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3 Temperature-color calibrations


3.1 Empirical calibrations at solar metallicity

 In the correction procedure defined in LCB97, the empirical temperature-color calibrations are the basic links between the model colors and the observations. First, they provide a crucial point of comparison, and secondly, they are used to define the empirical and theoretical pseudo-continua from which the correction functions will be defined in the following. Over a large range of temperatures, we used the empirical $T_{\mathrm{eff}}$-$(UBV)_{\rm J}(RI)_{\rm C}JHKL$ relations discussed in detail in Paper I, but the inclusion of the M dwarf models in the library now leads us to extend the previous calibrations to the bottom of the main sequence. Observations of very low-mass stars being still rather fragmentary, the construction of these temperature-color relations require indirect empirical methods which are now described.

3.1.1 Dwarfs

Over the temperature range 11500 K $\geq$ $T_{\mathrm{eff}}$ $\geq$ 4250 K, the empirical temperature-color sequences are based upon the temperature scale of Schmidt-Kaler (1982) for U-B and Flower (1996) for B-V, and the two-color relations compiled in the literature (FitzGerald 1970; Bessell 1979; and Bessell & Brett 1988).

For $T_{\mathrm{eff}}$ $\leq$ 4000 K, the temperature scale is very controversial, in particular because of the difficulty to accurately model the complex featured M dwarf spectra. Due to the lack of very reliable model-atmospheres, indirect methods such as blackbody or gray-body fitting techniques have been used to estimate effective temperatures of the intrinsically faintest stars (Veeder 1974; Berriman & Reid 1987; Bessell 1991; Berriman et al. 1992; Tinney et al. 1993). In practice, the temperatures derived from fitting to model spectra (e.g. Kirkpatrick et al. 1993; Jones et al. 1994) are systematically $\sim$ 300 K warmer than those estimated by empirical methods. Recently, a redetermination of the effective temperatures using the NextGen version of the AH95 model spectra has been proposed. Leggett et al. (1996) used observed infrared low-resolution spectra and photometry to compare with models. They found radii and effective temperatures which are consistent with estimates based only on photometric data. Their study shows that these updated models should provide, for the first time, a realistic temperature scale for M dwarfs. On the other hand, Jones et al. (1996), using a specific spectral region (1.16-1.22 $\mu$m) which is very sensitive to parameter changes of M dwarfs (Jones et al. 1994), have derived stellar parameters by fitting synthetic spectra for a limited sample of well-known low-mass stars. They found that the new models provide reasonable representations of the overall spectral features, with realistic relative strength variations induced by changes in stellar parameters.

Based on these promising - although preliminary - results of Leggett et al., providing closer agreement between theoretical and observational temperature scales, we adopted a mean relation constructed from a compilation of the results of Bessell (1991), Berriman et al. (1992), and Leggett et al. (1996). Figure 2 shows the different effective temperature scales adopted by these authors, as a function of I-K and V-K. All the IR photometric data have been transformed to the homogeneous JHKL system of Bessell & Brett. The solid line is a polynomial fit derived from these data. For comparison, we have also added in the figure the $T_{\mathrm{eff}}$-values estimated by Jones et al. (1996) for 6 stars (large open symbols) having VIK photometry data given in Jones et al. (1994). Due to the limited spectral range used, the temperatures estimated by Jones et al. (1996) are still uncertain, and hence have not been used to define the mean relation given in Fig. 2. For stars cooler than 3000 K, the discrepancies between the different temperature scales appear slightly more pronounced in I-K than in V-K. For this reason, V-K was preferred to I-K for establishing the mean temperature scale of M dwarfs. We found:
V-K = 1.18 \: 10^{-6}*(T_{\mathrm{eff}})^{2} -
0.01*T_{\mathrm{eff}}+ 28.49 \, .\end{displaymath} (2)

For practical purposes, a good approximation of the inverse relation is given by:
T_{\mathrm{eff}} = 18.27*(V-K)^{2} - 504.88*(V-K) + 5415 \, .\end{displaymath} (3)

Notice that this (V-K)-$T_{\mathrm{eff}}$ scale perfectly matches the Bessell (1991) calibration above 3000 K, and that the temperatures estimated by Leggett et al. are also in general agreement with this relation. The mean (V-K)-$T_{\mathrm{eff}}$ relation defined above was thus adopted as the basic scale for M dwarfs over the range 4000 K $\sim$ 2000 K.

For $T_{\mathrm{eff}}$ between 4000 K and 2600 K, we then used the $(BV)_{\rm J}(RI)_{\rm C}JHK$ photometric data given in Bessell (1991) and for stars hotter than $\sim$ 3000 K the U-B colors from FitzGerald (1970) in order to relate V-K to the other colors at a given temperature, via color-color transformations.

In order to establish the calibrations down to 2000 K, we used the two objects LHS2924 and GD165B with effective temperatures estimated by Jones et al. 1996 (2350 K and 2050 K, respectively), for which VIJHKL photometry is given in Jones et al. (1994). The $T_{\mathrm{eff}}$ of LHS2924 was found to be in very good agreement with our temperature scale (Fig. 2) whereas GD165B is the coolest object with available infrared photometry. Therefore, the cool tails of our empirical calibrations for the J-H, J-K, H-K and K-L colors were required to match these two extreme points. Nevertheless, since these two objects are good brown dwarf candidates, with potentially non-solar abundances ([Fe/H] $\sim -0.5$ for LHS2924 and +0.5 for GD165B, Jones et al. 1996), the empirical infrared stellar colors below 2300 K should be considered with caution. Furthermore, reliably accurate UBVRI photometry data for M dwarfs cooler than 3000 K (B-V $\sim$ 1.8) are also difficult to obtain - and are sparse indeed. Some relevant data can be found in the Gliese & Jahreiss (1991) catalog of nearby stars. Consequently, we extrapolated down to 2000 K the $T_{\mathrm{eff}}$-UBVRI calibrations defined above for hotter stars. Compared to the calibrations of Johnson (1966) and FitzGerald (1970), our (U-B)-(B-V) relation yields a better match of the extreme red points found in the Gliese & Jahreiss catalog (Fig. 3).

As before, the surface gravities along the dwarf sequence were defined at each $T_{\mathrm{eff}}$ from a ZAMS computed by the Bruzual & Charlot (1996, private communication) isochrone synthesis program.


\epsffile {fig2.eps}\end{figure} Figure 2: A comparison of different temperature scales of M dwarfs adopted by several authors. The solid line is a polynomial fit performed on all the data, except those of Jones et al. (1996) (large open symbols). See text for explanations

\epsffile {fig3.eps}\end{figure} Figure 3: U-B/B-V empirical sequences compared to the Gliese & Jahreiss (1991, CSN3) catalog of nearby stars

3.1.2 Giants

For cool giants, the Ridgway et al. (1980) (V-K)-$T_{\mathrm{eff}}$relation was used as the basic temperature scale. As in Paper I, the different $T_{\mathrm{eff}}$-color sequences were constructed by compiling the photometric calibrations from Johnson (1966), Bessell (1979), Bessell & Brett (1988), and recent observations of M giants given by Fluks et al. (1994). A 1 $M_{\odot}$ evolutionary track (Schaller et al. 1992) was used to define the $\log g$ of the red giants with $T_{\mathrm{eff}}$ in the interval 4500 K $\sim$ 2500 K.

3.2 Semi-empirical calibrations for non-solar abundances

 In LCB97, one of the basic assumptions made to define the correction process of the spectra is that the original model grids provide realistic color differences with respect to metal-content variations. These properties, established in particular for the K95 grid from Washington photometry by Lejeune & Buser (1996), can be naturally applied to define semi-empirical $T_{\mathrm{eff}}$-color calibrations at different metallicities. This is achieved by calculating at a given effective temperature the theoretical color differences due uniquely to a change in the chemical composition (hereafter called differential colors). These differences are then added to the colors provided by the empirical $T_{\mathrm{eff}}$-color calibrations for solar metallicity in order to fix the semi-empirical colors, $\stackrel{\sim}{c_{ij}}^{\rm emp}$, at a given metallicity:
\lefteqn{ \stackrel{\sim}{c_{ij}}^{\rm emp}\! \! \! \!
 ...\\  & & + \Delta
 c_{ij} (T,\log g_{\mathrm{seq}},\Delta [M/H])\,,\end{eqnarray}
where $\log g_{\mathrm{seq}}$ designates the surface gravity along the giant or the dwarf sequences, as defined by the empirical calibrations. The resulting empirical and semi-empirical $T_{\mathrm{eff}}$-color calibrations for the dwarfs and giants are given in Tables 1 to 10. Empirical colors for M dwarfs (2000 K $\sim$ 3500 K) are given in the range -3.5 < [M/H] < +0.5, while for M giants between 2500 K and 3500 K the calibrations are defined for -1.0 < [M/H] < +0.5. Theoretical bolometric corrections, BCV (see Eq. 1), are given as derived from the final grid of corrected spectra (see Sect. 4 and LCB97). While for completeness the semi-empirical calibrations are given for the largest possible ranges of colors and effective temperatures in Tables 1 to 10, we should emphasize here that the UBV magnitudes for the coolest M dwarf models - and hence the corresponding differential colors - are still rather uncertain, as will be shown in Sect. 4.2.

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