3. Comparison with empirical calibrations

In order to assess the reliability of the synthetic spectra, we now compare them to empirical temperature-color calibrations. Depending on the availability and the quality of calibration data, two basic calibration sequences will be used for the cooler giants and for the hotter main sequence stars, respectively.

3.1. Empirical color-temperature relations

3.1.1. Red giants and supergiants

Ridgway et al. (1980) derived an empirical temperature-(V-K) relation for cool giant stars. This relation is based on stellar diameter and flux measurements, and therefore on the definition of the effective temperature:

where is the apparent bolometric flux, is Boltzmann's constant, and is the angular diameter. Hence is almost entirely empirical.

Over the range , the Ridgway et al. calibration was adopted as the effective temperature scale for the V-K colors. We derived the color-temperature relations for V-I, J-H, H-K, J-K and K-L using the infrared two-color calibrations given by Bessell & Brett (1988) (hereafter referred to as BB88). For the U-B and B-V colors, we used the color-color relations established by Johnson (1966), and Bessell's (1979) calibration was adopted for the relation.

Because existing calibrations do not go below , we have used both observations and theoretical results published by Fluks et al. (1994) in order to construct semi-empirical -color calibrations down to the range . Synthetic V-K colors computed from their sequence of photospheric model spectra provide a very good match to the calibration by Ridgway et al., which could thus be extended to the range () by adopting the theoretical relation from Fluks et al.. We then used Fluks et al.'s compilation of observations of a large sample of bright M-giant stars in the solar neighbourhood to establish the mean intrinsic colors and standard deviations from their estimates of interstellar extinction within each photometric band. Finally, with these results, we derived mean intrinsic color-color relations - (V-I)-(V-K), (U-B)-(V-I), (R-I)-(V-K) and (B-V)-(R-I) -, which allowed us to translate to all these other colors the basic relation adopted above for red giants within the temperature range .

Examples of the adopted fits for the (V-I)-(V-K) and the (U-B)-(V-I) sequences are shown in Figs. 3 (click here) and 4 (click here).

  
Figure 3: Adopted (V-I)-(V-K) two-color calibration sequence for cool giants. Symbols represent mean values derived from observations given by Fluks et al. (1994). The solid line is a linear least-squares fit to the data for

  
Figure 4: Adopted (U-B)-(V-I) two-color calibration sequence for cool giants. Symbols represent the same as in Fig. 3 (click here). The solid line is a quadratic fit to the data for

For the infrared colors, the photometric data given by Fluks et al. are defined on the filter system, which is different from the filter system defined by BB88. Using the color equations relating the two systems and derived by BB88, transformed JHKL colors from the Fluks et al. data were computed. The resulting (V-K)-(V-J), (V-K)-(V-H), (V-L)-(V-J) and (V-H)-(V-J) sequences are well approximated by linear extrapolations of the two-color relations given by BB88. Furthermore, the model colors derived from the Fluks et al. synthetic spectra and the filter responses defined by BB88 also agree very well with these extrapolated relations (Fig. 5 (click here)). We therefore chose this method to derive the J-H, H-K, J-K and K-L colors over the range . However, the uncertainty implied by the extrapolation to the reddest giants is of the order of , indicating that for the coolest temperatures near 2500 K the resulting empirical calibration of the J-H, H-K, J-K, and K-L colors should be improved by future observations.

  
Figure 5: Adopted extrapolation of the (V-K)-(V-J) relation for cool giants (dashed line). Model colors (open squares) and observed data (crosses) from Fluks et al. (1994) are compared to the empirical sequence adopted. (See text for explanations)

Table 2 (click here) presents the final adopted temperature-color calibrations for red giants, which was supplemented by a sequence related to the effective temperature via an evolutionary track () calculated by Schaller et al. (1992).

 

U-B B-V V-I V-K R-I J-H H-K J-K K-L

4593 1.017⁵ 1.094⁵ 1.080¹ 2.500¹ 0.487⁶ 0.580¹ 0.100¹ 0.680¹ 0.080¹ 2.85

4436 1.187⁵ 1.173⁵ 1.170¹ 2.700¹ 0.530⁶ 0.630¹ 0.115¹ 0.740¹ 0.090¹ 2.50

4245 1.399⁵ 1.281⁵ 1.360¹ 3.000¹ 0.602⁶ 0.680¹ 0.140¹ 0.820¹ 0.100¹ 2.12

4095 1.566⁵ 1.364⁵ 1.479¹ 3.260¹ 0.672⁶ 0.730¹ 0.150¹ 0.880¹ 0.110¹ 1.82

3954 1.714⁵ 1.443⁵ 1.634¹ 3.600¹ 0.773⁶ 0.790¹ 0.165¹ 0.950¹ 0.120¹ 1.55

3870 1.784⁵ 1.489⁵ 1.768¹ 3.850¹ 0.859⁶ 0.830¹ 0.190¹ 1.010¹ 0.120¹ 1.39

3801 1.815⁵ 1.524⁵ 1.899¹ 4.050¹ 0.948⁶ 0.850¹ 0.205¹ 1.050¹ 0.130¹ 1.25

3730 1.812⁵ 1.552⁵ 2.053¹ 4.300¹ 1.058⁶ 0.870¹ 0.215¹ 1.080¹ 0.150¹ 1.15

3640 1.750⁵ 1.577⁵ 2.269¹ 4.640¹ 1.228⁶ 0.900¹ 0.235¹ 1.130¹ 0.170¹ 0.98

3560 1.651⁵ 1.590⁵ 2.472¹ 5.100¹ 1.568⁴ 0.930¹ 0.245¹ 1.170¹ 0.180¹ 0.83

3420 1.412⁵ 1.589⁵ 2.828¹ 5.960¹ 1.899⁴ 0.950¹ 0.285¹ 1.230¹ 0.200¹ 0.56

3250 1.019⁴ 1.527⁴ 3.309⁴ 6.840¹ 2.170⁴ 0.960¹ 0.300¹ 1.260¹ 0.256¹ 0.21

3126 0.645⁴ 1.499⁴ 3.709⁴ 7.830³ 2.391⁴ 0.950² 0.370² 1.320² 0.310² -0.05

2890 0.096⁴ 1.512⁴ 4.234⁴ 8.760³ 2.519⁴ 0.920² 0.400² 1.320² 0.420² -0.57

2667 -0.146⁴ 1.507⁴ 4.439⁴ 9.310³ 2.558⁴ 0.900² 0.410² 1.310² 0.510² -1.09

2500 -0.328⁴ 1.510⁴ 4.593⁴ 9.560³ 2.567⁴ 0.880² 0.420² 1.300² 0.550² -1.52

¹ Bessell & Brett (1988) two-color relation with Ridgway et al. (1980) to relate to V-K.
² Extrapolation of two-color relations from Bessell & Brett (1988).
³ Synthetic color indices from Fluks et al. (1994) models.
⁴ From mean two-color relations derived from the Fluks et al. (1994) observed data.
⁵ Empirical calibration from Johnson (1966).
⁶ Empirical calibration from Bessell (1979).  

3.1.2. Main sequence stars

To construct empirical -color sequences from 12000 K to 3600 K for the main sequence stars, we used different calibrations: Schmidt-Kaler (1982) was chosen to relate to U-B, B-V or R-I, and the two-color relations established by FitzGerald (1970), Bessell (1979), and BB88 were then used to derive the temperature scales for the remaining colors. This procedure should provide color-temperature calibrations with uncertainties of in color or in temperature (Buser & Kurucz 1992).

3.2. Comparison of theoretical and empirical color-temperature relations

3.2.1. Red giants and supergiants

In order to compare the models to the above color-temperature relations for red giants, model spectra were first interpolated in the theoretical libraries for appropriate values of surface gravity given by the relation defined by the evolutionary track from Schaller et al. (1992). Synthetic colors computed from these model spectra are then directly compared to the empirical color-temperature relations, as illustrated in Fig. 6 (click here).

  
Figure 6: Empirical color-effective temperature calibrations for solar-metallicity red giant stars (solid lines, according to Table 2 (click here)) compared to the corresponding theoretical relations calculated from original synthetic library spectra (symbols, according to key in insert). Note that different scales have been used for the different colors

It is evident that the color differences between equivalent models from the K- and the -libraries can be as high as 0.4 mag, while those between the theoretical library spectra and the empirical calibrations may be even larger, up to 1 mag.

Such differences - both between the original libraries and between these and the empirical calibrations - make it clear that direct use of these original theoretical data in population and evolutionary synthesis is bound to generate a great deal of confusion in the interpretation of results. In particular, applications to the integrated light of galaxies at faint magnitude levels, where effects of cosmological redshift may come into play as well, will provide rather limited physical insight unless the basic building blocks of the evolutionary synthesis - i.e., the stellar spectra - are systematically consistent with the best available observational evidence. Thus, our work is driven by the systematic consistency of theoretical stellar colors and empirical calibrations as a minimum requirement for the (future) standard library. As a viable operational step in this direction, a suitable correction procedure for the theoretical spectra will be developed in the following section.

3.2.2. Main sequence stars

The same procedure as for the giants was applied for the main sequence stars, except that a zero-age main sequence isochrone (ZAMS) compiled by Bruzual (1995) was used in the appropriate interpolation for the surface gravity . Again, synthetic photometry results obtained from the theoretical library (Kurucz) are compared to the empirical color-temperature relations in Fig. 7 (click here).

  
Figure 7: Empirical color-effective temperature calibrations for solar-metallicity dwarf stars (solid lines, see text for sources) compared to the corresponding theoretical relations calculated from original synthetic K-library spectra (symbols). Note that different scales have been used for the different colors

Note that the differences between the theoretical and the empirical colors are significantly smaller than those for the giants given in Fig. 6 (click here): for the hotter temperatures, they do not exceed 0.1 mag, while at cooler temperatures () differences of up to 0.3 mag in V-I between models and observations again indicate that the coolest K-library spectra still carry large uncertainties and should, therefore, be used with caution (e.g., Buser & Kurucz 1992). Thus, application of the same correction procedure as for the giant models appears warranted for the dwarf models as well.

Also note that the Kurucz spectra only go down to 3500 K. We are thus missing the low-luminosity, low-temperature main sequence M stars in the present library. However, we anticipate here that in a corollary paper (Lejeune et al. 1997, hereafter Paper II) the necessary extension is being provided from a similar treatment of the comprehensive grid of M-star model spectra published by Allard & Hauschildt (1995).