TRANSITING PLANET CANDIDATES BEYOND THE SNOW LINE DETECTED BY VISUAL INSPECTION OF 7557 KEPLER OBJECTS OF INTEREST

We analyzed the long-cadence fluxes of the Pre-search Data Conditioning component of the Kepler pipeline (PDCSAP) of 7557 KOIs, which were available on the NASA Exoplanet Archive at the time of 2015 June 4. We searched for STEs by visual inspection of all those PDCSAP fluxes and identified 28 STEs in 24 KOIs as shown in Table 1. Here the fading events that are not observed in the SAP data but only seen in the PDCSAP data are excluded because we find that the correction by the PDC pipeline sometimes leads to artificial dips in the light curves.⁶ Although it is admittedly difficult to quantify the detection limit of our visual inspection, we believe that transits deeper than $\sim 0.1\%$ and lasting longer than 5–50 hr have been detected.

Among the 28 STEs in Table 1, 14 have never been reported in the literature; they are marked with "new" in parentheses. KOI numbers designated by the Kepler team are listed for the other seven events. When more than one STE is found in one system, they are reported separately (two in KOI-847, KOI-1168, and KOI-6378; three in KOI-1032).

To further characterize the planet candidates causing STEs (hereafter "STE candidates"), we fit the STE light curves assuming that STE candidates are not due to contamination but orbiting the KOIs for which we found STEs on circular orbits. As discussed below, this assumption allows us to estimate the orbital periods of STE candidates even from only one transit. Here we exclude the systems designated as false positives in the KOI catalog and KOI-1032 exhibiting signatures of the CCD cross talk⁷ because the above assumption is less sound for them. We also examined the target pixel files of the remaining targets visually and excluded the STE of KOI-3145, whose depths are different in neighboring pixels and thus likely to be due to contamination from a nearby star (see also Wang et al. 2015). These criteria leave us with 16 STEs in 14 systems, which are listed in Table 2 along with their estimated parameters.

2.2.1. Principle

While we use the Markov-Chain Monte Carlo (MCMC) method to determine the system parameters, here we analytically show how the orbital period of the STE candidate is derived with the information on the mean stellar density either from the light curves of inner transiting planets with known orbital periods or from the follow up observations of the host star.

Assuming a circular orbit, the total and full transit durations (denoted by t_T and t_F, respectively) are given by (e.g., Winn 2010)

$$\begin{eqnarray}&&{t}_{T}\;=\;\displaystyle \frac{P}{\pi }{\mathrm{sin}}^{-1}\left(\displaystyle \frac{{R}_{*}}{a}\sqrt{\displaystyle \frac{{(1+k)}^{2}-{b}^{2}}{\mathrm{sin}i}}\right),\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{t}_{F}\;=\;\displaystyle \frac{P}{\pi }{\mathrm{sin}}^{-1}\left(\displaystyle \frac{{R}_{*}}{a}\sqrt{\displaystyle \frac{{(1-k)}^{2}-{b}^{2}}{\mathrm{sin}i}}\right),\end{eqnarray} \tag{ 2 }$$

where $k\equiv {R}_{p}/{R}_{*}$ indicates the ratio of the planetary radius to the stellar radius, i is the orbital inclination, and b is the impact parameter. Neglecting the terms of ${ \mathcal O }({({R}_{*}/a)}^{3})$ and higher, we obtain

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{*}}{a}\;=\;\displaystyle \frac{\pi }{2\sqrt{k}}\displaystyle \frac{\sqrt{{t}_{T}^{2}-{t}_{F}^{2}}}{P}.\end{eqnarray} \tag{ 3 }$$

Combined with Kepler's third law, Equation (3) yields the orbital period of an STE candidate in terms of its transit shape and the mean stellar density ${\rho }_{*}$ (Seager & Mallén-Ornelas 2003):

$$\begin{eqnarray}&&P\;=\;\displaystyle \frac{\pi G}{32}\displaystyle \frac{{M}_{*}}{{R}_{*}^{3}}{\left(\displaystyle \frac{{t}_{T}^{2}-{t}_{F}^{2}}{k}\right)}^{\tfrac{3}{2}}\simeq \displaystyle \frac{{\pi }^{2}G}{3}{\rho }_{*}{\left(\displaystyle \frac{T\tau }{k}\right)}^{3/2},\end{eqnarray} \tag{ 4 }$$

where M_* and R_* are the stellar mass and radius. We also defined $T\equiv \tfrac{1}{2}({t}_{T}+{t}_{F})$ and $\tau \equiv \tfrac{1}{2}({t}_{T}-{t}_{F})$ and assumed $\tau \ll T$.

If the inner transiting planet(s) is known in the system, Equation (4) can be used to constrain ${\rho }_{*}$ from its transit shape because P is already determined for the planet(s). In this case, the orbital period of the STE candidate P_s is given by

$$\begin{eqnarray}&&{P}_{s}\;=\;{\left(\displaystyle \frac{{k}_{i}{T}_{s}{\tau }_{s}}{{k}_{s}{T}_{i}{\tau }_{i}}\right)}^{\tfrac{3}{2}}{P}_{i},\end{eqnarray} \tag{ 5 }$$

where the subscripts s and i denote the quantities for the STE candidate and the inner planet, respectively. This is the case for the 10 systems except for KOI-1421, 1208, 1174, and 1096 in Table 2. We checked that the analytic estimate in Equation (5) is indeed consistent with the MCMC results for these systems. Even if the inner transiting planets are not known, the prior knowledge on ${\rho }_{*}$ from the color photometry can also be used to constrain P_s, although it may be less reliable than the dynamical value as obtained in the previous case; this method is adopted for KOI-1421, 1208, 1174, and 1096.

2.2.2. MCMC Fit to the Observed STE Light Curves

For the 10 systems except KOI-1421, 1208, 1174, and 1096, we fit the STE light curves simultaneously with the phase-folded transit light curves of the other planet candidates in the system. We basically analyzed the long-cadence PDCSAP fluxes except for the inner transits of KOI-671 and KOI-435, for which short cadence fluxes were used. We used the PyTransit package (Parviainen 2015) to generate the transit light curve based on the Mandel & Agol (2002) model for the quadratic limb darkening law. The effect of binning was taken into account by supersampling for the long-cadence data sampled at 30 minutes. Constraints on the parameters were obtained by the MCMC sampling using the emcee package (Foreman-Mackey et al. 2013) following the standard ${\chi }^{2}$ minimization using the Levenberg–Marquardt method (mpfit by Markwardt 2009). The likelihood for the MCMC sampling ${ \mathcal L }$ was computed as ${ \mathcal L }\propto \mathrm{exp}(-{\chi }^{2}/2)$, where ${\chi }^{2}$ is a sum of the standard chi-squared for each planet's transit, and we adopt non-informative priors unless otherwise noted.

The transits of the inner planets were processed as follows. We first detrended the light curves from each quarter using the second order spline interpolation after masking the known transits based on the ephemeris and duration in the KOI catalog. Baseline fluxes during the transit were determined by the linear interpolation between the two ends of the masked region. The detrended transits were folded at the orbital period given in the KOI catalog and averaged into one-minute bins. The values and errors of the binned flux are given by the mean and its standard deviation of the values in each bin. The smoothing parameter of the spline was chosen so that the depth of the phase-folded transit be consistent with the catalog value. The STE light curves were deterened in a similar manner except that the smoothing parameter was chosen to be $0.1\;\mathrm{days}$ and the endpoints of the linear interpolation were adjusted so that the resulting detrend light curve would not be too asymmetric.

The fitting parameters in this case are the sum of two coefficients for the quadratic limb darkening law ${u}_{1}+{u}_{2}$ and ${\rho }_{*}$ as the parameters common to all the planets in the same system; the time of transit center T_c, $\mathrm{cos}i$, k, and the normalization of the flux c for each of the inner transiting planets; and T_c, $\mathrm{cos}i$, k, c, and the orbital period P_s for STE candidates. The difference of the limb darkening coefficients ${u}_{1}-{u}_{2}$ were fixed to be the linearly interpolated values based on Sing (2010), and the orbital periods of the inner planets were fixed at the values given in the KOI catalog. In MCMC fitting, we adopt the prior distributions uniform in ${u}_{1}+{u}_{2}$, $\mathrm{log}{\rho }_{*}$, T_c, $\mathrm{cos}i$, $\mathrm{log}k$, c, and $\mathrm{log}{P}_{s}$. We assumed circular orbits for both inner candidates and STE candidates.

For KOI-1421, 1208, 1174, and 1096, we fit the STE light curve alone, imposing the Gaussian prior on ${\rho }_{*}$ based on the value at CFOP. The fitting parameters in this case are the two stellar parameters ${u}_{1}+{u}_{2}$ and ${\rho }_{*}$, and T_c, $\mathrm{cos}i$, k, c, and P_s of the STE candidate.

In the analyses above, we neglect the effect of possible stellar multiplicity. If the target star has an unidentified companion star, for instance, the planetary radii can be underestimated due to the dilution. We note, however, that this possibility is essentially ruled out for most of our main candidates discussed in Section 3.1.

The best-fit models and constraints on the parameters of STE candidates are summarized in Figure 2 and Table 2. All the transits of the inner candidates simultaneously fitted with STEs are listed in the Appendix. On the basis of the inferred orbital period and radius of STE candidates, we found that STEs in seven systems are consistent with the planetary transit, as will be detailed in Section 3. The architectures of these seven systems are illustrated in Figure 1. We also plotted each STE candidate on the ${R}_{p}-P$ plane with all known KOI candidates in Figure 3. The figure shows that STE candidates we found are all likely to be gas/ice giants beyond the snow line.

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Seven systems illustrated in Figure 1 exhibit STEs consistent with the planetary transit. Namely, (1) their transit light curves are consistently explained with those of the inner transiting planets or stellar density ${\rho }_{*}$ in the KIC catalog, (2) their inferred radii are less than about the radius of Jupiter, and (3) their inferred orbital periods are consistent with the absence of other transits in the exiting Kepler data (i.e., inferred P_s is longer than ${P}_{{\rm{min,Kepler}}}$ in Table 1). It is worth noting that most of the candidates in this category except for KOI-1421 are in multi-planetary systems and are thus likely to be genuine planets in the same systems (Lissauer et al. 2014).

3.1.1. KOI-847 (KIC 6191521)

3.1.2. KOI-671 (Kepler-208, KIC 7040629)

3.1.3. KOI-2525 (KIC 5942949)

3.1.4. KOI-1108 (KIC 3218908)

3.1.5. KOI-693 (Kepler-214, KIC 8738735)

3.1.6. KOI-435 (Kepler-154, KIC 11709124)

3.1.7. KOI-1421 (KIC 11342550)

For the STEs in these systems, the orbital periods inferred from our fitting are shorter than ${P}_{{\rm{min,Kepler}}}$ in Table 1, which is the minimum orbital period required for an STE candidate to be consistent with the absence of other transit signals in the existing Kepler data. While the discrepancy may imply that they are false positives, it is also possible that the orbits of these STE candidates are eccentric, as shown below. For this reason, we still consider these "misfit" singles as planet candidates, though less promising than those listed in the previous subsection.

For simplicity, we consider the case where only the STE candidates have non-zero eccentricities, while the inner transiting objects are all on circular orbits. For eccentric orbits, the right sides of t_T and t_F in Equations (1) and (2) are multiplied by $\sqrt{1-{e}^{2}}/(1+e\mathrm{sin}\omega )$, where e is the eccentricity and ω is the argument of periastron (measured from the sky plane) of the STE candidate (e.g., Equation (16) of Winn 2010). Thus, the orbital period of the STE candidate with non-zero eccentricity differs from the circular case by the factor of

$$\begin{eqnarray}&&{\alpha }_{{\rm{ecc}}}\;=\;{\left(\displaystyle \frac{1+e\mathrm{sin}\omega }{\sqrt{1-{e}^{2}}}\right)}^{3}\leqslant {\left(\displaystyle \frac{1+e}{1-e}\right)}^{3/2}\end{eqnarray} \tag{ 6 }$$

assuming the same ${\rho }_{*}$.⁹ Since ${\alpha }_{{\rm{ecc}}}$ is larger than the ratio of ${P}_{{\rm{min,Kepler}}}$ to P_s obtained from the circular fit (Table 2), i.e., ${\alpha }_{{\rm{ecc}}}\gt {P}_{{\rm{min,Kepler}}}/{P}_{s}$, Equation (6) yields the minimum value of eccentricity required to explain the observation:

$$\begin{eqnarray}&&{e}_{{\rm{min}}}\;=\;\displaystyle \frac{{({P}_{{\rm{min,Kepler}}}/{P}_{s})}^{2/3}-1}{{({P}_{{\rm{min,Kepler}}}/{P}_{s})}^{2/3}+1}.\end{eqnarray} \tag{ 7 }$$

The values of ${e}_{{\rm{min}}}$ and ${P}_{{\rm{min,Kepler}}}$ for each of the "misfit" candidates are listed in Table 3, along with their values computed for the most conservative case (see the note in the table).

Table 3.  List of the "Misfit" Singles and the Minimum Values of Eccentricity Required

		Fiducial Values		Most Conservative Values
KOI	Ps (days)	${P}_{{\rm{Kep,min}}}$ (days)	${e}_{{\rm{min}}}$	${P}_{{\rm{Kep,min}}}$ (days)	${e}_{{\rm{min}}}$
4307	${610}_{-400}^{+460}$	1311	0.25	335	...
3349	${510}_{-290}^{+480}$	1319	0.31	532	...
1870	${190}_{-85}^{+270}$	987	0.50	494	0.31
1208a	${65}_{-36}^{+73}$	1342	0.77	340	0.50
1174a	${310}_{-88}^{+370}$	1197	0.42	411	...
1096a	${700}_{-160}^{+190}$	1276	0.20	487	...

Notes. In the "most conservative values," ${P}_{{\rm{Kep,min}}}$ is computed by considering the possibility that all other transits of the STE candidate are hidden in the data gaps, although such a possibility is quite low in some cases, depending on the candidate.

^aNo transiting planet candidates other than STEs are known for these systems.

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If (some of) these candidates are confirmed to be eccentric LPGs by follow up observations, they can be interesting targets to understand the origin of hot Jupiters, as in the case of HD 80606 (Wu & Murray 2003). In this regard, the most promising target is KOI-1208, with a Kepler magnitude of 13.6 (Table 1), for which a possible nearby companion has been detected and the eccentricity is estimated to be $\gtrsim 0.8$.

The inferred radius of this STE candidate suggests that it is a stellar object rather than a planet. The good agreement between the parameters of the two STEs strongly implies that they are due to the same object. If this is indeed the case, this objects has ${P}_{s}\;=\;525.0216\pm 0.0006\;\mathrm{days}$. This system is also discussed in Wang et al. (2015), who also classified it as a likely false positive.

In Figure 1, the systems with multiple inner planets (KOI-671, 1108, 693, and 435) exhibit a gap-like signature between the inner and STE candidates, as seen between the terrestrial and giant planets in the solar system (top row). While we cannot completely exclude the existence of such a gap in our sample, we suspect that it is due to the decrease in the transit probability with increasing orbital periods. Because of the low transit probability of a distant planet, transiting objects on wide orbits (say $P\gtrsim {10}^{3}\;\mathrm{days}$) tend to be rare. This means that the system with one long-period transiting object detected, as in our sample, is unlikely to have another transiting object close to the detected one and exhibits a gap around the detected transiting object.

To demonstrate the above geometric effect, we performed a simple simulation of the transit detection of the multi-planetary system. We put 12 planets at a constant log interval as in the first row of Figure 4. The semimajor axis over the host star radius, a/R_⋆, of the outermost planet was chosen to be 420, which corresponds to $P={10}^{3}\;\mathrm{days}$ or $a=1.96\;\mathrm{au}$ in the solar system. We simulated the transit observation of this system from many different directions uniformly distributed in $\mathrm{cos}i$, sampling the mutual inclinations of the inner planets with respect to the outermost one from the Rayleigh distribution with $\sigma =1\_\_AMP\_\_fdg;8$ (Fabrycky et al. 2014) in each run. From the resulting sample of transiting systems, we chose the systems where the transit of the outermost planet was detected and plotted their apparent architecture below the horizontal dashed line in Figure 4. One can actually see the gap-like structure between the outermost planets (red circles) and the inner planets (blue circles) in many cases.

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As summarized in Table 4, the sample of 7557 KOIs we surveyed includes 695 systems with more than one inner transiting planet candidates, among which we detected four LPG candidates (see also Figure 1). These numbers can be used to estimate the occurrence rate of LPGs in compact, multi-transiting systems on the premise that LPG orbits are likely to be well aligned with those of the inner multiple planets. The premise is based on the following argument. The planets in a compact multi-transiting system presumably have well aligned orbits (Fabrycky et al. 2014), indicating that their orbital planes trace the original protoplanetary disk. Since an LPG in the same system formed in the same disk, its orbit is likely to be aligned with the inner ones as well.

The expected number of transiting LPGs, ${n}_{{\rm{tLPG}}}$, in ${N}_{{\rm{cmulti}}}$ compact multi-transiting systems is given by

$$\begin{eqnarray}&&{n}_{{\rm{tLPG}}}\simeq \displaystyle \frac{{T}_{{\rm{obs}}}}{{P}_{{\rm{LPG}}}}\;p(\mathrm{tra}| \mathrm{LPG},\mathrm{cmulti})\bar{n}(\mathrm{LPG}| \mathrm{cmulti}){N}_{{\rm{cmulti}}},\end{eqnarray} \tag{ 8 }$$

where $\bar{n}(\mathrm{LPG}| \mathrm{cmulti})$ is the average number of LPGs per system and $p(\mathrm{tra}| \mathrm{LPG},\mathrm{cmulti})$ is the transit probability of a given LPG, both under the existence of the inner compact multi-transiting system. The factor ${T}_{{\rm{obs}}}/{P}_{{\rm{LPG}}}$, where ${T}_{{\rm{obs}}}$ is the observing duration of Kepler and ${P}_{{\rm{LPG}}}$ is the orbital period of a given LPG, takes into account the probability that the single transit of the LPG falls into the mission lifetime of Kepler. Since $p(\mathrm{tra}| \mathrm{LPG},\mathrm{cmulti})$ depends on the mutual inclination of the LPG and the inner planets, the mutual inclination and occurrence rate $\bar{n}$ are usually degenerate (Tremaine & Dong 2012).

As far as the mutual inclination between the LPG and inner planet orbits is as small as $\sim {R}_{\star }/{a}_{{\rm{in}}}$, we approximately have $p(\mathrm{tra}| \mathrm{LPG},\mathrm{cmulti})\;=\;{a}_{{\rm{in}}}/{a}_{{\rm{LPG}}}$, where ${a}_{{\rm{LPG}}}$ and ${a}_{{\rm{in}}}$ are the typical semimajor axes of the LPG and inner multi-transiting planets, respectively (see Section 2.3 of Ragozzine & Holman 2010). Adopting ${a}_{{\rm{LPG}}}\;=\;2\;\mathrm{au}$ (corresponding to $P\simeq {10}^{3}\;\mathrm{days}$) and ${a}_{{\rm{in}}}=0.07\;\mathrm{au}$ (median of KOI candidates in multi-transiting systems), we obtain $p(\mathrm{tra}| \mathrm{LPG},\mathrm{cmulti})=0.035$, which yields $\bar{n}(\mathrm{LPG}| \mathrm{cmulti})\simeq 0.2$ for ${n}_{{\rm{tLPG}}}=4$, ${N}_{{\rm{cmulti}}}=695$, ${T}_{{\rm{obs}}}=4\;\mathrm{year}$, and ${P}_{{\rm{LPG}}}=2200\;\mathrm{days}$ (average of the seven systems in Figure 1) in Equation (8).

If the LPG actually has a larger mutual inclination relative to the inner planets than assumed here ($\gtrsim {R}_{\star }/{a}_{{\rm{in}}}\sim 4^\circ$), the above estimate underpredicts the true occurrence rate.¹⁰ In addition, the above discussion assumes that the transiting LPG is 100% detected as long as it transits the host star during the Kepler observation. For these reasons, we conclude that $\bar{n}(\mathrm{LPG}| \mathrm{cmulti})\simeq 0.2$ estimated above is a rough lower limit, and that about 20% or more of the compact multi-transiting systems host LPGs with $P\gtrsim {10}^{3}\;\mathrm{days}$.

Table 4 also shows that the fraction of transiting LPGs in KOIs with only one inner transiting candidate is smaller than the above multiple-candidate case by an order of magnitude. While we suffer from the small statistics, the fact suggests that the term $p(\mathrm{tra}| \mathrm{LPG},\mathrm{cmulti})\bar{n}(\mathrm{LPG}| \mathrm{cmulti})/{P}_{{\rm{LPG}}}$ in Equation (8) is smaller for (a part of) the single-candidate sample. This means that either (1) mutual inclination of the LPG relative to the inner planet may be larger, (2) the occurrence rate of the LPG may be smaller, or (3) the typical orbital period of the LPG may be longer in the systems with only one inner transiting planet. If (1) is actually the case, the result supports the scenario by Morton & Winn (2014) that a population of highly inclined multi-planet systems contributes the excess of single-transiting systems in the Kepler multiplicity statistics (Lissauer et al. 2011), which is known as the "Kepler dichotomy."