## 4. Luminosity function## 4.1. Construction of the observed LFIn order to reveal details in the LF of NGC 6611, we used a more sophisticated technique than the usual histogram construction. A non-parametric smoothing of the distribution was carried out with the kernel density estimation method using the Epanechnikov kernel (see Silverman, 1986 for more details of the method). The luminosity function was computed as where with a smoothing parameter and
## 4.2. Correction for the incompleteness due to variable absorptionAs we already noted above, the distribution of interstellar
absorption is highly inhomogeneous in the cluster corona where a
typical variation of absorption in the where , are the minimum and maximum values of the absorption observed in NGC 6611 and is the true distance modulus. The CC is complete down to and no cluster members with are included in the CC. Below we show how this effect and the corresponding correction to the LF can be evaluated. Let be a distribution function of absorption values for cluster stars with a normalization If is a "true" cluster LF then the number of members we may expect within given ranges of absolute magnitude and of absorption will be stars. Integrating this expression over where, is an incompleteness factor, and This means that at , at , and within varies in accordance with the distribution which is derived from the absorption data in the CC. From Paper I the maximum spread of the absorption reaches almost 7 magnitudes in NGC 6611, while the "half-width" of the absorption distribution is 2.5 magnitudes. In other words, the observed LF will be influenced by this kind of incompleteness at its faint end (at least 2.5 magnitudes). Therefore, we must compare the observations not with the "true" theoretical LF but with the "observed" theoretical LF . ## 4.3. Theoretical luminosity functionThe "observed" theoretical LF was computed according to with where and
are minimum and maximum ages of the
cluster stars, is the LF of stars
with age where is the initial mass
function. The mass- relation
and its derivative were calculated
along the isochrone of age We considered two representations of the IMF: -
a power-law -
a log-normal law
where , and are parameters. We assumed a constant SFR for NGC 6611: The resulting parameters for NGC 6611 were drawn from the best
fit of the theoretical and observed
LFs In order to construct theoretical isochrones and LFs which include both Post-MS and Pre-MS stages for ages typical to that of NGC 6611 (1 to 10 Myr), we combined Population I Pre-MS evolutionary tracks of Palla & Stahler (1993) for model masses from 0.6 to 6 , and Maeder' group Post-MS calculations (Schaller et al., 1992) for . The grids were properly tuned to provide a continuous transition from Pre- to Post-MS ages at the same mass as well as smooth and uniform mass-luminosity and mass-radius relations along the ZAMS. The models were reduced to the Population I chemical abundance ()=(0.30,0.02). The isochrones were computed from the grid of tracks by use of linear interpolation. In order to convert the theoretical coordinates of the Hertzsprung-Russell Diagram (HRD) to the observed values , we used bolometric corrections and relations from Schmidt-Kaler's (1982) tables for the luminosity classes I, III and V. © European Southern Observatory (ESO) 2000 Online publication: June 20, 2000 |