Crucial aspects of the initial mass function

II. The inference of total quantities from partial information on a cluster

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 European Southern Observatory, Casilla 19001, Santiago 19, Chile
⁵ Max Planck Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
⁶ Departamento de Astrofísica, Universidad de La Laguna (ULL), 38205, La Laguna, Tenerife, Spain
⁷ S. D. Astronomía y Geodesia, Fac. CC. Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

Received: 17 December 2012
Accepted: 21 February 2013

Abstract

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).