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, m max , 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 m max − M correlation within this context. We also examine the meaning of the equation used to derive a theoretical M −m max relationship, N × mup mmax φ(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,m max is a characteristic maximum stellar mass in the cluster, but not the actual maximum stellar mass. A M −m max correlation is a natural result of a probabilistic interpretation of the IMF; however, the distribution of observational data in the N (or M) − m max 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 m max − 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) − m max 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.
Context. In a probabilistic framework of the interpretation of the initial mass function (IMF), the IMF cannot be arbitrarily normalized to the total mass, M, or number of stars, N, of the system. Hence, the inference of M and N when partial information about the studied system is available must be revised (i.e., the contribution to the total quantity cannot be obtained by simple algebraic manipulations of the IMF). Aims. We study how to include constraints in the IMF to make inferences about different quantities characterizing stellar systems. It is expected that including any particular piece of information about a system would constrain the range of possible solutions. However, different pieces of information might be irrelevant depending on the quantity to be inferred. In this work we want to characterize the relevance of the priors in the possible inferences. Methods. Assuming that the IMF is a probability distribution function, we derive the sampling distributions of M and N of the system constrained to different types of information available. Results. We show that the value of M that would be inferred must be described as a probability distribution Φ M [M; m a , N a , Φ N (N)] that depends on the completeness limit of the data, m a , the number of stars observed down to this limit, N a , and the prior hypothesis made on the distribution of the total number of stars in clusters, Φ N (N).
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