{\rm He}$ is the Hei absorption peak in %, divided by the atmospheric scale height $H_{\rm eq}$ = $(k_{\rm B} T_{\rm eq})/(\mu g_{\rm p})$, where $k_{\rm B}$, $\mu$, and $g_{\rm p}$ are the Boltzmann constant, the mean molecular mass, and planet gravity, respectively. Figure 4 in Nortmann et al., 2018 presented a correlation between the Hei absorption signal and  ($\lambda$ = 5 – 504 Å), but with a sample of only five planets. Figure <ref> reproduces the same diagram with the current set of 43 Hei observations (detections and non-detections with  measurements). The trend is not that clear and there is no evident pattern between detections and non-detections, maybe due to the non-uniformity of the sample (e.g., different instruments and data analysis techniques).

Rather than using the strength of the  HeI absorption, Vissapragada et al., 2022, and later Zhang et al., 2023, explored the relation between the mass-loss rates derived from the observations () as a function of /$\rho_{\rm p}$ ($\rho_{\rm p}$ is the planet density). While Vissapragada et al., 2022 derived the observed mass-loss rates comparing the measured excess transit absorptions with a grid of Parker wind models, Zhang et al., 2023 estimated the observed mass-loss rates from an order-of-magnitude method. Furthermore, Zhang et al., 2023 found that the theoretical maximum energy-limited mass-loss rate () is proportional to /$\rho_{\rm XUV}$ ($\rho_{\rm XUV}$ is defined below). However, they reported very similar results when using /$\rho_{\rm p}$. Despite those differences, both works found a positive correlation between  and /$\rho_{\rm p}$, and, their energy-limited outflow efficiencies agree within uncertainties (Vissapragada et al., 2022: $\eta_0$ = 0.41$^{+0.16}_{-0.13}$; Zhang et al., 2023: $\eta_0$ = 0.31 $\pm$ 0.06).

Here, we focus on the relationship between  and   while adding the two new Hei detections presented in this work (TOI-1268 b and TOI-2018 b), which is the main difference with the study performed by Zhang et al., 2023. For consistency, we followed the indications from Zhang et al., 2022, Zhang et al., 2023 to compute the – diagram with our set of Hei observations.

We made the same assumptions as in Zhang et al., 2022, Zhang et al., 2023 to calculate $\dot{m}_{\rm obs}$  with the order-of-magnitude method as:

\begin{equation}
\dot{m}_{\rm obs} = \frac{m_{\rm e} \, m_{\rm He} \, c_{\rm s} \, c^2 \, \, EW \, R_{\star} }{0.25 \, f \, e^2 \, \lambda_0^2 \, \Sigma g_{l}f_{l}} \, \propto \, EW \cdot R_{\star}
\label{eq: mass-loss observed}
\end{equation}

Eq. <ref> is computed in cgs units, where $m_{\rm e}$ is the electron mass, $m_{\rm He}$ is the He atomic mass, $c_{\rm s}$ is the sound speed (assumed to be $c_{\rm s}$ = 10 km s$^{-1}$), $c$ is the speed of light, EW is the equivalent width of the Hei signal, $R_\star$ is the stellar radius, 0.25 comes from assuming that 25 % of mass out-flow is He (atoms or ions), $f$ is the fraction of He atoms in the metastable ground state (assumed to be $f$ = 10$^{-6}$), $e$ is the electron charge, $\lambda_0$ is the Hei wavelength (fixed to 10833.3 Å), $\Sigma g_{l}f_{l}$ is the sum of the product of the degeneracy and oscillator strength over the three lines of the Hei triplet which is 1.62 [Zhang et al., 2023].

The theoretical maximum energy-limited mass-loss rate ($\dot{m}_{\rm theory}$; Caldiroli et al., 2022, Zhang et al., 2023), assuming that all the XUV flux received by the planet is spent on evaporation, is computed as

\begin{equation}
\dot{m}_{\rm theory} = \frac{ \pi \, R^3_{\rm XUV} \, F_{\rm XUV} }{G \, M_{\rm p}} = \frac{3 \, F_{\rm XUV}}{4\, G \, \rho_{\rm XUV}} \, \propto \, F_{\rm XUV}/\rho_{\rm XUV}
\label{eq: mass-loss theory}
\end{equation}

where $F_{\rm XUV}$ is the XUV flux ($\lambda$ = 5 – 504 Å), $R_{\rm XUV}$ is the planetary XUV photosphere radius, G is the gravitational constant, $M_{\rm p}$ is the planet mass, and $\rho_{\rm XUV}$ is the planet density using the $R_{\rm XUV}$. We followed the equations and indications from Wang & Dai, 2018, Zhang et al., 2022, Zhang et al., 2023 to estimate $R_{\rm XUV}$ as

\begin{equation}
R_{\rm XUV} = \frac{ R_{\rm p} }{ 1 \, + \, \beta^{-1} \ln{(\rho_{\rm atm. XUV} / \rho_{\rm atm. phot} )} }
\label{eq: R XUV}
\end{equation}

\begin{equation}
\rho_{\rm atm. phot} = \frac{P \, \mu}{ k_{\rm B} \, T_{\rm eq} } \,\,\,\, ; \,\,\,\, \beta \equiv \frac{G \, M_{\rm c} \, \mu}{ R_{\rm p} \, k_{\rm B} \, T_{\rm eq} }
\label{eq: rho phot beta}
\end{equation}

In Eq. <ref> and <ref>, $P$ is the pressure at the white-light planet radius (assumed $P$ = 100 mbar), $\mu$ is the (dimensional) mean molecular mass (we assumed $\mu$ = 1.3 $\times$ $m_{\rm H}$; $m_{\rm H}$ is the H atomic mass), $k_{\rm B}$ is Boltzmann constant, $T_{\rm eq}$ is the equilibrium temperature, $M_{\rm c}$ is the planet mass core, and $\rho_{\rm atm. XUV}$ is the planet atmosphere's density at the $R_{\rm XUV}$ layer.
We approximated $M_{\rm c}$ $\simeq$ $M_{\rm p}$ and used $\rho_{\rm atm. XUV}$ = 10$^{-15}$ g cm$^{-3}$. Our computed values for $\rho_{\rm XUV}$ are consistent within errors with those presented in <cit.>.

Figure <ref> presents our computed observed and theoretical mass-loss rates, recovering the predicted positive correlation for the Hei detections. For a quantitative analysis, we fitted a power law as  = $\eta_0$ ()$^{\alpha}$. We used 's orthogonal distance regression to fit the equation $\log_{10}($) = $\alpha \log_{10}($) + $\log_{10}(\eta_0)$.
We obtained an energy-limited efficiency of $\eta_0$ = 0.35 $\pm$ 0.04, which is consistent with the previous derived values (Vissapragada et al., 2022: $\eta_0$ = 0.41$^{+0.16}_{-0.13}$; Zhang et al., 2023: $\eta_0$ = 0.31 $\pm$ 0.06). TOI-1268 b and TOI-2018 b positions in the diagram agree with the previous detections, and they are over the fitted line.
For the linearity between log mass-loss rates, we got $\alpha$ = 0.43 $\pm$ 0.05, also consistent with <cit.>. When using $\rho_{\rm p}$ instead of $\rho_{\rm XUV}$, we get slightly larger uncertainties for $\eta_0$ and $\alpha$ but consistent within uncertainties ($\eta_0$ = 0.39 $\pm$ 0.06, $\alpha$ = 0.42 $\pm$ 0.06).

If the evaporation on those planets was in the energy-limited regime, they should exhibit a linear relation between  and . Although we computed the mass-loss rates as an order-of-magnitude and with some approximations, $\alpha$ is $\sim$11$\sigma$ apart from 1.
We find a sub-linearity relation ($\alpha \sim 0.43 < 1$) and the efficiency of the  decreases with , as already pointed out by both Vissapragada et al., 2021 and Zhang et al., 2023. However, this is in agreement with the theory as models predict that as the  increases, the photo-evaporation escape begins to lose efficiency via radiative cooling (e.g., Murray-Clay et al., 2009, Caldiroli et al., 2022). The H/He ratio or different hydrodynamical regimes (e.g., Lampón et al., 2020, Lampón et al., 2021, Lampón et al., 2021, Lampón et al., 2023) may also contribute to derive a sub-linear relation.

Along the – sub-linearity, five planets (TOI-2134 b, TOI-2018 b, HD 235088 b, HAT-P-11 b, and HAT-P-18 b) show an unphysical $>$100 % photo-evaporation efficiency. , as it is computed in Eq. <ref>, assumes the photo-evaporation scenario.
However, there are other mechanisms that could enhance the atmospheric mass-loss, namely the core-powered and the Roche lobe overflow. The core-powered mechanism may contribute in the atmospheric escape of those planets as they are relatively young (except HAT-P-11 b and HAT-P-18 b). However, a core-powered contribution can not be confidently assumed in those planets, as other parameters as the H/He ratio, the stellar flux or the atmospheric heating efficiency may contribute to change the derived – relation and yield unrealistic efficiencies.
On the other hand, Roche lobe overflow is not expected, as these planets are not extremely close-in. Non-thermal processes (e.g., ion pick-up or sputtering) may also help to achieving a $>$100 % efficiency in those planets (Güdel et al., 2014).

§.§ Hei dependence with the Hill radius

Relationship between $R_{\rm He}/R_{\rm Hill}$  and the `theoretical' ($\dot{m}_{\rm theory}$) energy-limited mass-loss rate. We define XUV until the Hei ionisation range, $\lambda$ $\in$ 5 – 504 Å. The black line indicated the fitted relationship (shown in the legend) and the dashed area the 1$\sigma$ uncertainty.
Hei observations are coded as blue circles for detections, red crosses for non-detections, orange pointing-down triangles for non-conclusive. The colour coding is according to the results from Sect. <ref> (see Fig. <ref>).
We did not plot the error bars of $\dot{m}_{\rm theory}$ due to the very large uncertainties associated to the  values and its calculation.
Every planet has its name labelled, in black for detections and the rest is in grey.

Because the  is a critical factor for the detectability of extended He atmospheres via the Hei triplet in the NIR (Sanz-Forcada & Dupree, 2008, Oklopčić & Hirata, 2018), different relationships have been proposed, aimed at explaining the Hei detections, that take into account the  value, either directly (e.g., $\delta_{Rp}/H_{\rm eq}$–; see Fig. <ref>) or indirectly (e.g., –; see Fig <ref>).

Here, we explore other relationships that might explain the Hei signals but accounting also for the differences between system characteristics, as $\delta_{Rp}/H_{\rm eq}$.
We considered the dimensionless parameter $R_{\rm He}/R_{\rm Hill}$ which is the division between the apparent radius of the planet at the Hei triplet wavelength by the Hill radius of the planet. We computed the $R_{\rm He}$ as

\begin{equation}
R_{\rm He} = ( R_{\rm p}^{2} + D_{\rm He} \, R_{\star}^{2} )^{1/2}
\label{eq: R He}
\end{equation}

where $D_{\rm He}$ is the Hei absorption. The equation for the $R_{\rm Hill}$ is

\begin{equation}