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Astron. Astrophys. 348, 524-532 (1999)

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2. The Barnes-Evans relations for giants and dwarfs

Barnes & Evans (1976) defined the surface brightness as 1


where [FORMULA] is the apparent magnitude at wavelength [FORMULA] and [FORMULA] is the angular diameter of the star in mas. With the absolute magnitude [FORMULA] and the radius R of the star, Eq. (1) can be cast into the form (e.g. Bailey 1981)


Fig. 1 compares the visual surface brightness vs. Cousins [FORMULA] relationships for late-type giants and dwarfs. The giant sample includes 28 stars from Dumm & Schild (1998) with spectral types M0 to M7 ([FORMULA][FORMULA][FORMULA]) and nine non-variable K-giants from Dyck et al. (1998). Most of the stars are of luminosity class III with a few of class II. The V magnitudes and Cousins [FORMULA] values were taken from the on-line version of the HIPPARCOS catalogue (entries H5 and H40), the angular diameters are from Dumm & Schild (1998) and Dyck et al. (1998, and private communication). Most of the giants in the sample are located within the local bubble where the density of atomic hydrogen is low (Diamond et al. 1995, Thomas & Beuermann 1998). Inside the bubble, reddening is usually small and the standard reddening corrections based on distance and latitude (e.g. Fouqué & Gieren 1997) are of limited use. We opted not to apply reddening corrections, therefore. There may be a problem, however, with circumstellar absorption in some of the stars of latest spectral type (see e.g. the comments in Dumm & Schild 1998). We excluded one star from the giant sample, [FORMULA] Ser, which leaves us with 27 stars. For illustrative purposes, we show in Fig. 1 the reddening path for a standard interstellar absorption of [FORMULA] with [FORMULA] (Schlegel et al. 1998), but expect that none of the giants shows this much absorption. Since most of the M-giants are more or less variable, errors due to the non-simultaneity of the brightness and angular diameter measurements probably dominate the observed scatter in [FORMULA]. The error bars are, therefore, determined by quadratically adding the variability amplitude (1/2 of the difference in fields H50 and H49 in the HIPPARCOS catalogue), a minimum uncertainty in the V-magnitude, taken to be 0.02 mag, and the error in 5 log[FORMULA]. The standard error in [FORMULA] is taken from entry H41 of the HIPPARCOS catalogue. A linear fit for the 27 M-giants yields


which agrees with the fit of Dumm & Schild (1998, their Eq. [5]) within the 1[FORMULA] errors. The scatter in the giant data is a source of concern for the dwarf-giant comparison. We can not exclude that this comparison is affected by remaining systematic errors in the giant data, like selection effects, variability, and circumstellar reddening, which are difficult to estimate. It is comforting, though, that Eq. (3) is consistent with the red section of the original relation of Barnes & Evans (1976) which was given as a function of Johnson [FORMULA]. Our result is also consistent with the [FORMULA] vs. [FORMULA] relations for Cepheids, field giants and field supergiants discussed by Fouqué & Gieren (1997). Using SIMBAD K-magnitudes for 16 of our M-giants, we obtain


which is valid for [FORMULA] and agrees with the red part of the Fouqué & Gieren relation (see their Fig. 3).

[FIGURE] Fig. 1. Visual surface brightness [FORMULA] vs. Cousins [FORMULA] for dwarfs and giants. Dwarf data are for eight YD field stars from L96 ([FORMULA] ), the mean of the two components of the eclipsing binary YY Gem ([FORMULA]), and the Sun ([FORMULA]). Giant data are for 28 M-stars and 9 K-stars (+). Dashed and solid lines represent the fits of Eqs. (3) and (5). Error bars are included for the giant data. For the dwarfs, the error bars equal the size of the symbols. For illustrative purposes, the reddening path for [FORMULA] is indicated on the right. Also shown are the theoretical curves for dwarfs (solid curve) and for giants (dashed curve), displaced downward by two units for clarity (see text for the differences in slope).

For K-giants ([FORMULA] [FORMULA]), the relation in Fig. 1 steepens. This change of slope was already noted by Barnes & Evans (1976) and Barnes et al. (1977) who also showed that there is no difference in the surface brightness of giants and dwarfs for stars of spectral types B-G and found the Sun to fall on the giant relation. Tying the fit to the Sun at [FORMULA] = 0.70, [FORMULA] = 4.85, we obtain [FORMULA] = 2.86 + 2.84([FORMULA]) for [FORMULA][FORMULA]. All Barnes-Evans type relations for giants suggest that the slope of the relation changes near the transition from K to M-stars. In summary, the results for giants presented here coincide closely with those of other authors.

The dwarf data in Fig. 1 include eight young disk (YD) M-dwarfs from L96 and the YD eclipsing binary YY Gem of spectral type M1+M1. The angular diameters and effective temperatures [FORMULA] were derived by L96 from flux fitting of the Hauschildt et al. (1999) NextGen model atmospheres to the observed low-resolution optical/IR spectra. L96 consider the angular diameters obtained by this approach to be less fallible to errors in [FORMULA] than those derived from the observational bolometric magnitude and [FORMULA]. They quote an error of 2% in the derived angular diameters. An additional uncertainty in the radii may, however, arise from errors in the flux fitting procedure. The integrated luminosities of the best-fit models quoted by L96 tend to fall below the observed luminosity, i.e. [FORMULA][FORMULA]) + [FORMULA] [FORMULA] [FORMULA], with R and [FORMULA] as derived by L96 and the [FORMULA] sign implying "fainter than". For the eight L96 YD stars, the flux deficiency averages [FORMULA]0.15 mag relative to [FORMULA] = [FORMULA] + [FORMULA], with [FORMULA] the visual bolometric correction as given by L96 (their Table 6), and [FORMULA]0.10 mag relative to [FORMULA] = [FORMULA] + [FORMULA], with the bolometric correction in the K-band from Tinney et al. (1993). While it is well known that bolometric magnitudes are uncertain by as much as 0.10 mag (see e.g. the discussion in L96 and their Fig. 7), the amount of the flux deficiency suggests that either the temperatures of L96 are low by [FORMULA]%, their radii by [FORMULA]%, or both by correspondingly smaller percentages. Accepting the radii implies that the temperatures of the eight YD stars are on the average too low by 90 K which is well within the temperature errors, whereas accepting the temperatures leads to angular diameters (not radii which include the parallax errors) outside the quoted range. We assume, therefore, that the radii are basically correct and proceed to use them for calibrating our Barnes-Evans relation for dwarfs. The possible [FORMULA]% systematic uncertainty in the angular diameters corresponds to an uncertainty of 0.13 in [FORMULA] which may affect the absolute level of [FORMULA] but probably not the slope of the [FORMULA]-([FORMULA]) relation. The L96 YD dwarfs are well observed with an error in [FORMULA] less than [FORMULA]. Both error bars, in [FORMULA] and [FORMULA], are of the size of the dwarf data points in Fig. 1.

Finally, the mean angular diameter of the two components of YY Gem was calculated from their observed radii (Leung & Schneider 1978) and the HIPPARCOS parallax of 63.2mas. Note that there is no systematic difference between the surface brightness of YY Gem and that established for YD dwarfs by the radius scale of L96.

A linear fit to the data in Fig. 1, i.e. the eight YD dwarfs from L96 and to YY Gem, yields


which is valid for [FORMULA][FORMULA]. Comparison of Eqs. (3) and (5) shows that there is a difference in slope of [FORMULA] which is significant at the [FORMULA] level. The absolute levels of the giant and dwarf fits suggest that the visual surface brightness of early M-dwarfs is slightly higher than that of M-giants, reaching a separation of [FORMULA] mag at spectral type K7/M0 ([FORMULA] =1.65). Since there are no equally reliable radii for K-dwarfs, the extension to [FORMULA][FORMULA] is not covered. There is no difference in the surface brightness of giants and dwarfs for stars of spectral types B-G (Barnes & Evans 1976, Barnes et al. 1977), however, which suggests that the observational dwarf relation in Fig. 1 should connect to the Sun, that the dwarf/giant difference reaches a maximum for late K and early M stars, and that the break in the [FORMULA]([FORMULA]) relation near [FORMULA][FORMULA] is less pronounced for dwarfs than for giants. This observation suggests the presence of gravity effects in the surface brightness vs. colour relation.

Fig. 1 also compares the observational results with the predictions of recent theoretical work for late-type giants (Alibert et al. 1999, Hauschildt et al. in preparation) and late-type dwarfs (BCAH98), both of solar composition. The [FORMULA]([FORMULA]) relationship predicted for solar-metallicity ZAMS dwarfs with masses from 1.2 [FORMULA] down to 0.075 [FORMULA] is shown as solid curve (for clarity shifted downward by two units). The corresponding relationship for giants (dashed curve) is represented by the post main sequence evolutionary track of a 12 [FORMULA] star, evolved until central carbon ignition. Details of the calculations can be found in the recent work of Alibert et al. (1999) on Cepheids which shows a generally good agreement between models and recent observations in period-magnitude and period-radius diagrams. We note that tracks from 4 [FORMULA] to 12 [FORMULA] are very similar in the [FORMULA] vs. [FORMULA] diagram. This is in agreement with the results of Fouqué & Gieren (1997), who find that giant and supergiant surface brightness relations are indistinguishable. Therefore, the 12 [FORMULA] track shown in Fig. 1 is representative of the [FORMULA]([FORMULA]) relationship expected for giant and supergiants, with gravities log [FORMULA] and effective temperatures [FORMULA] = [FORMULA] K.

The differences in the slopes and normalizations of Eqs. (3) and (5) as well as the different strengths of the break for giants and dwarfs at the K/M transition ([FORMULA] = 1.65) agree quantitatively with those predicted by the stellar models. For the range in [FORMULA] covered here, the theoretical [FORMULA]([FORMULA]) relationships for giants and dwarfs reach a maximum separation of [FORMULA][FORMULA] = 0.35 mag at [FORMULA] = 1.65, very similar to the observed difference of 0.30 mag from Eqs. (3) and (5). Since the giant [FORMULA] is based on observed angular diameters and we consider the theoretical prediction of the difference in [FORMULA] reliable, the dwarf [FORMULA] and, hence, the dwarf radius scale can not be seriously in error either.

For an early M-star of given [FORMULA], a difference in [FORMULA] for giants and dwarfs of [FORMULA][FORMULA] = 0.30 mag corresponds to a difference in radius of 15%. I.e., if the giant calibration were used for dwarfs, the derived radii at [FORMULA] = 1.65 (spectral type K7) would be 15% too large.

For later spectral types, the difference becomes smaller and even reverses sign near [FORMULA][FORMULA], or spectral type M4.5. Models for giants of still later spectral type ([FORMULA] [FORMULA] K) are not yet available and the comparison between obervation and theory is restricted to spectral types earlier than M4. A quantitative comparison of the absolute values of [FORMULA] is limited by the fact that the theoretical V-magnitudes of M-stars calculated with the most recent models are still somewhat too bright and the colours involving V too blue (see also Fig. 2b which shows the corresponding effect in the colour [FORMULA]). This remaining error in the theory is suspected to be due to uncertainties in the molecular absorption coefficients for solar-metallicity stars in the visual passband (BCAH98). It causes the slopes of the [FORMULA]([FORMULA]) relations of both M-giants and M-dwarfs at [FORMULA][FORMULA] to be too steep, but should, to a first approximation, not affect the ordinate difference of the two curves which measures the gravity effect. Since the error in the theoretical V-magnitudes of late-type stars with near-solar metallicity reaches about 0.6 mag (see Fig. 3b) the corrected theoretical crossover between the [FORMULA]([FORMULA]) relations of giants and dwarfs occurs at [FORMULA][FORMULA] rather than [FORMULA][FORMULA], which is just as observed.

[FIGURE] Fig. 2. Barnes-Evans type relations [FORMULA] vs. [FORMULA] and [FORMULA] with the K-band magnitudes being on the CIT system. The data points indicate the L96 dwarfs: YD ([FORMULA] and [FORMULA]), OD ([FORMULA]), and H ([FORMULA]). The curves indicate the theoretical models of BCAH98 for ZAMS stars with solar metallicity [M/H]= 0 (solid curve), for solar-metallicity stars aged 0.1 Gyr (dotted), and for stars aged 10 Gyrs with metallicities [M/H] [FORMULA] (long dashes) and [M/H] [FORMULA] (short dashes). The dot-dashed line in the right-hand panel (b) represents the fit of Eq. (6) to the YD data.

[FIGURE] Fig. 3. a Top panel: Observationally determined radii of YY Gem and the L96 dwarfs (symbols as in Fig. 2) compared with the solar-metallicity ZAMS model radii of BCAH98 (solid curve). Also shown is the polynomial representation of the model curve by Eq. (7) (dot-dashed curve). b  Bottom panel: Colour-magnitude diagram [FORMULA] vs. [FORMULA]. The stars from Henry & McCarthy (1993) are shown as small solid and open circles (see text), the stars from Reid & Gizis (1997) as stars and GD165B as an asterisk. The dot-dashed curve represents the location of ZAMS stars with near-solar metallicity according to Eq. (8).

While our results show that a slight gravity effect in the [FORMULA]([FORMULA]) relation is present, the calibrations of Eqs. (3) and (5) are never further apart than 0.3 mag. This approximate equality is the observational and theoretical basis for Lacy's (1977) approach to determine the radii of late-type dwarfs from the Barnes-Evans relation for giants.

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Online publication: July 26, 1999