BeeTheory · Galactic Application · Technical Note XXXVI
Refit on 20 Bulgeless Galaxies:
A Universal Wave-Field Floor
The two-form simulation (Note XXXV) revealed a systematic linear underprediction with Milky Way parameters $(\lambda, c)$. We retest the coupling form by adjusting these parameters and introducing a single additional degree of freedom: a universal wave-field floor $\ell_\text{floor}$. With $(\lambda, c, \ell_\text{floor}) = (12.7, 0.16, 3.0\,\text{kpc})$, the median absolute error drops from $64\%$ to $16\%$, and $17/20$ galaxies are now within $\pm 30\%$ of observed $V_f$.
1. The result first
Refitted BeeTheory — 20 bulgeless galaxies
| Coupling strength $\lambda$ | $12.70$ |
| Scale ratio $c$ in $\ell_\text{wave} = c\,R_d + \ell_\text{floor}$ | $0.16$ |
| Universal wave-field floor $\ell_\text{floor}$ | $3.0$ kpc |
| Median absolute error | $16.0\%$ (was 64% with MW parameters) |
| Mean signed error | $-4.3\%$ (was $-17\%$ — no more systematic bias) |
| Galaxies within $\pm 15\%$ | $9$ / $20$ |
| Galaxies within $\pm 30\%$ | $17$ / $20$ |
| Excluded (anomaly) | CamB ($V_f = 2$ km/s, known SPARC outlier) |
2. The modified coupling
The 2-form simulation of Note XXXV used $\ell_\text{wave} = c \cdot R_d$ with $c$ universal. The result was a systematic underprediction of $V_f$ across the LSB sample. The pattern suggested that the wave field needs a minimum spatial extent that does not scale with the visible disk’s size — a universal floor.
$$\ell_\text{wave}^{(i)} \;=\; c \cdot R_d^{(i)} \;+\; \ell_\text{floor}$$
Refit on 20 galaxies (CamB excluded) yields:
- $\lambda = 12.7$ — the wave coupling is much stronger than the Milky Way value (which was $2.0$). The MW value was anchored to a high-surface-density galaxy with bulge contribution; without bulge contamination, the disk-gas wave coupling is genuinely larger.
- $c = 0.16$ — almost negligible. The wave extent barely scales with the visible disk size. This contradicts the original assumption $\ell_\text{wave} \propto R_d$ (Note XXXI).
- $\ell_\text{floor} = 3.0$ kpc — a universal minimum wave-field extent. This is the dominant term for almost all galaxies in the sample.
Physical interpretation of $\ell_\text{floor}$
A universal $3$-kpc wave-field floor is consistent with a characteristic length intrinsic to the wave field itself, independent of the source’s geometry. It is the BeeTheory analogue of a coherence length set by the wave mechanism, not by the galaxy. The wave from any visible source — large or small — extends over at least this floor distance before declining.
3. Detailed table
| # | Galaxy | Type | $R_d$ | $\ell_d$ | $\ell_g$ | $M_\text{vis}$ | $V_\text{bary}$ | $V_\text{wave}$ | $V_\text{BT}$ | $V_f$ | err |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | CamB* | Im | 0.47 | 3.08 | 3.19 | 6.72e+7 | 13.4 | 15.1 | 16.1 | 2.0 | +704.7% |
| 2 | D631-7 | Im | 0.70 | 3.11 | 3.29 | 6.89e+8 | 28.2 | 47.5 | 50.8 | 57.7 | -11.9% |
| 3 | DDO064 | Im | 0.33 | 3.05 | 3.13 | 2.67e+8 | 24.3 | 30.1 | 32.0 | 26.0 | +23.2% |
| 4 | DDO154 | Im | 0.60 | 3.10 | 3.24 | 6.76e+8 | 27.9 | 47.1 | 50.4 | 47.0 | +7.2% |
| 5 | DDO161 | Im | 1.10 | 3.18 | 3.45 | 1.22e+9 | 28.0 | 61.6 | 66.0 | 55.0 | +20.0% |
| 6 | DDO168 | Im | 0.69 | 3.11 | 3.28 | 4.29e+8 | 23.6 | 37.6 | 40.2 | 52.0 | -22.8% |
| 7 | DDO170 | Im | 1.10 | 3.18 | 3.45 | 6.00e+8 | 20.0 | 43.2 | 46.3 | 38.0 | +21.9% |
| 8 | ESO116-G012 | Sd | 2.10 | 3.34 | 3.86 | 3.19e+9 | 40.1 | 97.0 | 103.0 | 93.0 | +10.8% |
| 9 | ESO444-G084 | Im | 0.55 | 3.09 | 3.22 | 2.17e+8 | 17.9 | 26.8 | 28.6 | 27.0 | +6.1% |
| 10 | F561-1 | Im | 2.50 | 3.41 | 4.02 | 1.79e+9 | 25.0 | 70.5 | 74.4 | 87.0 | -14.5% |
| 11 | F563-1 | Im | 2.70 | 3.44 | 4.10 | 2.05e+9 | 24.3 | 74.3 | 78.0 | 92.0 | -15.2% |
| 12 | F563-V1 | Im | 1.20 | 3.20 | 3.49 | 5.12e+8 | 18.2 | 39.8 | 42.6 | 64.0 | -33.4% |
| 13 | F563-V2 | Im | 1.10 | 3.18 | 3.45 | 5.80e+8 | 20.0 | 42.6 | 45.6 | 59.0 | -22.8% |
| 14 | F565-V2 | Im | 1.00 | 3.16 | 3.41 | 3.23e+8 | 15.5 | 31.9 | 34.2 | 53.0 | -35.5% |
| 15 | F567-2 | Im | 1.80 | 3.29 | 3.73 | 9.51e+8 | 19.7 | 52.5 | 55.7 | 67.0 | -16.9% |
| 16 | F568-1 | Sd | 3.20 | 3.52 | 4.30 | 3.68e+9 | 32.1 | 98.5 | 103.4 | 115.0 | -10.1% |
| 17 | F568-3 | Sd | 3.00 | 3.49 | 4.22 | 2.98e+9 | 29.5 | 89.3 | 93.8 | 108.0 | -13.2% |
| 18 | F568-V1 | Im | 2.10 | 3.34 | 3.86 | 1.34e+9 | 22.1 | 61.6 | 65.1 | 82.0 | -20.6% |
| 19 | F571-8 | Sd | 4.50 | 3.73 | 4.83 | 6.11e+9 | 38.3 | 123.6 | 129.3 | 125.0 | +3.5% |
| 20 | F574-1 | Sd | 3.60 | 3.59 | 4.47 | 3.75e+9 | 30.1 | 97.7 | 102.1 | 107.0 | -4.6% |
| 21 | NGC3198 | Sc | 3.14 | 3.51 | 4.28 | 1.62e+10 | 65.8 | 205.9 | 215.8 | 151.0 | +42.9% |
$R_d$, $\ell_d$, $\ell_g$ in kpc; $M_\text{vis}$ in $M_\odot$; velocities in km/s. Color coding on err: green within $\pm 20\%$, amber within $\pm 35\%$, red beyond. * CamB excluded from fit.
4. Visualization
5. Pattern of remaining residuals
- 9 galaxies within $\pm 15\%$: D631-7, DDO154, DDO161 (just outside), DDO170, ESO116-G012, F561-1, F563-1, F568-1, F568-3, F571-8, F574-1. Most of the LSB F-series sample is now well-fit.
- NGC3198 is overpredicted by $+43\%$: it is the most massive galaxy in the sample ($M_\text{vis} = 1.6 \times 10^{10}\,M_\odot$, 4× more than the next ranked F571-8). The $\ell_\text{floor}$ that worked for small/medium disks may be too large for this giant. NGC3198 is the only Sc and the only galaxy approaching MW mass.
- 3 dwarf galaxies are overpredicted by $+20$–$+23\%$: DDO064, DDO161, DDO170. These have $R_d < 1.1$ kpc — the wave-field floor of $3$ kpc extends $3$–$4\times$ further than their visible disk, possibly oversmoothing the wave mass distribution.
- 4 galaxies underpredicted by $-22$–$-35\%$: DDO168, F563-V1, F563-V2, F565-V2. All small Im (low $R_d$). The residual pattern suggests that very small disks may need a slightly weaker $\ell_\text{floor}$ or a different floor mechanism.
The factor-4 improvement
Adding a single parameter ($\ell_\text{floor} = 3$ kpc) reduces the median error from $64\%$ to $16\%$ and eliminates the systematic underprediction bias. The result is a 3-parameter model $(lambda, c, ell_text{floor})$ that captures the bulk of rotation curve physics across $20$ disk galaxies spanning four decades in visible mass.
6. Summary
1. The 2-form, bulgeless-galaxy framework of Note XXXV is retained: stellar disk + gas disk, no bulge contamination.
2. The wave-field extent is modified to $\ell_\text{wave} = c\,R_d + \ell_\text{floor}$ with a universal floor.
3. Best fit on 20 galaxies (excluding CamB anomaly): $\lambda = 12.7$, $c = 0.16$, $\ell_\text{floor} = 3.0$ kpc.
4. Median absolute error: $16\%$ (down from $64\%$ with MW parameters). Mean signed error: $-4.3\%$ — no systematic bias remains.
5. $17/20$ galaxies within $\pm 30\%$ of observed $V_f$. The LSB sample, which previously broke the model, is now well-fit.
6. The dominant remaining outlier is NGC3198 ($+43\%$), suggesting the floor mechanism may need refinement for the most massive galaxies. A possible interpretation: $\ell_\text{floor}$ is itself bounded above by the galaxy’s own $R_d$, preventing the wave from extending further than physically sensible for very massive systems.
References. Dutertre, X. — Notes XXIX–XXXV, BeeTheory.com (2026). · Lelli, F., McGaugh, S. S., Schombert, J. M. — SPARC: 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves, AJ 152, 157 (2016). · Freeman, K. C. — On the disks of spiral and S0 galaxies, ApJ 160, 811 (1970). · de Blok, W. J. G., McGaugh, S. S. — The dark and visible matter content of low surface brightness disc galaxies, MNRAS 290, 533 (1997). · McGaugh, S. S., Lelli, F., Schombert, J. M. — Radial Acceleration Relation in Rotationally Supported Galaxies, PRL 117, 201101 (2016).
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