A Transiting, Temperate Mini-Neptune Orbiting the M Dwarf TOI-1759 Unveiled by TESS

Observations from TESS for TOI-1759 (TYC 4266-736-1, TIC408636441) were obtained during its second year of operation in its high-cadence, 2 minutes exposure mode in Sectors 16 (2019 September to October), 17 (2019 October to November), and 24 (2020 April to May—see Figures 1 and 2; the data are also presented in Table 6).

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The 2 minutes cadence data were processed in the TESS Science Processing Operations Center (SPOC; Jenkins et al. 2016) photometry and transit search pipelines (Jenkins 2002; Jenkins et al. 2010) at NASA Ames Research Center. The TESS data validation reports (Twicken et al. 2018; Li et al. 2019) on TOI-1759 (Guerrero et al. 2021) show detections of a transiting exoplanet candidate at a 37.7 days period (although the data were also consistent with a planet at half this period, i.e., 18.85 days) and a transit depth of about 2700 ppm. To perform further analyses on this target, we retrieved the Pre-Data-Conditioning (PDC)−corrected photometry (Smith et al. 2012; Stumpe et al. 2012, 2014) from all sectors from the Mikulski Archive for Space Telescopes archive ³⁷ , as this is the highest-quality photometry from the three TESS sectors mentioned above. After removing the transits of the planet candidate, we ran the Transit Least Squares (TLS; Hippke & Heller 2019) algorithm on these photometric time series and found no extra significant signals (i.e., signals with a signal-to-noise ratio >5) in the data.

We monitored TOI-1759 with the CARMENES ³⁸ instrument located on the 3.5 m telescope at the Calar Alto Observatory in Almería, Spain, from 2020 July 24 to 2021 January 17. Our data covered a time span of about 175 days, over which we were able to detect significant radial-velocity signals. The spectra were processed following the standard CARMENES data flow (Caballero et al. 2016) that has been extensively used by previous works (e.g., Zechmeister et al. 2018; Morales et al. 2019; Trifonov et al. 2020). In our analyses, we only used radial velocities from the visual (VIS) channel that had mean errors of 2.6 m s⁻¹. A total of 57 radial-velocity data points were used for our analysis, which are presented in Table 5. The spectra used to derive those have a median signal-to-noise ratio of 95 at 840 nm. Data from our infrared channel were not used, as their precision (mean error of 10 m s⁻¹) was not high enough to put meaningful constraints on the radial-velocity variations observed in the VIS channel.

Figure 3(a) shows the radial velocity as a function of time as observed through the VIS channel, which covers the spectral range 520–960 nm with a spectral resolution of $R=94,600$ (Quirrenbach et al. 2014, 2018). Our campaign allowed us to clearly detect a signal at about 18.5 days (consistent with half the period of the transiting exoplanet detected in the TESS photometry, already discussed in Section 2.1) on top of an additional long-term radial-velocity signal. We discuss the details of our analysis of these signals in Section 3.

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Ground-based photometric follow-up observations were performed as part of the TESS Follow-up Program's (TFOP) Subgroup 1 (SG1). Among the observations, a transit of TOI-1759 b in 2020 May 21 was captured by three independent telescopes/observatories: the OAA telescope of the Observatori Astronòmic Albanyà (Albanyà Spain; 4 hr of total observing time, per-point precision of 1140 ppm at 1 minutes cadence; R-filter observations), the RCO telescope of the Grand-Pra Observatory (Valais Sion, Switzerland; 6 hr of total observing time, per-point precision of 1080 ppm at 1.3 minutes cadence; i_p -filter observations), and the OMC telescope of the Montcabrer Observatory (Barcelona, Spain; 5 hr of total observing time, per-point precision of 1500 ppm at 1.9 minutes cadence; I_c -filter observations). Data reduction for the OAA and OMC observations was performed in a two-step process: the MaximDL image processing software was used to perform image calibration (bias, darks, flats), while differential photometry was obtained using the AstroImageJ software (Collins et al. 2017). For the RCO observations, image calibration and differential photometry were both performed using AstroImageJ. The data, along with a best-fit model transit after subtracting the best-fit systematics model for each data set (see Section 3 for details), are presented in Figure 4. The data are also presented in Table 6.

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The observed transits by these three independent observatories on 2020 May 21 not only confirmed that the event observed by TESS was on target (i.e., it happened on TOI-1759), but in practice confirmed that the real period of the event was 18.85 days (i.e., half the period proposed by the TESS data validation reports), with the duration and depth detected by those observatories being consistent with the duration and depth observed in the TESS transit events.

Long-term photometric monitoring was also performed from the ground using the 0.8 m Joan Oró telescope (TJO; Colomé et al. 2010) at the Montsec Observatory in Lleida, Spain and the 90 cm telescope at the Sierra Nevada Observatory (SNO; Amado et al. 2021). For the TJO observations, the data were obtained from 2020 June to 2021 April, spanning more than 300 days and covering 107 different nights. We obtained a total of 1331 images with an exposure time of 40 s using the Johnson R filter of the LAIA imager, a 4k × 4k CCD with a field of view of 30' and a scale of 04 pixel⁻¹. The SNO data were obtained from 2021 April to August, spanning 135 days and collecting observations on 55 different nights. Each night, 20 exposures per filter were obtained using both Johnson V and R filters, with exposure times of 60 and 40 s, respectively. The photometry from these exposures was averaged to obtain a single photometric value per filter each night. These data were obtained with a VersArray 2k × 2k CCD camera with a field of view of 13.2 × 13.2 arcmin² and a scale of 04 pixel⁻¹ as well.

The TJO CCD images were calibrated with darks, and bias and flat fields with the ICAT pipeline (Colome & Ribas 2006). The differential photometry was extracted with AstroImageJ (Collins et al. 2017) using the aperture size that minimized the rms of the resulting relative fluxes and a selection of the 30 brightest comparison stars in the field that did not show variability. Then, we used our own pipelines to remove outliers and measurements affected by poor observing conditions or presenting a low signal-to-noise ratio. This resulted in a total of 1087 measurements in the final data set with an rms of 6 parts per thousand (ppt).

In a similar way, the SNO resulting light curves were obtained by the method of synthetic aperture photometry. Each CCD frame was also corrected in a standard way for bias and flat-fielding. Different aperture sizes were tested in order to choose the best one for our observations. A number of nearby and relatively bright stars within the frames were selected as reference stars to produce differential photometry of TOI-1759. Finally, outliers due to poor observing conditions or very high airmass were removed. This resulted in a total of 1029 and 1027 individual data points in filters V and R, respectively, with rms values of 6.1 and 6.4 ppt.

Both the TJO and SNO data sets are presented in Table 7. An analysis of these data sets is presented in Section 3.2.

To help rule out stellar multiplicity and close blends with nearby stars and to obtain more precise planetary radii by accounting for close-in stellar blends (Ciardi et al. 2015; Schlieder et al. 2021), we observed TOI-1759 with both the 'Alopeke ³⁹ speckle imaging camera (Scott & Howell 2018) on the 8 m Gemini North telescope and the NIRC2 near-infrared (NIR) adaptive-optics fed camera on the 10 m Keck II telescope. The optical and NIR high-resolution imaging complement each other with higher resolution in the optical but deeper sensitivity (especially to low-mass stars) in the infrared.

'Alopeke obtains diffraction-limited imaging in two simultaneously imaged narrow bands centered at 562 and 832 nm. Due to the relative faintness of the target star at these wavelengths, we obtained five exposures in each channel, with integration times of 60 ms each. We reduced the data using standard techniques using the methods described by Matson et al. (2019). The resulting contrast curves and reconstructed 832 nm image are all shown in Figure 5. The optical speckle observations show no evidence of an additional stellar companion.

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TOI-1759 was observed with the NIRC2 instrument on Keck II behind the natural guide star AO system (Wizinowich et al. 2000). The observations were made on 2020 September 09 UT in the standard three-point dither pattern that is used with NIRC2 to avoid the left lower quadrant of the detector, which is typically noisier than the other three quadrants. The dither pattern step size was 3'' and was repeated twice, with each dither offset from the previous dither by 05. The camera was in the narrow-angle mode with a full field of view of ∼10'' and a pixel scale of approximately 00099 pixel⁻¹. The observations were made in the narrowband Brγがんま filter (λらむだ_o = 2.1686; Δでるたλらむだ = 0.0326 μみゅーm) with an integration time of 1 s with one coadd per frame for a total of 9 s on target.

The AO data were processed and analyzed with a custom set of IDL tools. The science frames were flat-fielded and sky-subtracted. The flat fields were generated from a median average of dark-subtracted flats taken on-sky, and the flats were normalized such that the median value of the flats is unity. Sky frames were generated from the median average of the nine dithered science frames; each science image was then sky-subtracted and flat-fielded. The reduced science frames were combined into a single combined image using a intrapixel interpolation that conserves flux, shifts the individual dithered frames by the appropriate fractional pixels, and median coadds the frames. The final resolution of the combined dithers was determined from the FWHM of the point-spread function, 0049. The sensitivities of the final combined AO image were determined by injecting simulated sources azimuthally around the primary target every 20° at separations of integer multiples of the central source's FWHM (Furlan et al. 2017). The brightness of each injected source was scaled until standard aperture photometry detected it with 5σしぐま significance. The resulting brightness of the injected sources relative to the target set the contrast limits at that injection location. The final 5σしぐま limit at each separation was determined from the average of all of the determined limits at that separation and the uncertainty on the limit was set by the rms dispersion of the azimuthal slices at a given radial distance. The final combined image and sensitivity curve are shown in Figure 6.

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Both the optical speckle and the near-infrared adaptive-optics observations find no additional stars (down to 01, dimmer by about 4–5 mag than the target in the optical and near-infrared) and so further strengthen the case for TOI-1759 b being a bona fide planet.

We obtained the photospheric parameters T_eff, $\mathrm{log}g$, and [Fe/H] of TOI-1759 following Passegger et al. (2019) by fitting PHOENIX synthetic spectra to the combined (coadded) CARMENES VIS spectrum described in Section 2.2, which has a signal-to-noise ratio in the VIS channel of ≈200. We used $v\sin i=2$ km s⁻¹, which was measured by Marfil et al. (2021) as an upper limit. We derived its luminosity following Cifuentes et al. (2020) by using the latest parallactic distance from Gaia EDR3 (Gaia Collaboration et al. 2021), and by integrating Gaia, 2MASS (Skrutskie et al. 2006), and AllWISE (Cutri et al. 2014) photometry covering the full spectral energy distribution with the Virtual Observatory Spectral energy distribution Analyzer (Bayo et al. 2008). The stellar radius follows from the Stefan−Boltzmann law and the stellar mass was determined using the linear mass–radius relation from Schweitzer et al. (2019).

TOI-1759's pseudo equivalent width of the Hαあるふぁ line as defined by Schöfer et al. (2019) is pEW'(Hαあるふぁ) < −0.3 Å (Marfil et al. 2021), classifying it as an Hαあるふぁ inactive star. Furthermore, Marfil et al. (2021) assign it to the Galactic thin disk population, which has a maximum age of about 8 Gyr (Fuhrmann 1998). Using these two properties as age indicators, we conclude that its age is between 1 Gyr (the typical minimum age for field stars) and 8 Gyr without being able to be more precise. All collected and derived parameters are presented in Table 1.

Table 1. Stellar Properties of TOI-1759

Parameter	Value	Reference
Names	TIC 408636441	TIC
	2MASS J1472477+6245139	2MASS
	TYC 4266-00736-1	Tycho-2
	WISEA J214724.51+624513.8	AllWISE
R.A. (J2000)	21h47m2439	Gaia EDR3
Decl. (J2000)	62°45'137	Gaia EDR3
Spectral type	M0.0 V	Lep13
${\mu }_{\alpha }\cos \delta$ (mas yr−1)	−173.425 ± 0.012	Gaia EDR3
μみゅーδでるた (mas yr−1)	−10.654 ± 0.011	Gaia EDR3
πぱい (mas)	24.922 ± 0.010	Gaia EDR3
d (pc)	40.112 ± 0.016	Gaia EDR3
GBP (mag)	11.7164 ± 0.0029	Gaia EDR3
G (mag)	10.8386 ± 0.0028	Gaia EDR3
T (mag)	9.9284 ± 0.0073	TIC
GRP (mag)	9.9174 ± 0.0038	Gaia EDR3
J (mag)	8.771 ± 0.043	2MASS
H (mag)	8.114 ± 0.059	2MASS
Ks (mag)	7.930 ± 0.020	2MASS
W1 (mag)	7.825 ± 0.027	AllWISE
W2 (mag)	7.886 ± 0.020	AllWISE
W3 (mag)	7.787 ± 0.018	AllWISE
W4 (mag)	7.643 ± 0.111	AllWISE
L⋆ (10−4 L⊙)	876.7 ± 6.3	This work
Teff (K)	4065 ± 51	This work
$\mathrm{log}g$ (dex)	4.65 ± 0.04	This work
[Fe/H] (dex)	0.05 ± 0.16	This work
$v\sin i$ (km s−1)	≤2	Mar21
M⋆ (M⊙)	0.606 ± 0.020	This work
R⋆ (R⊙)	0.597 ± 0.015	This work
Age (Gyr)	1–8	This work
ρろー⋆ (kg m−3)	3949 ± 323	This work

References—2MASS (Skrutskie et al. 2006), AllWISE (Cutri et al. 2014), Gaia EDR3 (Gaia Collaboration et al. 2021), Lep13 (Lépine et al. 2013), Mar21 (Marfil et al. 2021), TIC (Stassun et al. 2019), Tycho-2 (Høg et al. 2000).

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We performed a detailed analysis of the radial velocities described in Section 2.2 in order to constrain the possible signals arising from these data. To this end, we performed a suite of model fits to the radial-velocity data using juliet (Espinoza et al. 2019), in order to measure the evidence for a planet in the data using Bayesian evidences, Z = P(Model ∣ Data). The fits were performed using the Dynamic Nested Sampling algorithm implemented in the dynesty library (Speagle 2020).

We considered three main types of radial-velocity models. The first was a "no planet" model, namely, a set of models in which it is assumed there is no planetary signal present in the radial-velocity data, and which thus assumes the data are either consistent with a flat line or with correlated noise modeled through a Gaussian process (GP). The second class were "1 planet" models; these considered the presence of a planetary signal in the radial-velocity data (modeled as a circular orbit), and a suite of possible extra signals, such as linear or quadratic trends, or a (quasi-periodic) GP. Finally, we also considered the possibility that the data were best explained by a "2 planet" model, as a sum of two circular orbits and a suite of possible extra signals, such as a linear, quadratic, or GP trend.

We first performed "blind" fits to the data—that is, fits in which we assumed no strong prior knowledge on the signal(s) present on our radial velocities. For our GP, we assumed a quasi-periodic kernel of the form

$$\begin{eqnarray*}&&k({t}_{i},{t}_{j})={\sigma }_{{GP}}^{2}\exp \left(-\alpha {\tau }^{2}-{\rm{\Gamma }}{\sin }^{2}\left[\displaystyle \frac{\pi \tau }{{P}_{\mathrm{rot}}}\right]\right),\end{eqnarray*}$$

where τたう = ∣t_i − t_j ∣. We set log-uniform priors for σしぐま_GP , αあるふぁ, and Γがんま, with lower and upper limits of (0.01, 100) m s⁻¹, (10⁻¹⁰, 1) day⁻¹ and (0.01, 100) respectively, based on the experiments performed with this kernel in Stock et al. (2020a) and Stock et al. (2020b), and a uniform prior for P_rot between 0.5 (half the best sampling in our radial velocities) and 350 days (two times our time baseline). The linear and quadratic trends both had uniform priors on the coefficients of (10⁻³, 10³). As for the circular orbits, we set a uniform prior on the period of the first one from 0.5 to 50 days (so as to cover the 18 and 36 days periods that could possibly originate from the transiting exoplanet), and a uniform prior on the period of the second one from 50 to 350 days. Uniform priors were set for the time of inferior conjunction to cover the entire time baseline of our observations, the semiamplitude—between 0 and 100 m s⁻¹—and the systematic radial velocity—between −100 and 100 m s⁻¹. A jitter term, σしぐま_w was added to all of our fits with a log-uniform prior between 0.01 and 100 m s⁻¹.

In our "blind" fits, we found that all models considering a periodic component were consistent with a prominent signal at ∼18.5 days, which corresponds to the transit signal implied by the TESS photometry presented in Section 2.1 and the ground-based transits presented in Section 2.3. The model with the highest evidence in our set of fits was one composed of a sinusoid plus a quasi-periodic GP ($| {\rm{\Delta }}\mathrm{log}Z| \approx 4.3$, compared with the no-planet model). Given the high-resolution imaging data presented in Section 2.4, the ground-based transit detected on target presented in Section 2.3, and the fact that the period of the planetary signal for the model with the highest evidence (${18.49}_{-0.21}^{+0.23}$ days) agrees with the period implied by the photometric data (18.8480 ± 0.0010 days), we consider that this 18 days period signal in both photometry and radial velocities is, indeed, a bona fide transiting exoplanet.

Having concluded that the 18 days period signal is indeed a bona fide transiting exoplanet, we then focused on finding the best model that explains the radial-velocity data set. We considered the same class of models and priors as the ones presented above, but now for the first planet we fixed the period and transit center to the values defined by a photometric fit made to the data using juliet (see Section 3.3 for details on the priors of that fit): period P = 18.85008 ± 0.00018 days, and time-of-transit center t₀ = 2,458,745.4651 ± 0.0015 days. A compilation of the log-evidences for each of the fits we performed is presented in Table 2. As can be seen, the model with the highest evidence is once again a 1 planet + GP model. Interestingly, however, this model is in practice indistinguishable ($| {\rm{\Delta }}\mathrm{log}Z| \lt 2$; Trotta 2008) from most of the 2 planet models (except the 2 planet + linear trend model). It is also indistinguishable from all of those models considering either one or both of them having eccentric orbits.

Table 2. Log-evidence Differences ΔでるたZ between Different Models Considered for Our Radial-velocity-only Analysis

Model	$\mathrm{ln}{\rm{\Delta }}Z$
No planet models
Flat line	−9.4
GP	−6.6
1 planet models
1 planet + linear trend	−23.6
1 planet + quadratic trend	−9.1
1 planet	−8.8
1 planet + GP	0
2 planet models
2 planet + linear trend	−12.6
2 planet	−1.6
2 planet + GP	−1.4
2 planet + quadratic trend	−1.3

Note. It is assumed that one of the Keplerian signals has the same ephemerides as those implied by the observed transits. A flat-line model includes only a systematic radial velocity and a jitter term. The GP refers to a Gaussian Process with a quasi-periodic kernel. See text for details on the priors.

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It is interesting to note that the posterior distribution function of the GP rotation period of the 1 planet + GP model, P_rot, shows a bimodal distribution, with peaks at ∼80 days and >150 days. In the 2 planet + GP model, the latter long periodic signal is picked up by the second sinusoid and is constrained to be about 270 ± 60 days, while the quasi-periodic component of the GP shows again an accumulation of samples with a higher likelihood at ∼80 days. A Generalized Lomb−Scargle (GLS; Zechmeister & Kürster 2009) periodogram analysis of the residuals of each of those models is presented in Figure 7, in order to further explore the nature of these long-period signals. If the signal of the transiting planet is subtracted (second panel), a significant power excess is apparent at ∼270 days (which is above the baseline of our observations, which was 175 days), and next to it, even though it is insignificant, is a peak at ∼80 days. The latter is independent of the long-term variation, as it is still present after fitting the transiting planet together with the ∼270 days signal (third panel).

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To find out whether the ∼80 days signal or the ∼270 days power excess could be attributed to stellar activity, we investigated the activity indicators that are routinely derived from the CARMENES spectra (see Zechmeister et al. 2018; Schöfer et al. 2019; Lafarga et al. 2020, for the full list of indicators and how they are calculated). In Figure 8 we show the GLS periodograms of several activity indicators and the long-term photometry presented in Section 2.3. Almost all indicators, as well as the photometry, show consistent signals around 80 and/or 40 days, which would explain the ∼80 days seen in the radial velocities (RVs) as the stellar rotation period. If indeed the rotation period of the star is 80 days, the 40 days signal could be interpreted as spots at opposite longitudes and/or a byproduct of the not strictly periodic nature of the signal. Further, the TJO data, the BIS, and the Ca ii IRT b and c also show a long-term trend, which might be related with the ∼270 days power excess.

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Given that we could not rule out a stellar origin for the ∼80 days signal and the ∼270 days power excess, and that the 1 planet + GP model has the highest evidence for our RV data and can account for all significant signals in the data (see the residuals in the last panel of Figure 7), we consider this model for the global modeling of the data, which we present in the next subsection. We note that we also tested fitting our global model using the rest of the models in Table 2, which are indistinguishable from the one selected here, and all of them gave rise to similar constraints to the final parameters of the transiting exoplanet.

We performed a global modeling of the radial-velocity and photometric data using the juliet library (Espinoza et al. 2019) in order to jointly constrain the planetary properties from the photometry and radial-velocity data sets outlined in previous sections. As in the radial-velocity analysis presented in Section 3.2, we once again used the Dynamic Nested Sampling algorithm implemented in the dynesty library (Speagle 2020).

For the TESS photometry, we decided to use a GP to consider residual systematic trends in the PDC light curves used in this work. We used an Exponential−Matérn kernel (i.e., the product of an exponential and a Matérn 3/2 kernel) as implemented in the celerite library (Foreman-Mackey et al. 2017) via juliet, with hyperparameters (GP amplitude, σしぐま_GP , and two timescales,: one for the Matèrn 3/2 part of the kernel, ρろー, and another for the exponential part of the kernel, T), which are individual to each of the sectors; a jitter term was also added in quadrature to the covariance matrix for each sector. A quadratic law was used to constrain limb darkening, where the coefficients were shared between the different sectors; we used the parameterization of Kipping (2013) instead of fitting for the limb-darkening coefficients directly. For the ground-based photometry, we found that airmass was a very good predictor of the long-term trends in the data, and so we added this as a linear regressor in our fit—weighted by a coefficient θしーた, which was different for each instrument and was jointly fit with the rest of the parameters of the global fit. In addition, we observed that this linear regressor was insufficient to model all of the correlated noise leftover on the OMC and RCO data sets. We thus decided to fit those with an additional Matèrn 3/2 kernel. A linear limb-darkening law was used for all ground-based instruments, as a higher-order law was not necessary given the lower photometric precision (see, e.g., Espinoza & Jordán 2016). An individual jitter term was added to the diagonal of the covariance matrix on each of those data sets as well. For the radial velocities, following our results in Section 3.2, we considered a 1 planet model plus a quasi-periodic GP as the model to be fit in our joint analysis. We set a wide prior for the systemic radial velocity, as well as for the jitter term and the hyperparameters of the GP—in particular, for the period of the quasi-periodic GP P_rot, we use a wide period of between 20 and 350 days in order to cover the two possible periods for this parameter observed in Section 3.2. The full definition of the priors and corresponding posteriors of our joint fit are given in Table 3.

Table 3. Prior and Posterior Parameters of the Global Fit Performed on TOI-1759

Parameter	Prior	Posterior
Stellar and planetary parameters
P1 (days)	N(18.85, 0.12)	18.85019 ± 0.00013
t0,1 (BJD)	N(2,458,745.45, 0.12)	2,458,745.4654 ± 0.0011
Rp,1/R⋆	U(0.0, 1.0)	0.0483 ± 0.0010
${b}_{1}=(a/{R}_{\star })\cos (i)$	U(0.0, 1.0)	${0.21}_{-0.10}^{+0.09}$
K1 (m s−1)	U(0, 100)	${3.64}_{-0.51}^{+0.50}$
e1	fixed	0
ωおめが1	fixed	90
ρろー⋆ (kg m−3)	TN(3949, 3232; 1000, 10,000)	${3970}_{-233}^{+218}$
TESS photometry instrumental parameters
q1,TESS a	U(0, 1)	${0.32}_{-0.14}^{+0.24}$
q2,TESS a	U(0, 1)	${0.46}_{-0.25}^{+0.28}$
mflux,16 (ppm)	U(0, 105)	${851}_{-7770}^{+7916}$
mflux,17 (ppm)	U(0, 105)	$-{1598}_{-8089}^{+8062}$
mflux,24 (ppm)	U(0, 105)	${428}_{-7797}^{+7904}$
σしぐまw,16 (ppm)	$\mathrm{log}U(0.1,{10}^{5})$	${1.8}_{-1.5}^{+7.5}$
σしぐまw,17 (ppm)	$\mathrm{log}U(0.1,{10}^{5})$	${2.9}_{-2.5}^{+15.1}$
σしぐまw,24 (ppm)	$\mathrm{log}U(0.1,{10}^{5})$	${1.9}_{-1.6}^{+7.7}$
σしぐまGP,16 (ppm)	$\mathrm{log}U({10}^{-4},{10}^{2})$	${0.000104}_{-0.0000026}^{+0.000047}$
σしぐまGP,17 (ppm)	$\mathrm{log}U({10}^{-4},{10}^{2})$	${0.000106}_{-0.0000040}^{+0.000067}$
σしぐまGP,24 (ppm)	$\mathrm{log}U({10}^{-4},{10}^{2})$	${0.000100}_{-0.0000017}^{+0.000030}$
ρろーGP,16 (days)	$\mathrm{log}U({10}^{-3},{10}^{2})$	${73}_{-17}^{+16}$
ρろーGP,17 (days)	$\mathrm{log}U({10}^{-3},{10}^{2})$	${74}_{-18}^{+15}$
ρろーGP,24 (days)	$\mathrm{log}U({10}^{-3},{10}^{2})$	${76}_{-17}^{+14}$
TGP,16 (days)	$\mathrm{log}U({10}^{-3},{10}^{2})$	${0.0010}_{-0.000026}^{+0.000047}$
TGP,17 (days)	$\mathrm{log}U({10}^{-3},{10}^{2})$	${0.0011}_{-0.000045}^{+0.000081}$
TGP,24 (days)	$\mathrm{log}U({10}^{-3},{10}^{2})$	${0.0010}_{-0.000019}^{+0.000034}$
Ground-based photometry instrumental parameters
q1,OAA	U(0, 1)	${0.41}_{-0.24}^{+0.27}$
q1,OMC	U(0, 1)	${0.55}_{-0.31}^{+0.27}$
q1,RCO	U(0, 1)	${0.50}_{-0.29}^{+0.30}$
mflux,OAA (ppm)	U(0, 105)	$-{2148}_{-216}^{+214}$
mflux,OMC (ppm)	U(0, 105)	${7936}_{-6203}^{+6471}$
mflux,RCO (ppm)	U(0, 105)	$-{979}_{-5945}^{+6296}$
σしぐまw,OAA (ppm)	$\mathrm{log}U(0.1,{10}^{5})$	${3395}_{-140}^{+146}$
σしぐまw,OMC (ppm)	$\mathrm{log}U(0.1,{10}^{5})$	${3123}_{-190}^{+202}$
σしぐまw,RCO (ppm)	$\mathrm{log}U(0.1,{10}^{5})$	${4143}_{-158}^{+165}$
θしーたOAA (days)	U(−10, 10)	${0.00112}_{-0.00021}^{+0.00022}$
θしーたOMC (days)	U(−10, 10)	$-{0.0036}_{-0.015}^{+0.015}$
θしーたRCO (days)	U(−10, 10)	${0.013}_{-0.015}^{+0.015}$
σしぐまGP,OMC (ppm)	$\mathrm{log}U({10}^{-4},{10}^{2})$	${0.117}_{-0.012}^{0.022}$
ρろーGP,OMC (days)	$\mathrm{log}U({10}^{-3},{10}^{2})$	${0.438}_{-0.059}^{+0.040}$
σしぐまGP,RCO (ppm)	$\mathrm{log}U({10}^{-4},{10}^{2})$	${0.117}_{-0.011}^{0.019}$
ρろーGP,RCO (days)	$\mathrm{log}U({10}^{-3},{10}^{2})$	${0.438}_{-0.051}^{+0.039}$
Radial-velocity instrumental/activity parameters
μみゅーCARMENES (m s−1)	U(−100, 100)	${0.1}_{-4.0}^{+4.7}$
σしぐまw,CARMENES (m s−1)	$\mathrm{log}U(0.01,100)$	${1.85}_{-0.49}^{+0.50}$
σしぐまGP,CARMENES (m s−1)	$\mathrm{log}U(0.01,{10}^{2})$	${5.8}_{-2.5}^{+7.3}$
αあるふぁGP,CARMENES (days−1)	$\mathrm{log}U({10}^{-10},1)$	${0.000011}_{-0.000011}^{+0.000433}$
ΓがんまGP,CARMENES	$\mathrm{log}U(0.01,100)$	${0.5}_{-0.5}^{+2.8}$
PGP,Rot	U(20, 350)	${237}_{-103}^{+67}$

Note. For the priors, N(μみゅー, σしぐま²) stands for a normal distribution with mean μみゅー and variance σしぐま²; TN(μみゅー, σしぐま²; a, b) is a truncated normal distribution with lower and upper limits given by a and b, respectively; U(a, b) and $\mathrm{log}U(a,b)$ stand for a uniform and log-uniform distribution between a and b, respectively.

^a These parameterize the quadratic limb-darkening law using the transformations in Kipping (2013).

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The resulting best-fit models and corresponding credibility bands are presented in Figure 2 for the TESS photometry, in Figure 4 for the ground-based photometry, and in Figure 3 for the radial velocities. The constraints from our radial-velocity follow-up allowed us to obtain a precise (>5σしぐま above 0) measurement of the semiamplitude imprinted by TOI-1759 b on its star of $K={3.64}_{-0.51}^{+0.50}$ m s⁻¹, which is an over 7σしぐま detection of the semiamplitude. Joining the derived transit and radial-velocity parameters, along with the stellar properties presented in Table 1, we derive the fundamental parameters of TOI-1759 b in Table 4. As can be observed, TOI-1759 b is a relatively cool (443 K equilibrium temperature, assuming 0 albedo) sub-Neptune-sized exoplanet (R_p = 3.14 ± 0.10 R_⊕). Coupling these numbers with our estimated mass of M_p = 10.8 ± 1.5M_⊕, we derive a planetary bulk density (ρろー_p = 1.91 ± 0.32 g cm⁻³) and gravity (g_p = 10.7 ±1.6 m s⁻²) that are strikingly similar to those of Neptune (1.64 g cm⁻³ and 11.15 m s⁻², respectively). We discuss the properties of TOI-1759 b in the context of other discovered systems in the next section.