Binary Star Population with Common Proper Motion in Gaia DR2

Completeness of the created catalog of wide binary systems at the distance of 10 to 100 pc from the Sun is determined by several factors. First, it is limited by completeness of Gaia DR2. It is known that the Gaia DR2 catalog is substantially incomplete for stars brighter than approximately $G\approx 6^{m}$, while it is almost complete only for the magnitude range $12\leq G\leq 17$ [8]. In addition, the catalog may be incomplete for the stars with large proper motion (there are more such stars in the immediate vicinity of the Sun) and in the dense stellar fields. Inhomogeneous coverage of the celestial sphere by Gaia DR2 sources in the microscale (reflecting the scanning law), as well as the complete removal of the sources associated with the clusters from the created catalog, should not critically affect the properties of the obtained sample. In addition, the completeness of the resulting binary star catalog is affected by the method of selecting sources from the Gaia DR2: (i) we selected only stars with parallaxes; (ii) we applied the filters to select “astrometrically clean” solutions. In this case, the filter by the relative parallax error eliminates, predominantly, more distant stars. The probability of passing these filters (i.e., having a sufficiently high quality of astrometric and photometric solutions) particularly is lower than the average one for unresolved binary stars and binary components with pronounced orbital motion, and is equal to zero for pairs with angular distance between components less than or equal to 2 mas. For angular distances between components larger than 2 mas, the relation between limiting angular resolution and the brightness difference between components is expressed by the relation $\Delta G_{lim}\propto\log\rho$ for $\rho\geq 2"$ (see also a discussion in [13]). Obviously, this selection effect predominantly distinguishes pairs with a small brightness difference at small angular distances (See Fig. 4).

Even after excluding the regions of space in which the stars of the Hyades and Mamajek 1 clusters are located, the catalog contains a noticeable number of pairs, with one or both components as members of more than one pair. As the ratio of the distances between components in the groups of stars consist more than of one pair shows, they may be hierarchical multiple star systems; however, these are mainly stars of moving groups. About 400 pairs that have a common component with another pair were found, which were removed from the catalog in order to avoid distortions in the characteristic analysis of the ensemble of binary stars. After elimination of multiple stars, 9977 pairs remained in the sample.
Generally, one would expect that some of the components of the catalogued pairs ($10-13\%$, based on statistics on multiplicity from [14, 3]) can be represented by unresolved binary stars (see also the discussion of pairs with different radial velocities of the components in Section 2). On the other hand, the catalog compilation procedure for filtering sources according to the quality of the astrometric and photometric solutions should reduce this fraction. A study of solution quality indicators (RUWE, Flux Excess Factor) performed by the DR2 authors for known close binary stars ($\rho\leq 2$ mas), taken from the identifier catalog of binary and multiple stars ILB [15] showed the following: 3537 close pairs were identified as an unresolved binary star with Gaia DR2 sources for which the parallax values $\varpi\geq 10$ mas. Out of these sources, Gaia DR2 quality of the solution filter rejected 2180 or $62\%$. For comparison, only $8\%$ of objects do not pass the same filter when identifying Gaia DR2 sources with Hipparcos stars. This suggests that the use of recommended filters has particularly led to a significant reduction of the number of components that are unresolved binary stars.

Let us consider the distribution of sample pairs over the angular separation of components (Fig. 5a), over the distance between components in the projection onto the celestial sphere (Fig. 5b), and in the linear space (Fig. 5c). The distribution of the angular separations between components (Fig. 5a) in the logarithmic scale turns out to be flat in the $3\leq\rho\leq 20$ mas range and decreases at larger values. Although the parallax value is involved in the calculation of both linear distances, the accuracy of its determination is more significant for the distance in the three-dimensional space. This is reflected in the distributions: the projection distribution of the distances between components (Fig. 5b) is similar to the distribution in Fig. 5a, linearly decreasing in the logarithmic scale beyond 300–400 AU (consistently with the estimate of $10^{2.5}$ A.U. in [13]).The distribution over the distance between components in the three-dimensional space in the logarithmic scale increases to $\approx 0.5$ pc, or $10^{5}$ A.U.

Despite the accepted limitation of the relative parallax error by $10\%$, for the distances between components typical for binary stars, small relative parallax errors can lead to large relative errors in the determined distances between the stars. To separately consider the parameter distribution of a more refined, albeit less complete sample, a subsample of binary stars is introduced in which the nominal error estimate in the parallax distance does not exceed 0.1 pc: $\delta_{d}<0.1,\delta_{d}=\frac{1000}{\varpi-\delta_{\varpi}}-\frac{1000}{\varpi}$ (henceforth will be called sample I). On figure 5 distributions for sample I are represented by a grayfilled histogram. It can be seen that the introduction of a limit on the error of the distance significantly reduces the binary fraction with extremely large separations (more than 50 000 AU) and leads to a shift of the distribution maximum in the logarithmic scale to the lower values.
We use absolute magnitudes in the photometric band G and the color obtained from the difference in magnitudes in the BP and RP Gaia bands to construct the Hertzsprung-Russell diagram (Fig. 6). According to the position in the diagram, it is possible to distinguish the components populating the main-sequence for which we estimate the masses from absolute magnitudes in the band G, using Mamajek’s tables³³3http://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt. Moreover, we neglect the probability that the catalog contains unresolved binary stars and stars evolving to the main-sequence, although such objects may be present in the sample.
This allows us to proceed to the mass ratio distribution $q=\frac{mass(B)}{mass(A)}$ for pairs in which both components are supposedly located on the main-sequence (see Fig. 7). In the region $0.3\leq q\leq 0.9$ the distribution looks close to a planar one. In the region $q\geq 0.95$ an excess of binary stars is detected (the so-called twin stars, with components of close mass values). It is important to find out whether such a feature reflects the nature of wide binary stars or if it is related to the effects of sample selection.

As discussed above, selection effects that affect the sample are a combination of the Gaia DR2 selectivity function which has predetermined quality decision filters for single stars and the pair selectivity function. We create a sample of random star pairs from Gaia DR2 with $\varpi<15$ mas (to limit the sample size), selecting them with the same solution quality requirements that were imposed on the pair components in Section 2. Using Hertzsprung-Russell diagram, we select the pairs with both components belonging to the main sequence, estimate their component masses, and construct the ratio distribution of the masses of the weak (“secondary”) and bright (“main”) components. Figure 7 shows the mass distribution for the sample of wide binary stars in comparison with a similar distribution for the synthesized random sample. It is observed that the selectivity function of Gaia DR2 in combination with the applied filters for the solution quality of the components cannot be the reason for the appearance of the “peak of twins”.

In order to investigate how the pair selectivity function associated with the different visibility of stars with different magnitude contrasts at the same angular distance can distort the observed distribution of mass ratios, we will model this effect using binary population synthesis code [16] for various assumptions about the mechanisms “scenario”) of binary system formation [17]. Kroupa [18], initial mass function (IMF) was used and various scenarios of combining the component masses were assumed: independent distribution of them and/or distribution of the component mass sum according to the IMF. For scenarios assuming that the distribution of the component mass ratio $f(q)$, is a free parameter, various options were considered, including $f(q)\propto C$. For the distribution over semi-major axes of the orbits, a power function $f(a)\propto a^{\beta}$, was chosen. Since it was found that the value of the power does not affect the distribution shape over mass ratio of components, the value $\beta=-1$ was adopted. The distribution over eccentricity of the orbits was taken to be flat. For each of such combinations of initial conditions, the vicinity of the Sun within 100 pc was simulated. Star formation rate over 14 Gyr was assumed to be $15\,e^{-t/\tau}$, where $\tau=7$ Gyr [19].

Stellar evolution was described by approximate formulas of Hurley et al. [20]. Then, the simulated sample was compared with the part that would be “visible” under given conditions, including the requirements that both components are main-sequence stars, the restrictions on the apparent magnitude correspond to the Gaia DR2 restrictions $6^{m}\leq G\leq 17^{m}$, and the pair selectivity function (Section 3.1). In the studied scenarios, the catalog selectivity function for binary systems did not lead to a pronounced peak formation of close mass components. Moreover, if (in the scenarios of pair formation allowing an independent distribution of component mass ratios) a similar peak is introduced artificially, the adopted “observational” filters do not significantly affect the presence and shape of this peak. Figure 8 shows the normalized distributions of the model sample over mass ratios of components for the pairs “existing” in the model under given constraints for some combinations of initial conditions, compared to the “observable” ones. Presented are the following scenarios: RP scenario—“random pairing” (both components are formed independently with masses that are selected from the IMF, top panel, left); PCRP scenario—“primary constrained random pairing” (the main component is formed like in RP scenario, the secondary one – with the mass from the IMF, provided that it is not more massive than the main component, top panel, right); and PCP scenario – “primary constrained pairing” (the main component is formed like in RP scenario, the mass of the secondary is determined by an independently specified distribution over mass ratios of components, bottom panel for the case of a flat distribution by mass of components (left), and a flat distribution with a peak (step) at $q\geq 0.9$ for (right)). In these scenarios, the use of the catalog selectivity function that limits the pair visibility with close components with a large brightness difference does not lead to the formation of a “twins peak” in the distribution over component mass ratio if it was absent in the initial sample. The peak in the initial distribution according to the PCRP scenario disappears when the observations are simulated, due to the fact that it is formed by the least massive stars, rejected by the limit of the apparent magnitude. The peak of the twins in the initial distribution according to the PCP scenario with a flat distribution and superimposed peak remains in the observed distribution.

The excess of twin stars among wide binaries should be recognized as really existing. This result was obtained independently of a detailed study [21] and supports the conclusions drawn there.
The excess of twins becomes less pronounced with increasing separation of components: Fig. 9 shows how the distribution over mass ratios becomes flatter at large physical separations. To the right in the same Figure, the distribution for sample I with a restriction on the distance error is shown. In the sample with a more stringent restriction on the error of the distance between components, the peak of twins is more narrow and more pronounced. There can be several reasons for this, and it is possible that the resulting effect is achieved by their combined effect. First, the average distance between the components in the sample I is smaller (and at closer distances, the proportion of systems with twin components is higher). This may be due to the fact that the predominant formation mechanism of twin stars is not effective for the widest pairs. On the other hand, it can be expected that the introduction of a restriction on the error in determined distance for the catalogued stars leads to the decrease in the contamination of the sample by optical pairs, the distance between which is underestimated due to the parallax errors. The admixture of optical pairs in which the distribution of mass ratio falls in the range $0.4\leq q\leq 1$ (Fig. 7) should lead to the decrease in the star fraction with large q. Therefore, the smaller it is, the more pronounced the “twin peak” should be. A detailed review and discussion of possible channels for the formation of an excess of wide binary stars with close masses of components are presented in [21]and references therein. It is important to note that the most probable mechanism is the competitive accretion onto the forming pair components in a common protostellar disk, in which the component with an initially smaller mass, moving in the orbit with a large radius, can come up in mass with the initially more massive star. Currently, the problem of this mechanism is that the largest observable protostellar disks around forming binaries have a radius of the order of several hundred AU (e.g. [22]), while the excess of twin stars, though decreasing with increasing separation of components, continues to be significant up to the distances exceeding $3\cdot 10^{4}$ AU.

With the further increase of component separation, the star excess with close masses continues to decrease and completely disappears for pairs with a calculated separation larger than $8\cdot 10^{4}$ AU, for which the distribution over component mass ratio becomes flat over the entire range $0.3\leq q\leq 1$. Such a mass ratio distribution can be the result of a combination of a number of factors (the presence of initially binary stars whose orbit underwent dynamic broadening [23]; presence of a share of pairs formed during cluster decay [24]]; and admixture of optical pairs). Finally, in the analysis of the distribution dependence over component mass ratios on their separation, it should be considered that the accuracy of the separation estimates itself strongly depends on the determination accuracy of parallaxes.