Confidence limits of evolutionary synthesis models

Abstract

Context. 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). Aims. This paper presents a method of population synthesis that accounts for the distributed nature of stellar properties. Methods. 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). Results. With standard distribution theory, 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 luminosity function. 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: calibrating the SBF method, determining the luminosity function of globular clusters, comparing different isochrone sets, tracing the sLDF by means of resolved data, including fuzzy stellar properties in population synthesis, among others. Additionally, the algorithmic nature of our method makes it suitable for developing analysis tools for the Virtual Observatory. Conclusions. 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.

Using population synthesis of massive stars to study the interstellar medium near OB associations

“…Scalo 1986;Kroupa 2001) that also address the shape of the mass function at lower masses. We furthermore include an implementation of the statistical methods described in Cerviño & Luridiana (2006) to account for statistical variability of stellarmass weightings for smaller numbers of stars. This allows us to calculate analytically the statistical distribution of the output quantities, depending on the number of stars in the stellar population, similar to what can be done with Monte Carlo simulations.…”

“…The thin solid line was produced by using the atmosOLD atmosphere models, whereas the dashed line uses the LC06 stellar models and the atmosMS atmospheres. methods of Cerviño & Luridiana (2006). At a given time, these methods allow the analytical approximation of the probability distribution of a quantity.…”

“…With these we can estimate the distribution using a Gaussian multiplied by Edgeworth's series (see e.g. Blinnikov & Moessner 1998;Cerviño & Luridiana 2006), which, truncated to 2 terms, is:…”

“…Cerviño & Luridiana (2006) therefore derived analytical estimates of the necessary population size to allow the use of their approximation, and also when it is possible to use a Gaussian approximation. In Fig.…”

“…This is both done using analytical approximations and Monte Carlo simulations. To do this we implemented the statistical methods of Cerviño & Luridiana (2006). These enable us to determine when Monte Carlo simulations are mandatory in order to properly describe the distributions, and how many simulations are needed to obtain statistically robust results.…”