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Occurrence and core-envelope structure of 1–4× Earth-size planets around Sun-like stars
Edited by Adam S. Burrows, Princeton University, Princeton, NJ, and accepted by the Editorial Board April 16, 2014 (received for review January 24, 2014)

Significance
Among the nearly 4,000 planets known around other stars, the most common are 1–4× the size of Earth. A quarter of Sun-like stars have such planets orbiting within half an Earth’s orbital distance of them, and more surely orbit farther out. Measurements of density show that the smallest planets are mostly rocky while the bigger ones have rocky cores fluffed out with hydrogen and helium gas, and likely water, befitting the term ‘‘mini-Neptunes.’’ The division between these two regimes is near 1.5 R⊕. Considering exoplanet hospitality, 11% of Sun-like stars have a planet of 1–2× the size of Earth that receives between 1.0–4.0× the incident stellar light that our Earth enjoys. However, we remain ignorant of the origins of, and existence of, exobiology, leaving the location of the habitable zone uncertain.
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:
NASA’s Kepler mission astonishingly revealed a preponderance of planets having sizes between 1 and 4 times the diameter of Earth (1⇓⇓⇓–5). Our solar system has no planets larger than Earth and smaller than Neptune (3.9
This great population of sub-Neptune-mass exoplanets had first been revealed by precise Doppler surveys of stars within 50 parsecs (pc) (13, 14), a finding that Kepler ’s discoveries confirm. While most of the over 3,000 1–4
Occurrence Rates of 1–4 R⊕ Planets
Kepler is superior to RV surveys for measuring occurrence rates of planets down to 1
The occurrence rate of Earth-size planets is a major goal of exoplanet science. With three years of Kepler photometry in hand, two groups worked to account for the detection biases in Kepler planet detection caused by photometric noise, orbital inclination, and the completeness of the Kepler transiting-planet detection pipeline (4, 5, 17). They found that within 0.25 AU of solar-type stars, small planets of 1–3× the size of Earth orbit ∼30 ± 5% of Sun-like stars. In contrast, only 2 ± 1% have larger planets of Neptune size (4–6
A new planet search of nearly 4 y of Kepler photometry revealed planets as small as 1
Fig. 1 shows the resulting fraction of Sun-like stars having planets of different sizes (19) with orbital periods of 5–100 d. The lowest two bins show that 26.2% of Sun-like stars have a planets of size, 1–2
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
Fig. 2 shows the resulting occurrence rate of planets around Sun-like stars as a function of orbital period. The rate is about 15% at all orbital periods, within bins of multiples of orbital period (i.e., 10–20 d, 20–40 d, 40–80 d), as shown in Fig. 2. This constant planet occurrence with increasing orbital distance, in equal logarithmic bins, surely informs planet formation theory. Indeed, we know of no theoretical cause of major discontinuities in planet formation efficiency inside 1 AU. No phase changes of major planet-building material occur in that region. A smoothly varying occurrence rate, both observed and theoretically, supports mild extrapolations of planet occurrence rates beyond orbital periods of 100 d where the measured rates are empirically secure (19).
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).
Spectroscopy of the host stars of the Earth-size planets yields their luminosities, providing a measure of the incident stellar light fluxes falling on the planets. This analysis shows 11% of Sun-like stars have a planet of 1–2
Properties: Masses, Radii, and Densities
Although 1–4
Radii of exoplanets are measured based on the fractional dimming of host stars as planets transit and are known for all Kepler objects of interest. Planet masses require additional observations, and stem from Doppler-measured reflex motion of the host star or from variations in the cadence of the planet crossing in front of the star each orbit (transit-timing variations, TTV) caused by planets pulling gravitationally on each other.
To date, 33 planets of 1–4
Fig. 3 shows two representative applications of the Doppler technique to determine planet masses for Kepler-78 and Kepler-406. Each star reveals repeated dimmings in Kepler photometry due to their transiting planets with orbital periods of 8.5 h and 2.43 d (2), giving planet radii of 1.20 and 1.41
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
In the analysis that follows, we include both Doppler-determined and TTV-determined planet masses. It is worth noting that the TTV planet masses are mostly lower than the RV-determined masses for given radii (although Doppler and TTV measurements of the same planets agree), for reasons not understood (32). Perhaps multiplanet systems, which allow TTV measurements, survive dynamically only if the planet masses are low enough to limit catastrophic dynamical chaos.
All 33 transiting 1–4
Planet density vs. radius for all 33 known exoplanets smaller than 4
Planet mass vs. radius, including both the 33 known exoplanets smaller than 4
By contrast, the smallest planets (1–1.5
Among the prominent examples of planets with size 1.5–4.0
Even larger planets, 4–6
In contrast, the following planets with radii less than 2
Thus, we find a density dichotomy, with the dividing radius being near 1.5
Structure: Core-Envelope Model of 1–4 R⊕ Planets
The two domains of 1–4
In performing the weighted fit, we include all 22 exoplanets with radius and mass measurements, regardless of the quality of the mass measurement, to mitigate any bias in mass (32). This linear fit includes the four solar system rocky planets with uncertainties of 10% in density so that they do not dominate the fit. We note that both the exoplanets and solar system planets exhibit an increase in density with increasing radius. The mass−density dependence for exoplanets is anchored with Kepler-78b having R = 1.2
By including exoplanets having measured masses that are marginally significant, we promote a statistically useful representation of planets of all masses at a given planet radius (16, 32, 37). For all planets smaller than 1.5
For all planets larger than 1.5
Fig. 5 shows measured planet mass vs. radius for all 33 planets having a mass measurement better than 2 σ. As in Fig. 4, the dashed line shows the previously described linear fit to density vs. radius for all planets smaller than 1.5
With such a linear extrapolation of the density relation, we can make an approximate prediction of the interior structure of planets larger than 1.5
Thus, cloud of planets residing to the right of the ‘‘rocky’’ dashed line in Fig. 5 support a model of exoplanet structure with both rock and volatiles. These planets have larger radii (and volumes) than can be explained by a purely rocky interior. Therefore, these planets surely contain large amounts of gas and ices to account for their large size, given their mass. Clearly, the planets larger than 2
A core-envelope model follows from the expectation that the more dense material will sink (differentiate) toward the center of the planet. The argument presented here for large amounts of low-density material on a rocky core does not make use of any theoretical equation of state. The low-density material, presumably H and He gas, must exist in the planets larger than 2
Interiors, Formation, and Evolution
The range of sizes of rocky planets is visible in Figs. 4 and 5 as the observed rise in density and mass with increasing radius for planets smaller than 1.5
For those planets larger than 1.5
The existence of two planet domains on either side of 1.5
The spread in planet bulk densities at a given radius or mass may also be due to the subsequent photoevaporation of volatiles. Such evaporation may be germane because nearly all of the 1–4
Correlations with Heavy Element Abundance
The abundances of heavy elements in the protoplanetary disks around young stars may influence the efficiency of formation of the rocky cores made of such elements. Spectra of the brightest Kepler host stars of transiting planets were analyzed by Buchhave et al. (66) to yield their abundances of heavy elements relative to the Sun (‘‘metallicities’’). The planets with sizes greater than 3.5
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
One possible explanation for this correlation between planet size and the metallicity of the host star is that giant planets are created from a rocky core that accretes H and He gas from the protoplanetary disk. However, the gas in protoplanetary disks dissipates quickly (within a few million years). The heavy elements in the protoplanetary disk must form a rocky core quickly enough to accrete the gas before it vanishes. If so, the core can accrete H and He gas to form the low-density, gaseous planet. Those stars (and their protoplanetary disks) that have only modest metallicity (or less) form rocky cores more slowly, after most of the gas in the protoplanetary disk has vanished, leaving only rocky cores that are devoid of a gaseous envelope (66). If this explanation is roughly correct, the Earth resides at a planetary sweet spot, coming from a protoplanetary disk with inadequate heavy elements to grow quickly enough to grab huge amounts of gas, but adequate to initiate complex biochemistry.
Habitable Zone: Humility and Hubris
Scientific knowledge of complex systems is normally anchored by, and repeatedly tested by, experimental evidence. The planetary conditions necessary for biology certainly qualify as a complex physical, chemical, and biological problem. A common construct toward such discussions is the ‘‘habitable zone,’’ the orbital domain around a star where life can arise and flourish. Unfortunately, we have no empirical evidence of life arising, nor of it flourishing, around any other star.
Such lack of experimental evidence of life has not slowed the debate about the exact location of the habitable zone around stars of different types. The passion exhibited in this debate is worthy of some caution. We have no evidence of microbial life at any orbital location within our solar system beside the Earth. We have no empirical information about microbial life as a function of orbital distance from our Sun or from any other star. We also have no evidence of multicellular life around any other star, nor evidence of intelligent life.
Thus, we have no empirical knowledge about the actual domain of habitable zones, for any type of life, around any type of star. Moreover we have virtually no theoretical underpinnings about exobiology. We still do not know how biology started on Earth. We do not know the mechanisms that caused a transition from chemistry to biology, nor do we know the biochemical steps that spawn proteins, RNA, DNA, or cell membranes (67), although there has been recent progress (68). Indeed, we still have a poor definition of life (69).
Our ignorance about both the necessary planetary environments and the complex biochemical pathways for life should urge caution in predicting, with multiple significant digits, the location of the ‘‘habitable zones’’ around other stars. We can’t predict if Mars, Europa, or Enceladus have habitable environments any better than we can predict the weather in our hometown a week in advance.
What is needed is a census of biology among a sample of nearby stars, measuring the orbital locations and geological types of planets where biologies exist. A door-to-door census of life among stellar neighbors is needed to answer empirically and with credibility the true domain of habitability around other stars. That census can be carried out three ways: within our solar system among water-bearing planets and moons, by space-borne telescopes that perform chemical assays of resolved rocky planets, and by searches for transmissions from technological beings.
Acknowledgments
We thank Leslie Rogers, Eric Lopez, Jonathan Fortney, Dimitar Sasselov, Jack Lissauer, Eugene Chiang, Greg Laughlin, and Sara Seager for valuable conversations. We thank the many observers who contributed to the measurements reported here. The authors wish to extend special thanks to those of Hawai‘ian ancestry on whose sacred mountain of Mauna Kea we are privileged to be guests. Without their generous hospitality, the Keck observations presented herein would not have been possible. We thank the extraordinary group of engineers and scientists who worked tirelessly to produce the Kepler mission. Kepler was competitively selected as the tenth NASA Discovery mission. Funding for this mission is provided by the NASA Science Mission Directorate. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Keck Observatory was made possible by the generous financial support of the W. M. Keck Foundation. We thank the NSF Graduate Research Fellowship, Grant DGE 1106400. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.
Footnotes
- ↵1To whom correspondence should be addressed. E-mail: gmarcy{at}berkeley.edu.
Author contributions: G.W.M., L.M.W., E.A.P., H.I., A.W.H., and L.A.B. contributed to this work by acquiring data at telescopes, analyzing those data, and providing final data products or graphs of their results; and G.W.M. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission. A.S.B. is a guest editor invited by the Editorial Board.
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