On the relation between the mass-ratio distribution in binary stars and the mass function for single stars

Abstract

In the understanding of the formation of binary stars one valuable observational constraint is the distribution of mass ratios. Many attempts have been made to determine this elusive function but these have resulted in almost as many distributions. One possibility that must be considered is that both stars may be chosen independently from the same mass function. We derive the relationship between the mass-ratio distribution and the mass function and show how use of a Salpeter mass function has led to confusion. We also consider the expected distributions for spectroscopic binaries, both single- and double-lined, and explain some of the diversity of previous results while assuming independence and conclude that unknown selection effects severely limit the information that can be extracted from these samples. Complete samples, that avoid selection effects can be found and used to identify the finer details of the mass-ratio distribution.