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Astron. Astrophys. 338, 1031-1040 (1998)

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4. Starspots in RS CVn stars

Almost all RS CVn systems, for that matter all rotating variables, usually exhibit continuously varying light over the photometric cycle; flat-topped light curves are seldom observed. Even in the case of eclipsing RS CVn variables, with the inclinations of the rotational axis [FORMULA], the outside eclipse variation arising from the spotted nature of the active star is nearly sinusoidal. This implies that the spots which modulate the observed flux have a large longitudinal spread [FORMULA].

4.1. Latitudinal extent

Both the light curve modeling using large, discrete spots, and the Doppler imaging using weak photospheric lines suggest the presence of long-lived polar spots, which have no solar analogue, in active stars. However, recently, Unruh and Collier Cameron (1997) have found differences in the Doppler images obtained from the sodium, and calcium and iron line profiles - the images obtained from sodium D lines indicate less high latitude structure and give more reliable light curve predictions than the images derived from several weaker photospheric lines.

For an inclination of the rotational axis i the main contribution to the rotational modulation comes from spots present in the [FORMULA] latitudinal belt and the maximum possible amplitude increases drastically with the inclination. For example, for a spot-photosphere temperature difference of 1000 K an increase in i from [FORMULA] to [FORMULA] increases the maximum possible amplitude from 0.5 to 1.2 mag, a range of about 0.7 mag. For a temperature difference of 1500 K the range would be more than 1 mag. However, the spread in the maximum observed amplitudes for the four active, well-observed, single-lined systems - DM UMa (0.32 mag, Mohin & Raveendran 1994), II Peg (0.50 mag, Mohin & Raveendran 1993), HD 12545 (0.60 mag, Nolthenius 1991), and HD 81410 (Table 2) - is only around 0.25 mag. If spots are confined to [FORMULA] latitude the increase in amplitude when seen at [FORMULA] 60o[FORMULA]i = [FORMULA] is about 0.2 mag. The corresponding increase is around 0.3 mag if the spots are cooler than the photosphere by 1500 K. Hence it is tempting to conclude that the longitudinal asymmetry in the distribution of spots, which causes the observed light modulation, is largely restricted to around [FORMULA] in latitude. Such a conclusion, of course, depends on the following assumptions: (i) the inclinations of the above four active stars are different and they show at least a spread of about [FORMULA]; it is highly unlikely that all of them have the same inclination for the rotational axis, (ii) all these objects show similar levels of spot activity, which is quite possible in view of their similar spectral types, and (iii) spots form on both sides of the stellar equator with equal probability as in the case of the Sun.

Hall (1991) has found that the differential rotation is correlated strongly with the rotation period, in the sense that rapidly rotating stars approach solid body rotation. The existence of such a tight relationship as found by him indicates the possibility that the differential rotation in spotted stars is similar to that of the Sun, with the poles rotating slower than the equator. For a given spotted star with solar type differential rotation the photometric period would depend on the effective latitude of the spots or spot groups that produces the light modulation. The total range of photometric periods ([FORMULA]) derived from long-term photometry will be a rough measure of the total latitudinal extent of spots on its surface since the higher the latitude of spot occurrence, the larger will be the [FORMULA]. In Fig. 4 we have plotted the log([FORMULA]) against the corresponding [FORMULA] of nine well-observed objects - [FORMULA] And, [FORMULA] Gem, II Peg, V711 Tau, HR 7275, V350 Lac, V478 Lyr, BM Cam and V1149 Ori. The total range of rotational periods [FORMULA] and the average period P of these objects are taken from Henry et al. (1995), Strassmeier et al. (1994), Crews et al. (1995) and Hall et al. (1990, 1991, 1995). The positions of the Sun calculated with the rotational periods at [FORMULA] and [FORMULA] latitudes taken as the values of [FORMULA] are also indicated in the figure. It is quite interesting to see that [FORMULA] And, [FORMULA] Gem, HR 7275, BM Cam and V1149 Ori lie close to the position of the Sun with log ([FORMULA]) corresponding to [FORMULA] latitude, and that corresponding to [FORMULA] latitude is well above the mean position occupied by these objects. The effect of the weak correlation (at 2[FORMULA] significance level) of log ([FORMULA]) on the Roche-Lobe filling factor found by Hall (1991) is expected to be small around the periods of these stars. Again, it is tempting to conclude that in spotted stars, in general, the spots mainly occur within around [FORMULA] effective latitudes.

[FIGURE] Fig. 4. Plot of log ([FORMULA]) of the well-observed spotted stars against their respective [FORMULA]. The corresponding positions for the Sun are indicated as open circles. The orbital inclinations of the binaries are also indicated in the figure.

If there is no limit on the latitudinal extent for the formation of spots, the observed [FORMULA] should show a dependence on the inclination i in addition to the dependence on P discussed above. All the stars plotted in Fig. 4 have reliable estimation of inclination of rotational axis. The most likely values of i, sources of which are the same as those of the total range of rotational periods, are indicated against the corresponding points in Fig. 4. No dependence of [FORMULA] on i is evident from the figure, indicating that the latitudinal spread of the spots is similar in all the stars considered.

The spot activity, probably, is not always just restricted to within [FORMULA] latitudes. In the case of DM UMa ([FORMULA], Crampton et al. 1979) the total range in V observed, [FORMULA], [FORMULA](brightest) - [FORMULA](faintest), is 0.52 mag (Mohin & Raveendran 1994). Assuming that [FORMULA](brightest) corresponds to the unspotted magnitude and that spots are cooler than the surrounding photosphere by 1000 K, then such a range requires that even for 100% spot coverage spots should occur well beyond [FORMULA] in latitude in the hemisphere visible at light minimum at that epoch.

4.2. Limb-darkening effects

The theoretical light curve modeling available in the literature shows that the effect of wavelength dependence of limb-darkening on the colour variation over the photometric phase could be significantly larger than that of the temperature difference between the spot and photosphere. All the spot models that demonstrate such an effect assume circular spots which cross the centre of the projected stellar disc during the rotation of the variable (Poe & Eaton 1985; Eker 1994). The light and colour curves due to a rectangular spot bounded by the latitudes [FORMULA] and extending over the full range in longitude on the hemisphere visible at light minimum are plotted in Fig. 5. The angle between the rotational axis and line of sight was assumed to be [FORMULA], and the spot was assumed to be 1000 K cooler than the photosphere. The calculations were done for both (i) no limb-darkening (dashed line) and (ii) linear limb-darkening (solid line). The linear limb-darkening coefficients in UBVRI given in Strassmeier & Olah (1992) and Eker (1994), namely 0.96, 0.88, 0.75, 0.61 and 0.50, respectively, were used in the computations. Both the photospheric and spotted regions were assumed to follow the same limb-darkening law. From Fig. 5 it is clear that the light and colour curves corresponding to the two cases differ only slightly indicating that the colour variations are caused by the temperature effects rather than by the limb-darkening effects. The amplitudes of colour curves in both cases are nearly same. The difference is only in their shapes, with the colour curves with limb-darkening showing nearly flat maxima and minima.

[FIGURE] Fig. 5. Light and colour curves due to an equatorial band of spot with linear limb-darkening (LD) and with no LD (solid lines and dashed lines ), and due to an equivalent circular spot with linear LD (dotted lines )

A circular spot of [FORMULA] radius at a polar distance of [FORMULA] also produces a continuous light curve with an amplitude and shape similar to that of the rectangular spot considered above. Fig. 5 also shows the colour and light curves due to such a circular spot. The limb-darkening was assumed to be linear and the same as those quoted above. It is clear from the figure that the colour curves due a circular spot differs significantly, both in amplitude and shape, from that due to an extended rectangular spot. The former curves are flatter at maximum and show deeper minimum than the latter curves. The amplitudes of all colour curves due to the circular spot are larger than those due to the rectangular spot by more than 50%. We conclude that the net effect in the colours produced by the limb-darkening depends on the exact distribution of spots on the stellar surface; it could be even negligible for certain spot distributions.

4.3. Non-uniqueness of spot modeling

We have used a modified version of the computer program in Fortran developed by Mohin & Raveendran (1992) to solve for the spot parameters from a given light curve. In order to see how accurately the resulting equivalent spots reproduce the light curves being solved, the synthetic curves in UBVRI due to an equatorial band of spot described above and plotted in Fig. 5 were subjected to an analysis. A spot extended longitudinally instead of latitudinally was considered because the continuously varying light over the photometric phase usually observed in an RS CVn star indicates a large longitudinal spread for the spots.

Four different trials, corresponding to the cases of a single, two, three and four spots, were made, and in all cases the convergence of the solutions could be easily obtained. The temperatures of the spots were assumed to be the same for all the spots in the cases where the number of spots assumed were more than one. The final spot parameters for the four cases are given in Table 3. It is clear from the standard deviation of fit ([FORMULA]) given in Table 3 that the single-spot assumption gives a poor approximation to the light and colour curves while the two-spot assumption gives an acceptable fit to the synthetic data. The light and colour curves corresponding to the three- and four-spot assumptions are found to be mutually indistinguishable and in excellent agreement with the data.


[TABLE]

Table 3. Circular spot parameters derived for the synthetic light curve due to an equatorial band of spot.


Since the light continuously varies over the photometric cycle, the single-spot solution gives a large radius and a high latitude for the spot; half of the spot lies in the circumpolar region. As the number of spots assumed is increased their sizes become smaller and they shift towards the equator where the spot is assumed to be present while synthesizing the data; when their number becomes four the spots slightly overlap and produce an equatorial band with nearly uniform width. The area occupied by the spots as a fraction of the total area of the stellar surface is also given in Table 3. The fractional area of the equatorial spot is 0.16. Both the single- and two-spot assumptions give the fractional area as 0.10, about 50% smaller, where as the three-spot assumption gives almost the correct value. The spot temperature was assumed to be 3800 K, 1000 K cooler than the photosphere. It is interesting to see from the Table 3 that the total range in the spot temperatures derived is only around 150 K; the four-spot assumption gives the assumed temperature exactly.

The light minimum of the synthetic curve occurs at [FORMULA], corresponding to a longitude of [FORMULA]. However, in the case of the two-spot assumption the spots are separated by [FORMULA] in longitude and therefore the real light minimum occurs at their mean longitude. In the present case the spots are of equal size. In an actual case the spots could be of different sizes and the longitude estimated from the light curve would then correspond to a longitude weighted by the intensity distribution on the hemisphere visible at light minimum. Usually, the migration of the light minimum observed in RS CVn Systems is attributed to a migration of spot or spot group as a result of a difference between the actual period and the period used in folding the observations. If there are more than one prominent spot group, then the migration of the phase of the light minimum may not represent a true migration of a spot or spot group because the apparent shift in the phase of the light minimum may be arising from a change in the relative strengths of the various spot groups.

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© European Southern Observatory (ESO) 1998

Online publication: September 17, 1998
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