2 The multiparametric calibrations

The effective temperatures used for the present set of calibrations were uniformly obtained by means of the IRFM as described in Paper II. The sources of photometry are documented in Alonso et al. (1998; Paper I), critical attention has been paid to the homogenization of data extracted from different catalogues, and to the correction of interstellar reddening. It is worth noticing that the composition of the sample reflects a good balance between field and globular cluster giants.

 

 

Table 1: Effective temperature scale for class III stars. Data between parentheses mean that the number of stars considered in the average was too small as to provide a significant standard deviation

	$T_{{\rm eff}}$ (K)
Sp. type	$0.5~ \raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }~{\rm [Fe/H]} ~\raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }-1.0$	$-1.0~ \raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }~{\rm [Fe/H]}~ \raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }-2.0$	[Fe/H] $\raisebox{-0.6ex}{$\stackrel{\textstyle <}{\sim}$ }-2.0$
F0III	7046 $\pm$ 300	--	--
F2III	6804 $\pm$ 300	--	--
F5III	6255 $\pm$ 55	--	--
F6III	6190 $\pm$ 115	--	--
F8III	5805 $\pm$ 225	--	4920 $\pm$ 100
G0III	--	--	4875 $\pm$ 235
G2III	--	5306 $\pm$ 265	4753 $\pm$ 450
G4III	(5180)	--	--
G5III	5050 $\pm$ 155	4855 $\pm$ 220	4545 $\pm$ 250
G6III	5040 $\pm$ 100	--	--
G7III	4920 $\pm$ 155	--	--
G8III	4860 $\pm$ 130	4506 $\pm$ 235	--
G9III	4725 $\pm$ 230	--	--
K0III	4660 $\pm$ 180	--	(4550 $\pm$ 260)
K1III	4580 $\pm$ 245	--	--
K2III	4455 $\pm$ 190	--	--
K3III	4285 $\pm$ 160	--	(4235)
K4III	4195 $\pm$ 320	--	--
K5III	3950 $\pm$ 115	--	--

In Table 1, we present the effective temperature scale of giant stars (class III) from early F to late K, which results from averaging temperatures of the stars of the sample classified in each spectral type. The utility of this kind of table is limited since on the one hand it requires the knowledge of the spectral type of the problem star, which is not always available; on the other, the relation between temperature and spectral type depends also on metallicity, and only a restricted number of metal-poor giants have an accurate spectral type classification (the use of published spectral classifications is too coarse as proven by the size of errorbars in Table 1). For this reason we provide in sections below more useful calibrations of the temperature scale against direct photometric observables and [Fe/H].

From a practical standpoint we have fitted the data to polynomials of the form $\theta_{{\rm eff}} = P({\rm colour,[Fe/H]})$, where $\theta_{{\rm eff}}=5040/T_{{\rm eff}}$. In a preliminary step, a group of stars which departed from the mean tendency in several colour-colour diagrams were discarded from the sample used in the calibration. The least squares method was then iteratively applied discarding in successive steps stars which departed from the fit more than $2.5\sigma$ taking care that residuals were more or less normally distributed. Typically, four to seven iterations were enough depending on the colour used in the calibration. This approach has been previously applied with good results for the calibration of main sequence temperatures (Alonso et al. 1996; Paper III). In Table 2, we summarize the coefficients of the fits. In Table 3 we show the colour and metallicity ranges of applicability of each of the fits. Finally, we show in Table 4 the stars discarded in any of the fits.

 

 

Table 2: Coefficients for the fits of the form $\theta _{{\rm eff}}=a_0+a_1 X+a_2 X^{2}-a_3 X {\rm [Fe/H]}+a_4 {\rm [Fe/H]}+a_5{\rm [Fe/H]}^{2}$, where X stands for the colour (Col. 2). The corresponding standard deviations $\sigma (\theta _{{\rm eff}})$ and $\sigma (T_{{\rm eff}})$, together with the number of stars considered, are also shown. Column 1 contains the equation number assigned to each fit in the text

Eq. #	Colour	a0	a1	a2	a3	a4	a5	$\sigma (\theta _{{\rm eff}})$	$\sigma (T_{{\rm eff}})$ (K)	N. of stars
1	(U-V)	0.6388	0.4065	-0.1117	-2.308e-3	-7.783e-2	-1.200e-2	0.023	164	127
2	(U-V)	0.8323	9.374e-2	1.184e-2	2.351e-2	-0.1392	-1.944e-2	0.020	80	283
3	(B-V)	0.5716	0.5404	-6.126e-2	-4.862e-2	-1.777e-2	-7.969e-3	0.020	167	122
4	(B-V)	0.6177	0.4354	-4.025e-3	5.204e-2	-0.1127	-1.385e-2	0.024	96	416
5	(V-R)	0.4972	0.8841	-0.1904	-1.197e-2	-1.025e-2	-5.500e-3	0.021	150	248
6	(V-I)*	$\theta_{{\rm eff}}=0.5379+0.3981(V-I)+4.432$e-2 (V-I)2-2.693e-2(V-I)3						0.017	125	214
7	(R-I)	0.4974	1.345	-0.5008	-8.134e-2	3.705e-2	-6.184e-3	0.022	150	217
8	(V-K)	0.5558	0.2105	1.981e-3	-9.965e-3	1.325e-2	-2.726e-3	0.005	40	256
9	(V-K)	0.3770	0.3660	-3.170e-2	-3.074e-3	-2.765e-3	-2.973e-3	0.005	25	412
10	(J-H)	0.5977	1.015	-1.020e-1	-1.029e-2	3.006e-2	1.013e-2	0.023	170	505
11	(J-K)	0.5816	0.9134	-0.1443	0.0000	0.0000	0.0000	0.020	125	511
12	(V-L')*	$\theta_{{\rm eff}}=0.5641+0.1882(V-L')+1.890$e-2 (V-L')2-4.651e-3(V-L')3						0.009	65	122
13	(I-K)J	0.5859	0.4846	-2.457e-2	0.0000	0.0000	0.0000	0.018	130	213
14	(b-y)	0.5815	0.7263	6.856e-2	-6.832e-2	-1.062e-2	-1.079e-2	0.013	110	118
15	(b-y)	0.4399	1.209	-0.3541	8.443e-2	-0.1063	-1.686e-2	0.018	70	169
16	(u-b)	0.5883	0.2008	-5.931e-3	5.319e-3	-1.000 e-1	-1.542e-2	0.021	110	181

* The functional expression of the fit for this colour is explicitly shown, since it differs from the general expression adopted.

2.1 $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(U-V)~and\,{:}~(B-V)$

Johnson's UBV photometry is practically extended to all stars contained in astronomical catalogues of interest, for this reason the calibration of (B-V) and (U-V) as temperature indicators is useful in many studies. We have adopted (U-V) instead of (U-B), which is more commonly tabulated, because it is a better indicator of temperature and has a similar behaviour with metallicity.

The fits obtained for $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(U-V)$ are shown in Table 2 (Eqs. (1) and (2)); the corresponding ranges of application are shown in Table 3.

In the range of colour $1.2 \leq (U-V) \leq 1.5$ a linear interpolation of Eqs. (1) and (2) provides a good fit of the data ensuring continuity.

The fits obtained for $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(B-V)$ are shown in Table 2 (Eqs. (3) and (4)); the corresponding ranges of application are shown in Table 3.

In the range of colour $0.7\leq (B-V) \leq0.8$ a linear interpolation of Eqs. (3) and (4) provides a good fit and ensures continuity.

A caveat has to be pointed out about these calibrations in the range of low temperatures: At $(B-V)\approx 1.55$ and $(U-V)\approx 3.55$ temperature seems to drop abruptly $\sim$250 K. This effect could be real, probably related to the variation of surface gravity in this range. The ultimate reason being a variation of the flux balance in the UV/visible region linked to changes in the opacity sources with gravity. In this respect, a turn-over is observed in the colour:colour diagrams (V-K):(B-V) and (V-K):(U-V), which would imply constant colour with decreasing temperature (see Figs. 8 and 9 in Paper I). However, another possible explanation could be in the shortcomings of models below 4000 K. Unfortunately, temperatures derived by means of the IRFM in this range are affected by large errors which make difficult to ascertain if the effect is spurious or not. In any case, the polynomial fits used here are obviously unable to follow the described feature. Therefore, under 4000 K this point should be taken into account when applying the above calibrations.

Figure 1: $T_{{\rm eff}}$ against (U-V). The lines display the fit corresponding to Eqs. (1) and (2) for $\rm [Fe/H] = 0$ (solid line), $\rm [Fe/H] = -1$ (dashed line), $\rm [Fe/H] =-2$ (dotted line), $\rm [Fe/H] = -3$ (dashed-dotted line). Symbols stand for the following metallicity groups: Open triangles $\rm [Fe/H]$ >0, filled circles $0\geq$ $\rm [Fe/H] >-0.25$, asterisks $-0.25\geq {\rm [Fe/H]}>-1$, filled triangles $-1\geq {\rm [Fe/H]}>-2$, open circles $-2\geq {\rm [Fe/H]}>-2.5$, squares $-2.5\geq$ [Fe/H]. The horizontal long-dashed line delineates the region $T_{{\rm eff}}\leq$ 4000 K where temperatures derived by means of the IRFM have lower accuracy. The top panel of the figure shows the residuals of the fit

Figure 2: $T_{{\rm eff}}$ against (B-V). The lines display the fit corresponding to Eqs. (3) and (4) for [Fe/H] = 0 (solid line), $\rm [Fe/H] = -1$ (dashed line), $\rm [Fe/H] =-2$ (dotted line). Symbols stand for the same metallicity groups as in Fig. 1. The horizontal long-dashed line delineates the region $T_{{\rm eff}}\leq$ 4000 K where temperatures derived by means of the IRFM have lower accuracy. The top panel of the figure shows the residuals of the fit

We display in Figs. 1 and 2 the residuals of the fits. The observed dispersion is compatible with typical errors on $T_{{\rm eff}}$, [Fe/H], (U-V) and (B-V).

The mean variation $\Delta T_{{\rm eff}}/\Delta (U-V)$ amounts approximately to 18 K per 0.01 mag for (U-V)<1.0 and 6 K per 0.01 mag for (U-V)>1.0. At constant (U-V) temperature monotonically decreases with decreasing [Fe/H]. The gradient $\Delta T_{{\rm eff}}$/[Fe/H] depends slightly on colour and diminishes with decreasing [Fe/H], as expected from atmospheres theory. The value of saturation is out of the range of the present calibration, although extrapolation provides [Fe/H]  $\approx -3.5$. When using this calibration an error of 0.05 mag on measured (U-V) implies mean errors of $1.5-0.7\%$ in temperature. Equivalently, an error of 0.5 dex in [Fe/H] implies mean errors of $3.5-2.3\%$.

The mean variation $\Delta T_{{\rm eff}}/\Delta (B-V)$ amounts approximately to 42 K per 0.01 mag for (B-V)<0.8 and 15 K per 0.01 mag for (B-V)>0.8. At constant (B-V), temperature monotonically decreases with decreasing [Fe/H]. The gradient $\Delta T_{{\rm eff}}/\Delta$[Fe/H] depends on colour and tends to zero as [Fe/H] decreases (saturation occurring at $\rm [Fe/H]$  $\approx -3$). When using this calibration an error of 0.03 mag on (B-V) implies mean errors of $1.2-2.0\%$ in temperature. Equivalently, an error of 0.5 dex in [Fe/H] implies mean errors ranging $1.1-1.9\%$.

Figure 3: Comparison between the present calibrations and several published calibrations of $T_{{\rm eff}}$ against (U-V) and (B-V). a,b) Theoretical calibrations. Squares: calibration of Buser & Kurucz (1992; BK92) based on Kurucz models (Solid lines: $\rm [Fe/H] = 0$, dashed lines: $\rm [Fe/H] = -1$, dotted lines: $\rm [Fe/H] =-2$); circles: Calibration for $\rm [Fe/H] = 0$ of Bessel et al. (1998; BCP98) based on Kurucz models. Triangles: Calibration BCP97 based on NMARCS models. c) Empirical calibrations. Squares: Böhm-Vitense (1981) (Solid lines: $\rm [Fe/H] = 0$, dotted lines: $\rm [Fe/H] =-2$); Triangles: Calibration of Flower (1996); Circles: Calibrations of Blackwell & Lynas-Gray (1998; BL98); Crosses: Calibration of Montegriffo et al. (1998; M98) (Solid lines: metal-rich giants, dotted lines: metal-poor giants)

In Fig. 3, we show the comparison of calibrations (1), (2), (3) and (4) with several representative calibrations published previously.

We have included in our analysis the scale of Johnson (1996; J66) because, from a historic point of view, it is the first comprehensive calibration of the temperature scale, although his results have been superseded by more recent works. It is remarkable the good agreement found both with $T_{{\rm eff}}{:}~(B-V)$ and $T_{{\rm eff}}{:}~(U-V)$ calibrations, which only deviates significantly from ours in the red edge of colour axes.

Differences found with the calibration $T_{{\rm eff}}{:}~(B-V)$ based on the IRFM of Blackwell & Lynas-Gray (1998; BL98) for Population I stars is compatible with a zero point shift amounting to 70 K.

The calibration presented by Böhm-Vitense (1981) for stars with $z/z_{\odot}=0.01$is in very good agreement with our calibration for [Fe/H] = -2, however strong discrepancies are found for Population I calibration.

We provide also comparison with the scale of Montegriffo et al. (1998; M98). It must be pointed out that although it is not properly a homogeneous calibration but an amalgam based on previous calibrations, it takes into account the effect of metallicity although in a rough manner. Our temperatures for solar metalicity stars are $\sim$ 200 K larger than M98 ones, however our temperatures for metal-poor stars are $\sim$ 150 K smaller, the reason for these discrepancies is unclear, but they are similar to those found when comparing calibrations of temperature against other colours in Sects. 2.2 and 2.3.

In summary, differences observed are significant and illustrate the state of the art of temperature calibrations. Two conclusions may be extracted from the above analysis, on the one hand, our semi-empirical calibrations differ from theoretical ones in a systematic manner, discrepancies ranging $\pm$ 5% (Figs. 3a,b). On the other hand, a better agreement is found in general with semi-empirical calibrations. In this case, differences are within $\pm$ 2% in the range $8000>T_{{\rm eff}}> 4500$ K (Figs. 3a,c).

 

 

Table 3: Colour and metallicity ranges of applicability of temperature calibrations

Eq. #	Colour	Colour range	Metallicity range
1	(U-V)	$0.40 \leq (U-V) \leq 1.20$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$0.35 \leq (U-V) \leq 1.20$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$0.40 \leq (U-V) \leq 1.20$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$0.50 \leq (U-V) \leq 1.20$	$-2.5 \geq {\rm [Fe/H]} > -3.0$
2	(U-V)	$1.50 \leq (U-V) \leq 3.50$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$1.50 \leq (U-V) \leq 3.50$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$1.50 \leq (U-V) \leq 3.25$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
3	(B-V)	$0.20 \leq (B-V) \leq 0.80$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$0.35 \leq (B-V) \leq 0.80$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$0.35 \leq (B-V) \leq 0.80$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$0.50 \leq (B-V) \leq 0.80$	$-2.5 \geq {\rm [Fe/H]} > -3.0$
4	(B-V)	$0.70 \leq (B-V) \leq 1.90$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$0.70 \leq (B-V) \leq 1.80$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$0.70 \leq (B-V) \leq 1.35$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$0.70 \leq (B-V) \leq 1.00$	$-2.5 \geq {\rm [Fe/H]} > -3.0$
5	(V-R)	$0.15 \leq (V-R) \leq 1.70$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$0.45 \leq (V-R) \leq 1.50$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$0.50 \leq (V-R) \leq 1.00$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$0.55 \leq (V-R) \leq 0.85$	$-2.5 \geq {\rm [Fe/H]} > -3.0$
6	(V-I)	$0.20 \leq (V-I) \leq 2.90$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$0.80 \leq (V-I) \leq 2.00$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$0.85 \leq (V-I) \leq 2.20$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$1.00 \leq (V-I) \leq 1.70$	$-2.5 \geq {\rm [Fe/H]} > -3.0$
7	(R-I)	$0.15 \leq (R-I) \leq 1.40$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$0.25 \leq (R-I) \leq 0.80$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$0.35 \leq (R-I) \leq 0.70$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$0.40 \leq (R-I) \leq 0.65$	$-2.5 \geq {\rm [Fe/H]} > -3.0$
8	(V-K)	$0.20 \leq (V-K) \leq 2.50$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$1.00 \leq (V-K) \leq 2.50$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$1.20 \leq (V-K) \leq 2.50$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$1.70 \leq (V-K) \leq 2.50$	$-2.5 \geq {\rm [Fe/H]} > -3.0$
9	(V-K)	$2.00 \leq (V-K) \leq 4.90$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$2.00 \leq (V-K) \leq 4.60$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$2.00 \leq (V-K) \leq 3.40$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$2.00 \leq (V-K) \leq 2.80$	$-2.5 \geq {\rm [Fe/H]} > -3.0$
10	(J-H)	$0.00 \leq (J-H) \leq 0.90$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$0.20 \leq (J-H) \leq 0.80$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$0.30 \leq (J-H) \leq 0.70$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$0.35 \leq (J-H) \leq 0.65$	$-2.5 \geq {\rm [Fe/H]} > -3.0$
11	(J-K)	$0.00 \leq (J-K) \leq 1.10$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$0.20 \leq (J-K) \leq 1.00$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$0.30 \leq (J-K) \leq 0.90$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$0.40 \leq (J-K) \leq 0.80$	$-2.5 \geq {\rm [Fe/H]} > -3.0$
12	(V-L')	$0.40 \leq (V-L') \leq 5.00$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
13	$(I-K)_{\rm J}$	$0.00 \leq (I-K)_J \leq 1.90$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$0.50 \leq (I-K)_J \leq 1.60$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$0.70 \leq (I-K)_J \leq 1.50$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$0.80 \leq (I-K)_J \leq 1.20$	$-2.5 \geq {\rm [Fe/H]} > -3.0$
14	(b-y)	$0.00 \leq (b-y) \leq 0.55$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$0.30 \leq (b-y) \leq 0.55$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$0.35 \leq (b-y) \leq 0.55$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$0.40 \leq (b-y) \leq 0.55$	$-2.5 \geq {\rm [Fe/H]} > -3.0$
15	(b-y)	$0.50 \leq (b-y) \leq 1.00$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$0.50 \leq (b-y) \leq 0.90$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$0.50 \leq (b-y) \leq 0.80$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$0.50 \leq (b-y) \leq 0.70$	$-2.5 \geq {\rm [Fe/H]} > -3.0$
16	(u-b)	$1.60 \leq (u-b) \leq 4.00$	$+0.2 \geq {\rm [Fe/H]} > -0.5$
		$1.60 \leq (u-b) \leq 3.70$	$-0.5 \geq {\rm [Fe/H]} > -1.5$
		$1.60 \leq (u-b) \leq 3.40$	$-1.5 \geq {\rm [Fe/H]} > -2.5$
		$1.60 \leq (u-b) \leq 2.60$	$-2.5 \geq {\rm [Fe/H]} > -3.0$

2.2 $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(V-R)$, ${:}~(V-I)~and\,{:}~(R-I)$

The importance of VRI photometry is increasing, among other advantages, because of the better quantum efficiency of the first generation of CCD detectors in the optical-infrared wavelength range covered by these filters. Furthermore, a considerable part of the flux of red giant stars, and in general of cool stars, is emitted through R, I bands. As a consequence (V-R), (V-I) and (R-I) colours have revealed themselves as very useful stellar temperature indicators in many investigations. Although Cousins' system has been standardised even in the northern hemisphere, here we have adopted Johnson's system as a reference because most of the photometry of the stars in our sample is in this system. Transformations between Johnson, Cousins and many other systems are well determined by Bessell (1979) and Fernie (1983).

Figure 4: $T_{{\rm eff}}$ vs. (V-R). The lines display the fit corresponding to Eq. (5). Symbols and lines are the same as for Fig. 2

The fit obtained for $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(V-R)$ is shown in Table 2 (Eq. (5)); the corresponding ranges of application are shown in Table 3. We display in Fig. 4 the mean lines corresponding to Eq. (5), together with the residuals of the fit.

The mean variation $\Delta T_{{\rm eff}}/\Delta (V-R)$ amounts approximately to 58 K per 0.01 mag for (V-R)<0.65 and 16 K per 0.01 mag for (V-R)>0.65. At constant (V-R), temperature monotonically decreases with decreasing [Fe/H]. The gradient $\Delta T_{{\rm eff}}/\Delta{\rm [ Fe/H]}$ shows little dependence on colour and tends to zero as [Fe/H] decreases (saturation occurring at ${\rm [Fe/H]}\approx -2.5$). When using this calibration an error of 0.03 mag in (V-R) implies mean errors of 2.2-1.1% in derived temperature. Equivalently, an error of 0.5 dex in [Fe/H] may imply mean errors of 1.0%.

The fit obtained for $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(V-I)$ is shown in Table 2 (Eq. (6)); the corresponding ranges of application are shown in Table 3.

Figure 5: $T_{{\rm eff}}$ vs. (V-I). The lines display the fit corresponding to Eq. (6). Symbols and lines are the same as for Fig. 2

Figure 6: $T_{{\rm eff}}$ vs. (R-I). The lines display the fit corresponding to Eq. (7). Symbols and lines are the same as for Fig. 2

We display in Fig. 5 the mean lines corresponding to Eq. (6), together with the residuals of the fit.

The mean variation $\Delta T_{{\rm eff}}/\Delta (V-I)$ amounts approximately to 32 K per 0.01 mag for (V-I)<1.2 and 10 K for (V-I)>1.2. When using this calibration an error of 0.03 mag in (V-I) implies mean errors of 1.2-0.8% in temperature.

The calibration of giants' $T_{{\rm eff}}$ as a function of (V-I) is metallicity independent as in the case of dwarf stars (Paper III). This fact together with the relatively small value of $\Delta T_{{\rm eff}}/\Delta (V-I)$ makes of (V-I) an excellent temperature indicator for giants.

The fit obtained for $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(R-I)$ is shown in Table 2 (Eq. (7)); the corresponding ranges of application are shown in Table 3.

We display in Fig. 6 the mean lines corresponding to Eq. (7), together with the residuals of the fit.

The mean variation $\Delta T_{{\rm eff}}/\Delta (R-I)$ amounts approximately to 70 K per 0.01 mag for (R-I)<0.6 and 15 K per 0.01 mag for (R-I)>0.6. At constant (R-I), temperature monotonically increases with decreasing [Fe/H] in the blue range (R-I)<0.6, conversely it decreases with decreasing [Fe/H] in the red range (R-I)>0.6. The gradient $\Delta T_{{\rm eff}}/\Delta$[Fe/H] depends on colour and tends to zero as [Fe/H] decreases (saturation occurring at [Fe/H] $\approx -3$). When using this calibration an error of 0.03 mag in (R-I) implies mean errors of 2.7-1.1% in derived temperature. Equivalently, an error of 0.5 dex in [Fe/H] may imply errors as large as 1.9%. In Fig. 7 we show a comparison of relations (5) and (6) with several theoretical and empirical calibrations taken from literature.

Figure 7: a) Comparison between the present calibration and several published calibrations of $T_{{\rm eff}}$ against (V-R). When necessary colours have been transformed into Johnson system by using relations provided by Bessel (1979) and Fernie (1983). Stars: Calibration of Johnson (1966; J66). Circles: Calibration BCP98 based on Kurucz models. Triangles: Circles: Calibration BCP98 based on NMARCS models. Squares: calibration BK92 (Solid lines: $\rm [Fe/H] = 0$, dashed lines: $\rm [Fe/H] = -1$, dotted lines: $\rm [Fe/H] =-2$). Asterisks: Calibration of Bell & Gustaffson (1989; BG89) (Solid lines: $\rm [Fe/H] = 0$, dashed lines: $\rm [Fe/H] = -1$, dotted lines: $\rm [Fe/H] =-2$). b) Comparison between the present calibration and several published calibrations of $T_{{\rm eff}}$ against (V-I). Symbols are the same as for a) where only $\rm [Fe/H] = 0$ has been considered, and Crosses: Calibration of M98 (Solid lines: metal-rich stars, dotted lines: metal-poor stars)

Figure 8: $T_{{\rm eff}}$ vs. $(V-K)_{{\rm TCS}}$. The lines display the fit corresponding to Eqs. (8) and (9). Symbols and lines are the same as for Fig. 2. Recall that $(V-K)_{{\rm J}}=-0.05+1.007(V-K)_{{\rm TCS}}$

In general, theoretical calibrations (Bessell et al. (1998; BCP98) based on Kurucz models, and Buser & Kurucz (1992; BK92)) show strong systematic differences both with our $T_{{\rm eff}}{:}~(V-R)$ and $T_{{\rm eff}}{:}~(V-I)$ calibrations. A much better agreement, compatible with zero-point shifts, is found with BCP98 calibration based on NMARCS models. Concerning empirical calibrations, a fairly good agreement is found with the scale of J66. However differences with Bell & Gustaffson (1989; BG89) calibration show similar trends to those of theoretical calibrations (recall this calibration is based on IRFM temperatures corrected with synthetic colours). The comparison with M98 calibration yields again a contradictory result: On the one hand, M98 temperatures for metal-poor stars are systematically larger than ours (150 K on average), on the other, M98 temperatures for metal-rich stars are systematically smaller than ours (75 K on average for (V-I)>1.2), and differences increase dramatically for (V-I)<1.2.

The difficulty of model fluxes in the RI bands could account for a part of the observed differences, however it is possible that systematic errors persist in the transformations of colours of the different VRI photometric systems.

2.3 $T_{{\rm eff}}{:}~[{\rm Fe/H}]{:}~(V-K)$,:(J-H),:(J-K) and $~\!{:}~(V-L')$

The calibration of temperature against near IR colours is of increasing interest in many studies, since these kind of relations are marginally affected by blanketing, show a small dependence on surface gravity, and IR colours are less affected by reddening than UV and optical colours. Given that a specific subprogramme of near IR photometry (Paper I) has been carried out to apply the IRFM we have adopted the TCS system as reference. Transformations of this photometric system into/from Johnson, CIT and ESO systems are provided in Paper I and references therein.

The fits obtained for $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(V-K)$ are shown in Table 2 (Eqs. (8) and (9)); the corresponding ranges of application are shown in Table 3.

In the overlapping range, a linear interpolation of relations (8) and (9) is advisable in order to avoid discontinuity.

We display in Fig. 8 the mean lines corresponding to Eqs. (8) and (9), together with the residuals of the fit.

The mean variation $\Delta T_{{\rm eff}}/\Delta (V-K)$ amounts approximately to 17 K per 0.01 mag for (V-K)<2.2 and 5 K per 0.01 mag for (V-K)>2.2. At constant (V-K), temperature monotonically increases with decreasing [Fe/H] for stars with $(V-K)~\raisebox{-0.6ex}{$\stackrel{\textstyle <}{\sim}$ }~1.8$, and monotonically decreases with decreasing [Fe/H] for stars with $(V-K)~\raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }~1.8$. The gradient $\Delta T_{{\rm eff}}/\Delta$[Fe/H] depends on colour. It is small although non-negligible, negative for $(V-K)~\raisebox{-0.6ex}{$\stackrel{\textstyle <}{\sim}$ }~1.8$ and positive for $(V-K)~\raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }~1.8$. When using this calibration an error of 0.05 mag in (V-K) implies mean errors of 1.0 - 0.7% in temperature. Equivalently, an error of 0.5 dex in [Fe/H] implies at most errors of 0.7%. As a consequence, (V-K) is probably the best temperature indicator for giant stars.

The fit obtained for $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(J-H)$ is shown in Table 2 (Eq. (10)); the corresponding ranges of application are shown in Table 3.

Figure 9: $T_{{\rm eff}}$ vs. $(J-H)_{{\rm TCS}}$. The lines display the fit corresponding to Eq. (10). Symbols and lines are the same as for Fig. 2. Recall that $(J-H)_{{\rm J}}=0.011+1.062(J-H)_{{\rm TCS}}$ (Paper I)

We show in Fig. 9 the mean lines corresponding to Eq. (10), together with the residuals of the fit.

The mean variation $\Delta T_{{\rm eff}}/\Delta (J-H)$ amounts approximately to 85 K per 0.01 mag for (J-H)<0.4 and 32 K for (J-H)>0.4. At constant (J-H), temperature monotonically increases with decreasing [Fe/H]. The gradient $\Delta T_{{\rm eff}}/\Delta$[Fe/H] is small and saturation occurs at $\rm [Fe/H] =-2$. When using this calibration an error of 0.03 mag in (J-H) implies mean errors of 3-2.5% in temperature. Equivalently, an error of 0.5 dex in [Fe/H] implies at most errors of 1% in temperature.

The fit obtained for $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(J-K)$ is shown in Table 2 (Eq. (11)); the corresponding ranges of application are shown in Table 3.

Figure 10: $T_{{\rm eff}}$ vs. $(J-K)_{{\rm TCS}}$. The lines display the fit corresponding to Eq. (11). Symbols are the same as for Fig. 2. Recall that $(J-K)_{{\rm J}}=-0.009+1.099(J-K)_{{\rm TCS}}$ (Paper I)

Figure 11: $T_{{\rm eff}}$ vs. $(V-L')_{{\rm TCS}}$. The line displays the fit corresponding to Eq. (12) Symbols are the same as for Fig. 2. Recall that $L'=L_{\rm J}+0.04-0.016(J-K)_{\rm J}$ (Paper I)

Figure 12: Comparison between the present calibrations and several published calibrations of $T_{{\rm eff}}$ against (V-K) and (J-K). When necessary colours have been transformed into TCS system by using relations provided in Paper I and references therein. a) Theoretical calibrations of $T_{{\rm eff}}$ against (V-K): Circles BCP98 calibration based on Kurucz models; Triangles BCP98 calibration based on NMARCS models; Asterisks: BK92 calibration (Solid lines: [Fe/H] = 0, dashed lines: [Fe/H] = -1, dotted lines: [Fe/H] = -2). b) Empirical calibrations of $T_{{\rm eff}}$ against (V-K): Stars: J66 calibration; Circles: Direct calibration of Ridgway et al. 1980 (R80); Triangles: Blackwell & Lynas-Gray (1998; BL98); Squares: Di Benedetto (1998; DB98) (discontinuities observed in the line of differences are intrinsic to DB98 calibration). Crosses: M98 calibration (Solid lines: metal-rich giants, dotted lines: metal-poor giants). c) Calibrations of $T_{{\rm eff}}$ against (J-K): Stars: J66 calibration; Circles BCP98 calibration based on Kurucz models; Triangles BCP98 calibration based on NMARCS models; Squares: BG89 calibration; Crosses: M98 calibration (Solid lines: metal-rich giants, dotted lines: metal-poor giants)

We show in Fig. 10 the mean line corresponding to Eq. (11), together with the residuals of the fit.

The mean variation $\Delta T_{{\rm eff}}/\Delta (J-K)$ amounts approximately to 69 K per 0.01 mag for (J-K)<0.5 and 23 K for (J-K)>0.5. When using this calibration an error of 0.03 mag in (J-K) implies mean errors of $2.5-1.7\%$ in temperature. Notice that the calibration of giants' $T_{{\rm eff}}$ as a function of (J-K) has no dependence on metallicity as in the case of dwarf stars (Paper III).

The fit obtained for $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(V-L')$ is shown in Table 2 (Eq. (12)); the corresponding ranges of application are shown in Table 3.

We show in Fig. 11 the mean line corresponding to (12), together with the residuals of the fit.

The mean variation $\Delta T_{{\rm eff}}/\Delta (V-L')$ amounts approximately to 16 K per 0.01 mag for (V-L')<2.4 and 5 K for (V-L')>2.4. When using this calibration an error of 0.05 mag in (V-L') implies mean errors of 1.1-0.6% in temperature.

We show in Figs. 12a,b a comparison of relations (8) and (9) with calibrations described in previous works. As it can be appreciated, in the range (V-K)<1.2differences with other empirical and theoretical calibrations for solar metallicity are contained in a band of $\pm$ 100 K; In the range (V-K)>1.2 these differences increase as expected because of the uncertainty of bolometric flux measurements for early type stars. In any case, a better level of agreement is obtained for (V-K) than for other temperature indicators.

The agreement with theoretical calibrations BCP98 and BK92 is fairly good.

Concerning empirical works, the J66 calibration deviates from ours providing lower temperatures in the redder part of the colour axis.

It is remarkable, however, that there is fairly good agreement with the direct scale defined by Ridgway et al. (1980; R80) based on angular diameters measured by means of the lunar occultation method (Fig. 12b). The difference is practically a constant shift amounting to 30 K (our scale cooler). This fact provides a good test of the zero-point of our scale at least in the range from 3500 K to 4900 K.

The two independent calibrations of temperature versus (V-K) of BL98 present a slightly contradictory behaviour. One of them yields temperatures hotter than ours, and the other cooler ones. The size of the differences is small, but the reason for the inconsistency is unclear. In the range $(V-K)~\raisebox{-0.6ex}{$\stackrel{\textstyle >}{\sim}$ }~1.5$ ( $T~\raisebox{-0.6ex}{$\stackrel{\textstyle <}{\sim}$ }~6000$ K) our calibration provides temperatures approximately 20 K hotter (Fig. 12b) than those of the recent calibration of Di Benedetto (1998; DB98) based on the surface brightness technique. However in the blue range, differences increase to -350 K at 8500 K.

Differences with the scale of M98, show the same behaviour observed in previous sections but the size of discrepancies is somewhat small. Since M98 scale is calibrated versus optical CCD and IR array photometry, differences found could be caused by the uncertainties affecting the photometric calibration of these kind of data.

In Fig. 12c we show the comparison of our calibration $T_{{\rm eff}}{:}~(J-K)$ with other published scales. Differences are slightly larger than in the case of $T_{{\rm eff}}{:}~(V-K)$ probably due to the shrinkage of the colour axis.

2.4 $T_{{\rm eff}}{:}~[{\rm Fe/H}]{:}~(I-K)_J$

We provide the calibration of temperature against colour $(I-K)_{\rm J}$ for its interest in the study of late type giants.

The fit obtained for $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(I-K)_{\rm J}$ is shown in Table 2 (Eq. (13)); the corresponding ranges of application are shown in Table 3.

Figure 13: $T_{{\rm eff}}$ vs. $(I-K)_{\rm J}$. The lines display the fit corresponding to Eq. (13). Symbols are the same as for Fig. 2

Figure 14: $T_{{\rm eff}}$ vs. (b-y). The lines display the fit corresponding to Eqs. (14) and (15). Symbols and lines are the same as for Fig. 2

Figure 15: $T_{{\rm eff}}$ vs. (u-b). The lines display the fit corresponding to Eq. (16). Symbols and lines are the same as for Fig. 1

Figure 16: $T_{{\rm eff}}$ vs. BC(V). The lines display the fit corresponding to Eqs. (17) and (18). Symbols and lines are the same as for Fig. 2. The small box inside the graph contains a detailed view of the effect of metallicity on bolometric correction

Figure 17: Comparison between the present calibration of the bolometric correction to V and several theoretical and empirical calibrations previously published. Stars: J66 calibration; Circles: BCP98 calibration based on Kurucz models; Triangles BCP98 calibration based on NMARCS models; Asterisks: BK92 calibration (Solid lines: $\rm [Fe/H] = 0$, dashed lines: $\rm [Fe/H] = -1$, dotted lines: $\rm [Fe/H] =-2$); Crosses: M98 calibration (Solid lines: metal-rich giants, dotted lines: metal-poor giants); Open squares: F96 calibration; Full squares: Calibration of Di Benedetto (1993)

Figure 18: Comparison between intrinsic broad band colours of giant stars derived in the present work (Table 2) and calibrations of von Braun et al. (1998) (circles), and Bessell & Brett (1988) (squares). The difference $\Delta$ means our colours minus theirs. Solid lines: [Fe/H] = 0, dashed lines: [Fe/H] = -1, dotted lines: [Fe/H] = -2

Figure 19: Comparison between color-temperature calibration of giant stars derived in the present work (solid line) and calibrations of main sequence stars of Paper III (dotted line). Solar metallicity has been considered

We show in Fig. 13 the mean line corresponding to (13), together with the residuals of the fit.

The mean variation $\Delta T_{{\rm eff}}/\Delta (I-K)_{\rm J}$ amounts approximately to 39 K per 0.01 mag for $(I-K)_{\rm J}<0.8$ and 15 K for $(I-K)_{\rm J}>0.8$. When using this calibration an error of 0.03 mag in $(I-K)_{\rm J}$ implies mean errors of 1.5-1.0% in temperature. As in the case of (V-I) and (J-K), this colour is metallicity independent as temperature indicator.

2.5 $T_{{\rm eff}}{:}~[{\rm Fe/H}]{:}~(b-y)$ and $~\!{:}~(u-b)$

Although Strömgren system was not specifically tailored to study giant stars, its use has been extended to this class of luminosity, with internal accuracy $\raisebox{-0.6ex}{$\stackrel{\textstyle <}{\sim}$ }0.02$ mag, both for field stars (e.g. Anthony-Twarog & Twarog 1994) and for globular cluster stars (e.g. Davis Philip 1996). For this reason we present temperature calibrations based on colours of this system.

The fits obtained for $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(b-y)$ are shown in Table 2 (Eqs. (14) and (15)); the corresponding ranges of application are shown in Table 3.

In the range of colour $0.5\leq (b-y) \leq0.55$ a linear interpolation of Eqs. (14) and (15) provides a good fit avoiding discontinuity. We show in Fig. 14 the mean lines corresponding to (14) and (15), together with the residuals of the fit.

The mean variation $\Delta T_{{\rm eff}}/\Delta (b-y)$ amounts approximately to 62 K per 0.01 mag for (b-y)<0.45 and 26 K per 0.01 mag for (b-y)>0.45. At constant (b-y), temperature monotonically decreases with decreasing [Fe/H]. The gradient $\Delta T_{{\rm eff}}/\Delta{\rm [ Fe/H]}$ depends on colour and tends to zero as [Fe/H] decreases (saturation occurring at $\rm [Fe/H] \approx~-2.5$). When using this calibration an error of 0.02 mag in (b-y) implies mean errors of 1.5-0.8% in temperature. Equivalently, an error of 0.5 dex in [Fe/H] implies mean errors ranging 0.5-1.9%.

The fit obtained for $T_{{\rm eff}}{:}~{\rm [Fe/H]}{:}~(u-b)$ (considering only stars under 6000 K) is shown in Table 2 (Eq. (16)); the corresponding ranges of application are shown in Table 3. We show in Fig. 15 the mean lines corresponding to (16), together with the residuals of the fit.

The mean variation $\Delta T_{{\rm eff}}/\Delta (u-b)$ amounts approximately to 8 K per 0.01 mag. At constant (u-b) temperature monotonically decreases with decreasing [Fe/H]. The gradient $\Delta T_{{\rm eff}}/$[Fe/H] depends slightly on colour and diminishes as [Fe/H] decreases as expected from atmospheres theory. The value of saturation is out of the range of the present calibration, although extrapolation provides [Fe/H] $\approx -3.5$. When using this calibration an error of 0.03 mag in (u-b) implies a mean error of 0.5% in temperature. Equivalently, an error of 0.5 dex in [Fe/H] implies mean errors of 4.2-1.5%.

 

 

Table 4: Stars discarded in the final loops of the fit procedure. The values in parentheses refer to the number of standard deviations by which a given star departs from the corresponding fit

Star	Discarded colours	Star	Discarded colours
BD+042466	(2.5 BV)	SAO078681	(2.6 VR), (2.9 VI)
BD+092860	(3.0 BV), (2.6 JH)	SAO089549	(3.2 VI), (3.4 IK)
BD-180271	(3.6 VK)	SAO105082	(2.7 VI)
BS0219	(2.6 JK), (2.7 JH)	SAO144233	(2.6 RI)
BS0343	(2.6 RI)	SAO152644	(2.8 RI)
BS0911	(3.32 BV), (3.3 ub)	SAO158392	(2.5 RI)
BS2002	(2.8 VR), (2.8 VI)	SAO33445	(2.9 BV)
BS2990	(3.4 by)	M67-117	(2.6 JH), (2.8 JK)
BS3547	(2.9 VK)	M67-231	(2.7 JH)
BS3550	(2.6 by)	47Tuc-1414	(2.5 UV), (2.6 VK)
BS4336	(2.6 VR), (3.0 JH), (3.9 UV)	47Tuc-2416	(2.5 VK)
BS5301	(3.1 BV)	47Tuc-4417	(2.6 VK)
BS5480	(2.9 VR)	47Tuc-6502	(4.2 VK)
BS6469	(3.1 by)	47Tuc-7320	(2.7 UV), (3.0 JK), (3.8 JH)
BS7322	(2.7 VK)	M71-18	(3.2 BV)
BS7633	(2.6 VR)	M71-75	(3.0 JH)
BS7636	(3.5 VL')	M71-A5	(3.5 UV)
BS7776	(3.7 VI), (3.4 IK)	M71-N	(3.9 UV)
BS7928	(4.6 VL')	NGC 1261-81	(2.8 JK)
BS8649	(3.3 VR), (2.9 VI)	NGC 1261-9	(2.6 BV) (3.3 JK)
BS8866	(3.4 VR)	NGC 288-A260	(2.8 BV)
BS8878	(2.5 BV)	NGC 362-V2	(3.4 JH)
BS8905	(3.2 VR), (2.7 VI), (3.0 VL')	M3-33	(2.6 JH)
HD 03008	(2.9 BV), (3.8 JH), (3.9 JK)	M3-46	(3.2 VK), (3.5 IK)
HD 082590	(2.8 UV)	M3-53	(3.8 VK), (2.6 IK)
HD 108577	(3.3 UV), (3.6 JH)	M3-68	(2.9 JK)
HD 119516	(2.8 BV)	M3-72	(2.7 VK), (3.9 JK)
HD 126778	(2.8 UV), (2.9 ub)	M3-428	(2.9 JH)
HD 139641	(3.3 VK), (2.8 by)	M3-464	(2.9 VI)
HD 141531	(2.8 VK), (3.7 JH), (2.6 by)	M3-496	(3.1 JK)
HD 151937	(2.7 UV)	M3-525	(2.7 JH)
HD 165195	(2.7 IK)	M3-586	(2.5 JH)
HD 171496	(3.5 VK)	M3-627	(2.8 JH)
HD 199191	(2.7 UV)	M3-659	(3.7 JK)
HD 268518	(2.8 JK)	M3-675	(3.0 JH)
HD 7424	(2.6 BV)
SAO028774	(2.6 ub)
SAO054175	(2.7 VI)
SAO063927	(3.4 JK)
SAO069416	(2.9 JH)