4. Luminosity function
In 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 K is Epanechnikov kernel
with a smoothing parameter and i running through the data sample. The smoothing parameter was computed by minimizing the mean integrated square errors for the smoothed luminosity function.
As we already noted above, the distribution of interstellar absorption is highly inhomogeneous in the cluster corona where a typical variation of absorption in the V-band is about 2 or 3 magnitudes. Therefore, even if the CC is complete down to a given apparent magnitude , the observed luminosity function suffers from the incompleteness within a range of absolute magnitudes :
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 a, we derive an "observed" luminosity function
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 .
The "observed" theoretical LF was computed according to
where and are minimum and maximum ages of the cluster stars, is the LF of stars with age t (), is the star formation rate (SFR) at age t. For we have:
where is the initial mass function. The mass- relation and its derivative were calculated along the isochrone of age t by use of a cubic spline interpolation.
We considered two representations of the IMF:
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 1.
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