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Astron. Astrophys. 347, 225-234 (1999)

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4. Doppler imaging

4.1. The TempMap code

As in previous papers in this series, all maps were generated with the Doppler-imaging code TempMap of Rice et al. (1989). This code was originally developed for use with chemical-abundance inhomogeneities of Ap stars but was successively adapted for temperature recovery (Rice 1996; see Rice & Strassmeier 1998 for a brief update). In this paper, we chose a Maximum-Entropy regularization. A grid of 10 model atmospheres with temperatures between 3500 K and 6000 K and [FORMULA] were taken from the ATLAS-9 CDs (Kurucz 1993). For temperatures not within [FORMULA]10 K of one of the tabulated input atmospheres, we interpolate between them. Due to the narrow spectral lines of HD 12545, blending of the main mapping lines must be considered very carefully. We emphasize that TempMap uses all spectral line profiles in a particular spectral region for the "inversion", i.e. the main mapping line plus the blends, and thus even considers the (subtle) main-profile changes due to the spot-induced periodic changes of the blend-line profiles. The number of blends vary from spectral region to spectral region and were tabulated in previous papers in this series (e.g. Strassmeier 1997). We make use of six line regions centered at the following main spectral lines (the number in parenthesis is the number of included blends): Ca I 6439.075 Å (4), Fe I 6430.844 Å (8), Fe I 6421.349 Å (6), Fe I 6419.942 Å (7), Fe I 6411.649 Å (8), and Fe I 6393.602 Å (4). The adopted [FORMULA] values and lower excitation potentials (in eV) are [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA], respectively.

4.2. Doppler imaging at low spatial resolution

The 24-day rotation period of HD 12545, together with the need to cover all rotational phases within a single stellar rotation, makes access to high-resolution spectrographs at large telescopes practically impossible. With the current resolving power of [FORMULA]=38,000 (i.e. 0.19 Å at 6430 Å) and a full width of the lines at continuum level of [FORMULA] 0.87 Å, we have just 4.5 resolution elements across the stellar disk. According to the simulations of Piskunov & Wehlau (1990) this is the limit for a detailed recovery and their test inversions with artificial data still showed a fully correct recovery of the input image when five resolution elements were available. However, surface detail much smaller than a resolution element is being lost and the reconstructed detail depends on latitude (and stellar inclination) and on the phase coverage. Usually, low-latitude features are less reliable than high-latitude features. A similar conclusion was reached by Strassmeier & Rice (1998) from their test inversions with [FORMULA]=17.5 km s-1 to simulate the line profiles of the G2-dwarf EK Dra, while Hatzes (1993) concluded that Doppler imaging should be capable of recovering spots on stars rotating as slowly as 15 km s-1. HD 12545 shows a very large light-curve amplitude and thus presumably very large spots that, in turn, cause large spectral-line deformations, which makes the recovery more reliable. Phase smearing due to the one hour integration time amounts to just 0:p0017, or less than 1o at the stellar central meridian, and is negligible. The intrinsic (thermal) width of our mapping lines is always much smaller than the width of the instrumental profile and also does not further decrease the spatial resolution. Nevertheless, we caution not to overinterpret the details in the images (better data are always better).

4.3. The inclination of the stellar rotation axis

The revised mass function of 0.0100[FORMULA]0.0004, together with the primary mass of 1.8[FORMULA] [FORMULA], suggests an upper limit for the inclination angle of 78o[FORMULA]1o. Above that inclination, we would see an eclipse because then [FORMULA]. The very small mass function suggests a low-mass secondary star with masses of 0.37 [FORMULA] for [FORMULA]o and 1.02 [FORMULA] for [FORMULA]o. Because we do not see the secondary spectrum in red, nor blue or ultraviolet (Bopp et al. 1993) wavelengths, the secondary must be always fainter by at least [FORMULA]2:m 5. This excludes all subgiants hotter than [FORMULA]K2 and all stars hotter than late-F. Thus, the mass ratio primary/secondary must be larger than approximately 1.5, which sets a lower limit for the inclination to 25o. However, at this inclination the K giant would already overflow its Roche lobe and should show pronounced spectroscopic signs, e.g. systematic radial-velocity shifts in high-excitation chromospheric and transition-region line profiles, which it does not. It is thus likely that the lower limit of the inclination is more around 35-40o.

We tried inclination angles between 5o and 85o for the line profile inversion and found overal the largest reduction in the sum of the squares of the residuals (at maximized entropy) when inclinations between 50-80o were used (Fig. 4). Finally, we adopt [FORMULA]o as the most consistent inclination from the fits of four spectral regions and above considerations. The most likely secondary star is then a [FORMULA]0.4 [FORMULA] red dwarf star of spectral type M2. Note that our conclusions would not change significantly if the maps were reconstructed with inclinations of 50o or 70o.

4.4. Results

Figs. 5a-f show the Doppler images from the six available spectral regions along with the achieved line-profile and light-curve fits. The best [FORMULA]-level from profiles and light curves combined is around 0.06, thus slightly worse than what we usually achieve, e.g., for the stars in previous papers in this series (e.g. Strassmeier & Rice 1998). We attribute this to two effects; first - and most important - the increased influence of uncertainties of the atomic line parameters, the line synthesis, and the external errors of our spectra with the small rotational broadening of HD 12545 and, second, a variability time scale on the stellar surface that seems to be shorter than the rotational period of 24 days. The first effect is detectable by inconsistencies in the maps from spectral region to spectral regions. While four of the six regions agree very well, two maps (Fe I 6419 Å and 6421 Å) show some discrepancies concerning the recovered spot area, i.e. primarily the spot contrast. Both line regions turn out to be very complex blends of at least ten lines within two Å ngströms including temperature sensitive vanadium and titanium lines with more than uncertain transition probabilities (see, e.g., the VALD database compiled by Piskunov et al. 1995). Disentangling the individual blend's line strengths at the atmospheric conditions of HD 12545 is therefore not straightforward and we decided to exclude these two maps from the average map but plot them in Figs. 5e and 5f for comparison purpose.

[FIGURE] Fig. 5a-f. Doppler images of HD 12545 for six different spectral regions. a  Ca I 6439 Å, b  Fe I 6430 Å, c  Fe I 6411 Å, d  Fe I 6393 Å, e  Fe I 6421 Å, and f  Fe I 6419 Å, All maps were obtained with Johnson B (4340 Å), V (5500 Å) and Cousins IC (7900 Å) photometry, but only the VI maps are shown. The thick marks in the maps indicate the phase coverage.

[FIGURE] Fig. 5a-f. (continued)

[FIGURE] Fig. 5a-f. (continued)

The second effect is more subtle and is primarily indicated by the small seasonal light-curve variations visible in Fig. 1b rather than by the line profiles or the reconstruction process itself. Any current Doppler-imaging code must assume that surface features remain stable during the time of the observations. From the seasonal light-curve changes and the less-than-perfect [FORMULA]-level of our line-profile fits, we estimate that the amount of spot variations during one stellar rotation is still less than or comparable to the surface resolution of our spectra. It is thus unlikely that spurious surface features larger than 10-20o were introduced into our maps but the effect is that phase smearing due to a variable feature may have found its way into the data. However, we do not consider this a detectable effect from our data.

Despite these inherent difficulties - small rotational broadening and low spectral resolution, and a surprisingly short variability time scale - the maps from the six spectral regions show very similar features. All maps are dominated by a cool, high-latitude spot of elliptical shape and located at a longitude of [FORMULA]300o (phase [FORMULA]). Its average temperature is 1300[FORMULA]120 K below the nominal photospheric temperature of 4750 K and its area is approximately 11% of the entire stellar surface. With a stellar radius of 11.4 [FORMULA] from Sect. 3.2, the spot's linear size amounts to [FORMULA]12[FORMULA]20 [FORMULA] or 8[FORMULA]14 106 km. Even its projected size appears to be ten times larger than the solar diameter! Truly gigantic dimensions for a spot.

Besides this superspot, we reconstruct a second, but much smaller, cool spot at a longitude of [FORMULA]90o, a latitude of [FORMULA]50o and with an area of 2.3% (at [FORMULA] K), as well as an equatorial warm spot (or a conglomerate of several smaller spots) around [FORMULA]90-135o and with an area of 3.5%. The warm feature appears on the adjacent hemisphere of the big cool spot and "south" of the smaller cool spot and, if thought about in terms of opposite polarity of the underlying magnetic field for cool and hot features, possibly indicates violent interactions that could be the reason for the short variability time scale. The systematically changing maximum brightness 1997/98 to 1998/99, shown in Fig. 1a, along with a nearly constant minimum brightness indicates that the warm feature is the cause of the light curve changes. The warm spot's recovered temperature is at most 350[FORMULA]20 K above the effective photospheric temperature (see the temperature profiles later in Fig. 8) and is constrained by both the line profiles and the BVI-photometry. Its size and temperature, however, is different in the maps from the different spectral lines but is recovered even when no photometry is used for the inversion. We consider this feature to be reliable and needed by the data because, first, the light and color curves can not be fitted at all (simultaneous with and without the line profiles) if we restrict spots to [FORMULA], nor can the line profiles be consistently fitted to a comparable [FORMULA]-level if the photometry is completely excluded from the inversion. Note that the band of weak spots along a latitude of approximately -30o is due to the mirroring effect from the strong high-latitude features, a well-known shortcoming of Doppler imaging, and is purely artificial.

Figs. 6 and 7 show the average map from Ca I 6439, Fe I 6430, Fe I 6411 and Fe I 6393 Å, and the standard deviation map in mercator projection and pole-on projection, respectively. The pole-on projection style emphasizes surface detail at or near the visible pole as opposite to the Mercator projections which emphasize the equatorial regions. Finally, Fig. 8 is a plot of the average map in a more realistic spherical projection. It also shows the temperature profiles along longitudinal and latitudinal cuts through the large cool spot at [FORMULA] and the warm spot at [FORMULA], respectively.

[FIGURE] Fig. 6. Mercator views of HD 12545. a  Weighted average map in mercator projection. The maps from Fe I 6419 Å and 6421 Å gave slightly discordant results and were not included in the average map. b  Standard-deviation map. c , d  The binned temperature distributions as a function of stellar latitude and longitude for all individual maps, respectively. The dotted line shows the effective temperature from the photometric calibration, the dashed line the average surface temperature, and the thick lines show the distribution from the average map.

[FIGURE] Fig. 7. Pole-on views of HD 12545. a  Weighted average map in pole-on view from latitude -60o (circles indicate latitudes in steps of 30o). b  The standard-deviation map. As in Fig. 6b, dark areas correspond to small deviations (black=0K, white=60K), i.e the darker the grey scale the better the agreement between the individual maps.

[FIGURE] Fig. 8. Average temperature map of HD 12545 in spherical projection. The map is constructed from averaging four of the six individual maps from Fig. 5a-f. Note the comparison with the projected solar disk in the upper left corner. The small panels to the right plot the temperature profiles for the warm and cool spot along longitudinal and latitudinal cuts through their central position, respectively.

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© European Southern Observatory (ESO) 1999

Online publication: June 18, 1999