Dynamical masses, absolute radii and 3D orbits of the triply eclipsing star HD 181068 from Kepler photometry
Tamás Borkovits, Aliz Derekas, László L. Kiss, Amanda Király, Emese Forgács-Dajka, Imre Barna Bíró, Timothy R. Bedding, Stephen T. Bryson, Daniel Huber, Róbert Szabó
aa r X i v : . [ a s t r o - ph . S R ] O c t Mon. Not. R. Astron. Soc. , 1–23 (2012) Printed 9 November 2018 (MN L A TEX style file v2.2)
Dynamical masses, absolute radii and 3D orbits of thetriply eclipsing star HD 181068 from
Kepler photometry
T. Borkovits , , ⋆ , A. Derekas , , L. L. Kiss , , , A. Kir´aly , , E. Forg´acs-Dajka , I. B. B´ır´o , T. R. Bedding , S.T. Bryson , D. Huber , , R. Szab´o Baja Astronomical Observatory, H-6500 Baja, Szegedi ´ut, Kt. 766, Hungary Konkoly Observatory, MTA CSFK, H-1121 Budapest, Konkoly Thege M. ´ut 15-17, Hungary ELTE Gothard-Lend¨ulet Research Group, H-9700 Szombathely, Szent Imre herceg ´ut 112, Hungary Sydney Institute for Astronomy, School of Physics, University of Sydney, NSW 2006, Australia Astronomical Department of E¨otv¨os University, H-1118 P´azm´any P´eter stny. 1/A, Budapest, Hungary University of Vienna, T¨urkenschanzstrasse 17, 1180 Vienna, Austria NASA Ames Research Center, Moffett Field, CA 94035, USA
Accepted ??? Received ???; in original form ???
ABSTRACT
HD 181068 is the brighter of the two known triply eclipsing hierarchical triple starsin the
Kepler field. It has been continuously observed for more than 2 years with the
Kepler space telescope. Of the nine quarters of the data, three have been obtained inshort-cadence mode, that is one point per 58.9 s. Here we analyse this unique datasetto determine absolute physical parameters (most importantly the masses and radii)and full orbital configuration using a sophisticated novel approach. We measure eclipsetiming variations (ETVs), which are then combined with the single-lined radial velocitymeasurements to yield masses in a manner equivalent to double-lined spectroscopicbinaries. We have also developed a new light curve synthesis code that is used tomodel the triple, mutual eclipses and the effects of the changing tidal field on the stellarsurface and the relativistic Doppler-beaming. By combining the stellar masses from theETV study with the simultaneous light curve analysis we determine the absolute radiiof the three stars. Our results indicate that the close and the wide subsystems revolvein almost exactly coplanar and prograde orbits. The newly determined parametersdraw a consistent picture of the system with such details that have been beyond reachbefore.
Key words: stars: multiple – stars: eclipsing – stars: individual: HD 181068
The
Kepler space telescope, in addition to its primaryscience aims, has led to a new era in the investiga-tion of multiple star systems. Among the highlights wefind the discoveries of the first triply eclipsing triple sys-tems (Carter et al. 2011; Derekas et al. 2011) and someinteresting studies of multiple star systems (Steffen et al.2011; Feiden, Chaboyer, & Dotter 2011; Gies et al. 2012;Lehmann et al. 2012).Binary and multiple systems have an important rolein astrophysics. The most accurate way to measure stellarparameters is through eclipsing binaries, and their distancedetermination is also very accurate. Their light curves pro-vide essential information on the internal structure of the ⋆ E-mail: [email protected] (TB) components, their atmospheres and their magnetic activity.In the case of noncircular orbits and multiple systems, theorbital elements can change significantly, allowing detailedinsight into the time variation of these parameters. The spe-cial geometry of the very rare and new category of eclipsingsystems, namely the triply (or mutually) eclipsing triple sys-tems, enables us fast and easy determination of further char-acteristics that otherwise could only be studied with greateffort on a long time-scale.As an example, we refer to the spatial configura-tion of such hierarchical triple systems, which is a key-parameter in understanding their origin and evolution (seee. g. Fabrycky & Tremaine 2007, and references therein). Inthe absence of mutual eclipses, the two ways to determinethe mutual (or relative) inclination in a hierarchical systemare ( a ) astrometric (or, more rarely, polarimetric) measure-ments of the spatial orientations of the two orbits individu- c (cid:13) T. Borkovits et al. ally, or ( b ) indirect dynamical calculation from the measuredmutual gravitational perturbations of the bodies. The firstmethod requires long baseline optical (or very-long baselineradio) interferometric measurements for the most interest-ing close binaries, which typically have milli-arcsecond an-gular separations. It is therefore not suprising that, startingwith the pioneering work by Lestrade et al. (1993) on Algol,this method has only been applied to about a dozen bi-naries (see also Baron et al. 2012; Sanborn & Zavala 2012;Peterson et al. 2011; O’Brien et al. 2011, for more recent re-sults). The applicability of polarimetric measurements (al-though does not require high-category instruments) in thisfield is even more restricted (see e. g. Piirola 2010). The sec-ond method, the detection of gravitational perturbations, re-quires accurate, frequent and continuous photometric eclipsetime determination. This method will be described in detailin the next section.The situation is much easier in the case of mutualeclipses, where the shape of the light curve (especiallyaround the ingress and egress phases) contains direct andunique information about the system geometry. This is dis-cussed in detail by Ragozzine & Holman (2010) and P´al(2012). The former authors list several other values ofmulti-transiting systems, mainly in the context of multi-ple planetary systems. Their model has been succesfullyapplied to analysing complex light curves and determin-ing the corresponding geometrical and physical parameters(both for the orbits and the individual bodies) for differentmultiple-transiting planetary (Lissauer et al. 2011, Kepler-11, Doyle et al. 2011, Kepler-16, Welsh et al. 2012, Kepler-34b-35b, Carter et al. 2012, Kepler-36) and stellar systems(Carter et al. 2011, KOI-126).KOI-126 and HD 181068 are the first representatives ofthe new category of the triply eclipsing triple systems. Bothare also members of a very small group of compact hier-archical triple stellar systems. They contain a close binary,with orbital periods P KOI1 = 1 .
77 and P HD1 = 0 .
91 days, anda more distant component forming a wider binary with thecentre of mass of the close pair with periods P KOI2 = 33 . P HD2 = 45 .
47 days, respectively. The main speciality ofthe two systems is their triply eclipsing nature, which meansthat both the inner and the outer binaries show eclipses.They have other, very peculiar characteristics. Both belongto the most compact triple stellar systems, and there is onlyone known hierarchical triple system with a shorter outerperiod, namely λ Tau, with P = 33 .
03 days. Furthermore,these two systems are unusual even amongst the very fewsimilarly compact triples, in having reversed outer mass-ratio. In other words, in these two objects the wide, singlecomponent is the more massive star, and also the largest andbrightest. Before
Kepler , the highest known outer mass-ratiodid not reach 1.5, and for 97% of known hierarchical tripletsit remained under 1, i. e. almost in all the catalogized sys-tems, the total mass of the close binary exceeded the massof the tertiary component (see Tokovinin 2008). (The ques-tion of whether this comes from observational bias is notdiscussed here.) In contrast, the outer mass ratios of thesetwo new systems are q KOIAB ∼ .
0, and q HDAB ∼ .
9, respectively.Despite the similarities of KOI-126 and HD 181068 toeach other, there are remarkable differences between the twosystems. On one hand, KOI-126 consists of three nearlyspherical main sequence stars, where the members of the close binary have such a low surface brightnesses that theirlight curve modelling is largely equivalent to those of themultiple planetary systems. This is not true for HD 181068,where all the three stars are tidally distorted, have almostequal surface brightnesses and show evidence of intrinsiclight variations, all of which make light curve modelling ofHD 181068 more difficult than for KOI-126. On the otherhand, dynamical analysis of HD 181068 is much less complexthan for KOI-126, because of the much simpler and appar-ently constant orbital configurations. As a consequence, ourmethod of light-curve analysis is much closer to the tradi-tional eclipsing binary star light curve modeling methods(see Kallrath & Milone 2009, for a review) than the proce-dures applied for systems like KOI-126.In this paper, we analyse more than 2 years of
Kepler observations of HD 181068. We mainly concentrate on de-termining the fundamental astrophysical parameters of thethree stars and orbital elements of the close and wide orbits.These quantities by themselves carry very important infor-mation already about the system and their members’ originand evolution and, furthermore, give the necessary inputparameters for other forthcoming studies, for example for acomprehensive study of pulsations of the red giant compo-nent. Nevertheless, due to the uniqueness of the studied sys-tem, our aim is not simply to give a case study. The specificsof HD 181068 allow us to present methods never used be-fore. For example, in our period study (Section 3), which de-pends on the analysis of the eclipse timing variations (ETV)for both the close and the wide systems, we determine the(inclination-dependent) masses of the wide binary membersin a new manner. While the radial velocity curve of the mostmassive A component is known, the missing second radialvelocity curve of the spectroscopically unseen B component(i. e. the close binary itself) is replaced by the light-time or-bit of the B component deduced from the ETV analysis ofthe shallow eclipses. This method is fundamentally differentfrom the one followed by Steffen et al. (2011) for KOI-928,for example, because it does not use the dynamical part ofthe ETV, only the simple geometrical light-time contribu-tion. More details are given in Section 3. In Section 4, thelight curve analysis procedure is described in detail, whileSection 5 contains the discussion of the results. Finally, thedetails of our light curve synthesis and analysis code, andsome additional examples of calculations of certain quanti-ties purely in a photometrical and geometrical way from themutual eclipses, are given in the appendices.It is important to establish a clear notation for thissystem. In Derekas et al. (2011) the three components werelabelled A, B, and C (in order of decreasing masses and lu-minosities). Here, we use the more clarified and expressivedenotations, A , Ba , Bb . As before, A denotes the most mas-sive and luminous component (the main component of thewider A − B binary), while Ba and Bb refer to the membersof the close binary formed by the two red dwarfs, formerlydenoted by B and C . When referring to any physical quanti-ties of the individual stars, we use subscripts. For example, m A and m Ba denote the masses of the A and Ba compo-nents, respectively, but m B refers to the total mass of theclose binary, i.e. ( m Ba + m Bb ), and m AB stands for the totalmass of the hierarchical triple. With this notation we canavoid the confusion with the indices of the orbital param-eters of different orbits used for the period study. Namely, c (cid:13) , 1–23 ynamical masses, absolute radii and 3D orbits of HD 181068 following the common usage, the elements of relative orbit ofthe Bb component around its companion, Ba is subscriptedwith 1, whereas the relative orbit of the ternary compo-nent A , around the center of mass of the Ba − Bb subsys-tem (symbolically represented with B ) is associated with thesubscript 2. However, in terms of light-time and the radialvelocity, the absolute orbit (i.e. the orbit of some star aroundthe center of mass) is to be considered, rather than the rel-ative orbits. In these cases, those absolute orbital elements,which numerically differ from the corresponding relative or-bital element, were naturally denoted by the alphabetic signof the given star, or subsystem. The analysis is based on photometry from the
Kepler space telescope (Borucki et al. 2010; Gilliland et al. 2010;Koch et al. 2010; Jenkins et al. 2010a,b). The dataset is 775days long, observed in 6 quarters (Q1-Q6) at long cadence(time resolution of 29.4 min) and 3 quarters (Q7-Q9) atshort cadence (time resolution of 58.9 sec). Since HD 181068is a ∼ The 2.1 year-long observations cover ∼
885 orbital cycles ofthe close pair and 17 revolutions of the wide system. Approx-imately 10% of the eclipses of the close binary (hereafter werefer to them as shallow minima) occur during the eclipseevents of the wide system (hereafter deep minima), and can-not be observed. Additionally, a few hundred events escapedobservation due to data gaps. In all, 1177 of the 1770 shal-low minima were analysed. The analysis of these minima wasquite a complex task. As shown by Derekas et al. (2011), thered giant component shows oscillations on a time scale simi-lar to the half of the orbital period of the short period binary.In addition, there are long term variations, discussed in Sect.4, which slightly distort the shape of the shallow minima, asshown in Fig. 1. This distortion has a significant effect onthe measurement of the exact times of minima.To correct for these distortions, we applied the follow-ing method in determining the times of minima. We tookthe ± BJD - 2455642 (days) ∆ K e p l e r m a g Figure 1.
Example of a primary ( upper panel ) and a secondary( lower panel ) shallow minimum to illustrate the states before(black triangles) and after (red crosses) the detrending of theminima. The dashed line is the fit used for the detrending (seeSection 2.1). event individually with our newly developed simultaneouslight curve solution code. Both the code and the completelight curve analysis are described in Sect. 4.The determined times of minima are listed in Tables 1and 2 for the close and the wide pairs, respectively.
In order to study the eclipse timing variations (ETV), thefollowing linear ephemeris was calculated for the shallowminima:
MIN I − shallow [BJD] = 2 455 051 . . d × E, (1)where E is the cycle number. The corresponding ETV dia-gram is plotted in Fig. 2.We see a sinusoidal variation with a period identicalto the eclipsing period of the wide system. There is also asmaller, long-term variation, that might either be part of alonger period variation, or represent a secular trend, as isthe case with several close binary systems. First we analysethe periodic behaviour of the ETV, and then the possiblesecular (parabolic) term will also be discussed. To detect further periodicities, a discrete Fourier transformwas calculated for the ETV curve. The resulting amplitudespectrum shows that the odd harmonics of the fundamen-tal frequency are also present (see Fig. 3), while only thefirst even harmonic (i.e. 2 f ) exists, and its amplitude issmaller than that of the 3 f and 5 f components. To checkwhether this structure is a consequence of the non-uniformsampling (i.e., the missing data during the deep eclipses,when the eclipse-events of the close pair cannot be observed,see Fig. 4 below), we calculated a simple circular light-timeorbit solution (i.e. we first fitted a sine curve with the fun-damental frequency of the DFT spectrum). Sampling thissolution at the locations (i.e. cycle numbers) of the observed c (cid:13) , 1–23 T. Borkovits et al.
Table 1.
Times of minima for the close pairBJD σ Type BJD σ Type BJD σ Type BJD σ Type2454963.8399 0.0010 II 2454994.6312 0.0010 II 2455101.5010 0.0010 II 2455132.2946 0.0010 II2454964.2926 0.0010 I 2454995.0838 0.0010 I 2455101.9551 0.0010 I 2455132.7470 0.0010 I2454965.1967 0.0010 I 2454995.5359 0.0010 II 2455102.4099 0.0010 II 2455133.1999 0.0010 II2454965.6478 0.0010 II 2454995.9891 0.0010 I 2455102.8605 0.0010 I 2455133.6511 0.0010 I2454966.1021 0.0010 I 2454996.4429 0.0010 II 2455103.3130 0.0010 II 2455134.1046 0.0010 II2454966.5546 0.0010 II 2454996.8929 0.0010 I 2455103.7657 0.0010 I 2455134.5572 0.0010 I2454967.0071 0.0010 I 2454997.3488 0.0010 II 2455104.2174 0.0010 II 2455135.0101 0.0010 II2454967.4605 0.0010 II 2454997.8017 0.0010 I 2455104.6722 0.0010 I 2455137.2785 0.0010 I2454967.9144 0.0010 I 2454998.2542 0.0010 II 2455105.1258 0.0010 II 2455137.7282 0.0010 II2454968.3676 0.0010 II 2454998.7059 0.0010 I 2455105.5781 0.0010 I 2455138.1807 0.0010 I2454968.8194 0.0010 I 2454999.1610 0.0010 II 2455106.0310 0.0010 II 2455138.6355 0.0010 II2454969.2732 0.0010 II 2454999.6116 0.0010 I 2455106.4840 0.0010 I 2455139.0857 0.0010 I2454969.7251 0.0010 I 2455003.2343 0.0010 I 2455106.9385 0.0010 II 2455139.5388 0.0010 II2454970.1781 0.0010 II 2455003.6883 0.0010 II 2455107.3897 0.0010 I 2455139.9900 0.0010 I2454970.6311 0.0010 I 2455004.1399 0.0010 I 2455107.8435 0.0010 II 2455140.4443 0.0010 II2454971.0852 0.0010 II 2455004.5945 0.0010 II 2455108.2951 0.0010 I 2455140.8985 0.0010 I2454971.5369 0.0010 I 2455005.0453 0.0010 I 2455108.7484 0.0010 II 2455141.3563 0.0010 II2454971.9897 0.0010 II 2455005.4987 0.0010 II 2455109.2005 0.0010 I 2455141.8030 0.0010 I2454972.4439 0.0010 I 2455005.9514 0.0010 I 2455109.6543 0.0010 II 2455142.2567 0.0010 II2454972.8964 0.0010 II 2455006.4069 0.0010 II 2455110.1077 0.0010 I 2455142.7091 0.0010 I2454973.3487 0.0010 I 2455006.8569 0.0010 I 2455110.5593 0.0010 II 2455143.1628 0.0010 II2454973.8012 0.0010 II 2455007.3125 0.0010 II 2455111.0126 0.0010 I 2455143.6156 0.0010 I2454974.2544 0.0010 I 2455007.7626 0.0010 I 2455111.9171 0.0010 I 2455144.0683 0.0010 II2454974.7077 0.0010 II 2455008.2172 0.0010 II 2455114.6330 0.0010 I 2455144.5208 0.0010 I2454975.1600 0.0010 I 2455008.6681 0.0010 I 2455115.0886 0.0010 II 2455144.9733 0.0010 II2454975.6110 0.0010 II 2455009.1237 0.0010 II 2455115.9942 0.0010 II 2455145.4263 0.0010 I2454976.0664 0.0010 I 2455010.0269 0.0010 II 2455116.4469 0.0010 I 2455145.8797 0.0010 II2454976.5207 0.0010 II 2455010.4798 0.0010 I 2455116.8992 0.0010 II 2455146.3337 0.0010 I2454976.9719 0.0010 I 2455010.9336 0.0010 II 2455117.3522 0.0010 I 2455146.7850 0.0010 II2454981.0479 0.0010 II 2455011.8403 0.0010 II 2455117.8044 0.0010 II 2455147.2386 0.0010 I2454981.5003 0.0010 I 2455012.2914 0.0010 I 2455118.2563 0.0010 I 2455147.6920 0.0010 II2454981.9544 0.0010 II 2455012.7449 0.0010 II 2455118.7105 0.0010 II 2455148.1429 0.0010 I2454982.4048 0.0010 I 2455013.6507 0.0010 II 2455119.1635 0.0010 I 2455148.5975 0.0010 II2454982.8580 0.0010 II 2455014.1035 0.0010 I 2455119.6164 0.0010 II 2455149.0511 0.0010 I2454983.3116 0.0010 I 2455014.5572 0.0010 II 2455120.0681 0.0010 I 2455149.9553 0.0010 I2454983.7671 0.0010 II 2455015.0081 0.0010 I 2455120.5200 0.0010 II 2455150.4070 0.0010 II2454984.2167 0.0010 I 2455016.3699 0.0010 II 2455120.9754 0.0010 I 2455150.8602 0.0010 I2454984.6695 0.0010 II 2455016.8215 0.0010 I 2455121.4274 0.0010 II 2455151.3110 0.0010 II2454985.1226 0.0010 I 2455019.5391 0.0010 I 2455121.8792 0.0010 I 2455151.7693 0.0010 I2454985.5771 0.0010 II 2455019.9919 0.0010 II 2455122.3325 0.0010 II 2455152.2221 0.0010 II2454986.0275 0.0010 I 2455020.4449 0.0010 I 2455122.7838 0.0010 I 2455152.6729 0.0010 I2454986.4819 0.0010 II 2455020.8964 0.0010 II 2455123.2389 0.0010 II 2455153.1263 0.0010 II2454986.9334 0.0010 I 2455093.3498 0.0010 II 2455124.5962 0.0010 I 2455153.5789 0.0010 I2454987.3857 0.0010 II 2455093.8030 0.0010 I 2455125.0489 0.0010 II 2455154.0318 0.0010 II2454987.8395 0.0010 I 2455094.2549 0.0010 II 2455125.5019 0.0010 I 2455156.7506 0.0010 II2454988.2928 0.0010 II 2455094.7076 0.0010 I 2455125.9538 0.0010 II 2455157.2025 0.0010 I2454988.7449 0.0010 I 2455095.1606 0.0010 II 2455126.4076 0.0010 I 2455157.6564 0.0010 II2454989.1967 0.0010 II 2455095.6140 0.0010 I 2455126.8605 0.0010 II 2455160.3727 0.0010 II2454989.6504 0.0010 I 2455096.0667 0.0010 II 2455127.3131 0.0010 I 2455160.8254 0.0010 I2454990.1049 0.0010 II 2455096.5190 0.0010 I 2455127.7654 0.0010 II 2455161.2782 0.0010 II2454990.5561 0.0010 I 2455096.9724 0.0010 II 2455128.2180 0.0010 I 2455161.7294 0.0010 I2454991.0091 0.0010 II 2455097.4240 0.0010 I 2455128.6723 0.0010 II 2455162.1839 0.0010 II2454991.4619 0.0010 I 2455097.8796 0.0010 II 2455129.1244 0.0010 I 2455162.6355 0.0010 I2454991.9159 0.0010 II 2455098.3312 0.0010 I 2455129.5755 0.0010 II 2455163.0890 0.0010 II2454992.3675 0.0010 I 2455098.7829 0.0010 II 2455130.0306 0.0010 I 2455163.5420 0.0010 I2454992.8209 0.0010 II 2455099.2375 0.0010 I 2455130.4822 0.0010 II 2455163.9946 0.0010 II2454993.2726 0.0010 I 2455099.6907 0.0010 II 2455130.9330 0.0010 I 2455164.4462 0.0010 I2454993.7237 0.0010 II 2455100.1415 0.0010 I 2455131.3871 0.0010 II 2455164.8980 0.0010 II2454994.1785 0.0010 I 2455101.0502 0.0010 I 2455131.8408 0.0010 I 2455165.3521 0.0010 Ic (cid:13) , 1–23 ynamical masses, absolute radii and 3D orbits of HD 181068 Table 1.
Times of minima for the close pair (continued)BJD σ Type BJD σ Type BJD σ Type BJD σ Type2455165.8057 0.0010 II 2455196.5973 0.0010 II 2455229.2014 0.0010 II 2455262.2589 0.0010 I2455166.2571 0.0010 I 2455197.0506 0.0010 I 2455229.6530 0.0010 I 2455262.7143 0.0010 II2455166.7118 0.0010 II 2455197.5016 0.0010 II 2455234.1820 0.0010 I 2455263.1643 0.0010 I2455167.1625 0.0010 I 2455197.9563 0.0010 I 2455234.6354 0.0010 II 2455263.6172 0.0010 II2455167.6143 0.0010 II 2455198.8642 0.0010 I 2455235.0871 0.0010 I 2455264.0693 0.0010 I2455168.0689 0.0010 I 2455199.3154 0.0010 II 2455235.5403 0.0010 II 2455264.5227 0.0010 II2455168.5204 0.0010 II 2455199.7682 0.0010 I 2455235.9946 0.0010 I 2455264.9751 0.0010 I2455168.9745 0.0010 I 2455200.2232 0.0010 II 2455236.4474 0.0010 II 2455265.4305 0.0010 II2455169.4273 0.0010 II 2455200.6731 0.0010 I 2455236.8990 0.0010 I 2455265.8805 0.0010 I2455169.8793 0.0010 I 2455201.1280 0.0010 II 2455237.3518 0.0010 II 2455266.3335 0.0010 II2455170.3314 0.0010 II 2455201.5798 0.0010 I 2455237.8052 0.0010 I 2455266.7867 0.0010 I2455170.7846 0.0010 I 2455202.0321 0.0010 II 2455238.7127 0.0010 I 2455267.2403 0.0010 II2455171.2377 0.0010 II 2455202.4852 0.0010 I 2455239.1626 0.0010 II 2455267.6921 0.0010 I2455171.6922 0.0010 I 2455202.9387 0.0010 II 2455239.6170 0.0010 I 2455268.1446 0.0010 II2455172.1444 0.0010 II 2455205.6558 0.0010 II 2455240.0709 0.0010 II 2455268.5972 0.0010 I2455172.5969 0.0010 I 2455206.1085 0.0010 I 2455240.5226 0.0010 I 2455269.0530 0.0010 II2455173.0483 0.0010 II 2455206.5624 0.0010 II 2455240.9766 0.0010 II 2455269.9571 0.0010 II2455173.5019 0.0010 I 2455207.0141 0.0010 I 2455241.4291 0.0010 I 2455270.4094 0.0010 I2455173.9555 0.0010 II 2455207.4666 0.0010 II 2455241.8823 0.0010 II 2455270.8620 0.0010 II2455174.4083 0.0010 I 2455207.9207 0.0010 I 2455242.3343 0.0010 I 2455271.3125 0.0010 I2455174.8618 0.0010 II 2455208.3747 0.0010 II 2455242.7872 0.0010 II 2455274.0311 0.0010 I2455175.3120 0.0010 I 2455208.8256 0.0010 I 2455243.2405 0.0010 I 2455274.4849 0.0010 II2455176.2170 0.0010 I 2455209.2773 0.0010 II 2455243.6902 0.0010 II 2455274.9361 0.0010 I2455176.6701 0.0010 II 2455209.7301 0.0010 I 2455244.1458 0.0010 I 2455277.2007 0.0010 II2455177.1247 0.0010 I 2455210.1846 0.0010 II 2455244.5992 0.0010 II 2455278.1082 0.0010 II2455177.5766 0.0010 II 2455210.6363 0.0010 I 2455245.0515 0.0010 I 2455279.9181 0.0010 II2455178.0299 0.0010 I 2455211.0904 0.0010 II 2455245.5051 0.0010 II 2455280.8252 0.0010 II2455178.4846 0.0010 II 2455211.5409 0.0010 I 2455245.9575 0.0010 I 2455281.7304 0.0010 II2455178.9346 0.0010 I 2455211.9941 0.0010 II 2455246.4102 0.0010 II 2455282.6374 0.0010 II2455179.3877 0.0010 II 2455212.4471 0.0010 I 2455246.8628 0.0010 I 2455283.5419 0.0010 II2455179.8418 0.0010 I 2455212.8994 0.0010 II 2455247.3158 0.0010 II 2455284.4490 0.0010 II2455180.2948 0.0010 II 2455213.3535 0.0010 I 2455247.7698 0.0010 I 2455285.3552 0.0010 II2455180.7447 0.0010 I 2455213.8046 0.0010 II 2455248.2229 0.0010 II 2455286.2593 0.0010 II2455184.8249 0.0010 II 2455214.2586 0.0010 I 2455248.6748 0.0010 I 2455287.1656 0.0010 II2455185.2776 0.0010 I 2455214.7105 0.0010 II 2455250.9482 0.0010 II 2455288.0726 0.0010 II2455185.7273 0.0010 II 2455215.1632 0.0010 I 2455251.3928 0.0010 I 2455288.9782 0.0010 II2455186.1799 0.0010 I 2455215.6139 0.0010 II 2455251.8454 0.0010 II 2455289.8833 0.0010 II2455186.6346 0.0010 II 2455216.0699 0.0010 I 2455252.2979 0.0010 I 2455290.7894 0.0010 II2455187.0869 0.0010 I 2455217.4272 0.0010 II 2455252.7506 0.0010 II 2455291.6945 0.0010 II2455187.5412 0.0010 II 2455217.8792 0.0010 I 2455253.2041 0.0010 I 2455292.5995 0.0010 II2455187.9934 0.0010 I 2455218.3336 0.0010 II 2455253.6582 0.0010 II 2455293.5069 0.0010 II2455188.4450 0.0010 II 2455218.7867 0.0010 I 2455254.1080 0.0010 I 2455297.1281 0.0010 II2455188.8981 0.0010 I 2455219.2412 0.0010 II 2455254.5616 0.0010 II 2455298.0349 0.0010 II2455189.3515 0.0010 II 2455219.6907 0.0010 I 2455255.0146 0.0010 I 2455298.9417 0.0010 II2455189.8027 0.0010 I 2455220.1447 0.0010 II 2455255.4681 0.0010 II 2455299.8474 0.0010 II2455190.2577 0.0010 II 2455220.5967 0.0010 I 2455255.9214 0.0010 I 2455300.7532 0.0010 II2455190.7087 0.0010 I 2455221.0499 0.0010 II 2455256.3741 0.0010 II 2455301.6557 0.0010 II2455191.1655 0.0010 II 2455221.5019 0.0010 I 2455256.8255 0.0010 I 2455302.5648 0.0010 II2455191.6160 0.0010 I 2455221.9583 0.0010 II 2455257.2747 0.0010 II 2455303.4691 0.0010 II2455192.0681 0.0010 II 2455222.4097 0.0010 I 2455257.7299 0.0010 I 2455305.2800 0.0010 II2455192.5204 0.0010 I 2455222.8609 0.0010 II 2455258.1856 0.0010 II 2455306.1858 0.0010 II2455192.9764 0.0010 II 2455223.3136 0.0010 I 2455258.6384 0.0010 I 2455307.0881 0.0010 II2455193.4266 0.0010 I 2455223.7642 0.0010 II 2455259.0912 0.0010 II 2455309.8044 0.0010 II2455193.8803 0.0010 II 2455224.2188 0.0010 I 2455259.5424 0.0010 I 2455310.7133 0.0010 II2455194.3331 0.0010 I 2455224.6721 0.0010 II 2455259.9961 0.0010 II 2455311.6198 0.0010 II2455194.7853 0.0010 II 2455225.1257 0.0010 I 2455260.4492 0.0010 I 2455313.4275 0.0010 II2455195.2387 0.0010 I 2455226.0314 0.0010 I 2455260.8994 0.0010 II 2455314.3352 0.0010 II2455195.6946 0.0010 II 2455228.2937 0.0010 II 2455261.3547 0.0010 I 2455315.2384 0.0010 II2455196.1452 0.0010 I 2455228.7472 0.0010 I 2455261.8072 0.0010 II 2455316.1422 0.0010 IIc (cid:13) , 1–23
T. Borkovits et al.
Table 1.
Times of minima for the close pair (continued)BJD σ Type BJD σ Type BJD σ Type BJD σ Type2455317.0475 0.0010 II 2455377.7335 0.0010 II 2455408.0720 0.0010 I 2455439.7717 0.0010 I2455319.7697 0.0010 II 2455378.1862 0.0010 I 2455408.5241 0.0010 II 2455440.2272 0.0010 II2455320.6756 0.0010 II 2455378.6391 0.0010 II 2455411.2440 0.0010 II 2455440.6775 0.0010 I2455321.5787 0.0010 II 2455379.0916 0.0010 I 2455411.6947 0.0010 I 2455441.1326 0.0010 II2455322.4873 0.0010 II 2455379.5443 0.0010 II 2455412.1468 0.0010 II 2455441.5837 0.0010 I2455323.3916 0.0010 II 2455379.9967 0.0010 I 2455412.6000 0.0010 I 2455442.0395 0.0010 II2455324.2975 0.0010 II 2455380.4486 0.0010 II 2455413.0556 0.0010 II 2455442.4897 0.0010 I2455325.2031 0.0010 II 2455380.9024 0.0010 I 2455413.5075 0.0010 I 2455442.9436 0.0010 II2455326.1094 0.0010 II 2455381.3567 0.0010 II 2455413.9629 0.0010 II 2455443.3940 0.0010 I2455327.0152 0.0010 II 2455381.8100 0.0010 I 2455414.4120 0.0010 I 2455443.8493 0.0010 II2455327.9208 0.0010 II 2455382.2629 0.0010 II 2455414.8646 0.0010 II 2455444.3003 0.0010 I2455328.8275 0.0010 II 2455382.7152 0.0010 I 2455415.3159 0.0010 I 2455444.7546 0.0010 II2455329.7327 0.0010 II 2455383.1670 0.0010 II 2455415.7715 0.0010 II 2455445.2046 0.0010 I2455330.6358 0.0010 II 2455383.6215 0.0010 I 2455416.2234 0.0010 I 2455445.6593 0.0010 II2455331.5422 0.0010 II 2455384.0749 0.0010 II 2455416.6769 0.0010 II 2455446.1110 0.0010 I2455332.4492 0.0010 II 2455384.5267 0.0010 I 2455417.1288 0.0010 I 2455446.5625 0.0010 II2455333.3539 0.0010 II 2455384.9805 0.0010 II 2455417.5823 0.0010 II 2455447.0161 0.0010 I2455334.2614 0.0010 II 2455385.4313 0.0010 I 2455418.0353 0.0010 I 2455447.4660 0.0010 II2455335.1677 0.0010 II 2455385.8849 0.0010 II 2455418.4868 0.0010 II 2455447.9214 0.0010 I2455336.0729 0.0010 II 2455388.6028 0.0010 II 2455418.9404 0.0010 I 2455448.3760 0.0010 II2455337.8848 0.0010 II 2455389.0558 0.0010 I 2455419.3938 0.0010 II 2455448.8280 0.0010 I2455338.7889 0.0010 II 2455389.5085 0.0010 II 2455419.8471 0.0010 I 2455449.2839 0.0010 II2455339.6974 0.0010 II 2455389.9616 0.0010 I 2455420.3012 0.0010 II 2455449.7337 0.0010 I2455342.4122 0.0010 II 2455390.4144 0.0010 II 2455420.7525 0.0010 I 2455450.1861 0.0010 II2455343.3198 0.0010 II 2455390.8672 0.0010 I 2455421.2057 0.0010 II 2455450.6387 0.0010 I2455344.2237 0.0010 II 2455391.3201 0.0010 II 2455421.6579 0.0010 I 2455451.0936 0.0010 II2455345.1306 0.0010 II 2455391.7716 0.0010 I 2455422.1103 0.0010 II 2455451.5456 0.0010 I2455346.0355 0.0010 II 2455392.2254 0.0010 II 2455422.5636 0.0010 I 2455451.9975 0.0010 II2455346.9414 0.0010 II 2455392.6779 0.0010 I 2455423.0166 0.0010 II 2455452.4485 0.0010 I2455347.8465 0.0010 II 2455393.1300 0.0010 II 2455423.4695 0.0010 I 2455452.9030 0.0010 II2455348.7514 0.0010 II 2455393.5834 0.0010 I 2455423.9224 0.0010 II 2455453.3558 0.0010 I2455349.6589 0.0010 II 2455394.0365 0.0010 II 2455424.3746 0.0010 I 2455453.8064 0.0010 II2455350.5622 0.0010 II 2455394.4891 0.0010 I 2455424.8276 0.0010 II 2455454.2636 0.0010 I2455351.4699 0.0010 II 2455394.9404 0.0010 II 2455425.2819 0.0010 I 2455456.5287 0.0010 II2455352.3738 0.0010 II 2455395.3944 0.0010 I 2455425.7354 0.0010 II 2455456.9785 0.0010 I2455353.2807 0.0010 II 2455395.8469 0.0010 II 2455426.6409 0.0010 II 2455457.4319 0.0010 II2455354.1831 0.0010 II 2455396.2995 0.0010 I 2455427.0935 0.0010 I 2455457.8844 0.0010 I2455355.0909 0.0010 II 2455396.7526 0.0010 II 2455427.5467 0.0010 II 2455458.3384 0.0010 II2455355.9974 0.0010 II 2455397.2049 0.0010 I 2455427.9997 0.0010 I 2455458.7885 0.0010 I2455356.9013 0.0010 II 2455397.6620 0.0010 II 2455428.4520 0.0010 II 2455459.2414 0.0010 II2455357.8074 0.0010 II 2455398.1130 0.0010 I 2455428.9047 0.0010 I 2455459.6944 0.0010 I2455358.7129 0.0010 II 2455398.5619 0.0010 II 2455429.3569 0.0010 II 2455460.1483 0.0010 II2455359.6172 0.0010 II 2455399.0156 0.0010 I 2455429.8087 0.0010 I 2455460.6005 0.0010 I2455360.5266 0.0010 II 2455399.4689 0.0010 II 2455430.2654 0.0010 II 2455461.0525 0.0010 II2455361.4287 0.0010 II 2455399.9218 0.0010 I 2455430.7160 0.0010 I 2455461.5072 0.0010 I2455362.3342 0.0010 II 2455401.7331 0.0010 I 2455431.1699 0.0010 II 2455461.9624 0.0010 II2455365.0530 0.0010 II 2455402.1855 0.0010 II 2455433.8849 0.0010 II 2455462.4128 0.0010 I2455365.9592 0.0010 II 2455402.6390 0.0010 I 2455434.3388 0.0010 I 2455462.8674 0.0010 II2455366.8637 0.0010 II 2455403.0901 0.0010 II 2455434.7910 0.0010 II 2455463.7717 0.0005 II2455371.8469 0.0010 I 2455403.5441 0.0010 I 2455435.2442 0.0010 I 2455464.2261 0.0005 I2455372.2998 0.0010 II 2455403.9981 0.0010 II 2455435.6987 0.0010 II 2455464.6770 0.0005 II2455374.1114 0.0010 II 2455404.4514 0.0010 I 2455436.1498 0.0010 I 2455465.1315 0.0005 I2455374.5610 0.0010 I 2455404.9002 0.0010 II 2455436.6029 0.0010 II 2455465.5842 0.0005 II2455375.0148 0.0010 II 2455405.3552 0.0010 I 2455437.0560 0.0010 I 2455466.0369 0.0005 I2455375.4687 0.0010 I 2455405.8120 0.0010 II 2455437.5097 0.0010 II 2455466.4878 0.0005 II2455375.9223 0.0010 II 2455406.2622 0.0010 I 2455437.9610 0.0010 I 2455466.9428 0.0005 I2455376.3742 0.0010 I 2455406.7133 0.0010 II 2455438.4152 0.0010 II 2455467.3976 0.0005 II2455376.8286 0.0010 II 2455407.1677 0.0010 I 2455438.8682 0.0010 I 2455467.8483 0.0005 I2455377.2789 0.0010 I 2455407.6199 0.0010 II 2455439.3209 0.0010 II 2455468.3019 0.0005 IIc (cid:13) , 1–23 ynamical masses, absolute radii and 3D orbits of HD 181068 Table 1.
Times of minima for the close pair (continued)BJD σ Type BJD σ Type BJD σ Type BJD σ Type2455468.7542 0.0005 I 2455501.3575 0.0005 I 2455531.2464 0.0005 I 2455578.3420 0.0005 I2455469.2073 0.0005 II 2455501.8090 0.0005 II 2455531.6999 0.0005 II 2455578.7932 0.0005 II2455469.6603 0.0005 I 2455502.2632 0.0005 I 2455532.1515 0.0005 I 2455579.6986 0.0005 II2455470.1139 0.0005 II 2455502.7161 0.0005 II 2455532.6062 0.0005 II 2455580.1521 0.0005 I2455470.5665 0.0005 I 2455503.1681 0.0005 I 2455533.0574 0.0005 I 2455580.6049 0.0005 II2455471.0186 0.0005 II 2455503.6213 0.0005 II 2455533.5100 0.0005 II 2455581.0574 0.0005 I2455471.4718 0.0005 I 2455504.0741 0.0005 I 2455533.9625 0.0005 I 2455581.5105 0.0005 II2455471.9251 0.0005 II 2455504.5265 0.0005 II 2455534.4158 0.0005 II 2455581.9643 0.0005 I2455472.3781 0.0005 I 2455504.9801 0.0005 I 2455534.8681 0.0005 I 2455582.4163 0.0005 II2455472.8314 0.0005 II 2455505.4323 0.0005 II 2455535.3218 0.0005 II 2455582.8691 0.0005 I2455473.2831 0.0005 I 2455505.8865 0.0005 I 2455535.7741 0.0005 I 2455583.3218 0.0005 II2455473.7377 0.0005 II 2455506.3385 0.0005 II 2455536.2268 0.0005 II 2455583.7738 0.0005 I2455474.6430 0.0005 II 2455506.7915 0.0005 I 2455536.6788 0.0005 I 2455584.2276 0.0005 II2455475.0950 0.0005 I 2455507.2443 0.0005 II 2455537.1329 0.0005 II 2455584.6789 0.0005 I2455475.5475 0.0005 II 2455507.6974 0.0005 I 2455537.5842 0.0005 I 2455585.1325 0.0005 II2455476.0015 0.0005 I 2455508.6032 0.0005 I 2455538.0368 0.0005 II 2455585.5851 0.0005 I2455478.7173 0.0005 I 2455509.0573 0.0005 II 2455538.4896 0.0005 I 2455586.4906 0.0005 I2455479.1718 0.0005 II 2455509.5096 0.0005 I 2455538.9423 0.0005 II 2455586.9434 0.0005 II2455479.6237 0.0005 I 2455509.9625 0.0005 II 2455539.3957 0.0005 I 2455587.3954 0.0005 I2455480.0770 0.0005 II 2455510.4144 0.0005 I 2455539.8489 0.0005 II 2455587.8488 0.0005 II2455480.5302 0.0005 I 2455510.8680 0.0005 II 2455540.3005 0.0005 I 2455588.3015 0.0005 I2455480.9821 0.0005 II 2455511.3213 0.0005 I 2455540.7539 0.0005 II 2455588.7542 0.0005 II2455481.4349 0.0005 I 2455511.7738 0.0005 II 2455541.2068 0.0005 I 2455589.2072 0.0005 I2455481.8883 0.0005 II 2455512.2263 0.0005 I 2455541.6590 0.0005 II 2455589.6597 0.0005 II2455482.3405 0.0005 I 2455512.6809 0.0005 II 2455542.1120 0.0005 I 2455592.3773 0.0005 II2455482.7922 0.0005 II 2455513.1336 0.0005 I 2455542.5656 0.0005 II 2455592.8297 0.0005 I2455483.2465 0.0005 I 2455513.5859 0.0005 II 2455543.0184 0.0005 I 2455593.2826 0.0005 II2455483.7020 0.0005 II 2455514.0381 0.0005 I 2455543.4702 0.0005 II 2455593.7360 0.0005 I2455484.1517 0.0005 I 2455514.4902 0.0005 II 2455547.0937 0.0005 II 2455596.9066 0.0005 II2455484.6053 0.0005 II 2455514.9437 0.0005 I 2455547.5458 0.0005 I 2455597.3587 0.0005 I2455485.0578 0.0005 I 2455515.3971 0.0005 II 2455547.9990 0.0005 II 2455597.8120 0.0005 II2455485.5097 0.0005 II 2455515.8504 0.0005 I 2455548.4525 0.0005 I 2455598.2641 0.0005 I2455485.9627 0.0005 I 2455516.3035 0.0005 II 2455548.9056 0.0005 II 2455598.7184 0.0005 II2455486.4147 0.0005 II 2455516.7557 0.0005 I 2455549.3570 0.0005 I 2455599.1709 0.0005 I2455486.8686 0.0005 I 2455517.2081 0.0005 II 2455549.8100 0.0005 II 2455599.6232 0.0005 II2455487.3220 0.0005 II 2455517.6619 0.0005 I 2455550.2637 0.0005 I 2455600.0759 0.0005 I2455487.7737 0.0005 I 2455518.1156 0.0005 II 2455550.7169 0.0005 II 2455600.5298 0.0005 II2455488.2261 0.0005 II 2455518.5680 0.0005 I 2455551.1691 0.0005 I 2455600.9820 0.0005 I2455488.6789 0.0005 I 2455519.0206 0.0005 II 2455551.6238 0.0005 II 2455601.4349 0.0005 II2455489.1323 0.0005 II 2455519.4744 0.0005 I 2455552.0754 0.0005 I 2455601.8876 0.0005 I2455489.5846 0.0005 I 2455519.9260 0.0005 II 2455569.7379 0.0005 II 2455602.3424 0.0005 II2455490.0371 0.0005 II 2455520.3791 0.0005 I 2455570.1922 0.0005 I 2455602.7933 0.0005 I2455490.4899 0.0005 I 2455520.8313 0.0005 II 2455570.6430 0.0005 II 2455603.2462 0.0005 II2455490.9432 0.0005 II 2455521.2845 0.0005 I 2455571.0980 0.0005 I 2455603.6989 0.0005 I2455491.3953 0.0005 I 2455524.4545 0.0005 II 2455571.5494 0.0005 II 2455604.1543 0.0005 II2455491.8476 0.0005 II 2455524.9077 0.0005 I 2455572.0033 0.0005 I 2455604.6052 0.0005 I2455492.3012 0.0005 I 2455525.3594 0.0005 II 2455572.4559 0.0005 II 2455605.0599 0.0005 II2455492.7532 0.0005 II 2455525.8125 0.0005 I 2455572.9093 0.0005 I 2455605.5115 0.0005 I2455493.2069 0.0005 I 2455526.2669 0.0005 II 2455573.3596 0.0005 II 2455605.9647 0.0005 II2455494.5648 0.0005 II 2455526.7179 0.0005 I 2455573.8147 0.0005 I 2455606.4173 0.0005 I2455495.0172 0.0005 I 2455527.1722 0.0005 II 2455574.2668 0.0005 II 2455606.8715 0.0005 II2455495.4689 0.0005 II 2455527.6246 0.0005 I 2455574.7207 0.0005 I 2455607.3236 0.0005 I2455495.9226 0.0005 I 2455528.0772 0.0005 II 2455575.1715 0.0005 II 2455607.7765 0.0005 II2455496.3746 0.0005 II 2455528.5289 0.0005 I 2455575.6251 0.0005 I 2455608.2291 0.0005 I2455496.8289 0.0005 I 2455528.9816 0.0005 II 2455576.0767 0.0005 II 2455608.6823 0.0005 II2455497.2808 0.0005 II 2455529.4345 0.0005 I 2455576.5308 0.0005 I 2455609.1350 0.0005 I2455497.7340 0.0005 I 2455529.8870 0.0005 II 2455576.9824 0.0005 II 2455609.5880 0.0005 II2455498.1876 0.0005 II 2455530.3409 0.0005 I 2455577.4365 0.0005 I 2455610.0405 0.0005 I2455498.6395 0.0005 I 2455530.7942 0.0005 II 2455577.8889 0.0005 II 2455610.4941 0.0005 IIc (cid:13) , 1–23
T. Borkovits et al.
Table 1.
Times of minima for the close pair (continued)BJD σ Type BJD σ Type BJD σ Type BJD σ Type2455610.9466 0.0005 I 2455647.1719 0.0005 I 2455677.0593 0.0005 I 2455710.5712 0.0005 I2455611.3995 0.0005 II 2455647.6256 0.0005 II 2455677.5115 0.0005 II 2455711.0220 0.0005 II2455611.8529 0.0005 I 2455648.0790 0.0005 I 2455678.8702 0.0005 I 2455711.4773 0.0005 I2455612.3055 0.0005 II 2455648.5305 0.0005 II 2455679.3227 0.0005 II 2455711.9281 0.0005 II2455615.0236 0.0005 II 2455648.9834 0.0005 I 2455679.7759 0.0005 I 2455712.3825 0.0005 I2455615.4750 0.0005 I 2455649.4363 0.0005 II 2455680.2286 0.0005 II 2455712.8348 0.0005 II2455615.9271 0.0005 II 2455649.8897 0.0005 I 2455680.6813 0.0005 I 2455713.2877 0.0005 I2455616.3809 0.0005 I 2455650.3425 0.0005 II 2455683.3990 0.0005 I 2455713.7399 0.0005 II2455617.2863 0.0005 I 2455650.7951 0.0005 I 2455683.8511 0.0005 II 2455714.1931 0.0005 I2455617.7385 0.0005 II 2455651.2478 0.0005 II 2455684.3045 0.0005 I 2455714.6455 0.0005 II2455618.1924 0.0005 I 2455651.7015 0.0005 I 2455684.7566 0.0005 II 2455715.0986 0.0005 I2455618.6455 0.0005 II 2455652.1542 0.0005 II 2455685.2096 0.0005 I 2455715.5518 0.0005 II2455619.0977 0.0005 I 2455653.0602 0.0005 II 2455685.6613 0.0005 II 2455716.4582 0.0005 II2455619.5523 0.0005 II 2455653.5132 0.0005 I 2455686.1165 0.0005 I 2455716.9101 0.0005 I2455620.0035 0.0005 I 2455653.9665 0.0005 II 2455686.5683 0.0005 II 2455717.3626 0.0005 II2455620.4562 0.0005 II 2455654.4189 0.0005 I 2455687.0217 0.0005 I 2455717.8152 0.0005 I2455620.9085 0.0005 I 2455654.8719 0.0005 II 2455687.4731 0.0005 II 2455718.2668 0.0005 II2455621.3623 0.0005 II 2455655.3245 0.0005 I 2455687.9278 0.0005 I 2455718.7203 0.0005 I2455621.8147 0.0005 I 2455655.7771 0.0005 II 2455688.3801 0.0005 II 2455719.1725 0.0005 II2455622.2670 0.0005 II 2455656.2302 0.0005 I 2455688.8330 0.0005 I 2455719.6261 0.0005 I2455622.7196 0.0005 I 2455656.6842 0.0005 II 2455689.2853 0.0005 II 2455720.0793 0.0005 II2455623.1728 0.0005 II 2455657.1370 0.0005 I 2455689.7390 0.0005 I 2455720.5319 0.0005 I2455623.6254 0.0005 I 2455657.5885 0.0005 II 2455690.1923 0.0005 II 2455720.9837 0.0005 II2455624.0789 0.0005 II 2455660.7584 0.0005 I 2455690.6444 0.0005 I 2455721.4378 0.0005 I2455624.5310 0.0005 I 2455661.2120 0.0005 II 2455691.0981 0.0005 II 2455721.8891 0.0005 II2455624.9824 0.0005 II 2455661.6649 0.0005 I 2455691.5507 0.0005 I 2455722.3431 0.0005 I2455625.4362 0.0005 I 2455662.1179 0.0005 II 2455692.0032 0.0005 II 2455722.7959 0.0005 II2455625.8874 0.0005 II 2455662.5705 0.0005 I 2455692.4558 0.0005 I 2455723.2483 0.0005 I2455626.3419 0.0005 I 2455663.0229 0.0005 II 2455692.9090 0.0005 II 2455723.7019 0.0005 II2455626.7942 0.0005 II 2455663.4762 0.0005 I 2455693.3622 0.0005 I 2455724.1541 0.0005 I2455627.2478 0.0005 I 2455663.9279 0.0005 II 2455693.8154 0.0005 II 2455724.6072 0.0005 II2455627.6989 0.0005 II 2455664.3806 0.0005 I 2455694.2680 0.0005 I 2455725.0599 0.0005 I2455628.1529 0.0005 I 2455664.8344 0.0005 II 2455694.7206 0.0005 II 2455725.5131 0.0005 II2455628.6048 0.0005 II 2455665.2877 0.0005 I 2455695.1733 0.0005 I 2455725.9651 0.0005 I2455629.0582 0.0005 I 2455665.7389 0.0005 II 2455695.6255 0.0005 II 2455728.6829 0.0005 I2455629.5099 0.0005 II 2455666.1933 0.0005 I 2455696.0795 0.0005 I 2455729.1358 0.0005 II2455629.9642 0.0005 I 2455666.6453 0.0005 II 2455696.5309 0.0005 II 2455729.5884 0.0005 I2455630.4169 0.0005 II 2455667.0988 0.0005 I 2455696.9853 0.0005 I 2455730.0397 0.0005 II2455630.8700 0.0005 I 2455667.5509 0.0005 II 2455697.4391 0.0005 II 2455730.4938 0.0005 I2455631.3218 0.0005 II 2455668.0040 0.0005 I 2455697.8911 0.0005 I 2455730.9468 0.0005 II2455631.7755 0.0005 I 2455668.4563 0.0005 II 2455698.3443 0.0005 II 2455731.3993 0.0005 I2455632.2287 0.0005 II 2455668.9087 0.0005 I 2455698.7968 0.0005 I 2455731.8516 0.0005 II2455632.6800 0.0005 I 2455669.3622 0.0005 II 2455699.2494 0.0005 II 2455732.3060 0.0005 I2455633.1334 0.0005 II 2455669.8148 0.0005 I 2455699.7030 0.0005 I 2455732.7564 0.0005 II2455633.5857 0.0005 I 2455670.2678 0.0005 II 2455700.1542 0.0005 II 2455733.2114 0.0005 I2455634.0386 0.0005 II 2455670.7202 0.0005 I 2455700.6085 0.0005 I 2455733.6647 0.0005 II2455634.4916 0.0005 I 2455671.1723 0.0005 II 2455701.0623 0.0005 II 2455734.1170 0.0005 I2455641.7369 0.0005 I 2455671.6268 0.0005 I 2455701.5145 0.0005 I 2455734.5686 0.0005 II2455642.1894 0.0005 II 2455672.0785 0.0005 II 2455701.9651 0.0005 II 2455735.0218 0.0005 I2455642.6431 0.0005 I 2455672.5312 0.0005 I 2455702.4201 0.0005 I 2455735.4758 0.0005 II2455643.0961 0.0005 II 2455672.9824 0.0005 II 2455702.8724 0.0005 II 2455735.9284 0.0005 I2455643.5483 0.0005 I 2455673.4376 0.0005 I 2455703.3256 0.0005 I 2455736.3806 0.0005 II2455644.0018 0.0005 II 2455673.8893 0.0005 II 2455706.0422 0.0005 I 2455737.2865 0.0005 II2455644.4544 0.0005 I 2455674.3431 0.0005 I 2455706.4951 0.0005 II 2455737.7402 0.0005 I2455644.9086 0.0005 II 2455674.7954 0.0005 II 2455706.9482 0.0005 I 2455738.1930 0.0005 II2455645.3605 0.0005 I 2455675.2484 0.0005 I 2455708.3071 0.0005 II 2455738.6457 0.0005 I2455645.8139 0.0005 II 2455675.6992 0.0005 II 2455708.7601 0.0005 I2455646.2670 0.0005 I 2455676.1537 0.0005 I 2455709.2132 0.0005 II2455646.7209 0.0005 II 2455676.6071 0.0005 II 2455709.6667 0.0005 I c (cid:13) , 1–23 ynamical masses, absolute radii and 3D orbits of HD 181068 -0.004-0.0020.0000.0020.0040.006-100 0 100 200 300 400 500 600 700 800Cycle NumberMIN I = 2455051.23625 + 0.905677 E O - C i n da ys Q1-6 (LC)Q7-9 (SC)Q1-9 fitsecular trend
Figure 2.
Eclipse timing variations in the shallow minima. Triangles and circles mark the LC and SC data, respectively. The solid linestands for the Q − Q Table 2.
Times of minima for the wide systemBJD Cycle number a BJD Cycle number a − . − . − . − . − . − . − . − . − . − . − . − . − . . − . . − . . − . . − . . − . . − . . − . . a : half-integer values refer to secondary minima data, and calculating the DFT spectrum of this dataset, wefound that the two spectra have very similar structure (seeFig. 3), confirming our conjecture that the odd peaks are adata-sampling effect. Consequently, we restrict our analysison the main peak ( f ) and its second harmonic (2 f ).Considering the fundamental term, it is clear that itsmain source should be the gravitational interaction betweenthe inner, close binary, and the wider, more massive gi-ant star. This interaction has at least two consequences:( i ) the geometrical light-time effect (LITE), and ( ii ) a dy-namical effect, due to the gravitational perturbations of thethird body on the close, inner binary. In the case of LITE,the amplitude of the effect increases with the separation,as seen in dozens of systems (see e. g. Qian et al. 2012;Pop & Vamo¸s 2012, for most recent examples). Conversely,the amplitudes of the dynamical terms scale with ( P /P )which, due to various observational biases, makes this phe-nomenon difficult to detect with traditional ground-basedobservations. A detailed analysis of this topic can be foundin Borkovits et al. (2003, 2011). To our knowledge, the onlysystem in which the dynamical effect was clearly detected by A m p li t ude [ d ] Frequency [1/d] observed dataLITE-only solution+0.0004 d Figure 3.
The DFT amplitude spectrum of the ETV curve (lowersolid line). In order to illustrate the possible data-sampling originof the odd harmonics the spectrum of a similarly sampled sinefunction with f -frequency is also plotted (upper dashed line). classical ground-based, small-aperture photometric observa-tions, is IU Aurigae (Mayer 1990; ¨Ozdemir et al. 2003). Nev-ertheless, for compact systems like the recently discoveredKOI-126 (Carter et al. 2011), KOI-928 (Steffen et al. 2011),the amplitude ratio may be reversed, as it was clearly shownfor KOI-928 by Steffen et al. (2011).For HD 181068, we first consider the LITE contribution.Its shape and amplitude are: ET V
LITE = a B sin i c (cid:0) − e (cid:1) sin u B e cos v , (2) A LITE ≈ . d × − m A m / sin i P / (cid:0) − e cos ω B (cid:1) / , (3)where a B , i , e , ω B , P are the semi-major axis, inclination,eccentricity, argument of periastron, and period of the bi-nary’s orbit around the common centre of mass of the triplesystem. Furthermore, v is the true anomaly of the eclipsingpair in this orbit, u B = v + ω B is its true longitude mea-sured from the intersection of the orbital plane and the planeof the sky, and c is the speed of light. (Inclination, eccen-tricity, period and true anomaly are simply given subscript2, because their values are identical to those of the rela- c (cid:13) , 1–23 T. Borkovits et al. tive wider orbit, traditionally centered on the inner binary.)Note also that in Eq. (3) masses should be given in solarmasses, while period in days. Substituting the values foundby Derekas et al. (2011) (i.e., m A ≈ ⊙ , m AB ≈ . ⊙ , i ≈ . ◦ P ≈ . d e = 0), we get A LITE ≈ . × − d , (4)or ∼ . A dyn ∼ π m A m AB P P (cid:0) − e (cid:1) / (1 − e ) / , (5)(Borkovits et al. 2011). For the present system this resultsin A dyn ∼ . × − d , (6)which is similar to the LITE. However, as we now pointout, a more detailed analysis shows that the ETV curveshould be LITE-dominated. Although the harmonics of thefundamental frequency could arise from the eccentricity ofone of the orbits, there is strong evidence from the radialvelocity solution of (Derekas et al. 2011) that both orbitsare circular, which is further supported by the locations andshapes of the secondary minima with respect to the primaryminima in both the close and wide orbits (see next Section).Accepting that both orbits are nearly (or exactly) circu-lar, the LITE contribution is restricted to the fundamentalterm, and there is no dynamical addition to this term. Inthis situation, the only dynamical terms that can give non-vanishing contributions are as follows: ET V dyn = 38 π m A m AB P P (cid:8) sin i m sin 2( u − u m2 )+ 12 cot i sin i m [sin u m1 cos 2( u − u m2 )+ cos i m cos u m1 sin 2( u − u m2 )] (cid:9) (7)(see Eq. (46) Borkovits et al. 2003). As before, indices 1and 2 refer to the elements of the close and wide relativeorbits, respectively. Furthermore, i m denotes the mutual in-clination of the two orbital planes, while u m1 and u m2 standfor the angular distances of the intersection of the two or-bits from the plane of the sky, measured on the respectiveplanes (see Fig. A2 in Appendix A). We see that in the caseof coplanarity, all these terms vanish due to sin i m = 0. Forthe present situation, the second and third terms, arisingfrom nodal regression (the precession of the orbital planeof the close pair) can also be simply omitted independentlyfrom the mutual inclination, due to the almost edge-on viewof the orbital plane, as cot i = cot 87 . ◦ ≈ . . P -period compo-nent gives information about the physical dimensions of theclose binary’s orbit around the centre of mass of the triple We corrected here the erroneous negative sign in the nodal term(i.e. in front of cot i ). system. Combining this result with radial velocity measure-ments of the giant companion makes it possible to determinethe masses m A and m AB (as a function of the photometri-cally known sin i ), in a similar manner to a double-linedspectroscopic binary (SB2). Secondly, the P -period termmakes it possible to determine the relative (or mutual) in-clination of the two orbits, i.e. the spatial configuration ofthe triplet.Taking into account the above considerations, the ETVanalysis was carried out as follows. First, a general linearleast-squares method was applied to search for the best fitin the following form: f ( E ) = c + c E + c E + X j =1 ( a j sin jωE + b j cos jωE ) , (8)where the frequency was taken from the DFT analysis, andwas held fixed. Note that its physical meaning is ω = 2 π P e1 P e2 ,where P e1 and P e2 stand for the eclipsing periods of the closeand wide binaries. These quantities, strictly speaking, areneither equal to the anomalistic periods P and P (whichappear in the amplitudes of the dynamical terms) [e. g. for γ systematic velocity P e i = P i (cid:0) γc (cid:1) ], nor necessarily con-stant, especially when c = 0. Nevertheless, for our purposes,these differences are not significant.We carried out two fitting procedures: one for the com-plete data series, and another only for short-cadence Q − Q σ i = 0 . d σ i = 0 . d σ limitwere removed, and the procedure was reiterated. We list ourresults from the two data sets in Table 3, while the corre-sponding fitted curves are shown in Fig. 2. We also show thephased graph in Fig. 4. The polynomial terms (i.e., P c i E i )were subtracted from this latter curve. In Table 3, along withthe direct output of the least-squares fits, the derived physi-cal and geometrical quantities, and their standard errors arealso tabulated.Before analysing the individual Fourier-contributions,we should stress, however, that there is a discrepancy ofabout 0.05 days between the wide-orbit’s period obtainedhere from the LITE solution and the one determined fromthe deep eclipses directly (see later in Sect. 3.2). This is quitesignificant, as during the measured 17 cycle-long interval itwould result in a shift of about 0 .
85 days in the occurrence ofthe eclipse events. Our light curve solution (Sect. 4) clearlyshows that the correct period is the one obtained from thedeep minima times in Sect. 3.2, and not the present one.The origin of this discrepancy is unclear. It might be causedby the observations of shallow minima being absent aroundthe extrema of the LITE-orbit. A firm resolution will requirefurther investigations on a longer time interval. Fortunately,this period difference is too small to influence the analysisof the Fourier terms described below.
Considering the light-time contribution first, its most im-portant output is the physical size of the light-time orbit c (cid:13) , 1–23 ynamical masses, absolute radii and 3D orbits of HD 181068 -0.004-0.0020.0000.0020.0040.006 0 0.2 0.4 0.6 0.8 1Phase MIN I = 2455051.236232 + 0.905676802 E + 0.526 x 10 -9 E O - C i n da ys Q1-6 (LC)Q7-9 (SC)
Figure 4.
The phased ETV curve together with the best linearlsq-fit for the Q − Q of component B (at least as a function of inclination i ).Together with the semi-major axis of component A ’s orbit(obtained from radial velocity measurements), this yields thephysical masses of the wide binary (i.e., the mass of the gi-ant component and the total mass of the close binary). Notethat, as one can see in Table 3, the ratio m A /m AB has a sig-nificantly lower standard error than the masses individuallyand, furthermore, it does not depend on the inclination i .Nevertheless, there is clearly a significant discrepancy be-tween the mass ratios and the masses derived from the twosolutions. The mass ratio depends strongly on the amplitudeof the LITE term. However, the mass of the giant componentresulting from the pure, better-quality short-cadence dataaccurately confirms the value derived from previous resultsand astrophysical estimations of Derekas et al. (2011). Con-sequently, in the followings we adopt this second ( Q − Q . ± . d
16. By the use of the direct ETV-determinedephemeris of the wide binary (see Sect. 3.2) we measurephase φ = 0 . p
998 for this event, i. e., the φ = 0 phase oc-curred at BJD 55045 . Now we turn to the dynamical term. The correspondingFourier coefficients ( a , b ) are almost two orders of magni-tude smaller than those of the LITE terms, and they areclose to the standard errors. Consequently, the followingresults should be considered with great caution. From theamplitude we get sin i m ≈ .
05, which is large enough tomarginally verify the omission of the nodal contribution,but not large enough to give a numerically trustable output.From this result we obtain two different values for the rela-tive inclination. However, as will be shown in the Discussion, we can rule out the retrograde orientation photometrically.Therefore, the corresponding angles are calculated only forprograde relative orbits. By combining the mutual inclina-tion, the phase term ( u m2 ) and the visible inclination ( i )– the latter being known from the light curve solution – wecan calculate the complete 3D orbit of the triple system. InTable 3 we also give the difference of the longitudes of thenodes (∆Ω) on the sky, as well as the visible inclination i ofthe close system. Since i is also known from the light curvesolution, this result might help to resolve the Ω ambiguity,and also serves as an accuracy check for our solution.Both solutions seem to indicate a significant (13 ◦ − ◦ )misalingnment between the two orbital planes. If this factwere real, a precession of the two orbital planes would oc-cur around the invariable plane of the triple system. It canbe shown (see e. g. S¨oderhjelm 1975; Borkovits et al. 2007),that the orbital inlination of the close binary would thenvary cyclically with an amplitude of 28 ◦ − ◦ on a time-scale of 13 −
14 years. Furthermore, the fact that the phaseterm u m2 is close to 90 ◦ or 270 ◦ (i. e. the observable in-clinations ( i and i ) have very similar numerical values)shows that this hypothetical effect would produce the fastest i variations at the present epoch. This means that duringthe Q − Q ◦ variation in the visible inclination ( i ) of theclose pair. This variation would have resulted in significantchanges in the eclipse depths of the shallow minima. How-ever, according to our analysis (next Section) there is no signof any eclipse-depth variations in the close system, and sowe have to exclude this possibility. Consequently, the pres-ence of the first harmonic in the DFT-spectrum cannot beexplained by the non-coplanarity of the orbits.Having ruled out both the eccentricity of the orbit(s)and the noncoplanarity of the orbital planes, we examinedfurther possibilities by considering the effects of higher-orderdynamical terms. Although all the dynamical terms consid-ered e. g. by Borkovits et al. (2003, 2011) and Agol et al.(2005) disappear for coplanar and circular orbits, this hap-pens only within the frame of the applied approximation.The octuple and higher-order terms of the perturbationfunction cause non-vanishing contributions even in this case,as it was shown e. g. by S¨oderhjelm (1984); Ford et al.(2000). In order to check the magnitude of such forces, weintegrated the motion numerically and calculated the sim-ulated times of minima. In our integration both the New-tonian point-mass and the non-dissipative tidal terms wereincluded. The applied numeric integrator was described inBorkovits et al. (2004). An analysis of the DFT spectrum ofthis higher-order, numerically-generated (and evenly sam-pled) ETV curve revealed the presence of the first few har-monics of the orbital periods at a 90% significance level. Asthe amplitudes of these peaks are lower by approximatelytwo magnitudes than that of the questionable first harmonicin the observed curve, we can conclude that these higher-order effects are also insufficient to explain the structure ofthe Fourier space. Therefore, we cannot currently give anyplausible dynamically originated explanation for the P / c (cid:13) , 1–23 T. Borkovits et al.
Table 3.
Fitted and derived parameters (and their formal errorsin the last digits) from the general linear least-squares fit to theETV curve.Parameter Q − Q Q − Q f (cid:0) = P P (cid:1) . c − . . c − . − . c . × − . × − a . . b − . − . a − . . b . . T Bab − primin [BJD] 55051 . . P [day] 0 . . P [day/cycle] 1 . × − . × − P [day] 45 . . a B sin i [R ⊙ ] 54(1) 57(1)( u AB ) [ ◦ ] − − T AB − primin [BJD] 55045 . . a A sin i a [R ⊙ ] 33.43(5) m A /m AB . . m AB sin i [M ⊙ ] 4 . . m A sin i [M ⊙ ] 2 . . m A /m AB sin i m . . i b m [ ◦ ] 15(2) 13(3) u m2 [ ◦ ] 91(9) or 271(9) 95(15) or 275(15) i c [ ◦ ] 87.7 i b,d [ ◦ ] 88(2) or 88(2) 87(3) or 89(3)∆Ω b,c [ ◦ ] 15(2) or − − a : taken from Derekas et al. (2011); b : 180 ◦ − i m , ◦ − i , ◦ − ∆Ω give equivalent solutions; c : fixed from the light curve solution; d : The second values are valid for u m2 + 180 ◦ . As mentioned above, the ETV curve shows weak evidence forcontinuous orbital period changes with a contant rate duringthe whole observational interval. In order to investigate thisfeature, we consider the Q − Q Q − Q MIN I = 2 455 051 . . E +0 . × − E , (9)from which the rate of the constant period change is foundto be∆ PP ∼ ˙ P = 2 c c ∼ .
038 s / yr . (10)The origin of this variation is not clear. As we men-tioned, any orbital precession can be ruled out due to thealmost exact coplanarity. Due to the detached system geom-etry, none mass loss, mass exchange or magnetic cycles canbe considered, as a reason. Gravitational effects induced byan additional, more distant and faint companion, could beresponsible. Moreover, some interaction (e.g. tidal, magneticor other) with the giant component might also be the sourceof this phenomenon. Further observations and investigationsare needed to clarify the origin of the secular variations. For the deep minima the following linear ephemeris wasfound by a linear least-squares fit:
MIN I [BJD] = 2 455 499 . . d × E. (11)Due to the coverage of 17 orbital cycles only, and a largescatter of about 0 .
03 days, no periodic or secular trend canbe identified in the ETV curve. The relatively large scat-ter may arise from the irregular, intrinsic variations of thechromospherically active giant component. As it was shownby Kalimeris et al. (2002), starspots can alter the measuredmid-minimum times by ∼ .
01 days. Evidence for starspots(and even of eclipses of spotted regions) will be given inthe Discussion. Therefore, we conclude that during the 2.1year-long observed time interval, the period of the outer or-bit remained constant.
The light curve of HD 181086 has at least five different com-ponents:(i)-(ii) The eclipsing features of both the close inner ( Ba − Bb ),and the wide outer ( A − B ) binary subsystems. This cate-gory includes not only the eclipses themselves, but also othereffects coming from the close binarity, i. e., the ellipsoidalvariations arising mainly from the tidally distorted shape ofthe giant component A . As we will show below, relativisticDoppler-beaming also produces a contribution. The reflec-tion effect occurs in the close binary, but is negligible forthe wide system (c. f. Zucker et al. 2007). The characteris-tic time-scales of these variations are equal to the observedeclipsing periods P , P of the two subsystems. Note that theperiod ratio is almost exactly P : P = 5 : 251, hence, inevery fifth revolution on the wide orbit, the shallow eclipsesoccur at approximately the same orbital phases of the widesystem. Since the shape and the duration of the deep eclipsesare remarkably altered by the varying positions of the closebinary members, this resonance naturally defines five differ-ent deep eclipse patterns (or eclipse families, which are ana-loguous to the Saros cycles). Furthermore, considering twoconsecutive deep primary eclipses of a given “family” (whichoccur at cycle numbers E = n and E = n + 5, respectively),the intervening deep secondary eclipse of the same “family”(located at E = n + 2 .
5) has a similar egress and ingresspattern, but with a 0 . a and Fig. 5 b ): when the close binarytransits across a darker region, the minimum is shallower.The irregular variation seems to be continuous, showing cer-tain quasi-periodicities on a 1–2 month time-scale, and could c (cid:13) , 1–23 ynamical masses, absolute radii and 3D orbits of HD 181068 N o r m a li s ed F l u x A-B PhaseMIN I = 2455499.9950 a) N o r m a li s ed F l u x A-B PhaseMIN I = 2455272.6355 b) N o r m a li s ed F l u x A-B PhaseMIN II = 2455613.6734 c) N o r m a li s ed F l u x A-B PhaseMIN I = 2455545.4559 d) N o r m a li s ed F l u x A-B PhaseMIN I = 2455318.1113 e) N o r m a li s ed F l u x A-B PhaseMIN II = 2455204.4405 f) N o r m a li s ed F l u x A-B PhaseMIN I = 2455590.9390 g) N o r m a li s ed F l u x A-B PhaseMIN I = 2455363.5693 h) N o r m a li s ed F l u x A-B PhaseMIN II = 2455477.2681 i) Figure 5.
Examples for three of the five “families” of the outer eclipses. Solid curves are the raw (uncorrected) flux curves, while dottedones are corrected for the intrinsic variations. Solid and dashed vertical lines denote the small primary and secondary mid-minima,respectively. Note the flatness of the bottom of the primary minimum-curves in panel b (especially with respect to its counterpart inpanel a ), which might be the consequence of a transit in front of a spotted region. have some connection with the orbital and/or rotational pe-riods of the giant component.(iv) There are further, small amplitude oscillations in thelight curve with the half of the sinodic period of the closesystem with respect to the giant, which strongly indicates atidal origin.(v) Finally, flare events were also observed during some ofthe observational runs. If these transients have their originsin HD 181086 then, at least in one case, we can be sure thatit comes from the giant component, since the flare event atBJD 2 455 659 (in Q9) occurred during the secondary mini-mum of the wide system, i. e., when the close pair was totallyocculted (Fig. 6).In the present analyis, we mainly focus on the eclipsingfeatures [( i ) − ( ii )] of the light curve. As mentioned above,the presence of mutual eclipses in both subsystems makesit possible (at least theoretically) to infer some additional,otherwise unobtainable, physical and geometrical parame-ters from the light curve solution. For example, both the finestructure and the variable length of the ingress and egressphases of the deep minima reveal information on the mutual N o r m a li s ed F l u x Figure 6.
A possible flare event at BJD 55659 (in Q9) withina secondary deep minimum. The location and the amplitude ofthe eruption demonstrate clearly, that if it is a real flare event, itmust have occurred on the giant component. inclination of the two subsystems in such a way that eventhe usual i , 180 ◦ − i ambiguity can be resolved, i. e. we candecide whether the revolutions of the two subsystems areprograde or retrograde relative to each other. Furthermore, c (cid:13) , 1–23 T. Borkovits et al. the combination of the shallow and deep eclipses gives an in-dependent solution for the photometric mass-ratio in boththe close and in the wide systems. (In Appendix A, someexamples are given for mining the extra information codedinto the mutual eclipse geometry.)
In order to carry out this analysis, as a first step we had toseparate the different kinds of variations in the light curve.While the removal of the transients (or flares) was straight-forward, and the small-amplitude tidally generated oscilla-tions do not modify significantly the eclipsing structure, thesubtraction of the long-term intrinsic variations was a diffi-cult problem. We resorted to a step-by-step iterative process,in some steps very similar to a filtering in Fourier space.First, we obtained the averaged light curve of the close, Ba − Bb binary. Since one Kepler quarter covers ∼
100 cy-cles, we expect that those brightness variations which areindependent of the close binary’s orbital revolution wouldaverage out. We therefore binned and averaged the out-of-deep-eclipses parts of our light curves according to theeclipsing phase of the close binary. We applied this processfor six different datasets: the three short-cadence data-series( Q Q Q
9) were taken individually, and also together, thelong cadence Q − Q Q − Q Q − Q Q − Q ∼
17 orbital cycles, and there are alsosome gaps in the data. Therefore, we cannot expect a well-averaged light curve even for the full dataset. Furthermore,such an averaging smooths out the shoulders in the ingressand egress phases of the outer minima, which contain themost important geometric information.In order to recover this information, we calculated apreliminary net eclipsing and elliptical light curve for thewhole triple system. For this we developed a new light curvesynthesis code, which calculates the motions, gravitationalinteractions and mutual eclipses of the three stars simulta-neously. The main characteristics of our code are describedin Appendix B.For the computation of the synthetic curve, most of theinput parameters were taken from Derekas et al. (2011), re- fining their values with our results from the ETV analysisand the close binary’s PHOEBE light curve solution. Af-ter some very minor trial-and-error fine tunings we found aseemingly satisfactory fit. In Fig. 8 we show two versions ofthis synthetic curve (subjected to the same averaging pro-cess), one including the beaming effect, and the other with-out. We see that the curve which includes Doppler-beaming(in the order of 1 ppt) gives a better fit. Despite its prelimi-nary stage, the fit is quite satisfactory from the first contactof the deep primary minimum to the next quadrature. Thediscrepancy in the other portions is probably due to the inef-ficiency of the averaging. An averaged residual curve is alsoshown in Fig. 8.As a next step, we subtracted this synthetic light curvesolution from the raw data. This process was carried outindividually for each quarterly dataset. The raw Q Q − Q Q − Q Q − Q Q Q − Q Q ∗ − Q ∗ data, illustrates the effectiveness of this procedure. (The bot-tom right panel of the Figure also contains an indirect evi-dence for the lack of short-term variations in the inclination i : a change in the eclipse depth would imply an increase ofthe point-to-point scatter during the eclipses, which is notseen to occur.)In the next stage we made a grid-search analysis withour code on the detrended Q ∗ LC-dataset. We chose thisquarter because of its relatively regular, less-distorted shape.The fitted parameters were as follows: the two mass-ratios q , , the (fractional) stellar radii R A , Ba , Bb , temperatures ofthe close binary members T Ba , Bb , one of the three stellar lu-minosities in Kepler-band (the other two were calculated),the two orbital periods P , , two epochs T − , , two observ-able inclinations i , , and the relative longitude of the nodeof the two orbits on the sky ∆Ω, while other parameterswere kept as fix ones. Logarithmic limb-darkening formulaewere applied (equivalent with ld = 2 constraint of the WDand PHOEBE code), with coefficients taken directly fromPHOEBE code. The k j internal structure constants weretaken from the tables of Claret & Gim´enez (1992).In order to estimate the accuracy and reliability of theobtained parameters, we repeated our procedure for theother quarters. This enabled us to estimate the influence ofthe residual distorted, spotted features of the pre-processed c (cid:13) , 1–23 ynamical masses, absolute radii and 3D orbits of HD 181068 N o r m a li s ed F l u x S t d . D e v . -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6Ba-Bb Phase 1.00301.00401.00501.00601.0070 N o r m a li s ed F l u x S t d . D e v . -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6Ba-Bb Phase Figure 7.
Left panel:
The binned, averaged light curve of the Ba − Bb close binary for the Q − Q Right panel:
The binned, averaged light curve of the Ba − Bb close binary for the detrended Q ∗ − Q ∗ SCdata (upper blue circles) with a similarly processed typical solution curve yielded by our new synthetic code (red line), and the standarddeviations of the binned data with respect the average value of each individual cells (down) for both the detrended observed data (blue),and the solution one (red). Note that the bottom curves do not represent the residuals of the upper solution curves. N o r m a li s ed F l u x -0.004-0.002 0.000 0.002 0.004 R e s i dua l F l u x -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6A-B Phase Figure 8.
The binned, averaged light curve of the AB outerbinary for the total Q − Q light curves on the solutions. All the fixed and fitted param-eters, as well as their estimated errors, and some derivedquantities are listed in Table 4.Our final solution for Q We have determined a new set of physical parameters for allthree components in the system. Our results have roughly an order of magnitude lower random errors than was achievableafter the discovery by Derekas et al. (2011). Furthermore,we were able to exploit the unique geometry to infer newparameters that were previously beyond reach.For the previously determined parameters, we findexcellent agreement with the new values. For example,the primary’s radius, combining the Hipparcos parallaxwith CHARA/PAVO onterferommetry, was measured byDerekas et al. (2011) to be R A = 12 . ± . ⊙ . Now wehave determined R A = 12 . ± . ⊙ by combining thestellar masses from the ETV study with the simultaneouslight curve analysis. Similarly, the ETV analysis plus theSB1 radial velocity measurements yielded a primary massof m A = 3 . ± . ⊙ , which agrees with the estimated massfrom evolutionary tracks in Derekas et al. (2011). All in all,the derived physical parameters draw a consistent picture ofthe system, proving that despite the difficulties in the lightcurve modelling, our method yields robust results.A preliminary comparison with models from the BASTI(Pietrinferni et al. 2004) and Dartmouth (Dotter et al.2008) databases shows that the fundamental properties forall three components are consistent with solar-metallicityisochrones with ages ∼ −
500 Myr, although the dwarfradii appear to be significantly larger than expected. Moredetailed comparison using the near model-independent prop-erties presented here will allow powerful tests of stellar evo-lutionary theory, such as tidal effects on the mass-radiusrelation for low-mass stars in close-in binary systems (see,e. g., Kraus et al. 2011).One important question in relation to the giant pri-mary is its evolutionary stage, being located in a part ofthe H-R diagram where H -shell-burning stars ascending thefirst red giant branch overlap closely with He -core-burninggiants (in other words, there is an age uncertainty that can-not be resolved from the evolutionary tracks alone). Dy- c (cid:13) , 1–23 T. Borkovits et al. N o r m a li s ed F l u x -0.010-0.005 0.000 0.005 0.010 R e s i dua l f l u x N o r m a li s ed F l u x R e s i dua l F l u x N o r m a li s ed F l u x -0.010-0.005 0.000 0.005 0.010 R e s i dua l f l u x Figure 9.
The process of the removal of the intrinsic light curve variations from the raw data for the Q Left panel:
The subtraction of a preliminary synthesized eclipsing light curve (green) from the original Q Middle panel:
After a DFT-search of the significant frequencies in the residual curves,the intrinsic variations are represented by the corresponding Fourier polynomial (green), and this latter curve was subtracted from theoriginal data (upper red). The detrended Q ∗ data are plotted in the middle lower panel with red color. Right panel:
The final lightcurve solution (green) was fitted to this Q ∗ dataset (upper red). The residual curve can be seen in the bottom panel. A m p li t ude [ - d ] Frequency [1/d]raw (observed) datasynthetic eclipsing light-curveobserved minus synthetic light-curve 0.00.51.01.52.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A m p li t ude [ - d ] Frequency [1/d]raw (observed) datasynthetic eclipsing light-curveobserved minus synthetic light-curve 02468 2.12 2.14 2.16 2.18 2.2 2.22 A m p li t ude [ - d ] Frequency [1/d]raw (observed) datasynthetic eclipsing light-curveobserved minus synthetic light-curve
Figure 10.
Three different frequency-regions of the DFT spectra obtained for three different datasets of Q − Q Y -axes. See text for details. N o r m a li s ed F l u x -0.001 0.000 0.001 R e s i dua l F l u x N o r m a li s ed F l u x -0.001 0.000 0.001 R e s i dua l F l u x N o r m a li s ed F l u x -0.002-0.001 0.000 0.001 0.002 R e s i dua l F l u x
611 612 613 614 615 616BJD - 2455000
Figure 11.
Three zoom-ins into the Q Q Upper panels:
Points show the original Q Q Q ∗ and Q ∗ data, and the Fourier-modelled intrinsicvariations. Bottom panels:
The residual curves. The left and middle panels show small parts of the Q Q
8. c (cid:13) , 1–23 ynamical masses, absolute radii and 3D orbits of HD 181068 Table 4.
Stellar and orbital parameters derived from the com-bined ETV and synthetic light curve analysis. (The numbers inparantheses are the estimated errors in the last digits.)orbital parameterssubsystemBa–Bb A–B P [d] 0 . . T MINI [BJD] 2455051 . . a [R ⊙ ] 4 . . e . . ω − − i [deg] 86 . . . i m [deg] 0 . q . . L sec /L TOT . . r pole . . . r side . . . r point . . . r back . . . m [M ⊙ ] 0 . . . R [R ⊙ ] 0 . . . T eff [K] 5100(100) 4675(100) 5100(100) L bol [L ⊙ ] 0 . . . g [dex] 4 .
53 4 .
58 2 . k .
020 0 .
020 0 . β .
32 0 .
32 0 . A . . . x bol . . . y bol . . . x K . . . y K . . . namical considerations can help here, too, via comparingthe orbital configurations with theoretical tidal circulariza-tion time-scales. According to Eq. (7) in Verbunt & Phinney(1995), which was based on the works of Zahn (1977, 1989),a binary with the same parameters as HD 181068 A andB (=Ba+Bb) is expected to be circularized under a periodlimit of P circ ∼
15 days for H -shell burning primary. Withthe observed P ∼
45 days and the perfectly circular orbit,theory implies indirectly that the primary must be older, sothat in the He -core burning phase. The question, however,is more complicated because of the binary nature of the sec-ondary. This causes additional complications by the tidaloscillations that are expected to affect the convective enve-lope of the primary. It is not known if the tidal damping iseffective enough to shorten significantly the circularizationtime.Considering the other orbital parameters, our solution for the orbital inclination of the close binary ( i = 86 . ◦ ± . ◦ i = 87 . ◦ ± . ◦ Q i m = 1 ◦ curve is definitely separable from its retrograde counterpart i m = 179 ◦ . (Note, however, that this separation is only pos-sible when the masses or the radii are different in the closebinary.) A combination of the obtained ∆Ω parameter withthe two observable inclinations results in a mutual inclina-tion of i m = 0 . ◦ ± . ◦
4, which suggests an exact coplanarity.This is in accordance with the lack of the eclipse-depth vari-ation of the shallow eclipses.Finally, we briefly comment on the other features of thelight curves. First we consider the irregular, or semiregu-lar brightness changes, which likely originate from chromo-spheric activity. Evidence of the presence of spotted regionson the giant’s surface was shown in the previous section(see e. g. Fig. 5). Further characteristics can be deducedfrom the comparative investigation of the DFT spectra ofthe raw observed light curve, the synthetic and the residualones (Fig. 10). What can be seen well even at the first glanceis that in the low frequency domain (left panel), the spec-trum of the observed data remarkably departs from thatof the synthetic data. While in the synthetic eclipsing, el-lipsoidal data the dominant frequency corresponds to thehalf eclipsing period of the wide system, the highest peakof the original data is located about the eclipsing period it-self. Furthermore, this latter peak is clearly a double one,whose two peaks are already well separated in the spec-trum of the residual light curve (i. e. after the removal ofthe eclipsing and ellipsoidal features). In our interpretationthese two peaks might have a rotational origin. The goodcorrespondance of this pair of peaks with the orbital pe-riod proves the synchronised rotation of the primary, whileits splitting might give an evidence of differential rotation(see e. g. Ol´ah et al. 2003). Note, that the spectroscopi-cally obtained v rot sin i = 14 kms − (Derekas et al. 2011) for R A = 12 . ⊙ , and sin i = 87 . ◦ P rot = 45 . d f = 2 . − , f = 2 . − and c (cid:13) , 1–23 T. Borkovits et al. N o r m a li s ed F l u x SC data - Q7i m =45 o i m =90 o i m =179 o i m =1 o N o r m a li s ed F l u x SC data - Q7i m =45 o i m =90 o i m =179 o i m =1 o Figure 12.
Synthetic light curve for an outer secondary (left), and a primary (right) minimum, calculated with different mutualinclinations. See text for details. f = 2 . − , from which three, the first is exactlythe half of the eclipsing period of the close binary, whilethe other two are f = f − f , and f = f − f , where f = 0 . − corresponds to the half of the eclipsingperiod of the wide system. Such a way, the tidal origin ofthis small amplitude oscillation on the surface of the giantprimary is out of question. These oscillatory features will beinvestigated in details in a forthcoming paper. ACKNOWLEDGEMENTS
This project has been supported by the Hungarian OTKAGrants K76816, K83790 and MB08C 81013, ESA PECSC98090, the “Lend¨ulet-2009” Young Researchers Programof the Hungarian Academy of Sciences and the EuropeanCommunity’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 269194. AD and RSz hasbeen supported by the J´anos Bolyai Research Scholarship ofthe Hungarian Academy of Sciences. AD was supported bythe Hungarian E¨otv¨os fellowship. Funding for this DiscoveryMission is provided by NASA’s Science Mission Directorate.The Kepler Team and the Kepler Guest Observer Office arerecognized for helping to make the mission and these datapossible. TB thanks Professor R. E. Wilson, Drs. K. Ol´ahand Sz. Csizmadia for the valuable discussions on the ques-tions of light curve modelling. AD thanks Dr. A. Simon forthe technical assistance.
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In this Appendix we show examples of how several systemparameters can be determined from the geometry of thelarge, mutual eclipses. Strictly speaking, the most impor-tant condition for the validity of the following calculationsis not the mutuality of the shallow and the deep eclipsesthemselves, but rather the fact that due to the hierarchicalconfiguration of the triple system, the deep eclipses containsome mixtures of individual eclipses of the two members ofthe close pair with respect to the more distant giant com-ponent, which produce small changes in the deep eclipseconfigurations from eclipse to eclipse, and even between theingress and egress phases of the same event. Our algorithmis a natural extension of the well-known methodology of de-termination of the relative radii of the stars (with respect totheir separation, and as a function of their orbital inclina-tion) purely from the eclipse geometry, commonly used fromthe very beginning of eclipsing binary studies.The usual method in binaries with spherical compo-nents is well known: the sky-oriented distance of the stellardisc centers is R + R at the first and last contacts (i. e.,at the start of ingress and at the egress phases), and if theeclipses are total (either transit or occultation), the samedistance is R − R at the second and third contacts (i. e., atthe end of the ingress and at the start of the egress phases).Then expressing the projected distances with the orbital el-ements and time, and measuring the eclipse durations (boththe one from the first to the last contacts, and the totalitylength from the second to the third contacts), the individualfractional radii of the stars can be determined.For our triple star configuration, the egress and ingressphases of the deep eclipses show a complex pattern. Thetwo dwarf members of the close binary may enter in frontof or behind the giant’s disk individually, or even simul-taneously (see Fig. A1). Additionally, during an entry thestars’ velocities, directions and distances (both physical andprojected) relative to the giant component change continu-ously, producing variable length and shape in the egress andingress patterns. Anyhow, no matter how complex an egress or ingress pattern is in itself, every eclipse event contains oneand only one first, second, third, and fourth contacts. Andfurthermore, assuming that a given contact is not stronglyaltered by a just ongoing shallow eclipse event, we can sim-ply and unambiguously decide which member of the closebinary takes part in the given contact event. For example,in case of prograde revolution, the very first contact of a pri-mary transit is produced by the eclipser of the last shalloweclipse event, i. e., if the last event was a small secondaryminimum, then the very first contact of the large primarytransit is produced by the primary of the close pair.Let us consider the projected distances at the disk cen-tres in the moments of the contacts. In the present situationthe projected distance between the eclipser and the eclipsedstars no longer will be the projected radius vector of a Ke-plerian relative orbit, but comes from the superposition oftwo Keplerian orbits: the absolute orbit of the close binarymembers around their center of mass (CM), and the rela-tive orbit of this CM around the giant component. The mostconvenient and practical description of the present scenariouses Jacobian vectors. The first Jacobian vector ( ~ρ ) is di-rected from m Ba to m Bb , i. e., it is the radius vector of theclose binary’s relative orbit, while the second one ( ~ρ ) orig-inates from the CM of the close pair, and ends in m A , i. e.it is the radius-vector in the wide pair (see Fig. A2). Withthese notations, the position vectors connecting the threestars mutually are ~d BaBb = ~ρ , (A1) ~d BaA = ~ρ + q q ~ρ , (A2) ~d BbA = ~ρ −
11 + q ~ρ , (A3)where, as before, q denotes the mass ratio of the close pair.In the usual astrometric frame of reference the right-handed x and y coordinate axes lie in the plane of the sky, while the z axis points outwards from the observer. In the astrometricconvention, x points to the celestial north pole. In case ofphotometry, however, both the eclipsing light curve, and theradial velocity are invariant with respect to any rotation inthe plane of the sky, and so, in the absence of any additionalinformation on the spatial orientation of the intersection ofthe orbital plane and the sky (i. e., Ω), we are free to useany orientation for the x axis. In the context of modellingeclipsing binaries, the coordinate equations take their sim-plest form if one of the axes in the plane of the sky coincideswith the nodal line. In this case, the other axis gives thedirection of the projected orbital angular momentum vec-tor. In the present case, however, we cannot use this latterformal simplicity, because of the differing orbital planes ofthe close and the wide orbits.It is well known from the textbooks of celestial mechan-ics and/or astrometry that in such a frame of reference thecartesian coordinates of a Keplerian orbit can be written as x = r [cos( v + ω ) cos Ω − sin( v + ω ) sin Ω cos i ] , (A4) y = r [cos( v + ω ) sin Ω + sin( v + ω ) cos Ω cos i ] , (A5) z = r sin( v + ω ) sin i, (A6)What is important for us is the projected distances ontothe plane of the sky, instead of the spatial ones. In vectorialforms e. g. c (cid:13) , 1–23 T. Borkovits et al. N o r m a li s ed F l u x MIN I = 2455499.9950-0.200-0.100 0.000 0.100 0.200 D i s t an c e f r o m S t a r A [ d / a ] A-B PhaseBa Bb-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.985 0.990 0.995 1.000 1.005 N o r m a li s ed F l u x MIN II = 2455477.2681-0.200-0.100 0.000 0.100 0.200 D i s t an c e f r o m S t a r A [ d / a ] A-B Phase BaBb-0.54 -0.53 -0.52 -0.51 -0.5 -0.49 -0.48 -0.47 -0.46
Figure A1.
The “walzer” of the close pair in front of (left), and behind (right) the giant primary, projected on the sky. The upper panelsare identical with Figs. 5 i and a , respectively. The dashed horizontal lines denote the R A ± R Ba , b distances from the center of mass ofthe giant, i. e. the outer and inner contact places. Vertical lines connect the moments of the different contacts with the correspondinglight curve points. d xy BaA = s(cid:20) ~ρ + q q ~ρ − (cid:18) ~ρ · ~z + q q ~ρ · ~z (cid:19) ~z (cid:21) , (A7)or, with orbital elements, d xy BaBb = ρ p − sin i sin u , (A8) d xy BaA = ρ (cid:2) − sin i sin u +2 q q ρ ρ ( λ − sin i sin u sin i sin u )+ (cid:18) q q ρ ρ (cid:19) (cid:0) − sin i sin u (cid:1) / , (A9) d xy BbA = ρ (cid:2) − sin i sin u − q ρ ρ ( λ − sin i sin u sin i sin u )+ (cid:18)
11 + q ρ ρ (cid:19) (cid:0) − sin i sin u (cid:1) / , (A10)where u i = v i + ω i gives the true longitude of the givenobject measured from the node and furthermore, λ = cos w cos w + sin w sin w cos i m (A11)is the direction cosine between vectors ~ρ and ~ρ , in whichexpression w i = u i − u m i denotes the true longitude mea-sured from the intersection of the two orbital planes, while u m i is a nodal longitude-like quantity, namely the angulardistance of the intersection of the given orbital plane fromthe sky (cf. Fig. A2). It can also be seen in this figure thatthe three inclinations form angles of that spherical triangle,the sides of which are the three node-like arcs ∆Ω, u m1 and u m2 . Consequently, the two observable inclinations ( i , i ) and the difference of the nodes of the close and wide orbits(∆Ω = Ω − Ω ) unambiguously determine the remainingquantities (i. e., i m and u m -s) with the copious identifica-tions of the spherical triangles, from which some of the mostuseful ones in the present context are as follows:cos i m = cos i cos i + sin i sin i cos ∆Ω , (A12)sin i m cos u m2 = − cos i sin i + sin i cos i cos ∆Ω , (A13)sin i m sin u m2 = sin i sin ∆Ω , (A14)cos i = cos i cos i m − sin i sin i m cos u m2 , (A15)cos u m1 = cos u m2 cos ∆Ω + sin u m2 sin ∆Ω cos i . (A16)At this point we note that in case of coplanarityEqs. (A9,A10) become more simple, not only due to sin i =sin i , but also because then the direction cosine λ is simply λ = cos( u − u ) . (A17)In what follows, we assume that both the inner and theouter orbits are circular, as it happens to be in HD 181068.In this case ρ , ≡ a , , which means a substantial simplifi-cation in our treatment. At this point we are in the positionto give the functional dependencies of the stellar sizes fromthe orbital elements. Namely, for any contact of the deepeclipses R A ± R Ba,Bb = a f Ba,Bb ( i , i , ∆Ω , q , a /a ; u , u ) , (A18)where the plus and minus signs hold for outer and inner con-tacts, respectively. Moreover, for the partial shallow eclipsesfor the outer contacts we can also write R Ba + R Bb = a f Bab ( i ; u ) . (A19)In these equations the independent (time-like) variables are c (cid:13) , 1–23 ynamical masses, absolute radii and 3D orbits of HD 181068 Figure A2.
An illustration of the angles and other quantitiesused in the text. hidden in the u i -s, while the other parameters are constants.The u i longitudes are very closely related to the eclipsingphases. It is well known (see e. g. Gim´enez & Garcia-Pelayo1983) that for circular orbits in the moment of a mid-minimum u = 90 ◦ or u = 270 ◦ . Considering the shallowminima, in the present situation, as ~ρ points toward thesecondary of the close pair, i. e., u refers to the relative or-bit of the secondary, or Bb -component, therefore u = 90 ◦ for the secondary minimum, and u = 270 ◦ for the primaryone. So the connection of u with the eclipsing phase of theclose pair is very simple, u = 360 ◦ × φ − ◦ , or convertingit to time directly, u ( t ) = πP ( t − T ) − π , where T is amid-primary minimum time.The calculation of u requires a bit of extra care. First,we have to keep in mind that ~ρ is oriented from component B (i. e. the centre of mass of the close pair) towards com-ponent A , and so in the present formulae u refers to therelative orbit of the main, giant component A around thesmaller and fainter component B , i. e., the secondary of thewide system. Consequently, in this case u = 90 ◦ formallymeans not the secondary but the primary deep minima. Fur-thermore, the determination of a reference mid-minimumtime is not so simple. Because the positions of the close bi-nary members on their orbits are different at the beginningand at the end of a deep eclipse, the mid-minimum doesnot occur exactly at half-time between the first and the lastcontacts (similarly to the eccentric case). Nevertheless, bythe use of an averaged light curve (like the one in Fig. 8), orof a radial velocity curve, or from the average of several ap-proximately determined mid-minima times we can obtain asatisfactory reference mid-minimum moment, and then u ( t )can be calculated in the same way as u .In the next step we use the fundamental difference be-tween a traditional simple eclipsing binary and our triplesystem. In a single eclipsing binary all the eclipses are sim-ilar, i. e., all contacts occur at the same orbital phase, lon-gitude ( u ) and, furthermore, for circular case the eclipsesare geometrically symmetric in time around their midpoint.Due to the latter, for the first and fourth (last) contactssin u I = sin u IV , and a similar relation can be written for the inner contacts. As a consequence, we have only one equa-tion for R + R a and one for R − R a , and so, some extra wayis needed to resolve the inclination dependence. In opposi-tion, in case of our deep eclipses, the configuration of thefirst and last contacts, and the second and third contactsas well, generally vary from eclipse to eclipse, and they areeven different within the same event. Consequently, we canget separate sets of equations (A18) for different u and u ,from which the unknown parameters can in principle be de-termined by numerical methods.Furthermore, once a /a is known, q can also be cal-culated easily by the use of Kepler III: q q = (cid:16) a a (cid:17) (cid:16) P P (cid:17) . (A20)The individual relative radii of the three stars can also bederived from the combination of Eqs. (A18), even withoutthe use of Eq. (A19). On the other hand, these individaulradii can be determined also in the case when both the deepand the shallow minima are partial.Several difficulties arise, however, during the practi-cal application of this method. For example, in our case ofHD 181068, the light curve distortions of non-eclipsing origincause difficulties in the accurate determination of locations(and so times) of the contacts. Furthermore, our stars arenot exactly spherical, and finally, due to the 5 : 251 mean-motion resonance, we can get only a limited number of dif-ferent eclipse configurations. On the other hand, there aresome additional results that may serve as auxiliary sourcesof information. For example, Eq. (A19) could be used asan additional equation for i , or even a /a . Radial velocityand/or ETV results provide further constraints or equations.In the followings we give a practical example forHD 181068. To do this, we use our simulated light curve solu-tion. By this trick we avoid the practical problem of correctand accurate identification of contact times, since our pur-pose is simply to demonstrate the theoretical effectivenessof this method. Furthermore, our finding about the orbitalcoplanarity makes our formulae as simple as possible, there-fore the whole calculation can be performed analytically.In Table A1 we list the available contact moments forthe large secondary eclipses occuring within the Q − Q u ), one can see that its value may vary by a few tenths ofdegree for both the inner and outer contacts. Although thismay look like a small variation, note, however, that a typicalshift of 0 . ◦ u translates to a change in the occurrence ofthe corresponding event of δt ∼ . Ba , while for Bb these numbers are 2 and 5, respectively. We say different,as events E = − . E = 4 . E = − . E = 0 . u values into Eq. (A9), and subtrating their squares from each c (cid:13) , 1–23 T. Borkovits et al.
Table A1.
Contact times for outer secondary eclipse eventsNo contact star MBJD u u -0.5 I Ba 55476.1096 313 . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ . ◦ other we get three different equations, from which we needonly two. As such,sin i (cid:0) α + β x + γ x (cid:1) + δ x = 0 , (A21)sin i (cid:0) α + β x + γ x (cid:1) + δ x = 0 , (A22)where x = a Ba a , (A23)and α , = (sin u ) − (sin u ) , , (A24) β , = 2[(sin u ) (sin u ) − (sin u ) i (sin u ) , ] , (A25) γ , = (sin u ) − (sin u ) , , (A26) δ , = 2[cos( u − u ) , − cos( u − u ) , (A27)respectively. Eliminating the δ -terms and using the fact thatsin i = 0, we get a second order equation in x , i. e. theorbital ratio. Obtaining x , the inclination can simply be cal-culated from any of the two equations. A similar treatmentcan be applied to the other component Bb . In this case, aswe had only two outer contact times, we used the first twoinner contact moments for the second equation. (Note that,for Bb , the signs of β and δ coefficients should be changed!)When the two ratios a Ba /a and a Bb a are known, both q and a /a can be immediately calculated, and then q follows too. Finally, the fractional radii of the three starsare also easily detemined. Remaining at the present sample,we first determine R A and R Bb from the two (one outer andone inner) contact equations of component Bb , and then R Ba can be calculated from any of the outer contact equations for Ba -star, without using any inner contact moments. (On theother hand, we can also calculate inner contact moments forcomponent Ba , of course.) There is, however, another pos-sibility to determine all these quantities without using innercontact times. This is because we can also use the outercontact-equation of the shallow eclipses. Thus the theoret-ically minimal data needed are the moments of: one outer Table A2.
Analytic results from eclipse geometry and the origi-nal valuesparameter contacts used calculated original a Ba a (I,IV) − . ,I . .
023 0 . i (I,IV) − . ,I . . ◦ . ◦ . ◦ a Bb a II − . ,III . ,I . ,IV . .
027 0 . i II − . ,III . ,I . ,IV . . ◦ . ◦ . ◦ q .
85 0 . a /a .
049 0 . q .
439 0 . R A /a from Ba .
170 0 . R A /a from Bb .
139 0 . R Ba /a .
009 0 . R Bb /a .
010 0 . contact for a shallow eclipse, three outer contacts for deepeclipses of one of the components, and two outer contacts forthe other component. This means that we do not need anyinner contact times (so the method works for partial eclipses,too), the usually better measurable outer contact times aresufficient. In Table A2 we give our results. For comparisonwe also give the corresponding parameters of the syntheticlight curve.Finally, combining these results with the LITE solution,which returns a B sin i in physical units (see Sect. 2), we canalso calculate all the masses and stellar and orbital sizes inphysical dimensions. So, we can conclude that in the case oftriply eclipsing hierarchical systems (i. e., where all the threeobjects eclipse each other at least partially, but not necessar-ily simultaneously) a high-precision single-band photometryof the eclipses is at least in principle efficient for determiningall the above presented quantities in physical units.(The above calculations were made for coplanar andcircular orbits. In the non-coplanar case our equations canbe numerically solved in a similar way. The eccentric case ismore complicated, but the asymmetry of the eclipses bothin length and in phase gives all the required information too,so the difficulty is only practical.) APPENDIX B: LIGHT CURVE SYNTHESISCODE FOR HIERARCHICAL TRIPLESYSTEMS
The code is largely based on the well-known Wilson-Devinney program, which is being continuously developedfrom its first version (Wilson & Devinney 1971) up tonow (Wilson 2008; Wilson & Van Hamme 2009). (See alsoKallrath & Milone 2009, Chapters VI and VII.). ThePHOEBE Scientific Reference (Prˇsa 2006) was also used as acook book. Some of the subroutines were borrowed directlyfrom the Fortran code of the WD program (converting themfrom Fortran to C). However, a number of significant alter-ations were also applied. First, our code calculates the mo-tion and positions of all the three stars, and naturally, themutual eclipses (i.e. when an eclipse event of the close bi-nary occurs in front of the disk of component A , or duringthe egress or ingress phase of the wide eclipse events). Themutual tidal interaction of the three stars is computed for c (cid:13) , 1–23 ynamical masses, absolute radii and 3D orbits of HD 181068 every moment. In order to do this in a more simple way,instead of the usual two-mass point Roche-model, the stel-lar surfaces (and the local gravities as well) were calculatedfrom the first order (linear) series expansions of the potentialof a moderately distorted spherical body. In this formalismthe stellar radius can be written in the following form: r = R X j =2 f j + g ! , (B1)where the amplitudes of the first order tidal distortionscaused by star k on star i are f ( i ← k ) j = (cid:16) k ( i ) j (cid:17) m k m i (cid:18) R i ρ ik (cid:19) j +1 P j (cid:0) λ ′′ ik (cid:1) , (B2)while the amplitude of the rotational distortion of star i is g ( i )2 = − ω i R i Gm i P (cid:0) ν ′ i (cid:1) . (B3)In the equations above R i stands for the undistorted radius, k ( i ) j denotes the j -th internal structure constant of star i , ρ ik the distance of the two stars, ω i the rotational angularvelocity of the star, while the direction cosines in the argu-ments of the given Legendre polynomials P j are the anglebetween the radius vector of the given surface element andthe axis of the tidal bulge (practically the radius vector con-nects the centre of mass of the two stars) in the tidal terms( λ ′′ ), and the angle between the same surface element andthe axis of stellar rotation ( ν ′ ). See Kopal (1978), Chapter IIfor details.Strictly speaking, for strongly distorted systems thisformalism is less accurate than the closed form of the Roche-model, but in the present situation, due to the moderatedistortion of the present stars, it is adequate. Furthermore,besides the obvious advantage coming from linearity, it alsotreats the stars more realistically, as it no longer attributesinfinite central mass densities to them.Another improvement is the inclusion of the relativisticDoppler-beaming effect. The contribution to the total beam-ing is calculated for each surface cell of the three stars in-dividually, and the radial velocity contribution coming fromthe stellar rotation is also taken into account, so theoreti-cally the code is able to model the beaming analogous of theRossiter-McLaughlin effect too. (Regardless, in the presentsystem this is insignificant.) The beaming effect is well illus-trated in Fig. 8.Finally, we have also included the light-time effect. Itis applied only to the wide subsystem, meaning that thepositions of component A and component B were calculatedin different moments according to their different distancesfrom the observer, and furthermore, a second time-delay wasalso calculated in modelling the momentary tidal fields effecton each component.In its present version, the program has 3 ×
11 star-specific global physical parameters (masses, radii, tidal dis-tortion [ k .. ] parameters, effective temperatures, chemicalabundances, gravity darkening, two bolometric limb dark-ening coefficients, and bolometric albedos), 3 × × × m A ), and two mass ratios( q and q ) can also be used as input parameters. Similarly,instead of absolute radii, the use of fractional radii is morepractical for light curve solution, although if the mass of thetertiary ( m A ) and the outer mass-ratio ( q ) is known andfixed from ETV-solution, and of course, the orbital periodsare also fixed, the use of absolute or fractional radii is fullyequivalent. c (cid:13)000