PLANET SIZE DISTRIBUTION FROM THE KEPLER MISSION AND ITS IMPLICATIONS FOR PLANET FORMATION. Li Zeng1, Stein B. Jacobsen1, Eugenia Hyung1, Andrew Vanderburg2, Mercedes Lopez-Morales2, Dimitar D. Sasselov2, Juan Perez-Mercader1, Michail I. Petaev1,2, David W. Latham2, Raphaëlle D. Hay-wood2, and Thomas K. R. Mattson3. 1Department of Earth & Planetary Sciences, Harvard University, 20 Oxford Street, Cambridge, MA 02138 (astrozeng@gmail.com), 2Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, 3Sandia National Laboratories, PO Box 5800 MS 1189, Albuquerque, NM 87185.

Introduction: The overall size distribution of ex-

oplanets found by the Kepler mission so far appear to best fit a log-normal distribution, indicative of a popu-lation formed by a material-limited and time-limited growth process. Further detailed analysis of their size distribution with respect to that of the stellar fluxes they receive suggests a bimodal modification on top of the log-normal distribution - a division of planets into two groups: rocky planets (<2 R⊕) and water-rich plan-ets (>2 R⊕) with or without gaseous envelopes.

Planet Size Distribution from Observations: Log-Normal Distribution. Analyzing the overall

radius distribution of 5000+ Kepler planet candidates [1], we have obtained the best fit to a log-normal dis-tribution, indicative of growth process, as planets are formed by growing from small bits and pieces. In na-ture, things that grow from small increments where each increment is stochastic follows a log-normal dis-tribution (growth rate proportional to a certain power of its size or mass), such as the size distributions of apples, our finger nails, and pumpkins (yes, pumpkins, as occasionally we find pumpkins that grown really big, but those are rare), and the size distribution of cities on Earth [2].

0 1 2 3 4 50.0

0.1

0.2

0.3

0.4

0.5

0.6

r (R⊕)

Probability

Density

Figure 1. One-dimensional histogram of planet radii. Our galaxy is an orchard, in which stars and planets grow. They grow from small bits and pieces, thus obeying a log-normal distribution. Log-normal distri-bution maximizes the entropy, implying that the growth process is stochastic. Anything that grows by accumulation of random bits and pieces, with each increment stochastic, one would get a log-normal dis-tribution. There is a completeness correction that may need to be applied to planets <~1.5 R⊕ [3].

Bi-Modal Distribution. More careful analysis of plotting 2-D histogram of fluxes received Vs. radii reveals a bi-modal distribution on top of the log-normal. The modes are centered around 1.3 R⊕ and 2.3 R⊕, with a gap at 2 R⊕. This seems to be a universal feature throughout FGKM stars. These two modes could correspond to two types of planets in general, independent of the host star type (especially taking into account that M-Dwarf radius in Kepler Input Catalog (KIC) could be underestimated by ~20% [4]).

Figure 2. Two-dimensional smoothed histogram of planet radii versus stellar fluxes they receive.

Interpretations: Two types of Planets: rocky and water worlds.

These two distinct types of planets are rocky ones (Earths and super-Earths, planets made up of mostly silicates and metals with bulk composition similar to Earth), and volatile-rich ones. The volatile-rich planets should be water-worlds (made of a significant amount (>25%) of H-compounds: H2O, NH3, CH4, in addition to the silicates/metal and small amount of gas). They are NOT gas dwarfs (H2/He-dominated envelope di-rectly on top of a rocky core), because according to condensation/evaporation calculations, there is no way to deplete the less volatile O, N, C more than H2 and He, in fact they should be enriched. For example, in our solar system, the Carbon in Uranus and Neptune [5] is enriched ~50 times solar, and the O, N, C on Jupiter and Saturn are enriched a few up to ~10 times solar [5]. In three aspects: (1) volatility (characterized by equilibrium condensation temperature in the nebu-la), (2) density, and (3) cosmic abundance, the refrac-tory elements making up silicates and metals (Si, Fe, etc.), the H-compound forming elements (O, N, C), and H2/He always form a ladder or hierarchy, with the

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elements O, N, C always falling in the middle of this hierarchy:

Table 1. Hierachy of Planet-Building Elements Cosmic

Abundance (by mass)

Condensation Temperature (K)

Density (solid) (g/cc)

H2/He 1000 1~10 0.2 O, N, C 6+1+3 100~300 2 Mg-Silicates 2 1300~1400 4 Fe, Ni metal 1 1300~1400 8

There seems to be a problem with water line, as to whether the water worlds could form so close to the star. However, recent isotope dating of meteorites sug-gests that planetesimals form quickly after the first condensates from the nebula (<~1 million years), thus allowing wet accretion directly from the nebula before its dissipation [6].

Alternative View: photo-evaporation? An alterna-tive view is that the two populations (bi-modal distri-bution) is a result of photo-evaporation, but there are controversial results in the literature regarding XUV evaporation models, some predict a bimodal distribu-tion [7], while some do not [8].

100%Fe

50%Fe

MgSiO3(rock)50%H2O

100%H2O, NH3, CH45%H2/He

20%H2/He

50%H2/He

Cold H2/HeT=400K

@ 100bar

T=1000K@ 100bar

T=3000K@ 100bar

●●55 Cnc e

●●BD+20 594 b

●●CoRoT-1 b

●●CoRoT-10 b

●●CoRoT-11 b●●CoRoT-12 b

●●CoRoT-13 b

●●CoRoT-14 b●●CoRoT-16 b

●●CoRoT-17 b

●●CoRoT-18 b●●CoRoT-19 b

●●CoRoT-2 b

●●CoRoT-20 b

●●CoRoT-22 b

●●CoRoT-23 b

●●CoRoT-24 c

●●CoRoT-25 b

●●CoRoT-26 b

●●CoRoT-27 b●●CoRoT-28 b ●●CoRoT-29 b

CoRoT

●●CoRoT-4 b

●●CoRoT-5 b

●●CoRoT-6 b

●●CoRoT-7 b

●●CoRoT-8 b

●●CoRoT-9 b

●●EPIC 211351816 b

●●EPIC 211391664 b

●●GJ 1132 b

●●GJ 1214 b

●●GJ 3470 b

●●GJ 436 b

●●K2-30 b

●●K2-34 b

●●K2-38 b

●●K2-38 c

●●K2-39 b

●●K2-99 b

●●KOI-13 b

●●KOI-142 b

●●KOI-94 b

●●KOI-94 d

●●KOI-94 e

●●K-10 b

●●K-10 c

●●K-100 b

●●K-102 d

●●K-102 e

●●K-106 c●●K-106 e

●●K-11 b

●●K-11 c

●●K-11 d

●●K-11 e

●●K-11 f

●●K-113 b

●●K-12 b

●●K-131 b

●●K-131 c

●●K-14 b

●●K-15 b

●●K-16 b

●●K-17 b

●●K-18 b

●●K-18 c

●●K-18 d

●●K-20 b

●●K-20 c●●K-20 d

●●K-21 b

●●K-25 b

●●K-25 c

●●K-34 b●●K-35 b

●●K-4 b

●●K-40 b

●●K-406 b

●●K-406 c

●●K-41 b●●K-412 b

●●K-413 b

●●K-419 b●●K-420 b

●●K-422 b●●K-423 b

●●K-424 b

●●K-425 b

●●K-426 b

●●K-427 b

●●K-428 b

●●K-43 b ●●K-432 b

●●K-433 b

●●K-434 b

●●K-435 b

●●K-44 b

●●K-447 b

●●K-45 b

●●K-454 b

●●K-48 b

●●K-48 c

●●K-48 d

●●K-5 b

●●K-539 b

●●K-6 b

●●K-68 b

●●K-7 b

●●K-74 b

●●K-75 b

●●K-76 b

●●K-77 b

●●K-78 b

●●K-8 b

●●K-9 b●●K-9 c

●●K-93 b

●●K-94 b ●●K-95 b

●●K-96 b

●●K-97 b

●●K-98 b

●●K-99 b

●●Earth●●

Venus

●●Uranus

●●Neptune

●● Saturn

●●

Jupiter

flux(F/F⊕)

10

30

100

300

1000

3000T(K)

500

700

1000

1400

2000

0.4 0.60.81 1.5 22.53 4 5 6 789101215 20 30 4050 70 100 200 400 600 1000 2000 40000.80.91

1.21.41.61.82

2.5

33.54

5

6789101112131415

20

250.4 0.60.81 1.5 22.53 4 5 6 789101215 20 30 4050 70 100 200 400 600 1000 2000 4000

0.80.91

1.21.41.61.82

2.5

33.54

5

6789101112131415

20

25

m (M⊕)

r(R

⊕)

Figure 3. Mass-radius diagram of planets with RV-determined masses, color-coded by Tequilibrium.

“Main-Sequence” of Planets: If we restrict our samples to the planets with masses determined by the radial-velocity (RV) method (generally more robust than other methods [9]), then the distribution of planets on the mass-radius diagram form a main-sequence, that is a limited width array, similar to that of stars on Hertzsprung–Russell Diagram (Luminosity-Temperature Diagram). This is consistent with the picture of a material-limited, stepwise growth process, including two kinds of planet building materials – rocky/metallic component and ices.

In Figure 3, comparing the positions of planets to the mass-radius curves of various theoretical composi-tions (color curves as labelled): the first group of plan-

ets (<2 R⊕, red and yellow ones in the lower-left part) complies very well with that of silicates/metal (2:1 weight ratio) composition trend within uncertain-ty. This trend was first pointed out by [10] and later followed up by [11].

The second group (>2 R⊕, blue and green ones), on the other hand, complies reasonably well with the fol-lowing formation scenario: Stage (1) accrete met-al/silicate materials up to a few M⊕. Stage (2) accrete icy materials (H2O, NH3, CH4, i.e., Hydrogen-compounds), if they are available, for another few M⊕, following the trend of the dashed cyan curves, asymp-totically approaching 100% H2O, NH3, CH4 curve. Stage (3) then accrete H2/He gas, if they are available, following the trend of the dashed purple curves, as-ymptotically approaching 100% cold H2/He curve.

Other evidence. Contaminated white-dwarf spectra show H2O-rich planet debris. Although, the amount of debris is small compared to 1 Earth mass, it could indi-cate the existence of water worlds out there, if those debris come from the breakup of those planets [12, 13].

Conclusion: The planet formation process leads to three types of planets: rocky, water and gas worlds. They give well-defined fields in the mass-radius dia-gram as shown in Figure 4.

Rock

Water

Gas

Figure 4. Mass-radius diagram of the three types of planets that form in protoplanetary disks.

References: [1] NASA Exoplanet Archive. [2] Limpert E. et al. (2001) BioScience, 51, 5. [3] Petigura E. A. (2015) arXiv:1510.03902 [4] Newton E. R. et al. (2015) ApJ, 800, 85. [5] Atreya S. K. et al. (2016) arXiv:1606.04510 [6] Kleine et al. (2009) GCA, 73, 5150-5188. [7] Owen J. E. and Wu Y. (2013) ApJ, 775, 105. [8] Lopez E. D. and Fortney J. J. (2013) ApJ, 776, 2. [9] Hadden S. and Lithwick Y. (2016) ApJ, 828, 44. [10] Dressing C. D. et al. (2015) ApJ, 800, 135. [11] Zeng L. et al. (2016) ApJ, 819, 127 [12] Fa-rihi J. et al. (2013) Science, 342, 218. [13] Farihi J. et al. (2016) MNRAS, 463, 3186-3192.

1576.pdfLunar and Planetary Science XLVIII (2017)