5. Modeling of the light and colour curves of HD 81410
HD 81410 is a non-eclipsing binary and hence its orbital inclination i is unknown. An uncertainty in i will be directly reflected on the polar distance derived for the spots. Donati et al. (1997) derive a value of , where as Weber & Strassmeier (1998) estimate ; we assumed in the starspot modeling of the light curves of HD 81410.
The unspotted brightness of the star in each wavelength band observed is an important parameter which has a direct effect on all the spot parameters, , polar distance, radius and temperature. The problems associated with the assumption of the unspotted magnitudes have been discussed in detail by Poe & Eaton (1985). The light curves in U and u bands show a large scatter, and therefore only measurements in BVRI and vby bands were subjected to the method of spot modeling described in the previous section. The following values, which are the brightest observed so far in the corresponding bands, are assumed to represent the unspotted magnitudes: B = 8.240, V = 7.230, R = 6.650, I = 6.145, v = 8.860, b = 7.850 and y = 7.230 mags.
The photospheric temperature of HD 81410 was assumed to be 4750 K, consistent with its () colour index (Donati et al. 1997). Calculations were made with the same linear limb-darkening law for both the spotted and unspotted photospheric regions. Mohin & Raveendran (1992) have shown that the spot parameters derived under such an assumption and the more general assumption that spotted and unspotted regions follow different quadratic limb-darkening laws differ only marginally and are well within the uncertainties involved in their determination. The linear limb-darkening coefficients in BVRI bands were taken as the same as those mentioned earlier in connection with the synthetic light curves. The coefficients in vby bands, 0.91, 0.84 and 0.75, respectively, were derived from an interpolation of the values given by Claret & Gimenez (1990).
In order to study the short-term evolution of spots, we have modeled a sample of closely spaced light curves that showed drastic changes, and the results are given in Table 4. The smooth, continuous curves in Figs. 1a-h represent the computed light and colour curves. The computed curves in all cases closely approximate the observations; the factors which might be contributing to the disagreement ( 0.02 mag) include the blackbody assumption for the radiation emitted by both the spotted and unspotted regions, and the uncertainty in the unspotted magnitudes in various bands. In general the standard deviations of fit are comparable to the scatter in the light curves due to the folding of observations over several rotational cycles. We have seen in an earlier section that the colour curves depend to a large extent on the exact distribution of spots on the stellar surface because of the limb-darkening effects. Therefore, another factor which might be contributing to the disagreement between the computed and observed colour curves is the difference between the actual spot distribution on the star, which may be complex, and the assumed.
Table 4. Spot parameters derived from the light curves of HD 81410
The results presented in the previous section show that the spot parameters, polar distance and radius, and sometimes even longitudes derived from the observed light curves could misrepresent the actual situation, even when the light and colour curves computed from these parameters closely reproduce the observations; the only parameter less effected seems to be the spot temperature. This is amply demonstrated in Table 3.
The single- and two-spot solutions give almost similar overall fit to the observations obtained during 1988.41 (Fig. 1d). In the latter solution both the spots were assumed to be of the same radius. Again the spot temperatures in the two cases agree mutually very closely. The light curve obtained during 1989.20 (Fig. 1g) shows the smallest amplitude so far observed. It shows a broad and asymmetric light minimum. There is only a slight indication of a secondary minimum. In this case also two solutions were obtained, , with three- and four-spot assumptions. The corresponding of fit are 0.006 mag and 0.007 mag. Both sets of computed curves closely match the observations. The temperature obtained in the two cases again agree mutually. In both cases the spots were assumed to be of the same radius.
The observations (mean epoch 1989.12) of Cutispoto (1993) plotted in Fig. 1f were obtained about a month before the above mentioned observations. The corresponding light curve shows a slightly larger amplitude and a well-defined secondary minimum. The of fit is 0.006 mag, similar to those for the solutions obtained for the observations during 1989.20. It is interesting to see that the spot temperature derived from the 1989.20 data is higher by 1000 K than that derived from the 1989.12 data. We have seen above and also from the solutions of the synthetic data that the derived temperature of the equivalent circular spot(s) is less affected by the assumption on the number of spots. We interpret the higher temperature observed during 1989.20 as follows: The light modulation was caused by several small individual spots or spot groups, and they were more spread out during 1989.20 than during 1989.12, and hence the equivalent circular spots included a larger region of the unspotted photospheric region. Therefore, during epochs of smaller amplitudes for the light variation the spots are more spread out across the stellar surface. During these epochs the brightness at light maximum is invariably well below the unspotted magnitude, implying that spots are spread out both latitudinally and longitudinally. This sort of spot distribution at low light amplitudes is, probably, true for all the active RS CVn stars.
When their number is increased from three to four in the modeling of the 1989.20 data we see that the spots become smaller and shift towards the equator, as in the case of the circular spot solutions of the synthetic data corresponding to an equatorial band of spots given in Table 3. We have seen that as the number of spots is increased, the solutions approach more and more close to the real situation. Therefore, it is quite possible that the spots are distributed about the equator as in the case of the Sun, but with a latitudinal extent significantly larger than ; the solutions which indicate polar spots may be the result of limiting the number of spots in the modeling to just one, two or three.
© European Southern Observatory (ESO) 1998
Online publication: September 17, 1998