Non-stationarity of Open Star Clusters

Ministry of science and higher education of the Russian Federation Ural Federal University
named after the first President of Russia B. N. Yeltsin


Vladimir M. Danilov

Non-stationarity of
Open Star Clusters
Monograph






Ekaterinburg
Ural University Press


2021

UDK 524.4 D183


Translation from Russian by E. Shabalina and A. F. Seleznev

Reviewers:
           B. P. Kondratyev, Doctor of Science, Professor (MSU)
N. V. Emelyanov, Doctor of Science, Head of the Section of Celestial Mechanics SAI MSU;
V. E. Zharov, Doctor of Science, Professor, Head of the Department of Celestial Mechanics SAI MSU;
Yu. V. Khachay, Doctor of Science, Professor (Geophysics Institute of the Ural Branch of the Russian Academy of Sciences)


Danilov, V. M.
D183 Non-stationarity of Open Star Clusters : monograph / V. M. Danilov ; [trans. of E. Shabalina and A. F. Seleznev] ; Ministry of science and higher education of the Russian Federation. - Ekaterinburg : Ural University Press, 2021. - 352 p. - ISBN 978-5-7996-3173-4. - Text : electronic.

ISBN 978-5-7996-3173-4
DOI 10.15826/B978-5-7996-3173-4

         The monograph poses issues related to the study of the non-stationarity of open star clusters (OSCs), starting with an analysis of the properties of the trajectories of individual stars to the study of collective motion of stars. A discussion of the dynamics of correlations and wave processes in such clusters is presented. The mechanisms of the dynamic evolution of OSCs, the gravitational instability of OSC nuclei, the spectra of frequencies and wavenumbers for oscillations of numerical models of OSCs, astrophysical applications of the results of studies of the dynamics of OSCs are considered.
         We address the monograph to academic researchers (astronomers and physicists) who are interested in astrophysics, the dynamics of stellar systems, and PhD and senior students of relevant specialities.

UDK 524.4
ISBN 978-5-7996-3173-4                            © Данилов В. М., 2015
                 © Shabalina E. , Seleznev A.F., translation to English, 2021 © Mironov K. V., cover photo, 2011 © English Edition, typography Ural University Press, 2021

            Foreword













We live in the era of the Gaia satellite and the expectation of a revolution in the study of stellar systems. Its precise astrometry provides us with a sharp snapshot of the actual 3D structure of several nearby star clusters, while kinematics offer insights on their evolution. Detailed theory has to be developed, to interpret this wealth of data, and to cast light on processes like star formation and dynamical evolution of star clusters. The focus of this book is on open, or Galactic, star clusters, which populated the disc of the Milky Way, and are typically composed of less than 100 up to afew 1000 stars. Many of them survive little time and dissolve fast into the general Galactic field. Most (and extremely valuable) work has been done in the past in this field, essentially thanks to the efforts of Sverre Aarseth and collaborators, and the various releases of his N-body code. This book summarises another, independent and original, effort. The author, Vladimir M. Danilov, from Ural Federal University in Russia, has spent his scientific career to develop analytical and numerical methods to study open cluster dynamical evolution. The focus is on non-stationarity, which effectively evokes the idea of time evolution, sudden changes, phase mixing, re-organisation of trajectories of individual stars, collective processes, and so forth. Open clusters are lively entities, rapidly evolving and highly sub-structured. Most of the results of this life-long studies are included in papers published in Russian journals which, unfortunately, got little attention by foreign researchers. This comprehensive book, translated from the original in Russian, contains the most important results of Danilov work with a lot of details, and is intended for researchers in Galactic dynamics and N-body methods. It is not an easy book, but it offers to everybody the opportunity to hear a different voice, to consult an independent opinion from another world.


Padova, July 2020                                               Giovanni Carraro

iii

            Acknowledgements















Author expresses his gratitude to Miss E. Shabalina, who performed a huge work to translate to English a very complicated Russian text, and to Dr. Anton F. Seleznev, who finished and edited the translation and prepared the layout of the book.
   The work of V.M.D. was supported by the Ministry of Science and Higher Education of the Russian Federation, FEUZ-2020-0030 and also by the Act No. 211 of the Government of the Russian Federation, agreement No. 02. A03.21.0006.

v

            Contents













1   Introduction ........................................................ 1
     References ......................................................... 8

2   Motion of Halo Stars in the Numerical Dynamical Models of Open Star Clusters .......................................................... 11
     2.1  Introduction ................................................. 11
     2.2  DescriptionofOSCModels........................................ 12
     2.3  Small Parameter in the Equations of Stellar Motion ........... 16
     2.4  Stellar Motion Frequencies in OSC Models ..................... 19
     2.5  Conclusions .................................................. 21
     References ........................................................ 22

3   Properties of Stellar Trajectories in Numerical Dynamical Models of Open Stellar Clusters ............................................... 25
     3.1  Introduction ................................................. 25
     3.2  Basic Formulas and Notations ................................. 28
     3.3  The Lyapunov Times and Oscillation Periods of Stellar Trajectories 29
     3.4  Conclusions .................................................. 41
     References ........................................................ 42

4   Dynamic Models of the Stellar Motion at the Periphery of Open Star Clusters .......................................................... 45
     4.1  Introduction ................................................. 45
     4.2  Equations and Integrals of Stellar Motion .................... 47
     4.3  Stellar Motion in the Plane (£, ^) ........................... 51
     4.4  Phase Density Function Depending on the Integrals J1 and J2 .. 56
     4.5  Distribution of the Stellar Velocities in the Case of Three-Integral PDF ............................................................... 62
     4.6  Conclusions .................................................. 66
     References ........................................................ 67


vii

viii

Contents

5    Investigation of the Stellar Trajectories in the Two-Point Model of Open Star Cluster ........................................................ 69
     5.1  Introduction ....................................................... 69
     5.2  Two-Point Model of Non-Isolated Star Cluster ....................... 73
     5.3  Calculation of Separatrix .......................................... 77
     5.4  Canonical Form of Equations in Variations .......................... 78
     5.5  Construction of Poincare Sections................................... 81
     5.6  Discussion of The Properties of Stellar Trajectories ............... 84
     5.7  Conclusions ........................................................ 89
     References .............................................................. 90

6    Time Scales of Dynamic Evolution Mechanisms of Open Star Clusters 93
     6.1  Introduction ....................................................... 93
     6.2  Description of OSC Models .......................................... 96
     6.3  Basic Calculation Techniques ....................................... 98
     6.4  Changes in the Parameters of Stellar Velocity Distribution with
          Time. Instability of Phase Density to Small Initial Perturbations . 100
     6.5  Conclusions ....................................................... 110
     References ............................................................. 111

7    Analysis of Density Oscillations in Models of Open Star Clusters ....... 115
     7.1  Introduction ...................................................... 115
     7.2  Oscillations ofa Homogeneous Triaxial Gravitating Ellipsoid ....... 116
     7.3  Nonhomological Oscillations of Я-Ellipsoid..........................121
     7.4  Three-Integral Equilibrium Function of Phase Density............... 123
     7.5  The Boltzmann Equation ............................................ 125
     7.6  Instability of Phase Density Oscillations in the Cores of OSC
          Models. Estimates of Parameters and Consequences of Such Oscillations ...................................................... 129
     7.7  Conclusions ....................................................... 134
     References ............................................................. 136

8    On Phase Density Oscillations at the Centers of Six Open Star Clusters ................................................................ 137
     8.1  Introduction ...................................................... 137
     8.2  Structural Parameters of Six OSC .................................. 138
     8.3  Structural and Dynamic Parameters of Six OSC Models ............... 145
     8.4  Dynamic Parameters of Six OSC and Two OSC Models .................. 148
     8.5  Analysis of Dependencies Between the Parameters of Unstable Oscillations of Phase Density in the Centers of the Considered OSCs and Cluster Models ............................................ 153
     8.6  Conclusions ....................................................... 159
     References ............................................................. 160

Contents

ix

9    On the Dynamics of Open Star Clusters ............................... 163
     9.1  Introduction ................................................... 163
     9.2  Stellar Velocity Dispersion in the OSC Models .................. 165
     9.3  Instability Parameters of OSC and their Models ................. 170
     9.4  Structure of Regions of the PD Oscillation Instability in the Centers of Six OSCs ......................................................... 175
     9.5  Conclusions .................................................... 182
     References .......................................................... 183

10   Parameters of Nonstationarity of Open Star Clusters.................. 185
     10.1  Introduction .................................................. 185
     10.2  Contrast of Densities in the OSC Cores ........................ 187
     10.3  Homologous Oscillations of the OSC Models ..................... 190
     10.4  Oscillations in the Cores of OSC and OSC Models ............... 197
     10.5  Dynamic Dispersions of Stellar Velocities in OSC .............. 205
     10.6  Conclusions ................................................... 208
     References .......................................................... 209

11   On the Dynamics of Correlations in Models of Open Star Clusters . . . 213
     11.1  Introduction .................................................. 213
     11.2  Times and Radii of Correlations in OSC Models ................. 215
     11.3  Distributions of Correlations ................................. 224
     11.4  Fluxes of Correlations in the Space of у f .....................228
     11.5  Conclusions ................................................... 234
     References .......................................................... 235

12   Correlations, Spectra, and Instability of Phase-Density Oscillations
     in Models of Open Star Clusters ..................................... 239
     12.1  Introduction .................................................. 239
     12.2  Estimates of rc⁽/), vf, vᵣ, vᵥ in theOSC Models.................243
     12.3  Cross-Correlation Function .................................... 246
     12.4  Frequency Spectra and Dispersion Curves ....................... 248
     12.5  Conclusions ................................................... 257
     References .......................................................... 259

13   Smoothing of Force Functions and Oscillation Spectra of Open Star Cluster Model ............................................................ 263
     13.1  Introduction .................................................. 263
     13.2  Ona Connection Between the Parameters of the OSC Model and e . 268
     13.3  Cross-Correlation Functions ................................... 270
     13.4  Frequency Spectra and Dispersion Curves ....................... 272
     13.5  Conclusions ................................................... 279
     References .......................................................... 280

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Contents

14  Spectra of Wave Numbers of Oscillations in OSC Models ............. 283
     14.1 Introduction ................................................ 283
     14.2 Cross-Correlation Functions ................................. 288
     14.3 Spectra of Wave Numbers ..................................... 290
     14.4 Dependencies of 5(k) Spectra of OSC Model 1 on e..............296
     14.5 Conclusions ................................................. 299
     References ....................................................... 300

15  Dynamics of the Coronas of Open Star Clusters ..................... 303
     15.1 Introduction ................................................ 303
     15.2  Constructing Coronas of OSC models ......................... 307
     15.3 Parameters of Coronas of the OSC Models. Mechanisms of Corona Formation and Rate of Dissipation of Stars from Coronas of the OSC Models ....................................................... 308
     15.4  Conclusions ................................................ 321
     References ....................................................... 322

16  Appendix .......................................................... 327
     16.1 DescriptionofOSCModels....................................... 327
     16.2  Statistical comparison of OSC models ....................... 330
     16.3  Statistical Control of Calculations ........................ 332
     References ....................................................... 338

            Acronyms

















BS    Binary Star                             
CBS   Close Binary Star                       
CR3BP Circular Restricted Three-Body Problem  
CZVS  Critical Zero Velocities Surface        
DSO   Divergence of Stellar Orbits            
FFT   Fast Fourier Transform                  
FLI   Fast Lyapunov Indicator                 
GD    Gross-Dynamic                           
GMC   Giant Molecular Cloud                   
GSC   Gas-Star Complex                        
KDE   Kernel Density Estimator                
MC    Molecular Cloud                         
MLCE  Maximum Lyapunov Characteristic Exponent
OSC   Open Star Cluster                       
PCS   Phase Coordinates of the Stars          
PD    Phase Density                           
PDF   Phase Density Function                  
PS    Phase Space                             
QSS   Quasi-Stationary State                  
SC    Sinusoidal Component                    
SS    Single Star                             
ZAMS  Zero-Age Main Sequence                  
ZVS   Zero-Velocity Surface                   

xi

            Chapter 1
            Introduction












Abstract In the Introduction we briefly discuss the modern status of the problem of an analysis of the dynamical state of the open star clusters. The formulation of the problems necessary for such analysis is presented. Also, we list the most important results connected with this analysis and described in the present book.

   Open star clusters (OSC) are traditionally used by researchers in testing emerging theories, hypotheses, assumptions about the physical and dynamic evolution of stars and star clusters. Until the early 80’s of the last century the OSC were mainly considered as objects existing in the state close to virial equilibrium. In order to theoretically describe the OSC structure, the quasistationary (and even stationary) OSC models were used which did not take into account the possibility of significant deviation of OSC from the stationarity in a regular field. However, structural features of young and old OSC (distortion of the shape of cluster cores, splitting of cores, a mismatch of centers of density distributions of subsystems of stars with different masses in clusters, presence of step-like structures in the density profiles of clusters, etc.) found back in 1950-1960’s can not be explained without invoking the hypothesis of non-stationarity of clusters in regular field. It assumes an explicit time-dependence of the phase density functions (that is, a partial derivative of the phase density function is not equal to zero; see Chapters 4 and 7 of Binney and Tremaine (2008)). Completed in recent years, theoretical researches and numerical experiments on the modeling of a dynamics of non-isolated systems of N gravitating point masses showed that in clusters with a low density of the matter (close to the critical one in the field of external forces), the action of gravitational instability can lead to the development of a non-stationarity in the regular field and even to the loss of a significant part of stars during several intersections of the cluster by a star.
   In the framework of numerical experiments on the modeling of a dynamics of OSC, it was revealed that the velocity dispersion of stars in cluster cores (cold cores) was insufficiently high for the equilibrium state, and the dispersion of velocities of stars was increasing along with the distance from the center of the cluster, which indicates the lack of equilibrium and a non-stationarity of OSC (Danilov and Putkov, 2012a). In the models of OSC (Danilov and Dorogavtseva, 2008), density and

1

1 Introduction

regular field oscillations develop rapidly and spontaneously, and do not decay during 5-10 times of violent relaxation rᵥᵣ. Theoretical description of the phase density oscillations in cores of ellipsoidal OSC models was performed by Danilov (2008). Danilov (2010) showed the instability of the natural oscillations of the phase density in the centers of six OSC models and several real OSC and obtained the theoretical estimates of the periods and the growth increments of the homological oscillations of the phase density. Danilov and Putkov (2012b) determined the parameters of density, potential, and phase density waves in OSC models by the data on correlations for the values of the radius-vector absolute values, velocities, specific energies of stars, densities, and phase densities. They discovered the signs of formation of polarization clouds, when a series of distances between the stars is reached; the dominant flux of correlations from the region of strong correlations to the weak correlation region is detected, which leads to the appearance of kinetic energy flux to the center of the cluster. Danilov and Putkov (2012b, 2013b) revealed a number of indications of the formation of a weak turbulence in the motions of the core stars in the OSC model with the greatest degree of non-stationarity in a regular field.
   We emphasize that the average values of the matter density in the OSC are usually determined inaccurately and depend to a large extent on the observer’s estimate of the cluster radius. To analyze the observed manifestations of non-stationary nature of OSC, it is necessary to substantially refine the data on the dimensions, average and critical densities of the clusters in an external force field. In order to proceed with the analysis of the observable manifestations of non-stationary nature of the OSC, it is necessary to develop methods for estimating the degree of clusters’ non-stationarity, taking into account the action of the most important factors in the clusters (low peculiar velocities of the stars and small matter densities in the OSC, the encounters of the cluster-member stars, presence of external force fields).
   In this connection, the mass application of the statistical methods to the evaluation of the OSC sizes and the number of stars in clusters against the field star density fluctuations, as well as the development of means to assess the reality of existence of certain structural features of OSC (Danilov et al., 1985; Danilov and Seleznev, 1994), turned out to be very timely. The study of OSC structure using the method of significance levels (Danilov et al., 1985; Danilov and Seleznev, 1994) showed that the stars in young clusters are on average located deeper beneath the tidal surface in the Galactic force field than stars in old OSC. When analysing this phenomenon, it is necessary to take into account the joint effect on the young OSC of the force fields of the Galaxy and the gas-star complexes (GSC), in which these clusters are formed. An important place in this analysis belongs to the investigation of the stability of OSC in external force fields. Such studies provide information on the tidal size of OSCs and on certain characteristics of GSC containing these clusters (Danilov and Seleznev, 1995). An important role in the study of the OSC dynamics is also played by the observational data obtained by Danilov and Seleznev (1994) on the characteristics of the haloes and the cores of these clusters.
   Reliable observational data on the OSC structure makes it possible to formulate and solve a number of theoretical problems: