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Astron. Astrophys. 354, 537-550 (2000)

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4. Doppler imaging

Our spectroscopic observations covered 20 consecutive rotation cycles with a total of 52 spectra. For our study of the long-term variations, we divided these data into three subsets (S1, S2, and S3). The respective mid-points in time are 1996.860, 1996.912, and 1996.970. Later, in Sect. 4.3, we will adopt a different data-separation philosophy to investigate the short-term changes. To illustrate the phase coverage for both the spectroscopy and the photometry, we plot the spectroscopic phases versus Julian date in Fig. 4. The simultaneous photometric data and the boundaries of the subsets are indicated as well.

[FIGURE] Fig. 4. Phase coverage and time-resolution for the Doppler images in this paper. The left panel plots the phases of the spectroscopic observations versus Julian date according to Eq. (1) (time runs from top to bottom). In the right panel, we plot the simultaneous V-band photometry along with a continuous spot-model fit. The dashed, horizontal lines separate the individual data sets (S1-S3) for the three Doppler images.

4.1. Image-reconstruction parameters and mapping code

Our Doppler-imaging code for late-type stars was originally developed by Rice et al. (1989) to map chemical abundances of Ap-stars. It was adapted to map surface temperature by Rice (1993, priv. commun.) and is continuously updated, e.g. by Rice & Strassmeier (1998). Briefly, we first perform a full LTE spectrum synthesis in each of the wavelength regions used by using a grid of ten model atmospheres from ATLAS-9 (Kurucz 1993) as well as a pre-calculated table of elemental abundances. The latter are by default the most up-to-date (photospheric) solar abundances obtained from various sources. We then alter the abundances - if necessary - by fitting the observed spectrum and then proceed with the altered abundance table. For each iteration a discrepancy integral between model and data is computed and minimized with the help of conjugate gradients. Actually, the "discrepancy" integral is a vector for every surface pixel that is determined from the changes of the previous iterations. Once it converges to zero, i.e. the gradient doesn't change anymore, the final solution is found and then an error, i.e. basically a [FORMULA] statistics, between the last iterated model and the data is given. The latter is computed from the combined line profile and light-curve data.

The number of iterations is preset to 20 in this paper (it depends on the available number of line profiles and photometry) which is sufficient for correct convergence in the present case. Simultaneous inversions of up to twenty lines with either a maximum-entropy or a Tikhonov regularisation is possible but we usually include no more than 10 blends per main-mapping line to keep the CPU time down. For this paper we chose the maximum-entropy regularisation but, because the signal-to-noise ratio of our data is sufficiently high, we could have also chosen a different regularisation. Note that we invert the V and [FORMULA] light curves simultaneously and thus further constrain the Doppler image. More detail of our code and additional references can be found in previous papers of this series (e.g. Strassmeier et al. 1998) and in Piskunov & Rice (1993).

The astrophysical input parameters for V711 Tau are summarized in Table 3. The [FORMULA] value was obtained with good precision by minimizing the characteristic artificial features (dark or bright bands at the co-rotating latitude). The value obtained was 41[FORMULA]1 km s-1, in good agreement with the one found by Vogt et al. (1999) and Donati (1999).


[TABLE]

Table 3. Input astrophysical parameters of V711 Tau.


Fekel (1983) estimated an orbital inclination of [FORMULA] from a comparison of the age of the system and the masses and radii of the components. This value was also used by Vogt & Penrod (1983), Vogt et al. (1999) and Jankov & Donati (1995), but Donati (1999) preferred a value of [FORMULA]. The latter value is also suggested by our Ca-line test inversions that showed a minimum of the discrepancy integral at an error of 0.0854 1 at 40o as compared to 0.0915 at i=33o. Because such a small difference in inclination has no significant influence on the recovered spot distribution, we keep the inclination fixed at 40o for all further inversions. This is important because there are already many free parameters in Doppler imaging and correlations between parameters make it difficult to determine them independently. For example, a correlation exists between the transition probability ([FORMULA] value) of a spectral line, its elemental abundance, and the adopted microturbulence. The latter also affects the temperature range. In order to determine these values with better accuracy, we tried several combinations of these parameters within a reasonable range and found the best fits with the values listed in Table 3.

The initial atomic parameters for the blends in the main mapping-line region were obtained from Kurucz's (1993) line list. The transition probabilities in this list can sometimes be erroneous, especially for the weak lines of high-Z elements such as titanium, vanadium, and silicon. If no other value was available, e.g. from the Vienna atomic line database (VALD, Piskunov et al. 1995, Kupka et al. 1999), we first modelled a spectrum of the Sun and of Arcturus (K2III ) to refine the [FORMULA]'s of the lines in question. The values obtained are summarized in Table 4.


[TABLE]

Table 4. Atomic line parameters.


4.2. Images for 1996.860 (S1), 1996.912 (S2), and 1996.970 (S3)

Fig. 5 shows the Doppler images from the Ca I 6439-Å region for the three epochs S1-S3 along with the observations and the fits. The datapoints of each spectral line profile are shown as vertical bars and their length represents the [FORMULA]1 [FORMULA] error per pixel. The same is plotted in Fig. 6 for the Fe I -6430 region. It is important to realize that these errors are solely due to the per-pixel signal-to-noise ratio and do not include any external errors that, in our opinion, are much more severe than the photon noise and are almost impossible to correct for. These external errors are caused by mechanical flexure of spectrograph components resulting in, e.g., a time-variable optical misalignment of the transfer lens, linear and possibly non-linear drifts of the CCD in the camera dewar, focus and flat-field variations across the CCD due to ambient temperature and humidity variations, continuum-setting errors which may result in an unnoticed variable offset or in a residual slope of the continuum and, as mentioned earlier, due to the entire process of the secondary-star removal. Also, we must admit that operating a night-time spectrograph in a focus of a telescope that is heavily used during daytime is less than optimal. Beside these external data errors, we require our models to fit the spectral-line profiles simultaneously with absolute broad-band photometry in two bandpasses (i.e. a zero point for the bandpass difference plus the light-curve shape in two bandpasses). The use of photometry usually results in apparently less precise profile fits but gives a better handle on the average temperature contrast and is physically more sound if the photometry is secured simultaneously to the spectroscopy as in our case. Finally, our model must also reproduce the line-equivalent width and its variations due to the star being a variable, and not just the line-profile shape as some other approaches do. Meeting all these criteria means that the [FORMULA] from our line-profile fits is not driven by the S/N ratio of the spectra once S/N in excess of [FORMULA]200:1 is achieved but by the combined external errors of the spectra and the photometry. Therefore, we attribute some apparent line-profile misfits to the external errors in our data.

[FIGURE] Fig. 5. Ca I 6439-Å images for the three subsets S1, S2, and S3 (top row, from left to right). The maps are presented in a pole-on view from latitude -40o to 90o. The line profiles and their fits are shown in the middle row, the light curves and their respective fits are plotted in the bottom rows for V and [FORMULA], respectively. The phases of the observations are marked around the maps by arrows, latitudes are drawn in steps of [FORMULA].

[FIGURE] Fig. 6. Same as in Fig. 5, but for the Fe I 6430 line.

All images recover a structured polar spot which is off-centered with respect to the stellar rotation pole. This is in agreement with images from previous years (Vogt et al. 1999) and verifies the continuous existence of the polar spot since Doppler imaging of V711 Tau began in 1981/82 (compare with Fig. 1). Our Ca images also agree very well with a contemporaneous image obtained by Donati (1999) from data taken in December 24-29, 1996 (i.e. 1996.986) but obtained with a different mapping approach. Thus, there are now consistent maps for V711 Tau from three independent groups. All of them recover a polar spot which, in our opinion, is further evidence for its reality.

In Fig. 7, we plot the one-dimensional cross-correlation functions per latitude bin between the calcium and the iron maps. Perfect correlation is never reached because the temperature of the features recovered are not identical (but similar). This is due to the inherent uncertainties of our Doppler-imaging technique. Despite the fact that the Fe maps also recover an off-centered polar spot, there are features in the Fe maps that have no counterpart in the Ca maps and vice versa. This can be seen from a comparison of line-profile detail in Figs. 5 and 6, e.g. in the S1 data for phase 0:p585 where the Fe line has a much stronger bump in the red linewing as the Ca line. Although expected to a certain degree, because the equivalent width of the Ca line increases in the spot opposite to that of the Fe line, not all of it can be accounted for by the code, e.g., by lowering the spot temperature. For the phases between [FORMULA]0-0.5, we attribute such inconsistencies to the blending of the Fe-6430 line with the secondary's Fe II 6432-Å line and its subsequent subtraction from the composite spectrum. The strength of the secondary's Fe II line is approximated by the spectrum of the adopted template star (see Sect. 3.2) but, because it is never seen in a fully unblended phase, its subtraction remains uncertain. Similar difficulties do not exist for the Ca line despite that its mapping errors are always comparable to that for the Fe line ([FORMULA], excluding the photometry). Moreover, the Fe-6430 line contains two temperature-sensitive vanadium blends that change their strengths significantly with effective temperature. A slight spectral-type mismatch between the template star and the actual secondary spectrum could cause additional inconsistencies. Therefore, we consider the Fe results less reliable than the Ca results and base the further analysis solely on the results from the Ca line but will use the Fe data to confirm the basic findings from the Ca line.

[FIGURE] Fig. 7. The one-dimensional cross-correlation functions between the calcium and the iron maps as a function of latitude at zero phase shift. A correlation coefficient of unity represents perfect correlation, a coefficient of zero means no correlation. Note that the correlation is best for high latitudes and worst for latitudes at and below the equator.

Our images from late 1996 show, for the first time, the short-term evolution of individual surface features. Five distinct spots are identified in the S1 map: A large and asymmetric polar spot at a longitude, [FORMULA], of around 150o (phase 0:p42) and a latitude, b, of +65o and an average temperature contrast, [FORMULA], of 1100 K (referred to as spot A). A smaller and warmer spot at [FORMULA]=92o (0:p25), b=+55o and [FORMULA]700 K (spot B). Another small spot at [FORMULA]=348o (0:p95), b=+30o and [FORMULA]400 K (spot C), a moderate but elongated spot at [FORMULA]=306o (0:p84), [FORMULA]+65o and [FORMULA]700 K (spot D), and a weak fifth feature at [FORMULA]=248o (0:p68), [FORMULA]+40o and [FORMULA]400 K (spot E). These spots are also identified in the following S2 and S3 maps - except maybe spot C - and their quantities are summarized in Table 5. The equatorial regions suffer from the unavoidable mirroring effect due to the dominating polar features and for V711 Tau no reliable surface detail can be recovered at or below the stellar equator. Therefore, we consider the extended feature at phase 0:p45 and [FORMULA]o an artefact despite that it appears in all three images. A sixth feature at [FORMULA]40o (0:p1) and b=+35o appeared just in the 1996.860 map and may be real but, because it falls near a time of conjunction and is relatively weak in the average maps, it is not considered significant. However, it was also recovered in the image by Donati (1999) for 1996.99 (as were the five main features). Later in Sect. 4.3, we will identify it as spot B 2.


[TABLE]

Table 5. Spot parameters from the average Ca maps. [FORMULA] is longitude in degrees, b latitude in degrees, and T temperature in Kelvin.


We computed cross-correlation maps for the Ca maps for S1-S2 and S2-S3 by cross-correlating strips of constant latitudes of width 5o as described in our recent paper on HD 51066 in this series (Strassmeier et al. 1998, paper VIII). These maps can be used to search for a consistent longitudinal migration pattern as a function of latitude and indeed show a latitude-dependent phase migration for S1-S2 but not for S2-S3. This suggests that intrinsic spot changes took place and, as we will quantify later in Sect. 4.3, it happened that spots A and B , and possibly also C and D , had merged by the end of the time series. However, the surface cross-correlation functions are not adequate to describe latitudinal spot-migration patterns, e.g. due to differential rotation, once a particular feature moves in longitude and latitude or when its variability timescale is equal to or smaller than the time resolution of the images. In our case, it turns out that it is more accurate to identify consistent features and then measure their positions on the individual maps.

We also note that the combined positions of spots A plus B , as well as C plus D agree reasonably well with the positions from the photometric two-spot modelling in Sect. 3.3.

4.3. The motion pictures or 57 days in the life of V711 Tauri

Due to the low inclination of the stellar rotation axis, most spot features appear circumpolar and are in view all the time. Thus, the spots contribute to every line profile, independent of phase, though with variable impact on the profile. We may use this information to search for spot changes on an even shorter time scale than the time difference between our three separated maps in Sect. 4.2.

We start with a map from the first twelve observations listed in Table 1 from JD 2,450,391 through 2,450,401, i.e. profile numbers 1 through 12, and call it map 1. Then, we proceed with a map from another 12 profiles by adding only the subsequent observation and leaving away the first one, i.e. proceed with profile numbers 2 through 13, and continue by adding one profile at a time until profile number 52 is reached and included. This results in altogether 37 images (three blended phases are left away) that can be stacked one after the other to mimic a short movie. While the time steps between consecutive images are formally not equidistant due to weather losses and the excluded fully-blended phases, it is on average still approximately one day or 0:p4 stellar rotations.

Fig. 8 plots all 37 Ca images in a consistent manner and an animated gif movie can be viewed at our respective WWW homepages  2. Tracking the positions and temperature of the individual features, we detect the following qualitative spot behavior. First, spot B possibly consisted of two close spots at different latitudes and longitudes (maps 1-5) until they merged into a single feature from map 6 on (as was also recovered from the combined data). If real and not a mapping artifact, the time scale of this change was around one day. Unfortunately, the low-latitude regions appear significantly smeared along meridional directions, rendering the determination of the relative spot temperatures uncertain. Later though, from map 6 on, spot B became progressively cooler and larger, once even within [FORMULA]3 days, and started to merge with the previously dominating spot A at around map 7. The following maps still show two identifyable features (spots A and B , maps 8-30) but from map 31 on (approximately JD 2,450,418-429) only a single, though asymmetric, polar feature remained. This polar feature kept growing in size from the beginning of the observations and seemed to continuously cool the rest of the polar regions. We imagine that it fed the entire polar cap with magnetic flux that helped to suppress the convective motion and internal energy flow in its vicinity. The history of this merged feature shows that the original spot B migrated in latitude from [FORMULA]+50o to +60o while the most northern part of the large spot A possibly even crossed the pole at around map 28 (JD [FORMULA]2,450,425-432) and more or less remained there until the end of the observations (this is also suggested by the combined S3 map). During the same time, the coolest part of spot A remained at its original latitude but slightly migrated in the direction of the stellar rotation but slower than the orbit, i.e. clockwise on the pole-on plots, in agreement with the findings of Vogt et al. (1999) from long-term Doppler images. The original spot B , on the other hand, remained more or less at the same longitudinal position.

[FIGURE] Fig. 8. The full time series of Ca I -6439 images as described in the text. Each subset is plotted as a pole-on view (left image) and as a Mercator view (right image). Both projections show only the part of the stellar surface between latitudes of 0o and 90o. Time proceeds downwards starting at the upper left corner of the left column (map 1). The last panel is a dummy to show the surface grids and to identify the features from map 1. The phase coverage for the individual maps is indicated as thick marks below the Mercator maps.

The comparably much smaller spot E showed a smooth latitudinal migration from around +30o at JD 2,450,410 to +60o at JD 2,450,440. The other two small spots, C and D , may have also migrated toward the rotation pole but either merged earlier when they got too close to each other or, as is also possible, spot C is actually two separate short-lived features and only spot D showed a mild latitudinal migration. We are not absolutely sure about the reality of spot C because it is situated at a longitude of [FORMULA]340o and thus mostly seen near the conjunction phases where blending between the primary and secondary lines is most severe. Also, the appearance of the small spots is sometimes annoyingly different in the two spectral-line regions and the positions are consequently relative uncertain. Differences of up to 10o in latitude may exist but on average the residuals are below that.

Fig. 9 is an attempt to quantify the behavior described previously. At this point, the reader is reminded that such an undertaking can be biasing because the information content of a single image can not be easily parametrized into spot quantities. We chose to threshold the individual features instead of the entire image as done in previous papers by Vogt et al. (1999) and Hatzes (1998). The advantage is that weaker features that were reconstructed consistently are not lost by the thresholding. On the other hand, care must be taken not to include systematic artifacts that emerge, e.g. due to the north-south mirroring effect. We simply integrate the spot area within a pre-defined (constant) temperature contour that is chosen individually for each spot, and then average the temperatures within this area. The latitudinal and longitudinal positions are either the location of the coolest part of the spot or the central point of the integrated area. Both are not well determined if the spots are asymmetric in shape (which they mostly are) or are elongated in latitude due to coarse phase coverage.

[FIGURE] Fig. 9a-c. Spot migrations on V711 Tau in late 1996 from Ca I 6439. a A summary of the spot positions on the stellar surface. The individual spots are identified as A -E . A [FORMULA] is the position of the coolest part of spot A . All others plot the position of the central area (see text). b A "butterfly" diagram. Note the poleward migration of spot E around JD 2,450,420. c A plot of the longitudinal migrations. The lines are the linear least-squares fits described in the text.

The symbols in Fig. 9 show the longitudinal and latitudinal migrations obtained in this way. For the large polar spot, we also plot the positions obtained from its coolest part (denoted as [FORMULA]) for comparison purpose. The difference between [FORMULA] and [FORMULA] is only due to the variable temperature asymmetry of this particular feature but the recovered migrations are almost identical. We then proceed by simply fitting straight lines to the spot positions in an attempt to quantify the migration rates. Note that the fits are not plotted in Fig. 9b because they overload the diagram but are shown in panel c. The fit coefficients are listed in Table 6. The average latitudinal migration rate is +0.41[FORMULA]0.23 (rms) o/day from spots A to E (excluding the measurements for [FORMULA]). This compares well with the average longitudinal migration of +0.44[FORMULA]0.06 o/day found by Vogt et al. (1999). It also compares well with the [FORMULA] o/day for the K0III-IV RS CVn binary HU Vir (Strassmeier 1994). As a comparison, the solar longitudinal value is -2.87 o/day, thus approximately seven times higher and in the opposite direction, while the solar latitudinal migration within 35o of the equator changes direction and reaches peak values of [FORMULA]0.03 o/day (e.g. Howard & Gilman 1986), thus at least a factor of ten slower than on HR 1099.


[TABLE]

Table 6. Migration rates for spots A -E : Listed are the coefficients a and b from a linear-least squares fit to the Ca-6439 results of the form [FORMULA], where [FORMULA] is either longitude or latitude and t is the average Julian date minus 2,400,000 in days. [FORMULA] is the goodness of fit (n is the number of spot positions available).


Because our Doppler-imaging technique does not allow to give specific error bars for the positions and temperature of each surface feature, we try to verify our previous results by separating the entire data into two subsets and then produce two independent movies and compare the results. This will allow some estimation of the internal uncertainties. The data selection is done by using all odd-numbered and all even-numbered spectra from Table 1, respectively. Although this procedure leads to a degradation of the surface and the time resolution and thus possibly fails to resolve some small-scale short-time events, it allows an independent check on the reality of the features seen in the reconstructed images. With a time step of approximately twice as long as in Fig. 8, but with similar phase coverage i.e. twelve spectra per map, we end up with 13 images for each subset. Applying the same treshholding procedure described previously, we obtain Fig. 10a and Fig. 10b for the two subsets, respectively. As expected, the largest uncertainties reside with the spot latitudes, and thus their migration patterns, but both subsets clearly reproduce the poleward migration of spot E as well as the trends for the other spots. The latitudes of the larger spots are usually reproduced to within [FORMULA]2o but the uncertainties will rise to, say, [FORMULA]4-5o for spots near the equator. The longitudinal migration is reconstructed as well, even for spot C which identification is not unique after approximately JD 2,450,420. We conclude that the "even-odd" test verifies our basic findings from Fig. 9 but we caution that the many fine equatorial surface detail seen in the movie maps in Fig. 8 remain mostly spurious.

[FIGURE] Fig. 10a and b. As in Fig. 9b,c but now from only the spectra with odd-numbered phases (panel a ) and even-numbered phases (panel b ). The figure demonstrates the similarity of the migration patterns reconstructed from the two subsets of data and also agrees with our results from the combined data.

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Online publication: February 9, 2000
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