4. Comparison with non-fundamental stars
The relatively few fundamental stars means that non-fundamental stars are often used when testing model grids (e.g. Künzli et al. 1997; Castelli et al. 1997). Non-fundamental stars offer an alternative to the truly fundamental stars discussed above. However, there is always a very real danger of hidden model-dependent systematic errors that could bias any results. Hence, non-fundamental stars must be chosen carefully and the possible sources of bias identified.
4.1. Effective temperature
The Infrared Flux Method (IRFM) developed by Blackwell & Shallis (1977) allows for the simultaneous determination of and angular diameter. The method requires a measurement of the total integrated flux from the star and an observation of infrared flux. Model atmospheres are only required to determine the stellar surface infrared flux, but this is relatively insensitive to the actual choice of model atmospheres ( Blackwell et al. 1979, 1980). Hence, the IRFM has a clear advantage over other model-dependent methods (e.g. spectrophotometric flux fitting), in that the obtained can be regarded as semi-fundamental. In fact, it is the closest model-dependent method to a true fundamental determination.
Blackwell & Lynas-Gray (1994) presented a list of stars with determined from the IRFM. They used standard ATLAS9 models for the stellar infrared fluxes, but as stated above the effect of the choice of model should not be significant. Indeed, Blackwell & Lynas-Gray (1994) found very good agreement with the angular diameters obtained from interferometry. They stated that the values of should be accurate to 2%. This equates to 130 K at 6500 K, which is of the same order of accuracy as the fundamental of Procyon. Thus, the IRFM values should be of sufficient quality to be useful in the comparison of the different uvby grids.
Table 4 shows the IRFM stars considered here. The list is based on Table 9 of Künzli et al. (1997), which is primarily from Blackwell & Lynas-Gray (1994). Because the IRFM can be very sensitive to the presence of a binary companion (Smalley 1993b), we have excluded those stars noted as spectroscopic binaries by Blackwell & Lynas-Gray (1994).
Table 4. The obtained from the IRFM compared to the values obtained from the 3 grids. .
Fig. 4 shows a comparison between the IRFM values and those obtained from uvby for the 3 grids. Since the IRFM stars do not have formal error estimates we have adopted a typical uncertainty of 200 K, in order to calculate values. The CM models give values of that agree very well with those given by the IRFM, with mean difference of and an rms difference of 109 K. The three hottest stars, however, appear to indicate that the CM models give higher values of than the IRFM. Possibly, this is due to the IRFM underestimating the of these stars, since the fundamental stars are in good agreement (cf. Fig. 2). This would be consistent with underestimating the contribution from unobserved ultraviolet flux. Nevertheless, the agreement for the cooler stars supports that found from the fundamental stars, with = 0.312, giving a greater than 99.9% probabilty for a fit to within the 200 K error bars adopted above. The MLT noOV models are generally somewhat discrepant, with the obtained from the uvby colours being, on average, slightly hotter (mean difference of and rms = 160 K). In addition, there appears to be a distinct slope that develops for stars with 7000 K. This sort of slope was noted by Künzli et al. (1997) who used colour indices from the Geneva photometric system for their study. In addition, Castelli et al. (1997) found that the differences in from colours and the IRFM are of different sizes either side of = 6250 K (see their Table 5). Only for the coolest stars does the MLT noOV models begin to agree with the IRFM values, but the overall = 0.671, gives only a 87% probability for agreement with the IRFM points over the whole range. The MLT OV models are clearly discrepant, with all the stars hotter than the corresponding IRFM (mean difference of and an rms of 247 K). Indeed, the = 1.598 gives only a 4% probability for a fit!
Overall, the results from the IRFM agree with that found using the smaller sample of fundamental stars. The CM grid has the greatest success in recovering the obtained from both fundamental and non-fundamental methods.
4.2. Surface gravity
Open cluster stars can be used as surface gravity standards. Stellar evolutionary models enable the of cluster members to be determined by fitting isochrones to the cluster photometry. These are not truly fundamental values of , since they rely on the stellar evolutionary models. However, the values of are not directly dependent on the model-atmospheres considered here. Stellar interior calculations do, however, involve the use of convection theory and changes in the treatment of convection may influence the results of the evolutionary calculations (e.g. Stothers & Chin 1995; Canuto 1996a; Canuto et al. 1996). Nevertheless, we shall use the cluster surface gravity values to test the values obtained from the uvby colours for the 3 model grids. The Hyades have a metallicity of [M/H] = +0.125 (Boesgaard 1989). Linear interpolation in metallicity was used to obtain colours appropriate to the Hyades.
Fig. 5 shows a comparison between the obtained from the uvby colours and that given by evolutionary models for the Hyades (Künzli et al. 1997, Table 10). The actual numerical values are given in Table 5, which is available electronically from the CDS. Since the stars do not have formal error estimates we have adopted a typical uncertainty of 0.20 dex in order to calculate values. All three models exhibit the same general pattern in that there is a distinct change in behaviour of the differences below 7000 K. Above 7000 K the differences are essentially independent of . But, below 7000 K the trend in difference is non-linear, with a "bump" around 6500 K. This was discussed by Künzli et al. (1997) who showed that the observed Hyades main sequence clearly has a sudden change in slope when compared to model atmosphere colours. Evolutionary models are unable to account for the implied sudden change in . They concluded that something is missing in the atmosphere models what certainly requires further investigation, since the fundamental stars do not exhibit the same behaviour (cf. Fig. 3).
The above problems not withstanding, the CM and MLT noOV models both agree reasonably well with the Hyades values. The vast majority of the points are within 0.20 dex of the Hyades value, which is of the order of the typical error due to the uncertainties in the uvby colours alone. The CM models have a mean difference of and an rms of 0.14 dex, while the MLT noOV models give and an rms of 0.16 dex. Generally, the CM models give slightly better agreement with the Hyades than the MLT noOV models. However, the difference between the two is not significant, since both have values that give a greater than 99% probability for agreement. Interestingly, both models give values of which are slightly less than the Hyades values for the hotter stars, which is the opposite to that found using the fundamental stars (Fig. 3). This may indicate that the evolutionary calculations are not producing the correct values for the Hyades. Note in particular, that a decrease of obtained from the evolutionary calculations for hotter stars would increase in the same region and, hence, reduce the size of the "bump" around 6500 K. The MLT OV models are, yet again, somewhat discrepant, with a mean difference of and an rms of 0.27 dex. Both the difference and rms are greater than the adopted typical error of 0.20 dex, which indicates very poor agreement. In fact, = 1.800, which gives a less than 0.1% probability for agreement! The MLT OV models systematically underestimate the Hyades values.
Overall, the results from the Hyades values agree with that found using the fundamental stars, but with much more scatter and uncertainty. The model-dependent nature of the Hyades values means that they should not be used as the primary test of obtained from model grids; stars with fundamental values should always be preferred. Nevertheless, the CM grid has the greatest success in recovering the obtained from both fundamental and non-fundamental methods, with the MLT noOV models a very close second.
© European Southern Observatory (ESO) 1997
Online publication: March 24, 1998