4 Light curve analysis

The first light curve of the system was obtained by Menzies & Marang (1986). Assuming that the temperature of the primary star is $T_1=26\ 000~$K they found that T₂=4500 K, r₁ = 0.203 and r₂= 0.207. In the light curve analysis they used the Wilson Devinney (1971) differential correction programme. Using mass function they predicted the masses of components to be $0.25~M_\odot$ and $0.12~M_\odot$. Later, Wood et al. (1993) observed the system in UBVR and analyzed the light curves with the Wilson Devinney Code. In their analysis, although the radii of the components were consistent with the previous results, the temperatures of the primary star and the secondary one were predicted as $29\ 000 <T_1< 36\ 000$ K and $T_2\sim 3700$ K, respectively. In the same analysis, the masses of the components were given as $M_1=0.50~M_\odot$ and $M_2=0.15~M_\odot$.

Although it has already been stated, we feel compelled to repeat again that since the system is very faint and the photometer we used is insensitive to short wavelengths, our observations in U band were scattered, as we expected.

In the analysis of the light curves the normal points were performed in each band. The $B(\lambda~4200)$, $V(\lambda~5400)$ and $R(\lambda~7000)$ light curves were represented by 51, 65 and 69 normal points, respectively.

The mass of sdB stars is given as $0.50~M_\odot$ by Saffer et al. (1994). If we use this value in the mass function obtained by Hilditch et al. (1996), the mass of the companion star is found to be around $0.14~M_\odot$ as noted earlier by Hilditch et al. For this reason we adopt the value of 0.28 for the mass ratio. Limb darkening coefficients play a very important role in light curve analysis. These coefficients depending on the temperatures of the stars are theoretically evaluated for the stars hotter than 5500 K. To the best knowledge of the authors of the present study there is no reference giving those coefficients for stars cooler than 5500 K. In the previous analysis of HW Vir, limb darkening coefficients of the cooler component are taken as those of a star with an effective temperature of 5500 K. In the present paper, limb darkening coefficients of sdB star for B, V and R colour are taken from Rucinski (1985) as 0.240, 0.199 and 0.162, respectively. Corresponding values for cooler low mass star are derived from light curve analysis. Gravity darkening coefficients for hotter star for all three colours are taken as 1.0 and for cooler star as 0.32. Bolometric albedo for hotter star for all three colours is assumed to be 1.0. If the mass of the cooler component is taken as $M_1=0.14~M_\odot$ then its effective temperature should be 3300 K (Dorman et al. 1989). In the solution the above values are kept fixed.

The initial values of the parameters were performed by means of LC (Light Curve) program till a good approach to the observational data was obtained. Then the DC (Differential Corrections) was run iteratively until an acceptable stability of the solution was reached. The adjustable parameters were: orbital inclination (i), surface temperature of the hotter component (T₁), non-dimensional potentials $\Omega_{1,2}$ of both components, luminosity of hotter star, the limb darkening coefficient and the albedo of the cooler star. The results are given in Table 3.

 

Table 3: The results of the light curve analysis with WD Code

 

The computed light curves using the parameters given in Table 3 were plotted in Fig. 4 with those of the normal points.

  

Figure 4: The observed normal points in B, V and R are plotted against the orbital phase and compared with the computed curves. Note that the ordinate is intensity

The computed curves seem to fit well with the observed normal points.

Using the results of the radial velocity curve analysis we obtained the radius of the orbit as $0.89~R_\odot$. The three - colour light curves were analysed simultaneously to obtain the radii of the components. The mean fractional radius of hotter and cooler component stars are 0.218 and 0.201, respectively. The mass, radius and luminosity of the hotter primary star were calculated as 0.50, 0.21 and 55.628 solar units, whilst these parameters are 0.14, 0.20 and 0.0003 for the cooler secondary star. The critical Roche lobes of the components were calculated using the mass ratio of 0.28 and are shown in Fig. 5.

  

Figure 5: Roche model geometry of the HW Virginis system from our data. The quantities presented are based on the orbital parameters (DC solutions) and the assumed absolute dimension of this system. The radii correspond to spheres of equal volume

As it is seen from this figure, both of the components are smaller than those of the corresponding Roche lobes. However the cooler secondary star is closer to its Roche lobe than is the primary star.