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4. Photometric solutions

Based on the ephemeris (1), the observations of 1994 were combined into a single orbital cycle and the light curves of HL Aur in both V and B are shown in Fig. 2 (click here). The light curve of the system is essentially symmetric, although the average light level seems to be slightly brighter at phase 0.75 than at 0.25. The primary-ecilpse depth is 0.98 in V and 1.06 in B, the secondary-eclipse depth is 0.47 in V and 0.45 in B.

Figure 2: BV light curves and the synthetic light curves based on the photometric solutions of HL Aur. The synthetic light curves are shown with solid lines

In the following photometric analysis, the observations are further combined into 59 normal points in V and 51 points in B, respectively. The new, 1992 version of the Wilson-Devinney computing code (Wilson & Devinney 1971; Wilson 1990) is employed to perform the photometric solution of the B and V light curves simultaneously. In the Roche model, from the calibration of the statistical relations between stellar spectral/luminosity classes and effective temperature (de Jager & Nieuwenhuijzen 1987) the temperature of component 1 was fixed at tex2html_wrap_inline1115, corresponding to spectral type F4 (Götz & Wenzel 1961). The gravity-darkening coefficients are adopted to be tex2html_wrap_inline1117 (Lucy 1967), and the bolometric albedo to be tex2html_wrap_inline1119 (Rucinski 1969) in accordance with the assumed stellar convective envelope. From the tables in Al-Naimiy (1978) the limb-darkening coefficients are taken to be tex2html_wrap_inline1121, tex2html_wrap_inline1123 for the V band and tex2html_wrap_inline1127, tex2html_wrap_inline1129 for the B band. The adjustable parameters are: the inclination i, the polar temperature of the secondary component tex2html_wrap_inline1135, the nondimensional surface potentials tex2html_wrap_inline1137 and tex2html_wrap_inline1139, the luminosity of the primary (component 1) star tex2html_wrap_inline1141 and the mass ratio tex2html_wrap_inline1143. Unfortunately, no photometric or spectroscopic information on the mass ratio q is available. In search of an approximate mass ratio, we carried out test solution by assuming mass ratio tex2html_wrap_inline1147, 0.50, 0.60, 0.65, 0.70, 0.75, 0.80, 0.85, 0.90, 1.05, 1.45, 2.0, and 2.5.

The test solution for each assumed tex2html_wrap_inline1149 value started from the detached model (mode 2), but after a few runs in the iteration, the system converged into a semi-detached configuration with component 1 in contact with the inner critical equipotential surface of the binary system. Then computing mode 4 was used instead of mode 2 until a final convergent solution was reached. Figure 3 (click here) represents the sum of the weighted residuals, tex2html_wrap_inline1151 for each assumed mass ratio tex2html_wrap_inline1153. It is found that the mass ratio of the system is most likely somewhere around q=0.75, at which the test solution gives the smallest residuals in the tex2html_wrap_inline1157 diagram. With this probable mass ratio we started again and let it be adjusted freely along with other adjustable parameters. In the last few runs the parameters tex2html_wrap_inline1159, tex2html_wrap_inline1161 and tex2html_wrap_inline1163 in both V and B bands are also made adjustable. However, tex2html_wrap_inline1169, tex2html_wrap_inline1171 and tex2html_wrap_inline1173 seem to be not adjustable and no convergent values could be found for them. The final test converged into two slightly different solutions with nearly the same parameters and the same tex2html_wrap_inline1175 value at q= 0.722 for HL Aur, depending on the last mode (2 or 4) of the W-D program we used. Solution 1 indicates that the system is a detached binary with tex2html_wrap_inline1179 very close to tex2html_wrap_inline1181 while solution 2 tells us that the system has a semi-detached configuration with the primary in contact with its Roche lobe. The two solutions are set out in Table 6 (click here). The theoretical light curves based on solution 2 is shown in Fig. 2 (click here) and the configuration of the system is given in Fig. 4 (click here). Almost the same theoretical light curve and configuration as those in Figs. 2 (click here) and 4 (click here) could be found for HL Aur based on solution 1. Figure 2 (click here) displays a good, full-orbit fitting between the theoretical and observed light curves.

Table 6: Photometric solutions of HL Aur

Figure 3: The behaviour of tex2html_wrap_inline1287 as a function of the mass ratio q

Figure 4: The configuration of HL Aur

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