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

¹ Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la Astronomía s/n, 18008 Granada, Spain
e-mail: mcs@iaa.es
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

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, 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 − 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) − 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.