Peremennye Zvezdy ( Variable ¡Stars) 42, No. 4, 2022 Received 20 May; accepted 8 June.

DOI: 10.24412/2221-0474-2022-42-17-27

Photometric Elements, Apsidal Motion, Brightness Variations, and Light-time Effect in the Eclipsing Binary System V957 Cephei

V. S. Kozyreva1, A. V. Kusakin2, A. I. Bogomazov1, Ch. T. Omarov2, A. V. Krylov1

1 P.K. Sternberg Astronomical Institute, M.V. Lomonosov Moscow State University, 119234, Universitetskij prospect, 13, Moscow, Russia

2 V.G. Fesenkov Astrophysical Institute, National Space Agency, 050020, Observatory, 23, Almaty, Kazakhstan

We obtained several light curves of V957 Cep in 2009, 2011, 2013, 2016, and 2019 at the Crimean Astronomical Station of M.V. Lomonosov Moscow State University and at the Tien Shan Observatory of the V.G. Fesenkov Astrophysical Institute. Using the TESS satellite observations, we found brightness variations of V957 Cep with the period Pp = 0d664 and amplitude Ap = 0.0076m. For our light curves (2009-2019) and TESS light curves (2019-2020), orbital elements were computed and a new value of the apsidal motion rate du/dt = 1.47° ± 0.15° yr_1 was derived. We found a possible light-time effect that can indicate gravitational influence of one or several additional bodies in the system on the central binary. Based on the currently available data, the amplitude and period of the light-time effect still cannot be reliably estimated.

1 Introduction

V957 Cep was recognized as an eclipsing variable (its orbital period being 1"?98873) in an analysis of NSVS survey data on variable stars in the northern hemisphere (Wozniak et al., 2004), and it was included in the list of 50 new eclipsing stars with elliptical orbits found in data of ASAS, Hipparcos, and NSVS (Otero et al., 2006).

We obtained the variable's light curves in the V band at the Tien Shan Astronomical Observatory of the V.G. Fesenkov Astrophysical Institute (in 2009, 2011 using the 35-cm Ritchey-Chretien telescope with an ST-402 CCD; in 2013, 2016, 2019 using the 1-m Zeiss telescope with an Apogee 49000D9 CCD; in 2016, 2019 using the 60-cm Zeiss telescope with an Apogee Aspen CCD). Studies of the light curves, parameters of the components, orbital elements, and the determination of the apsidal motion rate were published by Kozyreva et al. (2012), Kozyreva & Kusakin (2014).

The system was also observed in two series of observations by the TESS space telescope in 2019-2020, the data were taken from the MAST archive1. The precision of TESS observations is about a few thousandths of a magnitude, it is several times better than the precision of ground-based observations. Figure 1 shows the combined light curve of the system obtained by TESS in 2019-2020.

The spectral types of the components were estimated as B7V-A3V using spectropho-tometric observations made in Barnesville near Washington using an 18-inch Newton telescope (Kozyreva et al., 2012).

1 https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html

0.2 0.4 0.6

Orbital phase

Figure 1. The combined TESS light curve of V957 Cep obtained in 2019-2020.

Table 1: Parameters of pulsations according to TESS observations

v, d"1 Pp, d Ap, mag p E, HJD a a0 9 1.505 0.664 0.0076 0.61 2458766.22 0.0043 0.0069 0.39 ±0.015 ±0.006 ±0.003 ±0.01 ±0.15

2 Analysis of TESS light curve pulsations

We analysed TESS observations using the PERDET code, based on the Fourier analysis (Breger, 1990), outside minima and obtained a frequency analysis of pulsations of the object's luminosity in the following form:

N

m = Ai sin(2n(vit + pi)) + c, (1)

i=1

where m is the differential magnitude of the object; N, the number of pulsations; t, time; Ai is the amplitude of pulsations; vi, the frequency of pulsations; pi is the initial phase of pulsations, and c is the normalizing constant.

Figure 2 shows the spectral amplitude, its highest peak corresponds to the frequency 1.505 ± 0.001 d"1 and amplitude A = 0m0076.

The reliability of the pulsations can be characterized by 9 = (a/a0)2, where ao is the root-mean-square error of initial observations, a is the root-mean-square error obtained after subtraction of the pulsation (Stellingwerf, 1978). The lower is this criterion, the higher is the reliability of the process. For the 1.505 ± 0.001 d"1 pulsation, the criterion is 9 = (0.0043/0.0069)2 = 0.39, so the reliability is reasonably high. Table 1 shows parameters of the pulsation.

The spectral window and spectral amplitude are shown in Figure 3. It can be seen from this plot that maxima of these functions are at different frequencies, and they are shifted

0.008 0.007

05

"§ 0.006

'c

rn 0.005 E

I 0.004

03

"55

oi 0.003

T3 13

H 0.002

E <

0.001 0.000

0 12 3

Frequency, d -1

Figure 2. The spectral amplitude of periodic pulsations in the TESS light curve.

by at least a half of their widths, so the pulsations found in the light curve seem to be real. Figure 4 shows the 1.505 ± 0.001 d_1 pulsation during 10 days between JD 2458766 and 2458777.

3 Photometric elements

Table 2 presents orbital elements and parameters of the system found from the light curves obtained from ground-based observations in 2009 at the Tien Shan observatory and from observations by TESS. We use the following designations: r1 and r2 are relative radii of the components in units of the semi-major axis of the system; i is the orbital inclination; e, eccentricity of the orbit; u, the periastron longitude of the orbit of the primary; L1 and L2 are luminosities of the components in units of the binary's total luminosity; L3 is the third light in the same units; u1 and u2 are limb darkening coefficients; a is the standard deviation. The model and method of minimization were described by Khaliullina & Khaliullin (1984), Kozyreva & Zakharov (2001). The parameters were calculated in a free search except the limb darkening coefficients that had been fixed at values taken from van Hamme (1993) taking into account the spectral types of components and the band of observations.

The model of spherical stars (that was used in the computer program) describes detached systems adequately. The binary under investigation, with relative radii of stars 0.2 and 0.15, is close to contact. This is a potential explanation of the difference of radii of stars that can be found between the "ground-based" and "space" observations.

4 Times of minima

Tables 3 and 4 present times of minima of V957 Cep, from ground-based observations (from the literature and from our data) as well as from space observations. Minima from our ground-based light curves and from TESS space light curves were calculated using the program by Kozyreva & Zakharov (2001) jointly with the photometric elements by

1.50 1.51 Frequency, d

1.55

Figure 3. A part of the spectral amplitude of periodic pulsations in the TESS light curve figure around the 1.505 frequency and the spectral window. The amplitude is multiplied by 20.

minimization of differences between the theoretical and observed light curves. In Table 3, (O-C)1 are residuals from light elements (2) (see Section 5) and (O-C)11, from light elements (4). In Table 4, (O-C)1 are residuals from light elements (3) (see Section 5) and (O-C)11, from light elements (5).

Due to features of the TESS duty cycle, an individual light curve within a particular minimum contains insufficient number of points, therefore the time of minimum can be calculated with a high uncertainty. Therefore, we were forced to combine two light curves in neighboring minima. The time of the minimum in Tables 3 and 4 belongs to the earliest of the two neighboring minima.

5 Apsidal motion

During recent nights of observations, u was close to 360°. In such a configuration (similarly to the case of u ~ 180°), a precise determination of the eccentricity e and the primary's periastron longitude u was very difficult, because widths of minima were very close to each other, the phase shift rate of the secondary minimum was strongly reduced (the orbit practically was "stopped" for the observer). In this configuration, the apsidal motion rate, calculated as the linear change of u with fixed eccentricity, has a very high uncertainty, so this method was not apt for this case.

An analysis that included all times of minima permitted a more precise determination of the apsidal motion rate for V975 Cep in such orientation of its orbit with respect to the observer. Minimizing residuals between observed and calculated times of the primary and secondary minima, O-C, we found the following parameters: E0i, E02, Pa, u0, du/dt. Ephemerides (2) and (3) were used as calculated (C) times (E is the number of cycles from the initial epoch):

Min I = 2458767.58602 + 1.9887331 x E, (2)

1.02

0.98

65 66 67 68 69 70 71 72 73

HJD - 2 458 700

Figure 4. A light curve of V957 Cep outside minima obtained by TESS.

74 75

Min II = 2458766.58990 + 1.9887331 x E. (3)

Ephemerides that include the apsidal motion of the system are the following:

Min I = Eoi + Pa x E — c x cos(^0 + ~r x E),

at

Min II = E02 + Pa X E + c x cos(ix>o + ^ x E).

at

(4)

(5)

The results are collected in Table 5 and shown in Fig. 5. The figure presents residuals (O-C) and theoretical curves that reflect sinusoidal variations of the times of minima due to apsidal motion. For clarity, we show the point where the primary's periastron longitude was 270° (close to the year 1960) and the phase of the secondary minimum was 0.5.

6 Light-time effect

Figures 6 and 7 show (O-C)II for minima of V957 Cep, where O is the observed time and C, the calculated time (equations (4) and (5)). The changes of these residuals (Fig. 6) for primary and secondary minima are synchronous, so the system exhibits the lighttime effect and thus we may expect the presence of one or several additional bodies. Parameters of the light-time effect are presented in Table 6. Figure 7 shows the recent part of observations, it contains two curves: "P1a+P1b" describes the sum of two periodical variations "P1a" and "P1b", "P2" describes the light-time effect as a single variation.

Attempts to compute parameters of the orbit of the third body by minimization of the light-time effect, as it was done by Kozyreva & Khaliullin (1999), show a period about 1240d ~ 3.4yr and eccentricity e ~ 0.8 (this solution is not shown in the figures). The duration of observations of this star from 2009 to 2021 is insufficient to reliably confirm this orbit. A lot of times of minima obtained by TESS only weakly influence

Table 2: Orbital elements of V957 Cep found using V data from ground-based observations ("ground") in 2009 and from TESS observations ("space") in 2019-2020. See text for details

Element "Ground" "Space"

ri 0.199 ± 0.005 0.2131 ± 0.0005

T2 0.151 ± 0.005 0.1545 ± 0.0005

i 86.1° ± 0.4° 83.6° ± 0.1°

e 0.131 ± 0.005 0.125 ± 0.005

uo 336.0° ± 0.3° 357.9° ± 1.5°

Li 0.615 ± 0.020 0.685 ± 0.02

L2 0.293 ± 0.020 0.298 ± 0.02

L3 0.090 ± 0.020 0.017 ± 0.02

u1 0.43 (fixed) 0.37 (fixed)

U2 0.45 (fixed) 0.42 (fixed)

a 0.0082 0.0033

the accuracy of the determination of parameters of the third body's orbit, because these minima compactly lie within a short time interval. The current set of observations makes it possible to more or less reliably estimate the period and amplitude of the light-time effect. To find these parameters, we used the PERDET code. Figure 8 shows the power spectrum of (O-C)11. The analysis of (O-C)11 residuals shows that the sum of two periods, P1 = 3090d and P2 = 187?5, satisfies the light-time effect with minimal 9 = 0.124 (marks "1" and "2" in Fig. 8). The contribution of the P1 period is most important

to reduce the mean-square scatter between the theoretical and observed times, (O-C)11 (9i = (0.00095/0.00247)2 « 0.15, see Table 6). If the process with only one period is considered, then the best 9 = 0.149 corresponds to the light-time effect with the period P = 2890d and amplitude 0d0028 ± 0d0005.

7 Conclusions

Using observations by the TESS satellite, we found pulsations of the V957 Cep brightness with the period 0d664 and amplitude 0m0076. We also derived new photometric elements of the system based on our light curve (2009-2019) and on the light curve by TESS (2019-2020), and re-estimated the apsidal motion rate du/dt = 1.47 ± 0.15° yr_1.

Analyzing all times of minima, we found evidence for the presence of light-time effect in the system. It can be caused by the gravitational influence of an additional body (or even additional bodies), gravitationally bound with the binary. Our calculations show a very high eccentricity of the third body (0.8), but this result needs confirmation with future observations, because a longer set of observations is required. Currently we can only indicate most probable values for the amplitude of the light-time effect (4 minutes) and for its period (8 years). It is important to precisely find spectral types of both components of the binary and to increase the number of observed times of minima in order to derive orbital parameters of the third body (probably also using the radial velocity curve that has not been obtained yet).

Table 3: Times of primary minima (Min I) of V957 Cep

HJD-2400000 (O-C)1 (O-C)11 Reference

51504.6660 0.0016 0.0020 Otero et al., 2006

54710.4925 -0.0021 -0.0004 Brat et al., 2008

55076.4176 -0.0032 -0.0010 Brat et al., 2008

55122.1580 -0.0036 0.0007 Kozyreva & Kusakin, 2014

55806.2827 -0.0021 -0.0017 Kozyreva & Kusakin, 2014

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56536.1511 0.0021 -0.0003 Kozyreva & Kusakin, 2014

57558.3524 -0.0045 -0.0023 ground

58123.1541 -0.0027 0.0016 ground

58125.1421 -0.0034 0.0009 ground

59141.3869 -0.0009 -0.0010 ground

58767.5053 -0.0008 0.0006 TESS

58771.4828 -0.0007 0.0005 TESS

58775.4601 -0.0009 0.0002 TESS

58781.4262 -0.0010 -0.0001 TESS

58785.4039 -0.0008 -0.0000 TESS

58793.3587 -0.0009 -0.0002 TESS

58797.3359 -0.0011 -0.0005 TESS

58805.2911 -0.0009 -0.0002 TESS

58809.2685 -0.0009 -0.0002 TESS

58813.2458 -0.0012 -0.0003 TESS

58958.4238 -0.0007 -0.0001 TESS

58962.4014 -0.0005 -0.0001 TESS

58966.3789 -0.0005 -0.0002 TESS

58970.3562 -0.0006 -0.0004 TESS

58974.3339 -0.0004 -0.0003 TESS

58978.3109 -0.0009 -0.0009 TESS

58986.2665 -0.0002 -0.0003 TESS

58990.2436 -0.0005 -0.0005 TESS

58994.2211 -0.0005 -0.0005 TESS

59000.1874 -0.0004 -0.0003 TESS

59006.1536 -0.0004 -0.0001 TESS

Table 4: Times of secondary minima (Min II) of V957 Cep

HJD-2400000 (O - C (O - C)U Reference

54741.4657 -0.0044 -0.0012 Brat et al., 2008

55089.4948 -0.0042 -0.0013 Brat et al., 2008

55093.4725 -0.0039 -0.0008 Brat et al., 2008

55113.3595 -0.0043 -0.0002 ground

55121.3147 -0.0040 0.0003 Kozyreva & Kusakin, 2014

55819.3656 0.0005 0.0013 Kozyreva & Kusakin, 2014

56533.3253 0.0042 0.0018 Kozyreva & Kusakin, 2014

57557.5182 -0.0014 0.0008 ground

58649.3333 -0.0013 0.0011 ground

58842.2403 -0.0014 0.0007 ground

58766.6693 -0.0005 0.0009 TESS

58770.6465 -0.0008 0.0005 TESS

58780.5907 -0.0003 0.0006 TESS

58784.5683 -0.0002 0.0006 TESS

58792.5228 -0.0006 0.0001 TESS

58796.5004 -0.0004 0.0002 TESS

58804.4556 -0.0002 0.0006 TESS

58808.4330 -0.0003 0.0005 TESS

58812.4101 -0.0006 0.0003 TESS

58957.5880 -0.0002 0.0003 TESS

58961.5652 -0.0005 -0.0001 TESS

58965.5426 -0.0006 -0.0004 TESS

58969.5202 -0.0005 -0.0003 TESS

58973.4978 -0.0004 -0.0003 TESS

58977.4754 -0.0002 -0.0002 TESS

58985.4302 -0.0004 -0.0004 TESS

58989.4073 -0.0007 -0.0007 TESS

58993.3852 -0.0003 -0.0003 TESS

58999.3513 -0.0004 -0.0003 TESS

59005.3174 -0.0005 -0.0002 TESS

Table 5: Parameters of the apsidal motion in ephemerides (4) and (5)

E01 E02 Pa c u du/dt du/dt

HJD HJD d d o 1 / Porb ◦/year

2458767.5860 2458766.5899 1.9887331 0.08 358.0 0.0081 1.47

±0.0004 ±0.0004 ±0.0000003 ±0.03 ±0.3 ±0.00001o ±0.15

0.08 . Min I, theory ---------as".-" ■

■ Min I, observations

0.06 ; Min I, theory _____________ ;

0.04 . Min I, observations -— .

0.02 ______—

- 'ZX'"

-0.02

-0.04

-0.06

-0.08 .....-

1950 1960 1970 1980 1990 2000 2010 2020 2030

Date

Figure 5. (O-C)1 for minima of V957 Cep, where O is the observed time and C is the calculated time (equations (2) and (3)).

Table 6: Parameters of the light-time effect

Parameter

P1a

P1b

P2

P3

Period, d Amplitude, d Initial phase

Initial epoch,

e

HJD

a

3090 ± 50 0.0022 ± 0.0005 0.87 ±0.01

187.5 ± 4 0.0016 ± 0.0008 0.81 ± 0.01 2455933 ± 5 00 0.00087 ("P1a+P1b")

do = 0.00247 (all models) 0.124

2890 ± 50 0.0028 ± 0.0005 0.665 ±0.005

0

0.000955

0.149

1240 ± 40 0.0060 ± 0.005

2455553 ± 40 0.8i0;21 0.00130

0.28

9

Acknowledgments

This research has made use of the SIMBAD database (operated at CDS, Strasbourg, France) and of NASA's Astrophysics Data System. Some of the data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). This research is funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP09259383).

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0.005 0.004 0.003 0.002 = 0.001 0.000 ^ -0.001 -0.002 -0.003 -0.004 -0.005

1995 2005 2015 2025

Date

Figure 6. (O-C)11 for minima of V957 Cep, where O is the observed time and C, calculated time (equations (4) and (5)). "Theory" is the light-time effect, see Table 6 ("P2").

References:

Brat, L., Smelcer, L., KuCakova, H., et al., 2008, Open European Journal on Variable stars, 94, 1

Breger, M., 1990, Delta Scuti Star Newsletter, 2, 21

Khaliullina, A.I., Khaliullin, K.F., 1984, Soviet Astronomy, 28, 228

Kozyreva, V.S., Khaliullin, Kh.F., 1999, Astronomy Reports, 43, 679

Kozyreva, V.S., Kusakin, A.V., Menke, J., 2012, Inform. Bull. Var. Stars, No. 6020

Kozyreva, V.S., Kusakin, A.V., 2014, Astrophysics, 57, 221

Kozyreva, V.S., Zakharov, A.I., 2001, Astronomy Letters, 27, 712

Otero, S.A., Wils, R., Hoogeveen, G., Dubovsky, P.A., 2006, Inform. Bull. Var. Stars, No. 5681

Stellingwerf, R.F., 1978, Astrophys. J., 224, 953.

Sterne, T.E., 1939, Monthly Notices Roy. Astron. Soc., 99, 451

van Hamme, W., 1993, Astron. J., 106, 2096

WoZniak, P. R., Vestrand, W. T., Akerlof, C. W., et al., 2004, Astron. J, 127, 2436

0.005 0.004 0.003 0.002 = 0.001 0.000 ^ -0.001 -0.002 -0.003 -0.004 -0.005

2019 2020 2021 2022 2023 2024 2025

Date

Figure 7. Same as in Fig. 6, shorter time interval, see Table 6 ("P1a+P1b", "P2").

0.0020 0.0018 0.0016 0.0014 § 0.0012 « 0.0010 0.0008

c/3

0.0006 0.0004 0.0002 0.0000

-0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Frequency, d -1

Figure 8. Spectral power of variations of the (O-C)11 residuals (O is the observed time and C, that calculated using equations (4) and (5)). "1" and "2" are variations respectively with the periods P1 = 3090d and P2 = 187d5.