=Paper= {{Paper |id=Vol-3036/paper16 |storemode=property |title=Star Clusters, Planets, Asteroids and Comets in the Light of Big Data |pdfUrl=https://ceur-ws.org/Vol-3036/paper16.pdf |volume=Vol-3036 |authors=Maria Sizova,Sergei Vereshchagin,Aleksandr Tutukov,Andrei Fionov |dblpUrl=https://dblp.org/rec/conf/rcdl/SizovaVTF21 }} ==Star Clusters, Planets, Asteroids and Comets in the Light of Big Data== https://ceur-ws.org/Vol-3036/paper16.pdf

Star Clusters, Planets, Asteroids and Comets
in the Light of Big Data

Maria Sizova1 , Sergei Vereshchagin1 , Aleksandr Tutukov1 , and Andrei Fionov2
1
Institute of Astronomy, Russian Academy of Sciences, Pyatnitskaya str., 48, 119017
Moscow, Russia sizova@inasan.ru
2
OCRV (Russian Railway), Triumphalnaya 1, Sochi, 354340 Russia
fionovs@mail.ru

Abstract. Planetary (exoplanetary) systems can lose comets, asteroids,
and planets due to their host stars close approaches. Such encounters may
occur during stars motion in the Galaxy disk. In this work, we calculate
the number of pairwise encounters of the stars inside the open star cluster
Hyades and in the selected volume of field stars in order to estimate the
number of interstellar small bodies in the Galaxy. The resulting catalog
was compiled with data obtained by the Gaia spacecraft. Gaia data
amount is increasing and systematically updating. These factors are fit
into the Big Data concept.

Keywords: Open star clusters · Solar System · Planets · Exoplanets ·
Comets · Big Data

1 Introduction
Recent researches of seemingly completely different phenomena, such as an in-
creasing number of exoplanets (for example, see [2]), rogue planets [9], exocomets
[10], interstellar comets [4] and the evolution of open star clusters (OSC) [17] al-
lowed a new look at the scale of the exchange of matter between celestial bodies
and systems. To estimate the scales of the phenomena, it is necessary to make
calculations based on available Big Data. In addition, the calculations become
statistically significant only when there are conclusions based on a large amount
of data.
If the star has a planetary system, thus, it has asteroids, comets, and planets
(ACP). In the paper [22], based on the analysis of the distribution of binary stars
by angular momentum, it was shown that nearly 30% of stars have planetary
systems. Modern observational data obtained with the Kepler spacecraft [11] do
not contradict this estimate.
The interstellar ACP may occur when stars approach each other. The first
idea to calculate the influence of close passages of stars on the Oort cloud comets
was proposed by [20]. The author showed that the effect of changing the speed of
comets in the Oort cloud of the Sun becomes significant at approaches distances
less than 1 pc. Nowadays, using modern Gaia spacecraft data, the investigation
of the close encounters of the field stars with the Solar system and resulting

Copyright © 2021 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).

213
effect on the Oort cloud comets is carrying out by many authors (for example,
see [21], [3]).

Purpose. We aim to estimate the volume of the population of small interstellar
bodies in the Galaxy generated by paired stellar encounters of the field stars
and encounters within OSCs. Then, we integrate stellar orbits backward up to
500 Myr, and calculate the close approaches of a limited sample of stars, then
extrapolate the obtained data to the Galaxy.

Data and Methods. We use the last available release of the Gaia spacecraft
catalog [6], which is currently best suited for the implementation of our task. To
integrate the stellar orbits, we use galpy [5] for python programming language.

Structure. In section 2 we explain how we proceed data and show our first
results of the calculations (pairwise encounters of the stars samples), section 3
explains the main result of our calculations (close approaches up to 1 pc) and
makes some predictions about the number of the ACP in the Galaxy. Work
summarizing presented in section 4.

2 Gaia Data Description
In order to statistically estimate the star paired approaches frequency in the
Galaxy disk and find stars that had close encounters up to 1 pc, we used the
most precise available data from Gaia Early Data Release 3 (Gaia EDR3 [7]).
We aim to compare interstellar ACP production by selected volume of the field
stars and OSC, hence we chosen the most studied open cluster Hyades.
Our task is to choose Gaia EDR3 stars for which is available data allowing
us to integrate the orbits in their motion around the Galactic Center. Thus, we
load data and filter stars with available positions on the sky (right ascension and
declination α, δ), proper motions (µα , µδ ), parallaxes (π) and radial velocities
(Vr ).

Data Quality and Quantity. The full astrometric Gaia solution (5 parame-
ters) – α, δ, µα , µδ , π – for around 1.468 billion sources, with a limiting magnitude
of about G ∼ 21 (mag hereinafter) and a bright limit of about G ∼ 3 (for more
details, see [1]).
The parallax uncertainties are
– 0.02 − 0.03 mas for G < 15
– 0.07 mas at G = 17
– 0.5 mas at G = 20
– 1.3 mas at G = 21
The proper motions uncertainties are

214
– 0.02 − 0.03 mas/yr for G < 15
– 0.07 mas/yr at G = 17
– 0.5 mas/yr at G = 20
– 1.4 mas/yr at G = 21
Radial velocities at Gaia EDR3 hence contains Gaia DR2 (Data release 2)
median radial velocities for about 7.21 million stars with a mean G magnitude
between ∼ 4 and ∼ 13. The overall precision of the radial velocities at the bright
end is of the order of ∼ 200−300 m/s while at the faint end, the overall precision
is ∼ 1.2 km/s.]

Data Filtering and Completeness. According to Gaia EDR3 data, we fil-
tered stars with distances closer to 20 pc to the Sun (π ≥ 50 mas). We obtained
2626 stars, including 677 stars with available radial velocity Vr (according to
Gaia DR2). We selected stars with Vr relative error less than 20%.
For the Hyades OSC, we match the latest available catalog of the Hyades
stars [13] with Gaia EDR3 catalog.
Both field stars and Hyades stars samples are presented in Figures 1,2 in the
projection on the Galaxy plane. Black dots represent the stars, color represent
levels of equal density (nlevels = 10).
For the field stars, we can estimate the completeness of the data by con-
sidering the dependence of the number of stars on the distance to them. The
completeness of the sample is sufficient until the moment when the number of
stars begins to decrease. Thus, as we can see in Figure 3, the completeness of
our stars sample is up to ∼19 pc.

Integration Method. We integrated backward on 500 Myr field stars and
Hyades stars during its motion in the Galaxy disk using galpy. The main as-
sumption is the representation of the stars as a point.
In galpy, Galaxy is represented with Milky Way potential (MWPotential2014),
described in [5]. This potential reproduces the Milky Way rotation curve and
includes the disk, bulge, and spherical (halo) components of the Galaxy. The disk
is given in the Miyamoto-Nagai expressions ([18]) and the spherically symmetric
spatial distribution of the dark matter density in the halo by the Navarro-Frank-
White profile ([19]).
To calculate the minimal distance dmin between each stars pair, we unloaded
the galactocentric cartesian coordinates X, Y, Z and applied an algorithm to
calculate the euclidian distance between each star pair. Minimal distance is a
distance of the minimal approach between pair of stars during their motion in
the Galaxy disk on a considered time interval. Then, for each pair we selected
dmin and corresponding time tmin . The algorithm avoids repetition, thus, we
get a complete statistical picture of the distances between all the stars we have
chosen on the considered time interval.

Calculation Results. As a result of backward orbit integration, we listed min-
imal distances dmin and corresponding times tmin of the stars pairs.

215
Fig. 1. Field stars sample according to Gaia EDR3 in Galacticentric cartesian coor-
dinate system. Directions of the coordinate axes: the X-axis is directed to the Galaxy
center, the Y-axis is in the direction of the Galaxy disk rotation, and the Z-axis is to
the North Pole of the Galaxy.

Fig. 2. Hyades OSC member stars sample according to Gaia EDR3 in Galacticentric
cartesian coordinate system. The white circle (r = 10 pc) shows the cluster core.

216
Fig. 3. Dependence of the number of field stars (ordinate or vertical axis) of our sample
(see Fig. 1) on the distance to them (abscissa or horizontal axis, pc).

Fig. 4. The result of the integration of the filed stars sample (see Fig. 1) The black
dots show pairs of stars, the minimum distances between which were dmin at the time
tmin on the considered time interval 500 Myr in the past.

217
Fig. 5. The result of the integration of the fieled stars sample for the time interval up
to -20 Myr.

Fig. 6. The result of the integration of the Hyades star cluster stars sample (see Fig. 2).
Here are clearly visible places of point concentration, or ”resonances”, arising due to
differences in the orbits of stars from circular orbits.

218
3 Close Star Pairs
Now, we consider such pairs that approach close enough to affect the outer
boundaries of the Oort clouds of each other. Results for field pairs and Hyades
pairs with minimal distance less than 1 pc are shown in Figure 7. Table 2 contains
list of the dmin –tmin and ID according to Gaia EDR3 for the filed and Hyades
stars (see Appendix A). Extremely close pairs (dmin < 0.05) in the Table 2
could be a binary stars, or an encountering pairs with accidentally unaccounted
calculation error.

Fig. 7. The result of the integration of the Hyades star cluster stars sample (see Fig. 2).

Affecting of the Input Parameters Errors. The integration output depends
not only on the orbit parameters but also on the Sun distance to the Galaxy
center and its circular velocity R0 and V0 . R0 is defined in dozens of publications,
different authors received values in the range of 7.4 to 8.7 kpc. In [14] was
investigated so-called ”majority merging effect” consisting in choosing closer to
previously published and expected values. It turned out that it is practically
impossible to choose the most reliable value of R0 . Changes in V0 lead to the
time shift: increasing V0 will lead to an earlier approach and vice versa. We used
R0 = 8.178 kpc [8] and V0 = 232.8 km s−1 [16].
To estimate the uncertanties in determination dmin and tmin , we calculated
dmin and tmin in extreme error values of the right ascention and declination (α,

219
δ), proper motions (µα , µδ ), and radial velocity (Vr ). As a result we obtained
− −
d+ +
min , tmin and dmin , tmin by integration with input values α ± σα , δ ± σδ ,
µα ±σµα , µδ ± σµδ , Vr ± σVr . Although it should be noted that Gaia errors of
the astrometric parameters have cross-correlations, which we did not take into
account here. For the stars in Figure 7 errors described below do not exceed
20%.

3.1 Estimation of the Interstellar Comets Number in the Galaxy

Obtained results allow us to estimate the number of events that can produce
interstellar small bodies. The list of values for calculations is presented in Table 1.
Columns contain time intervals we integrated for searching close pairs, number
of stars in the sample, spatial size and volume of the sample, number of pairs
for which we calculated the minimal distance, and number of close pairs that
can produce the interstellar small bodies. Stars sample at time moment t=0
presented at Fig. 1 for field stars and 2 for the Hyades star cluster.

Table 1. Parameters of the stars samples.

Field stars (Figure 1) Hyades / core (Figure 2)
Number of stars 642 280 / 150
X, Y, Z (pc) 36.9 × 38.4 × 39 90.3 × 163.6 × 139.5
Volume (pc3 ) 5.5·104 2.1·106 / 0.8·104
Number of pairs 205,760 26,948
Number of pairs (<1 pc) 64 62

Now we could estimate the rate of producing the interstellar small bodies by
field stars and by open clusters.
From the table, we see that the OSC produces 62 close pairs for 500 Myr, and
the field stars produce 64 close pairs for the same period for a considered volume.
There are 105 OSCs in the Galaxy, which means that the clusters replenish the
ACP population of the Galaxy by 62 × 105 objects in 500 Myr, or 1.24 × 104
ACP in a 1 Myr.
The volume of the Galaxy V is the volume of a cylinder with a height of 2
kpc and a base area S = 2πR, where R is the radius of the Galaxy of 16 kpc.
Thus, V = 2 × 2π × 16 = 201 kpc 3 . If for the volume calculated by us 5.5 × 10−5
kpc 3 the productivity is 64 close pairs, then for the entire Galaxy it will be
2.35 × 108 in 500 Myr, or 47 × 104 ACP in a 1 Myr.
Next, we calculate the density of the ACP in the Galaxy. The age of the
Galaxy is 1.35 × 1010 , thus, over the entire period of its existence were produced
10
0.67 × 1010 interstellar ACP. The ACP density is ρ = 0.67×10 201 = 3.3 × 108 ACP
3
per 1 kpc
It should be noted, that close flyby may not cause noticeable disruption of
the Oort cloud (see, for example, [15]).

220
Besides, note that our estimate does not take into account those stars that
left the circumsolar neighborhood and could also approach both other stars and
the Sun. Taking such encounters into account will increase the number of paired
encounters and the number of free ACP objects, respectively.

4 Conclusions

The main results of the work are as follows.

– Using filters to restrict Big Data of the Gaia EDR3 spacecraft, we composed
samples of the field stars (n = 642) and the Hyades cluster stars (n = 280).
Thus, only a small percentage of Gaia stars can still be used for our research.
To accumulate the growing amount of data, it is possible to use the Russian
Virtual Observatory.
– The integration of the motion of the stars around the Galaxy center in past
epochs was carried out. Using the developed algorithm, we calculated the
minimal distances dmin and corresponding time tmin of the star pairs of our
sample (Fig. 4, 5, 6), and considered pairs that could provide interstellar
small bodies (Fig. 7). Found pairs of stars approaching at a critical distance
that may cause a loss of comets from their Oort clouds.
– We estimated the frequency of the interstellar small bodies production over
the past 500 million years. Assuming that one close encounter delivers at
least one ACP object to interstellar space, we obtained that the density of
interstellar ACP is 3.3 × 108 ACP per 1 kpc 3 . Thus, about 10% of stars
in the Galaxy may provide an interstellar ACP. This estimate is consistent
with the observed data on the number of planetary systems in the Galaxy
[11].

As part of this task, we encountered technical limitations specific to working
with big data. First, the Gaia data contains terabytes of information, so we are
forced to artificially limit the sample of stars under our study (in our case, by
parallaxes, that is, by the distance from the Sun). Second, our computational
capabilities are also limited, which prevents us from performing a pairwise search
on a large amount of Gaia data. However, the Gaia data itself also has a number
of limitations - even with an unlimited power reserve, we would face the prob-
lem of a lack of information in terms of radial velocities (astrometry of stars is
much simpler and more accurate than spectral studies), as well as the limited
observable part of the Galaxy. Nevertheless, even these problems can be partially
solved using modern machine learning methods - the available sample of stars
is enough to use the so-called ”supervised learning” method, which would allow
us to calculate statistics of stellar encounters across the entire disk. It is worth
mentioning that machine learning is widely applied in astronomy, including open
clusters studying (see, for example, the paper on searching for solar siblings [23],
or on searching new open clusters in the Galaxy [12]).

221
Acknowledgments. This work presents results from the European Space Agency
(ESA) space mission Gaia. Gaia data are being processed by the Gaia Data Pro-
cessing and Analysis Consortium (DPAC). Funding for the DPAC is provided by
national institutions, in particular the institutions participating in the Gaia Mul-
tiLateral Agreement (MLA). The Gaia mission website is https://www.cosmos.
esa.int/gaia. The Gaia archive website is https://archives.esac.esa.int/gaia. The
authors thank the reviewers for their helpful comments. Authors acknowledge
the support of Ministry of Science and Higher Education of the Russian Feder-
ation under the grant 075-15-2020-780 (N13.1902.21.0039).

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A Table with dmin and tmin pf the close star pairs (see
section 3)

Table 2: Parameters of the stars samples.

Type Gaia EDR3 id1 Gaia EDR3 id2 dmin , t[ min],
pc Myr
field 1872046609345556480 1872046574983497216 0.00055 0
field 1022456139210632064 1022456104850892928 0.00056 0
field 5559265690666327168 5559265690666326016 0.00179 0
field 5443051537858529792 5443051537858529920 0.00186 0
field 386653747925624576 386653851004022144 0.00271 0
field 732856080807636224 732857558276385664 0.00358 0
field 4955395178633330432 4955395178633330304 0.00424 0

223
Table 2: Parameters of the stars samples.

Type Gaia EDR3 id1 Gaia EDR3 id2 dmin , t[ min],
pc Myr
field 4986970575602213632 4986970575602213504 0.00459 0
field 1237090738916392832 1237090738916392704 0.00500 0
field 4519789321942643072 4519789081415296128 0.00539 0
field 1071194431653425280 1071195187567668480 0.00660 0
field 4911306239828325632 4911306239828325760 0.00708 0
field 461701979229013376 461701979233050624 0.00774 0
field 263916742385357056 263916708025623680 0.00779 0
field 3936909723803146368 3936909723803146496 0.00847 0
field 3057712188691831936 3057712223051571200 0.00884 0
field 43335880716390784 43335537119385216 0.00925 0
field 4364527594192166400 4364480521350598144 0.00939 0
field 3712538811193759744 3712538708114516736 0.00969 0
field 5291028284195365632 5291028181119851776 0.00984 0
field 3812355328621651328 3812355294262255104 0.01760 0
field 4722111590409480064 4722135642226902656 0.01881 0
field 3498481592531208576 3498481519515679872 0.02295 0
field 853819947756949120 853820948481913472 0.02763 0
field 5951165616611763456 5951165616635298816 0.03612 0
field 3394298532176344960 3400292798990117888 0.06225 0
field 3837746380705638656 3837697972130323456 0.11509 0
field 4038724053986441856 579567598501642368 0.19464 -1.0010
field 3060788519149063680 3057712188691831936 0.30701 0
field 3060788519149063680 3057712223051571200 0.30728 0
field 1472718211053416320 1472903753640492160 0.34775 0
field 1408029436569383296 1359938520253565952 0.42527 0
field 1408029509584967168 1359938520253565952 0.42762 0
field 1408029509583934464 1359938520253565952 0.43257 0
field 3478160727866058368 646255212109355136 0.45622 -0.5005
field 3478127463341507072 3478160727866058368 0.48556 0
field 974536192658255104 974887555341533440 0.52180 0
field 5866992641380992256 5895265380327966464 0.58470 0
field 4706564427272810752 4706630501049679744 0.58553 0
field 1408029509584967168 4548562265603840768 0.61610 -0.5005
field 1092545710514654464 1093918828739878528 0.63606 0
field 4866978844438656768 4678664766393829504 0.65386 -0.5005
field 2287506148856660992 2299942278201276288 0.65398 0
field 4093301474693288960 4079684229322231040 0.65537 0
field 522863309964987520 426641955043723520 0.69301 0
field 892215482207937152 893214796542513280 0.69675 0
field 5945941905576552064 5925209583053212800 0.70644 0
field 5698188160215537920 5602386058511578368 0.74452 0
field 2964001014514075520 2395413147718069504 0.74830 -2.5025

224
Table 2: Parameters of the stars samples.

Type Gaia EDR3 id1 Gaia EDR3 id2 dmin , t[ min],
pc Myr
field 4805806449875760384 4805866957374888448 0.75198 0
field 4025850731201819392 4034171629042489088 0.80494 0
field 543789661932658432 551050046451136256 0.81364 0
field 5412250540681250560 5425628298649940608 0.81697 0
field 4247023886053586304 4248817876711932416 0.83519 0
field 1062935140823485184 1050678712910407424 0.84161 0
field 704967037090946688 6029992663310612096 0.84248 -0.5005
field 4270814637616488064 4270446404294208000 0.87358 0
field 522863309964987520 512167948043650816 0.93959 0
field 3828238392559860992 1237090738916392832 0.94686 -0.5005
field 2929062902976882304 2927791352138805504 0.95268 0
field 436648129327098496 438829629114680704 0.95995 0
field 6508401923473282432 6508776375901968640 0.97118 0
field 6673000841376349696 6697578465310949376 0.97479 0
field 5042734468172061440 5038817840251308288 0.98441 0
hyades 3313653894061044224 3313662896313355008 0.178 0
hyades 146160558879786624 3410640887035452928 0.192 -5.6306
hyades 3314063908819076352 3314079508140198528 0.254 0
hyades 146677879098433152 146687018788948224 0.299 0
hyades 3312644885984344704 3312575685471393664 0.324 -1.2512
hyades 3312136499294830848 3312197934506930944 0.344 -0.6256
hyades 3312899491645515776 3312951748510907648 0.410 0
hyades 3313778207594395392 3313689422030650496 0.419 0
hyades 3314109916508904064 3313689422030650496 0.446 -0.6256
hyades 3313662896313355008 3313689422030650496 0.459 0
hyades 146160558879786624 3312602348628348032 0.488 -158.90
hyades 3295883999449691264 3305871825637254912 0.521 -3.7537
hyades 3314079508140198528 144534724778235392 0.541 -389.13
hyades 3314212068010812032 3314109916508904064 0.543 0
hyades 145373377272257664 144534724778235392 0.559 -0.6256
hyades 3313653894061044224 3313689422030650496 0.561 0
hyades 3405113740864365440 3405909374965722368 0.572 0
hyades 3387573300585597184 3387381646261643776 0.589 -1.2512
hyades 3304412601908736512 3412605297699792512 0.607 -26.276
hyades 3311024789960504576 3310702770492165120 0.610 0
hyades 3313662896313355008 3313778207594395392 0.611 0
hyades 3314109916508904064 3313778207594395392 0.637 0
hyades 3312951748510907648 3312842042162993536 0.643 -0.6256
hyades 3313689422030650496 3312951748510907648 0.646 0
hyades 50327297200978176 50298125783225088 0.654 0
hyades 3307645131734449408 3307844864893938304 0.656 0
hyades 3308127405023027328 3307992336891315968 0.665 0

225
Table 2: Parameters of the stars samples.

Type Gaia EDR3 id1 Gaia EDR3 id2 dmin , t[ min],
pc Myr
hyades 3312644885984344704 3312613000147191808 0.673 -1.2512
hyades 3312575685471393664 3312564037520033792 0.676 0
hyades 3312575685471393664 108421608959951488 0.681 -595.59
hyades 3312899491645515776 3313689422030650496 0.681 0
hyades 3312575685471393664 3312613000147191808 0.694 -0.6256
hyades 47541100375011968 47813504381625088 0.695 -0.6256
hyades 3313778207594395392 3312951748510907648 0.718 0
hyades 3312899491645515776 3312842042162993536 0.744 0
hyades 3307844864893938304 50298125783225088 0.756 -152.65
hyades 3312613000147191808 3312842042162993536 0.759 0
hyades 3313653894061044224 3313778207594395392 0.769 0
hyades 145325548516513280 49231668222673920 0.784 -0.6256
hyades 3314212068010812032 3314063908819076352 0.793 0
hyades 3406823245223942528 144377803854541184 0.794 -153.90
hyades 3311179340063437952 51383893515451392 0.796 -153.90
hyades 3405988677241799040 3405113740864365440 0.852 0
hyades 3313259169388356608 3312709379213017728 0.855 -3.1281
hyades 3314063908819076352 3314109916508904064 0.859 0
hyades 3404850790083594368 3405127244241184256 0.861 0
hyades 3311492803955469696 3310640648085373824 0.884 0
hyades 3314109916508904064 3313662896313355008 0.888 -0.6256
hyades 3310702770492165120 3311492803955469696 0.892 0
hyades 47804394753757056 48203487411427456 0.896 0
hyades 3312951748510907648 3312613000147191808 0.898 0
hyades 3312136499294830848 45142206521351552 0.899 -1.2512
hyades 3313778207594395392 3312899491645515776 0.905 0
hyades 3312564037520033792 3312842042162993536 0.907 0
hyades 3307645131734449408 3310640648085373824 0.914 -1.2512
hyades 149313099234711680 144534724778235392 0.923 -8.7587
hyades 3406943091991364608 3405113740864365440 0.923 -35.660
hyades 3312575685471393664 3312899491645515776 0.941 -0.6256
hyades 47813504381625088 47345009348203392 0.949 -1.8768
hyades 3314109916508904064 3312951748510907648 0.951 0
hyades 3313689422030650496 3312842042162993536 0.971 0
hyades 3312951748510907648 3314213025787054592 0.990 0

226