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.

Triage of astrometric binaries – how to find triple systems and dormant black hole secondaries in the Gaia orbits

“…We reformulate the analysis by defining a unit-less observational parameter 'Astrometric Mass-Ratio Function' (AMRF), A, of a binary, based on primary-mass estimation, in addition to the astrometric parameters. This is similar to the spectroscopic reduced mass function of Shahaf et al (2017). We show how one can use this parameter for studying the mass ratio of a binary, provided the primary is an MS star.…”

Study of the Mass-Ratio Distribution of Spectroscopic Binaries. II. The Boundaries of the Brown-Dwarf Desert as Seen with the APOGEE Spectroscopic Binaries

“…Instead, the resulting distribution suggests a linear increase in log q, implying a tendency towards low-q values. Figure 5 compares the modified mass function distribution (Shahaf et al 2017) both with a mass-ratio distribution of a desert-shape and a linear distribution in log q. The figure demonstrates the capacity of the modified mass function to follow the mass-ratio distribution in both cases.…”

“…To overcome this problem one needs to apply statistical tools that utilize the assumed spherical symmetry of the binary inclinations (e.g., Boffin et al 1993;Curé et al 2015; Van der Swaelmen et al 2017). One of these tools is the modified mass function of single-lined spectroscopic binaries (SB1s), suggested recently by Shahaf et al (2017). They showed that this parameter, when derived for a sample of SB1s, follows the underlying mass-ratio distribution.…”

“…Although left with a relatively small sample, we try in Section 3 to draw the shape and location of the BDD in the period-secondary-mass parameter plane. This is done by applying the Shahaf et al (2017) algorithm to derive the mass-ratio distribution of the sample in two period bins, in 1 APOGEE: http://www.sdss3.org/surveys/apogee.php order to determine the mass limits of the BDD, and by drawing either a wedged or trapezoidal shape for the boundaries of the desert. Section 4 shortly discusses the meaning of the BDD in terms of planetary and binary formation.…”

Energy optimization in binary star systems: explanation for equal mass members in close orbits

“…The mass of the primary can be independently estimated, so that the observations determine the minimum value of the mass ratio, qmin, defined as the value of q that corresponds to an edge-on orbit (i = π/2). In contrast, for double-lined spectroscopic binaries, the additional data available allow for an estimate of the inclination angle and hence q itself (see also Shahaf et al 2017 for a detailed discussion of the observational complexities). The observational data from the aforementioned sample are presented in Figure 2, which plots the estimated mass ratios versus the semimajor axes of the systems.…”