It is currently common to use spatially unresolved multi-filter broad-band photometry to determine the masses of individual stellar clusters (and hence the cluster mass function, CMF). I analyze the stochastic effects introduced by the sampling of the stellar initial mass function (SIMF) in the derivation of the individual masses and the CMF and I establish that such effects are the largest contributor to the observational uncertainties. An analytical solution, valid in the limit where uncertainties are small, is provided to establish the range of cluster masses over which the CMF slope can be obtained with a given accuracy. The validity of the analytical solution is extended to higher mass uncertainties using Monte Carlo simulations and the Gamma approximation. The value of the Poisson mass is calculated for a large range of ages and a variety of filters for solar-metallicity clusters measured with single-filter photometry. A method that uses the code CHORIZOS is presented to simultaneously derive masses, ages, and extinctions. The classical method of using unweighted UBV photometry to simultaneously establish ages and extinctions of stellar clusters is found to be unreliable for clusters older than approx. 30 Ma, even for relatively large cluster masses. On the other hand, augmenting the filter set to include longer-wavelength filters and using weights for each filter increases the range of masses and ages that can be accurately measured with unresolved photometry. Nevertheless, a relatively large range of masses and ages is found to be dominated by SIMF sampling effects that render the observed masses useless, even when using UBVRIJHK photometry. A revision of some literature results affected by these effects is presented and possible solutions for future observations and analyses are suggested.
Deep Dive into Biases on initial mass function determinations. III. Cluster masses derived from unresolved photometry.
It is currently common to use spatially unresolved multi-filter broad-band photometry to determine the masses of individual stellar clusters (and hence the cluster mass function, CMF). I analyze the stochastic effects introduced by the sampling of the stellar initial mass function (SIMF) in the derivation of the individual masses and the CMF and I establish that such effects are the largest contributor to the observational uncertainties. An analytical solution, valid in the limit where uncertainties are small, is provided to establish the range of cluster masses over which the CMF slope can be obtained with a given accuracy. The validity of the analytical solution is extended to higher mass uncertainties using Monte Carlo simulations and the Gamma approximation. The value of the Poisson mass is calculated for a large range of ages and a variety of filters for solar-metallicity clusters measured with single-filter photometry. A method that uses the code CHORIZOS is presented to simultan
Subject headings: methods: analytical -methods: numerical -methods: statistical -open clusters and associations: general -globular clusters: general -galaxies: star clusters
This paper is the third one of a series where we explore the effects of different biases on the determination of the stellar and cluster mass functions (SMFs and CMFs, respectively). In paper I (Maíz Apellániz & Úbeda 2005) we analyzed the numerical biases induced by using bins of equal width when fitting power-laws to binned data (an effect that is more general than its application to the calculation of mass functions). Those biases can be eliminated in several ways, of which a simple one is by grouping the data in equal-number bins (as opposed to equal-width bins). In paper II (Maíz Apellániz 2008) I explored the effect of unresolved multiple systems, either physical or chance alignments, especially for the high-mass end of the stellar initial mass function (SIMF). In this paper I analyze the effect of random uncertainties in the mass determinations of individual stellar clusters and on the global properties of the obtained CMF as derived from spatially integrated (i.e. unresolved) photometry. I am currently working on the fourth paper of the series, which will explore the same issues as this one but referred to stellar instead of cluster masses.
The measurement of the masses of unresolved stellar clusters in external galaxies has become popular in the last decade, especially thanks to the availability of HST imaging (see Zhang & Fall 1999;Larsen 2002;de Grijs et al. 2005;Úbeda et al. 2007b;Dowell et al. 2008 for examples). Accurately measuring stellar cluster masses is crucial to understand their evolution and, more specifically, their destruction rates and mechanisms. This is usually done by analyzing a large ensemble of clusters within a galaxy and deriving the present-day CMFs as a function of age. Current results regarding how and at what rate stellar clusters are destroyed are inconclusive, with two alternative empirical models being proposed in the literature (Lamers 2008 and references therein).
One outstanding issue with the calculation of stellar cluster masses is SIMF sampling. In a series of papers, Miguel Cerviño and his collaborators (Cerviño & Luridiana 2006 and references therein) have shown that for clusters with less than a certain number of stars the SIMF is not well sampled and, as a consequence, there are added uncertainties in the derivation of cluster properties from unresolved photometric or spectroscopic data. Thus, two clusters of the same mass, age, and metallicity can have quite different integrated properties because of the differences in their initial stellar population caused by the stochastic nature of star formation1 . Undoubtedly, it is important to determine whether such effects are distorting the analysis of stellar cluster evolution by inducing biases in the observed CMFs.
The papers by Cerviño et al. deal mostly with the theoretical issues of SIMF sampling and provide a framework for its general study. The goal of this paper is more limited but at the same time more practical. It is more limited because I will concentrate on one observable, the cluster mass, and will deal with others (e.g. age) only as long as they influence the value of the observed mass. Furthermore, for the sake of simplicity I will restrict the analysis to solar-metallicity clusters. The practical character of this paper arises from its main aims, which are to provide [a] specific criteria to determine when cluster masses can be obtained, [b] methods for a more precise measurement of individual cluster masses, and [c] corrections to eliminate biases in both the individual masses and the CMF. In a sense, this paper can be understood as an unresolved-population equivalent to the resolved-stellar-population methods and tools used for extracting unbiased ages or star-formation histories from colormagnitude diagrams developed in the last decade (Harris & Zaritsky 2001;Dolphin 2002;Jørgensen & Lindegren 2005;Naylor & Jeffries 2006).
The paper is organized in the following way. I start by briefly describing the problem. Then, I select a simplified case (clusters with fixed single age, metallicity, and extinction) and derive an analytical solution for the effects of SIMF uncertainties on individual cluster masses and the CMF for the case where uncertainties are small and masses are derived from V -band photometry (an appendix provides the mathematical formalism). Later, I use Monte Carlo simulations to analyze arbitrary mass uncertainties and develop an approximation using the Gamma distribution to obtain the behavior of the observed mass distribution for a given arbitrary cluster mass. The simulations are then extended to other photometric bands and known ages. Finally, the cases where the age or the age and the extinction are unknown and have to be derived from the same multi-band photometry as the mass are considered, and a me
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