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Astron. Astrophys. 318, 841-869 (1997)

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7. [FORMULA] from colour indices ([FORMULA]), ([FORMULA]), and ([FORMULA])

In the previous section we showed that some models have a different structure depending whether the "overshooting" option is switched on or off. In this section we investigate the effect of the different model structure on the [FORMULA], [FORMULA], and [FORMULA] colour indices, which are often adopted for fixing effective temperatures for cool stars; furthermore, we will try to state whether the color indices from the "overshooting" models (COLK95) or those from the "no-overshooting" models (COLNOVER) give [FORMULA] closer to the values derived from the infrared flux method (IRFM), which is almost model independent.

7.1. The dependence of the synthetic colour indices on the convection

We computed grids of synthetic colours UBV, uvby, and RIJKL from models having the "overshooting" option switched off, microturbulent velocity [FORMULA] =2 km s-1, and metallicities [M/H]=0.0 and [M/H]=-3.0. For each gravity, we derived [FORMULA] by interpolating in the COLNOVER grids for the ([FORMULA]), ([FORMULA]), and ([FORMULA]) color indices of the COLK95 grids. In this way, we may estimate the effect of the convection on the effective temperatures derived from the color indices. The temperature differences [FORMULA] [FORMULA] = [FORMULA] over - [FORMULA] nover as function of [FORMULA] for the ([FORMULA]), ([FORMULA]), and ([FORMULA]) indices are shown for different gravities and solar metallicity in the upper panels of Fig. 13-15.

[FIGURE] Fig. 13. Differences between [FORMULA] from the COLK95 color grids and [FORMULA] from the COLNOVER synthetic grids for the same ([FORMULA]) index and the same gravity. Different curves in the upper plot are for different gravities, and the same metallicity [M/H]=0.0. The two curves in the lower plot are for the same gravity [FORMULA] =4, and the two different metallicities [M/H]=0.0 and [M/H]=-3.0
[FIGURE] Fig. 14. The same as Fig. 13, but for the ([FORMULA]) index
[FIGURE] Fig. 15. The same as Fig. 13, but for the ([FORMULA]) index

The largest [FORMULA] [FORMULA] differences are about 60 K, 100 K, and 170 K for the ([FORMULA]), ([FORMULA]), and ([FORMULA]) indices respectively. They occur for [FORMULA] and log g between 7500-8000 K and 4.0-4.5 for ([FORMULA]), 6750-7250 K and 3.0-4.0 for ([FORMULA]), 6500-7000 K and 3.0-4.0 for [FORMULA]. Temperatures from the color indices computed from the "no overshooting" models are lower than those from color indices computed from the "overshooting models". For all the three indices the value of [FORMULA] [FORMULA] weakly depends on gravity for [FORMULA] [FORMULA] 6500 K. For [FORMULA] [FORMULA] 6500 K the effect of the convection increases with increasing gravity.

The effect of metallicity on the temperature differences [FORMULA] [FORMULA] = [FORMULA] over - [FORMULA] nover as a function of [FORMULA] for the ([FORMULA]), ([FORMULA]), and ([FORMULA]) indices is shown in the lower panels of Fig. 13-15. The functions [FORMULA] [FORMULA] vs. [FORMULA] for the two metallicities [M/H]=0.0 and [M/H]=-3.0 are compared for [FORMULA]. [FORMULA] [FORMULA] for [M/H]=0.0 differs from [FORMULA] [FORMULA] for [M/H]=-3.0 no more than 40 K, 100 K, and 60 K when the ([FORMULA]), ([FORMULA]), and ([FORMULA]) indices are considered, respectively. In particular: for the ([FORMULA]) index and decreasing metallicity, [FORMULA] [FORMULA] decreases for [FORMULA] [FORMULA] 6500 K then, it increases for [FORMULA] [FORMULA] 6500 K. For [M/H]=-3.0, the largest [FORMULA] [FORMULA] is about 50 K, at 5750 K. For ([FORMULA]) and decreasing metallicity, [FORMULA] [FORMULA] increases for [FORMULA] [FORMULA] 5500 K, while [FORMULA] [FORMULA] does not change with metallicity for [FORMULA] [FORMULA] 5500 K. For [M/H]=-3.0, the largest [FORMULA] [FORMULA] is about 200 K, at 7250 K. For the ([FORMULA]) index and decreasing metallicity, [FORMULA] [FORMULA] slightly increases for [FORMULA] [FORMULA] 5750 K. For [M/H]=-3.0 the largest [FORMULA] [FORMULA] is about 190 K at 7250-7500 K. For [FORMULA] included between 4500 K and 5750 K, [FORMULA] [FORMULA] does not change with metallicity, then it increases for [FORMULA] [FORMULA] 4500 K. However, results for the "no overshooting" models with [FORMULA] [FORMULA] 4000 K have to be considered with caution, owing to some convergence problems in the computation of models with log g [FORMULA] 3.0

In summary: temperatures derived from the ([FORMULA]) indices are nearly independent of the different treatment of convection, whatever the metallicity; for decreasing metallicity from [M/H]=0.0 to [M/H]=-3.0 the largest temperature differences [FORMULA] [FORMULA] derived from the ([FORMULA]) index increase from about 100 K to 200 K, while for the ([FORMULA]) index the largest [FORMULA] [FORMULA] increase from about 170 K to 190 K. The "overshooting" option affects mostly the models with [FORMULA] [FORMULA] 5500 K. This analysis suggests that the difference in [FORMULA] derived from ([FORMULA]) or ([FORMULA]) color indices due to the "overshooting" option will never be larger than about 200 K for each gravity and each metallicity of the grids.

7.2. Comparison of [FORMULA] from the colour indices and from the infrared flux method (IRFM)

Although the largest difference between temperatures derived from ([FORMULA]), ([FORMULA]), and ([FORMULA]) indices computed using models with the "overshooting" option switched on and off is only 200 K, we tried to determine which kind of models yields [FORMULA] 's closer to those derived from methods almost model independent, as the infrared flux method (IRFM).

Blackwell & Lynas-Gray (1994) (BLG) derived [FORMULA] using the infrared flux method for a sample of stars with effective temperatures included between 4000 and 8500 K, gravities [FORMULA] from 1.5 to 4.5 and metallicities in the range from [M/H]= [FORMULA] and [M/H]=-0.5. Smalley & Dworetsky (1995) (SD) revised the effective temperatures derived by Code et al. (1976) from the measured angular diameter and measured total flux. We compared the BLG and SD temperatures for stars with [FORMULA] [FORMULA] 9000 K with those we derived from the colour indices computed from models with the "overshooting" option switched on and off respectively.

Effective temperatures obtained from the infrared flux method can be directly compared with [FORMULA] derived from color indices only if both methods are independent of gravity and metallicity or, viceversa, if the gravity and the metallicity of each star in the sample are well known quantities.

The IRFM method depends on models, and therefore on [FORMULA], [FORMULA], and [M/H], in that it makes use of the computed monocromatic flux [FORMULA] (T,g, [FORMULA],A) at a preselected infrared wavelength. To this purpose, BLG assumed for each star of their sample solar metallicity and an approximate [FORMULA]. Kurucz (1992) model atmospheres computed for solar metallicity were used. Mégessier (1994) showed that, in the framework of the Kurucz models, IRFM temperatures differ from 74 K at 8000 K to 0 K at 6200 K for gravity differences of 0.5 dex and that IRFM temperatues differ no more than 60 K for metallicity differences of 0.5 dex.

We investigated the effect of the gravity and metallicity on the ([FORMULA]), ([FORMULA]), and ([FORMULA]) indices. The only difference with a similar analysis performed by King (1993) is the use made by us of improved models (Kurucz, 1995) for cool stars. The results are shown in Fig. 16-18.

[FIGURE] Fig. 16. Upper plot shows the ([FORMULA])- [FORMULA] relation for solar metallicity and different gravities. The two curves in the lower plot show differences between [FORMULA] derived from ([FORMULA]) index for solar metallicity, [M/H]=0.0, and [FORMULA] derived from the same ([FORMULA]) index, but for [M/H]=-0.5 or [M/H]= [FORMULA] respectively. Uncertainties in [FORMULA] related with uncertainties in gravity and metallicity can be estimated
[FIGURE] Fig. 17. The same as Fig. 16, but for ([FORMULA])
[FIGURE] Fig. 18. The same as Fig. 16, but for ([FORMULA])

For gravity differences of [FORMULA] 0.5 dex, effective temperatures from the ([FORMULA]) index show differences ranging from [FORMULA] 100 K at [FORMULA] =8000 K to about 0 K for [FORMULA] [FORMULA] 6500 K. Therefore, for temperatures lower than 6500 K the effect of the gravity on the ([FORMULA]) index becomes negligible. For [FORMULA], metallicity differences of 0.5 dex yield differences in [FORMULA] ranging from a maximum of 70 K at 8500 K to a minimum of 0 K at [FORMULA] =5250 K. For [FORMULA] [FORMULA] 5250 K, the differences in [FORMULA] increase with decreasing temperature. They are about 20 K at 4250 K and about 120 K at 3750 K (Fig. 16).

For gravity differences of [FORMULA] 0.5 dex, effective temperatures from the ([FORMULA]) index show differences ranging from [FORMULA] 100 K at [FORMULA] =8000 K to about 0 K for 5250 K [FORMULA] [FORMULA] [FORMULA] 6250 K. For lower temperatures the largest difference is about 70 K at 4500 K and [FORMULA]. Therefore, for temperatures lower than 6250 K, the ([FORMULA]) index is nearly independent of gravity. The ([FORMULA]) index depends on metallicity more than the ([FORMULA]) index. For [FORMULA], metallicity differences of 0.5 dex yield differences in [FORMULA] on the order of 200 K for [FORMULA] included between 4750 K and 6750 K (Fig. 17).

The ([FORMULA]) index depends on gravity more than the ([FORMULA]) index. For gravity differences of 0.5 dex, the temperature differences are nearly 0 K only for [FORMULA] included between 5500 K and 6500 K. For larger and lower temperatures [FORMULA] [FORMULA] may be on the order of 120 K and 90 K respectively. The dependence of the ([FORMULA]) index on metallicity is different from that of the ([FORMULA]) index. For [FORMULA], metallicity differences of 0.5 dex yield differences in [FORMULA] on the order of 180 K at [FORMULA] =5000 K. Then [FORMULA] [FORMULA] decreases for lower and larger temperatures (Fig. 18).

Summarizing, the ([FORMULA]) index is nearly unaffected by both gravity and metallicity for the whole 4250-8500 K range of temperature, the largest uncertainty in [FORMULA] being on the order of 100 K for [FORMULA] [FORMULA] 6500 K. The ([FORMULA]) index is nearly unaffected by gravity for [FORMULA] [FORMULA] 6500 K, but it is affected by metallicity so that [FORMULA] [FORMULA] may be on the order of 200 K. The ([FORMULA]) index depends on both gravity and metallicity, except for the 5500-6500 K region where it is gravity independent.

The stars for which we compared [FORMULA] derived from color indices with [FORMULA] derived with the IRFM method are listed in Table 4.


[TABLE]

Table 4. The program stars


Since we have been not able to recover the gravity values left out by BLG in their Table 6, we used only the stars of their sample with a given value for the gravity or for which we found a value in the literature. For the seven stars that we extracted from the SD sample (indicated with an asterisk in Table 4) we adopted the gravity values provided by SD. For all the stars we adopted models with solar metallicity in agreement with the BLG analysis. The actual metallicity of each star taken from the literature is listed in column 10 of Table 4. Observed ([FORMULA]) indices were taken from BLG, except for Procyon, for which the value given by Steffen (1985) was adopted. ([FORMULA]) indices were taken from the Bright Star Catalog (Hoffleit, 1964), and [FORMULA] indices were taken from the Hauck & Mermilliod (1990) catalog. Observed ([FORMULA]), ([FORMULA]), and ([FORMULA]) indices are listed in columns 5, 6, and 7 respectively. The reddening [FORMULA] (column 3) was taken from BLG. We dereddened the ([FORMULA]) indices by means of the relation [FORMULA], where [FORMULA] was derived from [FORMULA] (Clementini et al., 1995). ([FORMULA]) indices were dereddened by using the above relation for [FORMULA], and ([FORMULA]) indices were dereddened by means of the relation [FORMULA] (Crawford & Mandwewala, 1976).

For each star, the effective temperatures corresponding to the assumed gravity and to the observed ([FORMULA]), ([FORMULA]), and ([FORMULA]) indices were derived by interpolation in the uvby, UBV, and RIJKL synthetic grids, respectively. These effective temperatures are listed in columns 11, 12, and 13 of Table 4. For each column there are two [FORMULA], the first one was derived from the COLK95 grids, the second one, in parenthesis, from the COLNOVER grids. For comparison, column 8 lists [FORMULA] derived by BLG or SD by using the IRFM method.

Fig. 19 shows the difference between temperatures obtained with the IRFM method and the temperatures derived from the ([FORMULA]), ([FORMULA]), and ([FORMULA]) respectively. The crosses correspond to [FORMULA] derived from the COLNOVER colors grids. The points, triangles, and squares indicate [FORMULA] derived from the COLK95 color grids. The squares indicate the SD stars, the points and the triangles indicate the BLS stars, where points are for stars with [M/H] [FORMULA] 0.2 in absolute value and triangles are for stars with [M/H] [FORMULA] 0.2 in absolute value. The line connecting each cross with the corresponding point, or square, or triangle is the difference [FORMULA] [FORMULA] between the effective temperatures derived from the COLK95 and the COLNOVER grids.

[FIGURE] Fig. 19. Comparison between [FORMULA] derived from observed and computed ([FORMULA]), ([FORMULA]), ([FORMULA]) indices and [FORMULA] derived by Blackwell & Lynas-Gray (1994)(BLG) and by Smalley & Dworetsky (1995) (SD) from the IRFM method. Crosses are for effective temperatures from COLNOVER indices; points, squares and triangles are for effective temperatures from COLK95 indices. The squares indicate stars from SD. Points and triangles indicate stars from BLG with [M/H] [FORMULA] 0.2 and [M/H] [FORMULA] 0.2 (in absolute value), respectively

As predicted by Fig. 13, 14, and 15, this difference is small for ([FORMULA]), and then increases for ([FORMULA]), and it is still larger for ([FORMULA]). For each index, [FORMULA] [FORMULA] is smaller for [FORMULA] [FORMULA] 6250 K than for [FORMULA] included between 6250 K and 7250 K. Then [FORMULA] [FORMULA] decreases again for temperatures larger than 7500 K. Effective temperatures from the COLK95 grids are always larger than effective temperatures from the COLNOVER grids, except for the ([FORMULA]) index and [FORMULA] [FORMULA] 5750 K, where, however, the differences are negligible.

Fig. 13 shows that the ([FORMULA]) index is almost independent of both gravity and metallicity for [FORMULA] included between 4000 K and 6250 K. Therefore, ([FORMULA]) indices of stars with [FORMULA] included in this range can be used to estimate the difference between [FORMULA] derived from models and [FORMULA] derived from the IRFM method. Fig. 19 shows that [FORMULA] derived from the ([FORMULA]) indices is 38 K larger, on average, than [FORMULA] derived from the IRFM method by BLG.

Fig. 19 suggests that [FORMULA] from the COLNOVER indices are, on average, closer to [FORMULA] from the IRFM method than [FORMULA] derived from the COLK95 indices. This is confirmed by the comparison of columns 2 and 3 and of columns 4 and 5 of Table 5, where the averages of the differences between [FORMULA] derived from color indices and [FORMULA] obtained with the IRFM method are listed. We computed two separated averages [FORMULA] [FORMULA] for [FORMULA] [FORMULA] 6250 K and for [FORMULA] [FORMULA] 6250 K. For each region, the differences between [FORMULA] from color indices and [FORMULA] from IRFM were computed by using both the COLK95 and the COLNOVER indices.


[TABLE]

Table 5. [FORMULA] from ([FORMULA]) indices and [FORMULA] profiles for 7 subdwarf stars


Finally, we investigated the consistency of [FORMULA] derived from the COLK95 and COLNOVER indices. Because the ([FORMULA]) index is nearly idependent of convection and the ([FORMULA]) index somewhat depends on it, the comparison of effective temperatures derived from the ([FORMULA]) and ([FORMULA]) indices, when both COLK95 and COLNOVER grids are used, should indicate which convection yields effective temperatures closer to each other.

We compared [FORMULA] derived from ([FORMULA]) indices with [FORMULA] derived from ([FORMULA]) indices for the stars of Table 4, having [FORMULA] included between 6000 K and 8000 K, because the models of this region are mostly affected by convection. We derived from Fig. 20 that the mean deviations of the points from the diagonal are 145 and 99 when the COLK95 and the COLNOVER indices are used respecticvely. This difference is not very large, however it indicates that, on average, NOVER models yield [FORMULA] from color indices more consistent each with other than K95 models do.

[FIGURE] Fig. 20. Comparison between [FORMULA] derived from ([FORMULA]) index with [FORMULA] derived from ([FORMULA]) index for the stars of Table 4 having [FORMULA] included between 6000 K and 8000 K. In the upper and lower plots synthetic indices are from the COLK95 and the COLNOVER grids, respectively
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Online publication: July 3, 1998
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