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3 Photometric solution

Photometric solutions of BH Vir were obtained by using the 1992 version of the Wilson-Devinney program (Wilson 1992). The individual observations were combined into 64 normal points in each color; the number of individual observations in each normal point was taken as the weight of each point. The two light curves were used simultaneously in deriving our solutions.

Comparing our observations with previous light curves of BH Vir, we can easily see that only a small decrease in the system brightness occurred around phase 0.65 where the systematic deviations by 0.023 mag (V) and 0.022 mag (B) are slightly greater than the intrinsic errors in 1991 although the light curves of the system in 1991 have the missing phases in the beginning of the secondary minimum. The basic astronomical and physical properties of BH Vir were summarized by Zeilik et al. (1990). The spectral classification of Abt (1965) is in agreement with that of Koch (1967) as for the secondary component (G2V) but there is a difference of about two sub-types in the case of the primary (G0V vs. F8IV-V). From our observations we can determine that the colour index (B-V) of BH Vir during primary eclipse is about $0 \hbox{$.\!\!^{\rm m}$ }65$. This value is in good agreement with the finding of Koch (1967) and Scaltriti et al. (1985). Therefore we adopted a temperature for star 2 (star eclipsed at Min II) of 5500 K. The other adopted parameters were: gravity-darkening exponents, g1, g2 = 0.32, the bolometric albedo of two components, A1, A2= 0.5. The linear limb darkening laws were used and the source of the values given by Al-Naimiy (1978). The reflection effect was computed with the detailed model of Wilson (1990). As the spectroscopic mass ratio was found to be q=m2 / m1 = 1.02 by Abt (1965), and q=0.968 by Zhai et al. (1990), we assumed a series of q values (0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2) to find the preliminary photometry mass ratio with the grid method. The adjustable parameters employed were: the orbitral inclination (i), the dimensionless potentials of star 1 and star 2, ($\Omega_1$ and $\Omega_2$), the mean temperature of star 1 (T1), and the monochromatic luminosity of star 1 (L1). The relative brightness of the secondary star was calculated by the blackbody radiation model.

Because BH Vir seems to be leaving the Main Sequence (see Zhai et al. 1990), the components are reasonably detached their Roche Lobes. It is very important for us to estimate the critical ages for circularization and synchronization for BH Vir. From Zhai et al. (1990) results the absolute parameters of BH Vir are $m_1= 0.854~M_{\odot}$, $m_2 = 0.826~M_{\odot}$, $R_1 = 1.166~R_{\odot}$, $R_2 =
1.048~R_{\odot}$, $A= 4.37~R_{\odot}$. Using the time scales given by Claret & Cunha (1997), we have found the critical ages for circularization and synchronization for BH Vir that $t_{\rm syn1} = 7.12\, 10^{9}$ yr, $t_{\rm syn2} = 1.16\, 10^{10}$ yr, $t_{\rm cir1}= 6.19\,
10^{9}$ yr and $t_{\rm cir2} = 1.17\, 10^{10}$ yr. According to stellar evolutionary theory, we have estimated that the system is older than $1.49\, 10^{10}$ yr. These results reveal that both components are essentially Main-Sequence stars, and they should be synchronized.

Firstly, a convergence test for the solution was carried out using a normal Roche model without spots. The $\Sigma_i (w_i$(O-C)i)2-q curve is shown in Fig. 1, where $\Sigma_i (w_i$(O-C)i)2 is the sum of weighted squares of residuals between the theoretical light curve and the observations. A minimum of $\Sigma_i (w_i$(O-C)i)2 is achieved at q=1.0. At this point, we adopted a dark, circular spot model for the active regions on the primary star and expanded the adjustable parameters to include q. The mass ratio converged to q=0.967 in the final solution, which is in very good agreement with the spectroscopic and photometric value determined by Zhai et al. (1990). Figure 2 shows the O-C residuals.

\par\includegraphics[width=7.8cm,clip]{5258f1.eps}\end{figure} Figure 1: Variance of the computed fit as a function of the mass ratio q

\par\includegraphics[width=8.8cm,clip]{5258f2.eps}\end{figure} Figure 2: (O-C) values of BH Vir in 1991 with the model of the spot on the primary star

The theoretical light curves do fit rather satisfactorily the observations, except around the secondary eclipse where the systematic deviations in the O-C by 0.020 mag (V) and 0.036 mag (B) are greater than the systematic errors (the computed light curves fainter than observations). This may be caused by some reason of the spot model being wrong, so we suggested that there was no spot on the primary in 1991. Then a cool spot placed on the surface of secondary star was tried (because the secondary star is likely to develop dark-spot activity of the type seen on the Sun). The DC program was used to obtain final spot parameters. The results are given in Tables 2 and 3. The theoretical light curves for the spotted solution are shown as solid lines in Fig. 3. The appropriate O-C residuals are shown in Fig. 4. The systematic deviations in the O-C residuals are decreased to be 0.003 mag (V) and 0.018 mag (B) which are less than the observations intrinsic errors.

\par\includegraphics[width=8.8cm,clip]{5258f3.eps}\end{figure} Figure 3: Light curves of BH Vir in 1991. Open circles represent the observations. Solid lines are theoretical light curves calculated from the parameters in Tables 2 and 3

\par\includegraphics[width=8.8cm,clip]{5258f4.eps}\end{figure} Figure 4: (O-C) values of BH Vir in 1991 with the model of the spot on the secondary star


Table 2: BH Vir photometric solution
Date 1963/64 1991
L1/ (L1 + L2) (V) $0.6716 \pm 0.0054$ $0.6350 \pm 0.0037$
L1/ (L1 + L2) (B) $0.6905 \pm 0.0053$ $0.6526 \pm 0.0036$
X1 (V) 0.64 0.64
X1 (B) 0.77 0.77
X2 (V) 0.66 0.66
X2 (B) 0.81 0.81
i $87\hbox{$.\!\!^\circ$ }43 \pm 0.17$ $87\hbox{$.\!\!^\circ$ }44 \pm 0.15$
$\Omega_1$ $4.873 \pm 0.007$ $4.858 \pm 0.016$
$\Omega_2$ $5.403 \pm 0.005$ $5.151 \pm 0.024$
g1 0.32 0.32
g2 0.32 0.32
q $0.967 \pm 0.018$ $0.967 \pm 0.009$
A1 0.5 0.5
A2 0.5 0.5
T1 (K) $6010 \pm 5.4$ $5945 \pm 9$
T2 (K) 5500 5500
r1 (pole) $0.2541 \pm 0.0015$ $0.2550 \pm 0.0010$
r1 (point) $0.2673 \pm 0.0019$ $0.2684 \pm 0.0013$
r1 (side) $0.2583 \pm 0.0016$ $0.2593 \pm 0.0011$
r1 (back) $0.2643 \pm 0.0018$ $0.2654 \pm 0.0012$
r2 (pole) $0.2193 \pm 0.0016$ $0.2324 \pm 0.0021$
r2 (point) $0.2265 \pm 0.0019$ $0.2417 \pm 0.0022$
r2 (side) $0.2217 \pm 0.0017$ $0.2355 \pm 0.0019$
r2 (back) $0.2251 \pm 0.0018$ $0.2398 \pm 0.0021$


Table 3: BH Vir starspots parameters
Data star latitude longitude ang.radius $T_{\rm s}/T_{\rm ph}$ $S_{\rm s}/S_{\rm hem}$
1963/1964: 1 $81\hbox{$.\!\!^\circ$ }10$ $271\hbox{$.\!\!^\circ$ }30$ $11\hbox{$.\!\!^\circ$ }50$ 0.71 2.5%
  1 $76\hbox{$.\!\!^\circ$ }50$ $91\hbox{$.\!\!^\circ$ }20$ $11\hbox{$.\!\!^\circ$ }00$ 0.72 2.3%
  2 $86\hbox{$.\!\!^\circ$ }90$ $180\hbox{$.\!\!^\circ$ }40$ $20\hbox{$.\!\!^\circ$ }10$ 0.71 7.6%
1991: 2 $81\hbox{$.\!\!^\circ$ }40$ $309\hbox{$.\!\!^\circ$ }50$ $11\hbox{$.\!\!^\circ$ }20$ 0.650 2.4%

Notes: ph = photosphere; hem = hemisphere.

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