>[!warning] >This content has not been peer reviewed. # SPARC Evaluation — Verified by Calculation > RST's gravity sector is tested against observational rotation curves. The acceleration threshold $a_0 = cH_0/(2\pi)$ is fixed by cosmology; the only free parameter per galaxy is the mass-to-light ratio $\Upsilon_{\text{disk}}$. --- ## Data Source **SPARC database** — 175 nearby galaxies with stellar surface photometry at 3.6 $\mu$m and rotation curves from HI/H$\alpha$ spectroscopy. **Citation:** Lelli, F., McGaugh, S. S., & Schombert, J. M. (2016). SPARC: Mass models for 175 disk galaxies with Spitzer photometry and accurate rotation curves. *The Astronomical Journal*, 152, 157. [arXiv:1606.09251](https://arxiv.org/abs/1606.09251) **Download:** [http://astroweb.cwru.edu/SPARC/](http://astroweb.cwru.edu/SPARC/) — rotation-curve data (e.g. rotmod format: radius, velocities, uncertainties). --- ## What Gets Tested | Parameter | Source | Value | |:---|:---|:---| | $a_0$ | Cosmology: $a_0 = cH_0/(2\pi)$ | $1.04 \times 10^{-10}$ m/s² | | $n$ | Calibrated from transition-zone galaxies | 1.25 | | $\Upsilon_{\text{disk}}$ | Fitted per galaxy (only free parameter) | — | **Method:** For each galaxy, compute Newtonian acceleration $g_N(r)$ from observed gas, disk, and bulge. Solve the RST field equation $\mu(q'/a_0,\, n)\, q' = g_N$ for $q'$, predict $v(r) = \sqrt{r \cdot q'}$. Fit $\Upsilon_{\text{disk}}$ by minimising reduced $\chi^2$. **No dark matter halo parameters** (mass, concentration, core radius) are used. **Reference:** [[Relational Substrate Theory (RST)#Part VIII Empirical Calibration — The SPARC Evaluation|RST Part VIII]]. --- ## Results (Fixed $n = 1.25$) ### Summary Statistics | Metric | Value | |:---|:---| | Galaxies fitted | 171 | | $\Upsilon_{\text{disk}}$ median | 0.52 | | $\Upsilon_{\text{disk}}$ mean | 0.68 | | $\chi^2_{\text{red}}$ median | 1.60 | | $\chi^2 < 2$ | 58% (99 galaxies) | | $\chi^2 < 5$ | 80% (136 galaxies) | | $\chi^2 < 10$ | 86% (147 galaxies) | ### RST Parameters Used | Parameter | Value | |:---|:---| | $a_0$ | $1.042 \times 10^{-10}$ m/s² | | $n$ | 1.25 | --- ## Interpretation 1. **$a_0$ anchor:** The cosmologically derived threshold produces rotation curve fits without requiring dark matter. If $a_0$ were systematically wrong, $\Upsilon_{\text{disk}}$ would be pushed to unphysical values. 2. **$\Upsilon_{\text{disk}}$:** Median 0.52 in this run; other runs report ~0.49–0.491. Both lie within the range expected from stellar population synthesis at 3.6 $\mu$m ($\sim$0.4–0.6). The spread is run-dependent; the median is the more robust indicator than the mean (which is elevated by galaxies hitting optimizer bounds). 3. **Fit quality:** 80% of galaxies have $\chi^2_{\text{red}} < 5$. Failures tend to be massive, high surface brightness spirals where the spherical algebraic shortcut deviates from the true 2D disk solution — a known geometric limitation, not a gravitational one. 4. **Zero dark matter:** All fits use a single free parameter per galaxy. RST reproduces the observed rotation curves without invoking a dark matter halo. --- ## Relation to Theory - **[[Relational Substrate Theory (RST)]]** — Part VIII documents the full calibration (fixed-$n$ and floating-$n$ runs). - **[[Resource Allocation Equation]]** — $\mu(q'/a_0, n)\, q' = g_N$ is the tested equation. - **[[Noise Floor]]** — $a_0$ is the gravitational noise floor; $a_0 = cH/(2\pi)$. - **[[about systems/expanded theory/python calculations/SPARC Evaluation - Code]]** — method and data source (citation + download link). Results summary: [[SPARC Evaluation Results]], [[python calculations results]].