Photometric analysis and evolutionary stages of the contact binary V2790 Ori

To understand the evolutionary status of this contact binary, we first calculated masses of both components using correlations between mass, orbital period and mass ratio (Gazeas 2009). We find that M₁ = 0.348±0.038 M_⊙ and M₂ = 1.020±0.112 M_⊙ for the secondary and primary components, respectively. According to the orbital period increase with a rate of dP/dt = 1.03 × 10⁻⁷ d yr⁻¹, described in Section 3, a mass transfer rate from the less massive hotter component to the other one can be estimated using Equation (3) (Singh & Chaubey 1986; Pribulla 1998). The mass transfer rate is obtained to be dM₂/dt = 6.31 × 10⁻⁸ M_⊙ yr⁻¹.

$$\begin{eqnarray}&&\frac{\dot{P}}{P}=3\left(\frac{{M}_{2}}{{M}_{1}}-1\right)\frac{{\dot{M}}_{2}}{{M}_{2}}.\end{eqnarray} \tag{ 3 }$$

Some W-type contact systems with increasing orbital period were collected to compare the value of dP/dt as listed in Table 3. The period increasing rate for V2790 Ori is found to be a typical value with respect to other W-type systems.

The photometric parameters in Section 4 are used to calculate radii and luminosities of the components, and the semimajor axis of the orbit, which are derived to be R₁ = 0.613±0.018 R_⊙, R₂ = 0.991±0.029 R_⊙, L₁ = 0.396±0.016 L_⊙, L₂ = 0.892±0.036 L_⊙ and a = 2.036±0.059 R_⊙, respectively. Masses of both components in the study are slightly greater than values in Michaels (2016)'s work because of the different method of calculations. In this study, we applied the three-dimensional correlations of physical parameters provided by Gazeas (2009), while Michaels (2016) employed the mass-period relation supplied by Qian (2003). Accordingly, our values of R₁, R₂, L₁, L₂ and a are slightly greater than values in the previous work as listed in Table 4.

The mass–luminosity and mass–radius diagrams in Figure 4 are plotted to compare evolutionary status for both components of V2790 Ori with the zero age main sequence (ZAMS) and terminal age main sequence (TAMS), constructed by the Hurley et al. (2002) binary star evolution code. The other well-known contact systems were obtained from the catalog of Yakut & Eggleton (2005). It is found that the primary component of V2790 Ori is located near the ZAMS, similar to the primary stars of other W-type systems, implying that its evolutionary stage remains in the main sequence phase, while the secondary component of V2790 Ori lies above the TAMS, indicating that the secondary component has evolved to be oversized and overluminous. We calculated the mean densities of the components using the equations taken from Mochnacki (1981)

$$\begin{eqnarray}&&{\bar{\rho }}_{1}=\frac{0.079}{{V}_{1}(1+q){P}^{2}},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\bar{\rho }}_{2}=\frac{0.079q}{{V}_{2}(1+q){P}^{2}},\end{eqnarray} \tag{ 5 }$$

where the relative volumes of the components V₁ and V₂ are normalized to the semimajor axis, q is the mass ratio and P is the orbital period. We obtained ${\bar{\rho }}_{1}$ = 2.121±0.259 g cm⁻³ and ${\bar{\rho }}_{2}$ = 1.474±0.179 g cm⁻³, which are quite close to the values in Michaels (2016)'s work. The mean density of the secondary component, ${\bar{\rho }}_{1}$, is less than the theoretical value of the ZAMS star, confirming that the component has evolved away from the ZAMS to become oversized, while the mean density of the solar-mass primary star, ${\bar{\rho }}_{2}$, is nearly the same value as the Sun, meaning that the primary component is still a main sequence star.

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With the assumption that mass transfer starts, when the secondary (initially more massive) component has evolved to be near or after the TAMS, the initial masses of both components are computed from Equations (6)–(8) (Yildiz & Doğan 2013):

$$\begin{eqnarray}&&{M}_{{\rm{Si}}}={M}_{{\rm{S}}}+\Delta M,\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{M}_{{\rm{Pi}}}={M}_{{\rm{P}}}-(\Delta M-{M}_{{\rm{lost}}})={M}_{{\rm{P}}}-\Delta M(1-\gamma ),\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\Delta M=2.50{\left[{({L}_{{\rm{S}}}/1.49)}^{1/4.216}-{M}_{{\rm{S}}}-0.07\right]}^{0.64},\end{eqnarray} \tag{ 8 }$$

where M_Si and M_Pi are the initial masses, and M_S and M_P are the current masses of the secondary and primary stars, respectively. Δ M is the total mass lost by the secondary, M_lost is the mass lost by the binary and γ is the ratio of M_lost to Δ M. L_S is the luminosity of the secondary star. The derived value for the initial mass of the secondary star is 1.535±0.148 M_⊙ and the mass decrease of the secondary is Δ M = 1.188±0.110 M_⊙. However, for W-type contact systems, the value of the fitting parameter γ, given by Yildiz & Doğan (2013), must be in the range of 0.500 < γ < 0.664. Consequently, the corresponding initial mass of the primary star must be a value between 0.426 and 0.620 M_⊙, depending on the value of γ. The precise value of γ needs to be assigned in Equation (7). We applied the minimum, average and maximum values for γ of 0.500, 0.582 and 0.664 respectively for three cases to estimate some possible values of the initial mass of the primary star and total initial mass of the binary. The values of mass lost by the system and mass transfer between the components were finally obtained. The possible mass parameters for the three cases are listed in Table 5. As shown in Figure 5, the total initial mass of V2790 Ori for the case of γ = 0.664 lies closer to the relation between total present mass (M_T) and total initial mass (M_Ti) for W-type (Yıldız 2014) than the other cases. Thus, the appropriate value of γ for V2790 Ori should be 0.664. The initial mass of the primary component is obtained to be 0.620±0.075 M_⊙. From the detached phase until the present time, some 0.789±0.073 M_⊙ of mass has been lost from the system, while there could be a mass transfer of 0.399±0.037 M_⊙ from the secondary component to the primary component, after the time of FOF.

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In general, the orbital angular momentum can be calculated by using the following well known equation

$$\begin{eqnarray}&&{J}_{{\rm{o}}}=\frac{q}{{(1+q)}^{2}}\sqrt[3]{\frac{{G}^{2}}{2\pi }{M}_{{\rm{T}}}^{5}P},\end{eqnarray} \tag{ 9 }$$

where J_o is the orbital angular momentum, q is the mass ratio, M_T is the total mass of the binary and P is the orbital period. The value of the present angular momentum of the binary is computed to be log J_o = 51.42±0.06 cgs. Figure 6 shows a diagram of the orbital angular momentum versus total mass of contact and detached binaries, separated by the quadratic border line (Eker et al. 2006). The sample of detached systems was collected from the catalogs of Eker et al. (2006), Yildiz & Doğan (2013) and Lee (2015). The sample of contact systems was obtained from the works of Eker et al. (2006) and Ibanoğlu et al. (2006). Location of the binary V2790 Ori at the present time (filled triangle) is found to be below the border line, meaning that the present angular momentum of the binary is less than all detached systems with the same mass. This is consistent with angular momentum and/or mass loss in the past during the detached phase causing the binary to evolve into the contact phase.

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During the formation of the contact binary V2790 Ori, the AML must continue from detached to semi-detached and contact phases. The orbital period and semimajor axis decrease lead to the components being close together. The angular momentum at the FOF, J_FOF, can be calculated from Equation (9). According to a mass transfer process starting at the FOF and the lifetime in the detached phase for W-type typically being negligibly short (Yıldız 2014), we assumed that masses of both components were not much different from their initial values. Thus, for the beginning of the semi-detached phase, initial mass parameters from Table 5 were implemented in calculating J_FOF, while the orbital period and semimajor axis at the FOF were computed using Equations (10)–(11) (Yıldız 2014)

$$\begin{eqnarray}&&{P}_{{\rm{FOF}}}=0.1159\sqrt{\frac{{a}_{{\rm{FOF}}}^{3}}{{M}_{{\rm{Pi}}}+{M}_{{\rm{Si}}}}},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{a}_{{\rm{FOF}}}=\frac{0.6{q}_{{\rm{i}}}^{2/3}+{\rm{ln}}(1+{q}_{{\rm{i}}}^{1/3})}{0.49{q}_{{\rm{i}}}^{2/3}}{R}_{{\rm{TAMS}}},\end{eqnarray} \tag{ 11 }$$

where P_FOF and a_FOF are the orbital period and semimajor axis at the FOF, respectively. R_TAMS is the Roche lobe radius filled by the massive component that evolved to reach TAMS. P_FOF, a_FOF and J_FOF are computed to be 1.083±0.049 d, 5.731±0.090 R_⊙ and 9.39(±0.73) × 10⁵¹ cgs respectively. These results demonstrate that the angular momentum has decreased from 9.39 × 10⁵¹ cgs at the FOF to 2.62 × 10⁵¹ cgs at the present time, concurrently with a mass lost by the system of 0.789 M_⊙ as depicted in Figure 6. Consequently, the orbital period and semimajor axis have reduced from 1.083 d and 5.731 R_⊙ to 0.2878420 d and 2.036 R_⊙, respectively.

Initially, the binary V2790 Ori in the detached phase consisted of two main sequence stars. The more massive component (the progenitor of the secondary component) has evolved to TAMS, leading to the resulting oversized envelope. In combination with AML, the Roche surface was filled by the evolved secondary component, causing mass transfer to begin. From the FOF until the present time, the orbit has been reduced by AML and mass loss.