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2 Data analysis

2.1 Observations and reduction

Spectroscopic observations have been performed with the REOSC échelle spectrograph at the 91-cm telescope of Catania Astrophysical Observatory - M. G. Fracastoro station (Serra La Nave, Mt. Etna). The spectrograph is fed by the telescope through an optical fibre (UV - NIR, 200$~\mu$m core diameter) and is placed in a stable position in the room below the dome level. Spectra were recorded on a CCD camera equipped with a thinned back-illuminated SITe CCD of 1024 $\times$ 1024 pixels (size 24 $\times$ 24 $\mu$m). The échelle crossed configuration yields a resolution of about 14 000, as deduced from the FWHM of the lines of the Th-Ar calibration lamp. The observations have been made in the red region. The detector allows us to record five orders in each frame, spanning from about 5860 to 6700 Å. In this spectral region there are several line pairs of low and high excitation potential, whose depth ratios are suitable for effective temperature determination.

The data reduction was performed by using the ECHELLE task of IRAF[*] package following the standard steps: background subtraction, division by a flat field spectrum given by a halogen lamp, wavelength calibration using the emission lines of a Th-Ar lamp, and normalization to the continuum through a polynomial fit. Particular care was paid to the continuum level definition. The IRAF task CONTINUUM was used for such a purpose and we chose a low-order (3rd-4th) Legendre polynomial to follow the continuum behaviour in each spectral order that is the result of the true spectral shape and of residual instrumental effects, like the blazing curvature, spectrograph sensitivity, etc., that are not fully removed by the reduction process.

The choice of a low-order polynomial guarantees a good definition of the continuum level at least on a spatial scale of a few tens of Å, much more than the typical wavelength separation of line pairs.

Table 1: Main parameters of observed active stars.
HD    Name Sp. Type V $v\sin i$   $P_{\rm rot}$
      (mag) (km s-1) (days)
209813 HK Lac K0III 6.52 15    24.4284
216489 IM Peg K2III-II 5.60 26    24.494
17433 VY Ari K3-4V-IV 6.9 6    16.1996

Observations were carried out from August 2000 to January 2001, on three selected targets, i.e. VY Ari, HK Lac, IM Peg, whose main parameters are reported in Table 1. The main requirement for the selection was the known strong spottedness and low $v\sin i$. The $v\sin i$ of the targets is in the 6-24 km s-1 range, so if the spots or the spotted area is large enough to feel the rotation broadening effect the width of the bumps caused by the spot visibility would be of the same order as the width of a spectral resolution element. The slow rotation combined with the relatively low spectral resolution will ensure the non-detection of the Doppler shift of the bump along the spectral lines due to the spot rotation.

In addition to the active stars, a number of giant and main sequence stars of spectral type in the range G2III-M0III and F8V-K7V, respectively, have been observed to establish the temperature scale of the line-depth ratios.

The average signal-to-noise ratio (S/N) at continuum in the spectral region of interest was 200-500 for the calibration stars and about 100-150 for the active stars.

2.2 Calibration stars and temperature scale

Table 2: Standard stars observed for the depth- $T_{\rm eff}$ calibrations.
HD    Name Sp. T. V $\pi$    B-V $T_{\rm eff}$
      (mag) (mas)   (K)
161239 84 Her G2IIIb 5.714 26.13 0.654 5732
196755 $\kappa$ Del G2IV 5.069 33.27 0.705 5583
161797 $\mu$ Her G5IV 3.417 119.05 0.752 5451
188512 $\beta$ Aql G8IV 3.715 72.95 0.855 5183
23249 $\delta$ Eri K0IV 3.527 110.58 0.922 5024
62345 $\kappa$ Gem G8IIIa 3.568 22.73 0.932 5001
216131 $\mu$ Peg G8III 3.488 27.95 0.934 4996
22796 12 Tau G6III 5.565 8.14 0.934 4996
28100 $\pi$ Tau G7IIIa 4.692 7.17 0.982 4891
197989 $\epsilon$ Cyg K0III 2.467 45.26 1.034 4783
74442 $\delta$ Cnc K0IIIb 3.937 23.97 1.082 4687
12929 $\alpha$ Ari K2III 2.009 49.48 1.153 4552
54719 $\tau$ Gem K2III 4.407 10.81 1.261 4350
43232 $\gamma$ Mon K1.5III 3.972 5.06 1.320 4239
49161 17 Mon K4III 4.758 6.73 1.394 4096
69267 $\beta$ Cnc K4III 3.532 11.23 1.481 3923
29139 $\alpha$ Tau K5III 0.868 50.09 1.537 3813
60522 $\upsilon$ Gem M0IIIb 4.058 13.57 1.539 3809
187691 54 Aql F8V 5.116 51.57 0.552 6045
22484 10 Tau F9IV-V 4.290 72.89 0.574 5976
157214 72 Her G0V 5.394 69.48 0.619 5837
186408 16 CygA G1.5Vb 5.960 46.25 0.645 5758
217014 51 Peg G2.5IV 5.463 65.10 0.665 5699
20630 $\kappa$1 Cet G5V 4.836 109.18 0.679 5658
10700 $\tau$ Cet G8V 3.496 274.17 0.727 5520
3651 54 Psc K0V 5.879 90.03 0.849 5197
22049 $\epsilon$ Eri K2V 3.726 310.75 0.882 5117
16160 HR 753 K3V 5.821 138.72 0.972 4912
201091 61 CygA K5V 5.224 287.13 1.169 4522
201092 61 CygB K7V 6.046 285.42 1.360 4162

To convert the depth ratio variation of our active stars into temperature variation we need to define a temperature scale for the measured line-depth ratios. We have then observed a number of single stars of different spectral type in the range from F8 to M0 and luminosity class from V to III. Main criteria for the selection were: (i) a low rotation velocity, (ii) a reasonably good parallax value, (iii) accurate B-V color index. Since the line-depth ratio is dependent on gravity also for some temperature-sensitive lines, we have observed main sequence and giant stars to correct the gravity effect and eventually set separate temperature scales to be used for active main sequence and giant stars. The calibration stars are listed in Table 2 together with their spectral type, V magnitude, parallax, B-V and effective temperature. Spectral types are from the Bright Star Catalogue (Hoffleit & Warren 1991), visual magnitudes V and B-V color indices are from the Geneva Web database (Mermilliod et al. 1997) and the parallaxes are from the Hipparcos Catalogue (ESA 1997).

Since effective temperatures are available for very few of our calibration stars, we used color indices B-V to set the effective temperature of each calibration star. Although interstellar reddening is not expected to be large, since all the stars in Table 2, with few exceptions, are closer than 100 pc, we applied an isotropic extinction correction to obtain the (B-V)0. We used $A_{\rm V}$ = 0.8 mag kpc-1 and a ratio of total to selective extinction of 3.3 as suggested by Henry et al. (2000). Conversion of (B-V)0 to effective temperature has been made through the empirical relation proposed by Gray (1992):

$\displaystyle \log T_{\rm eff}$=3.988 - 0.881(B-V)0 + 2.142(B-V)02
- 3.614(B-V)03 + 3.2637(B-V)04
- 1.4727(B-V)05 + 0.2600(B-V)06. (1)

This relation is quite accurate for $(B-V)_0~\leq~$1.5, i.e. within the color range of our calibration stars.

The metallicity effects can alter the B-V indices. Gray (1994) has investigated the influence of metallicity on color indices, finding an empirical relation between B-V and $\rm [Fe/H]$. The B-V color index is only very slightly dependent on $\rm [Fe/H]$, its maximum variation being of about $0\hbox{$.\!\!^{\rm m}$ }015$. According to the calibration relation given in Eq. (1), the corresponding temperature change is about 20-30 K. Given the uncertainties in the B-V values and in the setting of the temperature scale, such effects appears to be statistically not significant in our LDR-temperature calibrations.

2.3 Line identification in the 6100-6300 Å  range

Within the spectral range covered by our échelle frames, 5870-6700 Å, there are several pairs of lines suitable for temperature determination, the more frequently used being in the spectral region around 6200 Å (Gray & Johanson 1991; Gray & Brown 2001; Hatzes et al. 1998) and 6400 Å(Strassmeier & Fekel 1990; Strassmeier & Schordan 2000). We preferred to use lines in the 6100-6200 Å range because we were able to select a larger number of unblended pairs with separation smaller than 5 Å thus avoiding problems of different setting of the continuum, and less contamination from telluric lines that at our resolution is difficult to remove properly.

\par\includegraphics[width=8.8cm,clip]{} \end{figure} Figure 1: Sample of standard star spectra in the region around 6250 Å.

Figure 1 displays a portion of the 6200 Å region for a series of spectra of giant stars representative of spectral type from K0III to K4III. From this figure the strengthening of Fe I and V I lines with decreasing temperature is evident, while the $\lambda$6247 Fe II shows the opposite behaviour. Furthermore, the growth of low-excitation lines (like those of V I) is faster than that of iron lines. Altogether we identify 15 spectral lines forming 10 pairs suitable for line-depth ratios. These lines were identified through the solar spectral atlas (Moore et al. 1966), choosing the unblended lines. The only exception is the $\lambda$6243 V I line that is indeed composed of two very close V I lines of comparable intensities and with the same temperature dependence that appear as a single line at our resolution. Line identification and excitation potential, $\chi$, taken from Moore et al. (1966) and Bashkin & Stoner (1975) are listed in Table 3.

Table 3: Spectral lines used for LDR.
$\lambda$ Element $\chi$
(Å)   (eV)
6199.19 VI   0.29
6200.32 FeI   2.61
6210.67 ScI   0.00
6215.15 FeI   4.19
6215.22 TiI   2.69
6216.36 VI   0.28
6243.11 VI   0.30
6246.33 FeI   3.60
6247.56 FeII   3.89
6251.83 VI   0.29
6252.57 FeI   2.40
6265.14 FeI   2.18
6266.33 VI   0.28
6268.87 VI   0.30
6270.23 FeI   2.86
6274.66 VI   0.27

2.4 Measurement of line depth

The lines for each ratio are chosen to be close together in order to minimize errors in choosing the continuum. The lowest five points in the core of each measured line were fitted with a cubic spline and the minimum of this cubic polynomial was taken as the line depth. Writing the line depth d as

$\displaystyle d = \frac{S_{\rm c} - S_{\rm b}}{S_{\rm c}} = 1 - \frac{S_{\rm b}}{S_{\rm c}},$     (2)

where $S_{\rm c}$ and $S_{\rm b}$ are the signals in ADU (Analog to Digital Units) or in photons of continuum and bottom of the line, respectively, the fractional error on d can be expressed as
$\displaystyle \frac{\sigma_{d}}{d} = \frac{\sigma_{\frac{S_{\rm b}}{S_{\rm c}}}}{d} =
\frac{1 - d}{d}~\sqrt{\frac{1}{S_{\rm b}} + \frac{1}{S_{\rm c}}}\cdot$     (3)

This relation has been used to evaluate observational errors on line-depth ratios.

\par\includegraphics[width=8.8cm,clip]{} \end{figure} Figure 2: An example of LDR as a function of effective temperature. Different symbols refer to different luminosity classes. The solid line is the polynomial fit to evolved star LDR; the dashed line represents the fit to main sequence LDR.

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