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Astron. Astrophys. 331, 619-626 (1998)

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5. The effective temperature scale

The fourth power of the effective temperature of a stellar object is proportional to its surface brightness according to the Stefan-Boltzmann law on the black-body radiation. This relationship can be rewritten with convenient units and convenient observable quantities:

[EQUATION]

where [FORMULA] is the bolometric flux in units of [FORMULA] and [FORMULA] is the limb darkened diameter in milli-arcseconds. The stellar interferometer has provided values for the limb darkened diameters of our sample of giant stars. No photometric observations have been carried out simultaneously with interferometric ones. It is thus necessary to estimate these fluxes from other observations. Since our sample of sources are slightly variable, error bars on the flux must take into account their variability.

5.1. Bolometric fluxes and spectral types

5.1.1. Intrinsic stellar colors and spectral types

At several points below it will be useful to have a table of the intrinsic colors of giants later than M5. We have reviewed lists of bolometrically bright giants in the spectral type range M5-M8 as classified by Morgan & Keenan (1973), Wing (1967) and Bidelman (1981). From these lists we have excluded stars of large variability, based on either a classification as long period variables, or on observed large amplitude variability, [FORMULA] K [FORMULA] 0.5 (Two Micron Sky Survey (Gezari et al. 1993)). We have assembled available photometry for these stars, including the Two Micron Sky Survey, and numerous unpublished observations by one of the authors from Kitt Peak National Observatory. These observations have been used to estimate intrinsic colors, assuming that the brightest have no interstellar reddening, and ignoring possible circumstellar reddening. The colors are reported in Table 3.


[TABLE]

Table 3. Intrinsic colors of very cool giants.


It should be noted that the spectral classifications from the sources consulted are not entirely consistent, sometimes giving different or multiple classifications for the same star. Variations of 0.5 types are typical, and may reflect both the typing uncertainty and actual variability. Consequently, on the one hand, Table 3 may be more useful statistically than for individual stars. The colors in Table 3 are given to 0.01 magnitudes to facilitate smooth interpolation - not because they are significant at this level. But, on the other hand, we have found that spectral types derived from this color table lead to more consistent results for the temperature calibration. In fact, both Morgan & Keenan (1973) and Wing (1967) spectral types are based on stellar characreristics in the visual or very near infrared domains - to 1.1 µm. Due to high continuum and line opacities, and to the temperature dependence of the exponentiel tail of the Planck function, this spectral regime is very sensitive to fluctuations in stellar conditions, whereas in the [FORMULA] µm spectral region, the lower opacities reveal emission from a deeper, more stable part of the atmosphere. This suggests that a refined classification based on infrared colors computed from photometric data in bands redder than J should leave spectral types free of contamination from artifacts with basically visible characteristics that depend upon the phase when the sources are observed. The case of [FORMULA] Ser is very illustrative. Published spectral types are very discrepant and vary between M5 (Keenan & Mc Neil 1989) and M7 (Wing 1967). The observed colors lead to very consistent spectral types instead:

  • J-K: M7.3
  • H-K: M7.4
  • K-L: M7.6

We have adopted as an average M7.4 which clearly confirms the later spectral type. Spectral types for sources later than M6 in Table 2 were determined with this method.

5.1.2. Bolometric fluxes

The bolometric fluxes that we have used are of different origins. Some were determined from wide band photometry (V, R, I, J, H, K, L, M) made by one of us at Kitt Peak National Observatory. No observations were made in the U and B photometric bands and some observations were missing in a few cases. Some infrared magnitudes were found in Kerschbaum & Hron (1994) and Kerschbaum (1995). Other missing magnitudes have been deduced from Table 3 for spectral types from M5 to M8, complementary to that of Johnson (1966). For two stars we have used bolometric fluxes found in the literature. For [FORMULA] Boo the flux has been determined by Blackwell et al. (1986) and for [FORMULA] Tau it has been determined by Di Benedetto & Rabbia (1987). The bolometric flux is computed from the photometry by interpolating the spectral distribution between 0.36 µm and 5 µm. The integration is extended from [FORMULA] to [FORMULA] after having extrapolated the spectrum by a black body distribution matching the observed fluxes at 0.36 µm and 5 µm.

Although all sources are very bright and relatively close, it has been necessary to take interstellar reddening into account to compute the bolometric flux of all sources but [FORMULA]  Boo and [FORMULA]  Tau. The magnitudes of extinction in the visible are displayed in Table 4. A(V) for [FORMULA]  Oph and EU Del are from Fluks et al. (1994). The extinction for other stars was derived from reddening measured on nearby sources by Perry & Johnston (1982). We have assumed that the source of extinction is diffuse interstellar dust and we have used the extinction law of Mathis (1990) to compute extinction at any wavelength.


[TABLE]

Table 4. Photometric data and effective temperatures.


5.1.3. Error bars

Eq.  3shows that the error on the bolometric flux contributes less to the final error on temperature than the error on the angular diameter. But this is not negligeable and we must be as careful as for the evaluation of the final error bar on diameters. Since the photometric observations were not simultaneous with interferometric observations, error bars must include the variability of the sources. To do so, we have gathered the flux calibrations listed in the Catalog of Infrared Observations (Gezari et al. 1993). For each star, we have fitted the flux measurements by a black body curve with the least square method, although the spectrum of a cool star is not a black body because of a lot of lines blanketing the entire spectrum. Fluxes were weighted with the number of observations in the corresponding photometric band. This guarantees that each photometric band contributes with the same weight. This yields a temperature and a normalization constant. We have estimated the average range of variation (or uncertainty) for individual wide band calibrations by the square root of the ratio of the residual at the optimum and of the number of photometric bands minus 2 (two degrees of freedom). The least square formula is normalized by this average variance to form a [FORMULA]. The [FORMULA] is then varied to find the uncertainties on temperature and normalization coefficient. A Monte-Carlo Method yields the uncertainty on the bolometric flux. In all cases, this method leads to error bars that are larger than the error bars from the Kitt Peak photometry. This method has been applied to the stars later than M6 because these stars are likely to be slightly variable. Results are displayed in Table 4.

5.2. Effective temperatures

Effective temperatures are estimated with Eq.  3. Error bars are computed with a Monte-Carlo method. The estimates are presented in Table 4.

Effective temperature is plotted against spectral type in Fig. 3. The law determined from lunar occultation data
(Ridgway et al. 1980) is the dotted line and the laws determined from interferometric data, either with I2T (Di Benedetto & Rabbia 1987) or with IOTA (Dyck et al. 1996a), are the dashed and the semi continuous lines respectively. For spectral types earlier than M6 the FLUOR data are in good agreement with the mean laws except in the case of Arcturus which is about 200 K cooler than predicted by these laws for a K1.5 giant. It is probable that this discrepancy is due to the low metallicity of this halo star (Peterson et al. 1993) compared to the old disk stars.

[FIGURE] Fig. 3. Effective temperature versus spectral type. Dotted line: occultation data law (Ridgway et al. 1980). Semi continuous line: IOTA data law (Dyck et al. 1996a). Dashed line: I2T data law (Di Benedetto & Rabbia 1987). Solid line: FLUOR data law. Uncertainties on mean curves may be gauged from the difference between the various curves. The apparent discrepancy for the K1.5 III star [FORMULA]  Boo is believed to be due to its unusual metallicity as a halo star compared to old disk stars.

The temperatures of stars later than M6 are found to extend well the occultation data law. The solid line on Fig. 3 is our proposed extension of the effective temperature scale with the FLUOR points. Temperature for the M6 type is that found for EU Del. Temperature for the M8 type is the average of the temperatures of SW Vir and RX Boo. Temperature for the intermediate M7 type has been established by a linear fit of the FLUOR points for types between M6 and M8. The observations of Dyck et al. (1996a) are consistent with ours, though with smaller errors and a richer selection of very cool stars, we derive a slightly different extrapolation of the occultation calibration.

Since the lunar occultation is the most consistent with the old and new data, we have chosen to merge this scale with the FLUOR scale to produce a composite law. As there is only a 7 K discrepancy between the FLUOR and the lunar occultation scale at M6, we have adopted the FLUOR calibration and its error bar for the M6 temperature. The new merged scale is tabulated in Table 5.


[TABLE]

Table 5. Effective temperature scale for class III giants. Calibration for types earlier than M6 is from Ridgway et al. (1980). Types later than M6 are calibrated with the present data. Colors are from Johnson (1966) except for types later than M6 for which colors are derived from Table 3.


To be used, the spectral representation requires the knowledge of the spectral type, which may be a severe inconvenience. It may be useful to calibrate the temperature scale with a more direct observable like a color index. We have plotted effective temperatures against the [V-K] and [I-L] color indices in Figs. 4 and 5 and compared them with previous studies. Because all the sources are variable the most complex relation that can be derived from the FLUOR data is obtained with a fit by a linear law. In the [V-K] representation, our calibration is very consistent with both Ridgway et al. (1980) and Di Benedetto (1993) results. Consistency is not so obvious between the occultation data law and the FLUOR law in the [I-L] representation. Unfortunately no data are available to bridge the gap between [I-L]=2 and [I-L]=3.5 and it is difficult, with the present set of data, to comment on this discrepancy. The linear laws represented by solid lines in Figs. 4 and 5 are described by:

[EQUATION]


[FIGURE] Fig. 4. Effective temperature versus V-K. Dotted line: I2T data law (Di Benedetto 1993). Curved solid line: occultation data law (Ridgway et al. 1980). Straight solid line: FLUOR data law

[FIGURE] Fig. 5. Effective temperature versus I-L. Dotted line: occultation data law (Ridgway et al. 1980). Solid line: FLUOR data law

5.3. Discussion

It is now some decades since it was realized that the effective temperatures of cool stars deviate significantly from black body fits or color temperatures. First occultation, and more recently interferometric measurements, have progressively pushed the calibration of the giant effective temperatures to later spectral types. The current extension (perhaps the final one for the "normal" giants) represents a consistent extrapolation, almost an obvious best guess, from previous work on earlier spectral types.

The known variability of the stars in our sample, in magnitude, colors and spectral type, must contribute to the scatter in Figs. 3, 4 and 5. It is interesting and perhaps significant that the color index representation has less scatter than the spectral representation. A closer look at such relations may constrain the nature of the variability, which in these stars is currently uncertain.

The errors in the concluding calibrations presented here are clearly not negligible. It may be wondered what improvement might be expected. Of course further confirming observations, and contemporaneous spectral types and photometry, would be welcome for the current list of stars, as would an improved estimate or measure of the (small) interstellar extinction. However, at present we believe that the accuracy of the effective temperature calibration of at least the M-type stars may be approaching the limit justified by the consistency of spectral classification process, and the variability of the sources.

Furthermore, several major optical interferometer facilities are operating or under construction. It is now possible to plan scientific programs in which interferometric measurements are merely one component of an observing plan. We expect that in the future effective temperature measurements will be obtained more frequently as incidental results during the course of detailed interferometric studies of individual stars pertaining to, for example, limb darkening, evolution of surface structure, pulsation, and so forth.

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© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998
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