5 The determination of temperatures

According to the procedure described in the preceding sections, we have derived three (four) effective temperatures for each star in the sample by applying the IRFM at the IR wavelengths considered (Eq. 3). The individual values of T_J, T_H, T_K (and T_L') derived with their corresponding errors, are listed in Tables 6 and 7.

  

Figure 5: Differences $T_{J}-T_{\rm mean}$, $T_{H}-T_{\rm mean}$,$T_{K}-T_{\rm mean}$ and $T_{L'}-T_{\rm mean}$ versus effective temperature (only temperatures obtained from more than one filter are displayed). The dotted lines show the region $\vert\Delta T\vert\leq 1.5$%

  

Figure 6: Differences $T_{J}-T_{\rm mean}$, $T_{H}-T_{\rm mean}$,$T_{K}-T_{\rm mean}$ and $T_{L'}-T_{\rm mean}$ versus [Fe/H] (only temperatures obtained from more than one filter are displayed). The dotted lines show the region $\vert\Delta T\vert\leq 60$ K. The vertical accumulations correspond to the temperatures of the stars of the globular clusters contained in our sample

 

Table 7: Temperatures derived for the globular cluster stars of the sample. Column 1 Globular cluster. Column 2: Identification: M 3 nomenclature from Cohen et al. (1978) and Arribas & Martínez-Roger (1987); M 13, M 92 and M 67 nomenclature from Cohen et al. (1978); M 71 nomenclature from Frogel et al. (1979); 47 Tuc nomenclature from Frogel et al. (1981); NGC 288, NGC 1261, NGC 362 nomenclature from Frogel et al. (1983). Column 3: Metallicity. Column 4: Surface gravity. Column 5: Bolometric flux in $10^{-2}~{\rm erg\;cm}^{-2}\;{\rm s}^{-1}$. Column 6: Interstellar reddening. Column 7: Temperature derived in band J (units are K). Column 8: Error in T_J computed considering errors in $F_{\rm Bol}$, monochromatic fluxes, log(g) and [Fe/H]. Columns 9-10: The same as in Cols. 7-8 for temperature derived in band H. Columns 11-12: The same as in Cols. 7-8 for temperature derived in band K. Column 13: The weighted mean temperature derived from T_J, T_H and T_K. Column 14: Mean error computed by considering linear transmission of errors from Cols. 8, 10 and 12. Column 14. Number of temperatures considered in the average of Col. 13

Table 7: continued

Table 7: continued

Table 7: continued

The final temperature was derived as an average of T_J, T_H, T_K (and T_L') weighted with the inverse of their errors:

$$\overline{T_{\rm IRFM}}=\frac{\displaystyle\sum_{i=J,H,K,L}\... ...splaystyle\sum_{i=J,H,K,L}\frac{1}{\Delta T_{i}}\displaystyle}.$$ (6)

In order to estimate the error of the mean temperature, a linear transmission of the errors was considered, given that the errors in each band are not totally independent:

$$\Delta T_{\rm IRFM}=\frac{N}{\displaystyle\sum_{i=J,H,K,L}\frac{1}{\Delta T_{i}}\displaystyle},$$ (7)

where N is the number of bands considered and the error in the temperature of each band is defined by

$$\begin{eqnarray} (\Delta T_{i})^{2}=\left[\frac{\partial T_{i}}{\partial [q(\lam... ...partial T_{i}}{\partial \log(g)}\right]^{2}\;(\Delta \log(g))^{2}.\end{eqnarray}$$ (8)

The quantities in square brackets have been estimated by considering finite-difference interpolation with the help of grids of values similar to those displayed in Tables 1-4 with a finer spacing. Over 5000 K, the temperatures in the three (four) bands enter the average with similar weight. In that range, the assignment of weights automatically takes into account the uneven sensitivity of the IRFM in the different bands and the individual quality of IR photometry. However, below 5000 K only T_H, T_K and T_L' have been considered in the average, since R_J is a very insensitive indicator of temperature for the cooler stars. Under 4000 K, only T_K and T_L' have been considered. This is due to the fact that the coolest models show in the H band a local maximun of flux which is not observed in IR spectra (Lançon & Rocca-Volmerange 1992).

The mean error in the final temperatures is around 1-2%. Note however, that the uncertainties in the temperatures derived under 4000 K are greater than the errors determined from Eq. (8) due to the model imperfections in this range caused by the absence of important sources of opacity associated with certain molecules. Likewise, the IRFM is difficult to apply at temperatures above 8000 K because, as these stars emit a substantial proportion of energy at short wavelengths, the correction for insterstellar extinction and the determination of the bolometric flux are rather uncertain. For these reasons, the temperatures outside the range $4000\;{\rm K}<T_{\rm eff}<8000\;{\rm K}$ have a lower level of accuracy, and the error-bars quoted in Tables 6 and 7 have to be considered, in some cases, as lower estimates.

We show in Figs. 5 and 6 the difference between T_J, T_H, T_K and T_L', and the average temperature adopted. The individual residuals reveal that the dispersion is compatible with the estimated errors derived from the uncertainties in the input parameters of the IRFM. They follow approximately a normal distribution both with temperature and metallicity. As expected, the uncertainties are greater for temperatures obtained from R_J factors, due to the lower sensitivity of the IRFM in this band and the greater photometric error in the measurement of magnitude J. The consistency of T_J and T_H is good over 4500 K; however, under this temperature noteworthy discrepancies appear to be due to the fact that R_J- and R_H-factors lose their sensitivity to temperature in this range.

  

Figure 7: Differences between the temperatures derived in this work ($T_{\rm IRFM}$) and those derived by other authors. circles: Direct measurements: C76, R80, BR87, H89, M91, and WF87; squares: Bell & Gustafsson (1989), triangles: Arribas & Martínez-Roger (1987). stars: Blackwell & Lynas-Gray (1998). In the upper figure the lines corresponding to the mean internal error of the work ($\pm$ 1.5%) are shown

  

Figure 8: Differences between the temperatures derived in this work ($T_{\rm IRFM}$) and those derived by Frogel et al. (1979, 1981, 1983) and Cohen et al. (1978) ($T_{\rm CFP}$). Top: Differences against $T_{\rm IRFM}$.Bottom: Differences against metallicity. The differences are consistent with a zero-point offset amounting to 56 K (dotted line)

 

Table 8: Comparison between the temperatures derived in the present work (Col. 3) and those derived by direct methods (Col. 2). When several direct measurements were available we have considered the average value. The temperatures of the Sun and Procyon, measured in Paper I, are also listed. The mean difference $T_{\rm direct}-T_{\rm IRFM}$ is 3 $\pm$ 51 K

 

Table 9: Comparison between the temperatures derived in the present work (Col. 2) and those derived by Arribas & Martínez-Roger (1987) (Col. 3). The mean difference $T_{\rm IRFM}-T_{\rm AM89}$ is -1.3 $\pm$ 1.4% (without M3-53)