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Astron. Astrophys. 328, 349-360 (1997)

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3. Comparison with fundamental stars

The 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 [FORMULA] and those with fundamental values of [FORMULA]. Of all those available, only three ([FORMULA] CMa, [FORMULA] CMi, [FORMULA] Vir) have fundamental values of both [FORMULA] and [FORMULA]. They extended this by using 4 eclipsing binary systems, but the lower quality of the currently available spectrophotometry and uncertainties in distances, meant that these stars have much lower quality fundamental [FORMULA] values compared to those in Code et al. (1976).

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 uvby colours. Fundamental stars represent the only truly model-independent tests of the theoretical colours.

Throughout this paper, the observed uvby colours were obtained from Hauck & Mermilliod (1990). These were de-reddened, if necessary, using UVBYBETA (Moon 1985). This de-reddening process is based on the standard empirical relationships determined by Crawford (1975, 1979), which have been widely used with great success. For all the grids discussed here, any given pair of [FORMULA], [FORMULA] colours corresponds to a unique pair of [FORMULA], [FORMULA] values. The procedure is to locate the grid point ([FORMULA], [FORMULA]) closest to the observed [FORMULA], [FORMULA] colours. Then, parabolic interpolation is made within the grid in both the [FORMULA] and [FORMULA] directions to obtain the [FORMULA] and [FORMULA] that corresponds to the [FORMULA], [FORMULA] colours. Once [FORMULA] and [FORMULA] have been obtained, either parameter can be compared to a fundamental (this section) or non-fundamental (Sect. 4) value. Hence, [FORMULA] and [FORMULA] can be compared independently.

In order to assign an error on the [FORMULA] and [FORMULA] obtained from the grids, we used those appropriate to the uvby colours. The typical error on uvby colours is [FORMULA] 0.015 (see Relyea & Kurucz 1978). This value was propagated through the grid fitting process in order to obtain the error estimates for [FORMULA] and [FORMULA] from grids. The errors on the fundamental values were taken from Smalley & Dworetsky (1995), except for those improved by recent HIPPARCOS results (see below). These are the actual uncertainties due to observational errors. In comparing the grids with the fundamental values we assign a total error (obtained from the sum of the variances of the grid and fundamental values) to their difference.

In the comparisons that follow, in this and the next section, we use three statistical measures to compare the three grids:

  1. 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.
  2. A weighted root mean square of the differences, given by [FORMULA], where [FORMULA] are the weights as given by the square of the reciprocal of the errors, and [FORMULA] are the differences between grid and fundamental values.
  3. The reduced chi-square [FORMULA] and its associated probability, [FORMULA], 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 temperature

The observed uvby colours of the fundamental stars were used to obtain values of [FORMULA] and [FORMULA] from the 3 grids: CM, MLT noOV and MLT OV. The values of [FORMULA] obtained for the 3 grids were then compared to the fundamental [FORMULA] values (Table 2). Three of these fundamental stars are binary systems (HD 16739, HD 110379, HD 202275), whose [FORMULA] values are dependant on the adopted distances. As noted above, the [FORMULA] of these stars have been adjusted to take into account the significantly improved parallax measurements from HIPPARCOS. The [FORMULA] of HD 110379 is now over 500 K hotter than that obtained by Smalley & Dworetsky (1995). A full discussion on the revisions and extensions to the list fundamental stars is given in Smalley (1997). Note that HD 16739 appears to have a discrepant fundamental [FORMULA] value. Referring to Smalley & Dworetsky (1995) we see that the value was obtained without using any ultraviolet fluxes. Hence, we conclude that the fundamental [FORMULA] for HD 16739 is certainly too high. In fact, Smalley & Dworetsky (1995) obtained [FORMULA] = 6100 K from spectrophotometry and [FORMULA] = 6200 K from the H [FORMULA] profile. A temperature close to these values would remove the large discrepancy for all 3 grids. Therefore, HD 16739 will be excluded from the following discussion.


[TABLE]

Table 2. Comparison of fundamental and grid values of [FORMULA]. [FORMULA], with the error obtained from the sum of the variances on the fundamental and grid values.


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 [FORMULA] against [FORMULA]. The CM model is in very good agreement with the fundamental values, with a weighted mean difference of -36 [FORMULA] 111 K and a weighted rms difference of 71 K. The [FORMULA] = 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 [FORMULA] 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 [FORMULA] = 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 [FORMULA] 113 K and a weighted rms difference of 199 K, which is clearly not in good agreement with the fundamental values. In addition, the [FORMULA] = 0.770, which gives only a 54% probability of a good fit. This shows that the MLT OV model is not very satisfactory.

[FIGURE] Fig. 2. Comparison of difference between grid and fundamental [FORMULA] for the 3 grids: CM, MLT noOV and MLT OV. [FORMULA]. The CM results are in the best overall agreement with the fundamental stars.

Procyon (HD 61421, [FORMULA] CMi) is the fundamental star with the most tightly constrained value of [FORMULA]. 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 [FORMULA] 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 gravity

Fundamental [FORMULA] values are a fairly stringent test of the grids, since they are more numerous and generally less uncertain than fundamental [FORMULA] values. However, if we include the uncertainties in the values of [FORMULA] obtained from the grids due to uncertainties in uvby colours, the test becomes less stringent (Table 3). Nevertheless, the observed uvby colours were used to obtain values of [FORMULA] for the 3 model grids, which were then compared to the fundamental values. Note that HD 90242 has widely discrepant [FORMULA] values. The exact reason for this anomaly is not known, but could be due to problems with the uvby photometry, since all the grids give values of [FORMULA] [FORMULA] 4.5. Therefore, HD 90242 will be excluded from the following discussion.


[TABLE]

Table 3. Comparison of fundamental and grid values of [FORMULA]. [FORMULA], with the error obtained from the sum of the variances on the fundamental and grid values.


The results of the comparison of the [FORMULA] obtained from the various models with the fundamental values are shown in Fig. 3, as a function of [FORMULA] against [FORMULA] (obtained from the appropriate grid). With the exception of HD 90242 (see above), the CM model values of [FORMULA] agree with the fundamental values to within the error bars. The weighted mean difference is [FORMULA], and indicates that the CM models may, on average, very slightly overestimate [FORMULA]. 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 [FORMULA]. The MLT noOV models give results that are similar to the CM models. The weighted mean difference of [FORMULA], 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 [FORMULA] 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 [FORMULA] and weighted rms difference of 0.141. In this case [FORMULA] = 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 [FORMULA] with decreasing [FORMULA]. The MLT OV models underestimate [FORMULA] for cooler stars.

[FIGURE] Fig. 3. Comparison of difference between grid and fundamental [FORMULA] for the 3 grids: CM, MLT noOV and MLT OV. [FORMULA]. Both the CM and MLT noOV models are in good agreement, with very little of any trend with [FORMULA]. Note that the MLT OV models predicts surface gravities which are systematically too low, in particular for stars with lower temperatures.

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 [FORMULA] with [FORMULA]. The MLT noOV models give a hint of a discrepancy in the coolest fundamental stars. The MLT OV models are somewhat discrepant and underestimate [FORMULA] for the cooler stars.

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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