© The Authors 2025

1 Introduction

The first detection of an exoplanet around 51 Pegasi was made possible by the radial velocity (RV) method (Mayor & Queloz 1995) using the optical spectrograph ELODIE at the Haute-Provence Observatory in France (Baranne et al. 1996). Since then, advancements in optical instrumentation with instruments such as HARPS (Mayor et al. 2003) and ESPRESSO (Mégevand et al. 2014) have resulted in the identification of numerous exoplanetary systems (Faria et al. (2022); Suárez-Mascareño et al. (2023); to quote just a few of the latest results). Although historically they started in the optical domain, RV observations have recently extended to the near-infrared (nIR) domain with instruments such as GIANO (Carleo et al. 2016), HPF (Metcalf et al. 2019), CARMENES-NIR (Bauer et al. 2020), IRD (Hirano et al. 2020), SPIRou (Donati et al. 2020) and NIRPS (Thibault et al. 2022). The nIR does not benefit from the extensive research that has been conducted in the optical domain with respect to instrumental and data-reduction techniques. Most notably, the nIR comes with new challenges such as telluric correction, instrument stability at cryogenic temperatures, wavelength calibration, as well as persistence and other yet-to-be investigated effects of infrared arrays (Artigau et al. 2018; Cook et al. 2022). However, this spectral range provides a unique opportunity to explore the YJHK spectral bandpasses (Quirrenbach et al. 2018; Reiners & Zechmeister 2020), which are especially rich in molecular bands such as CO, OH, and H₂O, in addition to being the domain in which cool stars are the most luminous.

The nIR is thus of interest to better detect and characterize exoplanetary systems around M dwarfs (Cadieux et al. 2022; Martioli et al. 2022; Almenara et al. 2024). Those stars have attracted a great deal of interest due to their abundance in the solar neighborhood (Henry et al. 2006). They are of particular significance as they have lower masses, making them excellent candidates for detecting low-mass exoplanets since it increases their RV signatures in the m.s⁻¹ range (Anglada-Escudé et al. 2016; Bonfils et al. 2013; Faria et al. 2022; Suárez-Mascareño et al. 2023). Additionally, these stars are known to host a larger number of rocky planets than other star types (Bonfils et al. 2013; Dressing & Charbonneau 2015; Gaidos et al. 2016; Hsu et al. 2020; Sabotta et al. 2021).

On the other hand, M dwarfs are known to exhibit a high level of magnetic activity, which can lead to spurious RV signals (Bonfils et al. 2007; Gomes da Silva et al. 2012; Newton et al. 2016; Hébrard et al. 2016). However, this activity is wavelength-dependent (Reiners et al. 2013; Baroch et al. 2020) and less important in the nIR. Consequently, an example of the potential synergy between the optical and nIR is to conduct RV measurements in both domains so as to confirm or rule out the planetary nature of ambiguous signals (Cortes-Zuleta et al. 2024; Carmona et al. 2023).

Within this context, the SPIRou Legacy Survey Planet Search program (SLS-PS¹) is a large program whose aims are detecting and characterizing planetary systems around approximately ~50 nearby M dwarfs (Moutou et al. 2023). The program was allocated 153 nights of observation time at the Canada-France-Hawaii Telescope (CFHT) from February 2019 to June 2022. In this paper, we present the analysis of the nearby M dwarfs Gl 480 and Gl 382 using SPIRou data obtained as part of this program and other publicly available datasets.

This paper is organized as follows. In Section 2, we describe our targets and the RV data used in our analysis as well as an overview of the Wapiti correction method to mitigate systematics in the SPIRou data of our two targets. In Section 3 we present a separate analysis for each target, an discuss their characteristics and the determination of orbital parameters. We conclude in Section 4.

2 Observations

2.1 The SPIRou spectropolarimeter, the APERO reduction and the LBL algorithm

SPIRou is a high-resolution nIR spectropolarimeter installed in 2018 at the Cassegrain focus of CFHT. It has a spectral range going from 0.98 to 2.35 μ m at a resolving power of 70 000, and is stabilized in a vacuum chamber with milli-Kelvin temperature stability (Challita et al. 2018; Donati et al. 2020). A detailed presentation of the instrumental design and early measurement performance is provided in Donati et al. (2020).

All SPIRou spectra were reduced using the 0.7.288 version of the SPIRou data reduction system (DRS), called A PipelinE to Reduce Observations (APERO) detailed in Cook et al. (2022). APERO processes science frames to produce 2D and 1D spectra from the two science channels and the RV reference channel, and it uses a combination of exposures from a UNe hollow cathode lamp and a Fabry-Pérot etalon for wavelength calibration (Hobson et al. 2021). The Barycentric Earth radial velocity (BERV) was calculated using BARYCORRPY (Kanodia & Wright 2018) to shift the spectra to the barycentric frame of the Solar System. APERO also performs optimal extraction of the two science channels and the Fabry-Pérot RV reference channel, flat-fielding, a correction of thermal background, and a correction of the residual leak of the Fabry-Pérot RV reference spectrum onto the science channel. Based on the TAPAS model spectrum of the Earth atmosphere (Bertaux et al. 2014) and a library of hot-star spectra obtained with SPIRou in various conditions of airmass and water vapor, APERO corrects the science spectra from the telluric spectrum using a three-step telluric correction approach, which will be detailed in an upcoming publication (Artigau et al., in prep.). The final product is a wavelength-calibrated telluric corrected 2D spectrum.

Radial Velocities were subsequently computed from the telluric-corrected spectra using the 0.63.007 version of the Line-By-Line (LBL²) method. This method is based on the framework developed in Bouchy et al. (2001) and was first proposed by Dumusque (2018) then further explored by Cretignier et al. (2020) for optical RV observations with HARPS and more recently revisited by Artigau et al. (2022) in the context of nIR observations. The LBL algorithm requires a high signal-to-noise ratio (S/N) template spectrum of the observed star, simply called template in the following, and derived from the median of all collected spectra. The RVs are then computed by comparing each spectrum to the template and its first few derivatives. The LBL algorithm is applied to both the stellar extracted spectrum and the Fabry-Pérot spectrum in the RV reference channel, with the latter providing a precise measurement of the instrumental drift at the exact time of our observations. The final RV time series is corrected from this drift and from a nightly zero-point (NZP) correction in the time domain to eliminate long-term instrumental variations. The list of standard stars used to compute this NZP is GJ 905, GJ 699, GJ 1012, GJ 1289, GJ 3378, GJ 752 A, GJ 1002, GJ 1286, GJ 1103 and PMJ21463+3813 (reduced with the same APERO version).

2.2 Targets

G1 480 (Wolf 433) is a red dwarf star with spectral type M3.5V (Lépine et al. 2013) that is located 14.2625 ± 0.0060 pc away from Earth (Gaia Collaboration 2020). In Cristofari et al. (2022), its physical properties were determined from the analysis of SPIRou spectra and are summarized in Table 1. Additionally, time series of the longitudinal magnetic field (B_ℓ) of this star were obtained from circularly polarized spectropolarimetric sequences. However, no clear periodic variation was detected in the time series (Fouqué et al. 2023). In the study by Donati et al. (2023), which is a parallel analysis of the B_ℓ time series using different data reduction and analysis of the spectra, the authors identified a rotational period P_rot of 25.00 ± 0.36 d. However, a prior study by Feng et al. (2020) reported an activity-related signal occurring at a period of 49.3 ± 0.2 d. It is plausible that the 25 d signal observed in the B_ℓ data might actually represent the first harmonic of the true rotational period. Finally, a planet orbiting at a 9.567 ± 0.005 d orbital period was claimed to orbit G1 480 from the analysis of HIRES and HARPS data (Feng et al. 2020). The signal had a semi-amplitude K = 6.80 ± 0.87 m s⁻¹ with an estimated M p sin i of 13.2 ± 1.7 M_⊕.

Gl 382 (V* AN Sex) is a red dwarf of spectral type M2.0V (Kirkpatrick et al. 1991) located 7.707 ± 0.0015 pc away from Earth (Gaia Collaboration 2020). Its physical properties are also listed in Table 1. The time series of B_ℓ of this star were also obtained from Stokes V spectropolarimetric sequences and led to the detection and characterization of a signal at ${21.32}_{-0.03}^{+0.04}$ d (Fouqué et al. 2023) (21.91 ± 0.16 d in Donati et al. 2023). Its rotational period is well-established in the literature using other instruments such as ASAS photometric data (21.6 ± 2.2 d; Kiraga & Stepien 2011) and HARPS (21.7 ± 0.1 d; Suárez Mascareño et al. 2015). In Sabotta et al. (2021), a signal with a period of 10.65 d was detected in CARMENES optical RV data. However, the signal was automatically flagged as spurious due to this period being close to the first harmonic of the rotational period. This star was also part of the recent activity study conducted by Mignon et al. (2023) from HARPS spectra, and they did not find any signs of an activity cycle in the chromospheric activity indices even though a previous study had detected a cycle with a period of 13.6 ± 1.7 years at a 3σ level (Suárez Mascareño et al. 2016).

2.3 Radial Velocity data sets

2.3.1 Gl 480

SPIRou observations for Gl 480 span from November 13, 2019 to June 20, 2024, and consist of 130 data points from the SLS program and other programs (PI Carmona and PI Artigau)³. The airmass during these observations has a median value of 1.3, the signal-to-noise ratio per 2.28 km s⁻¹ pixel bin in the middle of the H band has a median value of 110.7 and the median of the error bars is 1.7 m s⁻¹.

We also retrieved RV time series collected with HARPS and CARMENES (optical data). We found 37 RV measurements from HARPS in Trifonov et al. (2020) (RMS = 6.1 m s⁻¹, error bar median value of 1.5 m s⁻¹) and 7 from CARMENES in Ribas et al. (2023) (RMS = 3.3 m s⁻¹, error bar median value of 1.2 m s⁻¹), which were both reduced using the SERVAL pipeline (Zechmeister et al. 2018).

2.3.2 Gl 382

SPIRou observations were again conducted using circular polarization sequences. Observations for Gl 382 were carried out from February 02, 2019 to June 07, 2023, and consist of 146 data points from the SLS program and other programs (PI Carmona and PI Artigau⁴. The median airmass during these observations was 1.4, the median signal-to-noise ratio per 2.28 km s⁻¹ pixel bin in the middle of the H band was 154.5 and the median of the error bars is 1.6 m s⁻¹.

In addition to the SPIRou data, RV time series were also retrieved from the HARPS, and CARMENES (optical data). We used 32 HARPS (RMS = 6.7 m s⁻¹, error bar median value of 1.0 m s⁻¹) and 77 CARMENES (RMS = 5.5 m s⁻¹, error bar median value of 1.5 m s⁻¹) RV measurements from Trifonov et al. (2020) and Ribas et al. (2023), respectively. These measurements were reduced using the SERVAL pipeline (Zechmeister et al. 2018).

2.4 Wapiti correction

The LBL method provides a mean RV value of the stellar spectrum and per-line RV time series on which it is possible to apply a weighted principal component analysis (wPCA⁵; Delchambre 2015). The Wapiti⁶ method (Ould-Elhkim et al. 2023) uses this approach and applies a wPCA to the LBL per-line RV time series in order to find proxy time series of any remaining systematics of various origins. In summary, it uses a wPCA in the per-line RV time series (corrected for drift and NZP) and automatically selects an appropriate number of components through an algorithm slightly different from the one outlined in Ould-Elhkim et al. (2023). In summary, we retained the permutation test described in this work, with the key difference that, instead of evaluating the significance of the explained variance of a component, we assessed how well it fit the RV. Specifically, we calculated the Bayesian information criterion (BIC; Schwarz 1978) for each component when fitted individually to the RV (including an offset). We then compared this to the BIC₀ value of a model containing only the offset. To assess the robustness of the fit, we tested the log Bayes factor between these two models, defined as: $$\log BF=\frac{BI{C}_0-BIC}{2}.$$(1)

Another modification to the original version of Wapiti is that we replaced the leave-p-out cross-validation and insted used a simple reordering of the principal components once they have been selected. Specifically, we rearranged the components in descending order of their BIC values. During this process, we identified the component that, when included, minimizes the BIC and we designated it as the new primary component. We then iteratively determined subsequent components: for each step, we added the next component that, when combined with the already selected components, further minimizes the BIC. This procedure produces a reordered sequence of principal vectors, denoted as ${\hat{V}}_i$. Using the mathematical framework established in Cretignier et al. (2023), this approach corresponds to calculating the BIC when fitting the following model to the data for the k-th component: $$RV(t)={\gamma }_{SPIRou}+{\displaystyle \sum _{i=1}^N}{a}_i{V}_i.$$(2)

Here we included an offset term γ_{S PIRou} and the coefficients a_i to fit the principal vectors V_i in the model. The parameter N represents the number of components used in the reconstruction, and it can be set to 0, corresponding to the data without applying Wapiti correction. We fitted these models through a weighted least-squares regression using the statsmodels module (Seabold & Perktold 2010), which also provided their BIC value.

To determine the final number of selected components, we simply chose the number of components that resulted in the lowest BIC value. This reordering method is similar to the χ² reordering technique employed in Collier Cameron et al. (2021), the only difference being our use of the BIC as the criterion to penalize using too many components.

We can detect the presence of periodic signals in the time series through a Bayesian periodogram (Delisle et al. 2018) that compares two models, one model μ₀ corresponding to Equation (2) and one model μ_P for each period P defined as $$RV(t)={\gamma }_{SPIRou}+{\displaystyle \sum _{i=1}^N}{a}_i{\hat{V}}_i+A\sin (2\pi t/P)+B\sin (2\pi t/P),$$(3)

and by computing their respective BIC (BIC₀ and BIC(P)) and the logarithm of the Bayes factor log $BF(P)=\frac{BI{C}_0-BIC(P)}{2}$. We computed this periodogram with a grid of 10 000 points ranging from 1.5 d to 10 000 d with a logarithmic spacing; a value above 5 was used as a threshold to consider a signal statistically significant (Delisle et al. 2018).

In the case of Gl 480, four outliers were removed from the dataset, prior to employing the Wapiti method. Only the fourth wPCA component was found to be appropriate for this target and the application of the BIC re-ordering was thus not necessary.

Regarding Gl 382, no evident outliers were detected in the RV time series. The components 1, 2, 5, 6 and 7 were found appropriate for this target and the application of the BIC reordering led to the use of four components (in order V₆, V₂, V₅ and V₇).

In the following section, a comprehensive analysis is presented of Gl 480 and Gl 382, using the SPIRou RV time series along with other available datasets. The analysis is conducted separately for each target. We begin by characterizing the stellar activity of the observed targets. Then, we perform an in-depth RV analysis of the targets over different combinations of the datasets and accurately determine the orbital parameters associated with their planetary companion, if any.

3 Analysis and results

3.1 Gl 480

3.1.1 Stellar activity

In their study, Feng et al. (2020) performed an activity analysis that detected a signal with a period of 49.3 ± 0.2 d in two stellar indicators, Hα and NaD1, in the HARPS data. We thus examined those indicators as well as the differential line width (dLW), which is another product obtained from SERVAL (Zechmeister et al. 2018), in the HARPS data. We analyzed the presence of the signals in those indicators using Bayesian periodograms and we found no evidence of activity at the reported period in dLW, while we did observe the reported signal in NaD1 and Hα even though not at a significant level in the former (see Figure 1).

Regarding SPIRou, we used three stellar activity indicators: dLW; B_ℓ; and a new proxy for activity, the differential effective temperature (dET; Artigau et al. 2024). This stellar activity proxy was obtained with the LBL analysis from the temperature variation of the star derived from the modulation of spectral line depths in SPIRou spectra. The periodograms can be seen in Figure 1, we found a strong signal at the reported period of Feng et al. (2020) in dLW and dET while B_ℓ appears to have a strong signal around its first harmonic.

Since the rotational period is the strongest in the dET time series, we decided to use it to obtain a characterization of the rotational period by applying a quasi-periodic Gaussian process (GP) in the following form: $$k(\tau )=A\exp (-\frac{{\tau }^2}{2{\lambda }^2}-\mathrm{\Gamma }{\sin }^2\left(\frac{\pi }{P}\tau \right)).$$(4)

The GP was computed using the george package (Ambikasaran et al. 2015) and to estimate the hyper-parameters we employed a Markov chain Monte Carlo (MCMC) approach with the emcee module (Foreman-Mackey et al. 2013) by generating 10 000 samples with a burn-in of 2500 samples and 100 walkers. We obtained a value for the rotational period of P_rot = ${50.6}_{-0.8}^{+0.8}$ d, consistent with the value of Feng et al. (2020). The corner plot of the hyper-parameters can be seen in the Appendix A.1.

Fig. 1 Bayesian periodograms of various stellar activity indicators of Gl 480. From left to right and top to bottom the dLW, Hα, and NaD1 index time series from the HARPS data, and the dLW, Bℓ and dET time series from the SPIRou data. The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the one-third rotation period (Prot/3). The significance level of log BF = 5 is indicated by the horizontal red dashed line.

3.1.2 SPIRou RV analysis

To search for signals in the SPIRou data of in Gl 480 we adopted an iterative approach using Bayesian periodograms where we systematically add significant signals in the model and then recompute a log BF periodogram. We continue this process while the maximum peak in the periodogram remains above 5 (Delisle et al. 2018). To be more precise, with N the number of Wapiti systematics used and N_p representing the number of fitted planets (which can be 0) of periods P_i, we compare the models: $$\begin{array}{cc}& M_0=\gamma +{\displaystyle \sum _{i=1}^N}{a}_i{\hat{V}}_i+{\displaystyle \sum _{i=1}^{N_p}}{A}_i\sin \left(2\pi t/P_i\right)+B_i\sin \left(2\pi t/P_i\right)\hfill \\ & M_P=M_0+A\sin (2\pi t/P)+B\sin (2\pi t/P).\hfill \end{array}$$(5)

Here M₀ represents the base model, which includes a constant term γ diferent for all instruments.

If at the N_p + 1 iteration a signal at period P is found significant, a circular model μ_C of the form: $$\begin{array}{cc}{\mu }_C=& \gamma +{\displaystyle \sum _{i=1}^N}{a}_i{\hat{V}}_i+{\displaystyle \sum _{i=1}^{N_p}}{K}_i\sin \left(2\pi \frac{t-T_i}{P_i}\right)\\ & +K\sin \left(2\pi \frac{t-T}{P}\right)\hfill \end{array}$$(6)

is fitted to the data where the parameters are the semi-amplitude of the signal K, the orbital period P and the time offset T. They were fitted by maximizing the maximum a posteriori (MAP) estimates of the parameters. Once no more significant signals are detected, we fitted the final parameter values by taking the median value of their posterior distribution using an MCMC algorithm to the data as we did for the dET time series.

As shown in Figure 2, the 9.5 d signal reported in Feng et al. (2020) is also present in the SPIRou data. Before fitting this signal, the component from the Wapiti correction was added to the model, which reduced the RMS of the time series from 5.84 to 5.37 m.s⁻¹ and removed a one-year signal from the data. Then we included this signal in the model, which led to a periodogram showing a long-term signal. This signal could be caused by a planet, the instrument, or a long-term magnetic cycle. To handle the long-term signal, we tested fitting it with a first- or seconddegree polynomial. The periodograms from these fits showed signals at 57 and 17.8 d, respectively. We interpreted these signals as being linked to stellar activity since they are close to the rotation period of the star P_rot and its harmonic (P_rot/3). Interestingly, when using a second-degree polynomial, only the 17.8 d signal appeared in the RV data, while no other harmonics were detected. However, this signal is also strong in the dLW and dET activity indicators (Figure 1). To further rule out the 17.8 d signal as a possible planet, we computed a stacked periodogram centered at this period plus or minus 1 day and also studied how its log BF changed with an increasing number of observations (Mortier & Collier Cameron 2017). If the signal were from a planet, its significance would increase steadily. Instead, as seen in Figure 2, its behavior is erratic.

To correct for stellar activity in the RV data, we used a quasi-periodic GP. We also stopped including a polynomial term in the model, as the GP can correct for long-term trends while being statistically preferred (see Table 2). To model stellar activity accurately, we applied the priors from Camacho et al. (2023) for the GP’s hyper-parameters (Priors I). We set the priors in the following way:

• For the amplitude and jitter (σ) of the GP, we used a modified log-uniform distribution (ℳℒ𝒰; Gregory 2005) with a “knee” value of y_σ (the standard deviation of the data) and an upper bound of 2y_ptp (twice the peak-to-peak difference of the RVs).

• For the smoothing parameter (Γ), we used a log-uniform distribution (ℒ𝒰) with a range from 0.1 to 5.

• For the decaying timescale (λ), the prior ranged from δt_av (the average time between observations) to 10t_span (ten times the total time span of the observations).

• For the rotational period (P_rot), we used a uniform distribution ranging from 1 to 1000 d.

The MAP estimate gave a P_rot value of 52 d, consistent with the star’s known rotation period. Running an MCMC analysis gave a median P_rot value of 26 d (one-half the rotational period) with a smoothing parameter of 2 probably due to the complex dynamics behind the 17.8 d signal and the various local minima. To ensure that P_rot better represents the actual rotation period, we opted to use a different set of priors for the P_rot, Γ and λ hyper-parameters by using their posterior distribution from the MCMC analysis of the dET time series (Priors II) or the dLW time series (Priors III). We chose these two indicators to create informative priors because the 17.8 d signal is present in both of them.

Once we obtained the hyper-parameters of the GP we kept searching for signals in the data by fixing its smoothing parameter Γ and decay time l while letting the amplitude coefficient vary freely. The reason for this choice is to balance accuracy and computational efficiency. Simply subtracting the GP from the data before searching for a signal would not be appropriate because the GP’s hyper-parameters would change when a signal with period P is added. Ideally, we should perform a MCMC analysis for each period in our grid, but this is too computationally intensive. Instead, we simplify the problem by keeping only the coefficient A as a free variable which linearizes the problem, making the computation faster. If we do find a significant signal, we then recompute all the GP hyper-parameters after adding the signal and we verify that the new model remains statistically significant.

As shown in Figure 2, after correcting for activity, a 6.43 d signal remains in the data for the three sets of priors. This period does not appear in the activity indicators and when modeled as a potential planetary signal, the MAP estimate of the semi-amplitude K is 1.50 m.s⁻¹. However, since adding this signal also updated the hyper-parameters of the GP, the log BF of this model compared to the previous value decreased to 2.31, 1.90 and 3.71 with respectively Priors I, II, and III which is below our significance threshold. As a result, we cannot confidently claim the detection of a new 6.43 d signal in Gl 480 based on the SPIRou data alone. A summary of the models used and their BIC (MAP estimate) values can be seen in Table 2.

Fig. 2 Iterative search of signals with Bayesian periodograms using SPIRou data of Gl 480. The top four panels display log BF periodograms for (a) raw data, (b) Wapiti correction, (c) Planet b model fit, and (d) long-term and/or stellar activity correction. The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the one-third rotation period (Prot/3). The significance level of log BF = 5 is indicated by the horizontal red dashed line. The bottom left panel shows the stacked periodogram with log BF power values, where log BF values below 5 are in gray. The bottom right panel presents the variation in log BF as a function of the number of observations for the most significant period at 17.8 d.

3.1.3 Optical RV analysis

For the optical RV analysis, we used a combination of the HARPS and CARMENES datasets, which slightly differs from the analysis in Feng et al. (2020), which did not use the seven CARMENES data points and because we discarded 20 HIRES RV measurements used in this study. The same iterative search for signals as for the SPIRou data was conducted. The known 9.5 d signal was recovered and fitted leading to the detection of a significant signal at 53.64 d near the rotational period of the star which can be interpreted as of stellar activity origin (Figure 3). Consequently we fitted a quasi-periodic GP to the data with a different instrumental jitter σ for each instrument and a MCMC was applied on this model using the set of priors I.

After removing stellar activity from the optical data, we found no other significant signals. The final MCMC analysis gives a semi-amplitude of ${5.0}_{-0.5}^{+0.5}$ m s⁻¹ for the 9.5 d signal (see Table 3), which is noticeably different from the 6.80 ± 0.87 m s⁻¹ value reported by Feng et al. (2020). Without using a GP, we found a semi-amplitude of ${5.5}_{-0.7}^{+0.7}$ m s⁻¹, suggesting that the difference in values is mostly due to excluding the HIRES data and adding a GP to our model. This semi-amplitude is also higher than the value found with SPIRou data alone of ${3.6}_{-0.5}^{+0.5}$ m s⁻¹. However, this discrepancy is not caused by either the Wapiti correction or the GP applied on SPIRou data, as not using both still gives a semi-amplitude of ${3.8}_{-0.7}^{+0.7}$ m s⁻¹. While this difference is within 5σ and not necessarily incompatible, it is worth noting as it could raise some uncertainty about the true planetary nature of this signal.

3.1.4 Joint nIR-optical RV analysis

Finally, we combined the SPIRou data with the optical data to perform the final iterative search for signals, as shown in Figure 5. After correcting for the Wapiti systematics and the 9.5 d signal, long-term signals appeared as four peaks at 958, 1211, 3006, and 7322 d.

However, as shown in Figure 5, the individual datasets from different instruments do not overlap, and their time baselines are not above 1000 d. This makes it difficult to accurately estimate the properties of such a long-term signal, as they are heavily influenced by the choice of zero points for the RV datasets. Furthermore, this long-term signal was only seen in the SPIRou data (which has a longer time span than both datasets) and not in the combined HARPS and CARMENES dataset, raising doubts about its planetary origin.

It is likely that this signal comes from the SPIRou data alone, caused either by an instrumental effect or a long-term magnetic cycle affecting nIR RV measurements. This conclusion is supported by the fact that applying a GP to the SPIRou data alone is sufficient to correct for these long-term signals while providing the best fit when compared to circular models, as indicated by the BIC values shown in Table 2.

We corrected for stellar activity using a GP with Priors I for the optical datasets and a different GP for the SPIRou data for which we tried the three sets of Priors. We found that Priors III gave the best BIC value and we again detected a 6.4 d signal in the residuals. When fitted, this signal had a log BF of 6.7, just above our detection threshold, indicating that adding the optical data slightly increased its significance and allowed a detection. However, we note that with both Priors I and Priors II, the signal fell below the significance threshold. This suggests that additional observations are needed to confirm or rule out this low-amplitude signal.

For completeness, we included the 6.4 d signal in the final model, but more data will be required to confidently confirm it as a new planetary signal orbiting Gl 480. Our final model, therefore, includes the 9.5 d planet, the 6.4 d signal and GPs to model activity using Priors III for SPIRou data and Priors I for the optical data. We also tested whether an eccentric model for the signals provided a better fit by using a Keplerian model μ_K of the form: $${\mu }_K=\gamma +K[\sin (\omega +\nu (t))+e\cos \omega ],$$(7)

where the parameters of the model includes the semi-amplitude of the signal (K), the orbital period (P), the eccentricity e, the argument of periastron ω, and the true anomaly ν. The radvel package (Fulton et al. 2018) was used to model the Keplerian and we found that considering an eccentric model did not improve the fit of our data. Therefore, we chose a circular model for both signals, as it had the lowest BIC.

Our combined nIR and optical radial velocity RV analysis of Gl 480 confirms the presence of the 9.5 d signal. Although the nIR and optical datasets individually yield slightly different semi-amplitude values, the final combined analysis results in a semi-amplitude of K = ${4.5}_{-0.3}^{+0.3}$ m s⁻¹. A detailed comparison of the semi-amplitude and period posterior distributions across the datasets is provided in Appendix A.1. Additionally, we detect a low-amplitude signal at 6.4 d in the combined dataset. This signal barely exceeds our detection threshold and it is only the case when a specific set of priors is applied, rendering it quite weak. Nonetheless, we include this signal in the final model for consistency with our detection criterion, with an associated semi-amplitude of K = ${1.6}_{-0.3}^{+0.3}$ m s⁻¹.

Although this signal was marginally detected only in the SPIRou data, it was entirely absent from the optical datasets. To investigate its non-detection in the HARPS and CARMENES datasets, we performed a simple injection-recovery test similar to the one conducted in Suárez-Mascareño et al. (2023). First, we removed the 9.5 d planetary signal from the data and injected 10 000 sinusoidal signals with a semi-amplitude K_sim ~ 𝒩 (1.6, 0.3), a period P_sim ~ 𝒩 (6.4349, 0.0006) (corresponding to the values obtained from the MCMC analysis in Table 3), and a random phase. We then subtracted the stellar activity using the same GP hyper-parameters as those measured for the one-planet model, while recomputing the activity model for each individual dataset.

Next, we computed the Bayesian periodogram at the injected period and assessed whether it could be recovered. The results, shown in Figure 4, display the histogram of the log BF values for the injected period. Only 19% of the simulations successfully detected the signal with a log BF above 5; the mean value of 2.0 is close to the value of 1.8 retrieved from the residuals of the actual datasets. Thus, our injection-recovery test indicates that the non-detection of the 6.4 d signal in the optical datasets seems compatible with the SPIRou candidate detection.

Finally, we estimated the minimum planet mass, M_p sin i, by using the star’s mass and its uncertainty obtained from Cristofari et al. (2022), which is listed in Table 1. To estimate the equilibrium temperature (T_eq), we used the star’s radius and effective temperature, both derived from the same study. The corner plot from the MCMC computed over the data sets are available in Appendix A.1. The final fit on the dataset using all instruments is presented in Figure 5, where we provide the phase-folded representation of the data and the residuals.

Fig. 3 Iterative search of signals with Bayesian periodograms using HARPS and CARMENES data of Gl 480. The three panels display log BF periodograms for (a) the raw data, (b) the Planet b model fit, and (c) the stellar activity correction. The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the one-third rotation period (Prot/3). The significance level of log BF = 5 is indicated by the horizontal red dashed line.

Fig. 4 Histogram of the maximum log BF values of the injected period in the injection-recovery simulations. The solid red line indicates the log BF value of 1.8 retrieved from the residuals of the real datasets, while the dashed red line marks the detection threshold at log BF = 5.

3.2 Gl 382

3.2.1 Stellar activity

For Gl 382 we employed the same analysis approach used for Gl 480. Initially, we examined the various indicators of stellar activity available for both HARPS and SPIRou, just as we did for Gl 480. Additionally, we had access to a larger dataset from CARMENES, comprising 77 data points, which allowed us to use multiple stellar activity indicators. These indicators included the dLW, the Hα and NaD1 indices, as well as the Ca II IRT1 (849.802,nm), Ca II IRT2 (854.209,nm), and Ca II IRT3 (866.214, nm) indices.

The periodograms of the stellar activity indicators are presented in Figure 6. In these figures, we have highlighted the rotational period in green, its first harmonic.

After examining of these indicators, we found that the CARMENES and HARPS data both exhibited power in their Hα periodograms near the known 21.3–21.9 day rotational period. Moreover, for the CARMENES data, the Ca II IRT 1, 2, and 3 indices also show a significant period corresponding to the rotational period. In the HARPS data the rotational period is also observed in the NaD1 at a more significant level than in the CARMENES data, a 50 d signal is also observed in the HARPS dLW and Hα activity proxies.

For the SPIRou indicators, we again used the dLW, B_ℓ and dET time series. The dLW time series is contaminated by a long-term signal but the B_ℓ and dET time series allowed us to detect the known rotational period. However we observed that in the dET time series, as for HARPS data, a 50 d signal is present at a significance similar to the rotational period; this signal is also highlighted by green dashed line in Figure 6. The origin of this signal is unclear; however it is unlikely to be due to an alias of the rotational period or its harmonics since otherwise it would also be observed in the B_ℓ time series, all the more so as it is also observed in HARPS indicators. To characterize the rotational period we launched a MCMC over the dET time series using quasi-periodic GP which gave us an estimated rotational period for Gl 382 of P_rot = ${21.41}_{-0.14}^{+0.14}$ d. The posterior distribution of the hyper-parameters can be seen in the corner plot in Appendix A.2.

3.2.2 SPIRou RV analysis

Starting with the analysis of our SPIRou data, Figure 7 shows significant long-term signals that are effectively removed after applying the Wapiti correction. This correction isolates a 10.4 d signal, which we attribute to stellar activity due to its closeness to the first harmonic of the star’s rotational period. Interestingly, no signal corresponding to the actual rotational period has been detected, making Gl 382 another example of a star exhibiting a stable rotational harmonic (Ahrer et al. 2021).

We corrected for stellar activity following the approach described in the previous section, employing a GP with the set of Priors I. After modeling the activity, we observed that the P_rot hyperparameter converged to a value of 20.8 d, which is slightly different from the known value of 21.14 d. We also detected a 51.1 d signal in our data (Figure 7). However, this signal is also present in ancillary activity indicators, including SPIRou’s dET time series and the Hα time series from HARPS. Additionally, a stacked periodogram centered around this period (Figure 7) shows that the signal’s significance does not gradually increase with additional observations but instead emerges abruptly due to the most recent data points. Given this behavior, we attributed it to stellar activity, even though its physical origin is unclear as mentioned earlier. To further correct for this activity, we applied an alternative prior set, Priors II, which incorporates the posterior distributions from the MCMC analysis of the dET time series as priors for the smoothing parameter Γ, and the decay timescale λ, which can be seen in Appendix A.2. For the rotational period, we left a uniform prior since we know the value should be close to 20.8 d, whereas the posterior distribution of the dET time series would incorrectly center it around 21.4 d. This set of priors successfully produced a GP model that effectively removed both the 10.4 d and 51 d signals from the data. Therefore, we adopted Priors II for the SPIRou dataset, and after applying this GP-based activity correction, no significant signals remained in the residuals.

Fig. 5 Final RV model for Gl 480. The top panel shows the raw RV data with offsets removed, plotted for the SPIRou, HARPS, and CARMENES instruments (RMS = 5.92 m.s−1). The second and fourth rows display the contributions of planets 1 and 2 to the RV signal, respectively, along with their phase-folded RV curves. The third row illustrates the correction for stellar activity using GPs, with individual GP contributions shown for each grouped instrument. The bottom panel presents the residuals after removing all modeled signals (RMS = 2.04 m.s−1). The rightmost column contains the log BF periodograms highlighting significant periods in the RV data, the significance level of 5 is indicated by the horizontal red dashed line.

3.2.3 Optical and joint nIR-optical RV analysis

In the analysis of the CARMENES and HARPS RV time series (Figure 8), we detected a prominent 10.4 d signal, which corresponds to the first harmonic of the star’s rotational period. Notably, in the optical data, we also observed a signal at 21.5 d. To correct for stellar activity, we applied a GP model using the set of Priors I. After this correction, a weak 43.4 d signal emerged in the residuals, with a log BF of 5.36. However, as mentioned earlier, this periodogram was generated for computational efficiency by allowing only the GP hyperparameter A to vary while keeping the others fixed. When we include this signal in our model and update all parameters, including the full set of GP hyper-parameters, the log BF decreases to 1.06, leading us to dismiss this signal. Interestingly, the 51 d signal observed in the SPIRou data is entirely absent from the RV residuals of this optical dataset, corroborating our claim that this signal was originating from stellar activity. Finally, we incorporated the SPIRou dataset to conduct a joint nIR and optical RV analysis. The SPIRou data were corrected for stellar activity using the set of Priors II. Despite combining all datasets, no significant signals remained in the residuals after correction. The BIC values of the different models used can be seen in Table 4.

The absence of a detected signal after correcting for stellar activity still allows us to estimate detection limits for potential additional companions that may have remained undetected in this study. To do so, we performed another injection-recovery test, similar to the one described for Gl 480, with the key difference that this time we used a semi-amplitude grid linearly spaced from 0.1 to 3 m.s⁻¹ and an orbital period grid linearly spaced from 40 to 150 days, both with 100 steps. For each combination of semi-amplitude and period, we injected ten signals with a phase linearly sampled from 0 to 2π.

We chose this period range because we estimated that the habitable zone (HZ) of this star extends from 0.19 to 0.41 AU (Kopparapu et al. 2014), corresponding to orbital periods of approximately 45.7 days and 134.3 days, respectively. We then assessed whether the injected signals could be recovered, with the results shown in Figure 9.

To estimate the upper limit on the detectable semi-amplitude of a potential companion in the habitable zone, we identified the semi-amplitude values for which the detection rate reached 100%, from which we calculated the corresponding minimum mass. The histograms, presented in Figure 9, indicate an upper limit on the semi-amplitude of a potential companion of K_upper = 1.3 ± 0.3 m.s⁻¹. This suggests that any potential companion within the habitable zone of Gl 382 would have a minimum mass below M_upper = 5.9 ± 0.8 M_⊕.

In summary, stellar activity significantly impacts the RV data of Gl 382, but its effect on the data differs between the optical and nIR domains. In the optical, stellar activity creates signals at both the rotational period and its first harmonic, with the latter being more pronounced. Conversely, in the nIR, only the first harmonic is strongly present. Furthermore, if stellar activity is not well corrected, an additional signal near 50 d, likely of stellar origin, can appear in the residuals. However, this does not necessarily imply that this effect occurs exclusively in the nIR, as the optical and nIR datasets are not simultaneous. The activity could still influence both in a similar manner, especially given that the GP amplitudes are comparable. After properly correcting for stellar activity across all RV datasets, no significant signals are present and the final model can be seen in Figure 10 and the parameter values in Table 5. However, the different ways in which stellar activity manifests in Gl 382 across different wavelengths is intriguing. Although exploring the origin of the 50 d signal is beyond the scope of this study, Gl 382 could be an interesting playground for developing and testing new methods for stellar activity correction.

Fig. 6 Bayesian periodograms of various stellar activity indicators of Gl 382. From left to right and top to bottom the Ca II IRT 1, 2, 3, dLW, Hα, and NaD1 index time series from the CARMENES data; the dLW, Hα, and NaD1 index time series from the HARPS data; and the dLW, Bℓ, and dET time series from the SPIRou data. The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at 51.1 d. The significance level of log BF = 5 is indicated by the horizontal red dashed line.

Fig. 7 Iterative search of signals with Bayesian periodograms using SPIRou data of Gl 382. The top four panels display log BF periodograms for (a) raw data, (b) Wapiti correction, and (c) stellar activity correction. The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the 51 d period. The significance level of log BF = 5 is indicated by the horizontal red dashed line. The bottom left panel shows the stacked periodogram with log BF power values, where log BF values below 5 are in gray. The bottom right panel presents the variation in log BF as a function of the number of observations for the most significant period at 51 d.

Fig. 8 Iterative search of signals with Bayesian periodograms using HARPS and CARMENES data of Gl 382. The top panels display log BF periodograms for (a) raw data and (b) stellar activity correction. The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the 51 day period. The significance level of log BF = 5 is indicated by the horizontal red dashed line.

4 Conclusion

In this study, we performed a detailed RV analysis of the nearby M dwarf stars Gl 480 and Gl 382 by combining nIR data from the SPIRou spectropolarimeter with non-simultaneous optical data from the HARPS and CARMENES spectrographs. Using advanced correction techniques such as the LBL method and the Wapiti algorithm, we effectively reduced instrumental noise and stellar activity effects, thus improving the detection of planetary signals.

For Gl 480, our analysis confirms the presence of a potential planet with an orbital period of 9.5537 ± 0.0005 d and a minimum mass of 8.8 ± 0.7 M_⊕. This result provides a more precise mass estimate and differs significantly from the previously reported value of 13.2 ± 1.7 M_⊕. A 17.8 d signal also appeared in the nIR data but was linked to incomplete correction of stellar activity. This signal, observed in the dET and dLW activity indicators, was suppressed only when the GP models used priors based on stellar activity.

Additionally, a low-amplitude signal with a period of 6.4 d was detected when combining all datasets. However, the statistical significance of this signal depends critically on the choice of prior distributions in our quasi-periodic GP models. Specifically, this signal reached detection significance only when the priors were informed by the dLW stellar activity indicator. In contrast, when using non-informative (Camacho et al. 2023) or dET-based priors, the signal did not surpass the detection threshold. We also studied the non-detection of this signal in the optical using an injection-recovery test and found that in 81% of our simulations, this signal could not be detected with these data. Further high-precision RV observations would be necessary to confirm or refute the planetary nature of this candidate signal and to better constrain its orbital parameters., this signal could not be detected with these data. If confirmed, this planet would be in a near 3:2 resonance with the 9.5 d planet and its minimum mass would be of 2.7 ± 0.6 M_⊕.

For Gl 382, despite employing the same rigorous methodology, we did not find any evidence for planetary companions. The RV variations in this system are dominated by stellar activity. Based on our combined dataset, we estimated that our detection limit for a potential companion orbiting Gl 382 within its HZ corresponds to an upper mass limit of M_upper = 5.9 ± 0.8 M_⊕.

The stellar activity in Gl 382 is particularly intriguing due to the distinct ways it manifests in the optical and nIR domains: while the optical data revealed both the rotational period and its harmonic, the nIR data predominantly exhibited the first harmonic as well as a 51 d signal (which was present in both SPIRou and HARPS proxies). This wavelength-dependent behavior of activity-induced signals highlights the complexity of magnetic phenomena in M dwarfs and suggests that Gl 382 could serve as a promising laboratory for studying stellar magnetic activity and refining activity correction methods in RV analyses.

The use of GP regression models with varying prior distributions played a critical role in our analysis. The priors were carefully constrained using stellar activity indicators such as the dET and dLW time series, which provided insights into the stellar rotational periods and activity timescales. This approach was crucial in identifying and mitigating activity-related signals in nIR RVs, such as the 17.8 d signal in Gl 480 and the 51.1 d signal in Gl 382, highlighting the importance of integrating activity diagnostics into RV analyses.

In contrast, the set of uninformative priors from Camacho et al. (2023) was sufficient to correct for stellar activity in the optical datasets. A similar difference in activity correction between optical and nIR data was also reported by Cortes-Zuleta et al. (2024), who suggested that this could indicate a different underlying process driving RV variations in the nIR, possibly linked to the Zeeman effect.

An alternative approach for incorporating such diagnostics would have been to use a multi-dimensional GP, as introduced by Rajpaul et al. (2015). This method enables the simultaneous modeling of stellar activity in RV time series by modeling both the RV data and activity indicators with a GP G and its time derivative $\dot{G}$. However, multi-dimensional GPs are not yet supported within the Wapiti framework, and implementing this capability could be the object of future work.

Interestingly, for both targets, the GP amplitude is similar in the optical and nIR, a result also reported by Cortes-Zuleta et al. (2024). This suggests that, for these two stars, observing in the nIR does not necessarily provide an advantage over the optical, which is expected for this type of M dwarf. As noted by Reiners et al. (2010), the improvement in precision when moving to the nIR is most significant for M4 and later spectral types.

Additionally, in the case of the nIR RVs of Gl 480, stellar activity was not clearly apparent, meaning that we did not observe a strong signal at either the rotational period or its first harmonic, as we did with Gl 382. However, the presence of the 17.8 d signal prompted us to model stellar activity, ultimately leading to our best-fit model. Nevertheless, in the absence of a clear activity-related signal, we would still recommend modeling stellar activity rather than disregarding it, as advised by Stock et al. (2023).

This study demonstrates the power of combining nIR and optical RV data to detect and characterize exoplanets around active M dwarf stars. The complementary nature of multi-wavelength observations is key to distinguishing true planetary signals from stellar activity. However, detecting low-amplitude signals remains challenging and depends heavily on how stellar activity is modeled.

Looking forward, extended RV monitoring of Gl 480 with higher cadence and precision will be essential in order to confirm the tentative 6.4 d signal and explore the possibility of additional planetary companions. Additionally, the complex and dynamic stellar activity observed in Gl 382 provides a valuable opportunity to refine techniques for stellar activity correction, with broader implications for improving exoplanet detection across diverse stellar types.

Fig. 9 Results of the injection-recovery test assessing the detectability of a planetary signal in the dataset. The top panel shows the detection rate as a function of orbital period and semi-amplitude; the colors represent the percentage of successful detections. The red vertical lines indicate the boundaries of the habitable zone. The bottom panels display histograms of Kupper (left) and Mupper (right), representing the semi-amplitude threshold at which the detection rate reaches 100% and its corresponding minimum planetary mass.

Fig. 10 Final RV model for Gl 382. The top panel shows the raw RV data with offsets removed, plotted for the SPIRou, HARPS, and CARMENES instruments (RMS = 5.79 m.s−1). The second row illustrates the correction for stellar activity using GPs, with individual GP contributions shown for each grouped instrument. The bottom panel presents the residuals after removing all modeled signals (RMS = 1.90 m.s−1). The rightmost column contains log BF periodograms highlighting significant periods in the RV data, the significance level of 5 is indicated by the horizontal red dashed line.

Acknowledgements

This study has ben partially supported through the grant EUR TESS N°ANR-18-EURE-0018 in the framework of the Programme des Investissements d’Avenir. J.F.D. and C.M. acknowledge funding from the European Research Council (ERC) under the H2020 research & innovation program (grant agreements #740651 NewWorlds). X.D. and A.C. acknoweldge funding by the French National Research Agency in the framework of the Investissements d’Avenir program (ANR-15-IDEX-02), through the funding of the “Origin of Life” project of the Université Grenoble Alpes. E.A., C.C., N.J.C. & R.D. acknowledge the financial support of the FRQ-NT through the Centre de recherche en astrophysique du Québec. E.M. acknowledges funding from FAPEMIG under project number APQ-02493-22 and research productivity grant number 309829/2022-4 awarded by the CNP, Brazil. J.G.S. acknowledges the support from FCT through the grant 2022.04416.PTDC. This paper makes use of data from the first public release of the WASP data (Butters et al. 2010) as provided by the WASP consortium and services at the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This research made use of the following software tools: NumPy; a fundamental package for scientific computing with Python. pandas; a library providing easy-to-use data structures and data analysis tools. Astropy; a community-developed core Python package for Astronomy. SciPy; a library for scientific computing and technical computing. RadVel; a python package for modeling radial velocity data. matplotlib; a plotting library for the Python programming language. tqdm; a library for creating progress bars in the command line. seaborn; a data visualization library based on matplotlib. wpca; a python package for weighted principal component analysis. emcee; a python package for MCMC sampling. PyAstronomy; a collection of astronomical tools for Python. celerite; a fast Gaussian process library in Python. george: a module for Gaussian process regression in Python.

Appendix A Supplementary material regarding the MCMC analysis

A.1 Gl 480 RV analysis

In Figures A.1 and A.2, we present the time series of the stellar activity indicators dET and dLW, respectively, along with their fitted GPs.

Fig. A.1 Top panel: the dET time series for GL 480, and the fitted GP model overlaid in red. Middle panel: Residuals after subtracting the GP model from the observations. Bottom panel: Bayesian periodograms for the activity signal (black curve) and residuals (red curve) plotted as a function of period (in days). The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the one-third rotation period (Prot/3). The significance level of log BF = 5 is indicated by the horizontal red dashed line.

Fig. A.2 Top panel: dLW time series for GL 480, and the fitted GP model overlaid in red. Middle panel: Residuals after subtracting the GP model from the observations. Bottom panel: Bayesian periodograms for the activity signal (black curve) and residuals (red curve), plotted as a function of period (in days). The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the one-third rotation period (Prot/3). The significance level of log BF = 5 is indicated by the horizontal red dashed line.

The corresponding corner plots, derived from the MCMC analysis of the dET and dLW time series, are shown in Figures A.3 and A.4.

Figure A.5 compares the posterior distributions of the semi-amplitude K and the orbital period P of Gl 480 across different datasets. Additionally, the posterior distribution of the rotational period retrieved from the SPIRou dataset is shown, using the posterior distribution of the same hyper-parameters from the dLW analysis as a prior. This distribution is compared to those derived from the activity indicators dLW and dET, highlighting the consistency and differences among the datasets.

Fig. A.3 Corner plot displaying the results of MCMC analysis on the dET time series of Gl 480.

The final corner plot from the MCMC analysis of the RV model is divided for clarity across Figures A.6, A.7, and A.8. These figures respectively display the posterior distributions of the instrumental and Wapiti parameters, the orbital parameters of the two planets, and the hyper-parameters for the two GPs applied to the SPIRou and optical datasets.

Fig. A.4 Corner plot displaying the results of MCMC analysis on the dLW time series of Gl 480.

Fig. A.5 Posterior distributions of the key parameters derived from the MCMC analysis for GL 480. The left panel shows the posterior distribution of the semi-amplitude K for the combined dataset (black), optical data (blue), SPIRou data (red), and the reference distribution from Feng et al. (2020). The middle panel presents the posterior distributions of the orbital period P, using the same color scheme. The right panel displays the posterior distributions of the stellar rotation period Prot derived from the dET (blue) and dLW (black) time series, and the SPIRou dataset (red). All histograms are normalized to unit area for comparison.

Fig. A.6 Corner plot displaying the results for the offsets, the Wapiti systematics and the jitters of the MCMC analysis on the RV time series of Gl 480.

Fig. A.7 Corner plot displaying the results for the orbital parameters of the MCMC analysis on the RV time series of Gl 480.

Fig. A.8 Corner plot displaying the results for the offsets, the Wapiti systematics and the jitters of the MCMC analysis on the RV time series of Gl 480.

A.2 Gl 382 RV analysis

In Figures A.9, we present the time series of the stellar activity indicators dET along with its fitted GPs. The corresponding corner plots, derived from the MCMC analysis of the dET and dLW time series, are shown in Figures A.10.

The final corner plot from the MCMC analysis of the RV model is divided for clarity across Figures A.11, and A.12. These figures respectively display the posterior distributions of the instrumental and Wapiti parameter, and the hyper-parameters for the two GPs applied to the SPIRou and optical datasets.

Fig. A.9 Top panel: dET time series for GL 382, and the fitted GP model overlaid in red. Middle panel: Residuals after subtracting the GP model from the observations. Bottom panel: Bayesian periodograms for the activity signal (black curve) and residuals (red curve), plotted as a function of period (in days). The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the 51 d period value. The significance level of log BF = 5 is indicated by the horizontal red dashed line.

Fig. A.10 Corner plot displaying the results of MCMC analysis on the dET time series of Gl 382.

Fig. A.11 Corner plot displaying the results for the offsets, the Wapiti systematics and the jitters of the MCMC analysis on the RV time series of Gl 382.

Fig. A.12 Corner plot displaying the results for the GP hyper-parameters applied on the SPIRou and optical data of the MCMC analysis on the RV time series of Gl 382.

References

All Tables

All Figures

Fig. 1 Bayesian periodograms of various stellar activity indicators of Gl 480. From left to right and top to bottom the dLW, Hα, and NaD1 index time series from the HARPS data, and the dLW, Bℓ and dET time series from the SPIRou data. The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the one-third rotation period (Prot/3). The significance level of log BF = 5 is indicated by the horizontal red dashed line.

In the text

Fig. 2 Iterative search of signals with Bayesian periodograms using SPIRou data of Gl 480. The top four panels display log BF periodograms for (a) raw data, (b) Wapiti correction, (c) Planet b model fit, and (d) long-term and/or stellar activity correction. The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the one-third rotation period (Prot/3). The significance level of log BF = 5 is indicated by the horizontal red dashed line. The bottom left panel shows the stacked periodogram with log BF power values, where log BF values below 5 are in gray. The bottom right panel presents the variation in log BF as a function of the number of observations for the most significant period at 17.8 d.

In the text

Fig. 3 Iterative search of signals with Bayesian periodograms using HARPS and CARMENES data of Gl 480. The three panels display log BF periodograms for (a) the raw data, (b) the Planet b model fit, and (c) the stellar activity correction. The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the one-third rotation period (Prot/3). The significance level of log BF = 5 is indicated by the horizontal red dashed line.

In the text

	Fig. 4 Histogram of the maximum log BF values of the injected period in the injection-recovery simulations. The solid red line indicates the log BF value of 1.8 retrieved from the residuals of the real datasets, while the dashed red line marks the detection threshold at log BF = 5.
In the text

Fig. 5 Final RV model for Gl 480. The top panel shows the raw RV data with offsets removed, plotted for the SPIRou, HARPS, and CARMENES instruments (RMS = 5.92 m.s−1). The second and fourth rows display the contributions of planets 1 and 2 to the RV signal, respectively, along with their phase-folded RV curves. The third row illustrates the correction for stellar activity using GPs, with individual GP contributions shown for each grouped instrument. The bottom panel presents the residuals after removing all modeled signals (RMS = 2.04 m.s−1). The rightmost column contains the log BF periodograms highlighting significant periods in the RV data, the significance level of 5 is indicated by the horizontal red dashed line.

In the text

Fig. 6 Bayesian periodograms of various stellar activity indicators of Gl 382. From left to right and top to bottom the Ca II IRT 1, 2, 3, dLW, Hα, and NaD1 index time series from the CARMENES data; the dLW, Hα, and NaD1 index time series from the HARPS data; and the dLW, Bℓ, and dET time series from the SPIRou data. The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at 51.1 d. The significance level of log BF = 5 is indicated by the horizontal red dashed line.

In the text

Fig. 7 Iterative search of signals with Bayesian periodograms using SPIRou data of Gl 382. The top four panels display log BF periodograms for (a) raw data, (b) Wapiti correction, and (c) stellar activity correction. The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the 51 d period. The significance level of log BF = 5 is indicated by the horizontal red dashed line. The bottom left panel shows the stacked periodogram with log BF power values, where log BF values below 5 are in gray. The bottom right panel presents the variation in log BF as a function of the number of observations for the most significant period at 51 d.

In the text

Fig. 8 Iterative search of signals with Bayesian periodograms using HARPS and CARMENES data of Gl 382. The top panels display log BF periodograms for (a) raw data and (b) stellar activity correction. The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the 51 day period. The significance level of log BF = 5 is indicated by the horizontal red dashed line.

In the text

Fig. 9 Results of the injection-recovery test assessing the detectability of a planetary signal in the dataset. The top panel shows the detection rate as a function of orbital period and semi-amplitude; the colors represent the percentage of successful detections. The red vertical lines indicate the boundaries of the habitable zone. The bottom panels display histograms of Kupper (left) and Mupper (right), representing the semi-amplitude threshold at which the detection rate reaches 100% and its corresponding minimum planetary mass.

In the text

Fig. 10 Final RV model for Gl 382. The top panel shows the raw RV data with offsets removed, plotted for the SPIRou, HARPS, and CARMENES instruments (RMS = 5.79 m.s−1). The second row illustrates the correction for stellar activity using GPs, with individual GP contributions shown for each grouped instrument. The bottom panel presents the residuals after removing all modeled signals (RMS = 1.90 m.s−1). The rightmost column contains log BF periodograms highlighting significant periods in the RV data, the significance level of 5 is indicated by the horizontal red dashed line.

In the text

Fig. A.1 Top panel: the dET time series for GL 480, and the fitted GP model overlaid in red. Middle panel: Residuals after subtracting the GP model from the observations. Bottom panel: Bayesian periodograms for the activity signal (black curve) and residuals (red curve) plotted as a function of period (in days). The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the one-third rotation period (Prot/3). The significance level of log BF = 5 is indicated by the horizontal red dashed line.

In the text

Fig. A.2 Top panel: dLW time series for GL 480, and the fitted GP model overlaid in red. Middle panel: Residuals after subtracting the GP model from the observations. Bottom panel: Bayesian periodograms for the activity signal (black curve) and residuals (red curve), plotted as a function of period (in days). The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the one-third rotation period (Prot/3). The significance level of log BF = 5 is indicated by the horizontal red dashed line.

In the text

	Fig. A.3 Corner plot displaying the results of MCMC analysis on the dET time series of Gl 480.
In the text

	Fig. A.4 Corner plot displaying the results of MCMC analysis on the dLW time series of Gl 480.
In the text

Fig. A.5 Posterior distributions of the key parameters derived from the MCMC analysis for GL 480. The left panel shows the posterior distribution of the semi-amplitude K for the combined dataset (black), optical data (blue), SPIRou data (red), and the reference distribution from Feng et al. (2020). The middle panel presents the posterior distributions of the orbital period P, using the same color scheme. The right panel displays the posterior distributions of the stellar rotation period Prot derived from the dET (blue) and dLW (black) time series, and the SPIRou dataset (red). All histograms are normalized to unit area for comparison.

In the text

	Fig. A.6 Corner plot displaying the results for the offsets, the Wapiti systematics and the jitters of the MCMC analysis on the RV time series of Gl 480.
In the text

	Fig. A.7 Corner plot displaying the results for the orbital parameters of the MCMC analysis on the RV time series of Gl 480.
In the text

	Fig. A.8 Corner plot displaying the results for the offsets, the Wapiti systematics and the jitters of the MCMC analysis on the RV time series of Gl 480.
In the text

Fig. A.9 Top panel: dET time series for GL 382, and the fitted GP model overlaid in red. Middle panel: Residuals after subtracting the GP model from the observations. Bottom panel: Bayesian periodograms for the activity signal (black curve) and residuals (red curve), plotted as a function of period (in days). The dashed green lines in the bottom panel mark periodicities at the stellar rotation period (Prot), at the one-half rotation period (Prot/2), and at the 51 d period value. The significance level of log BF = 5 is indicated by the horizontal red dashed line.

In the text

	Fig. A.10 Corner plot displaying the results of MCMC analysis on the dET time series of Gl 382.
In the text

	Fig. A.11 Corner plot displaying the results for the offsets, the Wapiti systematics and the jitters of the MCMC analysis on the RV time series of Gl 382.
In the text

	Fig. A.12 Corner plot displaying the results for the GP hyper-parameters applied on the SPIRou and optical data of the MCMC analysis on the RV time series of Gl 382.
In the text