Binary Star Population with Common Proper Motion in Gaia DR2

Abstract—

We describe a homogeneous catalog compilation of common proper motion stars based on Gaia DR2. A preliminary list of all pairs of stars within the radius of 100 pc around the Sun with a separation less than a parsec was compiled. Also, a subset of comoving pairs, wide binary stars, was selected. The clusters and systems with multiplicity larger than 2 were excluded from consideration. The resulting catalog contains 10 358 pairs of stars. The catalog selectivity function was estimated by comparison with a set of randomly selected field stars and with a model sample obtained by population synthesis. The estimates of the star masses in the catalogued objects, both components of which belong to the main-sequence, show an excess of “twins”, composed by stars with similar masses. This excess decreases with increasing separation between components. It is shown that such an effect cannot be a consequence of the selectivity function only and does not appear in the model where star formation of similar masses is not artificially preferred. The article is based on the talk presented at the conference “Astrometry yesterday, today, tomorrow” (Sternberg Astronomical Institute of the Moscow State University, October 14–16, 2019).

THE ALGORITHM FOR COMPILING THE LIST OF STELLAR PAIRS

In this section, we describe the algorithm developed to create a list of Gaia DR2 source pairs that are close enough to each other in three-dimensional space $\alpha$, $\delta$, $\varpi$. Creation of a list of binaries implies pairwise operations on the stars, which will take $C{{N}^{2}}$ of computer time the multitude of $N$ stars. This will make such calculations too long for any significantly large star sample. Of course, it makes sense to compare the parameters of those stars only that have at least some chance of being in a bound pair. There is no need to spend time to be sure that two stars located at a distance of 50 pc from the Sun in opposite parts of the celestial sphere are not a physical couple. It is possible to introduce a criterion that will consider whether it makes sense to perform detailed calculations for the given pair of stars (for example, a criterion can be a large projected angular separation), and to determine whether this criterion is fulfilled, skipping further calculations if it is not satisfied. However, this criterion calculation carried out for all pairs of stars in the sample is still a challenge of complexity $C{{N}^{2}}$: backtracking from further calculations if this criterion is not met, then C is reduced.

It seems more efficient to create an algorithm that would make a list of potential pairs in the first approximation. This algorithm can be optimized as much as possible to reduce the sample in which complexity $C{{N}^{2}}$ calculations would be performed. With a list of candidates for pairs instead of a star list, calculations can be performed in a row rather than pairwise over all stars, according to the existing list, which is much faster. The creation of such algorithm was the first (and significant) part of this study.

Thus, at the first stage of the study, the goal was to transform the task $C{{N}^{2}}$ into a task CN. All stars were selected as candidates for pairs located in three-dimensional space at a distance less than 1 pc to each other. The distance was calculated based on the celestial coordinates of the components ${{\alpha }_{i}}$, ${{\delta }_{i}}$ and the estimates of the distances to them, calculated as the reciprocal of the parallax $\varpi$. Using such estimates is permissible due to the fact that, in the ensemble under study, there are sufficiently close stars with characteristic determination error of the parallax of Gaia DR2 $\tfrac{{{{\varepsilon }_{\varpi }}}}{\varpi } < 10\%$.

The next problem was to determine which stars are subject to pairwise comparison in order to verify that the proximity criterion is met. Let us look for star pairs that have a distance from the Sun at least not less than some fixed value d. Obviously, starting with some angular separation between two stars θ, their physical separation cannot be less than the assigned allowed maximum. By solving a simple geometric problem, this angular separation is equal to $\theta = 2\arcsin(p{\text{/}}2d)$, where p is the maximum physical separation, in our case—1 pc. Using this, in the sky, we can select rectangular in coordinates ($\alpha$, $\delta$) regions $({{\alpha }_{1}},\;{{\alpha }_{2}};\;{{\delta }_{1}},\;{{\delta }_{2}})$ (hereinafter, we will call them “sections”), the width along both axes will be equal to the above-mentioned maximum separation. Thus, we would know that a star at one side of such a section (for example, $\alpha > {{\alpha }_{2}}$) cannot be a pair to a star on the opposite side ($\alpha < {{\alpha }_{1}}$). It is important to note that later in the Section on compiling the binary list, the expression “cannot be a pair” is used for convenience in the sense “cannot be closer than 1 pc from each other”. Partitioning into similar sections gives us an opportunity to not consider pairs located at the distance exceeding the width of a -section. It is worth noting that the size of such a rectangular section over α will be larger than over δ: for large declinations, the same angular separation corresponds to a greater distance over right ascension: ${\text{|}}{{\alpha }_{2}} - {{\alpha }_{1}}{\text{|}} = \theta {\text{/}}\cos(\delta )$.

Partitioning into sections is done within large areas equal to 1/4 of a complete circle in right ascension and about twenty degrees in declination. Within one large area, the correction ${\text{|}}{{\alpha }_{2}} - {{\alpha }_{1}}{\text{|}} = \theta {\text{/}}\cos(\delta )$) is accepted as the largest possible (i.e., it is selected for the largest declination value within the given area). Thus, each large area has its own rectangular grid of sections.

Upon the creation of the described partition, we check all possible pairs within one large area and obtain a gain in the computational time. Thus, for any star, a pair can be found either in the same region or in the neighboring one. Thus, two actions are accomplished: checking the distances between all stars in pairs within the same section, and checking the distances between all stars in the neighboring sections (that is, only the connections between the sections). The algorithm for performing these two actions is implemented in such a way that repetitions do not occur. The execution sequence is graphically depicted in a Fig. 10 example by one of the steps in the middle of the passage over a large area. The passage is made in the direction of increasing right ascension, and is shifted to the next section of the declinations array, when the right ascension passage for specific declination is completed. At each step, a section is selected, and the following are checked: the presence of star pairs within the site, the presence of star pairs of this site, and four out of the eight adjacent sites. The remaining four “neighborhoods” will be checked in the next steps (see Fig. 10).

Fig. 10.

Demonstration of sequential passage through the sections in order to find binaries. The square in the center with diverging arrows is the area checked at a certain step of the algorithm. Other shaded areas were already covered by the algorithm. Four arrows directed down and to the left connect the sections the presence of pairs of stars in which will be checked at this step. Gray circles connected by the lines represent examples of pairs that can be detected at this step, both inside the site and in the “adjacent” sites. The arrow to the right indicates which section will be the next, with the larger right ascension. Possible pairs located in the current section and in sections with the larger declination or larger right ascension will be checked at the later steps of the algorithm.

Within one large region, the algorithm ensures that all stars separated by less than 1 pc are added to the list of potential pairs. After this, the potential pairs formed by stars located on the opposite sides of the boundaries between large regions are checked separately. Again, only those stars that have separation not exceeding the maximum separation from the boundary of the regions pass the test. After combining all these regions for different declinations, only polar regions remain unchecked, in which the above-described algorithm is not suitable due to the tendency to infinity of the factor $1{\text{/}}\cos(\delta )$. Verification of the pairs in the circumpolar regions is implemented for the entire section with a declination radius selected in such a way that it exceeds $\theta$. The final result is a list of pairs for all parts of the celestial sphere.

The entire above-described procedure is performed for a “layer”, limited by two values of the distance from the tested stars to the Sun (calculated by Gaia parallax): ${{d}_{1}} < d < {{d}_{2}}$. Respectively, the maximum angular separation θ is determined by the smallest possible distance ${{d}_{1}}$. For this, the 10 pc $\leqslant d \leqslant$ 100 pc sample was divided into four layers: from 10 to 25 pc, from 25 to 50 pc, from 50 to 75 pc, and from 75 to 100 pc. For each layer, the values of $\theta$ and partition into sections are determined separately. Thus, despite the larger number of stars in the outer layers, we partially compensate for the increase in computational time by reducing the size of the sections. The pairs formed by stars located on the opposite sides of the boundaries of these spherical layers are checked separately: this is achieved by taking these layers with an overlap equal to the maximum selected physical separation (in our case, 1 pc) and removal of the pairs, both of which are in the overlapping region of one of these sets. Thus, we avoid including them in the list of pairs more than once. Due to the overlap, those pairs that are located on the opposite sides of the layer boundary are taken into account.