3 Bolometric correction of giant stars

Given the utility of BC(V) for the transformation of the luminosity axis of theoretical isochrones into observational M_V of colour-magnitude diagrams, we provide here its calibration.

 

$${\rm BC(V)}=\frac{-5.531~10^{-2}}{X} - 0.6177 + 4.420\; X \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$

$$- 2.669 \;X^2 +0.6943 \;X \;{\rm [Fe/H]} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$

$$\;\;\;\;\;\;\;\;-0.1071 {\rm [Fe/H]} -8.612~10^{-3} {\rm [Fe/H]}^2,\;\;\;\;\;\;\;\;\;\;\; (17)$$

$$\;\;\;\;\sigma{\rm (BC(V)})=0.024,\;\; (285\;\; {\rm stars})\;\;\;\;\;\;\;\;$$

$$3.50\leq {\rm log}(T_{{\rm eff}}) \leq 3.67 \;\; {\rm for} \;\; +0.2 \geq {\rm [Fe/H]} > -0.5,$$

$$3.56 \leq {\rm log}(T_{{\rm eff}}) \leq 3.67 \;\; {\rm for} \;\; -0.5 \geq {\rm [Fe/H]} > -1.5,$$

$$3.58 \leq {\rm log}(T_{{\rm eff}}) \leq 3.67 \;\; {\rm for} \;\; -1.5 \geq {\rm [Fe/H]} > -2.5,$$

$$3.61 \leq {\rm log}(T_{{\rm eff}}) \leq 3.67 \;\; {\rm for} \;\; -2.5 \geq {\rm [Fe/H]} > -3.0.$$

 

$${\rm BC(V)}=\frac{-9.930~10^{-2}}{X}+ 2.887~10^{-2} + 2.275\; X \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$

$$- 4.425 \;X^2 +0.3505 \;X \;{\rm [Fe/H]} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$

$$-5.558~10^{-2} {\rm [Fe/H]} -5.375~10^{-3} {\rm [Fe/H]}^2,\;\;\;\;\; (18)$$

$$\;\;\;\;\sigma{\rm (BC(V))}=0.009,\;\; (307\;\; {\rm stars)}\;\;\;\;\;\;\;\;$$

$$3.65\leq {\rm log}(T_{{\rm eff}}) \leq 3.96 \;\; {\rm for} \;\; +0.2 \geq {\rm [Fe/H]} > -0.5, \;\;\;\;\;\;\;\;$$

$$3.65 \leq {\rm log}(T_{{\rm eff}}) \leq 3.83 \;\; {\rm for} \;\; -0.5 \geq {\rm [Fe/H]} > -1.5, \;\;\;\;\;\;\;\;$$

$$3.65 \leq {\rm log}(T_{{\rm eff}}) \leq 3.80 \;\; {\rm for} \;\; -1.5 \geq {\rm [Fe/H]} > -2.5, \;\;\;\;\;\;\;\;$$

$$3.65 \leq {\rm log}(T_{{\rm eff}}) \leq 3.74 \;\; {\rm for} \;\; -2.5 \geq {\rm [Fe/H]} > -3.0. \;\;\;\;\;\;\;\;$$

where $X={\rm log}(T_{{\rm eff}})-3.52$.
As the distribution of residuals show, this functional expansion is adequate to fit the singular behaviour of BC(V) with temperature. However, the extrapolation of the present calibration is unsafe, especially in the range of low temperatures.

The following stars departed more than $2.5\,\sigma$ in either of the final fits:
BS4902 (2.8), BD-180271 (2.5), BS2286 (7.3), BS7523 (5.2),47Tuc-1421 (2.8), 47Tuc-3512 (3.4), 47Tuc-7320 (4.8), 47Tuc-1414 (2.5), 47Tuc-1518 (2.6), 47Tuc-4411 (3.1), 47Tuc-6509 (3.9), M71-B (2.9), HD 088609 (3.1), BS8930 (2.6), HD 171496 (4.7), BS3547 (2.8). We show in Fig. 16 the mean lines corresponding to (17) and (18), together with the residuals of the fit.

In the range $T_{{\rm eff}}<4500$ relation (18) is not suited to deriving accurate bolometric corrections since on the one hand the dispersion of the fit is around 2.5% (0.025 mag) due to the fact that cool temperatures have greater internal errors, and on the other, the variation of BC(V) with log $(T_{{\rm eff}})$ is very steep. A noticeable feature of the present scale is the significant variation of the bolometric correction with metallicity, especially in the range of higher temperatures (this point is emphasized in Fig. 16). Although a systematic bias in the data can never be completely discarded, the size of the effect found is not compatible either with photometric errors, or with the internal uncertainties in temperatures.

In Fig. 17, we show the comparison between the present calibration and several theoretical and empirical calibrations previously published. Differences in the zero-point caused by the adopted bolometric correction of the sun have been taken into account. In the range 8000 K $\geq T_{{\rm eff}}~\geq$ 6000 K our scale is systematically over the other scales considered ($\sim 0.05$ mag), in the range 6000 K $\geq T_{{\rm eff}} \geq$ 4000 K differences are symmetrically distributed in a band of $\pm 0.10$ mag. These differences show the existence of essential problems in deriving bolometric corrections. In the case of empirical calibrations the possible causes of discrepancies might be the absolute flux calibration, the way of fixing the zero-point and an insufficient discrimination of the metallicity effect. In the case of theoretical calibrations, the possible causes are drawbacks in the model atmospheres and/or difficulties in the synthesis of colours.