5 "Finale furioso''

In the final test in this paper, we add all previously discussed errors into a single forward computation of the artificial data and then run the image reconstruction on them. The choice of the errors are now such that a reasonable but still worst-case scenario is constructed. Extreme errors, like having the damping constants off by an order of magnitude or more, are not considered. We also proceed as if we had real stellar spectra and photometry and performed a spectrum synthesis to predetermine atmospheric parameters such as $T_{\rm eff}$ and $\log g$ and then adjust the elemental abundances so that the computed line equivalent width matches the observed one and the photometric zeropoint. The artificial data and its errors applied are in detail:

• S/N = 200:1;
• Fourteen unevenly distributed profiles;
• A single phase gap of 80 $\hbox {$^\circ $ }$ from 5 $\hbox {$^\circ $ }$ to 85 $\hbox {$^\circ $ }$ ;
• Continuum slope of $\pm$0.5%;
• Scattered light of 1%;
• Sum of the damping constants off by -0.05 dex;
• Radial macroturbulence is twice as large as the tangential macroturbulence;
• Rotational broadening is overestimated by 0.5 kms ;
• The inclination of the stellar rotation axis is 10 $\hbox {$^\circ $ }$ below the nominal value;
• Fourteen evenly spaced $VI_{\rm c}$ light-curve points are included in the inversion.

The result for Case 1 is shown in Fig. 16. For the recovery of this image the abundance of Fe was increased from nominal by 0.04 dex. Again notice that the phase gap is now between 5 $\hbox {$^\circ $ }$ and 85 $\hbox {$^\circ $ }$ (instead of 265 $\hbox {$^\circ $ }$ to 5 $\hbox {$^\circ $ }$ ), which clearly affects the reconstruction in this range. First, the polar appendage appears as a separated spot just barely linked to the polar cap and, secondly, an artificial spot appeared at a longitude of 90 $\hbox {$^\circ $ }$ and a latitude of +25 $\hbox {$^\circ $ }$ . Its contrast is very weak though, just 280 K below the photospheric temperature but probably would have been judged real if reconstructed from real data. We also notice that the polar cap had been recovered with smaller size than in the input map. This causes the steep temperature gradient seen in the difference map (Fig. 16c) at a latitude of $+65\hbox{$^\circ$ }-70\hbox{$^\circ$ }$. The small double spot at a longitude of $\ell\approx230$ $\hbox {$^\circ $ }$ is missed entirely while the hot spot at $\ell\approx120$ $\hbox {$^\circ $ }$ shows up in the reconstruction. Overall it is evident throughout these tests that while the visibility of features recovered in Doppler images may be under discussion, misplacement of the recovered feature is not a problem. As the reader can see from the figures presented, we can rely on positional information for the assessment of differential rotation or any other secular motion with confidence.