THE K2 & TESS SYNERGY: UNITING NASA’S PLANET HUNTERS By Erica A. Thygesen A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Astrophysics and Astronomy—Doctor of Philosophy 2025 ABSTRACT We have entered an era of studying the atmospheres of exoplanets in unprecedented detail, particularly through transmission spectroscopy of transiting planets using the James Webb Space Telescope (JWST). However, most of the 4300+ confirmed transiting planets are not currently accessible to JWST during its mission lifetime. This widespread problem is due mostly to ephemeris degradation: while the transit time and period of the planet may be known to a precision of minutes at discovery, the uncertainties compound with each successive transit, which can culminate in the projected time of future transits being off by hours to days when follow-up observations are being made years later. This costly problem can be alleviated by reobserving transits to greatly narrow down the future transit window before scheduling observations for characterization. Fortunately, NASA’s Transiting Exoplanet Survey Satellite (TESS) mission is observing most of the sky for transit signals, providing an efficient and timely avenue for refreshing the ephemerides of hundreds of planets. With this in mind, the K2 & TESS Synergy is a large scale effort to reanalyze planets originally discovered by NASA’s K2 mission with new observations from the ongoing TESS mission. We combine light curves obtained by both NASA missions along with archival radial velocities, Gaia parallaxes, and spectral energy distributions in global fits using EXOFASTv2, which not only allows us to update the ephemerides, but also build a self-consistent catalog of parameters for the planets and host stars. We present a reanalysis of 26 single-planet systems reobserved by TESS during its prime mission. For half of the planets, we improve the average 3𝜎 uncertainties by 2030 from the order of tens of hours down to under one hour. As a result of the faintness of some systems, 13 planets do not have transits detectable by TESS. In those cases, we exclude the TESS photometry from the global fits, resulting in a corresponding ephemeris improvement of 43.2 to 35.6 hours. This systematic approach also provides opportunity to amend ephemerides that were originally incorrect due to problems such as false positive transits in additional photometry used at discovery. We address one such case, that of K2’s first planet discovery, K2-2 b, where the period was 28.8 minutes (∼ 40𝜎) away from the true value at the time of discovery. In addition to the K2 and TESS light curves, we use a variety of other space- and ground-based photometry to hunt for the transit of K2-2 b. We successfully caught multiple transits of K2-2 b, allowing us to correct and refine the ephemeris such that the transit time uncertainty will be known to within <13 minutes by 2030. We continue the broader reanalysis to the top 50 planets for atmospheric characterization in the K2 catalog to ensure that JWST can be used to obtain transmission spectra for these planets. Seven of the planets in this sample have been part of the previous K2 & TESS Synergy analyses. Most planets in this sample are equally suitable for atmospheric characterization using JWST as other current targets. There are also many targets that would be useful for understanding the formation and evolution processes of sub-Neptunes and giant planets. We have completed analysis for 34 of these planets, with their average ephemeris uncertainties by 2030 improved from 17.4 hours to 16 minutes, enabling future targeted observations be scheduled. The culmination of the work in this thesis is updated global parameters for 54 planets and their hosts. Efforts like the K2 and TESS Synergy will ensure the accessibility of transiting planets for future characterization while leading to a self-consistent catalog of stellar and planetary parameters for future population efforts. Copyright by ERICA A. THYGESEN 2025 Dedicated to everyone who has supported me, and to my younger self who struggled to write even a sentence. A lady doctor? Has science come that far? — Johnny Bravo v ACKNOWLEDGEMENTS Never in my wildest dreams did I think I would pursue a PhD, let alone at the opposite side of the world to where I call home. For this, I thank Ryan. I could not have achieved this goal alone, and I truly appreciate all of the support and encouragement over the years from so many people in my life. To Mum and Dad, thank you for always giving me the best opportunities in life, and listening to my practice talks, even from the other side of the world. Dad - I’m sorry you didn’t get to see this, but I know you’re proud of me. Mum - thank you for all the years you put into homeschooling me. It finally paid off. To my siblings, thank you for keeping life fun. Thank you to my advisor, Joey Rodriguez, who gave me the opportunity to study exoplanets and tirelessly encouraged me to believe in myself. I am truly grateful to be following my dreams. Thank you to my thesis committee, Jay Strader, Wolfgang Kerzendorf, Seth Jacobson, and Kendall Mahn. You all made me feel like my work mattered. I could not have survived grad school without the friends who were going through it with me. To Josh and Katie, I am so thankful you were there with me right from the start. I would not have made it through the first few years without you. To Noah and Yue Yan, thank you for all the movie nights that kept me sane. I deeply cherish our friendship. To Kristen, for taking me under your wing from the beginning. To Jack, for driving to so many conferences and reassuring me the noises in the plane are all very normal. To Teresa, for always being down to commiserate any time of day. And so many more people - Bella, Rebecca, Emily, Alison Duck, Laura Chomiuk, Hailey, Brandon, CJ, Sierra, Sona, Susie, Kim, Rich. You all helped me get to where I am. Thank you Ryan for all the pep talks and for making so many roast potatoes to keep me happy over the years. I know it seemed like it would never end, but I made it. You’re not the only doctor any more. vi TABLE OF CONTENTS LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 1.1 A Brief History of Exoplanet Discovery . . . . . . . . . . . . . . . . . . . . . 1.2 Exoplanet Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Ephemeris Degradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Ephemeris Refinement Projects . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 The K2 & TESS Synergy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 . 17 1.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 2 . . Introduction . THE K2 & TESS SYNERGY II: REVISITING 26 SYSTEMS IN THE TESS MISSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 . 18 2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 . 22 2.3 Observations and Archival Data 2.4 Global Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 3 . . Introduction . THE K2 & TESS SYNERGY III: SEARCH AND RESCUE OF THE LOST EPHEMERIS FOR K2’S FIRST PLANET . . . . . . . . . . . . . 47 . 47 3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 . 49 3.3 Observations and Archival Data 3.4 Global Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 4 . THE K2 & TESS SYNERGY IV: K2’S TOP 50 ATMOSPHERIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 TARGETS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.1 Major JWST Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2 Target Selection . . . 4.3 Data and Global Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5 Example of Detailed Characterization Using the K2 & TESS Synergy . . . . . 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.6 Conclusion . . . . . CHAPTER 5 5.1 Summary of work . . 5.2 Future Work . SUMMARY AND FUTURE WORK . . . . . . . . . . . . . . . . . . . 108 . 108 . 110 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 vii LIST OF ABBREVIATIONS BJD C/O ELT ESA ESO H He HJ HST JWST MC Barycentric Julian Date Carbon/Oxygen Extremely Large Telescope European Space Agency European Southern Observatory Hydrogen Helium Hot Jupiter Hubble Space Telescope James Webb Space Telescope Monte Carlo MESA Modules for Experiments in Stellar Astrophyiscs MIST MJD NASA NEA PI PPM RV SED TESS TS UV VLT MESA Isochrones and Stellar Tracks Modified Julian Date National Aeronautics and Space Administration NASA Exoplanet Archive Principal Investigator Parts Per Million Radial Velocity Spectral Energy Distribution Transiting Exoplanet Survey Satellite Transmission Spectroscopy Ultraviolet Very Large Telescope viii CHAPTER 1 INTRODUCTION 1.1 A Brief History of Exoplanet Discovery The idea that extrasolar planets might exist has been considered for hundreds of years1. However, it was not until the 1990’s that the existence of planets orbiting stars other than the Sun was confirmed. The first exoplanet discovery was made in 1992, when two planets were found to orbit the pulsar PSR B1257+12 (Wolszczan & Frail, 1992), perhaps surprisingly in what is now considered an exotic system. Pulsar planets are detected through minuscule variations of the pulsar’s rotation, and despite being the first exoplanet discovery, are exceedingly rare, with only eight discovered to date. The first discovery of an exoplanet around a main sequence star came three years later with 51 Pegasi b (Mayor & Queloz, 1995). This detection was made by monitoring the radial velocities (RVs) of the host star, which had been a technique used for over a decade with the hopes to find an exoplanet signature and remains one of the most prolific discovery tools to this day. As a star and planet orbit their common center of mass, the spectrum of the host star is blue- or red-shifted along our line-of-sight based on the phase of the orbit, inclination of the system, and the mass of the planet in comparison to the star. The shift in wavelength of the spectral lines compared to rest wavelengths can be converted into a velocity as a function of time. A huge advantage of RVs over other exoplanet discovery techniques is that for the most part it does not require the system to be aligned in a particular way in relation to our point of view. A system can be oriented anywhere from perfectly edge-on to nearly face-on for an RV signal to be detectable. Combining Kepler’s third law with the measurement of RVs provides a way of measuring the mass of the companion(s) in the system from the semi-amplitude of the velocity in the form, (𝑀 3 𝑃 sin3 𝑖) (𝑀𝑃 + 𝑀∗)2 = 𝑃orb𝐾 3 2𝜋𝐺 (1 − 𝑒2)3/2 (1.1) 1The Italian philosopher Giordano Bruno floated the controversial idea of exoplanets in the 16th century, which at the time was heretical and contributed to his execution. 1 where 𝑀∗ and 𝑀𝑃 are the mass of the star and planet, respectively, 𝑖 is the orbital inclination, 𝑃orb is the orbital period, 𝐾 is the RV semi-amplitude, 𝑒 is the orbital eccentricity, and 𝐺 is the gravitational constant. In general, 𝑀∗ can be independently estimated through modeling of the stellar spectral energy distribution (SED) and parallax, from which intrinsic luminosity and stellar mass can be derived. 𝑃orb and 𝐾 can be directly measured from the RV curve, and the eccentricity of the orbit can be inferred from the deviation of the RV signal away from a pure sine wave. This leaves a degeneracy between the inclination of the system with 𝑀𝑃; with RVs alone, only a lower limit on the planetary mass can be found by assuming the system is being viewed edge on (i.e. 𝑖 = 90◦). For the most part 𝑀∗ >> 𝑀𝑃, meaning typical values for 𝐾 are on the order of several to several hundred m/s. For example, a Jupiter-like planet on a 10 day, edge-on orbit around a Sun-like star will produce an RV amplitude of ∼ 95m s−1. Successfully measuring radial velocities is highly dependent on the brightness of the host star and the strength of the spectral lines used in the measurement. The type of star generally determines the signal to noise of these lines, if they exist. RVs are particularly challenging to obtain for fast rotators (O and B type stars, and some M dwarfs) due to the severe line broadening, and for highly active stars (most M dwarfs), where flaring frequently interferes with the measurements. The faintness of M dwarfs can also pose a problem for obtaining high quality spectra. Therefore, the majority of RV planets have are around FGK stars. The RV community is currently pushing to achieve a precision of <10 cm s−1 - the expected signal of an Earth-Sun analog - with Extreme Precision Radial Velocities (EPRVs), which will provide a way of detailing small planets from the ground. RVs have been responsible for the direct discovery of hundreds of exoplanets, and have also been used to infer the presence of additional companions in planetary systems due to long-term trends. 51 Pegasi b was the first in a lot of aspects: not only was it the first exoplanet discovered via RVs and with a ‘normal’ stellar host, but, with a mass of > 0.47𝑀𝐽, was also the first example of a hot Jupiter - a giant planet on a <10 day period around its host star. While the idea that a planet could retain a large atmosphere so close to its host star was confounding at the time, hundreds of 2 hot Jupiters have been discovered since. Whether these planets formed at their current locations or migrated inward through dynamically active or quiet interactions still remains a highly pursued question in exoplanet formation and evolution. Radial velocities were the primary exoplanet discovery tool for several years following the discovery of 51 Pegasi b. However, this was to change drastically with the advent of the transit method, which had become possible with improvements on instrumentation. At its core, the transit method is an incredibly simple concept: monitor a star for periodic dips in the amount of light received from said star. The depth of the dip (i.e. transit) tells the ratio of the size of the planet to the host star, and the frequency of recurrence is the orbital period. Transits typically cause a sub-percent level of dimming of the star, meaning the photometric precision of telescopes used for this purpose needs to be on the order of at least ∼1000 parts per million (ppm). On their own, transits provide a measure of the radius of the planet, and can constrain parameters such as stellar density, eccentricity, orbital inclination, and the ratio of the semi-major axis to the host star’s radius. The first planet observed using the transit method was HD 209458 b (Charbonneau et al., 2000; Henry et al., 2000; Mazeh et al., 2000), which had previously been discovered via RVs (Henry et al., 2000) and found to have a 3.52 day period. This was the first time an exoplanet had been detected through multiple means. By combining the information contained in transit and RV datasets, the orbital eccentricity, argument of periastron, and density of the planet can be derived. HD 209458 b was the first planet for which both a radius and mass was measured (𝑀𝑃 = 0.63𝑀𝐽, 𝑅𝑃 = 1.27𝑅𝐽), yielding a bulk density of 0.38 g cm−3 - roughly half that of Saturn. This confirmed the idea that the emerging population of hot Jupiters can undergo significant inflation due to high levels of irradiation. A couple of years later, the first transiting exoplanet discovery was made using the Optical Gravitational Lensing Experiment (OGLE; Konacki et al. 2003; Udalski et al. 2002b,a). OGLE-TR-56 b is yet another hot Jupiter (𝑀𝑃 = 1.3𝑀𝐽, 𝑅𝑃 = 1.3𝑅𝐽) on a short, 1.2 day period. The transit method was solidified as a viable exoplanet-hunting tool, but it would take nearly a decade for it to rival RVs as the dominant discovery method (Figure 1.1). A major advantage of transits is that they do not require a planet to be tracked for a full orbit, unlike RVs where 3 Figure 1.1 Cumulative number of exoplanet discoveries since 51 Pegasi b using RVs (green) and transits (purple). RVs were the primary method until 2014, when Kepler’s transit discoveries exceeded the number of RV planets. Data taken from the NASA Exoplanet Archive (NEA; accessed February 9, 2025). each peak of the sine curve signature need significant coverage in order for the orbiting object to be confirmed. However, a downside to the transit method is the high rate of false positives. As objects with masses from 0.001 ∼ 0.1𝑀∗ (i.e. giant planets, brown dwarfs, or small stars) have a near-constant radius of ∼ 0.1𝑅∗, transits alone cannot always distinguish between the cases in this gray area of parameter space. It is therefore common to acquire RV measurements to determine the mass of the companion in order to rule out stellar binaries and confirm the object as planetary in nature. As the transit method requires near perfect alignment of the planet’s orbit with respect to Earth, there is a much smaller probability of observing planets than those found with RVs. The probability of the orbital plane of an exoplanet being oriented such that we could see it transit is 𝑃transit = 𝑅∗ + 𝑅𝑃 𝑎 (1.2) (Johnson, 2016). If the Solar System was being observed from afar, the probability of Earth transiting the Sun is ∼ 0.5%. 4 This only takes into account the geometry of the situation - in reality, the limited sensitivity of our telescopes means we do not detect all planets that are in transit alignment. Furthermore, planets need to be observed during the short window in which they transit their host star, which becomes more challenging for planets with longer orbital periods. As the chance of serendipitously observing a transit is minuscule, transit surveys monitoring large swaths of sky - or the entire sky - have become a cornerstone for exoplanet discovery. This began with a series of ground-based facilities and projects from ∼2000, the most prolific including the Wide Angle Search for Planets (WASP; 161 planets; Pollacco et al. 2006), the Hungarian Automated Telescope Network (HATNet; Bakos et al. 2004) and its southern counterpart (HATSouth; 140 planet discoveries combined Bakos et al. 2013), and the Kilodegree Extremely Little Telescope (KELT; 21 planets; Pepper et al. 2007). The first space-based transit hunter was CoRoT (Convection, Rotation and planetary Transits; Baglin et al. 2000; Auvergne et al. 2009), which operated from 2006 to 2013 and was used to discover 35 exoplanets, laying the foundation for space-based transit missions. The successful launch of NASA’s Kepler Mission (Borucki et al., 2010) in 2009 sparked a new era for exoplanet discovery. A dedicated space-based planet hunter, Kepler stared at one part of the sky for four continuous years and used the transit method to confirm thousands of exoplanets, including a handful of Earth-sized planets residing in their host star’s habitable zone (e.g. Kepler- 62 e and f; Borucki et al. 2013). The Kepler mission alone is still responsible for nearly half of the exoplanet discoveries to date, with 2778 confirmed planets and a further 1982 candidates yet to be validated. The leap in confirmed exoplanets allowed for the first population-level studies, which uncovered aspects of exoplanet demographics that have since become common knowledge in the field: the radius gap of small planets (Fulton et al., 2017), the prevalence of super-Earths and sub-Neptunes (and the peculiarity of our own Solar System not hosting either; e.g. Dressing & Charbonneau 2013), the existence of hot Jupiters (e.g. Dawson & Johnson 2018), to name a few. While Kepler was an invaluable resource and discovered hundreds of planets very quickly, follow-up observations of its targets can be difficult, if not impossible, due purely to the distance of the stars in its field of view. 5 Unfortunately, in 2013, two of the four reaction wheels on board the spacecraft had degraded to the point of being no longer functional, so Kepler could no longer maintain the stable pointing of the previous four years. As the telescope was otherwise still in working order, a concept to prolong the mission was developed: by aiming Kepler directly away from the Sun (i.e. along the ecliptic plane) and minimizing torque from radiation pressure, the two remaining reaction wheels could keep the telescope sufficiently stable to continue performing science-grade photometry. Two preliminary observing tests ensured the performance of the telescope (resulting in K2’s first discovery - see Section 3), and so the K2 mission was born (Howell et al., 2014). Over the next four years, K2 successfully executed 19 observing campaigns, each roughly three months long, and has been used to discover 547 planets to date, with a further 975 candidates to be validated. The systems observed by K2 were different to those in the original Kepler mission, as target selection was opened up to the community through guest observer programs. This allowed not only for a broader range of exoplanet demographics to be probed, but also for non-exoplanet science to be performed with observations of other sources such as supernovae, galaxies, and Solar System objects. The age of transits continued with the successor to Kepler, the Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015), which was launched in 2018. The purpose of TESS was to perform an all-sky survey to look for transiting planets (with a focus on Earth-sized ones) around nearby, bright stars that may be amenable to follow-up with the Hubble Space Telescope (HST) and the James Webb Space Telescope (JWST; Gardner et al. 2006; Beichman et al. 2020). TESS has a huge 24 by 90 degree field of view, which scans a single patch of sky for ∼27 days in an observing ‘sector’, before moving to the next position. The primary mission lasted for two years, in which it scanned ∼ 75% of the sky around the Northern and Southern hemispheres. TESS was approved for an extended mission lasting another two years, which included five sectors dedicated to covering ∼ 60% of the ecliptic plane (i.e. overlapping with K2 fields). The remainder of the ecliptic was not able to be targeted due to the high intensity of moonlight in this region at the time. TESS is currently in its second extended mission through to September 2025, which has seen another three sectors revisit the previous ecliptic fields, with planning for a third three-year extended mission in 6 Figure 1.2 Overlap between K2 campaigns and TESS sectors. Each point represents a K2 target, colored by the number of times it has been reobserved by TESS up to and including Sector 84, which concluded on October 26, 2024. the works. Figure 1.2 shows the overlap between the K2 and TESS fields as of Sector 84. The transit method has been wildly successful as a discovery tool, especially for blind surveys. The exoplanet community is still heavily invested in it, with many more transit-hunting facilities set to begin operations in the coming decades, including the European Space Agency’s (ESA) PLAnetary Transits and Oscillations of stars (PLATO, set to launch 2026; Rauer et al. 2014), NASA’s Nancy Grace Roman Space Telescope (formerly WFIRST; 2027 launch; Spergel et al. 2015), and various Extremely Large Telescope (ELT) class (>20m) ground-based facilities such as the European Southern Observatory’s (ESO) ELT (∼2028) and the Giant Magellan Telescope (GMT; early 2030s; e.g. Burgett et al. 2024). We have only seen the tip of the iceberg when it comes to the sheer number of exoplanets, with thousands more expected to be discovered in the near future. 7 1.2 Exoplanet Characterization With several thousands of exoplanets now confirmed, we have an abundance of worlds to study in more detail to better understand planet formation and evolution - and even search for life. A key observational diagnostic used in these studies is the chemical abundances in the planetary atmosphere. For transiting planets, the primary technique for making these measurements is transit spectroscopy, which involves observing the planet at different stages of its orbit to gain information during transit (transmission spectroscopy) or eclipse (emission spectroscopy). The focus from here on will be transmission spectroscopy, as the scheduling of these observations is the main motivator for this thesis. During a transit, light from the host star passes through the atmosphere of the planet. Interactions between the stellar light and planetary atmosphere alter the depth and shape of the transit at specific wavelengths depending on the molecules involved and their abundances. We can measure the amount of light blocked during transit as a function of wavelength, thus building a transmission spectrum. The first exoplanet transmission spectrum was that of HD 209458 b (Charbonneau et al., 2002), the same planet that had pioneered the transit method two years earlier. This was achieved using the HST Space Telescope Imaging Spectrograph (STIS), and revealed the existence of neutral sodium in the planet’s atmosphere in the form of the sodium doublet at 589.3 nm, which was later confirmed by other teams (e.g. Sing et al. 2008; Snellen et al. 2008; Albrecht et al. 2009). Although the presence of Na was consistent with models of hot Jupiters in thermodynamic equilibrium and had even been predicted for HD 209458 b (Seager & Sasselov, 2000; Brown et al., 2001; Hubbard et al., 2001), the line itself was weaker than expected. Several explanations were suggested as the cause of this, including non-local thermodynamic equilibrium (LTE) effects, clouds within the exoplanet atmosphere, photoionization from the host star, circulation due to extreme day and night temperatures if the planet was tidally locked, a non-primordial origin of the species, and even simply an inherently low sodium abundance (see Seager 2003 and references therein). More recent studies have found that the line feature is equally attributable to Na, TiO, or a combination of the two (Santos et al., 2020), or that the Na may not originate from the planetary atmosphere, but might 8 instead be an artifact of the Rossiter-McLaughlin effect and center-to-limb variations deforming the stellar spectrum (Casasayas-Barris et al., 2020, 2021; Sicilia et al., 2025). All of this is to say that detailed characterization of planetary atmospheres is incredibly complex and still in its early stages. Even with current state-of-the-art facilities, a single line feature of one of the most well-studied exoplanets (and most amenable to transmission spectroscopy) is still heavily debated: complete and detailed modeling of an atmosphere across a broad wavelength range is exponentially more challenging. Transmission spectroscopy will be an area of increasing focus and development over the coming decades, as many current and upcoming missions aim to improve on spectrograph sensitivity and resolution with the ultimate goal of accurately probing atmospheres of Earth analogs. Since obtaining the inaugural transmission spectrum over 20 years ago, HST has remained one of the predominant tools for studying atmospheres with the STIS (0.1-1.0 𝜇m; UV/optical) and Wide Field Camera-3 (WFC3; 0.8-1.6 𝜇m; near-infrared) instruments. Spitzer’s Infrared Array Camera (IRAC; Fazio et al. 2004) has also been used prolifically to probe longer wavelengths (3-8 𝜇m). From the ground, many high-resolution spectroscopy (HRS; R=25,000 - 100,000) instruments typically used for measuring RVs have also been used for transmission spectroscopy; e.g. CRIRES+ and ESPRESSO on the VLT (Holmberg & Madhusudhan, 2022; Pepe et al., 2021), the Keck Planet Imager and Characterizer (KPIC; Delorme et al. 2021), and the High Accuracy Radial velocity Planet Searcher (HARPS; Mayor et al. 2003). Transit spectroscopy was revolutionized with the much-anticipated launch of JWST (Gardner et al., 2006; Beichman et al., 2020), which boasts a wavelength range of 0.6-27.9 𝜇m across four instruments. The wide wavelength coverage allows simultaneous measurements of many molecular species of interest, particularly major hydrogen-, oxygen-, and carbon- bearing compounds. The mission has been abundantly fruitful, uncovering never-before-seen processes in exoplanet atmospheres and unexpected compositions. Major JWST results are discussed further in Chapter 4. Many other missions are in development to continue the advancement of transmission spectroscopy. ESA is developing the Atmospheric Remote-sensing Infrared Exoplanet Large-survey (Ariel) 9 mission (Tinetti et al., 2018), which has an expected launch during 2029 and nominal mission lifetime of four years. Ariel will be dedicated to studying ∼1000 pre-selected exoplanets to characterize their atmospheres in the 0.5 to 7.8 micron range. Twinkle (Edwards et al., 2019b) and Pandora (set to launch late 2025; Barclay et al. 2025) are both small satellites that will also observe several hundred planets across the visible to infrared, which will alleviate some of the over-subscription to larger missions such as JWST. As the field moves steadily towards uncovering the intricate properties of exoplanets, it is becoming increasingly clear that preparation is needed ahead of making these detailed observations. 1.3 Ephemeris Degradation A key factor in scheduling observations for transit spectroscopy is knowing when a planet will transit its host star. As with any measurement, there is uncertainty associated with the measured transit time of an exoplanet, which depends on the transit midpoint and orbital period of the planet at discovery. These uncertainties are typically on the order of minutes to a few hours, which in itself would not be an issue if it were not for the fact that they compound with every successive transit. The uncertainty on the transit time is simply √︃ (𝑛 × 𝜎2 𝑃) + 𝜎2 𝑇𝑐 (1.3) where 𝑛 is the number of transits that have occurred in a given time, 𝑇𝑐 is the transit midpoint in BJD, and 𝑃 is the orbital period of the planet. A planet with a 10 day period and an uncertainty of 10 minutes on both 𝑃 and 𝑇𝑐 will have a transit time uncertainty of ∼1 hour after a year. This may sound negligible, but most exoplanets were discovered more than eight years ago and have much larger uncertainties on 𝑃 and 𝑇𝑐, so in reality have ephemeris uncertainties on the order of tens of hours (Figure 1.3). Facilities like JWST are highly competitive and costly to run2, and any unnecessary overheads need to be minimized. In order to have an appropriate baseline and to fully capture the ingress and egress of the planet, it is typical for TS observations to include ∼an hour either side of the transit. 2The monetary cost of flagship missions is in the ballpark of 10-100k USD per hour to operate when considering the operational costs alone, not including design, construction, launch, etc. 10 Figure 1.3 The 3𝜎 uncertainties on predicted transit times for all transiting exoplanets by the year towards the end of the nominal JWST mission lifetime. Planets with uncertainties 2030, i.e. greater than 30 minutes are in the red shaded region, which is the ideal maximum when proposing for transmission spectrum observations. Most current transiting planets will be difficult, if not impossible, to observe for this purpose if their ephemerides are not updated. Values taken from the NEA, accessed February 5, 2025. In practice, this should be added to the uncertainty of the transit time to ensure the observation of a complete transit, which can significantly increase the amount of requested telescope time to unrealistic levels. Alternatively, if the compounded uncertainty is ignored and observations scheduled, it is possible that the transit is only partially captured, or missed altogether. Thus, an uncertainty of a few minutes from the discovery of the planet can quickly become an unsuccessful proposal or wasted opportunity. Ephemeris degradation is combated by reobserving transits to refine the precision on 𝑃 and 𝑇𝑐. The degree of improvement on the uncertainties increases with the length of the temporal baseline, i.e. number of transits that have occurred between observations. The solution to the ephemeris problem may be as simple as the problem itself, but a hidden complication arises when considering the resources needed to perform this on a large scale. Updating individual systems is achievable 11 with targeted observations, but is not feasible to address the hundreds of planets for which this is needed. Fortunately, the TESS mission is observing nearly the entire sky, with some regions of the sky being covered by multiple sectors. While TESS has discovered hundreds of new planets, it is also providing a timely opportunity to update transit times for hundreds, if not thousands, of already-known planets. The ephemeris problem is always going to be a part of utilizing the transit method. Even planets discovered by TESS will need to be revisited by future missions, as their ephemerides degrade significantly after only a year from discovery (Dragomir et al., 2020). As long as transiting planets are studied for their atmospheres, endeavors to keep ephemerides current will need to be carried out. 1.4 Ephemeris Refinement Projects Several efforts exist to update transit times for a large number of exoplanets, in some cases utilizing the power of citizen science. ExoClock3 (Kokori et al., 2021, 2022, 2023) is an open project designed to update ephemerides of most targets of the future Ariel mission. It began in September 2019 and has over 500 members, most of whom are citizen scientists/amateur astronomers, resulting in access to 450 telescopes worldwide. Ephemerides are updated using new data from these observations, as well as from the literature, the Exoplanet Transit Database (Poddaný et al., 2010), and from space missions (i.e. Kepler, K2, and TESS). Currently, the ephemerides of 450 planets have been updated through this effort. Exoplanet Watch4 is a NASA-led project heavily based on citizen science (Zellem et al., 2019, 2020). Similar to ExoClock, amateur astronomers from anywhere in the world can contribute with their own telescopes. This project has not only resulted in updated ephemerides for known planets, but has contributed to the discovery of an eccentric warm Jupiter (Sgro et al., 2024) and been used as part of undergraduate research experience (Hewitt et al., 2023). The Transit Ephemeris Refinement and Monitoring Survey (TERMS; Kane et al. 2009) is an ongoing effort to update ephemerides and system parameters for planets discovered through RVs. 3https://www.exoclock.space/ 4https://exoplanets.nasa.gov/exoplanet-watch/ 12 Orbital parameters for each planet are determined through the RV measurements, from which a transit window is calculated. This project highlights the importance of using both RVs and transits to better characterize exoplanets, and shows that observation techniques can be used to inform one another. All of these efforts will ensure the continued accessibility of hundreds of transiting exoplanets, however, as discussed previously, this will be an ongoing problem needing constant addressing. 1.5 The K2 & TESS Synergy The K2 & TESS Synergy is a large scale, dedicated effort to reanalyze known planets from the K2 mission with recently acquired TESS observations. The main difference setting it apart from the previously-mentioned projects is that it will result in a self-consistent catalog of not only transit times, but fully updated system parameters (for both the planets and host stars). This highlights another general problem in the field: many population studies to date use measurements made by different teams using various analysis methods, assumptions, and software, meaning artificial trends in the data may be introduced. In this section, we describe how the K2 & TESS Synergy project was started and continued with the work in this thesis. 1.5.1 Pilot Study The K2 & TESS Synergy began with a pilot study (predating the work in this thesis) by Ikwut- Ukwa et al. (2020) that considered four single-planet K2 systems (K2-114, K2-167, K2-237, and K2-261) for ephemeris renewal using TESS. This was a proof of concept to show the severity of ephemeris degradation for typical K2 targets (Figure 1.4), and how TESS would be a powerful tool for combating the problem on a large scale. At this time, TESS was still in its first year of operation and had only clipped the very edges of the K2 campaigns, which meant a single sector of data was available for each of the four planets. By performing global fits including the new TESS light curves, the ephemerides for all four planets were greatly improved: at a 3𝜎 level, the transit time uncertainty at 2030 went from 7.6 hours to 8 minutes for K2-114, 40.6 hours to 1.1 hours for K2-167, 55 minutes to 8 minutes for K2-237, and 3.5 hours to 30 minutes for K2-261. 13 Figure 1.4 Same as Figure 1.3, with K2 candidate and confirmed planets overlaid in black and purple, respectively. Most planets discovered by the K2 mission have severely degraded ephemerides, mainly due to being discovered prior to 2018 and having not been reobserved since. Values taken from the NEA, accessed February 5, 2025. 1.5.2 This Work With the success of the pilot study, the process was proven to be a viable method to achieve significant improvement on the transit times that could be extended to a larger sample as TESS sectors increasingly overlapped with the K2 footprint. It is this extension of the K2 & TESS Synergy that constitutes the work presented in this thesis. Beyond updating stale ephemerides, the systematic reanalysis of all K2 systems allows us to amend any incorrect values in the literature. Throughout this work, the same steps were followed to ensure consistency across all fits: acquire and prepare light curves from K2 and TESS for analysis, collate RVs from the literature, obtain parallaxes, stellar SEDs, and metallicities, and then run global fits. This process is detailed further in Sections 2.3 and 2.4. The reason for selecting the K2 catalog specifically for this project is manyfold. As most of the K2 targets were closer and brighter than those from the Kepler field, there are many overlapping targets with TESS. These targets included many M dwarfs, which are increasingly sought-after 14 systems when it comes to finding small planets within their habitable zones. Additionally, many K2 targets have not been reobserved since their original discovery, now 6 ∼ 10 years ago. The ephemerides for most these systems have uncertainties well beyond the limit for transmission spectroscopy observations (Figure 1.4). As our goal is to update ephemerides for targets amenable to future atmospheric characterization, we prioritize planets by their transmission spectroscopy metric (TSM; Kempton et al. 2018), which is an estimate of the expected signal to noise of a transmission spectrum. This ensures targets likely to be favorable for JWST observations will be updated first. With the updated global parameters obtained via the same analysis procedure, we can also perform appropriate population studies within the K2 sample in future work. 1.5.3 EXOFASTv2 The core philosophy of the K2 & TESS Synergy is to produce self-consistent global parameters for all of the systems that are addressed. This is where the choice of fitting software plays a major role. A plethora of modeling suites exists for exoplanet data, with many fitting either the photometry (e.g. Transit Analysis Package, Gazak et al. 2012; BATMAN, Kreidberg 2015; PYTRANSIT, Parviainen 2015) or radial velocities (e.g. RadVel, Gazak et al. 2012; Systemic, Meschiari et al. 2009), or both (e.g. Exonailer, Espinoza 2018; the Transit and Light Curve Modeller, Csizmadia 2020; Pyaneti, Barragán et al. 2019a; exoplanet, Foreman-Mackey et al. 2021; Juliet, Espinoza et al. 2019; allesfitter, Günther & Daylan 2021; ellc, Maxted 2016). Many of these fitting tools incorporate lower-level generalized statistical codes to sample the data (e.g. dynesty, Speagle 2020; emcee, Foreman-Mackey et al. 2013; celerite, Foreman-Mackey et al. 2023; george, Ambikasaran et al. 2015; PyMC3, Salvatier et al. 2016). The major differences between these codes are their approach to fitting (e.g. MCMC, nested sampling), and capabilities for fitting other phenomena present in the data (e.g. star spots and flares, transit timing variations). The fitting software of choice for the K2 & TESS Synergy is EXOFASTv2 (Eastman et al., 2013; Eastman, 2017; Eastman et al., 2019), which is, to our understanding, the most complete fitting tool for exoplanets. The major advantage of EXOFASTv2 compared to other available software is the 15 fact that it simultaneously fits parameters of the planet(s) and host star(s), meaning parameters are concurrently being fit while guiding the direction of the fit for related parameters. This is becoming increasingly pertinent, especially if we heed the popular adage of "know thy star, know thy planet". For example, the transit duration can help constrain the stellar density, while the planetary radius and mass rely on precisely knowing those of the host star. EXOFASTv2 uses a Differential Evolution (DE) MCMC to fit system parameters from the given data sets. The DE-MCMC approach allows for multiple chains to be run in parallel and ‘learn’ from each other. In this case, the number of chains is set to twice the number of dimensions (typically on the order of ∼ 40). EXOFASTv2 has the capability to simultaneously fit many data sets. Along with photometry and spectroscopy, it can fit for stellar parameters if a spectral energy distribution (SED) of the star is provided, in conjunction with stellar evolutionary models selected by the user, the default being the MESA Isochrones and Stellar Tracks (MIST; Dotter 2016; Choi et al. 2016). Additionally, the limb darkening coefficients are fit for each photometric band. Both additive and multiplicative detrending can be accounted for in the light curve and RV curve, along with dilution of the light curve due to contamination of the photometric aperture. EXOFASTv2 is able to model systems with multiple planets and/or stars. Other features of EXOFASTv2 that are not used within this work include modeling Doppler Tomography, and fitting for variations in transit timing, depth, and duration. The user is able to provide starting points and/or priors for each parameter that is being fitted. So as not to underestimate uncertainties, we limit the parameters that we place priors on to stellar metallicity, extinction, dilution, and parallax. Further details on the fitting process are outlined in Section 2.4. EXOFASTv2 is currently the most thorough fitting code for exoplanets, however, it comes with the inevitable high computation cost. Using a supercomputer, fits can take on the order of days to weeks to run, thus making the entire reanalysis a long-term endeavor. While it would certainly be faster to use a simpler fitting suite, or only fit photometry (as with the other large ephemeris projects), the use of EXOFASTv2 allows us to address two problems in one go - stale ephemerides and inconsistent system parameters. These outcomes are worth the time investment to prepare the 16 field for the coming years of precision observations. 1.6 Outline Chapter 2 is the published work of Thygesen et al. (2023), which included the first large batch of systems for the K2 & TESS Synergy. In this work, we addressed 26 single-planet systems that were observed by TESS in its primary mission. We were able to significantly improve the ephemerides of half of the systems, from an average 3𝜎 uncertainty by 2030 of 26.7 hours to 35 minutes. The remaining 13 planets did not have transits of high enough signal to noise to be detected in the TESS light curves, but we still ran the global fits with only the K2 light curves, which saw an average improvement of the ephemerides from 43.2 to 35.6 hours. In Chapter 3, we take a detour to investigate the lost ephemeris of K2’s first planet discovery, K2-2 b. This is the third installment of the K2 & TESS Synergy, published as Thygesen et al. (2024). A factitious transit in a secondary light curve originally used to confirm this planet meant the period was off by 28.8 minutes (∼ 40𝜎), which resulted in the transit being missed during targeted follow-up with HST and Spitzer. We found the true transit time with light curves obtained from ground- and space-based facilities over a seven year span, and uncovered a potential outer planet in the system via a long term trend in the radial velocities. Chapter 4 consists of the current work on the next installment of the K2 & TESS Synergy. This batch contains the top 50 K2 planets amenable to transmission spectroscopy, which includes seven planets previously analyzed as part of the K2 & TESS Synergy. We have currently improved the average 3𝜎 uncertainties on transit times by 2030 from 17.4 hours to 16 minutes for 34 of these planets, ensuring these planets can be targeted for future observations. Finally, the work in this thesis is summarized in Chapter 5, along with potential paths forward for the K2 & TESS Synergy project. 17 CHAPTER 2 THE K2 & TESS SYNERGY II: REVISITING 26 SYSTEMS IN THE TESS MISSION This chapter was published in Thygesen et al. (2023). Tables containing the results of the global fits are included as supplementary material. 2.1 Abstract The legacy of NASA’s K2 mission has provided hundreds of transiting exoplanets that can be revisited by new and future facilities for further characterization, with a particular focus on studying the atmospheres of these systems. However, the majority of K2-discovered exoplanets have typical uncertainties on future times of transit within the next decade of greater than four hours, making observations less practical for many upcoming facilities. Fortunately, NASA’s Transiting Exoplanet Survey Satellite (TESS) mission is reobserving most of the sky, providing the opportunity to update the ephemerides for ∼300 K2 systems. In the second paper of this series, we reanalyze 26 single-planet, K2-discovered systems that were observed in the TESS primary mission by globally fitting their K2 and TESS lightcurves (including extended mission data where available), along with any archival radial velocity measurements. As a result of the faintness of the K2 sample, 13 systems studied here do not have transits detectable by TESS. In those cases, we re-fit the K2 lightcurve and provide updated system parameters. For the 23 systems with 𝑀∗ ≳ 0.6 𝑀⊙, we determine the host star parameters using a combination of Gaia parallaxes, Spectral Energy Distribution (SED) fits, and MESA Isochrones and Stellar Tracks (MIST) stellar evolution models. Given the expectation of future TESS extended missions, efforts like the K2 & TESS Synergy project will ensure the accessibility of transiting planets for future characterization while leading to a self-consistent catalog of stellar and planetary parameters for future population efforts. 2.2 Introduction The past two decades have been fruitful for exoplanet discovery, with over 5000 exoplanets confirmed to date1. While new discoveries are still being made, we are simultaneously venturing into an era of exploring known systems in further detail, with a variety of dedicated efforts for 1https://exoplanetarchive.ipac.caltech.edu/ 18 Figure 2.1 Overlap between K2 campaigns and TESS sectors. The number of times each K2 target was observed in TESS sectors is indicated by the color, with gray indicating no TESS overlap as of Sector 46. The systems analyzed in this study are labeled. exoplanet characterization. Facilities that are operational or expected to be online in the next decade such as JWST (Gardner et al., 2006; Beichman et al., 2020), 39 m European Southern Observatory Extremely Large Telescope (ELT; Udry et al. 2014), Nancy Grace Roman Space Telescope (e.g. Carrión-González et al. 2021), Giant Magellan Telescope (Johns et al., 2012) and Atmospheric Remote-sensing Infrared Exoplanet Large-survey (ARIEL; Tinetti et al. 2018, 2021) will provide key information about the atmospheres of exoplanets, and insight into their formation and evolutionary processes. However, these ongoing and future endeavors to reobserve known transiting exoplanets heavily rely on precisely knowing the transit time, which is challenged by the degradation of the ephemeris over time. Most exoplanets and candidates found to date were originally discovered by the Kepler mission (Borucki et al., 2010). Kepler was launched in 2009 with the goal of understanding the demographics of transiting exoplanets. This mission was a success, having discovered ∼2700 confirmed planets 19 with a further ∼2000 candidates2, in addition to advancing our understanding of the host stars they orbit (e.g. Bastien et al. 2013; Berger et al. 2020a,b). However, by May of 2013 two of the four reaction wheels on the spacecraft had failed, severely limiting the pointing of Kepler, threatening to end the mission. A solution was conceived to point the spacecraft at the ecliptic to reduce torque from Solar radiation pressure, so that the remaining two reaction wheels, along with the thrusters, could maintain sufficient stability. This saw Kepler successfully reborn as the K2 mission (Howell et al., 2014). While Kepler continuously pointed at one region of sky, the necessity of K2 being aimed along the ecliptic opened up an opportunity to study different populations of stars. K2 continued on the path of exoplanet discovery, with currently ∼500 confirmed planets and another ∼1000 candidates found by the time the spacecraft retired in 2018 when fuel for the thrusters ran out (Vanderburg et al., 2016; Zink et al., 2021; Kruse et al., 2019; Pope et al., 2016; Livingston et al., 2018a; Crossfield et al., 2016; Dattilo et al., 2019). Unfortunately, many of the known planets discovered by the K2 mission have not been reobserved since their discovery, leading to future transit time uncertainties of many hours (Ikwut- Ukwa et al., 2020). This has recently changed with the launch of NASA’s Transiting Exoplanet Survey Satellite (TESS) mission in 2018 (Ricker et al., 2015), the successor to the Kepler and K2 missions. The two-year primary mission of TESS aimed to observe more than 200,000 stars at two-minute cadence across ∼75% of the sky. To date, TESS has found ∼280 confirmed planets and another ∼6100 candidates3. Even though K2 targeted the ecliptic plane and the TESS primary mission only skimmed the edges of some K2 fields, there are ∼30 systems that were observed by both (single- and multi-planet systems). This provides an opportunity to begin updating the ephemerides and parameters of K2 systems that have been reobserved by TESS. The first extended mission of TESS began during 2020, and includes sectors dedicated to the ecliptic plane, providing more substantial overlap of a further ∼300 systems with the K2 fields4 (Figure 2.1). With TESS scheduled to reobserve nearly the entire sky during its extended missions, it will be a useful tool 2https://exoplanetarchive.ipac.caltech.edu/ 3https://nexsci.caltech.edu/ 4https://heasarc.gsfc.nasa.gov/docs/tess/the-tess-extended-mission.html 20 for refreshing the ephemerides of thousands of transiting exoplanets. Currently, many known exoplanets do not have sufficiently accurate projected transit times to plan observations with future missions. Even TESS ephemerides will need to be updated as most TESS planets will have transit time uncertainties exceeding 30 minutes in the era of JWST (Dragomir et al., 2020). With the wealth of data coming from ongoing surveys like TESS and the ability to follow up many planets with small aperture (<1 m) telescopes (Collins et al., 2018), many efforts have begun to keep the ephemerides of transiting planets from going stale, like the ExoClock Project (Kokori et al., 2021, 2022) for future ARIEL targets and the K2 & TESS Synergy (Ikwut-Ukwa et al., 2020). Ephemeris refinement programs focused on citizen science (Zellem et al., 2019, 2020) and high-school students (e.g. ORBYTS; Edwards et al. 2019a, 2020, 2021) also provide opportunities to actively engage the public while contributing to an essential aspect of future exoplanet characterization. These efforts will be key to making a large number of systems accessible for future facilities. A continual renewal of ephemerides also presents an opportunity to create self-consistent catalogs of exoplanets and their parameters, which not only helps to plan for future missions, but also allows for appropriate population studies using data that have been uniformly prepared. While the vast amount of data available per system makes this a challenge, the advent of new exoplanet fitting suites to globally analyze large quantities of data, like Juliet (Espinoza et al., 2019), EXOFASTv2 (Eastman et al., 2013; Eastman, 2017; Eastman et al., 2019), Allesfitter (Günther & Daylan, 2021) and exoplanet (Foreman-Mackey et al., 2021), has made it possible to individually model the available observations for a large sample of exoplanetary systems. These types of studies are necessary to uncover large-scale trends or mechanisms that may play important roles in planet formation and evolution. A renowned example is the radius valley of small planets (Fulton et al., 2017), which was achieved through more accurate and consistent handling of host star parameters for over 2000 planets from the California-Kepler Survey. A case study for updating K2 ephemerides and system parameters with new TESS data was presented in the first paper of this series (Ikwut-Ukwa et al., 2020), where four K2-discovered 21 systems (K2-114, K2-167, K2-237 and K2-261) were reanalyzed by performing global fits using K2 and TESS lightcurves. This resulted in the uncertainties for the transit times of all four planets being reduced from multiple hours to between 3-26 minutes (at a one sigma level) throughout the expected span of the JWST primary mission, showcasing the value of combining the K2 and TESS data. We continue this work by reanalyzing a sample of 26 single-planet systems observed with K2 and the primary TESS mission (including refitting the original four systems for consistency), while also making use of archival radial velocities, Gaia parallaxes and any currently available lightcurves from the TESS extended mission. We focus on previously-confirmed single-planet systems, but future papers in this series are expected to reanalyze all K2 systems (including multi- planet systems) as part of an ongoing TESS guest investigator program (G04205, PI Rodriguez). Updated transit times will be made available to the community throughout this series through the Exoplanet Follow-up Observing Program for TESS (ExoFOP)5. In §2.3 we describe how we obtained and prepared the data used in our global fits. §2.4 outlines how we ran the EXOFASTv2 analysis, and §2.5 presents our results along with any peculiarities for specific systems. Our conclusions are summarized in §2.6. 2.3 Observations and Archival Data Given that most known K2-discovered exoplanet systems will have uncertainties larger than 30 minutes (see Figure 2.2), we take advantage of the high-quality data obtained with K2 and TESS, simultaneously fitting the photometry and archival spectroscopy to update system parameters for 26 K2 systems. Here we describe the techniques used to obtain and process K2 and TESS lightcurves, as well as radial velocities from existing literature. 2.3.1 K2 Photometry Each of these stars was observed by the Kepler spacecraft during its K2 extended mission (Howell et al., 2014). During K2, the spacecraft’s roll angle drifted significantly due to the failure of two reaction wheels, which introduced significant systematic errors into its lightcurves6. Over 5https://exofop.ipac.caltech.edu/tess/ 6The two remaining reaction wheels onboard K2 could control the position of the telescope’s boresight, but the roll angle could only be controlled by occasional firing of the thrusters about every 6 hours as radiation pressure caused 22 Figure 2.2 Uncertainty of the transit time (𝜎𝑇𝑐 ) for K2 candidate and confirmed planets at the year 2030, based on the discovery ephemeris. The majority of planets have uncertainties greater than 30 minutes (indicated by the red region) in the era of JWST, making these challenging to reobserve. Values taken from the NASA Exoplanet Archive (NEA) default parameter sets. the course of the mission, a number of different techniques and methods were developed to mitigate these errors (e.g. Aigrain et al. 2016; Barros et al. 2016; Luger et al. 2016; Lund et al. 2015; Pope et al. 2019). In this work, we used the methods of Vanderburg & Johnson (2014) and Vanderburg et al. (2016) to derive a rough systematics correction. In brief, these methods involve extracting raw lightcurves from a series of 20 different photometric apertures, correlating short timescale variations in the raw lightcurves with the spacecraft’s roll angle (which changes rapidly due to K2’s unstable pointing), and subtracting variability correlated with the spacecraft’s roll angle. The process of correlating and subtracting variability correlated with the roll angle is performed iteratively until the only remaining variations in the lightcurve are unrelated to the spacecraft’s roll. Finally, we select the aperture that produces the most precise lightcurve among the 20 originally extracted. Then, we refined the systematics correction by simultaneously fitting the transits for each planet along with the systematics correction and low-frequency stellar variability, prior to the the telescope to slowly roll about its long axis. 23 final global fit. Most of the data we analyzed were collected in 30-minute long-cadence data, but when available, we analyzed 1-minute short-cadence exposures for better time sampling. For all systems, we only included out-of-transit data from one full transit duration before and after each transit. This is to optimize the balance between having enough data points to establish the baseline flux of the star and lengthening the runtime of the fits due to having more data. 2.3.2 TESS Photometry While all 26 systems were initially observed by TESS in the primary mission, each was reobserved in at least one sector of the first extended mission. We therefore included TESS lightcurves from the primary and extended missions up to and including Sector 46 (as of February 1, 2022). This was the final sector dedicated to the ecliptic plane for the first extended mission. Future efforts in this series will analyze systems that were first observed by TESS during the first extended mission and beyond. We used the Python package Lightkurve (Lightkurve Collaboration et al., 2018) to retrieve TESS lightcurves from the Mikulski Archive for Space Telescopes (MAST). Three systems within the footprint of the TESS primary mission (K2-42, K2-132/TOI 2643 and K2-267/TOI 2461) did not have corresponding retrievable lightcurves, which is likely due to being too close to the edge of the detector, so we excluded these from the current analysis. For the TESS lightcurves, we used the Pre-search Data Conditioned Simple Aperture Photometry (PDCSAP) flux, which is the target flux within the optimal TESS aperture that has been corrected for systematics with the PDC module (Stumpe et al., 2012, 2014; Smith et al., 2012). Typically, observations for each sector are processed through the Science Processing Operations Center (SPOC) pipeline at the NASA Ames Research Center (Jenkins et al., 2016). The SPOC pipeline takes in the raw data and applies corrections for systematics, runs diagnostic tests and identifies transits, resulting in a calibrated lightcurve that can be used for analysis. TESS science observations are taken at 20-second and 2-minute cadences (the former only becoming available from the first extended mission), while the Full Frame Images (FFIs) are created every 30 minutes during the primary mission, and every 10 minutes since the first extended 24 mission. For our global analysis, (see §2.4) we used the shortest cadence available, preferentially using data processed through SPOC (Jenkins et al., 2016; Caldwell et al., 2020). The increased timing precision of short cadence observations is only valuable if there is a significant detection of the transit. For this reason, and since TESS is optimized for targets with brighter magnitudes than those of K2, we binned lightcurves observed at 20-second cadence to two minutes to increase signal-to-noise. If a TESS-SPOC FFI lightcurve was not available for a particular sector, we extracted the lightcurve using a custom pipeline as described in Vanderburg et al. (2019). The pipeline uses a series of 20 apertures from which lightcurves are extracted and corrected for systematic errors from the spacecraft by decorrelating the flux with the mean and standard deviation of the quaternion time series. Dilution from neighbouring stars within the TIC is corrected for within each aperture, which takes into account the TESS pixel response function. The final aperture used for the lightcurve extraction is selected as the one that minimized the scatter in the photometry. Recent efforts have compared this custom pipeline with other FFI pipelines (Rodriguez et al., 2022a), supporting our adoption of this pipeline. The list of available lightcurves (as of February 1, 2022) is shown in Table 3.2. 25 Table 2.1 Target list and data used in this analysis. TIC ID KID EPIC ID K2 Campaign TESS Sector K2-7 53210555 K2-54† 12822545 146799150 K2-57 435339847 K2-77 366568760 K2-97 366410512 K2-98 366576758 K2-114 7020254 K2-115 398275886 K2-147† 69747919 K2-167 366411016 K2-180 366528389 K2-181 366631954 K2-182 333605244 K2-203 248351386 K2-204 399722652 K2-208 399731211 K2-211 98677125 K2-225 176938958 K2-226 K2-237 16288184 98591691 K2-250 293612446 K2-260 281731203 K2-261 146364192 K2-265 404421005 K2-277 277833995 K2-321† 201393098 205916793 206026136 210363145 211351816 211391664 211418729 211442297 213715787 205904628 211319617 211355342 211359660 220170303 220186645 220225178 220256496 228734900 228736155 229426032 228748826 246911830 201498078 206011496 212357477 248480671 C1 C3 C3 C4 C5, C18 C5, C18 C5, C18 C5, C18 C7 C3 C5, C18 C5, C18 C5, C18 C8 C8 C8 C8 C10 C10 C11 C10 C13 C14 C3 C6 C14 (2 min) 9, 36, 45, 46 2, 42 2, 29 5′, 42′, 43′, 44′ 7′, 44′, 45′, 46′ 7, 34, 44, 45, 46 7, 44, 45, 46 7, 34, 45, 46 27 2, 28, 42, 34, 44, 45, 46 7, 44, 45, 46 34, 44, 45, 46 30, 42, 43 30, 42, 43 30, 42, 43 30, 42, 43 36, 46 36, 46 12, 39 36, 46 32, 43 9, 35, 45, 46 29, 42 10, 37′ 8′, 45′, 46′ K2 RV Ref. Ref. — 1 — 2 — 2 — 3 1, 2 4 3 5 4 6 4 6 — 7 — 3 5 8 — 3 6 9 — 3 — 3 — 3 — 7 — 3 — 3 10 7, 8 — 11 12 9 12 9 13 10 4 — 35′ (10min) — 14 (FFI) — — — — — — — — 13 — 7∗ — 7 3 3 3 3 10 10 — 10 5∗ — 2 — TESS SNR 5.62 1.76 1.99 13.52 16.76 20.93 134.03 88.19 2.91 13.82 12.03 5.74 32.39 3.17 5.44 4.76 2.90 2.93 3.82 129.83 3.79 98.44 83.58 6.01 8.75 9.21 Notes: TESS sectors in which transits had SNR≤7 and thus were too shallow to be recovered are colored red. † The host stars in these systems were classed as low mass (≲ 0.6 𝑀⊙), so we did not include the SEDs in the global fits. See Section 2.4 for details. ′ The full lightcurves for these were used to ensure the transit was able to be detected. All other lightcurves were sliced as discussed in Section 2.3. ∗ A custom pipeline was used to extract lightcurves for sectors without TESS-SPOC FFIs as discussed in Section 2.3.2. References for RV measurements: 1Grunblatt et al. (2016), 2Grunblatt et al. (2018), 3Barragán et al. (2016), 4Shporer et al. (2017), 5Korth et al. (2019), 6Akana Murphy et al. (2021), 7Soto et al. (2018), 8Smith et al. (2019), 9Johnson et al. (2018a), 10Lam et al. (2018) K2 references: 1 - Montet et al. (2015), 2 - Crossfield et al. (2016), 3 - Mayo et al. (2018), 4 - Livingston et al. (2018a), 5 - Barragán et al. (2016), 6 - Shporer et al. (2017), 7 - Adams et al. (2021), 8 - Korth et al. (2019), 9 - Akana Murphy et al. (2021), 10 - Soto et al. (2018), 11 - Livingston et al. (2018b), 12 - Johnson et al. (2018b), 13 - Lam et al. (2018), Castro González et al. (2020) 26 After retrieving the TESS lightcurves for our targets, we processed them further for our own analysis, assuming values for transit duration, time of conjunction (𝑇𝑐) and period from the NASA Exoplanet Archive (NEA). To flatten the out-of-transit lightcurve for fitting, we used keplerspline7, a spline-fitting routine to model and remove any variability from the star or remaining systematics (Vanderburg & Johnson, 2014). Within keplerspline, the spacing between breaks in the spline to handle discontinuities is optimized by minimizing the Bayesian Information Criterion (BIC) for different break points (see Shallue & Vanderburg 2018 for further methodology). We applied a constant per-point error for the photometry, calculated as the median absolute deviation of the out-of-transit flattened lightcurve, although this error is optimized within our analysis since EXOFASTv2 fits a jitter term. If any lightcurve had large outliers or features that may influence our transit fit, we used only the data that had no bad quality flags within Lightkurve (this was only the case for K2-250 and K2-260). To reduce the individual runtime for each system, we excluded the out-of-transit baseline of the TESS lightcurves from the EXOFASTv2 fit other than one full transit duration before and after each transit (as with the K2 lightcurves). However, for systems whose transits were not readily visually identified in the TESS data (K2-77, K2-97, K2-277 and K2-321; see Table 3.2), we included all out-of-transit photometry to account for any large uncertainties in the time of transits during the TESS epochs. 2.3.3 Archival Spectroscopy We identified spectroscopic observations from the literature for 10 of the 26 total targets (Figures 2.3 and 2.4; K2-97, K2-98, K2-114, K2-115, K2-180, K2-182, K2-237, K2-260, K2-261 and K2- 265; Grunblatt et al. 2016, 2018; Barragán et al. 2016; Shporer et al. 2017; Korth et al. 2019; Akana Murphy et al. 2021; Soto et al. 2018; Smith et al. 2019; Johnson et al. 2018a; Lam et al. 2018). We selected data sets with four or more RV measurements to ensure more degrees of freedom in the global fit, thus avoiding overfitting the data. For this reason we do not include RVs for K2-77 (Gaidos et al., 2017) and K2-147 (Hirano et al., 2018). Table 3.2 lists the analyses from which we obtained each set of RVs that we incorporated in the global analysis (see §2.4). All but one of 7https://github.com/avanderburg/keplerspline 27 Figure 2.3 Radial velocities for the 10 systems with archival spectroscopic measurements. The best fit model from EXOFASTv2 is shown in each subplot. Each set of RVs are phased using the best fit period and 𝑇𝑐 determined in the fit, and the residuals are shown below each dataset. The references for each set of RVs are listed in Table 2.1. the systems that have RVs also have significant TESS transits (see §2.4), which is an outcome of spectroscopic measurements preferentially targeting brighter stars. The archival RVs were obtained from the following instruments: the Levy spectrometer on the 2.4m Automated Planet Finder at Lick Observatory, the High Resolution Echelle Spectrometer (HIRES) on the Keck-I Telescope (Vogt et al., 1994), the FIbre-fed Echelle Spectrograph (FIES) on the 2.56m Nordic Optical Telescope at Roque de los Muchachos Observatory Frandsen & Lindberg (1999), the High Accuracy Radial velocity Planet Searcher (HARPS) spectrograph on the 3.6m telescope at La Silla Observatory (Mayor et al., 2003), HARPS-N on the 3.58m Telescopio Nazionale Galileo at the Roque de los 28 Figure 2.4 Radial velocities (continued). Muchachos Observatory (Cosentino et al., 2012), and the CORALIE spectrograph on the Swiss 1.2m Leonhard Euler Telescope at La Silla Observatory (Queloz et al., 2000). If any determination for the host star’s metallicity ([Fe/H]) was available, we included it as a prior in the fit to better constrain the host star parameters. For consistency, we used metallicity priors for most of the systems from spectra obtained using the Tillinghast Reflector Echelle Spectrograph (TRES; Fűrész 2008) on the 1.5m Tillinghast Reflector at the Fred L. Whipple Observatory (FLWO). Starting points were used for other stellar parameters where available, but no prior constraints were placed on any other values. We assumed the RV extraction and metallicity determination was done correctly in the discovery data. An RV jitter term is fit within the EXOFASTv2 analysis to ensure the uncertainties are properly estimated. In the cases of five or fewer RVs, we placed conservative uniform bounds on the variance of the jitter. The jitter variance for K2-114 and for the Soto et al. (2018) RVs for K2-237 were bounded to ±300 m/s, and for K2-98 the variance bounds were ±100 m/s for the FIES RVs, and ±4 m/s for HARPS and HARPS-N. For the HARPS RVs of K2-265, we removed three clear outliers that were included in the discovery paper based on visual inspection8. 8All parameters were within uncertainties when compared to an earlier fit including the outliers. 29 2.4 Global Fits To analyze the wealth of data for these 26 known K2 exoplanet systems, we used EXOFASTv2 (Eastman et al., 2013, 2019; Eastman, 2017) to perform global fits for our sample. EXOFASTv2 is an exoplanet fitting software package that uses MCMC sampling to simultaneously fit parameters for both the planets and host star. The K2 and TESS photometric observations (Figures 2.5 and 2.6), along with any archival RVs (Figures 2.3 and 2.4), were jointly analyzed to obtain best-fit parameters for planets and host stars. 2.4.1 Stellar parameters To characterize the host stars within each fit, we placed a uniform prior from 0 to an upper bound on line-of-sight extinction (𝐴𝑣) from Schlegel et al. (1998) and Schlafly & Finkbeiner (2011), and Gaussian priors on metallicity ([Fe/H]) and parallax (using Gaia EDR3 and accounting for the small systematic offset reported; Gaia Collaboration et al. 2016, 2021; Lindegren et al. 2021). This also included the spectral energy distribution (SED) photometry as reported by Gaia DR2 (Gaia Collaboration et al., 2018), WISE (Cutri et al., 2012) and 2MASS (Cutri et al., 2003). These values can be found in full in Thygesen et al. (2023). We excluded the WISE4 SED values for three systems that had this photometric measurement (K2-115, K2-225 and K2-237) due to the large uncertainties, and as there was a ≳ 2𝜎 discrepancy with the stellar model. Two other systems (K2-167 and K2-277) had WISE4 measurements that we used in the fits; these are consistent with the stellar models, but still have relatively large uncertainties. Within the EXOFASTv2 global fit, the MESA Isochrones and Stellar Tracks (MIST) stellar evolution models (Paxton et al., 2011, 2013, 2015; Choi et al., 2016; Dotter, 2016) are used as the base isochrone to better constrain the host star’s parameters. 2.4.2 Low-mass stars Stellar evolutionary models struggle to constrain low-mass stars (≲ 0.6 𝑀⊙; Mann et al. 2015) and are thus unreliable. For the three systems that fell into this category (K2-54, K2-147 and K2-321), we used the equations from Mann et al. (2015, 2019) that relate the apparent magnitude in the 𝐾𝑆 band (𝑀𝐾𝑠 ) to 𝑀∗ and 𝑅∗ to set a starting point with wide 5% Gaussian priors for these 30 parameters. We excluded the SEDs from these fits and did not use the MIST models, fitting only the lightcurves (these systems did not have RV measurements). For this reason, we caution that the stellar parameters for these systems are unreliable. We also did not use the limb-darkening tables from Claret (2017) for the low-mass stars, as is the default in EXOFASTv2 for fitting the 𝑢1 and 𝑢2 coefficients, but rather placed starting points based on tables from Claret & Bloemen (2011) (Eastman et al., 2013) with a conservative Gaussian prior of 0.2 (Patel & Espinoza, 2022). 2.4.3 Contamination For systems with TESS contamination ratios specified in the TESS input catalog (TICv8, Stassun et al., 2018) and a clear transit detected in both K2 and TESS, we fit for a dilution term9 on the TESS photometry with a 10% Gaussian prior. This accounts for any nearby sources that may contribute flux to the target aperture that were unknown at the time the TESS Input Catalog was created. Although the TESS PDCSAP lightcurves are corrected for contamination, fitting the dilution allows an independent check on the contamination ratio correction performed by the SPOC pipeline. Fitting a dilution term for only the TESS photometry assumes the K2 aperture has been correctly decontaminated or is comparatively uncontaminated, which is based on K2 having a significantly smaller pixel scale than TESS (4" and 21" for K2 and TESS, respectively). However, it is possible that there is still a level of contamination within the K2 aperture that might be identified through high-resolution imaging. We checked the K2 aperture for all of our targets to identify any major sources of contamination from the Gaia EDR3 catalog. We define contaminants as having flux ratios with the target star that are much larger than the uncertainties of the transit depth. To correct for the contaminating light, we followed the method from Rampalli et al. (2019) to account for the fraction of the flux within the aperture that belonged to our targets (𝐹star) as opposed to the contaminating stars based on the Gaia G-band fluxes. We found significant contamination for K2-54 (𝐹star ≈ 0.56) and K2-237 (𝐹star ≈ 0.98; the latter was originally discussed in Ikwut-Ukwa et al. 2020). Several other systems had potential faint contaminants, however, the global fit for the system with the next highest level of contamination (K2-250; 𝐹star ≈ 0.98) did not change within 9The starting point for dilution is calculated as D=C/(1+C). 31 uncertainties before and after flux correction, so we did not apply corrections to any systems other than K2-54 and K2-237. 2.4.4 Global fits We ran a short preliminary fit for each system to identify any potential issues, e.g. particularly shallow transits, and then ran a final fit to convergence. For a fit to be accepted as converged, we adopted the default EXOFASTv2 criteria of 𝑇𝑍 > 1000, where 𝑇𝑍 is the number of independent draws, and a slightly loose Gelman-Rubin value of < 1.02 due to some transits being very shallow in TESS, resulting is long runtimes for the global fits. Within EXOFASTv2, we opted to reject all flat and negative transit models, which ensured a more reliable recovery of marginal transits (Eastman et al., 2019). We did not fit for transit timing variations, but plan to explore this in future papers. Shallow transits clearly detected in K2 were not always evident in the TESS lightcurves as the latter are necessarily noisier due to the smaller collecting area of the telescope (see §2.5.6 for discussion). For these systems we ran a K2-only fit to convergence and a short preliminary fit (Gelman-Rubin of ∼ 1.1, 𝑇𝑧 ∼ 100). To assess whether it was advantageous to include the TESS lightcurves, we required certain criteria be met before running the K2 and TESS fit to convergence. Firstly, we compared the improvement on uncertainties for parameters such as period and 𝑇𝐶, and projected these to the year 2030. If the uncertainties were notably smaller when including TESS data, we continued by visually inspecting the transits modelled by EXOFASTv2. For extremely marginal transits, we further binned the phased lightcurves to determine whether the transit was indeed visible. If the transit in TESS was still not obvious, we inspected the probability distribution functions (PDFs) output by EXOFASTv2 for clearly non-Gaussian distributions for key parameters, particularly period. If the period was not well-constrained (e.g. multimodal) even with the increased baseline of TESS, we excluded the TESS lightcurve from the fit. A multimodal period indicates that the MCMC identified different transit solutions based on the TESS data, implying that the TESS transits are not securely enough detected to update the ephemeris. 32 Figure 2.5 K2 (gray) and TESS (purple) transits for all systems where TESS added significant value to the ephemeris projection. The phase-folded lightcurves include all data available across the K2 campaigns and TESS sectors for each system, and have the best-fit model from EXOFASTv2 overlaid (see Eastman et al. 2013, 2019; Eastman 2017 for how this is calculated). The system K2 identifier and orbital period of the planet are displayed in each subplot. The TESS lightcurves are shown binned to 12 minutes, and the K2 lightcurves are unbinned. For K2-237, the discreteness of the points is likely due to the period being an integer multiple of the exposure time. 33 Figure 2.6 K2 transits for systems that were not recoverable in TESS lightcurves. The darker points are binned to 30 minutes, and the EXOFASTv2 best-fit model is shown. 34 We ultimately excluded any TESS lightcurves where the transit has SNR ≲ 7. As these are all previously confirmed planets, we adopted a less conservative SNR for bona fide transits in TESS compared to what is required for initial planet verification. This SNR threshold was chosen because the first system below this cut (K2-265, SNR = 6.0) had a multimodal posterior for period, and all other systems with lower SNR exhibited similar issues. Conversely, the system just above this threshold (K2-277, SNR = 8.1) has a broad but Gaussian period posterior, with no other systems above this SNR having unreliable PDFs. Using this threshold, 13 of the 26 systems did not have recoverable TESS transits, so these were globally fit using only their K2 lightcurves (Figure 2.6). While these systems will not have as significant improvement on their ephemerides, we still provide the updated parameters to include them in our final catalog of self-consistent parameters. 2.5 Results and Discussion We updated the system parameters for 26 single-planet systems discovered by K2 and reobserved by TESS, four of which were part of the pilot study for the K2 & TESS Synergy (K2-114, K2-167, K2-237 and K2-261; Ikwut-Ukwa et al. 2020). Tables containing the parameters from the global fits are included as supplementary material, and can be found in full in Thygesen et al. (2023). Here, we address any points of interest for individual systems and for the sample as a whole. 2.5.1 Ephemeris improvement As addressed in §2.2, a major incentive for refitting all K2 and TESS systems is to update their ephemerides to provide the community with accurate transit times for observing with existing and upcoming facilities. Figures 2.7 and 2.8 show the projected transit timing uncertainties for our sample extrapolated to 2035, with markers indicating the expected launches for ongoing and future missions. The uncertainties on the transit times are calculated by standard error propagation, 𝜎𝑡trans = √︃ 𝜎2 𝑇0 + (𝑛trans × 𝜎𝑃)2 (2.1) where 𝜎𝑡trans is the uncertainty on future transit time, 𝜎𝑇0 is the uncertainty on the fitted optimal time of conjunction, 𝑛trans is the number of transits that occurred between timestamps and 𝜎𝑃 is 35 uncertainty on the period. For the future transit times using the results of the EXOFASTv2 global fits, we used the optimal time of conjunction in order to minimize the covariance between 𝑇𝐶 and 𝑃. However, 𝑇0 is not generally available for the K2 discovery parameters, so for the projected uncertainties on transit times for the original K2 values we used 𝑇𝐶. As expected, systems for which we excluded the TESS data due to shallow transits were not improved on the same scale as those with significant TESS transits. For the K2 and TESS systems, the updated global fits were able to reduce most uncertainties from hours to minutes within the scope of some of the major facilities in the near future (Figure 2.7). For the 13 systems with detected TESS transits, the average 3𝜎 uncertainty on the future transit time by the year 2030 was reduced from 26.7 to 0.35 hours (Table 2.2). Systems for which we only included the K2 lightcurves had significantly less improvement on the precision of predicted transit times. However, the ephemeris for K2-181 was considerably refined due to the addition of data from K2 Campaign 18, which was not included in any previous analysis of this system. Excluding K2-181, there was a slight reduction of the average 3𝜎 uncertainty from 43.2 to 35.6 hours (Table 2.3). The small improvement for some systems is likely due to using optimized K2 lightcurves obtained from the pipeline described in §2.3.1, in conjunction with our fits including both the planet and the host star. For systems with RV measurements, our ephemeris comparison uses uncertainties taken from previous analyses that included the RVs along with the K2 data. The uncertainties for systems without RVs are taken from the most recent study that included lightcurves from K2. There are a handful of exceptions to this rule: for K2-77 we use the values from Mayo et al. (2018) as Gaidos et al. (2017) only has three RV measurements which is insufficient for our EXOFASTv2 fits; for K2-97, we use the values from Livingston et al. (2018a) as no 𝑇𝑐 was presented in the analysis by Grunblatt et al. (2018) that included RVs; for K2-237 we use the less precise values from Soto et al. (2018) which are consistent with our results, rather than from Smith et al. (2019) which have a ∼ 4𝜎 discrepancy with our findings (this was also found in Paper I; Ikwut-Ukwa et al. 2020). 36 Table 2.2 Ephemerides as of discovery compared to our updated values for systems with K2 and TESS transits, with the 3𝜎 uncertainty on future transit time by the year 2030. 3𝜎2030 𝑇𝑐 (BJD) 𝑃 (days) TSM K2-77 Discovery Updated 8.199814+0.000364 −0.000367 8.2000844+0.0000086 −0.0000073 Discovery Updated 8.406726+0.001863 −0.001827 8.407115 ± 0.000023 Discovery Updated 10.13675 ± 0.00033 10.1367349+0.0000094 −0.0000092 Discovery Updated 11.39109+0.00018 −0.00017 11.39093100.0000031 −0.0000032 2457070.806480+0.001511 −0.001449 2457316.80766+0.00099 −0.00096 K2-97 2457142.04977+0.00888 −0.00854 2457722.1447+0.0027 −0.0026 K2-98 2457145.9807 ± 0.0012 2457662.95321+0.00077 −0.00074 K2-114 17.4 hr 22 min 27.3 84.8 hr 58 min — 12.6 hr 19 min 13.5 2457174.49729 ± 0.00033 2457687.08869 ± 0.00016 5.9 hr 6 min K2-115 Discovery Updated 20.273034+0.000036 −0.000037 20.2729914 ± 0.0000050 2457157.15701 ± 0.00025 2457522.07014 ± 0.00017 Discovery Updated 9.977481+0.001039 −0.001007 9.978541+0.000023 −0.000019 Discovery Updated 8.8665 ± 0.0003 8.8656630.000011 0.000010 K2-167 2456979.936780+0.002518 −0.002443 2457299.24650.0022 0.0023 K2-180 2457143.390 ± 0.002 22457489.15656+0.00078 −0.00076 K2-182 Discovery Updated 4.7369683 ± 0.0000023 4.7369696 ± 0.0000017 2457719.11517 ± 0.00028 2457652.79755+0.00027 −0.00028 K2-237 Discovery Updated 2.18056 ± 0.00002 2.18053332 ± 0.00000054 2457684.8101 ± 0.0001 2457706.61618+0.00003 −0.00003 K2-260 Discovery Updated 2.6266657 ± 0.0000018 2.62669762 ± 0.00000066 2457820.738135 ± 0.00009 2457894.284876+0.000060 −0.000059 K2-261 Discovery Updated 11.63344 ± 0.00012 11.6334681 ± 0.0000044 2457906.84084+0.00054 −0.00067 2458151.14394+0.00026 −0.00030 K2-277 Discovery Updated 6.326763+0.000355 −0.000361 6.326768+0.000015 −0.000012 2457221.22958+0.00221 −0.00217 2457303.4771 ± 0.0010 K2-321 42 min 5 min 40.8 hr 48 min 13.0 hr 26 min 10 min 8 min 3.2 hr 5 min 14 min 5 min 3.4 hr 7 min 21.5 hr 48 min — — 46.1 15.1 15.4 — — 85.5 35.6 Discovery Updated 2.298 ± 0.001 2.2979749+0.0000017 −0.0000019 2457909.17 2458141.26759+0.00064 −0.00068 144.0 hr 15 min — Notes: The discovery values are taken from the K2 references listed in Table 3.2. The 𝑇𝐶 for the updated values is 𝑇0 as determined by our global fits. 37 Table 2.3 Ephemerides as of discovery compared to our updated values for systems with only K2 transits, with the 3𝜎 uncertainty on future transit time by the year 2030. 𝑃 (days) 𝑇𝑐 (BJD) K2-7 Discovery Updated 28.67992 ± 0.00947 28.6781+0.0046 −0.0051 2456824.6155 ± 0.0149 2456853.2946+0.0046 −0.0042 K2-54 3𝜎2030 TSM 135.0 hr 68.8 hr 5.9 Discovery Updated 9.7843 ± 0.0014 9.7833+0.0013 −0.0012 Discovery Updated 9.0063 ± 0.0013 9.0073+0.0012 −0.0011 2456982.9360 ± 0.0053 2457002.5042 ± 0.0029 56.8 hr 50.6 hr — K2-57 2456984.3360 ± 0.0048 2457011.3568 ± 0.0023 57.3 hr 50.4 hr 10.8 K2-147 Discovery Updated 0.961918 ± 0.000013 0.961939 ± 0.000029 2457327.91683+0.00089 −0.00100 2457343.30907+0.00100 −0.00099 K2-181 5.0 hr 11.2 hr — Discovery Updated 6.894252+0.000430 −0.000426 6.893813 ± 0.000011 2457143.793550+0.002559 −0.002528 2457778.0262 ± 0.0012 Discovery Updated 9.695101+0.001285 −0.001334 9.6952 ± 0.0014 Discovery Updated 7.055784+0.000650 −0.000641 7.05576+0.00066 −0.00064 K2-203 2457396.638780+0.005765 −0.005844 2457435.4189+0.0037 −0.0036 K2-204 2457396.50862+0.00372 −0.00376 2457431.7872 ± 0.0022 K2-208 Discovery Updated 4.1909480.000230 −0.000248 4.19097 ± 0.00023 2457396.51164+0.00248 −0.00235 2457430.0390 ± 0.0016 K2-211 23.9 hr 0.5 hr 49.7 hr 52.7 hr 33.6 hr 33.6 hr 21.0 hr 20.0 hr Discovery Updated 0.669532 ± 0.000019 0.669561+0.000031 −0.000032 2457395.82322 ± 0.00160 2457432.6479 ± 0.0013 10.4 hr 17.2 hr K2-225 Discovery Updated 15.871455+0.002113 −0.001670 15.8723+0.0021 −0.0019 Discovery Updated 3.271106+0.000367 −0.000369 3.27109+0.00036 −0.00039 2457587.368230+0.004034 −0.004872 2457619.1111+0.0031 −0.0030 K2-226 2457584.026130+0.004436 −0.004366 2457620.0082 ± 0.0020 K2-250 Discovery Updated 4.01457+0.00062 −0.00057 4.01392 ± 0.00029 2457584.1212+0.0061 −0.0066 2457620.2535 ± 0.0015 K2-265 Discovery Updated 2.369172 ± 0.000089 2.369020+0.000058 −0.000059 2456981.6431 ± 0.0016 2457017.18078+0.00055 −0.00054 42.2 hr 44.3 hr 39.8 hr 40.3 hr 52.5 hr 25.4 hr 14.9 hr 9.8 hr 14.6 1.3 11.1 12.9 2.1 11.1 14.6 13.8 15.7 Notes: The discovery values are taken from the K2 references listed in Table 3.2. The 𝑇𝐶 for the updated values is 𝑇0 as determined by our global fits. 38 Figure 2.7 Projected uncertainties for transit times (𝜎𝑇𝑐 ) for systems with transits detected in both K2 and TESS. The shaded regions represent the 1, 2 and 3 𝜎 uncertainties, where gray is the uncertainty from the K2 ephemerides listed in Table 3.2 and purple is our updated version using EXOFASTv2. The vertical dashed lines show the expected or actual launch years for missions for which these systems would be prospective targets (JWST: red, NGRST: orange, ARIEL: yellow). Note the y-axis scale is different in each subplot. 39 Figure 2.8 Same as Figure 2.7 but for systems with transits only detectable in the K2 lightcurves. The shaded regions represent the 1, 2 and 3 𝜎 uncertainties, where gray is the uncertainty from the K2 ephemerides listed in Table 3.2 and green is our updated version using EXOFASTv2. The vertical dashed lines show the expected or actual launch years for missions for which these systems would be prospective targets (JWST: red, NGRST: orange, ARIEL: yellow). The ephemeris for K2-181 is significantly improved due to the inclusion of data from K2 Campaign 18. 40 As mentioned in §2.4.4, half of our sample did not have transits deep enough to be recovered by TESS. This presents a challenge for updating the transit times for these systems. If these systems are observed in future TESS sectors, it is possible that the SNR will increase sufficiently to include in a global fit. We will continue to monitor these and will include them in future releases, if this is the case. 2.5.1.1 K2-167 We note the use of an errant stellar metallicity prior used in the pilot study, where 0.45 instead of -0.45 (as reported by Mayo et al. 2018) was used as the Gaussian center. While this may have affected the solutions of stellar and planetary parameters, it would not have significantly altered the ephemeris. 2.5.1.2 K2-260 There is a clear discrepancy between the previously published ephemeris and our updated version (see Figure 2.7), well beyond a 3𝜎 level. To test whether this was an artifact of our global fit, we ran a fit using only the K2 lightcurves and compared the results to the original and K2 and TESS fits. Our K2-only fit was consistent with our K2 and TESS ephemeris, and still in disagreement with the original results, suggesting that our updated fit provides the optimal ephemeris. It is possible that the original lightcurves introduced systematics in the discovery analysis, or the inclusion of additional follow-up data affected the ephemeris, but this is not clear. In any case, the consistency between our K2-only and K2 and TESS ephemerides (and no other system showing similar issues) gives us confidence in our results. 2.5.1.3 K2-261 As discussed in the pilot study (Ikwut-Ukwa et al., 2020), the PDFs for some stellar parameters (particularly age and mass) of K2-261 exhibit distinct bimodality that is likely due to the star being at a main sequence transition point (and not associated with the poor fits of shallow transits discussed in §2.4.4), causing difficulties with fitting the MIST isochrones to the data to constrain age. We followed the same procedure from Ikwut-Ukwa et al. (2020), splitting the posterior at the minimum probability for 𝑀∗ between the two Gaussian peaks (at 𝑀∗=1.19 𝑀⊙; see Figure 5 of 41 Ikwut-Ukwa et al. 2020) and extracting two separate solutions for each peak. We use the low-mass solution for all figures as this has the higher probability. The different stellar mass solutions do not affect the ephemeris projection for this planet. 2.5.1.4 Comparison to pilot study The ephemerides were slightly improved for the four systems from the pilot study, the most significant being K2-167 (1.1 hours to 48 minutes) and K2-261 (30 minutes to 7 minutes). We did not expect to see major improvement because the baseline of new TESS sectors is relatively short compared to that of K2 and the TESS primary mission. 2.5.2 TSM We calculated the transmission spectroscopy metric (TSM; Kempton et al. 2018) for the planets in this sample to gauge the value of atmospheric follow-up (Tables 2.2 & 2.3; Figure 2.9). As the TSM is dependent on stellar parameters, we excluded the three systems for which we did not fit the host star (K2-54, K2-147, K2-321; see §2.4). The TSM is only valid for planets with 𝑅𝑝 < 10 𝑅⊕, which removes a further five planets from this calculation (K2-97, K2-114, K2-115, K2-237 and K2-260). Only one system, K2-261, has a TSM above the threshold suggested by Kempton et al. (2018), and falls between the second and third quartile for the corresponding mass bin (see Table 1 of Kempton et al. 2018.) Future work in this project to update ephemerides will prioritize planets with high TSMs relative to the entire K2 catalog. 2.5.3 The sample While the systems in this analysis span a broad range of stellar temperatures and planet masses, most planets have orbital periods ≲10 days and radii ≲5 𝑅⊕ (Figures 2.9 and 2.10). Planet masses range from 2.6 ∼ 639 𝑀⊕ and host stars include M dwarfs to F-type spectral classifications. This demonstrates the diversity of the original K2 sample as largely community-selected targets. Figure 2.10 shows how this sample compares to other known exoplanets. 2.5.4 TTVs We did not fit for transit timing variations (TTVs) in this study. We would expect these to manifest as a significant change in ephemeris over time, whereas all of the systems studied here 42 have updated ephemerides consistent to within 3𝜎 of the original K2 ephemeris (except K2-260; see §2.5.1.2). Therefore, any TTVs that may be present are currently too small to detect for these systems. Differences in the ephemerides on the 1 ∼ 3𝜎 level are likely due to the addition of the TESS lightcurves. 2.5.5 Candidate planets We note that a couple of the systems in our analysis have additional candidate planets (K2- 203 and K2-211). However, we ignore these for the purpose of updating ephemerides of known exoplanets that are more likely future targets for missions such as JWST, but plan to revisit these in a future paper addressing multi-planet systems. 2.5.6 K2 vs. TESS It is not surprising that relatively shallow K2 transits were not detected by TESS. Kepler and TESS were designed to observe different stellar demographics, resulting in different photometric capabilities. Kepler was built with the intent to explore the number of near-Earth-sized planets close to their respective habitable zones around distant stars with apparent magnitudes ≲ 16. The original Kepler mission could reach a precision of ∼ 20 parts per million (ppm), which was generally the same for the K2 mission (Vanderburg & Johnson, 2014; Vanderburg et al., 2016). On the other hand, TESS is focused on nearby, brighter stars with magnitude ≲ 12. The precision of TESS has a floor at ∼20 ppm at 1 hour for the brightest stars with 𝑇mag < 4, but is more realistically ≳100 ppm for the majority of stars. Due to the all-sky nature of the TESS missions, observing sectors last on average 27 days for efficient sky coverage. K2 campaigns were around 80 days in duration, meaning the same targets may have ∼3 times as many transits observed by K2. While TESS may not be able to recover all K2 systems, the ones it can detect will have vastly improved ephemerides as demonstrated in Figure 2.7. Our analysis indicates TESS transits with SNR ≳ 7 are recoverable, and while this places a limit on the scope of this reanalysis, we can potentially gain access for reobservation of at least half of known K2 planets. It is possible that future TESS missions that reobserve the planets with currently marginal transits (SNR ∼ 5 − 6) will increase the SNR enough for a significant detection. However, for transits with SNR≲ 5, it is 43 Figure 2.9 Architecture for each system showing the values from the global fits for the 26 systems in this analysis. The host stars are the left-most circles, with their temperatures indicated by color and relative radius shown by size. The right-most circles represent the planets, with size showing relative radius and color indicating their raw Transmission Spectroscopy Metric (TSM). The radius of the star and planet within each system is not scaled to each other. Systems for which we did not fit stellar parameters and planets that do not have a calculated TSM are represented by empty circles (see §2.5.2). An example of the Sun hosting a Jupiter planet with a 10-day period and TSM of 40 is shown. 44 Figure 2.10 Radius versus mass for all confirmed exoplanets (gray; values taken from the NEA) and those in our work (using the median values from the EXOFASTv2 output). The 10 systems with planetary masses measured through RVs are indicated by diamonds, while the planets without RVs that have masses obtained from the Chen & Kipping (2017) mass-radius relations are shown as crosses. The points are colored by the effective temperature of the host star, and are empty for the three systems without fitted stellar parameters. unlikely that more TESS observations will result in recoverable transits. 2.5.7 Future work With several major facilities able to characterize exoplanets in extensive detail planned to come online within the next decade, not having accurate and precise transit times is a relevant issue. The K2 & TESS Synergy aims to solve the problem of degrading ephemerides for all K2 systems reobserved by TESS (with clearly detectable transits as shown by this effort). Assuming TESS will reobserve all K2 systems throughout its extended missions, we expect to be able to update the ephemerides for around half of K2 planets (∼250 planets) with transits deep enough to be detected by TESS, based on this study. Over the next couple of years, we plan to reanalyze the remaining K2 systems with current TESS overlap, providing the updated parameters to the community. In future batches, we will place a focus on systems that are potentially well suited as 45 JWST targets for atmospheric studies based on their TSMs. While we do not see strong evidence for TTVs in the current work, we will make note of this in future for any systems with significant change in ephemeris, particularly for known multi-planet systems where this would be more readily detectable. 2.6 Conclusion Past efforts to create and analyze homogeneous populations of exoplanet parameters have led to great insight into major questions in planetary formation and evolution (Wang et al., 2014; Fulton et al., 2017; Fulton & Petigura, 2018). The K2 & TESS Synergy is uniting NASA’s planet hunting missions, and focuses on extending the scientific output of both telescopes by creating a self-consistent catalog for the K2 and TESS sample while providing the community with updated ephemerides to efficiently schedule future characterization observations with facilities like JWST (Gardner et al., 2006). As well as refreshing stale ephemerides, this provides a uniform way of addressing any inconsistencies between the original K2 ephemeris and the updated value from TESS. In this paper, we have presented updated parameters for 26 single-planet systems originally discovered by K2 and more recently reobserved by TESS during its primary and extended missions. Following from the success of the pilot study (Ikwut-Ukwa et al., 2020), we have significantly reduced the uncertainties on transit times for the 13 systems with transits detectable in TESS from hours down to minutes through the JWST operations window (∼2030). Assuming the current sample is representative of the entire K2 catalog, we expect significant improvement on ephemerides for about half of the systems revisited by TESS, with the goal of a ∼250-system catalog of parameters that will be publicly available. As TESS continues to reobserve large portions of the entire sky during its current and possible future extended missions, there will be a well-suited opportunity to conduct this analysis on all known exoplanets, possibly leading to key insights into the evolutionary processes of exoplanets. 46 CHAPTER 3 THE K2 & TESS SYNERGY III: SEARCH AND RESCUE OF THE LOST EPHEMERIS FOR K2’S FIRST PLANET This section is the work published in Thygesen et al. (2024). 3.1 Abstract K2-2 b/HIP 116454 b, the first exoplanet discovery by K2 during its Two-Wheeled Concept Engineering Test, is a sub-Neptune (2.5 ± 0.1 𝑅⊕, 9.7 ± 1.2 𝑀⊕) orbiting a relatively bright (K𝑆 = 8.03) K-dwarf on a 9.1 day period. Unfortunately, due to a spurious follow-up transit detection and ephemeris degradation, the transit ephemeris for this planet was lost. In this work, we recover and refine the transit ephemeris for K2-2 b, showing a ∼40𝜎 discrepancy from the discovery results. To accurately measure the transit ephemeris and update the parameters of the system, we jointly fit space-based photometric observations from NASA’s K2, TESS, and Spitzer missions with new photometric observations from the ground, as well as radial velocities from HARPS-N that are corrected for stellar activity using a new modeling technique. Ephemerides becoming lost or significantly degraded, as is the case for most transiting planets, highlights the importance of systematically updating transit ephemerides with upcoming large efforts expected to characterize hundreds of exoplanet atmospheres. K2-2 b sits at the high-mass peak of the known radius valley for sub-Neptunes, and is now well-suited for transmission spectroscopy with current and future facilities. Our updated transit ephemeris will ensure no more than a 13-minute uncertainty through 2030. 3.2 Introduction In the era of cutting-edge atmospheric characterization of transiting exoplanets, precise and accurate ephemerides are crucial for efficiently scheduling these expensive observations. However, over 80% of transiting exoplanets will have uncertainties on their future transit times greater than 30 minutes by the end of the decade (see Thygesen et al. 2023), rendering these systems extremely challenging to observe with JWST (Gardner et al., 2006; Beichman et al., 2020), major upcoming facilities such as the Atmospheric Remote-sensing Infrared Exoplanet Large-survey (ARIEL; Tinetti 47 et al. 2018, 2021), and 30m class telescopes like the Thirty Meter Telescope (TMT; Sanders 2013), Giant Magellan Telescope (Johns et al., 2012), and the 39 m European Southern Observatory Extremely Large Telescope (ELT; Udry et al. 2014). This problem can be solved by observing new transits of these planets with current facilities. Fortunately, NASA’s Transiting Exoplanet Survey Satellite (TESS) mission (Ricker et al., 2015) is observing the entire sky, providing a valuable opportunity to refine the transit ephemeris for most known planets. After a successful 4-year nominal mission, discovering thousands of exoplanets, the Kepler mission (Borucki et al., 2010) was repurposed due to a mechanical issue. Using the solar pressure to stabilize pointing of the Kepler spacecraft, the K2 mission was able to survey the ecliptic plane, finding hundreds of exciting new systems that are well-suited for detailed characterization (Howell et al., 2012; Vanderburg et al., 2016; Zink et al., 2021; Kruse et al., 2019; Pope et al., 2016; Livingston et al., 2018a; Crossfield et al., 2016; Dattilo et al., 2019). The K2 mission ended in 2019, with many of its newly-detected planets never being reobserved since their discovery campaign(s). The K2 & TESS Synergy project is an effort to provide the community with updated and accurate transit times and system parameters for exoplanets originally discovered by the K2 mission that have been recently observed by TESS (Ricker et al., 2015). Following a successful pilot study (Ikwut-Ukwa et al., 2020), the second paper in this series revisited 26 K2 single-planet systems that TESS reobserved during its prime mission (Thygesen et al., 2023). This work improved the average ephemeris uncertainties by multiple orders of magnitude due to the addition of new TESS transits. Additionally, we identified systems where the original ephemeris has been completely lost (See K2-260; Thygesen et al. 2023), which is similar to this work on K2-2 b, K2’s first exoplanet discovery. K2-2 b was identified during the Two-Wheeled Concept Engineering Test (campaign 0) of the K2 mission. K2-2 b is a sub-Neptune (2.5 ± 0.1 𝑅⊕, 9.7 ± 1.2 𝑀⊕) on a 9.1-day orbit around a bright (𝑉 = 10.2, 𝐽 = 8.6, HIP 116454) K-dwarf (Vanderburg et al., 2015). At discovery, a single clear transit was detected in the K2 observations, along with a marginal (∼3𝜎) detection from the Microvariablity and Oscillations of Stars (MOST) Space Telescope (Walker et al., 2003). 48 Follow-up observations were scheduled with Spitzer (P.I. Werner, AOR 57185280) and the Hubble Space Telescope (P.I. Bourrier, proposal I.D. 15127), however, the transit was not seen during the predicted window from the discovery ephemeris. It was then determined that the MOST transit was likely not a real transit of K2-2 b, having skewed the period enough to cause subsequent transits to be missed. In this work, we combine the discovery observations from Vanderburg et al. (2015) with new observations from NASA’s TESS mission, follow up ground-based photometry, and improved radial velocities to accurately measure the ephemeris of K2-2 b for the first time, proving the original detection from MOST to be a false positive. In Section 3.3 we describe the observations used and the relevant reduction and analysis methods, including the reanalysis of radial velocities from the High Accuracy Radial Velocity Planet Searcher-North (HARPS-N; Cosentino et al. (2012)) on the 3.58m Telescopio Nazionale Galileo at the Roque de los Muchachos Observatory. Section 3.4 outlines the methodology used in running the EXOFASTv2 global fit of all observations and archival information. We present our results and discuss the importance of ephemeris refinement in the context of future characterization of K2-2 b in Section 3.5. 3.3 Observations and Archival Data The discovery analysis for K2-2 b included a 47 day long light curve from MOST (Walker et al., 2003), which was thought to contain a marginal ∼ 3𝜎 detection of the transit, but future follow up attempts to reobserve the transit with Spitzer and HST showed no transit during or near the predicted window. This ultimately led to the idea that the MOST observations were not reliably constraining the transit ephemeris. While it is not clear why this happened, it is possible that Gaussian noise or satellite systematics caused an already marginal detection to be anchored to a different time of transit. Our new observations from MEarth, ULMT, Spitzer and TESS (Figure 3.1) confirm this hypothesis. In the near decade since its discovery, a variety of follow up observations have been conducted to better characterize the K2-2 system and to recover the transit ephemeris. In the following sections, we describe the new and archival observations used in our analysis. The magnitudes and literature values for K2-2 are listed in Table 3.1, and the photometric data sets we 49 used are outlined in Table 3.2. 3.3.1 Ground-based archival imaging At the discovery of of K2-2 b, Vanderburg et al. (2015) used multiple archival from the National Geographic Society–Palomar Observatory Sky Survey (POSS-I, van Leeuwen 2007) and Sloan Digital Sky Survey (SDSS, Abazajian et al. 2009), and newly acquired images from Robo- AO on Palomar (Baranec et al., 2014; Law et al., 2014) and Natural Guide Star Adaptive Optics (NGSAO) system on Keck to rule out nearby close companions that might be contaminating the K2 aperture. A nearby white dwarf with a separation of around 8′′ was identified to share a similar proper motion to K2-2, suggesting that they exist in a gravitationally bound system (this is discussed more in Section 3.5.2). The white dwarf is within the K2 aperture, but is 6-7 magnitudes dimmer than K2-2, which would not affect the final transit depth of K2-2 b. No other nearby companions were found to a 7𝜎 significance in the 𝐻 band to the limits of 3.0 mag at 0′′.1 separation, 9.2 mag at 1′′.0 and 12.7 mag at 5′′.0. 3.3.2 K2 Photometry A single transit of K2-2 b was observed at 30-minute cadence during the Kepler Two-Wheel Concept Engineering Test during February 2014. Due to the loss of two of the four reaction wheels on the spacecraft, significant systematics were introduced to the light curves of the K2 mission. We corrected for these using the methods described in Vanderburg & Johnson (2014) and Vanderburg et al. (2016), which utilize a series of 20 apertures to extract raw light curves used to perform the corrections. Short timescale variations in each of these light curves are correlated with the roll angle of the spacecraft, with the latter being subtracted from the light curves. This process is repeated iteratively until the light curve is free of any variations associated with the roll of the spacecraft. The most precise light curve out of the 20 following the corrections is selected for final analysis. We performed further corrections by fitting the transit and correcting for the systematics and any low-frequency stellar variability, prior to the global fit. 50 3.3.3 MEarth MEarth was used to initially recover the transit of K2-2 b and constrain the ephemeris, observing multiple partial and full transits. MEarth consists of 16 separate 0.4 m telescopes using custom 715 nm longpass filters designed to find Earth-sized planets around M dwarfs (Nutzman & Charbonneau, 2008; Irwin et al., 2015). Telescopes 1-8 are a part of the MEarth-North Observatory at Fred Lawrence Whipple Observatory (FLWO) on Mount Hopkins, Arizona, while the other eight telescopes (numbered as 11-18) are part of the MEarth-South Observatory located at Cerro Tololo Inter-American Observatory (CTIO) on Cerro Tololo, Chile. K2-2 was observed using a subset of four telescopes from each observatory (see Table 3.2) with 1 minute cadence on UT 2016 September 21 and 30, and UT 2016 October 09. Light curves from MEarth are automatically extracted through a pipeline (see Irwin et al. 2007; Berta et al. 2011) that calibrates the images using flat fields, dark current frames and bias exposures. We combined the light curves across multiple nights for each telescope, so within the global fit the variance can be determined independently for each instrument. We sliced the light curves such that we only included one full transit duration before and after the transit, and detrended against airmass in the global fit. While the original observations also included telescopes 4, 5 and 8, we did not use these in our analysis as the light curves did not contain full transits and would not contribute significant value to the global fit. The transit was also missed during the night of UT 2016 September 11 due to the incorrect ephemeris. These observations were the first use of the defocus observing mode of MEarth for transit follow-up, and served as the prototype for a large number of observations of TESS objects of interest done in later years. Here we describe the modifications made to the system to implement this mode. Prior to implementation of defocus, MEarth observations of bright stars were limited by scintillation noise due to the short maximum exposure times possible before detector saturation, combined with high overheads (approximately 15s, most of which was consumed by CCD readout and download over USB2 connection to the host computer), resulting in a low duty cycle. For scintillation limited observations of events of fixed duration such as transits, the overall transit- averaged photometric noise is determined by the duty cycle (e.g. Young 1967) so the goal of 51 implementing defocus was to improve this by substantially lengthening the exposure times possible before saturation. The scheduling and telescope control software were modified to allow each observation request to specify defocus as half flux diameter (HFD), in pixels. For these first observations of K2-2, we used HFD = 6.0 pixels, where the pixel scales are 0.76 arcsec/pix for MEarth-North and 0.84 arcsec/pix for MEarth-South. The telescope focus was offset by the scheduler prior to commencing observations of each target by the appropriate number of focus encoder counts, where the scaling factor was determined from the calibration curve of HFD versus focus encoder counts used by the standard automatic focus routine (normally used for focusing the telescope at the start of the night). MEarth did not have autoguiders, and guiding to stabilize the target star position on the detector (vital for precise transit work) had to be done using the science exposures themselves, which were 36s for K2-2. The standard MEarth target acquisition and guiding system for normal in-focus images consisted of astrometric analysis of the images after readout to determine their center in celestial coordinates, followed by offsetting of the telescope to center the target based on its calculated position. Target acquisition was done by applying the full offset, and guiding by passing these measurements into a standard proportional-integral-derivative (PID) control loop with an overall gain less than unity to provide damping and avoid overshoot and oscillation during guiding. To implement the defocus observing mode, the image analysis part of this astrometric routine was replaced with a custom source detection routine using a standard matched filter approach (e.g. Irwin 1985), where in the case of defocused images, rather than using a standard approximately Gaussian filter kernel, the filter kernel was instead a model of the defocused telescope PSF. This technique is appropriate for analysis of images with mild amounts of defocus, such as needed on MEarth. Previous work (e.g. McCormac et al. 2013) has usually concentrated on the case of severe defocus, where different analysis techniques are needed. The PSF model was constructed by approximating the telescope entrance pupil as a circular annulus, and introducing defocus by setting the complex phase of this function to a multiple of the 𝑍 0 2 Zernike mode. The resulting PSF was computed by taking the inverse Fourier transform of this 52 Figure 3.1 The discovery and follow up phase-folded transits of K2-2 b used in the EXOFASTv2 (see Section 3.4) analysis. The observations from K2 (black), TESS (purple), Spitzer (blue), MEarth (green), and ULMT (yellow) are shown in open colored circles with the solid colored line representing the EXOFASTv2 model for that dataset. The closed colored circles represent 30-minute bins. East transit is offset by a constant for clarity. 53 function. In practice, it was also convolved by a Moffat profile (Moffat, 1969) with parameters chosen based on standard in-focus MEarth observations to approximate seeing and any effects other than diffraction that contribute to the system’s normal in-focus PSF spot size. The relationship between the 𝑍 0 2 Zernike coefficient and HFD was determined empirically. The PSF model was also used to compute exposure times and set photometric aperture radii for the automatic extraction pipeline. We found that these theoretical estimates of exposure times based on the idealised PSF models were rather optimistic, and in practice it was necessary to use shorter exposures (or equivalently, somewhat more defocus for a given desired exposure time) to avoid the risk of saturation due to non-uniformity of the resulting defocused star image. This can be caused by atmospheric turbulence (particularly in short exposures), but also other optical aberrations affecting the defocused star image, such as coma, which causes an asymmetric distribution of brightness around the resulting ring shaped PSF, and can cause one side of the ring to become too bright. Being remotely operated robotic telescopes, it was not always possible to maintain optimal collimation of the MEarth telescope optics, and while this had minimal effect on the normal in-focus images used for the majority of the survey, it did noticeably affect the defocused PSFs. With an appropriate detection threshold, this source detection procedure was found to produce quite robust results, albeit at reduced sensitivity to faint sources, and with a practical upper limit to the defocus HFD of approximately 15 pixels. Given the field of view of the MEarth telescopes of approximately 27x27 arcmin the number of detected sources was found to still be sufficient for accurate multi-star guiding using the astrometric solutions on nearly all of the targets observed over several years of observations, including hundreds of TESS objects of interest. 3.3.4 ULMT Once the ephemeris was refined from the MEarth observations, an ingress of K2-2 b was observed using the University of Louisville Manner Telescope (ULMT; formerly MVRC) at the Mt. Lemmon summit of Steward Observatory, Arizona. The observation was made in the 𝑟′ band with 50 second exposure time on UT 2016 October 10. The setup used for the observation included a 0.6 m f/8 RC Optical Systems Ritchey–Chrétien telescope and SBIG STX-16803 CCD camera 54 Table 3.1 Literature values for K2-2. Other Identifiers TIC 422618449 2MASS J23354927+0026436 EPIC 60021410 WISE J233549.11+002641.9 Description Right ascension (R.A.) Declination (Dec.) Gaia EDR3 𝐺 mag Gaia EDR3 𝐵𝑃 mag Gaia EDR3 𝑅𝑃 mag TESS mag 2MASS 𝐽 mag 2MASS 𝐻 mag 2MASS 𝐾𝑆 mag WISE1 mag WISE2 mag WISE3 mag WISE4 mag Value 23:35:49.29 00:26:43.84 9.932 ± 0.020 10.393 ± 0.020 9.317 ± 0.020 9.374 ± 0.006 8.604 ± 0.021 8.140 ± 0.033 8.029 ± 0.021 7.996 ± 0.030 8.078 ± 0.030 8.019 ± 0.030 7.878 ± 0.199 Gaia p.m. in R.A. Gaia p.m. in Dec. Gaia parallax (mas) -232.90 ± 0.019 -187.± 0.017 16.004 ± 0.046 Parameter 𝛼J2000 𝛿J2000 𝐺 𝐺Bp 𝐺Rp 𝑇 𝐽 𝐻 𝐾𝑆 WISE1 WISE2 WISE3 WISE4 𝜇𝛼 𝜇𝛿 𝜋 Notes. The uncertainties of the photometry have a systematic error floor applied. Proper motions taken from the Gaia EDR3 archive and are in J2016. Parallaxes from Gaia EDR3 have a correction applied according to Lindegren et al. (2021). with a 4k×4k array of 9 𝜇m pixels, which yielded a 26.6’ × 26.6’ field of view and 0.39 pixel-1 plate scale. The images were calibrated and photometric data were extracted using AstroImageJ (Collins et al., 2017), and the light curves were detrended against airmass in the global fit. 3.3.5 Spitzer With the ephemeris more precisely constrained from the MEarth and ULMT transits, Spitzer was used to observe a single transit of K2-2 b on UT 2017 April 1 (P.I. M. Werner, observing program 13052, AOR 62428416; Werner et al., 2016). The observation was 10.5 hours long, and was taken with the InfraRed Array Camera (IRAC; Fazio et al. 2004) channel 2 (4.5 µm) with a 55 2-second exposure time. We used the technique described in Livingston et al. (2018c) to extract the light curve. In brief, we extracted an optimal light curve by selecting the photometric aperture that minimized both white and red noise, and then corrected for systematics using pixel-level decorrelation (PLD; Deming et al. 2015). As Spitzer can have correlated noise due to spacecraft systematics, we scaled the per point errors so that we did not underestimate the uncertainties. We followed the procedure from Winn et al. (2008), where a scaling factor, 𝛽, is applied to the measured standard deviation to account for time-correlated noise. We first calculated the out-of-transit standard deviation for the unbinned data, 𝜎1 (for this calculation we conservatively defined out-of-transit as being outside of a full transit duration centered at the transit midpoint). We then binned the out-of-transit data points to a series of 10 temporal bin widths ranging from 4.2 minutes to 8.8 minutes, increasing in equal steps of 0.46 minutes. The limits on the bin widths correspond to the 1𝜎 range of the ingress/egress duration based on a preliminary fit using K2 and TESS light curves. We then calculated the standard deviation for each set of binned data. In general, this should 𝑁 × √︁𝑀/(𝑀 − 1), where M is number of bins and N is data points per be equivalent to 𝜎𝑁 = 𝜎1/ √ bin, if there is no time-correlated noise. However, the measured 𝜎𝑁 can be larger than the expected value (by the factor 𝛽). We calculated this factor for each bin width, then used the mean value across all widths as the final value for 𝛽. Finally, we scaled the original unbinned, out-of-transit error bars by the factor 𝛽 = 1.19, which is used as the per point uncertainty in our global fit. 3.3.6 TESS Photometry A single transit was observed by the Transiting Exoplanet Survey Satellite (TESS) in each of Sectors 42 and 70. We used the 120 second cadence lightcurves in our global fits. We retrieved the light curve through the Python package Lightkurve (Lightkurve Collaboration et al., 2018), selecting the light curve processed through the Science Processing Operations Center (SPOC) pipeline at the NASA Ames Research Center (Jenkins et al., 2016), which corrects for various systematics and identifies transits. The light curves were created from the Pre-search Data Conditioned Simple Aperture Photometry (PDCSAP) flux, which uses the optimal TESS aperture to extract the flux and 56 Table 3.2 Photometry used in this analysis. Filter Kepler Observatory Date K2 February 6 2014 September 21 2016 𝑖′ MEarth South MEarth South, North September 30 2016 𝑖′ 𝑖′ MEarth North October 9 2016 𝑟′ ULMT October 10 2016 4.5𝜇m Spitzer April 1 2017 TESS August 21 2021 TESS September 21 2023 TESS TESS Cadence 30 min 1 min 1 min 1 min 50 sec 2 sec 2 min 2 min Notes: Each telescope caught one full transit, except for ULMT which observed the ingress and partial transit. Observations with MEarth North used Telescopes 1, 2, 3 and 6, while MEarth South included Telescopes 11, 12, 16 and 18. Figure 3.2 Archival HARPS-N radial velocities for K2-2 from Vanderburg et al. (2015) and Bonomo et al. (2023). The left panel shows the phased-folded RVs, and the right panel shows the long-term trend in the unphased RVs. corrects the target for systematics using the PDC module (Stumpe et al., 2012, 2014; Smith et al., 2012). To correct for stellar variability and any remaining systematics based on the out-of-transit photometry, we used the spline-fitting routine keplerspline1 (Vanderburg & Johnson, 2014). We applied an initial estimate on the per-point errors for the corrected light curves as being the median absolute deviation of the out-of-transit photometry. We note that the per-point error is optimized through a fitted jitter term in the EXOFASTv2 global fit (See Section 3.4). 1https://github.com/avanderburg/keplerspline 57 3.3.7 Archival Spectroscopy We included archival spectroscopy to determine the host star properties and to refine the mass measurement of K2-2 b. In particular, to better characterize the host star in the global fit, we used metallicity measurements of K2-2 from the Tillinghast Reflector Echelle Spectrograph (TRES; Fűrész 2008) on the 1.5m Tillinghast Reflector at the Fred L. Whipple Observatory (FLWO). This is in keeping with our procedure for the larger Synergy catalog, where we are using TRES metallicities where available. The stellar parameters using TRES spectra were derived using the Stellar Parameter Classification (SPC; Buchhave et al. 2012). Three measurements from TRES ([M/H] = -0.193 ± 0.086, -0.191 ± 0.08, 0.009 ± 0.08) were available through the ExoFOP website2. We used the mean value to place a Gaussian prior on metallicity ([Fe/H]) of -0.125 ± 0.08. We used a total of 105 spectra of K2-2, including those used in Vanderburg et al. (2015) and Bonomo et al. (2023), acquired using the High Accuracy Radial velocity Planet Searcher for the Northern hemisphere (HARPS-N) on the 3.6m Telescopio Nazionale Galileo (TNG) at the Roque de los Muchachos Observatory (Cosentino et al., 2012), in order to better characterize the mass of K2-2 b (Figure 3.2). Each observation had either 15 or 30 minutes exposure time, with a resolving power of 𝑅 = 115,000. We followed the procedure of Dumusque et al. (2021) to reduce the RVs that were used in our global fits. The observations occurred in two main blocks, separated by ∼ 2.5 years; the first run was from UT 2014 July 7 to December 6 2017, and the second from UT 2020 June 25 to 2023 November 27. The second series of RVs was significantly offset to the earlier measurements, which led us to apply post-processing systematics corrections to investigate whether the offset was instrumental or physical in nature. 3.3.7.1 YARARA processing to correct remaining systematics YARARA (Cretignier et al., 2021) is a post-processing methodology that aims to perform correction of the spectra by the analysis of the spectra time-series. While a more advanced version of the pipeline has been presented recently in Cretignier et al. (2023) (sometimes referred to as the YARARA V2 or YV2 datasets), the SNR of the target was too low to apply those advanced 2https://exofop.ipac.caltech.edu/tess/target.php?id=422618449 58 methods of correction (such as the SHELL presented in Cretignier et al. (2022)) and we remained with the YARARA V1 or YV1 version of the products. The corrections available in YARARA cover as much as the telluric lines, as instrumental systematics or stellar activity. The pipeline usually starts from the S1D order-merged spectra produced by official DRS that have been continuum normalized by RASSINE (Cretignier et al., 2020b). The method then consists of a multi-linear decorrelation by fitting a basis of vectors that are designed to correct for some dedicated effects, either obtained by optimized extraction (see e.g. Stalport et al. (2023)) or by principal component analysis (PCA) as initially presented in Cretignier et al. (2021). For a dataset around SNR ∼50, the main corrections that are possible to perform consist of removing cosmic, telluric lines, and the change of the instrumental PSF (Stalport et al., 2023). Even if a clear and strong emission is detected in the core of the CaII H&K lines, no reliable and precise extraction of the signal could be achieved and the stellar activity correction that mainly relies on this proxy (which contains most of the information from active regions (Cretignier et al., 2024)) was therefore skipped. The RVs were obtained with a cross-correlation function (CCF) on the corrected spectra using a line list optimised for the star following the line centre procedure described in Cretignier et al. (2020a). After the application of YARARA, we still detect the long-trend signal which discards any potential effects from telluric or change of the instrumental PSF at the precision level of our data. 3.3.7.2 CCF Activity Linear Model (CALM) to model stellar variability To model stellar variability in the radial velocities, we used activity indicators derived using the CCF Activity Linear Model (CALM) (de Beurs et al., 2024). CALM is a linear regression method which exploits the shape changes that stellar variability introduces into the cross-correlation functions (CCFs) computed from stellar spectra. Since CCFs represent an average of all line shapes in a star’s spectrum, CALM is especially sensitive to line shape changes that persist in most spectral lines. In this method, we do not include the entire CCF in our model since CCFs are comprised of 49-element arrays and we only have 105 RVs. Including the entire CCF would lead to overfitting. We experimented with sampling various fractions of the CCFs and across random locations within 59 Figure 3.3 Residual CCFs (ΔCCFs) computed from HARPS-N spectra. The residual CCFs are computed by subtracting a median CCF. The CALM model-predicted stellar activity signal is indicated by the color (red = redshifted RVs, blue = blue-shifted RVs). The 5 CCF indexes used in our stellar activity model are indicated by black lines. the CCF. We found that using 5 CCF locations provides a balance between preventing overfitting and optimizing goodness-of-fit. These 5 CCF locations are then used to decorrelate against in the global fit performed using EXOFASTv2. We visualize the CCFs for K2-2 and the specific 5 CCF locations in Figure 3.3, where we observe a clear pattern in the stellar variability and the CCF shape changes. This pattern allows us to use CALM to probe and predict stellar activity contributions to the RVs. In Figure 3.4, we plot the CALM model predicted stellar activity contributions to the RVs both in time and in the fourier domain. These activity indexes are able to probe both short- and long-term activity signals while preserving the planetary reflex motion. The ∼270 day signal that is predicted by the CCF4 parameter was also found by Bonomo et al. (2023) and they noted that this signal is also seen in the periodograms of s-index and FWHM. This suggests that this signal corresponds to stellar variability and may be on a timescale longer than the stellar rotation period for K2-2. 3.4 Global Fits Following the method described in Thygesen et al. (2023), we used the differential evolution Markov Chain Monte Carlo (DE-MCMC) exoplanet fitting software EXOFASTv2 (Eastman et al., 2013, 2019) to simultaneously fit the parameters of K2-2 b and its host star. For a global fit to 60 In the left Figure 3.4 Timeseries and periodograms of the CALM predicted stellar variability. panels, the DRS pipeline radial velocities and the stellar variability predictions from CCF index 1, 2, 3, 4, and 5 are plotted as a function of time. The location of these CCF indexes are indicated in Figure 3.3. On the right panel, the Lomb-Scargle periodograms of the corresponding RV timeseries are plotted. In yellow, the Keplerian period of K2-2 is indicated in the periodograms. We do not see signals at this planetary period, which provides reassurance that CALM is not absorbing or creating planetary signals. 61 be accepted as converged, we required that the Gelmin-Rubin statistic be less that 1.01 and the number of independent draws, 𝑇𝑧, greater than 1000. The global fits use MCMC sampling to find the best fit parameters for the system based on the photometric and spectroscopic data. We placed priors on several parameters as follows: a uniform prior from 0 to an upper bound of 0.09858 on the line-of-sight extinction (𝐴𝑣) from Schlegel et al. (1998) and Schlafly & Finkbeiner (2011); a Gaussian prior on parallax of 16.0044 ± 0.0456 from Gaia Early Data Release 3 (accounting for the small systematic offset; EDR3; Gaia Collaboration et al. 2016, 2021; Lindegren et al. 2021); and a Gaussian prior on metallicity ([Fe/H]) of -0.125 ± 0.08 based on measurements from TRES (see Section 3.3.7). The fit also included the spectral energy distribution (SED) photometry as reported by Gaia EDR3 (Gaia Collaboration et al., 2021), WISE (Cutri et al., 2012) and 2MASS (Cutri et al., 2003) (see Table 3.1). To better characerize the host star, the MESA Isochrones and Stellar Tracks (MIST) stellar evolution models (Paxton et al., 2011, 2013, 2015; Choi et al., 2016; Dotter, 2016) were used within the EXOFASTv2 fits. Within EXOFASTv2, limb darkening is constrained via priors derived from models by Claret & Bloemen (2011) and Claret (2017), with physical bounds from Kipping (2013) (see Section 3 of Eastman et al. (2019) for more details on how EXOFASTv2 constrains limb darkening). Although the TESS PDCSAP light curves generally have a correction applied for any contaminating sources, we fitted for a dilution term in case of any sources that may have been missed, based on the contamination ratio (CR) for K2-2 of 0.002101 as reported in the TESS input catalog (TICv8, Stassun et al., 2018). We used placed a 10% Gaussian prior on the dilution centered about CR/(1+CR) = 0.0021. However, the fitted dilution was consistent with zero in all the fits we ran. To account for any residual correlated noise in the systematics-corrected Spitzer data within the EXOFASTv2 fit (see Section 3.3.5), we followed the procedure outlined in §3 of Rodriguez et al. (2020). We scaled the uncertainties by the factor 𝛽 = 1.19 before using the light curve in the global fit. To ensure EXOFASTv2 did not reduce the per-point uncertainties on the Spitzer photometry within the fit, we enforced a lower bound on the variance of zero, otherwise the global fit could over-correct the scaled uncertainties to be consistent with pure white noise. 62 Table 3.3 Models tested for long-term RV trend. Model Description (i) (ii) (iii) (iv) (v) One RV season, linear and quadratic trend with time One RV season, linear trend with time Two RV seasons, no long-term trend Two RV seasons, linear trend with time Two RV seasons, linear and quadratic trend with time ΔBIC 0.0 0.72 49.75 55.15 68.21 3.4.1 RV model selection As the RVs still exhibited an offset in the second observing block after all processing (see Section 3.3.7), we compared five different models that attempt to model this long-term change and evaluated their goodness-of-fit with EXOFASTv2, while keeping all other inputs and priors the same. For each of these models, we first performed a fit using CALM since these long-term trends could be caused by stellar variability. We then took the initial CALM fit to the RVs for each model and ran a global fit with EXOFASTv2. The five models are listed in Table 3.3 and each include the CALM model, but differ in their modeling of the long-term trends where they include some combination of a linear ( (cid:164)𝛾) trend with time, a quadratic ( (cid:165)𝛾) trend with time, and/or an offset 𝐷 between the two observing blocks. In particular, our models include (i) a CALM model with a linear and quadratic trend with time that treats the RV timeseries as one RV observing season without an offsets between the two observing blocks, (ii) a CALM and linear trend model that treats the RV timeseries as one RV observing season without an offset, (iii) a CALM model with an offset 𝐷 between the two observing blocks, (iv) a CALM model with a linear trend and an offset 𝐷, and (v) a CALM model with a linear and quadratic trend and an offset 𝐷 . For the models where we treated the two observing blocks as separate seasons, this allows for different zero-points to be determined for each season. Comparing the Bayesian Information Criterion (BIC) of the models, we found that those including an offset component (i.e. two observing seasons) are heavily disfavored as seen in Table 3.3. The single-season models perform comparabley and we adopt the quadratic-trend model as it has the lowest BIC. 63 Figure 3.5 Projected difference in the time of transit for K2-2 b to the year 2030 using the original ephemeris (gray) and the new ephemeris from this work (purple). Shaded regions indicating up to the 3𝜎 level uncertainty are shown. The inset shows the updated ephemeris, zoomed in for clarity. 3.5 Results and Discussion In this work, we have combined multiple new observations with existing data available for K2-2 b to produce the most accurate and precise system parameters and transit ephemeris (transit time uncertainty <13 minutes in 2030). The period of K2-2 b has been updated to 9.1004157+4.1𝐸−06 −4.5𝐸−05 days and 𝑇0 to 2458072.29291+0.00062 −0.00061 BJD (Figure 3.5). The solutions for the stellar and planetary parameters are shown in Tables 3.4, 3.5, and 3.6. Table 3.7 contains the radial velocity parameters, including the detrending parameters we used, and Table 3.8 lists the parameters of the photometric models for each light curve. We included the MOST light curve in a preliminary fit, as the transit window was observed four times in the full light curve. However, this did not add value to the fit, and the transit was not detectable even with the updated ephemeris, so we did not include the MOST data in the final global fit. The discovery period (Vanderburg et al., 2015) we determined to be 28.8 minutes (∼40 𝜎) from the true period. For context, if someone attempted an observation in 2025 of a K2-2 b transit using the original ephemeris, it would be ∼200 hours from the correct time. We note that this would only result in an offset of ∼18 hours from a transit of K2-2 b since the offset would be quite close to the orbital period of the planet by then, resulting in catching the next adjacent transit. 64 Table 3.4 Median values and 68% confidence interval for K2-2 stellar parameters from the EXOFASTv2 global fit. Parameter Units Priors: Values 𝜋 . . . . . . Gaia parallax (mas) . . . . . . . . . . G [16.0044, 0.0465] G [−0.125, 0.080] [Fe/H] Metallicity (dex) . . . . . . . . . . . . U [0, 0.0985] 𝐴𝑉 . . . . 𝑉-band extinction (mag) . . . . . 0.800+0.033 𝑀∗ . . . . Mass (𝑀⊙) . . . . . . . . . . . . . . . . . −0.030 0.758+0.024 𝑅∗ . . . . . Radius (𝑅⊙) . . . . . . . . . . . . . . . . −0.022 0.3364+0.0100 𝐿∗ . . . . . Luminosity (𝐿⊙) . . . . . . . . . . . . −0.0096 2.757+0.081 𝐹𝐵𝑜𝑙 . . . Bolometric Flux ×10−9 (cgs) . −0.076 2.59+0.26 𝜌∗ . . . . . Density (cgs) . . . . . . . . . . . . . . . −0.24 4.582+0.030 log 𝑔 . . . Surface gravity (cgs) . . . . . . . . −0.031 𝑇eff . . . . 5048+79 Effective Temperature (K) . . . . −78 0.000+0.045 [Fe/H] Metallicity (dex) . . . . . . . . . . . . −0.039 0.000 ± 0.055 Initial Metallicity1 . . . . . . . . . . [Fe/H]0 5.5+5.0 𝐴𝑔𝑒 . . . Age (Gyr) . . . . . . . . . . . . . . . . . . −3.9 𝐸 𝐸 𝑃 . . 335+16 Equal Evolutionary Phase2 . . −34 0.045+0.035 𝐴𝑉 . . . . V-band extinction (mag) . . . . . −0.031 0.76+0.32 𝜎𝑆𝐸 𝐷 . . SED photometry error scaling −0.19 16.004 ± 0.046 𝜛 . . . . . Parallax (mas) . . . . . . . . . . . . . . 62.48 ± 0.18 𝑑 . . . . . . Distance (pc) . . . . . . . . . . . . . . . Notes. See Table 3 in Eastman et al. (2019) for a detailed description of all parameters. Gaussian and uniform priors are indicated as G [mean, 𝜎] and U [lower bound, upper bound], respectively. The metallicity prior is adopted from the average of three TRES measurements: [M/H] = -0.193, -0.191, 0.009 (see Section 3.3.7 for details). 1The metallicity of the star at birth. 2Corresponds to static points in a star’s evolutionary history. See §2 in Dotter (2016). 65 Table 3.5 Median values and 68% confidence interval for K2-2 b planetary parameters from the EXOFASTv2 global fit. Values 9.1004157+0.0000041 −0.0000045 2.469+0.10 −0.091 9.7 ± 1.2 Units Parameter 𝑃 . . . . . . Period (days) . . . . . . . . . . . . . . . . . . . . . . 𝑅𝑃 . . . . . Radius ( 𝑅E) . . . . . . . . . . . . . . . . . . . . . . . 𝑀𝑃 . . . . Mass ( 𝑀E) . . . . . . . . . . . . . . . . . . . . . . . . 𝑇0 . . . . . . Optimal conjunction Time1 (BJDTDB) 2458072.29291+0.00062 −0.00061 𝑎 . . . . . . Semi-major axis (AU) . . . . . . . . . . . . . . 𝑖 . . . . . . . Inclination (Degrees) . . . . . . . . . . . . . . . 𝑒 . . . . . . . Eccentricity2 . . . . . . . . . . . . . . . . . . . . . . 𝜔∗ . . . . . Argument of Periastron (Degrees) . . . 𝑇𝑒𝑞 . . . . . Equilibrium temperature3 (K) . . . . . . . 𝜏circ . . . . Tidal circularization timescale (Gyr) . 𝐾 . . . . . . RV semi-amplitude (m/s) . . . . . . . . . . . 𝑅𝑃/𝑅∗ . Radius of planet in stellar radii . . . . . 𝑎/𝑅∗ . . . Semi-major axis in stellar radii . . . . . (𝑅𝑃/𝑅∗)2 . . . . . . . . . . . . . . . . . . . . . . . . . 𝛿 . . . . . . 𝛿Kepler . . Transit depth in Kepler (fraction) . . . . 𝛿i′ . . . . . Transit depth in i’ (fraction) . . . . . . . . . 𝛿r′ . . . . . Transit depth in r’ (fraction) . . . . . . . . . 𝛿4.5𝜇m . . Transit depth in 4.5𝜇𝑚 (fraction) . . . . 𝛿TESS . . . Transit depth in TESS (fraction) . . . . . 𝜏 . . . . . . Ingress/egress transit duration (days) . 𝑇14 . . . . . Total transit duration (days) . . . . . . . . . 𝑇𝐹𝑊 𝐻 𝑀 . FWHM transit duration (days) . . . . . . 𝑏 . . . . . . Transit Impact parameter . . . . . . . . . . . 𝑏𝑆 . . . . . Eclipse impact parameter . . . . . . . . . . 𝜏𝑆 . . . . . . Ingress/egress eclipse duration (days) 𝑇𝑆,14 . . . Total eclipse duration (days) . . . . . . . . 𝑇𝑆,𝐹𝑊 𝐻 𝑀 FWHM eclipse duration (days) . . . . . . 0.0792+0.0011 −0.0010 88.91+0.68 −0.45 0.215+0.056 −0.094 88+19 −20 753.2+7.1 −6.9 1310+540 −430 3.54 ± 0.42 0.02981+0.00079 −0.00061 22.46 ± 0.72 0.000889+0.000048 −0.000036 0.001186+0.000052 −0.000050 0.001092+0.000035 −0.000034 0.001171+0.000056 −0.000051 0.000922+0.000046 −0.000040 0.001092+0.000039 −0.000038 0.00329+0.00088 −0.00036 0.1013+0.0015 −0.0014 0.0978 ± 0.0013 0.34+0.20 −0.22 0.51+0.15 −0.31 0.00540+0.00074 −0.00051 0.141+0.027 −0.028 0.135+0.027 −0.028 Notes. See Table 3 in Eastman et al. (2019) for a detailed description of all parameters. 1Optimal time of conjunction minimizes the covariance between 𝑇𝐶 and Period. 2Note that due to the low significance of the eccentricity, this is consistent with 𝑒 = 0 when considering the Lucy-Sweeney bias (Lucy & Sweeney, 1971). 3Assumes no albedo and perfect redistribution. 66 Table 3.6 Median values and 68% confidence interval for K2-2 b planetary parameters from the EXOFASTv2 global fit (continued). Parameter Units 𝛿𝑆,2.5𝜇𝑚 Blackbody eclipse depth at 2.5𝜇m (ppm) 𝛿𝑆,5.0𝜇𝑚 Blackbody eclipse depth at 5.0𝜇m (ppm) 𝛿𝑆,7.5𝜇𝑚 Blackbody eclipse depth at 7.5𝜇m (ppm) 𝜌𝑃 . . . . Density (cgs). . . . . . . . . . . . . . . . . . . . . . . . . 𝑙𝑜𝑔𝑔𝑃 . Surface gravity . . . . . . . . . . . . . . . . . . . . . . Θ . . . . . Safronov Number . . . . . . . . . . . . . . . . . . . . ⟨𝐹⟩ . . . Incident Flux (109 erg s−1 cm−2) . . . . . . . 𝑇𝑃 . . . . . Time of Periastron (BJDTDB) . . . . . . . . . . 𝑇𝑆 . . . . . Time of eclipse (BJDTDB) . . . . . . . . . . . . . 𝑇𝐴 . . . . . Time of Ascending Node (BJDTDB) . . . . 𝑇𝐷 . . . . Time of Descending Node (BJDTDB) . . . 𝑉𝑐/𝑉𝑒 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝑒 cos 𝜔∗ See footnote1 . . . . . . . . . . . . . . . . . . . . . . . . 𝑒 sin 𝜔∗ See footnote1 . . . . . . . . . . . . . . . . . . . . . . . . 𝑀𝑃/𝑀∗ Mass ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 𝑑/𝑅∗ . . Separation at mid transit . . . . . . . . . . . . . . Values 0.912+0.083 −0.073 15.33+1.00 −0.85 34.9+2.2 −1.7 3.53+0.63 −0.57 3.192+0.061 −0.065 0.0274+0.0035 −0.0034 0.0698+0.0032 −0.0029 2456689.01+0.30 −0.34 2456693.61+0.35 −0.36 2456705.54+0.20 −0.28 2456690.73+0.32 −0.22 0.810+0.086 −0.047 0.004+0.059 −0.060 0.205+0.057 −0.098 0.0000365+0.0000044 −0.0000043 17.9+2.3 −1.7 Notes. See Table 3 in Eastman et al. (2019) for a detailed description of all parameters. √ 1 Within the fits, these are parameterized as uniform prior on eccentricity. 𝑒 sin 𝜔∗, respectively, to ensure a 𝑒 cos 𝜔∗ and √ Table 3.7 Median values and 68% confidence interval for radial velocity parameters. Telescope Parameters: Systemic velocity (km/s) 𝛾sys . . 𝛾rel . . Relative RV Offset (m/s) . . . . (cid:164)𝛾 . . . . RV slope (m/s/day) . . . . . . . . . (cid:165)𝛾 . . . . RV quadratic term (m/s/day2) 𝜎𝐽 . . . RV Jitter (m/s) . . . . . . . . . . . . . 𝜎2 𝐽 . . RV Jitter Variance . . . . . . . . . 𝐶𝐶𝐹0 Additive detrending coeff. . . 𝐶𝐶𝐹1 Additive detrending coeff. . . 𝐶𝐶𝐹2 Additive detrending coeff. . . 𝐶𝐶𝐹3 Additive detrending coeff. . . 𝐶𝐶𝐹4 Additive detrending coeff. . . HARPS-N −2.91 0.02 ± 0.63 0.00239 ± 0.00039 0.00000133 ± 0.00000036 2.30+0.24 −0.22 5.27+1.2 −0.98 −3.23 ± 0.92 3.4 ± 2.6 2.0 ± 2.0 −6.6 ± 1.5 1.3 ± 1.0 Notes. Reference epoch = 2458561.069744 BJD. Five additive detrending parameters were included to account for stellar activity (see Section 3.4). 67 Table 3.8 Median values and 68% confidence intervals for the photometric models. Telescope Wavelength Parameters K2 MEarth (i’) Tel. 1 MEarth (i’) Tel. 2 MEarth (i’) Tel. 3 MEarth (i’) Tel. 6 MEarth (i’) Tel. 11 MEarth (i’) Tel. 12 MEarth (i’) Tel. 16 MEarth (i’) Tel. 18 ULMT (r’) Spitzer (4.5𝜇𝑚) TESS Sector 42 TESS Sector 70 𝑢† 1 0.57 ± 0.052 0.426+0.022 −0.023 0.426+0.022 −0.023 0.426+0.022 −0.023 0.426+0.022 −0.023 0.426+0.022 −0.023 0.426+0.022 −0.023 0.426+0.022 −0.023 0.426+0.022 −0.023 0.551 ± 0.054 0.077+0.047 −0.043 0.428 ± 0.038 0.428 ± 0.038 𝑢‡ 2 0.171 ± 0.051 0.205 ± 0.020 0.205 ± 0.020 0.205 ± 0.020 0.205 ± 0.020 0.205 ± 0.020 0.205 ± 0.020 0.205 ± 0.020 0.205 ± 0.020 0.183 ± 0.052 0.146 ± 0.050 0.21 ± 0.036 0.21 ± 0.036 𝜎2★(10−9) 2.45+0.56 −0.50 2360+320 −290 2220+290 −270 4830+510 −470 4700+460 −430 1310+200 −190 2060+230 −210 940+160 −150 1530+200 −180 3150.0+430.0 −380.0 5.7+9.2 −4.3 28.3+8.1 −8.0 6.1 ± 7.1 Transit Parameters 𝐹∗ 0 0.9999999 ± 0.0000047 1.00042+0.000096 −0.000097 1.000525 ± 0.000093 1.00033 ± 0.00012 1.00053 ± 0.00012 1.000213+0.000076 −0.000077 1.000355 ± 0.000080 1.000344 ± 0.000070 1.000253 ± 0.000077 0.99957 ± 0.00014 1.000003 ± 0.000036 1.0000131 ± 0.0000073 1.0000099 ± 0.0000065 Additive detrending coeff 𝐶0 — −0.00026 ± 0.00050 0.00001 ± 0.00049 −0.00142 ± 0.00057 −0.00076 ± 0.00058 −0.00147 ± 0.00042 −0.00091 ± 0.00045 −0.00003 ± 0.00040 −0.00098 ± 0.00045 0.00041 ± 0.00035 — — — Notes.†Linear limb-darkening coefficient. ‡Quadratic limb-darkening coefficient. ★Added variance. ∗ Baseline flux. 68 3.53+0.63 K2-2 b has a radius of 2.47+0.10 −0.09 𝑅⊕ and a mass of 9.7 ± 1.2 𝑀⊕. This yields a bulk density of −0.57 g cm−3, which is twice that of Neptune (1.638 g cm−3). According to the composition models from Zeng et al. (2016), it is likely K2-2 b has a high water content (Figure 3.6). While it is consistent with 100% water, a more physically motivated solution would be a rocky core with an extended envelope of volatiles including a H/He envelope. More observations are needed to place further constraints on the planetary composition. The mass of K2-2 b was updated in a recent in-depth radial velocity study of Kepler and K2 systems (Bonomo et al., 2023) to refine planet masses and identify cold Jupiters in systems containing small planets. Bonomo et al. (2023) refined important planetary parameters such as the period (to 9.0949 ± 0.0026 days) and mass (to 10.1 +1.2 −1.1 𝑀⊕), and did not find any long-term trends in the RVs that could correspond to a long-period companion. We used the same RV observations from this work (in addition to those from Vanderburg et al. 2015) but with improved precision from improved modeling of the stellar activity using the CALM technique (see Section 3.3.7.2) in our global fit, and when combined with the other photometric and spectroscopic data, we were able to refine these measurements and uncover a potential outer companion due to a long-term trend in the RVs. 3.5.1 RV trend As mentioned in Section 3.3.7, there is a long-term trend in the radial velocities (see Figure 3.2) after correcting for stellar variability. To test the possibility of a second planet or star within the system, we reran the fit described in Section 3.4 but allowed EXOFASTv2 to fit for a second planet within the RVs only. We note that there is no additional transit signal detected in any photometric data sets used in this analysis. However, preliminary fits did not converge nor provide any useful constraint on the period of a potential companion, even with improved constraints on K2-2 b. Figure 3.2 shows the long-term trend in the RVs, and our resulting best-fit model from EXOFASTv2. It is clear that the period of this secondary companion is much longer term than the extent of our RV data set from HARPS-N (∼2500 days). We instead model the long-term trend with a quadratic acceleration term. Our best-fit results find a linear slope in the RVs of 0.0024 ± 0.0004 m s−1 with 69 Figure 3.6 Mass-radius diagram for K2 sub-Neptunes (𝑅𝑃 = 2.0 − 3.0 𝑅⊕). The large black circle is K2-2 b, while the small gray circles are other sub-Neptunes with measured masses from the NASA Exoplanet Archive. The lines represent composition tracks from Zeng et al. (2016). a quadratic term of 1.33𝐸 − 06 ± 3.6𝐸 − 07 m s−1 day−2 to best represent the long-term RV trend. The observed RV trend may correspond to an additional companion to K2-2 with an orbital separation of several AU. Vanderburg et al. (2015) acquired high-resolution imaging observations of the star and did not detect any stellar companions between 0.1 − 5.0" (≈ 6 − 310 AU). This non-detection, combined with the relatively small amplitude of the RV acceleration, suggests that this outer companion could be a planet or a brown dwarf. As K2-2 was observed by Hipparcos, it is possible to place additional constraints on any outer companions using Hipparcos-Gaia astrometry (Brandt, 2018, 2021). If a massive companion exists at a separation of several AU from K2-2, it would likely generate a significant astrometric acceleration between Hipparcos and Gaia. However, no significant acceleration is detected in 70 the Hipparcos-Gaia astrometry, with 𝜒2 = 2.3 for a constant proper motion (Brandt, 2021). The astrometric precision for K2-2 is ∼0.07 mas yr−1, equivalent to ∼20 m s−1 at the 62.48 ± 0.18 pc distance of the system. This means that a net Hipparcos-Gaia velocity change greater than ≳100 m s−1 can be excluded at 5𝜎 confidence. This non-detection largely excludes the existence of massive companions (≳10 𝑀𝐽) orbiting K2-2 within several AU. However, a planetary-mass companion could be reconciled with the astrometric non-detection. Continued RV monitoring of the K2-2 system is needed to constrain the further evolution of the RV trend, providing some constraints on the fundamental parameters of the possible second planet in the system. 3.5.2 Future work The K2 mission was driven by the community, which led to planets orbiting much brighter host stars than the original Kepler mission, targets well suited for detailed characterization. Although characterization might be challenging with current facilities, K2-2 b is a worthwhile target for ongoing monitoring and targeted observations. Following the Kempton et al. (2018) prescription for the transmission spectroscopy metric (TSM), we find that K2-2 b has a TSM of 50.09.2 8.7, which falls just below the lowest value suggested for target prioritization for JWST. However, when compared to the other ∼ 160 sub-Neptunes (𝑅𝑃 = 2.0 − 3.0 𝑅⊕) in the K2 catalog, the TSM of K2-2 b is the fifth highest, suggesting that it is a suitable candidate for studying sub-Neptunes in closer detail. Monitoring the radial velocities of K2-2 would allow for more refined constraints on the stellar activity, and possibly uncover additional long-period and/or low-mass candidates in the system. The co-moving white dwarf (WD) companion to K2-2 provides an avenue to measure a precise age for the system if the mass and age for the WD can be determined. The stellar parameters were calculated as part of a catalog of all WDs within Gaia EDR33 by Gentile Fusillo et al. (2021). The mass, effective temperature, and surface gravity were determined for three different atmospheric compositions: pure H, pure He, and a mix of H and He (see Table 3.9). Assuming the highest 3Gaia EDR3 source_id 2645940445519931520 71 Table 3.9 Stellar parameters for the white dwarf companion of K2-2 from Gentile Fusillo et al. (2021). Composition H He H+He 𝑇eff (K) 7519 ± 195 7395 ± 189 7083 ± 167 log 𝑔 (cgs) Mass (𝑀⊙) 0.52 ± 0.04 7.88 ± 0.08 0.47 ± 0.02 7.82 ± 0.06 0.44 ± 0.03 7.71 ± 0.07 mass value from the models (pure-H, 0.52 ± 0.04 𝑀⊙), we find a lower limit on the cooling age of 1.13±0.13 Gyr. While this current age estimate does not constrain the system age further, more precise photometry and measuring the spectrum of the WD would constrain the mass (and system age) more reliably than Gaia photometry alone. 3.6 Conclusion With thousands of exoplanets discovered to date, some will inevitably be “lost" (unconstrained ephemerides) or forgotten as newer discoveries peak the interest of the community. Unfortunately, these lost planets may be excellent targets for detailed characterization with JWST (Gardner et al., 2006), but are not accessible due to large uncertainties in future transit times. K2-2 b was the first planet discovered during the Two-Wheeled Concept Engineering Test of the K2 mission (Howell et al., 2014), showing very quickly that K2 would be a successful repurposing of the Kepler spacecraft. By combining observations from multiple NASA missions along with key ground- based follow up that span nearly a decade, we have recovered the lost transit ephemeris of K2-2 b. In addition to being the first K2 planet, it is also well-suited for studying the atmosphere of a hot sub-Neptune as it orbits a bright (𝐾∼8.03) K-dwarf. This would be a valuable measurement since it sits on the high-mass peak of the sub-Neptune radius valley (Owen & Jackson, 2012; Fulton et al., ephemeris (𝑃 = 9.1004157+4.1𝐸−06 2017) and could provide insight to the formation and evolution of sub-Neptunes. Our updated −4.5𝐸−06 days, 𝑇0 = 2458072.29291+0.00062 −0.00061 BJD) confirms the false detection from the MOST satellite (Vanderburg et al., 2015) that led to a ∼40𝜎 offset to the true period. Systems like K2-2 show the importance of continued monitoring of exoplanet systems and dedicated ephemeris refinement efforts like the K2 & TESS Synergy project (Ikwut-Ukwa et al., 2020; Thygesen et al., 2023), ExoClock (Kokori et al., 2021, 2022, 2023), Exoplanet Watch (Zellem 72 et al., 2019, 2020), and ORBYTS (Edwards et al., 2019a, 2020, 2021). 73 CHAPTER 4 THE K2 & TESS SYNERGY IV: K2’S TOP 50 ATMOSPHERIC TARGETS The field of exoplanets has entered an age of detailed characterization. The success of JWST in its first few years of operation has already revolutionized the study of exoplanet atmospheres, but we are still only in the early stages of understanding the complexities of these distant worlds. All aspects of this topic will only continue to grow as we achieve greater observing precision with current and future missions, especially as we seek to characterize Earth analogs and search for evidence of biosignatures. In order for a large number of exoplanet atmospheres to be accessible for observations using transmission spectroscopy, there is an urgent need to have up-to-date ephemerides for known transiting planets. To aid in this effort, we continue the K2 & TESS Synergy to ensure that targets in the K2 catalog that are most amenable to transmission spectroscopy with JWST will be observable. This chapter presents the ongoing work for the fourth paper in the K2 & TESS Synergy series. The tables of fitted parameters are included as supplementary material. In Section 4.1, we outline some of the major discoveries using JWST for transmission spectroscopy, placing these in context of previous studies of the respective planets. This cutting-edge science cannot be done without knowing ephemerides to high precision beforehand. With this in mind, Sections 4.2-4.4 describe our ongoing reanalysis of the top 50 K2 planets for JWST atmospheric characterization. Section 4.5 then presents a hypothetical application of the results of the K2 & TESS Synergy, followed by our conclusions in Section 4.6. 4.1 Major JWST Results 4.1.1 WASP-96 b: JWST’s first spectrum The JWST Early Release Observations (ERO; Pontoppidan et al. 2022) contained the first images and spectra from the mission intended for public outreach purposes and to showcase the capabilities of the telescope. Among these was a transmission spectrum of WASP-96 b, a puffy hot Jupiter (∼ 0.48𝑀J, ∼ 1.20𝑅J) on a 3.4 day period around a Sun-like star (Hellier et al., 2014). WASP-96 b was an ideal first target as it has a short orbital period, its atmosphere had been previously studied, 74 and, having an atmosphere with a large scale height, would produce high S/N observations. Prior to JWST, the first transmission spectrum for WASP-96 b was obtained by Nikolov et al. (2018) with the Very Large Telescope (VLT) in the 0.36-0.82 𝜇m range, which uncovered pressure-broadened resonance doublets of both Na and K. This was a first, as only the cores of these particular lines had been seen previously in other exoplanet atmospheres, which can be explained by clouds and hazes obscuring the wings of these line profiles. Seeing the full profiles, then, indicated that deeper layers of WASP-96 b’s atmosphere were being probed. Their models resulted in a metallicity abundance of log(𝑍 𝑝/𝑍⊙) = 0.4+0.7 −0.5 (i.e. the relative abundance of elements heavier than helium) for WASP-96 b, and assuming chemical equilibrium, was consistent with a cloud-free atmosphere. They measured a sodium abundance of log(Na) = −5.1+0.6 −0.4, and the temperature of the atmosphere to be 𝑇 = 1710+150 −200K, which was higher than the equilibrium temperature. Several years later, Yip et al. (2021) curiously found a significant offset between new transmission spectra using HST (0.8-1.7 𝜇m) and Spitzer/IRAC (Fazio et al., 2004) compared to the previous one from the VLT. While the source of the offset was not determined, a correction was applied, and from this combined spectrum the atmosphere was found to have abundances of log(Na) = −3.88+1.05 −0.82, log(H2O)= −3.65+0.90 −0.94, log(NH3)= −6.51, and a temperature of 𝑇 = 954+198 with the previous study. The offset between HST and the VLT served as a reminder that extreme care −195K - in disagreement must be taken while using observations from different instruments (especially ground- vs. spaced- based) without overlapping wavelength coverage, making facilities like JWST with coverage across wide bands vital to advance the field. Nikolov et al. (2022) analyzed the same data, and found abundances in agreement with Yip et al. (2021), with a measured C/O of 0.35 (which can be used as an indicator of where the planet formed; see Section 4.5.1; Madhusudhan et al. 2014). A further study from McGruder et al. (2022) combined new measurements from the Magellan telescope in the range of 0.48-0.83 𝜇m with the previous studies. Their estimates of water and sodium abundances were consistent with the previous studies and, once again, the atmosphere was found to be consistent with cloud-free models. However, Samra et al. (2023) created 3D atmosphere models of WASP-96 b based on measured parameters of the system and the observed transmission 75 spectra, and found that cloudy models were consistent with the data, but that JWST observations would be able to constrain this further. During the ERO, a single transit of WASP-96 b was observed by JWST NIRISS/SOSS in the range of 0.6-2.8 𝜇m, with the goal of observing the H2O absorption at 1.4 𝜇m (Pontoppidan et al., 2022). Their measured abundances were log(H2O)= −3.59 ± 0.35, log(K)= −8.04+1.22 −1.71, log(CO2)= −4.38+0.47 −0.57, and an upper limit of log(CO)= −3.25. With these better constrained −0.37 and a metallicity of log(𝑍 𝑝/𝑍⊙) = −0.63+0.64 measurements, they found log(C/O)= −0.30+0.17 −0.44, placing these as comparable to Solar values. The abundance of H2O was consistent with the previous studies, with the precision refined by an order of ∼ 4 when compared to Yip et al. (2021) and ∼ 10 compared to McGruder et al. (2022). As the NIRISS wavelengths do not fully encapsulate the Na absorption line from the previous studies, robust constraints could not be placed on it. Finally, Taylor et al. (2023) found that the existence of inhomogeneous clouds and hazes best modeled the data, but that further observations, particularly at longer wavelengths with JWST/MIRI, would be able to constrain this further. The drastically improved abundances, along with the fact that the spectrum was achieved by observing a single transit, showcased JWST’s potential for exoplanet characterization. 4.1.2 WASP-39 b: unexpected detection of SO2 The ERO program was followed by several months dedicated to the Director’s Discretionary Early Release Science Programs (DD-ERS), a small selection of observations utilizing all the instruments on board JWST that would produce science-quality data. Included in these programs were transmission spectra of WASP-39 b which has since become the poster child for the cutting- edge science that can be done with JWST. WASP-39 b is a Hot Jupiter (∼ 0.28𝑀J, ∼ 1.27𝑅J) on a 4.06 day period around a G-type star (Faedi et al., 2011). Similar to WASP-96 b, it is highly irradiated and has a puffy atmosphere. The atmosphere of WASP-39 b had previously been well- studied, but the resulting chemical abundance measurements were contentious, making it a prime opportunity to test JWST’s capabilities. 76 Transmission spectra from HST/STIS (0.3-1.01 𝜇m; Sing et al. 2016; Fischer et al. 2016), HST/WFC3 (0.8-1.7 𝜇m; Wakeford et al. 2018), and the VLT (0.41-0.81 𝜇m, Nikolov et al. 2016) found pressure-broadened features of Na and K, inferring a clear atmosphere similar to WASP-96 b, as well as a strong H2O feature which allowed for an estimation on the metallicity to be 151+48 −46 × Solar (Wakeford et al., 2018). Tsiaras et al. (2019a) analyzed the same HST/WFC3 spectra, but included only the G141 grism data (1.1-1.6 𝜇m), which resulted in an abundance of log(H2O)= −5.94±0.61, equivalent to roughly 0.001× Solar: Fisher & Heng (2018) performed atmospheric retrievals on the same data, and obtained log(H2O)= −2.3+0.40 Pinhas et al. (2018) found an abundance of log(H2O)= −4.07+0.72 −0.67 or ∼ 8× Solar. Atmosphere retrievals from −0.78, i.e. around 0.1× Solar. Kirk et al. (2019) combined new transmission spectra from the ACAM instrument (Benn et al., 2008) on the William Herschel Telescope with all previous data sets, and ran retrievals resulting in a metallicity of 282+65 −58 × Solar. Across all of the analyses, the metallicity of WASP-39 b could be anywhere between 0.001-282× Solar, demonstrating not only the need for missions like JWST that simultaneously observe across a large band of wavelengths, but that chemical abundances rely heavily on the specific retrievals and assumptions made behind these. As part of the ERS program, WASP-39 b was observed using NIRISS (0.6-2.8𝜇m; Feinstein et al. 2022), NIRCam (2.42-4.03 𝜇m; Ahrer et al. 2022), NIRSpec G395H (2.7-5.2 𝜇m; Alderson et al. 2022), and NIRSpec PRISM (0.5-5.5 𝜇m; JWST Transiting Exoplanet Community Early Release Science Team et al. 2023; Rustamkulov et al. 2022), and was later observed with MIRI (5-12 𝜇m; Powell et al. 2024). The comprehensive and high-resolution spectral coverage finally allowed for a more robust measurement of multiple carbon- and oxygen-bearing species, and thus C/O and metallicity. The transmission spectra showed features of the typical species CO, CO2, H2O, Na, and K, a non-detection of CH4, and some gray cloud cover. However, the most notable discovery was that of SO2 (Rustamkulov et al., 2022; Alderson et al., 2022; Tsai et al., 2022; Powell et al., 2024) - a molecule never before seen in an exoplanet atmosphere, and one that is indicative of photochemical processes. The pathway to the creation of SO2 involves the photolysis of H2S and H2O from UV radiation (e.g. Zahnle et al. 2009) followed by the oxidation of sulfur, forming the 77 product of SO2. As such, it is expected to be seen in highly irradiated planets with 𝑇eq = 900 ∼ 1000 K, and to be produced more readily in planets with lower C/O values and/or higher metallicities (Polman et al., 2023). With the inclusions of this species in atmosphere retrievals, WASP-39 b was found to have C/O < 1 (i.e. sub-solar), and a metallicity between 7.1 − 8.0 ± 0.4× Solar depending on the reduction pipeline used for the data (Powell et al., 2024). These values suggest that WASP-39 b may have formed between the H2O and CO2 ice lines via core accretion, and became enriched with oxygen-bearing material as it migrated inward through the disk and accreted solids (Ahrer et al., 2022; Alderson et al., 2022). The precise measurements and resulting insight to the evolution of WASP-39 b established the unparalleled capabilities of JWST for exoplanet characterization. 4.1.3 K2-18 b: a controversial sub-Neptune The interpretation of JWST results has not been without controversy. Transmission spectra of K2-18 b sparked debate as to whether the signs of life had been detected for the first time. K2-18 b is a sub-Neptune (∼ 8.9𝑀⊕, ∼ 2.4𝑅J) orbiting an M dwarf on a 32.9 day orbit (Benneke et al., 2017; Cloutier et al., 2017; Sarkis et al., 2018). Using transmission spectra from HST/WFC3, Tsiaras et al. (2019b) and Benneke et al. (2019) identified H2O absorption, meaning that not only was K2-18 b the first small planet to have water detected in its atmosphere, but was also the first habitable-zone sub-Neptune to have a confirmed atmosphere. Due to the difficulties of observing atmospheres with small scale heights (compared to gas giants), as well as only having access to a narrow band of wavelengths, robust constraints could not be placed on the water abundance, with retrievals being consistent with an abundance up to 50% with a significant H2-He envelope (Tsiaras et al., 2019b). Models from Blain et al. (2021) and Charnay et al. (2021) found that it was possible for water ice clouds to be present in the planet’s atmosphere, particularly for metallicities >100× Solar. The internal structure of planets in this mass-radius regime is not fully understood, namely whether they have dense, rocky cores with extended H/He envelopes, or if they have significant water oceans with thin, H2-rich atmospheres (sometimes referred to as ‘Hycean’ worlds; Madhusudhan et al. 2021): Madhusudhan et al. (2020) found that the HST/WFC3 transmission spectrum of K2-18 b was consistent with both scenarios. It therefore became a much anticipated target for JWST, as the 78 broad wavelength coverage held the key to answering questions about the composition of K2-18 b, and whether or not it holds conditions amenable to life. Transmission spectra of K2-18 b were obtained using JWST/NIRISS and NIRSpec during the first JWST Guest Observer cycle (Madhusudhan et al., 2023), which revealed the strong presence of both CH4 and CO2, and a non-detection of H2O and NH3. The absence of water contradicted the previous findings (Tsiaras et al., 2019b; Benneke et al., 2019; Madhusudhan et al., 2020), which was attributed to the degeneracy between H2O and CH4 (Madhusudhan et al., 2023). This once again highlights the strides being made with JWST, even for planets that have been previously well studied. The most surprising find came, however, with the 2.4𝜎 detection of (CH3)2S (dimethyl sulfide; DMS). This was controversial, as DMS is a by-product of phytoplankton on Earth and was thus interpreted as a potential indicator of life on K2-18 b (Madhusudhan et al., 2023). Wogan et al. (2024) followed this up by simulating K2-18 b as both a Hycean world (testing inhabited and lifeless scenarios) and a mini-Neptune with an extended atmosphere. They found two models consistent with the current JWST data, that of a Hycean planet host to methanogenic life that could produce the observed levels of CH4, and a mini-Neptune with a deep atmosphere and no defined surface. Ultimately, they favored the latter as it is a much simpler model that reproduces the data equally well. Further investigation from Shorttle et al. (2024) found that models of a H2 atmosphere atop an ocean of silicate magma are consistent with the presence of CO2 and non-detection of NH3, although the models of Rigby et al. (2024) disputed this. Luu et al. (2024) suggested that K2-18 b is may host a supercritical water ocean with a temperature of 710 ∼ 1070 K, which would make it unlikely to harbor life. As of now, the mystery of K2-18 b’s composition and structure remains unsolved. While JWST observations clarified some aspects, they have sparked debate in others. Nevertheless, previous facilities have not been able to achieve nearly the same coverage and sensitivity as JWST, and as a result of this K2-18 b has risen to the forefront of potentially habitable exoplanets. More observations of this system, both with JWST and future missions, will potentially clear up the murkiness surrounding K2-18 b’s atmosphere. 79 4.1.4 Prospects for rocky planets with JWST While JWST has targeted many gas giants with large atmospheres that lend themselves to transmission spectra, a series of Earth and Venus-like planets have also been observed. Due to the significantly smaller atmospheric scale heights of rocky planets, it is often required to obtain more transmission spectra of the transits to build enough signal-to-noise to resolve spectroscopic features of a present atmosphere, or rule out an atmosphere altogether. Rocky planets orbiting M dwarfs in particular are increasingly sought-after for this purpose, not only for their optimal planet-to-star size ratio and relative abundance, but because they are the most promising targets in the immediate search for habitability (e.g. Charbonneau & Deming 2007). The habitable zones of M dwarfs being close to the host star make it easier to observe multiple transits of any planetary companions. However, it is not yet fully understood how the impacts of the often-active hosts may alter conditions on these worlds with their high X-ray and UV emission (e.g. Pizzolato et al. 2003; Peacock et al. 2019), on top of the fact that many of these planets are thought to be tidally-locked to their host stars (Kasting et al., 1993; Barnes, 2017). Such systems have therefore become the focus of multiple studies using JWST transmission spectra to investigate whether these planets can retain significant atmospheres. JWST/NIRSpec was used to validate its first planet and measure the atmosphere of the Earth- sized planet LHS 475 b (Lustig-Yaeger et al., 2023), which was found to have an orbital period of 2.02 days around its M dwarf host. Its spectrum was featureless, which is consistent with that of Venus with a high cloud deck, Mars with a very thin atmosphere, or Titan with no atmosphere at all (Lustig-Yaeger et al., 2023). A growing number of rocky planets with featureless atmospheres are being found, including the sub-Earth GJ 341 b (Kirk et al., 2024), L 169-9 b (Alam et al., 2025), and TOI-776 b (Alderson et al., 2025). As it stands, JWST has not yet detected significant atmospheric features on a rocky planet. Whether it is the case that more transits are needed to be able to reach the required signal-to-noise to uncover these absorption lines, or that there are simply no atmospheres to observe, for now remains a mystery. 80 4.2 Target Selection To identify the top 50 exoplanets discovered by K2 that are most well-suited for atmospheric studies, we used the confirmed catalog from the NASA Exoplanet Archive1 (NEA) and ranked them by transmission spectroscopy metric (TSM; Kempton et al. 2018). The TSM provides an estimate of the planet’s accessibility for transmission spectroscopy based on the planet’s mass, radius, and equilibrium temperature, and the host star’s radius and apparent J magnitude. We therefore prioritized planets with higher TSM values as they are more likely to result in high signal-to-noise observations. To converge on a feasible list for this batch, we made some alterations to the TSM list. We excluded all systems that have not been reobserved by TESS as of Sector 72 (some of these will be observed in upcoming sectors; see Section 5.2). As the number of planets being modeled in a system presents increasing complexity and therefore longer computational times, we limited our list to include only systems with one or two planets. Systems with > 2 planets will be addressed in future work of the K2 & TESS Synergy. If one planet in the remaining multiplanet systems met the TSM threshold but the companion did not, we still included both planets in the global analysis. Seven planets from the previous Synergy papers are in the top 50 (K2-2 b/HIP 116454 b, K2-167 b, K2-237 b, K2-260 b, K2-261 b, K2-277 b, K2-321 b). We included these systems in the current sample for completeness, and ran new global fits if new data were available (i.e. new TESS Sectors or RVs). The final list after applying the discussed filters is shown in Tables 4.1 and 4.2. Systems for which global fits have not yet been run are indicated. 4.3 Data and Global Fits We followed the same general process for data preparation and analysis as outlined in Chapters 2.3 and 2.4, and Thygesen et al. (2023), which is briefly summarized here. We used the methods described in Vanderburg & Johnson (2014) and Vanderburg et al. (2016) to process the K2 light curves, and accounted for any major contamination from other stars within the K2 aperture. For TESS light curves, we prioritized short-cadence observations when available, otherwise using 1https://exoplanetarchive.ipac.caltech.edu/. Accessed January 29, 2025. 81 FFIs, and used the PDCSAP flux (Stumpe et al., 2012, 2014; Smith et al., 2012) processed through the SPOC pipeline (Jenkins et al., 2016; Caldwell et al., 2020). We ran the TESS light curves through keplerspline2 to correct for remaining stellar variability (Vanderburg & Johnson, 2014). For systems with a quoted contamination ratio on ExoFOP, we allowed for a dilution term to be fitted to account for any remaining contamination in the TESS aperture. We scoured the literature for any previously published RVs for each system, only including data sets that have at least four measurements to protect from overfitting. The list of K2 Campaigns and TESS Sectors used in this analysis can be found in Tables 4.1 and 4.2, along with the discovery references and sources of RVs. For the global fits with EXOFASTv2 (Eastman et al., 2013; Eastman, 2017; Eastman et al., 2019), we placed Gaussian priors on stellar metallicity, preferentially using measurements from TRES (Fűrész, 2008) for consistency within our catalog. Extinction was taken into account by including uniform priors with bounds from 0 to an upper limit taken from Schlegel et al. (1998) and Schlafly & Finkbeiner (2011), and Gaussian priors for parallax were used from Gaia EDR3 (Gaia Collaboration et al., 2016, 2021), accounting for the small systematic offset from Lindegren et al. (2021). Each host star was modeled via MIST evolution models (Paxton et al., 2011, 2013, 2015; Choi et al., 2016; Dotter, 2016), except in the cases of low-mass stars (∼ 0.6𝑀⊙) where these models are not reliable (Mann et al., 2015). For these stars, we excluded the SED from the fit, set 5% Gaussian priors on 𝑀∗ and 𝑅∗ from their relationship with 𝐾𝑆 (Mann et al., 2015, 2019), and placed a Gaussian prior of 0.2 on the limb darkening coefficients (Patel & Espinoza, 2022). We considered a fit to be converged once it reaches the thresholds of 𝑇𝑍 > 1000 and a Gelmin-Rubin value < 1.02, as well as passing visual inspections of the resulting models, PDFs, and MCMC chains. One major difference in this sample compared to the previous batches is the inclusion of multiplanet and binary systems, the latter being a recent addition to the capabilities of EXOFASTv2. To account for multiple planets in a light curve, we masked all transits for all planets before flattening the light curve as described previously. For binary systems that are blended, the combined SED 2https://github.com/avanderburg/keplerspline 82 is modeled by including the broadband photometry for both stars in the global fit, allowing for parameters for each star to be modeled. 83 Table 4.1 Target list and data used in this analysis. TIC ID EPIC ID KID Planet TSM 301235044 380619414 68577662 56399553 350020859 203214081 7059054 38087018 292202337 366622912 16288184 363445121 39926974 434226736 281731203 437704321 456945304 26017005 301235044 443616612 149496868 332022997 365007485 307733361 301258470 K2-141∗ WASP-85 A∗ K2-232 K2-29∗ 246393474 201862715 247098361 211089792 248777106 K2-234/HD 89345 K2-313/G 9-40† 212048748 K2-121 211818569 K2-45 201345483 K2-406 249223471 K2-371 211399359 K2-237 229426032 K2-36∗ 201713348 K2-28† 206318379 K2-25† 210490365 K2-261 201498078 K2-55 205924614 K2-370 210797580 K2-30 210957318 K2-141∗ 246393474 K2-43† 201205469 K2-405 248874928 K2-275 212012119 K2-403 248758353 K2-100∗ 211990866 K2-329 246193072 c b b b b b b b b b b c b b b b b b b b b b b b b 1128.94 153.24 152.32 121.62 114.08 103.08 99.25 99.01 93.92 87.03 86.41 86.1 81.08 68.56 67.77 66.38 63.65 62.65 60.47 59.9 56.94 54.88 53.91 52.58 51.17 K2 Campaign 12, 19 1 13 4 14 16 5, 18 1 15 5, 18 11 1 3 4 14 1 13 4 12, 19 1 14 5, 18 14 5, 18 12, 19 TESS Sector 42, 70 45, 46, 72 43, 44, 71 43, 44, 70, 71 45, 46 44, 45, 46, 72 44, 46, 45, 71, 72 45, 46, 72 11, 38 7, 44, 45, 46, 72 12, 39 45, 46 42 44, 70, 71 9, 35, 45, 46, 72 28, 42 43, 44, 70, 71 42, 43, 44, 70, 71 42, 70 9, 36, 45, 46, 72 45, 46, 72 44, 45, 46, 72 45, 46, 72 44, 45, 46, 72 42 RV Ref. a, b a, c d, e f, g h, i j, k, l m K2 Discovery Ref. 1 2 3 4 5 6 7 8 9 9 10 11 12 13 14 8 9 15 1 8 9 16 9 17 18 † The host stars in these systems were classed as low mass (≲ 0.6 𝑀⊙), so we did not include the SEDs in the global fits. See Section 4.3 and 2.4 for details. ∗The global fits for these systems have not yet been run. References for RV measurements: (a) Yu et al. (2018), (b) Brahm et al. (2018), (c) Van Eylen et al. (2018), (d) Luque et al. (2022), (e) Stefansson et al. (2020), (f) Soto et al. (2018), (g) Smith et al. (2019), (h) Johnson et al. (2018c), (i) Knudstrup et al. (2024), (j) Johnson et al. (2016), (k) Lillo-Box et al. (2016), (l) Brahm et al. (2016), (m) Sha et al. (2021) K2 references: (1) Malavolta et al. (2018), (2) Močnik et al. (2016), (3) Brahm et al. (2018), (4) Santerne et al. (2016), (5) Van Eylen et al. (2018), (6) Stefansson et al. (2020), (7) Dressing et al. (2017), (8) Crossfield et al. (2016), (9) Christiansen et al. (2022), (10) Soto et al. (2018), (11) Sinukoff et al. (2016), (12) Hirano et al. (2016a), (13) Mann et al. (2016), (14) Johnson et al. (2018a), (15) Johnson et al. (2016), (16) Livingston et al. (2018a), (17) Mann et al. (2017), (18) Sha et al. (2021) 84 RV K2 Discovery Ref. a Table 4.2 Target list and data used in this analysis, continued. TIC ID EPIC ID KID Planet TSM 243244680 69747919 281748980 422618449 435339558 277833995 404421005 293612446 5882269 98669309 203289099 458686847 380255458 330687113 115010361 388804061 26078330 443616612 175261852 117275513 55315929 380884458 175261852 178217113 366683184 K2-236 K2-167 K2-320† HIP 116454/K2-2 K2-79∗ K2-321† K2-277 K2-260 K2-284 K2-253 K2-344† K2-117†∗ K2-140 K2-334 K2-353∗ K2-18†∗ K2-178∗ K2-43† K2-146†∗ K2-348∗ 211945201 205904628 201796690 60021410 210402237 248480671 212357477 246911830 247267267 228809550 212081533 211331236 228735255 211730024 251554286 201912552 210965800 201205469 211924657 212204403 251279430 K2-312/HD 80653 220522262 211924657 212110888 201635569 K2-281 K2-146†∗ K2-34∗ K2-14† b b b b b b b b b b b b b b b b b c b b b b c b b 51.05 50.43 49.48 48.48 46.69 43.03 41.29 40.92 39.62 38.72 38.37 37.75 37.56 37.54 35.97 35.43 34.27 34.04 33.76 33.49 33.24 32.53 31.73 31.04 30.84 K2 Campaign 5, 16 3 14 E 4 14 6 13 13 10 16 5, 18 10 16 17 1 4 1 5, 16 16 16 8 5, 16 5, 16, 18 1 TESS Sector 44, 45, 46, 72 2, 28, 42 45, 46, 72 42, 70 42, 43, 44, 70, 71 8, 35, 45, 46, 72 10, 37 5, 32, 43, 44, 71 43, 44, 45, 71 46 44, 45, 46, 72 34, 44, 45, 46, 72 46 44, 45, 46, 72 23, 46, 50 45, 46, 72 42, 43, 44, 70, 71 9, 36, 45, 46, 72 44, 46, 72 21, 44, 45, 46, 72 45, 46, 72 42, 70 44, 46, 72 44, 45, 46, 71, 72 46, 72 b, c d, e Ref. 1 2 3 4 5 3 6 7 8 9 10 11 12 10 10 13 2 14 15 10 16 6 17 18 13 † The host stars in these systems were classed as low mass (≲ 0.6 𝑀⊙), so we did not include the SEDs in the global fits. See Section 4.3 and 2.4 for details. ∗The global fits for these systems have not yet been run. References for RV measurements: (a) Chakraborty et al. (2018), (b) Vanderburg et al. (2015), (c) Bonomo et al. (2023), (d) Giles et al. (2018), (e) Korth et al. (2019) K2 references: (1) Chakraborty et al. (2018), (2) Mayo et al. (2018), (3) Castro González et al. (2020), (4) Vanderburg et al. (2015), (5) Crossfield et al. (2016), (6) Livingston et al. (2018a), (7) Johnson et al. (2018c), (8) David et al. (2018), (9) Livingston et al. (2018b), (10) de Leon et al. (2021), (11) Dressing et al. (2017), (12) Giles et al. (2018), (13) Montet et al. (2015), (14) Crossfield et al. (2016), (15) Hirano et al. (2018), (16) Frustagli et al. (2020), (17) Hamann et al. (2019), (18) Hirano et al. (2016b) 85 4.4 Results and Discussion The results presented in this section are preliminary, and will be refined for the next installment of the K2 & TESS Synergy series. 34 of the 50 planets have fits that are sufficiently converged (GR≲ 1.2, T𝑧 ≳ 300), the resulting parameters from which are included as supplementary material. We caution that not all of these are fully converged fits, and the final fits of all 50 planets will be included in the published version. Although K2-2 b/HIP 116454 b is included in the top 50 TSM planets, we do not display results for the system here, as Chapter 3 presents a much more detailed analysis of this system. The phase-folded light curves showing the transits in K2 and TESS are presented in Figures 4.1 - 4.6, and the RVs of systems that have them are shown in Figures 4.7 - 4.9. Using our preliminary results, we were able to improve the average 3𝜎2030 uncertainties on the ephemerides from 17.4 hours to 16 minutes (Figures 4.10-4.15). We compare the updated ephemerides to those from the discovery data sets from NEA to highlight the severity of ephemeris degradation when left unchecked. The significantly improved ephemerides mean these planets are accessible for JWST observations throughout the expected mission lifetime. The sample in this analysis consists of 34 planets with host stars ranging from M dwarfs through FGK spectral types, and planets ranging from Earth-like rocky planets all the way to gas giants (Figure 4.16). Figure 4.17 shows the TSM for JWST transmission spectroscopy targets and Synergy planets as a function of orbital period. The TSM values for Synergy systems are relatively high in this distribution, showing that these planets are as well-suited to characterization efforts as current JWST targets. 4.4.1 Low-mass (< 4𝑅⊕) planets Several super-Earths and sub-Neptunes in the sample would be interesting targets for follow-up. K2-313 b/G 9-40 b (5.9𝑀⊕, 2.1𝑅⊕) and K2-344 b (4.5𝑀⊕, 1.8𝑅⊕) are both sub-Neptunes on a 5.75 and 3.4 day period, respectively. They both orbit M dwarfs, and sit in the radius valley - an empirical sparsity of planets with radii ∼ 1.5 − 2𝑅⊕ (Fulton et al., 2017), thought to be a result of mass loss from photoevaporation. Both of thesis targets have favorable TSM values (see Tables 4.1 and 4.2). 86 Figure 4.1 Phase-folded light curves showing the transits in K2 (gray) and TESS (purple), with the period of each planet at the bottom right. The TESS light curves are shown both unbinned and binned to 12 minutes, while the K2 light curves are unbinned. All light curves are normalized and have an arbitrary offset for clarity. The best-fit transit model from EXOFASTv2 is shown as the sold line. The discovery references are listed in Tables 4.1 and 4.2. 87 Figure 4.2 Phase-folded light curves (continued). 88 Figure 4.3 Phase-folded light curves (continued). 89 Figure 4.4 Phase-folded light curves (continued). 90 Figure 4.5 Phase-folded light curves (continued). 91 Figure 4.6 Phase-folded light curves (continued). K2-43 is an M dwarf that is host to two sub-Neptunes (with another planet candidate to be confirmed) that straddle the higher-mass distribution of the radius valley. K2-43 c is on a 2.2 day period and is the inner planet of the system as it was validated more recently than K2-43 b, which has a 3.5 day period. K2-43 c (5.0𝑀⊕, 1.9𝑅⊕) is in the radius valley, while K2-43 b (14.6𝑀⊕, 3.6𝑅⊕) sits on the high-radius tail. Studying the atmospheres of these planets would allow for a unique comparison between sub-Neptune compositions that have likely undergone similar evolutionary histories. K2-2 b was part of a previous K2 & TESS Synergy paper (Chapter 3; Thygesen et al. 2024), and originally had an incorrect ephemeris due to a spurious transit in a second light curve used during validation. It sits at the peak of the sub-Neptune distribution of the radius valley (Fulton et al., 2017), and has the fifth highest TSM of all K2 sub-Neptunes (𝑅𝑃 = 2.0 − 3.0𝑅⊕). 92 Figure 4.7 Radial velocities for systems with archival spectroscopic measurements. The best fit EXOFASTv2 model is shown as the solid line. Each set of RVs are phased using the best fit period and 𝑇𝑐, and the residuals for each dataset are shown in each subplot. The references for each set of RVs are listed in Tables 4.1 and 4.2. 93 Figure 4.8 Radial velocities (continued). 94 Figure 4.9 Radial velocities (continued). 95 Figure 4.10 Projected uncertainties for future transit times in the current sample. The shaded regions represent up to the 3𝜎 level of uncertainty, which are shown for the discovery ephemeris (gray) and our updated ephemeris (purple). The discovery references are listed in Tables 4.1 and 4.2. 96 Figure 4.11 Projected uncertainties for future transit times in the current sample (continued). The shaded regions represent up to the 3𝜎 level of uncertainty, which are shown for the discovery ephemeris (gray) and our updated ephemeris (purple). 97 Figure 4.12 Projected uncertainties for future transit times in the current sample (continued). The shaded regions represent up to the 3𝜎 level of uncertainty, which are shown for the discovery ephemeris (gray) and our updated ephemeris (purple). 98 Figure 4.13 Projected uncertainties for future transit times in the current sample (continued). The shaded regions represent up to the 3𝜎 level of uncertainty, which are shown for the discovery ephemeris (gray) and our updated ephemeris (purple). 99 Figure 4.14 Projected uncertainties for future transit times in the current sample (continued). The shaded regions represent up to the 3𝜎 level of uncertainty, which are shown for the discovery ephemeris (gray) and our updated ephemeris (purple). 100 Figure 4.15 Projected uncertainties for future transit times in the current sample (continued). The shaded regions represent up to the 3𝜎 level of uncertainty, which are shown for the discovery ephemeris (gray) and our updated ephemeris (purple). K2-100 b (16.2𝑀⊕, 3.9𝑅⊕) is a hot Neptune with an orbital period of 1.7 days days around a G dwarf host star. Due to its close proximity to the star, K2-100 b is experiencing mass loss via to photoevaporation (Barragán et al., 2019b). Its high TSM makes in an ideal candidate for studying the impacts on atmospheric composition for planets undergoing rapid mass loss. 4.5 Example of Detailed Characterization Using the K2 & TESS Synergy One aspect of exoplanet evolution that can be studied with the aid of the K2 & TESS Synergy is giant planet migration. While not a focus of this thesis, this is an example of a pertinent question that can be investigated using our sample. Below, we discuss giant planet formation and evolution, and present a key scientific question and how updated results from the Synergy survey allow us to pursue future characterization. 101 Figure 4.16 Radius versus mass for all confirmed exoplanets (gray; values taken from the NEA, accessed 12 January, 2025) and those in the current Synergy sample with completed fits. Systems with measured masses are represented by diamonds, while those with masses from the Chen & Kipping (2017) mass-radius relations are shown as crosses. The markers are colored by the host star temperature. 4.5.1 Hot Jupiter Formation and Evolution Giant planets make up a third of all currently confirmed exoplanets, but there are still many open questions surrounding their origins and evolutionary mechanisms. Of the ∼1500 known giant planets, ∼550 are hot Jupiters (HJs), giant planets that orbit their host stars with periods of ≤ 10 days. These are widely thought to have formed outside of the ice line, then migrated inward to their current short-period orbits. Two predominant theories for the facilitation of migration are dynamical interactions and gas-disk migration. Gas giants that have experienced dynamical or secular interactions with other stars or planets will have their eccentricities increased, such that their periastron distances become small enough to be affected by tides, and thus get placed on short periods as their orbits circularize over time (this is known as “High Eccentricity Migration [HEM], Rasio & Ford, 1996; Fabrycky & Tremaine, 2007; Wu & Lithwick, 2011). HJs that have undergone these violent interactions tend to be the only known planets on short periods in their 102 Figure 4.17 TSM versus orbital period for all transmission spectroscopy targets with JWST as of the end of Cycle 3 (gray) and the current batch of planets in the Synergy sample (red). Values used to calculated the TSM for JWST targets were taken from the NEA, accessed April 9, 2025. systems (Huang et al., 2016), having disrupted planet formation and existing inner planets as they moved inward. In contrast, planetary migration through the gas disk is generally predicted to be smooth, maintaining fairly circular orbits throughout the process, and is expected to result in low eccentricity close-in orbits (D’Angelo et al., 2003). Using a combination of photometric and spectroscopic results, we can look for evidence that a planet has undergone HEM (e.g. high eccentricity, short orbital period, orbital misalignment), but since many mechanisms can influence these properties, it is not typically conclusive for any individual system. For example, high primordial eccentricities - evidence of dynamical migration - can be erased as tidal interaction circularizes the short-period (≲ 5-day) orbit (Adams & Laughlin, 2006). At longer periods (≳5 days), circularization timescales would be a few billion years, up to even longer than the age of the Universe. Therefore, the presence of highly-eccentric (e>0.3) hot Jupiters with periods longer than about 5 days would indicate that HEM is important for migration. Interestingly, recent work looking at the hot Jupiter population from NASA’s TESS mission is 103 Figure 4.18 The predicted C/O ratios as a function of distance from the host star for a migrating giant planet (Figure 8a from Madhusudhan 2019). The C/O ratio is shown for the abundance in the gas phase (solid line) and solid phase (dashed line), as well as enhancement due to pebble drift (dotted line). The shaded regions represent snowlines. Planets that form within or migrate through different regions of the disk should possess distinct C/O ratios. consistent with HEM as the dominant migration method (Rodriguez et al., 2021, 2022b; Yee et al., 2022a,b; Rice et al., 2022a,b). However, understanding the evolutionary history of any individual system has been a challenge. Although the population of hot Jupiters shows characteristics of being dominated by HEM, comparative studies of specific systems may distinguish individual evolutionary histories that can be informative for modeling planetary migration. Specifically, the atmospheric composition of a planet can provide constraints on the planet’s formation location, which, combined with its orbital architecture, can lead to a detailed understanding of a planet’s evolutionary pathway. A popular diagnostic that is feasible with current instrumentation is to measure the carbon-to-oxygen ratio (C/O) of the planetary atmosphere. Carbon and oxygen-bearing molecules (particularly CO, CO2, H2O and CH4) have readily observable line features and will freeze out at different distances from their host star (Figure 4.18). Therefore, the abundance of these molecules in the gas state within the atmosphere of the planet can place constraints on where the planet formed in relation to the respective snowlines, potentially suggesting some systems may have formed in- 104 situ. Although absolute measurements of carbon and oxygen do not entirely uncover the planet’s formation history, as they can depend on evolution models that are not yet well defined, we can now start to systematically compare these measurements for planets with different migration histories, as many HJs have been or are scheduled to be observed by JWST. Comparison of the atmospheric chemical composition of highly eccentric HJs to circular ones or those with nearby small companions can detect differences indicative of changes in their evolutionary history. Over the past decade, there have been many studies into the possible interpretations of different C/O ratios. The strength of the C/O ratio and the abundances of C and O compared to Solar (C/O=0.54; Asplund et al. 2009), along with metallicity (C/H and C/O, typically associated with the abundance of CO2 and H2O; Lodders & Fegley 2002; Zahnle et al. 2009; Moses et al. 2013; Madhusudhan & Seager 2011), are thought to be linked to the initial formation mechanism (core accretion or gravitational instability) paired with when and whether migration occurred (e.g. whether the protoplanetary disk still existed at the time of migration; Madhusudhan et al. 2014). Accretion of pebbles and planetesimals can also affect the atmospheric composition, as the molecules originally locked in the solid phase sublimate, while the planet migrates through successive snowlines. A planet that formed through core accretion and then migrated through the disk, subsequently accreting planetesimals that enriched the atmosphere, is expected to have a subsolar C/O ratio and supersolar metallicity (Madhusudhan et al., 2014; Öberg et al., 2011). Formation beyond the CO and CO2 snowlines followed by migration after the disk has dissipated is likely to result in C/O∼1 and subsolar metallicity (Madhusudhan et al., 2017). If gas accretion occurred close to the CO or CO2 snowlines, or if a significant amount of carbon-bearing grains was accumulated, both the resulting C/O ratio and metallicity are expected to be supersolar (Öberg et al., 2011). However, planets that underwent significant pebble accretion during disk migration, e.g. those that may have formed via gravitational instability far from the host star (≳20 AU), can display a range of both metallicity and C/O (Madhusudhan et al., 2014; Mordasini et al., 2016; Helled & Bodenheimer, 2010). Finally, hot Jupiters that formed within the H2O snowline and carbon-grain evaporation line 105 would likely have compositions reflecting that of the host star Öberg et al. (2011). 4.5.2 K2 & TESS Synergy Jupiters in the context of migration The K2 & TESS Synergy sample contains several ideal targets for future JWST observations with the goal of studying giant planet migration. There are 18 giant planets (> 4𝑅⊕) in the present K2 & TESS Synergy sample (excluding planets without complete fits). Five of these (HD 89345 b, K2-25 b, K2-232 b, K2-261 b, K2-329 b) have significant non-zero eccentricities considering the Lucy-Sweeney bias (Lucy & Sweeney, 1971), and have <12-day periods. Comparing the atmospheric compositions of the eccentric planets, which are likely undergoing HEM, to ones on circular orbits could potentially identify characteristics indicative of the differing migration histories. Figure 4.19 shows the eccentricity-period distribution of all planets with radii > 4𝑅⊕ that will be observed for transmission spectra by JWST the end of Cycle 3, compared to those in the K2 & TESS Synergy. Systems like TOI-1130 are ideal for the comparison between eccentric and circular HJs, as it is host to a sub-Neptune (0.06 𝑀𝐽, 0.3 𝑅𝐽) on a 4.1 day period, with an outer Jupiter (1.0 𝑀𝐽, 1.2 𝑅𝐽) on an 8.4 day period (Huang et al., 2020; Korth et al., 2023). The existence of two differently sized planets both on short-period, low-eccentricity (<0.2) orbits has the implication of a dynamically quiet migration history, as any major interactions should have disrupted this configuration. Beyond migration histories, the effects of parameters like stellar metallicity, insolation, and atmospheric inflation may influence the chemical abundances of HJs. JWST HJs in conjunction with the Synergy giant planets span a range of these parameter spaces, which adds to the value of future characterization of these planets. With updated ephemerides, it will be possible to observe the Synergy giant planets for their transmission spectra and compare them directly to current JWST targets. 4.6 Conclusion In this chapter, we presented the fourth installment of the K2 & TESS Synergy, reanalyzing 34 K2-discovered planets observed by TESS. These systems were ranked among the highest in terms of their TSM values, making it a timely effort to ensure the most up-to-date ephemerides for 106 Figure 4.19 Eccentricity as a function of orbital period for giant planets that will have JWST transmission spectra by the end of Cycle 3 (gray) and targets in the K2 & TESS Synergy (red). Circles with error bars have eccentricities measured through RVs, while those with arrows are upper limits. Values for JWST targets are adopted from the NEA, accessed April 9, 2025. characterization observations. Through running global fits for each system including photometry, spectroscopy, and stellar data, we improved the 3𝜎2030 uncertainties on the ephemerides for these planets from 17.4 hours to 16 minutes. We highlighted some of the most unique cases in this sample of planets that would be significant additions to planets with JWST transmission spectra. We also discussed a potential avenue for investigating the effects of migration on giant planets via atmospheric compositions, utilizing the updated parameters from the K2 & TESS Synergy and comparing these systems to those already observed using JWST. With so much yet to uncover about the formation and evolution of exoplanets, it is vital that we have up-to-date ephemerides for the planets that are most amenable to atmospheric characterization so that minimal resources are wasted during targeted observations. 107 CHAPTER 5 SUMMARY AND FUTURE WORK 5.1 Summary of work As we enter an era of characterizing exoplanets in unprecedented detail, we need to prepare ahead of time for performing the required observations. The work in this thesis aims to provide a comprehensive, self-consistent catalog for K2-discovered planets that have been reobserved by TESS, prioritizing those that are more amenable to characterization efforts. This catalog provides renewed ephemerides for accurately predicting future transit times, and updated global parameters which will aid in the selection process for atmospheric characterization and allow for appropriate population studies within the K2 target list. In the following sections, we summarize each chapter and explore possibilities for the K2 & TESS Synergy project going forward. 5.1.1 Chapter 2: Reanalysis of 26 TESS Prime Mission Targets In Chapter 2, we reanalyzed 26 single-planet systems that were observed during the TESS Prime Mission. Half of these had a transit S/N large enough to be detectable by TESS, which resulted in an average improvement on their ephemerides from 26.7 to 0.35 hours. The transits of the remaining 13 planets were not deep enough to be seen in the TESS light curves, but we were still able to reduce the average ephemeris uncertainties from 43.2 to 35.6 hours due to the light curve pipelines that we use and the nature of the global fits. We identified the inconsistency of the ephemeris for K2-260 b, where we believe an error was introduced for the literature transit time. This work was the first large batch to be published following the pilot study. 5.1.2 Chapter 3: Recovering K2’s First Planet Chapter 3 detailed our effort to recover the lost ephemeris of K2-2 b, which was the first planet discovery during the engineering phase of K2. The discovery of this planet included a spurious transit from a secondary light curve, which caused the calculated period to be 28.8 minutes (∼ 40𝜎) from the true period. This resulted in the transit being missed during targeted observations using HST and Spitzer, highlighting a real-world scenario where an inaccurate ephemeris led to the loss of data. 108 We used a series of ground- and space-based light curves from MEarth, ULMT, K2, TESS, and Spitzer to recover and refine the ephemeris of K2-2 b to within 13 minutes by 2030. Our analysis included new radial velocity measurements which were corrected for systematics with the YARARA post-processing tool, as well as using the new CALM method to remove any remaining stellar variability. The radial velocities of K2-2 showed a tentative long-term trend, suggesting there may be an outer companion in the system, and with astrometric analysis, we were able to constrain the potential planet to ≲ 10 𝑀J. Although challenging with current technology, K2-2 b would be a valuable addition to atmospheric characterization, as it has the fifth-highest TSM compared to all other K2 sub-Neptunes. K2-2 also has a comoving white dwarf companion, for which a spectrum would place significant constraints on the age of the entire system. 5.1.3 Chapter 4: K2’s Top Atmospheric Candidates Chapter 4 discussed the continuation of the K2 & TESS Synergy to the top 50 K2 targets for atmospheric characterization, seven of which were part of previous K2 & TESS Synergy installments. We followed the same methods as in the previous chapters for analyzing the sample, with the addition of multiplanet and binary star systems. We have currently run fits for 34 of the planets, and improved the average 3𝜎2030 ephemeris uncertainties from 17.4 hours to 16 minutes. The planets in this sample have TSMs similar to current JWST targets, showing that the K2 & TESS Synergy catalog can be used in practicality. The sample presented in this chapter consisted of multiple low- and high- mass planets that would make interesting targets for transmission spectroscopy. The low-mass sample (< 4𝑅⊕) included sub-Neptunes that sit in the radius valley (K2-313 b and K2-344 b), a hot Neptune that is potentially losing its atmospheres due to photoevaporation (K2-100 b), and a system with two sub-Neptunes on either side of the large-radius peak of the radius valley (K2-43). Each of these systems are pertinent in their own ways to the study of small planets with significant atmospheres, and would make excellent targets for transmission spectroscopy. 109 The planets in the high mass regime (> 4𝑅⊕) make an ideal selection of systems that could be used to study giant planet migration. We presented this as an example of how the K2 & TESS Synergy catalog can be used going forward to make significant contributions to the understanding of exoplanet evolution. The sample in this work consists of a variety of giant planets. The five planets with significantly eccentric orbits (HD 89345 b, K2-25 b, K2-232 b, K2-261 b, K2-329 b) are particularly interesting in terms of giant planet evolution, as they are likely actively undergoing high-eccentricity migration. Comparing the compositions of these planets to the giants on circular orbits could potentially uncover signatures of migration that are locked in the atmospheres of these planets. 5.2 Future Work We have been able to update the global parameters for 54 exoplanets, but the work is far from over. The immediate goal is to extend the K2 & TESS Synergy to all ∼500 remaining K2 planets. The success of this will depend on their transit signal to noise, and as we have so far prioritized those with larger transit depths, this will become increasingly challenging with future batches. Ever-increasing TESS coverage will help in this regard, and even future missions with higher photometric precision could be used to once again bring ephemerides into the current epoch. The footprints of several K2 campaigns have not yet been covered by TESS at the time of writing. These will be a part of Sectors 91 and 92 (planned dates April 9 - June 3 2025; Figure 5.1), completing the TESS ecliptic plane sectors, and will thus allow the completion of the main target selection for the K2 & TESS Synergy with 68 new planetary overlaps. Beyond refitting the remaining K2 systems, there are many paths this project could take going forward. We purposely did not fit for TTVs in the current work in order to reduce computation time for the global fits, however, EXOFASTv2 has the capability of fitting for TTVs, making this a natural extension of the project. TTVs can uncover planets undergoing orbital decay, as well as indicate the existence of other planets in the system. Transit depth and duration variations can also tell us about the structure of the system, although this might be challenging with the precision of TESS as the signal may be buried within the photometric noise. Another aspect for future study is 110 Figure 5.1 Overlap between K2 campaigns and TESS sectors. Each point represents a K2 target, which are colored by the number of times they have been reobserved by TESS up to and including Sector 92. Open circles are systems that will be observed for the first time in the upcoming ecliptic Sectors 91 and 92. the fitting of secondary eclipses (i.e. when a planet passes behind the host star), which can allow for the study of the temperature and brightness of the planet. At the time of writing, there are 975 candidate planets in the K2 catalog. A systematic search for recurring transits within the TESS light curves could allow for assessment of the candidates, while providing immediately usable ephemerides for follow-up on interesting systems. A delve into single-transit K2 systems may also result in new exoplanet confirmations. Another path forward would be to make use of the K2 & TESS Synergy catalog in terms of JWST spectroscopy. As presented in Chapter 4, the catalog contains a wide variety of planets that could be pursued for transmission spectroscopy to help answer certain aspects of planet formation and evolution. With the 54 planets from this work having up-to-date ephemerides, detailed characterization can be the next step for systems that are key to answering lingering 111 questions about exoplanet evolution. Ephemeris renewal will be an ongoing necessity in the age of atmospheric characterization. The K2 & TESS Synergy has used the power of (nearly) all-sky TESS observations, but this is unfortunately not the final answer to ephemeris deterioration. Dragomir et al. (2020) performed a study on real and simulated TESS data to see how long after new TESS discoveries are made it takes for ephemerides to become stale. They found that merely a year after being first observed, 81% of TESS planets will have transit time uncertainties >30 minutes at a 1𝜎 level, rendering them unfeasible for characterization. While this is alleviated somewhat by TESS revisiting the fields of previous sectors, it shows just how quickly planets can be lost, and missions that do not have the advantage of returning to the same fields years later will suffer more for this. Thousands more planets are expected to be discovered in the coming years with missions like the Nancy Grace Roman Space Telescope, Rubin Observatory, Plato, and ground-based ELTs, and it is imperative that we remember these planets will need to be reobserved to preserve their ephemerides. Ephemeris refinement may ultimately be a Sisyphean task, but it is a critical step in understanding the intricate details of exoplanets. 112 BIBLIOGRAPHY Abazajian, K. N., Adelman-McCarthy, J. K., Agüeros, M. A., et al. 2009, ApJS, 182, 543, doi: 10. 1088/0067-0049/182/2/543 Adams, E. R., Jackson, B., Johnson, S., et al. 2021, Planetary Science Journal, 2, 152, doi: 10. 3847/PSJ/ac0ea0 Adams, F. C., & Laughlin, G. 2006, ApJ, 649, 1004, doi: 10.1086/506145 Ahrer, E.-M., Stevenson, K. B., Mansfield, M., et al. 2022, arXiv e-prints, arXiv:2211.10489, doi: 10.48550/arXiv.2211.10489 Aigrain, S., Parviainen, H., & Pope, B. J. S. 2016, MNRAS, 459, 2408, doi: 10.1093/mnras/stw706 Akana Murphy, J. M., Kosiarek, M. R., Batalha, N. M., et al. 2021, AJ, 162, 294, doi: 10.3847/ 1538-3881/ac2830 Alam, M. K., Gao, P., Adams Redai, J., et al. 2025, AJ, 169, 15, doi: 10.3847/1538-3881/ad8eb5 Albrecht, S., Snellen, I., de Mooij, E., & Le Poole, R. 2009, in IAU Symposium, Vol. 253, Transiting Planets, ed. F. Pont, D. Sasselov, & M. J. Holman, 520–523, doi: 10.1017/S1743921308027105 Alderson, L., Wakeford, H. R., Alam, M. K., et al. 2022, arXiv e-prints, arXiv:2211.10488, doi: 10.48550/arXiv.2211.10488 Alderson, L., Moran, S. E., Wallack, N. L., et al. 2025, AJ, 169, 142, doi: 10.3847/1538-3881/ adad64 Ambikasaran, S., Foreman-Mackey, D., Greengard, L., Hogg, D. W., & O’Neil, M. 2015, IEEE Transactions on Pattern Analysis and Machine Intelligence, 38, 252, doi: 10.1109/TPAMI.2015. 2448083 Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481, doi: 10.1146/annurev. astro.46.060407.145222 Auvergne, M., Bodin, P., Boisnard, L., et al. 2009, A&A, 506, 411, doi: 10.1051/0004-6361/ 200810860 Baglin, A., Vauclair, G. X., Rard, & The COROT Team. 2000, Journal of Astrophysics and Astronomy, 21, 319, doi: 10.1007/BF02702417 Bakos, G., Noyes, R. W., Kovács, G., et al. 2004, PASP, 116, 266, doi: 10.1086/382735 Bakos, G. Á., Csubry, Z., Penev, K., et al. 2013, PASP, 125, 154, doi: 10.1086/669529 113 Baranec, C., Riddle, R., Law, N. M., et al. 2014, ApJL, 790, L8, doi: 10.1088/2041-8205/790/1/L8 Barclay, T., Quintana, E. V., Colón, K., et al. 2025, arXiv e-prints, arXiv:2502.09730, doi: 10. 48550/arXiv.2502.09730 Barnes, R. 2017, Celestial Mechanics and Dynamical Astronomy, 129, 509, doi: 10.1007/ s10569-017-9783-7 Barragán, O., Gandolfi, D., & Antoniciello, G. 2019a, MNRAS, 482, 1017, doi: 10.1093/mnras/ sty2472 Barragán, O., Grziwa, S., Gandolfi, D., et al. 2016, AJ, 152, 193, doi: 10.3847/0004-6256/152/6/193 Barragán, O., Aigrain, S., Kubyshkina, D., et al. 2019b, MNRAS, 490, 698, doi: 10.1093/mnras/ stz2569 Barros, S. C. C., Demangeon, O., & Deleuil, M. 2016, A&A, 594, A100, doi: 10.1051/0004-6361/ 201628902 Bastien, F. A., Stassun, K. G., Basri, G., & Pepper, J. 2013, Nature, 500, 427, doi: 10.1038/ nature12419 Beichman, C., Ygouf, M., Llop Sayson, J., et al. 2020, PASP, 132, 015002, doi: 10.1088/1538-3873/ ab5066 Benn, C., Dee, K., & Agócs, T. 2008, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7014, Ground-based and Airborne Instrumentation for Astronomy II, ed. I. S. McLean & M. M. Casali, 70146X, doi: 10.1117/12.788694 Benneke, B., Werner, M., Petigura, E., et al. 2017, ApJ, 834, 187, doi: 10.3847/1538-4357/834/2/ 187 Benneke, B., Wong, I., Piaulet, C., et al. 2019, arXiv e-prints, arXiv:1909.04642. https://arxiv.org/ abs/1909.04642 Berger, T. A., Huber, D., Gaidos, E., van Saders, J. L., & Weiss, L. M. 2020a, AJ, 160, 108, doi: 10.3847/1538-3881/aba18a Berger, T. A., Huber, D., van Saders, J. L., et al. 2020b, AJ, 159, 280, doi: 10.3847/1538-3881/ 159/6/280 Berta, Z. K., Charbonneau, D., Bean, J., et al. 2011, ApJ, 736, 12, doi: 10.1088/0004-637X/736/ 1/12 Blain, D., Charnay, B., & Bézard, B. 2021, A&A, 646, A15, doi: 10.1051/0004-6361/202039072 114 Bonomo, A. S., Dumusque, X., Massa, A., et al. 2023, A&A, 677, A33, doi: 10.1051/0004-6361/ 202346211 Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977, doi: 10.1126/science.1185402 Borucki, W. J., Agol, E., Fressin, F., et al. 2013, Science, 340, 587, doi: 10.1126/science.1234702 Brahm, R., Jones, M., Espinoza, N., et al. 2016, PASP, 128, 124402, doi: 10.1088/1538-3873/128/ 970/124402 Brahm, R., Espinoza, N., Jordán, A., et al. 2018, MNRAS, 477, 2572, doi: 10.1093/mnras/sty795 Brandt, T. D. 2018, ApJS, 239, 31, doi: 10.3847/1538-4365/aaec06 —. 2021, ApJS, 254, 42, doi: 10.3847/1538-4365/abf93c Brown, T. M., Charbonneau, D., Gilliland, R. L., Noyes, R. W., & Burrows, A. 2001, ApJ, 552, 699, doi: 10.1086/320580 Buchhave, L. A., Latham, D. W., Johansen, A., et al. 2012, Nature, 486, 375, doi: 10.1038/ nature11121 Burgett, W., Bernstein, R., Ashby, D., et al. 2024, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 13094, Ground-based and Airborne Telescopes X, ed. H. K. Marshall, J. Spyromilio, & T. Usuda, 1309417, doi: 10.1117/12.3020733 Caldwell, D. A., Tenenbaum, P., Twicken, J. D., et al. 2020, Research Notes of the American Astronomical Society, 4, 201, doi: 10.3847/2515-5172/abc9b3 Carrión-González, Ó., García Muñoz, A., Santos, N. C., et al. 2021, A&A, 651, A7, doi: 10.1051/ 0004-6361/202039993 Casasayas-Barris, N., Pallé, E., Yan, F., et al. 2020, A&A, 635, A206, doi: 10.1051/0004-6361/ 201937221 Casasayas-Barris, N., Palle, E., Stangret, M., et al. 2021, A&A, 647, A26, doi: 10.1051/0004-6361/ 202039539 Castro González, A., Díez Alonso, E., Menéndez Blanco, J., et al. 2020, MNRAS, 499, 5416, doi: 10.1093/mnras/staa2353 Chakraborty, A., Roy, A., Sharma, R., et al. 2018, AJ, 156, 3, doi: 10.3847/1538-3881/aac436 Charbonneau, D., Brown, T. M., Latham, D. W., & Mayor, M. 2000, ApJL, 529, L45, doi: 10.1086/ 312457 115 Charbonneau, D., Brown, T. M., Noyes, R. W., & Gilliland, R. L. 2002, ApJ, 568, 377, doi: 10. 1086/338770 Charbonneau, D., & Deming, D. 2007, arXiv e-prints, arXiv:0706.1047, doi: 10.48550/arXiv.0706. 1047 Charnay, B., Blain, D., Bézard, B., et al. 2021, A&A, 646, A171, doi: 10.1051/0004-6361/ 202039525 Chen, J., & Kipping, D. 2017, ApJ, 834, 17, doi: 10.3847/1538-4357/834/1/17 Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102, doi: 10.3847/0004-637X/823/2/102 Christiansen, J. L., Bhure, S., Zink, J. K., et al. 2022, AJ, 163, 244, doi: 10.3847/1538-3881/ac5c4c Claret, A. 2017, A&A, 600, A30, doi: 10.1051/0004-6361/201629705 Claret, A., & Bloemen, S. 2011, A&A, 529, A75, doi: 10.1051/0004-6361/201116451 Cloutier, R., Astudillo-Defru, N., Doyon, R., et al. 2017, A&A, 608, A35, doi: 10.1051/0004-6361/ 201731558 Collins, K. A., Kielkopf, J. F., Stassun, K. G., & Hessman, F. V. 2017, AJ, 153, 77, doi: 10.3847/ 1538-3881/153/2/77 Collins, K. A., Collins, K. I., Pepper, J., et al. 2018, AJ, 156, 234, doi: 10.3847/1538-3881/aae582 Cosentino, R., Lovis, C., Pepe, F., et al. 2012, in Ground-based and Airborne Instrumentation for Astronomy IV, ed. I. S. McLean, S. K. Ramsay, & H. Takami, Vol. 8446, International Society for Optics and Photonics (SPIE), 657 – 676, doi: 10.1117/12.925738 Cretignier, M., Dumusque, X., Aigrain, S., & Pepe, F. 2023, A&A, 678, A2, doi: 10.1051/ 0004-6361/202347232 Cretignier, M., Dumusque, X., Allart, R., Pepe, F., & Lovis, C. 2020a, A&A, 633, A76, doi: 10. 1051/0004-6361/201936548 Cretignier, M., Dumusque, X., Hara, N. C., & Pepe, F. 2021, A&A, 653, A43, doi: 10.1051/ 0004-6361/202140986 Cretignier, M., Dumusque, X., & Pepe, F. 2022, A&A, 659, A68, doi: 10.1051/0004-6361/ 202142435 Cretignier, M., Francfort, J., Dumusque, X., Allart, R., & Pepe, F. 2020b, A&A, 640, A42, doi: 10.1051/0004-6361/202037722 116 Cretignier, M., Pietrow, A. G. M., & Aigrain, S. 2024, MNRAS, 527, 2940, doi: 10.1093/mnras/ stad3292 Crossfield, I. J. M., Ciardi, D. R., Petigura, E. A., et al. 2016, ApJS, 226, 7, doi: 10.3847/0067-0049/ 226/1/7 Csizmadia, S. 2020, MNRAS, 496, 4442, doi: 10.1093/mnras/staa349 Cutri, R. M., Wright, E. L., T., C., & at al. 2012, VizieR Online Data Catalog, 2311, 0 Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, VizieR Online Data Catalog, 2246, 0 D’Angelo, G., Kley, W., & Henning, T. 2003, ApJ, 586, 540, doi: 10.1086/367555 Dattilo, A., Vanderburg, A., Shallue, C. J., et al. 2019, AJ, 157, 169, doi: 10.3847/1538-3881/ ab0e12 David, T. J., Mamajek, E. E., Vanderburg, A., et al. 2018, AJ, 156, 302, doi: 10.3847/1538-3881/ aaeed7 Dawson, R. I., & Johnson, J. A. 2018, Annual Review of Astronomy and Astrophysics, 56, 175, doi: 10.1146/annurev-astro-081817-051853 de Beurs, Z. L., Vanderburg, A., Thygesen, E., et al. 2024, arXiv e-prints, arXiv:2401.12276, doi: 10.48550/arXiv.2401.12276 de Leon, J. P., Livingston, J., Endl, M., et al. 2021, MNRAS, 508, 195, doi: 10.1093/mnras/stab2305 Delorme, J.-R., Jovanovic, N., Echeverri, D., et al. 2021, Journal of Astronomical Telescopes, Instruments, and Systems, 7, 035006, doi: 10.1117/1.JATIS.7.3.035006 Deming, D., Knutson, H., Kammer, J., et al. 2015, ApJ, 805, 132, doi: 10.1088/0004-637X/805/2/ 132 Dotter, A. 2016, ApJS, 222, 8, doi: 10.3847/0067-0049/222/1/8 Dragomir, D., Harris, M., Pepper, J., et al. 2020, AJ, 159, 219, doi: 10.3847/1538-3881/ab845d Dressing, C. D., & Charbonneau, D. 2013, ApJ, 767, 95, doi: 10.1088/0004-637X/767/1/95 Dressing, C. D., Vanderburg, A., Schlieder, J. E., et al. 2017, AJ, 154, 207, doi: 10.3847/1538-3881/ aa89f2 Dumusque, X., Cretignier, M., Sosnowska, D., et al. 2021, A&A, 648, A103, doi: 10.1051/ 0004-6361/202039350 117 Eastman, J. 2017, EXOFASTv2: Generalized publication-quality exoplanet modeling code, Astrophysics Source Code Library. http://ascl.net/1710.003 Eastman, J., Gaudi, B. S., & Agol, E. 2013, PASP, 125, 83, doi: 10.1086/669497 Eastman, J. D., Rodriguez, J. E., Agol, E., et al. 2019, arXiv e-prints, arXiv:1907.09480. https: //arxiv.org/abs/1907.09480 Edwards, B., Changeat, Q., Hou Yip, K., et al. 2019a, in EPSC-DPS Joint Meeting 2019, Vol. 2019, EPSC–DPS2019–595 Edwards, B., Rice, M., Zingales, T., et al. 2019b, Experimental Astronomy, 47, 29, doi: 10.1007/ s10686-018-9611-4 Edwards, B., Anisman, L., Changeat, Q., et al. 2020, Research Notes of the American Astronomical Society, 4, 109, doi: 10.3847/2515-5172/aba42b Edwards, B., Ho, C., Osborne, H., et al. 2021, ATOM - Astronomy: Theory, 2, 25, doi: 10.32374/ atom.2020.2.4 Espinoza, N. 2018, EXO-NAILER: EXOplanet traNsits and rAdIal veLocity fittER, Astrophysics Source Code Library, record ascl:1806.029 Espinoza, N., Kossakowski, D., & Brahm, R. 2019, MNRAS, 490, 2262, doi: 10.1093/mnras/ stz2688 Fabrycky, D., & Tremaine, S. 2007, ApJ, 669, 1298, doi: 10.1086/521702 Faedi, F., Barros, S. C. C., Anderson, D. R., et al. 2011, A&A, 531, A40, doi: 10.1051/0004-6361/ 201116671 Fazio, G. G., Hora, J. L., Allen, L. E., et al. 2004, ApJS, 154, 10, doi: 10.1086/422843 Feinstein, A. D., Radica, M., Welbanks, L., et al. 2022, arXiv e-prints, arXiv:2211.10493, doi: 10. 48550/arXiv.2211.10493 Fűrész, G. 2008, PhD thesis, University of Szeged, Hungary Fischer, P. D., Knutson, H. A., Sing, D. K., et al. 2016, ApJ, 827, 19, doi: 10.3847/0004-637X/ 827/1/19 Fisher, C., & Heng, K. 2018, MNRAS, 481, 4698, doi: 10.1093/mnras/sty2550 Foreman-Mackey, D., Agol, E., Ambikasaran, S., & Angus, R. 2023, celerite2: Fast and scalable Gaussian Processes in one dimension, Astrophysics Source Code Library, record ascl:2310.001 118 Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/ 670067 Foreman-Mackey, D., Luger, R., Agol, E., et al. 2021, The Journal of Open Source Software, 6, 3285, doi: 10.21105/joss.03285 Frandsen, S., & Lindberg, B. 1999, in Astrophysics with the NOT, ed. H. Karttunen & V. Piirola, 71 Frustagli, G., Poretti, E., Milbourne, T., et al. 2020, A&A, 633, A133, doi: 10.1051/0004-6361/ 201936689 Fulton, B. J., & Petigura, E. A. 2018, AJ, 156, 264, doi: 10.3847/1538-3881/aae828 Fulton, B. J., Petigura, E. A., Howard, A. W., et al. 2017, AJ, 154, 109, doi: 10.3847/1538-3881/ aa80eb Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1, doi: 10.1051/ 0004-6361/201629272 Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1, doi: 10.1051/ 0004-6361/201833051 —. 2021, A&A, 649, A1, doi: 10.1051/0004-6361/202039657 Gaidos, E., Mann, A. W., Rizzuto, A., et al. 2017, MNRAS, 464, 850, doi: 10.1093/mnras/stw2345 Gardner, J. P., Mather, J. C., Clampin, M., et al. 2006, SSRv, 123, 485, doi: 10.1007/ s11214-006-8315-7 Gazak, J. Z., Johnson, J. A., Tonry, J., et al. 2012, Advances in Astronomy, 2012, 697967, doi: 10.1155/2012/697967 Gentile Fusillo, N. P., Tremblay, P. E., Cukanovaite, E., et al. 2021, MNRAS, 508, 3877, doi: 10. 1093/mnras/stab2672 Giles, H. A. C., Bayliss, D., Espinoza, N., et al. 2018, MNRAS, 475, 1809, doi: 10.1093/mnras/ stx3300 Grunblatt, S. K., Huber, D., Gaidos, E. J., et al. 2016, AJ, 152, 185, doi: 10.3847/0004-6256/152/ 6/185 Grunblatt, S. K., Huber, D., Gaidos, E., et al. 2018, ApJL, 861, L5, doi: 10.3847/2041-8213/aacc67 Günther, M. N., & Daylan, T. 2021, ApJS, 254, 13, doi: 10.3847/1538-4365/abe70e 119 Hamann, A., Montet, B. T., Fabrycky, D. C., Agol, E., & Kruse, E. 2019, AJ, 158, 133, doi: 10. 3847/1538-3881/ab32e3 Helled, R., & Bodenheimer, P. 2010, Icarus, 207, 503, doi: 10.1016/j.icarus.2009.11.023 Hellier, C., Anderson, D. R., Collier Cameron, A., et al. 2014, MNRAS, 440, 1982, doi: 10.1093/ mnras/stu410 Henry, G. W., Marcy, G. W., Butler, R. P., & Vogt, S. S. 2000, ApJL, 529, L41, doi: 10.1086/312458 Hewitt, H. B., Simon, M. N., Mead, C., et al. 2023, Physical Review Physics Education Research, 19, 020156, doi: 10.1103/PhysRevPhysEducRes.19.020156 Hirano, T., Fukui, A., Mann, A. W., et al. 2016a, ApJ, 820, 41, doi: 10.3847/0004-637X/820/1/41 Hirano, T., Nowak, G., Kuzuhara, M., et al. 2016b, ApJ, 825, 53, doi: 10.3847/0004-637X/825/1/53 Hirano, T., Dai, F., Gandolfi, D., et al. 2018, AJ, 155, 127, doi: 10.3847/1538-3881/aaa9c1 Holmberg, M., & Madhusudhan, N. 2022, AJ, 164, 79, doi: 10.3847/1538-3881/ac77eb Howell, S. B., Rowe, J. F., Bryson, S. T., et al. 2012, ApJ, 746, 123, doi: 10.1088/0004-637X/746/ 2/123 Howell, S. B., Sobeck, C., Haas, M., et al. 2014, PASP, 126, 398, doi: 10.1086/676406 Huang, C., Wu, Y., & Triaud, A. H. M. J. 2016, ApJ, 825, 98, doi: 10.3847/0004-637X/825/2/98 Huang, C. X., Quinn, S. N., Vanderburg, A., et al. 2020, ApJL, 892, L7, doi: 10.3847/2041-8213/ ab7302 Hubbard, W. B., Fortney, J. J., Lunine, J. I., et al. 2001, ApJ, 560, 413, doi: 10.1086/322490 Ikwut-Ukwa, M., Rodriguez, J. E., Bieryla, A., et al. 2020, AJ, 160, 209, doi: 10.3847/1538-3881/ aba964 Irwin, J., Irwin, M., Aigrain, S., et al. 2007, MNRAS, 375, 1449, doi: 10.1111/j.1365-2966.2006. 11408.x Irwin, J. M., Berta-Thompson, Z. K., Charbonneau, D., et al. 2015, in Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun, Vol. 18, 18th Cambridge Workshop on Cool Stars, Stellar Systems, and the Sun, 767–772, doi: 10.48550/arXiv.1409.0891 Irwin, M. J. 1985, MNRAS, 214, 575, doi: 10.1093/mnras/214.4.575 120 Jenkins, J. M., Twicken, J. D., McCauliff, S., et al. 2016, in Proc. SPIE, Vol. 9913, Software and Cyberinfrastructure for Astronomy IV, 99133E, doi: 10.1117/12.2233418 Johns, M., McCarthy, P., Raybould, K., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8444, Ground-based and Airborne Telescopes IV, ed. L. M. Stepp, R. Gilmozzi, & H. J. Hall, 84441H, doi: 10.1117/12.926716 Johnson, J. A. 2016, How Do You Find An Exoplanet? Johnson, M. C., Gandolfi, D., Fridlund, M., et al. 2016, AJ, 151, 171, doi: 10.3847/0004-6256/ 151/6/171 Johnson, M. C., Dai, F., Justesen, A. B., et al. 2018a, MNRAS, 481, 596, doi: 10.1093/mnras/ sty2238 Johnson, M. C., Rodriguez, J. E., Zhou, G., et al. 2018b, AJ, 155, 100, doi: 10.3847/1538-3881/ aaa5af Johnson, M. C., Dai, F., Justesen, A. B., et al. 2018c, MNRAS, 481, 596, doi: 10.1093/mnras/ sty2238 JWST Transiting Exoplanet Community Early Release Science Team, Ahrer, E.-M., Alderson, L., et al. 2023, Nature, 614, 649, doi: 10.1038/s41586-022-05269-w Kane, S. R., Mahadevan, S., von Braun, K., Laughlin, G., & Ciardi, D. R. 2009, PASP, 121, 1386, doi: 10.1086/648564 Kasting, J. F., Whitmire, D. P., & Reynolds, R. T. 1993, Icarus, 101, 108, doi: 10.1006/icar.1993. 1010 Kempton, E. M. R., Bean, J. L., Louie, D. R., et al. 2018, PASP, 130, 114401, doi: 10.1088/ 1538-3873/aadf6f Kipping, D. M. 2013, MNRAS, 435, 2152, doi: 10.1093/mnras/stt1435 Kirk, J., López-Morales, M., Wheatley, P. J., et al. 2019, AJ, 158, 144, doi: 10.3847/1538-3881/ ab397d Kirk, J., Stevenson, K. B., Fu, G., et al. 2024, AJ, 167, 90, doi: 10.3847/1538-3881/ad19df Knudstrup, E., Albrecht, S. H., Winn, J. N., et al. 2024, A&A, 690, A379, doi: 10.1051/0004-6361/ 202450627 Kokori, A., Tsiaras, A., Edwards, B., et al. 2021, Experimental Astronomy, doi: 10.1007/ s10686-020-09696-3 121 —. 2022, ApJS, 258, 40, doi: 10.3847/1538-4365/ac3a10 —. 2023, ApJS, 265, 4, doi: 10.3847/1538-4365/ac9da4 Konacki, M., Torres, G., Jha, S., & Sasselov, D. D. 2003, Nature, 421, 507, doi: 10.1038/ nature01379 Korth, J., Csizmadia, S., Gandolfi, D., et al. 2019, MNRAS, 482, 1807, doi: 10.1093/mnras/sty2760 Korth, J., Gandolfi, D., Šubjak, J., et al. 2023, A&A, 675, A115, doi: 10.1051/0004-6361/ 202244617 Kreidberg, L. 2015, PASP, 127, 1161, doi: 10.1086/683602 Kruse, E., Agol, E., Luger, R., & Foreman-Mackey, D. 2019, ApJS, 244, 11, doi: 10.3847/ 1538-4365/ab346b Lam, K. W. F., Santerne, A., Sousa, S. G., et al. 2018, A&A, 620, A77, doi: 10.1051/0004-6361/ 201834073 Law, N. M., Morton, T., Baranec, C., et al. 2014, ApJ, 791, 35, doi: 10.1088/0004-637X/791/1/35 Lightkurve Collaboration, Cardoso, J. V. d. M. a., Hedges, C., et al. 2018, Lightkurve: Kepler and TESS time series analysis in Python. http://ascl.net/1812.013 Lillo-Box, J., Demangeon, O., Santerne, A., et al. 2016, A&A, 594, A50, doi: 10.1051/0004-6361/ 201628204 Lindegren, L., Bastian, U., Biermann, M., et al. 2021, A&A, 649, A4, doi: 10.1051/0004-6361/ 202039653 Livingston, J. H., Crossfield, I. J. M., Petigura, E. A., et al. 2018a, AJ, 156, 277, doi: 10.3847/ 1538-3881/aae778 Livingston, J. H., Endl, M., Dai, F., et al. 2018b, AJ, 156, 78, doi: 10.3847/1538-3881/aaccde Livingston, J. H., Dai, F., Hirano, T., et al. 2018c, AJ, 155, 115, doi: 10.3847/1538-3881/aaa841 Lodders, K., & Fegley, B. 2002, Icarus, 155, 393, doi: 10.1006/icar.2001.6740 Lucy, L. B., & Sweeney, M. A. 1971, AJ, 76, 544, doi: 10.1086/111159 Luger, R., Agol, E., Kruse, E., et al. 2016, AJ, 152, 100, doi: 10.3847/0004-6256/152/4/100 122 Lund, M. N., Handberg, R., Davies, G. R., Chaplin, W. J., & Jones, C. D. 2015, ApJ, 806, 30, doi: 10.1088/0004-637X/806/1/30 Luque, R., Nowak, G., Hirano, T., et al. 2022, A&A, 666, A154, doi: 10.1051/0004-6361/ 202244426 Lustig-Yaeger, J., Fu, G., May, E. M., et al. 2023, Nature Astronomy, 7, 1317, doi: 10.1038/ s41550-023-02064-z Luu, C. N., Yu, X., Glein, C. R., et al. 2024, ApJL, 977, L51, doi: 10.3847/2041-8213/ad9eb1 Madhusudhan, N. 2019, ARA&A, 57, 617, doi: 10.1146/annurev-astro-081817-051846 Madhusudhan, N., Amin, M. A., & Kennedy, G. M. 2014, ApJL, 794, L12, doi: 10.1088/2041-8205/ 794/1/L12 Madhusudhan, N., Bitsch, B., Johansen, A., & Eriksson, L. 2017, MNRAS, 469, 4102, doi: 10. 1093/mnras/stx1139 Madhusudhan, N., Nixon, M. C., Welbanks, L., Piette, A. A. A., & Booth, R. A. 2020, ApJL, 891, L7, doi: 10.3847/2041-8213/ab7229 Madhusudhan, N., Piette, A. A. A., & Constantinou, S. 2021, ApJ, 918, 1, doi: 10.3847/1538-4357/ abfd9c Madhusudhan, N., Sarkar, S., Constantinou, S., et al. 2023, ApJL, 956, L13, doi: 10.3847/ 2041-8213/acf577 Madhusudhan, N., & Seager, S. 2011, ApJ, 729, 41, doi: 10.1088/0004-637X/729/1/41 Malavolta, L., Mayo, A. W., Louden, T., et al. 2018, AJ, 155, 107, doi: 10.3847/1538-3881/aaa5b5 Mann, A. W., Feiden, G. A., Gaidos, E., Boyajian, T., & von Braun, K. 2015, ApJ, 804, 64, doi: 10.1088/0004-637X/804/1/64 Mann, A. W., Gaidos, E., Mace, G. N., et al. 2016, ApJ, 818, 46, doi: 10.3847/0004-637X/818/1/46 Mann, A. W., Gaidos, E., Vanderburg, A., et al. 2017, AJ, 153, 64, doi: 10.1088/1361-6528/aa5276 Mann, A. W., Dupuy, T., Kraus, A. L., et al. 2019, ApJ, 871, 63, doi: 10.3847/1538-4357/aaf3bc Maxted, P. F. L. 2016, A&A, 591, A111, doi: 10.1051/0004-6361/201628579 Mayo, A. W., Vanderburg, A., Latham, D. W., et al. 2018, AJ, 155, 136, doi: 10.3847/1538-3881/ aaadff 123 Mayor, M., & Queloz, D. 1995, Nature, 378, 355, doi: 10.1038/378355a0 Mayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20 Mazeh, T., Naef, D., Torres, G., et al. 2000, ApJL, 532, L55, doi: 10.1086/312558 McCormac, J., Pollacco, D., Skillen, I., et al. 2013, PASP, 125, 548, doi: 10.1086/670940 McGruder, C. D., López-Morales, M., Kirk, J., et al. 2022, AJ, 164, 134, doi: 10.3847/1538-3881/ ac7f2e Meschiari, S., Wolf, A. S., Rivera, E., et al. 2009, PASP, 121, 1016, doi: 10.1086/605730 Moffat, A. F. J. 1969, A&A, 3, 455 Montet, B. T., Morton, T. D., Foreman-Mackey, D., et al. 2015, ApJ, 809, 25, doi: 10.1088/ 0004-637X/809/1/25 Mordasini, C., van Boekel, R., Mollière, P., Henning, T., & Benneke, B. 2016, ApJ, 832, 41, doi: 10.3847/0004-637X/832/1/41 Moses, J. I., Line, M. R., Visscher, C., et al. 2013, ApJ, 777, 34, doi: 10.1088/0004-637X/777/1/34 Močnik, T., Clark, B. J. M., Anderson, D. R., Hellier, C., & Brown, D. J. A. 2016, AJ, 151, 150, doi: 10.3847/0004-6256/151/6/150 Nikolov, N., Sing, D. K., Gibson, N. P., et al. 2016, ApJ, 832, 191, doi: 10.3847/0004-637X/832/ 2/191 Nikolov, N., Sing, D. K., Fortney, J. J., et al. 2018, Nature, 557, 526, doi: 10.1038/ s41586-018-0101-7 Nikolov, N. K., Sing, D. K., Spake, J. J., et al. 2022, MNRAS, 515, 3037, doi: 10.1093/mnras/ stac1530 Nutzman, P., & Charbonneau, D. 2008, PASP, 120, 317, doi: 10.1086/533420 Öberg, K. I., Murray-Clay, R., & Bergin, E. A. 2011, ApJL, 743, L16, doi: 10.1088/2041-8205/ 743/1/L16 Owen, J. E., & Jackson, A. P. 2012, MNRAS, 425, 2931, doi: 10.1111/j.1365-2966.2012.21481.x Parviainen, H. 2015, MNRAS, 450, 3233, doi: 10.1093/mnras/stv894 Patel, J. A., & Espinoza, N. 2022, AJ, 163, 228, doi: 10.3847/1538-3881/ac5f55 124 Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192, 3, doi: 10.1088/0067-0049/192/1/3 Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4, doi: 10.1088/0067-0049/208/1/4 Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15, doi: 10.1088/0067-0049/220/1/15 Peacock, S., Barman, T., Shkolnik, E. L., Hauschildt, P. H., & Baron, E. 2019, ApJ, 871, 235, doi: 10.3847/1538-4357/aaf891 Pepe, F., Cristiani, S., Rebolo, R., et al. 2021, A&A, 645, A96, doi: 10.1051/0004-6361/202038306 Pepper, J., Pogge, R. W., DePoy, D. L., et al. 2007, PASP, 119, 923, doi: 10.1086/521836 Pinhas, A., Rackham, B. V., Madhusudhan, N., & Apai, D. 2018, MNRAS, 480, 5314, doi: 10. 1093/mnras/sty2209 Pizzolato, N., Maggio, A., Micela, G., Sciortino, S., & Ventura, P. 2003, A&A, 397, 147, doi: 10. 1051/0004-6361:20021560 Poddaný, S., Brát, L., & Pejcha, O. 2010, NewA, 15, 297, doi: 10.1016/j.newast.2009.09.001 Pollacco, D. L., Skillen, I., Collier Cameron, A., et al. 2006, PASP, 118, 1407, doi: 10.1086/508556 Polman, J., Waters, L. B. F. M., Min, M., Miguel, Y., & Khorshid, N. 2023, A&A, 670, A161, doi: 10.1051/0004-6361/202244647 Pontoppidan, K. M., Barrientes, J., Blome, C., et al. 2022, ApJL, 936, L14, doi: 10.3847/2041-8213/ ac8a4e Pope, B. J. S., Parviainen, H., & Aigrain, S. 2016, MNRAS, 461, 3399, doi: 10.1093/mnras/stw1373 Pope, B. J. S., White, T. R., Farr, W. M., et al. 2019, ApJS, 245, 8, doi: 10.3847/1538-4365/ab3d29 Powell, D., Feinstein, A. D., Lee, E. K. H., et al. 2024, Nature, 626, 979, doi: 10.1038/ s41586-024-07040-9 Queloz, D., Mayor, M., Weber, L., et al. 2000, A&A, 354, 99 Rampalli, R., Vanderburg, A., Bieryla, A., et al. 2019, AJ, 158, 62, doi: 10.3847/1538-3881/ab27c2 Rasio, F. A., & Ford, E. B. 1996, Science, 274, 954, doi: 10.1126/science.274.5289.954 Rauer, H., Catala, C., Aerts, C., et al. 2014, Experimental Astronomy, 38, 249, doi: 10.1007/ s10686-014-9383-4 125 Rice, M., Wang, S., & Laughlin, G. 2022a, ApJL, 926, L17, doi: 10.3847/2041-8213/ac502d Rice, M., Wang, S., Wang, X.-Y., et al. 2022b, AJ, 164, 104, doi: 10.3847/1538-3881/ac8153 Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003, doi: 10.1117/1.JATIS.1.1.014003 Rigby, F. E., Pica-Ciamarra, L., Holmberg, M., et al. 2024, ApJ, 975, 101, doi: 10.3847/1538-4357/ ad6c38 Rodriguez, J. E., Vanderburg, A., Zieba, S., et al. 2020, AJ, 160, 117, doi: 10.3847/1538-3881/ aba4b3 Rodriguez, J. E., Quinn, S. N., Zhou, G., et al. 2021, AJ, 161, 194, doi: 10.3847/1538-3881/abe38a Rodriguez, J. E., Quinn, S. N., Vanderburg, A., et al. 2022a, arXiv e-prints, arXiv:2205.05709. https://arxiv.org/abs/2205.05709 —. 2022b, arXiv e-prints, arXiv:2205.05709, doi: 10.48550/arXiv.2205.05709 Rustamkulov, Z., Sing, D. K., Mukherjee, S., et al. 2022, arXiv e-prints, arXiv:2211.10487, doi: 10.48550/arXiv.2211.10487 Salvatier, J., Wiecki, T. V., & Fonnesbeck, C. 2016, PyMC3: Python probabilistic programming framework, Astrophysics Source Code Library, record ascl:1610.016 Samra, D., Helling, C., Chubb, K. L., et al. 2023, A&A, 669, A142, doi: 10.1051/0004-6361/ 202244939 Sanders, G. H. 2013, s12036-013-9169-5 Journal of Astrophysics and Astronomy, 34, 81, doi: 10.1007/ Santerne, A., Hébrard, G., Lillo-Box, J., et al. 2016, ApJ, 824, 55, doi: 10.3847/0004-637X/824/1/ 55 Santos, N. C., Cristo, E., Demangeon, O., et al. 2020, A&A, 644, A51, doi: 10.1051/0004-6361/ 202039454 Sarkis, P., Henning, T., Kürster, M., et al. 2018, AJ, 155, 257, doi: 10.3847/1538-3881/aac108 Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103, doi: 10.1088/0004-637X/737/2/103 Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525, doi: 10.1086/305772 126 Seager, S. 2003, in Astronomical Society of the Pacific Conference Series, Vol. 294, Scientific Frontiers in Research on Extrasolar Planets, ed. D. Deming & S. Seager, 457–466, doi: 10. 48550/arXiv.astro-ph/0305338 Seager, S., & Sasselov, D. D. 2000, ApJ, 537, 916, doi: 10.1086/309088 Sgro, L. A., Dalba, P. A., Esposito, T. M., et al. 2024, AJ, 168, 26, doi: 10.3847/1538-3881/ad5096 Sha, L., Huang, C. X., Shporer, A., et al. 2021, AJ, 161, 82, doi: 10.3847/1538-3881/abd187 Shallue, C. J., & Vanderburg, A. 2018, AJ, 155, 94, doi: 10.3847/1538-3881/aa9e09 Shorttle, O., Jordan, S., Nicholls, H., Lichtenberg, T., & Bower, D. J. 2024, ApJL, 962, L8, doi: 10.3847/2041-8213/ad206e Shporer, A., Zhou, G., Fulton, B. J., et al. 2017, AJ, 154, 188, doi: 10.3847/1538-3881/aa8bb9 Sicilia, D., Malavolta, L., Scandariato, G., et al. 2025, A&A, 693, A316, doi: 10.1051/0004-6361/ 202451937 Sing, D. K., Vidal-Madjar, A., Lecavelier des Etangs, A., et al. 2008, ApJ, 686, 667, doi: 10.1086/ 590076 Sing, D. K., Fortney, J. J., Nikolov, N., et al. 2016, Nature, 529, 59, doi: 10.1038/nature16068 Sinukoff, E., Howard, A. W., Petigura, E. A., et al. 2016, ApJ, 827, 78, doi: 10.3847/0004-637X/ 827/1/78 Smith, A. M. S., Csizmadia, S., Gandolfi, D., et al. 2019, AcA, 69, 135, doi: 10.32023/0001-5237/ 69.2.3 Smith, J. C., Stumpe, M. C., Van Cleve, J. E., et al. 2012, PASP, 124, 1000, doi: 10.1086/667697 Snellen, I. A. G., Albrecht, S., de Mooij, E. J. W., & Le Poole, R. S. 2008, A&A, 487, 357, doi: 10.1051/0004-6361:200809762 Soto, M. G., Díaz, M. R., Jenkins, J. S., et al. 2018, MNRAS, 478, 5356, doi: 10.1093/mnras/sty1334 Speagle, J. S. 2020, MNRAS, 493, 3132, doi: 10.1093/mnras/staa278 Spergel, D., Gehrels, N., Baltay, C., et al. 2015, arXiv e-prints, arXiv:1503.03757, doi: 10.48550/ arXiv.1503.03757 Stalport, M., Cretignier, M., Udry, S., et al. 2023, A&A, 678, A90, doi: 10.1051/0004-6361/ 202346887 127 Stassun, K. G., Oelkers, R. J., Pepper, J., et al. 2018, AJ, 156, 102, doi: 10.3847/1538-3881/aad050 Stefansson, G., Cañas, C., Wisniewski, J., et al. 2020, AJ, 159, 100, doi: 10.3847/1538-3881/ab5f15 Stumpe, M. C., Smith, J. C., Catanzarite, J. H., et al. 2014, PASP, 126, 100, doi: 10.1086/674989 Stumpe, M. C., Smith, J. C., Van Cleve, J. E., et al. 2012, PASP, 124, 985, doi: 10.1086/667698 Taylor, J., Radica, M., Welbanks, L., et al. 2023, MNRAS, 524, 817, doi: 10.1093/mnras/stad1547 Thygesen, E., Ranshaw, J. A., Rodriguez, J. E., et al. 2023, AJ, 165, 155, doi: 10.3847/1538-3881/ acaf03 Thygesen, E., Rodriguez, J. E., de Beurs, Z. L., et al. 2024, AJ, 168, 161, doi: 10.3847/1538-3881/ ad60bf Tinetti, G., Drossart, P., Eccleston, P., et al. 2018, Experimental Astronomy, 46, 135, doi: 10.1007/ s10686-018-9598-x Tinetti, G., Eccleston, P., Haswell, C., et al. 2021, arXiv e-prints, arXiv:2104.04824. https: //arxiv.org/abs/2104.04824 Tsai, S.-M., Lee, E. K. H., Powell, D., et al. 2022, arXiv e-prints, arXiv:2211.10490, doi: 10.48550/ arXiv.2211.10490 Tsiaras, A., Waldmann, I. P., Tinetti, G., Tennyson, J., & Yurchenko, S. N. 2019a, Nature Astronomy, 451, doi: 10.1038/s41550-019-0878-9 —. 2019b, Nature Astronomy, 3, 1086, doi: 10.1038/s41550-019-0878-9 Udalski, A., Zebrun, K., Szymanski, M., et al. 2002a, AcA, 52, 115, doi: 10.48550/arXiv.astro-ph/ 0207133 Udalski, A., Paczynski, B., Zebrun, K., et al. 2002b, AcA, 52, 1, doi: 10.48550/arXiv.astro-ph/ 0202320 Udry, S., Lovis, C., Bouchy, F., et al. 2014, arXiv e-prints, arXiv:1412.1048. https://arxiv.org/abs/ 1412.1048 Van Eylen, V., Dai, F., Mathur, S., et al. 2018, MNRAS, 478, 4866, doi: 10.1093/mnras/sty1390 van Leeuwen, F. 2007, A&A, 474, 653, doi: 10.1051/0004-6361:20078357 Vanderburg, A., & Johnson, J. A. 2014, PASP, 126, 948, doi: 10.1086/678764 128 Vanderburg, A., Montet, B. T., Johnson, J. A., et al. 2015, ApJ, 800, 59, doi: 10.1088/0004-637X/ 800/1/59 Vanderburg, A., Latham, D. W., Buchhave, L. A., et al. 2016, ApJS, 222, 14, doi: 10.3847/ 0067-0049/222/1/14 Vanderburg, A., Huang, C. X., Rodriguez, J. E., et al. 2019, ApJL, 881, L19, doi: 10.3847/ 2041-8213/ab322d Vogt, S. S., Allen, S. L., Bigelow, B. C., et al. 1994, in Proc. SPIE, Vol. 2198, Instrumentation in Astronomy VIII, ed. D. L. Crawford & E. R. Craine, 362, doi: 10.1117/12.176725 Wakeford, H. R., Sing, D. K., Deming, D., et al. 2018, AJ, 155, 29, doi: 10.3847/1538-3881/aa9e4e Walker, G., Matthews, J., Kuschnig, R., et al. 2003, PASP, 115, 1023, doi: 10.1086/377358 Wang, J., Xie, J.-W., Barclay, T., & Fischer, D. A. 2014, ApJ, 783, 4, doi: 10.1088/0004-637X/ 783/1/4 Werner, M., Crossfield, I., Akeson, R., et al. 2016, Spitzer v. K2: Part II, Spitzer Proposal ID 13052 Winn, J. N., Holman, M. J., Torres, G., et al. 2008, ApJ, 683, 1076, doi: 10.1086/589737 Wogan, N. F., Batalha, N. E., Zahnle, K. J., et al. 2024, ApJL, 963, L7, doi: 10.3847/2041-8213/ ad2616 Wolszczan, A., & Frail, D. A. 1992, Nature, 355, 145, doi: 10.1038/355145a0 Wu, Y., & Lithwick, Y. 2011, ApJ, 735, 109, doi: 10.1088/0004-637X/735/2/109 Yee, S. W., Winn, J. N., Hartman, J. D., et al. 2022a, AJ, 164, 70, doi: 10.3847/1538-3881/ac73ff —. 2022b, arXiv e-prints, arXiv:2210.15473, doi: 10.48550/arXiv.2210.15473 Yip, K. H., Changeat, Q., Edwards, B., et al. 2021, AJ, 161, 4, doi: 10.3847/1538-3881/abc179 Young, A. T. 1967, AJ, 72, 328 Yu, L., Rodriguez, J. E., Eastman, J. D., et al. 2018, AJ, 156, 127, doi: 10.3847/1538-3881/aad6e7 Zahnle, K., Marley, M. S., Freedman, R. S., Lodders, K., & Fortney, J. J. 2009, ApJL, 701, L20, doi: 10.1088/0004-637X/701/1/L20 Zellem, R., Biferno, A., Ciardi, D. R., et al. 2019, BAAS, 51, 416. https://arxiv.org/abs/1903.07716 129 Zellem, R. T., Pearson, K. A., Blaser, E., et al. 2020, PASP, 132, 054401, doi: 10.1088/1538-3873/ ab7ee7 Zeng, L., Sasselov, D. D., & Jacobsen, S. B. 2016, ApJ, 819, 127, doi: 10.3847/0004-637X/819/2/ 127 Zink, J. K., Hardegree-Ullman, K. K., Christiansen, J. L., et al. 2021, AJ, 162, 259, doi: 10.3847/ 1538-3881/ac2309 130