TRANSIT TIMING OBSERVATIONS FROM KEPLER. IX. CATALOG OF THE FULL LONG-CADENCE DATA SET

More than four years of almost uninterrupted performance of the Kepler mission produced about 190,000 light curves with high precision, leading to the discovery of more than 8000 planet candidates (=KOIs)⁷ that show periodic shallow transits caused by small eclipsing objects. Many of the KOIs display shifts of the transit timings (O-Cs) relative to a strict periodicity, which is expected if the transiting planets were to move on a Keplerian orbit. These O-Cs, sometimes called transit time variations (=TTVs), can indicate a dynamical interaction with additional objects in the system, as was predicted by the seminal works of Holman & Murray (2005) and Agol et al. (2005).

Indeed, TTVs turned out to be a crucial tool in the study of systems with known multiple transiting planets (e.g., Holman et al. 2010; Cochran et al. 2011; Lissauer et al. 2011a; Fabrycky et al. 2012; Ford et al. 2012a; Steffen et al. 2012b; Weiss et al. 2013; Xie 2013, 2014; Yang et al. 2013; Jontof-Hutter et al. 2014; Masuda 2014; Ofir et al. 2014; Agol & Deck 2015). Furthermore, observed TTVs may indicate extra non-transiting planets through their dynamical interaction with the transiting ones (Ballard et al. 2011; Nesvorný et al. 2012, 2013, 2014; Dawson et al. 2014).

Therefore, it can be useful to perform a systematic TTV search of all KOIs, as was done by Ford et al. (2011, 2012b) and Steffen et al. (2012a) at the early stages of the mission. As Kepler released more data, additional systematic analyses were performed, using the longer light curves that became available (see Mazeh et al. 2013; Szabó et al. 2013). Based on the first 12 Kepler quarters, a global analysis of all KOIs, including the timing of the transits, was published recently by Rowe et al. (2015). In a follow-up publication Rowe & Thompson (2015) listed 258 KOIs with significant TTVs, based on the first 16 quarters.

The Kepler mission in its original mode of operation has been terminated after 17 quarters, and is now on its K2 mode (Howell et al. 2014), and we do not expect any additional Kepler TTVs for the KOIs identified during the original mission. Thus, here we analyze the whole data set of the mission and derive a complete catalog of the transit timings. Following the approach of Mazeh et al. (2013), we present here an analysis of 2599 KOIs, based on all 17 quarters of the Kepler data. The goal is to produce an easy-to-use catalog that can stimulate further analysis of interesting systems and a statistical analysis of the Kepler KOIs with significant long-term TTVs.

After presenting the details of our pipeline and the catalog itself in Section 2, we derive in Section 3 a few statistical characteristics of the timing series of each KOI which can identify significant variations. Sections 4 and 5 list and display 274 systems with significant TTVs, and Section 8 summarizes and discusses briefly the potential of the catalog.

The analysis presented here is based on the list of 4690 KOIs in the NASA Exoplanet Archive,⁸ as of 2013 November 23rd, ignoring KOIs listed as false positives. We did not analyze 2091 KOIs, for which at least one of the following is true:

• 1.  
  The folded light curve did not display a significant transit, either because the folded transit's SNR (defined as the transit depth of the model, divided by the median uncertainty of the individual points of the light curve, and multiplied by the square root of the number of measurements at the folded transit, including its ingress and egress) was smaller than 7.1, similar to the criterion used by Batalha et al. (2013), or where the p-value of the transit model exceeded 10⁻⁴, using an ${ \mathcal F }$-test relative to the no-transit assumption.
• 2.  
  The transit depth was larger than 10%; those KOIs were ignored in order to disregard eclipsing binaries in our analysis, with the price of leaving out some "legitimate" transits such as large planets around M-stars.
• 3.  
  The orbital period >300 days; those KOIs were ignored due to too few transits for a significant TTV analysis.
• 4.  
  KOIs identified as EBs, either listed in the Villanova eclipsing binary catalog,⁹ as of 2014 July, or by McQuillan et al. (2013).
• 5.  
  KOIs identified by this study as false alarm, listed in Table 1, with some evidence for stellar binarity or pulsation.

Following these cuts we were left with 2599 KOIs. We started by folding the PDC-MAP Kepler long-cadence¹⁰ data, with the BJD_TDB timings, using the ephemeris of NASA Exoplanet Archive in order to obtain a good template for the transit light curve (see below for details). We used the best-fit transit model to measure the timing of each individual transit (=TT) and derived its O-C—the difference between the TT and the expected time, based on a linear ephemeris. As in Mazeh et al. (2013), for KOIs with high enough SNR (see below), the TT derivation was performed while allowing the duration and depth of each transit to vary.

2.1. Detrending

2.2. The Transit Model

2.3. Deriving the Timing, Duration, and Depth of Each Transit

In order to derive the timing of a specific transit, we first searched through a grid of timings around the expected transit time, with a resolution of either one minute, or the estimated transit time uncertainty divided by five—whichever is lower. The transit time uncertainty was approximated to be (100 minutes)/(SNR of individual transit) (Mazeh et al. 2013, see also Figure 2). For each time shift on the grid we fitted the data with the KOI's transit model, while keeping the duration and depth fixed. The grid point with the lowest χ² served as a first guess for the transit time.

In our next step we divided the transits into two groups. Group 1 consisted of KOIs whose transits had a duration longer than 1.5 hr and an SNR per transit larger than 10. (The SNR per transit was defined as the transit depth of the model divided by the median uncertainty of the individual points of the light curve and multiplied by the square root of the typical number of measurements during one transit, including its ingress and egress.) Group 2 consisted of all other KOIs, which did not follow at least one of these two criteria. For both groups we performed a fine search for the best estimate of the transit timing, using MATLAB's FMINSEARCH function, which is based on the Nelder–Mead simplex direct search, allowing the duration and depth of the transit to vary in group 1, while in group 2 only the timing was varied.

For KOIs of group 1, we define the TDV and TPV of each transit as the relative change of the duration and depth found for that transit, respectively, with respect to the duration and depth of the transit model. Deriving the template depends on the folded light curve, which in turn is based on the adopted transit timings. Therefore, fitting of the model was performed only after the first derivation of the timings for each KOI, and then we derived a new set of timings with the new improved template. This process converged after three iterations.

We estimated the uncertainties of the three quantities (when available) from the inverted Hessian matrix, calculated at the identified minimum. The uncertainty of each individual Kepler measurement of a KOI was derived from the scatter of the light curves around the polynomial fit before and after all transits of that planet. When the Hessian matrix turned out to be singular, we assigned an uncertainty that was equal to the median of the other uncertainties derived for the KOI in question. Whenever this was the case, we marked the error with an asterisk in the table of transit timings (see Table 3).

In order to verify the obtained uncertainties for the transit timings, we computed for 2339 KOIs, which had at least 7 transit timings, the scatter of their O-C values, ${s}_{{\rm{O \mbox{-} C}}}$, defined as 1.4826 times the median absolute deviation (MAD) of the O-C series. We then compared this quantity with the typical error of each KOI, defined as the median of its timing uncertainties—${\bar{\sigma }}_{{\rm{TT}}}$. These two parameters should not be sensitive to timing and uncertainty outliers (see Table 3).

We expected the estimated scatter and the mean uncertainty values to be similar for systems with no significant TTV. This was indeed the case for most KOIs, as seen in Figure 1. KOIs with O-C scatter substantially larger than the typical timing uncertainty had significant TTVs (see below for a detailed analysis of those systems).

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Mazeh et al. (2013) found that for each KOI ${\bar{\sigma }}_{{\rm{TT}}}\sim$ (100 minutes)/SNR. Figure 2 shows that this relation still holds.

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2.4. Outliers and Overlapping Transits

2.5. The Catalog

As the main focus of this study is the TTVs of the KOIs, the next sections concentrate on the analysis of the derived O-Cs, leaving the analysis of the duration and depth variations for another study. In order to identify KOIs with significant long-term TTVs, we computed a few statistics, listed in Table 4, of the O-C series for 2339 KOIs with more than six timing measurements:

• 1.  
  The ratio between the median error, ${\bar{\sigma }}_{{\rm{TT}}}$, and the scatter of the O-Cs, ${s}_{{\rm{O \mbox{-} C}}}$, (see discussion above and Figure 1). The modified ratio of these two figures, ${s}_{{\rm{O \mbox{-} C}}}$/$(1.48{\bar{\sigma }}_{{\rm{TT}}})$, squared and multiplied by the number of measurements (see Table 3), gave us a modified χ² of the O-C series of each KOI. We used this figure to calculate a χ² p-value against the no-variation assumption, listed in the table. Very low p-value due to high values of ${s}_{{\rm{O \mbox{-} C}}}$ relative to ${\bar{\sigma }}_{{\rm{TT}}}$ might indicate a significant TTV, in particular because the MAD statistic was less sensitive to outliers than the rms of the O-Cs.
• 2.  
  A modified power spectrum (PS) periodogram of the O-Cs, presenting for each frequency the energy contained in its fundamental and first harmonic. We identified the highest peak in the periodogram and assigned a p-value to the associated periodicity in the data. This was done by calculating 10⁴ modified PS periodograms for different random permutations of the same O-Cs, and counting the number of permutations that had a PS peak higher than the peak of the real data. Table 4 quotes the estimated period and its p-value.
• 3.  
  An "alarm" ${ \mathcal A }$ score of the series, following the statistic of Tamuz et al. (2006), which is sensitive to the correlation between adjacent O-Cs. The value of ${ \mathcal A }$ is sensitive to the number of consecutive O-Cs with the same sign, without assuming any functional shape of the modulation (see Tamuz et al. 2006, for a detailed discussion). We assigned a false-alarm probability to the occurrence of the obtained score by calculating alarm scores for 10⁴ different random permutations of the same O-C series, and counting the number of permutations that had an alarm higher than the peak of the real data. Table 4 quotes the alarm score and its p-value.
• 4.  
  A long-term polynomial fit to the O-C series. A significant polynomial fit usually indicates a long-term modulation with a timescale longer than the data span. We searched for a fit with a polynomial with a degree lower than four, chose the best polynomial and tested its significance with an ${ \mathcal F }$-test. Table 4 quotes the best polynomial fit and its p-value.

Table 4.  Statistical Parameters of the O-Cs Series

KOI	${\bar{\sigma }}_{{\rm{TT}}}$ a	${s}_{{\rm{O \mbox{-} C}}}$ b	p-s/σc	PS	PS	p-PSf	${ \mathcal A }$ g	p-${ \mathcal A }$ h	Pol.	p-${ \mathcal F }$ j
				Periodd	Peake				Deg.i
	(minutes)	(minutes)	(log)	(days)	(log)	(log)		(log)		(log)
1.01	0.08	0.09	0.0	17.40	−9.54	0.0	0.188	−0.9	1	−0.2
2.01	0.23	0.24	0.0	4.43	−8.75	−0.6	0.036	−0.3	1	−1.3
3.01	0.24	0.35	−0.2	13.94	−8.11	−0.1	0.220	−0.6	2	−2.7
5.01	1.65	1.73	0.0	52.05	−6.81	−0.2	−0.064	−0.5	1	−0.1
7.01	3.54	3.28	0.0	8.31	−6.19	−0.2	0.038	−1.1	1	−0.3
10.01	0.60	0.59	0.0	26.03	−7.90	0.0	−0.278	−0.4	1	−0.2
12.01	0.37	1.51	<−16.0	1176.37	−6.24	−3.3	3.234	<−4.0	2	−3.9
13.01	0.13	0.16	0.0	5.72	−8.83	−1.1	−0.104	−2.7	1	−1.1
17.01	0.33	0.40	0.0	10.89	−8.10	−0.6	0.046	−0.6	1	−0.2
18.01	0.57	0.56	0.0	23.72	−7.83	−0.2	−0.104	−0.5	1	−0.1

Notes.

^aO-C uncertainty median. ^bO-C scatter (1.483 times the MAD). ^cThe p-value of the scatter divided by the uncertainty (see the text). ^dThe period with the highest peak in the power spectrum. ^eThe amplitude of the highest peak in the power spectrum, in days squared. ^fThe logarithm of the p-value of the ${ \mathcal F }$-test for the highest PS peak found. ^gAlarm score (see the text). ^hThe logarithm of the p-value of the alarm found. ⁱThe degree of the best fitting polynomial. ^jThe logarithm of the p-value of the ${ \mathcal F }$-test for the best polynomial fit.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  DataTypeset image

Each KOI with any of the aforementioned statistics yielding a p-value lower than 10⁻⁴ and a period longer than 100 days was identified as having a significant long-term TTV, provided the variation seemed real and not caused by some artifact. Table 5 lists 260 KOIs with long-term significant timing variation and summarizes their variability features, while Figures 3–28 display their TTVs. Of those, 73 KOIs did not pass the significance thresholds, but nevertheless had a relatively high significance and seemed to have a real long-term timing variation, and therefore were also included in the table.

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Table 5 lists the KOI number, the orbital period of the transiting planet, and the adopted model, either a Cosine function, "Cos," or a polynomial "Pol." For a cosine fit, we list the TTV period and its error, and the amplitude and its error. For the cases with parabolic fits, we list an estimated figure for the amplitude of the variability, based on half the difference between the maximum and minimum values of the parabola at the times of the TTVs. For both types of fitting, we list the scatter of the residuals relative to the adopted fit (which is not plotted). We also list the number of TTV measurements, the multiplicity of the system and references to previous studies, when available.

To derive the TTV periods and amplitudes and their errors we fitted the TTVs with a periodic cosine function superimposed on a linear trend. This was performed by Markov chain Monte Carlo runs, each of which used an ensemble of MCMC samplers (Goodman & Weare 2010; Foreman-Mackey et al. 2013). We summarize the marginal posterior distribution for each period by reporting its median value, along with an uncertainty based on a 68.26% interval ranging from the 15.87th to the 84.13th percentile. Out of 260 KOIs, 199 showed clear significant periodicity, with periods ranging from 100 to over 2000 days, and amplitudes of 1–1470 minutes.

In six special cases—KOI-142.01, -157.03, -417.01, -474.03, -984.01, and -2283.01, we fitted the TTVs with a straight line and two different cosine functions, both identified by the MCMC runs. In these cases, Table 5 lists the two periods, and the scatter of the residuals after the removal of the linear trend and the first periodicity, and also after the removal of the linear trend and the two modulations. The secondary periodicity is very clear only in two cases, KOI-142 and -984, as can be seen by comparing the residuals after removing the first and both periodic modulations. Nevertheless, we suggest that the second modulations in the other four cases are also real, especially because our MCMC samplers clearly showed two periodicities.

For 61 KOIs, the cosine function fit was poor, and we therefore fitted instead a simple parabola to the data. This probably meant that the timescale of the modulation was longer than the time span of the data.

The orbital periods listed in Table 2 for the 2599 KOIs were corrected by the slope of the linear fit to the TTVs. This approximation was good when there was no significant long-term TTV modulation. However, in the presence of a strong TTV periodic modulation, the linear fit by itself might not be good enough, and therefore might yield an inaccurate correction to the orbital period. Therefore, for each of the 199 KOIs that showed a periodic modulation, we have corrected the orbital period using the slope of the linear part of the model in Table 5, which included both a linear slope and a cosine modulation. Using the error of the linear slope from the MCMC model might underestimate the uncertainty of the derived orbital period. Instead, we used the difference between the orbital periods obtained by both methods (linear and linear + cosine models) as an error estimate. The new orbital periods and their estimated uncertainties are given in Table 6.

Following the approach of Mazeh et al. (2013), we identified 10 systems with highly significant short-period TTV modulations, in the range of 3–80 days. They were found by obtaining a PS peak with p-value lower than 3 × 10⁻⁴. The modulation amplitudes were relatively small, in the range of 0.07–80 minutes, and their detection was possible only due to the periodic nature of the signal and the long time span of the data relative to the periodicity.

For the sake of comparison, we included in this section plots for KOI-13.01 and -972.01 modulations, which were detected by Mazeh et al. (2013), and our analysis showed the same modulation but with a lower significance. Another seven KOIs that were listed in Mazeh et al. (2013) are not shown here, as they now appear to be false positives and not planet candidates. Two additional systems, KOI-895.01 and -1074.01, are shown because they exhibit TTVs induced by spot crossing events (Holczer et al. 2015).

Figures 29– 33 show the PS periodograms with their prominent peaks, and the phase-folded O-Cs of the 14 systems. Table 7 lists the periods and amplitudes found.

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As pointed out by Szabó et al. (2013) and discussed by Mazeh et al. (2013), not all detected short-period modulations are due to physical TTVs. An apparent TTV periodicity can be induced either by the long-cadence sampling of Kepler (the stroboscopic effect), or by an interference with a periodic stellar activity. To find the frequency of the presumed sampling-induced periodicity, we used for each KOI its P_orb from Table 2, and the pertinent P_samp. This was about 29.424 minutes for the long cadence, the exact value taken to be the median of the differences of the observed timings of that KOI. We searched for a stellar spot periodicity using the autocorrelation technique (e.g., McQuillan et al. 2013), and, if present, checked whether its frequency and/or one of its harmonics or aliases was equal to the TTV frequency. We mark the pertinent frequencies in Figures 29– 33 (see Holczer et al. 2015).

In each of the figures we marked the frequency of the highest peak, the stroboscopic frequency (due to sampling) and the rotational frequency, with its harmonics when relevant. For five systems, KOI-203.01, -217.01, -883.01, -895.01, and -1074.01, the rotational frequency and/or one of its harmonics coincided with the highest periodogram peak (see also Holczer et al. 2015). The periodicity in those cases could be induced by crossing the rotating stellar spots by the transiting planet. Two systems, KOI-883.01 and -13.01, showed a strong stroboscopic effect.

As discussed in the introduction, we expect most TTV periodic modulations to be caused by a dynamical interaction with an adjacent additional planet, near first-order resonance in particular. For those cases, we have prepared a theoretical infrastructure to calculate the expected TTV period, based on the orbital periods of the two planets (e.g., Lithwick et al. 2012; Hadden & Lithwick 2014). In the case that the adjacent planet is also a transiting one, we should be able to know its orbital period, and therefore to compare the predicted TTV periodicity with the observed one. Thus, we searched the known multiple KOIs systems to see if any additional planet candidate can account for the 199 TTV periodicities of Table 5.

We found 80 pairs of KOIs for which two KOIs resided in the same planetary system and at least one of them showed a clear periodicity as listed in Table 5. Following Xie (2013, 2014), for each of these KOI pairs we checked whether it was close to a first-order resonance by calculating for each of them the normalized distance to the closest first-order resonance, Δ:

$$\begin{eqnarray}&&{\rm{\Delta }}\equiv \displaystyle \frac{{P}_{2}}{{P}_{1}}\displaystyle \frac{j-1}{j}-1,\end{eqnarray} \tag{ 1 }$$

where P₁ and P₂ are the inner and outer orbital periods of the planets of the pair, respectively, and j is the resonance number. We set a cutoff at Δ = 0.1, ignored the pairs with higher values, and found 75 pairs of KOIs that were close to some first-order resonance.

For those pairs we derived the expected super-period P^j, given by

$$\begin{eqnarray}&&{P}^{j}\equiv \displaystyle \frac{1}{| j/{P}_{2}-(j-1)/{P}_{1}| },\end{eqnarray} \tag{ 2 }$$

and compared it with the observed TTV periodicities (two if available, otherwise only one observed periodicity) of the pair. We consider this derived super-period as a rough estimate, as the obtained orbital periods of the two transiting planets, P₁ and P₂, depend on their assumed TTV periodicities. We found 59 pairs, listed in Table 8, for which at least one of the TTV periodicities was consistent with the derived super-period. The table includes the two orbital periods of the pair, the resonance found, the distance to resonance, the calculated super-period, and the observed TTV periodicities with their uncertainties.

Table 8.  KOI Multies

KOI	KOI	P1a	P2b	Δc	j.j-1d	${P}_{{\rm{TTV,1}}}$ e	${\sigma }_{{P}_{1}}$ f	${P}_{{\rm{TTV,2}}}$ g	${\sigma }_{{P}_{2}}$ h	Pj i	References
(Inner)	(Outer)	(Inner)	(Outer)			(Inner)		(Outer)
		(days)	(days)			(days)	(days)	(days)	(days)	(days)
137.01	137.02	7.6416	14.8589	0.0278	2:1	269	1	268	1	270	(1),(19),(20) Kepler18
152.02	152.01	27.4024	52.0908	0.0495	2:1	963	47	615	13	530	(2), (3), (13), (20) Kepler79
152.01	152.04	52.0908	81.0643	0.0375	3:2	614	14	741	38	720	(3),(13),(20) Kepler79
152.03	152.02	13.4846	27.4024	0.0161	2:1	849	35	963	47	850	(2),(13),(20) Kepler79
157.06	157.01	10.3040	13.0249	0.0112	5:4	⋯	⋯	229	3	230	(4),(14),(20) Kepler11
157.03	157.04	31.9954	46.6857	0.0272	3:2	183	2	558	12	570	(4),(14),(20) Kepler11
157.02	157.03	22.6872	31.9954	0.0577	4:3	⋯	⋯	j139	2	140	(4),(14),(20) Kepler11
157.02	157.04	22.6872	46.6857	0.0289	2:1	⋯	⋯	558	11	810	(4),(14),(20) Kepler11
168.03	168.01	7.1070	10.7425	0.0077	3:2	452	12	443	6	470	(5),(19),(20) Kepler23
244.02	244.01	6.2385	12.7204	0.0195	2:1	326	3	329	4	330	(6),(19),(20) Kepler25
248.01	248.02	7.2039	10.9127	0.0099	3:2	368	3	370	4	370	(7),(20) Kepler49
250.01	250.02	12.2830	17.2512	0.0534	4:3	751	13	734	18	81	(6) Kepler26
262.01	262.02	7.8128	9.3766	0.0001	6:5	1060	24	664	11	10000	(7) Kepler50
271.03	271.02	14.4360	29.3933	0.0181	2:1	⋯	⋯	1086	65	810	(3) Kepler127
274.01	274.02	15.0894	22.8039	0.0075	3:2	954	29	922	22	1000	(2),(20) Kepler128
277.02	277.01	13.8487	16.2321	0.0047	7:6	439	2	449	1	500	(8) Kepler36
314.03	314.01	10.3120	13.7811	0.0023	4:3	⋯	⋯	1204	89	1500	(3), (9), (20) Kepler138
377.01	377.02	19.2452	38.9568	0.0121	2:1	1351	0	1353	0	1600	(10), (20) Kepler9
500.01	500.02	7.0535	9.5216	0.0124	4:3	190	2	193	1	190	(2),(17) Kepler80
500.04	500.01	4.6453	7.0535	0.0123	3:2	⋯	⋯	190	2	190	(2),(17) Kepler80
500.04	500.02	4.6453	9.5216	0.0249	2:1	⋯	⋯	193	1	190	(2),(17) Kepler80
520.01	520.03	12.7594	25.7526	0.0092	2:1	1294	54	1306	70	1400	(3),(20) Kepler176
620.01	620.03	45.1554	85.3168	0.0553	2:1	789	12	1084	67	770	(7),(20) Kepler51
620.03	620.02	85.3168	130.1781	0.0172	3:2	1084	67	⋯	⋯	2500	(7),(11),(20) Kepler51
730.01	730.03	14.7869	19.7257	0.0005	4:3	1292	79	1690	100	10000	(3) Kepler223
730.02	730.01	9.8479	14.7869	0.0010	3:2	⋯	⋯	1292	79	5000	(3) Kepler223
730.04	730.01	7.3840	14.7869	0.0013	2:1	⋯	⋯	1292	79	6000	(3) Kepler223
730.02	730.03	9.8479	19.7257	0.0015	2:1	⋯	⋯	1690	100	6500	(3) Kepler223
775.02	775.01	7.8774	16.3848	0.0400	2:1	206	1	⋯	⋯	200	(7),(20) Kepler52
806.03	806.02	29.3235	60.3251	0.0286	2:1	985	1	969	3	1100	(12), (15), (20) Kepler30
829.01	829.03	18.6493	38.5579	0.0338	2:1	554	20	496	15	570	(7),(20) Kepler53
834.01	834.05	23.6537	50.4472	0.0664	2:1	382	7	⋯	⋯	380	(2) Kepler238
841.01	841.02	15.3354	31.3302	0.0215	2:1	723	11	717	8	730	(6),(20) Kepler27
869.03	869.02	17.4608	36.2755	0.0388	2:1	⋯	⋯	597	8	470	(3) Kepler245
870.01	870.02	5.9123	8.9858	0.0132	3:2	230	2	230	3	230	(6),(19),(20) Kepler28
877.01	877.02	5.9549	12.0399	0.0109	2:1	⋯	⋯	535	12	550	(2),(20) Kepler81
880.01	880.02	26.4445	51.5383	0.0255	2:1	969	34	1213	13	1000	(2),(20) Kepler82
886.01	886.02	8.0108	12.0715	0.0046	3:2	850	5	852	6	870	(7),(20) Kepler54
935.01	935.02	20.8602	42.6341	0.0219	2:1	940	13	1162	42	970	(12), (20) Kepler31
935.02	935.03	42.6341	87.6476	0.0279	2:1	1162	42	⋯	⋯	1600	(12), (20) Kepler31
952.01	952.02	5.9013	8.7521	0.0113	3:2	267	4	⋯	⋯	260	(12), (19), (20) Kepler32
1102.02	1102.01	8.1451	12.3335	0.0095	3:2	440	8	413	7	430	(5),(19),(20) Kepler24
1203.01	1203.03	31.8838	48.6457	0.0171	3:2	⋯	⋯	821	38	950	(2),(20) Kepler276
1215.01	1215.02	17.3240	33.0067	0.0474	2:1	348	8	⋯	⋯	350	(2),(20) Kepler277
1236.01	1236.03	35.7353	54.4117	0.0151	3:2	1194	23	1254	21	1200	(2),(18) Kepler279
1241.02	1241.01	10.5012	21.4057	0.0192	2:1	542	6	⋯	⋯	560	(7),(20) Kepler56
1270.01	1270.02	5.7293	11.6092	0.0131	2:1	⋯	⋯	456	5	440	(7),(20) Kepler57
1426.01	1426.02	38.8684	74.9285	0.0361	2:1	1040	20	⋯	⋯	1000	(3),(16),(20) Kepler297
1426.02	1426.03	74.9287	150.0254	0.0011	2:1	⋯	⋯	1322	60	60000	(3),(16),(20) Kepler297
1529.02	1529.01	11.8682	17.9770	0.0098	3:2	⋯	⋯	618	9	610	(7) Kepler59
1576.01	1576.02	10.4158	13.0842	0.0050	5:4	517	24	481	20	530	(2),(20) Kepler307
1599.02	1599.01	13.6141	20.4117	0.0005	3:2	⋯	⋯	1487	37	15000
1783.01	1783.02	134.4793	284.0423	0.0561	2:1	1390	140	⋯	⋯	2500
1955.04	1955.02	26.2349	39.4572	0.0027	3:2	563	28	⋯	⋯	5000	(3) Kepler342
2038.01	2038.02	8.3053	12.5136	0.0045	3:2	1091	42	1008	28	930	(2),(18),(20) Kepler85
2038.02	2038.04	12.5136	25.2177	0.0076	2:1	1008	28	⋯	⋯	1700	(3),(18),(20) Kepler85
2086.01	2086.02	7.1316	8.9190	0.0005	5:4	⋯	⋯	715	28	3600	(7) Kepler60
2092.01	2092.03	57.6899	77.0870	0.0022	4:3	1280	130	⋯	⋯	9000	(3) Kepler359
2672.02	2672.01	42.9935	88.5081	0.0293	2:1	1516	32	1303	11	1500	(2),(18) Kepler396

Notes.

^aOrbital period of the inner planet. ^bOrbital period of the outer planet. ^cNormalized distance to resonance defined as ${\rm{\Delta }}\equiv \tfrac{{P}_{2}}{{P}_{1}}\tfrac{j-1}{j}-1$ (Lithwick et al. 2012). ^dResonance type. ^eThe TTV period of the inner planet (found by modeling the data). ^fThe TTV period uncertainty of the inner planet. ^gThe TTV period of the outer planet (found by modeling the data). ^hThe TTV period uncertainty of the inner planet. ⁱThe TTV super-period infered from the orbital periods: ${P}^{j}\equiv \tfrac{1}{| j/{P}_{2}-(j-1)/{P}_{1}| }$. ^jThe second strongest TTV period.

References.  (1) Cochran et al. (2011), (2) Xie (2014), (3) Rowe et al. (2014), (4) Lissauer et al. (2011a), (5) Ford et al. (2012a), (6) Steffen et al. (2012a), (7) Steffen et al. (2013), (8) Carter et al. (2012), (9) Kipping et al. (2014), (10) Holman et al. (2010), (11) Masuda (2014), (12) Fabrycky et al. (2012), (13) Jontof-Hutter et al. (2014), (14) Lissauer et al. (2013), (15) Sanchis-Ojeda et al. (2012), (16) Diamond-Lowe et al. (2015), (17) Ragozzine & Kepler Team (2012), (18) Yang et al. (2013), (19) Lithwick et al. (2012), (20) Hadden & Lithwick (2014).

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Out of the 59 systems, we have found 26 pairs for which the two observed TTV periodicities agreed with each other and also agreed with the calculated super-period, one pair (KOI-250) for which the two observed TTV periodicities agreed with each other but did not agree with their calculated super-period, and 32 pairs for which only one TTV periodicity agreed with the calculated super-period (the other planet did not show any significant periodicity or its TTV period did not agree with the calculated super-period). In one case, KOI-157.03, the derived TTV period was the second periodicity found.

To summarize this section, out of the 199 KOIs with periodic TTVs, 84 reside in systems where only one transiting planet is evident. Another 39 KOIs are found in multi-planet systems without an obvious adjacent planet that can explain the detected TTV periodicity. Another 23 KOIs are in multi-planet systems, for which the period of one adjacent planet yields a super-period that is consistent with the observed TTV periodicity. Finally, 53 KOIs show a TTV periodicity that is anti-correlated with TTV modulation of another adjacent planet in the system.

Before concluding, we compare our results with the published KOI catalog by Rowe et al. (2015, RoweI), which also contains the O-C measurements for 5751 KOIs, and with Rowe & Thompson (2015, RoweII) catalog,¹¹ which presents timing measurements for 258 KOIs that they identified as having significant TTV signals. (The criteria used were not given.)

A total of 2290 KOIs were analyzed by this paper and by either RoweI or RoweII. The reason 2014 is that RoweI and II analyzed an extended set of KOIs, including the new batch of identified planet candidates, while we considered the older list of KOIs released in 2013 November. Furthermore, we excluded from the list a set of KOIs that we found to be EBs. Therefore the intersection of the two studies is only 2290 KOIs. In those KOIs, RoweI obtained 275,551 transit timings, while we had a total of 287,428 measurements, and 208,650 timings that were not rejected as outliers. The TTV error estimates were very similar in both works.

We now compare the results of RoweII with the KOIs we found to have significant TTVs (Tables 5 and 7). Out of the 258 KOIs RoweII found with significant TTVs, we have analyzed only 208, out of which 124 KOIs appeared in Table 5, another 4 KOIs appeared in Table 7, and 80 KOIs that we did not identify as variables. On the other hand, we identified 137 KOIs as having significant variability (127/10 in Tables 5/7), which were analyzed by RoweI and were not marked as variables by RoweII. The three sets of timing series derived by the two analyses were carefully compared.

• 1.  
  80 KOIs labeled significant by RoweII only—we checked these systems and found most of them not to be significantly variable, both by eye and by our threshold statistics. Only two KOIs (KOI-94.02 and -209.02) actually did (barely) pass our statistic threshold, but we had removed them from the list, as the data did not seem significant enough.
• 2.  
  127 KOIs labeled significant by this work only—to check our analysis, we used RoweI timings and still found them to be significantly variable. Just to name a few: KOI-474.03, -841.01, -841.02, -1081.01, -1270.02, -1529.01, -1599.01, and -2061.01 (see Figures 9, 14, 16, 17, 19, 20 and 24).
• 3.  
  124 KOIs labeled significant by RoweII and by this work—in almost all cases the TTVs were comparable. Only KOI-806.03 displayed timing measurements that were significantly different in the two works. The two sets of derived TTVs are presented in Figure 34 (see also Sanchis-Ojeda et al. 2012). RoweI and RoweII probably missed the actual timings of this KOI because of the large TTVs.

The difference between the works may have resulted from the fact that this work was focused on the TTV variability, while both RoweI and RoweII were studying the general set of KOIs, with the goal of presenting a complete set of planet candidates with analyzed transits.

We presented here a new transit timing catalog of 2599 Kepler KOIs, using the PDC-MAP long-cadence light curves that cover the full seventeen quarters of the mission. The catalog included 69,914 transits of 779 KOIs with high enough S/N, for which we derived the best fitting timing, duration and depth for each transit, given the KOI's best transit template. For additional 225,273 transits of 1820 KOIs with lower S/N, we derived only the transit timing, keeping the duration and depth fixed for each KOI.

The catalog is available at ftp://wise-ftp.tau.ac.il/pub/tauttv/TTV/ver_112, where separate tables for the duration and depth variations are accessible too. In the future, we plan updating the catalog to include short-cadence data and better analysis of overlapping transits.

For each KOI we derived various statistics that can be used to indicate significant variations. Including systems found by previous works, we have found 260 KOIs that showed significant long-term TTVs, with periods longer than 100 days. Of those, 199 KOIs displayed well determined TTV periods and amplitudes. Another 61 KOIs have periods too long to be established without a doubt, and therefore only a parabola was fitted to the TTV series.

It is interesting to compare the analysis of Mazeh et al. (2013) with our results, and to see how the longer time span may change our assessment of the nature of the modulation. For example, after correcting for the TTVs, the deduced orbital periods of KOI-564.01, -1145.01, -1573.01, and -1884.01 were significantly changed; the TTVs of KOI-984.01 showed sinusoidal behavior in the previous catalog, and now display a non-sinusoidal shape with a sharp linear rise; the TTVs of KOI-1599.01 and -1426.01 showed only parabolic behavior in Mazeh et al. (2013), while now allowing for a sinusoidal fit.

For most of the KOIs, we need more data before the TTV period can be robustly determined. However, as the Kepler mission in its original mode of operation has been terminated, we do not expect any additional Kepler TTVs. The next space missions that will observe the Kepler field in a search for transiting planets (e.g., TESS, PLATO), or dedicated ground-based follow-up observations (e.g., Raetz et al. 2014), may be very useful for determining the true nature and TTV periodicity of these systems.

Most of the significant TTVs presented here are probably due to dynamical interaction with another planet(s), either transiting or still unknown. The present catalog includes 121 single KOIs with significant long-term modulations, probably caused by undetected additional planets, and therefore are multi-planet systems. The missing planets can be divided into two classes: (1) small planets that were not detected by the present analysis of Kepler's data, either because the transits were too shallow and/or the transit timings were shifted by the dynamical interaction with the known planet, so that the folding of the light curve could not reveal the shallow; and (2) planets with high enough inclinations, so they do not pass in front of their parent stars.

One could hope that the accumulating details of the observed TTVs could give some hints for the orbital elements of the perturbing unseen planet. However, as discussed already by Holman & Murray (2005) and Agol et al. (2005), the amplitude and periodicity of the TTV modulation depends on various parameters, in particular the mass and the orbital period of the unseen planet, and how close the orbits of the two planets are to some mean motion resonance (e.g., Lithwick et al. 2012). Therefore, it is quite difficult to deduce the parameters of the unseen planet. In selected cases, some stringent constraints can be derived, as was done by Ballard et al. (2011) and Nesvorný et al. (2012, 2013). We hope that our catalog will motivate a similar work on other single-KOI systems, as well as on the multi-planet systems, with significant TTVs.

Ten systems analyzed here showed significant, small amplitude modulations with periods shorter than 100 days. We presented these ten systems together with another two systems, whose modulations were detected by Mazeh et al. (2013) and still showed periodic, but less significant, modulations, and another two KOIs that displayed photometric slope correlated with the TTVs, found by Holczer et al. (2015). Out of the 13 systems found by Mazeh et al. (2013) with short-period modulation, 7 now appear to be false positives, and the other 6 are presented here.

The short-period modulations might not be due to dynamical interaction with another planet, but could be due to either the long cadence sampling of Kepler, or the stellar spots periodic activity. For five systems, the stellar rotational frequency and/or one of its harmonics coincide with the highest peak (see also Holczer et al. 2015) of the derived periodogram. Two systems, KOI-883.01 and -13.01, show a strong stroboscopic effect. The stars with TTVs related to the stellar spot activity are interesting, because the transit timings can shed some light on the stellar spot activity, as was shown very recently by Mazeh et al. (2015, 2015) and Holczer et al. (2015).

Finally, one could expect that the derived TTVs, for the single KOIs in particular, could help in constructing a statistical picture of the frequency and architecture of the population of the planetary multiple systems of the Kepler KOIs (e.g., Steffen et al. 2010; Ford et al. 2011, 2012b; Lissauer et al. 2011b; Fabrycky et al. 2014). To perform such a statistical analysis one needs to model the dependence of the detectability of long-term TTV coherent modulations on the parameters of the unseen perturbing planet. The present catalog can be used for such a study.

We are indebted to the referee for careful reading of the paper and for many thoughtful and helpful comments and suggestions. The research leading to these results has received funding from the European Research Council under the EU's Seventh Framework Programme (FP7/(2007-2013)/ERC Grant Agreement No. 291352), the ISRAEL SCIENCE FOUNDATION (grant No. 1423/11), and the Israeli Centers of Research Excellence (I-CORE, grant No. 1829/12). E.B.F. was supported in part by NASA Kepler Participating Scientist Program award # NNX14AN76G. All photometric data presented in this paper were obtained from the Mikulsky Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts.