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Astron. Astrophys. 330, 1029-1040 (1998)

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4. Doppler imaging

4.1. The line-profile inversion code

As for the previous papers in this series, all maps were generated with the Doppler-imaging code TEMPMAP (Rice et al., 1989), originally developed for use with chemical abundance inhomogeneities of Ap stars. For temperature mapping we use a more rigorous treatment of the local line profile than originally discussed in Rice et al. (1989) and also simultaneously solve for the relative continuum light in two photometric bandpasses (Rice, 1995; Piskunov & Rice, 1993). Local line profiles are computed for a series of limb angles from a solution of the equation of transfer through precomputed model atmospheres including updated opacities as implemented in the latest version of ATLAS-9 (Kurucz, 1993).

A grid of nine model atmospheres with [FORMULA] and temperatures from [FORMULA] = 3500 K to 5500 K in steps of 250 K was taken from the ATLAS-9 CDs (Kurucz, 1993). Using a [FORMULA] of 3.0 leads to similar line-profile fits, but the resulting metal abundances were even more deviant from the solar values in Table 4 than for [FORMULA] and we therefore adopted the latter value for the final maps. Solutions with [FORMULA] and 2.0 did not result in satisfactory fits.


[TABLE]

Table 4. Logarithmic elemental abundances relative to hydrogen ([FORMULA])


Due to the late spectral type combined with the wavelength coverage of our spectra, seven moderately blended lines could be used: FeI 6393, FeI 6400 (actually a close blend of two iron lines at 6400.000 and 6400.314 Å), FeI 6411, FeI 6419.94 and 6421.35 (refered to as FeI 6420), FeI 6430, CaI 6439, and FeI 6546 Å with [FORMULA] values between -3.9 and +0.47 and lower excitation potentials between 0.915 and 4.733 eV. Since all seven lines are blended to a certain degree, the number of lines which had to be synthesized was 6, 10, 7, 12, 8, 8, and 3 for the 6393, 6400, 6411, 6420, 6430, 6439, and 6546-line regions, respectively (Table 5). All these blends were included in the inversion and treated simultaneously with the primary mapping lines but only one spectral region can be handled per solution. In this paper we used a Maximum-Entropy regularization but in practice the program also allows a Tikhonov regularising functional (see, e.g., (Piskunov & Rice, 1993)).


[TABLE]

Table 5. Atomic parameters of the mapping lines


4.2. Redetermination of the rotational velocity and the inclination of the stellar rotation axis

Since the Doppler-imaging analysis is rather sensitive to the rotational velocity and to the inclination of the stellar rotation axis, it can be used to refine these two parameters with higher accuracy than with the methods described in Sect.  3(see also, e.g., Unruh (1996)). Changing these two parameters one at a time, while all others are held constant, yields a certain variation of the [FORMULA] from the resulting line-profile fits. The value of the parameter corresponding to the smallest [FORMULA] is the one we think is closest to the true value. The variation of [FORMULA] with the rotational velocity [FORMULA] and the inclination of the stellar rotation axis i are plotted in Fig. 4a and b, respectively. A minimum is seen in both cases: for the inclination around 50-60o and for the rotational velocity between 26 and 27 km s-1, except for the CaI 6439 line where it is near 28 km s-1. Since the latter velocity leads to a small bright band at the sub-observers latitude - usually the sign for too high a rotational velocity - the true rotational velocity may be closer to 26 than 28 km s-1. The adopted final values correspond to the grand minimum [FORMULA] and are [FORMULA] km s-1 and [FORMULA] o .

[FIGURE] Fig. 4. The variation of [FORMULA] as a function of the inclination of the stellar rotation axis (panel a), the projected equatorial rotational velocity (panel b), the iron abundance (panel c), and the calcium abundance (panel d). The adopted values are marked.

4.3. Finetuning the abundances

To determine more accurate elemental abundances we evaluate the run of the [FORMULA] of the line profile fits from a series of solutions starting with abundances of 0.5 dex below solar abundance and increasing that in steps of 0.05 dex. The transition probabilities, damping constants and laboratory wavelengths were kept constant. We then adopted the abundances that resulted in the minimum [FORMULA], i.e. [FORMULA] = 6.73 and [FORMULA] = 5.52 according to Fig. 4c and 4d. The same steps were then performed to determine the abundances of the line elements that are blended, leading to the values listed in Table 4. Although a consistent set of parameters, the abundances are mathematically not unique because test runs with different sets of fixed parameters (i.e. different [FORMULA] and [FORMULA]) resulted in similar Doppler maps but with individual abundances different by up to 0.2 dex.

4.4. Running TEMP MAP

Now that we have determined all necessary astrophysical input parameters we can generate the final Doppler images. One spectral region and two photometric bandpasses per run, alternately with VI and then with VR, are used to produce two maps for each spectral region. All computations are performed on a DEC-AXP 250/266 workstation and require between 25 to 60 min CPU time depending on the number of blends and the number of input model atmospheres. The resulting VI maps along with the observed and computed line profiles for 1994 and 1995 are plotted in Fig. 5 and in Fig. 6, respectively, while the observed and computed lightcurves for 1994 are shown in Fig. 7.

[FIGURE] Fig. 5a-c. Observed and computed line profiles for FeI 6393.602 Å (row a) with an average equivalent width of 270 mÅ , FeI 6400 Å with a combined equivalent width of 368 mÅ or [FORMULA] 184 mÅ each (row c), and FeI 6411.647 Å with 186 mÅ (row c). The plusses are the observations and the full lines are the fits. The right column shows the maps from the individual lines in pseudo-mercator projection. Note that the spectral lines are arranged from top to bottom according to increasing wavelength.

[FIGURE] Fig. 5d-f. As previously but for FeI 6419.639+6421.349 Å (row d) with 168 and 194 mÅ , respectively, FeI 6430.852 Å with 223 mÅ (row e) and CaI 6439.075 Å with 272 mÅ (row f).

[FIGURE] Fig. 6. FeI -6546 map from February-March 1995 and the corresponding line profile fits. The arrows indicate the times of observation.

[FIGURE] Fig. 7. Observed and computed VRI light curves for March 1994. The fits are from FeI 6393 Å , FeI 6400 Å , FeI 6411 Å , FeI 6420 Å , FeI 6430 Å and CaI 6439 Å . The plusses are the contemporaneous APT observations from Fig. 1c. The arrows indicate the phases of the spectroscopic observations. Note that the individual fits are almost identical.

4.5. Doppler maps for 1994 and 1995

The average map from all but the FeI 6420 spectral region is shown in Fig. 8. For the averaging the individual maps were given equal weight and the lower panel in Fig. 8 shows the distribution of the standard deviations from the mean. The average standard deviation is only around [FORMULA] 20 K per pixel while the peak deviations reach [FORMULA] 40 K. Thus, the individual maps are encouragingly similar, not only in morphology but also in absolute temperature. The only discordant map is the map from FeI 6420 Å being on average 200 K warmer than the others but still showing the same features than from the other lines. We note that the 6420-Å map was recovered from a larger than usual wavelength region containing not only two main mapping lines instead of one but also several temperature-sensitive vanadium blends with poorly determined atomic parameters. If not properly taken into account, these vanadium blends cause either additional or missing equivalent width that is then accounted for with a generally increased surface temperature.

[FIGURE] Fig. 8. A comparison of the average Doppler map from 1994 (top) with the surface distribution of its standard deviations (bottom). The average map is a combination of altogether 10 maps derived from five of the spectral regions shown in Fig. 5 (except FeI 6420), each one with both the VR and the VI light curves as additional constraint. Individual surface features are identified as described in the text. The arrows below each panel indicate the phases of the observations and the grey-scale bar plots the standard deviations along binned longitudes.

The average map from 1994 clearly reveals a cool polar spot (dubbed P in Fig. 8), three large and also cool spots close to the equator and at longitudes of [FORMULA] 40o , 75o , and 330o that we named A, B, and F, respectively, two further, also relatively cool features at longitudes of 200o and 270o (D and E) and one significantly warmer spot at [FORMULA] o (C). Spots A, B, and F have temperatures of about 600 K below the effective photospheric temperature while spots D and E are recovered at [FORMULA] 500 K and spot C with 300-400 K. The latitudinal elongation of some spots might not be real because features at low latitudes migrate through the line profile very quickly and thus their latitudinal extension is less well determined. Typical errors for the central latitudes of the spots are likely to be [FORMULA] 10o despite that the standard deviations for the consistent features from the average map are just "a few degrees". The major uncertainty for an error quotation of the latitude (and the longitude) of a particular spot arises simply from the imperfect definition of its centroid location. The values in Table 6 were measured off the combined map by trailing the temperature along latitudinal strips.


[TABLE]

Table 6. Detected surface features


There is also some detail in the individual maps in Fig. 5 which is not equally well seen in the combined map, despite that the spot morphology in the individual maps and in the combined map is basically identical. Five out of six spots in the average map always have a counterpart in the individual maps. Particularly unreliable is the latitude of feature D, which appears as an appendage of the polar spot in the FeI 6393, 6400, and 6411 maps and possibly with weaker contrast also in the 6420 map but as an isolated, high-latitude spot in the 6430 and CaI 6439 maps. This discrepancy can not be explained with the different line-formation depths because the lines in question are not simply either the weak or the strong ones but have mixed equivalent widths. Possibly, the inconsistent recovery of feature D is an artifact due to the external uncertainities of our spectra.

It is also interesting to compare the 1994 maps with the single-line map obtained from the FeI 6546-Å line one year later in 1995 (Fig. 6). The VRI light curves in Fig. 1 had already indicated a shift of about 0:p3 to smaller phases or 110o on the stellar surface. This can also be seen from the Doppler images and it is thus likely that the light-curve minimum in 1995 was caused by the same spot or groups of spots than in 1994. A cross correlation of the entire 1995 map with the average 1994 map leads to four maxima at [FORMULA] -110o, -50o, +25o, and +90o. The maximum at [FORMULA] -110o is the strongest and also corresponds best with the shift in the light curve and we use it to reidentify the individual spots in the 1995 map.

Spots A and F dominate the 1995 map and appear now at [FORMULA] o and 240o , respectively. They stretch from their original latitudinal position between 10-50o up to the polar spot above 60o in 1995. Spot B is only half the size than in the 1994 map and of significantly weaker contrast ([FORMULA] =600 and [FORMULA] =400 K) while spot C remained at approximately the same latitude and with similar contrast. Instead of the spot pair D and E in 1994, one single spot (marked E) appeared in the 1995 map accompanied by two smaller, adjacent spots of lower contrast and about 30o closer to the equator than spot E in 1994.

The reasonable consistently recovered latitude (and shape) of the individual spots - and the more or less stable overall spot morphology during the time span of the two maps (one year or approximately 27 stellar rotations) - may indicate that the lifetime of the individual spots is usually longer than the typically encountered variability time scale of spotted stars from continuous photometry ([FORMULA] 1 month). We take this as evidence that starspots can indeed be used as tracers for differential surface rotation despite that they are possibly made up of many little spots with individual lifetimes shorter than that of the entire spot region. The intrinsic scatter of our seasonal light curves in Fig. 1 could be attributed to such a scenario.

4.6. Differential surface rotation

In spite of the fact that our 1995 map is derived from a single spectral region that has not been available for the 1994 map, the two maps have nevertheless a surprisingly similar appearance. By cross correlating longitudinal strips from the two maps at sucessive latitudes we may derive the amount and the sign of the differential rotation on IL Hydrae if we assume that the individual surface features in 1995 are indeed the same as in 1994. As can be seen from the maps in Fig. 6 and Fig. 8 (top panel), isolated structures are identified up to a latitude of [FORMULA] 50-60o and the cross-correlation function reveals a significant peak but does indeed flatten out above 50o so that no single, strong peak is evident anymore. We then fitted a Gaussian to the peak of the cross-correlation function whenever well defined and plot its phase lag versus latitude in Fig. 9. The error bars on these lags were adopted to be proportional to the FWHM of the cross-correlation peak and were estimated from repeated measurements with different fitting routines within the IRAF package. The full line in Fig. 9 is a least-squares [FORMULA] fit to the phase lags versus stellar latitude b and leads to the following differential rotation law for IL Hya

[FIGURE] Fig. 9. The differential rotation profile on IL Hya. The points are the phase shifts between each constant-latitude strip of the maps from 1994 and 1995 as derived from least-square Gaussian fits to the cross-correlation function. Above a latitude of approximately 50o the cross-correlation peak becomes too weak due to the extent of the polar spot.

[EQUATION]

where [FORMULA] is the equatorial rotation rate and [FORMULA] is defined via the differential rotation coefficient [FORMULA]. The negative sign for the second term in Eq. (1) indicates that the equatorial regions of IL Hya rotate faster than the polar regions and that [FORMULA] is thus positive and of order [FORMULA], i.e. [FORMULA] days, smaller by a factor of 30 than the solar value. However, we caution that this differential rotation law is not only just based on two maps one year apart, i.e. 27 stellar rotations, but also suffers from the inevitable north-south mirroring effect in Doppler imaging, which affects mostly the "southern" latitudes below, say, -20o in case of [FORMULA] o .

Time-series Doppler images have now been obtained for four other RS CVn binaries (HR 1099, EI Eri, UX Ari, and HU Vir) and differential rotation was likely detected in all of them. The fastest rotator within the four, EI Eridani ([FORMULA] days), only showed a marginal amount of differential rotation and, if correct at all, then in the sense that the equatorial regions rotate faster than the polar regions (Hatzes & Vogt (1992)), in accordance with the solar picture. However, this result may be questioned because in a recent time-series Doppler-imaging study by Washüttl et al. (1997) with much higher time resolution than in Hatzes & Vogt (1992) the equatorial features on EI Eridani exhibited lifetimes of as short as 3 weeks. Another, probably similar, case is HR 1099, where Vogt & Hatzes (1996) had found very weak differential rotation but in the opposite sense than on the Sun, the poles rotating faster than the equator. The other two stars are slower rotators than EI Eri and HR 1099, i.e. [FORMULA] days for UX Ari (Vogt & Hatzes (1991)) and [FORMULA] days for HU Vir (Strassmeier (1994), Hatzes (1997)). For them the deduced differential-rotation rates are approximately a factor of 10 stronger than the two shorter-period stars (but still a factor of 10 weaker than on the Sun), but both in the sense that the poles rotate faster than the equator. IL Hydrae on the other hand, with its 12.7 day rotation period, shows only weak differential rotation with the poles rotating slower than the equator, thus being an exception compared to the other (long-period) RS CVn systems mentioned above.

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© European Southern Observatory (ESO) 1998

Online publication: January 27, 1998
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