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

Abstract

For single-lined spectroscopic binary stars (sbi), the mass ratio q = Msec=Mprim is calculated from the mass function f(m), which is determined from observations. For statistical investigations of the mass-ratio distribution, the term sin^3 i, that remains in the cubic equation from which q is solved, has to be dealt with. This paper compares the common practise of taking an average value for sin^3 i to a deconvolution scheme that takes into account the precise (expected) behaviour of P(sin^3 i)d sin^3 i. The behaviour of P(sin^3 i)d sin^3 i depends on how orbital planes of binary stars are oriented in space. For a random orientation of orbital planes, Pi(i)di = sini di. For the average value method, it is generally assumed that Pi(i)di = (4/pi) sin^2 i di. For verication purposes, the deconvolution scheme is applied to an observed sample of double-lined spectroscopic binary systems (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 that Pi(i)di = (4/pi) sin^2 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 sin^3 i. Application of the deconvolution method to the double-lined spectroscopic binary systems in the Eighth 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.