Figure 11. PSF photometry relative light curve (TOI-1452 to TIC 420112587 flux ratio) from OMM-PESTO on 2020 February 22. The dotted black lines represent the ingress and egress of the transit estimated from contemporaneous TESS sector 22 data. The light curve is normalized with the out-of-transit median. A flux deficit comparable to the TESS transit depth is detected on TOI-1452 during transit.
The intent here was not to produce a precise uncontaminated light curve but rather to detect any flux deficit (or excess) that would indicate from which star the transit originates. We thus inspected the TOI-1452 to TIC 420112587 flux ratio as a function of time, normalized to unity outside of transit. The resulting relative light curve is presented in Figure 11 and shows a flux deficit on TOI-1452 during transit. We did not fit a transit model on this light curve, as it is less precise than the one obtained using a combined circular aperture (Figure 3 ). We nonetheless measure a mean relative flux deficit of 2.33 ± 0.43 ppt, which is an approximation of the uncontaminated transit depth. This flux deficit is comparable in amplitude to the diluted corrected TESS depth (3.31 ± 0.19 ppt) and was detected with a confidence level sufficiently high (>5σ) to conclude that TOI-1452 was the source of the transit and justify an RV monitoring campaign on this star, starting with SPIRou in June 2020. Later, the MuSCAT3 photometry was able to resolve TOI-1452 and TIC 420112587 and unambiguously identify that the former star hosts a transiting object.
The objective of the first OMM-PESTO transit follow-up was to establish the origin of the TESS signal, particularly between the target (TOI-1452) and its companion (TIC 420112587). Standard aperture photometry ruled out any NEB in the FOV but was unable to isolate the transit between the two stars, as they were only partially resolved. We therefore had to rely on a different method using PSF fitting to extract the relative flux of both stars. We used the photutils (Bradley et al. 2020 ) package to perform the DAOPHOT (Stetson 1987 ) PSF photometry algorithm. This was achieved by fitting the PSFs with an effective PSF model (ePSF) generated in photutils using the six stars with the highest SNR in the FOV (excluding our targets), then integrating the best-fit models over pixels containing the stars' signal.
It is beyond the scope of this study to assess the exact cause of this strong and persistent signal, but we showed earlier that the SPIRou magnetic field constraints of TOI-1452 are inconsistent with a fast rotator and active object. Moreover, a 1.9 days rotation period for TOI-1452 would correspond to a of ∼7 km s −1 , readily detectable in the SPIRou combined spectrum. Instead, the mean line profile FWHM measured from the cross-correlation function (CCF) calculated in APERO suggests a slow rotator (i.e., km s −1 ). We repeated this step for the companion star from the single visit with SPIRou and also measured a FWHM consistent with km s −1 . Thus, the rotation of the companion star is also most probably not causing this photometric signal.
Our PDCSAP GP model consisted of the five hyperparameters above, plus an excess white noise term s. We sampled the posterior distributions of the parameters in their logarithmic form , , , , , using the Markov chain Monte Carlo (MCMC) package emcee (Foreman-Mackey et al. 2013 ) and a Bayesian formalism. We employed 100 walkers and performed 100,000 steps with a burn-in of 10,000. The number of steps was greater than 50 times the autocorrelation timescale for each parameter, which usually indicates a sufficient level of convergence (Sokal 1997 ; Foreman-Mackey et al. 2019 ). The adopted prior distributions and the posteriors median, 16th and 84th percentiles are reported in Table 6 . The resulting mean GP prediction is shown in Figures 2 and A1 superimposed on the original PDCSAP cadence. Even though the sinusoidal variations visually appear to repeat every ∼0.93 day, our model converged to a very well constrained primary oscillation of 1.8680 ± 0.0004 days, thus indicating significant power at the second harmonic.
where σ 1 , σ 2 are the standard deviations (amplitudes) of the primary and secondary modes, τ 1 , τ 2 are the damping timescales of the primary and secondary oscillations, and P GP is the undamped period of the primary mode. Note that these parameters differ slightly from the default RotationTerm kernel parameterization by making no assumptions on the relative amplitudes and quality factors between the two modes.
The GP regression was done with celerite2 (Foreman-Mackey et al. 2017 ; Foreman-Mackey 2018 ). We selected its RotationTerm kernel because it was specifically designed to model a range of quasiperiodic variability, from stellar rotation to pulsations. This kernel is the sum of two stochastically driven, damped harmonic oscillator (SHO) terms ( SHOTerm ) capturing both primary (P GP ) and secondary (P GP /2) modes in Fourier space. The Fourier transform of the covariance function, known as the power spectral density (PSD), takes the following form:
We started by masking the epochs of transit and removing outliers from the PDCSAP light curve with sigma clipping. It was determined that a 3.5σ clipping was robust enough to remove both obvious outliers and stellar flares. This sigma clipping removed less than 0.2% of the out-of-transit data. Parts of sectors 21 and 47 coinciding with TESS momentum dump events show large amplitude variations; those were considered to be nonastrophysical and were manually rejected. We also rejected data points in sectors 40 and 41, as they are isolated and have a median considerably different than unity. The data not considered in this analysis are either displayed in blue (transits) or in red (rejected) in Figures 2 and A1 . The cleaned out-of-transit PDCSAP data set was too large (N = 237 634) to be efficiently modeled with a GP. We therefore binned the data and instead used the corresponding 1 hour effective cadence light curve (N = 7924).
The TOI-1452 PDCSAP light curve (Figures 2 and A1 ) features stellar flares with amplitude of a few percents and ppt-level sinusoidal variations. A strong peak at 0.93 days appears in the Lomb–Scargle periodogram of the multiyear light curve, as well as in all individual sectors. However, computing the autocorrelation function, which is more reliable for accurate photometric rotation period determination (McQuillan et al. 2013 ), would often find a period of 1.9 days (2 × 0.93 days) depending on the sector. As the PDCSAP data are corrected for systematic trends, it is unlikely that such corrections significantly perturb those short-term flares and sinusoids. Regardless of the origin of these signals (TOI-1452, TIC 420112587, or any contaminating star), it is crucial to remove the periodic variations to accurately measure the transit parameters. To accomplish this, we adopted a sequential approach where we first correct the PDCSAP data using a Gaussian process (GP), then fit the 32 corrected transits with a model. The details of the GP regression are presented below, while the transit modeling is described in Section 4.3 .
In order to constrain the physical and orbital parameters of TOI-1452b, we conducted a joint analysis of the transits (TESS, OMM-PESTO, and MuSCAT3) and the RV data (SPIRou and IRD). The joint fit was performed using the juliet (Espinoza et al. 2019) package, which utilizes batman (Kreidberg 2015) to generate transit models and radvel (Fulton et al. 2018) to compute Keplerian RV models. The juliet framework implements nested sampling algorithms to sample posterior distributions, while also enabling model comparison via evaluations of the Bayesian log-evidence ( ). We chose the dynesty (Speagle 2020) dynamic nested sampling option in juliet . Standard nested sampling (Skilling 2006) was designed to estimate evidences, not posteriors, and thus struggles with parameter estimation for complex distributions. Dynamic nested sampling (Higson et al. 2019), on the other hand, adapts the number of live points based on the structure of the posteriors, providing parameter estimation comparable to MCMC algorithms.
The transit and RV components of the joint fit have four parameters in common: the orbital period P, the time of inferior conjunction t 0 , the eccentricity e, and the argument of periastron ω. For the transit modeling, we followed the parameterization from Espinoza (2018) of the impact parameter b and the planet-to-star radius ratio p = R p /R ⋆ to efficiently sample physically plausible values in the (b, p) space. Instead of fitting the scaled semimajor axis a/R ⋆ , we used the stellar density ρ ⋆ parameterization available in juliet . Fitting ρ ⋆ takes into account any prior information on the stellar mass and radius. We adopted a Gaussian prior on ρ ⋆ using the value and uncertainty in Table 3. Stellar limb-darkening effects in TESS, OMM-PESTO, and MuSCAT3 transits were modeled using the quadratic q 1 and q 2 parameters defined in Kipping (2013). For each instrument, we included in juliet a flux dilution factor D, a baseline flux M, and an extra jitter term σ. We set D TESS to 1 (no dilution), as the PDCSAP data are already corrected for crowding effects. The OMM-PESTO light curve combines the flux of TOI-1452 and TIC 420112587, which requires an adequate D PESTO factor to compensate for contamination. We thus constructed a Gaussian prior on D PESTO with a mean value calculated with Equation (6) of Espinoza et al. (2019) and flux ratio derived from TOI-1452 and TIC 420112587 magnitudes in the i band (from Tables 3 and 4). The adopted prior on D PESTO was , that is, with a 10% standard deviation to account for errors on the magnitudes and deviations between i and . We also explored fixing D PESTO to 0.564, while letting D TESS vary freely between 0 and 2. Both approaches yielded a consistent measurement of the planetary radius (within 1σ), indicating that the PDCSAP fluxes were in all likelihood properly corrected for contamination. The dilution in the MuSCAT3 light curves was a priori unknown. However, it is expected that the g' transit was more affected by dilution, as the seeing was worse for this filter (see Table 2). We adopted a conservative approach where a different D MuSCAT3 is applied for each filter, with uniform priors between 0.5 (twice the flux) and 1.
The parameters specific to the RV Keplerian component were the semi-amplitude K, per-instrument offsets γ, and extra white noise terms σ. We explored adding a global GP to model common stellar activity signal in the SPIRou and IRD data. For this, we used the GP implementation in juliet that runs celerite (Foreman-Mackey et al. 2017). We chose the Matérn-3/2 approximation kernel, which takes the following form:
where τ = ∣t i − t j ∣ is the time interval between data points i and j, A GP is the amplitude of the GP, , with ℓ GP the timescale of the GP, and is set to 0.01 (when → 0, k i,j converges to a Matérn-3/2 kernel). We did not fit a per-instrument A GP and ℓ GP due to the limited number of RV measurements from IRD. We also considered choosing a quasiperiodic kernel in celerite instead (Equation (56) of Foreman-Mackey et al. 2017). As no clear periodicity was detected in the B ℓ time series, or other activity indicators from the LBL such as the dLW metric (Zechmeister et al. 2018) or chromatic velocity slope changes, we applied a uniform prior on the stellar rotation period, namely, days. We found that the Matérn-3/2 kernel gave equivalent results with fewer hyperparameters needed (two instead of four) and that the quasiperiodic GP did not converge to a specific rotation period, showing no preference for a period of 0.93 days (or 2 ×0.93 days) as seen in TESS photometry. This is another indication that the sinusoidal signal in the out-of-transit PDCSAP data is probably not associated with TOI-1452 stellar activity.
We examined the change in Bayesian log-evidence for a suite of joint models ( ), all having an identical transit component. The "zero"-planet model ( ) has a K fixed to 0 m s−1, with only the RV offsets and extra white noise terms allowed to vary. This model tests whether the RV dispersion can be fully explained by white noise only, without questioning the transit detection. Single-planet models can either be with circular ( e = 0, ω = 90°) or eccentric orbits ( free e, ω). Two additional models include a global RV activity GP ( and ). To objectively assess the contribution from the IRD observations, we decided to apply this framework first on the SPIRou data individually and then using the full RV data set (SPIRou + IRD).
For two competing models, the difference in log-evidence ( ) informs on the probability that one model supports the data better than the other. To interpret the significance of the and select the "best" model, we followed the empirical scale introduced in Table 1 of Trotta (2008). A translates into "strong" evidence toward the model with the highest . A corresponds to "moderate" evidence, while shows "weak" evidence at best, i.e., neither model should be favored in that case.
Figure 12 shows the Bayesian log-evidence for different joint models and data sets. Note that the typical errors on the computed by dynesty were 0.5, so that the presented in Figure 12 have associated uncertainties of 0.7. We first observe that all planetary models are strongly favored ( ) compared to the "zero"-planet solution ( ), providing quantitative evidence that the TOI-1452b Keplerian signal is detected in velocimetry, in phase with transit. There is also compelling evidence for models with an RV activity GP ( and ), increasing by approximately 10 relative to and . However, considering only the SPIRou data yields similar or slightly larger for all compared to joint fits that include the seven IRD RVs. This suggests that the IRD observations do not significantly contribute to improve the Keplerian solution for TOI-1452b. The median RV uncertainty from IRD (4.03 m s−1) is nearly identical to SPIRou (4.00 m s−1), but the point-to-point scatter (rms) is much larger: 12.71 m s−1 for IRD and 5.76 m s−1 for SPIRou. The models fail to capture the extra rms in the IRD data and are instead converging to solutions with white noise term comparable to the overall scatter (see Table B1). This is apparent in Figure 13 showing the RV component of the joint fit (model ) using the full data set, with each instrument having their original error bar plotted. The IRD radial velocities were produced using the template spectrum of TOI-1452. As previously mentioned, this star has a small BERV excursion, which is not ideal for filtering out telluric lines. This may explain the increased dispersion in the resulting RVs, in this case, at a level much larger than the Keplerian signal. For this reason, we opted to present below the results using only the SPIRou RVs. We nonetheless provide all the relevant parameters of the RV modeling for the SPIRou only and SPIRou+IRD data sets in Table B1.
Figure 12. Bayesian log-evidence ( ) for different joint transit RV models ( ) and RV data sets. The typical uncertainty on the is 0.7. The "zero"-planet model has a fixed K = 0 m s−1. Single-planet models and correspond to an RV component with a circular (e = 0, ω = 90°) or eccentric (free e, ω) orbit, respectively. Models and add a Gaussian process to remove correlated noise in the RV data. Download figure: Standard image High-resolution image
Figure 13. RV time series from SPIRou and IRD with the best-fit Keplerian + activity GP (orange curve), activity GP-only (red dashed curve), and Keplerian-only (blue curve) models overplotted. The residuals below show an overall agreement between the SPIRou errors and the RV dispersion, which is difficult to assess for IRD given the limited number of measurements. Download figure: Standard image High-resolution image
The eccentric model produced the highest (Figure 12) but with a Bayesian evidence indistinguishable from the circular model ( ). We report an eccentricity of , with e < 0.32 at a 95% confidence, but argue that the simpler, circular model should be preferred at this point. The adopted priors and resulting posteriors of the fit are summarized in Table 7. The MuSCAT3 photometric parameters are given in another table (Table 8) to facilitate comparison between filters. We measure dilution factors D for MuSCAT3 consistent with no dilution for the i' transit, with a moderate level of contamination (∼30%) in the band. If we assume instead that the flux dilution was exactly zero for all filters, the uncorrected transit depths (δ uncorr. ) presented in Table 8 show no sign of strong chromaticity. The best-fit transit models of the TESS, OMM-PESTO, and MuSCAT3 photometry are shown in Figures 2, 3, and 4, respectively. The best circular ( ) and eccentric ( ) RV orbital fits of TOI-1452b are shown in Figure 14 in a phase-folded format.
Figure 14. Phase-folded SPIRou RV curve, with systemic velocity and activity GP removed. Binned RV measurements (0.1 phase bin) are marked with black circles. The best-fit Keplerian models (blue) are depicted with a solid curve (circular orbit) and a dashed curve (eccentric orbit). The residuals of the circular fit are presented below. The circular solution yields smaller residuals rms (3.58 m s−1) compared to the eccentric model (3.66 m s−1); yet both Bayesian evidences are indistinguishable (see Figure 12). Download figure: Standard image High-resolution image
Table 7. Prior and Posterior Distributions of the Joint Transit RV Fit for Model (Details in Section 4.3) Using Only the SPIRou Radial Velocities Parameter Prior a Posterior Description Fitted parameters ρ ⋆ (g cm−3) Stellar density P (days) 11.06201 ± 0.00002 Orbital period t 0 (BJD - 2 457 000) 1691.5321 ± 0.0015 Time of inferior conjunction r 1 0.46 ± 0.08 Parameterization b for R p /R ⋆ and b r 2 0.0555 ± 0.0014 Parameterization b for R p /R ⋆ and b K (m s−1) 3.50 ± 0.94 RV semi-amplitude e 0 (fixed) 0 Orbital eccentricity ω (°) 90 (fixed) 90 Argument of periastron A GP (ms−1) Amplitude of the GP ℓ GP (days) Timescale of the GP γ SPIRou (m s−1) −33985 ± 2 SPIRou RV systemic component σ SPIRou (m s−1) 2.3 ± 1.3 SPIRou RV extra white noise c q 1,TESS TESS limb-darkening parameter d q 2,TESS TESS limb-darkening parameter d D TESS 1.0 (fixed) 1.0 TESS dilution factor M TESS -0.00032 ± 0.00010 TESS baseline flux σ TESS (ppm) TESS extra white noise q 1,PESTO PESTO limb-darkening parameter d q 2,PESTO PESTO limb-darkening parameter d D PESTO 0.586 ± 0.040 PESTO dilution factor M PESTO 0.00013 ± 0.00018 PESTO baseline flux σ PESTO (ppm) 2287 ± 67 PESTO extra white noise ⋮ MuSCAT3 photometric parameters in Table 8 Derived parameters R p (R ⊕ ) ⋯ 1.672 ± 0.071 Planetary radius M p (M ⊕ ) ⋯ 4.82 ± 1.30 Planetary mass ρ (g cm−3) ⋯ Planetary bulk density a (au) ⋯ 0.061 ± 0.003 Orbital semimajor axis ⋯ 3.09 ± 0.16 Transit depth b ⋯ 0.19 ± 0.13 Transit impact parameter i (°) ⋯ 89.77 ± 0.16 Orbital inclination T eq (K) Equilibrium temperature ⋯ 326 ± 7 ⋯ 298 ± 6 ⋯ 226 ± 5 S (S ⊕ ) ⋯ 1.8 ± 0.2 Insolation Notes. a 2. b Parameterization from Espinoza ( c White noise term for single exposures within polarimetric sequences. d {q 1 , q 2 } are linked to the quadratic limb-darkening coefficients {u 1 , u 2 } through the transformations outlined in Kipping ( is the uniform distribution between values a and b.is the log-uniform (Jeffreys) distribution between values a and b.is the normal distribution with mean μ and variance σParameterization from Espinoza ( 2018 ). White noise term for single exposures within polarimetric sequences. {q, q} are linked to the quadratic limb-darkening coefficients {u, u} through the transformations outlined in Kipping ( 2013 ). Download table as: ASCIITypeset image