2. Surface brightness relations for cool giants and supergiants
2.1. Selection criteria
We have compiled a list of stars with accurate measures of angular diameters, mostly obtained by Michelson interferometry. Original sources are given by di Benedetto (1993), and Dyck et al. (1996). Diameters are corrected for limb-darkening. The relative precision of these measurements is generally better than 5%. When several measurements exist, a weighted mean has been adopted.
For sake of homogeneity in the photometry, only stars measured by Johnson et al. (1966) in V, , J and K bands have been retained. These authors attribute the following mean accuracy to their measurements: 0.015 in V, 0.030 in K, 0.018 in , and 0.025 in . Three double stars, namely HR 603, HR 1708 and HR 6406 have been rejected, because Johnson's V measurements refer to the whole system. Too few of these stars also have photometry in the VRI Cousins system (Cousins 1980) to allow an accurate calibration of the and surface brightness relations.
Forty stars are a priori available for calibrating the visual and infrared surface brightness variations with colour. They range from supergiants to dwarfs, and from F0 to M6 spectral types. We first reject variable stars. For this purpose, we use two criteria: the Bright Star Catalogue (Hoffleit & Jaschek 1982) gives the variable classification and its amplitude: we keep stars with less than 0.2 mag V amplitude; Cousins (1980) gives a raw estimate of the photometric variability by the number of decimal places: we reject stars whose V magnitude is given with only one significant decimal place. Both criteria generally agree, except for HR 8308, which is a flare star, and therefore may be considered as not variable for photometric purposes at a given epoch. HR 4902 and HR 8698 are rejected according to Cousins, although the Bright Star Catalogue classifies them as low amplitude variables (Lb 0.1 V for both). Applying this rejection criterion, 27 stars remain in our catalogue. Among these, 19 have high precision diameters (accuracy better than 6%). Remember that a 10% accuracy in angular diameter corresponds to 0.22 mag uncertainty in magnitude. Table 1 lists these 27 stars, among which 13 are giants (luminosity class III), and 13 are supergiants (luminosity classes I, I-II and II). The Sun has been added as a reference point, with photometry from Johnson (1965). HR number, star common name, spectral type and variability class and amplitude come from the Bright Star Catalogue (1982); the limb-darkened mean diameter (in milliarcsec) and its accuracy (in %) are from di Benedetto (1993, source 1) or Dyck et al. (1996, source 2): for details of the angular diameter measurements (technique, wavelength, limb-darkening correction), we refer the reader to these references; all the photometry comes from Johnson et al. (1966), followed by a cross if the star has also been measured by Cousins (1980); the source of the visual absorption value is given as a code in the last column, which is described in the Notes to the Table.
Table 1. Input parameters for the 28 stars with measured angular diameters
The next criterion deals with the amount of absorption (both interstellar and circumstellar) suffered by these stars. Various sources of are used, as detailed in the Notes to Table 1. According to the published values, 9 stars suffer more than 0.2 mag V of absorption, while 2 do not have an absorption determination. All but one are supergiants. Rejecting those stars drastically reduces our sample and, moreover, nearly eliminates all the supergiants. This is annoying, as the coefficients of the surface brightness relations may differ for giants and supergiants. We therefore adopt the following philosophy: for supergiants (luminosity classes I to II), we accept all stars and correct their magnitudes and colours for absorption, after discussing the choice of the reddening law. For giants (luminosity class III), we restrict our sample to those stars with a) a low absorption value ( mag), to avoid the dependence on an adopted reddening law (this eliminates only one star, namely HR 5340), and b) an accurate measure of diameter (, this eliminates only one additional star, namely HR 7951).
2.2. The giants surface brightness relations
The surface brightness (resp. ) is defined from the measured magnitude and angular diameter (in milliarcsec) as:
where the coefficient 4.2207 only depends on the bolometric absolute magnitude and the total integrated flux (solar constant) of the Sun, and on the Stefan-Boltzmann constant . Therefore, it does not depend on the chosen photometric band. It is given by:
As di Benedetto (1993) concludes that M giants follow a different relation than G to K giants, we a priori exclude the reddest giant, namely HR 5299 (M4 III). In spite of the low absorption values of the selected giants which make absorption corrections unimportant, we apply these corrections (as discussed in the following Sect. 2.3), for the sake of homogeneity. We then find the following dereddened relations for our giants sample, by linear least-squares fits, assuming all the errors in (resp. ):
with , , and .
with , , and . For comparison, di Benedetto (1993) gives for G0 to K5 giants.
with , , and .
Sources of visual absorption values: a: corresponds to Table 12 of di Benedetto (1993), original ref. e (di Benedetto & Ferluga 1990): absorption is computed from adopted distance and galactic latitude; we keep the original value, but without rounding it. b: corresponds to di Benedetto & Rabbia (1987), where absorption is computed from adopted distance (in kpc) and galactic latitude, according to:
For , this gives 1.4 mag per kpc. Other authors (Blackwell et al. 1990) adopt a smaller coefficient (0.8 or even 0.6 mag per kpc). For Boo, ref. d value is preferred. c: corresponds to the above method (ref. b), but computed by us (d and b are taken from the Bright Star Catalogue). For Gem, Table 12 of di Benedetto (1993) gives 0.1 under ref. d (Blackwell et al. 1990), but the original value is 0.00, because Blackwell et al. adopt 0.00 for pc; we therefore recompute the value from adopted distance and galactic latitude, which gives 0.08. d: corresponds to Dyck et al. (1996). e: corresponds to Table 12 of di Benedetto (1993), but original ref. is not given; adopted as such, except for Cyg, where ref. d value is preferred. f: from in Arellano Ferro & Parrao (1990), which gives .
2.3. Adopted reddening law
For the supergiants sample, a reddening law has to be adopted to convert values to , and . The agreement of published values for some of these ratios is not satisfying. For instance, in the case of M supergiants, the ratio is according to Lee (1970), but 0.84 according to Cardelli et al. (1989), using from Lee. We adopt an intermediate value of 0.97 from Hindsley & Bell (1989). For the mean value of Cepheids, namely 3.26 from Gieren & Fouqué (1993), this gives .
The agreement is better for the ratio: Lee gives for M supergiants, Cardelli et al. 3.17 for , and Laney & Stobie (1993) argue that for a large variety of grains, which leads to for . Adopting the Cardelli et al. law and for Cepheids gives .
For , Lee gives 0.59 for M supergiants, Cardelli et al. 0.64 for , while Laney & Stobie derive 0.485 for a zero-colour, zero-extinction star, which leads to 0.57 for . Again adopting the Cardelli et al. law with for Cepheids gives . In summary, we use for Cepheids:
2.4. The supergiants surface brightness relations
The following relations for supergiants, corrected for absorption as described above, are derived by linear least-squares fits, assuming all the errors in (resp. ):
with , , and . This equation is established on only 12 points, because of the rejection of discrepant data for HR 7735.
with , , and .
with , , and .
First, note that the supergiants relations are far less precise than the corresponding giants relations. Second, there is no evidence of different slopes for supergiants and giants, contrary to di Benedetto's claim (1993), who gives: .
In order to settle more definitely this question, let us compare these slopes from the giants and supergiants samples, with those directly determined from Cepheids. Indeed, short-period Cepheids are like type II supergiants, on the basis of their mean effective surface gravities, while long-period Cepheids are classified spectroscopically as type Ib, Iab, or - for the very longest periods - Ia.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998