TOI-3664 b, TOI-4034 b & TOI-6564 b: Three new hot Jupiters around stars approaching the terminal age main sequence2026TOI-3664 b, TOI-4034 b & TOI-6564 b: Three new hot Jupiters around stars approaching the terminal age main sequence–7

Matthew P. Battley Marina Lafarga Thanks: E-mail: m.battley@qmul.ac.uk Affiliation: Astronomy Unit, Queen Mary University of London, Mile End Road, London E1 4NS, UK Affiliation: Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland Edward Gillen Affiliation: Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK Affiliation: Centre for Exoplanets and Habitability, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK Monika Lendl Affiliation: Astronomy Unit, Queen Mary University of London, Mile End Road, London E1 4NS, UK Solène Ulmer-Moll Affiliation: Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland Cynthia S. K. Ho Affiliation: Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands Emilio Marfil Affiliation: Astronomy Unit, Queen Mary University of London, Mile End Road, London E1 4NS, UK Sergio Sousa Affiliation: Departamento de Ingeniería Topográfica y Cartografía, E.T.S.I. en Topografía, Geodesia y Cartografía, Universidad Politécnica de Madrid, 28031, Madrid, Spain Yolanda Frensch Affiliation: Instituto de Astrofisica e Ciencias do Espaco, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal Affiliation: Departamento de Fisica e Astronomia, Faculdade de Ciencias, Universidade do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal Dimitri Veras Affiliation: Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland François Bouchy Affiliation: Department of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK Affiliation: Centre for Exoplanets and Habitability, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK Affiliation: Centre for Space Domain Awareness, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK Yann Carteret Affiliation: Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland Ian J. M. Crossfield Affiliation: Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland Tyler Fairnington Affiliation: Department of Physics and Astronomy, University of Kansas, Lawrence, KS, USA Mathilde Houelle Affiliation: Centre for Astrophysics, University of Southern Queensland, Toowoomba QLD, 4350, Australia Affiliation: Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL, USA Dan Huber Affiliation: Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland Marziye Jafariyazani Affiliation: Institute for Astronomy, University of Hawai‘i, Honolulu, HI 96822, USA Affiliation: Sydney Institute for Astronomy, School of Physics, University of Sydney NSW 2006, Australia Léna Parc Affiliation: NASA Ames Research Center, Moffett Field, CA 94035, USA Affiliation: SETI Institute, Mountain View, CA 94043, USA Don Radford Affiliation: Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland TG Tan Affiliation: Brierfield observatory, Bowral, NSW, Australia Sara Tavella Affiliation: Perth Exoplanet Survey Telescope, Perth, Western Australia, Australia Rob Wittenmyer Affiliation: Observatoire Astronomique de l’Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland Duncan Wright

\And

George Zhou Affiliation: Department of Physics and Astronomy, University of Kansas, Lawrence, KS, USA Affiliation: Department of Physics and Astronomy, University of Kansas, Lawrence, KS, USA Affiliation: Department of Physics and Astronomy, University of Kansas, Lawrence, KS, USA

Accepted XXX. Received YYY; in original form ZZZ

Abstract

Studying the evolution of hot Jupiters requires a sample of well-characterised systems across all evolutionary states. We present three new gas giant exoplanets around stars approaching the end of the main sequence, a comparatively unexplored epoch of hot Jupiter evolution. These planets were discovered by TESS before being vetted and confirmed through dedicated spectroscopic follow-up programmes by CARMENES, CORALIE and MINERVA-Australis. TOI-3664 b has a period of 3.30 days, a radius of 1.22 $\pm$ 0.03 R_Jup and a mass of 0.36 $\pm$ 0.12 M_Jup. TOI-4034 b is a short-period hot Jupiter with a period of 1.80 days, a radius of 1.58 $\pm$ 0.02 R_Jup and a mass of 0.87 $\pm$ 0.16 M_Jup. Meanwhile TOI-6564 b has a period of 3.99 days, radius of 1.46 $\pm$ 0.02 R_Jup and mass of 0.70 $\pm$ 0.07 M_Jup. All three planets have radii larger than Jupiter but sub-Jupiter masses, in line with slight inflation as their hosts increase in luminosity towards the end of the main sequence. These exoplanets’ low densities and hosts’ advanced evolutionary states make them interesting planets with which to study the later stages of hot Jupiter evolution. Careful analysis was undertaken to determine the ages of each system, considering astrometry, gyrochronology, stellar isochrones and lithium abundance, yielding ages of 9.0 ${}^{+2.4}_{-2.1}$ Gyr, 5.7 $\pm 0.5$ Gyr and 4.0 $\pm 1.0$ Gyr for TOI-3664, TOI-4034 and TOI-6564 respectively, yet each system has a similar evolutionary state because of their differing stellar masses (0.98 $\pm$ 0.03, 1.19 ${}^{+0.13}_{-0.03}$ and 1.18 ${}^{+0.16}_{-0.03}$ $M_{*}$ ). These three planets add more steps to the "age-ladder" of exoplanetary evolution, building towards the community’s goal of understanding how planets evolve over time.

Keywords:

planets and satellites: detection – planets and satellites: gaseous planets – planets and satellites: individual: TOI-3664 b – planets and satellites: individual: TOI-4034 b – planets and satellites: individual: TOI-6564 b

1 Introduction

The discovery of 51 Pegasi b by 86 fundamentally changed the community’s understanding of how planets form and evolve, revealing a gas-giant exoplanet orbiting its host with a period of only four days, unlike anything seen in the Solar System. This seminal exoplanet, which orbits seven times closer to its star than Mercury, became the first in an enigmatic group of exoplanets called ‘hot Jupiters’, a rare but crucial group of exoplanets with masses of at least 0.25 Jupiter masses and orbital periods of up to ten days (33, following). Since this first discovery, over 650 hot Jupiters have been discovered,¹¹ 1 n.b. the number of hot Jupiters was drawn from the exoplanet archive, https://exoplanetarchive.ipac.caltech.edu/index.html, accessed on 24 November 2025 yet their formation and evolution are still not well understood (see 33 for a full review).

Three main formation pathways have been proposed to form hot Jupiters: in-situ formation, disk migration and high eccentricity migration. In-situ formation theories posit that hot Jupiters form in-place, already possessing their present-day orbital periods, either via gravitational instability in the disk (18, e.g.), or via core accretion, where a proto-planet accretes significant amounts of gas from its environment (98; 100, e.g.). While the high temperatures required for gravitational instability for such close-in exoplanets makes that theory less plausible (104, e.g. see ), core accretion may still be able to form these planets if sufficient solids can be built up very early in the planet formation (33).

Meanwhile, disk migration and high-eccentricity migration theories circumvent the challenges of in-situ formation by suggesting that hot Jupiters form further out in the disk and then migrate inwards. Under disk migration, torques acting upon the planet from the disk can slow the planet down, resulting in a decreased orbital separation, a.k.a inward migration (45; 81; 59, e.g. see ). Such migration would typically lead to planets with smaller eccentricities due to damping from the disk (93). On the other hand, in high-eccentricity migration, the planet loses angular momentum through eccentricity excitation by another body (typically via planet-planet scattering (105; 144; 28, e.g.) or secular interactions such as the Kozai-Lidov effect (71; 79), before losing orbital energy via tidal interactions with its host star (33, e.g.). This high-eccentricity migration pathway should initially lead towards higher eccentricities until these are damped away by tides at close orbital separations (60, e.g.).

In order to understand how these theories interact and determine the most common evolutionary pathways, it is imperative that the community builds a population of hot Jupiters across the entire extent of their evolution. While exoplanets spend the vast majority of their time orbiting stars in their main-sequence phase of evolution, this is book-ended by two dynamic periods of stellar (and indeed planetary) evolution: the pre-main sequence and the red giant branch. Young exoplanets around pre-main sequence stars (and those up to several hundred Myr) are particularly valuable as they probe eras of ongoing planet formation and the majority of the proposed disk/high-eccentricity migration, however true hot-Jupiter progenitors remain elusive despite several dedicated searches (89; 88; 14; 13; 84; 137, e.g.). On the other hand, exoplanets around stars at and beyond the terminal age main sequence offer an opportunity to explore the latter stages of exoplanet evolution, at a time where their host stars are dramatically increasing in luminosity and beginning to expand. Such changes can lead to reinflation of exoplanets (46; 47; 146; 97, e.g.), reduction in orbital separations and even planetary engulfment (140; 113, e.g.). Although the discovery of ‘hot’ ( $P$ < 10 days) planets around stars from the terminal age main sequence to red giant phase is challenged by the shorter time periods spent in these evolutionary regions and potential engulfment of shorter-period planets, a small number of such planets have been found, typically around sub-giant stars e.g. HD 185269 b (63), WASP-136 b (78), HD 202772A b (143), NGTS-12 b (22), TOI-954 b (115), TOI-1842 b (146), TOI-4377 b & TOI-4551 b (97). These exoplanets often appear to have inflated radii, low densities and may be imminently at risk of being engulfed by their host star (23); indeed 22 calculated that NGTS-12 b may be engulfed by its host star within only 500 Myr.

Two particularly crucial parameters for anchoring hot Jupiter evolution are stellar age and the equivalent evolutionary phase (EEP) of their hosts. In their construction of the MESA isochrones, 35 and 29 define EEP as a series of evolutionary points which correspond to physically-motivated stages of stellar evolution shared by most, if not all, stellar types. For example the terminal age main sequence, TAMS, represents the point at which hydrogen is exhausted in the core of the star, and the tip of the Red Giant branch, RGBTip, represents the point where the stellar luminosity is at its maximum (35). Hence, while age is a valuable parameter for considering the timespans available for processes such as migration and atmospheric evolution, EEP also considers the stellar mass, accounting for the fact that more massive, hotter stars evolve more quickly than their cooler counterparts. Using this proxy for stellar evolution thus allows for planetary systems to be compared more easily across different host star types.

This paper presents three new hot Jupiter exoplanets orbiting host stars approaching the very end of the main sequence, representing a sparsely populated but important region of hot Jupiter evolution. All three systems were discovered originally by TESS, and then were followed up with high-precision spectrographs (CARMENES, CORALIE and MINERVA-Australis) to determine their masses.

The remainder of this paper is organised as follows: section 2 discusses the photometric, spectroscopic and community observations of the three candidate systems and their stellar hosts, section 3 discusses the data analysis conducted to determine the stellar and planetary parameters, including a joint fit of each planetary system, and section 4 presents the results of the joint fits for these three new hot Jupiter exoplanets, before comparing the new exoplanets to the known population. The paper concludes in section 5.

2 Observations

2.1 TESS photometry

Refer to caption — Figure 1: Full TESS light-curve for TOI-6564, illustrating discovery of a repeated transit signal. Top: original TESS light-curve from Sector 65 (in blue), with joint model from section 3.5 over-plotted in black. Bottom: Residuals to the joint fit. Note that the light-curves for TOI-3664 and TOI-4034 are similar in vein, with clear transits and minimal stellar variability, but are not instructive to reproduce here due to the large numbers of sectors of available data.

All three planets considered in this work were originally detected by the Transiting Exoplanet Survey Satellite (106, TESS,). TESS is a space telescope conducting a near all-sky transit survey by observing a different 24^∘ x 96^∘ ‘sector’ of sky at approximately monthly intervals. The excellent photometric performance of the satellite and all-sky nature of TESS’s search has resulted in thousands of transiting exoplanet candidates,²² 2 7771 at time of writing; https://exoplanetarchive.ipac.caltech.edu/ - accessed 24 November 2025 or ‘TESS objects of interest’ (TOIs), including TOI-3664 b, TOI-4034 b and TOI-6564 b.

Because of the evolving survey strategy of TESS, each star on the sky receives a slightly different selection of observations. TOI-3664 (TIC 348437470) was observed by TESS only once in its primary mission, with long-cadence (30min cadence) observations occurring in Sector 18. However, following its elevation to a TOI (see below), it was included in the TESS short-cadence sample and observed in Sector 58 (2 min cadence only) and Sectors 85/86 (20s and 2 min cadence data). On the other hand, because of its northerly position close to the TESS continuous viewing zone, TOI-4034 (TIC 375654303) was observed in four Sectors in TESS’s Primary Mission (Sectors 17, 18, 24 and 25) with long cadence (30min data), and in nine sectors with short cadence (2 min data) in the extended mission (Sectors 52, 57, 58, 59, 77, 78, 79, 84 & 86). Finally, although TOI-6564 (TIC 453668803) fell on the TESS detectors in three sectors, in the first two sectors (Sectors 11 and 38), the star fell right on the edge of the TESS detectors, a region which is known to result in low-quality data dominated by systematics, where lightcurves are not routinely extracted. Hence light-curves for this target were generated for the first time in Sector 65, in year 3 of the TESS mission (200s and 2 min cadence data), meaning that only this sector is considered in this analysis. The full light-curve for TOI-6564 is shown in Figure 1 as an illustration of the transit discovery for all three targets.

Because of the brightness of its host star ( $T_{\mathrm{mag}}$ = 9.57 mag), the candidate planet signal around TOI-6564 was first identified as part of the main search pipeline implemented by the TESS Science Processing Operations Centre pipeline (SPOC; 62, located at NASA Ames Research Center), which includes data calibration and extraction from the raw TESS data, corrections for typical systematics through the Presearch Data Conditioning step, a dedicated transiting planet search and a suite of data validation steps (see 62, for full details). TOI-6564 b passed all of the internal SPOC vetting tests and was alerted as a TOI on 20 July 2023.

In contrast, planetary candidate signals around the dimmer stars TOI-3664 and TOI-4034 were first identified as part of the TESS Faint-star search (73), an expanded search of the TESS Primary Mission data conducted using the Quick Look Pipeline (55; 53; 54; 74, QLP,). This search focused on fainter targets with TESS magnitudes between 10.5 $\leq T_{\mathrm{mag}}\leq$ 13.5 and featured light-curve extraction using an independent pipeline based on difference imaging (55), a search for transits using the Box-Least-Squares algorithm (70) and a series of basic criteria to select the signals which passed the detection threshold (at least 5 points per transit, a signal to pink noise ratio of at least 9, and general signal to noise ratio of above 5 or 9 depending whether the TESS magnitude was above or below 12). Signals that passed these tests were deemed ‘threshold crossing events’ and passed through a series of additional vetting steps, including sine-wave identification, planet model template matching, depth and V-shape checks, centroid offset checks and human visual inspection of remaining candidates (74). The candidate signals around TOI-3664 and TOI-4034 passed each of these checks and were both elevated to TOIs on 23 June 2021.

2.2 CARMENES spectroscopy

The two northern hemisphere candidates, TOI-3664 & TOI-4034, were followed up spectroscopically by the Calar Alto high-Resolution search for M dwarfs with Exoearths with Near-infrared and optical Echelle Spectrographs, (103, CARMENES,), which was chosen due to its high spectral resolution (R = 94,600 in VIS; 80,400 in NIR), northern latitude (+37:13:25) and multi-colour wavelength coverage (visible and infra-red wavelengths). CARMENES is installed on the 3.5m telescope at the Calar Alto Observatory in Almería (Spain) and consists of two separate spectrographs (deemed the VIS and NIR channels respectively) covering a wavelength range of 0.55 - 1.7 $\mu m$ and fed by fibers from the Cassegrain focus of the telescope. Observation time on CARMENES was secured as part of the Opticon Radionet Pilot (ORP) Trans-National Access programme (Grant Reference Number 101004719) in semester 2024A; PI M. Battley, CAHA programme number 051. The observations were taken with one of the fibers on target and the other one fed by a Fabry-Pérot etalon for simultaneous calibration. The observations were reduced with the default CARMENES data reduction pipeline caracal (24).

Because the majority of candidates alerted by TESS typically have minimal spectroscopic observations, the radial velocity (RV) observations with CARMENES were split into two stages: an initial vetting stage to check for obvious false positives such as spectroscopic or eclipsing binaries, and an extended monitoring stage once it was determined that TOI-3664 and TOI-4034 were reliable targets.

2.2.1 Vetting

The initial vetting step was carried out by observing one stellar spectrum at each of the expected maximum and minimum radial velocity epochs, or as close to phases 0.25 and 0.75 in the orbit as possible, according to the photometric ephemeris from the TOI alert. Vetting observations carried out between expected maximum (phase 0.25) and minumum (phase 0.75) revealed offsets of only $\sim$ 100-150 m/s for both stars, and occured in phase with the planetary ephemeris, consistent with true planetary signals caused by orbiting bodies with sub-Jupiter masses. This meant that both TOI-3664 and TOI-4034 passed the vetting stage of the observations and continued to the longer monitoring stage described below.

Note that six additional targets were vetted as part of this CARMENES programme, but none were found to be suitable for further monitoring due to a range of false-positive scenarios and difficult observing constraints. Briefly, TOI-5459 and TOI-6382 were highlighted as spectroscopic binaries, TOI-6248 was found to be a likely eclipsing binary, TOI-2538 was found to have untenably wide spectral lines and the observations of TOI-3571 were found to be in anti-phase with the transit ephemeris, suggestive either of an eclipsing binary or complex stellar activity. Meanwhile due to its longer period, only one RV measurement was possible for TOI-4688, so conclusions on this object are limited. Further discussion of this wider vetting is presented in Appendix A.

2.2.2 Monitoring

Extended monitoring of TOI-3664 was carried out between 12 February 2024 and 5 March 2024 for a total of 20 observations, while observations of TOI-4034 occurred between 12 May 2024 and 29 June 2024 for a total of 22 observations. Observations were carried out in pairs, consisting of two 22.5 minute exposures taken in series with the VIS and NIR channels simultaneously to allow for either two independent measurements at similar phases or an averaged 45min exposure for higher signal to noise measurements. The raw data was analysed using both the raccoon (77) and serval (148) pipelines for comparison, however the final chosen radial velocities were those extracted by the serval pipeline, aligning with the bulk of other CARMENES discoveries. serval is a template matching pipeline that initially derives an approximate RV time series by performing a least squares fit between the spectra with the highest signal-to-noise ratio (S/N) and the rest of observations. These initial RVs are used to create a high S/N template by co-adding the RV-corrected observed spectra, which is subsequently used to derive new RVs in an iterative process (one iteration is usually sufficient). The template co-adding is performed by cubic B-spline regression. serval also accounts for the barycentric movement of the Earth and any instrumental drift corrections. The RVs are extracted order by order, and a final RV is computed as the weighted mean of the order RVs.

The full radial velocity results are plotted in Figures 2 and 3 for TOI-3664 and TOI-4034 respectively, and tabulated in Appendix B.

Figure 2: Overview of collected RV data for TOI-3664. Top: All VIS RV measurements are shown in orange, while the same observations binned nightly are shown in blue. All points are shown as empty circles, with their 1

\sigma

errors shown as straight lines. The mean stellar RV offset has been removed. Bottom: Lomb-Scargle periodogram for all collected RV observations of TOI-3664. The 1% and 10% false alarm probability (FAP) values are shown by the dotted grey lines. The highest peak at 3.3 days corresponds to the planetary period seen from the photometric data (illustrated by the red dotted line), while the additional peak which passes the 10% FAP value is an alias which disappears after fitting the main RV signal.

Figure 3: Overview of collected RV data for TOI-4034. Format is the same as Figure 2 except that the 0.1% FAP value is also shown on the periodogram due to the slightly stronger periodic signals present. The strongest peak at 1.8 days aligns with the photometric period, shown as the red dotted line

2.3 CORALIE spectroscopy

TOI-6564, the single southern-hemisphere target, was added to an existing programme on the CORALIE spectrograph (102) to follow up planet candidates from TESS. CORALIE is a echelle spectrograph installed on the Swiss 1.2 metre Leonhard Euler Telescope at ESO’s La Silla Observatory and is fed by two fibers to allow for simultaneous calibration with a Fabry-Pérot etalon. It covers a wavelength range of 0.39 - 0.68 $\mu m$ and has a spectral resolution of $\sim$ 60,000. Spectra were extracted from the detector using the standard CORALIE calibration reduction pipeline, before radial velocity measurements were determined through cross-correlation with a binary G2 mask (96). Similar to the CARMENES observations, several vetting spectra were taken at opposite phases initially to search for obvious false positive signals, before an extended monitoring campaign was undertaken to determine the mass of the candidate. In total, TOI-6564 was observed 22 times between 25 April 2024 and 3 March 2025. Note that a slight $\sim$ 10 m/s offset was introduced to the data between the 2024 and 2025 observing seasons due to a new calibration lamp being installed, so this is accounted for in the analysis by treating the pre- and post-2024 data as two separate datasets.

Full radial velocity data for TOI-6564 is plotted in Figure 4 and tabulated in Appendix B.

2.4 MINERVA-Australis spectroscopy

Independently to the CORALIE efforts, TOI-6564 was also followed up by MINERVA-Australis Observatory. MINERVA-Australis is an array of four identical 0.7 m telescopes located at Mount Kent Observatory in Queensland, Australia. The four telescopes are linked via fibre feeds to a single KiwiSpec echelle spectrograph at a spectral resolving power of $R\sim$ 80,000 over the wavelength region of 5000-6300Å (2). The array is wholly dedicated to radial-velocity follow-up of TESS planet candidates (90; 1; 146, e.g.). Two simultaneous fibres provide wavelength calibration and correct for instrumental variations. The calibration fibres are illuminated by a quartz lamp through an iodine cell, eliminating contamination by Ar lines. Each epoch consists of 30 to 60 minute exposures from up to four individual telescopes. The radial velocities from each telescope are treated as coming from separate instruments to account for small velocity offsets between the fibres.

MINERVA-Australis observed TOI-6564 between 11-25 August 2023 for a total of seven observations. Two different radial velocity values were extracted from each observation; low precision least-squares RVS and higher precision CCFRVs. Low precision RVs were derived from a least-squares deconvolution of the observed spectrum against a synthetic non-rotating template (similar to the CHIRON pipeline; e.g. see 64; 143; $\sim$ 20 m/s noise floor per telescope). CCFRVs are higher precision radial velocities derived from a cross correlation against an averaged spectrum of the target ( $\sim$ 5 m/s noise floor per telescope), but are only produced for low $v\sin{i}$ stars (<10km/s) that have received more than five observations. In this case the $v\sin{i}$ of TOI-6564 was found to be <1.0 km/s from the CORALIE spectra, so the higher-precision CCFRVs were extracted and used in this analysis.

The full MINERVA-Australis data are plotted alongside CORALIE data in Figure 4 and tabulated in Appendix B.

Figure 4: Overview of collected RV data for TOI-6564. Top: All collected RV observations for TOI-6564. MINERVA-Australis data is plotted in orange, before the two different seasons of CORALIE data are plotted in purple and blue respectively. All points are shown as empty circles, with their 1

\sigma

errors shown as straight lines. Inter-instrument offsets have been removed. Bottom: Lomb-Scargle periodogram for the collected RV observations. The 10%, 1% and 0.1% false alarm probabilities (FAP) are shown by the dotted grey lines. The highest peak corresponds to a period of 3.99 days, agreeing with the transit period shown as the red dotted line.

2.5 Community observations

Following elevation to TOIs, all planet candidates were uploaded to the Exoplanet Follow-up Observing Program website (ExoFOP),³³ 3 https://exofop.ipac.caltech.edu/tess/ where community members are able to upload independent follow-up observations. A variety of additional follow-up observations were carried out for all three targets for vetting purposes, so are discussed further here, split up by observation type.

2.5.1 Additional spectroscopy

Both TOI-3664 and TOI-4034 received one spectroscopic vetting observation with the High Resolution Echelle Spectrometer (142, HIRES), installed on the Keck 10m telescope in Hawaii. HIRES is a grating cross-dispersed echelle spectrometer with an overall wavelength range of between 0.3 - 1.0 $\mu$ m and a resolution of 25,000 to 85,000 depending on the slit plate used. The vetting observations were carried out with a spectral resolution of $R$ = 50,000 and a wavelength range of 0.36-0.90 $\mu$ m. The obtained spectra from these observations revealed single-star systems with no obvious false positives, alike to the CARMENES spectra, verifying the stars’ suitability for further monitoring.

2.5.2 Seeing-limited photometry

Both TOI-4034 and TOI-6564 were additionally followed up by members of the TFOP SG1 Sub-Group for seeing-limited photometry in order to confirm that the transit signals were on target and to check for chromaticity suggestive of an eclipsing binary.

TOI-4034 was followed up in visible & infra-red (V & I-bands; Nikita Chazov on MASTER-Ural (0.4m)/ApogeeAltaU16m), red (R-band; Tomas Popajewski on a Newtonian reflector (0.25 m)/ ASI294MM PRO) and blue (B-band; Adam Popowicz on SUTO-UZPW50-0m5 (0.5 m)/ MORAVIAN G4-900) filters. These observations together showed that the signal was on time, on-target, uncontaminated by other stars and without significant chromaticity, validating that the signal observed in TESS was commensurate with an on-target transit signal from a body of planetary size. However, because of the relatively dim magnitude of TOI-4034 (Vmag = 13.3), these additional photometric observations were of much lower signal to noise than the TESS observations (especially in the latter sectors of TESS observations), so are not included in the joint analysis below (section 3.5). Furthermore, because many sectors of TESS data are available on either side of these follow-up observations, their inclusion would not appreciably improve the transit ephemerides.

TOI-6564 was followed up by Brierfield Observatory and the Perth Exoplanet Survey Telescope. Brierfield Observatory is located near Bowral, N.S.W. Australia. The 0.36m Planewave CDK14 telescope is equipped with a 4096 x 4096 Moravian 16 803 camera. The image scale is 0.735 arcsec per pixel, dramatically better than the 21 arcsec pixels of TESS, resulting in a 50 arcmin x 50 arcmin field of view. The collected photometric data included a single full transit of TOI-6564 b observed using a Johnson Blue filter, split into 110 x 120-second exposures. Photometry was extracted on UTC 2024 May 23 using the AstroImageJ (AIJ) software package (31), utilising a circular photometric aperture with a 5.1 arcsec radius. Note that no additional detrending parameters were used. The observed transit, plotted in Figure 14, was on time based on the TESS ephemeris, and of comparable depth (9.5 ppt) to the TESS transits, despite its bluer filter.

The Perth Exoplanet Survey Telescope (PEST) is located near Perth, Australia. The 0.3 m telescope is equipped with a $5544\times 3694$ QHY183M camera. Images are binned 2x2 in software giving an image scale of 0 $\aas@@fstack{\prime\prime}$ 7 pixel^-1 resulting in a $32′\times 21′$ field of view. A custom pipeline based on C-Munipack⁴⁴ 4 http://c-munipack.sourceforge.net was used to calibrate the images and extract the differential photometry. This observation, also plotted in Figure 14, yielded an on-time transit in the red (rp) filter with a depth of 7.75 $\pm$ 0.32 ppt.

As well as proving that the transit signal was on target and achromatic, these two extra transits helped appreciably to improve the transit ephemeris because only one sector of TESS data was available for TOI-6564, and hence they were included in the joint fit.

2.5.3 SAI Speckle Polarimeter imaging

High-angular resolution speckle imaging is a valuable tool to check for nearby companions to stars which may contaminate TESS photometry, especially given the wide $\sim$ 21" pixel-scale of TESS. Nearby stellar sources can create two troublesome signals in TESS: background eclipsing binaries may mimic true planetary signals, while nearby non-eclipsing stars can result in excess flux in the TESS aperture, leading to underestimated planetary radii. A search for stellar companions near TOI-4034 was conducted on the night of 27 October 2021 UT by using the Speckle Polarimeter installed on the 2.5m telescope at the Caucasian Observatory of Sternberg Astronomical Institute (SAI) (107). The Cousins I-band, a bandpass very similar to that of the TESS camera, was used for the observations resulting in the image shown in Figure 5. This observation was sensitive to signals of a 5.6-magnitude fainter star at 1", but revealed no such stellar companions nearby to TOI-4034.

2.5.4 Gaia Photometry

TOI-3664 and TOI-4034 were also included in a study by 92 who utilised the high angular resolution of the Gaia satellite (43; 42) – which allow it to resolve stars with at least $\sim$ 0.7 arcsecond separation – and collected Gaia epoch photometry to circumvent TESS’s reasonably wide point spread function (PSF) to evaluate whether a large number of known TOI signals were on-target or caused by background eclipsing binaries. In order to do this, 92 compared photometric data taken by the Gaia mission for all available stars within the TESS PSF to the alerted TOI signal, and matched the transit depths, durations and ephemerides to the most likely Gaia sources. In the ideal case where the TOI-hosting star is the brightest star in the PSF and the same transit signal is observable for the star in the Gaia photometry, the background eclipsing binary hypothesis can be ruled out and the TOI signal can be confirmed as ‘on-target.’ This was the case for both TOI-3664 and TOI-4034, which were confirmed to be on-target in Phase II of the ongoing analysis programme.

3 Data analysis

3.1 Stellar parameter determination

Up to date stellar parameters for TOI-3664, TOI-4034 and TOI-6564 were determined through a combination of spectroscopic analysis of the collected CARMENES/CORALIE/HIRES spectra and fitting the Spectral Energy Distribution of each star.

Table 1: Stellar names, properties, and photometric magnitudes for all host stars

Parameter		Value		Source
Names
TOI	TOI-3664	TOI-4034	TOI-6564	TESS
TIC ID	348437470	375654303	453668803	TESS
2MASS ID	J02211405+5942425	J22203526+7146024	J13431631-5609229	2MASS
Gaia DR3 ID	459319848632063360	2230550759343923456	6064308867810722560	Gaia DR3
Astrometric properties
RA	02:21:14.04	22:20:35.28	13:43:16.29	Gaia DR3
Dec	+59:42:42.68	+71:46:02.45	-56:09:23.22	Gaia DR3
pmRA [mas/yr]	-7.5543 ± 0.0125	-0.1520 ± 0.0124	-12.1358 ± 0.0084	Gaia DR3
pmDec [mas/yr]	8.6675 ± 0.0155	3.3590 ± 0.0115	-15.0171 ± 0.0145	Gaia DR3
Parallax [mas]	2.1483 ± 0.0153	1.3056 ± 0.0102	5.1705 ± 0.0142	Gaia DR3
Distance [pc]	465 ${}^{+6}_{-4}$	760 ${}^{+8}_{-6}$	193 ± 1	Sect. 3.1
Magnitudes
TESS [mag]	12.91 ± 0.0061	12.5798 ± 0.0159	9.5687 ± 0.006	TESS
B [mag]	14.754 ± 0.048	14.168	10.946 ± 0.072	Tycho
V [mag]	13.883 ± 0.114	13.262 ± 0.092	10.236 ± 0.005	Tycho
G [mag]	13.5461 ± 0.0003	13.1082 ± 0.000365	10.0315 ± 0.000245	Gaia DR3
J [mag]	11.995 ± 0.022	11.825 ± 0.026	8.918 ± 0.019	2MASS
H [mag]	11.618 ± 0.028	11.523 ± 0.031	8.656 ± 0.046	2MASS
Ks [mag]	11.48 ± 0.021	11.459 ± 0.022	8.558 ± 0.021	2MASS
WISE1 [mag]	11.401 ± 0.024	11.38 ± 0.023	8.483 ± 0.023	WISE
WISE2 [mag]	11.439 ± 0.02	11.408 ± 0.021	8.523 ± 0.02	WISE
WISE3 [mag]	11.366 ± 0.161	11.183 ± 0.086	8.515 ± 0.02	WISE
WISE4 [mag]	8.911	9.499 ± 0.444	8.809 ± 0.269	WISE
Bulk properties
$T_{\mathrm{eff}}$ [K]	5593 ± 32	5891 ± 40	5943 ± 64	Sect. 3.1
$[\mathrm{Fe/H}]$ [dex]	0.17 ± 0.02	0.12 ± 0.02	0.19 ± 0.04	Sect. 3.1
log g [cm/s²]	4.37 ± 0.06	3.87 ± 0.08	4.04 ± 0.11	Sect. 3.1
$v\sin i$ [km/s]	2.4 $\pm$ 1.0	7.9 $\pm$ 1.0	$<$ 1.0	Sect. 3.1
M_⋆ [M_⊙]	0.98 ± 0.03	1.19 ${}^{+0.13}_{-0.03}$	1.18 ${}^{+0.16}_{-0.03}$	Sect. 3.1
R_⋆ [R_⊙]	1.11 ± 0.02	1.83 ± 0.02	1.70 ± 0.02	Sect. 3.1
$\rho_{*}$ (g/cm³)	1.014 $\pm$ 0.046	0.277 $\pm$ 0.026	0.395 $\pm$ 0.023	Sect. 3.5
Age [Gyr]	9.0 ${}^{+2.4}_{-2.1}$	5.7 $\pm$ 0.5	4.0 $\pm$ 1.0	Sect. 3.2
EEP [-]	407 ${}^{+13}_{-16}$	448 ${}^{+6}_{-52}$	445 ${}^{+8}_{-58}$	Sect. 3.1

References: TESS (128); 2MASS (119); Gaia DR3 (44); Tycho (58); WISE (147).

As a sanity check of the values provided in the TESS Input Catalogue (129; 128, TICv8.2,), initial stellar parameters for each star were determined by fitting their Spectral Energy Distributionss (SEDs) using the ARIADNE code (141). ARIADNE allows for fitting six different stellar models independently using the nested sampler dynesty (117; 118; 127), which can then be combined via Bayesian Model Averaging over the entire sample of models to derive the final set of parameters for each star. The models fit by ARIADNE are Phoenix v2 (57), BT-Settl, BT-Cond (5), BT-NextGen (5; 50), 76 and 26. In order to find the magnitudes to fit for each star, ARIADNE searches through a variety of literature sources for broad-band photometry, including the Two Micron All-sky Survey catalogue (119, 2MASS,), Gaia Data Release 2 (43; 42, Gaia DR2,), Tycho-2 catalogue (58), ALL-WISE (147), the All-Sky Compiled Catalog (ASCC) of 67, Pan-STARRS1 survey (27), Spitzer/GLIMPSE survey (30), GALEX catalogue (15), APASS photometric survey (52), Sloan Digital Sky Survey (4, SDSS DR12,), Strömgren-Crawford catalogue (94) and the TESS Input Catalog (129; 128, TICv8,).

Additional magnitudes known from literature but missed by the default ARIADNE search were then added by hand: Johnson B and V bands from TICv8 for TOI-3664 and TOI-4034, 2MASS_J, 2MASS_H and 2MASS_Ks bands from 2MASS (119) for all stars, and additionally the WISE RSR_W3 and RSR_W4 bands from the Wide-Field Infrared Survey Explorer (147) in the cases of TOI-3664 and TOI-6564. Note that in the case of TOI-4034 the Pan-STARRS r, i and z bands were removed due to very large errors.

For the purposes of the initial stellar parameter check, the default priors in ARIADNE were retained, before running the dynesty nested sampling using 500 live points, 4 threads and a dlogz of 0.5, implementing a random-walk sampling method. This initial SED fit revealed stellar parameters broadly consistent with those provided in the TESS Input Catalogue and Gaia DR3 (44).

The collected spectroscopic data gave the opportunity to refine these parameters further, by deriving precise stellar parameters for all stars from analysis of the spectra themselves.

In the case of TOI-3664 and TOI-4034, stellar parameters ( $T_{\rm eff}$ , $\log{g}$ and [Fe/H]) were obtained from the CARMENES VIS spectra by means of SteParSyn, a Bayesian implementation of the spectral synthesis method (133). Beginning with the raw CARMENES spectra, all CARMENES orders of the template FITS files were merged into a single spectrum for each star and corrected by the radial velocity shift of the targets with iSpec (16). Since both stars are F- to G-type, SteParSyn was run using the MARCS synthetic grid and the set of Fe i and Fe ii lines available in 133. The final results are displayed in Table 1 and agree well with those derived from SED fitting.

In the case of TOI-6564 however, spectroscopic stellar parameters were determined from the CORALIE data using the ARES+MOOG spectroscopic analysis as described in 110; 126; 122. We used the latest version of ARES⁵⁵ 5 The latest version, ARES v2, can be downloaded at https://github.com/sousasag/ARES (124; 123) to consistently measure the equivalent widths (EW) for the iron line list presented in 125. The best spectroscopic parameters are found using the ionization and excitation equilibrium. In this process, it is used for a grid of Kurucz model atmospheres (76) and the radiative transfer code MOOG (121). The results from this analysis are shown in Table 1, and once again agree well with the SED-based parameters.

As the initial $v\sin i$ values measured from the CARMENES and MINERVA-Australis data were found to be overestimated due to the somewhat low signal to noise spectra and wing-broadening of the stellar lines due to macroturbulence (36, e.g. see), we recomputed $v\sin i$ values using higher signal to noise spectra. For TOI-3664 and TOI-4034, $v\sin i$ was extracted from the Keck/HIRES spectra using the SpecMatch-Synth code (99), the standard pipeline for extracting stellar parameters from HIRES data, which characterises the star by fitting synthetic spectra to an observed spectrum. Meanwhile, a new $v\sin i$ measurement for TOI-6564 was determined from the Full-Width Half Maximum of the CORALIE Cross-Correlation Function, using the relation introduced by 109. This method is particularly valuable for $v\sin i$ determination because the width of the CCF at half its maximum is largely unaffected by macroturbulence. The extracted $v\sin i$ values are presented in Table 1.

In order to determine self-consistent stellar radii, masses, ages and Equivalent Evolutionary Phases (EEPs) for each star, a second ARIADNE run was completed for each star using the spectroscopically derived stellar parameters, which also included fitting the stars to their most likely isochrones using the isochrones code (35; 29). Priors were placed on $T_{\mathrm{eff}}$ , logg and [Fe/H] based on the outputs derived above from the spectra, but priors on distance, radius and extinction, $\mathrm{A}_{v}$ , were left as the default priors in ARIADNE, which sets the priors to the Gaia DR3 stellar radius, Gaia DR3 distance from the corrected 9 catalogue and the maximum line-of-sight extinction allowed by the SFD galactic dust map (114; 112). As before, the dynesty nested sampling was run using 500 live points, 4 threads and a dlogz of 0.5, implementing a random-walk sampling method. The combination of models fitted by ARIADNE led to results shown in Table 1 (and illustrated in Figure 6) and are consistent with, but more precise than, those within the TESS Input Catalogue.

Note however that isochrone fitting for both mass and age was complicated by the position of all three stars falling near the terminal main sequence, a region which is very sensitive to changes in stellar parameters. This was particularly challenging for TOI-4034 and TOI-6564, shown in Figure 6, which fell in the corner of the main-sequence/sub-giant branch and hence are highly susceptible to small changes in effective temperature and mass. Indeed, the initial ARIADNE results for TOI-4034 and TOI-6564 showed a dichotomy in age depending on the true stellar mass, largely due to the stars’ isochronal positions being consistent with both younger and older ages depending whether they were evolving on or off the main sequence or before/after the main sequence turn-off. The true age of the system is thus crucially important to the system parameters and will be discussed in detail in the next section.

3.2 Age determination

Determining the age of these systems through isochrone fitting and/or gyrochronology alone is complicated by their position near the terminal main sequence and the lack of clear stellar activity periods in the collected TESS data. Hence, additional stellar ageing methods are discussed here to determine the most reliable ages for each system.

3.2.1 Kinematic ages

One of the most precise methods of ageing stars is placing stars in stellar groups such as clusters and stellar associations. Both TOI-3664 and TOI-4034 were highlighted in 34 as high likelihood members of young clusters (TOI-3664: 67% member of ASCC 8, age $\sim$ 50Myr; TOI-4034: 85% member of Collinder 471, age $\sim$ 7 Myr), which originally motivated their follow-up with CARMENES. Similarly, TOI-6564 was included in a 32 sample of high-likelihood members of the $\sim$ 5-20 Myr Scorpius Centaurus association (expected 10-30% field-star contamination), based on a combination of spatial, kinematic and colour-magnitude cuts. However, these signs of young cluster membership contrasted with the lack of significant stellar activity signals seen in the collected photometric and spectroscopic data (see sections 3.3 and 3.4), motivating a more in-depth look into the ages of these systems.

The significantly improved astrometry delivered by the Gaia satellite (43; 42) offered the opportunity to revaluate cluster membership. Most notably, in a new study with Gaia DR2 data, 25 concluded that Collinder 471 was not a real cluster, and instead an asterism, debunking this claim of youth for TOI-4034. In order to examine whether the three host stars were likely associated with other co-moving groups of stars based on the more precise Gaia DR3 data, all three targets were analysed using the comove package,⁶⁶ 6 https://github.com/adamkraus/Comove a modified version of the FindFriends routine introduced in 135 which searches for comoving neighbours to a target star and returns useful information regarding their potential youth. In order to find the closest comoving stars to each target, comove was run using velocity limits of $<$ 5 km/s and distance limits of 30pc from the target. Although each star was in a crowded region of the sky, TOI-3664 and TOI-4034 were both found to have relatively few comoving companions within the 5km/s and 30pc limits (16 and 45 members from proper motion alone; 0 and 1 members including RV signals as well - although note that the number of targets with RV measurements in Gaia DR3 was limited by the magnitude of the stars), suggesting that they may be older background interlopers instead of true cluster members. This is supported by the relatively large distances found for these stars in section 3.1: TOI-3664: 407 ${}^{+13}_{-16}$ pc; TOI-4034: 760 ${}^{+8}_{-6}$ pc. On the other hand, TOI-6564 was found to have 208 comoving stars within the 5km/s and 30pc limits, of which at least 12 were also found to have consistent radial velocities (a number that is again limited by the available Gaia DR3 radial velocity measurements). A colour-magnitude diagram for these comoving stars is shown in Figure 7, compared to isochrones for some well-know young clusters. Although the placement of TOI-6564 on the plot gives some evidence as to why it might have been included in an Upper-Scorpius sample, as it was for 32, the wider population of comoving stars on the colour-magnitude diagram appears consistent with the main sequence, suggesting that TOI-6564 is evolving off the main sequence instead of on to it. It is also worth noting that the distance found for TOI-6564 of 193 $\pm$ 1 pc also places it behind the main Upper-Sco sample, which is know to be situated at $\sim$ 130 pc (51, e.g.).

3.2.2 Gyrochronology

Gyrochronology is a commonly used method to age field stars, based on the gradual spin down of stars as they evolve along the main sequence (65; 12; 19, e.g. see). However, gyrochronology is challenging for these three stars due to the difficulty in determining rotation periods for each target (as discussed in section 3.3), with a lack of activity in the TESS photometry for these G-type stars suggestive of rotation periods $>$ 10-20 days. Similarly, no additional significant periods at <10 days are present in the radial velocity measurements discussed in section 3.4. The $v\sin i$ values obtained from the spectra for TOI-3664 and TOI-6564 also suggest longer stellar rotation periods, with a measured $v\sin i$ value for TOI-3664 of 2.4 $\pm$ 1.0 km/s only slightly larger than the resolution-induced lower limit of 2.2 km/s for the HIRES spectrograph, and a $v\sin i$ measurement of <1.0 km/s from the CORALIE data of TOI-6564 also suggestive of a slowly-rotating star. This supports an older age hypothesis for both TOI-3664 and TOI-6564, however cannot provide very reliable age estimates due to the dependence on stellar inclination.

The situation for TOI-4034 is slightly more complex, with a measured $v\sin i$ value of 7.9 $\pm$ 1.0 km/s suggestive of a maximum rotation period of 11.7 days. Such a rotation period for a star with an effective temperature of 5891 $\pm$ 40 K would suggest an age on the order of 1 Gyr according to relations from 19, contrasting with the older ages discussed in the next two sections. However, inspection of other similar hot Jupiter exoplanetary systems ( $M_{p}$ > 0.25 $M_{\mathrm{Jup}}$ ) with very short periods (P $<$ 2 days) around similar G-type stars ( $5300<T_{\mathrm{eff}}<6000$ K) reveals several other ‘old’ systems with inflated $v\sin i$ values, for example HAT-P-23 b (10, age = 4.0 $\pm$ 2.0 Gyr; $v\sin i$ = 8.1 km/s) and Kepler-41 b (108; 17, age = 4.4 $\pm$ 1.3 Gyr; $v\sin i$ = 6 $\pm$ 2 km/s). This trend is reinforced by the work of 134 who found that hot Jupiter host stars on average spin faster than similar G-type stars which host small planets and/or giant planets on wider orbits (e.g. see Figure 5 of their paper). They propose that this ‘spin-up’ is caused by tidal torques transferring angular momentum from the closely orbiting hot Jupiter to the host star, supporting earlier works showing tidal spin-up for hot Jupiter hosts (21; 101; 85, e.g.). This suggests that gyrochonology is unreliable for stars with very closely orbiting hot Jupiter exoplanets, and hence does not place any useful constraints on the age of the TOI-4034 system.

3.2.3 Isochronal ages

Age estimates were also obtained via isochrone fitting as part of the ARIADNE stellar fits discussed in section 3.1. These fits, which considered the extracted stellar properties for each star, resulted in ages of 9.0 ${}^{+2.4}_{-2.1}$ , 5.7 ${}^{+0.5}_{-2.1}$ and 4.0 ${}^{+2.5}_{-0.6}$ Gyr for TOI-3664, TOI-4034 and TOI-6564 respectively, with the posterior distributions shown in Figure 8. While the posterior for TOI-3664 exhibits a normal (if wide) distribution, it is informative to examine the posterior distributions for TOI-4034 and TOI-6564 more carefully, as each star fell in a very dynamic part of the Hertzsprung-Russel diagram, near the corner of the Terminal Age Main Sequence (TAMS) - see Figure 6. Because of this location and the relationship between age and stellar mass, two distinct peaks are seen in both the age and mass posteriors for TOI-4034 and TOI-6564 (Figures 8 and 9 respectively), complicating a true age determination from isochrones alone. Nonetheless, the evolutionary state of each system is clearer for all three stars, with Equivalent Evolutionary Phase (EEP) values of 407 ${}^{+13}_{-16}$ , 448 ${}^{+6}_{-52}$ & 445 ${}^{+8}_{-58}$ respectively confirming that all systems are approaching the Terminal Age Main-Sequence. Note that for the MIST isochrones with which this isochrone fitting was carried out, an EEP number of 202 corresponds with the Zero-Age Main-Sequence (ZAMS), an EEP value of 353 denotes the Intermediate Age Main-Sequence (IAMS) and a value of 454 represents the TAMS.

In an attempt to overcome the dichotomy in mass/age for TOI-4034 and TOI-6564, an independent determination of the stellar mass was estimated by combining the output stellar density from the joint fit carried out in section 3.5 (largely driven by the shape of the transit) with the radius determined from the ARIADNE fit to the spectral energy distribution in section 3.1. This resulted in masses of 1.20 $\pm$ 0.12 $M_{\odot}$ for TOI-4034 and 1.38 $\pm$ 0.09 for TOI-6564, favouring the lower half of the mass posterior for TOI-4034 and the upper half for TOI-6564. These masses corresponded with the older age of 5.70 $\pm$ 0.5 Gyr for TOI-4034 and the slightly younger age of 4.0 $\pm$ 1.0 Gyr for TOI-6564, giving further weight to the determined averages of the ARIADNE fits.

3.2.4 Lithium ages

Another useful age indicator is photospheric lithium depletion, as photospheric lithium typically decreases with stellar age. While in less massive stars (T ${}_{\mathrm{{eff}}}<4500K$ ) lithium is often considered a youth indicator (indeed 61 show that lithium is usually depleted within 1 Gyr in stars cooler than 4500 K), in hotter stars, modest amounts of lithium may be present until much older ages, with some stars showing lithium signatures with equivalent width of up to 100 mÅ beyond 5 Gyr (83; 132, see e.g.).

61 have developed the EAGLES code⁷⁷ 7 https://github.com/robdjeff/eagles which allows the user to derive age estimates and posterior age probability distributions based on measurements of the equivalent width of the lithium doublet around 6708Å. Both TOI-3664 and TOI-6564 display clear lithium lines in their spectra, from which the lithium equivalent width (Li_EW) was determined by fitting a sum of three inverted Gaussian distributions to the spectra, centred on each of the lines in the lithium doublet and the nearby Fe I line which can become blended at lower spectrograph resolution or higher stellar activity levels. The results of this fitting revealed lithium equivalent width measurements of Li_EW = 31.3 $\pm$ 5.6 mÅ for TOI-3664 and Li_EW = 37.0 $\pm$ 1.5 mÅ for TOI-6564. Fitting these alongside the stellar parameters with the EAGLES code yielded the following age estimates: TOI-3664: 2.97 ${}^{+4.15}_{-1.85}$ Gyr; TOI-6564: 3.04 ${}^{+4.00}_{-2.23}$ Gyr, consistent with those from the isochronal fitting above. Note that the relatively large errors are driven by the fact that lithium abundances become much more disperse in older stars, leading to a wider range of plausible ages (61, see e.g.).

On the other hand, TOI-4034 did not display any notable Lithium lines around 6708 mÅ, so it could not be included in the lithium analysis. However, the lack of lithium provides a lower limit on the age of the system, suggesting that the target was sufficiently old that the bulk of its original lithium had already been consumed (e.g. likely $\gg$ 1 Gyr). The decreased lithium abundance compared to TOI-6564, which has a similar stellar mass, also provides some evidence that TOI-4034 is older than TOI-6564, aligning with the isochronal ages.

3.2.5 Aggregate ages

To summarise, although all three systems were initially suspected to be young due to inclusion in catalogues of young clusters/associations, the combination of better kinematic data, gyrochronology, isochronal fitting and lithium abundance reveal that these systems are actually much older, with the host stars approaching the terminal age main sequence. In the case of TOI-3664, a small $v\sin i$ value of 2.4 km/s and lithium age of 2.97 ${}^{+4.15}_{-1.85}$ Gyr supports the old isochronal age of 9.0 ${}^{+2.4}_{-2.1}$ Gyr, and so we adopt this value for the system. Meanwhile, although the gyrochronological age of such a close-in system as TOI-4034 appears unreliable due to tidal effects, the lack of observable lithium in the spectrum and isochronal age of 5.7 ${}^{+0.5}_{-2.1}$ Gyr supports an early G-type star approaching the very end of the main sequence. Finally, the isochronal age of 4.0 ${}^{+2.5}_{-0.6}$ Gyr for TOI-6564 is supported by a lithium age of 3.04 ${}^{+4.00}_{-2.23}$ Gyr, and a low $v\sin i$ value of 1.0km/s commensurate with an old, inactive star. One final complication is that both TOI-4034 and TOI-6564 display a mass/age degeneracy due to their position near the corner of the Terminal Age Main Sequence, but this degeneracy was solved by independently determining the stellar mass from the joint fit stellar density and the SED-driven radius, setting a narrower age of 5.7 $\pm$ 0.5 Gyr for TOI-4034 and 4.0 $\pm$ 1.0 Gyr for TOI-6564.

The most likely ages for these three systems are thus: TOI-3664: 9.0 ${}^{+2.4}_{-2.1}$ Gyr, TOI-4034: 5.7 $\pm$ 0.5 Gyr and TOI-6564: 4.0 $\pm$ 1.0 Gyr, representing a wide range of ages despite their similar evolutionary phases, driven by their different host star masses.

The discrepancy between the young literature ages and old ages found for all of these targets highlights the importance of careful age vetting of clusters and individual systems during the follow-up process of exoplanet candidates. Given the key importance of stellar/planetary ages for anchoring exoplanetary evolution theories, is recommended that the community uses multiple ageing methods wherever possible to determine the true age of exoplanetary systems.

3.3 Photometric analysis

Wherever possible, the light-curves analysed in this work were drawn from the standard TESS SPOC extraction pipeline (62), using the PDC_SAP flux (131; 130; 120), however where these were not available (e.g. during the Primary Mission) they were drawn from the QLP pipeline instead (55). After downloading from the Mikulski Archive for Space Telescopes (MAST),⁸⁸ 8 https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html individual sectors of data were cleaned using a custom-built cleaning pipeline to ensure that similar cleaning steps were applied to all sectors of data regardless of which data extraction pipeline had been used (e.g. SPOC 2 min or QLP 30min). Briefly, this pipeline removes poor-quality data based on the quality flags automatically generated by the SPOC and QLP pipeline, data-points close to spacecraft momentum dumps, epochs of excess scattered light and data at times where spacecraft pointing was compromised. Three additional epochs of particularly noisy data caused by scattered light from the Earth and Moon were removed in sectors 18 and 86, between 1791.37 to 1815.03 TBJD, 3637.90 to 3641.60 TBJD and 3651.46 to 3655.20 TBJD respectively. Note that in the case of TOI-3664, the excessive photometric scatter in Sector 86 was found to wash out all other signals and hence this sector was removed from this analysis. Following this cleaning step, the individual sectors of data were combined into a single light-curve for each target using the LightCurve object included within the lightkurve package⁹⁹ 9 https://lightkurve.github.io/lightkurve/index.html (80).

To make use of the increased data quantity available since the initial detection of the planet candidates, an updated planet-search through the light-curves for each system was undertaken, both with and without masking the known planet. This had the dual benefit of providing updated transit ephemerides for inputting into the joint fit (see Section 3.5), and allowing for signals due to potential additional planets in the system to be searched for. Planet searches were conducted using all sectors of available data for all three targets by first detrending the data using the custom Young Star Detrending (YSD) code described in 14, before searching for planets using the Box Least Squares algorithm (70) implemented as the BoxLeastSquares function in astropy (8; 6; 7). In all cases, no additional candidate signals were found in this search.

In addition to the transit search, a Lomb-Scargle (82; 111) search for sinusoidal periodicities was undertaken in the light-curves for all three targets to search for the most likely rotation period of each star. For the rotation-specific analysis, the QLP data was analysed because of its better performance at separating stellar signals in crowded regions (53; 54). Note that unlike in the main transit search and joint analysis, for the rotation period search the QLP Simple Aperture Photometry (SAP) flux was chosen to best preserve any stellar activity signals, as these are largely removed by the detrending applied to the Kepler-Spline SAP (KSPSAP) data. Two alternative period searches were carried out; one using all available light-curves and a second using only the longest continuous section of data (typically 1-2 sectors). Unfortunately, in all cases the TESS data was found to be dominated by systematic signals rather than reliable rotation signals (indeed the primary rotation period was found to vary on a sector by sector basis), suggesting that these stars are reasonably inactive and have stellar rotation periods longer than half a TESS sector.

As TESS is known to produce significantly less reliable stellar activity periods beyond periods of $\sim$ 12 days (20), a similar lomb-scargle search was carried out on data from the ASAS-SN survey (116; 69), however no significant rotation period peaks (i.e. those over the 10% false alarm probability level) were seen in the ASAS-SN periodograms.

Table 2: Planet parameters for all systems from juliet: median and 68% confidence interval

Parameter		Prior distribution^∗	TOI-3664 b	TOI-4034 b	TOI-6564 b
$P$ .	Period (days) .	$N$ ( $P_{photom}$ ,0.1) .	3.2974839 $\pm$ 0.0000041	1.8020932 $\pm$ 0.0000012	3.985413 $\pm$ 0.000011
$T_{0}$ .	Time of transit center (BJD_TDB) .	$N$ ( $t_{0,photom}$ ,0.1) .	2460661.83115 $\pm$ 0.00079	2460661.21859 $\pm$ 0.00047	2460071.37027 $\pm$ 0.00019
$b$ .	Impact parameter of the orbit .	$U$ (0,1) .	0.29 $\pm$ 0.19	0.620 $\pm$ 0.027	0.386 $\pm$ 0.048
$p=R_{p}/R_{*}$ .	Planet to star radius ratio .	$U$ (0,1) .	0.1132 $\pm$ 0.0018	0.08875 $\pm$ 0.00061	0.08843 $\pm$ 0.00052
$\sqrt{e}sin(\omega)$ .	Parametrisation for e and $\omega$ .	$U$ (-1,1) .	0.28 ${}^{+0.09}_{-0.17}$	N.A.	N.A.
$\sqrt{e}cos(\omega)$ .	Parametrisation for e and $\omega$ .	$U$ (-1,1) .	-0.01 ${}^{+0.30}_{-0.33}$	N.A.	N.A.
$K$ .	Radial velocity semi-amplitude (m/s) .	$U$ (0,200) .	51 ${}^{+16}_{-15}$	128 ${}^{+19}_{-21}$	80.3 ${}^{+2.0}_{-2.1}$
$e$ .	Orbital eccentricity .	N.A. or fixed.	0.159 $\pm$ 0.074	0	0
$\omega$ .	Argument of periastron (deg) .	N.A. or fixed.	93 ${}^{+53}_{-50}$	90	90
$i$ .	Inclination (deg) .	.	87.8 $\pm$ 1.3	80.76 $\pm$ 0.38	86.77 $\pm$ 0.39
$a$ .	Semi-major axis (AU) .	.	0.0431 $\pm$ 0.0010	0.03289 $\pm$ 0.00071	0.0542 $\pm$ 0.0012
$R_{p}$ .	Planetary radius ( $R_{\mathrm{Jup}}$ ) .	.	1.222 $\pm$ 0.030	1.580 $\pm$ 0.020	1.463 $\pm$ 0.019
$M_{p}$ .	Planetary mass ( $M_{\mathrm{Jup}}$ ) .	.	0.36 $\pm$ 0.12	0.87 $\pm$ 0.16	0.699 $\pm$ 0.066
$\rho_{p}$ .	Planetary density (g/cm³) .	.	0.248 $\pm$ 0.081	0.275 $\pm$ 0.050	0.277 $\pm$ 0.028
$T_{\mathrm{eq}}$ .	Equilibrium Temperature (K) .	.	1232 $\pm$ 12	1907 $\pm$ 22	1445 $\pm$ 21
$T_{\mathrm{dur}}$ .	Transit duration (hours) .	.	3.22 $\pm$ 0.16	3.215 $\pm$ 0.086	4.53 $\pm$ 0.11

*Notes: $U$ (a,b) denotes a uniform prior between a and b, while $N$ (a,b) denotes a normal distribution with mean of a and standard deviation b. Priors for $P$ and $T_{0}$ varied for each planet, situated around those computed in the photometric analysis described in section 3.3. Parameters without priors were derived from the fitted parameters and stellar parameters derived in section 3.1, sampling from a normal distribution for each parameter based on their derived means and standard deviations. N.A. denotes ‘not applicable for this fit’. For the eccentric fit of TOI-3664 b, $e$ and $\omega$ were derived from $\sqrt{e}sin(\omega)$ and $\sqrt{e}cos(\omega)$ , while for TOI-4034 b and TOI-6564 b they were fixed at 0 and 90 deg respectively, with the reparametrisations unused.

3.4 Spectroscopic analysis

Lomb-Scargle periodograms (82; 111) were also used to search for periodicity in the radial velocity measurements, yielding the results presented in Figures 2, 3 and 4. Because the CARMENES observing strategy employed two 22.5 min exposures in series each observing night, for TOI-3664 and TOI-4034 the period search was trialled first with all individual measurements treated separately and secondly by binning data night by night. While both configurations yielded similar results, overall the unbinned RV measurements showed the strongest periodic signals despite the slightly decreased signal to noise of the individual spectra, so the unbinned data were used throughout this analysis.

Although the strength of the signal was limited by the available observing time in the small programmes, in all cases the first or second strongest peak in the periodogram corresponded to the same period alerted by the TOI release and found in the photometric analysis above (TOI-3664 b = 3.30 days; TOI-4034 b = 1.80 days; TOI-6564 b = 3.99 days). In order to assess the statistical significance of these periodogram peaks, analytical False Alarm Probability (FAP) values were estimated for each target following the formalism of 11. While TOI-6564 b clearly passes even the 0.1% FAP threshold, both TOI-3664 b and TOI-4034 b were found to cross the 10% FAP threshold, and approach/cross the 1% FAP threshold, validating the period seen in photometry. Note that after the removal of the best-fitting RV signal in each system, no additional FAP-crossing signals remained. Meanwhile, periodograms computed for common activity indicators such as the Full-Width Half-Maximum of the cross-correlation function (CCF), CCF bisector and CCF contrast across all spectrographs (and additionally the Ca IRT and H-alpha indicators for the CORALIE data) did not display notable peaks at these periods.

As an independent check of the spectroscopic analysis, all three systems were also analysed with the kima package (39). The kima package uses Bayesian inference to model a sum of Keplerian signals in the combined data across all available spectroscopic instruments. Despite the relatively small number of RV observations per target, the maximum likelihood period in the case of both TOI-3664 b and TOI-6564 b was found to be consistent with that seen in the TESS photometry, with determined RV periods of P = 3.296 days for TOI-3664 b and P = 3.987 days for TOI-6564 b respectively. However, in the case of TOI-4034 b, the photometric period of P_phot = 1.802 days actually corresponded to the second strongest period in the kima analysis (P_kima = 1.803 days), with the strongest period from the kima analysis occurring at 2.149 days. This is echoed by the Lomb-Scargle analysis, with the two strongest periods also falling at 2.15 and 1.80 days respectively for TOI-4034 b (see Figure 3). This may hint at a second periodic signal in the data (e.g. activity), but additional spectroscopic monitoring is required to confirm this.

kima also allows for a direct comparison between the evidence for different system architectures, i.e. the most likely number of planets in each system. For TOI-6564 b, the candidate with the most RV follow-up, a 1-planet model was clearly preferred, with a 1-planet model having a log likelihood of log(Z) = -140.14 and a 0-planet model having a log likelihood of log(Z) = -176.19. However, the two other targets suffered from their small number of RV measurements and hence the 0-planet model was still preferred, despite many samples falling in the 1-planet (and in the case of TOI-4034, 2-planet) cases. Hence to confirm these latter two planets with radial velocities alone would require further observations, but when in support of the photometric data they validate the periods of the planetary systems quite effectively.

3.5 Joint fit

Global modelling for all three systems was completed using the juliet package (38). juliet uses Bayesian inference to model planetary systems by combining transit fitting using batman (72), RV fitting using radvel (41), Gaussian process modelling with george (40) and celerite (40), and nested sampling to simultaneously fit data from multiple photometric and spectroscopic datasets. Nested sampling was carried out using the dynesty package (127), using $>N^{2}$ live points (where $N$ was the number of parameters in each fit), and in each case the fits were run until the estimated uncertainty in the log-evidence was less than 0.1.

Note that because of the significantly higher cadence of the short 2 min vs long 30-min cadence data (and hence the much better coverage of individual transits), only the TESS short-cadence (2 min data) was included in the joint fit, but with Gaussian priors on t0 and planetary period drawn from the photometric analysis undertaken in Section 3.3 using all sectors of available TESS data. Both the radius ratio (p = R_p/R_∗) and impact parameter, b, were allowed to vary between zero and 1 uniformly.

A white-noise jitter term was added in quadrature to the error-bars of each photometric and RV dataset to account for underestimated uncertainties and/or additional white noise that was not captured by the model. In addition, to account for any leftover stellar activity and systematics caused by scattered light from the Earth/Moon and/or slight baseline shifts after data gaps in the TESS data, a Gaussian Process (GP) model was included in the photometric models using the Matérn-3/2 kernel in celerite (40). This GP is able to account from correlated noise from multiple sources, and ensured that a reliable baseline could be found for the out-of-transit data, enabling an accurately measured transit depth.

Realistic priors on the limb-darkening parameters were found using the LDCU package.¹⁰¹⁰ 10 https://github.com/delinea/LDCU LDCU is a modified version of the python routine implemented by 37 that computes the limb-darkening coefficients and their corresponding uncertainties using a set of stellar intensity profiles accounting for the uncertainties on the stellar parameters. The stellar intensity profiles are generated based on two libraries of synthetic stellar spectra: ATLAS (75) and PHOENIX (57). For each target, quadratic coefficients were derived from the LDCU code, based on the q1 and q2 parameterisation of (68), and used to set Gaussian priors on the limb-darkening parameters in the joint fit. For each set of TESS data, the TESS passband was used to calculate q1 and q2, while for the Brierfield Observatory the Bessel B-band was used and for PEST, uniform unconstrained priors of q1, q2 = [0,1] were used because LDCU did not contain a bandpass for the rp filter used by PEST. To account for any variations from the modelled limb-darkening in the real data, the errors found in LDCU were multiplied by five to give more conservative priors.

Because the Thoirum-Argon lamp in the CORALIE spectrograph was replaced in 2024, there are slight differences in both the baseline and properties of the pre-2024 and post-2024 data, so these were treated as two different datasets in the joint fit of TOI-6564 b, denoted CORALIE14 and CORALIE24 respectively. Additionally, any spectroscopic measurements that fell within the expected times of transit were removed.

For TOI-3664 and TOI-4034, the dilution value was fixed to one, as only the TESS 2-min data was used, which is already corrected for dilution. As discussed by 49, it is possible that additional systematic contamination caused by underestimation of stellar radii was not accounted for in this dilution correction, but tests with wide dilution priors of 0 to 1 were found to give unrealistically large radius values and much lower log evidence values, as was also cautioned by 38. Hence until higher resolution photometric follow-up is undertaken, fixing the dilution factor to one is the preferred approach for these two dimmer targets.

TOI-6564 however has data from three different photometric bands, of which only the TESS 2 min-cadence data is theoretically dilution corrected. The dilution value was thus allowed to vary uniformly between zero and one for the two follow-up observations carried out by Brierfield Observatory and PEST. Note that the end of the out-of-transit PEST data (t $>$ 2460390.35) was masked because it displayed an unexplained dip in flux which appeared systematic.

Circular and eccentric fits were trialled for each system and compared according to their Bayesian evidence. For the eccentric fits, the $\sqrt{e}\sin(\omega)$ and $\sqrt{e}\cos(\omega)$ allowed by 37 was employed, with uniform priors between -1 and 1, and an eccentricity limit of 0.8 to aid convergence of the model. Following 38 and 136, different models were compared using the natural log of the Bayesian evidence (ln $Z$ ), with $\Delta$ ln $Z$ = 2 the threshold between weak and moderate evidence for one model being preferred, and $\Delta$ ln $Z>$ 5 the threshold for strong evidence.

The final results of the joint photometric and spectroscopic fit, along with the priors used for all parameters, can be found in Tables 2 and 7, and discussed on a planet by planet basis in the next section.

4 Results and Discussion

4.1 Planetary fit results

4.1.1 TOI-3664 b

The joint fit of TOI-3664 b revealed a close-in (P = 3.30 days), slightly inflated planet with a radius of 1.22 $\pm$ 0.03 R_Jup and mass of only 0.36 $\pm$ 0.12 M_Jup, or roughly the mass of Saturn. This results in a reasonably low density of 0.25 $\pm$ 0.08 g/cm³.

Because of the low eccentricity suggested by the spectroscopic-only fit, two different models were trialled: one with a circular orbit (e = 0, $\omega$ = 90 deg), and another with an eccentric orbit, using the discussed $\sqrt{e}sin(\omega)$ and $\sqrt{e}cos(\omega)$ reparametrisation. The comparative log-evidence values for these two fits were ln $Z_{\mathrm{circ}}$ = 140421.670 and ln $Z_{\mathrm{ecc}}$ = 140428.264, giving $\Delta$ ln $Z$ = 6.59 in favour of the eccentric fit, which suggests strong evidence of the eccentric fit being preferred. However, with a calculated eccentricity of e = 0.16 $\pm$ 0.07, this eccentricity should be treated with a little caution. To get additional confirmation of the true eccentricity, additional RV monitoring with higher-precision instruments is recommended.

The folded photometric data showing the transit fit for TOI-3664 b can be seen in Figure 10, with the fitted RVs in Figure 11.

4.1.2 TOI-4034 b

The joint fit of TOI-4034 b revealed a planet with an even closer orbit (P = 1.80 days), a radius of 1.58 $\pm$ 0.02 R_Jup and mass of 0.87 $\pm$ 0.16 M_Jup, indicative of a more typical hot Jupiter exoplanet.

In this case, the circular orbit was marginally preferred in the model comparison, with a ln $Z_{\mathrm{circ}}$ = 397184.588 and ln $Z_{\mathrm{ecc}}$ = 397182.586, giving $\Delta$ ln $Z$ = 2.0 in favour of the circular fit. Given that for such a short orbital period the tidal circularisation timescale is also much shorter than the stellar age (e.g. 60, c.f. $t_{\mathrm{circularisation}}\sim$ 820 Myr, age = 5.7 ${}^{+0.5}_{-0.5}$ Gyr) we model the system as a simpler circular orbit. The folded photometric data showing the transit fit for TOI-4034 b can be seen in Figure 12, with the fitted RVs in Figure 13. The CARMENES residuals to this fit are slightly less well behaved than those from the TOI-3664 b fit, which could indicate the presence of additional signals in the RV data, however investigating this further would require additional RV monitoring of this system, which it outside the scope of the current analysis.

4.1.3 TOI-6564 b

TOI-6564 hosts a hot planet with super-Jupiter radius ( $R_{p}$ = 1.46 $\pm$ 0.02 R_Jup) and sub-Jupiter mass ( $M_{p}$ = 0.70 $\pm$ 0.07 M_Jup). It is the furthest-out system of the three, with a period of 3.99 days.

For this system, a circular fit was significantly preferred to an eccentric one, with a $\Delta$ ln $Z$ = 17.8 towards a circular fit. Furthermore, the eccentric fit resulted in an eccentricity value of only 0.015 $\pm$ 0.016, consistent with zero, so the simpler circular model was chosen.

To test the effect of including the ground-based photometric data alongside the more extensive TESS data, an initial fit was conducted using just the TESS data and spectroscopic data, before additional fits were carried out by adding the two supporting photometric datasets (Brierfield and PEST) one by one. In each case, the addition of the ground-based data improved the overall log-evidence further and significantly improved the achieved precision in the transit ephemerides (t0 and P), so the fit including all photometric data is preferred.

The final photometric and spectroscopic fits can be seen in Figures 14 and 15 respectively. Note that the somewhat sparse RV sampling was caused by the near-integer day period (P = 3.99 days), which meant that many days were required to sample different regions of the phase curve.

4.2 Comparison to the known population

These three new planets are particularly interesting for two main reasons: their evolutionary phase near the Terminal Age Main Sequence and their relatively low densities.

All three planets fall towards the upper end of the distribution of known exoplanets, with reasonably large radii but average to below-average masses compared to the rest of the gas-giant distribution, as shown in Figure 16. Because of the masses and radii of the three exoplanets studied here, they all exhibit relatively small densities compared to the rest of the population, as illustrated in Figure 17. Planets in this figure are coloured by their insolation flux, with TOI-4034 b clearly possessing the largest planetary insolation due to its large host and close orbit (P=1.80 day). Such low densities are often seen for planets evolving off the main sequence, possibly due to re-inflation of planetary atmospheres as their host stars brighten (46; 146, e.g. see).

Figure 16 also shows the equivalent evolutionary phase (EEP) number for every star which hosts a precisely characterised known exoplanet ( $\Delta{\rm R}_{\rm p}/{\rm R}_{\rm p}$ <10% and $\Delta\mathrm{M}_{\rm p}/\mathrm{M}_{\rm p}$ of <30%) by way of the colour-bar. The EEP for each known exoplanet host star was calculated using the MESA isochrones (35; 29) via the isochrones package¹¹¹¹ 11 https://github.com/timothydmorton/isochrones (87) by interpolating from the closest MESA isochrones based on the stellar radius, metallicity and log(age) of each star from the Exoplanet Archive¹²¹² 12 https://exoplanetarchive.ipac.caltech.edu/ (3). This plot suggests that these three new planets are around some of the most evolved stars currently in the high precision exoplanet sample, appearing lighter green than the bulk of the sample. This is corroborated by the planetary radius vs EEP plot in Figure 18, clearly illustrating that TOI-4034 and TOI-6564 sit right at the terminal age main sequence boundary and also host two of the largest radii exoplanets around high-EEP host stars.

4.3 Evolutionary state and planetary fate

The evolutionary state of these three host stars, so close to the terminal age main sequence, raises an interesting question: what is the likely future evolution of these planetary systems as their hosts begin their journey onto the red giant branch?

As these planets’ host stars leave the main sequence and enter the red giant branch, they will increase dramatically in radius, eventually expanding to well beyond the orbit of these short-period planets, leading to the engulfment of the planets (139; 91, e.g). Before this engulfment, the planets will be subject to the competing effects of an inward tidal force (105; 60; 139, e.g.) and a mass-loss-induced orbital expansion caused by stellar winds (48; 138). For close-in planets like hot Jupiters, the inward tidal force typically far outstrips the mass-loss-induced orbital expansion, leading to a decrease in orbital separation with time.

The fate of each planet can be investigated through equations 3-9 of 139 given time evolution profiles of stellar mass and radius from 56. These equations are suitable for the subgiant and red giant branch phases, and hence are suitable for all three of these planets, because none of them will survive beyond the red giant branch phase. Each of these planets will be engulfed by the star and destroyed, as their masses are far too low to survive residence within a common envelope (91). The equations from 139 allow us to simultaneously evolve the semi-major axis and eccentricity of the planets.

We find that in all three systems, the orbital radii at which the engulfment occurs ( $\approx 0.025$ au for TOI-3664 b, $\approx 0.028$ au for TOI-4034 b and $\approx 0.033$ au for TOI-6564 b) are smaller then the planets’ pericentre values at the terminal main sequence. Hence, all three planets are expected to be drawn into the star as the stellar radius was increases. The timescale for the planets to be engulfed, as measured from the end of the main sequence lifetime, are $\sim 1200$ Myr for TOI-3664 b, $\sim 650$ Myr for TOI-4034 b and $\sim 700$ Myr for TOI-6564 b. Further, the orbit of TOI-3664 b does not completely circularise before the planet is engulfed: at the moment of engulfment, the planet’s eccentricity will have reduced to $\approx 0.05$ .

The very short period of TOI-4034 b (P = 1.80 days) may also accelerate this process through tidal destruction. 145 found that for planets with periods of <2 days, non-linear damping of tidally driven g-modes in the host stars during the sub-giant phase could lead to orbital decay timescales of $\leq$ 10 Myr for stars around the size of TOI-4034 or larger ( $M_{\odot}\geq 1.20$ ).

Observationally, 23 also noted a steep drop-off in the occurrence rate of planets with periods <2 days around post-main sequence stars, even as early as the sub-giant phase. Hence as TOI-4034 passes the terminal age main sequence onto the sub-giant branch proper, its planet is likely to be swiftly destroyed by its host. Although this drop-off is less steep for planets with periods above 2 days, 23 also saw a notable decrease in the occurrence rate of planets around sub-giant and early red-giant phases of stellar evolution up to periods of 12.5 days, supporting the idea that the lifetimes of the short-period TOI-3664 b and TOI-6564 b planets will also be limited upon leaving the main sequence proper.

The evolutionary state of these planets’ host stars hence makes them very useful windows into the later stages of hot Jupiter evolution, and warrants follow-up over longer timescales to search for tell-tale signs of orbital decay.

4.4 Prospects for atmospheric characterisation

The interesting evolutionary stage, short periods and large radii of these three new planets make them intriguing targets for atmospheric characterisation. Two commonly used parameters to ascertain the suitability of different targets for atmospheric characterisation are the transmission spectroscopy metric (66, TSM,) and the atmospheric scale height, $H_{b}$ . The TSM for each target can be determined directly from the mass, radius and equilibrium of the planet, alongside the brightness and radius of the host star, while the atmospheric scale height can be calculated using the equation $H_{b}=kT_{eq}/(\mu g_{b})$ , where $k_{b}$ is the Boltzmann constant, $\mu$ is the mean molecular mass (assumed to be 2.3 amu here for a H/He atmosphere), and $g_{b}$ is the planet surface gravity based in the derived mass and radius. Because of its host’s brightness, TOI-6564 b was found to be particularly attractive for atmospheric characterisation, with TSM of 163 and an atmospheric scale height of 645km, roughly ten times that of Jupiter. For Jupiter-sized planets, 66 recommend a cut-off for follow-up efforts of TSM = 96, with TSM = 159 their cut-off for planets in the first quartile, meaning that TOI-6564 b is a very high priority target for atmospheric characterisation. Indeed, by comparison to other giant planets in Figure 19, TOI-6564 can be seen to be one of the brightest targets with such a large scale height. On the other hand, TOI-3664 b and TOI-4034 b were found to have transmission spectroscopy metrics of 90 and 49 respectively (with $H_{b}$ = 745 km and 798km), which suggests that they may be of lower priority for follow-up despite having large scale heights, largely driven by their lower observed brightnesses.

5 Conclusions

TOI-3664 b, TOI-4034 b and TOI-6564 b are three new hot Jupiter planets orbiting host stars towards the very end of their main-sequence lifetimes. These planets, all with sub-Jupiter masses (0.36 $\pm$ 0.12, 0.87 $\pm$ 0.16 & 0.70 $\pm$ 0.07 M_Jup) and super-Jupiter radii (1.22 $\pm$ 0.03, 1.58 $\pm$ 0.02 & 1.46 $\pm$ 0.02 R_Jup) represent three key opportunities to explore the latter stages of planetary evolution, before their eventual engulfment as their host stars expand along the red giant branch. All three planets have reasonably low densities and moderate to high insolation flux due to their short orbital periods ( $P=$ 3.30, 1.80 and 3.99 days respectively). TOI-6564 b is also particularly promising for atmospheric characterisation, with a TSM of 163 and atmospheric scale height of 649 km.

Because of their position near the terminal main sequence, the ages of the systems are somewhat degenerate, so care is taken to determine the correct ages through consideration of a variety of age-dating methods. This resulted in ages of 9.0 ${}^{+2.4}_{-2.1}$ , 5.7 $\pm$ 0.5 Gyr and 4.0 $\pm$ 1.0 Gyr respectively, adding three important points to the evolutionary distribution of well characterised hot-Jupiters. Modelling of the evolution of the stellar envelopes and semi-major axes of the planets and their host stars suggests that all three systems will be engulfed by their hosts on the order of only 1 Gyr from the end of the main sequence.

The results of a wider vetting effort of TESS planet candidates conducted in the same CARMENES programme in which TOI-3664 and TOI-4034 were monitored are also presented to aid the community.

Acknowledgements

The authors would like to thank the anonymous referee for their comments which improved the quality and robustness of this paper.

MPB and EG gratefully acknowledge support from UK Research and Innovation (UKRI) under the UK government’s Horizon Europe funding guarantee for an ERC starting grant [grant number EP/Z000890/1]. The contributions of MPB, SU-M, YF, FB, YC, MH, LP, and ST have been carried out within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation under grants 51NF40_182901 and 51NF40_205606. ML acknowledges support of the Swiss National Science Foundation under grant number PCEFP2_194576. MLafarga gratefully acknowledges support from UK Research and Innovation (UKRI) under the UK government’s Horizon Europe funding guarantee [grant number Grant EP/X027562/1].

This paper includes data collected with the TESS mission, obtained from the MAST data archive at the Space Telescope Science Institute (STScI). Funding for the TESS mission is provided by NASA’s Science Mission directorate. We acknowledge the use of public TOI Release data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.

This paper was based on observations collected at Centro Astronómico Hispano en Andalucía (CAHA) at Calar Alto, operated jointly by Junta de Andalucía and Consejo Superior de Investigaciones Científicas (IAA-CSIC).

This publication makes use of The Data & Analysis Center for Exoplanets (DACE), which is a facility based at the University of Geneva (CH) dedicated to extrasolar planets data visualisation, exchange and analysis. DACE is a platform of the Swiss National Centre of Competence in Research (NCCR) PlanetS, federating the Swiss expertise in Exoplanet research. The DACE platform is available at https://dace.unige.ch.

MINERVA-Australis is supported by Australian Research Council LIEF Grant LE160100001, Discovery Grants DP180100972 and DP220100365, Mount Cuba Astronomical Foundation, and institutional partners University of Southern Queensland, UNSW Sydney, MIT, Nanjing University, George Mason University, University of Louisville, University of California Riverside, University of Florida, and The University of Texas at Austin.

We respectfully acknowledge the traditional custodians of all lands throughout Australia, and recognise their continued cultural and spiritual connection to the land, waterways, cosmos, and community. We pay our deepest respects to all Elders, ancestors and descendants of the Giabal, Jarowair, and Kambuwal nations, upon whose lands the Minerva-Australis facility at Mt Kent is situated.

Data Availability

TESS data for all three systems are publicly available from the Mikulski Archive for Space Telescopes (MAST).¹³¹³ 13 https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html Additional photometry for TOI-6564 are also publicly available on the TESS ExoFOP platform.¹⁴¹⁴ 14 https://exofop.ipac.caltech.edu/tess/ Radial velocity measurements for each system are presented here in full, and are included in machine-readable form in the supplementary material alongside the online copy of this article. Additional intermediate data products underlying this article will be shared on reasonable request to the corresponding author.

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Appendix A CARMENES vetting

Table 3: Overview of CARMENES vetting results. Note that the RV values were taken as close to phases 0.25 and 0.75 in the expected orbit in order to probe the approximate magnitude of the radial velocity signal. N.b. EB = Eclipsing Binary and SB = Spectroscopic Binary

Name	Time 0.25 [BJD]	RV 0.25 [km/s]	Time 0.75 [BJD]	RV 0.75 [km/s]	$\Delta$ RV [m/s]	Vetting Result
TOI-2538	2460312.5278260	762.817 $\pm$ 0.044	2460311.4602998	277.610 $\pm$ 0.042	485206.81	Broad CCFs
TOI-3571	2460311.3410344	29.759 $\pm$ 0.085	2460313.2636574	29.469 $\pm$ 0.078	289.87	Anti-phase
TOI-3664	2460312.3571130	-16.915 $\pm$ 0.032	2460317.3912960	-16.969 $\pm$ 0.060	54.43	Passed
TOI-4034	2460311.3235430	-6.648 $\pm$ 0.056	2460317.3242420	-6.820 $\pm$ 0.081	172.45	Passed
TOI-4688	Missed	Missed	2460311.3837339	76.684 $\pm$ 0.051	N/A	Missed
TOI-5459	2460312.4723960	-86.415 $\pm$ 0.121	2460313.4496347	-81.991 $\pm$ 0.079	-4423.80	SB
TOI-6248	2460311.2984069	54.995 $\pm$ 0.093	2460313.2844601	71.419 $\pm$ 0.099	-16423.87	EB
TOI-6382	2460325.4066204	Fit failed	2460311.4844659	36.353 $\pm$ 0.097	N/A	Likely SB

In addition to the CARMENES follow-up carried out above for TOI-3644 and TOI-4034, six additional TOIs with evidence of youth in a variety of literature catalogues were vetted in the same programme. The results of this vetting are presented here to support the community’s wider efforts.

Each of the eight targets was vetted by taking one spectrum at each of the expected maximum and minimum radial velocity epochs, or as close to phases 0.25 and 0.75 in the orbit, according to the photometric ephemeris presented on ExoFOP.¹⁵¹⁵ 15 https://exofop.ipac.caltech.edu/tess/

With only two exceptions, vetting observations were carried out between 1-7 January 2024, in order to quickly decide which targets warranted further monitoring. Following each observation, the raw data was analysed using the raccoon pipeline (77), in order to check the spectra, extract measurements of the radial velocity, and to inspect the Cross Correlation Functions (CCFs).

Two simple vetting tests were carried out to evaluate their suitability to further monitoring: 1) Inspecting the shape of the extracted CCFs and 2) evaluating the approximate amplitude of the radial velocity signal by calculating the difference between the measurements at phases 0.25 and 0.75.

During this process TOI-5459 and TOI-6382 were found to have CCFs with more than one dip, suggestive of spectroscopic binaries, while TOI-2538 was found to have untenably wide spectral lines (likely due to fast stellar rotation), preventing precise and reliable radial velocity measurements from being extracted. Similarly, in the case of TOI-6248, the difference between the RV measurements at phases 0.25 and 0.75 was in excess of 16 km/s, suggesting that this candidate signal was caused by an eclipsing binary, not an exoplanet. Meanwhile, although the RV amplitude measured for TOI-3571 was reasonable for a Jupiter-sized planet around its host star, the two collected radial velocity measurements actually occurred at opposite times to what was expected from the TESS ephemeris, suggestive of either an eclipsing binary system or complex stellar activity, so this target too was dropped from the programme. Unfortunately due to scheduling and weather issues, the observation of TOI-4688 at phase 0.25 could not be carried out in the programme, so nothing can be said definitively about this target.

A full overview of the collected vetting observations and their final dispositions is presented in Table 3.

Appendix B Radial Velocity Measurements

The full radial-velocity measurements analysed in this paper are presented in Tables 4, 5 and 6 and are also available in machine-reading format alongside the online version of this paper.

Table 4: RV data for TOI-3664

Time	RV	RV Error	Instrument
BJD	[ms^-1]	[ms^-1]	-
2460312.357	-16917.48849	42.15306	CARMENES
2460317.391	-16961.87207	78.05758	CARMENES
2460353.294	-17008.20173	105.88694	CARMENES
2460353.31	-16970.19956	104.07584	CARMENES
2460354.294	-16893.94325	49.56977	CARMENES
2460354.31	-16904.92121	49.06538	CARMENES
2460355.292	-16969.35916	42.11380	CARMENES
2460355.308	-17022.11931	54.28219	CARMENES
2460358.296	-16996.66665	60.66222	CARMENES
2460358.313	-16916.99678	68.83403	CARMENES
2460359.409	-17006.97492	38.70351	CARMENES
2460360.338	-16958.79632	30.45402	CARMENES
2460360.356	-16995.74236	35.08230	CARMENES
2460361.322	-16955.73644	49.14355	CARMENES
2460361.34	-16893.47251	49.14944	CARMENES
2460371.324	-16944.68772	40.73604	CARMENES
2460371.342	-16947.42867	46.67448	CARMENES
2460375.321	-17019.4548	36.70504	CARMENES
2460375.339	-17031.02445	34.50057	CARMENES

Table 5: RV data for TOI-4034

Time	RV	RV Error	Instrument
BJD	[ms^-1]	[ms^-1]	-
2460311.323543	-6648.02671	78.63116	CARMENES
2460317.32424	-6818.88258	120.78882	CARMENES
2460317.36384	-6635.13833	149.92357	CARMENES
2460442.6425	-6520.82385	43.18636	CARMENES
2460442.65876	-6460.91182	42.83671	CARMENES
2460443.62291	-6704.88316	53.18989	CARMENES
2460443.63993	-6720.41098	44.56097	CARMENES
2460447.61369	-6593.71121	86.24849	CARMENES
2460447.63136	-6494.60292	136.48119	CARMENES
2460449.60282	-6407.10461	90.82587	CARMENES
2460449.61965	-6293.06404	69.70656	CARMENES
2460455.64061	-6510.34728	51.14255	CARMENES
2460455.65621	-6485.58294	48.23511	CARMENES
2460458.62617	-6507.06771	37.72920	CARMENES
2460458.64236	-6507.2148	37.60009	CARMENES
2460460.61675	-6411.06859	38.62132	CARMENES
2460484.61761	-6661.2072	48.96500	CARMENES
2460484.63632	-6649.54783	53.69958	CARMENES
2460485.6254	-6568.54907	39.20842	CARMENES
2460485.64819	-6507.46114	40.74366	CARMENES
2460490.60015	-6581.23589	275.60896	CARMENES
2460490.61761	-6514.34127	267.9552	CARMENES

Table 6: RV data for TOI-6564

Time	RV	RV Error	Instrument
BJD	[ms^-1]	[ms^-1]	-
2460167.888321	-8525.299	18.765	MINERVA
2460167.920541	-8549.063	19.279	MINERVA
2460175.919779	-8552.413	16.954	MINERVA
2460177.899948	-8379.442	19.882	MINERVA
2460178.919678	-8477.514	16.528	MINERVA
2460179.920636	-8521.412	17.732	MINERVA
2460181.902220	-8393.795	15.221	MINERVA
2460425.636623240076	-7557.43355	7.752599	CORALIE14
2460440.692218369804	-7529.524807	11.376738	CORALIE14
2460466.56458572997	-7678.742939	15.927452	CORALIE14
2460492.58155064983	-7534.667595	6.336502	CORALIE14
2460498.64336684998	-7685.137505	6.492417	CORALIE14
2460505.56996307988	-7584.443779	8.201031	CORALIE14
2460507.51850314997	-7645.053542	6.952943	CORALIE14
2460510.538426990155	-7677.498125	7.186331	CORALIE14
2460514.55492101982	-7691.279090	17.339981	CORALIE14
2460515.49542242987	-7627.871626	10.112589	CORALIE14
2460516.520176849794	-7534.520062	8.443043	CORALIE14
2460519.50077578006	-7642.663497	7.441667	CORALIE14
2460542.52890574001	-7695.595509	16.942226	CORALIE14
2460569.50403237995	-7600.939399	8.343578	CORALIE14
2460667.84078198997	-7557.476035	6.366845	CORALIE24
2460696.80468448019	-7589.693804	7.265243	CORALIE24
2460699.82876972994	-7554.467131	6.213132	CORALIE24
2460707.75943840016	-7548.908761	6.451489	CORALIE24
2460711.8147889399	-7546.974675	5.724207	CORALIE24
2460723.84198427992	-7547.294835	6.556601	CORALIE24
2460736.83497559978	-7600.098830	6.373488	CORALIE24
2460737.86745380005	-7705.868519	5.763901	CORALIE24

Appendix C Instrumental fitted parameters

System and instrumental fitted parameters for all systems are presented in Table 7.

Table 7: Instrumental parameters for each fit from juliet: median and 68% confidence interval

Parameter		Prior distribution^∗	Value
$m_{\mathrm{dilution,TESS,all}}$ .	Dilution parameter .	fixed .	1.0
TOI-3664
$q_{1,\,TESS}$ .	Quadratic limb-darkening parametrization .	$N$ (0.360,0.069) .	0.368 ${}^{+0.057}_{-0.062}$
$q_{2,\,TESS}$ .	Quadratic limb-darkening parametrization .	$N$ (0.29,0.13) .	0.25 ${}^{+0.11}_{-0.10}$
$m_{\mathrm{flux,TESS}}$ .	Photometric offset (relative flux) .	$N$ (0,0.1) .	-0.00046 ${}^{+0.00023}_{-0.00023}$
$\sigma_{\mathrm{TESS}}$ .	Photometric jitter in data (ppm) .	log $U$ (1e^-5,1e³) .	0.028 ${}^{+5.6}_{-0.028}$
$\sigma_{GP,\mathrm{TESS}}$ .	Photometric GP amplitude (relative flux) .	log $U$ (1e^-6,1e⁶) .	0.00213 ${}^{+0.00015}_{-0.00014}$
$\rho_{GP,\mathrm{TESS}}$ .	Photometric GP time-scale (days) .	log $U$ (1e^-3,1e³) .	0.362 ${}^{+0.037}_{-0.032}$
$\mu_{\mathrm{CARMENES}}$ .	Systematic RV offset (km/s) .	$U$ (-20,-10) .	-16.977 ${}^{+0.012}_{-0.011}$
$\sigma_{\mathrm{CARMENES}}$ .	RV jitter (m/s) .	$U$ (0.001,100) .	11.1 ${}^{+11.5}_{-7.7}$
TOI-4034
$q_{1,\,TESS}$ .	Quadratic limb-darkening parametrization .	$N$ (0.337,0.057) .	0.289 ${}^{+0.048}_{-0.047}$
$q_{2,\,TESS}$ .	Quadratic limb-darkening parametrization .	$N$ (0.251,0.135) .	0.180 ${}^{+0.087}_{-0.100}$
$m_{\mathrm{flux,TESS}}$ .	Photometric offset (relative flux) .	$N$ (0,0.1) .	-0.000381 ${}^{+0.000036}_{-0.000044}$
$\sigma_{\mathrm{TESS}}$ .	Photometric jitter (ppm) .	log $U$ (1e^-5,1e³) .	0.013 ${}^{+3.352}_{-0.013}$
$\sigma_{GP,\mathrm{TESS}}$ .	Photometric GP amplitude (relative flux) .	log $U$ (1e^-6,1e⁶) .	0.000586 ${}^{+0.000034}_{-0.000029}$
$\rho_{GP,\mathrm{TESS}}$ .	Photometric GP time-scale (days) .	log $U$ (1e^-3,1e³) .	0.473 ${}^{+0.062}_{-0.044}$
$\mu_{\mathrm{CARMENES}}$ .	Systematic RV offset (km/s) .	$U$ (-10,-5) .	-6.572 ${}^{+0.017}_{-0.017}$
$\sigma_{\mathrm{CARMENES}}$ .	RV jitter (m/s) .	$U$ (0.001,100) .	25 ${}^{+22}_{-17}$
TOI-6564
$m_{\mathrm{dilution,Bfield}}$ .	Dilution parameter for Brierfield data .	$U$ (0,1) .	0.961 ${}^{+0.027}_{-0.038}$
$m_{\mathrm{dilution,PEST}}$ .	Dilution parameter for PEST data .	$U$ (0,1) .	0.931 ${}^{+0.050}_{-0.098}$
$q_{1,\,TESS}$ .	Quadratic limb-darkening parametrization .	$N$ (0.335,0.057) .	0.302 ${}^{+0.041}_{-0.044}$
$q_{2,\,TESS}$ .	Quadratic limb-darkening parametrization .	$N$ (0.250,0.147) .	0.213 ${}^{+0.064}_{-0.060}$
$q_{1,\,Bfield}$ .	Quadratic limb-darkening parametrization .	$N$ (0.666,0.025) .	0.664 ${}^{+0.023}_{-0.024}$
$q_{2,\,Bfield}$ .	Quadratic limb-darkening parametrization .	$N$ (0.423,0.073) .	0.375 ${}^{+0.061}_{-0.061}$
$q_{1,\,PEST}$ .	Quadratic limb-darkening parametrization .	$U$ (0,1) .	0.20 ${}^{+0.28}_{-0.14}$
$q_{2,\,PEST}$ .	Quadratic limb-darkening parametrization .	$U$ (0,1) .	0.42 ${}^{+0.33}_{-0.30}$
$m_{\mathrm{flux,TESS}}$ .	Photometric offset for TESS data (relative flux) .	$N$ (0,0.1) .	-0.000007 ${}^{+0.000076}_{-0.000067}$
$m_{\mathrm{flux,Bfield}}$ .	Photometric offset for Brierfield data (relative flux) .	$N$ (0,0.1) .	-0.0007 ${}^{+0.0093}_{-0.0225}$
$m_{\mathrm{flux,PEST}}$ .	Photometric offset for PEST data (relative flux) .	$N$ (0,0.1) .	-0.000081 ${}^{+0.00068}_{-0.00061}$
$\sigma_{\mathrm{TESS}}$ .	Photometric jitter for TESS data (ppm) .	log $U$ (1e^-5,1e³) .	247 ${}^{+17}_{-18}$
$\sigma_{\mathrm{Bfield}}$ .	Photometric jitter for Brierfield data (ppm) .	log $U$ (1e^-5,1e³) .	0.48 ${}^{+100}_{-0.48}$
$\sigma_{\mathrm{PEST}}$ .	Photometric jitter for PEST data (ppm) .	log $U$ (0.1,1e⁵) .	3570 ${}^{+150}_{-140}$
$\sigma_{GP,\mathrm{TESS}}$ .	Photometric GP amplitude for TESS data (relative flux) .	log $U$ (1e^-6,1e⁶) .	0.000344 ${}^{+0.000047}_{-0.000038}$
$\sigma_{GP,\mathrm{Bfield}}$ .	Photometric GP amplitude for Brierfield data (relative flux) .	log $U$ (1e^-6,1e⁶) .	0.013 ${}^{+0.065}_{-0.012}$
$\sigma_{GP,\mathrm{PEST}}$ .	Photometric GP amplitude for PEST data (relative flux) .	log $U$ (1e^-6,1e⁶) .	0.00159 ${}^{+0.00058}_{-0.00038}$
$\rho_{GP,\mathrm{TESS}}$ .	Photometric GP time-scale for TESS data (days) .	log $U$ (1e^-3,1e³) .	0.54 ${}^{+0.10}_{-0.08}$
$\rho_{GP,\mathrm{Bfield}}$ .	Photometric GP time-scale for Brierfield data (days) .	log $U$ (1e^-3,1e³) .	2.60 ${}^{+13.78}_{-2.24}$
$\rho_{GP,\mathrm{PEST}}$ .	Photometric GP time-scale for PEST data (days) .	log $U$ (1e^-3,1e³) .	0.0143 ${}^{+0.0087}_{-0.0049}$
$\mu_{\mathrm{MINERVA}}$ .	Systematic RV offset for MINERVA-Australis data (km/s) .	$U$ (-10,-5) .	-8461.3 ${}^{+7.1}_{-7.6}$
$\mu_{\mathrm{CORALIE14}}$ .	Systematic RV offset for CORALIE14 data (km/s) .	$U$ (-10,-5) .	-7608.6 ${}^{+2.2}_{-2.2}$
$\mu_{\mathrm{CORALIE24}}$ .	Systematic RV offset for CORALIE24 data (km/s) .	$U$ (-10,-5) .	-7624.7 ${}^{+2.4}_{-2.5}$
$\sigma_{\mathrm{MINERVA}}$ .	RV jitter (m/s) for MINERVA-Australis data .	$U$ (0.001,100) .	0.16 ${}^{+3.17}_{-0.15}$
$\sigma_{\mathrm{CORALIE14}}$ .	RV jitter (m/s) for CORALIE14 data .	$U$ (0.001,100) .	0.6 ${}^{+1.8}_{-0.5}$
$\sigma_{\mathrm{CORALIE24}}$ .	RV jitter (m/s) for CORALIE24 data .	$U$ (0.001,100) .	0.09 ${}^{+0.93}_{-0.09}$

*n.b. similar to Table 2, $U$ (a,b) denotes a uniform prior between a and b, while $N$ (a,b) denotes a normal distribution with mean a and standard deviation n. In addition, log $U$ (a,b) denotes a log-uniform prior between a and b.