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3 Photometric solutions

Photometric solutions were obtained by means of the WD program which includes a new reflection treatment, the option of using non-linear limb-darkening laws and the ability to adjust spot parameters. All the observations were used in computing the solutions. The convergence of the minimization procedure was obtained by means of the method of multiple subsets (Wilson & Biermann 1976).


Table 5. The photometric solutions of FG Hya

parameters
1962's obs.   1982's obs.   1999's obs.

color
V B V B V
L1/(L1+L2) 0.8621  $\pm~{0.0008}$ 0.8637  $\pm~{0.0010}$ 0.8636  $\pm~{0.0009}$ 0.8616  $\pm~{0.0011}$ 0.8648  $\pm~{0.0.0007}$
x1=x2 0.640 0.780 0.640 0.780 0.640
q=m2/m1 0.1306  $\pm~{0.0010}$   0.1273  $\pm~{0.0008}$   0.12889  $\pm~{0.0009}$
i 86.60  $\pm~{0.86}$   87.12  $\pm~{0.63}$   86.78  $\pm~{0.52}$
A1=A2 0.500   0.500   0.500
g1=g2 0.320   0.320   0.320
$\Omega$ 1.9908  $\pm~{0.0026}$   1.9834  $\pm~{0.0022}$   1.9821  $\pm~{0.0026}$
f 0.69   0.72   0.74
$\Sigma$ 0.005321   0.004657   0.003768

component
1 2 1 2 1 2
T(K) 5900 5836 $\pm~{16}$ 5900 5811 $\pm~{18}$ 5900 5852 $\pm~{19}$

r(pole)
0.5277  $\pm~{0.0007}$ 0.2182  $\pm~{0.0009}$ 0.5335  $\pm~{0.0005}$ 0.2217  $\pm~{0.0006}$ 0.5380  $\pm~{0.0006}$ 0.2179  $\pm~{0.0008}$
r(side) 0.5872  $\pm~{0.0011}$ 0.2286  $\pm~{0.0010}$ 0.5963  $\pm~{0.0008}$ 0.2332  $\pm~{0.0007}$ 0.6031  $\pm~{0.0008}$ 0.2293  $\pm~{0.0009}$
r(back) 0.6100  $\pm~{0.0013}$ 0.2762  $\pm~{0.0025}$ 0.6202  $\pm~{0.0009}$ 0.2914  $\pm~{0.0022}$ 0.6265  $\pm~{0.0018}$ 0.2893  $\pm~{0.0025}$


The adopted parameters in the solutions are described as follows: a temperature of 5900 K for Star 1 (the star eclipsed at Min.I), Claret et al.'s (1990) values of the limb darkening coefficient ( x1=x2=0.640), Lucy's (1967) values of the gravity darkening coefficients ( g1=g2=0.320) and Rucinski's (1969) values of the albedo ( A1=A2=0.500), which corresponds to the spectral type of G0 for FG Hya. The adjustable parameters were the orbital inclination, i, the mean temperature of Star 2, T2, the potential of the components, $\Omega
_1$ and $\Omega _2$, and the monochromatic luminosity of Star 1, L1(the Planck function was used in computing the luminosity).

Solutions were made for a series of fixed values of the mass ratio q=m2/m1(0.10, 0.15, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 1.50, 2.00, 2.50, 3.00, 4.00, 5.00 and 6.00). Assuming that initially it was a detached system, the differential corrections started from mode 2, but the converged solutions were always obtained at the contact mode 3. The resulting sum $%
\Sigma $ of the square deviations of the converged solutions for each value of q indicates that the fitting is best for q=0.15. At this point, the set of the adjustable parameters was expanded to include q. The mass ratio converged to a value of q=0.1289 in the final solution. This solution indicates that FG Hya is an A-type W UMa binary, in agreement with the results published by Mochnacki & Doughty (1972), Twigg (1979), Lafta & Grainger (1986), Jabbar et al. (1986) and Yang et al. (1991). The photometric parameters are listed in Table 5, where Star 1 indicates a more massive component and Star 2 is one of less mass. The computed light curve by using the parameters in Table 5 is shown as the solid line in Fig. 3. As shown in Fig. 3, the fit between the theoretical light curve and the observations is very good.

In order to compare the present solution with the previous ones, the light curves in B and V bands obtained by Binnendijk (1963) in 1961-62 and Yang et al. (1991) in 1982 were also re-analyzed in the same way. The results are also listed in Table 5, from which one can see that the solutions obtained from different observations are in agreement with each other despite the considerable change in the shape of the light curves.

We tried also to analyze the light curves obtained by Mahdy et al. (1985), but a reasonable solution could not be found. In the course of finding a solution by means of the DC program of the WD code, we found all obtained converged solutions difficult to interpret. The solutions indicated either orbital inclinations that were too small or mass ratios substantially larger than the spectroscopically determined one. When spot models were attempted, the solutions indicated that FG Hya would have to be classified as a W-type system. It is possible that FG Hya is an example of a small mass ratio binary that fluctuates between the A- and W-type configuration, depending upon its current state of spot coverage. This is certainly not without precedence, as the similar system V677 Cen (Kilmartin et al. 1987) was found to possess equal temperatures for both components, i.e., on the boundary between A-type and W-type. Other W UMa binaries (e.g., TZ Boo, VW Cep) have displayed significantly changing depths of eclipses such that the normally less deep secondary eclipse was sometimes deeper than the primary one. Therefore we abandoned our attempt to find a final solution from Mahdy et al.'s observations. We suspect that either their observations are in question or the distortion of the light curves is so much because of the substantial spot that it is difficult to find a true solution.


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