Physical limits to the validity of synthesis models - The Lowest Luminosity Limit

In this paper we establish a necessary condition for the application of stellar population synthesis models to observed star clusters. Such a condition is expressed by the requirement that the total luminosity of the cluster modeled be larger than the contribution of the most luminous star included in the assumed isochrones, which is referred to as the Lowest Luminosity Limit (LLL). This limit is independent of the assumptions on the IMF and almost independent of the star formation history. We have obtained the Lowest Luminosity Limit for a wide range of ages (5 Myr to 20 Gyr) and metallicities (Z = 0 to Z = 0 : 019) from the Girardi et al. (2002) isochrones. Using the results of evolutionary synthesis models, we have also obtained the minimal cluster mass associated with the LLL, M-min, which is the mass value below which the observed colors are severely biased with respect to the predictions of synthesis models. We explore the relationship between M-min and the statistical properties of clusters, showing that the magnitudes of clusters with mass equal to M-min have a relative dispersion of 32% at least (i.e., 0.35 mag) in all the photometric bands considered; analogously, the magnitudes of clusters with mass larger than 10 x M-min have a relative dispersion of 10% at least. The dispersion is comparatively larger in the near infrared bands: in particular, M-min takes values between 10(4) and 10(5) M-circle dot for the K band, implying that severe sampling effects may affect the infrared emission of many observed stellar clusters. As an example of an application to observations, we show that in surveys that reach the Lowest Luminosity Limit the color distributions will be skewed toward the color with the smallest number of effective sources, which is usually the red, and that the skewness is a signature of the cluster mass distribution in the survey. We also apply our results to a sample of Globular Clusters, showing that they seem to be affected by sampling effects, a circumstance that could explain, at least partially, the bias of the observed colors with respect to the predictions of synthesis models. Finally, we extensively discuss the advantages and the drawbacks of our method: it is, on the one hand, a very simple criterion for the detection of severe sampling problems that bypasses the need for sophisticated statistical tools; on the other hand, it is not very sensitive, and it does not identify all the objects in which sampling effects are important and a statistical analysis is required. As such, it defines a condition necessary but not sufficient for the application of synthesis models to observed clusters.