On the Probability of the Astrometric Resolution of Spectroscopic Binaries into Components

Abstract

Resolved spectroscopic binaries (RSBs) are an extremely valuable class of objects, being the only (apart from trigonometric) supplier of dynamical stellar parallaxes. This circumstance, as well as the comparative paucity of studied RSBs, makes the problem of identifying binary systems potentially capable of being added to the list of known RSBs extremely urgent. In this paper, we propose a methodology for estimating the probability of a spectroscopic binary system to be resolved into components, perform the first step of its application to the SB9 catalogue, and present preliminary results, in particular, a list of the most promising RSB candidates.

1. Introduction

A significant fraction of stars in the Milky Way are binary or multiple stars. The studies conducted in [1,2,3,4,5] allow us to conclude that F-G-K-M stars may have an observed binary fraction of 20–40%, while that of O- and B-type stars may reach 100%.

A very interesting observational class of binary systems is resolved spectroscopic binaries (RSBs). Like some other observational classes (for example, double-lined eclipsing binaries, DLEBs), they are suppliers of dynamic stellar masses and other component parameters, but, most importantly, they provide us with the only way (apart from trigonometric parallax) to directly determine (rather than estimate) the distances to stars. Ref. [6] was one of the first to draw attention to this fact and compiled a list of about forty RSB systems. Later, Refs. [7,8,9,10] presented more representative lists, and, finally, Ref. [11] recently published a catalogue containing presumably all RSB systems studied to date. The catalogue in [11] is the most comprehensive list of RSBs to date, and it contains orbital elements (the orbital period P (yr), time of periastron $T0$ (JD), eccentricity e, inclination of the orbital plane i (deg), position angle of the ascending node $\Omega$ (deg), periastron longitude $\omega$ (deg), semi-major axis a (mas), and semi-major axis a (AU)), masses, orbital and trigonometric parallaxes, spectral classification, multicolour photometry, and reddening for 173 objects.

The exceptional importance of this class of binary systems makes the task of discovering new RSBs extremely urgent. One of the possibilities of replenishing the list of known RSBs is an astrometric study of the most “promising” (primarily the widest) spectroscopic binaries with lines of both components in the spectrum (so-called SB2). The results of such a study were presented, for example, in [12].

Currently, the most complete and constantly updated catalogue of spectroscopic binaries is presented in [13] (hereafter, SB9). In this paper, we focus on estimating the probability of the astrometric resolution of spectroscopic binaries into components. We develop an original methodology for estimating this probability and conduct a pilot application of it to the SB9 catalogue. This paper is structured as follows: The methodology for estimating the resolution probability function of spectroscopic binary stars is described in Section 2. Results are presented and discussed in Section 3. Finally, in Section 4, we draw our conclusions.

2. Methodology for Estimating the Resolution Probability Function of Spectroscopic Binary Stars

Suppose that, in order for an unresolved spectroscopic binary star (SB) to be spatially resolved (RSB), it must have parameter values characteristic of other resolved spectroscopic binaries, i.e., be close to them in some parameter space. We also assume that the parameter space, for which the first assumption is correct, is formed by the observational parameters of the spectroscopic binaries, defined in the Catalogue of Resolved Spectroscopic Binaries [14] (hereafter, RSBcat) and the SB9 catalogue [13], which can be found in Table 1. The subset of SB9 objects identified with [14] is denoted by SB9R ($\mathrm{SB}9\mathrm{R}=\mathrm{SB}9\cap \mathrm{RSBcat}$), and the subset not identified with [14] (i.e., unresolved) is denoted by SB9U. Note that $\mathrm{SB}9\mathrm{R}\ne \mathrm{RSBcat}$, as 24 of the 173 RSBcat objects are not included in the SB9 catalogue. Accordingly, the set of all catalogue objects is SB9A ($\mathrm{SB}9\mathrm{A}=\mathrm{SB}9\mathrm{R}\cup \mathrm{SB}9\mathrm{U}$). Note also that we consider only double-lined spectroscopic binary (SB2) systems from the SB9 catalogue, as only those systems can be fully solved for the orbital parameters (and thus the orbital parallax).

Table 1. Parameters of spectroscopic binary stars in the RSBcat and SB9 catalogues. (*) The values of these parameters were added from the SIMBAD database for stars in the RSBcat and SB9 catalogues.

Parameter	Designation	Unit of Measurement	Availability in RSBcat	Availability in SB9
Orbit semi-major axis	a	mas	✓
Orbital inclination	i	deg	✓
Orbital parallax	${\varpi }_{\mathrm{orb}}$	mas	✓
Longitude of node	$\Omega$	deg	✓
Mass of primary	$m_1$	$M_{\odot }$	✓
Mass of secondary	$m_2$	$M_{\odot }$	✓
Coordinates	$\alpha$, $\delta$	deg	✓	✓
Period	P	yr	✓	✓
Periastron time	$T_0$	JD	✓	✓
Orbital eccentricity	e	–	✓	✓
Argument of periastron	$\omega$	deg	✓	✓
Trigonometric parallax (*)	${\varpi }_{\mathrm{trig}}$	mas	✓	✓
B-flux (*)	$F_{\mathrm{B}}$	mag	✓	✓
V-flux (*)	$F_{\mathrm{V}}$	mag	✓	✓
G-flux (*)	$F_{\mathrm{G}}$	mag	✓	✓
J-flux (*)	$F_{\mathrm{J}}$	mag	✓	✓
H-flux (*)	$F_{\mathrm{H}}$	mag	✓	✓
K-flux (*)	$F_{\mathrm{K}}$	mag	✓	✓
Velocity amplitude of primary	$K_1$	km/s		✓
Velocity amplitude of secondary	$K_2$	km/s		✓
Systematic velocity	$\gamma$	km/s		✓
Magnitude of component 1	$V_1$	mag		✓
Magnitude of component 2	$V_2$	mag		✓

The essence of the proposed method is extremely simple: to use multidimensional distances in the space of observational parameters given a unified scale to estimate the proximity of spectroscopic binaries from SB9U and SB9R and, on the basis of the value of the obtained parametric distances, to estimate the prospectivity of the observations of objects from SB9U in terms of the probability of being resolved. We denote by $\overrightarrow{x}={\left\{x_k\right\}}_{k=1}^{N_{\mathrm{k}}}$ the ordered set of SB9R object parameters, and we denote by $\overrightarrow{y}={\left\{y_k\right\}}_{k=1}^{N_{\mathrm{k}}}$ the same set of SB9U object parameters. From the parameters given in Table 1, we choose the following $N_{\mathrm{k}}=15$ values to estimate the parametric distances: $\left\{x_k\right\},\left\{y_k\right\}=($$logP$, e, $\omega$, ${\varpi }_{\mathrm{trig}}$, $F_{\mathrm{B}}$, $F_{\mathrm{V}}$, $F_{\mathrm{G}}$, $F_{\mathrm{J}}$, $F_{\mathrm{H}}$, $F_{\mathrm{K}}$, $K_1$, $K_2$, $\gamma$, $V_1$, $V_2)$. Hereafter, k is the number of the parameter (component of the parametric vector).

Here, we use the vector notation to illustrate the idea of the method, but, of course, the quantities compiled in this way are not true vectors. Since all observational parameters have distinct scales of values, it is first necessary to renormalise all components $x_k$, $y_k$ to the domain of values of the corresponding parameters for all SB9A objects and bring them to a unit value interval to equilibrate all parameters with each other. For the purpose of demonstrating the method, we assume that all $N_{\mathrm{k}}$ parameters are equally important and have the same weight while realising that this assumption can be false in some other cases, and the unequal weighting of the parameters can be easily accounted for by normalising the components differently. In order to renormalise $x_k$ and $y_k$, at first, we have to calculate the bounding box in the parameter space that would contain both SB9U and SB9R sets of objects. Hereafter, we imply that every i-th object in SB9R and every j-th object in SB9U have their own $x_{k,i}$ and $y_{k,j}$ values, so omitting i and j object indices means that the expression is applicable for all objects.

The lower bound $s_k$ is determined by the expression

$$s_k=min\left(\underset{i=1}{\overset{N_{\mathrm{SB}9\mathrm{R}}}{min}}\left(x_{k,i}\right),\underset{j=1}{\overset{N_{\mathrm{SB}9\mathrm{U}}}{min}}\left(y_{k,j}\right)\right)$$

(1)

where i is the number of the object in the SB9R set, and j is the number of the object in the SB9U set. The upper bound $u_k$ is determined accordingly by the expression

$$u_k=max\left(\underset{i=1}{\overset{N_{\mathrm{SB}9\mathrm{R}}}{max}}\left(x_{k,i}\right),\underset{j=1}{\overset{N_{\mathrm{SB}9\mathrm{U}}}{max}}\left(y_{k,j}\right)\right)$$

(2)

Given that $r_k=u_k-s_k$ is the range of values of the k-th parameter, we finally obtain the expression for the renormalised parameters:

$${\tilde{x}}_k={\displaystyle \frac{x_k-s_k}{r_k}},{\tilde{y}}_k={\displaystyle \frac{y_k-s_k}{r_k}}$$

(3)

In the following, we omit the notation of the dimensionless value (^~) and assume that all components of vectors $\overrightarrow{x}$ and $\overrightarrow{y}$ are already renormalised.

The mean values of $⟨\overrightarrow{x}⟩$ and $⟨\overrightarrow{y}⟩$ point to the centres of the sets of objects SB9R and SB9U, respectively, in the observation parameter space. The vector $⟨\overrightarrow{z}⟩=⟨\overrightarrow{y}⟩-⟨\overrightarrow{x}⟩$ defines the principal direction in the parameter space between the SB9U and SB9R sets.

When calculating the mean vectors $⟨\overrightarrow{x}⟩$ and $⟨\overrightarrow{y}⟩$, one has to take into account that, for every k-th parameter, there may be different numbers of objects for whom this parameter is measured and known from SB9cat. Therefore, the definition of the mean vectors can be written as follows:

$$⟨\overrightarrow{x}⟩={\left\{⟨x_k⟩\right\}}_{k=1}^{N_k},⟨\overrightarrow{y}⟩={\left\{⟨y_k⟩\right\}}_{k=1}^{N_k}$$

(4)

$$⟨x_k⟩={\displaystyle \frac{1}{n_{k,i}}}\sum _{i=1}^{N_{\mathrm{SB}9\mathrm{R}}}\left\{\begin{array}{c}{x}_{k,i},\mathrm{if}{x}_{k,i}\mathrm{is}\mathrm{defined}\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.n_{k,i}=\sum _{i=1}^{N_{\mathrm{SB}9\mathrm{R}}}\left\{\begin{array}{c}1,\mathrm{if}{x}_{k,i}\mathrm{is}\mathrm{defined}\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.$$

(5)

$$⟨y_k⟩={\displaystyle \frac{1}{n_{k,j}}}\sum _{j=1}^{N_{\mathrm{SB}9\mathrm{U}}}\left\{\begin{array}{c}{y}_{k,j},\mathrm{if}{y}_{k,j}\mathrm{is}\mathrm{defined}\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.n_{k,j}=\sum _{j=1}^{N_{\mathrm{SB}9\mathrm{U}}}\left\{\begin{array}{c}1,\mathrm{if}{y}_{k,j}\mathrm{is}\mathrm{defined}\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.$$

(6)

Also, for the principal direction vector $⟨\overrightarrow{z}⟩$, we have

$$⟨\overrightarrow{z}⟩=⟨\overrightarrow{y}⟩-⟨\overrightarrow{x}⟩,⟨\overrightarrow{z}⟩={\left\{⟨z_k⟩\right\}}_{k=1}^{N_{\mathrm{k}}},⟨z_k⟩=⟨y_k⟩-⟨x_k⟩,$$

(7)

and normalising this vector gives the weights indicating the significance of the parameters with regard to the difference between the SB9U and SB9R sets.

$$⟨\overrightarrow{\tilde{z}}⟩={\displaystyle \frac{⟨\overrightarrow{z}⟩}{|⟨\overrightarrow{z}⟩|}}={\left\{{\displaystyle \frac{⟨z_k⟩}{\sqrt{{\sum }_{k=1}^{N_{\mathrm{k}}}{⟨z_k⟩}^2}}}\right\}}_{k=1}^{N_{\mathrm{k}}}$$

(8)

Values for the principal direction vector components are provided in Table 2 in the “SB9U-SB9R mean diff.” and “Normalised mean diff.” columns.

Table 2. Properties of the parameter distributions of spectroscopic binary stars from SB9R and SB9U sets.

	SB9U				SB9R				SB9U-SB9R	Normalised
$x_k,y_k$	Min	Max	Mean	Stddev	Min	Max	Mean	Stddev	Mean Diff.	Mean Diff.
$logP$	−1.273	4.972	3.332	2.131	−0.056	5.056	4.440	2.527	−1.107	−0.203
e	0	0.972	0.423	0.315	0	0.975	0.739	0.457	−0.316	−0.376
$\omega$	−134.3	360.0	248.1	171.5	0	357.1	331.1	198.7	−83.04	−0.268
${\varpi }_{\mathrm{trig}}$	0.001	379.2	18.99	17.90	0.538	742.1	72.33	74.69	−53.34	−0.083
$F_{\mathrm{B}}$	−1.460	23.40	18.10	9.485	0.720	13.26	13.10	6.882	5.004	0.233
$F_{\mathrm{V}}$	−1.460	22.77	17.10	9.028	0.010	11.52	11.84	6.228	5.267	0.252
$F_{\mathrm{G}}$	2.089	20.94	17.32	9.204	1.714	9.934	12.08	6.223	5.237	0.316
$F_{\mathrm{J}}$	−2.300	18.63	14.84	7.990	−1.454	7.563	9.761	5.109	5.085	0.282
$F_{\mathrm{H}}$	−3.220	17.68	14.19	7.696	−1.886	7.209	9.218	4.848	4.979	0.276
$F_{\mathrm{K}}$	−3.510	17.49	13.95	7.584	−2.008	7.121	9.049	4.764	4.906	0.271
$K_1$	0.000	593.0	87.09	69.15	1.870	135.6	61.16	39.01	25.92	0.051
$K_2$	0.000	457.0	219.4	139.9	2.960	201.0	75.29	48.52	144.1	0.366
$\gamma$	−460.0	617.0	−6.448	47.30	−88.27	55.67	1.435	24.51	−7.883	−0.001
$V_1$	−1.470	22.50	17.24	9.096	−0.010	11.48	12.02	6.303	5.221	0.253
$V_2$	2.260	19.70	19.82	10.34	1.330	9.832	15.03	7.845	4.794	0.303

The proximity of the SB9U object to the SB9R region is determined by the distance $l_{\mathrm{m}}=|\overrightarrow{y}-⟨\overrightarrow{x}⟩|$. The most interesting measure of parametric proximity is the distance to the nearest SB9R object, $l_{\mathrm{x}}={min}_{i=1}^{N_{\mathrm{SB}9\mathrm{R}}}|\overrightarrow{y}-{\overrightarrow{x}}_i|$, since its evaluation allows us to directly generate a list of candidates from SB9U to RSB and point to specific comparison objects. The smaller the $l_{\mathrm{m}}$ and $l_{\mathrm{x}}$ of an object from SB9U, the higher the probability that it can be resolved. Thus, for each SB9U object, we have a proximity criterion for the whole SB9R set ($l_{\mathrm{m}}$) and for an individual SB9R object ($l_{\mathrm{x}}$).

When calculating the parametric distances $l_{\mathrm{m}}$ and $l_{\mathrm{x}}$, it should be taken into account that, for angular components, their arithmetic difference will not be the minimum angle difference, required for distance estimation, and the corresponding summands in the norm calculation should be calculated as ${\Delta }_k=min\left(\right|y_k-x_k|,|{360}^{\circ }/r_k-|y_k-x_k\left|\right|)$. In addition, it should be considered that, since the values of all parameters are not known simultaneously for all objects, for each object from SB9U or a pair of objects from SB9U and SB9R, when computing $l_{\mathrm{m}}$ and $l_{\mathrm{x}}$, the expression for the norm vector will only include the $n_k$ of the known parameter values out of the total number of $N_{\mathrm{k}}$ parameters. Hence, the resulting value of $l_{\mathrm{m}}$ and $l_{\mathrm{x}}$ is not distorted downward when $n_k<N_{\mathrm{k}}$ due to the use of fewer identically normalised components, i.e., so that the parametric distance between the objects does not become formally smaller just because fewer parameters are known about them from observations, and the values of $l_{\mathrm{m}}$, $l_{\mathrm{x}}$ must be corrected by the factor $c_k=N_{\mathrm{k}}/n_k$.

With all these considerations, we can rewrite $l_{\mathrm{m}}$ and $l_{\mathrm{x}}$ in a more detailed form. First, we define an absolute minimum difference between the k-th components as the $\mu (y_k,x_k)$ function:

$$\mu (y_k,x_k)=\left\{\begin{array}{c}|y_k-x_k|,\mathrm{if}{y}_k\mathrm{and}{x}_k\mathrm{are}\mathrm{not}\mathrm{angles}\hfill \\ min\left(|y_k-x_k|,\left|{\displaystyle \frac{{360}^{\circ }}{r_k}}-|y_k-x_k|\right|\right),\mathrm{if}{y}_k\mathrm{and}{x}_k\mathrm{are}\mathrm{angles}\hfill \\ 0,\mathrm{if}\mathrm{either}{y}_k\mathrm{or}{x}_k\mathrm{is}\mathrm{undefined}\hfill \end{array}\right.$$

(9)

Then, the parametric distance between two objects is just an Euclidean distance corrected for undefined quantities:

$$l(\overrightarrow{y},\overrightarrow{x})={\displaystyle \frac{N_{\mathrm{k}}}{n_k}}\sqrt{\sum _{j=1}^{N_{\mathrm{k}}}{\left(\mu (y_k,x_k)\right)}^2},n_k=\sum _{j=1}^{N_{\mathrm{k}}}\left\{\begin{array}{c}1,\mathrm{if}\mathrm{both}{y}_k\mathrm{and}{x}_k\mathrm{are}\mathrm{defined}\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.$$

(10)

Then, the distance $l_{\mathrm{m}}$ to the SB9R set centre for the j-th SB9U object is simply

$$l_{\mathrm{m}}\left(j\right)=l({\overrightarrow{y}}_j,⟨\overrightarrow{x}⟩)$$

(11)

and the distance $l_{\mathrm{x}}$ to the nearest SB9R object for the j-th SB9U object is

$$l_{\mathrm{x}}\left(j\right)=l({\overrightarrow{y}}_j,{\overrightarrow{x}}_m)$$

(12)

where m is the number of an SB9R object that has minimal distance l to the j-th SB9U object

$$m\left(j\right):l({\overrightarrow{y}}_j,{\overrightarrow{x}}_m)=\underset{i=1}{\overset{N_{\mathrm{SB}9\mathrm{R}}}{min}}\left(l({\overrightarrow{y}}_j,{\overrightarrow{x}}_i)\right)$$

(13)

3. Results and Discussion

An analysis of the distribution of observational parameters for the SB9U and SB9R sets shows that SB9R includes, on average, brighter stars with a larger orbital period and eccentricity and, correspondingly, smaller amplitudes of radial velocities. The lowest contribution to the difference between SB9R and SB9U is given by the radial velocity of the system $\gamma$, and the difference in the mean parallaxes ${\varpi }_{\mathrm{trig}}$ and radial velocities $K_1$ of the primary component of the binary system is somewhat more significant, but the contribution of all other quantities is approximately the same. The properties of the distributions of the observational parameters are given in Table 2: the columns SB9R (min, max, mean, stddev) and SB9U (min, max, mean, stddev) are given for SB9R and SB9U, respectively, containing the parameter value interval boundaries, mean values, and their standard deviations; the column “SB9U-SB9R” contains components of the vector $⟨\overrightarrow{z}⟩$, i.e., the difference between the mean values SB9U.mean-SB9R.mean; and the “Normalised mean diff.” column contains the values of the components of the normalised vector $⟨\overrightarrow{z}⟩/\left|⟨\overrightarrow{z}⟩\right|$ to estimate their relative contribution.

As a result of the above methodology, we obtained lists of SB9U objects ordered by the parametric distances $l_{\mathrm{m}}$ and $l_{\mathrm{x}}$. The most promising objects for observation, corresponding to the smallest values of the parametric distances, are listed in Table 3 and Table 4. Figure 1 (red points) shows the location of SB9U objects in $(l_{\mathrm{x}},l_{\mathrm{m}})$ coordinates. For comparison, the figure also shows the position of SB9R objects (red crosses).

Table 3. The most promising SB9U systems in terms of parametric distance $l_{\mathrm{m}}$ to the mean value $⟨\overrightarrow{x}⟩$ in the SB9R (to the centre of the SB9R object cloud). The RSBcat HIP column contains the Hipparcos number of the parametrically nearest (by $l_{\mathrm{x}}$) SB9R object.

SB9 ID	RSBcat HIP	V, mag	SpType	P. day	$\mathit{\varpi }$, mas	${\mathit{l}}_{\mathbf{m}}$	${\mathit{l}}_{\mathbf{x}}$
CD -49 11403	103055	10.849		1318.2	1.1176 ± 0.0197	0.322	0.381
G 9-27	83064	12.050		1018.53	3.71 ± 0.1113	0.380	0.538
HIP 109767	102545	10.100	K0	650.19	11.0174 ± 0.1428	0.451	0.248
HIP 64787	77725	9.190	G0V	3751.0	8.6413 ± 0.055	0.480	0.293
HIP 70534	10952	8.610	K0	26,000.0	2.2475 ± 0.0402	0.514	0.188
BD +17 2603	77725	9.800	G6V	9278.0	7.6548 ± 0.0267	0.523	0.303
HIP 77122	83064	8.950		4290.0	9.8175 ± 0.0579	0.552	0.348
BD +51 2091	102545	9.580	G5	247.08	3.5091 ± 0.024	0.554	0.276
HIP 13498	83064	7.730	G1V?	18,794.0	11.9650 ± 0.3677	0.555	0.177
HIP 17102	77725	9.050	K2	183.113	25.4343 ± 0.0319	0.565	0.310
BD +22 2581	77725	9.590	G0V	15,000.0	6.9482 ± 0.4934	0.570	0.333
GAIADR2 45574	102545	11.020	F9	18.4359535	2.7714 ± 0.0135	0.579	0.380
02515188596224
HIP 56229	77725	9.750	M0	186.3	22.7378 ± 0.0511	0.580	0.278
HIP 56622	100287	8.320	G7V	1690.15	21.1416 ± 0.0165	0.615	0.165
HIP 11537	77725	9.450	G5V	1146.44	15.5178 ± 0.4379	0.616	0.129
HIP 101472	77725	9.380	G8V	354.88	11.3768 ± 0.0977	0.621	0.224
HIP 88639	102545	10.600	K7V	25.7631	24.694 ± 0.0225	0.625	0.236
HIP 105879	98416	7.180	F7V	2935.6	12.4373 ± 0.2139	0.626	0.203
Gl 220	117415	9.918G	M1.5V	721.0	47.7313 ± 0.5115	0.641	0.399
HIP 117666	77725	8.730	G5V	10,984.0	13.3973 ± 1.2719	0.643	0.231
2E 3709	102545	12.230	K7	35.95	5.9041 ± 0.3012	0.651	0.391
HIP 107731	117415	8.640	G1/G2V	469.92	14.9306 ± 0.2675	0.652	0.224
HIP 2563	19508	8.720		2127.82	17.315 ± 0.1009	0.660	0.245
BD +28 387	102545	11.000	G0	41.2013	3.6606 ± 0.2174	0.669	0.359
BD +29 2313	102545	10.140	G8V	17.778584	7.5141 ± 0.0185	0.669	0.325
HIP 79979	98416	6.820	F9V	1083.16	25.5345 ± 0.232	0.671	0.165
BD +29 2313	102545	10.140	G8V	17.7786	7.5141 ± 0.0185	0.673	0.325
HIP 28432	4889	7.940	Am	60.6378	5.7711 ± 0.0371	0.674	0.338
RX J0532.1-0732	102545	12.570	K2-K3	46.85	2.7331 ± 0.0131	0.674	0.517
HIP 91935	102545	8.310	F5	34.8634	8.0971 ± 0.0169	0.675	0.151
HIP 3428	102545	10.970	K7V	97.25	22.8103 ± 0.0695	0.677	0.303
GAIADR2 30174	34603	11.130	F9IV	118.34	3.5561 ± 0.0279	0.690	0.568
02443157072768
ADS 8861	77725	9.510	dM1.5	200.26	72.9733 ± 0.3217	0.690	0.262
GAIADR2 33168	77725	11.890		352.49	6.8078 ± 0.0557	0.690	0.515
44608086144128
HIP 105585	19508	8.100	G8V	87,658.0	30.24 ± 4.85	0.690	0.364
Cl* Melotte	102545	12.400	K0III	94.4384	3.4746 ± 0.0166	0.695	0.583
111 AV 390
HIP 61732	77725	9.180	G5	595.18	6.852 ± 0.3285	0.697	0.307
HIP 84129	100287	7.140	F5	556.213	14.3755 ± 0.0255	0.699	0.229

Table 4. The most promising SB9U systems in terms of the parametric distance $l_{\mathrm{x}}$ to the nearest SB9R object, whose HIP number is given in the RSBcat HIP column.

SB9 ID	RSBcat HIP	V, mag	SpType	P. day	$\mathit{\varpi }$, mas	${\mathit{l}}_{\mathbf{m}}$	${\mathit{l}}_{\mathbf{x}}$
HIP 89925	99404	5.630	A5m	5.5146126	17.9191 ± 0.0489	1.264	0.050
HIP 64478	20284	6.350	G0V	4.23345	23.1989 ± 0.0946	1.246	0.053
HIP 105660	46704	6.160	A2V	6.9463	5.5323 ± 0.0768	1.207	0.063
HIP 3572	52980	5.630	A1V	4.4672235	10.6528 ± 0.0511	1.278	0.065
HIP 4572	52980	5.968	A0III	6.82054	9.1239 ± 0.0375	1.227	0.067
HIP 61727	117186	7.590	F5	54.87864	10.1467 ± 0.0382	1.033	0.073
HIP 92117	52980	5.900	A2Vm	4.7653	10.8978 ± 0.0549	1.236	0.073
HIP 66438	16042	5.630	F6V	8.0714	27.99 ± 0.58	1.380	0.074
HIP 63143	52980	5.820	A5m	5.1259	11.54 ± 0.51	1.877	0.075
HIP 95176	57565	4.610	B8p+F2pe	137.9343	2.0869 ± 0.1716	1.287	0.077
HIP 42871	111170	6.543	F7V	700.3	13.3908 ± 0.0374	1.014	0.079
HIP 31850	52980	6.360	F8IV	5.6983	27.6925 ± 0.0393	1.223	0.084
HIP 107238	46704	6.220	Am	6.3702	8.9617 ± 0.0285	1.216	0.085
HIP 759	18277	7.170	G2V	23.49785	19.3683 ± 0.0238	0.972	0.086
HIP 68064	46704	6.790	F2V	4.9918	9.3946 ± 0.0234	1.192	0.087
HIP 10961	46704	6.880	F5	4.2220168	16.0685 ± 0.0202	1.200	0.089
HIP 29746	20284	6.360	F6V	2.184952	13.6168 ± 0.8743	1.254	0.091
HIP 48273	20284	6.230	F6V	3.05459836	21.1803 ± 0.1565	1.399	0.091
HIP 43496	52980	5.540	A4m	2.8952	7.3105 ± 0.1725	1.420	0.092
HIP 41564	52980	6.380	A5m	5.9771	13.41 ± 0.79	1.202	0.092
HIP 10285	46704	7.170	A7m	4.1212	7.4227 ± 0.0192	1.176	0.093
BD +45 172	111528	8.840	G0	5556.0000	11.1479 ± 0.252	0.787	0.094
HIP 113770	46704	7.390	F8	5.21465	15.1122 ± 0.0242	1.164	0.095
HIP 88581	46704	6.870	O6.5V	6.14	0.7893 ± 0.0297	1.135	0.095
HIP 25776	22407	6.190	A0V	8.569	8.0907 ± 0.0381	1.482	0.095
HIP 32067	46704	6.370	O6e	3.078098	0.7632 ± 0.0417	1.227	0.096
HIP 48218	20284	6.090	A8IV	4.1467465	5.22 ± 0.63	1.242	0.096
HIP 84586	20284	6.670	G5IV	1.6817	32.9511 ± 0.0184	1.264	0.097
HIP 115500	22407	6.660	F8	6.0499	15.0601 ± 0.0228	1.626	0.097
HIP 2888	20284	6.780	G3V	3.7418	22.5227 ± 0.0179	1.212	0.098
HIP 81305	99404	5.470	O9Ia	9.81475	0.7601 ± 0.0709	1.273	0.099

Figure 1. Parametric distances from the SB9U (blue points) and SB9R (red crosses) objects to the parametric centre (mean position) of the SB9R set ($l_{\mathrm{m}}$) and to its nearest SB9R object ($l_{\mathrm{x}}$). Circles mark the first 30 objects with the smallest $l_{\mathrm{m}}$ (green) and smallest $l_{\mathrm{x}}$ (cyan) distances.

A complete list of promising candidates is available upon request.

As one can see in Table 3 and Table 4, the smallest $l_{\mathrm{x}}$ values are significantly lower than the smallest $l_{\mathrm{m}}$ values, and both tables have no common objects. At first glance, this looks like an inconsistency in the proposed methodology, as a solid result would have had a single list of objects, but, in fact, this indicates a more complex problem.

The value of $l_{\mathrm{m}}$ represents the distance to the SB9R population average, while the value of $l_{\mathrm{x}}$ represents the distance to the nearest neighbour in SB9R. Both of these measures are valuable but in different ways, and there is no fair reason to prefer one over the other. The distance $l_{\mathrm{x}}$ is more precise, as it reveals what SB9R object is the nearest neighbour to an SB9U object; additionally, since the smallest values of $l_{\mathrm{x}}$ are much lower than the $l_{\mathrm{m}}$ values, these particular SB9U and SB9R objects are parametrically closer than other objects in general, and, for that reason, both should probably be resolved. The distance $l_{\mathrm{m}}$ is a rough measure; the important thing here is not its value for a particular object but rather its range for all SB9R objects, as every SB9U object that falls within that range should probably be resolved. The basic idea behind the $l_{\mathrm{m}}$ values and Table 3 is that the closer to the centre of an SB9R cloud an SB9U object is, the higher the probability of its resolution should be; however, considering the SB9R mean position $⟨\overrightarrow{x}⟩$ an optimal parameter combination in terms of binary resolvability is, of course, a gross oversimplification.

Table 3 shows that, for the smallest $l_{\mathrm{m}}$ values, the corresponding $l_{\mathrm{x}}$ values are comparable. This means that, nearer the centre of the SB9R cloud, the population density of SB9U and SB9R is about the same. In comparison, Table 4 shows pretty large $l_{\mathrm{m}}$ values corresponding to the smallest $l_{\mathrm{x}}$ values, and this means that the population density of SB9U is much higher at the edge of the SB9R cloud than at its centre. This difference in population density in the parameter space is the main reason why a single list of best candidate objects cannot be produced within this methodology (at least not now).

If we limit ourselves to a subset of SB9U objects that have $l_{\mathrm{x}}\ge \min \left(l_{\mathrm{m}}\right)$, then it is possible for common objects to appear in both tables, but this would be a bad choice since a huge number of better nearest neighbour candidates would drop out. Alternatively, we can explicitly merge tables and sort objects by the $\min (l_{\mathrm{x}},l_{\mathrm{m}})$ value or by the $L_2$ norm of these measures $l_{\ast }=\sqrt{l_{\mathrm{x}}^2+l_{\mathrm{m}}^2}$; either way, such a result would not contain $l_{\mathrm{x}}<\min \left(l_{\mathrm{m}}\right)$ candidates.

Therefore, we propose to use Table 4 as the main list of candidates and Table 3 as a secondary guide and examine these spectroscopic binaries by other means.

It should be remembered that the SB9 catalogue contains orbital solutions rather than spectroscopic systems (multiple solutions are often given for a single spectroscopic system). In total, SB9 contains 5099 entries, of which 339 (7%) are SB9R and 4760 (93%) are SB9U. This means that the number of candidate stars is somewhat smaller than the number of candidate records. In addition, obviously, the probability of a spectroscopic binary to be resolved into components is also affected by other parameters: the angular distance between the components, the brightness of the system, the magnitude difference of the components, the orbit orientation, and so on. However, we believe that the values that we obtained can be used as rough estimates.

It should be added that another promising source of RSB candidates is the LAMOST survey [15,16]. Spectroscopic binary studies using LAMOST are being carried out quite intensively in [17,18,19,20]. This source also deserves the closest attention.

4. Conclusions

Resolved spectroscopic binaries (RSBs) are the only way (besides trigonometric parallax) to determine the dynamical, hypothesis-free distances to the stars of the Galaxy. Today, we know very few (less than two hundred) RSBs, so each new object makes a significant contribution to our knowledge of stellar distances.

We developed a method for estimating the probability of resolving the components of double-lined spectroscopic binaries (SB2), which depends on orbital element values and component parameters. The main points of this work are summarised as follows:

• The probability of the resolution of unresolved objects could be inferred from the similarity to resolved objects;
• Distances in the space of observational parameters could be used as a basic and straightforward measure of similarity;
• The highest similarity is achieved between individual objects, and it is loosely connected to the parameter space occupied by resolved objects.

The method was applied to the principal catalogue of spectroscopic binaries, SB9, and the results of this application are shown in Figure 1 and Table 1. A complete list of promising candidates is available upon request. Obviously, these results will be useful for organising astrometric/interferometric observations of spectroscopic binary systems in order to resolve them into components and add them to the list of resolved spectroscopic binaries.