6 Conclusions

Doppler imaging of stellar surface structure is a novel technique with similarities to medical brain tomography (instead of a fixed brain and a rotating scanner, astronomers have a fixed spectrograph and a rotating brain, well alright, star of course). The number of free (internal) parameters is at first glance of the order of the number of surface grid points, i.e. $\approx$2600, and is constrained by the number of input data points (for our test cases 1638 profile points and 24 light-curve points.) This obviously underdetermined situation makes some astronomers uneasy about Doppler imaging but modern inversion algorithms with penalty functions of the form of maximum entropy or Tikhonov etc. together with the surface Doppler-shift constraint from the stellar rotation restricts the freedom of the surface grid points so that we are left with only 11 parameters that are in any sense mathematically free (the eight listed in Table 1 plus three atomic parameters; $\log gf$, damping, and central wavelength). Yet these 11 parameters are still not completely independent or free of physical constraint. For example if $v_{\rm eq}\sin i$ is measured and i fixed, $v_{\rm eq}$ is no longer a free parameter. The same is true if we fixed the gravity and the line strength, then the microturbulence is no longer completely free because the equivalent width must be reproduced. This shrinks our problem basically to the common range of uncertainties of stellar astrophysical parameters and atomic line data.

In this paper, we visualized the quantitative influence of the uncertainties of many of the stellar quantities and atomic data. We demonstrated the extreme robustness of our Doppler-imaging code to even exaggerated assumptions, e.g. a recovery with a phase gap of 100 $\hbox {$^\circ $ }$ (0 $.\!\!^{\scriptscriptstyle\rm p}$28) with moderate S/N still correctly recovered the spots located within the phase gap. Having simultaneous photometry in two bandpasses not only gives a better handle on the overall temperature of the stellar surface but when used with the line equivalent width the photometry provides a powerful additional constraint so that we are forced to make adjustments in factors such as element abundance to compensate for uncertainty in atomic parameters or $\log g$. This works to minimize the impact on the recovered image when errors in adopted line parameters such as the damping constants occur. We conclude that our stellar maps published so far are relatively free of systematic artifacts and should be considered as the best possible solution for the data available.

Acknowledgements

JBR acknowledges financial support from the Natural Science and Engineering Research Council of Canada (NSERC). KGS is very grateful to the Austrian Science Foundation (FWF) for support under grants S7301-AST (APT) and S7302-AST (Doppler imaging). We would like to thank the referee, Dr. Collier-Cameron for a very careful and thoughtful report on this paper. His comments were very useful and supportive.