4 Time-series Doppler imaging

  
4.1 The line-profile inversion code

As for all previous papers in this series, the maps were calculated with the Doppler-imaging code TEMPMAP (Rice et al. 1989; Piskunov & Rice 1993; Rice 1996). See e.g. Paper X (Strassmeier et al. 1999b) for an updated description of the program and Paper VI (Strassmeier & Rice 1998) and Rice & Strassmeier (2000) for numerical tests with artificial data.

Because of the small wavelength coverage of our NSO spectra, we could only use three main mapping lines: Fe I 6421, Fe I 6430 and Ca I 6439 with $\log gf$ values of -2.23, -2.0 and +0.47 and lower excitation potentials of 2.279, 2.176 and 2.526 eV, respectively. Since all three lines are blended to a certain degree - the 6421-Å region seems to be especially vulnerable - the number of lines which had to be synthesized was 6, 6, and 7 for the 6421, 6430 and 6439-line regions, respectively. All these blends are included in the inversion and treated simultaneously with the primary mapping lines but only one spectral region can be handled per solution. We employed a maximum-entropy regularization for the Doppler maps presented in this paper, but the program also allows a Tikhonov regularizing functional (for a comparison see, e.g., Piskunov & Rice 1993). An appropriate set of model atmospheres was taken from the ATLAS-9 CD (Kurucz 1993). All radiation transfer calculations are done under the assumption of LTE.

   
4.2 Finetuning the rotational velocity and the inclination of the stellar rotation axis

Figure 4: Optimization of some stellar parameters for Rotation 1. a) The inclination in degrees, b) the projected rotational velocity, c) the calcium abundance in units of $\log n\rm (H)=12.0$ and d) the iron abundance. The full line is for the Ca I 6439 line, the dashed line for Fe I 6430, and the dotted line for Fe I 6421. The vertical axis plots $\chi ^2$ in relative units.

Because the Doppler-imaging analysis is 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 method described in Sect. 3 (see also, e.g., Unruh 1996). Changing these two parameters one at a time, while all others are kept constant, yields various values for the misfit between the data and the model ($\chi ^2$). The value of the parameter corresponding to the smallest $\chi ^2$ is the one we believe is closest to the true value. The variation of $\chi ^2$ with the inclination i and the rotational velocity $v\sin i$ are plotted in Figs. 4a and b, respectively. A minimum is seen in both cases: for the inclination between $40\hbox{$^\circ$ }$ and $60\hbox{$^\circ$ }$, and for the projected rotational velocity between 27 and 29 kms^-1. We obtain $v\sin i=28.2\pm0.7$ kms^-1 and $i=50\hbox{$^\circ$ }\pm 10\hbox{$^\circ$ }$ as our final values, according to the unweighted average minimum in Figs. 4a and b.

  
4.3 Finetuning the elemental surface abundances

To determine approximate elemental abundances, we evaluate the run of the $\chi ^2$ from a series of inversion solutions with different initial abundances. A straightforward computation of the local line profiles with solar abundances and from model atmospheres with gravities between $\log g=2.0{-}3.0$, and micro- and macroturbulences between 0.5-4.0 kms^-1, already indicated a relative underabundance of calcium and iron with respect to the Sun. Our test inversions were thus started with solar abundances and decreased in steps of 0.1 dex (Figs. 4c and d). The transition probabilities, damping constants, and laboratory wavelengths were kept constant at the values specified in Sect. 4.1 and taken partially from the VALD database (Kupka et al. 1999) and our previous papers in this series. We then adopted the abundances that resulted in the least $\chi ^2$ as the most probable values for the surface abundances, i.e. $\log n(\mbox{Fe}) = 7.03$ and $\log n(\mbox{Ca}) = 5.32$ (on the $\log n(\mbox{H}) = 12.00$scale). Although an internally consistent set of parameters of high precision, these abundances are of low accuracy because test inversions with different sets of fixed parameters (i.e. mostly different $\log g$ and microturbulence) resulted in similar Doppler maps but with individual abundances different by up to $\pm$0.2 dex.

4.4 Doppler maps for two consecutive stellar rotations in 1996/97

Figures 5a,b shows the average maps from three spectral regions and for two consecutive stellar rotations, respectively. One spectral region and two photometric bandpasses, V and $I_{\rm C}$, are used simultaneously to produce a single map. Three such maps from Ca I 6439.08, Fe I 6430.84 and Fe I 6421.35 are then used to create an average map by averaging the temperature in each pixel. Each map is given equal weight because the overall $\chi ^2$ achieved is similar. We caution though that the averaging may lead to an overemphasizing of the strongest lines because of their larger intrinsic line width and thus a more "smeared-out'' temperature distribution relative to a weaker line. Therefore, we also present the individual maps in Figs. 6 and 7 for the two stellar rotations, respectively. All computations are performed on a DEC-Alpha 500/400 workstation and require between 20 to 30 min of CPU time depending on the number of spectral blends, the number of input model atmospheres, and the number of photometric data points.

Figure 5: Combined Ca and Fe maps for Rotation 1 a) and Rotation 2 b). The bottom map c) is computed from all spectral data combined and represents the time average for the two rotations. The rotation period is 25 days and the arrows below each map indicate the phases of the spectra.

Figure 6: Individual Doppler images for Rotation 1. a) Ca I6439.08 Å with an equivalent width of 272 mÅ, b) Fe I6430.84 Å with 270 mÅ, and c) Fe I6421.35 Å with 221 mÅ. Temperature maps from the individual lines are plotted in a pseudo-mercator (top row) projection where the arrows indicate the spectroscopic phase coverage. The observed and computed line profiles are shown in the middle row, and observed and computed V and $I_{\rm C}$-lightcurves in the bottom row. The pluses and the bars are the observations and the full lines are the fits. The size of the bars correspond to a $\pm$1$\sigma$ error per data point.

Figure 7: Individual Doppler images for Rotation 2. Otherwise as in Fig. 6.

 

 

Table 3: Table of surface features A-F from the average maps ($\ell$ longitude, b latitude, $T_{\rm spot}$ spot temperature).

Spot	Rotation 1			Rotation 2
	$\ell$	b	$T_{\rm spot}$	$\ell$	b	$T_{\rm spot}$
A	20	35	4200	50	40	4050
B	85	35	4250	110	35	4250
Ca	160	45-75	3900	160	30-75	3850
D	220	40	4050	230	35	4100
E	285	40	4250	280	35	4100
F	340	30	4150	330	45	4100

^aThis feature appears to be connected with the polar spot in both maps; its range in latitude is listed.

A visual comparison of the two consecutive maps in Figs. 5a and b reveals a comparable spot morphology as well as consistent absolute temperatures. A cool polar spot with an appendage centered around a longitude of $\ell\approx 160\hbox{$^\circ$ }$ and a latitude of ${\approx} 60\hbox{$^\circ$ }$- $70\hbox{$^\circ$ }$is seen in both maps (referred to as spot C in Table 3). Its lower end reaches almost down to the equator in Rotation 1, but we believe that this is partly due to latitude smearing in the image reconstruction than a real single appendage and it is entirely possible that the feature consists of a high-latitude appendage and a low-latitude spot at the same longitude (the narrower Fe I6421 line shows actually two features well separated). The Fe I6421 map for the second rotation does not indicate the low-latitude spot anymore but instead show an enhanced polar appendage. One possible explanation could be that the lower spot had merged with the polar appendage within a single stellar rotation.

Additionally to the polar spot there seem to be five spots at low-to-medium latitudes (named A, B, D, E, F from left to right in Fig. 5c). See Table 3 for a summary of the spot positions on both maps. Note that these numbers are measured off the maps by treating the Doppler image as a CCD image in the CCD-photometry package digiphot in IRAF. We define a box around the approximate location of a feature and then use the IRAF routine phot to find a local (temperature) minimum within this box. The resulting coordinates have sub-pixel precision but a significantly worse accuracy that is determined by a large variety of external sources (see Rice & Strassmeier 2000). We estimate an average of $\pm$5$^\circ$ in latitude and somewhat less than that in longitude. The coolest of these spots has a temperature of about 800 K below the effective photospheric temperature of 4700 K, and the warmest is approximately 450 K cooler. Typical errors for spot temperatures from TEMPMAP Doppler images are 70-100 K.

The equivalent widths of Ca I6439 Å and Fe I6430 Å are identical within the observational errors but the Fe I6421 line is approximately 20% weaker. It is severely affected by the nearby Fe I6419 line and its large number of blends that affect the blue wing of the Fe I6421 line. Its fit to the observed light curves of the second rotation is also noticeably different to the fits from the other two lines. Despite this inconsistency, all spectral lines indicate a cool polar cap. The main difference in the maps is that the polar spot from the Fe I6421 line appears 100-150 K cooler than in the other two maps while the low-latitude spots appear warmer by that amount and, consequently, the polar spot's enhanced contrast seems eye catching. It is a known problem of line-profile inversions that a polar spots' size and contrast depends strongly on the fit to the line wings (see Unruh et al. 1996) and thus the contrast of the 6421-line is not a big surprise.

Nevertheless, to disentangle such inconsistencies, we compute a rotation-averaged map by combining all spectral data into one data set and invert it as usual. This map is shown in Fig. 5c; it is already the average from all three spectral regions. To quantify the differences between the fits of the individual rotations and the rotation-averaged fit, we used the average map to calculate synthetic line profiles in a forward solution and compare them with the inverse solutions from the two rotations. The difference of their $\chi ^2$ is then an indirect measure for the reality of surface changes from one rotation to the next. Figure 8a summarizes these changes. Each individual point in this figure is the squared minimum of the residuals between a single observation and the rotation-averaged fit. We see, firstly, that the misfit is on average larger in the second rotation and, secondly, that there are two phases in both rotations with consistent $\Delta\chi^2$ peaks from all three spectral regions (at $\approx$100$^\circ$ and $\approx$300$^\circ$). The plots of the difference maps in Figs. 8b and c finally show the changes on the stellar surface. We find that the asymmetry of the polar-spot appendage at $\approx$200$^\circ$ had vanished in the second rotation and that two high-latitude regions appeared at $\approx$40$^\circ$ and $\approx$300$^\circ$. Two or maybe three of the lower-latitude spots seemed to have migrated. We interpret these differences to be due to real changes on the stellar surface and will investigate them further in the next chapter. For the Fe I6421 line profiles, the $\chi ^2$ values in Fig. 8a are sometimes double than those for the two other lines. We thus consider the severely blended Fe I6421-line map more uncertain than the other two, albeit its overall $\chi ^2$ is comparable and it recovers the features at approximately the same locations.

Figure 8: a) The normalized goodness of fit for each observed line profile of the rotation-averaged Doppler-imaging solution. We use a $\chi ^2$ statistic based on the three wavelength regions to obtain a single value for each spectral line profile. The photometry was not considered for this test. The grey area emphasizes the average from the three wavelength regions. A systematic deviation from zero is seen around two longitudes at $\approx$100$^\circ$ and $\approx$300$^\circ$, that suggests surface changes from one rotation to the next. b) and c) show the difference maps Rotation 1 minus rotation-averaged map and Rotation 2 minus rotation-averaged map, respectively (shown is just the average from all spectral lines).

A further test of our Doppler images is to divide the spectra into "even'' and "odd'' data sets for each rotation, i.e. using first even-numbered spectra for an inverse solution and then the odd-numbered spectra, and then compare the resulting maps. The images from the rotation-averaged sets correlate very well, but the odd and even images for the individual rotations do not. The reason is that the phase coverage is just too sparse to derive results of the same quality as for the combined data set. Thus, the even-odd test is unfortunately not overly useful to verify the cross-correlation analysis applied in the next chapter.