Graduate Seminar: Compact Objects in Dense Stellar Systems
Omega Centauri Society Problem Sets
Problem Set 1 — Stellar Kinematics & Mass Estimation
Estimated time: 3–4 hours Total marks: 40 Tools: velocity-dispersion · anisotropy-degeneracy-explorer omegacentauri.me/pset-1-kinematics.html
Learning objectives
  1. Apply the M–σ relation to predict a cluster's central velocity dispersion from an assumed IMBH mass
  2. Assess how velocity anisotropy (β) can mimic the kinematic signature of a central point mass
  3. Quantify the mass resolution needed to distinguish the Häberle (2024) and Baumgardt (2017) models
  4. Evaluate what follow-up observations would resolve the current kinematic tension

Background

Omega Centauri (NGC 5139) hosts the strongest current evidence for an intermediate-mass black hole (IMBH) in any globular cluster. Häberle et al. (2024) identified seven stars within 0.08 pc of the cluster center with velocities implying a central dark mass of M = 8,200 M☉, while Baumgardt (2017) placed a 3σ upper limit of M < 3,000 M☉ from N-body modeling of the velocity dispersion profile. This tension is the central unresolved issue in the field.

The M–σ relation for globular clusters relates IMBH mass M to the cluster's global velocity dispersion σ. In OC, Baumgardt & Hilker (2018) measured σ0 = 16.8 km/s from HST proper motions.

The Keplerian velocity at projected radius R from a point mass M is:

v_K(R) = sqrt(G M / R) G = 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻² 1 M☉ = 1.989 × 10³⁰ kg 1 pc = 3.086 × 10¹⁶ m

The spherical Jeans equation (isotropic, power-law density profile) gives the projected velocity dispersion as a function of anisotropy parameter β (0 = isotropic, β > 0 = radially anisotropic, β < 0 = tangentially anisotropic). The OCS Anisotropy Degeneracy Explorer computes this numerically.

Problems

1. M–σ prediction at two mass hypotheses

The M–σ relation for intermediate-mass black holes in globular clusters is calibrated at:

σ_pred ≈ σ_ref × (M / M_ref)^(1/4) σ_ref = 16.8 km/s at M_ref = 8,200 M☉ [Baumgardt & Hilker 2018]
  1. Using the OCS Velocity Dispersion tool (pre-filled link below), record σ_pred for M = 8,200 M☉. Does the tool's prediction agree with the Baumgardt & Hilker (2018) measurement of 16.8 km/s? [5 marks]
  2. Repeat for M = 3,000 M☉ (Baumgardt 2017 N-body upper limit). What predicted σ do you get? By how many km/s does this differ from the M = 8,200 M☉ prediction? [5 marks]
  3. Given that HST proper-motion surveys achieve σ measurement precision of roughly ±0.5 km/s, is the σ difference between the two mass hypotheses observationally distinguishable at present? Explain. [5 marks]
↗ Velocity Dispersion tool — pre-filled M = 8,200 M☉, rh = 4.8 pc
Show solution
Part (a): The tool should return σ_pred ≈ 16.8 km/s for M = 8,200 M☉, consistent with the Baumgardt & Hilker (2018) value of 16.8 km/s. This is essentially exact because the tool is calibrated to that reference point.

Part (b): For M = 3,000 M☉, the (M/M_ref)1/4 factor is (3000/8200)0.25 ≈ 0.757. So σ_pred ≈ 16.8 × 0.757 ≈ 12.7 km/s. The difference is ~4.1 km/s.

Part (c): A 4.1 km/s difference far exceeds the ±0.5 km/s precision of HST proper-motion surveys. In principle the two mass hypotheses are distinguishable from the global σ profile. However, the Baumgardt (2017) N-body analysis did use this measurement and still preferred low mass — the discrepancy arises because the N-body fit uses the full radial profile, not just the central σ value. The global σ predicted by M–σ is not directly falsifying either model without higher-resolution central measurements.
2. Keplerian velocities of the fast-moving stars

Häberle et al. (2024) report the seven fast-moving stars lie within a projected radius R ≈ 0.04–0.08 pc of the cluster center. Use the Keplerian velocity formula above (assuming circular orbits and M = 8,200 M☉).

  1. Compute v_K at R = 0.04 pc and at R = 0.08 pc. Express your answers in km/s. [5 marks]
  2. Häberle et al. report peak 3D velocities of ~50 km/s. Is this consistent with your Keplerian estimate for a bound orbit at these radii? [5 marks]
  3. If instead M = 3,000 M☉, what Keplerian velocity would you predict at R = 0.04 pc? Can this explain the observed ~50 km/s stars? [5 marks]
Show solution
Part (a): M = 8200 × 1.989×10³⁰ = 1.631×10³⁴ kg R₁ = 0.04 × 3.086×10¹⁶ = 1.234×10¹⁵ m R₂ = 0.08 × 3.086×10¹⁶ = 2.469×10¹⁵ m v_K(R₁) = sqrt(6.674e-11 × 1.631e34 / 1.234e15) = sqrt(8.823e8) ≈ 29,700 m/s ≈ 29.7 km/s v_K(R₂) = sqrt(6.674e-11 × 1.631e34 / 2.469e15) = sqrt(4.411e8) ≈ 21,000 m/s ≈ 21.0 km/s So: ~30 km/s at 0.04 pc, ~21 km/s at 0.08 pc.

Part (b): The Keplerian circular-orbit speed at 0.04 pc is ~30 km/s. The reported ~50 km/s is for the 3D space velocity; for a circular orbit seen at some inclination angle, the projected velocity can differ. A 50 km/s 3D speed at 0.04 pc requires M ~ (50/v_K)² × 8200 ≈ 2.8 × 8200 ≈ 23,000 M☉ for a purely circular orbit — more than the Häberle point estimate. However, for eccentric orbits, stars can reach higher velocities near periapsis than a circular orbit of the same semi-major axis, so 50 km/s 3D velocities are consistent with M ≈ 8,200 M☉ on eccentric orbits at small periapsis.

Part (c): For M = 3,000 M☉: v_K(0.04 pc) = 29.7 × sqrt(3000/8200) ≈ 29.7 × 0.605 ≈ 18.0 km/s This is far below 50 km/s. A 3,000 M☉ central mass cannot produce bound stars at 0.04 pc with 3D velocities of ~50 km/s; such stars would need to be unbound (escaping the cluster), which is dynamically implausible in large numbers.
3. The anisotropy degeneracy

A centrally-concentrated population of stellar-mass black holes, or radial velocity anisotropy (β > 0), can produce a central σ cusp that mimics an IMBH. Use the OCS Anisotropy Degeneracy Explorer.

  1. Set MIMBH = 0 and Mrem = 0. Increase β from 0 to 0.4 in steps of 0.1. At what β does the predicted central σ approach the value produced by MIMBH = 8,200 M☉ with β = 0? [5 marks]
  2. The IMBH hypothesis predicts a Keplerian σ ∝ R−1/2 cusp inside the IMBH sphere of influence (rh = G M / σ0²). Anisotropy produces a much shallower or flat central profile. What observational signature would most cleanly distinguish the two cases? [5 marks]
↗ Anisotropy Degeneracy Explorer — pre-filled M = 0, β = 0
Show solution
Part (a): The exact answer depends on the model implementation, but typically β ≈ 0.3–0.4 is needed to reproduce the central σ enhancement that an 8,200 M☉ IMBH produces at isotropic β = 0. Students should report the value they read from the tool and note that this β value is physically plausible (many globular clusters show mild radial anisotropy).

Part (b): The clearest distinguishing signature is the radial profile of σ in the innermost 0.05 pc. An IMBH produces a σ ∝ R−1/2 cusp (Keplerian rise) only within its sphere of influence rh = G M / σ0² ≈ G × 8200 M☉ / (16.8 km/s)² ≈ 1.8 pc. Anisotropy flattens the profile outside the innermost ~0.01 pc. Individual star proper motions at R < 0.05 pc (achievable with ELT/MICADO) would show a tangential-to-radial velocity ratio consistent with isotropic Keplerian motion for the IMBH case, but radially-biased anisotropy for the no-IMBH case.

References