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Astron. Astrophys. 336, 587-603 (1998)

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5. Doppler imaging of HD 51066

5.1. The line-profile inversion code "TempMap"

Our Doppler-imaging code is based on the Ap-star code by Rice et al. (1989) and Rice (1991). Its late-type star variant has been described and tested in previous papers of this series (e.g. Rice & Strassmeier 1998, Strassmeier & Rice 1998) and we refer the reader to these papers. Briefly, our approach includes a full spectrum synthesis via a solution of the equation of transfer through a set of Kurucz (1993) model atmospheres with 72 depth points and for effective temperatures between 5750 and 3500 K in each of the observed wavelength regions at all limb angles. Simultaneous inversion of up to twenty lines as well as two photometric bandpasses using either a maximum-entropy or a Tikhonov regularisation is possible. For this paper we chose maximum entropy. All computations for HD 51066 were either performed on a DEC Alpha 500 workstation in Vienna or on a Sun Ultra-2 at Konkoly Observatory and required on average 20-40 CPU-minutes for one run with 9 blends and 15 iterations.

5.2. Astrophysical parameters for the HD 51066 system

The parallax measured by Hipparcos (ESA 1997) results in a distance of 275[FORMULA]45 pc and, with a maximum V magnitude of 6:m 935[FORMULA]0:m 005 in 1996/97, in an absolute magnitude of [FORMULA]=-0:m 26[FORMULA]0:m 34 (interstellar extinction is assumed to be negligible at [FORMULA]). This clearly confirms the giant luminosity classification already suggested from the H&K emission-line reversals (Strassmeier 1994, see also Fig. 5a) and from the combination of rotation period and line broadening (Fekel & Balachandran 1994). The relevant astrophysical data of HD 51066 are summarized in Table 4.

[FIGURE] Fig. 5a-d. Representative spectra of HD 51066 for several wavelengths of interest (thick lines). a the singly ionized calcium H and K lines, b Balmer H[FORMULA], c the Li I 6708-Å line region and, d the 6420-Å region with the photospheric lines selected for Doppler imaging. The thin lines are spectra of MK-standard stars; o Vir (G8IIIa) in panel b and d, and 16 Vir (K0.5III) in c.


[TABLE]

Table 4. Astrophysical data for HD 51066


Our Doppler-imaging procedure allows to redetermine the equatorial rotational velocity with better accuracy than with any other method because one takes into account all line blends within the broadened stellar line profile during the disk integration as well as the line deformations due to spots and a misfit from a wrong [FORMULA] shows up as an excessively dark or bright band at the corotating latitude. Zeeman broadening, however, is neglected. Our best value for HD 51066 is [FORMULA]1.0 km s-1 in very good agreement with the recent measurement of 46.5[FORMULA]2-3 km s-1 by Fekel (1997). Together with the photometric period this results in a minimum stellar radius of 14.9[FORMULA]0.4 [FORMULA]. With [FORMULA] being fixed we solve for the best inclination by recomputing a series of maps with inclinations between [FORMULA] and [FORMULA]. A plot of the sum of the squares of the residuals from all profiles and the photometry versus inclination, along with a consistency check of the computed maps, then yielded the most probable inclination of [FORMULA] and thus the most probable radius of 17.2[FORMULA] [FORMULA]. Note that the [FORMULA]-values are in this case not error bars but merely give the range within each value is equally likely.

The Hipparcos B-V color of 0:m 943[FORMULA]0:m 007 would be consistent with a G7.5III giant with [FORMULA] of 4970 K, [FORMULA]0:m 7, and [FORMULA] [FORMULA] or 12.5 [FORMULA] according to the tables of Gray (1992) or Schmidt-Kaler (1982), respectively. The effective temperature from the bolometric brightness-radius relation

[EQUATION]

is 4720[FORMULA] K (adopting B.C.=-0.33 from Flower (1996), [FORMULA]0:m 26 from the parallax, and the most probable radius from above). The maximum temperature from Eq. (2) is then 5080 K in case the minimum radius is inserted, while Flower (1996) lists a formal value of 4955 K for B-V=0:m 943. A comparison of all these values strongly indicate that HD 51066 is, first, intrinsically brighter than a G7.5III star by almost a magnitude and has a luminosity of 124[FORMULA] [FORMULA] (on the [FORMULA]4:m 64 scale; Schmidt-Kaler 1982), second, has a 20-50% larger minimum radius than a G7.5 giant and, third, is slightly cooler. We suggest that a G8IIIa-IIb classification (following the notations of Keenan & McNeil 1989) is more appropriate and also agrees with the fact that [FORMULA]=2.5 atmospheres produce a significantly better fit to the photospheric absorption line profiles than [FORMULA]=3.0 atmospheres. The latter gravity would also require an overabundance for Fe and Ca of 0.2 dex to match the observed line strength. Furthermore, a single blue high-resolution spectrum shows a strong Sr II  4077-Å line typically indicative of a bright giant spectrum (Gray & Garrison 1989). A straightforward comparison of the position of HD 51066 in the H-R diagram with the evolutionary tracks of Schaller et al. (1992) for solar metallicity suggests a mass within 3.1[FORMULA]0.1 [FORMULA].

With the primary's mass fixed, we may use the preliminary orbital elements from Table 3 to estimate fundamental stellar parameters for the secondary star. If we furthermore assume that the rotational axis of the primary is perpendicular to the orbital plane, we may also adopt the inclination angle of [FORMULA] that is obtained from the Doppler imaging as the orbital inclination. The mass function then implies a secondary mass of 1.3 [FORMULA] and [FORMULA]. This mass is consistent with either a F7V, F2III, or F9III star according to the tables of Gray (1992). Since the primary is G8III-II and relatively young (see next paragraph) it is likely that the secondary had not enough time yet to significantly leave the main sequence and we adopt F7V for its spectral classification but emphasize again that the secondary's parameters are based on a preliminary orbit. The mass ratio and the approximate [FORMULA] from Table 3 combine then to [FORMULA] km (i.e. 7.7 AU or [FORMULA]). From a distance of 275 pc this would be at most 0:0003 in the tangential plane and hardly detectable. However, a spectrum at ultraviolet wavelengths might show the secondary's continuum.

With the new parallax and proper motions from Hipparcos and our revised average radial velocity, we determine the (U,V,W) space-motion components of HD 51066 relative to the Sun in a right-handed coordinate system to be (+4.2[FORMULA]2.5, -15.1[FORMULA]0.9, -15.6[FORMULA]1.5) km s-1 . The error in U is mostly due to the uncertainty of the systemic radial velocity given in Table 3. According to the (U,V)-plane classifications of Eggen (1989) these space motions are consistent with the young disk population. Note, however, that HD 51066 appears 125[FORMULA]20 pc above the galactic plane. The relatively young age is also consistent with the existence of a moderately strong Li I 6707 Å line as shown in Fig. 5c. After subtraction of a velocity shifted and broadened spectrum of the inactive K0.5III star 16 Vir we measure an equivalent width for Li I of 120[FORMULA]10 mÅ , compared to 230[FORMULA]10 mÅ for the nearby Ca I 6717-Å line. The appropriate non-LTE curve of growth from Pavlenko & Magazzú (1996) converts this to an abundance of [FORMULA](Li)=2.0 (on the usual [FORMULA](H)=12.00 scale) while their LTE version results in a slightly larger abundance. Our full spectrum synthesis of the 6704-6710 Å region with solar [FORMULA] values gives the best fit with a Li abundance of [FORMULA](Li)=2.0 with an estimated error of [FORMULA]0.1, in excellent agreement with the curves-of-growth method but only in fair agreement with the value of 1.5 obtained by Fekel & Balachandran (1994) who adopted [FORMULA]=4800 K and [FORMULA]=3.0.

Finally, we also need to know the values for micro- ([FORMULA]) and macroturbulence ([FORMULA]) but, since HD 51066 is a rapidly rotating star, their effects will be of minor nature. We simply adopt the typical values for a late G giant as given in Gray (1992), i.e. [FORMULA]=1.0 km s-1 and [FORMULA]=3.0 km s-1. Several test inversions with microturbulences between 0.5 and 2.2 km s-1 indicated lowest [FORMULA] and the most consistent maps when using above values. Within the known shortcomings of our LTE treatment, the assumption of solar abundances as given by Grevesse & Anders (1991), and the assumption of equal radial and tangential components for [FORMULA], a value for [FORMULA] as large as 2.0 km s-1 must be nevertheless excluded. From the test inversions we estimate the internal error in [FORMULA] to be [FORMULA]0.2 km s-1 .

5.3. Atomic parameters

The basic atomic parameters for our mapping lines were taken from the Kurucz-(1993) linelist but were adjusted by fitting synthetic profiles to a spectrum of the Sun. Several [FORMULA] values as well as wavelengths had to be changed to obtain a good fit. For the actual numbers we refer to previous papers in this series (e.g. Strassmeier & Rice 1998 and Strassmeier et al. 1997c) and also to Johns-Krull & Hatzes (1997) for the 6400-Å line. The latter authors noted that they had trouble fitting the Fe I 6411-Å line due to unknown blends and omitted this line from their analysis of the T Tauri star Sz 68. For the present paper we re-examined this wavelength region and obtained a reasonable good fit to the solar line when using altogether 7 blends including two Ti I lines and one V I line. A plot of this wavelength region for the solar Fe I  6411 Å line has already been shown in Strassmeier (1996).

Fig. 5d shows a representative spectrum of the entire 6420-Å region of HD 51066 with the five main Doppler-imaging lines marked: Ca I 6439.075Å ([FORMULA] =290 mÅ , n=8), Fe I 6393.602Å ([FORMULA] =280 mÅ , n=7), Fe I 6430.844Å ([FORMULA] =220 mÅ , n=8), Fe I 6411.647Å ([FORMULA] =205 mÅ , n=8), and Fe I 6400.000 + 6400.314Å (combined [FORMULA] =335 mÅ , n=6), where [FORMULA] is the measured equivalent width for the main mapping line and n the total number of blends included in the local-line profile computation. An overplot with the G8IIIa M-K standard star o Vir ([FORMULA] km s-1 ) visualizes the expected amount of blending. Note that the 6400-Å line is actually a very close blend of two, almost equally strong Fe lines but of quite different excitation potential. We emphasize that all of the blends are included in the inversion simultaneously, but that the five wavelength regions are treated separately.

5.4. Images for dataset April 1997

For this season 12 spectra and 20 VRI light-curve points were available. The spectroscopy was obtained within 14 consecutive nights while the photometry was combined from altogether 32 nights centered around the spectroscopy. Fig. 6a-e show the Doppler images from all five lines plus the observations and respective fits. The figure allows a visual comparison of the line-to-line consistency of the reconstructed surface temperature along with the unavoidable differences from "real" spectra of limited S/N and basically unknown external errors.

[FIGURE] Fig. 6a-e. Doppler images of HD 51066 from April 1997. Each panel presents the results from one spectral region and shows a pseudo-Mercator projection of the reconstructed surface temperature (top), the observed line profiles and their fits (middle), and the light curve in two bandpasses and their respective fits (bottom). The [FORMULA] of the line-profile fit is given below each map. Rotational phase is indicated to the left of each line profile and also marked with arrows. The main spectral lines are identified at the top of each map and are arranged in order of decreasing equivalent width from panel [FORMULA] through [FORMULA].

HD 51066 had several significant spots in 1997 with [FORMULA] K and centered at longitudes of [FORMULA], [FORMULA], [FORMULA], and possibly also near [FORMULA]. The latter spot covers [FORMULA] in longitude and its contrast is not reconstructed equally from the individual lines and, consequently, the formal average temperature is not as well constrained as for the other main features. We thus regard this feature as uncertain. In one image (Fe I 6400 Å) the data require a bright feature at [FORMULA] which obviously stems from the very distorted profile at phase 0:p597 but disappears if this line profile is given zero weight in the inversion. Despite that this profile's distortion is qualitatively the same as in the other lines, it appears significantly exaggerated in Fe I 6400 Å and does not repeat properly in the subsequent phase. We have nevertheless chosen to keep this profile for the inversion since there is nothing obviously wrong with this exposure and the Doppler-imaging code is very good in differentiating between phase-dependent and purely time-dependent distortions.

There is no dominant cap-like polar spot in 1997 as seen on many evolved RS CVn binaries. The one image (Fe I 6430 Å) that possibly does show a weak polar feature with [FORMULA]250 K is reconstructed from a blend with the strong Fe II 6432.68-Å line that is not only located 1.8 Å to the red of the main mapping line and thus prone to produce high-latitude artefacts (see, e.g., Unruh & Collier Cameron 1995) but also singly ionized and thus sensitive to electron pressure and some caution is advised. We thus conclude that there is no polar cap-like spot on HD 51066 in 1997 but a single high-latitude ([FORMULA]-[FORMULA]) spot at longitude [FORMULA]. The remaining features group along an average latitude of [FORMULA].

5.5. Images for data set January 1996

In 1996 11 spectra and 24 BV light-curve points were available. The photometry was combined from a 48-day interval centered around the mid time of the spectroscopic observations while the spectroscopy was obtained within 14 consecutive nights. Fig. 7a-e visualizes the data and the mapping results.

[FIGURE] Fig. 7a-e. Individual Doppler images for January 1996. Otherwise as in Fig. 6a-e. Note the splitted feature near a longitude of [FORMULA] (e.g. in panel e ) and how it is recovered from the individual lines. Compared to the 1997 maps in the previous figure (e.g., again with panel e ) suggests its persistent existence for over 28 stellar rotations.

As in 1997, no dominating polar cap-like spot is seen. A single, high-latitude feature at [FORMULA]=[FORMULA] with [FORMULA] K is recovered from the three strong lines but is not so obvious from the two weaker lines (Fe I 6411 and 6400 Å) where its contrast is only about 250-300 K below the nominal photospheric value of 4950 K. Several but slightly cooler spots ([FORMULA] K) are seen at latitudes between [FORMULA] and [FORMULA] and at longitudes of [FORMULA]=[FORMULA], [FORMULA], [FORMULA], [FORMULA] and around [FORMULA]. Note that the two spots, termed "splitted spot" in the figure caption, that were dominating the 1997 Fe I 6400-Å map (compare with Fig. 6e) are strikingly similar to the feature in this year's map, even in shape. The whole group appears to be shifted by [FORMULA] toward smaller longitudes compared to 1997 though. In Sect. 5.8, we try to quantify the likelihood of similarities between the annual images by means of cross correlation.

5.6. Images for data set February-March 1995

For 1995 just 6 spectra and no simultaneous light-curve points were available. The photometry closest in time was 70 days (more than four stellar rotations) prior to the spectroscopic observations and could not be used for the imagery. While the spectroscopic observations are fairly well distributed in phase and taken within a single stellar rotation we must regard the 1995 map as less reliable than the others. To estimate such (external) uncertainties we inverted all spectral-line regions in 1995 also with photometric input but gave lower weight to the photometry of 0.10 relative to the line profiles. As expected there were no obvious differences in the reconstructed surface features, indicating that the photometric variations must have been relatively constant, but the maximum temperatures were on average 100 K cooler than without photometric input.

Fig. 8a-e shows the data and the mapping results without photometric input. However, since our Doppler-imaging code works on absolute rather than relative photometry we still need to supply a phase averaged B-V color. Its value is indicated in the light-curve panels in Fig. 8a-e as a straight line and was adopted from the BV photometry of Henry et al. (1995b) in December 1994.

[FIGURE] Fig. 8a-e. Individual Doppler images from February-March 1995. No photometric input was used for this season but is nevertheless plotted in the respective light-curve panels. The straight line is the adopted average magnitude for this season. See text. Note that the radial structures in the maps are due to the sparse phase coverage. Otherwise as in Fig. 6a-e.

Again, our maps show a high-latitude feature at [FORMULA] and at longitude of [FORMULA]. Two of the lines (Ca I 6439 and Fe I 6430 Å) require a small polar cap-like spot of significant temperature difference with the high-latitude feature at [FORMULA] and another region near [FORMULA] as its appendages. The large number of low-latitude features of lesser contrast ([FORMULA] K) suggest that even smaller features, say with projected area [FORMULA]1%, can be also very persistent. We note that the time scale for the stability of any spot pattern is an uncertainty that cannot be resolved on the basis of even annual Doppler maps but there is abundant evidence that polar spots are long lived and equatorial spots short lived (e.g. Vogt & Hatzes 1996). Multicolor photometric monitoring of HD 51066 throughout 1997/98 (see Sect. 3) showed the star with a relatively constant light curve shape (Fig. 3 a) but sudden amplitude variations did occur, e.g. in early 1994 .

5.7. Images for data set March 1994

In 1994 altogether 16 spectra and 15 BV light-curve points were available. The photometry was combined from a 29-day interval centered around the mid time of the spectroscopic observations which was obtained within 14 consecutive nights. Fig. 9a-e shows again the data and the results.

[FIGURE] Fig. 9a-e. Individual Doppler images from March 1994. Otherwise as in Fig. 6a-e.

Contrary to the years 1995-97, all line regions in 1994 recovered a moderately size cap-like polar spot with [FORMULA] K, i.e. at least 200 K cooler than the coolest feature seen in the other data sets. This is an interesting result because, first, a polar spot never has been seen to develop or vanish and, second, it explains the long-term brightness changes in Fig. 1. Of course, one is also tempted to propose the same cause for the many spotted stars having large long-term brightness variations (periodic or not) and we suspect that polar spots are a common feature on active stars.

Besides the polar feature we see again several low-latitude spots of moderate temperature difference ([FORMULA] K). Two elongated features at [FORMULA] and [FORMULA] even reach the polar spot. The respective profiles at phases around 0:p5 and 0:p65 reveal two bumps in the line core, which give rise to the two latitudinally elongated features.

5.8. Average maps and image correlations

In Fig. 10 we compare the average maps from the four years. The averaging has the advantage of suppressing spurious features from the individual lines but emphasizing features that were reconstructed consistently. The only shortcoming when interpreting an average map is that it is not directly constrained from observations. We find cool spots with an average temperature difference with respect to the unspotted photosphere of [FORMULA]500 K, the strongest feature being the polar spot in 1994 with [FORMULA] K. The smaller structures are mostly located to within a latitude of [FORMULA] and [FORMULA]. However, this region is near the sub-solar line at [FORMULA] which is most sensitive to line-profile variations and, if a particular phase region is undersampled, the code tends to put spots preferably along this latitude. We can not rule out that the many small, low-latitude features are actually arranged symmetrically along the stellar equator.

[FIGURE] Fig. 10a-d. The evolution of surface features on HD 51066. Shown are the average temperature maps for the years 1994 through 1997. Each map is the non-weighted average from the five maps already shown individually in Figs. 6a-e-9a-e. The bar below each map plots the standard deviations in Kelvin along binned longitudes. Their grey scale is also indicated. Note the weak polar spot in 1994 and possibly in 1995, the single high-latitude spot at [FORMULA] in 1996, and the absence of the polar spot in 1997.

By cross correlating strips from different maps at sucessive latitudes, we may derive the amount and the sign of an eventual phase lag in time, which could then be interpreted as differential surface rotation (Donati & Collier Cameron 1997, Weber & Strassmeier 1998). And, when comparing different latitudinal strips, we may additionally obtain information on the appearance and subsequent latitudinal movement of particular features, e.g. a poleward drift as in the Doppler images of HR 1099 (Vogt & Hatzes 1996). Our maps are made up by 72[FORMULA]36 pixels of size five-by-five degrees, starting from the "southern" limb of the stellar disk at a latitude of [FORMULA] up to the visible pole. The annual cross-correlation maps in Fig. 11 were derived by sucessively cross correlating two maps consecutive in time and at constant latitudes. They already indicate the tendency of a curved latitude-dependent correlation function. The pixelshift then indicates the shift in phase (or longitude) that was required to find the optimum correlation, the latter expressed with a standard linear correlation coefficient (0 means no correlation, 1 is perfect correlation). All cross correlations were computed with the IRAF fxcor routine (see IRAF V2.11 manual, http://iraf.noao.edu ).

[FIGURE] Fig. 11a-c. Surface cross-correlation functions. The annual variations for each latitude bin are cross-correlated against each other. a from 1994 to 1995, b from 1995 to 1996 and, c: from 1996 to 1997. Pixel size is [FORMULA] in longitude and a grey scale is plotted at the given values of the linear correlation coefficient (the higher the correlation coefficient the darker the grey scale).

From the annual maps in Fig. 10, we can already see numerous surface features up to a latitude of [FORMULA] and the cross-correlation functions mostly reveal more than one significant peak per latitude bin. The portion of the stellar surface below [FORMULA] is especially prone to artifacts and its cross correlation is thus less reliable than on other parts of the surface. To obtain a single estimate on the phase lag per latitude slice, we fit a Gaussian to the most significant peak of the cross-correlation function whenever well defined and plot the results in Fig. 12a-c. The error bars on these lags are typically only a few pixels and were adopted to be proportional to the FWHM of the cross-correlation peak and estimated from repeated measurements with different fitting routines within the IRAF package. The full lines in Fig. 12a-c are then least-squares [FORMULA] and [FORMULA] fits to these phase lags versus stellar latitude [FORMULA], and may represent several variants of surface differential rotation (Table 5).

[FIGURE] Fig. 12a-c. The distribution of the cross-correlation peaks versus phase and their fits with [FORMULA] (full lines) and [FORMULA] (dotted lines) terms. [FORMULA] is the stellar latitude. The points are from a Gaussian fit to the most significant cross-correlation peak shown in Fig. 11. The individual panels are again for the annual cross correlations between 1994-1995, 1995-1996, and 1996-1997.


[TABLE]

Table 5. Coefficients for the cross-correlation fits


As already pointed out by Donati & Collier Cameron (1997) and by Unruh et al. (1995) for the rapidly-rotating K-dwarf AB Dor (P=0.5 days), we caution that it is entirely possible that the spot distribution has completely re-arranged within our annual maps, and that the interpretation of the phase shifts with differential rotation could therefore be spurious. However, the long rotational period of 16 days together with a low-gravity atmosphere, and the relative constancy of the seasonal light curves from 1997/98 in Fig. 3 argue against a complete re-arrangement of spots. Also, our maps from 1996 show at least one feature - dubbed the "splitted spot" in Fig. 7a-e - that is also seen in 1997. Note though that there were 22, 20, and 28 stellar rotations between the 1994-1995, 1995-1996, and 1996-1997 maps, respectively. We conclude that there is good evidence for latitude dependent phase shifts on the surface of HD 51066 but whether these are due to differential rotation remains to be seen once higher time-resolution maps are available.

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Online publication: July 20, 1998
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