DENSITY AND ECCENTRICITY OF KEPLER PLANETS

The spectacular success of the Kepler mission (Borucki et al. 2010a, 2011; Batalha et al. 2013, hereafter B12) opens our eyes to the world of low-mass planets (also see radial velocity discoveries, e.g., Mayor et al. 2011). The Kepler mission has uncovered a surprising abundance of such planets close to their host stars and to each other. While some planets are inferred to have high densities and therefore are terrestrial-like, some have low densities, indicating the presence of a substantial gaseous envelope (see references in Table 3). How do these low-mass planets form and migrate? What gives rise to the range in planet radius and density? What is the internal composition of these planets? More measurements of planet density may help resolve these puzzles. In particular, ascertaining whether Kepler planets contain a significant amount of water would shed light on the site of their construction (see Lopez et al. 2012 for a detailed discussion).

In addition to masses, the planets' eccentricities and inclinations are valuable clues. Relative inclinations in Kepler systems are inferred to be only a few degrees (e.g., Tremaine & Dong 2012; Fabrycky et al. 2012; Figueira et al. 2012). This flat configuration invokes images of proto-planetary disks or planetesimal disks. Eccentricities, on the other hand, are relatively poorly constrained (see, e.g., Moorhead et al. 2011). Since eccentricities can be excited by planet interactions and damped by planet–disk interactions or tidal interactions with the host stars, they are a fossil record of past dynamical events. In Lithwick et al. (2012, hereafter Paper I), we analyze the transit time variation (TTV) signals of six planet pairs and show that the TTV phases are consistent with most planets having in general small eccentricities. More eccentricity determinations will help us rewind the clock and infer the dynamical past of these planets.

With these two goals in mind, we analyze here the TTV signals for pairs of transiting planetary candidates that lie near first-order mean-motion resonances (MMRs) using the Q0–Q6 Kepler table of transit times in Ford et al. (2012). We also incorporate some results from Xie (2012). We then apply our analytical TTV expressions (derived in Paper I) to infer planet properties. We introduce a new element in this work by showing how true planet density can be inferred from the nominal density measured by TTV by using statistical information from a large sample. We outline our sample selection criteria in Section 2 and discuss our results in Sections 3–5. First we recap some results from Paper I that are of relevance here.

For a pair of transiting planets that lie near a first-order MMR, mutual gravitational perturbations cause each planet's transit timing to vary sinusoidally with a sizable amplitude. The periods, amplitudes, and phases of the sinusoids encode physical properties of the planet pair.

• 1.  
  The periods of the two sinusoids are the same and we call them the super-period:

  $$\begin{equation} P^j = {1\over { |{j/P^{\prime }-(j-1)/P}|}} = {{P^{\prime }}\over {j|\Delta |}}\,, \end{equation} \tag{ 1 }$$

  where P and P' are orbital periods of the inner and outer planets, respectively, j stands for the j:j − 1 resonance, and Δ ≡ P'/P(j − 1)/j − 1 is the fractional distance to resonance. Most Kepler pairs with clear TTV signals have |Δ| of the order of a few percent.
• 2.  
  The amplitudes (|V| and |V|') are functions of both planet mass and orbital eccentricity, roughly as (disregarding coefficients of the order of unity)

  $$\begin{eqnarray} {{|V|}\over {P}} & \sim & {\mu ^{\prime }\over |\Delta |}\left(1+{|Z_{\rm free}|\over |\Delta |}\right), \nonumber \\ {{|V^{\prime }|}\over {P^{\prime }}}& \sim & {\mu \over |\Delta |}\left(1+{|Z_{\rm free}|\over |\Delta |}\right). \end{eqnarray} \tag{ 2 }$$

  Here, μ is the mass ratio between the inner planet and the star and μ' that of the outer planet. Z_free is a weighted sum of the complex free eccentricity in both planets,

  $$\begin{equation} Z_{\rm free}= f z_{\rm free} + g z^{\prime }_{\rm free}, \end{equation} \tag{ 3 }$$

  where f and g are sums of Laplace coefficients with order-unity values, as listed in Table 3 of Paper I. The free eccentricity, z_free, is the excess in eccentricity over what is forced by the resonance ("forced eccentricity"). We shall often shorten "free eccentricity" to "eccentricity," since the forced parts are mostly small for the pairs we consider (∼10⁻³). These expressions show that amplitude ratio of the two sinusoids is roughly related to the ratio of planet masses. They also highlight an inherent problem in TTV analysis: mass and eccentricity are difficult to disentangle. However, if they could be disentangled,⁴ TTV could be used to probe orbital eccentricity with a sensitivity of a couple percent (∼|Δ|). This is superior to other techniques like radial velocity or transit shape.We define nominal masses for the two planets based on the observed TTV amplitudes,

  $$\begin{eqnarray} m_{\rm nom}& \equiv & M_* \left|{V^{\prime }\Delta \over P^{\prime }g}\right|\pi j\,, \nonumber \\ m^{\prime }_{\rm nom}& \equiv & M_* \left|{V\Delta \over Pf}\right|\pi j^{2/3}(j-1)^{1/3} \,. \end{eqnarray} \tag{ 4 }$$

  The true mass for the inner planet is m = m_nom/|1 − Z_free/(2gΔ/3)|, and analogously for the outer planet. Hence, the nominal masses are typically upper limits: they are the true masses at zero eccentricity, but usually lie above the true masses when eccentricity is non-zero (Paper I).
• 3.  
  Phases of the TTV sinusoids encode information about the free eccentricity. TTV phases are defined relative to the time when the longitude of conjunction points at the observer. The two phases ($\phi _{\rm ttv},\phi ^{\prime }_{\rm ttv}$) should lie at (0, π) for zero eccentricity,⁵ while for most other eccentricities, the two phases can take other values but remain largely anti-correlated ($\phi ^{\prime }_{\rm ttv} \approx \phi _{\rm ttv} + \pi$). Equivalently, if the phases are close to (0, π), then the pair likely has a small eccentricity. We show below that TTV phase can be used to infer planet eccentricity (Section 3), and consequently, genuine planet masses (Section 4).

Here we describe how our sample of 22 TTV pairs are assembled.

Transit times for quarters 0–6 are published by Ford et al. (2012) for a large number of Kepler candidates. We select from this list all pairs within |Δ| ⩽ 8% from a first-order MMR (2:1, 3:2, 4:3, 5:4). For each pair, we obtain average orbital periods and TTV amplitudes and phases using the fitting procedure described in Paper I. We then calculate the periodogram for each TTV series and include a pair for further consideration only when both planets have periodograms that show a clear excess of power, by visual inspection, at the desired super-period (Equation (1)). We only include pairs with super-period shorter than 1000 days to ensure that more than half of the TTV sinusoids have been observed. Amplitudes and phases of the TTV sinusoids are measured at this super-period using a least-squares fit (Paper I). Weeding out pairs whose sinusoid phases are not roughly out of phase with each other, we find a total of 20 pairs that pass these thresholds, 6 of which are the confirmed systems analyzed in Paper I.

Compared to the exquisite TTV data for KOI 137.01/02 (Kepler-18c/d; Cochran et al. 2011), TTV data for typical Kepler candidates are less accurate since planets are typically smaller in size and sampled at a lower cadence. Our stringent weeding criteria are necessary to overcome random noise, and to isolate perturbations from other planets. The latter is a prominent issue in Kepler data, as planet pairs are often accompanied by other planets with near-resonant period ratios (e.g., KOI 82, KOI 500, and KOI 2169).

In Figure 1, we present the sinusoidal fits, error bars, and periodograms for six of the above pairs, KOI 156.01/03, KOI 775.02/01, KOI 841.01/02, KOI 1215.01/02, and KOI 1241.02/01. KOI 157.01/02, KOI 775.02/01, and KOI 1241.02/01 have also been identified as TTV pairs by Lissauer et al. (2011a) and Steffen et al. (2013) previously. For pairs that also appear in Xie (2012), we adopt his TTV fits for these systems as they are obtained using a longer baseline (through quarter 9). We also include two new pairs from that work, KOI 829.01/03 and KOI 1336.01/02, which our shorter duration data failed to pick up. However, we do not use three objects from that paper: KOI 877.01/02, KOI 880.01/02, and KOI 869.03/02; the first two because their TTV phases are not roughly anti-correlated, which may indicate pollution, and the last one because of an unphysically large density (and mass) for the inner planet: it requires a Jupiter mass for a 2.7 R_⊕ planet. It is also likely contaminated by a nearby planet.

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Thus, together with the 6 planet pairs analyzed in Paper I, we have a total of 22 convincing TTV pairs (19 published, 3 new). TTV solutions for these pairs, including their nominal planet masses, are listed in Table 1 and depicted in Figure 2.

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Table 1. TTV Analysis of 22 Planet Pairs

KOI	Δ	TTV Amplitudes (minutes)		TTV Phases (deg)		Inner Planet			Outer Planet
		|V|	|V'|	ϕttv	$\phi ^{\prime }_{\rm ttv}$	P	R	mnom(M⊕)	P'	R'	$m^{\prime }_{\rm nom} (M_\oplus)$
137.01/02	−.028	5.27(± 7%)	4.06(± 10%)	−4.3(± 4.0)	168.7(± 5.0)	7.6420	5.49	20.20 ± 2.02	14.8590	6.98	17.17 ± 1.20
157.06/01	0.011	12.99(± 40%)	4.85(± 52%)	97.3(±25.2)	237.0(±34.3)	10.3040	1.89	3.87 ± 2.01	13.0250	2.92	13.69 ± 5.48
157.03/04	−.027	7.10(± 48%)	20.55(± 68%)	25.7(±38.8)	159.3(±52.0)	31.9960	4.37	9.68 ± 6.58	46.6890	2.60	5.14 ± 2.47
168.03/01	0.008	43.98(± 37%)	18.95(± 23%)	−70.1(±19.3)	123.6(±18.2)	7.1070	1.90	14.65 ± 3.37	10.7420	3.20	55.35 ± 20.48
244.02/01	0.020	3.80(± 20%)	1.04(± 36%)	5.7(±11.8)	200.1(±21.4)	6.2390	2.80	7.91 ± 2.85	12.7200	5.30	14.59 ± 2.92
870.01/02	0.013	11.76(± 24%)	12.41(± 28%)	−46.0(±14.7)	128.5(±16.6)	5.9120	2.50	12.44 ± 3.48	8.9860	2.35	19.39 ± 4.65
952.01/02	−.011	8.88(± 26%)	11.15(± 33%)	45.2(±15.5)	227.8(±18.3)	5.9010	2.22	7.11 ± 2.34	8.7520	2.01	8.94 ± 2.32
1102.02/01	0.009	41.03(± 19%)	37.60(± 25%)	−3.9(±9.5)	189.5(±14.3)	8.1460	2.74	28.39 ± 7.10	12.3330	2.90	50.58 ± 9.61
148.01/02	0.012	4.00(± 27%)	3.31(± 24%)	−23.5(±17.6)	199.5(±18.5)	4.7780	2.14	13.96 ± 3.35	9.6740	3.14	8.93 ± 2.41
*152.03/02	0.016	7.92(± 33%)	26.21(± 19%)	−29.6(±28.6)	123.2(±8.7)	13.4850	2.59	65.59 ± 12.46	27.4030	2.77	10.22 ± 3.37
248.01/02	0.010	9.22(± 16%)	20.02(± 8%)	54.1(±9.3)	218.0(±5.5)	7.2040	2.72	9.07 ± 0.73	10.9130	2.55	6.83 ± 1.09
500.01/02	0.012	8.50(± 14%)	7.06(± 22%)	−9.1(±9.0)	178.0(±13.3)	7.0530	2.64	6.31 ± 1.39	9.5220	2.79	10.88 ± 1.52
*829.01/03	0.034	16.70(± 36%)	22.32(± 21%)	−33.9(±11.1)	135.9(±13.0)	18.6490	2.89	92.43 ± 19.41	38.5590	3.17	30.90 ± 11.12
898.01/03	0.028	7.34(± 29%)	10.80(± 43%)	63.8(±15.0)	213.2(±23.3)	9.7710	2.83	44.89 ± 19.30	20.0900	2.36	14.34 ± 4.16
*1270.01/02	0.013	3.17(± 36%)	37.15(± 8%)	96.6(±18.5)	281.7(±4.7)	5.7290	2.19	134.49 ± 10.76	11.6090	1.55	6.03 ± 2.17
1336.01/02	0.016	18.00(± 35%)	26.93(± 28%)	−61.7(±17.3)	125.6(±15.8)	10.2190	2.78	24.33 ± 6.81	15.5740	2.86	26.86 ± 9.40
1589.01/02	−.016	12.82(± 35%)	29.38(± 26%)	21.3(±26.3)	184.6(±14.1)	8.7260	2.23	29.45 ± 7.66	12.8820	2.36	20.12 ± 7.04
156.01/03	−.024	2.07(± 97%)	2.00(± 55%)	3.7(±74.8)	168.4(±33.9)	8.0410	1.60	2.37 ± 1.30	11.7760	2.53	3.79 ± 3.68
*775.02/01	0.040	11.81(± 24%)	13.94(± 38%)	−153.1(± 12.7)	40.6(±23.2)	7.8770	2.10	94.79 ± 36.02	16.3850	1.84	33.78 ± 8.11
841.01/02	0.021	29.38(± 30%)	20.13(± 70%)	30.5(±13.2)	198.1(±28.6)	15.3360	5.44	57.49 ± 40.24	31.3280	7.05	41.59 ± 12.48
*1215.01/02	−.047	9.86(± 81%)	25.93(± 54%)	29.9(±56.7)	183.8(±29.1)	17.3240	2.92	107.18 ± 57.88	33.0060	3.36	28.74 ± 23.28
*1241.02/01	0.019	149.21(± 17%)	58.46(± 39%)	125.9(±12.3)	360.7(±13.1)	10.5040	5.17	273.59 ± 106.70	21.4010	10.57	353.11 ± 60.03

Notes. A list of our sample of 22 TTV pairs. Here Δ is the distance to the nearest first-order MMR, |V| is the measured TTV amplitude (in minutes) for the inner planet, ϕ_ttv is its TTV phase (1-σ error bars in parentheses), P is its orbital period (in days), R is its radius (in R_⊕), and m_nom is its nominal mass (in M_⊕). The primed quantities are the same except for the outer planets. The first eight pairs are confirmed systems (corresponding to Kepler 18c/d, 11b/c, 11e/f, 23 b/c, 25 b/c, 28 b/c, 32 b/c, and 24 b/c, respectively), six of which were analyzed in Paper I. TTV parameters for the next eight pairs are adopted from Xie (2012), and TTV sinusoids for the last five pairs are reported in Figure 1. See Section 2.2 for the stellar parameters we adopt. Pairs marked with the * sign are suspected to have significant free eccentricity (Section 3). Typical error bars in planetary radii are of the order of 10%, smaller than those in TTV amplitudes. These are not included in calculating uncertainties in nominal densities.

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2.1. Selection Effects

2.2. Uncertainties in Stellar Parameters

One of the most important results from Paper I is the utility of the TTV phase (ϕ_ttv) for inferring the value of the free eccentricity. If a significant fraction of systems have ϕ_ttv ≈ 0, it can be inferred that most of those have |Z_free| ≲ |Δ|. In Paper I, we find that the TTV phases of the six analyzed pairs are clustered near zero, indicating that in general the free eccentricity is small (|Z_free| ⩽ Δ).

Now armed with 22 pairs, we explore the eccentricity distribution further. We find that for this larger sample too the TTV phases are not uniformly distributed, but cluster around zero (Figure 3), again indicating that the general population possesses small free eccentricity. However, there are a number of pairs that buck this trend. These are marked in Figure 4, based on three indices: TTV phase, nominal density, and transit duration. Both the TTV phases and nominal densities (compared to other planets of similar sizes) for KOI 775, 1241, and 1270 are large, suggesting high eccentricities. Since for the high-e population, ϕ_ttv is uniformly distributed between (− π, π), we expect to find pairs that have high-e but low TTV phases. Indeed, pairs KOI 152, 829, and 1215, while showing small phases, have high nominal densities. The transit durations for these planets are often longer than expected for circular orbits. We categorize these six pairs as the high-e population and analyze them in Section 5.1. It is interesting to note that all planets with nominal densities above ∼15 g cm⁻³ are now classified as high-e. The true densities of these planets are lower than the nominal values by some factors, depending on the actual eccentricities (Equation (2)). Unfortunately, our current strategy cannot uncover the actual eccentricities for these planets, but only lower limits.

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The remaining 16 pairs (∼75%) are consistent with low eccentricity. We assume them to be a homogeneous population satisfying a single eccentricity distribution. TTV phases depend on Z_free, a weighted sum of the two planets' free eccentricities (Equation (3)). Splitting into real and imaginary parts,

$$\begin{equation} z_{\rm free} = e_x + i e_y\,, \end{equation} \tag{ 5 }$$

and adopting a one-dimensional Gaussian distribution for these components (for both the inner and outer planets), P(e_x) = P_σ(e_x), P(e_y) = P_σ(e_y), with

$$\begin{equation} P_\sigma (e_x) = {1\over {\sqrt{2 \pi }}\sigma } \exp \left(- {{e_x^2}\over {2\sigma ^2}}\right), \end{equation} \tag{ 6 }$$

we can calculate the resulting distribution of TTV phases. We assume that e_x and e_y are uncorrelated, i.e., that the phases of the complex eccentricities are random, as they would be due to secular or general relativistic (GR) precession (Paper I). TTV phases are either uniformly distributed or clustered around zero, depending on the relative ratio of σ/|Δ|, and on the resonance of relevance. Generating a mock catalog that has the same period ratios as our low-e sample, we produce different phase distributions for different values of σ (Figure 5). The current sample lead us to conclude that a value of σ ≈ 0.007 provides the best fit, with an uncertainty in σ of the order of 20%. This value corresponds to an rms value of 0.01 in eccentricity ($e = \sqrt{e_x^2 + e_y^2}$). Thus, it appears that the low-e population has free eccentricities that are of order (but somewhat smaller than) the typical resonance offset (|Δ| ∼ 0.02).⁶

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This exercise demonstrates the exceptional sensitivity of TTV measurements to even small values of eccentricity. We discuss the implication of these results in Section 5.1.

Employing our newly found eccentricity distribution, we perform statistical simulations to correct for the eccentricity effect, and thereby obtain true planet densities for the low-e population, subject to statistical and measurement errors. Densities of the high-e sample are not recoverable.

For each planet pair, we generate complex free eccentricities ($z_{\rm free}, z^{\prime }_{\rm free}$) weighted by the intrinsic distribution inferred above, with σ = 0.007 (Equation (6)). Each pair of eccentricities produces correction factors relating the true and nominal masses (see below Equation (4)). Keeping only those generated eccentricity pairs that reproduce the observed TTV phases ($\phi _{\rm ttv}, \phi ^{\prime }_{\rm ttv}$) within the 68% error bars, the median correction factor then gives the median true density. This procedure introduces an uncertainty that we quantify by the 68% bounds in the distribution of correction factors. For the total uncertainty in density measurement, we add these bounds quadratically to the error bars in TTV amplitudes. The results are listed in Table 2. The ratio of median to nominal density depends on the observed TTV phases, but is typically of the order of 0.5. Even for planets with TTV phases consistent with zero, the correction factor falls below unity. This is because the most likely eccentricity value lies not at zero, but at e ∼ σ.

Table 2. Density after Correcting for Eccentricity Effect

KOI	ρ	ρ'	${{\rho }\over {\rho _{\rm nom}}}$	${{\rho ^{\prime }}\over {\rho ^{\prime }_{\rm nom}}}$	${{e+e^{\prime }}\over {2}}$
	(g cm−3)	(g cm−3)
137.01/02	$0.4^{+ 74\%}_{- 52\%}$	$0.2^{+ 29\%}_{- 28\%}$	0.66	0.82	0.007
157.06/01	$3.2^{+ 52\%}_{- 52\%}$	$3.1^{+ 40\%}_{- 40\%}$	1.00	1.00	0.000
168.03/01	$5.6^{+ 96\%}_{- 65\%}$	$3.7^{+107\%}_{- 73\%}$	0.47	0.40	0.008
244.02/01	$0.9^{+ 96\%}_{- 70\%}$	$0.4^{+ 38\%}_{- 37\%}$	0.43	0.72	0.007
870.01/02	$2.6^{+ 67\%}_{- 54\%}$	$4.2^{+ 73\%}_{- 55\%}$	0.57	0.50	0.008
952.01/02	$1.9^{+ 76\%}_{- 59\%}$	$2.8^{+ 82\%}_{- 59\%}$	0.51	0.45	0.008
1102.02/01	$3.3^{+ 80\%}_{- 57\%}$	$4.4^{+ 90\%}_{- 59\%}$	0.43	0.38	0.008
148.01/02	$2.2^{+ 95\%}_{- 63\%}$	$0.9^{+ 53\%}_{- 47\%}$	0.27	0.55	0.008
156.01/03	$2.6^{+ 82\%}_{- 68\%}$	$1.0^{+123\%}_{-107\%}$	0.80	0.76	0.007
157.03/04	$0.5^{+ 87\%}_{- 78\%}$	$1.3^{+ 82\%}_{- 64\%}$	0.83	0.79	0.008
248.01/02	$1.3^{+ 90\%}_{- 59\%}$	$1.1^{+105\%}_{- 65\%}$	0.53	0.47	0.008
1336.01/02	$4.7^{+ 77\%}_{- 58\%}$	$4.2^{+ 89\%}_{- 65\%}$	0.74	0.65	0.008
500.01/02	$1.0^{+ 76\%}_{- 53\%}$	$1.3^{+ 85\%}_{- 54\%}$	0.52	0.48	0.007
841.01/02	$0.6^{+ 81\%}_{- 79\%}$	$0.4^{+ 36\%}_{- 37\%}$	0.32	0.63	0.009
898.01/03	$3.7^{+ 49\%}_{- 50\%}$	$4.7^{+ 32\%}_{- 34\%}$	0.33	0.77	0.012
1589.01/02	$9.6^{+ 75\%}_{- 53\%}$	$5.1^{+ 93\%}_{- 63\%}$	0.65	0.60	0.007

Notes. Planet densities after applying the statistical model for the eccentricity distribution. The column ρ lists the median density for the inner planet (and primed quantities for the outer planet), a product of the median correction factor (ρ/ρ_nom) and the nominal density. Error bars on the correction factor reflect the width within which 68% of all possible solutions fall. Error bars on the median density are a quadratic sum of uncertainties in the correction factor and uncertainties in nominal mass (measurement error in TTV amplitude). The last column lists the average of the two median eccentricities as a rough indicator for the magnitude of eccentricity in the pair. Our values for KOI 137.01/02 (Kepler 18c/d) agree with previous determinations of ρ = 0.59  ±  0.07, ρ' = 0.27  ±  0.03 (Cochran et al. 2011). We do not perform a correction for KOI 157.06/01 (Kepler 11b/c) as its TTV phase is polluted by other resonances.

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The median densities and their error bars are plotted in Figure 6. Combining our results with a sample of low-mass planets that have previously determined masses (Table 3; mostly through radial velocity), we find a best-fit mass–radius relation of

$$\begin{equation} M \approx 3 \,M_\oplus \, \left({R\over {R_\oplus }}\right)\,, \end{equation} \tag{ 7 }$$

or a density ρ ≈ 3ρ_⊕(R/R_⊕)⁻² where the density of the Earth is ρ_⊕ = 5.5 g cm⁻³. This best fit differs from the one that applies for solar system planets (Lissauer et al. 2011b), M = (R/R_⊕)^2.06 M_⊕, or ρ ≈ ρ_⊕(R/R_⊕)⁻¹.

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It is somewhat premature to discuss the significance of this difference for a number of reasons. First, we do not yet know what gives rise to the solar system scaling, or if there is anything deep behind it. Second, our mass–radius relation is only an empirical fit to a select group of planets that are situated around their host stars for 10 days, have radii ⩾1 R_⊕, and masses ∼10–20 M_☉. It may not be universally applicable. We do, however, note that the above mass–radius relation corresponds to one that has a constant surface escape velocity of 20 km s⁻¹. This is fortuitously close to the sound speed of hydrogen plasma at 10⁴ K (13 km s⁻¹), suggesting that the process of photoevaporation of hydrogen may be involved in some way.

In the next section, we discuss further details of our findings as well as their implications.

We first divide planets into two groups: "mid-sized" (those with R ⩾ 3 R_⊕) and "compact" (those smaller). This division line is partly inspired by an observation of the overall Kepler sample (B12): there is a drastic falloff in planet numbers around 3 R_⊕ (right panel of Figure 8), suggesting that a transition occurs. This division is also motivated by the distinction in density between these two groups of planets: the "mid-sized" ones are lighter than water planets, while "compact" ones are of order or denser. We aim to discover the nature of this transition.

5.1. Dynamical Excitation

5.2. Mid-sized Planets, R ⩾ 3 R_⊕

5.2.1. Structure and Photoevaporation

Mid-sized planets are a minority. They comprise 23% of the total Kepler planet candidates.⁹ Figure 6 shows that mid-sized planets are less dense than water and must have extensive H/He envelopes. Based both on the evolutionary calculations of Fortney et al. (2007) and on our simple model (see below), we conclude that the gas envelopes of these planets are comparable in mass to their solid cores (also see Gillon et al. 2007; Bakos et al. 2010; Baraffe et al. 2008 for RV planets). The actual composition of their solid cores does not affect this conclusion as the core sizes are small compared to the planet sizes.

We calculate the fractional masses in their hydrogen envelopes that are required to account for their observed densities and sizes. We assume that the cores of these planets are made of pure rock, with the core radii and masses related by Equation (7) of Fortney et al. (2007). We assume that the photospheres of all planets are at a pressure of 1 bar, and a temperature that is solely determined by irradiation, T = T_eq, where T_eq = T_*(R_*/2a)^1/2 is the equilibrium temperature of a blackbody at a distance a around a star with temperature T_* and radius R_*. We integrate the thermal structure of a solar-composition gas inward toward the core surface, using the equation of state from Saumon et al. (1995) and opacity from Allard et al. (2001). The temperature structure we obtain is nearly isothermal near the surface and adiabatic in the deep interior. The core mass is adjusted until the observed total mass is reproduced. The result depends on the assumed internal luminosity since the latter affects the temperature gradient and therefore the pressure scale height at the base of the envelope. For a luminosity of 10⁻¹¹L_☉ (the current value on Neptune), the mid-sized planets require hydrogen envelopes of order 2%–30% in mass (Figure 7). The fractions rise to 10%–90% when the internal luminosity drops by a factor of 100.

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The mid-sized planets exhibit surface escape velocities comparable to the sound speed of an H ii region (10⁴ K, 13 km s⁻¹), suggesting that photoevaporation is important for their evolution. This is corroborated by the absence of mid-sized planets with periods shorter than a few days (see Figure 11). Here, we investigate this possibility quantitatively.

We estimate mass loss due to photoevaporation following Murray-Clay et al. (2009) for X-ray and ultraviolet (XUV) photoevaporation. For more detailed treatments, see Owen & Jackson (2012); Lammer et al. (2013). Let the stellar XUV luminosity be L_UV. When this flux is absorbed by the planetary atmosphere, it drives a thermal expansion and outflow. In the limit of low flux, the outflow rate can be approximated by the so-called energy-limited formula,

$$\begin{eqnarray} {\dot{M}} & \approx & \epsilon {{L_{\rm UV} \pi R_p^2}\over {4 \pi a^2 \, v_{\rm esc}^2}}\, \sim 1.2\times 10^{10} \,{\rm g}\,\,{\rm s}^{-1} \left({{L_{\rm UV}}\over {5\times 10^{-6} L_\odot }}\right)\, \nonumber \\ & & \times \left({{0.1 {\rm AU}}\over {a}}\right)^2 \left({{R_p}\over {5 \,R_\oplus }}\right)^3 \left({{15 \,M_{\oplus }}\over {M_p}}\right), \end{eqnarray} \tag{ 8 }$$

where we have scaled the XUV luminosity by 5 × 10⁻⁶L_☉, roughly the current solar value, and have assumed an efficiency () of unity.¹⁰ We have verified that the radiation/recombination-limited mass-loss rates, relevant when heat loss is significant (Murray-Clay et al. 2009), do not apply: they lie above the energy-limited rate for all our objects (compact and mid-sized). We further note that the formally defined sonic radius, $r_s = G M_p/2 c_s^2$ (Parker 1965), frequently lies inside the physical radius for the low surface-escape planets we consider here.

Estimates for the fractional mass loss, integrated over 5 Gyr, are presented in Figure 7. We have uniformly taken the XUV luminosity to be 5 × 10⁻⁶ of the bolometric value, and have adopted values for stellar radius and effective temperature from the B12 catalog for the TTV planets (see Section 2.2), and from discovery papers for planets in Table 3.¹¹ We find that mid-sized planets could shed between a few percent to a few tenths of their masses, comparable to the hydrogen mass fractions inferred to be present on these planets (gray symbols in Figure 7). Our estimates for the mass-loss rate are for the current systems. The rates were likely higher in the past when the stars were more chromospherically active and brighter in the FUV (see, e.g., Lammer et al. 2004; Ribas et al. 2005), and also when the planets were younger and fluffier (also see Lopez et al. 2012). However, this active phase typically lasts ∼100 Myr, a short interval compared to the stellar age. In addition, the fractional XUV flux may be higher for stars lower in mass than the Sun. We therefore conclude that photoevaporation has significantly sculpted these mid-sized planets. More detailed photoevaporation calculations are presented in Owen & Wu (2013).

In the following, we further argue that photoevaporation may explain the mass–radius relation for mid-sized planets (Equation (7)), and naturally produces planets with comparable core/envelope masses (also see discussions in Lopez et al. 2012).

Photoevaporation proceeds healthily, as described by Equation (8), as long as the surface escape velocity of the planet remains low. When the latter rises, the sonic radius (for a wind of ionized hydrogen) moves much beyond the planet surface. This throttles the photoevaporation rate as the radiation/recombination-limited mass-loss, now relevant, decreases exponentially with the expansion of the sonic surface (Murray-Clay et al. 2009). Now imagine a gas-rich planet that was exposed to the strong XUV flux from its young host star before it has thermally contracted, either because it was formed in situ or was migrated very quickly (Equation (8); also see Rogers et al. 2011). If such a planet started with much more gas than that in its core, evolutionary calculations (Fortney et al. 2007) show that its size will expand during mass loss, leading to further evaporation. This process continues until the mass of the solid core contributes significantly to gravity. At this point, the planet shrinks when it loses mass until the escape speed from its surface exceeds the thermal speed of ionized hydrogen. Planets thus sculpted are expected to have envelopes that are lower or comparable in mass to their cores. These arguments highlight the need for a more accurate model, where one accounts for the time evolution of both the XUV luminosity and planet structure (see, e.g., Owen & Jackson 2012).

Regardless of the evolution, the mid-sized planets almost certainly acquired their massive hydrogen envelopes by accreting from their proto-planetary disks.

5.2.2. Correlation with Stellar Mass

We focus here on one remarkable correlation between mid-sized planets and their host properties, discovered in the published Kepler catalog. In Figure 8, we display the distribution of planet sizes as a function of their host mass. Mid-sized planets occur almost exclusively around stars more massive than 0.8 M_☉, or T_eff ⩾ 5000 K on the main sequence. Quantitatively, the fraction of mid-sized planets, 23% of the overall sample, is nearly zero for stars less massive than 0.8 M_☉, and of the order of half for stars more massive than 1.1 M_☉. In contrast, compact planets are more equitably distributed around stars of all spectral types, albeit with a trend that average planet sizes decrease around lower mass stars.

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This result differs from that in Howard et al. (2012) where they concluded that, albeit with weak statistics, the occurrence of planets with radii larger than 4 R_⊕ does not correlate with T_eff. The difference may stem from the fact that while they included all candidates into their study, Figure 8 only profiles systems with multiple transiting planets. We suggest that their conclusion may be contaminated by false positives that are more populous among single transiting systems. Further studies are required to settle this difference.

The fact that K/M dwarfs do not host mid-sized, gas-rich planets¹² has not been noted previously. We offer two possible explanations but no final resolution. First, photoevaporation around lower mass stars could have been more severe than around F/G dwarfs as a result of more potent and more prolonged chromospheric activity. Second, planetary cores may take longer to form around lower mass stars, so that they cannot accrete gas before the proto-planetary disks disperse. Both scenarios can also explain why compact planets tend to be smaller around lower mass stars.

5.3. Compact Planets, R < 3 R_⊕

We turn now to the compact planets. The measured densities of compact planets range from ∼1 to ∼10 g cm⁻³ (Figure 6). The largest compact planets (R ∼ 2–3 R_⊕) have densities that hover around the water line (ρ ∼ 2 g cm⁻³). At any given size, planets span almost a decade in density. While the mid-sized planets are most likely cores overlaid with massive H/He envelopes, the structure of compact planets is less certain. They can either be water–rock mixtures, or rocky cores enveloped by a small amount of hydrogen, or a mixture of the two. It is important to resolve this degeneracy because a water-laden planet would favor the scenario where the planet was constructed outside the ice line and then migrated inward; a rocky core would be consistent with in situ formation. Although density measurements of individual planets cannot break this inherent degeneracy (see, e.g., Adams et al. 2008), we attempt to break it with our statistical sample. Fortuitously, the sample straddles the region where hydrogen photoevaporation can significantly alter planet sizes.

5.3.1. Density Correlation with Temperature

Planet densities exhibit a remarkable correlation with environment: planets with hotter equilibrium temperatures are in general denser (Figure 9). In Figure 9, we scale each planet's density to the density of a pure-water (or pure-rock) planet of the same mass (see the theory curves in Figure 6).¹³ Cold planets (T_eq ≈ 600 K) are compatible with being pure water, while hot planets (T_eq ⩾ 1500 K) are compatible with being pure rock. Similar behavior has been noted by Lopez et al. (2012).

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A second piece of evidence for this correlation is furnished by studying the density ratio within planet pairs (top panel of Figure 10). Unlike individual planet mass, the mass ratio of two planets within a pair can be largely determined without knowledge of the free eccentricity. Thus here we simply plot the density contrast as the ratio of their nominal densities. We find that the inner planets in pairs near 2:1 resonance are always denser by a factor of a few compared to the outer ones, while pairs near closer resonances (5:4, 4:3, and 3:2) have more comparable densities.¹⁴ Thus some process appears to be at work to systematically compactify the inner planets.

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The last piece of evidence comes from studying sizes of compact planets in the B12 catalog. The bottom panel of Figure 10 shows the size ratio within a planet pair as a function of its period ratio. We only plot pairs with inner period P < 10 days. Inner planets tend to be smaller than their companions. This trend is more significant for more widely spaced pairs. It is absent for pairs orbiting too far from the star (P > 10 days). We also plot planet sizes versus orbital period or equilibrium temperature in Figure 11. One observes that compact planets farther away from their stars can be larger than those closer to their stars (also see, e.g., Figure 8 of B12). The median size for planets inward of 5 days is R ∼ 1.5 R_⊕ compared with R ∼ 2.5 R_⊕ for planets between 5 and 20 days. These results suggest that planets inward of ∼10 days, or ∼1000 K around a Sun-like star, show signs of compactification.

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After the submission of this paper, Weiss et al. (2013) published a related study based on radial velocity data. They showed that, for planets in the mass range from a few to 20 M_⊕, planet densities correlate with stellar incident flux. This is consistent with the correlation we discover here.

5.3.2. Rock or Water?

A naive explanation for Figures 9 and 10 is that all compact planets started out as water/rock mixtures with water being the dominant component, but the hottest ones have experienced total water removal and became rocky spheres.

This, however, seems difficult on theoretical grounds. It is impossible to photoevaporate water (or any other high molecular weight material). Even if heated to 10⁴ K, water cannot escape from the surface of the observed planets because its sound speed would be ∼2 km s⁻¹, too low compared to the escape velocity.¹⁵ Since the mass loss rate is determined by the smaller of the energy-limited and radiation/recombination-limited rates, and since the latter is exponentially suppressed when the sound speed is smaller than the escape speed (Murray-Clay et al. 2009), the mass loss will be severely inhibited.¹⁶ This conclusion differs from that of Valencia et al. (2010) who adopt the energy-limited mass-loss rate for water and hence find significant water loss.

Another scenario, one that may have happened on Venus, is to photodissociate water molecules into H₂ and oxygen and then remove H₂ via either Jeans escape or hydrodynamical outflow driven by photoevaporation. This would require a reductive agent to absorb the free oxygen. If this reductive agent is atomic iron (see, e.g., Elkins-Tanton & Seager 2008), the original planet would have to contain 70% iron and 30% water in mass to absorb all free oxygen into Fe₂O₃. Such a planet, at a mass of 10 M_⊕, would have a density of ∼6 g cm⁻³ (cf. the theoretical curves in Figure 2; Fortney et al. 2007), much denser than the cold planets we observe. In addition, for the newly produced hydrogen to be photoevaporated, it cannot be well mixed with the other heavy molecules. Thus we consider this option ineffective in compactifying Kepler planets.

Instead we propose that the observed correlations can be explained if all compact planets started out as rocky cores overlaid with various amounts of hydrogen. Planets experiencing the strongest radiation would have their hydrogen completely depleted, exposing dense rocky cores. We quantitatively investigate this possibility here.

As a useful rule of thumb, at T_eq = 1000 K, a hydrogen envelope of 1% in mass will expand the size of its planet by 25%–40%, depending on the core composition and mass.¹⁷ Another useful rule of thumb is the timescale to erode 1% of the planetary mass. We recast Equation (8) as

$$\begin{eqnarray} {{1\% M_p}\over {\dot{M}}} & \approx & {5\,{\rm Gyr}}\, \left({{L_{\rm UV}}\over {5\times 10^{-6} L_\odot }}\right)^{-1} \nonumber \\ & & \times \left({{0.1 {\rm AU}}\over {a}}\right)^{-2} \left({{R_p}\over {2.5 \,R_\oplus }}\right)^{-3} \left({{7.5 \,M_\oplus }\over {M_p}}\right)^{-2}\!\!, \end{eqnarray} \tag{ 9 }$$

where we have scaled the variables using typical parameters for compact planets. Since typical planets in our sample orbit with period ∼10 days, they just straddle the range where photoevaporation can make an order-unity change to their radii.

For the compact planets, mass fractions in the hydrogen envelope run from 0% to ∼1% (gray symbols in Figure 7; see assumptions for the calculations in Section 5.2.1). Figure 7 also shows that smaller compact planets in our sample could have experienced evaporation loss that ranges from 1% to 20% in mass, while larger compact planets typically suffer loss of the order of 1% in mass or lower. This then makes a self-consistent story. Larger planets are larger because they can hold on to their hydrogen envelope, while smaller ones have lost it all, mostly as a result of their close proximity to the stars.

We can stretch the calculation a little further. Assuming that the planetary cores are not fully rocky (as assumed above) but are instead made up of a half-water/half-rock mixture, we would then require hydrogen mass fractions of the order of 10⁻³ or less to explain the observed planets. Such a thin atmosphere has no chance of survival for most of the planets in our sample. The situation is worse if the cores are made up of pure water. We conclude that the cores of these planets are likely composed of rock (or even denser material). This statement is substantially weakened, however, if the internal luminosity of the planet is much lower than that of Neptune.

Thus we are able to resolve the structural degeneracy in compact planets, helped by the fact that typical planets in our sample have T_eq ≈ 1000 K. Our proposal, that compact planets are made up of rocky cores overlaid with hydrogen ⩽1% in mass, explains the rock-like density of hot planets (e.g., Kepler-10b, CoRoT-7b, and 55 Cnc e), explains the factor of ∼2 difference in size between hot and cold compact planets in the Kepler catalog, and explains the density contrast within 2:1 pairs and the lack of density contrast in other closer pairs.

The hydrogen mass fractions we infer for many of the compact planets are of the order of 1%. This can be primordial in origin. Alternatively, as suggested by Rogers et al. (2011) and Elkins-Tanton & Seager (2008), outgassing and the subsequent breakdown of water could produce a hydrogen envelope ⩽1% in mass. The former option implies a fast formation timescale, while the latter implies a significant amount of primordial water (⩾10% in mass) when the planet was formed.

5.4. Rock or Water: The Lopez et al. (2012) Study

5.5. Compact versus Mid-sized

Based on two samples of low-mass planets—22 planet pairs characterized by the TTV method and a dozen planets characterized by the radial velocity technique—we reach the following conclusions.

• 1.  
  TTV phases for most of our near-resonant pairs lie close to zero. We conclude that the majority of these pairs are consistent with an eccentricity distribution that has a root-mean-squared value of e ∼ 0.01. About a quarter of the pairs, on the other hand, have eccentricities as large as 0.1–0.4. True planet masses for the low-eccentricity population can be recovered statistically, based on their inferred eccentricity distribution.
• 2.  
  Masses of planets are roughly proportional to their radii, such that the best-fit solution corresponds to a population that has a constant surface escape velocity of 20 km s⁻¹.
• 3.  
  Mid-sized planets (R ⩾ 3 R_⊕) in our sample invariably have such low densities that they have to contain substantial H/He envelopes. Their current rates of photoevaporation suggest that their masses and radii may have been limited by this process and that masses in their cores should be at least comparable to those of their gaseous envelopes.
• 4.  
  Mid-sized planets, some 23% of the B12 sample, show up exclusively around stars more massive than 0.8 M_☉. Perhaps, relatedly, planets around lower mass stars tend to have smaller sizes. We do not have an explanation for these observations.
• 5.  
  Densities of compact planets (R ⩽ 3 R_⊕) fall between that of pure rock and pure water, with some even less dense than water. Planets with higher equilibrium temperatures tend to be denser and smaller. Planets in our sample fortunately straddle the region where photoevaporation could significantly erode a 1% hydrogen envelope. This allows us to break the degeneracy in internal composition and show that it is possible to explain the observed correlations if these planets are mostly likely rocky cores overlaid with a layer of hydrogen that is ⩽1% in mass. These planets are likely not water worlds.
• 6.  
  The data suggest that 3 R_⊕ is a dividing line between "super-Earths" (largely solid) and "hot Neptunes" (extensive gaseous envelopes). The atmospheres on mid-sized planets were most likely accreted, while those on compact planets may have come from accretion or outgassing.

We foresee a number of directions to pursue in the future. First, more pairs are needed. We are currently restricted to pairs in the immediate vicinity of resonances, and ones that orbit with periods of 5–12 days. Transit timing data of a longer span or a higher quality would allow one to probe many more planet pairs. It will be of interest to see whether the density–T_eq correlation (Figure 9) extends to larger ranges. More pairs will also provide a more accurate eccentricity distribution. At the moment, uncertainties in mass determination arise both from measurement errors in TTV amplitudes, and from statistical uncertainties when converting nominal mass to genuine mass.

Our conclusion that gas-rich planets appear to have suffered significant photoevaporation helps explain why there are so few of them close to the star. It also predicts that more of these gas-rich planets will show up at longer periods. The extended Kepler mission should be able to resolve the issue. It is also of interest to follow self-consistent thermal and photoevaporation evolution for these planets in order to determine their initial states. It remains puzzling how such low-mass planets could have accreted so much hydrogen, so close to the star. Lastly, the origin of hydrogen on compact planets is an interesting question to pursue. If it was accreted directly from the disk, these planets had to form before the disks disperse. If it was outgassed from broken-down water, these planets had to have water-rich interiors when they formed.

Most of our planets, gas-rich or compact, have masses ≲ 20 M_⊕, a decade lower in mass than the Jovian planets at the same period range. This mass, incidentally, is roughly the gap-opening mass in a disk with a scale height H/R ≈ 0.03. This may be a valuable clue for planet formation.

Moreover, other processes not considered in this study may be important. We argue that an icy object cannot be converted into a rocky ball. But we have not considered ice removal by bombardment of planetesimals or proto-planets.

Lastly, we need to understand the source of eccentricity in multiple low-mass planet systems.

We thank Doug Lin, Eugene Chiang, James Owen, Jonathan Fortney, Eric Lopez, and Daniel Huber for interesting discussions. Y.W. acknowledges the hospitality of KIAA-PKU, where part of this work was performed. This research is supported by NSERC and the province of Ontario, as well as NSF Grant AST-1109776. This study would not have been possible without the amazing work by the Kepler team.