ABSTRACT Title of Dissertation: REVEALING UNIQUE EXOPLANET ATMOSPHERES WITH MULTI-INSTRUMENT SPACE TELESCOPE TRANSIT AND ECLIPSE SPECTROSCOPY Kyle Sheppard Doctor of Philosophy, 2021 Dissertation Directed by: Professor Drake Deming Department of Astronomy Atmospheres act as windows into their host planets, containing measurable in- formation on their planets? chemistry, climate, and atmospheric physics. The bulk properties of planets outside of the Solar System (exoplanets) prove to be much more varied than the Solar System, allowing the ability to test atmospheric models over a range of temperatures, radii, and host star properties. Modeling and observing exo- planet atmospheres provides a better understanding of both atmospheric processes and planetary diversity, and it places the Solar System in a greater context to un- derstand how unique it is, if at all. I take a broad approach, analyzing both transit and emission spectroscopy of 5 exoplanets populating the edges of parameter space, ranging from cool, Earth-sized planets (T?500K, R=0.8R?) up to massive, ultra-hot Jupiters (T?2500K, M=10MJup). I use my publicly available, open source Python 3 analysis pipeline DEFLATE to process telescope data and produce verifiable spectra. I then retrieve atmospheric properties using a forward model + Bayesian sampler retrieval tool, exploring how both inter- and intra- modeling assumptions impact results. I retrieve unexpected atmospheres, including: evidence of stellar activity mimicking water vapor features in two terrestrial planets in the multi-planet L9859 system; evidence of a clear atmosphere and a superstellar atmospheric metallicity and water abundance (5? detection) in the hot Jupiter HAT-P-41b (R=1.65RJup, Teq=1950 K); a potentially non-TiO driven thermal inversion and a photometric CO detection (6?) in the ultrahot Jupiter WASP-18b; and a water absorption feature (2.8?) and non-inverted T-P profile in the water-dissociation-vulnerable hot Jupiter WASP-19b (R=1.4RJup, Teq=2120 K). Overall, these results expand already exten- sive diversity of exoplanet atmospheres. Revealing Unique Exoplanet Atmospheres with Multi-instrument Space Telescope Transit and Eclipse Spectroscopy by Kyle Sheppard Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2021 Advisory Committee: Professor Drake Deming, Chair/Advisor Professor Kayo Ide, Dean?s Representative Professor Eliza Kempton Dr. Avi Mandell Professor Derek Richardson ? Copyright by Kyle Sheppard 2021 Preface Disclaimer: I originally published the majority of Chapter 3 in the Astronom- ical Journal (Sheppard et al., 2021). I published the majority of the WASP-18b- related analyses in Chapter 4 in Astrophysical Journal Letters (Sheppard et al., 2017). Even though I was first author and in charge of writing and organizing each paper, I also worked with collaborators, who contributed a significant amount. There are sections of the dissertation where a collaborator performed a retrieval or data analysis and my contribution was limited to discussing, interpreting, and writ- ing. These sections are necessary to include for a complete understanding of each chapter. To clearly designate sections where a collaborator contributed significantly to the analysis, I use ?we?, ?a collaborator?, or ?they?. ii Acknowledgments No one achieves anything alone. I have been surrounded by supportive people which was not only helpful, but also necessary for me to attain a PhD. First and foremost, I want to thank my parents, Amy Feinberg Sheppard and Paul Sheppard, for always encouraging, supporting, and believing in me ? both in my education and in life in general. The same goes for my older siblings Mike and Brianna, who helped guide me through the early years. I owe a lot to my girlfriend Megan, who was beside me the entire way. She made the lows of my time in graduate school tolerable, and the highs immeasurably better. I?d like to thank my extended family and Megan?s family who have been on my side every step of the way. I?d also like to thank my friends, who entertained and distracted me enough to keep me sane during the more stressful moments. I want to acknowledge my graduate classmates, who were always welcoming and made everything from classes to quals easier. I owe a lot to my advisors, Drake Deming and Avi Mandell, both of who consistently advocated for me. They helped guide my research, coauthored my papers, and connected me to other researchers outside of UMD. In particular, Avi spent countless hours reading over my drafts, video chatting with me, helping me develop code, and mentoring me in general, and for that I?m grateful. iii Thanks to my thesis committee for helping me refine my research and for their instrumental feedback on my dissertation. I want to acknowledge my collaborators ? especially Nikolay, Madhu, Luis, and Sid ? who were great to work with (both as scientists and people). Special thanks to Cole Miller for long discussions on Bayesian statistics that were indispensable to my research. iv Table of Contents Preface ii Acknowledgements iii Table of Contents v List of Tables viii List of Figures ix List of Abbreviations xi Chapter 1: Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Atmospheric Retrievals . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Hot Jupiter Climates . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Hot Jupiter Composition and Chemistry . . . . . . . . . . . . . . . . 19 1.5 Earth-sized Exoplanet Atmospheres . . . . . . . . . . . . . . . . . . . 24 1.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Chapter 2: Investigating a Potential Terrestrial Atmosphere in the L98-59 Multi-planet System 31 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.3 Data Preprocessing and Analysis . . . . . . . . . . . . . . . . . . . . 36 2.4 Light Curve Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.4.1 Modeling the Light Curves . . . . . . . . . . . . . . . . . . . . 49 2.4.2 White light Results . . . . . . . . . . . . . . . . . . . . . . . . 57 2.4.3 Transit Spectra Derivation . . . . . . . . . . . . . . . . . . . . 60 2.5 Exploratory Analysis of Potential Atmospheres . . . . . . . . . . . . . 67 2.5.1 Likelihood and Characteristics of a Hydrogen-rich Atmosphere on Rocky Planet L9859c . . . . . . . . . . . . . . . . . . . . . 68 2.5.2 Discussion and Impact of Stellar Activity . . . . . . . . . . . . 72 Chapter 3: A Metal-rich Atmosphere for the Inflated Hot Jupiter HAT-P-41b 79 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 v 3.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.3.1 HST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.3.2 Spitzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.3.3 Photometric Monitoring Observations . . . . . . . . . . . . . . 86 3.4 Stellar Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.4.1 Analysis of Stellar Variability . . . . . . . . . . . . . . . . . . 88 3.4.2 Stellar Parameters . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.5.1 STIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.5.2 WFC3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.5.3 Spitzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.6 Atmospheric Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.6.1 PLATON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.6.2 AURA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.6.3 A note on metallicity and C/O . . . . . . . . . . . . . . . . . 115 3.7 PLATON Retrieval Analysis . . . . . . . . . . . . . . . . . . . . . . . 118 3.7.1 Fiducial Model . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.7.2 More Complex Models . . . . . . . . . . . . . . . . . . . . . . 124 3.7.3 Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.8 Results for the Favored PLATON Model . . . . . . . . . . . . . . . . 143 3.8.1 Summary of Retrieved Parameters . . . . . . . . . . . . . . . 143 3.8.2 Evidence of Water and Optical-Wavelength Absorbers . . . . . 146 3.9 AURA Retrieval Analysis and Results . . . . . . . . . . . . . . . . . 148 3.9.1 Evidence of Water and Optical-Wavelength Absorbers . . . . . 149 3.9.2 Possible offsets in the data . . . . . . . . . . . . . . . . . . . . 155 3.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 3.10.1 Comparison Between Retrieval Methods . . . . . . . . . . . . 159 3.10.2 Comparison to Interior Modeling Metallicity Constraints . . . 167 3.10.3 Implications for Planet Formation . . . . . . . . . . . . . . . . 168 3.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 3.12 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Chapter 4: Constraining the Dayside Thermal Structure of Hot Jupiters from Secondary Eclipse Observations 174 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.4 HST Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.4.1 Light Curve Analysis . . . . . . . . . . . . . . . . . . . . . . . 180 4.5 Spitzer Re-analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.6 WASP-18b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 4.6.1 Atmospheric Retrieval . . . . . . . . . . . . . . . . . . . . . . 189 4.6.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 4.6.3 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 4.7 WASP-19b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 vi 4.7.1 Atmospheric Retrieval . . . . . . . . . . . . . . . . . . . . . . 201 4.7.2 WASP-19b Results . . . . . . . . . . . . . . . . . . . . . . . . 202 Chapter 5: Summary and Future Work 209 Appendix A: Chapter 3 Supplementary Material 219 A.1 L9859b White Light Curves . . . . . . . . . . . . . . . . . . . . . . . 219 A.1.1 MCMC Validation Figures . . . . . . . . . . . . . . . . . . . . 221 A.2 L9859 Transit HST Spectrophotometric Light Curve Fits . . . . . . . 223 A.3 Red Noise Diagnostic Figures . . . . . . . . . . . . . . . . . . . . . . 228 Appendix B: Chapter 4 Supplementary Material 233 B.1 HAT-P-41b Transit HST Spectrophotometric Light Curve Fits . . . . 233 Appendix C: Facilities and Software 238 Bibliography 240 vii List of Tables 2.1 L9859 Transit Observation Details . . . . . . . . . . . . . . . . . . . . 37 2.2 L9859 Stellar and Planetary Properties . . . . . . . . . . . . . . . . . 47 2.3 L9859 Planets WFC3 White light Curve-Derived Transit Parameters 57 2.4 L9859 Transmission Spectra . . . . . . . . . . . . . . . . . . . . . . . 61 2.5 L9859c Model Priors . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.1 AIT Photometric Observations of HAT-P-41 . . . . . . . . . . . . . . 88 3.2 HAT-P-41 Photometry Results . . . . . . . . . . . . . . . . . . . . . 89 3.3 System Parameters for HAT-P-41 . . . . . . . . . . . . . . . . . . . . 91 3.4 HAT-P-41 Host Star Elemental Abundances . . . . . . . . . . . . . . 92 3.5 Transit Parameters for HAT-P-41b . . . . . . . . . . . . . . . . . . . 96 3.6 HAT-P-41b Transit Spectrum . . . . . . . . . . . . . . . . . . . . . . 97 3.7 HAT-P-41b Spitzer Transit Analysis Results . . . . . . . . . . . . . . 106 3.8 AURA Retrieval Parameters and Priors . . . . . . . . . . . . . . . . . 116 3.9 PLATON Fiducial Model Priors . . . . . . . . . . . . . . . . . . . . . 119 3.10 PLATON More Complicated Model Priors . . . . . . . . . . . . . . . 126 3.11 PLATON Retrieval Results Across Model Assumptions . . . . . . . . 140 3.12 PLATON Species Detection Evidences . . . . . . . . . . . . . . . . . 147 3.13 AURA Retrieval Results Across Model Assumptions . . . . . . . . . . 151 4.1 WASP-18b Thermal Emission Spectrum . . . . . . . . . . . . . . . . 190 viii List of Figures 1.1 Exoplanet Detections . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Transit and Eclipse Geometry . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Summary of Atmospheric Retrieval Process . . . . . . . . . . . . . . . 10 1.4 Cross-section Estimates for Hot Jupiters . . . . . . . . . . . . . . . . 15 1.5 Opacity Estimates for Ultrahot Jupiters . . . . . . . . . . . . . . . . 18 1.6 Opacity Estimates for Typical Transiting Hot Jupiters . . . . . . . . 21 1.7 Inferred Exoplanet Mass-Atmospheric Metallicty Relationship . . . . 23 1.8 Opacity Estimates for Super-Earth Transits . . . . . . . . . . . . . . 28 2.1 Data Preprocessing Flow Chart . . . . . . . . . . . . . . . . . . . . . 38 2.2 Example WFC3 IR exposure at different stages of data processing. . . 39 2.3 Example G141 Wavelength Calibrations . . . . . . . . . . . . . . . . 45 2.4 L9859c and L9859b visit 03 white light curves . . . . . . . . . . . . . 52 2.5 L9859c White Light Curve Corner Plot . . . . . . . . . . . . . . . . . 55 2.6 L9859c Transit Spectral Light Curves for visit 00 . . . . . . . . . . . 58 2.7 L9859b Weighted Transit Spectrum . . . . . . . . . . . . . . . . . . . 60 2.8 L9859c Derived Transit Spectrum . . . . . . . . . . . . . . . . . . . . 62 2.9 L9859c red noise diagnostic figure: Bin RMS Analysis . . . . . . . . . 63 2.10 L9859c red noise diagnostic figure: Autocorrelation Function . . . . . 64 2.11 L9859c Retrieval Results . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.12 L9859 Star Spot Retrievals . . . . . . . . . . . . . . . . . . . . . . . . 76 3.1 Transit White Light Curve . . . . . . . . . . . . . . . . . . . . . . . . 100 3.2 WFC3 Correlated Noise Analysis . . . . . . . . . . . . . . . . . . . . 103 3.3 Spitzer -derived a/Rstar-inclination Joint Posterior . . . . . . . . . . . 106 3.4 Spitzer Transit Light Curves and Correlated Noise Analyses . . . . . 108 3.5 PLATON Fiducial Model Median Retrieved Spectrum . . . . . . . . . 120 3.6 PLATON Fiducial Model Retrieved Corner Plot . . . . . . . . . . . . 122 3.7 PLATON Partial Cloud and Parametric Scattering Model Retrieved Corner Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.8 PLATON Partial Cloud and Mie Scattering Model Retrieved Corner Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.9 Transit PLATON Marginalized Posterior Distributions for Metallic- ity, Temperature, and C/O . . . . . . . . . . . . . . . . . . . . . . . . 139 3.10 PLATON Favored Model Median Retrieved Spectrum . . . . . . . . . 144 3.11 PLATON Favored Model Retrieved Corner Plot . . . . . . . . . . . . 145 ix 3.12 AURA Favored Model Median Retrieved Spectrum . . . . . . . . . . 150 3.13 AURA Favored Model Retrieved Corner Plot . . . . . . . . . . . . . . 152 3.14 AURA Retrieval Spectral Impact of Chemical Species . . . . . . . . . 153 3.15 AURA Simple Model Corner Plot . . . . . . . . . . . . . . . . . . . . 154 3.16 AURA Offset Models Median Retrieved Spectra . . . . . . . . . . . . 158 3.17 Impact of Chemical Equilibrium Assumption on Abundance Profiles . 162 3.18 Comparison of Median Retrieved 0.3?10?m Spectrum Between All Three Retrieval Methods . . . . . . . . . . . . . . . . . . . . . . . . 165 4.1 WASP-19b Eclipse Processed Exposures . . . . . . . . . . . . . . . . 180 4.2 WASP-18b Eclipse Light Curve Detrending Examples . . . . . . . . . 182 4.3 WASP-19b Eclipse Light Curve Detrending . . . . . . . . . . . . . . . 184 4.4 WASP-18b WFC3 Eclipse Spectra for All 4 Visits . . . . . . . . . . . 185 4.5 WASP-19b Eclipse Correlated Noise Analysis . . . . . . . . . . . . . 188 4.6 WASP-18b Median Retrieved Eclipse Spectrum and T-P Profile . . . 192 4.7 WASP-18b Eclipse Retrieved Corner Plot . . . . . . . . . . . . . . . . 195 4.8 WASP-18b JWST Prediction . . . . . . . . . . . . . . . . . . . . . . 201 4.9 WASP-19b Eclipse Retrieval Results . . . . . . . . . . . . . . . . . . 204 4.10 WASP-19b Corner Plot . . . . . . . . . . . . . . . . . . . . . . . . . . 206 5.1 Sample of this Dissertation in Context . . . . . . . . . . . . . . . . . 210 5.2 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 A.1 Additional L9859b White Light Curves . . . . . . . . . . . . . . . . . 220 A.2 L9859bc White Light Curve MCMC Fit . . . . . . . . . . . . . . . . 221 A.3 L9859c Transit MCMC Full Corner Plot . . . . . . . . . . . . . . . . 222 A.4 L9859c Transit MCMC Proof of Convergence . . . . . . . . . . . . . . 223 A.5 L9859b Transit Spectral Light Curves for visit 00 . . . . . . . . . . . 224 A.6 L9859b Transit Spectral Light Curves for visit 01 . . . . . . . . . . . 225 A.7 L9859b Transit Spectral Light Curves for visit 02 . . . . . . . . . . . 226 A.8 L9859b Transit Spectral Light Curves for visit 03 . . . . . . . . . . . 227 A.9 L9859b Visit 00 Red Noise Diagnostic Figures . . . . . . . . . . . . . 229 A.10 L9859b Visit 01 Red Noise Diagnostic Figures . . . . . . . . . . . . . 230 A.11 L9859b Visit 02 Red Noise Diagnostic Figures . . . . . . . . . . . . . 231 A.12 L9859b Visit 03 Red Noise Diagnostic Figures . . . . . . . . . . . . . 232 B.1 HAT-P-41b Transit Spectral Light Curves for STIS G430L, Visit 83 . 234 B.2 HAT-P-41b Transit Spectral Light Curves for STIS G430L, Visit 84 . 235 B.3 HAT-P-41b Transit Spectral Light Curves for STIS G750L . . . . . . 236 B.4 HAT-P-41b Transit Spectral Light Curves for WFC3 G141 . . . . . . 237 x List of Abbreviations Astronomical Symbols and Units: AU Astronomical Unit (1.496 ? 1011 m) M? Earth mass (5.972 ? 1024 kg) MJupiter Jupiter mass (1.899 ? 1027 kg) M Solar mass (1.988 ? 1030 kg) R? Earth radius (6.378 ? 106 m) R 7Jupiter Jupiter radius (7.149 ? 10 m) R Solar radius (6.957 ? 108 m) Chemical Symbols and Formulae: AlO Aluminum Oxide Ca Calcium C/H Carbon-to-Hydrogen ratio CH4 Methane CO Carbon Monoxide CO2 Carbon Dioxide C/O Carbon-to-Oxygen ratio Fe Iron FeH Iron Hydride H? Negative Hydrogen Ion H2 Molecular Hydrogen H2O Water He Helium K Potassium MgSiO3 Pervoskite Na Sodium Ni Nickel xi O/H Oxygen-to-Hydrogen ratio SiO Silicon Monoxide Ti Titanium (chemical symbol) TiO Titanium Oxide VO Vanadium Oxide Statistical Symbols: U(min, max) Uniform distribution N (mean, width) Normal distribution LU(min, max) Log-uniform distribution Z Bayesian Evidence O Odds Ratio BF Bayes Factor Acronyms: AIC Akaike Information Criterion BIC Bayesian Information Criterion ESPs Earth-sized Planets GO General Observer HAT Hungarian-made Automated Telescope HST Hubble Space Telescope HZ Habitable Zone IR Infrared IRAC Infrared Array Camera JWST James Webb Space Telescope KELT Kilodegree Extremely Little Telescope LD Limb-Darkening LTE Local Thermodynamic Equilibrium MAST Mikulski Archive for Space Science MCMC Markov Chain Monte Carlo xii NASA National Aeronautics and Space Administration NICMOS Near Infrared Camera and Multi-Object Spectrometer NIR Near Infrared NIRISS Near InfraRed Imager and Slitless Spectrograph NIRSpec Near InfraRed Spectrograph NUV Near Ultraviolet OOT Out-Of-Transit PanCET Panchromatic Comparative Exoplanet Treasury PASP Publications of the Astronomical Society of the Pacific PI Principal Investigator PLATON PLanetary Atmospheric Tool for Observer Noobs RMS Root-mean-square, or standard deviation S/N Signal-to-Noise SNR Signal-to-Noise Ratio SOSS Single Object Slitless Spectroscopy STIS Space Telescope Imaging Spectrograph STScI Space Telescope Science Institute TESS Transiting Exoplanet Survey Satellite TIC TESS Input Catalog TICv8 TESS Input Catalog version 8 T-P Temperature-Pressure TRAPPIST TRAnsiting Planets and PlanetesImals Small Telescope TRES Tillinghast Reflector Echelle Spectrograph TSM Transmission Spectroscopy Metric UMD University of Maryland UV ultraviolet VLT Very Large Telescope WASP Wide-Angle Search for Planets WFC3 Wide Field Camera 3 UVIS WFC3/Ultraviolet and Visible Light Z Metallicity xiii Chapter 1: Introduction Atmospheres are windows into their host planets? characteristics. They con- tain measurable information on their planets? chemistry and climate, which directly informs atmospheric processes and can even provide insight into the formation and evolution of a planet. Understanding these physics is an interesting endeavor, since planets are more diverse and not as ?neat? as stars. We do not have to look further than the Solar System to see that there is no obvious main sequence for planetary atmospheres. Despite similar radii, similar orbits, and the exact same formation conditions (in terms of host star and protoplanetary disk), Earth (N2 and O2) and Venus (CO2) evolved to have two wildly different atmospheres. The rest of the Solar System also exhibits astonishing diversity, with cold ice giants, hydrogen dominated gas giants, and atmosphere-free terrestrial planets. Fortunately, thanks to observing missions such as Kepler and TESS, we have access to thousands of test cases in the form of planets outside of the Solar System (extrasolar planets, or exoplanets). The bulk properties of exoplanets prove to be much more varied than the Solar System, allowing the ability to test atmospheric models over a range of temperatures, radii, and host star properties. By modeling and observing exoplanet atmospheres, we can better understand both atmospheric processes and planetary diversity, and place the 1 Solar System in a greater context to understand how unique it is, if at all. 1.1 Background The first step in characterizing the atmosphere of a planet is detecting the planet. Figure 1.1 gives context for the extreme increase in the detections in the last few years (left panel), as well as a sample of roughly characterizable planets (right panel). These planets are all in the Milky way, typically on the order of 100 parsecs away. This makes them close enough to observe, but too far away to spatially resolve from their host star. Atmospheres are observable, but only indirectly, primarily utilizing two detection methods. In transit detections (used by Kepler and the Transiting Exoplanet Survey Satellite, TESS), the light from stars in a wide angle of sky is monitored for a long time, and periodic dips indicate that a planet is periodically orbiting in front of a star and blocking a small fraction of light. The size of the dips reveal, among other properties, the radius of the planet relative to the star. Larger planets with smaller orbits and smaller host stars are most easily observed in transit. Radial velocity (RV) detections (e.g, High Accuracy Radial velocity Planet Searcher HARPS) monitor wide angles of the sky for star ?wiggles? (i.e, doppler-shifted spectra), which are indicative of the star orbiting a joint planet- star center of mass. RV measurements provide the mass of the planet. High mass planets with low mass host stars are more easily observed in RV. The typically path to atmospheric characterization of a planet is: detected by transit (radius), followed up with RV measurements (mass), which combined provide density, gravity and 2 Figure 1.1: Exoplanet detection information from the NASA Exoplanet Archive. Left: Cumulative number of confirmed planets each year. Right: Sample of planets with both mass and radius measured.a ahttps://exoplanetarchive.ipac.caltech.edu/ amenability to atmospheric characterization. We then follow up promising planets with transit or eclipse spectroscopy. The infographic in Figure 1.2, taken from Kreidberg (2018), helps demonstrate the ge- ometry of this process. Given the distances involved, we observe the combined planet-star light as a single point. The idea behind transit and eclipse spectroscopy is to use relative, wavelength-dependent changes in the combined planet-star light to indirectly infer the spectrum of the planet. Primary transit occurs when a planet passes in front of its host star. If a solid, atmosphere-free planet passes in front of a star, it will be equally opaque at every wavelength (since solids block all light in the NUV-IR range typical of observations) and so the planet will look the same size regardless of wavelength. However, if it has a gaseous atmosphere, that gas will interact with light in different ways depending on its composition. Therefore, the atmosphere can then leave its imprint on light that passes through it. Depending on its composition and physical properties, different amount of lights will be blocked 3 Figure 1.2: Transit and Eclipse Geometry, from Kreidberg (2018) at varying heights for different wavelengths. Since atomic and molecular gaseous species have different spectral signatures, we can relate the changes in depth with specific atoms/molecules. An atmosphere of all water will look bigger at 1.4?m, where water is opaque, than at 0.5?m, where it has a relatively low opacity. The amount of light blocked is known as a transit depth, (Rp/R 2 s) . We can match observed transit depth variation with wavelength with spectral signatures to in- fer composition and physics in the atmosphere. Transit spectroscopy measures the limb of the atmosphere, otherwise known as the day-night terminator, and typically probes high in the atmosphere at pressures around 1 mbar. In eclipse spectroscopy, we observe the system immediately before the planet goes behind the star, and thus get a total flux of the system. When the planet passes behind the star, we measure only stellar flux. We compare this flux ratio to total flux to infer emergent flux from the planet itself. The change in flux corresponds to the composition of the atmosphere and, unlike transits, is also highly sensitive to the 4 thermal structure of the atmosphere. Eclipses probe the dayside of the atmosphere (i.e, the half of the planet facing the host star, which is often permanent since ? due to circularization ? many planets at orbits this close are inferred to be tidally locked), at pressures deeper in the atmosphere around 100 mbar. Phase curves, where entire orbits of an exoplanet are observed to better inform climate and atmospheric dynamics, are another popular characterization method. Space-based instruments are, with good reason, the most common way to per- form transit and eclipse spectroscopy. Though ground-spaced spectroscopy is useful, particularly for high-resolution cross-correlated spectra (Birkby, 2018), it faces an extra challenge of correcting for Earth?s time-variable atmosphere. In addition to avoiding this complication, space-based instruments are able to observe in wave- lengths at which Earth?s atmosphere is opaque (e.g, water bands are too difficult too observe from the ground), and they have much cooler thermal background noise, which is especially important for IR observations. Consequently, space-based instru- ments HST/WFC3 (1.1?1.7?m; near-IR), HST/STIS (0.3?0.9?m; near-UV-optical), and Spitzer IRAC (3.6?8?m; mid-IR) have been the most productive instruments for characterizing exoplanet atmospheres (e.g, Benneke & Seager, 2013; Deming et al., 2013; Mandell et al., 2013; Kreidberg et al., 2014b; Stevenson et al., 2014, among countless others). These instruments cover separate wavelength regimes which are able to probe different pressures, physics, and spectroscopically active chemical species. Com- bining observations from multiple instruments maximizes the spectral baseline and allows for the most complete characterization of exoplanet atmospheres (Nikolov 5 et al., 2014; Sing et al., 2016; Beatty et al., 2017; Mansfield et al., 2018; Chachan et al., 2019a). Further, combining optical and IR spectra can break potential de- generacies between molecular abundances and planetary properties (Griffith, 2014; Line & Parmentier, 2016; Welbanks & Madhusudhan, 2019). One must be careful combining data from different instruments since the absolute depth is typically less well constrained then the relative change in depth (i.e, the shape of an instruments spectrum). Systematic errors could potentially bias the depths of an instrument relative to other instruments, and that should be considered in analysis (Garhart et al., 2020). 1.2 Atmospheric Retrievals To derive atmospheric properties of an exoplanet from transmission or emission spectroscopy, we must solve the inverse problem of ?what chemical composition and physics produce the observed spectrum?? Atmospheric retrieval is the process of retrieving chemical and physical information about an atmosphere based on an observed spectrum (Irwin et al., 2008; Madhusudhan & Seager, 2009; Lee et al., 2012; Line et al., 2013; Benneke & Seager, 2013; Waldmann et al., 2015b). The two primary components of atmospheric retrieval are forward models (a spectrum generated given exactly known atmospheric properties) and a parameter estimation method (for a given model, which properties lead to the best fit to the data?). In total, the observed spectrum is compared to many simulated spectra (from forward models) in order to constrain the possible values of parameters of interest (such 6 as planet radius, temperature, or water abundance). This is not a novel process: Rodgers (2000) used atmospheric retrieval to analyze remote sensing data of Earth, and Irwin et al. (2008) similarly used atmospheric retrieval on Solar System planets. An infographic summary from a recent review paper (Madhusudhan, 2018) is shown in Figure 1.3. First, I summarize the forward modeling of exoplanet transit and eclipse spec- tra. Through both quantum mechanics and laboratory experiments, we understand how light interacts with many different molecules and atoms (Barber et al., 2006; Rothman et al., 2009, 2010; Tennyson & Yurchenko, 2012; Allard et al., 2019). This is important, since knowing the spectral signatures of species can allow us to learn about the composition of an atmosphere from an observed spectrum. Atoms generally interact via spectral lines with shapes dictated by energy level transitions (and various sources of broadening). Molecules typically form ?bands?, which are combinations of millions of electronic, vibrational, and rotational transi- tions (Tennyson & Sutcliffe, 1982). The shapes and magnitudes of these molecular bands or atomic lines constitute the spectral feature for a given species. The shape of these spectral features, for a given temperature and pressure, are constantly be- ing updated in libraries of line lists such as EXOMOL (Tennyson & Yurchenko, 2012; Tennyson et al., 2016; Tennyson & Yurchenko, 2018; Tennyson et al., 2020), HITEMP (Rothman et al., 2009), and HITRAN (Rothman et al., 2010). These li- braries are particularly relevant to exoplanet atmospheres since they include spectral features of species at higher temperatures. Included species are informed by chemistry (e.g, the carbon and hydrogen are 7 common elements, so H2, CH4, and C2H2 should be considered), as well as obser- vations of Solar System planet atmospheres (e.g, Hubbard et al., 2002), cool stellar atmospheres (e.g, Allard et al., 1996; Tsuji et al., 1996), and, importantly, simi- lar temperature brown dwarf atmospheres (e.g, Burrows & Sharp, 1999). Though hundreds of species are chemically expected, many are at such low abundances in chemical equilibrium, or have such low opacity in telescope wavelength bands, that they can be neglected in analyses (Kempton et al., 2017). This generally results in about 30 relevant species. Of special importance are water, CO, CO2, CH4, Na, K, TiO, and VO. I emphasize that these are all gaseous species. In an exoplanet atmospheric context, water always means water vapor. If the temperature was low enough (at a given pressure) for a species to condense into a cloud, it would interact with light in a much different, grayer (constant with wavelength) manner. Forward models utilize line lists for these species in order to derive the opacity (effective area of an interaction per unit mass of material) for each molecule and atom. Continuum opacity sources such as collision induced H2-He absorption, H- bound-bound and bound-free interactions, and Rayleigh scattering are also generally included as important opacity sources. Finally, opacity due to aerosols is consid- ered. This is broken down into two groups: clouds (similar to rain clouds on Earth; condensates which form due to thermochemical equilibrium) and hazes (small, com- plex particles likely created by the combination of photochemical byproducts of molecules). Cloud and haze opacities are often treated as a flat line (at a cloud top pressure) and a scaled version of Rayleigh scattering, respectively, since a full microphysical treatment is computationally expensive. 8 With opacity sources well understood, it is possible to model a transit (or emission spectrum). In modeling exoplanet atmospheres, a useful philosophy to have is summarized by a quote famous statistician George Box: ?All models are wrong, but some are useful.? A completely self-consistent 3D model that simulta- neously accounts for disequilibrium photochemistry would be a relatively accurate atmospheric model, but it is prohibitively computationally expensive to be use- ful for retrievals, which often require hundreds of thousands of model evaluations. Accordingly, approximations, such as 1D atmospheres (the temperature and abun- dances at each longitude and latitude are uniform, only pressure (height) matters) are necessary. Fortunately, we can check approximations both against full treat- ments and better quality Solar System data. For example, the atmospheric physics used for exoplanets are derived from stellar atmospheres, and have been validated on stars (Allard et al., 1996; Tsuji et al., 1996), brown dwarfs (Burrows & Sharp, 1999) and Solar System planets (Hubbard et al., 2002). Further, the exoplanet at- mosphere modeling community is good at self-regulating. Madhusudhan & Seager (2009) validated that their temperature-pressure (T-P) profile parameterization is able to fit the T-P profile of each Solar System planet. Both Burrows et al. (2010) and Fortney et al. (2010) compared 1D models to 3D general circulation models (GCMs) and found tolerable agreement. Blecic et al. (2017) similarly benchmarked the performance of 1D models to more updated 3D GCM models for several data resolutions. Line et al. (2013) found consistent results between their five-parameter T-P profile to the level-by-level approach used in Earth analyses. As a final example, Kempton et al. (2017) showed that a transit spectrum generated from an isother- 9 mal temperature profile sufficiently matched one generated from a self-consistent radiative-convective T-P profile. In summary, the bulk of the literature finds the error caused by computation- ally useful approximations to be generally negligible for transit and eclipse analyses as compared to current observational precision. Additionally, the physics used to predict model atmospheres successfully describes higher-quality observations from stars, brown dwarfs, and Solar System planet atmospheres. Figure 1.3: Summary of the Atmospheric Retrieval Process, from Madhusudhan (2018). Forward models as used in retrievals are summarized in the rightmost panel of Figure 1.3. They generally work as follows: assume a 1-D, plane-parallel atmosphere; set a T-P profile; set abundances at each T-P point for each important species (po- tentially using chemical equilibrium); calculate opacity of each species for a given temperature and pressure for each wavelength; determine physical height associated with each pressure by solving hydrostatic equilibrium (Eq 1.1; ? is the mean molec- ular weight of the atmosphere); perform an abundance-weighted sum and combine with Rayleigh scattering to get total opacity at each atmospheric layer. 10 dP ?GM ?mamuP = (1.1) dr r2 kT The final step is a radiative transfer calculation, where an incident ray from the star is traced through atmospheric layers and towards the observer at Earth. The optical depth experienced by a ray of a given impact parameter is determined by the ?width? and total opacity of each layer it passes through. The amount of light blocked by the atmosphere at each wavelength is determined by the optical depth experienced by rays at different impact parameters. The apparent size of the planet due to this blocked light gives the transit depth at that wavelength. This is formalized in Equation 1.2. D? is the transit depth at wavelength ?, ?? is the optical depth at wavelength ?. The transit depth at different wavelengths gives the modeled transit spectrum. ? ? r D? = (Rbottom/R 2 s) + 2 (1? e???) dr (1.2) R2Rbottom s A similar process is used to model eclipses, but instead of light blocked by the atmosphere now flux emitted by the planet at each wavelength (and stellar flux) is important. Additionally, the T-P profile is more important, as emitted light is more sensitive to temperature gradients (Kempton et al., 2018). Self-consistent models are computationally expensive, so profiles are often parameterized such that different profile slopes can be captured (Madhusudhan & Seager, 2009; Guillot, 2010; Line et al., 2012). For a given set of T-P parameters, a T-P profile is generated. Again, abundances are set at each pressure layer, and a height is assigned to each 11 pressure via hydrostatic equilibrium. Total opacity is similarly determined for each wavelength, and the emergent planetary flux is determined by assuming the source function is the Planck function (i.e, neglect scattering, which is too computationally expensive to model) and summing contributions over each pressure layer for entire planetary disk. Stellar flux is often determined from a stellar spectral library. The exact eclipse depth (D?) is given in Equation 1.4 and depends on the planetary flux (Fp,?) calculated in Equation 1.3 (Zhang et al., 2020a). B?(??) is the Planck function (B) at optical depth ? , and it is integrated over both optical depth and the viewing angle (?). ( )2 Rp,? Fp,? D? = (1.3) Rs Fs,? ? ? ? 1 ??? F = B ?p,? ?(??)e d? d?? (1.4) 0 0 The parameter estimation component is best accomplished with a Bayesian sampler. As temperature, composition, and other physical parameters such as plan- etary mass vary, so will the output spectrum, but in a non-linear way; tempera- tures impact opacities, as well as abundances (e.g, condensation). Gaseous species abundances impact both optical depth and scale height. Further, there are many degeneracies (Seager & Sasselov, 2000; Madhusudhan & Seager, 2009; Line & Par- mentier, 2016; Welbanks & Madhusudhan, 2019). Accordingly, Bayesian inference is a necessary tool to properly retrieve values and uncertainties of important param- eters, such as temperature and water abundance. This involves setting reasonable 12 priors on parameters of interest, and using an intelligent sampler, such as Markov Chain Monte Carlo (MCMC Goodman & Weare, 2010) or nested sampling (Skilling, 2004), to fully explore prior parameter space and finely sample the posterior near its peak values. Bayesian inference can provide both credible intervals for parameters and the Bayesian evidence, which is useful for model comparison. As an example, the ratio of Bayesian evidences of a model with water opacity to one without water opacity gives the strength of a water detection. 1.3 Hot Jupiter Climates Due to their high temperatures and large radii, planets in the hot Jupiter archetype are the most amenable to characterization via emission spectroscopy (eclipse signal ? R2p ? Tp). The first thermal emission from an exoplanet was de- tected by Deming et al. (2005) and Charbonneau et al. (2005), and since then WFC3 and Spitzer data have been jointly used to constrain the dayside atmosphere of tens of hot Jupiters. Early theories on hot Jupiter atmospheres were generally driven by two connections: extrapolating models of brown dwarf atmospheres to lower tem- peratures and adding stellar irradiation, or assuming they are higher temperature analogues of Solar System gas giant atmospheres. These early predictions were iter- atively tested against observations, and more complicated physics would be invoked if the models could not describe the data. This process contextualizes the current state of the field. Hot Jupiters typically lie on orbits which are 20% the orbital distance of Mer- 13 cury, making them both tidally-locked and subject to intense irradiation. This strongly influences their thermal structure, which is also sensitive to many other fac- tors (such as opacity and advection) and are complicated to model (Fortney et al., 2021). There is an additional sub-branch of hot Jupiters known as ultrahot Jupiters (typically Teq >2300 K). For context on important gaseous species, I show emission spectrum cross sections at typical hot Jupiter temperature 2000 K in Figure 1.4. A cross-section is a temperature, pressure, and wavelength-dependent ?effective area? of a particular photon-particle interaction. It is a measure of the likelihood of a photon interacting with a single particle via that interaction. The cross sections are derived from the model presented in Gandhi & Madhusudhan (2018), though the figure is from private communications. 14 Figure 1.4: Relevant opacity sources in typical hot Jupiter emission spectroscopy. Taken from Gandhi & Madhusudhan (2018), this only covers WFC3 wavelength range. Note these are cross-sections, not opacities, meaning it does not take abundance into account. For example, H2O and NH3 have similar cross sections, but water is almost always at a greater abundance and thus typically dominates opacity. Recently, GCMs ? which couple fluid dynamics and radiative transfer ? have been used to estimate the self-consistent 3-D structure of hot Jupiters via the meteorology primitive equations (Lewis et al., 2014; Kataria et al., 2015; Carone et al., 2020). These revealed that an eastward-shifted hotspot is the norm, and that large day-night temperature contrasts are typical. This is consistent with the Cowan & Agol (2011) inference of poor heat recirculation efficiency in hot Jupiters based on high dayside temperatures. GCMs are computationally expensive, and so 1-D temperature-pressure (T-P) profiles are often used in practice, especially with atmospheric retrievals which require many model evaluations. These rely on the decent approximation that the integrated dayside hemisphere can be described by 15 a single T-P profile, which probes the change in temperature with height. They are typically parameterized to capture three profile shapes: isothermal, decreasing (temperature decreases with height), or a thermal inversion. Of particular interest are stratospheric thermal inversions, where temperature increases with height. Hubeny et al. (2003) hypothesized that, given that TiO and VO are observed in brown dwarfs, and given their relatively high opacity in the optical at low pressures, that they would absorb the intense stellar irradiation more quickly than IR opacity could radiate, causing temperature to increase to maintain radiative equilibrium. Fortney et al. (2008) built off this by classifying planets by both temperature and TiO/VO presence, and predicted TiO and VO to be gaseous and spectroscopically active above 1600K. However, this classification is complicated by the non-ubiquity of TiO/VO evidence and the many mechanisms to take it out of the atmosphere. It could fall victim to a cold trap (vertical (Spiegel et al., 2009) or nightside (Parmentier et al., 2013)) for temperatures roughly below 1900K. It could be photodissociated by the UV light from an active host star (Knutson et al., 2010). On the opposite end, it is predicted to be thermally dissociated at temperatures above 3200 K (Lothringer et al., 2018). This leaves a relatively small ?goldilocks zone? for TiO and VO to be spectroscopically active. Lothringer et al. (2018) predicted that regardless of TiO/VO, metallic atom and ion opacity would be enough in planets above 3200 K to drive a thermal inversion. Though this would be more akin to a thermospheric inversion (which is typical, though not often probed in exoplanet transits) being pushed down to observable pressures rather than a stratospheric inversion. Finally, Mollie?re et al. (2015) hypothesized that 16 in atmospheres with bulk carbon-to-oxygen ratios (C/O) close to one, the water abundance would drop resulting in limited IR opacity, and so the lack of efficient cooling would drive an inversion. Another interesting question relating to hot Jupiter climates is the paucity of water vapor in emission spectroscopy. Water has been detected as an absorption feature several times, typically in cooler hot Jupiters (Teq < 1500 K) (Crouzet et al., 2014; Kreidberg et al., 2014b; Line & Parmentier, 2016; Beatty et al., 2017). How- ever, it has only been tenatively observed in emission in two hot Jupiters (Haynes et al., 2015; Evans et al., 2017). The thermal structure dictates the type of feature: decreasing thermal profiles appear as absorption dips (since the higher opacity water band is sampling Planck function higher in the atmosphere where it is cooler), in- versions appear as emission bumps, and isothermal atmospheres appear flat. Given that ultra-hot Jupiters are expected to host thermal inversions, and that water is well mixed in hot Jupiter atmospheres (Madhusudhan, 2019a), emission features were predicted to be more common. In completely featureless spectra explanations range from the precision being too low (Wilkins et al., 2014) to an isothermal region of the atmosphere being sampled (Nikolov et al., 2018). When water is not seen in conjunction with a CO detection, a high C/O is inferred (Stevenson et al., 2014), since oxygen would be sequestered in CO and a limited amount would be available for water (Madhusudhan, 2012; Moses et al., 2013). More recently, Lothringer et al. (2018) and Parmentier et al. (2018) hypothesized that, in ultrahot Juipters, molec- ular dissociation and H- opacity become significant. At high enough temperatures, H- opacity can mask water opacity and water can be thermally dissociated, resulting 17 in water features being muted in both the WFC3 and Spitzer bandpasses. Addi- tionally, these changes in opacity impact the thermal structure of the atmosphere, cuasing water to only be spectroscopically active in the deep, isothermal layers of the atmosphere. The net result is a featureless blackbody spectra with only CO being a potential feature in the MIR since its strong molecular bonds prevent it from being thermally dissociated. For context, I share Figure 4 from Parmentier et al. (2018), which details relevant opacity sources in emission in the WFC3-Spitzer wavelength range at a typical ultra-hot Jupiter temperature (3100 K). Figure 1.5: Relevant opacity sources in typical ultrahot Jupiter emission spectroscopy. Taken from Parmentier et al. (2018). 18 1.4 Hot Jupiter Composition and Chemistry Bulk density measurements (mass and radius of entire planet, not just the atmosphere) reveal that hot Jupiters are H2/He-dominated, similar to Jupiter (Lis- sauer & Stevenson, 2007). This matches predictions of core-accretion formation theory, where a rocky planetessimal core accretes enough mass to trigger runaway gas accretion (Pollack et al., 1996). This formation model predicts a bulk mass- metallicity trend amongst planets (similar to the solar system; Mordasini, 2014), and that prediction was born out in observations (Thorngren et al., 2016). The for- mation mechanism of hot Jupiters is not perfectly known. The three most prominent theories are in-situ formation (form at their current location), disk-migration (form far from host star but migrate inwards through planetary disk), and disk-free migra- tion (form far away, become perturbed onto elliptical orbit after disk dissipates and migrate inward via tidal dissipation) (Fortney et al., 2021). In the core-accretion paradigm, proto-hot Juptiers cannot accrete enough gas to reach observed masses and radii ? the feeding zone is too small. Inward migration is more likely, and its possible both formation mechanisms are common. For example, some planets have orbits aligned with their host stars spin (naively expected for disk-migration), but others have a significant misalignment (naively expected for disk-free migra- tion). The formation history of a planet, though highly stochastic, still causally impacts the planet?s eventual atmosphere, and so the atmosphere may contain hints to formation history (Mordasini et al., 2016). Due to their formation mechanism, planets are thought to approximate the el- 19 emental abundances of their host stars. This is only to the first order though, since disk-migration and the evolution of the disk itself (e.g, snow-lines) impact the even- tual atmospheric properties of an accreting planet (O?berg et al., 2011; Madhusudhan et al., 2014a; Mordasini, 2014). On top of that, the relative gas/ice accretion, peb- ble accretion, and pebble drift all directly impact the atmospheric metallicity (ratio of ?metals?1 to hydrogen; typically O/H in planets? atmospheres) and C/O ratio (Mordasini et al., 2016; Madhusudhan, 2019a). In particular, pebble drift, which describes inward-drifting pebbles sublimating as they cross snow-lines and enriching the metallicity of gas, can allow for metal-enriched planets with a variety of C/O ratios (Booth et al., 2017). C/O ratios are a common parameterization of exoplanet atmospheres (Mad- husudhan, 2012; Moses et al., 2013). This is because 1) C and O are the two most common metals 2) most spectroscopically active species in the optical to MIR are C- or O-based and 3) the abundances profiles in chemical equilibrium differ drastically as C/O ratio approaches and crosses one. It is estimated in exoplanet atmospheres either from chemical equilibrium assumptions or directly if a C-bearing and O- bearing molecular abundance is determined. Though C/O ratios are often debated and there are very few definitive C-bearing molecule detections (not coincidentally because common molecules CO and CO2 have strongest cross-sections by photomet- ric Spitzer points), hot Jupiters are primed for abundance determination. Their temperatures are high enough to vaporize most species and to keep them aloft and 1Astronomers define metals as any element heavier than hydrogen and helium. This contrasts with the chemical/physical definition (an electrical conductor). For example, though carbon and oxygen are classified as non-metals in the periodic table, they are considered metals in an astro- nomical context. 20 well-mixed in the atmosphere. In fact, the relative coldness of the Solar System gas giants means that we know more about water abundance in planets hundreds of parsecs away than we do about those in the Solar System (though recent in-situ probes have helped even the playing field, such as Galileo (e.g, von Zahn et al., 1998; Owen et al., 1999) and Juno (e.g, McComas et al., 2017; Li et al., 2020)). Figure 1.6: Relevant opacity sources in typical hot Jupiter transit spectroscopy. Taken from Pinhas et al. (2018) (left) and Kreidberg (2018) (right). Note the Pinhas et al. (2018) values are cross-sections, meaning they do not take the abundance of each species into ac- count. The Kreidberg (2018) values are abundance-weighted opacities for a representative atmosphere, and cover optical wavelengths. Potentially important opacities TiO and VO are absent since they are expected to condense out of the atmosphere below 1600K. The cross-sections of relevant molecules for typical transiting exoplanet tem- peratures are shown in Figure 1.6, taken from Pinhas et al. (2018); Kreidberg (2018). Many of these species have been detected in hot Jupiter atmospheres: water (Fraine et al., 2014; Evans et al., 2016, and many more), Na (Charbonneau et al., 2002; Nikolov et al., 2014, etc.), K (e.g, Sing et al., 2011), TiO (Sedaghati et al., 2017, though contested), and CO2 (Morley et al., 2017). These detections can lead to abundance constraints, which ? under the assumption of that the atmospheres is in a state of chemical equilibrium ? can be used to estimate the amount of ele- 21 mental C or O (and thus C/O) in an atmosphere. Abundance constraints are much more reliable in multi-instrument spectra, since optical and IR spectra can break degeneracies between abundance and clouds, hazes, or the reference pressure (Line & Parmentier, 2016; Welbanks & Madhusudhan, 2019). Similar to eclipse spectroscopy, in which water features were not observed as frequently as predicted, WFC3 transit observations commonly found water features to be smaller in size than expected based on the slant path of stellar irradiation through the atmospheric limb (Deming et al., 2013; Stevenson, 2016; Fu et al., 2017). Water features have been found to typically cover only 2 scale heights instead of the 5?10 predicted for a saturated spectral feature in a clear atmosphere (Madhusudhan, 2019a). Though the cause was not clear, this effect was not necessarily unexpected, as early models predicted that atmospheric clouds or hazes could result in the muting of atomic and molecular gaseous absorption features (e.g, Brown, 2001; Fortney, 2005). Still, viable alternative explanations such as oxygen depletion (i.e, a high C/O ratio; Madhusudhan, 2012; Crouzet et al., 2014) or a high mean molecular weight (Line & Parmentier, 2016) are also plausible. Sing et al. (2016) presented ten optical-to-IR spectra to argue that hazes and clouds were responsible for the majority of muted features. However, the cause is still unclear. A recent population study by Welbanks et al. (2019) argued that although atoms Na and K are typically supersolar, water is generally depleted as compared to Solar System extrapolations, even when accounting for clouds (Fig- ure 1.7). Separately, Line & Parmentier (2016) demonstrated that, without optical data, there is a degeneracy between partial cloud coverage (i.e, clouds only covering 22 a fraction of the planet?s limb) and abundance. A high mean molecular weight can also describe the subdued feature sizes. Clouds (typically treated as grey opacity condensates at a fixed height) and hazes (which manifest as a steep slope increasing in the optical) are a unique problem since their opacity tends to mute or even hide gaseous features. Though there are many ideas on the make-up on these clouds and hazes (e.g, perovskite, corundum, or silicate clouds and hydrocarbon soot hazes in hot Jupiters), the microphysics is complicated and the source of clouds remains elusive (Kreidberg, 2018). These are often modeled parametrically when retrieving atmospheric properties (Zhang et al., 2019). Figure 1.7: Extrapolating the Solar System mass-atmospheric metallicity using several different tracers for metallicity, from Welbanks et al. (2019) (left) and Wakeford et al. (2017) (right). Welbanks et al. (2019) find supersolar Na and K, but a consistent relative underabundance of water. Wakeford et al. (2017) suggested exoplanets generally follow the Solar system trend. This is highly dependent not only on retrieval method, but also on molecular abundance-to-metallicity conversion method. Similar to bulk properties, a mass-atmospheric metallicity trend has been pro- posed for exoplanets (Kreidberg et al., 2014b; Wakeford et al., 2017; Mansfield et al., 2018). This would match what is seen in the Solar System planets and is predicted ? albeit with significant intrinsic scatter ? by formation models (Fortney et al., 2013). Figure 1.7 shows two recent observational investigations of this trend from 23 Welbanks et al. (2019) and Wakeford et al. (2017). While Welbanks et al. (2019) does a systematic study (re-deriving abundances for each planet with the same re- trieval methodology), there is much variation on abundance constraints between the two studies (e.g, the median HAT-P-11b abundance differs by a factor of 1250 be- tween the two plots). Additionally, as Heng (2018) points out, converting water to O/H to compare to solar is a pressure and temperature-dependent (e.g, there is not a single solar water abundance for which to compare), so derivation of metallicities may be flawed. Still, populating this plot (and cross-checking multiple methodolo- gies) is useful in determining if such a trend is real, and this is possible through transit spectroscopy. 1.5 Earth-sized Exoplanet Atmospheres Earth-sized planets (sometimes referred to as super-Earths or mini-Neptunes) are the most commonly discovered planet type over the last several years (Bean et al., 2021). Though often grouped together, there is a well studied radius valley that splits this planets into two categories: R>1.75R? (sub-neptunes) and R<1.75R? (Earth-sized planets) (Fulton et al., 2017). In this dissertation, I primarily discuss the Earth-sized planets. The most interesting Earth-sized planets (ESPs) are found around M-dwarfs, the most common star in the galaxy, because they provide the best opportunity for identifying a terrestrial atmosphere due to their small radii (Charbonneau et al., 2009; National Academies of Sciences & Medicine, 2018; Ben- neke et al., 2019b). 24 This archetype of planet is comparable to Earth in size and is the same order- of-magnitude mass, making them likely terrestrial. The key difference is that they orbit their (cooler) host stars at distances roughly 3% of Earth?s orbit. Despite this lower host star temperature, the orbital distance is small enough to make their equilibrium temperatures and incident stellar irradiation greater relative to Earth?s. Still, these are most direct gateway to biosignatures and understanding habitable planets, who are their cooler cousins (pun intended). In this planetary regime, we really only know about Earth and Venus. There is a question about how common each atmosphere type is: is Earth unique and Venus the norm, or vice versa, or neither? Or is the Mars ? with no significant atmosphere ? the norm? Understanding the range and degree of ubiquity of atmospheres for terrestrial planets contextualizes our Solar System. Unlike the massive hot Jupiters in Chapters 3 and 4, atmospheres are not definitively the norm for rocky planets. Looking at our own Solar System, only 50% of the rocky planets possess obvious, thick atmospheres (sorry Mars). Atmosphere detections are even rarer for ESPs (Pidhorodetska et al., 2021). This is partially due to atmospheric escape (Watson et al., 1981). In the core-accretion model, isolation mass cores form from the build up of solids into planetessimals (Safronov & Zvjagina, 1969; Wetherill & Stewart, 1993; Pollack et al., 1996). These planetessimals collide violently and do not accrete any primordial atmosphere if the isolation mass is not reached until disk gas has already dissipated. Even if an atmosphere was accreted, giant impacts (like the one that formed the moon) would likely remove primordial gas (Howe et al., 2020). However, Misener & Schlichting (2021) argue that super 25 Earths both form in the presence of disk gas and maintain residual H/He from their primordial atmosphere. This is because the initial removal of the majority of the primordial atmosphere decreases the cooling time scale of the remaining atmosphere enough for it to efficiently cool before being stripped. Further, this primordial H/He atmosphere would influence any secondary atmosphere (vulcanism/outgassing after H2 is stripped, or volatile deposits from comets (Swain et al., 2021; Mugnai et al., 2021)) and potentially lower the mean molecular weight, making the planet more amenable to atmospheric characterization. Additionally, (Howe et al., 2020) argues that an alternative formation mechanism ? pebble accretion ? would allow super Earth cores to assemble quickly enough to accrete gas before the disk dissipates. Impact erosion, the typically dominant atmosphere loss process, would be much less effective in the pebble accretion paradigm. A primordial H2-dominated atmosphere for an ESP is thus plausible and should not be dismissed a priori. More likely, however, are secondary atmospheres. Swain et al. (2021) argued that, on the terrestrial exoplanet GJ1132b, a secondary H2 atmosphere was possible via hydrogen dissolving in vulcanic magma, allow it to be stored, then later being released back into the atmosphere due to tidal heating. Observationally, Moran et al. (2018) found that a hydrogen-rich atmosphere with high altitude clouds can explain observations of several terrestrial planets in the TRAPPIST system (Gillon et al., 2017; de Wit et al., 2018). Expected signal size is another reason for the lack of a convincing atmospheric detection for an ESP. Atmospheres are most clearly detected via atomic or molec- ular absorption features; atmospheres may exist, but the signal size is such that 26 features are undetectable with the precision of current ground-based telescopes and HST WFC3 (e.g, Diamond-Lowe et al., 2018; Pidhorodetska et al., 2021). There is also a selection bias, since the majority of characterizable terrestrial planets orbit M-dwarfs, which emit a greater UV flux than Solar-type G stars. This intense UV irradiation makes low mean molecular weight, H2-dominated atmospheres ? which produce the largest features and are the easiest to detect ? unlikely. Additionally, the increased stellar variability of M-dwarfs ? and the fact that water absorption is clearly present in their spectra ? also complicates atmospheric detection (Rackham et al., 2018; Deming & Sheppard, 2017). High-altitude clouds may also mask at- mospheres by making spectra appear as featureless flat lines (Diamond-Lowe et al., 2018). Figure 1.8, taken from (Kempton et al., 2017), provides opacities for sev- eral species at 1000 K. This is above the typical equilibirum temperature of around 500 K, but it is a better approximation of important species opacities than the previous opacity figures. 27 Figure 1.8: Approximation of relevant opacity sources in typical Earth-sized planet (ESP) transit. This is Figure 4 from Kempton et al. (2017). GJ1214b (Berta et al., 2012), HD97658b (Kreidberg et al., 2014a), and the TRAPPIST system (de Wit et al., 2018) are ESPs which have all been observed with WFC3, and all display flat spectra. Though there is no molecular detection, this is at least informative since it can rule out a cloud-free, hydrogen dominated atmosphere. Swain et al. (2021) claimed a secondary H2 atmosphere with an HCN and CH4 detection in GJ1132b, but follow up data analysis by Mugnai et al. (2021) and Libby-Roberts et al. (2021) found a flat spectrum, consistent with earlier op- tical observations by Diamond-Lowe et al. (2018). Flat spectra are the norm for ESPs, since essentially the only easily detectable atmosphere is a clear, hydrogen- dominated one. Flatness could be caused by clouds, photochemical hazes, a high mean molecular weight secondary atmosphere (similar to Venus), even a giant spot or faculae on an active host star, which can act to cancel out spectral features (Rack- 28 ham et al., 2018). The ability of ESPs to host atmospheres ? and their detectability ? is an active question in the super-Earth sub-field. 1.6 Outline This dissertation takes a broad approach, analyzing both transit and emission spectroscopy in different sections of parameter space. This includes a range of plan- ets from cool, Earth-sized planets (T?500K, R=0.8R?) up to massive, ultra-hot Jupiters (T?2500K, M=10MJup). I emphasize clarity and sensitivity tests in data analysis and light curve fitting. I also emphasize properly contextualized statistics and how to combine data from different instruments. Deep dives into individual planets are important to properly account for uncertainty in model choice, to con- textualize how results depend on assumptions, and to accurately represent results. This dissertation explores how different modeling assumptions impact results, both within a single model paradigm and between paradigms. In Chapter 2, I derive and the transit spectra for 5 visits of two likely rocky ex- oplanets, L9859b and L9859c. I detail my custom analysis pipeline DEFLATE, which is the basis of all WFC3 data analysis in this thesis. I investigate the likelihood that the structure in the spectra are indicative of the first convincing rocky planet atmospheres, and if it is plausible for stellar activity to implant that structure onto a flat transmission spectrum. In Chapter 3, I perform a detailed multi-instrument, multi-retrieval transit analysis of the hot Jupiter HAT-P-41b. I explore dozens of retrieval assumptions and relate any disparities directly to ? sometimes equally 29 valid ? modeling choices. The analysis reveals a 5? water detection and a signifi- cantly super-stellar atmospheric metallicity in almost every single retrieval model. Finally, in Chapter 4, I analyze the thermal structure of two exoplanets, both on the boundary of potentially important chemistry in gravity-temperature parame- ter space. I find a thermal inversion for ultra-hot WASP-18b, water absorption in hot WASP-19b. I explore how both results provide insight into dominant physical processes on hot Jupiter atmospheres. 30 Chapter 2: Investigating a Potential Terrestrial Atmosphere in the L98-59 Multi-planet System 2.1 Introduction The effort to understand the atmospheres of terrestrial exoplanets is a major goal of exoplanet science and a priority for JWST. However, the total number of cur- rently characterizable Earth-sized exoplanets is in the single-digits (Pidhorodetska et al., 2021). Though there have been no convincing atmospheric detections, HST WFC3 observations of potentially rocky, Earth-size planets have been informative. Notably, they have ruled out cloud and haze-free hydrogen-rich atmospheres (de Wit et al., 2018), and shown that cloudy H2 rich atmospheres or heavier molecules (e.g, CO2 or H2O) could explain observed transit spectra (Moran et al., 2018; Wakeford et al., 2019; Mugnai et al., 2021; Libby-Roberts et al., 2021). L9859 is a small (R=0.31R ) and bright (Teff=3400K) M3V-dwarf roughly 10 parsecs away. TESS recently discovered it to be the host star in a multi-planet system, making it the second closest transiting multi-planet system to Earth (Kostov et al., 2019). Kostov et al. (2019) discovered three Earth-sized planets, and follow up HARPS RV observations (Cloutier et al., 2019) derived masses and concluded 31 that the two innermost planets ? L9859b (R=0.8R?, M<1M?, Teq=550K) and L9859c (R=1.35R?, M=2.4M?, Teq=470K) ? have bulk densities consistent with terrestrial planets. Their orbital periods are both on the order of a few days, placing them at an orbital distance of roughly 3% of Earth?s orbital distance. Given this close orbit, the planets receive much more irradiation than Earth, and are in the ?Venus Zone? (Pidhorodetska et al., 2021). Cloutier et al. (2019) calculated the atmospheric detection index (Kempton et al., 2018) as greater than the previous highest for a pre-TESS terrestrial planet, up to 1.6? that of GJ1132b. The expected signal size from two of the planets ? inner- and-smaller b, and outer-and-bigger c ? thus merited follow up observations and 5 HST WFC3 transits were awarded to a collaborator, PI Tom Barclay. Pidhorodet- ska et al. (2021) highlighted that HST WFC3 is the most favorable instrument to conduct transit spectroscopy of the L9859 in the near-term. The atmospheres are worth investigating. Pidhorodetska et al. (2021) further classified potentially characterizable atmospheric scenarios for L9859c. Of course, it could have no atmosphere ? or at least a non-detectable one. Pidhorodetska et al. (2021) showed that 1 transit of L9859c and 4 transits for L9859b with HST WFC3 are informative: the precision could potentially detect a H2-dominated atmosphere, and a water feature in a steam atmosphere, respectively. They also put forward runaway greenhouse (CO2 dominated, similar to Venus; Kasting, 1988), and O2- dessicated as plausible atmospheres, though neither would be detectable with HST WFC3?s precision. Further, TESS monitoring showed no obvious stellar flares or ac- tivity (Kostov et al., 2019), potentially simplifying analysis (Wakeford et al., 2019). 32 However, see Section 2.5.2 for full stellar activity discussion. Though Kostov et al. (2019) postulated that an H-rich atmosphere akin to larger gas giant planets was unlikely due to atmospheric escape, Pidhorodetska et al. (2021) showed that L9859c could retain a secondary H2/He-dominated atmosphere (since its equilibrium temperature, depending on albedo, is plausibly lower than the hydrogen escape temperature 510 K). Pidhorodetska et al. (2021) also demonstrated that atmospheric escape would prevent the lower gravity L9859b from retaining an H2-dominated atmosphere, though higher mean molecular weight atmospheres such as runaway greenhouse gas (CO2 dominated, like Venus) or steam (H2O-dominated) atmospheres are possible (Kopparapu et al., 2013). Further, Pidhorodetska et al. (2021) predicted a typical L9859b water feature in a clear, steam atmosphere would be detectable in as few as 3 transits. In this chapter I analyze 4 transits of L98b (R=0.8 R?, M2.7-? detection) due to Na, AlO and/or VO/TiO, though no individual species is strongly 79 detected. Both retrievals determine the transit spectrum to be consistent with a clear atmosphere, with no evidence of haze or high-altitude clouds. Interior model- ing constraints on the maximum atmospheric metallicity (log10 Z/Z < 1.7) favor the AURA results. The inferred elemental oxygen abundance suggests that HAT- P-41b has one of the most metal-rich atmospheres of any hot Jupiters known to date. Overall, the inferred high metallicity and high inflation make HAT-P-41b an interesting test case for planet formation theories. 3.2 Introduction Transit spectroscopy has been fundamental in understanding the physics and chemistry of hot exoplanet atmospheres. Transit observations with the Hubble Space Telescope (HST) and the Spitzer Space Telescope have been especially fruitful in illuminating the composition and atmospheric structure of close-in planets, starting with the first measurements of sodium absorption (Charbonneau et al., 2002) and the first detection of thermal emission (Deming et al., 2005) for the atmosphere of HD209458b. The installation of the Wide Field Camera 3 (WFC3) instrument and the refurbishment of the Space Telescope Imaging Spectrograph (STIS) on HST opened up a new era of transit spectroscopy measurements for hot Jupiters. WFC3 has provided the first repeatable and well-validated detections of the presence of water vapor (Deming et al., 2013; Huitson et al., 2013; Wakeford et al., 2013; Mandell et al., 2013), and has opened the field to population studies looking at H2O abundance 80 and metallicity as a function of stellar and planetary properties (Sing et al., 2016; Tsiaras et al., 2018; Pinhas et al., 2019; Welbanks et al., 2019). The upgraded STIS instrument has been a key contributor in illuminating the critical role that aerosols play in driving the continuum opacity for transit measurements of hot planets (Pont et al., 2013; Nikolov et al., 2014; Sing et al., 2016; Chachan et al., 2019b). One of the most intriguing topics from these studies is the question of at- mospheric metallicity. Studies of individual planets suggested a wide diversity of atmospheric metallicity as a function of planetary mass (e.g., Madhusudhan et al., 2014c; Kreidberg et al., 2014b; Wakeford et al., 2017, 2018). However, recent ho- mogenous statistical analyses of many planets reveal that a paucity of water vapor in hot planet atmospheres is the norm (Barstow et al., 2017; Pinhas et al., 2018; Wel- banks et al., 2019). One strategy to investigate the relationships between mass and atmospheric metallicity is to study the best targets within the Saturn and Jupiter mass range, in order to achieve high S/N and leverage the expectation of a high primordial gas fraction and large transit signals. First discovered in 2012 (Hartman et al., 2012), the inflated hot Jupiter HAT- P-41b (Teq=1940K, P=2.7 days) is a strong candidate to inform these trends. It is among the most inflated hot Jupiters (R=1.69RJup, M=0.8MJup), and it orbits a relatively inactive mid-F dwarf (R=1.68R , Teff = 6390 K). HAT-P-41b?s extended atmosphere and its host star?s lack of significant variability make it highly amenable to characterization through transit spectroscopy. Johnson et al. (2017) determined the spin-orbit misalignment of the system to be moderate (-22?), while the host star appears to be part of a multi-stellar system, with a wide-orbit late-type companion 81 discovered at ? 1000 AU (Hartman et al., 2012; Wo?llert & Brandner, 2015; Evans et al., 2016). Tsiaras et al. (2018) retrieved the WFC3 G141 grism spectrum (1.1?1.7?m) with ? -Rex (Waldmann et al., 2015a,b), confirming an 4.2? atmospheric detection and finding no evidence of contributions from either high-altitude clouds or photo- chemical hazes (e.g., Zahnle et al., 2009a). Tsiaras et al. (2018) also found evidence of and abundance constraints for water vapor (log10(XH2O) = ?2.77? 1.09), though due to narrow wavelength coverage abundance uncertainties are large. Still, they are able rule out upper atmospheric water depletion. Fisher & Heng (2018) built upon this result by retrieving on the same dataset with a focus on cloud opacity and other near-infrared opacity sources (NH3, HCN). They find a water abundance of ?0.9+0.28?1.20 which they note is generally consistent with that of Tsiaras et al. (2018). However, this reported abundance is for a cloud-free model with only H2O and NH3 as opacity sources, and this simplified treatment is not necessarily directly compa- rable with more comprehensive atmospheric models. They also find weak evidence of NH3, though they are unable to favor the NH3 and H2O model over a model with grey clouds and H2O. Wide spectral baselines provide the potential for a more complete and con- strained understanding of atmospheric properties (Benneke & Seager, 2013; Griffith, 2014; Welbanks & Madhusudhan, 2019). For example, Line & Parmentier (2016) demonstrated how individual WFC3 spectra are unable to constrain mean molecular weight due to a degeneracy with partial clouds. Furthermore, for a fully homoge- neous cloud cover the cloud-top pressure is degenerate with the chemical abundance 82 (e.g., Deming et al., 2013). Welbanks & Madhusudhan (2019) showed that optical data help alleviate such degeneracies and improve the precision with which plane- tary radius, cloud properties, and molecular/atomic abundances are inferred. As a practical example, Sing et al. (2016) utilized optical-to-infrared spectra to jointly constrain cloud, haze, and chemistry parameters for a sample of ten hot Jupiters. STIS data have been specifically useful in constraining the atmospheric metallicity of giant exoplanets (Wakeford et al., 2017, 2018; Chachan et al., 2019a). Bayesian spectral retrievals are the most reliable way to interpret exoplanet spectra, due to their flexibility in describing diverse exoplanet atmospheres and their ability to evaluate the full posterior distribution of a forward model?s parameters (Madhusudhan, 2018). This allows for understanding not only the properties of an exoplanet?s atmosphere, but also the uncertainties on those properties. Con- sequently, such retrieval codes are common in atmospheric characterization (Mad- husudhan & Seager, 2009, 2010a; Lee et al., 2012; Benneke & Seager, 2013; Line et al., 2013; Amundsen et al., 2014; Waldmann et al., 2015b; Barstow et al., 2017, and many others). Nested sampling (Skilling, 2004) is a particularly powerful Bayesian sampler, as it naturally determines the Bayesian evidence of the fitted model (with the posterior distribution being a byproduct), which is necessary for correct model comparison (e.g., justifying more complicated models, reporting correct detection significances). Despite their ubiquity, each retrieval code is necessarily unique given the as- sumptions and modeling choices that must be made. Though these retrievals gen- erally agree, subtle discrepancies can lead to different conclusions for the same data 83 (Kilpatrick et al., 2018; Fisher & Heng, 2018; Barstow et al., 2020). Examples include different chemical parameterizations (i.e., enforcing chemical equilibrium), cloud parameterizations, opacity sources, and prior assumptions. Therefore, it is important to understand the effect of modeling assumptions on the retrieved at- mospheric parameters (e.g., Welbanks & Madhusudhan, 2019). Testing a suite of models for a given retrieval code ? and, even better, different modeling paradigms altogether ? more accurately captures the uncertainty in the atmospheric param- eters. It is important to be transparent about the assumptions made in a retrieval analysis in order to best contextualize and understand the results. In this chapter I derive the 0.3?5?m transit spectrum of HAT-P-41b using transit observations from HST/STIS, HST/WFC3, and Spitzer (Sec. 3.3). Sec- tion 3.4 characterizes both the variability (incorporating both X-ray and visible photometric monitoring; Sec. 3.4.1) and the parameters (Sec. 3.4.2) of the host star. Section 3.5 describes the data analysis to derive the transit spectrum. Sec. 3.6 describes that we use two different retrieval methods. First, I use a chemical- equlibirum framework (PLATON Zhang et al., 2019, Sec. 3.7), and those results are described in Sec. 3.8. We also explore a more flexible free-chemistry retrieval us- ing the AURA framework (Pinhas et al., 2018, Sec. 3.9). The two retrieval analyses were independently done by different members of the team to allow for an unbiased comparison. I conclude that a high, supersolar atmospheric metallicity (on the or- der of 30?200? solar O/H) best describes the observed spectrum, and ? though median retrieved values differ ? this result is not sensitive to model assumptions. Sec. 3.10 discusses the comparison between retrievals (Sec. 3.10.1, the comparison to 84 interior modeling constraints (Sec 3.10.2, and the implications for planet formation (Sec 3.10.3). Sec. 3.11 provides a summary of my conclusions. 3.3 Observations 3.3.1 HST We observed one transit of HAT-P-41b with the WFC3 instrument on HST and three transits with the STIS instrument as part of the PanCET Program (ID 14767, P.I. Sing). The WFC3 observations were taken on October 16, 2016 using the G141 prism, which covers a wavelength range of approximately 1.1-1.7?m with a spectral resolving power of R?150. The STIS observations were taken with the G430L and G750L grisms, which cover a wavelength range of approximately 0.3- 1.0?m with a spectral resolving power of R?500. The STIS data were acquired on September 4 2017 (G430L, visit 83), May 7 2018 (G430L, visit 84) and June 11 2018 (G750L, visit 85). For each visit, the target was observed for 7 hours over five consecutive HST orbits. An HST gyro issue prevented the acquisition of the third orbit for visit 85. For the WFC3 observations, data were taken in spatial scan spectroscopic mode with a forward scanning rate of 0.065 arcsec s?1 along the cross-dispersion axis, resulting in scans across approximately 46 pixel rows. The observations utilized the 256 ? 256 pixel subarray and the SPARS-10 sampling sequence, with 12 non- destructive reads (NSAMP = 12) resulting in a total integration time of 81 seconds for each exposure. HST obtained a total of 17 exposures in the first HST orbit 85 following acquisition and 19 exposures in each subsequent HST orbit. Typical peak frame counts were ?33,000 electrons per pixel, which is within the linear regime of the WFC3 detector. For the STIS observations, each visit consisted of 5 orbits, with ? 45 min gaps due to Earth occultations. We utilized the wide 52 ? 2 arcsec slit to minimize slit light losses and an integration time of ?253 sec for each exposure, for a total of 48 spectra for each visit. Data acquisition overheads were minimized by reading-out a subarray of the CCD with a size of 1024 ? 128 pixels. 3.3.2 Spitzer The Spitzer Infrared Array Camera (IRAC) observations were taken in January and February 2017 as part of Program 13044 (P.I. D. Deming). A single transit of HAT-P-41b was observed in each of the IRAC1 (3.6?m) and IRAC2 (4.5?m) channels. Each transit was preceded by a 30-minute peakup sequence that also mitigates the steepest portion of a temporal ramp due to the detector. The transit was observed over ?12 hours, with equivalent in-transit and out-of-transit coverage. 338 exposures were obtained for each transit, and each exposure consisted of 64 subarray frames of 32x32 pixels, using an exposure time of 2.0 seconds per frame. 3.3.3 Photometric Monitoring Observations To better diagnose the likelihood of stellar variability impacting the transit spectrum, I complemented the transit observations with monitoring observations at 86 both visible (AIT) and X-ray wavelengths (XMM-Newton). XMM-Newton observed HAT-P-41 on 2017-April-07 with an overall 17 ks exposure time (Proposal ID 80479, P.I. J. Sanz-Forcada). The target was not detected in any of the EPIC detectors; I discuss the implications of this in Section 3.4.1. A collaborator obtained nightly ground-based photometric observations of HAT-P-41 during its 2018 and 2019 observing seasons with the Tennessee State University Celestron 14-inch (C14) automated imaging telescope (AIT) located at Fairborn Observatory in the Patagonia Mountains of southern Arizona (see, e.g., Henry, 1999; Eaton et al., 2003). The AIT is equipped with an SBIG STL-1001E CCD camera; observations were made through a Cousins R filter. Details of the ob- serving, data reduction, and analysis procedures are described in Sing et al. (2015). They collected a total of 207 successful nightly observations (excluding a few isolated transit observations) over the two observing seasons. The observing activ- ities at Fairborn must come to a halt each year during the southern Arizona rainy season, which typically lasts from approximately July 1 to September 10. Since HAT-P-41 comes to opposition around July 18, each observing season is broken into two intervals, which we designate as intervals A and B. Information for a portion of the AIT observations are shown in Table 3.1; the full table is available in the electronic edition of ApJ. 87 Table 3.1: AIT photometric observations of HAT-P-41 Hel. Julian Date Delta R Sigma (HJD ? 2,400,000) (mag) (mag) 58175.0255 ?0.56702 0.00122 58176.0214 ?0.56986 0.00109 58180.0102 ?0.56636 0.00052 58181.0158 ?0.56444 0.00293 58182.0022 ?0.56479 0.00185 58184.9980 ?0.56390 0.00182 NOTE ? This table shows a sample of the full set of observations. 3.4 Stellar Properties 3.4.1 Analysis of Stellar Variability The results of analysis of the AIT photometric observations (Sec. 3.3.3) are given in Table 3.2. The low numbers of observations in 2018 B, 2019 A, and 2019 B are the result of the unusually cloudy weather at Fairborn for the past two years. This cloudy weather pattern continues to the present. Column 4 of the table gives the standard deviation of the individual observations with respect to their cor- responding seasonal mean. The standard deviations range between 0.00224 and 0.00394 mag for the four observing intervals. This is near the limit of the nightly measurement precision with the C14, as determined from the constant comparison stars in the field. Periodogram analyses of the four intervals reveal no significant periodicities. The scatter in the seasonal means given in column 5 is consistent with the expected photometric precision considering the small number of observations in the last three intervals and the marginal photometric conditions prevalent at 88 Table 3.2: Results of the analysis of photometric monitoring observations for HAT- P-41 Observing Date Range Sigma Seasonal Mean Season Nobs (HJD ? 2,400,000) (mag) (mag) 2018 A 110 58175?58295 0 00224 ?0.56496? 0.00021 2018 B 34 58386?58451 0.00291 ?0.56979? 0.00051 2019 A 42 58577?58657 0.00264 ?0.56966? 0.00041 2019 B 21 58756?58802 0.00394 ?0.56602? 0.00088 Fairborn Observatory over the past two years. Therefore, HAT-P-41 appears to be constant on night-to-night and year-to-year timescales to the limit of the telescope?s precision. Additionally, HAT-P-41 was not detected in any of the XMM Newton?s EPIC detectors. Given the distance of the object I can set an upper limit of LX = 1?1029 erg s?1 on the stellar X-ray luminosity. This implies a value of logLX/Lbol < ?5.2, indicating that the star has a moderate activity level at most (Wright et al., 2011). The photometric observations of HAT-P-41 describe a relatively quiet star. Furthermore, the Ca II chromospheric activity index (S = 0.18 Duncan et al., 1991) and the corresponding estimated parameter flux logR?HK (-5.04) for HAT-P-41?s spectral type (B-V = 0.29) are not indicative of high activity (Hartman et al., 2012; Noyes et al., 1984) and may indicate instead a basal-level activity (Isaacson & Fischer, 2010). Rackham et al. (2019) show that for all but the most active F- dwarfs variability does not result in any detectable change to the transit spectrum. Specifically, the impact of potential complications such as false TiO/VO detections, false water detections, and optical offsets are all determined to be less than? 10ppm. Therefore, I conclude that stellar variability is unlikely to contaminate HAT-P-41b?s 89 transit spectrum. 3.4.2 Stellar Parameters Inferred atmospheric planetary parameters are directly dependent on host star parameters. For my analyses, I incorporate the stellar parameters from TESS Input Catalog ? version 8 (TIC-8; Stassun et al., 2019; Stassun, 2019). TIC-8 provides reliable stellar parameters for planetary host stars based primarily on Gaia Data Re- lease 2 (GDR2) point sources (Gaia Collaboration et al., 2016, 2018). The algorithm for HAT-P-41?s parameters is as follows: distance is first derived from Gaia DR2 parallax, using a correct inference procedure (Bailer-Jones et al., 2018). HAT-P-41?s galactic longitude (-10.6) puts it in a region where uncertainty on reddening makes determining effective temperature from Gaia photometry difficult. As a result, a spectroscopically-derived effective temperature (from the PASTEL catalog Soubi- ran et al., 2016) is preferred. The stellar radius and mass are then self-consistently derived from the distance and effective temperature (Andrae et al., 2018). Finally, log gs is calculated from the stellar radius and mass. It is important to recalculate Rp, Mp, and semi-major axis a based on the Rs value from TIC-8, since those are derived in the discovery paper assuming a certain value for Rs. As a simple example, Rp is derived by constraining Rp/Rs in transit and multiplying by Rs. To re-derive the planetary parameters, I follow the methodology of Stassun et al. (2017). The resulting values and 1-? ranges are shown in Table 3.3 along with the values from the discovery paper (Hartman et al., 90 Table 3.3: System Parameters for HAT-P-41 Parameter TIC-8a Discovery Paperb R [R +0.08 +0.058s Sun] 1.65?0.06 1.683?0.036 Ms[MSun] 1.32 +0.25 ?0.16 1.42? 0.047 log gs [cgs units] 4.12 +0.11 ?0.06 4.14? 0.02 T +100s,eff [K] 6480?100 6390? 100 Rp[RJup] 1.65 +0.08 1.685+0.076?0.07 ?0.051 Mp[MJup] 0.76 +0.14 ?0.12 0.80? 0.10 ?p [g cm ?3] 0.21? 0.05 0.20? 0.03 T +40p,eq [K] 1960?35 1940? 38 a [AU] 0.0418+0.0021?0.0019 0.0426? 0.0005 Distance [pc] 348? 4.5 344+12?8 a Provided by or derived from Tess Input Catalog Stassun et al. (2019) b Hartman et al. (2012) 2012). I favor the TIC-8 stellar parameters over the discovery paper values (derived using isochrones and high-resolution spectroscopy; Hartman et al., 2012) since they are based on more recent and comprehensive data. I emphasize that the two sets of parameters are consistent to better than 1-?, and using the discovery paper values has no impact on the conclusions of this chapter. A planet?s composition is directly linked to its host star?s composition. Brewer & Fischer (2018a) determined the stellar abundance of 15 different elements for HAT-P-41 as part of the Spectral Properties of Cool Stars (SPOCS) catalog. Ta- ble 3.4 gives the abundances, relative to solar, for the relevant elements. Brewer & Fischer (2018a) find an effective temperature, metallicity, and log gs consistent with 91 Table 3.4: HAT-P-41 Host Star Elemental Abundances Elemental ratio Abundance (log solar unit) [O/H] 0.37? 0.04 [C/H] ?0.08? 0.03 [Na/H] 0.17? 0.01 [Ti/H] 0.22? 0.01 [V/H] 0.09? 0.03 [Al/H] 0.07? 0.03 [M/H] 0.18? 0.01 [C/O] ?0.45? 0.05 NOTE?All values from Brewer & Fischer (2018b) both TIC-8 and the discovery paper. HAT-P-41 is a metal-enriched star, and no- tably has an elemental oxygen abundance of 2.3? solar. Carbon is the only depleted element at ? 0.8? solar, resulting in a subsolar C/O ratio of 0.19 (0.36? solar). 3.5 Data Analysis 3.5.1 STIS 3.5.1.1 Data Reduction The STIS data analysis procedures follow the general methodology detailed in Nikolov et al. (2014, 2015). A collaborator commenced analysis from the flt.fits files, which were reduced (bias-, dark- and flat-corrected) using the latest version of the CALSTIS pipeline and the latest calibration frames. We used median combined difference images to identify and correct for cosmic-ray events in the images as described by Nikolov et al. (2014). We found ? 4 percent of the detector pixels were affected by cosmic ray events. We also corrected pixels identified by CALSTIS 92 as bad with the same procedure, which together with the cosmic ray identified pixels resulted in a total of ?14 percent interpolated pixels. We performed spectral extraction with the IRAF procedure APALL using aperture sizes in the range from 6 to 18 pixels with a step of 0.5. The best aperture for each grating was selected based on the resulting lowest light-curve residual scatter after fitting the white light curves. We found that aperture sizes 13.5, 13.5 and 10.5 pixels satisfy this criterium for visits 83, 84 and 85, respectively. We cross-correlated and interpolated all spectra with respect to the first spec- trum to account for sub-pixel wavelength shifts in the dispersion direction. The STIS spectra were then used to extract both white-light spectrophotometric time series and custom wavelength bands after summing the appropriate flux from each bandpass. The raw STIS light curves exhibit instrumental systematics on the spacecraft orbital time-scale, which are attributed to thermal contraction/expansion (referred to as the ?breathing effect?) as the spacecraft warms up during its orbital day and cools down during orbital night. We take into account the systematics associated with the telescope temperature variations in the transit light-curve fits by fitting a baseline function depending on various parameters. 3.5.1.2 Light Curve Analysis White and spectroscopic light curves were created from the time series of each visit by summing the flux of each stellar spectrum along the dispersion axis. We 93 fit each transit light curve using a two-component function that simultaneously models the transit and systematic effects. The transit model was computed using the analytical formulae given in Mandel & Agol (2002), which are parameterized with the mid-transit times (Tmid), orbital period (P) and inclination (i), normalized planet semimajor axis (a/R?), and planet-to-star radius ratio (Rp/R?). Stellar limb-darkening was accounted for by adopting the four parameter non- linear limb-darkening law with coefficients c1, c2, c3 and c4, computed using a three-dimensional stellar atmosphere model grid (Magic et al., 2015). We adopted the closest match to the effective temperature, surface gravity, and metallicity values for HAT-P-41 determined by Hartman et al. (2012). As in their past STIS studies, my collaborator applied orbit-to-orbit flux cor- rections by fitting for a fourth-order polynomial to the spectrophotometric time series phased on the HST orbital period and a linear time term. We also used a low-order polynomial (up to a third degree with no cross terms) of the spectral dis- placement in the dispersion and cross dispersion direction. The first exposures of each HST orbit exhibit lower fluxes and have been discarded in the analysis. Similar to our past HST STIS analyses (Nikolov et al., 2014; Sing et al., 2016), we intended to discard the entire first orbit to minimize the space craft thermal breathing trend, but found that for two of the three HST visits, a few of the exposures taken toward the end of the first orbit can be used in the analysis. We then generated systematics models that spanned all possible combinations of detrending variables and performed separate fits using each systematics model included in the two-component function. The Akaike information criterion (AIC; 94 Akaike, 1974) was calculated for each attempted function and used to marginalize over the entire set of functions following Gibson (2014b). My choice to rely on the AIC instead of the Bayesian information criterion (BIC; Schwarz, 1978) was determined by the fact that the BIC is more biased toward simple models than the AIC. The AIC therefore provides a more conservative model for the systematics and typically results in larger or more conservative error estimates, as demonstrated by Gibson (2014b). Marginalization over multiple systematics models assumes equal prior weights for each model tested. For the white light curves, we fixed the orbital period, inclination and a/Rs to the values reported in Table 3.5 and fit for the transit mid-time and planet-to-star radius ratio. We find central transit times TC [MJD]= 58000.6958 ? 0.0029 (visit 83), Tc[MJD]= 58245.85414? 0.00039 (visit 84), Tc[MJD]= 58280.87484? 0.00036 (visit 85). We derive the white light transit depths to be 10200 ? 104 ppm and 10320? 85 ppm for G430L and G750L, respectively. For the spectroscopic light curves, a common-mode systematics model was established by simply dividing the white-light curve by a transit model (Berta et al., 2012; Deming et al., 2013). We computed the transit model using the orbital period, inclination and a/Rs from Table 3.5 and the central times for each orbit from the whit-light analysis. The common-mode factors from each night were then removed from the corresponding spectroscopic light curves before model fitting. We then performed fits to the spectroscopic light curves using the same set of systematics models as in the white-light curve analysis and marginalized over them as described above. For these fits, Rp/Rs was allowed to vary for each spectroscopic 95 Table 3.5: Transit parameters for HAT-P-41b Parameter Value Rp/Rs 0.1028? 0.0016 a/R 5.44+0.09s ?0.15 i [Degrees] 87.7? 1.0 Tc [BJD] 2454983.86167? 0.00107 log gp [cgs units] 2.84? 0.06 P [days] 2.694047? 4? 10?6 All values from discovery paper, Hartman et al. (2012) channel, while the central transit time and system parameters were fixed. We as- sumed the non-linear limb-darkening law with coefficients fixed to their theoretical values, determined in the same way as for the white-light curve. The detrended spectrophotometric light curves are shown in Figures B.1, B.2, B.3. The derived STIS transit spectrum is shown along with the entire transit spectrum in Table 3.6. 3.5.2 WFC3 3.5.2.1 Data Reduction The WFC3 data reduction closely follows the data processing described in de- tail in Section 2.3. I briefly summarize it here. I download the ?ima? data files from the HST archive, and remove background contamination following the ?difference reads? methods of Deming et al. (2013), which allows us to easily resolve and remove potential contamination from the nearby companion (Evans et al., 2016). I deter- mine a wavelength solution by taking the zero-point from the F140W photometric 96 Table 3.6: HAT-P-41b Transit Spectrum Instrument ? [?m] Depth [ppm]a Instrument ? [?m] Depth [ppm] STIS G430Lb 0.290?0.350 10091 ? 230 STIS G750L 0.711?0.731 10499 ? 364 0.350?0.370 10006 ? 249 0.731?0.750 10038 ? 302 0.370?0.387 10397 ? 198 0.750?0.770 10142 ? 224 0.387?0.404 10273 ? 163 0.770?0.799 10124 ? 281 0.404?0.415 9999 ? 137 0.799?0.819 10618 ? 317 0.415?0.426 9980 ? 185 0.819?0.838 9910 ? 239 0.426?0.437 10324 ? 145 0.838?0.884 10252 ? 238 0.437?0.443 10428 ? 287 0.884?0.930 10217 ? 298 0.443?0.448 10245 ? 168 WFC3e 1.122?1.141 10297 ? 107 0.448?0.454 10411 ? 165 1.141?1.159 10620 ? 115 0.454?0.459 10367 ? 192 1.159?1.178 10347 ? 130 0.459?0.465 10227 ? 145 1.178?1.196 10479 ? 113 0.465?0.470 10640 ? 164 1.196?1.215 10497 ? 111 0.470?0.476 10429 ? 165 1.215?1.233 10644 ? 111 0.476?0.481 10453 ? 144 1.233-1.252 10289 ? 100 0.481?0.492 10584 ? 125 1.252?1.271 10360 ? 107 0.492?0.498 10224 ? 192 1.271?1.289 10488 ? 122 0.498?0.503 10289 ? 166 1.289?1.308 10405 ? 97 0.503?0.509 10459 ? 149 1.308?1.326 10396 ? 94 0.509?0.514 10519 ? 183 1.326?1.345 10581 ? 92 0.514?0.520 10506 ? 168 1.345?1.364 10684 ? 105 0.520?0.525 10429 ? 186 1.364?1.382 10622 ? 113 0.525?0.531 10558 ? 128 1.382?1.401 10477 ? 112 0.531?0.536 10247 ? 174 1.401?1.419 10689 ? 98 0.536?0.542 10451 ? 180 1.419?1.438 10535 ? 108 0.542?0.547 10476 ? 177 1.438?1.456 10626 ? 113 0.547?0.552 10422 ? 206 1.456?1.475 10564 ? 121 0.552?0.558 10794 ? 153 1.475?1.494 10686 ? 113 0.558?0.563c 9221 ? 279 1.494?1.512 10799 ? 129 0.563?0.569 10444 ? 188 1.512?1.531 10551 ? 124 STIS G750Ld 0.526?0.555 10356 ? 220 1.531?1.549 10566 ? 111 0.555?0.575 10497 ? 278 1.549?1.568 10515 ? 134 0.575?0.594 10317 ? 202 1.568?1.587 10436 ? 110 0.594?0.614 10061 ? 236 1.587?1.605 10492 ? 145 0.614?0.633 10386 ? 142 1.605?1.624 10340 ? 122 0.633?0.653 10542 ? 219 1.624?1.642 10338 ? 142 0.653?0.672 10418 ? 276 1.642?1.661 10331 ? 138 0.672?0.692 10182 ? 225 Spitzer IRAC1 3.2?4.0 10191 ? 102 0.692?0.711 10168 ? 192 Spitzer IRAC2 4.0?5.0 10679 ? 145 a Transit depth = R2 2planet/Rstar b Typical STIS G430L bin size = 0.0055?m (median resolution ? 350) c Outlier bin strongly affected by systematics and ignored in retrieval analyses d Typical STIS G750L bin size = 0.0196?m (median resolution ? 130) e WFC3 bin size = 0.0186?m (median resolution ? 75) 97 observation and fitting for the wavelength coefficients that allow an out-of-transit spectrum to match the appropriate ATLAS stellar spectrum (Castelli & Kurucz, 2004). I then divide the background-subtracted ?ima? science frame by the WFC3 flat-field calibration file, and return the dark-, bias-, and flat-field-corrected flux array, in units of electrons. The uncertainty of the flux at each pixel is taken from the ?ima? file?s error frame, which accounts for gain-adjusted Poisson noise, read noise, and noise from dark current subtraction. This is further adjusted via error propagation for the added uncertainty from background removal and flat-field correction. I use the ?ima? file?s data quality frame to mask pixels (i.e., give zero weight) that are flagged as bad in every exposure in the time series. I then correct for cosmic rays using a conservative time series sigma-cut of 8-?, while accounting for changes in flux that occur due to uneven scan rates and the transit itself, and set the affected pixels to the median value of that pixel in the time series. The average amount of pixels either impacted by cosmic ray events or flagged as bad pixels is 1.8% of all pixels in a exposure. The reduced exposures are summed over the spatial scan direction to give a 1-D spectrum at each observation time. 3.5.2.2 Light Curve Analysis The light curve analysis follows that described in detail in Section 2.4. I briefly summarize it here. I use a similar marginalization light curve analysis as Sheppard et al. (2017), applied to transit curves. This is a Bayesian model averaging method, 98 first described by Gibson (2014b) and applied by Wakeford et al. (2016a), with further de-trending by use of band-integrated (white light) residuals in spectral light curve fitting (Mandell et al., 2013; Haynes et al., 2015). I first analyzed the band-integrated light curve to simplify the spectral light curve analysis, then I fit the spectral light curves to derive the WFC3 transit spectrum. For the transit model I assume nonlinear limb darkening and derive the co- efficients by interpolating the 3-D values from Magic et al. (2015) to the central wavelength of WFC3 (1.4?m). The limb darkening derivation is consistent with that used in the STIS analysis. I only fit for transit depth and central transit time, since the incomplete coverage of HST makes it difficult to improve constraints on other transit parameters, such as a/Rs or inclination. I fix these values in the light curve analyses of each instrument (STIS, WFC3, and Spitzer), which ensures con- sistent orbital parameters are used when analyzing different datasets. The transit and system parameters are shown in Table 3.5. I fit the each systematic model in the grid, weight each model by its Bayesian evidence ? approximated by the Akaike information criterion (AIC) ? and marginal- ize over the model grid to derive the light curve parameters and uncertainties while inherently accounting for uncertainty in model choice. The normalized raw light curve, the de-trended light curve, and the residuals from the highest-weight sys- tematic model are shown in Figure 3.1. I derive the white light depth to be 10490? 51 ppm. To derive the transit spectrum, I bin the 1D spectra from (1.12?1.66?m), and use bins of width 0.0186?m (4 pixels) to maximize resolution. I note that the 99 Figure 3.1: Top panel: pre- processed band-integrated light curve for WFC3 observations. This is the band-integrated flux versus planet phase derived from the reduced data. The first orbit is excluded, as it is dominated by instrumental sys- tematics. Middle panel: band- integrated light curve divided by the highest weighted system- atic model (i.e., the detrended light curve). Bottom panel: residuals between light curve data and highest-weighted joint transit and systematic model. The reduced ?2 for the highest weighted model is 1.17, which is consistent with the model being a good fit for 67 degrees of free- dom. atmospheric retrieval is not sensitive to the choice of bin size. The wavelength range, transit depth, an depth uncertainty for each WFC3 bin is shown in Table 3.6. The shape of my derived spectrum is in excellent agreement with the literature spectrum (Tsiaras et al., 2018), though it is shifted to higher depths by ? 90 ppm, indicating that I derive a larger white light depth. This difference persists even if I derive the spectrum without using white light residuals. This could plausibly be due to different limb darkening treatments or different systematic modeling choices. This difference emphasizes the importance of considering offsets between instruments in retrieval analyses (Sec. 3.6.1.2). Further, the white light depth is subject to the choice of orbital parameters, which are typically fixed in spectroscopic light 100 curve fits. I run sensitivity tests to determine that accounting for orbital parameter uncertainties increases the scatter between HST STIS and HST WFC3 by roughly 60 ppm for HAT-P-41b. I capture this scatter by including it in WFC3?s depth uncertainty, increasing it from 50 ppm to 80 ppm. As a check, I performed retrievals using the published spectrum from Tsiaras et al. (2018) in combination with the derived STIS and Spitzer depths and found differences well within the 1-? uncertainties for the retrieved parameters. The major results and conclusions in this chapter are not sensitive to the WFC3 spectrum choice. I verify the derived spectrum following the process outlines in Section 2.4.3.1. First, I check ?2. For both the band-integrated and spectral light curves (66 and ? 88 degrees of freedom, respectively), the acceptable reduced ?2 range is roughly 0.7?1.3. The band-integrated analysis (?2? = 1.17) and all but one of the spectral bins (median ?2? = 0.9) fall within this range. The only light curve that doesn?t fall in this range (1.299?m) has a reduced ?2 of 0.56. Like L9859c, this bin is not flagged by the normality or correlated noise analyses, and fitting the light curve without incorporating white light residuals finds a consistent depth with a more reasonable ?2? = 0.71. Further, the derived depth and uncertainty exhibit good agreement with the published (Tsiaras et al., 2018) transit depth at this wavelength (accounting for the white light offset). Consequently, I include it in the transit spectrum. Next, I check normality. Normality is rejected at the 5% significance level only for the band-integrated residuals and the 1.243?m spectral bin residuals. In both cases, normality is ruled out due to a single outlier in the time-series. Normality is 101 rejected at 3% significance for the band-integrated residuals, due entirely to the first exposure in the first orbit. When this exposure is ignored, I recover a consistent depth and uncertainty and the residuals are consistent with normality, and so I keep this exposure in the analysis. A possible cause of the spectral bin?s outlier is a minor cosmic ray event or bad pixel that was small enough to both avoid the detection by the sigma-cut and not affect the band-integrated curve, but large enough to impact the much smaller bin flux. Removing this spectral bin from the retrieval had no noticeable effect on the results. Further, the derived depth and uncertainty are consistent with literature values (Tsiaras et al., 2018). I include this bin in the retrieval. Finally, I test for correlated noise in the residuals using the time-averaging test as in 2.4.3.1. I find no evidence of correlated noise (Figure 3.2). I emphasize that removing any of the flagged bin spectra has no affect on the retrieval. Together, these tests support the validity of the derived transit depths and uncertainties in the WFC3 bandpass. 3.5.3 Spitzer 3.5.3.1 Data Reduction The Spitzer data consists of cubes of 64 subarray frames in each band, each of size 32?32 pixels. A collaborator extracted aperture photometry for each frame, totaling 21,632 frames at both 3.6 and 4.5?m. To extract photometry, we used 11 numerical apertures with radii ranging from 1.6 to 3.5 pixels, and we centered those 102 1.0 1.131 m 1.150 m 1.168 m 1.0 1.187 m 1.206 m 1.224 m 1.0 1.243 m 1.261 m 1.280 m 1.0 1.299 m 1.317 m 1.336 m 1.0 1.354 m 1.373 m 1.391 m 1.0 1.410 m 1.429 m 1.447 m 1.0 1.466 m 1.484 m 1.503 m 1.0 1.522 m 1.540 m 1.559 m 1.0 1.577 m 1.596 m 1.614 m 1.0 Expected 1.633 m 1.652 m Data RMS Band-integrated 0.1 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Exposures Per Bin Figure 3.2: Correlated noise analysis for each of the 29 WFC3 spectral bins and the band-integrated light curve. For each bin?s light curve, I find the RMS of the residuals for an increasing number of exposures per bin. Pure white noise would scale with the black line, while correlated noise would increase with binning. Though not exact given the gaps between WFC3 data, this is a useful heuristic to search for correlated noise. See Cubillos et al. (2017) and Pont et al. (2006) for more details. 103 Normalized RMS apertures on the position of the host star determined using both a 2-D Gaussian fit to the stellar point spread function, and also a center-of-light calculation. Since HAT-P-41 has a companion star 3.6 arc-sec distant (Hartman et al., 2012; Evans et al., 2016), we measured the flux of the companion scattered into each of our numerical apertures, using the method described by Garhart et al. (2020). We adopted the magnitude difference in the Spitzer bands as deduced by Garhart et al. (2020). Accounting for our different aperture radii than was used by Garhart et al. (2020), we derive dilution correction factors of 1.0171 and 1.0106 at 3.6- and 4.5?m, respectively. The transit depths are then multiplied by those factors in order to correct for the presence of the companion star. 3.5.3.2 Light Curve Analysis A collaborator fit transit curves to the 22 sets of photometry at each wave- length (eleven apertures, each with two centering methods). Their default fitting procedure fixes the orbital parameters at the values in the discovery paper by Hart- man et al. (2012), fitting only for the central time and depth of the transit. The shape of the Spitzer transits is well matched when fixing the orbital parameters to those values. However, we also explored including the orbital inclination and a/Rs in the fit (see below), those being the orbital parameters that most strongly affect the shape of the transit. We adopt quadratic limb darkening coefficients calculated by least squares for the Spitzer bands by Claret et al. (2013), using 2 km/sec mi- croturbulence. We choose the values for Teff = 6400 K and logg = 4.0, without 104 interpolation. We fix those coefficients in the fitting process, and we deem these choices to be appropriate given that the limb darkening is minimal at these infrared wavelengths. Our fits to the transit account for the intra-pixel sensitivity variations of the Spitzer photometry using pixel-level decorrelation (PLD, Deming et al., 2015), including a linear baseline (ramp) in time. We use a Bayesian information criterion to decide between a linear versus quadratic ramp. The details of the PLD fit are the same as described for secondary eclipses by Garhart et al. (2020), except that we are fitting transits, so we include limb darkening. Briefly, the fitting code bins the photometry and pixel basis vectors to various degrees, and selects the optimal bin size, aperture radius, and centering method, based on the smallest difference from an ideal Allan deviation relation (Allan, 1966). The Allan deviation relation expresses that the standard deviation of the residuals should decrease as the square-root of the bin time. Operating on binned data allows the PLD algorithm to concentrate on the longer time scales that characterize the red noise (and also the transit duration), as opposed to the 0.4-second cadence time of the raw photometry. We determine the errors on the transit depths and times using an MCMC procedure, with a burn-in phase of 10,000 steps, followed by 800,000 steps to ex- plore parameter space. We calculate multiple chains for each transit, and verify convergence using a Gelman-Rubin (GR) statistic (Gelman & Rubin, 1992). The GR values for the PLD fits are very close to unity, being 1.0027 at 3.6?m and 1.0004 at 4.5?m, indicating good convergence. The transit depths and times are listed in Table 3.7. The derived transit times are in excellent agreement with measurements of the 105 Table 3.7: HAT-P-41b Spitzer Transit Analysis Results Wavelength BJD(TDB) R2p/R 2 s (ppm) a 3.6?m 2457772.20440 ? 0.00033 10020 ? 100 4.5?m 2457788.36860 ? 0.00032 10568 ? 135 a These are ?as observed? transit depths, not corrected for dilu- tion by the companion star. To correct for dilution, multiply the depth by 1.0171 at 3.6?m, and by 1.0106 at 4.5?m. The corrected values are shown with the rest of the transit spectrum in Table 3.6. Figure 3.3: Likelihood distribution from the fit to the 3.6?m transit of HAT-P-41b, based on an MCMC us- ing uniform priors, and shown ver- sus a/Rs and the orbital inclina- tion. These two orbital parameters are degenerate when using only a Spitzer transit, and the values de- rived by Hartman et al. (2012) are indicated by the point with error ranges. The Spitzer transits at both 3.6- and 4.5?m are fully consistent with a/Rs and i from Hartman et al. (2012). same Spitzer transits by Wakeford et al. (2020). Specifically, using our uncertainties, our fitted times differ by 1.1? and 0.6? at 3.6- and 4.5?m, respectively. Wakeford et al. (2020) use 8 HST transits as well as the discovery results and the Spitzer transits to derive a new ephemeris. Our fitted times (Table 3.7) differ from that ephemeris by insignificant amounts (0.2 and 7.8 seconds). We explored the effect of uncertainties in the orbital parameters, since those can affect the derived transit depth (Alexoudi et al., 2018). Adopting uniform pri- ors for a/Rs and inclination, we find that they are degenerate when fitting only the 106 Spitzer transits. That degeneracy is illustrated in Figure 3.3, where it is appar- ent that inclination and a/Rs can trade off to maintain the observed transit dura- tion, and the sharp ingress/egress that characterizes the Spitzer transits. Changing the inclination changes the chord length across the stellar disk, and (when limb- darkening is minimal) that can be compensated by changing a/Rs to maintain the same transit duration. The orbital solution from Hartman et al. (2012) is entirely consistent with the derived likelihood distribution for those parameters, as shown in Figure 3.3 for 3.6?m (4.5?m is similar). We therefore freeze the orbital parameters at the Hartman et al. (2012) values when fitting the Spitzer transits. Figure 3.4 illustrates the transits at 3.6 and 4.5?m. The residuals from the best-fit model are included in the figure, and the right panel shows the residuals binned over increasing time scales, a so-called Allan deviation relation (Allan, 1966). The slopes of those relations are close to the -0.5 value expected for photon noise. 3.6 Atmospheric Retrieval There are two common frameworks to retrieve physical parameters from trans- mission spectra. The first is by assuming chemical equilibrium, where the abundance of a molecule is only dependent on local temperature, local pressure, and global ele- mental abundances such as O/H and C/H (e.g., Kreidberg et al., 2015). The second is by instead fitting for molecular abundances based on observed spectral features, then determining global elemental abundances from the molecular abundance values (e.g., Madhusudhan, 2012). Since carbon and oxygen-based molecules are typically 107 Figure 3.4: Left: Spitzer transit light curves of HAT-P-41b at 3.6 and 4.5?m after correction of the intra-pixel effects of the detector and temporal ramps. The data are binned to 100 points per transit for clarity of illustration. The residuals (data minus fitted model) are shown below the transit curves, and have error bars added. Right: Allan deviation relations for the binned residuals, i.e. standard deviation of the residuals when the original data are binned over an increasing number of points, N. The solid lines project the single-point (N = 1) scatter to larger bin sizes with a slope of -0.5 as expected for photon noise. 108 the most spectroscopically active species over the wavelengths covered by HST and Spitzer, the elemental abundances are commonly parameterized by metallicity ? defined as the enhancement of metal elements relative to hydrogen compared to solar values (see Sec 3.6.3 for more detail) ? and carbon-to-oxygen ratio (C/O). Some retrievals improve flexibility by allowing other elements ? such as sodium or vanadium ? to also vary from their solar ratios (Amundsen et al., 2014; Tremblin et al., 2015, 2016; Sing et al., 2016). The open source code PLATON1 (Zhang et al., 2019) is able to perform quick retrievals which assume chemical equilibrium, whereas AURA (Pinhas et al., 2018) is able to capture possible disequilibrium chemistry by not assuming chemical equi- librium. We retrieve the atmospheric parameters with both frameworks to see how interpretations compare, and to explore how sensitive the conclusions are to retrieval assumptions. 3.6.1 PLATON PLATON is a fast, open-source retrieval code developed by Zhang et al. (2019). Like many retrieval codes, it comprises a forward model and an algorithm for Bayesian inference. Though there are some differences, it essentially uses the same forward model as Exo-Transmit (Kempton et al., 2017). Here, I summarize the forward model: To calculate a spectrum, it first determines the abundances of 34 potentially relevant chemical species for a given atmospheric metallicity and C/O. These include Na and K as well as molecules CH4, CO, CO2, HCN, H2O, MgH, NH3, 1https://github.com/ideasrule/platon 109 TiO, and VO (see Kempton et al. (2017) for complete list). The metallicity and C/O provide elemental abundances, which are combined with a temperature-pressure grid as input into GGchem (Woitke et al., 2018) to compute equilibrium molecular and atomic abundances at each pressure layer in the atmosphere, accounting for the ef- fects of condensation on equilibrium abundances. PLATON allows for a grey cloud deck, below which no light can penetrate, and the abundance-temperature-pressure grid facilitates the determination of total opacity at each pressure layer in the atmo- sphere which lies above this cloud top. PLATON includes the same opacity sources as Exo-Transmit, and accounts for opacity from gas absorption, H2-He collision- induced absorption (CIA), and scattering (either parametric Rayleigh scattering or Mie scattering). The forward model converts the opacity-pressure grid to a opacity- height grid using hydrostatic equilibrium with Pref = 1 bar, which is then used as an input to a radiative transfer code to determine the uncorrected transit depths. After correcting for possible stellar activity, due to either unocculted spots or facu- lae, PLATON?s forward model outputs the corrected transit spectrum. The largest source of computational uncertainty is opacity sampling error, which is a source of white noise from using a relatively low resolution (R=1000) that cannot resolve in- dividual lines (Zhang et al., 2019). Accounting for opacity sampling for the transit spectrum of HAT-P-41b typically increases the depth uncertainty by 1.5% (2.5ppm), which is sufficiently small such that it does not affect interpretation. For more de- tails on PLATON, see Zhang et al. (2019). I note that the version of PLATON I describe and use in this chapter is Platon 3.1. A newer version, PLATON 5.1, has since been released with additional features as described in Zhang et al. (2020a). 110 This newer version is used is Section 2.5.1. My PLATON analysis does not retrieve individual abundances. Instead, it fits for the isothermal temperature, atmospheric metallicity as a multiple of solar values for atomic species, and C/O ratio; the retrieved equilibrium abundances for atomic and molecular absorbers are a natural consequence of those values. This is in contrast with the AURA analysis (Sec. 3.6.2), which retrieves individual molecular and atomic abundances. The algorithm PLATON uses for Bayesian inference is nested sampling (Skilling, 2004). Specifically, PLATON uses multimodal nested sampling from the Python im- plementation nestle.2 Like MCMC samplers, nested sampling efficiently samples posterior distributions with dimensionalities typical of atmospheric retrievals (n=5? 20), and so it is effective at atmospheric parameter estimation. Unlike MCMC routines, it automatically calculates the Bayesian evidence for a model, which is necessary for model comparison. The Bayesian evidence intrinsically accounts for overfitting by punishing too much model structure and thus determines if extra parameters are warranted. I use this to justify excluding parameters which add structure and do not significantly improve the fit. Nested sampling also has a well- defined stopping criteria, so there is no need to check for convergence. For an excellent write up on this algorithm, especially about using it in practice, see the documentation of Dynesty3 (Higson et al., 2019). In addition to the standard set included in PLATON?s forward model, I added 2https://github.com/kbarbary/nestle 3https://dynesty.readthedocs.io/en/latest/ 111 four new fittable parameters: a partial cloud parameterization and three instrumen- tal transit depth bias parameters (henceforth referred to as instrumental offsets). 3.6.1.1 Partial Clouds The partial cloud parameter is motivated by work by Line & Parmentier (2016) and MacDonald & Madhusudhan (2017), which showed that if the grey cloud deck were inhomogenous, then the spectrum I observe (D) would be a weighted average of the clear atmosphere transit spectrum (Dclear) and the cloudy atmosphere spectrum (Dcloudy) with weights given by the cloud fraction (fc). I implement this as D = fc?Dcloudy+(1?fc)?Dclear. Since high altitude grey clouds are seen in spectra as flat lines, averaging a spectrum with features with this line will shrink the features and can mimic the effect of a high mean-molecular mass, small scale height atmosphere. Thus, including this parameter allows us to account for this possible degeneracy and prevents us from overconfidently claiming a high-metallicity atmosphere. 3.6.1.2 Instrumental Offsets The instrumental offsets are nuisance parameters that can capture the extent to which transit depths from STIS G430L, STIS G750L, or WFC3 are biased rel- ative to the depths from the other instruments. This is motivated by the use of common-mode corrections in the light curve analysis of each instrument, which can potentially introduce a uniform bias for the depth at each spectral bin for that instrument. Offsets are also able to account for inter-instrumental transit depth 112 scatter introduced by uncertainty in orbital parameters a/Rs and inclination (Sec- tion 3.5.2.2). I explore three offset scenarios. The first is physically-motivated. In this scenario, I use Gaussian priors with sigmas determined by the uncertainties in the band-integrated transit depths to try to reflect the correlated uncertainty that exists between spectral bins for each instrument, whether due to common mode correc- tions or orbital parameter uncertainties. This essentially propagates the white light depth uncertainty into the retrieval. Derived in Section 3.5, these uncertainties are 105 ppm, 85 ppm, and 80 ppm for STIS G430L, STIS G750L, and WCF3, respec- tively. The second scenario investigates the potential impact of unknown sources of bias by setting a large, uniform offset prior for each instrument?s offset. The third scenario extends this, by setting two large, uniform priors: a WFC3 offset and a single offset for both STIS instruments. The third scenario allows the absolute depths at STIS to vary while preserving the optical spectral shape. I caution that offsets ? especially penalty-free, uniform prior offsets ? can cloak missing physics in a model. I do not think they should act as a safety net to achieve a good fit to a spectrum, and the inferred atmospheric properties should be understood in context. However, offsets offer a way to both investigate potential instrumental biases and account for absolute depth uncertainty for each instrument. It is valuable to include offsets as model parameters and marginalize over these possible values in order to understand how the uncertainty in the absolute transit depth for each instrument affects the marginalized posterior distributions of the other model parameters. 113 3.6.2 AURA A collaborator complements my analysis of HAT-P-41b by performing re- trievals on the STIS, WFC3 and Spitzer observations without the assumption of chemical equilibrium. They employ an adaptation of the retrieval code AURA (Pin- has et al., 2018), as described in (e.g., Welbanks & Madhusudhan, 2019). The code computes line by line radiative transfer in a transmission geometry and assumes hydrostatic equilibrium. We consider a one-dimensional model atmo- sphere consisting of 100 layers uniformly spaced in log10(P) from 10 ?6-102 bar. The pressure-temperature (P-T) profile in the atmosphere is retrieved using the P-T parameterization of Madhusudhan & Seager (2009). The measured radius of the planet Rp is assigned to a reference pressure level in the atmosphere through a free parameter Pref . The model atmosphere assumes uniform mixing ratios for the chemical species and treats them as free parameters. We consider sources of opacity expected to be present in hot Jupiter atmospheres (e.g., Madhusudhan, 2012) and include H2O (Rothman et al., 2010), Na (Allard et al., 2019), K (Allard et al., 2016), CH4 (Yurchenko & Tennyson, 2014), NH3 (Yurchenko et al., 2011), HCN (Barber et al., 2014), CO (Rothman et al., 2010), CO2 (Rothman et al., 2010), TiO (Schwenke, 1998), AlO (Patrascu et al., 2015), VO (McKemmish et al., 2016), and H2-H2 and H2-He collision induced absorption (CIA; Richard et al., 2012). The opacities for the chemical species are computed following the methods of Gandhi & Madhusudhan (2017). The CO2 abundance is restricted to remain below the H2O and CO abun- 114 dances as expected at these temperatures for H-rich atmospheres (Madhusudhan, 2012). We allow for the presence of clouds and/or hazes following the parameter- ization in Line & Parmentier (2016); MacDonald & Madhusudhan (2017). Non- homogenous cloud coverage is considered through the parameter ??, correspond- ing to the fraction of cloud cover at the terminator. Hazes are incorporated as ? = a?0(?/?0) ?, where ? is the scattering slope, a is the Rayleigh-enhancement factor, and ?0 is the H2 Rayleigh scattering cross-section (5.31 ? 10?31 m2) at the reference wavelength ?0 = 350 nm. Opaque regions of the atmosphere due to clouds are included through an opaque (gray) cloud deck with cloud-top pressure Pcloud. Lastly, we allow for the same three instrumental offset scenarios as described in Section 3.6.1.2. In these model runs, a constant offset in transit depth is applied to the data set of choice. The offset priors for each scenario are given in Table 3.8. In summary, the AURA retrievals consist of up to 25 parameters: 11 chemical species, 6 parameters for the P-T profile, 1 for the reference pressure, 4 for clouds and hazes, and up to 3 extra parameters for instrumental shifts. Table 3.8 shows the parameters and priors used in the AURA retrievals. 3.6.3 A note on metallicity and C/O Atmospheric metallicity is a broadly used term that does not always have the same definition or assumptions built into its derivation (Madhusudhan et al., 2014c; Kreidberg et al., 2014b). Here I define C/O and global atmospheric metallicity ex- 115 Table 3.8: Parameters and priors used in the AURA retrievals. Description Parameter Symbol Prior Distribution General Fractional Abundances Xi LU(?12,?1) Reference Pressurea Pref LU(?6, 2) T-P Profilea Zero-point Temperature T0 U(800, 2000) K Gradients ?1,2 U(0.02, 2.00) K?1/2 Gradient Pressures P1,2 LU(?6, 2) bar Isothermal Pressure P3 LU(?2, 2) bar Cloud/Haze Scattering Factor a LU(?4, 10) Scattering slope ? U(?20, 2) Cloudtop pressure Pcloud LU(?6, 2) bar Cloud fraction ?? U(0, 1) Instrument STIS - U(?500, 500) ppm Offsetsb STIS G430L - U(?500, 500)/N (0, 105) STIS G750L - U(?500, 500)/N (0, 85) WFC3 - U(?500, 500)/N (0, 80) a Section 3.6.2 defines the temperature-pressure profile parameters. b Instrumental offsets were employed in a subset of the retrievals and had either uniform or Gaussian priors as explained in section 3.9.2. 116 plicitly. For AURA, the abundances of different elements are derived independently from the corresponding gaseous absorbers - O/H from oxygen-bearing molecules such as H2O, Na from gaseous Na, and so on. This retrieval approach allows for different elements to be enhanced or depleted in different quantities. As such, there is no single metric for metallicity in this approach. Nonetheless, as described below, I use the O/H ratio from the retrieved H2O abundance using AURA as a proxy for metallicity in order to facilitate comparisons with PLATON retrievals in which all the elements are enhanced by a single metallicity parameter. In PLATON, metallicity is a factor that scales the solar elemental abundances, denoted M/H. The ratios between metals (e.g., Fe/O, Ti/V) are fixed to solar metal ratios, but the solar metal-to-hydrogen ratio is allowed to vary. Thus, all metals are scaled by the same factor. Then, the elemental carbon abundance is determined by C/O ? metallicity. Therefore, only carbon is allowed to differ from its solar ratio compared to other metals. While allowing carbon instead of oxygen to vary is arbitrary, C/O variation is motivated: it is the only metal-to-metal ratio for which predicted molecular abundances are typically sensitive to in the wavelength range covered by HST/Spitzer transit spectra. In this paradigm, metallicity is effectively a heuristic for O/H, since that dominates the retrieval both because of molecular opacity (e.g., H2O, TiO, VO) and because over 99% of the mean molecular weight is due to C, O, and H. In reality I retrieve O/H and C/O, then set all other elements to X = X O/H ? . Therefore, PLATON?s derived metallicity is reasonably comparableO/H to AURA?s derived O/H. 117 3.7 PLATON Retrieval Analysis The relative importance of each physical process that affects an observed tran- sit spectrum is not clear ahead of time. PLATON, though less flexible than free- chemistry retrievals or retrievals that allow elemental abundances to vary from solar ratios, is able to quickly perform chemically-constrained retrievals (?30 minute run- time for fiducial model retrieval on full data set). This makes it well suited for testing an array of models, which is important in order to determine how different model assumptions impact the conclusions of a retrieval. To explore this, I choose the fiducial model to be the set of parameters necessary to fully describe the simplest physical processes that I know affect the spectrum: opacity from gas absorption, CIA, and Rayleigh scattering. I then detail the effect of incorporating more com- plicated physics. In Section 3.7.3, I use both physical and statistical arguments to determine the ?best? model. However, it is important to show the sensitivity of the results to each model assumption to not provide overconfident constraints and to be able to predict how new observations might affect the conclusions. 3.7.1 Fiducial Model The priors for the fiducial model are shown in Table 3.9. The metallicity of the atmosphere, the temperature of the limb, and the C/O ratio are necessary to include in order to calculate molecular and atomic abundances, which determine the gas absorption, CIA, and Rayleigh scattering opacities. While I cannot im- prove constraints on Mp and Rs, it is best practice to include them as parameters 118 Table 3.9: Priors for parameters used in all PLATON retrievals. Parameter Symbol Prior Distribution Default Value Planet Radius Rp U(0.83, 2.48)a 1.65 RJup Limb Temperature T U(850, 2550)a 1700K Carbon-oxygen ratio C/O U(0.2, 2.0) 0.53b Metallicity Z LU(?1, 3) 1 Z Planet Mass Mp N (0.76, 0.14) 0.76MJup Stellar Radius Rs N (1.65, 0.08) 1.65 R Cloudtop Pressure Pcloud LU(?3, 8) 1 Pa a Range is 50?150% of the default value. b Solar C/O with Gaussian priors in order to propagate the uncertainties on those measurements (Zhang et al., 2019). Otherwise, I would mistakenly assume that Mp and Rs are precisely known. I also include the cloud top pressure of a grey cloud deck in the fiducial model. I fix the reference pressure to 1 bar and retrieve the planetary radius at that pressure. Note that Welbanks & Madhusudhan (2019) demonstrated that it is justified to assume a reference pressure and retrieve the planetary radius without affecting the ability to constrain atmospheric composition. Although Rp and Mp are both allowed to vary independently in PLATON retrievals, their uncertainties are not independent: the uncertainties for Mp are derived from log(gp) (from transit observations) and Rp (derived from Rp/Rs from transit and Rs from TIC-8). PLATON does not constrain Mp and R 2 p to match log(gp) a priori, however this is only an issue if regions of high likelihood extend to combinations of values that should not be allowed (i.e., more than 3-? from observed log(gp)). I re-derive log(gp) using values at Rp and Mp at the edge of significant likelihood and find good agreement with the prior, well within 3-?). 119 1.10 1.08 1.06 1.04 1.02 1.00 Median Model Solar Metallicity Model 0.98 Binned Median Model Observed 0.3 0.4 0.5 1.0 1.5 2.0 3.0 4.0 5.0 Wavelength [ m] Figure 3.5: Median retrieved model with 1-? and 2-? uncertainty contours for the fiducial model. Also shown is the median retrieved model when metallicity is fixed to solar. The median model and uncertainties are derived by generating 100 samples from the correctly-weighted posterior and calculating the depth at each bin for each sample. The contours are given by the 2nd, 16th, 50th, 84th, and 98th-percentile depths at each bin. The continuous model is smoothed with a Gaussian filter with ? = 15, which approximates the resolution of HST WFC3 (Zhang et al., 2019). I use uninformative priors where appropriate in order to fully explore the pos- sible parameter space. For quantities that can range over many orders of magnitude, such as the cloud top pressure or the metallicity, this means a log-uniform prior is necessary to avoid bias towards higher values. Otherwise ? for Tlimb, Rp, and C/O ? I use uniform priors with limits either set by the functionality of the code (e.g., C/O) or conservatively derived from previous observations. Widening the prior for any parameter in the fiducial model has no significant effect on the result of the retrieval. The retrieved median fiducial model with uncertainty contours is shown with the observed spectrum in Fig 3.5. The model is an excellent fit (reduced ?2=1.09; 120 Transit depth [%] consistent with the ?2 of the true model for 70 degrees of freedom to 1-?). I clearly detect water vapor (>5-? significance) via the 1.4?m water band in the WFC3 data. The bump in the STIS data is indicative of TiO, and the lack of any optical slope or flat-line indicates that grey clouds and scattering haze do not contribute signifi- cant opacity in the planet?s spectroscopically active region. The difference between the two Spitzer points is attributed to CO2, though since they are photometric observations I do not resolve any feature. The posterior probability distribution is represented by the corner plot (Foreman- Mackey, 2016)4 in Fig 3.6. This figure provides the marginalized posterior distri- bution for each parameter (with median, 16th, and 84th percentile values indicated by vertical dashed lines), as well as every two-dimensional projection of the poste- rior (with 0.5-, 1-, 1.5, and 2-? contours) to reveal covariances. Well-constrained parameters have narrow distributions with clear peaks, and slanted or diagonal shapes are indicative of correlated sets of parameters (e.g., the Rp-Rs shape). I find a super-solar metallicity (259+174?114? solar metallicity, henceforth Z ), a likely sub- solar C/O (C/O < 0.6) that is consistent with stellar C/O (0.19), a clear atmosphere (P +70cloud > 0.5mBar), and Tlimb = 1650?120K. The temperature is driven primarily by the STIS data, mostly because PLATON interprets the bump in the STIS data as a metallic oxide, which is only the dominant opacity source above ?1500K in chemical equilibrium. Below ? 1500K the optical spectrum would be dominated by an atomic sodium line, and this is not seen in the data. The upper limit on C/O is related mainly to the H2O: in equilibrium chemistry for T?1650K and P?1 bar, 4https://github.com/dfm/corner.py 121 Rs/R = 1.66+0.050.06 Mp/MJup = 0.76+0.130.12 1.0 5 .900 5 0.7 0.6 0 Rp/RJup = 1.60+0.050.05 1.7 0 1.6 5 1.6 0 1.5 5 Tp/103K = 1.65+0.070.13 1.8 1.6 1.4 1.2 logZ/Z = 2.41+0.220.25 2.5 2.0 1.5 1.0 C/O = 0.38+0.160.12 0.6 0 5 0.4 0 0.3 logP /Pa = 4.60+2.13cloud 2.22 7.5 6.0 4.5 3.0 1.5 0 1.6 1.6 5 .70 51 1.7 0.6 0 0.7 5 0 5 0.9 1.0 1.5 5 .60 .65 .70 1.2 1.4 1.61 1 1 1 .8 1.0 1.5 2.0 2.5 .30 .45 .60 1.5 .0 .50 0 0 3 4 6 .0 7.5 Rs/R Mp/MJup Rp/RJup Tp/103K logZ/Z C/O logPcloud/Pa Figure 3.6: Corner plot illustrating posterior probability distributions from PLA- TON for the fiducial model. The 16th, 50th, and 84th percentile values are indicated by vertical dashed lines and stated in the title of each parameter?s 1D marginalized posterior distribution. The contours indicate the joint 0.5-, 1-,1.5-, and 2 ? ? lev- els for each 2D distribution. The 1-? metallicity range is log10 Z/Z = 2.41 +0.23 ?0.25 (145?437? solar metallicity), the isothermal limb temperature is well constrained around 1650 K, and the C/O ratio is likely subsolar. The spectrum is consistent with a cloud-free atmosphere, and the marginalized posterior distributions of Mp and Rs are dominated by their priors. The slight correlations between T?logZ and Mp?logZ are due to their relation in the scale height equation. 122 logPcloud/Pa C/O logZ/Z Tp/103K Rp/RJup Mp/MJup H2O opacity decreases exponentially when C/O > 0.6 (Madhusudhan, 2012). Any model with C/O > 0.6 struggles to capture the water feature and relatively high infrared baseline opacity (compared to the optical) and is thus a poor fit to the data. The high metallicity is constrained by the size of both the STIS and WFC3 features, as well as the lack of a significant Rayleigh scattering slope. While the metallicity affects chemistry, it is primarily constrained via its effect on the mean molecular mass of the atmosphere. Increasing metallicity increases the ratio of metals to hydrogen by definition, which increases the atmosphere?s mean molecular mass. This lowers the atmospheric scale height and consequently decreases the predicted feature size. The equation for approximate feature size (??; Kreidberg, 2018), where ? is the mean molecular mass, clarifies its dependencies: ? TRp TR3p ?? ? (3.1) ?gpR2s ?M R 2 p s PLATON can decrease the feature size by changing Rp or Rs, but both are well- constrained by the continuum baseline as well as priors and thus are relatively fixed. It can also be lowered by decreasing the temperature, increasing the metallicity (and thus the mean molecular weight), or by increasing Mp. The temperature is strongly constrained by chemistry, and Mp is constrained by previous observations, so only metallicity can vary enough to explain the observed feature sizes. Note that this relationship explains the correlations between Mp, Tp, and metallicity seen in Fig 3.6: as mass increases or temperature decreases, metallicity decreases since a lower value is necessary to achieve the scale height that predicts the observed feature sizes. For 123 reference, the median derived mean molecular weight is 5.8 AMU and the derived scale height is roughly 322km. At solar metallicity, the model predicts features that are much larger than what the data shows. Consequently, solar metallicity atmospheres in the fiducial model can only explain the observed feature by invoking a cloud to mute the troughs of the features. The median retrieved model for metallicity fixed to solar is shown in Figure 3.5. Note that this model is dispreferred by 3.5-?, since the cloud leads to a poor fit to the bluest transit depths. The same metallicity value is an excellent fit for both the water feature in the WFC3 data and the TiO feature in the STIS data. Retrieving only on the STIS data or only on the WFC3 data recovers supersolar atmospheric metallicities. Addition- ally, due to predicting a greater abundance of CO2, it is better than low-metallicity solutions at explaining the large change in depth between the Spitzer points. Ob- servations from all three instruments support the high metallicity solution. 3.7.2 More Complex Models In this section I incorporate additional model parameters to explore if more complex physics impacts the inferred atmospheric parameters. I demonstrate the insensitivity of my results to model assumptions. 124 3.7.2.1 Partial Cloud Coverage Line & Parmentier (2016) showed that partial cloud coverage (i.e., clouds at the same height but not uniformly covering the limb azimuthally) could mimic the effect of a high mean molecular mass atmosphere for WFC3 spectra. When partial clouds are present, the observed spectrum would be the weighted average of the cloudy and clear spectra. The transit depth of a grey-cloud dominated atmosphere does not vary with wavelength, and so the cloudy spectrum is a straight line. Averaging a clear spectrum with molecular features and a cloudy, flat spectrum reduces the size of the features by an amount proportional to the cloud fraction. Given that I find a significantly supersolar metallicity, I investigate if this possible mean molecular mass-cloud fraction degeneracy affects the results from the fiducial case. When fit independently and allowing the cloud fraction to vary, both WFC3 and STIS spectra retrievals no longer constrain the metallicity to be supersolar. However, fitting the infrared and optical data jointly breaks this degeneracy, as predicted by Line & Parmentier (2016). Effectively, a low-mean molecular mass and non-uniform cloud solution should be impacted by Rayleigh scattering in the optical data, especially the bluest six wavelength bins. The dominance of gas absorption opacity over Rayleigh scattering opacity in the STIS data disallows this solution, breaking the degeneracy in favor of the high mean molecular mass solution. However, it is possible that removing assumptions made in the fiducal model ? such as fixed Rayleigh scattering or no instrumental offsets ? could muddle this decisive degeneracy break and allow for a low-metallicity solution. I investigate this 125 Table 3.10: Priors for parameters used in more complicated PLATON models. Parameter Symbol Distribution Default Cloud fraction fc U(0, 1) 1 Scattering slope ? U(?2, 20) 4 Scattering factor a0 LU(?4, 8) 1 WFC3 offset - U(?500, 500)/N (0, 80) 0 ppm STIS G430L offset - U(?500, 500)/N (0, 105) 0 ppm STIS G750L offset - U(?500, 500)/N (0, 85) 0 ppm STIS offset - U(?500, 500) 0 ppm Stellar Effective Temperature Tstar Fixed 6480 K Faculae Temperature T afac Fixed 6580 K Faculae covering fraction ffac U(0, 0.1) 0 Mie Particle Size rpart LU(?2, 0) ?m 0.1?m Mie Number Density n LU(1, 15) m?3 105 m?3 Fractional Scale Height Hcloud LU(?1, 1) 1.0 Hgas a Tfac = Tstar + 100K (Rackham et al., 2019) below. 3.7.2.2 Parametric Rayleigh Scattering The fiducial model assumes Rayleigh scattering. In lieu of complicated micro- physics, PLATON allows parametric scattering, in which the slope and the magni- tude of Rayleigh scattering vary in order to capture the possible signature of many hazes. For a more detailed explanation, see Zhang et al. (2019). Though there is no obvious signature of haze in the optical data (i.e., no linear slope decreasing with increasing wavelength), it is worth exploring if loosening the assumption of exact Rayleigh scattering affects the results. Allowing the full scattering parameter space (see Table 3.10) has little effect: the clear lack of slope in the STIS data conclusively leads to a haze-free atmosphere. Further, the median scattering factor is 0.01, implying that the data is easiest to 126 fit when opacity from Rayleigh scattering is muted. This complicates the mean molecular weight-cloud fraction (? ? fc) degeneracy. Lower values of ? are now possible, since the model no longer expects scattering opacity to be important at optical wavelengths. The lower the magnitude of Rayleigh scattering ? and the shallower the scattering slope ? the lower ? can be. This is because decreases in Rayleigh opacity allow for gas absorption to still be dominant at larger scale heights. As a result, a patchy cloud and low metallicity solution is viable. Though possible, the low ? solution requires a specific combination of cloud top pressure, scattering slope, scattering factor, and cloud fraction, and does not improve the fit. Therefore, it is much less likely than the high metallicity solution. The marginalized posterior probability distribution for metallicity has the same maximum likelihood value as the fiducial model. The difference is that the distribution has a tail extending to lower metallicities (Fig 3.7). The resulting median log metallicity and 1-? range (as determined by the 16th and 84th percentile values) is log10 Z/Z = 2.34 +0.27 ?0.64. Though all cloud fractions and cloud pressures are allowed, the posterior is consistent with a clear atmosphere due to the likelihood-desert in the upper-left corner of the cloud fraction-cloud top pressure pairs plot: clouds are only seen above the altitude corresponding to the ? 10 Pa pressure level at fractions below 0.50. 127 M /M = 0.76+0.14p Jup 0.13 T 3p/10 K = 1.62+0.090.22 1.8 1.6 1.4 1.2 a0 = 1.67+2.211.59 3.0 1.5 0.0 1.5 3.0 = 2.61+4.753.03 12 8 4 0 logZ/Z = 2.35+0.260.64 2.4 1.6 0.8 0.0 logPcloud/Pa = 2.61+3.553.12 7.5 5.0 2.5 0.0 .5 f = 0.45+0.312 c 0.27 0.8 0.6 0.4 0.2 .45 .60 .75 .90 .05 1.2 1.4 1.6 1.8 3.0 1.5 0.0 1.5 3.0 0 4 8 12 0.0 0.8 1.6 2.4 2.5 0.00 0 0 0 1 2 .5 5.0 7.5 0.2 0.4 0.6 0.8 Mp/MJup T /103p K a0 logZ/Z logPcloud/Pa fc Figure 3.7: Corner plot illustrating the posterior probability distribution from PLA- TON for the partial cloud and parametric scattering case. Rp, Rs, and C/O are not shown for clarity, since their marginalized posterior distributions are the same as in the fiducial case. The 1-? metallicity range is shifted down to log10 Z/Z = 2.34 +0.27 ?0.64. Note that at fc = 0 or logP 2.5 cloud > 10 , I recover the fiducial marginalized posteri- ors. 128 fc logPcloud/Pa logZ/Z a0 Tp/103K 3.7.2.3 Instrumental Offsets I model an instrumental offset as a constant value added to the forward model?s binned transit depth in the wavelength range of the instrument of interest. For the physically motivated scenario (Scenario 1 from Section 3.6.1.2) , I set the priors for STIS G430L, STIS 750L, and WFC3 to be Gaussians centered on zero ppm with widths set to the uncertainty on the transit depth from their white light curves (105, 85 and 80 ppm, respectively; see Table 3.10). The retrieved median WFC3 offset is non-trivial, with a median of about 1.5? the white light uncertainty (130 ? 50 ppm). The offsets in the STIS G430L and G750L are less significant, at 58? 58 ppm and ?65? 55 ppm, both well within their white light uncertainties. However, there is a significant median offset of ? 120 ppm between the two instruments. This is driven primarily by the retrieval attempting to align transit depths in the overlapping wavelength region between the instruments. The Spitzer 3.6?m point drives the WFC3 offset: shifting the WFC3 depths down necessitates a smaller radius ratio, which better captures the relatively low transit depth at 3.6?m. The ability to better capture the Spitzer 3.6?m results in a higher evidence, indicating that this model is strongly preferred over the fiducial model (for a more detailed discussion, see Section 3.7.3). When combined with partial clouds, the instrumental offsets cause a small decrease in the retrieved median metallicity (log10 Z/Z = 2.33 +0.23 ?0.25). This is for similar reasons as explained in the parametric scattering section; whereas parametric scattering justified the absence of an optical scattering slope in the low-metallicity 129 solution by effectively removing Rayleigh scattering opacity, the instrumental offset model can decrease the WFC3 depths relative to the STIS depths to artificially allow for it. Note that increasing STIS depths (instead of decreasing WFC3 depths) has the same affect on Rayleigh opacity and thus metallicity. However, it is not a viable solution since ? unlike decreasing WFC3 depths ? it does not improve the forward model?s ability to capture the low Spitzer 3.6?m point. Allowing Gaussian-prior instrumental offsets had no significant effect on the results. However, it is possible that there is some unknown wavelength-independent systematic that biases the absolute transit depths of the instruments relative to one another. Though unlikely, to explore this I allowed offsets in the STIS G430L, STIS G750L, and WFC3 data to vary by about 5% (500 ppm) in either direction (Scenario 2 from Section 3.6.1.2). Due to the model preferring a lower radius ratio to best explain the Spitzer 3.6?m point, the median WFC3 offset is a 250 ppm decrease, about 3? the white light uncertainty. Surprisingly, this large offset does not significantly change the 1-? ranges for metallicity (log10 Z/Z = 2.26 +0.24 ?0.40). The size of the molecular features and the large differential between Spitzer photometric points drive the supersolar metallicity in this case. Regardless of the magnitude of the offset, I retrieve a high metallicity. The two large, uniform offset case, where both STIS instruments are offset by the same amount (Scenario 3 from Section 3.6.1.2), retrieves effectively identical posterior distributions as Scenario 2. While it is worthwhile to understand the effect on the retrieval, there is no reason to expect such large instrumental offsets for HAT-P-41b. A transit depth 130 offset can be caused by the necessity of analyzing each instrument differently. For example, not handling limb darkening consistently and not using consistent orbital parameters (i.e., inclination) for each analysis might cause an offset, but this is easily fixed and is not an issue for my dataset. Since the instruments? observations are from different dates, it is also possible that stellar variability could cause an offset. However, I have long-term photometry (Sec 3.3.3) that shows no such variability. Additionally, the STIS depths are in good agreement with HST UVIS observations (Wakeford et al., 2020). There is no indication that this particular observation is biased in any way, and unresolved companions are confidently ruled out (Evans et al., 2018). The most plausible source is unaccounted for uncertainties or bias from the spectral analysis, as the WFC3 spectrum derived in this chapter is shifted up ? 90 ppm relative to the literature spectrum (Tsiaras et al., 2018), as noted in Section 3.5.2.2. However, this is still well below the 250 ppm value preferred by the large, uniform offsets models. I determine that offsets beyond the physically motivated values are unlikely. 3.7.2.4 Stellar Activity Section 3.4.1 demonstrated that HAT-P-41 is consistent with a quiet star and stellar activity is not expected to impact the transit spectrum. However, to be conservative, I investigated if allowing for greater stellar variability impacted my conclusions. The typical signature of un-occulted, cool starspots is to mimic a haze-like 131 slope in the transit spectrum, and such a signature is clearly absent in the derived transit spectrum of HAT-P-41b. On the other hand, the signature of hot faculae is a steep optical drop-off towards shorter wavelengths (Rackham et al., 2019). Given that I see a drop in transit depths in the optical, the retrieval could plausibly be affected if faculae dominate over star spots, and so I focus on a faculae overabun- dance. I assumed that the temperature of the stellar photosphere equals the stellar effective temperature. I modeled the faculae following the prescription from Rack- ham et al. (2019) and, accordingly, fixed the faculae temperature to Tphot + 100K. PLATON weights the contributions from the different temperature regimes via the fractional coverage parameter, which represents the overabundance of faculae in the unocculted regions. Rackham et al. (2019) states that moderately active F5V-dwarfs will have around 1% faculae coverage, and up to about 7% on the more active end. This is the faculae fraction, which is likely much higher than the faculae overabun- dance. However, I set a conservative uniform prior on the fractional coverage of 0-10% in order to determine if high activity would significant alter my conclusions. I find that including stellar activity has no effect on the posterior probability distribution. It may seem that the STIS data could be explained by a featureless flat line and stellar activity instead of a TiO feature. However, the overabundance of faculae necessary to explain the drop in bluest six points (0.32?0.42?m) produces a poor fit to the rest of the STIS data. Therefore, even when including stellar variability, TiO is necessary to explain the STIS depths. Allowing a wider range of faculae temperatures also had no effect. 132 In summary: there is no evidence of stellar variability from prior observations, and allowing for activity does not affect the retrieval. 3.7.2.5 Mie Scattering Benneke et al. (2019a) recently invoked Mie scattering to explain anomalously low Spitzer transit depths. Given the relatively low value of HAT-P-41b?s Spitzer 3.6?m depth relative to the rest of the spectrum ? the fiducial model?s predicted depth at 3.6?m is about 3.3-? away from the observed depth ? I included Mie scattering in this analysis. In PLATON, Mie scattering can be used in lieu of parametric Rayleigh scattering. Each condensate is described by a wavelength dependent complex refractive index, n-ik, where n is the real part and k is the imaginary part of the index. This index explains how that particular condensate interacts with light with wavelengths similar to the particle size. PLATON assumes log-normal distribution in particle size with geometric standard deviation 0.5 to determine the abundance of different radii condensates for a given mean particle size (Zhang et al., 2019). The other relevant factors are cloud height (condensates are only relevant above that pressure; below it gray cloud opacity dominates), particle density at the cloud top pressure, and condensate scale height. The condensate scale height is parameterized as a fraction of the gas scale height, and it describes how the abundance of Mie scattering particles decreases with height. The refractive index is fixed for a given condensate, and the other four parameters are fit for in the retrieval (see 3.10). 133 PLATON tests one Mie scattering species at a time. Only a few species ex- pected to form clouds in hot Jupiter atmospheres have condensation temperatures above HAT-P-41b?s limb temperature (T ? 1600K) (Wakeford & Sing, 2015). Those five (SiO2, Al2O3, CaTiO3, FeO, and Fe2O3) fall into two phenotypes: ?low-n? with real refractive index n ? 1.5 and ?high-n? with n ? 2.5. Though the k values vary more significantly, I find that they do not have a significant impact on the absorption cross section of the condensates. I use n and k values from Kitzmann & Heng (2018), which I average over the relevant wavelength range (0.3?5?m). The n values are flat over this range, and so the average is an excellent approximation. I tested retrievals with both the low-n (corundum; Al2O3) and high-n (hematite; Fe2O3) phenotypes. The priors for the fittable parameters are shown in Table 3.10. The prior for cloudtop pressure is the same as the fiducial model. Since the condensate radii must be such that they cause a relative drop in opacity around 4?m (i.e., increase opacity more in the near-UV by more than around 4?m), I can constrain the mean particle size reasonably well. I set the prior to be log-uniform with a range that contains all plausible values. The number density is not known ahead of time, so I set an uninformative log-uniform prior; widening the prior further did not affect the retrieval. Finally, it is unclear what physical constraints there are on condensate scale height. Fortney (2005) finds that condensate scale heights can be one third of the gaseous scale height for hot Jupiters, and Benneke et al. (2019a) found Hpart/Hgas ? 3 for a sub-Neptune. Using these values as guides, I set a conservative uniform prior on the fractional scale height and constrain it to be in 134 the range 0.1?10. Including Mie scattering opacity does not noticeably affect the results of the retrieval, and the Mie scattering parameters are not constrained by the retrieval. The inferred small gaseous scale height ? which dampens features and is necessary to explain STIS and WFC3 feature sizes ? makes it difficult to explain the large variations in the radius ratio. Combining Mie scattering with partial clouds ? phys- ically, an atmosphere with patchy clouds and Mie scattering particles distributed only above those clouds ? alleviates the issue of explaining the large transit depth variation. Since partial clouds allow for higher scale heights, Mie scattering could then cause a larger drop in transit depth near Spitzer without needing to invoke an unreasonably high fractional scale height. Figure 3.8 shows the corner plot for this model. Though the Mie scattering parameters are not constrained, at number densities above ? 108 m?3, particle radii around 0.15?m, and condensate scale heights greater than the gaseous scale height, lower metallicity and temperature values are possible. This is because added Mie opacity tends to mute features near its peak opacity. This provides a physical reason to expect smaller spectral features, and so a less small scale height is necessary to fit the features. The net impact is a decreased ? but still supersolar ? median metallicity of log10 Z/Z = 2.27 +0.30 ?0.55 135 logn = 9.08+6.805.24 logrpart/m = 6.96+0.590.66 6.4 6.8 7.2 7.6 H /H = 0.05+0.67part gas 0.62 0.8 0.4 0.0 0.4 0.8 Mp/MJup = 0.75+0.130.13 5 1.0 0.9 0 0.7 5 0 0.6 45 Rp/RJup = 1.59 +0.07 . 0.070 1.6 8 1.6 0 1.5 2 4 1.4 Tp/10 3K = 1.61+0.090.17 0 1.8 .651 0 1.5 1.3 5 .201 logZ/Z = 2.27+0.290.55 2.5 2.0 1.5 1.0 logPcloud/Pa = 3.27+3.133.50 7.5 5.0 2.5 0.0 2.5 fc = 0.35 +0.29 0.21 0.8 0.6 0.4 0.2 4 8 12 16 7.6 7.2 6.8 6.4 0.8 0.4 0.0 0.4 0.8 0.4 5 .60 .75 00 0 0.9 1.0 5 1.4 4 .52 .60 .68 0 51 1 1 1.2 1.3 1.5 0 1.6 5 1.8 0 1.0 1.5 2.0 2.5 2.5 0.0 2.5 5.0 7.5 0.2 0.4 0.6 0.8 logn logrpart/m Hpart/Hgas Mp/MJup Rp/RJup T 3p/10 K logZ/Z logPcloud/Pa fc Figure 3.8: Corner plot illustrating posterior probability distribution from PLATON for the fiducial plus Mie scattering and partial clouds model. Rs, and C/O are not shown for clarity, since their marginalized posterior distributions are the same as in the fiducial case. The 1? metallicity range is shifted down a bit to log10 Z/Z = 2.27+0.30?0.55. Note that I recover the fiducial marginalized posteriors when any of the mean particle size, the particle number density, or the condensate scale height are too small. 136 fc logPcloud/Pa logZ/Z Tp/103K Rp/RJup Mp/MJup Hpart/Hgas logrpart/m 3.7.3 Model Selection Section 3.7.2 stepped through the PLATON retrieval for increasingly complex models, examining both how each additional parameter affected the posterior and why it affected it in that way. While knowing the effect of each model assumption is useful, it is important to determine a preferred model in order to effectively convey the results. In this section, I use Bayesian model comparison to select the best model. Model selection is as important as parameter estimation in atmospheric re- trievals. I determine the preferred model by a combination of physical arguments and Bayesian statistics. Specifically, I check if it is necessary to consider more com- plicated physics using the odds ratio, which is the Bayes factor between models (de- fined as the ratio of their evidences) multiplied by their prior probability ratio. The prior probability ratio is typically assumed to be one (i.e., the models are assumed to be equally likely). The odds ratio determines if one model should be preferred over another by intrinsically rewarding better fits while punishing overcomplicated structure (Trotta, 2008). This is entirely data-and-model defined, assuming appro- priately uninformative priors are used. I compare the Bayesian evidences of each model in order to determine which should be favored. The Bayesian evidences and 1-? metallicity ranges for every model discussed in Section 3.7 are shown in Table 3.11. The 1-? range is represented by both the median metallicity with quantiles (i.e., the central 68% of metallicity values) The 1-? metallicity ranges are included to illustrate the uncertainty caused by model 137 choice. The retrieved atmospheric metallicities are remarkably consistent across the models, and a supersolar metallicity is ubiquitous. This demonstrates that under PLATON?s assumptions, supersolar metallicity is a robust conclusion. Figure 3.9 emphasizes the insensitivity of the atmospheric parameters to model assumptions. This shows the one dimensional marginalized posterior distributions for metallicity, temperature, and C/O for five of the models I examined. These specific models are shown because they are ?interesting? in that they differ from the fiducial model?s posteriors the most. I emphasize that these are these are the models that most differ from the fiducial case. While including instrumental offsets tends to flatten the distributions, the peaks of all of the models are consistent. I define the model selection-relevant columns here: ? lnZ is the natural log of the Bayesian evidence. A higher value indicates the model is better able to describe the data without overfitting. ? O is the odds ratio in favor of a model over the fiducial model. It is the product of their Bayes factor and their prior probability ratio. The prior probability ratio is often assumed to be one, as is the case here. This can be directly interpreted: an odds ratio of 100 indicates 100:1 odds in favor of the more complex model. Values less than one indicate evidence against the corresponding model. ? Interpretation This is the empirically derived interpretation of odds ratio based on the Jeffreys? scale (Trotta, 2008). Table 3.11 contains every notable model I considered. I did not do an iterative 138 Fiducial 1.75 F + PC + Parametric ScatteringF + PC + 3 Uniform Offsets F + PC + Mie F + PC + Gaussian Offsets 1.50 1.25 1.00 1.2 1.4 1.6 1.8 2.0 2.2T/103K 0.75 0.50 0.25 0.00 1 0 1 2 3 0.2 0.4 0.6 0.8 1.0 logZ/Z C/O Figure 3.9: Marginalized posterior distributions for metallicity, temperature, and C/O from select models compared to solar values. Note that stellar C/O = 0.19 (Table 3.4). Though some models have low metallicity tails, the 68% credible interval for metallicity is robust (Table 3.11). Offsets allow for higher temperatures and C/O ratios, while both parametric and Mie scattering allow for lower temperatures. combination of every model scenario, for two primary reasons. Most importantly, I am weary of overfitting the data. The fiducal model is already an excellent fit to the data (?2? = 1.09), so I must be careful about adding complications. Layering multiple parameter physical processes, such as Mie scattering and stellar activity, involves an extra five parameters and significantly overcomplicates the fit. Instead, I only combine complications when there is a physically motivated reason to do so, e.g. partial clouds. The second reason is computational difficulty. Some model combinations have enough free parameters to describe the data with many different 139 PDF PDF PDF Table 3.11: Evidence and metallicity ranges for each plausible model. Only the models including instrumental offsets are preferred over the fiducial model. No model assumption changes the conclusion of a supersolar atmospheric metallicity. Model log a b10Z Z lnZc Od Interpretatione Fiducial (F) 2.41+0.23?0.25 145?437 551.6 Ref Default Model F + Partial Clouds (PC) 2.38+0.22?0.37 102?398 551.7 1.1 Inconclusive F + PC + Parametric Scattering 2.34+0.27?0.64 50?407 550.7 0.4 Inconclusive F + PC + Stellar Activity 2.41+0.27?0.41 100?479 551.8 1.3 Inconclusive F + Mie Scattering 2.41+0.24?0.29 132?447 551.0 0.6 Inconclusive F + PC + Mie 2.27+0.3?0.55 52?372 551.8 1.3 Inconclusive F + PC + 3 Gaussian Offsets 2.33+0.23?0.25 120?363 556.4 122 Strongly preferred F + PC + 3 Uniform Offsets 2.26+0.24?0.4 72?316 556.9 213 Strongly preferred F + PC + 2 Uniform Offsets 2.30+0.26?0.39 81?363 554.9 29 Moderately preferred a Median log metallicity with 16% and 84% quantiles, in units of log solar metallicity b 68% credible interval for metallicity, in units of solar metallicity c Natural log of Bayesian Evidence d Odds ratio between model and the fiducial model e According to Jeffreys? scale (Trotta, 2008) combinations, and so the retrieval does not converge on the timescale of weeks. Mie scattering combined with offsets falls in this category. However, given the overfitting concerns and the lack of evidence for just Mie scattering, I do not think this is a worrying omission. It is generally best practice to assume the simplest model unless there is ev- idence in favor of extra parameters. That is why I list the fiducial model as the reference, and determine the evidence of the more complicated models. If the ev- idence of the model with extra parameters is not significantly greater, it means that the ability to explain to data was not improved enough to justify the added complexity. This essentially quantifies Occam?s razor. Following this logic, I determine the ?fiducial + partial clouds + 3 Gaussian offsets? model to be the best model. Only the Gaussian offset and uniform offset 140 models are preferred over the fiducial model. While the three uniform offsets model has the highest evidence/weight, the odds ratio between that and the Gaussian offsets model is 1.75. This is inconclusive on the Jeffrey?s scale, meaning I am unable to distinguish between the models by evidence alone. Instead, I favor the Gaussian offsets model as more plausible, since its Gaussian priors are physically motivated by common-mode corrections. The evidence for partial clouds is inconclusive, however partial clouds are more plausible than assuming 100% cloud coverage, as argued by MacDonald & Mad- husudhan (2017); Welbanks & Madhusudhan (2019). Therefore, to be conservative, I choose the model which account for partial clouds as the ?best? model. The odds ratio only works in a direct comparison of two models and is not a statement on the absolute goodness-of-fit. The reduced chi-squared test statistic is a useful sanity check to ensure that the model is able to explain the variance in the data. The value for the best model is an ideal ?2? = 1.0. The results section (Section 3.8) ? and the abstract values ? are based on parameter estimation from the ?fiducial + partial cloud + Gaussian offsets? model. 3.7.3.1 Bayesian Model Averaging Instead of model selection, it is possible to take a weighted average of the results from each model and therefore automatically take their respective evidences into account (Gibson, 2014b; Wakeford et al., 2016a, 2018). The benefit of Bayesian model averaging is the ability to quantify uncertainty in model selection, as well 141 as avoiding having to arbitrarily choose between models with slightly different evi- dences. However, it requires a few assumptions: it is only valid if the set of models comprises the full model space, i.e, at least one model is a good description of the data. The weight-averaged uncertainties assume Gaussian-distributed posteriors, which is not strictly correct. However, it is useful in combining information from every model. Here, I show the assumptions I make to use Bayesian model averaging. The ?2? values for the models I tested are clustered around one, so it is fair to assume that a ?correct? model is contained in the set. Figure 3.9 shows that although the posteriors are not perfectly Gaussian, they have sharp, unimodal peaks, and so the uncertainty derived from marginalization is informative. The model weights are defined by Eq 3.2 (adapted from Gibson (2014b)). Wq is the weight assigned to model q, P (Mq|D) is the the likelihood of model q given the data, and P (D|Mq) is the likelihood of the data given model q, which is equivalent to the Bayesian evidence of model q, Eq. The denominator is a normalization term, summed over N models. I assume a conservative prior that each model is equally likely (P (Mi) = 1 for all i). ?P (Mq|D) ?P (D|Mq)P (Mq) EqP (Mq)Wq = = = ? (3.2)N i=1 P (Mi|D) N i=1 P (D|M N i)P (Mi) i=1EiP (Mi) The marginalized log metallicity with 1-? uncertainties is calculated from equa- tions 15 and 16 from Wakeford et al. (2016a). The result is log Z/Z = 2.29+0.2410 ?0.36 (194+144?109? Z ). As expected, the highest weighted models are the offset models. 142 Bayesian model averaging demonstrates that in PLATON?s chemical equilibrium framework, a supersolar metallicity is the most likely result even after accounting for uncertainty in model selection. The marginalized metallicity is useful as a reference, but it is valuable to give the metallicity distribution for each specific model assumption. Marginalization is most appropriate when the specific model parameters are unimportant, however I am interested in the impact that modeling assumptions have on the atmospheric parameters. I emphasize that even for apparently ?data-defined? methods, many assumptions have to be made and those should be explicitly stated for an appropriate interpretation. 3.8 Results for the Favored PLATON Model In Section 3.7.3 I argued that the best PLATON model scenario is the fiducial model with partial clouds and physically motivated, Gaussian prior instrumental offsets added. The retrieved median spectrum with uncertainty contours is shown in Figure 3.10. It is an excellent fit to the data, with ?2? = 1.0. In this section I discuss the details of the retrieved atmospheric parameter values. 3.8.1 Summary of Retrieved Parameters The posterior distribution corner plot is shown in Figure 3.11. Both atmo- spheric metallicity and temperature are well constrained, and the C/O ratio, though relatively flat, has a strict upper limit. The median retrieved metallicity is super- 143 1.10 1.08 1.06 1.04 1.02 1.00 Median Model 0.98 Binned Median Model Observed 0.3 0.4 0.5 1.0 1.5 2.0 3.0 4.0 5.0 Wavelength [ m] Figure 3.10: Median retrieved model with 1-? and 2-? uncertainty contours for the favored PLATON model (fiducial model with partial clouds and instrumental shifts with physically-motivated Gaussian priors). solar (log10 Z/Z = 2.33 +0.23 ?0.25), and solar metallicity is inconsistent to 3-? (lower limit 4.8? Z ). As noted in Section 3.6.3, PLATON?s derived metallicity is a proxy for [O/H], enabling comparison with its host star?s oxygen abundance ([O/H]=0.37; Table 3.4). PLATON determines HAT-P-41b to be metal-enriched relative to its host star ((log Z/Z = 1.97+0.2310 star ?0.25), and it is inconsistent with the stellar metal- licity to to 3-? (lower limit 2.1? Zstar). The planetary C/O (0.44+0.18?0.15) has a 3-? upper limit of 0.83. Though the planetary C/O is technically inconsistent with the stellar C/O to 1-? (0.19; Table 3.4), the comparison is not valid as the planetary C/O prior had a computational lower limit of 0.20, and the posterior has signifi- cant likelihood at that limit. This ?piling? at the prior boundary implies that the planetary C/O is consistent with the stellar C/O. The median isothermal limb tem- perature (Tlimb = 1710 +100 ?80 K) is close to the equilibrium temperature of the planet 144 Transit depth [%] R /R = 1.59+0.06p Jup 0.06 Tp/103K = 1.71+0.100.08 1.9 5 .801 .651 0 1.5 1.3 5 logZ/Z = 2.33+0.220.25 2.5 2.0 1.5 1.0 C/O = 0.44+0.180.15 0.7 5 .600 .450 0.3 0 logPcloud/Pa = 3.07+3.292.87 7.5 5.0 2.5 0.0 .5 f = 0.44+0.332 c 0.28 0.8 0.6 0.4 0.2 WFC shift = 132.64+50.8948.09 0 60 12 0 801 24 0 G430 shift = 58.13+57.6857.76 60 0 60 0 12 0 18 G750 shift = 66.26+54.6854.57 18 0 12 0 60 0 60 .50 .56 .62 .68 .35 .50 .65 0 5 .0 .5 .0 .5 0 5 0 5 .5 .0 .5 .0 .5 .2 .4 .6 .8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1.8 1.9 1 1 2 2 0.3 0.4 0.6 0.7 2 0 2 5 7 0 0 0 0 24 18 12 6 18 12 6 6 6 6 12 18 Rp/RJup T 3p/10 K logZ/Z C/O logPcloud/Pa fc WFC shift G430 shift G750 shift Figure 3.11: Corner plot for the best PLATON model (fiducial with partial clouds and Gaussian offsets). Rs and Mp are prior dominated and are excluded for clarity. The offsets are given in parts per million; for example, the median WFC3 offset indicates the retrieval favors shifting the WFC3 depths down by ? 132 ppm. 145 G750 shift G430 shift WFC shift fc logP 3cloud/Pa C/O logZ/Z Tp/10 K (Teq = 1960 K), which implies an efficient heat recirculation. These parameters lead to a high mean molecular weight (? ? 5.5 AMU) atmosphere with a scale height of about 320 km. The retrieved results are consistent with a clear atmosphere. Though cloud top pressure and cloud fraction are unconstrained, their joint marginalized poste- rior is constrained. A uniform grey cloud is only allowed deeper than ? 10 Pa (0.1 mBar), and clouds above that pressure are only possible if they cover less than about 40% of the limb. Hazes are dispreferred by model selection, and the median scattering opacity was 50? weaker than Rayleigh scattering in the model which allowed parametric scattering. The retrieved relative shift between the STIS G430L and G750L instruments is 120ppm, due in part to the model attempting to align their overlapping regions. A downshift for the WFC3 data is preferred (WFC3 offset = ?132? 50 ppm). The stellar radius, the planetary mass, and the planetary radius are consistent with the prior values. The planetary mass and stellar radius are, as expected, dominated by their priors. The planetary radius (Rp = 1.59? 0.06) is at the reference pressure of 1 Bar, and when calculated at the planet?s photosphere it is consistent with the planetary radius derived based on stellar parameters from TIC-8. 3.8.2 Evidence of Water and Optical-Wavelength Absorbers While the spectral features in STIS, WFC3, and Spitzer are attributed by PLATON to TiO, H2O, and CO2, respectively, the retrieval only robustly detects 146 Table 3.12: PLATON species detection evidences Species Oa Detection Significanceb H2O 46630 5.0? TiO 2.1 1.9? VO 2.3 1.9? TiO/VO 9.4 2.7? Na 1.1 1.2? CO2 3.3 2.1? CO 0.4 N/A a Odds ratio between model and the preferred PLATON model (fiducial model with partial clouds and Gaussian-prior instrumental shifts included) b Benneke & Seager (2013) H2O - the H2O abundance is constrained by observations, while the abundances of other species are primarily constrained by the assumption of chemical equilibrium. I note that while CO is more abundant than CO2, CO2 has a much larger cross section at 4.5?m, such that even with a smaller abundance its opacity dominates over that of CO at the temperatures and C/O ratios inferred by the retrieval. I determine if a species is detected by finding the odds ratio between the best model with and without opacity from a particular species. This breaks the assump- tion of chemical equilibrium, so it is not strictly correct, but it is a useful heuristic nonetheless. A species is considered detected only when the odds ratio significantly favors the model with the species? opacity. Table 3.12 shows the odds ratios ? and their more familiar frequentist analog, the detection significances (Benneke & Seager, 2013) ? for several relevant spectroscopically active species. The odds ratio in favor of H2O is? 46630, indicating that the model with water is 46630? more likely than the model without water opacity. This is equivalent to 147 a 5.0-? detection in frequentist terms. The odds ratio in favor of CO2 is 3.3, which is barely enough evidence to claim a weak detection. PLATON finds no evidence of Na, and CO is dispreferred. The odds ratios for TiO and VO are 2.1 and 2.3, respectively, and these are not favored enough to claim detections (less than 2-?). However, TiO and VO are only seen as non-detections because they have similar cross-sections. When TiO opacity is ignored, the retrieval can compensate because VO opacity is able to describe the STIS feature just as well as TiO. If I ignore both VO and TiO then the model cannot describe the STIS data as well, and so the odds ratio in favor of TiO/VO is 9.4 (2.7-?). Therefore, I find suggestive evidence of metallic oxide opacity, but I am unable to discern if it is due to TiO or VO. Based on the assumption of chemical equilibrium at the retrieved temperatures, PLATON attributes the STIS feature to TiO because it is more abundant and opaque in the spectrscopically active region for a solar Ti/V ratio. 3.9 AURA Retrieval Analysis and Results A collaborator performed a second, complementary atmospheric retrieval anal- ysis: a series of free-chemistry retrievals on HAT-P-41b using AURA (3.6.2) to con- strain the atmospheric properties at the day-night terminator of the planet while allowing for deviations from chemical equilibrium. First, we consider the presence of different chemical species in the atmosphere of HAT-P-41b using its full broad- band spectrum. Then, we consider the presence of possible transit depth offsets between data sets and their possible impact in the derived chemical abundances 148 and associated metallicities. 3.9.1 Evidence of Water and Optical-Wavelength Absorbers We perform a full retrieval on the broadband spectrum of HAT-P-41b and present the observations and retrieved median spectrum in Figure 3.12. The full re- trieval provides constraints on the presence of H2O, and provides indications for the presence of Na and/or AlO in the optical. The full retrieval finds log10(XH2O) = ?1.65+0.39 +1.03?0.55, log10(XNa) = ?3.09?1.83 and log (X ) = ?6.44+0.6610 AlO ?0.91. While the re- trieval with PLATON prefers TiO/VO to explain the STIS observations, the re- trieval with AURA does not, and instead prefers a combination of Na and AlO. The retrieved TiO abundance is low and unconstrained (log (X ) = ?9.58+1.3710 TiO ?1.50). Neither the CO nor CO2 abundances are constrained by the retrieval. While the cloud/haze parameters are not tightly constrained, the retrieval indicates a coverage fraction of ?? = 0.25+0.26?0.16 consistent with a mostly clear atmosphere. The tempera- ture profile of the atmosphere is mostly unconstrained. We infer the temperature near the photosphere, at 100mbar, to be T = 1345+349?206 K. The posterior distributions for the relevant parameters are shown in Figure 3.13. I utilise this full retrieval as a reference model to perform a Bayesian analysis and assess the impact of not considering some of these parameters in the models. This change in model evidence is then converted to its more familiar frequentist counterpart, a detection significance (DS) following Benneke & Seager (2013). Ta- ble 3.13 shows the different models considered, their model evidence, DS, and ??2. 149 1.10 Retrieved Model Smoothed 1? 2? 1.08 1.05 1.03 1.00 0.98 0.3 0.4 0.5 0.6 0.7 0.80.91.0 2.0 3.0 4.0 Wavelength (?m) Figure 3.12: Retrieved spectrum of HAT-P-41b using STIS, WFC3 and Spitzer data. Observations are shown using blue markers. The retrieved median spectrum is shown in red while the 1-? and 2-? regions are shown using the shaded purple areas. We find a robust detection of H2O at a 4.89-? confidence. There is suggestive ev- idence of Na and/or AlO with confidence levels of 2.09-? and 2.58-?, respectively. The removal of TiO from the models results in an increase in the model evidence, indicating a disfavor for this molecule to be present in these models. VO is similarly undetected. However, removing opacity from the three primary metal oxides (TiO, VO, and AlO), finds a moderate-to-strong ?detection?, with 3.59-? confidence. This is similar to PLATON, which did not find evidence of TiO or VO individually, but found weak-to-moderate evidence of their combined presence (Sec. 3.8.2). This can be interpreted as follows: AURA is confident (to 3.6-?) that the sharp dip in the blue STIS data (0.4?0.5?m) is a real molecular feature due to a metallic oxide. The retrieval finds that the most likely candidate for the metallic oxide is AlO, as shown by it?s 2.6-? preference, whereas TiO and VO are individually dispreferred. 150 Transit Depth (%) Table 3.13: Impact of AURA model assumptions on retrieved abundances. Model log10(XH2O) log10(XNa) log10(XAlO) log a 10 Z/Z ln(Z) ??2 DS Full model ?1.65+0.39?0.55 ?3.09 +1.03 +0.66 +0.39 ?1.83 ?6.44?0.91 1.72?0.55 559.1 0.93 Ref. No H2O N/A ?2.41+0.99 +0.99?2.99 ?5.71?1.39 N/A 548.9 1.37 4.89 No Na ?1.62+0.42?0.67 N/A ?6.90 +0.84 ?1.05 N/A 558.0 0.95 2.09 No AlO ?1.49+0.35?0.70 ?4.32 +1.88 ?4.31 N/A N/A 557.0 1.03 2.58 No TiO ?1.70+0.41 ?2.97+0.95 ?6.39+0.66?0.56 ?1.25 ?0.88 N/A 559.7 0.92 N/A No Metal Oxides ?1.52+0.38 +1.28?0.91 ?3.59?1.47 N/A N/A 554.2 1.21 3.59 Simpler model ?1.65+0.40?0.63 ?2.60 +0.94 +0.51 +0.40 ?1.10 ?5.81?0.66 1.72?0.63 560.0 0.89 N/A Gaussian shiftsc ?1.91+0.53?0.68 ?2.38 +0.81 ?1.33 ?6.64 +0.70 1.46+0.53?0.96 ?0.68 562.0 0.90 N/A Uniform shifts (3)c ?2.96+0.98 ?2.43+0.84 ?7.05+0.75 0.40+0.98?0.88 ?1.34 ?0.94 ?0.88 561.8 0.88 N/A Uniform shifts (2)d ?3.34+1.00?0.86 ?3.43 +1.35 ?2.19 ?6.98 +0.77 +1.00 ?0.78 0.03?0.86 560.7 0.89 N/A aThe metallicity is approximated from water abundance (see Section 3.9.1 for details). bDetection significance (DS) of excluded species as compared to full model. Only valid if evidence of model is less than that of the full model. cShift applied to each of the three gratings (WFC3 G141, STIS G430L, STIS G750L; see Sec 3.9.2). dShift applied to each spectroscopic instrument (STIS, WFC3) We assess the retrieved H2O abundance relative to expectations from thermo- chemical equilibrium for solar elemental compositions (Asplund et al., 2009). As- suming a solar composition and 50% of the available oxygen in H2O, the retrieved H2O abundance corresponds to a log metallicity ([O/H]) of log10 Z/Z = 1.72 +0.39 ?0.55 (metallicity of 53+82?38? Z ). I also compare the retrieved H2O abundance to the stellar metallicity of the host star ([O/H]=0.37, Table 3.4) and obtain a value of log Z/Z = 1.35+0.3910 star ?0.55 (metallicity 23 +33 ?17? Zstar) We consider the possibility of fitting the data using a simpler model consisting mainly of the parameters that are reasonably constrained by the full model. The sim- pler model considers the chemical abundances of H2O, Na, CO, AlO, an isothermal pressure-temperature profile, and a clear atmosphere. The retrieved median fit and confidence contours are shown in Figure 3.14. The simplified model retrieves values 151 Figure 3.13: Posterior distributions of the relevant parameters for the full retrieval (Model 1 in Table 3.13) using STIS, WFC3 and Spitzer data. The abundances of H2O, Na and AlO are constrained, while the cloud and haze parameters are not constrained. The parameter T0, the temperature at the top of the atmosphere (10?6bar) is shown as a subset of the P-T parameters used in the model. 152 No H2O No AlO 1? No Na Retrieved Model Smoothed 2? 1.10 No CO 1.05 1.00 0.3 0.4 0.5 0.6 0.7 0.80.91.0 2.0 3.0 4.0 Wavelength (?m) Figure 3.14: Retrieval of HAT-P-41b using a simplified model compared with the fiducial parameter set (see Section 3.9.1). Observations are shown using blue mark- ers. The retrieved median spectrum is shown in red while the 1-? and 2-? regions are shown using the shaded purple areas. Forward models using the retrieved median parameters show the contributions to the spectra due to individual chemical species. The forward models shown exclude absorption due to H2O (blue), Na (orange), CO (cyan), and AlO (brown). consistent with the full model. The retrieved values are log10(XH2O) = ?1.65+0.40?0.63, log (X ) = ?2.60+0.9410 Na ?1.10, log10(XAlO) = ?5.81+0.51 +0.40?0.66, and log10 Z/Z = 1.72?0.63 . The retrieved isothermal temperature is T= 1120+170?140 and consistent with the inferred temperature at 100 mbar from the full retrieval. The posterior distribution for the retrieved parameters is shown in Figure 3.15. We use these retrieved parameters to generate a set of forward models to assess the spectroscopic contribution from each chemical species. Figure 3.14 shows that the WFC3 observations are better explained by the H2O absorption feature at ?1.4?m driving its strong detection in the spectrum of HAT-P-41b. On the other hand, a series of chemical species in the optical can provide some degree of fit to the STIS observations. In the optical, between ?0.5?0.7?m, the broadened wings of Na 153 Transit Depth (%) Figure 3.15: Full posterior distributions for the simple AURA model. 154 along with its absorption peak provide a fit to observations. AlO provides some fit to the substructure present in the STIS observations, particularly the increased transit depth between 0.4?0.5?m. Lastly the abundance of CO is not constrained and its contribution to the spectrum is minimal. CO is responsible for small changes in the optical and infrared that are well within the error bar of the observations. The AURA retrieval analysis of the broadband transmission spectrum of HAT- P-41b provides excellent fits to the data; using its fiducial model (Model 1) we obtain a best-fit ??2 of 0.93 and ln(Z) = 559.1. I note that we do not require additional continuum opacity sources (e.g., H?) in order to explain the data, as recently claimed by Lewis et al. (2020). 3.9.2 Possible offsets in the data Lastly, we consider the presence of offsets in the data and their effect on the retrieved atmospheric properties. We consider the three scenarios from Sec- tion 3.6.1.2. We note that these retrieved offsets are relative to the atmospheric model and that in all scenarios the Spitzer observations remain unchanged. We consider both Gaussian and uniform priors, as seen in Table 3.8. We present the results of considering the presence of three offsets with Gaus- sian priors informed by the analysis of the white light transit curves (Model 8; Scenario 1 from Section 3.6.1.2). These priors are shown in Table 3.8. The re- trieved shifts are ?52+61?63 ppm for G430L, 80+59?56 ppm for G750L and ?91+48?50 ppm for WFC3. Similar to PLATON, the retrieval generally prefers to increase the 155 G750L depths, primarily motivated by aligning the transit depths in the overlap- ping wavelength region between G430L and G750L. The retrieval also prefers to decrease WFC3 depths in order to better capture the Spitzer 3.6?m depth. The retrieved abundances are log (X ) = ?1.91+0.5310 H2O ?0.68, log10(XNa) = ?2.38+0.81?1.33, and log10(XAlO) = ?6.64+0.70?0.96. Although the retrieved H2O abundance corresponds to a lower metallicity estimate, the derived range log Z/Z = 1.46+0.5310 ?0.68 is consistent with the fiducial model and describes a metal-rich atmosphere. The median metal- licity is superstellar (log10 Z/Zstar = 1.09 +0.40 ?0.63 ), though it is consistent with stellar metallicity to within 2-?. Second, we present the results for the case with three uniform shifts between HST-STIS G430L, HST-STIS G750L, and HST-WFC3 observations (Scenario 2 from Section 3.6.1.2). The retrieved G430L shift is consistent with 0 (1+144?156 ppm), but the retrieval prefers a large positive G750L offset (176+151?160 ppm) and a large negative WFC3 offset (?189+91?94 ppm). In addition to aligning the overlapping G430L-G750L region, it is possible that this large G750L shift is due to the model forcing the data to match features it finds easier to explain. This uncertainty is the danger in using uniform prior offsets, especially in an already-flexible free-chemistry re- trieval. The retrieved abundances are shown in Table 3.13 as Model 9 and are log10(X +0.98 H2O) = ?2.96?0.88, log10(X ) = ?2.43+0.84Na ?1.34, and log10(XAlO) = ?7.05+0.75?0.94. While the retrieved abundances for these three species are consistent within 1- ? with the full unshifted model, the retrieved H2O abundance corresponds to a lower metallicity estimate consistent with solar, sub-solar, and sub-stellar values log +0.9810 Z/Z = 0.40?0.88 . 156 Lastly, we present the results accounting for offsets in the STIS and WFC3 observations using a uniform prior, while keeping the Spitzer observations unshifted (Scenario 3 from Section 3.6.1.2). The retrieval results in a shift in the STIS data of 90+167?157 ppm and a shift in the WFC3 data of ?204+97?98 ppm. While the retrieved value for the STIS observations is consistent with no shift, the WFC3 observa- tions preferentially retrieve a negative offset. The derived abundances, shown as Model 10 in Table 3.13, are log +1.00 +1.3510(XH2O) = ?3.34?0.86, log10(XNa) = ?3.43?2.19, log (X ) = ?6.98+0.7710 AlO ?0.78. The H2O abundance, like Model 9, corresponds to a metallicity consistent with solar and sub-solar values: log10 Z/Z = 0.03 +1.00 ?0.86. Figure 3.16 shows the retrieved median models and confidence contours along with their respectively shifted observations for the cases described in this Section (Models 8, 9, and 10). The models considering instrumental shifts are all preferred over the fiducial model at above the 2-? level. The model with Gaussian priors has a preference at the 2.9-? level, followed by the model with three uniform shifts at a 2.8-? level. The model with two uniform shifts is preferred over the fiducial model at 2.3-?. We note that while both models with three offsets are similarly preferred over the fiducial model, the associated metallicity ranges are different. The model with three uniform shifts retrieves an H2O abundance corresponding to a metallicity estimate consistent with substellar and stellar values. On the other hand, the model with Gaussian priors retrieves an associated metallicity range mostly superstellar and in agreement with the fiducial model. These results highlight the sensitivity of the inferred metallicity ranges to possible large offsets between instruments. Model 157 1.10 Retrieved Model Smoothed 1? 2? 1.08 1.05 1.03 1.00 0.98 0.3 0.4 0.5 0.6 0.7 0.80.91.0 2.0 3.0 4.0 Wavelength (?m) Retrieved Model Smoothed 1? 2? 1.10 1.08 1.05 1.03 1.00 0.98 0.3 0.4 0.5 0.6 0.7 0.80.91.0 2.0 3.0 4.0 Wavelength (?m) Retrieved Model Smoothed 1? 2? 1.10 1.05 1.00 0.3 0.4 0.5 0.6 0.7 0.80.91.0 2.0 3.0 4.0 Wavelength (?m) Figure 3.16: Retrieved spectrum of HAT-P-41b allowing for offsets in the STIS and WFC3 data sets. Observations are shown using blue markers and are shifted according to the models? retrieved median shifts. The retrieved median spectrum is shown in red while the 1-? and 2-? regions are shown using the shaded purple areas. Top: Three shifts with Gaussian priors (Model 8) and retrieved median offsets of ? ?50 ppm for STIS G430L,? 80 ppm for G750L, and ? ?90 for WFC3. Middle: Three shifts with uniform priors (Model 9) and retrieved median offsets of ? 0 ppm for STIS G430L,? 180 ppm for G750L, and ? ?190 for WFC3. Bottom: Two shifts with uniform priors (Model 10) and retrieved median offsets of ? 90 ppm for STIS and ? ?200 ppm for WFC3. 158 Transit Depth (%) Transit Depth (%) Transit Depth (%) comparisons suggest a preference for the models considering offsets, though it is inconclusive between these models. I favor the more physically plausible Gaussian prior model (i.e., Model 8) as the reference for my discussion (Section 3.10). 3.10 Discussion 3.10.1 Comparison Between Retrieval Methods 3.10.1.1 Results Comparison In this Section, I compare the results from the preferred PLATON and AURA models. These include the fiducial model with partial clouds and Gaussian instru- mental offsets for PLATON (Section 3.7.2.3) and the Gaussian instrumental offset model for AURA (Model 8; Section 3.9.2). The similarities reveal the most robust conclusions of my analysis, since they are retrieved despite the many different assumptions that went into each method. Notably, both retrievals robustly find a metal-rich atmosphere with metallicity (de- fined as O/H) inconsistent with the solar metallicity at >2-?. Both methods find a decisive (>4.8-?) water vapor detection, and at least a moderate detection (>2.7-?) of a non-haze gas absorption feature in the optical. Further, both PLATON and AURA retrievals are consistent with a mostly clear atmosphere, with neither finding strong evidence of haze or uniform, high-altitude grey clouds. Though the atmospheric properties derived from PLATON and AURA are similar, there are noteworthy differences. AURA infers a cooler limb temperature at 159 100 mbar (1320+270?200 K compared to 1710 +100 ?80 K for PLATON) as well as a lower metal- licity of log Z/Z = 1.46+0.5310 ?0.68 compared to log +0.23 10 Z/Z = 2.33?0.25 for PLATON, a difference of 1.3-?. This translates to 29+69 +149?23?Z for AURA and 214?88 ?Z . The optical absorber also differs: AURA determines the best description of the STIS feature to be absorption from sodium and AlO, whereas PLATON prefers some combination of TiO and VO absorption. Finally, AURA makes no claim on the C/O ratio as it is a free retrieval framework and no C-bearing species are detected nor meaningfully constrained. On the other hand, the chemical equilibrium assump- tion allows PLATON to find a 3-? upper limit on the C/O ratio of C/O< 0.83. To further contextualize the results, I added the functionality to retrieve the abundance profiles of relevant molecules in PLATON. I show abundance profiles for six spectroscopically relevant species from AURA which are also included in PLATON ? H2O, CO, CO2, Na, TiO, and VO ? in Figure 3.17. I emphasize the enforcing chemical equilibrium narrows the abundance constraints, and I am not reporting these abundances. Instead, they should be interepreted as the ex- pected abundance profiles under the conditions of stable chemical equilibrium for the reported temperature, metallicity, and C/O ratio. As an example, I find no observational constraint on CO, but its abundance is well defined under chemical equilibrium for the temperatures and metallicities that I do observationally constrain via the water feature. Still, these profiles provide a useful baseline for comparison to free-chemistry retrieval abundances. The abundance profiles for the optical-wavelength absorbers reflect the dis- agreement on the primary gas absorber: AURA prefers Na, and so it retrieves 160 ? 10? more Na than PLATON and significantly less TiO and VO. Note that the decreasing TiO and VO abundance with increasing pressure for PLATON is due to those molecules condensing out of the atmosphere. PLATON?s inferred water abundance is typically a few times greater than AURA?s, reflecting the difference in inferred metallicities. CO2 and CO are unconstrained by AURA, while PLATON finds a high abundance of CO2 is consistent with the Spitzer observations. This dif- ference is expected, given that PLATON finds weak evidence of CO2 while AURA found none. Interestingly, this may relate to AURA?s only chemical constraint, which is that CO2 must be less abundant than CO and H2O due to the inferred temperatures (Section 3.6.2). In total, AURA finds a cooler atmosphere with less oxygen but a large sodium enrichment to explain the optical absorption, while PLATON finds a hotter, higher oxygen-abundance atmosphere with TiO/VO absorption in the optical. 3.10.1.2 Impact of Retrieval Model Assumptions The differences between a free-chemistry retrieval (AURA) and one constrained by chemical equilibrium (PLATON) are the natural result of the different assump- tions made by each method. I therefore consider the PLATON and AURA retrievals to be two orthogonal analyses. I examine the impact of the differences by first ex- plicitly listing the notable assumptions in each method, and then by providing the rationale for which assumptions are driving the differences. The relevant methodological differences for PLATON as compared to AURA 161 6 6 PLATON 6 5 AURA 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 1 1 1 2 4 3 2 1 2 4 3 2 1 2 10 8 6 4 2 logXH2O logXCO logXCO2 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 1 1 1 2 6 5 4 3 2 2 14 12 10 8 6 2 13 12 11 10 9 8 7 logXNa logXTiO logXVO Figure 3.17: Abundance profiles for PLATON (red) and AURA (blue) for six rele- vant gaseous species. PLATON abundance profile distributions are derived by sam- pling the posterior 200 times, calculating the abundance profiles for each species for each sample, and finding the median value (solid black line) with 1-? uncertainties at each pressure layer. AURA assumes abundances to be constant with pressure. The median retrieved value (dashed black line) and 1-? uncertainty range are shown. are 1) the assumption of chemical equilibrium, 2) fixing the elemental ratios between all metals other than carbon to their solar values, 3) assuming an isothermal profile for the atmosphere, 4) not including opacity from AlO, and 5) not including the Allard et al. (2019) H2-broadened Na line profile. I find that the Allard et al. (2019) H2-broadened Na line profile is the key driver in the differences between the retrievals, and flexible element abundances and chemical equilibrium also play roles. AURA is the more flexible retrieval, so I first describe its solution before addressing why PLATON differs. AURA?s lower temperature solution is preferred for being able to explain the 162 P [Bar] P [Bar] P [Bar] P [Bar] P [Bar] P [Bar] H2O feature in the WFC3 spectrum, while also explaining the STIS data with H2-broadened Na absorption and capturing the Spitzer data. AURA is able to provide a fit to the STIS data by independently increasing the Na abundance and by also invoking AlO at relatively low temperatures. At this lower temperature (T ? 1300 K), the amount of oxygen necessary for the water abundance and scale height to explain the observed water feature is about 29?Z , with a mean molecular weight of about 2.7 AMU and a scale height of about 440 km. Since PLATON has not yet incorporated the H2-broadened Na line profile, the low temperature solution is a relatively poor fit to the STIS data. Instead, TiO/VO are needed to explain the STIS absorption feature, and these are only abundant enough in chemical equilibirum (with fixed metal ratios) at around 1650 K. At this higher temperature, a higher mean molecular weight is required for the same scale height, which must be small enough to explain the molecular feature sizes as well as the dominance of TiO/VO absorption over Rayleigh scattering. The atmospheric metallicity necessary to achieve the higher mean molecular weight is the much higher ? 200? Z . Therefore, the differences make sense in light of the stricter assumptions. To provide more support to this idea, I compare results with a those of a third retrieval method, ATMO (Amundsen et al., 2014; Tremblin et al., 2015, 2016, 2017; Sing et al., 2016), which acts as a middle ground between PLATON and AURA. ATMO?s spectral retrievals can further help to gain insight into the effect of retrieval assumptions as it includes the Allard et al. (2007) pressure-broadened sodium line but also has the added flexibility of performing a free-element equilibrium-chemistry 163 retrieval. With this assumption for the chemistry, the elemental abundances for each model are freely fit and calculated in equilibrium on the fly. Four elements were selected to vary independently, as they are major species which are also likely to be sensitive to spectral features in the data, while the rest were varied by a trace metallicity parameter ([Ztrace/Z ]). By separately varying the carbon, oxygen, sodium and vanadium elemental abundances ([C/C ], [O/O ], [Na/Na ], [V/V ]) it allows for non-solar compositions but with chemical equilibrium imposed such that each model fit has a chemically-plausible mix of molecules given the retrieved temperatures, pressures and underlying elemental abundances. The resulting retrieved atmospheric parameters describe an atmosphere most consistent with the one described by AURA. ATMO prefers a temperature of 1190+170?120 K and a metallicity (as defined by the oxygen abundance) of log10O/O = log10 Z/Z = 1.53+0.55?0.67, in excellent agreement with AURA?s values, and consistent with PLA- TON?s metallicity to 1.3-?, though the retrieved temperatures differ significantly. Like AURA, ATMO finds an enhanced sodium abundance, though uncertainties are large (log10Na/Na = 1.40+0.75?1.80). This supports the idea that the inclusion of H2- broadened sodium line profiles and the flexibility of non-solar metal ratios ? and not necessarily the equilibirum chemistry constraint ? allow for the low-temperature, lower oxygen abundance solution found by AURA. The metallicities on all three retrievals indicate a metal-rich atmosphere and agree at the ?1.3-? level. Like PLATON (Section 3.8), ATMO also finds a subsolar C/O ratio (C/O = 0.17+0.53?0.16 consistent with stellar (C/O = 0.19), though carbon is not well constrained so the uncertainties are large. The 3-? upper limit of 0.94 is in good agreement with 164 1.10 1.08 1.06 1.04 1.02 1.00 PLATON AURA 0.98 ATMO Observed 0.3 0.4 0.5 0.6 0.8 1.0 1.5 2.0 3.0 4.0 5.0 7.0 9.0 Wavelength [ m] Figure 3.18: Comparison of the median retrieved model for each retrieval method?s fiducial model. 1-? and 2-? uncertainty contours are included for PLATON and AURA. PLATON and AURA are smoothed with a Gaussian filter with ? = 15 for clarity. The chemical equilibrium assumption used by PLATON and ATMO allows for meaningful predictions at unobserved wavelengths, and so those models are shown out to 10?m. PLATON?s 0.83 upper limit. However, unlike PLATON or AURA, ATMO finds no evidence of optical absorbers beyond Na, and instead prefers a haze and Na to explain the STIS optical data. Figure 3.18 elucidates the differences in retrievals by showing the median re- trieved fiducial model for PLATON (red), AURA (black), and ATMO (green) from 0.3?10?m. The 1- and 2-? uncertainty contours are shown for PLATON and AURA, both of which are smoothed with a Gaussian filter with ? = 15 for clarity. The AURA predictions are only shown up to 5?m - as a free-chemistry retrieval, AURA retrieving on the 0.3?5?m data does not place meaningful constraints on multiple molecules with significant opacity in the 5?10?m range. Therefore, a prediction is not warranted. 165 Transit depth [%] While there are subtle differences, such as PLATON and ATMO?s preference for CO2 at 4.5?m and sodium?s prominence at 0.6?m in the AURA and ATMO retrievals, the most obvious difference is below 0.5?m, where ATMO prefers a haze instead of a metallic oxide feature. Though ATMO does not include AlO as an opacity source, this difference is likely due to different condensation schemes. PLATON uses GGchem?s prescription (Woitke et al., 2018) such that species con- dense out when it is energetically favorable. AURA is a free-chemistry retrieval, so there are no restrictions on oxides being in the gaseous phase. ATMO, however, includes rainout chemistry (Goyal et al., 2019), such that if a species condenses at a higher pressure, that then depletes the element above that layer. It is plausible that although PLATON?s condensation scheme allows TiO/VO to be in the gas phase around 1700K, ATMO?s scheme does not, making the metallic oxide feature difficult to capture. In total, I tentatively favor AURA?s derived atmospheric parameters over PLA- TON?s, for two main reasons. First, the inclusion of the most up-to-date sodium line profiles and AlO opacity impact the retrieval. Second, constraints from inte- rior modeling (Section 3.10.2), though not necessarily decisive, are consistent with AURA and in tension with PLATON. Overall, this paints a picture of an atmo- sphere with a supersolar ? but not necessarily superstellar ? metallicity, sodium enrichment, possible disequilibrium metallic oxides (e.g., circulated from dayside, dredged up due to vertical mixing), and a planet with a well-mixed interior and a limb temperature lower than the equilibrium temperature. 166 3.10.2 Comparison to Interior Modeling Metallicity Constraints Though they both describe metal-rich atmospheres, the 1-? retrieved atmo- spheric metallicities ranges from AURA and PLATON are inconsistent (log10 Z/Z = 0.78?1.99 and log10 Z/Z = 2.08?2.56, respectively). Further, it is questionable whether such supersolar metallicities ? especially those retrieved by PLATON ? are physically reasonable. I check the viabilitiy of these values by comparing them to atmospheric metallicity constraints from interior structure models. Thorngren & Fortney (2019) demonstrated how interior models can constrain atmospheric metallicity. Essentially, this is a three step process: 1) Determine what range of bulk metallicities are necessary for structure models to explain the observed radius, taking into account the planet?s mass, age, heating efficiency, and parameter uncertainties, 2) set the maximum bulk metallicity to be the 3-? upper limit of the derived posterior distribution, and 3) set the maximum atmospheric metallicity to be equal to the maximum bulk metallicity. The third step assumes that the atmospheric metallicity cannot be greater than the core?s metallicity for significant timescales due to convection or Rayleigh- Taylor instability. They define metallicity as the ratio of all metals to hydrogen compared to the ratio in the Sun?s photosphere. This is a good proxy for O/H, and so it is a valid comparison to the retrieved atmospheric metallicities. For more details on the derivation, see Thorngren & Fortney (2019). Using stellar parameters from Hartman et al. (2012) (Table 3.3), the interior structure model fit yields a bulk metal abundance ratio of Z/Z = 33.7? 9.1, corre- 167 sponding to a maximum atmospheric metallicity of 50?Z (D. Thorngren, private communication). There is no significant uncertainty on this number, as it is the 3-? upper limit of the distribution. This is consistent with the metallicity from the AURA retrieval, but it is in tension with PLATON?s retrieved metallicity ? 50?Z falls outside PLATON?s 1-? range (but within 2-?, as the metallicity distri- bution is asymmetric PLATON?s 2-? lower limit is 37?Z ). This could indicate that the ?true? atmospheric metallicity falls in the lower range of PLATON?s retrieved metallicity, or it could be interpreted as slight evidence in support of AURA over PLATON. Either way, the atmospheric metallicity approaching the bulk metallicity indicates a well-mixed interior. Such vertical mixing could allow for micron-sized particles to stay afloat in the atmosphere, potentially facilitating gaseous metal oxide survival and Mie scattering. 3.10.3 Implications for Planet Formation The atmospheric metallicity I retrieve for HAT-P-41b provides important con- straints on the formation and migration history of the planet. At the outset, the super-solar metallicity (O/H) of ?30?200? Z requires substantial accretion of solids, beyond several Earth masses of H2O ice, during the planet?s evolutionary history. It is unlikely that such a large amount of volatile accretion is possible at the planet?s current orbit. Therefore, the planet is unlikely to have formed in-situ (Batygin et al., 2016) but instead formed far out beyond the H2O snow line and migrated inward. The formation location and migration path of a giant planet can 168 significantly affect its chemical composition. Beyond the H2O snow line the gas in the protoplanetary disk is depleted in oxygen whereas the solids are enriched in oxygen (O?berg et al., 2011). Therefore, planets with high enrichment of oxygen re- quire predominant accretion of H2O-rich planetesimals while forming and migrating through the protoplanetary disk. The high metallicity (specifically O/H) of HAT-P-41b, therefore, supports the migration of the planet through the disk via viscous torques (Madhusudhan et al., 2014b). This is in contrast to other hot Jupiters with low O/H abundances which have been suggested to be caused by insufficient solid accretion, e.g. via disk-free migration (Madhusudhan et al., 2014b) or formation via pebble accretions whereby the oxygen-rich solids are locked in the core (Madhusudhan et al., 2017). The fact that HAT-P-41b?s orbit is moderately misaligned to the host star?s rotation axis is also in tension with the disk migration hypothesis, since spin-orbit misalignments are considered to be evidence of disk-free migration and planet-planet interactions (Winn et al., 2010). In principle, instead of disk migration, super-solar elemental abundances could be caused by accreting gas whose metallicity has been enhanced due to pebble drift (O?berg & Bergin, 2016; Booth et al., 2017). But while pebble drift can cause metal enhancements up to ? 10? Z , much larger enhancements as constrained in the present case are unlikely to be explained by this process. More importantly, such enhancements due to pebble drift are also expected to cause high C/O ratios (?1), which may be at odds with the high H2O abundace and the low C/O ratio retrieved for the planet. Overall, the most plausible explanation for the potentially high atmospheric 169 metallicity inferred for HAT-P-41b is formation outside the H2O snowline and mi- gration inward while accreting substantial mass in planetesimals. If confirmed, this would be a departure from other hot Jupiters observed hitherto which have generally shown low H2O abundances, indicative of the low accretion efficiency of H2O-rich ices that is possible for disk-free migration mechanisms (Madhusudhan et al., 2014b; Pinhas et al., 2019; Welbanks et al., 2019). Such an abundance is also a substantial departure from expectations based on Solar System giant planets. The metallicity of Jupiter in multiple elements is ?1?5? Z Atreya et al. (2016); Li et al. (2020). With the mass of HAT-P-41 b being similar to that of Jupiter, its higher metallicity would indicate an even higher amount of solids accreted than that of Jupiter in the Solar System. 3.11 Summary I have conducted a comprehensive, multi-pronged Bayesian retrieval analysis of the 0.3?5?m transit spectrum of HAT-P-41b derived from HST STIS (previously unpublished; Section 3.5.1), HST WFC3 (re-analysis; Section 3.5.2), and Spitzer (independent analysis; Section 3.5.3) transit observations. I determined the host star has, at most, a low level of stellar activity (logLX/Lbol < ?5.2) using both visible and X-ray photmetric monitoring observations (Section 3.4.1). We performed two complementary retrieval analyses: a relatively strict PLA- TON analysis (Section 3.6.1, Section 3.8) assuming chemical equilibrium and solar metal ratios (except carbon), and a more flexible AURA free-chemistry retrieval 170 (Section 3.6.2, Section 3.9.1). Both methods? fiducial models are excellent fits to the entire transit spectrum. I further tested an array of more complicated models (Sections 3.7 and 3.9.2), including instrumental transit depth biases (offsets), para- metric rayleigh scattering, partial cloud coverage, Mie scattering (PLATON only), and stellar activity (PLATON only). I find the conclusions to be insensitive to model choice within a paradigm. Despite PLATON and AURA?s differing model assumptions, priors, and even opacity sources, I find several shared conclusions between the two methods (Sec- tion 3.10.1). Both PLATON and AURA retrieve a high atmospheric metallicity (O/H) that is inconsistent with Z to greater than 2-? (log Z/Z = 1.46+0.53 10 ?0.68 compared to log10 Z/Z = 2.33 +0.23 ?0.25, respectively). They also both are consistent with a haze-free and cloud-free atmosphere, and both find a decisive water vapor detection and at least suggestive evidence of an optical absorption feature. We fur- ther confirm the result by performing a middle-ground retrieval, ATMO, and find results generally consistent with AURA?s (Section 3.10.1.2). I determine the inclu- sion of H2-broadened sodium opacity impacts the retrieved metallicities. While I consider AURA to be more physically plausible due to its consistency with interior modeling constraints and inclusion of H2-broadened sodium opacity, I present the results from both PLATON and AURA as assumption-dependent orthogonal anal- yses. Overall, this study emphasizes the importance of comparative retrievals with different forward modeling, prior, and model selection assumptions in order to best contextualize presented results. 171 3.12 Addendum A separate group?s analysis of HAT-P-41b was released concurrently with mine (Sheppard et al., 2021), releasing on arXiv simultaneously. Lewis et al. (2020) used UVIS observations (analyzed in the Part I paper Wakeford et al. (2020)) in conjunc- tion with their own derived Spitzer and WFC3 transit depths to conduct transit spectroscopy on HAT-P-41b. Their UVIS observations and analysis are the first ap- plication of UVIS to exoplanet transit spectroscopy. Similar to AURA, PLATON, and ATMO retrievals, their retrievals needed to invoke something to explain the unexpectedly small size of the clear water feature in the WFC3 data. They perform several retrievals and fin their UVIS and WFC3 data is best described by an orders- of-magnitude overabundance of H- as compared to expectations in equilibrium chem- istry, which they argue is plausible via photochemistry (Lavvas et al., 2014). Their median metallicity is superstellar, and interestingly find a water abundance con- sistent with the AURA retrieval but convert from H2O to O/H differently causing the inferred metallicities to disagree by a factor of two. This type of disequlibrium process is not capturable by my PLATON retrieval, but even for a free-chemistry AURA retrieval with H- opacity we ran we did not detect H-. We emphasize that both PLATON and AURA fit the data with reduced ?2 values consistent with one. Additionally, Espinoza & Jones (2021) applied a version of CHIMERA (assumes chemical equilibrium, Line et al., 2013) to the exact data in Sheppard et al. (2021) and retrieved a superstellar metallicity of 100? solar (logZ/Z ? 2.00? 0.30, con- sistent with the PLATON retrievals. I conclude that the differences in conclusions 172 are due to UVIS-STIS data differences, and not modeling differences. 173 Chapter 4: Constraining the Dayside Thermal Structure of Hot Jupiters from Secondary Eclipse Observations 4.1 Overview In this chapter, I derive the HST WFC3 emission spectra for two highly ir- radiated hot Jupiters (WASP-18b and WASP-19b) and retrieve their atmospheric properties. Most notably, I find evidence for a strong thermal inversion in the day- side atmosphere of the highly irradiated hot Jupiter WASP-18b (Teq = 2411K, M = 10.3MJ) based on emission spectroscopy from Hubble Space Telescope sec- ondary eclipse observations and Spitzer eclipse photometry. I demonstrate a lack of water vapor in either absorption or emission at 1.4?m. However, I infer emission at 4.5?m and absorption at 1.6?m that I attribute to CO, as well as a non-detection of all other relevant species (e.g., TiO, VO). The most probable free-chemistry atmospheric retrieval solution indicates a C/O ratio of 1 and a high metallicity (C/H=283+395?138? solar). However, water dissociation and H- opacity could explain the spectrum without necessitating a super-solar metallicity. The derived composi- tion and T/P profile suggest that WASP-18b is the first example of a planet with a non-oxide driven thermal inversion. I find moderate evidence (2.8?) of water ab- 174 sorption with non-depleted abundance (logXH2O=-3.64 +1.44 ?0.72; consistent with stellar) and a likely sub-stellar C/O (C/O<0.63), consistent with transit analyses. I also retrieve a non-inverted T-P profile in WASP-19b, which, at an equilibrium temper- ature of 2120 K, is at the border of where both TiO-driven inversions and water dissociation in hot Jupiter atmospheres are expected to become important. 4.2 Introduction Hot Jupiters have been vital in revealing the structural and atmospheric di- versity of gas-rich planets (see recent reviews by Crossfield, 2015; Madhusudhan et al., 2016; Deming & Seager, 2017). Since they are exposed to extreme condi- tions and relatively easy to observe through transit and eclipse spectroscopy, hot Jupiters provide a window into a unique part of parameter space, allowing us to better understand both atmospheric physics and planetary structure. An outstanding question that has emerged for highly irradiated planets is the presence and origin of stratospheric thermal inversions, which have been detected in several extremely irradiated hot Jupiters (Haynes et al., 2015; Evans et al., 2017). Hubeny et al. (2003) predicted that thermal inversions in highly-irradiated atmo- spheres would be caused by the presence of optical absorbers (e.g. TiO and VO) high in the atmosphere, but there may be other causes such as insufficient cooling (Mollie?re et al., 2015) or sulfur-based aerosols (Zahnle et al., 2009b). It is also unclear what conditions are necessary for TiO or VO to exist in the gaseous form and drive inversions. There is reason to expect a correlation with 175 planet temperature (cold traps, dissociation) and gravity (vertical cold traps), but observational evidence is limited and the exact temperatures and gravities where these processes dominate is unclear (Parmentier et al., 2013; Beatty et al., 2017; Parmentier et al., 2018). H- opacity and molecular dissociation is expected to impact the thermal structure of ultra-hot Jupiters and mask water features (Parmentier et al., 2018; Lothringer et al., 2018), though the magnitude and prevalence of this effect is uncertain. Huitson et al. (2013) originally observed WASP-19b in transit with WFC3 and STIS (optical) and found a clear water detection, no evidence of TiO/VO, and likely a low C/O ratio. This was confirmed by Sing et al. (2016), who also found no evidence of CO or CO2 in the Spitzer transit data. Additionally, Benneke (2015) re- ported a water abundance of 0.2?30x solar (logX =-3+1.2H2O ?1.0). However, Sedaghati et al. (2017) observed WASP-19b with VLT and claimed a 7? detection of TiO in transit (and 7.5? detection of water), in strong contrast with previous results. Espinoza et al. (2019) then challenged this claim, showing that observations over a similar optical range with the ground-based telescope Magellan/IMACS found no evidence of TiO. Finally, Sedaghati et al. (2021) followed up the planet with high-resolution cross-correlation spectra from VLT/ESPRESSO, finding a ?barely significant? peak of TiO. Since TiO is predicted to drive thermal inversions, measur- ing the thermal profile of WASP-19b can provide insight into the likelihood of TiO in the atmosphere. Though photometric and ground-based secondary eclipse obser- vations exist (e.g, Anderson et al., 2013), no high quality space-based spectroscopic eclipse observations have been analyzed. 176 Constraints on the structure and composition of exoplanetary atmospheres allow us to test, refine, and generalize planetary formation models. Volatile ices are expected to play an important role in planet formation; thus a constraint on the composition of a hot planet?s atmosphere gives us insight on how and where it was formed (O?berg et al., 2011; Madhusudhan et al., 2014a). In our Solar System there is an inverse mass vs. atmospheric metallicity relationship, and whether or not it extends to exoplanets is informative to planetary formation and migration models. There is some evidence that the trend holds (Kreidberg et al., 2014), however that parameter space is not yet sufficiently populated to enable firm conclusions. In this chapter I use Hubble Space Telescope (HST) spectroscopy and Spitzer IRAC photometry of secondary eclipses to explore the thermal structure and com- position of the dayside atmosphere of two hot Jupiters. First I analyze WASP-18b, an extremely hot (Teq = 2411K) and massive (M = 10.3MJup) hot Jupiter orbiting an F-type star with an orbital period of less than one day (Hellier et al., 2009). Then I analyze WASP-19b (Teq = 2120K, M=1.11MJup), a hot Jupiter orbiting an active G8-star, also with a period of less than a day. WASP-19b is particularly interesting given that its gravity and temperature place it on the boundaries of the parameter space where processes such as water dissociation and TiO cold traps are expected to impact the chemistry and thermal structure of hot Jupiter atmospheres. 177 4.3 Observations I used Wide Field Camera-3 (WFC3) observations of five secondary eclipses of WASP-18b from the HST Treasury survey by Bean et al. (Program ID 13467). WFC3 obtains low resolution slitless spectroscopy from 1.1 to 1.7?m using the G141 grism (R=130), as well as an image for wavelength calibration using the F140W filter. Grism observations were taken in spatial scan mode (Deming et al., 2013) with a forward-reverse cadence (Kreidberg et al., 2014). The first three visits, taken between April-June 2014, are single eclipse events. Visit 4, taken in August 2014, contains two eclipses in an orbital phase curve, and I extract those eclipses and analyze them separately. I also extract a single eclipse of WASP-19b from the HST program GO-13431 phase curve observation (PI: C. Huitson), also taken in spatial scan mode with a uni-directional reverse cadence. A collaborator re-analyzes two eclipse observations of WASP-18b taken in the 3.6?m and 4.5?m channels of the Spitzer Space Telescope?s IRAC instrument (Program ID 60185). The 3.6 ?m observation was performed on 2010 January 23, while the 4.5 ?m observation was taken 2010 August 23. Both observations were taken using an exposure time of 0.36s in subarray mode, and were first analyzed in Maxted et al. (2013). Spitzer eclipse depths for the WASP-19b are taken from Garhart et al. (2020). Depths from cold Spitzer (IRAC3 and IRAC4) are taken from Nymeyer et al. (2011, WASP-18b) and Anderson et al. (2013, WASP-19b). 178 4.4 HST Data Analysis The data analysis is an earlier version of the DEFLATE analysis described in Section 2.3. I briefly summarize it here. My grism spectroscopy analysis utilized HST ?ima? data files. I separated the data by scan direction, removed background flux, and corrected for cosmic rays and bad pixels. I removed background flux via the ?difference frames? method outlined in the appendix of Deming et al. (2013), and set the aperture to maximize the amount of source photons in my analysis. All corrections are propagated to the flux errors, which are retrieved from the ?ima? error extension and intrinsically account for read noise and bias. The end result is two reduced light curves - one forward scan and one reverse scan - for each eclipse, which I analyze separately. The F140W photometric image determines the location of the zero-point, which I used to assign a wavelength to each column. I confirmed the wavelengths by fitting an appropriate ATLAS stellar spectrum (e.g, for WASP-18b, T=6400K, log g=4.3, [Fe/H]=0.1) (Castelli & Kurucz, 2004), multiplied by the grism sensitivity curve, to an observed in-eclipse spectrum. I note that the archival data on WASP-19b is technically from a ?failed? phase curve observation. This is because the visit relied on gyro guiding instead of FGS due to an issue with the guide star, resulting a large horizontal shift over the course of the observation that is difficult to deconstruct. However, that is for the entire days- long phase curve. Over the course of a 6-hour eclipse, the shift is almost negligible, amounting to almost a pixel over the course of the observation. This sub-pixel shift 179 Figure 4.1: First and last exposures of WASP-19b eclipse observation. The color- scale is exaggerated to make the location of bad pixels (black), which are constant on the detector, more visible. The detector shift during eclipse is no more than a single pixel, which is not significant enough to impact the quality of the data. is easily accounted for in systematic modeling. Further, each wavelength bin is 6 pixels, and in low-resolution spectroscopy trying to capture molecular features, such shifts have minimal impact. For a 6 pixel bin size, a ?typical? water feature has a width on the order of ?42 pixels. Thus, these eclipses still contain useful information and our worth retrieving on. 4.4.1 Light Curve Analysis The light curve analysis is the root of what would become DEFLATE (described in Section 2.4). I summarize it below on WASP-18b, which was a cominbation of four observations. I used the same process on WASP-19b. Empirical methods are necessary to correct for non-astrophysical systematic effects in WFC3 spectroscopy (Berta et al., 2012; Haynes et al., 2015). Correction methodology is especially important in emission spectroscopy, where the magnitude of systematic effects can be greater than the eclipse depth (Kreidberg et al., 2014). I thereby combined two strategies: initial removal of systematic trends using paramet- ric marginalization (Gibson, 2014a; Wakeford et al., 2016b), and further detrending by subtraction of scaled band-integrated residuals from wavelength bins (Mandell et al., 2013; Haynes et al., 2015). This method accounts for uncertainty in instru- 180 ment model selection, and residuals from the band-integrated analysis allow us to utilize the normally excluded first orbit of each HST data set in the spectroscopic analysis. Fitting a band-integrated light curve provides residuals that I use to remove unidentified systematics from the spectrally resolved light curves. I calculate the HST phase (parameter for ramp and HST breathing), planetary phase (parameter for visit-long slope), and a wavelength shift derived by cross correlating each spec- trum with the last spectrum for the visit (parameter for jitter) for each exposure in a time series. The grid of systematic models comprises a combination of a linear planetary phase correction and up to four powers of HST phase and wavelength shift. These models are then multiplied by a Mandel & Agol (2002) eclipse model. I simultaneously fit for the eclipse depth, all systematic coefficients, and - for two light curves with ingress and egress points - the center of eclipse time. All other system parameters are fixed to literature values. For WASP-19b, which has good egress coverage, I also fit for a/Rstar. All other system parameters are fixed to literature values (Hebb et al., 2010; Tregloan-Reed et al., 2013; Wong et al., 2016; Sedaghati et al., 2017). I use a Levenberg-Markwardt (L-M) least squares minimization algorithm (Markwardt, 2009) to determine the parameter values. An example band-integrated light curve with systematic effects removed using the best-fitting model is shown in the leftmost panel of Figure 4.2. For WASP-18b, the scatter (RMS) of the residu- als of the band-integrated curves ranges from 1.3-5.5? the photon noise, indicating that there is excess noise beyond the photon limit present. Excess noise in the 181 band-integrated curves is also shown by comparisons of the cumulative distribu- tions of residuals with those of a photon-limited Gaussian (see bottom-left panel of Figure 4.2). However, the structure of this excess noise does not change with wavelength, allowing for its removal from the corresponding spectral light curves. x10-4 -4 -4HST Band-Integrated Time Series x10 HST Spectral Bin (1.37 ?m) x10 Spitzer IRAC1 (3.6 ?m) 5 Raw Light Curve 20 100 0 0 -5 -20 0 -10 -40 -15 -60 -100 -20 Error: -80 Error: Error: 10 15 30 8 6 Light curve 10 20 4 with systematics 5 removed 0 102 0 0Error: -5 Error: Error: Residuals 4 1 5 2 0 0 0 -2 -1 -5 -4 -2 0.4 0.5 0.6 0.7 -10 0.3 0.4 0.5 0.7 0.2 0.4 0.5 0.6 Phase Phase Phase 1.5 20 1.0 Normality of Residuals RMS 5 10 0.5 0 ? 0 0ph -0.5 RMS = 67 ppm RMS = 209 ppm -5 -10 RMS = 680 ppm -1.0 RMS/?ph= 2.02 RMS/?ph= 1.00 RMS/?ph= 1.01 -1.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 -200.0 0.2 0.4 0.6 0.8 1.0 CDF CDF CDF Figure 4.2: An example of the detrending process for an HST band-integrated light curve (left), a light curve for an HST spectral bin (middle), and a Spitzer/IRAC photometry light curve (binned for clarity). The HST band-integrated results fall within 1.3?5.5? the photon noise limit, while both the HST spectral bins and the Spitzer data typically achieve close-to-photon-limited results. The bottom row compares the cumulative distribution function (CDF; red dots) of the residuals to that of a Gaussian with dispersion equal to the photon noise (black line). Good agreement is obtained for the HST spectral and Spitzer residuals, while excess scatter is observed for the HST band-integrated residuals. For the latter, the CDF of a Gaussian with dispersion equal to the residual RMS is also plotted for comparison. To derive the emission spectrum, I bin the exposures in wavelength between 182 Obs-Model Obs-Model Normalized Flux Normalized Flux the steep edges of the grism response and fit these spectrally resolved light curves. I remove wavelength-dependent systematics by fitting each spectral bin separately in a process that mimics the band-integrated process, with three exceptions. First, the eclipse mid-time is now fixed to the value determined by the band-integrated analysis. Second, it is possible that shifts on the detector are wavelength-dependent, so the jitter parameter is recalculated for each wavelength bin using only that portion of the spectrum in the cross-correlation procedure. Third, each systematic model now incorporates the residuals from the band-integrated fit of the same model as a decorrelation variable. The amplitude of the residuals is a free parameter, although the shape is assumed to be constant in wavelength. This removes any remaining wavelength-independent trends in the data. An example result of a reduced spectral bin light curve is shown in the central panel of Figure 4.2. 183 WASP-19b Band-Integrated Time Series WASP-19b Spectral Bin (1.47 ?m) Data Raw Light curve Model 0.3 0.4 0.5 0.2 0.3 0.4 0.5 Light curve with Systematics Removed 0.3 0.4 0.5 0.2 0.3 0.4 0.5 Residuals RMS: 223 ppm RMS: 777 ppm RMS/?ph: 1.263 RMS/?ph: 0.985 0.3 0.4 0.5 0.2 0.3 0.4 0.5 Figure 4.3: Band-integrated light curve (left) and a sample spectral bin light curve (right) for HST WFC3 observations of WASP-19b. The top panels show the raw data with the best-fit model overplotted, and the middle panels show the light curve and model after detrending systematic effects. The bottom panels show the difference between the observations and the best-fit model at each point in the time series and give the standard deviation of those residuals (RMS). Finally, eclipse depths from the multiple visits are combined via an inverse- variance weighted mean, giving the emission spectrum for WASP-18b. The spectra for all visits are shown in Figure 4.4. For WASP-18b, the average RMS of the systematic-reduced spectroscopic light curves is 1.04? the photon noise and the median RMS is 0.97? the photon noise, 184 Comparison of Spectra From All Visits April Weighted Mean, All visits August Error=26.3ppm 1400 June Weighted Mean, May Excluded May Error=28.8ppm 1200 1000 800 600 1.2 1.3 1.4 1.5 1.6 Wavelength [?m] Figure 4.4: Spectra for all of the HST visits, horizontally offset for clarity, with the weighted mean overplotted. Depths from both the forward and reverse scan light curves are plotted for each eclipse. The May data receives a low weight due to the large uncertainties, and therefore does not impact the results beyond the individual uncertainties, as shown by the dashed grey line. Values for the individual data points are available from the authors upon request. indicating that shot noise is the dominant error source. The close agreement between the cumulative distributions of residuals and those of a Gaussian with a width determined by the photon noise provides further evidence that the analysis achieved photon-limited results for the vast majority of spectral curves (see bottom-center panel of Figure 4.2).The remaining spectral curves have residuals with an RMS greater than 1.5? the photon limit, indicating that excess noise is present. These only constitute 6% of all spectral bins, and every one is from the single eclipse observation taken in May. I explored removing the May dataset due to this increased noise, but the exclusion of these data did not affect the variance-weighted spectrum, 185 Eclipse depth [ppm] and I chose to include this visit in subsequent analyses. Figure 4.4 contains the emission spectra from every visit, demonstrating the consistency of the structure of the spectrum. My analysis routine finds that the outlier depths from the May visit have very high errors due to the presence of correlated noise, and so they contribute very little to the weighted spectrum. For WASP-19b the exposures are separated into wavelength bins six pixels (0.028 ?m) in size and fit individually to derive the emission spectrum. The RMS of the residuals of the band-integrated light curve is 1.26x the theoretical photon- noise limit, which indicates there is some excess noise present. Further, there is an indication of red noise in the white light curve. An underestimation of the ramp systematic seems to cause a majority of points in the third orbit (bottom left panel of Figure 4.3) to be positive residuals. However, the average and median RMS of the residuals of the spectroscopic light curves are 1.02x and 1.01x the theoretical limit, indicating that no additional noise is present in the binned data after subtracting the band-integrated residuals (see right half of Figure 4.3). An analysis of the RMS of the residuals as a function of binning in time further demonstrates that shot noise is the dominant error source (see Figure 4.5). This indicates that the structure that the model grid was unable to capture in the white light model was successfully used to remove the same structure in the spectral bins. The relative shape of the spectrum is then reliable, only the absolute depth is suspect. I followed up on this by deriving the spectrum independently with the physical charge-trap model from Zhou et al. (2017). Given that the ramp is the dominant systematic in this light curve, this model is an excellent fit to the white light curve, finding a 186 depth about 75ppm deeper than marginalization. However, the derived spectra are in excellent agreement. I emphasize that the emission spectrum is not dependent on methodology. To further check my methodology, I reanalyzed published emission spectra for WASP-43b (Kreidberg et al., 2014b), WASP-103b(Cartier et al., 2017), and WASP-121b (Evans et al., 2017). I find an agreement to the published spectra, with a mean point-by-point variation (difference / uncertainty) of 89%, 23%, and 50% for the three data sets, respectively, demonstrating the consistency of my analysis pipeline with those published by other authors. 4.5 Spitzer Re-analysis Spitzer secondary eclipse measurements of WASP-18b were reported by Maxted et al. (2013), and a collaborator has re-analyzed key portions of those data. We confine our re-analysis to the 3.6 and 4.5?m bands, because the instrumental sys- tematic errors are greatest in those bands, and there are new methods to correct those systematics. We use an updated version (Tamburo et al., 2017) of the Pixel-Level Decorre- lation framework (Deming et al., 2015). Our photometry uses 11 different circular aperture sizes (with radii ranging from 1.6 to 3.5 pixels). We decorrelate the instru- mental systematics while simultaneously fitting for the eclipse depth, using binned data, as advocated by Deming et al. (2015) and Kammer et al. (2015). The fitting code selects the optimal aperture and bin size, and obtains an initial estimate of the 187 WASP-19b Correlated Noise Analysis 100 1.136 m 1.163 m 1.191 m 10 1 Expected Data RMS 100 1.219 m 1.246 m 1.274 m 10 1 100 1.302 m 1.329 m 1.357 m 10 1 100 1.385 m 1.412 m 1.440 m 10 1 100 1.468 m 1.495 m 1.523 m 10 1 100 1.551 m 1.578 m 1.606 m 10 1 100 101 100 101 100 101 Points Per Bin Figure 4.5: Analysis of the temporal correlated noise in each spectral bin for WASP-19b. The data is binned up in time and the RMS of the light curve residuals is calculated; the results are then normalized by the RMS of the light curve with minimal binning (i.e, one point per bin)?and compared with the predicted trend assuming there is no correlation in time (RMS0/ N), where N is the number of exposures per bin). The light curves show no evidence of correlated noise; the most extreme deviations (bin 1.329 ?m) are consistent with white noise within 1-2?. 188 Normalized RMS eclipse depth and the pixel basis vector coefficients using linear regression. We then implement an MCMC procedure (Ford, 2005) to explore parameter space, refine the best-fit values, and determine the errors. At each step, we allow the central phase, orbital inclination, and eclipse depth to vary, but lock all other orbital parameters to the values used in the WFC3 analysis. We also vary the multiplicative coefficients of our basis pixels (see Deming et al., 2015) and visit-long quadratic temporal baseline coefficients at every step. Our best fits use aperture radii of 2.0 and 2.5 pixels, and bin sizes of 76 and 116 points at 3.6 and 4.5?m, respectively. The scatter in the binned data, after removal of the best-fit eclipse, is 1.01 and 0.95? the photon noise at 3.6 and 4.5?m, respectively, those ratios being statistically indistinguishable from unity. We ran three chains of 500,000 steps for both bands, confirming their conver- gence through the Gelman-Rubin statistic (Gelman & Rubin, 1992). We combine all chains of eclipse depth into a unified posterior distribution for each band, and fit a Gaussian to this distribution to determine the error on eclipse depth. Our results are included in Table 4.1, and exhibit excellent agreement with Maxted et al. (2013), but with smaller errors. 4.6 WASP-18b 4.6.1 Atmospheric Retrieval I use the WFC3 spectrum along with the Spitzer and ground-based Ks band photometry to constrain the composition and temperature structure of the dayside 189 Table 4.1: WASP-18b Thermal Emission Spectrum Instrument ? [?m] Depth [ppm] Instrument ? [?m] Depth [ppm] WFC3 G141 1.118?1.136 818 ? 28 1.434?1.452 1105 ? 25 1.136?1.155 847 ? 26 1.452?1.471 1107 ? 25 1.155?1.173 858 ? 24 1.471?1.489 1088 ? 24 1.173?1.192 784 ? 25 1.489?1.508 1155 ? 28 1.192?1.211 944 ? 26 1.508?1.527 1159 ? 28 1.211?1.229 885 ? 26 1.527?1.545 1162 ? 28 1.229?1.248 913 ? 25 1.545?1.564 1077 ? 30 1.248?1.266 927 ? 25 1.564?1.582 1139 ? 30 1.266?1.285 900 ? 24 1.582?1.601 1130 ? 28 1.285?1.304 919 ? 25 1.601?1.620 1045 ? 34 1.304?1.322 957 ? 24 1.620?1.638 1019 ? 31 1.322?1.341 961 ? 23 1.638?1.657 1014 ? 38 1.341?1.359 1022 ? 25 IRIS2 Ks 2.0?2.3 1300 ? 300a 1.359?1.378 1029 ? 29 Spitzer IRAC1 3.2?4.0 2973 ? 70 1.378?1.396 1066 ? 26 Spitzer IRAC2 4.0?5.0 3858 ? 113 1.396?1.415 1097 ? 25 Spitzer IRAC3 5.0?6.4 3700 ? 300b 1.415?1.434 1145 ? 25 Spitzer IRAC4 6.4?9.6 4100 ? 200b NOTE?WFC3 bin size = 0.0186?m a Anglo-Australian Telescope (Zhou et al, 2015) b Nymeyer et al, 2011 190 atmosphere of WASP-18b. We use the HyDRA retrieval code (Gandhi & Madhusud- han, 2018), which comprises a thermal emission model of an atmosphere coupled with a nested sampling algorithm for Bayesian inference and parameter estimation. The forward model, based on standard prescriptions for retrieval (Madhusudhan & Seager, 2009; Madhusudhan et al., 2011), computes line-by-line radiative transfer in a plane parallel atmosphere under the assumptions of hydrostatic equilibrium and local thermodynamic equilibrium. The pressure-temperature (P -T ) profile and chemical compositions are free parameters in the model. The model includes 14 free parameters. For the P -T profile, we use the parametrisation of (Madhusudhan & Seager, 2009) which involves six free parame- ters. The atmosphere comprises 100 layers equally spaced in log-pressure between 10?6 bar and 102 bar. For the atmospheric composition we consider several species expected to be prevalent in very hot Jupiter atmospheres and with significant opac- ity in the observed spectral range (Madhusudhan, 2012; Moses et al., 2013; Venot & Agu?ndez, 2015). This includes H2O, CO, CH4, CO2, HCN, C2H2, TiO, and VO. The uniform mixing ratio of each species are free parameters in the model. We assume an H2/He rich atmosphere with a solar He/H2 ratio of 0.17. We consider line absorption from each of these species and collision-induced opacity from H2-H2 and H2-He. The sources of opacity data are described in Gandhi & Madhusud- han (2017); the molecular linelists are primarily from EXOMOL (Tennyson et al., 2016) and HITEMP (Rothman et al., 2010), and the CIA opacities are from Richard et al. (2012). The retrieval explores model parameter space with Bayesian nested sampling using the MultiNest code via the Python wrapper, PyMultiNest (Skilling, 191 CO Emission Figure 4.6: Observed spectrum and retrieved solutions. WFC3 and Spitzer data are shown in green. The median retrieved spectrum, with the uncertainty envelopes, is shown in red. The binned median model, in yellow, with ?2red = 3.67 is an unambiguously better fit than a blackbody (?2red = 15.2). A fiducial model with solar-abundance H2O absorption is shown in blue to demonstrate the lack of an H2O feature in the data. The results favor a thermal inversion, and the only spectral features detected are those of CO at 1.6 and 4.5 ?m. The retrieved P-T profile with error contours is shown in the lower-right inset along with normalized contributions functions at 1.6 and 4.5 ?m. 2004; Feroz et al., 2013; Buchner et al., 2014). We sample the multi-dimensional parameter space using 4,000 live points for a total of more than one million model evaluations. The best-fit retrieval requires a strong thermal inversion in the dayside at- mosphere. The bottom inset of Figure 4.6 shows the retrieved P -T profile with confidence contours, indicating an upper atmospheric temperature increase. The 192 requirement of a thermal inversion is guided by the strong emission inferred in the 4.5 ?m Spitzer IRAC band, with a brightness temperature of 3100?50 K, which is significantly higher than the rest of the data. This can be explained by the presence of a thermal inversion in the atmosphere along with the presence of either CO or CO2, which both exhibit pronounced spectral features in the 4.5?m band (Burrows et al., 2007; Fortney et al., 2008; Madhusudhan & Seager, 2010b). We break this degeneracy by requiring that CO2 be less than H2O as expected for hot Jupiter atmospheres (Madhusudhan, 2012; Heng & Lyons, 2016). Another subtlety is the apparent minor trough near ? 1.6?m, which we attribute to CO absorption below the inversion layer (? 1-10 bar), where temperature decreases outward. Emission in the 4.5?m band is due to CO in the 0.001 - 0.1 bar range which contains the thermal inversion. As part of the nested sampling analysis, we compute the Bayesian evi- dence value for the retrieved spectrum. By comparing this value with that obtained for a model without a thermal inversion, we conclude that a thermal inversion is favored at the 6.3? significance level. Similarly, comparison to a model lacking CO implies that the presence of CO is favored at the 6.1? level. Interestingly, the tran- sition point of the inversion occurs at 0.1 bar which is characteristic of all planets in the Solar System with inversions as well as models of hot Jupiters (Madhusudhan & Seager, 2009; Robinson & Catling, 2014). Figure 4.7 shows the posterior probability distributions of all the model pa- rameters. The data require a CO volume mixing ratio of 19+18?8 % in the atmosphere, which is 380+360?160? the amount expected for a solar abundance atmosphere at this temperature in thermochemical equilibrium. The high CO abundance is primarily 193 constrained by the emission required to explain the 4.5?m IRAC point as well as the absorption trough in the WFC3 band at 1.6-1.7?m. We detect no other chemi- cal species (see Figure 4.7). In particular, the non-detection of H2O at both 1.4?m and 6?m provides a robust 3? upper-limit of 10?6 on the volume mixing ratio. The sum-total of constraints on the chemical species lead to a super-solar metallicity in the planet (C/H = O/H = 283+395?138? solar O/H) and a C/O ratio of ?1. We also conducted free-chemistry retrievals with no priors on the CO2 abun- dance and find the same key results. For both cases, the data require a strong thermal inversion, a C/O ratio of ?1, and a super-solar metallicity. 4.6.2 Discussion The constraints on the chemical abundances are consistent with expectations for a high C/O ratio atmosphere in the high temperature regime of WASP-18b (Mad- husudhan, 2012; Moses et al., 2013) where chemical equilibrium are expected to be satisfied. At high temperatures, H2O is expected to be the most dominant oxygen- bearing molecule for a solar-abundance elemental composition (e.g. with a C/O = 0.5) (Madhusudhan, 2012; Moses et al., 2013). In contrast, the low-abundance of H2O observed is possible only if the overall metallicity and O abundance were low, or if the C/O ratio were high. Given the high abundance of CO we retrieve, the only plausible solution is both a high oxygen abundance and a high C/O ra- tio. The constraints on all the other species are also consistent with this scenario. While I cannot rule out a contribution from CO2 emission in the 4.5?m Spitzer 194 2 Pinhas et al. Parameter Retrieved value+1??1? log(XH2O) 2.7-? detection) due to Na, AlO and/or VO/TiO, though no individual species is strongly detected and the two methods disagree on which species drives the feature. Both retrievals determine the transit spectrum to be consistent with a clear atmosphere, with no evidence of haze or high-altitude clouds. Interior modeling constraints on the maximum atmospheric metallicity (log10 Z/Z < 1.7) favor the AURA results (Sec 3.10.2). Future Work: A parallel study examined the HST UVIS spectrum of HAT- P-41b (Lewis et al., 2020), and a natural extension would be to combine the entire transit data set (UVIS, STIS, WFC3, and Spitzer) into a single analysis. Similarly, there are archival HST WFC3 and Spitzer eclipse data on HAT-P-41b, which can be used to further discern between the environments inferred by the different retrieval methods. A joint retrieval ? where the transit and eclipse spectra are constrained simultaneously ? would be especially informative. More generally, it would be inter- esting to re-analyze literature multi-instrument spectrum to determine if including 215 an instrumental bias parameter significantly impacts conclusions. WASP-18b and WASP-19b: I derived the HST WFC3 emission spectra for these two highly irradiated hot Jupiters and combined it with Spitzer eclipse data to retrieve their atmospheric properties (Sec 4.4). Most notably, I found robust evidence for a strong thermal inversion on the dayside atmosphere of the massive, ultra-hot WASP-18b (Sec 4.6). I found a non-detection of water, TiO, and VO, but a moderate detection of a CO via both an emission feature at 4.5 ?m and a less convincing absorption feature at 1.6 ?m. The derived composition and T/P profile suggest that WASP-18b is the first example of a planet with a non-oxide driven thermal inversion. In WASP-19b, I found moderate evidence (2.8?) of water absorption and a decreasing T-P profile (Sec 4.7). Additionally, I derived a water abundance (logX =-3.64+1.44H2O ?0.72) and C/O ratio (C/O<0.63) which are consistent with both host star abundances and those found in the transit analysis (Benneke, 2015). My constraints based on the spectroscopic HST eclipse data overturn the results of previous eclipse analyses, which relied only on four Spitzer and two ground-based photometric points (Madhusudhan, 2012; Line et al., 2014). The presence of water and decreasing thermal profile have interesting implications on TiO-driven inver- sions (Fortney et al., 2008) and water dissociation (Parmentier et al., 2018), since WASP-19b is hot enough (2100 K) that both are expected to play a role. Future work: Like the other planets in this dissertation, analysis of additional data is a natural extension. For WASP-19b, that involves a joint transit/eclipse spectra retrieval, which can leverage the extra information to achieve additional 216 constraints (e.g, Kreidberg et al., 2014b). For both, JWST observations will provide vital insight into their atmosphere?s natures, and Figure 4.8 gives an example of how JWST?s high resolution will easily break current degeneracies. In the big picture, a uniform analysis of the roughly 30 eclipse observations on the MAST archive can add to these case studies and best constrain the atmospheric questions most relevant to eclipses. An example would be to update the Knutson et al. (2010) study, which correlated stellar activity to thermal inversions based on Spitzer photometric data alone. The additional resolving power of spectral HST data allows for more strict constraints on population-wide trends. Analysis Pipeline: Finally, I developed the codes I used to analyze data in this dissertation into a Python 3 pipeline, nicknamed DEFLATE, which is downloadable on Github2. I described the pipeline in detail in Sections 2.3 and 2.4. Within the dissertation, I validated my pipeline against literature spectra and verified my derived depths with a suite of quality-of-it diagnostics. The pipeline is highly customizable, allowing for exploration of the impact of both data processing and light curve fitting assumptions on the derived transit spectrum. It converts telescope fits image files to light curves (non-trivial for spatial scan mode), and uses marginalization to fit those light curves to both determine orbital properties (namely radius, transit/eclipse time, and optionally linear limb- darkening coefficient, inclination, or a/Rs) and to derive a transit spectrum. Is also provides a suite of diagnostics to verify light curve fits. Future work: Given the flexibility of the systematic model grid, this pipeline 2https://github.com/AstroSheppard/WFC3-analysis 217 can be readily expanded to work with JWST data once available, and will be es- pecially useful in determining which auxiliary parameters (such as orbital phase or wavelength shift) are relevant to JWST light curve analyses. More immediately, the flexible and customizable nature of the code make it ideal for creating a WFC3 exoplanet spectral library. While spectra are published in the literature, a uni- form analysis approach is necessary to ensure minimal bias due to analysis method in population studies. A uniform spectral library will give modelers access to an unprecedented amount of spectroscopic exoplanet data, enabling more frequent com- parative exoplanetology. 218 Appendix A: Chapter 3 Supplementary Material A.1 L9859b White Light Curves These figures show the band-integrated data for each of the three L9859b observations not shown in the chapter (Section 2.4). The band-integrated light curve is calculated by summing the integrated flux of every pixel in the final.fits exposures, which gives the total photons observed over the course of the exposure. The de-trended light curve is the data divided by the highest-weight systematic model. The bottom panel shows the residuals between the data and the highest- weight light curve model. For each L9859b observation, the first orbit and first few exposures of each orbit are removed. 219 1e8 visit00 1e8 visit01 9.16 9.18 9.14 (a) 9.16 (a) 9.12 9.10 9.14 9.08 9.12 9.06 9.04 9.10 9.02 9.08 9.00 0.06 0.04 0.02 0.00 0.02 0.04 0.06 0.04 0.02 0.00 0.02 0.04 +1 0.0002 1.0000 (b) 0.0000 (b) 0.9998 0.0002 0.9996 0.0004 0.9994 0.0006 0.9992 0.0008 0.04 0.02 0.00 0.02 0.04 0.06 0.04 0.02 0.00 0.02 0.04 0.06 200 100 (c) 150 (c) 50 100 0 50 50 0 100 50 100 150 150 0.04 0.02 0.00 0.02 0.04 0.06 0.04 0.02 0.00 0.02 0.04 0.06 Phase Phase 1e8 visit02 9.17 9.16 (a) 9.15 9.14 9.13 9.12 9.11 9.10 9.09 0.06 0.04 0.02 0.00 0.02 0.04 +1 0.0002 0.0000 (b) 0.0002 0.0004 0.0006 0.0008 0.04 0.02 0.00 0.02 0.04 200 150 100 (c) 50 0 50 100 150 200 0.04 0.02 0.00 0.02 0.04 Phase Figure A.1: Visualization of white light curve fit for the highest weighted systematic model for L9859b visits 00, 01, and 02. Panel (a) shows the band-integrated light curve. Panel (b) shows the de-trended light curve and the best fitting transit model. Note that this is illustrative ? the instrumental effect and transit model parameters are fit for simultaneously. Panel (c) shows the residuals between the data and the best-fitting model. 220 Obs - Model [ppm] Normalized Flux Raw Flux Normalized Flux Obs - Model [ppm] Raw Flux Normalized Flux Obs - Model [ppm] Raw Flux A.1.1 MCMC Validation Figures These are the additional figures from the L9859c whitelight fit. First, I visu- alize the result in Figure A.2 by projecting random samples from the posterior onto the space of the data. 1.0010 1.0005 1.0000 0.9995 0.9990 0.9985 0.9980 0.9975 0.100 0.125 0.150 0.175 0.200 0.225 0.250 0.275 +5.8946e4 Figure A.2: Distribution of model (black points) based on parameters derived in MCMC fit compared to data (blue). Y-axis is normalized flux and x-axis is MJD time. Next, I provide the full corner plot. Transit depth has no strong dependence or correlation with any systematic parameter. Finally, I prove covergence via autocorrelation. In ensemble MCMC samplers like emcee, the Gelman-Rubin statistic is not valid since the chains are not indepen- dent. Instead, for adequate sampling to achieve a small enough computational error, it is a good rule of thumb to run chains for at least 50?? , the autocorrelation time. This is not known a priori and must be estimated, as in Figure A.4. This process is 221 Depth = 1620.14+10.7410.44 +5.89462e4 Epoch = 58946.21+0.000.00 0 .00 7 0 9 .00 6 0 68 0.0 0 06 7 0.0 6 HST1 = 0.01+0.006 0.00 0.0 0 10 0.0 09 0.0 08 0.0 07 0.0 06 HST2 = 0.05 +0.00 0 0.000. 32 0.0 0 0.0 4 .04 8 0 6 0.0 5 .06 4 0 HST3 = 0.10 +0.01 0.01 .15 0 0 25 0.1 0 0.1 0 5 0.0 7 50 HST4 = 0.08+0.01.0 0.010 25 0.0 .05 0 0 75 0.0 .10 0 0 25 0.1 sh1 = 0.05+0.010.01 90 0.0 75 0.0 60 0.0 5 0.0 4 30 0.0 sh2 = 3.52+0.760.77 1.5 3.0 4.5 6.0 sh3 = 511.25+69.9572.19 0 30 504 60 0 75 0 sh4 = 17442.80+1841.151909.49 00 0 12 00 0 16 00 0 20 00 0 24 +1 fnorm = 1.00+0.000.00 00 4 0.0 0 .00 0 0 04 0.0 0 8 .00 0 0 flinear = 0.02+0.000.00 19 0.0 20 0.0 1 0.0 2 2 0.0 2 +1 rnorm = 1.00+0.000.00 01 5 0.0 0 000 0.0 0 01 5 0 0.0 0 00 3 0.0 80 00 20 40 60 66 67 68 69 70 06 07 08 09 10 64 56 48 40 32 50 75 00 25 50 25 00 75 50 25 30 45 60 75 90 6.0 4.5 3.0 1.5 50 00 0 0 0 0 0 0 8 4 0 4 2 1 0 9 0 55 6 6 6 6 0 0 0 0 0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 5 0 0 0 0 0 0 0 0 0 2 2 2 1 3 1 0 0 15 1 1 1 1 1 0.0 0.0 0.0 0.0 0.0 0 0 0 0 0 0 0 0 0 0 0 0 0 .1 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7 6 4 3 40 00 60 0 0 0 0 0 0 0 0 0 0 0 0 02 2 1 12 0.0 . . . . 0.0 0.0 0.0 0 0 0 0 .00 .00 .00 .00 Depth Epoch+5.89462e4 0 0 0 0 HST1 HST2 HST3 HST4 sh1 sh2 sh3 sh4 fnorm +1 flinear rnorm +1 Figure A.3: L9859c Transit MCMC Full Corner Plot 222 rnorm flinear fnorm sh4 sh3 sh2 sh1 HST4 HST3 HST2 HST1 Epoch explained in this tutorial https://emcee.readthedocs.io/en/stable/tutorials/autocorr/. At each chain length, the autocorrelation time is estimated for each parameter. When these estimates flatten out and cross the N/50 line, then the autocorrelation estimate becomes reliable. It is most important that depth converges, but ideally all parameters will. In this case, the autocorrelation time is about 100, meaning 5000 samples is the minimum for convergence. I run the chain for 20000 steps for 2.5?nDimensions walkers, and conservatively remove the first 2000 as burn-in. Depth Epoch HST1 HST2 HST3 HST4 102 sh1sh2 sh3 sh4 fnorm flinear rnorm N/50 101 102 103 104 Chain Length Figure A.4: L9859c Transit MCMC Proof of Convergence A.2 L9859 Transit HST Spectrophotometric Light Curve Fits The figures in this section show the de-trended light curve data and best-fit transit model for every spectral bin for each observation. These are analogous to panel (b) of the figures in Section A.1. They illustrate the fit used to derive the transit spectrum (Section 2.4). 223 Autocorrelation time estimate 1.143 m 1.0000 1.190 m 0.9975 1.236 m 1.282 m 0.9950 1.329 m 1.375 m 0.9925 1.422 m 0.9900 1.468 m 1.514 m 0.9875 1.561 m 1.607 m 0.9850 0.02 0.00 0.02 0.04 0.06 Orbital Phase Figure A.5: Spectral light curves for L9859b, visit 00. 224 Normalized Flux - Constant 1.145 m 1.000 1.191 m 0.998 1.237 m 0.996 1.284 m 1.330 m 0.994 1.377 m 0.992 1.423 m 0.990 1.470 m 1.516 m 0.988 1.562 m 0.986 1.609 m 0.984 0.02 0.00 0.02 0.04 0.06 Orbital Phase Figure A.6: Spectral light curves for L9859b, visit 01. 225 Normalized Flux - Constant 1.146 m 1.0000 1.192 m 0.9975 1.239 m 1.285 m 0.9950 1.331 m 1.378 m 0.9925 1.424 m 0.9900 1.471 m 1.517 m 0.9875 1.563 m 1.610 m 0.9850 0.02 0.00 0.02 0.04 Orbital Phase Figure A.7: Spectral light curves for L9859b, visit 02. 226 Normalized Flux - Constant 1.144 m 1.0000 1.191 m 0.9975 1.237 m 1.284 m 0.9950 1.330 m 1.377 m 0.9925 1.423 m 0.9900 1.470 m 1.516 m 0.9875 1.562 m 1.609 m 0.9850 0.02 0.00 0.02 0.04 Orbital Phase Figure A.8: Spectral light curves for L9859b, visit 03. 227 Normalized Flux - Constant A.3 Red Noise Diagnostic Figures The figures in this section visualize correlated noise analysis. See Section 2.4.3.1 for detailed discussion on how to interpret these figures. 228 1.0 Expected Data RMS 1.143 m 1.190 m 1.236 m 0.1 1.0 1.282 m 1.329 m 1.375 m 0.1 1.0 1.422 m 1.468 m 1.514 m 0.1 1.0 Band-integrated Figure A.9: Correlated 1.561 m 1.607 m 0.1 Noise Diagnostic Figures for 1 2 3 4 5 6 789 1 2 3 4 5 6 789 1 2 3 4 5 6 789 Exposures Per Bin L9859b Visit 00. Top: Bin- ning analysis for each spec- tral bin (see Section 2.4.3.1. Bottom: Autocorrelation 0.3 function for each spectral 0.2 bin. 0.1 0.0 2 sigma range 0.1 0.2 1.143 m 1.190 m 1.236 m 0.3 0.2 0.1 0.0 0.1 0.2 1.282 m 1.329 m 1.375 m 0.3 0.2 0.1 0.0 0.1 0.2 1.422 m 1.468 m 1.514 m 0.3 0.2 0.1 0.0 0.1 0.2 1.561 m 1.607 m Band-integrated 0.3 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Lag 229 Autocorrelation Normalized RMS 1.0 Expected Data RMS 1.145 m 1.191 m 1.237 m 0.1 1.0 1.284 m 1.330 m 1.377 m 0.1 1.0 1.423 m 1.470 m 1.516 m 0.1 1.0 Band-integrated Figure A.10: Correlated 1.562 m 1.609 m 0.1 Noise Diagnostic Figures for 1 2 3 4 5 6789 1 2 3 4 5 6789 1 2 3 4 5 6789 Exposures Per Bin L9859b Visit 01. Top: Bin- ning analysis for each spec- tral bin (see Section 2.4.3.1. Bottom: Autocorrelation 0.3 function for each spectral 0.2 bin. 0.1 0.0 2 sigma range 0.1 0.2 1.145 m 1.191 m 1.237 m 0.3 0.2 0.1 0.0 0.1 0.2 1.284 m 1.330 m 1.377 m 0.3 0.2 0.1 0.0 0.1 0.2 1.423 m 1.470 m 1.516 m 0.3 0.2 0.1 0.0 0.1 0.2 1.562 m 1.609 m Band-integrated 0.3 0 20 40 60 0 20 40 60 0 20 40 60 Lag 230 Autocorrelation Normalized RMS 1.0 Expected Data RMS 1.146 m 1.192 m 1.239 m 0.1 1.0 1.285 m 1.331 m 1.378 m 0.1 1.0 1.424 m 1.471 m 1.517 m 0.1 1.0 Band-integrated Figure A.11: Correlated 1.563 m 1.610 m 0.1 Noise Diagnostic Figures for 1 2 3 4 5 6789 1 2 3 4 5 6789 1 2 3 4 5 6789 Exposures Per Bin L9859b Visit 02. Top: Bin- ning analysis for each spec- tral bin (see Section 2.4.3.1. Bottom: Autocorrelation 0.3 function for each spectral 0.2 bin. 0.1 0.0 2 sigma range 0.1 0.2 1.146 m 1.192 m 1.239 m 0.3 0.2 0.1 0.0 0.1 0.2 1.285 m 1.331 m 1.378 m 0.3 0.2 0.1 0.0 0.1 0.2 1.424 m 1.471 m 1.517 m 0.3 0.2 0.1 0.0 0.1 0.2 1.563 m 1.610 m Band-integrated 0.3 0 20 40 60 0 20 40 60 0 20 40 60 Lag 231 Autocorrelation Normalized RMS 1.0 Expected Data RMS 1.144 m 1.191 m 1.237 m 0.1 1.0 1.284 m 1.330 m 1.377 m 0.1 1.0 1.423 m 1.470 m 1.516 m 0.1 1.0 Band-integrated Figure A.12: Correlated 1.562 m 1.609 m 0.1 Noise Diagnostic Figures for 1 2 3 4 5 6789 1 2 3 4 5 6789 1 2 3 4 5 6789 Exposures Per Bin L9859b Visit 03. Top: Bin- ning analysis for each spec- tral bin (see Section 2.4.3.1. Bottom: Autocorrelation 0.3 function for each spectral 0.2 bin. 0.1 0.0 2 sigma range 0.1 0.2 1.144 m 1.191 m 1.237 m 0.3 0.2 0.1 0.0 0.1 0.2 1.284 m 1.330 m 1.377 m 0.3 0.2 0.1 0.0 0.1 0.2 1.423 m 1.470 m 1.516 m 0.3 0.2 0.1 0.0 0.1 0.2 1.562 m 1.609 m Band-integrated 0.3 0 20 40 60 0 20 40 60 0 20 40 60 Lag 232 Autocorrelation Normalized RMS Appendix B: Chapter 4 Supplementary Material B.1 HAT-P-41b Transit HST Spectrophotometric Light Curve Fits 233 Figure B.1: Spectral light curves for STIS G430L, visit 83. 234 Figure B.2: Spectral light curves for STIS G430L, visit 84. 235 Figure B.3: Spectral light curves for STIS G750L, visit 85. 236 Figure B.4: Spectral light curves for single WFC3 visit. 237 Appendix C: Facilities and Software 1. Archival Data Used in Thesis All data are from the MAST Archive1, and are provided with an associated proposal ID (GO), Principal Investigator (PI), and Digitial Object Identifier (DOI). ? HAT-P-41b: HST STIS and WFC3 (GO 14767, PI Sing) and Spitzer (GO 13044, PI Deming) transit data: DOI 10.17909/t9-fg9z-er59 ? L9859 System: HST WFC3 (GO 15856, PI Barclay) transit data: DOI 10.17909/t9-xf60-w063 ? WASP-18b: HST WFC3 (GO 13467, PI Bean) eclipse data: DOI 10.17909/t9- 4pmh-fd65 ? WASP-19b: HST WFC3 (GO 13431, PI Huitson) eclipse data: DOI 10.17909/t9-xnwm-hd84 ? WASP-18b and WASP-19b Spitzer (GO 60185, PI Maxted) eclipse data: DOI 10.17909/t9-kavg-yj38 2. Open source software used in thesis 1https://archive.stsci.edu/ 238 ? IRAF (Tody, 1986, 1993) ? SciPy (Jones et al., 2001?) ? Matplotlib (Hunter, 2007) ? nestle (https://github.com/kbarbary/nestle) ? dynesty (Higson et al., 2019) ? BATMAN (Kreidberg, 2015) ? Kapetyn (Terlouw & Vogelaar, 2015) ? Corner.py (Foreman-Mackey, 2016) ? PLATON (Zhang et al., 2019, 2020b) ? NumPy (Harris et al., 2020) ? Pandas (The Pandas Development Team, 2020) ? mc3 (https://github.com/pcubillos/mc3) 3. Software I developed used in thesis ? 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