3.1. Conversion: theoretical observational plane
The transformation of the bolometric magnitude to the K-magnitude scale in Fig. 2 is obtained from the period-luminosity (PL- and PK-) relation of carbon Miras (Groenewegen & Whitelock 1996, hereafter referred to as GW96). Transformed to the ESO photometric system this gives: MK = 1.37Mbol - 1.58. The actual transformation from the bolometric magnitude to a K-magnitude in the models shown in Fig. 2 is achieved with the relation provided by Suntzeff et al. (1993):
The relation is valid for 0 3 2 3 and was obtained from a fit to the (J-K, ) relation given by Frogel et al. (1980). The bolometric magnitude was obtained from the numerical integration of the broadband flux distributions of the galactic carbon stars observed by Mendoza & Johnson (1965). The same bolometric corrections are applied to non-carbon stars. Fig. 2 from Frogel et al. (1980) shows that the differences are at most 0 2.
The conversion from to (J-K)0 is established from an empirical relation derived by Ng et al. (1997), based on giant stars with a spectral type ranging from late G to late M. The effective temperatures for these stars were obtained from angular diameter measurements. The empirical /(J-K)0 relation takes into account a small shift in colour as a function of metallicity. The relations, provided for the metallicity Z = 0.004 and Z = 0.024, are logarithmically interpolated for Z 0.004 and the Z = 0.004 relation is applied for lower metallicities. Although some uncertainty is present due to the fact that this relation is not based on carbon stars, the differences will not be too large as long as the colours are not too red. The empirical relation covers conveniently the observed colour range of the carbon stars.
In Figs. 2a - c some isochrones from Bertelli et al. (1994) are displayed. Fig. 2a shows the RGB for an age of 10 Gyr for various metallicities and Fig. 2b and 2c show the AGB for different age and metallicities. Fig. 2a demonstrates that if one assumes a fixed age then a large metallicity spread could be present among the SDG stars. Given the uncertainties, the GB stars likely have a metallicity of Z = 0.008, comparable to the LMC. Fig. 2b demonstrates that difficulties are present to distinguish 1 & 10 Gyr isochrones for the AGBs with Z = 0.001 from those with a 0.1 & 1 Gyr age for Z = 0.008. The isochrones in Fig 2c show that the colour difference between 5 and 10 Gyr populations with the same metallicity is quite small compared to the colour difference due to a large metallicity range.
© European Southern Observatory (ESO) 1997
Online publication: March 24, 1998