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“Science and innovations 2021: development directions and priorities” Conference, Melbourne, Australia
DOI 10.34660/INF.2021.44.49.022
STOCHASTIC MODELLING FOR EVOLUTION OF GLOBULAR STAR
CLUSTER OMEGA CENTAURI
P.V.Vasilev1
1Candidate of Technical Sciences, Associate Professor, email:vassiliev@bsu.edu.ru
Belgorod National Research University, Russia
Abstract. The preliminary stochastic model for the evolution of the globular
star cluster NGC 5139 is presented, including technique to simulate ecosystems
associated with parent stars having terrestrial exoplanets. The approach is based
on the Monte Carlo method using generation of the rate of star formation in discrete
time intervals in accordance with the available data on the prevalence of spectral
classes of stars within the modeling area. A mechanism for recursive correction of
the star formation rates based on the comparison of the model structure with
astronomical data has been developed. The equation of convolution functions for
birth and death of stars with ecosystems passing through cosmological filters in
connection with a composition of the nearest stars in the tetrahedral network is
introduced. The proposed stochastic model can be used to evaluate the
transmissions between technospheres in habitable zones of cluster. The
transportation or signal propagation along the edges of the tetramesh is an
alternative method compared to the Landis approach based on percolation theory
for the numerical solution of the Fermi paradox. The Delaunay 4D mesh and
Voronoi's mosaic can be used to outline the location hulls of cluster regions with
biosignatures where technospheres are likely in state of communicative phases and
may be detected by SETI programs.
Keywords: stochastic model, globular cluster, star evolution, convolution,
spectral class, Delaunay tetrahedralization, Voronoi mosaic, SETI, biosignature.
Introduction
Star Cluster NGC 5139 Omega Centauri is the largest in our Galaxy, the
brightest and most massive globular cluster known. It is 15000lyr distant from
Earth, making it one of the closest clusters. The cluster is about 150lyr across and
includes about stars, the mass of the cluster is . The density near the
center of the cluster is about . Most of the stars are main sequence
stars in the HertzsprungRussell diagram. There are a number of red giants  stars
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in the final stages of evolution. Recently radio pulsars have been found in the center
of the cluster.
To evaluate planetary products and astrobiological aspects of evolution with the
wellknown energy classification of cosmic civilizations by Kardashev Н.С. [2] we
are using also the next classification based on communicative opportunities of
possibile intelligent life signatures inside the boarders of their planet Ns –
noosphere, in the star planetosphere Ts – technosphere, in interstellar space inside
the volume of the Galaxy Gs  galosphere and in the metagalaxy or in the universe
Ms  metasphere. These characteristics are also taken into account as probabilistic
parameters in the model of cluster evolution based on the stochastic approach using
the Monte Carlo method, Markov chains [1], and the equation for the convolution
of the birth and death functions of stars.
Purpose of the study – to establish the stochastic evolutionary model of
globular star cluster NGC 5139 in connection with the star formation rates and
spectral composition of the population throughout the lifetime of the cluster. One
of the favorable features of the Omega Centauri cluster is that the stars in it consist
of mainsequence spectral types and have different ages and metallicities, while in
most clusters they form almost simultaneously and practically do not differ in
chemical composition. The stars in the cluster are generally up to 10 Gyr in age,
that is, they were not formed uniformly but during at least two bursts of star
formation. Taking into account the peculiarities of the object Fig. 1 shows a block
diagram of stochastic modeling.
Fig.1. Block diagram of stochastic evolution modeling star cluster NGC 5139
An important point is setting the rate of star formation, SFR. If the Milky Way
is currently forming up to masses of the Sun per year and per 0.1 Gyr,
then in the globular cluster NGC 5139, the activity of star formation is about 5
orders of magnitude less intense. As a first approximation, the model uses a uniform
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distribution over time intervals. Stellar dynamics based on taking into account
gravitational forces using the Nbody method [4] and the kinetics of fragmentation
of gas clouds during compaction of matter can be included in the model to increase
the detail of the evolution of the cluster at all stages of its formation.
Materials and methods
The Gaia mission data release (DR3) provides the positions and mean proper
motions for and more then 150 Milky Way globular clusters with a typical
uncertainty of 0.05 mas limited systematic errors. We use also some variants
of Monte Carlo Markov Chain (MCMC) technique and common parameters from
[5]. Table 1 shows the initial initialization settings for the stochastic model.
Table 1. Initialization parameters for the stochastic model
Spectral star class
Star formation rate /
0.1 Gyr
Fraction, units
Lifetime, Gyr
O
0.00001
0.01
B
0.00109
0.10
A
0.0519
1.00
F
0.037
3.50
G
0.06
10.0
K
0.12
50.0
M
0.73
200.0
The algorithm for simulating the evolution of the globular cluster includes:
Step 1. Setting the number of stars in a representative sample of input data .
The modeling area is a sphere with a diameter of 150 light years with a given
distribution of stars, chosen depending on the generation method. To record and store
dataset the OmegaCentauri.sqlite tables of SQLite data base is implemented.
Step 2. The coordinates within the framework of the model and the spectral class
of the main sequence stars are set using the Monte Carlo method. Nonmain sequence
star types for classes W, L, T, Y, C, S and D may be generated optionally. The lifetime
of stars is installed in relative units within each spectral class.
Cluster age is installed in the range from 0 to 10 Gyr. In the population, 20% of stars
are considered binaries and triples in different proportion.
Step 3. After the end of the life span of stars in classes O, B, A, F, G, new stars of
one of the spectral types are generated in a random position or, in the case of the third
generation, at a distance of one light year from the previous position to maintain a
constant number and density in generations of stars. The processes of the formation of
new stars from gasdust clouds and stellar dynamics in the model are partially taken in
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account by according to the data of their own velocities. The probabilities of transition
from blue giants to red supergiants or from blue giants to red giants and so on with a
relatively short lifetime can be taken into account as optional parameters.
Step 4. Stars of all classes except O, upon reaching the age form
planetospheres with the probability , stars of all classes, except for O and B,
with an age exceeding can have planets with lithospheres (with an earth
similarity index ) with a probability .
Step 5. Stars of classes F, G, K, M at age can form earthlike planets
having biospheres with probability . The lifetime of stars of the K and M
classes is much longer than the entire age of the cluster.
Step 6. Stars of classes G, K, M with age exceeding can have terrestrial
planets with noospheres on probability , and stars of classes G, K, M with age
exceeding with probability can have terrestrial exoplanets with
technospheres.
Step 7. The marking of stars with lithospheres, biospheres, noospheres and
technospheres is performed in the database tables in accordance with the established
probabilities.
Step 8. Construction Delaunay and Voronoi diagrams to compare options for a)
isotropic transmission of communication signals and b) transmission of SETI messages
with signal amplifiers and repeaters located at the nodes of the tetrahedral network.
Step 9. Calculation of integral values according to the discrete equation of
convolution for the birth and death functions of the stars population and statistical
indicators for the expansion in the search to the peripheral regions of the cluster.
When constructing tetra models of DT/VD stellar arrays, Delaunay
tetrahedralization and Voronoi diagram the mesh generator TetGen [10] is used. In the
calculation of the local cosmic filter of evolution , the data on the nearest
neighbors of each star, their spectral classes, are determined by the tetrahedral Delaunay
network. After comparing the model with the actual data on the spectral classes of the
cluster, there is also a change in the Markov chain of transition probabilities and a
restructuring of the star formation rate over discrete time periods.
Based on the available data that the stars composition for the observed cluster
NGC 5139 is close to the ones of the stars for the main sequence of the Milky Way,
it is possible to estimate the formation probability of planetospheres and exoplanets
in the habitable zones of stars. Random processes are taken into account by
specifying the values of the Markov chain in the form of star formation intensity rates
for the evolution intervals. The resulting convolution of the functions for
creation and death of stars in the discrete case was taken as the following formula
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where the summation is performed over spectral classes S with characteristic
radiation wavelength λ and over time intervals τ within each time interval. The
population density is estimated. Thus for each spectral star class Markov chains of
transition probabilities for the stages with planetospheres to ones with metaspheres
are specified, as shown in Table 2.
Table 2. The matrix of markov chains for transition probabilities between stages
Fig.2. The graph of Markov chain for transition probabilities
between stages of parent stars and filters of life evolution
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The nearest neighbors are determined by the tetrahedral network of the Delaunay
diagram. Expansion is possible to stars of classes F, G, K, M if they have formed
planets with lithospheres. Shortlived blue giants, pulsars, magnetars, red giants like
Aldebaran, and supergiants like Betelgeuse are not suitable for inclusion in the sphere
of influence. In contrast to the Landis percolation model [6], the colonization process
is modeled using the Delaunay tetrahedral stellar grid, rather than a twodimensional
grid of cells.
To simulate expansion, transport and colonization, the probability of the
expansion of technospheres and metaspheres to neighboring stars is set with the
conditional probability per unit time. The movements are limited by the
speed of light and can take place along the optimal paths along the edges of the
tetrahedral grid for bypassing areas with dangerous objects of the Galaxy. When the
Nbody stellar dynamics mechanism is included in the model [4], the effect of the
global and local filters on evolution can be enhanced by increasing
the probability of collisions of stars and ejected from the system or destabilized
planetospheres in the central regions of the cluster.
The habitable zone SHZ of the cluster is not assessed as such, but for all stars the
average distance between them should be at least , otherwise, the
formation of planetospheres around this pair of stars is an unlikely event. Delaunay
tetrahedral networks and polyhedral networks of Voronoi mosaics are constructed
using the TetGen mesh generator (https://wiasberlin.de/software/tetgen/ index.html)
as a whole for the entire set of stars, including stars outside the main sequence of the
HertzsprungRussell diagram and separately for each spectral class. It is assumed that
if a higher probability of the formation of technospheres in Kclass stars is revealed,
the network of communications and transport will develop mainly among stars of this
type.
The visualization was performed using the GLScene graphic engine (https://
github.com/GLScene/GLScene) with Fermi Paradox Simulator software, where
different cadencer components were used to process the discrete age intervals for
each spectral class of stars. The color palette for the visual spectral types
corresponded to the colors of stars of the Harvard scale to allow comparison of the
resulting images of the model and astronomical images obtained with satellite
telescopes. Figure 2 shows images of the cluster and the Delaunay tetrahedral
network of its central part.
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Fig.2. The globular star cluster Omega Centauri NGC 5139 (left, credit ESO) and
Delaunay tetrahedral mesh for Gclass stars near central core (right)
In the center of the cluster, the distance between the stars is about 0.2 ly. When
visualizing the evolution of the model in a time period from 0 to 10 Gyr, discrete time
intervals of the timer are selected in the range from a millisecond to an hour. Color
of glow sprites for stars with clOlive biospheres, clGreen noospheres and clLime
technospheres.
The stellar evolution is spawned and closely related to the likelihood of the
formation of planetospheres with Earthlike exoplanets. In this regard, to reflect the
connection with the spectral types of stars, it is proposed to use the Drake equation
[3] in the following modification:
where – number of stars with technospheres; – number of technospheres
actually known currently 1;  the number of stars in the cluster is estimated as ;
– fraction of stars in a safe zone or outside close proximity;
 the abundance
of stars of the spectral type;
 the probabilities of the formation of stars of
various classes of ecospheres on terrestrial planets with the index ; e.g.
 the probability of the formation for biospheres,
– the average
stellar class lifetime, – the average time to start bioevolution;

the formation likelihood for the noospheres and technospheres.
Results and discussion
The Monte Carlo method with markov chain matrices and descrete convolution
equations for star’s birth and death in different spectral classes the new treatment to
solving Fermi paradox was implemented. Parallel computational software project
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SFPS (Stochastic Fermi Paradox Simulator) has included two evolutional filters for
the dynamic convolution as a global cosmologically strong and a local
planetary weak . Markov triangular matrix of transition probabilities in
conjunction with dynamics and relaxation method could be used for
procedural generation of planetary worlds which are connected with appropriate
spectral star classes. The L/T coefficient of the probability to find a technosphere in
the communicative phase, that is, the ratio of the length of the message transmission
period to the age of the cluster, is not used, since it is not the total number of
technospheres in history that is determined, but their possible number at the current
moment. The selfdestruction of technospheres or the scenario of space wars are not
considered; therefore, the upper optimistic threshold for the growth of the number of
semipotential signal sources is estimated. Figure 3 shows the growth curves of the
population of stars at a fixed rate of star formation and the growth curves of
ecosystems in the form of biospheres and technospheres associated with parent stars.
To characterize the communication potential of a cluster for SETI, one can use
the volumetric coefficient for the prevalence of technospheres in the signal
transmission phase. The potential estimated is the ratio of the volume for the cells
of the Voronoi mosaic with communicative technospheres to the total volume
of the network of polyhedra in the Voronoi diagram of the NGC 5139 cluster.
Thus, the network potential to find technospheres according to the optimistic scenario
is in the current time interval for stars in the globular Omega
Centauri cluster.
Based on the simulation results, the stochastic model gives an estimate of the
number of biospheres in a cluster of almost 18200, while the number of noospheres
does not exceed 170 entities and technospheres presumably near 7. But if we take
into account the possibility for terrestrial exoplanets to evolve in binary and triple
systems, then these figures can be higher. Assuming that the radius of reliable
registration of bio and technosignatures by spectroscopic methods is no more than
20 pc, and the range of isotropic signal propagation is not over 50 pc, then it is
possible that technospheres of the cluster have established contacts with each other
for information exchange.
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Fig.3. The star formation rates and associated potential numbers of
bio noo and technospheres
Conclusion
The stochastic approach to simulate star cluster noovolution on the sample of
NGC Omega Centauri is presented in the paper. A new approach is outlined to follow
the known star locations together with stochastic procedural generations of
mainsequence stars for the cluster from the beginning of life evolution by a Monte
Carlo technique. An advanced convolution function for the birth and death of stars
was used in descrete form in accordance with Markov chain matrix of transitive
probabilities. The Delaunay 4D tetrahedral meshes and Voronoi diagram mosaics are
required to take in account nearest star neighbours and find distances for isotropic
transport propagations or network communications with cosmic restrictions. The
parallel computational method can be further extended to generate an efficient and
realistic models for large star clusters. This results of the simulation of star evolution
in the local Sun vacinity [9] coud be applied also as alternative to Landis percolation
approach to quantify the Fermi paradox. In the article [10] is argued that the factor L
as lifetime of communicative civilization in Drake formula [3] is in fact the most
important regarding the practical implications SETI, because it determines the
maximal extent of the "sphere of influence" of any technological civilization or
technospheres in our names. In the described above stochastic model only lifetime L
of technospheres is limited by the age of stars of the corresponding spectral classes
but not associated and more advanced entities.
In future work the simulator can be verified with a more accurate adjustment of
astrobiological parameters and values for the probabilities of the formation of
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terrestrial exoplanets, both earthlike and superearth ones based upon data processing
from groundbased telescopes and new survey DR3 from the mission of GAIA
satellite telescope [5]. The probabilistic evaluations for the rates of evolution for
biospheres, noospheres and technospheres in more then 150 globular clusters of our
Galaxy can represent an additional justification in solving the Fermi paradox [7, 8]
by numerical methods, taking into account the metallicity and star types of various
spectral classes. References
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3. Maccone, Claudio. (2012). Mathematical SETI. edn. Springer, 723p.
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APRIL 21, 2021
MELBOURNE, AUSTRALIA
INTERNATIONAL SCIENTIFIC CONFERENCE
Proceedings of the International Scientific Conference
“Science and innovations 2021: development directions and priorities”. Part 1
(April 21, 2021. Melbourne, Australia)
ISBN 9780645102437 (AUSPUBLISHERS, online)
ISBN 9785905695018 (Infinity publishing, print)