Statistical Approaches to Investigating Star Formation

We know that many stars form in binary systems and their properties provide us with some understanding of how stars form. Binary stars are the only means of directly determining stellar mass. The mass of a star is an important factor for understanding the formation, evolution and observable characteristics of stars. The first study seeks to estimate the fraction of binary systems in stellar clusters with different underlying densities and morphologies. A nearest neighbour method is developed that retrieves close to the true binary fraction when tested against Monte Carlo simulations of star clusters. In Study 2, attention is shifted to the statistics of the Initial Mass Functions (IMF), a fitted distribution to observed stellar mass data. Many modern studies of the IMF focus on a particular functional form and the parametric fits of those forms. In this chapter, the implications of the parametric fits are examined in more detail. It is shown that the combination of small sample sizes, heavy-tailed distributions, and the use of parametric methods together lead to an inability to detect quite significant differences in the underlying IMF, even when those differences are explicitly modelled and simulated. It is shown that alternative non-parametric methods can do a better job of allowing us to detect key properties of the IMF. Study 3 reconsiders visual binaries, but now considers the impact of observational selection effects. For visual binaries, an instantaneous observation provides us with their projected separation and the brightness of the member stars. However, they can only be detected if they pass observational selection effects i.e. the projected separation between member stars is wide enough with respect to the luminosity difference. This observational selection effect makes it difficult for us to uncover the true properties of the binary systems in a cluster. In Study 3, a method is developed to compare sample distributions to simulated data to recover true properties in the absence of selection effects. However, while this method is quite successful without selection effects, it is shown that as selection effects grow stronger the ``shape'' of the sample distribution changes and affects our results. This helps to illustrate some fundamental limitations in distribution comparison methods. In the end, the analysis is extended to show how improvements in measurement techniques such as Adaptive Optics and interferometry mitigate selection effects just enough to drastically improve the accuracy of this method.