Appendix A

Derivation of absolute magnitudes and colour indexes from theoretical evolutionary tracks

  In this Appendix we will briefly describe the method that we have used to transform the K-completeness absolute magnitude given in Table 1 into an estimate of the lowest mass detectable in each field, the so-called minimum mass. The discussion follows closely the method described in Meyer (1996).

Firstly, we need to convert the PMS evolutionary tracks of D'Antona & Mazzitelli (1994) from the theoretical HR diagram Log $L_\star$, Log $T_{\rm eff}$ into the observable one M_K, (V-K), where M_K is the absolute K magnitude. We have used the calibrations of Schmidt-Kaler (1981), Bessel (1979, 1991), Bessel & Bret (1988) and Koornneef (1983) for dwarf stars to compile tables of the bolometric corrections at V band (BC_V) and the (V-K) colour index as a function of the effective temperature ($T_{\rm eff}$). These relations are shown in Fig. 48. For each value of $L_\star$ and $T_{\rm eff}$, we can then compute M_K as: where 4.725 is the assumed absolute V magnitude of the Sun. The colour-magnitude (V-K, M_K) diagram resulting from the transformation of the D'Antona & Mazzitelli (1994) "CM Alexander'' (their Fig. 3) tracks is shown in Fig. 50.

 

Figure 50:  Colour-magnitude diagram with the D'Antona & Mazzitelli (1994) tracks. The dotted lines show the evolutionary tracks for 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.7, 1.0, 1.2, 1.5, 2.0, and $2.5~M_\odot$, respectively. Continuous lines show the isochrones at 0.1, 1, 10, and 100 Myr

 

Figure 51:  Top panel: absolute K magnitude as a function of time for $M_\star/M_\odot=2.5$, 2.0, 1.5, 1.2, 0.9, 0.7, 0.5, 0.3, 0.2, 0.1, respectively (from bottom to top); bottom panel: $M_\star$ versus MK for isochrones in the interval 0.1-10 Myr. Given the K absolute magnitude and the age of a star it is possible to derive the corresponding mass, using the appropriate isochrone

We can now derive for each stellar mass the run of the K absolute magnitude with time. This is shown in the upper panel of Fig. 51 for masses in the interval 2.5 - 0.1 $M_\odot$. The peak in M_K that appears for the more massive stars at Log(age) $\sim\, 6.2-7.4$is due to the transition from the convective to the radiative section of the evolutionary tracks (cf. Fig. 50 and Fig. 3 of D'Antona & Mazzitelli 1994). We can see that in the range of ages (t<10 Myr) and minimum masses considered in this paper (see Tables 1 and 2), the M_K of a star of a given mass is a monotonically increasing function of time. For this reason, given a K absolute completness magnitude, the minimum mass detectable is a function of time: as the age of the cluster increases we loose sensitivity on the lowest mass members. In graphical form this is presented in the lower panel of Fig. 51, where the masses corresponding to M_K are plotted for isochrones between 0.1 to 10 Myr.

We have used this last figure to derive the minimum mass in each field from the de-reddened K limiting magnitude $M_K^{\rm c}-A_K$ and the age of the Herbig AeBe star for two values of A_K=0 and 2 mag.