Technical Note XXXVIII
First test of BeeTheory against the Radial Acceleration Relation
BeeTheory.com — Xavier Dutertre — Technoplane S.A.S. — 20 May 2026
Result. We confront BeeTheory’s 3-parameter model (\(\lambda = 12.696\), \(c = 0.163\), \(\ell_{\rm floor} = 3.00\) kpc) with the Radial Acceleration Relation (RAR) of McGaugh, Lelli & Schombert (2016). Evaluating the predicted \(g_{\rm obs}(g_{\rm bar})\) on 909 points sampled across 101 SPARC bulgeless galaxies, the BeeTheory cloud has the correct overall shape — it follows the McGaugh curve at low \(g_{\rm bar}\) — but is biased high by +0.25 dex on average, with a dispersion of 0.39 dex (vs. 0.13 dex empirically). The bias is not uniform: at low acceleration (\(g_{\rm bar} < 10^{-12}\) m/s²) BeeTheory agrees with the RAR, while at high acceleration (around and above \(g_\dagger\)) it overpredicts by 0.4 to 0.5 dex. The model captures the asymptotic MOND-like behaviour but fails to recover the \(g_{\rm obs} \to g_{\rm bar}\) Newton limit at small radius. This is a partial success and a clear pointer to the next refinement.
1. The test
The Radial Acceleration Relation expresses the strikingly tight empirical correlation between two quantities measured on rotation curves: \(g_{\rm bar}(R)\), the Newtonian acceleration that would be produced by the visible matter alone, and \(g_{\rm obs}(R)\), the total centripetal acceleration inferred from the rotation velocity. McGaugh, Lelli & Schombert (2016) showed that ~150 SPARC galaxies fall on a single curve well fit by:
\[ g_{\rm obs}(g_{\rm bar}) \;=\; \frac{g_{\rm bar}}{1 – \exp\!\left(-\sqrt{g_{\rm bar}/g_\dagger}\right)},\qquad g_\dagger = (1.20 \pm 0.02)\times 10^{-10}\;{\rm m/s}^2 \]
The dispersion around this curve is only ~0.13 dex — barely larger than the observational uncertainties. Any modified-gravity theory that claims to replace dark matter must reproduce this relation point by point. The RAR is the strictest empirical filter on the market.
BeeTheory’s prediction is straightforward. At every radius \(R\) within a galaxy, the visible matter generates the wave field whose enclosed mass adds to the baryonic enclosed mass with coupling \(\lambda\):
\[ \begin{aligned} g_{\rm bar}(R) &= G\,M_{\rm bar}(<\!R)\,/\,R^2 \\ g_{\rm obs}^{\rm BT}(R) &= g_{\rm bar}(R) + G\,\lambda\,M_{\rm wave}(<\!R)\,/\,R^2 \\ M_{\rm wave}(<\!R) &= M\,\bigl[\,1 – (1 + x + x^2/2)\,e^{-x}\bigr],\quad x = R/\ell_{\rm wave} \\ \ell_{\rm wave} &= c\,R_d + \ell_{\rm floor} \end{aligned} \]
We use the same baryonic decomposition as the calibration: \(M_{\rm disk} = \Upsilon\cdot 2\pi\,\Sigma_d\,R_d^2\) with \(\Upsilon = 0.5\), \(M_{\rm gas} = 1.33\,M_{\rm HI}\), \(R_{d,{\rm gas}} = 2.5\,R_{d,\star}\). All three universal parameters are held at their stable values. The cloud is generated by sampling each galaxy at nine radii from \(0.5\,R_d\) to \(10\,R_d\), yielding 909 points across the 101 disks.
2. The diagram
Three features stand out.
(a) The observed points lie close to the McGaugh curve. The dispersion of the SPARC \(V_f\) values around the empirical RAR is 0.17 dex with a small +0.10 dex bias — comparable to the 0.13 dex reported by McGaugh+2016. Our 101-galaxy sample is healthy and consistent with the published RAR. This was not guaranteed: it confirms that the SPARC-derived \((g_{\rm bar}, g_{\rm obs})\) pairs we use here trace the same empirical relation.
(b) The BeeTheory cloud globally tracks the McGaugh curve. It is not flat (it does not collapse onto Newton), and it bends in the right direction at low \(g_{\rm bar}\). The functional form is qualitatively correct.
(c) The cloud lies systematically above the curve, with median offset +0.25 dex and total dispersion 0.39 dex — three times the empirical scatter. BeeTheory in this form overpredicts \(g_{\rm obs}\).
3. Where BeeTheory departs from the RAR
The structure is not random. At the lowest \(g_{\rm bar}\) (around \(3\times 10^{-13}\) m/s², i.e. the outermost regions of LSB disks), BeeTheory residuals cluster within \(\pm 0.15\) dex of zero — the model lands on the McGaugh curve within the empirical dispersion. This is the regime where the calibration was performed (\(V_f\) is reached at \(R \approx 5\,R_d\) where \(g_{\rm bar}\) is low), so consistency here was expected.
The trend then climbs steadily with \(g_{\rm bar}\). Around \(g_\dagger = 1.2\times 10^{-10}\) m/s² (the transition acceleration in the empirical RAR), the binned median sits at +0.45 to +0.50 dex. In linear units, \(g_{\rm obs}^{\rm BT}\) is roughly 3 times the McGaugh value at the corresponding \(g_{\rm bar}\). This is the inner regions of the galaxies, where \(R\) is small compared to \(R_d\).
Why it fails inward. The empirical RAR requires that at high \(g_{\rm bar}\) (deep inside the disk), \(g_{\rm obs} \to g_{\rm bar}\) — visible matter dominates, and the wave field’s contribution must become subdominant. In the current BeeTheory parameterization, \(\ell_{\rm wave} = c\,R_d + \ell_{\rm floor}\) is approximately 3 kpc for almost all galaxies (because \(c = 0.16\) is small). At \(R \ll \ell_{\rm wave}\), \(x = R/\ell_{\rm wave}\) is small and \(M_{\rm wave}(<\!R) \approx M\,x^3/6\) — the wave mass grows with \(R\), but it grows from a coupling \(\lambda = 12.7\). The product \(\lambda \cdot M_{\rm wave}(<\!R)\) does not vanish fast enough to recover Newton at small \(R\). The model “leaks” wave contribution inward.
4. Synthesis and next steps
| Metric | BeeTheory cloud | Observed (\(V_f\) points) | McGaugh+2016 |
|---|---|---|---|
| Median offset (dex) | +0.25 | +0.10 | 0.00 (definition) |
| Dispersion \(\sigma\) (dex) | 0.39 | 0.17 | ~0.13 |
| Number of points | 909 (101 gal.) | 101 | ~2700 (~150 gal.) |
| Shape qualitatively correct? | Yes | Yes | — |
| Newton limit at high \(g_{\rm bar}\)? | No (+0.5 dex) | — | Yes (built in) |
Reading. BeeTheory passes the RAR asymptotically at low acceleration — the regime that drives flat rotation curves, where the calibration was anchored. It fails the high-acceleration regime, where the RAR converges to Newton and BeeTheory does not. This is a clean, localized defect — not a wholesale incompatibility. The 3-parameter form was never required to reproduce the inner-disk behaviour because \(V_f\) is set at \(R \sim 5\,R_d\). Encountering the RAR forces the model to be correct at every radius, not just at \(V_f\).
Paths to explore
(i) Saturating \(\ell_{\rm wave}\) at small \(R\). A natural refinement is to make the effective wave-field extent a function of local conditions, e.g. \(\ell_{\rm wave}(R)\) growing with \(R\) rather than fixed at \(c\,R_d + \ell_{\rm floor}\). This would suppress wave coupling at small \(R\) and let Newton dominate where \(g_{\rm bar}\) is high.
(ii) \(g\)-dependent coupling. If \(\lambda\) itself depends on \(g_{\rm bar}\) — for instance, \(\lambda \to \lambda \cdot f(g_{\rm bar}/g_\dagger)\) with \(f \to 0\) at high \(g_{\rm bar}\) and \(f \to 1\) at low — the model could reproduce the McGaugh curve exactly. The challenge is to motivate such a dependence from the wave-function microphysics, not impose it phenomenologically.
(iii) Bound on \(\ell_{\rm floor}\) for massive galaxies. The same defect that produces the high-\(g_{\rm bar}\) bias here may be related to the +43% over-prediction on NGC3198 in calibration and the +25–45% over-predictions on several Sc/Sbc blind galaxies. A saturating \(\ell_{\rm floor}\) (or one that depends on \(M_{\rm visible}\)) could simultaneously fix both.
Note on the methodology. All computations: 101 SPARC bulgeless galaxies (\(T \geq 4\)), 909 \((R, g_{\rm bar}, g_{\rm obs})\) triplets at \(R/R_d \in \{0.5, 1.0, 1.5, 2.0, 3.0, 4.0, 5.0, 7.0, 10.0\}\). Stable parameters \((\lambda, c, \ell_{\rm floor}) = (12.696,\, 0.163,\, 3.00\ {\rm kpc})\). McGaugh empirical RAR with \(g_\dagger = 1.20\times 10^{-10}\) m/s². Comparison with the published McGaugh+2016 individual SPARC measurements (not reproduced here from the original rotation-curve samples) is a natural next step.
BeeTheory.com — First test against the Radial Acceleration Relation · Initial generation: 2026-05-20 with Claude.ai · © Technoplane S.A.S. 2026