Universiteit van Amsterdam MSc Artificial Intelligence Master Thesis Characterizing Exoplanet Transits and Stellar Activity with Scalable Gaussian Processes by Victoria Foing 11773391 October 17, 2020 36 ECTS February 2020 - October 2020 Main Supervisor: Ana M. Heras (European Space Agency) Co-Supervisor: Patrick Forré (University of Amsterdam) Abstract Exoplanets are planets orbiting around stars other than our sun. The field of exoplanet science has boomed since the 1990s [1]. Space missions like CoRoT, Kepler and TESS, and ground-based observatories have collected millions of lightcurves and detected more than 3,200 exoplanets using transit photometry, an approach that involves measuring the brightness of a star over time and looking for periodic dips in the brightness caused by a planet repeatedly passing in front of the star. Once a transit is detected, astronomers seek to characterize the transit signal in the lightcurve to learn about properties of the planet. This can be challenging due to noise from stellar activity, namely rotational modulation patterns in the lightcurve caused by starspots and plages appearing in and out of view as the host star rotates. In the past decade, Gaussian Processes have become popular models for modelling both instrumental and astrophysical noise in lightcurves to aid the detection and characterization of exoplanet transits. Gaussian Processes are well-suited to this task because they are flexible, interpretable, and probabilistic estimates of the model parameters can be obtained with Markov Chain Monte Carlo sampling. However, they have been disadvantaged by slow computation speeds which has precluded their application to the vast datasets provided by Kepler and TESS. Recently, a new software, celerite, has introduced special kernel functions that are not only well-suited to model stellar activity but also scale linearly with the dataset by taking advantage of semi-separable matrices [2]. In this work, we apply scalable Gaussian Process models using the software celerite in order to characterize exoplanet transits and stellar activity in Kepler and TESS lightcurves. By "charac- terizing", we mean the retrieval of accurate transit and rotation parameters, with a focus on the radius of the planet and the rotation period of the star. We develop a pipeline for preprocessing the lightcurves, defining prior distributions using Physics techniques, building a Gaussian Process model with a transit mean function and a quasi-periodic Rotation kernel and a noise kernel, and sampling from the model with Markov Chain Monte Carlo. To assess the benefits of jointly mod- elling the rotation and the transits, we compare the performances of three Gaussian Process models: RotGP (only rotation), ExoGP (only transit), and ExoRotGP (rotation and transit). The mod- els are applied to 15 TESS stars and 9 Kepler stars with confirmed planets and rotation periods, and the parameter estimates obtained are compared to those from refereed publications. For the lightcurves classified as having unambiguous rotation periods by previous authors, our best GP models obtain rotation periods within one one day of the reference rotation periods. For four of the TESS lightcurves classified as having ambiguous rotation periods by previous authors, the GP models are able to find rotation periods, outperforming the analysis of other researchers using tra- ditional techniques. For the recovery of planet radii, our best GP models are able to estimate the planet radii within one standard deviation of the radii for all Kepler planets and all but two TESS planets. Moreover, we highlight the importance of preprocessing decisions, demonstrating that the pipeline correction method can worsen the retrieval of the rotation period and temporal binning can dramatically worsen the retrieval of planet radii. Finally, we discover that joint modelling improves rotation period estimates for a few stars, due to the extra noise kernel, but only marginally improves the planet radii estimates, suggesting that celerite’s noise kernel can be effective for removing stellar activity. Our results further indicate that the joint model could be a promising tool for the charac- terisation of small exoplanets, with sizes between Neptune and Earth. Our method provides a solid basis for a future application and extension of the model to these objects. Code: https://github.com/victoriafoing/ExoRotGP 1 Acknowledgements I would like to express my sincere gratitude to my supervisor at the European Space Agency, Ana M. Heras, for her wonderful guidance and excellent support throughout the research project. I would also like to thank my supervisor at the University of Amsterdam, Patrick Forre, for the great feedback about the Artificial Intelligence side of the project. I would like to thank my family for their unwavering support and advice, in particular Bernard Foing. Furthermore, I would like to thank the authors of celerite software, which I used throughout the project, in particular Dan Foreman-Mackey, who was very helpful in answering my questions and guiding me in the right direction. Lastly, I would like to thank my boyfriend Adam, my friends, and my classmates in the AI masters program for teaching me a lot throughout the AI master and making this experience very enjoyable. 2 Contents 1 Executive Summary 5 2 Introduction 7 2.1 Problem Statement . .7 2.2 Motivation . .7 2.3 Research questions . .8 2.4 History of Exoplanet Science . .9 2.4.1 Exoplanet Transits . 10 2.4.2 Stellar activity . 10 2.5 Challenges . 11 2.5.1 Detection of Signals . 11 2.5.2 Noise: photon, instrumental, astrophysical . 11 2.5.3 Stellar Limb Darkening . 11 3 Literature Review 12 3.1 Traditional Methods in Physics . 12 3.1.1 Transit Fitting . 12 3.1.2 Recovering Rotation Periods . 13 3.2 Applications in Machine Learning . 14 3.2.1 Gaussian Processes . 14 3.3 Position in the State of the Art . 15 4 Methodology 16 4.1 Bayesian Inference . 16 4.2 Gaussian Processes . 17 4.3 Celerite . 17 4.3.1 Stochastically-Driven Dampled Harmonic Oscillator (SHO) Kernel . 19 4.3.2 Rotation Kernel . 19 4.3.3 Semi-Separable Matrics . 20 4.4 Transit models . 22 4.5 Markov Chain Monte Carlo . 23 5 Algorithm 24 5.1 Algorithm . 24 5.1.1 Initial estimates . 24 5.1.2 Three Models: ExoGP, RotGP, ExoRotGP . 26 5.1.3 Sampling . 30 6 Experiments 31 6.1 Data . 31 6.2 Selection of Targets . 31 6.3 Preprocessing . 32 6.4 Experimental Design . 35 6.5 Experimental Evaluation . 36 7 Results and Discussion 38 7.1 Simulated Experiment . 38 7.2 Rotation Periods . 39 7.2.1 Kepler . 39 7.2.2 TESS . 40 7.2.3 Observation Window . 42 3 CONTENTS CONTENTS 7.3 Transit Characterization . 43 7.3.1 Kepler . 43 7.3.2 TESS . 44 7.4 Computation Time . 46 7.5 Discussion . 47 7.5.1 Kepler vs. TESS . 48 7.5.2 How could our method be improved? . 48 7.5.3 Challenges . 49 8 Conclusion and Future Work 50 8.1 Contributions . 50 8.2 Future work . 51 8.3 Concluding Remarks . 52 References 54 Appendix A 59 A.1 Summary Statistics . 59 A.2 Power spectrum of Celerite kernel and SHO . 60 A.3 Example of Document Outputted by Pipeline: WASP-173 A . 60 CHAPTER 1 Executive Summary In this work, we apply scalable Gaussian Process models using the software celerite in order to characterize exoplanet transits and stellar activity in Kepler and TESS lightcurves. Our goal is to use this machine learning method to obtain accurate characteristics of the planet and its orbit, and of the stellar light variations, with a focus on the radius of the planet and the rotation period of the star. We begin by presenting our research questions: (1) How accurately can we model the stellar activity with Gaussian processes in Kepler and TESS lightcurves? (2) How accurately can we model the exoplanet transits with Gaussian processes in Kepler and TESS lightcurves? (3) Does joint modelling improve the characterization of rotation and transit parameters in Kepler and TESS lightcurves? In the motivation section, we outline the motivation for studying the transit signals and rotation signals in Kepler and TESS lightcurves. The motivation is to help astronomers study and understand the planetary systems observed by these missions and to investigate one of the most fundamental questions in science: is our solar system unique? We further highlight how important finding accurate transit parameters is for deriving the properties of planets, which can give clues about the habitability of the planet. Modelling stellar activity can provide valuable insights about stellar physics (i.e. the evolution, age, and magnetic fields of stars) and the potential habitability of orbiting planets. Stellar activity models can also be used to detrend lightcurves, which can improve the detection of smaller Earth-sized planets, which are more likely to be habitable, and improve the characterization of transit parameters. Next, we explore the motivation for using Gaussian Process to solve the problem of finding accurate transit and rotation parameters. In the past decade, Gaussian Processes have become popular models for modelling both instrumental and astrophysical noise in lightcurves because they are flexible and interpretable. However, they have been largely limited by slow computation speeds. Recently, a new software, celerite, has introduced special kernel functions that are not only well-suited to model stellar activity but also scale linearly with the dataset by taking advantage of semi-separable matrices [2]. We provide a brief summary of the history of exoplanet science and an introduction of the two astronomy concepts that we will study with.