A scale-invariant framework and reproducible dataset for comparing black holes across the mass spectrum (Nagy 2025).

Stephen L. Nagy · Independent Researcher · September 2025 - Current

A Scale-Invariant Ruler for Black Holes: From Stellar-Mass to Ultra-Massive with Unified Uncertainties
License: MIT

BHRuler implements the Black Hole Ruler framework for scale-invariant, reproducible comparison of black holes from stellar to ultra-massive regimes. It provides code and data to compute gravitational units and ISCO landmarks, spin-aware Kerr corrections, dual accretion prescriptions (η-bridge and ADAF/RIAF), Blandford–Znajek jet-power estimates, environmental metrics (σ, r_infl, r_infl/R_e), and a TDE module with a logistic capture boundary—plus a versioned catalog schema for cross-scale studies.

The single-file CLI, bhruler.py, now also supports direct ingestion of the Survey V2 CSV format, including fields such as Name, Class, Mass_Msun, Spin_a, Spin_Known, and Regime, while preserving the standard BHRuler schema for pipeline use.

• Native Survey V2 CSV support in bhruler.py
• New core branch for mass/spin/timing/ISCO/TDE calculations without requiring accretion inputs
• Flexible CSV parsing across case, spaces, underscores, and punctuation
• Survey-compatible output aliases written alongside canonical BHRuler columns:
  • Name
  • Mass_Msun
  • Spin_a
  • Grav_Time_tg_s
  • Schwarzschild_Rad_km
  • Orbital_Period_s
  • Invariant_fISCO_tg
• Survey metadata pass-through for Class, Spin_Known, and Regime
• Direct Survey V2 workflow:

  python bhruler.py --input 50_BH_Survey_V2_verified.csv --branch core --output survey_out.csv

• Five publication-quality figures generated from computed data (cross-scale invariant, spin diagnostics, M87* sensitivity, 10-object atlas, worked uncertainty propagation)
• Expanded reference list (45 citations anchoring every equation to its theoretical source)
• Worked uncertainty example for M87* with both analytic error propagation and a 50,000-draw Monte Carlo, showing κ dominates the P_BZ error budget at 41% of variance
• Strengthened case studies with direct comparisons to observed jet powers (Cyg X-1), flare timescales (Sgr A*), and EHT constraints (M87*)
• Complete equation-to-column appendix mapping every bhruler.py output to its paper equation
• Physical constants appendix with CODATA/IAU sources
• Removed unimplemented TDE rate equation from the paper body (retained as future-work item)
• TDE spin coefficient explicitly framed as tunable heuristic with guidance to calibrate against Kesden (2012) and Servin & Kesden (2017)
• B_H normalization anchored to Tchekhovskoy+ (2011) and McKinney+ (2012) GRMHD MAD simulations

• Why BHRuler?
• Core Equations
• Features
• Repository Layout
• Quick Start
• Data Products
• Schema
• Reproducing Results
• Quality Control & Provenance
• Contributing
• Citation
• License

• Scale-invariant: Everything is expressed in gravitational units so times, lengths, and frequencies map across 10+ orders of mass.
• Universal invariant: f_ISCO × t_g = 1/(6^{3/2} 2π) ≈ 0.01083 provides a built-in sanity check, verified to machine precision across the mass scale.
• Model-transparent: Dual accretion prescriptions (η-bridge vs ADAF/RIAF) make model dependence explicit for jet-power predictions.
• Cross-scale validation: Unifies X-ray binaries, AGN, EHT imaging, and GW remnants on the same ruler.
• Survey-ready: The same single-file code now accepts both standard BHRuler inputs and Survey V2 tables directly.
• Uncertainty-aware: Analytic error propagation formulas plus a worked Monte Carlo example for M87*.
• Reproducible: Versioned catalog schema with per-row provenance and QC flags.

Gravitational units

t_g = GM/c³,    r_g = GM/c²,    r_s = 2r_g

Schwarzschild ISCO

r_ISCO = 6 r_g
f_ISCO = c³ / (6^{3/2} 2π GM)
f_ISCO × t_g = 1 / (6^{3/2} 2π)    [mass-invariant]

Kerr ISCO (equatorial, prograde/retrograde) Using Bardeen–Press–Teukolsky (1972):

f_ISCO(a*) = (c³ / 2π GM) × 1 / (r_ISCO^{3/2}(a*) ± a*)

Accretion prescriptions

• η-bridge: ṁ = λ_Edd / η_eff, B_H ∝ ṁ^{1/2} M^{-1/2}, P_BZ ∝ a*² (λ_Edd/η_eff) M
• ADAF/RIAF: λ_Edd = κ ṁ² ⟹ ṁ = (λ_Edd/κ)^{1/2}, B_H ∝ (λ_Edd/κ)^{1/4} M^{-1/2}, P_BZ ∝ a*² (λ_Edd/κ)^{1/2} M

Blandford–Znajek (order-of-magnitude; normalization from Tchekhovskoy+ 2011 GRMHD MAD simulations)

P_BZ ≈ 10⁴⁵ erg/s × (a*/0.9)² × (B_H/10⁴ G)² × (M/10⁹ M☉)²

Environment & TDE

r_infl = GM/σ²
t_fb ≈ 41 d × (M/10⁶ M☉)^{1/2} × (R*/R☉)^{3/2} × (M*/M☉)⁻¹

TDE capture boundary: logistic S(M; a*) with tunable midpoint and width.

• Ruler core: t_g, r_s, f_ISCO, plus invariant check
• Spin-aware: r_ISCO(a*), f_ISCO(a*) via BPT
• Accretion toggle: core, η-bridge, ADAF/RIAF, or both
• Jet power: B_H estimates and P_BZ ranges (MAD-motivated scaling)
• Environment: σ → r_infl, r_infl/R_e
• TDE: disrupt-vs-swallow flag plus t_fb
• Survey support: direct ingestion of Survey V2 CSVs without manual column renaming
• Flexible parser: handles alternate naming conventions across cases, spaces, underscores, and punctuation
• Schema + QC: versioned, with provenance fields and automated checks

• bhruler.py — single-file CLI implementation of the Black Hole Ruler
• Using_BHRuler.md — full user guide, schema notes, examples, and troubleshooting
• CHANGELOG.md — version history and survey integration notes
• Paper/ — manuscript source and figure-generation scripts
• data/ — reproducible catalog products and supporting tables

Requirements: Python ≥ 3.8, pandas

pip install pandas

python bhruler.py --output trio_out.csv

This writes a 3-row CSV (Cygnus X-1, Sgr A*, M87*) using both accretion branches by default.

python bhruler.py --input objects.csv --branch both --output derived.csv

python bhruler.py --input 50_BH_Survey_V2_verified.csv --branch core --output survey_out.csv

• core — mass/spin/timing/ISCO/TDE quantities only
• eta_bridge — efficiency-bridge accretion prescription
• adaf — ADAF/RIAF accretion prescription
• both — computes both accretion branches

For survey-style catalogs that contain mass and spin but not accretion inputs, --branch core is the recommended mode.

See Using_BHRuler.md for the full user guide, accepted aliases, worked examples, and troubleshooting.

import bhruler as br

row = dict(
    name="Test",
    M_Msun=1e8,
    a_star=0.7,
    lambda_Edd=0.01,
    eta_eff=0.1,
    kappa=0.1,
    sigma_kms=200,
    Re_kpc=4.0
)

out = br.compute_row(row, branch="both", tde_mcrit=3e7, tde_width=0.15)

Core inputs: name, M_Msun, a_star, lambda_Edd, eta_eff, kappa, sigma_kms, Re_kpc, branch

Name / identifier

• name
• Name
• Object
• object

Mass

• M_Msun
• Mass_Msun
• M
• mass_Msun
• Mass [M_sun]
• M [M_sun]
• M [Msun]

Spin

• a_star
• Spin_a
• spin
• Spin a_* (prograde)
• a_* (assumed/est.)

Eddington ratio

• lambda_Edd
• lambda
• L/L_Edd
• L/L_Edd (assumed)

Optional metadata

• Class
• Spin_Known
• Regime

Core: t_g_s, r_s_km, f_ISCO_Schw_Hz, fISCO_tg_invariant

Spin-aware: r_ISCO_rg, f_ISCO_Kerr_Hz, f_ratio

Environment: rinfl_pc, rinfl_over_Re, t_fb_days

TDE: tde_possible

Accretion/Jets: B_H_G_eta, P_BZ_erg_s_eta, B_H_G_adaf, P_BZ_erg_s_adaf

When survey-style input is detected, bhruler.py also writes:

• Name
• Mass_Msun
• Spin_a
• Grav_Time_tg_s
• Schwarzschild_Rad_km
• Orbital_Period_s
• Invariant_fISCO_tg

This preserves compatibility with survey-style tables while keeping the canonical BHRuler columns in the same output file.

cd Paper/figures
python fig1_invariant.py
python fig2_spin.py
python fig3_m87.py
python fig4_atlas.py
python fig5_uncertainty.py

Requires: matplotlib, numpy.

cd Paper
pdflatex Black_Hole_Ruler_V2_2025.tex
pdflatex Black_Hole_Ruler_V2_2025.tex

import pandas as pd

# Verify invariant
check = pd.read_csv("data/bh_ruler_check_2025.csv")
print(check[["Object", "Mass [M_sun]", "f_ISCO * t_g (dimensionless)"]])

# M87* sensitivity
sens = pd.read_csv("data/M87_RIAF_sensitivity_2025.csv")
print(sens[["Model", "mdot (dotM/dotM_Edd)", "P_BZ [erg/s]"]])

• Invariant verification: f_ISCO × t_g ≈ 0.01083 is checked automatically
• Flexible schema handling: alternate column names are normalized before parsing
• Survey metadata preservation: Class, Spin_Known, and Regime are retained when present
• Accretion warnings: missing eta_eff or kappa produce explicit warning columns for the affected branch
• Every table row can include provenance-style metadata for traceability

Issues and pull requests are welcome. Please:

1. Keep numeric changes reproducible.
2. Update documentation when adding or changing accepted input schema.
3. Run the invariant check (f_ISCO × t_g ≈ 0.01083) before submitting.
4. Keep bhruler.py self-contained, since the single-file CLI is the main public interface.

If you use BHRuler or the accompanying datasets in your work, please cite:

Nagy, S. L. (2025). A Scale-Invariant Ruler for Black Holes: From Stellar-Mass to Ultra-Massive with Unified Uncertainties. Black Hole Ruler.

BibTeX

@software{Nagy2025BHRuler,
  author       = {Stephen L. Nagy},
  title        = {{A Scale-Invariant Ruler for Black Holes:
                   From Stellar-Mass to Ultra-Massive with Unified Uncertainties}},
  year         = {2025},
  version      = {2.1.1-survey},
  url          = {https://github.com/SNagy3/BHRuler},
  note         = {BHRuler software, data, and paper}
}

Topics

• black-holes
• astrophysics
• astronomy
• general-relativity
• gravitational-waves
• agn
• x-ray-binaries
• accretion
• relativistic-jets
• kerr-metric
• isco
• python
• scientific-computing
• scientific-python
• data-science
• simulation
• reproducible-research
• research
• space
• cosmology

This project is licensed under the MIT License. See LICENSE for details.