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
Instituto de Astrofísica de Andalucía (IAA-CSIC),
Glorieta de la Astronomía s/n,
2 Instituto de Astrofísica de Canarias, c/ vía Láctea s/n, 38205 La Laguna, Tenerife, Spain
3 Instituto de Astronomía, Universidad Académica en Ensenada, Universidad Nacional Autónoma de México, Ensenada BC, 22860 Mexico, Mexico
4 Departamento de Astrofísica, Universidad de La Laguna (ULL), 38205 La Laguna,Tenerife, Spain
5 European Southern Observatory, Casilla 19001, Santiago 19, Chile
6 Max Planck Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
7 S. D. Astronomía y Geodesia, Fac. CC. Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Received: 28 April 2012
Accepted: 21 February 2013
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 − relationship, 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, is a characteristic maximum stellar mass in the cluster, but not the actual maximum stellar mass. A ⟨ℳ⟩ − correlation is a natural result 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, ΦN(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 Φℳ(ℳ) and ΦN(N) distributions; hence it depends on the star formation process and the assumed definition of stellar cluster.
Key words: stars: statistics / Galaxy: stellar content / methods: data analysis
© ESO, 2013