Statistical Analysis of Binary Stars from the Gaia Catalog Data Release 2

In this paper we analyze the recent data from the new catalog DR2 Gaia Collaboration et al. (2018) obtained by the Gaia-ESA mission. If compared with the previous DR1 catalog Gaia Collaboration et al. (2016a, 2016b), the DR2 contains the cleaner data complemented with parallaxes and data on the 2D proper motion. Parallaxes allow us to determine the distance of the stars, so we can substantially enlarge our former DR1 analysis (Zavada & Píška 2018) and work with the 3D patterns of moving stars.

In the present study, we focus on the statistical analysis of the presence of binaries. This topic is related to the recent studies of various aspects of binaries with the use of the catalogs DR1 (Oelkers et al. 2017; Oh et al. 2017) and DR2 (Ziegler et al. 2018; Jiménez-Esteban et al. 2019). These authors, apart from their own results, present an up-to-date overview of important findings on binary stars. Other important papers exploring wide binaries can be cited from the era before Gaia, for example, Caballero (2009) and Close et al. (1990). However, our approach and objectives are rather different, so the results obtained are complementary.

Methodology for 2D analysis has been described in detail in our above-quoted paper. In Section 2 we repeat its essence and perform generalization for the 3D case. For 2D analysis in Section 3 we take the same region in the DR2 catalog as we used in the DR1, so we can compare results from both corresponding data sets.

Principal results are obtained from the 3D analysis of a sample of DR2 data and are presented in Section 4. This part deals with two issues: the analysis of 3D separations and the analysis of proper motion of  pairs of sources. The combination of both insights provides essential information about the statistical set of binaries. Obtained results are discussed in Section 5. Here we define the probabilistic function β(Δ), which is important for discussion on the occurrence of binaries. Our present catalog of binary candidates is described in Section 6, where we also shortly discuss its content and overlap with the catalog JEC—Jiménez-Esteban et al. (2019).

A brief summary of the paper is presented in Section 7. The Appendix is devoted to the derivation of some relations important for our statistical approach. The most important are distributions of separations of random sources uniformly distributed inside circles or spheres of unit diameter. The significant role of these functions for our approach is explained in Section 2.

The methods are designed for analysis of the distribution of stars inside circles or spheres covering the chosen region of the sky, as sketched in Figure 1. These 2D and 3D star patterns are called events. Input data for the generation of the event grids are supposed in the galactic reference frame. So, the position of a source is defined by spherical coordinates L, l, and b (distance from the Sun, galactic longitude and latitude):

In the center of circles or spheres we define a local orthonormal frame defined by the basis:

where defines angular position of the event center. Unit vector  k_l is perpendicular to  k_r and has direction of increasing l, see Figure 2. Unit vector  k_b is defined as and has direction of increasing b. Vector  k_r has radial direction, perpendicular vectors  k_b and  k_l lies in the transverse plain.

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2.1. Definition of Events

The 2D event of the multiplicity M is a set of stars with angular positions  n_i inside a circle with the event center  n₀ and a small angular radius ρ₂:

With the use of event local basis (2), the local coordinates are defined as

We define the 2D event as the set:

Since the DR2 catalog involves also data on parallaxes, we can similarly also generate the 3D events—patterns of the M sources with position  L_i inside the spheres with the center  L₀ and  radius ρ₃:

With the use of the star positions (1) and local basis (2) we define local coordinates {X_i, Y_i, Z_i} as

where p_i is parallax, and L_i is distance of the star. The 3D event is defined as the set:

2.2. 2D Methods

The first method is based on the Fourier analysis of 2D events, where we have introduced characteristic functions Θ_n(M) depending on the event multiplicity M. These functions are generated by a set of events and measure statistical deviations from uniform distribution of stars , for instance a tendency to clustering or anticlustering . Details of the method are described in Zavada & Píška (2018), in this paper we will present only the result.

With the use of a second complementary method, we analyze distributions of angular separations of sources inside the 2D events (5). Distribution is generated from the set of events. We use either absolute separations

or scaled ones

The suitable unit of the parameters x_i, y_i, ρ₂ will be for our purpose 1 as (1″). Distribution of scaled separations generated by Monte Carlo (MC) for a uniform distribution of stars in the sky is shown in Figure 3. The exact shape of normalized MC distributions reads

where the functions EllipticK (EllipticE) are complete elliptic integrals of the first (second) kind. The proof is given in the Appendix. These distributions do not depend on the event multiplicity and radius, this is an advantage of the scaled separations. Obviously, we have always . These exact functions replace their approximations resulting from the MC calculation applied in the previous paper.

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2.3. 3D Methods

Similarly, as in the 2D case, we shall work with absolute separations

and/or with the scaled ones

The suitable unit of the parameters X_i, Y_i, Z_i, ρ₃ is for our purpose 1 pc. Distribution of scaled separations generated by MC from the uniform distribution of stars in the 3D region of the sky is shown in Figure 4. Exact shapes of these normalized MC distributions read

as proved in the Appendix. Shapes of these distributions similarly to (12) and (13) do not depend on event multiplicity and radius. The analysis with the use of characteristic functions Θ_n(M) could be in the 3D case done separately in the plains XY, YZ, and ZX. However, such analysis is not the aim of the present paper.

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2.4. Aims

Here we present the results obtained from regions of the DR2 catalog listed in Table 1. The regions are shown in Figure 5. The corresponding events are created with the same angular radius as in Zavada & Píška (2018), which allows us the consistent comparison of results from the DR1 and DR2 catalogs. First, we checked the events covering the regions N and S. Their nonuniformity defined by the characteristic functions Θ_n(M) is demonstrated in Figure 6. The clear result Θ_n(M) > 1 indicates the presence of clustering.

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Table 1.  Analyzed Regions in the DR2 Catalog, Where ρ₂ Is the Angular Radius of the Events, Are Average Distance and Event Multiplicity, and N_e Is the Total Number of Events

	2D Region: l × b(deg2)	ρ2(as)			Ne
N and S		72	1290	3.21	2055,674
C		18	1912	2.75	3588,183

Note. Only sources in distance 1–5000 pc are taken into account. The analysis is done for events 2 ≤ M ≤ 15.

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Corresponding distributions of angular separations are shown in Figure 7 together with curves (12) and (13). These results can be compared with those in Figures 7 (lower panels), 10, and 11 in the former paper. We observe:

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(i) The peaks at small angular separations in the DR2 corresponding to binaries are clearer and more pronounced than in the DR1 catalog. Panels (e) and (k) in Figure 7 demonstrate the double stars separated by are absent because such close pairs are not resolved in the DR2 data set as reported in Arenou et al. (2018). In both catalogs, we observe an excess of binaries in the region N and S for or equivalently for d_ij ≲ 8.6 as. For greater separations inside the event, we observe perfectly uniform distributions of stars. Note the data and curves are equally normalized for . That is why the strong peak in panels (g)–(i) is balanced by a small reduction of distribution outside the peak. Brighter stars (G ≤ 15, panels (g)–(l)) show evidently stronger peaks than the sample without any cut on magnitude (panels (a)–(f)). A similar tendency was already observed in the catalog DR1.

(ii) A more pronounced presence of binaries is demonstrated also in Figure 6. The slopes of lines in DR2 are greater than in DR1—clustering is more obvious. For M ≤ 8 the slopes are ≈4%(8%) for the DR1(DR2) data. In Figure 8 we have shown some results obtained in a more populated region C. Also here we can observe a clear peak at small angular separations of sources of the magnitude G ≤ 15, which proves the presence of binaries. Panel (b) again demonstrates the absence of double stars separated by due to insufficient resolution. Different scales of in Figures 7(l) and 8(c) are due to different radii ρ₂ of events from N and S and C regions. Similar plots could be presented for whole spectrum of magnitudes in region C; however, elevation above the red line due to binaries is much less than that in Figure 8(c). The reason can be that denser region C with all magnitudes generates higher background of the optical doubles and a consequently lower relative rate of binaries.

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We present the results obtained from the 3D region defined in Table 2. The parallax and angular components of the star proper motion are the parameters, which substantially enrich the recent Gaia data. We work with the 3D events (9).

Table 2.  Region of 3D Analysis is Cube Centered at the Origin of the Galactic Reference Frame

Region				Ne
Cube of edge 400 pc	2	188	6.8	727,744

Note. Only sources of positive parallax are included. ρ₃ is the radius of events, are average distance and event multiplicity, and N_e is the total number of events. The analysis is done for events 2 ≤ M ≤ 15.

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4.1. Analysis of 3D Separations

The summary results of the analysis obtained from all magnitudes G are shown in Figure 9. Distributions of scaled separations in panels (a)–(c), (g), and (h) perfectly match the uniform distribution of sources, but with the exception of the first bin in (a), (b), and (h). Apparent excess of very close pairs in planes XY, YZ, and ZX is seen also in panels (d)–(f). However, the sharp peak can be observed only in the plane XY (panel (d) and its magnified version (j)). Smearing in the direction of Z (difference of radial positions, panels (c), (e), and (f)) is due to lower accuracy in measuring of parallaxes. The errors of local coordinates depend on the errors of separations and with the use of definitions (7) and (8) are calculated as

and similarly

Note that , where p is parallax. Obviously δZ_ij ≫ δX_ij, δY_ij since . That is why we prefer distributions of Δ_ij to D_ij for obtaining precise results. Maximum values of separations X, Y, and Z is 4 pc, which follows from the event radius ρ₃ = 2 pc. The excess of close pairs is obvious most explicitly from the distribution of Δ_ij in panel (k) that represents a magnified version of (h). The important ratio of the distribution P(Δ_ij) to the uniform simulation (red curve in panel (h) rescaled to ) is shown in logarithmic scale of Δ_ij in panel (i).

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The same distributions but for brighter sources G ≤ 15 are shown in Figure 10. Similarly, as in the 2D case, the peaks are stronger for brighter sources and distributions outside the peaks confirm uniformity of the star distribution. Again due to equal normalization of data and red curves for the strong peak in panels (g) and (h) is balanced by a small reduction of distribution beyond the peak. The excess of close pairs observed in both figures again convincingly indicates the presence of binaries.

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For quantitative estimates, the important panel is (i), which displays the ratio data/simulation. This is more accurate than only displaying peaks with some undefined background. Panel (i) suggests that separations of binary systems in the analyzed region meet very approximately

We observe only a tail of distribution corresponding to more separated binaries. Closer pairs are absent due to the limited angular resolution in DR2 data. This result is compatible with the older data reported in Close et al. (1990). In Section 5 further discussion is devoted to the probability of the binary separation above this limit. We have checked that sampling with events generated by spheres of different radius (ρ₃ = 5 pc) does not change the approximate result (22).

4.2. Proper Motion of Binaries

The proper motion of the stars in DR2 is defined by two angular velocities

in directions of the R.A. and decl. in the ICRS. So the corresponding transverse 2D velocity  U is given as

where L is distance of the star calculated from the parallax (8). For the pair of stars we can define:

where α_ij is the angle between both transverse velocities. The corresponding errors read

where

Note that relative error δv_ij/v_ij can be large, since v_ij is small compared with U_ij and the errors and are the same. In Figure 11(a) we show the distribution of the velocities U of the stars from the region defined in Table 2. Distribution of the corresponding pair angles α_ij is presented in Figures 11(b)–(d) for different regions of Δ_ij.  In Figure 12(a) we have shown the correlation of pair transverse separations and angles . We observe a very narrow peak in the region of small Δ_ij and .¹ The peak is connected with presence of binaries as follows. The transverse velocities of two gravitationally coupled stars are

where V is transverse velocity of their center of gravity and v_i, v_j are transverse projections of instantaneous orbital velocities, they have always opposite direction. Dominance of very small α_ij means that

so for binaries in our Δ window (Δ_min is given by resolution of two close sources and Δ_max = 0.1 pc by (22)) we have

For comparison, we have generated an MC plot from uniform distributions of positions and velocity directions, which is shown in Figure 12(b). Obviously, uniform distribution contradicts Figure 12(a), which reflects the presence of binaries (peak at small α_ij and  ). The corresponding α_ij peak is also observed in Figures 11(b)–(d) on the background of the collective motion of stars (dominance of α_ij < 90°). The peak is suppressed for Δ_ij ≳ 1 pc. Selection of comoving systems is a basis of the methodology applied in the catalog JEC.

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In Figure 13 we show correlations of the velocities V and v_ij with Δ_ij in the region of small separations, where the binaries are present. With the use of v_ij and Δ_ij we try to roughly estimate the orbital period of the binary star. To simplify the calculation, we assume the space binary orbits are circular and the star 3D separation is a (semimajor axis). There are the extreme cases:

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(A) M₁ ≈ M₂, then

Orbital period is

where w is the space orbital velocity and a equals the diameter of the orbit.

(B) M₁ ≫ M₂, then

but the orbital period is different

since the separation a equals to the orbit radius.

At the same time, Kepler’s law implies for orbital period T_g:

where G is gravitational constant and is mass of the star system. For units , and we have

This relation also allows us to estimate the period. If the plane of orbit is perpendicular to the line of sight (axis Z), like the orbit N in Figure 14(a), then it is possible to simply substitute:

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in (32), in (34) and in (36), (32), and (34).

Then we get:

The orbit reference frame in the figure is similar to the local 3D event frame defined by the basis (2) and coordinates (7). But its origin  L₀ is defined by the actual position of the center of mass of the binary. How does one deal with the orbits whose plane is not perpendicular to the line of sight like the orbit K in the same figure? The orbits in the figure are defined as follows:

where r = a/2 for the case (A), r = a for (B), and ϕ is azimuthal angle in the plane XY. The orbit K inclined at an angle θ is observed only in its projection P. Corresponding observed separation between the stars is Δ_ij:

Random angles ϕ, θ generate distribution of ε₁ shown in Figure 14(b). The MC distribution demonstrates smearing of the real separation a due to random ϕ and slope θ of the orbit. Similarly, the ratio a/w is distorted as

Since velocity  w is perpendicular to  r, there is exchange in denominator. Corresponding distribution of ε₂ is shown in Figure 14(c). The mean values are

and represent a scale of distortion of real orbital periods, if replaced by relations (37). More accurate estimate of the periods in some region of Δ_ij can be obtained by rescaling of these relations:

We have estimated the average periods from the maximum in Figure 13. If we take the sources roughly in the region of half width of the maximum,

then

and one can check that equality implies estimation M_tot ≈ 0.8 M_⊙.

The separation of a pair of stars and the similarity of their movements can serve as two signatures of the binaries. We can compare them:

(i) Distributions of the 2D and 3D separations are studied in Sections 3 and 4.1. The procedure is simple, the distribution of separations P(Δ) (within the defined circles or spheres) is compared with the corresponding separations P_bg(Δ) generated by uniformly distributed sources representing background

where P₀ is given by (12) or (18) and N_bg is the corresponding number of the background pairs in the data events. Its accurate calculation is described below, see Equations (47)–(49). The binary distribution reads

and the probability that the pair is a real binary is given as

The function displayed in Figures 7(f) and (l), 8(c), 9(i), and 10(i) is another representation of the probabilistic function :

Consider the distribution from Figures 9(k) and 10(k). The corresponding function is shown in Figure 15(a), where the red curve represent fit of the function:

The result of the fit is shown in the first row (all pairs) of Table 3. The longer tail corresponding to the second term describes the small probability of bound pair at greater separations: Δ ≳ 0.15 pc. This tail is not visible in panels (i) in Figures 9 and 10. Two exponential terms in may correspond to two different classes of binaries. The question is to what extent the excess of wide binaries consists of stable bound systems. Part of the excess may be an image of widening pairs that were less separated but weakly bound in the past. The accuracy of the method is based on three conditions:

1. 1.  
   precise separation measurement in a suitably selected statistical set of events that generates ,
2. 2.  
   precise modeling of the background defining P_bg(Δ), and
3. 3.  
   relatively high peak and low background giving probability close to 1 in the peak region.

(ii) In principle, a similar approach could be applied to comoving pairs. However, it is obvious that meeting the conditions above is more difficult for velocities or their differences. The precision of velocity measurement is lower than separation. The distribution of velocities is far from being simply uniform, we do not know the exact form of the velocity background. Figure 11(b) suggests a more complicated collective motion in the background and a relatively low peak of binaries. That is why we prefer the primary signature based on the spatial separation, where we work with exactly defined background and relatively high peaks, like panels (i) and (k) in Figures 9 and 10. On the other hand, one can expect that a combination of position and velocity data will suppress background and improve the selection of binaries in terms of the function . A selection based on such a combination is described below. In Section 4.2 we have shown that combination of the space separation with proper motion allows us to estimate the orbital period.

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We denote by and the numbers of real and false (background) binaries in some domain D of separations Δ. Correspondingly we denote , where is the number of pairs in a given data set. We have

where P_bg is distribution defined by (18) and renormalized in such a way that

where the integration is over the domain E safely outside the peak of binaries. So we calculate

where P₀ is defined by (18) with the substitution . The same procedure can be applied for the domains . Since we can assume that Δ and α are not correlated in the background distribution

then

If we choose then the second ratio is 1. In this way, we can calculate N_bg^D without the knowledge of The selected domains are shown, together with the results β ≈ N_bin/ N₂ in Figure 16. For the calculation, we take the common interval pc, where N_bin^E should be zero. In accordance with Figure 12(a) we observe the highest rate of binaries in the domain deg. The binaries are well observable also in the neighboring domain deg. For G ≤ 15 the number of binaries in the domains C is zero within statistical errors. Presence of binaries in the domain D requires further analysis.

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The probabilistic function corresponding to the united domain AB is shown in panels (b)–(d) in Figure 15 for different intervals of the magnitude G. Obviously, the function (46) fitted with the parameters listed in Table 3 can be rather approximate. We observe that function β is getting wider with decreasing G. This may suggest that brighter and therefore statistically more massive stars can form stable bound systems even at greater separations. Obviously, for Δ ≳ 1 pc, the probability β is compatible with zero. This also corresponds to an absence of a binary peak in Figure 11(d), which relates to Δ > 1 pc. Total numbers of binaries are listed in Table 4. The results suggest that in the Gaia DR2 data in the region defined by Table 2, the number of binaries can represent ≈2% of all stars in this region.

In this section, we describe the catalog of binary candidates, which we have created from the events of multiplicity 2 ≤ M ≤ 15 defined by Table 2. For the first version of the catalog we accept only the candidates from domain A shown in the first panel in Figure 16. So, we do not accept all candidates, but only candidates with a high probability to be the true binary. The candidates meet the following conditions:

(1) Projection of separation

We accept the pairs, which satisfy

In general, the projection of separation Δ depends on the reference frame. In the paper, we worked with the local reference frame defined by the event center, where projection Δ into the local plane XY is given by (15). In the catalog we do not use local frames. The cut (52) is applied to Δ_cat, which is defined as the length of the arc

where L_α,  n_α are defined in Section 2.1. The separations Δ and Δ_cat are not exactly equal, but we have checked that in our conditions their difference is small, pc. Then the sharp cut on means only a slightly smeared cut of the distribution of Δ.

(2) Projection of collinearity

The pairs must meet the condition

Both conditions define the domain A in the first panel of Figure 16. The panel shows that the average probability of a binary star is β ≈ 84%. If the stars are brighter, (second and third panel), then the average β is almost 100%.

(3) Radial separation

In fact, the radial separation is not explicitly used in our algorithm for selection of binaries. The reason is a rather low precision of radial separations, as explained in the discussion of panels (c), (e), and (f) in Figures 9 and 10. The only constraint is given by the diameter of our events (4 pc). Separation selection is based solely on Δ_ij. Additional cuts on inaccurate radial separation would eliminate many of the real binaries and invalidate the function β calculated for Δ_ij. We obtained high β even without a cut on radial separation.

Furthermore, it is evident that spherical events fill the space only partially (≈52%). In this way, half the stars are lost for analysis. We also lose binaries between two adjacent events when each star falls into another. In order to recover these losses, we work with the  modified coverage:

• (i)  
  The event spheres are replaced by cubes of edge 4 pc with no gaps between them. In each cube, we search for the pairs meeting the conditions (52) and (54).
• (ii)  
  The procedure is repeated with the same cubes centered in the corners of the former cubes and the search results are merged.

The catalog is represented by a matrix that is defined as follows. Each line represents one star and the following data are in the columns:

1–2: Group ID and Group size to match stars with the group they belong to.²

3–96:  Copy of the original entry for the star from Gaia-DR2 archive,³ according to the documentation.⁴

97–98: Minimum and maximum angular separation of the star from other stars in the group [as].

99–100: Minimum and maximum projected physical separation of the star from other stars in the group [pc].

The summary data from our catalog of binary candidates (I), along with the data extracted from the catalog JEC - Jiménez-Esteban et al. (2019) (II) is shown in Table 5. The number of candidates N₂ in this table correspond to N₂ in the domains A in Figure 16 after increasing with the repeated covering. The comparison of the two catalogs shows the following:

• (a)  
  We can only compare sources of magnitude G ≤ 13, because (II) does not contain less bright stars. Of the total number 3055 binaries in (II), only 301 lie in the (I) cube . Increasing the edge of this cube by the factor 2.15 would increase volume 10 times with the number of binaries comparable to the total (II).
• (b)  
  The number 301(II) could be compared with the corresponding number of candidates C₂ = 9082(I). However, in the JEC catalog, apart from the cut G ≤ 13 many other restrictions and selections are made. Definition of the binary is not the same in both catalogs. In our opinion, this is the reason for the difference between (I) and (II).
• (c)  
  (I) and (II) have 108 common binary candidates.
• (d)  
  86(II) candidates are absent in (I) since the separation Δ exceeds 0.15 pc. These candidates do not contradict our general criteria, but the corresponding probability can be lower as shown in Figure 15.
• (e)  
  54(II) candidates are absent in (I) since their spatial separation exceeds 4 pc (event diameter). Such candidates may not contradict our criteria; however, due to a great background can be extremely low.
• (f)  
  18(II) candidates are absent in (I) since these candidates are located in a dense area generating the high multiplicity events, which exceed our currently set limit.
• (g)  
  The last 35(II) candidates are absent in (I), mainly due to the fact that even after the second coverage some couples (II) remain separated in two neighboring event cubes.

Table 5.  Summary Table of Binaries from the Catalogs (I) and (II), See the Text

Catalog		G
(I): Domain A		All
		≤15
		≤13
(II):  Total  Nbin = 3055		≤13

Note. N₂ is the number of binary candidates, N_bin is the real expected number of binaries. The data in the second column are related to the full cube region (Table 2).

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The total number of the binary candidates of all Gaia magnitudes in the catalog (I) is 80,560, which corresponds to the expected real number of binaries 67,670. The full current catalog (I) in the csv form is available on the website https://www.fzu.cz/~piska/Catalogue/. We plan to further develop and optimize our catalog methodology.

We have proposed a general statistical method for analysis of finite 2D and 3D patterns. In the present study, the method has been applied to the analysis of binary star systems in different regions of the Gaia catalog DR2.

Results on 2D statistical analysis were compared with our former results obtained from the previous catalog DR1. The new results give in the distribution of angular separations more clear evidence of binaries. Independent signature follows from the characteristic functions , which clearly indicate a tendency to clustering. However, the most important results are obtained from the 3D analysis introduced in the present paper. We have analyzed about 5 × 10⁵ of events inside the cube of edge 400 pc centered at the origin of the galactic reference frame. In distributions of pair separations, we observe the sharp peaks at small separations corresponding to binaries, which are more striking for brighter sources, G ≤ 15.

The important result of the analysis is probabilistic function , which depends on the separation Δ of a pair of stars and indicates the probability that the pair constitutes a bound system. The function suggests that brighter, more massive binary stars have on average a greater separation. With increasing separation the function falls rapidly. We obtained the ratio binaries/singles ≈2%.

Furthermore, we had shown that a combined analysis of 3D separations with the proper motion of the pairs of sources gives a clear picture of the binaries with two components of the motion: parallel and orbital. The analysis allowed us to estimate the average orbital period and mass of the binary star system in the chosen statistical ensemble.

The highest probability of the binary is observed at smallest separations Δ and angles α between proper motions. From the corresponding domain deg, we have created the catalog involving 80,560 binary candidates, which represents 67,670 of the true binaries.

This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular, the institutions participating in the Gaia Multilateral Agreement. The work was supported by the project LTT17018 of the MEYS (Czech Republic). Furthermore, we are grateful to J. Grygar for deep interest and many valuable comments and J. Palouš and O. Teryaev for very useful discussions and inspiring comments.

(i) Relation (12)

First we consider two random points on a segment L. The probability that the points are separated by interval l reads:

Furthermore, we suppose a circle of diameter with a chords L(y) involving random segment l (Figure 17). We have

The probability of interval l reads

This distribution is generated by random pairs on the chords parallel to axis x. For arbitrary random pairs separated by l inside the circle, we integrate distributions (57) over all directions in 2D and replace

which gives distribution

Relation (12) is its normalized form.

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(ii) Relation (13)

Distribution (59) can be modified

Calculation of integral

with the use of Wolfram Research, Inc. (2019), and after replacement and normalization gives relation (13).

(iii) Relation (17)

Now instead of circle 2R = 1, we consider the sphere of the same radius. The procedure is a modification of the case (i). Now instead of integral (57) we get

where y means radius of a cylinder of parallel chords. The additional y in the integral means that we integrate chords on surfaces of cylinders of different radii. Then instead of (58) we use

since the integration of chords is over all directions in 3D. The resulting distribution reads

which after normalization gives relation (17).

(iv) Relation (19)

Probability of random segments inside the sphere can be expressed as

which together with (64) gives

The probability that segment l has projection l_x is given as

The last integral (equally for l_y, l_z) can be after inserting from (66) easily calculated, and after normalization gives relation (19).

(v) Relation (18)

In a similar way, the probability that segment l has projection is given as

which after inserting from (66) and integration with the use of Wolfram Research, Inc. (2019) implies relation (18).