Signals of exomoons in averaged light curves of exoplanets
aa r X i v : . [ a s t r o - ph . E P ] O c t Mon. Not. R. Astron. Soc. , 1–9 (2010) Printed 12 August 2018 (MN L A TEX style file v2.2)
Signals of exomoons in averaged light curves of exoplanets
A. E. Simon , ⋆ , Gy. M. Szab´o , † , L. L. Kiss , and K. Szatm´ary Konkoly Observatory of the Hungarian Academy of Sciences, PO. Box 67, H-1525 Budapest, Hungary Department of Experimental Physics and Astronomical Observatory, University of Szeged, 6720 Szeged, Hungary Sydney Institute for Astronomy, School of Physics A28, University of Sydney, NSW 2006, Australia
Accepted Received; in original form
ABSTRACT
The increasing number of transiting exoplanets sparked a significant interest indiscovering their moons. Most of the methods in the literature utilize timing analysis ofthe raw light curves. Here we propose a new approach for the direct detection of a moonin the transit light curves via the so called Scatter Peak. The essence of the method isthe evaluation of the local scatter in the folded light curves of many transits. We testthe ability of this method with different simulations: Kepler “short cadence”, Kepler“long cadence”, ground-based millimagnitude photometry with 3-min cadence, and theexpected data quality of the planned ESA mission of PLATO. The method requires ≈
100 transit observations, therefore applicable for moons of 10-20 day period planets,assuming 3-4-5 year long observing campaigns with space observatories. The successrate for finding a 1 R Earth moon around a 1 R Jupiter exoplanet turned out to be quitepromising even for the simulated ground-based observations, while the detection limitof the expected PLATO data is around 0.4 R Earth . We give practical suggestions forobservations and data reduction to improve the chance of such a detection: (i) transitobservations must include out-of-transit phases before and after a transit, spanningat least the same duration as the transit itself; (ii) any trend filtering must be done insuch a way that the preceding and following out-of-transit phases remain unaffected.
Key words: planetary systems — planets and satellites: general — techniques:photometric — methods: numerical
The number of known transiting exoplanets is rapidly in-creasing, which has recently inspired significant interest asto whether they can host a detectable moon. Although therehas been no such example where the presence of a satellitewas proven, several methods have already been investigatedfor such a detection in the future. The most important meth-ods evaluate the timing of transits, e.g. barycentric TransitTiming Variation, TTV Sartoretti & Schneider 1999, Kip-ping 2008, photocentric Transit Timing Variation, TTVp,Szab´o et al. 2006, Simon et al 2007, Transit Duration Vari-ation, TDV, Kipping, 2009, Time-of-Arrival analysis of pul-sars, Lewis et al. 2008). There are further photometric meth-ods for observing rings of exoplanets (Ohta et al. 2009, DiStefano et al. 2010) starspots in transits (Silva 2003, Silvaet al. 2010 and references therein), or even, transits of alienspacecrafts (Arnold, 2005).Here we propose a photometric method for the detection ⋆ E-mail: [email protected] † E-mail: [email protected] of the moon directly in the raw transit light curves. Whenthe moon is in transit, it puts its own fingerprint on the in-tensity variation. In realistic cases, this distortion is too littleto be detected in the individual light curves. Simply takingthe boxcar average of a folded light curve that consists ofmany transits, is not a powerful solution because it resultsin a significant amount of correlated (“pink”) noise. Thesmooth variation of this correlated noise can mimic/hide thereal distortions of the light curve due to the moon. Here weintroduce the scatter of the folded light curve as an appro-priate estimator for the presence of a moon. The stability ofthe method relies on its robust nature, i.e. the scatter willbe estimated in a boxcar that is comparable, or even longer,than the transit duration.Here we show that a careful analysis of the scatter curveof the folded light curves enhances the chance of detectingthe exomoons directly. Our aim is to present a detectiontechnique that is very specific, i.e. when the test is posi-tive, the presence of an exomoon is probable. With carefulpre-processing of the light curves (e.g., by recentering thetransits) signals that can mimic exomoons are largely sup-pressed. Consequently, the Scatter Peak method can be con- c (cid:13) A. E. Simon, Gy. M. Szab´o, L. L. Kiss and K. Szatm´ary ∆ t a mm a ν R star −a t t −D/2 p p t +D/2 moon planetm Observer x Figure 1.
Transit geometry of a star–planet–satellite system. sidered both as a tool (i) for quickly finding systems thatwarrant more detailed analyes and (ii) for confirming thepresence of an exomoon when suspected from TTV and/orTDV analyses.
Here we describe a very simple model to illustrate the con-cept of the Scatter Peak. For the sake of clarity, we considera special configuration (Fig. 1) that can be handled ana-lytically. Generic transit light curves with a moon will beexamined later in numerical simulations.Let us assume a moon on circular orbit, and therefore,we a priori know the shape of the light curve componentof the moon (it is similar to that of the planet in shapeand duration, but with shallower transit). In this case, theorbital inclination of the moon is 90 ◦ . We assume furtherthat the moon orbits slowly, i.e. the moon – planet geometrydoes not change significantly during the transit while thetransits of the moon appear somewhat earlier or later thanthe planet’s transit. We also make use of the knowledge thatTTV was initially removed from data by transforming thetime (i.e. shifting the derived transit to the predicted valueby recentering, Sect. 4.1).Let f ( x ) be the density function of x , the moon’s pro-jected position. Here x := a m ∗ sin ν , where ν is the anomalyand a m is the semi-major axis of the moon (Fig. 1). Be-cause the orbit is circular, ν follows uniform distribution.After some calculus (see Appendix for the details) we getthe density function of x which is f ( x ) dx ∝ √ a m − x dx. (1) The projected distribution of the moon around the planetfollows an / √ a m − x distribution. Because both the transit light curve of the moon, lc ( t )and the averaged light curve is a function of time, f ( x ) dx must be rewritten to time domain. If the projected positionof the moon is x apart from the planet, the time delay be-tween the transits of the moon and the planet is x/v orb ,where v orb is the orbital velocity of the planet. The ap-propriate transformation to the time domain is therefore∆ t = x/v orb . Here the relative transit time of the moon∆ t = t − t p , where t p is a time of the transit of the planet.With this notation, the distribution of the transit time ofthe moon is f (∆ t ) dt ∝ p ( a m /v orb ) − (∆ t ) dt. (2)In every case, when the moon is observed in an indi-vidual light curve, the occulted flux will be the sum of thetransit light curve of the planet centred at t p , and the tran-sit light curve of the moon, centred at t p + ∆ t . The lightcurve of a single event is a convolution of the transit lightcurve with two Dirac delta functions with different weights,representing the planet and the moon at t p and t p + ∆ t .The average light curve of many events is the expectationflux, taking all ν values into account. At this step, the planetcomponent can be subtracted, and the residual of the moonwill remain alone. Since we average many events, the manyindividual Dirac delta functions representing the moon willfollow the distribution of f (∆ t ), thus the many delta func-tions in the summation can simply replaced by a convolutionwith f (∆ t ). Consequently, the lc m (∆ t ) light curve compo-nents due to the moon will be averaged to lc m (∆ t ), whichis lc m (∆ t ) = f (∆ t ) ⊗ lc (∆ t ) , (3)where ⊗ represents a convolution. The presence of a moon at a given ∆ t transit time follows adistribution with a local probability defined in Eq. 2. Pro-vided that the moon is in fact at the given position, lc (∆ t ),the light curve component associated to the moon will beknown everywhere. From a set of ∆ t positions distributed ac-cording to Eq. 2, one can infer the successive distribution oflight loss at each generic time τ . In the general case, this dis-tribution will be of the multinomial family, with a non-trivialshape (i.e. if the transit parameters are such that ingress-egress phases are shallower/steeper, there will be more/lessprobability to detect just little light hidden by the moon). Ofcourse simulations can easily support parameter-dependentdistributions, but for the theoretical framework it is moreprudent to consider a very simple light curve shape: a sim-ple box with the duration ( D ) and the depth ( δm moon ) asfree parameters.Within this framework, the light occulted by the moonat a generic τ time will be equally δm moon if the moon’sposition is closer to τ than D ( | ∆ t − τ | < D/ δm moon light is occulted by themoon with a probability p expressed by a convolution c (cid:13) , 1–9 ractical suggestion on detecting exomoons Figure 2.
Simulations of 109 transits of an 1 R Jupiter size planet with Kepler short cadence sampling and noise. Left panel: simulationswithout a moon; Right panel: scatter peak of a 1.0 R Earth sized moon. Each column shows the input light curves, the noisified lightcurves, the median filtered light curves, the residuals to the median and the rms scatter of the residuals (the inserts plot the ingressphase of the exomoon; tick-step is 50 ppm). p (∆ t ) = Z τ + D/ τ − D/ f (∆ t ) dτ ≡ f (∆ t ) ⊗ I ( | ∆ t − τ | < D/ τ . Here I ( C ) is the identity function that is1 whereas C is true and 0 elsewhere. With this formulation,the light curve components due to the moon will be bino-mially distributed, and the local scatter of light curves canbe estimated using the standard deviation of the binomialdistribution rms = δm m p p (1 − p ) , (5)which is the scatter curve of the moon’s transit, and δm m is the expected light loss if the single moon is in transit.Because precise measurement of scatter requires theanalysis of many data points, the light curves will be evalu-ated in a very wide long boxcar in practice, to derive precisescatter values. This sampling will behave as a convolutionkernel acting on the local values of the scatter, and finally,the shape of scatter curves will be reduced to a simple widepeak (the “Scatter Peak”) around the transit time of theplanet.In the followings, we examine the Scatter Peak in realsimulations. We made realistic numerical simulations with our image-level simulator (Simon et al. 2009) to examine the discovery probabilities of large exomoons in light curves of differentphotometric qualities. The reliability analysis (Sect. 4.1.) in-vokes an accurate estimation of the light curve scatter andmoreover, the scatter of the scatter. Because these statisti-cal variables are highly fluctuating, there is a demand for ≈
100 transit data for convincing results. Additionally, thesame length of data set is required without a moon, inter-preting the null signal event. This will be the reference inmaking the decision whether the detection of the moon issignificant. Therefore, all subsets incorporated 109 individ-ual transit light curves. Continuous datasets are delivered byspace observatories and even the longest ones from
Kepler will span about 3 to 5 years at most. Hence, the Scatter Peakis restricted for exomoons around planets with P orb . P orb =10 days for the modelplanet. This is also favourable because of the distribution ofthe currently known exoplanets: the majority of them hasorbital periods in this order of magnitude. However, we re-call again that the most relevant parameter is not the periodof the planet, but the number of transits which we are ableto observe.The planet was a hot Jupiter with 0.7 M Jupiter , 1.0 R Jupiter mass and radius on a circular orbit with a planet =0 .
09 AU. The model moon orbited at a moon = 6 . × km,82% of the Hill-radius, and had a period of P moon =4 . c (cid:13) , 1–9 A. E. Simon, Gy. M. Szab´o, L. L. Kiss and K. Szatm´ary prehended. The central star was a solar analogue. Samplelight curves of such a system are shown in the top rows ofFig. 2.Transit light curves of four different qualities were sim-ulated. One dataset represented the best quality ground-based (GB) photometry with 178 second sampling and 0.23mmag standard deviation of the light curve points (0.7mmag error, closely mimicking what has been achieved bySouthworth et al., 2010). Space measurements were repre-sented by Kepler space telescope short cadence (SC) andlong cadence (LC) samplings and the bootstrap noise ofnon-variable stars. The quality of future space observatorieswas represented by the anticipated data quality of ESA’splanned PLATO mission. For the “PLATO” quality datasetwe assumed 25.13 sec sampling and an accuracy of 0.12mmag (data taken from Catala et al. 2011). In such way,16 subsets of light curves were calculated, each representingindividual systems with different moons (0.5 to 1.0 R Earth for ground-based, Kepler LC and SC quality, and 0.4 to 1.0 R Earth for PLATO quality).
The secure detection of a moon relies on four importantsteps. After pre-processing the data, the detection parame-ters have to be fine-tuned, then applied to the observationsand finally, we make a decision on whether the Scatter Peakis significant. The detailed recipe of the entire process is asfollows.
During pre-processing, the transits must be recentered andstretched to have zero TTV and TDV. This is because thereare other sources of TTV and TDV than moons, e.g. per-turbing planets. If the folded light curves are still allowed toexhibit such variation of timings and duration, these eventswill of course result in a Scatter Peak but not due to a moon.However, if TTV and TDV are removed, and the ScatterPeak still survives, one will have an evidence that there areslight variations in the shape of the transit and in its vicin-ity. Some other processes can lead to similar results, i.e.the stochastic occultation of individual starspots by a singleplanet. In suspicion of some process leading to systemic lightcurve variations, the variations must be modeled specificallybefore applying the Scatter Peak evaluation (see also Sect.4.4). However, it has to be stressed again that most scenar-ios with a Scatter Peak but without a moon can be excludedby recentering, therefore, this step of pre-processing is themost important ingredient of the method.In this paper, we used simulations with zero TTV andTDV because the mass of the exomoon was forced to be zeroin our light curve simulator. Thus, all detections reflect thephotometric effects of the moon itself.
The average transit shape is derived from a boxcar median ofthe folded light curves. The length of the boxcar is a sensitiveparameter that must be preset with care. Too little boxcarscontain too few points, thus the scatter in the folded light curve, and the scatter of the scatter cannot be determinedaccurately enough. Too large boxcars, on the other hand,cover a longer part of the light curve with significant lightvariation, therefore, a false Scatter Peak emerges just be-cause the blurred template differs a lot from the measuredlight curve. Moreover, the boxcar size will depend on thelength and sampling of data, and on the parameters of theplanet.In every case, the boxcar length must be set manuallywith numerical experiments. A large number of planet tran-sits must be simulated with the same sampling as the dataand varying noise. Then, the largest boxcar must be definedwhich does not produce a false Scatter Peak with no-moonsimulations in the input.
We experienced optimal boxcar lengths of 249, 25, 749, and1749 photometric points for the GB, LC, SC, and PLATOdata, respectively. This means that the optimal boxcar was ≈
400 seconds long, regardless of the sampling rate. (Thisboxcar size corresponds to 1/2200 orbital phase.) Longerboxcars tend to blur the light curve of the planetary transittoo much, while shorter boxcars give too noisy results. Theuse of median is necessary because the signal is little, andthe possible outliers have to be eliminated effectively. Forsuch data distributions (e.g. Gaussian noise with distortedwings), the median is a more stable estimate than the mean(Lupton, 1995). We have checked the stability of our meth-ods utilizing the mean as the local estimate of light curvesand we indeed experienced that the median is more stable,especially for the length of the boxcar.In the next step, the median light curve shape mustbe subtracted from the observations, leading to the scat-ter of the light curves (that is partly due to the signal ofthe moon if it exists). The Scatter Peak is in there already,but the data distribution is too noisy for an identification.Therefore a smoothing is needed in another boxcar whichcan be similarly optimized as described above. In our simu-lations, the second boxcar consisted of 14999, 1499, 44777,and 104999 points (LC, SC, GB, and PLATO data, respec-tively); meaning 1.3-times the transit duration. Surprisingly,so long boxcar is necessary to determine the scatter valuewith appropriate accuracy. When the boxcar is centred tothe mid-time of the transit, it can measure the ingress andegress phases of the moon, which may occur well before andwell after the transit of the planet, depending on the instantgeometrical configuration.If the exoplanet hosts a moon, a well-defined peak oflight curve scatter appears at the phase of the planetarytransits. The height of the peak expresses how significantlythe scatter will be increased by systemic light curve distor-tions. We normalize the height to the scatter level of the out-of-transit phase. The Scatter Peak increases with the size ofthe moon, but its height also depends on the quality of dataacquired. In Fig 3, a set of simulations is shown with the dif-ferent pre-defined data qualities (in successive columns) andwith increasing moon sizes (in successive rows). From thisfigure it can be suspected that as large moons as 1 R Earth can very likely be detected via the Scatter Peak. c (cid:13) , 1–9 ractical suggestion on detecting exomoons Kepler LC Kepler SC PLATO
Figure 3.
Normalized scatter peaks due to moon transits in sample simulations. Each consecutive row shows 10 curves with Keplershort cadence, long cadence and expected PLATO simulations. Figure lines show simulations of an 1 R Jupiter planet with a moon of0.5, 0.7, 0.8, 0.9, 1.0 R Earth , respectively.
In the final step we decide whether there is a convincinglyhigh Scatter Peak in the data. Even in the no-moon case, thesmoothed residuals can mimic a Scatter Peak just by chance,because of numerical fluctuations. A convincingly high Scat-ter Peak means such a peak value which infrequently (FalseAlarm Probability,
F AP ) evolves from random fluctuations.A Scatter Peak is convincing if the specificity, 1 − F AP , isclose to 1.The most simple strategy is to observe whether theobserved scatter curve exceeds a pre-set threshold level atthe time of the transit (i.e., the local scatter is significantlyhigher than the average scatter plus a few the scatter of thescatter, which means that the scatter has really increased,and we do not see the result of simple numerical fluctu-ations). A lower threshold increases the sensitivity and de-creases the size limit, but the false alarm probability worsensif the value is set too low. Balancing between sensitivity andspecificity sets up the lowest appropriate threshold level. Todo this, we first decide the specificity of the desired detectionrate, and then simulate and evaluate many (thousands) ofno-moon events. The threshold level belonging to the givenspecificity is the upper bound of the lowest 1 − F AP pro-portion of scatter peaks. If the observed height of the peakexceeds the threshold level we accept the positive detection.If one suspects the act of any other process which canlead to a Scatter Peak, this process must be modelled andincorporated in selecting a threshold level. E.g., in the case ofan active star, a spotted stellar model can be fitted. The nullevent has to be simulated with this spotted stellar model,and the threshold level has to be determined in reference tothese light curves.
A modified implementation of this method involves the ap-propriate weighting of photometric residuals, instead of aboxcar smoothing. This will be necessary whenever data ofdifferent quality is available. When smoothing in the boxcar(Sect. 4.4), the expectation for the mean value of the scatteris calculated as, of course,sc e atter boxcar = s N X ∀ i in boxcar r i , (6)where r i is the residuals inside the boxcar, N is the numberof data points, and e denotes a estimate.If the errors of different data points differ, this formu-lation requires weighting to keep the least-square propertyof our estimator. In this case the scatter in the boxcar willbe estimated as:sc e atter boxcar = vuut X ∀ i in boxcar σ i ! − X ∀ i in boxcar r i σ i , (7)where σ i is the error of the i th datapoint. Although Eq. 7is proportional to the statistic which is usually tested with χ distribution, we keep suggesting a non-parametric eval-uation of the weighted scatter, such as described in Sect.4.4. This is because χ evaluation assumes that data pointscome from normal distributions. This is not strictly true inthe general case. This improper assumption introduces a lit-tle bias, which may easily hide the little signal that we arelooking for, or may result in a false alarm. The detectionthreshold must always be derived from the statistics of theout-of-transit scatter. c (cid:13)000
A modified implementation of this method involves the ap-propriate weighting of photometric residuals, instead of aboxcar smoothing. This will be necessary whenever data ofdifferent quality is available. When smoothing in the boxcar(Sect. 4.4), the expectation for the mean value of the scatteris calculated as, of course,sc e atter boxcar = s N X ∀ i in boxcar r i , (6)where r i is the residuals inside the boxcar, N is the numberof data points, and e denotes a estimate.If the errors of different data points differ, this formu-lation requires weighting to keep the least-square propertyof our estimator. In this case the scatter in the boxcar willbe estimated as:sc e atter boxcar = vuut X ∀ i in boxcar σ i ! − X ∀ i in boxcar r i σ i , (7)where σ i is the error of the i th datapoint. Although Eq. 7is proportional to the statistic which is usually tested with χ distribution, we keep suggesting a non-parametric eval-uation of the weighted scatter, such as described in Sect.4.4. This is because χ evaluation assumes that data pointscome from normal distributions. This is not strictly true inthe general case. This improper assumption introduces a lit-tle bias, which may easily hide the little signal that we arelooking for, or may result in a false alarm. The detectionthreshold must always be derived from the statistics of theout-of-transit scatter. c (cid:13)000 , 1–9 A. E. Simon, Gy. M. Szab´o, L. L. Kiss and K. Szatm´ary - F AP de t e c t i on p r obab ili t y threshold/ σ E E E E - F AP de t e c t i on p r obab ili t y threshold/ σ E E E E - F AP de t e c t i on p r obab ili t y threshold/ σ E E E E - F AP de t e c t i on p r obab ili t y threshold/ σ E E E E Figure 4.
Detection probabilities and specificity levels (1 - false alarm probability) at different threshold levels above the backgroundsignal with σ standard deviation. Figures show performance of Kepler long cadence data (top left), best quality ground-based observations(top right), Kepler short cadence data (bottom left) and expected quality of PLATO (bottom right). To illustrate how smearing impairsthe detection statistics, in the top panels we plot results from unsmeared reference curves with dashed lines. Smearing does not affectshort cadence sampling (bottom panels). Different sized moons are colour-coded; note that the change in moon sizes in the bottom rightpanel. In Fig 4. we show simulated detection probabilities andspecificity (1 − F AP ) estimates, expected for ground-based,Kepler long cadence, short cadence, and PLATO-qualitysimulated observations. The threshold level is represented inthe ordinate, in units of the standard deviation of the scattercurve belonging to null signal (i.e. out-of-transit). Decreas-ing curves (in gray colour) represent the detection probabil-ities belonging to different size moons, while the black curveplots specificity. We have deduced that for a clear detec-tion (false alarm rate < σ range must be chosen (Fig. 4), almost independently of thequality of the data (sampling rate and scatter).Somewhat surprisingly, top quality ground-based obser-vations promise a 30% discovery rate for Earth-sized ex-omoons, having a Scatter Peak above the 4 . σ thresholdlevel. Yet we have not got sub–mmag quality observations of ≈
100 full transits of the same planet, but the increasingnumber of transit observations and the increasing accuracyof data promises this possibility in the future.Space-telescopes offer a better detection performanceonly with short cadence sampling. Selecting 4 . σ thresh-old, practically all moons of 1 R Earth size will be discoveredin SC (detection rate is 99%.) The 0.9 and 0.8 Earth-sizedmoons can be discovered with 70% and 20% in
Kepler
SCdata, respectively. The detection limit with Kepler is around0.7 Earth radius. These are such large moons which do notexist in the Solar System, but they may be found elsewhere.If such moons exist, they should be discovered in Keplerdata, and a possible negative result will be a significant im-plication for the lack of so large moons around hot Jupiters.Somewhat surprisingly, detection statistics rapidlyworsens with longer cadence. We will show that this is pri-marily a smearing effect (Kipping 2010) rather than a sam-pling effect. In the top left panel of Fig. 4, we compared c (cid:13) , 1–9 ractical suggestion on detecting exomoons no r m a li z ed sc a tt e r pea k pea k he i gh t/ σ a moon /a Hill
Roche-radius 0.7 R E E E Figure 5.
Detection statistics of close-in moons. The peaks oc-curring by random fluctuations are plotted with various dashedand dotted lines, while the distribution of peak heights in 15model simulations is plotted by the symbols with error bars. Theleft axis plots the normalized height of the peak (level of 1 repre-sents the out-of-transit fluctuations), while the right axis plots theheight of the peaks above the out-of-transit fluctuations, scaledby σ , the standard deviation of the blurred out-of-transit fluctua-tions. The threshold level of 4.5 times of the out-of-transit scatteris represented by the solid line. the detection statistics with the Kepler
LC cadence curveswith instantaneous sampling of the unsmeared light curves(1 R Earth size moon; plotted with dashed line), and thesmeared light curve (that is the integrated brightness overa the half hour long exposure; plotted with solid squaresand error bars). Selecting a 4.4 σ threshold, the detectionrate of our model exomoon would be more than 90% withhalf hour cadence and without smearing (instantaneous sam-pling), while it decreases to ≈
15% if smearing is also in-culded in the model light curves. The detection statisticsof smaller exomoons is identical to the distribution of falsepositives, so in these cases we do not expect success. Thestriking impairment of the detections is simply due to thesevere smearing on the light curve wings, which blurs thelightcurve of the planet, suppressing the little light varia-tions of the moon itself.A real breakthrough is expected by PLATO mission,which is expected to have significantly lower detection lim-its (bottom right panel in Fig 4). PLATO should be ableto discover the most exomoons which are larger than 0.6 R Earth with very low FAP rates. Setting the threshold levelto 4 . σ , we expect to discover 40% of the of the moons with0.5 R Earth radius, and 7–8% of exomoons with 0.4 R Earth .This experiment will be conclusive in the field of quest forexomoons: we do know that moons greater than 0.4 R Earth exist: 3 moons in our Solar System exceeds this size limit.
Besides the size of the moon, the detectability also dependson the orbital radius of the moon. Light curve effects ofclose-in moons are limited to the close vicinity of the tran-sit, shortening the time interval when the light curve distor-tion is present. This will decrease the scatter of the residu- als, and somehow deteriorate the detection statistics. (N.B.close-in moons suffer similar observational limitations withthe other methods.) However, the Scatter Peak method isable to detect at least a part of these kind of moons. Todemonstrate this, we illustrate how the detection statisticsworsens for a certain configuration and a single instrument.The complete analysis of close-in moons is a complex multi-parametric problem and lies beyond the scope of this paper(see Kipping 2011 for a detailed discussion of a such config-uration).We designed systems consisting of the same planet as inthe previous simulations, and systematically decreased theorbital radius of the moons to values of 10%, 28%, 46%, 64%,82% of the Hill-radius. The selected data quality was thePLATO-kind sampling and noise, while we observed moonsof 0.7, 0.6, 0.5 R Earth . In Fig. 5, we plot the normalizedheight of the scatter peak, and compare it to the high-est peaks by numerical fluctuations in the no-moon case.The heights of the false alarm peaks are plotted with non-continuous lines, while the 4.5 σ threshold is denoted by thesolid horizontal line. To the left, we plotted the Roche-limit,assuming the moon has a densitiy of 3 g/cm . The symbolsshow the detection statistics of the 15 probed configuration.Here, one symbol and the interval lines represent a wholedistribution of detection success. Therefore, the interval cov-ered by the “error bar” is the most informative: wheneverthe error bar goes higher then the threshold level, there aresome correct positive detections above 4.5 σ level, regardlessof the position of the symbol itself.The plots demonstrate that the detection rate of 0.5 R Earth sized moons decreases with decreasing orbital dis-tance, however, the “error bar” above the solid line expressesthat there will be chance for a detection even at 25–30% ofthe Hill-radius (depending on how lucky configurations oc-cur during the observations). On the other hand, detectionsuccess does not decrease for a 0.6 R Earth moon (althoughthe significance does decrease; but all systems are detected,since all of them remains above the 4.5 σ threshold). Theclose-in orbits of moons influence the detection statisticsonly for the moons near the size limit of an observationalconfiguration, and does not affect the detection success oflarger moons. The observational bias near the low-end ofmoon sizes can be determined with similar numerical exper-iments, making use of the relevant parameters of the certainsystem and the quality of the observations as input. Besides the Scatter Peak, several other methods have beenproposed which do have the potential of discovering an exo-moon (e.g. TTV, TDV, Time-of-arrival analysis of pulsars).A significant limitation for such an application is that thetransit configuration of the planet can also vary becauseof perturbations. Hence, the detected variations have to beanalysed further and the perturbations must be excludedas the origin of variations. Another limitation is the require-ment of having ≈
100 transits at least for the analysis, whichmakes the method to be applicable only for planets with or-bital periods of less than 10–20 days. However, this limita-tion is due to the planned 3-4-5 year lifetime of space ob-servatories; and we anticipate that it will be applicable for c (cid:13) , 1–9 A. E. Simon, Gy. M. Szab´o, L. L. Kiss and K. Szatm´ary longer period planets if homogeneous datasets will be avail-able for some transiting systems. Another possible limita-tion is of physical nature, i.e. more massive moons are morerapidly removed by tidal forces (see Barnes and O’Brien2002 for a detailed description), while moons beyond ≈ a priori assumptions of theshape of the transit light curve do not influence the result.We remark that the method is now tested for different moonsizes and orbital radii, while a more general testing (also forinclined and non-circular orbits) is the task of a forthcom-ming paper. However, the current level of testing does notinfluence the suggestion that the Scatter Peak method canhelp a lot in discovering the exomoons.ESA’s planned PLATO mission will offer a great op-portunity for the detection of exomoons, because of thelarge number of the targeted stars ( ≈ , • All light curves must be stacked in such way that the transit time of the planet exactly coincide in each of theanalysed light curves. • Transit observations must include the out-of transitphases before and after the transit of the planet, where thescatter due to the moon is the highest. The wings must spanat least as long as the transit duration. • Trend filtering of the light curves must be carried outin such a way that small deviations immediately before andafter the transit of the planet shall remain unaffected.
ACKNOWLEDGMENTS
This project was supported by the Hungarian OTKA GrantsK76816 and MB08C 81013, and the “Lend¨ulet” Young Re-searchers’ Program of the Hungarian Academy of Sciences.We thank the referee D. M. Kipping for valuable comments,that helped to impove the paper.
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We know that ν follows a uniform distribution and thequestion is the density function of x = sin ν (we assumelength is measured in the unit of a m for the sake of sim-plicity). Let F ( ν ) = p ( ν ′ > ν ) and F ( x ) = p ( x ′ > x ) bethe cumulative distribution of the same set of positions,parametrized by ν and x , respectively (here p representsprobabilities, ν ′ and x ′ are generic running parameter).Since ν is uniformly distributed, dF ( ν ) dν = 1 π (A1)where ν is element of the interval [ − π/ , π/ x m = sin ν . The dif-ferentials of this transformation are dx m = cos νdν , andbecause cos ν = √ − x m , dν = dx m / √ − x m . Substi-tuting dν with dx leads to the result, dF ( ν ) dx = 1 π √ − x (A2)which is Eq. 1 of the paper after a m is written explicitly torepresent the appropriate length scale. Although the ex-pression is singular at the end points ( x/a m = ± c (cid:13)000