SECURE MASS MEASUREMENTS FROM TRANSIT TIMING: 10 KEPLER EXOPLANETS BETWEEN 3 AND 8 M_⊕ WITH DIVERSE DENSITIES AND INCIDENT FLUXES

We simulate planetary orbits with an eighth-order Dormand–Prince Runge–Kutta integrator (Fabrycky 2011; Lissauer et al. 2011a, 2013; Jontof-Hutter et al. 2014, 2015) and compare simulated transit times to the observed transit times. TTVs are expressed as the difference between the observed transit time and a calculated linear fit to the transit times (O–C). In each case, we have assumed coplanar orbits since TTV amplitudes change only to second order in mutual inclinations (Lithwick et al. 2012; Nesvorný & Vokrouhlický 2014).

For planet pairs with large orbital period ratios, TTVs can be sensitive to mutual inclinations, but mutual inclinations do not make a major contribution to TTVs of planets near low-order mean motion resonances (Payne et al. 2010), which dominate the TTVs in our sample. Furthermore, the distribution of transit duration and orbital period ratios among Kepler's multiplanet systems imply that typical inclinations in Kepler's population of multitransiting systems are a few degrees or less (Fabrycky et al. 2014).

We assume that all the TTVs are attributable to mutual perturbations between the known transiting planet candidates. In some multiplanet models, we exclude transiting planets that have no observed and no expected TTVs given the ratio of their orbital periods with the planets that have detected TTVs. In these cases we performed integrations including all known planets to verify the accuracy of this approximation.

For any multiplanet model, we fit five parameters per planet: the orbital period (P), the time of the first transit (T₀) after our chosen epoch (BJD –2,455,680), $k=e\mathrm{cos}\omega$, $h=e\mathrm{sin}\omega$, and the ratio of the planet masses to their host, which we refer to as "dynamical mass" throughout. We express dynamical masses in the form $\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$ for easy conversion to Earth masses for any estimate of stellar mass.

2.2.1. Likelihood Function

We evaluated the goodness of fit to the transit timing data (${\boldsymbol{x}}$) for each simulated model ${ \mathcal M }$ for a choice of parameters ${\boldsymbol{\theta }}$ using the likelihood function

$$\begin{eqnarray}&&{ \mathcal L }(\theta | {\boldsymbol{x}},{ \mathcal M })=\displaystyle \prod _{i=1}^{n}\displaystyle \frac{\mathrm{exp}\left[-\frac{1}{2}{\left(\frac{{y}_{i,\theta }-{x}_{i}}{{\sigma }_{i}}\right)}^{2}\right]}{\sqrt{2\pi {\sigma }_{i}^{2}}},\end{eqnarray} \tag{ 1 }$$

where there are n observed transit times, and ${\boldsymbol{y}}$ are simulated transit times associated with the model parameters ${\boldsymbol{\theta }}$. To minimize numerical round-off error in this product, we calculate the familiar ${\chi }^{2}={\sum }_{i=1}^{n}{\left(\frac{{y}_{i,\theta }-{x}_{i}}{{\sigma }_{i}}\right)}^{2}$ statistic for all model evaluations:

$$\begin{eqnarray}&&{\chi }_{\mathrm{eff}}^{2}=-2\mathrm{log}({ \mathcal L })=\displaystyle \sum _{i=1}^{n}\mathrm{log}(2\pi {\sigma }_{i})+{\chi }^{2}.\end{eqnarray} \tag{ 2 }$$

In comparing any two models, the ratio of log-likelihoods is proportional to the difference in χ², and the summation in Equation (2) cancels.

2.2.2. Priors

2.3.1. Eccentricity Priors

For our primary results in Table 11, we have assumed a Gaussian prior on each of the eccentricity vector components ($e\mathrm{sin}\omega$, $e\mathrm{cos}\omega$) with a mean of zero and variance 0.1. This corresponds to a Rayleigh distribution with scale width 0.1 as the prior for scalar eccentricity. This is consistent with the distribution of eccentricities among Kepler's exoplanets found by Moorhead et al. (2011) using an independent analysis of measured transit durations. This wide distribution is also consistent with the known eccentricities of giant RV planets (Kane et al. 2012; Plavchan et al. 2014). A narrower eccentricity distribution of characteristic width ∼0.05 was found by Van Eylen & Albrecht (2015) by comparing a sample of well-characterized transiting exoplanet hosts with their inferred photometric densities.

An even narrower eccentricity distribution (of characteristic width σ = 0.02) was found for a sample of compact multiplanet systems with TTVs by Hadden & Lithwick (2014). While this is the most relevant sample for our purposes, we adopt the wider prior to test where the TTVs can provide tight constraints on eccentricity. Where eccentricities are poorly constrained by the TTVs, we compare our dynamical masses with solutions from a narrower eccentricity prior (with scale length 0.02), to test whether our masses and orbital solutions are robust against the choice of prior for eccentricity.

In most cases, we found that individual eccentricities are weakly constrained, with the posterior for the eccentricity vector components being strongly affected by our prior. However, we have inferred tight constraints on relative eccentricity vectors since the vector components are highly correlated. This correlation is expected for near-first-order mean motion resonant TTVs, as can be seen from the analytical solutions derived by Lithwick et al. (2012). The solution breaks the eccentricity vector up into the sum of two component vectors: a free component that is constant for timescales that are short compared to apsidal precession, and a forced component that varies over the coherence time of near-resonant perturbations. Each planet's free eccentricity vector can be expressed in complex notation (${z}_{\mathrm{free}}=k+{ih}={e}_{\mathrm{free}}\mathrm{cos}\omega +{{ie}}_{\mathrm{free}}\mathrm{sin}\omega$), although the periodic TTV signal is not sensitive to these directly. Rather, the amplitude of a sinusoidal TTV signal caused by a mutually interacting pair near first-order resonance depends on the mass of the perturber, the orbital periods of both planets, and the conjugate of the complex sum of free eccentricities, Z_free, where, given the free eccentricity of an inner planet ${z}_{\mathrm{free}}={e}_{\mathrm{free}}\mathrm{exp}(i\omega )$, and an outer planet ${z}_{\mathrm{free}}^{\prime }={e}_{\mathrm{free}}^{\prime }\mathrm{exp}(i{\omega }^{\prime })$,

$$\begin{eqnarray}&&{Z}_{\mathrm{free}}={{fz}}_{\mathrm{free}}+{{gz}}_{\mathrm{free}}^{\prime }.\end{eqnarray} \tag{ 3 }$$

Here the coefficients f and g are the sums of Laplace coefficients of order unity and solved to first order by Lithwick et al. (2012) as a function of orbital period ratios. The TTVs are sensitive to the complex conjugate of the expression in Equation (3), ${Z}_{\mathrm{free}}^{*}$. Since f ≈ −1, ${Z}_{\mathrm{free}}^{*}$ is the difference between eccentricity vector components scaled by the coefficients f and g and hence approximately proportional to the relative eccentricity. This leads to an eccentricity–eccentricity degeneracy in TTVs, where the relative eccentricities between planets may be well constrained, but their absolute eccentricity vector components are poorly constrained. With ${Z}_{\mathrm{free}}^{*}$ constant, the derivatives of the vector components of the outer planet with respect to the same component of the inner planet depend only on Laplace coefficients. The gradients are

$$\begin{eqnarray}&&\displaystyle \frac{d({e}^{\prime }\mathrm{cos}{\omega }^{\prime })}{d(e\mathrm{cos}\omega )}=\displaystyle \frac{d({e}^{\prime }\mathrm{sin}{\omega }^{\prime })}{d(e\mathrm{sin}\omega )}=\displaystyle \frac{-f}{g}.\end{eqnarray} \tag{ 4 }$$

Hence, if the orbital periods are known precisely, one can easily predict the correlation between the eccentricity vector components of neighboring planets that have detected TTVs.

2.3.2. Non-Gaussian Uncertainties in Transit Times

Figure 1 shows the distribution of residuals for all measured and model transit times in this study at the best-fit solution for each planet, with comparisons to three normalized and symmetric statistical distributions, a Gaussian, and Student's t distributions with 2 and 4 degrees of freedom, respectively.

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The central peak slightly exceeds a Gaussian distribution, consistent with moderate overfitting in our best-fit models, but unlikely to affect our Markov Chain Monte Carlo (MCMC) posteriors. On the other hand, the wings of the distribution reveal far more outliers than expected from Gaussian uncertainties on measured transit times. These data have more leverage in model fitting. Hence, to check the robustness of our results against these outliers, we repeated our analysis assuming that measurement uncertainties followed a Student's t distribution with 2 degrees of freedom. In this case, the likelihood function is

$$\begin{eqnarray}&&{ \mathcal L }(\theta | {\boldsymbol{x}},{ \mathcal M })=\displaystyle \prod _{i=1}^{n}\displaystyle \frac{1}{2\sqrt{2}{\sigma }_{i}}{\left(1+\displaystyle \frac{{({y}_{i,\theta }-{x}_{i})}^{2}}{2{\sigma }_{i}^{2}}\right)}^{-3/2},\end{eqnarray} \tag{ 5 }$$

and the log of the likelihood becomes

$$\begin{eqnarray}&&\mathrm{log}({ \mathcal L })=n\mathrm{log}\left(\displaystyle \frac{1}{2\sqrt{2}{\sigma }_{i}}\right)-\displaystyle \sum _{i=1}^{n}\displaystyle \frac{3}{2}\mathrm{log}\left(1+\displaystyle \frac{{({y}_{i,\theta }-{x}_{i})}^{2}}{2{\sigma }_{i}^{2}}\right).\end{eqnarray} \tag{ 6 }$$

2.3.3. Independently Measured Transit Times

The systems were found by identifying appropriate period ratios of all planet pairs in the online exoplanet database at the NASA Exoplanet Archive. ⁹ We chose suitable systems for measuring dynamical masses by performing maximum-likelihood fits to the long-cadence transit timing data of all known systems with one or more pairs of planets with near-resonant orbital periods, using the Levenberg–Marquardt algorithm, and with measured transit times from Rowe & Thompson (2015). The Levenberg–Marquardt algorithm efficiently converges to a local minimum near the initial values for free parameters. One can estimate uncertainties from the local curvature of the χ² surface at the local minimum. However, a major limitation of this method for estimating uncertainties is that it approximates the χ² surface as parabolic and unimodal in the dimensions of all free parameters. We tested this assumption for our TTV models by searching for regions of high likelihood using repeated Levenberg–Marquardt minimization starting with initial values of $e\mathrm{sin}\omega$ and $e\mathrm{cos}\omega$ along a grid for each planet. We used a single set of values for the initial orbital periods and phases since these quantities are tightly constrained by the transit times. In all cases, we found multiple but statistically consistent solutions near the overall best fit.

Performing initial fits on this sample of systems, we sought systems with likely detections of low-mass exoplanets. We selected systems with a wide range of orbital periods, including systems with apparent synodic chopping (including Kepler-26 and Kepler-177). We rejected systems where known stellar rotation periods are nearly commensurate with a planetary orbital period, including Kepler-128 (KOI-274), whose 13.6 day rotation period has a near 5:3 commensurability with the orbital period of Kepler-128 c, since this could cause spurious TTV signals that would affect our mass measurements.

We rejected systems where the posterior did not provide useful lower bounds on planetary masses. However, we included the three-planet system Kepler-105 that included two weak detections, because the third planet was well constrained in our Levenberg–Marquardt fits, and the planets have particularly short orbital periods compared to most planets with TTV analyses, which may make this system a viable target for RV follow-up. Kepler-26, Kepler-29, Kepler-49, and Kepler-57 were among the first systems to have planets confirmed via anticorrelated TTV signals (Fabrycky et al. 2012; Steffen et al. 2012b, 2013). We identified Kepler-177 as having a particularly strong chopping signal, deep transits, and low masses, indicative of likely extremely low planetary densities. We also noted strong TTVs at Kepler-60, possibly due to librations within a three-body Laplace-like resonance chain (Goździewski et al. 2016).

To appropriately estimate relevant parameters with their uncertainties, we explored regions of interest that had been found via our initial fits using a Differential Evolution Markov Chain Monte Carlo algorithm (DE-MCMC; Ter Braak 2006; Nelson et al. 2014; Jontof-Hutter et al. 2015). Using MCMC allows for the exploration of a region of interest without becoming trapped in shallow local minima. Of the many variants of MCMC, the DE-MCMC algorithm is particularly efficient for exploring posteriors of correlated variables, which, as we shall see, is common in the high-dimensional modeling of TTV systems. It employs multiple "walkers" exploring the region of interest in parallel, with proposals calculated from the displacement vectors between other walkers chosen at random, increasing the likelihood that a proposal will be roughly parallel to the direction of correlation between variables. We employed three times as many walkers as free parameters in our model fitting, i.e., 30 walkers for the two-planet systems and 45 walkers for the three-planet systems.

For planet pairs near first-order mean motion resonances, the TTVs are dominated by the sinusoidal signal expressed in the approximations of Lithwick et al. (2012), from which we identify the degeneracy in mass and eccentricity in TTV inversion. In some cases, however, the degeneracy can be broken where high-frequency TTVs are detected, like the so-called chopping at synodic frequencies (Nesvorný & Morbidelli 2008; Agol & Deck 2015; Deck & Agol 2015a; Hadden & Lithwick 2015).

For two systems that we include here, Kepler-26 and Kepler-177, we test our results with the analytical solutions of Agol & Deck (2015). These are accurate to first order in planet–star mass ratios, first order in eccentricity, and assume co-planarity. Accuracy to first order in eccentricities is satisfied if eccentricities are ≲0.1, which is likely given the narrow dispersion in eccentricities among Kepler's multiplanet systems (Fabrycky et al. 2014; Hadden & Lithwick 2014). Where there is a second-order dependence on eccentricity, e.g., near-second-order mean motion resonances, the solutions of Deck & Agol (2015b) provide an even more general analysis of TTV frequencies. ¹⁰ Our analytical fits used an affine-invariant MCMC routine in order to determine parameter estimates and uncertainties (Goodman & Weare 2010).

We integrated samples of our posteriors for 1 Myr using the HNBODY symplectic integrator code (Rauch & Hamilton 2012) to determine whether the requirement of long-term stability places additional constraints on the masses and orbits of the planetary systems. Here we make a compromise between testing stability as long as numerically feasible for one solution (e.g., Lissauer et al. 2013; Jontof-Hutter et al. 2014) and testing for stability for a sample from our posteriors. For this study, since many of the eccentricity posteriors were poorly constrained, we chose to integrate samples of 50 sample solutions for the two-planet systems and 45 solutions for the three-planet models for 1 Myr (from the last generation of walkers in the DE-MCMC chains). We defined systems as unstable if any planet was expelled during the simulation, but none were found to be unstable.

Our summary statistics from TTV analysis are the medians and the 68.3% and 95.5% credible intervals for dynamical masses, with high and low posterior tails of equal likelihood excluded from the credible intervals. We also report calculated radii from analysis of light-curve transit profiles. We converted dynamical mass posteriors to actual mass posteriors by drawing from posteriors of the mass of each planet's host star, and the mean and variance of planetary radii are calculated by sampling posteriors of stellar radii. The resulting planetary mass, radius, and density medians and credible intervals are summarized in Table 11. We also report credible intervals on incident flux, following sampling of stellar effective temperature and a/R_⋆ posteriors measured from light-curve fitting.

Kepler-26 has four known transiting planets (in ascending order of orbital period, they are named Kepler-26 d, b, c and e), although only the intermediate pair (Kepler-26 b and c) show significant TTVs (Steffen et al. 2012b). The period ratio between the innermost pair is 3.46 and that for the outermost pair is 2.71, which are far enough from any first- or second-order mean motion resonance to make any detectable TTVs unlikely. We therefore expect no TTVs in Kepler-26 d or e, and we detect none. Furthermore, as these are the smallest two of the four planets, at 1.2 and 2.1 R_⊕, respectively, they are likely to be less massive than the interacting pair, and hence we exclude them from our nominal TTV model. The intermediate pair of planets orbit near the second-order 7:5 mean motion resonance, with an expected TTV period of 658 days, easily discernible in Figure 2.

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In Table 1, we include our adopted parameters from model fitting and the results of various tests on our measured dynamical masses, following posterior sampling with an alternative eccentricity prior, robust fitting with a Student's t₂ distribution on uncertainties, a four-planet model that includes the two planets that show no TTVs, analytical modeling of the chopping signal, and using the alternative measured times of the Holczer catalog. With one exception, all tests showed close agreement with our adopted parameters. We found that the analytical formula of Agol & Deck (2015) yielded solutions that are discrepant at the 1σ level. However, the first-order formula for this system is likely inadequate because the planets are near the 7:5 (second-order) mean motion resonance. This requires terms at order e² to model correctly. Using an analytical model specific for the second-order resonant terms (Deck & Agol 2015b), we found close agreement with the dynamical fits.

Table 1. TTV Solutions for Kepler-26 b and c (KOI-250), the Results of Various Tests on Dynamical Masses, and an Analytical Model

Adopted Parameters in Bold with 1σ (2σ) Credible Intervals
Planet	P (days)	T0 (days)	$e\mathrm{cos}\omega$	$e\mathrm{sin}\omega$	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$ (±2σ)
b	12.2800 ± 0.0003	791.2497 ± 0.0006	−0.027 ${}_{-0.052}^{+0.053}$	−0.005 ${}_{-0.052}^{+0.055}$	9.40 ${}_{-1.05}^{+1.09}({}_{-1.63}^{+2.22})$
c	17.2559 ± 0.0006	790.1830 ${}_{-0.0008}^{+0.0007}$	−0.013 ± 0.044	0.010 ${}_{-0.043}^{+0.046}$	11.39 ${}_{-1.08}^{+1.10}$ (${}_{-1.72}^{+2.20}$)
	Test 1	Test 2	Test 3	Test 4	Analytic Model
Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$
b	9.48 ${}_{-1.04}^{+1.07}$	9.62 ${}_{-1.30}^{+1.32}$	9.44 ${}_{-0.98}^{+1.10}$	${8.06}_{-0.70}^{+0.73}$	9.78 ± 1.36
c	11.52 ${}_{-1.07}^{+1.08}$	11.97 ${}_{-1.31}^{+1.35}$	11.40 ${}_{-0.95}^{+1.11}$	12.20 ${}_{-0.84}^{+0.85}$	11.92 ± 1.38

Note. Test 1: an alternative eccentricity prior. Test 2: robust fitting. Test 3: a four-planet model. Test 4: the Holczer catalog. The last column in the bottom section gives an analytical result using the approximation of Deck & Agol (2015b) in close agreement with the dynamical fits.

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The joint posteriors show tight constraints on dynamical masses but poor constraints on orbital eccentricity, as shown in Figure 3. In this case, the orbital eccentricities may be limited only by our prior and hence are not useful. Nevertheless, the inferred masses are independent of the wide range of eccentricities that fit the data.

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To assess whether our adopted solutions for Kepler-26 b and c are long-term stable, we performed long-term integrations for this system with all four planets included. For Kepler-26 d and e, we estimated masses from their measured radii using an empirical mass–radius relation for the planets of the solar system (in Earth units, ${M}_{p}={R}_{p}^{2.06};$ Lissauer et al. 2011b) and set their initial eccentricities to zero. We sampled our MCMC chains to integrate a range of masses and eccentricities from our posteriors. All simulations were stable to 1 Myr, and we adopt these as reliable dynamical masses. Hadden & Lithwick (2015) performed an independent analysis of the TTVs of Kepler-26 using transit times measured from long-cadence data. Their measured dynamical masses are closely consistent with ours.

Kepler-29 was confirmed with TTVs by Fabrycky et al. (2012), and upper limits were placed on the masses of the planets, both by the requirement of Hill Stability in the limit of low eccentricities and with dynamical fits to the first 500 days of transit data. In that study, the TTV signal was well fitted by a quadratic function, and the long-term TTV periodicity could not be discerned. The orbital period ratio, $\frac{{P}_{c}}{{P}_{b}}=1.2853$, has its nearest first-order resonance at 5:4 with an expected TTV period of 94 days, commensurable with synodic chopping (93 days), and can be discerned by eye for the best-fit model shown in Figure 4. The pair are much closer to and may be in the 9:7 second-order mean motion resonance, as identified by Lissauer et al. (2011b). With the full data set, we still only observe a fraction of the long TTV period, although the signal appears like a cubic function, and mass and orbital parameters are better constrained.

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Table 2 displays the results of our dynamical fits for Kepler-29. We performed three tests on our adopted results for consistency, as shown in the lower rows of Table 2: (i) an alternative (narrower) prior on eccentricity, (ii) robust fitting with an assumed Student's t₂ distribution on transit timing uncertainties, and (iii) fits against an alternative set of measured times from the Holczer catalog. We also tested the long-term stability of a sample of our MCMC chains, and all were stable to 1 Myr. The joint posteriors of dynamical masses and eccentricity vector components are shown in Figure 5.

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Table 2. TTV Solutions for Kepler-29 b and c (KOI-738), and the Results of Various Tests on Dynamical Masses

Adopted Parameters in Bold with 1σ (2σ) Credible Intervals
Planet	P (days)	T0 (days)	$e\mathrm{cos}\omega$	$e\mathrm{sin}\omega$	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$ ($\pm 2\sigma$)
b	10.3384 ± 0.0003	785.7544 ± 0.0014	−0.008 ± 0.072	−0.032 ± 0.072	4.59 ${}_{-1.47}^{+1.43}$ $({}_{-2.28}^{+2.88})$
c	13.2884 ± 0.0005	782.7818 ± 0.0019	0.006 ${}_{-0.062}^{+0.063}$	−0.023 ${}_{-0.062}^{+0.063}$	4.06 ${}_{-1.29}^{+1.25}({}_{-2.03}^{+2.51})$
Test 1		Test 2		Test 3
Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$
b	4.69 ± 1.44	b	5.89 ${}_{-1.92}^{+1.86}$	b	3.35 ${}_{-1.71}^{+1.79}$
c	4.16 ${}_{-1.27}^{+1.26}$	c	5.11 ${}_{-1.61}^{+1.54}$	c	2.58 ${}_{-1.31}^{+1.39}$

Note. Test 1: an alternative eccentricity prior. Test 2: robust fitting. Test 3: Holczer catalog.

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Since these results pass all tests, we adopt our nominal dynamical masses as being secure for Kepler-29 b and c.

Kepler-49 has four transiting planets. While the intermediate pair were confirmed by Steffen et al. (2013) with anticorrelated TTVs at the expected periodicity, the innermost and outermost planets awaited validation by Rowe et al. (2014) and have been named Kepler-49 d and Kepler-49 e, respectively. The period ratio of the innermost pair is 2.80, far from any first- or second-order resonance. The outer pair have a period ratio of 1.7 close to the 5:3 resonance. However, being second order, this resonance is fairly weak in the regime of low orbital eccentricities and is unlikely to induce strong TTVs. To test this, we compared four-planet models to models of just the middle pair, Kepler-49 b and c. The expected TTV periodicity of Kepler-49 b and c is around 370 days, very close to the period detected by Steffen et al. (2013). With dynamical fitting using a four-planet model, using Levenberg–Marquardt minimization over a grid of input parameters, our best-fit models do not usefully constrain the masses for the innermost and outermost planets, but we do find strong mass detections for the intermediate planets. We include the credible intervals of the masses of the middle pair from our four-planet model in Table 3 and adopt the two-planet model as our nominal result.

Table 3. TTV Solutions for Kepler-49 b and c (KOI-248) and the Results of Various Tests on Dynamical Masses

Adopted Parameters in Bold with 1σ (2σ) Credible Intervals
Planet	P (days)	T0 (days)	$e\mathrm{cos}\omega$	$e\mathrm{sin}\omega$	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$(±2σ)
b	7.2040 ± 0.0002	780.4529 ± 0.0006	0.011 ${}_{-0.067}^{+0.074}$	0.037 ${}_{-0.068}^{+0.062}$	9.16 ${}_{-3.46}^{+3.75}({}_{-5.31}^{+8.03})$
c	10.9123 ± 0.0006	790.3470 ± 0.0011	0.006 ${}_{-0.054}^{+0.059}$	0.027 ${}_{-0.057}^{+0.051}$	5.91 ${}_{-2.33}^{+2.65}({}_{-3.52}^{+5.88})$
Test 1		Test 2		Test 3		Test 4
Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$
b	9.65 ${}_{-3.19}^{+3.29}$	b	9.26 ${}_{-3.50}^{+4.11}$	b	15.35 ${}_{-3.96}^{+5.03}$	b	11.12 ${}_{-2.78}^{+5.93}$
c	6.28 ${}_{-2.18}^{+2.36}$	c	7.08 ${}_{-2.80}^{+3.55}$	c	14.39 ${}_{-4.24}^{+5.71}$	c	7.32 ${}_{-2.21}^{+4.49}$

Note. Test 1: an alternative eccentricity prior. Test 2: robust fitting. Test 3: Holczer catalog. Test 4: a four-planet model.

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Our best-fit two-planet TTV model is displayed in Figure 6, and the results of our MCMC analysis are given in Table 3. Once again, the eccentricities are poorly constrained, although we infer strong constraints on dynamical masses. Figure 7 shows that the masses of the two planets are correlated and that the eccentricity vector components are likely limited only by their prior, although the relative eccentricities are tightly constrained.

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Our solutions were insensitive to the choice of prior and the outliers, although we find higher masses with the Holczer catalog of transit times. We also found that a four-planet model allowed agreement at the 1σ level for the dynamical masses of Kepler-49 b and c in a two-planet model. Both transit time catalogs were plagued with outliers for Kepler-49. Furthermore, our light-curve analysis of Kepler-49 c yields an inconsistent measure of the stellar bulk density compared to the three other transiting planets, as discussed in more detail in Section 5. Since the TTVs of this discrepant transit model were used in this TTV analysis, we flag this system as one requiring more analysis to obtain reliable results. Hence, we exclude Kepler-49 b and c from our list of reliably measured planetary masses.

Two transiting planets with anticorrelated TTVs are known to orbit Kepler-57 (Steffen et al. 2012b; Hadden & Lithwick 2014). Given their proximity to the first-order 2:1 mean motion resonance, their expected TTV period of 456 days is easily seen in Figure 8.

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Our results in Table 4 and the joint posteriors shown in Figure 9 show that the dynamical mass of Kepler-57 b is less well constrained than that of its neighbor "c." We also note that the bimodal posterior for the eccentricity vector components makes the TTV modeling for this system rather slow to converge. Our nominal solutions here result in bulk densities for both planets that are consistent with rock. However, our results are not robust against outliers, and there is moderate disagreement in inferred dynamical masses from our independent data sets of measured transit times. In addition, as noted in Section 5, transit models for Kepler-57 b and c give an estimate for the stellar density that is inconsistent with spectral observations, perhaps owing to a low-mass stellar companion. We therefore omit these planets from the mass–radius diagram.

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Table 4. TTV Solutions for Kepler-57 b and c (KOI-1270) and the Results of Various Tests on Dynamical Masses

Adopted Parameters in Bold with 1σ (2σ) Credible Intervals
Planet	P (days)	T0 (days)	$e\mathrm{cos}\omega$	$e\mathrm{sin}\omega$	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$(±2σ)
b	5.7295 ± 0.0006	781.9966 ± 0.0006	0.018 ${}_{-0.012}^{+0.009}$	−0.019 ${}_{-0.010}^{+0.026}$	27.81 ${}_{-10.61}^{+11.40}({}_{-15.85}^{+22.73})$
c	11.6065 ± 0.0006	786.7562 ${}_{-0.0021}^{+0.0020}$	0.030 ${}_{-0.055}^{+0.014}$	−0.036 ${}_{-0.020}^{+0.076}$	6.62 ${}_{-2.80}^{+3.10}$ $({}_{-4.14}^{+5.97})$
Test 1		Test 2		Test 3
Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$
b	33.79 ${}_{-10.83}^{+10.25}$	b	16.97 ${}_{-9.54}^{+12.67}$	b	17.24 ${}_{-7.33}^{+9.93}$
c	7.80 ${}_{-2.83}^{+2.85}$	c	3.98 ${}_{-2.39}^{+3.66}$	c	3.22 ${}_{-1.43}^{+2.01}$

Note. Test 1: an alternative eccentricity prior. Test 2: robust fitting. Test 3: Holczer catalog.

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Kepler-60 has three confirmed planets in a very compact configuration (Steffen et al. 2012a). The inner pair orbit near the 5:4 mean motion resonance, and the outer pair are near the 4:3 resonance, leading to a near 5:4:3 chain of commensurability in mean motions, where the resonant quantity $| {n}_{b}-2{n}_{c}+{n}_{d}| \;\lesssim$ 0005 day⁻¹. The TTVs for all three planets are plotted in Figure 10.

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The results of our MCMC analysis for the TTVs of Kepler-60 are in Table 5. Our results were robust against the choice of eccentricity prior and outlying transit times. The Holczer catalog does not include measured transit times for all three planets of Kepler-60, although a three-planet model is required to study this system. Hence, we have only the tests shown in Table 5 for this system. Goździewski et al. (2016) have performed an independent TTV analysis of this system, using long-cadence data. With short-cadence data, we find more precise bounds on the dynamical masses of the planets. Our results are closely consistent, with differences at the 1σ level only for the middle planet, Kepler-60 c. We have also significantly improved on the precision of the stellar parameters for this system and place these planets on the mass–radius diagram in Section 6.

Table 5. TTV Solutions for Kepler-60 b, c, and d (KOI-2086) and the Results of Various Tests on Dynamical Masses

Adopted Parameters with 1σ (2σ) Credible Intervals
Planet	P (days)	T0 (days)	$e\mathrm{cos}\omega$	$e\mathrm{sin}\omega$	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$(±2σ)
b	7.1334 ± 0.0001	782.2796 ${}_{-0.0033}^{+0.0034}$	0.023 ${}_{-0.069}^{+0.067}$	0.008 ${}_{-0.059}^{+0.060}$	4.02 ${}_{-0.42}^{+0.43}$ $({}_{-0.66}^{+0.88})$
c	8.9187 ± 0.0002	786.5827 ± 0.0032	−0.003 ${}_{-0.063}^{+0.062}$	0.034 ${}_{-0.053}^{+0.054}$	3.70 ${}_{-0.73}^{+0.72}$ $({}_{-1.16}^{+1.44})$
d	11.8981 ± 0.0002	780.2779 ${}_{-0.0037}^{+0.0038}$	0.021 ${}_{-0.053}^{+0.052}$	0.002 ${}_{-0.046}^{+0.047}$	4.00 ${}_{-0.66}^{+0.73}$ $({}_{-1.01}^{+1.58})$
Test 1		Test 2
Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$
b	4.02 ${}_{-0.42}^{+0.44}$	b	4.07 ${}_{-0.59}^{+0.61}$
c	3.57 ${}_{-0.74}^{+0.72}$	c	3.93 ${}_{-0.92}^{+0.88}$
d	3.95 ${}_{-0.66}^{+0.73}$	d	4.14 ${}_{-0.92}^{+1.11}$

Note. Test 1: an alternative eccentricity prior. Test 2: robust fitting.

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Joint posteriors for dynamical masses and eccentricity vector components are shown in Figure 11. Unlike in all two-planet interactions analyzed with TTVs, the dynamical masses of neighboring planets in Kepler-60 are moderately anticorrelated.

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While the coefficients of Equation (4) match the gradient seen in the joint posteriors of eccentricity vectors, we see that there is an offset in the correlation, excluding the possibility of zero relative eccentricity. The offset in the joint posteriors of eccentricity vector components relative to the lines passing through the origin shown in Figure 11 gives us a minimum relative eccentricity for each adjacent planet pair at Kepler-60 of 0.02, i.e., $| {{\boldsymbol{e}}}_{c}-{{\boldsymbol{e}}}_{b}| \gt 0.02$ and $| {{\boldsymbol{e}}}_{d}-{{\boldsymbol{e}}}_{c}| \gt 0.02$.

We tested what fraction of our posterior samples show stable librations in the 3:4:5 resonance. We tested whether the solutions were in stable librations in the Laplace-like 3:4:5 resonant chain, where the resonant argument is defined as $\phi \equiv {\lambda }_{b}-2{\lambda }_{c}+{\lambda }_{d}$. We integrated 50 solutions from our low-eccentricity posterior samples for 1 Myr and found that both librating and nonlibrating solutions were all stable. Hence, whether or not the system is trapped in libration is still uncertain. We show examples of libration and nonlibration over 10,000 days in Figure 12. Of our sample, we found that 40 of 50 (80%) are in libration in the Laplace-like resonance. All but one of these librated around the resonant argument ϕ = 45°, as found by Goździewski et al. (2016). The exception librated about ϕ = 225°. The mean and standard deviation of the dynamical masses of the librating sample were ${m}_{b}\ =4.08\pm 0.11\;{M}_{\oplus }\frac{{M}_{\odot }}{{M}_{\star }}$, ${m}_{c}=3.69\pm 0.34\;{M}_{\oplus }\frac{{M}_{\odot }}{{M}_{\star }}$, and ${m}_{d}\ =3.96\pm 0.35\;{M}_{\oplus }\frac{{M}_{\odot }}{{M}_{\star }}$. Of the sample that was not in libration, the dynamical masses were ${m}_{b}=3.81\pm 0.37\;{M}_{\oplus }\frac{{M}_{\odot }}{{M}_{\star }}$, ${m}_{c}\ =4.03\pm 0.43\;{M}_{\oplus }\frac{{M}_{\odot }}{{M}_{\star }}$, and ${m}_{d}=3.99\pm 0.65\;{M}_{\oplus }\frac{{M}_{\odot }}{{M}_{\star }}$. These are in close agreement with the sample in libration, and hence we conclude that the condition of being in the Laplace-like resonance imposes no significant additional constraints on the masses.

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There are three planet candidates in a compact configuration at Kepler-105, orbiting every 3.44, 5.41, and 7.12 days. Since TTVs scale with orbital period, this makes the detection of TTVs among short-period planets quite rare. Nevertheless, our preliminary fits indicate a detection of perturbations due to the outer planet in the transit times of the middle planet, Kepler-105 b. In orbital period, Kepler-105 b orbits closer than other low-mass planets with published TTV analyses. Kepler-18 c, with an orbital period of 7.6 days, has strongly detected TTVs (Cochran et al. 2011) that constrained the mass of its neighbor orbiting every 14.9 days. WASP-47 b, a Jovian-mass planet with an orbital period of 4.2 days, has both strongly detected TTVs and a measurable mass from the TTVs it induces on its outer neighbor (Becker et al. 2015). Here we explore whether a TTV model can usefully constrain the mass of a planet just 1.3 R_⊕ in size with an orbital period of 7.12 days.

The innermost candidate of Kepler-105 (KOI-115.03) has a transit signal-to-noise ratio of just 9 and hence remains unconfirmed. It is likely very small (0.7 R_⊕) and has a transit depth of just 23 ppm. Thus, KOI-115.03 has poorly constrained individual transit times. The inner pair have a period ratio of 1.57, far from any first-order mean motion resonance, while the outer pair have a period ratio of 1.32, close enough to the first-order 4:3 mean motion resonance for an expected TTV period of 145 days. The inner candidate is close to the 2:1 resonance with the outer planet.

The expected periodicity of this near resonance is 97 days, and it is expected to have a low amplitude. We estimated the expected minimum amplitude of the TTVs induced between planet pairs of Kepler-105, using the equations of Lithwick et al. (2012) and assuming circular orbits. For masses, we assumed the mass–radius relation of Lissauer et al. (2011b), with ${M}_{p}/{M}_{\oplus }={({R}_{p}/{R}_{\oplus })}^{2.06}$ for planets larger than Earth (Kepler-105 b and c) and ${M}_{p}/{M}_{\oplus }={({R}_{p}/{R}_{\oplus })}^{3}$ for planets smaller than Earth (KOI-115.03).

We compared these expected amplitudes to the likely uncertainty of transit times in phase-folded TTV curves with 10 bins. We take this effective uncertainty as ${\sigma }_{\mathrm{eff}}=\frac{{\sigma }_{\mathrm{med}}}{\sqrt{N/10}}$, where σ_med is the median timing uncertainty for the planet and N is the number of measured transit times.

The estimates for all six possible TTV-inducing interactions between the planets at Kepler-105 are tabulated below. These estimates indicate that the interactions between the outer pair should be readily detectable in the measured transit times. While detectable interactions with the inner planet appear unlikely from our estimates in Table 6, we nevertheless include all three planets in our dynamical fits, since the inner planet is near 2:1 with the outer planet and there are many uncertainties in the approximations used to estimate the signal strength.

Table 6. Expected Information Content in Phase-folded TTV Signals of Planet Pairs at Kepler-105

Interaction	${\sigma }_{\mathrm{med}}$ (minutes)	${\sigma }_{\mathrm{eff}}$ (minutes)	Expected Min. TTV Ampl. (minutes)
$0.03\to {\rm{b}}$	2.6	0.5	0.04
${\rm{b}}\to 0.03$	43	6.9	0.3
${\rm{b}}\to {\rm{c}}$	7.4	1.7	3.7
${\rm{c}}\to {\rm{b}}$	2.6	0.5	0.9
$0.03\to {\rm{c}}$	7.4	1.7	0.02
${\rm{c}}\to 0.03$	43	6.9	0.16

Note. The first column lists the perturber and perturbee in each interaction. The remaining columns give the median measurement uncertainty in the perturbee, ${\sigma }_{\mathrm{med}};$ the effective uncertainty in the times of the perturbee given the number of transits, ${\sigma }_{\mathrm{eff}};$ and the minimum expected TTV amplitude, all measured in minutes. Where the expected signals in the last column exceed the effective uncertainties in the third column, strong upper and lower limits on perturbing masses are likely obtainable.

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We plot the observed TTVs and our best-fitting model in Figure 13. The model shows a reasonable detection at Kepler-105 b and what may be a weak signal in Kepler-105 c, although there are several outlying transit times in Kepler-105 c that may cause a spurious detection, notwithstanding the promising expected TTV amplitude for this planet shown in Table 6. In this case, the underestimated uncertainties in Kepler-105 c (as indicated by the excess of outliers) may cause our estimate of the effective timing uncertainty for binned data to be too low also.

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To test whether the TTV period in Kepler-105 b could be the result of fitting noise (middle panel of Figure 13), we phase-folded the TTV data at the expected TTV period from the date of the first transit and binned at 10 phases. The resulting phase TTVs and their uncertainties are plotted in Figure 14. In this case, the shape of the phase curve matches the simulated TTVs shown in Figure 13(b). This agreement is consistent with the mass detection of Kepler-105 c shown in the posteriors in Figure 15.

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We find strong upper limits for all three planets, but the upper limit on the dynamical mass of KOI-115.03 is not useful. The most useful constraints here are the strong upper and lower limits on Kepler-105 c.

The eccentricity joint posteriors show the expected degeneracy and slope described by Equation (4), with a slight offset indicating that relative eccentricities must be nonzero.

Our results for Kepler-105, as shown in Table 7, indicate agreement at the 1σ level between our nominal transit times and the Holczer catalog for the outermost planet only. This is consistent with the detection of TTVs in the middle planet only. Hence, we include the outermost planet as having a robust mass measurement, but we exclude the inner two planets from the mass–radius diagram.

Table 7. TTV Solutions for KOI-115.03, Kepler-105b, and Kepler-105 c and the Results of Various Tests on Dynamical Masses

Adopted Parameters in Bold with 1σ (2σ) Credible Intervals
Planet	P (days)	T0 (days)	$e\mathrm{cos}\omega$	$e\mathrm{sin}\omega$	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$($\pm 2\sigma$)
0.03	3.4363 ± 0.0003	780.3187 ± 0.0021	−0.015 ${}_{-0.052}^{+0.059}$	−0.050 ± 0.054	1.28 ${}_{-0.71}^{+1.25}$ $({}_{-1.00}^{+3.10})$
b	5.4119 ± 0.0001	780.5529 ${}_{-0.0004}^{+0.0003}$	−0.042 ${}_{-0.040}^{+0.041}$	−0.020 ${}_{-0.043}^{+0.044}$	4.06 ${}_{-1.97}^{+1.97}$ $({}_{-3.02}^{+3.93})$
c	7.1262 ± 0.0002	784.5992 ± 0.0006	−0.036 ${}_{-0.035}^{+0.035}$	−0.028 ${}_{-0.037}^{+0.038}$	4.78 ${}_{-0.89}^{+0.91}$ $({}_{-1.37}^{+1.84})$
Test 1		Test 2		Test 3		Test 4
Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$
0.03	2.09 ${}_{-1.07}^{+1.74}$	0.03	${1.59}_{-1.02}^{+1.93}$
b	3.67 ${}_{-1.81}^{+1.82}$	b	${4.21}_{-2.32}^{+2.22}$	b	3.51 ${}_{-1.96}^{+2.06}$	b	6.85 ${}_{-2.56}^{+2.49}$
c	4.79 ${}_{-0.91}^{+0.91}$	c	${4.93}_{-0.97}^{+1.01}$	c	4.79 ${}_{-0.93}^{+0.91}$	c	5.96 ${}_{-0.95}^{+0.96}$

Note. Test 1: an alternative eccentricity prior. Test 2: robust fitting. Test 3: two-planet model. Test 4: Holczer catalog.

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Steffen et al. (2012b) identified significant TTVs at Kepler-177. Its planets have the longest orbital periods among the planets of the systems in this study. Figure 16 shows the TTV signal and our best-fit solution for Kepler-177. The TTV cycle is not complete over the 4 yr Kepler baseline. However, strongly detected chopping and a low mass for the inner planet make this an interesting system for TTV analysis.

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The joint posteriors for dynamical masses and eccentricity vector components are shown in Figure 17. In this case, there are tight constraints on the masses of both planets. To assess the long-term stability of our TTV solutions, we performed long-term integrations for this system. All simulations were stable to 1 Myr.

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Table 8 lists our results for Kepler-177, including the tests that we have performed. In this case, our results were robust against the eccentricity prior and the measurement uncertainies. Our analytical fits are accurate to first order in eccentricity (given the proximity of the planets to the first-order 4:3 resonance) and show close agreement with our dynamical fits. We obtained higher dynamical masses with the Holczer transit timing catalog. We found that with fewer transits and an incomplete TTV cycle, this system was more sensitive to the methodology of the light-curve analysis than others. We therefore omit these planets from the mass–radius diagram, although we note that fits against both sets of transit times agree that these planets have relatively low mass given their sizes, and the upper limits on the mass of Kepler-177 c imply an extremely low bulk density (see Section 6).

Table 8. TTV Solutions for Kepler-177 b and c (KOI-523) and the Results of Various Tests on Dynamical Masses

Adopted Parameters in Bold with 1σ (2σ) Credible Intervals
Planet	P (days)	T0 (days)	$e\mathrm{cos}\omega$	$e\mathrm{sin}\omega$	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$($\pm 2\sigma$)
b	36.8590 ${}_{-0.0017}^{+0.0019}$	809.0397 ${}_{-0.0030}^{+0.0032}$	−0.027 ${}_{-0.075}^{+0.074}$	−0.015 ${}_{-0.068}^{+0.065}$	5.75 ${}_{-0.81}^{+0.84}$ $({}_{-1.24}^{+1.72})$
c	49.4096 ± 0.0010	822.9979 ± 0.0013	−0.029 ${}_{-0.065}^{+0.064}$	−0.015 ${}_{-0.059}^{+0.056}$	14.58 ${}_{-2.53}^{+2.69}$ $({}_{-3.94}^{+5.50})$
Test 1		Test 2		Test 3		Analytical Model
Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$
b	5.72 ${}_{-0.79}^{+0.82}$	b	5.59 ${}_{-0.99}^{+1.03}$	b	8.68 ${}_{-0.81}^{+0.83}$	b	5.15 ± 0.71
c	14.66 ${}_{-2.56}^{+2.69}$	c	11.74 ${}_{-2.76}^{+3.31}$	c	${21.91}_{-3.92}^{+4.09}$	c	12.64 ± 2.19

Note. Test 1: an alternative eccentricity prior. Test 2: robust fitting. Test 3. Holczer catalog. The last column in the bottom section gives an analytical result using the approximation of Agol & Deck (2015) in close agreement with the dynamical fits.

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Two of the planetary candidates that transit KOI-1576 were validated by Rowe et al. (2014) and Lissauer et al. (2014). They were also confirmed by virtue of their detected TTVs by Xie (2014). The two confirmed planets, Kepler-307 b and Kepler-307 c, orbit every 10.4 days and every 13.1 days, respectively. There is a third candidate with an orbital period of 23.34 days that is significantly smaller than the two validated planets with a period 1.78 times that of Kepler-307 c. It is far from any first- or second-order mean motion resonance. Therefore, we include just the two confirmed planets, near the 5:4 resonance, in our dynamical modeling. Figure 18 shows the TTVs of Kepler-307 b and Kepler-307 c. The predicted 727 day TTV period due to their near resonance is easily discernible in the data. The joint posteriors of dynamical mass and eccontricity vector components of Kepler-307 b and Kepler-307 c are shown in Figure 19.

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We list the results of our TTV analysis in Table 9. We obtained secure solutions for the dynamical masses of Kepler-307 b and Kepler-307 c.

Table 9. TTV Solutions for Kepler-307 b and c (KOI-1576) and the Results of Various Tests on Dynamical Masses

Adopted Parameters in Bold with 1σ (2σ) Credible Intervals
Planet	P (days)	T0 (days)	$e\mathrm{cos}\omega$	$e\mathrm{sin}\omega$	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$($\pm 2\sigma$)
b	10.4208 ${}_{-0.0008}^{+0.0009}$	784.3157 ± 0.0006	0.011 ${}_{-0.035}^{+0.038}$	−0.040 ${}_{-0.058}^{+0.053}$	8.16 ${}_{-0.89}^{+0.97}({}_{-1.36}^{+2.02})$
c	13.0729 ± 0.0012	785.2666 ± 0.0010	0.004 ${}_{-0.032}^{+0.034}$	−0.029 ${}_{-0.052}^{+0.048}$	4.02 ${}_{-0.62}^{+0.68}({}_{-0.96}^{+1.44})$
Test 1		Test 2		Test 3
Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$	Planet	$\frac{{M}_{p}}{{M}_{\oplus }}\frac{{M}_{\odot }}{{M}_{\star }}$
b	8.63 ${}_{-0.65}^{+0.65}$	b	8.69 ${}_{-1.02}^{+1.09}$	b	9.24 ${}_{-0.91}^{+0.98}$
c	4.30 ${}_{-0.58}^{+0.58}$	c	3.52 ${}_{-0.73}^{+0.80}$	c	4.81 ${}_{-0.62}^{+0.68}$

Note. Test 1: an alternative eccentricity prior. Test 2: robust fitting. Test 3: Holczer catalog.

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Table 10. Adopted Stellar Parameters

Kepler #	KOI	M⋆ (M⊙)	R⋆ (R⊙)	Teff (K)	[Fe/H] (dex)	Reference
Kepler-26	250	0.544 ± 0.025	0.512 ± 0.017	3914 ± 119	−0.13 ± 0.13	This work
Kepler-29	738	0.979 ± 0.052	0.932 ± 0.060	5701 ± 102	−0.04 ± 0.12	This work
Kepler-49	248	0.55 ± 0.04	0.52 ± 0.04	${3838}_{-74}^{+111}$	−0.02 ± 0.14	Swift et al. (2015)
Kepler-57	1270	0.884 ± 0.066	0.791 ± 0.179	5324 ± 166	−0.04 ± 0.28	CFOP
Kepler-60	2086	1.041 ± 0.077	1.257 ± 0.094	5905 ± 144	−0.09 ± 0.10	This work
Kepler-105	115	0.961 ± 0.046	0.894 ± 0.044	5827 ± 94	−0.19 ± 0.10	This work
Kepler-177	523	1.259 ± 0.108	1.399 ± 0.097	6189 ± 183	−0.09 ± 0.50	This work
Kepler-307	1576	0.907 ± 0.034	0.814 ± 0.024	5367 ± 94	0.19 ± 0.10	This work

Note. In most cases, we have improved constraints on stellar properties with our light-curve analysis following TTV analysis. In the case of Kepler-49, we identified an anomalous transit profile that may indicate spot-crossing events. Hence, we adopt the stellar parameters of Swift et al. (2015). For Kepler-57, the stellar density from light-curve analysis is inconsistent with spectroscopic observations of the star, and we adopt the spectroscopic parameters listed on Kepler's community follow-up program (CFOP), as explained in the text.

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The TTVs tightly constrain the dynamical masses of Kepler-307 b and c. A sample of 50 solutions from our posteriors were integrated for 1 Myr, and all were found to be stable. Hadden & Lithwick (2015) performed an independent analysis of the TTVs of Kepler-307 using transit times measured from long-cadence data, and their dynamical results are consistent with ours, with mass (and radius) estimates for Kepler-307 b and c at ${8.6}_{-1.4}^{+1.6}\;{M}_{\oplus }$ (${3.2}_{-0.5}^{+1.2}\;{R}_{\oplus }$) and ${3.7}_{-0.8}^{+1.0}\;{M}_{\oplus }$ (${2.8}_{-0.4}^{+1.0}\;{R}_{\oplus }$), respectively. However, our precise constraints on the stellar parameters imply slightly lower masses for the planets, smaller radii, and higher densities, as shown in Section 5.

We thus have tight constraints on the dynamical masses of 10 transiting planets. We note that the ratios of the 2σ lower bounds on the planet masses to 1σ values cluster just above 1.5, and none exceed 2; the negative uncertainties are almost always less than the positive uncertainties. Hence, zero mass (or nondetection of TTVs) is much more strongly rejected by the data than would be implied with assumed Gaussian distributions about the median.

Although we have not recovered precise eccentricities, we have tightly constrained the relative eccentricities between neighboring planets in all of these systems. We examine the posteriors of relative apses for planet pairs below.