CHEOPS Performance for Exomoons: The Detectability of Exomoons by Using Optimal Decision Algorithm

During the past 20 years, astronomers have discovered a large set of exoplanets, which has sparked excitement in the community as to whether these planets may host a detectable and/or habitable exomoon. The number of confirmed exoplanets exceeds 1000, but so far there is not a single case where the existence of an exomoon could be demonstrated. Here we have to note that, although Bennett et al. (2014) detected a microlensing event, which was interpreted as signal from an exomoon orbiting a gas giant, the interpretation of the result is doubtful.

Sartoretti & Schneider (1999) argued that a moon around an exoplanet can cause a measurable timing effect in the motion of the planet and therefore in its transit timing (known as TTV). Since then, many efforts have been made to develop and test such algorithms that can help us answer the question of whether an exomoon can be discovered around exoplanets. The series of dedicated studies started with the photocentric transit timing variation (PTV, TTV_p in Szabó et al. [2006] and Simon et al. [2007]), followed by the detection of mutual transit events between a planet and its moon(s) (Cabrera & Schneider 2007), the transit duration variation (TDV; Kipping 2009b), the influence of lunar-like satellites on the infrared orbital light curves (Moskovitz et al. 2009), and direct photometric detection of moons and rings (Tusnski & Valio 2011). Recent studies include the scatter analysis of phase-folded light curves (Simon et al. 2012), the HEK Project (direct detection of photometric light curve distortions and measurements of timing variation like TTV and TDV; Kipping et al. 2012, 2014), and the analysis of orbital sampling effects (Heller 2014). In addition, there has been a plethora of other techniques identified that can play important role in confirming such detections. These include the Rossiter–McLaughlin effect (Simon et al. 2009, 2010; Zhuang et al. 2012), timing variations in pulsars' signal (Lewis et al. 2008), microlensing (Han & Han 2002), excess emission due to a moon in the spectra of distant Jupiters (Williams & Knacke 2004), direct imaging of tidally heated exomoons (Peters & Turner 2013), plasma tori around giant planets by volcanically active moons (Ben-Jaffel & Ballester 2014), and modulation of planetary radio emissions (Noyola et al. 2014).

Also, there were attempts to identify the potential Kepler candidates that could host a possible exomoon (Szabó et al. 2013; Kipping et al. 2012, 2014; the HEK Project). Even though Kepler is theoretically capable of such a detection, there is no compelling evidence for an exomoon around the KOI (Kepler Objects of Interest) targets so far.

In this paper, we turn to the next space telescope, CHEOPS, and characterize chances of an exomoon detection by using the method of the photocentric transit timing variation (PTV, TTV_p; Szabó et al. 2006; Simon et al. 2007). Our aim is to explore the capabilities of CHEOPS in this field and calculate how many transit observations are required to a firm detection in the case of a specific range of exoplanet–exomoon systems. We did this according to the CHEOPS science goals and planets to be observed, without investigating the theoretically expected occurrence rate and probability of exomoons generally. We test a decision algorithm, which gives the efficiency of discovering exomoons without misspending expensive observing time. We perform simulations and bootstrap analysis to demonstrate the operation of our newly developed decision algorithm and calculate detection statistics for different planet–moon configurations. Finally, we introduce a folding technique modified by the PTV to increase the possibility of an exomoon detection directly in the phase-folded light curve, which is similar to the idea presented by Heller (2014).

First, for the sake of clarity, we have to note that the photocentric transit timing variation (PTV) differs from the conventional transit timing variation (TTV). Because of their blurred meanings (PTV and TTV were called as TTV_p and TTV_b, respectively, by Simon et al. [2007]), we briefly clarify the differences and show their usability and limitations in the following.

Sartoretti & Schneider (1999) suggested that a moon around an exoplanet can be detected by measuring the variation in the transit time of the planet due to gravitational effects. In their model, the barycenter of the system orbits the star with a constant velocity, and transits strictly periodically. As the planet revolves around the planet–moon barycenter, its relative position to the barycenter is varying, so the transit of the planet starts sometimes earlier, sometimes later (Fig. 1). This time shift is the conventional TTV:

where m_s and m_p are the masses of the moon and the planet, and χ and ϑ are the ratios (moon/planet) of the densities and radii, respectively. We note that TTV can be caused not only by an exomoon itself. Several other processes; e.g., an additional planet in the system (e.g., Agol & Steffen 2005; Nesvorn et al. 2014), exotrojans (Ford & Gaudi 2006), and periastron precession (Pál & Kocsis 2008) can also cause TTVs.

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The traditional TTV simply measures the timing variation of the planetary transit and does not consider the tiny photometric effect of the moon. Szabó et al. (2006) and Simon et al. (2007) argued against this simplification and proposed a new approach for obtaining the variation of the central time of the transit and calculated the geometric central line of the light curve by integrating the time-weighted occulted flux. They derived a formula which showed that there is a fixed point on the planet–moon line, which is the so called photocenter (PC in Fig. 2). In this photocenter, an imaginary celestial body causes the same photometric timing effect as the planet and the moon combined together. The motion of this visual body around the planet–moon barycenter leads to the variation of the transit time, which is the photocentric transit timing variation (PTV in Fig. 2):

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This model takes into account both the photometric and barycentric effect of the moon.

Both methods have advantages and disadvantages. The traditional TTV uses parametric model light curve fitting (e.g., Mandel & Agol 2002) to derive the timing variations of the transit. This does not need equally sampled data and the result does not significantly depend on the number of the measurements. Furthermore, it allows characterizing the star-planet pair immediately and gives a physical interpretation of the system. However, the results depend on the model adopted to fit the data. For distorted light curves, blind model fitting can lead to nonphysical parameter combinations and ultimately misleading results; hence, model-independent tests are always very important. Moreover, mapping large parameter spaces for fitting extensive sets of observations can be extremely time-consuming, making a full, detailed analysis nearly impossible. In contrast, the technique of the PTV is a so-called nonparametric method which can provide fast reduction of the light curve shapes even for large sets of data. There is no need to make assumptions on the analytic light curve model, because PTV uses only a numerical summation of time-weighted fluxes in a window with certain width; hence, there is no need to increase the number of model parameters for distorted transit curves. It is true for both the TTV and the PTV that their usage is limited by the spatial configuration of the systems. Independently of their size, for close-in moons, mutual eclipses occur during the transit; therefore, due to the opposite motion of the moon, the apparent timing variation will be largely canceled out.

Another difference between the techniques is demonstrated in Figure 3, where the green light curve shows a transiting system with a leading moon, while the red one corresponds to the opposite case, when the moon is in a trailing position. As can be seen, the two effects are opposite in sign and the PTV can produce a larger signal. Interestingly, one can show from the derived formula of the TTV and the PTV that the former is more sensitive to the mass of the moon, while the latter gives better constraints on the radius of the moon (Simon et al. 2007).

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To measure the entire PTV effect, the evaluation window should be longer than the transit duration. The size of this window depends on the parameters of the systems (Simon et al. 2012), mainly on the semimajor axis of the moon, and therefore on the Hill sphere as well. For our simulation the choice of a window with triple duration width was enough to measure the tiny drops due to the moon for all cases. Even if this small brightness decline on the left or right shoulder of the main light curve cannot be seen directly, they can cause measurable time shift in the transit time (Simon et al. 2007).

We note that the magnitude of these effects depends on the density ratio of the companions. For example, if the density of the moon is double that of the planet, which can be the case for a gas giant-rocky (or icy) moon system, the TTV dominates only if the size ratio is higher than 0.25. If the moon-planet density ratio is about 0.5 for an Earth-Moon system, the PTV always surpasses the TTV (Fig. 4).

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Hereafter, we will take the advantages of the PTV method to investigate the expected performance of the CHEOPS space telescope. However, to put the results into broader context, we will also compare PTV and TTV in selected cases.

In this paper, we used our existing code for precise light curve calculations of transiting exoplanet with an exomoon (for a detailed description, see Simon et al. [2009, 2010]). Our code takes into account all the dynamical, kinematic, and photometric effect, such as nonlinear limb darkening, planet–moon dynamical motion, mutual transit, transit timing variation (TTV), photometric timing variation (PTV), transit duration variation (TDV), light curve asymmetry, light curve distortion due to the moon, smearing effect (in progress: rotating star, starspots), etc., for calculating reliable exomoon light curves and radial velocity curves with the Rossiter–McLaughlin effect.

Although our code is capable of including the full photodynamical model, a comprehensive analysis of discovering exomoons in simulated data is beyond the scope of this paper. The PTV technique kept our investigation as simple as necessary, but yielded fast and robust results for estimating the detectability of exomoons via CHEOPS, and defined the parameter space of systems which could be plausible targets for such a detection. A search for exomoons in real data, such as the HEK Project (Kipping et al. 2014), needs a more thorough analysis where a full photodynamical fit also has to be taken into account in addition to recent techniques (PTV, TTV, etc.).

After some recent performance improvements in the calculation speed of our code, we developed a decision algorithm with which we can generate and handle large data sets and make detection statistics for such systems. In devising our decision algorithm, we considered the CHEOPS scientific objectives. The main goal of the CHEOPS mission, to be launched by late 2017, is to characterize exoplanets with typical sizes ranging from Neptune down to Earth, orbiting bright stars. The target list will consist of stars which are already known to host planets, previously detected by accurate Doppler surveys. The planned duration of the mission will be 3.5 years and about 500 target radii will be measured with precision down to 10% (Broeg et al. 2013).⁵

3.1. Parameter Space

For the sake of simplicity, we assumed circular orbit for the companions and set the impact parameter (b) to zero. The semimajor axes were calculated from the third Kepler law.

To reduce the large parameter space during our simulations, we provided estimates for some physical variables and defined their intervals. For the sake of simplicity, we first chose one solar mass and one solar radius for the host star. We divided our simulations into five main bins with planet sizes ranging from 2 to 6 Earth radii, and with increments of one Earth-size. The sizes of the moons were chosen from the interval 0.7 and 1.5 Earth-size, but the planet always had to be larger by a factor of 3 in radius, except for the smallest planet, where the ratio was 2.86 due to rounding. This restriction in size is necessary to avoid investigating double planets. For PTV calculations, we needed the density ratios of the companions. In order to stay within physical reasonable parameter range, we took planetary data in the Solar System (Fig. 5). We have considered the Earth–Moon pair, the system of Neptune and its five largest moons, and combined the data for Jupiter and Saturn and their nine largest satellites. The result is a correlation between the planet size and moon density, which was used in the simulations: for a fixed planet radius, the density of the moon was calculated using the correlation we see in the Solar System. The actual numbers in the simulations correspond to a moon mass range of 1–50 Earth masses.

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Before the detailed simulations, we performed some trial investigations for a few representative parameter combinations. These preliminary investigations all agreed in their conclusions that we need at least 15–20 simulated transits to make efficient characterization of a reliable detection statistics for the entire parameter space.

Then we had to set the most important observational constraint, the orbital period of the planet. The upper limit was fixed by the planned operation time of CHEOPS, which is 3.5 years. In order to be able to observe 15–20 transits, the orbital period must be shorter than 70–80 days. Hence we chose 75 days.

For the lower limit of the orbital period, we took into account the fact that the moons must be both stable for a billion years and large enough to produce a detectable signal. Barnes & O'Brien (2002) discussed the tidal stability of satellites and presented a detailed analytic description of the relationship between the planet's semimajor axis, the moon's maximum mass, and its long-term stability. This can be easily converted into a relationship between the orbital period of the planet and the radius of the moon (Fig. 6). For example, if we select a 1.5 Earth-sized moon (the largest one in our simulations) and assume for at least 1 billion years around a Super-Neptune (6 Earth-size), the orbital period of the planet must be longer than 55 days (black curve). Considering this 55 days, the maximum size of a stable moon around a Super-Earth (2 Earth-size) is approximately 0.9 Earth-size (red curve); therefore, the choice of 0.7 Earth-sized moon fulfilled this criterion. The rectangular area in Figure 6 shows the investigated parameter range.

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We note that the actual numbers depend on many physical parameters of the system and are only valid for a one solar mass central star and the assumptions of the Barnes & O'Brien (2002) study, with 0.51, 10^-5, and 0.36 set for the tidal Love number, the tidal dissipation factor, and the constant fraction, respectively (see also references in Barnes & O'Brien [2002]). The largest uncertainty in longevity of exomoons lies in the tidal dissipation factor (Q_p). While a (Q_p = 10⁵) is commonly used (Weidner & Horne 2010), Cassidy et al. (2009) suggested values as high as 10¹³ for exoplanets. The choice of time for stability has less effect on the calculation; therefore, we chose 1 billion years, which is the minimum desirable value for calculating longevity of exomoons (Sasaki & Barnes 2014) for determining the range of our parameter space. We also note that Sasaki et al. (2012) found that the inclusion of the effect of lunar tides on planet rotation allows significantly longer lifetimes for massive moons. Despite the large uncertainties in the stability calculations, our simulations give a representative view of what we can expect from CHEOPS; for real systems, one must perform as closely matching calculations as possible.

The orbital period of the moon was chosen randomly, where a logarithmic distribution was used. At this point, to exclude the short-time escape of the moon, the system has to fulfill the criterion on the Hill-sphere and the moon's orbit must be beyond the Roche-limit. From the definition of the Hill-sphere, one can determine that if we want the moon to orbit closer than the Hill-radius, then the ratio of the orbital periods of the planet and the moon can never drop below the value of (Kipping 2009a). The minimum value was 2.5 in our simulations, while the maximum ratio was 20, which fulfilled the criteria on the Roche-limit for every case, too. By choosing the value of 20 for the maximum ratio, we can neglect close-in moons, such as Jupiter-Io, but for very large ratios the transit duration and the moon's orbital period will be in the same order of magnitude and mutual eclipses during the transit can occur; hence, TTV and PTV will diminish.

The duration of the simulations was equally the total time span of 20 transit (orbits), and therefore, the shortest and longest simulations ran for 1100 and 1500 days, respectively. The cadence rate was set to 60 s. The total number of differently configured systems was 260,000. This number came from five main planet–moon combinations (see Table 1), where parameters were randomly chosen from the next parameter space: the size of the moon and the ratio of the orbital periods. In these main bins, the number of simulations were proportional to the width of the interval of the moon size, where every 0.1 step in the moon size means 10,000 runs. To make the detection statistics for each configuration, we performed another 3000 simulated observations by using the bootstrapping technique. The only difference among these runs was in the time sequence of the noise, and half of these simulations were performed without moons, which led to the null events, i.e., the detection limit for our decision algorithm (see below).

3.2. Noise Model

Another important component of the simulations is the applied noise model. The noise was generated by the CHEOPS Noise Simulator, which was still in development during this study, and therefore, a preliminary noise model was used. We note that as the design of the mission progresses, noise estimates will become available. Because the detailed model description and documentation of the CHEOPS Simulator is not available for public use, here we can give only a short description of its current state and the considered noise budget.

First, the CHEOPS Simulator produces images by calculating the expected PSF of a star. At this point, the spectral type of the star, the temperature of the CCD, and the telescope throughput are considered. The measurement points are obtained by performing a simple aperture photometry. At the time of the simulations, the CHEOPS Noise Simulator could consider the following noise sources: shot noise; read-out noise; quantization noise; sky background (Zodiacal light); temperature-induced variability of the dark current; Quantum Efficiency variation of the detector with temperature; gain variation of the proximity electronics; flat-field errors; bad pixels; stray light; cosmic ray effects; and jitter due to the pointing accuracy. The stellar noise was simplified by adding a value drawn randomly from a Gaussian distribution. The first four were treated as white noise. The other sources are related to systematic or environmental noises, which are nonrandom and likely to be predominantly periodic, possibly synchronized with the period of the chosen orbit. The Fourier spectrum of the applied noise can be seen in Figure 7, where a peak around 430 1/day shows a periodic component indicating that our simulated noise differ somewhat from the Gaussian distribution. We note that the predominant source per exposure is the shot noise; however, its time average decreases rapidly and the environmental and systemic noises can have more significant effects.

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Lewis (2011, 2013) studied more thoroughly the effect of a realistic stellar noise for the detectability of exomoons via the PTV method. It was found that in the case of a filtered solar noise the size of a detectable moon increases by a factor of 1.5 (Lewis 2011). Our analysis showed that the inclusion of the systematic and environmental noises also results in a growth in the moon size of 10–20% compared to the white Gaussian noise. Despite this sensitivity to noises of the PTV method, this technique can still produce a signal, which can usually be larger than those of the other methods (TTV, TDV, etc.) and can offer a statistically fast and still robust evaluation. Therefore, using PTV is still a reasonable choice for analyzing huge simulated data sets, selecting targets, and estimating the possible detection rate of exomoons via CHEOPS.

Because the development of the CHEOPS Simulator is still in progress, the implementation of the more realistic stellar noise is also under development. We had to adjust the preliminary noise by multiplying by a certain factor to meet the requirements of the photometric precision, which is 20 ppm during 6 hr of integration time (CHEOPS Definition Study Report).⁶ Given that the planned magnitudes of the CHEOPS targets range from 6 to 12, we fixed the apparent brightness of the simulated signal at 9th magnitude, with the corresponding noise characteristics.

It is known that there are several other unknown factors that can contribute to the noise budget, such as the celestial position, the brightness, and the real stellar noise, of a target. Therefore, the present approximation of the adjusted noise is the main limitation factor of our investigations. Because the primary aim of this investigation is not to discover and confirm exomoons in the real data, such as the HEK Project (Kipping 2014), but to present a feasibility study for the CHEOPS space telescope, which is still under development, and to propose possible targets with certain detection rates and to determine whether this instrument is capable of an exomoon detection, the more sophisticated modeling of the total noise budget is not a goal of this paper.

The discovery of an exomoon requires multiple transit measurements to reduce the noise level without wasting time to systems that have no signal or are clearly detected. Consequently, the search must be optimized. We integrated our exomoon simulation code (Simon et al. 2009, 2010) into a newly developed decision algorithm that is capable of searching the most efficient detection index from simulated data along preset false alarm levels and desired lower limits. The goal of this algorithm was speeding up the process of deciding whether the tested system is a potential candidate or if the detected signal cannot be distinguished from the statistical noise. In addition, this has to be done for a limited number of transit observations (Fig. 8).

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We used the bootstrapping technique to determine the threshold levels for detection/rejection based on the measured PTV values. For this, we performed 3000 simulated observations for every given system configurations as follows. First, we computed the PTV for the null event from 1000 runs, where only planetary transits without moons were simulated. Consequently, the "measured" PTV comes purely from the statistical noise. After sorting these 1000 PTV values, the algorithm selects the top nth point as the detection threshold according to the desired a posteriori output. Second, we performed another 1000 bootstrapped simulations with the same planet + moon configurations to determine the minimum PTV amplitude that can be produced by and expected from the given configuration. After sorting the measured PTV values, the algorithm selects here the bottom mth value as the rejection threshold. Finally, we performed another 500 runs without moons and 500 runs with moons randomly to calculate the detection statistics for the actual configuration. With the capability of adjusting threshold levels we can balance between the "speed" of the decision and the number of false alarms. The lower detection threshold means less observing time and higher detection rate, but more false alarms. On the other hand, pushing the rejection threshold higher makes the chances of losing a true system higher. Therefore, both the detection rate and the number of false positives is lower. We note that despite the adjustable threshold levels (i.e., the freedom of choosing different n and m values), we have chosen fixed values, namely, n = 980 and m = 20, to speed up the algorithm and decrease the computation time for 260,000 systems. These choices translate to a false alarm probability of 2%, which corresponds to about 2.5σ (one sided) if we assume a normal detection statistics. The given significance level is an important input parameter in determining the detection strategy, and can be set to an arbitrary level (e.g., 4σ) if the problem requires it.

During the evaluation process for a system, the algorithm computes the central time for each transit light curve and fits a linear model to the set of observed minus expected central times. The residuals of the linear fit define the PTV values. Then it determines the rejection and the detection thresholds with the bootstrapping technique, and subsequently, evaluates the simulated observations. The rejection threshold, which is the expected minimum amplitude of the PTV for the given system, is calculated as the maximum deviation from the linear fit. As expected, this value increases with the number of observed transits as more and more points are measured, and the period of the PTV becomes apparent. While determining the detection threshold, we expect null signal for the amplitude of the PTV, and therefore, zero variation of the differences between the expected and observed central times. (A planet without any external influences from other companions, such as other planets and moons, can be considered strictly periodic within the studied time-scale of 3.5 years.) To determine the maximum signal due to the statistical noise, we searched for only the maximum deviation from zero. Since the noise level was constant during the analysis, the maximum differences between the expected and observed central times, thus the PTV, are independent of the number of observed transits, and therefore, the detection threshold remains constant (Fig. 9).

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By increasing the number of measured transits, the algorithm reevaluates the PTV and compares that to the thresholds (Fig. 9). If a system exceeds the detection limit or drops below the rejection limit, it will be either classified as a potential candidate for hosting a moon or rejected after a certain number of observed transits. If the target needed more observations to be classified, it remained in the undecided state. Two examples are shown in Figure 9. The upper panel refers to systems with moons, where the evolution of the PTV amplitudes shows ascending trends, while the systems without moons evolve in a horizontal direction, except two false detections indicated by thicker lines.

The final step of the algorithm arrives when a certain system is placed into one of the three categories: detection, rejection, and undecided. To optimize the observing strategy, we need to know how much of the limited instrument time should be spent in order to get an unambiguous classification. For this, we took the average classification time for the 500 simulations with and without a moon. This approach is analogous to the case when the probability of the existence of a moon is 50% and we expect a clear yes/no/indeterminable answer from the algorithm (see T₅₀ in Table 1).

Here we note for the sake of interest: In the first part of our simulations, we analyzed general cases by splitting the size of planets into five bins. For each case, the number of the simulations was proportional to the width of the interval of the moon radius, and therefore, the total number of simulated configurations was 260,000 (Table 1). Each run consisted of the 3000 simulated observations (see above) and every simulation had 20 transits. This means in total about 15.6 billion calculations of the PTVs.

Our investigation of the potential CHEOPS performance in detecting exomoons consisted of two parts. The first part was done in a general way, in which the systems were divided into five main bins by defining the size of the planets and randomizing the other physical parameters. The second part consisted of candidates which already have radial velocity measurements (data taken from The Extrasolar Planets Encyclopaedia⁷) by fulfilling our simulation criteria and the predefined restrictions for the physical properties of the companions. To make assumptions of the boundaries of expected radii of these planets, we used observational and theoretical results from Weiss & Marcy (2014) and Swift et al. (2012), who studied the correlation between the masses and the radii of exoplanets. Then, we simulated transit observations by taking values randomly for the moon sizes and for the ratios of the orbital periods.

The probability of detecting exomoons around both the simulated systems and the selected candidates was obtained by averaging the detection statistics of individual systems. This a posteriori output highly depends on the preset levels of the threshold during the bootstrapping analysis. Forcing fast decisions can yield a large number of false hits, while choosing high-confidence levels for detection and rejection can leave a lot of systems in undecided states, thereby wasting observing time.

5.1. Detection Statistics for Simulated Observations

We summarized the physical parameters (first six columns) of the simulations and their detection statistics (last four columns) in Table 1. The number of discoverable exomoons (column 7) increases with the size of the planet and almost reaches 90% in the case of 4 Earth-sized planets at less than 2% false alarm rate (column 8). This result is a bit surprising because we expect smaller PTV values in the cases of larger planetary sizes according to equation (2), and therefore fewer detections. As our probability tests showed, the simple explanation is that the signal-to-noise ratio (in this case the ratio of the transit depth and the magnitude of the noise) is increasing with the radius of the planet and this effect is stronger than the decrease of the PTV, so we get more hits even if the value of the PTV is smaller. For a planet with 4 Earth-size, the average observing time (column 9) in order to detect the signal of an existing moon decreases with the planetary size, and is about 6.7 days. (We note that for most of the systems in the explored parameter range, the average transit duration is about 8 hr; hence one full transit observation requires 1 day of monitoring. Therefore, there is a close correlation between the required number of observed transits and the total number of days spent on a target.)

Figure 10 shows how the detection statistics change with the size of the moon and the planet. It is surprising that we have a more than 70% chance for detection of a 0.8 Earth-sized exomoon if the host planet is bigger than 3 Earth-size. The observing time starts dropping rapidly below Earth-sized moons but does not exceed 10 days (less than 1% of the expected operation time of CHEOPS) for planets that are bigger than Neptune.

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Another aspect of the detection statistics is shown in Figure 11, where the variations are plotted against the ratio of the orbital periods for three representative planetary sizes. The figure clearly shows that success rate increases with the size of the planet (indicated by colors), while the required observing time decreases. The large spread within the same colors is caused by the fact that the simulated moons have different sizes, while the planet sizes were fixed. Also, there is a clear repetitive pattern for the orbital resonances, when the periods are commensurable. In such cases there is a drop in the detection rate, which is caused by those configurations in which the planet and the moon transit the stellar disk almost always in the same relative phase, so that there is no PTV effect. For this reason, 100% cannot be reached for the average detection rate ("adr" in Table 1). Another finding is that there is a decreasing trend for larger period ratios, i.e., for the short-period close-in moons. This agrees well with the result of Simon et al. (2012).

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We have to note that the detection rate cannot be interpreted alone without considering the false alarm rate and the time needed for the detection. Neither the false alarm rate nor the observing time are constant for different configurations. For example, the variation of the false alarm rate can be seen in the middle panel of Figure 11, where the values change from 0% to 12%. In the case of commensurable orbital periods, it is much higher than elsewhere. When such a configuration of an orbital resonance is being examined, the rejection level will be much lower because the expected minimum value for the PTV is zero, and hence it does not differ from the statistical noise. This means that many false signals can survive the first steps, which increases the chances of their detection in the later steps. This can be noticed as peaks at certain ratios of the orbital periods in the middle and lower panels of Figure 11. The level of the rejection threshold can be pushed upward by choosing a higher m value or feeding back the false alarms to the algorithm, which can automatically raise the rejection threshold and making a new evaluation as many times as the false alarm reaches the desired level. Our code is capable of such training but for the sake of saving computation time we simplified our investigations by setting the thresholds to a fixed nth and mth value according to the result of the bootstrapping.

To illustrate the relationship between the false alarm rate and the number of observed transits, we made a separate simulation, where the parameters of the systems were the same, and only the level of the false alarm rate was varied. The results showed that pushing the false alarm rate for the detections toward lower values increases the observing time nearly linearly until 2%, and then below this, the observing time increases nearly exponentially (Fig. 12). Repeating this simulation for other configurations, the relationship was found to be similar. Only the number of transits to be observed varied.

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5.2. Known Exoplanets with Radial Velocity Measurements

We repeated our simulations by using real physical parameters from The Extrasolar Planets Encyclopaedia, where 428 planetary systems are listed which were discovered by radial velocity measurements. Because the results of the simulated observation showed that 6–10 transit measurements is enough to reach a decision in most of the cases, we selected planets that have orbital periods between 55 and 100 days, so we will not exceed the planned observing time in the case of the longest orbital period. We chose those planets which had masses larger than 5 but did not exceed 70 Earth-masses. After applying these restrictions, 10 systems remained, mostly more similar in apparent brightness range than what was assumed previously. To estimate the expected sizes of the planets we used the result of Weiss & Marcy (2014), who studied mass-radius correlation of 65 exoplanets smaller than 4 Earth-size and with orbital periods shorter than 100 days, and that of Swift et al. (2012), who derived relations for different classes of material in a wider range of mass and radius. The other parameters of the moon have been chosen randomly as described previously. The properties of the systems and the results for their detection statistics are listed in Table 2.

Despite the small number of the examined exoplanets and the shorter observing time (not more than 10 light curves), we got a more than 80% successful detection rate with less than 1.5% false alarm probability and with no more than 7 days of observing time for half of the cases (5, 6, 8, 9, 10 in Table 2). This means that if we do not stick to this strict restriction (allowing larger moons, more observing time) the success rate could be very promising, so that CHEOPS could have good chances for an exomoon detection. Nevertheless, we have to note here that we assumed that we can detect transit, but obviously that is not possible in every case, because the planets to be observed were discovered by Doppler method and the spatial configuration of the systems can be different from the ideal case.

We notice that two planets (5,6) are members of multiple systems, and therefore they can show TTVs, which can bias the signal of the PTV. A possible solution for getting unbiased PTV values is to make a time correction by shifting the measurement points of the transit with the appropriate values of the TTV. If the signal of the PTV is still present, then there should be another signal in the light curve which is due to an exomoon or sign similar to an exomoon. To decide the origin of the detected PTV signal, we need to consider additional detection techniques, which need to show the same results as the PTV did to confirm a presence of probable exomoon. In the following, we present two semi-independent methods.

Although there is no consensus about the "best" method for an exomoon discovery, the classical approach can be used, where independent techniques provide independent confirmations. Here we propose two simple semi-independent methods for confirming if the detected PTV signals are due to an exomoon. We use the term "semi-independent" because all of these techniques quantify different indirect signals from an exomoon but use the same photometric measurements and are based on the analysis of the same transit light curves. We note that for a reliable confirmation of an exomoon in the real data, we have to consider all possible detection techniques, including the photodynamical fit.

Since we already discussed the differences between the traditional TTV and the PTV in § 2, here we only emphasize again that the amplitudes of the produced signals differ from each other. A previous analysis showed that the PTV can produce stronger signal in opposite phase to the TTV. To illustrate this, we simulated a transiting system with a half Jupiter-sized planet and a 1.2 Earth-sized moon around a star with a solar mass and radius. The orbital periods of the planet and the moon were 64.973 and 30.862 days, respectively. Both timing variations can be seen in Figure 13, which shows that the PTV's signal has a much larger amplitude and runs oppositely in sign. The latter is because the photocenter is on the opposite side of the barycenter than the center of the planet (compare Figs. 1 and 2). The PTV measures the timing variation of the photocenter, which can be derived by integrating the brightness decline both from the moon and the planet. Although the planet can produce deeper light curve, the moon is located far from the planet and transits much later (earlier), so the longer time basis can amplify its small photometric effect. In contrast, the TTV measures only the shift of the main light curve, which is linked to the distance between the barycenter and the center of the planet. If both variations and their opposite behavior can be measured then it can provide good evidence for the existence of an exomoon. We note that because the effects of the moon in both sides of the light curve need to be measured, the evaluation window of the PTV technique is much larger than just the main transit of the planet. In practice, the light curve should be evaluated in a window that is at least three times wider than the transit duration.

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The other method is similar to Heller's (2014), which presented a sensitive method of detecting exomoons. This approach is based on the fact that moons appear more often at larger separation from their planet. In that case, a photometric orbital sampling effect appears in the phase-folded light curve, causing a small decline in stellar brightness immediately before and after the planetary transit. In our technique, the PTV actually measures the delay in the central time of the transit compared to the expected value, which is defined by the orbital period of the planet. Depending on the sign of the PTV, the tiny distortion from the moon appears on the opposite side of the light curves. After mirroring the appropriate light curves and folding them, the tiny effects of the moon gather on one side of the light curve. After applying a simple moving average to the phase-folded data, one can reveal and confirm whether there is a tiny brightness drop caused by the exomoon (Fig. 14).

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In this paper, we investigated the performance of the CHEOPS space telescope in searching for and detecting exomoons. In all of our simulations, we used the technique of photocentric timing variation (PTV), which was introduced by Szabó et al. (2006) and Simon et al. (2007). By comparing to the traditional TTV, we concluded that both methods measure the variation of the central time of the transit, but the PTV is more sensitive to the tiny light curve distortion due to a moon and produces opposite signal than the TTV. This anticorrelated behavior occurs only when an exomoon is present, and therefore, can help the confirmation process by itself as well as at the mirrored folding technique.

We developed a decision algorithm and integrated it to our existing code (Simon et al. 2009), with which we generated billions of light curves and calculated the same amount of PTV values to determine the number and the ratio of systems with detectable exomoons. The idea was to develop code that can make a fast decision about any observed system, can recognize potential moon-host candidates, and can reject those which exhibit only signals that cannot be distinguished from statistical noise.

After applying some restrictions to the physical parameters of the planets and moons by fulfilling the criteria of the CHEOPS mission, we divided the possible configurations into five general bins and, on other hand, selected appropriate candidates from the database of The Extrasolar Planets Encyclopaedia. Taking into account stable moon configurations, we randomized the moon parameters in predefined intervals and monitored how the maximum of the PTV was evolving with the observing time, how many observed transits were necessary to detect or reject a system, and what their ratios were. Our results showed that above Neptune-sized planets and Earth-sized moons, the expected ratio of the detected systems exceeds 85% with less than 2% false alarm rate and less than 8 days observing time. Among the ten analyzed candidates, which were discovered by Doppler-method, we found five, for which the presence of a moon would be detected with 80% probability and with less than 1.5% false alarms and less than 6.5 days (six transit observations) measurement time.

In the last years, there were several attempts to identify signals from exomoons in the Kepler data (Szabó et al. 2013; Kipping et al. 2012, 2014; HEK Project), but there is still no firm evidence of such a detection so far. The next space telescope that is designed to observe exoplanetary transits will be CHEOPS. Although its light collecting capability will be much smaller than that of Kepler, the dedicated observing strategy and brighter targets rate will make it at least equivalent or better photometric instrument Kepler was. Our results indicate that there is good reason for optimism regarding the detection of exomoons using CHEOPS.