DX Cygni: A triple system with mass transfer

New Astronomy 76 (2020) 101336

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DX Cygni: A triple system with mass transfer ⁎,a

b,c

a

M. Wolf , M. Mašek , P. Zasche , H. Kučáková

a,b,d,e

d

, K. Hornoch

T

a

Astronomical Institute, Faculty of Mathematics and Physics, Charles University Prague, CZ-180 00 Praha 8, V Holešovičkách 2, Czech Republic Variable Star and Exoplanet Section, Czech Astronomical Society, Czech Republic Institute of Physics, Czech Academy of Sciences, Na Slovance 1999/2, Praha 8 CZ-182 21, Czech Republic d Astronomical Institute, Academy of Sciences, Fričova 298, Ondřejov CZ-251 65, Czech Republic e Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, Opava CZ-746 01, Czech Republic b c

ARTICLE INFO

ABSTRACT

Keywords: Stars: Binaries: Eclipsing Stars: Individual: DX cyg Stars: Fundamental parameters

A dozen of new precise times of eclipses were measured for the eclipsing binary DX Cygni as a part of our longterm observational project for studying neglected eclipsing binaries with a short orbital period. Based on a current O C diagram, we found for the first time that its period is increasing (dP / P = 1.68 × 10 7 day/years) and that times of minima show also significant cyclical changes with a period of about 16 years, caused very probably by a third body orbiting the eclipsing pair. The minimal mass of this companion is 0.49 M⊙. The light curve solution in PHOEBE results to the typical Algol-type semidetached configuration where the secondary fills its Roche lobe. The temperature of primary component was fixed to T1 = 5300 K according to its spectral type, which gives us T2 = 3330 ± 20 K for the secondary. The photometric mass ratio was estimated q = 0.504 ± 0.012 . We also compare orbital parameters of selected known Algol-type eclipsing binaries with proven mass transfer and a third body.

1. Introduction Although the main aspects of single star formation are well understood, the origin and evolution of close binaries is less known because of process responsible for bringing two or more stars together. On the other hand, the study of triple and multiple stellar systems serves as an important observational test of stellar evolution in such systems as well as a laboratory of celestial mechanics. Algol-type close binaries are semi-detached interacting systems in which one type of interaction is mass transfer between the component stars by means of a gas stream. They have been known as good astrophysical laboratories for studying accretion processes because a number of them are relatively bright. They are usually in the slow phase of mass transfer with dM / dt = 10 11 10 7 M⊙/yr and do not undergo violent eruptions that interfere with the accretion process (Richards and Albright, 1999). The eclipse-timing variation of hierarchical triple systems with short-period eclipsing binary in the Galactic Bulge was recently presented by Hajdu et al. (2019). They identified nearly 1000 potential triple (or multiple) candidates exhibiting light-time effect. Many of them shows additional period changes possibly caused also by a mass transfer. The similar period analysis of semi-detached Algol-type binaries with third component was presented also by Erdem et al. (2007a,

⁎

2007b, 2010) and Zasche et al. (2008). The eclipsing nature of the 14 mag northern Algol-type binary DX Cygni (AN 177.1928, 2MASS J19260444+2922545) was discovered by Hoffmeistr (1928) on photographic plates of the Sonnenberg Observatory. Later (Wachmann, 1961) presented the first photographic light curve of the star and the original light elements with a period of about 33 hours:

Pri. Min. = HJD 24 34603.3720(13) + 1.37032596(150)·E , He also estimated D1 = 2 hours for the deep and narrow primary minimum (A1 ≃ 1.83 mag) and plotted the O C diagram with a rapid period change. Later, (Hegedüs, 1988) selected this system as a possible candidate for the study of apsidal motion, but no eccentricity of the orbit was found. To our knowledge no modern period, photometric nor spectroscopic study of this short-period Algol-type eclipsing binary exists so far. 2. Observations Our long-term CCD photometry of DX Cyg was obtained mostly at Ondřejov observatory, Czech Republic. The Mayer 0.65-m (f/3.6) reflecting telescope with the CCD cameras SBIG ST-8, Apogee AP7p or Moravian Instruments G2-3200 and the R photometric filter were used

Corresponding author. E-mail address: [email protected] (M. Wolf).

https://doi.org/10.1016/j.newast.2019.101336 Received 13 August 2019; Received in revised form 23 October 2019; Accepted 31 October 2019 Available online 05 November 2019 1384-1076/ © 2019 Elsevier B.V. All rights reserved.

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during the period 2003–2018. The FRAM robotic telescope (0.25-m SC telescope (f/6.3) and CCD camera MII G2-1000BI) located at the Cherenkov Telescope Array, La Palma, was used during the 2019. The CCD observations were reduced in a standard way. APHOT, a synthetic aperture photometry and astrometry software developed by M. Velen and P. Pravec at the Ondřejov observatory, was routinely used for reduction of these CCD images obtained at Ondřejov. C-MUNIPACK 1 was used to reduce our CCD times series obtained on the FRAM telescope. Data were dark-subtracted and flat-fielded and heliocentric correction was applied. Time-series were constructed by computing the magnitude difference between the variable and a nearby comparison star GSC 2137-0528 (V = 12.9 mag). The new times of primary and secondary minima and their errors were generally determined by fitting the light curve by Gaussians or polynomials of the third or fourth order; we used the least-squares method. The Zwicky Transient Facility (ZTF)2 and the All-Sky Automated Survey for Supernovae (ASAS-SN)3 were successfully searched and several new reliable mid-eclipse times of DX Cyg were derived from this data set. The epochs in Table 1 were computed according to the following new linear light elements:

Table 1 New precise times of minimum light of DX Cyg. JD Hel.-2400000

Epoch

Error [day]

Filter

Observatory

52853.44338* 52877.4272* 53256.31491* 55662.62414 55984.65568 56113.46380 56135.38945 56527.31021 56737.66154 58204.6161 58218.3186 58307.3893 58332.0570 58600.64075 58666.41322 58674.63475 58679.4293

1582.0 1599.5 1876.0 3632.0 3867.0 3961.0 3977.0 4263.0 4416.5 5487.0 5497.0 5562.0 5580.0 5776.0 5824.0 5830.0 5833.5

0.0001 0.0005 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0007 0.0005 0.0005 0.0010 0.0010 0.0001 0.0002 0.0001 0.0005

R R R R R R R R R ZTFg ZTFg V V C C C C

Ondřejov Ondřejov Ondřejov Ondřejov Ondřejov Ondřejov Ondřejov Ondřejov Ondřejov ZTF ZTF ASAS-SN ASAS-SN FRAM FRAM FRAM FRAM

Note: * published also in IBVS No. 5676

Pri. Min. = HJD 24 50685.5694(8) + 1.37033485(9)·E , The current O C diagram is plotted in Fig 1. It is clearly visible that current mid-eclipse times do not follow a simple linear ephemeris. 3. Orbital period study The period analysis of DX Cygni was performed using all available mid-eclipse times found in the literature (O C gateway 4, (Paschke and Brát, 2006)) as well as our newly measured times of minimum light. Variations in the period of DX Cyg are immediately apparent from the O C diagram on Fig. 1 and can be represented as a superposition of a secular period change and cyclic variations. We tried to solve the increase of period together with the light-time effect (hereafter LITE) simultaneously. In this case the deviation of the observed values (O C )obs from the linear ephemeris is given by a superposition of the period increase and by the LITE caused by a third body. Using the leastsquares method, the O C diagram on Fig. 1 can be described by quadratic ephemeris:

Pri.Min. = HJD2450685.5701(3) + 1.37033454(7)· E + 2.76(8)·10

Fig. 1. The complete O C diagram for the times of minimum of DX Cyg. The individual primary and secondary minima are denoted by circles and triangles, respectively. The dashed curve represents the slow period increase, the sinusoidal full curve the LITE. Larger symbols correspond to the CCD measurements, which were used in our calculation of LITE. The error bars are indicated, for the precise measurements the bars are smaller than the symbols.

10 · E 2.

The quadratic term Q of the ephemeris indicates a long-term and continuous period increase at a rate of + (1.68 ± 0.3) × 10 7 days/yr. Because DX Cyg is very probably a semi-detached system with less massive secondary component filling its inner Roche lobe, its geometry permits mass transfer from the secondary to the more massive primary star. The continuous increase of the period was subtracted on the O C diagram and the corresponding residuals are plotted in the Fig. 2. The oscillation caused probably by an orbiting third body is clearly visible. 4. Light-time effect The theory of the third body motion and the LITE analysis in eclipsing binaries was reviewed several times in the literature, see e.g. the original paper by Irvin (1952) or more useful relations by Mayer (1990). The light travel time is given by (O

C )LITE =

1

1 e32 A [ sin(v + 3) + e3 sin 3], e32 cos2 3 1 + e3 cos v

(1)

where e3 is the eccentricity of the third-body orbit, ω3 the longitude of periastron and v the mean anomaly. The observed semi-amplitude A of Fig. 2. The O C diagram in detail for the current precise times of minimum of DX Cyg. For clarity, the quadratic term was subtracted. The sinusoidal curve represents the LITE with a period of 16 years and a semi-amplitude of about 17 minutes.

1

http://c-munipack.sourceforge.net. https://www.ztf.caltech.edu/. 3 https://asas-sn.osu.edu/. 4 http://var2.astro.cz/ocgate. 2

2

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Table 2 The LITE elements of DX Cyg.

Table 3 The photometric elements of DX Cyg, the semidetached configuration.

Parameter

Unit

Value

Parameter

Primary

T0 Ps A P3 P3 e3 ω3 T3 f(m) M3, min K

HJD day days days years – deg JD M⊙ M⊙

24 50685.5705 (9) 1.3703343 (5) 0.0111 (4) 5979 (25) 16.37 (15) 0.33 (15) 148.8 (3.5) 24 52751 (20) 0.030 0.49 3.8

i [deg] q = M2/ M1 T1,2 [K] Ω1,2 X, Y r(pole) r(side) r(point) r(back) L3 [%]

89.77 (0.2) 0.504 (12) 5300 (fixed) 2.887(5) 0.249, 0.456 0.2957(8) 0.3018(7) 0.3109(8) 0.3077(7) 8.8 (0.5)

km s

1

Secondary

3330 (20) 2.585(5) 0.019, 0.594 0.2969(10) 0.3094(12) 0.3886(14) 0.3396(10)

the light-time curve (in days) is

A=

a12 sin i3 173.15

1

e32 cos2

3,

(2)

where a12 is the semi-major axis of the relative orbit of the eclipsing pair around the common centre of mass (in AU) and i3 is the inclination of the third-body orbit. There are eight independent variables to be determined in this procedure: (T0, Ps, Q), for the mass transfer and (A, T3, P3, e3, ω3) for the LITE. A possible alternative to the third body and LITE hypothesis assumes the period modulation in Algols connected with magnetic activity of stars. Applegate (1992) proposed a model which explains orbital period modulations as a consequence of the magnetic activity changes of one of the components. According to this model, the cooler component, which has a magnetic activity cycle, can show a period change of P / P 10 5. Numerous eclipse timings for DX Cyg have been reported in the literature. Besides those minima given in Table 1, we used previous times of minimum obtained by Caton (2005); Hübscher (2005); Hübscher et al. (2005, 2006, 2010); Kotková and Wolf (2006); Brát et al. (2007), and Smith and Caton (2007). Mid-eclipse timings obtained visually by Mr. Kurt Locher and listed in the BBSAG database/ Bulletins were not used due to large scatter of these data. A total of 66 times of minimum light were used for determination of the long term period increase. Selected 23 precise CCD times including 7 secondary eclipses were used for the LITE analyses only. The computed LITE parameters and their internal errors of the least-squares fit are given in Table 2, the current O C diagram in detail is shown in Fig. 2. Assuming a coplanar orbit of the third body (i3 ≃ 90∘) and the total mass of the eclipsing pair M1 + M2 1.5 M⊙, we can estimate a lower limit for the mass of the third component M3, min. This value, as well as the mass function f(m), and the amplitude of the systemic radial velocity K are also given in Table 2. The probable third component may be a main-sequence star of a spectral type K8 - K9 with a bolometric magnitude of about +7.5 mag (Pecaut and Mamajek, 2013) 5 produces a detectable third light of L3 ≃ 10%.

Fig. 3. The light curves of DX Cyg and their solution in PHOEBE. The dots represent our unfiltered CCD measurements (upper curve), the triangles denote the V observations taken form the ASAS-SN. For clarity, the curves are shifted. The error bars are smaller than the individual symbols.

Fig. 4. The geometry of DX Cyg at orbital phase 0.25. The semidetached configuration, where secondary fills its Roche lobe.

curves of eclipsing binaries as a standard tool. Because DX Cyg belongs to late-type binaries, we adopted the bolometric albedos and gravity darkening coefficients as A1 = A2 = 0.5 and g1 = g2 = 0.32, which corresponds to the convective envelopes. Synchronous rotation for both components of the system and a circular orbit (e = 0 ) were adopted. We used the square-root limb-darkening law with the coefficients adopted from van Hamme (1993) tables. The TESS broad band filter was used formally in the initial setup of the unfiltered photometric data. The temperature of the primary component was fixed to the value of 5300 K given in the GAIA DR2 Archive. This value is in good agreement with the color index J H = 0.406 given in the SIMBAD database and corresponding to the effective temperature of 5170 K (Pecaut and

5. Light curve solution No photometric solution has been reported for DX Cyg so far. The unfiltered and time-compact light curves of DX Cyg obtained at FRAM observatory in 2019 as well as the All-Sky Automated Survey for Supernovae (ASAS-SN) Vlight curve were simultaneously analyzed using the well-known PHOEBE code (Prša and Zwitter, 2005; Prša et al., 2016), which is based on the Wilson-Devinney algorithm (Wilson and Devinney, 1971) and is widely used for modeling the photometric light 5 http://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_ Teff.txt.

3

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solution with a lower cost function for the detached configuration than for the semidetached one. The final light-curve solution is given in Table 3. Therefore, DX Cyg is probably an Algol-type binary where the secondary components is close to its Roche lobe. The computed light curve based on derived parameters is compared with our measurements in Fig. 3. The geometrical representation of DX Cyg at phase 0.25 is displayed in Fig. 4. As one can see, the agreement between the theoretical and observed light curve is relatively good. 6. Discussion The period changes of Algols with convective secondaries were studied e.g. by Zavala et al. (2002). Here we compare our results for DX Cyg with other similar eclipsing binaries. The Table 4 summarizes from the literature altogether 25 selected and relatively well-known Algoltype eclipsing binaries with mass-transfer, increasing orbital period and a possible third body. As one can see, the spectral type of all these systems is typically between A0 and F0 and their orbital period is up to 5 days. Detected periods of the third bodies are up to 80 yr, which is probably a result of selection effect given by the total length of photometric observations. We also plotted the P3 Ps diagram on Fig. 6. A weak dependence of the third-body period on the binary orbital period is visible. This correlation can be formalized as

Fig. 5. Results of the q-search showing the relation between the PHOEBE cost function and the photometric mass ratio q. The minimum value was obtained for q = 0.504 .

Mamajek, 2013). More than 103 trials of PHOEBE in different modes using the different setup of initial parameters were evaluated. The results as well as the cost function was recorded. In the absence of the spectroscopic mass ratio, the q-search process was performed to find a corresponding photometric mass ratio. A series of solutions were made using fixed mass ratios from 0.30 to 0.75 with a step of 0.05 or smaller. This qsearch had minimum value at about q = 0.504 and was used as the initial mass ratio for the final solution attempts for each data set (see Fig. 5). The adjustable parameters were the inclination i, the effective temperature of the secondary component T2, luminosity and dimensionless potentials of both components Ω1, Ω2. The previous LITE analyses of the O C diagram allowed us to include the third light L3 to the light curve solution as a free parameter. We obtained a little better

P3 = 9.12 Ps + 29.8, where P3 is in years and Ps in days. The correlation coefficient was found to be R2 = 0.425. 7. Summary The new observations of DX Cyg were used to determine the photometric parameters of the components and investigate the orbital period changes. Our results indicate that DX Cyg is an Algol-type triple

Table 4 Short-period Algol-type eclipsing binaries with a third body and mass transfer. System

Spectral type

M1 [M⊙]

P [days]

dP/dt [10

TT And XZ And XZ Aql* ZZ Aur CL Aur EG Cep RW Cet SW Cyg DX Cyg W Del Z Dra RR Dra SX Dra* TZ Eri BO Gem SZ Her VX Lac † DG Lac RW Leo UU Leo TZ Lyr SW Oph DI Peg WY Per V723 Per UV Vir

A2 A1 A2 A7 A0 A7 A5 A2 – B9.5 A5 A2 A7 A5 A2 F0 F0 A5 A3 A2 F5 A0 F0 A0 F0 A8

2.15 2.15 2.5 – 2.24 1.65 1.9 2.5 – 2.1 1.47 2.15 1.75 1.97 2.15 1.56 1.57 1.9 – 2.54 1.5: 2.4 1.4 2.4 1.65 –

2.765 1.357 2.139 0.601 1.244 0.545 0.975 4.573 1.370 4.806 1.357 2.831 5.169 2.606 4.068 0.818 1.074 2.206 1.683 1.680 0.529 2.446 0.712 3.327 0.796 1.811

8.2 5.37 7.82 0.23 1.48 0.34 4.72 7.65 1.68 12.8 2.05 32.5 40.4 5.05 6.84 0.31 2.14 4.79 1.78 4.64 0.72 6.57 0.20 2.16 76.3 13.7

7

d/yr]

A [days]

P3 [years]

Ref.

0.049 0.0368 0.0116 0.0053 0.0133 0.0037 0.0215 0.109 0.0121 0.050 0.033 0.0731 0.115 0.0423 0.114 0.0055 0.0210 0.0346 0.033 0.0273 0.040 0.098 0.0101 0.056 0.0392 0.0397

78 32.3 36.9 26.4 21.6 35.8 38.7 72.8 16.4 53.4 60.2 84.3 81.5 48.8 65 30 68.2 44.3 38.4 54.5 45.5 68 55.0 47 17.4 62.3

1 2 3 4 5, 6 7 8 9 10 11 12 14 13 14 8 15 14, 16 8 17 18 19 8 20 8 21 22

Remarks: * δ Sct component, † possible quadruple system. References: 1 - Erdem et al. (2007a), 2 - Yang (2013b), 3 - Soydugan et al. (2016), 4 - Oh et al. (2006), 5 - Wolf et al. (2007), 6 - Lee et al. (2010), 7 - Zhu et al. (2009), 8 - Erdem et al. (2010), 9 - Erdem et al. (2007b), 10 - this paper, 11 - Hanna (2006), 12 - Khaliullina (2016), 13 - Soydugan and Kacar (2013), 14 Zasche et al. (2008), 15 - Hosseinzadeh et al. (2015), 16 - Yilmaz et al. (2017), 17 - Qian (2003), 18 - Yang (2013a), 19 - Yang and Yin (2007), 20 - Hanna and Amin (2013), 21 - Tian and Zhu (2019), 22 - Zhang et al. (2009), 4

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professional astronomers and the Czech Astronomical Society, Variable Star and Exoplanet Section. This work is supported by MEYS (Czech Republic) under the projects MEYS LM2015046, LTT17006 and EU/ MEYS CZ.02.1.01/0.0/0.0/16_013/0001403. The research of MW and PZ was supported by the project Progres Q47 Physics of the Charles University in Prague. HK and KH were supported by the project RVO:67985815. The authors would also like to thank L. Kotková, Ondřejov observatory, and J. Vraštil, Charles University Prague, for their important contribution to photometric observations. The following internet-based resources were used in research for this paper: the SIMBAD database operated at CDS, Strasbourg, France, the NASA’s Astrophysics Data System Bibliographic Services, and the O-C Gateway of the Czech Astronomical Society (http://var.astro.cz/ocgate/). References Applegate, J.H., 1992. ApJ 385, 621. Brát, L., Zejda, M., Svoboda, P., 2007. OEJV 74, 1. Caton, D.B., 2005. IBVS 5595. Erdem, A., Dogru, S.S., Bakis, V., Demircan, O., 2007b. AN 328, 543. Erdem, A., Dogru, S.S., Soydugan, F., et al., 2010. New Astron. 15, 628. Erdem, A., Soydugan, F., Dogru, S.S., et al., 2007a. New Astron. 12, 613. Hajdu, T., Borkovits, T., Forgács-Dajka, E., et al., 2019. MNRAS 485, 2562. van Hamme, W., 1993. Astron. J. 106, 2096. Hanna, M.A., 2006. J. Korean Astr. Soc. 39, 129. Hanna, M.A., Amin, S.M., 2013. J. Korean Astr. Soc. 46, 151. Hegedüs, T., 1988. Bull. Inf. CDS 35, 15. Hoffmeistr, C., 1928. AN 233, 33. Hosseinzadeh, B., Pazhouhesh, R., Yakut, K., 2015. New Astron. 35, 79. Hübscher, J., 2005. IBVS 5643. Hübscher, J., Lehmann, P.B., Monninger, G., et al., 2010. IBVS 5941. Hübscher, J., Paschke, A., Walter, F., 2005. IBVS 5657. Hübscher, J., Paschke, A., Walter, F., 2006. IBVS 5731. Irvin, J.B., 1952. ApJ 116, 211. Khaliullina, A.I., 2016. Astron. Rep. 60, 517. Kotková, L., Wolf, M., 2006. IBVS No. 5676. Lee, J.-W., Kim, C.-H., Kim, D.-H., et al., 2010. AJ 139, 2669. Mayer, P., 1990. Bull. Astron. Inst. Czech. 41, 231. Oh, K.-D., Kim, C.-H., Lee, W.B., et al., 2006. MNRAS 366, 1243. Paschke, A., Brát, L., 2006. OEJV 23, 13. Pecaut, M.J., Mamajek, E.E., 2013. ApJS 208, 9. Prša, A., Conroy, K.E., Horvat, M., et al., 2016. ApJS 227, 29. Prša, A., Zwitter, T., 2005. ApJ 628, 426. Qian, S.B., 2003. PASJ 55, 289. Richards, M.T., Albright, G.E., 1999. ApSS 123, 537. Smith, A.B., Caton, D.B., 2007. IBVS 5745. Soydugan, E., Kacar, Y., 2013. Astron. J. 145, 87. Soydugan, E., Soydugan, F., Alicavus, F., Erdem, A., 2016. New Astron. 46, 10. Tian, X.-M., Zhu, L.Y., 2019. PASJ 71, 66. Wachmann, A.A., 1961. Astr. Abhandlungen Hamburg Sternwarte 6, 1. Wilson, R.E., Devinney, E.J., 1971. ApJ 166, 605. Wolf, M., Kotková, L., Brát, L., et al., 2007. IBVS 5780. Yang, Y.G., 2013a. Research in Astron. Astrophys. 13, 1471. Yang, Y.G., 2013b. New Astron. 25, 109. Yang, Y.-G., Yin, X.G., 2007. Chinese J. Astron. Astrophys. 7, 258. Yilmaz, M., Nelson, R.H., Senavci, H.V., et al., 2017. Rev. Mex. A&A 53, 29. Zasche, P., Liakos, A., Wolf, M., Niarchos, P., 2008. New Astron. 13, 405. Zavala, R., McNamara, B.J., Harrison, T.E., et al., 2002. Astron. J. 123, 450. Zhang, J., Qian, S.-B., Boonrucksar, S., 2009. Chin. Astron. Astrophy. 33, 279. Zhu, L.-Y., Qian, S.-B., Liao, W.P., et al., 2009. PASJ 61, 529.

Fig. 6. The P3 Ps diagram for selected triple Algol-type systems with the third body and mass transfer. The best linear fit to the data is plotted as a full line.

eclipsing system showing the mass transfer as well as the LITE caused probably by a third body orbiting with the relatively short period of 16 years. The combination of the mass transfer with the LITE in this multiple system serves as an excellent laboratory of multiple stellar evolution models. Moreover, DX Cyg probably belongs to the important group of similar semi-detached eclipsing systems with mass transfer and a short third-body orbital period (see Table 4). The secular O C changes were explained by the orbital period increase at a rate of 15 s per century requiring mass flow from the less massive component to more massive one. Assuming the conservative mass transfer and the masses of the components M1 = 1.0 M⊙, M2 = 0.5 M⊙ we can obtain a rate 1.5·10 10 M⊙ per year. The third light computed from the light-curve solution (9%) is in good agreement with the value derived form the minimal mass of the third body. New high-accuracy timings of this eclipsing binary are necessary to improve the LITE parameters derived in this paper. It is also highly desirable to obtain new, high-dispersion and high-S/N spectroscopic observations for this system and to apply modern disentangling methods to obtain the radial-velocity curves of all three components and, therefore, derive accurate masses for this important system. Declaration of Competing Interest None. Acknowledgments This investigation is part of an ongoing collaboration between

5