Atmospheric Circulation of Eccentric Extrasolar Giant Planets

Transcription

1 Atmospheric Circulation of Eccentric Extrasolar Giant Planets Item Type text; Electronic Dissertation Authors Lewis, Nikole Kae Publisher The University of Arizona. Rights Copyright is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 15/6/218 :8:33 Link to Item

2 ATMOSPHERIC CIRCULATION OF ECCENTRIC EXTRASOLAR GIANT PLANETS by Nikole Kae Lewis A Dissertation Submitted to the Faculty of the DEPARTMENT OF PLANETARY SCIENCES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA 212

3 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Nikole Kae Lewis entitled Atmospheric Circulation of Eccentric Extrasolar Giant Planets and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. Adam P. Showman Date: 16 April 212 Roger V. Yelle Date: 16 April 212 Caitlin A. Griffith Date: 16 April 212 Yancy L. Shirley Date: 16 April 212 Jonathan J. Fortney Date: 16 April 212 Final approval and acceptance of this dissertation is contingent upon the candidate s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. Dissertation Director: Adam P. Showman Date: 16 April 212

4 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: Nikole Kae Lewis

5 4 ACKNOWLEDGEMENTS I am grateful to everyone who supported me in the efforts that lead up to this dissertation. I owe a deep debt of gratitude to my advisor, Adam Showman, for all of his input and support with research, grant and paper writing, and award nominations. I would like thank Roger Yelle for his support as I started my graduate student career and his continued support as I pursued my interest in exoplanet atmospheres. I am also grateful to my non-official advisors, Jonathan Fortney and Heather Knutson, for all of their support with my research efforts and beyond. Also many thanks to Caitlin Griffith and Yancy Shirley for agreeing to serve on my committee and share their expertise as observers. I am especially grateful for my husband, Matthew Lewis, who has been nothing but patient and supportive while I pursued my dreams and my parents, Alison and Terry Howard, who are always proud to have me as a daughter. Nikole Lewis

6 5 TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES ABSTRACT CHAPTER 1 INTRODUCTION CHAPTER 2 ATMOSPHERIC CIRCULATION OF ECCENTRIC HOT NEPTUNE GJ 436B Introduction Model Results Thermal Structure and Winds: Dependence on Metallicity Effect of eccentricity and rotation rate Light Curves and Spectra Discussion Conclusions CHAPTER 3 ORBITAL PHASE VARIATIONS OF THE ECCENTRIC GI- ANT PLANET HAT-P-2B AT 3.6 AND 4.5 MICRONS Introduction Observations Photometry Noise Pixel Parameter Intrapixel Sensitivity Correction Transit and Eclipse Fits Phase Curve Fits Stellar Variability Atmospheric Model Thermal Structure and Winds Theoretical Light Curves and Spectra Results Orbital and Transit Parameters Phase Curve and Secondary Eclipse Parameters Discussion

7 6 TABLE OF CONTENTS Continued Orbital and Transit Parameters Secondary Eclipse Depths Phase Curve Fits Conclusions CHAPTER 4 ATMOSPHERIC CIRCULATION OF THE HIGHLY ECCEN- TRIC EXTRASOLAR PLANET HD 866B Introduction Model Results Global Scale Winds and Temperatures Dissecting HD 866b s Shock Theoretical Light Curves Discussion Conclusions CHAPTER 5 CONCLUSION AND FUTURE WORK Dissertation Findings Future Work in Characterizing Exoplanet Atmospheres APPENDIX A LONGITUDE OF SUBSTELLAR AND EARTH POINTS FOR ECCENTRIC ORBITS REFERENCES

8 7 LIST OF FIGURES 1.1 Planetary mass and eccentricity as a function of semimajor axis µm light curve and brightness map for HD b Pressure temperature profiles for each metallicity case of GJ 436b Orbital diagram for GJ 436b RMS velocity as a as a function of pressure and simulated time for the 1 and 5 solar cases of GJ 436b Temperature and winds for the 1 and 3 solar metallicity cases of GJ 436b at 1 mbar, 3 mbar, and 1 bar Zonal-mean zonal winds for the 1, 3, 1, 3, and 5 solar metallicity case of GJ 436b assuming a pseudo-synchronous rotation Zonal-mean temperatures as a function of pressure and latitude for the pseudo-synchronous 1 and 3 solar cases of GJ 436b RMS velocity (colorscale) as a function of pressure and simulated time for the 1 and 5 solar cases of GJ 436b Temperature and winds for the 1 and 3 solar metallicity cases of GJ 436b assuming synchronous rotation at the 3 mbar level Zonal-mean zonal winds for the synchronous rotation (P rot = P orb ) cases at 1 and 3 solar metallicity GJ 436b atmospheres Planet/Star flux ratio as a function of orbital phase in each of the Spitzer bandpasses for the 1 and 5 solar metallicity cases of GJ 436b Flux per unit frequency as a function of wavelength for the 1 and 5 solar metallicity cases of GJ 436b Flux per unit wavelength as a function of wavelength for the 1 and 5 solar metallicity cases of GJ 436b RMS vertical velocity as a function of pressure for both the 1 and 5 solar metallicity cases of GJ 436b at secondary eclipse Raw 3.6 µm HAT-P-2 photometry Raw 4.5 µm HAT-P-2 photometry Standard deviation of the residuals and required mapping time as a function of number of nearest neighbors Orbital diagram for HAT-P-2b Average temperature and RMS velocity as a function of time from periapse from the HAT-P-2b atmospheric model

9 8 LIST OF FIGURES Continued 3.6 Temperature and winds at 1 mbar level of the HAT-P-2b atmospheric model Temperature as a function of pressure and longitude from the HAT- P-2b atmospheric model near periapse and secondary eclipse Theoretical light curves in the Spitzer bandpasses from the HAT-P-2b atmospheric model Emitted flux as a function of wavelength and orbital phase from the HAT-P-2b atmospheric model Observed 3.6 and 4.5 µm phase curves for the HAT-P-2 system HAT-P-2b secondary eclipse observations at 3.6 and 4.5 µm HAT-P-2b transit observations at 3.6 and 4.5 µm Theoretical light curves for HAT-P-2b at 3.6 and 4.5 µm assuming a range of atmospheric opacities Pressure-Temperature profiles as function of orbital phase from the HAT-P-2b atmospheric simulation Average temperature and RMS horizontal velocity as a function of time relative to periapse passage Zonal-mean zonal winds for HD 866b near apoapse and periapse Temperature and horizontal winds at the 3 mbar level of our HD 866b model Temperatures and vertical wind speeds for the atmospheric shock Mach numbers for the atmospheric shock Planet/Star flux ratio as a function of time from periapse passage for each rotation period case A.1 Geometry of eccentric orbit

10 9 LIST OF TABLES 2.1 GJ436A/b parameters Orbital parameters for HAT-P-2b derived from joint fit of 3.6 and 4.5 µm data Phase curve and secondary eclipse parameters

11 1 ABSTRACT This dissertation explores the three-dimensional coupling between radiative and dynamical processes in the atmospheres of eccentric extrasolar giant planets GJ 436b, HAT-P-2b, and HD 866b. Extrasolar planets on eccentric orbits are subject to time-variable heating and probable non-synchronous rotation, which results in significant variations in global circulation and thermal patterns as a function of orbital phase. Atmospheric simulations for the low eccentricity (e =.15) Neptune sized planet GJ 436b reveal that when Neptune-like atmospheric compositions are assumed day/night temperature contrasts and equatorial jet speeds are significantly increased relative to models that assume a solar-like composition. Comparisons between our theoretical light curves and recent observations support a high metallicity atmosphere with disequilibrium carbon chemistry for GJ 436b. The analysis of full-orbit light curve observations at 3.6 and 4.5 µm of the HAT-P-2 system reveal swings in the planet s temperature of more than 9 K during its significantly eccentric (e =.5) orbit with a four to six hour offset between periapse passage and the peak of the planet s observed flux. Comparisons between our atmospheric model of HAT-P-2b and the observed light curves indicate an increased carbon to oxygen ratio in HAT-P-2b s atmosphere compared to solar values. Atmospheric simulations of the highly eccentric (e =.9) HD 866b show that flash-heating events completely alter planetary thermal and jet structures and that assumptions about the rotation period of this planet could affect the shape of light curve observations near periapse. Our simulations of HD 866b also show the development an atmospheric shock on the nightside of the planet that is associated with an observable thermal signature in our theoretical light curves. The simulations and observations presented in this dissertation mark an important step in the exploration of atmospheric circulation on the more than 3 exoplanets known to posses significantly non-zero eccentricities.

12 11 CHAPTER 1 INTRODUCTION The number of planets detected outside of our solar system has increased at an almost exponential rate over the past 1 years. To date, more than 7 exoplanets have been detected through radial velocity and transit surveys predominately. The current exoplanet population differs in many ways from the planets in our own solar system. There exists a large population of exoplanets with masses similar to that of Jupiter that orbit close to their host stars at distances less than.1 AU as shown in top panel of Figure 1.1. These close-in Extrasolar Giant Planets (EGPs) are generally assumed to have roughly circular orbits and synchronous rotation rates as the result of tidal interactions with their host star (Halbwachs et al., 25). However, 2% of close-in EGPs have orbital eccentricities greater than.1. Nearly 5% of the current exoplanet population has an orbital eccentricity greater than.1. The bottom panel of Figure 1.1 shows the great diversity in planetary eccentricity as a function of semimajor axis for the current exoplanet population. In our own solar system Mercury is the only planet to posses an orbital eccentricity greater than.1. We therefore cannot look to solar system analogs to understand the atmospheric properties of the vast majority of the current exoplanet population. Instead, we must rely on a combination of observations and theoretical models to understand the atmospheric radiative, dynamical, and chemical properties of these strange new worlds. Among the current exoplanet population there are over 2 exoplanets that have been determined to transit their host stars as seen from earth. There are generally two key phases to the orbit of a transiting exoplanet, the primary and secondary eclipses, which occur when the planet passes in-front of and behind its host star respectively. The primary eclipse, or transit, allows the observer to determine the radius of the planet, which when combined with mass estimates from radial-velocity

13 Planet Mass (M J ) Semimajor Axis (AU) Eccentricity Semimajor Axis (AU) Figure 1.1 Top: Planetary mass (log scale) in units of Jupiter masses (M J ) as a function of orbital semimajor axis. Bottom: Orbital eccentricity as a function of orbital semimajor axis. Data taken from current exoplanet population as reported on and limited to planets with semimajor axes smaller than 5 AU.

14 13 measurements gives an estimate of planetary density. Transmission spectra can also be obtained during the transit event to probe the atmospheric composition near the limb of the planet. The secondary eclipse is especially favorable to determining the contribution of the planet s flux to the overall flux from the system at infrared wavelengths, which can then be used to determine the planet s dayside temperature. The overall shape of an infrared light curve can also be useful in finding the longitudinal position of the peak brightness of the planet. The first system for which the brightness of the planet was measured as a function of longitude is HD b. The infrared light curve for HD b from Knutson et al. (27) shows that the hottest region of the planet is 3 east of the substellar point (Figure 1.2). This shift in the thermal profile of the planet is a clear indication of the presence of winds in the atmosphere of HD b. Light curves for several other close-in EGPs have also been obtained and all show varying degrees of thermal homogenization by winds (e.g. Cowan et al. (27); Harrington et al. (26)).

15 Figure 1.2 Left: 8 µm Spitzer light curve for HD b from Knutson et al. (27). The secondary eclipse of the planet occurs at.5 phase. Right: A map of brightness estimates along the surface of the planet derived from the light curve. Note that the brightest (hottest) point on the planet is displaced from the substellar point. 14

16 15 Atmospheric circulation models of these close-in EGPs help us understand what role winds play in these observed thermal variations. To date, most atmospheric circulation models have focused on the specific cases of HD 29458b and HD b (Showman and Guillot, 22; Showman et al., 28, 29; Cho et al., 23, 28; Cooper and Showman, 25, 26; Langton and Laughlin, 27; Dobbs-Dixon and Lin, 28; Dobbs-Dixon et al., 21; Menou and Rauscher, 29; Rauscher and Menou, 21). The study by Langton and Laughlin (28) was the first to explore atmospheric circulation for exoplanets besides HD 29458b and HD b, specifically exoplanets on eccentric orbits. Although there are similarities between these models, there are also significant differences. Current atmospheric circulation models of close-in EGPs all predict that a small number ( 3) of broad jets will develop and that there will be a significant day/night temperature contrast in the tidally locked case. Each of these models use different radiative forcing and radiative transfer schemes that result in differences in the overall global temperature distribution and the range of temperatures from the day to the night side. The two-dimensional models (Cho et al., 23, 28; Langton and Laughlin, 27, 28) tend to produce a strong westward jet at the equator and polar vortices, while the three-dimensional models (Showman and Guillot, 22; Showman et al., 28, 29; Cooper and Showman, 25, 26; Dobbs-Dixon and Lin, 28; Menou and Rauscher, 29; Rauscher and Menou, 21) tend to produce an eastward equatorial jet with no polar vortices. The inclusion of non-zero obliquity and eccentricity (Langton and Laughlin, 27, 28) results in significant changes in the the thermal profile of the planet and appears to introduce more turbulent structures to the atmosphere. The development of these models, even with their differences in approaches and results, have helped to describe the atmospheric dynamics on these close-in EGPs The primary goals of the research presented here are: 1) Determine how changes in planetary orbital, rotational, and compositional characteristics affect atmospheric circulation patters on close-in giant exoplanets; 2) Explain current visible and infrared observations and guide future observations of close-in giant exoplanets. We have met these goals by focusing our efforts on three specific members of eccentric

17 16 transiting exoplanet population: GJ 436b (e =.15), HAT-P-2b (e =.5), and HD 866b (e =.9). These planets probe a wide range in planetary mass, radius, and effective temperature as well as orbital eccentricity. A more specific goal of this project was to determine how the time-variable heating and non-synchronous rotation rates experienced by exoplanets on eccentric orbits affect global atmospheric circulation patterns and emergent flux as a function of orbital phase. We have developed atmospheric models for all three of these planets using an advanced threedimensional coupled radiative transfer and general circulation model. In addition to our modeling efforts, we have analyzed the Spitzer full orbit light curves for the HAT-P-2 system at 3.6 and 4.5 µm to determine planetary flux as a function of orbital phase. Chapter 2 presents our study of GJ 436b, which focuses on how atmospheric chemistry affects global circulation and thermal patterns. Chapter 3 gives the results from our analysis of the 3.6 and 4.5 µm full orbit light curves and compares these results with theoretical predictions from our atmospheric model of HAT-P-2b. Chapter 4 details our HD 866b atmospheric model and explores how rapid heating of the planet and assumptions about planetary rotation rate affect global circulation patterns and planetary flux as a function of orbital phase. Chapter 5 summarizes the finding of this dissertation and discusses future work related to exoplanet atmospheric modeling. We note that the majority of the content in Chapter 2 first appeared in the Astrophysical Journal as Lewis et al. (21).

18 17 CHAPTER 2 ATMOSPHERIC CIRCULATION OF ECCENTRIC HOT NEPTUNE GJ 436B 2.1 Introduction The hot Neptune GJ 436b was first discovered by Butler et al. (24) and later determined to transit its host star as seen from earth by Gillon et al. (27b). Since the discovery of its transit and subsequent secondary eclipse, GJ 436b has become a popular target for Hubble (Bean et al., 28; Pont et al., 29a) and Spitzer (Gillon et al., 27a; Deming et al., 27; Demory et al., 27; Stevenson et al., 21) observations as well as modeling efforts (Spiegel et al., 21; Madhusudhan and Seager, 211). Given GJ 436b s mass (M p =.729M J ) and radius (R p =.3767R J ), its interior must contain significant quantities of heavy elements in addition to hydrogen and helium (Adams et al., 28; Figueira et al., 29; Rogers and Seager, 21; Nettelmann et al., 21). This raises the possibility that, like Uranus and Neptune, whose atmospheric C/H ratios lie between 2 and 6 times solar (Gautier et al., 1995), the atmosphere of GJ 436b is highly enriched in heavy elements. This makes GJ 436b an excellent case study for atmospheric chemistry, radiative transfer, and global circulation that should differ significantly from the well studied hot Jupiters HD 29458b and HD b. Observations of HD b using the Spitzer Space Telescope provided the first clear evidence for atmospheric circulation on an extrasolar planet (Knutson et al., 27, 29). Most efforts to model atmospheric circulation for extrasolar planets have focused on hot Jupiters, specifically HD189733b and HD29458b (Showman and Guillot, 22; Showman et al., 28, 29; Cho et al., 23, 28; Cooper and Showman, 25, 26; Dobbs-Dixon and Lin, 28; Menou and Rauscher, 29; Rauscher and Menou, 21). Only a handful of studies have specifically investigated the effects of non-synchronous rotation (Cho et al., 28; Showman et al.,

19 18 29), non-zero obliquity (Langton and Laughlin, 27), and non-zero eccentricity (Langton and Laughlin, 28). The possible effect of atmospheric composition, and hence opacity, on circulation patterns that may develop on extrasolar planets has been investigated to some extent by Dobbs-Dixon and Lin (28) and Showman et al. (29), but largely ignored in most of the current two-dimensional and three-dimensional atmospheric models. Atmospheric composition is key in determining opacity and radiative timescales that play a crucial role in the development of circulation on these planets. Here we present three-dimensional atmospheric models for GJ 436b that incorporate both equilibrium chemistry and realistic non-gray radiative transfer. Although the actual composition of GJ 436b s atmosphere is likely to deviate from an equilibrium chemistry solution as shown from the secondary eclipse observations of Stevenson et al. (21), our investigation still serves to explore the effect metallicity can play in controlling the atmospheric circulation not only on GJ 436b but for a broad range of gaseous extrasolar planets in a similar temperature range. Our models are not constrained to match measured chemical abundances and temperatures, but instead provide a systematic look at how changes in atmospheric metallicity over the range from 1 times to 5 times solar values affects the basic thermal and dynamical structure of the planet s atmosphere. We do not expect that spectra and light curves from our model will provide a match to observational data, but instead illuminate some of the underlying atmospheric physics responsible for current observations and suggest areas of focus for future observations. Section 2.2 gives an overview of the three-dimensional coupled radiative transfer and atmospheric dynamics model used in this study. Section 2.3 presents the global thermal structures and winds that develop in each of our models along with predicted light curves and emission spectra. Sections 2.4 and 2.5 provide a brief discussion of the results and final conclusions.

20 Model The atmospheric model used in this study is a three-dimensional (3D) coupled radiative transfer and dynamics model that was specifically developed with the study of extrasolar planetary atmospheres in mind. The Substellar and Planetary Atmospheric Radiation and Circulation (SPARC) model is described in detail in Showman et al. (29) as applied to HD189733b and HD29458b. A basic overview of the SPARC model along with the specific changes made to the model setup for GJ 436b are presented here for completeness. The SPARC model employs the MITgcm (Adcroft et al., 24) to treat the atmospheric dynamics using the primitive equations, which are valid in stably stratified atmospheres where the horizontal dimensions of the flow greatly exceed the vertical dimension. For GJ 436b, the horizontal length scale of the flow is 1 7 m while the vertical scale height of the atmosphere is 3 km. The simulations presented here take advantage of the cubed-sphere grid (Adcroft et al., 24) at a resolution of C32 (roughly in latitude and longitude) to solve the relevant dynamic and energy equations. The vertical dimension in these simulations spans the pressure (p) range from 2 bar to 2 µbar with 47 vertical levels, evenly spaced in log(p). The boundary conditions in our simulations are an impermeable surface at the bottom and a zero pressure surface at the top both of which are free slip in horizontal velocity. We have coupled the MITgcm to the non-gray radiative transfer model of Marley and McKay (1999) to realistically determine the magnitude of heating/cooling at each grid point. The radiative transfer model, a two-stream version of the Marley and McKay (1999) plane-parallel code, assumes local thermodynamic equilibrium and includes intensities over the wavelength range from.26 to 3 µm. The opacity at each pressure-temperature-wavelength grid point is tabulated using the correlated-k method (Goody et al., 1989). Our extensive opacity database is described in Freedman et al. (28). The chemical mixing ratios, which are computed assuming thermochemical equilibrium, are calculated as in Lodders and Fegley (22, 26). Calculated opacities assume a gaseous composition without particu-

21 2 late matter and account for the possibility of chemical rainout. Because GJ 436b plausibly has an atmospheric chemistry that is enhanced in heavy elements, we developed opacity tables for 3 times (3 ), 1 times (1 ), 3 times (3 ), and 5 times (5 ) solar metallicity in addition to the 1 times (1 ) solar metallicity opacity table. In the enhanced metallicity opacity tables, all elements other than hydrogen and helium are assumed to be enhanced by the same factor over current solar values. The opacity databases of Freedman et al. (28) were updated to include the opacity effects of CO 2, which is an important carbon bearing species at higher metallicities. The full opacity tables are divided into 3 wavelength bins as outlined in Showman et al. (29). This binning of opacities allows for greater computational efficiency while only introducing small (< 1%) deviations from the net radiative flux calculated with higher resolution opacity tables. For each model, the winds are assumed to initially be zero everywhere and each column of the grid is assigned the same pressure-temperature profile. This initial pressure-temperature profile is derived from one-dimensional radiative-equilibrium calculations performed using the radiative transfer code in the absence of dynamics. Figure 2.1 shows the pressure-temperature profiles derived for each metallicity case of GJ 436b. These pressure-temperature profiles were derived using the methodology presented in Fortney et al. (28, 25). The physical properties assumed for GJ 436b and its host star (GJ436A) are presented in Table 2.1. Using these planetary and stellar parameters, the effective temperature (T eff ) of GJ 436b is calculated to be 649 K, assuming planet-wide redistribution of the incoming stellar flux. This T eff corresponds to a mean photospheric level 1 of 1 to 1 mbar depending on the assumed metallicity of the atmosphere (Figure 2.1). Because GJ 436b is known to have an eccentric orbit, we incorporated the effects of non-synchronous rotation and time-varying distance from the host star into the SPARC model. The most probable rotation rate for GJ 436b was determined using 1 Defined in this context as the atmospheric pressure where the local temperature equals the effective temperature.

22 Figure 2.1 Initial pressure temperature profiles assumed for each metallicity case of GJ 436b. Lines of equal abundance for CH 4 vs CO and NH 3 vs N 2 are shown to highlight the dominant carbon and nitrogen bearing species at each pressure level for each metallicity case. As the metallicity in the atmosphere is increased form 1 to 5 solar, the dominant carbon bearing species changes from CH 4 to CO. Diamonds represent the level of the mean photosphere (T = T eff ), which decreases in pressure as the metallicity is increased. Profiles assume planet-wide redistribution of absorbed incident energy. 21

23 22 Table 2.1 GJ436A/b parameters. Parameter Value a R p (R J ).3767 M p (M J ).729 g (m s 2 ) a (AU).2872 e.15 ϖ (deg) 343 P orb (days) P rot (days) R (R ).464 M (M ).452 T eff (K) 335 a Planetary and stellar parameters taken from Torres et al. (28). Values for e and ϖ were taken from Deming et al. (27). the following pseudo-synchronous rotation relationship presented in Hut (1981): [ ] (1 + 3e P rot = P e4 )(1 e 2 ) 3/2 orb e e4 + 5 (2.1) 16 e6 where P rot is the planetary rotation rate, P orb is the orbital period of the planet, and e is the eccentricity of the planetary orbit. obliquity of the planet is assumed to be zero. In all cases considered here the The time-varying distance of the planet with respect to its host star, r(t), is determined using Kepler s equation (Murray and Dermott, 1999) and used to update the incident flux on the planet at each radiative timestep. A diagram of GJ43b s orbit is presented in Figure 2.2. To test the impact of pseudo-synchronous rotation and time-varying stellar insolation, additional simulations for the 1 and 3 solar metallicity cases were performed assuming synchronous rotation and zero eccentricity. In our models, for computational efficiency, the radiative timestep used to update the radiative fluxes is longer than the timestep used to update the dynamics. Generally, as we increased the metallicity of the atmosphere, progressively shorter radiative and dynamical timesteps were needed to maintain stability. For the 1

24 Figure 2.2 Orbit of GJ 436b. The true anomaly, f, represents the angular distance of the planet from periapse. Assuming a longitude of pericenter, ϖ, of 343 from Deming et al. (27), transit occurs at f = 17 and secondary eclipse occurs at f = 73. Dots along the orbital path represent points where data was extracted to produce Figures 2.1, 2.11, and Colored dots represent points near periapse (red), transit (orange), apoapse (green), and secondary eclipse (blue), which correspond to the colored spectra presented in Figures 2.11 and Figure is to scale with the small purple dot after periapse representing the size of GJ 436b in relation to its host star and orbit. 23

25 24 and 3 solar metallicity cases a dynamic timestep of 25 s and a radiative timestep of 2 s were used. The 1 and 3 solar metallicity cases required a dynamic timestep of 2 s and a radiative timestep of 1 s while the 5 solar case required a dynamic timestep of 15 s and a radiative timestep of 6 s. Timestepping in our simulations is accomplished through a third-order Adams-Bashforth scheme (Durran, 1991). We applied a fourth-order Shapiro filter in the horizontal direction to both velocity components and the potential temperature over a timescale equivalent to twice the dynamical timestep in order to reduce small scale grid noise while minimally affecting the physical structure of the wind and temperature fields at the large scale. We integrated each of our models until the velocities reached a stable configuration. Figure 2.3 show the root mean square (RMS) velocity as a function of pressure and simulated time, calculated according to: (u2 + v V RMS (p) = 2 ) da (2.2) A where the integral is a global (horizontal) integral over the globe, A is the horizontal area of the globe, u is the east-west wind speed, and v is the north-south wind speed. The high-frequency variations in the RMS velocity seen in the upper levels of both the 1 and 5 solar cases are largely due to variation in the incident stellar flux associated with the eccentric orbit of GJ 436b. Notice that, in the observable atmosphere (pressures less than 1 mbar), the orbit-averaged winds become essentially steady within 25 Earth days for solar metallicity and 1 Earth days for 5 solar metallicity. RMS wind speeds typically reach 1 km s 1 at photosphere levels. Any further increases in wind speeds will be small and confined to pressure well below the mean photosphere so as not to affect any synthetic observations derived from our simulations. As outlined in Showman et al. (29) the energy available for the production of winds is limited largely by the global available potential energy within the atmosphere and to some extent energy losses due to the Shapiro filter which acts as a hyperviscosity. A full discussion of the energetics of our simulated GJ 436b-like atmosphere is left for a future paper.

26 25 1x solar pressure (bar) simulated time (Earth days) 5x solar (m s 1 ) pressure (bar) (m s 1 ) simulated time (Earth days) Figure 2.3 RMS velocity (colorscale) as a function of pressure and simulated time for the 1 (top) and 5 (bottom) solar cases of GJ 436b. The RMS velocity at each pressure level is calculated from the instantaneous wind speeds recorded every s. Simulations are continued until the winds reach a relatively flat profile at all pressure levels.

27 Results The following sections overview the key results from the study of GJ 436b s atmospheric circulation at 1, 3, 1, 3, and 5 solar metallicity. Both the thermal structure and winds in these simulations have a strong dependence on the assumed composition of the atmosphere for GJ 436b. Additionally, theoretical light curves and spectra are produced from our 3D model atmospheres and compared with available data Thermal Structure and Winds: Dependence on Metallicity Figure 2.4 presents snapshots of the temperature and wind fields at three pressure levels in the atmosphere for the 1 and 3 solar cases near secondary eclipse when the full day-side of the planet faces Earth (Figure 2.2). Overall, the 3 solar case is significantly ( 1 K) warmer than the 1 solar case at each pressure. The increased atmospheric opacity that comes with metallicity enhancements leads to an upward shift in the pressure-temperature profiles. This effect is self-consistently generated in the three-dimensional model integrations but can also be seen in the one-dimensional radiative-equilibrium solutions shown in Figure 2.1. Overall, the day/night temperature contrast in the upper layers of the atmosphere ( 1 mbar) and the equator/pole temperature contrast deeper in the atmosphere ( 3 mbar) increase with atmospheric metallicity. However, because the pressure at a given optical depth is smaller at high metallicity than low metallicity, the regions that develop significant day/night temperature contrasts shifts to higher altitude as metallicity increases. At the 1 bar level, the equator/pole temperature contrast in the 3 solar case is smaller than that in the 1 solar case. This lack of a strong temperature contrast at 1 bar in the 3 solar case occurs because this pressure is at a greater optical depth in the 3 solar case than in the 1 solar case, and thus occurs below the levels with the strongest heating/cooling. It is also informative to compare the flow patterns indicated by the arrows in Figure 2.4 between the 1 and 3 solar cases. The overriding feature in all sim-

28 27 8 1x solar, 1 mbar x solar, 1 mbar latitude ( ) latitude ( ) longitude ( ) (K) longitude ( ) (K) 1x solar, 3 mbar 3x solar, 3 mbar latitude ( ) latitude ( ) longitude ( ) 44 (K) longitude ( ) (K) 55 1x solar, 1 bar 3x solar, 1 bar latitude ( ) latitude ( ) longitude ( ) (K) longitude ( ) (K) Figure 2.4 Temperature (colorscale) and winds (arrows) for the 1 (left) and 3 (right) solar metallicity cases of GJ 436b. For both 1 and 3 solar cases the thermal structure and winds are shown at the 1 mbar (top), 3 mbar (middle), and 1 bar (bottom) levels of the simulation. The longitude of the substellar point is indicated by the solid vertical line in each panel. Each panel is a snap shot of the atmospheric state taken near secondary eclipse (f = 73, Figure 2.2). The horizontal resolution of these runs is C32 (roughly in longitude and latitude) with 47 vertical layers.

29 28 ulations is the development of a prograde (eastward) flow at low pressure. The flow patterns in the 3 solar case exhibit clear wavelike structures outside of the equatorial region at the 1 bar, 3 mbar, and 1 mbar levels. At the 1 bar level, the flow is predominately westward in the 3 solar case and predominately eastward in the 1 solar case. In both the 1 and 3 solar metallicity cases the strength of the winds indicated by the length of the wind vectors in Figure 2.4 is a stronger function of latitude than longitude, except at the highest levels (1 mbar) in the 3 solar case, which shows a significant day/night temperature contrast similar to what was seen in the simulations of the tidally locked hot Jupiters HD189733b and HD29458b from Showman et al. (29). We find that the atmospheric metallicity plays a key role in determining the jet structure for a planet with temperatures similar to those expected on GJ 436b. This is demonstrated clearly in Figure 2.5, which shows the zonal-mean zonal wind 2 versus latitude and pressure for each of the five atmospheric metallicities for GJ 436b considered in this study. In the 1 solar case, strong high-latitude jets develop in the atmosphere with a weaker equatorial jet. Increasing the metallicity of the planet to 3 solar causes a strengthening of the equatorial jet and a weakening of the high-latitude jets. Once the metallicity of the atmosphere is increased to 1 solar or more, the high-latitude jets disappear and the equatorial jet becomes dominant. Overall, the maximum zonal wind speed increases with metallicity from roughly 13 m s 1 in the 1 solar case to over 2 m s 1 in the 5 solar case. The flow is subsonic everywhere in our 1, 3, and 1 solar metallicity cases. In our 3 and 5 solar metallicity cases, the winds are subsonic throughout most of the domain, but they become marginally supersonic at the very top of the domain (at pressures 1 mbar) in the equatorial region. Hydraulic jumps similar to those seen in the HD189733b and HD29458b cases presented in Showman et al. (29) are present in the supersonic regions of the atmosphere in the 3 and 5 solar metallicity cases (see 1 mbar level of the 3 solar case in Figure 2.4). It is important 2 That is, the longitudinally averaged east-west wind, where eastward is defined positive and westward negative. See Holton (24).

30 29 1x solar 3x solar pressure (bar) pressure (bar) latitude ( ) (m s 1 ) latitude ( ) (m s 1 ) 2 1x solar x solar pressure (bar) latitude ( ) (m s 1 2 ) pressure (bar) 5x solar latitude ( ) (m s 1 ) pressure (bar) latitude ( ) (m s 1 ) Figure 2.5 Zonal-mean zonal winds for the five atmospheric metallicities considered in this study for GJ 436b assuming pseudo-synchronous rotation. The wind speeds presented here represent 1 day averages of the zonal winds taken after each simulation was considered to have reached an equilibrium state. The colorbar shows the strength of the zonally averaged winds in m s 1. Contours are spaced by 1 m s 1. Positive wind speeds are eastward, while negative wind speeds are westward. Note the significant change in the jet structure as a function of atmospheric metallicity.

31 3 to note that the flow in all of the metallicity cases are predominately eastward at pressure less than 1 bar and predominately westward at pressures greater than 1 bar. Momentum conservation requires that the eastward momentum in the upper atmosphere of these simulations comes from the deeper atmospheric layer, which requires the development of the mean westward flow at depth. The detailed mechanisms responsible for this momentum and the jet pumping mechanisms themselves will be discussed in a future paper. In rapidly rotating atmospheres, atmospheric temperature gradients are linked to winds by dynamical balances, so it is interesting to next examine the atmospheric temperature structure in our simulations. Figure 2.6 shows the zonal-mean atmospheric temperature versus latitude and pressure for the 1 and 3 solar cases. In both cases, the deepest isotherms (for temperatures exceeding 12 K) are flat, but isotherms between 6 and 11 K are bowed upward, indicating a warm equator and cool poles. The relationship between this structure and the winds can be understood with the thermal-wind equation, which relates the latitudinal temperature gradients to the zonal wind and its derivative with pressure: ( ) 2u tan φ u + 2Ω sin φ a ln p = R T a φ, (2.3) where u, φ, a, Ω, p, R and T are the zonal wind speed, latitude, planetary radius, planetary rotation rate, pressure, specific gas constant, and temperature, respectively. The latitudinal temperature gradient on the right-hand side is evaluated at constant pressure. This relationship derives from taking a vertical (pressure) derivative of the meridional momentum equation for a flow where the predominant zonalmean meridional momentum balance is between the Coriolis, pressure-gradient, and curvature terms (called gradient-wind balance; see Holton 24, p ). From Equation (2.3), one expects that, away from the equator, regions exhibiting vertical shear of the zonal wind must also exhibit latitudinal gradients of temperature. Comparing Figures 2.5 and 2.6 confirms that this is indeed the case: in the midlatitudes, the 1 solar case exhibits the greatest vertical shear of the zonal wind in the pressure range of.1 to 3 bars (Figure 2.5), and this is the same pressure

32 31 1x solar pressure (bar) latitude ( ) 3x solar 6 6 (K) pressure (bar) latitude ( ) (K) 8 6 Figure 2.6 Zonal-mean temperatures (colorscale) as a function of pressure and latitude for the pseudo-synchronous 1 (top) and 3 (bottom) solar cases of GJ 436b. Contours represent isotherms and are spaced by 5 K. Gradients in temperature along an isobaric surface tend to drive zonal winds according to equation (2.3). The 3 solar case shows stronger thermal gradients at lower pressures that extend into lower latitudes when compared with the 1 solar case. This change in the thermal gradient structure can be directly related to the zonal wind profiles seen in Figure 2.5.

33 32 range over which the greatest latitudinal temperature gradients occur (Figure 2.6). For the 3 solar case, in the mid-latitudes, the regions exhibiting significant wind shear are shifted upward, occurring from.1 to less than 1 bar, and likewise this is the pressure range where significant latitudinal temperature gradients exist. The upward shift in the temperature gradients (and hence winds) at greater metallicity is the direct result of enhanced atmospheric opacities, which lead to shallower atmospheric heating (Fortney et al., 28; Dobbs-Dixon and Lin, 28). It is interesting to characterize the variations in wind speed that occur throughout the eccentric orbit due to the time-variable incident stellar flux. Figure 2.7 shows the RMS wind speed as a function of pressure and simulated time with outputs every two hours for a ten Earth day period after the simulations had reached equilibrium. Overall, wind speeds in these simulations of GJ 436b vary with a frequency roughly equal to the orbital period at pressures above the mean photospheric level 3 (roughly 1 mbar in the 1 solar case and 1 mbar in the 5 solar case). Wind speeds are fairly constant at pressures below the mean photosphere for each metallicity case of GJ 436b. The vertical lines in Figure 2.7 indicate the time of periapse passage, which are followed by a peak in the RMS wind speeds. The variation in the wind speeds between periapse and apoapse is several times larger in the 5 solar case compared with the 1 solar case. The higher atmospheric metallicity cases show greater variability in wind speeds as function of orbital phase, which could affect light curves especially if hotter or more eccentric systems are considered. 3 In Figure 2.3, the high-frequency fluctuations appear to occur on periods of tens of Earth days, but this is an artifact that results from aliasing of the sampling frequency of s with the orbital period.

34 33 1x solar pressure (bar) simulated time (Earth days) 5x solar (m s 1 ) pressure (bar) simulated time (Earth days) (m s 1 ) Figure 2.7 RMS velocity (colorscale) as a function of pressure and simulated time for the 1 (top) and 5 (bottom) solar cases of GJ 436b as calculated from highcadence simulation outputs every 72 s. Vertical lines indicate periapse passage. Overall, wind speeds in the atmosphere vary with a period equal to the orbital period. The peak in the wind speeds typically occurs 4-8 hours after periapse passage.

35 Effect of eccentricity and rotation rate The models presented in Section assume an eccentric orbit (hence time-variable stellar irradiation) and adopt the pseudo-synchronous rotation rate given in Table 2.1. Here, we explore the effect of eccentricity and rotation rate on the circulation by comparing our standard cases (Section 2.3.1) to cases with synchronous (rather than pseudo-synchronous) rotation rates and zero eccentricity. Figure 2.8 presents the thermal structure and winds at the 3 mbar level for the 1 and 3 solar cases assuming synchronous rotation (P rot = P orb ). The top panels of Figure 2.8 assume the nominal eccentric orbit of GJ 436b while the bottom panels assume a circular orbit with the nominal semimajor axis of GJ 436b (Table 2.1). The assumption of synchronous rotation and/or a circular orbit has little effect on the overall thermal structure and wind patterns that develop in these simulations. This is presumably because the eccentricity of GJ 436b s orbit is modest (e =.15) and changes in the average stellar flux and pseudo-synchronous rotation rate from a circularized and tidally locked orbit are small. Figure 2.9 presents the zonal-mean zonal winds for these same four cases (1 and 3 solar metallicity, with eccentricity of or.15, all using the synchronous rotation period). Overall, the jet structures differ little from the nominal cases. It is interesting to note that the eastward equatorial jet in the 3 solar synchronous rotation cases has a maximum wind speed that is 2 m s 1 faster than what is seen in the non-synchronous case. Because GJ 436b has a relatively small eccentricity orbit the effects of non-synchronous rotation and time-variable heating are only small perturbations on the synchronous rotation and circular orbit cases. Planets with higher eccentricities are likely to show a larger variation in the circulation patterns that develop compared with circularized and synchronous cases.

36 35 8 1x solar, e=.15, 3 mbar x solar, e=.15, 3 mbar latitude ( ) longitude ( ) (K) 44 latitude ( ) longitude ( ) (K) 1x solar, e=, 3 mbar 3x solar, e=, 3 mbar latitude ( ) latitude ( ) longitude ( ) (K) longitude ( ) (K) Figure 2.8 Temperature (colorscale) and winds (arrows) for the 1 (left) and 3 (right) solar metallicity cases of GJ 436b assuming synchronous rotation at the 3 mbar level. The top panels represent simulations where synchronous rotation was assumed, but the nominal eccentric orbit was maintained. The bottom panels represent simulations where synchronous rotation and a circular orbit with the nominal semimajor axis (Table 2.1) were assumed. In the eccentric cases (top) the panels represent a snap shot taken near secondary eclipse (f = 73, Figure 2.2). The solid vertical line in each panel represents the longitude of the substellar point.