An Independent Analysis of the Six Recently Claimed Exomoon Candidates
DDraft version August 11, 2020
Typeset using L A TEX twocolumn style in AASTeX62
AN INDEPENDENT ANALYSIS OF THE SIX RECENTLY CLAIMED EXOMOON CANDIDATES
David Kipping
1, 2 Department of Astronomy, Columbia University, 550 W 120th Street, New York, NY 10027, USA Center for Computational Astophysics, Flatiron Institute, 162 5th Av., New York, NY 10010, USA
ABSTRACTIt has been recently claimed that KOIs-268.01, 303.01, 1888.01, 1925.01, 2728.01 & 3320.01 areexomoon candidates, based on an analysis of their transit timing. Here, we perform an independentinvestigation, which is framed in terms of three questions: 1) Are there significant excess TTVs? 2)Is there a significant periodic TTV? 3) Is there evidence for a non-zero moon mass? We appliedrigorous statistical methods to these questions alongside a re-analysis of the Kepler photometry andfind that none of the KOIs satisfy these three tests. Specifically, KOIs-268.01 & 3220.01 pass none ofthe tests and KOIs-303.01, 1888.01 & 1925.01 pass a single test each. Only KOI-2728.01 satisfies two,but fails the cross-validation test for predictions. Further, detailed photodynamical modeling revealsthat KOI-2728.01 favours a negative radius moon (as does KOI-268.01). We also note that we finda significant photoeccentric for KOI-1925.01 indicating an eccentric orbit of e > (0 . ± . M S /M P < .
4% [2 σ ] at a similar relative semi-major to thatof the Earth-Moon. Keywords: planets and satellites: detection — methods: statistical INTRODUCTIONIt has been recently proposed that six Kepler Ob-jects Interest (KOI) host candidate exomoons in Fox &Weigert (2020). Given the paucity of these objects in theliterature, this would represent a major increase in thenumber of known candidates, as of the time of writing.For this reason, we here provide an independent analy-sis of the moon hypothesis for these six: KOIs-268.01,303.01, 1888.01, 1925.01, 2728.01 & 3320.01.It has been proposed that exomoons could be discov-ered through a myriad of approaches, such as pulsar tim-ing (Lewis et al. 2008), microlensing (Han & Han 2002)and spectroscopy (Williams & Knacke 2004), but thetransit method is somewhat unique in offering the abil-ity to measure the mass and radius of potential moons(see review by Heller et al. 2014). The mass is avail-able by the study of transit timing effects imparted bythe moon upon the planet, which include transit timing
Corresponding author: David [email protected] variations (TTVs; Sartoretti & Schneider 1999), veloc-ity induced transit duration variations (TDV-Vs; Kip-ping 2009a), transit impact parameter induced transitduration variations (TDV-TIPs; Kipping 2009b), andingress/egress asymmetries (Kipping 2011). Whilst allof these are generally present, TTVs typically offer themost detectable signal and are the more commonly cat-aloged timing effect (e.g. see Mazeh et al. 2013).The case for TTVs is strengthened when one consid-ers that they appear common amongst KOIs (Holczer etal. 2016), potentially indicating a large number of unre-vealed exomoons. Indeed, it was recently shown that of2416 KOIs with a model preference for a periodic TTV,2198 of them exhibit TTVs and TDVs consistent withan exomoon (Kipping & Teachey 2020). In that pa-per, amongst the 2198 aforementioned cases, one findsthat KOIs-268.01, 303.01, 1888.01, 1925.01, 2728.01 and3320.01 are indeed all listed in their Table 1 as beingfully consistent with an exomoon. Although, the au-thors refrained from describing these as exomoon candi-dates, nor indeed any of the other 2192 cases. a r X i v : . [ a s t r o - ph . E P ] A ug Kipping
A basic reason for this is that although the TDVswere consistent with an exomoon, no significant detec-tion of them had been made; only often very tentativeevidence for TTVs existed. Certainly, in many fieldsa single type of observational information can be suffi-cient to securely claim a detection, but the unique chal-lenge facing TTVs is that a considerable number of non-exomoon phenomena can equally cause TTVs. Theseinclude, but are not limited to, exotrojans (Ford & Hol-man 2007), parallax effects (Scharf 2007), eccentricityvariations (Kipping 2008), apsidal precession (Jord´an& Bakos 2008), star spots (Alonso et al. 2008), stellarproper motion (Rafikov 1999), planetary in-fall (Hellieret al. 2009), the Applegate effect (Applegate 1992; Wat-son & Marsh 2010), stellar binarity (Montalto 2010),cadence stroboscoping (Szab´o et al. 2013), horseshoecompanions (Vokrouhlick´y & Nesvorn´y 2014), planet-planet conjunctions (Nesvorn´y & Vokrouhlick´y 2014),and, near mean motion resonant planets (Agol et al.2005; Holman & Murray 2005). Planet-planet inter-actions are particularly common, given the abundanceof packed planetary systems found amongst the
Kepler sample (Winn & Fabrycky 2015). On this basis, the ex-istence of TTVs, even periodic TTVs, can be perilousground upon which to solely base an exomoon claim.The six candidates claimed by Fox & Weigert (2020)seem to exhibit TTVs, and thus are indeed consistentwith one observational effect of exomoons then. Fur-ther, the authors selected targets for exomoons that areplausibly dynamically stable, and honed in on the high-est signal-to-noise transits available. That latter pointis particularly important since one might expect theseKOIs to enable particularly sensitive searches for exo-moons. A search for exomoons is thus interesting aroundthese KOIs in its own right.Accordingly, in this work, we will interrogate the claimof Fox & Weigert (2020) for each of the six KOIs arguedto be exomoon candidates. TARGET DATA2.1.
Background
In Fox & Weigert (2020), the authors relied on a cata-log of transit timing measurements presented in Holczeret al. (2016). As the
Kepler light curves upon whichthese transit times are derived are publicly available,and the number of objects is fairly small, we elected toderive our own own transit timing estimates.There are several reasons for doing this. First, theHolczer et al. (2016) transit times are the product of anautomated analysis, which made several approximationsto expedite their calculation. For example, uncertaintiesare assigned using an empirical relation rather than ac- tually being formally determined for each object (Hol-czer et al. 2016). Second, the analysis was conductedprior to the final
Kepler
Data Release, DR25, and thusdoes not use the most up to date reduction of the
Kepler light curves (nor indeed short cadence data where avail-able). Third, the magnitude of the the claim of Fox &Weigert (2020) warrants a careful independent analysisto interrogate their hypothesis.2.2.
Method marginalized light curve detrending
To detrend the
Kepler light curves, we follow the ap-proach of Teachey & Kipping (2018) and detrend thelight curve multiple ways through method marginalizeddetrending. Of the five detrending approaches used inTeachey & Kipping (2018), we use the same set hereexcept we drop the median filtering approach, as it wasfound to be the least reliable in that work (see theirFigure S7).We obtained the Simple Aperture Photometry (SAP)and Pre-Data search Conditioning (PDC) DR25 pho-tometric time series from MAST for each KOI. Short-cadence (SC) data is used with preference over long-cadence (LC), whenever available. We first applied a re-moval of outliers based on any error flags in the fits file,or 4 σ deviations from a 21-point rolling median. Eachtransit epoch is then detrended independently (wherean epoch is centered on the time of transit minimumand spans ± . P P ) with all four algorithms, on boththe SAP and PDC time series, giving a total of eightlight curves per epoch per target (see Appendix for lightcurves).We next generated 1000 fake light curves for eachmethod and epoch assuming pure Gaussian noise, in or-der to test how whitened each light curve is. First, wecomputed a simple Durbin-Watson statistic (Durbin &Watson 1950) and rejected any light curves which ex-hibit autocorrelation more than 2 σ deviant from white,as characterized by the fake light curve population. Sec-ond, we binned the light curves into progressively largerbins, computed an RMS, and then fitted a gradientthrough the log-log plot of bin size versus RMS. Thiswas done for every fake light curve as well as the real,allowing us to again reject any light curves for which thebinning properties are more than 2 σ deviant from whitenoise behaviour.The ≤ N INDEPENDENT ANALYSIS OF THE SIX RECENTLY CLAIMED EXOMOON CANDIDATES ANALYSISThe claim of Fox & Weigert (2020) is that KOIs-268.01,303.01, 1888.01, 1925.01, 2728.01 and 3320.01 are“exomoon candidates”, which is based upon an analysisof the transit times published by Holczer et al. (2016).Exomoons of transiting planets will also transit theirparent star, presenting an additional piece of informa-tion that may be used to infer their presence (e.g. Kip-ping et al. 2013). Nevertheless, we focus on transit tim-ing in what follows since that is the basis upon whichthe claim of Fox & Weigert (2020) was made.To this end, we consider three basic questions for eachof the six KOIs under consideration:
Q1]
Are there statistically significant TTVs?
Q2]
Is there a statistically significant periodic TTV?
Q3]
Do the observations support a statistically signifi-cant non-zero moon mass?In the following three subsections, we tackle each ofthese questions in-turn, and then apply the same teststo a previously announced exomoon candidate, Kepler-1625b in the final part of this section.3.1.
Q1 - Are there significant TTVs?
Inferring the transit times
To derive TTVs, we first modeled the transit lightcurve using the Mandel & Agol (2002) formalism withquadratic limb darkening (using the q - q parameteriza-tion of Kipping 2013), and the light curve integrationscheme of Kipping (2010a) to account for LC smearing.Two versions of this model were considered against thedata. Model P assumes a linear ephemeris character-ized by an orbital period P and reference time of transitminimum, τ . Model T allows for TTVs by giving eachepoch a unique time of transit minimum, τ i .The models are regressed to the method marginalizedlight curves using a multimodal nested sampling algo-rithm, MultiNest (Feroz & Hobson 2008; Feroz et al.2009), providing marginal likelihoods and posterior sam-ples. Priors were set to be uniform for any ephemerisparameters, to within ± . ρ (cid:63) ) used a log-uniform prior from10 − g cm − to 10 +3 g cm − , impact parameter ( b ) wasuniform from 0 to 2, the ratio-of-radii ( p ) was uniformfrom 0 to 1, as were the limb darkening coefficients q & q . Formally, the orbit is circular but the ability for the stellar density to vary effectively allows for eccen-tric orbits since this allows the velocity of the planet tovary. The photometry was modeled with a normal like-lihood function, which is justified on the basis that ourdetrending pre-whitened the data with explicit tests forgaussianity (see Section 2.2).Since model T assigns a unique τ i to each epoch, thiscan lead to short-period planets having a large numberof total free parameters to explore, which impedes pa-rameter exploration. To circumvent this, we segmentedsuch fits into two subsets of ∼
10 epochs, which was nec-essary for KOIs-303.01, 1925.01 & 2728.01.3.1.2.
Comparison to times used by Fox & Weigert (2020)
From this process, we obtained marginalized posteriordistributions for the τ i parameters for each KOI, whichare summarized in the Appendix (Tables 2-7) and madeavailable at this URL. We derived summary statistics foreach epoch by computing the median and ± P . For all KOIs, the bulk of the TTVs points exhibiteddeviations no larger than approximately half an hour,and thus we excluded any points greater than an hour asoutliers. These TTVs are shown in Figure 1, alongsidethose of Holczer et al. (2016) (and used by Fox & Weigert2020) for comparison.There are noticeable differences between our TTVsand those of Holczer et al. (2016). For every KOI ex-cept KOI-1925.01, we find that the ∆ χ improvementof a best sinusoidal fit versus a linear ephemeris is de-creased when using our TTVs (see values inset in panelsof Figure 1) - thus largely attenuating the significanceof any TTVs.3.1.3. The challenge of defining TTV significance
Equipped with our new transit times, let us askwhether there are statistically significant variations - asubtle and non-trivial task. One might consider a met-ric such as the reduced chi-squared, as utilized by Fox &Weigert (2020), but since the model is non-linear thenthat metric is inappropriate (Andrae et al. 2010).One might consider comparing the marginal likeli-hoods evaluated from
MultiNest for models T and P .However, as noted earlier, model T over-parameterizesthe problem here leading to overly conservative esti-mates for model T . Indeed, for all six KOIs, model P would be favoured using this approach. Whilst this over-parameterization is indeed an issue formarginal likelihoods, it’s highly useful for posterior inference, sincethe approach is agnostic as to the cause/shape of possible TTVsand thus lacks any strong model conditionality.
Kipping - - TT V [ m i n s ] Δχ = 11.3 Δχ = 25.1N = 12 KOI-268.01 - - TT V [ m i n s ] KOI-303.01
N = 22 Δχ = 18.0 Δχ = 20.7 - - TT V [ m i n s ] N = 10 Δχ = 13.3 Δχ = 30.4 KOI-1888.01 - - - TT V [ m i n s ] KOI-1925.01
N = 17 Δχ = 63.4 Δχ = 22.2 - - TT V [ m i n s ] KOI-2728.01
N = 22 Δχ = 24.6 Δχ = 37.1 - - TT V [ m i n s ] KOI-3220.01
N = 14 Δχ = 21.8 Δχ = 45.5 Figure 1.
TTVs for the six claimed exomoon candidate hosts of Fox & Weigert (2020), with our own measurements inblack and those of Holczer et al. (2016) in brown. We overplot the best-fitting sinusoid with the χ improvement shown inthe lower-left corner (and similarly for that of Holczer et al. (2016) in the lower-right, although we do not plot the associatedsinusoid). Instead then, one might consider evaluating somestatistical measures on the derived TTVs, such as theBayesian Information Criterion (BIC;Schwarz 1978).However, those numbers are summary statistics derivedfrom a posterior, and thus applying statistical teststo them is a) lossy, and b) demands certain approxi-mate assumptions. It is lossy because when one adoptssummary statistics of a marginalized distribution, oneignores the full, rich detail of the joint posterior shapes.To avoid such losses, it is preferable to make inferenceson the rawest data product which is practical (e.g. seeHogg et al. (2010) for an analogous problem with eccen-tricities), which in our case would be the photometriclight curves. Applying a test like the BIC to summarystatistics is also approximate , because it requires anestimate of the maximum likelihood of a hypothesizedmodel, and if that likelihood is derived from summarystatistics, then some approximation about the likeli-hood function describing those summary statistics isnecessary (e.g. independent Gaussians). 3.1.4.
TTV significance tests with the photometry directly
A better solution, then, is to apply model comparisontests on the light curve products, but to avoid using themarginal likelihood due to the parameterization prob-lem of model T (ultimately that issue is resolved withthe photodynamics analysis in the next subsection). An-other important limitation is that T has no predictivepower for a held-out epoch, and thus we cannot directlyuse cross-validation either, at least for Q1.A basic quantity we can rely on is the maximum likeli-hood of the light curve fits, ˆ L . Since P is, by definition, anested model of the more complex model T , then ˆ L willalways be greater for T - the real question is whether theimprovement outweighs the expense of the additionalcomplexity that model entails. A common tool for as-sessing this is the BIC, given by k log n − L , where k is the number of parameters estimated by the model and N INDEPENDENT ANALYSIS OF THE SIX RECENTLY CLAIMED EXOMOON CANDIDATES n is the number of data points . Whilst ˆ L and n are well-defined, we again run into an obstacle with k . If we set k as the genuine number of free parameters in the model(i.e. with each epoch requiring an additional parame-ter), it will again over-parametrize the problem. The hy-pothesis of Fox & Weigert (2020) is an exomoon, whoseinfluence on the TTVs can be fully parameterized by justsix additional parameters (Kipping 2011), assuming acircular orbit moon as expected due to rapid tidal circu-larization (Porter & Grundy 2011). All six KOIs underconsideration here include more than six transit epochsand thus employ more parameters than is necessary toexplain an exomoon (or indeed a perturbing planet). Onthat basis, we would expect a much more optimistic casefor TTVs by using k T = 6 + k P , and indeed a solutionwell-motivated by our understanding of orbital param-eterization. Accordingly, we argue that tests for TTVsignficance, when a single orbital component is hypothe-sized, should use k T + k P → min[ k T + k P , k P ], wherethe minimum function accounts that in some cases wehave less than 6 epochs.Proceeding as described, we find that that BIC P < BIC T for KOIs-268.01, 303.01, 1888.01 & 3220.01, indi-cating no TTVs, whereas KOI-1925.01 & KOI-2728.01do (see Table 1).We note a peculiarity about the two positive cases,though. Both are examples of where it was necessaryto segment the epochs into two groups, and so we arealso able to apply our statistical tests to each segmentindependently. In doing so, we find that - for bothKOIs - one segment shows positive evidence but theother does not. For example, for KOI-1925.01, we ob-tain BIC P−T = − . . P−T = 30 . − .
6. Sinceexomoons are expected to be strictly periodic signals,it is peculiar for the significance to change versus time,implying a time-dependent amplitude.3.2.
Q2 - Is there a significant periodic TTV?
Having discussed whether the is statistical evidencefor TTVs, we now ask whether there is periodic
TTVembedded, as expected for exomoons (Sartoretti &Schneider 1999).We first note that model T has no predictive capacityfor a missing epoch, since every epoch is defined with a And also note that log is natural, unless stated otherwise. unique τ i independent of the others. As a result, cross-validation - a powerful tool for model selection - is notpossible. Although cross-validation cannot be appliedto model T directly, there is a way one can employ it.To do so, we work with the marginalized transit timesproduced by model T , rather than the original photom-etry. This is less preferable for reasons described ear-lier, but by doing so we can propose a simple sinusoidalmodel against the derived transit times and use cross-validation to assess its merit. In particular, we proposethe following 5-parameter model for the transit times: τ ( i ) = τ + iP + A TTV sin( ν TTV i + φ TTV ) , (1)where the TTV subscript terms control the sinusoidalfeature of the model. Note, that we applied our model tothe transit times, not a list of TTVs. TTVs are definedas deviations from a linear ephemeris, whose parametersare themselves uncertain and indeed degenerate with thesinusoid, especially for slow ν TTV .Of course, one could use this model on the photometryitself too (e.g. see Ofir et al. 2018). However, cross-validation generally varies the choice of training andhold-out sets, performing many realizations and theninspecting the ensemble for the purposes of model com-parison. Since our the photometric fits take around aweek to complete on ∼
200 cores, it is not practical toexplore this approach in a reasonable time frame.For our cross-validation, we defined a 20% hold-out setfrom the available epochs. We then took the 80% train-ing set and ran a weighted Lomb-Scargle periodogram(Lomb 1976; Scargle 1982) uniform in frequency. Weselected the lowest- χ period and record the associatedparameters. We also performed a second fit with a sim-ple linear ephemeris as the null model. We then appliedboth models to the hold-out set and ask which one leadsto the best prediction in a χ -sense . We then repeatedthe entire process, choosing another random group ofhold-out data, and continue 10 times.The cross-validation results are listed in Table 1. Asummary is that that none of the KOIs yield cross-validation results where more than half of the sinusoidalpredictions out perform the linear ephemeris model,with the exception of KOI-303.01, which is marginal at54%. However, KOI-303.01 was found earlier to not sta-tistically favour the existence of TTVs in a more generalsense. This is because a) the cross-validation results aremarginal here and give almost even weight to the com-peting hypotheses, and, b) the earlier test treats the This implicitly means we approximate the transit time poste-riors as being Gaussian.
Kipping
Table 1.
Statistical tests for evidence for exomoons, using gravitational effects only. Answers to the three questions posed atthe start of Section 3 are provided in columns 2, 3 & 4, where the first number denotes the statistic used to assess each question,and the mark in square brackets is a simple yes/no summary of the posed question. The final column gives a mass ratio upperlimit derived. We also show the same tests for Kepler-1625b (Teachey & Kipping 2018), although the cross-validation test is notpossible due to the limited number of samples, and a mass upper limit is not provided since this case corresponds to a detection.KOI BIC
P−T [Q1] % of good TTV predictions [Q2] log K M : X [Q3] ( M S /M P ) at 60 R P [2 σ ]268.01 − . X ] 0% [ X ] − . X ] < . − . X ] 54% [ (cid:88) ] − . X ] < . − . X ] 17% [ X ] +0 . (cid:88) ] < . . (cid:88) ] 41% [ X ] − . X ] < . . (cid:88) ] 28% [ X ] +0 . (cid:88) ] < . − . X ] 40% [ X ] − . X ] < . . . degrees of freedom as being equal to that of an orbitingmoon, but here the dimensionality is more restricted.KOIs-1925.01 & 2728.01 are worth commenting onsince those appeared to exhibit significant TTVs (seeSection 3.1). As noted in the previous subsection, thecase for TTVs seems disparate between the first/secondhalves of the data set for both objects and indeed thepoor cross-validation results make sense in this context.If there are stochastic TTVs (e.g. due to stellar activ-ity), a deterministic model such as a sinusoid or exo-moon will indeed fail to make useful predictions, despitethe fact that large and significant variations exist.3.3. Q3 - Are there moon-like timing variations?
The third and final question requires a model for thedynamical effect of exomoons on the observations. It isnot enough for a KOI to exhibit some kind of TTVs, ora periodic TTV signal. This is because exomoons pro-duce more subtle and complex effects into the light curvethan the approximate theory of Sartoretti & Schneider(1999), Kipping (2009a) or Kipping (2009b). As explic-itly noted in Kipping (2011), expressions for the TTV(and TDV) waveform caused by an exomoon, are ap-proximate and depend upon several assumptions. Forexample, the moon and planet are assumed to experi-ence no acceleration during the transit duration, whichrequires that P S (cid:29) T . Given that the KOIs in ques-tion have durations up to (cid:39)
12 hours, this implies moonsless than few days orbital period would fail this criteria.Further, exomoons induce other dynamical effects on thelight curve besides TTVs - such as TDV-Vs (Kipping2009a), TDV-TIPs (Kipping 2009b) and ingress/egressasymmetry (Kipping 2011). Whilst TTVs are a soundplace to start an investigation, a detailed considerationof exomoon candidacy should - in our opinion - considerthe full details of the hypothesized model.To address this then, we recommend a photodynami-cal analysis of the light curve, which allows us to a) use full photometric time series, rather than lossy derivativeproducts; and b) fully model the subtle effects exomoonscan impart on the light curve.Photodynamics models the light curve at each timestep by evolving a N -body system and calculating thefraction of stellar flux occulted to create a light curve(e.g. see Barros et al. 2015; Almenara et al. 2018;Borkovits et al. 2019). In this work, we use the LUNA al-gorithm (Kipping 2011) which is optimised for exomoonfits and extends the Mandel & Agol (2002) formalism.The claim of Fox & Weigert (2020) is that these sixKOIs exhibit transit timing effects indicative of an ex-omoon. Transit timing effects are only sensitive to themass of an exomoon, not its radius; and thus, if theclaim of Fox & Weigert (2020) holds, then there shouldbe some positive evidence for a non-zero exomoon mass.The Fox & Weigert (2020) claim does not address exo-moon radius and so, even though that can be includedin our photodynamical model, we leave its inferred valueaside for the time being and focus on the photodynam-ically inferred exomoon mass.Our moon model included the seven parameters frommodel P ( P , τ , p , b , ρ (cid:63) , q & q ) as well as seven ad-ditional satellite (“S”) parameters ( M S /M P , R S /R P , a S /R P , P S , φ S , cos i S , Ω S ). Note, that only six of thesepertain to the gravitational influence on the planet, andwere thus counted as penalized terms earlier in Sec-tion 3.1, since TTVs are not functionally dependent on R S /R P . We adopted uniform priors for all terms exceptfor P S , which has a log-uniform prior, and consider or-bits out to 100 planetary radii. Models were regressedto the light curve using MultiNest , as before.If there are statistically significant transit timing ef-fects (not just TTVs) that were caused by an exomoon,then the exomoon mass in a photodynamical fit wouldfavour a non-zero value. In our fits, we were careful tonot impose any constraint on the exomoon density so
N INDEPENDENT ANALYSIS OF THE SIX RECENTLY CLAIMED EXOMOON CANDIDATES 𝒫 = 20.2lim M S /M P → π = 1lim M S /M P → M S / M P p r o b . d e n s i t y ( π / 𝒫 ) = 20.2lim M S /M P → K 𝒳 : ℳ = 1/20.2log K 𝒳 : ℳ = -3.0 Bayes factorzero-mass moon model moon model
Figure 2.
Example of how we calculated the SD ratio ofthe zero-mass moon model, here for the case of KOI-1925.01.The histogram is calculated from the marginalized posteriordistribution, and this is generalized to a continuous functionusing KDE (red line). We then evaluated the density in thelimit of zero mass and compare it to that of the prior. Forexample, here we find a 20:1 Bayes factor in favor of a zero-mass moon model. that the posteriors can explore masses tending to zerowithout penalization for unphysical satellite densities.The planetary density, derived using the method of Kip-ping (2010b), is constrained to be 0.03 g cm − < ρ P <
150 g cm − to prevent the code from exploring unphysi-cal combinations of P S and a SP .Mass is a positive definite quantity leading to tradi-tional measures, such as the median, to be become pos-itively skewed, and thus posing a challenge to straight-forwardly assessing its significance away from zero. Toresolve this, one might first consider using somethinglike a Lucy & Sweeney (1971) test, but a more rigorousBayesian approach is decribed in Jontof-Hutter et al.(2015) via the Savage-Dickey (SD) ratio Dickey (1971),and we follow that approach here. We evaluated the SDratio by comparing the posterior density at M S /M P = 0versus the prior (uniform) with an example illustratedin Figure 2.The SD ratio allows for an estimate of the Bayes fac-tor for nested models. Here, then, we compare the orig-inal full moon model, dubbed model M , against thesame model but with no mass effects (dubbed X ). Sincethe Fox & Weigert (2020) claim concerns transit timingeffects due to an exomoon, implaying an non-zero ex-omoon mass, then this act directly evaluates the casefor their claim in a Bayesian framework with a self-consistent, photodynamical model.Table 1 shows the results of this exercise, where wefind a preference for zero-mass moon models for KOIs-268.01, 303.01, 1925.01 & 3220.01 and very marginalpreferences for a positive mass for KOI-1888.01 andKOI-2728.01. 3.4. Other insights from the photodynamical fits
Some other notable aspects of the results are brieflydiscussed. For KOIs 268.01, 303.01, 2728.01 & 3220.01,the agreement between the light curve stellar densityand that from an isochrone analysis are within 2 σ . ForKOI-1888.01, it’s a little worse at 3 σ . But for KOI-1925.01 the difference is pronounced, with the log of theratio between them found to be log( ρ (cid:63), LC /ρ (cid:63), isochrones ) =2 . ± .
3, implying a minimum orbital eccentricity viathe photoeccentric effect (Dawson & Johnson 2012) of0 . ± .
06 - which would pose a significant challengefor an exomoon due the truncation of the Hill sphereat periapse (Domingos et al. 2006). We also verifiedthis by taking the results from model T , evaluating aKDE of each segment’s density ratio posterior, takingthe product of the two, numerically normalizing, andthen evaluating the median and standard deviation togive log( ρ (cid:63), LC /ρ (cid:63), isochrones ) = 2 . ± .
2. On this basis,we assert with confidence that the densities are in ten-sion for KOI-1925.01 and the object likely maintains aneccentricity in excess of 0 . Application to Kepler-1625b
For completion, we decided to apply these tests todata used to claim an exomoon candidate by Teachey& Kipping (2018). The results are shown in Table 1.Kepler-1625b passes Q1 and Q3 but we are able to evalu-ate Q2. The reason for this is that with just four epochs,the act of regressing the five parameter model given byEquation (1) to the data leads to an over-determinedsystem. This is exacerbated if we drop an epoch forcross-validation purposes. Nevertheless, we find that theresults of the tests described here pose no challenge tothe candidacy of Kepler-1625b i. DISCUSSIONIn this work, we have conducted an independent exam-ination of the claim of Fox & Weigert (2020) that KOIs-268.01, 303.01, 1888.01, 1925.01, 2728.01 and 3320.01are “exomoon candidates”. As the claim is based ontransit timing effects only, we have primarily framed ourinvestigation in those same terms. This is achieved by using the Gaia DR2 parallax,
Kepler mag-nitude and Mathur et al. (2017) DR25 stellar atmospheric prop-erties of each KOI into isochrones (Morton 2015).
Kipping (cid:1)(cid:1)(cid:2)(cid:2) - - - a SP [ R P ] M S / M P KOI-268.01 KOI-303.01KOI-1888.01 KOI-1925.01KOI-2728.01 KOI-3220.01 (cid:1)(cid:1)(cid:2)(cid:2) - - - a SP [ R P ] M S / M P (cid:1)(cid:1)(cid:2)(cid:2) - - - a SP [ R P ] M S / M P (cid:1)(cid:1)(cid:2)(cid:2) - - - a SP [ R P ] M S / M P (cid:1)(cid:1)(cid:2)(cid:2) - - - a SP [ R P ] M S / M P (cid:1)(cid:1)(cid:2)(cid:2) - - - a SP [ R P ] M S / M P (cid:1)(cid:1)(cid:2)(cid:2) - - - a SP [ R P ] M S / M P (cid:1)(cid:1)(cid:2)(cid:2) - - - a SP [ R P ] M S / M P (cid:1)(cid:1)(cid:2)(cid:2) - - - a SP [ R P ] M S / M P (cid:1)(cid:1) (cid:2)(cid:2) - - - a SP [ R P ] M S / M P (cid:1)(cid:1) (cid:2)(cid:2) - - - a SP [ R P ] M S / M P (cid:1)(cid:1) (cid:2)(cid:2) - - - a SP [ R P ] M S / M P σ σ σ Figure 3.
Mass limits to exomoons for KOIs-268.01, 303.01, 1888.01, 1925.01, 2728.01 & 3320.01. Although we find noevidence for exomoon candidates, the high signal to noise of these transits permits for strong upper limits. We denote theposition of Pluto-Charon and the Earth-Moon on the diagram for context.
We structure our investigation in terms of three basicquestions: 1) Are there significant TTVs? 2) Is there asignificant periodic TTV? 3) Is there a statistically sig-nificant non-zero exomoon mass? It’s worth noting thatthe third criterion is a standard test used by the “Huntfor Exomoons with Kepler” (HEK) project, namely cri-terion B2a (Kipping et al. 2013, 2015). Rather than relyon the catalog transit times of Holczer et al. (2016), weelected to infer our own times using method marginal-ized detrending of the latest
Kepler data products andincorporating short-cadence time series where available.The results of these three questions/tests are summa-rized in Table 1. We find that KOIs-268.01 & 3220.01result in a “no” for all three questions. KOIs-303.01,1888.01 & 1925.01 pass a single test each, although adifferent one in each case. The analysis of this workthus concludes that these five KOIs are not exomooncandidates. Only KOI-2728.01 passes two of the three, failing thecross-validation test when we ask if the periodic TTVhas predictive capability. Specifically, when we split thetransit times into an 80:20 training:holdout set, we findthat the hypothesis of a periodic sinusoid defeats thepredictions of the null hypothesis (a linear ephemeris)in only 28% of the draws. One explanation would bethat the TTVs are significant but are stochastic, per-haps caused by stellar activity (Alonso et al. 2008), thusfailing the periodic prediction test. As an additionalpoint of concern, KOI-2728.01 favours a negative-radiusmoon when fit with a photodynamical exomoon model.On this basis, we do not consider there to be a good casefor KOI-2728.01 being an exomoon candidate.It is important that we continue to search for exo-moons, but they are unquestionably very challenging ob-jects to detect; not only at the hairy edge of
Kepler ’s sen-sitivity, but also plagued by a myriad of false-positiveswhen considering a single observable quantity, such as
N INDEPENDENT ANALYSIS OF THE SIX RECENTLY CLAIMED EXOMOON CANDIDATES
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N INDEPENDENT ANALYSIS OF THE SIX RECENTLY CLAIMED EXOMOON CANDIDATES
Figure 4.
Method marginalized detrended light curves for KOI-268.01 and KOI-303.01. The eight colored continuouslines show the eight different independent detrendings of each epoch, which are then combined together to form the methodmarginalized time series (black points). Kipping
Figure 5.
Method marginalized detrended light curves for KOI-1888.01 and KOI-1925.01. The eight colored continuouslines show the eight different independent detrendings of each epoch, which are then combined together to form the methodmarginalized time series (black points).
Table 2.
Transit timing of KOI-268.01 derived in this work using model T . Transit times are quoted as BJD UTC − , , P . Central values rethe median of themarginalized posterior distribution and the uncertainties represents the ± τ i TTV i [mins]0 8 . +0 . − . . +2 . − . . +0 . − . − . +3 . − . . +0 . − . − . +3 . − . . +0 . − . . +3 . − . . +0 . − . − . +1 . − . . +0 . − . . +2 . − . . +0 . − . . +2 . − . . +0 . − . . +1 . − . . +0 . − . . +1 . − .
10 1112 . +0 . − . − . +5 . − .
11 1223 . +0 . − . − . +3 . − .
12 1333 . +0 . − . . +7 . − . N INDEPENDENT ANALYSIS OF THE SIX RECENTLY CLAIMED EXOMOON CANDIDATES Figure 6.
Method marginalized detrended light curves for KOI-2728.01 and KOI-3220.01. The eight colored continuouslines show the eight different independent detrendings of each epoch, which are then combined together to form the methodmarginalized time series (black points). Kipping
Table 3.
Transit timing of KOI-303.01 derived in this work using model T . Transit times are quoted as BJD UTC − , , P . Central values rethe median of themarginalized posterior distribution and the uncertainties represents the ± τ i TTV i [mins]0 6 . +0 . − . − . +3 . − . . +0 . − . − . +3 . − . . +0 . − . − . +2 . − . . +0 . − . . +3 . − . . +0 . − . − . +3 . − . . +0 . − . . +2 . − . . +0 . − . . +2 . − .
10 615 . +0 . − . . +2 . − .
11 676 . +0 . − . − . +5 . − .
12 737 . +0 . − . − . +3 . − .
13 798 . +0 . − . . +3 . − .
14 859 . +0 . − . . +2 . − .
15 920 . +0 . − . . +3 . − .
16 981 . +0 . − . − . +2 . − .
17 1042 . +0 . − . − . +2 . − .
18 1103 . +0 . − . . +2 . − .
19 1164 . +0 . − . − . +2 . − .
20 1224 . +0 . − . − . +2 . − .
21 1285 . +0 . − . . +2 . − .
22 1346 . +0 . − . − . +3 . − .
23 1407 . +0 . − . . +2 . − . Table 4.
Transit timing of KOI-1888.01 derived in this work using model T . Transit times are quoted as BJD UTC − , , P . Central values rethe median of themarginalized posterior distribution and the uncertainties represents the ± τ i TTV i [mins]0 − . +0 . − . − . +5 . − . . +0 . − . . +7 . − . . +0 . − . . +5 . − . . +0 . − . − . +4 . − . . +0 . − . . +4 . − . . +0 . − . . +4 . − . . +0 . − . . +5 . − .
10 1167 . +0 . − . − . +4 . − .
11 1287 . +0 . − . − . +5 . − .
12 1407 . +0 . − . . +5 . − . N INDEPENDENT ANALYSIS OF THE SIX RECENTLY CLAIMED EXOMOON CANDIDATES Table 5.
Transit timing of KOI-1925.01 derived in this work using model T . Transit times are quoted as BJD UTC − , , P . Central values rethe median of themarginalized posterior distribution and the uncertainties represents the ± τ i TTV i [mins]0 12 . +0 . − . − . +3 . − . . +0 . − . − . +4 . − . . +0 . − . . +3 . − . . +0 . − . . +7 . − . . +0 . − . . +3 . − . . +0 . − . . +4 . − . . +0 . − . . +3 . − . . +0 . − . − . +3 . − . . +0 . − . − . +3 . − .
10 701 . +0 . − . . +5 . − .
13 908 . +0 . − . . +2 . − .
14 977 . +0 . − . − . +1 . − .
15 1046 . +0 . − . . +1 . − .
16 1115 . +0 . − . − . +1 . − .
17 1184 . +0 . − . . +2 . − .
18 1253 . +0 . − . . +2 . − .
19 1322 . +0 . − . . +1 . − . Kipping
Table 6.
Transit timing of KOI-2728.01 derived in this work using model T . Transit times are quoted as BJD UTC − , , P . Central values rethe median of themarginalized posterior distribution and the uncertainties represents the ± τ i TTV i [mins]2 49 . +0 . − . . +7 . − . . +0 . − . . +7 . − . . +0 . − . − . +6 . − . . +0 . − . − . +6 . − . . +0 . − . − . +5 . − . . +0 . − . − . +7 . − . . +0 . − . . +6 . − .
12 472 . +0 . − . . +7 . − .
13 515 . +0 . − . . +8 . − .
15 599 . +0 . − . . +5 . − .
16 642 . +0 . − . . +6 . − .
17 684 . +0 . − . . +8 . − .
18 726 . +0 . − . . +6 . − .
21 854 . +0 . − . . +9 . − .
23 938 . +0 . − . − . +6 . − .
24 981 . +0 . − . . +5 . − .
25 1023 . +0 . − . . +5 . − .
26 1065 . +0 . − . . +6 . − .
31 1277 . +0 . − . . +4 . − .
33 1362 . +0 . − . − . +4 . − .
34 1404 . +0 . − . − . +6 . − . Table 7.
Transit timing of KOI-3220.01 derived in this work using model T . Transit times are quoted as BJD UTC − , , P . Central values rethe median of themarginalized posterior distribution and the uncertainties represents the ± τ i TTV i [mins]0 − . +0 . − . − . +4 . − . . +0 . − . − . +3 . − . . +0 . − . . +6 . − . . +0 . − . . +4 . − . . +0 . − . . +4 . − . . +0 . − . − . +4 . − . . +0 . − . − . +3 . − .
10 808 . +0 . − . . +3 . − .
11 889 . +0 . − . . +4 . − .
14 1133 . +0 . − . − . +3 . − .
15 1215 . +0 . − . − . +3 . − .
16 1296 . +0 . − . . +3 . − .
17 1377 . +0 . − . . +4 ..
17 1377 . +0 . − . . +4 .. − ..