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Register a new userNo Energy Equipartition in Globular Clusters
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M. Trenti
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It is widely believed that globular clusters evolve over many two-body relaxation times toward a state of energy equipartition, so that velocity dispersion scales with stellar mass as sigma ~ m^{-eta} with eta = 0.5. We show that this is incorrect, using direct N-body simulations with a variety of realistic IMFs and initial conditions. No simulated system ever reaches a state close to equipartition. Near the center, the luminous main-sequence stars reach a maximum eta_{max} ~ 0.15 pm 0.03. At large times, all radial bins convergence on an asymptotic value eta_{infty} ~ 0.08 pm 0.02. The development of this partial equipartition is strikingly similar across our simulations, despite the range of initial conditions employed. Compact remnants tend to have higher eta than main-sequence stars (but still eta < 0.5), due to their steeper (evolved) mass function. The presence of an intermediate-mass black hole (IMBH) decreases eta, consistent with our previous findings of a quenching of mass segregation under these conditions. All these results can be understood as a consequence of the Spitzer instability for two-component systems, extended by Vishniac to a continuous mass spectrum. Mass segregation (the tendency of heavier stars to sink toward the core) has often been studied observationally, but energy equipartition has not. Due to the advent of high-quality proper motion datasets from the Hubble Space Telescope, it is now possible to measure eta. Detailed data-model comparisons open up a new observational window on globular cluster dynamics and evolution. Comparison of our simulations to Omega Cen observations yields good agreement, confirming that globular clusters are not generally in energy equipartition. Modeling techniques that assume equipartition by construction (e.g., multi-mass Michie-King models) are approximate at best.
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In the construction of multi-mass King-Michie models of globular clusters, an approximated central energy equipartition between stars of different masses is usually imposed by scaling the velocity parameter of each mass class inversely with the stellar mass, as if the distribution function were isothermal. In this paper, this isothermal approximation (IA) has been checked and its consequences on the model parameters studied by a comparison with models including central energy equipartition correctly. It is found that, under the IA, the temperatures of a pair of components can differ to a non-negligible amount for low concentration distributions. It is also found that, in general, this approximation leads to a significantly reduced mass segregation in comparison with that given under the exact energy equipartition at the centre. As a representative example, an isotropic 3-component model fitting a given projected surface brightness and line-of-sight velocity dispersion profiles is discussed. In this example, the IA gives a cluster envelope much more concentrated (central dimensionless potential W=3.3) than under the true equipartition (W=0.059), as well as a higher logarithmic mass function slope. As a consequence, the inferred total mass (and then the global mass-to-light ratio) results a factor 1.4 times lower than the correct value and the amount of mass in heavy dark remnants is 3.3 times smaller. Under energy equipartition, the fate of stars having a mass below a certain limit is to escape from the system. This limit is derived as a function of the mass and W of the giants and turn-off stars component.
We explain dark energy with equipartition theorem in string landscape.
We compare the results of a large grid of N-body simulations with the surface brightness and velocity dispersion profiles of the globular clusters $omega$ Cen and NGC 6624. Our models include clusters with varying stellar-mass black hole retention fractions and varying masses of a central intermediate-mass black hole (IMBH). We find that an $sim 45,000$ M$_odot$ IMBH, whose presence has been suggested based on the measured velocity dispersion profile of $omega$ Cen, predicts the existence of about 20 fast-moving, $m>0.5$ M$_odot$ main-sequence stars with a (1D) velocity $v>60$ km/sec in the central 20 arcsec of $omega$ Cen. However no such star is present in the HST/ACS proper motion catalogue of Bellini et al. (2017), strongly ruling out the presence of a massive IMBH in the core of $omega$ Cen. Instead, we find that all available data can be fitted by a model that contains 4.6% of the mass of $omega$ Cen in a centrally concentrated cluster of stellar-mass black holes. We show that this mass fraction in stellar-mass BHs is compatible with the predictions of stellar evolution models of massive stars. We also compare our grid of $N$-body simulations with NGC 6624, a cluster recently claimed to harbor a 20,000 M$_odot$ black hole based on timing observations of millisecond pulsars. However, we find that models with $M_{IMBH}>1,000$ M$_odot$ IMBHs are incompatible with the observed velocity dispersion and surface brightness profile of NGC 6624,ruling out the presence of a massive IMBH in this cluster. Models without an IMBH provide again an excellent fit to NGC 6624.
The classical theory of cluster relaxation is unsatisfactory because it involves the Coulomb logarithm. The Balescu-Lenard (BL) equation provides a rigorous alternative that has no ill-defined parameter. Moreover, the BL equation, unlike classical theory, includes the clusters self-gravity. A heuristic argument is given that indicates that relaxation does not occur predominantly through two-particle scattering and is enhanced by self-gravity. The BL equation is adapted to a spherical system and used to estimate the flux through the action space of isochrone clusters with different velocity anisotropies. A range of fairly different secular behaviours is found depending on the fraction of radial orbits. Classical theory is also used to compute the corresponding classical fluxes. The BL and classical fluxes are very different because (a) the classical theory materially under-estimates the impact of large-scale collectively amplified fluctuations and (b) only the leading terms in an infinite sum for the BL flux are computed. A complete theory of cluster relaxation likely requires that the sum in the BL equation be decomposed into a sum over a finite number of small wavenumbers complemented by an integral over large wavenumbers analogous to classical theory.
Internal rotation is considered to play a major role in the dynamics of some globular clusters. However, in only few cases it has been studied by quantitative application of realistic and physically justified global models. Here we present a dynamical analysis of the photometry and three-dimensional kinematics of omega Cen, 47 Tuc, and M15, by means of a recently introduced family of self-consistent axisymmetric rotating models. The three clusters, characterized by different relaxation conditions, show evidence of differential rotation and deviations from sphericity. The combination of line-of-sight velocities and proper motions allows us to determine their internal dynamics, predict their morphology, and estimate their dynamical distance. The well-relaxed cluster 47 Tuc is very well interpreted by our model; internal rotation is found to explain the observed morphology. For M15, we provide a global model in good agreement with the data, including the central behavior of the rotation profile and the shape of the ellipticity profile. For the partially relaxed cluster omega Cen, the selected model reproduces the complex three-dimensional kinematics; in particular the observed anisotropy profile, characterized by a transition from isotropy, to weakly-radial anisotropy, and then to tangential anisotropy in the outer parts. The discrepancy found for the steep central gradient in the observed line-of-sight velocity dispersion profile and for the ellipticity profile is ascribed to the condition of only partial relaxation of this cluster and the interplay between rotation and radial anisotropy.
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