Modeling Exoplanet Interiors from Host Star Elemental Abundances a Thesis Presented to the Faculty of the College of Arts and Sc

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Modeling Exoplanet Interiors from Host Star Elemental Abundances

A thesis presented to

the faculty of

the College of Arts and Sciences of Ohio University

In partial fulfillment

of the requirements for the degree

Master of Science

Brandi B. Hamilton

December 2019

This thesis titled

Modeling Exoplanet Interiors from Host Star Elemental Abundances

BRANDI B. HAMILTON

has been approved for

the Department of Geological Sciences

and the College of Arts and Sciences by

Keith A. Milam

Associate Professor of Geological Sciences

Florenz Plassmann

Dean, College of Arts and Sciences

ABSTRACT

HAMILTON, BRANDI, M.S., December 2019, Geological Sciences

Modeling Exoplanet Interiors from Host Star Elemental Abundances

Director of Thesis: Keith A. Milam

The overall dimensions and internal structures of potential Earth-sized rocky exoplanets are modeled based on the relative photospheric abundances of Mg, Si, and Fe in their host stars. The modeling assumes that the planet has these same relative abundances and that all iron resides in a liquid core surrounded by a magnesiosilicate mantle comprising enstatite orthopyroxene and forsterite olivine or their high-pressure transformation products. The Mg/Si ratio sets the olivine/orthopyroxene fractions of the upper mantle and the Mg-perovskite/MgO fractions of the lower mantle. The amounts of volatile elements are not considered aside from the oxygen required by the silicate and oxide minerals.

The planets are assumed to have formed under conditions where silicon remained in condensed form during accretion and volatile elements such as hydrogen, nitrogen and carbon did not. The core mass fraction is determined from the amount of Fe relative to

Mg, Si, and O.

Following the approaches of Valencia et al. (2006) and Sotin et al. (2007), the self-compression of a planet is modeled using published experimental observations or theoretical calculations of densities, bulk moduli, and thermal expansion coefficients of the silicate and iron components. Assuming a specific planet mass and an initial estimate of radius, compression is calculated from the surface downward in increments of 1-km layers, taking into account reconstructive phase changes as pressure and temperature

4 increase. The core-mantle boundary is reached when the underlying (i.e., remaining) mass equals the core mass fraction of the planet. The self-compression is modeled iteratively, varying planet radius until the gravitational acceleration at the center converges to zero. In this research, effort is made to avoid using potentially Earth-unique assumptions regarding the core mass fraction of the planet, or the partitioning of iron into mantle silicates, instead relying primarily on data from observational astronomy.

Potential terrestrial planets were modeled for fifteen F or G class stars, each with an

24 assumed mass equal to that of Earth: 5.97 x 10 kg (1 ME). The resulting planet radii varied by over 200 km due to differing elemental ratios, with the strongest influence being the core (iron) mass fraction according to R = 6980 km - 1751 km (Fe mass fraction). The Mg/Si ratio had a second-order effect, with a higher ratio producing a larger planet, as reported by Sotin et al. (2010). These radii are lower bounds in that the partitioning of Fe into the mantle (assumed to be zero here) would produce smaller cores and larger planets.

DEDICATION

This work is dedicated to my family, and all the bright minds who vow to never stop

learning.

ACKNOWLEDGMENTS

I would like to extend my sincere thanks to Dr. Douglas Green for taking me on as a graduate student and allowing me the opportunity to pursue this area of research. He was instrumental to this project, and I hope he enjoys retirement as much as I enjoyed having him as an advisor. I would also like to acknowledge Dr. Keith Milam for stepping in as my final advisor and helping me complete this process. And finally, I would like to thank my committee members, both past and present, who have assisted me in this endeavor.

TABLE OF CONTENTS

Page

Abstract ...... 3 Dedication ...... 5 Acknowledgments...... 6 List of Tables ...... 8 List of Figures ...... 9 Chapter 1: Introduction ...... 10 Detection of Exoplanets ...... 10 Terrestrial Planet Formation ...... 12 Elemental Composition ...... 17 Stellar Abundances ...... 18 Chapter 2: Methods ...... 20 Mineral Assemblages and Internal Structure ...... 21 Geophysical Calculations...... 24 Temperature Profile ...... 26 Sol Model Earth Mass Trial ...... 31 Sol Model Venus Mass Trial ...... 35 Modeling Other Target Stars ...... 36 Chapter 3: Results ...... 39 Chapter 4: Discussion ...... 44 References ...... 50 Appendix A ...... 54 Appendix B ...... 56 Appendix C ...... 57

LIST OF TABLES

Page

Table 1 Sol Abundance Data ...... 21

Table 2 PREM vs. Sol Model Values for 1 ME Planet ...... 32 Table 3 Target Stars ...... 38

Table 4 Fe mass% and Resulting Core and Planet Radii for 1ME Planets ...... 42 Table 5 Modeling Results for Earth-Moon System ...... 46 Table 6 Modeling Results for Mercury and Mars ...... 47

LIST OF FIGURES

Page

Figure 1. Confirmed Exoplanets by Type ...... 12 Figure 2. Condensation Temperatures of Elements...... 13 Figure 3. Diagram of Earth’s Differentiated Layers...... 16 Figure 4. Abundances of the Solar Atmosphere and Carbonaceous Chondrites...... 17 Figure 5. Temperature Profile Used in the Model...... 29 Figure 6. Self-Compression Flow Chart...... 30 Figure 7. Sol Model vs. PREM Density Plot...... 33 Figure 8. Sol Model vs. PREM Gravity Plot ...... 34 Figure 9. Hertzsprung-Russell Diagram...... 37 Figure 10. Modeled Planet Radii vs. Mg/Si Ratio...... 40 Figure 11. Modeled Planet Radii vs. Iron Mass Percent...... 41

CHAPTER 1: INTRODUCTION

Thousands of planets have recently been discovered, and now most (99.8%) of the planets we are aware of orbit stars other than the Sun. Few characteristics are known about these exoplanets, but missions such as the Kepler Space Telescope and the James

Webb Telescope are designed to learn more (Borucki et al., 2003; Gardner et al., 2009).

Given the challenges presented to even detect these small, distant bodies, it is crucial to apply our understanding of planetary accretion and layering, derived from our studies of the Solar System, to exoplanets in order forward research of these bodies which are currently difficult to directly observe.

The purpose of this project was to test whether an accurate interior model of a terrestrial planet could be made using only known geophysical constraints derived from the Solar System and the elemental abundance data from the host star. In particular, this project sought to estimate internal boundaries and overall radii of small (1 Earth-mass) planets based on temperature constraints and the presence of Si, Mg, O, and Fe.

Detection of Exoplanets

Few “rocky” terrestrial Earth-like planets have been detected. Of the 3912 exoplanets that have been confirmed, only 156 of them, less than 4%, have been classified as terrestrial planets (Figure 1). Additionally, less than one half of one percent of the confirmed exoplanets are estimated to have masses within 0.1-10 times that of

Earth (Brennan and Walbolt, 2019).

The prevalence of giant planets among the discovered exoplanets lies largely in the bias of detection methods. Compared to gas giants or Neptune-like planets, which are over 17 times more massive than Earth, the number of terrestrial planets is generally

11 small. Traditional planet detection methods favor exoplanets with short orbital periods, large masses, and/or large diameters, such that the planet temporarily masks the light from a star or affects the motion of the star to the extent that Doppler shifts can be detected (Sotin, et al., 2007; Lissauer et al., 2000; Marcy et al., 2014). Nearly all of the confirmed exoplanets (96.1%) have been detected due to radial velocity/wobble or transit observations (in which the star’s light is dimmed a measurable amount by the presence of a planet passing in front of it) (Brennan and Walbolt, 2019). The remaining 3.9% of confirmed exoplanets were discovered using microlensing, imaging techniques, transit time variations, eclipse timing variations, orbital brightness modulation, pulsar timing variations, or astrometry (Brennan and Walbolt, 2019). Although detection of the smaller, terrestrial planets is more challenging, these planets are of particular scientific interest, since Earth-like planets within the habitable zone of their host stars are thought to be the most likely places in the universe where liquid water, an assumed requirement for life, may be found.

Confirmed Exoplanets by Type

Unknown 12

Terrestrial 156

Super Earths 877

Neptune-like 1660

Gas Giants 1207

0 500 1000 1500 2000 Number of Confirmed Exoplanets

Figure 1. Confirmed exoplanets by type (adapted from Brennan and Walbolt, 2019).

Terrestrial Planet Formation

All bodies within a planetary system, including stars, planets, and smaller bodies, are assumed to form from an initial cloud of gas and dust, known as a nebula (Reddy et al., 2003). A nebula may contain molecules such as CO, OH, H2O, H2CO, NH3, SiO and

3 SO2, occurring at densities of a few thousand molecules per cm (De Pater and Lissauer,

2001). Nebular hypothesis states that when an area in the nebula becomes so dense that its self-gravity exceeds its turbulent, thermal, and magnetic support, it begins to collapse

(De Pater and Lissauer, 2001). Once the collapse is initiated, densities continue to increase and the collapse proceeds more quickly. Conservation of momentum leads to a rotation in the cloud, and pressures and temperatures increase as collapse ensues. As this process unfolds a flat disc shape is formed, with a proto-star forming at the dense center of the disc. While much of the motion at this point is related to material moving into and rotating within the disc, bipolar outflows have been observed in the area of the forming

13 star, ejecting material perpendicular to the plane of the disc. The star is considered mature when the outflows have ceased, accretion is complete, and the star has achieved hydrogen fusion. At this point, the increase in stellar temperature such that it halts contraction and the star becomes relatively stable. The basic elemental components remaining in the disc are the building-block materials that will condense to form any bodies that will orbit the host star (De Pater and Lissauer 2001; Lissauer, 1993;

Raymond, 2008).

The material within the disc is thought to initially be relatively homogeneous, as observed in chemical analysis of 181 thin disc systems studied by Reddy et al. (2003).

But as the disc begins to cool the remnant material within the disc becomes sorted based on its condensation temperature (Figure 2). Temperatures are cooler further from the nebular midplane and with increasing distance from the star, so the materials with lower condensation temperatures solidify further from the star (>3 AU in the Solar System)

(Prockter, 2005; De Pater and Lissauer, 2001).

2000 Metals and silicates condense Rocky planets too warm (Iron, nickel, rocks) 1500 Mercury for hydrogen-bearing ices to condense.

"Volatiles" condense 1000 Venus (Copper zinc, mercury, etc.) Earth Temperature(K) 500 Icy and gaseous planets "Extreme Volatiles" Jupiter cold enough to consist of (Ices: methane and ammonia) hydrogen-bearing ices. 0 H and He never condense

Figure 2. Condensation temperatures of elements adapted from Lodders, 2003.

Initially, temperatures in the disc are thought to be ~1500 K at 1 AU and ~100 K at 10 AU (De Pater and Lissauer, 2001). As the disc cools, the more refractory elements, such as metals and silicates, begin to condense into microscopic grains closer to the star, at temperatures down to ~1300 K (Lodders, 2003). More volatile materials, such as water-ice, carbon dioxide, methane, and ammonia, sublimate near the star, and require much lower temperatures (<300K) to condense, and thus are found solidified in larger quantities further out in the circumstellar disc (Lodders, 2003; Prockter, 2005). The theoretical boundary separating the inner areas where silicates and metals condense from those where water and other ices freeze is referred to as the “snow line”. Beyond the snow line gas giants, or Jovian planets, their satellites and other small bodies, form from an abundance of condensed volatiles and smaller percentages of silicates and metals.

Inside the snow line, terrestrial planets form using larger fractions of condensed refractory elements (De Prater and Lissauer, 2001).

Molecules within the disc are orbiting in roughly a Keplerian manner around the proto-star when they begin to cool and condense into solids. These grains then begin to interact and collide, and although some collisions are erosive, some result in accretion.

Inter-grain forces which may aid in the early stages of accretion include electrostatic interaction with positively charged silicate dielectric grains, van der Waals attraction, magnetic attraction between mutually oriented grains, and partial fusion of grains during impact (Lewis, 1995). Through these interactions, the grains may grow from microns to millimeters in size and eventually form planetesimals on the order of 1 km or larger in diameter. Because they generally are travelling in coplanar orbits, as these bodies grow they continue to interact and gravitationally perturb one another, leading to collisions,

15 increasingly larger bodies, and the formation of planetary embryos. A planetary embryo is a body that has reached a mass large enough to collapses under its own self-gravity into a spherical shape, in order to reach hydrostatic equilibrium. Although it varies due to the density of the material, in planetary formation this is thought to occur once a rocky object reaches a diameter >1000 km, or more specifically when:

2 4 Pc = 3GM /8πR (1) where PC is the pressure at the object’s center, G is the universal gravitational constant,

M is the mass, and R is the radius (Lewis, 1995).

Embryos grow more quickly than planetesimals due to their larger gravitational focusing factors (Chambers, 2010). Constructive collisions between bodies not only add mass to the growing planetary embryos, but also add energy in the form of accretional heat. When the materials in the body become heated, they may undergo partial melting, which allows denser and relatively less refractory elements to sink toward the center of the body and less dense materials to remain near the surface, a process termed differentiation. Once a planetary embryo has experienced differentiation it is reclassified as a protoplanet and is no longer considered relic or primitive material. Protoplanet interiors may evolve to show distinct interior layering, with iron and nickle-rich alloys such as kamacite and taenite, forming a core overlain by silicate minerals such as olivine and pyroxene. Once the development of the protoplanet is also sufficient to clear its orbital path of debris and other bodies, it may be reclassified as a planet (Margot, 2015).

A mature planet retains a great deal of heat from accretion, but also gains additional heat from radioactive isotopes within it, such as uranium-238 (238U), uranium-

235 (235U), thorium-232 (232Th) and potassium-40 (40K) as they begin to decay. If the

16 planet becomes heated such that it achieves at least a partial melting, it will continue to differentiate and may eventually have multiple distinct layers, such as those observed in the Earth (Figure 3).

Not all materials become incorporated in stars, planets, or smaller bodies. Within our Solar System such material is known to exist as asteroids, comets, and meteorites.

Chondritic meteorites contain spheres of minerals that retain properties from when they were first formed because they show little evidence of experiencing partial or complete re-melting that is associated with the differentiation process (Lewis, 1995). Several classes of chondrites exist, and while some show evidence of having experienced hydrothermal activity or thermal metamorphism, chondrites are thought to represent undifferentiated material that existed in the disc, with carbonaceous chondrites being the most primitive (Wood, 2011).

Figure 3. Diagram illustrating Earth’s differentiated layers (values from Sotin et al., 2010).

Elemental Composition

An interesting property of carbonaceous chondrites is that their bulk chemical composition shows strong similarity with the Solar photosphere for all but the most volatile elements (Figure 4) (Ringwood, 1979: Anderson, 1989; Sotin et al., 2010; Wood,

2011). This similarity in chemical composition reinforces the notion that carbonaceous chondrites are representative of the initial primitive materials in the disc from which the bodies in the Solar System including the Sun formed (Wood, 2011).

Given this correlation, chondritic/solar abundances are assumed for the bulk composition of the Earth and other planets for the Solar System, at least for moderate and highly refractory elements (Sotin et al., 2010; Montmerle et al., 2010).

Figure 4. Abundances for the solar photosphere plotted against the abundances of carbonaceous chondrites, normalized to silicon (adapted from Ringwood, 1979). Elements of interest for this project are Fe, Mg, Si, and O.

Stellar Abundances

The chemical abundances present in the star’s photosphere can be detected using spectroscopy. Each element has distinct fine structure lines that appear in two or more wavelengths, which give it a specific spectral signature. These fine structures occur when an atom emits light while changing from one energy state to another, and are unique to that element. By carefully analyzing the light from a star, astronomers can determine what elements are present in the photosphere, and in what relative abundances

(Lewis, 1995).

Although there are problems detecting certain elements in stellar spectra, most of the 83 naturally occurring elements are detectable in the Solar photosphere, as seen in

Figure 4. Lodders (2003) reported on 56 elements that have Solar abundance comparisons. Of those 56 elements, 31 demonstrated that the photospheric and chondritic abundances agree within 10%, and 41 of 56 abundances agree within 15%

(Lodders, 2003). Additionally, Asplund et al. (2005) determined the abundances of 64 elements in both CI chondrites and the Solar photosphere and found that 50 agreed within measurement uncertainties, and 60 of the 64 agreed within 10% (data available in

Appendix A).

These observations reinforce the nebular hypothesis and when applied to other stars allow the inference that planets, moons, asteroids, comets and meteorites that formed in the same planetary system have similar bulk compositions as their host stars.

Therefore, although direct observation or sampling is not yet possible, it is logical to move forward under the assumption that the same principles governing the formation and

19 differentiation of terrestrial planets in our system can be applied to exoplanets of another main sequence star (Sotin et al. 2007).

This research follows the work of Valencia et al. (2006) and Sotin et al. (2007), which use derivations of the Adams Williamson equation and relationships established therein to model large “Super-Earths” (10-15 times the mass of Earth) and water-rich ocean planets. In contrast to their work, this study does not use any Earth values for composition, core size, or differentiation of Fe into mantle silicates, and focuses the modelling on terrestrial planets of smaller masses. The goal of this study is to model the structure and size of terrestrial Earth-mass exoplanets by enlisting established geodynamic and thermodynamic relationships and using only the available elemental abundance data of the host stars as input.

CHAPTER 2: METHODS

The model uses a series of simultaneous calculations in order to simulate self- compression and thermal expansion processes in a terrestrial planet. Major minerals are estimated based on the relative elemental abundances of the host star. Physical property values for these minerals, such as density and bulk modulus, are used as input for the model. Calculations are carried out in a spreadsheet program beginning at the planet surface and continuing inward, with values determined at depth increments of 1 km, from the surface to the planet center. Every row of the spreadsheet represents a 1 km thick layer within the planet, and properties including pressure, temperature, gravity, density, bulk modulus, and the underlying mass are calculated for each layer. For this project, the

24 mass was fixed at 1 ME = 5.97 x 10 kg (Williams, 2017) for all modeling, unless specifically noted.

Because the planet radius is unknown, an initial guessed radius is used, and adjusted iteratively until a best fit radius is found. The best fit is ascertained by observing the behavior of gravity at the center of the planet. If the assumed planet radius is too small, gravity will diverge (positively) in the core and if the planet radius too large, gravity will become negative before reaching the center. A best-fit radius for the planet is found when the calculated gravity converges on zero at the center of the planet.

The upper and lower mantle boundary in the model is defined as the depth coincident with the pressure requirement for the phase transitioning of upper mantle minerals to perovskite (pv), which occurs near 23.5 GPa (Hirose et al., 2017). The core mantle boundary is determined by the mass of core material, which is iron for this project, that is available to constitute the core. The goal of this approach is to yield a

21 modeled planet radius, as well as internal boundary layers, using no adjustable parameters.

Mineral Assemblages and Internal Structure

Spectroscopists analyze visible light from target stars and can detect the presence and proportions of many elements by identifying corresponding absorption. This information is then used to calculate stellar elemental abundances, which are generally reported relative to hydrogen (H) (Asplund et al., 2005; Bensby, 2014). Abundances from previous studies are used as input for this project, and are then used to estimate mineral assemblages of a modeled terrestrial exoplanet in a specific star system. The proportion of each element which composes the planet can be found by converting the stellar abundance data and atomic weights. An example of the Solar abundances from

Asplund et al. (2005) and the corresponding mass fraction of a corresponding hypothetical 1 ME planet is shown in Table 1.

Table 1 Solar abundance data NEl (from Asplund et al., 2005) was used to calculate the following molar ratios for a 1 ME planet. Values are relative to (Si = 1.0 x 106).

Element NEl Molar % of a 4 Component Exoplanet Si ≡1.00E+06 16.60% Mg 1.05E+06 17.43% O 3.10E+06 51.47% Fe 8.73E+05 14.49% Core mass% 32.23%

Planetary accretion involves high temperatures, due to both high velocity impacts and in larger bodies, compression due to gravity. Solid particles (dust and larger) are

22 more effectively accreted than are gas molecules due to their higher density (Lewis,

1995). Consequently, volatile elements that are in gaseous compounds during accretion can be “lost” from the growing planet due to its low gravity and higher temperatures.

These elements include hydrogen, helium, nitrogen, carbon, and oxygen, among others.

For this reason, the modeling here does not attempt to account for these elements and is restricted to the primary refractory elements that make up nearly all the planet’s mass: silicon, magnesium, and iron. Although it may seem limiting to use only four elements

(O, Si, Mg, and Fe), previous research has shown that these are the only elements necessary to describe the bulk of a terrestrial planet as they comprise over 95% of the mass fraction of Earth (Sotin et al., 2010; Brown and Mussett, 1993). Because Si and Mg primarily exist in the form of silicates, oxygen is required to form these silicates. In fact, approximately half of the atoms in Earth are oxygen (Brown and Mussett, 1993). For this modeling method, the planet’s oxygen abundance is set by the proportions of silicates/oxides present, which is fixed by the stellar Mg/Si ratio.

Because this project focuses on modeling terrestrial planets, it is concerned with the major refractory, silicate-forming, elements of silicon (Si), magnesium (Mg), and iron

(Fe). These elements, with oxygen (O), form olivine (ol), orthopyroxene (opx), silicate- perovskite (pv) (or “ringwoodite”), oxides (per), and core material (Fe). This model does not make any assumptions regarding chemical differentiation of a crustal layer, therefore the material from the surface to the lower mantle is a mixture of forsterite (ol) and enstatite (opx), and is referred to as upper mantle material. The fraction of each mineral found in the upper mantle is estimated using the Mg/Si ratio of the star, and the following formulas:

Olivine (Forsterite): Mg2SiO4 (2)

Orthopyroxene (Enstatite): MgSiO3 (3)

Olivine fraction (ol) = Mg/SiStar -1 (4)

Orthopyroxene fraction (opx) = 1-ol (5)

Examples of the derivations used in this project may be found in the Appendix B.

When subjected to higher pressures, caused by increasing depth, the upper mantle minerals experience phase changes. Although multiple phase changes have been found to occur, this project focuses on one primary transition in the mantle, in which olivine and orthopyroxene transition into perovskite and periclase (MgO), which make up the lower mantle. This transition defines the Upper-Lower-Mantle-Boundary (ULMB). No

Transition Zone, such as the one indicated between Earth’s upper and lower mantle, is included in this model.

The transition defining the ULMB occurs at a pressure near 23.5 GPa (Hirose et al., 2017).

MgSiO3 (opx) → 23.5 GPa Pressure → MgSiO3 (pv)

Mg2SiO4 (ol) → 23.5 GPa Pressure → MgSiO3 (pv) + MgO (per)

The amount of oxygen (O) incorporated relative to silicon is calculated to be the amount required to complete the chemical formulas of the silicates, as previously discussed. The oxygen required per silicon atom to complete the olivine component of the mantle is given by the product (olivine fraction of the upper mantle)*(the number of oxygen per magnesium = 2)*(the number of magnesium per silicon in the planet), or ol x

2(Mg/SiStar). Similarly, the required oxygen for the pyroxene component is opx x

3(Mg/SiStar). The sum of these two gives the required oxygen for the planet, referenced to silicon. With this, the relative abundance and mass fractions can be determined for each of the four elements assumed to comprise the planet: Mg, Si, O, and Fe.

All iron (Fe) present is modeled as residing in a liquid core, therefore the Fe mass is equal to the theoretical planet’s core mass. Fe mass fraction is found using the previously calculated values and the atomic weights of the elements O, Mg, Si and Fe:

16, 24, 28, and 56, respectively.

MFe/ (1 ME) = (Fe/Si x 56)/[(Fe/Si x 56) + (O/Si x 16) + (Mg/SiStar x 24) + (1 x 28)] (6)

Fe mass of planet (core mass fraction) = 1 ME x (M-Fe/1 ME) (7)

Geophysical Calculations

Calculations for this model are modified for each layer of the planet, as the materials vary, but the principles guiding the calculations remain consistent throughout.

Surface gravity is initially calculated using the planet mass (M), radius (R), and the universal gravitational constant (G). Pressure (P) is calculated according to depth (d) and density (ρ). Density increases with d, according to the P, bulk modulus (KP), temperature

(T), and the thermal expansion coefficients (αT). The underlying mass (mu), ρ, g, P, KP, and thermal expansion factors are automatically calculated at every 1 km thick layer from the surface inward, using the values from the above layer, based on the following relationships:

Δρ = ρ((ΔP/(KP + ΔKP)) - αTΔT) (8)

2 g = Gmu/(R-d) (9)

ΔP = ρgΔh (10)

ΔKP = 5ΔP (11)

Δρ = α/(ρ1/ρ0)*ΔT (12)

These equations are used in conjunction with the mineral physical properties from published research, which are first entered at the surface of the planet and then updated at each boundary as the mineral proportions and types change.

Minerals change with increasing pressure and temperature at depth by transforming into more compact crystalline structures or polymorphs. In the Earth, there is a Transition Zone, which is the depth at which olivine transitions to the more compact structure of spinels, and then to Si-perovskite and periclase. Orthopyroxene also transitions to the perovskite structure, with the high-pressure mineral equivalent of these two parent minerals (ol and opx) being perovskite and periclase (Tange et al., 2012).

This process has been simplified in the model by eliminating the intermediate (spinel) phase changes and implementing an olivine to perovskite + periclase transition (as well as an orthopyroxene to perovskite transition at the same depth). This simplification makes the planet less dense within the Transition Zone (~400-670 km), and consequently, slightly larger. Thus, within the model, this first major phase change in the minerals occurs at a pressure of 23.5 GPa and defines the Upper and Lower Mantle Boundary

(ULMB). Since the new crystalline structures have different physical properties, the ρ,

KP, α, and proportion of each new mineral (pv and per) is entered at this depth and calculations using these inputs continue toward the center of the exoplanet.

When the underlying mass, calculated at every 1 km increment of depth, is equal to the Fe mass for the entire planet, the Core Mantle Boundary (CMB) has been reached.

As with the ULMB, the ρ, KP, and α must be changed at the CMB to reflect the physical properties of the new material, liquid iron. This model does not simulate an inner and

26 outer core system, the entire core is modeled as liquid iron from the CMB to the center of the planet.

The depths of the ULMB and CMB are used to modify the temperature profile.

The model is then run again with the new temperature profile and changes in the ULMB and CMB depths are noted. When these depths and the temperature profile are stable, modeling is complete.

Temperature Profile

The temperature profile of a planet is set by the latent heat from formation, radioactive heat production, the relative sizes of the planet’s mantle and core, and its surface temperature. Proportions of the primary long-lived radioisotopes, U and Th, are likely to be the same as those of the planet’s host star, but these elements are poorly if at all resolvable in stellar spectra. Residual heat is difficult to estimate, even for the Earth, but is thought to comparable to radioactive heat production (Brown and Mussett, 1993).

Surface temperature is also an unknown for most exoplanets, so a typical habitable zone surface condition (280 K) is assumed for this model. The relative size of the mantle and core are determined by the self-compression modeling, these boundaries along with a no- heat-flow constraint across the center of the planet (i.e. zero temperature gradient), make up the firmest constraints on the temperature profile. Because the other factors are so poorly known, a heat-flow regime similar to Earth’s is assumed: lithospheric heat conduction (and surface temperature and gradient), with adiabatic gradients for the convective parts of the mantle (T>1500 K). A 200-km thick conductive thermal boundary layer is placed at the base of the mantle with a temperature gradient set to match heat flow out of the core. Fortunately, temperature has far less of an influence on

27 the sizes of planetary layers than does pressure, so the errors associated with these temperature assumptions will be second order.

Terrestrial exoplanets of significant interest to science are those that are potentially habitable. With this in mind, a temperature profile was designed to model a terrestrial exoplanet within the habitable zone. The inner boundary of the habitable zone

(closest to the star) is set by temperature, with the maximum surface temperature 395 K.

Above this temperature any water present will experience a drop in the dielectric constant, resulting in a change in solubility, and damage biological molecules by way of lipid layer destabilization (McKay, 2014). The outer boundary of the habitable zone

(furthest from the star) is generally set at 273 K, the temperature at which pure water freezes. Many organisms, however, have intercellular saline solutions which allow them to live and reproduce at lower temperatures, or can survive in extreme environments with only thin films of water that exists between ice and grains of soil (McKay, 2014). When factoring in the reproductive temperature ranges of these extremophiles, the minimum temperature to support life can be extended to 258 K (McKay, 2014). Using these qualifiers, the habitable surface temperature range of a planet is between 258 K and 395

K. The surface temperature of the model is set at 280 K. This temperature profile is

Earth-like in many ways and is adjusted iteratively as described in the previous section to create a unique profile tailored to the subject exoplanet.

The model assumes three distinct layers: the upper mantle, lower mantle, and core. It also assumes five rheological layers: lithosphere (T<1500 K), asthenosphere

(T>1500K), lower mantle, thermal boundary layer (conductive base of mantle), and liquid core (Figure 5). The lithosphere extends from the surface to the depth where the

28 temperature reaches 1500 K and has an average lithospheric temperature gradient of 11

K/km. The depths to which the upper and lower mantle extend are calculated by the model and are refined with multiple iterations. A flow chart of the general modeling process is presented in Figure 6, and a copy of the model is available from the author.

The model has an adiabatic temperature gradient from the base of the lithosphere to the

Thermal Boundary Layer (TBL), which is set to be 200 km above the CMB. The depth to the CMB is calculated by the model. Although other temperatures have been explored, the temperature of the CMB for this model is set at 4000 K. Heat flow is matched across the CMB, and the thermal profile has a zero gradient across the center. Like other aspects of the model, calculating the change in temperature with depth requires initial mineral input in the form of thermal expansion coefficients for the materials which are entered at the surface, ULMB, and CMB.

The adiabatic gradient in viscoelastic silicates decreases with increasing pressure and temperature (Lewis, 1995). This is reflected in the calculated temperature profile for the convecting area of the mantle, where the temperature gradient decreases from 2 K/km at the base of the lithosphere to 0.5 K/km at the top of the thermal boundary layer.

Figure 5. Each of the five sections within the temperature profile shown above are chemically or mechanically distinct. The lithosphere, upper lower mantle boundary (ULMB), thermal boundary layer (TBL), and core areas are labeled, and the solid black line illustrates how temperature increases with depth.

Figure 6. Flow chart depicting the general modeling process.

Sol Model/Earth Mass Trial

In order to test the model’s effectiveness in calculating planet radius and depths to internal boundaries, a test of the model was performed using abundance data from the

Sun (Sol). Results were compared to Earth values for average radius and depth to internal boundaries taken from Preliminary Reference Earth Model (PREM) data

(Dziewonski and Anderson, 1981). PREM calculations are the result of analyzing seismic wave travel times and it is recognized as the leading model for the Earth’s interior structure and physical properties. PREM values for the Earth’s radius, depth to

ULMB, and depth to CMB were reported as 6371 km, 670 km, and 2891 km, respectively.

The model input was a planet mass = 1 ME and Solar photospheric abundance data from Asplund et al. (2005). Abundance data was then used to calculate the amount of Si, Fe, and Mg present for a terrestrial planet of this mass in the Sol system. Mineral assemblages for the mantle were determined and the amount of oxygen incorporated in the planet was calculated, as well as the mass percent of Fe (Table 1).

The olivine fraction of the mantle was modeled as 5.0% and the orthopyroxene fraction was 95.0%. Oxygen incorporated in the mantle was calculated based on the chemical formulas of the minerals, with 4 oxygen atoms allotted for each olivine molecule and 3 oxygen atoms to complete each orthopyroxene structure. The core mass percent was 32.23%, which for a 1ME planet with all Fe in a liquid iron core, would equal a core mass of 1.92 x 1024 kg.

Once abundance data had been converted into mineral information and the core mass had been established, physical property inputs and percentages for each mineral

32 were entered into the model (referenced physical property input data can be found in

Appendix C). Olivine and orthopyroxene densities, bulk moduli, and thermal expansion coefficients were obtained from published mineral physics research, with the surface olivine and orthopyroxene information originating from Anderson et al. (1991), Duffy and Anderson (1989), and Reynard and Price (1990). Values for perovskite and Mg- oxides from Tange et al.(2012) were entered for calculations starting at the ULMB. Core values for liquid iron from Vocadlo et al. (2003) were input for calculations starting at the

CMB, which was calculated to be the point at which the underlying mass equals 1.92 x

1024 kg, the Fe core mass of the planet.

An initial guessed radius of 6400 km was entered in order to begin calculations and the model was run in an iterative process until a best fit radius and depth to the internal boundaries were found. The best fit radius was found to be 6413 km, depth to

ULMB was 690 km, depth to CMB was 3044 km, and the rotational moment of inertia

(I/MR2) was 0.3252. These values are compared to those of PREM in Table 2. Figures 7 and 8 show the final gravity and density profiles of both PREM and the Sol Model. The model overestimates the planetary radius by 42 km and underestimates the planet’s core size and rotational inertia factor, or moment of inertia (MOI).

Table 2 PREM vs. Sol Model values for a 1 ME planet.

PREM Sol Model* Deviation Planet Radius 6371 km 6413 km ±42 km Depth to ULMB 670 km 690 km ±20 km Depth to CMB 2891 km 3044 km ±153 km Rot. Inertia I/MR2 0.3308 0.3252 -0.0056 *uses abundance input from Asplund et al., 2005.

Figure 7. Comparison of the density profiles of the Sol Model and PREM. This plot illustrates how the densities of both the Sol Model and PREM compare with depth.

Figure 8. Comparison of gravity profiles for the Sol Model and PREM.

Some deviations from the PREM data may be due to the model not partitioning iron into the mantle, however any iron going into the mantle materials would decrease the core size, which was already smaller than PREM. It is likely that Earth’s supersolar iron abundance, which contributes to it having a larger than average core size, was also contributing to the discrepancies. It has long been known that the Earth has a higher than

Solar abundance of Fe (Rubie et al., 2003; Sotin, et al., 2010). One possible cause for

Earth’s supersolar iron content and larger core is believed to be its interaction with the protoplanet Theia, with which it collided with and pirated material from ~4.5 Ga (Lewis,

1995; Young et al., 2016). Following this interaction, some material coalesced with

Earth and other materials formed the Moon. In an effort to test how much of the deviation was due to iron partitioning into the mantle versus due to Earth’s unique formational history, another trial was ran using Sol data on a Venus mass planet.

Sol Model/Venus Mass Trial

The same abundance input from Asplund et al., (2005) that was used above was also used as Sol abundances for this trial, the planet mass however, was changed to be equal to Venus’ mass of 4.87 x 1024 kg (Williams, 2017). Comparative data for the radius of Venus was found in Williams (2017) which a reported the average radius of

6052 km. Definitive information regarding internal boundaries of Venus are not available, so the rotational inertia factor (I=MR2), which is dependent upon the correct distribution of mass within a body, was calculated and compared to that of Venus.

Using the Asplund et al. (2005) Sol abundances, estimated upper mantle mineral modes were calculated to be olivine 5% and orthopyroxene 95% (as before), with a core size of 1.57 x 1024 kg. The model was run until a best fit radius of 6054 km was found.

Results indicated a planet radius differing by only 2 km from Venus’ average planetary radius and reproduced the known rotational inertia factor of 0.33 (Williams, 2017).

Given that modeling of exoplanets may use mass-radius relationships with relative uncertainties of 5-15% (Lissauer et al. 2011), the relative uncertainties of these two Sol modeling results (1.4% and 9.4%) are well within reasonable planetary modeling standards.

Modeling Other Target Stars

After the initial trials were run, additional star targets were selected for modeling.

Abundances for fourteen F and G main sequence stars were obtained from Bensby et al.

(2014) via the VizieR astronomical cataloguing access tool maintained by the Centre de

Donnees astronomique (VizieR Catalogue J/A+A/562/A71, CDS & ESA-ERIN

Information Systems Division, 2015). Main sequence stars maintain stability in their photospheres for billions of years, which make them ideal targets for this research

(Sackmann et al., 1993). Additionally, higher metallicity star systems are thought to have a greater propensity for the formation of terrestrial planets (Cumming, 2010; Greaves et al., 2005; Wyatt et al., 2013), and for this reason F and G class stars were sought for this project (Figure 9). The Bensby et al., 2014 data set contains elemental abundances for

714 F and G class stars, from which study targets were selected according to the completeness of their abundance set, their effective temperatures, and proximity. Stars with incomplete data sets, or with very large uncertainties were not considered. Stars with temperatures below 5370 K were also eliminated, since Bensby et al. (2014) stated that with effective temperatures lower than ~5400 K uncertainties became greater. The stars selected in this study have effective temperatures ranging from 5372 (±108) to 6024

(±57) K. From the remaining list of stars in the data set preference was given to those with a closer proximity to Sol, with the final list consisting of stars that are ~19.5 to 491 light years away. A list of the fourteen target stars that were selected and some of their attributes can be found in Table 3.

Figure 9. Hertzsprung-Russell Diagram. Main sequence dwarf stars are relatively stable and will spend ~10 Gyr on the main sequence, during which time their atmosphere is untouched by internal nuclear processes (Sackmann, et al., 1993; Greaves et al., 2005; Bensby et al., 2014).

Table 3. Target stars, listed by their Hipparcos (HIP) catalogue numbers (ESA, 1997), and some of their attributes (Bensby et al., 2014). The relative abundance values listed are the log values of the ratios between stellar and solar values as reported by Bensby et al. (2014).

Effective Star HIP Age Distance Mass Temp [Fe/H] [Mg/Fe] [Si/Fe] Identifier Gyr kpc (MSun) ˚K 15510 13.1 5372 0.0060 0.78 -0.46 0.33 0.22 2021 5.3 5852 0.0075 1.21 -0.07 0.08 0.03 115577 10.2 5648 0.0079 0.95 0.04 0.11 0.09 64924 13.1 5469 0.0086 0.89 -0.03 0.06 0.07 1599 6.4 5932 0.0086 1 -0.19 0.05 0.05 113044 3.7 6024 0.0326 1.29 0.26 0.05 0.03 64459 8.6 5839 0.0361 1 -0.31 0.15 0.05 100970 8.1 5748 0.0385 1.06 0.03 0.07 0.01 12048 6.2 5751 0.0389 1.15 0.12 0.12 0.02 5301 9.3 5686 0.0548 0.94 -0.11 0.02 0.00 95262 9.6 5759 0.0580 0.94 -0.51 0.27 0.16 113386 6.5 5736 0.0620 1.14 -0.06 0.06 0.03 1931 1.5 5990 0.1028 1.16 0.34 0.02 0.02 61802 8.9 5379 0.1506 0.87 -1.25 0.38 0.17

CHAPTER 3: RESULTS

Variations in elemental abundances of the target stars resulted in differing compositions for terrestrial exoplanets modeled for each of the systems. Variations in

Mg/Si ratios influenced the mineral assemblages of the mantle, with the range being between 1.05 (pyroxenite mantle, 97% pyroxene) and 1.70 (peridotite mantle, 70% olivine) (Figure 10). Consequently, the lower mantles of each of these end member models was also quite different with the former and the latter having 98% and 65% perovskite in the lower mantle, respectively.

Differences in the Fe mass percent, which ranged between 20% and 32.2% resulted in differing core sizes in the models as well (Figure 11 and Table 4). The model with the smallest Fe mass percent of 20% had a core radius of 2897 km, while the model with the largest Fe mass percent of 32.2 % had a core radius of 3369 km. These variations resulted in modeled 1 ME planets having radii ranging between 6413 and 6635 km, with corresponding surface gravity values between 9.72 and 9.05 m s-2.

The temperature profile applied to each exoplanet included a 280 K surface temperature and 4000 K CMB, with the thermal profile being customized for each planet as previously discussed. The effect of temperature was found to be secondary to that of pressure, the dominant factor determining the depth at which phase changes in the minerals occurred and boundaries of the internal structures began.

Figure 10. Radii for all modeled 1 ME planets plotted against their Mg/Si ratio.

Figure 11. Radii for all modeled 1 ME planets plotted against their Fe mass percent.

Table 4. Fe mass% and resulting core and planet radii for 1ME planets.

Star Planet Core Planet Upper Mantle Planet Identifier Fe Radius Mg/Si Proportion of Radius Mass% (km) opx/ol (km) HIP 15510 19.97 2897 1.35 .65/.35 6622 HIP 61802 20.09 2899 1.70 .30/.70 6635 HIP 95262 22.27 3001 1.35 .65/.35 6586 HIP 64459 27.20 3216 1.32 .68/.32 6512 HIP 115577 27.33 3208 1.10 .90/.10 6499 HIP 12048 28.59 3268 1.32 .68/.32 6490 HIP 64924 29.04 3263 1.05 .95/.05 6465 HIP 2021 29.37 3290 1.18 .82/.18 6471 HIP 1599 29.73 3286 1.05 .95/.05 6456 HIP 113386 29.89 3291 1.13 .87/.13 6455 HIP 100970 30.07 3296 1.21 .79/.21 6454 HIP 113044 30.16 3300 1.10 .90/.10 6450 HIP 1931 31.19 3334 1.05 .95/.05 6431 HIP 5301 31.63 3349 1.10 .90/.10 6425 Sol 32.22 3369 1.05 .95/.05 6413

Earth and Venus-mass planets in our Solar system were able to be well modeled using 4 major elements and the modeling process covered in this research. Subsequently, fourteen additional F and G class stars were analyzed and the model was run for a 1 ME planet in each system. The dominant control on planet size was found to be the Fe mass percent, which showed a strong linear relationship with R= 6980 km – 17.51 km (Fe mass%). The Mg/Si values accounted for second-order deviations with higher Mg/Si ratios increasing the overall planet radius by at most 11 km in these trials (Figures 10 and

11).

The highest Mg/Si ratio of this group was 1.70 (HIP 61802), and the second highest Mg/Si ratio was 1.35 (HIP 15510 and HIP 95262). The lowest Mg/Si was 1.05

(HIP 64824, HIP 1599, and HIP 1931), with the average Mg/Si was 1.20. Although the

Mg/Si does alter the mineral modes of the mantle, the range of variations in Mg/Si ratios in this study and the limits of the minerals used resulted in a much smaller affect on the overall size of the planet than the iron content incorporated in a planet, with the modeled planets with the highest Fe mass percents having the smallest radii, and those with smaller cores and thicker mantles having larger radii (Figure 10).

CHAPTER 4: DISCUSSION

As exoplanets in other systems may have formed under differing conditions than those of terrestrial planets in our own Solar System, this model was designed to be able to adapt to a number of different modeling situations. The previous results presented demonstrate that a 1 ME planet, with chondritic bulk composition and a habitable thermal regime can be modeled such that it falls well within acceptable relative uncertainties, but this set of circumstances certainly does not apply to all planet systems. There are many cases that fall outside of these constraints, and as such, there is a need to look at other possible modeling situations and outcomes. It is not the objective of this discussion to provide a comprehensive review of each of these adjustable variables, but rather to discuss other potential scenarios, how the model created for this research may be adapted to better correspond to other specific cases, and what some limitations of it may be.

Using the method outlined in this study a strong linear relationship was found to exist in 1ME planets between the Fe Mass % and the planet radius, with R= 6980 km –

17.51 km (Fe mass%) (Figure 11). This is a generalized way of modeling exoplanets as no iron was incorporated in mantle mineral assemblages. Therefore, the radii listed in these results are considered to be a conservative estimation of a planet radius, as any partitioning of iron into mantle materials would result in a smaller core and a larger planet radius.

The method used in this study assumed an (exo)chondritic bulk composition for all planets in a system, and test modeling has shown that due to the unique formational histories of different planets some exoplanets may no longer share this composition.

Unlike some of the terrestrial planets in our Solar System, the Earth has a natural

45 satellite, the Moon, and interactions within this Earth-Moon system during formation have been hypothesized to have resulted in compositional changes in both, with both bodies ultimately ending up with relative abundances of elements differing from Solar.

Since we have radii and mass data for both the Earth and Moon, another version of this model could be run by Green (personal communication, November 2, 2019) assuming

Solar/chondritic abundances in the combined Earth-Moon system. Using these assumptions a Moon-mass body was modeled with a 3% iron mass percent, and the lunar radius of 1738 km was matched. This was a sub-Solar amount of iron incorporated in the

Moon-mass model, and the excess iron was subsequently added to the abundances for the

Earth-mass model. Although this Earth-Moon combination altered the iron mass in each body the results showed little variance from the previous reported Solar/chondritic Earth- mass models presented in this study. Even when modeling a more extreme version of the

Earth-Moon system, with a Moon with no iron core or Fe partitioning, and all Fe from the

Earth-Moon system incorporated in the Earth, the overall outcome was only minimally altered (Table 5). While some of these values improved slightly using this method the differences were minimal, suggesting there is another factor influencing this system beyond average Solar/chondritic values (Green, personal correspondence, November 2,

2019).

Table 5. Results of Earth-mass planet modeling with Solar abundances from this study compared to those for an Earth-Moon system scenario with no lunar core from Green (personal communication, November 2, 2019). PREM values from Dziewonski and Anderson (1981). Planet Depth to Depth to Moment of Radius ULMB CMB Inertia Factor

1 ME modeled planet* 6413 km 690 km 3044 km 0.3252 Deviation from PREM ±42 km ±20 km ±153 km -0.0056

1 ME planet E-M system** 6407 km 688 km 3027 km 0.3248 Deviation from PREM ± 36 km ± 18 km ± 136 km -0.006 * modeling data from this study. ** modeling data from Douglas Green (personal communication November 2, 2019).

Taking a look at other terrestrial planets in our Solar System, Douglas Green

(personal communication, November 2, 2019) reported that when using the assumptions outlined in this research to model Mercury and Mars-mass planets with Solar abundance input, this method it overestimated the radius of a Mercury-mass planet (0.33011 x 1024 kg) by 225 km and underestimated its rotational moment of inertia(0.35) by -0.012

(Mercury data from Williams, 2018). This method underestimated the radius of a Mars- mass planet by 82 km, and underestimated its rotational moment of inertial by

-0.029 (Table 6).

Table 6. Modeling results for Mercury and Mars-mass planets with chondritic compositions and no Fe-partitioning into the mantle from Green (personal communication, November 2, 2019). Mercury and Mars measurements taken from Williams (2018).

24 Mercury (MMerc = 0.33011 x 10 kg) Actual Modeled Deviation Planet radius (km) 2440 2665 225 Moment of Inertia Factor 0.35 0.338 -0.012

24 Mars (MMars = 0.64171 x 10 kg) Actual Modeled Deviation Planet radius (km) 3396 3314 -82 Moment of Inertia Factor 0.366 0.337 -0.029

In order to investigate iron partitioning into the mantle of terrestrial planets, iterations of the model were run assuming Solar elemental abundances but fixing the known radii and masses for Mercury, Venus, Earth, and Mars. The addition of iron into mantle mineral assemblages increased the densities of mantle materials and decreased the core mass percent of the planet until a best fit was found for partitioning in each. Zero partitioning of iron into the mantle was found to be the best modeled fit for Mercury, with the results matching those in Table 5 (R = 2665, Rot. Inertia = 0.338), suggesting

Mercury has a super-Solar abundance of iron. The best fit for Venus was also found to be zero percent Fe-partitioning with Solar abundances, which matched the results of this study (R = 6053, Rot. Inertia = 0.33) and was a good match to the known values for

Venus. Because the depth to interior boundaries are known for Earth, these boundaries were also used for comparison for its Fe-partitioning model, with the results showing

10% Fe-partitioning into the mantle gave a best fit that closely matched (-.0008) the

Earth’s rotational inertia of 0.3308. This variation matched the ULMB depth well at 660 km but estimated the CMB to be at a depth of 3088 km, a difference of -197 km from

Earth’s CMB as reported by PREM (Dziewonski and Anderson, 1981). When the radius and mass of Mars was fixed and run using the same method, 40% Fe-partitioning was found to be the best fit, with this amount resulting in an overestimation (+0.0161) of

Mar’s rotational inertia (0.366) (Williams, 2018). Although this is a very small sample, and much further investigation is needed to derive any conclusions, excepting Mercury

(which could not be matched with Solar/chondritic abundances), the results of the modeling by Green (personal communication, November 2, 2019) suggests an interesting trend of increased iron partitioning in the terrestrial planets with greater distance from the host star.

The super-Solar abundance of iron in Mercury suggests that due to variables in the planet formation process, not all terrestrial planets may share the same relative abundances of Fe (or other elements). Another way of investigating this variability was to run iterations of this model using a fixed known mass and radius, with zero partitioning of Fe into the mantle, and adjusting the overall Fe mass percent of the planet until a convergence was found for each. The preliminary results of this method of modeling done by Green (personal communication, November 2, 2019) found best fit Fe mass percent for Mercury = 40%, Venus = 32% (matching Solar abundance), Earth =

40%, and Mars = 23%. With the exception of the Earth system, the overall trend in these results is a decreasing Fe mass percent with increasing distance from the Sun.

More information is becoming available about exoplanets as we continue searching for them with missions such as the Kepler Space Telescope and the James

Webb Space Telescope (the latter is slated to launch in 2021) (Borucki et al., 2003;

Gardner et al., 2009). The research presented in this thesis is the foundation for future

49 terrestrial exoplanet modeling endeavors, in that this model is meant to be adaptable in order to examine many additional interior modeling possibilities beyond the ones discussed here. While further research would be needed to address confidences in these areas, additional examples of scenarios this model could be applied to include modeling larger planet masses (up to 10 ME), experimenting with different temperature profiles, further investigating incorporating Fe-partitioning into mantle materials, and the inclusion of more elements, mantle minerals, and phase changes within the transition zone of the mantle.

Research is also currently in progress exploring ways to refine this model by the addition of a post-perovskite layer at the base of the mantle, and incorporation of minerals such as stishovite to model planets with low Fe or low Mg/Si ratios (Jahl and

Green, 2018).

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APPENDIX A

Sol and CI chondrite abundance data from Asplund et al. (2005):

Sol photospheric and CI chondrite abundances from Asplund et al. (2005). Elements of interest for this study are indicated with bold labels.

APPENDIX B

Mineralogical composition calculations were based on similar work by Sotin et al. (2007) that calculated mineral modes from star abundance. Mg = 1, if no Fe is partitioned into the mantle.

This example uses abundance values for HIP 61802 (serving as the host star) from

Bensby et al. (2014), and Solar values from Asplund (2005). Planet mass = 1 ME = 5.97 x 1024 kg Star (HIP 61802) ValuesSolar Absolute Abundance Ratio

[Mg/Fe]: 0.38 (Mg/Si)Sol: 1.05

[Si/Fe]: 0.17 (Fe/Si)Sol: 0.871

Mg/Si is calculated for the star (Mg/SiStar). [Mg/Fe]-[Si/Fe] Mg/SiStar = 10^ x (Mg/Si)Sol

Mg/SiStar = 1.70

opx fraction is 2 - (Mg/Si)Star = 0.3 ol fraction is 1 - opx = 0.7 O incorporated in the planet is calculated based on the relative amounts of olivine

(Mg2SiO4) and orthopyroxene (MgSiO3).

O/Siopx = (opx) x (3*O/Mg)opx x (Mg/Si)Star

O/Siopx = 1.53

O/Siol = (ol) x (2*O/Mg)ol x (Mg/Si)Star

O/Siol = 2.38 Total O required, per silicon = 3.91 The amount of Fe per silicon is:

(Fe/Si)Star = 10^(-[Si/FeStar]) x (Fe/Si)Sol

(Fe/Si)Star = 0.5889 These values are used along with the atomic masses of the elements to calculate the core (Fe) mass fraction of a theoretical 1 ME planet:

MFe/(1 ME) = (Fe/Si x 56)/[(Fe/Si x 56) + (O/Si x 16) + (1 x 28) + (Mg/Si x 24)]

Fe mass of planet (core mass fraction) = 1 ME x (MFe/1 ME) 24 Core mass (Fe) for a 1 ME planet = 1.198 x 10 kg

APPENDIX C

Mineral properties referenced in this study.

From Anderson et al. (1991): Properties of Mg2SiO4, including densities (ρ), thermodynamic coefficients (α), adiabatic bulk modulus (KS) and absolute value of the temperature derivative (KT) at differing temperatures.

T ρ α KS KT (K) (g/m3) (10-6/K) (Gpa) (Gpa) 300 3.222 27.2 128.7 129 400 3.213 30.3 127.2 127 500 3.203 32.2 125.5 123 600 3.192 33.6 123.8 121 700 3.181 34.8 122.0 119 800 3.170 35.9 120.3 116 900 3.159 37.0 118.4 114 1000 3.147 38.1 116.7 112 1100 3.135 39.2 114.9 110 1200 3.122 40.5 113.0 107 1300 3.109 41.6 111.2 105 1400 3.096 42.7 109.3 102 1500 3.083 43.9 107.6 100 1600 3.069 45.0 105.7 98 1700 3.055 46.2 103.8 95

From Duffy and Anderson (1989): Summary of the elastic properties of adiabatic bulk modulus (KS), shear modulus (G), pressure derivative of the bulk modulus (K’S), pressure derivative of the shear modulus (G’), absolute value of the temperature derivative of the adiabatic bulk modulus(Ḱ), and absolute value of the temperature derivative of the shear modulus (Ǵ) for the phases of Mg2SiO4 plus starting values for MgSiO3.

ρ KS, G, K'S G' KS, G, Study (g/m3) Gpa Gpa Gpa/K Gpa/K

Mg2SiO4 α Phase Duffy and Anderson [1989] 3.222 129 82 5.1 1.8 0.016 0.013 Bina and Wood [1987] 129 81 5.2 1.8 0.014 0.014 Irifune [1987] 129 81 5.2 1.79 0.014 0.013 Weidner [1986] 129 80 4.7 1.6 0.016 0.014

Mg2SiO4 β Phase Duffy and Anderson [1989] 3.472 174 114 4.9 1.8 0.018 0.014 Anderson [1988] 4.8 2 0.017 0.015 Bina and Wood [1987] 174 114 4.8 1.8 0.016 0.023 Irifune [1987] 174 114 4 1.1 0.011 0.011 Weidner [1986] 174 114 4.3 0.9 0.02 0.014

Mg2SiO4 γ Phase Duffy and Anderson [1989] 3.548 184 119 4.8 1.8 0.017 0.014 Anderson [1988] 5 2 0.017 0.015 Bina and Wood [1987] 184 119 4.8 1.8 0.015 0.024 Irifune [1987] 184 119 4.8 1.1 0.012 0.011 Weidner [1986] 184 119 4.3 0.9 0.02 0.014

Al2MgO4 Spinel Sumino and Anderson [1984] 197 108 4.9 0.016 0.009 MgSiO3 Orthopyroxene Duffy and Anderson [1989] 3.204 104 77 5.0 2.0 0.012 0.011

From Reynard and Price (1990): Calculated and measured values of volume, thermal expansion, bulk modulus, and Gruneisen parameter (γ) at 300 K and 1 bar for four minerals.

From Tange, et al. (2012): Thermoelastic Properties of MgSiO3 Perovskite.*

P (GPa) 1000 K 2000 K 3000 K Density ρ (g/cm3) 25 4.398 (4.401) 4.294 (4.299) 4.179 (4.184) 50 4.697 (4.701) 4.611 (4.620) 4.519 (4.532) 75 4.954 (4.956) 4.881 (4.889) 4.802 (4.816) 100 5.184 (5.181) 5.118 (5.123) 5.048 (5.060) 125 5.393 (5.384) 5.333 (5.332) 5.270 (5.277) Grüneisen parameter γ 25 1.438 (1.388) 1.476 (1.438) 1.521 (1.498) 50 1.337 (1.257) 1.364 (1.290) 1.395 (1.328) 75 1.261 (1.162) 1.282 (1.186) 1.305 (1.213) 100 1.200 (1.087) 1.217 (1.105) 1.235 (1.126) 125 1.149 (1.026) 1.163 (1.041) 1.178 (1.057) Thermal expansion α (10−5/K) 25 2.219 (2.153) 2.551 (2.528) 2.881 (2.931) 50 1.716 (1.605) 1.933 (1.834) 2.109 (2.033) 75 1.408 (1.279) 1.573 (1.441) 1.687 (1.563) 100 1.198 (1.061) 1.334 (1.188) 1.416 (1.271) 125 1.046 (0.905) 1.162 (1.010) 1.226 (1.072)

Adiabatic bulk modulus KS (GPa) 25 346.2 (343.7) 327.7 (321.7) 308.4 (299.0) 50 436.3 (437.3) 419.2 (416.5) 401.6 (395.1) 75 521.1 (527.4) 505.0 (507.4) 488.3 (486.8) 100 602.4 (615.1) 586.9 (595.7) 570.8 (575.6) 125 680.8 (700.9) 665.8 (682.0) 650.3 (662.3)

Isobaric heat capacity CP (J/K·mol) 25 122.0 (121.6) 132.4 (132.1) 140.4 (140.4) 50 119.7 (119.2) 129.3 (128.6) 134.9 (133.9) 75 117.9 (117.6) 127.4 (126.7) 131.9 (130.8) 100 116.5 (116.3) 126.2 (125.5) 130.1 (129.0) 125 115.4 (115.3) 125.3 (124.6) 128.9 (127.7)

* Values are based on an EOS model using Vinet equation and values in parentheses are based on that using third‐order Birch‐Murnaghan equation.

From Tange, et al. (2012): Thermoelastic Properties of MgO.*

P (GPa) 1000 K 2000 K 3000 K Density ρ (g/cm3) 25 3.976 (3.980) 3.867 (3.874) 3.750 (3.758) 50 4.334 (4.338) 4.244 (4.254) 4.151 (4.165) 75 4.629 (4.631) 4.550 (4.559) 4.467 (4.483) 100 4.887 (4.883) 4.814 (4.819) 4.738 (4.751) 125 5.118 (5.106) 5.049 (5.048) 4.978 (4.986) Grüneisen parameter γ 25 1.357 (1.304) 1.375 (1.332) 1.399 (1.368) 50 1.314 (1.229) 1.323 (1.224) 1.333 (1.261) 75 1.293 (1.185) 1.298 (1.195) 1.304 (1.206) 100 1.280 (1.157) 1.283 (1.163) 1.287 (1.171) 125 1.272 (1.136) 1.274 (1.141) 1.277 1.147 Thermal expansion α (10−5/K) 25 2.609 (2.525) 2.914 (2.866) 3.236 (3.241) 50 1.990 (1.856) 2.146 (2.040) 2.292 (2.193) 75 1.657 (1.498) 1.786 (1.629) 1.862 (1.716) 100 1.442 (1.271) 1.552 (1.376) 1.605 (1.436) 125 1.289 (1.113) 1.389 (1.204) 1.431 (1.249)

Adiabatic bulk modulus K S (GPa) 25 255.0 (252.9) 241.8 (237.7) 227.1 (221.5) 50 346.4 (346.8) 335.9 (333.5) 324.6 (319.6) 75 431.4 (436.7) 422.4 (424.4) 412.9 (411.6) 100 512.3 (524.8) 504.3 (512.3) 495.9 (500.2) 125 590.1 (609.5) 582.9 (598.2) 575.4 (586.5)

Isobaric heat capacity C P (J/K·mol) 25 49.7 (49.6) 53.4 (53.2) 56.5 (56.3) 50 48.8 (48.7) 52.1 (51.8) 54.2 (53.8) 75 48.1 (48.0) 51.5 (51.1) 53.2 (52.7) 100 47.6 (47.5) 51.1 (50.7) 52.6 (52.1) 125 47.1 (47.1) 50.7 (50.4) 52.2 (51.6)

* Values are based on an EOS model using Vinet equation and values in parentheses are based on that using third‐order Birch‐Murnaghan equation.

From Vocadlo et al. (2003): Properties of liquid iron (Fe).

Pressure (in Gpa) calculated as a function of temperature (T) and density for liquid Fe. T (K) ρ (kg m-3) 9540 10700 11010 12130 13300 3000 60 4300 132 5000 140 6000 90 151 170 251 360 7000 161 181 264 375 8000 172 191 275 390

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