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Astron. Astrophys. 322, 511-522 (1997)

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7. A parameter study of the Doppler images of IN Comae

The following parameter study has been carried out, first, to refine critical input parameters that remained fixed during the inversion and, second, to better understand their influence on the surface maps. This will still only lead to better "estimates" because the individual stellar parameters depend upon each other in a very non-linear way but nevertheless allows to judge the reliability of our Doppler maps.

7.1. The rotation period

Despite that we have good evidence now that 5.9 days is the rotation period of IN Comae and not its alias of 1.2 days, we nevertheless computed Doppler maps from data phased with the 1.20-day period and compared the results with the correct maps. The Ca I -6439 map with the 1.2-day period has already been shown in Hubl & Strassmeier (1995) and for a visual comparison we refer the reader to this (poster)paper. Note that the maps in the present paper supercede the previous results. The most surprising result is that all three 1.2-day maps show a small but relatively cool polar spot while two of the three 5.9-day maps do not. However, we must reject this polar spot as an artifact because the fits to both the line profiles and the photometry were much poorer than with the 5.9-day period.

7.2. Gravity

Gravities of [FORMULA] and 3.0 result in images that are not significantly different if the elemental abundances are also adjusted. For example, the Ca map with [FORMULA] and with [FORMULA] decreased by 0.2 dex but otherwise as in Table 3, is almost identical to the map shown in Fig. 6. A similar adjustment is impossible when model atmospheres with [FORMULA] and 3.5 are used; then, a hot and a cool band appear at high and low latitude, respectively and the [FORMULA] 's climb to unacceptable values. Because the initial elemental abundances are not known a priori we cannot specify the gravity to better than, say, 0.25 dex. A value of [FORMULA] was finally adopted for IN Comae and is in good agreement with the result of Jasniewicz et al. (1996) of 2.7 [FORMULA] 0.5 from a comparison with model spectra in the H [FORMULA] -wavelength region.

7.3. Effective temperature

The effective temperature of the "quiet" photosphere is used as the initial temperature for the map since we do not constrain the temperature to remain at this initial value anywhere on the map. We adopted initial temperatures between 4800 and 5600 K in steps of 200 K with a fixed model-atmosphere grid with effective temperatures between 3500 K and 6000 K in steps of 250 K taken, as usual, from the Kurucz (1993) CD-ROM. Fig. 7a shows the run of the achievable fits with the adopted [FORMULA]. We see that the iron-line inversions only marginally depend on the adopted initial temperature while the Ca I -6439 fits show a clear minimum between 5000 and 5200 K. The Fe I -6411 fits probably have also a weak minimum at 5000 K but the Fe I -6393 line shows practically no dependence.

[FIGURE] Fig. 7. Results from a parameter study of the IN Comae images. All panels plot one specific input parameter versus the residuals of the achieved fit expressed as the sum of the squared [FORMULA] 's from all line profiles and the two photometric bandpasses. Panels e and f are for the Ca I 6439 line only. The parameters in the individual panels are: the unspotted surface temperature [FORMULA] (panel a), the inclination of the stellar rotation axis i (panel b), the equatorial rotation velocity [FORMULA] (panel c), the radial-tangential macroturbulence [FORMULA] (panel d), the elemental abundance for Ca for three values of microturbulence (panel e), and the microturbulence [FORMULA] for three abundance values of Ca (panel f). The arrow in each panel indicates the adopted value in the final solution.

However, the [FORMULA] values alone are not sufficient to choose the correct [FORMULA]. We also need to look at the reconstructed maps and their overall consistency. Examining the reconstructed minimum and maximum surface temperatures as well as their differences [FORMULA] for each adopted [FORMULA] we find that, in general, the resulting maximum map temperature increases with higher input temperature for all three lines. The minimum values, however, show a rather different behavior for the three lines: while for Fe I 6411 the temperature minimum becomes lower the higher the adopted input temperature, the Ca I -6439 map shows exactly the opposite behavior; the lower [FORMULA] the lower [FORMULA]. The minimum temperature from the Fe I -6393 line first decreases with increasing effective temperature but then increases once [FORMULA] is above 5000 K.

Our choice for the best value for [FORMULA] is now based on the consistency of the reconstructed maps and their temperature range [FORMULA] (not shown here). This is achieved at an effective temperature of 5200 [FORMULA] 100 K and results in [FORMULA] values between 820 K and 880 K for the three spectral lines.

7.4. Inclination of the stellar rotation axis

Fig. 7b shows the dependence of the achievable fit on the adopted value of the inclination of the stellar rotation axis i. Images were run for all three lines with inclinations between [FORMULA] and [FORMULA]. The best-fit inclinations are between [FORMULA] and [FORMULA] but the maps remain more or less identical within [FORMULA] and [FORMULA]. Significant differences are only seen between the [FORMULA] and [FORMULA] maps in that the reconstructed minimal temperature continuously increases by about 100 K for Fe I 6393 as well as for Ca I 6439 but decreases by about 350 K for Fe I 6411. Also, at [FORMULA] mirroring effects between the north and the south hemispheres become obvious.

7.5. Equatorial rotational velocity

The equatorial velocity was altered between 89 and 103 km s-1 while the inclination was kept constant at [FORMULA]. This equals a [FORMULA] range between 63 and 73 km s-1. Fig. 7c shows the resulting [FORMULA] 's as a function of the equatorial velocity. The minima for all three lines are between 97 and 100 km s-1.

Table 5 lists the achieved [FORMULA] s and the minimum and maximum surface temperature and their difference [FORMULA] as a function of the adopted equatorial velocity. Once again, it is the overall consistency between the individual maps, e.g. expressed as [FORMULA] in the last column of Table 5, that leads us to adopt [FORMULA] km s-1 to be the most likely value for the unprojected rotational velocity.


[TABLE]

Table 5. [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] for various equatorial rotational velocities [FORMULA] (only for Ca I 6439)


7.6. Macroturbulence

Fig. 7d shows the results from varying the amount of the symmetric radial-tangential component between 1.0 and 7.0 km s-1. Obviously, the reconstructed maps do not significantly depend on the correct value for the macroturbulence.

7.7. Microturbulence and abundance

Contrary to macroturbulence, the sensitivity on the adopted microturbulence is very strong because it directly influences the line equivalent width. Therefore, any change in microturbulence is to be accompanied with an appropriate change in the chemical abundances in order to match the observed profile. Otherwise no fit is possible at all. Fig. 7e shows the reconstructions for Ca I 6439 as a function of logarithmic abundance and for three fixed values for the microturbulence, while Fig. 7f shows the same but as a function of microturbulence and for three fixed values for the Ca abundance.

At first glance the [FORMULA] 's get smaller the higher the Ca abundance and the lower the microturbulence. However, with increasing abundance (or decreasing microturbulence) the minimum and maximum temperatures increase not uniformly but create a hot band at a latitude of [FORMULA] with an average temperature that is up to 600 K above the unspotted temperature, thus a very pronounced effect. Minimizing this effect leads us to the adopted values in Table 3 ([FORMULA] km s-1 and [FORMULA]). Tables 6 and 7 list the [FORMULA] 's and minimum and maximum temperatures as a function of calcium abundance and microturbulence, respectively.


[TABLE]

Table 6. [FORMULA], [FORMULA] and [FORMULA] from Ca I 6439 with fixed [FORMULA]  km/s but for various Ca abundances (relative to [FORMULA])



[TABLE]

Table 7. [FORMULA], [FORMULA] and [FORMULA] from Ca I 6439 with fixed [FORMULA] but for various microturbulences [FORMULA]


7.8. Regularising functional

Finally, we compared two different regularising functionals for the profile inversion: a maximum-entropy function (MEM) and a Tikhonov function. A detailed formulation and description of these algorithm has been given by Piskunov et al. (1990), Collier-Cameron (1992), and recently by Piskunov & Rice (1993) and we refer the reader to these papers. Principally, the MEM reconstruction favors an image with no pixel-to-pixel dependence while the Tikhonov image favors a solution with a smooth image. We found no significant difference between the two regularisation functionals.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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