The observed () and fitted theoretical (, ) LFs are shown in Fig. 9. The smoothed observed LFs drawn with the solid lines were computed for two sets of data. First, the observed LF (called herewith restricted LF) was constructed with stars down to what corresponds to the completeness limit of the CC estimated in Paper I. The second data set for the LF construction includes all selected members and extends down to the limiting magnitude of the CC (full LF). Respectively, the best-fitted theoretical LFs () were computed for the restricted and full samples which are reproduced by the dotted and dotted-dashed curves in Fig. 9. Additionally, a "true" (incompleteness free) theoretical LF based on the same parameters as the restricted theoretical LF is drawn with the dashed line.
The "true" theoretical LF and "observed" theoretical LF for the restricted sample differ only due to the incompleteness factor which was derived from the data on absorption given in the CC. An excellent agreement at faint magnitudes () between and the observed restricted LF gives an evidence that the observations reflect the H-feature appearing at even though the data are not complete at the faint end (see also the discussion below).
In contrast, we consider another detail (a small dip) in the observed LF between and to be caused by statistical fluctuations due to the low number of stars in this magnitude range (see e.g. Fig. 10).
In Table 3 we present the cluster parameters derived from the fit of the restricted LF. From the comparison of the results obtained with the restricted and full samples, we found that the corresponding cluster parameters do not differ significantly. Therefore, we may consider the derived set of the cluster parameters in Table 3 to be unbiased with respect to the incompleteness of the data used. The accuracy of the parameters was evaluated numerically in the vicinity of the minimum of -distribution at the 99% significance level. We found that our estimates of the age parameters are accurate within 25% and the IMF parameters have mean errors of about 0.1. It should be stressed that these estimates refer to the accuracy of the fitting method only and do not take into account such important sources of systematic errors as sampling and dereddening errors, inadequacy of stellar models, and therefore, should be regarded as a lower limit for the mean errors of the derived parameters.
Table 3. NGC 6611 parameters from the LF fitting
In addition to the IMF parameters which are traditionally deduced from the LF, we were also able to infer a set of descriptors of the star formation history () in the cluster. These evolutionary parameters can be derived for young clusters from the comparison of their observed and theoretical LFs provided that an age-dependent detail (H-feature) can be identified in the cluster LF (Piskunov & Belikov 1996). The feature occurs due to the presence of pre-MS stars in young clusters and was found both in the luminosity functions published in the literature (Belikov & Piskunov 1997) and in the Pleiades LF already studied by the authors (Belikov et al. 1998, 1999a).
According to Fig. 9, the observed LF based on the restricted sample is affected by the variable absorption at and is insufficiently deep to describe completely the H-feature occurring in the theoretical LF () at . In contrast, the observed LF from the full sample is in agreement with the "true" theoretical LF down to . This can be considered as an evidence of the completeness of the full sample down to . We can conclude that the theoretical calibration for the derived cluster parameters is confirmed by the observations and that a significant decrease of the corresponding observed LF at is related to the fine structure of the LF and is not caused by an incompleteness of the data. However, the limit of the CC is not sufficient to outline the H-feature completely. This will require much deeper observations.
From the results given in Table 3, the LF analysis provides a reasonable age of cluster stars which is in good agreement both with independent estimates (from 2 to 6 Myr) by Sagar et al. (1986), Hillenbrand et al. (1993), Massey et al. (1995), De Winter et al. (1997) and with our CMD analysis described below. Also, a considerable age spread among the cluster stars (up to 5 - 6 Myr) derived by Massey et al. (1995) and De Winter et al. (1997) could be confirmed by our results, whereas Hillenbrand et al. (1993) found an age of 1 - 2 Myr for the bulk of their stars (both MS and Pre-MS). Let us now consider the IMF parameters derived from the LF fit. The slope of the log-normal mass function i.e. the logarithmic derivative of the IMF could be calculated as
The power-law approximation gives the IMF slope which is common for the whole mass range under study and comparable with for the log-normal IMF at . According to Table 3, the parameter b of a log-normal IMF has a value at the level of its accuracy. Therefore, we may conclude that a power-law approximation (i.e., the case ) is sufficient to represent the mass function of NGC 6611 within the studied mass range. The results can be directly compared with the IMF slopes published for NGC 6611.
During the last decade, the mass spectra of NGC 6611 stars have been studied by several authors. Due to the large distance of the cluster, the investigations were mainly restricted to massive stars. Analysing homogeneous photoelectric UBV data and proper motion membership, Sagar et al. (1986) constructed a mass spectrum of a few tens of cluster stars located in the cluster core. The masses were estimated with respect to the theoretical tracks. Within the range , the IMF slope was found to be if no mass loss was taken into account, and in the case of a moderate mass loss. Hillenbrand et al. (1993) used optical CCD photometry and spectroscopic data for the construction of IMF in NGC 6611. The masses were also derived with respect to the theoretical tracks. The slope of the IMF was determined for stars with masses above (44 stars) where the mass function could be considered as complete and well defined. The slope was derived to be for all 44 stars and for a sample of stars with the best photometric data. However, re-reducing the same data by using another mass calibration, different theoretical models as well as different scales of effective temperature and bolometric correction, Massey et al. (1995) derived a slope of in a mass range of [7, 75] . This result is similar to the value by Sagar et al. (1986) but differs considerably from the IMF slope suggested for field stars.
In contrast to the published results, we can support our conclusions with about one order of magnitude larger sample of cluster stars. Further, we could extend the investigations to a larger mass range and involve both MS- and PMS models. The slope we derived for IMF of the NGC 6611 cluster is steeper than that by Sagar et al. (1986) and Massey et al. (1995) but it confirms the results by Hillenbrand et al. (1993) and is comparable with the slope found by Scalo (1998) for the IMF of field stars.
Now let us discuss whether our conclusions on the star formation history (e.g., cluster age and age spread) drawn from the LF analysis can be supported by the cluster CMD. In Fig. 10 the diagram of NGC 6611 is plotted with superimposed isochrones for the corresponding ages (see Table 3). In order to get an idea on the stellar mass scale, Post-MS tracks are also plotted.
From Fig. 10, a considerable age spread can be concluded among the cluster stars. The and isochrones embrace the cluster stars, both around the turn-off point as well as in the vicinity of the turn-on point near . Although a fainter Pre-MS branch is not clearly outlined due to increasing incompleteness below , there is a considerable population of stars located above the main sequence in a wide range of absolute magnitudes (). These objects could be either stars with underestimated reddening or Pre-MS stars still contracting to the main sequence. In the last case, we should suggest a considerable (of order of several Myr) age spread in NGC 6611 and an existence of massive () Pre-MS stars at in the cluster. This statement is in agreement with Sagar et al. (1986), Hillenbrand et al. (1993), and de Winter et al.(1997) who also proposed a presence of massive Pre-MS stars in the cluster. In contrast to Hillenbrand et al. (1993) who found that a typical age of cluster stars is about 1 - 2 Myr, we believe that the cluster stars display the ages within the whole range . The question whether this spread results from local effects in star formation history which is probably different in different regions of the cluster or, on the other hand, the star formation history is uniform over the cluster needs an additional careful investigation and is proposed for a future study.
According to the CMD (Fig. 10), a sequential formation of cluster stars is not evident from the present data. Both the most massive MS stars and the Pre-MS stars demonstrate the whole range of ages from to .
© European Southern Observatory (ESO) 2000
Online publication: June 20, 2000