Determination of the mass-ratio distribution, I: Single-lined spectroscopic binary stars

Abstract

For single-lined spectroscopic binary stars (SBI), the mass ratioq=M _sec/M _prim is calculated from the mass functionf(m), which is determined from observations. For statistical investigations of the mass-ratio distribution, the term sin³ i, that remains in the cubic equation from whichq is solved, has to be dealt with.

This paper compares the common practise of taking an average value for sin³ i to a deconvolution scheme that takes into account the precise (expected) behaviour ofP (sin³ i) d sin³ i. The behaviour ofP(sin³ i) d sin³ i depends on how orbital planes of binary stars are oriented in space. For a random orientation of orbital planes,P _i(i) di=sini di. For the average value method, it is generally assumed thatP _i(i) di=(4/π) sin² i di.

For verification purposes, the deconvolution scheme is applied to an observed sample of double-lined spectroscopic binary stars (SBII), and to a synthetic sample of SBI systems, produced by a numerical model. In both cases, the scheme produces better results with the assumption thatP _i(i) di=(4/gp) sin² i di, rather than a purely random orientation of orbital planes. In the case of the synthetic sample of SBI systems, the deconvolution scheme does not produce better results than the method that assumes an average value for sin³ i.

Application of the deconvolution method to the double-lined spectroscopic binary systems in theEighth Catalogue of the Orbital Elements of Spectroscopic Binary Stars, provides results which are compatible with the assumption that the orbital planes of these systems are oriented randomly in space.