[Paper Review] Rotation, Statistical Dynamics and Kinematics of Globular Clusters

D. Lynden–Bell|arXiv (Cornell University)|Jul 10, 2000

Stellar, planetary, and galactic studies3 citations

TL;DR

This paper investigates self-similar core collapse in globular clusters with mass segregation and rotation, proposing that core evolution follows ρ₀ ∝ r_c^−α with α ≈ 1/2. It introduces a statistical equilibrium model without violent relaxation, derives a dynamical main sequence where binary energy balances core expansion, and presents a method to detect tidal streams using radial velocities and orbital energy to infer angular momentum and proper motions.

ABSTRACT

Evolution with mass segregation and the evolution of the rotation of cores are both discussed for self-similar core collapse. Evolution with angular velocity proportional to the square root of the density is predicted. On the Dynamical Main Sequence of globular clusters the energy emission from binaries balances the energy expended in expanding the halo. Newton's exactly solved N-body problem is then given, along with recent generalisations, all of which have no violent relaxation, but a new type of statistical equilibrium is discussed. Finally, we set the creation of streams in the Galaxy's halo in the historical context of their discovery.

Motivation & Objective

• To resolve long-standing problems in stellar dynamics, particularly self-similar core collapse with mass segregation and rotation in globular clusters.
• To develop a predictive theory for the Dynamical Main Sequence, where binary energy generation balances core expansion.
• To identify and model tidal streams in the Galactic halo using orbital energy and angular momentum from radial velocity data.
• To establish a statistical equilibrium framework for N-body systems without violent relaxation, based on Newton's exact solution.

Proposed method

• Uses self-similar solutions with ρ(r,t) = ρ₀(t)ρ∗(r∗), r∗ = r/rc(t), to model core collapse and derive ρ₀ ∝ rc^−α.
• Applies dimensional analysis and the Local Variational Principle (Inagaki & Lynden-Bell, 1990) to solve multidimensional problems in energy, angular momentum, time, and mass.
• Adopts Michie-King models with modified Maxwell-Boltzmann distributions to account for mass segregation and avoid unphysical escape velocities.
• Derives the relaxation time T_c⁻¹ ∝ G²β³/²ρ₀lnΛ⟨m⁷/²⟩/⟨m⟩, showing evolution depends on the 7/2 power of mass.
• Uses radial velocity data to plot Er = ½v_r² − ψ(r) against r⁻² to infer specific energy E and angular momentum h² from linear fits.
• Applies iterative methods to correct for non-galactocentric radial velocities and identify stream members via consistent orbital parameters.

Experimental results

Research questions

• RQ1 Can self-similar core collapse occur in clusters with a continuous distribution of stellar masses?
• RQ2 How does rotation evolve during core collapse in weakly rotating clusters, and does self-similarity persist?
• RQ3 Can a predictive theory be developed for the Dynamical Main Sequence where binary energy balances core expansion?
• RQ4 Can tidal streams from disrupted satellites be detected and their orbital parameters reconstructed from radial velocity data?
• RQ5 What is the role of statistical equilibrium without violent relaxation in pulsating N-body systems?

Key findings

• Self-similar core collapse predicts ρ₀ ∝ r_c^−α with α ≈ 1/2, implying core density increases as core radius shrinks.
• The relaxation time in the core depends on the 7/2 power of mass, making evolution sensitive to mass segregation.
• The Dynamical Main Sequence is maintained when binary energy generation balances core expansion, suggesting a stable equilibrium state.
• Newton’s exact N-body solution allows for statistical equilibrium without violent relaxation, even under macroscopic radial pulsation.
• Radial velocity data plotted against r⁻² yield straight lines whose intercept and slope give E and h², enabling stream detection and proper motion prediction.
• The method successfully identifies streams such as the Magellanic and Sagittarius Streams, with potential to trace early Galactic formation via fossilized orbital invariants.