THE POSSIBLE MOON OF KEPLER-90g IS A FALSE POSITIVE

We begin our investigation by exploring whether the full set of photometric observations of Kepler-90g can be explained by a planet-moon configuration, which was not attempted by Cabrera et al. (2014). There are two reasons why one might suspect this is not possible. First, there is apparently only one high-significance piece of evidence for Kepler-90g.01, namely, the companion transit to Kepler-90g in Q6 (quarter 6), as shown in Figure 1. This raises the question as to whether it is dynamically possible for Kepler-90g.01 to avoid imparting a second transit elsewhere for the other five transits of Kepler-90g. Second, the transit of Kepler-90g.01 is ∼21.5 hr later than that of Kepler-90g, implying a large planet-moon separation. Indeed, this is the principal reason why Cabrera et al. (2014) do not consider the moon hypothesis likely, arguing the moon would be on the edge of Hill stability.

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The seven-planet system, Kepler-90, was observed continuously by the Kepler spacecraft from Q1 to Q17, except for Q8, Q12, and Q16 where the target fell off the CCDs. We use all of the available data for this target, including the short-cadence observations where available (from Q9 onward). The Pre-search Data Conditioning (PDC) time series (Smith et al. 2012) was downloaded and detrended after first excluding all transits and outliers, and manually removing discontinuities (e.g., ramps), in the same way described in our previous papers (e.g., see Kipping et al. 2014). Long-term detrending for the transit photometry of Kepler-90g was handled by the CoFiAM algorithm (Kipping et al. 2013b), protecting timescales less than or equal to the full duration of Kepler-90g (12.593 hr; Table 3 of Cabrera et al. 2014).

Kepler-90g exhibits a ∼1 day transit timing variation (TTV) for its sixth and final transit, which is far too large to be due to an exomoon alone (Kipping 2009). For this reason, we model the six transits of Kepler-90g using the photodynamical planet-moon code LUNA (Kipping 2011) plus six freely fitted transit times, which are likely due to planet-planet interactions. By fitting the transit times freely, any TTVs due to an exomoon are absorbed by these parameters and thus would be degenerate with the moon's mass. This is particularly salient since TTVs due to an exomoon should dominate a mass determination over transit duration variations (TDVs; Kipping 2009), due to the visually obvious large separation between the transits of Kepler-90g and Kepler-90g.01, implying a large planet-moon separation. For these reasons, the mass of the exomoon is simply set to zero in the photodynamical model (but the moon radius is still freely fitted). As with our previous works, we regress our model to the detrended photometry using the MultiNest package (Feroz et al. 2008, 2009), with our priors following those of Kipping et al. (2014).

Using the Bayesian evidence computed by MultiNest, we find a planet+moon+TTVs model is favored over a planet+TTVs model with a formal significance of 5.2σ ($\Delta \ln \mathcal {Z} = 15.3\,\pm\, 0.2$). Furthermore, we find that the six transits of Kepler-90g are well-explained by this model, with Kepler-90g.01 imparting two dominant signals on the light curve, the originally identified signal during Q6 and a second event preceding the first transit of Kepler-90g (Q1), as shown in Figure 1.

The Q6 transit of Kepler-90g.01 is supported by a change in χ² of Δχ² = 49.0 over 26 data points. The Q1 transit is far less convincing, imposing a modest improvement of Δχ² = 11.1 over 15 data points, during a period of increased light curve scatter. We also note a very minor exomoon feature during the Q10 transit of negligible significance. We therefore conclude that the case for Kepler-90g.01 rests solely on the reality of the Q6 moon-like transit.

The parameters for the planet-moon system, assuming R_⋆ = (1.2  ±  0.1) R_☉ (Cabrera et al. 2014), indicate a (7.96 ± 0.65) R_⊕ planet orbited by a (1.88 ± 0.21) R_⊕ exomoon at a semi-major axis of $(a_{{\rm SP}}/R_P)=69.4_{-6.4}^{+7.3}$ planetary radii. We find short-period moons (≲ 10 days) are prohibited, but otherwise the moon's period posterior distribution is broad, leading to a wide range of possible planet densities via the technique described in Kipping (2010) of ρ_P < 21.4 g cm⁻³ to 95% confidence. In the case of Kepler-90g, a more useful constraint on the planet's density comes from demanding the moon orbit within the Hill sphere, which requires

$$\begin{eqnarray} &&\left (\frac{M_P}{M_{\star }}\right) > 3 \left (\frac{ (R_P/R_{\star }) (a_{{\rm SP}}/R_P) }{ (a/R_{\star }) } \right)^{3}, \end{eqnarray} \tag{ 1 }$$

where the bracketed terms on the right-hand side are all observables from our light curve model. Using our posterior distributions, this constraint imposes $(M_P/M_{\star }) > (8.4_{-2.1}^{+3.1})\times 10^{-5}$. Dynamically speaking, we estimate the seven-planet system is stable if Kepler-90g has a mass (M_P/M_⋆) ≲ 2 × 10⁻⁴, implying the Hill sphere can plausibly extend to encompass this moon signal.

Since ρ_P = ρ_⋆(M_P/M_⋆)(R_P/R_⋆)⁻³, then our limit may be expressed as $\rho _P > 0.39_{-0.10}^{+0.14}$ g cm⁻³, where we have used ρ_⋆ = (1.036 ± 0.033) g cm⁻³, based on averaging the light curve derived stellar densities from the seven planets in the system. Densities greater than this sub-Saturn mean density are certainly plausible for an 8 R_⊕ planet. Additionally, for a retrograde moon, Domingos et al. (2006) argue moons are stable up to nearly the edge of the Hill sphere (∼93%).

Based on our light curve analysis, we draw the following conclusions. First, the light curve can be well-explained by Kepler-90g hosting a single large exomoon. Although the formal significance is 5.2σ, we find that the significance is dominated by the single moon-like transit in Q6, which indicates the need for a careful inspection of this event. Second, we find that Kepler-90g.01 can be within the Hill sphere of Kepler-90g if Kepler-90g has a mean density in the range of 0.39 → 0.93 g cm⁻³ (upper limit based on seven-planet stability), which is certainly plausible given the radius of 8 R_⊕.

We have now established that the case for an exomoon around Kepler-90g now rests solely on the reality of the Q6 moon-like transit. The usual detection criteria used by the HEK project (Kipping et al. 2013b) are less useful for this system, since for example we cannot conduct double-likelihood tests by omitting some of the transits, given that there is just one moon-like signal. Low amplitude transit-like events are sometimes imparted into the Kepler time series due to various sources of false positives, ranging from astrophysical to instrumental (Tenenbaum et al. 2014).

We confirmed that this signal is coherent in both the PDC and Simple Aperture Photometry data products, using both CoFiAM and polynomial detrendings of each. The moon-like signal is therefore not an artifact of the PDC pipeline, which can occur in rare instances (Christiansen et al. 2013). Using the Pixel Response Function (PRF; see Bryson et al. 2010), precise centroids of each Kepler target are made available as a standard data product. These may be used to identify certain types of false positive scenarios, such as a background eclipsing binary, and this has become a standard component of the Data Validation (DV) process by the Kepler team (Bryson et al. 2013). During a transit, a large centroid shift indicates that the true eclipsing object is offset from the total flux centroid.

In Figure 2, we show the time series of the x (column) and y (row) centroids for Kepler-90 during Q6, after detrending with CoFiAM. The highlighted areas depict the times when Kepler-90g and Kepler-90g.01 are transiting. Both signals appear reasonably consistent with the out-of-transit centroids at a level <3σ (the typical threshold in DV reports). We find no compelling reason to reject the possible moon Kepler-90g.01 based on this information.

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We wished to understand where on the CCD the transits occur and how localized the transit signals were relative to the point-spread function. In order to investigate this, we considered a new type of centroid we dub the "transit centroid." Whereas a conventional centroid calculates the mean pixel position weighted by the flux of each pixel, the transit centroid instead weights by the signal-to-noise ratio (S/N) of the transit depth on each pixel. Since the flux drops away from the flux centroid, causing greater photon noise, the S/N should be located at the same position as the flux centroid. If high S/N transits are found far away from the flux centroid, this is an indication that the transit is not localized at the expected flux centroid position. We define the transit centroid position as

$$\begin{eqnarray} x_{\mathrm{transit}} &= \frac{ \sum _{i=1}^N x_i (\delta _i/\sigma _i) }{ \sum _{i=1}^N (\delta _i/\sigma _i) } \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} y_{\mathrm{transit}} &= \frac{ \sum _{i=1}^N y_i (\delta _i/\sigma _i) }{ \sum _{i=1}^N (\delta _i/\sigma _i) }, \end{eqnarray} \tag{ 3 }$$

where {x_i, y_i} is the Cartesian location of the center of the ith pixel, δ_i is the transit depth recorded on that pixel and σ_i is the associated uncertainty on that transit depth. For the depth calculation, we simply compute the mean flux within the durations found by our earlier photodynamical fits and use the standard error as the uncertainty.

The S/N of each pixel for the transits of Kepler-90g and Kepler-90g.01 during Q6 are color coded on Figures 3(b) and (c), respectively, which may be compared to the total flux centroid shown in Figure 3(a). Centroid positions are provided at the top of each panel in Figure 3, with uncertainties estimated using a bootstrap of each pixel's light curve residuals. Although the Kepler-90g.01 transit centroid is not significantly offset (<2 σ) from the planetary transit, the S/N pattern for Kepler-90g.01 is quite distinct. Specifically, it is clear that the moon-like signal is not localized at the center of the postage stamp, unlike Kepler-90g, with even pixels on the edges showing high S/N levels.

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We considered the possibility that the fact Kepler-90g.01 is such a low amplitude transit is responsible for the transit centroid's delocalization, i.e., the data are not good enough to pin down the location. To test this, we created a fake Q6 time series by taking the pixel-level light curves of Kepler-90g's transit, reducing the depth to equal that of Kepler-90g.01, and injecting it in place of the original Kepler-90g.01 transit for each pixel. Since the durations of the events are similar (12.1 hr for g and 10.4 hr for g.01), this process should produce a realistic estimate of the transit centroid pattern produced if Kepler-90g.01 were a genuine exomoon. Recalculating the transit centroids (see Figure 4), we find that of the fake moon to be highly localized, as visible in Figure 4(c), and the transit centroid is now within 1 σ of both Kepler-90g and the total flux. This test indicates that the delocalized structure seen in Figure 3(c) is strong evidence that the Kepler-90g.01 transit is spurious.

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On this basis, we identify Kepler-90g.01 as a false positive exomoon. The underlying origin for the anomalous transit centroid is unclear, but may be due to a cosmic ray hit causing Sudden Pixel Sensitivity Dropout (SPSD) at this time (see Christiansen et al. 2013). As a further check, we tried detrending the Q6 photometry using the Trend Filtering Algorithm (TFA) described in Huang et al. (2013), which searches for common mode behaviors on the same CCD module. We find that the moon-like transit persists in the TFA light curve, indicating that the other stars on this module do not exhibit the event. For the purposes of this short letter it is unimportant what the actual origin is; the key conclusion is that the signal is unlikely to be that of an exomoon.