The compact multiple system HIP 41431
T. Borkovits, J. Sperauskas, A. Tokovinin, D. W. Latham, I. Csányi, T. Hajdu, L. Molnár
MMNRAS , 1–19 (2019) Preprint 30 May 2019 Compiled using MNRAS L A TEX style file v3.0
The compact multiple system HIP 41431
T. Borkovits , (cid:63) , J. Sperauskas † , A. Tokovinin ‡ , D. W. Latham § ,I. Cs´anyi , T. Hajdu , , , L. Moln´ar , Baja Astronomical Observatory of Szeged University, H-6500 Baja, Szegedi ´ut, Kt. 766, Hungary Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Hungarian Academy of Sciences,H-1121 Budapest, Konkoly Thege Mikl´os ´ut 15-17, Hungary Vilnius University Observatory, ˇCiurlionio 29, 03100 Vilnius, Lithuania Cerro Tololo Inter-American Observatory, Casilla 603, La Serena, Chile Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA E¨otv¨os Lor´and University, Department of Astronomy, H-1118 Budapest, P´azm´any P´eter stny. 1/A, Hungary MTA CSFK Lend¨ulet Near-Field Cosmology Research Group, H-1121, Budapest, Konkoly Thege Mikl´os ´ut 15-17, Hungary - ABSTRACT
The nearby (50 pc) K7V dwarf HIP 41431 (EPIC 212096658) is a compact 3-tier hierarchy. Three K7V stars with similar masses, from 0.61 to 0.63 solar, make atriple-lined spectroscopic system where the inner binary with a period of 2.9 days iseclipsing, and the outer companion on a 59-day orbit exerts strong dynamical influ-ence revealed by the eclipse time variation in the
Kepler photometry. Moreover, thecentre-of-mass of the triple system moves on a 3.9-year orbit, modulating the propermotion. The mass of the 4-th star is 0.35 solar. The Kepler and ground-based pho-tometry and radial velocities from four different spectrographs are used to adjust thespectro-photodynamical model that accounts for dynamical interaction. The mutualinclination between the two inner orbits is 2 . ◦ ± . ◦
11, while the outer orbit is inclinedto their common plane by 21 ◦ ± ◦ . The inner orbit precesses under the influence ofboth outer orbits, causing observable variation of the eclipse depth. Moreover, thephase of the inner binary is strongly modulated with a 59-day period and its line ofapsides precesses. The middle orbit with eccentricity e = 0 .
28 also precesses, causingthe observed variation of its radial velocity curve. Masses and other parameters ofstars in this unique hierarchy are determined. This system is dynamically stable andlikely old.
Key words: binaries: spectroscopic – binaries: eclipsing – stars: individual: HIP41431
Study of stellar hierachies containing three or more bodieshelps to understand their origin, still a matter of controversyand debate. Although the main aspects of star formation arewell understood, the genesis of stellar systems, particularlyclose binaries, is obscure because the mechanisms reponsi-ble for bringing together two or more stars, initially formedat a much larger separation, are not identified or modelled.From the observational side, establishing a reliable statisticsof hierarchies is a basis for testing theoretical predictions. (cid:63)
E-mail: [email protected] † E-mail: julius.sperauskas@ff.vu.lt ‡ E-mail: [email protected] § E-mail: [email protected]
However, individual systems with rare and/or extreme char-acteristics are equally enlightening, as such objects challengethe formation theories and extend the boundaries of the ex-plored parameter space. This is the case under study here.We investigate an interesting low-mass hierarchical stel-lar system, HIP 41431 (GJ 307). Basic data on this star col-lected with the help of Simbad are assembled in Table 1. Thisobject came to the attention of the present authors indepen-dently as a triple-lined spectroscopic system (D.L. and J.S.)and as an eclipsing binary, exhibiting fast and large ampli-tude eclipse timing variations (ETV) of likely dynamical ori-gin (T.B. and T.H.). Its architecture is illustrated in Fig. 1,where the three spectroscopically visible components withcomparable masses and luminosities are designated as A, B,and C. The fourth star D was discovered by the modulation c (cid:13) a r X i v : . [ a s t r o - ph . S R ] M a y Borkovits et al.
A,BA AB,C C
HIP 41431
B D0.63P =2.9d P =3.9yrABC,D0.35
P =59d0.63 0.61
Figure 1.
Architecture of the quadruple system HIP 41431.Brown circles denote stars A to D, the numbers are their masses,green circles are subsystems.
Table 1.
Main characteristics of HIP 41431Parameter ValueIdentifiers HIP 41431, GJ 307EPIC 212096658Position (J2000, Gaia DR2) 08:27:00.91, +21:57:24.7PM µ α , µ δ (mas yr − , UCAC4) +9.2 ± ± ± B , V , G (mag) 13.01, 10.84, 10.15Infrared photometry J , H , K (mag) 8.02, 7.37, 7.19Spatial velocity U, V, W (km s − ) 8.1, 7.7, − of radial velocity (RV) of the centre of mass of the innertriple and from the residuals of the dynamical, three-bodyETV model, and confirmed by its astrometric signature inthe Gaia catalog. All orbits seem to be close to one planeand have small eccentricities, resembling in this sense a so-lar system, like the “planetary” quadruple star HD 91962(Tokovinin et al. 2015). In these systems, the moderate pe-riod ratios on the order of 20 favor dynamical interactionbetween inner and outer orbits, so the motion cannot bemodelled as a superposition of independent Keplerian or-bits. However, compared to HD 91962, HIP 41431 is muchmore compact and fast.The paper begins with a short description of the datain Section 2. Then in Section 3 we present and discuss spec-troscopic orbits and determine the preliminary components’masses. Global dynamical modelling of the
Kepler
K2 andground-based photometry and the RVs is presented in Sec-tion 4. Its results are confronted with empirical and theo-retical stellar properties in §
5. Observed effects associatedwith dynamical interaction between the orbits are coveredin Section 6. We summarize and discuss our findings in Sec-tion 7.
Kepler K2 photometry
HIP 41431 was observed with
Kepler spacecraft (Borucki etal. 2010) in long cadence (LC) mode during Campaigns 5, 16and 18 of K2 mission. Furthermore, in Campaign 18 shortcadence (SC) data were also collected. Eclipses with a periodof 2.93291 days in the C5 data were reported by Barros etal. (2016). Figure 2 shows the K2 photometry and its modeldiscussed below.We determined the mid-time of each observed eclipseand generated the ETV curves. The method we used is de-scribed in detail by Borkovits et al. (2016). The times ofminima are listed in Table B1, while the ETV curves areshown in Fig. 3. The amplitude of the cyclic variation withthe 60-d period is 0.007 d, an order of magnitude largerthan the light-time delay in the outer orbit. This variationis caused primarily by the interaction with the star C thatmodulates the orbital elements of the inner orbit, includ-ing its period. Moreover, the inner orbit has a fast apsidalrotation which is also forced dynamically by star C.In such a compact, strongly interacting triple system,even marginally (by ≈ − ◦ ) misaligned inner and outerorbital planes produce fast precession of the inner orbit and,hence, eclipse depth variations. Therefore, we checked the K2 lightcurves for such features. Raw K2 LC data are pro-cessed and corrected with different pipelines, resulting insomewhat different lightcurves. We downloaded from theBarbara A. Mikulski Archive for Space Telescopes (MAST) and investigated the PDCSAP lightcurves obtained with theKepler/K2 pipeline and the K2 self-flat-fielding ( K2SFF )pipeline of Vanderburg & Johnson (2014). Regarding thePDCSAP lightcurves, normalizing the flux levels of eachof the three datasets to their out-of-eclipse averages re-veals that the eclipses in the C16 and C18 data are deeperby about ≈
6% and 1 . K2SFF lightcurves, too. This finding, however, does not mean auto-matically that the eclipse depth variation is real. Differentlocations of the target on the
Kepler ’s CCDs and differentaperture masks used for the photometry in these datasetsmay produce apparent eclipse depth variation because ofdifferent amounts of contaminating fluxes from other starswithin the apertures. However, there are no stars brighterthan G = 19 . . (cid:48) radius from our target in the Gaia
DR2 catalog. We conclude that slight eclipse depthvariation during the observing window of K2 photometry ispossible, although it cannot be proven conclusively. There-fore, we decided to apply our photodynamical modellingpackage both for the uneven and the uniform eclipse depthlightcurve. For the latter, we transformed the C5 and C18lightcurves to have equal eclipse depths to the C16 datawhich exhibit the deepest eclipses. http://archive.stsci.edu/k2/data_search/search.php In what follows, we will refer to those two kinds of lightcurvesand the corresponding photo-dynamical solutions as the unevenand uniform eclipse depths scenarios. MNRAS000
DR2 catalog. We conclude that slight eclipse depthvariation during the observing window of K2 photometry ispossible, although it cannot be proven conclusively. There-fore, we decided to apply our photodynamical modellingpackage both for the uneven and the uniform eclipse depthlightcurve. For the latter, we transformed the C5 and C18lightcurves to have equal eclipse depths to the C16 datawhich exhibit the deepest eclipses. http://archive.stsci.edu/k2/data_search/search.php In what follows, we will refer to those two kinds of lightcurvesand the corresponding photo-dynamical solutions as the unevenand uniform eclipse depths scenarios. MNRAS000 , 1–19 (2019) ompact multiple HIP 41431 R e l a t i v e F l u x -0.01 0.00 0.01 58100 58150 58250 5830057150 57200 R e s i dua l F l u x BJD - 2400000
EPIC 212096658 AB Campaign 5 T =57143.470410 P=2.9329243 d R e l a t i v e F l u x -0.002 0.000 0.002 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 R e s i dua l F l u x Phase
Figure 2.
The K2 lightcurves of HIP 41431 (EPIC 212096658). Left panel : The Campaigns 5, 16 and 18 long cadence PDCSAPlightcurves (blue circles) indicate moderate eclipse depth variations from campaign to campaign (see text for details). Red lines showour spectro-photodynamical model solution (Sect. 4).
Right panel:
The phase-folded, binned, and averaged Campaign 5 K2 -lightcurve ofthe innermost binary. The phased averages of the observed flux near the eclipses are plotted by the blue circles (these data were usedfor the joint spectro-photodynamical analysis), while the out-of-eclipse flux is plotted by grey circles. The red curve is the folded, binnedand averaged lightcurve of the cadence-time corrected photodynamical model solution calculated at the time of each observation; theresiduals to the model are also shown in the bottom panels. In order to monitor the possible quick eclipse depth vari-ations and to lengthen the interval of the available ETVdata suitable for the study of the dynamical evolution ofthe system, we carried out additional eclipse event observa-tions with the 0.5 m telescope of Baja Astronomical Obser-vatory of Szeged University located at Baja, Hungary, andequipped with an SBIG ST-6303 CCD detector. The targetwas observed on 7 nights between Jan 14 and Apr 18, 2019,which led to the determination of 6 additional times of min-ima data (see also in Fig. 3 and Table B1). The usual datareduction and photometric analysis were performed using
IRAF routines.As shown below in Sect. 6, these observations confirmednot only the existence, but even the rate of the eclipse depthvariations that was predicted by the uneven eclipse depthmodel solution. High-resolution spectroscopy was conducted independentlyusing several facilities. The primary goal was measurementof RVs for orbit determination. Stellar parameters such asrotation, metallicity and gravity can be determined as wellfrom the spectra. We tabulate the measured radial velocitiesin Table B2.
This nearby star was observed with two identical CfA DigitalSpeedometers (Latham 1985, 1992) from 1999.3 till 2008.3. IRAF is distributed by the National Optical Astronomy Obser-vatories, which are operated by the Association of Universities forResearch in Astronomy, Inc., under cooperative agreement withthe National Science Foundation.
Seven observations were carried out with the instrument in-stalled at the 1.5-m Wyeth Reflector at the Oak Ridge Ob-servatory in the town of Harvard, Massachusetts. The otherspectra were obtained with the 1.5-m Tillinghast Reflectorat the Whipple Observatory on Mount Hopkins, Arizona.A total of 102 observations were collected. The RVs weremeasured by correlations of the single echelle order centeredon the Mg b triplet near 519 nm, with a wavelength windowof 4.5 nm and resolving power of 35 000. As the spectrumis triple-lined, the
TRICOR algorithm was used, analogousto
TODCOR (Zucker & Mazeh 1994). A correction of +0.14km s − must be added to these RVs to put them on the IAUsystem. D.L. found the flux ratio C:A:B of 1:0.94:0.78 at5187˚A.In 2009, the new fibre-fed Tillinghast Reflector EchelleSpectrograph (TRES; Szentgyorgyi & Fur´esz 2007) was usedto obtain an additional spectrum, followed by five more spec-tra taken in 2014. We measured the RVs by cross-correlatingthese spectra with the binary mask and applied the zero-point correction of − .
62 km s − appropriate for this instru-ment. One of the authors (J.S.) has been conducting a long-termRV survey of nearby low-mass stars using several spectrom-eters. Most data are obtained at the 1.65-m telescope at theMoletai observatory in Lithuania (Sperauskas et al. 2016).A CORAVEL-type spectrometer was used to measure theRVs with a typical accuracy of the order of km s − and aspectral resolution around 20000. HIP 43431 was observedwith CORAVEL at Moletai several times in the period from2000 to 2014. Owing to the relative faintess of the star andthe complex multi-line nature of its spectrum, the CCF dipsare noisy and often blended. In this paper, we do not usethese CORAVEL observations.In 2015, a modern fibre-fed echelle spectrometer VUES(Jurgenson et al. 2016) was commissioned at Moletai. We MNRAS , 1–19 (2019)
Borkovits et al.
EPIC 212096658ABT =2457143.470410 P=2.9329243 d E T V [ i n da ys ] -0.002 0.000 0.002 57200 57400 57600 57800 58000 58200 58400 58600 R e s i dua l BJD - 2400000
Figure 3.
Eclipse timing variations of the innermost, eclipsing pair. Red circles and blue boxes stand for the primary and secondaryETVs, respectively, calculated from the observed eclipse events, while black upward and green downward triangles show the correspondingprimary and secondary ETV, determined from the spectro-photodynamical model solution. Furthermore, orange and lightblue linesrepresent approximate analytical ETV models for the primary and secondary eclipses. The residuals of the observed vs photodynamicallymodelled ETVs are plotted in the bottom panel.
Figure 4.
CCFs of HIP 41431 recorded with VUES, with verticalshifts. The reduced Julian dates are indicated on the right. took spectra of HIP 43431 with a resolution of 30 000 in thewavelength range from 400 to 880 nm. The spectra recordedby the CCD detector are extracted and calibrated in thestandard way. The RV is determined by numerical cross-correlation of the spectrum with a binary mask, emulatingthe CORAVEL method in software (Fig. 4). Compared toCORAVEL, the RVs delivered by VUES are more accurate;their rms residuals from the orbits are, typically, from 0.2 to 0.3 km s − . For each observing night, the instrumentalvelocity zero point and its drift were checked by observa-tions of a few RV standard stars. The mean RV zero-point,calculated using 186 measurements of the standard stars, is∆RV=0.09 ± − , and the standard deviation is 0.19km s − . One can suspect a small drift of the zero-point from0.04 to 0.14 km s − in about three years. Practically thesame value of ∆RV=0.08 ± − (rms 0.18 km s − , n = 15) is obtained using telluric lines in the spectra of HIP41431 as the RV reference. Seven spectra of HIP 41431 were taken at the 1.5-m tele-scope located at Cerro Tololo (Chile) and operated by theSMARTS consortium. Observations were conducted by thetelescope operator in the service mode. The optical echellespectrometer CHIRON (Tokovinin et al. 2013) was used inthe slicer mode with a spectral resolution of 80000. Oneach visit, a single 10-minute exposure of the star wastaken, accompanied by the spectrum of the comparison lampfor wavelength calibration. The data were reduced by thepipeline written in IDL.The RVs are derived from the reduced spectra by cross-correlation with a binary mask based on the solar spec-trum, similarly to the CORAVEL RVs. More details are See
MNRAS000
MNRAS000 , 1–19 (2019) ompact multiple HIP 41431 massA BC Centreof CC F Figure 5.
CCF of HIP 41431 using the CHIRON spectrum takenon JD 2458443.8. provided by Tokovinin (2016). Only the spectral range from4500˚A to 6500˚A, relatively free from telluric lines, was usedfor the CCF calculation. The RVs delivered by this proce-dure should be on the absolute scale if the wavelength cal-ibration is good. A comparison of CHIRON RVs with sev-eral RV standards revealed a small offset of +0.16 km s − (Tokovinin 2018c); in the following this offset is neglected.Figure 5 illustrates the 3-component cross-correlationfunction (CCF) derived from the CHIRON spectrum. Thestrongest dip belongs to the component C; the dip of A isalmost equal, while B is obviously weaker. The relative dipareas of C:A:B are 1:0.95:0.71. The dips are narrow andcorrespond to the projected rotational velocities of 5.5, 5.1,and 4.2 km s − according to the calibration of Tokovinin(2016). The rotation of A and B is almost two times slowerthan synchronous (10.1 and 9.9 km s − for the primary andthe secondary, respectively. The first four sets of RVs mea-sured with CHIRON are plotted in Fig. 6 together with thespectro-photodynamical model curves (see Sect. 4). In an effort to extend the time coverage, we consulted theESO archive and found eight high-resolution spectra takenwith UVES at the 8-m VLT telescope in December 2017,on two nights, in the framework of the program 0100.D-0282(A) to study chromospheric activity of inactive main-sequence stars (PI A. Santerne). The data recently becamepublic. We measured the RVs using only the red-arm spec-tra (wavelength range 5655–9463˚A) by correlation with thebinary mask. No zero-point correction was applied. TheseRVs serve primarily to confirm the 4-year modulation of thecentre-of-mass RV induced by the star D. The
Gaia data release 2, DR2 (Gaia collaboration 2018),provides accurate parallax (see Table 1) and proper motion(PM) of HIP 41431. However, the reduced goodness-of-fitparameter gofAL of 27.19 and the statistically significantexcess noise of 0.29 mas show that the single-star model http://archive.eso.org/cms.html adopted in DR2 is not adequate. The photo-centre positionis modulated with the 59-day and 4-year periods of the mid-dle and outer orbits, and the future data releases will hope-fully provide the astrometric elements of these orbits.Comparison of the DR2 position with the second Hip-parcos data reduction (van Leeuwen 2007) allows us to com-pute the average PM of ( µ α , µ δ ) mean = (+10 . , +21 . − . This long-term PM agrees well with the ground-based PM of (+9 . ± . , +21 . ± .
1) mas yr − given inUCAC4 (Zacharias et al. 2012), but differs very signifi-cantly from the “instantaneous” PM measured by Gaia :∆ µ DR2 − mean = ( − . ± . , − . ± .
14) mas yr − . Asimilar, although less significant, difference ∆ µ HIP2 − mean =( − . ± . , − . ± .
5) mas yr − is found between the Hip-parcos and long-term PM. So, HIP 41431 is an astrometricbinary of the ∆ µ type. We show below that the measured∆ µ is explained by the photocentric motion induced by theouter orbit. The star was observed in 2018.97 in the I band usingspeckle camera at the 4.1-m Southern Astrophysical Re-search (SOAR) telescope. The angular resolution (minimumdetectable separation) was 50 mas, and the dynamic range(maximum magnitude difference) was about 4 mag at 0 . (cid:48)(cid:48) ∼ Dynamical interaction between the inner and outer orbitsmeans that the observed RVs cannot be accurately modelledas superposition of two Keplerian orbits. Dynamical mod-elling using both RVs and ETV is presented in the followingSect. 4. However, fitting Keplerian orbits to the subsets ofRVs provided important insights and led to the discoveryof the fourth star, D. Table 2 lists spectroscopic orbital el-ements of the inner and middle systems derived from threeindependent sets of RVs coming from different instruments.Both orbits were fitted simultaneously using the orbit3.pro
IDL code (Tokovinin & Latham 2017). Some elements werefixed (they are listed with asterisks instead of errors). TheRV amplitudes of the inner pair A,B are denoted as K and K , the RV amplitudes in the outer orbit are K (center ofmass of AB) and K .The orbits in the first column of Table 2 were computedby D. L. in 2008 based on 30 RVs measured with the CfAfrom 2007.2 to 2008.3, using his own code for fitting two or-bits simultaneously. Incomplete phase coverage of the outerorbit likely explains the slight disagreement of the RV ampli-tudes K and K and corresponding masses with the recent MNRAS , 1–19 (2019)
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Table 2.
Provisional orbital elements of the inner tripleElement CfA VUES CHIRON P (d) 2.93326 ± τ (BJD − ± ± ± e ± ± ω (deg) 23.5 ± ± K (km s − ) 78.54 ± ± ± K (km s − ) 80.45 ± ± ± M A , B sin i A , B ( M (cid:12) ) 0.618, 0.603 0.631, 0.621 0.652, 0.620 P (d) 58.819 ± τ (BJD − ± ± ± e ± ω (deg) 112.9 ± ± K (km s − ) 25.23 ± ± ± K (km s − ) 49.79 ± ± ± γ (km s − ) − ± − ± − ± σ A , B , C (km s − ) 2.03, 2.60, 1.66 0.46, 1.20, 0.31 0.14, 0.17, 0.27 M AB , C sin i AB , C ( M (cid:12) ) 1.517, 0.769 1.278, 0.648 1.266,0.641 -100 -50 0 50 100 R ad i a l V e l o c i t y [ k m / s e c ] -1 0 158465 58470 58475 58480 58485 58490 58495 R e s i dua l BJD - 2400000
Figure 6.
A recent, one-month-long section of the RV curvesof all the three spectroscopically visible components. Red circles,blue squares and black triangles represent the observed data ofstars A, B and C, respectively, while the lines with similar colorsshow the full spectro-photodynamical model solution (see Sect. 4).In the bottom panel the residual values are plotted. The largestresiduals correspond to spectra with blended lines. orbits based on VUES and CHIRON data. For the latter,we fixed the periods and the outer eccentricity to their val-ues determined photometrically. The VUES RVs measuredbefore 2018.2 were corrected for the offsets due to the outerorbit (see below).The most striking disagreement between those orbitsconcerns the centre-of-mass velocity γ . The velocity zeropoints of respective spectrographs are carefully controlled,hence the effect is real. The centre-of-mass velocity V canbe computed for each individual observation independentlyof the orbital elements as V = ( M A V A + M B V B + M C V C ) / ( M A + M B + M C ) (1)using relative component’s masses derived from the RV am-plitudes. We adopted provisionally M A : M B : M C = 1 :0 .
98 : 1 .
007 and applied eq. 1 to the observations where allthree RVs are measured from the same spectrum, excludingspectra with blended lines. The correct choice of the relative -15-10 -5 R ad i a l V e l o c i t y [ k m / s e c ] EPIC 212096658ABC RV curve T =2457768.4262 P=1435.68 d -2 0 20.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R e s i dua l Phase
Figure 7.
RV curve of the centre-of-mass ABC correspondingto the outer orbit with P = 3 . masses is verified by the absence of correlation between V and RVs of the individual components.A plot of V vs. time clearly shows its variation with aperiod of ∼ V ( t ) variation are given in Table 3, the RV curve is plottedin Fig. 7. The RV amplitude in the outer orbit is denotedby K . We adopted the erors of 1 km s − for the CfA data,0.5 km s − for TRES, VUES and UVES, and 0.1 km s − forCHIRON. The latter RVs are distinguished by the tight se-quence of black squares around the phase 0.5, showing the V trend in just two months. The global weighted rms resid-uals are 0.38 km s − . The long and extensive coverage ofthe CfA data is essential for constraining the outer orbit.The middle and inner orbits computed from the CfAdata of 2007–2008 happen to be near the maximum of theRV curve in Fig. 7, hence γ CfA = − . − ; the trendduring this period was small. Similarly, most VUES observa-tions cover the minimum of this curve, hence γ VUES = − . MNRAS000
RV curve of the centre-of-mass ABC correspondingto the outer orbit with P = 3 . masses is verified by the absence of correlation between V and RVs of the individual components.A plot of V vs. time clearly shows its variation with aperiod of ∼ V ( t ) variation are given in Table 3, the RV curve is plottedin Fig. 7. The RV amplitude in the outer orbit is denotedby K . We adopted the erors of 1 km s − for the CfA data,0.5 km s − for TRES, VUES and UVES, and 0.1 km s − forCHIRON. The latter RVs are distinguished by the tight se-quence of black squares around the phase 0.5, showing the V trend in just two months. The global weighted rms resid-uals are 0.38 km s − . The long and extensive coverage ofthe CfA data is essential for constraining the outer orbit.The middle and inner orbits computed from the CfAdata of 2007–2008 happen to be near the maximum of theRV curve in Fig. 7, hence γ CfA = − . − ; the trendduring this period was small. Similarly, most VUES observa-tions cover the minimum of this curve, hence γ VUES = − . MNRAS000 , 1–19 (2019) ompact multiple HIP 41431 Table 3.
Elements of the outer orbitElement Value P (d) 1427 ± τ (BJD − ± e ± ω (deg) 52 ± K (km s − ) 3.963 ± γ (km s − ) − ± km s − (the first VUES observation does not match the innerorbits without an offset correction).Given that the inner pair is eclipsing, the factorsin i A , B ≈
1, so the spectroscopic masses are close to thetrue masses of the components. The mass sum of 1.9 M (cid:12) forA+B+C and the RV amplitude of the outer orbit then leadto M D > . M (cid:12) . If the inclination of the outer orbitwere substantially different from 90 ◦ , the large resulting M D would contradict its non-detection in the spectra, sowe adopt M D = 0 . M (cid:12) (this is confirmed below by thefull modelling). The total mass and the period define thesemimajor axis of the outer orbit, 3.26 au or 67 mas on thesky. At the Gaia
DR2 epoch, 2015.5, the star D was reced-ing from ABC, moving toward maximum separation. Theprojected speed of the mean orbital motion during the timeinterval of 2015.5 ± µ orb = 40 mas yr − , di-rected away from the primary. Comparing this speed to theobserved ∆ µ = 7 . − and neglecting the light of thestar D allows a direct measurement of the outer mass ratio q from the relation ∆ µ/µ orb = q / (1 + q ). Hence, q = 0 . M D = 0 . M (cid:12) . This confirms our assumption thatthe outer orbit has a large inclination. The direction of ∆ µ suggests that the companion D was at the position angleof ∼ ◦ in 2015.5. Without actually resolving the outer bi-nary, we already know approximately all its orbital elementsand can compute the positions. As a consequence of the compactness of this quadruple sys-tem (the period ratios are P /P ∼ . P /P ∼ . Lightcurvefactory (see Borkovits et al.2019, and further references therein) of which the latest ver-sion is now able to handle quadruple systems both in 2+2and 2+1+1 configurations. The relevant modifications of theorbital equations to be numerically integrated in this newversion are discussed in Appendix A.Apart from the inclusion of the fourth star forming thethird, outermost “binary” with the centre of mass of theABC components, this complex analysis was carried out ina very similar manner as described in Sect. 7 of Borkovits etal. (2019) and, therefore, here we discuss only the basic steps briefly. We carried out a joint Markov Chain Monte Carlo(MCMC) parameters search for the following data series:(i) Two sets of long cadence K2 lightcurves;(ii) The RVs of components A, B, and C;(iii) The ETV curves of the innermost EB (for both primaryand secondary minima).Regarding item (i), we consider two variants of pro-cessed K2 lightcurves, with constant and variable eclipsedepth, as described in Sect. 2.1.1. In both cases, we use onlya narrow window of width ∼ .
12 d centered on each eclipse(blue points in Fig. 2, right panel). The out-of-eclipse bright-ness variations are negligible, and the omission of these datasaves a significant amount of computational time. Most dy-namical information coded in the lightcurves is containedin the fine structure and timings of the eclipses. Note also,that for the ∼ . Kepler , weapply a cadence time correction on the model lightcurves(see Borkovits et al. 2019, for details). Considering that theeclipse depth variation in the
Kepler data has been con-firmed by our ground-based photometry, we discuss belowonly the uneven eclipse depth solution.Turning to the RV curves, we emphasize that insteadof fitting the usual analytical formulae, our numerical inte-grator calculates for all time instances the 3D velocity vec-tors of all four bodies, the v z components of those vectorsgive directly the RVs relative to the centre of mass of thequadruple system. The systemic velocity, γ , is then calcu-lated a posteriori by a simple linear regression minimizingthe χ residuals between measured RVs and the model.We found only minor zero-point differences amongst the RVinstruments (see Sect. 2.2) and neglected them.In principle, the K2 lightcurves carry the same timinginformation as the ETV curves, making the latter redun-dant. However, the advantages of using both the lightcurvesand the ETV curves together have been explained inBorkovits et al. (2019). Similarly to our previous work, theETV curves were used to preset the period ( P ) and phaseterm ( T ) of the innermost binary for each new set of thetrial parameters. The latest ETV points from the ground-based photometry were also added to the data set.During our analysis, we carried out several dozens ofMCMC runs and tried different sets and combinations ofadjustable parameters. We also applied some additional re-lations to constrain some of the parameters in order to re-duce the degrees of freedom in our problem. For example,while the masses of all four stars can be deduced from thejoint dynamical analysis of the ETV and RV curves and,combining these results with the outputs of the lightcurveanalysis, the physical dimensions of the eclipsing stars canalso be determined, none of the observational data used forthe photodynamical modelling carry information on the radiiof the stars C and D. Similarly, only the temperature ra-tio of stars B and A ( T B /T A ) can be constrained by thelight curve, while the effective temperature of one star inthe inner binary should be taken from an external source.The photodynamical model can say nothing on the effectivetemperatures of the stars C and D. Their net flux in the Ke-pler band is manifested only as extra flux for the lightcurvemodel. In order to get reliable information on these param-eters we applied different options. Regarding T A , in someruns we constrained it with a Gaussian prior centered to the MNRAS , 1–19 (2019)
Borkovits et al. temperature given in
Gaia
DR2 ( T eff = 3978 K), while inanother series of runs the code calculated internally the tem-perature in each trial step from the stellar mass m A with theuse of the mass–temperature relations of Tout et al. (1996),valid for zero age main sequence (ZAMS) stars. During ouranalysis the radii of the two outer stars ( R C , D ) and alsothe effective temperature of the star D ( T D ) were also con-nected internally to their masses via the relations of Toutet al. (1996). (For these calculations solar metallicity wasassumed.) Applying these three constraints, the fourth re-maining parameter, i. e. T C , takes the role of the extra lightparameter ( l x ) and, therefore, there is no need to use thislatter one.Besides the above mentioned constraints, in most of ourruns we adjusted the following parameters:(i) Three parameters related to the orbital elements of theinner binary: eccentricity ( e ), the phase of the secondaryeclipse relative to the primary one ( φ sec , ) which constrainsthe argument of periastron ( ω , see Rappaport et al. 2017),and the inclination ( i ) ;(ii)-(iii) Two times six parameters related to the orbital el-ements of the middle and the outermost orbits: P , ,( e sin ω ) , , ( e cos ω ) , , i , , the times of the periastron pas-sages of stars C and D along their revolutions on the middleand the outermost orbits, respectively ( τ , ), and the posi-tion angles of the nodes of the two orbits (Ω , ) ;(iii) Four mass-related parameters: the mass of the compo-nent A, m A , and the mass ratios of all three orbits q , , ;(iv) and, finally, four other parameters which are related(almost) exclusively to the lightcurve solutions, as follows:the duration of the primary eclipse (∆ t ) pri closest to epoch t (which is an observable that is strongly connected to thesum of the fractional, i. e. scaled by the inner semi-majoraxis, radii of stars A and B, see Rappaport et al. 2017),the ratio of the radii of stars A and B ( R B /R A ), and thetemperature ratios of T B /T A and T C /T A .Turning to other, lightcurve-dependent parameters, weapplied a logarithmic limb-darkening law, where the coeffi-cients were interpolated from the pre-computed passband-dependent tables in the Phoebe software (Prˇsa & Zwitter2005). The
Phoebe -based tables, in turn, were derived fromthe stellar atmospheric models of Castelli & Kurucz (2004).Due to the nearly spherical stellar shapes in the inner binary,an accurate setting of gravity darkening coefficients has noinfluence on the lightcurve solution and, therefore, we simplyadopted a fixed value of g = 0 .
32 which is appropriate forlate-type stars according to the traditional model of Lucy(1967). We also found that the illumination/reradiation ef-fect was quite negligible for the eclipsing binary; therefore,in order to save computing time, this effect was neglected.On the other hand, the Doppler-boosting effect (Loeb &Gaudi 2003; van Kerkwijk et al. 2010) was included into ourmodel. Furthermore, in the absense of any other informa-tion, we assumed that the equatorial planes of stars A and Note, again, that P and ( T ) are constrained through theETV curves. Strictly speaking, as we set Ω = 0 ◦ at epoch t for all runs,adjusting the other two Ω , -s is practically equivalent to theadjustment of the differences of the nodes (i. e., ∆Ω-s), which arethe truly relevant parameters for dynamical modelling. B are aligned with the innermost orbital plane. The pro-jected rotational velocities of stars A, B and C were set totheir spectroscopically obtained values (see Sect. 2.2.3). The orbital and astrophysical parameters derived fromthe ‘uneven eclipse depth scenario’ spectro-photodynamicalanalysis are tabulated in Table 4, and will be discussed inthe subsequent Sections 5 and 6. The corresponding modellightcurves are presented in Fig. 2, while the different RVcurves are shown in Figs. 6, 7, and 8. Finally, the modelETV curve plotted against the observed ETVs is shown inFig. 3.As a sanity check for the photodynamical solution, wecalculate the maximum photometric distance of HIP 41431by combining the total V magnitude of the system (see thepenultimate row in Table 4) with the apparent V magni-tude (see Table 1). This results in a photometric distance of d phot ≤ ± Gaia parallax.
The masses obtained from the spectro-photodynamicalmodel are in good agreement with masses deduced from thepreliminary RV analysis and the
Gaia astrometry (Sect. 3).Similarly, the relative fluxes of the visible stars A, B, and Cin the V band deduced from the model, 0.95:0.79:1, matchwell their relative fluxes measured spectroscopically (seeSect. 2.2.1). The minor contribution of the faint star D tothe V -band flux is accounted for by assuming that it is anMS dwarf. The V − K colors of the stars are computed fromtheir effective temperatures listed in Table 4 using standardrelations for MS stars (e.g. Pecaut & Mamajek 2013) andadjusted to match the measured combined V − K s colorin Table 1. This allows us to place the stars on the color-magnitude diagram (CMD) in Fig. 9 using the Gaia parallax.The masses, colors, and absolute magnitudes of the visi-ble stars A, B, and C match well both the empirical relationsof Pecaut & Mamajek (2013) and the theoretical isochronefor solar metallicity, while the star D of 0.35 M (cid:12) contributesonly 0.01 to the total light in the V band and 0.35 in the Ke-pler band. The empirical relations of Benedict et al. (2016)for M-type dwarfs predict the V − K colors from 3.61 to3.66 mag for stars with masses of the components A, B, andC, and match the observed combined color V − K s = 3 . V magnitudes are brighter thanthose of Benedict et al. by ∼ Gaia gives T eff = 3978 K) and gravity log g = 4 . − . These settings are irrelevant for the lightcurve modelling ofsuch almost spherical stars, but matter for the longer-term dy-namical studies discussed in Sect. 6. MNRAS000
Gaia astrometry (Sect. 3).Similarly, the relative fluxes of the visible stars A, B, and Cin the V band deduced from the model, 0.95:0.79:1, matchwell their relative fluxes measured spectroscopically (seeSect. 2.2.1). The minor contribution of the faint star D tothe V -band flux is accounted for by assuming that it is anMS dwarf. The V − K colors of the stars are computed fromtheir effective temperatures listed in Table 4 using standardrelations for MS stars (e.g. Pecaut & Mamajek 2013) andadjusted to match the measured combined V − K s colorin Table 1. This allows us to place the stars on the color-magnitude diagram (CMD) in Fig. 9 using the Gaia parallax.The masses, colors, and absolute magnitudes of the visi-ble stars A, B, and C match well both the empirical relationsof Pecaut & Mamajek (2013) and the theoretical isochronefor solar metallicity, while the star D of 0.35 M (cid:12) contributesonly 0.01 to the total light in the V band and 0.35 in the Ke-pler band. The empirical relations of Benedict et al. (2016)for M-type dwarfs predict the V − K colors from 3.61 to3.66 mag for stars with masses of the components A, B, andC, and match the observed combined color V − K s = 3 . V magnitudes are brighter thanthose of Benedict et al. by ∼ Gaia gives T eff = 3978 K) and gravity log g = 4 . − . These settings are irrelevant for the lightcurve modelling ofsuch almost spherical stars, but matter for the longer-term dy-namical studies discussed in Sect. 6. MNRAS000 , 1–19 (2019) ompact multiple HIP 41431 -80-60-40-20 0 20 40 60 R ad i a l V e l o c i t y [ k m / s e c ] -5 0 5 51400 51600 51800 52000 52200 R e s i dua l [ k m / s e c ] -80-60-40-20 0 20 40 60 R ad i a l V e l o c i t y [ k m / s e c ] -5 0 5 54200 54400 54600 54800 55000 R e s i dua l [ k m / s e c ] -80-60-40-20 0 20 40 60 R ad i a l V e l o c i t y [ k m / s e c ] -5 0 5 57600 57800 58000 58200 58400 R e s i dua l [ k m / s e c ] BJD - 2400000
Figure 8. γ velocity. Note variations both inthe shape and orientation of the RV orbits. These are consequences of the quick apsidal motion of the middle orbit due to the strongdynamical interactions of the four stars.MNRAS , 1–19 (2019) Borkovits et al.
Table 4.
Orbital and astrophysical parameters from the joint photodynamical lightcurve, three RV curves and ETV solutionorbital elements a subsystemA–B AB–C ABC–D P [days] 2 . ± . . ± .
006 1441 . ± . a [R (cid:12) ] 9 . ± .
054 78 . ± .
48 700 . ± . e . ± . . ± . . ± . ω [deg] 175 . ± .
32 332 . ± .
51 293 . ± . i [deg] 87 . ± .
032 86 . ± .
136 84 . ± . τ [BJD - 2400000] 57142 . ± . . ± .
089 57768 . ± .
1Ω [deg] 0 . − . ± . − . ± . i m [deg] − . ± .
107 21 . ± . q = m sec /m pri ] 0 . ± .
009 0 . ± .
009 0 . ± . K pri [km s − ] 79 . ± .
593 23 . ± .
318 3 . ± . K sec [km s − ] 80 . ± .
599 46 . ± .
404 20 . ± . γ [km s − ] − − . ± . R/a ] 0 . ± . . ± . . ± . . ± . Kepler -band] 0 . . . . m [M (cid:12) ] 0 . ± .
010 0 . ± .
012 0 . ± .
016 0 . ± . R [R (cid:12) ] 0 . ± .
012 0 . ± .
012 0 . ± . b . ± . b T eff [K] 4043 ± c ±
60 4064 ±
83 3373 ± c L bol [L (cid:12) ] 0 . ± . . ± . . ± . . ± . M bol . ± .
08 7 . ± .
08 7 . ± .
10 9 . ± . M V . ± .
11 8 . ± .
12 8 . ± .
14 12 . ± . g [dex] 4 . ± .
02 4 . ± .
02 4 . ± .
03 4 . ± . M V ) tot . ± . d [pc] < . ± . Notes. a : Instantaneous, osculating orbital elements, calculated for epoch t = 2457143 .
395 (BJD); b : Calculated from the mass–radiusrelations of Tout et al. (1996); c : Calculated from the mass–temperature relations of Tout et al. (1996); d : Photometric maximumdistance, see text for details. (HD 201092) as an example. Its parameters and position onthe CMD happen to be similar to the components A, B,and C of HIP 41431. The measurements of [Fe/H] for the 61Cyg A and B listed in Simbad have a large scatter, illustrat-ing the difficulty of the spectroscopic analysis of late-typedwarfs. Most measurements give [Fe/H] ≈ − . (Palacios et al. 2010). We made this comparison forthe [Fe/H] values ranging from − . − T eff or with a binary mask, near the CaII infraredtriplet. The resulting correlation function does not containthe expected details. The star D could have a fast rotationor could be a close spectroscopic pair. See http://pollux.oreme.org
MNRAS000
MNRAS000 , 1–19 (2019) ompact multiple HIP 41431 CA B DABC61 CygB
Figure 9.
Color-magnitude diagram. The lines are 1-Gyrisochrones from Bressan et al. (2012) for metallicity [Fe/H] of0 and − . B C A + observedmodeldifference
Figure 10.
Observed spectrum (red crosses) is compared tothe synthetic spectrum with T eff = 4000 K, log g = 4 .
7, and[Fe/H]= − As mentioned previously, the four stars in this compactquadruple system interact dynamically and therefore, thethree orbits are subjected to strong and fast dynamical per-turbations. Spectacular manifestations of these interactionsare nicely visible even on the four-year-long data train of themeasured ETVs (Fig. 3) of the innermost, eclipsing pair.First, we note the 59-day sine-like modulation of theETV, similar (but not identical) for the primary and sec-ondary ETVs. The dominating contributor to this modu-lation is the third-body perturbation from the star C thatalternates the mean motion (and also the orbital elements)of the innermost binary on the timescale of the period of the middle orbit (Borkovits et al. 2015). The contributionof the classic light-travel time effect (LITE) to this 59-dayvariation is only about 10%.Second, the crossing of the two ETV curves reveals afast, dynamically forced apsidal motion. According to ournumerical integration, which was a substantial part of thephotodynamical solution, during this four years the majoraxis of the innermost orbit has turned by ≈ − ◦ ,i. e., has made almost half a revolution (see Fig. 11). There-fore, the current period of the apsidal motion of the in-nermost orbit is U ≈ P ∼ . − days around BJD 2 458 150and 2 458 520, i. e. during the late campaign (C16 and C18)observations of the K2 mission. In this moment we cannotdecide whether these small, but systhematic discrepancies,which however do not exceed the estimated accuracies ofthe individual ETV points have physical origins indicatingsome inaccuracies in the parameters of the expected four-body model, or they are consequences of some instrumentaleffects.Turning to the RV data, a more spectacular, and almostuniquely observed manifestation of the apsidal rotation ofthe middle orbit can be seen in Fig. 8, where we plot the RVsof the three visible stars A, B, and C (after subtracting theorbit of the eclising pair), together with the correspondingspectro-photodynamical model for the whole, 20-year-longtime span of our observations. The apsidal rotation of themiddle orbit results in the notable variation of the shape ofthe RV curves. Orbital inclination is another key observable in aneclipsing binary. Our photodynamical solution has revealedsmall, but definitely non-zero relative (mutual) inclinationbetween the two inner orbital planes: i mut1 − = 2 . ◦ ± . ◦ The mutual inclination between the outermost and the in-ner and middle orbits is larger, although its uncertaintly issubstantial: i mut1 − = 22 ◦ ± ◦ and i mut2 − = 20 ◦ ± ◦ .The non-coplanarity of the orbits triggers precession of allthree orbital planes, illustrated in the right panel of Fig. 11,where the variations of the three observable orbital incli-nations (i. e. the angles between the orbital planes and theplane of the sky) are plotted. While spectroscopically detected apsidal motions were pre-viously reported for other close binaries (see e. g. Ferrero et al.2013) and even for an exoplanet (Csizmadia et al. 2019), too, weare not aware of any other systems where such a significant frac-tion of a complete apsidal revolution period was covered with RVdata so densely as in the present case. Note, that the combination of the dynamical and geometri-cal effects on the light- and ETV curves break the degeneracybetween prograde and retrograde solutions, therefore an almostcoplanar, but retrograde solution can be ruled out with high con-fidence.MNRAS , 1–19 (2019) Borkovits et al. ω , , [ i n deg ] BJD - 2400000Mean orbital phase of the medium orbit
83 84 85 86 87 88 89 90 52000 54000 56000 58000-100 -75 -50 -25 0 I n c li na t i on [ i n deg ] BJD - 2400000Mean orbital phase of the medium orbit
Figure 11.
Variations of the osculating (observable) arguments of periastron ( ω , , ) and inclinations ( i , , ) of the three orbits between1998.5 and 2020.4, as calculated from our spectro-photodynamical solution. Black, red and blue lines denote the orbital elements of theinnermost, middle and outer orbits, respectively. The elements are averaged for the period of the corresponding orbits. Vertical brownline represents the beginning of the Kepler observations. R e l a t i v e F l u x BJD - 2400000
Figure 12.
The model lightcurve of the star between 1998.5 and2020.4 (thin black curve). The eclipse depth variations caused bythe orbital plane precession are well visible. The K2 lightcurvescollected during Campaigns 5, 16 and 18 are also plotted with redcircles. While the dynamical effects of such orbital misalign-ments are expected to occur only on very long times scales,their observational consequences, however, are manifestedalmost promptly, in dramatic eclipse depth variations. InFig. 12 we plot the model lightcurve of the system since thebeginning of the spectroscopic observations. The ≈ . i ≈ . ◦ amplitude, short-termvariation of the inclination ( i ) of the innermost orbit, isclearly visible. Moreover, another (on this timescale lin-ear) effect is also well visible; it corresponds to the longertime-scale and larger-amplitude precession triggered by themore inclined outermost orbit. As a consequence, if the pho-todynamical solution is correct, in the forthcoming decades The precession period is in perfect agreement with the an-alytically calculated period within the framework of the stellarthree-body problem (see, e. g., S¨oderhjelm 1975, Eq. 27). This factillustrates that on short time scales, the effects of the third andfourth bodies remain almost independent, at least, from a dynam-ical point of view. one can expect that the mean visible inclinations of the in-nermost and middle orbits (i. e., i and i averaged overthe ≈ . ≈ ◦ around 2040, eclipses ofthe component C should also become observable for severalyears.We emphasize, however, that the relative nodal angleof the outermost orbit (Ω ) is obtained only with a largeuncertainty and thus, the corresponding two outer mutualinclinations are only weakly constrained. Therefore, theseresults should be considered as tentative. The reason of thisuncertainty is that only the eclipse depth variation of theeclipsing pair is strongly sensitive to the rate of the (visible)inclination variation of the innermost pair and, therefore,only the K2 observations, which cover a small fraction ofthe ≈ Kepler might be de-batable. Therefore, follow-up observations and continuousmonitoring of the eclipse depth variations are crucial.We added this star to the long-term eclipse monitoringprogramme of Baja Observatory, Hungary, as a top prior-ity target. Unfortunately, due to the bad weather conditions(which are usual in the winter season), so far we were able toobserve only four primary and two secondary eclipses. Fur-thermore, owing to the poor sky conditions, two primaryminima were observed in unfiltered mode, and only foureclipses were observed with a standardized Kron-Cousins R C filter. Normally, unfiltered minima observations are usefulfor the times of minima determination but unfit for study-ing the eclipse depth variation. Therefore, we consider onlythe R C -band eclipse observations. We generated the R C -band model lightcurve for those nights and compared it tothe observations (see Fig. 13). As one can see, the agreementfor the primary eclipse is almost perfect. For the secondaryeclipse, a minor systematic deviation can be seen. However,the decrease of the eclipse depths is beyond doubt. This factconfirms not only the ongoing precession of the innermostorbit but, retrospectively, justifies the physical origin of theeclipse depth variations observed in the different K2 cam-paigns.In order to check the long time-scale dynamical evolu- MNRAS000
The model lightcurve of the star between 1998.5 and2020.4 (thin black curve). The eclipse depth variations caused bythe orbital plane precession are well visible. The K2 lightcurvescollected during Campaigns 5, 16 and 18 are also plotted with redcircles. While the dynamical effects of such orbital misalign-ments are expected to occur only on very long times scales,their observational consequences, however, are manifestedalmost promptly, in dramatic eclipse depth variations. InFig. 12 we plot the model lightcurve of the system since thebeginning of the spectroscopic observations. The ≈ . i ≈ . ◦ amplitude, short-termvariation of the inclination ( i ) of the innermost orbit, isclearly visible. Moreover, another (on this timescale lin-ear) effect is also well visible; it corresponds to the longertime-scale and larger-amplitude precession triggered by themore inclined outermost orbit. As a consequence, if the pho-todynamical solution is correct, in the forthcoming decades The precession period is in perfect agreement with the an-alytically calculated period within the framework of the stellarthree-body problem (see, e. g., S¨oderhjelm 1975, Eq. 27). This factillustrates that on short time scales, the effects of the third andfourth bodies remain almost independent, at least, from a dynam-ical point of view. one can expect that the mean visible inclinations of the in-nermost and middle orbits (i. e., i and i averaged overthe ≈ . ≈ ◦ around 2040, eclipses ofthe component C should also become observable for severalyears.We emphasize, however, that the relative nodal angleof the outermost orbit (Ω ) is obtained only with a largeuncertainty and thus, the corresponding two outer mutualinclinations are only weakly constrained. Therefore, theseresults should be considered as tentative. The reason of thisuncertainty is that only the eclipse depth variation of theeclipsing pair is strongly sensitive to the rate of the (visible)inclination variation of the innermost pair and, therefore,only the K2 observations, which cover a small fraction ofthe ≈ Kepler might be de-batable. Therefore, follow-up observations and continuousmonitoring of the eclipse depth variations are crucial.We added this star to the long-term eclipse monitoringprogramme of Baja Observatory, Hungary, as a top prior-ity target. Unfortunately, due to the bad weather conditions(which are usual in the winter season), so far we were able toobserve only four primary and two secondary eclipses. Fur-thermore, owing to the poor sky conditions, two primaryminima were observed in unfiltered mode, and only foureclipses were observed with a standardized Kron-Cousins R C filter. Normally, unfiltered minima observations are usefulfor the times of minima determination but unfit for study-ing the eclipse depth variation. Therefore, we consider onlythe R C -band eclipse observations. We generated the R C -band model lightcurve for those nights and compared it tothe observations (see Fig. 13). As one can see, the agreementfor the primary eclipse is almost perfect. For the secondaryeclipse, a minor systematic deviation can be seen. However,the decrease of the eclipse depths is beyond doubt. This factconfirms not only the ongoing precession of the innermostorbit but, retrospectively, justifies the physical origin of theeclipse depth variations observed in the different K2 cam-paigns.In order to check the long time-scale dynamical evolu- MNRAS000 , 1–19 (2019) ompact multiple HIP 41431 R e l a t i v e F l u x -0.02 0.00 0.02 58542.4 58542.5 58542.6 R e s i dua l F l u x BJD - 2400000 R e l a t i v e F l u x -0.02 0.00 0.02 58567.3 58567.4 58567.5 R e s i dua l F l u x BJD - 2400000
Figure 13.
Upper left panel:
Primary eclipse of HIP 41431 measured in R C -band on the night of 27/28 Feb 2019 (blue circles) andthe corresponding photodynamical model lightcurves both for the uneven and uniform K2 eclipse depths scenarios (red, and grey lines,respectively). As one can see, the measured eclipse depth is in perfect agreement with the predictions of the uneven eclipse depthsscenario and therefore, it confirms the physical origin of the eclipse depth differences amongst the different K2 campaigns. Upper rightpanel:
The same for the secondary eclipse measured also in R C -band on the night of 24/25 March 2019. Lower panels:
Observations vs.uneven eclipse depth model lightcurve residuals. tion and stability of our quadruple system, we carried outfurther numerical integration on a timescale of 10 yr. Theintegrator was the same as in the case of the photodynam-ical analysis. Therefore, beyond the four-body point-massforces, tidal forces acting upon in the innermost binary werealso considered, including the Eulerian equations of the ro-tations of stars A and B. Furthermore, for some additionalruns tidal dissipation (within the framework of the equilib-rium tide approximation), and relativistic apsidal motionwere also included (see Appendix A for details). The addi-tional parameters necessary for these integrations were setas follows. The inner structure (or apsidal motion) constantsof both stars A and B were set to k = 0 .
02 which, accord-ing to Torres et al. (2010), is appropriate for such low-massstars. Furthermore, the dissipation rates for both stars wereset to λ = 2 × − . This type of dissipation rate was definedby Eq. (13) in Kiseleva et al. (1998). It is connected to thesmall tidal lag time through the formula:∆ t = − (cid:114) R Gm (1 + 2 k ) λ (2)(see Borkovits et al. 2004, Eq. 25). The choosen numericalvalues of λ correspond to tidal lags of ∆ t ≈ − × − daysfor both stars. The integrations did not reveal any dramaticvariations in the orbital elements of the three orbits. There-fore, we conclude that the orbital configuration is stable upto the nuclear evolution time scales.On the other hand, the numerical method allows us tostudy the spin evolution of the innermost two stars. This isespecially interesting in the present case, as the most un-usual characteristic of this system is the slow axial rotationof the stars comprising the inner pair. For HIP 41431, thestandard assumption that the axes are perpendicular to theorbit is not trivial. The likely non-coplanarity of the outer-most orbit forces significant orbital plane precession, which The Referee, however, noted that Fig. 6 of Lurie et al. (2017)contains other eclipsing binaries with short periods and substan-tially sub-synchronous rotation measured from starspots. may lead to spin-orbit misalignement, as suggested e. g. byBeust et al. (1997) in the case of TY CrA. Furthermore, asfound by Correia et al. (2016), the secular evolution of thespins in hierarchical triple systems when viscous tidal forcesare present might be affected strongly by secular resonancesbetween orbital and spin precessions and, therefore, chaoticrotation might occur.In what follows we discuss briefly some results of threedifferent integrator runs. Dissipative forces were taken intoaccount in all three runs. For the run ‘A’, the spin axes ofstars A and B are parallel to the orbital spin vector of theinnermost orbit at the epoch t used in the photodynamicalmodel. Furthermore, the spin rates are set according to thespectroscopically measured projected rotation velocities. Inother words, apart from the dissipation terms, this numericalintegration is a simple extention of the accepted photody-namical model over a much longer time scale. For the run‘B’, the initial orbital elements were the same, but the ori-entation and magnitude of stellar spins are set arbitrarily.Finally, for the run ‘C’, the initial parameters are the sameas in run ‘A’, but the outermost, fourth body is removed,i. e., a three-body integration was carried out.In Fig. 14 we plot the variations of the orbital inclina-tions of the innermost orbit and of the equatorial planesof stars A and B on different time scales. The ∆ i ≈ ◦ peak-to-peak amplitude, ≈ MNRAS , 1–19 (2019) Borkovits et al. some kind of spin-orbit resonances. We plot the evolution ofthe stellar spin rates in Fig. 16. As one can see, due to thedissipative forces the originally sub- or super-synchronousrotation periods quickly relax to the orbital period. How-ever, in this case various spin-orbit resonances may occur.As a consequence, the stellar spins can again desynchronizeand, furthermore, large amplitude equatorial plane preces-sion may also happen. The investigation of these phenomenais beyond the scope of the present paper; they were studied,e. g., by Correia et al. (2016). In the context of the presentpaper, we conclude that the measured low projected rota-tional velocities of stars A and B probably offer observationalevidence for the presence of strong spin-orbit coupling. Thequestion whether the stars have strongly inclined spin axesor rotate slowly (or both) cannot be answered at present. The triple system HIP 41431 is remarkable in several re-spects. First, it is very compact, with a 3-tier (3+1) hier-archy fitting inside the 3.3-au outer orbit. Second, all or-bits are close to one plane (mutual inclinations of 2 . ◦ ± . ◦ ◦ ± ◦ ), while the period ratios are similar (20.17 and24.4). The orbits interact dynamically.The spatial velocity of this system ( U, V, W ) =(8 . , . , − .
4) km s − does not distinguish it from the olddisk population and does not match known kinematicalgroups of young stars in the solar neighbourhood. The spec-tra do not have the lithium 6708˚A line or emissions in H α typical of young stars and no variability associated withchromospheric activity or star spots was found in the K2 data. We conclude that this multiple system is not young.The most unusual characteristic of this system is theslow axial rotation of stars comprising the short-period in-ner pair, expected to be tidally synchronzed. However, thisapparent paradox might be caused by the spin-orbit cou-pling and resonances triggered by the dynamically interact-ing third and fourth stellar companions, leading to chaoticrotation.We looked for similarly compact hierarchies in the Mul-tiple Star Catalog (MSC) (Tokovinin 2018a) . The currentversion of the catalog contains 29 triples with outer peri-ods P out <
150 d (not counting the present system). AllMS triples except one have primary components of earlierspectral type than HIP 41431 (likely an observational selec-tion effect). There are only six known triples, however, withthe outer periods shorter than 59 d. While the absolute di-mensions of the orbits (and, therefore, the orbital periods)are very important parameters from the point of view ofthe effectiveness of the tidal forces and also of the systemformation scenarios, the period ratios are more significantindicators of the strength of dynamical interactions betweenorbits. In this regard, the period ratios of ∼
20 found in thetwo subsystems of HIP 41431 are far from being extreme. Inthe small mutual inclination regime, such period ratios are We made a period search of the residuals of
Kepler photom-etry to our photodynamical model for potential signal caused bystarspots, and have not found any significant periods differentfrom the orbital period and its harmonics. well within the stability region of hierarchical triple stars(see, e. g. Mardling & Aarseth 2001).No quadruple systems of 3+1 hierarchy as compact asHIP 41431 were known previously. However, there are atleast three compact triple systems with short outer periodsamong the
Kepler ’s prime misson EBs where the systematicresiduals of the four-year-long ETV data might indicate thepresence of a fourth component (Borkovits et al. 2016).Our work has contributed an interesting system thatchallenges the theories of star formation. Compact andcoplanar hierarchical stellar systems like HIP 41431 areprobably a result of migration in massive disks that arepresent at the time of star formation. The first two stel-lar embryos condense from gas, accrete mass, and migrateinward, while outer components condense later in the sameaccretion flow (e. g. by disc fragmentation) and, in theirturn, migrate inward. No other scenario can plausibly ex-plain the origin of such well-organized, planetary-like hierar-chies. However, further discussion of formation mechanismsis beyond the scope of this paper.
ACKNOWLEDGMENTS
We thank the Referee, D. Gies, for useful suggestions andcorrections. T. B. acknowledges the financial support of theHungarian National Research, Development and InnovationOffice – NKFIH Grants OTKA K-113117 and KH-130372.L. M. was supported by the Premium Postdoctoral Programof the Hungarian Academy of Sciences. The research lead-ing to these results has received funding from the LP2017-8Lend¨ulet grant of the Hungarian Academy of Sciences.We used the Simbad service operated by the Centredes Donn´ees Stellaires (Strasbourg, France) and the ESOScience Archive Facility services (data obtained under re-quest number 396301). This work has made use of datafrom the European Space Agency (ESA) mission
Gaia ,processed by the Gaia
Data Processing and Analysis Con-sortium (DPAC, ). Funding for the DPAC has been pro-vided by national institutions, in particular the institutionsparticipating in the
Gaia
Multilateral Agreement.This paper includes data collected by the K2 mission.Funding for the K2 mission is provided by the NASA Sci-ence Mission directorate. Some of the data presented in thispaper were obtained from the Mikulski Archive for SpaceTelescopes (MAST). STScI is operated by the Associationof Universities for Research in Astronomy, Inc., under NASAcontract NAS5-26555. Support for MAST for non-HST datais provided by the NASA Office of Space Science via grantNNX09AF08G and by other grants and contracts. REFERENCES
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Figure 14.
The evolution of the orbital inclination of the innermost binary and the orientation of the stellar equators of stars A andB during three different integration runs.
Left panel:
A 3000-year-long zoom into the variation of the angles. Solid lines (with threedifferent shades of red) represent the variation of the orbital inclination and the equatorial angles of the two stars during run ‘A’. Theshort-period, small amplitude fluctuations in the inclination reflects the precession due to the star C, while the longer period, largeramplitude variation is the precession caused by the star D. Dotted lines (in blue colors) show the same parameters for the run ‘B’ (i.e.inclined spin axes), while the dash-dotted green lines represent run ‘C’, i.e. the originally aligned 3-body model.
Right panel: R e l a t i v e s p i n - o r b i t i n c li na t i on s [ i n deg ] Million Years
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As it was mentioned above, the numerical integrator which was used in the spectro-photodynamical code is an upgradedversion of the 3-body integrator described in Borkovits et al. (2004). Further details of the practical implementation of anumerical integrator coupled to the lightcurve emulator were discussed in the appendix of Borkovits et al. (2019). Here wediscuss the additional modifications introduced into the code to handle quadruple systems with 3+1 hierarchy.Similar to the previous hierarchical triple star case, the Jacobian vector formalism is consistently used. In order to describethe motion of the fourth body and its effect on the inner three stars, now we introduce the third Jacobian vector which pointsto the outermost component (star D) from the centre of mass of the inner triple subsystem (stars A, B, and C).Let us denote by (cid:126)r i the barycentric radius vector of the component i and by (cid:126)r ij = (cid:126)r j − (cid:126)r i the vector between components j and i . Then, the first three Jacobian vectors are as follows: (cid:126)ρ = (cid:126)r , (A1) (cid:126)ρ = (cid:126)r − m m (cid:126)r − m m (cid:126)r = (cid:126)r − m m (cid:126)r = (cid:126)r + m m (cid:126)r , (A2) (cid:126)ρ = (cid:126)r − m m (cid:126)r − m m (cid:126)r − m m (cid:126)r , (A3)while the mutual distances between the components are: (cid:126)r = (cid:126)ρ , (A4) (cid:126)r = (cid:126)ρ + m m (cid:126)ρ (A5) (cid:126)r = (cid:126)ρ + m m (cid:126)ρ + m m (cid:126)ρ (A6) (cid:126)r = (cid:126)ρ − m m (cid:126)ρ (A7) (cid:126)r = (cid:126)ρ − m m (cid:126)ρ + m m (cid:126)ρ (A8) (cid:126)r = (cid:126)ρ − m m (cid:126)ρ . (A9)Then, the point-mass ( U ), tidal ( T ), and rotational ( R ) components of total potential take the following forms: U = Gm m r + Gm m r + Gm m r + Gm m r + Gm m r + Gm m r , (A10) T = Gm m r
12 4 (cid:88) j =2 (cid:40) m m k (1) j (cid:18) R r (cid:19) j (cid:18) R r d (cid:19) j +1 P j ( λ ) + m m k (2) j (cid:18) R r (cid:19) j (cid:18) R r d (cid:19) j +1 P j ( λ ) (cid:41) , (A11) T = (cid:88) i =1 4 (cid:88) (cid:96) =3 Gm i m (cid:96) r i(cid:96) m − i m i k ( i )2 (cid:18) R i r i(cid:96) (cid:19) (cid:18) R i r (cid:19) P ( λ i(cid:96) ) + Gm m r
12 2 (cid:88) i =1 4 (cid:88) (cid:96) =3 m (cid:96) m i k ( i )2 (cid:18) R i r (cid:19) (cid:18) R i r i(cid:96) (cid:19) P ( λ i(cid:96) ) , (A12)and, furthermore, R = Gm i m (cid:96) r i(cid:96) (cid:88) i =1 4 (cid:88) (cid:96) =1 (cid:96) (cid:54) = i (cid:40) k ( i )2 R i Gm i (cid:34) ω z (cid:48) i r i(cid:96) − ( (cid:126)r i(cid:96) · (cid:126)ω z (cid:48) i ) r i(cid:96) (cid:35)(cid:41) . (A13)(These expressions were deduced with the 2+1+1 case generalization of the Eqs. (10–13) of Borkovits et al. 2004, based on thetreatment of Kopal 1978.) The tidal and rotational terms are calculated only for stars A and B (denoted here by indices 1 and2), i. e. for the members of the innermost binary. In these expressions, R i denotes the radius of the i -th star, k ( i )j stands for the j -th apsidal motion constant of the i -th star (practically only k -s were used). Furthermore, in the T term which describesthe mutual interaction between the close binary members, r d i is the distance between the two stars, taken into account thetidal lag time of the component i , and λ i denotes the direction cosine between the radius vector and the tidal bulge of the i -th star. For the non-dissipative case, which was used for the spectro-photodynamical runs, d = d = r and λ = λ = 1.The terms T out give the tidal contributions of stars C and D to the motion of the innermost binary. Finally, in the lastterm ( R ), which describes the contributions of the rotational oblateness of stars A and B, (cid:126)ω z (cid:48) i stands for the uni-axial spinangular momentum vector of the i -th component. MNRAS , 1–19 (2019) Borkovits et al.
With the use of these potential terms, the equations of the motions to be integrated take the following form:¨ (cid:126)ρ = − Gm ρ (cid:126)ρ + Gm (cid:18) (cid:126)r r − (cid:126)r r (cid:19) + Gm (cid:18) (cid:126)r r − (cid:126)r r (cid:19) − Gm ρ (cid:40) (cid:88) i =1 (cid:40) (cid:88) j =2 m − i m i j + 1) k ( i ) j (cid:18) R i ρ (cid:19) j (cid:18) R i r d i (cid:19) j +1 (cid:126) P j ( λ i ) + k ( i )2 R i Gm i (cid:40)(cid:34) ω z (cid:48) i ρ − (cid:126)ρ · (cid:126)ω z (cid:48) i ) ρ (cid:35) (cid:126)ρ + 2 (cid:126)ρ · (cid:126)ω z (cid:48) i ρ (cid:126)ω z (cid:48) i (cid:41)(cid:41)(cid:41) + (cid:88) (cid:96) =3 m (cid:96) (cid:88) i =1 ( − i k ( i )2 R i m i r i(cid:96) (cid:40)(cid:34) ω z (cid:48) i − (cid:126)r i(cid:96) · (cid:126)ω z (cid:48) i ) r i(cid:96) (cid:35) (cid:126)r i(cid:96) + 2( (cid:126)r i(cid:96) · (cid:126)ω z (cid:48) i ) (cid:126)ω z (cid:48) i (cid:41) + (cid:88) (cid:96) =3 Gm (cid:96) (cid:88) i =1 (cid:40) ( − i m − i k ( i )2 R i m i r i(cid:96) ρ (cid:26)(cid:20) (cid:126)r i(cid:96) · (cid:126)ρ ) ρ r i(cid:96) − (cid:21) (cid:126)r i(cid:96) − (cid:126)r i(cid:96) · (cid:126)ρ ρ (cid:126)ρ (cid:27) − m k ( i )2 R i m i r i(cid:96) ρ (cid:26)(cid:20) (cid:126)r i(cid:96) · (cid:126)ρ ) ρ r i(cid:96) − (cid:21) (cid:126)ρ − (cid:126)r i(cid:96) · (cid:126)ρ r i(cid:96) (cid:126)r i(cid:96) (cid:27)(cid:41) , (A14)¨ (cid:126)ρ = − Gm m (cid:18) m r (cid:126)r + m r (cid:126)r (cid:19) + Gm m (cid:18) m r (cid:126)r − m r (cid:126)r − − m r (cid:126)r (cid:19) − m m (cid:40) (cid:88) i =1 k ( i )2 R i r i (cid:40)(cid:40)(cid:34) ω z (cid:48) i − (cid:126)r i · ω z (cid:48) i ) r i (cid:35) (cid:126)r i + 2( (cid:126)r i · (cid:126)ω z (cid:48) i ) (cid:126)ω z (cid:48) i (cid:41) + 3 Gm − i ρ (cid:26)(cid:20) (cid:126)ρ · (cid:126)r i ) ρ r i − (cid:21) (cid:126)r i − (cid:126)ρ · (cid:126)r i ρ (cid:126)ρ (cid:27)(cid:41)(cid:41) − m m (cid:40) (cid:88) i =1 k ( i )2 R i r i (cid:40)(cid:40)(cid:34) ω z (cid:48) i − (cid:126)r i · ω z (cid:48) i ) r i (cid:35) (cid:126)r i + 2( (cid:126)r i · (cid:126)ω z (cid:48) i ) (cid:126)ω z (cid:48) i (cid:41) + 3 Gm − i ρ (cid:26)(cid:20) (cid:126)ρ · (cid:126)r i ) ρ r i − (cid:21) (cid:126)r i − (cid:126)ρ · (cid:126)r i ρ (cid:126)ρ (cid:27)(cid:41)(cid:41) . (A15)¨ (cid:126)ρ = − m m (cid:26) Gm r (cid:126)r + Gm r (cid:126)r + Gm r (cid:126)r + (cid:88) i =1 k ( i )2 R i r i (cid:40)(cid:40)(cid:34) ω z (cid:48) i − (cid:126)r i · ω z (cid:48) i ) r i (cid:35) (cid:126)r i + 2( (cid:126)r i · (cid:126)ω z (cid:48) i ) (cid:126)ω z (cid:48) (cid:41) + 3 Gm − i ρ (cid:26)(cid:20) (cid:126)ρ · (cid:126)r i ) ρ r i − (cid:21) (cid:126)r i − (cid:126)ρ · (cid:126)r i ρ (cid:126)ρ (cid:27)(cid:41)(cid:41) , (A16)where (cid:126) P ( λ i ) = P ( λ i ) (cid:126)ρ + λ i ρ r d i ( (cid:126)r d i × (cid:126)ρ ) × (cid:126)ρ = 12 (cid:34) (cid:126)ρ · (cid:126)r d i ) ρ r i − (cid:35) (cid:126)ρ − (cid:126)ρ · (cid:126)r d i r i (cid:126)r d i , (A17) (cid:126) P ( λ i ) = P ( λ i ) (cid:126)ρ + 32 5 λ i − ρ r d i ( (cid:126)r d i × (cid:126)ρ ) × (cid:126)ρ = (cid:34)
10 ( (cid:126)ρ · (cid:126)r d i ) ρ r i − (cid:126)ρ · (cid:126)r d i ρ r d i (cid:35) (cid:126)ρ − (cid:34) (cid:126)ρ · (cid:126)r d i ) ρ r i − ρ r d i (cid:35) (cid:126)r d i , (A18) (cid:126) P ( λ i ) = P ( λ i ) (cid:126)ρ + 58 7 λ i − λ i ρ r d i ( (cid:126)r d i × (cid:126)ρ ) × (cid:126)ρ = 18 (cid:34)
70 ( (cid:126)ρ · (cid:126)r d i ) ρ r i −
45 ( (cid:126)ρ · (cid:126)r d i ) ρ r i +3 (cid:35) (cid:126)ρ − (cid:34) (cid:126)ρ · (cid:126)r d i ) ρ r i − (cid:126)ρ · (cid:126)r d i r i (cid:35) (cid:126)r d i . (A19)Note, however, that for non-dissipative cases (cid:126)ρ = ± (cid:126)r d , and thus, Eqs. (A17–A19) reduce simply to (cid:126) P j ( λ i ) = (cid:126)ρ . (A20)The spin angular momentum vectors of stars A and B ( (cid:126)ω z (cid:48) , ) may take any arbitrary orientations and magnitude. Theirevolution is also numerically integrated simultaneously via the Eulerian equations of rotation. The corresponding expressionsare given in Eqs. (B.11–B.13) of Borkovits et al. (2004) and, therefore, we do not repeat them here. APPENDIX B: SUPPLEMENTARY MATERIALB1 Times of eclipsing minima for ETV studies
In this subsection we tabulate the times of eclipsing minima of HIP 41431. The full list is available online only and, for reader’sconvenience, it is provided in machine readable format.
MNRAS000
MNRAS000 , 1–19 (2019) ompact multiple HIP 41431 Table B1.
Times of minima of HIP 41431 (EPIC 212096658).BJD Cycle std. dev. BJD Cycle std. dev. BJD Cycle std. dev. − d ) − d ) − d )57140.546616 -1.0 0.000745 57212.416959 23.5 0.000101 58165.600764 348.5 0.00034457142.030814 -0.5 0.000244 57213.863215 24.0 0.000159 58167.071306 349.0 0.00033457143.477973 0.0 0.000170 58096.686632 325.0 0.000451 58168.532472 349.5 0.00029457144.962745 0.5 0.000078 58098.149278 325.5 0.000653 58170.002961 350.0 0.00088657146.409726 1.0 0.000214 58099.617845 326.0 0.000084 58171.464520 350.5 0.00078357147.894165 1.5 0.000074 58101.080692 326.5 0.001293 58172.935285 351.0 0.00081957149.340890 2.0 0.000046 58102.549156 327.0 0.001777 58174.396859 351.5 0.00391357150.825786 2.5 0.000077 58104.012053 327.5 0.000132 58252.136475 378.0 0.00182857152.272733 3.0 0.000028 58105.480644 328.0 0.000211 58253.596266 378.5 0.00059257153.757104 3.5 0.000192 58106.943575 328.5 0.000624 58255.071344 379.0 0.00053257155.204030 4.0 0.000019 58108.412067 329.0 0.000687 58256.529671 379.5 0.00010857156.688796 4.5 0.000063 58109.875621 329.5 0.000159 58258.004748 380.0 0.00345557158.135547 5.0 0.000160 58111.343495 330.0 0.001087 58259.462186 380.5 0.00088157159.620606 5.5 0.000113 58112.807034 330.5 0.000873 58260.937413 381.0 0.00009357161.066880 6.0 0.000127 58114.275684 331.0 0.000479 58262.394033 381.5 0.00012557162.552206 6.5 0.000171 58115.739490 331.5 0.000912 58263.869333 382.0 0.00115357163.998782 7.0 0.000354 58117.208752 332.0 0.000199 58265.325674 382.5 0.00130557165.483852 7.5 0.000130 58118.672846 332.5 0.001196 58266.801169 383.0 0.00085357166.930611 8.0 0.000351 58120.143383 333.0 0.000865 58268.257329 383.5 0.00004357168.415775 8.5 0.000106 58121.607314 333.5 0.000263 58269.732583 384.0 0.00265157169.863229 9.0 0.000172 58123.079612 334.0 0.000102 58271.188833 384.5 0.00037657171.348066 9.5 0.000161 58124.543101 334.5 0.000998 58272.664065 385.0 0.00055057172.796124 10.0 0.000167 58126.015778 335.0 0.000776 58274.120305 385.5 0.00020457174.281235 10.5 0.000057 58127.478111 335.5 0.001329 58275.595426 386.0 0.00241757175.730707 11.0 0.000100 58128.949092 336.0 0.000484 58277.051804 386.5 0.00236957177.216441 11.5 0.000347 58130.412798 336.5 0.000113 58278.526897 387.0 0.00012357178.666657 12.0 0.000198 58131.883669 337.0 0.002043 58279.983353 387.5 0.00018357180.153887 12.5 0.000225 58133.348268 337.5 0.000071 58281.458184 388.0 0.00068657181.603000 13.0 0.000106 58134.819223 338.0 0.000474 58282.914900 388.5 0.00773757183.090425 13.5 0.000073 58136.282722 338.5 0.000920 58284.389716 389.0 0.00048157184.537118 14.0 0.000056 58137.753601 339.0 0.009431 58285.846670 389.5 0.00259457186.024259 14.5 0.000231 58139.215825 339.5 0.000868 58287.321398 390.0 0.00006157187.471259 15.0 0.000578 58140.686940 340.0 0.000300 58288.778231 390.5 0.00036057188.958197 15.5 0.000172 58142.148125 340.5 0.000455 58290.253313 391.0 0.00076757190.406446 16.0 0.000021 58143.619238 341.0 0.011193 58291.710867 391.5 0.00495557191.892233 16.5 0.000169 58145.079962 341.5 0.000609 58293.186097 392.0 0.00004557193.340629 17.0 0.000077 58146.551272 342.0 0.001466 58294.643694 392.5 0.03632657194.825643 17.5 0.000281 58148.011626 342.5 0.000309 58296.120404 393.0 0.00108057196.273525 18.0 0.000091 58149.482848 343.0 0.008323 58297.577843 393.5 0.00005757197.758109 18.5 0.000103 58150.943127 343.5 0.000941 58299.055921 394.0 0.00048157199.205659 19.0 0.000257 58152.414341 344.0 0.000714 58300.513091 394.5 0.00090057200.690248 19.5 0.000385 58153.874627 344.5 0.000127 58301.992113 395.0 0.00333757202.137571 20.0 0.000208 58155.345636 345.0 0.002435 58498.506621 462.0 0.00003157203.622066 20.5 0.000147 58156.806057 345.5 0.001051 58542.496480 477.0 0.00002957205.069162 21.0 0.000148 58158.276984 346.0 0.001531 58548.366974 479.0 0.00003157206.553610 21.5 0.000239 58159.737593 346.5 0.000636 58567.409913 485.5 0.00002257208.000524 22.0 0.000127 58161.208348 347.0 0.000067 58570.341545 486.5 0.00002257209.485395 22.5 0.000282 58162.669118 347.5 0.001053 58592.350572 494.0 0.00002257210.931931 23.0 0.000050 58164.139718 348.0 0.000398 Notes.
Integer and half-integer cycle numbers refer to primary and secondary eclipses, respectively. Most of the eclipses (cycle nos. − . Kepler spacecraft. The last six eclipses were observed at Baja Astronomical Observatory.
B2 Radial velocity data
The columns give the BJD date of observation, the RVs of the components A, B, and C in km s − , their residuals tothe spectro-photodynamical model, and the instrument code (see Sect. 2.2.). In fitting the RVs, we adopt the errors of 2.0km s − for CfA, 0.5 km s − for TRES, UVES, VUES, and CHIRON. For reader’s convenience, the full, online available list isprovided in machine readable format. MNRAS , 1–19 (2019) Borkovits et al.
Table B2.
Radial velocity data of the three components of HIP 41431.BJD RV A ∆RV A RV B ∆RV B RV C ∆RV C instr. − − ) (km s − ) (km s − )51295.7154 -92.50 +3.08 42.20 -0.38 28.90 +0.23 CfA51325.6699 -43.20 -1.63 66.40 +1.49 -43.20 +1.09 CfA51471.9757 -103.90 +0.60 55.50 +0.77 31.00 +0.23 CfA51503.0469 77.40 +1.06 -47.80 +0.99 -47.80 -2.24 CfA51505.0016 29.10 -1.44 8.20 +2.61 -52.90 +0.19 CfA51506.9915 -54.50 +0.78 97.70 -1.27 -60.50 -1.44 CfA51539.9870 -46.30 -1.74 6.20 +3.25 22.40 -0.57 CfA51566.7523 56.70 -2.64 -13.30 +2.19 -60.00 +0.38 CfA51568.7912 -59.00 -2.95 102.10 -0.06 -58.90 +1.94 CfA51595.8217 -30.20 -0.91 -12.30 +4.73 27.90 +0.44 CfA51620.8426 6.50 +3.17 21.30 -4.08 -45.30 -0.20 CfA51622.7819 89.20 -0.61 -52.60 +2.12 -52.50 +0.01 CfA51623.7550 17.60 +5.07 21.10 -6.01 -56.20 -0.46 CfA51647.6866 -92.60 +2.98 46.60 +0.70 32.30 +0.99 CfA51648.6177 -34.00 -2.48 -14.90 +4.44 31.30 -0.17 CfA51649.6840 40.60 +1.42 -91.80 -0.80 32.30 +0.96 CfA51652.6459 37.40 +0.45 -88.50 -1.59 28.80 -0.74 CfA51684.6315 98.40 -2.04 -60.00 -1.84 -60.10 +0.30 CfA51686.6514 -1.40 -3.51 44.90 +2.58 -61.10 -0.05 CfA51834.9496 -28.10 +3.43 -16.20 -3.19 18.30 -1.09 CfA51856.0416 -66.20 +1.16 89.20 -0.46 -45.50 -0.21 CfA51883.8218 40.50 -1.58 -98.90 +2.12 29.10 -0.37 CfA51884.0374 50.10 +0.29 -107.80 +1.07 29.70 +0.23 CfA51884.8582 -42.80 +2.61 -15.20 -3.11 28.20 -1.12 CfA51885.0429 -71.10 +3.47 15.10 -2.48 30.10 +0.84 CfA51885.8325 -87.00 +0.11 32.10 +1.45 29.80 +0.90 CfA51886.0330 -60.70 -1.33 6.00 +3.41 29.30 +0.52 CfA51886.8373 47.30 -0.21 -103.20 +2.19 27.30 -0.90 CfA51886.9558 51.00 +0.60 -107.00 +1.23 28.20 +0.09 CfA51887.8637 -53.70 +3.00 -2.30 -3.63 26.70 -0.57 CfA51887.9953 -73.70 +2.77 18.90 -2.64 26.40 -0.74 CfA51888.8238 -80.90 -2.46 26.40 +1.94 26.20 -0.00 CfA51889.9595 52.50 +1.16 -104.60 +1.25 24.70 -0.04 CfA51890.8406 -57.60 +4.43 9.20 -1.34 23.70 +0.24 CfA51891.0314 -83.90 +3.64 34.90 -1.84 22.20 -0.96 CfA51937.7043 -47.10 +5.40 -5.50 -3.63 25.50 +2.18 CfA51938.7805 -69.30 -2.90 13.20 +3.32 26.20 +0.68 CfA51942.7051 48.50 +0.12 -109.30 +0.77 30.30 +1.73 CfA51944.6573 -67.40 -1.37 8.90 +2.32 29.50 +1.51 CfA51962.6696 19.90 -0.81 -42.10 +2.99 -9.00 -1.00 CfA51967.6505 -82.00 +0.70 76.50 -0.25 -24.30 +0.14 CfA51997.6097 -44.40 -4.04 -17.00 +2.01 23.40 -1.03 CfA52007.6703 20.20 +5.58 -76.40 -2.77 20.30 -2.41 CfA52008.7070 -107.60 +0.34 52.00 -0.40 21.50 +0.35 CfA52009.6658 11.50 -2.85 -67.00 +3.15 18.20 -1.34 CfA52010.7163 2.90 +3.10 -54.90 -1.44 16.70 -0.93 CfA52011.6707 -103.80 +0.73 53.80 -0.56 16.00 +0.20 CfA52032.6531 -27.90 -2.23 42.80 +2.74 -49.90 -1.49 CfA52034.6525 -37.30 +3.87 59.70 -3.39 -55.00 +0.49 CfA52038.6589 2.00 -3.19 29.50 +3.68 -65.60 -0.49 CfA52211.9704 40.90 -3.20 -23.10 +2.00 -60.20 -1.13 CfA52237.9679 -69.10 -1.98 2.90 +3.50 25.60 -0.13 CfA52242.0083 41.30 +0.20 -108.00 -1.13 21.30 -0.97 CfA52244.9651 42.90 +1.03 -103.00 -0.12 17.10 -0.61 CfA52270.9681 79.60 -2.61 -61.90 +1.64 -61.80 -2.67 CfA52271.8541 21.30 +3.59 3.90 -0.69 -61.80 -0.16 CfA52273.8364 78.30 -2.53 -52.50 +2.74 -65.40 +0.30 CfA52274.8196 17.20 +2.13 11.50 -0.99 -67.40 -0.78 CfA52277.8443 0.80 +3.08 22.00 -3.36 -61.80 +0.22 CfA52278.8399 -53.70 -4.45 69.40 +1.02 -55.30 +2.23 CfA52297.8375 32.40 +0.45 -100.50 -0.09 24.80 -0.76 CfA52298.8232 -103.80 +1.59 40.10 +0.24 25.40 +0.59 CfA MNRAS000
Radial velocity data of the three components of HIP 41431.BJD RV A ∆RV A RV B ∆RV B RV C ∆RV C instr. − − ) (km s − ) (km s − )51295.7154 -92.50 +3.08 42.20 -0.38 28.90 +0.23 CfA51325.6699 -43.20 -1.63 66.40 +1.49 -43.20 +1.09 CfA51471.9757 -103.90 +0.60 55.50 +0.77 31.00 +0.23 CfA51503.0469 77.40 +1.06 -47.80 +0.99 -47.80 -2.24 CfA51505.0016 29.10 -1.44 8.20 +2.61 -52.90 +0.19 CfA51506.9915 -54.50 +0.78 97.70 -1.27 -60.50 -1.44 CfA51539.9870 -46.30 -1.74 6.20 +3.25 22.40 -0.57 CfA51566.7523 56.70 -2.64 -13.30 +2.19 -60.00 +0.38 CfA51568.7912 -59.00 -2.95 102.10 -0.06 -58.90 +1.94 CfA51595.8217 -30.20 -0.91 -12.30 +4.73 27.90 +0.44 CfA51620.8426 6.50 +3.17 21.30 -4.08 -45.30 -0.20 CfA51622.7819 89.20 -0.61 -52.60 +2.12 -52.50 +0.01 CfA51623.7550 17.60 +5.07 21.10 -6.01 -56.20 -0.46 CfA51647.6866 -92.60 +2.98 46.60 +0.70 32.30 +0.99 CfA51648.6177 -34.00 -2.48 -14.90 +4.44 31.30 -0.17 CfA51649.6840 40.60 +1.42 -91.80 -0.80 32.30 +0.96 CfA51652.6459 37.40 +0.45 -88.50 -1.59 28.80 -0.74 CfA51684.6315 98.40 -2.04 -60.00 -1.84 -60.10 +0.30 CfA51686.6514 -1.40 -3.51 44.90 +2.58 -61.10 -0.05 CfA51834.9496 -28.10 +3.43 -16.20 -3.19 18.30 -1.09 CfA51856.0416 -66.20 +1.16 89.20 -0.46 -45.50 -0.21 CfA51883.8218 40.50 -1.58 -98.90 +2.12 29.10 -0.37 CfA51884.0374 50.10 +0.29 -107.80 +1.07 29.70 +0.23 CfA51884.8582 -42.80 +2.61 -15.20 -3.11 28.20 -1.12 CfA51885.0429 -71.10 +3.47 15.10 -2.48 30.10 +0.84 CfA51885.8325 -87.00 +0.11 32.10 +1.45 29.80 +0.90 CfA51886.0330 -60.70 -1.33 6.00 +3.41 29.30 +0.52 CfA51886.8373 47.30 -0.21 -103.20 +2.19 27.30 -0.90 CfA51886.9558 51.00 +0.60 -107.00 +1.23 28.20 +0.09 CfA51887.8637 -53.70 +3.00 -2.30 -3.63 26.70 -0.57 CfA51887.9953 -73.70 +2.77 18.90 -2.64 26.40 -0.74 CfA51888.8238 -80.90 -2.46 26.40 +1.94 26.20 -0.00 CfA51889.9595 52.50 +1.16 -104.60 +1.25 24.70 -0.04 CfA51890.8406 -57.60 +4.43 9.20 -1.34 23.70 +0.24 CfA51891.0314 -83.90 +3.64 34.90 -1.84 22.20 -0.96 CfA51937.7043 -47.10 +5.40 -5.50 -3.63 25.50 +2.18 CfA51938.7805 -69.30 -2.90 13.20 +3.32 26.20 +0.68 CfA51942.7051 48.50 +0.12 -109.30 +0.77 30.30 +1.73 CfA51944.6573 -67.40 -1.37 8.90 +2.32 29.50 +1.51 CfA51962.6696 19.90 -0.81 -42.10 +2.99 -9.00 -1.00 CfA51967.6505 -82.00 +0.70 76.50 -0.25 -24.30 +0.14 CfA51997.6097 -44.40 -4.04 -17.00 +2.01 23.40 -1.03 CfA52007.6703 20.20 +5.58 -76.40 -2.77 20.30 -2.41 CfA52008.7070 -107.60 +0.34 52.00 -0.40 21.50 +0.35 CfA52009.6658 11.50 -2.85 -67.00 +3.15 18.20 -1.34 CfA52010.7163 2.90 +3.10 -54.90 -1.44 16.70 -0.93 CfA52011.6707 -103.80 +0.73 53.80 -0.56 16.00 +0.20 CfA52032.6531 -27.90 -2.23 42.80 +2.74 -49.90 -1.49 CfA52034.6525 -37.30 +3.87 59.70 -3.39 -55.00 +0.49 CfA52038.6589 2.00 -3.19 29.50 +3.68 -65.60 -0.49 CfA52211.9704 40.90 -3.20 -23.10 +2.00 -60.20 -1.13 CfA52237.9679 -69.10 -1.98 2.90 +3.50 25.60 -0.13 CfA52242.0083 41.30 +0.20 -108.00 -1.13 21.30 -0.97 CfA52244.9651 42.90 +1.03 -103.00 -0.12 17.10 -0.61 CfA52270.9681 79.60 -2.61 -61.90 +1.64 -61.80 -2.67 CfA52271.8541 21.30 +3.59 3.90 -0.69 -61.80 -0.16 CfA52273.8364 78.30 -2.53 -52.50 +2.74 -65.40 +0.30 CfA52274.8196 17.20 +2.13 11.50 -0.99 -67.40 -0.78 CfA52277.8443 0.80 +3.08 22.00 -3.36 -61.80 +0.22 CfA52278.8399 -53.70 -4.45 69.40 +1.02 -55.30 +2.23 CfA52297.8375 32.40 +0.45 -100.50 -0.09 24.80 -0.76 CfA52298.8232 -103.80 +1.59 40.10 +0.24 25.40 +0.59 CfA MNRAS000 , 1–19 (2019) ompact multiple HIP 41431 Table B2.
Continued BJD RV A ∆RV A RV B ∆RV B RV C ∆RV C instr. − − ) (km s − ) (km s − )52331.7892 -25.80 -2.75 50.70 +1.72 -63.50 -0.01 CfA52332.8058 90.30 -1.07 -66.00 -0.61 -66.00 -0.78 CfA52334.7602 -19.20 -3.05 46.30 +1.69 -66.20 -0.25 CfA52335.6767 91.00 -0.83 -64.40 +1.92 -66.90 -2.20 CfA52336.6512 -23.70 +1.91 47.90 -2.33 -61.40 +0.68 CfA52337.8307 -1.20 -2.44 20.40 +2.82 -56.90 -0.01 CfA52338.7667 80.50 -0.50 -70.60 -1.41 -50.00 +1.32 CfA54188.7421 3.40 -1.71 -51.20 +0.90 21.70 -0.20 CfA54216.6301 -19.70 +1.37 48.90 -2.27 -52.30 -1.60 CfA54218.6682 96.60 +1.23 -66.00 -0.52 -51.90 +0.24 CfA54221.6492 94.00 +0.91 -64.00 +1.30 -47.10 +2.88 CfA54224.6592 84.30 +0.23 -67.60 -0.81 -38.20 +1.48 CfA54227.6731 68.30 +1.08 -71.20 +0.00 -18.70 +0.22 CfA54422.0143 -69.10 +1.09 20.90 -0.18 31.90 +1.35 CfA54423.0277 -52.90 -0.52 8.10 +2.09 28.50 +0.83 CfA54423.9282 56.30 -1.45 -102.90 +0.18 24.90 -0.14 CfA54425.0500 -77.30 +1.52 38.80 -0.30 21.30 -0.39 CfA54425.9291 -51.30 +1.20 16.90 +1.73 19.70 +0.67 CfA54430.0339 57.90 +0.01 -84.00 -0.18 6.40 -0.07 CfA54456.9898 3.00 -0.01 28.70 +0.13 -46.40 +1.37 CfA54457.9095 -52.50 -0.28 82.30 -0.35 -47.50 -1.63 CfA54458.8985 83.60 +0.12 -56.70 +1.41 -40.20 +2.78 CfA54459.9317 -2.70 +0.16 24.20 -1.04 -38.00 +0.85 CfA54461.9290 80.70 -0.76 -71.90 +0.28 -28.40 -0.86 CfA54462.8983 -15.90 +1.76 18.00 -3.04 -19.40 +1.00 CfA54481.8592 -23.80 +0.52 -23.70 -1.10 28.10 +0.18 CfA54482.9574 38.10 +2.55 -79.80 +0.21 25.60 +0.91 CfA54483.8767 -92.50 +1.27 54.80 +0.61 21.90 -0.00 CfA54484.8492 -10.40 -0.45 -27.30 +0.53 18.20 -0.72 CfA54485.8154 49.20 +1.02 -84.50 -0.74 16.30 +0.33 CfA54486.9101 -92.90 +0.54 61.90 -1.66 12.60 +0.01 CfA54513.8629 -24.10 -0.76 58.30 +1.37 -49.60 -0.07 CfA54518.8041 -36.90 -0.70 58.60 -0.24 -36.90 +1.98 CfA54520.8225 80.70 -2.24 -75.70 -1.72 -27.70 -0.15 CfA54545.8346 -88.60 -0.51 59.20 +1.26 11.60 -0.57 CfA54546.6854 32.10 -0.51 -61.60 +0.27 7.60 -1.92 CfA54550.7700 -3.10 +3.16 -11.10 -1.64 -3.10 -0.14 CfA54573.6194 93.80 +0.26 -64.00 -0.92 -48.70 +0.48 CfA54578.7001 16.90 -0.89 -1.80 +0.26 -33.40 +0.64 CfA54579.6250 74.00 -0.00 -66.40 -1.39 -30.00 -1.57 CfA54968.6548 -17.118 -0.967 6.827 +1.334 -28.088 -1.064 TRES56704.8416 -45.116 +1.620 11.782 +1.546 -1.342 +0.481 TRES56705.8337 55.089 +0.054 -100.501 +0.486 5.987 +0.218 TRES56706.7765 -72.918 +0.479 18.370 -3.383 11.915 -1.241 TRES56708.6662 46.884 +0.762 -113.727 +0.511 27.562 +0.315 TRES56709.8773 -102.803 +0.541 ... ... 33.533 -1.143 TRES57799.3637 58.296 -0.378 -56.802 -3.732 -44.185 -2.694 VUES58107.8212 76.968 -0.111 -76.082 +0.229 -40.637 -0.139 UVES58107.8251 76.893 +0.006 -75.790 +0.332 -40.485 +0.006 UVES58107.8291 76.695 +0.012 -75.599 +0.323 -40.551 -0.066 UVES58107.8336 76.348 -0.098 -75.448 +0.243 -40.603 -0.126 UVES58107.8376 76.228 -0.002 -75.197 +0.282 -40.581 -0.111 UVES58107.8415 76.029 +0.014 -75.023 +0.243 -40.513 -0.050 UVES58107.8462 75.898 +0.150 -74.485 +0.518 -40.358 +0.096 UVES58116.7032 56.575 -0.424 -85.526 -0.113 -12.247 -0.282 UVESMNRAS , 1–19 (2019) Borkovits et al.
Table B2.
Continued BJD RV A ∆RV A RV B ∆RV B RV C ∆RV C instr. − − ) (km s − ) (km s − )58126.4989 -111.622 +0.398 29.312 +1.119 44.053 +0.099 VUES58182.3530 -112.058 -0.120 43.677 -0.809 28.887 -1.194 VUES58184.3302 13.689 +0.176 -93.940 -0.083 40.624 -0.036 VUES58217.3253 -50.538 +3.945 65.509 -0.898 -45.330 +0.272 VUES58221.3042 2.615 +1.835 ... ... -44.353 -0.241 VUES58222.2875 74.486 -0.741 -67.480 +0.074 -43.692 -0.443 VUES58222.3162 72.634 -0.267 -65.650 -0.429 -43.571 -0.351 VUES58222.3445 69.667 -0.680 -62.825 -0.170 -43.815 -0.623 VUES58222.3729 67.198 -0.335 -59.450 +0.376 -43.773 -0.609 VUES58443.8394 -53.674 +0.516 64.815 -0.293 -35.167 +0.043 VUES58467.8078 15.795 +0.418 -25.682 -1.106 -15.481 +0.522 CHIRON58468.8109 47.319 -0.392 -61.295 +0.388 -11.990 -0.126 CHIRON58480.7873 -19.796 -0.214 -57.005 +0.641 50.977 +0.724 CHIRON58494.7661 71.759 +0.063 -84.678 +0.105 -12.084 +0.009 CHIRON58508.7648 12.523 +1.703 6.620 -1.002 -40.437 +0.498 CHIRON58526.6719 46.943 +0.654 -54.765 -0.217 -15.882 -0.663 CHIRON58541.6351 38.625 +0.142 -114.347 +0.389 51.417 +0.805 CHIRONMNRAS000