The stellar-dynamics engine that links ω Cen's stellar census to its central dark mass. Computes the relaxation time, the mass-segregation timescale for heavy remnants, the predicted central concentration after a given evolution time, and whether the runaway-merger IMBH formation channel is open.
In a self-gravitating stellar system, two-body encounters drive the cluster toward equipartition of kinetic energy between species of different mass. Because kinetic energy goes as ½ m v², equipartition requires the heavier objects to move more slowly — and slower objects sink toward the cluster centre. The characteristic timescale is the half-mass relaxation time t_rh. For very heavy populations (m_heavy / <m_star> greater than a critical ratio of ~10), Spitzer 1969 showed that no equipartition exists at all — the heavy subsystem decouples and core-collapses on its own.
It sets every dynamical evolution timescale in a globular cluster. Two-body relaxation drives mass segregation, core collapse, BH-binary hardening, dynamical BH ejection via three-body encounters, and ultimately the runaway-merger channel for IMBH formation. If t << t_rh the cluster is dynamically young — its present stellar distribution still reflects initial conditions. If t >> t_rh the cluster has forgotten its initial state and is fully dynamically processed.
For ω Cen's default parameters (M ≈ 4×10⁶ M☉, r_h ≈ 4.8 pc, ⟨m⟩ = 0.5 M☉) the half-mass relaxation time comes out around 5.8 Gyr (Spitzer 1987 Eq. 2.62) — somewhat shorter than the cluster's age. ω Cen is therefore only marginally relaxed at its current age. The central dark mass observed today is the integrated result of roughly one t_rh of mass segregation acting on an initially heavy-remnant-rich population (the progenitor dwarf galaxy was metal-poor, so produced relatively more black holes via the low-Z high-mass-cutoff channel).
Portegies Zwart et al. (Nature 428:724, 2004) showed numerically that in very dense young clusters, stellar-mass BHs can repeatedly merge into a runaway central object, growing an IMBH within ~10⁷ yr — before stellar feedback disrupts the dense core. The channel requires two thresholds simultaneously: high central density (greater than ~10⁶ M☉/pc³) and a large population of heavy seeds (more than ~10³). The verdict panel checks whether your current slider parameters satisfy both.
The Spitzer 1987 derivation assumes a single-component cluster; real multi-mass cases need full N-body integration. The "BHs concentrate into the inner 10% of r_h" assumption used for the core-density calculation is a fitting-formula approximation from N-body simulations (Wang, Spurzem, Aarseth et al. 2016, MNRAS 458:1450); real numerical results vary by a factor of ~2 depending on initial conditions, primordial binary fraction, and natal-kick prescription. The retained-BH fraction f_ret is itself uncertain at the factor-of-2 level — direct N-body work (Breen & Heggie 2013) suggests deep-potential massive clusters like ω Cen retain ~30–50% of their natal BHs, but the value depends strongly on the assumed kick distribution.
Spitzer 1987, Dynamical Evolution of Globular Clusters (Princeton University Press) — canonical reference for t_rh and segregation. Spitzer 1969 (ApJ 158:L139) — segregation instability for very heavy populations. Breen & Heggie 2013 (MNRAS 432:2779) — BH retention in dense clusters. Portegies Zwart et al. 2004 (Nature 428:724) — runaway-merger IMBH channel. Wang, Spurzem, Aarseth et al. 2016 (MNRAS 458:1450) — N-body fitting formulas for segregated BH distributions. Bañares-Hernández et al. 2025 (A&A 693:A104) — dark-cluster upper limit ~6,000 M☉ for ω Cen central remnant population.
The retained BH count and total mass computed here are the natural inputs to the Dark Cluster Alternative tool. The verdict feeds the central-mass row in the IMBH Constraint Stacker. For the orthogonal central-mass estimate from kinematics, see the M-σ tool.