BeeTheory · Foundations · Technical Note XXI
Twenty-Three Galaxies, One Coherence Length:
The Simplified Model at Scale
The simplified BeeTheory formalism of Note XX — single universal coherence length $\ell_0$, single global coupling $\lambda$, four baryonic components summed in the plane — is now applied to all twenty-three test galaxies. Both $\ell_0$ and $\lambda$ are fitted jointly on the twenty-two SPARC galaxies; the Milky Way is evaluated with the same parameters as an independent check. Three diagnostic plots reveal what the model does well, and where it tightens or breaks.
1. The result first
Joint fit on the 22 SPARC galaxies
Best parameters across the 22-galaxy calibration set:
$\ell_0 = 2.45$ kpc, $\lambda = 0.203$
22 SPARC: median $|\text{err}| = 15.0\%$, mean signed err $= +29.1\%$, 14/22 within $20\%$, 18/22 within $30\%$.
Milky Way: err = $+61.2\%$ at $R = 5\,R_d$ with the same $\ell_0$ and $\lambda$.
A trade-off is now visible
Imposing a single $ell_0 = 2.45$ kpc across all 22 SPARC galaxies introduces a systematic bias: the mean signed error is $+29%$, meaning that on average the simplified model over-predicts the flat velocity. The Milky Way is the most over-predicted single case ($+61%$). This is the cost of universalising $\ell_0$ across galaxies spanning six decades in baryonic mass. The diagnostic plots below identify where this bias concentrates.
2. What was computed
For each of the 23 galaxies, the simplified pipeline runs as follows:
(a) Build the baryonic density in the plane. The four components are projected onto $z = 0$ and summed:
$$\Sigma_\text{bar}(R) \;=\; \Sigma_\text{bulge,proj}(R) + \Sigma_\text{disk}(R) + \Sigma_\text{gas}(R) + \Sigma_\text{arm}(R)$$
(b) Convolve once with the universal kernel. One coherence length $\ell_0$, no per-component scale:
$$\Sigma_\text{wave}(R) \;=\; \lambda \int_0^{R_\text{max}} \Sigma_\text{bar}(R’) \cdot \langle\mathcal{K}\rangle(R,R’)\,2\pi R’\,dR’, \quad \langle\mathcal{K}\rangle = \frac{K_0}{\pi}\int_0^\pi\frac{e^{-D/\ell_0}}{D^2}\,d\phi$$
(c) Compute the enclosed wave mass and the rotation velocity.
$$M_\text{wave}(
$ell_0$ and $lambda$ are fit by minimising the median absolute prediction error on the 22 SPARC galaxies at $R = 5,R_d$. The Milky Way is then evaluated with the resulting parameters as a separate, independent check.
3. Graph 1 — Rotation curves of all 23 galaxies
The predicted rotation curve of each of the 23 galaxies, plotted in absolute units. Each curve is the full $V(R)$ from the centre to the outer disk, with the observed flat velocity $V_f$ marked as a dot at $R = 5\,R_d$. Colour by Hubble type, the Milky Way in thick red.
Reading the absolute view
The curves are well organised by class: massive Sb–Sbc (red, top), then Sc–Scd (gold), then Sd–Im dwarfs (blue, bottom). All curves rise from $R \sim 0$ to a peak at $R \sim 4$–$8$ kpc, then decline. The Milky Way (thick red) reaches $sim 290$ km/s at peak — higher than its observed $V_f sim 230$ km/s — reflecting the $+61%$ over-prediction noted above. NGC 2841 (red, $V_f = 278$) and NGC 3198 (gold, $V_f = 151$) sit in their expected places. The qualitative morphology is correct; the quantitative scaling overshoots for some galaxies.
4. Graph 2 — Normalised by observed velocity
To remove the absolute scale and see only the prediction error structure, each curve is divided by the observed flat velocity $V_f$ of its galaxy, and the radius is scaled by $R_d$. A perfect prediction would place all curves on the horizontal line $y = 1$ at large $R/R_d$.
A wide envelope, with a clear bias above unity
At $R/R_d = 5$, most galaxies cluster between $y = 0.7$ and $y = 1.6$. The median is around $y = 1.15$ — the $+29\%$ mean signed error. A few outliers stretch to $y \approx 1.8$ (massive spirals with high mass) and a few sit near $y = 0.6$ (dwarfs with low surface density). The Milky Way (thick red) reaches $y approx 1.6$ — consistent with its $+61%$ over-prediction. The envelope of curves at small $R/R_d$ is much wider than at large $R/R_d$, indicating that the central region is where the model struggles most with the simplified single $\ell_0$.
5. Graph 3 — Prediction error per galaxy
The error of each galaxy, individually, sorted by disk scale $R_d$ (smallest at left, largest at right). Galaxies in the green band have $|\text{err}| < 20\%$, in the gold band $20 \leq |\text{err}| < 30\%$, beyond the bands $|\text{err}| > 30\%$.
A bias structure remains
The error distribution is not centred on zero: most bars point upward, with a median around $+12\%$. Compact dwarfs at small $R_d$ (left) tend to be moderately over-predicted. Mid-scale spirals (centre) cluster within $\pm 20\%$ of the target. The largest galaxies at right — including NGC 2841 and the Milky Way — show the largest positive errors.
This is qualitatively the same pattern documented in Note XI (sorted by $R_d$, the error grows with $R_d$): the simplified single-$\ell_0$ formulation has not made this pattern disappear — it has only changed its quantitative character.
6. Detailed reflection — what works, what does not
What the simplified model does well
(i) The shape is now correct. Every curve in Graph 1 rises, peaks, and declines — the same morphology as observed rotation curves. The chronic over-prediction at large $R$ that plagued Notes XIV–XIX is gone. The short coherence length $ell_0 approx 2.5$ kpc forces the wave field to follow the visible baryons locally.
(ii) The model is mass-blind in the right way. Across six decades in baryonic mass, the median error stays at $15%$ — the same number whether the galaxy is a $10^8,M_odot$ dwarf or a $5 times 10^{10},M_odot$ Milky Way. The wave mechanism is intrinsically scale-free.
What the simplified model does not do well
(iii) A systematic positive bias. The mean signed error is $+29\%$. The model over-predicts on average, particularly for the most massive galaxies in the sample. The Milky Way at $+61%$ is the most over-predicted single galaxy. This is the price of using a single $\ell_0$ for galaxies of very different sizes.
(iv) The residual still correlates with $R_d$. Graph 3 sorted by $R_d$ shows the same trend identified in Note XI — large $R_d$ galaxies are over-predicted, small ones tend toward under-prediction. The simplification has not removed the structural defect: the single $\ell_0$ cannot adapt to the different physical scales of different galaxies.
Tension with the Milky Way
In Note XX, the Milky Way alone fit Gaia 2024 with $\ell_0 = 1.59$ kpc and $\lambda = 0.098$. Here, fitting the 22 SPARC galaxies yields $\ell_0 = 2.45$ kpc and $\lambda = 0.203$. The two parameter sets differ significantly:
| Parameter | MW alone (Note XX) | 22 SPARC joint (this note) | Ratio |
|---|---|---|---|
| $\ell_0$ (kpc) | $1.59$ | $2.45$ | $1.54$ |
| $\lambda$ | $0.098$ | $0.203$ | $2.07$ |
The Milky Way “prefers” a tighter coherence length and weaker coupling. The SPARC sample, dominated by dwarfs and intermediate spirals with longer disks, “prefers” a longer coherence length and stronger coupling. A truly universal $(\ell_0, \lambda)$ does not yet exist with this formulation — there is residual physics that depends on a galaxy’s structural properties (surface density, mass), as already identified in Note XI.
7. Comparison with previous formulations
| Quantity | 5-component (Note XV) | Simplified (this note) |
|---|---|---|
| Theory parameters | 5 | 3 |
| Coherence lengths | 5 different per galaxy | 1 universal |
| Median $|\text{err}|$ on 22 SPARC | $14.6\%$ | $15.0\%$ |
| Mean signed err on 22 SPARC | $-4.7\%$ | $+29.1\%$ |
| 14/22 within $20\%$? | Yes | Yes (14/22) |
| Within $30\%$ | 18/22 | 18/22 |
| MW error at $R = 5\,R_d$ | $+15\%$ | $+61\%$ |
| Shape of rotation curve at large $R$ | Over-flat | Declines correctly |
A genuine simplification with mixed numerical performance
The simplified model matches the original in median accuracy ($15\%$) and in the fraction of galaxies within $20\%$ and $30\%$, while using only three theory parameters instead of five. It also corrects the qualitative shape of rotation curves at large $R$. The cost is a larger positive bias on the most massive galaxies, including the Milky Way. This trade-off must be considered when deciding whether to keep the simplified formulation or to reintroduce some flexibility — for example, via a density-dependent $\ell_0$ as suggested by Note XI.
8. Summary
1. The simplified BeeTheory formalism — single universal coherence length, single global coupling, four baryonic components — is applied to all 23 test galaxies.
2. Joint fit on the 22 SPARC galaxies yields $\ell_0 = 2.45$ kpc and $\lambda = 0.203$, with median $|\text{err}| = 15\%$.
3. The rotation curve shape is now correctly reproduced for all galaxies: rising, peaking, declining — the qualitative defect of Notes XIV–XIX is gone.
4. Quantitatively, the model over-predicts on average ($+29\%$ mean signed error). The Milky Way is the most over-predicted single galaxy ($+61\%$ at $R = 5\,R_d$).
5. The Milky Way alone (Note XX) was best fit at $\ell_0 = 1.59$ kpc, $\lambda = 0.098$ — significantly tighter and weaker than the SPARC-derived values. A truly universal $(\ell_0, \lambda)$ does not exist with this purely geometric formulation.
6. The residual error correlates with $R_d$ (and indirectly with $\Sigma_d$ as identified in Note XI), suggesting that $\ell_0$ should depend on the local baryonic density. The next refinement is to introduce $\ell_0 = \ell_0(\Sigma_d)$ explicitly.
References. Lelli, F., McGaugh, S. S., Schombert, J. M. — SPARC: Mass Models for 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves, AJ 152, 157 (2016). · Ou, X. et al. — The dark matter profile of the Milky Way, MNRAS 528, 693 (2024). · McGaugh, S. S. — The third law of galactic rotation, Galaxies 2, 601 (2014). · Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023).
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