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5 Completeness corrections

\includegraphics [width=8.8cm,clip]{}\end{figure} Figure 11: Completeness corrections as a function of absolute visual magnitude MV in six of the most cluster-rich galaxies in the sample of galaxies in Paper I. The upper panel shows the corrections applicable to point sources, while the lower panel is for extended objects with $R_{\rm e} = 20$ pc

After the discussion of software tools we now return to YMCs. In Paper I the total numbers of clusters detected in each of the 21 galaxies in our sample were listed, and we also attempted to correct the raw numbers for completeness effects. Here the completeness corrections will described in a bit more detail than in Paper I.

The fraction of clusters that are not detected may be estimated by adding artificial clusters to the images using a programme like mksynth and then checking how many of them are recovered by DAOFIND. This kind of completeness test is well documented in the literature for globular clusters in elliptical galaxies (Forbes 1996; Kissler-Patig et al. 1996) and also in other contexts. The artificial objects are usually added at random positions, which is a reasonable thing to do because this is also how the actual sources under study are distributed.

However, in our case the problem is somewhat different because YMCs are not distributed at random within their host galaxies. Instead, the clusters (in particular those in the "blue'' group, i.e. the youngest ones) tend to be located in or near the spiral arms where they are much more likely to drown in background fluctuations, and hence a completeness test based on objects distributed at random is likely to underestimate the completeness correction.

Our approach has been to add the artificial objects "close'' to the detected objects, adding 5 clusters at random positions within a radial range of 20-70 pixels from each detected cluster. The objects were added to the V, B and U band frames using the mksynth task, and it was then required that they were refound by daofind and that phot was able to obtain magnitudes in all three filters. The completeness tests were carried out for magnitudes down to the MV = -8.5 limit at intervals of 0.5 magnitudes, and a completeness correction was then calculated for each magnitude interval. The procedure was repeated twice, for point sources and for extended objects modeled as a MOFFAT15 profile with an effective radius corresponding to $R_{\rm e} = 20$ pc convolved with the PSF. Finally, the number of clusters actually detected in each magnitude interval was corrected by the completeness correction calculated for that interval, and the corrected numbers are given in Paper I for the point-source as well for the extended-source corrections.

Completeness tests were carried out only for the galaxies with more than 20 clusters. Many of the cluster-poor galaxies contained only a handful or fewer clusters, and it makes little sense to attempt to apply completeness corrections to such small numbers.

The estimated completenesses for sources in the 6 galaxies with more than 20 clusters are shown graphically in Fig. 11 as a function of absolute visual magnitude. Note the striking difference between point sources (upper panel) and extended sources (lower panel), in particular in NGC 2997. The number of detections in NGC 2997 in the extended source experiment actually drops to 0 at MV = -9.0, which indicates that in NGC 2997 we simply cannot make a realistic estimate of how many objects the galaxy contains down to a limit of MV = -8.5. The other galaxies in our sample were either more nearby, or they were observed at the NOT rather than the 1.54 m telescope (and therefore with better image quality), so with the exception of two more galaxies (NGC 7424 and NGC 1493, see Paper I) the incompleteness corrections should be smaller than those for NGC 2997.

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