Exoplanetary Atmospheres PRINCETON SERIES IN Edited by David N. Spergel

Theory of Rotating Stars, by Jean-Louis Tassoul

Theory of Stellar Pulsation, by John P. Cox

Galactic Dynamics, Second Edition, by James Binney and Scott Tremaine

Dynamical Evolution of Globular Clusters, by Lyman S. Spitzer, Jr.

Supernovae and Nucleosynthesis: An Investigation of the History of Matter, from the Big Bang to the Present, by David Arnett

Unsolved Problems in Astrophysics, edited by John N. Bahcall and Jeremiah P. Ostriker

Galactic , by James Binney and Michael Merrifield

Active Galactic Nuclei: From the Central to the Galactic Environment, by Julian H. Krolik

Plasma for Astrophysics, by Russell M. Kulsrud

Electromagnetic Processes, by Robert J. Gould

Conversations on Electric and Magnetic Fields in the Cosmos, by Eugene N. Parker

High-Energy Astrophysics, by Fulvio Melia

Stellar Spectral Classification, by Richard O. Gray and Christopher J. Corbally

Exoplanet Atmospheres: Physical Processes, by Sara Seager

Physics of the Interstellar and Intergalactic Medium, by Bruce T. Draine

The First Galaxies in the Universe, by Abraham Loeb and Steven R. Furlanetto

Exoplanetary Atmospheres: Theoretical Concepts and Foundations, by Kevin Heng Exoplanetary Atmospheres

Theoretical Concepts and Foundations

Kevin Heng

PRINCETONUNIVERSITYPRESS

PRINCETONANDOXFORD Copyright c 2017 by Princeton University Press

Published by Princeton University Press 41 William Street, Princeton, New Jersey 08540

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ISBN 978-0-691-16697-1 (cloth) ISBN 978-0-691-16698-8 (paperback)

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10987654321 Dedications

Dick McCray, for showing me that is to be enjoyed Helmer Aslaksen, who believed in me when no one else did Scott Tremaine, for teaching me how to read the literature Rashid Sunyaev, for giving me a chance to learn from greatness Sara Seager, for inspiring me to think big Willy Benz, for teaching me the business of science Ray Pierrehumbert, for being a fountain of energy and creativity Claudia, my little black cat, the dearest creature I have ever known Felix Hetzenecker-Heng, who forced the first draft 7 months before deadline Stefanie Hetzenecker, my wife and best friend, who stood by me in all times

Contents

Foreword by Sara Seager xi

Preface xiii

1 Observations of Exoplanetary Atmospheres: ATheorist’sReviewofTechniquesinAstronomy 1 1.1 The birth of exoplanetary science ...... 1 1.2 Transits and occultations ...... 2 1.3 Radial velocity measurements ...... 8 1.4 Direct imaging ...... 11 1.5 Gravitational microlensing ...... 12 1.6 Futuremissionsandtelescopes ...... 12

2 Introduction to Radiative Transfer 14 2.1 The optical depth: The most fundamental quantity in radiative transfer...... 14 2.2 Basic quantities in radiative transfer ...... 16 2.3 Theradiativetransferequation...... 20 2.4 Simple solutions of the radiative transfer equation ...... 20 2.5 A practical checklist for radiative transfer calculations ...... 23 2.6 Clouds ...... 24 2.7 Atmosphericretrieval ...... 27 2.8 Problemsets...... 31

3 The Two-Stream Approximation of Radiative Transfer 35 3.1 Whatisthetwo-streamapproximation?...... 35 3.2 The radiative transfer equation and its moments ...... 36 3.3 Two-stream solutions with isotropic scattering ...... 39 3.4 Thescatteringphasefunction ...... 45 3.5 Two-stream solutions with non-isotropic scattering ...... 46 3.6 Different closures of the two-stream solutions ...... 49 3.7 The diffusion approximation for radiative transfer ...... 51 3.8 Problemsets...... 53 viii CONTENTS

4 Temperature-Pressure Profiles 56 4.1 A myriad of atmospheric effects: Greenhouse warming and anti- greenhousecooling...... 56 4.2 The dual-band or double-gray approximation ...... 57 4.3 The radiative transfer equation and the scattering parameter . . 58 4.4 Treatmentofshortwaveradiation ...... 60 4.5 Treatmentoflongwaveradiation ...... 64 4.6 Assembling the pieces: Deriving the general solution ...... 65 4.7 Exploration of different atmospheric effects ...... 67 4.8 Milne’s solution and the convective adiabat ...... 71 4.9 Problemsets...... 72

5 AtmosphericOpacities: HowtoUseaLineList 74 5.1 From spectroscopic line lists to synthetic spectra ...... 74 5.2 TheVoigtprofile...... 76 5.3 Thequantumphysicsofspectrallines ...... 78 5.4 The million- to billion-line radiative transfer challenge ...... 81 5.5 Differenttypesofmeanopacities...... 88 5.6 Problemsets...... 89

6 Introduction to Atmospheric Chemistry 92 6.1 Why is atmospheric chemistry important? ...... 92 6.2 Basic quantities: Gibbs free energy, equilibrium constant, rate coefficients ...... 93 6.3 Chemical kinetics: Treating chemistry as a set of mass conser- vationequations ...... 101 6.4 Self-consistent atmospheric chemistry, radiation and dynamics: A formidable computational challenge ...... 106 6.5 Problemsets...... 107

7 AHierarchyofAtmosphericChemistries 110 7.1 A hierarchy of models for understanding atmospheric chemistry 110 7.2 Equilibrium chemistry with only hydrogen ...... 110 7.3 Equilibrium C-H-O chemistry: Forming methane, water, carbon monoxideandacetylene...... 113 7.4 Equilibrium C-H-O chemistry: Adding carbon dioxide ...... 115 7.5 Equilibrium C-H-O chemistry: Adding ethylene ...... 121 7.6 Problemsets...... 122

8 Introduction to Fluid Dynamics 123 8.1 Why is the study of fluids relevant to exoplanetary atmospheres? 123 8.2 What exactly is a fluid? ...... 124 8.3 The governing equations of fluid dynamics ...... 124 8.4 Potential temperature and potential vorticity ...... 128 8.5 Dimensionless fluid numbers ...... 130 CONTENTS ix

8.6 Problemsets...... 132

9 Deriving the Governing Equations of Fluid Dynamics 135 9.1 Preamble...... 135 9.2 The mass continuity equation (mass conservation) ...... 135 9.3 The Navier-Stokes equation (momentum conservation) . . . . . 136 9.4 The thermodynamic equation (energy conservation) ...... 138 9.5 The conservation of potential vorticity ...... 139 9.6 Various approximate forms of the governing equations of fluid dynamics...... 143 9.7 Magnetohydrodynamics...... 147 9.8 Problemsets...... 151

10 The Shallow Water System: A Fluid Dynamics Lab on Paper 155 10.1 A versatile fluid dynamics laboratory on paper ...... 155 10.2 Deriving the shallow water equations ...... 156 10.3 Gravity as the restoring force: The generation of gravity waves . 158 10.4 Friction in an atmosphere: Molecular viscosity and Rayleigh drag 160 10.5 Forcing the atmosphere: Stellar irradiation ...... 162 10.6 Like plucking a string: Alfv´en waves ...... 163 10.7 Rotation: The generation of Poincar´eand Rossby waves . . . . 165 10.8 Generalcouplingofphysicaleffects ...... 167 10.9 Shallow atmospheres as quantum harmonic oscillators ...... 168 10.10 Shallow water systems and exoplanetary atmospheres ...... 174 10.11Problemsets...... 175

11 The de Laval Nozzle and Shocks 182 11.1 WhatisthedeLavalnozzle? ...... 182 11.2 Whatareshocks? ...... 184 11.3 What does the de Laval nozzle teach us about shocks? . . . . . 187 11.4 Applications to, and consequences for, exoplanetary atmospheres 191 11.5 Problemsets...... 192

12Convection, Turbulence and Fluid Instabilities 196 12.1 Fluid motion induced by physically unstable configurations . . . 196 12.2 Hot air rises and cold air sinks: Schwarzschild’s criterion for convective stability ...... 196 12.3 A simplified “theory” of convection: Mixing length theory . . . 199 12.4 Implementing convection in numerical calculations: Convective adjustmentschemes ...... 200 12.5 A simple “theory” of turbulence: The scaling laws of Kolmogorov 202 12.6 Water over oil: The Rayleigh-Taylor instability ...... 204 12.7 Shearing fluids: The Kelvin-Helmholtz instability ...... 206 12.8 Weather at mid-latitudes: The baroclinic instability ...... 207 12.9 Problemsets...... 209 x CONTENTS

13 Atmospheric Escape 211 13.1 TheKnudsennumberandJeansparameter ...... 211 13.2 Jeansescape ...... 213 13.3 The classical Parker wind solution ...... 213 13.4 Non-isothermal Parker winds: Using the nozzle solutions . . . . 216 13.5 Detailed processes: Photo-ionization, radiative cooling and non- thermalmechanisms...... 218 13.6 Problemsets...... 221

14 Outstanding Problems of Exoplanetary Atmospheres 223

AppendixA:SummaryofStandardNotation 228

Appendix B: Essential Formulae of Vector Calculus 233

Appendix C: Essential Formulae of Thermodynamics 235

Appendix D: Gibbs Free Energies of Various Molecules and Re- actions 237

Appendix E: Python Scripts for Generating Figures 240

Bibliography 250

Index 271 Foreword by Sara Seager

The field of research of exoplanet atmospheres is flourishing in observation and theory. Both the quality and quantity of observations are increasing rapidly—for a variety of planet types including transiting planets and directly imaged giant planets. With the James Webb Space Telescope on the horizon, the promise of large numbers of a huge variety of exoplanet atmospheres with observations at high-precision, high-spectral resolution will finally be fulfilled. As a result, more and more researchers are entering the field, especially students from many disci- plines. Observations alone are not enough—theory to interpret the observations and to guide future observing strategy is key. The theory of exoplanets atmospheres must draw on a vast body of physics, chemistry, and atmospheric science. Yet, exoplanet atmospheres is truly its own discipline, requiring the topics to be tied together in a way not normally taught in any standard physics or atmospheric science class. Further, remote sensing of distant exoplanets means we will always be limited in data extent and quality, as compared to Solar System planets which can be visited directly by orbiters and landers. So, different techniques and different applications of theory are needed over traditional planetary science. But more importantly, exoplanets have a huge diversity, appearing in nearly all masses, sizes, and orbits physically plausible. We anticipate this diversity will extend to the planet atmospheres— and indeed already have evidence in this regard for hot Jupiters observed with the Hubble Space Telescope. But just how does one tie all of the topics together? Dr. Heng succeeds in providing an insightful and comprehensive treatise threading together key elements of exoplanet atmospheres, starting with radia- tive transfer and through opacities, chemistry, fluid dynamics, convection, and touching on atmospheric escape. Both deep and broad, Dr. Heng’s book goes beyond a standard graduate-level textbook to provide a very thorough foun- dation for those wanting to perform research in the field. For example, early in the book Dr. Heng treats the two-stream approximation to the radiative transfer equation in meticulous detail and later, over three chapters, gives an impressive treatment of atmospheric fluid dynamics. The best way to learn and gain intuition is to apply—in this field by implementing equations into one’s own computer code and then experimenting with it. To this end—and a fresh ingredient of many chapters—Dr. Heng outlines a recipe checklist for implemen- tation, including pitfalls. One of my favorite features of the book are comments throughout on how to simplify a complicated equation, when simplification is appropriate, and what the dangerous caveats to the simplification are. The book is the most thorough text on exoplanet atmospheres to date. It dives deeper and therefore builds upon my book “Exoplanet Atmospheres,” the first on the subject and published in 2010 also by Princeton University Press. Indeed, the author intends his new book to be a logical continuation of the xii

first. The book is complementary to the range of existing scholarly books on exoplanets in general. Dr. Heng is one of the world’s foremost experts on exoplanet atmospheres and has encapsulated his expertise into the book. His book will take you through the complexities and simplicities, the elegance and challenges of exoplanet at- mospheres. Enjoy the journey!

Sara Seager Professor of Planetary Science Professor of Physics Massachusetts Institute of Technology Cambridge, MA, U.S.A. Preface

Why did I feel the need to write this book in the first place? I started in astrophysics studying supernova remnants for my doctoral thesis. Almost im- mediately after obtaining my doctorate, I dropped the topic and embarked on a period of exploration, eventually chancing upon the subject of exoplanetary atmospheres in the same year that exoplanet pioneer Sara Seager’s “Exoplanet Atmospheres” textbook was published. The topic intrigued me, because study- ing and understanding the atmospheres of exoplanets is an indispensable step on the path towards addressing one of the oldest questions posed by humanity: Are we alone? And how do we scan, from afar, this myriad of worlds to find out? It became readily apparent that atmospheric science is an unavoidably inter- disciplinary endeavor. Undeniably, it has its roots in the Earth —and why would it not, since we actually live in this particular atmosphere? Yet it has long had an eye towards the stars, as the Solar System provides several examples of atmospheres that have been carefully scrutinized by planetary sci- entists. To understand the structure and appearance of an atmosphere requires a working understanding of fluid dynamics, radiation and its passage through matter, chemistry, the influence of aerosols and clouds, thermodynamics and phenomenology. To place the atmosphere of an exoplanet in a broader con- text, beyond the confines of our Solar System, requires that we understand the birth and death of stars other than our Sun, the properties of stellar popula- tions and how substellar objects known as brown dwarfs bridge the continuum between exoplanets and stars [230]. To do my own work, I had to reconcile techniques, terminology and modeling philosophies from numerous monographs and papers, across the atmospheric, oceanic and climate sciences, astronomy and astrophysics, planetary science, geophysics and chemistry. Even the term “model” means different things to different communities. Thus, the idea was planted in my mind: the need for a single book that explained the concepts and principles needed to embark on theoretical research in the atmospheres of exoplanets. The original intention was to write it in the style of “exoplanetary atmo- spheres for dummies,” based on the notion that one would not have to wade through pages of equations or prose to get at the salient features of an idea. This has evolved into something slightly more sophisticated: to explain a con- cept using the shortest and/or most intuitive approach possible and relegating the standard or classical explanations to the problem sets. This approach has borne unexpected fruit, as some of the alternative derivations are either novel or cleaner, but without necessarily sacrificing on technique. It is my firm belief that one does not really understand physics until one has to apply the concepts acquired to examples; about a sixth of the textbook is devoted to problem sets. xiv

In a departure from the traditional approach, some of the puzzles described in the problem sets are qualitative and quasi-open-ended, designed to provoke thought and encourage debate. To tackle a vast and interdisciplinary field of inquiry like exoplanetary at- mospheres, it is not enough to check that its theoretical foundations are not Earth-centric—and, if so, to generalize them. It also helps to understand the history behind some of these ideas and appreciate what the different communi- ties of scientists accept as being standards of proof. Thus, I return to my earlier worry: what is a model? To purists, a model is an approximate description of Nature based on a governing equation that is itself derived from a conservation law—of mass, momentum, energy or some other, more generalized quantity such as the potential vorticity. I favor this approach. To others, it may be a mathe- matical function describing how some aspect of the atmosphere behaves as the different parameters describing it are varied—but derived entirely from fitting this ad hoc function to experimental data; the function itself is “pulled out of a hat,” unconstrained by any law of Nature. To compound the issue, the use of simulations to address scientific questions is coming into its own as a third way of establishing scientific truth, bringing with it an entire set of epistemological and metaphysical concerns [95]. To engage constructively in conversations about exoplanetary atmospheres, across the spectrum of disciplines, requires that one is perpetually aware of what commonly-used terms mean to different scientists. The way that data are measured, collected and interpreted in the Earth and planetary sciences differs somewhat from astronomy. Data of the Earth’s at- mosphere are abundant and rich in detail. Measurements of the atmospheric properties of the planets and moons of our Solar System are a little less rich in detail, but still of a quality that would make any exoplanet astronomer en- vious. It is important to recognize that these fields of inquiry are collecting a wealth of information on only a handful of objects. Astronomers studying exoplanetary atmospheres are in the other regime: the data are sparse, usually incomplete, and reveal fragmentary information about any given atmosphere, but the number of objects being studied is vastly larger. The challenge is to reconcile the possible benchmarking opportunities offered by the Solar System with the statistical trends revealed by exoplanets. In this debate, it is wise to not mistake precision for accuracy. We will always know more about the Solar System in detail, but it remains unclear if we are stuck in the situation of looking for our metaphorical keys under the nearest lamp—the astronomical data are already hinting at this possibility. Assigning importance to scientific questions based on the cosmic proximity of the object being studied distracts from the bigger, broader questions exoplanetary science is attempting to address. I do not see how one can conduct research in exoplanetary science without a working understanding of the data collected by astronomers (rather than by Earth or planetary scientists). It is no accident that the first chapter contains a broad review of observational techniques and the data they produce. This brings us to a technique loved by astrophysicists and loathed by the other disciplines that are used to enjoying more precision: understanding a con-