Can We Predict the Composition of an Exoplanet?

Can we predict the composition of an exoplanet?

A Thesis

Presented in Partial Fulﬁllment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University

Joseph Gregory Schulze, B.S.

Graduate Program in School of Earth Sciences

The Ohio State University

2020

Master’s Examination Committee:

Dr. Wendy Panero, Advisor Dr. Joachim Moortgat Dr. Ji Wang c Copyright by

Joseph Gregory Schulze

2020 Abstract

Is the Earth unique in its habitability, or is it just one of many life-hosting planets in the universe? The key to answering this question lies in determining how rocky planets form and evolve, which can be inferred from how similar/dissimilar the compositions of such planets are to their host stars. For instance, the terrestrial planets formed from a disk made of the same material as the Sun. The compositions of Earth, Venus, and Mars are consistent with the relative amounts of the major rock- building materials (Fe, Mg, Si) found in the Sun. Mercury, however, is much more iron-enriched relative to the Sun. This implies that it underwent a different formation/evolutionary pathway, likely a single or series of catastrophic mantle-stripping collisions during the late stages of its formation. In short, Mercury and Earth have two very different histories, which is realized in how their compositions deviate from the Sun’s. Therefore, understanding the relative importance of various formation and evolution processes for rocky exoplanets hinges on first determining how consistent the compositions of these planets are with their host stars. In this work, I develop a statistical framework for assessing the degree to which super-Earth’s re- flect the refractory compositions of their host stars. I implement this framework on the 8 best-measured super-Earths for which the host star’s composition is known and explore the possibility of secondary atmospheres to explain super-Earths with an apparent iron-depletion relative to their host.

ii Acknowledgments

I ﬁrst want to thank my advisor, Dr. Wendy Panero. I am exceptionally grateful for the freedom Wendy has given me to explore the topics I ﬁnd most exciting while also preventing me from going down endless rabbit holes. This thesis would not have been possible without her. I am excited to continue to grow as a scientist under her supervision as a Ph.D. student. I would also like to extend thanks to my committee members, Dr. Ji Wang and Dr. Joachim Moortgat, for their knowledge and insights provided during this research.

I would also like to thank the members of the Panero Group and the Planet Group for their interest in my work. Working with these talented scientists has greatly expanded my research and exposed me to new ways of thinking about the same research questions.

I wish to thank my parents, Tammy and Bob, and my siblings, Brad, Jen, Gabbert, and Millie for sitting through several practice talks and listening to my many exoplanet rambles. I would like to give a special shout-out to my Dad for ﬁrst piquing my interest in exoplanets by giving me my ﬁrst astronomy book, binge-watching and re-watching The Universe with me, and many, many late-night discussions. Thank you, Mr. Dad.

Last, I want to thank my fellow SES graduate students for helping me navigate the waters of grad school and for providing a fun social outlet. I would especially like to thank Sam and her four-legged companion (aka The Beast) for their great presentation and writing suggestions, for ﬁnishing and restructuring my sentences, and for some much needed comedic relief during conﬁnement. Now back to chorin’.

iii Vita

June 11, 1994 ...... Born - Columbus, OH, USA

2017 ...... B.S. Physics & Astronomy, The Ohio State University 2018-2019 ...... University Fellow

2019-2020 ...... Graduate Teaching Associate, The Ohio State University.

Publications

Research Publications

Page, M. R., McCullian, B. A., Purser, C. M., Schulze, J. G., Nakatani, T. M., Wolfe, C. S., Childress, J. R., McConney, M. E., Howe, B. M., Hammel, P. C. and Bhallamudi,V. P., Optically detected ferromagnetic resonance in diverse ferromag- nets via nitrogen vacancy centers in diamond. Journal of Applied Physics, 126, 12.

Fields of Study

Major Field: Earth Sciences

iv Table of Contents

Page

Abstract ...... ii

Acknowledgments ...... iii

Vita...... iv

List of Tables ...... vii

List of Figures ...... viii

1. Introduction ...... 1

2. Do the compositions of rocky exoplanets reﬂect their star’s refractory abundances? ...... 9

2.1 Introduction ...... 10 2.2 Sample Selection ...... 11 2.3 Planetary Structure Calculations ...... 14 2.4 Hypothesis testing ...... 16 2.5 Results ...... 19 2.6 Discussion and Conclusions ...... 25

3. ExoLens – A Compositional Calculator for Rocky Worlds ...... 28

3.1 Introduction ...... 28

v 3.2 Analytical Expression for Core Mass Fraction as a Function of Plan- etary Observables, Mass and Radius ...... 29 3.3 Parameter Fitting ...... 31 3.4 Inputs and Outputs ...... 33 3.5 Comparison with Zeng et al. (2016a) ...... 37

4. A Simple Adiabatic Model for Secondary Atmospheres ...... 41

4.1 Introduction ...... 41 4.2 The physics of atmospheric escape ...... 42 4.3 Atmospheric Modeling ...... 45 4.4 Reproducing Earth and Venus’ Atmospheres ...... 47 4.5 Discussion ...... 48

Appendices 53

A. Calculation of CMF and σCMF ...... 53

vi List of Tables

Table Page

2.1 Selected sample of well-characterized exoplanets. M-R sources: (1) Dai et al. (2019), (2) Ligi et al. (2019), (3) Espinoza et al. (2019), (4) Bonomo et al. (2019). Spectroscopy sources: (1) Santerne et al. (2018). (2) Santos et al. (2015). (3) Hypatia Catalogue (median values) (Hinkel et al., 2014). (4) Hellier et al. (2012). (5) Espinoza et al. (2019). (6) Bonomo et al. (2019)...... 13

2.2 Inferred properties of the selected small, well-characterized exoplanets, classiﬁed as S.E., super-Earth, S.M. super-Mercury, or S.F. super- Fluﬀ. P.R.C. = Previously Reported Class. Sources: (1) Santerne et al. (2018), (2) Brugger et al. (2017), (3) Dorn et al. (2019), (4) Espinoza et al. (2019). (5) Bonomo et al. (2019)...... 21

4.1 Initial Parameters for Earth and Venus from Earth Fact Sheet and Venus Fact Sheet, respectively...... 52

4.2 Summary of Results...... 52

A.1 The eﬀects of Mp, Rp, σRp and σMp on CMFρ and σCMFρ ...... 54

A.2 The eﬀects of Fe/Mg, Si/Mg, σFe/Mg and σSi/Mg on CMF? and σCMF? . 55

vii List of Figures

Figure Page

2.1 Ternary diagrams of the CMF-MMF-WMF solution space for all planets in our sample with known stellar Fe/Mg and Si/Mg ratios. All solutions are subject to the constraint that CMF+MMF+WMF = 100% 20

2.2 CMFρ and CMF? uncertainty maps as a function of observational uncertainties. (a) Uncertainty in CMFρ as a function of mass and radius uncertainties. We plot the values for Mp = 5.0M⊕ and Rp = 1.545R⊕, whereas values in Table 2.2 are exact. (b) Uncertainty in CMF? as a function of Fe/Mg and Si/Mg uncertainties. We use Fe/Mg = 0.98 and Si/Mg = 1.0. These choices give a central value of 0.35

for both CMFρ and CMF?. Uncertainties for additional planets with measured masses and radii without stellar Fe/Mg and Si/Mg are also included. Color labels are for clarity...... 24

3.1 CMFρ vs. Rp curves for Mp = 0.01 (cyan) to 11M⊕ (magenta). Exo- Plex data points are denoted by squares. Fits to Equ. 3.9 are shown as solid lines. Fit residuals are shown in the top panel. While there is a clear systematic trend in the residuals, the amplitude of this trend is, at most, 0.02% which is 3 orders of magnitude smaller than the

current uncertainties in CMFρ due to observational uncertainties in planetary mass and radius...... 32

3.2 Fit to mean core density as a function of planetary mass...... 34

3.3 Fit to mean mantle density as a function of planetary mass...... 35

viii 3.4 Diﬀerence between CMFρ calculated from ExoPlex and ExoLens for 13 likely rocky exoplanets (Rp < 1.9R⊕) with σM < 20% and σR < 10%. R.M.S. = 1.4%...... 36

3.5 Schematic overview of ExoLens...... 38

3.6 Sample ExoLens output for K2-229b. Black solid lines correspond to nine equally-spaced conﬁdence intervals between 10 and 90%. The dashed cyan lines are the 68% and 95% conﬁdence intervals. (Bot-

tom left) Contour lines (white) correspond to lines of constant CMFρ values. Negative values indicate M-R combinations that cannot be ex-

plained by a purely MgSiO3 composition and require either an outer volatile layer or enrichment in ultra-refractory materials. (Bottom

right) likelihood functions for CMFρ and CMF?...... 39

3.7 Comparison plot between Z16 and ExoLens. (Left) Z16 (dashed) and ExoLens (solid) M-R curves for varying core mass fractions. (Right)

Diﬀerence in CMFρ between Z16 and ExoLens for Mp = 1.0 and 10.0M⊕. 40

4.1 Atmospheric particle mass where vthermal = vescape as a function of planet mass for 300, 1000, and 2000 K. Solid symbols for Earth, Venus, and 55 Cnc e are plotted according to their equilibrium temperature according to their orbital distance (no greenhouse). The open symbol for Venus reﬂects its present surface temperature...... 46

4.2 My model of Earth’s atmosphere (solid) compared with the 1976 U.S. standard atmosphere model (dashed)...... 49

4.3 My model of the Venetian atmosphere compared with Justus & Braun (2007) ...... 50

ix Chapter 1: Introduction

The solar system contains 8 planets which can be broadly divided into two compositional categories: the rock-dominated terrestrial planets and the light-element dominated gas/ice giants. Mercury, Venus, Earth, and Mars are the innermost planets, located between ∼ 0.4 - 1.5 AU (60 - 255 million km) from the Sun, and contain nearly all of their mass in rock. While all of the terrestrial planets are rocky, they are diverse in their compositions and geological activity. Earth is built of 32.5% iron and 67.5% silicates by mass. Venus and Mars have similar relative amounts of Fe, Mg, and Si to the Earth. However, global-scale plate tectonics does not appear to be present on either planet, and Venus has a CO2 atmosphere that is roughly ninety times thicker than the Earth’s. Last, Mercury has a composition that is much more Fe-rich than the other terrestrial planets with an iron core that makes up 70% of the planet’s mass. Mercury’s iron-enrichment is believed to be the result of a series of mantle stripping collisions (Asphaug & Reufer, 2014).

Further out, between ∼ 5 - 30 AU (0.75 - 4.5 billion km), we have the gas giants, Jupiter and Saturn, and the ice giants, Uranus and Neptune. The gas giants are the largest planets in the solar system, making up ∼ 75% of the total planetary mass. These planets consist of predominantly hydrogen and helium in similar proportions to that of the Sun. The gas giants are have substantial outer gas envelopes and metallic hydrogen mantles separated by a liquid hydrogen layer. Jupiter and Saturn also likely have rock-ice cores making up ∼ 3 − 6% and ∼ 1% of their total masses, respectively. The ice giants also contain signiﬁcant amounts of hydrogen and helium with substantial outer gas layers and rocky cores. However, they diﬀer from gas

1 giants compositionally in that they are more enriched in heavier elements like carbon, oxygen, nitrogen, and sulfur. As opposed to metallic hydrogen, their mantles consist of water, ammonia, and methane ices.

The first confirmed detection of a planet outside of the solar system (exoplanet) came in 1992 when Alex Wolszczan and Dale Frail discovered two ∼ 4M⊕ planets in orbit around the neutron star PSR B1257+12 with evidence for a third which was later confirmed (Wolszczan & Frail, 1992). In 1995, the first exoplanet around a main-sequence star was discovered by Michel Mayor and Didier Queloz (Mayor & Queloz, 1995). This planet, 51 Pegasi b, has a mass that is roughly half the mass of Jupiter with an orbital period of 4.2 days. This planet was dubbed a ’Hot Jupiter’ due to its high mass and proximity to its host star. To date, over 4000 exoplanets have been discovered, with more than 5000 additional candidates (NASA Exoplanet Count). While there are several techniques to detect exoplanets, the R.V. method and transit photometry account for over 95% of confirmed discoveries (NASA Exoplanet and Candidate Statistics).

The R.V. method exploits a physical phenomenon called the Relativistic Doppler eﬀect. Similarly to the shift in the frequency of sound waves emitted by a train as it moves towards or away from an observer, as a planet and star both orbit their mutual center of mass the light from the host star will shift from short, or higher frequency, wavelengths, to longer, lower frequency, wavelengths. The amplitude of this eﬀect is directly related to the gravitational pull of the planet on the star. If the mass of the host star, orbital period, and orbital geometry are known, then the mass of the planet can be directly measured using this technique.

Transit photometry utilizes the fact that planets are essentially opaque relative to their host stars. Similarly to the transits of Venus and Mercury, if a planet passes in front of its star relative to an observer as it orbits, some amount of the host star’s light will be blocked. The shape and duration of this dip in measured light is directly related to the star-planet size ratio, orbital eccentricity, and orbital period of the planet. Thus, if the size of the host star, the planet’s orbital period, and the

2 orbital geometry can be determined via other methods, then the radius of the planet can be measured.

While the R.V. method and transit photometry can be used independently to one another to detect an extrasolar planet, both mass and radius are needed to determine if a planet is rock dominated, like the terrestrial planets, or gas/ice dominated like the Jovian planets. Surprisingly, the majority of detected exoplanets look unlike anything in the solar system, falling between the Earth and Neptune in size. Further- more, within this size range, there appear to be two distinct planetary populations with similar occurrence rates separated by a minimum, or gap, between 1.5 and 2.0

Earth radii (R⊕) (Fulton et al., 2017).

Planets with 1.0R⊕ < R < 1.5R⊕ are referred to as super-Earths because, in general, their mean densities are consistent with that of the Earth’s and are thus likely rocky.

Conversely, planets with R > 2.0 − 4.0R⊕ are called mini- or sub-Neptunes as they have much lower average densities that are more consistent with a rocky interior and a light extended envelope. In contrast to gas and ice giants, the masses of sub- Neptunes are still dominated by their rocky interiors. Additionally, in systems where more than one super-Earth or sub-Neptune has been detected, the planets appear to be tightly packed near their hosts, with most having orbits interior to that of Mercury’s.

Why does nature favor these compact systems of super-Earths and sub-Neptunes? Where does our planetary system ﬁt into the broader picture? How likely are “Earth- like” planets outside of the solar system? To answer such questions, we ﬁrst need to take a look at how planets form and evolve. While the details of planetary formation/evolution are still an active area of research, astronomers and planetary scientists have a good handle on the basic physics that drive these processes.

All planets form within a gas-dust disk made of the same material as their host star(s). This disk consists of predominantly H/He gas with small amounts of heavier elements such as oxygen and refractory materials like magnesium, silicon, iron. Thus,

3 the relative amounts of Fe, Mg, and Si in the disk reﬂect that of the host star. This is commonly quantiﬁed by the molar ratios of these elements, Fe/Mg and Si/Mg.

Rocky planets form in the inner, hot regions of this disk where hydrogen and helium are unable to condense and remain in the gas phase during the entire lifetime of the disk. As the disk cools, the higher condensation temperatures of refractory elements allow them to condense and form planetary embryos. If these embryos grow rapidly enough, and there is enough material to feed their growth, then they can reach approximately 1M⊕. At this mass, the planetesimals begin to migrate rapidly inwards. As these embryos migrate towards the inner edge of the disk, they continue to grow via accretion and embryo-embryo collisions. Under reasonable disk parameters, they can reach sizes between 1−10M⊕. This represents the super-Earth formation pathway (see Lambrechts et al. (2019) and references therein for more details on planet formation).

The most massive of these proto-planets may gravitationally bind significant amounts of H/He gas and become a sub-Neptune. How much of its initial H/He a planet can retain throughout its life is a function of how massive it is and how much radiation it receives (Jin & Mordasini, 2018a). Stellar radiation, or insolation, can strip some or all of a sub-Neptune’s envelope. Thus, a super-Earth may form intrinsically rocky, meaning it simply never grew massive enough to retain a significant envelope (like the terrestrial planets). It may also form as a sub-Neptune and have its H/He envelope eroded. The radius-gap represents the transition zone between sub-Neptunes and the barren rocky interiors of such planets, which would now fall into the super-Earth classification.

The solar system, instead, probably started with a modest-sized gas-dust disk. Rocky embryos in the solar system grew to between the Moon and Mars in mass before their growth halted before the end of the gas disk phase when no more material was able to be accreted. These proto-planetary masses were insuﬃcient for them to obtain H/He envelopes or migrate substantially, which is why the solar system is not compact like many of the super-Earth/sub-Neptune systems.

4 The disk phase lasts approximately 3 million years, at which time the proto-star at the center becomes hot enough to begin hydrogen fusion. This new source of energy causes the star to radiate signiﬁcantly more, pushing out what remains of the dust- gas disk. With the pressure support of the disk now gone, orbits become unstable, and the system enters a growth stage dominated by embryo-embryo collisions. This stage can last many tens of millions of years. It is during the later stages of this growth regime that the Moon-forming impact likely occurred (Canup & Asphaug, 2001) and when Mercury probably lost much of its mantle (Benz et al., 1988).

In all cases, the dominant parameter that determines the number and size of planetary embryos a system will have after the disk phase is how much material is initially available as this determines how much material is available for proto-planets to ac- crete. The number, size, and distribution of proto-planets then determine how the system will evolve after the gas disk phase. However, planetary formation typically takes no more than 100 million years, an extremely short amount of time compared to the lifetime of planetary systems. Therefore, the likelihood of a planetary system being in the formation stage at any given time is extremely small (approximately 1-2% for a Sun-like star).

Instead, we need to look at the large number of mature planetary systems that nature has given us to constrain such models and better understand our own system’s place within them. Speciﬁcally, we need to look at the compositional distribution of these planets. While the mass distribution of planets can give us an idea of the range of reasonable initial disk masses, the compositional distribution will help to constrain the relative importance of dynamical formation and evolutionary processes. For example, the relative number of planets found to be iron-enriched will yield information about the frequency and scale of planetary collisions. It will also allow us to determine how many super-Earth planets are likely to be ‘Earth-like’, and, thus, potentially habitable.

One fundamental reason the Earth is habitable is its ability to undergo and sustain geological processes like plate tectonics. The fact that a subducting plate sinks on Earth is a result of the Earth’s composition. Planets with diﬀerent compositions may

5 have plates that are unable to sink resulting in no global-scale tectonics (Unterborn et al., 2017). Thus, to understand how unique the habitable nature of the Earth is and which planets are the most promising candidates to host life, we must be able to determine their compositions.

Unfortunately, we cannot directly observe the structures and compositions of exoplanets. Instead, we must infer these using the planet’s bulk density calculated from mass and radius. The current uncertainties in both observables, however, leave significant uncertainty in density. Less than 70 potential super-Earths have both mass and radius measurements. Approximately 65% percent of the planets have mass and radius uncertainties of >20% and >10%, respectively, meaning it is difficult to determine if these planets are actually super-Earths, or if they are, instead, sub-Neptunes. Even for planets with measurement errors below these thresholds, their uncertainties are still large enough to make it difficult to distinguish between a super-Earth with a Mercury-like composition (Fe-dominated) and a super-Earth with a thick non-primordial atmosphere (like Venus’ thick CO2 atmosphere). Thus, given the current precisions in mass and radius, it is extremely challenging to determine if a given small exoplanet is Earth-like.

Since planets form from the same materials as their host star, exoplanetary scientists commonly assume that relative amounts of Fe, Mg, and Si in a planet should directly reﬂect that of its host star. Within the solar system, the Fe/Mg and Si/Mg ratios of Venus, Earth, and Mars are consistent with the solar values. However, Mercury has an Fe/Mg ratio that is more than 200% greater than that of the Sun. Given that this assumption fails even in the solar system, how valid is it outside of the solar system?

In this work, I present a method for determining how well we can predict the composition of a rocky planet from its host star. I develop an open-source, computationally cheap, and easy-to-use implementation of this method for use with the statistically large samples expected in the near future. For likely rocky planets whose densities are too low to be explained with a purely rocky composition, I explore core oxidation,

6 outer water layers, and thick carbon-dioxide atmospheres as possible explanations for their density deﬁcits.

In Chapter 2, I test how consistent the inferred compositions of 8 of the best characterized likely rocky exoplanets (σRp < 10% and σMp < 20%) are with their host stars, all of which have known Fe/Mg and Si/Mg. We use ExoPlex (Unterborn et al., 2018a), the open-source self-consistent mass-radius software, to solve for the allowed compositions that satisfy a given planet’s mass and radius and compare this planetary-composition parameter space with the range of compositions expected from the host star. We find that the current uncertainties only allow us to conclusively determine that one of these planets is statistically different from what is expected from its host star with ≥ 95% confidence.

In Chapter 3, I present ExoLens, a new python-based open-source compositional calculator for rocky planets based oﬀ of ExoPlex. This calculator can estimate the core mass fraction of a 0.1-10M⊕ planet from mass and radius to within 1-2% of ExoPlex in a fraction of the time. If the host star’s Fe/Mg and Si/Mg ratios are provided, ExoLens also calculates the likelihood that a planet’s composition inferred from Mass and Radius is statistically diﬀerent than what is expected from its host star. As large-scale exoplanet surveys like the Transiting Exoplanet Survey Satellite (TESS; Ricker et al. (2014)) discover many more small, likely rocky exoplanets, a quick and accurate means to infer planetary composition for a large sample will be imperative.

Last, in Chapter 4, I present an atmospheric model that is able to reproduce the atmospheres of Earth and Venus. My model assumes that atmospheres are chemically homogeneous and have an adiabatic temperature proﬁle. My goal is to integrate this software into ExoPlex as an outer gas layer. Currently, ExoPlex can model a purely rocky planet with or without an outer water/ice layer. Including an atmospheric layer opens up another region of the compositional parameter space I can model. I aim to use this additional region of parameter space to take a closer look at planets whose densities are lower than can be explained with a purely rocky composition

7 and bound the minimum and maximum amounts of atmosphere such a planet would need to explain its density.

8 Chapter 2: Do the compositions of rocky exoplanets reﬂect their star’s refractory abundances?

Manuscript in preparation for publication in The Planetary Science Journal (AAS)

Joseph G. Schulze1, Ji Wang2, Jennifer A. Johnson2 Cayman T. Unterborn3, Wendy R. Panero1

Abstract

Not always. Planetary mass and radius are the primary direct observables of planets from which bulk density can be calculated. From the bulk density, planet structure and composition can be constrained. However, relative proportions of iron core, rocky mantle, and gaseous envelopes, are degenerate for a given density. The solution, however, for rocky planets without significant gaseous envelope do not suffer from the same degeneracy as they have only two layers: a differentiated iron core and rocky mantle. As a result, the core mass fraction (CMF) is conventionally used as a first-order description of a planet’s bulk composition. A rocky planet’s CMF may be derived from the bulk density, or by assuming the planet reflects the major refractory element abundance (Fe, Mg, and Si) of the host star. The two assumptions

1School of Earth Sciences, The Ohio State University, 125 South Oval Mall, Columbus, OH 43210 USA 2Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus, OH 43210 USA 3School of Earth and Space Exploration, Arizona State University 781 Terrace Mall Tempe, AZ 85287, USA

9 may not be necessarily consistent for all rocky planets. The (in)consistency sheds light on the diversity of the outcomes of planet formation due to various processes, e.g., mantle stripping, out-gassing, and/or volatile delivery at late stages of planet formation. Here, we present an analysis that quantifies the (in)consistency of the two assumptions in the presence of measurement uncertainties of planet mass and radius and stellar chemical abundances. We apply the analysis to a sample of 8 well-characterized exoplanets around FGK-type stars with known Fe, Mg, and Si abundances. We find that Kepler-107c is the only planet in our sample at a 95% confidence that shows with a CMF calculated from bulk density that is significantly greater than the inferred CMF from its host star abundance, and is, therefore, iron- enriched.

2.1 Introduction

The Earth’s bulk refractory element composition is roughly matched by that of the Sun. A simple inference from this observation is that the Earth formed from material condensed from a disk of Solar composition. This relationship is supported by CI- chondrites, which are thought to be the most chemically primitive bodies in the solar system and reﬂective of the initial refractory composition of the solar nebula. Refractory elements, those elements that solidify at high temperatures include Fe, Mg, and Si, whereas volatile elements are those with low condensation temperatures. Indeed, CI-chondrites have 39 refractory element abundances that are within ± 10% of the relative abundances found in the Sun (Lodders, 2003). Their Fe/Mg and Si/Mg ratios reﬂect the solar photospheric ratios to within 2 and 4%, respectively (Putirka & Rarick, 2019).

Earth and Mars have molar Fe/Mg ratios to within 10% of the Sun’s abundance (Lodders, 2003; McDonough, 2003; Wanke & Dreibus, 1994), and while the Fe/Mg ratio for Venus is poorly constrained, it is consistent with the value of the Earth (Zharkov, 1983). Thus, the bulk chemical compositions of Venus, Earth, and Mars appear to be consistent with the hypothesis that Venus, Earth, and Mars initially

10 formed from chondrites, and are thus reﬂective of the initial relative abundances of refractory elements of the solar photosphere.

Mercury is anomalous, however, with an Fe concentration ∼200% - 400% greater than expected relative to silicates (Nittler et al., 2019). Therefore, not all rocky planets in the Solar System formed with refractory Solar abundances. Whether these diﬀerences are due to formation or impact processes, condensation chemistry, or radial mixing within the disk is a matter of debate.

This ambiguity in the relationship between rocky planets and host-star abundances leads to diﬃculty in ﬁnding guidance from the Solar System to interpret the state and formation of rocky exoplanets. Furthermore, interpretations of mass and radius models are degenerate with respect to composition. Therefore, stellar compositions of major, refractory, rocky planet-building elements (Mg, Si, Fe) have been used as proxies to break the degeneracy, implicitly assuming that rocky planet compositions mirror their host star’s refractory abundances. Where presumed rocky planets are inconsistent with stellar abundances, they are suggested to be water worlds (Unterborn et al., 2018b), CAI planets (Dorn et al., 2019), or core-free planets (Elkins-Tanton & Seager, 2008) to explain lower-than-expected density. In contrast, iron enrichment relative to magnesium and silicon is invoked to explain higher-than-expected density (Santerne et al., 2018).

In this work, we present a straightforward method for determining the likelihood that rocky planets satisfy the null hypothesis, H0, that their Fe/Mg and Si/Mg ratios directly mirror their host star’s Fe/Mg and Si/Mg. For those planets that do not satisfy the null hypothesis, we discuss the range of possible compositions, including whether such a planet requires a super-stellar iron abundance (H1) or smaller-than- expected core or demands outer volatile layer (H2).

2.2 Sample Selection

To test our hypotheses, we focus on those planets most likely to be rocky, with both measured masses and radii. Using the NASA Exoplanet Archive, we identify 734

11 planets with both mass and radius measurements. We limit our sample selection to planets whose masses and orbital distances make them unlikely to retain significant primordial H/He envelopes due to both their low gravity and the excessive radiation received from their host stars (Jin & Mordasini, 2018b). This reduces the significant degeneracy between planet structure models so that we can focus on the impact of the refractory composition. We therefore identify transiting exoplanets with Rp ≤ 1.9R⊕ on short-period orbits (<15 days). We find 62 planets meeting our radius criteria. Adding in the period constraint brings this number to 57.

Next, we limit our sample to planets with uncertainties of . 20% and . 10% in planetary mass and radius, respectively. These uncertainty criteria further reduce our sample to just 20 planets. Lastly, we limit our sample to those around FGK-type (Sun-like) stars and whose host star has known stellar Fe, Mg, and Si abundances from which we derive molar ratios of (Fe/Mg)? and (Si/Mg)? using the solar abundances from Lodders et al. (2009). We identify 8 planets meeting all our criteria (Table 2.1).

Of this set, the observational uncertainties in mass and radius range from 4% to 19% and 1.6% to 10.5%, respectively. The associated uncertainty in bulk density of these planets ranges from 10% (WASP-47e) to 24% (K2-229b).

12 Planet Mp[M⊕] Rp[R⊕] M-R Source Star Type Period (d) (Fe/Mg)? (Si/Mg)? Spect. Source +0.42 +0.045 K2-229 b 2.49−0.43 1.197−0.048 1 K 0.58 0.78±0.05 1.0±0.1 1 +0.51 +0.023 Kepler-10 b 3.57−0.53 1.489−0.021 1 G 0.84 0.63±0.15 0.83±0.22 2 HD 219134 b 4.27±0.34 1.500±0.057 2 K 3.09 0.69±0.23 0.98± 0.38 3 HD 219134 c 3.96±0.34 1.415±0.049 2 K 6.76 0.69±0.23 0.98± 0.38 3 +0.81 +0.049 WASP-47 e 6.91−0.83 1.773−0.048 1 G 0.79 0.76±0.14 1.35±0.25 4 +0.37 +0.044 55 Cnc e 7.74−0.30 1.897−0.046 1 K 0.74 0.76± 0.25 0.87±0.34 3 13 HD 213885 b 8.83±0.66 1.745±0.052 3 G 1.008 0.81±0.13 0.98±0.24 5 Kepler-107 c 9.39±1.77 1.597±0.026 4 G 4.9 0.75± 0.14 0.96±0.14 6

Table 2.1: Selected sample of well-characterized exoplanets. M-R sources: (1) Dai et al. (2019), (2) Ligi et al. (2019), (3) Espinoza et al. (2019), (4) Bonomo et al. (2019). Spectroscopy sources: (1) Santerne et al. (2018). (2) Santos et al. (2015). (3) Hypatia Catalogue (median values) (Hinkel et al., 2014). (4) Hellier et al. (2012). (5) Espinoza et al. (2019). (6) Bonomo et al. (2019). 2.3 Planetary Structure Calculations

We test hypothesis H0, that the composition of a rocky planet is related to the stellar abundances, by comparing the fraction of the planet that must be explained by a metallic core to the relative iron abundances in the star. Practically, we calculate the core mass fraction (CMF) in two ways, (1) by the mass fraction of the core as predicted by the refractory composition of the star, CMF?, and (2) by the fraction of core required to explain the average density of the planet, CMFρ. The hypothesis H0 is refuted when these two measures for CMF diﬀer given the limits of the observational data.

A signiﬁcant limitation to an analysis comparing planetary composition to stellar abundances is the observational uncertainties of mass and radius, often leading to bulk densities with more than 15% uncertainty. We, therefore, quantify the relationship between these observational uncertainties and the associated uncertainties in planetary structure as described by the relative proportions of rocky mantle and metallic core.

The core mass fraction expected from the refractory abundance ratios of the host,

CMF?, is

Fe Mg ?mFe CMF? = Fe Si , (2.1) Mg ?mFe + Mg ?mSiO2 + mMgO Fe Si where Mg ? is the stellar molar fraction of Fe relative to Mg, Mg ? is the stellar molar fraction of Si relative to Mg, mFe is the molar mass of iron, mSiO2 is the molar mass of silicon dioxide, and mMgO is the molar mass of magnesium oxide.

The uncertainty in CMF? with respect to Fe/Mg and Si/Mg is found through a propagation of uncertainties in Eq.2.1,

v u 2 2 u ∂CMF? Fe ∂CMF? Si δCMF? = t δ + δ . (2.2) Fe Mg Si Mg ∂ Mg ∂ Mg

14 Complementary to the inferred CMF?, is the core mass fraction inferred from the mean density of the rocky planet, CMFρ. We use the thermodynamically self- consistent ExoPlex mass-radius software (Unterborn et al., 2018b) to solve for CMFρ and the distribution in CMFρ given the mass and radius, and their observational uncertainties. ExoPlex solves five coupled differential equations: the mass within a sphere, hydrostatic equilibrium, adiabatic temperature profile, Gauss’s law of gravity in one dimension, and the thermally-dependent equation of state. We fix the planetary mass and set a radius convergence criterion of 0.01% of the planet’s radius, 2-3 orders of magnitude smaller than the current planetary radius uncertainties.

Our H0 models consist of a pure solid Fe core and a fully oxidized silicate mantle

(MgSiO3). As the goal of these calculations is to determine the uncertainty in planetary structure instead of mantle mineralogy, we assume the mantle to have a fixed molar ratio of Si/Mg = 1. No star in our sample set has a Si/Mg ratio distinguish- able from 1, and even 100% variations in this ratio affect the calculation in planet mass by no more than 2%, far less than the observational uncertainties (Dorn et al., 2015; Unterborn et al., 2016). We also do not include minor mantle elements (i.e. Ca and Al) in our models as these also do not significantly affect inferred masses. We adopt the iron Vinet equation of state from Smith et al. (2018) for the core and the equation of state developed in Stixrude & Lithgow-Bertelloni (2005) for the mantle.

In the simpliﬁed, two-layer model of a rocky planet, CMFρ is the mass of iron, mFe, divided by the mass of the planet, Mp: CMFρ = MFe/Mp.

For each mass and radius of the planets in our sample (Table 2.1), we calculate the

1σ and 2σ uncertainties in CMFρ resulting from the uncertainty in their mass and radius measurements using the joint Mass-Radius elliptical distribution. We sample π 12 Mass-Radius pairs with angular separation 6 along the 1-σ ellipse and the 2-σ ellipse. For each of these ellipses, we average the 6 mass-radius pairs that yield a

CMFρ that is less than the best-ﬁt CMFρ to solve for the lower 1-σ and 2-σ bounds on CMFρ and the 6 mass-radius pairs that yield a CMFρ greater than the best-ﬁt

CMFρ to ﬁnd the upper 1-σ and 2-σ bounds. CMFρ is, therefore, a function of mass, radius, and the uncertainty in each.

15 2.4 Hypothesis testing

Solutions where CMFρ overlaps with CMF? within uncertainty correspond to the regions where the structure and composition of the planet are consistent with what is expected from its host star, consistent with H0. We calculate the probability that the observed CMFρ and CMF? come from the same distribution by calculating the joint probability normalized by the null hypothesis that both distributions have the same mean values,

R 100% 0 0 N (CMFρ, σCMFρ )N (CMF?, σCMF? )dCMF P (H ) = P (CMFρ = CMF?) = R ∞ −∞ N (σCMFρ )N (σCMF? )dCMF (2.3) where N (CMF, σCMF) is the probability of a given CMF with assumed Gaussian distribution with width σCMF.

This approach incorporates the mutual uncertainties of mass, radius, and host star abundances, in which the probability of overlap increases with increasing uncertainties, that is, large uncertainties decrease the likelihood that a planet will be inconsistent with H0. We then classify planets at both the 1σ and 2 signiﬁcance levels.

For example, if P(CMFρ = CMF?)≤32% then we say this planet deviates from what is expected at the 1σ signiﬁcance level. Similarly, if P(CMFρ = CMF?)≤5% then this planet deviates from its host star at the 2σ signiﬁcance level. While we classify planets at the 1σ level, we only consider planets that deviate from their host stars at the 2σ level to be statistically inconsistent with the null hypothesis.

0 For such planets that deviate from H , we identify two regimes, (1) in which CMF? <

CMFρ, suggesting a planet with a larger core than expected, and (2) in which

CMF? > CMFρ, which then suggests a region of non-unique solutions, which we explore through a combination of core oxidation and surface volatile layers.

H1: the planet has an apparent excess mass relative to that predicted by its host star refractory abundances.

16 For H1, we evaluate the likelihood that a planet has a greater-than-expected core size as there are few other explanations for increasing the density of a planet beyond excess iron. As cores are generally expected to be mostly liquid and to contain some amount of light elements, our core density in evaluating H0 is a systematic overestimate of the likely core density. An overestimate of core density leads to a reduced core volume and greater volume fraction of mantle material. Thus, assuming an iron core systematically underestimates CMFρ for a given planetary mass and radius. We opt for a pure Fe core, as no liquid iron equation of states exist that are reliable at super-Earth core pressures. We compare Stixrude & Lithgow-Bertelloni (2005) with the liquid Fe EoS of Anderson & Ahrens (1994) (valid to 10 Mbar) and estimate that the diﬀerence in CMFρ between a solid and liquid core is, at most, ∼ 2-4% over the mass and core mass fraction ranges considered in this work.

Conversely, the starting assumption that all iron is in the core with none in the mantle systematically overestimates the core mass fraction for a ﬁxed planet mass and radius because mantle FeO increases the average density of the mantle. This overestimate is approximately 2-4% per 10 mol % iron oxidized in the mantle. However, any degree of iron oxidation requires a higher Fe/Mg ratio than for no iron-oxidation to conserve both planetary mass and radius. For example, with 10% oxidation of iron (by mole) requires an Fe/Mg ratio that is ∼ 8% greater than for the scenario that all iron is in the core. Thus, for both assumptions, the probability assigned to H0 is an upper bound in the case CMFρ > CMF?.

H2: the planet has an apparent mass deﬁcit relative to that predicted by its host-star refractory abundance.

For H2, we consider the three compositional variations that have been proposed to satisfy this hypothesis: H2a, volatile outer layers, H2b, oxidized core such that the planet’s mantle contains signiﬁcant oxidized iron, or H2c, a calcium-aluminum oxide dominated planet that formed from only those highest-temperature refractory materials.

17 In the case of H2a, we assume that the volatile outer layer is water, and quantify the necessary water mass fraction (WMF) of the planet to satisfy the combined constraints of the stellar refractory composition and the planet density. We use the thermally dependent water equation of state from Stixrude & Bukowinski (1990) for the liquid water phase and an approximate EoS for ice I, ice VI and ice VII for ice phases (Unterborn et al., 2018b). As all planets considered here are on short-period orbits, any surface water will be vaporized, such that these estimates for WMF are strict upper bounds, likely overestimating necessary WMF by up to an order of magnitude.

We also consider the required mass fraction of iron (Femantle/Fecore) necessary to be oxidized and stored in the mantle in H2b. Planets that require a signiﬁcant fraction of oxidized iron inform nebular chemistry, cooling rates, and formation location (Gross- man et al., 2008). This model is likely for planets with a lighter than expected bulk density as iron oxidation within the solar system is abundant (Putirka & Rarick, 2019), with approximately 31% and 11% of the total iron content of Mars (Taylor, 2013) and Earth (McDonough & Sun, 1995), respectively, is oxidized.

The expected density for the nearly core-free planets that are assumed in H2c is as much as 20% less than would be expected of a planet with its stellar refractory abundances, as explored by Dorn et al. (2019), comparable to density deﬁcits achievable by oxidizing the core. We do not, however, consider H2c further because there is an insuﬃcient mass of Al and Ca present within the protoplanetary disk available to produce the observed masses of the massive Ca-Al-rich planets hypothesized in Dorn et al. (2019). This calculation assumes the planets formed from a minimum mass solar nebula Kuchner (2004), which likely overestimates the disk mass available to planets forming around the G-dwarf, 55 Cnc. We do not consider these Ca-Al planets likely, then, without a factor of ∼3.5 (0.55 dex) and ∼2 (0.3 dex) increase in the already super-Solar Ca and Al abundances of 55 Cnc, respectively.

We note the degeneracy of H2 due to multiple processes leading to a lower density planet means we cannot assess the degree to which a planet with lower than expected density deviates from the expected refractory composition (H2c), is highly oxidized

18 (H2b), simply has retained an outer volatile envelope with or without stellar refractory abundances (H2a), or a combination of each.

2.5 Results

At the 2σ level, we identify one planet, Kepler-107c, that is statistically inconsistent 0 0 with H . Kepler-107c has a 1% likelihood of satisfying H , with a CMF? < CMFρ (Fig. 2.1). Instead, we ﬁnd this planet consistent with H1 and is thus a likely “super-

Mercury”, a planet with R > 1.0R⊕ and a statistically significant iron-enrichment relative to its host. We find that it has a mean iron mass fraction of 70%, nearly identical to Mercury’s. At the 1σ significance level, we identify 6 super-Earths (Table 2.2), 1 super-Mercury (Kepler-107c), and 1 “super-Fluff” (55 Cnc e), a planet with

R < 1.9R⊕ and a statistically signiﬁcant lower than expected CMFρ relative to

CMF?.

19 Figure 2.1: Ternary diagrams of the CMF-MMF-WMF solution space for all planets in our sample with known stellar Fe/Mg and Si/Mg ratios. All solutions are subject to the constraint that CMF+MMF+WMF = 100%

20 0 Planet CMFρ (%) CMF? (%) P (H ) (%) CMFratio P.R.C. 1σ Class 2σClass +16 +9 +1.2 1 K2-229 b 56.5−20 27−13 40 2.1−1.0 S.M. S.E. S.E. +15 +0.55 2 Kepler-10 b 13−13 28 ± 6 66 0.47−0.48 S.E. S.E. S.E. +15 +0.7 3 HD 219134 b 29−15 28 ± 4 100 0.95−0.6 S.F. S.E. S.E. +13 +0.85 3 HD 219134 c 42−14 28 ± 4 69 1.4−0.50 S.E. S.E. S.E. +14 +0.55 3 WASP-47 e 15.5−15 26 ± 5 78 0.60−0.60 S.F. S.E. S.E. +10 3 55 Cnc e 0.4<0 31 ± 9 7 [< 0 − 0.47] S.F. S.F. S.E. 21 +0.44 4 HD 213885 b 42 ± 9 31 ± 5 61 1.34−0.39 S.E. S.E. S.E. +10 +0.70 5 Kepler-107 c 70−12 30 ± 4 1 2.3−0.55 S.M. S.M. S.M.

Table 2.2: Inferred properties of the selected small, well-characterized exoplanets, classified as S.E., super- Earth, S.M. super-Mercury, or S.F. super-Fluff. P.R.C. = Previously Reported Class. Sources: (1) Santerne et al. (2018), (2) Brugger et al. (2017), (3) Dorn et al. (2019), (4) Espinoza et al. (2019). (5) Bonomo et al. (2019). K2-229b was previously described as a super-Mercury with a mean core mass fraction of 68% (Santerne et al., 2018). However, our analysis of K2-229b does not support this classification. Using updated mass and radius measurements (Ligi et al., 2019), +16 0 we find a CMFρ of 56.5−20 and a P (H ) value of 40%. While our calculated core mass fraction suggests that this planet may be iron-enriched, we are unable to classify it at either the 1σ or 2σ significance levels as a result of the current observational uncertainties. The uncertainties in both mass and radius (Fig. 2.2) are the primary limiting factor in further constraining the composition. We classify this planet as indistinguishable from a super-Earth.

The only planet in our sample beside Kepler-107c that violates H0 with statistical significance is 55 Cnc e. This planet has been previously reported in the literature as a super-Fluff (Dorn et al., 2019). Using the updated mass and radius values for this planet reported by Ligi et al. (2019), we find a mean CMFρ of 0.4% and a P (H0) = 7%. Thus, our analysis supports the hypothesis that 55 Cnc e is a super- Fluff, or consistent with H2, at the 1σ but not 2σ level. In contrast to K2-229b, the compositional constraint from the host star (Fig. 2.2) is as significant a contributor to the uncertainty as mass and radius. Therefore, this planet is a potential, but not conclusive, super-Fluff candidate.

2a +8 If 55 Cnc e is, in fact, a super-Fluﬀ, it can be explained by H with a 16.6−7% outer water/ice layer assuming the constraint that the inner rocky portion of this planet has the same Fe/Mg and Si/Mg ratios as its host star. The bulk density of 55 Cnc e cannot be explained by H2b even with 100% oxidation of iron using the Fe/Mg and Si/Mg ratios of the host. Finally, explaining the lower than expected bulk density of 55 Cnc e via H2c, assuming a minimum solar mass nebula, requires all of the disk’s Ca and Al and an additional 3.5 and 2 Earth masses of Al and Ca, respectively.

HD 219134b has been previously reported as a super-Fluff. However, using the recently reported mass and radius values for this planet from Ligi et al. (2019), our analysis does not support the classification of this planet as a super-Fluff. Instead, our models show that, within its current uncertainties, HD 219134b is the most

22 consistent with its host star at the current uncertainties. We calculate a mean CMFρ +15 value of 29−15% which is very near its expected value of 28 ± 4%. Using mass and radius values from Ligi et al. (2019) for the second innermost planet in the HD 219134 system, HD 219134c, we find that this planet is also broadly consistent with its host star with P (H0) = 69%. In contrast to planet b, planet c has +13 a mean CMFρ value of 42−14%. The uncertainties on CMFρ for HD 219134c do not permit classification of this planet as anything other than a super-Earth at the 1σ or 2σ significance levels. As with 55 Cnc-e, the uncertainties of the mass, radius, and host-star compositions are nearly equal contributors to our inability to distinguish these planets from H0. Therefore, we classify both HD 219134b and HD 219134c as super-Earths in this work.

WASP-47e is another previously reported super-Fluff planet (Dorn et al., 2019). However, using mass and radius values reported by Dai et al. (2019), we do not support this classification at either the 1σ or 2σ significance levels. While the mean

CMFρ value of 15.5 we calculate for this planet is lower than what is expected from its host star, WASP-47e’s range of possible CMFρ values are still consistent with what is expected from its host and lead to a P (H0) of 78%.

Lastly, Kepler-10b and HD 213885b have both previously been reported as super- Earths. In this work, we ﬁnd both of these planets to be consistent with the super- Earth classiﬁcation as well with a P (H0) of 66% for Kepler-10b and 61% for HD 213885b.

Across the radius and mass range of the analysis (2.5-9.7 M⊕ and 1.1-1.9 R⊕), we

ﬁnd that the uncertainty in CMFρ is dominated by the observational uncertainty in planetary mass and radius. The uncertainties in our results due to modeling uncertainties in the equations of state are negligible relative to the uncertainties due to mass and radius. For example, Unterborn & Panero (2019) show that for a 4 M⊕ planet, the radius uncertainty would need to be ∼0.25%, or an order of magnitude smaller than the current best-case radius errors, for the modeling uncertainties to be of the same order as the observational uncertainties.

23 Figure 2.2: CMFρ and CMF? uncertainty maps as a function of observational uncertainties. (a) Uncertainty in CMFρ as a function of mass and radius uncertainties. We plot the values for Mp = 5.0M⊕ and Rp = 1.545R⊕, whereas values in Table 2.2 are exact. (b) Uncertainty in CMF? as a function of Fe/Mg and Si/Mg uncertainties. We use Fe/Mg = 0.98 and Si/Mg = 1.0. These choices give a central value of 0.35 for both CMFρ and CMF?. Uncertainties for additional planets with measured masses and radii without stellar Fe/Mg and Si/Mg are also included. Color labels are for clarity.

24 For a planet with a central CMFρ value of 0.35, a 20% uncertainty in mass leads to a 50% uncertainty in CMF (Figure 2.2a, Table A.1). That is, constraining a planet to CMFρ = 0.35 ± 0.15. The observational uncertainties in the planet radius have the greatest impact on inferred CMF, in which a 10% radius uncertainty leads to an uncertainty in CMFρ nearly equal the expected central value, that is, CMFρ = 0.35 ± 0.35.

We find that the observational uncertainties in stellar refractory abundances have a significantly smaller effect on CMF? (Table A.2). A 40% uncertainty in molar Fe/Mg leads to a < 10% CMF uncertainty (Figure 2.2b). As previously explored in other ways, we find that the uncertainty in molar Si/Mg has minimal impact, such that even a 50% observational uncertainty leads to an inferred CMF uncertainty of 7%.

2.6 Discussion and Conclusions

Our hypothesis testing approach is limited by the combined uncertainties of planet mass, radius, and host-star [Fe/H], [Mg/H], and [Si/H]. Improving mass and radius uncertainties for small planets is a resource-intensive process, which will beneﬁt from increased signal-to-noise ratio for bright hosts discovered by the TESS satellite (Ricker et al., 2015), along with parallax measurements from Gaia (Gaia Collabora- tion et al., 2016). These may improve mean planet density observational uncertainty to as little as 4% (Stevens et al., 2018a). From the stellar perspective, direct analysis for [Fe/Mg] and [Si/Mg] circumvents compounding eﬀects of covariances and permits a reduction in uncertainties relative to calculating abundances from [Fe/H], [Mg/H], and [Si/H]. With such increases in observational precisions for a few optimal targets, we will be able to resolve apparent iron enrichment at the 2σ level in an exoplanet if CMFρ ≥ 1.5 CMF?. We note that this limited data set of eight planets can be expanded by targeted measurements of stellar abundances for those small planets with existing well-constrained masses and radii, K2-141, Kepler 36, and improving the stellar abundance measurements for 55 Cnc and HD 219134.

25 We note that a reanalysis of K2-229b (Dai et al., 2019) with new mass and radius uses an empirical, Earth-based model (Zeng et al., 2016b) to determine the CMF of the planet. This model is shown to over-predict core mass fraction for a given density (Unterborn & Panero, 2019). This model, extrapolated both in composition and pressure, predicts a CMF of 64% ± 26%, both greater CMF and with larger uncertainties than in Table 2.1, but with nearly identical probability (40%) of being consistent with the null hypothesis.

In this work, we present an agnostic planetary characterization scheme that assesses how statistically consistent a planet’s composition and structure inferred from its mass and radius is with what is expected from its host star’s relative major refractory elemental abundances. This approach is complementary to the Bayesian approach in Dorn et al. (2015). In contrast to the hypothesis testing approach taken here, the Bayesian approach instead tests the degree to which a terrestrial planet’s Fe/Mg and Si/Mg could be assessed from planet mass and radius measurements, ﬁnding that stellar abundances were required to reduce degeneracy. We instead test the hypothesis directly and demonstrate that for several planets, the mass and radius cannot be described as terrestrial planets with stellar refractory abundances. Once the conditions under which a terrestrial planet is describable by its stellar abundances are understood, a more complete Bayesian analysis of planetary composition and structure will be warranted.

We note that several mechanisms have been proposed to explain super-Mercury planets but each model identiﬁes mechanisms by which planets become relatively Fe enriched relative to the host star. These models include a single giant impact (Leinhardt & Stewart, 2011; Marcus et al., 2010), a series of smaller impacts (Chau et al., 2018; Swain et al., 2019), the mantle evaporation of hot planets (Santerne et al. (2018) and references therein), iron-enrichment in the inner regions of planet-forming disks due to iron condensing at a higher temperature than silicate material (Lewis, 1972) or via photophoresis (Ebel & Stewart (2019) and references therein), and mantle stripping via planet-star tidal interactions (Jia & Spruit, 2016) or planet-planet tidal interactions (Deng, 2020). Likewise, planets such as 55 Cnc-e have been suggested to

26 contain almost entirely ultrarefractory elemental abundances due to a lack of mixing in the inner disk during formation (Al, Ca Dorn et al., 2019). This mechanism also suggests a separation of refractory materials within the planetary formation disk. It is only once planets can be quantitatively distinguished from their host stars that an appropriate model for planetary formation and compositional sculpting can be assessed.

Aknowledgements

CTU acknowledges the support of Arizona State University through the SESE Explo- ration fellowship. The results reported herein beneﬁted from collaborations and/ or information exchange within NASA’s Nexus for Exoplanet System Science (NExSS) research coordination network sponsored by NASA’s Science Mission Directorate. JGS acknowledges the support of The Ohio State School of Earth Sciences through the Friends of Orton Hall research grant. We acknowledge partial support to WRP from the National Science Foundation under Grant No. EAR-1724693. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.

27 Chapter 3: ExoLens – A Compositional Calculator for Rocky Worlds

3.1 Introduction

Zeng et al. (2016a) (hereafter Z16) presented an analytical equation for calculating planetary radius from core mass fraction and planetary mass based on the Prelim- inary reference Earth model (PREM; Dziewonski & Anderson (1981)). In short, PREM is a 1D radial proﬁle of the elastic properties and densities within the Earth built by inverting a large data set of seismological observations. Z16 uses PREM to build ‘semi-empirical’ equations of state for the planetary mantles and cores, valid over the mass range of ∼ 1 − 8M⊕ and for core mass fractions between ∼ 0 − 40%.

With over 120 citations, the Z16 M-R relationship is heavily used in the literature to build rocky planetary mass and radius curves for varying core mass fractions and infer the compositions of observed potentially rocky exoplanets. This function is so widely used because it is easy to implement and computationally fast. However, this function is frequently extrapolated well outside of the CMF and pressure ranges over which it is valid.

In this chapter, I show that while the Z16 function is in good agreement with ExoPlex for Earth-like compositions over 1-10M⊕, as expected, it is not reliable for planets with core mass fractions that diﬀer from the Earth’s. As the number and diversity of small exoplanets grow rapidly, a quick and accurate function to infer planetary composition beyond just Earth-like is becoming increasingly imperative.

28 Here I present ExoLens, a python-based open-source compositional calculator for rocky planets based on ExoPlex. This calculator can accurately estimate the iron mass fraction of a 0.1-10M⊕ planet to within 1-2% of ExoPlex in a fraction of the time. ExoLens also simultaneously assesses the probability that a given planet satis- ﬁes the null hypothesis, H0, presented in Chapter 2. Additionally, ExoPlex requires both an upper and lower mantle compositional grid. Building these grids is the most time-intensive step in running ExoPlex and requires the user to have a priori knowledge of the pressure and temperature limits for the planet being modeled. ExoLens requires no such grids or a priori knowledge making it much more user friendly. ExoLens is freely available on GitHub

3.2 Analytical Expression for Core Mass Fraction as a Func- tion of Planetary Observables, Mass and Radius

At the heart of ExoLens is an analytical expression for core mass fraction (CMFρ) as a function of the planetary observables, mass (Mp) and radius (Rp). To derive

CMFρ(Mp,Rp) start with the deﬁnition of CMF,

M CMF = c . (3.1) Mp

Here, Mc is the mass of a planet’s iron core. Next, assuming a characteristic density for the core, ρc, and using the deﬁnition for the mass of a sphere with constant density, Mc can be rewritten as

Mcore = ρcVc. (3.2)

Since the volume of the core, Vc, is unknown, this variable needs to be expressed in terms of observables and ﬁtting parameters. This can be accomplished in a few algebraic steps by ﬁrst looking at an expression for the mass of the mantle, Mm, and assuming that, similarly to the core, the mantle can be represented by some mean

29 density, ρm. The mass of the mantle can be expressed as the total mass of the planet minus the mass of the core, which can then be rewritten in terms of volumes,

4 M = M − M = ρ V = πρ (R3 − R3). (3.3) m p c m m 3 m p c 3 Solving Equ. 3.3 for Rc leads to

3 3 3 Rc = Rp − (Mp − Mc). (3.4) 4πρm

Substituting Equ. 3.4 into Equ. 3.2, solving for Mc, and simplifying leads to

ρc ρc −1 Mc = (ρcVp − Mp)(1 − ) . (3.5) ρm ρm

Since the parameters of rocky planets are commonly normalized relative to Earth values, it is useful to rewrite this expression in the same manner. Scaling this equation to the Earth gives,

ρc Vp (ρc/ρ⊕) (ρc/ρ⊕) −1 Mc = ( − Mp)(1 − ) (3.6) ρ⊕ V⊕ (ρm/ρ⊕) (ρm/ρ⊕)

To get this expression in terms of planetary mass and radius, we can use

4 3 3 Vp 3 πRp Rp = 4 3 = , (3.7) V⊕ 3 πR⊕ R⊕ which reduces Eq. 3.6 to

3 ρc Rp ρc − Mp ρ⊕ R⊕ ρm M = (3.8) c 1 − ρc ρm

30 Finally, plugging 3.8 into the deﬁnition of core mass fraction (Eq. 3.1) yields

3 ρc Rp ρc − Mp ρ⊕ R⊕ ρm CMF (M ,R ) = . (3.9) ρ p p M (1 − ρc ) p ρm

Alas, Equ. 3.9 gives CMF in terms of planetary observables and unknown ρc and

ρm. To remain consistent with the notation in Chapter 2, I call this CMFρ to denote that it is the iron mass fraction inferred from the planet’s bulk density. Here, Rp and

Mp are independent variables, but values for ρc and ρm have yet to be determined.

The exact values of these coefficients depend on Mp. This arises from the fact that materials are compressible, and therefore these fitting coefficients need to be written in terms of Mp. Thus, a more general form of 3.9 is

ρc 3 ρc(Mp) (Mp)R − Mp ρ⊕ p ρm(Mp) CMF(Mp,Rp) = . (3.10) M (1 − ρc(Mp) ) p ρm(Mp)

3.3 Parameter Fitting

To obtain the expressions for ρc(Mp) and ρm(Mp), I ﬁrst generate CMF vs Rp curves for 0.01 ≤ Mp ≤ 11M⊕ corresponding to a range of masses between that of the Moon and a conservative upper bound on the expected masses of super-Earths. These curves are shown in Figure 3.1.

Next, I ﬁt both ρc(Mp) and ρm(Mp) to a power law with a constant oﬀset

a2 ρx Mp = ao + a1 . (3.11) ρ⊕ M⊕

These fits and coefficient values are shown in Fig. 3.2, and Fig. 3.3. All densities in this chapter are normalized to the mean density of the Earth. In other words, substituting the ρc coefficients and 1 M⊕ into Eq. 3.11 yields ρc(1M⊕) = 2.16ρ⊕, which can easily be converted to g/cc by using ρ⊕ = 5.51 g/cc. This gives ρc(1M⊕) =

31 Figure 3.1: CMFρ vs. Rp curves for Mp = 0.01 (cyan) to 11M⊕ (magenta). ExoPlex data points are denoted by squares. Fits to Equ. 3.9 are shown as solid lines. Fit residuals are shown in the top panel. While there is a clear systematic trend in the residuals, the amplitude of this trend is, at most, 0.02% which is 3 orders of magnitude smaller than the current uncertainties in CMFρ due to observational uncertainties in planetary mass and radius.

32 2.16ρ⊕ = 2.16 ∗ 5.51g/cc = 11.8g/cc which is nearly identical to the average density of the Earth’s core.

With these coefficients for my ρc(Mp) and ρm(Mp) functions, I can reproduce core mass fraction values calculated by ExoPlex with a standard deviation of 1.3% (abso- lute). This standard deviation value of 1.3% is significantly smaller than the uncertainties in CMF due to the current uncertainties in mass and radius. For simplicity in this work, I opt for a global ρm fitting function, despite the apparent mantle phase transition at 4.5 − 5.0M⊕ seen in Fig. 3.3. As this fit still allows me to reproduce

CMFρ values calculated by ExoPlex to well below the current uncertainties, I do not consider this a major limitation. However, I am still actively working to account for this phase transition in my modeling to further reduce the discrepancies between ExoLens and ExoPlex.

For a planet with an Earth-like CMF, a 1% radius uncertainty and a 4% mass uncertainty leads to a CMF uncertainty of approximately 5-6%. Currently, only three small exoplanets (R < 2.0Rp) have mass uncertainties of 4-5% and four have a radius uncertainties of 1-2%, and none have both mass and radius measured to these levels. However, Stevens et al. (2018b) show that such precisions are possible in the TESS era. At these M-R precisions, the uncertainties in core mass fraction due to mass and radius uncertainties are still 4x as large as the disagreement between

ExoLens and ExoPlex. Fig. 3.4 shows the diﬀerence in CMFρ between ExoPlex and

ExoLens for 13 of the most precisely measured planets with R < 1.9R⊕ including the planets analyzed in Ch. 2.

3.4 Inputs and Outputs

ExoLens is built to test the hypothesis that a given planet’s composition is consistent with the Fe/Mg and Si/Mg ratios of its host star. Thus, the main ExoLens function takes in the planetary oberservables and their uncertainties, Rp ± σRp and Mp ± σMp , Fe Si and the stellar refractory abundance ratios, ( )? ± σ Fe and ( )? ± σ Si . Mg ( Mg )? Mg ( Mg )?

33 Figure 3.2: Fit to mean core density as a function of planetary mass.

34 Figure 3.3: Fit to mean mantle density as a function of planetary mass.

35 Figure 3.4: Diﬀerence between CMFρ calculated from ExoPlex and ExoLens for 13 likely rocky exoplanets (Rp < 1.9R⊕) with σM < 20% and σR < 10%. R.M.S. = 1.4%.

36 ExoLens has two main outputs: (1) CMFρ ± σCMFρ : planetary core mass fraction inferred from planetary mass and radius, and (2) P (H0): likelihood that the planetary core mass fraction is consistent with what is expected from stellar Fe/Mg and Si/Mg ratios. CMFρ is calculated from Equ. 3.10. The uncertainty in CMFρ is calculated by pulling a large number of mass-radius pairs from the joint planetary mass-radius 1 σ ellipse. Equ. 3.10 is used again to calculate the corresponding core mass fractions for each mass-radius pair. The core mass fractions that are greater than the mean

CMFρ are then averaged to obtain the upper uncertainty. Similarly, the core mass fractions that are less than the mean CMFρ value are averaged to ﬁnd the lower uncertainty. A schematic overview of ExoLens is provided in Fig. 3.5.

If speciﬁed by the user, ExoLens will also generate a series of four subplots for each planet. These plots are designed to show the range of possible planetary core mass fractions as a function of observables and the overlap between the CMFρ solution space and the CMF? solution space. An example is shown in Fig. 3.6.

3.5 Comparison with Zeng et al. (2016a)

For Earth-like core mass fractions (25-35%), Z16 is in good agreement with ExoLens, as expected. Over this CMFρ range, Z16 diﬀers from ExoLens by 1% for a 1M⊕ planet and < 4% for a 10M⊕, both of which are well below the current uncertainties in CMFρ (Fig. 3.7). However, for a CMFρ of 50%, Z16 overpredicts planetary core mass fraction by 6% and 10% for a 1 and 10 M⊕ planet, respectively. A 6-10% diﬀerence in CMFρ is comparable to the CMFρ uncertainties in Chapter 2 and larger than the uncertainties associated with 1% uncertainty in radius and 4% in mass.

At a 70% core mass fraction, Z16 over estimates CMFρ by 16% for a 1 M⊕ planet and 23% for a 10 M⊕ planet. This disagreement is larger than almost all of the

CMFρ uncertainties in Chapter 2. Thus, for exo-Merucuries, even at the current observational precisions, Z16 is wholly inadequate.

37 Figure 3.5: Schematic overview of ExoLens.

38 Figure 3.6: Sample ExoLens output for K2-229b. Black solid lines correspond to nine equally-spaced conﬁdence intervals between 10 and 90%. The dashed cyan lines are the 68% and 95% conﬁdence intervals. (Bottom left) Contour lines (white) correspond to lines of constant CMFρ values. Negative values indicate M-R combinations that cannot be explained by a purely MgSiO3 composition and require either an outer volatile layer or enrichment in ultra-refractory materials. (Bottom right) likelihood functions for CMFρ and CMF?.

39 Figure 3.7: Comparison plot between Z16 and ExoLens. (Left) Z16 (dashed) and ExoLens (solid) M-R curves for varying core mass fractions. (Right) Diﬀerence in CMFρ between Z16 and ExoLens for Mp = 1.0 and 10.0M⊕.

40 Chapter 4: A Simple Adiabatic Model for Secondary Atmospheres

4.1 Introduction

In Chapter 1, we remained agnostic on processes to explain apparent excesses or depletion of iron for rocky planets relative to their hosts. In this chapter, I take the initial steps to break this agnostic stance to explore thick secondary atmospheres as an explanation for planets satisfying H2 (super-Fluﬀs): the planet has an apparent mass deﬁcit relative to that predicted by its host star’s refractory abundance. The term secondary atmosphere is used to describe atmospheres that form after the initial accretion phase of planet formation. Secondary atmospheres are generally denser than primordial atmospheres and form in two ways: (1) intense stellar radiation can cause H/He gas to escape from a planet’s atmosphere, while it retains its higher mass atmospheric particles, or (2) from the release of interior gases through volcanism, commonly referred to as outgassing. (1) and (2) do not need to be mutually exclusive.

Secondary atmospheres appear to be the most promising explanation for super-Fluﬀs since they do not require more material than is available in the disk, which is required for the explanation of such planets being built primarily from Ca and Al minerals (Dorn et al., 2019). In the case of 55 Cnc e, the mass and expected CMF fraction of this planet yield an expected radius of 1.746 R⊕. However, the average measured radius of this planet is 1.897 R⊕, meaning there is an approximately 1000 km discrepancy between the expected radius and what is observed that needs to be

41 accounted for if this planet is found to be statistically inconsistent with its host star in the future.

While a thick secondary atmosphere on 55 Cnc e has not yet been confirmed, the data suggests this as a possibility (Angelo & Hu, 2017; Miguel, 2019), which could explain the observed radius inflation. Furthermore, secondary atmospheres are com- mon for the rocky planets in the solar system. Earth, Venus, and Mars all have secondary atmospheres that were formed by outgassing. In contrast to the Earth, the atmospheres of Venus and Mars are dominated by CO2, with >95% CO2 by mass. Here, I present the initial steps to creating a secondary atmosphere model. I show that this model performs well for Solar System objects like Venus. I then outline steps necessary to extend my model to thicker atmospheres as would be expected for potential super-Fluffs, such as 55 Cnc e.

The potential super-Fluﬀs explored in Chapter 2 have masses and orbital periods such that any primordial H/He envelope would have been stripped via stellar irradiation from their host stars (Jin & Mordasini, 2018b). However, heavier elements or molecules, like CO2, may still be retained in a gas envelope because of their higher masses and lower kinetic energies, even if the stellar irradiation is suﬃcient to remove extended hydrogen and helium envelopes from a planet.

I use an adiabatic atmosphere model to reproduce the atmospheres of Venus and Earth and argue it is a valid approximation for super-Fluﬀ atmospheres given the current radius uncertainties. I will integrate such a model into ExoPlex as an outer gas layer. The aim of this integration is to constrain the thickness and mass of atmosphere needed to explain the lower than expected densities of potential super- Fluﬀs.

4.2 The physics of atmospheric escape

For an atmospheric particle with mass, m, the total energy of the particle, Etotal, can be written as the sum of its kinetic, Ek, and potential, Ep, energies. The kinetic

42 1 2 energy of a particle can be expressed simply as Ek = 2 mvk, where vk is the kinetic velocity of the particle. The potential energy is due to the gravitational force between GMpm the particle and its host planet and, thus, can be expressed as Ep = r , where G is the gravitational constant, Mp is the mass of the planet, and r is the distance of the particle from the center of the planet. With these deﬁnitions, the total energy of the particle can be expressed as

1 GM m E = E + E = mv2 − p . (4.1) total k p 2 k r If the kinetic energy of the particle is greater than the potential energy, the particle will escape the gravitational pull of the planet. This gives the escape criterion that 1 2 GMpm Ek > Ep or 2 mvk > r . The minimum velocity required for a particle to escape 1 2 GMpm is when Ek = Ep or 2 mvk = r . Solving this expression for the minimum escape velocity, vescape gives r 2GM v = p , (4.2) escape r

The speed distribution of a given element or particle species (i.e. H2, O, CO2, etc.) in an ideal gas can be described by a Maxwell-Boltzmann distribution, where the most probable speed is

r 3kT v = v = . (4.3) k thermal m

Here, k is the Boltzmann constant and T is the average temperature of the gas.

Thus, particle species in a gas with vthermal > vescape will escape from the planet’s gravitational pull, and species with vthermal < vescape will be retained by the planet.

Since vthermal has an inverse square root dependence on the mass of the particle, it fol- lows that lighter elements will have higher speeds, on average, than heavier particles. As hydrogen and helium are the two lightest elements, they are the most challenging particle types for a planet to gravitationally retain. One the other hand, at a given temperature, heavier molecules have lower speeds and are more easily retained by a

43 planet. Therefore, a planet can have an atmosphere made up of molecules and atoms heavier than hydrogen and helium, like CO2, even under intense stellar irradiation.

CO2 gas, water vapor, and other elements/molecules heavier than hydrogen and helium can be released from planetary interiors through outgassing. As long as a planet has active volcanism then gases from the interior of a planet will continue to be released at the surface. Thus, even under high stellar radiation, provided the rate of atmospheric loss does not exceed the rate of outgassing and volcanism does not cease, then a planet can sustain a secondary atmosphere over geological time scales. As gas is much less dense then rock, a planet with a substantial secondary atmosphere will have a lower bulk density than a similar planet that is completely barren, even if astronomers can not directly detect the presence of this atmosphere.

The equilibrium temperatures and masses of Venus, Earth, and 55 Cnc e permit these planets to retain CO2 gas, but not H/He gas, as illustrated in Fig. 4.1. For this simplistic calculation, I assume that the atmospheric temperatures of these planets are equal to their equilibrium temperatures since this parameter can be directly calculated for exoplanets with a known orbital period or distance, while the actual temperature cannot be directly measured.

It is worth noting that the thermal velocity in Eq. 4.3 represents the most probable speed of a given particle or molecular species according to a Maxwell-Boltzmann distribution. Since the speeds of a particle type follow a distribution, some will have speeds lower than vthermal and others will have speeds greater than vthermal. Thus, these particles do not instantaneously evaporate from a planet’s atmosphere, but, rather, they “leak” away from the planet with a decay (or growth) constant set by the outgassing rate minus the escape rate for a particle species.

Venus’s equilibrium temperature is ∼ 227 K. Due to the greenhouse eﬀect, its surface temperature is actually closer to 735 K. While Venus’s equilibrium temperature places it below the CO2 retention line in Fig. 4.1, its actual surface temperature places Venus above this line. Why does Venus still have an atmosphere? Venus is losing its atmosphere. However, there is strong evidence that Venus was volcanically

44 active in its recent past (< 2.5 Mya) or that it is presently active (Filiberto et al., 2020). Even in the case of a presently inactive Venus, if the volcanic outgassing rate was greater than or equal to the atmospheric escape rate then Venus’s atmosphere would have continued to grow or replenish until volcanism ceased. Now this atmosphere is slowly escaping from the planet.

Similarly, if 55 Cnc e does have an atmosphere, then the equilibrium temperature I use (Demory et al., 2011) underestimates the surface temperature of the planet. If its actual temperature places it above the CO2 retention line, as long as its outgassing rate is greater or equal to the atmospheric escape rate then it is plausible that this planet still has an atmosphere.

4.3 Atmospheric Modeling

I assume that atmospheres can be approximated as an ideal adiabatic gas. Given the current radius uncertainties for exoplanets, a more complicated atmospheric model beyond purely adiabatic is not warranted. I calculate the mass and height of the atmosphere by solving the ﬁve coupled equations: the mass within a sphere:

dm(r) = 4πr2ρ(r), (4.4) dr the equation of hydrostatic equilibrium:

dP (r) = −g(r)ρ(r), (4.5) dr the adiabatic temperature proﬁle:

dT (r) γ − 1 µg(r) = − , (4.6) dr γ R Gauss’s law of gravity in one dimension:

1 dg(r) r2 = 4πGρ(r), (4.7) r2 dr

45 Figure 4.1: Atmospheric particle mass where vthermal = vescape as a function of planet mass for 300, 1000, and 2000 K. Solid symbols for Earth, Venus, and 55 Cnc e are plotted according to their equilibrium temperature according to their orbital distance (no greenhouse). The open symbol for Venus reﬂects its present surface temperature.

46 and the density of an ideal gas:

P (r)µ ρ(r) = , (4.8) RT (r) where r is radius, m(r) is the mass, ρ(r) is the gas density, P (r) is the pressure, g(r) is the acceleration due to gravity, T (r) is the temperature, γ is the ratio of speciﬁc heats for a gas, µ is the mean molecular weight, and R is the ideal gas constant. I start with some surface radius (Rrock), surface pressure (Psurface), mass of the rocky portion of the planet (Mrock), temperature (Tsurface), and mean molecular weight (µ), and solve these equations at each radius step, dr, until the pressure is nominally zero.

Here, I describe atmospheres using two bulk parameters: (1) the atmospheric scale height, Hatm, and (2) the total mass of the atmosphere, matm. Hatm is deﬁned as the height above the surface where the pressure has decreased by 1/e of the surface pressure.

I test my model of the atmosphere against the Earth and Venus to assess the validity of my adiabatic approximation for use in exoplanetary structure calculations. For the initial parameters, I use the values reported by NASA in the Earth Fact Sheet and the Venus Fact Sheet (Table 4.1).

4.4 Reproducing Earth and Venus’ Atmospheres

At the current observational uncertainties, the adiabatic approximation provides an adequate first-order estimate for atmospheric parameters. For the Earth, I find an atmospheric mass of 5.3 x 1018 kg (Table 4.2, Figure 4.2) which is within 4% of the value reported by NASA. I find an atmospheric scale height of 7.7 km which is within 10% of the actual value. This difference is ∼ 0.01% the radius of the Earth, or over two orders of magnitude smaller than the best-case exoplanetary radius uncertainties.

For Venus, I am able to reproduce the atmospheric mass of 4.8 x 1020 kg (Table 4.2, Figure 4.3). I calculate a scale height of 14.3 km. This is 1.6 kilometers smaller than

47 the value reported by NASA. However, this 1.6 km diﬀerence corresponds to only ∼0.03% of the radius of Venus which, again, is insigniﬁcant relative to the current planetary radius uncertainties.

4.5 Discussion

While my model can reproduce the pressure and density proﬁles, and thus the scale heights and masses, of Earth’s and Venus’s atmosphere reasonably well, I fail to reproduce the temperature proﬁles of both of these planets as shown in Fig. 4.2 and Fig. 4.3, respectively. This is simply because implicit in my model is the assumption that atmospheres are homogeneous with respect to molecular species. I describe the molecules in an atmosphere by a single parameter, the mean molecular weight. However, atmospheres are not homogeneous, but layered. For example, in the case of the Earth, at approximately 25 km above the surface, in the Troposphere, there is a layer of ozone (O3) which absorbs UV light from the Sun which causes an increase in temperature in this layer. Despite my model being inadequate for detailed temperature modeling, I do not consider this a major limitation, as the main parameters I wish to constrain are the mass and thickness of the atmosphere.

To make my atmospheric model relevant to super-Fluﬀs like 55 Cnc e, I will integrate it into ExoPlex as an outermost gas layer. As mass and radius are the primary direct observables of small planets, I will need to adapt my model to constrain the height and mass of an atmosphere from planetary mass and radius inputs. To do this, I will assume a near-massless atmosphere with all Fe, Si, and Mg contained within the rocky planet, and solve for the relative radius proportions of rocky planet and gaseous atmosphere.

The assumption that the atmosphere is essentially massless is valid within the current observational uncertainties since even in the case of the Venetian atmosphere, which is 100x as massive as the Earth’s, it accounts for < 0.01% of the total planetary mass. Furthermore, Crida et al. (2018a) explored the possibility of a gas layer on 55 Cnc e and found that, at most, it could have an atmosphere mass fraction of

48 Figure 4.2: My model of Earth’s atmosphere (solid) compared with the 1976 U.S. standard atmosphere model (dashed).

49 Figure 4.3: My model of the Venetian atmosphere compared with Justus & Braun (2007)

50 ∼ 0.002%. Signiﬁcantly larger atmospheric mass fractions (. 0.1%) than this are probably not the result of outgassing, but more likely to be extended primordial H/He atmospheres. Such planets will, thus, be sub-Neptunes, which are outside of the scope of this work.

According to my model presented in this chapter, 99% of the mass of Venus’s atmosphere is within ∼ 50 km of its surface. However, there is an opaque cloud layer on Venus extending to approximately 70 km above the surface and the ionosphere extends to approximately 240 km above the surface. While the measured radii of a planet with a thick atmosphere are dependent on the wavelength range used, I can use these values to bound the measured atmospheric thickness giving a range of ∼ 0.8-4% the radius of Venus. In other words, if astronomers were to observe an exo-Venus, then its atmosphere would increase the measured radius by 0.8-4% depending on the wavelength range used for observations.

The difference in CMFρ between a barren planet and one with an atmosphere that makes up 4% of its measured radius is 16%. This is of the same order as the CMFρ uncertainties presented in Ch. 2. Thus, at the current precisions, such a difference in CMFρ is not statistically significant. However, for radius and mass uncertainties of 1 − 2% and 4 − 5%, respectively, that are achievable using the methods proposed in Stevens et al. (2018a), the CMFρ value of a barren Venus compared Venus plus its atmosphere will be statistically different from one another, albeit at the 1 but not 2-sigma significance level. However, for super-Fluffs like 55 Cnc e that may host atmospheres that may make up closer to ∼ 8% of their measured radii (Crida et al., 2018b), this corresponds to closer to a 30% difference in CMFρ. With mass and radius uncertainties of 1−2% and 4−5%, respectively, such a difference in CMFρ will be resolvable at better than the 2-sigma level, and would conclusively characterize such a planet as a super-Fluff, further motivating the atmospheric models presented in this chapter.

51 Planet µ (kg/mol) Rrock(km) Mrock(kg) Psurface(bar) Tsurface(K) Venus 0.044 6052 4.867*1024 90 735 Earth 0.029 6371 5.972*1024 1 288

Table 4.1: Initial Parameters for Earth and Venus from Earth Fact Sheet and Venus Fact Sheet, respectively.

Planet Hatm (km) Hatm (km) matm (kg) matm (kg) Calculated Actual Calculated Actual Earth 7.6 8.5 5.3 x 1018 5.1 x 1018 Venus 14.3 15.9 4.8 x 1020 4.8 x 1020

Table 4.2: Summary of Results.

52 Appendix A: Calculation of CMF and σCMF

CMFρ, CMF?, and their uncertainties are calculated based on planetary mass, radius, host star Fe/Mg and Si/Mg as well as the uncertainty in each parameters. Example models in Table A.1 and A.2 were calculated using ExoLens, an open- source compositional calculator for rocky planets based on ExoPlex. This calculator estimates the core mass fraction of a 0.1-10M⊕ planet from mass and radius to within 1-2% of ExoPlex in a fraction of the time of ExoPlex, as well as automatically calculating P (H0). ExoLens is freely available on GitHub.

53 Mp σMp (%) Rp σRp (%) CMFρ σCMFρ,upper σCMFρ,lower 5 10 1.54 1 35 8.8 10.3 5 10 1.54 2.5 35 11.6 13.0 5 10 1.54 10 35 31.8 38.7 5 5 1.54 5 35 16.7 18.5 5 10 1.54 5 35 18.1 20.3 5 20 1.54 5 35 22.0 27.7 1 10 0.99 5 35 19.2 21.8 2.5 10 1.28 5 35 18.7 21.1 7.5 10 1.71 5 35 17.7 19.8 10 10 1.84 5 35 17.4 19.5 5 10 1.65 5 10 >10 24.7 5 10 1.59 5 25 19.7 22.0 5 10 1.50 5 45 16.5 18.6 5 10 1.37 5 70 12.7 14.2 5 10 1.25 5 90 9.6 10.8

Table A.1: The eﬀects of Mp, Rp, σRp and σMp on CMFρ and σCMFρ

54 Fe/Mg σFe/Mg (%) Si/Mg σSi/Mg (%) CMF? σCMF? 1.0 20 1.0 10 35.8 4.8 1.0 20 1.0 20 35.8 5.4 1.0 20 1.0 30 35.8 6.2 1.0 20 1.0 40 35.8 7.2 1.0 20 1.0 50 35.8 9.2 1.0 20 0.5 20 44.3 5.4 1.0 20 0.8 20 38.7 5.4 1.0 20 1.25 20 32.7 5.2 1.0 20 2.0 20 25.8 4.8 1.0 10 1.0 20 35.8 3.6 1.0 30 1.0 20 35.8 7.4 1.0 40 1.0 20 35.8 9.6 1.0 50 1.0 20 35.8 11.8 0.5 20 1.0 20 21.8 3.6 0.8 20 1.0 20 30.8 7.4 1.25 20 1.0 20 41.0 9.6 2.0 20 1.0 20 52.7 11.8

Table A.2: The eﬀects of Fe/Mg, Si/Mg, σFe/Mg and σSi/Mg on CMF? and σCMF?

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