A comprehensive study of the Kepler triples via eclipse timing Free

Abstract

1 INTRODUCTION

The analysis of eclipse time variations (ETVs) via O−C (observed minus calculated) diagrams is a powerful tool for the investigation of period variations in eclipsing binary (EB) systems, and, therefore, has been used in many EB studies over more than a century. ETVs may arise from different causes that act on various time-scales with various amplitudes. It follows that O−C diagrams may show a wide range of variational forms. The causes may be either physical, i.e. connected to a real variation of the orbital period, or merely apparent.

Long-term physical ETVs mainly occur as a result of evolutionary effects such as mass exchange between the binary components, wind-driven mass-loss, magnetic braking, tidal dissipation, or even gravitational radiation. Often the characteristic time-scale of the phenomenon substantially exceeds the entire period of human EB observations. Generally in each such case, the ETVs are manifest as a slow, constant-rate variation of the orbital period which results in a quadratic O−C pattern (for the strict analytic forms of some of the listed effects on the ETV, see Nanouris et al. 2011, 2015). Shorter time-scale physical ETVs can arise, e.g., from magnetic activity (see, e.g., Hall 1989; Applegate 1992; Lanza & Rodonò 2002) or from the dynamical effects of a close companion star on a binary orbit (Söderhjelm 1975; Borkovits et al. 2015). These shorter time-scale effects tend to produce periodic or, at least, quasi-periodic ETV behaviour.

The two most well-known classes of apparent orbital period changes leading to ETVs are the light-travel time effect (LTTE), caused by the changing distance of a binary in a hierarchical multiple-star system, and the apsidal motion effect (AME) which may be seen in eccentric EBs. Apart from the extremely compact triples which were investigated by Borkovits et al. (2015), these two phenomena often result in quasi-sinusoidal monoperiodic O−C diagrams. In the case of AME, the O−C curve formed from the secondary minima anticorrelates with the curve formed from the primary minima, while in the case of LTTE the two kinds of minima must vary in the same manner. Additional apparent orbital period changes inducing ETVs may arise, in theory, from the precession of the orbital plane of the EB due to the perturbations induced by either a third-star companion revolving in an inclined orbit or the non-aligned rotation of each or both stars. Such ETVs are not yet known to have been observed.

In addition to the above effects, erratic variations have been observed as well. They may indicate physical effects such as variable mass transfer rates or currently unidentified apparent timing effects.

Finally, when a light curve is distorted by the effects of, e.g., stellar spots or pulsations, the measurement process tends to yield spurious ETVs that may include periodic or quasi-periodic components (see e.g. Kalimeris, Rovithis-Livaniou & Rovithis 2002; Tran et al. 2013; Balaji et al. 2015, for spots and Borkovits et al. 2014, for stellar oscillations).

The almost continuous 4-yr-long set of high-precision photometric observations from the Kepler mission (Borucki et al. 2010) offers an unprecedented opportunity to study ETVs in thousands of EBs and ellipsoidal variables (ELVs). Among a wide range of possibilities, these data are especially suited for searches for short-period third-star companions of these binaries. Third-star companions to binaries are interesting from several perspectives. Third stars may be particularly significant in the formation of close binaries; this has been discussed and investigated intensively over the past two decades (for a short summary, see Fabrycky & Tremaine 2007). The statistically significant lack of short (P₂ < 1000 d) outer period ternaries amongst solar or lower mass binaries (Tokovinin 2014b) makes such investigations especially important.

The first, preliminary, systematic search of Kepler ETV data for hierarchical triples was carried out by Gies et al. (2012), who identified possible long-term ETVs in 14 of 41 EBs but did not find any evidence of short-period companions (P₂ < 700 d). Later, Rappaport et al. (2013) surveyed the whole available Q0–Q13 data set for some 2100 EBs. They found 39 candidate triple systems in the short-outer-period domain (48 d <P₂ < 960 d), for which they presented combined LTTE+dynamical effect solutions. This was the first systematic study of the dynamical effect in EBs using the Kepler data. Nearly contemporaneously, Conroy et al. (2014) determined eclipse times for all the short-period EBs, most of which are overcontact systems, and ELVs, and identified 236 systems for which the ETVs could be compatible with the LTTE. However, the majority of these were observed for less than one complete outer (third-body) orbital period. More recently, Borkovits et al. (2015) investigated 26 Kepler-field eccentric EBs which feature ETVs that are dominated by dynamical perturbations rather than LTTE. This work featured the simultaneous analysis of both the primary and secondary eclipses so as to break a number of degeneracies in the solutions. In a report published in 2015 June, during the preparation of the present paper, Zasche et al. (2015) published light-curve and ETV analyses of 10 detached or semi-detached Kepler-field EBs. For most of these 10 systems, the authors were able to extend the duration of the flux time series by including ground-based timing measurements. Finally, after the submission of the present paper, Gies et al. (2015) published an improved analysis on the same subset of 41 EBs which was previously investigated in their earlier work (Gies et al. 2012). They now provide third-body LTTE solutions for seven EBs. Additional studies of a possible third body affecting the ETVs of individual EBs in the Kepler field have also been reported in Steffen et al. (2011, for KOI-928 (=KIC 09140402)), Borkovits et al. (2013, for HD 181068 (=KIC 05952403)), Lee et al. (2013, for KIC 02856960), Lee et al. (2014, for V404 Lyr (=KIC 03228863)), and Lee, Hong & Hinse (2015, for KIC 05621294). Most recently, Baran et al. (2015) reported the detection of a planet-mass companion in the sdB+dM EB 2M1938+4603 (=KIC 09472174).

In the present paper, we regenerate and reanalyze the ETVs of all the previously investigated triple-body candidate EBs, with the exception of the 26 systems investigated in Borkovits et al. (2015), and we extend our analysis to longer period systems which were excluded from the study of Conroy et al. (2014). While the new study of the previously investigated systems is natural because of the significantly longer time span of the Q0–Q17 Kepler observations, there are additional reasons to further investigate the EBs listed in Conroy et al. (2014). First, our method for determination of times of minima gives results for semi-detached or detached systems, i.e. systems with relatively sharp and deep minima, that are significantly more accurate than the times for these systems used in Conroy et al. (2014). Secondly, for overcontact EBs and ELVs, we also analyse quadrature timing variations (QTV), i.e. O−C times of maxima. Thirdly, we checked the individual LTTE solutions in detail with particular attention to whether the inferred masses could be reliable, and, in the cases where further treatment was indicated, we modelled the effects of dynamical perturbations of the binaries. Finally, for the minority of the investigated EBs for which pre-Kepler ground-based times of minima were available, we also included these data in our analysis. In such a way, we were also able to improve the reliability of the LTTE solutions for previously investigated systems.

In what follows, in Section 2 we briefly describe the LTTE and dynamical perturbation effects. Then, in Section 3 we outline the method of calculating accurate times of eclipse and non-eclipse minima as well as our method for searching for ETV solutions. We introduce the idea of determining times of light-curve maxima, and utilize these so-called quadrature timing curves as diagnostics to weed out false positives. Section 4 discusses the use of supplementary ground-based timing data for extending the overall span of the observations for a small subset of our triples. Section 5 gives an overview of the 230 systems that we investigated. This includes a plot of each ETV curve with fitted solution as well as an extensive set of tables listing fitted system parameters. In Section 6, we discuss our findings from a number of different perspectives, and we draw some conclusions from this substantial statistical collection of triple-star systems. Finally, we summarize our work in Section 7.

2 EFFECTS OF A THIRD BODY ON THE ETV

The coefficients c₀ and c₁ were adjusted simultaneously with the physical terms in all analyses. The quadratic coefficient c₂ was allowed to be non-zero only for originally parabolic ETVs or when the LTTE fitting yielded parabolic residuals; in these cases, the quadratic term was determined simultaneously with all other included terms. The coefficient of the cubic term was set to zero except in five cases wherein at least three full outer periods were observed; this yielded reduced-size O−C residuals without substantially altering the orbital parameters. The parameters of the LTTE-term (see below) were adjusted in all cases. Dynamical ETV contributions were considered for a subset of our sample where there was some indication that a pure LTTE solution would not be adequate. Finally, the apsidal motion contributions were also taken into account for a few eccentric EBs.

As will be discussed in Section 3, for systems with significant ellipsoidal light variations, we also measure and analyse the times of the ellipsoidal maxima. As most of the systems with well-measured ellipsoidal variations – being overcontact or semi-detached systems – revolve in circular orbits, the maximum brightnesses occur near quadrature phases (i.e. ϕ = 0.25 and 0.75) and, therefore, we refer to the O−C times of the maxima as QTV. The QTV curves for LTTE and quadratic variations must have the same form as given by equation (1). The dynamical contribution and the AME term would, however, be different for quadratures, but, practically speaking, these effects would have needed considerable extra care only for ‘heart-beat’ binaries (Thompson et al. 2012), of which only one, KIC 03766353, is covered in this paper. Note also that generalizing the natural convention that the epoch or cycle number (E) is integer for primary and half-integer for secondary minima, we calculate it as E + 0.25 for the first quadrature (at ϕ ∼ 0.25, i.e. after the primary minima) and E + 0.75 for the second quadrature.

The mathematical form and other properties of the different ETV contributions were discussed comprehensively in Borkovits et al. (2015). Here we discuss briefly, and from a bit different point of view, only the two main effects which were applied in this work.

2.1 The light-travel time effect

General criteria for the plausibility of an LTTE model of ETVs have been given by Frieboes-Conde & Herczeg (1973). The criteria may be summarized as follows. (1) The shape of the ETV curve must follow the analytical form of an LTTE solution. (2) The ETV of the secondary minima must be consistent in both phase and amplitude with the primary ETV. (3) The estimated mass or lower limit to the mass of the third component, derived from the amplitude of the hypothetical LTTE solution via the mass function – see below, must be in accord with photometric measurements or limits on third light in the system. (4) Variation of the system radial velocity should be in accord with the LTTE solution. While these criteria do not look very restrictive, none of their candidate systems fulfilled all of them. More than 50 years after the first mathematical description of the problem, there was only one system, Algol itself, where the LTTE was identified clearly via its ETV curve. Even over the ensuing decades, the number of confirmed LTTE cases has grown very slowly. The reason is as follows.

Over the last several decades, the advent of CCD detectors and other advances has led to the acquisition of much new and relatively accurate EB timing data that have, in turn, made it possible to tentatively or definitely detect LTTE in hundreds of EBs. Most of these LTTE solutions reveal companions with orbital periods longer than a decade. Third stars were found in shorter period orbits only for a very limited number of EBs. Before the era of the Kepler space mission, IU Aurigae was the only EB system in which there was a detection of LTTE due to a third-star companion with a period shorter than 1 yr (Mayer 1983). All the other tertiaries with periods less than 1 yr had been discovered spectroscopically in accord with the fact that spectroscopic detection is much more effective for short-period outer orbits (see, e.g., Mayer 1990; Tokovinin 2014a). However, spectroscopy requires much more light and, therefore, larger instruments as well as exposure time for a given system than photometry. Thus, the majority of the EBs are too faint to be suited for spectroscopic third-body detection.

In such a way, the Kepler mission offers an unprecedented opportunity for the discovery of short-period companions orbiting EBs, including also lower mass systems, such as, e.g., the majority of overcontact binaries, which are usually too faint for spectroscopic investigations. Furthermore, in contrast to the earlier, ground-based observations, which were inhomogeneous, and generally restricted to small portions of the EBs’ light curves around their eclipsing minima, Kepler observations provide almost continuous and highly homogeneous light curves over intervals as long as four years. As a consequence, we are now in a position to extend our timing investigations to the out-of-eclipse parts of the light curves. Accordingly, an additional criterion of reliable LTTE solution can be introduced, as (5) the times of the maxima of the ellipsoidal variations, at least in EBs that have circular orbits, should be in accord both in phase and amplitude with the ETVs.

Another never-seen-before feature is the presence, in a small number of Kepler light curves, of outer eclipses. For such systems, a further natural criterion for identifying the outer eclipsing body with the source of the observed LTTE is that (6) the LTTE should exhibit the same period as the extra eclipses, and the latter should occur around the extrema of the LTTE. In Sections 3 and 6.3, we illustrate the applications of these new criteria.

2.2 Dynamical perturbations of a third body on the ETV

If an EB has a more or less distant companion, its binary motion no longer remains purely Keplerian since time-dependent perturbations affect all six orbital elements. Naturally, the occurrence times of the eclipses are also affected. The perturbations of the ETVs were first studied in this context by Söderhjelm (1975) and Mayer (1990). Later, the third-body effects were elaborated in full in a series of papers by Borkovits et al. (2003, 2011, 2015), and, in the context of transit timing variations of exoplanets, by Agol et al. (2005).

Expressions for the dynamical perturbation ETVs for EBs with elliptical inner orbits (e₁ > 0) are much more complicated (see Borkovits et al. 2011, 2015). In particular, the amplitude of the ETVs depends sensitively on the eccentricities of the binary and third-star orbits and on the mutual inclination of the two orbits. Therefore, even for a given mass and period ratio, the amplitude may take a value within a wide range, as was illustrated, e.g., in fig. 3 of Borkovits et al. (2011).

The analysis itself was carried out in the same manner and with the same code that was described in detail in Borkovits et al. (2015).

3 SYSTEM SELECTION AND DATA PREPARATION

We use the present version of the Kepler EB catalogue and light-curve files available at the Villanova website⁵ (Slawson et al. 2011; Matijevic et al. 2012; Conroy et al. 2014; LaCourse et al. 2015). All the light-curve files for the sources in the original Kepler field were downloaded, and, using the first (BJD), seventh (detrended relative flux), and eighth (uncertainty of the latter) columns of these files, O−C diagrams were formed in an automated way. For a significant portion of the systems to be investigated, some quarters of the observed data sets (Q4 and/or Q12–13) were not available at the Villanova site; in most of those cases, we downloaded the missing data directly from the MAST data base operated by the Space Telescope Science Institute,⁶ converted it into the proper format, and merged it with the Villanova-derived data set. We then selected those systems which either were mentioned in the context of having third components in previous literature or had interesting preliminary O−C curves. For the selected systems, we calculated more accurate eclipse times in a somewhat more sophisticated, semi-automated manner. Our method, which is based on forming folded, binned, and averaged light curves for the whole data set of each EB, then constructing polynomial templates for intervals around the minima of these averaged light curves, and finally using these templates for fitting individual minima, was described in detail in section 4 of Borkovits et al. (2015). Therefore, here we note only some subtleties and variations specific to the present work.

This procedure yielded some 400 ETVs for further analysis. The majority of these systems definitely show ellipsoidal variations, which makes it possible to calculate not only times of the eclipses, but also of the maxima in the light curves. The latter set of QTVs was produced in the same manner as the ETVs. We found that in the case of overcontact EBs and most of the ELV binaries, with the exception of a few eccentric ELVs, it was satisfactory to set the phase limits for building up minima and maxima templates to ϕ_p = [−0.15; 0.15], ϕ_s = [0.35; 0.65], for primary and secondary minima, and ϕ_q1 = [0.10; 0.40] and ϕ_q2 = [0.60; 0.90], for the first and second quadratures (maxima), respectively. For semi-detached and detached systems with definite and sharper eclipses, narrower phase limits were set for the minima. However, in so far as the out-of-eclipse section of a folded, binned light curve also exhibited ellipsoidal light variations, and the latter property of the averaged curve was not altered by any cycle-to-cycle variations (see below) on the original light curve, we again calculated quadrature (or maxima) templates, applying mostly the same phase constraints. (Note, however, that in the case of a few eccentric systems, we departed from the above-listed phase constraints in also calculating quadrature templates, in accordance with the properties of the given individual light curve.)

Then, having obtained templates, times of individual minima and maxima were determined in exactly the same manner as described in Borkovits et al. (2015). In such a way, we have obtained 1–2 ETV and 0–2 QTV curves for each system. In several cases, these curves are obviously distorted by the effects of stellar spots, pulsations, or oscillations. Fortunately, as was shown in Tran et al. (2013), stellar spots in general distort the primary and secondary ETVs in an anticorrelated way. Similarly, the distortions in the two QTVs due to stellar spots also anticorrelate with each other and, furthermore, they are shifted by |$\pm 90\deg$| in phase from the respective ETVs (see Fig. 1). Therefore, the effects of starspots can be significantly reduced by averaging the primary and secondary ETVs, and the two QTVs as well. Thus, we also calculated averaged ETVs and QTVs. This process was carried out by interpolating the times of the primary ETVs to the times of the corresponding secondary eclipses with the help of a cubic spline. The same was done for the QTV curves.

Figure 1.

The highly anticorrelated, quasi-periodic ETV and QTV curves of the overcontact EB KIC 06431545. This type of ETV variations, which is likely attributable to large spotted areas on the stellar surface(s), was first reported in Tran et al. (2013), and was also investigated by Balaji et al. (2015). Left-hand panel: the individual primary (red circles) and secondary (blue boxes) ETV, and first quadrature (directly after the primary eclipses; magenta upward triangles) and second quadrature (black downward triangles) QTV curves. Right-hand panel: the averaged ETV (red) and QTV (black) curves show only some low-amplitude residuals, while the difference curves between the two ETVs (blue) and QTVs (magenta) exhibit a phase shift of one-fourth of a period between the two sets.

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In our experience, this averaging process is most effective for overcontact EBs and low-eccentricity or circular orbit ELVs where the two minima, and also the two maxima, are comparable in both amplitude and duration. Therefore, on the one hand, the times of mid-minima and mid-maxima can be determined with approximately the same accuracy, while on the other hand, they are affected by the distortions more or less at the same level (see Fig. 2). Another benefit of forming averaged ETVs (and QTVs) is the reduced scatter in the O−C curves with respect to the original ones for several systems and, therefore, in these cases we used the averaged curves instead of the individual ETV curves for LTTE fitting.

Figure 2.

The highly irregular O−C curves of the low-amplitude, short-period, possibly overcontact EB KIC 02715417. Left-hand panel: the individual primary (red circles) and secondary (blue boxes) ETV, and first quadrature (directly after the primary eclipses; magenta upward triangles) and second quadrature (black downward triangles) QTV curves. Right-hand panel: the averaged ETV (red) and QTV (black) curves reveal some (quasi-)periodic variations similar both in magnitude and phase for the two curves; this indicates that the LTTE curve could be due a low-mass (or very low inclination) third companion.

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Another method useful for reducing or eliminating the influences of intrinsic brightness variations on the times of minima is local smoothing of the light curves. This method was applied by fitting a low-order (typically fourth-order) polynomial to a portion of each light curve centred on each minimum but excluding the minimum itself, i.e. usually in the intervals [−0.25; ϕ_{p, sleft} − 0.02] and [ϕ_{p, sright} + 0.02; +0.25]. This polynomial was then subtracted from the entire interval ([−0.25; 0.25]; see the left-hand panels of Fig. 3). This method yielded excellent results for several systems affected by starspots and even for systems affected by stellar oscillations. Some examples are shown in the right-hand panels of Fig. 3. An additional example of the oscillating EB system KIC 08560861 can be found in fig. 1 of Borkovits et al. (2014). Local smoothing was found to be effective mainly for detached systems with definite and sharp eclipses, but we could also use it even for some semi-detached binaries. For overcontact EBs and ELVs, however, this algorithm cannot be used.

Figure 3.

Two examples of the workings and efficiency of local smoothing with fourth-order polynomial fits. Both KIC 05216727 (upper row) and KIC 09711751 (bottom row) are Algol-type EBs, and exhibit likely rotational variations due to starspots. Left-hand panels show small segments of their detrended Kepler long-cadence light curves (red) and the corresponding locally smoothed curves (blue). Right-hand panels give the ETV curves obtained from both the original unsmoothed and the locally smoothed light curves. One can see that the method is more effective, and eliminates nearly perfectly the effects of the (rotational) distortions for the deeper primary minima. As to the shallower secondary minima, some residual distortions survive, but the magnitude has been substantially reduced. (For better visibility, the ETV curves for the secondary minima are shifted down from the primary ETV curves.)

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The use of QTVs and the averaging and smoothing techniques made it possible not only to find and determine lower amplitude LTTE solutions, but also to apply more stringent criteria for filtering out false positive systems. An example is KIC 11247386, a possible overcontact EB (P₁ = 0.394 d) with a remarkable O'Connell effect.⁷ Conroy et al. (2014) give an LTTE solution with a period P₂ = 71.2 ± 0.1 d, which would be the shortest outer period in their sample. As one can see in the left-hand panel of Fig. 4, this periodicity is definitely present in the primary ETV and the second QTV with quite different amplitudes, is hardly visible in the first QTV, and is even weaker in the secondary ETV. Such amplitude ratios are not typical of LTTE induced by a third body. There are a few additional systems where the averaged QTVs behave similarly to the averaged ETVs, but there are minor discrepancies in the amplitudes. A typical example is shown in the right-hand panel of Fig. 4. In the latter cases, we accept the LTTE solution, and note the amplitude discrepancy in the tabulated results.

Figure 4.

Examples of discrepant ETVs and QTVs. Left-hand panel: a clearly false positive case: the ETV and QTV curves of KIC 11247386. The P₂ = 71.2 ± 0.1 d periodicity attributed to LTTE by Conroy et al. (2014) is readily visible. The different amplitudes for the different curves clearly show, however, that the origin of this feature cannot be LTTE due to a third body. Therefore, we categorized this system as a false positive in the sense that there is no evidence for this being a triple-star system. (Note that a careful inspection of the curves also reveals that some phase discrepancies also occur between the curves.) Right-hand panel: the case of a marginally acceptable LTTE solution: the averaged ETV and QTV curves of KIC 11246163. Although the amplitudes of the two curves differ slightly, we do not rule out a possible LTTE origin.

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Another group of false positives comprises those objects where, although the ETV may suggest an LTTE solution, the Kepler target was erroneously classified as either an EB or as an ELV binary. For example, δ Scuti-type oscillating variables can easily be misclassified as ELVs or even low-amplitude overcontact EBs. This is quite likely especially in systems with P ≲ 0.15 d. In the absence of radial velocity measurements which would be able to confirm or reject the binary hypothesis for candidate ELV binaries (see, e.g., Tal-Or, Faigler & Mazeh 2015), we could make decisions on the nature of such systems based on temperature or colour information when available. Instead, our decisions are based on the characteristics of the folded, binned light curves and the ETV and QTV data.

There are light-curve-based checks that may reveal whether a target is actually a physical binary. For example, the light curve of an ELV binary is dominated by a sinusoidal component with a period that is half of the orbital period. The two sections of the folded and binned light curve in the phase ranges [0.0; 0.5] and [0.5; 1.0] typically differ noticeably from each other due to Doppler boosting (see, e.g., van Kerkwijk et al. 2010) and reflection effects (see, e.g., Zucker, Mazeh & Alexander 2007), not fully averaged-out spot effects, or even, in the case of detached ELVs, orbital eccentricity. On the other hand, if the variations originate in stellar oscillations or pulsations, the underlying oscillation period will be half of the inferred orbital period and the light curve will be more or less identical in the two phase intervals. Furthermore, in such cases the times of consecutive minima (or maxima) tend to consistently follow one ETV (or QTV) curve. By contrast, as was shown in Tran et al. (2013) and Balaji et al. (2015), in the case of ELVs and overcontact systems, especially those which are formed by spotted stars, consecutive minima and maxima timings may show different O−C patterns. Therefore, in accord with the suggestion of Tran et al. (2013), all the sources which produce significantly different pairs of ETVs (and/or pairs of QTVs) are not likely to be due to oscillations or pulsations. In summary, if the two sections of a light curve differ, the system may be a binary. If the two parts of the light curve happen to be identical, and ETVs and QTVs also look very similar, we take the source to be a false positive binary with a high likelihood. Examples of these checks may be found in Fig. 5.

Figure 5.

One verification and two rejections. Three systems where, at first, the classifications as binaries are ambiguous. Left-hand panels: the folded and binned long-cadence light curves and their 0.5 d phase-shifted versions (red and black, respectively). Middle panels show the individual O−C curves belonging to the purported primary and secondary ETVs as well as the first and second QTVs. Right-hand panels: the average and the difference of the two ETVs and QTVs are plotted. In the case of KIC 01873918, which has been flagged as a false positive in the Kepler EB catalogue (first row), the light curve shows alternating maxima and minima that are slightly different in amplitude, thereby indicating that this is not a sinusoidal pulsator. The quasi-anticorrelated behaviour in the ETV curves, and also in the QTV curves, adds confidence to this being a binary. Therefore, we conclude that this system is indeed a binary within a triple system. In the case of the ultrashort-period KIC 10855535 (middle row) and the longer period KIC 08099615 (bottom row), the alternating maxima and minima of the light curves look completely the same, suggesting another type of variability with half of the given period. Furthermore, the two ETVs and also the QTVs track each other, which further strengthens the false binary hypothesis. Independent of this fact, the presence of the LTTE effect in the ETV and QTV curves of KIC 10855535 seems very clear, and thus we may conclude that this system is actually a wide binary (instead of being a triple) with a period of P_LTTE = 411.9 ± 0.2 d. For KIC 08099615, the large-amplitude, peculiar ETV (and QTV) might have a different origin.

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After obtaining pre-processed ETV and QTV curves and weeding out likely false positives in the above manner, the next task was to decide whether a pure LTTE solution, or a combined dynamical and LTTE solution, for a given system should be sought. In most cases, the decision was evident as, on one hand, some of the ETVs had shown features typical of dynamical perturbations (for a detailed discussion, see Borkovits et al. 2015), or, on the other hand, a large P₂/P₁ ratio indicated that dynamical contributions would be negligible. Extra care was necessary, however, for systems with relatively sinusoidal ETVs and moderate P₂/P₁ ratios. Therefore, in all cases, when a pure LTTE solution was obtained, we also estimated the possible relative contribution of the dynamical perturbations. For this, the binary mass was approximated by 2 M_⊙ and then by the use of the mass function f(m_C) obtained from the LTTE solution, a minimum mass of the third body was calculated, as well as the minimum value of the dynamical amplitude (equation 12). Then, when the estimated ratio |$\mathcal {A}_\mathrm{dyn}/\mathcal {A}_\mathrm{LTTE}$| exceeded ∼25 per cent, we also calculated a combined LTTE+dynamical solution.

4 SUPPLEMENTAL GROUND-BASED ECLIPSE TIMING

Before giving an overview of our results, we briefly discuss the use of pre- or post-Kepler ground-based eclipse measurements which are available for a small number of our triples. As was discussed in Section 2, because of their limited accuracy, ground-based timing measurements of eclipses and light-curve minima are generally not suitable for third-body ETV studies in the period range of P₂ ≲ 1–2 yr. Even for systems where the outer period is comparable to the length of the Kepler data set, supplementary ground-based times of minima collected over a somewhat wider time span may serve mainly to confirm or reject a possible solution rather than to quantitatively improve it.

Supplementary ground-based times of minima are most useful for those systems that were discovered as EBs well before the Kepler era when times of minima are available over a time span that is much longer even by order(s) of magnitude than the Kepler data train itself. However, our sample includes only seven EBs for which there are times of minima taken over a time span longer than a decade. Some dozens of our sample EBs, however, were observed a few years before the beginning of the original Kepler mission in the photometric surveys of ASAS (Pigulski et al. 2009), HATNET (Hartman et al. 2004), TRESS (Devor et al. 2008), and SuperWASP (Pollacco et al. 2006). Unfortunately, the times of minima obtained from these data bases often offer only lesser benefits because of the restricted extension of the data span and, in several cases, the sampling rate of the observations of each EB was so infrequent that the data do not yield times of individual eclipses with useful accuracy. For these reasons, we did not determine and utilize the times of minima from the observations of the above-listed surveys for all systems; we make use of the data only for those EBs for which the eclipse times were determined by Lee et al. (2014) and Zasche et al. (2015).

Ground-based times of minima obtained from targeted eclipse observations of individual binaries are particularly helpful. In most cases, these were collected from the Lichtenknecker-Database of the Bundesdeutsche Arbeitsgemeinschaft für Veränderliche Sterne e. V. (BAV),⁸ rather than from the journal literature. The sole exceptions are a few recently observed post-Kepler times of minima published in Zasche et al. (2015). Some of the oldest times of minima in the extended ground-based data sets are based on visual brightness estimations which have a highly limited accuracy of 5–10 min. Despite this, we decided to keep these observations (with the exception of the evident outliers) in the cases where they could substantially extend the overall span of the data.

Given the available data and the above considerations, we were able to extend our timing data sets with ground-based measurements for about a dozen systems. In some cases, however, the ground-based minima evidently contradict the Kepler observations. In the case of KIC 092883826 (=V2366 Cyg), we found two ground-based times of minimum which were obtained from observations during the Kepler era but were inconsistent with the Kepler times. In the case of KIC 09101279 (=V1580 Cyg), three ground-based times of minimum would have extended the data span by a factor of 3, but they did not match our ETV solution from the Kepler data and, therefore, were not considered further. In a case yielding an opposite conclusion, for KIC 010581918 (=WX Dra) we rejected the Kepler LTTE solution, and therefore deleted the EB from our sample because of the contradictory characteristics of the relatively numerous ground-based data.

In summary, we kept all or a part of the ground-based times of minima for eight EBs. In the 221 panels of Fig. 6, we plot the O−C curves of almost all of the investigated EBs (see Section 5). For the eight systems where ground-based minima were also incorporated for the third-body solution, these ground-based minima are plotted together with their uncertainties. As one can see, these uncertainties in some cases are larger than the full amplitude of the third-body ETV feature for these systems. In other cases, however, the extended data set was found to be suitable for confirming, or even improving, the third-body solutions as will be discussed in the next section.

Figure 6.

ETVs with third-body solutions. ETV curves calculated from Kepler observations of primary and secondary minima, and the average of the two, are denoted by red circles, blue boxes, and orange diamonds, respectively. We display and fit the ETV curves for both the primary and secondary eclipses only when the data quality warrants a joint analysis and the binary is eccentric. If the primary and secondary ETV curves are of comparable quality and the binary eccentricity is nearly zero, we display and fit only the average of the two ETV curves. If the quality of the primary ETV curve is significantly better than that of the secondary curve or if only primary eclipses are present, we present only the plot and the fit for the primary eclipses. Ground-based minima (taken from the literature, and available only for a few systems) are denoted by upward red triangles (primary) and downward blue triangles (secondary); their estimated uncertainties are also indicated. Pure LTTE solutions are plotted with black lines, while combined dynamical and LTTE solutions are drawn with grey lines. (Note that the use of quadratic or cubic terms is not indicated; for these and other details, see Table 2.) The complete figure covering 221 ETV curves is available in the online version of the journal.

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5 OVERVIEW OF THE INVESTIGATED SYSTEMS

We give LTTE and/or dynamical solutions for 230 Kepler systems. Some parameters of these systems are given in Table 2, where the basic properties of the generated ETV and QTV curves, the types of our solutions, and relevant references are also listed.