=Paper=
{{Paper
|id=Vol-2277/paper19
|storemode=property
|title=
Insight into Binary Star Formation via Modelling Visual Binaries Datasets
|pdfUrl=https://ceur-ws.org/Vol-2277/paper19.pdf
|volume=Vol-2277
|authors=Oleg Malkov,Dmitry Chulkov,Dana Kovaleva,Alexey Sytov,Alexander Tutukov,Lev Yungelson,Yikdem Gebrehiwot,Nikolay Skvortsov,Solomon Belay Tessema
|dblpUrl=https://dblp.org/rec/conf/rcdl/MalkovCKSTYGST18
}}
==
Insight into Binary Star Formation via Modelling Visual Binaries Datasets
==
https://ceur-ws.org/Vol-2277/paper19.pdf
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
malkov@inasan.ru chulkov@inasan.ru dana@inasan.ru
sytov@inasan.ru atutukov@inasan.ru lry@inasan.ru
© Yikdem Gebrehiwot
Entoto Observatory and Research Center, Addis Ababa, Ethiopia
Mekelle University, College of Natural and Computational Sciences, Mekelle, Ethiopia
yikdema16@gmail.com
© Nikolay Skvortsov
Institute of Informatics Problems, Federal Research Center "Computer Science and Control",
Russian Academy of Sciences, Moscow, Russia
nskv@mail.ru
© Solomon Belay Tessema
Ethiopian Space Science and Technology Institute,
Entoto Observatory and Research Center Astronomy and Astrophysics, Addis Ababa, Ethiopia
tessemabelay@gmail com
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 ≤ 𝜌𝜌 ≤ 200 arcsec, 𝑉𝑉1 ≤ 9.5𝑚𝑚 , 𝑉𝑉2 ≤
11.5𝑚𝑚 , ∆𝑉𝑉 ≤ 4𝑚𝑚 , which can be considered as free of observational incompletness effects. We present some
preliminary results which allow already to reject initial distributions of binaries over semi-major axes of the
orbits more steep than ∝ 𝑎𝑎−1.5 .
Keywords: binary stars, stellar formation, modeling
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 𝑀𝑀1 , mass
Among fundamental problems aimed by these studies ratio of components 𝑞𝑞, and semi-major axes of orbits 𝑎𝑎
is the one of initial distributions of binary stars over their [25]. It was suggested by Vereshchagin et al. (1988) [35]
main parameters: that BF for visual binaries has the form
• mass of the primary component 𝑀𝑀1 ,
• mass ratio of components 𝑞𝑞 = 𝑀𝑀2 ⁄𝑀𝑀1 , d3 𝑁𝑁 ∝ 𝑀𝑀1−2.5 d 𝑀𝑀1 ∙ d log 𝑎𝑎 ∙ 𝑞𝑞 −2.5 d𝑞𝑞 𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 −1 (1)
• and semi-major axis 𝑎𝑎 of component orbits,
where M1 and 𝑎𝑎 are expressed in solar units. As a
“minor” characteristics, we consider eccentricity of
Proceedings of the XX International Conference orbits 𝑒𝑒. The aim 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,
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 𝑎𝑎
synthetic dataset differs significantly due to the change was drawn from one of the following distributions: ∝
of initial fundamental distributions, and how the change 𝑎𝑎−1 , ∝ 𝑎𝑎 −1.5 , ∝ 𝑎𝑎−2 , and eccentricity 𝑒𝑒 was distributed
of every distribution affects it. For this purpose, we assuming following options: (i) all orbits are circular, (ii)
compare synthetic populations for different pairing eccentricities obey thermal distribution 𝑓𝑓𝑒𝑒 (𝑒𝑒) = 2𝑒𝑒, and
functions and particular sets of fundamental functions. (iii) equiprobable distribution 𝑓𝑓𝑒𝑒 (𝑒𝑒) = 1. We adopt
We attempt to find whether certain initial distributions or random orbit orientation. Mass ratio 𝑞𝑞, when needed, is
combinations of initial distributions result in synthetic randomly drawn from ∝ 𝑞𝑞 𝛽𝛽 distribution, where 𝛽𝛽 is
datasets incompatible with observational data at certain adopted to be −0.5, 0 or −0.5. The lower limit for 𝑞𝑞 is
significance level and, on the contrary, whether certain determined by mass limits [0.08 ⋯ 100] 𝑀𝑀⊙ . Certain
initial distributions or combinations of initial pairing functions, such as RP, PCRP, PSCP and TPP, do
distributions provide synthetic dataset best compatible not allow independent random distribution 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 𝑠𝑠,
[9]. 𝑚𝑚, 𝑞𝑞, 𝑎𝑎, 𝑒𝑒 in Table 2 and regarding that 𝑠𝑠0 and 𝑠𝑠5
Besides, our model allows to obtain estimates for the scenarios do not imply independent initial distribution
fraction of binary stars that remains unseen for different over 𝑞𝑞 (see Table 1). Any combination of distributions
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, 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−t/7 , where the time t is expressed in Gyr (Yu &
recovering actual multiplicity fraction, mass hidden in
Jeffery 2010 [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 𝑀𝑀⊙ ) stars, 50 To normalize the number of simulated objects, we
use estimates of stellar density in the solar neighborhood,
pc for high-mass (≥ 10 𝑀𝑀⊙ ) stars, and linear |z| −
based on recent Gaia results [4]. The data for A0V-K4V
log 𝑀𝑀 relation for intermediate masses.
stars presented by Bovy (2017) [4] give 0.01033 stars
For such a small volume we can neglect the radial
per pc3. This means that in the 500 pc sphere we generate
gradient (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(M1 , M2 , [M𝑚𝑚𝑚𝑚𝑚𝑚 ⋯ M𝑚𝑚𝑚𝑚𝑚𝑚 ]);
sort(M1 , M2 );
calc(𝑞𝑞).
PCRP Primary Constrained Random rand(M1 , [M𝑚𝑚𝑚𝑚𝑚𝑚 ⋯ M𝑚𝑚𝑚𝑚𝑚𝑚 ]);
Pairing rand(M2 , [M𝑚𝑚𝑚𝑚𝑚𝑚 ⋯ M𝑚𝑚𝑚𝑚𝑚𝑚 ], M1 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 until M2 < M1 );
calc(𝑞𝑞).
PCP Primary Constrained Pairing rand(M1 , 𝑞𝑞);
calc(M2 ).
SCP Split-Core Pairing rand(M𝑡𝑡𝑡𝑡𝑡𝑡 , [2M𝑚𝑚𝑚𝑚𝑚𝑚 ⋯ 2M𝑚𝑚𝑚𝑚𝑚𝑚 ]);
rand(𝑞𝑞);
calc(M1 , M2 ).
PSCP Primary Split-Core Pairing rand(M𝑡𝑡𝑡𝑡𝑡𝑡 , [2M𝑚𝑚𝑚𝑚𝑚𝑚 ⋯ 2M𝑚𝑚𝑚𝑚𝑚𝑚 ]);
rand(M1 , [0.5(M1 +M2 ) ⋯ M𝑚𝑚𝑚𝑚𝑚𝑚 ], until M1 < M𝑡𝑡𝑡𝑡𝑡𝑡 );
calc(M2 );
calc(q).
TPP Total Primary Pairing rand(M𝑡𝑡𝑡𝑡𝑡𝑡 , [2M𝑚𝑚𝑚𝑚𝑚𝑚 ⋯ 2M𝑚𝑚𝑚𝑚𝑚𝑚 ]);
rand(M1 , [M𝑚𝑚𝑚𝑚𝑚𝑚 ⋯ M𝑚𝑚𝑚𝑚𝑚𝑚 ], until M1 < M𝑡𝑡𝑡𝑡𝑡𝑡 );
calc(M2 );
sort(M1 , M2 );
calc(q).
Note: M𝑡𝑡𝑡𝑡𝑡𝑡 , M1 , M2 – total mass of the binary, primary mass and secondary mass, respectively; M𝑚𝑚𝑚𝑚𝑚𝑚 , M𝑚𝑚𝑚𝑚𝑚𝑚 –
lower (0.08 𝑀𝑀⊙ ) and upper (100 𝑀𝑀⊙ ) limits set for masses; 𝑞𝑞 = 𝑀𝑀2 /𝑀𝑀1 – mass ratio. The meaning of
abbreviations is the following: “rand” – randomizing, “calc” – calculation, “sort” – sorting.
Table 2 Summary of applied initial distributions
sN Scenario mN IMF qN Mass ratio aN Semi-major axis eN Eccentricity
(𝑠𝑠) (𝑚𝑚) (𝑞𝑞) (𝑎𝑎) (𝑒𝑒)
0 RP 0 Salpeter 0 flat, 𝑓𝑓 = 1 0 power, 𝑓𝑓~𝑎𝑎−1 0 thermal, 𝑓𝑓 = 2𝑒𝑒
−1.5
1 Kroupa 1 power, 𝑓𝑓~𝑎𝑎 1 delta, 𝑓𝑓 = 𝛿𝛿(0)
2 PCP 2 power, 𝑓𝑓~𝑎𝑎−2 2 flat, 𝑓𝑓 = 1
3 SCP
4 power, 𝑓𝑓~𝑞𝑞 −0.5
5 TPP 5 power, 𝑓𝑓~𝑞𝑞 0.5
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 𝜌𝜌 < 200 arcsec, primary component visual magnitude
components on the main-sequence), and appended by 𝑉𝑉1 < 9.5𝑚𝑚 , secondary component visual magnitude 𝑉𝑉2 <
parallaxes. A refined dataset for comparison was selected 11.5𝑚𝑚 , magnitude difference Δ𝑉𝑉 ≡ |𝑉𝑉2 − 𝑉𝑉1 | ≤ 4𝑚𝑚
from the data, so as to avoid regions of observational (henceforth, “synthetic dataset”). For the purposes of
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 $𝑓𝑓(𝑒𝑒) ∼ 2𝑒𝑒
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 distribution
magnitude, magnitude difference, projected separation, of wide binaries contains more orbits with 𝑒𝑒 < 0.2 and
parallax. less orbits with 𝑒𝑒 > 0.8 (Tokovinin & Kiyaeva 2016
Then we compare the synthetic distributions with [34]) or a flat distribution in the 𝑒𝑒 = [0.0 ⋯ 0.6] range
refined observational ones using 𝜒𝜒 2 two-sample test. We and declining one for larger 𝑒𝑒 [30].
deem, the better result of comparison, the closer our Having in mind the difficulties hampering
assumptions on pairing scenarios, initial distributions of determination of eccentricities 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
𝑁𝑁 = 1089 stars. To compare them properly with results valued with 𝑒𝑒 = 0 for all stars.
of our simulations we need to use histograms with 𝑛𝑛 = The very selection of fundamental parameters for
5 log 𝑁𝑁 bins [33], i. e., 15 ones. initial distribution is arguable. For instance, primary and
secondary masses were considered as fundamental
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
𝑓𝑓(𝑀𝑀), 𝑓𝑓(𝑎𝑎), 𝑓𝑓(𝑒𝑒) and then stepped aside from them to 5 Results and discussion
test, whether the algorithm would be able to feel
difference at all. 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
we applied as yet only commonly used power law fundamental variables and changes between 0 and about
parametrization, with the particular case of a log-log flat 15000; an exception is distribution over 𝑒𝑒 which affects
distribution known as “Öpik’s law” [26]. Validity of 𝑓𝑓𝑎𝑎 ∝ the volume of simulated datasets only mildly. Thus, if
𝑎𝑎−1 law up to 𝑎𝑎 ≈ 4600 AU, which is close to 𝑎𝑎𝑚𝑚𝑚𝑚𝑚𝑚 of our accepted normalization [4], along with other used
our refined sample of visual binaries, was confirmed by assumptions regarding spatial distribution of visual
Popova et al [27] and Vereshchagin et al [35] 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
Poveda et al [28], found that Öpik's distribution matches present observational dataset volume (1089 pairs) is
with high degree of confidence binaries with 𝑎𝑎 ≲ 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, 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 𝜒𝜒 2 criteria. Nevertheless, one can definitely
separations, since it is hard to envision currently a star- reject those combinations of initial distributions that lead
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 𝑁𝑁𝑜𝑜𝑜𝑜𝑜𝑜 − �𝑁𝑁𝑜𝑜𝑜𝑜𝑜𝑜 ≈ 1056 as lower limit).
101
Figure 1 represents how the resulting 𝜒𝜒 2 statistics are
distributed versus number of pairs in the synthetic
datasets. The results do not allow us to select “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
𝑁𝑁𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠ℎ , 𝜒𝜒 2 correspond to different initial scenarios (𝑠𝑠0,
𝑠𝑠2, 𝑠𝑠4, 𝑠𝑠5, see Table 1, Table 2), IMFs (𝑚𝑚0, 𝑚𝑚1), mass
ratio initial distribution (𝑞𝑞0, 𝑞𝑞4, 𝑞𝑞5, applicable solely for
the 𝑠𝑠2, 𝑠𝑠3 scenarios), and distribution over semi-major
axes 𝑎𝑎0, 𝑎𝑎1, 𝑎𝑎2. Scenarios 𝑠𝑠0 and 𝑠𝑠5 do not involve
independent distribution over 𝑞𝑞; it is generated as an
outcome of the pairing function and IMF, this is why the
𝑞𝑞-panel contains less dots than the other ones.
Figure 1 Distribution of resulting 𝜒𝜒 2 statistics over
number of pairs in the synthetic dataset. Every set of
initial distributions of the 144 processed ones results in Figure 2 Distribution of resulting 𝜒𝜒 2 statistics for
4 dots of different colour in this plot. The dashed line magnitude difference Δ𝑉𝑉 vs. number of pairs in the
marks 5% confidence level of the null-hypothesis (the synthetic datasets, depending on various initial
dots over it correspond to the sets of initial distributions distributions, from top to 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 𝑠𝑠2, 𝑠𝑠3), and semi-major
axes.
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 Δ𝑉𝑉 for the observational dataset serves
as a benchmark.
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, 𝑠𝑠0 and 𝑠𝑠5 (see Table 1, Table 2),
respectively, produce a group of results that seems
acceptable in respect of the number of “observed”
binaries in the synthetic dataset and, simultaneously,
leads to acceptable 𝜒𝜒 2 values at least for two observable
distributions (𝑉𝑉1 and Δ𝑉𝑉).
None of the probed 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 𝜒𝜒 2 statistics allows definite choice
between them. Kroupa IMF looks slightly more
promising, than Salpeter 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 𝑀𝑀⊙ . The comparison in low-
mass region is needed here.
Also, we can not make definite conclusion on the
mass ratio 𝑞𝑞 distribution. The 𝑞𝑞-distributions that we
have analyzed in the present study show significant
difference only in the low-𝑞𝑞 region (below 𝑞𝑞 < 0.5). In
the compilative sample of visual binaries used to
construct our benchmark dataset, however, binaries with
large magnitude differences (and, thus, low 𝑞𝑞) 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 𝑞𝑞-distribution based
on this observational sample.
As to the semi-major axes (𝑎𝑎) distribution, we have
found that power law functions steeper than 𝑎𝑎 −1.5 can be
excluded from further consideration. Figures 2 and 3
demonstrate that initial distribution 𝑎𝑎2 (𝑓𝑓~𝑎𝑎−2 , Table 2)
Figure 3 Distributions of resulting synthetic datasets leads to inappropriately low volume of synthetic dataset.
over magnitude difference Δ𝑉𝑉 for the combinations of It was found also that eccentricity distribution does
initial distributions differing only in (top to bottom) not influence significantly the resulting distributions.
pairing scenarios (see Table 1, Table 2), IMFs, and semi-
major axes. The distribution over Δ𝑉𝑉 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. projected separation 𝜌𝜌 between them, binary star can be
observed as two, one or no source of light, i.e., a part of
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 𝜌𝜌 (the limiting value is selected based Besides the 𝜒𝜒 2 two sample test, we plan to consider
on analysis of the WDS catalogue), and vary limiting other statistical methods (e.g., Kolmogorov-Smirnov two
magnitude 𝑉𝑉𝑙𝑙𝑙𝑙𝑙𝑙 . We consider a pair to be invisible if its sample test) for more reliable interpretation of
total brightness magnitude exceeds 𝑉𝑉𝑙𝑙𝑙𝑙𝑙𝑙 , and to be comparison of our simulation results with observations.
photometrically unresolved if its 𝜌𝜌 does not exceed 0.1 Finally, we aim to consider other parameters as
arcsec. We do not pose any restriction to the component fundamental for initial distributions, e.g., total mass of
magnitude difference. Then, comparing primary and the binary, angular momentum of a pair, and so on.
secondary magnitude with 𝑉𝑉𝑙𝑙𝑙𝑙𝑙𝑙 , we decide, both or only
one component can be observed. 7 Acknowledgments
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.
𝑉𝑉𝑙𝑙𝑙𝑙𝑙𝑙 , nor on initial distributions over 𝑀𝑀, 𝑞𝑞 and 𝑒𝑒. Kouwenhoven, A. Malancheva and D. Trushin for
However, it severely depends on the initial 𝑎𝑎- helpful discussions and suggestions. The work was
distribution: the ratio of unresolved binaries to all visible partially supported by the Program of fundamental
(as two or one source of light) binary stars equals to about researches of the Presidium of RAS (P-28). This research
0.59 ± 0.01 and 0.967 ± 0.003 for 𝑓𝑓𝑎𝑎 ∝ 𝑎𝑎 −1 and 𝑓𝑓𝑎𝑎 ∝ has made use of the VizieR catalogue access tool and the
SIMBAD database operated at CDS, Strasbourg, France,
𝑎𝑎−1.5 , respectively.
Fraction of simulated pairs, visible as two sources of the Washington Double Star Catalog maintained at the
U.S. Naval Observatory, NASA's Astrophysics Data
lights, hereafter 𝐹𝐹𝑃𝑃𝑃𝑃 , strongly depends both on 𝑎𝑎-
System Bibliographic Services, Joint Supercomputer
distribution and 𝑉𝑉𝑙𝑙𝑙𝑙𝑙𝑙 . For 𝑓𝑓𝑎𝑎 ∝ 𝑎𝑎 −1 , 𝐹𝐹𝑃𝑃𝑃𝑃 (depending on
Center of the Russian Academy of Sciences, and data
𝑞𝑞, 𝑚𝑚 and 𝑒𝑒 distributions) varies from 0.01 to 0.19 for
from the European Space Agency (ESA) mission Gaia
𝑉𝑉𝑙𝑙𝑙𝑙𝑙𝑙 = 16𝑚𝑚 and from 0.04 to 0.26 for 𝑉𝑉𝑙𝑙𝑙𝑙𝑙𝑙 = 20𝑚𝑚 . 𝐹𝐹𝑃𝑃𝑃𝑃
(https://www.cosmos.esa.int/gaia), processed by the
values are about ten times lower for 𝑓𝑓𝑎𝑎 ∝ 𝑎𝑎−1.5 .
Gaia Data Processing and Analysis Consortium
Finally, the fraction of simulated stars observed as a
(DPAC,https://www.cosmos.esa.int/web/gaia/dpac/cons
single source of light, depends on the 𝐹𝐹𝑃𝑃𝑃𝑃 as follows:
ortium).
0.4 − 𝐹𝐹𝑃𝑃𝑃𝑃 for 𝑓𝑓𝑎𝑎 ∝ 𝑎𝑎−1 and 0.03 − 0.7 × 𝐹𝐹𝑃𝑃𝑃𝑃 for 𝑓𝑓𝑎𝑎 ∝
𝑎𝑎−1.5 , with no significant dependence on other
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