This is the third paper of a series of our works on the self-similar orbit-averaged Fokker-Planck (OAFP) equation. The first paper provided an accurate spectral solution of the equation for isotropic pre-collapse star clusters and the second detailed the physical feature of the model. Based on the works, the present work applies the solution to the observed structural profiles of Galactic globular clusters. For fitting to the profiles, the most fundamental (quasi-)stationary model, the King model, and the variants have shown successful results while they can not apply to core-collapsing and core-collapsed clusters at the late stage of the relaxation evolution. We propose an energy-truncated self-similar OAFP model that can apply to clusters at both the early and late stages of the evolution. This new model fits the structural profiles of at least half of Galactic globular clusters while it also applies to core-collapsed stars with resolved cores. As a main result, we provide the completion rate of core collapse against concentration for the clusters. Also, we show our new model can apply to the globular clusters even in a broad range of radii (0.01$\sim$10 arcminutes). However, since our model includes polytrope (elongated outer halo), the tidal radius of the model becomes unrealistically large for some clusters. To avoid the issue, we also propose an approximated form of the new model. Lastly, we report that Milky Way globular clusters with low concentrations have the same spatial structures as stellar polytropes and discuss whether such polytropic cluster is a reasonable concept.
We improve the standard theory of collisional stellar systems by considering the presence of a continuous mass distribution. The calculus of the diffusion coefficients is generalized and a new expression of the Fokker-Planck equation is found for multi-mass systems. A King-like distribution function, which validates the basic assumptions of most multi-mass models for Globular Clusters existing in literature, is obtained.
This is the first paper of a series of our works on the self-similar orbit-averaged Fokker-Planck (OAFP) equation for distribution function of stars in dense isotropic star clusters. At the late stage of relaxation evolution of the clusters, standard stellar dynamics predicts that the clusters evolve in a self-similar fashion forming collapsing cores. However, the corresponding mathematical model, the self-similar OAFP equation, has never been solved on the whole energy domain $(-1< E < 0)$. The existing works based on kinds of finite difference methods provide solutions only on the truncated …
This is the second paper of a series of our works on the self-similar orbit-averaged Fokker-Planck (OAFP) equation and details physical properties of isotropic pre-collapse solution. The fundamental core collapse process at the late stage of relaxation evolution of spherical star clusters can be described by the self-similar OAFP equation. The accurate spectral solution was found recently in the first paper. The present work details the thermodynamic aspects of the model based on the stellar DF obtained from the solution. Our calculation shows the following local properties (i) Equation of sta…
This paper presents a new scheme to treat escaping stars in the orbit-averaged Fokker-Planck models of globular star clusters in a galactic tidal field. The existence of a large number of potential escapers, which have energies above the escape energy but are still within the tidal radius, is taken into account in the models. The models allow potential escapers to experience gravitational scatterings before they leave clusters and thus some of them may lose enough energy to be bound again. It is shown that the mass evolution of the Fokker-Planck models are in good agreement with that of N-body…
We have developed a set of dynamically evolving Fokker-Planck models for the collapsed-core globular star cluster M15, which directly address the issue of whether a central black hole is required to fit Hubble Space Telescope (HST) observations of the stellar spatial distribution and kinematics. As in our previous work reported by Dull et al., we find that a central black hole is not needed. Using local mass-function data from HST studies, we have also inferred the global initial stellar mass function. As a consequence of extreme mass segregation, the local mass functions differs from the glob…
Despite the importance of their size evolution in understanding the dynamical evolution of globular clusters (GCs) of the Milky Way, studies are rare that focus specifically on this issue. Based on the advanced, realistic Fokker-Planck (FP) approach, we predict theoretically the initial size distribution (SD) of the Galactic GCs along with their initial mass function and radial distribution. Over one thousand FP calculations in a wide parameter space have pinpointed the best-fit initial conditions for the SD, mass function, and radial distribution. Our best-fit model shows that the initial SD …
Evaporative evolution of stellar clusters is shown to be relaxation limited when the number of stars satisfies $N>>N_c$, where $N_c\simeq 1600$. For a Maxwell velocity distribution that extends beyond the escape velocity, this process is {\em bright} in that the Kelvin-Helmholtz time scale, $f_H^{-1}t_{relax}$, is shorter than the Ambartsumian-Spitzer time scale, $f_N^{-1}t_{relax}$, where $f_H>f_N$ denote the fractional changes in total energy and number of stars per relaxation time, $t_{relax}$. The resulting evaporative lifetime $t_{ev}\simeq 20.5 t_{relax}$ for isolated clusters is consist…
We present a cosmological zoom-in simulation of a Milky Way-like galaxy used to explore the formation and evolution of star clusters. We investigate in particular the origin of the bimodality observed in the colour and metallicity of globular clusters, and the environmental evolution through cosmic times in the form of tidal tensors. Our results self-consistently confirm previous findings that the blue, metal-poor clusters form in satellite galaxies which are accreted onto the Milky Way, while the red, metal-rich clusters form mostly in situ or, to a lower extent in massive, self-enriched gala…
We present results of a series of Monte Carlo simulations investigating the imprint of a central intermediate-mass black hole (IMBH) on the structure of a globular cluster. We investigate the three-dimensional and projected density profiles, and stellar disruption rates for idealized as well as realistic cluster models, taking into account a stellar mass spectrum and stellar evolution, and allowing for a larger, more realistic, number of stars than was previously possible with direct N-body methods. We compare our results to other N-body and Fokker-Planck simulations published previously. We f…
The multitude of binary black hole coalescence detections in gravitational waves has renewed our interest on environments that can be the cradle of these mergers. In this work we study merger rates of binary black holes in globular clusters that are among the most dense stellar environments and a natural place for the creation of black hole binaries. To model these systems with all their variations we rely on the observational properties of the known Milky Way globular clusters. We consider direct capture events between black holes, as well as soft interactions of black hole binaries with star…
Secular evolution of binaries driven by an external (tidal) potential is a classic astrophysical problem. Tidal perturbations can arise due to an external point mass, as in the Lidov-Kozai (LK) theory of hierarchical triples, or due to an extended stellar system (e.g. galaxy or globular cluster) in which the binary resides. For many applications, general-relativistic (GR) apsidal precession is important, and has been accounted for in some LK calculations. Here we generalise and extend these studies by exploring in detail the effect of GR precession on (quadrupole-level) tidal evolution of bina…
I summarize our recent results to use the orbits of globular clusters (GCs) in the Fornax dwarf spheroidal (dSph) galaxy to learn more about dark matter (DM) properties. Our focus is on clarifying how dynamical friction (DF) from the DM halo is modified from the different microscopic properties of DM, which may alter $both$ the scattering processes responsible of DF and the DM profiles (in particular generating a core), which also modifies DF. We consider: $(i)$ fermionic degenerate dark matter (DDM), where Pauli blocking should be taken into account in the dynamical friction computation; $(ii…
We recently found that massive cluster elliptical galaxies have strong Na I 8183,8195 and FeH 9916 Wing-Ford band absorption, indicating the presence of a very large population of stars with masses <~ 0.3 M_sun. Here we test this result by comparing the elliptical galaxy spectra to those of luminous globular clusters associated with M31. These globular clusters have similar metallicities, abundance ratios and ages as massive elliptical galaxies but their low dynamical mass-to-light ratios rule out steep stellar initial mass functions (IMFs). From high quality Keck spectra we find that the dwar…
We determine the metallicities for globular clusters (GCs) in the WLM and IKN dwarf galaxies, using VLT/UVES and Keck/ESI spectroscopy. For the WLM GC we also measure detailed abundance ratios for a number of light, alpha, Fe-peak and n-capture elements. We find low metallicities of [Fe/H]=-2.0 and -2.1 for the WLM GC and the GC IKN-5, respectively. We estimate that 17%-31% of the metal-poor stars in WLM belong to the GC, and IKN-5 may even contain a similar number of metal-poor stars as the whole of the IKN dwarf itself. These high ratios of metal-poor GCs to field stars are in tension with G…
We investigate what the orbits of globular clusters (GCs) in the Fornax dwarf spheroidal (dSph) galaxy can teach us about dark matter (DM). This problem was recently studied for ultralight dark matter (ULDM). We consider two additional models: (i) fermionic degenerate dark matter (DDM), where Pauli blocking should be taken into account in the dynamical friction computation; and (ii) self-interacting dark matter (SIDM). We give a simple and direct Fokker-Planck derivation of dynamical friction, new in the case of DDM and reproducing previous results in the literature for ULDM and cold DM. ULDM,…
We report a systematic and statistically significant offset between the optical (g-z or B-I) colors of seven massive elliptical galaxies and the mean colors of their associated massive metal-rich globular clusters (GCs) in the sense that the parent galaxies are redder by 0.12-0.20 mag at a given galactocentric distance. However, spectroscopic indices in the blue indicate that the luminosity-weighted ages and metallicities of such galaxies are equal to that of their averaged massive metal-rich GCs at a given galactocentric distance, to within small uncertainties. The observed color differences …
We study the escape fraction of ionizing photons (f_esc) in two cosmological zoom-in simulations of galaxies in the reionization era with halo mass M_halo~10^10 and 10^11 M_sun (stellar mass M*~10^7 and 10^9 M_sun) at z=5 from the Feedback in Realistic Environments project. These simulations explicitly resolve the formation of proto-globular clusters (GCs) self-consistently, where 17-39% of stars form in bound clusters during starbursts. Using post-processing Monte Carlo radiative transfer calculations of ionizing radiation, we compute f_esc from cluster stars and non-cluster stars formed duri…
Large Magellanic Cloud Globular Clusters in the Near-infrared. I. RR Lyrae in Reticulum
Inter-University Centre for Astronomy and Astrophysics (IUCAA)（大学间天文学与天体物理学中心）；European Southern Observatory（欧洲南方天文台）；INAF-Osservatorio Astronomico di Capodimonte（意大利国家天体物理研究所卡波迪蒙特天文台）；Istituto Nazionale di Fisica Nucleare (INFN)—Sezione di Napoli（意大利国家核物理研究院那不勒斯分部）；Graduate Institute of Astronomy, National Central University（中央大学天文研究所）
Context. NGC 1851 is one of several globular clusters for which multiple stellar populations of the subgiant branch have been clearly identified and a difference in metallicity detected. A crucial piece of information on the formation history of this cluster can be provided by the sum of A(C+N+O) abundances. However, these values have lacked a general consensus thus far. The separation of the subgiant branch can be based on age and/or A(C+N+O) abundance differences. Aims. Our main aim was to determine carbon, nitrogen, and oxygen abundances for evolved giants in the globular cluster NGC1851 in…
Orbital evolution of binary systems in dense stellar clusters is important in a variety of contexts: origin of blue stragglers, progenitors of compact object mergers, millisecond pulsars, and so on. Here we consider the general problem of secular evolution of the orbital elements of a binary system driven by the smooth tidal field of an axisymmetric stellar cluster (globular, nuclear, etc.) in which the binary orbits. We derive a secular Hamiltonian (averaged over both the inner Keplerian orbit of the binary and its outer orbit within the cluster) valid to quadrupole order for an arbitrary clu…
We use N-body simulations to explore the influence of orbital eccentricity on the dynamical evolution of star clusters. Specifically we compare the mass loss rate, velocity dispersion, relaxation time, and the mass function of star clusters on circular and eccentric orbits. For a given perigalactic distance, increasing orbital eccentricity slows the dynamical evolution of a cluster due to a weaker mean tidal field. However, we find that perigalactic passes and tidal heating due to an eccentric orbit can partially compensate for the decreased mean tidal field by energizing stars to higher veloc…
Fluctuations in a stellar system's gravitational field cause the orbits of stars to evolve. The resulting evolution of the system can be computed with the orbit-averaged Fokker-Planck equation once the diffusion tensor is known. We present the formalism that enables one to compute the diffusion tensor from a given source of noise in the gravitational field when the system's dynamical response to that noise is included. In the case of a cool stellar disc we are able to reduce the computation of the diffusion tensor to a one-dimensional integral. We implement this formula for a tapered Mestel di…
