EIGHT PLANETS IN FOUR MULTI-PLANET SYSTEMS VIA TRANSIT TIMING VARIATIONS IN 1350 DAYS

Kepler is a landmark space telescope designed to search for Earth-sized planets in and near the habitable zone of Sun-like stars (Borucki et al. 2010). A transit signature can be detected by Kepler due to its high photometric precision of 20 ppm (Koch et al. 2010). Kepler has scored significant achievements since it was launched on 2009 March 6. Using the first 4 months of data (Borucki et al. 2011), 1235 candidates were identified and these numbers rose to over 2300 as the observation time increased to 16 months (Batalha et al. 2013). So far, 138 planets have been confirmed by Kepler. Recently, a new set of data has been released (Q0–Q15 from MAST, http://archive.stsci.edu/kepler) and the observation duration of Kepler has extended to ∼1350 days. The number of candidates has reached as many as 3548, including 1475 candidates in multiple systems (as of 2013 July). Due to longer observation times, more long-period candidates are found, e.g., KOI-3946.01, with a period of 308.545 days. Some single transiting systems are seen to be multiple systems when more transiting candidates are found, e.g., KOI-255.¹

For multi-planet systems, the interactions between planets will cause variations of their transit midtimes (Agol et al. 2005; Holman & Murray 2005). Kepler-9 was successfully confirmed as a multi-planet system via transit timing variations (TTVs) by Holman et al. (2010). Even non-transiting planets can be detected via TTVs (Ballard et al. 2011; Nesvorný et al. 2012). TTVs for a two-planet system will generally be anti-correlated because of conservation of energy. By recognizing such anti-correlation and computing the amplitudes of their TTVs, one can infer the mass upper limits of these candidates and therefore confirm them as planets (Steffen et al. 2013; Xie 2012). A series of papers by the Kepler group have described the theory in detail (Ford et al. 2012; Steffen et al. 2012a, 2012b, 2013; Fabrycky et al. 2012a; Mazeh et al. 2013). They have confirmed 48 planets in 23 systems by combining the anti-correlation method and the dynamical simulation of orbital stability. Xie (2012) also confirmed 12 planet pairs via TTVs. The masses of three planet pairs are exactly calculated, while the mass upper limits of others are inferred via the amplitudes of their TTVs.

Hitherto, all the confirmed planet pairs via TTVs are within an observational time span of less than 900 days (Steffen et al. 2013; Carter et al. 2012). Although some planets with long periods can be confirmed via TTVs, e.g., Kepler-30 d, with a period about 143.2 days (Fabrycky et al. 2012a), confirmations of these planets are less convincing, and their masses are quite uncertain due to limited number of observed transits. In this paper, we utilize the recent data released on 2013 July 1 (Q0–Q15). With a time span as long as 1350 days, we can search for planet pairs near the first-order mean-motion resonances (MMRs) with long-period TTVs. Their mass upper limits can be inferred via the amplitude described by Lithwick et al. (2012).

The arrangement of this paper is as follows. We present how we obtain TTVs of planet candidates and describe the confirmation method in Section 2, i.e., recognizing anti-correlations, calculating false alarm probabilities (FAPs), and estimating mass limits. In Section 3, the properties of the eight confirmed planets in four systems are listed and compared with planets previously confirmed via TTVs. Furthermore, we show the Gaussian-fit masses in Section 4 and compare them with current planet systems in Section 5. We summarize and discuss our results in Section 6.

We use the Kepler data in long cadence to calculate TTVs. We analyze all Kepler objects of interest (KOIs) flagged as "CANDIDATE." We can obtain the corresponding orbital parameters and stellar properties from the NASA Exoplanet Archive (http://exoplanetarchive.ipac.caltech.edu; Akeson et al. 2013).

We compute TTVs for all candidates following the steps described by Xie (2012), but some improvements are made. When we calculate TTVs, we cut the light curves into small segments. Each segment centers around a transit midtime and has a length of four times the transit duration. For KOIs with periods smaller than one month, we eliminate segments contaminated by other candidate(s) to obtain accurate observed midtimes. For KOIs with periods longer than one month, the segments are inevitably easily contaminated due to their long transit durations. However, if transits of different candidates in one contaminated segment are not overlapped with a distinguishable separation, we still estimate the transit midtime for each candidate. We detrend the wings of every segment with first- to third-order polynomials. The fit with the lowest chi-square is adopted. Since Kepler data have a high precession of 20 ppm, we can verify TTV signals with amplitudes of several minutes.

To confirm planet pairs in a multiple system, two criterions must be satisfied. First, they must be anti-correlated so as to infer that the two candidates are in the same system. The anti-correlation can exclude the conditions in which the transits are caused by some external reasons, e.g., background eclipsing binaries or background stars transited by planets, etc. To estimate the probability that the observed anti-correlation is caused by random fluctuations, we also check the FAPs for candidate pairs. Only with a low FAP of less than 10⁻³ are the candidate pairs considered to interact with each other reliably. Second, the physical parameters of the candidate pairs must be in acceptable ranges and in particular their masses must be less than 13 M_J (Spiegel et al. 2011). We are interested in candidate pairs near first-order MMRs; namely, the period ratio P₂/P₁ for each pair is around j: j − 1. P₁ and P₂ are the periods of the inner and outer candidate. We use a normalized distance to resonance parameter Δ to select out such candidate pairs. Δ is defined as

$$\begin{equation} \Delta = \frac{P_2}{P_1} \frac{j-1}{j} - 1. \end{equation} \tag{ 1 }$$

All confirmed planets via TTVs near first-order MMRs satisfy |Δ| < 0.06, as shown in Figure 12(b). We adopt this threshold to select candidate pairs. We check the anti-correlated periods of the adopted candidate pairs, calculate their FAPs and set a threshold of 10⁻³, and estimate their mass upper limits via TTV amplitudes. These steps will be described in detail in the following subsections.

2.1. TTV Anti-correlations and FAPs

TTV anti-correlation can indicate the perturbations between two planets in the same system because of the conservation of energy. Transit midtimes of two planets will vary simultaneously and oppositely. Many multiple systems show such characteristics. The anti-correlation method was described first by Steffen et al. (2012a). We modify it by considering a linear trend mixed in long-period TTVs. When computing TTVs, a simple linear fitting of transit midtimes has been applied. However, it will not completely remove the secular trend of long-period TTVs. Although an untreated linear trend in the TTV signal does not affect the anti-correlation tendency, it will affect TTV amplitude and cause an inaccurate estimation of the mass upper limit. Thus, we fit TTVs for each KOI using the following function:

$$\begin{equation} f = A {\rm {sin}} \left(\frac{2 \pi t}{P} \right) + B {\rm {cos}} \left(\frac{2 \pi t}{P} \right) + C t + D, \end{equation} \tag{ 2 }$$

where A, B, C, and D are model parameters with test period P. We vary P with a wide range from 100 to 1500 days to get a series of A, B, C, and D together with their uncertainties σ_A, σ_B, σ_C, and σ_D. An anti-correlation parameter Ξ(P) is calculated as

$$\begin{equation} \Xi (P) = - \left(\frac{A_{1} A_{2}}{{\sigma }_{A_{1}} {\sigma }_{A_{2}}} + \frac{B_{1} B_{2}}{{\sigma }_{B_{1}} {\sigma }_{B_{2}}} \right), \end{equation} \tag{ 3 }$$

where the subscripts "1" and "2" represent the two planets. The maximum value Ξ_max represents the strongest anti-correlation among all the test periods. We select out candidate pairs with strong and significant anti-correlations by ranking their Ξ_max.

FAPs are checked for these candidate pairs. For each pair, we scramble their TTVs randomly and obtain a similar $\Xi _{\rm max} ^{\prime }$. We use a superscript of " $\acute{}$ " to distinguish it from Ξ_max, which is obtained from the nominal data. The same process repeats 10⁴ times. The proportion of $\Xi _{\rm max} ^{\prime }$ larger than Ξ_max represents the expected FAP. Following the approach of Steffen et al. (2012a), only candidate pairs with FAP < 10⁻³ are accepted (Steffen et al. 2012a).

2.2. Mass Upper Limit

Masses and free eccentricities of planet pairs near (but not in) first-order MMRs can be estimated via TTV amplitudes (Lithwick et al. 2012; Xie 2012). We will estimate the mass upper limits of the accepted candidate pairs theoretically instead of simulating their stability (as done by the Kepler group).

For a system near j: j − 1 MMR, the amplitudes of a planet pair are (Lithwick et al. 2012):

$$\begin{eqnarray} \vert V_1 \vert & = & P_1 \mu _2 \left| \frac{f}{\Delta } \right| \frac{1}{\pi j^{2/3} (j-1)^{1/3} } \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} \vert V_2\vert & = & P_2 \mu _1 \left| \frac{g}{\Delta } \right| \frac{1}{\pi j }, \end{eqnarray} \tag{ 5 }$$

where V is the TTV amplitude, P is the orbital period, μ is the ratio of the candidate to the star, subscripts "1" and "2" represent the inner and outer candidate, respectively, and g and f are Laplace coefficients.

In Equations (4) and (5), we have set |Z_free| = 0 to obtain the mass upper limit. Z_free is a parameter related to the complex eccentricities of the two planets. We use the model parameters corresponding to Ξ_max to calculate TTV amplitudes. Resonance order can be obtained on the basis of the ratio of the periods.

Before we calculate the mass limit, we check the anti-correlated period P_anti and the theoretical synodic period P_syn for each pair. P_anti is the test period when Ξ = Ξ_max. P_syn for a system near j: j − 1 MMR is defined as

$$\begin{equation} P_{\rm syn} = \frac{1}{|\frac{j}{P_2}-\frac{j-1}{P_1}|}. \end{equation} \tag{ 6 }$$

For a multiple system with only two candidates, P_anti should be equal to P_syn under ideal conditions. However, for systems containing more than two candidates or observed TTVs with large errors, the two periods will not be exactly equal. Especially for long-period candidates with a limited number of transit measurements, the P_anti will be less accurate. When we compute mass upper limits, we will use P_syn instead of P_anti to avoid its large error.

We confirm eight planets in four KOI systems. They are all near first-order MMRs, including one pair near 2:1 MMR (KOI-2672) and three pairs near 3:2 MMR (KOI-1236, KOI-1563, and KOI-2038). We compute the P_syn of the four planet pairs. The results are listed in Table 1. As mentioned in Section 2.2, we can see that KOI-1563 and KOI-2672 have planet pairs with almost the same P_anti and P_syn, while KOI-1236 and KOI-2038 do not. Their P_anti and P_syn are analyzed in the following section by combing frequency spectrums. KOI-2038 was also confirmed independently by J.-W. Xie (2013, private communication). We will describe these planet pairs individually. Their properties are shown in Table 2.

3.1. KOI-1236.01 and 1236.03

KOI-1236 has three candidates moving around a star with a radius of 1.27 R_☉. The radii of the three candidates are 4.30 ± 1.80, 2.60 ± 1.10, and 3.10 ± 1.30 R_⊕, respectively. The orbital periods of KOI-1236.01 and KOI-1236.03 are 35.74113 and 54.3995 days near 3:2 MMR. The inner most, KOI-1236.02, has an orbital period of 12.309717 days.

Figures 1(a) and (d) illustrate distinct anti-correlation between KOI-1236.01 and 1236.03 with a low FAP = 0.0006 (see Table 1). However, their P_syn is unequal to P_anti as shown in Figure 1(c). Combining the frequency spectrums of all the candidates in this system, we can see that only KOI-1236.01 and KOI-1236.03 agree well with the theoretical synodic periods P_syn ∼ 1234 days, while KOI-1236.02 has a much smaller effect. Therefore, it can be inferred that KOI-1236.01 and KOI-1236.03 interact with each other. The perturbations from the innermost KOI-1236.02 are very limited on both KOI-1236.01 and 1236.03. Fortunately, although the synodic period of KOI-1236.01 and KOI-1236.03 is as long as ∼1234 days, it is still less than our observational duration.

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We fit the TTV amplitudes of KOI-1236.01 and KOI-1236.03 with the theoretical P_syn via Equation (2) to obtain their mass upper limits. Amplitudes of 91.585 minutes for KOI-1236.01 and 191.462 minutes for KOI-1236.03 are fitted here. Their residuals after TTV fitting are 19.245 minutes and 44.983 minutes, respectively. The residuals might be caused by the perturbations of KOI-1236.02. According to Equations (4) and (5), we obtain the mass upper limits of 62 ± 14 M_⊕ for KOI-1236.01 and 49 ± 10 M_⊕ for KOI-1236.03.

3.2. KOI-1563.01 and 1563.02

Four candidates are moving around star KOI-1563, which has a radius of 0.87 R_☉. Some properties of the host star are listed in Table 3. The radii of the four candidates are 3.60 ± 1.30, 3.30 ± 1.10, 2.16 ± 0.77, and 3.70 ± 1.30 R_⊕, respectively. The periods of KOI-1563.01 and KOI-1563.02 are 5.487006 and 8.29113 days, which are near 3:2 MMR. The other candidates have periods of 3.205322 days for KOI-1563.03 and 16.73826 days for KOI-1563.04.

Table 3. Stellar Properties of Confirmed Multi-planet Systems

KOI	KIC	Kp	Teff	log (g)	R*	M*	R.A.	Decl.
		(mag)	(K)	(cm s−2)	(R☉)	(M☉)	(J2000)	(J2000)
1236	6677841	13.659	6779	4.35	1.27	1.31	19 09 33.889	+42 11 41.40
1563	5219234	15.812	4918	4.51	0.87	0.89	19 56 53.840	+40 20 35.46
2038	8950568	14.779	5666	4.57	0.84	0.95	19 23 53.621	+45 17 25.16
2672	11253827	11.921	5565	4.33	1.04	0.84	19 44 31.875	+48 58 38.65

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Figures 2(a) and (d) shows that KOI-1563.01 and KOI-1563.02 have significant anti-correlation with a low FAP = 0.0002 (see Table 1). Their P_syn is approximately equal to P_anti as shown in Figure 2(c). Therefore, it can be inferred that the perturbations of KOI-1563.03 and KOI-1563.04 on both KOI-1563.01 and KOI-1563.02 are small.

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To obtain the mass upper limits of these two planets, we fit the TTV amplitudes of KOI-1563.01 and KOI-1563.02 with the theoretical P_syn via Equation (2). We have almost three complete synodic cycles during the observation time. Therefore, we can obtain a very accurate result. Amplitudes of 6.675 minutes for KOI-1563.01 and 12.704 minutes for KOI-1563.02 are fitted here. The corresponding residuals are 3.604 minutes and 7.608 minutes. Since KOI-1563.02 and 1563.04 are near 2:1 MMR with a small Δ = 0.009, we can infer that the mass of the outermost candidate is less than 7.6 M_⊕. According to Equations (3) and (4), we obtain the mass upper limits of 9.0 ± 5.4 M_⊕ for KOI-1563.01 and 7.7 ± 4.1 M_⊕ for KOI-1563.02.

3.3. KOI-2038.01 and 2038.02

Star KOI-2038, of radius 0.84 R_☉, has four candidates. Some properties of this star are listed in Table 3. The radii of the candidates are 1.99 ± 0.86, 2.20 ± 0.95, 1.56 ± 0.68, and 1.61 ± 0.70 R_⊕, respectively. The periods of KOI-2038.01 and KOI-2038.02 are 8.305992 and 12.51217 days, respectively, which are near 3:2 MMR. Other candidates have periods of 17.91304 days for KOI-2038.03 and 25.21767 days for KOI-2038.04.

Figures 3(a) and (d) show that KOI-2038.01 and KOI-2038.02 have significant anti-correlation with a low FAP <10⁻⁴ (see Table 1). However, their P_syn is unequal to P_anti as shown in Figure 3(c). Combining the frequency spectrums of all the candidates in this system, we can see only KOI-2038.01 and KOI-2038.02 agree with P_syn ∼ 977 days the most, while others have much smaller powers. Therefore, it can be inferred that KOI-2038.01 and KOI-2038.02 clearly interact with each other. The perturbations of KOI-2038.03 and KOI-2038.04 are very limited on both of them.

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To obtain the mass upper limits of these two planets, we fit the TTV amplitudes of KOI-2038.01 and KOI-2038.02 with the theoretical P_syn via Equation (2). Amplitudes of 40.541 minutes for KOI-2038.01 and 51.382 minutes for KOI-2038.02 are fitted here. The corresponding residuals are 11.206 minutes and 14.850 minutes. The perturbations of KOI-2038.03 and 2038.04 may be the main reason for the residuals. Assuming the residuals of KOI-2038.02 are mainly caused by KOI-2038.03, the mass of KOI-2038.03 must be less than 42 M_⊕. Otherwise, if the residuals of KOI-2038.02 are mainly caused by KOI-2038.04, the mass of KOI-2038.04 must be less than 8.6 M_⊕. According to Equations (4) and (5), we obtain the mass upper limits of 15 ± 4.3 M_⊕ for KOI-2038.01 and 19 ± 5.2 M_⊕ for KOI-2038.02.

3.4. KOI-2672.01 and 2672.02

KOI-2672, of radius 1.04 R_☉, has only two candidates around. Other parameters of KOI-2672 are listed in Table 3. The radii of the two candidates are 5.30 ± 2.10 and 3.50 ± 1.40 R_⊕, respectively. The periods of KOI-2672.01 and KOI-2672.02 are 88.51658 and 42.99066 days, respectively, which are near 3:2 MMR.

Figures 4(a) and (d) show that KOI-2672.01 and KOI-2672.02 have significant anti-correlation with a low FAP <10⁻⁴ (see Table 1). Their P_syn is approximately equal to P_anti as shown in Figure 4(c). The strongest powers in their frequency spectrums also coincide with P_syn. Although the synodic period is as long as ∼1500 days, our observational duration still covers almost a complete cycle so we can proceed with our analysis.

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To obtain the mass upper limits of KOI-2672.01 and 2672.02, we fit their TTV amplitudes with the theoretical P_syn via Equation (2). Amplitudes of 77.687 minutes for KOI-2672.01 and 28.952 minutes for KOI-2672.02 are fitted here. According to Equations (4) and (5), we obtain the mass upper limits of 17 ± 1.8 M_⊕ for KOI-2672.01 and 80 ± 3.5 M_⊕ for KOI-2672.02. After removing our best-fit results via Equation (2), the remaining residuals have a scatter of about 3.394 minutes for KOI-2672.01 and 3.115 minutes for KOI-2672.02. The small residuals indicate that there are no large perturbations in this system. Assuming another planet outside are near 2:1 MMR with KOI-2672.01, we infer from Equation (4) that the mass of the assumed planet must be less than 1.7 M_⊕.

We scan the masses of the four planet pairs and simulate their TTVs. For each inner planet, we set the scanning range of M_in from 0.1 to 1.5 times the mass upper limit. For a given M_in, we can obtain the mass of the outer planet M_out0 via Equations (4) and (5). Then we set the scanning range of M_out from 0.5 to 1.5 times M_out0. The wide scanning ranges adopted for M_in and M_out are large enough to tolerate observational uncertainties. The mean anomaly of the inner planet is fixed at 0, while the mean anomaly of the outer planet is changed from 0 to 350 deg with a step of 10 deg. The eccentricities, the argument of pericenter ω, and the longitude of ascending node Ω are all fixed at 0.

Finally, we obtain a series of chi-squares χ² by comparing the simulated TTVs with the observed TTVs. We can easily find the minimum chi-square $\chi ^2_{\rm min}$ and the mass associated with the minimum chi-square for each planet M_cmin. For each pair of M_in and M_out, we have 36 cases because we changed the mean anomaly from 0 to 350 deg. We consider the fraction of simulations with $\chi ^2 < \chi ^2_{\rm min}+\Delta \chi ^2$. The values of $\chi ^2_{\rm min}$ and Δχ² in each system are listed in the caption of Figure 5. By applying a two-dimensional Gaussian fit to the fractions, we can obtain the Gaussian-fit masses M_fit and standard deviations σ_fit. Since M_fit is the mathematical expectation of all the simulated masses with small chi-squares, we adopt it as the most-likely mass of the planet. As shown in Figure 5, the differences between M_cmin and M_fit are all less than 1σ_fit. Figure 6 shows Gaussian-fit masses ("+") and the minimum χ² masses ("o") of all planets. Figures 7–10 show the minimum chi-squared fit for each system.

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Using the Gaussian-fit masses, we can estimate the density of each planet as shown in Table 2. The densities of KOI-1236.01 and KOI-1236.02 are similar to rock planets. KOI-1563.01 and KOI-1563.02 seem to be sub-Neptunes, according to their radius. However, the densities of the two planets are relatively low, and therefore they might contain a lot of gas or some other light materials. KOI-2038.01 and KOI-2038.02 are sub-Neptunes, and their densities are similar to Earth. KOI-2672.01 and KOI-2672.02 show a large density contrast.

We also check the stabilities of the four two-planet systems. According to Kokubo & Ida (2002) and Zhou et al. (2007), the stable timescale depends on the separation between planets scaled by mutual hill radius. We calculate the separation of planets Δa = |a₁ − a₂| in each system and obtain a scaled separation k = Δa/R_Hill, where the mutual Hill radius R_Hill = (μ₁ + μ₂/3)^1/3(a₁ + a₂/2), μ₁, and μ₂ are the mass ratios between the planets and the host star. a₁ and a₂ are the semi-major axes of the two planets. Using the M_fit obtained above, k = 7.2, 10.4, 10.9, and 9.8 for KOI-1236, 1563, 2038, and 2672 respectively. Obviously, the scaled separation k > 3.5 for all four systems, which means they are stable in circular coplanar 3-body systems (Gladman 1993). Using the mercury integration package, we test the stability of each planet pairs with the best-fit masses. The hybrid symplectic algorithm is adopted here. All the four systems are stable in at least 10 Myr.

We have confirmed eight planets via TTVs and list the parameters of them in Table 2. We improve the fitting function of TTVs (Steffen et al. 2013) by adding a secular term. We have found a planet with a period of 88.51658 days, which is the second longest periodical planet confirmed via TTVs under Neptune size, as shown in Figure 11. The longest confirmed via TTVs is Kepler-30 d, which is near 5:2 MMR with Kepler-30 c (Fabrycky et al. 2012a). Although Kepler-30 d has only five TTV data, it can also be confirmed by analyzing the TTVs of 30 b and 30 c using a transiting starspot model (Sanchis-Ojeda et al. 2012). However, KOI-2672.01, confirmed in this paper, is presently the longest periodical planet near a first-order MMR. In addition, this pair also has a large density contrast. A similar system is Kepler-36 (Carter et al. 2012). As pointed by Lopez & Fortney (2013), the large density contrast of Kepler-36 can be interpreted as a result of photo evaporation of the inner planet. However, this interpretation may not suitable for the KOI-2672 system. The mass of the inner planet KOI-2672.02 is much larger than that of a rocky planet. We infer that the less dense planet KOI-2672.01 has a small core mass and may have formed inside the "snow line," while KOI-2672.02 may have formed outside and might experience obvious collisions to become so dense and massive. After dynamical evolution, especially migrations and collisions, they are driven into their present architecture. We also note that this system has only two detected planets.

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We show all the Kepler planets near MMRs, as shown in Figure 12, and compare our results with them. Adding one pair near 2:1 MMR and three pairs near 3:2 MMR, there are 20 total pairs near 2:1 MMR and 17 total pairs near 3:2 MMR. Only 10 pairs are in other first-order MMRs and another 10 pairs in the second-order MMRs. Theoretical works show that, assuming these configurations were through type I migration of planetary embryos with mass up to super-Earth (Goldreich & Tremaine 1979; Tanaka et al. 2002; Ward 1997), the final architectures are mainly determined by the migration speed, e.g., low-speed migration can stall at 2:1 MMR, while high-speed migration passes 2:1 MMR and will stall at j: j − 1 with higher j (e.g., Zhou 2010; Szuszkiewicz & Papaloizou 2010; Wang et al. 2012). Supposing all the planet pairs are initially located outside the 2:1 MMR, the frequent outcome of planet pairs at 2:1 and 3:2 MMR (Figure 12) is consistent with the migration scenario. Figure 12 presents Δ as defined by Equation (1) and the mass ratio of the planet pair of each system. It is obvious that 75% of the planet pairs have a period ratio that are a little larger than the exact value j: (j − 1). This is revealed by Fabrycky et al. (2012b) and interpreted by Lithwick & Wu (2012) and Lee et al. (2013) via tidal dissipation.

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We have found four planet pairs near first-order MMRs via TTVs with a time span as long as ∼1350 days, i.e., KOI-1236.01 and KOI-1236-03 near 3:2 MMR, KOI-1563.01 and KOI-1563.02 near 3:2 MMR, KOI-2038.01 and KOI-2038.02 near 3:2 MMR, and KOI-2672.01 and KOI-2672.02 near 2:1 MMR. We also estimate their mass upper limits via Equations (4) and (5), as shown in Table 2. KOI-2672 has only two planets with a long theoretical P_syn of about 1500 days, and the outer planet has the longest period among the planet pairs near first-order MMRs.

By simulating the TTVs of planet pairs, we show the minimum chi-squared masses and Gaussian-fit masses of the planet pairs. All the two-planet systems with the Gaussian-fit masses are stable for at least 10 Myr. We calculate the density of each planet according to their best-fit mass: planets in the KOI-1236 and KOI-2038 systems seem like rock planets, while planets, in KOI-1563 system have a relative low density. Considering the large difference between the densities of 2672.01 and 2672.02, we infer that they formed in different regions and evolved in current orbital architecture due to migration or some other mechanism.

We also compare our new confirmed planets with other Kepler planets, especially the planet pairs near MMRs. Adding the new four pairs near MMRs, we find that about 36% and 30% of the planet pairs are near 2:1 MMR and 3:2 MMR, respectively. A few more planet pairs are near 2:1 MMR than are near 3:2 MMR. If this trend continues, it is important constraining factor to the migration rate of planets during the evolution of multi-planet systems (Ogihara & Kobayashi 2013). Planet pairs prefer to keep an architecture where the outside planets are slightly outside of the exact positions with period ratios of j: j − 1, which is consistent with Fabrycky et al. (2012b).

In this paper, we only consider planet pairs near first-order MMRs. Additionally, planet pairs near higher-order MMRs have a weaker TTV signal than planets near first-order MMRs, thus it is much harder to estimate their mass upper limits. However, the anti-correlations of their TTVs and small FAPs are also available to confirm these gravitational pairs. The fraction of planet pairs near high-order MMRs is relatively small; only 10 pairs are confirmed by the observations from Kepler.

Here we use the TTVs spanning up to about 1350 days. TTVs in a longer timescale can provide more information about the secular influences in planet pairs near or in MMRs (Ketchum et al. 2013; Boué et al. 2012). Subsequent observations of transiting planets or Kepler candidates provide us more information about their TTVs to investigate the dynamical properties of multi-planet systems, and this might confirm planets with longer periods in the habitable zone.

We appreciate NASA and the Kepler group for their great work on exoplanet detection. This work is supported by the Key Development Program of Basic Research of China (No. 2013CB834900), the National Natural Science Foundations of China (Nos. 10925313, 11003010 and 11333002), the 985 Project of the Ministry of Education, and the Superiority Discipline Construction Project of Jiangsu Province. We are also grateful to the High Performance Computing Center (HPCC) of Nanjing University for the catalog refinement process.