A Bayesian Approach to the Understanding of Exoplanet Populations and the Origin of Life

Transcription

1 A Bayesian Approach to Understanding of Exoplanet Populations and Origin of Life Jingjing Chen Submitted in partial fulfillment of requirements for degree of Doctor of Philosophy in Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2018

2 c 2018 Jingjing Chen All rights reserved

3 ABSTRACT A Bayesian Approach to Understanding of Exoplanet Populations and Origin of Life Jingjing Chen The study of extrasolar planets, or exoplanets for short, has developed rapidly over last decade. While we have spent much effort building both ground-based and space telescopes to search for exoplanets, it is even more important that we use observational data wisely to understand m. Exoplanets are of great interest to both astronomers and general public because y have shown varieties of characteristics that we couldn t have anticipated from planets within our Solar System. To properly analyze exoplanet populations, we need tools of statistics. Therefore, in Chapter 1, I describe science background as well as statistical methods which will be applied in this sis. In Chapter 2, I discuss how to train a hierarchical Bayesian model in detail to fit relationship between masses and radii of exoplanets and categorize exoplanets based on that. A natural application that comes with model is to use it for future observations of mass/radius and predict or measurement. Thus I will show two application cases in Chapter 3. Composition of an exoplanet is also very much constrained by its mass and radius. I will show an easy way to constrain composition of exoplanets in Chapter 4 and discuss how more complicated methods can be applied in future works. Of even greater interest is wher re is life elsewhere in Universe. Although

4 future discovery of extraterrestrial life might be totally a fluke, a clear sketched plan always gives us some directions. Research in this area is still very preliminary. Fortunately, besides directly searching for extraterrestrial life, we can also apply statistical reasoning to first estimate rate of abiogenesis, which will give us some clue on question of wher re is extraterrestrial life in a probabilistic way. In Chapter 5, I will discuss how different methods can constrain abiogenesis rate in an informatics perspective. Finally I will give a brief summary in Chapter 6. Probabilistic forecasting of masses and radii of or worlds, J. Chen & D. Kipping, The Astrophysical Journal 834 (1), 17 Forecasted masses for 7000 Kepler Objects of Interest, J. Chen & D. Kipping, Monthly Notices of Royal Astronomical Society 460 (3), Forecasting detectability of known radial velocity planets with upcoming CHEOPS mission, J. S. Yi, J. Chen, D. Kipping, Monthly Notices of Royal Astronomical Society 475(3), A HARDCORE model for constraining an exoplanet s core size, G. Suissa, J. Chen, D. Kipping, Monthly Notices of Royal Astronomical Society 476 (2), On rate of abiogenesis from a Bayesian informatics perspective, J. Chen, D. Kipping, arxiv preprint arxiv:

5 Contents List of Figures List of Tables Acknowledgements v viii ix 1 Introduction Population Properties of Exoplanets and Prediction of Missing Features Extraterrestrial Life The Bayesian Method Bayes Rule Metropolis Algorithm Hierarchical Bayesian Model Information Gain Forecasting Masses and Radii of Or Worlds Introduction Model Choosing a Model Data Selection Probabilistic Broken Power-Law Hierarchical Bayesian Modeling i

6 2.2.5 Continuous Broken Power-Law Model Hyper Priors Two Different Categories of Local Parameters Inverse Sampling Total Log-Likelihood Analysis Parameter Inference with Markov Chain Monte Carlo Model Comparison The Effect of our Data Cuts Injection/Recovery Tests Classification Classification with an MR relation Naming Classes T (1) : The Terran-Neptunian Worlds Divide T (3) : The Jovian-Stellar Worlds Divide T (2) : The Neptunian-Jovian Worlds Divide Forecasting Forecaster: An Open-Source Package Forecasting Radius Forecasting Mass Examples: Kepler-186f and Kepler-452b Discussion Applications of Forecaster Forecasted masses for 7000 Kepler Objects of Interest Forecasting KOI Masses Data Requirements Transit Posteriors Stellar Posteriors ii

7 3.2.4 KOI Radii Posteriors Predicting Masses with Forecaster Implication & Limitations Densities, Gravities and RV Amplitudes Observed Patterns Promising Small Planets for RV follow-up Discussion The detectability of known radial velocity planets with upcoming CHEOPS mission Forecasting Radii from Masses Probabilistic predictions with Forecaster Accounting for measurement uncertainties Approximate form for posterior distributions Treating minimum mass as being equal to true mass An Overview of Results Constraining an Exoplanet s Composition Introduction Boundary Conditions on Core Outline and Assumptions A parametric interpolative model for CRF min A parametric model for CRF max Solving for CRF Limits From forward- to inverse-modeling The Earth as an example Sensitivity analysis for an Earth Generalized sensitivity analysis Discussion On Rate of Abiogenesis from a Bayesian Informatics Perspective 122 iii

8 5.1 Introduction Bayesian Model A uniform rate model for abiogenesis Time for life to emerge as an exponential distribution Accounting for selection bias Accounting for observation time Likelihood function for λ Multiplanet likelihood function for λ Choosing priors for λ and τ Sampling method Using information gain to compare different posteriors Results Experiment 1: Reducing t obs Experiment 2: Reducing λ max Future Evidence from Exoplanets Discussion Conclusion Overview of Exoplanets Probabilistic Mass and Radius Relation Overview of Exoplanets Composition Overview of Abiogenesis Rate On Bayesian V.S. Frequentist Methods Bibliography 162 iv

9 List of Figures 1.1 Graphical representation of hierarchical Bayesian models Graphical model of HBM used to infer probabilistic MR relation in this work. Yellow ovals represent hyper-parameters, white represent true local parameters and gray represent data inputs. All objects on plate have N members Triangle plot of hyper-parameter joint posterior distribution (generated using corner.py). Contours denote 0.5, 1.0, 1.5 and 2.0 σ credible intervals The mass-radius relation from dwarf planets to late-type stars. Points represent 316 data against which our model is conditioned, with data key in top-left. Although we do not plot error bars, both radius and mass uncertainties are accounted for. The red line shows mean of our probabilistic model and surrounding light and dark gray regions represent associated 68% and 95% credible intervals, respectively. The plotted model corresponds to spatial median of our hyper parameter posterior samples Each sub-panel shows residuals of a hyper-parameter in our model, as computed between ten injected truths and corresponding recovered values. The black square denotes recovered posterior median and dark & light gray bars denote 1 & 2 σ credible intervals. The green horizontal bar marks zero-point expected for a perfect recovery v

10 2.5 Posterior distributions of radius (measured) and mass (forecasted) of two habitable-zone small planets, with predictions produced by Forecaster (triangle plots generated using corner.py) Forecasted masses (top) and radial velocity semi-amplitudes (bottom) for a circular orbit, as a function of observed KOI radius. We here only show objects with a NEA disposition of being eir a candidate or validated planet. Error bars depict 68.3% credible intervals Orbital periods (top) and host star masses (bottom) as a function of planetary radius for 2936 KOIs with modal class probability of being Neptunian and not dispositioned as a false-positive on NEA. The trend between period and radius is most easily explained as being due to detection bias, which in turn gives rise to RV plateau seen for Neptunians in Figure 3.1. Median (black line) and 68.3% credible interval (purple) shown using a top-hat smoothing kernel of 0.14 dex Forecasted radial velocity amplitudes for small planets. The colored region contains 68.3% credible interval of maximum likelihood forecasted K values from a moving window. Points plotted use 68.3% uncertainties and black points are those more than 1.64 σ (90%) above moving median (black) Corner plot of our resulting posteriors for HD 20794c an illustrative example. Covariances are evident between parameters, including between mass and radius Ternary diagram of a three-layer interior structure model for a solid planet. All points along red line lead to a M = 1 M and R = 1 R planet. Although indistinguishable from each or with current observations, all points satisfy having a core-radius fraction exceeding 43%, a boundary condition we exploit in this work. The largest iron core size allowed is depicted by lowest sphere, where volatile envelope contributes negligible mass vi

11 4.2 Upper: Contour plots depicting a-posteriori standard deviation on minimum (left), marginalized (center) and maximum (right) CRF, as a function of fractional errors on mass and radius. Lower: Re-parameterization of above plots by combining mass and radius axes into a single density term, demonstrating a strong dependency in all cases Proposed graphical model of HBM used to infer composition fractions. Yellow ovals represent hyper-parameters, white represent true local parameters, and gray represent data inputs. All objects on plate have N members The left panel shows information gain calculated by KLD as t obs gets smaller. The right panel gives detailed posterior distribution of λ for fixed λ max of 10 2 Gyr 1. Line A, B, C, D represents t obs of 10 0 Gyr, 10 1 Gyr, 10 2 Gyr and 10 3 Gyr respectively The left panel shows information gain calculated by KLD as λ max gets smaller. The right panel gives detailed posterior distribution of λ for fixed t obs of 10 1 Gyr. Line a, b, c, d represents λ max of 10 3 Gyr 1, 10 2 Gyr 1, 10 1 Gyr 1 and 10 0 Gyr 1 respectively The three plots show predicted ratio of planets with lives given situation on Earth. From left to right panel, λ min = 10 3 Gyr 1, Gyr 1, and Gyr 1. In each panel, we vary t obs (x-axis) and λ max (y-axis) The left panel shows information gain calculated as we increase number of observed exoplanets. The right panel gives detailed posterior distribution of λ for different ratios Summary chart of some key general results found in this work vii

12 List of Tables 2.1 Masses and radii used for this study Description and posterior of hyper parameters.the prior distributions [ ] of hyper parameters are C (1) U( 1, 1); S (1 4) N(0, 5); log 10 σ (1 4) R U( 3, 2); T (1 3) U( 4, 6) Data flags assigned to 7104 KOIs considered in this analysis. 0 denotes no problems, 1 denotes that stellar posteriors were missing and 2 denotes transit posteriors were missing. Only a portion of table is shown here, full version is available on GitHub repository chenjj2/forecasts Final predicted properties for 6973 KOIs using Forecaster. Quoted values are [ 2, 1, 0, +1, +2] σ credible intervals. First letter of flag column gives modal planet classification ( T =Terran, N =Neptunian, J =Jovian & S =Stellar). Second letter of flag column denotes NEA KOI disposition where f is a false-positive, c is a candidate and v is validated/confirmed. Only a portion of table is shown here, full version is available on GitHub repository chenjj2/forecasts List of planetary candidates with maximum likelihood radii between 0.5 to 4.0 Earth radii and unusually high forecasted RV semi-amplitudes viii

13 ACKNOWLEDGMENTS Five years ago, in late summer of 2013, I arrived in New York and started my journey of PhD studies. I need to first thank Astronomy department of Columbia University, especially admission committee board, for giving me chance of becoming a graduate student here. What I have experienced in my graduate study is very different from what I used to. It opened up my eyes. I am most grateful to my advisor, Prof. David Kipping, who has so many brilliant ideas, encourages debate, listens to my point of view, and always puts my success in center our relation. I couldn t have finished my PhD studies so fruitfully without his help. I offer my special thanks to my family, my mor, far, my boyfriend, my cats, and my dog. You always treasure me, adore me, and put everything aside when I need you. You are most important ones in my life. I also thank my first and second year research advisors Prof. David Schiminovich and Prof. Greg Bryan, my sis committee and my sis panel, every or professor who had helped me or taught me, department secretaries Millie and Ayoune, and my folk graduate students. I have learned from and been supported by everyone of you and I truly appreciated it. 2018, New York City ix

14 Chapter 1 Introduction 1.1 Population Properties of Exoplanets and Prediction of Missing Features The study of exoplanets, planets outside Solar System, is currently one of most active research areas in astronomy. In 1992, very first exoplanetary system was found around a millisecond pulsar, PSR , using precise timing measurements (Wolszczan & Frail 1992). Over past decade, astronomers have developed a diverse array of detection methods such as transits, radial velocity, micro-lensing etc. By May 2018, more than 3,700 confirmed exoplanets are cataloged (Exo 2018), and number is still increasing. The great advance in exoplanet discoveries is largely due to survey projects such as ground-based HARPS (High Accuracy Radial Velocity Planet Searcher), HAT (HATNet and HATSouth Projects), OGLE (Optical Gravitational Lensing Experiment), space mission COROT (Convection, Rotation and planetary Transits), Kepler, K2, and re 1

15 are more to come in near future such as TESS (Transiting Exoplanet Survey Satellite), CHEOPS (Characterizing ExOPlanets Satellite), and JWST (James Webb Space Telescope). Thanks to se efforts in searching for and garing information of exoplanets, we are now able to perform statistical analyses on exoplanet populations and gain insights of ir population properties. In recent years, along with rise of data science, statistical methods and models have become more and more widely used in astronomy. Besides some basic statistical techniques like least-square regressions, astronomers have delved deeper into realms such as Bayesian models, survival analysis, and machine learning. A Bayesian approach has become widely adopted in exoplanetary research as it easily incorporates updated information from previously conducted measurements, and gives posterior distribution estimates for model parameters as opposed to point estimates by frequentist methods. Hierarchical Bayesian models (HBM), by adding more complexities to basic Bayesian models, have been applied to many problems, inferring, for example, eccentricty distribution of exoplanets (Hogg et al. 2010), obliquities of Kepler stars (Morton & Winn 2014), abundance of Earth analogs (Foreman-Mackey et al. 2014), albedos of super-earths (Demory 2014), mass and radius relationship (Wolfgang et al. 2016) etc. In this sis, I will focus on using Bayesian approach to build models to understand properties of exoplanets as an ensemble and use models to make predictions on unobserved features. By using Bayesian methods, Chapter 2 describes how I train a probabilistic model to fit a hierarchical relationship between mass and radius features of exoplanets. This model not only tells us underlying population properties of exoplan- 2

16 ets, but also can be used to predict an unknown feature provided measurements of related features. The prediction can n be used to estimate range of unknown feature and help with follow-up observations. 1.2 Extraterrestrial Life One of reasons behind fever of searching for exoplanets is to answer longstanding question on our mind are we alone? Actual detections of extraterrestrial life might be only way that can answer this question with certainty. Many life explorers had been launched within Solar System, especially on Mars, such as NASA s Curiosity rover. And re are more to come in next decade, such as NASA s Mars 2020 rover and ESA s ExoMars rover. These previous life detection missions have been fruitful in finding life related evidence such as methane (Krasnopolsky et al. 2004; Atreya et al. 2007), organic matter (Grotzinger 2014; Eigenbrode et al. 2018), and water (Orosei et al. 2018). But direct evidence is still lacking. Outside Solar System, people have researched on a wide range of topics from oretically how life originates and sustains (Pepe et al. 2011; Quintana et al. 2014; Khodachenko et al. 2007; Lammer et al. 2007), to observationally finding signatures of life (Segura et al. 2005). For example, assuming that life is produced photochemically with UV light, Rimmer et al. (2018) calculates where abiogenesis zone lies and where it overlaps with habitable zone. In this sis, I will show how one can use statistical methods to estimate rate of abiogenesis, or how often life spontaneously originates from environment, on Earth- 3

17 like planets. If abiogenesis rate turns out to be high, it is more likely that extraterrestrial life exist. However, at current stage, we have very limited information on abiogenesis rate. First, abiogenesis events greatly depend on specific environment and evolution history of given planet. Besides, since Earth is only known example with life, we have very little data upon which to condition our inference. Thus, result of previous work showed that any estimate on abiogenesis rate can be dramatically affected by assumptions (Spiegel & Turner 2012). In or words, additional information is required for meaningful estimates. In this sis, I speculate three scenarios, inspired by possible results from or areas of research, which might provide some additional information to help better constrain abiogenesis rate under Bayesian statistic framework. I n show how se scenarios affect final inferred distribution of abiogenesis rates. 1.3 The Bayesian Method In previous two sections, I briefly introduced science background of sis. In this section, I will mainly talk about methodology background. The Bayesian framework has three elements: pre-existing information, which is called prior; new information, which is called data; and result, which is called posterior. For a parameter of interest, Bayesian inference uses newly observed data to update prior belief of parameter s distribution and results in posterior distribution of parameter. For example, in wear forecast, tomorrow s probability of rain can be interpreted as a summary of rain history updated with currently observed cloud 4

18 pattern in a Bayesian framework. In this section, I will briefly introduce Bayesian methods used throughout sis. For a more detailed explanation, readers should refer to some standard textbooks in Bayesian statistics, such as Bayesian Data Analysis by Andrew Gelman (Gelman 2004) Bayes Rule The idea of Bayesian analysis is to update uncertainty in parameters of interest from ir previous estimates conditioned on newly obtained data. The Bayes rule can be described mamatically as following equation, P(θ y) = P(y θ)p(θ), P(y) where θ is parameter of interest and y is observation data. The left hand side represents posterior distribution of θ, which is probability distribution of parameter θ conditioned on observation y. On right hand side, P(y) can be considered as a constant because y is observed and has no randomness. P(θ) is called prior distribution of parameter θ because it represents our assumption on θ prior to observation of y. P(y θ) means probability of obtaining data y given θ, which is also known as likelihood when considered as a function of θ. The specific forms of P(θ) and P(y θ) depend on mamatical model we want to use and details of problem. 5

19 1.3.2 Metropolis Algorithm Most of time in real-world applications, posteriors cannot be easily calculated with an analytical form. In those cases, we need simulation methods to derive an empirical distribution of posterior. The Metropolis algorithm via random walk is one of most widely used simulation methods. The algorithm generates a chain of samples step by step, which converges to true distribution (Roberts et al. 1997). The algorithm operates as follows (Gelman 2004) to find posterior distribution of θ. 1. Initialization: sample a random starting point θ 0 from prior distribution of θ. 2. Jumping: for each step t (n = 1, 2,...), sample a proposed θ from a symmetric jumping distribution J(θ θ t 1 ). By symmetric, it means that probability of θ jumping from θ to θ t 1 is equal to probability of jumping from θ t 1 to θ. 3. Acceptance/Rejection: accept proposed θ with probability min(r, 1), where r = P(θ y) P(θ t 1 y) The algorithm will end up with a chain of θ. The chain has two different phases delineated by a burn-in point. Before burn-in, chain is going in one direction in general to search for area where final samples will lie in. After burn-in, chain becomes steady and converges to posterior distribution. In practice, initial samples of θ before burn-in will be thrown away. 6

20 1.3.3 Hierarchical Bayesian Model Hierarchical models add anor layer of parameters called hyper-parameters, that control distribution of original parameters, which are called local-parameters in a hierarchical model. Hierarchical Bayesian Models (HBM in short) are usually used in analysis of a population of objects. In application, hyper-parameters are usually used to describe population properties while local-parameters are used to describe each single object. The goal of using hierarchical models is to fit both hyper and local parameters in a single framework. A general model can be described as following. P(α, θ y) = P(y α, θ)p(α, θ) P(y) where α, θ, y represents hyper-parameters, local-parameters and data respectively. Because observation y is only directly related to θ, P(y α, θ) can be written as P(y θ). P(α, θ) can be written as P(α)P(θ α). So to build a hierarchical model we need to specify three things, prior distribution of hyper parameters, distribution of localparameters conditioned on hypers, and distribution of data conditioned on locals. Graphical models are usually used to help explain hierarchy in practice. Throughout this sis, as is shown in Figure 1.1, hyper-parameters will be represented in yellow ovals, local-parameters will be in white ovals, and data will be in grey ovals. Within box, re will be N samples of data and ir corresponding local parameters. 7

21 Hyper Local Data N Figure 1.1 Graphical representation of hierarchical Bayesian models Information Gain In Bayesian inference, posterior distribution is considered to be updated beliefs of a parameter with data from prior beliefs. Therefore we can ask question, how much has belief changed from previous one? In information ory, characteristics of a distribution are quantified through entropy of distribution, or average amount of information carried in distribution. The entropy of a continuous random 8

22 variable X with distribution p(x) is defined as H(X) = p(x)logp(x)dx Relative entropy, also known as Kullback-Leibler divergence (KLD for short), between distributions P and Q is defined as D KL (P Q) = p(x)log p(x) q(x) dx As entropy measures amount of information, relative entropy measures information gain. Relative entropy can be applied to compare posterior distribution with prior distribution and quantify impact of observation which plays in between. 9

23 Chapter 2 Forecasting Masses and Radii of Or Worlds Mass and radius are two of most fundamental properties of an astronomical object. Increasingly, new planet discoveries are being announced with a measurement of one of se terms, but not both. This has led to a growing need to forecast missing quantity using or, especially when predicting detectability of certain follow-up observations. We present an unbiased forecasting model built upon a probabilistic mass-radius relation conditioned on a sample of 316 well-constrained objects. Our publicly available code, Forecaster, accounts for observational errors, hyper-parameter uncertainties and intrinsic dispersions observed in calibration sample. By conditioning our model upon a sample spanning dwarf planets to late-type stars, Forecaster can predict 0 This chapter is a reproduction of a paper that has been published by The Astrophysical Journal. It can be found at The article has been reformatted for this section. 10

24 mass (or radius) from radius (or mass) for objects covering nine orders-of-magnitude in mass. Classification is naturally performed by our model, which uses four classes we label as Terran worlds, Neptunian worlds, Jovian worlds and stars. Our classification identifies dwarf planets as merely low-mass Terrans (like Earth), and brown dwarfs as merely high-mass Jovians (like Jupiter). We detect a transition in mass-radius relation at M, which we associate with divide between solid, Terran worlds and Neptunian worlds. This independent analysis adds furr weight to emerging consensus that rocky Super-Earths represent a narrower region of parameter space than originally thought. Effectively, n, Earth is Super-Earth we have been looking for. 2.1 Introduction Over last two decades, astronomers have discovered thousands of extrasolar worlds (see exoplanets.org; Han et al. 2014), filling in parameter space from Moon-sized planets (e.g. Barclay et al. 2013) to brown dwarfs many times more massive than Jupiter (e.g. Deleuil et al. 2008). Over 98% of se detections have come from radial velocity, microlensing or transit surveys, yet each of se methods only directly measures mass (M) or radius (R) of a planet, not both 1. This leads to common situation where it is necessary to forecast what missing quantity is based on or. A typical case would be when one needs to predict detectability of a potentially observable effect for a resource-intensive, time-competitive observing facility, which in some way depends upon missing quantity. For exam- 1 Except for rare cases of systems displaying invertible transit timing variations. 11

25 ple, TESS mission (Ricker et al. 2014) will soon start detecting hundreds, possibly thousands, of nearby transiting planets for which radius, but not mass, will be measured. Planets with radii consistent with Super-Earths will be of great interest for follow-ups and so radial velocity facilities will need to forecast detectability, which is proportional to planet mass, for each case. Vice versa, CHEOPS mission (Broeg et al. 2013) will try to detect transits of planets discovered with radial velocities, necessitating a forecast of radius based upon mass. In those two examples, objective was to forecast missing quantity in order to predict feasibility of actually measuring it. However, value of forecasting mass/radius for purposes of predicting detectability extends beyond this. As anor example, exoplanet transit spectroscopy is expected to be a major function of upcoming JWST mission (Seager et al. 2009). At first-order level, detectability of an exoplanet atmosphere is proportional to scale height, H, which in turn is proportional to 1/g R 2 /M. Given limited supply of cryogen onboard JWST, discoveries of future Earth-analog candidates may be found with insufficient time to reasonably schedule a radial velocity campaign first (if even detectable at all). Therefore, re will likely be a critical need to accurately forecast scale height of new planet discoveries from just eir mass or (more likely) radius. Forecasting mass/radius of an object, based upon or quantity is most obviously performed using a mass-radius (MR) relation. Such relations are known to display sharp changes at specific locations, such as transition from brown dwarfs to hydrogen burning stars (e.g. see Hatzes & Rauer 2015). These transition points can 12

26 be thought of as bounding a set of classes of astronomical objects, where classes are categorized using features of inferred MR relation. In this case n, it is apparent that inference of MR relation enables both classification and forecasting. Classification is more than a taxonomical enterprise, it can have dramatic implications in astronomy. Perhaps most famous example of classification in astronomy is Hertzsprung-Russell (HR) diagram (Hertzsprung 1909; Russell 1914) for luminosity versus effective temperature, which revealed distinct regimes of stellar evolutions. A common concern in classification is that very large number of possible features against which to frame problem can be overwhelming. Mass and radius, though, are not random or arbitrary choices for framing such a problem. Rar, y are two of most fundamental quantities describing any object in cosmos and indeed represent two of seven base quantities in International System of Units (SI). The value of classification extends beyond guiding physical understanding, it even affects design of future instrumentation. As an example, boundary between terrestrial planets and Neptune-like planets represents a truncation of largest allowed habitable Earth-like body. The location of this boundary strongly affects estimates of occurrence rate of Earth-like planets (η ) and thus in-turn design requirements of future missions needed to characterize such planets (Dalcanton et al. 2015). To illustrate this, using occurrence rate posteriors of Foreman-Mackey et al. (2014), η decreases by 42% when altering definition of Earth-analogs from R < 2.0 R to R < 1.5 R. In order to maintain same exoearth yield for proposed HDST mission, this change corresponds to a 27% increase in required mirror diameter (using yield equation in 13

27 3.5.4 of Dalcanton et al. 2015). We refore argue that both forecasting and classification using masses and radii of astronomical bodies will, at very least, be of great utility for present/future missions and may also provide meaningful insights to guide our interpretation of se objects. Accordingly, primary objective of this work is to build a statistically rigorous and empirically calibrated model to forecast mass/radius of an astronomical object based upon a measurement of or, and for classification of astronomical bodies based upon ir observed masses and/or radii. The layout of this paper is as follows. In Section 2.2, we outline our model for MR relation, which is used for forecasting and classification. In Section 2.3, we describe regression algorithm used to conduct Bayesian parameter estimations of our model parameters. The results, in terms of both classification and forecasting are discussed separately in Sections 2.4 & 2.5. We summarize main findings of our work in Section

28 2.2 Model Choosing a Model We begin by describing rationale behind model used in this work. As discussed in Section 2.1 (and demonstrated later in Section 2.3), two primary goals of this paper are both achievable through use of a MR relation and this defines approach in this work. Broadly speaking, such a relation can be cast as eir a parametric (e.g. a polynomial) or non-parametric model (e.g. a nearest neighbor algorithm). Parametric models, in particular power-laws, have long been popular for modeling MR relation with many examples even in recent literature (e.g. Valencia et al. 2006; Weiss et al. 2013; Hatzes & Rauer 2015; Wolfgang et al. 2016; Zeng et al. 2016). In our case, we note that such models are more straightforward for hierarchical Bayesian modeling (which we argue to be necessary later), since y allow for a simple prescription of Bayesian framework. Moreover, based on those earlier cited works, power-laws ostensibly do an excellent job of describing data and greater flexibility afforded by non-parametric methods is not necessary. Accordingly, we adopt power-law prescription in this work. As noted earlier, use of power-laws to describe MR relation is common in literature. However, many of assumptions and model details in se previous implementations would make forecasts based upon se relations problematic. We identify three key aspects of model proposed in this work which differentiate our work from previous studies. 15

29 [1] Largest data range: Inferences of MR relation often censor available data to a specific subset of parameter space (for example Wolfgang et al consider R < 8 R exoplanets). Whilst it is inevitable that certain subjective choices will be made by those analyzing MR relation, a more physically-motivated choice for parameter limits can be established. Ideally, this range should be as large as possible such that forecasting is unlikely to encounter extrema, leading to truncation errors. A natural lower bound is an object with sufficient mass to achieve hydrostatic equilibrium leading to a nearly spherical shape and thus a well-defined radius (a planemo), which would encompass dwarf planets. As an upper bound, late-type stars take longer than a Hubble time to leave main-sequence and should exhibit a relatively tight trend between mass and radius. [2] Fitted transitions: As a by-product of using such a wide mass range, several transitional regions are traversed where MR relation exhibits sharp changes. For example, onset of hydrogen burning leads to a dramatic change in MR relation versus brown dwarfs (Hatzes & Rauer 2015). In previous works, such transitional points are often held as fixed, assumed locations (e.g. Weiss & Marcy 2014 assume a physically motivated, but not freely inferred, break at 1.5 R ). In contrast, we here seek to make a more agnostic, data-driven inference without imposing any assumed transition points from ory or previous data-driven inferences. In this way, uncertainty in se transitions is propagated into inference of all or parameters defining our model, leading to more robust uncertainty estimates for both forecasting and classification. Accordingly, in this work, MR relation is described by a broken power-law with freely fitted transition 16

30 points (in addition to or parameters). [3] Probabilistic modeling: Whilst mass can be considered to be primary influence on size of an object, many second-order terms will also play a role. As an example, rocky planets of same mass but different core mass fractions will exhibit distinct radii (Zeng et al. 2016). When viewed in MR plane n, a particular choice of mass will not correspond to a single radius value. Rar, a distribution of radii is expected, as a consequence of numerous hidden second-order effects influencing size. Statistically speaking n, MR relation is expected to be probabilistic, rar than deterministic. A probabilistic model fundamentally relaxes assumption that underlying model (in our case a broken power-law) is correct or true description of data, allowing an approximate model to absorb some (although it can never be all) of error caused by model misspecifications (in our case via an intrinsic dispersion). Naturally, closer one s underlying model is to truth, smaller this probabilistic dispersion need to be, and in ultimate limit of a perfect model probabilistic model tends towards a deterministic one. Since we do not make claim that a broken power-law is true description of MR relation, probabilistic model is essential for reliable forecasting, as it enables predictions in spite of fact our model is understood to not represent truth. Whilst each of se three key features has been applied to MR relations in some form independently, a novel quality of our methodology is to adopt all three. For example, Wolfgang et al. (2016) inferred a probabilistic power-law conditioned on masses and radii of 90 exoplanets with radii below 8 R. This range crosses expected divide 17

31 between solid planets and those with significant gaseous envelopes at R (Lopez & Fortney 2014) and so authors tried truncating data at 1.6 R as an alternative model. In this work, we argue that transitional points can actually be treated as free parameters in model, enabling us to infer (rar than assume) ir locations and test oretical predictions. Additionally, data need not be censored at < 4 R and wider range makes a forecasting model less susceptible to truncation issues at extrema (we point out that Wolfgang et al. (2016) did not set out to develop a forecasting model explicitly, and thus this is not a criticism of ir work, but rar just an example of how our work differs from previous studies) Data Selection Having broadly established motivation (see Section 2.1) and requirements (see Section 2.2.1) for our model, we will use rest of Section 2.2 to provide a more detailed account of our methodology. To begin, we first define our basic criteria for a data point (a mass and radius measurement) to be included in what follows. Since our work focuses on MR relation, all included objects must fundamentally have a well-defined mass and radius. Whilst former is universally true, latter requires that object have a nearly spherical shape. Low mass objects, for example comet 67P/Churyumov-Gerasimenko, may not have sufficient self-gravity to overcome rigid body forces and be assumed as in hydrostatic equilibrium shape (i.e. nearly spherical). The corresponding threshold mass limit should lie somewhere between most massive body which is known to not be in hydrostatic equilibrium (Iapetus; kg; Sheppard 2016) and least massive body 18

32 confirmed to be in hydrostatic equilibrium (Rhea; kg; Sheppard 2016). This leads us to adopt a boundary condition of M > kg for all objects considered in this work. As for upper limit, we choose maximum mass to be that of a star that must still lie on main-sequence within a Hubble time. The lifetime of a star is dependent upon its mass and luminosity, to first-order. Given that Sun will spend 10 Gyr on main-sequence and L M 7/2, lifetime τ (M/M ) 5/2 10 Gyr. This results in an upper limit of M < 0.87 M ( kg) for τ = H 1 0 Gyr (where we set H 0 = 69.7 km/s; Planck Collaboration et al. 2014). Therefore, between our lower and upper limits, re is a difference of nine orders-of-magnitude in mass and three orders-of-magnitude in radius. We performed a literature search for all objects within this range with a mass and radius measurement available. For Solar System moons, we used The Giant Planet Satellite and Moon Page (Sheppard 2016) which is curated by Scott Sheppard (Sheppard & Jewitt 2003; Sheppard et al. 2005, 2006) and for planets we used NASA Planetary Fact Sheet (Williams 2016). For extrasolar planets, we used TEPCat catalog of wellstudied transiting planets, curated by John Southworth (Southworth 2008, 2009, 2010, 2011, 2012). Brown dwarfs and low-mass stars were drawn from a variety of sources, which we list (along with all or objects used in this work) in Table 2.1. In order to later fit se data sources to an MR model, it is necessary to define a likelihood function of each datum. We later (see 2.2.9) make assumption that for a quoted mass (or radius) measurement of M = (a ± b), one can reasonably approximate M N(a, b). This assumption is a poor one for low signal-to-noise data, especially for upper limit constraints only, where M (or R) is more likely to follow an asymmetric profile 19

33 centered near zero. Without knowledge of correct likelihood function, we argue that such data should be best excluded in what follows. For this reason, we apply a 3 σ cut to both mass ((M/ M) > 3) and radius ((R/ R) > 3). In what follows, we assume that both mass and radius measurements follow normal distributions, which are symmetric. For those data which have substantially asymmetric errors ( + ) n, we only use cases where errors differ by less than 10% (i.e. ( + )/( 1( )) 0.1). Toger, se cuts remove 16% of initial data, which, as discussed later in 2.3.3, do not bias (or even noticeably influence) our final results.. Next, we take average of both errors 1( 2 ++ ) as standard deviations of normal distributions. In end, we have 316 of objects in total which are listed in Table 2.1. The data spans a diverse range of environments, with a variety of orbital periods, insolations, metallicities, etc. Since se terms are not used in our analysis, results presented here should be thought of as a MR relation marginalized over all of se or terms. Once again, we stress that effects of se terms are naturally absorbed by probabilistic framework of our model, meaning that forecasts may be made about any new data, provided that it can be considered a representative of data used for our analysis. 20

34 Table 2.1: Masses and radii used for this study. Name Mass Radius Reference NGC6791 KR V20 (0.827 ± 0.004) M (0.768 ± 0.006) R Torres et al. (2010) HD (0.854 ± 0.003) M (0.830 ± 0.004) R Torres et al. (2010) Parenago 1478 (0.727 ± 0.010) M (1.063 ± 0.011) R Torres et al. (2010) HD 7700 (0.764 ± 0.004) M (0.835 ± 0.018) R Torres et al. (2010) BD (0.814 ± 0.013) M (0.838 ± 0.011) R Torres et al. (2010) TYC (0.869 ± 0.004) M (0.964 ± 0.004) R Torres et al. (2010) GU Boo A (0.610 ± 0.006) M (0.627 ± 0.016) R Torres et al. (2010) GU Boo B (0.600 ± 0.006) M (0.624 ± 0.016) R Torres et al. (2010) YY Gem A (0.599 ± 0.005) M (0.619 ± 0.006) R Torres et al. (2010) YY Gem B (0.599 ± 0.005) M (0.619 ± 0.006) R Torres et al. (2010) CU Cnc A (0.435 ± 0.001) M (0.432 ± 0.006) R Torres et al. (2010) CU Cnc B (0.399 ± 0.001) M (0.392 ± 0.009) R Torres et al. (2010) CM Dra A (0.231 ± 0.001) M (0.253 ± 0.002) R Torres et al. (2010) Continued on next page 21

35 Table 2.1 continued from previous page Name Mass Radius Reference CM Dra B (0.214 ± 0.001) M (0.240 ± 0.002) R Torres et al. (2010) MOTESS-GNAT A (0.527 ± 0.002) M (0.505 ± 0.011) R Kraus et al. (2011) MOTESS-GNAT B (0.491 ± 0.001) M (0.471 ± 0.011) R Kraus et al. (2011) MOTESS-GNAT A (0.567 ± 0.002) M (0.552 ± 0.014) R Kraus et al. (2011) MOTESS-GNAT B (0.532 ± 0.002) M (0.532 ± 0.009) R Kraus et al. (2011) MOTESS-GNAT A (0.584 ± 0.002) M (0.560 ± 0.004) R Kraus et al. (2011) MOTESS-GNAT B (0.544 ± 0.002) M (0.513 ± 0.008) R Kraus et al. (2011) MOTESS-GNAT A (0.499 ± 0.002) M (0.457 ± 0.007) R Kraus et al. (2011) MOTESS-GNAT B (0.443 ± 0.002) M (0.427 ± 0.006) R Kraus et al. (2011) MOTESS-GNAT A (0.557 ± 0.001) M (0.569 ± 0.023) R Kraus et al. (2011) MOTESS-GNAT B (0.535 ± 0.001) M (0.500 ± 0.014) R Kraus et al. (2011) MOTESS-GNAT A (0.469 ± 0.002) M (0.441 ± 0.003) R Kraus et al. (2011) MOTESS-GNAT B (0.382 ± 0.001) M (0.374 ± 0.003) R Kraus et al. (2011) NSVS A (0.870 ± 0.074) M (0.983 ± 0.030) R Çakirli et al. (2010) Continued on next page 22

36 Table 2.1 continued from previous page Name Mass Radius Reference NSVS B (0.607 ± 0.053) M (0.901 ± 0.026) R Çakirli et al. (2010) KOI-686b (103.4 ± 4.8) MJ (1.216 ± 0.037) RJ Díaz et al. (2014) KOI-189b (78.0 ± 3.4) MJ (0.998 ± 0.023) RJ Díaz et al. (2014) OGLE-TR-123 B (0.085 ± 0.011) M (0.133 ± 0.009),R Pont et al. (2006) GJ 570 A (0.802 ± 0.040) M (0.739 ± 0.019) R Demory et al. (2009) GJ 845 (0.762 ± 0.038) M (0.732 ± 0.006) R Demory et al. (2009) GJ 879 (0.725 ± 0.036) M (0.629 ± 0.051) R Demory et al. (2009) GJ 887 (0.503 ± 0.025) M (0.459 ± 0.011) R Demory et al. (2009) GJ 551 (0.118 ± 0.012) M (0.141 ± 0.007) R Boyajian et al. (2012) SDSS B (0.090 ± 0.010) M (0.110 ± 0.004) R Parsons et al. (2012b) NN Ser B (0.111 ± 0.004) M (0.149 ± 0.002) R Parsons et al. (2010) GK Vir B (0.116 ± 0.003) M (0.155 ± 0.003) R Parsons et al. (2012a) OGLE-TR-106 B (0.116 ± 0.021) M (0.181 ± 0.013) R Pont et al. (2005) HAT-TR B (0.124 ± 0.010) M (0.167 ± 0.006) R Beatty et al. (2007) Continued on next page 23

37 Table 2.1 continued from previous page Name Mass Radius Reference SDSS B (0.132 ± 0.003) M (0.165 ± 0.001) R Parsons et al. (2012a) GJ 699 (0.146 ± 0.015) M (0.187 ± 0.001) R Boyajian et al. (2012) SDSS B (0.158 ± 0.006) M (0.200 ± 0.004) R Pyrzas et al. (2012) SDSS B (0.173 ± 0.027) M (0.181 ± 0.015) R Pyrzas et al. (2009) RR Cae B (0.183 ± 0.013) M (0.209 ± 0.014) R Maxted et al. (2007) 2MASS B (0.190 ± 0.020) M (0.210 ± 0.010) R Hebb et al. (2006) HATS B (0.17 ± 0.01) M (0.18 ± 0.01) R Zhou et al. (2014a) Moon M R Sheppard (2016) Io M R Sheppard (2016) Europa M R Sheppard (2016) Ganymede M R Sheppard (2016) Callisto M R Sheppard (2016) Rhea M R Sheppard (2016) Titan M R Sheppard (2016) Continued on next page 24

38 Table 2.1 continued from previous page Name Mass Radius Reference Titania M R Sheppard (2016) Oberon M R Sheppard (2016) Triton M R Sheppard (2016) Eris M R Sheppard (2016) Mercury M R Williams (2016) Venus M R Williams (2016) Earth M R Williams (2016) Mars M R Williams (2016) Jupiter M R Williams (2016) Saturn 95.2 M 9.45 R Williams (2016) Uranus 14.5 M 4.01 R Williams (2016) Neptune 17.1 M 3.88 R Williams (2016) Pluto M R Williams (2016) 55-Cnc e ( ± 0.001) MJ ( ± ) RJ Demory et al. (2016) Continued on next page 25

39 Table 2.1 continued from previous page Name Mass Radius Reference CoRoT-01 b (1.03 ± 0.1) MJ (1.551 ± 0.064) RJ Southworth (2011) CoRoT-02 b (3.57 ± 0.15) MJ (1.46 ± 0.031) RJ Southworth (2012) CoRoT-03 b (21.96 ± 0.703) MJ (1.037 ± 0.069) RJ Southworth (2011) CoRoT-06 b (2.96 ± 0.34) MJ (1.185 ± 0.041) RJ Southworth (2011) CoRoT-07 b ( ± ) MJ ( ± ) RJ Barros et al. (2014) CoRoT-08 b (0.216 ± 0.036) MJ (0.712 ± 0.083) RJ Southworth (2011) CoRoT-09 b (0.826 ± 0.083) MJ (1.037 ± 0.082) RJ Southworth (2011) CoRoT-10 b (2.78 ± 0.14) MJ (0.941 ± 0.085) RJ Southworth (2011) CoRoT-11 b (2.34 ± 0.39) MJ (1.426 ± 0.057) RJ Southworth (2011) CoRoT-12 b (0.887 ± 0.078) MJ (1.35 ± 0.075) RJ Southworth (2011) CoRoT-13 b (1.312 ± 0.096) MJ (1.252 ± 0.076) RJ Southworth (2011) CoRoT-14 b (7.67 ± 0.49) MJ (1.018 ± 0.079) RJ Southworth (2011) CoRoT-18 b (3.27 ± 0.17) MJ (1.251 ± 0.083) RJ Southworth (2012) CoRoT-20 b (5.06 ± 0.36) MJ (1.16 ± 0.26) RJ Southworth (2012) Continued on next page 26

40 Table 2.1 continued from previous page Name Mass Radius Reference CoRoT-21 b (2.26 ± 0.33) MJ (1.27 ± 0.14) RJ Pätzold et al. (2012) CoRoT-26 b (0.52 ± 0.05) MJ (1.26 ± 0.13) RJ Almenara et al. (2013) CoRoT-27 b (10.39 ± 0.55) MJ (1.007 ± 0.044) RJ Parviainen et al. (2014) CoRoT-28 b (0.484 ± 0.087) MJ (0.955 ± 0.066) RJ Cabrera et al. (2015) CoRoT-29 b (0.85 ± 0.2) MJ (0.9 ± 0.16) RJ Cabrera et al. (2015) EPIC c (0.085 ± 0.022) MJ (0.698 ± 0.064) RJ Petigura et al. (2015) EPIC b (1.774 ± 0.079) MJ (1.06 ± 0.35) RJ Grziwa et al. (2015) EPIC b (0.038 ± 0.009) MJ (0.138 ± 0.014) RJ Sinukoff et al. (2015) GJ-0436 b ( ± ) MJ (0.366 ± 0.014) RJ Lanotte et al. (2014) GJ-1214 b ( ± ) MJ (0.254 ± 0.018) RJ Harpsøe et al. (2013) GJ-3470 b ( ± ) MJ (0.346 ± 0.029) RJ Biddle et al. (2014) HAT-P-01 b (0.525 ± 0.019) MJ (1.319 ± 0.019) RJ Nikolov et al. (2014) HAT-P-02 b (8.74 ± 0.27) MJ (1.19 ± 0.12) RJ Southworth (2010) HAT-P-03 b (0.584 ± 0.027) MJ (0.947 ± 0.03) RJ Southworth (2012) Continued on next page 27

41 Table 2.1 continued from previous page Name Mass Radius Reference HAT-P-05 b (1.06 ± 0.11) MJ (1.252 ± 0.043) RJ Southworth et al. (2012c) HAT-P-06 b (1.063 ± 0.057) MJ (1.395 ± 0.081) RJ Southworth (2012) HAT-P-07 b (1.87 ± 0.03) MJ (1.526 ± 0.008) RJ Benomar et al. (2014) HAT-P-08 b (1.275 ± 0.053) MJ (1.321 ± 0.04) RJ Mancini et al. (2013c) HAT-P-09 b (0.778 ± 0.083) MJ (1.38 ± 0.1) RJ Southworth (2012) HAT-P-11 b (0.084 ± ) MJ ( ± ) RJ Southworth (2011) HAT-P-12 b (0.21 ± 0.012) MJ (0.936 ± 0.012) RJ Lee et al. (2012) HAT-P-13 b (0.906 ± 0.03) MJ (1.487 ± 0.041) RJ Southworth et al. (2012a) HAT-P-14 b (2.271 ± 0.083) MJ (1.219 ± 0.059) RJ Southworth (2012) HAT-P-15 b (1.946 ± 0.066) MJ (1.072 ± 0.043) RJ Kovács et al. (2010) HAT-P-16 b (4.193 ± 0.128) MJ (1.19 ± 0.037) RJ Ciceri et al. (2013) HAT-P-17 b (0.534 ± 0.018) MJ (1.01 ± 0.029) RJ Howard et al. (2012) HAT-P-18 b (0.196 ± 0.008) MJ (0.947 ± 0.044) RJ Esposito et al. (2014) HAT-P-19 b (0.292 ± 0.018) MJ (1.132 ± 0.072) RJ Hartman et al. (2011a) Continued on next page 28

42 Table 2.1 continued from previous page Name Mass Radius Reference HAT-P-20 b (7.246 ± 0.187) MJ (0.867 ± 0.033) RJ Bakos et al. (2011) HAT-P-21 b (4.063 ± 0.161) MJ (1.024 ± 0.092) RJ Bakos et al. (2011) HAT-P-22 b (2.147 ± 0.061) MJ (1.08 ± 0.058) RJ Bakos et al. (2011) HAT-P-23 b (2.07 ± 0.12) MJ (1.224 ± 0.037) RJ Ciceri et al. (2015b) HAT-P-30 b (0.711 ± 0.028) MJ (1.34 ± 0.065) RJ Johnson et al. (2011) HAT-P-32 b (0.86 ± 0.164) MJ (1.789 ± 0.025) RJ Hartman et al. (2011b) HAT-P-33 b (0.762 ± 0.101) MJ (1.686 ± 0.045) RJ Hartman et al. (2011b) HAT-P-35 b (1.054 ± 0.033) MJ (1.332 ± 0.098) RJ Bakos et al. (2012) HAT-P-36 b (1.852 ± 0.095) MJ (1.304 ± 0.025) RJ Mancini et al. (2015a) HAT-P-37 b (1.169 ± 0.103) MJ (1.178 ± 0.077) RJ Bakos et al. (2012) HAT-P-40 b (0.615 ± 0.038) MJ (1.73 ± 0.062) RJ Hartman et al. (2012) HAT-P-42 b (1.044 ± 0.083) MJ (1.28 ± 0.153) RJ Boisse et al. (2013) HAT-P-50 b (1.35 ± 0.073) MJ (1.288 ± 0.064) RJ Hartman et al. (2015b) HAT-P-51 b (0.309 ± 0.018) MJ (1.293 ± 0.054) RJ Hartman et al. (2015b) Continued on next page 29

43 Table 2.1 continued from previous page Name Mass Radius Reference HAT-P-52 b (0.818 ± 0.029) MJ (1.009 ± 0.072) RJ Hartman et al. (2015b) HAT-P-53 b (1.484 ± 0.056) MJ (1.318 ± 0.091) RJ Hartman et al. (2015b) HAT-P-54 b (0.76 ± 0.032) MJ (0.944 ± 0.028) RJ Bakos et al. (2015) HAT-P-55 b (0.582 ± 0.056) MJ (1.182 ± 0.055) RJ Juncher et al. (2015) HAT-P-56 b (2.18 ± 0.25) MJ (1.466 ± 0.04) RJ Huang et al. (2015) HATS-02 b (1.345 ± 0.15) MJ (1.168 ± 0.03) RJ Mohler-Fischer et al. (2013) HATS-03 b (1.071 ± 0.136) MJ (1.381 ± 0.035) RJ Bayliss et al. (2013) HATS-04 b (1.323 ± 0.028) MJ (1.02 ± 0.037) RJ Jordán et al. (2014) HATS-05 b (0.237 ± 0.012) MJ (0.912 ± 0.025) RJ Zhou et al. (2014b) HATS-06 b (0.319 ± 0.07) MJ (0.998 ± 0.019) RJ Hartman et al. (2015a) HATS-09 b (0.837 ± 0.029) MJ (1.065 ± 0.098) RJ Brahm et al. (2015b) HATS-13 b (0.543 ± 0.072) MJ (1.212 ± 0.035) RJ Mancini et al. (2015b) HATS-15 b (2.17 ± 0.15) MJ (1.105 ± 0.04) RJ Ciceri et al. (2015a) HATS-16 b (3.27 ± 0.19) MJ (1.30 ± 0.15) RJ Ciceri et al. (2015a) Continued on next page 30

44 Table 2.1 continued from previous page Name Mass Radius Reference HATS-17 b (1.338 ± 0.065) MJ (0.777 ± 0.056) RJ Brahm et al. (2015a) HD b (3.262 ± 0.113) MJ (1.065 ± 0.035) RJ Southworth (2011) HD b (4.114 ± 0.155) MJ (1.003 ± 0.027) RJ Southworth (2011) HD b ( ± ) MJ ( ± ) RJ Van Grootel et al. (2014) HD b (0.368 ± 0.014) MJ (0.813 ± 0.027) RJ Carter et al. (2009) HD b (1.15 ± 0.039) MJ (1.151 ± 0.038) RJ Southworth (2010) HD b (0.714 ± 0.017) MJ (1.380 ± 0.017) RJ Southworth (2010) HD b ( ± ) MJ ( ± ) RJ Motalebi et al. (2015) K2-02 b (0.037 ± 0.004) MJ (0.226 ± 0.016) RJ Vanderburg et al. (2015) K2-19 b (0.138 ± 0.038) MJ (0.666 ± 0.068) RJ Barros et al. (2015) KELT-03 b (1.477 ± 0.066) MJ (1.345 ± 0.072) RJ Pepper et al. (2013) KELT-04 b (0.902 ± 0.06) MJ (1.699 ± 0.046) RJ Eastman et al. (2015) KELT-07 b (1.28 ± 0.18) MJ (1.533 ± 0.047) RJ Bieryla et al. (2015) KELT-15 b (1.196 ± 0.072) MJ (1.52 ± 0.12) RJ Rodriguez et al. (2015) Continued on next page 31

45 Table 2.1 continued from previous page Name Mass Radius Reference Kepler-07 b (0.453 ± 0.068) MJ (1.649 ± 0.038) RJ Southworth (2012) Kepler-08 b (0.59 ± 0.12) MJ (1.381 ± 0.037) RJ Southworth (2011) Kepler-09 b (0.142 ± 0.005) MJ (0.990 ± 0.009) RJ Dreizler & Ofir (2014) Kepler-09 c (0.098 ± 0.003) MJ (0.955 ± 0.009) RJ Dreizler & Ofir (2014) Kepler-14 b (7.68 ± 0.38) MJ (1.126 ± 0.049) RJ Southworth (2012) Kepler-15 b (0.696 ± 0.099) MJ (1.289 ± 0.054) RJ Southworth (2012) Kepler-18 c (0.054 ± 0.006) MJ (0.49 ± 0.023) RJ Cochran et al. (2011) Kepler-18 d ( ± ) MJ (0.623 ± 0.029) RJ Cochran et al. (2011) Kepler-25 c (0.077 ± 0.018) MJ (0.464 ± 0.008) RJ Marcy et al. (2014) Kepler-26 b ( ± 0.002) MJ (0.248 ± 0.01) RJ Jontof-Hutter et al. (2015) Kepler-26 c ( ± ) MJ (0.243 ± 0.011) RJ Jontof-Hutter et al. (2015) Kepler-29 b ( ± ) MJ (0.299 ± 0.02) RJ Jontof-Hutter et al. (2015) Kepler-29 c ( ± ) MJ (0.28 ± 0.018) RJ Jontof-Hutter et al. (2015) Kepler-30 b ( ± ) MJ (0.35 ± 0.02) RJ Sanchis-Ojeda et al. (2012) Continued on next page 32

46 Table 2.1 continued from previous page Name Mass Radius Reference Kepler-30 d ( ± ) MJ (0.79 ± 0.04) RJ Sanchis-Ojeda et al. (2012) Kepler-34 b (0.22 ± 0.011) MJ (0.764 ± 0.045) RJ Welsh et al. (2012) Kepler-35 b (0.127 ± 0.02) MJ (0.728 ± 0.014) RJ Welsh et al. (2012) Kepler-39 b (19.1 ± 1.) MJ (1.11 ± 0.03) RJ Bonomo et al. (2015) Kepler-40 b (2.16 ± 0.43) MJ (1.44 ± 0.12) RJ Southworth (2012) Kepler-41 b (0.56 ± 0.08) MJ (1.29 ± 0.02) RJ Bonomo et al. (2015) Kepler-43 b (3.09 ± 0.21) MJ (1.115 ± 0.041) RJ Bonomo et al. (2015) Kepler-44 b (1. ± 0.1) MJ (1.09 ± 0.07) RJ Bonomo et al. (2015) Kepler-45 b (0.5 ± 0.06) MJ (0.999 ± 0.069) RJ Southworth (2012) Kepler-48 c (0.046 ± 0.007) MJ (0.242 ± 0.012) RJ Marcy et al. (2014) Kepler-51 d ( ± ) MJ (0.865 ± 0.045) RJ Masuda (2014) Kepler-56 b (0.069 ± 0.012) MJ (0.581 ± 0.025) RJ Huber et al. (2013) Kepler-56 c (0.569 ± 0.066) MJ (0.874 ± 0.041) RJ Huber et al. (2013) Kepler-60 c ( ± ) MJ (0.17 ± 0.013) RJ Jontof-Hutter et al. (2015) Continued on next page 33

47 Table 2.1 continued from previous page Name Mass Radius Reference Kepler-74 b (0.63 ± 0.12) MJ (0.96 ± 0.02) RJ Bonomo et al. (2015) Kepler-75 b (10.1 ± 0.4) MJ (1.05 ± 0.03) RJ Bonomo et al. (2015) Kepler-76 b (2.18 ± 0.42) MJ (1.25 ± 0.08) RJ Faigler & Mazeh (2015) Kepler-77 b (0.43 ± 0.032) MJ (0.96 ± 0.016) RJ Gandolfi et al. (2013) Kepler-78 b ( ± ) MJ (0.107 ± 0.008) RJ Grunblatt et al. (2015) Kepler-79 e ( ± ) MJ (0.311 ± 0.012) RJ Jontof-Hutter et al. (2014) Kepler-87 b (1.02 ± 0.028) MJ (1.203 ± 0.049) RJ Ofir et al. (2014) Kepler-87 c ( ± ) MJ (0.548 ± 0.026) RJ Ofir et al. (2014) Kepler-88 b ( ± ) MJ (0.337 ± 0.035) RJ Nesvorný et al. (2013) Kepler-89 d (0.333 ± 0.035) MJ (1.005 ± 0.095) RJ Weiss et al. (2013) Kepler-93 b ( ± ) MJ ( ± ) RJ Dressing et al. (2015) Kepler-94 b (0.034 ± 0.004) MJ (0.313 ± 0.013) RJ Marcy et al. (2014) Kepler-95 b (0.041 ± 0.009) MJ (0.305 ± 0.008) RJ Marcy et al. (2014) Kepler-99 b (0.019 ± 0.004) MJ (0.132 ± 0.007) RJ Marcy et al. (2014) Continued on next page 34

48 Table 2.1 continued from previous page Name Mass Radius Reference Kepler-101 b (0.161 ± 0.016) MJ (0.515 ± 0.076) RJ Bonomo et al. (2014) Kepler-102 e (0.028 ± 0.006) MJ (0.198 ± 0.006) RJ Marcy et al. (2014) Kepler-105 c ( ± ) MJ (0.117 ± 0.006) RJ Kostov et al. (2015) Kepler-106 c (0.033 ± 0.01) MJ (0.223 ± 0.029) RJ Marcy et al. (2014) Kepler-117 c (1.84 ± 0.18) MJ (1.101 ± 0.035) RJ Bruno et al. (2015) Kepler-131 b (0.051 ± 0.011) MJ (0.215 ± 0.018) RJ Marcy et al. (2014) Kepler-289 c (0.013 ± 0.003) MJ (0.239 ± 0.015) RJ Schmitt et al. (2014) Kepler-289 d (0.415 ± 0.053) MJ (1.034 ± 0.017) RJ Schmitt et al. (2014) Kepler-307 b ( ± ) MJ (0.217 ± 0.008) RJ Jontof-Hutter et al. (2015) Kepler-406 b (0.02 ± 0.004) MJ (0.128 ± 0.003) RJ Marcy et al. (2014) Kepler-412 b (0.939 ± 0.085) MJ (1.325 ± 0.043) RJ Deleuil et al. (2014) Kepler-420 b (1.45 ± 0.35) MJ (0.94 ± 0.12) RJ Santerne et al. (2014) Kepler-422 b (0.43 ± 0.13) MJ (1.15 ± 0.11) RJ Endl et al. (2014) Kepler-423 b (0.595 ± 0.081) MJ (1.192 ± 0.052) RJ Gandolfi et al. (2015) Continued on next page 35

49 Table 2.1 continued from previous page Name Mass Radius Reference Kepler-433 b (2.82 ± 0.52) MJ (1.45 ± 0.16) RJ Almenara et al. (2015) Kepler-435 b (0.84 ± 0.15) MJ (1.99 ± 0.18) RJ Almenara et al. (2015) Kepler-447 b (1.37 ± 0.16) MJ (1.65 ± 0.2) RJ Lillo-Box et al. (2015) Kepler-454 b ( ± ) MJ (0.211 ± 0.012) RJ Gettel et al. (2016) KOI-188 b (0.25 ± 0.08) MJ (0.978 ± 0.022) RJ Hébrard et al. (2014) KOI-192 b (0.29 ± 0.09) MJ (1.23 ± 0.21) RJ Hébrard et al. (2014) KOI-195 b (0.34 ± 0.08) MJ (1.09 ± 0.03) RJ Hébrard et al. (2014) KOI-372 b (3.25 ± 0.2) MJ (0.882 ± 0.088) RJ Mancini et al. (2015c) KOI-830 b (1.27 ± 0.19) MJ (1.08 ± 0.03) RJ Hébrard et al. (2014) KOI-1474 b (2.6 ± 0.3) MJ (0.96 ± 0.12) RJ Dawson et al. (2014) LHS-6343 b (62.1 ± 1.2) MJ (0.783 ± 0.011) RJ Montet et al. (2015) OGLE-TR-010 b (0.68 ± 0.15) MJ (1.72 ± 0.11) RJ Southworth (2010) OGLE-TR-056 b (1.41 ± 0.17) MJ (1.734 ± 0.058) RJ Southworth (2012) OGLE-TR-111 b (0.55 ± 0.1) MJ (1.011 ± 0.038) RJ Southworth (2012) Continued on next page 36

50 Table 2.1 continued from previous page Name Mass Radius Reference OGLE-TR-113 b (1.23 ± 0.2) MJ (1.088 ± 0.054) RJ Southworth (2012) OGLE-TR-132 b (1.17 ± 0.15) MJ (1.229 ± 0.075) RJ Southworth (2012) OGLE-TR-182 b (1.06 ± 0.15) MJ (1.47 ± 0.14) RJ Southworth (2010) Qatar-1 b (1.294 ± 0.052) MJ (1.143 ± 0.026) RJ Collins et al. (2015) Qatar-2 b (2.494 ± 0.054) MJ (1.254 ± 0.013) RJ Mancini et al. (2014c) TrES-1 b (0.761 ± 0.051) MJ (1.099 ± 0.035) RJ Southworth (2010) TrES-2 b (1.206 ± 0.049) MJ (1.193 ± 0.023) RJ Southworth (2011) TrES-3 b (1.899 ± 0.062) MJ (1.31 ± 0.019) RJ Southworth (2011) TrES-5 b (1.79 ± 0.068) MJ (1.194 ± 0.015) RJ Barstow et al. (2015) WASP-02 b (0.88 ± 0.038) MJ (1.063 ± 0.028) RJ Southworth (2012) WASP-04 b (1.249 ± 0.052) MJ (1.364 ± 0.028) RJ Southworth (2012) WASP-05 b (1.595 ± 0.052) MJ (1.175 ± 0.055) RJ Southworth (2012) WASP-06 b (0.485 ± 0.028) MJ (1.23 ± 0.037) RJ Tregloan-Reed et al. (2015) WASP-07 b (0.98 ± 0.13) MJ (1.374 ± 0.094) RJ Southworth (2012) Continued on next page 37

51 Table 2.1 continued from previous page Name Mass Radius Reference WASP-11 b (0.492 ± 0.024) MJ (0.99 ± 0.023) RJ Mancini et al. (2015a) WASP-13 b (0.512 ± 0.06) MJ (1.528 ± 0.084) RJ Southworth (2012) WASP-15 b (0.592 ± 0.019) MJ (1.408 ± 0.046) RJ Southworth et al. (2013) WASP-16 b (0.832 ± 0.038) MJ (1.218 ± 0.04) RJ Southworth et al. (2013) WASP-17 b (0.477 ± 0.033) MJ (1.932 ± 0.053) RJ Southworth et al. (2012b) WASP-18 b (10.52 ± 0.32) MJ (1.204 ± 0.028) RJ Maxted et al. (2013b) WASP-19 b (1.139 ± 0.036) MJ (1.41 ± 0.021) RJ Mancini et al. (2013a) WASP-20 b (0.311 ± 0.017) MJ (1.462 ± 0.059) RJ Anderson et al. (2015a) WASP-21 b (0.276 ± 0.019) MJ (1.162 ± 0.054) RJ Ciceri et al. (2013) WASP-24 b (1.109 ± 0.054) MJ (1.303 ± 0.047) RJ Southworth et al. (2014) WASP-25 b (0.598 ± 0.046) MJ (1.247 ± 0.032) RJ Southworth et al. (2014) WASP-26 b (1.02 ± 0.033) MJ (1.216 ± 0.047) RJ Southworth et al. (2014) WASP-28 b (0.907 ± 0.043) MJ (1.213 ± 0.042) RJ Anderson et al. (2015a) WASP-29 b (0.244 ± 0.02) MJ (0.776 ± 0.043) RJ Gibson et al. (2013) Continued on next page 38

52 Table 2.1 continued from previous page Name Mass Radius Reference WASP-31 b (0.478 ± 0.029) MJ (1.549 ± 0.05) RJ Anderson et al. (2011) WASP-32 b (3.6 ± 0.07) MJ (1.18 ± 0.07) RJ Maxted et al. (2010) WASP-35 b (0.72 ± 0.06) MJ (1.32 ± 0.05) RJ Enoch et al. (2011) WASP-36 b (2.303 ± 0.068) MJ (1.281 ± 0.029) RJ Smith et al. (2012) WASP-38 b (2.691 ± 0.058) MJ (1.094 ± 0.029) RJ Barros et al. (2011) WASP-39 b (0.28 ± 0.03) MJ (1.27 ± 0.04) RJ Faedi et al. (2011) WASP-41 b (0.977 ± 0.026) MJ (1.178 ± 0.018) RJ Southworth et al. (2015c) WASP-42 b (0.527 ± 0.028) MJ (1.122 ± 0.039) RJ Southworth et al. (2015c) WASP-43 b (2.034 ± 0.052) MJ (1.036 ± 0.019) RJ Gillon et al. (2012) WASP-44 b (0.869 ± 0.081) MJ (1.002 ± 0.038) RJ Mancini et al. (2013b) WASP-45 b (1.002 ± 0.062) MJ (0.992 ± 0.038) RJ Ciceri et al. (2016) WASP-46 b (1.91 ± 0.13) MJ (1.174 ± 0.037) RJ Ciceri et al. (2016) WASP-47 b (1.13 ± 0.06) MJ (1.134 ± 0.039) RJ Becker et al. (2015) WASP-47 c (0.038 ± 0.012) MJ ( ± ) RJ Dai et al. (2015) Continued on next page 39

53 Table 2.1 continued from previous page Name Mass Radius Reference WASP-48 b (0.907 ± 0.085) MJ (1.396 ± 0.051) RJ Ciceri et al. (2015b) WASP-49 b (0.378 ± 0.027) MJ (1.115 ± 0.047) RJ Lendl et al. (2012) WASP-50 b (1.437 ± 0.068) MJ (1.138 ± 0.026) RJ Tregloan-Reed & Southworth (2013) WASP-52 b (0.46 ± 0.02) MJ (1.166 ± 0.088) RJ Swift et al. (2015) WASP-54 b (0.636 ± 0.025) MJ (1.653 ± 0.09) RJ Faedi et al. (2013) WASP-56 b (0.571 ± 0.035) MJ (1.092 ± 0.035) RJ Faedi et al. (2013) WASP-57 b (0.644 ± 0.062) MJ (1.05 ± 0.053) RJ Southworth et al. (2015b) WASP-58 b (0.89 ± 0.07) MJ (1.37 ± 0.2) RJ Hébrard et al. (2013) WASP-59 b (0.863 ± 0.045) MJ (0.775 ± 0.068) RJ Hébrard et al. (2013) WASP-60 b (0.514 ± 0.034) MJ (0.86 ± 0.12) RJ Hébrard et al. (2013) WASP-61 b (2.06 ± 0.17) MJ (1.24 ± 0.03) RJ Hellier et al. (2012) WASP-62 b (0.57 ± 0.04) MJ (1.39 ± 0.06) RJ Hellier et al. (2012) WASP-64 b (1.271 ± 0.068) MJ (1.271 ± 0.039) RJ Gillon et al. (2013) WASP-65 b (1.55 ± 0.16) MJ (1.112 ± 0.059) RJ Gómez Maqueo Chew et al. (2013) Continued on next page 40

54 Table 2.1 continued from previous page Name Mass Radius Reference WASP-66 b (2.32 ± 0.13) MJ (1.39 ± 0.09) RJ Hellier et al. (2012) WASP-67 b (0.406 ± 0.035) MJ (1.091 ± 0.046) RJ Mancini et al. (2014a) WASP-69 b (0.26 ± 0.017) MJ (1.057 ± 0.047) RJ Anderson et al. (2014b) WASP-71 b (2.242 ± 0.08) MJ (1.46 ± 0.13) RJ Smith et al. (2013) WASP-72 b (1.461 ± 0.059) MJ (1.27 ± 0.2) RJ Gillon et al. (2013) WASP-74 b (0.95 ± 0.06) MJ (1.56 ± 0.06) RJ Hellier et al. (2015) WASP-75 b (1.07 ± 0.05) MJ (1.27 ± 0.048) RJ Gómez Maqueo Chew et al. (2013) WASP-77 b (1.76 ± 0.06) MJ (1.21 ± 0.02) RJ Maxted et al. (2013a) WASP-78 b (0.89 ± 0.08) MJ (1.7 ± 0.11) RJ Smalley et al. (2012) WASP-79 b (0.9 ± 0.08) MJ (2.09 ± 0.14) RJ Smalley et al. (2012) WASP-80 b (0.562 ± 0.027) MJ (0.986 ± 0.022) RJ Mancini et al. (2014b) WASP-84 b (0.687 ± 0.033) MJ (0.976 ± 0.025) RJ Anderson et al. (2015b) WASP-85 b (1.265 ± 0.062) MJ (1.24 ± 0.03) RJ Močnik et al. (2015) WASP-87 b (2.18 ± 0.15) MJ (1.385 ± 0.06) RJ Anderson et al. (2014a) Continued on next page 41

55 Table 2.1 continued from previous page Name Mass Radius Reference WASP-89 b (5.9 ± 0.4) MJ (1.04 ± 0.04) RJ Hellier et al. (2015) WASP-90 b (0.63 ± 0.07) MJ (1.63 ± 0.09) RJ West et al. (2016) WASP-96 b (0.48 ± 0.03) MJ (1.2 ± 0.06) RJ Hellier et al. (2014) WASP-97 b (1.32 ± 0.05) MJ (1.13 ± 0.06) RJ Hellier et al. (2014) WASP-98 b (0.83 ± 0.07) MJ (1.1 ± 0.04) RJ Hellier et al. (2014) WASP-100 b (2.03 ± 0.12) MJ (1.69 ± 0.29) RJ Hellier et al. (2014) WASP-101 b (0.5 ± 0.04) MJ (1.41 ± 0.05) RJ Hellier et al. (2014) WASP-103 b (1.47 ± 0.11) MJ (1.554 ± 0.045) RJ Southworth et al. (2015a) WASP-104 b (1.272 ± 0.047) MJ (1.137 ± 0.037) RJ Smith et al. (2014) WASP-108 b (0.892 ± 0.055) MJ (1.284 ± 0.047) RJ Anderson et al. (2014a) WASP-109 b (0.91 ± 0.13) MJ (1.443 ± 0.053) RJ Anderson et al. (2014a) WASP-110 b (0.51 ± 0.064) MJ (1.238 ± 0.056) RJ Anderson et al. (2014a) WASP-111 b (1.83 ± 0.15) MJ (1.443 ± 0.094) RJ Anderson et al. (2014a) WASP-112 b (0.88 ± 0.12) MJ (1.191 ± 0.049) RJ Anderson et al. (2014a) Continued on next page 42

56 Table 2.1 continued from previous page Name Mass Radius Reference WASP-120 b (5.01 ± 0.26) MJ (1.515 ± 0.083) RJ Turner et al. (2015) WASP-121 b (1.183 ± 0.064) MJ (1.865 ± 0.044) RJ Delrez et al. (2015) WASP-122 b (1.372 ± 0.072) MJ (1.792 ± 0.069) RJ Turner et al. (2015) WASP-123 b (0.92 ± 0.05) MJ (1.327 ± 0.074) RJ Turner et al. (2015) WASP-135 b (1.9 ± 0.08) MJ (1.3 ± 0.09) RJ Spake et al. (2015) WTS-2 b (1.12 ± 0.16) MJ (1.363 ± 0.061) RJ Birkby et al. (2014) XO-1 b (0.924 ± 0.077) MJ (1.206 ± 0.041) RJ Southworth (2010) XO-2 b (0.597 ± 0.021) MJ (1.019 ± 0.031) RJ Damasso et al. (2015) XO-3 b (11.83 ± 0.38) MJ (1.248 ± 0.049) RJ Southworth (2010) XO-5 b (1.19 ± 0.031) MJ (1.142 ± 0.034) RJ Smith (2015) 43

57 2.2.3 Probabilistic Broken Power-Law We elect to model MR relation with a probabilistic broken power-law, for reasons described in By probabilistic, we mean that this model includes intrinsic dispersion in MR relation to account for additional variance beyond that of formal measurement uncertainties. This dispersion represents variance observed in nature itself around our broken power-law model. To put this in context, a deterministic MR power-law would be described via R = C ( M ) S, (2.1) R M where R & M are mass and radius of object respectively and C & S are parameters describing power-law. However, it is easy to conceive of two objects with exact same mass but different compositions, reby leading to different radii. For this reason, we argue that a deterministic model provides an unrealistic description of MR relation. In probabilistic model, for any given mass re is a corresponding distribution of radius. In this work, we assume a normal distribution in logarithm of radius. The mean of distribution takes result of deterministic model, and standard deviation is intrinsic dispersion, a new free parameter. A power-law relation can be converted to a linear relation by taking logarithm on both axes. In practice, we take logarithm base ten of both mass and radius in Earth units, and use a linear relation to fit m. In what follows, we will use M, R to represent mass and radius, and M, R to represent log 10 (M/M ) and log 10 (R/R ). The power-law relation turns into 44

58 R = C + M S, (2.2) where R = log 10 (R/R ), M = log 10 (M/M ), and C = log 10 C. In what follows, we will use N(µ, σ) as normal distribution, where µ is mean and σ is standard deviation. The corresponding probabilistic relation in log scale becomes R N(µ = C + M S, σ = σ R ) (2.3) On a logarithmic scale, data still approximately follow normal distributions, because logarithm of a normal distribution is approximately a normal distribution when standard deviation is small relative to mean, which is true here since we made a 3 σ cut in both mass and radius. The original data, M N(M t, M), will turn into M ob N(M t, M ob ), where M t = log 10 (M t /M ) and M ob = log 10 (e)( M/M). We consider it more reasonable to assume that intrinsic dispersion in radius will be a fractional dispersion, rar than an absolute dispersion. For example, dispersion of Earth-radius planets might be O[0.1 R ] but for stars it should surely be much larger in an absolute sense. Since a fractional dispersion on a linear scale corresponds to an absolute dispersion on logarithmic scale, this assumption is naturally accounted for by our model. To implement probabilistic model, we employ a hierarchical Bayesian model, or HBM for short. 45

59 2.2.4 Hierarchical Bayesian Modeling The difference between an HBM and more familiar Bayesian method is that HBMs have two sets of parameters; a layer of hyper parameters, Θ hyper, on top of local parameters, Θ local (see Hogg et al for a pedagogical explanation). The local parameters usually describe properties of each individual datum, whilst hypers describe overall ensemble properties. For example, in this work, local parameters are true log 10 (M/M ), log 10 (R/R ) (or M t, R t ) of all objects, and hyper parameters, Θ hyper, are those that represent broken power-law. This hierarchical structure is illustrated in Figure 2.1, which may be compared to analogous graphical model shown in Figure 1 of Wolfgang et al. (2016). Some of first applications of this method are Loredo & Wasserman (1995), Graziani & Lamb (1996), and Hogg et al. (2010) (in exoplanets research). For local parameters, we define a mass, M t, and radius, R t, term for each object, giving 632 local variables. In practice, R t local parameters are related to M t term through broken power-law and each realization of hyper parameters. In total n, our model includes 632 local parameters and a compact set of hyper parameters, as described later in MCMC subsection Continuous Broken Power-Law Model Plotting masses and radii on a log-log scale, (as shown later in Figure 2.3), it is clear that single, continuous power-law is unable to provide a reasonable description of data. For example, one might reasonably expect that Neptune-like planets follow a 46

60 C (1) S (1 4) (1 4) R T (1 3) M (i) ob M (i) t R (i) t R (i) ob M (i) ob R (i) ob N Figure 2.1 Graphical model of HBM used to infer probabilistic MR relation in this work. Yellow ovals represent hyper-parameters, white represent true local parameters and gray represent data inputs. All objects on plate have N members. different MR relation from terrestrial planets, since voluminous gaseous envelopes of former dominate ir radii (Lopez & Fortney 2014). This refore argues in favor of using a segmented (or broken) power-law. At least three fundamentally distinct regimes are expected using some simple physical insights; a segment for terrestrial planets, gas giants, and stars. Indeed, MR data clearly shows distinct changes in power-index, corresponding to transition points between each segment. However, a visual inspection also reveals a turn-over in MR 47

61 relation at around a Saturn-mass. Therefore, from one roughly Saturn-mass to onset of stars, re is a strong case for a fourth segment which we consequently include in our model. Later, in Section 2.3.2, we perform a model comparison of a three- versus four-segment model to validate that four-segment broken power-law is strongly favored. Our favored model consists of 12 free hyper parameters; 1 offset (C (1) ), 4 slopes (S (1 4) ), 4 intrinsic dispersions (σ (1 4) ), and 3 transition points (T (1 3) ). Critically n, we actually R fit for locations of transition points and include an independent intrinsic dispersion for each segment (making our model probabilistic). Also note that slopes in log-log space are power-law indices in linear space. The hyper parameter vector is refore Θ hyper = {S (1), S (2), S (3), S (4), σ (1) R, σ(2) R, σ(3) R, σ(4) R, T (1), T (2), T (3), C (1) }. (2.4) There is only one free parameter for offset since we impose condition that each segment of power-law is connected, i.e. a continuous broken power-law. By requiring that two segments meet at transition point between m, we can derive offsets for rest of segments. At each transition point T (j), C (j) + S (j) T (j) = C (j+1) + S (j+1) T (j) for j = 1, 2, 3. (2.5) 48

62 We can now iteratively derive or offsets as, C (j+1) = C (j) + (S (j) S (j+1) ) T (j) for j = 1, 2, 3. (2.6) Hyper Priors The hyper priors, or priors on hyper-parameters, are selected to be sufficiently broad to allow an extensive exploration of parameter space and to be identical for each segment. Uniform priors are used for location parameters, namely offset, C, and transition points, T. For scale parameters, namely intrinsic dispersion σ R, we adopt log-uniform priors. For slope parameters, we don t want to constrain m in a specific range, so we use normal distribution with a large variance. This leads to a prior distribution which is approximately uniform in any small region yet loosely constrains MCMC walkers to relevant scale of data. A detailed list of priors is provided in Table

63 Table 2.2: Description and posterior of hyper parameters.the prior distributions of hyper parameters [ ] are C (1) U( 1, 1); S (1 4) N(0, 5); log 10 σ (1 4) R U( 3, 2); T (1 3) U( 4, 6). Θhyper Description Credible Interval 10 C R Power-law constant for Terran (T-class) worlds MR relation R S (1) Power-law index of Terran worlds; R M S S (2) Power-law index of Neptunian worlds; R M S S (3) Power-law index of Jovian worlds; R M S S (4) Power-law index of Stellar worlds; R M S σ (1) Fractional dispersion of radius for Terran MR relation % R 0.64 σ (2) Fractional dispersion of radius for Neptunian MR relation % R 1.3 σ (3) Fractional dispersion of radius for Jovian MR relation % R 0.45 σ (4) Fractional dispersion of radius for Stellar MR relation % R T(1) M Terran-to-Neptunian transition point M 0.59 Continued on next page 50

64 Table 2.2 continued from previous page Θhyper Description Credible Interval 10 T(2) M Neptunian-to-Jovian class transition point M J 10 T(3) M Jovian-to-Stellar class transition point M

65 2.2.7 Two Different Categories of Local Parameters The local parameters in our model are formally M t and R t, although in practice R t doesn t need to be fitted explicitly since it is derived from realization of broken power-law (as described in more detail later). Even for M t though, re are two categories that we must distinguish between. Objects within Solar System tend to have very precise measurements of ir fundamental properties such that ir formal uncertainties are negligible relative to uncertainties encountered for extrasolar objects, for which we must account for measurement uncertainties in our model. For objects with negligible error, we simply fix M t = M ob and R t = R ob, since M ob, R ob M M, R R = 0. For objects in second category, M t are set to be independently uniformly distributed in [ 4, 6]. Throughout paper, we will use U(a, b) to denote a uniform distribution, where a and b are lower and upper bounds of distribution. So M (i) t U( 4, 6) for i = 1, 2,..., 316. (2.7) Inverse Sampling We use inverse sampling method to sample parameters M t and Θ hyper. By inverse sampling, we mean that walkers directly sample in probability space, rar than parameter space itself. By directly walking in prior probability space with Gaussian function as our proposal distribution, inverse sampling is more efficient than walking in 52

66 parameter space plus likelihood penalization (see Devroye 1986 furr details on inverse sampling method). For each jump in MCMC chain, we sample a probability, p, for each parameter with U(0, 1). We n determine this parameter s cumulative distribution from its prior probability distribution. With p and cumulative distribution, we can n calculate corresponding sample value of parameter. The equations of prior distributions of M t and Θ hyper are already shown in Table 2.2 and Equation (2.7). With inverse sampling, effects of priors have already been accounted for, meaning that we do not need to add prior probabilities of parameters into total log-likelihood function Total Log-Likelihood As discussed above, since M t and Θ hyper are drawn with inverse sampling, re is no need to add corresponding penalty terms to log-likelihood function. The total loglikelihood is now based on how we sample R t from M t and Θ hyper, and relations between M t, R t, and data. The relations are given by M (i) ob N( M (i) t, M (i) ob). (2.8) When M (i) ob = 0, above equation can be interpreted as M(i) ob = M(i) t, which corresponds to case where measurement errors are zero. This is also true for R (i) ob, such that 53

67 R (i) ob N( R (i) t, R (i) ob), (2.9) and R (i) t N ( f (M (i) t, Θ hyper ), σ R), (2.10) where we define ( f (M (i) t,θ hyper ), σ R) = ( C (1) + M (i) t S (1), σ (1) R ( C (2) + M (i) t S (2), σ (2) R ( C (3) + M (i) t S (3), σ (3) R ( C (4) + M (i) t S (4), σ (4) R ) ) ) ) M (i) t T (1) T (1) < M (i) t T (2) T (2) < M (i) t T (3) T (3) < M (i) t. (2.11) Combining Equation (2.9) and (2.10), we have R (i) N( ) f (M (i) ob t, Θ hyper ), ( R (i) ob )2 + (σ R )2. (2.12) Equation (2.12) shows that if we have already sampled M t and Θ hyper, we don t need to sample R t anymore since R ob can be directly related to M t and Θ hyper. From Equation (2.8) and (2.12), we can see that total log-likelihood of model is 54

68 2 log L = N i=1 N i=1 N i=1 ( M (i) ob M(i) t M (i) ob )2 + N ( M (i) ob )2 + i=1 ( R (i) f ob (M(i) t, Θ hyper ) ) 2 ) 2 + ( R (i) ob ) 2 + ( σ R log [( ) R (i) 2 ( 2 ] ob + σ R). (2.13) Note that in above, we assume mass and radius have no covariance, which is almost always true given independent methods of ir measurement. 2.3 Analysis Parameter Inference with Markov Chain Monte Carlo We used Markov Chain Monte Carlo (MCMC) method with Metropolis algorithm (Metropolis et al. 1953) to explore parameter space and infer posterior distributions for both hyper and local parameters. The Metropolis algorithm uses jumping walkers, proceeding by accepting or rejecting each jump by comparing its likelihood with likelihood of previous step. Since we have 12 hyper parameters and 316 data points (corresponding to 316 M t ), walker jumps in a probability hyper cube of (12+316) dimensions. We begin by running 5 independent initial chains for 500,000 accepted steps each, seeding parameters T (1 3) from 0.5, 2, and 4 with Gaussian distributions of sigma one 55

69 (but keeping all or terms seeded randomly from hyper priors). We identify burn-in point by eye, searching for instant where local variance in log-likelihood (with respect to chain step) stabilizes to a relatively small scatter in comparison to initial steps. This burn-in point tended to occur after 200, 000 accepted steps, largely driven by fact that both hyper and local parameters were not seeded from local minimums (with exception of T (1 3) ) and refore required a substantial number of steps to converge. Combining se initial chains, we chose 10 different realizations which have highest log-likelihoods but also not too close to each or. We n start 10 new independent chains, where each chain is seeded from one of top 200 log-like solutions found from stacked initial chains. We run each of se 10 chains for 10 7 trials with acceptance rates 5% (i.e. 500,000 accepted steps each) and find, as expected, that each chain is burnt-in right from beginning. To check for adequate mixing, we calculated effective length, defined as length of chain divided by correlation length, where correlation length is defined as l corr = min lag { AutoCorrelation(chain, lag) < 0.5} (Tegmark et al. 2004). We find that sum of effective lengths exceeds 2000 (i.e. is 1), indicating good mixing. We also verified that Gelman-Rubin statistic (Gelman & Rubin 1992) dropped below 1.1 (it was 1.02), indicating that chains had converged. Finally, we thinned 10 chains by a factor of 100, and stacked m toger, which gives a combined chain of length of The hyper-parameter posteriors, available at this URL, are shown as a triangle plot in Figure 2.2. We list median and corresponding 68.3% credible interval of each hyper 56

70 parameter posterior distribution in Table 2.2. Our model, evaluated at spatial median of hyper parameters, is shown in Figure 2.3 compared to data upon which it was conditioned. The spatial median simply finds sample from joint posteriors which minimizes Euclidean distance to all or samples Model Comparison The model with four segments was at first selected by visual inspection of data. Two of three transition points, T(1) and T(3), occur at locations which can be associated with physically well-motivated boundaries (planets accreting volatile envelopes, Rogers 2015, and hydrogen burning, Dieterich et al. 2014), whereas T(2) transition is not as well studied. In order to demonstrate that this model is statistically favored over three-segment model, we repeated all of fits for a simpler three-segment model. We seed remaining two transition points from approximate locations of T(1) and T(3) found from four-segment model fit. We label this model as H 3 and four-segment model used earlier as H 4. For this simpler model, H 3 uses only two transition points which break data into three different segments. To implement this model, only difference is that hyper parameters vector Θ hyper becomes Θ hyper = (C, S(1 3), σ (1 3) R, T (1 2) ). (2.14) We find that maximum log-likelihood of H 3 is considerably less than that of H 4, 57

71 Figure 2.2 Triangle plot of hyper-parameter joint posterior distribution (generated using corner.py). Contours denote 0.5, 1.0, 1.5 and 2.0 σ credible intervals. 58

72 Figure 2.3 The mass-radius relation from dwarf planets to late-type stars. Points represent 316 data against which our model is conditioned, with data key in top-left. Although we do not plot error bars, both radius and mass uncertainties are accounted for. The red line shows mean of our probabilistic model and surrounding light and dark gray regions represent associated 68% and 95% credible intervals, respectively. The plotted model corresponds to spatial median of our hyper parameter posterior samples. 59

73 less by (corresponding to χ 2 = 68.3 for 316 data points) at gain of just three fewer free parameters. The marginal likelihood cannot be easily computed in very high dimensional parameter space of our problem, and Bayesian and Akaike information criteria (BIC, Schwarz 1978 & AIC, Akaike 1974) are also both invalid for such high dimensionality. Instead, we used Deviance information criterion (DIC, Spiegelhalter et al. 2002), a hierarchical modeling generalization of AIC, to compare two models. When comparing two models with DIC, smaller value is understood to be preferred model. We find that DIC(H 4 ) = and DIC(H 3 ) = 333.5, indicating a strong preference for model H The Effect of our Data Cuts As discussed earlier in 2.2.2, our data cuts removed 16% of initial data considered. Since se points are low SNR data, y, by definition, have weak effects on likelihood function. As is evident in Figure 2.3, re is an abundance of precise data constraining slope parameters in each segment and none of segments can be described as residing in a poorly constrained region. Given that transition points are defined as intercepts of slopes, y too are well constrained by virtue of construction of our model. Critically, n, a paucity of data at actual transition point locations (as is true for T(1)) has little influence on our inference of ir locations. In order for results of this work to be significantly affected by exclusion of se low SNR data, se points would have to have modified inference of slope parameters. To demonstrate this effect is negligible, we consider Neptunian segment in iso- 60

74 lation, since it strongly affects critical transition T(1) and features largest fraction of excluded points (24%). Since excluded data were due to lossy mass measurements, we ignore radius errors and perform a simple weighted linear least squares regression with and without excluded data, where we approximate observations to be normally distributed. We find that slope parameter, S(2), changes from ± to ± by re-introducing excluded data, illustrating negligible impact of se data Injection/Recovery Tests In order to verify robustness of our algorithm, we created ten fake data sets and blindly ran our algorithm again on each. The data sets are generated by making random, fair draws from our joint posteriors (both local and hyper parameters), ensuring that each draw is from an different effective chain. We n re-ran our original algorithm as before, except that number of steps in final chain is reduced by a factor of ten for computational expedience. We computed one and two-sigma credible intervals on each hyper-parameter and compare m to injected truth in Figure 2.4. As evident from this figure, we are able to easily recover all of inputs to within expected range, validating robustness of main results presented in this work. 61

75 points and points include anpoints independent intrinsic dispersion and include anbegin independent intrinsic dispersion and include anmodel. independent intrinsic dispersion points and include an independent intrinsic dispersio We by running 5Secindependent initial we consequently include in our model. Later, in cha Se we consequently we consequently include we in consequently our include in our include Later, model. in in our SecLater, model. in Later, in Secan cube of (12+316) dimensions. turally accounted unted mption is naturally sponding toaccounted 316 hyper M ), walker jumps in a probability posteriors, availab t for each segment (making our model probabilistic). Also for each segment (making our model probabilistic). Also for each segment (making our model probabilistic). Also for each segment (making our model probabilistic). 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AtTeach rest of parameters, segments. rest segments. rest each of transition At segments. point transition At Teach, point transition, point T (j), point T (j), parameters, as of At rs, as of hyper as power-law. This hierarchical This chicalhierarchical 2.6. Hyper 2.6. Priors Hyper 2.6. Priors Hyper2.6. Priors Hyper Priors bsection. MCMC subsection. gure For pa local paal or pa1. local 2.4 Each shows residuals of(j) a(j) hyper-parameter in our model, (j) (j)(j+1) (j+1) (j) (j+1) (j) as (j) (j)figure (j) (j)(j+1) (j) (j) (j+1) (j) (j+1) (j)(j) (j+1) (j+1) mass, Mt, computed and R, C +S T = C +S T s, and,+s,+s CRtradius, T CR(j) =between C T Csub-panel +S +S = C T T +S = C for T j = +S 1, 2, for 3. T j = 1, 2, for 3. j = 1, 2, for 3. j = 1, 2, t tradius, ten injected truths and corresponding recovered values. The black ving 632 localdenotes variables. ( ables. local variables. (5)& light gray(5) (5) square recovered posterior median and dark bars denote 1now & 2We σare credible intervals. The now green horizontal marks expected for ao sets as, lers parameters related We cano sets now iteratively derive or lated arecan related We iteratively can now We derive iteratively can or derive iteratively baror derive as, o sets zero-point or as, o sets as, perfect recovery. h power-law broken power-law r-law ken he In hyper parameters. In rs. parameters. (j) (j+1) (j)in (j+1) (j) (j)(j+1) (j+1) (j) (j) (j)(j+1) (j) (j) (j+1) (j) (j)(j+1) C (j+1) = C (j) +(S S ) T j = 1, 2, 3. ( C = C C +(S = C S C +(S ) T = C S +(S for ) T j = S 1, 2, for 3. ) T (6) j = 1, 2, for 3. (6) j = 1, 2, for 3. (6) ncludes local paramaram2 local 632 paramof as hyper parameters, as rs, parameters, as Hyper Priors 2.6. Hyper 2.6. Priors Hyper 2.6. Priors Hyper 2.6. Priors MCMC subsection. bsection. 62

76 2.4 Classification Classification with an MR relation A unique aspect of this work was to use freely fitted transitional points in our MR relation. As discussed earlier, se transitional points essentially classify data between distinct categories, where class boundaries occur in mass and are defined using feature of dr/dm. Such classes are evident even from visual inspection of MR data (see Figure 2.3), but our Bayesian inference of a self-consistent probabilistic broken power-law provides statistically rigorous estimates of se class boundaries. In what follows, we discuss implications of inferred locations of class boundaries (T (1), T (2) and T (3) ) Naming Classes Rar than refer to each class as segments 1, 2, 3 and 4, we here define a name for each class to facilitate a more physically intuitive discussion of observed properties. A naming scheme based on physical processes is appealing but ultimately disingenuous since our model is deliberately chosen to be a data-driven inference, free of physical assumptions about mechanics and evolution sculpting se worlds. We consider it more appropriate, n, to name each class based upon a typical and well-known member. For segment 2, Neptune and Uranus are typical members and are of course very similar to one anor in basic properties. We refore consider this class a sub-sample of Neptune-like worlds, or Neptunian worlds more succinctly. Similarly, we identify Jupiter as a typical member of segment 3, unlike Saturn which lies close to a transitional 63

77 point. Accordingly, we define this sub-sample to be representative of Jupiter-like worlds, or Jovian worlds. For hydrogen-burning late-type stars of segment 4, se objects can be already classified by ir spectral types spanning M, K, and late-type G dwarfs. Rar than refer to m as M/K/late-G class stars, we simply label m as stars for sake of this work and for consistency with worlds taxonomy dub m Stellar worlds. Finally, we turn to segment 1 which is comprised largely of Solar System members and thus all of which are relatively well-known. The objects span from dwarf planets to terrestrial planets, from silicate worlds to icy worlds, making naming this broad class quite challenging. Additionally, calling this class Earth-like worlds would be confusing given usual association of this phrase with habitable Earth-analogs. For consistency with naming scheme used thus far, we decided that dubbing se objects as Terran worlds to be most appropriate T (1) : The Terran-Neptunian Worlds Divide From masses of 10 4 M to a couple of Earth masses, we find that a continuous powerlaw of R M 0.279±0.009 provides an excellent description of se Terran worlds. No break is observed between dwarf planets and planets. If Terrans display a constant mean density, we would expect R M 1/3, and so slightly depressed measured index indicates modest compression with increasing mass (ρ M 0.16±0.03 ). Our result is in close agreement with oretical models, which typically predict R M 0.27 for Earth-like compositions (e.g. see Valencia et al. 2006). 64

78 We find first transition to be located at (2.0 ± 0.7) M, defining transition from Terrans to Neptunians. After this point, density trend reverses with ρ M 0.77±0.13, indicating accretion of substantial volatile gas envelopes. This transition is not only evident in power-law index, but also in intrinsic dispersion, which increases by a factor of (3.6 ± 0.9) from Terrans to Neptunians. This transition point is of major interest to community, since it caps possibilities of rocky, habitable Super-Earth planets, with implications for future mission designs (e.g. see Dalcanton et al. 2015). Our result is compatible with independent empirical and oretical estimates of this transition. Starting with former, we compare our result to Rogers (2015), who sought transition in radius rar than mass. This was achieved by identifying radii which exceed that of a solid planet, utilizing a principle first proposed by Kipping et al. (2013). Assuming an Earth-like compositional model, radius threshold was inferred to be R (Rogers 2015). Our result may be converted to a radius by using our derived relation. However, since our model imposes intrinsic radius dispersion (i.e. probabilistic nature of our model), uncertainty in radius is somewhat inflated by this process. Neverless, we may convert our mass posterior samples to fair radii realizations using our Forecaster public code (described later in Section 2.5). Accordingly, we find that transition occurs at R, which is fully compatible with Rogers (2015). A comparison to ory comes from Lopez & Fortney (2014), who scale down compositional models of gaseous planets to investigate minimum size of a H/He rich sub-neptune. From this oretical exercise, authors estimate that 1.5 R is minimum radius of a H/He rich sub-neptune, which is also compatible with our measurement. 65

79 Therefore, despite fact that we do not impose any physical model (unlike Lopez & Fortney 2014 & Rogers 2015), our broken power-law model recovers transition from Terrans to Neptunians T (3) : The Jovian-Stellar Worlds Divide Anor well-understood transition is recovered by our model at (0.080 ± 0.008) M, which we interpret as onset of hydrogen burning. Like Terran-Neptunian worlds transition, we may compare this transition to or estimates of critical boundary. In recent work of Dieterich et al. (2014), authors performed a detailed observational campaign around this boundary. Inspecting T eff -R plane, authors identified a minimum at R, which corresponds to M, with 5 Gyr isochrones 2 of Baraffe et al. (1998). Based on this, we conclude that result is fully compatible with our own prediction. From stellar modeling, estimates of minimum mass for hydrogen-burning range from 0.07 M to 0.09 M (Burrows et al. 1993, 1997; Baraffe et al. 1998; Chabrier et al. 2000; Baraffe et al. 2003; Saumon & Marley 2008). Therefore, both independent observational studies and oretical estimates are consistent with our broken power-law estimate T (2) : The Neptunian-Jovian Worlds Divide We find strong evidence for a transition in our broken power-law at (0.41 ± 0.07) M J, corresponding to transition between Neptunians and Jovians. Whilst this transition 2 Although point slightly precedes first point in Baraffe et al grid, requiring a small linear extrapolation to compute. 66

80 has been treated as an assumed, fixed point in previous works (e.g. Weiss et al adopt a fixed transition at 150 M, or 0.47 M J ), our work appears to be first instance of a data-driven inference of this transition. A plausible physical interpretation of this boundary is that a Neptunian world rapidly grows in radius as more mass is added, depositing more gaseous envelope to its outer layer. Eventually, object s mass is sufficient for gravitational self-compression to start reversing growth, leading into a Jovian world. The existence of such a transition is not unexpected, but our model allows for an actual measurement of its location. We infer significance of this transition to be high at nearly 10 σ (see 2.3.2), motivating us to propose that this transition is physically real and that a class of Jovians is taxonomically rigorous in mass-radius plane. A defining feature of Jovian worlds is that MR power-index is close to zero ( 0.04 ± 0.02), with radius being nearly degenerate with respect to mass. We find that brown dwarfs are absorbed into this class, displaying no obvious transition (also see Figure 2.3) at 13 M J, as was also argued by Hatzes & Rauer (2015). When viewed in terms of mass and radius n, brown dwarfs are merely high-mass members of a continuum of Jovians and more closely resemble planets than stars. The fact that Neptunian-to-Jovian transition occurs at around one Saturn mass is generally incompatible with oretical predictions of a H/He rich planet, such as Saturn. Calculations by Zapolsky & Salpeter (1969) predict that a cold sphere of H/He is expected to reach a maximum size somewhere between 1.2 M J to 3.3 M J. The suite of models produced by Fortney et al. (2007) for H/He rich giant planets, for various insolations and 67

81 metallicities, peak at masses between 2.5 M J to 5.2 M J. Neverless, Jupiter and Saturn have similar radii (within 20%) of one anor despite a factor of three difference in mass, crudely indicating that Jovians commence at a mass less than or equal to that of Saturn. 2.5 Forecasting Forecaster: An Open-Source Package Using our probabilistic model for MR relation inferred in this work, it is possible to now achieve our primary objective: to forecast mass (or radius) of an object given radius (or mass). Crucially, our forecasting model can not only propagate measurement uncertainty on inputs (easily achieved using Monte Carlo draws), but also uncertainty in model itself thanks to probabilistic nature of our model. Thus, even for an input with perfect measurement error (i.e. none), our forecasting model will still return a probability distribution for forecasted quantity, due to (i) our measurement uncertainty in hyper-parameters describing model; and (ii) intrinsic variability seen in nature itself around imposed model. To enable community to make use of this model, we have written a Python package, Forecaster 3 (MIT license), which allows a user to input a mass (or radius) posterior distribution and return a radius (or mass) forecasted distribution. Alternatively, one can simply input a mean and standard deviation of mass (or radius), and package

82 will return a forecasted mean and standard deviation of radius (or mass). This code works for any object with mass in range of [ M, M (0.87 M )], or radius in range of [0.1 R, 100 R (9 R J )]. We present details of how we use MR relation we obtained to forecast one quantity from or below Forecasting Radius Predicting radius given mass is straightforward from our model. If input is mean and standard deviation of mass, Forecaster will first generate a vector of masses, {M (i), i = 1, 2,..., n}, following a normal distribution truncated within mass range. Orwise, code will accept input mass posterior distribution as {M (i), i = 1, 2,..., n}. Forecaster will n randomly choose n realizations of hyper parameters from hyper posteriors derived in this work. A radius will be drawn for each M (i) with each set of hyper parameters Θ (i) hyper, as R (i) N( f (M (i), Θ (i) hyper ), σ(i) R ). (2.15) The output in this case is a vector of radii {R (i), i = 1, 2,..., n}. It is worth pointing out that since our model uses a Gaussian distribution, it is possible that predicted radius for a given mass turns out to be so small that no current physical composition model can explain. However we choose not to truncate prediction with any oretical model and let our code users to choose what s suitable for m. 69

83 2.5.3 Forecasting Mass Mass cannot be directly sampled given {R (i), i = 1, 2,..., n} with our model. To sample mass, Forecaster first creates a grid of masses as {M (j), j = 1, 2,..., m} in whole mass range grid of our model. Similarly, n we randomly chose n sets of hyper parameters from hyper posteriors of our model. For each radius R (i), Forecaster calculates probability {p (j) grid, j = 1, 2,..., m} of R(i) given M (j) with Θ (i) hyper. Finally, Forecaster samples M(i) from {M (j), j = 1, 2,..., m} with {p(j), j = 1, 2,..., m}. The output in this case is a vector of masses grid grid {M (i), i = 1, 2,..., n} Examples: Kepler-186f and Kepler-452b To give illustrative examples of Forecaster, we here forecast masses of arguably two most Earth-like planets discovered by Kepler, Kepler-186f and Kepler-452b. Kepler-186f was discovered by Quintana et al. (2014), reported to be (1.11 ± 0.14) R and receiving % insolation received by Earth. A re-analysis by Torres et al. 4 (2015) refined radius to (1.17 ± 0.08) R and we use radius posterior samples from that work as our input to Forecaster. As shown in Figure 2.5, we predict a mass of M, with 59% of samples lying within Terrans category. Therefore, in agreement with discovery paper of Quintana et al. (2014), we also predict that Kepler-186f is most likely a rocky planet. Kepler-452b was discovered by Jenkins et al. (2015) and was found to have a very similar insolation to that of Earth, differing by a factor of just The reported 0.22 radius of R means that Kepler-452b would be unlikely to be rocky using 70

84 5.1.being Forecaster: An Open-Source Package d inprobability this way, of future objects can be prob object aalso Terran world. this key transition is largeir ( 33%) due to masses paucity divide between Terran and Neptunian worlds is much radii. (j) upon observed R(i), Forecaster calculates and/or probability {pgrid, j = mass/radius ofaspect an object, based upon cally classified too, which is anor feature of The classification of our work, which is essenof objects with > 3 precision masses and radii than canonical 10 M limit. in Recently, Simp-(i) Using our probabilistic model for MR relationlower inferred (i) (j) 1, 2,awork...,Bayesian m}could of Rargument given M hyper. Finally, ster. tially free by-product of our ntity is amost obviously performed using a Earth-mass regime. hopefully im- with (2016) Future made using populain this work, itapproach, is possibleprovides to now some achieve ourson primary (j) The layout of this paper is as follows. In 2, we outl (i) scussed in 1, expected applications may include interesting insights: prove precision 10% by using a larger tion bias to to infer that inhabited, Terran worlds should Forecaster samples Msample, from {M objective: to forecast mass (orto radius) of an object MR) relation. Such relations are known grid, j = 1, 2,..., m} y discovered transiting planet candidate could (j) coming which will inevitably be in years, or by which is used ena our radii model for1.2 relation, have of found R with < 95% confidence. given radius (or mass). Crucially, {prgridto, jmr = 1, 2,..., m}. TheAssuming output in this case is a There is no discernible change in MR rela- our forecasting changes at specific locations, such s massiforecasted in order to estimate de-as extending our method to include lossier measurements an Earth-like core-mass fraction, this limit corresponds model can not only propagate measurement uncertainty (i) forecasting and classification. from Jupiter to brown dwarfs. Brown dwarfs vector of mass {M, i = 1, 2,..., n}. with tion radial velocities. Vice versa, a newly and upper limits by directly original 2.0 M (Zeng re-fitting et al. 2016). This is obseralso compatible mitybrown dwarfs tohydrogen stars on inputsplanets, (easilyburning achieved using Monte Carloto draws), are merely high-mass when classified uspr(solid) = 13% red planet found via radial velocities may be vations (which was beyond scope of this work). In e ectively with our again argues for butand also uncertainty in points model itself thanks to measurement 5.4.and Examples: Kepler-186f and Kepler-452b s & Rauer 2015). These transition ing ir size mass. red for transit follow-up and our code can preany case, se divides are unlikely sharp, with countera paucity of Super-Earths. It may be, n, that Pr(solid) = 59% atprobabilistic nature of our model. Thus, for an detectable all). Therefore, I even to forecast mass/radius of an ob- we An illustrative example of astronomical Forecaster in action, teven of as bounding apresent set of classes of astrodetectability given constraints. Anexamples such as M -mass Neptunian world KOII There is noinput discernible change in MR relation Earth is Super-Earth we have been looking for all with perfect measurement error (i.e. none), our Kepler-186f here forecast masses of two xample might be classes to to forecast scale height of using need accurately foreject based upon a measurement ofarguably or, andmost (Kipping et al. 2015). from dwarf planets to Earth. Dwarf planets along. s,critical where are categorized forecasting model will still return a314c probability distriearth-like planets discovered by Kepler, Kepler-186f and planet by TESS for atmospheric followup Kepler-452b A to wholly independent are merely low-mass when classified using = line 59%of thinking can also be of newfound planet discoveries just bution forplanets, from forecasted quantity, due (i) ourpr(terran) meafwst, inferred MR relation. In this case Kepler-452b. where Forecaster would also calculate shownitofor support provocative hyposis that ir size and mass. Pr(Terran) = 13% classification of astronomical bodies based surement uncertainty in hyper-parameters describore radius. discovered lity oflikely) object being a Terran world. parent that inference of MR relation divide between Terran and Kepler-186f Neptunian was worlds is muchby Quintana et al. (2014), ing model; and (ii) intrinsic variability seen ir in 2. MODEL upon observed and/or reported to masses berecently, a (1.11 ±Simp0.14) R radii. and receiving % classification of aspect of our work, which upon is essenlower than canonical 10 M limit. ss/radius an object, based lassification and forecasting. nature herself around imposed model. insolation received by Earth. A re-analysis by free by-product of our approach, provides some son (2016) made a Bayesian argument using popula2.1. Choosing a radius Model obviously performed using a to makeit To enable community use of this, we have nmost isinsights: more than a taxonomical enterprise, Torres et al. (2015) refined In (1.17±0.08) ing tionthe bias to infer that inhabited, Terran worlds should layout of this paper is as follows. 2,towe outliner written aare pieceknown of Python code, Forecaster, which alation. Such relations to and we use radius posterior samples from that work have radii of R < 1.2 R toby 95% confidence. Assuming matic in astronomy. Perhaps We begin describing rationale behind mo here isimplications no discernible MR relalowschange a user in to input a mass (or radius) posterior and our model for MR relation, which is used enable as our input to Forecaster. As shown in Figure 4, we an Earth-like core-mass fraction, this limit corresponds sus at specific locations, such as example classification astronomy on from Jupiter of to brown dwarfs usedalin this work. As discussed in 1 (and demonstar return dwarfs. a radiusbrown (orinmass) forecastedforecasting distribution. predict a mass of 1.74 to 2.0 M (Zengand et al.classification. 2016). This is also compatible 0.60 M, with 59% of samples redwarfs merely high-mass planets, when classified usnrung-russell to hydrogen burning stars ternatively, one can simply input a mean and standard (HR) diagram (Hertzsprung later=in13% 3), two primary goalsclassification. of this paper are b lying within for rocky worlds Therefore, with our measurement and again argues e ectively g ir size and mass. deviation of mass (or radius), and Pr(solid) package will reuer 2015). These transition points in agreement with discover paper of Quintana et al. t a paucity of Super-Earths. It may be, n, that 1914) for luminosity versus e ective tem- Pr(solid) achievable through use of a MR relation and turn an forecasted mean and standard deviation of = 59% (2014), we also predict that Kepler-186f is most likely a here is no discernible change in of MRastrorelation Earth is Super-Earth we have been looking for all bounding a setradius of classes h revealed distinct regimes of stellar (or mass), This code works for any object with rocky defines approach in this work. Broadly speaki world. Kepler-186f om dwarf planets to Earth. Dwarf planets 4 along. mass range ofusing [3using 10 M, Pr(Terran) such M =(0.87 e merely classes are categorized common concern inin when classification is that Kepler-452b discovered by Jenkins et al. (2015) (e a relation can bewas cast as eir a parametric re low-mass planets, classified 59% Kepler-452b M )], or [0.1 Rthis, 100case R (9 RJ )]. and was found to have a very similar insolation toathat nferred MR relation. In heir size and mass. Pr(Terran) = 13% number of possible which a polynomial) or non-parametric model (e.g. near We features present against details of how we use MR relaof Earth, di ering by a factor if just The that inference of MR relation we obtained to foreast one quantity or roblem can betion overwhelming. Mass and from neighbour algorithm). 2. radius MODEL reported of R means that Kepler-452b below. ation and forecasting. would be unlikely to be a rocky world using result h, are not random and arbitrary choices Parameteric models, in particular power-laws, hav 2.1. Choosing Model of (Rogers 2015). ausing this radius with Forecaster re than a taxonomical enterprise, it ch a problem. Rar, y are tworadius of long been popular for modeling MR relation w 5.2. Forecasting predicts that M = M, with only 13% of samples 1.5 mplications in astronomy. Perhaps We begin by describing rationale behind model amental quantities describing anymass object many evenin recent literature (e.g. Predicting radius given is straightforward fromexamples lying within rocky world classification (see Figure 4). V mple of classification in Ifastronomy inlencia thisdework. As2006; discussed inet 1 (and demonstarted our model. input is base meanused and standard Therefore, in contrast to al. discover paper of Jenkins and indeed represent two of seven et al. Weiss 2013; Hatzes & Ra viation of mass, Forecaster will first generate a vecet al. (2015), we predict that Kepler-452b is unlikely ussell (HR) diagram (Hertzsprung later in 3), two primary goals of this paper are both he International System of Units (SI). 2015; Wolfgang et al. 2015; Zeng et al. 2016). to In tor of masses, {M (i), i = 1, 2,..., n}, following a normal be a rocky world. or luminosity versus e ective temachievable through use of a MR relation and this Figure 2.5 Posterior distributions of range. radius (measured) andsuch massmodels (forecasted) of twostraightforw of classification extends beyond guiding case, we note that are more distribution truncated within mass Orhabitable-zone small planets, with predictions produced by Forecaster (triangle plots 6. DISCUSSION aled distinct regimes of hierarchical approach inbayesian this work. Broadly speaking, wise, code willstellar accept ofinput mass posterior as standing, it even a ects design fu-defines for modeling (which we argue (i) generated using corner.py). {M, i = 1, 2,..., n}. Forecaster will n randomly In this work,as we eir have developed a new package, called n concern in classification is that such a relation can be cast a parametric ntation. As an example, boundary later), since y allow for a (e.g. simple p chose n realizations of hyper parameters be fromnecessary Forecaster, to predict mass (or radius) of an object of possible features against which a polynomial) or non-parametric model (e.g. a nearest trial-like rockyhyper worlds andderived Neptune-like scription ofbased upon Bayesian network. Moreover, posteriors in this work. A radius will be radius (or mass). Our code uses abased new (i) can be overwhelming. Mass and drawn for each M with each set of hyper parameters neighbour algorithm). probabilistic mass-radius relation which has been condirepresents adefinition truncation of from largest althose earlier cited works, power-laws ostensbily do (i) analysis of Rogers (2015). Using reported radius with hyper. resulting tioned upon masses of power-laws, radii of 316 objects spanning not random and arbitrary choices Parameteric models, in particular have a o le Earth-like body. The location of this excellent job of describing data and ability dwarf planets to late-type stars. Aside from enabling +2.9 roblem. Rar, y predicts are twothat of, (i) (i) M long been popular13% for of modeling MR relation with (i) Forecaster = 3.9, with samples within non-parametric ngly a ects estimates of method tolying explain complex patterns R N (foccurrence (M (i)m (15) only forecasting, this excercise naturally performs classificar ) hyper ), 1.5 l quantities anyin-turn object de-many examples even recent literature Vation ofin observed population, since we(e.g. fit forby tran- la ike planets ( describing )The and thus pears superfluous (a conclusion reenforced (i) output in thisfigure case is a2.5). vectortherefore, of radius {Rin, contrast i= Terran worlds (see to discovery paper of Jenkins sitional points. Since observed population has been eed of seven base lencia et al. 2006; Weiss al. 3). 2013; Hatzes & we Rauer nts represent of future two missions results of this work,etsee Accordingly, adopt 1, 2,..., n}. needed to characclassified in this way, future objects can also be probrnational System ofal.units (SI). 2015; Wolfgang etprescription 2015; Zeng et work. al. 2016). In our nets (Dalcanton 2015). To illustrate in this et al. et (2015), we predict that Kepler-452b ispower-law unlikely to beal. a rocky planet. abilistically classified too, which is anor feature of 5.3. Forecasting Mass ification extends beyond guiding that Forecaster. such models straightforward occurrence rate posteriors of Foreman-case, we (i)note As noted earlier, are usemore of power-laws to descr Mass cannot be directly sampled given {R, i = As discussed in 1, expected applications may include ng, it even a ects design of fufor hierarchical Bayesian modeling (which we argue to (2014), increases when MR relation is common in literature. Howev 1, 2,..., n}by with72% our model. To altersample mass, Forecaster a newly discovered transiting planet candidate could (j). As an example, boundary be necessary later), since y allow ion of Earth-analogs from R of<mass 1.5 R many of have assumptions and model inprese first creates a grid as {Mto its mass forecasted in for ordera details tosimple estimate de- p grid, j = 1, 2,..., m} ke worlds andwhole Neptune-like scription ofn Bayesian Moreover, in rangeexoearth of our model. Similarly, tectability network. with would radial velocities. Vice based versa, based a on newlyup In rocky order to maintain mass same vious implementations make forecasts ents a truncation of this largest al- corre-those earlier cited works, power-lawsweostensbily do ankey oposed HDST mission, change se relations problematic. identify three h-like body. Therequired location of this job of of describing data and work ability of di er a % increase in mirror diameterexcellent pects model proposed in this with ects of Dalcanton occurrence to explain complex uationestimates in of et al. 2015).non-parametric tiate our method work from previous studies.patterns apnets ( )that and both thus in-turn de(a conclusion reenforced by later e argue forecasting and clas-pears superfluous [1] Largest data range: Inferences of MR re 71 future missions needed to characresults of this work, see 3). Accordingly, we adopt g masses and radii of astronomical tion often censor available data to a specific subse alcanton et al. 2015). To illustrate prescription this work. Wolfgang et al c t very least, be of great utility forpower-law parameter space in (for example rence rateand posteriors of provide Foremanof power-laws to describe missions may also meaning- As noted sider earlier, R < 4 Ruseexoplanets). Whilst it is inevita

85 2.6 Discussion In this work, we have developed a new package, called Forecaster, to predict mass (or radius) of an object based upon radius (or mass). Our code uses a new probabilistic mass-radius relation which has been conditioned upon masses and radii of 316 objects spanning from dwarf planets to late-type stars. Aside from enabling forecasting, this exercise naturally performs classification of observed population, since we fit for transitional points. Since observed population has been classified in this way, future objects can also be probabilistically classified too, which is anor feature of Forecaster. As discussed in Section 2.1, expected applications may include a newly discovered transiting planet candidate could have its mass forecasted in order to estimate detectability with radial velocities. Vice versa, a newly discovered planet found via radial velocities may be considered for transit follow-up and our code can predict detectability given present constraints. Anor example might be to forecast scale height of a small planet found by TESS for atmospheric follow-up with JWST, where Forecaster would also calculate probability of object being a Terran world. The classification aspect of our work, which is essentially a free by-product of our approach, provides some interesting insights: There is no discernible change in MR relation from Jupiter to brown dwarfs. Brown dwarfs are merely high-mass planets, when classified using ir size and mass. There is no discernible change in MR relation from dwarf planets to Earth. 72

86 Dwarf planets are merely low-mass planets, when classified using ir size and mass. The transition from Neptunians to Jovians occurs at M J, meaning that Saturn is close to being largest occurring Neptunian world. This is first empirical inference of this divide. The transition from Terrans to Jovians occurs at M, meaning that Earth is close to being largest occurring solid world. Rocky Super-Earths, n, can be argued to be a fictional category. This latter point may seem remarkable given that Super-Earths have become part of astronomical lexicon. The large number of 2-10 M planets discovered is often cited as evidence that Super-Earths are very common and thus Solar System s makeup is unusual (Haghighipour 2013). However, if boundary between Terran and Neptunian worlds is shifted down to 2 M, Solar System is no longer unusual. Indeed, by our definition, three of eight Solar System planets are Neptunian worlds, which are most common type of planet around or Sun-like stars (Foreman-Mackey et al. 2014). As shown earlier, whilst our value is lower than previous estimates, it is fully compatible with previous estimates from both ory (e.g. see Lopez & Fortney 2014) and independent population studies (e.g. see Rogers 2015). The uncertainty on our inference of this key transition is large ( 33%) due to paucity of objects with > 3 σ precision masses and radii in Earth-mass regime. Future works could hopefully improve precision to 10% by using a larger sample size, which will inevitably be found in coming years, or by extending our method to include lossier measurements and upper 73

87 limits by directly re-fitting original observations (which was beyond scope of this work). In any case, se divides are unlikely sharp, with counter-examples such as M -mass Neptunian world KOI-314c (Kipping et al. 2015). A wholly independent line of thinking can also be shown to support provocative hyposis that divide between Terran and Neptunian worlds is much lower than canonical 10 M limit. Recently, Simpson (2016) made a Bayesian argument using population bias to infer that inhabited, Terran worlds should have radii of R < 1.2 R to 95% confidence. Assuming an Earth-like core-mass fraction, this limit corresponds to 2.0 M (Zeng et al. 2016). This is also compatible with our determination and again argues for effectively a paucity of Super-Earths. It may be, n, that Earth is Super-Earth we have been looking for all along. 74

88 Chapter 3 Application of Forecaster In previous chapter, I used hierarchical Bayesian method to fit masses and radii of exoplanets and built a predictive model called Forecaster. In this chapter, I will show two cases of application of Forecaster. From Section 3.1 to Section 3.4, I use Forecaster to predict masses of 7000 exoplanet candidates which have ir radii measured with transit survey Kepler. From Section 3.5 to Section 3.7, I briefly discuss vice versa application, predicting radii from measured masses. 3.1 Forecasted masses for 7000 Kepler Objects of Interest Recent transit surveys have discovered thousands of planetary candidates with directly measured radii, but only a small fraction have measured masses. Planetary mass is 0 This chapter is a reproduction of two papers published by The Monthly Notices of Road Astronomical Society, which can be found at and The articles have been reformatted for this section. 75

89 crucial in assessing feasibility of numerous observational signatures, such as radial velocities (RVs), atmospheres, moons and rings. In absence of a direct measurement, a data-driven, probabilistic forecast enables observational planning. So here we compute posterior distributions for forecasted masses of approximately seven thousand Kepler Objects of Interest (KOIs). We find that mass forecasts are unlikely to improve through more precise planetary radii, with error budget presently dominated by intrinsic model uncertainty. Our forecasts identify a couple of dozens KOIs near Terran-Neptunian divide with particularly large RV semi-amplitudes which could be promising targets to follow up, especially in near-ir. With several more transit surveys planned in near-future, need to quickly forecast observational signatures is likely to grow and work here provides a template example of such calculations. 3.2 Forecasting KOI Masses Data Requirements The principal objective of this work is to compute self-consistent and homogeneous a- posteriori distributions for predicted mass of each KOI. The predicted mass of each KOI will be solely determined by its radius and empirical, probabilistic forecasting model of Chen & Kipping (2017). Because of this conditional relationship, masses and radii will certainly be covariant, along with any or derived terms based on se quantities. We refore aim to derive joint posteriors for all parameters of interest, which 76

90 will encode any resulting covariances. To accomplish this goal, we first require posterior distributions for KOI radii. The transit light curve of each KOI enables a measurement of planet-to-star radius ratio, p, with precision depending upon photometric quality, number and duration of observed transit events and modest degeneracies with or covariant terms describing transit shape (Carter et al. 2008). With quantity p in-hand, it may be combined with stellar radius, R, to infer true planetary size, R P. Therefore, to make progress we need homogenous posterior probability distributions for 1) basic transit parameters of each KOI 2) fundamental stellar properties of each KOI Transit Posteriors Basic transit parameters of each KOI are provided in NASA Exoplanet Archive (NEA, Akeson et al. 2013), but se are summary statistics rar than full posterior distributions, as required for this work. Fortunately, Rowe & Thompson (2015) provide posterior distributions for almost every KOI, with 100,000 samples for each obtained via a Markov Chain Monte Carlo (MCMC) regression of a Mandel & Agol (2002) light curve model to Kepler photometric time series (we direct reader to Rowe & Thompson 2015 for details). We downloaded all available posteriors and found 7106 object files. Except for two KOIs, all of se objects appear in currently listed NEA database (comprising of 9564 KOIs), with exceptions being KOI and KOI We deleted se two objects from our sample in what follows, giving us 7104 KOIs. The or 2460 KOIs were not modeled by Rowe & Thompson (2015) and thus are not considered 77

91 furr in what follows Stellar Posteriors For stellar properties, we again note that summary statistics are available on NEA but posteriors are not directly available. As one of our criteria is a homogenous set of posteriors, and we wish to calculate masses for as many KOIs as possible, inferences for particular subsets of KOI database are not as useful for this work. Instead, we used publicly available posteriors from Mathur et al. (2017) who used information such as colors, spectroscopy, and asteroseismology to fit Dartmouth isochrone models (Dotter et al. 2008) for each Kepler star, giving 40,000 posterior samples per star. We attempted to download stellar posteriors for all 7104 KOIs in Rowe & Thompson (2015) database, but found that posteriors were missing for 93 KOIs spanning 87 stars. These KOIs are flagged with a 1 in Table 3.1 and were not considered furr for analysis KOI Radii Posteriors We next combined se distributions toger to generate fair realizations of KOI radii. We do this by consecutively stepping through each row of Mathur et al. (2017) samples and drawing a random row from corresponding Rowe & Thompson (2015) posterior samples for p. This is possible because two posteriors are completely independent and share no covariance. This process results in 40,000 fair realizations for radius of each KOI, where we 78

92 Table 3.1 Data flags assigned to 7104 KOIs considered in this analysis. 0 denotes no problems, 1 denotes that stellar posteriors were missing and 2 denotes transit posteriors were missing. Only a portion of table is shown here, full version is available on GitHub repository chenjj2/forecasts. KOI data flag report radii in units of Earth radii (R = km), which are made available in public repository at this URL ( During this process, we found 38 KOIs that could not locate corresponding MCMC file for Rowe & Thompson (2015) transit parameters. It is unclear why se were missing but given ir relativity small number, we simply flagged m with a 2 in Table 3.1 and did not consider m furr for analysis. At this point, we are left with 6973 KOIs for which we have been able to derive radius posteriors, of which 2283 are dispositioned as CONFIRMED on NEA, 1665 are CANDIDATE, and 3025 are FALSE POSITIVE (se dispositions are also provided in Table 3.2). 79

93 3.2.5 Predicting Masses with Forecaster The next step is to take radius posterior of each KOI and, row-by-row, predict a corresponding mass with Forecaster. We direct reader to Chen & Kipping (2017) for a full description of Forecaster. We also direct reader to Wolfgang et al. (2016) for ir prior and alternative discussion of probabilistic mass-radius relations. Forecaster is fundamentally probabilistic, by which we mean that model includes intrinsic dispersion in mass-radius relation to account for additional variance beyond measurement uncertainties. This dispersion represents variance observed in nature itself. Anor characteristic of Forecaster is that it was only trained within a specific (albeit broad) mass range, from M to M, corresponding to dwarf planets to late-type stars. However, some extreme KOIs have radii which fall outside of expected corresponding radius range. To enable a homogeneous analysis, we simply extend first and last part of broken power-law relation, so that it could cover a semi-infinite interval. In practice, this is only necessary for very large KOIs exceeding a Solar radius, for which KOI cannot be a planet in any case, and thus se extrapolations simply highlight unphysical nature of se rare cases. As discussed earlier, model was applied to each radius posterior sample for each KOI ( million individual forecasts). It is important to stress that probabilistic nature of Forecaster means that re-running same script again on same KOI would lead to a slightly different set of posterior samples for planetary mass, although y would still of course be fair and representative samples. 80

94 The 68.3% credible intervals of forecasted masses are depicted in upper panel of Figure 3.1 and listed along with 95.5% credible intervals in Table 3.2. To perform analysis, we used function Rpost2M in Forecaster. To be clearer, a summary of steps taken in function is listed as below. 1. generates a grid of mass in allowed mass range; 2. calculates probabilities of measured radius given each mass in grid; 3. uses above probabilities as weights to redraw mass. For more details, we direct reader to original Forecaster paper (Chen & Kipping 2017). 81

95 A. Mass-Radius Degeneracy mass [M ] B. Homoscedastic 10-2 RV semi-amplitude, lim K [m/s] e Neptunian Stellar radius [R ] 10 2 Terran Jovian C. Neptunian Flattening radius [R ] Figure 3.1 Forecasted masses (top) and radial velocity semi-amplitudes (bottom) for a circular orbit, as a function of observed KOI radius. We here only show objects with a NEA disposition of being eir a candidate or validated planet. Error bars depict 68.3% credible intervals. 82

96 Table 3.2 Final predicted properties for 6973 KOIs using Forecaster. Quoted values are [ 2, 1, 0, +1, +2] σ credible intervals. First letter of flag column gives modal planet classification ( T =Terran, N =Neptunian, J =Jovian & S =Stellar). Second letter of flag column denotes NEA KOI disposition where f is a falsepositive, c is a candidate and v is validated/confirmed. Only a portion of table is shown here, full version is available on GitHub repository chenjj2/forecasts. KOI flag log 10 (RP [R ]) log 10 (MP [M ]) log 10 (K [m s 1 ]) Jv [1.08,1.10,1.11,1.13,1.15] [1.83,2.33,3.20,4.14,4.53] [1.51,2.02,2.88,3.82,4.19] Jv [1.16,1.18,1.20,1.22,1.24] [1.86,2.23,2.95,4.51,4.67] [1.44,1.82,2.54,4.08,4.22] Nv [0.66,0.67,0.69,0.70,0.71] [0.81,1.07,1.32,1.58,1.83] [0.45,0.71,0.96,1.21,1.46] Jc [0.85,0.92,1.00,1.08,1.17] [1.34,1.72,3.04,4.13,4.46] [0.90,1.27,2.55,3.64,3.95] Nc [0.83,0.85,0.87,0.89,0.91] [1.13,1.40,1.67,2.00,4.28] [0.64,0.91,1.18,1.51,3.77] Sf [1.21,1.38,1.60,1.91,2.10] [2.10,4.77,5.05,5.39,5.48] [1.83,4.46,4.69,4.96,5.03] Nv [0.58,0.60,0.62,0.65,0.67] [0.70,0.96,1.22,1.47,1.73] [0.31,0.57,0.82,1.07,1.32] Nf [0.19,0.22,0.26,0.32,0.38] [0.26,0.44,0.66,0.90,1.17] [0.06,0.24,0.45,0.70,0.97] Jf [0.99,1.06,1.15,1.27,1.37] [1.77,2.26,3.27,4.56,4.80] [1.32,1.80,2.83,4.08,4.24] Jv [1.11,1.14,1.17,1.20,1.23] [1.85,2.27,3.03,4.23,4.63] [1.43,1.85,2.61,3.81,4.17] Sf [0.77,0.93,1.18,1.51,1.76] [1.32,1.86,3.78,4.95,5.24] [0.97,1.51,3.43,4.52,4.75] Jv [1.09,1.13,1.17,1.21,1.27] [1.86,2.27,3.04,4.31,4.67] [1.14,1.56,2.34,3.59,3.91] Jv [1.05,1.12,1.21,1.32,1.50] [1.87,2.29,3.35,4.71,4.95] [1.42,1.85,2.93,4.21,4.36] Nf [0.55,0.62,0.69,0.80,0.90] [0.77,1.05,1.35,1.65,2.04] [0.29,0.57,0.85,1.14,1.52] Sf [1.45,1.70,2.02,2.28,2.46] [2.01,4.96,5.40,5.48,5.48] [1.42,4.40,4.75,4.83,4.86] Sf [0.79,1.01,1.64,1.94,2.13] [1.36,2.49,5.09,5.42,5.48] [1.06,2.15,4.70,4.97,5.02] Jv [1.08,1.10,1.13,1.15,1.19] [1.82,2.32,3.14,4.14,4.56] [1.44,1.93,2.76,3.75,4.14] Jv [1.12,1.14,1.18,1.21,1.24] [1.86,2.27,3.01,4.29,4.64] [1.41,1.81,2.56,3.83,4.15] Sf [0.96,1.12,1.32,1.45,1.56] [1.77,2.39,4.67,4.88,5.03] [1.55,2.16,4.40,4.59,4.70] Jv [1.11,1.16,1.23,1.30,1.38] [1.86,2.25,3.25,4.68,4.82] [1.43,1.82,2.82,4.20,4.31]

97 3.3 Implication & Limitations Densities, Gravities and RV Amplitudes With a joint posterior for both stellar and planetary fundamental parameters in hand, we can compute or derived quantities of interest, such as surface gravity, bulk density, and radial velocity (RV) amplitude (se are given in Table 3.2). Once again, we stress that se values should be treated as forecasts and not measurements. We have computed forecasted masses for all KOIs where available, irrespective of wher KOI is a known false-positive or confirmed planet. False-positives frequently have extreme radii associated with m, giving rise to anomalous derived parameters. For example, in case of KOI , a known false-positive, we obtain a radius of R P = R 500 (for comparison NEA report R P = R ), implying that Rowe & Thompson (2015) MCMC chains ultimately diverged to a very high p. We caution that this effect also appears for some KOIs which are not dispositioned as false-positives. For example in case of KOI , which is listed as a candidate on NEA, we obtain R P = R (NEA also report an anomalously large radius of R P = R ). Despite this, se cases are straight-forward to identify and essentially represent cases where MCMC diverged, indicating poor quality light curves or false-positives. The density forecasts are computed assuming spherical shapes for planets and are listed in Table 3.2. Density forecasts are particularly useful in cases where one wishes to predict Roche radii around KOIs for estimating potential ring radii (Zuluaga et al. 2015). 5 84

98 The surface gravity forecasts are also made assuming spherical planets. These estimates are particularly important for climate and atmospheric modeling (Heng & Vogt 2011), as well as predicting scale height of exoplanetary atmospheres (Seager & Sasselov 2000) when planning observational campaigns. Finally, radial velocity amplitudes are computed assuming i π/2, which is appropriate for transiting planets, and additionally under explicit assumption of a circular orbit, such that lim K = e 0 ( 2πG P )1/3 M P. (3.1) (M + M P ) 2/3 We do not assume M P M and ensure that uncertainties in stellar mass are propagated correctly into calculation of K, although we assume negligible uncertainty in orbital period, P. Circularity is assumed to provide a tight fiducial value rar than marginalizing over poorly constrained eccentricity distributions for se worlds. These predictions should help observers decide which KOIs have potentially detectable signatures from ground. We highlight that our predicted masses are calculated homogeneously for all KOIs, regardless to wher system is known to exhibit strong transit timing variations (TTV) or not. This is relevant since planetary masses detected through TTV method, rar than RV method, are subject to distinct selection effects leading to possibility of ostensibly distinct mass distributions Steffen (2016). Furr, original Chen & Kipping (2017) calibration of Forecaster is dominated by planets with masses measured through 85

99 RVs. As this issue continues to be investigated, it will be interesting to test if subsequent TTV detected masses are offset from predictions in this work Observed Patterns From studying results, we highlight three noticeable patterns, which are shown in Figure 3.1. Observation A: masses (and indeed RV predictions) follow a relatively tight relation up to 10 R after which point uncertainties greatly increase. Naively, one might assume that uncertainties should be smaller here, since larger planets give rise to deeper transits and thus we acquire a higher relative signal-to-noise. However, at around a Jupiter-radius, degeneracy pressure leads to an almost flat mass-radius relation all way until deuterium burning starts in stars, which occurs at M = M (Chen & Kipping 2017). As a result of this degeneracy, mass predictions end up spanning almost entire Jovian range, leading to much larger credible intervals. Observation B: we note that radius-mass diagram in Figure 3.1 shows that credible interval of Neptunians and even many of Terrans have approximately same width (in logarithmic space), i.e. logarithmic variance is homoscedastic. Ultimately, se uncertainties are a combination of measurement error in observed radii and intrinsic dispersion in mass-radius relation. Accordingly, fact that uncertainties appear approximately homoscedastic indicates that y are dominated by intrinsic dispersion term, rar than measurement errors on R. This, in turn, implies that it will not be possible to forecast noticeably reduced credible intervals in future by using more precise planetary radii (for example by using more precise 86

100 stellar radii e.g. Johnson et al. 2017). Therefore, only way to forecast improved credible intervals would be to train a revised Forecaster model, perhaps using additional variables treated as latent in current version (e.g. insolation). Until that happens though, forecasted masses for R 10 R presented here can be treated as most precise possible regardless to observational errors. Observation C: we observe a flattening in gradient of predicted RV amplitude as a function of planetary radius for Neptunian planets (see Figure 3.1). This is somewhat surprising because RV amplitude is proportional to planetary mass, yet radius-mass diagram shows a steeper dependency in Neptunian range. The implication from this is that large Neptunians detected by Kepler are predicted to have almost same RV amplitudes as small, detected Neptunians. Given that M P M in this regime, and that we ve assumed circular orbits, re are only two ways to explain this: i) larger Neptunians tend to have longer orbital periods, ii) larger Neptunians tend to have higher mass host stars (or a combination of two effects). Both effects are also detection biases for Kepler (Kipping & Sandford 2016), since long-period planets transit less frequently and high-mass stars tend to be larger, giving rise to smaller transit depths. To investigate which effect dominates, we plot orbital periods and host star masses as a function of planetary radii for Neptunian planets in Figure 3.2. Figure 3.2 reveals that re is indeed an apparent trend between planetary radius and orbital period for Kepler Neptunians, which is most easily explained as being due to detection biases. We find no such trend for stellar mass objects, although Kepler had a strong selection bias towards Sun-like stars and thus sample is limited for low-mass 87

101 planetary radius, R P [R ] period [d] planetary radius, R P [R ] parent star mass [M ] Figure 3.2 Orbital periods (top) and host star masses (bottom) as a function of planetary radius for 2936 KOIs with modal class probability of being Neptunian and not dispositioned as a false-positive on NEA. The trend between period and radius is most easily explained as being due to detection bias, which in turn gives rise to RV plateau seen for Neptunians in Figure 3.1. Median (black line) and 68.3% credible interval (purple) shown using a top-hat smoothing kernel of 0.14 dex. 88

102 stars Promising Small Planets for RV follow-up We here demonstrate perhaps most useful application of this work, identifying promising small planets for RV follow-up. In this work, we have applied Forecaster to Kepler planets but this should be treated as a demonstration of what could be done with upcoming TESS survey too (Ricker et al. 2015). In particular, narrow-but-deep nature of Kepler combined with fact that RV amplitudes are enhanced for low-mass stars means that most favorable targets (in terms of absolute RV amplitude) will be around faint stars. The all-sky nature of TESS will lead to brighter targets and here Forecaster can clearly play a direct role in identifying promising follow-up targets. We list 28 promising targets in Table 3.3, where we have down-selected on KOIs not dispositioned as false-positives on NEA, with radii between one half to four times that of Earth and with unusually high RV amplitude predictions. For latter criterium, we specifically used ˆK K med > 1.64( ˆK K lower ), where ˆK is median forecast for K, K lower is 15.9% lower quantile of forecasted K distribution and K med is running median of median forecasts of K with same window size as used before (note that 1.64 σ = 90%). These KOIs are visualized versus ensemble of small KOIs in Figure 3.3. As can be seen from Table 3.3, this sample of 28 KOIs all have Kepler bandpass magnitudes K P > 14. The 1.35 R planetary candidate KOI is brightest at K P = 14.1, for which we predict a 2 m/s amplitude. This sample of targets also highlights great potential of near-infrared spectrographs (e.g. Cersullo et al. 2017), where apparent 89

103 Table 3.3 List of planetary candidates with maximum likelihood radii between 0.5 to 4.0 Earth radii and unusually high forecasted RV semi-amplitudes. KOI R P [R ] K [m/s] M [M ] P P [days] K P

104 radial velocity semi-amplitude [m/s] planetary radius, R P [R ] Figure 3.3 Forecasted radial velocity amplitudes for small planets. The colored region contains 68.3% credible interval of maximum likelihood forecasted K values from a moving window. Points plotted use 68.3% uncertainties and black points are those more than 1.64 σ (90%) above moving median (black). 91

105 magnitude is significantly brighter. Despite ir faintness, very short-period nature of se objects (as evident from Table 3.3) enables repeated monitoring and potentially easier disentanglement of spurious signals originating from stars (Pepe et al. 2013; Howard et al. 2013). Thus, despite challenges, se targets may be worth pursuing for Kepler and indeed one can expect this approach to yield far more suitable targets in TESS-era. 3.4 Discussion In this work, we have used a data-driven and probabilistic mass-radius relation to forecast masses of approximately seven thousands KOIs with careful attention to correctly propagate parameter covariances and uncertainties (see Wolfgang et al and Chen & Kipping 2017). We report 1 and 2 σ credible intervals for each KOI in Table 2, including derived parameters such as density, surface gravity and radial velocity semi-amplitude. Full joint posterior samples are made publicly available at this URL( We stress that se results should be treated as informed, probabilistic, and model-conditioned predictions, but not measurements. Forecaster has already seen numerous applications (e.g. see Foreman-Mackey et al. 2016; Obermeier et al. 2016; Dressing et al. 2017; Rodriguez et al. 2017) to individual systems or small subsets, but we believe this work to be first application to a large (several hundred) ensemble of planets ensuring homogeneous methodology. This work serves as a demonstration of ensemble predictions for planetary masses based on transitderived radii. In near future, one can expect number of planetary candidates 92

106 discovered through transits to continue to rapidly grow, exacerbating demands on follow-up resources, for which planetary mass is frequently a key driver for various observational signatures (e.g. radial velocities, atmospheric scale heights, ring/moon stabilities). For example, TESS is expected to discover 5, 000 transiting planets (Bouma et al. 2017), LSST a furr O[10 3 ]-O[10 4 ] (Lund et al. 2015) and O[10 4 ] from PLATO (Rauer et al. 2014). In choosing subset of targets for follow-up, forecasted masses will be beneficial, and thus analysis provided in this work can be considered as a template for se future surveys too. As an example, we have shown how this approach can quickly identify small KOIs for which radial velocity amplitudes are unusually high, increasing chances for detections (see Section 3.3.3). The combination of forecasted mass, low parent star mass, and short orbital period maximizes K. However, deep-sky nature of Kepler and bias towards low-mass stars mean that se targets are faint - a problem likely to be resolved with all-sky surveys of TESS and PLATO. Of course, realistic target selections will depend upon more than just signal amplitude, likely factoring in expected noise, planet properties, and or program-specific questions of interest (e.g. see Kipping & Lam 2017 for an example of target selections based on forecasted multiplicity). Future improvements to forecasts made in this work are unlikely to result from simply obtaining more precise planetary radii, since predicted mass uncertainties are observed to be dominated by intrinsic dispersion inherent to model itself (see Section 3.3.2). Future work could attempt to build an updated probabilistic model using more data and likely a second dependent variable, such as insolation. Until that time, 93

107 forecasts made here are likely to be most credible estimates available without direct observations. 3.5 The detectability of known radial velocity planets with upcoming CHEOPS mission The Characterizing Exoplanets Satellite (CHEOPS) mission is planned for launch next year with a major objective being to search for transits of known RV planets, particularly those orbiting bright stars. Using an empirically calibrated probabilistic mass-radius relation, Forecaster, we predict a catalog of 468 planets discovered via radial velocities. Our work aims to assist CHEOPS team in scheduling efforts and highlights great value of quantifiable, statistically robust estimates for upcoming exoplanetary missions. The purpose of this and following sections are to exemplify application of Forecaster in predicting radii of planets. As code execution was performed by Joo Sung Yi, a high school student who I co-advised during summer of 2017, I won t go into too many details of methodology and results. 3.6 Forecasting Radii from Masses Probabilistic predictions with Forecaster A basic requirement of this work is to predict transit depth, and thus radius, of each of few hundred exoplanets discovered through radial velocity method. A major 94

108 challenge facing any attempt to convert masses to radii, or vice versa, is that massradius relation of exoplanets displays intrinsic spread. This spread, likely due to intrinsic variations in composition, chemistry, environment, age, and formation mechanism, means that simple deterministic models are unable to reliably predict range of plausible radii expected for any given mass. One solution to this problem is to model exoplanetary masses and radii with a probabilistic relation. This treats each mass as corresponding to a probability distribution of radius, rar than a single, deterministic estimate. Chen & Kipping (2017) inferred such a relation for masses spanning nine orders-of-magnitude, both providing a comprehensive scale for inversion, as well as calibrating relation as precisely as possible by utilizing full dynamic range of data available. An additional key quality of Forecaster model is that relationship is trained on actual measurement posteriors of each training sample via a hierarchical Bayesian model, meaning that measurement errors of se data are propagated into Forecaster predictions. For se reasons, Forecaster is a natural tool for predicting radii of radial velocity planets Accounting for measurement uncertainties Whilst Forecaster accounts for intrinsic dispersion displayed by nature in massradius relation, as well as measurement uncertainties of its training set, a third source of error exists that requires accounting for on our end in this work. Specifically, for each RV planet, re exists often sizable uncertainty in mass of each body. 95

109 To account for this, we technically require a-posteriori probability distribution of each planet s mass, m. For n samples randomly drawn from such a mass posterior, Pr(m), a robust forecast of planetary radius can be made by proceeding through samples, row-by-row, and executing Forecaster at each line. The compiled list of predicted radii represents covariant posterior distribution for forecasted radius. A similar process is utilized in reverse-direction application described in Chen & Kipping (2018). A practical challenge to implement scheme above is that posterior distributions are rarely made available in papers announcing or studying RV planets 1. Fortunately, of all parameters describing RV model, RV semi-amplitude (K) posterior rarely exhibits multi-modality or extreme covariance (Ford 2006) and can often, in practice, be reasonably approximated as Gaussian (e.g. see Tuomi 2012 and Hou et al. 2014). By extension, since K m for m M, we will assume that planetary mass can be described as Gaussian, such that m N[µ m, σ m ] in what follows in order to make progress Approximate form for posterior distributions Naively using Gaussians for m can be problematic though for two reasons. First, Gaussians have non-zero probability density at negative values and thus negative masses will occasionally be sampled from a Gaussian distribution. This can be solved by performing a truncation at zero for Gaussian distribution, preventing distribution from drawing negative samples. Second, Gaussians are perfectly symmetric yet literature quoted credible intervals for m may include asymmetric uncertainties, e.g. m = (µ m ) +σ m+ σ m. But 1 This is in contrast to our earlier work predicting masses from radii in Chen & Kipping (2018), where transit-derived posteriors are often available. 96

110 in practice, we note that none of reported planetary masses used for this work is asymmetric Treating minimum mass as being equal to true mass We assume that minimum mass derived from radial velocity measurements equals true mass of exoplanet, in or words sin i 1. Whilst one might naively assume sin i to be, a-priori, isotropically distributed, our work primarily concerns itself with transiting planets. Within range of inclination angles which lead to transits, we can safely assume sin i 1 to be a reasonable approximation. 3.7 An Overview of Results We predicted posterior distribution for radius as described earlier. To illustrate a specific result, we show a corner plot of posteriors produced in this work for planet HD 20794c in Figure 3.4. We mark median, as well as 68.3% and 95.5% credible intervals. Predictions in this work are validated by extracting RV planets in our sample for which planetary radius has been directly measured via a transit detection. We compare predictions versus observations, which reveals that our predictions are fully compatible with observed values. Only one of our predicted radii falls outside of 1 σ credible interval (HD b), yet this object falls within 2 σ prediction. This is not statistically surprising, since amongst a sample of nine objects, it should not be 97

111 +0.43 m sin(i) [M ]= HD c forecast RP [R ]= % credible interval 95.5% credible interval SNR (log-scaling) δ [ppt] (log-scaling) R [R ] RP [R ] R [R ]= δ [ppt]= m sin(i) [M ] R P [R ] R [R ] δ [ppt] (log-scaling) SNR (log-scaling) SNR= Figure 3.4 Corner plot of our resulting posteriors for HD 20794c an illustrative example. Covariances are evident between parameters, including between mass and radius. 98

112 generally expected that all nine fall within 68.3% probability interval. 99

113 Chapter 4 Constraining an Exoplanet s Composition The interior structure of an exoplanet cannot be directly measured with today s observational techniques. However interior structure plays an important role in determining evolution and environment of an exoplanet, such as predicting habitability of Earth analogs. With measurements of only two bulk parameters: mass and radius, inference of interior structure with any multi-layer model suffers from a fundamental degeneracy. In this work, we show that although problem is indeed degenerate, re exist two boundary conditions that enable one to infer minimum and maximum core radius fraction, CRF min & CRF max. These hold true even for planets with light volatile envelopes, but require planet to be fully differentiated and that layers denser than 0 This chapter is a reproduction of a paper that has been published by The Monthly Notices of Royal Astronomical Society. It can be found at I mentored undergraduate student Ms Suissa in completing this work and guided project design and interpretation. The article has been reformatted for this section, and some additional commentary has been appended. 100

114 iron are forbidden. In end, we also propose using hierarchical Bayesian model to derive typical fraction of each layer given a sample of similar exoplanets. 4.1 Introduction Despite recent strides in our ability to characterize exoplanets (Kaltenegger 2017), knowledge regarding internal structure of distant worlds remains almost entirely lacking (Spiegel et al. 2014; Baraffe et al. 2014). Unlike search for exoplanetary atmospheres (Burrows 2014), moons (Kipping 2014), or tomography (McTier & Kipping 2018), our remote observations do not have direct access to what we seek to infer - planet s interior. The habitability of an Earth-like planet, in particular via likelihood of plate tectonics, is likely to be strongly influenced by internal structure (Noack et al. 2014) and thus community is strongly motivated to infer what lies beneath, as part of broader goal of understanding our own planet s uniqueness. In general, only information we have about an exoplanet, which is directly affected by internal structure, is bulk mass and radius of planet 1. Aside from this, we highlight that re are some special cases where additional information about planetary interior can become available. For example, Kaltenegger et al. (2010) argue that volcanism and planetary outgassing could be detectable using atmospheric characterization techniques. Certain dynamical configurations of planetary systems, such as tidal fixed points (Batygin et al. 2009), can also enable inference of planetary tidal properties, which in turn constrain internal compositions (see also Kramm et al. 2012). Finally, direct 1 Quantities such as bulk density and surface gravity are of course derivative of mass and radius 101

115 measurements of oblateness may also provide constraints on internal structures (Seager & Hui 2002; Carter & Winn 2010; Zhu et al. 2014). Whilst re is some hope of identifying outgassing of exoplanets, providing clues to mantle composition (Kaltenegger et al. 2010), and measuring tidal dissipation constants in special cases (Batygin et al. 2009), full structure inference will likely be limited to indirect methods based on oretical models. In this approach, one takes basic observables we do have access to, in particular planetary mass and radius, and compares m to oretical models in an effort to find families of compatible solutions. Since oretical models depend on not only masses and radii, but also factors such as chemical compositions (Seager et al. 2007), ultraviolet environments (Lopez & Fortney 2013; Batygin & Stevenson 2013), and ages (Fortney & Nettelmann 2010), problem is degenerate, in a general sense. Since we do not have direct access to interior of exoplanets, interior structure is generally modeled by assuming several key chemical constituents. In case of solid exoplanets, extrapolation from Solar System implies that y should be comprised of three primary chemical ingredients, namely water, H 2 O, enstatite, MgSiO 3, and iron, Fe. If we assume that planet is not young and has thus become fully differentiated, equations of state of se three layers can be solved to provide oretical estimates of mass and radius of solid bodies (Zeng & Sasselov 2013). A fourth layer describing a light volatile envelope can be placed on top to capture behavior of mini-neptunes, where light envelope is assumed to have negligible relative mass and thus only affects bulk radius but not mass (Kipping et al. 2013; Wolfgang et al. 2016). 102

116 These four constituents can be combined in multiple ways to re-create same combination of mass and radius. Even in absence of a volatile envelope this degeneracy persists, leading to common use of ternary diagrams to illustrate ir symplectic yet degenerate loci of solutions (e.g. see Seager et al. 2007). This degeneracy is a major barrier to inferring unique solution for planetary interior, leading authors to eir switch out to simpler and non-degenerate two-layer models (e.g. Zeng et al. 2016) or add a chemical proxy from parent star (e.g. Dorn et al. 2017) to break degeneracy. Whilst se are both certainly promising avenues for tackling interior inference, in this work we focus on a third approach based on boundary conditions. The possibility of exploiting boundary conditions was first highlighted in Kipping et al. (2013), where authors focused on concept of minimum atmospheric height. The method works by first predicting maximum allowed radius of a planet without any extended envelope given its measured mass. This atmosphere-less planet is typically assumed to be a pure water/icy body, for which detailed models are widely available. If observed radius exceeds this maximum limit, some finite volume of atmosphere must sit on top of planetary interior, and difference in radii represents minimum atmospheric height (MAH). The approach refore formally describes a key boundary condition of a general four-layer exoplanet. In this chapter, we explore or extreme, asking question under what conditions would observed mass and radius definitively prove some finite iron-core to exist, and what is minimum radius fraction that core must comprise? Going furr, we argue that maximum core radius fraction is anor boundary condition in 103

117 problem and thus can be derived to provide a complete bounding box of a planet s core size in a general four-layer model framework. We introduce concept of minimum core size in Section 4.2, as well as our fast parametric model to interpolate Zeng & Sasselov (2013) grid models. Section 4.3 discusses our approach to inverting relation to solve for core radius fraction directly, as well as much more straightforward method for inferring maximum core size. Finally, we discuss applying hierarchical models on composition fractions of exoplanets in Section Boundary Conditions on Core Outline and Assumptions Exoplanets are generally expected to display a diverse range of physical characteristics, owing to ir presumably distinct formation mechanisms, chemical environments, and histories, as three examples. Even if mass and radius of an exoplanet were known to infinite precision, and body was known to be definitively solid, se two observed parameters are insufficient to provide a unique solution for relative fractions of water, silicate, and iron typically assumed to represent major constituents of solid planets (Kipping et al. 2013). In or words, mass and radius alone cannot confidently reveal an exoplanet s CRF or CMF (core radius fraction or core mass fraction, respectively). As a concrete example, a planet composed of water and iron can have same mass and radius as an iron-silicon planet, and thus have very different CRFs (as illustrated in 104

118 CRF= M 1.0 R MgSiO Fe CRF=0.60 CRF= M 1.0 R 1.0 M 1.0 R CRF>0.43 } compositions and CRF<0.77 compositions CRF= M 1.0 R H20 Fe + H/He Figure 4.1 Ternary diagram of a three-layer interior structure model for a solid planet. All points along red line lead to a M = 1 M and R = 1 R planet. Although indistinguishable from each or with current observations, all points satisfy having a core-radius fraction exceeding 43%, a boundary condition we exploit in this work. The largest iron core size allowed is depicted by lowest sphere, where volatile envelope contributes negligible mass. Figure 4.1). As touched on in Section 4.1, four-layer oretical models of solid planets are degenerate for a single mass-radius observation. However, re exists a boundary condition when composition is pure silicate and iron. At this point, CRF takes smallest value out of all possible models, since second-layer ( mantle) is now as heavy as it can be, being pure silicate ( second densest material). Therefore, for any given mass-radius pair, we can solve for corresponding CRF of a pure silicate-iron model (which is not a degenerate problem) and define that this CRF must equal minimum 105

119 CRF, CRF min, allowed across all models. Similarly, anor boundary condition we can exploit is to consider maximum allowed core size. As depicted in Figure 4.1, this occurs when all of mass is located within a pure iron core, padded by a second layer of a light volatile envelope. The core can t possibly exceed this fractional size else mass would be incompatible with observed value. Note that although we refer to CRF rar than CMF here in what follows, once armed with a CRF, mass and radius, it is easy to convert back to CMF. Note too that here and throughout what follows, we refer to CRF strictly in terms of an iron core. Although technically we acknowledge that a water-silicate body could be described as having a finite sized silicate core, that core would not qualify as being a core in this work. Before continuing, we highlight some key assumptions of our model, for sake of transparency: The planet is fully differentiated and is not recently formed. The outer volatile envelope has insufficient mass and thus insufficient gravitational pressure to significantly affect equation-of-state of inner layers. The densest core permitted is that of iron i.e. heavy-element (e.g. uranium) cores are forbidden. We have accurate models for a planet s mass and radius given a particular compositional mix. 106

120 We stress that what has been described thus far includes possibility of a light volatile envelope, and is not limited to some special case of two- or three-layer conditions, as discussed earlier. Under se assumptions, limiting core radii fractions should be determinable, although violating any of assumptions listed above would invalidate our argument. An obvious one is that oretical models used are invalid or inaccurate, for example because ir assumed equations of state are wrong. In this work, we primarily use Zeng & Sasselov (2013) model but we point out that should a user believe an alternative model to be superior, it is straight-forward to reproduce methods described in this paper using model of ir preference. The actual existence of a boundary condition remains true. Anor more serious flaw would be if planetary body in question has a significant mineral fraction based on some heavier element than iron, for example a uranium core. Such a body could feasibly have a significantly smaller core than that derived using our approach. If evidence for such cores emerges in coming years, we advice against users employing model described in this work. The remaining two assumptions, a differentiated, non-young planet and a light volatile envelope, mean that young systems are not suitable and gas giants would not be eir. In general n, model described is expected to be valid for most planets smaller than mini-neptunes. 107

121 4.2.2 A parametric interpolative model for CRF min For any combination of mass and radius, we need to be able to predict what corresponding CRF would be for a silicate-iron two-layer model, in order to determine CRF min. We use models of Zeng & Sasselov (2013), which are made available as a regular grid of oretical points. Whilst we could simply perform a nearest neighbor look-up, this is unsatisfactory since our precision will be limited to grid spacing and resulting posteriors would be rasterized to same grid resolution. Instead, we seek a means to perform an interpolation of grid. The first successful literature interpolation of Zeng & Sasselov (2013) models comes from Kipping et al. (2013), who found that for a specific fixed CRF, each of various two-layer models of Zeng & Sasselov (2013) is very well-approximated by a seventh-order polynomial of radius with respect to logarithm of mass. In this work we used a similar approach by training a suite of seventh-order polynomials with varying CRFs assuming two-layer iron-silicate models of Zeng & Sasselov (2013), and validated model with all of original training data Zeng & Sasselov (2013) but omitting a random single datum each time, serving as a holdout validation point. The model, hardcore, was made publicly available at this URL ( A parametric model for CRF max Determining CRF max is far more straight-forward than CRF min. One may simply take 100% iron mass-radius models, in our case from Zeng & Sasselov (2013), and directly 108

122 compute expected radius of a pure iron planet given an observed mass, R iron (M obs ). 4.3 Solving for CRF Limits From forward- to inverse-modeling Thus far we have described a method to solve for CRF max but not for CRF min. The silicateiron interpolative model is a forward model in which we begin with knowledge of both planet s mass and minimum core radius fraction to compute corresponding radius. In practice, however, we are interested in inverse model, where we wish to determine CRF min from mass and radius. The nested coefficient structure makes problem non-linear with respect to CRF min, yet it is one dimensional and found to be unimodal in practice. For se reasons, it is amenable to a large number of possible optimization algorithms, but in what follows we adopted Newton s method, since we are able to directly differentiate our functions thanks to ir parametric nature. Specifically, in our implementation we minimize following cost function, J, with respect to one degree of freedom, CRF: J = (R Fe Si (CRF; M obs ) R obs ) 2, (4.1) where M obs and R obs are observed mass and radius of planet, and R Fe Si (CRF; M) 109

123 is radius of a two-layer iron + silicate model with core radius fraction CRF and mass M. The latter function is determined using smooth parametric interpolation model described in Section Our inversion algorithm, starts at an initial guess of CRF min = 0.5 and n iterates by computing CRF i+1 = CRF i J(CRF i ) [dj/dcrf](crf i ). (4.2) To improve speed and stability, we impose a check as to wher R obs is below that of a pure iron planet of mass M obs, in which case we fix CRF min = 1, or if radius exceeds that of an pure silicate planet of mass M obs (where again we use interpolative model of Kipping et al. 2013), in which case we fix CRF min = The Earth as an example Let us use Earth itself as an example of our method. We took a 1 M and 1 R planet and used methods described earlier to solve for CRF min and CRF max, giving CRF min = 0.43 and CRF max = In reality, Earth is not perfectly described by Zeng & Sasselov (2013) model and core in particular is only 80% iron, with nickel and or heavy elements comprising rest. The mantle-core boundary occurs at a radius of 3480 km relative to Earth s mean radius of 6371 km, meaning that its actual CRF = Accordingly, our CRF bounds correctly bracket true solution, as expected. 110

124 We go furr by treating se limits as being bounds on a prior distribution for CRF. Adopting least informative continuous distribution for a parameter constrained by only two limits corresponds to a uniform distribution. Sampling from this distribution yields a marginalized CRF of CRF marg = ± 0.098, which is again fully compatible with true value. To test our inversions in a probabilistic sense, we decided to create a mock posterior distribution of an Earth-like planet where M N[1.0M, 0.01M ] and R N[1.0R, 0.01R ] (we also apply a truncation to distributions at zero to prevent negative masses/radii). This is clearly an optimistic assumption but a more detailed investigation of sensitivity for different relative errors is tackled later in Section Generating 10 5 samples, we inverted each sample as described earlier to produce a posterior for CRF min and CRF max. Our experiment returns near-gaussian distributions for both terms with mean and standard deviation given by CRF min = (0.43 ± 0.04) and CRF max = ( ± ). This establishes that inversions are stable against perturbations around physical solutions. As a brief aside, we argued earlier in Section that principle of exploiting boundary condition of oretical models to infer CRF min does not explicitly require solid planets and works for mini-neptunes too. To demonstrate this point with a specific example, let us return to earlier thought experiment of Earth as a gaseous planet consisting of a solid iron core surrounded by a light H/He envelope. We consider that mass and radius of planet remain same as real Earth, and that envelope significantly influences radius but has negligible mass. Using 100% iron-model of Zeng & Sasselov (2013) and corresponding 7 th order polynomial interpolation of 111

125 Kipping et al. (2013), we estimate that a 1 M iron core would have a radius of 0.77 R and refore remaining 0.23 R is given by light, H/He envelope as depicted earlier in Figure 4.1. Accordingly, such a body would have a core radius fraction exceeding our inferred minimum value of (which was CRF > 0.43), which is expected and self-consistent with our definition of CRF min. We highlight that counter-example described above is highly unphysical though; 1 R planets are not expected to retain significant volatile envelopes in mature systems, both from a oretical perspective (Lopez & Fortney 2014) and an observational one (Rogers 2015; Chen & Kipping 2017; Fulton et al. 2017). Planets exceeding 1.5 R, on or hand, are certainly at risk of retaining large envelopes (e.g. see Kipping et al. 2015) and we generally advise against using our model for such large planets which cannot be reasonably assumed to be solid Sensitivity analysis for an Earth A basic and important question to ask is what kind of precisions on a planet s mass and radius lead to meaningful constraints on CRF min. In or words, what is correspondence we might expect between {( M/M), ( R/R)} and ( CRF min /CRF min ). This is key for designing future observational surveys, where primary science objectives may center around inferring internal compositions. To investigate this, we repeated retrieval experiment described in Section 4.3.2, but varied fractional error on mass and radius away from fixed 1% value previously assumed. In total, we generated 81 2 = 6561 experiments, where for each one we generated a new 112

126 mock posterior of 10 5 mass-radius samples, which was n converted into a posterior of CRF min and CRF max. For each experiment, we record standard deviation of resulting posteriors as CRF min and CRF max. The errors on mass and radius were independently varied with a fractional error given by 10 x, where x was varied across a regular grid from -4 to 0 in steps of size 0.05, giving 81 grid points in each dimension, and thus 6561 across both. In all experiments, underlying mass and radius posteriors are generated assuming a mean of µ M = 1 M and µ R = 1 R. 113

127 Δρ/ρ ΔCRFmin αmin (Δρ/ρ) αmin, = ΔCRFmarg ΔCRFmarg αmarg, αmarg, = Δρ/ρ CRFmarg Δρ/ρ ΔCRFmax αmax (Δρ/ρ) αmax, = CRFmax Figure 4.2 Upper: Contour plots depicting a-posteriori standard deviation on minimum (left), marginalized (center) and maximum (right) CRF, as a function of fractional errors on mass and radius. Lower: Reparameterization of above plots by combining mass and radius axes into a single density term, demonstrating a strong dependency in all cases. ΔCRFmin CRFmin ΔCRFmax