QPO Mass-Spin Estimator — Omega Centauri Society

Constrain IMBH mass and spin from observed quasi-periodic oscillation frequencies — ISCO orbital model and 3:2 epicyclic resonance (Török et al. 2011)

Observed QPO Frequency

Observed quasi-periodic oscillation frequency from X-ray timing data

QPO frequency ν_obs (mHz)11.76

0.01 mHz10 mHz1 Hz1 kHz

Frequency uncertainty σ_ν / ν (%)10

QPO MODEL

Spin Constraint

Set spin to compute the implied mass, or scan the full M–a space

Spin parameter a0.50

0 (Schw.)0.998 (max)

Implied M at this a

—

M☉

r_ISCO at this a

—

× r_g

ν_ISCO (this M, a)

—

mHz

Period at ISCO

—

seconds

—

Mass-Spin Constraint Contour

Allowed (M, a) pairs under the selected QPO model. OC constraint window overlaid.

      — QPO constraint band
      ▬ Häberle ≥8,200 M☉
      ▬ Bañares ≤6,000 M☉
      ✦ Current (M, a)
    

Mass Range Over Full Spin Domain

Min/max mass implied by the QPO across all allowed spins (0 → 0.998)

M at a=0 (Schwarzschild)

—

M☉

M at a=0.998 (max spin)

—

M☉

In OC window (6k–8.2k)?

—

at some spin a

Required a for M=8,200

—

spin parameter

What this tool does

Quasi-periodic oscillations (QPOs) are modulations in the X-ray flux of accreting black holes at frequencies associated with orbital motion near the innermost stable circular orbit (ISCO). Because the ISCO frequency depends on both mass and spin, a detected QPO frequency constrains the (M, a) parameter space.

QPO models implemented

ISCO orbital frequency: The Keplerian orbital frequency at the ISCO is ν_ISCO = c³/(2πGM × x_ISCO^(3/2)), where x_ISCO = r_ISCO/r_g is computed from the Kerr metric (Bardeen et al. 1972). For a=0, x_ISCO = 6; for a=0.998, x_ISCO ≈ 1.24. A QPO at ν_obs interpreted as ν_ISCO directly gives a (M, a) constraint.

3:2 epicyclic resonance (Török et al. 2011): Many QPO pairs are observed with frequency ratio 3:2. The parametric resonance condition occurs at r_res where Keplerian ν_K and radial epicyclic ν_r are in 3:2 ratio. For Schwarzschild: r_res at x=10.8 r_g. The upper QPO maps as M ≈ 912×10³ M_sun / (ν_U in mHz) (at a=0), and the lower as M ≈ 608×10³ M_sun / (ν_L in mHz). Both shift with spin.

Nodal precession (Lense-Thirring): Frame-dragging causes a precession of the orbital plane at frequency ν_prec = 2GJ/(c²r³) = aGM/(πr³c). For M = 10,000 M☉ and a = 0.5, ν_prec at r = 5 r_g is ≈ 0.4 mHz. Useful for detecting very low-frequency QPOs.

Observational context

Bian et al. (2025, Nature Astronomy) reported an 85-second quasi-periodicity (ν = 11.76 mHz) from a TDE by a candidate IMBH, constraining the mass to 9,900–16,000 M☉ and spin to 0.26–0.36 using spectral plus timing analysis combined. This tool's 3:2 Keplerian resonance model alone gives M ≈ 77,000–95,000 M☉ for 11.76 mHz — a factor ~6–10 higher than the Bian 2025 result because the paper uses additional spectral constraints not captured by the Keplerian resonance formula alone.

OC constraint window

The grey band on the contour plot shows the OC IMBH constraint window: Häberle et al. 2024 lower bound ≥ 8,200 M☉ (amber) and Bañares-Hernández et al. 2025 upper bound ≤ 6,000 M☉ (red). If a QPO were detected in OC's core, identifying the spin value where the mass constraint intersects this window would be the key scientific result.

References

Török, G., Kotrlová, A., & Šrámková, E. (2011). A&A 531:A59. DOI: 10.1051/0004-6361/201015549 Abramowicz, M. A. & Kluzniak, W. (2001). A&A 374:L19. DOI: 10.1051/0004-6361:20010791 Bian, W. H. et al. (2025). Nature Astronomy. DOI: 10.1038/s41550-025-02502-0 Häberle et al. (2024). Nature 631:285. DOI: 10.1038/s41586-024-07511-z Bañares-Hernández et al. (2025). A&A 693:A104. DOI: 10.1051/0004-6361/202451763

v1.0 — 2026-06-02 · Tool content may be revised as scientific knowledge evolves.