An exomoon survey of 70 cool giant exoplanets and the new candidate Kepler-1708 b-i

Target selection

The focus of this work is the population of cool, giant transiting exoplanets observed by Kepler. The most comprehensive catalogue of Kepler transiting planet candidates comes from the NASA Exoplanet Archive (NEA53) and thus we began by downloading this catalogue at the start of this investigation (27 March 2018). Not all Kepler objects of interest in the catalogue are viable planet candidates though, and so we applied a cut to remove any objects that have been dispositioned as likely false positives by the NEA.

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Both the terms ‘cool’ and ‘giant’ are somewhat subjective and thus require a clear definition for the purposes of target selection. For giant, we elected to use a cut of (hat{{R}_{mathrm{P}}} > 0.5) RJ, where (hat{{R}_{mathrm{P}}}) is the most probable radius value listed in the NEA. This choice is motivated to minimize the number of sub-Neptunes that make it into our sample, and thus focus on Jupiter-sized worlds.

The definition of cool is again subjective, but here we are primarily interested in planets in environments cooler than that of the Earth. Using a simple instellation or temperature cut alone is inadequate though, as these values depend upon the stellar parameters, which have been subject to substantial revision over time54. On this basis, they may not be reliable in isolation as a means of capturing all of the cool giants. Instead, we apply three different definitions of cool and accept giant planets that satisfy any of the three. These criteria are (hat{{S}_{mathrm{P}}} < 1) S⊕, (hat{{T}_{mathrm{P}}} 400) d—where TP is the blackbody equilibrium temperature of the planet.

Using these cuts, 48 Kepler objects of interest were identified from the NEA. However, we noted that a subset of these had suspiciously large radii. We thus applied an additional cut to remove any with best reported planetary radii in excess of 2 Jupiter radii (removing KIC-3644071.01, KIC-6426592.01, KIC-6443093.01, KIC-9025662.01, KIC-9011955.01, KIC-8240617.01 and KIC-8868364.01). This left us with 41 unique planetary candidates. Of particular note is KIC-5437945, which possesses two cool giants associated with a single source.

At this point, we introduce another cut: that at least two transits have been observed by Kepler. Without the two transits, the orbital period cannot be precisely measured and this in turn makes it impossible to measure the eccentricity of the planets photometrically—a test we will depend on later in our exomoon analysis. Of the 41 cool giants from the NEA, 5 were found to only exhibit a single transit in the available Kepler data (KIC-2162635.01, KIC-3230491.01, KIC-3962440.01, KIC-11342550.01 and KIC-11709124.02) and thus were excluded. This leaves us with 36 cool giants.

Although the NEA is the most complete catalogue available, long-period planets are more challenging to find than their shorter-period counterparts, and thus numerous independent studies have identified long-period exoplanetary candidates that were not present in the NEA. In particular, we identified several additional studies13,14,15,16, which we inspected in an effort to locate any additional possible planets missed thus far. In what follows, we attempt to apply the same filters as before, such as exhibiting at least two transits, but note that in some cases the radius and instellation values had not been computed by the original authors.

From the work of Wang and colleagues13, KIC-8012732, KIC-9413313 and KIC-11465813 were flagged as exhibiting three or four visible transits and have long periods (431 d, 440 d and 671 d) and large radii (9.8 RJ and 13.8 RJ). KIC-5437945 and KIC-7619236 also satisfy the criteria but are already included in the NEA catalogue. KIC-5652983 is long period and large but has been argued to be a likely false positive13 due to the observation of large radial-velocity variations. Amongst the two-transit cases, KIC-5732155, KIC-6191521 and KIC-10255705 are also added, which leads to six new cool giants from this sample.

From the work of Uehara and colleagues14, only KIC-10460629 exhibits at least two transits, separated by 525 d. Although the radius is not computed, the grazing, deep (2%) transit is plausibly a giant around a diminutive star and thus was included in what follows.

From the work of Foreman-Mackey and colleagues15, only three of the candidates have more than two transits: KIC-3239945, KIC-8410697 and KIC-8800954. However, KIC-3239945 is already found in our NEA catalogue and KIC-8800954 has a radius of 0.39 RJ, making it too small for our sample. Accordingly, we only add KIC-8410697, of period 1,047 d and 0.70 RJ.

From the work of Wheeler and Kipping16, we pick an extra transiter of 3% depth but unreported radius (KIC-8508736) with a period of 681 d.

Combining these targets with the NEA sample yielded 45 cool giants.

As our study progressed, spanning multiple years, the possibility of new cool-giant detections appearing in the NEA grew. To address this, we re-ran our NEA search on 12 June 2019, which identified 24 new cool giants. Of these, six were single-transit systems (KIC-2162635, KIC-3230491, KIC-3644071, KIC-3962440, KIC-11342550 and KIC-11709142) and were thus removed. In addition, upon querying KIC-8308347 in the Mikulski Archive for Space Telescopes, we noted that it had been flagged as an ‘eclipsing binary: likely false positive’ under condition flag and thus rejected it in what follows. These new inclusions raise our total number of cool giants to 62.

As a final addition to our catalogue, a recently published study17 (although quite far into our own analysis) presented 23 long-period Kepler planetary candidates. Of these, 14 were suitable for our study and all but one were double-transit systems. However, three of these were already included in our sample by this point (KIC-3756801, KIC-9663113 and KIC-10460629), meaning that the ref. 17 sample added 11 cool giants to our ensemble. Of the new objects, KIC-5351250 is of particular note since it represents the fifth planetary candidate in the Kepler-150 system31. Together, this brings our final catalogue of cool giants up to 73 planetary candidates. Three of these were found to exhibit unacceptably high correlated noise structure in the light curves and were thus rejected, as described later in this section. The remaining 70 are listed in Supplementary Table 1.

Data preparation

For each target, the light-curve files were downloaded from the Mikulski Archive for Space Telescopes, using primarily the long-cadence data but short cadence where available. However, given the long-period nature of our transiting bodies, the value of short cadence is considerably less than that of typical transiters55. In all cases, the data were processed as part of the 25th and final data release issued by the Kepler science team56, dubbed DR25 hereafter.

For all light curves, we trimmed any points with an error flag equal to anything other than zero—thus removing points known to be afflicted by effects such as reaction-wheel zero crossings57. Additional outliers (for example unidentified cosmic-ray hits58) were removed independently for the ‘simple aperture photometry’ (SAP) and ‘pre-search data conditioning’ (PDC59) light curves, by flagging points more than 3σ deviant from a moving median of bandwidth 20 long cadences.

Kepler light curves exhibit modulations in intensity due to myriad effects. Ultimately, the short-term modulations corresponding to a transit are of central interest to this study, but longer-term variability is also present and introduces sizable trends that require correction. Such variability could originate from the instrument (for example focus drift60) or the parent star (for example rotational modulations61). In what follows, we describe our approach for detrending these effects.

As a brief aside, we note that short-term variability on the same timescale as the transits can also be present (for example pulsations in evolved stars62) and is generally much more difficult to remove, since it is not separable in the frequency domain. Consequently, attempts to remove such noise come at grave risk of distorting a transit signal of interest. Given that our primary objective is to look for exomoons, which manifest as small undulations on this timescale, it was considered an unacceptable risk to attempt to remove such short-term variability—since such efforts may in fact introduce false positives into the time series. Instead, the philosophy in what follows is to use statistical tests to identify light curves corrupted by such noise and simply discard them. This naturally comes at the expense of increasing our false-negative rate, since such systems are not even analysed further.

For each cool-giant target, we detrend the light curves of the individual transit epochs individually, rather than imposing the idea that the noise in one quarter need be representative of others. This is largely motivated by the fact that the spacecraft rolls every quarter and thus sources appear on different silicon with different optimal apertures, blend contaminations and CCD (charge-coupled device) behaviours. In addition, we adopt the approach of detrending each transit epoch in eight different ways. The reasoning here is that, although we generally consider each of the eight different methods to be fairly accurate (else we would not be using them), we cannot guarantee that any of them are going to work in every situation. From experience, peculiarities in particular light curves can interact with detrending algorithms in unanticipated ways, leading to anything from a complete failure to a subtle residual trend. Fundamentally, any claim we make about the presence of an exomoon needs to be robust against choices made in this detrending stage, and a path to achieving this is to simply use multiple detrending methods and compare.

The details of the different detrending algorithms used are presented shortly, but once in hand they are combined into a single data product (per transit epoch) known as a method-marginalized light curve21. In this work, we generate such light curves by simply taking the median of the multiple detrended intensities at each time stamp. The formal uncertainty on each photometric data point is also inflated by adding it in quadrature to 1.4286 multiplied by the median absolute deviation between the methods. Median statistics are used throughout to mitigate the influence of failed detrending(s). In this way, we increase the robustness of our light products against detrending choices and also inflate the errors to propagate the uncertainty in the detrending procedure itself.

As an additional safeguard against poorly detrended light curves, we compute two light-curve statistics to measure their Gaussianity. If any of the eight light curves fail this test, they are rejected before the method-marginalization procedure. For the first test, we bin the light curves (after removing the transits) into ever larger bins and compute the s.d. versus bin size, against which we then fit a linear slope in log–log space. For such a plot, the slope should be minus one-half, reflecting the behaviour of Poisson counting of independent measures. However, time-correlated noise structure will lead to a shallower slope that can be used to flag such problematic sources63. We thus generate 1,000 light curves of precisely the same time sampling and pure Gaussian noise and measure their slopes in this way. This allows us to construct a distribution of expected slope values. If the real slope deviates from the Monte Carlo experiments with a P value exceeding 2σ, the light curve is flagged as non-Gaussian.

For the second test, we compute the Durbin–Watson64 statistic of the unbinned light curves (after removing the transits). This is essentially a test for autocorrelation at the timescale of the data’s cadence, where uncorrelated time series should yield a score of 2. As before, we test for non-Gaussian cases by generating 1,000 fake Gaussian light curves at the same time sampling and scoring their Durbin–Watson metrics. If the real light curve is deviant from this distribution by more than 2σ, the light curve is rejected.

The above describes how we combine eight light curves detrended independently, but we have yet to describe how these eight light curves are generated in the first place—to which we turn in what follows. In total, four different detrending algorithms are used, which are then applied to the SAP and PDC data to give eight light curves. The four algorithms are described now.

CoFiAM. Cosine Filtering with Autocorrelation Minimization (CoFiAM) builds upon the cosine filtering approach previously developed for CoRoT65 data. Cosine filtering is attractive because it behaves in a predictable manner in the frequency domain, unlike the other methods used here, which leak power across frequency space. Fourier decomposition of the transit morphology reveals dominant power at the timescale of the transit duration and higher frequencies66. Thus, by only removing frequencies substantially lower than this, we can ensure that the morphology of the transit is not distorted by the detrending process itself. On the other hand, cosine filtering is problematic in that we could regress a very large number of cosines to the data. Much like fitting high-order polynomials, predictions from such model become unstable at high order. In our case, we train on the out-of-transit data (in fact the entire quarter) and interpolate the model into the transit window, thus introducing the possibility of high-order instabilities here.

This is where our implementation deviates from that used for CoRoT65, to account for this effect. We detrend the light curve in up to 30 different ways, in each case choosing a different number of cosine components to include. The simplest model is a single cosine of frequency given by twice the baseline of available observations (thus looking like a quadratic trend)—known as the basic frequency. At each step, we add another cosine term of higher frequency to the function (equal to a harmonic of the basic frequency), train the updated model, detrend the light curve and compute statistics concerning the quality of the detrending. We continue up to 30 harmonics, or until we hit 1.5 times the reported transit duration. From the 30 options, we pick the one that leads to the most uncorrelated light curve—as measured from the Durbin–Watson statistic evaluated on the data surrounding (but not including) the transit (specifically to within six transit durations either side). These local data are then exported with the data further away from transit trimmed at this point. We direct the reader to our previous work67 for more details of this approach, including the underlying formulae used.

PolyAM. Polynomial Detrending with Autocorrelation Minimization (PolyAM) is similar to the above except that the basis function is changed from a series of harmonic cosines to polynomials. As before 30 different possible maximum polynomial orders are attempted from 1st to 30th order, and as before for each epoch the least autocorrelated light curve is selected as the accepted detrending on a transit-by-transit basis.

local. The next approach again uses polynomials, and again up to 30th order, but this time the final accepted polynomial order is that which leads to the lowest Bayesian information criterion (BIC)68 as computed on the data directly surrounding the transit (specifically to within six transit durations). This is arguably the simplest of the four algorithms attempted and is a fairly typical strategy in the analysis of short-period transiters69.

GP. Finally, we implemented a Gaussian process (GP) regression to the light curve. As with all of the methods above, the transits are masked during the regression by using the best available ephemeris. We implemented the regression using a squared exponential kernel where the hyperparameters (for example length scale) are optimized for each epoch independently. For consistency, we only export the data that is within six transit durations of the transit, although technically the entire segment (±0.5 orbital periods of each transit) is detrended.

As a final note, recall that during the method-marginalization process we perform checks to identify detrended light curves that do not conform to Gaussian behaviour. If all eight detrendings of a given epoch fail these tests, then there will be no accepted light curves to combine and thus the epoch is dropped. In some cases, this can reduce the number of available transits (after detrending) to fewer than two—thereby making it fail our basic criteria of presenting two epochs or more. We found that this happened for three objects in our sample, KIC-10255705, KIC-11513366 and KIC-6309307. This removal of these three targets reduces our sample size from 73 to 70 (which are the ones listed in Supplementary Table 1).

Isochrone analysis

The 70 transiting planet candidates are associated with distinct stars from one another, with the exception of one pair associated with KIC-5437945, leading to 69 unique stars. To derive physical dimensions for the planetary candidates, it is necessary to first derive stellar parameters. This is accomplished using an isochrone analysis, which compares observable quantities associated with a star against a grid of stellar models, assuming different masses, radii, ages and so on. In this way, best-matching solutions can be inferred in a Bayesian framework to derive posterior distributions for the stellar properties.

Given that our stars are broadly FGK type, we elected to use the Dartmouth stellar isochrone models70 to describe these stars. For each target, we took the Gaia DR2 parallax71, the Kepler bandpass apparent magnitude, and the stellar atmosphere properties reported in the Kepler DR25 catalogue54, and appended them to a star.ini file along with their associated errors. These were then fed into the isochrones package72 to obtain a posteriori fundamental stellar parameters, including the mean density of the host star (ρ⋆). These fundamental parameters are reported in Supplementary Table 2 and were used later in our analysis for deriving planet/moon radii/masses.

Due to the particular interest of the target KIC-7906827, we updated our isochrone analysis to include the Gaia DR3 parallax when it became available (reducing the parallax uncertainty by 25%). We also elected to use the stellar atmosphere properties from the transit detection work17 (({T}_{{{{rm{eff}}}}}=6{,}15{7}_{-202}^{+231}) K, ({{mathrm{log}}},g=4.3{7}_{-0.05}^{+0.04}), Fe/H (=0.{0}_{-0.2}^{+0.2})) rather than the DR25 catalogue54 (Teff = 5,977 ± 176 K, log g = 4.39 ± 0.12, Fe/H = −0.08 ± 0.26), although we note that these values are clearly very similar. The system parameters listed in Table 1 correspond to these revised choices, for which the associated fundamental stellar parameters are M⋆ = 1.088 ± 0.072 M⊙, R⋆ = 1.117 ± 0.064 R⊙, log10[A (yr)] = 9.50 ± 0.31, log10[L⋆ (L)] = 0.182 ± 0.082, d = 1,712 ± 75 pc and log10[ρ⋆ (g cm−3)] = 0.042 ± 0.065.

We note that these are not the same fundamental stellar parameters for KIC-7906827 as listed in Supplementary Table 2, which originate from the original Kepler DR25 and Gaia DR2 inputs. Once again though, we note that there is very little difference between the two, with M⋆ = 1.056 ± 0.067 M⊙, R⋆ = 1.098 ± 0.095 R⊙, log10[A (yr)] = 9.61 ± 0.33, log10[L⋆ (L)] = 0.140 ± 0.087, d = 1,750 ± 100 pc and log10[ρ⋆ (g cm−3)] = 0.05 ± 0.11. As a final point of comparison, both sets of values are in good agreement with the independent analysis (using Gaia DR2) by Berger and colleagues73, who find ({M}_{star }=1.06{1}_{-0.079}^{+0.073}) M⊙, ({R}_{star }=1.14{1}_{-0.066}^{+0.073}) R⊙, log10[L⋆ (L)] = 0.140 ± 0.087 and d = 1,640 ± 100 pc.

Light-curve fits

For planets exhibiting three or more transits, at least three different light-curve models, or hypotheses, were proposed to explain the data. The first is model ({{{mathcal{P}}}}), which represents the null hypothesis of a transiting planet orbiting its star on a strictly Keplerian orbit. In this case, the Mandel–Agol74 light-curve algorithm is used. The second hypothesis, model ({{{mathcal{T}}}}), expands upon the first by adding TTVs. This is formally accounted for using same algorithm again but allowing each transit epoch to have a unique time of transit minimum, τi. The third model considered is that of a planet–moon system, ({{{mathcal{M}}}}), generated using the LUNA photodynamic algorithm36. For planets exhibiting just two transits, TTVs cannot be distinguished from a linear ephemeris and thus model ({{{mathcal{T}}}}) was not used.

In each model, the limb darkening of the star is modelled with a quadratic limb darkening law using the q1–q2 reparameterization75. Since the majority of light curves are long cadence, the potentially compelling light-curve smearing effect is accounted for by employing the numerical resampling method55 (with Nresamp. = 30). Finally, contaminated light from nearby sources is tabulated in the Kepler fits files as ‘CROWDSAP’ and this value is used in a blend correction to each quarter’s light curve using a previously published method76. We also note that the models formally assume circular orbits, although elliptical planets are almost perfectly described by these models too77, with the exception that the fitted stellar density will be skewed away from the true value78. Exomoon orbits are also treated as circular, which is justified on the basis of the expected rapid circularization timescales79.

Regressions were executed using the multimodal nested sampling algorithm MultiNest80 with 4,000 live points. The advantage of using MultiNest over conventional Markov chain Monte Carlo (MCMC) methods is the ability to sample disparate modes and efficiently evaluate the marginal likelihood of the proposed hypotheses, which is used later in Bayesian model selection.

For model ({{{mathcal{P}}}}), seven parameters fully describe the light-curve model and thus are the free parameters in these fits. These are (1) P, (2) τ, (3) p, the ratio of radii between the planet and the star, (4) b, (5) ρ⋆, (6) q1, the first limb-darkening coefficient, and (7) q2, the second limb-darkening coefficient. Uniform priors are adopted for all except the light-curve-derived stellar density (under the assumption of a circular orbit), ρ⋆,circ, for which we use a log-uniform prior between 10−3 g cm−3 and 103 g cm−3.

For model ({{{mathcal{T}}}}), we have 5 + N parameters, where N is the number of available transit epochs. The first five terms are the same as those of model ({{{mathcal{P}}}}) except for P and τ. The extra N terms are the individual times of transit minimum for each epoch.

Finally, for model ({{{mathcal{M}}}}) we have 14 free parameters. The first seven are simply the same as for model ({{{mathcal{P}}}}) but the last seven describe the exomoon. Specifically, these are (1) PS, (2) aSP/RP, (3) RSP, the ratio of radii between the satellite and the planet, (4) MSP, the ratio of masses between the satellite and the planet, (5) ϕS, the orbital phase of the satellite at the instant of planet–star inferior conjunction during the reference epoch, (6) cos(iS) and (7) ΩS. As before, uniform priors are adopted for all with the exception of PS, which uses a log-uniform prior from 75 min to the period corresponding to one Hill radius. The semimajor axis of the satellite has a uniform prior from 2 to 100 planetary radii.

For all models, a normal likelihood function is adopted. The only addition made to this is that we applied a likelihood penalty to model ({{{mathcal{M}}}}), which explored unphysical parameter combinations. Specifically, we calculate the satellite and planet density using previously published expressions81 and reject any samples for which ρS > 20 g cm−3 or ρP > 150 g cm−3 or ρP < 0.5 g cm−3, in an effort to keep the sampler in the region of physically plausible solutions. Additionally, solutions where the satellite period exceeds 93.09% of the Hill sphere are unstable even for retrograde orbits28, and are thus rejected.

Initial checks for exomoon candidacy

The primary objective of this work is to search for new possible exomoon candidates amongst Kepler’s cool-giant sample. One of the first observable effects predicted for exomoons originates from TTVs imparted to the planet by the moon’s gravitational influence7. Thus, a basic criterion one might suggest is that TTVs should be present. However, for 25 of our targets only two transits were available and thus TTVs cannot be inferred. For the remainder, we apply a statistical test for TTVs as described later.

The other observational consequence of an exomoon is that its shadow can eclipse either the star or planet (that is, a syzygy), leading to additional changes in flux on top of the conventional transit signature36. These changes can occur in or out of transit and impart complex shapes. Furthermore, the limited number of transits available in our sample means that stacking approaches82,83 will not be statistically valid. However, our photodynamical planet–moon model (model ({{{mathcal{M}}}})) computed using LUNA does fully account for the moon’s subtle influence on the light curve. Since MultiNest provides marginal likelihoods, we can evaluate the Bayes factor between models ({{{mathcal{P}}}}) and ({{{mathcal{M}}}}), while correctly penalizing the moon model for its greater complexity, to determine the statistical evidence for a moon. We thus demand that the Bayes factor formally favours the planet–moon model over the planet model to be considered further.

Finally, we require that the planet has an orbit that is consistent with a circular path. Elliptical orbits can be produced through planet–planet scattering84, which is expected to strip exomoons29. Even if the eccentricity is produced through some other effect, the dynamical region of stability is severely truncated by non-zero eccentricity28. Thus, although an elliptical orbit does not prohibit exomoons, we consider it a priori improbable and thus reject any planets exhibiting eccentric orbits.

If these 2(+1) criteria are satisfied, the object is promoted for further checks, which we refer to as secondary tests (discussed in the next subsection). We describe the details of the initial tests in the following paragraphs and highlight that the results are listed in Supplementary Table 1.

For the eccentricity test, we require some formal criterion to evaluate if a planet has an eccentric orbit or not. Since we assume a circular orbit in all of our light-curve models, the derived mean stellar density will be offset from the true value if the orbit is in fact eccentric85. If we have an independent and unbiased measure of the mean stellar density in hand, this offset can be detected and thus used to constrain the orbital eccentricity86. Again, the philosophy here is to minimize the exomoon false-positive rate at the expense of the false-negative rate.

Our eccentricity test thus begins by taking the a posteriori parameter samples from either model ({{{mathcal{P}}}}) or ({{{mathcal{T}}}})—specifically, we default to ({{{mathcal{P}}}}) unless we conclude the system is ‘TTV hot’ as defined by our TTV test described later. Naturally, for two-transit planets we always use model ({{{mathcal{P}}}}).

Next, we need an independent measure of the true stellar density and here this originates from an isochrone analysis. This is described in a dedicated section later and results are summarized in Supplementary Table 2.

To determine an eccentricity posterior for each planet, we begin with the asterodensity profiling relationship85, which relates ρ⋆,circ to the true value, ρ⋆:

$${rho }_{star ,{{{rm{circ}}}}}simeq {rho }_{star }{{varPsi }},$$

(1)

where

$${{varPsi }}equiv frac{{(1+esin omega )}^{3}}{{(1-{e}^{2})}^{3/2}}.$$

(2)

Since ρ⋆,circ and ρ⋆ are inferred independently, we construct a Ψ posterior by simply drawing random samples from our light-curve-derived density and dividing them by random samples from the isochrone density. This Ψ posterior sadly does not trivially lead to an eccentricity posterior because of the joint dependence on ω, the argument of periastron. Thus, we need to again sample the parameter space. To achieve this, we used kernel density estimation (KDE) on the log Ψ posterior with a Gaussian kernel and a bandwidth optimized for using least-squares cross-validation. The KDE distribution was then used as a log-likelihood function for the purposes of an MCMC exploration in {e, ω} parameter space.

Since transiting planets are more likely to be eccentric, a priori, as a result of geometric bias87,88, it is necessary to account for this selection bias during the inference. This in turn requires a prior for the eccentricity distribution, which cannot be strictly uniform to avoid infinities89. We thus assume that Pr(e), the prior on eccentricity, is a beta distribution with shape parameters α = 1 and β = 3, broadly matching the long-period radial velocity population90. The selection effect inherent to the transit method is then accounted for using the joint prior, Pr(e, ω∣transiting) as derived for eccentric planets89.

We then sampled the {e, ω} parameter volume 110,000 times, burning out the first 10,000 steps. Note that MCMC sampling suffers from biases at boundary conditions, such as e > 0, and this can lead to an artificial positive skew in eccentricity91. This can be overcome by reparameterizing92 to √e sin ω and √e cos ω, which we use here. Once the eccentricity posterior has been evaluated, we next perform Bayesian model selection by evaluating the Savage–Dickey ratio93. This simply evaluates the posterior density at e = 0 versus the prior, where the ratio provides a direct estimate of the Bayes factor of an eccentric versus circular orbit (in the case of nested models such as here). In our case, any instance where the posterior density is less than the prior at e = 0 is hereby labelled as ‘eccentric’, otherwise ‘circular’. The prior density is analytic and thus trivial to evaluate at zero89 but for the posterior density we apply a KDE to the posterior to evaluate the density at zero. Because of the boundary condition at e = 0, we mirror the posterior samples around zero and combine them with the originals, and then apply a Gaussian KDE to the combined sample. The density at zero is then equal to twice the density of this KDE at zero, as a result of the doubling of the sample volume.

Thus far, the eccentricity test described above has been applied to either the planet-only or planet-with-TTVs light-curve model, depending on whether we classified the planet as TTV hot (see next paragraphs). However, we also repeated this a second time applied to the planet–moon posteriors from model ({{{mathcal{M}}}}). If the planet appears incompatible with a circular orbit only after the moon component is introduced, we mark this with the † symbol in Supplementary Table 1 and the object is not considered further as a viable moon candidate. Such cases essentially mean that the required moon solution demands a light-curve shape that is inconsistent with the derived stellar density unless eccentricity or blending is introduced94.

We emphasize that transit planets preferring eccentric orbit solutions are identified via the existence of a ‘photoeccentric effect’, which describes an apparent tension between the light-curve-derived stellar density and an independently inferred value86. However, blends and starspots can also cause substantial tension, and both would lead to an elevated risk of exomoon false positives, motivating their exclusion. Finally, although we exclude these systems from this study, that does not mean they are necessarily devoid of moons, any more than hot Jupiters are necessarily devoid of moons. However, in both cases, physical arguments suggest that they are not the most suitable environment. Therefore, although we elect to avoid such systems in this study, efforts by other teams to survey such objects are by no means futile and we encourage such work.

For the TTVs, as noted earlier, 25 of our targets have only two transits available and thus cannot be tested. This is because TTVs are defined as an excursion from a linear ephemeris fit, but a linear ephemeris model (governed by two free parameters) will always provide a perfect fit to two arbitrary transit times (two data points). For the other cases, we can search for TTVs as an indication of an exomoon, as well as providing some novel insights about the propensity of cool giants to exhibit TTVs more broadly.

Testing for TTVs through periodogram searches is impractical for the vast majority of our sample. This is because 51/72 of our planets have three transits or fewer and thus will offer just three data points for a regression. For a sinusoidal TTV, the simplest periodic function, five unknown parameters describe the ephemeris (the period and reference time of transit minimum, as well as three sinusoid parameters—period, phase, amplitude). Even in a grid-search periodogram, which removes one parameter, TTV period, we still have fewer data than free parameters. Note that if we possess four data points the system becomes constrained but fits are typically ‘perfect’, although in such cases we can apply regularization techniques on the amplitude term to make progress, as has been done, for example, for Kepler-1625 b95.

Instead of trying to seek a periodic TTV, we simply ask whether there is evidence for a TTV. To this end, we follow earlier work96 and apply their first test that addressed this question. This takes the maximum-likelihood light-curve fits of models ({{{mathcal{P}}}}) and ({{{mathcal{T}}}}) and compares their log-likelihood through a BIC evaluation68. By working with the light curves directly, rather than derived products such as marginalized transit times, we are able to extract as much information from the light curve as possible. Any planet with a BIC preference for model ({{{mathcal{T}}}}) is labelled as TTV hot, else TTV cold, unless only two transits exist, in which case we use ‘TTV grey’.

Secondary checks for exomoon candidacy

If a planet passes the basic checks described in the last section, we apply additional checks to evaluate the plausibility of an exomoon. In total, 11 of the 70 cool giants satisfy this criterion. First, we require that the planet’s eccentricity, as determined from model ({{{mathcal{M}}}}), also favours a circular path. Following the same method as described in the last section, we find that all 11 indeed appear consistent with circular after applying this test.

Next, we regressed a new moon model to the data, model ({{{mathcal{X}}}}), which is identical to model ({{{mathcal{M}}}}), except that negative- and zero-radius moons are permitted. Negative radii correspond to inverted transits and are simply implemented by flipping the signals. Zero-radius moons are formally forbidden in model ({{{mathcal{M}}}}) since we impose the density constraint that ρS < 20 g cm−3 and a zero-radius moon has infinite density. Thus, to enable this we relax this condition by simply commenting out this check in our code. Using the posteriors of model ({{{mathcal{X}}}}), we apply three statistical additional tests to the 11 objects.

The first of these, which could be labelled test 4 by this point, is that we computed a Savage–Dickey ratio at the location of MS/MP = 0 to evaluate the statistical evidence in favour of a non-zero exomoon mass. If the system had three or more transits, we imposed the constraint that the Bayes factor from this calculation must exceed 10 in favour of a finite mass (that is, strong evidence27). Next, test 5, we computed a Savage–Dickey ratio at the location of RS/RP = 0 and demanded that, for all objects, the Bayes factor preference for a non-zero radius exceeds 10. Finally, test 6, we counted up how many of the RS/RP samples were negative versus positive and demanded that the positive:negative ratio exceeded 5. This final test catches the possibility that RS/RP is offset from zero but has substantial weight in the unphysical negative-radius regime.

After applying these cuts, three objects emerged as possible candidates: KIC-8681125, KIC-7906827 and KIC-5351250 (aka Kepler-150).

Vetting of KIC-8681125.01

Advanced vetting of the KIC-8681125.01 planetary moon candidate begins by visual inspection of the transit light-curve fits to better understand what type of moon signal is seemingly detected. As shown in Extended Data Fig. 2, the signal is remarkable for featuring no clear moon-like transit. Instead, the main difference observed is a transit depth change from (3{,}59{0}_{-130}^{+160}) ppm to (3{,}03{0}_{-110}^{+140}) ppm. This is explained by the moon model by placing the moon transit on top of the planetary transit in the first epoch, but then the moon avoids transiting the star altogether in the next epoch.

This situation was immediately suspicious and appeared somewhat convoluted and fine-tuned, particularly when we compare it with typical planet–moon models generated in simulation work36. One possibility is that a nearby contaminant source is more prominently included within the aperture of the second epoch than the first, thus diluting the second’s transit depth. However, if the source landed on the same silicon with the same aperture used each time, this would clearly be excluded as a possibility.

To investigate this, we used the Kepler target pixel files to inspect the photometry at the pixel level. Since the first epoch occurs in quarter 10, but the second in quarter 16, the spacecraft has rolled into a distinct position between the epochs (every four quarters it returns to the same position). As a result, the source is on different silicon between the two epochs. However, KIC-8681125 is located near the centre of the entire detector array, within module 13, and thus ends up remaining within this module even after the roll, since the roll is itself uses an axis with an origin close to the centre of the detector array. Despite this, it does indeed end up on different silicon moving, from quadrant 4 to quadrant 2 between the two epochs.

The optimal aperture used by the Kepler pipeline is also quite distinct between the epochs, as shown in Supplementary Fig. 1. Epoch 1 has a simple 2 × 2 square pixel centred on the source, but epoch 2 uses a ‘+’ shaped aperture with an extra pixel included in one corner. In total, six pixels are used in the second aperture, thus increasing the chance of a contaminant falling within the aperture. On this basis, we consider the hypothesis of a contaminant driving the depth change as being highly plausible.

To investigate further, we fitted the light curve with a model that was identical to the planet-only model except for the fact that the second epoch had a unique blend factor associated with it, γ. The maximum likelihood of this fit did not exceed the moon model, but it led to a major Δχ2 = 50 improvement over the planet-only model. Since the model only requires one extra parameter over the planet-only model, whereas the planet–moon model needs seven, the blend model outperforms the moon model in terms of the marginal likelihood. As a result, it is formally the preferred model by a Bayes factor of 6.8.

While the blend hypothesis seems to naturally resolve this system, we highlight that problems still remain with this idea. Unfortunately, no high-resolution imaging has been previously obtained, but Gaia can resolve sources more than 1–2 arcsec away. The closest source (id 2127184090671914880) is 11.8 arcsec away and 1.7 mag fainter. Given the pixel scale of Kepler of 4.0 arcsec, this is probably too far away to explain the relatively large depth change, as well being somewhat fainter than expected to explain the depth change. Another possibility is that an unseen source resides closer, within approximately 1 arcsec of the source, evading Gaia. However, this is also not satisfactory, as the contaminant should then be sufficiently close as to be included in both epoch apertures. The contaminant hypothesis is thus challenged by the lack of an obvious known source.

We also considered the possibility that the star may be covered in spots, and between the two epochs the spot coverage varies to manifest the depth change. However, high spot coverage appears incompatible with the Kepler photometry, which is relatively flat. To explore this, we ran a Lomb–Scargle periodogram on each quarter and found that the amplitude is consistently below 200 ppm (Supplementary Fig. 2).

Other possibilities still remain, such as uncorrected stray-light video cross-talk, for example, but it will be difficult to make further progress in the absence of high-resolution imaging, which we encourage at this time. However, given our generally conservative approach of seeking reasons to reject moon candidates rather than keep them, sufficient reason for scepticism exists about this object that we do not consider it further as an exomoon candidate.

Vetting of KIC-5351250.06

In vetting the planetary moon candidate of KIC-5351250.06/Kepler-150 f, we begin by noting that the star is unusually active amongst the sample considered. This is apparent from simple inspection of the light curves but also it has been previously reported35 as a rotationally active star with a periodicity of 17.6 d and an amplitude of 10.9 mmag (approximately 1%). Since the transit depth of Kepler-150 f is ~1.5 mmag, this implies that the area of the stellar surface covered by spots is larger than the sky-projected area of the planetary disk. Accordingly, it is quite possible for the planet to cross over one or more spots during the transit and induce upward flux undulations34 that mimic the signature of star–planet–moon syzygies36.

If the spots are much colder than the photosphere, then the spot crossings can be up to the entire transit depth. In practice, this is somewhat rare for even the most active stars97, requiring both a very cold spot and a perfect alignment of the spot and planetary transit chord98. On this basis, we proceeded with caution given the enhanced possibility of false positives.

To investigate further, we ran a Lomb–Scargle periodogram of the PDC Kepler data, quarter by quarter. As shown in Supplementary Fig. 3, we confirm the ~1% level activity reported previously35 and note that the activity seems greater in Q12 (corresponding to the second transit epoch of Kepler-150 f) versus Q5 (the first epoch). This indicates that spots are more likely to corrupt the second transit than the first.

Inspection of the transit light curve itself, shown in Extended Data Fig. 3, reveals an apparent transit depth change from epoch 1 to 2, going from (1{,}35{0}_{-190}^{+260}) ppm to (1{,}10{0}_{-100}^{+90}) ppm. Closer inspection reveals that the trough of the second transit is not uniformly higher, but rather bounces up and down sporadically—consistent with the behaviour expected for spot crossings98. Given that this transit coincides with an episode of high activity, this begins to cast doubt on the reality of the exomoon signal.

To go further, we fitted just the first epoch in isolation with a planet-only model and then used its maximum-likelihood solution as a template for adding starspots for the second transit. If the planet is passing over a spotty, dark patch—as we hypothesize—then the transit will also be diluted in depth because it is only now blocking out a relatively dim region of the star’s total intensity98. Thus, the second transit is modified in two ways: (1) the addition of a dilution factor, γ, and (2) the inclusion of N spot-crossing events. Since we are not particularly interested in the properties of the spots themselves, just whether they can fit the light curve better than a moon, we adopt a simple heuristic model for the crossings. Specifically, we add on a Gaussian of width σ, amplitude A and central time ({mathbb{T}}), thus meaning we have three parameters per spot.

In total, we regressed four different versions of this model to the second epoch: (1) no spot crossings but a contamination factor (one extra parameter), (2) one spot crossing and the contamination factor (four extra parameters), (3) two spot crossings and the contamination factor (seven extra parameters) and (4) three spot crossings and the contamination factor (ten extra parameters). Since the planet-only model has seven native parameters, the final model includes 17 variables altogether. The results of these fits are shown in Extended Data Fig. 3, along with the fits from the planet-only and planet–moon models.

For the planet-only and planet–moon models, we have been thus far comparing models using the Bayesian evidence. However, here, we seek an alternative model selection method. To see why, consider that in the cases of the planet–moon and planet-only models the model parameters are have physical meaning and thus have well defined parameter limits. For example, the moon’s orbital period is bounded by the inner Roche limit and the outer Hill sphere. In contrast, our heuristic model has no clear bounds on the parameters of interest. Thus, we could just increase the widths of the priors somewhat arbitrarily, which would then dilute the Bayesian evidences. Accordingly, the model selection results become highly subjective when using marginal likelihoods for heuristic models, and we instead prefer to use a model selection metric that compares the maximum-likelihood solutions, for which there is no sensitivity to prior widths.

The two most commonly used maximum-likelihood model comparison metrics are the BIC68 and the Akaike information criterion, or AIC99. The AIC—motivated from information theory—is more appropriate when none of the models are considered truly correct, but we are ranking them in terms of their ability to approximate the truth, which is certainly true for heuristic models. Further, the BIC includes a penalty term, which depends on the number of data points, and this introduces a degree of subjectivity into the model selection process since it depends on how much we window the data around each transit mid-point. For these reasons, we used the AIC to rank these different models.

In doing so, we found that the two-spot model is favoured with AIC improvements versus model (1) of 2.6, 8.0 and 4.9 for models (2) to (4) respectively. For the two-spot model, the χ2 score when computed on both epochs is 702.89, whereas the planet–moon model achieved 712.07. In other words, the two-spot model is a better match to the light curve than the planet–moon model by Δχ2 = 9.2, despite using the same number of free parameters.

At this point, we could go further and introduce astrophysical spot models, coupled to the rotational modulations, but for the purposes of this work—seeking exomoons—this is simply beyond the scope of our objectives. Although we cannot fully reject the hypothesis of an exomoon, for the reasons described there is now sufficient basis to reject this particular candidate as a compelling object.

Robustness of KIC-7906827.01’s moon signature against detrending choices

A possible concern with any claimed moon-like signal is that it is sensitive to the choices of detrending method used to process the data. In this work, we use the method-marginalized light curves, computed as described earlier, for the model comparison tests. Since this uses the median of eight different light-curve detrendings, it is possible that the signal is present in the majority of these, but not all. This does not necessarily indicate that the moon-like signal is spurious, but it would certainly motivate a deeper investigation as to why this is happening and increases the possibility of a spurious origin.

We therefore decided to inspect the individual detrendings for evidence that the signature of the exomoon candidate was not a global feature. This is complicated by the fact that the moon-like signal is not a single event, but rather presents itself in both transits through subtle distortions. Although a visual inspection of the light curve reveals broadly consistent morphologies across all methods (Extended Data Fig. 4), we sought a more quantitative metric to assess this.

To this end, we took the maximum a posteriori fit of the planet–moon model conditioned upon the method-marginalized light curve as a template, and compared it with each of the eight detrended light curves. For reference, we also took the maximum a posteriori planet-only fit. Crucially, we do not refit any of these eight light curves; we simply ask how well these templates agree with the data in hand. In every case, we find that the planet–moon model yields superior agreement, indicating that the specific signature of the hypothesized moon (and not some generic moon) is present in all detrendings. Furthermore, the Δχ2 values obtained are consistent with the value obtained from the method-marginalized light curve (Δχ2 = 23.2), yielding 22.2, 27.1, 23.4, 25.0, 23.9, 31.4, 23.8 and 15.0. This list has a median of 23.9, and a mean of 24.0 ± 4.6—consistent with the value obtained from the method-marginalized light curve.

On this basis, we conclude that the moon-like signature is robust against choice of detrending method.

Pixel-level analysis of KIC-7906827.01

We analysed the pixel-level data of KIC-7906827 to look for anything out of the ordinary that might suggest that the moon-like signal is spurious. To this end, we largely follow the approach outlined in a previous work37, where a putative exomoon around Kepler-90 g was shown to be a likely false positive. This also builds upon the tests already shown for KIC-8681125.01 discussed previously.

We begin by extracting the individual raw light curves of each pixel within the postage stamp of the target and for times directly surrounding the two transits of KIC-7906827.01. Specifically, we extracted light curves of ±2.5 transit durations around the two known events. Each light curve was then detrended using the local method described earlier.

Next, we measure the planet SNR in each pixel by simply calculating the weighted mean of the detrended pixel light curves inside/outside the transit region (where we use the duration as determined from the full planet-only fits found earlier). The s.d. values (divided by the square root of the number of data points in each section) are used to compute an error (through quadrature), which then forms the SNR. The result is illustrated in the middle panel of Extended Data Fig. 5. In comparing with the mean flux counts of each pixel (shown in the left-hand panel of Extended Data Fig. 5), we see good agreement between the location of the highest flux and the location of the highest transit SNR. The planetary transit thus shows no sign of being dislocated from the target or any other strange pixel behaviour.

Turning now to the moon-like signal, we seek to replicate the SNR test, but this is challenged by the fact that the moon signature is not a simple box but rather displays features across the light curve, and in different positions in each transit. The SNR can instead be measured by asking, in each pixel, how much better the maximum a posteriori planet–moon light-curve model template is versus that of the planet-only model. Here, ‘template’ refers to the solution obtained by regressing to the method-marginalized light curves. To quantify what we mean by ‘better’, we evaluate the Δχ2 between the two templates, such that positive numbers indicate that the planet–moon model leads to improved agreement.

As the moon-like signature inherently has much lower SNR than the planetary signal, the SNR map is correspondingly noisier, but it clearly shows a concentration of the SNR on top of the target, as expected for a genuine signal. We highlight that it is precisely this point that the moon candidate of Kepler-90 g failed to pass37. On this basis, we find no evidence in the pixel-level data to suspect that the moon-like signature (1) is associated with a contaminating offset source, (2) is caused by a global dimming of the detector postage stamp pixels (for example due to stray light) or (3) is caused by a single pixel triggering a false positive through anomalous behaviour.

Centroid analysis of KIC-7906827.01

From the KIC-7906827 fits files, we extracted the flux-weighted centroid columns and inspected the time-series behaviour of the X and Y positions within the vicinity of the two transit epochs of KIC-7906827.01. Masking the transits themselves, and filtering on only data within six transit durations of the eclipses, we fitted a series of polynomials through the centroids of increasing complexity. Scoring with the BIC68, we identified the most favourable model for each transit in both X and Y and used this to remove the long-term trend caused by pointing drift.

We then evaluated the mean position in and out of the transit event, using the s.d. to estimate uncertainty, to find that the centroids exhibit a {−0.52 ± 0.06, +0.62 ± 0.05} millipixel shift in the {X, Y} directions (Supplementary Fig. 4). Given the presence of nearby stars observed by Gaia, a centroid shift of some kind is not surprising but it can also indicate that the transit is not associated with the target39—which would open the door to KIC-7906827.01 being a false-positive planet.

To investigate the possibility that one of the other known stars was in fact the host, we created and modelled difference images for the high-SNR transit events in quarters 8 and 16 for KIC-7906827. The results of this very strongly show that the observed transit signal cannot be due to any stars in the Gaia catalogue except the target star, KIC-7906827.

We created the difference image as described in previous work39. Assuming that all flux change is due to the transit event, the difference image will show a star-like image at the location of the transit signal source. For each quarter, cadences were chosen in the transit event and the pixel values were averaged over these cadences, creating an average in-transit image. The same number of cadences was chosen on both sides of the transit event, and averaged to create an average out-of-transit image. These observed images are compared in Supplementary Figs. 5 and 6. The similarity between the out-of-transit and difference images very strongly indicates that KIC-7906827 is the source of the transit.

Even greater confidence in this arrives via modelling of the point response function (PRF). We modelled the scene using the Kepler PRF and stars returned by a Gaia catalogue cone search with radius 12.8 arcsec as described in earlier work100. This search returned five stars, as dim as gmag = 21.0. The Gaia proper motion-corrected positions of these stars are plotted in all figures. The stars are placed at pixel locations using Kepler’s raDec2Pix code (https://github.com/stevepur/Kepler-RaDex2Pix).

Supplementary Figs. 7 and 8 compare the observed and modelled pixels, demonstrating the quality of the PRF modelling. Supplementary Figs. 9 and 10 compare the observed difference image (top left) with the modelled difference image assuming that the transit is on each of the five stars in the cone search. These simulated difference images were created by subtracting simulated scenes similar to Supplementary Figs. 7 and 8, with the in-transit scene reducing the flux of the modelled star by a fitted depth.

It is clear from Supplementary Figs. 9 and 10 that a transit on the target star is the only one that remotely matches the observed difference image. The other stars in the Gaia catalogue cannot reproduce the observed signal.

While this analysis excludes the possibility that a different known star hosts the transit signal, it does not address the possibility of an unknown star blended with the target. To investigate this, we first measured the position of the target star by performing a multistar PRF MCMC fit to the average out-of-transit image, and the position of the transit signal source by performing a single-star PRF MCMC fit of the difference image. These fits computed the posterior distribution of pixel position and flux for each star consistent with the data, and used a Gaussian likelihood for each pixel with width given by the propagated per-pixel uncertainty of the fitted image. These measurements are differenced to give the distance of the transit source from the target star. Measuring both the target star source and transit source with PRF fitting mitigates possible bias due to PRF error, because the same bias probably occurs for both stars.

The blend probability is computed using equation (14) of Morton’s earlier work101, which gives the probability of a blend that can mimic any planet within 2 arcsec of the target star as a function of the star’s Galactic latitude and Kepler magnitude (caution: the columns in Table 1 of ref. 101, which gives the coefficients for their equation (14), are reversed). For our star, the probability of a planet-mimicking blend within 2 arcsec is 3.08 × 10−4. We compute the 3σ radius of the target star on the basis of the 68th-percentile credible interval from the fit to the difference image, and scale the blend probability by the ratio of the 3σ circle to a 2 arcsec circle.

The results are summarized in Supplementary Table 3 for quarters 8 and 16. The transit depth is recovered by taking the ratio of the fitted fluxes of the difference image to the target star from the out-of-transit image, demonstrating the success of the fit. The transit source is about 70 mas from the target star, which is just over 1σ. The resulting blend probability is about 2.6 × 10−6.

Statistical validation of Kepler-1708 b

Our centroid analysis establishes that the transit signal is associated with the target star and that a blend is highly improbable given current observations. This, in isolation, provides a compelling case that KIC-7906827.01 is a genuine planet. This possibility, often dubbed PRF contamination, dominates the catalogue of known Kepler false positives; for example, 1,587 of the 1,859 false positives identified through ephemeris matching102 to known eclipsing binaries originate from PRF contamination56. However, eclipsing binaries can also occasionally cause false positives without PRF contamination, via column anomalies, cross-talk and reflections102. However, we note that KIC-7906827.01 has already been tested for an eclipsing binary ephemeris match in the aforementioned work56 and no matches were found, further strengthening the case that KIC-7906827.01 is a genuine planet. To finalize this, we took the shape of the transit light curve, in combination with the stellar parameters, to independently validate KIC-7906827.01.

To this end, we used the vespa package developed for precisely this task40. Here, the shape of the transit light curve is compared with a suite of models including both planet and false-positive scenarios, to evaluate the statistical probability of each. The a priori probability of blending, on the basis of the star’s position and fundamental properties constrained from spectroscopy and Gaia, is used to weigh these scenarios appropriately in the final evaluation. One additional piece of information that can be helpful in this task is the existence and upper limit of an occultation event. A long-period planet like this should not produce a detectable occultation, and so its existence would put pressure on the planet hypothesis.

An occultation event is generally expected to be of approximately the same duration as the transit, so we can exploit this feature to provide a non-parametric means of detrending all of the Kepler quarters. Specifically, we use a median filter where the bandwidth is set to three times the transit duration of KIC-7906827.01, which essentially acts as a low-cut filter removing all variability on timescales greater than this threshold. The detrended light curve was then phase folded onto the ephemeris of the transiting planet modulo a half-period shift.

For a circular-orbit planet, the occultation should occur at a folded time of precisely zero. However, orbital eccentricity effects cause the occultation to shift away from zero. Since the eccentricity is unknown, especially if we remain agnostic about whether the transit signal is truly associated with the target star, then the shift is also unknown. Accordingly, we created a uniform grid of possible times across the entire orbit, spaced by one-tenth of the transit duration.

At each grid point, representing a possible time of occultation, we first evaluated the s.d. within an interval equal to the transit duration. This number was then divided by the square root of the number of data points minus one, and thus represents the achievable precision on an occultation event of similar duration to the transit as a function of orbital phase. Although this precision score exhibits fluctuations as a result of data gaps and sampling effects, we find that it centres around a value of 62 ppm. Repeating using the median deviation as a more robust variance estimator yields 59 ppm. If no occultation effect is detected then, we would estimate a 3σ limit of <180 ppm. For much shorter-duration occultation events, this would be overly optimistic though, since the smaller number of data points would inflate the uncertainty. While this essentially approaches infinity for infinitesimal-duration events, we adopt an upper limit of 101/2 times shorter, which corresponds to <330 ppm.

The above explicitly assumes no occultation event, which we have to demonstrate. To this end, we took each grid point and evaluated the SNR of an occultation at 20 different trial durations (0.1 to 2.0 times the transit duration in 0.1 steps). From these, we select the highest-SNR duration as the saved solution and continue to move through the grid of possible occultation times. In this way, a genuine detection would manifest as a high-SNR bump within the grid, where we define the SNR as the mean out-of-occultation minus the in-occultation intensity divided by the uncertainty on that mean (as computed using the s.d.).

For Kepler photometry, eclipses generally need to have SNR > 7 to be considered compelling38, and we find no values near this level. The highest recorded SNR amongst 9,225 realized positions with more than two data points within the interval was 3.0. We thus find no evidence for an occultation event of KIC-7906827.01. If KIC-7906827.01 were a real planet, this would be the expected result, since its long-period nature means it would be far too dim to be detected photometrically. From the grid, we can also estimate an upper limit on the occultation depth in an alternative way. Specifically, we evaluated max(δocc, 0) + 3σocc at each grid point, where δocc is the occultation depth and σocc is the uncertainty. We then evaluated the median of this array and added three times the s.d. of the array. This is technically overkill as a 3σ limit, since we have used a 3σ limit twice, but nevertheless it yields <350 ppm as an upper limit. This is in good agreement with our <330 ppm value from earlier and thus we adopt 350 ppm as a 3σ upper limit in what follows.

Using this constraint with the light curve, stellar atmosphere properties and Gaia parallax, we used vespa to calculate the statistical probability of a false-positive scenario. Eclipsing binary, hierarchical eclipsing binary and blended eclipsing binary scenarios are all highly disfavoured and lead to a planet FPP of 1 in 4,237, or 0.024% (Supplementary Fig. 11). Combining this with the similar independent conclusion from the centroid analysis, we conclude that KIC-7906827.01 is a genuine planet to high confidence and thus refer to it as Kepler-1708 b in what follows.

Exploring the possibility of alternative astrophysical models for Kepler-1708 b-i

The case for an exomoon rests upon the light-curve analysis of the Kepler photometry. In particular, the Bayes factor of 11.9 for the planet–moon model versus the planet-only model drives the exomoon candidacy, as it surpasses the strong-evidence threshold of >10 adopted in this work and recommended by previous works27. Bayes factors are influenced by the likelihood function and the priors. In this case, the priors do not have arbitrary bounds but rather well motivated physical limits (for example the longitude of the ascending node lives on a circle from 0 to 2π rad). Further, the case for an exomoon signal remains compelling when viewed in a purely likelihood-based framework, with a Δχ2 = 23.2 improved fit, indicating a 4.8σ effect. On this basis, we argue that the likelihood function drives this result and is the place where we might rightfully apply sceptical interrogation.

The likelihood function can be wrong in two circumstances: (1) the forward model is wrong; (2) the noise model is wrong. We consider each of these in turn but in this section address the former.

Regarding the forward modelling, the models in question are those of a planet transiting a limb-darkened star versus a planet–moon transiting a limb-darkened star. We might well wonder if some other model is ignored that is truly responsible. In general, the asymmetric and short-term time-variable nature of the transit shape is difficult to explain with some other localized astrophysical effect associated with the planet. For example, a ring system103 would need high obliquity, precession and many times greater physical extent than Saturn’s rings to explain the data. Further, such an extensive ring system would compellingly distort the light-curve-derived mean stellar density from the true value in a manner not observed here104.

We performed an additional check to see if the timing of the two inferred moments of exomoon transit were suspicious or improbable. Exomoon transits should be located close to the planetary event, moving back and forth ostensibly randomly with a range governed by their semimajor axis around the planet. The probability distribution of times is expected to follow an arcsine distribution83. Although we only have two such times available, it is possible to evaluate a P value (‘surprisingness’ score), which might indicate tension with our choice of model (that is, the planet–moon model). To investigate this, we took our maximum a posteriori planet–moon and regenerated the light curve but randomized the phase of the exomoon. Repeating 1,000 times, we were able to determine that the moon transits could have occurred up to ±0.25 d either side of the transit, with a spread broadly following the arcsine distribution as expected. This may be compared with the observed times of exomoon transit minima, of −0.226 d and +0.136 d. Adopting the arcsine distribution, we can evaluate the formal likelihood of obtaining the two observed times, which was ({{mathrm{log}}},{{{mathcal{L}}}}=1.82). To put this number in some context, we repeated the above but drew two random times from the arcsine distribution, evaluated their likelihood and built up a distribution of likelihoods under the null hypothesis. The distribution is shown in Supplementary Fig. 12, where one can see that the real likelihood score sits very close to the centre of the expected distribution and is thus not remotely surprising. Accordingly, the timing of the observed moon transits does not appear suspicious or offer grounds to reject the planet–moon hypothesis.

Aside from a localized effect, the light-curve model could be wrong if some other non-localized phenomena simply coincidentally occurred during the time of transit of Kepler-1708 b. The most obvious example would be a second transiting planet in the system. Given the local window used of ±6.2 d, the probability of this occurring is min(12.4/Pc, 1) (depicted in Supplementary Fig. 13 by the green dashed line) and thus improbable for Pc ≫ 12.4 d. We note that there are no other known planetary candidates or even threshold-crossing events56 reported for this source. Nevertheless, this remains a possibility if the hypothetical planet were simply too small to have been reliably detected by the Kepler pipeline. Given the depth of the observed deviations, the planet would need to be 2.6 R⊕ in radius at some unknown period—so how possible is it that such a planet is hiding in the existing Kepler data?

To explore this possibility, we first regressed a two-planet transit model to the same data as used for the planet-only and planet–moon fits. Note that these data only locally detrend the time series to within ±6.2 d of the transit events—which we dub ({{{{mathcal{D}}}}}_{{{{rm{local}}}}}) in what follows. We let the second planet have an unknown period with a log-uniform prior from 10 to 1,000 d and uniform priors for impact parameters, transit time within the first epoch’s window, and ratio of radii. Using MultiNest to explore the parameter space, the best-fitting solution yielded a χ2 substantially lower than that of the planet–moon model, by Δχ2 = −16.2. Further, the two-planet model is only modestly improved over the one-planet model despite being a nested model with four additional free parameters (thus demanding an improved χ2) with Δχ2 = +7.0. Indeed, this leads to the model having a worse marginal likelihood than the planet-only model, with ({{mathrm{log}}},{{{{mathcal{Z}}}}}_{{{{rm{2-planet}}}}}-{{mathrm{log}}},{{{{mathcal{Z}}}}}_{{{{rm{2-planet}}}}}=-0.94). We note that this is well approximated by evaluating the AIC between the two models using the χ2 difference (yielding −1.05). Thus, we find that the local transit photometry, ({{{{mathcal{D}}}}}_{{{{rm{local}}}}}), does not support the two-planet hypothesis.

We find that the posterior distribution for Pc almost replicates the prior of a log-uniform form. Thus, in any given log P window, we have approximately the same number of posterior samples. Exploiting this, we group the posterior into eight evenly spaced bins in log P space, with approximately 4,000 samples in each window. From these, we evaluate the maximum-likelihood solution amongst the subset. Since the AIC well approximates the marginal likelihood here, we use it to evaluate the Bayes factor as a function of log P at these eight grid points, which we then spline-interpolate to create a continuous function. The result is shown in Supplementary Fig. 13 (red dotted line), where we can see that long-period solutions are in greatest tension with the ({{{{mathcal{D}}}}}_{{{{rm{local}}}}}) data.

The above only uses the photometry local to the Kepler-1708 b transits, but the broader complete Kepler time series would also be expected to exhibit transit signatures if the signal were caused by an interior transiting planet.

To support planet occurrence estimates from the DR25 Kepler planet-candidate catalogue56, the sensitivity for detecting a planet of a given period and radius was previously measured in detail105,106. The planet detection sensitivity was measured through Monte Carlo transit signal injection and recovery experiments107,108. In previous work105, a model was generated for planet detection sensitivity that depends on the stellar properties and noise characteristics of the Kepler flux time series based on fits to the database of transit signal injections. The planet detection sensitivity model can be calculated for any given Kepler target from the data products hosted by the NEA (https://exoplanetarchive.ipac.caltech.edu/docs/Kepler_completeness_reliability.html) and the accompanying KeplerPORTs Python software package (https://github.com/nasa/KeplerPORTs). Example uses of KeplerPORTs, in the context measuring planet occurrence rates, have been previously published109,110,111.

To calculate a planet detection contour for Kepler-1708, we use stellar parameters as updated in this study given in Supplementary Table 2. We adopt stellar limb-darkening parameters for Kepler-1708 (0.428, 0.4356, −0.1019, −0.0394; four-parameter nonlinear limb-darkening parameterization) by adopting the limb-darkening values from the target with the closest match in stellar properties in the DR25 Kepler stellar catalogue54. The photometric noise properties for Kepler-1708 are provided as supplemental columns in the DR25 stellar catalogue. Kepler-1708 was observed for 1,459 d with a duty cycle of 87%. The values of CDPP (combined differential photometric precision112) slope at short and long durations (−0.70477, −0.3524, respectively105) indicate a well behaved flux time-series data series for Kepler-1708 with minimal amounts of non-Gaussian noise. The window function and 1σ depth function data113 for Kepler-1708 were downloaded from the NEA.

The resulting planet detection contour from KeplerPORTs for Kepler-1708 is shown in Supplementary Fig. 14. If an additional planet had the proper inclination to transit, the detection contour provides the probability that a particular signal of a given period and radius would have been classified as a planet candidate in the final DR25 Kepler planet-candidate catalogue. As expected, the detection contour degrades toward small planets, as they have smaller transit depths, and at long orbital periods, as fewer transits are available to stack and enhance the detection SNR. The standard version of KeplerPORTs publicly available represents the recoverability of planet signals due to the Kepler pipeline by itself. An additional reduced sensitivity to planet signals can result from the vetting procedure56,106,114. The results shown in Supplementary Fig. 14 take into account the additional loss of sensitivity due to the vetting procedure. The vetting degradation was measured following a procedure similar to the description in section 4.2 of a previous analysis111.

While Supplementary Fig. 14 depicts the entire range of possible radii, for this study we are most interested in the radius slice corresponding to that necessary to explain the exomoon-like signal, namely 2.6 R⊕. Supplementary Fig. 13 (blue solid line) shows a slice in the detection contour plane at this radius as a function of orbital period. The discrete changes in detection probability (at 10, 60, 100, 200, 400 and 700 d) result because the pipeline detection probability and vetting recoverability probability are fitted independently over orbital period regions. The detection probability model fits are not required to be continuous across orbital period region boundaries.

The results described thus far can be combined to evaluate the overall probability that the observed moon-like deviations were caused by a second, previously undetected transiting planet. This is evaluated by taking the product of the three probabilities described thus far: (1) the probability that the second planet coincidentally transits during the local window used to regress Kepler-1708 b, (2) the probability that the two-planet model better explains the local data versus the one-planet model (in a Bayesian sense) and (3) the probability that a second planet of the required radius evaded detection from the Kepler pipeline. This combined probability as a function of orbital period is shown in solid black in the right-hand panel of Supplementary Fig. 13. As shown, the probability does not exceed 1% in the 10–1,000 d region considered. This result is somewhat overoptimistic in that does assign a prior probability for a such a planet existing in the first place and thus tacitly it is unity in the above calculation. On this basis, we find that the probability of a second transiting planet explaining the observed effects is ≲1%.

Noise properties of the detrended Kepler-1708 b photometry

As discussed in the previous subsection, the likelihood function could be wrong if the forward model is wrong or the noise model is wrong. With the former investigated, we now turn to the latter. The data used to infer the exomoon candidate span ±6.2 d of the two transit events. These data were already prewhitened by virtue of the detrending process described earlier—specifically method-marginalized detrending. Since our likelihood function assumes independent Gaussian noise, the likelihood function adopted (and thus inferences thereafter) would be technically wrong if the noise were not described by independent Gaussian noise.

In reality, no observations are ever purely Gaussian. The removal of instrumental and astrophysical trends cannot ever be a perfect process. However, we hope to reach a state where the non-Gaussian component is much smaller than the Gaussian noise, and thus the likelihood function adopted can be treated as an excellent approximation. In this subsection, we thus investigate to what extent this appears to be true.

Using the method-marginalized time series we excluded the data within ±0.55 d of the times of transit minima, to trim the region where the planet–moon transits occur. The remaining data should now be described by a normal distribution. One of the simplest tests of this is to plot a histogram of the normalized intensities (upper panel of Supplementary Fig. 15). On this same figure, we plot the probability density function of a normal distribution centred on unity with an s.d. governed by the measurement uncertainties (that is, this is not a fit). As expected for Gaussian noise, we find excellent agreement. Further, a Kolmogorov–Smirnov test of the Gaussianity reports a P value of 0.10—consistent with expectation. Finally, the χ2 of these data against a flat-line model is equal to 1,073.8, in close agreement with the expected value given by the number of data points, 1,074. On this basis, the data do indeed appear normal.

Time-correlated noise can be difficult to identify using the tests described thus far and a better evaluation comes from looking at the temporal properties of the noise. Because of the jumps between each data segment, we split the data into four sections; epoch 1 pretransit, epoch 1 post-transit, epoch 2 pretransit and epoch 2 post-transit. The pretransit moon feature in epoch 1 corresponds to approximately eight cadences and thus we first tried a simple autocorrelation test at lag-8 on these four sections, which finds no compelling autocorrelation (P values of 0.54, 0.11, 0.73 and 0.28). We next tried a classic root mean squared binning test, where we bin the data sections into progressively larger bins and evaluate how the scatter evolves. Supplementary Fig. 15 (lower panel) shows the results compared with the expected behaviour of independent Gaussian noise, where again we find no clear evidence for time correlation.

On this basis, we conclude that the detrended time series appears consistent with independent Gaussian noise and thus the adopted likelihood function is appropriate.

FPP of the exomoon signal Kepler-1708 b-i

An advantage of seeking exomoons is that the null hypothesis is well defined and can be injected into real photometric time series. Briefly, we can take the best-fitting planet-only model parameters, generate a template model and inject this into the SAP or PDC photometry as desired, and repeat the same detection process as outlined in this work. This allows us to directly calculate the FPP of detecting an exomoon-like signal.

We emphasize that this is not the same FPP as used earlier when validating Kepler-1708 b. There, the reality of the transit signal was unambiguous, but the underlying cause was uncertain. Here, the situation is somewhat reversed. The reality of the exomoon-like signal is unclear—and the FPP in this section seeks to address this. However, the underlying cause of the signal (assuming it to be true) is not addressed by such a calculation. This has already been addressed earlier, where we concluded that a second unseen transiting planet was the most likely astrophysical false positive, but even this has a probability of ≲1% of explaining the observations. In what follows, we focus on the FPP of the signal itself being astrophysical. Although we refer to this as ‘astrophysical’ hereafter, this is technically somewhat of a misnomer since we are really addressing the probability of time-correlated noise causing a false positive, which could in fact be considered astrophysical if due to stellar activity.

To inject fake planet signals, we took the maximum a posteriori parameters from the planet-only model fits of Kepler-1708 b to define a null-hypothesis template. Quarters 1 to 17 long-cadence data are available for KIC-7906827 as possible times into which to inject the signal. In some of these quarters, we observed discontinuities in the SAP time series (for example, due to pointing tweaks) and we went through and located these, saving to a library file. We then injected two transits of Kepler-1708 b, using the template with the only difference being that τ is randomized. The injection is performed by simply multiplying the SAP and PDC flux values by the template model (=1 outside transits and <1 inside). In this way, the time-correlated noise structure of the data is preserved.

There are several cases where the injections were rejected and reattempted. For example, if one of the transits is injected into a data gap (for example between quarters) the realization was rejected and retried. Our specific criteria for a ‘good’ injection were the following.

1. 1.

   The injected transits must occur more than three transit durations away from the real transit (to avoid signal overlap).

2. 2.

   The injection has to occur at least half a transit duration after the beginning of a quarter’s start time, and at least half a duration before the end of a quarter’s start time.

3. 3.

   The trimmed (see below) transit epoch files have to contain at least 530 data points in each (ensuring injections have ≳90% of the data volume of the real signals).

The real transits were removed from the photometry, excluding data within two transit durations of the best-fitting transit times. The real analysis trimmed the photometry to within six transit durations of the central times, and thus we use the same trimming here. As with the original analysis, if a discontinuity occurs somewhere within the time series of interest, we only consider the continuous segments surrounding the transit in question. Finally, the number of 530 points was chosen since the original data have 591 (Q8) and 586 (Q16) points. We wish to ensure that the injections contain at least 90% of the smallest of these two (586), which in principle is 527 data points. However, we found that out outlier rejection algorithm rejects approximately 2% of the time series, and by this expectation we need 530 points to ensure the 90% threshold.

In total, we created 200 random injections (and thus 400 injected transits). We next applied the same method-marginalized detrending algorithm to all 200, with the only difference being that the GP method was dropped for computational expediency. The method-marginalization algorithm performs a final check for the Durbin–Watson statistic and root mean squared versus bin size behaviour, evaluating a P value against bootstrapped experiments. In some rare cases, this led to an epoch being rejected if none of the methods were able to produce sufficiently whitened time series. If this occurred, and fewer than two detrended transits were outputted, the realization was rejected and restarted with a new random seed.

The detrended light curves were then fitted using MultiNest coupled to LUNA, as before, trying both the planet-only and planet–moon models with identical priors (except that τ is shifted onto the new ephemeris). If the Bayes factor between the two models exceeded 10, denoting strong evidence, it was flagged as a possible candidate as with the real analysis, as shown in Fig. 3. For such cases, of which we found just three instances, the next step was to perform the negative-moon-radius test. Two of the three aforementioned cases (injections 103 and 161) pass this test, and we consider these to be ‘false positives’ from the suite of 200 injections. Their signal shapes are shown in Supplementary Fig. 16.

By requiring these signals to be viable moon candidates, in other words signals that our planet–moon model can explain as being physically sound, not all dips and bumps in the light curve trigger a false positive, only the plausible ones. For example, Fig. 2 shows a small deviation around BJD 2456341 that was not interpretable by our fits as a moon signal and thus does not constitute a false positive by this definition.

On this basis, we conclude that the false-positive rate of the exomoon-like signal of Kepler-1708 b is (1.{0}_{-1.0}^{+0.7})% (uncertainty from counting statistics).

Interpreting the FPP

Given that we looked at 70 exoplanetary candidates in this survey, one success from 70 with a 1% false-positive rate might at first seem to fully explain this event without invoking an exomoon. Certainly, this is a valid concern, and one we share. However, this tacitly assumes that the 1% false-positive rate computed for Kepler-1708 b holds for all of the exoplanetary candidates surveyed, which neither has been demonstrated nor can be reasonably assumed. Each case will have its own bespoke FPP dependent upon its specific time-correlated noise properties. Further, 45 of the 70 surveyed objects have three or more transits (unlike Kepler-1708 b) and thus have to also pass a TTV test, which would lower their false-positive rates by virtue of the extra check.

However, let us assume that the FPP (as caused by time-correlated noise) was indeed 1%, for the sake of making progress. In this case, it is important to stress that while one false positive is not an unexpected outcome, nor is zero false positives. Indeed, the likelihood ratio of the binomial distribution for n = 70 samples and P = 0.01 between X = 1 false positives and X = 0 false positives is 0.707. In other words, it is marginally more likely that a survey of 70 objects would produce no false positives than one. However, even this does not address what we really care about, which is the probability that this is an exomoon given the signal. Let us step back from the ensemble and look at Kepler-1708 b in isolation once again.

Let E stand for ‘has a Kepler-1708 b-i-like exomoon’ (and (bar{E}) means it does not), and Q denotes ‘passes our battery of tests’ (and (bar{Q}) does not). With this notation, we can express the probability we seek as

$$Pr (E| Q)=frac{Pr (Q| E)Pr (E)}{Pr (Q)}$$

(3)

where Pr(E) is the a priori probability of an exoplanetary candidate in our sample having a Kepler-1708 b-i-like exomoon (that is, the underlying occurrence rate of such moons in the target sample). The denominator can be expanded as

$$Pr (Q)=underbrace{Pr (Q| E)}_{begin{array}{c} = {{{rm{TPP}}}}end{array}}Pr (E)+underbrace{Pr (Q| bar{E})}_{begin{array}{c} = {{{rm{FPP}}}}end{array}}Pr (bar{E}),$$

(4)

where we have denoted the FPP and true-positive probability (TPP) explicitly. With some rearranging, we can show

$$frac{Pr (E| Q)}{Pr (bar{E}| Q)}=frac{{{{rm{TPP}}}},Pr (E)}{{{{rm{FPP}}}}(1-Pr (E))}.$$

(5)

The TPP is essentially the completeness, and an accurate assessment is challenged by defining what we even mean by a ‘Kepler-1708 b-i-like exomoon’. However, given that the signal had a 4.8σ significance, we should generally expect a high TPP for such signals (TPP ≃ 1). Put another way, it would be odd if we missed these. A detailed calculation of the TPP is beyond the scope of this work and we argue largely unnecessary for the reasons stated above, as well as the fact that Pr(E) dominates our uncertainty in the calculations. If we set TPP → 1, then we have

$$begin{array}{ll}mathop{lim }limits_{{{{rm{TPP}}}}to 1}frac{Pr (E| Q)}{Pr (bar{E}| Q)}&=frac{1}{{{{rm{FPP}}}}}left(frac{Pr (E)}{1-Pr (E)}right)\ mathop{lim }limits_{Pr (E)ll 1}mathop{lim }limits_{{{{rm{TPP}}}}to 1}frac{Pr (E| Q)}{Pr (bar{E}| Q)}&=frac{Pr (E)}{{{{rm{FPP}}}}}.end{array}$$

(6)

From this, we estimate that (Pr (E| Q) > Pr (bar{E}| Q)) if Pr(E) > FPP. Here, then, if 1% or more of our sample host Kepler-1708 b-i-like exomoons, we should expect that the detected signal is most probably a real exomoon rather than a false positive. This calculation reveals the catch-22 conundrum facing the interpretation of this detection. In isolation, it is not possible to reliably assess the odds that it is real since we do not know the underlying occurrence rate of similarly sized moons around cold Jupiters.

Zooming back out to the ensemble, the total number of detections of Kepler-1708 b-i-like exomoons should be

$$mathop{sum }limits_{i=1}^{70}{{{{rm{TPP}}}}}_{i}Pr (E)+{{{{rm{FPP}}}}}_{i}.$$

(7)

In principle, we could define a likelihood function from this to infer Pr(E) based on our one success and see if it is consistent with zero—which would favour Kepler-1708 b-i being a false positive. However, assuming TPPi ≃ 1 for all 70 is not well motivated here due to the different noise properties of each source, and similarly the FPPs will be distinct, as already discussed.

As this section establishes then, an accurate calculation of the probability of Kepler-1708 b-i being genuine is marred with challenges, stemming from the unknown occurrence rate of exomoons and the individual target FPP/TPP properties. This also extends to considerations of specific parameters of our retrieved fit versus false-positive scenarios. In principle, the FPPs and TPPs could be determined with far more extensive computational runs than done here, although we highlight that this study has already taken several years to complete and leveraged supercomputing time throughout (although not continuously). The enormous computational challenge, human time and CO2 production associated with such an endeavour has to be weighed against the benefits, or the simple act of just reobserving Kepler-1708 b in the future to more straightforwardly (and less ambiguously) address this question.

In conclusion, in considering the exomoon-like signal associated with Kepler-1708 b, we can find no firm grounds to reject it as a candidate at this time. Future supporting evidence could be found by detecting TTVs, predicted in the main text to have an amplitude between 1.2 and 77.0 min (95% confidence). In isolation, this would not be sufficient to confirm the moon due to the possibility of perturbing planets. In practice, we argue that the only real way to confirm/deny the existence of the moon convincingly would be high-precision transits of several future epochs, with the next event due on 24 March 2023 (BJD 2460027.86).

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An exomoon survey of 70 cool giant exoplanets and the new candidate Kepler-1708 b-i

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