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Accepted to ApJ Letters on 30 January, 2013. Preprint typeset using LATEX style emulateapj v. 5/2/11

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THE MASS-RADIUS RELATION FOR 65 EXOPLANETS SMALLER THAN 4 EARTH RADII

Lauren M. Weiss1,† & Geoffrey W. Marcy1

1B-20 Hearst Field Annex, Astronomy Department, University of California, Berkeley, CA 94720

Accepted to ApJ Letters on 30 January, 2013.

ABSTRACT

We study the masses and radii of 65 exoplanets smaller than 4R⊕ with orbital periods shorter than 100 days. We calculate the weighted mean densities of planets in bins of 0.5 R⊕ and identify a density maximum of 7.6 g cm−3at 1.4 R⊕. On average, planets with radii up to RP = 1.5R⊕ increase in density with increasing radius. Above 1.5 R⊕, the average planet density rapidly decreases with increasing radius, indicating that these planets have a large fraction of volatiles by volume overlying a rocky core. Including the solar system terrestrial planets with the exoplanets below 1.5 R⊕, we find ρP = 2.43 + 3.39 (RP/R⊕) g cm

−3 for RP < 1.5R⊕, which is consistent with rocky compositions.

For 1.5 ≤ RP/R⊕ < 4, we find MP/M⊕ = 2.69 (RP/R⊕) 0.93

. The RMS of planet masses to the fit between 1.5 and 4 R⊕ is 4.3 M⊕ with reduced χ

2 = 6.2. The large scatter indicates a diversity in planet composition at a given radius. The compositional diversity can be due to planets of a given volume (as determined by their large H/He envelopes) containing rocky cores of different masses or compositions.

1. INTRODUCTION

The Kepler Mission has found an abundance of planets with radii RP < 4R⊕ (Batalha et al. 2013); the most re- cent head-count indicates 3206 planet candidates in this size range (NASA exoplanet Archive, queried 15 Jan. 2014), most of which are real (Morton & Johnson 2011). Although there are no planets between the size of Earth and Neptune in the solar system, occurrence calculations that de-bias the orbital geometry and completeness of the Kepler survey find that planets between the size of Earth and Neptune are common in our galaxy, occurring with orbital periods between 5 and 50 days around 24% of stars (Petigura et al. 2013). However, in many sys- tems, it is difficult to measure the masses of such small planets because the gravitational acceleration these plan- ets induce on their host stars or neighboring planets is challenging to detect with current telescopes and instru- ments. We cannot hope to measure the masses of all planets in this size range discovered by Kepler. Ob- taining measurements of the masses of a subset of these planets and characterizing their compositions is vital to understanding the formation and evolution of this pop- ulation of planets. Many authors have explored the relation between

planet mass and radius as a means for understanding exoplanet compositions and as a predictive tool. Sea- ger et al. (2007) predict the mass-radius relationship for planets of various compositions. The mass-radius rela- tion in Lissauer et al. (2011), which is commonly used in literature to translate between planet masses and radii, is based on fitting a power law relation to Earth and Saturn only. Other works, such as Enoch et al. (2012); Kane & Gelino (2012); Weiss et al. (2013), determine em- pirical relations between mass and radius based on the exoplanet population. Recent mass determinations of small planets motivate

† Supported by the NSF Graduate Student Fellowship, Grant DGE 1106400.

a new empirical mass-radius relation. Restricting the empirical mass-radius relation to small exoplanets will improve the goodness of fit, allowing better mass pre- dictions and enabling a superior physical understanding of the processes that drive the mass-radius relation for small planets. One challenge in determining a mass-radius relation

for small planets is the large scatter in planet mass. At 2R⊕, planets are observed to span a decade in density, from less dense than water to densities comparable to Earth’s. This scatter could result from measurement un- certainty or from compositional variety among low-mass exoplanets. In this paper, we investigate mass-radius relationships

for planets smaller than 4 Earth radii. We explore how planet composition–rocky versus rich in volatiles– influ- ences the mass-radius relationship. We also investigate the extent to which system properties contribute to the scatter in the mass-radius relation by examining how these properties correlate with the residuals of the mass- radius relation.

2. SELECTING EXOPLANETS WITH MEASURED MASS AND RADIUS

We present a judicious identification of small transit- ing planets with masses or mass upper limits measured via stellar radial velocities (RVs) or numerically modeled transit timing variations (TTVs). The only selection cri- terion was that the exoplanets haveRP < 4R⊕ and either a mass determination, a marginal mass determination, or a mass upper limit. There were no limits on stellar type, orbital period, or other system properties. We include all 19 planets smaller than 4R⊕ with

masses vetted on exoplanets.org, as of January 13, 2013. Twelve of these masses are determined by RVs, but the masses of four Kepler-11 planets, Kepler-30 b, and two Kepler-36 planets are determined by TTVs (Lissauer et al. 2013; Sanchis-Ojeda et al. 2012; Carter et al. 2012). We include five numerically-determined planet masses from TTVs not yet on exoplanets.org: three KOI-152

http://arxiv.org/abs/1312.0936v4

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(Kepler-79) planets (Jontof-Hutter et al. 2013), and two KOI-314 planets (Kipping et al. 2014). We also include all 40 transiting planets with RV follow-up in Marcy et al. (2014) that are smaller than 4R⊕, and the RV- determined mass of KOI-94 b (Weiss et al. 2013), none of which yet appear on exoplanets.org. 55 Cnc e, Corot-7 b, and GJ 1214 b have been studied

extensively, and we had to choose from the masses and radii reported in various studies. For 55 Cnc e, we use MP = 8.38± 0.39, RP = 1.990± 0.084 (Endl et al. 2012; Dragomir et al. 2013a); for Corot-7 b, we use MP = 7.42± 1.21, RP = 1.58± 0.1 (Hatzes et al. 2011), and for GJ 1214 b, we use MP = 6.45 ± 0.91, RP = 2.65 ± 0.09 (Carter et al. 2011). Histograms of the distributions of planet radius, mass, and density are shown in Figure 1, and the individual measurements of planet mass and radius are listed in Table 1. The exoplanets all have P 3σ) masses, imposing a significance crite- rion will bias the sample toward more massive planets at a given radius. This bias is especially pernicious for small planets, for which the planet-induced RV signal can be small (∼ 1m s−1) compared to the noise from stellar ac- tivity (∼ 2m s−1) and Poisson photon noise (∼ 2m s−1). We must include the marginal mass detections and non- detections in order to minimize bias in planet masses at a given radius. Marcy et al. (2014) employ a new technique for in-

cluding non-detections. They allow a negative semi- amplitude in the Keplerian fit to the RVs and report the peak and 68th percentiles of the posterior distribu- tion of the semi-amplitude. The posterior distribution peak often corresponds to a “negative” planet mass, al- though the wings of the posterior distribution encompass positive values. Although planets cannot have negative masses in nature, random fluctuations in the RVs from noise can produce a velocity curve that is low when it should be high, and high when it should be low, mimick- ing the RV signature of a planet 180◦ out of phase with the transit-determined ephemeris. Since the planetary ephemeris is fixed by the transit, Marcy et al. (2014) al- low these cases to be fit with a negative semi-amplitude solution in their MCMC analysis. Reporting the peak of the posterior distribution is statistically meaningful be- cause there are also cases where the fluctuations in RVs from the random noise happen to correlate with the plan- etary signal, artificially increasing the planet mass. We include non-detections (as negative planet masses and low-significance positive planet masses) to avoid statisti- cal bias toward large planet masses at a given radius. Including literature values, which typically only report

planet mass if the planet mass is detected with high con- fidence, slightly biases our sample toward higher masses at a given radius. We include the literature values to provide a larger sample of exoplanets.

3. THE MASS-RADIUS RELATIONS

In Figure 2, we show the measured planet densities and planet masses for RP < 4R⊕. In addition, we show the weighted mean planet density and mass in bins of 0.5 R⊕. The weighted mean densities and masses guide the eye, demonstrating how the ensemble density and mass change with radius. We also include the solar system planets. Examining the solar system terrestrial planets and the weighted mean density at 1.5 R⊕, we see that planet density increases with increasing radius up to 1.5 R⊕. For planets between 1.0 and 1.5 R⊕, the weighted mean density achieves a maximum at 7.6 ± 1.2g cm−3, and the weighted center of the bin is at 1.4 R⊕. Above 1.5 R⊕, planet density decreases with increasing radius. The break in the density-radius relation motivates us to explore different empirical relations for planets smaller and larger than 1.5 R⊕. Exoplanets smaller than 1.5 R⊕ mostly have mass un-

certainties of order the planet mass, except for Kepler-10 b, Kepler-36 b, Kepler-78 b, and Kepler-406 b (KOI- 321 b). Because there are so few planets with well- determined masses in this regime, we include the terres- trial solar system planets (Mercury, Venus, Earth, Mars) in a fit to the planets smaller than 1.5 R⊕. We impose uncertainties of 20% in their masses and 10% in their radii so that the solar system planets will contribute to, but not dominate, the fit. Because the solar system plan- ets appear to satisfy a linear relation between density and radius, we choose a linear fit to planet density vs. radius. We find:

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