TRANSIT TIMING OBSERVATIONS FROM KEPLER. IV. CONFIRMATION OF FOUR MULTIPLE-PLANET SYSTEMS BY SIMPLE PHYSICAL MODELS

The light curve of Kepler-29 is plotted in Figure 2, and the phase curves of the two Neptune-size planets is plotted in Figure 3. Their periods of 10.3376(2) days and 13.2907(4) days form a ratio 1.28566 ± 0.00005, which lies right at a 9:7 (=1.28571) orbital resonance. Being fractionally closer than 8 × 10⁻⁵ of a second-order resonance in this region would happen only 0.5% of the time by chance, and similar for other resonant orders. Being so close to a low-order resonance—consistent with dynamical libration—suggests dynamical interaction helped to establish their orbital architecture. Thus, from the period ratio alone, we have an indication that these objects are in the same system.

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Furthermore, since the ratio is not significantly offset from the resonance, the nominal integrations result in an extremely long-timescale variation in O − C (Figure 4), even slightly longer than the sample 10 year integration used to find it. Therefore, on the timescale of our data, the predicted sinusoid would look only like a quadratic in O − C, as is observed (Figure 4, left-hand side). The data for the shorter-period planet (Kepler-29b) indicate this quite robustly (FAP ≃ 0.02%) and hint at oppositely directed curvature for the longer-period planet (Kepler-29c has FAP ≃ 1.5%). On the basis of the former we confirm the system really is composed of two planets.

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However, a parabolic signature is not specific enough to uniquely specify the other planet as the perturber, so we carried the investigation one more step. The nominal prediction for year-timescale O − C (Figure 4, right-hand side) is not dominated by a parabola, but is instead a "chopping" signal on the frequency at which the planets come back to the same configuration,

$$\begin{equation} P_{{\rm TTV}} = 9 \times 10.3376 d = 7 \times 13.2907 d = 93 d. \end{equation} \tag{ 1 }$$

Thus, we search for that timescale. After subtracting the curvature models shown in Figure 4 (left-hand side), we computed the amplitude of the best-fitting sinusoid at the period of Equation (1). For Kepler-29b and Kepler-29c, the amplitude was 5 and 12 minutes, respectively, quite similar to that predicted by Figure 4. Then we scrambled the data to find an FAP: it was 0.6% and 1.1% for the two planets, respectively. To emphasize this point, we plot in Figure 5 the cross-correlation statistic defined in Paper III: a spike is seen at ∼90 days, at positive values of Ξ (i.e., anticorrelation at the predicted timescale). Therefore, the signal at the expected chopping timescale is a third, albeit modestly significant, indication that these two objects are really planets in the same system.

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Kepler-30 is a remarkable system of three planets, all with extremely significant TTVs. Its light curve is shown in Figure 6 and each planet's phase curve is shown in Figure 7. A Neptune-size inner planet (P_b = 29.2 days) lies just interior to a 2:1 resonance with a Jupiter-size planet (P_c = 60.3 days). Their mutual perturbation causes a TTV with a frequency equal to the difference of the orbital frequencies from the resonance:

$$\begin{equation} P_{{\rm TTV}} = 1 / | 2/ 60.3 d - 1 / 29.2 d | = 900 d. \end{equation} \tag{ 2 }$$

Since one of the participating planets has deep transits and thus is presumably massive, the signal is expected to be large. The theory predicts a sinusoid with a period a bit longer than the data. Indeed, the data show an enormous signal consistent with these expectations (Figure 8). Thus, we confirm Kepler-30b and Kepler-30c as objects orbiting the same star.

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Since their orbits are spaced a bit wider than the other planets confirmed in this paper, dynamical stability is not as strong a constraint on their masses. We find a mass upper limit (see Section 4.1) of about 17 M_Jup. However, eccentricity can also account for the large transit timing signals, so neither planet is necessarily nearly this massive (in particular, the small-radius Kepler-30b should not be multiple Jupiter masses). The deviations from a constant ephemeris for planet b are roughly parabolic and span ±10 hr, nearly a day, by far the largest timing variations for a multiple-planet system yet detected (but not quite rivaling the circumbinary planet Kepler-16(AB)b; Doyle et al. 2011). However, there is more signal in this TTV curve than just the large parabolic trend. The rising branch of both the observed signal and the theoretical signal show a short-timescale "chopping" effect for Kepler-30b, where the even and odd transits times show structure, due to the 2:1 resonance. A similar effect was seen previously in Kepler-9b (Holman et al. 2010). A mass limit that takes into account the observed transit times will be discussed in Section 4.2. It results in a much tighter constraint, but it is less robust because it is not based on an exhaustive exploration of parameter space.

We can also confirm Kepler-30d is likely a planet in the same system, as TTVs with a ∼2 hr timescale are seen in it (Figure 9). However, the interpretation is slightly less clear, as only four transits of Kepler-30d (P = 143.2128 days) have been recorded so far, so the shape of the variation is poorly determined. This perturbation is anticorrelated with the swing seen in its neighbor (Kepler-30c), but that swing might be predominantly due to Kepler-30c's near-resonant interaction with Kepler-30b. According to the nominal integration, the timescale for the interaction between planets d and c should be ∼400 days (Figure 9 and Table 5), similar to the span of the data set. When the data set is doubled, this might be discerned from the longer timescale (Equation (2)) interaction between the inner two planets. Even without a detailed analysis, we determine from transit timing that Kepler-30d is an object in the same system. We carry out preliminary transit-timing modeling of the three planets in Section 4.2.

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Due to the large timing variations and deep transits found in this system, we find it instructive to show individual transits to make several other points. In Figure 10 we plot the transits of Kepler-30b. Each portion of the light curve is centered where the transit would be if it followed the best fit linear ephemeris. Instead, we see that the timing of the planet varies by of order a day. The standard Kepler pipeline (Jenkins et al. 2010) assumes near-periodic transits, but the transits are deep enough for detection despite its strong acceleration. If the transits were considerably shallower, it may not have yielded a significant signal before the phase changed by more than a transit duration (García-Melendo & López-Morales 2011). To address this, the transit-timing subteam of Kepler is currently developing search algorithms that assume either a quadratic, a sinusoidal, or a non-specified quasi-periodic ephemeris.

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Finally, consider Figure 11, the individual transits of Kepler-30c. The residuals (with a 3× zoom) are shown beneath each transit. The average transit light curve does not capture some of the variations from transit-to-transit, at the level of ∼10% of the planet's depth. We suggest this is due to the planet transiting over starspots, which are clearly visible in the out-of-transit curve at the level of a few percent (Figure 6). To extract the spot period from the light curve, we used a slightly modified version of the Discrete Correlation Function of Edelson & Krolik (1988) (see also White & Peterson 1994), as was recently applied to the CoRoT-7 data (Queloz et al. 2009), and obtained P_rot = 15.25 ± 0.25 days. Modeling the spot-crossing signal during transits may allow the measurement of the spin orientation of the host star relative to the planets' orbits in this particularly valuable multiple-planet system (Sanchis-Ojeda et al. 2011; Nutzman et al. 2011; Désert et al. 2011). Kepler-30c (also known as KOI-806.02) was recently detected in transit from the ground, by Tingley et al. (2011), who made the point that it will serve as an interesting system to continue following even after Kepler finishes its mission.

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Kepler-31 is a system of four planet candidates spaced just wide of a 1:2:4:8 resonant configuration. Its light curve is shown in Figure 12 and its phase curves in Figure 13. From the nominal models, only the middle two (Kepler-31b and Kepler-31c) are expected to show a >1 min variation, big enough to be detected in Kepler data. The timescale due to this near-resonance is predicted as

$$\begin{equation} P_{{\rm TTV}} = 1 / | 2/ 42.6330 d - 1 / 20.8609 d | = 980 d. \end{equation} \tag{ 3 }$$

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Thus, we have the now-familiar situation that a long-timescale variation is predicted due to the interaction. In this case we supplemented the data set with Quarters 7 and 8 as well, which were able to capture more of this long-timescale variation. The data for Kepler-31b indeed clearly show a variation matching the prediction (Figure 14, top panels), with a low false alarm probability. We are seeing its orbital period increase due to the torque exerted by a 2:1 mean-motion resonance. The magnitude of this torque depends on both the planets' masses and eccentricities, so it is not surprising the theory underpredicted the amplitude by a factor of ∼3. Thus we confirm that these two objects are interacting, which combined with the mass limits set in Section 4.1 confirms them as planets.

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The hint (FAP = 3.4%) of TTVs of Kepler-31c due to Kepler-31b does not itself satisfy our statistical criterion. Nevertheless, the clear detection of TTVs in Kepler-31b consistent with the timescale predicted due to Kepler-31c provides evidence that they are in the same physical system.

The system also has a long-period candidate KOI 935.03 and an inner candidate KOI 935.04, but these are not visibly interacting.

Kepler-32 boasts five planet candidates, one of the richest systems among the Kepler discoveries (Lissauer et al. 2011b). Its light curve and phase curves are plotted in Figures 15 and 16, respectively. The light curve has variations suggesting a starspot signal, in which case we determine P_rot = 37.8 ± 1.2 days using an autocorrelation technique as above.

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Candidates .01 and .02 have short orbital periods and lie close to a 3:2 mean-motion resonance: P_.02/P_.01 = 1.483. Therefore, we expect a short TTV timescale:

$$\begin{equation} P_{{\rm TTV}} = 1 / | 3/ 8.7522 d - 2 / 5.9012 d | = 260 d. \end{equation} \tag{ 4 }$$

The observed timescale of TTVs is consistent with this expectation from the nominal N-body integrations (Figure 17). The amplitude is somewhat larger than predicted by the nominal model (∼2.6 and 3.3×). This is not unexpected as even modest eccentricities can significantly increase the amplitude and completely change the phase of the predicted TTVs. The amplitudes also scale with the planet masses, which could be larger than those adopted in our nominal model. The ratio of TTV amplitudes for Kepler-32b and Kepler-32c agrees with the predictions of the nominal model to ∼20%. Despite these uncertainties, the requirement of dynamical stability provides firm upper limits on the planet masses. Thus, we confirm this pair of planets and rename them as Kepler-32b and Kepler-32c.

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Based on the nominal N-body integrations of the five planets, we expect shorter timescales and smaller amplitudes for transit variations of KOI-952.03, .04, and .05. Indeed, no significant transit time variations were seen in them, nor do they induce detectable variation in Kepler-32b or Kepler-32c, so we cannot confirm them via TTVs at this time.

Kepler-29 has low contamination from surrounding stars. The nearest stars to Kepler-29, listed in the Kepler Input Catalog (KIC), are KID 10358756, a Kp = 17.5 mag star located 6.5 arcsec to the south, and KID 10358742 located 9 arcsec away. We examined the UKIRT J-band image and obtained a new image from Faulkes Telescope North in the Sloan-r band (Figure 18); neither showed evidence of additional companions to a magnitude difference of ∼5 in to ∼1 arcsec. We therefore adopt the aperture contamination calculated using stars in the KIC alone.

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The planets pass the centroid test: the photocenter out-of-transit and during transit are consistent to within (2σ, 1σ) for planets (b, c) respectively. This rules out background stars as the planet host if they are farther than Rc = (0.7, 0.8) arcsec from the target, where Rc is the 3σ radius of confusion. Thus none of the background stars in the image can be the host. Next we computed the space density of background stars, to determine the chance that the planetary system is actually around a star other than the target. That is, we assume that the TTV signature robustly indicates that this system is planetary, but we wish to see the probability that it is around a star other than the one we have characterized.

Stability limits (Section 4.1) require that the planets cannot be too massive relative to their host, otherwise they would be unstable. In this case, the mass ratio of both planets to their host must be ≲ 2 × 10⁻⁴. With such low masses, we consider 1 R_Jup to be a robust upper limit for the radii of the planets. This sets the constraint on the maximum magnitude difference that the blending star can have. Any star fainter than this limit would not contribute enough to the combined flux to generate an eclipse as deep as observed (1204 ppm). Based on the U-shaped light curve, we assumed that grazing transits were excluded, such that (R_p/R_⋆)² is a good approximation of the (undiluted) depth. We used Yonsei–Yale isochrones (3 Gyr, solar metallicity) to relate the maximum allowed magnitude difference (Δ Kp) to the spectral type and mass of the blending object.

The background host hypothesis splits into two distinct cases: the planets are larger and (1) orbit a physically unassociated star, or (2) orbit an unseen binary companion of the target.

For the former, we used the Besancon model of the galaxy (Robin et al. 2003) to estimate the number of background stars within the region allowed by the centroid shift result and the region in the allowed magnitude difference (or stellar mass). The probability of having a background star in this region is 0.6%. If we assume (conservatively) that the more massive planets on the background star are as likely, a priori, as less massive planets on the target star, then the odds ratio that the target star hosts the planets rather than a background star is ∼170: 1.

Next we calculate the probability of the latter case, that the planetary system is orbiting an unseen stellar companion to the target star. The blends involving stars less massive than a 0.6 M_☉ star would need large planets, which in the range of known exoplanets would be too massive to be stable. In the range 0.6–0.8 M_☉, we can use the (Sloan) r − (2MASS) K color of the target to rule out such blends, as they would not correspond to the observed color. A blended star more massive than 0.8 M_☉ is allowed, and we call this scenario a "twin star" blend. It is possible to put limits on such blends because we do not see two stars in the images, nor do we see two sets of lines in the spectrum. The only remaining case is that the two twins have a large separation along the line of sight (a small velocity difference) but not in the plane of the sky. If there were such a case, it would correspond to twice the light, so the undiluted depth of transits would be twice as large, so the planets would be ${\sim }\sqrt{2}$ larger.

Even without employing that constraint, we use Raghavan et al. (2010) and Duquennoy & Mayor (1991) to determine the distribution (frequency/mass ratio) of binaries and find the probability of having an unseen companion star in the allowed mass range (0.8–1.0 M_☉) is 7%.

For this target, which has a clean centroid and closely packed planets that would go unstable if they were gas-giant masses, we find that background hosts are much less likely than the target star being the host.

We obtained an I-band image of Kepler-30 from Lick 1-m telescope, and found no stars besides those in the KIC, the closest of which is Kp = 19.8 mag KIC 3832477, 7.6 arcsec to the east–northeast. A star ∼3 arcsec to the east is marginally detected in J band with UKIRT, suggesting it is ≳ 5 mag fainter than the target in Kp. It cannot host the 2% deep transit of Kepler-32c, and we neglect its additional contribution to the dilution. Furthermore, since planets c and d produce deep transits, their centroid information is particularly valuable. Their transits have a consistent centroid with the target to ∼1σ in either case, and they limit the distance of a putative blend hosting the system to be within Rc = 0.2 arcsec of the target.

As a routine part of vetting planet candidates, the depth for odd-numbered transits was compared to the depth for even-numbered transits. For the other candidates reported in this paper, there is agreement to ∼3σ; however in this system, they disagree by 16σ for c and 4σ for d. In some cases, that would point to a near-twin blended eclipsing binary causing the events. In this case, we have already identified via Figure 11 that transiting over starspots is the probable cause of these depth variations.

Using the same procedure as for Kepler-29, we find the probability that an unassociated background star hosts the planets is ∼2 × 10⁻⁴. In this case, a physical binary companion cannot host the planets, as then their depths would be large despite dilution, and their inferred radii would be larger than any planetary radii thus far measured.

Kepler-31 has low contamination from surrounding stars; no stars are seen within 8 arcsec in a UKIRT J-band image, and the closest comparably bright star is KIC 9347893, 9.4 arcsec to the west. Moreover, the centroid information has all transits coincident within 1σ of the target. The transits cannot be hosted by a background star farther than Rc = (0.3, 0.5, 0.8) arcsec in the case of Kepler-31b, Kepler-31c, and KOI-935.03, respectively. For KOI-935.04, the transits are too shallow for a constraining centroid analysis.

Again pursuing probability calculations as above, the chance of a star unassociated with the target being the actual host is only ∼3 × 10⁻⁴. The probability of a physical companion hosting the planets is ∼0.04.

A J-band image from UKIRT shows the nearest star to be KID 9787232, ∼66 to the west, resulting in rather low contamination.

The centroids during transit for Kepler-32b and Kepler-32c differ from those out-of-transit by only ∼2σ, roughly consistent with measurement uncertainties. The ∼3σ radii of confusion Rc are 05 for Kepler-32b and 08 for Kepler-32c. For KOI-952.03, .04, and .05, the transits are too shallow for a constraining centroid analysis.

The host star is an M-dwarf and therefore of special interest. The Kepler Follow-up Program has obtained two spectra of Kepler-32: one spectrum from McDonald Observatory and one from Keck Observatory. Both spectra are weak due to the faintness of the star (Kp = 15.8). The cross-correlation function between the observed spectra and available models is maximized for temperatures of ∼3900 K and ∼3600 K, respectively. However, the atmospheric parameters are not well determined, as the star is cooler than the library of atmosphere models available. Both spectra are consistent with the KIC classification as a cool dwarf (T_eff = 3911 K, log g = 4.64, [M/H] = 0.172). We conservatively adopt these values of T_eff and log g with uncertainties of 200K and 0.3 dex and a [M/H] of 0 ± 0.4 based on the KIC (Brown et al. 2011). By comparing to the Yonsei–Yale isochrones, we derive values for the stellar mass (0.58 ± 0.05 M_☉) and radius (0.53 ± 0.04 R_☉) that are slightly larger than those from the KIC. We estimate a luminosity of 0.06 ± 0.02 L_☉ and an age of ⩽9 Gyr.

Muirhead et al. (2011) have also obtained high-resolution IR spectrum of Kepler-32=KOI-952, finding a stellar T_eff = 3726⁺⁷³₋₆₇, [Fe/H] = 0.04^+0.08_−0.10. Interpreting their data via Padova models (Girardi et al. 2002), they inferred a considerably less massive and smaller star. We encourage further detailed analyses of the host star properties, as these have considerable uncertainties that directly affect the sizes and masses for the planets.

The probability of a star unassociated with the target being the actual host is only ∼3 × 10⁻³. The probability of a physical companion hosting the planets is ∼0.34. This latter number is relatively large in this case because all the transit depths are small, so they could in principle be much larger planets hosted by a star which is dramatically diluted. This opens up the possibilities for a very large range of companions (down to masses as low as ∼0.1 M_☉) that could host one or more of these objects, as long as transits near apocenter are invoked to match the durations (Figure 1).