Study of the mass-ratio distribution of spectroscopic binaries. I. A novel algorithm

Abstract

We developed a novel direct algorithm to derive the mass-ratio distribution (MRD) of short-period binaries from an observed sample of single-lined spectroscopic binaries (SB1). The algorithm considers a class of parameterized MRDs and finds the set of parameters that best fits the observed sample. The algorithm consists of four parts. First, we define a new observable, the `modified mass function', that can be calculated for each binary in the sample. We show that the distribution of the modified mass function follows the shape of the underlying MRD, turning it more advantageous than the previously used mass function, reduced mass function or reduced mass function logarithm. Second, we derive the likelihood of the sample of modified mass functions to be observed given an assumed MRD. An MCMC search enables the algorithm to find the parameters that best fit the observations. Third, we suggest to express the unknown MRD by a linear combination of a basis of functions that spans the possible MRDs. We suggest two such bases. Fourth, we show how to account for the undetected systems that have an RV amplitude below a certain threshold. Without the correction, this observational bias suppresses the derived MRD for low mass ratios. Numerous simulations show that the algorithm works well with either of the two suggested bases. The four parts of the algorithm are independent, but the combination of the four turn the algorithm to be highly effective in deriving the MRD of the binary population.