From a cluster's total mass and half-light radius, the virial theorem gives a velocity dispersion. The M-σ scaling relation then predicts an IMBH mass. The interesting result is the gap between that prediction and what's actually been measured.
Two stages. First, the virial theorem gives a rough velocity dispersion: σ² ≈ G M_cluster / r_h. Second, the Gültekin et al. (2009) M-σ relation predicts a central black hole mass: M_BH ≈ 8.5×10⁶ M☉ · (σ / 200 km/s)^4.38. These are independent published results; this tool just composes them and compares the result to actual OC IMBH measurements.
The M-σ exponent is 4.38 — a factor-of-3 change in σ becomes a factor of 3^4.38 ≈ 130 in predicted mass. So the gap between OC's observed central σ (≈ 20 km/s) and the naive virial estimate (~60 km/s for OC) translates to a ~10⁴ difference in predicted M_BH. This is why we expose σ as an override slider: small input differences blow up massively in the output.
The M-σ relation was calibrated against galactic nuclei (10⁶ to 10⁹ M☉ supermassive holes) — a regime several orders of magnitude above OC's candidate IMBH. Whether a single power law extends down to globular cluster scales is empirically open. The Gültekin paper itself doesn't claim it does. Treat the predicted M_BH as a "what would a naive scaling-relation extrapolation say?" — useful for context, not as a measurement.
OC's central velocity dispersion has been re-measured many times. Modern estimates from oMEGACat (Sommer et al. 2025, ApJ) cluster around 20 km/s but vary with the radial range chosen and whether rotation is included. The "virial estimate" computed here (~60 km/s for the OC default inputs) is much higher than observed because σ² ≈ GM/r_h ignores structural factors — real cluster density profiles have form-factor corrections of order ~0.1 to 0.2.
The M-σ prediction line in the IMBH Constraint Stacker uses the same calculation with σ = 20 km/s hardcoded. This tool exposes the inputs.
The black hole mass / velocity dispersion relation was discovered independently in Ferrarese & Merritt 2000 (ApJL 539:L9) and Gebhardt et al. 2000 (ApJL 539:L13), then refined by Gültekin et al. 2009 (ApJ 698:198) using ~50 well-measured galactic BHs spanning 10⁶ to 10¹⁰ M☉. The exponent ≈ 4.38 is steep but the intrinsic scatter is real: ~0.3 dex in M_BH at fixed σ. McConnell & Ma 2013 (ApJ 764:184) and later updates push to a sample of ~100 BHs with slightly different normalisation but similar slope. Crucially, all calibrations were done on galactic-nucleus BHs — not GC IMBHs.
OC's central line-of-sight velocity dispersion has been re-measured many times. oMEGACat VI (Sommer et al. 2025) reports the central σ_los ≈ 19–21 km/s from 1.4 million HST proper motions out to the half-light radius, with anisotropy parameter β consistent with isotropy in the inner core. Earlier studies (Pancino 2007, Sollima 2009) found similar values from line-of-sight RVs alone. The "virial estimate" σ ≈ √(GM/r_h) ≈ 60 km/s for OC defaults is three times the measured value because the simple formula ignores the cluster's structural form factor (~0.1–0.2 for King profiles), the projection from 3D to LOS, and the radial profile of σ(r) itself.
With exponent 4.38, a factor of 3 in σ becomes a factor of 3⁴·³⁸ ≈ 130 in M_BH. The gap between the virial σ ≈ 60 km/s (giving M_BH ≈ 4×10⁴ M☉, right at the Noyola detection) and the observed σ ≈ 20 km/s (giving M_BH ≈ 350 M☉, well below Häberle's lower bound) is a 100× difference in predicted IMBH mass. This is not a tool bug — it's the actual reason the M-σ extrapolation to GCs is contested. The relation is steep enough that small uncertainties in σ blow up massively in predicted M_BH.