# Fundamental parameters#

The best fit process aims at estimating the distributions for a set of the cluster’s fundamental parameters. The process basically consists in comparing the observed cluster’s photometric diagram with the diagrams of many synthetic clusters generated with known parameters’ values.

The process by which synthetic star clusters (SSC) are generated by ASteCA involves several steps, described in the sections that follow.

The result of these steps is a SSC with given metallicity, age, extinction, distance, mass and binarity values, affected by a maximum magnitude cut, completeness star removal and photometric errors mimicking those of the input observed cluster.

# IMF sampling#

The initial mass function (IMF) is the distribution of initial masses for a population of stars. For a population of $$N$$ stars with masses $$m_i$$ and a total mass of $$M_T$$:

$\begin{split}IMF \, &\rightarrow \, \xi(m)=\frac{dn}{dm} \, \rightarrow \,dn = \xi(m)dm \\ M_T &= \sum_{i=1}^N m_i \, \rightarrow \, M_T = C\int_{m_l}^{m_h} m(n)dn = \\ &= C\int_{m_l}^{m_h} m\xi(m)dm\end{split}$

where $$m_l$$ and $$m_h$$ are the mass limits for the IMF ($$m_h$$ is fixed to $$100 M_{\odot}$$ in the code) and $$C$$ is a normalization constant. Setting the total mass to unity, $$M_T=1 M_{\odot}$$, allows us to obtain the normalization constant $$C_1$$ and treat the normalized IMF as a PDF:

$M_T=1M_{\odot}\, \rightarrow \, C_1 = \frac{1}{\int_{m_l}^{m_h} m\xi(m)dm}$

and thus the normalized IMF can be written as:

$PDF(m) = \xi(m)_{norm} = C_1 \xi(m)$

This is the first step, performed by the get-IMF-PDF function for a given selected IMF (Chabrier 2001, Kroupa et al. 1993, Kroupa 2002)

Once the PDF is generated, every time a new synthetic cluster is created the get-mass-dist function is called from within synth-cluster. This former function takes the PDF and samples a number of masses randomly from it, following the probabilities distribution given by the PDF, until the mass fixed by the total-mass parameter is achieved.

The get-mass-dist function thus returns a distribution of masses probabilistically sampled from a certain IMF, whose masses sum up to a total cluster mass.