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

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4. Astrophysical parameters of HD 199178

4.1. HIPPARCOS-based absolute parameters

The Hipparcos spacecraft measured a trigonometric parallax of 0.01068" (ESA 1997) corresponding to a distance of 94[FORMULA]6 pc. The brightest V magnitudes observed between 1975 and 1996 were 7:m 05 in 1986.84, and 7:m 09 in 1993.44 and 1993.71 (Jetsu et al. 1999b). Because the 7:m 05 value is from a single V light curve being on average 0:m 1 brighter than a light curve taken just one month before, we conservatively adopt the 7:m 09 brightness as the unspotted (or least-spotted) magnitude. Combined with the distance, this gives an absolute visual magnitude of +2:m 22[FORMULA]0:m 14. Adopting the bolometric correction of -0.184 from Flower (1996), and neglecting interstellar absorption, we find a bolometric magnitude of +2:m 04 and a luminosity of 10.9[FORMULA]2.6 [FORMULA] (based on [FORMULA] of +4:m 64). These properties are typical for (sub)giants of spectral class mid G. The position of HD 199178 relative to the evolutionary tracks of Schaller et al. (1992) for solar metallicity suggests a mass of 1.65 [FORMULA] with a formal uncertainty of [FORMULA]0.1 [FORMULA].

4.2. Spectral classification and photospheric temperature

The spectral type was already determined by Herbig (1958) to be G5III-IV. More recently, Huenemoerder (1986) suggested that a classification of G7III-IV would be a better match for the optical spectrum. Although the Hipparcos-based luminosity is consistent with the G5III-IV classification, we point out that there exists considerable uncertainty for the effective surface temperature and gravity of a star of this type. While Bell & Gustafsson (1989) list [FORMULA] K and [FORMULA]=3.5 for a G5III-IV star and [FORMULA] K for a G8III-IV star (each based on a single star: [FORMULA] CrB and [FORMULA] Aqr, respectively), Jetsu et al. (1990a) estimated values for HD 199178 in the range of 5300 to 5450 K from broad-band photometry. Recently, Donati et al. (1995) found that the quiet atmospheric structure of [FORMULA] And (G8III-IV) has [FORMULA] K and [FORMULA]. We experimented with all these temperatures and found the best fits for the line profiles and the BV(RI)c photometry when using a nominal photospheric temperature in the range 5400-5500 K and [FORMULA]=2.5 but allowing the inversion program to freely choose temperatures of up to 6000 K. Finally, we adopted [FORMULA] K as the photospheric temperature that is most consistent with the observed colors and the G5III-IV classification. Note though, that this is not a constraint on the obtainable surface temperature distribution in the mapping process but is the default temperature of the unspotted photosphere.

Jetsu et al. (1990a, 1999a) observed systematic changes of the B-V color between extrema of 0:m 73 in 1987 and 0:m 83 in 1981. After 1988/89, the B-V color remained constant to within [FORMULA]0:m 02 and agrees with the Hipparcos B-V of 0:m 785[FORMULA]0:m 015. Average values for U-B, B-V, V-Rc, and V-Ic during the times of our spectroscopic observations in 1988-1990 are 0:m 30, 0:m 78, 0:m 45, and 0:m 86, respectively (Heckert & Stewart 1992, Heckert 1994). Comparing these average colors with the theoretical color library generated by Buser & Kurucz (1992) we find reasonable good matches with [FORMULA] = (5000-5250, 2.5-3.0, -0.5-0.0), but not for 5500 K. In fact, the observed colors do not agree with any [FORMULA]=2.5-3.0 Buser & Kurucz model hotter than [FORMULA] K, even if metals were overabundant by 0.5 dex. However, newer tables of (B-V) vs. [FORMULA] from the Kurucz (1993) CD-ROM seem to agree much better: for a giant with [FORMULA] and 5500 K a B-V of 0.77 is listed, in good agreement with the observations.

4.3. The quest for the correct rotation period

An uneven spot distribution on the surface of a rotating star allows, in principle, a precise measurement of the stellar rotation period. However, in case the stellar surface is not rigidly rotating the photometric period is a function of the a priori unknown latitudinal position of the spots. Differential surface rotation could alter the photometric period by, say, plus-minus several percent. In case of a synchronized binary one can circumvent this problem by using the orbital period as a timekeeper. In case of a single star there is no orbital period and one usually adopts the average photometric period as the stellar rotation period.

Previous determinations for HD 199178 yielded following periods: 3.337[FORMULA]0.001 days (Bopp et al. 1983) from Cloudcroft data taken between June and August 1980; 3.289 days (Nations & Seeds 1986) from Phoenix APT data from 144 nights in late 1985, and 3.337484 [FORMULA]0.000043 days from a combined data set spanning from 1975 to 1989 (Jetsu et al. 1990a). Instead of the traditional constant period ephemeris, Jetsu et al. (1999a) adopted time-restricted data sets due to sudden phase shifts ("flip-flops") and determined seasonal periods from a large collection of data between 1975 and 1996. Their analysis yields a large and homogeneous set of photometric periods that can be used to phase our spectroscopic data from 1988-1990. The 1996/97 photometry from our own APTs is used to determine the period and light-curve minimum for April 1997 (Strassmeier et al. 1999).

The spectroscopic and photometric data sets in this paper are therefore phased with the following ephemeris: data-set August 1988 :

[EQUATION]

data-sets April 1989 and May-June 1989 :

[EQUATION]

data-set May 1990 :

[EQUATION]

and data-set April 1997 :

[EQUATION]

where the period is the seasonal photometric period and the initial epoch is a time of minimum light.

4.4. Rotational, radial, and space velocities

Herbig (1958) was the first to measure the projected rotation velocity, [FORMULA], of HD 199178 and found 80 km s-1. This value was confirmed by Huenemoerder (1986) from comparably low-resolution spectra. More recently, however, Dempsey et al. (1992) obtained [FORMULA] = 74[FORMULA]2 km s-1 from a cross-correlation study using a subset of the NSO spectra from this paper while Fekel (1997) obtained 65.4[FORMULA]5 km s-1 from a calibration of FWHM and [FORMULA].

The inversion of line-profile shapes allows to determine [FORMULA] even more accurately than with a cross-correlation technique. This is because the line asymmetries due to the spots are modeled explicitly. A wrong [FORMULA] would produce a pronounced, artificial band encircling the star being either too bright or too dark depending upon whether the adopted [FORMULA] was too large or too small, respectively (e.g. Vogt et al. 1987). Minimizing such a feature and including the blends from Table 2 yields our adopted value for the projected rotational velocity of 71.5[FORMULA]1.0 km s-1.

The mean radial velocity of HD 199178 in the 11 observations from 1997 was -26.8 km s-1, with a dispersion of 1.4 km s-1. The adopted velocity of the reference star [FORMULA] Gem was 3.30 km s-1. Together with the Hipparcos data, the revised space velocities of HD 199178 relative to the Sun in a right-handed coordinate system are then (U,V,W)=([FORMULA], [FORMULA], [FORMULA]) km s-1.

4.5. Inclination of the rotation axis

Using the rotationally modulated linear Stokes parameters from UBVRI photometry, Huovelin et al. (1987) had derived i=79o for the inclination of the rotation axis of HD 199178. This value was later "confirmed" by Jetsu et al. (1990a) by applying the same polarisation model as Huovelin et al. but to a larger data set. Jetsu et al. found inclination angles between 64o and 88o from the first and second-order Fourier fits to the polarimetric variations, respectively. The weighted mean cited by Jetsu et al. (1990a) was 82.2o[FORMULA]1.2o.

We tried inclination angles between 5o and 85o for the line profile inversion and found a significantly larger reduction of the sum of the squares of the residuals (at maximized entropy) when an inclination of 30-50o was used (Fig. 2) instead of 80o as obtained from the polarization measurements. A low value such as this for i has already been suggested by Huenemoerder (1986) on the basis that the equatorial rotational velocity of HD 199178 is less than or equal to that of FK Comae (160 km s-1) and that the minimum radius is still in agreement with a G5-7 giant or subgiant spectrum. Our revised values for [FORMULA] and [FORMULA] in 1997 (Table 3) result in a minimum radius for HD 199178 of [FORMULA] [FORMULA]. If [FORMULA]40o is the correct inclination as indicated in Fig. 2, the unprojected radius of HD 199178 is [FORMULA]7.3 [FORMULA], i.e. fully consistent with the G5III-IV classification. The unprojected equatorial rotational velocity [FORMULA] would then be 111 km s-1, which is also consistent with HD 199178 being a (rapidly-rotating) FK Comae star.

[FIGURE] Fig. 2. The dependence of the normalized goodness of fit ([FORMULA]) on the adopted stellar inclination angle as discussed in Sect. 4.5. The different line styles are for the different data sets: 1997 (full line), 1990 (dashed line), 1989b (dashed-dotted), and 1989a (dotted line). The inclinations within the horizontal bar give overally the largest reduction of the residuals and we adopt [FORMULA]o as the most likely inclination for HD 199178.

Both previous polarimetric determinations of the inclination angle were based on the polarization model of Brown et al. (1978). This model is only applicable to pure Thompson or Rayleigh scattering or to a combination of these sources. However, as pointed out by Jetsu et al. (1990a), the wavelength dependence of the amplitude of the rotational modulation of polarization in HD 199178 does not support pure Thompson or Rayleigh scattering but could be accounted for by the Zeeman effect in magnetic surface regions. Also, Jetsu et al.'s values for i derived from the first and second order Fourier fits deviated systematically from each other. As Jetsu et al. noted in their paper, "This discrepancy could be a sign of a yet unexplained pitfall in the method".

Therefore, we believe that the observed polarization variations in HD 199178 are likely caused by a combination of scattering and the Zeeman effect due to magnetic surface features rotating in and out of view. Such detections are now being made almost routinely with the technique of Zeeman-Doppler imaging (Donati et al. 1997) and could be used to verify the nature of the polarization variability of HD 199178. Furthermore, there is still a strong controversy about the reliability of broad-band linear polarisation measurements on active stars (Leroy & LeBorgne 1989).

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© European Southern Observatory (ESO) 1999

Online publication: June 18, 1999
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