1 Introduction↩︎

Following its launch in 2018, the Transiting Exoplanet Survey Satellite (TESS) mission [1] has discovered many thousands of new exoplanet candidates. As per its mission mandate, most of these targets orbit stars bright enough for detailed follow up characterization. While TESS’s nearly full-sky coverage and bright object target list are undeniably valuable qualities, they do come with drawbacks. One of the most notable is its limited temporal coverage of a given patch of sky. TESS’s observational strategy has been to shift its viewing angle every 25–30 days to a new sector. A portion of the sky experiences field overlap between sectors, but a large fraction (\(\sim\)​63%) receives only month-long baseline coverage. This is obviously detrimental for the detection of planets with orbital periods longer than \(\sim\)​30 days. At best, TESS might catch one single transit in these regions before moving on to the next sector. Returning to the field in subsequent sectors can help, but does not guarantee another transit detection. Even catching a second transit detection typically leaves many possibilities for the orbital period depending on how many transits may have occurred during the unobserved time interval [2]. Without knowledge of the period, certain intrinsic system parameters remain unobtainable or strongly correlated. In particular, the semi-major axis and the period (both of which affect transit duration) are largely degenerate. As such, determination of stellar irradiation is unavailable. Attempted measurements of eccentricity and the argument of pericentre are also mostly uninformative. Lacking a clear picture of the orbital structure makes quantifying the system quite challenging. In addition, without strong constraints on the period via multiple transit detections or extensive radial velocity (RV) follow up, scheduling any sort of additional transit-based observations (e.g., transmission/emission spectroscopy, Rossiter-McLaughlin effect, etc.) becomes nearly impossible.

This is unfortunately the fate of most long-period single-transiting planet candidates in the TESS catalogue. To date, more than 98% of the 6000+ TESS Objects of Interest (TOIs) with known periods are on orbits shorter than \(50\) days (Exoplanet follow up Observing Program; doi:), and many of those that have longer reported orbits are poorly characterized and require further verification. Even if a rough period estimate can be established with RV measurements, the timing uncertainty of future transits grows with each subsequently unobserved transit (\(\sigma_{T_n} \propto n\sigma_P\), where \(n\) is the number of transits since the period uncertainty, \(\sigma _P\), was calculated). Generally, multiple transit observations are needed to provide tight constraints on the period and keep future timing uncertainties small.

Both transit and RV detection methods suffer observation and detection biases against long-period planets. In transit surveys, such planets require much longer baseline to capture sufficient events [3]. With finite data sets, their folded multi-transit signal-to-noise ratio (SNR) builds more slowly, making shallower transit events especially hard to detect. Their wide orbital geometries naturally lead to lower transit probabilities, reducing the number of expected events in a given search sample. RV measurements, which are complementary to transit observations, are also hindered by wider orbits. The signal amplitude shrinks and it takes longer to cover a full orbit.

Despite and because of these challenges, there is real value in improving our catalogue of longer-period planets with their cooler equilibrium temperatures [4]. Due to these biases and accompanying investment required to study them, long-period planets tend to fall by the wayside and become underrepresented in exoplanet catalogues. Orbital periods of about 50 days mark a notable boundary in our confirmed planet databases. Given that every planet in our own solar system orbits with a period \(>50\) days, the restriction to our known exoplanet population is quite staggering. By confirming and cataloguing these wide-orbit planets we build up our understanding of the physical and orbital characteristics within this sparsely measured population.

Though they are few in number, our solar system giants provide detailed data on large cold planets, even allowing in-situ measurements [5]. Hot, giant, transiting exoplanets also comprise a high-quality data set due to their large sample size and relative ease of detection. Temperate transiting planets in between these extremes require particular effort to observe due to their adverse observational biases. However, diligent confirmation studies can still accomplish precise measurements of radius, mass, and orbital structure. These studies provide information to better understand the long-period planets as individuals, and as a population.

Another advantage of these cool giants is that they are more directly comparable to our well-studied cold solar system giants in that they do not exhibit the hot-Jupiter radius anomaly [6], [7]. The relative simplicity of modeling them acts as a valuable control group for understanding hot-Jupiter inflation. Though RV surveys have measured masses for many cool/cold (\(T_{\rm eq} \lesssim 500\) \(K\)) planets, few of them exhibit transits and few of those have reasonably bright hosts, severely limiting their potential for atmospheric characterization. It is a challenge to create generalized chemical or structural atmospheric models that can span the broad temperature range of the giant planet population without having a solid testing ground in the intermediate range [8]. Chasing down the longest-period targets in the TESS sample helps bridge this gap [9]. In terms of atmospheric chemistry, the cooler atmospheres may contain disequilibrium by-products that would serve as valuable probes of atmospheric physics [4]. Spectroscopic endeavours can use these lower-insolation targets to tease apart the composition transition between the very cold and very hot giant planet atmospheres.

Stellar insolation can also have many complex effects on a planet’s atmosphere. The question of X-ray and ultraviolet (XUV)-driven mass loss frequently arises in the context of the super-Earth and sub-Neptune populations [10]–[12]. Irradiation levels are also important for general structure and evolution models as well as atmospheric circulation and photochemistry in cool planets [13]. Insolation ought to push the radiative–convective boundary deeper, but also seems to drive the radius anomaly which pushes the boundary back up along with the planet radius [14]. Given the inverse-square law of stellar irradiation, wider-orbit planets will be significantly less affected by XUV-driven mass loss, preserving more of their primordial composition. Building a sample of planets with reduced insolation will help with the creation of more broadly applicable planetary models.

Long-period planets also provide test cases for system dynamics. Models describing the formation and migration processes thought to be responsible for the hot-Jupiter population are generally of two categories: protoplanetary disk torques [15]–[18] or high-eccentricity migration [19]–[22]. With the end products often being very similar, knowledge of intermediate-separation transiting planets, their companions, and their environments will help distinguish between these types of models. Transit and RV surveys are always biased against long-period planets, so filling out the eccentricity distribution of long-period giants still requires additional effort [23]. Obliquity measurements (using either the RM effect or doppler tomography) have also been done almost exclusively on short-period giants thus far.

TOI-2010 (details in Table [tab:host95info]) was flagged as containing a transiting planet candidate after a single transit was detected on 2019 August 16. The transiting body, designated TOI-2010.01, did not exhibit a retransit in the remainder of TESS’s primary mission. In this manuscript we confirm and characterize its planetary nature, and so we will hereafter refer to it by the designation TOI-2010 b in accordance with standard planetary nomenclature. Its initial single-transit status inspired an intensive RV campaign which constrained the period well enough to make feasible a photometric search for a second transit. While both of these efforts were successful, a much later TESS sector fortuitously revealed an additional transit during the late stages of this manuscript preparation, confirming our findings and providing even tighter parameter determination.

We present the various data and observations that contributed to this planet confirmation in Section 2. We then describe the various analyses carried out to characterize the star, planet, and system as a whole in Section 3. Sections 4 and 5 contain descriptions and discussion of our findings, respectively. Finally, a brief summary of the entire study is presented in Section 6.

2 Data and Observations↩︎

Numerous observations of this planet candidate were made by the TESS follow up Observing Program (TFOP). Some of them are overlapping in their coverage or scope, and many were intermediate steps of target validation used to green-light more intensive observations. We list all the contributions in the subsections below for completeness and recognition, but note that not every data set is included in the analysis that follows.

2.1 Discovery and Sky-Monitoring Photometry↩︎

2.1.1 TESS↩︎

In the early stages of the preparation of this manuscript, TOI-2010 had been observed at 2 minute cadence for Sectors 14, 15, 16, and 40 and the image data were reduced and analyzed by the Science Processing Operations Center [24] at NASA Ames Research Center. A single transit was detected at the beginning of Sector 15 using an adaptive, wavelet-based matched filter [25]–[27] on 2019 September 21. The signal was also independently discovered by the Visual Survey Group [28] around the same time and forwarded to the attention of the TESS Single Transit Planet Candidate (TSTPC) working group for follow up. It was alerted by the TESS Science Office as a Community TOI (CTOI) on 17 June 2020. This single transit event, though unambiguous due to its high SNR, posed a validation challenge due to its lack of period constraint.

During the late stages of manuscript preparation, TESS reobserved TOI-2010 in Sectors 54, 55, and 56 as part of its extended missions. An additional transit was detected in the Sector 56 light curve and included in our final analysis. A search by the SPOC of Sectors 14–56 reported a difference image centroiding result [29] constraining the host star location to within \(2\farcs54 \pm 2\farcs9\) of the difference image centroid. This is in agreement with the low contamination by reported nearby Gaia stars.

We acquired the Pre-search Data Conditioning Simple Aperture Photometry [30]–[32] flux data from the Milkuski Archive for Space Telescopes (MAST) for the 2 minute Sector 14, 15, 16, and 40 data (doi:) and the 20 s Sector 54, 55, and 56 data (doi:).

Before applying any light curve fitting, we use the lightkurve [33] software package’s built-in flatten routine to remove any remaining PDCSAP variability. This applies a Savitzky–Golay filter [34] to the light curves, fitting a low-order polynomial to a rolling subset of the data to remove low-frequency trends. The transit regions were masked during this process and the subset windows were chosen to be longer than the transit duration. We apply the same flatten routine to the 20 s cadence Sector 56 TESS data, applying an additional rolling sigma-clip routine (3\(\sigma\) from the median in a window of \(\pm\)​100 minutes) to remove outliers. The portions of the flattened light curve containing the transits and used for analysis are shown in Figure [fig:TESS95transit].

With the current Year 5 plan for the TESS mission, there are no scheduled visits to this region of the sky after Sector 56.

2.1.2 WASP↩︎

The field containing TOI-2010 was observed by the Wide Angle Search for Planets (WASP) transit-search survey [35] from 2008 to 2010. In each year the observing season spanned \(\sim\) 130 nights, with the SuperWASP-North camera array observing the field on clear nights with a typical 15 minute cadence. A total of 32 000 photometric data points were obtained using 200 mm, \(f\)/1.8 Canon lenses backed by 2k\(\times\)​2k CCDs. TOI-2010 is the only bright star in the 48extraction aperture.

The WASP data are dominated by systematics and red noise. While the transit depth is likely sufficient to show up in the WASP light curves, the survey mission relies on multiple repeated events to distinguish transits from noise. As a single-transit target at the time, WASP did not detect any events on the target. Even with the benefit of hindsight and a firm orbital ephemeris, the WASP coverage only overlaps with one predicted transit. The 4 hr span of data lies in the middle of a 8.7 hr transit, and the light curve shows no convincing transit-like features.

2.2 Candidate Vetting↩︎

Once established as a TOI, a number of vetting observations were undertaken. They were used to search for false positive indications, and to assess the target’s suitability for further follow up observations.

2.2.1 Keck/HIRES Spectra↩︎

We obtained a spectrum of TOI-2010 with the High Resolution Echelle Spectrometer [36] on the Keck I telescope at W. M. Keck Observatory to explore false positive explanations for the single-transit event, to assess the quality of the host as a target for Doppler spectroscopy, and to conduct a basic spectral characterization of the host. Initial processing of the spectrum with SpecMatch-Emp [37] determined the stellar parameters. The results, along with the star’s bright magnitude indicated that it would likely be a suitable target for Doppler spectroscopy.

This Keck/HIRES measurement was taken under excellent seeing conditions and produced a spectrum with SNR \(\sim\) 200. Given this data quality it was used as the template spectrum with which the Levy RV measurements were extracted (see Section 2.3). Similarly, we favour the Keck/HIRES extracted stellar parameters over those from LCOGT/NRES and FLWO/TRES (Sections 2.2.2 and 2.2.3) due to the quality of the spectrum, though we note the close agreement of most parameters. Table [tab:spec95params] provides a comparison of these stellar parameters.

2.2.2 LCOGT/NRES Spectra↩︎

We scheduled spectroscopic observations for TOI-2010 on the Las Cumbres Observatory Global Telescope (LCOGT; [38]) Network of Robotic Echelle Spectrographs [39]. NRES comprises four identical echelle spectrographs in different observatories, covering a range of longitudes in the Northern and Southern Hemispheres. The resolving power of the echelle spectrographs is \(R\sim 53,000\) covering the wavelength range 3900–8600 Å. We obtained four good-quality (SNR 18–56) spectra with the NRES unit at the Wise Observatory between 2020 June 19 and 28. We used the BANZAI-NRES pipeline [40] to reduce the spectra and extract RVs, and the SpecMatch-Synthetic code for the stellar parameterization [41], [42].

While the NRES observations provided helpful early vetting of the system, we have chosen to exclude the four RV measurements from the analysis due to their much lower precision (\(>\)​20 m s\(^{-1}\)). The derived stellar parameters are generally in close agreement with the Keck/HIRES values (Table [tab:spec95params]).

2.2.3 FLWO/TRES Spectra↩︎

Three reconnaissance spectra of TOI-2010 were obtained on 2020 July 9, 18, and 27 with the Tillinghast Reflector Echelle Spectrograph [43]. TRES is an optical (390–910 nm) spectrograph with a resolving power of \(R\sim 44\,000\) mounted on the 1.5 m Tillinghast Reflector telescope at the Fred Lawrence Whipple Observatory (FLWO). The spectra, with SNR in the range of 25–35, were extracted using the TRES standard pipeline [44] and the stellar parameters were derived using the Stellar Parameter Classification [45], [46] tool. SPC cross correlates the observed spectra against a grid of synthetic spectra based on Kurucz atmosphere models [47] deriving stellar effective temperature, surface gravity, metallicity, and rotational velocity.

In the same sense as the NRES spectra described above, these data were useful in the early classification of the star and ruling out of false positives, enabling more detailed measurements to be carried out. The FLWO/TRES stellar parameters are also generally in close agreement with the Keck/HIRES analysis (Table [tab:spec95params]).

2.2.4 Gemini-N/’Alopeke Imaging↩︎

Using the ’Alopeke instrument mounted on the Gemini-North telescope we acquired high-contrast imaging of TOI-2010 on 2020 June 7 (Program ID: GN-2020A-Q-132). This observation was a part of the exoplanet follow up campaign by [48]. With ’Alopeke’s design, it can simultaneously capture imagery at both 562 nm and 832 nm. The resulting images were processed using the pipeline of [49], and the resulting contrast curves are shown in Figure 2. Due to the clearly superior performance of the 832 nm filter, we use the red contrast curve for all analyses (e.g. Section 3.7).

One previously unknown nearby source was detected, seen in the lower left corner of the Figure 2 inset. This object is separated by 1 at a position angle of 138\(^\circ\) east of north. Brightness uncertainty on this neighbour is somewhat elevated as it lies outside the speckle correlation radius of \(\sim\)​1. Speckle decorrelation begins to set in beyond this separation when the rays do not pass through the same atmospheric path. Our photometric estimate of the source places it at a \(\Delta {\rm mag} = 5 \pm 0.5\).

This source is not to be confused with the neighbouring Gaia star discussed further in Section 4. The Gaia source lies at a separation of 1 and a position angle of 33\(^\circ\) east of north, beyond the field of view of our ’Alopeke image. The 1 source found in our ’Alopeke image does not appear in Gaia’s DR2 or DR3 catalogues.

2.3 Radial Velocities↩︎

The reconnaissance spectra from Keck/HIRES, LCOGT/NRES, and FLWO/TRES were able to place sufficient constraints on the stellar parameters to identify TOI-2010 as a suitable candidate for precise RV measurements. We collected a total of 110 RV measurements (Figure 3, values in the Appendix) to make up our combined RV data set. These measurements come from three separate instruments, span 992 days, and uniformly sample the phase space of the 142 day periodic signal that stands out in the data.

2.3.1 Levy↩︎

In February of 2020 (BJD 2458887), we began to gather spectra on the target for RV measurements. We started with the Levy spectrograph installed on the 2.4 m Automated Planet Finder (APF) telescope at Lick Observatory in California, acquiring 70 spectra over a 2 year period, carried out by the dynamic queue scheduler [50]. The Levy spectrograph is a high-resolution (\(R\sim114\,000\)) slit-fed optical echelle spectrometer [51] that has previously been used to refine the orbital period and mass of single-transit planet candidates identified by TESS [9]. We gathered spectra with exposure times of 20–25 minutes (mostly 25), achieving signal-to-noise ratio (SNR) values of 50–100 at around 550 nm. An iodine cell in the light path allows for wavelength calibration and the forward modeling of the stellar RV for each spectrum [52], [53]. This forward modeling process relies on having a high-SNR spectrum that is used as a template. The HIRES spectrum described in Section 2.2.1, which had a SNR of roughly 200, was used to create this template spectrum for the extraction of the Levy RVs which were obtained with uncertainties of 4–7 m s\(^{-1}\).

We look for correlations in the \(\log R'_{HK}\) activity index (computed from the S-index using PyAstronomy routines) with RVs to determine if stellar activity may be biasing the measurements. We determine a correlation coefficient of \(0.10 \pm 0.05\) and a \(p\)-value of \(0.43 \pm 0.23\), indicating no evidence of correlation.

2.3.2 Tull↩︎

We also gathered high-precision RV observations at the McDonald Observatory using the Tull coudé spectrometer 2 (TS2) on the 2.7 m Harlan J. Smith Telescope [54]. This cross-dispersed echelle white-pupil spectrometer was used in its “TS23" mode (indicating the third focus) with an entrance slit of \(1\farcs2 \times 8\farcs2\), giving a spectral resolving power of \(R = 60\,000\) over most of the visible spectrum. A temperature-stabilized \(I_2\) gas absorption cell in front of the spectrograph entrance aperture provided the velocity calibration. An exposure meter recorded the time series of flux entering the spectrograph, enabling us to compute the flux-weighted barycentric correction. A wave front sensor was used for telescope focus to optimize pupil illumination stability and throughput. We obtained the measurements with 20–30 minute exposures, achieving an SNR per pixel of 62–96 (mean \(\sim\) 75). The spectra are recorded on a \(2048 \times 2048\) pixel Tektronix CCD. All spectra were reduced and 1D spectra were extracted using standard IRAF routines [55], [56]. In all, a total of 16 spectra of TOI-2010 were obtained between 2020 December 8 and 2022 October 26. RVs were computed using the AUSTRAL code [57], resulting in uncertainties of 9–12 m s\(^{-1}\).

We conduct a similar activity–RV correlation search as was done with the Levy, resulting in a coefficient of \(0.02 \pm 0.20\) and a \(p\)-value of \(0.59 \pm 0.26\) for Tull. Again, there is no indication of RV correlation.

2.3.3 SOPHIE↩︎

We started observing TOI-2010 with the Spectrographe pour l’Observation des Phénomènes des Intérieurs stellaires et des Exoplanètes (SOPHIE) in July of 2020, securing 25 spectroscopic measurements up to September of 2022. SOPHIE is a stabilized échelle spectrograph dedicated to high-precision RV measurements in optical wavelengths on the 193 cm Telescope at the Observatoire de Haute-Provence, France [58], [59]. We used the SOPHIE high-resolution mode (resolving power \(R=75\,000\)). Depending on the weather conditions, the exposure times ranged from 11 to 30 minutes (typically 18 minutes) and their SNR per pixel at 550 nm ranged from 24 to 55 (typically 46). The corresponding RVs were extracted with the standard SOPHIE pipeline using cross-correlation functions [59] and including CCD charge transfer inefficiency correction [60]. Following the method described, e.g., in [61] and [62], we estimated and corrected for the moonlight contamination using the second SOPHIE fiber aperture, which is targeted on the sky while the first aperture points toward the star. We estimated that four of the 25 spectra were significantly polluted by moonlight; one of which was too contaminated and was excluded. The other three contaminated measurements were corrected, with corrections below 20 m s\(^{-1}\). Thus our final SOPHIE data set included 24 measurements showing RV uncertainties ranging 3–9 m s\(^{-1}\).

SOPHIE \(\log R'_{HK}\) activity measures similarly show no correlation with RV values. We determine a correlation coefficient of \(0.22 \pm 0.14\) and a \(p\)-value of \(0.38 \pm 0.27\).

2.4 follow up Photometry↩︎

The collective RV campaign was able to map out a clear planetary signal, but the period uncertainty was only constrained to the order of a few days. This was insufficient for reliable scheduling of transit observations, so we undertook a few efforts to catch a subsequent transit and fine-tune the period.

2.4.1 GMU↩︎

We observed TOI-2010 with the George Mason University Observatory’s 0.8 m Ritchey–Chretien telescope on the nights of the 2021 July 21 and 22 to capture a second transit. We imaged in R with an SBIG-16803 CCD with exposure times of 30 s repeated for a duration of \(\sim\)​5 and 2.5 hr each night, respectively. Both nights were impacted by intermittent clouds, and single measurement precisions of 6.5 and 7.5 ppt were obtained per 30 second exposure. Data was reduced and plate-solved using a custom python code alnitak¹ and aperture photometry, reference star selection, and systematic detrending were performed with AstroImageJ [63].

This attempt was prompted due to a predicted transit (from preliminary RV fits) occurring very near the end of TESS Sector 40, and motivated by an absence of TESS coverage in Sector 41. Unfortunately, no transit was detected on either night. Given the broad transit timing uncertainty at the time, the narrow available observing windows, and the very long transit duration, the odds of detecting the transit here were quite low. Unbeknownst at the time, the transit occurred 1.8 days after the second observation. These data provided initial constraining power for refining the RV period, but do not benefit the global orbital model. As such, they are not included in the modeling of Section 3.

2.4.2 NEOSSat↩︎

The Near-Earth Object Surveillance Satellite (NEOSSat) is a small spacecraft operated jointly by the Canadian Space Agency (CSA) and Defence Research and Development Canada (DRDC). It has a 15 cm telescope aperture and is capable of precision relative photometry [64]. NEOSSat’s clear-filter effective bandpass is approximately 400–900 nm.

As the RV data accumulated, preliminary joint fits (see Section 3 for details) of the RVs and TESS sector 15 transit revealed a roughly 142 day period, albeit with broad uncertainties. The extended transit duration (\(\sim\)​8.7 hr), wide timing uncertainty (on the order of a week), and long period made observing a subsequent transit from the ground extremely challenging. While NEOSSat had previously proven its capability with short-period exoplanet follow up [65], TOI-2010 b marked the first attempt at using the instrument to recover a long and uncertain period. With its space-based vantage point, NEOSSat has the capability to stare continuously at a target for an extended duration, interrupted only by Earth-eclipse events and technical satellite operations. In mid-December of 2021 we employed NEOSSat to observe a \(\sim\)​6 day (\(2\sigma\)) window around the predicted transit. The telescope imaged TOI-2010 as continuously as was feasible during this time.

We reduced the raw images and extracted aperture photometry using a custom python pipeline developed for NEOSSat, available on GitHub.² With photometry in hand, we applied a principal component analysis (PCA) procedure to the raw photometry using other in-frame stars as reference to calculate a normalized relative flux light curve of the target. The PCA process removes time-varying trends in the photometry that are common across many stars in the frame. At this point there remained some residual variability for which the PCA could not account. This low-frequency variability was removed using a spline fit (discussed in Section 3.3). Even against this variability the deep transit event was clearly visible roughly 15 hr after the RV-predicted midpoint (well within the 3 day, \(1\sigma\) timing uncertainty).

This detection by NEOSSat provided the first precise period measurement for TOI-2010 b, and prompted the preparation of this manuscript. The much later transit detected in TESS’s Sector 56 data agrees completely with the refined period. The NEOSSat light curve is displayed in Figure 4.

3 Analysis↩︎

As the RV campaign progressed, we made preliminary fits using the TESS light curve and the available RV data to place initial constraints on the orbital period. This allowed us refine the ephemeris enough to plan our follow up search for a subsequent transit event. Once they became available, the additional transit detections (NEOSSat and TESS Sector 56) allowed for much more precise period determination.

For our final global fit, we use the IDL software package EXOFASTv2 [66]. EXOFASTv2 provides an integrated framework to jointly analyze multiple exoplanet data sets. Drawing from the IDL astronomy library [67], it simultaneously fits for wide ranges of stellar, planetary, orbital, and instrumental parameters in a self-consistent manner that leverages the rich complementarity of modern data sets.

The details listed below in Sections 3.1–3.3 pertain to the final fit, including archival SED measurements; RVs from Levy, Tull, and SOPHIE; and light curves from the initial TESS Sector 15 transit, the subsequent NEOSSat detection, and also the much later Sector 56 detection by TESS. The parameter posterior results are listed in Table [tab:TOI2010]. The fits converged fully by two different statistics: the Gelman–Rubin statistic, R\(z\), and the number of independent samples, T\(z\). We set very stringent thresholds of R\(z<1.01\) and T\(z>1000\). We provide a brief description of the steps involved, but for precise details on the internal operations of EXOFASTv2 please consult the primary paper by [66].

Beyond EXOFASTv2, we conduct several other independent analyses. We model the bulk metallicity of planet b with a custom software, and we analyse the photometric modulation of TOI-2010 to assess the stellar rotation. In discovering a slight acceleration across the RV measurements, we also conduct a search of mass–orbit parameter space to determine what type of additional companion could be the cause.

3.1 EXOFASTv2: SED/MIST Stellar Modeling↩︎

EXOFASTv2 fetches archival photometry from Galaxy Evolution Explorer [68], Tycho-2 [69], UCAC4 [70], APASS [71], the Two Micron All Sky Survey [72], the Wide-field Infrared Survey Explorer [73], Gaia [74], the Kepler INT Survey [75], the UBV Photoelectric Catalog [76], and the Stroemgren–Crawford \(uvby\beta\) photometry catalog [77], as well as extinctions from [78] and [79] and parallaxes from Gaia DR2 [80]. Allowing photometric uncertainties to be inflated in case of underestimation, it then fits an SED model to this archival photometry using the parallax value and a library of stellar atmospheres. The stellar physics are constrained from either the empirical relations laid out by [81], the Yonsie Yale stellar evolutionary models [82], or the MIST evolutionary models [83], [84], which itself is built using MESA [85]–[88]. Stellar atmospheric models from NextGen [89]), ATLAS [90], and PHOENIX [91] underlie several aspects of the code.

We are able to impose Gaussian priors on the stellar effective temperature (\(T_{\rm eff}\)) and metallicity ([Fe/H]) in the fit, originating from the Keck/HIRES spectra (Table [tab:spec95params]). The \(F_{\rm bol}\) and \(T_{\rm eff}\) parameters have enforced error floors representative of the systematic uncertainties between stellar evolution models to prevent unrealistic precision [92].

3.2 EXOFASTv2: RV Modeling↩︎

The multiple instrument RV data sets are simultaneously fit to a Keplerian model, retaining separate jitter and systemic offset terms. We measured typical \(\log R'_{HK}\) activity index measures of \(-5.1\) to \(-4.7\) that were uncorrelated with RV values. This activity level could induce stellar jitter up to \(\sim\)​10 m s\(^{-1}\), but is unlikely to affect the derived parameters given such a strong planetary signal.

Within EXOFASTv2 the exoplanet mass radius relation from [93] can be referenced to estimate the mass or radius of the exoplanet (and all relevant derived parameters) in the absence of an RV data set or transit, respectively. In this case, however, the RV data constrain the mass while the transit data constrain the radius.

A single long-term linear drift parameter is included in the model. We have excellent temporal overlap of the data across instruments, so there is no large correlation between the trend parameter and systemic RV offset parameters. The fitted RV model is shown in Figure 3.

3.3 EXOFASTv2: Transit Modeling↩︎

EXOFASTv2’s transit model is generated using [94] and [95] with limb-darkening parameters constrained by [96] and [97]. We pass it the TESS coverage of the initial Sector 15-detected transit, a stretch of flat light curve from a Sector 40 that narrowly missed another transit, as well as the later detection in Sector 56. We also include the entire 6 day NEOSSat light curve with its detection. Limb-darkening parameters and transit depths are allowed to differ between instruments. Each instrument also gets its own jitter parameter and out-of-transit offset value. We impose no additional transit-specific priors for this portion of the fit. The period and other orbital element constraints arise from a simultaneous fit of the transit and RV data.

We incorporate a spline fit in the EXOFASTv2 modeling of the NEOSSat data, based on the keplerspline³ [98] designed to handle long-term variability in long Kepler light curves. We used a knot spacing of 1.1 days (roughly 3\(\times\) the transit duration) to model the low-frequency variation.

3.4 Bulk Planetary Composition↩︎

To infer the bulk composition of the planet, we use the modeling and retrieval approach of [99], which we will briefly summarize. Forward models parameterize the thermal state of the planet by the envelope specific entropy, which we evolve from a hot initial state using the atmosphere models of [100]. This requires that we know the radius and temperature structure of the planet at a given specific entropy. We calculate this using a 1D static model of the planet which solves the equations of hydrostatic equilibrium, conservation of mass, and the equation of state (EOS). We use the H/He EOS from [101], and a 50/50 rock/ice mixture for the metals [102], combining these using the additive volumes approximation. For a given mass, metallicity, and stellar insolation, this yields evolution tracks of the radius with time. To match these models to TOI-2010 b, we use a Bayesian statistical model with the true mass, bulk metallicity, and true age as model parameters and fit them against the observed mass, radius, and age from the EXOFASTv2 fit (Table [tab:TOI2010]). Because TOI-2010 b is much cooler than the hot-Jupiter inflation threshold [6], we do not include any additional heating in the planet.

3.5 TESS Light Curve Modulation↩︎

Even in the PDCSAP TESS data, with a degree of its systematics removed, we noticed some low-level variability. As a secondary measure of stellar rotation we looked at the star’s long-term light curve modulation. A simple normalization was applied in order to concatenate the TESS 2 minute cadence SAP light curves from Sectors 14, 15, and 16. Data points that were flagged as poor quality, greater than 5\(\sigma\) outliers, or during the TOI-2010 b transit were removed. Stitching together multiple sectors of observations (even when observed continuously) can introduce systematics into the concatenated light curve that could produce a spurious signal in a periodogram search. Therefore, we searched for periodic photometric variability using the TESS systematics-insensitive periodogram tool, TESS-SIP⁴ [103], which uses PCA to account for spacecraft systematics while simultaneously performing a periodogram search. In the periodogram search from 1 day to half the baseline of the continuous TESS observations (which are \(\sim\)​32 days), we identify a periodic signature in the light curve at \(19.0 \pm 3.0\) days, albeit at a low normalized Lomb–Scargle power (\(<\)​0.001). Assuming a small stellar obliquity, this 19 day signal corresponds to a \(\sim\)​2.8 km s\(^{-1}\) stellar rotation rate, in general agreement with the spectroscopic estimates (see Table [tab:spec95params]). A short-period periodogram search (0.01–13 days) was also performed separately on the PDCSAP photometry from Sectors 14, 15, and 16 following the procedure described in [104]. A small-amplitude (\(\lesssim\) 0.2 ppt), 5.7 day signal was identified in the TESS photometry but we note this is consistent with being attributed to spacecraft systematics due to its location in power–period space relative to other stars in these TESS sectors. With its low SNR and similarity to known systematics, we do not consider this signal physically relevant.

3.6 WASP Light Curve Modulation↩︎

We searched each season of WASP data for a rotational modulation using methods discussed in [105]. We find a significant and persistent modulation at a period of 20 \(\pm\) 1 days. The modulation is weak, with an amplitude of only 1–2 mmag, but the overall false-alarm likelihood is below 1%. This closely matches the TESS photometric modulation, and also likely reflects a stellar rotation rate of \(\sim\)​2.8 km s\(^{-1}\).

3.7 Mass–Orbit Possibilities for an Additional Companion↩︎

A small residual acceleration is detected in the collective RV data. To constrain potential objects on very long orbits that could cause this acceleration, we adopt the process described in [106]. In essence, we step through a 2D grid of semi-major axes and object masses. In each cell, we draw a set of planet b parameters from our fitted posteriors of Table [tab:TOI2010], generating an RV model. This model is subtracted from the RV measurements to reveal a residual slope. A model for object 2 is created by drawing \(M\) and \(a\) values from the current cell, and \(i\) and \(e\) values from motivated distributions. In this case, \(i\) is drawn randomly from a uniform \(\cos i\) distribution and \(e\) from a \(\beta\) distribution [106]. The remaining \(T_0\), \(T_{\rm peri}\), and \(\omega\) parameters are determined by fitting the drawn object 2 model to the residuals, as well as an RV offset. The offset accounts for any change to the assumed systemic velocity caused by the second object.

This process is repeated 500 times per cell. For each cycle, the \(\chi^2\) is calculated and stored, as well as the probability that the object is visible. This probability is based on the inferred magnitude of the object and the fraction of its orbital period that it would spend at a detectable separation from the host, set by the Gemini-N/’Alopeke contrast curve (Figure 2). The data cubes are marginalized over the third axis to create a 2D probability density grid over the object’s \(a\) and \(M\). These results are displayed in Figure 5.

With the allowable mass–orbit space mapped out, we investigate the potential for the known nearby stars to cause the acceleration. The plotted mass uncertainties for the two nearby sources in Figure 5 reflect the magnitude uncertainty of the source only. They do not account for the intrinsic uncertainty of the precalculated stellar evolution grids⁵ [107] from which they were interpolated.

To place a tentative uncertainty on the semi-major axis of the neighbouring stars (under the assumption they are bound), we follow the steps of [108] given an observed projected separation, unknown orbital orientation, and assumed eccentricity distribution. We do not adopt their analytic approximation (Equation (A2)) which roughly matches the numerical distribution arising from a simple \(f(e) = e^2\) eccentricity distribution, but instead carry out the full Monte Carlo approach to create a nonanalytic distribution. We use \(f(e) = e^{0.4}\) where \(e \in [0,0.8]\) is extracted from the observation of binaries with Sun-like primaries [109].

4 Results↩︎

Based on the results of the global transit, RV, and SED fit, we confirm TOI-2010 b as a temperate Jovian exoplanet around a Sun-like star. We find TOI-2010 b to have a mass of \(M_P=1.286^{+0.055}_{-0.057}\;M_{\rm J}\) and a radius of \(R_P=1.054\pm0.027\;R_{\rm J}\). It orbits with a period of \(P=141.834025^{+0.000065}_{-0.000066}\) days and an eccentricity of \(e=0.212^{+0.022}_{-0.021}\). A full list of the fitted and calculated parameters and their uncertainties are displayed in Table [tab:TOI2010]. We determine that the contribution to the uncertainty of \(M_P\) is almost evenly split between the uncertainties on \(K\) and \(M_\star\), and that the uncertainties of \(i\), \(e\), and \(P\) have a negligible impact.

The reported \(T_{\rm eq}\) of Table [tab:TOI2010] comes from an assessment using a single representative star–planet distance. However, with nonnegligible eccentricity, several calculated parameters, \(T_{\rm eq}\) included, are subject to variation with orbital phase. Table [tab:orbit] displays certain of these parameters at four key points in the planet’s orbit.

TOI-2010 b’s moderate eccentricity falls in the \(\sim\) \(75^{\rm th}\) percentile of giant planets with well-known masses and radii. It is distinctly above the cluster of planets with very low eccentricities, but not so high as to stand out among the population of high-eccentricity planets.

Between the SED/MIST model fitting within EXOFASTv2 and the spectral analysis from various instruments, we determine many of TOI-2010’s stellar parameters. In the global EXOFASTv2 posteriors, we see no sign of the bimodality commonly seen between stellar mass and age. This can arise when a star is slightly evolved and the MIST stellar evolution models experience some degeneracy near the subgiant branch [23].

From the Keck/HIRES spectral measurement, we find the stellar radius, mass, and effective temperature to be \(R_\star = 1.084^{+0.028}_{-0.027}\,R_\odot\), \(M_\star = 1.107^{+0.050}_{-0.057}\,M_\odot\), and \(T_{\rm eff} = 5917\pm75\,K\), respectively. TOI-2010 has a surface gravity of \(\log g = 4.412^{+0.023}_{-0.026}\) and a metallicity of \([{\rm Fe/H}] = 0.169^{+0.055}_{-0.056}\). We used these Keck/HIRES stellar values in the global fit as this spectral measurement has the highest resolution and SNR. The stellar parameters determined via spectral fitting of the Keck/HIRES, LCOGT/NRES, and FLWO/TRES data sets (Table [tab:spec95params]) show close agreement in general. However, the instruments report distinct values for the stellar rotation (\(v\sin i_\star\)), though all indicate a slowly rotating star (\(<\)​5 km s\(^{-1}\)). The discrepancies between the instruments may be due to differences in spectral resolution and SNR, or potentially the slow rotation of the star itself. For slowly rotating stars, the effects of rotationally induced line broadening can be of similar magnitude to other broadening mechanisms (e.g., thermal, pressure, and turbulence). Disentangling them becomes challenging, and thus \(v\sin i_\star\) may be inflated if some mechanisms are not properly considered. We therefore report the spectrally derived rotation rates as rough upper limits (Table [tab:spec95params]).

We note that the photometric modulations seen in the TESS and WASP light curves both suggest an equatorial rotation of \(\sim\)​2.8 km s\(^{-1}\), falling in the middle of the spectral \(v\sin i_\star\) values. The precise value is not central to any key findings of our study, but it does have some bearing on estimates of a potential RM signal for future endeavours. We adopt the modulation-derived value for RM calculations in Section 5.2 as it bypasses the line-broadening issues of a slow rotator.

Available data on the local stellar environment reveals two faint neighbours to TOI-2010. We use tpfplotter⁶ [110] to jointly visualize the TESS aperture and Gaia positional information (Figure 6). The only Gaia star of note within the TESS aperture is 5 mag fainter and separated from TOI-2010 by 1 at a position angle of 33\(^\circ\) east of North (Gaia ID: 2136815881247621760). The PDCSAP flux used in our light curve analysis accounts for this minor dilution so as not to affect the radius estimate of the planet.

Though absent from the Gaia catalogue, we photometrically detect a second neighbour in the immediate vicinity of TOI-2010 using ’Alopeke high-contrast imaging. It is of similar brightness to the Gaia star (\(\Delta {\rm mag} = 5 \pm 0.5\)) and was detected at a separation of 1, 138\(^\circ\) east of North. This star does not appear in the Gaia catalogue. The flux dilution from the ’Alopeke star is small enough to cause \(<1\%\) deviation in the \(R_p/R_\star\) measurement.

Spectral investigation for evidence of blended binaries also shows no indication of significant contamination. In the SOPHIE spectra, the corresponding bisectors of the cross-correlation functions do not show any significant variation nor correlation with the RV. This means there are no indications for RV variations induced by blend configurations or stellar activity. We also computed cross-correlations using masks characteristic of different spectral types: all produce similar RV variations, suggesting against the presence of a blend of stars with different spectral types. Similarly, the Keck template spectrum was run through the ReaMatch [111] software to check for blended stellar spectra. The analysis revealed no hint of any such blended components in its cross-correlation routines, and limits any unresolved sources to well below 1% of TOI-2010’s flux.

With the lack of any photometric or spectroscopic evidence for significant or problematic nearby or blended stars, we conclude that the measurements of TOI-2010 are free of any significant stellar contamination. The planet radius assessment of TOI-2010 b is therefore robust.

The bulk metallicity results are shown in Figure 7. The bulk metal mass fraction of \(Z_P = 0.11\) corresponds to 45 Earth masses of heavy elements. There appears to be a small degeneracy between age and metallicity in this analysis. This arises because leftover heat from formation in a young star (\(<1\) Gyr) is compensated in the model with extra metal.

In this case we used a fully mixed planet model. Using a “core+envelope" model would mean replacing compressible gas in the core with less compressible metals, requiring more metals to achieve the same radius. This model would require \(\sim\)​20% (or \(\sim\)​1\(\sigma\)) extra, according to [7]. The result is similar when using a moderate number of layers making up a semiconvective staircase”core," but when considering thousands of layers [112] cooling slows down and even more metal is required. However, simulations suggest that small layers merge quickly as the planet evolves [113], [114], so we would not assume such an extreme case without more evidence.

The RV fit reveals a residual acceleration of \(\dot{\gamma} = 0.0185 \pm 0.0055\) m s\(^{-1}\) day\(^{-1}\). This \(\sim\) \(3\sigma\) slope detection is suggestive of some additional distant planetary or stellar companion acting on the system. In running the RV data through a generalized Lomb–Scargle periodogram [115]–[117] within the astropy package [118]–[120], the obvious 142 day signal stands out with indisputable significance. Removing the best-fit Keplerian RV model of planet b (Figure 3) and rerunning the periodogram on the residuals produces no peaks with false-alarm probabilities better than \(\sim\)​10%. These findings are shown in Figure 8. There appear to be no other periodic signals present in our data set for \(P \lesssim 1000\) days.

We also check the TESS light curves for additional transits. Using the astropy.stats.BoxLeastSquares (BLS) function, we scan the available photometry for periodic transit-like signals. With two known TOI-2010 b transits in the data, the procedure flags the 1134 day separation along with accompanying aliases (including the 142 day true period). Removing the two known transits leaves a very flat light curve, and a second BLS pass detects nothing above the noise level. The BLS algorithm is sensitive only to repeated events, and so single transits (e.g., due to very long periods or transits falling in observing gaps) would not be detected here. A visual inspection of the light curve reveals no obvious transits to indicate additional bodies in the system.

To explore the possible very-long-period scenarios, we employed a 2D grid search of semi-major axis and companion mass combinations that might produce the observed RV acceleration [106]. The resulting relative probability map marks correlated boundaries on the probable configurations (Figure 5). At the low-mass end, the system could harbour an object of 0.4 \(M_{\rm J}\) orbiting at 2.6 au. Anything interior and/or less massive than that struggles to match the observed acceleration. For sources remaining below the photometric detection threshold, the most probable configuration at the high-mass end is a 475 \(M_{\rm J}\) (0.46 \(M_\odot\)) object orbiting at \(\sim\)​22 au, however the allowable configuration space becomes quite broad. As we are now in the range of self-luminous low-mass main-sequence stars, the Gemini-N/’Alopeke speckle imaging would likely detect anything more massive or more separated (i.e. above the white contours).

In determining the plausibility of each close neighbour star causing the RV acceleration, we estimate some of their relevant properties. We determine the 1 ’Alopeke star to have a mass of \(M = 0.525^{+0.045}_{-0.049}\,M_\odot\) and a semi-major axis of \(a = 162.16^{+75.33}_{-41.43}\) au, while the 1 Gaia star has \(M = 0.5226^{+0.0001}_{-0.0001}\,M_\odot\) and \(a = 205.21^{+95.33}_{-52.43}\) au. These estimates, though rough, allow us to place both stars on the Figure 5 grid.

If neither of these nearby stars are the cause, the most likely candidate is a yet-unseen planetary or low-mass stellar object along the high-probability region in Figure 5. This region through the explored parameter space roughly follows the trend of \(M/[M_J] \approx 0.015\,(a/{\rm [au]})^{3.2}\) where \(a > 3\) au.

5 Discussion↩︎

5.1 TOI-2010 b in Context↩︎

With physical and orbital parameters of the TOI-2010 b system properly constrained, we can place it in the context of other known exoplanets. Using planet data gathered using ExoFile⁷ from the [121], Figure 9 shows several properties for the population of confirmed giant planets (\(R>0.5\,R_{J}\)). As can be seen in the top panel, TOI-2010 b is deep in the low-insolation wings of the population. Few other confirmed giants can boast such low stellar input, and fewer still have magnitudes bright enough to enable detailed spectroscopic follow up. TOI-2010 b stands out as valuable addition to this corner of parameter space.

The bottom panel of Figure 9 locates TOI-2010 b in mass and radius space. A distinction is made between strongly and weakly irradiated planets (dotted line in the top panel and marker type in the bottom panel), given that their mass–radius relationship changes. We find TOI-2010 b to be a fairly typically proportioned giant planet, akin to Jupiter though quite a bit warmer. Its moderate eccentricity may suggest a dynamic history either in its formation or due to ongoing interaction with unseen neighbours.

5.2 Future observation potential↩︎

We determine the expected signal strengths for a number of potential observations that might be made on this target in the future (Figure 10). We calculate and report the [123] Transmission Spectroscopy Metric (TSM), though we note it was calibrated for smaller planets. The TSM nominally provides an SNR estimate for fixed-duration observations made with JWST/NIRISS. However, without the small-planet-calibrated scaling factor, the specific values of Figure 10a may be better interpreted by their relative strengths, rather than absolute value. We use TOI-2010 b’s transit-phase equilibrium temperature (450 K) for this calculation, and in doing so we find that it has moderate transmission spectroscopy potential with TSM \(\sim 26\). As a relative measure, it lands at the \(\sim\)​23\(^{\rm rd}\) percentile for the population plotted in Figures 9 and 10.

We can also look at the Emission Spectroscopy Metric (ESM) of [123]. Similarly to the TSM, the ESM estimates the SNR achieved with a mid-infrared secondary eclipse detection by JWST. We find a more promising scenario in emission than with transmission. Using the eclipse-phase equilibrium temperature (364 K), TOI-2010 b has one of the strongest predicted emission signals (ESM \(\sim 60\)) among cool giants \(\lesssim 750\) K. Even against giant planets as a whole, TOI-2010 b falls near the median value. The ESM does not include an empirical calibration like the TSM, and so the values indicate the expected SNR of a JWST secondary eclipse detection with the MIRI instrument. TOI-2010 b may provide a very interesting test bed for certain atmospheric properties. The \(T_{\rm eq}\) range of 360–450 K across its orbit spans a transition regime where disequilibrium chemistry may be evident. Models by [4] predict this temperature range to exhibit marked changes in CO/CH\(_4\) and N\(_2\)/NH\(_3\) ratios between equilibrium and disequilibrium conditions. Such detections may go a long way toward connecting models across the Jupiter–exoplanet–brown dwarf continuum.

We also compute the potential for making an obliquity measurement using the RM effect. [124] provides an approximation for calculating the \(\Delta\)RV amplitude expected during transit for given planet/star size ratio, impact parameter, and projected equatorial velocity (\(v\sin i_\star\)) of the star. Adopting \(v\sin i_\star = 2.8\) km s\(^{-1}\), derived from the modulations observed in both the TESS and WASP light curves, we calculate an RM amplitude of 27.5 m s\(^{-1}\) (Figure 10 (c)).

5.3 Single-transit Planets↩︎

Any transit survey mission is eventually bound to produce single-transit targets. It comes as a direct consequence of having only finite monitoring time for a particular star, and the potential for exoplanets to have very long orbits. With TESS’s month-long baseline coverage for most of the sky, single-transit targets are not infrequent. Passively detected retransits in subsequent sectors can help narrow down options, but long gaps between detections leave many possible period aliases. The survey’s schedule of reobserving a target is also not always compatible with the planet’s orbit. For example, the second detected transit of TOI-2010 b in Sector 56 was very nearly missed. If the period had turned out even 0.6% longer (\(\sim\)​19 hr), the additional TESS transit would have been missed entirely.

Without substantial active follow up effort, single-transit planets often remain unviable for further study. Any phase/timing-related endeavours cannot be scheduled without a firm ephemeris, and attempting an RV study of the system requires careful vetting even before investing the sizeable observing program for the RVs themselves. Happily, the TFOP has a wide variety of researchers, infrastructure, and resources to put toward the effort. If a candidate proves suitable for RV follow up (as was the case for TOI-2010), the mass measurement and orbital refinement come packaged together. This provides a lot of value for the RV investment. That being said, RV-derived periods generally have uncertainties on the scale of hours or days for single-transit targets (which tend to have longer orbits), and the predictive timing uncertainty on future transits only grows worse as transits go undetected. This ephemeris is generally insufficiently refined to plan precisely timed observations (e.g., transits and eclipses), but it does open the door for the last step needed to constrain the system neatly.

The 3\(\sigma\) uncertainty window of an RV-predicted transit can easily span up to 20 days or more. Photometric instruments that are able to locate a transit within such a wide window can refine the period uncertainties to the order of minutes. The space-based vantage point provided by NEOSSat and other small space telescopes [125] is ideal for this application. Such facilities can monitor this star over the whole time frame, and their detections do not even have to be of particularly high SNR. The single high-quality TESS measurement is generally sufficient to constrain the transit shape, and so a low SNR additional transit can simply supply timing information beyond the precision of the RVs. Being able to point at a target at any time also allows space-faring instruments to quickly narrow down possible period aliases if two widely spaced transits have been detected.

It is a major and ongoing challenge to carry out successful retransit searches for TESS single-transit targets. The nature of the mission’s sector-by-sector and hemisphere-by-hemisphere observing strategy leaves plenty of room for longer-period planets to fall through the cracks. Small space telescopes are uniquely suited in providing support observations to pull these long-period planets back from the edge of obscurity by firmly establishing their ephemerides.

5.4 Cause of the RV Acceleration↩︎

We believe that the 1 Gaia neighbour is unlikely to be the cause of the residual RV slope for two reasons. Firstly, the Gaia DR3 parallax distances of TOI-2010 and this faint neighbour differ by 1.5 pc. This does place the source as a close neighbour in interstellar terms, but given their respective parallax uncertainties their distances are more than \(2\sigma\) discrepant. It is therefore very likely to be physically separated from TOI-2010 at the parsec scale and thus could not cause the observed acceleration given its low magnitude-inferred mass. Secondly, even if we assume some bias on the parallax measure and that they are indeed at the same distance as one another, this companion does not fall in a favourable location of the Figure 5 plot. The source’s inferred mass and orbital separation place it away from the high-probability region. The specific relative probability of its location is only \(\sim\)​5% of the global peak probability, and \(\sim\)​7% of the highest-probability region at its particular mass. The combination of these two reasons disfavour the Gaia star as the source of the acceleration.

The 1 source found in the ’Alopeke image may be a slightly better candidate, though it is missing some crucial information. With a similar inferred mass but tighter separation, this source is closer to the high-likelihood region of the Figure 5 plot. Its particular cell is \(\sim\)​12% of the peak probability for this mass. The uncertainty on its semi-major axis allows for it to intersect a bit deeper into the high-probability region. However, with an unknown parallax, it is entirely possible that this is a background or foreground object and wholly unassociated with the system.

Based on their low-probability locations in the search grid, coupled with a parsec-scale difference in distance between the Gaia neighbour and TOI-2010 and lack of parallax information on the ’Alopeke neighbour, it appears unlikely that either star is responsible for the RV acceleration.

The presence of these two sources certainly does not rule out the additional possibility of a hidden lower-mass object. The ’Alopeke source may be approaching the right region of parameter space, but the unknown nature of its 3D location relative to TOI-2010 precludes any certainty for now. Further characterization of the nearby sources may offer more clarity, such as refinement of the Gaia star’s parallax, or a check on the ’Alopeke star’s proper motion in a few years time. Additional RV coverage of TOI-2010 may even reveal some clear curvature to the residual acceleration, which would add strong constraints on the high-mass/wide-orbit end of the currently allowed parameter space.

6 Summary↩︎

In the course of this study, we have confirmed the planetary nature of the exoplanet TOI-2010 b. A wide range of data sets from the TESS mission and the TFOP working group were collected in this effort. Most notably, the initial single transit discovered in the TESS Sector 15 data provided strong transit morphology constraints, but no information on the period. A substantial RV campaign involving several observatories mapped the RV curve, determined a rough period, and predicted a subsequent transit to an uncertainty of a few days. Using NEOSSat, we observed a continuous week-long window and caught this transit, refining the period down to just a few minutes uncertainty. A fortuitous catch in TESS’s Sector 56 light curve revealed an additional transit detection at a late stage of this manuscript’s preparation.

We carried out a global model fit using EXOFASTv2 to determine the system’s physical and orbital parameters by simultaneously fitting time-series light curves, RVs, and historical photometric data. TOI-2010 b turns out to be Jupiter-like in size, about 30% more massive, and its equilibrium temperature may fluctuate between roughly 360–450 K given its moderately eccentric orbit.

Our bulk metallicity analysis also suggests a fairly Jupiter-like metal mass fraction, i.e. modestly lower than the general trend given its mass [7]. The host star is very slightly super-solar in terms of mass, radius, luminosity, and temperature.

We find evidence of a small-amplitude residual acceleration in the RV data set once TOI-2010 b’s signal has been removed, potentially indicative of an outer companion in the system. Searching a broad grid of potential mass and semi-major axis values, we determine the relative probability that such companions could cause the observed acceleration. We also determine which of these simulated systems would be visible in our high-contrast imaging. Among hidden objects (too faint and/or close to TOI-2010 to be detected), we find a correlated allowed parameter space ranging 0.4–475 \(M_{\rm J}\) in mass (475 \(M_{\rm J} \approx 0.45 M_\odot\)) and 2.6–23 au in semi-major axis along its highest-probability region. Smaller masses and orbits cannot reproduce the observed RV slope, while more massive (i.e. brighter) objects on wider orbits would be observable in our Gemini-N/’Alopeke speckle imaging.

We make note of two nearby sources that could potentially be connected to the acceleration. A Gaia source, 1 away, is at a similar distance to the TOI-2010 system, but perhaps not close enough to be considered a binary capable of producing the RV slope. Also, its inferred mass and semi-major axis do not fall in a likely region of the parameter space. A second source, discovered in our high-contrast imaging 1 away, has an inferred mass and semi-major axis that are slightly more likely to produce the acceleration. However, with no parallax information, this source could easily be just a projected neighbour.

In refining TOI-2010 b’s period, we have enabled future transit/eclipse-based research. In large part due to its cool temperature, the transmission spectroscopy potential of this target is somewhat poor. However, its predicted signal strength is much better for emission spectroscopy and RM measurements.

TOI-2010 b turns out to be very typical member of the population of known Jovian exoplanets. However, the planet’s unique value and interest stem from its observability and low insolation/effective temperature. Currently, amongst giant planets with reliable radius and mass measurements, only \(\sim\)​20 are at comparable or lower stellar insolation levels. Among those, only two are bright targets.

TOI-2010 b is a successful case of searching for and catching additional transits for a single-transit candidate planet. This process is often expensive and challenging, but it allows us to build up the confirmed exoplanet catalogue where it is only sparsely populated.

C.R.M. and D.L. acknowledge funding from the Trottier Family Foundation in their support of Trottier Institute for Research on Exoplanets (iREx). They also acknowledge individual funding from the Natural Sciences and Engineering Research Council (NSERC) of Canada. P.A.D. acknowledges support by a 51 Pegasi b Postdoctoral Fellowship from the Heising-Simons Foundation and by a National Science Foundation (NSF) Astronomy and Astrophysics Postdoctoral Fellowship under award AST-1903811.

S.D. is funded by the UK Science and Technology Facilities Council (grant No. ST/V004735/1).

X.D. and T.Fo. acknowledge funding from 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 Grenoble-Alpes University.

E.M. acknowledges funding from FAPEMIG under project number APQ-02493-22 and research productivity grant No. 309829/2022-4 awarded by the CNPq, Brazil.

D.D. acknowledges support from the NASA Exoplanet Research Program grant 18-2XRP18_2-0136, and from the TESS Guest Investigator Program grants 80NSSC22K1353 and 80NSSC22K0185.

T.Fe. acknowledges support from the University of California President’s Postdoctoral Fellowship Program.

K.A.C. and D.W.L. acknowledge support from the TESS mission via subaward s3449 from MIT.

The authors would like to thank the on-duty telescope observers Patrick Newman, Owen Alfaro, Ben Chang, and William McLaughlin for their contribution in gathering the George Mason University Observatory data.

This paper made use of data collected by the TESS mission and are publicly available from the Mikulski Archive for Space Telescopes (MAST) operated by the Space Telescope Science Institute (STScI). Funding for the TESS mission is provided by NASA’s Science Mission Directorate.

We acknowledge the use of public TESS data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center.

Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products.

We would like to thank the PIs of the TESS Guest Investigator programs that put TOI-2010 on the 2 minute (Steven Villanueva – G04195, Diana Dragomir – G04231, Andrej Prsa – G04171, Andrew Mayo – G04242, and James Davenport – G04039) and 20 s (Guadalupe Tovar Mendoza – G05121, and Daniel Huber – G05144) cadence lists.

This research has made use of the Exoplanet follow up Observation Program (ExoFOP; doi:) website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.

We would like to thank and acknowledge the efforts of the TESS Single Transit Planet Candidate working group for working to keep tabs on and improving our understanding of long-period targets.

This work makes use of observations from the LCOGT network. Part of the LCOGT telescope time was granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is funded by the NSF.

Observations in the paper (Program ID: GN-2021A-LP-105) made use of the high-Resolution imaging instrument ’Alopeke. ’Alopeke was funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. ’Alopeke was mounted on the Gemini-North telescope of the international Gemini Observatory, a program of NSF’s NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea).

This work is based on observations collected with the SOPHIE spectrograph on the 1.93 m telescope at the Observatoire de Haute-Provence (CNRS), France. We thank the staff of the Observatoire de Haute-Provence for their support at the 1.93 m telescope and on SOPHIE. We also thankfully acknowledge grants from CNES and the CNRS “Programme National de Planétologie."

This research has made use of 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 work has made use of data from the European Space Agency Gaia (https://www.cosmos.esa.int/gaia), processed Data Processing and Analysis Consortium (DPAC; https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

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