| Abstract
| - Context. Our understanding of stellar systems depends on the adopted interpretation of the initial mass function, IMF φ( m). Unfortunately, there is not a common interpretation of the IMF, which leads to different methodologies and diverging analysis of observational data. Aims. We study the correlation between the most massive star that a cluster would host, mmax, and its total mass into stars, ℳ, as an example where different views of the IMF lead to different results. Methods. We assume that the IMF is a probability distribution function and analyze the mmax − ℳ correlation within this context. We also examine the meaning of the equation used to derive a theoretical ℳ − \hbox{$\hat{m}_\mathrm{max}$}ℳ − m̂max relationship, \hbox{${\cal N} \times \int_{\hat{m}_\mathrm{max}}^{m_{\rm up}} \phi(m)\,\mathrm{d}m = 1$}𝒩×∫m̂maxmupφ(m)dm=1 with N the total number of stars in the system, according to different interpretations of the IMF. Results. We find that only a probabilistic interpretation of the IMF, where stellar masses are identically independent distributed random variables, provides a self-consistent result. Neither ℳ nor the total number of stars in the cluster, N, can be used as IMF scaling factors. In addition, \hbox{$\hat{m}_\mathrm{max}$}m̂max is a characteristic maximum stellar mass in the cluster, but not the actual maximum stellar mass. A ⟨ℳ⟩ − \hbox{$\hat{m}_\mathrm{max}$}⟨ℳ⟩ − m̂max correlation is a naturalresult of a probabilistic interpretation of the IMF; however, the distribution of observational data in the N (or ℳ) − mmax plane includes a dependence on the distribution of the total number of stars, N (and ℳ), in the system, Φ N( N), which is not usually taken into consideration. Conclusions. We conclude that a random sampling IMF is not in contradiction to a possible mmax − ℳ physical law. However, such a law cannot be obtained from IMF algebraic manipulation or included analytically in the IMF functional form. The possible physical information that would be obtained from the N (or ℳ) − mmax correlation is closely linked with the Φ ℳ(ℳ) and Φ N( N) distributions; hence it depends on the star formation process and the assumed definition of stellar cluster.
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