Apsidal Motion Study of Close Binary System CW Cephei
 Author: Han Wonyong, Jeong MinJi, Yoon JohNa, Kim HyounWoo, Kim Yonggii, Kim ChunHwey
 Publish: Journal of Astronomy and Space Sciences Volume 32, Issue4, p341~348, 15 Dec 2015

ABSTRACT
New observations for the times of minimum lights of a wellknown apsidal motion star CW Cephei were made using a 0.6 m wide field telescope at Jincheon station of Chungbuk National University Observatory, Korea during the 2015 observational season. We determined new times of minimum lights from these observations and analyzed
O C diagrams together with collected times of minima to study both the apsidal motion and the Light Time Effect (LTE) suggested in the system. The new periods of the apsidal motion and the LTE were calculated as 46.6 and 39.3 years, respectively, which were similar but improved accuracy than earlier ones investigated byHan et al. (2002) ,Erdem et al. (2004) andWolf et al. (2006) .

KEYWORD
clipsing variables , apsidal motion , photometry

1. INTRODUCTION
CW Cep is a wellknown apsidal motion binary system observed by many investigators (Abrami & Cester 1960; Nha 1975; Han et al. 2002; Wolf et al. 2006) since its discovery by Petrie (1947) as a double lined spectroscopic eclipsing binary. The spectral types of two components of CW Cep were determined as B0.5 + B0.5, and masses were calculated as 11.7 M_{⊙} and 11.0 M_{⊙} by Popper (1974). The light curves were analyzed by many authors using well observed fulllight curves with different models for the parameters of the binary system, related with its apsidal motion (Cester et al. 1978) with the WINK model, Clausen & Giménez (1991) by the EBOP model, and Terrell (1991), and Han et al. (2002) by the WD model. While Popper & Hill (1991) discussed the radial velocity curve based on the spectroscopic observations by Popper (1974), high resolution IUE observations for the radial velocity curve were made by Stickland et al. (1992). Most of these studies suggested that CW Cep is a detached main sequence binary system, composed of two stars of slightly different mass and radius.
With regard to the apsidal motion study of the close eclipsing binary systems, CW Cep showed one of the shortest apsidal period, among similar binary systems, which leads particular interests for this binary system. Based on intensive photoelectric observations with
UBV filters, Nha (1975) calculated the apsidal motion period of CW Cep as 39 years, while Han (1984) derived the apsidal motion period as 43.1 years with more times of minimum lights based on theUBV photoelectric observations. After their studies, more times of minimum lights were available by many investigators recently, according to the new observations. Therefore, the apsidal motion period of CW Cep was improved in time order as 45.5 years by Gimenez et al. (1987), 45.6 years by Clausen & Giménez (1991), and 45.7 years by Han et al. (2002). The last authors, for the first time, noticed a new type of complication confusing the apsidal motion behavior, which shows about 34 years periodic variation with an amplitude of 0.003 days. Erdem et al. (2004) calculated the apsidal motion period of CW Cep as 46.1 years through the variation of the longitude of periastron (ω ) of the system's orbit, with new observations usingBVR filters. More recently, with new observations by Wolf et al. (2006), they suggested the apsidal motion period of CW Cep as 46 years, which is similar to those of other previous authors (Clausen & Giménez 1991; Han et al. 2002). They confirmed an extra periodic change suggested by Han et al. (2002) and proposed a Light Time Effect (LTE) with the period of 38.5 years and the small semiamplitude of 0.002 days caused by an unseen third body.It is nearly 10 years since the observations by Wolf et al. (2006), and this indicates the necessity of new observations for this binary system to examine more accurate apsidal motion period. In line with this, we made new observations for CW Cep in the 2015 season and obtained three new times of minimum lights to confirm the apsidal motion period of this system, together with available minimum times from the published literatures.
2. NEW OBSERVATIONS
New CCD photometric observations presented in this work were made in the 2015 observational season at the Jincheon station of Chungbuk National University Observatory (CBNUOJ), Korea. The altitude of the Jincheon station is 87 m sea level and the 60cm reflecting telescope is installed by Korea Astronomy and Space Science Institute (KASI), and operated by CBNUOJ. An electronically cooled SBIG STX 16803 4K CCD camera is equipped with a Johnson standard
BVRI filter set, and is attached at the reflector f/2.9 prime focus of the 60 cm wide field telescope. Since our observations are focused on the acquisition of accurate eclipse timings, only B filter is used for fast timeresolved photometry for the first two nights. The observations of the last night were made withBVR filters to derive the weighted average of each filters for higher accuracy. The field of view of the CCD camera is 72’ × 72’ with a pixel scale of 1.05 arcsec/pixel, which is wide enough to cover the many field stars around CW Cep including the comparison star GSC 0428200330, and differential photometry was applied for this observation. All of the observed CCD frames were exposed 15 ~ 80 seconds depending on the observational circumstances, and corrected with bias, dark and flatfield images for the differential photometry. The IRAF/DIPHO software package was applied to process the observed CCD frames. The details of the data reduction were extensively described by Kim et al. (2014) based on the same observational circumstances. We obtained three times of minimum lights, two for primary and one secondary minima, and determined the times of minimum lights by conventional Kwee & van Woerden (1956) method as listed in Table 1 together with other times of minima determined by previous investigators.3. MINIMUM LIGHTS AND
O C DIAGRAMWith our new times of minimum lights of CW Cep, we collected all published times of minimum lights from the literature to analyze the apsidal motion of this binary system. Among the available times of minimum lights, the visual and photographic data observed at the early stage of apsidal motion were excluded in this study because those data could increase uncertainty of period variations. Only photoelectric and CCD times of minimum lights were used in our analysis and listed in Table 1, including our timings.
These times of minima were simultaneously analyzed with the apsidal motion plus the lighttime effect ephemeris represented by:
where
Ps is the sidereal period,τ_{ap} andτ _{3} denote the apsidal motion and the LTE terms, respectively.τ_{ap} is a function of four parameters (anomalous periodP_{a} , eccentricitye , longitude of the periastronω _{0}, angular velocity of the line of apsides ) The formalism forτ _{3} was given up to sixth power of the eccentricity Giménez & Bastero (1995). Irwin (1952) gave an exact form forτ _{3} with five unknown parameters which are the semiamplitudeK , the eccentricitye _{3}, the periodP _{3}, the periastron passage timeT _{3}, the longitude of periastronω _{3} for the light orbit of mass center of the eclipsing pair around the mass center of the triple system.All the timings were fitted to Eq. (1) by using the code (available at http://sirrah.troja.mff.cuni.cz/~zasche/Programs.html) given by Zasche et al. (2009) and Zasche & Wolf (2013). We made small corrections of the code by considering the inclination of the eclipsing pair in it involving up to sixth power of the eccentricity. With the initial values of the apsidal motion and LTE parameter given by Wolf et al. (2006), we improved iteratively the apsidal motion and LTE parameters. the calculation we fixed the inclination of the eclipsing pair as 82.5° derived by Clausen & Giménez (1991) and Han et al. (2002). The final solution was listed in the sixth column of Table 2, together with those et al. (2002), Wolf et al. (2006) and Bulut et al. (2011) for comparison. The resultant OC diagram was drawn with the linear term of our result in the top of Fig. 1 where the solid continuous curves denote the combination of
τ_{ap} andτ _{3} and the dashed curve represents onlyτ _{3} term. In the middle the residuals from the linear plus the apsidal motion ephemeris were plotted with the theoretical LTEτ _{3} term. The residuals from the full contribution of Eq. (1) were plotted in the bottom panel. The standard deviation of the residuals with a relatively large scatter was calculated to be ±0.0025 days which is slightly larger than the small semiamplitude of 0.0021 days for the LTE orbit as listed in Table 2.Because the Zasche code we used was based on a differential leastsquare method Zasche et al. (2009), our solution of Eq. (1) may be a solution with a local minimum rather than a global solution. It would be very necessary to check whether our solution is globally valid for all parameter space. In line with this validity, we subsequently attempted a Monte Carlo simulation to obtain optimized values of the parameters both of the apsidal motion and the LTE orbit as performed by Lee et al. (2015). After 800 simulations all the apsidal motion parameters showed a quick convergence to nearly the same values listed in Table 2, while the LTE parameters show relatively wide variation as shown in the histograms of Fig. 2. Particularly, the distribution of
e _{3} showed remarkably wide variation, indicating that the proposed LTE orbit is not well defined with reliable accuracy by present observational data.4. DISCUSSION OF APSIDAL MOTION PERIOD
In this study, the new times of minimum lights were presented by the observations of CBNUOJ, together with the collected times of minimum lights as shown in Table 1. The
OC diagram of a differential leastsquare fit to the minimum lights were shown in Fig. 1. We calculated the new apsidal motion period as 46.27 ± 0.04 years. The apsidal parameters are listed in Table 2, which shows similar apsidal motion period with those of earlier investigators (Han at al 2002; Wolf et al. 2006), however with much improved accuracy. The existence of a third body in the CW Cep system was also suggested by many previous investigators (Han et al. 2000; Wolf et al. 2006; Bulut et al. 2011). Based on the analysis of currently available minimum lights in Table 1, we calculated the period of the LTE caused by the third body as 39.3 ± 4.4 years as shown dash line in Fig. 1. However, the LTE caused by the third body may not be convinced due to a wide variation of orbital eccentricity (e _{3}) in Fig. 2 and a small semiamplitude compatible to a large scatter band of the residuals in Fig. 1.The masses and radii of CW Cep by the photometric and spectroscopic parameters suggested by Clausen & Giménez (1991) are
M _{1}=11.82 ± 0.14M _{⊙},M _{2}=11.09 ± 0.14M _{⊙},R _{1}=5.48 ± 0.12R _{⊙}, andR _{2}=4.99 ± 0.12R _{⊙}. With these parameters, we calculated the internal structure constant of the system, as logk _{2}_{,obs} =2.106 that is slightly small value rather than the theoretical internal structure constant 2.08 suggested by Wolf et al. (2006). If the third body component exists around CW Cep, the minimum mass of the third can be calculated by the mass function (f(M _{3})) of 4×10^{5}. The results show that the third body component should be 0.279 M_{⊙} wheni _{3} is 90°, and in the case ofi _{3} is 30°, the mass is 0.562 M_{⊙}. These are approximately 2% range of mass of CW Cep binary system, which is too small compared to CW Cep and the masses are close to red dwarfs range. Therefore detecting the third body by photometry and spectroscopy would be very difficult. Apsidal motion period is very important because it can provide the important constant for the stellar internal structure. In this regards, many investigators have paid attention to the apsidal motion of CW Cep, and more accurate observations will be necessary to analyze the apsidal motion of the binary system and periodic variations by the third body.

[Table 1.] The times of minimum lights of CW Cep

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[Fig. 1.] Eclipse timing variation diagram with theoretical lines calculated with the parameters in Table 2. Open and cross symbols refer to primary and secondary minimum timings, respectively, where circle and square represent Photoelectric (PE) and CCD timings, respectively. The black and grey solid lines on the top panel represent the theoretical apsidal motion plus LTE terms for primary and secondary eclipses, respectively. The dash lines on the top and middle panels also represent the theoretical LTE (τ3). The bottom panel indicates the residual from Eq (1).

[Table 2.] Parameters of the apsidal motion and LTE for CW Cep

[Fig. 2.] Histograms for distributions of the third body orbit parameters. The black lines represent the calculated elements of the third body orbit.