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

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5. Radii of main-sequence M-dwarfs

In this section, we test our results on the 12 YD/OD stars of L96 which serve as calibrators and then apply them to a sample of 60 single or presumed single YD/OD main-sequence stars. The colour-magnitude diagram of these stars is shown in Fig. 3b. Six K-stars are from Reid & Gizis (1997) (stars) and 53 M-stars from Henry & McCarthy (1993) (small solid and open circles). The L-star GD165B represents the transition to the brown-dwarf regime (asterisk).

We consider the calibrator stars first. In Table 2, we compare the L96 radii with those obtained from the Barnes-Evans type relations [FORMULA][FORMULA] and [FORMULA]([FORMULA]) and from [FORMULA]:

  • Column 9: Radii of YD or [M/H][FORMULA] stars derived from the observed [FORMULA] with Eqs. (2) and (6). Radii of OD stars obtained correspondingly, but with the theoretical [FORMULA][FORMULA] relation for [M/H][FORMULA] of Table 1 instead of Eq. (6).

  • Column 10: Radii obtained from the observed [FORMULA] with the theoretical [FORMULA]([FORMULA]) relations of Table 1 for the [M/H] = 0 or the [M/H] = -0.5 stars.

  • Column 11: Radii derived from [FORMULA] with Eq. (7) without regard of metallicity, assuming the stars to be on the ZAMS.

The radii in Column 11 and the YD radii in Column 9 agree with the L96 radii within 5% or [FORMULA]log[FORMULA] which reflects the goodness of the fits. The small differences between the radii in Columns 10 and 8, on the other hand, demonstrate the close agreement between theory and observation.


Table 2. Radii of YD and OD dwarfs from L96. The columns indicate (1) the name, (2) the spectral type, (3) the kinematic population class, (4) the parallax in mas from L96, (5) [FORMULA], (6) [FORMULA], (7) the absolute K-magnitude, in case of binaries the mean of the two components, (8) the logarithm of the radius in units of [FORMULA] as derived observationally by L96, (9) the radius obtained from [FORMULA] in Column 5 with Eqs. (2) and (6) for the [M/H][FORMULA] stars and with [FORMULA][FORMULA] in Table 1 for the [M/H][FORMULA] stars, (10) the radius derived from [FORMULA] in Column 5 with [FORMULA]([FORMULA]) in Table 1 and Eqs. (2), (11) the radius from Eqs. (7), and (12) the radius given by Lacy (1977) corrected to the parallax in Column 3.
1) AB refers to the mean of the two binary components.
2) Following L96, the contribution by the faint secondary is neglected.

Application of the Barnes-Evans type relations to the complete sample of 60 K/M dwarfs requires at least a rough estimate of their metallicity. For this purpose, we divide the sample into a brighter and a fainter subsample, containing stars within [FORMULA] mag of the bright limit in [FORMULA] (small solid circles in Fig. 3b) and stars within [FORMULA] mag from the bright limit (small open circles), respectively. This subdivision can be interpreted in terms of different metallicity if age is not an influencing factor. Assuming all stars to be close to the ZAMS, the two subsamples correspond to stars of near-solar metallicity and a metallicity reduced by [FORMULA], respectively (e.g. L96). In deriving the radii, we proceed as above for the L96 stars. Table 3 provides the observed properties and the derived radii. The restriction to single stars is important in order to avoid the larger radii falsely assigned to unrecognized binaries.


Table 3. Radii of single or presumed single field M-dwarfs from the list of Henry & McCarthy (1993). The columns indicate (1) the name, (2) the spectral type, (3) the kinematic population class if available (4) the parallax, (5) [FORMULA], (6) [FORMULA], (7) the absolute K-magnitude, (8) the logarithm of the radius in units of [FORMULA] as derived from Eqs. (2) and (6), (9) same as derived from the theoretical [FORMULA]([FORMULA]) relation of BCAH98 for [M/H] = 0 in Fig. 2a, (10) same as derived from Eq. (7), and (11) the radius as given by Lacy (1977), adjusted to the parallax in Column 4.


Table 3. (continued)

Column 12 of Table 2 and Column 11 of Table 3 list the radii given by Lacy (1977) adjusted to the parallaxes used here. Compared with Lacy's results, our radii are smaller by up to [FORMULA]%. Fig. 4 shows that there is a systematic trend for the difference between Lacy's radii and those determined by L96 or, for the addtional stars, from our Barnes-Evans type relations [FORMULA][FORMULA]. Very similar pictures obtain for the radii derived from the Barnes-Evans type relation [FORMULA]([FORMULA]) (Table 3, Column 9) or directly from [FORMULA] (Table 3, Column 10, both not shown). The deviation of Lacy's from our radii assumes a maximum at spectral type K7 and vanishes for early K and for late M dwarfs. Much of this is due to the different surface brightness calibrations for giants (used by Lacy) and dwarfs (used here) which reach a maximum separation at [FORMULA] [FORMULA], [FORMULA] [FORMULA], or spectral class K7 (Fig. 1). The remainder is due to differences in the individual giant relations used by Lacy (Barnes & Evans 1976) and by us (Eq. [3] and Dumm & Schild 1997). Lacy discussed a deviation of similar magnitude and colour-dependence between the surface brightness of giants and the theoretical ZAMS dwarf models of Copeland et al. (1970). He interpreted it as being entirely due to inadequacies of the models. We now know that (i) the surface brightness of mid-K to mid-M dwarfs is, in fact, higher than that of giants of the same colour and (ii) the recent dwarf models (BCAH98) predict somewhat larger radii than the early models which reduces the discrepancy noted by Lacy (1977).

[FIGURE] Fig. 4. Differences in log(R/[FORMULA]) between the radii given by Lacy (1977) and here: (i) L96 YD/OD stars (Table 2, Column 8, large symbols as in Fig. 2) and (ii) stars in Table 3, radii derived from [FORMULA][FORMULA] (Column 8, small symbols as in Fig. 3).

Finally, we discuss the systematic errors in our radius calibration which is tied to the observational results of L96 and to the theoretical predictions of BCAH98. Our radius scale for stars of near-solar metallicity is based on the results of L96 which may still be in error by some [FORMULA]% or 0.03 in log R (radii too small, see Sect. 2). The difference of this radius scale to the BCAH [M/H] = 0 model is [FORMULA]%. Clemens et al. (1998), on the other hand, quote radii for stars later than M2 at given mass which are larger than those predicted by BCAH98 by [FORMULA]20%. Part of this discrepancy may be due to remaining uncertainties in the bolometric corrections (Sect. 2) and part to the transformations used by them. Clemens et al. adopt the [FORMULA] scale and the temperature scale of L96 to deduce radii which are higher by [FORMULA]% than those quoted by L96. They employ the mean observational [FORMULA](M) relation of Henry & McCarthy (1993) in order to convert absolute visual magnitudes [FORMULA] to masses M. For [FORMULA] [FORMULA], this relation presently relies on 10 stars only (Henry et al. 1999). It shows substantial scatter which may be caused by a spread in ages and metallicity and/or still by the inclusion of erroneous masses. The masses of the YD binary Wolf 424 ([FORMULA] and [FORMULA] [FORMULA]) were recently re-determined with the HST Fine Guidance Sensors (Torres et al. 1999) and agree perfectly with the predictions of BCAH98. We expect that a better definition of the mass-radius relation will become available soon (Henry et al. 1999) and allow the remaining discrepancies to be resolved.

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Online publication: July 26, 1999