Baryonic Omega Analysis

Upgrading the empirical omega velocity correction model (Flynn & Cannaliato 2025) from point-mass approximations to full baryonic mass decomposition using the Corbelli method.

Overview

This project fits the omega parameter to galaxy rotation curves using decomposed baryonic velocity components (gas, disk, bulge) from the SPARC database, rather than relying on Keplerian point-mass approximations.

Core model (Flynn & Cannaliato 2025 — Linear):

$$V_{model}(R) = V_{bary}(R) + \omega \cdot R$$

Extended model (Schneider 2026 — Rational Taper):

$$V_{model}(R) = V_{bary}(R) + \frac{\omega \cdot R}{1 + R / R_t}$$

The taper introduces a transition radius $R_t$ where the linear correction saturates, producing flat rotation at large radii. We find that $R_t \approx k \cdot R_d$, where $R_d$ is the disk scale length and $k \approx 2.8$ is a candidate universal coupling constant.

Key Results

1. Mechanism: Kinematic, Not Dynamic

BIC model comparison on the M33 calibration target strongly favors linear velocity addition over quadrature force addition ($\Delta\text{BIC} = -4559$). The omega correction acts as a velocity boost — not a dark matter-like potential. This rejects the standard DM halo addition mechanism.

2. The Tapered Model

The pure linear model diverges at large radii. The rational taper saturates the correction at a characteristic radius $R_t$, reducing M33 RMSE by 69% (31.1 → 9.5 km/s) and yielding $\chi^2_\nu = 4.6$ vs. 72.9 for the linear form.

M33 — the calibration target. The Flynn-Cannaliato (2025) linear model (red dashed, $\omega = 6.97$) diverges beyond ~8 kpc while the Schneider (2026) tapered model (orange, $\omega = 42.97$, $R_t = 2.0$ kpc) tracks the flat observed rotation curve. RMSE drops from 31.1 to 9.5 km/s ($\Delta$BIC = 3896).

NGC 3198 — an intermediate spiral showing the same pattern across galaxy types. The linear model overshoots at large radii while the tapered model tracks the flat rotation curve with RMSE = 6.4 km/s.

3. Universal Coupling Constant

Batch analysis of 118 SPARC galaxies reveals a candidate scaling law: the transition radius $R_t$ is proportional to the disk scale length $R_d$ with a median coupling factor $k = 2.81$ (full sample) or $k = 2.55$ among the 89 galaxies with $\chi^2_\nu < 5$.

Distribution of the coupling constant $k = R_t / R_d$ across 118 SPARC galaxies. The primary peak near $k \approx 2$ and the secondary pile-up at $k = 20$ (the parameter bound) reveal two distinct populations: galaxies where the taper is well-constrained (84%) and those that exhibit extended linear-like rise before tapering (16%).

4. Two Populations: Interior vs. Boundary Solutions

The $k$ distribution is bimodal. Galaxies splitting into two populations:

• Interior solutions ($k < 20$, N=99, 84%): The taper is well-constrained. These are predominantly lower-luminosity, lower surface brightness systems.
• Boundary solutions ($k = 20$, N=19, 16%): The optimizer hits the parameter bound — these galaxies prefer the pure linear model. They are systematically brighter, more massive, and have higher surface brightness.

Boxplots comparing luminosity, central surface brightness, and flat velocity between the two populations. Boundary-solution galaxies (linear-preferred) are systematically more luminous and denser — a physically meaningful distinction, not random fitting noise.

5. Robustness

Mass-to-light ratio sensitivity testing ($\Upsilon_d \in {0.3, 0.5, 0.8}$) shows that $\omega$ is most stable for gas-dominated (LSB) systems (28% variation for DDO 161) and most sensitive for disk-dominated (HSB) systems (134% for NGC 2841). The model's strength lies in the LSB regime where baryonic uncertainties are smallest.

6. Model Gallery

Head-to-head comparison of the Flynn & Cannaliato (2025) linear model vs. the Schneider (2026) tapered model across diverse galaxy types:

Galaxy	$\Sigma_0$ ($L_\odot$/pc$^2$)	Prediction	Preferred (BIC)	Verdict
DDO 154	62	Tapered (LSB)	Tapered	Correct
DDO 161	59	Tapered (LSB)	Tapered	Correct
NGC 0300	152	Tapered (LSB)	Tapered	Correct
NGC 3198	618	Transition	Tapered	—
NGC 2841	2260	Linear (HSB)	Tapered	Incorrect
NGC 7331	1583	Linear (HSB)	Linear	Correct
M33	—	—	Tapered	—

The surface brightness predictor achieves 80% accuracy (4/5 testable cases). The NGC 2841 failure suggests the HSB threshold needs refinement or that the tapered model is more broadly applicable than the population split implies.

7. Full-Catalog BIC Analysis (Phase III — 175 Galaxies)

Applying both models to the complete SPARC catalog confirms the Tapered model's superiority at scale. 171 of 175 galaxies converge cleanly (97.7% success rate).

Model Preference	Count	Fraction
Tapered	127	74.3%
Linear	27	15.8%
Indistinguishable	17	9.9%

Median $\Delta\text{BIC} = +49.9$ ("very strong" evidence on the Kass & Raftery scale). The strongest individual wins are UGC02953 ($\Delta$BIC = 19024, RMSE 41.7 → 21.9 km/s) and NGC2403 ($\Delta$BIC = 16435, RMSE 18.6 → 5.4 km/s).

8. $R_t$–$R_d$ Scaling (Full Catalog)

The $R_t \propto R_d$ correlation holds across the full 171-galaxy sample:

$$\log_{10}(R_t) = 0.794 \cdot \log_{10}(R_d) + 0.448 \quad (R^2 = 0.135,; p = 7.8 \times 10^{-7},; N = 171)$$

Median coupling constant: $k = R_t / R_d = 2.42$ (IQR: 0.85–8.60). The statistical significance of the $R_t$–$R_d$ correlation across an independent 171-galaxy sample confirms that the taper scale is physically set by the baryonic disk, not a fitting artifact.

9. Surface Brightness Regime (Full Catalog)

A Mann-Whitney test finds no statistically significant difference in $\Delta$BIC between LSB and HSB regimes ($p = 0.171$). The Tapered model is broadly preferred across all surface brightness classes (LSB: 75%, Transition: 70%, HSB: 75%), superseding the Phase II prediction of a strong LSB/HSB dichotomy. The model is more universal than initially expected.

Documentation

• PHASE_III_RESULTS.md — Full Phase III results: full-catalog BIC analysis, $R_t$–$R_d$ scaling, and complete rotation-curve gallery (171 galaxies)
• PHASE_II_RESULTS.md — Full Phase I & II results with figures and analysis
• METHODOLOGY.md — Detailed methods, equations, and fitting procedures
• CLAUDE.md — Project plan, phases, and developer guidelines

Quick Start

# Install dependencies
pip install -r requirements.txt

# Initialize database
python src/database.py --init

# Ingest M33 from Corbelli 2014 Table 1
python src/ingest.py --m33

# Ingest all SPARC galaxies from MRT file
python src/ingest.py --mrt data/raw/MassModels_Lelli2016c.mrt --metadata data/raw/SPARC_Lelli2016c.mrt

# Run omega fit on a single galaxy
python src/fit.py --galaxy M33 --plot

Notebooks

#	Notebook	Description
01	M33 Calibration	Pipeline validation against Corbelli 2014 data
02	Linear vs Quadrature	Mechanism test: kinematic boost vs. force addition
03	Tapered Models	Rational taper and tanh taper on M33
04	SPARC Batch	118-galaxy batch fit with $k \cdot R_d$ parameterization
05	Phase II Analysis	Population split, density coupling, $\Upsilon$ sensitivity
06	Model Gallery	Head-to-head Linear vs. Tapered across galaxy types
07	Full Catalog Analysis	Phase III: BIC selection, $R_t$–$R_d$ scaling, $\Sigma_0$ regime test on all 175 SPARC galaxies
08	Full Gallery	Rotation-curve gallery for all 171 quality-controlled galaxies (29 pages, sorted by $\Delta$BIC)

Project Structure

• src/ — Core Python modules (database, physics, ingestion, fitting)
• tests/ — Pytest test suite
• notebooks/ — Analysis and visualization notebooks
• data/raw/ — Original SPARC data files
• data/extracted/ — Data extracted from published papers (e.g., Corbelli 2014 Table 1)
• data/processed/ — SQLite database
• results/figures/ — Publication-quality plots
• results/tables/ — Summary statistics and fit results (CSV)
• docs/ — Methodology, results, and internal documentation

Data Sources

• SPARC Database (Lelli, McGaugh, & Schombert 2016): 175 disk galaxies with Spitzer photometry and rotation curves. http://astroweb.cwru.edu/SPARC/
• M33 Calibration (Corbelli et al. 2014): Surface density profiles from Table 1, converted to velocity components via Casertano (1983) thin-disk method.

References

1. Flynn, D. C. & Cannaliato, J. (2025). "A New Empirical Fit to Galaxy Rotation Curves."
2. Corbelli, E. & Salucci, P. (2000). MNRAS, 311, 441. "The Extended Rotation Curve and the Dark Matter Halo of M33."
3. Corbelli, E., et al. (2014). A&A, 572, A23. "Dynamical signatures of a $\Lambda$CDM-halo and the distribution of the baryons in M33."
4. Casertano, S. (1983). MNRAS, 203, 735. "Rotation curve of the edge-on spiral galaxy NGC 5907."
5. Kass, R. E. & Raftery, A. E. (1995). JASA, 90, 773. "Bayes Factors."
6. Lelli, F., McGaugh, S. S., & Schombert, J. M. (2016). AJ, 152, 157. "SPARC: Mass Models for 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves."