BeeTheory · Galactic Application · Technical Note XXXII
A Case Where the Method Fails:
F568-1 with Milky Way Parameters
Applying the exact methodology of Note XXXI — geometric decomposition into sub-elements, visible mass and wave mass calculated ring by ring, universal parameters $(\lambda, c) = (2.00, 1.85)$ — to F568-1, a low surface brightness Sd galaxy. The result: $V_\text{max}^\text{predicted} = 37$ km/s versus $V_f^\text{observed} = 115$ km/s, an underestimation of $-68\%$. We document this failure in detail because it reveals the structural limits of universal parameters and points toward what BeeTheory must include to handle LSB galaxies.
1. The result first
F568-1 with universal parameters — failure documented
| Galaxy type | LSB (low surface brightness), Hubble Sd, T=8 |
| Disk scale length $R_d$ | $3.2$ kpc |
| Central surface density $\Sigma_d$ | $40\,L_\odot/\text{pc}^2$ (very low) |
| Total visible mass $M_\text{bar}$ | $3.68 \times 10^9\,M_\odot$ (18× less than MW) |
| Observed $V_f$ (SPARC) | $115$ km/s |
| BeeTheory $V_\text{max}$ predicted | $37$ km/s (universal MW parameters $\lambda=2.00$, $c=1.85$) |
| Error | $-68\%$ — major underestimation |
The methodology of Note XXXI applied identically to F568-1 produces a rotation velocity less than one third of the observed value. The universal Milky Way parameters do not extrapolate to this LSB galaxy. The reason is structural and informative.
2. Step 1 — Geometric decomposition into sub-elements
Per Note XXX, every visible mass element carries its own wave function. To compute the galactic wave field, we decompose F568-1 into discrete rings — 10 for the stellar disk, 10 for the gas disk — each treated as an independent source.
Stellar disk — exponential profile $\Sigma_\star(R) = \Sigma_{d,0}\,e^{-R/R_d}$ with $R_d = 3.2$ kpc, integrated with $\Upsilon = 0.5\,M_\odot/L_\odot$ at $3.6\,\mu$m:
$$M_\star \;=\; \Upsilon \cdot 2\pi\,\Sigma_{d,0}\,R_d^2 \;=\; 1.29 \times 10^9\,M_\odot$$
| Ring $i$ | $R_i$ (kpc) | $\Sigma_\star(R_i)$ ($L_\odot/\text{pc}^2$) | $dM_{\star,i}$ ($M_\odot$) |
|---|---|---|---|
| 0 | 0.96 | 29.6 | $1.72 \times 10^8$ |
| 1 | 2.88 | 16.3 | $2.83 \times 10^8$ |
| 2 | 4.80 | 8.9 | $2.58 \times 10^8$ |
| 3 | 6.72 | 4.9 | $1.99 \times 10^8$ |
| 4 | 8.64 | 2.7 | $1.40 \times 10^8$ |
| 5 | 10.56 | 1.5 | $9.4 \times 10^7$ |
| 6 | 12.48 | 0.8 | $6.1 \times 10^7$ |
| 7 | 14.40 | 0.4 | $3.9 \times 10^7$ |
| 8 | 16.32 | 0.2 | $2.4 \times 10^7$ |
| 9 | 18.24 | 0.1 | $1.5 \times 10^7$ |
| Sum | — | — | $1.28 \times 10^9$ (99.7% of $M_\star$) |
Gas disk — extended exponential with $R_{d,\text{gas}} = 2.5\,R_d = 8.0$ kpc (gas reaches further than stars), total mass $M_\text{gas} = 1.33 \cdot M_{\text{HI}} = 2.39 \times 10^9\,M_\odot$ (He correction included). Decomposition into 10 rings up to $R = 48$ kpc.
3. Step 2 — Wave mass generated by each sub-element
For each ring $i$ of mass $dM_i$, the BeeTheory wave field carries an additional mass $dM_{\text{wave},i} = \lambda \cdot dM_i$ with $\lambda = 2.00$. The spatial extent of each ring’s wave function is $ell_text{wave} = c cdot R_d$ where $c = 1.85$ is taken from the Milky Way calibration.
| Component | $R_d$ | $\ell_\text{wave} = c\,R_d$ | $M_\text{visible}$ | $M_\text{wave} = \lambda\,M_\text{visible}$ |
|---|---|---|---|---|
| Stellar disk | 3.2 kpc | 5.9 kpc | $1.29 \times 10^9\,M_\odot$ | $2.57 \times 10^9\,M_\odot$ |
| Gas disk | 8.0 kpc | 14.8 kpc | $2.39 \times 10^9\,M_\odot$ | $4.78 \times 10^9\,M_\odot$ |
| Total | — | — | $3.68 \times 10^9\,M_\odot$ | $7.34 \times 10^9\,M_\odot$ |
4. Step 3 — Rotation curve from sub-element summation
The total circular velocity at each radius $R$ combines the baryonic Freeman contribution (visible stars + gas) and the wave-field contribution (collective wave mass):
$$V^2(R) \;=\; V_\text{baryon}^2(R) + V_\text{wave}^2(R) \quad\text{with}\quad V_\text{wave}^2(R) = \frac{G\,\lambda\,M_\text{wave,enc}(R)}{R}$$
| $R$ (kpc) | $V_\text{baryon}$ (km/s) | $V_\text{wave}$ (km/s) | $V_\text{total}$ (km/s) |
|---|---|---|---|
| 2.0 | 19.4 | 5.6 | 20.2 |
| 4.0 | 27.2 | 10.0 | 29.0 |
| 6.0 | 30.6 | 13.5 | 33.4 |
| 8.0 | 31.9 | 16.1 | 35.7 |
| 10.0 | 32.1 | 18.2 | 36.8 |
| 12.0 | 31.7 | 19.7 | 37.3 |
| 15.0 | 30.6 | 21.4 | 37.3 |
| 20.0 | 28.5 | 22.9 | 36.5 |
| 25.0 | 26.4 | 23.4 | 35.3 |
| 30.0 | 24.5 | 23.5 | 33.9 |
5. Why does it fail? — A structural problem, not a calibration one
The failure on F568-1 is not a small numerical error to be tuned away. It is a $-68\%$ underestimation that exposes a fundamental property of the formulation.
In the universal-parameter framework, the relation between the observed plateau velocity and the visible mass takes a definite form. For a system in the asymptotic regime, the total dynamical mass enclosed is $M_\text{dyn} = M_\text{visible}(1+\lambda)$, and:
$$V_f^2 \;\approx\; \frac{G\,(1+\lambda)\,M_\text{visible}}{R_\text{plateau}} \quad\Rightarrow\quad V_f \;\propto\; \sqrt{M_\text{visible}}$$
The universal-parameter prediction is therefore $V_f \propto M_\text{vis}^{1/2}$. But observations across hundreds of galaxies (the baryonic Tully-Fisher relation) give:
$$V_f^4 \;\propto\; M_\text{visible} \quad\Rightarrow\quad V_f \;\propto\; M_\text{visible}^{1/4}$$
This is a different power law. A model with universal $\lambda$ and $c$ cannot simultaneously match galaxies that span four decades in visible mass. The Milky Way ($M_text{vis} sim 7 times 10^{10}$) and F568-1 ($M_text{vis} sim 4 times 10^9$) differ by a factor 18 in mass — under $V_f propto sqrt{M}$, this gives a factor $sqrt{18} approx 4.2$ in velocity, while the observed ratio is only $V_f^text{MW}/V_f^text{F568-1} = 229/115 approx 2$.
The diagnostic
The Milky Way parameters $(lambda, c) = (2.00, 1.85)$ embed information specific to a massive Sbc galaxy with substantial bulge and high central surface density. For an LSB galaxy with the same baryonic mechanism but much lower surface density, the wave-mass response must be stronger — either $\lambda$ must scale, or $c$ must scale, or both. In its current form, BeeTheory with universal parameters cannot span the full SPARC sample.
6. What does this tell us about BeeTheory?
The F568-1 case is not a refutation of BeeTheory — it is a constraint on its physical content. Three observations follow naturally:
- The wave coupling cannot be a single number. Either $\lambda$ depends on local surface density $\Sigma_d$, or $\ell_\text{wave}$ depends on it, or both. LSB galaxies, with diffuse visible matter, must generate a relatively stronger wave field per unit visible mass than HSB galaxies.
- This is consistent with a wave-field physical mechanism. A more diffuse source spreads its wave function over a larger volume; constructive interference between widely separated source elements is geometrically different from interference in a dense, compact disk. The coherence length is a property of the source’s geometry, not of the source itself.
- The Radial Acceleration Relation (RAR) of McGaugh et al. (2016) already encodes this empirically: the relation $g_text{obs} = nu(g_text{bar}),g_text{bar}$ is universal across galaxy types, where $nu$ depends on the local baryonic acceleration. BeeTheory must reproduce this in detail, which requires the wave-field response to scale with local $\Sigma_d$ — not with global $\lambda$.
The failure on F568-1 is therefore informative: it tells us that BeeTheory’s two-parameter universal form is incomplete, and points toward a refinement where the wave coupling depends on local surface density.
7. Summary
1. F568-1 was selected as a representative LSB galaxy from the SPARC calibration sample.
2. The exact Note XXXI methodology was applied: 10 stellar rings + 10 gas rings, each ring carrying visible mass $dM_i$ and wave mass $\lambda\,dM_i$, with $\ell_\text{wave} = c\,R_d$ universal.
3. The total predicted rotation velocity peaks at $V_\text{max} = 37$ km/s, against $V_f^\text{obs} = 115$ km/s. Error: $-68\%$.
4. The failure follows from the implicit scaling $V_f \propto \sqrt{M_\text{vis}}$ of the universal model, which contradicts the empirical baryonic Tully-Fisher relation $V_f \propto M_\text{vis}^{1/4}$.
5. BeeTheory with universal $\lambda$ and $c$ cannot span the four-decade mass range of the SPARC sample. The wave coupling must depend on local surface density — a refinement that the next note will introduce and test on the full 23-galaxy set.
6. The failure is structural and informative: it identifies where the current formulation lacks physical content, and points to a specific path forward — surface-density-dependent coupling — that is both physically motivated and empirically constrained by the RAR.
References. Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023). · Notes XXX–XXXI — 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). · McGaugh, S. S., Schombert, J. M., Bothun, G. D. — The cosmological constraints on low surface brightness galaxies, AJ 109, 2019 (1995). · McGaugh, S. S., Lelli, F., Schombert, J. M. — Radial Acceleration Relation in Rotationally Supported Galaxies, PRL 117, 201101 (2016). · Freeman, K. C. — On the disks of spiral and S0 galaxies, ApJ 160, 811 (1970).
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