TRANSIT TIMING OBSERVATIONS FROM KEPLER. I. STATISTICAL ANALYSIS OF THE FIRST FOUR MONTHS

A combination of pipelines is used to identify KOIs as described in Latham et al. (2011). We measure TTs based on the long cadence (LC), optimal aperture photometry performed by the Kepler SOC pipeline version 6.2 (Jenkins et al. 2010). The pipeline produces both "calibrated" light curves (photometric analysis, PA, data) for individual analysis and "corrected" light curves (pre-search data conditioning, PDC) which are used to search for transits. This paper presents a bulk set of TTs measured from PDC data. For a detailed analysis of an individual system, we strongly recommend that users inspect both PA and PDC data to determine which is best suited for their target star and applications. For example, the TTV detections of Kepler-9 b and c and Kepler-11 b–f were based on PA data (Holman et al. 2010; Lissauer et al. 2011a).

We fold light curves at the orbital period reported in B11. Next, we fit a limb-darkened transit model to the folded light curve, allowing the epoch, planet–star radius ratio, transit duration, and impact parameter to vary. We use the best-fit transit model as a fixed template to measure the TT of each individual transit (while holding other parameters fixed). The best-fit TTs are determined by Levenberg–Marquardt minimization of χ² (Press et al. 1992). In a few cases, we iterate the procedure, aligning transits based on the measured period in order to generate an acceptable light curve model. For a small fraction of candidates (KOI 2.01, 403.01, 496.01, 508.01, 559.01, 617.01, 625.01, 678.01, 687.01, 777.01), we measure TTs during Q1 only, due to complications in the Q2 light curves (e.g., stellar noise and/or grazing short-period events). Two other types of complications merit some explanation. For bright stars that saturate the CCD, electrons bleed into neighboring pixels. When the spacecraft rotates each quarter, a target moves from one CCD to another and the aperture mask specifying which pixels are downloaded changes. If the aperture mask does not capture all the electrons, this can cause the transit light curve to vary from quarter to quarter, breaking the code used for measuring TTs. Another complication that interfered with measuring TTs during Q2 arises since the Q2 photometry for most targets is not as high quality as the Q1 photometry, due to a variety of technical issues (e.g., replacing guide stars that were discovered to be variable or binary resulted in the use of even more problematic guide stars, causing high-frequency spacecraft motion). For a few KOIs with small signal to noise per transit, this prevented accurate measurement of TTs during Q2. The photometric quality in subsequent quarters has improved significantly, as demonstrated in publically available light curves for Kepler-9, 10, and 11.

We estimate the uncertainty in each TT from the covariance matrix. For most planet candidates, typical scatter in its TT relative to a linear ephemeris is comparable to the median timing uncertainty (see Figure 1). Even when the transit model used for TT measurements is not ideal, good TTVs and errors can be extracted using this technique, since the TT is the only free parameter and only a roughly correct shape is needed to identify accurate times and errors. Subsets of the observed times reported in Table 1 were tested with different TT estimation codes, which employ different techniques. While a more thorough analysis can sometimes reduce the timing uncertainty or minimize the number of apparent TTV outliers, the results are consistent across different algorithms.

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For target stars where multiple transiting planet candidates have been identified, we sequentially fit each transit separately. Prior to fitting for TTVs, we remove the best-fit transit models for additional candidates present in the light curve. For example, if there are three planet candidates in the system, then the fitting procedure would first fit for the template light curves in the following order: (1) fit candidate.01, (2) remove.01 from the light curve, (3) fit candidate.02 from residuals, (4) remove.02 from light curve, (5) fit candidate.03 from residuals, and (6) remove.03 from light curve. In most cases, the sequence.01,.02, ... is from the highest to the lowest signal-to-noise (integrated over all transits). Next, we would measure the TTs according to the following plan: (1) remove.02 and.03 from original light curve, (2) fit for.01 template, (3) measure TTs for candidate.01, (4) remove.01 and.03 from original light curve, (5) fit for.02 template, (6) measure TTs for candidate.02, (7) remove.01 and.02 from original light curve, (8) fit for.03 template, and (9) measure TTs for candidate.03. In some cases, we repeat the measurement of TTs by aligning the transits using the first set of TTs before generating the template.

Once each set of individual TTs has been measured, we calculate multiple sets of TTVs by comparing them to multiple ephemerides. The TTVs reported in Table 1 are measured relative to the linear ephemeris published in B11 (E_L5). A positive TTV corresponds to a transit occurring later than the ephemeris. Second, we considered the TTVs relative to the best-fit linear ephemeris calculated from TTs in Q0-2 (E_L2; Table 2). The E_L5 ephemerides differ from the E_L2 ephemerides in that epoch ($\hat{E}_0$) and period ($\hat{P}$) were determined using Kepler data up to and including Q5, rather than Q2. Finally, we also considered TTVs relative to the best-fit quadratic ephemeris calculated from TTs in Q0-2 (E_Q2). The quadratic ephemeris is given by

$$\begin{equation} \hat{t}_n = \hat{E}_0 + n \hat{P} (1 + n \hat{c}), \end{equation} \tag{ 1 }$$

where $\hat{E}_0$ is the best-fit time of the zeroth transit, $\hat{P}$ is the best-fit orbital period, and $\hat{c}$ is the best-fit value of the curvature. For a linear ephemeris, $\hat{c}=0$.

First, we calculated $X^2 \equiv \sum _{n\in Q0-2} \left(\left(t_n - \hat{t}_n \right)/ \sigma _n \right)^2$ for $\hat{t}_n$ calculated from both the E_L2 and E_L5 ephemerides. As X² closely resembles a χ² variable, we first apply a χ² test, but based on X². If we assume that X² follows a χ² distribution, then a χ² test based would result in a p-value less than 0.001 for 15%–17% of candidates. As expected, the fraction is lower if we use the E_L2 ephemerides, since the same times are being used for the model fit and the calculation of X². If the errors in the TT measurements, normalized by the estimated timing uncertainty, were accurately described by a standard normal distribution, then one would expect only a few false alarms. However, we note that a disproportionate fraction (42%) of the planet candidates with seemingly significant scatter in their TTs have an average signal-to-noise ratio (S/N) in each transit of less than 3, while planet candidates with such a small S/N per transit represent only (∼23%) of planet candidates considered. While it is possible that planets with low S/N transit are more likely to have significant TTVs, we opt for a more conservative interpretation that the distribution of errors in our TTs measurements for low S/N transits is not accurately described by a normal distribution with a dispersion given by the estimated timing uncertainties. In particular, when measuring the times of low S/N transits, the X² surface as a function of t_n can be bumpy due to noise in the observed light curve, resulting in a much greater probability of measuring a significantly discrepant TT than assumed by our normal model.

To avoid a high rate of false alarms due to random noise, we focus our attention on planet candidates where the typical S/N in a transit exceeds 3. If we assume that X² follows a χ² distribution, then results in ∼11%–13% (86 or 103/805) of the remaining planet candidates having a p-value less than 0.001. We observe that for many of these cases, the contribution to X² is dominated by a small fraction of the TTs. Based on the inspection of many light curves around these isolated discrepant TTs, we find that occasionally the measured TTs can be significantly skewed by a couple of deviant photometric measurements. In principle, this could occur due to random noise, but for high S/N transits, stellar variability or improperly corrected systematic errors are the more likely culprits. For example, systematic errors often appear after events such as reaction wheel desaturations, reaction wheel zero crossings, and data gaps (see data release notes at MAST for details, http://archive.stsci.edu/kepler/data_release.html).

To further guard against a few discrepant TTs leading to a spurious detection of TTVs, we calculate a statistic that is similar to X², but one that is more robust to outliers,

$$\begin{equation} X^{\prime 2} \equiv \frac{\pi N_{{\rm TT}} \left(\mathrm{MAD}\right)^2}{2\sigma _{{\rm TT}}^2}, \end{equation} \tag{ 2 }$$

where N_TT is the number of TTs measured in Q0-2 and σ_TT is the median TT measurement uncertainty. Here, MAD is the median absolute deviation of the measured TTs from an ephemeris, given by

$$\begin{equation} \mathrm{MAD} \equiv \mathrm{median}_{i=1}^{N_{{\rm TT}}}(|t_i-\hat{t}_i |), \end{equation} \tag{ 3 }$$

where $\hat{t}_i$ is the time predicted by a given ephemeris. The X'² statistic can be viewed as the ratio of a robust variance of the TTs to the square of the typical measurement uncertainty for TTs. We calculate alternative p-values for rejecting the null hypothesis that there are no deviations from a linear ephemeris by replacing χ² with X'². Since the distribution of X'² may deviate from a χ² distribution, the p-values and resulting significance levels are not reliable. Our X'² test merely provides a means of filtering the list of KOIs to identify those with potentially significant TTVs. The advantage of X'² for our application is that X'² is much less sensitive to the assumption that measurement errors are well characterized by a normal distribution. A small fraction of significant outliers can greatly increase X² but have a minimal effect on X'². While this is more robust than a standard χ² test, the statistical significance will not be accurately estimated if the measurement errors are severely non-Gaussian. Of course, the increased robustness comes at a price: X'² is not useful for recognizing TTV signals where only a small fraction of TTs deviate from a standard linear ephemeris. We expect this is a good trade, given the predicted TTV signatures (Veras et al. 2011) and the characteristics of our TT measurements.

We identify planet candidates showing strong indications of a TTV signal based on an ad hoc threshold which would correspond to a p-value of p ⩽ 10⁻³, if X'² followed a χ² distribution. Restricting our attention to planet candidates for which the individual transits are detected in the light curve with an S/N greater than 3, this test identifies roughly ∼4%–5% (29 or 39/805) as showing excess scatter relative to a linear ephemeris. We find the larger fraction (∼5%) when comparing to the E_L2 ephemerides (rather than the E_L5 ephemerides). We speculate that this may be due to the E_L5 ephemerides being more robust to a small number of outlying TTs in Q2 which would have a greater effect on the E_L2 ephemerides. We discuss these candidates further in Section 4.1. A summary of these and other summary statistics for all planet candidates considered appears in Table 3.

Table 3. Metrics for Transit Timing Variations of Kepler Planet Candidates

		(EL5,T2)				(EL2,T2)
KOI	σTT	MADa	WRMSb	MAXc	$p_{X^{\prime 2}}$d	MADa	WRMSb	MAXc	$p_{X^{\prime 2}}$d	$\frac{\left|\Delta P\right|}{\sigma _P}$e	$\frac{\left|\Delta E\right|}{\sigma _E}$f
	(min)	(min)	(min)	(min)		(min)	(min)	(min)
1.01	0.2	0.1	0.1	2.4	0.99999727	0.1	0.1	2.5	0.99989265	1.3	1.1
2.01	0.3	0.2	0.3	0.7	0.66745223	0.1	0.3	0.8	0.99935567	1.4	1.4
3.01	0.7	0.3	0.4	3.2	0.99982414	0.3	0.4	3.1	0.99929283	1.2	0.6
4.01	3.5	2.4	3.5	8.6	0.84167045	2.8	3.4	9.4	0.35265519	0.0	2.4
5.01	1.9	1.4	1.6	3.1	0.711099	1.7	1.5	3.0	0.11421086	2.2	0.6

Notes. ^aMedian absolute deviation of transit times in Q0-2 from ephemeris. ^bWeighted root mean square deviation of transit times in Q0-2 from ephemeris. ^cMaximum absolute deviation of transit times in Q0-2 from ephemeris. ^dp-value for a χ²-like test assuming X'² follows a χ² distribution, as described in Section 3.1. ^eAbsolute value of difference of best-fit periods for E_L2, T2 and E_L5, T2 ephemerides normalized by formal uncertainty. ^fAbsolute value of difference of best-fit transit epochs for E_L2, T2 and E_L5, T2 ephemerides normalized by formal uncertainty.

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.

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We also considered calculating X², but based on "clipped" TTs from Q0-2, where we reject TTs with either a formal uncertainty greater than twice the median timing uncertainty or with an absolute deviation greater than three times the MAD of TTs. This results in ∼4% of planet candidates with at least three transits during Q0-2 and an S/N per transit greater than 3 as showing excess scatter relative to the E_L2 ephemeris. While the rate of TTV candidates is consistent with the results of the X'² statistic described above, the exact systems flagged differ from those identified based on X'² test. Roughly half of these are cases with less than 10 transits during Q0-2, in which case the threshold for clipping is poorly defined. Therefore, we prefer the tests based on X'² for the present data set.

3.2.1. Difference in Best-fit Orbital Periods

3.2.2. Difference in Best-fit Epochs

3.2.3. $\mathcal {F}$-test for Comparing Quadratic and Linear Models

While TTV signatures can be quite complex, the timescale for resonant dynamical interactions among planetary systems is typically orders of magnitude longer than the orbital period. Thus, we expect that the dominant TTV signature of many resonant planetary systems can be well approximated by a gradual change in the orbital period over timescales of months to years. Indeed, such a pattern led to the confirmation of Kepler-9 b and c (Holman et al. 2010). (There is also a pattern of alternating transit arriving earlier and later than the quadratic ephemeris. However, this "chopping" signature occurs on an orbital timescale, but a much smaller amplitude.) Thus, we fit both linear and quadratic ephemerides (see Section 2.2) to the Q0-2 data, calculate $X_{E_L2}^2$ and $X_{E_Q2}^2$ for the two models, respectively. We perform a variant of the F-test to assess whether including a curvature term significantly improves the quality of the fit. We use a test statistic as $\mathcal {F} \equiv (N_{{\rm TT}}-3) X_{E_L2} / \left[ (N_{{\rm TT}}-2) X_{E_Q2} \right]$. This is similar to an F-statistic, but we use $\mathcal {F}$ since it is based on a ratio of X statistics that do not necessarily follow χ² distributions. This method has the advantage of being less sensitive to the accuracy of the TT uncertainty estimates, as they affect both the numerator and denominator of the $\mathcal {F}$ statistic similarly. If we assumed that $\mathcal {F}$ followed an F distribution, then an F-test would not result in a p-value less than 10⁻³ for any of the KOIs that are still viable planet candidates. For our application, the $\mathcal {F}$-test has limited power, since we fit only TTs measured in Q0-2, resulting in a shorter time span for a gradual change in the period to accumulate. We note that the $\mathcal {F}$-test does strongly favor the quadratic model for two KOIs previously identified as being due to stellar binary or triple systems (KOI-646.01, D. Fabrycky et al. 2011, in preparation; 1153.01), where the curvature is sufficiently large that the period changes appreciably during Q0-2.

Next, we consider systems for which $\mathcal {F}$ would imply a p-value of 0.05 or less, if $\mathcal {F}$ were to follow an F distribution. Quadratic ephemerides for these KOIs are given in Table 4. Approximately 1% of KOIs considered are identified. Half of these correspond to candidates with only four or five transits in Q0-2, so it is impractical to assess the quality of a three-parameter model accurately. Of the remaining systems, KOI 142.01 and 227.01 were already identified by the X'² test, while KOI-528.01 and 1310.01 were not. Upon visual inspection, 528.01 appears to be a plausible detection of TTVs, but 1310.01 does not, as the TTs and uncertainties are also consistent with a linear ephemeris. These candidates are discussed further in Section 4.1.

Table 4. Quadratic Ephemerides for Kepler Objects of Interest Based on Transit Times during Q2

KOI	Ea	σE	P	σP	cb	σc	MADc	WRMSd	MAXe	$p_{X^{\prime 2}}$f	$p_{\mathcal {F}}$g
	(d)	(d)	(d)	(d)			(d)	(d)	(d)
42.01	149.89207	0.002317258	17.835806	0.000999096	−7.93E-05	4.83E-05	0.000224292	0.000465821	0.00087804	0.98156568	0.053661725
124.02	107.5387	0.004856891	31.713798	0.003447988	0.000155781	9.53E-05	0.000101866	8.70E-05	0.000177141	0.96675223	0.018503953
142.01	120.58229	0.0014794	10.912549	0.000260135	6.28E-05	7.86E-06	0.001685495	0.002572359	0.00727168	0.67069648	0.005875349
227.01	122.54014	0.001542628	17.678654	0.000469767	−7.68E-05	1.55E-05	0.000969901	0.001511303	0.003312192	0.82842292	0.028645506
314.01	124.63071	0.001690889	13.780802	0.000429775	3.71E-05	1.52E-05	0.002035933	0.001290098	0.003126117	0.35704363	0.078718429
467.01	151.46194	0.001891538	18.007959	0.000842961	−4.92E-05	2.47E-05	6.71E-05	6.55E-05	0.000102345	0.93992411	0.029226395
528.01	138.41866	0.003407732	9.578232	0.000663483	−5.97E-05	2.13E-05	0.002067159	0.003516547	0.007456802	0.99547272	0.035870139
649.01	139.37393	0.010461445	23.447238	0.004043237	−0.000165971	0.000148364	0.000121626	0.000120244	0.000228574	0.98082178	0.017380622
878.01	130.40553	0.007549669	23.615695	0.006189283	−0.000667528	0.000214225	0.001208953	0.001048118	0.001790873	0.79282356	0.067398588
935.01	133.86998	0.003139609	20.860536	0.001445591	8.34E-05	5.89E-05	6.38E-05	0.000120015	0.000214038	0.99919724	0.002730505
941.03	146.68417	0.00254628	24.664449	0.002533906	0.000158955	9.16E-05	0.000255548	0.000235014	0.000381217	0.87610092	0.07150007
960.01	125.95369	0.000313363	15.800727	0.000146537	1.07E-05	5.90E-06	0.000135125	0.000123596	0.000210589	0.88030747	0.054869155
1308.01	126.11344	0.008972545	23.597302	0.004072063	0.00039199	0.000147879	0.003606992	0.003070563	0.005514918	0.73324413	0.061761557
1310.01	129.81925	0.009257043	19.130778	0.00267195	6.13E-05	8.77E-05	0.000511397	0.001120656	0.002320763	0.99997768	0.020817298

Notes. ^aEpoch. ^bCurvature; see Equation (1). ^cMedian absolute deviation of TTs from quadratic ephemeris. ^dWeighted root mean square deviation of TTs from quadratic ephemeris. ^eMaximum absolute deviation of TTs from quadratic ephemeris. ^fp-value for a χ²-test based on X'² relative to quadratic ephemeris, as described in Section 3.1. ^gp-value for an F-like test that compares linear and quadratic ephemerides, assuming that $\mathcal {F}$ follows an F distribution, as described in Section 3.1.

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Over long timescales, dynamical interactions in two planet systems can result in complex TTV signatures characterized by many frequencies. However, on short timescales, TTV signatures can often be well described by a single periodic term. For example, for closely packed, but non-resonant, planetary systems, the TTV signature is often dominated by the reflex motion of the star due to other planets. In this case, we would expect a typically small TTV signal on an orbital timescale.

Given the relatively short time span of the observations, we search for planet candidates with a TTV signature that can be approximated by a single sinusoid,

$$\begin{equation} \hat{t}_{n} = \hat{E}_0 + n \hat{P} + \hat{A} \sin (2\pi n\hat{P} / \hat{P}_{{\rm TTV}}) + \hat{B} \cos (2\pi n\hat{P} /\hat{P}_{{\rm TTV}}), \end{equation} \tag{ 4 }$$

where $\hat{A}$ and $\hat{B}$ determine the amplitude and phase of the TTV signal, while $2\pi /\hat{P}_{{\rm TTV}}$ gives the frequency of the model TTV perturbation. For a given $\hat{P}_{{\rm TTV}}$, finding the best-fit (i.e., minimum X²) model is a linear minimization problem. Thus, we perform a brute force search over $\hat{P}_{{\rm TTV}}$ to identify the best-fit simple harmonic model (Ford et al. 2011). Assessing whether the best-fit harmonic model is significantly better than a standard linear ephemeris is notoriously difficult (e.g., Ford & Gregory 2007), even when the distribution and magnitude of measurement uncertainties are well understood. Therefore, we plotted the cumulative distribution of summary statistics and recognized a break in the distribution corresponding to a tail that included approximately 2% of the planet candidates considered, which also included several KOIs which have since been moved to the false positives list. One-quarter of the KOIs in this tail had eight or fewer TTs measured in Q0-2, so it was not practical to assess the quality of a five-parameter fit accurately. Of the remaining KOIs with a possible periodic signal, 90% have some other evidence suggesting that they are a stellar binary. Three KOIs remain KOI-258.01 (most significant), 1465.01, and 1204.01 (least significant). The first two were also identified by the X'² analysis described in Section 3.1. KOI-258.01 appears to show a periodic pattern with amplitude ∼40 minutes and timescale of either ∼28 or 58 days (see Figure 2), however there is also a hint of a secondary eclipse for KOI-258.01. For KOI-1465.01 the combination of a short transit duration and LC observations might have resulted in the PDC data having artifacts that render the TTs unreliable. For KOI-1204.01, there is only a hint of a periodicity and additional observations will be necessary to assess the significance of the putative signal.

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We provide a summary of the planet candidates identified by our statistical tests in Table 5. For completeness sake, the table includes all planet candidates from B11, with some indication of TTVs and at least three transits in Q0-2. However, we regard many as weak candidates. For a strong candidate, we typically require (1) an S/N per transit of at least 4, (2) at least five TTs observed in Q0-2 for most tests, and (3) no indications of potential difficulties measuring the TTs (e.g., near data gap, short duration, PDC artifacts, heavily spotted star). When comparing the best-fit epochs, we require only three transits during Q0-2. Many of these candidates were identified by the X'² statistic which appears to provide a balance of sensitivity and robustness when searching for excess scatter in the present data set. Those detected by other tests have already been discussed in Sections 3.2 and 3.3 or are indicated with a flag of 6 in Table 5. As the number of transits observed by Kepler increases, it is expected that model fitting will eventually provide superior results (Ford & Holman 2007). For example, Kepler-9 b and c, and Kepler 11 b–f were confirmed by TTVs, but only Kepler-9 c shows TTVs in the Q0-2 data. For Kepler-9 b, the TTVs are not detectable during Q0-2, as the libration timescale (∼10 years) is much greater than the time span of observations (Holman et al. 2010). The TTVs of planets in Kepler-11 are much smaller and demonstrate the benefits of fitting a dynamical model to TTs of all planets simultaneously (Lissauer et al. 2011a).

The analysis presented in this manuscript considers each candidate individually. For multi-candidate systems, it is possible that there exists a significant anti-correlation in TTV signals (E. B. Ford et al. 2011, in preparation; J. Steffen et al. 2011, in preparation), even though the individual signals do not reach the various significance thresholds given in Section 3. If the relationship between TTVs of multi-candidate systems can be proven to be non-random, it significantly weakens the probability that the candidate signals are due to false positives (Ragozzine & Holman 2010), even if the properties of the planets cannot be directly inferred. This method of confirming planets will become much stronger with additional data.

4.1.1. False Positives

4.1.2. Weak TTV Candidates

4.1.3. Strong TTV Candidates

Based on Q0-2 data, the largest timing variations for active planet candidates are KOIs 227.01, 277.01, 1465.01, 884.02, 103.01, 142.01, and 248.01 (starting with largest magnitude of TTVs). Each of these has well-measured TTs due to a high S/N in each transit, a transit duration of over 3 hr (minimizing complications due to LC and PDC corrections), and a star with limited variability. Each shows a clear trend of TTVs, indicating at least a period change between Q0-2 and Q0-5. Figure 2 (top row) shows the TTVs of two examples, KOI 103.01 and 142.01. For comparison, Figure 3 shows transit time measurements for four planet candidates for which we do not detect transit timing variations. For KOIs 142.02 and 227.01, the period derivative can be measured from Q0-2 data alone (see Table 4). Each of these candidates has been tested with centroid motion tests, high-resolution imaging (except 884 and 1465), and at least some spectroscopic observations. Only KOI 142 shows any indications of the KOI being due to a blend with an eclipsing binary star. Given the large magnitude of the period derivatives, the transiting body must be strongly perturbed, by a companion that is massive, close and/or in a mean-motion resonance (MMR). Despite the clear timing variations, we do not consider these confirmed planets, out of an abundance of caution. Confirming them would require excluding nearly all known possible false positives, as was done for Kepler-9 d (Torres et al. 2011) and Kepler-10 c (Fressin et al. 2011).

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For example, it is possible that some (or all) of these may be examples of physically bound triple systems consisting of the bright target star and a low-mass eclipsing binary. ETVs are not uncommon in the Kepler Binary Star Catalog (Prsa et al. 2011; Slawson et al. 2011; J. A. Orosz et al. 2011, in preparation). If this is the case, then it may eventually be possible to detect eclipses of additional bodies due to orbital precession, as observed in the triply eclipsing triple system, KOI-126 (Carter et al. 2011). Indeed, some of the first KOIs investigated by the Kepler Transit Timing Working Group have timing variations of a similar magnitude and timescale but turned out to be triple systems (KOIs 646, 928). Since these systems were selected to have the largest amplitude TT variations during Q0-2, it would not be surprising if they were atypical of multiple transiting planet candidate systems. The Kepler Science Team will investigate these systems further to determine which are indeed planets and which are multiple star systems. If they are planets, then the planet radii are ∼2–3 R_⊕, assuming stellar radii from the Kepler Input Catalog (Brown et al. 2011). Even if all were to turn out to be false positives, the timing variations will play a critical role in understanding the nature of these KOIs.

Several other planet candidates were identified as having significant scatter in their TTs by the X'² test or a difference between our E_L2 ephemeris and the E_L5 ephemeris of B11. Further information for these is given in Table 5. A few, such as KOI 151.01 and 270.01, appear to have hints of a pattern in the TTs, but large timing uncertainties make interpretation difficult at this time. We expect that further TT observations will clarify which are real signals and enable confirmation and/or complete dynamical model modeling. One potential source of astrophysical false positives that could masquerade as planets with dynamical TTVs is an isolated star+planet (or eclipsing binary) combined with stellar activity and/or spots. This is a particular concern for planet candidates that appear to have excessive scatter of their TTVs, rather than a long-term trend or periodic pattern. The Kepler Follow-up Observation Program will be conducting a variety of follow-up measurements to help eliminate potential false positives and likely confirming many of the multiple transiting planet candidates.

4.1.4. Strong TTV Candidates in Multiple Transiting Planet Candidate Systems

We discuss a few particularly interesting systems with multiple transiting planet candidates. In order to form an order-of-magnitude estimate of the magnitude of TTVs that are likely to arise in systems with multiple transiting planets, we performed an N-body integration for each of these systems. We assume coplanar and circular initial conditions and assign masses according to

$$\begin{equation} M_p/M_\oplus = (R_p/R_\oplus)^{2.06}, \end{equation} \tag{ 5 }$$

as described in Lissauer et al. (2011b). As an example, the TTs predicted by the baseline model for the four of the planet candidates of KOI 500 are presented in Figure 4. The magnitude of TTVs predicted by this baseline model for all stars with multiple transiting planet candidates is reported in Table 6. As the TTV signature can be very sensitive to masses and initial conditions (Veras et al. 2011), we do not expect that these simulations will accurately model the TTV observations. However, they can help us develop intuition for interpreting TT observations. For example, we can often predict a timescale and associated amplitude of TTVs to within a factor of ∼2, but the phase of the associated TTVs is more sensitive to the detailed initial conditions. For systems with at least one eccentric planet, there can be additional TTV frequencies with much larger amplitudes.

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4.2.1. Non-detection of TTVs in Multiple Transiting Planet Candidate Systems

4.2.2. Dynamical Instability and Multiple Transiting Planet Candidate Systems

4.2.3. Specific Systems

Our analysis of TT measurements during Q0-Q2 demonstrate that Kepler is capable of providing precise TTs which can be expected to enable the dynamical confirmation of transiting planet candidates and detection of non-transiting planets. After discarding those planet candidates with S/N per transit of less than 4 or just a few transits in Q0-2, the MAD of TTs from a linear ephemeris during Q0-2 is as small as ∼20 s for Jupiter-size planets, and 1.5 minutes for Neptune-size and super-Earth-size planets. The median (taken over planet candidates) MAD of TTs is approximately ∼11 minutes for super-Earth-size planets, ∼6.5 minutes for Neptune-size planets, and ∼1.5 minutes for Jupiter-size planets. Since large planets can be detected around fainter stars at low signal to noise, each class includes some candidates with relatively poor timing precision (up to an hour).

All the TT measurements presented in this paper were based on LC data only and were performed in a semi-automated fashion. Experience with systems subjected to detailed TTV studies suggests that TT precision could often be significantly improved, if individual attention to devoted to mitigating systematic effects and choosing which transits yield the best templates. This work can serve as a broad, but shallow survey of Kepler planet candidates. Our results can help scientists choose the most interesting systems for follow-up work to perform more detailed light curve modeling (as well as other types of follow-up such as observations from other observatories, theoretical investigations, and observations using short cadence mode).

In addition to LC observations, Kepler collects photometry in short cadence mode (≈ 1 minute samples) for a small fraction of its targets. In some cases, short cadence data may further improve timing precision (e.g., Batalha et al. 2011; Carter et al. 2011). It is expected that the differences in the times and errors inferred from the two data types are small for purely white, Gaussian noise and adequate transit phase coverage. However, when either precondition is not met, the accuracy of TTs can depend on whether LC or SC data are used.

Short cadence times show improved accuracy and smaller errors for situations where the ingress/egress phases are short in duration and largely unresolved in the LC photometry. In the case of KOI-137, this improvement is significant, with timing errors differing by nearly a factor of two at some epochs. In a similar vein, the short cadence times are more robust against deterministic (as opposed to stochastic) trends that have characteristic timescales less than the LC integration time. In particular, brightening anomalies occurring during transit associated with the occultation of a cool spot on the star by the planet will be unresolved with LC photometry and will likely result in a timing bias.

For several planet candidates, the orbital period is a near multiple of the LC integration time. The result is sparse transit phase coverage that is not improved as the number of observed epochs increases. In these cases, the short cadence photometry provides a more complete phase coverage and, subsequently, times with smaller error bars.

For low signal-to-noise transit events, stochastic temporally correlated noise may dominate on short cadence timescales. As a result, the short cadence timing uncertainties may be unrealistically optimistic if they were inferred assuming a "white" noise model. Here, the LC photometry will give more reliable errors when assessed with the same noise model by effectively averaging over high-frequency noise. Short cadence times estimated for Kepler-10 b seem to be affected by correlated noise, under the expectation of a linear ephemeris, given the relatively small timing errors and the relatively large deviates (compared to the LC times).

While TTVs have been used to confirm transiting planet candidates, TTVs have yet to provide a solid detection of a non-transiting planet. In principle, one strength of the TTV method is that it is sensitive to planets which do not transit the host star (Agol et al. 2005; Holman & Murray 2005). A longer time baseline will be required before TTVs yield strong detections. Yet, we can already use the multiple transiting planet candidate systems and the number of preliminary TTV signals to estimate the frequency of multiple planet systems.

Doppler planet searches find that systems with multiple giant planets are common (⩾28%; Wright et al. 2009). Kepler is probing new regimes of planet mass and orbital separation, so it will be interesting to compare the frequency of multiple planet systems among systems surveyed by Doppler and Kepler observations. B11 estimate the false alarm rate of Kepler planet candidates to range from ⩽2% for the confirmed planets ("vetting flag"=1) to ⩽20% for well-vetted candidates (vetting flag=2) to ⩽40% for those that are yet to be fully vetted (vetting flag=3 or 4). Morton & Johnson (2011) use the specifications for Kepler to predict that even incomplete vetting could result in a false positive rate of less than ∼10% for planet candidates larger than 1.3 R_⊕ and a typical host star. The most common mode of false positive involves a blend of multiple stars which are closely superimposed on the sky (Torres et al. 2011). Contriving such scenarios for KOIs with multiple planet candidates is more difficult. The odds of three or more physically unassociated stars being blended together are extremely small. In many cases, requiring dynamical stability excludes the possibility of multiple transit-like events being due to a multiply eclipsing star systems. The most common types of false positives for a candidate multiple transiting systems are expected to be either a blend of two stars each with one transiting planet or a blend of one star with a transiting planet and one eclipsing binary (Torres et al. 2011). Even these scenarios become highly improbable for pairs of planets that are close to an MMR. Thus, candidate multiple transiting planet systems, and especially near-resonant candidate multiple planet systems, are expected to have very few false positives (Holman et al. 2010; Latham et al. 2011; Lissauer et al. 2011b).

Given the low rate of expected false positives, we can use the frequency of stars with multiple transiting planet candidates to estimate a lower bound on the frequency of systems with multiple planets (with sizes and orbits detectable by Kepler using the present data), assuming that all systems are coplanar. Table 7 lists the number of stars with at least one transiting planet candidate (N_st) and the number of transiting planet candidates (N_tr), separated by the number of candidates per star (N_cps). The probability that at least two coplanar candidates transit for a randomly positioned observer is simply R_⋆/a₍₂₎, where a₍₂₎ is the semimajor axis of the planet with the second smallest orbital period. Among the planet candidates in B11, the average a/R_⋆ for single planet candidates is ∼30. For stars with two (multiple) transiting planet candidates, the average a₍₂₎/R_⋆ is 44 (39). Thus, the difference in the detection rates due to purely geometric considerations would be modest if the two planets are strictly coplanar. The fraction of Kepler planet candidate host stars with at least two (exactly two) transiting planets (with sizes and orbits detectable by Kepler using the present data) is at least ∼23% (17%). For stars with at least three transiting planet candidates, the average a₍₃₎/R_⋆ is 50, so purely geometric considerations are more significant even if the orbits are strictly coplanar. While only ∼5.6% of Kepler planet candidate host stars have at least three candidates, adopting a minimum geometric correction factor of 50/30 yields a fraction of Kepler planet candidate host stars with at least three similar transiting planets of at least ∼9.4%. Such a high occurrence rate of multiple candidate systems requires that there be large numbers of systems with one transiting planet where additional (more distant) planets are not seen to transit, even in the most conservative coplanar case. Of course, the true rates of multiplicity could be much higher, if systems have significant mutual inclinations. Lissauer et al. (2011b) provide more detailed analysis of the observed rate of multiple transiting planet systems and its implications for their inclination distribution and multiplicity rate.

Table 7. TTV Candidates and Transiting Planet Candidates

$N_{\mathrm c{\rm ps}}$a	$N_{\mathrm s{\rm t}}$b	Ntrc	Ntrd	Ntre	Ntrf	NTTVg	NTTVh	NTTVi	% TTVj	% TTVk	% TTVl
			s4n3	s3n5	s4n5	s4n3	s3n5	s4n5	s4n3	s3n5	s4n5
1	809	809	462	453	371	95	59	45	21%	13%	12%
2	115	230	121	110	83	24	16	13	20%	15%	16%
3	45	135	71	67	57	14	8	6	20%	12%	11%
4	8	32	15	14	10	2	1	1	13%	7%	10%
5	1	5	2	3	2	1	1	0	50%	33%	0%
6	1	6	4	3	3	0	0	0	0%	0%	0%
All	979	1217	675	650	526	136	85	65	20%	13%	12%

Notes. Number and fractions of systems with evidence showing for TTVs separated by the number of transiting planet candidates in the system. ^aNumber of candidates per star. ^bNumber of stars. ^cNumber of transiting planet candidates (only those used for this analysis). ^dNumber of transiting planet candidates with an S/N per transit ⩾ 4 and at least three transits in Q0-2. ^eNumber of transiting planet candidates with an S/N per transit ⩾ 3 and at least five transits in Q0-2. ^fNumber of transiting planet candidates with an S/N per transit ⩾ 4 and at least five transits in Q0-2. ^gNumber of transiting planet candidates from s4n3 sample that show indications of TTVs. ^hNumber of transiting planet candidates from s3n5 sample that show indications of TTVs, excluding epoch offset test. ⁱNumber of transiting planet candidates from s4n5 sample that show indications of TTVs, excluding X'² test. ^j% of planets from s4n3 sample that show indications of TTVs. ^k% of planets from s3n5 sample that show indications of TTVs, excluding the epoch offset test. ^l% of planets from s4n5 sample that show indications of TTVs, excluding X'² test.

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Next, we make use of the fact that TTV observations are sensitive to non-transiting planets. Table 7 includes multiple values of the number of planet candidates (N_TTV) that were identified as likely having TTVs by various sets of tests described in Section 3 and the number of planet candidates which these tests were applied to (N_tr). We evaluate the robustness of our results by applying various sets of tests for TTVs to different samples of planet candidates. In Table 7 the columns labeled sSnN refer only to planet candidates with a single transit S/N of at least S and at least N TTs measured during Q0-2. Some planet candidates in the s4n3 sample have too few transits to apply the X'² test, $\mathcal {F}$-test, and period comparison test. Similarly, for the epoch and period comparison tests, we required an S/N per transit of at least 4, so these tests were not applied to all the planet candidates in the s3n5. The s4n5 sample is the smallest, but is the least likely to result in false alarms when searching for TTV signals. To further reduce the risk of false alarms, we do not include the X'² test for excess scatter when analyzing the s4n5 sample. The columns labeled "% TTV" give the fraction of planet candidates that were identified by the tests for TTVs that were applied to the given sample.

We find that ∼11%–20% of planet candidates suitable for TTV analysis show some evidence for TTVs, depending on the tests applied (see Table 7). We obtain similar rates when we consider only systems with N_cps = 1–3 transiting planet candidates. For N_cps ⩾ 4, the accuracy of the resulting rates is limited by small number statistics.

Of planet candidates which are near a 1:2 MMR with another planet candidate, 25% show some evidence for a long-term trend. This could be due to planets near the 1:2 MMR being more likely to have large TTVs, but we caution that this results is based on a small sample size and an early estimate for the frequency of TTVs.

Regardless of which sample and tests are chosen, we do not find significant differences in the fraction of planet candidates which show TTV signals as a function of N_cps. This suggests stars with a single transiting planet are nearly as likely to have additional planets that cause TTVs, as stars with multiple transiting planets are to have masses and orbits that result in detectable TTVs. Since large TTVs most naturally arise for systems that are densely packed and/or have pairs of planets in or near an MMR, a system with a single transiting planet that shows TTVs is likely to have a significant dispersion of orbital inclinations. A dispersion of inclinations has the effect of increasing the probability that a randomly located observer will observe a single planet to transit and decreasing the probability of observing multiple planets to transit. Thus, a dispersion of inclinations may help explain the relatively small frequency of systems with two transiting planet candidates relative to the frequency of systems with one transiting planet candidate. However, simply increasing the inclination dispersion to match the ratio of two transiting planet candidate systems to one transiting planet candidate systems fails to produce the observed rate of systems with three or more transiting planet candidates. This suggests that a single population model is insufficient to explain the observed multiplicity frequencies (Lissauer et al. 2011b).

The small fraction of systems with very large TTVs is consistent with the notion that the Kepler planet candidate list has a small rate of false positives. In particular, physically bound triple systems are one of the most difficult types of astrophysical false positives to completely eliminate (i.e., an eclipsing binary that is diluted by light from a third star). In many cases, wide triple systems would be recognized based on centroid motion during the transit (B11). For KOIs that are not near the threshold of detection there is a relatively narrow range of orbital periods that would escape detection by the centroid motion test and be dynamically stable (for stellar masses). In many cases, such a triple system would exhibit ETVs, as are often seen in the Kepler binary star catalog (Prsa et al. 2011; Slawson et al. 2011; J. A. Orosz et al. 2011, in preparation). We identify only a handful of systems with large period derivatives that are consistent with a stellar triple system. This suggests that the Kepler planet list contains few physical triple stars with eclipsing timing variations and that the current Kepler planet candidates are rarely in a tight binary systems.

We identify over 60 transiting planet candidates that show significant evidence of TTVs, even on relatively short timescales. Even for the early TTV candidates identified here, an increased number of transits and time span of Kepler observations will be necessary before TTVs can provide secure detections of non-transiting planets. Additionally, follow-up observations to determine the stellar properties and reject possible astrophysical false positives will also be important for confirming planets to be discovered by TTVs.

We expect the number of TTV candidates to increase considerably as the number and time span of Kepler observations increase. For non-resonant systems (e.g., Kepler-11), TTVs typically have timescales of order the orbital period, but relatively small amplitudes (Nesvorný 2009; Veras et al. 2011). In this case, Kepler will be most sensitive when the planets are closely spaced (e.g., Kepler-11). Our analysis of TTs during Q0-2 identified no periodic signals at a confidence level of ⩽0.01. When searching for a simple periodic signal in time series, the minimum detectable signal decreases dramatically as the number of observations increases beyond ∼12 observations and continues to decrease faster than classical ∼N^−1/2 scaling even as the number of observations grows to ∼40 observations. Thus, over the 3.5 year nominal lifetime of Kepler the increased number of TT observations will significantly improve Kepler's sensitivity to closely spaced, non-resonant systems. In this regime, Kepler will be most sensitive to TTVs of short-period planets, since they will provide enough transits during the mission lifetime to detect a periodic signal. Some TTV candidates identified by Kepler will become targets for ground-based follow-up in the post-Kepler era.

For planetary systems near an MMR (e.g., Kepler-9), both the amplitude and timescale of the TTVs can be quite large. Fortunately, continued observations provide the double benefit of increased number of observations and an increasing signal size. To illustrate this point, we used N-body integrations to predict the rms TTV of multiple transiting planet candidate systems identified in B11 for a nominal circular, coplanar model (see Figure 5). During Q0-2 less than 2% had a TTV signature with an rms more than 10 minutes, but the fraction grows to over 10% over the 3.5 year mission lifetime (see Table 6). Both this result and the ∼12% of suitable planet candidates showing evidence for a long-term drift in TTs suggest that the TTV method will become a powerful tool for detecting non-transiting planets as well as confirming transiting planets. For systems with multiple transiting planets, the additional information makes the interpretation of TTVs even more powerful for confirming their planetary nature and that they orbit the same host star (e.g., Holman et al. 2010; Lissauer et al. 2011a). With continued observations, TTVs become very sensitive to the planet mass and orbital parameters.

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Based on the distribution of orbital periods of Kepler transiting planet candidates in B11, at least ∼16% of multiple transiting planet candidate systems contain at least one pair of transiting planets close to a 2:1 period commensurability (1.83 ⩽ P_out/P_in ⩽ 2.18). If we assume that planets near the 1:2 MMR are nearly coplanar, then true rate of detectable planets near the 1:2 MMR (if both had a favorable inclination) is at least ∼ 25%. If a significant fraction of these systems are not in the low inclination regime, then the true rate of pairs of planets near the 1:2 MMR would be even larger (Lissauer et al. 2011b). This rate is approximately double the rate of systems that show TTVs based on our initial analysis of early Kepler data, implying that the number of systems with TTVs could double over the course of the mission.

In conclusion, transit timing is extremely complementary to Doppler observations form confirming planets. On one hand, transit timing only works for planets with detectable TTVs. On the other hand, TTVs can be quite sensitive to low-mass planets that are extremely challenging for Doppler confirmation. Additionally, TTVs are likely to be particularly useful for confirming some of the Kepler planet candidates with host stars that are problematic for confirmation via Doppler observations (e.g., faint stars, active stars, hot and/or rapidly rotating stars).

Of particular interest for the Kepler mission is whether the transit timing method might be able to discover or confirm rocky planets in the habitable zone. Due to the TT uncertainties for Earth-size planets, a detection of TTVs of the rocky planet itself would require a large signal which is only likely if the small planet is near a resonance with another planet. Further complicating matters, planets in the habitable zone will have only a few transits for solar-mass stars. Fortunately, Kepler is detecting many systems with multiple transiting planets, which would open the door to TT measurements of both the planet in the habitable zone and an interior planet (see Figure 6). Indeed, Kepler has identified over a dozen planet candidates that are in or near the habitable zone and are associated with stars that have multiple transiting planet candidates. In most cases, the period ratio between the transiting planet candidates is large, so the TTVs could be small (unless there are additional non-transiting planets). However, N-body integrations suggest that continued Kepler observations of several pairs could prove very useful for confirming (or rejecting) these planet candidates based on TT. Of course, many other stars with a planet candidate in or near the habitable zone may harbor additional non-transiting planets. The distribution of period ratios of Kepler transiting planet candidates shows that planets near the 1:2, 2:3, and 1:3 MMRs are not uncommon. For reference, Kepler-9 b and c were confirmed on the basis on 9(b)+6(c) = 15 TT observations. Obtaining 15 TTs for two planets in a 1:2 MMR requires observing for 5–6 times the orbital period of the outer planet. Thus, it is feasible that a transiting planet in the habitable zone identified by Kepler could be confirmed using TTVs, provided that there is another transiting planet near an interior MMR and that the Kepler mission were extended to ⩾6 years. The prospects improve significantly for stars less massive than the Sun, thanks to the shorter orbital period at the habitable zone.

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