Crucial aspects of the initial mass function (I): The statistical correlation between the total mass of an ensemble of stars and its most massive star

Abstract: Our understanding of stellar systems depends on the adopted interpretation of the IMF, phi(m). Unfortunately, there is not a common interpretation of the IMF, which leads to different methodologies and diverging analysis of observational data.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. 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-char_mmax relationship, N x int[Char_mmax-mup] phi(m) dm = 1 with N the total number of stars in the system, according to different interpretations of the IMF. 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 N, can be used as IMF scaling factors. In addition, Char_mmax is a characteristic maximum stellar mass in the cluster, but not the actual maximum stellar mass. A <M>-Char_mmax correlation is a natural result of a probabilistic interpretation of the IMF; however, the distribution of observational data in the N (or M)-cmmax plane includes a dependence on the distribution of the total number of stars, N (and M), in the system, Phi(N), which is not usually taken into consideration. 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 Phi(M) and Phi(N) distributions; hence it depends on the star formation process and the assumed.