THE FIRST PHOTOMETRIC STUDY AND ORBITAL SOLUTION/PERIOD ANALYSIS OF THE MISCLASSIFIED BINARY SYSTEM V380 CAS

V0380 Cas (= TYC 4307–1121-1, SV* BV 329) was discovered to be a variable star (α_J2000 R.A.: 00^h 30^m56^s decl.: +73°40'06'') by Bauernfeind in 1899 who reported its first minimum (Table 4) based on plates of the Bamberg Observatory and its eclipsing behavior was studied by Strohmeier & Knigge (1961) and Strohmeier & Bauernfeind (1968). It was identified as having an EA-type light curve with a deep primary eclipse but without a recognizable secondary minimum with a period of 1.357270 days. The spectral type of the primary star was classified as A0 by Meinunger (1965). Orbital elements for V0380 Cas were first estimated by Brancewicz & Dworak (1980) using the Bamberg period. In the General Catalogue of Variable Stars (GCVS), the total photographic magnitude of the system is about 10.4–11.2 mag (Samus et al. 2011) and in the catalogs of eclipsing variables (Budding 1984; Malkov et al. 2006) the reported depth of the primary eclipse is 0.7–0.8 mag with a duration of 0.2 phase and duration of totality 0.08 phase. Nevertheless, its status is characterized against a normal semi-detached classification (sd = 0.3) and it "likely [has] an erroneous occultation assumption."

Although times of minimum light have been published by numerous researchers, we could not find any published photometric light curve of the system in the literature, except a reference to the fact that "the secondary is not yet observed." We show below that the original orbital period is entirely wrong beginning in the first decimal place, leaving open the question of whether the system contains a very shallow secondary minimum or an identical primary and secondary minimum. In the first case, the system must be very evolved like Algol itself and in the second the components of the system should be very similar in mass, radius, and luminosity. In this paper, we present the first extensive monitoring of V380 Cas in B, V, R_c, I_c passbands and the first analysis of the O − C diagram in order to obtain improved ephemeris and clarify its nature.

First time CCD photometric observations were obtained with the 35.5 cm f/6.3 Schmidt–Cassegrain telescope at the University of Patras Observatory. This telescope was equipped with an SBIG ST-10 XME CCD camera which has 2184 × 1472 pixels. The size of each pixel is 0.57'', resulting in an effective field of view of 14 × 20 arcmin². The standard Johnson–Cousin–Bessel set of BVR_cI_c filters was mounted. The FWHM of the stellar image varied from 2.5 to 5 pixels during the observations. Beginning on 2009 July, the observations of the first season were made for five successive nights with duration of 6 hr each in order to verify the reported period of the system and continued for 13 more nights from 2009 July to November. The observations of the second season were made on 16 nights from 2010 July to August. The exposure times were 40–130 s for the B band, 20–130 s for the V band, 15–90 s for the R_c band, and 40–140 s for the I_c band and either the 2 × 2 or 1 × 1 mode was selected depending on weather conditions.

The reduction method has been described by Christopoulou et al. (2011). Selecting a proper comparison star has turned out to be quite a difficult task. The field of V380 Cas is not very dense and all of the closer stars are too faint or have spectral types different from the variable itself. GSC 04307: 00632 (α_J2000 = 00^h31^m38^s, δ_J2000 = +73°44'0402, V_mag = 9.73) and GSC 04307: 00699 (α_J2000 = 00^h31^m16^s, δ_J2000 = +73°45'31'', V_mag = 11.5) were used as comparison and check star, respectively, for the observations of 2009, but changed to GSC 04307: 00699 (α_J2000 = 00^h31^m16^s, δ_J2000 = +73°45'31', V_mag = 11.5) and GSC 04307: 00837 (α_J2000 = 00^h31^m02^s, δ_J2000 = +73°45'40'', V_mag = 12.6) as comparison and check star, respectively, for the photometry after the observations of 2010. The differential atmospheric corrections extinction among the three stars were ignored, since the target star is very close to the comparison and the observations have never been obtained below 30° above horizon. The 1σ values of the dispersions of the magnitude differences between them are about ±0.01 mag for all bandpasses.

A total of 5468 individual observations were obtained among the four bandpasses (2090 in B, 1742 in V, 725 in R_c, and 911 in I_c) and a sample of them in the form of heliocentric Julian dates (HJD) versus magnitude differences between the variable star and the comparison (Δm) is listed in Table 1. Times of minimum light were calculated with the software Minima25c (Nelson 2005) using K–W (Kwee & van Woerden 1956). From our new CCD observations we obtained 11 new times of minimum light listed in Table 2 with their errors. During 2009 July, we monitored the system for five successive nights in order to find the first minimum and verify the period given in the ephemeris of Kreiner (2004), whereas during the second season our goal was to cover the full light curve of V380 Cas. The complete light BVR_cI_c curves of the system are plotted in Figure 1 as magnitude difference Δm versus phase. The phases were calculated using our first minimum and the period of Kreiner (2004)

$$\begin{equation} {\rm Min}\;I = 2455038.544360 + 1\fd 3572698 \times E. \end{equation} \tag{ 1 }$$

As shown in Figure 1 the greater part of the phase is fully covered, sometimes twice.

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As shown in Figure 1, our observations clearly indicate that the light curve morphology of V380 Cas is characterized by the complete absence of secondary minimum. To understand the geometrical structure and the physical parameters of the system, our BVR_cI_c curves were solved simultaneously by using the PHOEBE program (Prša & Zwitter 2005) based on the Wilson–Devinney algorithm (Wilson & Devinney 1971). At first we used the mode where the "semi-detached binary secondary component fills its Roche Lobe" (in Wilson–Devinney mode 5) and tried to simulate the set of parameters of Brancewicz & Dworak (1980) or Malkov et al. (2006). This set has an A0 primary star with effective temperature T₁ = 9600 K and a mass ratio q = 0.68. Second, the photometric mass ratio was investigated via the q-search method from a value of 0.1–1.0 with a step of 0.1. This approach was applied in mode 2 (detached system mode) and mode 5 in order to find the most suitable value of the photometric mass ratio q. As the dimensionless potential value of the secondary component reached its limit we concluded that Mode 5 is the configuration of the system. The χ² value reached its minimum for the semi-detached configuration for q = 0.65 which is in agreement with the ratio used by Brancewicz & Dworak (1980). Due to the absence of secondary minimum the initial value for the mean temperature of star 2 was set to T₂ = 3250 K. The corresponding linear limb-darkening coefficients, x_bol = 0.6694, x₂ = 0.4882, y_bol = 0.0732, y_bol = 0.2732, were the interpolated logarithmic law from the van Hamme (1993) tables. Following Lucy (1967) and Rucinski (1973) the gravity-darkening exponent g₁ = 1 and the bolometric albedo A₁ = 1.0 were assumed for the early-type primary component with a radiative envelope, whereas the values of g₂ = 0.32 and A₂ = 0.5 were used for the late-type secondary component with a convective envelope.

As it turned out, this model did not give an acceptable fit to the observed light curves since both eclipse depths were wrong (not shown). The best result is listed in Column 2 of Table 3 (Model 1) and in Figure 1 (solid line) where we had to enter the input value of A₂ = 0.1. Even in this case, the use of model parameters of Brancewicz & Dworak (1980) as initial parameters gave a poor fit to the observed light curves since the model's primary minimum was too shallow and its secondary minimum too deep (not shown). The same defects are seen in the light curves of the other bands. In addition problems arose in the representation of the out-of-eclipse light variation. In order to improve the fitting we included a hot spot on the surface of the primary as the result of the impact of the gas stream from the cooler, less massive secondary star. The spot parameters are listed in Table 3. Separate trials for a cool spot on the secondary star were not as successful as for the hot spot model. In all these trials a possible third light source (l₃) was considered, but its contribution remained negligible within its error so we fixed l₃ = 0 during the final light curve analysis. In a formal sense, as shown by ΣW(O − C)², the hot spot model does improve the light curve fit in almost all bands. Nevertheless, it is considered too sophisticated due to its complexity (A₂ = 0.1). The main reason was that close inspection of BVR_cI_c light curves around phase 0.5 shows no brightness decrease as expected during the occultation of the secondary star, but rather a conspicuous increase. This observational hint led us to double the period given by Brancewicz & Dworak (1980). In this way, the increase of light is shifted around phase 0.25 as expected from the largest separation between the components of the binary system.

This is common in EBs with only one detectable eclipse which can be modeled in two alternative ways as suggested by Devor et al. (2008) in the analysis of the data from the Trans-atlantic Exoplanet Survey (TrES) near the field of Cassiopeia (R.A.: 00^h39^m09.8941^s, decl.: +49°21'16.519 in radius 30''). One way is to assume very unequal stellar components that have a very shallow undetected secondary eclipse, assuming that they have circular orbits. The other way is to assume that the period is twice the correct value and that the components are nearly equal. Resolving the ambiguity may not always be possible without spectroscopic data, but in some cases we are able to do so, using either a morphological—as in this case—or a physical approach.

The doubling of the initial reported period was also the case for WY Hya (Struve 1950), BS Dra (Popper 1971), AY Cam (Williamon et al. 2004), V505 Per (Demircan et al. 1997), BK Peg (Popper & Dumont 1977), and WZ Oph (Clausen et al. 2008). In most of these cases, the reported period was based on the photographic minima, but the photoelectric and/or spectrographic study showed that this period must be doubled or be half (V396 Cas; Lacy et al. 2004).

The new period changed the light curves completely and the new phases were computed according to the ephemeris described in Equation (2). As shown in Figure 2, although there are some gaps, the new light curves show no asymmetries larger than the scatter in the data. The eclipses are clearly partial (i.e., d = 0.012 P) and the duration of the eclipses is about 0.12 P. The eclipses are so nearly equal in depth that the choice of primary component is very difficult in all filters.

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To understand the geometrical structure and the physical parameters of the system the new light curves were solved simultaneously using PHOEBE and an extensive q-search procedure. The surface temperature of the primary star was fixed at 9790 K according to its spectral type A0 from Cox (2000). For our solution, Mode 2 (for detached stars) was chosen based on the general appearance of the light curves. The q-search of PHOEBE converged and showed acceptable photometric solutions for detached configuration near q_phot = 1.6. The corresponding bolometric coefficients x₁ = 0.6694, x₂ = 0.66868, y₁ = 0.0732, and y₂ = 0.07352 were interpolated for the logarithmic law from the Van Hamme (1993) tables. Following Lucy (1967) and Rucinski (1973) the gravity-darkening exponent g₁ = g₂ = 1 and the bolometric albedo A₁ = A₂ = 1.0 were assumed for both components with a radiative envelope. With the initial parameters (T₁, q_phot, i), we solved the light curves simultaneously until the solution converged. Finally, we included the q_phot as an adjustable parameter and after some differential corrections the solution gave the final mass ratio of q_phot = 1.604 ± 0.013. Solution parameters including standard errors of adjusted parameters are presented in the third column of Table 3 (Model 2). The theoretical light curves computed with these parameters are plotted in Figure 2 as solid lines. The fill-out factors for both components imply that V380 Cas is a detached system. Circular orbit and synchronous rotation were assumed. Third light was tested for and found to be insignificant in accordance with the O − C analysis. Non-zero eccentricity was also tested, but the value was too low and it was ignored.

We have to note that since we did not have any information about which component (if either) had the greater surface brightness and should be termed the "primary," phase zero was set arbitrarily, for the ephemeris adopted from Equation (2), to the epoch of minimum light at which the more massive, more luminous, but not necessarily hotter, star is eclipsed. However, our analysis revealed that the primary star eclipsed at our phase zero has the greater surface brightness and this star is the hotter, larger, less massive, more luminous one. Thus, we had to shift the light curve by 0.5P to obtain an agreement between the synthetic light curve produced by PHOEBE and observations. Despite the equality of surface brightness of the two components, our analysis showed that the two components can differ significantly in mass and luminosity. Therefore, in Table 3, Model 2, the star considered the secondary has the greater surface flux and it is more massive, larger, and more luminous. Our solution shows that the two components are very similar and the distinction between transit and occultation for the primary minimum is just a matter of formal choice.

Nevertheless, since the mass ratio is not based on spectroscopic data and for a detached binary the mass ratio cannot be determined from photometry alone, we have generated a model with q = 1.09 because this value minimized the q-ratio in individual filters and especially in the I_c passband without deviating too far from the initial parameters of Model 2. Solution parameters for this model, including standard errors of adjusted parameters, are presented in the fourth column of Table 3 along with its error (Model 3). In order to discriminate between the two candidate models we have to take into account the error reported between the observed and the synthesized light curves. There are about 1742 data points in the light curve of the V band and the standard deviation of the comp-check Δm is about 0.01 mag. Hence, a "perfect" light curve fit would be expected to give a chi-square of about 1742 × (0.01)² = 0.17. From our results both solutions (Model 2 and Model 3) are acceptable.

Throughout Table 3 the subscripts p and s represent a primary (eclipsed at phase zero) and a secondary component, respectively. Additionally, we can calculate the fill-out factor (f_p = 〈r〉/r_lobe, which is the ratio of the mean radii to the volume radius of the Roche lobe (Eggleton 1983).

In order to investigate the variations of orbital period of V380 Cas we compiled all available light minimum timings up to date, from 1899 to 2011, that together with our 12 new CCD observations consist of a total of 167 (15 visual, 127 photoelectric, and 25 CCD) times of minimum light which are listed in the first column of Table 4. The O − C values of all times of minimum light are computed with the following ephemeris:

$$\begin{equation} {\rm Min}\;I = 2455038.544360 + 2.\!\!^{\rm d}7145396 \times E \end{equation} \tag{ 2 }$$

using our first minimum and the double period and are listed in the third column of Table 4. The corresponding O − C diagram in Figure 3 shows that the data are distributed around a straight line, but the photographic and visual data show a large scatter from the straight line due to their low quality. A linear ephemeris was used to fit the O − C values by using only the reliable CCD and photoelectric minima. A least-squares fit to the data gave the first precession ephemeris that is plotted in Figure 3 with a solid line:

$$\begin{equation} {\rm Min}\;I = 2455410.43823 + 2.\!\!^{\rm d}714539884 \times E. \end{equation} \tag{ 3 }$$

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Since we have to take into account the shift of 0.5 phase as suggested by our photometric solution, we have to add half a period to the above HJD_o. There is neither any indication of any period change dP/dt nor any indication of a sinusoidal oscillation that may indicate the presence of a third companion.

5.1. Parameters

V380 Cas appears to be a well-detached system with the more and less massive stars filling 65.5% and 43% (Model 2) or 59.2% and 47.4% (Model 3) of their limiting Roche lobes, respectively. Since the spectroscopic data for V380 Cas are deficient, it is necessary to adopt an approximate method to infer the physical parameters of the system.

With the continuous development of observational techniques, more and more high-precision mass data and multiple-band luminosity data are determined, and we tried to obtain our own empirical mass–luminosity (ML) relation. Nevertheless, a reliable empirical ML relation must be obtained from a sample of main-sequence component stars in detached systems. By searching the literature as completely as possible, we have chosen the published bibliographic catalog of detached MS double-lined eclipsing binaries (DLEBs) by Malkov (2007) containing all available data on the observed DLEBs. The catalog contains the values of masses, radii, effective temperatures, and bolometric luminosities for 217 components, along with their uncertainties. By fitting linearly this catalog's observational data in the effective temperature–luminosity plot (log L on log m) we obtained β = 3.87 ± 0.03. The larger value of β comes from the change of slope of the ML relation at about m = 2.31 M_☉ as noted by Fang & Yan-Ning (2010). We have attempted to refine our ML relation with the more recent compilation of Torres et al. (2010) which contains 190 stars (94 eclipsing systems, and a Centauri) of 95 detached systems for which the mass and radius of both stars are known to ±3% or better. All are non-interacting systems, so the stars should have evolved as if they were single. Figure 4 represents this catalog's observational data in the log L–log m plot, excluding the giant stars (TZ For star A, Al Phe star A, and OGLE 051019). By fitting Figure 4 we obtain β = 3.95 ± 0.03. We investigate the following possibilities.

• 1.  
  Both components of V380 Cas belong to main sequence. Solving the system's equations that describe the geometric and physical parameters of the components of an eclipsing binary relatively to log m₁ we obtain

  $$\begin{eqnarray} \log m_1 &=& 0.203\,[2\log P + 6\log T_1 + 3\log r_1 \nonumber\\ &&+ \log (1 + q) + 1.8722], \end{eqnarray} \tag{ 4 }$$

  where 1 refers to the massive component, m₁ (M_☉), T₁ (T_☉), and r₁ are the mass, effective temperature, and relative radius obtained from the light curve solution, respectively, P is the period in days, and q is the mass ratio.Thus, for Model 2 we obtain m_p = 1.85 M_☉ and m_s = 2.97 M_☉ (from the mass ratio) and relative radii r_p = 2.49 R_☉ and r_s = 3.04 R_☉, respectively. The derived physical parameters of the system are listed in Table 5.
• 2.  
  If only one component belongs to the main sequence and the second to the other luminosity class. If the main-sequence component is the more massive one, then the solution is the same as 1. If the less massive component belongs to the main sequence then we obtain from Equation (4), by putting 1/q instead of q, for Model 2, m_p = 2.92 M_☉ and consequently m_s = 4.69 M_☉. The corresponding radii are R_p = 2.89 R_☉ and R_s = 3.54 R_☉, respectively.
• 3.  
  If both components belong to the main sequence, for Model 3 we obtain m_p = 2.81 M_☉ and consequently m_s = 3.09 M_☉. The corresponding radii are R_p = 2.72 R_☉ and R_s = 3.30 R_☉. Solutions with either star as the primary component produce nearly the same fit error so we could not discriminate between these.

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Table 5. Astrophysical Parameters of V380 Cas

Parameters		Model 2		Model 3
		Primary	Primary	Primary
		Secondary	Secondary	Secondary
	Cox 2000	CASE A	CASE B
		q = 1.6	q = 1.6	q = 1.09
Sp Type	A0V
M (M☉)	2.90	1.85 ± 0.06	2.92 ± 0.06	2.81 ± 0.09
		2.97 ± 0.1	4.69 ± 0.01	3.09 ± 0.01
R (R☉)	2.40	2.49 ± 0.012	2.89 ± 0.01	2.72 ± 0.01
		3.04 ± 0.015	3.54 ± 0.02	3.30 ± 0.02
log g (cgs)	4.139	3.645 ± 0.035	3.892 ± 0.024	3.891 ± 0.034
		3.944 ± 0.035	4.011 ± 0.025	3.975 ± 0.034
T (K)	9790	9790	9790	9790
		9716 ± 6.6	9716	9778 ± 6.7
L (L☉)	48	51.06 ± 1.02	69.2 ± 1.4	60.12 ± 1.22
		73.99 ±1.48	100.28 ± 2.01	87.86 ± 1.8
Mbol (mag)	0.35	0.48 ± 0.02	0.16 ± 0.02	0.29 ± 0.02
		0.08 ± 0.02	−0.25 ± 0.02	−0.13 ± 0.02
Mv (mag)	0.65	0.78	0.46	0.59
		0.38	0.05	0.17
Mvtotal (mag)		−0.19	−0.52	−0.39
Distance (pc)		1032 ± 43	1176 ± 50	1110 ± 47

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In the next step, we can calculate the bolometric magnitudes and the luminosities of both components using the well-known equations (Equations (5) and (6)) from the Stefan–Boltzman law,

$$\begin{equation} M_{{\rm bol - p}{\rm.s}} = 4.\!\!^m75 - 5\log (R_{{\rm p}{\rm.s}} /R_ \odot) - 10\log (T_{{\rm p}{\rm.s}} /T_ \odot) \end{equation} \tag{ 5 }$$

$$\begin{equation} L_{{\rm p}{\rm.s}} = (R_{{\rm p}{\rm.s}} /R_ \odot)^2 (T_{{\rm p}{\rm.s}} /T_ \odot)^4. \end{equation} \tag{ 6 }$$

The results for all cases are listed in Table 5 along with the estimate of the surface gravity. In the process of computing the parameters of the system we have adopted the nominal parameters of T_☉ = 5778.57 K, M_bol☉ = 4.75 mag, as suggested by Harmanec & Prša (2011) in order to reduce the systematic errors induced in the calculations. Assuming both stars are of spectral type A0V and using the table of the main-sequence properties given in Allen's astrophysical quantities (Cox 2000), from the bolometric correction BC = −03, the absolute magnitudes of the components can be estimated.

It should be noted that we obtained the above parameters under the assumption that the components of any given eclipsing system behave as single, non-rotating stars and we neglect the effects of limb darkening, reflection, and mass exchange. With these assumptions and β = 3.95 in the mass–luminosity relation we calculate the parameters for a star being a component of an eclipsing binary system and for that reason we need only 5% intrinsic accuracy in the solution of basic equations.

5.2. Evaluation of Distance

Using the intrinsic color (B − V)₀ = −0.02 for A0 spectral type (Cox 2000) and the observed Johnson colors (B − V)_J = 0.228 (ESA 1997) or 0.293 (Kharchenko & Roeser 2009) we can obtain the color excess as E(B − V) = (B − V) − (B − V)₀ = 0.208 or 0.273 and can find the interstellar extinction in the V filter as A_V = 3.1 × E(B − V) = 0.69 or 0.76. With those values and with the values of V_J = 10.56 mag from ESA (1997), M_Vtotal from Table 5, and the interstellar extinction, we can estimate the distance to the system from Equation (7):

$$\begin{equation} d({\rm pc}) = 10^{((V - A_V - M_{{\rm total},V} + 5)/5}. \end{equation} \tag{ 7 }$$

The calculated distance for all cases are listed in Table 5 and although it seems to be in agreement with the reported value by Brancewicz & Dworak (1980) as d = 909 ± 142 pc, it cannot be compared as the latter is based on the parallax from the Dworak (1975) catalog, where it was estimated from absolute magnitudes of their components M₁ = 1.3 and M₂ = 1.4. These values are completely different from the M₁ and M₂ values obtained by us and listed in Table 5.

Additionally, it should be noted that the above estimation is indicative of the mean system color index and interstellar reddening can be in error by as much as +5% but also, because we do not know the phase at which the measurements were made, we cannot be sure that they refer to the maximum light. For this reason the error in distance determination might be as much as 20%–30%. Using the results of Table 4 we obtain the fundamentals parameters listed in Table 5 along with those from the properties of zero-age main-sequence (ZAMS) stars from Cox (2000). As can be seen from Table 5, in all cases, both stars are overluminous for the (solar abundant) ZAMS (Cox 2000) by factors 1.1–2, respectively, as in the case of V364 Cas (Nelson 2009).

5.3. Evolutionary Status

In many cases, when the two minima in the light curve are very nearly equal in depth and one might expect the components to be similar in their properties, the photometric analyses agree with the spectrographic evidence that the two stars of nearly equal surface temperature differ appreciably in mass, radius, and luminosity with the more massive component apparently well evolved while the less massive is not (BS Dra, BK Peg; Popper & Etzel 1981).

Due to the absence of radial velocity curves the resulting absolute parameters in Table 5 should not be final. However, they should define a first picture of the system. It is interesting to see in this picture that V380 Cas is formed with two components with different masses (Model 2) or with very identical masses (Model 3) which should be main-sequence stars. In order to understand the evolutionary status of the components using the adopted temperatures and masses of Table 5 we plot the preliminary dimensions of Models 2 and 3 on the log m − log L diagram presented in Figure 4 together with the catalog of detached main-sequence stars by Torres et al. (2010) and ZAMS (Girardi et al. 2000). As can be seen from Figure 4, the problem with both solutions of Model 2 is that in the first case the lower mass star seems to be more evolved and in the second case the more massive star seems to be below ZAMS. Even in the log T − log L diagram (not presented here), together with the evolutionary tracks of single stars having masses 3 M_☉ and 1.8 M_☉ or 5 M_☉ and 3 M_☉ (Hurley et al. 2002) and Z = 0.0192 (solar metallicity), both stars lie in the main-sequence band between ZAMS and TAMS lines, but they are not located at the evolutionary tracks expected for their mass. For the solar-abundant models, the primary is larger and brighter, but the secondary and more massive one is dimmer than expected from its mass.

According to Malkov (2003), A- and F-type main-sequence eclipsing binaries have larger radii and/or higher temperatures than single stars. It is because of this that we have reconstructed the same diagram in Figure 5 with the ZAMS and TAMS lines constructed from the evolutionary code of Hurley et al. (2002), but the picture was the same. According to Torres et al. (2010) and their mass–radius diagram (their Figure 2), the large range in radius for a given mass clearly shows the effect of stellar evolution up to the main-sequence band, which in this diagram moves a star up along a vertical line as it evolves, if no significant mass loss occurs. Thus, any point below the theoretical ZAMS curve would be interpreted as that star having a lower metallicity than the models and a binary with a mass ratio sufficiently different from unity can constrain the range in Z of acceptable models. A match to a theoretical model can often be obtained by adjusting the metal and/or helium abundance (equal for the two stars) and/or the mixing-length parameter.

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From the above analysis it is obvious that the initially assumed main-sequence character of the detached system V0380 Cas with q = 1.6 has no meaning physically. Thus, the most likely picture of the system is that of Model 3.

Figure 6 shows the location of the two stars of Model 3 in a theoretical HR diagram that contains the evolutionary tracks for 2.55, 2.75, and 3 M_☉ computed by Hurley et al. (2002) for stars with solar chemical composition. Although we did not put any error bars in Figure 6, we note that the errors are larger than those listed in Table 5. Although in the light solution the mean surface temperature of the primary component is not an adjusted parameter, we can obtain a subjective estimate for the uncertainty in log L by assigning an error of 100 K in effective temperature. Figure 6 also shows the 0.2, 0.316, 0.355, and 0.4 Gyr isochrones. Both stars seem to be located close to the evolutionary tracks expected for their mass and between the 0.315 and 0.355 Gyr isochrones. Therefore, we estimate the age of V380 Cas to be approximately 0.34 Gyr.

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5.4. The log J_o–log P and log M–log P Diagrams

Because orbital angular momentum (J_orb) and mass (M) are basic physical quantities determining orbital size (α) and period (P), the log J_orb–log M diagram is a natural choice to study dynamical evolution of binary orbits. The expression for orbital angular momentum is

$$\begin{equation} J_{{\rm orb}} = 2.525 \times 10^{52} \;M^{5/3} \;P^{1/3} \;q(1 + q)^{ - 2} \end{equation} \tag{ 8 }$$

with J_orb in cgs units where M = M₁ + M₂. The sample of our study is made again by the detached main-sequence stars listed in the catalog of Torres et al. (2010; calculated parameters not presented here) together with 61 semi-detached systems of Ibanoğlu et al. (2006; their Table 2). The plot of log J_o − log M is displayed in Figure 7, respectively, where the position of Model 3 is marked. We neglect the spin angular momentum of the components because it is smaller compared to the total orbital angular momentum of the system. We have to point out that during the construction of Figure 7 we noticed that 45 of the 74 detached Algol systems included by Ibanoğlu et al. (2006) are among the detached main-sequence stars listed in the catalog of Torres et al. (2010). Nevertheless these systems are selected by Torres et al. (2010) because they fulfill the first requirement that excludes Algols ("systems with past or ongoing mass exchange").

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As shown in Figure 7 marking several positions on the border line between detached and semi-detached stars, the following linear equation was produced:

$$\begin{equation} \log \,J_{{\rm lim}} = 1.7723\;\log \,M + 51.6, \end{equation} \tag{ 9 }$$

where M is the total mass in solar units and J_lim is in cgs. The physical significance of this line is that for the detached systems marks a border in their evolution from the detached stage to the semi-detached by losing mass and orbital angular momentum, but for V380 Cas the moderate fill-out factors (i.e., f_p,s < 60%) indicate that the binary is far from such an evolution.

The BVR_cI_c light curves of the system V380 Cas are presented for the first time along with the correction of its period. Although the mass ratio was fixed to the value adopted from the relevant q-search method, the light curves do not discriminate between two candidate model parameters for this system so the discrimination is based on the evolutionary status of the components.

The influence of the photometric mass ratio was incorporated in estimating the final parameter errors. For this we made trial runs using the highest and lowest possible values and we adopted the highest error for the parameter values listed in Table 3. But still the real parameter uncertainties may be greater than those listed since a significant contribution to the errors may come from the uncertainty of the effective temperature of the hotter star which was fixed on the basis of its spectral type. Although the mass ratio of the system was set according to the photometric solution, we know that there is a discrepancy from the spectroscopic mass ratio, especially for binaries undergoing partial eclipses and for this we adopted two models with different mass ratios. The parameters of the system are only known with low precision based on various assumptions. Despite this, our first time analysis throws new light into this interesting system.

The system V380 Cas turned out to be a detached system with moderately evolved main-sequence components and represents another example of the ambiguity of purely photometric data even from four-color light curves. Although we did not find any tidal dissipation between the components, there are not enough reliable minima to investigate further the possible variation of its period. High-resolution spectroscopy will determine the astrophysical parameters and evolutionary status of the system better than is possible with photometry alone.

Additionally, it is necessary to observe similarly classified binary systems with unobservable secondaries (e.g., RZ Aur) by photometry and spectroscopy in order to examine the validity of older proposed models.

The authors thank an anonymous referee for valuable comments and suggestions which helped us improve the paper.