An Exo-Jupiter Candidate in the Eclipsing Binary FL Lyr
V. S. Kozyreva, A. I. Bogomazov, B. P. Demkov, L. V. Zotov, A. V. Tutukov
aa r X i v : . [ a s t r o - ph . S R ] D ec An Exo-Jupiter Candidate in the Eclipsing Binary FL Lyr
V. S. Kozyreva , A. I. Bogomazov , B. P. Demkov , ,L. V. Zotov , A. V. Tutukov Sternberg Astronomical Institute, Lomonosov Moscow State University,Universitetskii pr. 13, Moscow, 119991 Russia “IT Project”, Savelkinskii proezd 4, Zelenograd, Moscow, 124482 Russia All-Russian Research Institute of Physical, Technical,and Radio Technical Measurements,Mendeleevo, Moscow Region, 141570 Russia Institute of Astronomy, Russian Academy of Sciences,ul. Pyatnitskaya 48, Moscow, 119017 Russia
Light curves of the eclipsing binary FL Lyr acquired by the Kepler space telescope are ana-lyzed. Eclipse timing measurements for FL Lyr testify to the presence of a third body in thesystem. Preliminary estimates of its mass and orbital period are & M J and & Extra-solar planetary systems remained hypothetical objects until the 1990s, whenmodern means for their detection were developed. Since then, some 10 candidate ex-oplanets have been discovered using various methods; the existence of many of theseexoplanets has been reliably confirmed [1]. The vast majority of the discovered planetsorbit single stars or individual components of wide multiple systems.Currently, we know of eight exoplanets in five stellar systems and two candidate planetsthat simultaneously orbit both components of binaries, with both stars on the mainsequence. The first planet discovered in such a binary was Kepler-16 (AB)b [2]. Othersinclude Kepler-34 (AB)b and Kepler-35 (AB)b [3], Kepler-38 (AB)b [4], Kepler-47 (AB)bKepler-47 (AB)c [5], PH1-Kepler-64 b [6], Kepler-413 (AB)b [7, 8], a possible third planetin the Kepler-47 [9] system and the candidate planet KIC 9632895 (AB)b [10]. Severalplanets near cataclysmic variables and a planet near the young star FW Tau [1] have alsobeen discovered.Searching for planets in binary systems is important for a number of reasons. Thoughit follows from [11] that planetary orbits in binaries exhibit long-term stability, it remainsto be confirmed from observations that planets can survive in systems with various pa-rameters. The systems known up to now have very similar parameters. The presence orabsence of planets in binary systems and the systems‘ parameters are very important forour understanding of the processes of star and planet formation (e.g., [12]). In addition,1inary systems are more favorable for harboring life than single stars, and could in princi-ple have several inhabited planets [13]. This makes searches for planets in various binarysystems very important for searches for extraterrestrial, possibly even intelligent, life. Alist of binary stars suitable for planetary searches can be found in [14], and includes theFL Lyr system.In 2009-2014, the area of the sky containing FL Lyr was in the field of view of theKepler space telescope [15], which was launched into near-solar orbit with the aim ofsearching for exoplanets. During these years, the telescope carried out a continuous pho-tometric sky survey, during which a large amount of observing material was accumulatedfor FL Lyr. The Kepler observations can be used to study various scientific problems.In particular, a third body orbiting an eclipsing variable star gives rise to periodic shiftsof the system’s center of mass with respect to the observer, causing the observed orbitalperiod of the binary to vary about a certain value. The aim of our current study is tostudy the light-time effect in the FL Lyr system . The eclipsing variable FL Lyr was discovered on photographic plates in 1935 [16]. Itsminima are deep, with the change in the star’s brightness at the primary and secondaryminimum being different by a factor of two: m max = 9 m . m min I = 9 m . m min II =9 m .
52. According to the “General Catalog of Variable Stars” [17], P orb = 2 . d M = 1 M ⊙ , a relative radius r = 0 .
132 and a relative luminosity in the V filter L = 0 . M = 1 . M ⊙ , r = 0 . L = 0 . L = 0 . In other words, to perform timing of the minima of the FL Lyr light curve. Expressed in fractions of the orbital semi-major axis of FL Lyr. Expressed in fractions of the system’s combined luminosity. e · cosω ≤ . . m − . m
05. It is possible that all these distortions have a random character,due to systematic errors in the particular part of the FL Lyr light curve studied in [20].In 1986, Popper et al.[21] obtained photoelectric light curves and spectroscopic radial-velocity curves of FL Lyr. They derived a new photometric solution of the light curve( r = 0 . r = 0 . i = 86 ◦ . L = 0 . L = 0 . . M ⊙ , 0 . M ⊙ , F8+G8. The binary orbit is circular withhigh accuracy. Comparing the observed parameters of the system to those determinedfrom theoretical evolutionary tracks of stars of the same mass, they estimated the age ofthe FL Lyr system to be 5 . · –3 . · yrs, with the most likely age being 2 . · yrs[22]. Since the MS lifetime of a star with a mass of 1 . M ⊙ is approximately 4 . · (Eq.6 in [23]), neither component of FL Lyr has left the MS. No third light was detected in[21]. When calculating the photometric parameters, an upper limit k ≤ k = r /r (the ratio of the radii of the secondary and the primary). Thisexcludes all solutions for which the two stars form an Algol-type system, i.e., a systemwith a reversed component-radius ratio. However, Popper et al. [21] suggest that thecorrectness of their derived parameters is supported by the lack of systematic deviationsbetween the calculated and observed values for the brightness difference as a function oftime. The parameters of the stars differ considerably from the solution found by Cristaldi[19]. Among the characteristic features of the light curves, Popper et al. [21] noted abrightness modulation (∆ m = 0 . m We studied data obtained with Kepler. The main goal of the Kepler project wasto search for exoplanets using observations of their transits. We used the Kepler datafor eclipse-timing measurements (determining the light-time effect) for FL Lyr. Detailedinformation on the Kepler space telescope can be found in [24].The Kepler data we used can be found in the Barbara A. Mikulski Archive for SpaceTelescopes [25], which is supported by the Space Telescope Science Institute. The identi-fication number of FL Lyr in the Kepler Input Catalog is 9641031. Detailed informationon the search and retrieval of data from the archive can be found in [24].The Kepler archive consists of data files in FITS format. Two versions of these filesare provided: LC (long cadence) and SC (short cadence). LC is the main version; these3ata were collected once each 30 minutes. One LC FITS file contains observations ofone object over one quarter . SC (short cadence) SC is a complementary version of thedata (intended for variability and asteroseismology studies); these data were collectedonce each minute. A single SC FITS file provides data for one month for a single object.Because of the design of Kepler, SC data were not accumulated during every quarter ofthe telescope’s operation. SC data for FL Lyr are fully available only for the observingquarters 7, 8, 13, 14, 15, and 16. To improve the accuracy of our study, we used theFITS files obtained in SC mode. We converted the FITS files to a form convenient forthe analysis using the IRAF software with the PyRAF extension (the kepconvert routine,which converts FITS files to text files). One of the methods that can be used to detect a third body in an eclipsing systemis to search for the light-time effect. The periods of the primary and secondary minimawill oscillate if the distance between the center of the solar system and the center of theeclipsing system varies. Any third body in a binary system makes the system’s centerof mass move with the period of this body’s orbit . The comparatively short period ofthe stars’ orbit about the common center (about two days) makes it possible to identifya large number of light curves within minima in the Kepler observations. Our aim is tolook for the light-time effect in the FL Lyr system; i.e., to search for shifts in the observedtimes of minima relative to the calculated values.The orbit of a binary system rotates due to tidal forces between the two stars andgeneral relativistic effects. In the case of an elliptical orbit, this rotation is manifestthrough apsidal motion, which gives rise to a shift of the observed relative to the calculatedtimes of minima. For FL Lyr, with its practically circular binary orbit ( e ≤ . ≈ . m
6) –more than twice as deep as the secondary minima – increasing the accuracy of the timingof the minima by the same factor. We identified 600 Kepler light curves within primaryminima of FL Lyr.Kepler observations possess systematic errors (see, for instance, section 7.1 in [24])– so-called linear trends, which can reach several hundredths of a magnitude during theduration of a minimum in the FL Lyr light curve. Our study of the FL Lyr light curvesalready corrected for this linear trend using correction factors shows that the trend was The light curves within the FL Lyr minima contain only five to six data points in the LC mode. The stability of planetary orbits in binary systems was studied in [11]; according to Table 6 in [11],the orbit of a planet around the central binary will be stable if the semi-major axis of the planet’s orbitis approximately a factor of four or more larger than the semi-major axis of the binary orbit; i.e., if theorbital period of the planet is longer than the orbital period of the central binary star by a factor of 10or more, as follows from Kepler’s third law. Thus, the conditions for the long-term survival of planets inthe FL Lyr system are satisfied for planetary orbital periods exceeding 20 days. ≈ . The minimization functional contains the sumof the squared differences between the observed and theoretical magnitudes at each point,including simple and linear limitations for the parameter values we seek. Because of theirvery weak influence on the light curves, we did not vary the limb-darkening coefficients u and u , and fixed them in accordance with the spectral types of the binary components(F8V + G8V [21]). Values for the theoretical coefficients u and u corresponding towavelengths in the middle of the instrumental range were taken from [32]. Some of theparameters we determined in our free search for the orbital elements and parameters differconsiderably from those obtained by Popper [21]; this is especially true for componentluminosities. This can partially be explained by the different spectral ranges used. Inour current study, we are interested in the set of elements only as a tool for deriving atheoretical curve that most closely approaches the observed curves at the primary minima.Times of minima we collected from the literature are presented in Fig. 2 and Table2. The scatter of the data points in Fig. ±
15 minutes. The scatter of the photoelectrictimes of minima can reach ± . ; the only free parameter was the shift of theprimary minimum, with all other parameters being fixed at the values indicated in column3 of Table 1. The criterion for our solution was a symmetric position of the deviations(between the observed and calculated light-curve points) relative to zero phase. Wechecked this by determining the linear trend in the O-C residuals, with the result beingconsidered satisfactory only in the absence of any trend. This procedure was performed forall primary minima of FL Lyr observed with Kepler; the times of minima are collected inthe first column of 3, while the second column contains the O-C residuals: the differencesbetween the observed times of minima and the theoretical times of minima calculatedwith the ephemeris (1).Searching for the light-time effect requires as accurate as possible knowledge of thebinary’s orbital period, on which the parameters of the light-time effect depend. We usedthree values of the orbital period of FL Lyr. The period P = 2 . d was taken The same algorithm was used earlier in [27]–[31], resulting in the discoveries of brown dwarfs in theHP Aur and AS Cam systems. See [27]–[31] for details. P = 2 d . d data.Finally, the orbital period P = 2 d . . The system ephemerides for these threeperiods are M in I ( HJ D ) = 2438221 . . × E ; (1) M in I ( HJ D ) = 2438221 . . × E ; (2) M in I ( HJ D ) = 2438221 . . × E ; (3)The gray triangles in Fig. 2 are the O-C values calculated with the ephemeris (1) forthe times of minima from the literature; the black circles are our values calculated usingthe Kepler observations. The shift in the times of minima we are seeking is clearly visible.We carried out our further analysis of the data obtained for the three periods ( P , P , P ). The large scatter of the O-C deviations limits our ability to obtain manyparameters of the light-time effect. To minimize the number of parameters, we adoptedthe simple hypothesis that the third body undergoes circular motion about the eclipsingbinary. Using a Fourier expansion , we analyzed the O-C residuals obtained for each ofthe periods and the calculated parameters of the best-fit sine curve approximating thetime dependence of the times of minima (the light-time effect). Table 4 presents theamplitudes and periods of this theoretical curve. Figure 3 presents the power spectrumfor the O-C residuals calculated using the ephemeris 1, and Fig. 4 displays a part of 3 ona larger scale. The peak near a period of ≈ P , P , P ) demonstrate systematic deviations that can be explained as a light-timeeffect with a period somewhat larger than the entire time interval covered by the Keplerobservations. Figures 6, 7 and 8 show the O-C deviations of the times of minima as afunction of the orbital phase of the third body.If less than a half of the period of the light-time effect elapsed during the time coveredby the Kepler observations, an alternative explanation for the observed systematic shiftscould be a variation of the close-binary period ( dP ∼ − days/year). The systemhas already been observed for a long time, and period variations of this kind shouldalready have been detected from the parabolic shape of the O-C curve. During the timeinterval of the observations (almost 60 years), the FL Lyr period variations would havealready accumulated in the fourth place after the decimal point, and the period shouldbe increasing, while all the previously measured period values [34]-[50] are not lower than So that the space data would not dominate the other measurements, we took three Kepler timesof minima for 2009 and three for 2014. The 2009 times of minima are HJD-2400000 = 54965.02424,54967.20240, and 54969.38054, and the 2014 times are HJD-2400000 = 56385.18082, 56387.35900, and56389.53716. We applied the PERDET (PERiod DETermination) code [33].
In the general case, accurately determining the mass of a body in a binary, and es-pecially a multiple, system can require a dedicated, complex study. There is no sense incarrying out such estimates in the framework of our current study, since the orbital periodof the third body has not been accurately determined, and is longer than the time coveredby the Kepler observations. Moreover, we were not able to derive the orbital inclinationof the third body relative to the orbital plane of the system. Therefore, we have obtaineda simple lower limit for the third body’s mass.Since the orbital period of the third body is much longer than the orbital period ofthe central binary, we can use Kepler’s third law to obtain a simple mass estimate P orb = 0 . a / M / , (4)where P orb is the orbital period of the third body in days, a the semi-major axis of thethird bodys orbit in solar radii, and M the combined mass of the two components of FLLyr and the third body in solar masses.The sum of the component masses in the FL Lyr system is ≈ M ⊙ [21]. The orbitalperiod of the third body with the ephemeris 1 is & & R ⊙ .The amplitude of the light-timeeffect with the same ephemeris is 4.8 s. During this time, light traverses half the distanceof the periodic shift of the FL Lyr binary due to the third body; i.e., the semi-major axisof the orbit of the FL Lyr system about the center of mass of the FL Lyr-third bodysystem is approximately 2 R ⊙ . Thus, the ratio of the third body’s mass to the mass ofthe FL Lyr binary is ≈ / M ⊙ / ≈ M J . If the orbital period of the third body proves to be longer thanour estimate, the estimated mass of this body will be lower; at the same time, the orbitalinclination of the third body will increase its estimated mass. Note that the orbital planesfor all eight known exoplanets in orbits around binaries are very close (within 1 ◦ ) to theorbital planes of their parent binaries. Thus, our rough estimate of the planet’s mass mayprove to be close to the actual mass of this planet. We have analyzed Kepler light curves for the eclipsing binary FL Lyr and detectedthe light-time effect, indicating that the system probably contains a body with a massof about four Jupiter masses, with an orbital period around the close binary of ≥ The orbital period of the third body is 7 years or more, compared to the 2-day orbital period ofFL Lyr; stable orbits admitting application of Kepler’s third law (for rough estimates, since there willdefinitely be perturbations of the third body’s orbit) would begin with an orbital period of 20 days [11].
The authors thank A.I. Zakharov, S.E. Leontyev, and V.N. Sementsov, who developedsoftware for the computation of photometric elements of eclipsing binaries.
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Johnson, eprint arXiv:1410.4192.11able 1: Parameters derived from the Kepler light curves and adopted in the calculationsof the theoretical light curve of FL Lyr r , r are the radii of the primary and the secondaryin units of the semi-major axis of FL Lyr; e the orbital eccentricity; ω the longitude ofperiastron; L , L the luminosities of the primary and secondary in units of the systemsluminosity; L the “third light” in units of the system’s luminosity; u , u limb-darkeningcoefficients for the primary and secondary; and σ the standard (O-C) deviation. (O-C).Parameters Value for Values adoptedFL Lyr in thecomputations r r i ◦ . − ◦ . ◦ . e ω ◦ L L L u u σ O − C - 0 . m able 3 – continuation Moment, HJD-2400000 (O-C), days55056.50666 -0.0012055060.86302 -0.0011555063.04122 -0.0011055065.21935 -0.0011355067.39750 -0.0011355069.57563 -0.0011555071.75385 -0.0010955073.93203 -0.0010655076.11020 -0.0010555078.28829 -0.0011155080.46645 -0.0011155082.64459 -0.0011255084.82276 -0.0011055087.00089 -0.0011355091.35710 -0.0012355093.53539 -0.0010955095.71356 -0.0010855097.89167 -0.0011255100.06984 -0.0011155102.24797 -0.0011355104.42610 -0.0011555106.60426 -0.0011555108.78242 -0.0011455110.96060 -0.0011255113.13873 -0.0011455115.31692 -0.0011155117.49504 -0.0011455119.67321 -0.0011355121.85135 -0.0011455124.02950 -0.0011455126.20765 -0.0011555128.38578 -0.0011755132.74208 -0.0011855134.92022 -0.0012055137.09838 -0.0011955139.27653 -0.0011955141.45468 -0.0012055143.63282 -0.0012155145.81099 -0.0012055147.98918 -0.0011655150.16731 -0.0011955152.34548 -0.00117continuation in the next page15 able 3 – continuation
Moment, HJD-2400000 (O-C), days55156.70175 -0.0012155158.87990 -0.0012155161.05809 -0.0011855163.23632 -0.0011055165.41448 -0.0011055167.59263 -0.0011055169.77094 -0.0009555171.94888 -0.0011655174.12705 -0.0011555176.30523 -0.0011255178.48340 -0.0011055180.66164 -0.0010255187.19601 -0.0011155189.37422 -0.0010655191.55238 -0.0010555193.73047 -0.0011155195.90868 -0.0010655198.08687 -0.0010255200.26502 -0.0010355202.44317 -0.0010355204.62133 -0.0010355206.79947 -0.0010455208.97767 -0.0010055211.15577 -0.0010555213.33390 -0.0010755215.51200 -0.0011355217.69013 -0.0011555219.86829 -0.0011555222.04642 -0.0011755224.22459 -0.0011655228.58090 -0.0011655235.11540 -0.0011255237.29356 -0.0011155239.47174 -0.0010955243.82801 -0.0011355246.00621 -0.0010855248.18439 -0.0010555250.36247 -0.0011355252.54061 -0.0011455254.71876 -0.0011555256.89700 -0.0010655259.07514 -0.00108continuation in the next page16 able 3 – continuation
Moment, HJD-2400000 (O-C), days55261.25326 -0.0011155263.43145 -0.0010855265.60958 -0.0011055267.78771 -0.0011255269.96587 -0.0011255272.14398 -0.0011655274.32208 -0.0012255278.67836 -0.0012555280.85650 -0.0012655283.03465 -0.0012755285.21281 -0.0012655287.39099 -0.0012355289.56916 -0.0012255291.74733 -0.0012055293.92549 -0.0012055296.10362 -0.0012255298.28180 -0.0012055300.45997 -0.0011855302.63812 -0.0011855304.81631 -0.0011555306.99439 -0.0012255311.35074 -0.0011855313.52889 -0.0011955315.70703 -0.0012055317.88525 -0.0011455320.06334 -0.0012055322.24152 -0.0011755324.41968 -0.0011755326.59804 -0.0009655328.77600 -0.0011655330.95426 -0.0010555335.31062 -0.0010055337.48874 -0.0010455339.66684 -0.0010955344.02316 -0.0010855346.20128 -0.0011155348.37940 -0.0011555350.55753 -0.0011755352.73568 -0.0011855354.91386 -0.0011555357.09203 -0.0011355359.27017 -0.00115continuation in the next page17 able 3 – continuation
Moment, HJD-2400000 (O-C), days55361.44831 -0.0011655363.62648 -0.0011555365.80460 -0.0011855367.98278 -0.0011655372.33909 -0.0011655374.51723 -0.0011755376.69534 -0.0012155378.87356 -0.0011555381.05167 -0.0011955383.22987 -0.0011555385.40802 -0.0011555387.58616 -0.0011755389.76436 -0.0011255391.94264 -0.0010055394.12078 -0.0010155396.29897 -0.0009755398.47699 -0.0011155400.65524 -0.0010155402.83331 -0.0011055405.01151 -0.0010555407.18966 -0.0010655409.36777 -0.0011055411.54600 -0.0010255413.72416 -0.0010255415.90226 -0.0010755418.08049 -0.0010055420.25857 -0.0010755422.43676 -0.0010455424.61487 -0.0010855428.97120 -0.0010655431.14933 -0.0010855433.32744 -0.0011355435.50560 -0.0011255437.68373 -0.0011555439.86184 -0.0011955442.04001 -0.0011855444.21819 -0.0011555446.39635 -0.0011555448.57448 -0.0011755450.75262 -0.0011855452.93078 -0.0011855455.10892 -0.00119continuation in the next page18 able 3 – continuation
Moment, HJD-2400000 (O-C), days55457.28713 -0.0011455459.46531 -0.0011155461.64348 -0.0011055463.82164 -0.0010955465.99978 -0.0011055468.17791 -0.0011355470.35605 -0.0011455472.53419 -0.0011655474.71229 -0.0012155476.89043 -0.0012355479.06858 -0.0012355481.24671 -0.0012655483.42489 -0.0012355485.60302 -0.0012555487.78119 -0.0012455489.95944 -0.0011455494.31569 -0.0012055496.49389 -0.0011655498.67206 -0.0011455500.85025 -0.0011155503.02841 -0.0011055505.20660 -0.0010655507.38474 -0.0010855509.56290 -0.0010755511.74103 -0.0011055513.91917 -0.0011155516.09734 -0.0011055518.27549 -0.0011055520.45362 -0.0011255522.63181 -0.0010955524.80995 -0.0011055526.98808 -0.0011355529.16624 -0.0011255531.34442 -0.0011055533.52251 -0.0011655535.70065 -0.0011855537.87880 -0.0011855540.05696 -0.0011755542.23514 -0.0011555546.59147 -0.0011355548.76963 -0.0011255550.94781 -0.00110continuation in the next page19 able 3 – continuation
Moment, HJD-2400000 (O-C), days55570.55110 -0.0012055572.72927 -0.0011855574.90746 -0.0011455577.08559 -0.0011755579.26373 -0.0011855581.44194 -0.0011355583.62010 -0.0011255585.79823 -0.0011555587.97637 -0.0011655590.15452 -0.0011755592.33267 -0.0011755598.86711 -0.0011955601.04526 -0.0012055603.22343 -0.0011855605.40162 -0.0011555607.57978 -0.0011455609.75796 -0.0011255611.93609 -0.0011455614.11423 -0.0011555616.29246 -0.0010855618.47067 -0.0010255620.64884 -0.0010155622.82710 -0.0009055625.00513 -0.0010355627.18320 -0.0011155629.36134 -0.0011255631.53942 -0.0012055633.71776 -0.0010155642.43022 -0.0011755644.60847 -0.0010855646.78657 -0.0011355648.96475 -0.0011055651.14298 -0.0010355653.32105 -0.0011155655.49912 -0.0012055657.67740 -0.0010755659.85545 -0.0011855662.03376 -0.0010255664.21186 -0.0010855666.39007 -0.0010255668.56813 -0.0011155670.74631 -0.00109continuation in the next page20 able 3 – continuation
Moment, HJD-2400000 (O-C), days55672.92436 -0.0011955677.28069 -0.0011755679.45889 -0.0011355681.63700 -0.0011755683.81515 -0.0011755685.99329 -0.0011955688.17145 -0.0011855690.34958 -0.0012155692.52779 -0.0011555694.70594 -0.0011655696.88411 -0.0011455699.06225 -0.0011655701.24044 -0.0011255703.41853 -0.0011855705.59667 -0.0012055707.77467 -0.0013555709.95298 -0.0012055712.13117 -0.0011655714.30934 -0.0011555716.48749 -0.0011555718.66564 -0.0011655720.84373 -0.0012255723.02189 -0.0012155727.37829 -0.0011255729.55646 -0.0011155731.73468 -0.0010455733.91276 -0.0011255738.26903 -0.0011555740.44726 -0.0010855742.62547 -0.0010255744.80362 -0.0010355746.98179 -0.0010155749.15991 -0.0010555751.33807 -0.0010455753.51623 -0.0010455755.69433 -0.0010955757.87245 -0.0011255760.05061 -0.0011255762.22869 -0.0011955764.40680 -0.0012455766.58501 -0.0011855768.76315 -0.00120continuation in the next page21 able 3 – continuation
Moment, HJD-2400000 (O-C), days55770.94133 -0.0011755773.11953 -0.0011355775.29767 -0.0011455777.47579 -0.0011755779.65396 -0.0011655781.83212 -0.0011555784.01024 -0.0011955786.18851 -0.0010755788.36667 -0.0010755790.54489 -0.0010055792.72297 -0.0010755794.90112 -0.0010855797.07925 -0.0011055799.25740 -0.0011155801.43556 -0.0011055803.61371 -0.0011155805.79183 -0.0011455807.96999 -0.0011455810.14815 -0.0011355812.32630 -0.0011355814.50448 -0.0011155816.68261 -0.0011355818.86073 -0.0011755821.03886 -0.0011955823.21710 -0.0011155825.39518 -0.0011855827.57334 -0.0011855829.75152 -0.0011555831.92960 -0.0012255836.28598 -0.0011555838.46413 -0.0011655840.64225 -0.0011955842.82045 -0.0011555844.99861 -0.0011455847.17675 -0.0011555849.35494 -0.0011255851.53301 -0.0012055853.71115 -0.0012255855.88933 -0.0011955858.06744 -0.0012455860.24565 -0.0011855862.42379 -0.00120continuation in the next page22 able 3 – continuation
Moment, HJD-2400000 (O-C), days55864.60193 -0.0012155866.78009 -0.0012055868.95822 -0.0012355871.13645 -0.0011555873.31456 -0.0012055875.49279 -0.0011255877.67097 -0.0011055879.84907 -0.0011555882.02721 -0.0011755884.20530 -0.0012355886.38349 -0.0011955888.56164 -0.0012055890.73973 -0.0012655892.91789 -0.0012655895.09595 -0.0013555897.27421 -0.0012555899.45237 -0.0012455901.63047 -0.0012955905.98673 -0.0013455908.16495 -0.0012855910.34310 -0.0012855912.52125 -0.0012955914.69954 -0.0011555916.87759 -0.0012655919.05580 -0.0012055921.23397 -0.0011855923.41208 -0.0012355925.59032 -0.0011455927.76846 -0.0011655929.94663 -0.0011455934.30273 -0.0013555936.48115 -0.0010955938.65905 -0.0013455940.83741 -0.0011355943.01557 -0.0011355945.19372 -0.0011355947.37191 -0.0011055949.55006 -0.0011055951.72812 -0.0012055953.90642 -0.0010555956.08457 -0.0010555958.26278 -0.00100continuation in the next page23 able 3 – continuation
Moment, HJD-2400000 (O-C), days55960.44078 -0.0011555962.61903 -0.0010655964.79718 -0.0010655966.97532 -0.0010855969.15349 -0.0010655971.33167 -0.0010455973.50978 -0.0010855975.68789 -0.0011255977.86600 -0.0011755980.04415 -0.0011755982.22232 -0.0011655984.40043 -0.0012055990.93486 -0.0012455993.11305 -0.0012055995.29120 -0.0012055997.46934 -0.0012255999.64750 -0.0012156001.82566 -0.0012156004.00383 -0.0011956006.18199 -0.0011956008.36016 -0.0011756010.53822 -0.0012656012.71645 -0.0011956014.89457 -0.0012256017.07265 -0.0013056019.25077 -0.0013356021.42895 -0.0013156023.60710 -0.0013156025.78530 -0.0012756027.96348 -0.0012456030.14168 -0.0011956032.31980 -0.0012356034.49793 -0.0012556036.67620 -0.0011456038.85433 -0.0011656041.03247 -0.0011856043.21064 -0.0011656045.38880 -0.0011656047.56695 -0.0011656049.74509 -0.0011756051.92322 -0.0012056054.10136 -0.00121continuation in the next page24 able 3 – continuation
Moment, HJD-2400000 (O-C), days56056.27953 -0.0012056058.45768 -0.0012056060.63583 -0.0012156062.81401 -0.0011856064.99215 -0.0011956067.17031 -0.0011956069.34844 -0.0012156071.52663 -0.0011856073.70473 -0.0012356075.88309 -0.0010356080.23923 -0.0012056082.41739 -0.0011956084.59558 -0.0011556086.77377 -0.0011256088.95187 -0.0011756091.13003 -0.0011756093.30819 -0.0011656095.48629 -0.0012256097.66443 -0.0012356099.84256 -0.0012656102.02071 -0.0012656104.19885 -0.0012756108.55520 -0.0012356110.73337 -0.0012256112.91154 -0.0012056115.08969 -0.0012156117.26788 -0.0011756119.44606 -0.0011456121.62421 -0.0011556130.33683 -0.0011556132.51495 -0.0011856134.69314 -0.0011556136.87126 -0.0011856141.22754 -0.0012156143.40564 -0.0012656145.58383 -0.0012356147.76194 -0.0012756149.94015 -0.0012256152.11831 -0.0012156154.29644 -0.0012456156.47455 -0.0012856158.65285 -0.00113continuation in the next page25 able 3 – continuation
Moment, HJD-2400000 (O-C), days56160.83101 -0.0011356163.00918 -0.0011156165.18734 -0.0011156167.36547 -0.0011356171.72170 -0.0012156176.07805 -0.0011756178.25616 -0.0012156182.61266 -0.0010256184.79079 -0.0010556186.96889 -0.0011056189.14696 -0.0011956191.32514 -0.0011656193.50333 -0.0011256195.68142 -0.0011956197.85966 -0.0011056200.03782 -0.0011056202.21585 -0.0012256206.57218 -0.0012056208.75036 -0.0011856210.92850 -0.0011956213.10672 -0.0011256215.28484 -0.0011656217.46300 -0.0011556219.64117 -0.0011456221.81930 -0.0011656223.99745 -0.0011756226.17558 -0.0011956228.35372 -0.0012056230.53189 -0.0011956232.71004 -0.0011956234.88821 -0.0011856237.06636 -0.0011856239.24460 -0.0011056241.42268 -0.0011756243.60089 -0.0011256245.77903 -0.0011356252.31344 -0.0011856254.49160 -0.0011856256.66976 -0.0011756258.84793 -0.0011656261.02618 -0.0010656263.20430 -0.00110continuation in the next page26 able 3 – continuation
Moment, HJD-2400000 (O-C), days56265.38245 -0.0011056267.56055 -0.0011556269.73872 -0.0011456271.91687 -0.0011456274.09496 -0.0012156276.27306 -0.0012656278.45124 -0.0012456280.62939 -0.0012456282.80746 -0.0013256284.98561 -0.0013356287.16373 -0.0013656289.34193 -0.0013256291.52008 -0.0013256293.69826 -0.0013056295.87646 -0.0012556298.05464 -0.0012356300.23279 -0.0012356302.41098 -0.0011956306.76730 -0.0011856308.94545 -0.0011956322.01433 -0.0012356324.19246 -0.0012656326.37065 -0.0012256328.54883 -0.0012056330.72696 -0.0012256332.90509 -0.0012556335.08323 -0.0012656337.26137 -0.0012756339.43956 -0.0012456341.61770 -0.0012556343.79586 -0.0012556345.97401 -0.0012556348.15220 -0.0012256350.33032 -0.0012556352.50855 -0.0011856354.68662 -0.0012656356.86482 -0.0012156361.22111 -0.0012356363.39919 -0.0013156365.57734 -0.0013156367.75560 -0.0012156369.93379 -0.00117continuation in the next page27 able 3 – continuation
Moment, HJD-2400000 (O-C), days56372.11196 -0.0011656374.29005 -0.0012256376.46825 -0.0011756378.64637 -0.0012156380.82450 -0.0012356383.00266 -0.0012356385.18082 -0.0012256387.35900 -0.0012056389.53716 -0.0011928able 4: Amplitude of the theoretical curve and the orbital period of the third bodyobtained from a Fourier expansion using three values of the orbital period of FL LyrOrbitalperiod of FLLyr, days Light-time effectamplitude, s Light-time effectperiod, years2.17815440 4.8 7.22.17815408 9.9 12.42.17815414 7.6 11.329 rbital phase of FL Lyr
Figure 1: Light curve of FL Lyr compiled from Kepler observations between HJD55031.54198 and HJD 55042.44777. It includes the times of primary minimumHJD 55032.54697, HJD 55034.72509, HJD 55036.90328, HJD 55039.08140, and HJD55041.25957. All HJD times actually correspond to HJD2400000.30 ime line, years d a y s Figure 2: Times of minima of FL Lyr. The black circles show Kepler data (Table 3), andthe gray triangles data from ground-based observations (Table 2 ). The x axis plots thedates of the observations and the y axis the differences between the observed times ofminimum and times of minimum calculated using 1.31 r equency, 1/day Sp e c t r a l p o w e r Figure 3: Power spectrum of FL Lyr in relative units calculated using the ephemeris 1.32 r equency, 1/day Sp e c t r a l p o w e r Figure 4: Same as Fig. 3 on a larger scale, for the part at low frequencies..33 u t y c y c l e Fr equency, 1/day Figure 5: Duty cycle as a function of the signal frequency.34 rbital phase of the thir d (cid:0)(cid:1)(cid:2)(cid:3) d a y s Figure 6: Light-time effect for the third body. The ephemeris 1 was used in the calcula-tions. See Table 4 for the period and amplitude of the light-time effect. The solid curveshows the theoretical curve, and the gray circles the observations.35 rbital phase of the thir (cid:4) (cid:5)(cid:6)(cid:7)(cid:8) d a y s Figure 7: Same as Fig. 6 using the ephemeris 2.36 rbital phase of the thir (cid:9) (cid:10)(cid:11)(cid:12)(cid:13) d a y ss