4 Input-parameter errors

4.1 Confusion in recovery of hot vs. cool spots

For this test, we moved the single hot spot in our original input map (Fig. 1) to different latitudes to see whether we can still recover its correct location and whether it influences the recovery of the cool spots. Two tests are made, one with the hot spot close to the stellar equator spanning the latitude range 0 $\hbox {$^\circ $ }$ to 25 $\hbox {$^\circ $ }$ , and the other with the hot spot close to the rotation pole spanning latitudes from 60 $\hbox {$^\circ $ }$ to 85 $\hbox {$^\circ $ }$ . Figure 11 shows the results. The lower part of the figure shows slightly higher temperatures than the 5000 K original temperature of the model because the initial trial temperature for TEMPMAP was set at 5200 K to simulate more realistic situations where the temperature of the unspotted photosphere is unknown. In those "southern'', rarely observed parts of the star the program has little incentive to modify the temperature from the starting value and limited regions with slightly elevated temperature can be seen. No area in the originally unspotted part of the star was recovered at a temperature of more than 50 K above the recovered temperature of the surrounding unspotted regions of the original model except immediately adjacent to very dark regions such as the polar extension where the temperature "rebounds'' to about 150 K above the original (although threshold effects in the figures occasionally exaggerate the differences in adjoining regions). A test with a model completely without a hot spot on the surface showed no recovered temperature above 5100 K (or 100 K above the original) anywhere above a latitude of -40 $\hbox {$^\circ $ }$ in Case 1 except, again, immediately adjacent to the polar extension where the temperature reached 5150 K. We conclude then that Fig. 11 shows an excellent recovery for both cases and this then strengthens our findings of warm spots on the WTTS V410 Tau (Strassmeier et al. 1994; Rice & Strassmeier 1998) and on the K0 giant XX Tri (Strassmeier 1999). Note though that for Case 2 (the lower inclination case) the high-latitude hot spot affects the temperature distribution within the cool polar cap by up to 1000 K in the immediate vicinity but not so for Case 1. This is a geometric projection effect because the latitudinal location of the hot spot appears the more elongated towards the back hemisphere of the star the lower the inclination.

4.2 Atmospheric parameters

It has been argued (e.g. Byrne 1996) that an active star's atmosphere undergoes large amounts of non-radiative heating which may influence their upper photospheres. This, in turn, could give rise to a filling-in of the line cores of strong photospheric lines. Recent non-LTE synthesis of the Ca I 6439-Åline (the most commonly used spectral line) showed that its flat-bottomed profile structure can not be synthesized with chromospheric activity at the visible rotation pole and that the results are practically identical for strong and weak lines and thus agree with the standard case of a LTE assumption (Bruls et al. 1998). A direct comparison of Doppler images of AB Dor from the strong sodium D lines with several weaker photospheric lines also did not show an artificial polar spot (Unruh & Collier Cameron 1997). Nevertheless, we agree with the arguments of Byrne (1996) above and perform tests in which we adopt certain atmospheric input parameters for the forward problem but then invert the artificial data with wrongly chosen atmospheric parameters.

4.2.1 Radial-tangential macroturbulence

The first test deals with macroturbulence and its radial and tangential isotropy across the stellar disk. As is usual in applications to solar-type stars (e.g. Gray 1992), we adopt a radial-tangential macroturbulence velocity with equal radial and tangential components as our standard case (listed in Table 1). The forward computations are now done with a purely tangential velocity component of 5 kms but the reconstructions are performed with the parameters of the standard case, i.e. 3.5 kms and isotropic. In Fig. 12a, we show the maps recovered for Case 1 and Case 2, respectively, together with their difference maps with respect to the input map in Fig. 1. We choose the highest possible S/N ratio for the artificial data in order to clearly isolate the effect.

Figure 11: Recovered maps with the original hot spot placed on a) the equator and b) close to the pole. A correct recovery is possible, even for hot spots. Left panels are for Case 1, right is for Case 2. Note the local brightening of the polar spot in the vicinity of the hot spot (panel b), right map)

Figure 12b is similar to Fig. 12a but that the forward model is computed with a purely radial macroturbulence model and then recovered with the standard-case parameters. Both figures thus represent the most extreme influence possible and cause the generation of a dark band at intermediate latitude for Case 1. A test with more moderate values for the anisotropy and also for smaller differences of the amount of macroturbulence resulted in maps so similar to the originals that we do not show them separately.

4.2.2 Gravity

For the radiative transfer calculation within TEMPMAP, we usually input up to 10 model atmospheres of different temperature but, of course, the same gravity. Our goal now is to estimate the influence of a set of atmospheres with wrongly chosen $\log g$.

The value for $\log g$ is usually pre-estimated from a trial-and-error fit of a synthesized spectrum to one or several particularly gravity-sensitive lines like e.g. Ca I 6439

Å.Its uncertainty usually amounts to $\pm$0.5 dex for giants and $\pm$0.25 for G- and early-K dwarfs (see, e.g. Strassmeier & Rice 1998a).

Figure 13 shows the two maps and light curves for Case 2 reconstructed with wrong $\log g$ atmospheres. Figure 13a is a test with ten input atmospheres with $\log g$=4.5 (instead of 4.0 as in the forward computation), and Fig. 13b with $\log g=3.5$. Both recoveries have no problem of finding the correct spot geometry, including the polar spot as well as the smaller low-latitude spots. The misfit to the light curve zeropoints (in our case the $V-I_{\rm c}$ color), however, shows up immediately. The higher equivalent width of the local line profile for the $\log g=4.5$ atmospheres additionally results in a bright band with $\Delta T\approx240$ K encircling the star between a latitude of $30\hbox{$^\circ$ }-60\hbox{$^\circ$ }$, while the correspondingly lower value for the $\log g=3.5$ atmosphere results in an overall cooler surface by $\Delta T\approx200$ K. If the gravity mismatch would go unnoticed in a real application, we would alter the abundances of that particular element to increase or decrease the equivalent width by an amount that removes such artifacts and agrees with the color zeropoint.

Figure 12: Effects of the anisotropy of macroturbulence velocity. Panel a) shows the case of a purely tangential macroturbulence but recovered under the assumption of an isotropic radial-tangential distribution. Panel b) is the recovery from a purely radial macroturbulence distribution but otherwise identical assumptions. See text

Figure 13: Tests with model atmospheres of different gravity. a) The input gravity was set to $\log g=4.5$ instead of 4.0 as in the forward computation. b) Here it was assumed $\log g=3.5$. Notice the inability to fit the color-curve zeropoint. Only Case 2 is shown

Figure 14: A test with different line-damping constants. a) A change of +0.15 dex of the nominal value of all three damping constants, b) a change of -0.15 dex. The left column is always for Case 1 and the right column for Case 2. For each test, we show the reconstructed map and the V and $I_{\rm C}$ lightcurves and their fits. Note the increasing misfit when all three damping coefficients are altered. A change of $\pm$0.15 dex is also the range of values where our code can react in terms of lowering and raising the overall surface temperature. Any change above this value results in a significant misfit of the line equivalent width and spurious hot and cool spots

  
4.3 Atomic line parameters

Many atomic parameters of a particular transition contribute to a stellar spectrum. Most notably the central wavelength, the excitation potential, i.e. its sensitivity to temperature, the damping constants, i.e. the population level and its lifetime, and the transition probability, i.e. the strength of a line. In the past, we found that Doppler imaging is mostly prone to wrong values for the transition probability and, to a lesser degree though, to the various damping constants. We will thus concentrate our tests on these two parameters.

4.3.1 Transition probability

The number of electrons per second that will spontaneously jump from an upper level to a lower level is proportional to the Einstein probability coefficient for spontaneous emission or absorption, to the population of the upper or lower level, and to the energy $h\nu$thatseparates the two levels. These quantities are linked to the emission or absorption coefficient and thus to the specific intensity of the line. In practice, a single value, the so-called $\log(gf)$ value, determines the line intensity. It is an input parameter in our Doppler imaging and either taken from atomic-line databases such as VALD (Kupka et al. 1999) and/or obtained by fitting the appropriate parts of the solar spectrum (e.g. Valenti & Piskunov 1996; Strassmeier et al. 1999). It is an important parameter and was tested earlier in Strassmeier (1996) for Ca I 6439, and we refer to that paper since the conclusions did not change for the line used in the present paper. We note that the effects of a wrong $\log gf$ on the maps are indistinguishable from the effects of wrong elemental abundances.

Figure 15: Reconstruction of Case 1 with inclination angles between 5 $\hbox {$^\circ $ }$ and 85 $\hbox {$^\circ $ }$ . The vertical axis is $\chi ^2$ which here is just the sum of the squared difference between the model profile points and the 91 artificial profile data points. A clear minimum near the correct inclination is obtained

4.3.2 Damping constants

TEMPMAP automatically computes three damping constants; one for natural (radiative) damping, one for Stark damping, and one for van der Waals damping. The user has the choice of automatic computation by using the Unsöld (1955) expressions for $\gamma_{\rm rad}$, C₄, and C₆or presets his/her own favorite values in the input file. The automatic computation of the three relevant damping constants for the Fe I 6411.649-Åline that are used for the tests in this paper are (in parenthesis are the Kurucz-(1993) values for comparison purposes) 7.732 (7.905), -5.096 (-5.455), and -7.670 (-7.622) for radiative, Stark, and van der Waals broadening, respectively. We found that for the temperatures and densities of active G- and K-stars and their cool spots, the natural and the van der Waals broadening are about equal sources of damping, and we thus alter just the radiative damping constant for some of our tests and the sum of all three for the remaining tests.

Figure 14 shows two sets of tests. One has the sum of the damping constants set to +0.15 dex of the nominal value, and the other has the sum set to -0.15 dex. Both tests are made for the high-inclination case (Case 1, i=65 $\hbox {$^\circ $ }$ ) and the low-inclination case (Case 2, i=30 $\hbox {$^\circ $ }$ ). A value of $\pm$0.1-0.2 dex represents the "worst-case'' uncertainty of this parameter and was estimated from the spread of damping constants in VALD for several neighbouring lines. A $\pm$1.0-dex variation in the radiative damping alone turned out to be the amount of change that is required to recognize some significant changes in the maps.

Figure 16: Test reconstruction with all previous errors included. Case 1 with phase gap between $5\hbox {$^\circ $ }-80\hbox {$^\circ $ }$. a) Spherical projection, b) Mercator projection, c) difference map between input and reconstructed map in Mercator projection, d) light curve in V and $I_{\rm C}$ and the fits, e) Cross correlations between input and reconstructed map. Top panel: correlation in longitude given in surface pixels, bottom panel: in latitude ( xwidth is the FWHM of the cross-correlation function in longitudinal direction in surface pixels, ywidth in latitudinal direction). The grid is 72$\times$36 pixels. f) is the artificial line profiles and the fits

The tests in Fig. 14 show significantly hotter and cooler reconstructions for higher and lower damping, respectively. Still, the color zero point can not be fitted at all and an offset of approximately $\pm$0 $.\!\!^{\rm m}$1 in $V-I_{\rm c}$ for $\mp$0.15 dex, respectively, remains. This would be easily spotted in real photometric data. Furthermore, the line equivalent width that is added (or removed) in order to fit the observations would be compensated for by adjusting the elemental abundances by hand. This is a regular procedure before the actual line-profile inversion with TEMPMAP.

The tests with $\pm$0.15 dex of the radiative damping constant alone show practically identical reconstructions compared with the nominal value except maybe for lower damping for Case 2. There, some structure appears in the polar cap that is not in the input map. The only recognizable misfit for both cases is a slight zero-point offset in $V-I_{\rm c}$ of $-0\hbox{$.\!\!^{\rm m}$ }01$ (i.e. the reconstructed image got hotter by 30 K). By the time the radiative damping constant is increased by 1.0 dex, the color offset amounts to a full 0 $.\!\!^{\rm m}$1 (at +1.5 dex even 0 $.\!\!^{\rm m}$25), thus we would observe a severe observational mismatch. However, it can be compensated by adjusting the elemental abundance by just -0.17 dex. We do not show the really extreme cases with damping increased by more than 1 dex because they are totally unrealistic and usually would be spotted in the data even before the mapping because the reconstructed equivalent width becomes about half the observed. Lowered values for the radiative damping have a slightly less pronounced effect on the maps and almost none on the broad-band colors because the van der Waals broadening keeps up the line-equivalent width.

4.4 Inclination

In this test, we assume a wrong inclination in the reconstruction and run TEMPMAP on artificial data with S/N=3000:1 in steps of 5 $\hbox {$^\circ $ }$ from $i=5\hbox{$^\circ$ }$ to $i=85\hbox{$^\circ$ }$. Figure 15 demonstrates that the correct inclination can be recovered just from the run of the $\chi ^2$ as a function of i. The width of the minimum in the curve in Fig. 15 would allow one to adopt inclination values within a few degrees of the nominal value with only a small variation in the goodness of fit. However, there are a lot of other adjustable parameters that need to be optimised at each value of the inclination before a formal error on the inclination can be estimated. Since this was not done in the present test, a $1-\sigma$ error quote is likely too optimistic for real applications but nevertheless illustrates the usefullness of this method of estimating i.

In an earlier paper, Hatzes (1996) had shown that low-inclination stars had a more pronounced flat-bottomed shape compared to high-inclination stars. The line-profile shape is thus an indicator for the inclination of the stellar rotation axis and the quality of our fits to real data is such that we can achieve an inclination to within $\pm$10-15 $\hbox {$^\circ $ }$.