Development of Multilayer Models of Globular Star Clusters and Study of Their Evolution

Usually, models of globular star clusters are created by analyzing their luminosity and other observation parameters. The goal of this work is to create stable models of globular clusters based on the laws of mechanics. It is necessary to set the coordinates, velocities and masses of the stars so that as a result of their gravitational interaction the globular cluster is not destroyed. This is not an easy task, and it has been solved in this paper. Using an exact solution of the axisymmetric gravitational interaction of N-bodies, single-layer spherical structures were created. They are combined into multilayer models of globular clusters. An algorithm and a program for their creation is described. As a result of solving the problem of gravitational interaction of N bodies, evolution of 5-, 10-, and 15-layer structures was studied. During the inter-body interaction, there proceeds a transition from the initial specially organized structure to a structure with bodies, uniformly distributed in space. The number of inter-body collisions decreases, and the globular cluster model passes into the stable form of its existence. The collisions of bodies and the acquisition of rotational motion and thermal energy by them are considered. As a result of the passage to scaled dimensions, the results were recalculated to the conditions of globular star clusters. The periods of rotation and the temperatures of merged stars are calculated. Attention is paid to a decreased central-body mass in the analyzed models of globular star clusters.

The fractions of post-binary-interaction stars and evolved blue straggler stars on the red giant branch of globular clusters

The red giant branch(RGB)of globular clusters(GCs)is home to some exotic stars,which may provide clues on the formation of multiple stellar populations in GCs.It is well known that binary interactions are responsible for many exotic stars.Thus,it is important to understand what fraction of stars on the RGB of GCs is the result of binary interactions.In this paper,we performed a binary population synthesis study to track the number of post-binary-interaction(post-BI)stars that appear on the RGB,with particular emphasis on the evolved blue straggler stars(E-BSSs).Assuming an initial binary fraction of nearly 50%,we find that about half of the objects on the RGB(called giants)underwent the binary interactions,and that E-BSSs account for around 10%of the giants in our standard simulation.We also compare the properties of post-BI giants that evolved from different channels.We find that the initial orbital period and mass ratio distributions significantly affect the fraction of post-BI giants.Our results imply that the non-standard stars from binary interactions provide a non-negligible contribution to the RGB stars in GCs,which should be considered in future investigations of the origin of multiple stellar populations.

Isochrone Fitting of Galactic Globular Clusters—Ⅵ.High-latitude Clusters NGC 5024 (M53),NGC 5053,NGC 5272 (M3),NGC 5466,and NGC7099 (M30)

We fit various color–magnitude diagrams(CMDs) of the high-latitude Galactic globular clusters NGC 5024(M53),NGC 5053,NGC 5272(M3),NGC 5466,and NGC 7099(M30) by isochrones from the Dartmouth Stellar Evolution Database and Bag of Stellar Tracks and Isochrones for α–enrichment [α/Fe] = +0.4.For the CMDs,we use data sets from Hubble Space Telescope,Gaia,and other sources utilizing,at least,25 photometric filters for each cluster.We obtain the following characteristics with their statistical uncertainties for NGC 5024,NGC 5053,NGC 5272,NGC 5466,and NGC 7099,respectively:metallicities [Fe/H] =-1.93 ± 0.02,-2.08 ± 0.03,-1.60 ± 0.02,-1.95 ± 0.02,and-2.07 ± 0.04 dex with their systematic uncertainty 0.1 dex;ages 13.00 ± 0.11,12.70 ± 0.11,11.63 ± 0.07,12.15 ± 0.11,and 12.80 ± 0.17 Gyr with their systematic uncertainty 0.8 Gyr;distances(systematic uncertainty added) 18.22 ± 0.06 ± 0.60,16.99 ± 0.06 ± 0.56,10.08 ± 0.04 ± 0.33,15.59 ±0.03 ± 0.51,and 8.29 ± 0.03 ± 0.27 kpc;reddenings E(B-V) = 0.023 ± 0.004,0.017 ± 0.004,0.023 ± 0.004,0.023 ± 0.003,and 0.045 ± 0.002 mag with their systematic uncertainty 0.01 mag;extinctions AV= 0.08 ± 0.01,0.06 ± 0.01,0.08 ± 0.01,0.08 ± 0.01,and 0.16 ± 0.01 mag with their systematic uncertainty 0.03 mag,which suggest the total Galactic extinction AV= 0.08 across the whole Galactic dust to extragalactic objects at the North Galactic Pole.The horizontal branch morphology difference of these clusters is explained by their different metallicity,age,mass-loss efficiency,and loss of low-mass members in the evolution of the core-collapse cluster NGC 7099 and loose clusters NGC 5053 and NGC 5466.

p-capture reaction cycles in rotating massive stars and their impact on elemental abundances in globular cluster stars: A case study of O, Na and Al

Elemental abundance patterns of globular cluster stars can provide important clues for understanding cluster formation and early chemical evolution.The origin of the abundance patterns,however,still remains poorly understood.We have studied the impact of p-capture reaction cycles on the abundances of oxygen,sodium and aluminium considering nuclear reaction cycles of carbon-nitrogenoxygen-fluorine,neon-sodium and magnesium-aluminium in massive stars in stellar conditions of temperature range 2×107to 10×107K and typical density of 102gm cc-1.We have estimated abundances of oxygen,sodium and aluminium with respect to Fe,which are then assumed to be ejected from those stars because of rotation reaching a critical limit.These ejected abundances of elements are then compared with their counterparts that have been observed in some metal-poor evolved stars,mainly giants and red giants,of globular clusters M3,M4,M13 and NGC 6752.We observe an excellent agreement with[O/Fe]between the estimated and observed abundance values for globular clusters M3 and M4 with a correlation coefficient above 0.9 and a strong linear correlation for the remaining two clusters with a correlation coefficient above 0.7.The estimated[Na/Fe]is found to have a correlation coefficient above 0.7,thus implying a strong correlation for all four globular clusters.As far as[Al/Fe]is concerned,it also shows a strong correlation between the estimated abundance and the observed abundance for globular clusters M13 and NGC 6752,since here also the correlation coefficient is above 0.7 whereas for globular cluster M4 there is a moderate correlation found with a correlation coefficient above 0.6.Possible sources of these discrepancies are discussed.

Fractions of Compact Object Binaries in Star Clusters:Theoretical Predictions

The binary population in field stars and star clusters contributes to the formation of gravitational wave(GW)sources.However,the fraction of compact-object binaries(CBs),which is an important feature parameter of binary populations,is still difficult to measure and very uncertain.This paper predicts the fractions of important CBs and semi-compact object binaries(SCBs) making use of an advanced stellar population synthesis technique.A comparison with the result of N-body simulation is also presented.It is found that most CBs are formed within about 500 Myr after the starburst.The fractions of CBs and SCBs are demonstrated to correlate with stellar metallicity.The higher the metallicity becomes,the smaller the fraction of black hole binaries(BHBs),neutron star binaries(NSBs) and SCBs.This suggests that the GW sources of BHBs and NSBs are more likely to form in metal-poor environments.However,the fraction of black hole-neutron star binaries is shown to be larger for metalrich populations on average.

The Study of Helium Variations in Star Clusters Using China Space Station Telescope

The China Space Station Telescope(CSST)is a 2 m space-based optical-UV telescope.Its primary goal is to carry out a high-resolution photometric imaging survey of a 17,500 square degree sky area using the on board Survey Camera.With its wide field of view(1.1 square degrees)and a mosaic imager containing 640 million pixels,studying the different populations of stars within star clusters is highly feasible.The aim of this study is to assess the CSST's ability to distinguish between stellar populations with varying helium abundance levels,with the help of Modules for Experiments in Stellar Astrophysics.The results of the CSST's photometry for these different populations are presented by transferring the models into the CSST Survey Camera photometric system.The findings confirm that helium-enriched stellar populations will have unique patterns in the color-magnitude diagrams under the CSST photometric system,compared to normal stellar populations.The CSST,with its filters and wide field of view of the Survey Camera,provides a new avenue for the study of multiple populations in star clusters.

Discovery of high-velocity EHB stars in the globular cluster ωCentauri(NGC 5139)

Abstract We report the discovery of 45 high-velocity extreme horizontal branch （EHB） stars in the globular cluster Omega Centauri （NGC 5139）. The tangential ve- locities of these EHB stars are determined to be in the range 93-313 km s^-1, with an average uncertainty of -27 km s^-1. The central escape velocity of the cluster is determined to be in the range 60~105 km s^-1. These EHB stars are significantly more concentrated toward the cluster core compared with other cluster members. The formation mechanisms of these EHB stars are discussed. Our conclusions can be sum- marized as follows： （1） A comparison of the tangential velocities of these EHB stars to the central escape velocity of the cluster shows that most if not all of these EHB stars are unbound to the cluster; （2） These EHB stars obtained high velocities in the central cluster region no longer than - 1 Myr ago and may be subsequently ejected from the cluster in the next -1 Myr; （3） If the progenitors of these EHB stars were single stars, then they may have experienced a fast mass-loss process. If the progen- itors were in close binaries, then they may have formed through disruptions by the intermediate-mass black hole in the cluster center.

The Long-term Survival Chances of Young Massive Star Clusters

We review the long-term survival chances of young massive star clusters （YMCs）, hallmarks of intense starburst episodes often associated with Violent galaxy interactions. We address the key question as to whether at least some of these YMCs can be considered protoglobular clusters （GCs）, in which case these would be expected to evolve into counterparts of the ubiquitous old GCs believed to be among the oldest galactic building blocks. In the absence of significant external perturbations, the key factor determining a cluster＇s long-term survival chances is the shape of its stellar initial mass function （IMF）. It is, however, not straightforward to assess the IMF shape in unresolved extragalactic YMCs. We discuss in detail the promise of using high-resolution spectroscopy to make progress towards this goal, as well as the numerous pitfalls associated with this approach. We also discuss the latest progress in worldwide efforts to better understand the evolution of entire cluster systems, the disruption processes they are affected by, and whether we can use recently gained insights to determine the nature of at least some of the YMCs observed in extragalactic starbursts as proto-GCs. We conclude that there is an increasing body of evidence that GC formation appears to be continuing until today; their long-term evolution crucially depends on their environmental conditions, however.

A Fermi-LAT Study of Globular Cluster Dynamical Evolution in the Milky Way:Millisecond Pulsars as the Probe

Using archival Fermi-LAT data with a time span of~12 yr,we study the population of Millisecond Pulsars(MSPs)in Globular Clusters(GlCs)and investigate their dependence on cluster dynamical evolution in the Milky Way.We show that theγ-ray luminosity(L_(γ))and emissivity(i.e.,ε_(γ)=L_(γ)/M,with M the cluster mass)are good indicators of the population and abundance of MSPs in GlCs,and they are highly dependent on the dynamical evolution history of the host clusters.Specifically speaking,the dynamically older GlCs with more compact structures are more likely to have larger L_(γ)andε_(γ),and these trends can be summarized as strong correlations with cluster stellar encounter rateΓand the specific encounter rate(Λ=Γ/M),with L_(γ)∝Γ^(0.7±0.11)andε_(γ)∝Λ^(0.73±0.13)for dynamically normal GlCs.However,as GlCs evolve into deep core collapse,these trends are found to be reversed,implying that strong encounters may have lead to the disruption of Low-Mass X-ray Binaries and ejection of MSPs from core-collapsed systems.Besides,the GlCs are found to exhibit largerε_(γ)with increasing stellar mass function slope(ε_(γ)∝10^((0.52±0.1)α)),decreasing tidal radius(ε_(γ)∝R_(t)^(-10±0.22))and distances from the Galactic Center(GC,ε_(γ)∝R_(gc)^(-1.13±0.21)).These correlations indicate that,as GlCs losing kinetic energy and spiral in toward the GC,tidal stripping and mass segregation have a preference in leading to the loss of normal stars from GlCs,while MSPs are more likely to concentrate to cluster center and be deposited into the GC.Moreover,we gaugeε_(γ)of GlCs is~10-1000 times larger than the Galactic bulge,the latter is thought to reside thousands of unresolved MSPs and may be responsible for the GC 7-ray excess,which supports that GlCs are generous contributors to the population of MSPs in the GC.

A Possible γ-Ray Pulsation from PSR J1740-5340B in the Globular Cluster NGC 6397

Recently, a new radio millisecond pulsar(MSP) J1740-5340B, hosted in the globular cluster(GC) NGC 6397,was reported with a 5.78 ms spin period in an eclipsing binary system with a 1.97 days orbital period. Based on a modified radio ephemeris updated by tool tempo2, we analyze the ~15 yr γ-ray data obtained from the Large Area Telescope on board the Fermi Gamma-ray Space Telescope and detect PSR J1740-5340B's γ-ray pulsation at a confidence level of ~4σ with a weighted H-test value of ~26. By performing a phase-resolved analysis, the γ-ray luminosity in on-pulse interval of PSR J1740-5340B is L_(γ)~ 3.8 × 10^(33) erg s^(-1) using NGC 6397's distance of 2.48 kpc. And γ-rays from the on-pulse part of PSR J1740-5340B contribute ~90% of the total observed γ-ray emissions from NGC 6397. No significant γ-ray pulsation of another MSP J1740-5340A in the GC is detected.Considering that the previous four cases of MSPs in GCs, more data in γ-ray, X-ray, and radio are encouraged to finally confirm the γ-ray emissions from MSP J1740-5340B, especially starving for a precise ephemeris.

A binary study of color-magnitude diagrams of 12 globular clusters

Binary stars are common in star clusters and galaxies, but the detailed ef- fects of binary evolution are not taken into account in some color-magnitude diagram （CMD） studies. This paper studies the CMDs of twelve globular clusters via binary- star stellar populations. The observational CMDs of the star clusters are compared to those of binary-star populations, and then the stellar metallicities, ages, distances and reddenings of these star clusters are obtained. The paper also tests the different effects of binary and single stars on CMD studies. It is shown that binaries can better fit the observational CMDs of the sample globular clusters compared to single stars. This suggests that the effects of binary evolution should be considered when modeling the CMDs and stellar populations of star clusters and galaxies.

New Aspects of Extragalactic Globular Clusters

I present a review on the new aspects of extragalactic globular clusters revealed from recent observations with ground-based telescopes and the Hubble Space Telescope.Extragalactic globular clusters are ideal targets to study with large telescopes and space telescopes.

Probing dark contents in globular clusters with timing effects of pulsar acceleration

We investigate pulsar timing residuals due to the coupling effect of the pulsar transverse acceleration and the Romer delay.The effect is relatively small and usually negligible.Only for pulsars in globular clusters,it is possibly important.The maximum residual amplitude,which is from the pulsar near the surface of the core of the cluster,is about tens of nanoseconds,and may hardly be identified for most globular clusters currently.However,an intermediate-mass black hole in the center of a cluster can apparently increase the timing residual magnitudes.Particularly for pulsars in the innermost core region,their residual magnitudes may be significant.The high-magnitude residuals,which are above critical lines of each cluster,are strong evidence for the presence of a black hole or dark remnants of comparable total mass in the center of the cluster.We also explored the timing effects of line-of-sight accelerations for the pulsars.The distribution of measured line-of-sight accelerations are simulated with a Monte Carlo method.Two-dimensional Kolmogorov-Smirnov tests are performed to reexamine the consistency of distributions of the simulated and reported data for various values of parameters of the clusters.It is shown that the structure parameters of Terzan 5 can be constrained well by comparing the distribution of measured lineof-sight accelerations with the distributions from Monte Carlo simulations.We find that the cluster has an upper limit on the central black hole/dark remnant mass of~6000 M⊙.

Spectroscopic study of globular clusters in the halo of M31 with the Xinglong 2.16 m telescope Ⅱ: dynamics, metallicity and age

In Paper I, we performed spectroscopic observations on 11 confirmed glob- ular clusters （GCs） in M31 with the Xinglong 2.16 m telescope. We mainly focused on the fitting method and the metallicity gradient for the M31 GC sample. Here, we analyze and further discuss the dynamics, metallicity and age, and their distributions, as well as the relationships between these parameters. In our work, eight more con- firmed GCs in the halo of M31 were observed, most of which lack previous spectro- scopic information. These star clusters are located far from the galactic center at a projected radius of ~ 14 to ~ 117 kpc, which is more spatially extended than that in the previous work. Firstly, we measured the Lick absorption-line indices and the ra- dial velocities. Then the ages and metallicity values of [Fe/H] and [a/Fe] were fitted by comparing the observed spectral feature indices and the Single Stellar Population model of Thomas et al. in the Cassisi and Padova stellar evolutionary tracks, respec- tively. Our results show that most of the star clusters in our sample are older than 10 Gyr except B290, which is ~ 5.5 Gyr, and most of them are metal-poor with metallicity [Fe/H] 〈 - 1, suggesting that these clusters were born at the early stage of the galaxy＇s formation. We find that the metallicity gradient for the outer halo clusters with rp 〉 25 kpc may have an insignificant slope of -0.005 ＋ 0.005 dex kpc-1 and if the outliers G001 and H11 are excluded, the slope does not change significantly, with a value of -0.002 ~ 0.003 dex kpc-1. We also find that the metallicity is not a function of age for the GCs with age 〈 7 Gyr, but for the old GCs with age 〉 7 Gyr, there seems to be a trend that the older ones have lower metallicity. Additionally, we plot metallicity distributions with the largest sample of M31 GCs so far and show the bimodality is not significant, and the number of metal-poor and metal-rich groups be- comes comparable. The spatial distributions show that the metal-rich group is more centrally concentrated but the metal-poor group occupies a more extended halo. In ad- dition, the young population is centrally concentrated but the old population is more spatially extended towards the outer halo.

The NGC 7742 star cluster luminosity function： a population analysis revisited

We re-examine the properties of the star cluster population in the circumnuclear starburst ring in the face-on spiral galaxy NGC 7742, whose young cluster mass function has been reported to exhibit significant deviations from the canonical power law. We base our reassessment on the clusters＇ luminosi- ties （an observational quantity） rather than their masses （a derived quantity）, and confirm conclusively that the galaxy＇s starburst-ring clusters--and particularly the youngest subsample, log（t yr-1） ≤7.2- show evidence of a turnover in the cluster luminosity function well above the 90% completeness limit adopted to ensure the reliability of our results. This confirmation emphasizes the unique conundrum posed by this unusual cluster population.

Determination of Distance,Extinction,Mass,and Age for Stars in LAMOST DR7

Large scale spectroscopic surveys such as that using Large-sky Area Multi-Object Fiber Spectroscopic Telescope(LAMOST) have collected spectra of millions stars in the Milky Way.Utilizing this huge sample of stars to study the assembling history and structure of our Galaxy requires accurate estimates of distance,extinction,age,and mass for individual stars.Combining the parallax constraint from Gaia EDR3 with Bayesian inference,we have estimated the distance and extinction for stars observed in LAMOST DR7,as well as the stellar mass and age for evolved stars in this data release.We validated the accuracies of the stellar parameters by comparing our results against various measurements,including the star-pair technique,asteroseismology,globular clusters,and isochrone fits to main sequence stars and subgiants.This is a valuable catalog of stellar parameters under a Bayesian framework estimated using the data from Gaia EDR3 and LAMOST spectroscopic data.With this data set we explored the stellar population of the Galactic massive substructure Gaia-Sausage-Enceladus(GSE).The kinematically selected members of GSE have a median metallicity of [Fe/H] =-1.29 and a median age of 11.6 Gyr.

Variable Star Status of Two Stars:V204 and I-I-39 in M3

The star 1-1-42 （=vZ1390）, a cluster member in M3, located near the red edge of the instability strip of the horizontal branch, was discovered by Roberts and Sandage as a low amplitude variable, it was designated as V204 in the ＂second catalogue of variable stars in globular clusters＂, but its coordinates given in all versions of this catalogue are wrong since 1955. We argue that V204 is indeed a low amplitude HB variable star, located near to the red edge of the instability strip, with a period of 0.74785d and an amplitude of about 0.04 mag in V. We also find that the red cluster member star 1-1-39 is a low amplitude variable with a period of 1.16^d and amplitude of about 0.03 mag in V which might be pulsating at the second overtone.

The Analyses of Globular Cluster Pulsars and Their Detection Efficiency

Up to 2022 November,267 pulsars had been discovered in 36 globular clusters(GCs).In this paper,we present our studies on the distribution of GC pulsar parameters and the detection efficiency.The power law relation between average dispersion measure(■)and dispersion measure difference(ΔDM)of known pulsars in GCs is lgΔDM∝1.52lg■.The sensitivity could be the key to finding more pulsars.As a result,several years after the construction of a large radio telescope facility,the number of known GC pulsars will likely be increased accordingly.We suggest that currently GCs in the southern hemisphere could have higher possibilities for finding new pulsars.

Two particular EA-type binaries in the globular cluster ω Centauri

We analyzed the B and V light curves of two EA-type binaries V211 and NV358 for the first time using the Wilson-Devinney code. Our analysis shows that V211 is a typical Algol-type binary and NV358 is a well detached binary system. As the two binaries are definite members of ω Centauri due to their proper motion, we estimated their physical parameters, obtaining M1 = 1.13 4±0.03 M⊙, R1 = 0.98±0.01 R⊙, M2 = 0.33 ±0.01 M⊙ and R2 = 0.92 ±0.01 R⊙ forV211; M1 = 1.304-0.05 M⊙, R1 = 1.03±0.01 R⊙, M2 = 0.58±0.02 M⊙ and R2 = 0.78±0.01 R⊙ for NV358. On the color-magnitude diagram of ω Centauri, V211 is located in the faint blue straggler region and its primary component is more massive than a star at the main-sequence turnoff. Therefore, V211 is a blue straggler and was formed by mass transfer from the secondary component to the primary. The age of NV358 is less than 1.93 Gyr, indicating that it is much younger than the first generation of stars in ω Centaufi. Like NV364 in ω Centaufi, NV358 should be a second-generation binary.

Metal abundance and kinematical properties of the M81 globular cluster system

We present metal abundance properties of 144 globular clusters associated with M81. These globulars represent the largest globular cluster sample in M81 till now. Our main results are： the distribution of metallicities is bimodal, with metallicity peaks at [Fe/H] -1.51 and -0.58, and the metal-poor globular clusters tend to be less spatially concentrated than the metal-rich ones; the metal-rich globular clusters in M81 do not demonstrate a centrally concentrated spatial distribution like the metalrich ones in M31 do; like our Galaxy and M31, the globular clusters in M81 have a small radial metallicity gradient. These results are consistent with those obtained from a small sample of M81 globular clusters. In addition, this paper shows that there is evidence that a strong rotation of the M81 globular cluster system around the minor axis exists, and that rotation is present in the metal-rich globular cluster subsample, but the metal-poor globular cluster subsample shows no evidence of rotation. The most significant difference between the rotation of the metal-rich and metal-poor globular clusters occurs at intermediate projected galactocentric radii. Our results confirm the conclusion of Schroder et al. that M81＇s metal-rich globular clusters at intermediate projected radii are associated with a thick disk of M81.