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‎README.md‎

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# Baryonic Omega Analysis
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# Baryonic Coupling in Galaxy Rotation Curves
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Upgrading the empirical omega velocity correction model (Flynn & Cannaliato 2025) from point-mass approximations to full baryonic mass decomposition using the Corbelli method.
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[![Status: Under Review](https://img.shields.io/badge/Status-Under_Review-yellow.svg)]()
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[![License: MIT](https://img.shields.io/badge/License-MIT-blue.svg)](LICENSE)
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## Overview
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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.
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**Core model (Flynn & Cannaliato 2025 — Linear):**
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$$V_{model}(R) = V_{bary}(R) + \omega \cdot R$$
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**Extended model (Schneider 2026 — Rational Taper):**
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$$V_{model}(R) = V_{bary}(R) + \frac{\omega \cdot R}{1 + R / R_t}$$
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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.
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## Key Results
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### 1. Mechanism: Kinematic, Not Dynamic
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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.
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### 2. The Tapered Model
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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.
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![M33 Linear vs Tapered](results/figures/M33_linear_reanalysis.png)
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_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)._
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![NGC 3198 Gallery](results/figures/gallery_NGC3198.png)
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_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._
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### 3. Universal Coupling Constant
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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 $\chi^2_\nu < 5$.
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![Coupling Constant Distribution](results/figures/k_distribution.png)
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_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%)._
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### 4. Two Populations: Interior vs. Boundary Solutions
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The $k$ distribution is bimodal. Galaxies splitting into two populations:
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- **Interior solutions** ($k < 20$, N=99, 84%): The taper is well-constrained. These are predominantly lower-luminosity, lower surface brightness systems.
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- **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.
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![Population Split](results/figures/split_populations.png)
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_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._
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This repository contains the data, fitting pipeline, and full results table for the manuscript:
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**"A Baryonically-Coupled Rational Taper Model for Galaxy Rotation Curves: Evidence from the Full SPARC Catalog"** (Schneider, 2026; _Awaiting submission_).
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### 5. Robustness
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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.
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### 6. Model Gallery
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Head-to-head comparison of the Flynn & Cannaliato (2025) linear model vs. the Schneider (2026) tapered model across diverse galaxy types:
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| Galaxy | $\Sigma_0$ ($L_\odot$/pc$^2$) | Prediction | Preferred (BIC) | Verdict |
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| -------- | ----------------------------: | ------------- | --------------- | --------- |
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| DDO 154 | 62 | Tapered (LSB) | Tapered | Correct |
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| DDO 161 | 59 | Tapered (LSB) | Tapered | Correct |
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| NGC 0300 | 152 | Tapered (LSB) | Tapered | Correct |
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| NGC 3198 | 618 | Transition | Tapered | — |
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| NGC 2841 | 2260 | Linear (HSB) | Tapered | Incorrect |
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| NGC 7331 | 1583 | Linear (HSB) | Linear | Correct |
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| M33 | — | — | Tapered | — |
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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.
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### 7. Full-Catalog BIC Analysis (Phase III — 175 Galaxies)
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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).
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| Model Preference | Count | Fraction |
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|---|---|---|
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| Tapered | 127 | 74.3% |
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| Linear | 27 | 15.8% |
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| Indistinguishable | 17 | 9.9% |
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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).
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![BIC Histogram](results/figures/phase_iii_bic_histogram.png)
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### 8. $R_t$–$R_d$ Scaling (Full Catalog)
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The $R_t \propto R_d$ correlation holds across the full 171-galaxy sample:
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## Overview
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$$\log_{10}(R_t) = 0.794 \cdot \log_{10}(R_d) + 0.448 \quad (R^2 = 0.135,\; p = 7.8 \times 10^{-7},\; N = 171)$$
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This project explores a phenomenological extension to the empirical velocity correction model proposed by Flynn & Cannaliato (2025). By upgrading from point-mass approximations to full baryonic mass decompositions using the SPARC database, we test two kinematic models:
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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.
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**1. Linear Model (Flynn & Cannaliato 2025):**
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$$V_{model}(R) = V_{bary}(R) + \omega R$$
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![Rt vs Rd](results/figures/phase_iii_Rt_vs_Rd.png)
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**2. Rational Taper Model (Schneider 2026):**
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$$V_{model}(R) = V_{bary}(R) + \frac{\omega R}{1 + R / R_t}$$
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### 9. Surface Brightness Regime (Full Catalog)
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The Tapered model introduces a transition radius $R_t$ where the linear correction saturates. Across 171 quality-controlled SPARC galaxies, we find a statistically significant empirical scaling relation where the saturation scale couples to the baryonic disk: $R_t \approx 2.4 R_d$.
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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.
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## Reproducing the Manuscript Figures
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![Sigma0 Regime](results/figures/phase_iii_sigma0_regime.png)
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Reviewers and readers can reproduce the exact figures and statistical analyses found in the manuscript using the provided Jupyter Notebooks:
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## Documentation
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## Notebooks
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- [**PHASE_III_RESULTS.md**](docs/PHASE_III_RESULTS.md) — Full Phase III results: full-catalog BIC analysis, $R_t$–$R_d$ scaling, and complete rotation-curve gallery (171 galaxies)
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- [**PHASE_II_RESULTS.md**](docs/PHASE_II_RESULTS.md) — Full Phase I & II results with figures and analysis
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- [**METHODOLOGY.md**](docs/METHODOLOGY.md) — Detailed methods, equations, and fitting procedures
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- [**CLAUDE.md**](CLAUDE.md) — Project plan, phases, and developer guidelines
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| # | Notebook | Description |
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| --- | -------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------ |
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| 01 | [M33 Calibration](notebooks/01_m33_calibration.ipynb) | Pipeline validation against Corbelli 2014 data |
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| 02 | [Linear vs Quadrature](notebooks/02_linear_vs_quadrature_comparison.ipynb) | Mechanism test: kinematic boost vs. force addition |
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| 03 | [Tapered Models](notebooks/03_tapered_linear_model.ipynb) | Rational taper and tanh taper on M33 |
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| 04 | [SPARC Batch](notebooks/04_sparc_batch_analysis.ipynb) | 118-galaxy batch fit with $k \cdot R_d$ parameterization |
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| 05 | [Phase II Analysis](notebooks/05_phase2_density_coupling.ipynb) | Population split, density coupling, $\Upsilon$ sensitivity |
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| 06 | [Model Gallery](notebooks/06_model_gallery.ipynb) | Head-to-head Linear vs. Tapered across galaxy types |
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| 07 | [Full Catalog Analysis](notebooks/07_full_catalog_analysis.ipynb) | Phase III: BIC selection, $R_t$–$R_d$ scaling, $\Sigma_0$ regime test on all 175 SPARC galaxies |
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| 08 | [Full Gallery](notebooks/08_full_gallery.ipynb) | Rotation-curve gallery for all 171 quality-controlled galaxies (29 pages, sorted by $\Delta$BIC) |
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## Quick Start
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The pipeline is written in Python and uses SQLite for data management.
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```bash
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# Clone the repository
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git clone [https://github.com/JustinSchneider/baryonic-omega-analysis.git](https://github.com/JustinSchneider/baryonic-omega-analysis.git)
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cd baryonic-omega-analysis
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# Install dependencies
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pip install -r requirements.txt
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# Initialize database
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# Initialize database and ingest SPARC/M33 data
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python src/database.py --init
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# Ingest M33 from Corbelli 2014 Table 1
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python src/ingest.py --m33
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# Ingest all SPARC galaxies from MRT file
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python src/ingest.py --mrt data/raw/MassModels_Lelli2016c.mrt --metadata data/raw/SPARC_Lelli2016c.mrt
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# Run omega fit on a single galaxy
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python src/fit.py --galaxy M33 --plot
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# Run the comparative fit on a single galaxy (e.g., NGC 3198)
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python src/fit.py --galaxy NGC3198 --plot
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```
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## Notebooks
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| # | Notebook | Description |
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| --- | -------------------------------------------------------------------------- | ---------------------------------------------------------- |
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| 01 | [M33 Calibration](notebooks/01_m33_calibration.ipynb) | Pipeline validation against Corbelli 2014 data |
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| 02 | [Linear vs Quadrature](notebooks/02_linear_vs_quadrature_comparison.ipynb) | Mechanism test: kinematic boost vs. force addition |
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| 03 | [Tapered Models](notebooks/03_tapered_linear_model.ipynb) | Rational taper and tanh taper on M33 |
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| 04 | [SPARC Batch](notebooks/04_sparc_batch_analysis.ipynb) | 118-galaxy batch fit with $k \cdot R_d$ parameterization |
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| 05 | [Phase II Analysis](notebooks/05_phase2_density_coupling.ipynb) | Population split, density coupling, $\Upsilon$ sensitivity |
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| 06 | [Model Gallery](notebooks/06_model_gallery.ipynb) | Head-to-head Linear vs. Tapered across galaxy types |
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| 07 | [Full Catalog Analysis](notebooks/07_full_catalog_analysis.ipynb) | Phase III: BIC selection, $R_t$–$R_d$ scaling, $\Sigma_0$ regime test on all 175 SPARC galaxies |
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| 08 | [Full Gallery](notebooks/08_full_gallery.ipynb) | Rotation-curve gallery for all 171 quality-controlled galaxies (29 pages, sorted by $\Delta$BIC) |
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## Repository Structure
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## Project Structure
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- `src/` — Core Python modules (database, physics, ingestion, fitting)
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- `tests/` — Pytest test suite
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- `notebooks/` — Analysis and visualization notebooks
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- `data/raw/` — Original SPARC data files
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- `data/extracted/` — Data extracted from published papers (e.g., Corbelli 2014 Table 1)
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- `data/processed/` — SQLite database
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- `results/figures/` — Publication-quality plots
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- `results/tables/` — Summary statistics and fit results (CSV)
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- `docs/` — Methodology, results, and internal documentation
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- `src/` — Core Python modules (database, physics, ingestion, fitting pipeline)
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- `notebooks/` — Analysis and visualization notebooks mapped to manuscript figures
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- `data/` — Raw SPARC data (LMS16) and the compiled SQLite database
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- `results/figures/` — Publication-quality plots generated by the notebooks
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- `results/tables/` — Full fit parameters and summary statistics (CSV)
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- `docs/` — Internal documentation and mathematical methodology
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## Data Sources
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- **SPARC Database** (Lelli, McGaugh, & Schombert 2016): 175 disk galaxies with Spitzer photometry and rotation curves. http://astroweb.cwru.edu/SPARC/
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- **M33 Calibration** (Corbelli et al. 2014): Surface density profiles from Table 1, converted to velocity components via Casertano (1983) thin-disk method.
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## References
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## Citation
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(Citation details will be updated upon publication. For preprint inquiries, please reference the GitHub URL).
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## License
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1. Flynn, D. C. & Cannaliato, J. (2025). "A New Empirical Fit to Galaxy Rotation Curves."
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2. Corbelli, E. & Salucci, P. (2000). MNRAS, 311, 441. "The Extended Rotation Curve and the Dark Matter Halo of M33."
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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."
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4. Casertano, S. (1983). MNRAS, 203, 735. "Rotation curve of the edge-on spiral galaxy NGC 5907."
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5. Kass, R. E. & Raftery, A. E. (1995). JASA, 90, 773. "Bayes Factors."
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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."
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This project is licensed under the MIT License - see the [LICENSE](https://www.google.com/search?q=LICENSE) file for details.

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