We present a new method to detect planetary transits from time-series photometry, the transit least squares (TLS) algorithm. TLS searches for transit-like features while taking the stellar limb darkening and planetary ingress and egress into account. We have optimized TLS for both signal detection efficiency (SDE) of small planets and computational speed. TLS analyses the entire, unbinned phase-folded light curve. We compensated for the higher computational load by (i.) using algorithms such as "Mergesort" (for the trial orbital phases) and by (ii.) restricting the trial transit durations to a smaller range that encompasses all known planets, and using stellar density priors where available. A typical K2 light curve, including 80 d of observations at a cadence of 30 min, can be searched with TLS in ∼ 10 s real time on a standard laptop computer, as fast as the widely used box least squares (BLS) algorithm. We perform a transit injection-retrieval experiment of Earth-sized planets around sun-like stars using synthetic light curves with 110 ppm white noise per 30 min cadence, corresponding to a photometrically quiet K P = 12 star observed with Kepler. We determine the SDE thresholds for both BLS and TLS to reach a false positive rate of 1 % to be SDE = 7 in both cases. The resulting true positive (or recovery) rates are ∼ 93 % for TLS and ∼ 76 % for BLS, implying more reliable detections with TLS. We also test TLS with the K2 light curve of the TRAPPIST-1 system and find six of seven Earth-sized planets using an iterative search for increasingly lower signal detection efficiency, the phase-folded transit of the seventh planet being affected by a stellar flare. TLS is more reliable than BLS in finding any kind of transiting planet but it is particularly suited for the detection of small planets in long time series from Kepler, TESS, and PLATO. We make our python implementation of TLS publicly available.

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The unprecedented light curves of the Kepler space telescope document how the brightness of some stars pulsates at primary and secondary frequencies whose ratios are near the golden mean, the most irrational number. A nonlinear dynamical system driven by an irrational ratio of frequencies generically exhibits a strange but nonchaotic attractor. For Kepler 's "golden" stars, we present evidence of the first observation of strange nonchaotic dynamics in nature outside the laboratory. This discovery could aid the classification and detailed modeling of variable stars.

Context. Transit photometry of the Jupiter-sized exoplanet candidate Kepler-1625 b has recently been interpreted to show hints of a moon. This exomoon, the first of its kind, would be as large as Neptune and unlike any moon we know from the solar system. Aims. We aim to clarify whether the exomoon-like signal is indeed caused by a large object in orbit around Kepler-1625 b, or whether it is caused by stellar or instrumental noise or by the data detrending procedure. Methods. To prepare the transit data for model fitting, we explore several detrending procedures using second-, third-, and fourthorder polynomials and an implementation of the Cosine Filtering with Autocorrelation Minimization (CoFiAM). We then supply a light curve simulator with the co-planar orbital dynamics of the system and fit the resulting planet-moon transit light curves to the Kepler data. We employ the Bayesian Information Criterion (BIC) to assess whether a single planet or a planet-moon system is a more likely interpretation of the light curve variations. We carry out a blind hare-and-hounds exercise using many noise realizations by injecting simulated transits into different out-of-transit parts of the original Kepler-1625 light curve: (1) 100 sequences with three synthetic transits of a Kepler-1625 b-like Jupiter-size planet and (2) 100 sequences with three synthetic transits of a Kepler-1625 b-like planet with a Neptune-sized moon. Results. The statistical significance and characteristics of the exomoon-like signal strongly depend on the detrending method (polynomials versus cosines), the data chosen for detrending, and on the treatment of gaps in the light curve. Our injection-retrieval experiment shows evidence of moons in about 10 % of those light curves that do not contain an injected moon. Strikingly, many of these false-positive moons resemble the exomoon candidate, i.e. a Neptune-sized moon at about 20 Jupiter radii from the planet. We recover between about a third and half of the injected moons, depending on the detrending method, with radii and orbital distances broadly corresponding to the injected values. Conclusions. A ∆BIC of −4.9 for the CoFiAM-based detrending is indicative of an exomoon in the three transits of Kepler-1625 b. This solution, however, is only one out of many and we find very different solutions depending on the details of the detrending method. We find it concerning that the detrending is so clearly key to the exomoon interpretation of the available data of Kepler-1625 b. Further high-accuracy transit observations may overcome the effects of red noise but the required amount of additional data might be large.