Spectroscopic and photometric studies of four W UMa-type eclipsing binaries

Abstract

We performed new photometric and spectroscopic studies on four W UMa-type eclipsing binaries (V400 Lyr, V574 Lyr, V1033 Her and V1062 Her). The main aims were to determine the orbital parameters and period variations of these four systems. We classified V574 Lyr, V1033 Her and V1062 Her spectral types based on LAMOST spectra, and used a spectra subtraction technique to measure the properties of the H α, H β and Ca ii H&K lines. We updated the ephemerides of these four systems, and conducted analyses of period variation for three systems (V574 Lyr, V1033 Her and V1062 Her). The period of V400 Lyr shows a continuous decrease, while that of V1033 Her continuously increases. Meanwhile, V574 Lyr and V1062 Her show a cyclic variation, caused by the light-time effect via the third body or magnetic activity cycle. The orbital and star-spot parameters of these four systems were obtained using the Wilson–Devinney program. This is the first time that precise orbital parameters have been obtained for V574 Lyr and V1062. We also concluded that these four objects are shallow contact eclipsing binaries with a cool spot on the primary. We found star-spot variabilities of about one month for V574 Lyr, and on long (years) time-scales for V1033 Her and V1062 Her. There are also obvious variations in the differences of light-curve maxima between 0.25 and 0.75 phases.

binaries: eclipsing, stars:fundamental parameters, stars: individual (V400 Lyr, V574 Lyr, V1062 Her, V1033 Her)

1 INTRODUCTION

W UMa eclipsing binaries are usually composed of two late-type stars with spectral types of F, G and K, both of which are filled with Roche lobes and share a common convective envelope. W UMa eclipsing binaries are common binaries with short periods (shorter than 1 d) and low temperatures. As of 2018 December 17, there were about 56 020 EW-type binaries (EWs) listed in the international variable star index (VSX¹). Among these 56 020 EWs, Qian et al. (2017) determined the stellar atmospheric parameters of 5363 EWs based on the spectroscopic observations. The photometric study found that W UMa eclipsing binaries have some unique properties: (1) The light change is continuous, and the depth of the two eclipses is almost equal, indicating that the surface temperatures of the two components are similar (Eggen 1967; Lucy 1968b). (2) According to the characteristics of the light curve and velocities, W UMa eclipsing binaries can be divided into A-subtype and W-subtype (Binnendijk 1970). The primary minima of the A-subtype are produced by more massive components being eclipsed by less massive components; the more massive components are hotter, while W-subtype are those in which less massive components are eclipsed and more massive components are cooler (Whelan 1972; Rucinski 1974; Mochnacki 1981; Linnell 1986, 1987). (3) The light curves of W UMa eclipsing binaries usually show asymmetries, which is called the O’Connell effect (O'Connell 1951), and some change with time. The presence of star-spots can cause distortion of the luminosity (Linnell 1991), and many researchers use the sunspot model to explain the O’Connell effect, such as GSC 02038-0293 (Dal, Sipahi & Ozdarcan 2012), GN Boo (Wang et al. 2015), LP UMa (Guo et al. 2016), and AR CrB (Alton & Nelson 2018). Astronomers theoretically simulate the evolution of W UMa eclipsing binaries, trying to explain their physical processes (e.g. Lucy 1968a, 1976; Biermann & Thomas 1972; Flannery 1976; Robertson & Eggleton 1977; Kähler 2002). Li et al. found that the regions of energy transfer, angular momentum loss and the spin angular momenta have a significant influence on the structure and evolution of low-mass W UMa eclipsing binaries.

Stellar activity, i.e. chromospheric and coronal activity, and stellar light variation are considered to be a manifestation of convective motions coupled with rapid rotation, which drives the dynamo mechanism to produce magnetic activity (Senavci et al. 2018). A chromospherically active binary is a type of star with a spectral type later than F (Eker et al. 2008), and most W UMa-type eclipsing binaries should be chromospherically active binaries. van’t Veer (1991) and Applegate (1992) studied the effect of magnetic activity on period variations of W UMa-type eclipsing binaries. Dryomova & Svechnikov (2006) discussed the physical characteristics of the period variation of 73 W UMa-type eclipsing binaries. W UMa eclipsing binaries are good vehicles to explore the presence of third bodies. When additional companions rotate around the binaries, they can transfer the angular momentum of binary systems and affect the evolution of the binaries (Kiseleva, Eggleton & Mikkola 1998; Eggleton & Kiseleva-Eggleton 2001; Pribulla & Ruckinski 2006). At the same time, the orbital period of the binaries may change periodically due to the light-time effect by the third body (Irwin 1952). Therefore, it is possible to find and study the third body through comprehensive light-curve and orbital period analysis of binaries. Studies have shown that the existence of additional companion stars (mainly the third bodies) may be a common phenomenon in short-period close binaries (Tokovinin et al. 2006; Eggleton & Tokovinin 2008). Pribulla & Ruckinski (2006) found that about 59 per cent of W UMa-type eclipsing binaries in the northern hemisphere of the sky have additional companions. Long-term spectroscopic and photometric observations of stellar activity of W UMa eclipsing binaries are necessary.

The variability of V400 Lyr was first discovered by Miller (1969). Blättler & Diethelm (2000a) obtained a full-phase unfiltered light curve of six nights, showing it to be a W Ursae Majoris (EW) type star. They also updated the linear ephemeris and obtained an updated period of 0.253 4306 d. Marino (2011) conducted the first photometric analysis of the BVR_CI_C band using phoebe software (Prsa & Zwitter 2005), and the result indicated that it is a W-type W UMa contact binary. Marino (2011) revised the linear ephemeris and plotted an O −C diagram, which showed the period decreasing. Nelson (2014) carried out a quadratic fitting for V400 Lyr and the rate of decrease of the orbital period with a coefficient of correlation of 0.992 was given as dP/dt = −2.61(± 0.23) × 10⁻⁷ d yr⁻¹.

V574 Lyr, V1033 Her and V1062 Her were discovered to be variable from the ROTSE-1 CCD survey (Akerlof et al. 2000). Blättler & Diethelm gave the light curves of V574 Lyr, V1033 Her and V1062 Her, showing them to be of type W UMa, and they also obtained the linear ephemerides of these three systems (Blättler & Diethelm 2000b,c,d). Bradstreet et al. (2009) carried out a preliminary orbital analysis and published it as a poster. For V574 Lyr and V1062 Her, the period variations and the orbital solutions have not been analysed. Before this work, there had been no spectroscopy studies on these four systems. Previously known basic parameters of the four W UMa binaries are listed in Table 1.

Table 1.

Previously known basic parameters of the four W UMa binaries.

Star name	RA	Dec.	Orbital period	B−V	J−H	H−K	Primary depth	Reference
	(2000)	(2000)	(d)	(mag)	(mag)	(mag)	(mag)
V400 Lyr	19h13m52|${^{\rm s}_{.}}$|6	+ 38°06′54|${^{\prime\prime}_{.}}$|6	0.253427	0.743(±0.026)	0.466	0.117	–	[1], [2]
V574 Lyr	18h27m12|${^{\rm s}_{.}}$|2	+ 36°14′36|${^{\prime\prime}_{.}}$|2	0.273126	0.867(±0.019)	0.517	0.090	–	[1], [2]
V1033 Her	16h50m39|${^{\rm s}_{.}}$|9	+ 27°44′23|${^{\prime\prime}_{.}}$|0	0.298051	–	0.386	0.101	0.60	[2], [3]
V1062 Her	17h34m54|${^{\rm s}_{.}}$|2	+ 44°11′52|${^{\prime\prime}_{.}}$|5	0.251436	–	0.473	0.118	0.59	[3], [4]

Note: References. [1] Terrell et al. (2012); [2] Hoffman et al. (2009); [3] Malkov et al. (2006); [4] Cutri et al. (2003).

In this paper, we present new CCD photometric light curves for four W UMa eclipsing binaries (V400 Lyr, V574 Lyr, V1033 Her and V1062 Her). We will also present new orbital and star-spot parameters and discuss their periodic variation. To better understand chromospheric activities, we will analyse the spectral data of V574 Lyr, V1033 Her and V1062 Her from the LAMOST spectroscopic survey.

2 OBSERVATION AND DATA REDUCTION

New photometric observations for these four W UMa-type eclipsing binaries were carried out from 2015 to 2018 (V400 Lyr: 2017 July 30, 31 and August 13; V574 Lyr: 2017 July 26 and August 16; V1033 Her: 2017 June 25, 26, 27 and August 21, 23, 25 and 2018 May 24, 27, 29; V1062 Her: 2015 May 25 and July 4, 2017 July 2, 3, 4 and 2018 May 30). Four pieces of equipment were used: the Holcomb telescope, which is a 94-cm Cassegrain reflecting telescope with a focal ratio of f/6.1 and a 2048 × 2064 CCD; the SARA 91.4-cm telescope at the Kitt Peak National Observatory (SARA KP) in Arizona; and the 60-cm and 85-cm telescopes at Xinglong station of the National Astronomical Observatories of China (NAOC). The camera of SARA KP is a 2048 × 2048 CCD pixel. Since we chose 2 × 2 binning mode, the effective field of view becomes 1024 × 1024 CCD pixel. The 60-cm telescope at Xinglong is equipped with a focal ratio of f/4.23, a CCD of PI 1024, and a field of view of 18′ × 18′. The 85-cm telescope is equipped with a focal ratio of f/3 and a field of view of 32′ × 32′. The cameras of the 60-cm and 85-cm telescopes are 1024 × 1024 and 2048 × 2048 pixel CCD, respectively. We used the standard Johnson–Cousins B, V, R, I filters in all observations.

We reduced our data using M axI m DL software, including corrections using bias, dark and flat (Blackwell, Sliier & Wood 2005). A summary of photometric observation information is provided in Table 2, including object stars, comparison stars, check stars, observed time and exposure time. For V1062 Her, due to the different fields of view of these three telescopes, the comparison stars and check stars that we chose are different in order to take advantage of the best comparison stars in the available field of view. All magnitudes were determined using M axI m DL software. We list some of the data in Table 3. All photometric observational data for these four systems are available in the online version of this paper.

Table 2.

Basic observation information for the four low-mass W UMa-type eclipsing binaries.

Star name	Comparison	Check star	Exposure time	Telescope
V400 Lyr	2MASS J19140129 + 3804043	TYC 3121-1335-1	90 s, 60 s, 60 s, 60 s (BVRI)	Holcomb
V574 Lyr	2MASS J18270798 + 3618110	TYC 2636-1638-1	90 s, 60 s, 40 s, 40 s (BVRI)	Holcomb
	BD + 27 2701	TYC 2066-1075-1	70 s, 40 s, 25 s, 25 s (BVRI)	Holcomb
V1033 Her	BD + 27 2701	TYC 2066-1075-1	80 s, 60 s, 35 s, 35 s (BVRI)	85 cm
	BD + 27 2701	TYC 2066-1075-1	60 s, 30 s, 20 s, 10 s (BVRI)	60 cm
	2MASS J17340658 + 4408496	2MASS J17341237 + 4409339	90 s, 60 s, 60 s (VRI)	Holcomb
V1062 Her	TYC 3099-1581-1	2MASS J17342101 + 4419258	100 s, 30 s, 30 s, 30 s (BVRI)	60 cm
	2MASS J17350517 + 4419515	TYC 3099-1315-1	120 s, 60 s, 60 s (BVI)	SARA KP

Table 3.

|$B V R \& I$| photometric observational data for the four eclipsing binaries.

Star name	HJD|$\_B$|	ΔB	HJD|$\_V$|	ΔV	HJD|$\_R$|	ΔR	HJD|$\_I$|	ΔI
	245 7960.5955	0.284	245 7960.5964	−0.156	245 7960.5971	−0.452	245 7960.5976	−0.72
	245 7960.5984	0.255	245 7960.5993	−0.173	245 7960.6000	−0.462	245 7960.6005	−0.723
V547 Lyr	–	–	–	–	–	–	–	–
	245 7981.8159	0.193	245 7981.8168	−0.169	245 7981.8174	−0.201	245 7981.8151	−0.783
	245 7981.8188	0.088	245 7981.8197	0.581	245 7981.8203	0.58	245 7981.8179	0.141
	245 7964.6778	−0.066	245 7964.6787	0.068	245 7964.6798	0.382	245 7964.6799	0.132
	245 7964.6808	−0.1	245 7964.6817	0.053	245 7964.6828	0.381	245 7964.6830	0.131
V400 Lyr	–	–	–	–	–	–	–	–
	245 7978.8545	−0.186	245 7978.8593	−0.033	245 7978.8620	0.299	245 7978.8565	0.065
	245 7978.8574	−0.106	245 7978.8621	0.034	245 7978.8649	0.36	245 7978.8594	0.118
	245 7929.6569	1.792	245 7929.6573	1.654	245 7929.6576	1.567	245 7929.6579	1.485
	245 7929.6582	1.796	245 7929.6586	1.658	245 7929.6589	1.576	245 7929.6591	1.499
V1033 Her	–	–	–	–	–	–	–	–
	245 7990.7216	1.315	245 7990.7256	1.231	245 7990.7242	1.116	245 7990.7245	1.048
	245 7990.7233	1.325	–	–	245 7990.7259	1.108	245 7990.7262	1.071
	–	–	245 7207.6593	2.235	245 7207.6681	2.231	245 7207.6729	2.153
	–	–	245 7207.6633	2.23	245 7207.6721	2.228	2457207.6769	2.082
V1062 Her	–	–	–	–	–	–	–	–
	–	–	245 7908.8932	1.517	245 7908.8963	1.491	2457908.8991	1.51
	–	–	245 7908.8954	1.569	245 7908.8985	1.551	–	–

Note: We only show a short table here and provide the full table as part of the electronic version of this paper.

The Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST) (also known as the Goushoujing Telescope) provides a large amount of low-dispersion spectral observations. It provides a great opportunity to study spectral and chromospheric properties of eclipsing binaries (Zhang et al. 2017, 2018). Our spectroscopic data of V574 Lyr, V1033 Her and V1062 Her with R ∼ 1800 were downloaded from the LAMOST DR5 website²(Luo et al. 2012; Zhao et al. 2012). The spectral regions covered were |$3700\!-\!9000\, \mathrm{\mathring{\rm A} }$|⁠, and we list observed time, temperature and the signal-to-noise ratio (S/N) with the I band in Table 4. We normalized all observed spectra with the iraf³ package and plotted them in Fig. 1 (black solid lines). Then, we used the hammer program (Covey et al. 2007; West et al. 2011) to reclassify the spectra. We determined that the spectra of V1033 Her and V1062 Her are both K2 (±2 subtype) types, while the spectrum of V574 Lyr is the K4 (±2 subtype) type.

Figure 1.

LAMOST spectra for V574 Lyr, V1033 Her and V1062 Her in the H α, H β and Ca ii H&K and IRT lines. The green (upper) spectra are the subtracted spectra.

Table 4.

LAMOST spectroscopic parameters and EWs of V574 Lyr, V1033 Her and V1062 Her.

Parameters	V400 Lyr	V1033 Her	V1062 Her
Observed time	2017 Jun. 4	2016 Feb. 25	2013 May 3
Spectra type	K4 ± 2	K2 ± 2	K2 ± 2
Temperature (K)	4910.61	5426.42	5010.64
S/N	227.05	330.69	75.97
H α (Å)	0.9209 ± 0.0089	1.0719 ± 0.1189	0.7448 ± 0.0392
H β (Å)	0.3200 ± 0.0020	–	–
Ca ii H (Å)	1.3766 ± 0.0284	1.7691 ± 0.1939	–
Ca ii K (Å)	1.7507 ± 0.0153	1.5007 ± 0.0683	–

3 ORBITAL PERIOD STUDY

From our new observations, we used the method of Kwee & Van Woerden (1956) with the polynomial fitting program of Nelson (2007) to fit the light curves of all the bands and obtained new times of light minima and their corresponding uncertainties. To analyse the orbital periods of these four eclipsing binaries, we collected all light minima, and list them in Tables 5 to 8. In Tables 5 to 8, we also list the types (p: primary or s: secondary), the methods (CCD: charge-coupled, vis: visual or pe: photometric) and epochs of these times of light minima. During the fitting of period variation, we have assigned weights of individual eclipsing times according to their errors. If there were no errors, we used the average of the errors of other minima with the same observational method.

Table 5.

Minimum light times and relevant parameters of V400 Lyr.

JD(Hel.)24, +	Error	Min	Method	Cycle	(O − C)1	(O − C)2(poly)	Ref.
51274.8713	0.0012	s	CCD	−26 401.5	0.0334	0.0036	2
51294.3822	0.0018*	s	CCD	−26 324.5	0.0296	0.0000	1
51294.5095	0.0018*	p	CCD	−26 324.0	0.0228	−0.0036	1
51300.3410	0.0018*	p	CCD	−26 301.0	0.0340	−0.0104	1
51300.4656	0.0018*	s	CCD	−26 300.5	0.0273	0.0007	1
51308.4473	0.0018*	p	CCD	−26 269.0	0.0244	−0.0059	1
51308.5750	0.0018*	s	CCD	−26 268.5	0.0276	−0.0088	1
51321.8727	0.0018*	p	CCD	−26 216.0	0.0227	−0.0056	1
51321.8742	0.0010	p	CCD	−26 216.0	0.0227	−0.0104	2
51327.4544	0.0018*	p	CCD	−26 194.0	0.0273	−0.0104	1
...	...	...	...	...	...	...	...

Note: All minimum light times and relevant parameters are available online. The symbol `*’ represents the average of all CCD errors.

References. [1] Data from the O − C gateway (http://var.astro.cz/ocgate/); [2] Diethelm (2001b); [3] Zejda (2004); [4] Agerer & Hübscher (2002); [5] Brát, Zejda & Svoboda (2007); [6] Diethelm (2003); [7] Diethelm (2004); [8] Hübscher, Steinbach & Walter (2009); [9] Diethelm (2007a); [10] Diethelm (2007b); [11] Diethelm (2008); [12] Marino et al. (2010); [13] Arena et al. (2011); [14] Diethelm (2010a); [15] Hübscher (2011); [16] Hübscher, Lehmann & Walter (2012); [17] Hoňková et al. (2013); [18] this work.

3.1 V400 Lyr

For V400 Lyr, we collected all the available minimum CCD light times in addition to the five minimum times that we obtained, resulting in a total of 112 data points (listed in Table 5). We obtained an updated linear ephemeris:

\begin{eqnarray*} \mathrm{HJD(Min.I)} = 245\,7965.6947(16)+0^{d}.253\,425\,93(8) \times E, \end{eqnarray*}

(1)

where E is the epoch number. We calculated the values of (O − C)₁ and plotted them in Fig. 2. We can see from the figure that the (O − C)₁ values show a downward parabolic variation. Using least-squares fitting, we obtained the following quadratic ephemeris:

\begin{eqnarray*} \mathrm{HJD(Min.I)} &=& 245\,7965.6892(11)+0^{d}.253\,422\,73(6)\nonumber \\ &&\times E-0.717(2) \times 10^{-10} \times E^{2}. \end{eqnarray*}

(2)

The (O − C)₂ values are listed in Table 5. This ephemeris indicates that there is a continuous decrease in the orbital period of V400 Lyr. According to equation (2), the rate of decrease of the orbital period is dP/dt  = -2.06(± 0.09) × 10⁻⁷ d yr⁻¹.

Figure 2.

O − C diagrams of V400 Lyr and V1033 Her. The solid lines represent their polynomial fit.

3.2 V1033 Her

For V1033 Her, we collected a total of 50 CCD light minima times and nine new times that we obtained, and listed them in Table 6. Using the least-squares method, we obtained an updated linear ephemeris for V1033 Her:

\begin{eqnarray*} \mathrm{HJD(Min.I)} = 245\,7929.8095(9)+0^{d}.298\,051\,28(7) \times E, \end{eqnarray*}

(3)

where E is the epoch number. The (O − C)₁ values listed in Table 6 are plotted in Fig. 2. As can be seen from Fig. 2, the values of (O − C)₁ show an upward parabolic variation, which means that the orbital period of V1033 Her has an increasing trend. Fitting O − C data to a quadratic function, we obtained the following quadratic ephemeris:

\begin{eqnarray*} \mathrm{HJD(Min.I)} &=& 245\,7929.8141(13)+0^{d}.298\,052\,42(8)\nonumber \\ && \times E+0.510(4) \times 10^{-10} \times E^{2}. \end{eqnarray*}

(4)

The (O − C)₂ values are listed in Table 6. The residuals after non-linear fitting are showed in the lower panel of the right-hand figure in Fig. 2. According to equation (4), the rate of increase of the orbital period is dP/dt  = 1.25(± 0.02) × 10⁻⁷ d yr⁻¹.

Table 6.

Minimum light times and relevant parameters of V1033 Her.

JD(Hel.)24, +	Error	Min	Method	Cycle	(O − C)1	(O − C)2(poly)	Ref.
51265.8329	0.000 04	s	CCD	−22 358.5	−0.0070	−0.0014	2
51265.9804	0.000 04	p	CCD	−22 358.0	−0.0085	−0.0029	2
52056.4124	0.0012*	p	CCD	−19 706.0	−0.0078	−0.0004	1
52056.4147	0.0012*	p	CCD	−19 706.0	−0.0055	0.0018	3
52056.5648	0.0012*	s	CCD	−19 705.5	−0.0044	0.0029	1
52058.5009	0.0012*	p	CCD	−19 699.0	−0.0057	0.0016	1
52065.5059	0.0012*	s	CCD	−19 675.5	−0.0048	0.0025	1
52073.4019	0.0012*	p	CCD	−19 649.0	−0.0072	0.0001	1
52073.5545	0.0012*	s	CCD	−19 648.5	−0.0036	0.0037	1
52075.4910	0.0012*	p	CCD	−19 642.0	−0.0045	0.0028	1
...	...	...	...	...	...	...	...

Note: All minimum light times and relevant parameters are available online. The symbol `*’ represents the average of all CCD errors.

References. [1] Data from the O − C gateway (http://var.astro.cz/ocgate/); [2] Diethelm (2001a); [3] Diethelm (2003); [4] data from https://www.aavso.org/bob-nelsons-o-c-fles; [5] Diethelm (2005); [6] Hübscher et al. (2006); [7] Diethelm (2007a); [8] Hübscher (2007); [9] Diethelm (2007b); [10] Hübscher et al. (2009); [11] Hübscher, Lehmann & Monninger (2010); [12] Diethelm (2010a); [13] Hübscher & Monninger (2011); [14] Hübscher et al. (2012); [15] Diethelm (2011); [16] Demircan et al. (2012); [17] Hübscher, Braune & Lehmann (2013); [18] Hübscher & Lehmann (2013); [19] this work.

3.3 V574 Lyr

We collected a total of 122 data points (3 vis and 119 CCD) of minimum light times and listed them in Table 7. Combining with our new minima, we updated the linear ephemeris:

\begin{eqnarray*} \mathrm{HJD(Min.I)} = 245\,7960.8201(6)+0^{d}.273\,126\,33(2) \times E, \end{eqnarray*}

(5)

where E is the epoch number. Using this new ephemeris, we calculated the (O − C)₁ values and plotted them in Fig. 3. An obvious periodic fluctuation is visible and the orbital period of the V574 Lyr may have a periodic change. There are two possible mechanisms for this change. One of the mechanisms is the light-time effect (LITE) by the gravitational interaction between a third body and the eclipsing binaries. Based on LITE (Irwin 1952), fitting the (O − C)₁ values gives:

\begin{eqnarray*} \tau = \frac{a_{12}\sin i}{c}[\frac{1-e^{2}}{1+e\cos v}\sin (v+\omega)+e\sin \omega ], \end{eqnarray*}

(6)

where A  = a₁₂sin i/c is the semi-amplitude of the LITE, c is the speed of light, and a₁₂ is the semi-major axis of the eclipsing pair orbiting the mass centre with the third body. The other parameters of the third body are i, v, ω and e, which represent the inclination, true anomaly and eccentricity of the periastron, and longitude, respectively. As can be seen from equation (6), in order to calculate the light-time effect of the third body, five parameters must be determined simultaneously. We used the L–M algorithm (Levenberg–Marquardt algorithm) (Press et al. 1992; Yang et al. 2011) to fit the values, and listed the resulting parameters in Table 9, which also contains the mass function of the third component f(m), the observable semi-amplitude of systemic radial velocity K_RV and the time of periastron passage of the third body in its orbit T₀. The values of (O − C)₂ are listed in column 7 of Table 7 and plotted in Fig. 3. The period of the third body is P₃  = 12.373(±0.006) yr.

Figure 3.

O − C diagram of V574 Lyr and V1062 Her. The (O − C)I values were computed by using our newly determined linear ephemeris. The solid lines represent the fit on a cyclic variation (the third body or magnetic cycle).

Table 7.

Minimum light times and relevant parameters of V574 Lyr.

JD(Hel.)24, +	Error	Min	Method	Cycle	(O − C)1	(O − C)2(3rd body)	(O − C)2(sine)	Ref.
51260.8816	0.0011*	s	CCD	−1818.5	−0.0023	0.0044	0.0026	1
51288.8797	0.0011*	p	CCD	−1716.0	0.0002	0.0067	0.0049	1
51757.4276	0.0011*	s	CCD	−0.5	−0.0006	0.0023	0.0011	1
51757.5643	0.0011*	p	CCD	−0.0	−0.0002	0.0027	0.0015	1
51768.4894	0.0011*	p	CCD	40.0	−0.0003	0.0025	0.0013	1
51773.4061	0.0011*	p	CCD	58.0	0.0005	0.0033	0.0021	1
51781.4619	0.0011*	s	CCD	87.5	−0.0018	0.0009	−0.0002	1
52116.4516	0.0011*	p	CCD	1314.0	−0.0017	−0.0013	−0.0018	1
52443.3857	0.0011*	p	CCD	2511.0	−0.0003	−0.0013	−0.0014	1
52783.4267	0.0013	p	CCD	3756.0	−0.0021	−0.0035	−0.0035	2
...	...	...	...	...	...	...	...	...

Note: All minimum light times and relevant parameters are available online. The symbol `*’ represents the average of all CCD errors.

References. [1] Data from O − C gateway (http://var.astro.cz/ocgate/); [2] Diethelm (2003); [3] Diethelm (2004); [4] Hübscher, Paschke & Walter (2005); [5] Brát et al. (2007); [6] Diethelm (2007a); [7] Hübscher et al. (2006); [8] Diethelm (2007b); [9] Hübscher (2007); [10] Hübscher et al. (2009); [11] Brát et al. (2011); [12] Diethelm (2010b); [13] Hoňková et al. (2013); [14] Hübscher et al. (2012); [15] Banfi et al. (2012); [16] Hübscher & Lehmann (2013); [17] Diethelm (2012); [18] Hübscher et al. (2013); [19] Hoňková, Juryšek & Lehký (2015); [20] Hübscher (2016); [21] this work.

Another mechanism is magnetic activity. Using the least-squares method, we adopted the linear  + sine function to fit (O − C)₁ and the obtained the equation as follows:

\begin{eqnarray*} {(O-C)}_{1} &=& 6.1(2) \times 10^{-4}+0.0040(2)(d)\times \sin [0.000\,37(1)(\mathrm{rad}^{-1})\nonumber \\ && \times E-5.78(8)(\mathrm{rad})]. \end{eqnarray*}

(7)

From this formula, we calculated the period for this oscillation T  = 2π × P/ω  = 12.70(±0.30) yr, where P and ω denote the orbital period in years and the coefficient of E, respectively. This formula also reveals the amplitude of periodic change A  = 0.0080(±0.0004) d. We list the values of (O − C)₂ in column 8 of Table 7 and plot them in Fig. 3.

3.4 V1062 Her

We collected all the available minimum light times and listed them in Table 8 (45 existing and 10 new data points). We amended the linear ephemeris of V1062 Her:

\begin{eqnarray*} \mathrm{HJD(Min.I)} = 245\,7906.6500(7)+0^{d}.251\,435\,77(5) \times E, \end{eqnarray*}

(8)

where E is the epoch number. We calculated the (O − C)₁ values by the above formula and then plotted them in Fig. 3. As can be seen from Fig. 3, the (O − C)₁ values of V1062 Her show a clear periodic oscillation. The periodic variation indicates that there is a high probability of a third body around V1062 Her. Based on equation (6) to fit the (O − C)₁ values, we obtained a set of parameters for the third body and listed them in Table 9 and the corresponding residuals (O − C)₂ are listed in column 7 of Table 8 and plotted in Fig. 3. The period of the third body is P₃  = 14.377(±0.003) yr. Additionally, based on the assumption that this period change is caused by magnetic activities, we also adopted the linear  + sine function to fit (O − C)₁ and obtained the following:

\begin{eqnarray*} {(O-C)}_{1} &=& -6.3(3) \times 10^{-4}+0.0033(4)(d)\nonumber \\ && \times \sin [0.000\,28(2)(\mathrm{rad}^{-1})\times E-4.09(4)(\mathrm{rad})]. \end{eqnarray*}

(9)

According to the above formula, the period for this oscillation T is calculated to be 15.45(±0.04) yr, and the amplitude is 0.007(8) d. The values of (O − C)₂ are listed in column 8 of Table 8 and plotted in Fig. 3.

Table 8.

Minimum light times and relevant parameters of V1062 Her.

JD(Hel.)24, +	Error	Min	Method	Cycle	(O − C)1	(O − C)2(3rd body)	(O − C)2(sine)	Ref.
51305.7095	0.0015*	p	pe	−1753.0	0.0038	−0.0020	0.000 059	1
51311.8712	0.0015*	s	pe	−1728.5	0.0053	−0.0005	0.001 568	1
51746.4764	0.0015*	p	CCD	0.0	0.0034	−0.0015	0.000 643	1
51752.3868	0.0015*	s	CCD	23.5	0.0051	0.0001	0.002 361	1
51752.5136	0.0015*	p	CCD	24.0	0.0061	0.0011	0.003 362	1
51767.3485	0.0015*	p	CCD	83.0	0.0063	0.0013	0.003 608	1
51773.3793	0.0015*	p	CCD	107.0	0.0026	−0.0022	−0.000 073	1
51773.5052	0.0015*	s	CCD	107.5	0.0028	−0.0020	0.000 127	1
51781.4265	0.0015*	p	CCD	139.0	0.0039	−0.0009	0.001 252	1
51781.5518	0.0015*	s	CCD	139.5	0.0035	−0.0013	0.000 853	1
...	...	...	...	...	...	...	...	...

Note: All minimum light times and relevant parameters are available online. The symbol `*’ represents the average of all CCD errors.

References. [1] Data from the O − C gateway (http://var.astro.cz/ocgate/); [2] data from https://www.aavso.org/bob-nelsons; [3] Diethelm (2003); [4] Diethelm (2004); [5] Hübscher et al. (2005); [6] Hübscher et al. (2009); [7] Diethelm (2007a); [8] Diethelm (2007b); [9] Hübscher (2007); [10] Hübscher (2010); [11] Hübscher & Lehmann (2012); [12] Diethelm (2010a); [13] Hübscher & Monninger (2011); [14] Hoňková et al. (2013); [15] Hübscher & Lehmann (2013); [16] this work.

Table 9.

Parameters of assumed third bodies for V574 Lyr and V1062 Her.

Parameters	V574 Lyr	V1062 Her
A (d)	0.0041(±0.0001)	0.0046(±0.0002)
P3(yr)	12.373(±0.006)	14.3768(±0.003)
e	0.205(±0.014)	0.538(±0.009)
ω (°)	6.429(±0.054)	9.586(±0.014)
T0 (HJD)	246 5338.5449	246 1826.0668
a12sin i (au)	0.7099	0.7965
KRV (km s−1)	1.746(±0.027)	1.956(±0.030)
f(m) (M⊙)	0.002 337(±0.000 006)	0.002 445(±0.000 010)

4 PHOTOMETRIC ANALYSES

We use the updated version of the Wilson–Devinney program (W–D) (Wilson & Devinney 1971; Wilson 1979, 1990, 1994; Wilson & Van Hamme 2004) to solve for the photometric orbital solution of these four systems.

4.1 Orbital parameters

The effective temperature was calculated by the formula (Collier Cameron et al.2007):

\begin{eqnarray*} T_\mathrm{eff} = -4369.5(J-H)+7188.2, 4000\,\mathrm{K}\,\lt \,T_\mathrm{eff}\,\lt\, 7000\,\mathrm{K}, \nonumber \\ \end{eqnarray*}

(10)

where J − H was obtained from the 2MASS All Sky Catalog⁴ (Skrutskie et al. 2006). The temperatures of the primary components of our four systems are listed in Table 10. For V574 Lyr, V1033 Her, and V1062 Her, compared with the temperatures of primary components derived by the LAMOST survey, the differences between the temperatures derived by these two methods are similar. Since the systems are late-type eclipsing binaries with common convective cladding, we set the bolometric albedo as A₁  = A₂  = 0.5 (Rucinski 1969) and the gravity-darkening coefficients as g₁  = g₂  = 0.3 (Lucy 1967). The limb-darkening coefficients are interpolated by the table from van Hamme (1993). Since these four systems are contact eclipsing binaries, we chose mode3 (contact binary) to solve for the orbital solutions. The adjusted parameters are the inclination (i), the mass radio (q = m₂/m₁), the temperature of the secondary component (T₂), the potentials of both components (Ω_{1, 2}), and the monochromatic luminosity of the primary component (L₁). The mass radio q is a key parameter. We determined its value in the following way. For each q_i value in a given range, we run the W–D program and calculate the fitting residual defined as the sum of the weighted square deviation (⁠|$\Sigma (O-C)_{i}^{2}$|⁠) and and plot the residual versus q in Fig. 4. We choose the values with the smallest (⁠|$\Sigma (O-C)_{i}^{2}$|⁠) as the best mass ratio of these four systems. For V400 Lyr and V1033 Her, the mass radios have been obtained previously (Marino 2011; Bradstreet et al. 2009); we adjusted them again on these bases and obtained the mass radio with smaller |$\Sigma (O-C)_{i}^{2}$| values. For total eclipse, the accuracy of W–D fitting is higher than that for partial eclipse. Internal errors of the W–D fitted q will underestimate the true uncertainty of q. Radial velocities will be needed to narrow down the q values.

Figure 4.

Relationships between the sum of the squares of the residuals and the mass ratios q for four eclipsing binaries.

Table 10.

Theoretical orbital parameters for the four eclipsing binaries.

Parameters	V400 Lyr	V574 Lyr	V1033 Her	V1062 Her
T1 (K)	5152 ± 153	4929 ± 149	5501 ± 175	5121 ± 130
q (M2/M1)	3.70 ± 0.1	0.95 ± 0.05	3.90 ± 0.05	2.47 ± 0.02
i (°)	87.942 ± 0.875	75.961 ± 0.111	84.275 ± 0.045	87.494 ± 0.225
T2 (K)	4833 ± 26	4690 ± 44	5137 ± 11	4715 ± 13
Ωin	7.5284	3.6682	7.7843	5.8085
Ωout	6.9007	3.1477	7.1546	5.2008
Ω1  = Ω2	7.370 ± 0.007	3.616 ± 0.003	7.713 ± 0.0002	5.652 ± 0.006
L1/(L1 + L2) (B)	0.3324 ± 0.0032	0.6061 ± 0.0014	0.3233 ± 0.0009	–
L1/(L1 + L2) (V)	0.3071 ± 0.0026	0.5875 ± 0.0012	0.2987 ± 0.0007	0.4243 ± 0.0018
L1/(L1 + L2) (R)	0.2946 ± 0.0021	0.5706 ± 0.0009	0.2836 ± 0.0006	0.3976 ± 0.0015
L1/(L1 + L2) (I)	0.2844 ± 0.0018	0.5510 ± 0.0008	0.2725 ± 0.0005	0.3820 ± 0.0013
r1 (pole)	0.2636 ± 0.0015	0.3671 ± 0.0007	0.2542 ± 0.0003	0.3046 ± 0.0021
r1 (side)	0.2758 ± 0.0018	0.3870 ± 0.0009	0.2651 ± 0.0003	0.3211 ± 0.0026
r1(back)	0.3174 ± 0.0034	0.4212 ± 0.0012	0.3013 ± 0.0005	0.3728 ± 0.0053
r1(average)	0.2856 ± 0.0005	0.3918 ± 0.0009	0.2735 ± 0.0004	0.3328 ± 0.0033
r2(pole)	0.4733 ± 0.0013	0.3585 ± 0.0007	0.4722 ± 0.0002	0.4521 ± 0.0017
r2(side)	0.5130 ± 0.0019	0.3775 ± 0.0009	0.5109 ± 0.0003	0.4878 ± 0.0023
r2(back)	0.5406 ± 0.0024	0.4122 ± 0.0012	0.5363 ± 0.0004	0.5232 ± 0.0033
r2 (average)	0.5090 ± 0.0019	0.3079 ± 0.0009	0.5065 ± 0.0003	0.4877 ± 0.0024
f (%)	25.2(± 0.2)	10.0(± 1.3)	11.3(± 0.9)	25.7 (± 1.0)
The residuals	1.4199	0.2089	0.0385	0.0842
Absolute parameters⋆
|$M_{1}(\, \mathrm{M}_{\odot }$|⁠)	0.79 ± 0.02	0.69 ± 0.02	0.74 ± 0.05	0.74 ± 0.05
|$M_{2}(\, \mathrm{M}_{\odot }$|⁠)	2.92 ± 0.09	0.65 ± 0.02	2.89 ± 0.12	1.83 ± 0.12
a (R⊙)	2.54 ± 0.8	1.95 ± 0.06	2.89 ± 0.12	2.30 ± 0.15

Note: ⋆ absolute masses of the primaries (⁠|$M_{1}(\, \mathrm{M}_{\odot }$|⁠)) were determined from their spectral types and the mass–spectral type relationship in Cox (2000), and absolute masses of the secondaries (⁠|$M_{2}(\, \mathrm{M}_{\odot }$|⁠)) were determined from q (M₂/M₁). We calculated the semi-major axis (a(R_⊙)) of each system from the total mass using Kepler's third law.

4.2 Star-spot parameters

There are asymmetries in the light curves of the four systems. We used spot models to explain the asymmetries. Since there are no simultaneous spectroscopic observations for the four systems, we cannot determine whether a spot is on the primary or secondary component. Since the equatorial spots on the primary star have the smallest temperature contrast and size to fit the light-curve distortion, we assumed that the spots are on the primary with a latitude of 90°. Then, we adjusted the other three parameters (longitude, radius and temperature). Finally, after repeated runs, we obtained the photometric solutions and the corresponding star-spot parameters, and listed them in Tables 10 and 11. We plotted observational and theoretical light curves of these four systems in Fig. 5, and plotted their configurations and spot distributions in Fig. 6.

Figure 5.

Observational and theoretical light curves of the four eclipsing binaries. The points and solid lines represent the observational and theoretical light curves, respectively.

Figure 6.

Configurations and spot distributions of the four eclipsing binaries.

Table 11.

Spot parameters of the four eclipsing binaries.

Star	HJD(24, +)	Longitude	Radius	T(s)
V400 Lyr	57964.7	101.4°±2.2°	12.5°±1.1°	4324 K±201 K
V400 Lyr	57965.7	103.0°±1.9°	15.7°±0.7°	4188 K±147 K
V400 Lyr	57978.7	101.2°±1.0°	9.9°±0.5°	4210 K±214 K
V574 Lyr	57930.6	88.9°±0.2°	12.4°±0.2°	3062 K±135 K
V574 Lyr	57988.6	100.6°±0.1°	15.5°±0.03°	3068 K±219 K
V1033 Her	57988.6	39.4°±0.9°	19.5°±0.3°	3993 K±93 K
V1033 Her	58263.1	47.3°±0.5°	21.1°±0.3°	3994 K±15 K
V1033 Her	58268.1	46.0°±0.4°	20.1°±0.1°	3825 K±34 K
V1062 Her	57168.1	99.8°±2.2°	20.5°±0.8°	3545 K±147 K
V1062 Her	57207.7	101.4°±1.3°	20.3°±0.3°	2622 K±1391 K
V1062 Her	57907.6	80.6°±0.1°	20.3°±1.4°	3545 K±92 K
V1062 Her	57908.6	80.6°±0.3°	20.3°±1.5°	3545 K±60 K

5 DISCUSSIONS AND CONCLUSIONS

We will discuss the period variations, orbital parameters and chromospheric activity of the four eclipsing binaries.

5.1 Period variations

5.1.1 V400 Lyr and V1033 Her

In general, for contact binaries, constant change in the orbital period is likely due to mass transfer between the two components. In the case that the total mass and the total angular momentum of the system are conserved, the mass transfer rate is given by (Kwee 1958):

\begin{eqnarray*} \frac{\dot{M}_{1}}{M_{1}} = \frac{q}{3(1-q)}\frac{\dot{P}}{P}. \end{eqnarray*}

(11)

The masses of the primary stars are listed in Table 10; these were determined by the relationship between mass and spectral type (Cox2002 ). Based on our mass ratios q, we calculated the masses of the secondary stars and listed them in Table 10.

V400 Lyr exhibits a continuous decrease in its orbital period at rates dP/dt = −2.06(± 0.09) × 10⁻⁷ d yr⁻¹. Our period change rate for V400 Lyr is similar to that obtained by Nelson (2014). If the mass transfer is indeed the mechanism causing the continuous period decrease, the mass should be transferred from the more massive secondary star to the less massive primary. Combining these values, from equation (11), we obtained the mass transfer rates to be |$\dot{M}_{1} = 2.93(\pm 0.05)\times 10^{-7}\, \mathrm{M}_{\odot }$| yr⁻¹.

While V1033 Her exhibits a continuous increase at dP/dt = 1.25(± 0.02) × 10⁻⁷ d yr⁻¹, this continuous increase was explained by mass transfer from the less massive primary star to the more massive secondary. Using the same method as for V400 Lyr, we also obtained the mass transfer rate of V1033 Her to be |$\dot{M}_{1} = -1.39(\pm 0.02)\times 10^{-7}\, \mathrm{M}_{\odot }$| yr⁻¹.

5.1.2 V574 Lyr and V1062 Her

For V574 Lyr and V1062 Her, their periodic oscillations maybe caused by the LITE of the third body or magnetic activity. The semi-axis of the eclipsing binary a₁₂ can be determined by the formula a₁₂ sin i  = A × c, where i is the orbital inclination of the eclipsing binary, A and c correspond to the semi-amplitude of the periodic variation and the speed of light in vacuum, respectively. The mass function of the third body f(m) can be calculated from the well-known formula:

\begin{eqnarray*} f(m) = \frac{4\pi ^{2}}{GT^{2}}\times (a^{\prime }_{12}\sin i^{\prime })^{3} = \frac{(M_{3}\sin i_{3}^{\prime })^{3}}{(M_{1}+M_{2}+M_{3})^{2}}. \end{eqnarray*}

(12)

The parameters needed in the above mass function are listed in Table 9. With these parameters, we calculated the mass of the third body at different orbital inclinations. Fig. 7 shows their variation curves. When the orbital inclination is 90°, the minimal mass of the third body M_{3, min} is 0.0286 |$\, \mathrm{M}_{\odot }$| for V574 Lyr and 0.0475 |$\, \mathrm{M}_{\odot }$| for V1062 Her. According to Allen's astrophysical quantities (Cox 2000) and assuming that the third bodies of V574 Lyr and V1062 Her are both dwarfs, the spectral types for the third bodies can be estimated as M8–M9 for V574 Lyr and V1062 Her. The very low mass and luminosity of the third component may simply make the object so dim that its contribution to the total flux of the system is undetectable. For this case, it is difficult to find direct evidence to prove the existence of the third body through photometric and spectroscopic observations.

Figure 7.

Relationships between the mass of the third body and the orbital inclinations for V574 Lyr and V1062 Her.

V574 Lyr and V1062 Her are both late-type eclipsing binaries, with spectral types estimated to be K4 ± 2V and K2 ± 2V, respectively. At the same time, photometric studies show that there are active spots on the surface, so V574 Lyr and V1062 Her should have strong magnetic activities. The cyclic oscillation can also be explained by magnetic activity cycles. According to the Applegate mechanism (Applegate 1992), magnetic activity cycles can cause changes in the moment of inertia and quadrupole moment, resulting in orbital period changes. We used the formula (Applegate 1992):

\begin{eqnarray*} \Delta Q = -\frac{\Delta P}{P}\times \frac{Ma^{2}}{9}, \end{eqnarray*}

(13)

where M is the mass and a is the semi-major axis estimated by Kepler’s third law (M₁ + M₂ = 0.0134a³/P²), and listed the results in Table 10. The size of the periodic variation ΔP/P could be calculated by the equation |$\Delta P/P = A\times \sqrt{2[1-\cos (2\pi \times P/T)]}/P$| (Rovithis-Livaniou et al. 2000), where A is the amplitude of the sine function, and T is its period of magnetic cycle. Combining the absolute physical parameters listed in Table 10, we can calculate the variations required in the quadrupole moments V574 Lyr and V1062 Her:

\begin{eqnarray*} \mathrm{ V574\ Lyr:} \,\Delta Q_{1} = 2.60\times 10^{49}\, \mathrm{g\, cm}^{2},\ \Delta Q_{2} = 2.47\times 10^{49}\, \mathrm{g\, cm}^{2}, \end{eqnarray*}

\begin{eqnarray*} \mathrm{ V1062\ Her:} \,\Delta Q_{1} = 3.38\times 10^{49}\, \mathrm{g\, cm}^{2},\ \Delta Q_{2} = 8.13\times 10^{49}\, \mathrm{g\, cm}^{2}, \end{eqnarray*}

where ΔQ₁ and ΔQ₂ represent variations required in quadrupole moments of the primary and secondary stars, respectively. The variations in the quadrupole moments for V574 Lyr and V1062 Her are both lower than the typical values (10⁵¹–10⁵² g cm²) of active binaries (Lanza & Rodonò 1999). Therefore, we prefer the LITE by a third body as the mechanism for causing the cyclic variations in their orbital periods, although we cannot exclude magnetic activity mechanism.

5.2 Orbital parameters and light-curve variations

Using the W–D program, we obtained the orbital parameters of the four systems (see Table 10). This is the first time that such parameters were obtained for V574 Lyr and V1062 Her. With the less massive secondary slightly cooler than the primary, V574 Lyr could be classified as an A subtype W UMa system. V400 Lyr, V1033 Her and V1062 Her are typical W subtypes where the less massive secondary components have higher temperatures. We calculated the contact factor |$f({{\ \rm per\ cent}})$| for these four systems using f  = (Ω_in-Ω)/(Ω_in-Ω_out), and listed the results in Table 10. We found that|$f({{\ \rm per\ cent}})$| for these four systems are less than 30 per cent, indicating that they are in shallow contact. For V400 Lyr, the mass ratio of 3.70 is larger than 2.97 (Marino 2011), and our inclination (87.942°) is similar to the result (89.6°) of Marino (2011). The reason for this may be that they did not consider the star-spot effect. We believe that our results are more precise than the previous results (Marino 2011 ). For V1033 Her, the mass ratio of 3.90 is slightly larger than 3.489 (Bradstreet et al. 2009) and the inclination (84.275°) is similar to 88.40° by Bradstreet et al. (2009).

In this work, we found that the two maxima in the light curves of these four systems are not equal and show asymmetry. To study the change in the light curves over time, we calculated the difference between Max(I) (0.25 phase) and Max(II) (0.25 phase) and listed the results in Table 12. We can see Max(II) is lower than Max(I) for V400 Lyr and V574 Lyr, showing a positive O’Connell effect. For V1062 Her, we plotted the values of Max(I)–Max(II) versus HJD in Fig. 8. From Fig. 8, we found that all bands in HJD 245 8269 show Max(II) being larger than Max(I), which is in contrast to the results of our own photometric observations at earlier dates and the published light curve by Blättler & Diethelm (2000d). For V1033 Her, we found that our data are greater than zero, which is the opposite of Blättler & Diethelm (2000c). We explain these variations by a spot model. Since these binaries are solidly over-contact, it is safe to assume that their components are tidally locked and rotate synchronously with their orbital motion. The increase in longitude is then solely due to the migration of the spot across the stellar surface. For V400 Lyr, there is no variation for the longitude of a star-spot from 2017 July 30 to August 13. For V574 Lyr, the longitude of a star-spot was 88.9° on 2017 July 26, while it became 100.6° on 2017 August 26. For V1033 Her, the longitude of a star-spot was 39.4° in 2017 July. However, the longitude of a star-spot became about 47° in 2018 May. For V1062 Her, the longitude of a star-spot was about 100° in 2015, becoming about 80° in 2017. We plotted spot longitudes of the four systems in Fig. 9. From Fig. 9 and Table 11, we can see star-spot variability on short (about one month or even shorter) time-scales for V574 Lyr, and long (years) time-scales for V1033 Her and V1062 Her. However, due to a lack of observed data, it is difficult to further analyse magnetic activities.

Figure 8.

Magnitude difference Max(I)(0.25)–Max(II)(0.75) versus observing time (HJD) for V1062 Her. Each band is represented by a different symbol.

Figure 9.

Spot longitudes versus observing times (HJD). Each binary is represented by a different symbol.

Table 12.

Values of Max(I)(0.25)–Max(II)(0.75) for these four eclipsing binaries.

Star	HJD(24, +)	White band (mag)	B band (mag)	V band (mag)	R band (mag)	I band (mag)	Reference
	51801	0.010	–	–	–	–	[1]
V400 Lyr	57964	–	0.020	0.023	0.023	0.024	[5]
	57965	–	0.054	0.036	0.028	0.024	[5]
	57978	–	0.033	0.022	0.025	0.020	[5]
	51757	0.050	–	–	–	–	[2]
V574 Lyr	57960	–	0.062	0.059	0.049	0.038	[5]
	57981	–	0.034	0.048	0.051	0.037	[5]
	52056	−0.050	–	–	–	–	[3]
	57930	–	0.003	0.007	0.007	0.003	[5]
V1033 Her	57988	–	0.013	0.020	0.023	0.004	[5]
	58268	–	0.026	0.022	0.015	0.027	[5]
	58263	–	0.036	0.027	0.019	0.023	[5]
	51746	0.060	–	–	–	–	[4]
	57168	–	–	0.040	–	–	[5]
V1062 Her	57207	–	0.061	0.047	–	0.034	[5]
	57907	–	–	0.046	0.045	0.031	[5]
	57908	–	–	0.043	0.045	0.037	[5]
	58269	–	−0.002	−0.015	−0.015	−0.005	[5]

Note: References. [1] Blättler & Diethelm 2000a; [2] Blättler & Diethelm 2000b; [3] Blättler & Diethelm 2000d; [4] Blättler & Diethelm 2000c; [5] this work.

5.3 Chromospheric activity of V574 Lyr, V1033 Her and V1062 Her

We analysed our normalized V574 Lyr, 1033 Her and V1062 Her spectra using the spectral subtraction technique by the starmod program (Barden 1985; Montes et al. 1995). We downloaded the inactive-star spectra of spectral types G9 to K5 as templates from the LAMOST website. We constructed synthesized spectra using the inactive spectra. Finally, we found that the most suitable template star for V574 Lyr and V1033 Her is BD+09 1627, and that for V1062 Her is BD + 39 2723. We obtained the subtracted spectra by subtracting the synthesized spectra from the observed spectra, and plotted them in Fig. 1. From Fig. 1, we can see that the H α, H β and Ca ii H&K and IRT lines have weak emissions in the subtracted spectra. We calculated their equivalent widths (EWs) for V574 Lyr, V1033 Her and V1062 Her in the subtracted spectra for the H α, H β and Ca ii H&K lines, and listed them in Table 4.

The subtracted spectra of V574 Lyr and V1033 Her show that the H α and Ca ii H&K and IRT lines show weak emissions, and the H β line in V574 Lyr also exhibits a weak emission. The H α, H β and Ca ii H&K lines of V574 Lyr, V1033 Her and V1062 Her are greater than 0 Å, from which we can infer that V574 Lyr, V1033 Her and V1062 Her are active. Our photometric solutions for these four eclipsing binaries show star-spots on them, confirming that these four binaries are indeed active.

SUPPORTING INFORMATION

Table 3. |$B V R \& I$| photometric observational data for the four eclipsing binaries.

Table 5. Minimum light times and relevant parameters of V400 Lyr.

Table 6. Minimum light times and relevant parameters of V1033 Her.

Table 7. Minimum light times and relevant parameters of V574 Lyr.

Table 8. Minimum light times and relevant parameters of V1062 Her.

Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.

ACKNOWLEDGEMENTS

This work was supported by the Joint Research Fund (U1631236, U1431114 and U1631109) under cooperative agreement between the NSFC and CAS. The spectral data were obtained by the Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST). We would like to thank Dr Frank H. Levinson for his generous financial support that enabled Butler University to join SARA and upgrade the Holcomb telescope.

Footnotes

1

http://www.aavso.org/vsx/

2

http://dr5.lamost.org

3

iraf is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.

4

This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.