Insight Into Binary Star Formation Via Modelling Visual Binaries Datasets

Insight into Binary Star Formation via Modelling Visual Binaries Datasets

© Oleg Malkov © Dmitry Chulkov © Dana Kovaleva © Alexey Sytov © Alexander Tutukov © Lev Yungelson Institute of Astronomy, Russian Academy of Sciences, Moscow, Russia [email protected] [email protected] [email protected] [email protected] [email protected] [email protected]

Abstract. We describe the project aimed at finding initial distributions of binary stars over masses of components, mass ratios of them, semi-major axes and eccentricities of orbit, and also pairing scenarios by means of Monte-Carlo modeling of the sample of about 1000 visual binaries of luminosity class V with Gaia DR1 TGAS trigonometric parallax larger than 2 mas, limited by 2 200 arcsec, 9.5 , 11.5 , 4 , which can be considered as free of observational incompletness effects. We present𝑚𝑚 some 1 2 preliminary𝑚𝑚 results𝑚𝑚 which allow already to reject initial distributions of≤ bin𝜌𝜌aries≤ over semi-major𝑉𝑉 ≤ axes of𝑉𝑉 the≤ orbits more∆𝑉𝑉 ≤steep than . . Keywords: binary stars,−1 5 stellar formation, modeling ∝ 𝑎𝑎 which, combined, we will call “the birth function” 1 Introduction (henceforth, BF). Most important, BF is, first, a benchmark for the Majority of stars accessible for detailed observational theories of star formation and, second, the base for the study appear to be binary ones. Interaction between estimates of the number of objects in the models of binary star components in the course of their evolution different stellar populations and model rates of various results in a rich variety of astrophysical phenomena and events, e. g., supernovae explosions etc. objects. Study of the structure and evolution of binary In the present study, we assume that BF is defined by stars is one of the most actively developing fields of the three fundamental functions describing distribution of modern astrophysics. stars over initial mass of primary component , mass Among fundamental problems aimed by these studies ratio of components , and semi-major axes of orbits 1 is the one of initial distributions of binary stars over their [25]. It was suggested by Vereshchagin et al. (1988)𝑀𝑀 [35] main parameters: that BF for visual binaries𝑞𝑞 has the form 𝑎𝑎 • mass of the primary component , • mass ratio of components = , . . 1 d d d log d (1) • and semi-major axis of component𝑀𝑀 orbits, 3 −2 5 −2 5 −1 2 1 1 1 𝑞𝑞 𝑀𝑀 ⁄𝑀𝑀 where 𝑁𝑁M ∝ 𝑀𝑀and are𝑀𝑀 ∙expressed𝑎𝑎 ∙ 𝑞𝑞 in solar𝑞𝑞 𝑦𝑦𝑦𝑦 units.𝑦𝑦𝑦𝑦 As a 𝑎𝑎 “minor” characteristics, we consider eccentricity of 1 Proceedings of the XX International Conference orbits . The aim𝑎𝑎 of the current paper is presentation of “Data Analytics and Management in Data Intensive preliminary results of the assessment of BF by means of Domains” (DAMDID/RCDL’2018), Moscow, Russia, 𝑒𝑒 October 9-12, 2018

98 comparison of results of Monte-Carlo model of the local the short summary of the used pairing functions. population of field visual binaries with their observed Masses of the of components or total masses of the sample. binaries were drawn randomly from Salpeter [32] or We probe, for a given type of stars, whether the Kroupa [21] initial mass functions (IMF), separation synthetic dataset differs significantly due to the change was drawn from one of the following distributions: of initial fundamental distributions, and how the change , . , , and eccentricity was distributed𝑎𝑎 of every distribution affects it. For this purpose, we assuming−1 −1 following5 −2 options: (i) all orbits are circular, (ii)∝ compare synthetic populations for different pairing 𝑎𝑎eccentricities∝ 𝑎𝑎 obey∝ 𝑎𝑎 thermal distribution𝑒𝑒 ( ) = 2 , and functions and particular sets of fundamental functions. (iii) equiprobable distribution ( ) = 1. We adopt 𝑒𝑒 We attempt to find whether certain initial distributions or random orbit orientation. Mass ratio , 𝑓𝑓when𝑒𝑒 neede𝑒𝑒 d, is 𝑒𝑒 combinations of initial distributions result in synthetic randomly drawn from distribution𝑓𝑓 𝑒𝑒 , where is datasets incompatible with observational data at certain adopted to be 0.5, 0 or 0.5𝛽𝛽 . The lower𝑞𝑞 limit for is significance level and, on the contrary, whether certain determined by mass limits∝ 𝑞𝑞 [0.08 100] . Certain𝛽𝛽 initial distributions or combinations of initial pairing functions,− such as −RP, PCRP, PSCP and TPP,𝑞𝑞 do ⊙ distributions provide synthetic dataset best compatible not allow independent random distributi⋯ 𝑀𝑀on of mass with observational data, hopefully, at certain significance ratios, it is calculated from masses of components. level. Table 2 contains short summary on initial The model also accounts for star formation rate, distributions used in the modelling. Some cells are empty stellar evolution and takes into account observational because the pre-planned distributions are not selection effects. The model is compared to the dataset implemented as yet. The total number of possible compiled as described by Kovaleva et al [18] with combinations of initial distributions considered as yet is addition of the data on parallaxes from Gaia DR1 TGAS 144, equal to the number of possible combinations of , [9]. , , , in Table 2 and regarding that 0 and 5 Besides, our model allows to obtain estimates for the scenarios do not imply independent initial distribution𝑠𝑠 fraction of binary stars that remains unseen for different 𝑚𝑚over𝑞𝑞 𝑎𝑎(see𝑒𝑒 Table 1). Any combination of distributions𝑠𝑠 𝑠𝑠 reasons and is observed as single objects and to listed in Table 2 can be conveniently referred, for investigate how these fractions depend on the initial instance,𝑞𝑞 as “s2m0q5a1e0”. distributions of parameters. Such estimates are To account for star formation rate we adopt SFR(t) = important, for instance, as an approach toward 15 e / , where the time t is expressed in Gyr (Yu & recovering actual multiplicity fraction, mass hidden in Jeffery−t 72010 [36]). Disc age is assumed to be equal to 14 binaries, as well as toward models of different stellar Gyr. populations. Currently, we consider the following stellar The model and observational data are described in evolutionary stages: MS-star, red giant, white dwarf, chapters 2 and 3, respectively. Some considerations on neutron star, black hole. The objects in the two latter the choice of theoretical models are described in chanter stages do not produce visual binaries (though they 4. Results and conclusions are presented in chapter5. In contribute to the statistics of pairs, observed as single chapter 6 we outline the plans of future studies. stars, see Section 5.2 below). We do not consider brown dwarfs and pre-MS stars here, as they are extremely 2 The model rarely observed among visual binaries and their Visual binaries are observed, mostly, in the immediate multiplicity rate is substantially lower than for more solar vicinity. Therefore, we consider them to be massive stars (Allers 2012) [1]. As we deal with wide distributed up to the distance of 500 pc in radial direction pairs only, we assume the components to evolve and according to a barometric function along z. The scale independently. To calculate evolution of stars and their height z for the stars of different spectral types and, observational properties we used analytical expressions respectively, masses was studied, e. g., in derived by Hurley et al [15] and assumed solar [3,10,12,20,31]. Synthesizing results of these studies, we metallicity for all generated stars. assume |z| = 340 pc for low-mass ( 1 ) stars, 50 To normalize the number of simulated objects, we use estimates of stellar density in the solar neighborhood, pc for high-mass ( 10 ) stars, and linear |z| ⊙ based on recent Gaia results [4]. The data for A0V-K4V log relation for intermediate masses.≤ 𝑀𝑀 ⊙ stars presented by Bovy (2017) [4] give 0.01033 stars For such a small≥ volume𝑀𝑀 we can neglect the radial− per pc3. This means that in the 500 pc sphere we generate gradient𝑀𝑀 (Huang et al. 2015) [14]. We also ignore about 43300 pairs of stars. interstellar extinction. For the generated objects, we determine To simulate stellar pairs we use different pairing observational parameters, in particular, the brightness of functions (scenarios), mostly taken from components, their evolutionary stage and projected Kouwenhowen's list [16]. separation. Then we apply a filter to select a sample of It includes random pairing and other scenarios, where stars, which can be compared with observational data two of the four parameters (primary mass, secondary (see the next section). mass, total mass of the system, mass ratio) are randomized, and other are calculated. Table 1 contains

99 Table 1 Summary of considered pairing functions (scenarios)

Abbreviation Full name Scheme RP Random Pairing rand(M , M , [M M ]); sort(M , M ); 1 2 𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚 calc( ). ⋯ 1 2 PCRP Primary Constrained Random rand(𝑞𝑞M , [M M ]); Pairing rand(M , [M M ], M = until M < M ); 1 𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚 calc( ). ⋯ 2 𝑚𝑚𝑚𝑚𝑚𝑚 ⋯ 𝑚𝑚𝑚𝑚𝑚𝑚 1 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 2 1 PCP Primary Constrained Pairing rand(𝑞𝑞M , ); calc(M ). 1 𝑞𝑞 SCP Split-Core Pairing rand(M2 , [2M 2M ]); rand( ); 𝑡𝑡𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚 calc(M , M ). ⋯ 𝑞𝑞 PSCP Primary Split-Core Pairing rand(M1 ,2[2M 2M ]); rand(M , [0.5(M +M ) M ], until M < M ); 𝑡𝑡𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚 calc(M ); ⋯ 1 1 2 𝑚𝑚𝑚𝑚𝑚𝑚 1 𝑡𝑡𝑡𝑡𝑡𝑡 calc(q). ⋯ 2 TPP Total Primary Pairing rand(M , [2M 2M ]); rand(M , [M M ], until M < M ); 𝑡𝑡𝑡𝑡𝑡𝑡 𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚 calc(M ); ⋯ 1 𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚 1 𝑡𝑡𝑡𝑡𝑡𝑡 sort(M , M ); ⋯ 2 calc(q). 1 2 Note: M , M , M – total mass of the binary, primary mass and secondary mass, respectively; M , M – lower (0.08 ) and upper (100 ) limits set for masses; = / – mass ratio. The meaning of 𝑡𝑡𝑡𝑡𝑡𝑡 1 2 𝑚𝑚𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚𝑚𝑚 abbreviations is the following: “rand” – randomizing, “calc” – calculation, “sort” – sorting. 𝑀𝑀⊙ 𝑀𝑀⊙ 𝑞𝑞 𝑀𝑀2 𝑀𝑀1

Table 2 Summary of applied initial distributions

sN Scenario mN IMF qN Mass ratio aN Semi-major axis eN Eccentricity ( ) ( ) ( ) ( ) ( )

0 RP 𝑠𝑠 0 Salpeter𝑚𝑚 0 flat, =𝑞𝑞1 0 power, 𝑎𝑎~ 0 thermal,𝑒𝑒 = 2 −1. 1 Kroupa 𝑓𝑓 1 power, 𝑓𝑓~𝑎𝑎 1 delta, =𝑓𝑓 (0)𝑒𝑒 −1 5 2 PCP 2 power, 𝑓𝑓~𝑎𝑎 2 flat, 𝑓𝑓= 1 𝛿𝛿 −2 3 SCP 𝑓𝑓 𝑎𝑎 𝑓𝑓 4 power, ~ . −.0 5 5 TPP 5 power, 𝑓𝑓~𝑞𝑞 0 5 𝑓𝑓 𝑞𝑞 incompleteness in the space of observational parameters. 3 Observational data for comparison The procedure of dataset compilation and analysis described in details in [17,18] was improved due to use To compare our simulations with observational data, we of new trigonometric parallaxes from TGAS DR1 Gaia use the most comprehensive list of visual binaries WCT [9] that allowed to re-obtain constraints to avoid regions [17], compiled on the base of the largest original of observational incompleteness. catalogues WDS [24], CCDM [5] and TDSC [8]. These Out of simulated objects we select pairs, satisfying data were refined or corrected for mistaken data, optical the same observational constraints, as the refined pairs, effects of higher degrees of multiplicity, sorted by observational set does, namely: projected separation 2 < luminosity class (primarily, to select pairs with both < 200 arcsec, primary component visual magnitude components on the main-sequence), and appended by < 9.5 , secondary component visual magnitude < parallaxes. A refined dataset for comparison was selected 11𝜌𝜌 .5 , 𝑚𝑚magnitude difference | | 4 1 2 from the data, so as to avoid regions of observational (henceforth,𝑉𝑉 𝑚𝑚 “synthetic dataset”). For the purposes𝑉𝑉 of𝑚𝑚 Δ𝑉𝑉 ≡ 𝑉𝑉2 − 𝑉𝑉1 ≤

100 correct comparison, we also limit refined set of As for the eccentricity distribution, from physical observational data by 500~pc distance. point of view, one usually prefers in theoretical simulations the “thermal” law $ ( ) 2 We construct distributions of synthetic datasets over (Ambartsumian 1937) [2], though in observational the following parameters: primary and secondary datasets one finds, e. g., that the eccentricity distribution𝑓𝑓 𝑒𝑒 ∼ 𝑒𝑒 magnitude, magnitude difference, projected separation, of wide binaries contains more orbits with < 0.2 and parallax. less orbits with > 0.8 (Tokovinin & Kiyaeva 2016 Then we compare the synthetic distributions with [34]) or a flat distribution in the = [0.0 𝑒𝑒 0.6] range refined observational ones using two-sample test. We and declining one𝑒𝑒 for larger [30]. deem, the better result of comparison,2 the closer our Having in mind the diff𝑒𝑒iculties ⋯hampering assumptions on pairing scenarios,𝜒𝜒 initial distributions of determination of eccentricities𝑒𝑒 from observations and masses, mass ratio, separation and eccentricity are to numerous selection effects, we probe three quite reality. The refined set of observational data contains different model distributions: “thermal”, flat, and single = 1089 stars. To compare them properly with results valued with = 0 for all stars. of our simulations we need to use histograms with = The very selection of fundamental parameters for 5𝑁𝑁log bins [33], i. e., 15 ones. initial distribution𝑒𝑒 is arguable. For instance, primary and 𝑛𝑛 secondary masses were considered as fundamental 𝑁𝑁 parameters for MS binaries by Malkov [23] and pre-MS 4 Some reflections concerning selection of binaries by Malkov and Zinnecker [22], while Goodwin models [11] has argued that system mass is the more fundamental physical parameter to use. We do not reject In the selection of trial initial distributions for the model possibility to choose and investigate other parameters as we adopted the following approach: we started with well fundamental ones in the course of further work. established or widely used in the literature functions for ( ), ( ), ( ) and then stepped aside from them to 5 Results and discussion test, whether the algorithm would be able to feel 𝑓𝑓difference𝑀𝑀 𝑓𝑓 𝑎𝑎 at𝑓𝑓 𝑒𝑒all. We preferred simple analytical 5.1 Star formation function expressions, supposing we would pass to more complicated ones later if we find it necessary. Comparison of our simulations with observational data Thus, we use traditional Salpeter's IMF [32] along allows us to make the following preliminary conclusions with the much more recent and generally accepted on initial distributions. Kroupa's one [21]. In spite of the statement by Duchêne Even before application of statistical tests, we should and Kraus [7] that no observed dataset agrees with meet a strong and evidently important criterion of random pairing scenario, we use the latter among other validity of the tested combination of initial distributions, ones. namely the agreement between the number of binaries in the simulated datasets and the observed number of visual On the other hand, for semi-major axis distribution pairs. This number depends on initial distributions of 0 we applied as yet only commonly used power law fundamental variables and changes between and about parametrization, with the particular case of a log-log flat 15000; an exception is distribution over which affects distribution known as “Öpik’s law” [26]. Validity of the volume of simulated datasets only mildly. Thus, if 𝑒𝑒 law up to 4600 AU, which is close to of our accepted normalization [4], along with other used 𝑎𝑎 assumptions regarding spatial distribution of visual our−1 refined sample of visual binaries, was confirmed𝑓𝑓 by∝ 𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎Popova et al 𝑎𝑎[27] ≈ and Vereshchagin et al [35]𝑎𝑎 who binaries in solar vicinity is valid, we can exclude certain analyzed the data in the amended 7th Catalog of combinations of initial distributions, based purely on the Spectroscopic Binaries [19] and IDS, respectively. number of binaries in synthetic dataset. However, our 1089 Poveda et al [28], found that Öpik's distribution matches present observational dataset volume ( pairs) is with high degree of confidence binaries with 3500 limited to binaries having MK spectral classification. AU (but we note, that selection effects which hamper Thus, we are careful and do not rely exclusively on this discovery of the widest systems were not considered,𝑎𝑎 ≲ criterion because we allow certain freedom due to contrary to the abovementioned studies). We also stress, simplifications and possible incomplete account of after Heacox [13], that Gaussian distribution of selection effects while constructing the refined separations encountered in the literature (e. g., observational dataset, as well as to vagueness of Duquennoy & Mayor 1991 [6], Raghavan et al 2010 [30] theoretical notions on solar vicinity population. This is is an artefact of data representation. Like Poveda et al why we do take into account both number and two [28], we reject Gaussian distribution of stellar sample criteria. Nevertheless, one can definitely separations, since it is hard to envision currently a star- reject those2 combinations of initial distributions that lead 𝜒𝜒 formation process leading to such a distribution. to the number of binary stars in a synthetic observational dataset significantly less than 1000 (taking present dataset volume 1056 as lower limit).

𝑁𝑁𝑜𝑜𝑜𝑜𝑜𝑜 − �𝑁𝑁𝑜𝑜𝑜𝑜𝑜𝑜 ≈

101 Figure 1 represents how the resulting statistics are distributed versus number of pairs in 2the synthetic datasets. The results do not allow us to select𝜒𝜒 “the best” initial distributions over every parameter, but rather to prefer some combinations of initial distributions to others. One may see that no combination leading to acceptable number of pairs in synthetic dataset would give acceptable distribution over angular distances between components, while magnitude difference and, in some cases, distribution over primary magnitudes, are reproduced better for the same initial conditions. Below there are some figures providing examples of how the same distribution over certain parameter, in different combinations with other initial distributions, leads to better or worse agreement with the observational dataset.

Figure 2 represents an example of how different combinations of initial distributions change resulting synthetic datasets and their agreement with observational one. Four figures demonstrate, in turn, which values of , correspond to different initial scenarios ( 0, 2 4 52 0 1 𝑠𝑠𝑠𝑠, 𝑠𝑠𝑠𝑠ℎ, , see Table 1, Table 2), IMFs ( , ), mass ratio𝑁𝑁 initial𝜒𝜒 distribution ( 0, 4, 5, applicable solely 𝑠𝑠for the𝑠𝑠 𝑠𝑠2, 𝑠𝑠3 scenarios), and distribution over𝑚𝑚 semi𝑚𝑚 -major axes 0, 1, 2. Scenarios𝑞𝑞 𝑞𝑞 0 𝑞𝑞and 5 do not involve independent𝑠𝑠 𝑠𝑠 distribution over ; it is generated as an outcome𝑎𝑎 of𝑎𝑎 the𝑎𝑎 pairing function𝑠𝑠 and IMF,𝑠𝑠 this is why the -panel contains less dots than the𝑞𝑞 other ones.

𝑞𝑞

Figure 1 Distribution of resulting statistics over number of pairs in the synthetic dataset.2 Every set of initial distributions of the 144 processed𝜒𝜒 ones results in Figure 2 Distribution of resulting statistics for 4 dots of different colour in this plot. The dashed line magnitude difference vs. number of2 pairs in the marks 5% confidence level of the null-hypothesis (the synthetic datasets, depending on 𝜒𝜒various initial dots over it correspond to the sets of initial distributions distributions, from top Δ𝑉𝑉to bottom: pairing scenarios (see that are rejected at the level of 95%, based on the used Tables 1, Table 2), IMFs, distributions over mass ratio observational sample). (applicable solely for scenarios 2, 3), and semi-major axes. 𝑠𝑠 𝑠𝑠

102 Figure 3 shows how the distribution over observational parameter magnitude difference changes with the change of one initial distribution (pairing scenarios, IMF, distribution over semi-major axes). The distribution over for the observational dataset serves as a benchmark. Δ𝑉𝑉 Based on combination of the two (number and statistical) criteria, we may (very preliminary) state the following. For the considered observational dataset, RP and TPP pairing scenarios, 0 and 5 (see Table 1, Table 2), respectively, produce a group of results that seems acceptable in respect𝑠𝑠 of the𝑠𝑠 number of “observed” binaries in the synthetic dataset and, simultaneously,

leads to acceptable values at least for two observable distributions ( and 2 ). 𝜒𝜒 1 None of 𝑉𝑉the Δ𝑉𝑉probed combinations of initial distributions can reproduce observational distribution over angular distance between components adequately (see Figure 1). The cause may lay either with selection effects, that still remain unaccounted for (and then the reconsideration of observational sample is necessary), or in need of other initial distributions. Kroupa and Salpeter IMF's lead to different number of pairs in the synthetic dataset, however, neither this difference nor statistics allows definite choice between them. Kroupa2 IMF looks slightly more promising, than 𝜒𝜒Salpeter one, however, more accurate conclusion should be postponed, as these two IMF differ actually only in the low-mass region, and the majority of visual binaries in our observational dataset presumably have masses around 1 to 3 . The comparison in low- mass region is needed here. ⊙ 𝑀𝑀 Also, we can not make definite conclusion on the mass ratio distribution. The -distributions that we have analyzed in the present study show significant difference only𝑞𝑞 in the low- region𝑞𝑞 (below < 0.5). In the compilative sample of visual binaries used to construct our benchmark dataset,𝑞𝑞 however, binaries𝑞𝑞 with large magnitude differences (and, thus, low ) are severely underrepresented. This is why we limit refined observational sample so that pairs with low 𝑞𝑞 are excluded. For this reason we can not come to a definite conclusions concerning selection of -distribution 𝑞𝑞based on this observational sample. As to the semi-major axes ( ) distribution,𝑞𝑞 we have found that power law functions steeper than . can be excluded from further consideration.𝑎𝑎 Figures−1 25 and 3 demonstrate that initial distribution 2 ( ~ 𝑎𝑎 , Table 2) Figure 3 Distributions of resulting synthetic datasets leads to inappropriately low volume of synthetic−2 dataset. over magnitude difference for the combinations of It was found also that eccentricity𝑎𝑎 distribution𝑓𝑓 𝑎𝑎 does initial distributions differing only in (top to bottom) not influence significantly the resulting distributions. pairing scenarios (see TableΔ𝑉𝑉 1, Table 2), IMFs, and semi- major axes. The distribution over for the 5.2 Simulation of visibility of binary stars observational dataset plotted by the bold red line serves Depending on the brightness of components and as a benchmark. Δ𝑉𝑉 projected separation between them, binary star can be observed as two, one or no source of light, i.e., a part of 𝜌𝜌

103 binaries can appear as single stars or remain invisible at moderate-mass stars), to consider more distant objects, all. We involve in our simulations the following and to involve final stages of stellar evolution into observational states: “both observed”, “primary only”, consideration. Having a number of Monte-Carlo “secondary only”, “photometrically unresolved”, and simulations representing various observational datasets, “invisible”. To estimate fraction of simulated pairs, we should be able to check if the approximate formula which fall into listed states, we take 0.1 arcsec as a (1) needs reconsideration of remains valid. minimum limit for (the limiting value is selected based Besides the two sample test, we plan to consider on analysis of the WDS catalogue), and vary limiting other statistical methods2 (e.g., Kolmogorov-Smirnov two magnitude . We𝜌𝜌 consider a pair to be invisible if its sample test) 𝜒𝜒for more reliable interpretation of total brightness magnitude exceeds , and to be comparison of our simulation results with observations. 𝑙𝑙𝑙𝑙𝑙𝑙 photometrically𝑉𝑉 unresolved if its does not exceed 0.1 Finally, we aim to consider other parameters as 𝑙𝑙𝑙𝑙𝑙𝑙 arcsec. We do not pose any restriction to𝑉𝑉 the component fundamental for initial distributions, e.g., total mass of magnitude difference. Then, comparing𝜌𝜌 primary and the binary, angular momentum of a pair, and so on. secondary magnitude with , we decide, both or only one component can be observed. 7 Acknowledgments 𝑉𝑉𝑙𝑙𝑙𝑙𝑙𝑙 Results of our simulation show that the fraction of We thank our reviewers, whose comments greatly helped photometrically unresolved binaries depends neither on us to improve the paper. We are grateful to T. , nor on initial distributions over , and . Kouwenhoven, A. Malancheva and D. Trushin for However, it severely depends on the initial - helpful discussions and suggestions. The work was 𝑙𝑙𝑙𝑙𝑙𝑙 distribution:𝑉𝑉 the ratio of unresolved binaries𝑀𝑀 to 𝑞𝑞all visible𝑒𝑒 partially supported by the Program of fundamental (as two or one source of light) binary stars equals to about𝑎𝑎 researches of the Presidium of RAS (P-28). This research 0.59 ± 0.01 and 0.967 ± 0.003 for and has made use of the VizieR catalogue access tool and the . SIMBAD database operated at CDS, Strasbourg, France, , respectively. −1 𝑎𝑎 𝑎𝑎 the Washington Double Star Catalog maintained at the −1Fr5 action of simulated pairs, visible𝑓𝑓 as∝ two 𝑎𝑎 sources𝑓𝑓 of∝ U.S. Naval Observatory, NASA's Astrophysics Data lights𝑎𝑎 , hereafter , strongly depends both on - System Bibliographic Services, Joint Supercomputer distribution and . For , (depending on 𝑃𝑃𝑃𝑃 Center of the Russian Academy of Sciences, and data , and distributions)𝐹𝐹 varies −1from 0.01 to 0.19 for𝑎𝑎 𝑙𝑙𝑙𝑙𝑙𝑙 𝑎𝑎 𝑃𝑃𝑃𝑃 from the European Space Agency (ESA) mission Gaia = 16 and 𝑉𝑉from 0.04𝑓𝑓 to∝ 0. 𝑎𝑎26 for𝐹𝐹 = 20 . . (https://www.cosmos.esa.int/gaia), processed by the values𝑞𝑞 𝑚𝑚 are 𝑚𝑚𝑒𝑒about ten times lower for . 𝑚𝑚 𝑙𝑙𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙𝑙𝑙 𝑃𝑃𝑃𝑃 Gaia Data Processing and Analysis Consortium 𝑉𝑉 Finally, the fraction of simulated stars𝑉𝑉 observed−1 5 as𝐹𝐹 a 𝑎𝑎 (DPAC,https://www.cosmos.esa.int/web/gaia/dpac/cons single source of light, depends on 𝑓𝑓the∝ 𝑎𝑎 as follows: ortium). 0.4 for and 0.03 0.7 × for . 𝑃𝑃𝑃𝑃 , with no significant−1 dependence𝐹𝐹 on other 𝑃𝑃𝑃𝑃 𝑎𝑎 𝑃𝑃𝑃𝑃 𝑎𝑎 parameters.−1−5 𝐹𝐹 𝑓𝑓 ∝ 𝑎𝑎 − 𝐹𝐹 𝑓𝑓 ∝ References 𝑎𝑎 [1] Allers, K. N. Brown Dwarf Binaries. In: We should note that simple compatibility of synthetic International Astronomical Union. From data from initial distributions with observational data is Interacting Binaries to Exoplanets: Essential not an ultimate evidence of an adequate modelling, since Modeling Tools, M. T. Richards, I. Hubeny (eds), the observational data are far from being comprehensive. vol. 7, iss. 282, pp. 105-110 (2012). doi: However, the initial distributions we use in our 10.1017/S1743921311027086. simulations are obtained, checked and used by many [2] Ambartsumian, V. On the statistics of double stars. other authors, and this suggests that our conclusions are Translated by D. W. Goldmith. In: fairly reliable. Astronomicheskii Zhurnal, vol. 14, p. 207, Leningrad (1937). 6 Future plans [3] Bahcall, J. N., Soneira, R. M. The Universe at We presented here very preliminary results proving that Faint Magnitudes. I. Models for the Galaxy and we need more thorough investigation of the models and the Predicted Star Counts. In: Astrophysical comparison with observations to accomplish the task. Journal Supplement Series, vol. 44, p. 73-110 To make confident conclusions on BF of binary stars, (1980). doi: 10.1086/190685 we need to make comparison of our simulations with [4] Bovy, J. Stellar inventory of the solar other sets of observational data for wide binaries, to neighbourhood using Gaia DR1. In: Monthly cover wider regions of stellar parameters. Also, one of Notices of the Royal Astronomical Society, vol. the important further steps is to extend our study to closer 470, iss. 2, pp. 1360–1387 (2017). doi: binary systems. We will involve basic ideas on evolution 10.1093/mnras/stx1277. of interacting binaries in our simulations, and, [5] Dommanget, J., Nys, O. CCDM (Catalog of consequently, will take into consideration other types of Components of Double & Multiple stars), VizieR binaries for comparison. It will allow us to make more On-line Data Catalog: I/274 ( 2002). definite conclusions on BF of more massive stars (as, simulating visual binaries, we deal mostly with

104 [6] Duquennoy, A., Mayor, M. Multiplicity among Physical Parameters of Spectroscopic Binary solar-type stars in the solar neighbourhood. II - Stars. In: Bulletin d'Information du Centre de Distribution of the orbital elements in an unbiased Donnees Stellaires, vol. 19, p. 71 (1980) sample. In: Astronomy and Astrophysics, vol. 248, [20] Kroupa, P. The distribution of low-mass stars in no. 2, pp. 485-524 (1991). the disc of the Galaxy. Cambridge University, UK [7] Duchêne, G., Kraus, A.. Stellar multiplicity. In: (1992) Annual Review of Astronomy and Astrophysics., [21] Kroupa, P. On the variation of the initial mass vol. 51, p. 269 (2013). function. In: Monthly Notices of the Royal [8] Fabricius, C., Høg, E., Makarov, V. V., Mason, B. Astronomical Society, vol. 322, no. 2, p. 231-246 D., Wycoff, G. L., Urban, S. E. The Tycho double (2001) star catalogue. In: Astronomy & Astrophysics, vol. [22] Malkov, O.Yu., Piskunov, A.E., Zinnecker, H. On 384, iss..1, p. 180-189 (2002). doi: 10.1051/0004- the luminosity ratio of pre-main sequence binaries. 6361:20011822 In: Astronomy and Astrophysics, vol. 338, p. 452- [9] Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., 454 (1998) et al. The Gaia mission. In: Astronomy & [23] Malkov, O.Yu., Zinnecker, H. Binary stars and the Astrophysics, vol. 595, p. A1:1-36 (2016). doi: fundamental initial mass function. Monthly 10.1051/0004-6361/201629272 Notices of the Royal Astronomical Society, vol. [10] Gilmore, G., Reid, N. New light on faint stars – 321, no. 1, p. 149-154 (2001) III. Galactic structure towards the South Pole and [24] Mason, B. D., Wycoff, G. L., Hartkopf, W. I., the Galactic thick disc. In: Monthly Notices of the Douglass, G. G., Worley, C. E. The Washington Royal Astronomical Society, vol. 202, iss. 4, p. Visual Double Star Catalog. VizieR Online Data 1025-1047 (1983) Catalog: B/wds (2014) [11] Goodwin, S.P. Binary mass ratios: system mass [25] Massevich, A., Tutukov A. Stellar eEvolution: not primary mass. In: Monthly Notices of the Theory and Observations. Moscow: Nauka (1988) Royal Astronomical Society: Letters, vol. 430, iss. (in Russian) 1, p. L6-L9 (2013). doi: 10.1093/mnrasl/sls037 [26] Öpik, E. Statistical Studies of Double Stars: On [12] Gould, A., Bahcall, J.N., Flynn, Ch. Disk M dwarf the Distribution of Relative Luminosities and luminosity function from HST star counts. In: The Distances of Double Stars in the Harvard Revised Astrophysical Journal, vol. 465, p. 759 (1996) Photometry North of Declination -31 deg. In: [13] Heacox, W. D. Heacox, W. D. Of Logarithms, Publications of the Tartu Astrofizica Observatory, Binary Orbits, and Self-Replicating Distributions. 25 (1924) In: Publications of the Astronomical Society of the [27] Popova, E. I., Tutukov, A. V., Yungelson, L. R. Pacific, vol. 108, p. 591-593 (1996) Study of physical properties of spectroscopic [14] Huang, Y., Liu, X., Zhang, H., Yuan, H., Xiang, binary stars. In: Astrophysics and Space Science M., Chen, B., et al. On the metallicity gradients of vol. 88, no .1, p. 55-80 (1982) the Galactic disk as revealed by LSS-GAC red [28] Poveda, A., Allen, C., Hernández-Alcántara, A. clump stars. In: Research in Astronomy and Binary Stars as Critical Tools & Tests in Astrophysics, vol.15, no. 8, p. 1240 (2015) Contemporary Astrophysics, 240, 417 [15] Hurley, J.R., Pols, O.R., Tout C.A. [29] Poveda, A., Allen, C., Hernández-Alcántara, A. Comprehensive analytic formulae for stellar Halo Wide Binaries and Moving Clusters as evolution as a function of mass and metallicity. In: Probes of the Dynamical and Merger History of Monthly Notices of the Royal Astronomical our Galaxy. In: Hartkopf, B., Guinan, E. Society, vol. 315, no. 3, p. 543-569 (2000) Harmanec, P., Eds. Binary Stars as Critical Tools [16] Kouwenhoven, M. B. N., Brown, A. G. A., and Tests in Contemporary Astrophysics, IAU Goodwin, S. P., Portegies Zwart, S. F., & Kaper L. Symposium, no. 240, p. 405. Cambridge Pairing mechanisms for binary stars. In: University Press, Cambridge (2007) Astronomische Nachrichten: Astronomical Notes, [30] Raghavan, D., McAlister, H. A., Henry, T. J., et al. vol. 329, no. 9‐10, p. 984-987 (2008) A survey of stellar families: multiplicity of solar- [17] Kovaleva, D.A., Malkov, O.Yu., Yungelson, L.R., type stars. In: The Astrophysical Journal Chulkov, D.A., Yikdem, G.M. Statistical analysis Supplement Series, vol. 190, p. 1 (2010) of the comprehensive list of visual binaries. In: [31] Reed, B.C. New estimates of the scale height and Baltic Astronomy, vol. 24, p. 367-378 (2015) surface density of OB stars in the solar [18] Kovaleva, D.A., Malkov, O.Yu., Yungelson, L.R., neighborhood. In: The Astronomical Journal, vol. Chulkov, D.A. Visual binary stars: data to 120, no. 1, p. 314 (2000) investigate the formation of binaries. In: Baltic [32] Salpeter, E.E. The luminosity function and stellar Astronomy, vol. 25, p. 419-426 (2016) evolution. In: The Astrophysical Journal, vol. 121, [19] Kraitcheva, Z., Popova, E., Tutukov, A., & p. 161-167 (1955) Yungelson, L. Kraitcheva, Z., et al. Catalogue of

105 [33] Shtorm, R. Probability theory. Mathematical visual binaries. In: Colloquium on Wide statistics. Statistical quality control. 368 p. Mir, Components in Double and Multiple Stars. Moscow (1970) (in Russian) Astrophysics and Space Science, vol. 142, no. 1-2, [34] Tokovinin, A., Kiyaeva, O. Eccentricity p. 245-254 (1988) distribution of wide binaries. In: Monthly Notices [36] Yu, S., Jeffery, C.S. The gravitational wave signal of the Royal Astronomical Society, vol. 456, no. 2, from diverse populations of double white dwarf p. 2070-2079 (2016) binaries in the Galaxy. Astronomy & [35] Vereshchagin, S., Tutukov, A., Yungelson, L., Astrophysics, vol. 521, p. A85 (2010) Kraicheva, Z., Popova, E. Statistical study of

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