6 Critical comparison with stellar models

CU Cnc is only the third M-type eclipsing binary for which accurate absolute dimensions (masses and radii) have been determined. In addition, we have been able to set rather stringent constraints on the effective temperatures of the components as well as on the age and chemical composition of the system. With this information in hand, we are in a position to carry out a critical evaluation of the stellar model predictions. For the first time we can provide observational checks on M stars of about 0.4 $M_{\odot }$ for which no reliable empirical information has been available to date. The procedure followed here is very similar to that described in TR02 for YY Gem, but with a significant difference. While the two components of YY Gem are almost identical, this is not the case for CU Cnc where the components differ by almost 10% in mass. Since the two stars in the binary system are coeval, a single isochrone expected to fit both components simultaneously. This implies that not only the position in the observational diagrams but also the slopes of the isochrones come into play. Thus, the models must reproduce both the absolute and the relative location of the components.

For our comparison we have considered nine different sets of theoretical calculations: Swenson et al. (1994), D'Antona & Mazzitelli (1997), Siess et al. (1997), Baraffe et al. (1998), Palla & Stahler (1999), Charbonnel et al. (1999), Girardi et al. (2000), Yi et al. (2001), and Bergbusch & VandenBerg (2001). These include essentially all of the modern models for low mass stars, even though some of them do not quite reach down to the masses of CU Cnc. We have, nevertheless, incorporated them so that their performance at higher masses can be compared with the rest of the models. Isochrones for an age of 320 Myr and Z = 0.018 have been interpolated for all models except those by Palla & Stahler (1999), with an oldest age available of 100 Myr (this has no effect whatsoever on our conclusions). The models of Siess et al. (1997) and Baraffe et al. (1998) employ the most sophisticated physics (equation of state) and boundary conditions (non-grey model atmospheres), so, in principle, they would be expected to provide the best fit to the physical properties of CU Cnc.

Figure 4: The location of CU Cnc's components on several observational planes is compared with isochrones from nine different theoretical models (see legends and references in text). The age for the isochrones is 320 Myr and the metallicity adopted is Z=0.018, except for the isochrones by Palla & Stahler (1999) that are for an age of 100 Myr and solar abundance. In addition to the components of CU Cnc (represented with filled circles), we show the locations of YY Gem's mean component at a mass of 0.6 $M_{\odot }$ and the secondary component of V818 Tau with a mass of 0.76 $M_{\odot }$ (both represented as open circles).

We have compared the observational data for CU Cnc with the model predictions at different planes. The plots are shown in the eight panels of Fig. 4. For clarity reasons, we have plotted the models of Siess et al. (1997) and Baraffe et al. (1998) separated from the rest in the right panels. Note that we have also included the observational data on YY Gem AB and V818 Tau B (both from TR02), which will help to understand the comparison in a wider context. The best determined parameters of our analysis are the masses and the radii, which are purely empirical, whereas the effective temperatures and absolute magnitudes hinge to some extent on external calibrations. Thus, the mass-radius diagram in Fig. 4a constitutes the most reliable check on the models, also because the error bars of the measurements are very small. As can be seen, some of the models yield reasonably good, albeit not perfect, fits to the data. The three isochrones that exhibit the best performance are those by Swenson et al. (1994), Palla & Stahler (1999), and Yi et al. (2001). However, among these, only the models of Swenson et al. (1994) succeed in reproducing at the same time the relative location (i.e. slope) of the binary components. The models of Siess et al. (1997) and Baraffe et al. (1998), in spite of using state-of-the-art physical ingredients, do not yield a good fit to the data, although the latter perform slightly better. The radii that these models predict at the CU Cnc's masses are about 10-14% smaller than observed. (Note that this difference amounts to $\sim$7$\sigma$.) The plots indicate that the huge discrepancies between model predictions and observations encountered by TR02 at slightly higher masses (YY Gem AB and V818 Tau B) are not quite as dramatic at the mass regime of CU Cnc, and some of the models produce reasonable fits to the data.

The panels in Fig. 4b introduce the effective temperature in the comparison. Here the internal discrepancies among the seven models grow larger as we move towards lower masses. At around 0.4 $M_{\odot }$differences of up to 20% exist in the effective temperatures predicted by the different isochrones. Interestingly, most of the models seem to indicate temperatures that are significantly (10-15%) higher than observed for CU Cnc. The isochrones of Swenson et al. (1994) and Yi et al. (2001) clearly stand out among all and appear to produce a fairly close fit. But again, only the isochrone computed from the models by Swenson et al. (1994) reproduces the relative location of the components to achieve an outstanding fit to CU Cnc and also YY Gem, although V818 Tau B seems significantly cooler than model predictions.

The mass-luminosity diagram in Fig. 4c presents a situation which is analogous to that described above. Most of the isochrones predict M_V's for CU Cnc that are too bright by up to 2 mag. For example, the models of Siess et al. (1997) and Baraffe et al. (1998) yield close fits to both V818 Tau B and YY Gem while huge differences are found with the observed magnitudes of CU Cnc. The models by Swenson et al. (1994) are able to both reproduce the absolute magnitude of CU Cnc and YY Gem, while also achieving a quite close fit to V818 Tau B. At the same time, the slope of the isochrone matches perfectly the relative configuration of the CU Cnc components.

Finally, the H-R diagrams in Fig. 4d do not reserve any surprises. Most of the models tend to overestimate the effective temperature at a given absolute magnitude. The best agreement is found for the models of Swenson et al. (1994) that produce an isochrone with the correct slope that reproduces well the observed parameters of CU Cnc and YY Gem. Note that the relatively close fit by the models of Baraffe et al. (1998) is fictitious because it occurs for masses that are far off from those determined for CU Cnc.