1 Introduction

Eclipsing binaries are ideal testbeds for a number of astrophysical studies. Based on high-quality radial velocity curves and photometric light curves of eclipsing binaries, one can determine fundamental mass and radius measurements for both components. According to the shape of the light curves, eclipsing binaries can be divided into three types: EA, EB and EW. EW type eclipsing binaries show continuous light variation, and the difference between the depths of the two minima is very small. Although such binaries have been analyzed a lot, a satisfactory theory for their origin, structures and evolution is not complete. More observations and studies for such systems are needed.

V441 Lac was first identified to be an eclipsing binary by Agerer (2001) when he started a photometric study of IU Lac. Agerer concluded that the light variability of V441 Lac shows an EW type and the orbital period is 0d.308894. After that, a lot of eclipsing times were determined by many investigators (e.g., Agerer & Hubscher 2002, 2003; Hubscher 2011). Recently, Wang et al. (2015) analyzed multiple color light curves and the orbital period changes of V441 Lac. They found that V441 Lac is a semi-detached binary with the less massive component filling the inner Roche lobe and that the orbital period of V441 Lac is secular increasing at a rate of 5.67 × 10−7 d/yr. In this paper, we present the BVR c I c light curves and the orbital period variation of V441 Lac.

2 Observations

CCD photometric observations of V441 Lac were carried out on August 28 and September 4, 2014 using the 1.0-m Cassegrain telescope at Weihai Observatory of Shandong University. The Andor DZ936 camera attached to the telescope was used during the observations. The effective field-of-view is about \(12^{\prime } \times 12^{\prime }\). The information of the 1.0-m Cassegrain telescope can be identified in Hu et al. (2014). The standard Johnson–Cousins–Bessel BVR c I c filters were used and the corresponding exposure times for each filter are 80, 60, 40 and 30 s, respectively. During the observations, the readout noise is 7, the gain is 2, and the readout time is about 2 s. The observed data were processed using the IMRED and APPHOT packages in IRAFFootnote 1 procedure, including bias and flat-field correction, and aperture photometry. The information of the comparison and check stars is listed in Table 1. The determined light curves with photometric errors in the four bands are shown in the upper panel of Fig. 1, and the differences between the comparison and the check stars are displayed in the lower panel of Fig. 1. The standard deviations of the differences between the comparison and the check stars are 0.0067 in B band, 0.0036 in V band, 0.0047 in R c band, and 0.0042 in I c band, respectively. The phases were calculated using the following equation:

$$\begin{array}{@{}rcl@{}} \text{Min.~I} = \text{HJD}2456905.08114 + 0.^{d} 30891501E. \end{array} $$
(1)

Two new times of light minimum were determined to be 2456898.13018 ±0.00028 and 2456905.08114 ±0.00049.

Figure 1
figure 1

The BVR c I c light curves of V441 Lac. The open circles represent the data determined on August 28, while the solid ones display the data determined on September 4. The photometric errors are also shown in this figure. The error bars are not clear because they are less than the size of the symbols.

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Table 1 Informations of V441 Lac, the comparison, and the check stars.
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3 Orbital period variation

The period variation of V441 Lac was first studied by Wang et al. (2015). As in Wang et al. (2015), the O − C times were firstly calculated by the linear ephemeris published by Agerer (2001). Using the least squares method, they determined a new linear ephemeris and constructed a new O − C diagram. A long term period increase at a rate of 5.67 × 10−7 d/yr was derived by them. By combining the times of light minimum in the literature and our two new determined ones, we analyzed the O − C behavior of V441 Lac. The same linear ephemeris taken from Wang et al. (2015) was used to calculated the O − C values. All eclipsing times and the O − C values are listed in Table 2, and the corresponding O − C diagram is displayed in the upper panel of Fig. 2.

Table 2 Times of minimum light for V441 Lac.
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Figure 2
figure 2

The O − C diagram of V441 Lac. The open circles represent the eclipsing times from literatures, while the solid ones display the eclipsing times given by us. The minima that have no errors were set as 0.0010.

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As seen in Fig. 2, the O − C values show an upward parabolic trend. Using the least squares method, we determined the following quadratic ephemeris:

$$ \text{Min.~I}=\text{HJD}2451817.5488(2)+0^{d}.308911589(5)E+2.484(3)\times10^{-10}E^{2}. $$
(2)

Based on the quadratic term of this ephemeris, a secular orbital period increase rate of dP/dt = 5.874(±0.007) × 10−7 d yr−1 was obtained. The residuals from the full ephemeris are displayed in the lower panel of Fig. 2, where no regularity can be found.

4 Light curve solutions

The light curve of V441 Lac has been analyzed by Wang et al. (2015). They determined that V441 Lac is a semi-detached binary with the less massive secondary component filling the inner Roche lobe and has a mass ratio of q = 0.15. We used the Wilson–Devinney (W–D) code (Wilson & Devinney 1971; Wilson 1990, 1994) to investigate our new derived four light curves simultaneously. W–D code is based on the Roche model and is a tool for the modeling of eclipsing binaries by real photometric and radial velocity data. Different sets of photometric data of the same object gives different parameters. The spots on the solution can affect the values of the parameters. Many investigations of binaries have confirmed this, such as the studies of GSC 03526-01995 (Liao et al. 2012), EP And (Lee et al. 2013), and EQ Tau (Li et al. 2014).

The effective temperature of primary component was fixed at T 1 = 7906 K, which is taken from Wang et al. (2015). According to von Zeipel (1924), the gravity-darkening coefficient and the bolometric albedo of the two components were set to be 1.0 for radiative atmospheres (T eff>7200 K). The logarithmic limb darkening law was adopted and limb darkening coefficients were taken from Van Hamme (1993) according to the filter and temperatures of the components. These parameters were kept constant during all solutions. We adjusted orbital inclination i, temperature of the secondary component T 2, the dimensionless potentials Ω1,2 and luminosity L 1 of the primary component during the solutions.

Since no spectroscopic mass ratio has been obtained for V441 Lac, a q-search method was used to determine the mass ratio, meaning that we calculated a series of models with assumed values of mass ratio q. During the solutions, the W–D code was started with mode 2 (detached configuration). We found that the solutions can be converged at both mode 4 (the more massive component filling its inner Roche lobe) and mode 5 (the less massive component filling its inner Roche lobe). The weighted sum of the squared residuals, \(\sum W_{i}(O-C)_{i}^{2}\), for mode 4 and mode 5 are displayed in Fig. 3. Solid circles represent mode 4, while open ones show mode 5. The minimum value of \(\sum W_{i}(O-C)_{i}^{2}\) is found at q = 0.10 of mode 5, meaning that solutions with mode 5 can achieve the best fit to the observations. According to the orbital period study, the orbital period of V441 Lac is continuously increasing, revealing a mass transfer from the less massive component to the more massive one. Mode 5 with the less massive component filling its inner Roche lobe is consistent with the result that was gained by the orbital period investigation. Solutions with mode 5 were performed by setting q = 0.10 as the initial value and an adjustable parameter. The derived photometric elements are listed in Table 3. As shown in Fig. 1, the two light maxima of V441 Lac have different light levels, indicating that the spot mode of the W–D program should be used in as many other active eclipsing binaries, AD Cnc (Qian et al. 2007), PY Vir (Zhu et al. 2013) and EQ Tau (Li et al. 2014). Iterative testing shows that two dark spots on the primary components with a mass ratio of q = 0.093±0.001 leads to the best fit. The corresponding solution results are listed in Table 3 and the spots parameters are shown in Table 4. The value of \(\sum W_{i}(O-C)_{i}^{2}\) for the solution with spots is much smaller than that without spot. Therefore, the solution with two spots is chosen as the final result. The corresponding synthetic light curves are displayed in Fig. 1 and the geometric structure is plotted in Fig. 4.

Figure 3
figure 3

\(\sum - q\) relation of V441 Lac. Solid circles represent mode 4, while open ones show mode 5.

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Figure 4
figure 4

The geometric structure of V441 Lac.

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Table 3 Photometric solutions for V441 Lac determined by analyzing the four color light curves simultaneously.
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Table 4 Spot parameters for V441 Lac.
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5 Results and conclusion

The newly observed BVR c I c light curves of the EW type eclipsing binary V441 Lac were presented. The four color light curves were simultaneously analyzed by the W–D program. The solution shows that V441 Lac is an extremely low mass ratio (q = 0.093) semi-detached binary with the less massive secondary component filling the inner Roche lobe. The primary component fills 94% of its inner Roche lobe. The asymmetric light curves are explained by two dark spots on the primary components. Our results are very different from that determined by Wang et al. (2015), such as the mass ratio, the orbital inclination and the effective temperature of the secondary component. The values of Max. I–Max. II, Min. I–Min. II, Min. I–Max. I, and Min. II–Max. II of V441 Lac are listed in Table 5. As shown in this table, the level of Max. I and Max. II is almost the same as in Wang et al. (2015). No spot was needed to fit their light curves. However, the values of Max. I–Max. II in B and V bands of our light curves are larger than 0.01. Spot modes should be used to fit our asymmetric light curves. The differences between Min. I and Min. II of Wang et al. (2015) is larger than that of our light curves, leading to the difference in the effective temperatures between the two components determined by Wang et al. (2015) which is bigger than that derived by us. In addition, the changes of the values of Max. I–Max. II, Min. I–Min. II, Min. I–Max. I and Min. II–Max. II indicate that V441 Lac shows strong magnetic activity.

Table 5 The magnitude differences of V441 Lac.
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The reasons that caused the difference of photometric elements between that of Wang et al. (2015) and ours are as follows: (1) the shape of the light curve is changed a lot, the depths of the primary minimum and the secondary minimum are different, and the light curves obtained by Wang et al. (2015) are almost symmetric whereas ours are asymmetric; (2) V441 Lac is a partially eclipsing binary, making the mass ratio determined only by the photometric light curves less confident (Terrell & Wilson 2005). We suggest that our results are more reliable. Firstly, our results are determined by four color light curves with higher precision,secondly, the orbital inclination derived by us is larger which reveals more reliable photometric elements due to the result of Terrell & Wilson (2005).

The mass of the primary component can be estimated to be 1.84M ⊙ based on the spectral type of A7 (determined by the effective temperature T 1 = 7906 K, Cox 2000) assuming that the primary component is a normal main sequence star. The mass of the secondary can be determined to be 0.17M ⊙ according to the mass ratio q = M 2/M 1 = 0.093. The other physical parameters of V441 Lac can be determined as follows: a = 2.43R ⊙, R 1 = 1.34R ⊙, R 2 = 0.53R ⊙, L 1 = 6.29L ⊙ and L 2 = 0.71L ⊙. Using the period–color relation determined by Wang (1994),

$$\begin{array}{@{}rcl@{}} (B-V)_{0}=0.077-1.003\log P ~(\text{day}). \end{array} $$
(3)

We can calculate that the color index of V441 Lac is (B−V)0 = 0.59. Then, by using the following equation taken from Rucinski & Duerbeck (1997),

$$\begin{array}{@{}rcl@{}} M_{v} =-4.44\log P + 3.02(B-V)_{0} + 0.12. \end{array} $$
(4)

We determined the absolute magnitude M V = 4.17 mag. Based on Cox (2000), the V band magnitude of V441 Lac can be estimated to be m V = 12.4 mag. Accordingly, the distance modulus is calculated to be m V −M V = 8.23 mag, and the corresponding distance of about 442.6 pc can be derived.

Based on all available times of light minimum, we analyzed the orbital period variation of V441 Lac, and long term increase at a rate of dP/dt = 5.874(±0.007) × 10−7 d yr−1 was derived. This result is very similar with that determined by Wang et al. (2015). Using the well-known equation

$$\begin{array}{@{}rcl@{}} {\dot{P}\over P}=-3\dot{M}_{1}\left( {1\over M_{1}}-{1\over M_{2}}\right), \end{array} $$
(5)

the mass transfer from the secondary component to the primary can be calculated to be dM 1/dt = 1.187(±0.001) × 10−7 M ⊙ yr−1. The positive sign reveals that the more massive primary component is the receiving mass. Table 6 shows some EW type binaries with low mass ratio (q < 0.2) and increasing orbital period. V441 Lac is very different from these systems. These systems have contact configurations, but V441 Lac is semi-detached. This makes V441 Lac a very interesting target.

Table 6 Some EW type with low mass ratio (q < 0.2) and increasing orbital period.
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In order to estimate the evolutionary state of V441 Lac, the two components of this system are displayed on the mass–luminosity diagram in Fig. 5, where the open circle represents the primary component, while the solid one refers to the secondary component. The solid and dotted lines show the Zero Age Main Sequence (ZAMS) and the Terminal Age Main Sequence (TAMS) lines constructed by the BSE Code (Hurley et al. 2002). As shown in Fig. 5, the more massive primary component is located below the ZAMS line, while the less massive secondary component is above the TAMS line, indicating that the primary is under-luminous and the secondary is over-luminous and over-sized. This is corresponding to Algol type eclipsing binaries.

Figure 5
figure 5

Mass–luminosity diagram of V441 Lac. The open circle represents the primary component, while the solid one shows the secondary component. The solid and dotted lines show the ZAMS and TAMS lines, constructed by the BSE code (Hurley et al. 2002).

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V441 Lac is an EW type eclipsing binary. Normally, EW type eclipsing binaries are contact binaries. But our solutions show that V441 Lac is a semi-detached Algol type binary. The formation and evolution of such a system is worth studying. A possible formation process for this system is: this object is originally a W-subtype shallow contact binary, and the less massive secondary component is transferring mass to the more massive primary; with the mass transferring, this binary will evolve to the more shallow contact state and expand and finally the contact configuration destroys as predicted by the Thermal Relaxation Oscillation (TRO) models (e.g., Lucy 1976; Flannery 1976; Robertson & Eggleton 1977; Rahunen 1981). As predicted by the TRO theory, close binaries undergo oscillations around the state of marginal contact. Each oscillation contains a contact and a semi-detached phase. V441 Lac is undergoing the semi-detached phase. This system is a very good target to prove the TRO theory. In future, spectroscopic and photometric observations are needed to confirm the mass ratio and to determine the orbital evolution and the possibility of third components.