Astron. Astrophys. 347, 212-224 (1999)
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 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 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 2.6
(based on
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
with a formal uncertainty of
0.1
.
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 K and
=3.5 for a G5III-IV star and
K for a G8III-IV star (each based on
a single star: CrB and
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
And (G8III-IV) has
K and
. 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
=2.5 but allowing the inversion
program to freely choose temperatures of up to 6000 K. Finally, we
adopted 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
0:m 02 and agrees with the Hipparcos
B-V of 0:m 785 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 =
(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
=2.5-3.0 Buser & Kurucz model
hotter than K, even if metals
were overabundant by 0.5 dex. However, newer tables of (B-V) vs.
from the Kurucz (1993) CD-ROM seem
to agree much better: for a giant with
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 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 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]](img53.gif)
data-sets April 1989 and May-June 1989 :
![[EQUATION]](img54.gif)
data-set May 1990 :
![[EQUATION]](img55.gif)
and data-set April 1997 :
![[EQUATION]](img56.gif)
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, , 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 =
74 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 5 km s-1 from a
calibration of FWHM and .
The inversion of line-profile shapes allows to determine
even more accurately than with a
cross-correlation technique. This is because the line asymmetries due
to the spots are modeled explicitly. A wrong
would produce a pronounced,
artificial band encircling the star being either too bright or too
dark depending upon whether the adopted
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 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
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)=( ,
,
) 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 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 and
in 1997 (Table 3) result in a
minimum radius for HD 199178 of
. If
40o is the correct
inclination as indicated in Fig. 2, the unprojected radius of
HD 199178 is
7.3 ,
i.e. fully consistent with the G5III-IV classification. The
unprojected equatorial rotational velocity
would then be 111 km s-1,
which is also consistent with HD 199178 being a (rapidly-rotating)
FK Comae star.
![[FIGURE]](img69.gif) |
Fig. 2. The dependence of the normalized goodness of fit ( ) 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 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).
© European Southern Observatory (ESO) 1999
Online publication: June 18, 1999
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