## 3. Comparison with fundamental starsThe ultimate test of any model colours is to compare them to the
colours of stars whose atmospheric parameters have been determined by
direct, model-independent, methods. Unfortunately, such fundamental
stars are relatively few in number, mainly due to the difficulty in
obtaining the necessary observations. The current best list was
discussed by Smalley & Dworetsky (1995). They reviewed the list of
stars with fundamental values of and those with
fundamental values of . Of all those available,
only three ( CMa, CMi,
Vir) have fundamental values of Fundamental stars were used by Smalley & Dworetsky (1995) to
investigate the accuracy of the Kurucz (1991b) models. Clear
inadequacies were found, which warranted further investigations. In
this paper we use the same fundamental stars to compare the various
treatments of convection and their effects on calculated Throughout this paper, the observed In order to assign an error on the and
obtained from the grids, we used those
appropriate to the In the comparisons that follow, in this and the next section, we use three statistical measures to compare the three grids: - A weighted mean of the differences between the grid and fundamental values, in order to determine which grid is in closest overall agreement with the fundamental values.
- A weighted root mean square of the differences, given by , where are the weights as given by the square of the reciprocal of the errors, and are the differences between grid and fundamental values.
- The reduced chi-square and its associated probability, , as a measure of the goodness of agreement between the grid and fundamental values.
These three measures, together with a visual inspection, enable us to fully compare the three grids, in order to determine which gives the best overall agreement. ## 3.1. Effective temperatureThe observed
The results of the comparison of the various model colours with those of the fundamental stars are shown in Fig. 2, as a function of against . The CM model is in very good agreement with the fundamental values, with a weighted mean difference of -36 111 K and a weighted rms difference of 71 K. The = 0.102 which gives a 98% probability that the model fits to the fundamental points. The MLT noOV model has a weighted mean difference of 75 115 K and a weighted rms difference of 100 K. This agreement is not as good as that for the CM model, but still acceptable to within the error bars. Indeed, the = 0.190 implies a 94% probability of a good fit, which is very good, but not quite as good as the CM model. The MLT OV model, however, has a weighted mean of 175 113 K and a weighted rms difference of 199 K, which is clearly not in good agreement with the fundamental values. In addition, the = 0.770, which gives only a 54% probability of a good fit. This shows that the MLT OV model is not very satisfactory.
Procyon (HD 61421, CMi) is the fundamental star with the most tightly constrained value of . As such, this star ought to be a stringent test of the different grids. Inspection of Table 2 shows that the CM grid is in excellent agreement with the fundamental value. The MLT noOV models are somewhat discrepant, but still just within the error bars. The MLT OV models are clearly discrepant and well outside the error bars. Hence, from this comparison alone, we expect that the CM models should be the more realistic. Overall, the CM models are in best agreement with the fundamental stars. The MLT noOV models are in less agreement, but still agree to within the error bars. The MLT OV models, however, are clearly discrepant. ## 3.2. Surface gravityFundamental values are a fairly stringent
test of the grids, since they are more numerous and generally less
uncertain than fundamental values. However, if
we include the uncertainties in the values of
obtained from the grids due to uncertainties in
The results of the comparison of the obtained from the various models with the fundamental values are shown in Fig. 3, as a function of against (obtained from the appropriate grid). With the exception of HD 90242 (see above), the CM model values of agree with the fundamental values to within the error bars. The weighted mean difference is , and indicates that the CM models may, on average, very slightly overestimate . The weighted rms difference of 0.075 shows that the points are clustered very tightly around the fundamental value. There is no evidence that the difference varies systematically with . The MLT noOV models give results that are similar to the CM models. The weighted mean difference of , which is in better formal agreement than the CM models, but there is slightly more scatter (weighted rms difference = 0.085). This larger scatter is primarily due to a discrepancy which appears to be developing at the cool end. Both the CM and MLT noOV models are in agreement with the fundamental stars to within the error bars and have values that give a probability in excess of 99.9% for a good fit! The same cannot be said of the MLT OV models which have a weighted mean difference of and weighted rms difference of 0.141. In this case = 0.445, which gives a 95% probability for a good fit. Certainly the MLT OV points just agree to within the error bars, but the agreement is nowhere near as good as those for the CM and MLT noOV models. In addition, there is a distinct trend of decreasing with decreasing . The MLT OV models underestimate for cooler stars.
To conclude, there is very little difference between the results from the CM and MLT noOV models. The CM models give slightly better results, since there is slightly less scatter and no evidence of any systematic trends in with . The MLT noOV models give a hint of a discrepancy in the coolest fundamental stars. The MLT OV models are somewhat discrepant and underestimate for the cooler stars. © European Southern Observatory (ESO) 1997 Online publication: March 24, 1998 |