3 The temperature line-depth ratio calibration

Using the measured depths we are now able to construct temperature line-depth ratio (LDR) calibration for the different line pairs. We select the $\lambda$6269 V I-$\lambda$6270 Fe I to illustrate the analysis. The other nine pairs are treated in a similar way. The plot of the observed LDR (Fig. 2) versus effective temperature displays two separate dependences: the main-sequence star dependence (squares) and the evolved star dependence, sub-giants (circles) and giants (triangles). For this line pair a significant gravity effect on the LDR is apparent.

In principle we need to determine the temperature variation of active giants as well as main-sequence active stars, so we need to set appropriate temperature scales. To use all the observations we have in hand, we proceeded in the following way: (i) first we defined separated temperature scales by doing low-order polynomial fits through main sequence and giant stars, (ii) then we determined the difference of the observed LDR with respect to the fits. Defining as gravity index the absolute magnitude difference $\Delta M_{\rm V}$ with respect to the ZAMS magnitude for the star's temperature we plot in Fig. 3 the LDR difference with respect to the giant fit, $\Delta {\rm Ratio}_{\rm GIA}$, and in Fig. 4 the LDR difference for the MS fit, $\Delta {\rm Ratio}_{\rm MS}$, as a function of $\Delta M_{\rm V}$. In both cases a clear correlation is apparent. We have fitted this dependence with a linear regression function (continuous line in Figs. 3 and 4). The absolute-magnitude corrected line-depth ratio for main sequence and giant are then

$${\rm LDR}_{\rm MS,G} = {\rm LDR} - (a_{\rm MS,G} + b_{\rm MS,G} \Delta M_{\rm V})$$ (4)

with b and a the slope and the intercept of regression line of $\Delta {\rm Ratio}$ on $\Delta M_{\rm V}$-magnitude correlation lines. With few exceptions, the scatter of the points around the correlation line is consistent with the error of line-depth ratio determination, thus giving confidence in the correction procedure and in the adoption of $\Delta M_{\rm V}$ as the gravity index.

Figure 3: Residuals of LDRs with respect to the polynomial fit to MS data of Fig. 2 plotted as a function of the gravity indicator  $\Delta M_{\rm V}$ (dots). The continuous line represents a linear fit to the data.

Figure 4: Residuals of LDRs with respect to the polynomial fit to Giant-stars data of Fig. 2 plotted as a function of the gravity indicator  $\Delta M_{\rm V}$ (dots). The continuous line represents a linear fit to the data.

Plots of the temperature calibration as a function of corrected LDRs are displayed in Figs. 5 and 6. Data points for stars of different luminosity class mix very well, leading to well-defined unique correlations, alternatively applicable to MS and Giant stars. The spread around the polynomial fits (continuous lines in the figures) is greatly reduced and is consistent with the uncertainties and errors in the temperature and LDR determination. The rms of the fits in this case are of 89 K and 73 K for MS and Giant-stars calibrations, respectively. This values are comparable with the uncertainty on the setting of the temperature scale (see e.g. Gray 1992). Similar values are found for most of the LDRs.

Figure 5: Effective temperature as a function of corrected LDR for MS stars. The solid line represents a polynomial fit to all data. The dotted line is the polynomial fit to LDR data corrected for Giant-star calibration (solid line in Fig. 6).

Figure 6: Effective temperature as a function of corrected LDR for Giant stars. The solid line represents a polynomial fit to all data. The dotted line is the polynomial fit to LDR data corrected for MS-star calibration (solid line in Fig. 5).

New low-order polynomial fits were placed through the plots of temperature versus LDRs to obtain separate temperature calibrations for MS and giant stars. Final fits of temperature versus LDR for the other line pairs are illustrated in Fig. 7. Some of the line pair ratios display a clear gravity effect like the $\lambda$6211 Sc I-$\lambda$6215 Fe I+Ti I or the $\lambda$6275 V I-$\lambda$6270 Fe I, that becomes very strong for the two ratios $\lambda$6243 V I-$\lambda$6247 Fe II and $\lambda$6246 Fe I-$\lambda$6247 Fe II. Other ratios, like the $\lambda$6252 V I-$\lambda$6253 Fe I or $\lambda$6266 V I-$\lambda$6265 Fe I, do show a very marginal gravity effect. In all cases we have made separate temperature calibrations.

For the two LDRs involving the $\lambda$6247 Fe II line, the gravity dependence is so high that the correction method described above could not be applied to derive average relations for the MS stars and for Giant stars, using all standard stars.

In order to measure the temperature sensitivity of each line-depth ratio we have calculated the slopes of the polynomial fits $\frac{{\rm d}T}{{\rm d}r}$ at temperatures of 4500 K, 5000 K and 5500 K (typical of most of active RS CVn binaries), for a 0.01 variation in LDR, which represents the typical uncertainty for the LDR determination in well exposed spectra. From single LDR-dependence we derived typical sensitivities of 10-20 K, and in some case even smaller, for a 0.01 variation in r (see Table 4).

Figure 7: $T_{\rm eff}$-LDR calibrations for Giant stars. The meaning of symbols and curves is as in Figs. 5 and 6.

   

Table 4: Temperature sensitivity of LDRs.

Lines pair	$\Delta T$ for $\Delta r$ = 0.01			$\Delta T$ for $\Delta r$ = 0.01
	4500 K			4500 K
		DWARFS			GIANTS
6199/6200	10.4	11.4	14.1	11.7	13.0	15.3
6211/6215	10.5	12.4	-	10.4	12.3	-
6216/6215	18.4	15.3	20.2	18.8	15.6	20.5
6243/6246	9.1	11.1	14.3	9.9	12.1	15.7
6243/6247	-	1.2	91.8	2.0	5.4	6.0
6246/6247	-	1.3	87.7	7.2	20.1	11.8
6252/6253	22.1	23.5	25.0	21.9	23.2	24.7
6266/6265	17.1	18.1	19.1	16.2	16.5	16.8
6269/6270	5.8	7.6	11.0	6.0	8.2	12.8
6275/6270	8.5	9.7	11.4	8.6	9.9	11.8