On the distribution of initial masses of stellar clusters inferred from synthesis models

The fundamental properties of stellar clusters, such as the age or the total initial mass in stars, are often inferred from population synthesis models. The predicted properties are then used to constrain the physical mechanisms involved in the formation of such clusters in a variety of environments. Population synthesis models cannot, however, be applied blindy to such systems. We show that synthesis models cannot be used in the usual straightforward way to small-mass clusters (say, M < few times 10**4 Mo). The reason is that the basic hypothesis underlying population synthesis (a fixed proportionality between the number of stars in the different evolutionary phases) is not fulfilled in these clusters due to their small number of stars. This incomplete sampling of the stellar mass function results in a non-gaussian distribution of the mass-luminosity ratio for clusters that share the same evolutionary conditions (age, metallicity and initial stellar mass distribution function). We review some tests that can be carried out a priori to check whether a given cluster can be analysed with the fully-sampled standard population synthesis models, or, on the contrary, a probabilistic framework must be used. This leads to a re-assessment in the estimation of the low-mass tail in the distribution function of initial masses of stellar clusters.Comment: 5 pages, 1 figure, to appear in ``Young Massive Star Clusters - Initial Conditions and Environments'', 2008, Astrophysics & Space Science, eds. E. Perez, R. de Grijs, R. M. Gonzalez Delgad

On surface brightness fluctuations: probabilistic and statistical bases I: Stellar population and theoretical SBF

This work aims to provide a theoretical formulation of Surface Brightness Fluctuations (SBF) in the framework of probabilistic synthesis models, and to distinguish between the different distributions involved in the SBF definition. RESULTS: We propose three definitions of SBF: (i) stellar population SBF, which can be computed from synthesis models and provide an intrinsic metric of fit for stellar population studies; (ii) theoretical SBF, which include the stellar population SBF plus an additional term that takes into account the distribution of the number of stars per resolution element psi(N); theoretical SBF coincide with Tonry & Schneider (1998) definition in the very particular case that psi(N) is assumed to be a Poisson distribution. However, the Poisson contribution to theoretical SBF is around 0.1% of the contribution due to the stellar population SBF, so there is no justification to include any reference to Poisson statistics in the SBF definition; (iii) observational SBF, which are those obtained in observations that are distributed around the theoretical SBF. Finally, we show alternative ways to compute SBF and extend the application of stellar population SBF to defining a metric of fitting for standard stellar population studies. CONCLUSIONS: We demostrate that SBF are observational evidence of a probabilistic paradigm in population synthesis, where integrated luminosities have an intrinsic distributed nature, and they rule out the commonly assumed deterministic paradigm of stellar population modeling.Comment: A&A accepte

Confidence limits of evolutionary synthesis models. IV Moving forward to a probabilistic formulation

Synthesis models predict the integrated properties of stellar populations. Several problems exist in this field, mostly related to the fact that integrated properties are distributed. To date, this aspect has been either ignored (as in standard synthesis models, which are inherently deterministic) or interpreted phenomenologically (as in Monte Carlo simulations, which describe distributed properties rather than explain them). We approach population synthesis as a problem in probability theory, in which stellar luminosities are random variables extracted from the stellar luminosity distribution function (sLDF). We derive the population LDF (pLDF) for clusters of any size from the sLDF, obtaining the scale relations that link the sLDF to the pLDF. We recover the predictions of standard synthesis models, which are shown to compute the mean of the sLDF. We provide diagnostic diagrams and a simplified recipe for testing the statistical richness of observed clusters, thereby assessing whether standard synthesis models can be safely used or a statistical treatment is mandatory. We also recover the predictions of Monte Carlo simulations, with the additional bonus of being able to interpret them in mathematical and physical terms. We give examples of problems that can be addressed through our probabilistic formalism. Though still under development, ours is a powerful approach to population synthesis. In an era of resolved observations and pipelined analyses of large surveys, this paper is offered as a signpost in the field of stellar populations.Comment: Accepted by A&A. Substantially modified with respect to the 1st draft. 26 pages, 14 fig

On emission-line spectra obtained from evolutionary synthesis models I. Dispersion in the ionising flux and Lowest Luminosity Limits

(abriged) Stellar clusters with the same general physical properties (e.g., total mass, age, and star-formation mode) may have very different stellar mass spectra due to the incomplete sampling of the underlying mass function; such differences are especially relevant in the high-mass tail due to the smaller absolute number of massive stars. The dispersion in the number of massive stars also produces a dispersion in the properties of the corresponding ionising spectra. In this paper, we lay the bases for the future analysis of this effect by evaluating the dispersion in the ionising fluxes of synthetic spectra. As an important consequence, we found that the intensities of synthetic fluxes at different ionisation edges are strongly correlated, a fact suggesting that no additional dispersion will result from the inclusion of sampling effects in the analysis of diagnostic diagrams; this is true for HII regions on all scales. Additionally, we find convincing suggestions that the He II lines are strongly affected by sampling, and so cannot be used to constrain the evolutionary status of stellar clusters. We also establish the range of applicability of synthesis models set by the Lowest Luminosity Limit for the ionising flux, that is the lowest limit in cluster mass for which synthesis models can be applied to predict ionising spectra. This limit marks the boundary between the situations in which the ionising flux is better modeled with a single star as opposed to a star cluster; this boundary depends on the metallicity and age, ranging from 10^3 to more than 10^6 Mo. As a consequence, synthesis models should not be used to try to account for the properties of clusters with smaller masses.Comment: Replaced with accepted versio

Towards an integrative framework of brand country of origin recognition determinants : a cross-classified hierarchical model.

To propose a framework integrating the types and levels of the determinants of brand CO recognition and to provide evidence on Internet users’ brand CO recognition rates using a sample of multi-regional and global brands from a variety of product categories and countries. We integrate 'level-1' consumer and brand characteristics and 'level-2' product category and country effects in a single framework. Data obtained through an original on-line survey hosted by Yahoo provide the basis for the empirical analysis. Seven hypotheses are tested using a two-level cross-classified random-effect model (‘HCM2’) : (a) Education is positively related with brand CO recognition; (b) experience with brands is positively related with brand CO recognition; (c) integration between the consumer and the country of a foreign brand is positively related with brand CO recognition; (d) Internet users’ classification performance is significantly better for domestic than for foreign brands; (e) brand-name congruence with true brand origin is positively related with brand CO recognition; (f) brand equity explains brand CO recognition, and (g) product categories with higher consumer involvement enhance brand CO recognition. Brand CO recognition performance by Internet users is in line with classification performance rates reported in other studies dealing with well-know and global brands. The main limitation is the cross-sectional study design. The research implications suggest that scholars should consider level-2 product category and country characteristics in their models, and that the level of brand CO recognition must be understood as inherently associated to the kind of brands under study. Managers would benefit from considering product category and country aspects of their most valuable brands. Policy makers should encourage firms to promote a clear association between brands and countries (when these countries have a positive image) and discriminate between high and low involvement product categories. We contribute to the brand CO awareness literature by integrating consumer and brand characteristics in a theoretical model, and identifying level-2 product category features and CO effects previously disregarded in brand CO recognition frameworks. In addition, our study positively contrasts with previous research by providing empirical evidence on brand CO recognition from the largest set of global brands (109), countries of origin (19) and product categories (15) ever investigated.Brand awareness; Country of origin; Brand CO recognition; Cross-classified hierarchical model; International marketing;

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

(abriged) In this paper we establish a Lowest Luminosity Limit (LLL) for the use of synthesis codes. The limit is defined by the following statement: The total luminosity of the cluster modeled must be larger than the individual contribution of any of the stars included in the assumed isochrones. This limit is independent of the assumptions on the IMF and almost independent on the star formation history. We have obtained the LLL for a wide range of ages (5 Myr to 20 Gyr) and metallicities (Z=0 to Z=0.019) from Girardi et al. (2002) isochrones. Using the LLL and the results of evolutionary synthesis models we have also obtained the minimal cluster mass, Mmin, for which the results of synthesis models may suffer from a severe bias in the computation of colors. We show that the results of synthesis models for clusters with mass equal to Mmin have a relative dispersion about or larger than 32% (i.e. a dispersion of 0.35 mag) in the corresponding photometric bands. This effect is more important for near infrared bands (except in the case where Asymptotic Giant Branch stars dominate the luminosity in optical bands). In particular, Mmin takes values between 10^4 and 10^5 Mo for the K band. From the observational point of view, we show that in surveys that reach luminosities near the LLL, the color distributions will be skewed toward the color with lower effective sources (toward the red in general), and that the skewness is a signature of the distribution of initial cluster masses in the survey. We also apply the LLL to Globular Clusters showing that sampling effects are relevant in these clusters and that they can explain (at least partially) the bias of the observed colors with respect to the predictions of synthesis models.Comment: Replaced by accepted version: 14 pages, 6 Figures, A&A in pres

Quadratic forms and their theta series - infinitesimal aspects

We study the theta map which assigns to a real quadratic form its theta series. We introduce two invariants reflecting whether the differential of the theta map vanishes or is degenerate. We provide examples of lattices where this differential is zero. These invariants turn out to be modular forms for integral lattices. We illustrate this in the rank two case.Comment: 11 page

MASSCLEAN - MASSive CLuster Evolution and ANalysis Package - Description, Tests, and Results

MASSCLEAN is a new, sophisticated and robust stellar cluster image and photometry simulation package. This package is able to create color-magnitude diagrams and standard FITS images in any of the traditional optical and near-infrared bands based on cluster characteristics input by the user, including but not limited to distance, age, mass, radius and extinction. At the limit of very distant, unresolved clusters, we have checked the integrated colors created in MASSCLEAN against those from other simple stellar population (SSP) models with consistent results. Because the algorithm populates the cluster with a discrete number of tenable stars, it can be used as part of a Monte Carlo Method to derive the probabilistic range of characteristics (integrated colors, for example) consistent with a given cluster mass and age. We present the first ever mass dependent integrated colors as a function of age, derived from over 100,000 Monte Carlo runs, which can be used to improve the current age determination methods for stellar clusters.Comment: 4 pages, 5 figures, Proceedings International Astronomical Union Symposium No. 266, Star Clusters - Basic Galactic Building Blocks throughout Time and Space, Editors: Richard de Grijs & Jacques R. D. Lepin