Occurrence and core-envelope structure of 1-4× Earth-size planets around Sun-like stars

Abstract

Small planets, 1-4× the size of Earth, are extremely common around Sun-like stars, and surprisingly so, as they are missing in our solar system. Recent detections have yielded enough information about this class of exoplanets to begin characterizing their occurrence rates, orbits, masses, densities, and internal structures. The Kepler mission finds the smallest planets to be most common, as 26% of Sun-like stars have small, 1-2 R⊕ planets with orbital periods under 100 d, and 11% have 1-2 R⊕ planets that receive 1-4× the incident stellar flux that warms our Earth. These Earth-size planets are sprinkled uniformly with orbital distance (logarithmically) out to 0.4 the Earth-Sun distance, and probably beyond. Mass measurements for 33 transiting planets of 1-4 R⊕ show that the smallest of them, R < 1.5 R⊕, have the density expected for rocky planets. Their densities increase with increasing radius, likely caused by gravitational compression. Including solar system planets yields a relation: ρ = 2:32 + 3:19 R=R ⊕ [g cm(-3)]. Larger planets, in the radius range 1.5-4.0 R⊕, have densities that decline with increasing radius, revealing increasing amounts of low-density material (H and He or ices) in an envelope surrounding a rocky core, befitting the appellation ''mini-Neptunes.'' The gas giant planets occur preferentially around stars that are rich in heavy elements, while rocky planets occur around stars having a range of heavy element abundances. Defining habitable zones remains difficult, without benefit of either detections of life elsewhere or an understanding of life's biochemical origins.

Keywords: SETI; astrobiology; extrasolar planets.

Fig. 1.

The size distribution for planets around Sun-like stars. The fraction of Sun-like stars (G- and K-type) hosting planets of a given planet radius are tallied in equal logarithmic bins. Only planets with orbital periods of 5–100 d (corresponding to orbital distances of 0.05–0.42 AU) are included. Together, the lowest two bins show that 26% of Sun-like stars have planets of 1–2 $R_{\oplus }$ orbiting within ∼0.4 AU. The occurrences of Neptune-size planets (2.8–4 $R_{\oplus }$) and gas-giant planets (8–11 $R_{\oplus }$) are 5.9% and 0.9%, respectively, more rare than Earth-size planets (19).

Fig. 2.

The fraction of Sun-like stars having planets larger than Earth and within ∼0.4 AU, as a function of the planets’ orbital periods (log scale). The occurrence of planets is roughly constant, ∼15%, in period bins sized by equal factors of 2 in orbital period between 12 and 100 d. Thus, planet occurrence is roughly constant with orbital distance, dN/dlog a = constant, in the inner regions of planetary systems (19).

Fig. 3.

Doppler measurements made during the orbits of the exoplanets Kepler-78 (Left) and Kepler-406 (Right), stars that harbor planets with radii of 1.20 and 1.41 $R_{\oplus }$, respectively. The Doppler measurements show a sinusoidal periodicity, yielding masses corresponding to densities of 5.3 ± 1.8 g cm⁻³and 9.2 ± 3.3 g cm⁻³, implying rocky compositions (16, 36).

Fig. 4.

Planet density vs. radius for all 33 known exoplanets smaller than 4 $R_{\oplus }$ that have 2-σ mass determinations. Venus, Earth, Mars, Uranus, and Neptune are included (diamonds). The radius of ∼1.5 $R_{\oplus }$ has the highest densities, and marks the transition between rocky planets (smaller size, at left) and planets with increasing amounts of low density material (larger size, at right) (28, 32, 37). For radii 0–1.5 $R_{\oplus }$, density increases with planet radius, consistent with a purely rocky constitution. In the radius range of 1.5–4.0 $R_{\oplus }$, density decreases with radius, indicating increasing amounts of H and He gas or water. The transition radius at 1.5 $R_{\oplus }$ has a density maximum near ∼7.6 g cm⁻³(weighted average). A linear fit including all planets (including sub-2-σ densities, not shown) for $R<1.5$ $R_{\oplus }$ (dashed line) yields: $\rho \left(R\right)=2.32+3.19R/R_{\oplus }$ in units of g cm⁻³. A fit for $R>1.5$ $R_{\oplus }$ (solid line) yields a density law: $\rho \left(R\right)=2.69{\left(R/R_{\oplus }\right)}^{0.93}$ in g cm⁻³, consistent with a characteristic core mass of roughly 10 $M_{\oplus }$ (28, 32).

Fig. 5.

Planet mass vs. radius, including both the 33 known exoplanets smaller than 4 $R_{\oplus }$ with 2-σ mass determinations (circles) and the solar system planets (diamonds). Planet mass is correlated with radius in the domain $R<1.5$ $R_{\oplus }$. The dashed line marked ‘‘rocky’’ represents the linear density−radius relation from Fig. 4, projected into mass−radius space. The points residing near that dashed line represent planets that must be mostly rocky. The points residing to the right of the ‘‘rocky’’ dashed line represent planets with radii too large to be purely rocky. For such planets, the dashed line represents a simple approximation of the dividing line between a rocky core and a low-density envelope: The horizontal distance to the left of the dashed line (dark gray) represents the radius of the rocky core, while the horizontal distance to the left of the dashed line (light gray) represents the extra radius from the low-density material (H and He or water) in the envelope, which contributes extra size but negligible mass; see refs. , , , , , , and . As an example, the additional size, on top of the rocky core, contributed by the H and He or H₂O envelopes for GJ 1214b and for Kepler-94b are indicated by dotted lines. Planets of 1–4 $R_{\oplus }$ are well modeled by a rocky core containing most of the mass plus a low-density envelope, if any, that enlarges the planet’s radius.

Fig. 6.

Abundance of heavy elements (metallicity) of the host star vs. planet radius for over 400 stars as a function of the size of the Kepler planet orbiting it. The planets with sizes larger than 4 $R_{\oplus }$ have host stars relatively rich in heavy elements. In contrast, the smaller planets orbit stars that are roughly solar-like in metallicity. The explanation may be that high metalicity in the proplanetary disk allowed rocky cores to form quickly, before the gas in the disk vanished, allowing the cores to gravitationally accrete that gas to make gas-rich planets.