Appendix A: Comments on the different stellar parameters of Tau
1. McWilliam (1990) based his results on high-resolution spectroscopic observations with resolving power 40000.
The effective temperature was determined by empirical and semi-empirical results found in the literature and from broad-band Johnson colours. The gravity was ascertained by using the well-known relation between g, , the mass M and the luminosity L, where the mass was determined by locating the stars on theoretical evolutionary tracks. So, the computed gravity is fairly insensitive to errors in the adopted L. High-excitation iron lines were used for the metallicity [Fe/H] in order that the results are less spoiled by non-LTE effects. The author refrained from determining the gravity in a spectroscopic way (i.e. by requiring that the abundance of neutral and ionized species yield the same abundance) because `a gravity adopted by demanding that neutral and ionized lines give the same abundance, is known to yield temperatures which are K higher than found by other methods. This difference is thought to be due to non-LTE effects in Fe I lines.' By requiring that the derived iron abundances, relative to the standard 72 Cyg, were independent of equivalent width, the microturbulent velocity was found.
2. Lambert & Ries (1981) have used high-resolution low-noise spectra. The parameters were ascertained by demanding that the spectroscopic requirements (ionization balance, independence of the abundance of an ion versus the excitation potential and equivalent width) should be fulfilled. The effective temperature was found from the Fe I excitation temperature and the model atmosphere calibration of the excitation temperature as a function of . As quoted by Ries (1981), Harris et al. (1988) and Luck & Challener (1995) their and g are too high and should be lowered by 240 K and 0.40 dex, respectively. The isotopic ratio was taken from Tomkin et al. (1975), while the luminosity was estimated from the K-line visual magnitude given by Wilson (1976) and the bolometric correction BC by Gustafsson & Bell (1979). The abundances of carbon, nitrogen and oxygen were based on C2, [O I] and the red system CN lines respectively. Luck & Challener (1995) wondered whether the nitrogen abundance [N/Fe] = -0.20 quoted reflects a typographic error and should rather be [N/Fe]=+0.20, resulting in (N)=7.86, which is more in agreement with being a red giant branch star.
3. The effective temperature and the angular diameter given by Blackwell et al. (1990) were determined by the infrared flux method (IRFM), a semi-empirical method which relies upon a theoretical calibration of infrared bolometric corrections with effective temperature. One expects that the IRFM should yield results better than 1% for the effective temperature and 2-3% for the angular diameter. The final effective temperature is a weighted mean of T(Jn), T(Kn) and T(Ln), with Jn at 1.2467 µm, Kn at 2.2135 µm and Ln at 3.7825 µm.
4. - 5. The stellar parameters quoted by Tsuji (1986, 1991) were based on the results of Tsuji (1981), in which the temperature was determined by the IRFM method. A mass of 3 was assumed to ascertain the gravity. Tsuji (1986) has used high-resolution FTS spectra of the CO (first overtone) lines to determine (C) and by assuming that the abundance should be independent of the equivalent width of the lines. In Tsuji (1991) CO lines of the second overtone were used, where a standard analysis yielded results of (C) = 8.39 and a linear analysis of weak lines led to (C) = 8.31.
6. High-resolution spectra of OH (), CO(), CO() and CN() were obtained by Smith & Lambert (1985). They have used () colours and the calibration provided by Ridgway et al. (1980) to determine . Using the spectroscopic requirement that (Fe I) = (Fe II) yields g = 0.8 dex, which is too low for a K5III giant. They suggested that the reason for this low value is the overionization of iron relative to the LTE situation. They then have computed the surface gravity by using a mass estimated from evolutionary tracks in the H-R diagram. The metallicity was taken from Kovács (1983) and for the microturbulence they used Fe I, Ni I and Ti I lines, demanding that the abundances are independent of the equivalent width. Using the molecular lines, they determined (C), (N), (O) and .
7. Harris & Lambert (1984) have taken , g and determined by Dominy, Hinkle & Lambert in 1984, a reference which we could not trace back. The isotopic ratio was adopted from Tomkin et al. (1975). The carbon abundance was found by fitting weak 12C16O lines at 1.6, 2.3 and 5 µm.
8. Kovács (1983) obtained observations at the 1.52 m telescope of ESO at La Silla. By using different IR colour indices (, , , , , for the broad-band photometry and , , , , for the narrow-band photometry) the effective temperature was found. The gravity and microturbulent velocity were determined in a spectroscopic way, where the abundance derived from strong and medium lines should equal the abundance derived from weak lines for the right value of . The ratio was taken from Tomkin et al. (1976). Using a parallax of and of 24 mas, resulted in a radius of , which corresponds to a luminosity L of , while and a bolometric correction taken from Johnson (1966) yields L = 413 .
9. Lambert et al. (1980) took model parameters from published papers which constrained the ratio. The parameters for Tau were based on Tomkin et al. (1976) and Lambert (1976). Tomkin et al. (1976) have used IR colours for the determination of and the microturbulence was ascertained by fitting the theoretical curve of growth to the 12CN curve of growth (12CN(2-0) around 8000 , weak 12CN(4-0) lines around 6300 and weak 12CN(4-2) lines around 8430 ). Together with 13CN lines around 8000, the isotopic ratio computed. The gravities were estimated from the effective temperature, the mass and the luminosity.
10. By fitting the Engelke function (Engelke 1992) to the template of Tau, Cohen et al. (1996b) could determine and . A gravity of 2.0 was adopted, though in Cohen et al. (1992) a value of 1.5 - taken from Smith & Lambert (1985)- was used.
11. By taking the mean value of the different temperatures derived by calibrations with photometric indices (, , , , , , ) Fernández-Villacañas et al. (1990) have fixed the effective temperature. For the gravity, the DDO photometry indices and were used. The Fe I lines served for the determination of .
12. van Paradijs & Meurs (1974) have adopted the angular diameter from Currie et al. (1974). To determine the effective temperature they used several continuum data, the curve of growth with the line strengths of Fe I lines and the surface brightness. Requiring that the neutral and ionized lines of Fe, Cr, V, Ti and Sc gave the same abundance, yielded the gravity. Gravity, parallax and angular diameter yielded the mass, while the luminosity was determined from the effective temperature, the parallax and the angular diameter. van Paradijs & Meurs (1974) quoted that Wilson (1972) found [Fe/H] = -0.69.
13. Tomkin & Lambert (1974) obtained photoelectric scans of the red CN lines with a resolution of 0.05 for the red scans and 0.09 for the infrared scans. The effective temperature and gravity were adopted from Conti et al. (1967), who have ascertained the effective temperature from the absolute magnitude derived from the K emission-line width and the gravity from the effective temperature, the mass and the luminosity. The microturbulent velocity was determined using a curve of growth technique and the ratio was obtained directly from the horizontal shifts between the curves of growth for 12CN and 13CN lines.
14. Also Luck & Challener (1995) have determined the effective temperature using photometric data (DDO and , Geneva , Johnson , , and ). Two methods were used to fix the surface gravity. The first one determined the `physical gravity' by using the mass M and the radius R, with the mass M determined by , L(, BC) and theoretical evolutionary tracks and the radius R by and L(, BC). The `spectroscopic gravity' is based on the ionization balance of Fe I and Fe II lines. The difference between these two gravities was very large, being 1.0 dex! When the Fe II oscillator strengths were modified to reflect a solar Fe abundance of 7.50 (instead of the used 7.67), then the spectroscopic gravity scale would rise by +0.25 dex (still far less than 1.0 dex) resulting in an increase of [Fe/H] of dex. They have quoted both pluses and minuses for the spectroscopic gravity and a single plus for the physical gravity. The microturbulence was ascertained by forcing no dependence of abundance (derived from individual Fe I lines) upon the equivalent width. In determining the carbon abundance using the C2 Swan system lines and [C I] lines, the nitrogen abundance and isotopic ratio using CN lines and the oxygen abundance from [O I] lines, they always found a better trend with temperature when the spectroscopic gravity was used and also a better agreement between the two carbon indicators. When comparing their results with the ones of Lambert & Ries (1981), they found that Tau was always the most discrepant star. Their very low spectroscopic gravity, compared with the results of other authors using different methods and taking into account possible non-LTE effects, may be an indication that a value of g 1.5 is in better agreement with the real gravity of Tau.
16. Bonnell & Bell (1993) constructed a grid based on the effective temperature of Manduca et al. (1981). They used ground-based high-resolution FTS spectra of OH and [O I] lines. The requirement that the oxygen abundances determined from the [O I] and OH line widths agree amounts to finding the intersection of the loci of points defined by the measured widths in the ([O/H], g) plane. For Tau they found large discrepancies in the [O I] oxygen abundance, which leads to a spread of dex in g. The determination of was based on the OH-lines, but was hampered by a lack of weak lines from this radical. Using a least square fit, they found for the OH() lines and for the OH() lines, reflecting that the OH() sequence lines are formed at greater average depth relative to the OH() lines. For the [O I] lines a microturbulent velocity of 1.5 or 2.0 was used.
17. Ridgway et al. (1982) determined the limb-darkening-corrected angular diameter using the lunar occultation technique.
18a-b. Di Benedetto & Rabbia (1987) used Michelson interferometry by the two-telescope baseline located at CERGA. Combining this angular diameter with the bolometric flux (resulting from a directed integration using the trapezoidal rule over the flux distribution curves, after taking interstellar absorption into account) they found an effective temperature of 3970 K, which is in good agreement with the results obtained from the lunar occultation technique. Di Benedetto (1998) calibrated the surface brightness-colour correlation using a set of high-precision angular diameters measured by modern interferometric techniques. The stellar sizes predicted by this correlation were then combined with bolometric flux measurements, in order to determine one-dimensional (T, V-K) temperature scales of dwarfs and giants.
19. Mozurkewich et al. (1991) used the MarkIII Optical Interferometer. The uniform-disk angular diameter had a residual of 1% for the 800 nm observations and less than 3% for the 450 nm observations. The limb-darkened diameter was then obtained by multiplying the uniform-disk angular diameter with a correction factor (using the quadratic limb-darkening coefficient from Manduca 1979).
20. Quirrenbach et al. (1993) have determined the uniform-disk angular diameter in the strong TiO band at 712 nm and in a continuum band at 754 nm with the MarkIII stellar interferometer on Mount Wilson. Because limb darkening is expected to be substantially larger in the visible than in the infrared, the measured uniform-disk diameters should be larger in the visible than in the infrared. This seems, however, not always to be the case. Using the same factor as Mozurkewich et al. (1991) we have converted their continuum uniform-disk value (19.80 mas) into a limb-darkened angular diameter, yielding a value of 22.73 mas, with a systematic uncertainty in the limb-darkened angular diameter of the order of 1% in addition to the measurement uncertainty of the uniform-disk angular diameter (Davis 1997).
21. Volk & Cohen (1989) mentioned a distance of pc. The effective temperature was directly determined from the literature values of angular diameter measurements and total flux observations (also from literature). The distance was taken from the Catalogue of Nearby Stars (Gliese, 1969) or from the Bright Star Catalogue (Hoffleit & Jaschek 1982).
22. Blackwell et al. (1991) is a revision of Blackwell et al. (1990) where the H- opacity has been improved. They investigated the effect of the improved H- opacity on the IRFM temperature scale and derived angular diameters. Also here, the mean temperature is a weighted mean of the temperatures for , and . Relative to Blackwell et al. (1990) there was a change of temperature up to 1.4% and an decrease by 3.5% in .
24. Bell & Gustafsson (1989) first determined the temperature from the Johnson K band at 2.2 µm by use of the IRFM method. By comparison with temperatures deduced from the colours Glass , , and K; Cohen, Frogel and Persson , , and K; Johnson , , and K; Cousins , ; Johnson and Mitchell 13-colour and Wing's near-infrared eight-colour photometry, they found that (IRFM) was K too high, by which they corrected the temperature. The gravity and the metallicity were adopted from Kovács (1983).
26. Burnashev (1983) has determined , g and [Fe/H] from narrow-band photometric colours in the visible part of the electromagnetic spectrum.
27. Fracassini et al. (1988) have made a catalogue of stellar apparent diameters and/or absolute radii, listing 12055 diameters for 7255 stars. Only the most extreme values are listed. References and remarks to the different values of the angular diameter and radius may be found in this catalogue. Also here these angular diameter values are given in italic mode when determined from direct methods and in normal mode for indirect (spectrophotometric) determinations.
28. Perrin et al. (1998) have derived the effective temperatures for nine giant stars from diameter determinations at 2.2 µm with the FLUOR beam combiner on the IOTA interferometer. This yielded the uniform-disk angular diameter of Boo and Tau of our sample. The averaging effect of a uniform model leads to an underestimation of the diameter of the star. Therefore, they have fitted their data with limb-darkened disk models published in the literature. The average result is a ratio between the uniform and the limb-darkened disk diameters of 1.035 with a dispersion of 0.01. This ratio could then also be used for the uniform-disk angular diameters of Dra and Peg, listed by Di Benedetto & Rabbia (1987), and Cet, which was based on a photometric estimate. Several photometric sources were used to determine the bolometric flux, which then, in conjunction with the limb-darkened diameter, yielded the effective temperature.
29. Engelke (1992) has derived a two-parameter analytical expression approximating the long-wavelength (2-60 µm) infrared continuum of stellar calibration standards. This generalized result is written in the form of a Planck function with a brightness temperature that is a function of both observing wavelength and effective temperature. This function is then fitted to the best empirical flux data available, providing thus the effective temperature and the angular diameter.
30. Blackwell & Shallis (1977) have described the Infrared Flux Method (IRFM) to determine the stellar angular diameters and effective temperatures from absolute infrared photometry. For 28 stars (including Car, Boo, Cma, Lyr, Peg, Cen A, Tau and Dra) the angular diameters are deduced. Only for the first four stars the corresponding effective temperatures are computed.
31. Scargle & Strecker (1979) have compared the observed infrared flux curves of cool stars with theoretical predictions in order to assess the model atmospheres and to derive useful stellar parameters. This comparison yielded the effective temperature (determined from flux curve shape alone) and the angular diameter (determined from the magnitudes of the fluxes). The overall uncertainty in is probably about 150 K, which translates into about a 9% error in the angular diameter.
32. Manduca et al. (1981) have compared absolute flux measurements in the 2.5-5.5 µm region with fluxes computed for model stellar atmospheres. The stellar angular diameters obtained from fitting the fluxes at 3.5 µm are in good agreement with observational values and with angular diameters deduced from the relation between visual surface brightness and () colour. The temperatures obtained from the shape of the flux curves are in satisfactory agreement with other temperature estimates. Since the average error is expected to be well within 10%, the error for the angular diameter is estimated to be in the order of 5%.
33. Smith & Lambert (1990) have determined the chemical composition of a sample of M, MS and S giants. The use of a slightly different set of lines for the molecular vibra-rotational lines, along with improved gf-values for CN and NH, lead to a small difference in the carbon, nitrogen and oxygen abundance compared to Smith & Lambert (1985).
Appendix B: The approximate absorption coefficient
Kjærgaard et al. (1982) examined two cases for which they derived equations for the approximate absorption coefficient . In case (a) most of the carbon and nitrogen are still considered to be free atoms, assumed for 5000 K, while in case (b) almost all the C and N are in the form of CO and N2, assumed for 4500 K. Looking at the relatively high partial pressure of CN, it seems that the assumption of complete association of N into N2 is not valid for our giant models. Attributing all depletion to the formation of N2 seems to be more valid in a dwarf model (Bell & Tripicco 1991). The formula can however still give a qualitative view of the influence of the different parameters. For case (b) Kjærgaard et al. deduced an approach for CN:
Using the approach of Kjærgaard et al., similar equations for CO, SiO, OH, H2O and NH are derived.
For CO, we have
In the assumption that all carbon is locked into CO,
with the approximation of n(e) and n(H)/n(e) in the line-forming region being (Kjærgaard et al., 1982)
The approximate absorption coefficient of SiO is defined as
For oxygen-rich giants, n(O I) may be approximated by
The equilibrium of silicon is dominated by SiO and Si I
(Si II is present in the deeper layers as well). For example, in
the model with = 4050 K,
g = 1.00, z = 0.00, M = 1.5
2.0 , the proportion of the number
densities are given in Table B.1. Since the ratio
changes rapidly in the outer layers
of the photosphere, a rough approximation to
, valid in the region where
, is then
Using the same notations, (OH) is written as
Also for H2O we obtain
The approximate absorption coefficient of NH is written as
The equilibrium of the nitrogen species is dominated by the N2 formation. So,
© European Southern Observatory (ESO) 2000
Online publication: December 15, 2000