Astron. Astrophys. 319, 535-546 (1997)

3. Stellar properties of IN Virginis

Since little is known about this star I briefly rediscuss some of its inferred properties in the light of the new data presented in this paper. This seems important in order to evaluate the reliability of Doppler maps in general and because we need fundamental stellar parameters prior to the mapping analysis, such as the rotation period and the rotation velocity, an estimated photospheric temperature and gravity, the abundances of chemical elements and, last but not least, the approximate inclination of the stellar rotation axis.

3.1. The active chromosphere of IN Virginis

IN Vir has been discovered to be a coronally active star from its detection as a microwave emission source (Slee et al. 1987) and its moderate X-ray emission (Giommi et al. 1991). No published ultraviolet spectrum seems to be available for this star. Examples of the most prominent chromospheric-activity indicators in the optical spectrum of IN Virginis are shown in Fig. 1. Most obvious are the exceptionally strong H&K emission lines of Ca II, a signature of a chromospherically very active star. Also in emission is the Balmer H line redwards of the Ca II -H line. Our three H&K spectra show a variable emission strength with a total range of about 20% in emission equivalent width (see Fig. 9a in Sect. 5). Following the calibration procedure of Linsky et al. (1979b) we obtain absolutely calibrated emission-line fluxes in the H and K lines and convert them to radiative losses by subtracting the appropriate flux from a radiative equilibrium model atmosphere (i.e. without a chromosphere). The radiative losses in the (H/K) lines are (1.56/2.07) 10⁶ erg cm^-2 s^-1, (1.05/1.38) 10⁶ erg cm^-2 s^-1, and (1.33/1.77) 10⁶ erg cm^-2 s^-1 for JD 2,450,103, JD 2,449,781, and JD 2,449,774, respectively.

Fig. 1 also shows the strong emission in the cores of two of the infrared triplet lines of singly ionized calcium (Ca II  8542 and 8498 Å). Again, the width and the strength of these emission lines are within the range seen in other RS CVn-type binaries (e.g. Dempsey et al. 1993). Using the method described in Linsky et al. (1979a) we obtain an absolute emission-line flux of 2.1 10⁶ erg cm^-2 s^-1 in a 1 Å band around the line center of the 8542-Å line. These fluxes are typical for active RS CVn stars of the late spectral type of IN Virginis.

The complex profile of H is reminiscent of the "inverse P-Cygni"-type profile seen in other active stars with spatially inhomogeneous chromospheres and coronae. Just recently, Hatzes (1995a, 1995b) obtained simultaneous Doppler images and H line profiles for the RS CVn binary DM UMa and the WTT star V410 Tau. The typical DM UMa profile appeared to be composed of two Gaussian-like emission components: a narrow and constant component and a broad and highly variable component. However, their origin is not clear and interpretations include a corotating H emitting shell, a nonuniform and/or variable wind, an expanding chromosphere, mass flow in a gigantic coronal loop, and large hydrogen flares associated with significant mass flow. Even more dramatic H changes are seen in V410 Tau where the coincidence of maximum H and He I D₃ emission during the time when the large polar appendage transits the disk indicates that chromospheric activity such as plages and flares cause, at least part of, the H emission. The IN Vir H spectra will be discussed in more detail in Sect. 5.

We also plot in Fig. 1 a spectrum of the lithium-line region at 6708Å because it is supposed to be a crude indicator of stellar age and thus indirectly also of chromospheric activity. This spectrum shows a weak but clearly observable absorption feature at the lithium wavelength. The insert in Fig. 1e is the residual spectrum after subtraction of a reference star of identical M-K classification and reveals a weak Li I 6708 line consistent with previous observations of active stars, singles and binaries (e.g., Fekel & Balachandran 1993).

3.2. The rotation period

We applied a multiple period search program (Breger 1990) to our APT photometry covering 106 nights in 1994 and 70 nights in 1995. Fig. 2a shows the periodogram from the V data (lower panel) and the corresponding window function (upper panel). The greatest reduction of the sum of the squares of the residuals is obtained with a period of 8.232 0.003 days ( in Fig. 2a) in very good agreement with the periods derived by Cutispoto et al. (1992, 1994, 1996). Fig. 2a also shows several aliases of comparable but smaller amplitude, most noticable at frequencies of and , but also at a.s.o.. A frequency of ( 16 days) produces only half the amplitude of less than 0.038 mag in V. The primary reason for these aliases is the one-observation-per-night windowing of the APT observing schedule.

3.3. Orbital elements

The range of velocities in Table 1 already indicates that IN Virginis is a single-lined spectroscopic binary. Altogether, we obtained 23 high-precision radial velocities and use them to compute a preliminary orbit. A period search on just those 10 velocities from our run in 1994 suggested a period of 8.22 days, very close to the photometric period but final orbital elements, including an improved orbital period, were derived with all 23 velocities and with the differential-correction program of Barker et al. (1967). A first run with the preliminary period converged at an eccentricity so close to zero that a formal zero-eccentricity solution was adopted. The standard error of an observation of unit weight is 1.9 km s^-1, but two O-C residuals were as large as 5.0 km s^-1 and were given half weight in the orbit computation. The elements are given in Table 2 and the computed velocity curve is plotted in Fig. 3 along with the observations.

Fig. 3. Observed and computed radial velocity curve. Dots are the velocities from Table 1 and the line is the newly determined orbital solution from the elements in Table 2. Note that a zero-eccentricity orbit was adopted.

3.4. Rotation velocity, spectral type, and inclination of the stellar rotation axis

Tagliaferri et al. (1994) obtained a high-resolution spectrum of the Li I 6708-Å wavelength region of IN Virginis and measured a projected rotational velocity () of 22 km s^-1. This value is in good agreement with our own initial measure of 23.0 1.5 km s^-1 from cross-correlating a well-exposed IN Vir spectrum with 16 Vir and taking into account a radial-tangential macroturbulence of 4 km s^-1. Our final value of 24.0 1.0 km s^-1 for the was obtained from a series of test solutions with the Ca I -6439 profiles and the Fe I -6421 profiles with fixed inclination but different equatorial velocities.

Tagliaferri et al. (1994) also conducted a spectrum-synthesis analysis based on grids of model atmospheres mostly taken from Gustafsson et al. (1975), and determined a relatively low lithium abundance of and a metallicity of [Fe/H] using a 25-Å region around 6708 Å . Their conclusion was that IN Vir is more likely a K4IV+G8V binary instead of a single K5V star as proposed earlier by Cutispoto et al. (1992) from multi-color photometry (the assumed spectral type of the unseen secondary was recently revised to G7V by Cutispoto et al. 1996).

The late-K subgiant classification is basically consistent with our observations, just that various line ratios in the 6430-Å region indicate a slightly earlier spectral type for the visible star. A spectrum synthesis with several reference stars in the range G5 to K4 and luminosity classes III, IV, and V gives the best fit with  Ser (=HR 5940). The spectral classification of  Ser is listed as K1IV in Gray & Nagar (1985) but Fekel (1996) assigned a K2-3IV type from high-resolution spectra and the classification criteria of Strassmeier & Fekel (1990). We note that its color of 1.14 would be slightly too red for K1 anyway and fits the K2-3IV classification better. The moderately broad wings of strong absorption lines like Ca I 6439 Å confirm the subgiant luminosity classification of  Ser.

Gray & Nagar (1985) determined the projected rotational velocity of  Ser to 1.1 km s^-1 and its radial-tangential macroturbulence to  4 km s^-1. The line broadening in our single  Ser spectrum is marginally larger, on average 0.29-Å FWHM, and results in a value of 1.7 0.5 km s^-1 when we take into account a macroturbulence of 4 km s^-1.

Table 3. Stellar parameters for IN Virginis

Having the rotational velocity, the rotational period, and the luminosity class of IN Vir fixed we could, in principle, determine the inclination of the stellar rotation axis from the relation - if there were not the large range of radii for an evolved star. The Landolt-Börnstein tables (Schmidt-Kaler 1982) list radii for a K IV star between 2 and 10 . Nevertheless, the above relation still allows to compute a definite minimum stellar radius from observed quantities and we find =3.77 0.18 in good agreement with the M-K luminosity class IV inferred from the spectrum morphology. We note that none of our class III reference stars reproduced the IN Vir spectrum nearly as well as  Ser and we tend to rule out a class III classification.

We can also estimate an upper limit of the inclination of the stellar rotation axis because we do not see eclipses in the light curve and thus must be less than . If we adopt the G7-8V estimate from Tagliaferri et al. (1994) and Cutispoto et al. (1996) for the (unseen) secondary star, thus from the Landolt-Börnstein tables, we obtain the upper limit for the inclination of . Since any hot and thus more massive secondary star of, e.g., spectral-type F and main-sequence luminosity would be inconsistent with the observed colors, we may also estimate a lower limit for the inclination of the stellar rotation axis from our new mass function of and the fact that no secondary lines are visible in high-resolution red-wavelength spectra. Adopting masses between 0.79 - 0.92 (according to G5V to K0V) for the secondary star and masses in the range of 1.0 - 1.2 for the primary star, we obtain the lower limit for the inclination of . Thus, our best estimate for the inclination of the stellar rotation axis of IN Virginis is 62 and we adopt for our Doppler-imaging analysis and emphasize that the given range is not an error estimate but that all values in the given range are equally likely.

3.5. Average spot temperature

Compare a line-depth ratio of a particular line pair in which one line is temperature sensitive and the other not, and monitor this line ratio over one rotational cycle of a spotted star, then the changing average hemispheric temperature should modulate mainly the temperature-sensitive line but not the other, thus modulating the easy measurable line-depth ratio. This was pioneered by Gray & Johanson (1991), and an improved calibration for several spectral-line ratios in the 6160-Å region against effective temperature (actually color) was derived by Gray (1994) and recently reviewed by Gray (1996).

Unfortunately, such a calibration is not as straightforward as one might hope, because the lines are rotationally broadened, blended, perturbed by velocity fields, differentially abundant, and differently saturated if of different strength. A recent study of the influence of macroscopic velocity fields on line-depth ratios by Stift & Strassmeier (1995) also showed that only if the two lines in question are of comparable strength and do not differ radically in their broadening parameters, will the line-depth ratio not depend on stellar rotation. All of this will eventually just allow an estimate of the (average) surface temperature, but is nevertheless an additional - and independent - constraint for Doppler imaging.

Figs. 4a-d show the observed line-ratio variations for two line pairs and their calibration with color. For IN Virginis with 4600 K we chose following line pairs in the 6430-Å region: the V I line at 6413.509 Å (excitation potential = 1.35 eV) and the close blend Ni I 6414.581 (4.15 eV) + Si I 6414.980 (5.87 eV), and Y I 6435.004 Å (0.07 eV) + V I 6435.158 (1.94 eV) and the Fe I line at 6436.411 Å (4.19 eV). Their variations are in phase with the broad-band light curve (shown as a dotted line in Figs. 4a and 4c) in the sense that larger line ratios occur when the light curve shows a minimum, i.e. when a spot is in view. The observed, full amplitudes are 0.21 0.05 and 0.73 0.09 for the two line-pair ratios, respectively. Their uncertainties are estimated from the whole range of repeated measurements with both a Gaussian fit to the individual profiles using appropriate IRAF routines and by simply identifying the deepest point in the absorption profile.

Fig. 4. Line-depth ratio variations of IN Virginis (upper panels) and their respective calibrations from luminosity-class III and IV M-K standard stars against observed color (lower panels). The dots are the measured line ratios and their error bars indicate the whole range of values from repeated measurements with different techniques. The dotted line in the upper panels is the scaled, simultaneous V -band light curve and emphasizes the relation with the line-ratio variations due to the common cause. Note that the line ratios are chosen with temperature-sensitive line over temperature-insensitive line, therefore, the larger the line ratio the stronger was the temperature-sensitive line, and thus the cooler the average surface temperature. The crosses in the lower panels are the standard-star observations and the lines are the fits with the second-order polynomials in Eq. (2).

The lower panels in Fig. 4 present the observations of the two line pairs in a set of 68 Morgan-Keenan standard stars obtained with the same telescope and instrumental set-up as for IN Vir. A second-order polynomial fit to these data yields the following calibrations,

where means and the ratio . Together with the standard - relation of Bell & Gustafsson (1989) and the respective calibrations in Eq. (2) the observed line-ratio amplitudes of 0.21 0.05 and 0.73 0.09 imply temperature variations between phase 0.3 and 0.8 of 150 20 K (4400-4550 K) and 400 30 K (4350-4750 K) from the two line ratios, respectively. Obviously, the temperatures from both line ratios at phase 0.3 agree within their formal uncertainties but the temperatures for phase 0.8 differ by 200 K.

Errors for the absolute temperatures are larger than those for the variations because of errors in our calibration in Eq. (2) as well as in the - relation of Bell & Gustafsson (1989). Furthermore, by using Eq. (2) we implicitly assumed the same for all our calibration stars (and IN Virginis), although there is some evidence that the temperature difference between spots and photosphere depends on gravity (Saar et al. 1995). Altogether, we estimate the above relative temperature variations to be probably no better than 50 K.

Another possibility to estimate the surface temperature is to model the broad-band color curves. The -color curve of IN Vir in Fig. 2b shows an average seasonal amplitude of 0.050 0.007 mag and a maximum value for of 1.22 0.01 mag at phase 0.2. The corresponding values are 0.025 0.007 and 0.65 0.01 mag ¹, respectively. Although part of this amplitude is due to differential limb darkening in the V and bandpasses, a light and color-curve fit with wavelength-dependent limb darkening and the model of Strassmeier & Bopp (1992) with two spotted regions yields a temperature difference between photosphere and spots of 1000 200 K.

3.6. Chemical surface abundances

The equivalent widths of most metal lines in the red spectrum of IN Vir are larger by approximately 10-20 % when compared to HD 81410 - another RS CVn binary with a qualitatively very similar spectrum (classified as K1III by Bidelman & MacConnell 1973), almost identical and a rotation period of around 12 days. An average difference of about 10% is still obvious when compared to the inactive star  Ser, which is of identical M-K classification as IN Vir (K2-3IV). Differences of the line strengths are most noticeable for the iron lines, e.g., Fe I 6392.538, 6393.602, 6408.016 as well as our one mapping line at 6421.349 Å, but also for the Ni I -Si I blend at 6414.8 and several vanadium lines. We interpret this as evidence that the surface abundances of IN Vir deviate from solar values. Therefore, we first need to obtain specific elemental abundances before attempting to map the surface temperature.

The determination of chemical abundances for stars of low effective temperature is rather prone to blending by weak lines and thus requires detailed spectrum synthesis. Because of limited computing time we synthesize only a relatively small wavelength portion (6416-6442 Å) but at high wavelength resolution that enables to utilize lines down to a limit of 3 mÅ . The upper panel in Fig. 5 compares a rotationally unbroadened theoretical spectrum with the observed IN Vir spectrum and identifies most line contributions. The lower panel in Fig. 5 presents the achieved fit with the new abundances listed in Table 4. Our theoretical spectra use pre-computed Kurucz (1993) model atmospheres with a microturbulence of 2.0 km s^-1, pre-specified chemical compositions, and a modified line list including improved transition probabilities. The applied synthesis code has been written in Ada (Stift & Könighofer 1996) and is based on the original Fortran code of Baschek et al. (1966). The code allows the synthesis of blends over a large wavelength range and includes the effects of micro- and macroturbulence, rotation, and pulsation if present.

Fig. 5. Synthetic and observed spectra for IN Virginis. The upper panel compares an unbroadened synthetic spectrum (thin line) with the observed IN Vir spectrum (thick line). It demonstrates the amount of blending evident at the late spectral type of IN Vir. The line identifications on the top include, in addition to the element and the rest wavelength, our new value for the transition probability. The lower panel repeats the observed spectrum (thick line) and shows the obtained fit (thin line) when appropriate rotation is included. The three strongest lines are the spectral lines used for Doppler-imaging (Ca I 6439 Å , Fe I 6430 Å and Fe I 6421 Å).

Table 4. Logarithmic elemental abundances relative to hydrogen ()

For the purposes of this paper abundances are only needed for the blends within our mapping lines and we will only focus on these elements. Before solving for a new set of abundances for IN Vir we perform a check on the atomic line data by fitting the observed solar spectrum in the same wavelength region as for IN Vir (6390-6455 Å) but with "known" abundances from Kurucz's (1991) solar model. Then, the only remaining uncertainties in the synthesis of the solar spectrum are the transition probabilities () for the individual spectral lines. Similar work was already done for the wavelength region around the Doppler-imaging line Ca I 6439 in the G5III-IV FK Comae star HD 199178 (Strassmeier et al. 1997) and around the 6240-Å region in the G8III-IV RS CVn binary  And (Donati et al. 1995). Recently, Linnell et al. (1996) presented a spectrum synthesis of the metallic-line A3m binary EE Peg in the same wavelength region as in this paper.

The second step includes a fit to IN Vir with effective temperature as the only parameter. Only the two model atmospheres with 4500 K and 4750 K produce reasonable good fits while, at the same moment, can be varied between 3.5 and 4.0 and still reproduce the IN Vir spectrum. The "observed" spectrum is thereby always that spectrum that is closest to the light-curve maximum (phase 0.174) and thus represents the least spotted phase. Our fits are therefore just for an average photospheric spectrum consisting of a convolution of the normal photospheric spectrum plus a (weak) spot spectrum. The spots' influence on the iron line strength, however, is very small and is neglected in our abundance study (but not in the mapping). Temperature-sensitive lines like vanadium likely have a significant spot contribution, but this is also neglected in our synthesis approach because only weak vanadium lines are present. The obtained photospheric temperature is then mainly constrained by the relative strength of the Fe I 6430 to the Fe II 6431 lines and yields an effective temperature for IN Vir of 4600 70 K.

The gravity determination relies mostly on the pressure sensitive wings of the strong Ca I -6439 line. Unfortunately, there are several blends in the wings of this line, e.g. Eu II and Y I, that strengths and chemical abundances are not known a priori. Consequently, our determination is somewhat uncertain but must be in the 3.5-4.0 range.

The remaining final step now is to reproduce the observed spectrum by adjusting the chemical abundances. A simple trial-and-error approach is sufficiently effective and leads to the fit in the lower panel of Fig. 5 and the abundances in Table 4. We caution, however, that these abundances are only approximative because just a small portion of the optical spectrum was used to determine them, sometimes even just from a single line (e.g. europium). Nevertheless, we note that only lines from the heavier elements (above = silicon) had to be adjusted to fit the observed spectrum.

Beside the spectrum synthesis in the 6430-Å region we also measured the equivalent width of the Li I 6708-Å line from the residual spectrum shown in Fig. 1e to 33 3 mÅ . Using the curves-of-growth published by Pallavicini et al.(1987) for and , we find a logarithmic lithium abundance of , somewhat larger than the 0.3 value obtained by Tagliaferri et al. (1994).

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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