Astron. Astrophys. 347, 212-224 (1999)

3. Data preparation, atomic input data, and mapping procedure

3.1. Line-profile preparation for Doppler imaging

Once the rotational phases of the spectra are known, we extract the mapping line profiles from each spectrum in the proper sequence. Since HD 199178 is a single star, it represents the simplest possible case at this stage of data preparation - there are no radial velocity variations nor depressions of the spot signatures due to the continuum of a secondary star.

Our next step is to remove the instrumental profile from each of the extracted profiles. This is done with a Gaussian approximation to the true instrumental profile. We adopt the average nightly FWHM of several weak lines from a Th-Ar comparison lamp as the FWHM of the instrumental profile. Furthermore, we remove the very high frequency noise much above the Nyquist frequency (based on the instrumental profile) by using a filter determined from the FWHM of the instrumental profile in diodes at the sample spacing in the original stellar spectrum. The noise removal is done before the instrumental profile is removed (see Gray 1992). This procedure is more accurate since the convolution in the forward process would be done in the data domain (not in the Fourier domain) and since the data points are always very widely spaced compared to the instrumental profile, sampling for the convolution would have to involve some interpolation which would be somewhat ad hoc.

3.2. Spectrum synthesis of the 6439-Å wavelength region

Unruh & Collier-Cameron (1995) demonstrated that neglecting blends in the wings of the primary mapping line will lead to spurious banding in the reconstructed image. These artificial bands could show up at higher latitudes the further a blend is away from the mapping-line center. From detailed simulations, Unruh & Collier-Cameron estimate that the ratio of wavelength displacement of the blend to the of the star had to be of the order of 1/3 for artifacts to appear.

Our approach to handling this problem is to synthesize a strip of spectrum that includes all known line blends down to an equivalent width of 3 mÅ. This spectrum is then used for the inversion instead of a single line profile. Thus, all the blends will be mapped simultaneously with the dominant main mapping line (Ca I 6439 Å in our case). This requires knowledge of the correct line transition probabilities (-values) for each included blend and we first fit a very high resolution spectrum of the Sun ( 600,000; Delbouille et al. 1973) with Kurucz's (1991) own solar model at the wavelengths of our interest (Fig. 1b), and then revise the tabulated transition probabilities from Kurucz (1993) if necessary. Table 2 is a list of the adopted transition probabilities for the lines marked in Fig. 1.

Table 2. Adopted values

Fig. 1a-d summarizes the results from such a synthesis analysis based on pretabulated ATLAS-9 model atmospheres and an updated line-synthesis program written in Ada (Stift 1995) which is based on the original code by Baschek et al. (1966). A fit to the spectrum of the cool star Arcturus (K1.5III, Peterson et al. 1993), having an effective temperature similar to the average spot temperature on HD 199178, is used to modify the 's for the temperature-sensitive lines that do not show up properly in the solar spectrum, mostly from vanadium, but also to verify the revised 's from the solar fit for a cooler and lower-gravity atmosphere. We thereby slightly modify some of the elemental abundances (Eu, V, and Si) determined by Peterson et al. (1993). A final consistency check is made by fitting the spectrum of a normal M-K standard star that might represent the unspotted photosphere of HD 199178; in our case we used CrB (G5III-IV Fe-1, , Keenan & McNeil 1989). We emphasize though that HD 199178 lies in the Hertzsprung gap, where it is difficult to find a star with a spectrum that can be termed "normal" (see Keenan & McNeil 1989). The CrB fit is shown separately in Fig. 1c. CrB is a mildly chromospherically active, single star and a rotationally modulated light variable with a period of around 60 days (Fernie 1991, Choi et al. 1995). It is also somewhat metal deficient, and several new abundances were estimated in the course of this work (using only the spectral range shown in Fig. 1). We found (Ca) = 6.000.05, (Fe) = 7.550.03, (V) = 4.850.05 and (Eu) = 0.450.02 (based on (H) = 12.00).

The most noticeable wrongly synthesized lines in Fig. 1 are the Si I lines at 6440.566 and 6442.777 Å. Further, the observed spectrum of CrB shows some very weak lines that can not be synthesized properly (e.g. Ti I 6439.705 and Ce I 6439.964 Å) while the unblended Fe I 6436.411 line is clearly seen in CrB and in our synthetic spectrum but is apparently absent, or much weaker, in HD 199178. Only the Eu II 6437.640 + V I 6438.088 blend seems to be present in all spectra in Fig. 1 having a combined equivalent width of 19 mÅ in the synthesized HD 199178 spectrum and 25 mÅ in the CrB spectrum.

Finally, the referee brought to our attention the problem of gravitational darkening of a rotationally deformed star (see Hatzes et al. 1996). In principle, it introduces a temperature gradient from pole to equator, the poles being slightly hotter than the equator, that possibly mimics an equatorial belt of cool spots. If we adopt von Zeipel's (1924) gravity darkening law and its extention to stars with a convective envelope (Lucy 1967), we find a temperature difference between pole and equator for HD 199178 of 120 K. This transforms into a change of equivalent width in our Ca I mapping line of 5-6% from pole to equator. We believe that such a small change is below our temperature resolution and remains burried in the modelling. However, we caution the reader that the recovery of a very low contrast belt of equatorial spots may be an artefact due to gravitational darkening.

3.3. Mapping procedure

The inverse problem for stars with cool spots amounts to solving the integral equation relating the surface temperature distribution to the observed line profiles and light and color curve variations, while controlling the effects of noise in the data through a regularizing functional. We solve for the photometric continuum variations simultaneously with the line profiles, but can handle only two continuum bandpasses per solution. If the spectroscopic phase coverage contains gaps of more than 30^o on the stellar surface more weight is shifted to the photometry.

For all maps in this and subsequent papers in this series we apply the Doppler-imaging code described by Rice et al. (1989) and reviewed by Piskunov & Rice (1993) and Rice (1996). We do not claim that our maps are true, e.g., maximum-entropy maps since the generated maps can not simultaneously fit the geometric constraints and get the error of fit down to the level where the O-C is a Gaussian distribution with a sigma of the size of the formal photon statistical error mainly because the external errors are often much larger and more systematic. The current version of the code includes continuous opacity calculations adjusted for temperature variations across the stellar surface and that allows the use of the latest model atmospheres in the calculation of the local line profiles. Further, the code works from local profile tables that are synthesized strips of spectrum of up to 7 Å so that several lines and line blends (up to 20) can be fit simultaneously.

The code fits either relative or absolute color variations for continuum light in two photometric bandpasses simultaneously with fitting the line profiles. See Rice & Strassmeier (1998) for a more detailed description and a first application. If the absolute photometry switch is turned on, a calibration between the various model atmospheres and the observed broad-band colors must be supplied to the code. Currently, we adopted the B-V vs. calibration of Flower (1996). For all cool and all low-gravity models this is not without significant additional uncertainties for the absolute spot temperatures (see, e.g., Smalley & Kupka 1997or Buser & Kurucz 1992). Fortunately, if the atomic line parameters are basically correct, it is the line strength that determines the surface temperature, not the photometry.

Collier-Cameron (1995) noticed a strong dependence of the recovered total spotted area when the of the solution was pushed progressively lower. This is simply because the MaxEnt code will start to overfit the line profiles and produce spurious features on the surface once a critical level has been past. Our code is less prone to this problem because we remove the very high frequency noise prior to inversion and therefore do not pre-set the level. Solving also for the light curve, as we do, helps to minimize this effect. Although we do not input the overall light and continuum level of the star when "relative" photometry is used, the overall area of the spots is in any case mainly determined from the line shape and strength and while the result - including the total spotted area - is subject to external errors in the line profile data, it should be reasonably correct.

Our program is divided into two main sections. The first block contains the computation of the local line profiles and the minimization and geometry routines. We compute local line profiles from a solution of the equation of transfer from LTE model atmospheres. For HD 199178 a grid of model atmospheres between = 3500 and 6000 K in steps of 250 K and = 2.5 were taken from the ATLAS-9 CD (Kurucz 1993). The gravity of = 2.5 was chosen because a recent determination of the quiet atmospheric structure of the G8III-IV component of the RS CVn binary And by Donati et al. (1995) yielded such a low value. Maps computed with = 3.0 differ only by their average surface temperature being 30 K warmer. For each model atmosphere local line profiles are computed with a wavelength spacing of 0.008 Å. The adopted values for microturbulence and radial-tangential macroturbulence are listed in Table 3 and were taken primarily from the work of Gray (1992). Wavelength-dependent limb darkening is implicitly accounted for during the disk integration.

Table 3. Stellar parameters for HD 199178.

The second program block is for solving the inverse problem by using either a Maximum-Entropy penalty function or a Tikhonov regularization (for a comparison see, e.g., Collier-Cameron 1992). For all maps in this paper we chose a Maximum-Entropy reconstruction but a Tikhonov reconstruction would be equally suited. The grid spacing for the disk integration is chosen such that each grid element on the stellar surface has equal angular extent in longitude and latitude; in our case 5^o5^o.

© European Southern Observatory (ESO) 1999

Online publication: June 18, 1999
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