The Atmospheres and Orbital Dynamics of Hot Jupiters

THE ATMOSPHERES AND ORBITAL DYNAMICS OF HOT JUPITERS

Matthew Pryal Archbald, Pennsylvania

B.S. Astronomy & Astrophysics, Penn State University, 2015

M.S. Astronomy, University of Virginia, 2017

A Dissertation Presented to the Graduate Faculty of the University of Virginia in Candidacy for the Degree of Doctor of Philosophy

Department of Astronomy

University of Virginia December 2020

Committee Members: Phillip L. Arras Zhi-Yun Li Tim Bastian Ajay B. Limaye c Copyright by Matthew Pryal

December 18, 2020 To my family and friends. Whom without, I would have ﬁnished this thesis much, much earlier. iii Abstract

The detection of the ﬁrst exoplanets in the early 1990s brought along many questions about their characteristics, including on their atmospheric structure and the dynamics of their planetary systems. One of the most puzzling types of exoplanets were the so-called ‘hot Jupiters.’ Hot Jupiters are Jupiter sized objects, orbiting very close to their host star, that absorb a large amount of radiation, making their atmospheres very hot. Their short orbital periods and hot, extended atmospheres are some of the many characteristics that make these objects unlike any planet in our Solar System. In this thesis, I present models of the electrical conductivity of the upper atmosphere of hot Jupiters as well as the migration of hypothetical moons orbiting hot Jupiters due to tidal friction eﬀects.

In Chapter1, I give an introduction and provide context to the research outlined in the thesis. Speciﬁcally, I give an overview of the detection of exoplanets and discuss HD 189733b, the hot Jupiter at the focus of my atmosphere model. I also introduce the interaction of a stellar wind with a planet’s magnetosphere which is central towards setting the current that can run through the atmosphere, which is then aﬀected by the atmosphere’s electrical conductivity. Finally, I overview the dynamics of star- planet-moons systems and tidal friction as it relates to the hypothetical moons of hot Jupiters.

In Chapter2, I detail the star-planet interaction between the charged particles of the stellar wind with a planetary magnetic ﬁeld that set the current in the atmosphere. Importantly, the electrical conductivity of the atmosphere governs the rate at which the charged particles can ﬂow and release energy into the atmosphere through collisions. I also review the formulas for electrical conductivity of an atmosphere and discuss the collision rates between charged particles and other species of the atmosphere which is then implemented into the HD 189733b atmosphere model in Chapter 3.

In Chapter3, I present a hydrostatic model of the dayside upper atmosphere of hot Jupiter HD 189733b. With this model I compute its electrical conductivity and conductance, which determines the amount of energy that can be transferred to the atmosphere through Joule heating. I find that the conductance of hot Jupiters is much greater than the conductance of the Earth, and therefore Joule heating does not significantly impact the thermal structure of the atmosphere for Jupiter sized magnetic fields (B ∼ 10 G).

Finally, in Chapter4, I examine the orbital migration of hypothetical moons of hot iv

Jupiters. I ﬁnd that, even in the case of synchronous rotation and circular orbits of the moon, gravitational perturbations from the star create a forced eccentricity in the moon that makes tidal friction never cease. This tidal friction then causes the inward migration of the moon towards the planet where it can be tidally disrupted. In the case of hot Jupiter systems, this inward migration is fast enough to explain the lack of large moons detected orbiting hot Jupiters. Additionally, I place an upper limit on the mass of moons that are able to survive this orbital migration eﬀect within 5 Gyr, to be many orders of magnitude smaller than the mass of the Earth’s moon. v Acknowledgements

My love for Astronomy began in a ninth grade Earth and Space Science class, that I was warned by my older siblings was taught by a ‘crazy alien guy’ – who I would later find out preferred to go by Mr. Durkin. Over the course of that class, I realized that Mr. Durkin’s ‘craziness,’ was actually child-like wonder and passion for a science that can be appreciated by anyone simply by looking up at the night sky. Unknown to him, Mr. Durkin’s passion was passed to me, and over my years in Astronomy, I realized that I wanted to become that ‘crazy alien guy’ for others. It is truly unbelievable to think how much Mr. Durkin’s passion about Astronomy has impacted my path in life – taking me from a four bedroom house in tiny Archbald, PA (at one time shared by 10 people!), to Penn State University, and Rome, on cross country trips, to the University of Virginia, and Hawaii and somehow to the stars and back. I urge everyone to embrace the passions in your life and share them with as many people as you can. You never know how much of an impact your passion can have on others. At each milestone I reach in my life, I can’t help but think of all the little things that those in my life have taught me to get me to where I am. Grad school has admittedly been emotionally hard. The year 2020 has been especially stressful. But with everything I’ve been taught by family and close friends, we’ve been able to reach this finish line together. Thank you to my Mom and Dad for filling me with love and being proud of me every step of the way. For teaching me empathy, kindness, how to deal with sadness, and embrace my passions. Thanks, to my siblings Bridget, Sarah, Robbie, Lizzy, and Emma for teaching me how to be a family. To learn how to deal with anger (don’t punch walls), to be more outgoing, to love unconditionally, and handle disagreements. One of the few good things about growing up, has been becoming closer to all of you each and every year and I can’t wait for our relationships to continue to grow. Being the first person in my family to get a Ph.D., I can’t help but think of my Grammie, Grampie, Grandma, and Grandpa. Thank you for creating a better life for all those around you and laying the groundwork for your loved ones to be able to dream big today. My thanks to my family could extend forever but since acknowledgements are supposed to be shorter than the thesis, I’ll just say thank you to rest of my extended family for making my life as full as it can be. Thanks to my Brohana, my cohort, old Charlottesville friends, and the rest of those at UVA and in Charlottesville for making my time in grad school fun. Thanks for grilling and chilling, having tailgates, chatting around a fire, for going to Nelly’s or Escafé,going to basketball games, playing poker, for Foxfield, and Raven banquets, for Festy and for everything else in between. Grad school would have been unbearable without all of you in it. Thank you to the Awesome Astronomy Enthusiasts of 2015 and everyone else at vi

Penn State for teaching me that life’s about 8 things and 8 things only. For being a bunch of nerds who embraced our shared weirdness, for really knowing how to have fun, and for always saying hi to me in the elevator. Integrity. Thank you to my NEPA friends for being a part of some of the most fundamental parts of my life and for eﬀectively helping me grow up – although I wouldn’t say we’re there yet, and I’m not sure if we’ll ever quite make it. Thanks to my thesis committee and thesis advisor Phil for making my research as great as it could be over the past 5.5 years. And to the rest of the UVA and Penn State Astronomy departments – thanks for your kindness and support along the way. And a special thanks to all those who I’ve had the pleasure of doing outreach with over the years. Especially to all the members of Dark Skies, Bright Kids, Astronomy on Tap - C’ville, those at the Science Museum of Virginia, and everyone else. You’ve helped make the journey worthwhile and I certainly would have been much crazier without you in my life. And last, but certainly not least (except in physical size) – thank you Molly for making one of the most stressful parts of my life, somehow the happiest. I was only able to get to this ﬁnish line because of your support these past few years. I eagerly await any stressful days ahead, as long as we’re facing them together. Also, feather.

With love, The other crazy alien guy Table of contents

List of Figures xiv

List of Tables xvi

1 Introduction1 1.1 Exoplanets...... 1 1.1.1 The Discovery of Exoplanets...... 2 1.1.2 The Detection of Exoplanets...... 3 1.1.3 Hot Jupiters...... 9 1.2 The Upper Atmosphere of HD 189733b...... 11 1.2.1 Spectroscopic Observations of Exoplanet Atmospheres.... 12 1.2.2 Chemical Species in the Atmosphere of Hot Jupiters...... 13 1.2.3 Upper Atmosphere Models of Hot Jupiters...... 15 1.3 Planetary Magnetospheres and Ionospheres...... 16 1.3.1 Overview...... 17 1.3.2 The Solar Wind and Magnetic Reconnection...... 18 1.4 The Dynamics of Star-Planet-Moon Systems...... 20 1.4.1 Introduction...... 22 1.4.2 The Lack of Moons in the Inner Solar System...... 23 1.4.3 Tidal Friction in Two Body Systems...... 23 1.4.4 The Three-Body Problem...... 27 1.4.5 Model of a Star-Planet-Moon System...... 28 1.5 Overview...... 29

2 Electric Currents and Conductivity in Planetary Atmospheres 31 2.1 Context...... 31 2.2 Historical Overview...... 32 2.3 Electric Currents Overview...... 34 2.4 Joule Heating...... 36 2.4.1 Height Integrated Currents and Conductivity...... 36 2.4.2 Parallel Currents and AlfvénConductance...... 37 2.4.3 Reflection of AlfvénWaves...... 39

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2.4.4 Joule Heating Calculation...... 41 2.5 Electrical Conductivity Overview...... 43 2.6 Deriving the Conductivity Formulas...... 45 2.6.1 Conductivity Across a B-field...... 45 2.6.2 Conductivity Parallel to the B-field...... 48 2.7 Height Integrated Conductivity...... 50 2.7.1 Qualitative Features of the Conductivity Profile...... 50 2.7.2 Analytic Estimate of Σ...... 51 2.8 Collision Rates...... 53 2.8.1 General Collision Parameters...... 53 2.8.2 Ion/Electron-Neutral Non-resonant Collisions...... 54 2.8.3 Resonant Charge Exchange...... 55 2.8.4 Coulumb Collisions...... 56 2.8.5 Location of Peaks in the Conductivity...... 57 2.9 Summary...... 60

3 Electrical Conductance of the Upper Atmosphere of Hot Jupiter HD 189733b 62 3.1 Summary...... 62 3.2 Introduction...... 63 3.3 The Model...... 66 3.3.1 Determining the Pressure Limits...... 67 3.3.2 Basic Atmosphere Structure...... 69 3.3.3 Differential Equations...... 76 3.3.4 Balance of Production and Loss Processes...... 77 3.3.5 Charge Neutrality...... 80 3.3.6 Thermal Balance...... 81 3.4 Results...... 90 3.4.1 Density and Temperature Profiles...... 90 3.4.2 Conductivity Profile...... 95 3.4.3 Magnetic Field Effect on Conductivity...... 98 3.4.4 Joule Heating in the Upper Atmosphere of HD 189733b.... 100 3.5 Discussion...... 102 3.5.1 Effect of Molecules on the Conductance...... 102 3.5.2 Relative Size of the Pederson Conductance...... 103 3.5.3 Significance of Joule Heating...... 105 3.5.4 Location of Joule Heating in the Atmosphere...... 108 3.6 Conclusions...... 109

4 Tidal Disruption of Moons in Star-Planet-Moon Systems 111 4.1 Summary...... 111 4.2 Introduction...... 112 ix

4.3 Tidal Evolution of Two-Body Systems...... 117 4.4 The Star-Planet-Moon System...... 119 4.5 Equations of Motion...... 120 4.6 Torque Equilibrium...... 125 4.7 Solutions at First Order...... 127 4.7.1 Ignoring Perturbations of the Star...... 127 4.7.2 Ignoring Tidal Friction in the Planet...... 128 4.7.3 First Order Gravitational Perturbations...... 129 4.7.4 First Order Tidal Friction Perturbations...... 131 4.7.5 Full First Order Perturbations...... 133 4.8 Equation for a at Second Order...... 135 4.8.1 Moon Migration Timescale...... 138 4.8.2 Comparison to Other Studies...... 139 4.9 Applications and Discussion...... 140 4.9.1 Migration of Moons around hot Jupiters...... 141 4.9.2 Migration of Moons around Mercury and Venus...... 143 4.9.3 Scaling of Migration Time with Tidal Q...... 145 4.9.4 Constraining Maximum Moon Sizes...... 147 4.10 Conclusion...... 149

5 Conclusions 150 5.1 Summary...... 150 5.2 Future Applications...... 151 List of Figures

1.1 Planetary mass (in Jupiter masses) vs semi-major axis (in astronomical units – 1 AU is the distance from the Earth to the Sun) for all of the confirmed exoplanets as of Nov 2020. Colors designate the exoplanet detection method used for each discovery as follows: red – transit; blue – radial velocity; green – microlensing; yellow – direct imaging. The images designate the location that each Solar System planet would fall on the plot in order from left to right of: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. The plot is from exoplanets.org (Han et al. 2014)...... 4 1.2 A cartoon of the four major exoplanet detection methods (N. Bathalla – NASA Ames)...... 8 1.3 Face-on (left) and side (right) view of an exoplanet transiting its host star with an atmosphere. The face-on view shows the core of the planet in black, which blocks all the light from the star and a surrounding atmosphere which allows some transmission. The side view shows the effects that different impact parameters, b, have on the transmission of star light through the atmosphere. The more atmosphere that the light passes through, the more it gets absorbed...... 14 1.4 Observed transmission spectrum of the Na I D doublet lines in the upper atmosphere of HD 189733b from Wyttenbach et al.(2015). Top panel: out of transit spectrum from the host star normalized to the maximum flux observed in the wavelength regime noted by the x-axis. A ‘telluric’ spectrum expresses the absorption from the Earth’s atmosphere to identify (e.g.) water lines. Bottom panel: the transmission spectrum as a ratio of the flux observed in transit versus out of transit. The red lines show a Gaussian fit of the spectral bins, which clearly show absorption at the wavelengths expected from the Na I D doublet. 14

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1.5 The left side of this plot shows the temperature of the Earth’s atmosphere as a function of altitude. Dashed lines designate the different regimes of the atmosphere. The right side of the plot shows where the ionosphere of the Earth is in relation to other regions of the atmosphere by plotting electron density as a function of height. The peak in the electron density designates the location of the ionosphere. Image is adapted from: https://en.wikipedia.org/wiki/Ionosphere...... 19 1.6 The shape of the magnetic field around the Earth from Dungey(1961). 21 1.7 Tides on a body. Each line represents the differential gravitational force experienced at that location. Panel (a) is for a two body system without any tidal friction. Points A denote where a bystander on the planet would experience high tide and points B denote where a bystander would experience low tide. Subfigure (b) highlights the effects that tidal friction and tidal lag would have on the differential gravitational forces. The deviation from a central line connecting the center of the two bodies causes a tidal torque which drives systems towards tidal locking. Image reference: https://en.wikipedia.org/wiki/Tide.. 24 1.8 Lagrange points (left) and Roche limit (right) for a two body system. On the left, lines denote points of gravitational equipotential. Point L1 denotes the location where the gravity of the star, dominates the gravity of the planet at the Hill radius. The right subfigure exaggerates the tidal effects from a planet on a moon, as a moon migrates inwards until it is fully disrupted at the Roche limit. Image credits: (left) NASA, (right) Astronoo.com...... 26 1.9 A cartoon of the star-planet-moon model that I consider in Chapter4. Each body in the system orbits in nearly circular orbits where the only deviations are due to the mutual gravitational perturbations between all the bodies in the system...... 28

2.1 Examples of the Northern Lights (Aurora Borealis) and Southern Lights (Aurora Australis) as pictured from different locations on the Earth. Photo credit: https://en.wikipedia.org/wiki/Aurora...... 33 2.2 A schematic of two closure paths for the field-aligned current, shown in red as presented in Ganushkina et al.(2018). The left most path connects directly to the magnetopause while the right most path connects to the solar wind through the nightside plasma sheet...... 35 2.3 A diagram of the three vectors of current flow in an ionosphere with a given B and E field. The Pederson (σP ), Hall (σH ), and parallel (σk) conductivities are derived in Chapter 2.6...... 46 2.4 Diagram of a simple collision between a large particle and a background gas of constant volume of smaller particles (Schunk & Nagy 2004).. 54 xii

3.1 Flux density of stellar spectrum vs wavelength...... 73 3.2 Number density versus pressure. The top of this plot is the top of the atmosphere. Except when noted by the legend, solid lines represent neutral species and dashed lines represent ionized species. All other species are specifically represented in the legend with electrons denoted by ‘e’ and protons denoted by ‘p.’...... 92 3.3 (a) Temperature vs pressure as calculated by the thermal balance equation until 10 µbar below which point the temperature is prescribed to be that from Huitson et al.(2012). (b) Radius vs pressure as calculated by hydrostatic balance (Equation 3.9)...... 93 3.4 The density fraction of ions, neutral atoms, and molecules as a function of pressure. The density fraction is calculated by summing the densities of each particle of the given species and then dividing that value by the total number density of all species in the atmosphere. The transition from ionized to atomic to molecular becomes more clear in this representation than compared to Figure 3.2...... 93 3.5 The major heating and cooling rates in the atmosphere plotted against pressure until the prescribed temperature profile from Huitson et al. (2012) takes over at 10−5 bar. All cooling rates are expressed as dashed or dotted lines and all heating rates are expressed as solid lines. The dashed lines represent a sum of the line cooling for the specified species. Rates that are included in the model but do not significantly contribute to setting thermal balance are not plotted. A balance of the heating and cooling rates sets the temperature profile in Figure 3.3...... 94 3.6 Pederson, Hall, and parallel conductivities as a function of pressure.. 96 3.7 Individual contributions to the Pederson conductivity for electrons, protons, and ions. Each line represents the Pederson conductivity solely from the species denoted in the legend. The solid black line is a sum of all individual contributions...... 97 3.8 Individual contributions to the Hall conductivity for electrons, protons, and ions. Each line represents the Hall conductivity solely from the species denoted in the legend. Note that because the Hall conductivity is dependent on the charge of the species, the absolute value of the electron contribution is represented by the dotted line. The black line is a sum of all individual contributions...... 97 3.9 Pederson conductivity versus pressure as a function of magnetic field strength, B, given in units of Gauss...... 99 3.10 A running height integral of the Pederson conductivities in Figure 3.9 versus pressure. The final number that each line approaches at the highest pressures can be thought of as the conductance in the atmosphere for the given B field...... 99 xiii

3.11 Pederson conductivity vs. pressure for models with and without molecules. The solid lines show the total Pederson conductivity (black) and CII contribution (blue) to the Pederson conductivity for our standard atmosphere model. The dashed lines show the total Pederson conductivity (black) and CII contribution (blue) to the Pederson conductivity for the same atmosphere model without molecules included. Since carbon does not get tied into CO for the dashed lines model it is able to generate an anomalous peak in the Pederson conductivity at P ≈ 10−6 bar...... 103 3.12 A comparison of the total heating in the atmosphere (shown in red) over the pressure regions where we solve the thermal balance equation compared to the Joule Heating. We calculate Joule Heating from Equation 3.34 and from the Pederson conductance values at different magnetic field sizes as given in Table 3.12. The black lines represent the Joule heating rate calculated for each magnetic field tested.... 107

4.1 A cartoon of the migration and disruption of a theoretical moon in a hot Jupiter system. As show in (a) the planet and moon will be in circular orbits about the host star. In (b) tidal friction for forced eccentricity will cause inward migration of the moon. In (c) the moon will migrate within the Roche limit and become tidally disrupted. Finally in (d), the ring system caused by the tidal disruption of the moon will dissipate and no evidence of the moon will remain...... 115 4.2 Hut(1981) model for the tides raised on a primary object with mass M by an outside perturber with mass m. Hut(1981) approximates the tides raised by the companion as point masses, µ, to compute the perturbing tidal force on the companion with separation r, moving around the companion as θ˙...... 118 4.3 Geometry from the reference point of the central planet. Eccentricities are highly exaggerated to highlight the diﬀerences in longitude of pericenters ($) between the two objects. The deﬁnitions of each variable are highlighted in Section 4.4...... 121 4.4 Migration of a hypothetical moon orbiting HD 189733b, HD 209458b, Mercury, and Venus. On the x-axis is time since the beginning of the integration in years and on the y-axis is the semi-major axis of the moon (a1) divided by the Hill radius of the individual system. Each system is integrated until the semi-major axis becomes less than the Roche limit for tidal disruption at which point a moon would theoretically be destroyed. If the Roche limit is within the planetary radius, then the moon will collide with the atmosphere of the planet, before being tidally disurpted...... 144 xiv

4.5 A plot of tidal destruction time (in years) of Earth-size moons as a function of star-planet semi-major axis (a2). The destruction times are calculated via Equation 4.70. Each black line represents a tidal Q as given by the legend. We assume the planet mass is equal to the mass of HD 189733b, the mass of the star is equal to the mass of HD 189733A, an apsidal motion constant of k = 0.1, ρ = ρ⊕, and that the moon is originally located at half the hill radius of the system. The dashed lines highlight the orbital location of HD 189733b and Mercury. 146 4.6 Here we highlight maximum moon mass to survive tidal disruption within 5 × 109 years versus semi-major axis of the star-planet system. Each line represents the mass of the planet from 0.1 MJup − 10 MJup as given in the legend. We calculate the lines by applying Equation 4.70 where we assume Q1 = 10, ρ1 = ρ⊕, k1 = 0.1, and that each moon is initially placed at half the hill radius of the system. Systems that fall within the colored regions above each line are additionally able to survive tidal disruption...... 148 List of Tables

2.1 Relevant neutral gas polarizabilities. These values were taken from Table 4.1 in Schunk & Nagy(2004)...... 55 2.2 Relevant collision frequencies for resonant ion-netural collisions. These values were taken from Table 4.5 in Schunk & Nagy(2004). All temperatures are in K and densities (n) in cm−3...... 56

3.1 Major collision frequencies. Note that SN refers to Schunk & Nagy (2004)...... 68 3.2 Adopted properties for HD 189733A and HD 189733b...... 69 3.3 Hydrogen Chemistry. In the text, these reactions are denoted as RH#. Photoionization and photodissociation rates are optically thin values in units of s−1. They are calculated with the fits for the cross sections given as the reference in the table and the ionizing flux from the MUS- CLES spectrum. Three body reaction rate coefficients are in units of cm6 s−1 and two body reaction rate coefficients are in units of cm3 s−1. Rate 13a/13b is calculated using 11a/11b, the Saha equation, and the law of mass action...... 79 3.4 Carbon and Oxygen Chemistry. In the text, these reactions are denoted as RCO#. NIST refers to the National Institute of Standards and Technology Chemical Kinetics Database...... 81 3.5 Metal Chemistry: Photoionization/Recombination. In the text, these 4 reactions are denoted as RM#. Note: T4 ≡ (T/10 K). Note: The photoionization rates of all the metals are calculated from the cross section fits of Verner & Ferland(1996) and Verner et al.(1996).... 82 3.6 Alkali Metal Chemistry: Three Body Reactions. In the text, these reactions are denoted as RM#...... 83 3.7 Additional Molecule Chemistry. In the text, these reactions are denoted as RM#...... 83 3.8 Relevant species and heats of formation (in kJ mol−1) as listed in Le Teuff et al.(2000)...... 85

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3.9 Major Metal Cooling Lines. Notes: Clu refers to the collision frequency given by the source and CHIANTI refers to the Atomic Database for Spectroscopic Diagnostics of Astrophysical Plasmas which can be found at: https://www.chiantidatabase.org...... 86 4 3.10 Minor Metal Cooling Lines. Note: T4 ≡ 10 K...... 87 3.11 Stellar wind parameters near HD 189733b from Kavanagh et al.(2019). 101 3.12 Values of the Pederson conductance and the maximum Joule heating of the atmosphere of HD 189733b for diﬀerent sized magnetic ﬁelds. 102

4.1 HD 189733 system with hypothetical Earth-like moon. The circularization time is calculated via Equation 4.1. We use k = 0.1 for each body for all systems and the tidal Q is taken from Goldreich & Soter(1966). Note that the initial semi-major axis (a0) for the moon is located at half the Hill radius of the star-planet system...... 141 4.2 HD 209458 system with hypothetical Earth-like moon...... 142 4.3 Mercury system with hypothetical Europa-like moon...... 142 4.4 Venus system with hypothetical Europa-like moon...... 142 4.5 The Roche limit and tidal disruption time (Roche time) for each of the four systems considered as deﬁned above. Note that we assume 6 an initial nratio = 37.1α for each of the systems. Additionally note that the Roche limits for system A and B are within the planetary radius and therefore collision with the atmosphere will occur before tidal disruption...... 142 1

Chapter 1

Introduction

1.1 Exoplanets

Context: The study of extrasolar planets, or exoplanets, is central towards the core research in this thesis. Specifically, in Chapter3 an exoplanet atmosphere model is developed to understand the temperature, density, and conductivity profile for a type of exoplanet known as a hot Jupiter. Hot Jupiters are an exoplanet unlike anything observed in our Solar System and so in this introduction I describe the historical context of the detection of exoplanets. In doing this I outline the various exoplanet detection methods, which provide context to the place of hot Jupiters among other types of detected exoplanets and paint a picture of how the various exoplanet detection methods will inform my atmosphere model. The outline of the two major exoplanet detection methods (the transit method and radial velocity method) is intended to put in perspective how hot Jupiters were able to be detected and how we obtain information on their characteristics which are then implemented into my models. Additionally, an understanding of the types of exoplanet detection methods is needed to understand why we haven’t yet detected a moon orbiting an exoplanet, 2 which is important for the context of the research described in Chapter4. In this introductory Chapter, I also describe the specific hot Jupiter on which I base my atmosphere model – HD 189733b. Additionally, in Chapter4, I develop an analytic equation that can be applied to describe the orbital migration of hypothetical moons of hot Jupiters, or any planetary system that matches the criteria outlined within Chapter4. Therefore, background on the types of exoplanets observed and how they relate to objects in my models is needed to put into context the significance of the research described throughout this thesis.

1.1.1 The Discovery of Exoplanets

The year 1992 marked the first discovery of a planet outside our Solar System, when two exoplanets were detected orbiting a pulsar located 2300 light years away in the constellation of Virgo (Wolszczan & Frail 1992). Until then, all of our knowledge on the structure, dynamics, formation, and evolution of planetary bodies was limited entirely to the study of our own Solar System. The existence of exoplanets was entirely expected at the time, but this first detection immediately put our understanding of planets to the test. Exoplanets were originally only expected to be found around main-sequence stars (Wolszczan 2018) and not around the rapidly rotating and highly magnetized neutron stars that are pulsars, which form in the supernova explosion that marks the ‘death’ of a massive main-sequence star. This unexpected discovery foreshadowed the surprises that would come with subsequent exoplanet discoveries through the years. It wasn’t until 1995 that an exoplanet was finally observed orbiting a main- sequence star (Mayor & Queloz 1995). The discovery of exoplanet 51 Pegasi B, orbiting a Sun-like star located 50 light years away in the constellation of Pegasus, 3 opened the door to a new field of study that remains at the forefront of modern astronomical research. The detection and understanding of exoplanets is still rapidly changing, and as of November 2020 there have now been over 3000 exoplanets confirmed and almost 2500 candidate exoplanets (exoplanets.org – Han et al. 2014).

1.1.2 The Detection of Exoplanets

Figure 1.1 shows the majority of detected exoplanets as of November 2020 in a plot of planetary minimum mass vs distance from the planet’s host star (semi-major axis). Images of each of the Solar System planets are over-plotted on Figure 1.1 to highlight how the detected exoplanets compare to planets in our Solar System. Additionally, the colors in Figure 1.1 designate the observational method that was used to detect each conﬁrmed exoplanet with red, blue, green, and yellow designating the ‘transit,’ ‘radial velocity,’ ‘microlensing,’ and ‘direct imaging’ methods respectively. These methods are outlined in more detail below and a cartoon of the four major detection methods is shown in Figure 1.2.

The Transit Method

The transit method for exoplanet detection has been the most successful detection method to date, largely in part to the Kepler space telescope and more recently from the Transiting Exoplanet Survey Satellite (TESS). For a review on this detection method see Deeg & Alonso(2018). The transit method was ﬁrst successfully em- ployed in 2000 to conﬁrm the planetary status of HD 209458b (Henry et al. 2000; Charbonneau et al. 2000), which had already been detected via the radial velocity method. As a planet orbits around its host star and moves in the line of sight between its 4

Fig. 1.1.— Planetary mass (in Jupiter masses) vs semi-major axis (in astronomical units – 1 AU is the distance from the Earth to the Sun) for all of the confirmed exoplanets as of Nov 2020. Colors designate the exoplanet detection method used for each discovery as follows: red – transit; blue – radial velocity; green – microlensing; yellow – direct imaging. The images designate the location that each Solar System planet would fall on the plot in order from left to right of: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, and Neptune. The plot is from exoplanets.org (Han et al. 2014). 5 host star and our telescope, light from the star is periodically blocked resulting in a decrease in observed brightness (Figure 1.2). The time interval between dimming events is the same as the time it takes the planet to revolve once around its host. The amount of light blocked from the star is dependent on the relative size of the planet to the star. For example, the transit by HD 209458b results in a decrease in visual brightness of about 2% (Henry et al. 2000), but most transits of confirmed exoplanets are much smaller than this since HD 209458b is relatively large in relation to its host star than most exoplanets are to their host. If the change in brightness of the star is large enough to be detected, the observation becomes a candidate exoplanet. Exoplanet candidates can then be confirmed via follow up observations using another method to rule out false positives (Brown 2003; Santerne et al. 2013) due to stellar eclipsing binaries, star spots, or background eclipsing stellar binaries. Exoplanets detected via this method are also typically confirmed via follow-up observations that employ the radial velocity method. By detecting an exoplanet via the transit method one can gain insights on many exoplanet characteristics such as orbital period, semi-major axis (average distance from the star), and planetary radius. Additionally, many stars can be observed simultaneously to look for these periodic dips in stellar brightness as is done with Kepler and TESS. A drawback of the method is that planetary orbits need to be aligned with the observer’s line of sight and large enough to block a sizable amount of star light for an observable transit to occur. Additionally, rates of false positives are much higher than any other method and for some surveys can be as high as 35% (Santerne et al. 2012). For this reason, the transit method typically results in the detection of planets that are physically large in radius, very close to their host star, or a combination of the two as can be seen by the red dots in Figure 1.1. Despite 6 this, the transit method has been essential in our ability to detect large amounts of exoplanets, which has resulted in a better characterization and understanding of the types of exoplanets that exist in our Galaxy.

The Radial Velocity Method

Until the launch of Kepler in 2009, the radial velocity method for exoplanet discovery was the most productive method for detecting and confirming exoplanets. In fact, the first exoplanet detected around a Sun-like star, 51 Pegasi B, was detected with this method (Mayor & Queloz 1995). For an in depth review of the radial velocity method see Wright(2018). As an exoplanet orbits around its host star, it will gravitationally accelerate the star towards and away from the observer. This acceleration will shift the relative velocity of the star towards and away from the observer, and as a result the light emitted from the star will be Doppler-shifted to higher and lower frequencies (shorter and longer wavelengths). A common daily life example of the Doppler shift phenomenon occurs in the change of the pitch of an ambulance as it moves towards and away from a listener. See Figure 1.2 for a cartoon of this effect. The size of the shift in the stellar spectral features give insights into the planet’s minimum mass, orbital period, and semi-major axis. Only the minimum mass of a planet can be determined via this method, as the size of the Doppler shift depends on the line of sight velocity of the star, which is dependent on relative inclination of the star-planet orbit to the observer. More massive planets will gravitationally perturb their star more, resulting in a larger Doppler shift. For this reason, the radial velocity method tends to find exoplanets that are more massive than the types of exoplanets detected via the transit method. Some advantages to this method are that physical shifts of light from the Doppler 7 effect are independent of distance to the planetary system and the exoplanet orbit does not need to be precisely aligned with the observer as is required with the transit method. This results in planets being able to be detected further from their star than with the transit method. Despite the physical shift of light being distance independent, statistically significant detections do require a high signal-to-noise ratio of the light because of the many corrections that need to occur to constrain the planetary gravitational perturbation. Because of this, exoplanets detected via this method are somewhat limited in physical distance from the Earth. While exoplanets can be detected in large surveys using the radial velocity method (e.g. Troup et al. 2016 with APOGEE), comparatively larger surveys such as Kepler and TESS have detected many more exoplanets. For this reason, the radial velocity method has been used more frequently to confirm the detection of exoplanets via the transit method and to gain more insights in their planetary characteristics.

Microlensing, Direct Imaging, and Timing

Microlensing, direct imaging and timing variation round out the other major methods for detecting exoplanets. Their relevance towards the exoplanets examined in this thesis is outside the scope of this work and so I will not go into more detail on them other than listing them for completion of the major detection methods. Their immediate signiﬁcance can be highlighted by the fact that the ﬁrst exoplanet discov- ered was detected via pulsar timing variations (Wolszczan & Frail 1992) and some of these methods are more capable of detecting planets orbiting very far from their host star as seen by the orange and green points in Figure 1.1. For a review of these exoplanet detection methods see Batista(2018) for microlensing, Pueyo(2018) for direct imaging, and Kramer(2018), Hermes(2018) and Agol & Fabrycky(2018) for 8

Fig. 1.2.— A cartoon of the four major exoplanet detection methods (N. Bathalla – NASA Ames). timing variations. Examination of Figure 1.1 highlights that the majority of exoplanets detected are unlike the planets in our Solar System. They can be closer to their host than Mercury is to our Sun, frequently exist in the mass range between the Earth and Neptune, can be more massive than Jupiter, and can have large masses and be very close to their host star. This does not indicate that our Solar System is inherently unique, but rather shows the selection biases of each of the exoplanet detection methods. These methods inherently detect very massive planets or planets very close to their host star which are unlike any of the planets in our Solar System. In order to get a full census of the types of exoplanets in our Galaxy, we would need to bridge the gaps in our observational methods to detect planets of all sizes and orbital distances. So far, the primary methods we have that can bridge this gap are through microlensing and direct 9 imaging. Further bridging this gap will be a very diﬃcult task due to the observational limitations we currently face, but will be vital in fully characterizing planets in our Galaxy and the place that our Solar System stands in our characterization. After all, the inspiration of much Astronomical research, and especially exoplanet research, is in determining our place in the Universe. From the planets we are able to detect though, there are certainly objects unlike anything we see in our Solar System. For example, in Figure 1.1 the large amount of planets between the masses of Earth and Uranus/Neptune are typically referred to as ‘Super Earths’ or ‘Mini Neptunes.’ Most relevant to the scope of this thesis though are the planets located in the upper left quadrant of Figure 1.1. These planets are about the mass of Jupiter, but located at orbital distances much closer to their host star than Jupiter is to the Sun, making their temperatures much hotter than those observed in Jupiter. Therefore, these objects are often referred to as ‘hot Jupiters.’ These object are discussed in more detail in the following section. For an overview on all of the types of exoplanets that have been detected so far see Winn & Fabrycky (2015).

1.1.3 Hot Jupiters

The detection of Jupiter sized exoplanets orbiting very close to their host star was entirely unexpected and put into question our understanding of the origins, formation, and evolution of planets and stellar system as a whole (see Dawson & Johnson(2018) for a review on the origins of hot Jupiters). As outlined in Dawson & Johnson(2018) and references therein, a leading theory of planet formation is that planets form in the disk of gas and dust surrounding a new star. Rocky planets (like Mercury, Venus, Earth, and Mars) can form in the inner region of the disk and gas giants 10

(like Jupiter, Saturn, Uranus, and Neptune) formed in the outer region of the disk (beyond the ‘snow line’) – where the presence of more solids allowed larger cores to form which could then gravitationally attract more gases and form large atmospheres. Once the planets were assembled they would maintain their orbits about their host star where they were formed. This planetary formation model was based on the only planets we knew of at the time – those located in our own Solar System. Historically, our understanding of our place in the Universe has been upended time and time again. Earth is not the center of our Solar System. The Sun is not at the center of our Galaxy. Our Galaxy is not the only galaxy in the Universe. It turns out our understanding of exoplanets is no diﬀerent – our Solar System is not the standard for observed planetary systems. For this work, and following the approach of Dawson & Johnson(2018), hot Jupiters are deﬁned as planets with masses greater than or equal to 0.25 Jupiter masses and orbital periods of less than 10 days. The existence of these objects immediately brought up a fundamental question: How did they get there? There are three leading ideas: (1) they formed near the locations from the stars that they are dis- covered, (2) they formed beyond the ‘snow line’ of their stellar system and migrated inwards due to interactions with the gas disk, or (3) they migrated inwards from interactions with another large planet that caused large eccentricities that settled into the close orbits we observe due to tidal friction. Regardless of their formation mechanism, hot Jupiters are one of the most studied types of exoplanets and the understanding and modeling of their atmospheres is a primary focus of this thesis. 11

HD 189733b

The focus of Chapter 3 of this thesis is on modeling the upper atmosphere of the hot Jupiter HD 189733b. It was detected in 2005 via the transit method (Bouchy et al. 2005) and is the nearest transiting hot Jupiter, at a distance of about 65 light years. Its close distance to the Earth makes it one of the most observed and studied hot Jupiters to date. HD 189733b has a semi-major axis of 0.03 AU around a K2V star and an orbital period of just 53 hours – meaning that it is likely tidally locked to its host star as discussed in Chapter3. The planet has a mass about 1.2 times the mass of Jupiter and a radius about 1.1 times the radius of Jupiter.

1.2 The Upper Atmosphere of HD 189733b

Context: In Chapter3, I develop a hot Jupiter atmosphere model to understand the electrical conductivity within the upper atmosphere of HD 189733b. The species in the model, thermal processes considered, and dynamics of the atmosphere are constrained by observations of hot Jupiter atmospheres by observers. Therefore, in this section I outline how hot Jupiter atmospheres are observed, how the species in those atmospheres are able to be detected through spectroscopy, and outline the specific species that have been observed in hot Jupiters atmospheres, with an emphasis on HD 189733b – the hot Jupiter that I model. Additionally, a hot Jupiter atmosphere model cannot be fully encompassing of every physical process that occurs in the atmosphere due to computational and observational constraints and so assumptions must be made to model different effects in the atmosphere. Therefore, in this section I outline the different types of atmosphere models that have been created to model hot Jupiters. This discussion adds context to why I choose the various methods 12 outlined in Chapter3 to specifically understand the electrical conductivity in the upper atmosphere of HD 189733b.

1.2.1 Spectroscopic Observations of Exoplanet Atmospheres

Due to their immense distance from the Earth, almost all the information on exoplanet atmospheres is gathered through transmission spectroscopy (Roberge & Seager 2018) of transiting exoplanets. As an exoplanet transits its host star, part of the light interacts with particles in the planet’s atmosphere (see Figure 1.3). Stellar light can excite electrons in atoms and molecules in the planetary atmosphere at distinct wavelengths corresponding to discrete transitions of energy between two bound states. In this process, light at these discrete wavelengths in this spectrum, when compared to the out of transit stellar spectrum will appear to be ‘missing.’ By comparing the light observed in transit to the light from the host star out of transit, we can learn which particles are blocking the light and gain insights on the species in the planetary atmosphere. Figure 1.4, for example, shows a transmission spectrum for HD 189733b. This observation shows the presence of Na D doublet lines in the transmission spectrum, which requires Na atoms to be in the upper atmosphere of HD 189733b to absorb the starlight at those wavelengths. These observations also allow us to characterize the temperature and density profile of exoplanets, which provides us the information needed to test atmosphere models and theories of hot Jupiters (Roberge & Seager 2018). Since many exoplanets, such as hot Jupiters, orbit their stars very close to their host star, this allows us to test atmosphere models for many different conditions, such as for models of Na and other metals in its atmosphere. While exoplanet atmospheres have been detected via transmission spectroscopy on a handful of hot Jupiters, including the first exoplanet atmosphere 13 detection of the Na doublet line in HD 209458b (Charbonneau et al. 2002), I focus my discussion on the observations and models of HD 189733b. Additionally, HD 209458b is the most observed hot Jupiter and so I highlight the species detected in its upper atmosphere as well.

1.2.2 Chemical Species in the Atmosphere of Hot Jupiters

The atmosphere of HD 189733b was first detected from the ground by Redfield et al. (2008) from the Na D doublet resonance lines using the High Resolution Spectro- graph on the Hobby-Eberly Telescope (HET). This detection was a breakthrough at the time as it was the first detection of an exoplanet atmosphere from a ground-based observatory, which is typically difficult due to the seeing of the Earth’s atmosphere. HD 189733b is the closest hot Jupiter observed to the Earth, making this ground- based spectroscopic observation possible. Na has subsequently been detected in follow up observations by many teams including Jensen et al.(2011); Czesla et al.(2015); Cauley et al.(2016); Barnes et al.(2016), with the Space Telescope Imaging Spectro- graph (STIS) on board the Hubble Space Telescope (Huitson et al. 2012), and with the HARPS echelle spectrograph on the ESO 3.6 meter telescope (Wyttenbach et al. 2015). Additionally, the strong Hα line has also been detected with the HET by Jensen et al.(2012) and with the High Resolution Echelle Spectrometer (HiRES) on the Keck telescope in Hawaii (Cauley et al. 2015, 2016) – where they also reported observing Hβ and Hγ lines. Lyα absorption has been reported by Lecavelier Des Etangs et al. (2010) using the low-resolution spectrograph on board Hubble (STIS). Observations of metals in the atmosphere of HD 189733b have been detected as well including

OI (Ben-Jaﬀel & Ballester 2013), KI (Pont et al. 2013), MgI (Cauley et al. 2016), 14

Fig. 1.3.— Face-on (left) and side (right) view of an exoplanet transiting its host star with an atmosphere. The face-on view shows the core of the planet in black, which blocks all the light from the star and a surrounding atmosphere which allows some transmission. The side view shows the eﬀects that diﬀerent impact parameters, b, have on the transmission of star light through the atmosphere. The more atmosphere that the light passes through, the more it gets absorbed.

Fig. 1.4.— Observed transmission spectrum of the Na I D doublet lines in the upper atmosphere of HD 189733b from Wyttenbach et al.(2015). Top panel: out of transit spectrum from the host star normalized to the maximum flux observed in the wavelength regime noted by the x-axis. A ‘telluric’ spectrum expresses the absorption from the Earth’s atmosphere to identify (e.g.) water lines. Bottom panel: the transmission spectrum as a ratio of the flux observed in transit versus out of transit. The red lines show a Gaussian fit of the spectral bins, which clearly show absorption at the wavelengths expected from the Na I D doublet. 15 and a minimal detection of CaII (Barnes et al. 2016; Czesla et al. 2015), and HeI (Salz et al. 2018) lower in the atmosphere. In addition to atoms, molecules have also been detected including H2O(Birkby et al. 2013), CO, CO2, and a minimal detection of CH4 (Swain et al. 2008), the latter three suggesting the presence of C in the atmosphere. As for HD 209458b, there have been observations of all the atoms and molecules listed above in addition to the presence of CI and OI in its extended upper atmosphere

(Vidal-Madjar et al. 2004), more signiﬁcant MgI detections (Vidal-Madjar et al. 2013) and recent FeII observations (Cubillos et al. 2020). Combined, these observations of HD 189733b and HD 209458b deﬁne the major species that must be included in atmosphere models of hot Jupiters. I use these observations to motivate the species included in my model of HD 189733b, as discussed in more detail in Chapter3.

1.2.3 Upper Atmosphere Models of Hot Jupiters

Hot Jupiter atmosphere models have typically been created to investigate their physical and chemical processes and to explain the observed spectral features as discussed above. For example, models for the upper atmosphere of HD 209548b have been created to study atmospheric escape due to the intense heating from its host star (Yelle 2004; Garc´ıaMuñoz 2007; Murray-Clay et al. 2009; Koskinen et al. 2013; Salz et al. 2016) and caused by Roche Lobe overflow (Gu et al. 2003; Li et al. 2010; Lai et al. 2010; Ehrenreich & Désert 2011). Additionally, hydrodynamic models have been created to study the interaction of colliding planetary and stellar winds (Trem- blin & Chiang 2013; Christie et al. 2016) and magnetohydrodynamic (MHD) models have been created to further investigate this effect when considering the planetary magnetic field (Trammell et al. 2011, 2014). Comparisons to observations for HD 16

209458b of H Lyα,OI, SiIII, NaI, and KI lines have been done by (e.g.) Fortney et al. (2003); Koskinen et al.(2013); Lavvas et al.(2014) and for HD 189733b by Huang et al.(2017). Atmospheric species in these models are often calculated assuming local thermodynamic equilibrium (LTE) (Burrows & Sharp 1999; Seager et al. 2000; Sharp & Burrows 2007; Fortney et al. 2008; Visscher et al. 2010), slight deviations from LTE (Lodders & Fegley 2002; Fortney et al. 2003, 2006; Visscher et al. 2006; Showman et al. 2009; Visscher et al. 2010), photo-chemical models (Yelle 2004; Garc´ıaMuñoz 2007; Koskinen et al. 2013; Lavvas et al. 2014; Huang et al. 2017), or a combination of the two (Moses et al. 2011). The details of my model for the upper atmosphere of HD 189733b are discussed in detail in Chapter3. There, I develop a model that transitions from the ionized region of the upper atmosphere, through the region dominated by atoms, and to deeper pressures where molecules begin to dominate the density profile. I consider photo-chemical processes to form and destroy species in the atmosphere and model thermal balance to determine the temperature profile. The resulting density profile is applied towards understanding the electrical conductivity in the upper atmosphere of HD 189733b and the implications that the conductivity profile can have on changing the thermal balance and density profile over this region. Applications of the conductivity are often dependent on the planet’s magnetic field and interactions with a stellar wind as outline below and in Chapter2.

1.3 Planetary Magnetospheres and Ionospheres

Context: In Chapter3, I apply a hot Jupiter atmosphere model to understand the electrical conductivity throughout the upper atmosphere and ionosphere of HD 189733b. Energy can be transferred to an atmosphere via collisions of charged parti- 17 cles with neutrals, which is quantified by the conductivity of the atmosphere. In that Chapter, I examine the significance of the heating from these collisions (often referred to as Joule heating) towards impacting the thermal balance of the atmosphere. Ad- ditionally, in Chapter2, I review the origin of the electrical conductivity. Therefore, understanding the flow of currents in the atmosphere is central towards understanding Chapter2 and Chapter3 of this thesis. The current at the top of the atmosphere is set by the interaction of charged particles with a planetary magnetic field at the magnetopause. Therefore, in this section, I introduce the concept of planetary magnetospheres and ionospheres. I also introduce the role that the charged particles in the ‘solar wind’ plays in the interaction with the planetary magnetosphere and how through ‘magnetic reconnection’ charged particles from a star are able to enter the atmospheres of planetary systems.

1.3.1 Overview

Planetary magnetospheres and ionospheres are vital in setting the current that can flow through a planetary atmosphere and the atmospheric conductivity research outlined in Chapter3 of this thesis. The magnetosphere of a planet is defined as the region surrounding a planet where its magnetic field pressure dominates the gas pressure (Kivelson & Russell 1995). The ionosphere is the relatively thin region of the atmosphere (for the Earth) of maximum charge density (Kivelson & Russell 1995). All planets that have a magnetic field, have a magnetosphere, and all planets that have a neutral-gas atmospheric envelope, have an ionosphere (Kivelson & Russell 1995). Figure 1.5 shows the location of Earth’s ionosphere in relation to other regions of the atmosphere as well as the electron density profile in the atmosphere. For more detailed reviews of magnetospheres and ionsopheres see Kivelson & Russell(1995) or 18

Schunk & Nagy(2004). The source of electrons and ions in the upper atmosphere is photoionization of neutral particles (Schunk & Nagy 2004). Ions can additionally undergo other chemical processes, such as recombination, chemical reactions with neutrals, diffusion deeper into the atmosphere, or can be transported by zonal winds or turbulence in the atmosphere to set the full profile of plasma in this region. In the ionosphere, electrons can no longer freely flow along planetary magnetic field lines due to collisions with ions and neutral species (Schunk & Nagy 2004). These collisions create a finite conductivity through this region of the atmosphere as opposed to the much larger conductivity in regions above the ionosphere. I discuss important interactions with the magnetosphere and ionosphere below and how they relate towards applications of the conductivity profile of hot Jupiter atmospheres.

1.3.2 The Solar Wind and Magnetic Reconnection

In the case of the Earth, charged particles can enter the Earth’s atmosphere along planetary magnetic field lines and interact with particles in the Earth’s ionosphere (Frahm et al. 1997; Galand & Richmond 2001; Fox et al. 2008). The source of these charged particles is the Sun and they emanate away from the Sun in what is called the solar wind. High temperatures in the outer atmosphere, or corona, of the Sun (∼ 106 K) allow charged particles to regularly form and continuously outflow radially from the Sun along its magnetic field lines (Kivelson & Russell 1995). Through a process known as ‘magnetic reconnection,’ solar magnetic field lines can connect to the Earth’s field lines, creating a source of charged particles which impact the Earth’s atmosphere (see Yamada et al. 2010 for a more recent review of this process). 19

Fig. 1.5.— The left side of this plot shows the temperature of the Earth’s atmosphere as a function of altitude. Dashed lines designate the diﬀerent regimes of the atmosphere. The right side of the plot shows where the ionosphere of the Earth is in relation to other regions of the atmosphere by plotting electron density as a function of height. The peak in the electron density designates the location of the ionosphere. Image is adapted from: https://en.wikipedia.org/wiki/Ionosphere 20

Dungey(1961) first applied magnetic reconnection in the context of the Earth-Sun system. Without reconnection, the Earth’s magnetic field would be nearly a dipole and an entirely closed system, meaning that no charged particles would be able to freely enter into the Earth’s atmosphere. In reality, solar wind plasma interacts with the Earth’s magnetosphere and compresses the magnetic field on the day side of the Earth and produces an extended tail on the night side as shown in Figure 1.6(Dungey 1961). The boundary region where the magnetic field strength of the solar wind and the Earth is equal, is called the magnetopause. Across the magnetopause, charged particles transition from a weak magnetic field and dense/compressed region to a region of low density that is strongly magnetized in the Earth dominated regime. From this interaction of the solar wind impacting the Earth’s magnetic field at the magnetopause, the Earth’s magnetic field connects to the solar wind flow on the day side on magnetic field lines originating from very near the poles, and magnetic reconnection occurs. In this setup, plasma originating from the Sun can penetrate into the Earth’s atmosphere and interacts mainly through collisions in the ionosphere. Central towards dictating the flow of these charged particles in the ionosphere is the atmospheric conductivity. I outline the conductivity in regards to my research and detail the calculations that go into calculating atmospheric conductivity in Chapter 2.

1.4 The Dynamics of Star-Planet-Moon Systems

Context: In Chapter4, I examine the orbital migration and tidal disruption of moons in star-planet-moon systems. The research in Chapter4 was motivated by the lack of moons orbiting Mercury and Venus and the lack of detected moons around exoplanets, and so I also brieﬂy outline previous research that has been done to explain 21

Fig. 1.6.— The shape of the magnetic field around the Earth from Dungey(1961). these observations here. Specifically, In Chapter4, I derive an analytic equation for the change in the semi-major axis of a moon in a star-planet-moon system where all bodies in the system have zero free eccentricity (meaning they orbit in nearly circular orbits). Gravitational perturbations from each body in the system, however, cause slight deviations from zero eccentricity for each body (known as the forced eccentricity) that allows tidal friction to never cease, even in the case of circularized orbits and synchronous rotation. This tidal friction effect transfers energy from the orbital motion of the bodies into heat, causing orbital migration. Therefore, in this section, I discuss the history of Solar System dynamics as it relates to my model and describe the basic physics of gravitational perturbations and tidal friction in planetary systems. I also outline the types of planetary systems to which my model will be applicable. 22 1.4.1 Introduction

The study of Solar System dynamics, or the motion of bodies due to the influence of gravitational forces, has a detailed history dating all the way back to the earliest of Astronomers (Murray & Dermott 1999). Through the pursuit of trying to understand the regular motion of objects in the night sky, Astronomers have derived laws to better explain this motion and to make order out of the movement of celestial bodies. These early breakthroughs include some of the most fundamental results in Astronomy including Kepler’s laws of planetary motion and Newton’s universal law of gravitation. A more ‘modern’ breakthrough includes the prediction of Neptune by Urbain Le Verrier in 1846 to explain the irregularities in Uranus’ orbit caused by an eternal gravitational perturber (Le Verrier 1846; for a historical overview of this prediction, see Krajnović 2016). While the problems that those currently working on Solar System dynamics may not at first appear as ‘grand’ as the works of the early pioneers, significant theories and predictions are still being developed to this day as there are still many outstanding questions about the dynamics of our Solar System. For example, see the prediction of ‘Planet Nine,’ a large Neptune sized planet orbiting beyond the Kuiper Belt of our Solar System (Batygin & Brown 2016). With the advancement of computer technology, dynamical simulations of Solar System bodies have provided many answers to these questions. One of those questions that I take an approach to better understand, is on the lack of moons orbiting the rocky planets of the inner Solar System which is similar in nature to the lack of moons detected orbiting any exoplanets. 23 1.4.2 The Lack of Moons in the Inner Solar System

The gas giants of our Solar System all have many moons orbiting them compared to the planets of the inner Solar System. As of Nov 2020, Jupiter has 79 detected moons, Saturn has 82, Uranus has 27, and Neptune has 141. On the other hand, Earth only has a single very large moon, Mars has two small moons in Phobos and Deimos, and Mercury and Venus both have no moons. While this phenomenon was originally believed to be a result of the formation of our Solar System (Canup & Ward 2002), there have been theories that the lack of moons in the inner Solar System is a more recent phenomenon caused by tidal friction (Burns 1973; Ward & Reid 1973; Burns 1977). A familiar example of the eﬀects that tidal friction can have on a moon’s orbit can be found in the Earth-Moon system. Thanks to reﬂectors placed on the surface of the Moon by the Apollo astronauts (Bills & Ray 1999), we know that the Moon is currently drifting away from the Earth at a rate of 3.8 cm (or 1.5 inches) per year. This migration is caused by tidal friction from tides raised on the Earth by the Moon (Bills & Ray 1999). Tidal friction is described in more physical terms below.

1.4.3 Tidal Friction in Two Body Systems

In daily life, tides are most familiar from the rise and fall of the ocean twice per day. Tides on the Earth are primarily caused by the differential acceleration felt across the surface of the Earth due to the Moon’s gravity. This differential acceleration is expressed by the vectors in Figure 1.7. In this setup, as the Moon orbits the Earth, the differential acceleration will be largest, and outward from the Earth, on both sides of the planet in the line directly between the center of the Earth and the

1https://solarsystem.nasa.gov/moons/overview/ 24

Moon (denoted by points A in Figure 1.7) – causing what we know as high tide. The diﬀerential acceleration is smallest, and inward on the Earth, on the points farthest away from the line connecting the Earth and the Moon (denoted by points B in Figure 1.7) – causing what we know as low tide. As the Moon orbits the Earth, the tidal ‘bulge’ of high tide will follow the Moon causing the rise and fall of water levels on the Earth twice a day. Everything on the Earth experiences this eﬀect, including the solid body of the planet, but it is most easily observed in the rise and fall of water levels on the surface.

(a) (b)

Fig. 1.7.— Tides on a body. Each line represents the differential gravitational force experienced at that location. Panel (a) is for a two body system without any tidal friction. Points A denote where a bystander on the planet would experience high tide and points B denote where a bystander would experience low tide. Subfigure (b) highlights the effects that tidal friction and tidal lag would have on the differential gravitational forces. The deviation from a central line connecting the center of the two bodies causes a tidal torque which drives systems towards tidal locking. Image reference: https://en.wikipedia.org/wiki/Tide

In reality, the tides we experience on the Earth are much more complex than this basic picture. The Earth also experiences significant tides from the differential acceleration felt across its surface due to gravity from the Sun. When the tidal bulges due to the Moon and the Sun line up (during full and new moons), the sea level changes are their most extreme and referred to as ‘spring’ tides. When the tidal bulges from the Moon and Sun are exactly opposite of each other (during first or last quarter moons), the sea level changes are their smallest and referred to as ‘neap’ 25 tides. Furthermore, the time that high and low tides on the Earth occurs actually ‘lags’ behind the time that would be expected from the line connecting the Earth and the Moon in Figure 1.7. This tidal lag, in regards to the ocean, is caused by friction between water and the Earth’s surface as the water moves over the Earth’s surface during high and low tides over the course of a day. The more friction there is in the system, the greater the tidal lag. This tidal friction also inputs heat into the Earth, resulting in a loss of energy from the orbital motion and the Moon slowly migrating away from the Earth. The transfer of heat is dependent on the density of objects, so by studying this tidal lag and tidal friction we can also gain insights on the interior structure of the Earth and the Moon (Peale et al. 1979). If the Moon continues to migrate away from the Earth, it will eventually reach a location where the Sun’s gravity overcomes the Earth’s and the Moon will be ‘stripped’ from the Earth-Moon system. This is set by the ‘Hill radius’ of a two-body system (Murray & Dermott 1999) and the limit is shown by L1 in Figure 1.8. For the Earth- Moon system, the limit for tidal stripping is about four times the current distance between the Earth and the Moon. Alternatively, if the Moon was migrating inwards towards the Earth, it would eventually reach a point where the stretching of the Moon due to tides from the Earth would overcome its own self-gravitation and rip the Moon apart (see Figure 1.8). The limit where acceleration from tides overcomes self-gravitation is known as the Roche limit and in the Earth-Moon system would occur at about 2% of the current distance to the Moon. Saturn’s ring system and other ring systems of the gas giants may have been caused by destroyed moons that got too close to their planet (Canup 2010; Black & Mittal 2015; Hesselbrock & Minton 2019). Additionally, this is likely to be the fate of the Martian moon Phobos as it 26 migrates towards Mars (Yoder 1982; Black & Mittal 2015). Perhaps moons that were orbiting the inner planets during the early Solar System were stripped or destroyed due to this tidal friction effect.

Fig. 1.8.— Lagrange points (left) and Roche limit (right) for a two body system. On the left, lines denote points of gravitational equipotential. Point L1 denotes the location where the gravity of the star, dominates the gravity of the planet at the Hill radius. The right subﬁgure exaggerates the tidal eﬀects from a planet on a moon, as a moon migrates inwards until it is fully disrupted at the Roche limit. Image credits: (left) NASA, (right) Astronoo.com

An important consequence of tidal lag and friction in a two-body system is that it can lead to circularization of orbits and tidal locking. Circularization is the process by which eccentric orbits evolve into circular orbits due to tidal friction. Tidal locking occurs when the gravitational forces of tides, including friction, force the rotation of an object to synchronize with its revolution about its host. The phenomenon is already seen from the surface of the Earth in that one side of the Moon always faces the Earth. If the Earth was also tidally locked to the Moon such that only one side of the Earth faced the Moon, then tides would not move over the surface of the Earth and tidal friction would stop dissipating energy. Therefore, when objects become circularized and tidally locked, one would think that tidal friction effects would completely turn 27 off. However, when a third body is introduced into the system, it can cause gravitational perturbations that create a short-period eccentricity or non-synchronous rotation effect that can never go away and allows tidal friction to always be present. I model the effects that this tidal friction can have on three-body systems in Chapter 4 and apply that model to understand the lack of moons in the inner Solar System and lack of moons detected orbiting exoplanets.

1.4.4 The Three-Body Problem

The generalized three-body problem is a subject that has a significant amounts of literature on it (e.g. Szebehely(1967) and Marchal(1990)). I, however, seek to analytically derive the migration of moons in nearly circular orbits about their host planet due to the tidal friction effect described above and employ the circular, three body problem to create my model. The circular, three-body problem is defined (e.g. Mur- ray & Dermott 1999) for a system where two of the bodies move in circular, coplanar orbits about a center of mass and the third body is too small to significantly affect the motion of the other bodies. While real bodies in the Solar System are often not in circular and coplanar orbits, the restricted three-body problem can be applicable in many unique situations, such as a star-planet-moon system. If the motions of two bodies are dominated by a central body (e.g. a star), then the secondary bodies will orbit in ellipses with only small deviations due to gravitational perturbations of the other bodies in the system. These small deviations lead to the constant tidal friction effect previously discussed and can be analyzed by the ‘disturbing function’ which is discussed in more specific terms in Chapter4. 28

Fig. 1.9.— A cartoon of the star-planet-moon model that I consider in Chapter4. Each body in the system orbits in nearly circular orbits where the only deviations are due to the mutual gravitational perturbations between all the bodies in the system.

1.4.5 Model of a Star-Planet-Moon System

My model in Chapter4 describes a three-body system with a moon in a nearly circular orbit about a planet, that itself is in a nearly circular about a star. These bodies have zero relative inclination to each other and the deviations from the circular orbit are solely due to the gravitational perturbations each body feels from each other. The system is deﬁned such that the planet is the central mass and the moon and star are the ‘inner’ and ‘outer’ bodies respectively. A cartoon of the model is shown in Figure 1.9. 29

Other Applicable Systems

The star-planet-moon model, as discussed above, can be applied to any system that matches the criteria, not just Solar System objects. In particular, we can apply this system to exoplanet systems and determine the likelihood that they can dynamically maintain any moons. To date, there have been no strong detections of moons orbiting exoplanets (exomoons), and only a single minor detection of a Neptune sized moon orbiting a hot Jupiter (Teachey & Kipping 2018). Under current observational capabilities, it has been shown that relatively large exomoons (e.g. the size of the Galilean moons or larger), could be possible to detect, but very diﬃcult with current exoplanet detection methods (Sartoretti & Schneider 1999; Kipping 2009a,b). There- fore, there is a chance that moons in these exoplanet systems would be dynamically unstable and they would not have any moons orbiting around them similar to the inner planets in our Solar System.

1.5 Overview

In this thesis I model two problems: (1) the upper atmosphere of hot Jupiter HD 189733b with a focus on calculating its electrical conductivity profile, as well as (2) a model for the tidal evolution of a moon in a star-planet-moon system. Specifi- cally, in Chapter2, I review the equations that are applied to calculate the electrical conductivity of my hot Jupiter atmosphere model. Additionally, I outline how the magnetosphere surrounding a planet constrains the flow of current into the atmosphere. I also discuss how this current leads to heating of the atmosphere through the interaction of charged particles with other species in the atmosphere. In Chapter 3, I detail the hot Jupiter atmosphere model developed and apply the electrical con- 30 ductivity equations and current flow to determine the Joule heating in the atmosphere of HD 189733b. In Chapter4, I detail the model developed to explain the tidal evolution of moons in three body systems that are orbiting in coplanar and circular orbits. I also apply my derived analytical expression to theoretical moons of hot Jupiters, Mercury, and Venus to try to understand the lack of moons in these systems. In Chapter5, I summarize the work throughout the thesis and discuss applications of this work for future studies. 31

Chapter 2

Electric Currents and Conductivity in Planetary Atmospheres

2.1 Context

For Solar System planets, electric currents play an important role in the dynamics and energetics of atmospheres – most notably in the ionosphere (Paschmann et al. 2002; Schunk & Nagy 2004). One type of current system induced in the atmosphere is due to the interaction of the solar wind with the planetary magnetosphere. The size of the current which can flow through the atmosphere is dependent on the collisions of charged particles with other species in the atmosphere, which is parameterized by the electrical conductivity. A consequence of the collision of charged particles in the electric current with other species in the atmosphere is through the transfer of energy by collisions, more commonly referred to as Joule heating. Through studies of the electric currents and conductivity in Solar System planets, models can be developed for more general objects and applied, for example, to exoplanet atmosphere models. In Chapter3 I develop a hot Jupiter atmosphere model for HD 189733b with the 32 purpose of determining the electrical conductivity in its upper atmosphere. Once the conductivity profile is known, I then apply parameters for the stellar wind of its host star and magnetic field strengths to set the current at the top of the atmosphere and then determine the significance of Joule heating in HD 189733b. Therefore, in this chapter, I detail how currents in planetary atmospheres are set by the stellar wind- magnetosphere interaction, outline the calculation of Joule heating in a planetary atmosphere, and review the formulae for electrical conductivity.

2.2 Historical Overview

The most beautiful example of charged particles from a stellar wind interacting with a planetary magnetic field is in the visible auroras near the polar regions of the Earth (Figure 2.1). We now know that auroral light shows are caused by solar wind particles moving along magnetic field lines connecting the solar wind to the Earth and colliding, primarily, with oxygen in the Earth’s ionosphere. When collisions occur, electrons can be excited into a higher energy state and then release energy at very specific wavelengths (primarily corresponding to green light) as they de-excite back into the ground state. This understanding, however, eluded humans for centuries. There are written notes on auroras dating back to at least 600 B.C. (Ch. 1 – Kivelson & Russell 1995 and references therein) and even possible cave drawings as early as 10,000 B.C., but it wasn’t until the 1700s when astronomers began gaining a physical understanding of the phenomenon. Edmund Halley first suggested that auroras depended on the direction of the Earth’s magnetic field and in the 1730s, Anders Celsius published findings that magnetic needles reacted strongly to auroral displays. The first theory that auroras were connected to particles emanating from the Sun was by H. Becquerel in 1878. In the early 1900s Kristian Birkeland 33

Fig. 2.1.— Examples of the Northern Lights (Aurora Borealis) and Southern Lights (Aurora Australis) as pictured from different locations on the Earth. Photo credit: https://en.wikipedia.org/wiki/Aurora 34 began obtaining extensive data on auroras and concluded that large electric currents flowed along magnetic field lines during auroral storms, which was later supported by mathematician Carl Størmer. Advances in photography, spectroscopy, and the radio transmitter accelerated the understanding of auroras. In 1918, Sydney Chapman and collaborators suggested that a single charged beam of plasma from the Sun caused magnetic disturbances on the Earth which laid the foundation for our modern understanding of solar wind- magnetosphere interaction. Finally, advances in rockets and space probes in the 1950s allowed direct in situ confirmations of the solar wind, the magnetosphere, and their connection to auroras – most notably done by James Van Allen. In subsequent years, astronomers began observations of auroras on other planets in the solar system and our modern physical models of stellar wind-magnetosphere interaction were developed. We are currently at the early stages of developing models for the stellar wind- magnetosphere interactions of exoplanets but the historical context of auroras is important towards understanding the basic picture of how charged particles form a star can interact with the magnetic field and atmosphere of a planet. In this thesis I apply the understanding of stellar wind-magnetosphere interaction to set the currents that can flow in hot Jupiter atmospheres. A general overview of electric currents is discussed in more detail below.

2.3 Electric Currents Overview

An electric current is defined as the net flow of charge from one location to another. See Ganushkina et al.(2015) for an expert overview of current systems in the Earth’s magnetosphere and Ganushkina et al.(2018) for a more elementary overview on 35 currents in planetary atmospheres. Most relevant to this thesis is the flow of electrons, protons, and ions which originate in a star and move through a planetary atmosphere while interacting with the magnetic field and other species in the atmosphere. If we assume that a planet has a strong dipolar magnetic field, stellar wind particles will collide with the magnetic field, compressing it on the side facing the star (‘the dayside’) at the magnetopause and elongating it on the side away from the star (‘the nightside’) as shown in Figure 1.6. The dayside currents are most applicable to the research outlined in Chapter3 and the discussion throughout this chapter is, therefore, focused on these currents. While there are many types of dayside and nightside currents in planetary magnetospheres as outlined in Ganushkina et al.(2015, 2018), the type of current most applicable to the Joule heating discussed in Chapter 2.4 is the current flowing parallel to the planetary magnetic field, appropriately called the ‘field-aligned current’ or ‘Birkeland current,’ and first suggested by Birkeland(1908). A schematic of this current with respect to the magnetopause of a planet is shown in Figure 2.2.

Fig. 2.2.— A schematic of two closure paths for the ﬁeld-aligned current, shown in red as presented in Ganushkina et al.(2018). The left most path connects directly to the magnetopause while the right most path connects to the solar wind through the nightside plasma sheet. 36

This field-aligned current on the left side of Figure 2.2 connects to the dayside magnetopause and is driven by the solar wind, which acts like a generator through dayside reconnection (Cowley 2000; Iijima & Potemra 1976; Siscoe et al. 1991; Stern 1983; Xu & Kivelson 1994). This circuit then connects with one end in the ionosphere and the other in the solar wind as shown by Stern(1983). This current flows into and closes in the ionosphere as described by (e.g.) Iijima & Potemra(1976) and then returns to space (Kivelson & Russell 1995; Gombosi 2004). The stellar wind interaction with the magnetosphere can then set the current that can flow in the ionosphere that can then generate Joule heating from the conductivity profile as outlined in the next section.

2.4 Joule Heating

Joule heating is the process by which an electric current, such as the field-aligned current, generates heat. A familiar example of Joule heating is in the heating of the filament of an incandescent light bulb as current passes through it. Kivelson & Ridley(2008) discuss how to calculate the Joule heating of an atmosphere given a conductance and dependent on the stellar wind-magnetosphere interaction that sets the field-aligned current through the atmosphere. The equations introduced in Kivelson & Ridley(2008) and Cohen et al.(2018) for Joule heating, are a consequence of current flows in the ionosphere of a planet and the physical mechanism behind the currents and the heating is discussed below.

2.4.1 Height Integrated Currents and Conductivity

The region where currents in the ionosphere ﬂow is a very thin region compared the entirety of the magnetosphere (Paschmann et al. 2002). Thus, the ionosphere can, in 37

effect, be treated as a thin conducting layer in which vertical currents or electric fields can be ignored. This formalism allows us to refer to ‘height-integrated’ currents and conductivities for the ionosphere. We save the detailed discussion on height integrated conductivities for Section 2.7 but define here the height integrated values of electrical conductivity and current to be given by:

Z Σ ≡ σdz (2.1)

Z j ≡ Jdz (2.2) where σ is the electrical conductivity in the ionosphere, Σ is the height integrated conductivity which is called the conductance, J is the current density, and j is the height integrated current density of the ionosphere over altitude z. In calculating these height-integrated values, we assume that the electric and magnetic ﬁelds do not vary much over the ionosphere. The use of the height-integrated concept of currents in the ionosphere can be used to describe the closure of parallel currents (like the ﬁeld-aligned current) in this region. Parallel currents that close in the ionosphere can then transfer heat to the atmosphere through Joule heating.

2.4.2 Parallel Currents and Alfv´enConductance

Parallel currents that close in the atmosphere are created as a result of Alfvénwaves generated by the movement of planetary magnetic field lines due to the stellar wind- magnetosphere interaction introduced in Chapter 1.3. Alfvénwaves, which are a type of magnetohydronamic (MHD) wave generated as the response of charged particles to a restoring force from tension in a magnetic field line, travel at the Alfvénvelocity 38 given by: B v = √ (2.3) A 4πρ where B is the magnetic field strength of the field line and ρ is the mass density of the charged particles. The electric field from the current can be written in terms of the magnetic field perturbation which generates the Alfvén wave. This perturbation can be expressed as: 1 1 k B2 δE = − δv × B = z δB × eˆ (2.4) c c ω 4πρ z

Here, the δ denotes the perturbations to the various vectors as a result of the perturbation to the magnetic field, kz and ω are the wave numbers of the Alfvénwave propagating along magnetic field lines, ρ is the mass density of the charged particles, and e denotes the unit vector in the z direction. We can substitute in for the Alfvén speed above to get the perturbation to the electric field to be:

v δE = ± A δB × eˆ (2.5) c z

2 2 2 since ω = k vA for the propagating wave. The perturbation to the electric current therefore only has anx ˆ andy ˆ component. We can then determine the parallel current in the z-direction from Ampere’s law as: c 2 c ∂δBy ∂δBx c Jz = (∇ × δB)z = − = ± ∇⊥ · δE⊥ (2.6) 4π 4π ∂x ∂y 4πvA

2 We can deﬁne a quantity called the ‘Alfv´enconductance’ to be ΣA ≡ c /(4πvA) and 39 therefore the current in the z direction is:

Jz = ±ΣA∇⊥ · δE⊥ (2.7) where a positive value means the current moves along +ˆz and a negative value means the current moves along −ˆz. Note that this current is the current that is ‘incident’ on the ionosphere due to the Alfv´enwave.

2.4.3 Reﬂection of Alfv´enWaves

At all boundaries where two plasmas have different values of Alfvénvelocity, part of the wave is reflected and part of the wave is transmitted. This is also the case for the Alfvénwave created from the stellar wind-magnetosphere reaction when it impacts the ionosphere which has a conductance given by ΣP . Here the ‘P’ denotes the ‘Pederson’ conductance which is discussed in more detail in the following sections. The field-aligned current density in the ionosphere can be written as:

Jz = ΣP ∇⊥ · E⊥ (2.8) when the Pederson conductance is dominant. Physically, more than just the Pederson conductivity governs the ﬂow of charged particles through the ionosphere. The Hall conductivity can also play a role as shown by Equation 3.7 in Section 3.1.1 of Paschmann et al.(2002). However, for the simplest scenario where ionospheric conductivity is uniform over the plasma region, as we are assuming is the case for the height integrated conductivities of the ionosphere, only the Pederson current contributes to closing ﬁeld-aligned currents and generating Joule heating in this region. The height-integrated approximation cannot be done to fully 40

calculate the instantaneous Joule heating over all regions of the atmosphere. In this case, full conductivity, collision, and current profiles would need to be known from the top of the atmosphere through the base of the model and an MHD model would need to be carried out to get the full effect. We, therefore, choose to follow the simplest approach outlined in Section 3.4 of Paschmann et al.(2002) and discussed in Kivelson & Ridley(2008) to calculate the Joule heating in our model, which is independent of the Hall conductivity discussed in more detail in the following sections. We can now determine the reflection coefficient of the Alfvénwave by considering the parallel current variation generated by the difference of the Pederson and Alfvén conductance. For the Alfvéncurrent, the incident current will have a plus sign, while the reflected (upgoing) wave will has a minus sign. The sum of the two Alfvénwaves is thus:

inc ref inc ref Jinc + Jref = ΣA∇⊥ · E⊥ − E⊥ = ΣP ∇⊥ · E⊥ + E⊥

inc ref ⇒ ∇⊥ · (ΣA − ΣP )E⊥ − (ΣA + ΣP )E⊥ = 0 (2.9)

By integrating the above equation we can define a reflection coefficient to describe the current in the ionosphere due to the effects of the Pederson conductance. This is then given as: E Σ − Σ R ≡ ref = A P (2.10) Einc ΣA + ΣP

Similarly we can deﬁne a transmission coeﬃcient (T ) given by

2Σ T ≡ 1 + R = A (2.11) ΣA + ΣP

With a physical understanding of how currents can be generated due to the stellar 41

wind-magnetosphere reaction and the implications of the height integrated currents and conductivities, we can now calculate the Joule heating from the parallel currents as done in Kivelson & Ridley(2008) and Cohen et al.(2018).

2.4.4 Joule Heating Calculation

The amount of current that can be generated by the magnetosphere is set by the

Alfv´enconductance (ΣA) which is dependent on the Alfv´enspeed (vA) and the size

of the magnetic field of the stellar wind (Bsw). As described above, once generated, the remaining amount of current that can flow through the atmosphere is set by the conductance of the ionosphere (ΣP ). Once the ‘transmitted’ electric field from the charged particles in the stellar wind through the atmosphere is known, we can determine the size of the current and then the size of the amount of Joule heating. This is quantitatively expressed below. The volumetric heating heating of the atmosphere from a current J is give by:

dE J 2 J 2 = → x (2.12) dV dt σP σP

since the current that contributes to Joule heating moves horizontal to the planet. This gives us the heat generated in energy (E) per volume (V) per time (t). Plugging in for the current J we get:

2 2 dE (σP Ex) Jx 2 = → = σP Ex (2.13) dV dt σP σP

We can apply the argument from above where the ionosphere is considered a very thin region and therefore we can take the height integrated value of conductivity to represent the eﬀects from conductivity. Therefore, plugging in for height integrated 42

values we get: dE Z = dzσE2 = Σ E2 (2.14) dA dt x P x

where from the height integral we get the heat in terms of energy per area (A) per time. We can also plug in for the height integrated current density, j, to get:

2 2 dE jx jx = ΣP 2 = (2.15) dA dt ΣP ΣP

We now combine the transmitted current we found due to the Alfv´enwave discussion above, with the Joule heating rate to determine the Joule heating rates set by the current generated due to the stellar wind-magnetosphere interaction. The stellar wind at the location where the ﬁeld-aligned current connects to the magnetosphere

has velocity, vsw, and magnetic ﬁeld strength Bsw. Therefore the electric ﬁeld (in cgs units) is: 1 E = − v × B (2.16) sw c sw sw and the magnitude is given by:

1 |E | = |v ||B | (2.17) sw c sw sw

Applying the Alfvénconductance and Alfvénspeed given above, the reflection and transmission of the incoming wave electric field is given by:

−1 −1 ΣP − ΣA |Er| = |Ei| −1 −1 = |Ei|R (2.18) ΣP + ΣA

where ΣP is the conductance of the atmosphere where the ﬁeld-aligned current ﬂows. 43

The transmitted electric ﬁeld is then given by:

−1 ΣP |Et| = |Ei| + |Er| = 2|Esw| −1 −1 = |Esw|T (2.19) ΣP + ΣA

The transmitted electric ﬁeld felt by the ionosphere will be largest when the Alfv´en and Pederson conductance are equal. The heat then transferred to the atmosphere due to this conductance and given by Cohen et al.(2018) is:

2 dE j 2 −1 −2 = qJH = = ΣP |Et| [erg s cm ] (2.20) dA dt ΣP

where j is the height-integrated current over the region that heat is dissipated. Note

that Cohen et al.(2018) gives the Joule heating, qJH, as an energy ﬂux, which is diﬀerent than the standard for Joule heating given as J · E. We choose to denote the

standard Joule heating as QJH. The volumetric heating can be given by:

2 2 dE vswBsw 2ΣA = QJH = σP (2.21) dV dt c ΣA + ΣP

We apply this equation for Joule heating in Chapter3 to our hot Jupiter atmosphere model after we determine its electrical conductivity proﬁle. A discussion on electrical conductivity and its derivation is included below.

2.5 Electrical Conductivity Overview

Planetary atmospheric conductivity has primarily been researched in regards to the Earth’s magnetosphere and ionosphere. The Earth’s height integrated conductivity, or conductance, has been modelled for various applications as overviewed in Brekke 44

& Moen(1993), and more recently in the review of ionized plasmas by Ballester et al.(2018). The relationship between the conductivity and currents flowing through Earth’s magnetosphere has, also, been more generally quantified by Ridley et al. (2004). The conductivity of planetary atmospheres other than the Earth’s has also been examined. For example, Hill(1979) has shown how the Pederson conductivity (discussed in detail in the following sections) of Jupiter’s atmosphere can explain the observed co-rotation lag of the magnetosphere with the rotation frequency of the planet surface. Goodman(1995) outlines how to implement the conductivity of an atmosphere towards setting the boundary conditions of MHD planet-stellar wind simulations. Badman et al.(2015) examined the ionospheric response to ion and electron precipitation for giant planets in the Solar System and Coates et al.(2011) reviewed the effect of photoelectrons in the atmospheres of the terrestrial bodies of our Solar System. For exoplanets, recent studies by Cohen et al.(2018, 2020) have detailed the effects that an electrical conductivity can have on the energy balance of a habitable zone atmosphere through Joule heating. They find that stellar wind induced currents can generate heat in the atmosphere that, when compared to other sources of heat, is equal to ∼ 1% of the stellar bolometric flux. This means that stellar wind induced currents can be important in setting the temperature profile of the atmosphere. A focus of my study in Chapter3 is in determining the Joule heating of exoplanets, specifically for hot Jupiters. Rather than use an approximation for the conductance based on Solar System assumptions, I model the full conductivity profile in the upper atmospheres of these planets to get a more accurate prediction of the Joule heating of exoplanets. The derivation of electrical conductivity in ionospheres and the relevant 45

terms that go into its derivation are outlined below.

2.6 Deriving the Conductivity Formulas

For electric currents in strongly magnetized atmospheres like, for example, the Earth’s or Jupiter’s, charged particles can flow in directions along and across the magnetic field, B, in response to an electric field. The geometry of this current is shown in Figure 2.3. The speed at which these charged particles can flow can be calculated via momentum conservation from collisions as the current moves through the atmosphere.

2.6.1 Conductivity Across a B-ﬁeld

The momentum equation for a single species is (Schunk & Nagy 2004):

v × B q E + i = v m ν (2.22) c i i i

P where νi ≡ νis is the sum of collisions over all other species. The velocity of the ions, vi, gives us the ﬂow of the charged particles through the atmosphere.

Additionally, we can deﬁne Ωci ≡ (qB)/(mic) to be the cyclotron frequency for a given ion and magnetic ﬁeld strength, B. Our equation now becomes:

q Ωci E⊥ + vi × b = vi (2.23) miνi νi

where E⊥ = E + (1/c)v × B is an effective electric field due to the moving neutral particles and b ≡ B/B. We can break this ion velocity into its individual vector ˆ ˆ ˆ components (vi = vxi + vyj + vzk) by utilizing Cartesian coordinate identities where ˆ ˆ ˆ E⊥ is defined to be along the x-axis (E⊥ = E⊥i+0j+0k) and b is defined to be along 46

Fig. 2.3.— A diagram of the three vectors of current ﬂow in an ionosphere with a given B and E ﬁeld. The Pederson (σP ), Hall (σH ), and parallel (σk) conductivities are derived in Chapter 2.6. 47

the z-axis (b = 0î + 0ˆj + Bkˆ). We are looking for the velocity vector components that flow across the B field in response to an electric field and we can find this by taking the cross product of each side of Equation 2.23 with B. The vector equation then becomes:

Ωci ˆ q Ωci ˆ ˆ ˆ ˆ ˆ − vxB i − E⊥B + vyB j + 0k = vyBi − vxBj + 0k (2.24) νi miνi νi

Algebraically solving for the individual velocity components, {vx, vy, vz}, we get:

q νi vx = 2 2 E⊥ (2.25) mi νi + Ωi

q Ωci vy = 2 2 E⊥ (2.26) mi νi + Ωci

vz = 0 (2.27)

and Equation 2.23 becomes:

q νi Ωci vi = 2 2 E⊥ − 2 2 b × E⊥ (2.28) mi νi + Ωci νi + Ωci

We can then multiply each side of the equation by niq to get our equation more intuitively in terms of current densities, J, rather than velocities. The equation in terms of current density is:

2 νi νiΩci Ji = σi 2 2 E⊥ − 2 2 b × E⊥ (2.29) νi + Ωci νi + Ωci

2 where σi ≡ (niqi )/(miνi) is the ‘ion conductivity’ (or electron conductivity). Ex- 48

amination of Equation 2.29 shows that there is a current density in the E direction and in the b × E direction. We can then deﬁne conductivities in each of these directions to get:

Ji,⊥ ≡ σP E⊥ + σH b × E (2.30)

where σP is the ‘Pederson conductivity’ and σH is the ‘Hall conductivity’ as deﬁned as: 2 X nke νk σP = 2 2 (2.31) mk ν + Ω k k ck

2 X nke Ωck σH = 2 2 (2.32) mk ν + Ω k k ck over all species ‘k.’

2.6.2 Conductivity Parallel to the B-ﬁeld

Parallel to the B-field, electron drift velocities are much greater than the ion or neutral velocities due to the difference in masses (Schunk & Nagy 2004), and so we can approximate the current density to be entirely dominated by electrons. Mathemat- ically, in the B-field direction the v × B/c term drops out and we can additionally ignore ρg and ∇P forces in the accelerations since they will be negligible compared to pressure and electric forces (Schunk & Nagy 2004). Therefore, the parallel component to the B-field momentum conservation becomes:

X nemeνesve,k = −eneEk (2.33) s

Assuming that the field-aligned current density is Jk = qkneve and νe is defined to be the sum of the electron collision frequencies with all other species in the atmo- 49 sphere, this equation simplifies to:

Jk = σkEk (2.34)

Here, σk is the ‘parallel electrical conductivity’ as given by:

2 nee σk = (2.35) meνe

Note that this conductivity is independent of magnetic ﬁeld.

Summary

As shown above, the Pederson conductivity (σP ) relates current density J and the electric ﬁeld transverse (E⊥) to the magnetic ﬁeld. Similarly the Hall conductivity

(σH ) relates J and E fields in the E×B direction (or transverse to both the magnetic and electric fields). The parallel conductivity (σk) gives the rate at which charged particles can flow parallel to the magnetic field. Therefore, in order to calculate the Pederson and Hall conductivity of an atmosphere you must know the particles contained in the atmosphere. Specifically, you must have values for: (1) the electron density and ion density throughout, (2) the neutral densities, (3) the magnetic field strength, and (4) the collision frequencies. My atmosphere model therefore must contain all major sources that can contribute to the conductivity of the atmosphere. The Joule heating of the atmosphere, as shown above and in Paschmann et al. (2002), can be significantly impacted by the effects from the Pederson conductance and so I focus my research outlined in Chapter 3 on calculating this value. The collision frequencies that are input into the conductivity formulas are outlined below. 50 2.7 Height Integrated Conductivity

An important aspect of the conductivity proﬁle of the atmosphere, and the value used in most applications of electrical conductivity of atmospheres, is the height integrated conductivity, often referred to as the conductance and denoted as Σ. Since the ionosphere can be treated as a thin layer, we can height integrate the conductivity σ and only the value of the conductance Σ is needed for applications. The equation for conductance is Z Σ = σdr (2.36)

This equation can be applied to the Pederson and Hall conductivity but breaks down for the parallel conductivity.

2.7.1 Qualitative Features of the Conductivity Proﬁle

Further examination of Equations 2.31 and 2.32 shows contributions to the conductance integral given by Equation 2.36. Since the conductance is determined by a height integral of the conductivity, any peaks in the conductivity will dominate the conductance calculation. For example, the Pederson conductance is calculated as

Z 2 ! X nie νi Σ = dz (2.37) P m ν2 + Ω2 i i i ci

Peaks in the conductivity proﬁle can occur (1) when there are peaks in the density of the charged species ne,i and (2) when ν = Ω and creates a peak similar to a Lorentzian function. Therefore, in order to properly calculate the conductance, we must create an atmosphere model that encompasses all of these peaks of conductivity in the atmosphere. The peaks in the conductivity due to density depend on the chemical species and reactions in the model but the Lorentzian peaks, which depend 51

on B and collision frequency, can be estimated analytically as shown in Chapter 2.8.5.

2.7.2 Analytic Estimate of Σ

We can make an analytic estimate for the conductance of a plane parallel atmosphere below. For simplicity, we assume constant electron/ion density and constant temperature in our atmosphere. For now, let’s assume the only ion present are protons and calculate the Pederson conductance of this atmosphere. Subbing Equation 2.31 into Equation 2.36:

Z ν2 ν2 Σ = dr σ i + σ e (2.38) P i ν2 + ω2 e ν2 + ω2 i ci e ce

Focusing on the electron half of the integral and plugging in for σe we get:

Z 2 nee νe ΣP,e = dr 2 2 (2.39) me νe + ωce

We can assume that the collision rate, ν, is dependent linearly on the density of

neutrals given by νe = αnn where α is a constant. Since we have a plane parallel

−z/H atmosphere we can estimate nn = n0e where n0 is a reference density and H =

kBT/µmpg is the scale height of the atmosphere. Here we deﬁne z as the location

where the electron collision rate is equal to the cyclotron frequency (Ωe) since this is

−z/H where we will have a peak in the integral. Therefore νe = αnn = Ωee . Plugging all this into Equation 2.39 and pulling out constants we get:

2 Z ∞ −z/H nee Ωee ΣP,e = dz 2 −2z/H 2 (2.40) me −∞ Ωe e + Ωe 52

let x ≡ z/H 2 Z ∞ −x nee H e ΣP,e = dx −2x (2.41) meΩe −∞ e + 1

therefore: 2 π nee H ΣP,e = (2.42) 2 meΩe

This process can be repeated for ions to ﬁnd that the full Pederson conductance, after substituting in Ωe = eB/mec can be analytically estimated as:

π ec Σ = H (n + n ). (2.43) P 2 B e i

This exercise shows the important result in that for this atmosphere model, the Pederson conductance is inversely proportional to magnetic field strength. This relation of how Pederson conductivity scales with magnetic field size can be physically understood by considering how charge flows through the atmosphere as a result of collisions of charged particles. As shown in Figure 3.1 of Paschmann et al. (2002), a disruption of E × B drift occurs via collisions and results in the Pederson and Hall currents. With no collisions, electrons and ions will cycle around the magnetic field and move only in the E × B direction apart from the spiraling caused by the magnetic field. As collisions occur though, electrons and ions no longer drift together but are affected due to the collisions with species in the atmosphere which are governed by the conductivity equations detailed above. The path of the current then, in the directions described by the Pederson and Hall conductivity, effectively moves by the Larmor radius via each collision which scales inversely with magnetic field size. Therefore, for larger sized magnetic fields, the Larmor radius will be smaller, and the drift of the particles in the directions governed by the Pederson and Hall conductivity will also scale inversely with magnetic field size. 53 2.8 Collision Rates

The collision frequencies are set by the interactions of electrons, protons, neutral species, and ions. Collisions can be either elastic or inelastic. In an elastic collision, momentum and translational kinetic energy are conserved, while inelastic collisions result in a change in internal energy. The nature of the collisions depends on the relative kinetic energy of the colliding particles and the type of particles. The speciﬁc collisions that can be important in ionospheres include non-resonant ion-neutral and electron-neutral interactions, resonant charge exchange, and Coulumb collisions. I review the collision frequencies used in Chapter3 below. Chapter 4 of SN04 outlines in more detail how to calculate the collision frequencies between these particles.

2.8.1 General Collision Parameters

To introduce collision parameters, consider the scenario shown in Figure 2.4 (see

Chapter 4.1 of SN04). Here, a test particle with radius r0 is moving through a gas of target particles with a constant density n. Ignoring random motions of the particles, the test particle moving with speed v over a time ∆t, will trace out a volume

2 V = σ(v∆t), where σ = πr0 is the cross-section for interaction. The number of target particles in this volume is simply N = V n = (σv∆t)n. Therefore the frequency of collisions, ν can be given by: ν = vσn (2.44)

The time between collisions is τ = 1/ν, and the ‘mean-free-path,’ or the length that a particle can travel on average before colliding with another particle, is λmfp = τv = 1/(σn). In reality, a number of factors can eﬀect this simple picture of collisions including thermal motions of the colliders and the size of the cross-section as a function 54

Fig. 2.4.— Diagram of a simple collision between a large particle and a background gas of constant volume of smaller particles (Schunk & Nagy 2004)

of velocity (σ = σ(∆v)). The exact forms of the collision frequencies are shown below.

2.8.2 Ion/Electron-Neutral Non-resonant Collisions

For ion-neutral and electron-neutral collisions, ‘Maxwell molecule’ collisions are a reasonable approximation for elastic (nonresonant) ion–neutral or electron-neutral interactions (see Chapter 4.3 in SN04). As the ion approaches the neutral, the neutral becomes polarized and the collision can be treated as between the ion and an induced dipole, for which the interaction potential (in cgs units) is:

γ e2 V = − n (2.45) 2r4

Here, γn is the neutral polarizability and values are given in Table 2.1 for the 55

relevant neutral gases. The collision frequency can be expressed as:

2 1/2 nnmn γne νin = 2.21π (2.46) mi + mn µin

where µin = mimn/(mi + mn) is the reduced mass of the ion-neutral collision. This formula can be applied to collisions of ions and electrons with all neutral species in the atmosphere.

2.8.3 Resonant Charge Exchange

In addition to non-resonant collisions as described above, ions can also undergo resonant charge exchange with their parent neutral atom or molecule (see SN04 Section 4.8). This typically dominates reactions at temperature T > 300 K (SN04), which is well within the range of the high temperatures in the hot Jupiter ionospheres that I am considering. In charge exchange reactions, as an ion and neutral of the same species approach each other, an electron can transfer from the neutral to the ion resulting in a large transfer of momentum and energy between the particles. These types of reactions are ‘pseudo-elastic’ in that a very small amount of energy is lost in the collision process. Charge exchange can also occur for species with similar ionization potentials. This reaction is particularly important for protons with OI and OII with atomic H. I list the important collision frequencies for resonant ion-neutral

−24 −3 −24 −3 Species γn(10 cm ) Species γn(10 cm ) H 0.67 H2 0.82 He 0.21 H2O 1.48 CO 1.97 N 1.13 Na 2.70 O 0.77

Table 2.1: Relevant neutral gas polarizabilities. These values were taken from Table 4.1 in Schunk & Nagy(2004) 56

interactions in Table 2.2.

−1 Species νin(s ) + −10 1/2 2 H ,H 2.65 × 10 nHT (1 − 0.083 log T ) + −11 1/2 2 He , He 8.73 × 10 nHeT (1 − 0.093 log T ) + −11 1/2 2 O ,O 3.67 × 10 nOT (1 − 0.064 log T ) + −11 1/2 2 H ,O 6.61 × 10 nOT (1 − 0.047 log T ) + −12 1/2 O ,H 4.63 × 10 nH(17T/16) + −11 1/2 2 CO , CO 3.42 × 10 nCOT (1 − 0.085 log T ) Table 2.2: Relevant collision frequencies for resonant ion-netural collisions. These values were taken from Table 4.5 in Schunk & Nagy(2004). All temperatures are in K and densities (n) in cm−3.

2.8.4 Coulumb Collisions

The collision frequency of two charged particles is set by Coulumb collisions. SN Equation 4.140 and 4.56 gives us the general form of this equation of species ‘s’ colliding with species ‘t’ to be:

√ −3/2 2 2 16 π ntmt 2kBT eset νst = 2 ln Λ (2.47) 3 ms + mt µst µst

where ln Λ is the Coulumb logarithm given by:

3/2 4π kBT Λ = 9 ne 2 (2.48) 3 4πnee

For ionospheres, the Coulumb logarithm has typical values of ln Λ ∼ 15 (Schunk & Nagy 2004). I consider collisions of ion and electrons with all other charged particles in the atmosphere, including self-collisions, utilizing this formula.

The sum of the collision frequencies described above for all charged particles are input into the conductivity formula to calculate the electrical conductivity in the 57

atmosphere.

2.8.5 Location of Peaks in the Conductivity

I’ve already discussed how peaks in the conductivity are vital towards setting the conductance of the atmosphere and now that we know the major collision frequencies, I now discuss how to analytically estimate the location of the electron, proton, and ion peaks in the Pederson and Hall conductivities due to collision frequencies equaling the cyclotron frequencies. The locations where these peaks are estimated to occur will deﬁne how deep an atmosphere must be modeled to consider the full aﬀect of the

conductivity. We are looking for the location in the atmosphere where νk = Ωc,k.

Electrons

At the top of the atmosphere, the collision frequencies of electrons with other particles in the atmosphere are dominated by non-resonant electron-H collisions and electron-

proton Coulomb collisions. This means that νe = νeH +νep where νeH is determined by

Equation 2.46 and νep is determined by Equation 2.47. The major collision frequencies and electron cyclotron frequency are:

n + n P 2500 K ν = 10.0 s−1 H H2 = 2.48 × 105 s−1 (2.49) eH 108 cm−3 1 µbar T

n 104 K3/2 ν = 3700.0 s−1 p (2.50) ep 108 cm−3 T eB 8 −1 B Ωce = = 1.75 × 10 s (2.51) mec 10 G

Here, I’ve assumed that the pressure in this region of the atmosphere is primarily

set by atomic hydrogen and molecular hydrogen (P = (nH + nH2 )kBT ). At magnetic 58

ﬁeld strengths B > 0.01 G, Ωce νep and so the electron collision frequency is dominated by H in this region. Therefore, we can expect a peak in the conductivity

to occur when νeH ≈ Ωce, which occurs at a pressure (Peq,e) of:

B T P = 7.07 × 10−4 bar (2.52) eq,e 10 G 2500 K

which is linearly proportional to B and T .

Protons

In the case of protons, resonant charge exchange reactions with atomic hydrogen completely dominate the collision frequencies of protons with all other species in the atmosphere. This collision rate is listed in Table 2.2 for the species of H+ with H. Therefore, for protons, the major collision frequencies and cyclotron frequency is:

n T 1/2 P 2500 K1/2 ν ≈ 1.2 s−1 H = 4.35 × 103 s−1 pH 108 cm−3 104 K 1 µbar T (2.53) eB 4 −1 B Ωcp = = 9.58 × 10 s (2.54) mpc 10 G

Setting these frequencies equal to each gives us a location for a peak in the proton contribution towards conductivity of:

B T 1/2 P = 7.78 × 10−6 bar (2.55) eq,p 10 G 2500 K which is linearly proportional to B and scales as T 1/2 for temperature. 59

Ions

In the atomic and molecular dominated region of an atmosphere, ion collision frequencies are mainly set by non-resonant collisions with the dominant neturals (H and

H2). Therefore we can apply Equation 2.46 to get estimates for the collision frequency

of an ion with H or H2. Again, I make use of the estimate that P ≈ (nH + nH2 )kBT . The collision frequency equation becomes:

2 1/2 (nH + nH2 )mH γHe νin ≈ 2.21π (2.56) mi + mn µiH

Here I make use of the fact that γH ≈ γH2 . In the case of ions colliding with atomic or molecular H, mi is often much greater than mn and µin ≈ mn. Therefore, this equation simpliﬁes further to:

2 1/2 (nH + nH2 )mH γHe −1 mH nH + nH2 νin ≈ 2.21π = 0.032 s 8 −3 (2.57) mi mn mi 10 cm

−1 mH P 2500 K ⇒ νin = 927.5 s mi 1 µbar T

and the cyclotron frequency is simply:

eB mH 4 −1 mH B Ωci = = Ωcp = 9.58 × 10 s (2.58) mic mi mi 10 G

Setting νin = Ωci highlights an interesting phenomenon where the Lorentzian peak is independent of ion mass and the peaks for each of these species occurs at roughly the same location. This Lorenztian peak occurs at:

B T P = 1.03 × 10−4 bar (2.59) eq,i 10 G 2500 K 60

The location of the Lorentzian peaks in the Pederson and Hall conductivities when

νk = Ωk give us insights into the locations where an atmosphere model that seeks to fully capture Σ must be developed. For Jupiter sized magnetic fields (B ∼ 10 G) and temperatures that are expected near the transition from atomic dominated to molecular dominated for HD 189733b model of T ∼ 2500 K (e.g. Huang et al. 2017), the proton peak occurs highest in the atmosphere (at lowest pressure), followed by the ion peak and the electron peak deeper in the atmosphere. We can also see the peak location scales linearly with magnetic field strength for each species meaning that higher magnetic field strengths will result in peaks occurring deeper in the atmosphere. Similarly, the location of the peak scales at T for electrons and ions and as T 1/2 for protons due to the strong dependence on the charge exchange reaction. For this reason, I seek to fully encapsulate the Lorentzian peaks in conductivity in my atmosphere model and so aim to create the model to pressures approaching 1 bar.

2.9 Summary

In this chapter, I’ve over-viewed the historical context of stellar wind-planet interactions and highlighted how it relates to my hot Jupiter atmosphere model in Chapter 3. I’ve discussed the important currents that can run through atmospheres due to the stellar wind-magnetosphere interaction and outlined a calculation of Joule heating in atmospheres. I’ve reviewed the general conductivity equations for the Pederson, Hall, and parallel conductivities (Equations 2.31, 2.32, and 2.35) and outlined the relevant collision frequencies that are dominant in the ionospheres. I’ve shown the expression to calculate conductance (Equation 2.36), and highlighted the importance that the conductance can have on the dynamics of an ionosphere. Additionally, I’ve analytically shown the regions of the atmosphere that I must model to include peaks in the 61

Pederson and Hall conductance (P ∼ 1 bar). This discussion on conductivity is the basis for the model outlined in Chapter3 for HD 189733b and informs what species I need to include and the types of reactions considered. 62

Chapter 3

Electrical Conductance of the Upper Atmosphere of Hot Jupiter HD 189733b

3.1 Summary

We present a hydrostatic model of the dayside upper atmosphere of the hot Jupiter, HD 189733b and compute its electrical conductivity. Height integrated Pederson conductivities are computed in order to estimate Joule heating due to electrical currents from the magnetosphere which close in the ionosphere. We find that for Jupiter sized magnetic fields (B ∼ 10 G), peaks in the Pederson conductivity occur near the region of the atmosphere where molecules begin to dominate atoms. Therefore, it is vital to model this transition region to properly calculate its conductance as the atomic species that dominate the conductivity form molecules there. We calculate the conductance for a range of magnetic fields from 0.1 − 100.0 G, as the peak in the conductivity is strongly dependent on magnetic field strength. We find the Pederson 63

16 20 −1 conductance for the atmosphere of HD 189733b to be ΣP = 10 − 10 cm s over the range B = 100 − 0.1 G, which is much greater than the Pederson conductance

10 11 −1 of the Earth which is ΣP,⊕ ∼ 10 − 10 cm s . We find that at these conductance values, Joule heating does not contribute significantly to the thermal structure of the atmosphere, but could become important if magnetic fields can be very large (B > 100 G).

3.2 Introduction

The focus of our study is to determine the Joule heating of exoplanets, specifically for hot Jupiters. As discussed in Chapter2, electrical conductivities can play an important role in the dynamics and energy balance in the ionosphere of a planetary atmosphere, which can impact other regions of the atmosphere. Currents can be generated in the magnetosphere due to the stellar wind interaction and as these charged particles move through the planetary atmosphere they will collide with other species, exchanging momentum and energy with the atmosphere. The momentum exchange that these particles experience due to collisions is quantified by the atmosphere’s electrical conductivity. For planets that have a relatively strong magnetic field, currents can flow parallel to or perpendicular to a B field due to an imposed

electric ﬁeld that governs the ﬂow of the current. The Pederson conductivity (σP )

describes current ﬂowing along a transverse electric ﬁeld (E⊥), the Hall conductivity

(σH ) describes current in the E × B direction (or transverse to both the magnetic and electric fields), and the parallel conductivity (σk) describes the current flowing parallel to the magnetic field. As outlined in Chapter 2.6, planetary atmospheric conductivity has primarily been studied with respect to the Earth’s magnetosphere and ionosphere but conductivities 64 have also been studied for other Solar System planets in order to understand their current systems. Most applicable to the research outline in this Chapter, though, are the recent studies by Cohen et al.(2018, 2020) on Joule Heating in exoplanets. They find that, with an Earth-like conductance, the stellar wind induced currents can dissipate a large amount energy to the atmosphere through Joule heating and when compared to other heating rates in the atmosphere can be as large as ∼ 1% of the stellar bolometric flux to the planet’s atmosphere. Rather than use Solar System planet values for the conductance, we model the full conductivity structure in the upper atmospheres of these planets to get a more accurate estimate of the Joule heating of exoplanets. The first detection of an exoplanet atmosphere was the sodium doublet transit signal of the hot Jupiter HD 209458b (Charbonneau et al. 2002), which has led to many detailed works to model the structure of exoplanet atmospheres. This hot Jupiter atmosphere discovery has opened the door for investigation of more extreme environments for planetary atmospheres than are found in our Solar System. Due to the intense radiation hot Jupiters receive from their host star (∼ 104 times greater than the Solar System gas giants), their atmospheres can differ significantly from the gas giant planets in our Solar System. Models of the most studied hot Jupiters, HD 209458b and HD 189733b, have shown that the upper atmospheres of these planets likely have significant ionization fraction. The observations of mainly neutral species in hot Jupiter atmospheres are outlined in Chapter 3.3. We set out to combine the observations of atoms and other species in the atmosphere of hot Jupiters, with a detailed model of the conductivity structure throughout the upper atmosphere of these hot Jupiters. Hot Jupiter atmosphere models have typically been created with a primary focus 65

of investigating their physical/chemical processes or to explain the observed spectral features in the transmission spectrum for transiting exoplanets. For example, models for the upper atmosphere of HD 209548b have been created to study atmospheric escape due to the intense heating from its host star (Yelle 2004; Garc´ıaMuñoz 2007; Murray-Clay et al. 2009; Koskinen et al. 2013; Salz et al. 2016) or Roche Lobe overflow (Gu et al. 2003; Li et al. 2010; Lai et al. 2010; Ehrenreich & Désert 2011). Addi- tionally, hydrodynamic models have been created to study the interaction of colliding planetary and stellar winds (Tremblin & Chiang 2013) and magnetohydrodynamic (MHD) models have been created to further investigate this effect when including the planetary magnetic field (Trammell et al. 2011, 2014; Christie et al. 2016). Compar-

isons to observations of HD 209458b of H Lyα,OI, SiIII, NaI, and KI lines have been done by Fortney et al.(2003); Koskinen et al.(2013); Lavvas et al.(2014) and for HD 189733b by Huang et al.(2017). Atmospheric abundances of each species in these models are often calculated via local thermodynamic equilibrium (LTE) models (Burrows & Sharp 1999; Seager et al. 2000; Sharp & Burrows 2007; Fortney et al. 2008; Visscher et al. 2010), models with slight deviations from LTE (Lodders & Fegley 2002; Fortney et al. 2003, 2006; Visscher et al. 2006; Showman et al. 2009; Visscher et al. 2010), photo-chemical models (Yelle 2004; Garc´ıaMuñoz 2007; Koskinen et al. 2013; Lavvas et al. 2014; Huang et al. 2017), or a combination of the two (Moses et al. 2011). We focus our model on the photo-chemical processes in HD 189733b. In this chapter, we specifically model the composition and temperature profiles of the upper atmosphere of HD 189733b with the goal of determining its electrical conductivity structure and the applications to Joule heating due to stellar wind interaction. We show the Joule heating rates are sensitive to the size of the planetary 66 magnetic field. We discuss the relevant equations to calculate the electrical conductivity and go into more details on determining the conductivity as well as discussing the model created to determine atmospheric composition and temperature in Chap- ter 3.3. We list the results of our atmosphere model and the electrical conductivity structure in Chapter 3.4 and discuss the applications of these results in Chapter 3.5.

3.3 The Model

Following a similar approach to Christie et al.(2013) and Huang et al.(2017) we have constructed a spherically symmetric and hydrostatic atmosphere model for the day-side upper atmosphere of hot Jupiter HD 189733b. We model the regions of the upper atmosphere from mainly ionized gas to the molecular gas dominated regions at pressures greater than 1 mbar. Since the atmosphere is not in local thermodynamic equilibrium, the structure of the atmosphere is modeled by solving the following system of rate equilibrium equations at each level: (1) a balance of the chemical production and loss of each species, (2) a reservoir equation in which the abundance of each element is assumed to be solar abundance, (3) a thermal equilibrium equation set by balancing heating and cooling mechanisms, (4) a charge balance of all charged species, and (5) the equation of state. At each step of the model we additionally calculate the pressure, radius, and column of each species by solving differential equations integrated inwards. The roots for the system of equations are determined using the ‘scipy.optimize.root’ package (Virtanen et al. 2020). Specifically, we found that the ‘hybr’ method (More et al. 1980) is the fastest and most accurate of the root solvers for the problem as we’ve set it up. The ‘hybr’ method determines the roots using a modified Powell method (Press et al. 2007) similar to the more familiar Newton-Raphson method. 67

The atomic region of the model is validated via direct comparison to the similar Huang et al.(2017) model for HD 189733b. In that work, the authors constructed a hydrostatic model for the dayside upper atmosphere of HD 189733b, and compared their calculated density and temperature structure to observations of HD 189733b. Specifically, they found good agreement with Na D and Hβ transmission spectra of HD 189733b. We find good agreement in our density profile, to that modelled by Huang et al.(2017). The molecular region of HD 189733b has not been observed and so direct comparison to observations is not possible. Instead, we discuss the impacts that the molecular layer will have on calculating conductivity in Section 3.5 and compare our model to work by other authors, as mentioned above, who have modelled the molecular layer of hot Jupiters.

3.3.1 Determining the Pressure Limits

We seek to model the region of the atmosphere where electrical conductivity can potentially eﬀect the dynamics of the atmosphere through Joule heating. As shown in Ionospheres by Schunk & Nagy(2004) in Section 5.11 and here in Chapter2, the Pederson, Hall and parallel conductivities are given by:

2 X nke νk σP = 2 2 (3.1) mk ν + Ω k k ck

2 X nke Ωck σH = 2 2 (3.2) mk ν + Ω k k ck

2 nee σk = (3.3) meνe

Here Ωc is the cyclotron frequency for a given species where Ωck = qkB/mkc and 68

Deﬁnition Description Reference 2 1/2 nnmn γne νin = 2.21π Polarization SN Equation (4.88) mi+mn µin 2 1/2 ν = 2.21πn γne Polarization SN Equation (4.88) eH n me 3/2 −1 np 104 K νep = 3700.0s 108 cm−3 T Coulumb SN Equations (4.56) and (4.140)

−1 nH T 1/2 νpH = 1.2 s 108 cm−3 104 K Charge Exchange SN Table 4.5 Table 3.1: Major collision frequencies. Note that SN refers to Schunk & Nagy(2004).

νk is the collision frequency for species k summed over all other collision partners P (i.e. νa = b νab, where b is all other species). Note that the charge of electrons, qk, is negative. Collisions between all species are calculated by the equations given in Section 4 of Schunk & Nagy(2004) and here in Table 3.1. We discuss how the collision frequencies are determined in more detail in Chapter 2.8. The parallel conductivity is dominated by electrons and so we ignore the ion contribution in Equation 3.3. An important aspect of the conductivity structure of the atmosphere, and the value used in most applications of electrical conductivity of planetary atmospheres, is the height integrated conductivity, often referred to as the conductance and calculated as Σ = R σdr (see Chapter 2.7). The conductance has been shown to be important (e.g.) in calculating the co-rotation lag in Jupiter (Hill 1979), setting the bounds of MHD simulation (Goodman 1995), and most notably for us, in calculating the Joule heating in an atmosphere (Cohen et al. 2018). Further examination of Equations 3.1 and 3.2 highlights where the electrical conductivity will contribute most to the conductance. Since the conductance is determined by a height integral of the conductivity, any peaks in the conductivity will dominate the conductance calculation. Peaks in the conductivity structure can occur

(1) when there are peaks in the density of the charged species ne,i and (2) when ν = Ω and creates a peak similar to a Lorentzian function. Therefore, in order to properly calculate the conductance, we must create an atmosphere model that encompasses 69

Stellar Type K2V 33 Star Mass M∗ = 1.60 × 10 g 10 Star Radius R∗ = 5.60 × 10 cm Semimajor axis a = 0.031 AU 30 Planet Mass Mp = 2.17 × 10 g 9 Planet Radius Rp = 8.137 × 10 cm 2 3 −2 Planet surface gravity gp = GMp/Rp = 2.2 × 10 cm s Table 3.2: Adopted properties for HD 189733A and HD 189733b. all of these peaks of conductivity in the atmosphere.

3.3.2 Basic Atmosphere Structure

The basic properties we assume for HD 189733b and its host star are given in Table

−11 3.2. We solve our model over the pressure range Ptop = 5 × 10 bar to Pb =0.1 bar (note that 1 µbar = 10−6 bar = 1 dyne cm−2) is the equivalent cgs unit for pressure). This pressure range is to ensure that we fully model the transition from the ion dominated region to the molecular dominated region as this is likely the vital region for peaks in the electrical conductivity. We assume the base of the model, Pb, occurs at the pressure of the optical photosphere as given in Sharp & Burrows(2007)

9 and at a radius of Rb = 8.137 × 10 cm. The pressure-radius relation is determined by the hydrostatic balance equation. We take inward radius steps for the integration equivalent to 0.01 of the scale height at the previous level. The pressure-radius relation can be found in Figure 3.3. Near the outer boundary, the densities we find will be inaccurate as physical effects such as out-flowing gas from the planet, interaction with the stellar wind, magnetic forces, radiation pressure and stellar tidal forces have been neglected. Specifically, Yelle(2004) created a model of hot Jupiter HD 209458b which shows that due to the high temperatures (T > 104 K) in the upper atmosphere from extreme ultraviolet 70

(EUV) flux, the atmosphere escapes rapidly and is primarily cooled by adiabatic expansion. In regards to our model, this effect would be most significant in the ionized region of the upper atmosphere. Yelle(2004) did not consider cooling due to atomic resonance lines which will become more important than adiabatic cooling once atoms begin to dominate. As shown in Huang et al.(2017), these effects are most important above the P ∼ 10−9 bar region and less important at higher pressures. Therefore, in the regions where conductivity is most important as shown in Section 3.4, the density is so high that these effects are negligible. Hence, the hydrostatic model is sufficient for the purposes of this study.

Chemical Structure of the Atmosphere

We employ a chemical scheme of 35 atoms, ions, and molecules and 70 reactions. Our full chemical scheme consists of electrons; hydrogen based species: protons, H, H2,

+ + H2 ,H3 ; HeI and HeII; metals and their ﬁrst ionized state: CI,CII,OI,OII, NaI, NaII,

MgI, MgII, SiI, SiII,SI,SII,KI,KII, FeI, and FeII; the second ionized state of some species: OIII, MgIII, SiIII, and SIII; and molecules: CO, OH, H2O, NaH, and SiO. The reactions considered are shown in Tables 3.3-3.7. We choose species to ensure we account for:

1. All major particles in the atmosphere (electrons, protons, H, H2, He)

2. All metals (and their ions) that have been signiﬁcantly observed on hot Jupiters, to ensure we account for all major conductivity contributors.

3. All species that make important contributions to the heating and cooling rates

+ (e.g. atomic line cooling; H3 , CO, and H2O ro-vibrational cooling)

4. All major molecules that incorporate atoms that would otherwise dominate the 71

electrical conductivity.

5. Species that signiﬁcantly absorb ﬂux from the host star at the pressures considered.

6. Other minimal species that can play important roles in the chemical production or loss of the major species in the atmosphere or the major sources for conductivity at the temperatures and pressures considered.

HD 189733b and HD 209458b are the two most significantly observed and studied hot Jupiters and we base our decision on atmospheric metal species included in the model on observations of these exoplanets. The first detection of an exoplanet atmosphere was of the Na doublet observed in HD 209458b using the Hubble Space Telescope (HST) (Charbonneau et al. 2002). The atmosphere of HD 189733b was first significantly detected by HST observations of the Lyα transition (Lecavelier Des

Etangs et al. 2010). Hα,Hβ, and Hγ lines as well as absorption lines from NaI and potentially MgI were later observed in the atmosphere of HD189733b (Redﬁeld et al. 2008; Jensen et al. 2011, 2012; Cauley et al. 2015, 2016; Wyttenbach et al. 2015; Kha- laﬁnejad et al. 2017). Further observations of HD 189733b with HST have suggested the presence of OI, NaI, and KI (Jensen et al. 2011; Pont et al. 2013; Huitson et al.

2012; Ben-Jaﬀel & Ballester 2013). There are also more recent observations of HeI in HD 189733b’s atmosphere (Salz et al. 2018). Multiple molecules have also been

detected in the atmosphere of HD 189733b including H2O(Birkby et al. 2013), CO,

CO2, and a minimal detection of CH4 (Swain et al. 2008), the latter three suggesting the presence of C in the atmosphere. As for HD 209458b, there have been observations of all the atoms and molecules

listed above in addition to the presence of CI and OI in its extended upper atmosphere

(Vidal-Madjar et al. 2004), more signiﬁcant MgI detections (Vidal-Madjar et al. 2013) 72

and recent FeII observations (Cubillos et al. 2020). Combined, these observations of HD 189733b and HD 209458b deﬁne the major metal species we include in our model atmosphere. We assume solar abundance (Asplund et al. 2009) for all elements in the atmosphere. This choice is consistent with similar studies of exoplanet atmospheres. The choice of abundances is important in determining the structure throughout.

Stellar Spectrum and Photoionization Rates

The host star of HD 189733b is a K2V star located at a distance of about 20 parsecs from the Earth. We apply the stellar spectrum of another K2V star, Eridani to our model. The Eri star system, located only 3.2 pc from the Earth, is the third closest star system visible to the naked eye and therefore has much more precise observations of its stellar spectrum. The stellar spectrum of Eri we employ is given by the MUSCLES treasury survey 2.1 (France et al. 2016, Loyd et al. 2016, Youngblood et al. 2016). The MUSCLES team reports their spectral information from Epsilon Eridani from about 4 to 55000 Angstroms (3100 - 0.2 eV) in 54996 bins

with each bin of size 1 Angstrom. We normalize the energy flux (Fλ) observed from Eri to a distance of the semimajor axis of HD 189733b (a = 0.031 AU), as shown in Figure 3.1. A dominant reaction for many species in the model includes photo-absorbing processes. These reactions all absorb incoming stellar flux that can work to create electrons, heat the atmosphere, and significantly change the ionization level throughout. We consider photoionization (A + hν → A+ + e) of all atomic species as well as some dissociative ionization (AB + hν → A+ + B + e) and photodissociation (AB + hν → A + B) for some molecules. These reactions are given in Table 3.5 and 73

105

104

103 ) 1 A 2 102 m c 1 1 s 10 g r e ( 0

F 10

10 1

10 2

101 102 103 104 Wavelength (Angstrom)

Fig. 3.1.— Flux density of stellar spectrum vs wavelength.

discussed in more details in Section 3.3.4.

The photoionization or photodissociation rate of species i is denoted Γi and has

−1 units of s . The general equation for Γi is

Z λth λF λ −τi,λ Γi = σi(λ) e dλ (3.4) 0 hc

Here, σi, is the cross-section for photoionization or photodissociation of species i,

Fλ is the flux density, and τ is the optical depth integrated from the top of the model to that layer in the atmosphere, all evaluated at wavelength λ. Additionally, λth is the threshold wavelength for ionization or dissociation to occur for that species. The Eri spectrum is provided in 1 Angstrom wavelength bins and so in practice we turn these integrals to compute Γi into sums over the photon number flux in each wavelength bin. The photon number flux, Φi, is simply Φi = λFλ,i/(hc) and our Γi is calculated for each species and photon reaction as: 74

X −τi,λ Γi = Φiσi(λi)e ∆λ (3.5)

λi

where λi is the center of the wavelength bin of the spectrum and ∆λ is the size of the wavelength bin (here: ∆λ = 1 A˚). The optical depth for photo-ionizing and dissociating is calculated as

X τi,λ = σs(λ)Ns(λ) (3.6) s where the sum takes place over all species, s, that can absorb at wavelength λ.

The column of each species, Ns is calculated as outlined in Section 3.3.3 and Equation 3.10. We list the photoionization, dissociative ionization, and photodissociation rates (see Equation 3.5) at τ = 0 (optically thin case) in the chemical reaction tables (Tables 3.3-3.7).

The photoionization rates for H, H2, and He are slightly more complex than the metals as electrons ejected from these photo-ionizing reactions can create ‘secondary ionizations’ in all species. The metals can also undergo secondary ionizations, but their density is small enough that the eﬀect in negligible and not considered in our model. Photoelectrons that are generated in the atmosphere at energies much higher

than the threshold energies of H (13.6 eV), H2 (18.6 eV), and He (24.6 eV) can create further ionizations through collisions before sharing their energy to the atmosphere through photoelectric heating. If the ratio of electrons to the neutral component of

the atmosphere (xe ≡ ne/[nH + nH2 + nHe]) is small, this eﬀect can become non-

negligible. This eﬀect will therefore increase the total photoionization rate of H, H2, and He, while simultaneously decreasing the photo-electron heating eﬃciency through photoelectric heating as discussed in the ‘photoelectric heating’ subsection below. 75

Yelle(2004) and Murray-Clay et al.(2009) considered this effect by applying a constant efficiency while Koskinen et al.(2013) applied an ionization fraction independent term for the secondary ionization. Instead, we follow the similar method of Huang et al.(2017) in applying an efficiency dependent on local ionization fraction, xe, and photoelectron energy (E = h(ν − νth)). Equation 13.6 in Chapter 13.3 of

Draine(2011) gives us this secondary ionization eﬃciency factor, here denoted ke,H for atomic hydrogen in terms of secondary ionization per ionizing photon. Given an incoming photon frequency ν and ionization energy E the equation applied is

E − 15 eV 1 − xe/1.2 ke(E) = 0.8 (3.7) 35 eV 1 + 18xe / ln(E/35 eV)

As outlined in Draine(2011), in Equation 3.7 we assume that E > 50 eV and xe < 1.2. If E > 50, we assume that the secondary ionization contribution is zero. The photoionization rates for (e.g.) H can then be written more precisely as:

Z ∞ 1st 2nd Fν −τ ΓH = ΓH + ΓH = σH (ν)(1 + ke) e dν (3.8) νth hν

Here, Γ1st and Γ2nd stand for the photoionization rate (Equation 3.4) and the secondary electron rate respectively. The full photoionization rate for H, H2, and He is calculated in our model by applying Equations 3.8 and 3.5. For H2 and He we assumed that this effect is the same as for H discussed above and given by Dalgarno et al.(1999), with the energy of the photoelectrons applied to the proper ionization threshold energies for H2 and He. We discuss the impact that the secondary ionization effect has on photoelectric heating rate in Section 3.3.6. 76 3.3.3 Differential Equations

In addition to solving the set of rate equations at each level of our atmosphere model, we also integrate the hydrostatic balance and equation for the column of each species inwards from the top of the atmosphere. The equation for hydrostatic balance is

dP ρGM = − p , (3.9) dr r2 and the columns are determined by the solving the diﬀerential equation

dN i = −n (3.10) dr i

These equations are solved using a fourth order Runge-Kutta integrator (Press et al. 2007) based on the radius step of 1% of the scale height of the previous level in the model. Here, the subscript i designates the species and we calculate the columns for all species in the model. We assume an ideal gas law for our equation of state:

X P = nikBT (3.11) i

−3 where ni number density (in units of cm ), kB is the Boltzmann constant, and T is the temperature. The mass density applied in Equation 3.9 is

X ρ = mini (3.12) i where mi is the mass of species i. We integrate these diﬀerential equations inwards given a temperature and mass 77

density found by solving the system of rate equation with starting values at the top

−11 −2 of the atmosphere of Ptop = 5 × 10 bar and Ni = 0 cm . As discussed in Section 3.3.2 we integrate our model until the base values for P and R are met.

3.3.4 Balance of Production and Loss Processes

At each level in the atmosphere, the pressure and column of each species is set by the boundary conditions discussed in 3.3.2 and by Equations 3.9 and 3.10. At each level we then determine the temperature by the thermal balance equation discussed in Section 3.3.6 and update the particle densities as we integrate the model to deeper pressures. The particle densities are determined by equating the production and loss of each species for the reactions considered in Tables 3.3–3.7 and by a reservoir equation that requires solar abundance of all elements. Processes considered include photoionization, photodissociation, radiative recombination, charge exchange, and collisional reactions between species and the references for each reaction rate is listed in Tables 3.3–3.7. The speciﬁc reactions for each species are discussed in their sub- sections below. The photoionization rates of all non-molecular species are from Verner & Ferland

(1996) and Verner et al.(1996). The photoionization and photodissociation of H 2 and H2O are from Schunk & Nagy(2004). The rates of Osterbrock & Ferland(2006) and Le Teuﬀ et al.(2000) are used to describe the radiative recombination rates of H and H2, respectively, with Pequignot et al.(1991) for that of C and O, Shull & van Steenberg(1982) for that of Mg, Si, S, Ca, and Fe, and Verner & Ferland(1996) and Landini & Fossi(1991) for that of Na and K, respectively. To better understand the production/loss balance equations, consider a toy model 78

of species A, B, C, and D. Given the following reactions:

(RA1) A + B → C + D

(RA2) C + D → A + B the production/loss balance equation for species A would be:

kA2nCnD = kA1nAnB (3.13)

where k is the rate coeﬃcient for reactions RA1 and RA2 to occur – often dependent on temperature. Here, RA1 is a destruction reaction for A and B and RA2 is a formation reaction for species A and B. This general framework is followed for each species in the model to set their own individual production/loss balance equation that is solved with the system of equations for species density. In actuality the production/loss balance equations are complex sums of all formation and destruction mechanisms for each species and reaction considered in Tables 3.3–3.7.

Hydrogen Chemistry

Atomic and molecular hydrogen have the two most complex production/loss balances of all the species in the model. The reactions, rate coeﬃcients, and references for these species are listed in Tables 3.3, 3.4, and 3.7. Our list of reactions are pulled primarily from Yelle(2004) and Garc´ıaMu˜noz(2007).

+ + For the molecular ions H2 and H3 we follow the chemical framework of Yelle (2004) and Garc´ıaMu˜noz(2007) and all relevant reactions are in Table 3.3. For a

+ purely H atmosphere, H3 was shown to potentially be a signiﬁcant coolant for HD

+ 209458b (Yelle 2004). The formation and destruction of H3 is strongly inﬂuenced by 79

Rate Reaction Rate coefficient Reference + −4 0 H2 + hν → H2 + e 9.49 × 10 Yan et al.(1998) 1 → H+ + H + e 4.08 × 10−5 Yan et al.(1998) 2 H + hν → H+ + e 1.62 × 10−3 Verner & Ferland(1996) + −8 0.4 3 H2 + e → H + H 2.3 × 10 (300/T ) Le Teuff et al.(2000) 4 H+ + e → H + hν 2.6 × 10−13(104/T )0.8 Osterbrock & Ferland(2006) + −7 −0.30 5 H3 + e → H2 + H 1.38 × 10 T Le Teuff et al.(2000) 6 → H + H + H 4.15 × 10−7T −0.30 Le Teuff et al.(2000) + + −9 7 H2 + H2 → H3 + H 2.08 × 10 Le Teuff et al.(2000) + + −10 8 H2 + H → H + H2 6.40 × 10 Le Teuff et al.(2000) + + −9 −21240/T 9 H + H2 → H2 + H 2.00 × 10 e Le Teuff et al.(2000) + + −9 10 H3 + H → H2 + H2 2.00 × 10 Yelle(2004) - estimated −5 −1.4 −52530/T 11a H2 + H → H + H + H 7.60 × 10 T e Tsang & Hampson(1986) −5 −1.4 −52530/T 11b H2 + H2 → H + H + H2 7.60 × 10 T e Tsang & Hampson(1986) + + −29 12 H + H2 + H2, H → H3 + H2, H 3.20 × 10 Miller et al. (1968) 13a H + H + H → H2 + H kH11aφsaha(T ) See caption 13b H + H + H2 → H2 + H2 kH11aφsaha(T ) See caption

Table 3.3: Hydrogen Chemistry. In the text, these reactions are denoted as RH#. Photoionization and photodissociation rates are optically thin values in units of s−1. They are calculated with the fits for the cross sections given as the reference in the table and the ionizing flux from the MUSCLES spectrum. Three body reaction rate coefficients are in units of cm6 s−1 and two body reaction rate coefficients are in units of cm3 s−1. Rate 13a/13b is calculated using 11a/11b, the Saha equation, and the law of mass action.

+ H2 (which is a product of H2 photoionization) and so both ions must be considered to properly account for their combined chemistry.

He and the Metals

The production and loss of all atoms besides H is primarily set by the balance of photo ionization and recombination in the atomic region of the atmosphere with the one exception being that the balance of O is primarily set by charge exchange (RCO37 and RCO71 given in Table 3.4). Each atomic species additionally has production and loss reactions that occur with major molecules that the atom can be formed into. The full list of atomic and molecular reactions considered is given in Tables 3.5–3.6. For an example of how the abundance of each species is applied, consider the C reservoir equation. Using the abundance of C in the Sun (ξC ) given by Asplund et al. 80

(2009) the C reservoir equation would become:

ξCnH,tot = nCI + nCII + nCIII + nCO (3.14)

where nH,tot is the total density of all hydrogen atoms in the atmosphere. A reservoir equation is given for all elements in the model where atoms, ions, and molecules are considered for each element. We then combine the production/loss balance equations for all species with the reservoir equations ensuring solar abundance to solve our system of equations.

Molecules

We consider major molecules for all the ionized atoms in the model that can contribute significantly towards the electrical conductivity. Additionally, we consider molecular species that can significantly absorb ionizing flux for certain atoms. The reactions and rate coefficients for the formation and destruction of the molecules considered is given in Tables 3.4, 3.7, and 3.6. The production/loss balance equations are constructed similar to the other species in the model and each molecule is also considered in their relevant reservoir equations.

3.3.5 Charge Neutrality

The charge balance equation ensures that there is local charge neutrality. This is set by balancing the density of the electrons with the density of all other charged species. We consider singly ionized atoms of all species and second ionized atoms of ionized species that can signiﬁcantly absorb the stellar ﬂux, so the charge balance equation becomes: 81

Rate Reaction Rate coefficient Reference 37 H+ + O → O+ + H 1.97 × 10−10T 0.23e−226/T Le Teuff et al.(2000) 71 O+ + H → H+ + O 7.26 × 10−11T 0.36e8.6/T Le Teuff et al.(2000) −9 −52900/T 76 H2O + H2 → H + OH + H2 5.80 × 10 e Tsang & Hampson(1986) −16 1.90 −9265/T 77 H2O + H → H2 + OH 1.03 × 10 T e Tsang & Hampson(1986) −17 2 −1490/T 78 OH + H2 → H2O + H 1.06 × 10 T e Tsang & Hampson(1986) −21 2.8 −1950/T 79 OH + H → H2 + O 8.21 × 10 T e Tsang & Hampson(1986) −16 1.4 200/T 80 OH + OH → O + H2O 3.50 × 10 T e Tsang & Hampson(1986) −20 2.8 −2980/T 81 O + H2 → OH + H 1.80 × 10 T e Tsang & Hampson(1986) −15 1.3 −8600/T 82 O + H2O → OH + OH 7.59 × 10 T e Tsang & Hampson(1986) −26 −2 84 OH + H + H2, H → H2O + H2, H 6.10 × 10 T Tsang & Hampson(1986) −29 −1 85 O + H + H2, H → OH + H2, H 1.30 × 10 T Tsang & Hampson(1986) −9 −50900/T 87 OH + H2, H → O + H + H2, H 6.00 × 10 e Le Teuff et al.(2000) −34 88 C + O + H2, H → CO + H2, H 2.00 × 10 Le Teuff et al.(2000) −4 −3.1 −1.291E5/T 89 CO + H2, H → C + O + H2, H 1.52 × 10 (T/298) e NIST Table 3.4: Carbon and Oxygen Chemistry. In the text, these reactions are denoted as RCO#. NIST refers to the National Institute of Standards and Technology Chemical Kinetics Database.

X np + nHeII + (nMII + 2nMIII ) = ne (3.15) i where ‘M’ denotes all ﬁrst and second ionized atoms considered.

3.3.6 Thermal Balance

We solve for the temperature of the atmosphere by balancing heating and cooling rates at each level until 10−5 bar, and then transitioning to a prescribed temperature- pressure proﬁle given by Huitson et al.(2012) for deeper regions in the atmosphere. The reasoning for this transition is discussed in more detail below. For the thermal balance equation, we consider heating from photoelectric heating and exothermic chemical reactions. We consider cooling from metal lines, radiative recombination, free-free cooling, endothermic chemical reactions, and molecular vibrational and rotational line cooling. We discuss each of these terms that go into the thermal balance equation below. 82

Rate Reaction Rate coefficient Reference 0 He + hν → He+ + e 4.41 × 10−4 See caption 1 He+ + e → He + hν Verner fits Verner & Ferland(1996) 2 C + hν → C+ + e 7.55 × 10−3 See caption + −13 −0.6192 1.2910 3 C + e → C + hν 5.068 × 10 T4 /(1 − 0.0815T4 ) Pequignot et al.(1991) 4 O + hν → O+ + e 3.09 × 10−3 See caption + −13 −0.6880 1.7070 5 O + e → O + hν 3.201 × 10 T4 /(1 − 0.0174T4 ) Pequignot et al.(1991) 6 Na + hν → Na+ + e 9.72 × 10−4 See caption 7 Na+ + e → Na + hν Verner fits Verner & Ferland(1996) 8 Mg + hν → Mg+ + e 6.33 × 10−4 See caption + −13 −0.855 9 Mg + e → Mg + hν 1.40 × 10 T4 Shull & van Steenberg(1982) 10 Si + hν → Si+ + e 1.27 × 10−2 See caption + −13 −0.601 11 Si + e → Si + hν 5.90 × 10 T4 Shull & van Steenberg(1982) 12 S + hν → S+ + e 2.15 × 10−2 See caption + −13 −0.630 13 S + e → S + hν 4.10 × 10 T4 Shull & van Steenberg(1982) 14 K + hν → K+ + e 2.13 × 10−3 See caption + −13 −0.8023 15 K + e → K + hν 2.762 × 10 T4 Landini & Fossi(1991) 16 Fe + hν → Fe+ + e 4.09 × 10−3 See caption + −13 −0.891 17 Fe + e → Fe + hν 1.42 × 10 T4 Shull & van Steenberg(1982) 18 O+ + hν → O++ + e 7.12 × 10−4 See caption ++ + −13 −0.6413 19 O + e → O + hν 4.092 × 10 (T4/4) Pequignot et al.(1991) 20 Mg+ + hν → Mg++ + e 6.93 × 10−5 See caption ++ + −13 −0.838 21 Mg + e → Mg + hν 8.80 × 10 T4 Shull & van Steenberg(1982) 22 Si+ + hν → Si++ + e 1.87 × 10−4 See caption ++ + −12 −0.786 23 Si + e → Si + hν 1.00 × 10 T4 Shull & van Steenberg(1982) 24 S+ + hν → S++ + e 2.26 × 10−4 See caption ++ + −12 −0.686 25 S + e → S + hν 1.80 × 10 T4 Shull & van Steenberg(1982) Table 3.5: Metal Chemistry: Photoionization/Recombination. In the text, these re- 4 actions are denoted as RM#. Note: T4 ≡ (T/10 K). Note: The photoionization rates of all the metals are calculated from the cross section fits of Verner & Ferland(1996) and Verner et al.(1996).

Photoelectric Heating

Photoelectric heating of H, He, and metals are the dominant heating source at the top of the atmosphere. Photoelectric heating adds energy to the atmosphere by the interaction of an energetic photoelectron ejected from an atom from photoionization colliding with particles in the atmosphere and transferring its energy. Additionally, dissociative ionization and photodissociation can contribute to heating of the atmosphere through a similar process where the dissociated particles have extra energy after unbinding the molecule. The photoelectric heating rate (in units of erg s−1) is 83

Rate Reaction Rate coeﬃcient Reference + −14 −3.77 30a Na + e + H2 → Na + H2 3.43 × 10 T Lavvas et al.(2014) and Su et al. (2001) + 30b Na + e + H → Na + H k30a Lavvas et al.(2014) and Su et al. (2001) + 31a K + e + H2 → K + H2 Estimated as k30a Lavvas et al.(2014) and Su et al. (2001) + 31b K + e + H → K + H k31a Lavvas et al.(2014) and Su et al. (2001) + 15 1.506 59630/T 32a Na + H2 → Na + e + H2 2.288 × 10 T e Lavvas et al.(2014) and Su et al. (2001) + 32b Na + H → Na + e + H k32a Lavvas et al.(2014) and Su et al. (2001) + 33a K + H2 → K + e + H2 Estimated as k32a Lavvas et al.(2014) and Su et al. (2001) + 33b K + H → K + e + H k33a Lavvas et al.(2014) and Su et al. (2001) Table 3.6: Alkali Metal Chemistry: Three Body Reactions. In the text, these reactions are denoted as RM#.

Rate Reaction Rate coefficient Reference −1 0.636 −29990/T 40 Na + H2 → NaH + H 1.160 × 10 T e Lavvas et al.(2014) −25 −2.21 −41/T 41 H + Na + H, H2 → NaH + H, H2 1.9 × 10 T e Lavvas et al.(2014) −12 0.69 −2360/T 42 NaH + H → Na + H2 2.38 × 10 T e Lavvas et al.(2014) 19 0.912 −20940/T 43 NaH + H, H2 → H + Na + H, H2 4.873 × 10 T e Lavvas et al.(2014) 56 OH + Si → SiO + H 1.00 × 10−10 Le Teuff et al.(2000) 60 SiO + C+ → Si+ + CO 5.40 × 10−10(T/300)−0.50 Le Teuff et al.(2000)

Table 3.7: Additional Molecule Chemistry. In the text, these reactions are denoted as RM#.

Z ∞ Fν −τ Qi = Epeσi(ν) e dν (3.16) νth hν

Here Epe = Ephoton − Eth stands for the energy of the photoelectron. This photoelectron energy is the photon energy minus the binding energy. As with the photoionization rate, we can turn this integral into a discrete sum.

X −τi,λ Qi = (E − Eth)σi(λi)Φie ∆λ (3.17)

λi

As shown with the photoionization of H, H2, and He in Section 3.3.2, secondary ionization eﬀects play a factor in the photoelectric heating rate. We can re-write the heat deposited by a single photoelectron in the case of hydrogen to:

Z ∞ Fν −τ QH = ησi(ν) e dν (3.18) νth hν 84

where η is the factor to account for secondary ionizations (Dalgarno et al. 1999):

η(E) = E − (13.6 eV + 10.2 eV)ke (3.19)

This same equation is then applied for H2 and He, but substituting in the proper threshold energy for each reaction.

Chemical Heating/Cooling

Chemical reactions can also contribute to the heating or cooling of the atmosphere. Following a similar approach to Garc´ıa Mu˜noz(2007), this energy change of the atmosphere can be calculated as the rate of the reaction, multiplied by the energy change of the reaction and expressed as Qch = fch∆Ech.

For a two body chemical reaction of species A and B, with rate coeﬃcient kr, the rate of the reaction is just fch = krnAnB. Similarly, for a three body reaction with an additional species C, the rate for the reaction to occur is just fch = krnAnBnC .

The heat of reaction, ∆Ech, is positive for endothermic reactions and negative for exothermic reactions. We use the heats of formation listed in Le Teuﬀ et al.

(2000) for our model, and they are listed in Table 3.8. As an example, Qch for rate

3 RH13a (H + H + H → H2 + H) is calculated as Qch = kH13anH(3 × 216.0 − [0.0 × 216.0])(kJ mol−1). Note that the heat of formation for H is 216.0 kJ mol−1 and for

−1 H2 is 0.0 kJ mol . Chemical heating from all non photoionization/photodissociation reactions is calculated using this method and included in the thermal equilibrium equation. 85

p 1528.0 e 0.0 H 216.0 H2 0.0 + + H2 1488.3 H3 1107.0 HeI 0.0 HeII 2372.0 CI 711.2 CII 1797.6 OI 246.8 OII 1560.7 NaI 107.6 NaII 603.4 MgI 146.5 MgII 884.2 SiI 446.0 SiII 1233.0 SI 274.7 SII 1272.0 KI – KII – FeI 414.0 FeII 1173.0 + OH 38.4 CO -113.8 H2O -238.9 H2O 977.9 NaH -57.0 SH 136.5 H2S -17.6 SiO -101.6

Table 3.8: Relevant species and heats of formation (in kJ mol−1) as listed in Le Teuﬀ et al.(2000).

Atomic Line Cooling

The main source of cooling of the atmosphere above the molecular dominated region is due to radiative line cooling from H and metal species. When an atom is impacted by another particle, it can be raised to an excited state and emit radiation through de-excitation back into the ground state. If this radiation can escape the atmosphere before being absorbed, this process will work to cool the atmosphere. Following the assumptions of Huang et al.(2017), we assume throughout our entire atmosphere model that our atmosphere is optically thin to this metal line radiation and can freely escape the atmosphere. This assumption is likely valid in the upper atmosphere but will fail when molecules begin to dominate the density structure. At this point however, molecular cooling (as discussed in Section 3.3.6) will begin to dominate the thermal balance equation and so metal line cooling will no longer be important. We consider metal line cooling transitions following the method of Huang et al. (2017) and striking a balance between low excitation energies, ∆E, and large radiative decay rate Aul for abundant atoms in the model. The metals in the atmosphere were chosen in our model as discussed in Section 3.3.2. The major cooling processes are take from Table 4 and Table 5 of Huang et al.(2017). We list the metal cooling lines used in our model in Tables 3.9 and 3.10. 86

−3 3 −1 Transition ∆E (ev) Aul (s) ncrit (cm ) Λ (erg cm s ) Source 8 15 −19 0.18 −5.04/T4 MgI 2852 4.350 4.91 × 10 6.7 × 10 3.4 × 10 T e van Regemorter(1962) 9 −16 −3.15/T4 −8 MgI 4571 2.712 254 8.5 × 10 1.0 × 10 e /(254 + 3.0 × 10 ne) Osorio et al.(2015) 8 14 −12 (e) MgII 2803 4.422 2.57 × 10 8.6 × 10 7.1 × 10 Clu CHIANTI 8 14 −12 (e) MgII 2796 4.434 2.60 × 10 2.2 × 10 7.1 × 10 Clu CHIANTI 7 14 −12 (e) NaI 5890/5896 2.104 6.16 × 10 4.4 × 10 3.4 × 10 Clu Igenbergs et al.(2008) 7 14 −19 0.18 −1.87/T4 KI 7665/7699 1.615 3.78 × 10 1.8 × 10 3.7 × 10 T e van Regemorter(1962)

Table 3.9: Major Metal Cooling Lines. Notes: Clu refers to the collision frequency given by the source and CHIANTI refers to the Atomic Database for Spectroscopic Diagnostics of Astrophysical Plasmas which can be found at: https://www.chiantidatabase.org

As discussed in Huang et al.(2017), the metal line cooling rates require a model for the excited-state densities. We highlight the important parameters here, but see Huang et al.(2017) for an in depth discussion. For a two-level system, rate equilibrium between upward and downward rates can be written

(e) (e) nl(neClu + BluJlu) = nu(Aul + neCul ) (3.20)

Here, Aul designates the spontaneous decay rate, Blu is the Einstein coeﬃcient for the upward transitions, nlu is the density for the lower and upper electron states respectively, and C is the electron impact rate from the lower to upper state (lu) or upper to lower state (ul). Collisional excitation is a sink of thermal translation energy, while collisional de-excitation is a source. Thus, the cooling rate of this two- level system is

! (e) (e) (e) Aul − BluJlu(nl/nu)eq ∆Ene(Clu nl − Cul nu) = nenl∆EClu (e) ≡ Λnenl (3.21) Aul + neCul

where (nl/nu)eq = gl/gu exp(∆E/(kBT )) and Λ(T ) is the cooling function.

Alternatively, Huang et al.(2017) states that for forbidden transitions, ne is much larger than the critical density ncrit above which collisional de-excitation dominates 87

3 −1 Transition ∆E (ev) Eu (eV) Aul (s) Λ (erg cm s ) Source −3 −14 −2.28/T4 OI 6300 1.957 1.967 7.45 × 10 1.3 × 10 /nee Equation 3.22 −4 −16 −1.47/T4 CI 9850 1.260 1.264 1.45 × 10 3.3 × 10 /nee Equation 3.22 −21 −3.12/T4 −8 CI 8727 1.420 2.684 0.599 3.6 × 10 e /(0.599 + 2.39 × 10 ne) Pequignot & Aldrovandi(1976) −3 −15 −0.91/T4 SiI 16454 0.762 0.781 2.72 × 10 1.8 × 10 /nee Equation 3.22 −13 −2.22/T4 SiI 10991 1.128 1.909 1.00 2.0 × 10 /nee Equation 3.22 −2 −15 −2.22/T4 SiI 6527 1.899 1.909 2.74 × 10 9.3 × 10 /nee Equation 3.22

4 Table 3.10: Minor Metal Cooling Lines. Note: T4 ≡ 10 K. radiative de-excitation. In this limit, we can approximate the level population with the Boltzmann distribution, and the cooling rate is simply

nu Λnenl = ∆EnlAul (3.22) nl eq

which we note is independent of ne and collision rates.

Recombination and free-free Cooling

Recombination of H, H2, and the metals acts to cool the gas by radiating the binding energy and the free electron’s kinetic energy when the free electron recombines with an ion. As shown in Chapter 3.3 of Osterbrock & Ferland(2006), the energy radiated away for metal species ‘m’ in this process can be expressed as

X ion Qrr = kBT αmnenm (3.23) m where α is the recombination rate of the metal as listed in Table 3.5 The radiative recombination cooling from H is slightly diﬀerent than the metals in that the mean kinetic energy can play a more signiﬁcant factor and must be considered in the H radiative recombination cooling equation. As discussed in Huang et al.(2017) the radiative recombination cooling for H is QH,rr = hErriαBnenp. Here, hErri is the 88 mean kinetic energy of recombining electrons given in Draine(2011) as

hErri = [0.684 − 0.0416 ln(T4)]kBT (3.24)

4 where T4 = T/10 K. Free-free cooling is typically only important in a pure H gas and cools the gas by emitting a continuous spectrum due to the acceleration of an electron by an ion. We calculate free-free cooling using Chapter 3.4 of Osterbrock & Ferland(2006) and assuming a Gaunt factor of 1.3 as

−27 1/2 −3 −1 Qﬀ = 1.85 × 10 T nenp (erg cm s ) (3.25)

The cooling due to radiative recombination (Qrr) and free-free cooling (Qﬀ ) is never dominant throughout the atmosphere but is included for consistency and for direct comparison to Huang et al.(2017).

Radiative Cooling by Molecules

Molecules can play an important role in cooling the atmosphere by releasing energy through rotational and/or vibrational transitions. In our atmosphere, this cooling is

+ most signiﬁcantly seen in H3 ,H2O, and CO. Yelle(2004) has shown that for a purely H atmosphere model of HD 209458b,

+ H3 can be a dominant coolant in the regions where H2 begins to dominate H. We

+ consider the ro-vibrational cooling of H3 by using ﬁts derived in Miller et al.(2013).

The cooling function, Λ + , is given by their Equation 7 as H3

! 7 X n −1 Λ + = 4π × 10 exp CnT [erg s ] (3.26) H3 n 89

+ where Cn are their ﬁts to the H3 cooling function expressed in Table 5 of Miller et al.

(2013). We use ﬁts in the 1800-5000 K region of Cn = {−55.7672, 0.0162530, −7.68583× 10−6, 1.98412 × 10−9, −2.68044 × 10−13, 1.47026 × 10−18}. The contribution to the

thermal balance equation can then be found by multiplying Λ + by n + . H3 H3

We implement rotation and vibrational cooling due to CO and H2O as done in Neufeld & Kaufman(1993). Neufeld & Kaufman(1993) derive ro-vibrational cooling

functions for CO and H2O as a function of column that the radiation must travel through. We make the assumption though, that at the pressures where we are solving the thermal balance equation, radiation is still able to freely escape the atmosphere. As pressures become greater, this assumption will no longer hold but by that point we will have transferred to solving for the temperature using the pressure-temperature proﬁle adapted from Huitson et al.(2012). Neufeld & Kaufman(1993) also derive the cooling rates as molecular energy level populations deviate from LTE. For simplicity, we assume that the molecules are in LTE at the pressure and temperature where cooling can be signiﬁcant and therefore take the LTE cooling functions from Neufeld &

Kaufman(1993) as our ro-vibrational cooling rates for CO and H 2O. The vibrational cooling functions are given by:

−11 −3080K/T 3 −1 ΛCO,vib = 1.58 × 10 e [erg cm s ] (3.27)

−11 −2325K/T 3 −1 ΛH2O,vib = 1.58 × 10 e [erg cm s ] (3.28)

We then calculate rotational cooling rates from the ﬁts in Neufeld & Kaufman(1993) to be:

−24 T/2941K 3 −1 ΛCO,rot = 7.08 × 10 10 [erg cm s ] (3.29)

−24 T/2631K 3 −1 ΛH2O,rot = 5.50 × 10 10 [erg cm s ] (3.30) 90

We can then prescribe all these cooling rates into our thermal balance equation.

Prescribed Temperature Proﬁle Deep in the Atmosphere

We solve for the temperature using thermal balance of the heating and cooling rates described above until a pressure of 10−5 bar. At this point in the atmosphere, the gas density becomes optically thick enough that the major sources of cooling (metal line cooling and molecular cooling) are unlikely to freely escape the atmosphere as we assume for the upper atmosphere. Our model does not include a detailed solution of the transfer equation or opacities for optical continuum radiation. Therefore, the thermal balance equation is not properly accounting for all sources of heating and cooling when pressures become large enough. To account for this, we merge our temperature structure solved by thermal balance until 10−5 bar to the empirically determined pressure-temperature profile by Huitson et al.(2012) for deeper pressure of HD 189733b. Huitson et al.(2012) used transmission spectra of the Na I doublet in HD 189733b taken with the Space Telescope Imaging Spectrograph (STIS) aboard the Hubble Space Telescope to constrain the temperature at different pressure regions. The full temperature profile, including the contribution at greater pressure from Huitson et al.(2012) is shown in Figure 3.3.

3.4 Results

3.4.1 Density and Temperature Proﬁles

In Figure 3.2, we show a plot of the densities of all the species versus pressure. This plot is a result of the solution to the rate equations and a function of the temperature-pressure and radius-pressure proﬁles which are both shown in Figure 91

3.3. We note that at the top of the atmosphere, the density is dominated by protons and electrons. Eventually neutral hydrogen (H) begins to dominate at P ∼ 10−8 bar and it remains dominant until P ∼ 10−6 bar when the temperature becomes small enough for molecular hydrogen to dominate the pressure. The transition of the atmosphere from ionized-atomic-molecular is also shown more succinctly in Figure 3.4. Besides protons, the major ion in the ionized and atomic regions of the atmosphere is He. As the atmosphere becomes more molecule dominated, ionized magnesium and sulfur become the dominant ions and set the electron density. Eventually the alkali ions begin to take over as the dominant ions (P > 10−4 bar) due to their higher photoionization rate at deeper pressures. At the location where molecules begin to dominate, atomic densities rapidly decrease as these elements are incorporated into molecules. There is a very abrupt reduction in carbon atoms when CO becomes large enough to dominate the reservoir equation of carbon. As H2O becomes larger, it eventually starts to dominate the reservoir equation of oxygen, and contains the majority of the oxygen. Other molecules, such as SiO, dominate the abundances of Si and therefore the atomic species decreases. The main atoms that remain at higher pressure are atomic Na and K, which can then set the electron and ion densities at the deepest region. This shows the important feature that when a molecule baring one of the major atomic ions becomes large enough, it will begin to dominate the abundance equation of that atom and the neutral and ionized atomic densities will decrease quickly. In Figure 3.5 we show the dominant rates towards setting the thermal balance as a function of pressure in the pressure range where the thermal balance equation is being solved (P < 10 µbar). At the top of the atmosphere, the temperature is 92

e Si p S H K 10 9 H2 Fe He OH C H2O O CO Na SiO 10 7 Mg

10 5 Pressure (bar)

10 3

102 104 106 108 1010 1012 1014 1016 Density (cm 3)

Fig. 3.2.— Number density versus pressure. The top of this plot is the top of the atmosphere. Except when noted by the legend, solid lines represent neutral species and dashed lines represent ionized species. All other species are speciﬁcally represented in the legend with electrons denoted by ‘e’ and protons denoted by ‘p.’ 93

9 10 10 9

7 10 10 7

5 10 10 5 Pressure (bar) Pressure (bar)

10 3 10 3

10 10 10 10 2000 4000 6000 8000 10000 12000 14000 16000 1010 2 × 10 3 × 10 4 × 10 6 × 10 (a) Temperature (K) (b) Radius (cm)

Fig. 3.3.— (a) Temperature vs pressure as calculated by the thermal balance equation until 10 µbar below which point the temperature is prescribed to be that from Huitson et al.(2012). (b) Radius vs pressure as calculated by hydrostatic balance (Equation 3.9).

atoms ions 9 10 molecules

10 7

10 5 Pressure (bar)

10 3

10 4 10 3 10 2 10 1 100 Density Fraction

Fig. 3.4.— The density fraction of ions, neutral atoms, and molecules as a function of pressure. The density fraction is calculated by summing the densities of each particle of the given species and then dividing that value by the total number density of all species in the atmosphere. The transition from ionized to atomic to molecular becomes more clear in this representation than compared to Figure 3.2. 94

4 + 10 H Mg H3 QH QHe Recomb K H2O QMetals Chemical Free-free Na CO QH2 ) 1 10 5 s

3 m c 6

g 10 r e (

e t a 7 R

10 g n i t a e 8 H

/ 10 g n i l o o

C 10 9

10 10 10 9 10 8 10 7 10 6 10 5 Pressure (bar)

Fig. 3.5.— The major heating and cooling rates in the atmosphere plotted against pressure until the prescribed temperature profile from Huitson et al.(2012) takes over at 10−5 bar. All cooling rates are expressed as dashed or dotted lines and all heating rates are expressed as solid lines. The dashed lines represent a sum of the line cooling for the specified species. Rates that are included in the model but do not significantly contribute to setting thermal balance are not plotted. A balance of the heating and cooling rates sets the temperature profile in Figure 3.3. 95 set by a balance of Hydrogen photoelectric heating and radiative line cooling from neutral hydrogen. Photoelectric heating of H and He continues to dominate at the top of the atmosphere until P ∼ 10−6 bar when photoelectric heating from neutral metals dominates the heating. The dominant cooling from pressures of 10−8 − 10−5 bar is set by Na and Mg line cooling. Specifically, cooling from the Na D doublet and the 2852 Angstrom line from Mg is the dominant coolant. Eventually, molecular cooling begins to sharply increase at the region where we transition to determining the temperature structure based on the temperature-pressure profile as reported in Huitson et al.(2012). This is expected as rotational and vibrational cooling should be a major coolant at lower temperatures.

3.4.2 Conductivity Proﬁle

The primary focus of this chapter is in determining the conductivity structure throughout the atmosphere with application to Joule heating. For the Pederson and Hall conductivity this is dependent on magnetic field strength as discussed in Chapter 2.6. We choose a fiducial magnetic field strength of B = 10 G. Figure 3.6 shows the total Pederson, Hall and parallel conductivities throughout the atmosphere as a function of pressure. Figures 3.7 and 3.8 then show the individual ion or electron contributions to the Pederson and Hall conductivities versus pressure for the same field. Examination of the conductivity in Figures 3.6-3.8 shows that there are distinct peaks in the conductivity throughout the atmosphere as discussed in Chapter 2.6. Peaks in the conductivity can occur when there are peaks in the density structure of a conductivity setting species or when the cyclotron frequency (Ωc) is equal to the major collision frequency (ν). There are three major peaks in the Pederson and Hall conductivities of Figure 3.6: 96

Conductivity (S m 1) 10 4 10 3 10 2 10 1 100

HD 189733b B = 10.0G 10 9

10 7

10 5 Pressure (bar)

10 3

Pederson Hall Parallel

105 106 107 108 109 1010 Conductivity (s 1)

Fig. 3.6.— Pederson, Hall, and parallel conductivities as a function of pressure. 97

Pederson Conductivity (S m 1) 10 7 10 5 10 3 10 1

B = 10.0G

10 9

7 10 Total NaII

OII KII

CII HeII

MgII FeII Si p 10 5 II

Pressure (bar) SII e

10 3

101 103 105 107 109 Pederson Conductivity (s 1)

Fig. 3.7.— Individual contributions to the Pederson conductivity for electrons, protons, and ions. Each line represents the Pederson conductivity solely from the species denoted in the legend. The solid black line is a sum of all individual contributions.

Hall Conductivity (S m 1) 10 7 10 5 10 3 10 1

B = 10.0G

10 9

7 10 Total NaII

OII KII

CII HeII

MgII FeII Si p 10 5 II

Pressure (bar) SII e

10 3

101 103 105 107 109 Hall Conductivity (s 1)

Fig. 3.8.— Individual contributions to the Hall conductivity for electrons, protons, and ions. Each line represents the Hall conductivity solely from the species denoted in the legend. Note that because the Hall conductivity is dependent on the charge of the species, the absolute value of the electron contribution is represented by the dotted line. The black line is a sum of all individual contributions. 98

(1) at P ≈ 10−8 bar, (2) at P ≈ 10−5 bar and (3) at P ≈ 10−3 bar for Pederson and P ≈ 10−4 bar for Hall. Comparison of Figure 3.6 to Figure 3.2 highlights that peak 1 occurs at a region where the proton and electron densities reach a peak in the atmosphere. Peak 2 occurs near the atomic-molecular transition, but more importantly this location also corresponds to peaks in the density of many of the ions which significantly influence the Pederson conductivity. Coincidentally, this peak also occurs at the location where the cyclotron frequency equals the ion collision frequencies as estimated by Equation 2.59. Finally, peak 3 corresponds to the Lorentzian peak for electrons. With the density structure now set and all major peaks in Pederson conductivity contained within the atmosphere model, we can calculate the conductance and Joule heating for different sized magnetic fields and stellar wind parameters.

3.4.3 Magnetic Field Eﬀect on Conductivity

Figure 3.9 shows the dependence of conductivity on magnetic field. There are two major effects as magnetic field strength changes: (1) The magnitude of the peak in the Pederson conductivity decreases for larger magnetic field and (2) the location of the Lorentzian peak (Ωc = ν) occurs at higher pressures for higher magnetic field. The cyclotron frequency increases for larger magnetic fields and since the collision rate increases as pressure increases, the location of the conductivity peak in the atmosphere also occurs at higher pressures as B increases. The implications of this are discussed in more details in Chapter 3.5. Figure 3.10 shows a running height integral of the Pederson conductivities in Fig- ure 3.9 over the same magnetic fields strengths. The number that each of these height integrals approach is equal to the conductance over the entire region as will be calculated by Equation 2.36. This plot shows the important result that as magnetic 99

Conductivity (S m 1) 10 3 10 2 10 1 100 101

B = 0.1 B = 10.0 B = 1.0 B = 100.0 10 9

10 7

10 5 Pressure (bar)

10 3

107 108 109 1010 1011 Conductivity (s 1)

Fig. 3.9.— Pederson conductivity versus pressure as a function of magnetic ﬁeld strength, B, given in units of Gauss.

Running Conductivity Height Integral (mho) 105 106 107 108 109 1010

B = 0.1 B = 10.0 B = 1.0 B = 100.0 10 9

10 7

10 5 Pressure (bar)

10 3

1015 1016 1017 1018 1019 1020 Running Conductivity Height Integral (cm s 1)

Fig. 3.10.— A running height integral of the Pederson conductivities in Figure 3.9 versus pressure. The ﬁnal number that each line approaches at the highest pressures can be thought of as the conductance in the atmosphere for the given B ﬁeld. 100

field size increases, the conductance decreases. The Pederson conductance decreases approximately as 1/B, which agrees with our analytic estimate for Pederson conductance from Chapter 2.7. Figure 3.10 also shows the importance of modelling each peak in the conductivity structure of the atmosphere. The final result in the conductance is effectively set by the largest peak in the conductivity at each B. Now that we have results for the densities, temperature, radius, and conductivity of the atmosphere, we can calculate the conductance and make estimates for the Joule heating in HD 189733b.

3.4.4 Joule Heating in the Upper Atmosphere of HD 189733b

Cohen et al.(2018) discuss how to calculate the Joule heating of an exoplanet atmosphere, following Kivelson & Ridley(2008). The derivation of the Joule heating calculation is also discussed in detail in Chapter 2.4. We ﬁnd that the Joule heating of an atmosphere is given by:

2 QJH = σP |Et| (3.31)

where σP is the Pederson conductivity and Et is the ‘transmitted’ electric ﬁeld driving the current running from the magnetosphere through the atmosphere. The speciﬁc form of Et is: −1 ΣP |Et| = 2|Esw| −1 −1 (3.32) ΣP + ΣA where ΣA is the ‘Alfven conductance’ at the top of the atmosphere:

2 2√ c c 4πρsw ΣA ≡ = (3.33) 4πvA 4πBsw

and |Esw| = vswBsw/c is the electric ﬁeld of the stellar wind and vA is the Alfven speed. Note that the density of the stellar wind, can be calulated as ρsw = nswmp. 101

−3 −1 nsw (cm ) vsw (cm s ) Bsw (G) 3.8 × 106 2.2 × 107 5.56 × 10−2

Table 3.11: Stellar wind parameters near HD 189733b from Kavanagh et al.(2019).

Therefore in order to calculate the Joule heating QJH we need stellar wind parameters at the location of HD 189733b for Bsw, vsw, and nsw. We can compare the Joule heating directly to the heating and cooling rates in our atmosphere which are shown in Figure 3.5. Our full Joule heating equation is:

2 2 vswBsw 2ΣA −1 −3 QJH = σP [erg s cm ] (3.34) c ΣA + ΣP

Additionally, we ﬁnd that when ΣP ΣA this equation simpliﬁes further to:

2 2 c vswnswmp QJH = σP 2 (3.35) π ΣP which is independent of B, except for the effect that B has on the Pederson conductance. Therefore for a larger conductance, the Joule heating will be smaller. We obtain values for the stellar wind parameters from Kavanagh et al.(2019). They generate a stellar wind model at the location of HD 189733b by incorporating stellar magnetic field maps (Vidotto et al. 2015; Fares et al. 2017) as boundary conditions to the 3D magnetohydrodynamics (MHD) code BATS-R-US (Powell et al. 1999; Tóthet al. 2012) for the host star. BATS-R-US solves the set of ideal MHD equations for conservation of mass, magnetic flux, momentum, and energy. Our adopted stellar wind parameters, as averaged from their Table 3 values are given in Table 3.11. We can now calculate the Joule heating rate in Equation 3.34 based on the Peder- son conductance for different sized magnetic fields as plotted in Figure 3.10. The values for the Pederson conductance and maximum Joule heating rates for HD 189733b 102 for magnetic field sizes of 0.1, 1.0, 10.0, and 100.0 G are given in Table 3.12.

−1 −1 −3 B (G) ΣP (cm s )QJH (erg s cm ) 0.1 1.89 × 1020 7.38 × 10−12 1.0 9.37 × 1018 7.98 × 10−10 10.0 3.11 × 1017 8.00 × 10−8 100.0 4.53 × 1016 9.37 × 10−6

Table 3.12: Values of the Pederson conductance and the maximum Joule heating of the atmosphere of HD 189733b for diﬀerent sized magnetic ﬁelds.

3.5 Discussion

3.5.1 Eﬀect of Molecules on the Conductance

From Figure 3.7 we note that there is a peak in the ion contributions to the Pederson conductivity for a B = 10 G field very near the location where the atmosphere becomes dominated by molecules (P ∼ 10−6 − 10−5 bar). When molecules begin to dominate atoms, neutral atoms will be incorporated into molecules. As the atomic neutral abundance decreases, the atomic ion abundance will also decrease which will result in a decrease in the ion contribution towards the Pederson conductivity. We show this effect in Figure 3.11. That figure presents the Pederson conductivity for a B = 10 G magnetic field for a standard run of our atmosphere model, and a second run where we remove molecules from the atmosphere. The model run without molecules is done to show the significance on the conductivity in properly modelling the transition from the atomic region of the atmosphere to the molecular region of the atmosphere. Had, we considered a model of the upper atmosphere up until the molecular region, the conductivity structure and thus the conductance, would be incorrect and slightly higher than the actual value of the conductance as shown in Figure 3.10. Therefore, 103

Pederson Conductivity (S m 1) 10 7 10 5 10 3 10 1

B = 10.0G

10 9

10 7

10 5 Pressure (bar)

10 3

Total (with molecules) CII (with molecules)

Total (without molecules) CII (without molecules)

101 103 105 107 109 Pederson Conductivity (s 1)

Fig. 3.11.— Pederson conductivity vs. pressure for models with and without molecules. The solid lines show the total Pederson conductivity (black) and CII contribution (blue) to the Pederson conductivity for our standard atmosphere model. The dashed lines show the total Pederson conductivity (black) and CII contribution (blue) to the Pederson conductivity for the same atmosphere model without molecules included. Since carbon does not get tied into CO for the dashed lines model it is able to generate an anomalous peak in the Pederson conductivity at P ≈ 10−6 bar.

when constraining the conductance of hot Jupiter atmospheres, it is important to model the molecular layer for B > 1 G sized fields. For fields smaller than 1 G, the major peak of the conductivity is constrained in the atomic region as seen in Figure 3.9, and so the conductance of an atmosphere modelled up to the molecular layer is more accurate for smaller fields.

3.5.2 Relative Size of the Pederson Conductance

For Earth sized magnetic fields (B ∼ 1 G), we calculate a Pederson conductance of 9.37 × 1018 cm s−1 for HD 189733b. The atmosphere of the Earth, however, has been 104 measured to have a conductance of about 1010 − 1011 cm s−1 (e.g. Kivelson & Ridley 2008). From Equation 3.1 and the subsequent height integral associated with the conductance we see that the source of the ∼ 108 difference between the conductance of HD 189733b and the Earth, despite comparable cyclotron frequencies, is due to (1) the difference in the peak of the electron density, (2) the relative scale heights of both atmospheres, and (3) the relative size of the collision frequency of ions and electrons with neutrals. We estimate the deviations from Earth conductance expected in HD 189733b considering these three points below. Considering point (1) above, in Figure 3.2 we note that the peak of the electron density for HD 189733b is ≈ 109 cm−3. The peak of the electron density in the ionosphere of the Earth, however, is only ≈ 106 cm−3 (see Figure 1.5). Since the Pederson conductivity in Figure 3.7 is primarily dominated by the electron contribution, this accounts for 3 orders of discrepancy between the Pederson conductance of the two planets. The peak of the electron density of HD 189733b is much greater than the Earth due its closer distance to the host star and the intense radiation which allows greater amounts of photoionizaiton in the upper atmosphere resulting in a higher abundance of charged particles. Considering point (2), the relative scale height of the atmospheres come into effect when taking a height integral of the conductivity as shown in Section 2.7. The scale height of the Earth is ≈ 106 cm, whereas the scale height of HD 189733b is ≈ 108 cm. Therefore, the relative scale heights of the two planets account for 2 more orders of discrepancy between the conductance. The scale height of HD 189733b is greater than the Earth’s primarily due to the higher temperatures in its upper atmosphere. The final three order of magnitude discrepancy between the conductance of the Earth and the conductance of HD 189733b comes down to the relative size of ions/electrons 105

with neutral species. At the region where the Earth’s ionosphere is located, the

neutral density is primarily dominated by O and N2 with densities of the neutral

10 −3 nN ∼ 10 cm (Cravens 1997). The ﬁrst peak in the Pederson conductivity for HD

189733b occurs in a region where the neutral density is dominated by H and H2 with

14 −3 a neutral density of nN ∼ 10 cm . From Table 3.1, the electron collisions with neutral species scales approximately linearly with a slight dependence on the polarizability of the neutral species. Therefore, the ratio of collision frequencies between the ionosphere of the Earth and HD 189733b would lead to four more orders of magnitude difference in the expected conductance of the two planets. The exact values for the conductance are also dependent on ions and specific collision frequencies but this estimation highlights the reasons for the deviation between the conductance of the Earth and of HD 189733b. In order to achieve an Earth like conductance in HD 189733b, the size of the magnetic field would need to be ∼ 107 G. Kao et al.(2018) has shown than kilogauss sized magnetic fields can exist in planet like objects as they report a 3 kG sized field

in a brown dwarf with a mass of 13.7 MJup.

3.5.3 Signiﬁcance of Joule Heating

We report maximum Joule heating values for HD 189733b as shown in Table 3.12. For smaller magnetic ﬁelds the conductance is larger and therefore the Joule heating

decreases as shown in Equation 3.35. In Figure 3.12 we compare the values for QJH to the heating rates in Figure 3.5. We see that the Joule heating is much smaller than photoelectric heating or chemical heating at the pressures where the conductivity peaks (P > 10−8 bar) for B < 100 G. This suggests that Joule heating cannot significantly change the temperature structure of HD 189733b on the dayside. For 106 larger magnetic fields (B ≥ 100 G), the Joule heating begins to approach the peak in the heating in the thermal balance equations. Therefore, Joule heating likely does not have a large impact on the thermal balance in HD 189733b for Jupiter sized magnetic fields at the pressures we considered in our model. Cohen et al.(2018) reports that for the Trappist system, the Joule heating rates can be as large as ∼ 1% of the bolometric stellar irradiation of the atmosphere and therefore be important in setting the thermal structure. Compared to HD 189733b, the heating in the upper atmosphere of the Trappist planets is much less than that of HD 189733b for the photoelectric heating of (e.g.) H and He. Therefore, for the Trappist planets, the Joule heating values do not need to be as large to impact the upper regions of the atmosphere. If HD 189733b had comparable stellar irradiation and heating in its upper atmosphere to the Trappist planets, then the Joule heating rates reported here would be large enough to have a significant impact on the thermal structure even if transferred over a length of a scale height. Another important consideration on the impact that Joule heating can have on the atmosphere is the size of the region where Joule heating can occur. Kivelson & Ridley(2008) and Cohen et al.(2018) argue that the location where Joule heating can occur is over the polar cap region where magnetic field lines from the planet are connected with the stellar wind. For the case of the Trappist planets, Cohen et al. (2018) states that the size of the polar cap region can be over the entire planet and therefore Joule heating can occur anywhere in the atmosphere. For the case of HD 189733b, Kavanagh et al.(2019) has created MHD models which suggest that the polar cap for a 10 G planetary magnetic field occurs at a co-latitude of 33◦, which corresponds to a 46% coverage factor over the planet. Therefore, for HD 189733b, the significance of the Joule heating as discussed here, can be most significant in this 107

10 5 ) 1 s 7 3 10 m c

g r e (

e t 10 9 a R

g n i t a

e Total heat H 10 11 B = 0.1 G B = 1.0 G B = 10.0 G B = 100.0 G 10 13 10 8 10 7 10 6 10 5 Pressure (bar)

Fig. 3.12.— A comparison of the total heating in the atmosphere (shown in red) over the pressure regions where we solve the thermal balance equation compared to the Joule Heating. We calculate Joule Heating from Equation 3.34 and from the Pederson conductance values at different magnetic field sizes as given in Table 3.12. The black lines represent the Joule heating rate calculated for each magnetic field tested. 108 polar cap region which is dependent on the size of the unknown planetary magnetic field. Note that for smaller sized magnetic fields, the polar cap region will be smaller. Through examination of Equation 3.35 we can also approximate how the size of Joule heating will scale with semi-major axis. This equation gives the relation

2 that QJH ∝ vswnsw/σP . Therefore, larger stellar wind velocities and number densities result in a higher Joule heating rate, whereas a higher conductivity results in a lower Joule heating rate. As semi-major axis increases, stellar wind velocity will also increase until it asymptotes to a constant velocity. Therefore, holding all other variables constant, a planet located farther away from the star will result in a higher Joule heating rate. However, we note that as semi-major axis increases, the density of the stellar wind decreases as a−2 and the size of the conductivity will also decrease as we have seen when comparing the conductivity of HD 189733b to that of (e.g) the Earth. Therefore, for a given system there is a distance which will result in a maximum possible Joule heating rate in the atmosphere, which is dependent on the stellar wind parameters and the ionizing ﬂux which sets the conductivity of the ionosphere.

3.5.4 Location of Joule Heating in the Atmosphere

We’ve shown that Joule heating likely does not contribute significantly to the thermal balance of the pressure regions considered in our model for HD 189733b when B < 100 G. However, if the location where the field-aligned current (see Chapter 2.3) turns around in the atmosphere occurs very deep (e.g. P > 1 bar), then Joule heating may be more important. For example, for B > 100 G sized magnetic fields, the electron peak of the conductivity would occur P > 1 bar which is below the atmosphere cutoff of our model. Kao et al.(2018) has reported a likely brown dwarf with a mass of 12.7

3 MJup to have a magnetic field B ∼ 10 G. Perhaps hot Jupiters, or other Jupiter sized 109 objects can have comparable magnetic fields to the brown dwarf reported by Kao et al.(2018). We note, however, that the thin layer approximation of the ionosphere breaks down at deep pressures in hot Jupiters due to the rising ionization fraction as shown by Lavvas et al.(2014) for HD 209458b. A detailed MHD model will likely be needed to map the current flow in these deeper regions to determine Joule heating. If the electron density of the atmosphere is constantly increasing as you go to deeper pressures for hot Jupiters, and current is able to flow into these deep pressure regions, then Joule heating can become more significant than the upper atmosphere depending on precise heating and cooling mechanisms. However, Joule heating will likely never compete with radiative heating near the 0.1 − 1 bar level, because stellar bolometric flux, which is ∼ 105 times larger than the upper heating in the upper atmosphere (e.g. Lavvas et al. 2014), is deposited near this region. More models on the consequences of Joule heating very deep in the atmosphere of hot Jupiters are needed to test how significant this effect can be.

3.6 Conclusions

We have developed a model of the upper atmosphere of the hot Jupiter HD 189733b, with the primary purpose of determining the electrical conductivity and Pederson conductance of the upper atmosphere. We then apply that conductance to determine the significance of Joule heating over the region of the atmosphere that we model. We find that for Jupiter sized magnetic fields (B ∼ 10 G), the upper atmosphere of HD 189733b has a much higher conductance than that of the Earth by a factor of 108. We also find that for Solar System sized magnetic fields (B ∼ 0.1 − 10 G), Joule heating does not have a large impact on the thermal structure of HD 189733b. For larger magnetic fields (B ∼ 100 − 1000 G), Joule heating can have a larger effect. 110

We show that peaks in the electrical conductivity, which are vital towards setting the conductance, occur very near the transition of the atmosphere from dominated by atoms to dominated by molecules. The inclusion of molecules into the atmosphere model reduces the size of the ion contribution to the electrical conductivity, which effects the conductance. Additionally, the density of ions determines the density of the electrons which are the main contributor to electrical conductivity. This effect is most notable for Earth sized magnetic fields and larger (B > 1 G). Therefore, it is vital to model the molecular region, in addition to the atomic region for hot Jupiter atmosphere models that seek to properly account for electrical conductivity effects. It is possible that for extremely large magnetic fields (B > 103 G), Joule heating could be important very deep in the atmosphere if current is able to reach pressures greater than 1 bar. For future studies, we suggest testing the atmosphere of other hot Jupiters and distances from host stars to determine when Joule heating can become more significant. It is also important to consider large magnetic fields and determine if electric current can travel very deep into the atmosphere and transfer heat at these deep pressures via Joule heating. 111

Chapter 4

Tidal Disruption of Moons in Star-Planet-Moon Systems

4.1 Summary

Despite the large number of exoplanets known today, there has yet to be a confirmed discovery of a moon orbiting an exoplanet. Due to the biases of exoplanet detection methods, the majority of exoplanets are observed to be orbiting very close to their host star and often much closer than Mercury is to our Sun. Similarly, there is a distinct lack of moons orbiting planets in the inner Solar System as both Mercury and Venus have no moons. Perhaps there is a physical mechanism connecting the hypothetical star-planet-moon system of exoplanets and Mercury and Venus that can explain the lack of moon in these systems. Here, we set out to explain this phenomenon by analytically modelling the motion of moons in these systems due to gravitational and tidal friction forces. Specifically, the long-term evolution of a coplanar, star-planet-moon system with zero free eccentricity (circular orbits) is studied. Even for zero free eccentricity, and 112 synchronous rotation, tidal evolution of a moon’s orbit still occurs due to radial and angular motions induced by forced eccentricity effects from an outside gravitational perturber. Tidal friction acting on these perturbations transfers energy from the orbital motion into heat within the moon, giving rise to expansion or contraction of the orbits. Lagrange’s planetary equations and Burns’ equations are solved including gravitational perturbations as well as the tidal friction force, in the equilibrium-tide and lag time approximations. An equation for the time-averaged change of the semi- major axis is derived, including all possible terms to second order in perturbation theory, and proportional to the lag time. In these systems, tides raised in the moon by the planet give rise to large perturbations as moons orbit near the Hill radius. We find that in star-planet-moon systems, long term tidal friction from forced eccentricity effects due to the gravitational perturbation of the host star results in a decay of the moon’s orbit. In the case of hypothetical Earth-like moons of hot Jupiters HD 189733b and HD 209458b the decay time is great enough to explain the lack of observed moons in these systems as moons will migrate towards the tidal disruption Roche limit or within the planetary radius within 106 years. These equations can additionally be applied to hypothetical moons of Mercury and Venus where we find the decay time to be over 1 Gyr and thus unable to explain the lack of moons in the inner Solar System.

4.2 Introduction

To date, there have been no strong detections of moons orbiting exoplanets, hereafter exomoons, and only a single debated detection of a Neptune sized exomoon orbiting a Jupiter sized exoplanet (Teachey & Kipping 2018; Teachey et al. 2020). Even before the Teachey & Kipping(2018) paper, various studies have attempted to explain the 113 lack of exomoons including examining the stability of closely packed planetary systems (Payne et al. 2013), the survival rate of exomoons due to planet-planet scattering (Gong et al. 2013), and the removal of exomoons during planetary migration due to resonance effects (Spalding et al. 2016). In general, these studies have found that a majority of the exoplanet systems observed have the ability to sustain a relatively large moon (Cassidy et al. 2009) that could be detected through other exoplanet detection methods. See Heller(2018) for an overview on the exomoon detection methods. Recent studies on the ability of hot Jupiters to retain moons have suggested that if hot Jupiters reach their current state via orbital migration, they are unlikely to retain a moon due to tidal friction effects during the circularization phase (Trani et al. 2020). Another explanation for the lack of exomoons is that tidal friction effects have caused orbital expansion or decay of the moon. If a moon were to move too far from its host star, past approximately half of the Hill radius of the star-planet-moon system (Payne et al. 2013), gravitational effects from the star will begin to dominate and the moon’s orbit will become unstable until it either is stripped or crashes into its host planet (Hong et al. 2018). Alternatively, if the moon were to move too close to its host planet, within the Roche limit, tidal effects from the planet will be strong enough to tidally disrupt the moon creating a ring system similar to what is observed in the gas giants of the Solar System. An example of a ring system existing outside of our Solar System has been suggested by Mamajek et al.(2012) to explain the observed light curve of a star with a complex series of deep eclipses. If a ring system forms and is able to eventually dissipate, then there would effectively be no evidence of a moon ever existing in the system. This effect is expressed in Figure 4.1. Additionally, if the Roche limit of a planet-moon system is within the planetary radius, then a 114 decaying moon will be absorbed into the atmosphere of the planet, before it can be tidally disrupted and form a ring system. This additionally will result in no dynamical evidence of the moon existing in the system. This effect is also represented in Figure 4.1, with the stage of the ring system not occurring in this scenario. For three or more bodies in a system, both the mutual gravitational perturbations as well as tidal friction forces can cause evolution in the orbits. However, contrary to the case of a two-body system, tidal friction can never cease. We analytically model this tidal friction force for a system of two objects in nearly circular orbits about a central body as shown in Figure 4.3. We apply this three body system to a theoretical star-planet-moon system with the planet as the central body of the coordinate system and the star and moon as orbiting perturbers. As discussed more in Section 4.7, we neglect the tide on the moon’s body due to the star and only include tides on the body of the moon due to the planet. Even if the objects are placed in circular orbits with zero free eccentricity, the perturbations from the companions cause radial and angular perturbations to the circular orbit, inducing a time-changing tide on the moon’s orbit (by the star), and this tidal dissipation draws energy out of the orbits and deposits it as heat in the perturbed moon. This causes long-term expansion or contraction of the moon orbit. Our model of moons located in circular and co-rotating orbits can be thought of as the most likely situation in which a planet, specifically a hot Jupiter, can retain a moon. In the case of non-circular orbits or non-synchronous rotation, tidal friction effects can cause significant migration of the moon’s orbit and potentially result in the moon being removed from the system. If a planet is able to retain a moon after the circularization and synchronization phase, we show that migration due to tidal friction effects can still occur. 115

Fig. 4.1.— A cartoon of the migration and disruption of a theoretical moon in a hot Jupiter system. As show in (a) the planet and moon will be in circular orbits about the host star. In (b) tidal friction for forced eccentricity will cause inward migration of the moon. In (c) the moon will migrate within the Roche limit and become tidally disrupted. Finally in (d), the ring system caused by the tidal disruption of the moon will dissipate and no evidence of the moon will remain. 116

The goal of this chapter is to derive the time-averaged rate of orbital expansion or contraction, and to apply this formula to explain the lack of moons in the inner Solar System (Burns 1973; Ward & Reid 1973; Burns 1977; Reynolds et al. 1987; Williams et al. 1997; Sartoretti & Schneider 1999). Additionally this model can be applied to theoretical moons orbiting close-in exoplanets (Barnes & O’Brien 2002; Doyle & Deeg 2004; Scharf 2006; Szabóet al. 2006; Domingos et al. 2006; Cassidy et al. 2009; Kipping 2009a,b) or to Kepler multi-planet systems (Lithwick & Wu 2008, 2012; Batygin & Morbidelli 2013). In Section 4.3, the tidal evolution of a two-body system is described and the circularization timescale of an eccentric orbit is defined. The circularization timescale constrains which systems for which we can apply our analytical equation. In Sec- tion 4.4 the set up of the star-planet-moon model and the variables considered for the derivation are defined. In Section 4.5 the full equations of motion are defined considering both gravitational and tidal friction perturbations. Section 4.6 includes an important constraint on the spin rate of the planets due to torque equilibrium. Section 4.7 outlines the perturbations to each variable at first order in perturbation theory from gravity and tidal friction and in Section 4.8, the change in semi-major axis for each orbiting body is derived to second order applying proper perturbation theory techniques. In Section 4.8.2 we compare this general tidal perturbation work to more specific studies by Lithwick & Wu(2012) and Batygin & Morbidelli(2013). Finally in Section 4.9 we discuss the applications of the second order changes in semi-major axis and discuss the results before concluding in Section 4.10. Note that throughout the majority of this chapter we will use the customary language “planet-planet” perturbations to refer to the gravitational perturbations from each orbiting body in the star-planet-moon system. 117 4.3 Tidal Evolution of Two-Body Systems

Hut(1981) discusses tidal evolution of a binary system using the equilibrium tide approximation for the tidal motions, and the lag-time approximation to model the dissipative eﬀects as shown in Figure 4.2. For Hut, orbit averaging was used to derive evolution equations for the semi-major axis (a), free eccentricity (e) and spin rate (Ω). In the case of a binary system, when the moment of inertia of the perturbed body is much smaller than that of the orbit, the system tends to rapidly become pseudo- synchronized, with spin rate Ω ' Ωeq(e), so that the time-average of the torque is zero over an orbit. Subsequently, over a much longer timescale, e is damped away, with associated changes in a and Ω ' Ωeq(e) in order to keep orbital angular momentum (Ω ∝ a(1 − e2)) roughly constant. Since the spin rate does change, as e changes, there must be nonzero torques, however they are very small compared to the values they would have for very non-synchronous spin. Hence the zero time-average torque is a good approximation during the circularization phase. The end result of tidal evolution, if merger is avoided, is circular orbits with synchronized spins. Once that is achieved for a two body system, the tidal force is no longer time-dependent, and tidal friction shuts oﬀ. For pseudo-synchronous rotation, the circularization timescale for free eccentricity, including both radial and tangential tidal friction forces, is (Hut 1981)

5 1 dek 1 21 m0 m0 GmkPk Rk − = = kk 1 + 8 ek dt tcirc 4 mk mk Qk ak Q −1 k m −2/3 m −1 R 5 P −13/3 ∼ 2.7 × 107 yr−1 k k 0 k k k . 10 0.1 M 10 M⊕ 2 R⊕ 5 days (4.1)

Here the subscript k = 0, 1, 2 denotes the object (0 for the planet, 1 for the moon, and 118

Fig. 4.2.— Hut(1981) model for the tides raised on a primary object with mass M by an outside perturber with mass m. Hut(1981) approximates the tides raised by the companion as point masses, µ, to compute the perturbing tidal force on the companion with separation r, moving around the companion as θ˙.

2 for the star in the star-planet-moon system), and kk is the apsidal motion constant of body k. The apsidal motion constant (Brooker & Olle 1955) is a measure of the gravitational perturbation produced by perturbing the mass of an object through tides. It is larger for more uniformly dense objects and smaller for objects with most of their mass at the center. The masses of the bodies are given by mk, where m0 is

the mass of the planet, and mk are the moon/star masses. The radii of the bodies

are denoted Rk and the orbital period of the star/planet is given by Pk. Lastly, Qk is the ratio of energy stored in the tide to energy dissipated in one cycle (Goldreich

& Soter 1966). On timescales longer than tcirc, free eccentricity is damped away, and the orbits are circular, aside from small gravitational perturbations. 119 4.4 The Star-Planet-Moon System

Consider a system of two coplanar objects of masses m1 and m2 orbiting a central body m0 (the planet in the star-planet-moon case), where “0” denotes the central body (the planet), “1” denotes the inner object (the moon), and “2” denotes the outer object (the star). The orbits of each of these bodies is assumed nearly circular and the deviations from the circular orbit are due to the gravitational perturbations each body feels from each other. The body’s spins are synchronized on timescales ∼ (R/a)2 shorter than the circularization timescale, where R is the body’s radius. Hence the spins will be assumed to be pseudo-synchronous. A cartoon of the model is shown in Figure 4.3 and the relevant variables that deﬁne the system, which are important in the equations that govern the system, are listed below. Important variables:

• a – semi-major axis: one half of the larger axis of the orbital ellipse

• e – eccentricity: the amount by which an ellipse deviates from a perfect circle. The eccentricity of a circle is 0 and the eccentricity of a parabola is 1.

• θ – true longitude: angle on an ellipse measured from a reference point. Note that in celestial mechanics, the term ‘longitude’ is used when referring to an angle measured with respect to a reference line ﬁxed in inertial space.

• $ – longitude of pericenter (the symbol is pronounced ‘po-mega’ or ‘curly pi’): longitude at which the closest approach to the central body occurs. This is constant for two-body problems but can vary with time in three-body problems.

• M – mean anomaly: this has no simple geometrical interpretation but is the time in units of orbital period that has elapsed since the orbiting body passed pericenter and expressed as an angle

• λ – mean longitude: the angular distance a body would have moved from the reference direction if it moved with uniform speed. λ = M + $

• – mean longitude at epoch: the mean longitude at the reference time 120

• f – true anomaly: angle of the orbiting body as measured with respect to pericenter. f = θ − $.

For a more general discussion on the variables and equations that are used in Solar System dynamics, see Brouwer & Clemence(1961) or Murray & Dermott(1999). For nearly circular orbits it is convenient to switch from the two real quantities e and $ to the complex quantity z ≡ e exp(i$). The evolution equations then are then in terms of {a, z, λ}.

The unperturbed orbits are taken to be circular orbits of orbital separation ak,0

q 3 about the central object, with mean motion nk,0 = G(m0 + mk)/ak,0. The eccen-

tricity ek of the unperturbed orbit is zero so that zk,0 = 0, and $k,0 is undeﬁned. The mean longitude of the unperturbed orbits is then

λk,0 = nk,0t + k,0. (4.2)

Here the mean longitude at epoch, k,0, is a constant denoting the value at time t = 0. For the remainder of the chapter, the “0” subscripts denoting the unperturbed orbit will be suppressed for brevity.

4.5 Equations of Motion

The equations of motion including both planet-planet gravitational interactions and tidal friction forces are

da da da k = k c + k c (4.3) dt dt pp dt tf dz dz dz k = k c + k c (4.4) dt dt pp dt tf dλ dλ dλ k = k c + k c (4.5) dt dt pp dt tf 121

Fig. 4.3.— Geometry from the reference point of the central planet. Eccentricities are highly exaggerated to highlight the diﬀerences in longitude of pericenters ($) between the two objects. The deﬁnitions of each variable are highlighted in Section 4.4 122

where “pp” and “tf” refer to planet-planet and tidal friction forces, respectively. The planet-planet terms are given by Lagrange’s planetary equations as (Murray & Dermott 1999)

dak 2 ∂Rk cpp = , (4.6) dt nkak ∂λk √ √ √ ∗ ∗ ∗ dzk 2i 1 − zz ∂Rk 1 − zz 1 − 1 − zz ∂Rk cpp = 2 ∗ − zk 2 ∗ , (4.7) dt nkak ∂zk nkak zz ∂λk

dλ 2 ∂R k c = n − k c dt pp k n a ∂a √ k k k √ ∗ ∗ 1 − zz 1 − 1 − zz ∗ ∂Rk ∂Rk + 2 ∗ z ∗ + z , (4.8) nkak zz ∂zk ∂zk

2 2 ∗ where ek = |zk| and z represents the complex conjugate of z. The ‘disturbing function,’ R, represents the gravitational potential due to a third body in the system. The disturbing function in coordinates centered on the primary object (in the star-planet-moon case, the planet) to O(e) for the planet-planet forces is (Murray & Dermott 1999)

G m1m2 X (k) (k) (k) R = f eij(λ2−λ1) + g z∗ + h z∗ ei∆λj + c.c. (4.9) k 2a m j j 1 j 2 2 k j

where c.c. represents the complex conjugate of the terms already listed in the bracket. The sum can be done over all values of j from −∞ to ∞, but we will show that signiﬁcant contributions can only occur from −5 < j < 5. The diﬀerence in mean longitudes is

∆λj ≡ jλ2 − (j − 1)λ1, (4.10) and ﬁrst order mean-motion resonances are contained in this order of perturbation 123 theory. The “f” terms give rise to changes in a and λ. The “g” and “h” terms may cause perturbations in a and λ, but more importantly can give rise to forced eccentricity at leading order.

(j) The f, g, and h coefficients contain Laplace coefficients (bs (α)) and their derivatives arising in the expansion of the disturbing function, and are given in Brouwer & Clemence(1961). Laplace coefficients are defined as the coefficients in the Fourier expansion of the disturbing function. For the inner body

1 1 f (1) = b(j) − αδ (4.11) j 2 1/2 2 |j|,1 1 1 3 g(1) = (−2j − αD)b(j) − αδ + αδ (4.12) j 2 1/2 2 j,−1 2 j,1 1 h(1) = (−1 + 2j + αD)b(j−1) − 2αδ (4.13) j 2 1/2 j,2

while for the outer body

1 1 f (2) = b(j) − δ (4.14) j 2 1/2 2α2 |j|,1

1 2 g(2) = (−2j − αD)b(j) − δ (4.15) j 2 1/2 α2 j,−1 1 3 1 h(2) = (−1 + 2j + αD)b(j−1) + δ − δ . (4.16) j 2 1/2 2α2 j,0 2α2 j,2

Here, α ≡ a1/a2 and D = d/dα and the Kronecker delta functions, δ, are used to represent the indirect terms arising because, for the chosen coordinates, the origin of the coordinate system is centered on the primary object, which undergoes accelerated motion. The radial and tangential components of the tidal friction force using the lag time 124

assumption is given by (Hut 1981)1 to be:

r˙k ar,k = −3γk 8 (4.17) rk

˙ (θk − Ωk) aθ,k = −γk 7 (4.18) rk

where γk is deﬁned as

m0(m0 + mk) 5 γk ≡ 3G Rkkkτk. (4.19) mk

Here rk is the separation for object k from the central body. For simplicity in the later derivation, we deﬁne

7γk Γk ≡ 8 (4.20) 2ak

which has the units of s−1 and is equal to the inverse of the circularization time

−1 (Γk = tcirc,k) as given by Equation 4.1. Burns(1977) showed how equations for the time derivatives of the orbital elements can be derived using elementary dynamics, which we will refer to as “Burns’

Equations.” The radial (ar) and tangential (aθ) components of the tidal friction force can then give rise to the following changes in the orbital elements (Murray & Dermott 1999, Ch. 2; Brouwer & Clemence 1961, Ch. 11)

da 2 k c = [a e sin f + a (1 + e cos f )] (4.21) dt tf p 2 r,k k k θ,k k k nk 1 − ek

dz p1 − e2 2 + e cos f e k k ifk k k k ctf = zk e −iar,k + aθ,k + aθ,k dt eknkak 1 + ek cos fk 1 + ek cos fk (4.22)

1Equation 8 in (Hut 1981) has been divided by the reduced mass to get this acceleration. 125

2 dλk 2 1 − ek ctf = nk − ar,k dt nkak 1 + ek cos fk 2 p 2 ek 1 − ek 2 + ek cos fk + 2 1/2 −ar,k cos fk + aθ,k sin fk (4.23) 1 + (1 − ek) nkakek 1 + ek cos fk

The proper changes in orbital elements cannot be approximated to second order

until a pseudo-synchronous value for Ωk, and thus aθ,k, is known. This is discussed in more detail in Section 4.6.

4.6 Torque Equilibrium

2 The moment of inertia of the moon’s spin is ∼ mkRk while the moment of inertia

2 of the orbit is ∼ mkak, which is much larger. Hence for equal and opposite torques applied to spin and orbit, the spin will be synchronized on a timescale much shorter than the orbit expansion timescale. The central planet, however, is not necessarily synchronized to the moon and is considered friction-less in our model as discussed more in Section 4.7. Naively it would be expected that synchronous rotation of nearly circular orbits implies Ωk = nk. However, the gravitational perturbation caused by the star can give an additional shift in the mean angular velocity as if the planet’s mass was slightly shifted. This shift must be carefully accounted for when computing the tidal dissipation due to the tangential friction force aθ,k. Ignoring this constant shift in spin rate would lead to a spurious tidal dissipation contribution. Physically, tidal dissipation can only arise from a time-dependent tidal acceleration acting on the planet, which is due to time-dependent radial and angular motions. This “pseudo-synchronization” eﬀect results in the time average torque, N, of the moon to be zero. This does not mean that tidal friction from the torque will have 126

no eﬀect on orbital migration, but rather that constant shifts in rotational angular frequency causing tidal evolution can be ignored by an appropriate choice of Ω. We therefore need to take the time average torque of the tidal friction forcing from Hut (1981) in order to obtain an appropriate choice of Ω. This process is discussed in more detail below. The torque per unit reduced mass for each orbit is

˙ ! θk − Ωk Nk ≡ raθ,k = −γk 6 (4.24) rk

where γk is the constant given in Equation 4.19.

Setting the time average hNki = 0, where the brackets hi denote the time average, gives the torque-equilibrium rotation rate

−6 ˙ hrk θki Ωeq,k = −6 . (4.25) hrk i

This rotation rate is now the value of Ωk that can be plugged into Equation 4.18 to solve the time dependent tidal changes to the equations of motion of the two outer bodies considered.

From Murray & Dermott(1999) we can obtain the following expansions of rk ˙ 2 and θk to O(e ) that we can plug into the torque-equilibrium rotation rate equation. These equations are:

2 r 1 − e 1 1 1 2 = ≈ 1 − z∗e−iλ + zeiλ + zz∗ − z∗ e−2iλ + z2e2iλ (4.26) a 1 + e cos f 2 2 4

n 3 2 3 1 θ˙ = (1 + e cos f)2 ≈ n 1 + z∗eiλ + ze−iλ + z∗ e2iλ + z2e−2iλ − zz∗ (1 − e2)3/2 2 2 2 (4.27) 127

We can now use these equations to calculate the pseudo-synchronous rotation rate that will allow us to properly apply Burns’ equations.

4.7 Solutions at First Order

For first order perturbations it is important to determine the “small parameter” that defines the order to which the equations of motion need to be expanded, to properly consider all relevant planet-planet and tidal terms. In the case of a star-planet- moon system, defining a single small parameter is not trivial as it is a combination of variables that define our small parameter. For the change in orbital parameters

3 in the moon, Lagrange’s planetary equation scale as µ2α to leading order and as

2 6 (m2/m0) α to ﬁrst order. For Burns’ equation, the leading order scaling goes as

3 2 6 γ1(m2/m0)α and as γ1(m2/m0) α to ﬁrst order. We will therefore, derive full equa-

2 6 tion of motion to terms that have (m0/m1) α terms, including those that scale as

2 6 γ1(m2/m0) α in Burns’ equations.

4.7.1 Ignoring Perturbations of the Star

In order to make the following derivation easier to follow, we ignore perturbations to the orbital parameters of the star and instead hold a2, z2, and 2 constant and assume e2 = 0. Internally, we complete the full derivation, including perturbations to the moon and star orbital parameters for completeness. We ﬁnd that in the end, the migration of the moon, due to gravitational perturbations in the star are negligible. This allows us to present the ﬁnal result here in a more straightforward manner. The end result of negligible contribution to the migration of the moon from perturbations of the star orbits can be seen as follows. From the disturbing function in Equa- ˙ tion 4.9, changes in the orbital parameters of the star will scale as {a˙2, z˙2, λ2} ∝ m1, 128 ˙ whereas changes in the orbital parameters of the moon will scale as {a˙1, z˙1, λ1} ∝ m2.

The star mass (m2) is much greater than the moon mass (m1) and therefore, changes in the orbital parameters of the star are negligible compared to changes in the moon, by many orders of magnitude. Additionally, energy can be removed from the system due to tidal friction as discussed in Section 4.3. The energy that can be removed scales as the inverse of circularization time in Equation 4.1. This fact is expressed later in Section 4.7.4. The circularization time of the star around the planet, when compared to the moon around the planet, is much greater and therefore the energy input into the star is also much smaller than that into the moon. The combination of these two eﬀects gives us the result that perturbations to the orbital parameters of the star are negligible to the tidal migration of the moon in our system.

4.7.2 Ignoring Tidal Friction in the Planet

We additionally ignore tidal friction in the planet in deriving the tidal migration of the moon in our system. Tidal friction of each of the three bodies is the system can remove energy from the three-body system causing changes to the orbital parameters. We’ve already shown above that tidal friction in the star is negligible. Additionally, we can assume that tidal friction in the planet is negligible when compared to tidal friction and orbital changes of the moon. Cassidy et al.(2009) examined tidal heating from satellites around hot Jupiter-like planets. They show that gas giants are much less dissipative than rocky moons due to their tidal Q values of 105 or greater and therefore for our purposes are negligible compared to the moon in removing energy from the system. Speciﬁcally, Cassidy et al.(2009) shows that tides in the planet raised by the star result is a Q ∼ 105 and that tides in the planet raised by the moon 129 result in Q ∼ 1012.

4.7.3 First Order Gravitational Perturbations

Applying Lagrange’s planetary equations as outlined in Section 4.5, the perturbations to the orbital elements at ﬁrst order considering planet-planet interaction are:

da1 X (1) c = a n µ α −ijf eij(λ2−λ1) + c.c. (4.28) dt pp 1 1 2 j j

dz1 X h (1) i c = iµ n α g ei∆λj (4.29) dt pp 2 1 j j

dλ1 X h (1) i c = n − n αµ αDf eij(λ2−λ1) + c.c (4.30) dt pp 1 1 2 j j where

mk µk ≡ , (4.31) (m0 + m1m2/mk)

∆λj ≡ ∆jt + j2 − (j − 1)1 (4.32) and the detuning is

∆j ≡ jn2 − (j − 1)n1. (4.33)

The dak/dt and dzk/dt equations can easily be integrated to obtain the first order planet-planet effects. In order to apply perturbation theory properly, first order perturbations to nk in the dλk/dt equations must be included. Therefore, the first order perturbation to ak is required to obtain the proper relation. The ak and zk equations are thus

n h i 1 X (1) ij(λ2−λ1) a1(t)pp = a1µ2α fj e + c.c. (4.34) n1 − n2 j6=0 130

X (1) n1 z (t) = µ α g ei∆λj (4.35) 1 pp 2 j ∆ j j

In order to find nk to first order in masses, we must consider perturbations of ak to first order in masses as nk is only at leading order. The first order perturbations to all variables will be designated by δ, therefore through a Taylor expansion to first order in masses

s G(m0 + mk) 3 δak nk ≈ 3 1 − , (4.36) ak 2 ak

and from Equation 4.34

δa n h i 1 1 X (1) ij(λ2−λ1) cpp = µ2α fj e + c.c. (4.37) a1 n1 − n2 j6=0 To ﬁrst order in α and masses the complete changes to the mean longitudes from planet-planet interactions are

ij(λ2−λ1) −ij(λ2−λ1) dλ1 X (1) n1 2 (1) e + e cpp = −µ2n1 −3αfj + 2α Dfj dt n2 − n1 2 j6=0 (4.38)

Integrating the λk equations we then obtain the oscillating parts

ij(λ2−λ1) −ij(λ2−λ1) n1 X 1 (1) n1 2 (1) e − e δλ1,pp = −µ2 −3αfj + 2α Dfj n2 − n1 j n2 − n1 2i j6=0 (4.39) We can then use this equation to plug into the full second orbit tidal migration equation. 131 4.7.4 First Order Tidal Friction Perturbations

Here we show how to calculate the second order terms for ar and aθ that can then be applied to the ﬁrst order or second order perturbations. From the torque equilibrium equation we get2

2! (1) 33 (1) 23 X n1 (1) Ω = n 1 + 2µ αg + (µ αg )2 + µ2α2 g (4.40) eq,1 1 2 0 2 2 0 4 2 ∆ j j j

With the pseudo-synchronous Ω value now known, we can ﬁnd the tidal friction accelerations (Equation 4.17, 4.18). Complete to ﬁrst order in eccentricity for ar and second order in eccentricity for aθ, these values are:

3 a = iΓ n a z∗eiλk − z e−iλk (4.41) r,k 7 k k k k k

h i (1) ∗ iλ1 −iλ1 (1) aθ,1 = a1Γ1n1 −2µ2αg0 + z1e + z1e µ2αg0 (4.42)

Note that the leading order terms for ar,k are already at ﬁrst order in mass, therefore the perturbations to orbital elements from tidal friction, as given by Burns’ equations become

da 2 4 k ∗ iλk −iλk ctf = aθ,k = − Γkak z e + ze (4.43) dt nk 7

dz 1 k iλk iλk ctf = −ie ar,k + 2e aθ,k (4.44) dt nkak

dλk 2 ctf = nk − ar,k (4.45) dt nkak 2Author’s note: As this work is still in progress we note that the torque equilibrium expression reported is provisional. A full expansion to second order in eccentricity must be calculated to get the full eﬀect. Look for a future publication on the full eﬀect, entirely complete to second order in perturbation theory. 132

To ﬁnd da/dt, we apply perturbation theory by substituting in the zpp values at ﬁrst order from gravitational perturbations. Therefore for the inner and outer bodies, this becomes:

da 4 n 1 X (1) 1 ij(λ2−λ1) −ij(λ2−λ1) ctf = − Γ1a1µ2α gj e + e (4.46) dt 7 ∆j j6=0

Integrating these equations gives us (to ﬁrst order in perturbation theory):

ij(λ2−λ1) −ij(λ2−λ1) δa1 4 n1 X (1) Γ1 e − e ctf = − µ2α gj (4.47) a1 7 n2 − n1 ∆j ij j6=0

Note that similar to the δλ1,pp equations we will see that when applying δa1,tf to the full second order equation, all j−1 terms will cancel out. Now, substituting in the friction terms into the dz/dt equation gives (at ﬁrst order in perturbation theory):

dz 3 4 1 c = Γ z∗e2iλ + z + Γ z∗e2iλ − z = Γ z∗e2iλ − z (4.48) dt tf 7 7 7

Where z = zk(t)pp from above. Plugging in for the zpp terms for the inner and outer body we get: dz1 X (1) n1 1 c = µ αΓ g − ei∆λj + ei∆λ−j (4.49) dt tf 2 1 j ∆ 7 j j Therefore, the complex eccentricity at ﬁrst order in perturbation theory is:

i∆λj i∆λ−j X (1) n1 1 e e z (t) = µ αΓ g − + (4.50) 1 tf 2 1 j ∆ 7 i∆ i∆ j j j −j

As for the mean longitudes we get (ignoring terms that are absorbed into the 133

constant change in λn due to the unperturbed mean motion):

dλ 3 δa 6 k k ∗ iλk −iλk ctf = − nk − Γki zke − zke . (4.51) dt 2 ak 7

We can again, plug in zpp, and δa for the first order gravitational and tidal effects, to get the change in the mean longitude to first order in perturbation theory for the inner and outer body to be:

ij(λ2−λ1) −ij(λ2−λ1) dλ1 X (1) Γ1 n1 2 e − e ctf = −iµ2n1 gj 2 − (4.52) dt ∆j n2 − n1 j 2 j6=0

Therefore, the ﬁrst order equations for the mean longitude that we will plug into the second order equations are:

ij(λ2−λ1) −ij(λ2−λ1) n1 X 1 (1) Γ1 n1 2 e + e δλ1,tf = −µ2 gj 2 − n2 − n1 j ∆j n2 − n1 j 2 j6=0 (4.53) We can now calculated the full ﬁrst order perturbations considering both planet- planet and tidal friction forces.

4.7.5 Full First Order Perturbations

We now apply perturbation theory in the full {z,˙ a,˙ λ˙ } equations to ﬁrst order, as Γ µnα3 in the leading order approximation. The changes in the semi-major axis to full ﬁrst order in eccentricity are:

δa1 n1 X (1) = −µ α f eij(λ2−λ1) + e−ij(λ2−λ1) a 2 n − n j 1 2 1 j

ij(λ2−λ1) −ij(λ2−λ1) 4 (1) Γ1 e − e + gj (4.54) 7 ∆j ij 134

By applying complex trigonometric identities, we can write this equation more intuitively in terms of sines and cosines. Therefore,

δa1 n1 X (1) 8 (1) Γ1 = −µ2α 2fj cos[j(λ2 − λ1)] + gj sin[j(λ2 − λ1)] (4.55) a1 n2 − n1 7 j∆j j6=0

The complex eccentricities to ﬁrst order in e are

i∆λj i∆λ−j X (1) n1 1 e e z (t) = µ α g ei∆λj − Γ + Γ (4.56) 1 2 j ∆ 7 1 i∆ 1 i∆ j j j −j

The oscillating parts of the full mean longitude equation become:

ij(λ2−λ1) −ij(λ2−λ1) n1 X 1 (1) n1 2 (1) e − e δλ1 = −µ2 −3αfj + 2α Dfj n2 − n1 j n2 − n1 2i j6=0

ij(λ2−λ1) −ij(λ2−λ1) 1 (1) Γ1 n1 2 e + e + gj 2 − (4.57) j ∆j n2 − n1 j 2

Again, we can apply complex trigonometric identities to more intuitively write δλ1 as:

n1 X 1 (1) n1 2 (1) δλ1 = −µ2 −3αfj + 2α Dfj sin[j(λ2 − λ1)] n2 − n1 j n2 − n1 j6=0 1 (1) Γ1 n1 2 + gj 2 − cos[j(λ2 − λ1)] (4.58) j ∆j n2 − n1 j

We can now use the ﬁrst order perturbations to the orbital parameters to calculate the migration of the moon to second order in perturbation theory following the methods outlined in Ch. 11 of Brouwer & Clemence(1961). 135 4.8 Equation for a at Second Order

To get the full equations to second order in µ using perturbation theory we must consider the disturbing function to ﬁrst order in eccentricity as we know that all the orbital parameters already have one order of µ and the disturbing function also is at order µ. We additionally note that the change in semi-major axis due to tidal friction must be expanded to second order in eccentricity.3 The reported equation for the change in semi-major axis of the moon to second order is:

da1 1 δa1 X (1) = n µ a α 1 + −ijf eij(λ2,0−λ1,0) (1 − ijδλ ) dt 1,0 2 1,0 0 2 a j 1 1,0 j (1) ∗ (1) ∗ i∆λj,0 − i(j − 1) gj z1 + hj z2 e + c.c. 4 δa 1 ∗ iλ1,0 −iλ1,0 − Γ1a1,0 1 + z1e + z1e (4.59) 7 a1,0 where the ‘0’ subscript denotes the leading order perturbations to that variable which we had previously suppressed for brevity and the ‘δ’ denotes ﬁrst order perturbations to that variable from gravitational and tidal friction perturbations as outlined in Chapter 4.7. For each of the ‘z’ variables, we can plug in the z(t) equations calculated in Chapter 4.7.

3 Author’s note: At the time of submission, the full second order eﬀect fora ˙ tf was not complete. While likely not large enough of an eﬀect to drastically change the results, the complete equation must consider this term to second order in eccentricity. Look for a future published paper that will have the dervation fully complete to second order in eccentricity. 136

In terms of trigonometric functions, Equation 4.59 can be written as:

da1 X (1) (1) δa1 = n1µ2a1α 2jfj sin[j(λ2 − λ1)] − jfj sin[j(λ2 − λ1)] dt a1 j6=0

(1) δa1 2 (1) + 2jαDfj sin[j(λ2 − λ1)] − 2j fj δλ1 cos[j(λ2 − λ1)] a1 (1) −i∆λj ∗ i∆λj + i(j − 1)gj z1e − z1e (4.60)

Note that here, the ‘0’ subscript has again been removed for brevity.

We now take the time average of da1/dt to determine the constant migration of the moon due to the planet-planet and tidal friction perturbations. The time averaged terms that will remain constant are most easily understood by considering Equation 4.60. The only terms that will not time average to zero are those that go

as sin A sin A, cos A cos A, or the ‘z1’ terms that can cancel the ∆λj terms. This sin and cos cancellations can be understood physically as in phase terms will have secular eﬀects but terms that are 90◦ out of phase cannot contribute to secular migration. Now plugging in for each of the variables, the full equations to second order in perturbation theory, considering both planet-planet and tidal perturbations and ignoring eﬀects that time average to zero gives:

da1 2 2 X 8 n1 (1) (1) n1 (1) n1 (1) = Γ1a1µ2α −fj + 2αDfj gj + g−j dt 7 n2 − n1 ∆j ∆−j j6=0 n1 (1) (1) n1 n1 1 (1) n1 n1 1 − 2 fj gj 1 − − g−j 1 − n2 − n1 ∆j n2 − n1 j ∆−j n2 − n1 j n1 (1) n1 (1) 1 n1 (1) − 2(j − 1) gj gj + g−j (4.61) ∆j ∆j 7 ∆−j

We can simplify this equation even further since the Laplace coeﬃcients can be approximated for small α as is the case for a star-planet-moon system. We also note 137 that the largest contributions to the da1/dt equation occurs when j = ±2. This is

(1) (1) due to the fact that g1 = g−1 = 0. The resulting equation is:

da 1 ≈ −Γ a µ2α2n (4.62) dt 1 1 2 ratio where we deﬁne the unit-less sum within the brackets of Equation 4.61 to be

X 8 n1 (1) (1) n1 (1) n1 (1) n ≡ −f + 2αDf g + g ratio 7 n − n j j ∆ j ∆ −j j=±2 2 1 j −j n1 (1) (1) n1 n1 1 (1) n1 n1 1 − 2 fj gj 1 − − g−j 1 − n2 − n1 ∆j n2 − n1 j ∆−j n2 − n1 j n1 (1) n1 (1) 1 n1 (1) − 2(j − 1) gj gj + g−j (4.63) ∆j ∆j 7 ∆−j

(1) (1) 2 (1) (1) (1) 2 and note that f2 = f−2 = (3/8)α , Df2 = Df−2 = (3/4)α, g2 = (−9/4)α , and

(1) 2 6 g−2 = (3/4)α . Therefore the migration of the moon is proportional to α asa ˙ ∝ α . If we place an object at half the Hill radius of the system, as we initially do with our

6 model, then n1/n2 ≈ 5 and nratio ≈ 37.1α . This gives the full second order change in the semi-major axis of the inner object which we can then apply to applicable star-planet-moon systems such as for theoretical moons of hot Jupiters and the inner Solar System planets. We ﬁnd that, in this limit, the change in the semi-major axis of the moon is negative, meaning that the moon will migrate towards the planet due to this tidal friction eﬀect. Additionally this change in the semi-major axis is proportional to the tidal damping in the moon

(a ˙1 ∝ Γ1). Therefore, if a moon were to migrate to the Roche limit of the planet- moon system, tidal disruption can occur and the moon would be destroyed. If this tidal disruption can occur quick enough, it can explain the lack of moons observed in systems where the outlined model is applicable. 138 4.8.1 Moon Migration Timescale

To ﬁnd the migration time to the Roche limit we can now plug in for all the variables in Equation 4.62 and solve for a(t). Doing this we get:

5 2 da1 21 m0 p R1 k1 m2 1/2 = − π G(m0 + m1) 6 nratio a1 (4.64) dt 2 m1 a2 Q1 m0 + m1

Now integrating this equation with respect to a1 we get:

βt2 a (t) = a − (4.65) 1 1,0 2

where 5 2 21 m0 p R1 k1 m2 β ≡ π G(m0 + m1) 6 nratio (4.66) 2 m1 a2 Q1 m0 + m1

We can rewrite Equation 4.65 and solve for the time it would take to reach the tidal disruption distance at the Roche limit. For the moon, the distance for tidal disruption at the Roche limit as given in Chapter 4.8 of Murray & Dermott(1999) is: 1/3 3m0 aRoche = R1 (4.67) m1 and the Hill radius is: 1/3 m0 aHill = a2 (4.68) 3m2

This implies that the timescale for tidal disruption is:

2 r1 t = a − a (4.69) decay |β| 2 Hill Roche

We can now apply this equation to various systems to determine the tidal disruption timescale. We additionally note that if the Roche limit for a planet-moon system is 139 within the radius of the planet, then the moon will be absorbed into the atmosphere of the planet before disruption occurs. For gas giant planets, the location of the Roche limit for rocky moons is often within the radius of the planet and therefore collision with the upper atmosphere of the planet will occur before tidal disruption. The equations calculated here assume that the moon is able to reach the Roche limit before collision with the planetary atmosphere, therefore this decay timescale can be thought of as an upper limit because if the Roche limit is within the planetary radius, collision with the atmosphere will occur before the moon can be tidally disrupted. To get a qualitative understanding of the tidal disruption time, we can assume that aHill aRoche. Therefore, the tidal disruption time becomes:

−1 −2/3 2/3 −13/6 2 Q1 k1 m1 m0 m2 tdecay = 2.8 × 10 yr 10 0.1 M⊕ MJup M a 1/2 a 13/2 ρ 5/3 1 2 1 (4.70) 0.5 aHill 0.031 AU ρ⊕

We’ve plugged in parameters similar to the HD 189733b system for direct comparisons to other similar hot Jupiter system. The a1 value chosen is equal to half the Hill radius of the HD 189733 system. The consequences of the scalings for the tidal disruption time are discussed in more details in Section 4.9.

4.8.2 Comparison to Other Studies

As a test on the accuracy of our deviations, we compare our ﬁnal equation to Lithwick & Wu(2012). In their study, they explained the excess of Kepler planet pairs that lie just wide of resonance by invoking similar equations of motion presented here and tidal heating rates due to radial forces outlined in Hut(1981). They argued that the dominant interaction between planets in a similar system that we consider is due 140

to the 2:1 resonance. The 2:1 resonance in our equations corresponds to the j = 2 harmonic. Therefore the equations derived in Lithwick & Wu(2012) to ﬁrst order in

eccentricity for dak/dt, in our notation, for the 2:1 resonance become

2 d ln a1 C µ2 (1) 2 (1) (2) = − 2 Γ1(g2 ) C + Γ2h2 h2 (4.71) dt 2 ∆2

2 d ln a2 µ1 (2) (1) (2) 2 = 2 Γ1g2 g2 C + Γ2(h2 ) . (4.72) dt ∆2

where µ ra C ≡ 1 2 (4.73) µ2 a1

It is clear that near resonance, the case that Lithwick & Wu(2012) were concerned

2 2 with, the extra tidal heating forcing can be ignored as n2 ∆2. However, as the planets are repelled from each other and exact commensurability, this additional tidal

2 2 heating term can become signiﬁcant when n2 ∼ ∆2. Their equation for the change in the semi-major axis of the ‘inner body’ of the system considered agrees with ours for the j = 2 harmonic. Interestingly, for the inner planet, this additional tidal heating term has the opposite sign of the resonant forcing term and can therefore potentially balance this forcing away from resonance. This can possibly explain the excess of planets near resonance where none would be expected as shown by theory in Figure 1 of Lithwick & Wu(2012).

4.9 Applications and Discussion

We can now apply the equations derived in Chapter 4.8 to systems like the model outlined in Chapter 4.4. Each of the star-planet-moons systems considered are outlined 141 in Tables 4.1-4.4.

4.9.1 Migration of Moons around hot Jupiters

The first cases we consider is for a hypothetical Earth-like moon of hot Jupiters HD 189733b and HD 209458b. If a hot Jupiter were to have a significantly large enough exomoon (e.g. the size of the Earth), then it could be detected via the exoplanet detection methods outlined in Chapter 1. In the case of current exoplanets detected, the vast majority of them (∼ 95%) are Earth-sized or larger (Han et al. 2014). As an estimate of current observational constraints, the smallest exoplanet detected is approximately Mercury sized (Barclay et al. 2013). Considering these observational constraints, we therefore take an Earth-like moon, and place it at half the Hill radius of the star-planet system to see what effect this constant tidal friction can then have on the system. As outlined in Tables 4.1-4.2, the hot Jupiters and Earth-like moons would circularize fairly quickly (within 1 Gyr), meaning that our model and analytically derived equation is applicable. We can then assume that tidal disruption for each system occurs at the Roche limit as given in Table 4.5 for each system. The change in the semi-major axis of the moon, due to this tidal friction from being in a three-body system, is shown in Figure 4.4. Applying Equation 4.65 for HD 189733b and HD 209458b, this tidal disruption would occur in 3.7 × 102 and 3.1 × 103 years

System A Mass (g) a0 (cm) R (cm) Q tcirc (yr) Star – HD 189733A 1.69 × 1033 – 5.63 × 1010 106 – Planet – HD 189733b 2.28 × 1030 4.49 × 1011 7.97 × 109 106 2.10 × 108 Moon – Earth-like 5.97 × 1027 1.72 × 1010 6.37 × 108 101 2.09 × 102

Table 4.1: HD 189733 system with hypothetical Earth-like moon. The circularization time is calculated via Equation 4.1. We use k = 0.1 for each body for all systems and the tidal Q is taken from Goldreich & Soter(1966). Note that the initial semi-major axis (a0) for the moon is located at half the Hill radius of the star-planet system. 142

System B Mass (g) a0 (cm) R (cm) Q tcirc (yr) Star – HD 209458 2.29 × 1033 – 8.75 × 1010 106 – Planet – HD 209458b 1.35 × 1030 7.03 × 1011 9.45 × 109 106 6.18 × 108 Moon – Earth-like 5.97 × 1027 2.04 × 1010 6.37 × 108 101 1.40 × 103

Table 4.2: HD 209458 system with hypothetical Earth-like moon.

System C Mass (g) a0 (cm) R (cm) Q tcirc (yr) Star – Sun 1.99 × 1033 – 6.96 × 1010 106 – Planet – Mercury 3.30 × 1026 5.80 × 1012 2.44 × 108 102 1.57 × 1015 Moon – Europa-like 3.30 × 1025 1.10 × 1010 1.73 × 108 102 2.35 × 107

Table 4.3: Mercury system with hypothetical Europa-like moon.

System D Mass (g) a0 (cm) R (cm) Q tcirc (yr) Star – Sun 1.99 × 1033 – 6.96 × 1010 106 – Planet – Venus 4.87 × 1027 1.05 × 1013 6.05 × 108 101 1.10 × 1015 Moon – Europa-like 3.30 × 1025 4.91 × 1010 1.73 × 108 102 7.25 × 109

Table 4.4: Venus system with hypothetical Europa-like moon.

System Roche limit (cm) Roche time (yr) A 6.6 × 109 3.7 × 102 B 5.6 × 109 3.1 × 103 C 5.4 × 108 9.2 × 108 D 1.3 × 109 2.2 × 1011

Table 4.5: The Roche limit and tidal disruption time (Roche time) for each of the four 6 systems considered as deﬁned above. Note that we assume an initial nratio = 37.1α for each of the systems. Additionally note that the Roche limits for system A and B are within the planetary radius and therefore collision with the atmosphere will occur before tidal disruption. 143 respectively as expressed in Table 4.5. This short time-span, which would be much shorter than the age of the hot Jupiter stellar systems (∼ 1 Gyr), suggests that Earth-size moons would not be able to exist around hot Jupiters.

4.9.2 Migration of Moons around Mercury and Venus

The second cases we consider are of a hypothetical moon orbiting Mercury and Venus. Mercury and Venus are the only planets of the Solar System without moons and we apply our migration equation to see if this tidal friction eﬀect can explain the lack of moon in these systems as well. For both Mercury and Venus, we consider a ‘Europa-like’ moon orbiting at half the Hill radius of each system, with properties given in Tables 4.3 and 4.4. We assume a Europa-like moon to test if a relatively large moon of the Solar System would be able to survive in this system. We note that for both Mercury and Venus the theoretical circularization time is much too large for the planets to circularize around the Sun within the age of the Solar System. This is most distinctly seen in the free eccentricity of Mercury which is measured to be 0.205. Venus, however, has one of the smallest free eccentricities of any Solar System body at 0.0068. We have derived our equation under the assumption of circular orbits and so we can treat the case of applying our model to Mercury and Venus as a test on the lower limit of what the tidal migration time would be. In reality, eccentric orbits can result in greater migration of moons either towards or away from the planet depending on a variety of factors including spin-orbit ratio and rotation direction relative to the orbit direction. Applying our migration equation as a lower limit on the tidal evolution of moons around Mercury and Venus, we ﬁnd that tidal disruption would occur within 9.23×108 and 2.19 × 1011 year for Mercury and Venus respectively as listed in Table 4.5 and 144

0.5

0.4 l

l 0.3 i H r / 1 a

0.2

HD 189733b Moon 0.1 HD 209458b Moon Mercury Moon Venus Moon

100 102 104 106 108 1010 1012 Time (years)

Fig. 4.4.— Migration of a hypothetical moon orbiting HD 189733b, HD 209458b, Mercury, and Venus. On the x-axis is time since the beginning of the integration in years and on the y-axis is the semi-major axis of the moon (a1) divided by the Hill radius of the individual system. Each system is integrated until the semi-major axis becomes less than the Roche limit for tidal disruption at which point a moon would theoretically be destroyed. If the Roche limit is within the planetary radius, then the moon will collide with the atmosphere of the planet, before being tidally disurpted. 145 plotted in Figure 4.4. The age of the Solar System is about 5 Gyr and so a moon of Mercury would be on the edge of surviving in the Mercury system, whereas a moon of Venus would easily be able to survive. This suggests that the tidal friction force from perturbed orbits cannot explain the lack of moons orbiting Venus but is large enough to explain the lack of moons orbiting Mercury. In reality, the Mercury system is not directly applicable to our model since it is not circularized or in a pseudo-synchronous orbit but suggests that a detailed analysis of the tidal migration of a moon around Mercury can be worthwhile.

4.9.3 Scaling of Migration Time with Tidal Q

In Figure 4.5 we show how the tidal destruction time for an Earth sized moon of a star– planet system, scales with semi-major axis of the planet and the tidal Q parameter of the moon. To create the lines, we assume a planet mass of HD 189733b and star mass of HD 189733A as given in Table 4.1. We assume Earth-like parameters for the moon, an apsidal motion constant of k = 0.1, and set the tidal disruption time to be

9 5×10 years. The destruction time as a function of a2 is then calculated via Equation 4.70. In Figure 4.5 the dashed lines represent the size of the semi-major axis for HD 189733b and Mercury for a comparison of different types of systems. This plot highlights that for a range of tidal Q parameters at the semi-major axis of HD 189733b, Earth-sized moons would be destroyed within a million years. This figure also highlights the fact that for larger tidal Q, the destruction time of the moon scales linearly. This is due to the fact that the tidal Q parameter is effectively a calculation of the ability of tidal distortions to generate heat into the moon and therefore remove energy from the orbit of the planet-moon system. At the location of the orbit of Mercury, Earth-sized moons orbiting HD 189733b-like planets would 146

Orbit of Mercury

) 10 U A (

2 a

HD 189733b

Q = 1 Q = 10 Q = 100 10 2 Q = 1000

100 102 104 106 108 1010 1012 1014 Destruction time (yr)

Fig. 4.5.— A plot of tidal destruction time (in years) of Earth-size moons as a function of star-planet semi-major axis (a2). The destruction times are calculated via Equation 4.70. Each black line represents a tidal Q as given by the legend. We assume the planet mass is equal to the mass of HD 189733b, the mass of the star is equal to the mass of HD 189733A, an apsidal motion constant of k = 0.1, ρ = ρ⊕, and that the moon is originally located at half the hill radius of the system. The dashed lines highlight the orbital location of HD 189733b and Mercury. 147

still not be able to survive unless they have a relatively small tidal Q.

4.9.4 Constraining Maximum Moon Sizes

From Equation 4.70, we can place upper limits on the size of moons that would be able to survive tidal disruption for a given star-planet system. The tidal disruption

−2/3 time is proportional to moon mass as tdisrupt ∝ m1 . Therefore, holding all other variables constants, as moon mass increases, the tidal disruption time would decrease. This relation is expressed in Figure 4.6. In this ﬁgure we plot lines of maximum moon mass versus the semi-major axis of the star–planet system for diﬀerent sized central planets. To create these lines we assume that tidal disruption occurs in 5 × 109 years and that the mass of the star is equivalent to the mass of the Sun. Additionally we assume tidal Q values equivalent to the Earth (Q = 10), an apsidal motion constant of k = 0.1, and a mass density of

−3 the moon equivalent to the mass density of the Earth (ρ⊕ = 5.5 g cm ). The shaded regions (located up and to the left of each line) then express the type of systems that would be able to have a moon survive within 5 Gyr. Therefore, planets that are

approximately the mass of Jupiter (MJup = 318M⊕) would only be able to have an Earth sized moon survive in its system at a semi-major axis of approximately 0.3 AU. This further shows that hot Jupiter are unlikely to have Earth-sized moon survive tidal disruption over the course of the lifetime of the system. In fact, the size of the moons that would be able to survive around hot Jupiters is much smaller than even

−8 the size of Mercury and have a mass of M ≈ 10 M⊕. 148

Orbit of Mercury ) U

A 1 (

10 2 a

m0 = 0.1 MJup HD 189733b m0 = 1.0 MJup

m0 = 10.0 MJup

10 10 10 8 10 6 10 4 10 2 100 102

Maximum Moon Mass to Survive (MEarth)

Fig. 4.6.— Here we highlight maximum moon mass to survive tidal disruption within 5 × 109 years versus semi-major axis of the star-planet system. Each line represents the mass of the planet from 0.1 MJup − 10 MJup as given in the legend. We calculate the lines by applying Equation 4.70 where we assume Q1 = 10, ρ1 = ρ⊕, k1 = 0.1, and that each moon is initially placed at half the hill radius of the system. Systems that fall within the colored regions above each line are additionally able to survive tidal disruption. 149 4.10 Conclusion

We have derived the change in the semi-major axis of a moon in a star-planet-moon system with the planet and moon in circular and co-planar orbits about the host star considering both gravitational perturbation and tidal friction forces to second order in masses. Our equation shows that perturbations creating forced eccentricity will make tidal friction never cease and cause inward migration of moons towards their host planet. When applied to hypothetical Earth-sized moons of hot Jupiters, the migration time of the moon is quick enough to explain the lack of observed large exomoons orbiting hot Jupiters as moons will migrate to within the Roche tidal disruption limit. For HD 189733b, the maximum moon mass to survive tidal

−8 disruption or collision with the planet within 5 Gyr is M ≈ 10 M⊕. When the moon collides with the planet, it will be absorbed into its atmosphere and there will be no evidence of the moon existing in the system. If the Roche limit of the planet-moon system is outside the radius of the planet then the moon becomes destroyed and will create a ring system around its host planet similar to the ring systems observed around all the gas giants of the Solar System. If the ring system is able to dissipate, or is not big enough to be observationally detected, there will also be no evidence of a moon ever existing in the star-planet system. Mamajek et al.(2012) have suggested that a ring system can explain the complex series of deep eclipses observed around a pre-main sequence K5 star and so perhaps, similar rings systems around hot Jupiters could be detected. A cartoon of the tidal destruction effect is shown in Figure 4.1. We additionally applied our migration equation to theoretical moons of Mercury and Venus. We find that this effect is not large enough to explain the lack of moons in the inner Solar System and other effects must be considered. 150

Chapter 5

Conclusions

5.1 Summary

In this thesis, I explored the atmospheres and orbital dynamics of hot Jupiters. This work was motivated by (1) the need to better understand the magnitude of Joule heating in hot Jupiter upper atmospheres as a product of current running through the atmosphere and the electrical conductivity structure, and (2) the lack of large moons detected around observed hot Jupiters. In Chapter2, I reviewed the stellar wind-magnetosphere interaction that drives currents in the atmospheres of planets and provided context to the calculation of electrical conductivity in the atmosphere. This current leads to Joule heating in the atmosphere as a result of momentum and energy transfer from collisions of charged particles in the current to the atmosphere. I detailed the calculation of the Pederson, Hall, and parallel conductivities and discussed the types of collisions between species that are important in the electrical conductivity calculations. The height integrated Pederson conductivity (Pederson conductance) importantly sets the size of the energy input through Joule heating. I therefore make estimates for the size of conductance 151 in the atmosphere, discuss peaks in the conductivity that set the conductance, and discuss how the conductance scales with magnetic field. In Chapter3, I detailed a hydrostatic model for the dayside upper atmosphere of hot Jupiter HD 189733b. I solved rate equations to set the density structure and temperature profile for HD 189733b and applied the model to calculate the electrical conductivity throughout the atmosphere. I utilized prior MHD models to estimate the current flowing through the atmosphere of HD 189733b and then determined the conductance and Joule heating of the atmosphere for different sized magnetic fields. I found that for Jupiter-sized magnetic fields (B = 10 G), the large conductance of the atmosphere of HD 189733b results in Joule heating rates that are smaller than other major sources of heating in the atmosphere, such as photoelectric heating. In Chapter4 I modeled the migration of hypothetical moons of hot Jupiters in star-planet-moon systems. Even for circular and pseudo-synchronous orbits, tidal friction in the moon can never cease. I find that, even if tidal friction in the star and planet are ignored, the tidal friction acting on the gravitational perturbations due to the star still causes energy to be removed from the star-planet-moon system, and goes into heating the moon. The energy removed from the system results in the migration of the moon towards the planet. In the case of hypothetical large moons of hot Jupiters, this migration is fast enough (within 1 Myr) to explain the lack of large moons observed around hot Jupiters.

5.2 Future Applications

The hot Jupiter atmosphere model developed for the research described in Chapter3 can be applied to other systems. For example, the electrical conductivity of other hot Jupiters can be examined over a wide range of system parameters. We can solve the 152 rate equations for different types of stellar spectra and different star-planet systems. This will allow us to determine the size of Joule heating over a suite of systems and gain a better understanding of all types of hot Jupiter atmospheres. The thermal structure of the atmosphere model was only solved to regions of the atmosphere where molecules begin to dominate atoms. Molecular cooling becomes important in this region and so a future addition to the model can be to add the major coolants throughout the entirety of the molecular region. This will allow us to determine the conductivity of the atmosphere to even deeper regions than we previously modeled. For extremely large magnetic fields, currents can potentially be driven very deep into the atmosphere and Joule heating may be more important in this region. This would require currents to flow through the bulk of the planet. A model of the atmosphere to deeper pressures than developed in Chapter3 is needed to estimate the size of the Joule heating in this high pressure region and if currents in these regions are possible. The equation for orbital decay time of a moon in a star-planet-moon system derived in Chapter4 can be applied to all hot Jupiter systems. Therefore, this equation can be further applied to all hot Jupiters detected to place a limit on the minimum moon mass that can survive in each system. We can further constrain the model by detailing tidal friction in the planet and star. For hot Jupiters, this tidal friction is negligible, but for other exoplanets like super Earths or mini-Neptunes, tidal friction in the planet of circular and pseudo-synchronous three body systems can be more important. This would allow us to determine the tidal migration time of moons around a wider suite of exoplanets. Additionally, the equations can be applied to star-planet-planet systems if we take the star to be the central body with two planets orbiting around the star in circular 153 orbits as done in Lithwick & Wu(2012). We would need to consider tidal friction in both planets of the system as well as gravitational perturbations from both bodies, as the ratio of the masses for star-planet-planet systems can be much less extreme than for star-planet-moon systems. Finally, these analytical equations can also be tested via numerical methods. All together, research on the atmospheres and orbital dynamics of hot Jupiters is vital towards understanding these objects that were some of the most puzzling discoveries of the 1990s. As technology advances, more detailed observations of hot Jupiters will be able to be made to confirm or reject the models, such as those in this thesis, that seek to understand their properties. Regardless, models of these systems continue to be vital towards understanding our Solar System’s place amongst other planetary systems in the Universe. References

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