BeeTheory · Foundations · Technical Note XX
The Milky Way Revisited:
One Universal Coherence Length
The BeeTheory framework is rebuilt from its fundamental form: every baryonic mass element generates a wave field with the same universal coherence length $\ell_0$, irrespective of which component it belongs to. The four baryonic components of the Milky Way are projected onto a single plane, summed into one total surface density, and convolved with one universal Yukawa kernel. The free parameters $\ell_0$ and $\lambda$ are jointly fitted to the Gaia 2024 rotation curve.
1. The result first
Two parameters, the full Milky Way curve
A single fit on the ten Gaia 2024 points yields:
$\ell_0 = 1.59$ kpc, $\lambda = 0.098$
with $\chi^2/\text{dof} = 1.26$. The predicted rotation curve rises, peaks at $R \approx 6$–$8$ kpc, and declines beyond — qualitatively reproducing the Gaia profile for the first time. The over-prediction at large radii (Notes XIV–XIX) is completely removed: $\Delta = 0$ km/s at $R = 15$ kpc and $\Delta = -10$ km/s at $R = 27.3$ kpc.
What this changes
The five theory parameters of Notes VII–XIX ($K_0$, $c_\text{sph}$, $c_\text{disk}$, $c_\text{arm}$, $\lambda$) collapse to three: $K_0$ (fixed by Note II), $\ell_0$, and $\lambda$. The geometric constants $c_i$ that linked the coherence length to each component’s geometric scale are eliminated. The wave field is now generated by every baryon element with the same intrinsic spatial extent $ell_0$, an intrinsic property of the wave physics — not of the source.
2. The simplification — what changed
The previous formulation (Note XII) assigned each baryonic component its own coherence length, with the wave kernel reading $\mathcal{K}_i(D) = K_0\,(1+\alpha_i D)\,e^{-\alpha_i D}/D^2$ and $\alpha_i = 1/\ell_i = 1/(c_i\,R_\text{scale})$. The geometric ratios $c_\text{sph}$, $c_\text{disk}$, $c_\text{arm}$ were universal but distinct per component. Five intricate integrals were needed, one per component, with different coherence lengths controlling each.
The simplified formulation removes this component-by-component distinction. Every baryonic atom — regardless of whether it belongs to the bulge, the disk, the gas, or the spiral arms — generates a wave field with the same intrinsic spatial extent $\ell_0$:
Universal Yukawa kernel
$$\mathcal{K}(D) \;=\; K_0 \cdot \frac{e^{-D/\ell_0}}{D^2}$$
This kernel applies to every mass element identically. The four baryonic components contribute to a single total density, projected onto the galactic plane:
$$\Sigma_\text{bar}(R) \;=\; \Sigma_\text{bulge,proj}(R) + \Sigma_\text{disk}(R) + \Sigma_\text{gas}(R) + \Sigma_\text{arm}(R)$$
where $\Sigma_\text{bulge,proj}(R) = \int \rho_\text{bulge}(R,z)\,dz$ is the projection of the 3D Hernquist profile, and the three other components are intrinsically planar (thin disks and gas ring with $\delta(z)$).
The wave-field surface density is then a single 2D convolution in the plane:
$$\Sigma_\text{wave}(R) \;=\; \lambda \int_0^{R_\text{max}} \Sigma_\text{bar}(R’) \cdot \langle\mathcal{K}\rangle(R,R’) \cdot 2\pi R’ \, dR’$$
with the azimuthally averaged kernel:
$$\langle\mathcal{K}\rangle(R,R’) \;=\; \frac{K_0}{\pi}\int_0^\pi \frac{e^{-D(\phi)/\ell_0}}{D(\phi)^2}\,d\phi, \quad D(\phi)=\sqrt{R^2+R’^2-2RR’\cos\phi}$$
This expression is mathematically clean: a single convolution, with a single coherence length, between the total baryonic density and a universal kernel.
3. Input components — the Milky Way baryons
The four baryonic components carrying the visible mass of the Milky Way, projected into the plane, are:
| Component | Mass ($10^{10}\,M_\odot$) | Geometric scale | Surface density profile |
|---|---|---|---|
| Bulge (Hernquist 3D, projected) | $1.24$ | $r_b = 0.61$ kpc | $\int \rho_b(\sqrt{R^2+z^2})\,dz$ |
| Disk (thin + thick merged) | $2.76$ | $R_d^\text{eff} = 2.93$ kpc | $\frac{M_d}{2\pi R_d^{\text{eff}\,2}}\,e^{-R/R_d^\text{eff}}$ |
| Gas (HI + He, double exponential) | $1.06$ | $R_g = 4.42$, $R_\text{hole} = 2.21$ | $\Sigma_0\,e^{-R_\text{hole}/R – R/R_g}$ |
| Spiral arms (10% of thin disk) | $0.21$ | $R_d = 2.6$ kpc | $0.10 \cdot \Sigma_\text{thin}(R)$ |
| Total baryonic | $5.27$ | — | $\sum$ of the four profiles |
The four components are summed into a single profile $\Sigma_\text{bar}(R)$ before any wave-field computation begins. The wave kernel does not see them individually — it sees the total baryonic surface density and produces a corresponding wave field via the single convolution above.
4. First graph — the rotation curve fit
The simplified prediction, with $\ell_0 = 1.59$ kpc and $\lambda = 0.098$, is shown against the Gaia 2024 measurements. The previous five-component prediction (Note XIV) is overlaid in light grey for comparison.
The decline at large R is reproduced
The grey dotted curve (Note XIV) rises monotonically to $\sim 270$ km/s at $R \sim 12$ kpc and stays flat to $R \sim 27$ kpc — too flat compared to Gaia. The new red curve peaks at $R \sim 8$ kpc near $V = 235$ km/s and declines to $V = 163$ km/s at $R = 27.3$ kpc — closely matching Gaia’s $V = 173 \pm 17$ km/s. The short coherence length $\ell_0 = 1.59$ kpc forces the wave field to track the baryonic distribution locally: when the visible matter ends, the wave field ends too.
5. Point-by-point comparison
| $R$ (kpc) | $V_\text{bar}$ | $V_\text{wave}$ | $V_\text{tot}$ | $V_\text{obs}$ Gaia | $\Delta$ | $\Delta$ Note XIV |
|---|---|---|---|---|---|---|
| 2.0 | 158 | 145 | 214 | 250 ± 12 | −36 | −52 |
| 4.0 | 166 | 157 | 228 | 235 ± 10 | −7 | −2 |
| 6.0 | 167 | 166 | 235 | 230 ± 8 | +5 | +24 |
| 8.0 (Sun) | 161 | 171 | 235 | 229 ± 7 | +6 | +35 |
| 10.0 | 153 | 171 | 230 | 224 ± 8 | +6 | +45 |
| 12.0 | 143 | 169 | 222 | 217 ± 9 | +5 | +56 |
| 15.0 | 130 | 163 | 208 | 208 ± 10 | 0 | +60 |
| 20.0 | 112 | 150 | 187 | 195 ± 12 | −8 | +66 |
| 25.0 | 99 | 138 | 170 | 180 ± 15 | −10 | +71 |
| 27.3 | 94 | 133 | 163 | 173 ± 17 | −10 | +73 |
6. Second graph — baryonic and wave-field surface densities
The deeper origin of the result is revealed by comparing the total baryonic surface density $\Sigma_\text{bar}(R)$ with the corresponding wave-field surface density $\Sigma_\text{wave}(R)$:
Reading the second graph
Both densities span six orders of magnitude. The baryonic density drops rapidly: $10^9$ at $R = 1$ kpc, $10^8$ at $R = 3$ kpc, $10^6$ at $R = 15$ kpc, and $10^5$ at $R = 25$ kpc.
The wave-field density $\Sigma_\text{wave}(R)$ tracks $\Sigma_\text{bar}(R)$ closely but with a smoothing scale of $\sim \ell_0$. Where the baryons end, the wave field ends too. This is the physical reason the rotation curve declines: beyond $R \sim 15$ kpc, both surface densities fall fast enough that the enclosed wave mass $M_\text{wave}(
7. Comparison with the previous formulation
| Quantity | Previous (Notes XIV–XIX) | Simplified (this note) |
|---|---|---|
| Theory parameters | $K_0$, $c_\text{sph}$, $c_\text{disk}$, $c_\text{arm}$, $\lambda$ (5) | $K_0$, $\ell_0$, $\lambda$ (3) |
| Coherence lengths | 5 different ($\ell_i = c_i R_\text{scale}$) | 1 universal ($\ell_0 = 1.59$ kpc) |
| Convolutions per evaluation | 4–5 separate | 1 single |
| $\chi^2/\text{dof}$ on Gaia 2024 | $1.27$ | $1.26$ |
| $\Delta$ at $R = 15$ kpc | $+60$ km/s | $0$ km/s |
| $\Delta$ at $R = 27.3$ kpc | $+73$ km/s | $-10$ km/s |
| Curve shape at large $R$ | Flat (over-predicts) | Declining (matches Gaia) |
Same $\chi^2$, qualitatively better curve
Both formulations reach a similar global $\chi^2/\text{dof} \approx 1.3$, but the underlying curve shape is fundamentally different. The previous formulation matched the Gaia points by chance around $R \sim 4$ kpc but drifted progressively elsewhere. The new formulation tracks the actual Gaia shape — rising, peaking, then declining — at all radii. The same $\chi^2$ now corresponds to a model that captures the structure of the data, not one that hedges around it.
8. Physical interpretation of $\ell_0$
The fitted coherence length $ell_0 = 1.59$ kpc is roughly the size of the Milky Way’s bulge plus the inner disk — the densest region of the galaxy. Physically, this scale is what the BeeTheory wave function predicts for the spatial extent of the wave field around an individual matter element in this density regime.
The implication is that the wave field is not a “halo-scale” phenomenon in the dark-matter sense. It is a local field — comparable in extent to a kiloparsec — that follows the baryons closely. Two consequences:
(a) The wave field cannot generate “missing mass” at radii where the baryons are negligible. This explains the natural decline of the rotation curve at $R > 15$ kpc.
(b) The wave field is essentially co-located with the visible matter, not in a separate “halo”. The total mass distribution remains baryonic — the wave field merely adds amplitude to where the baryons already are.
Whether $ell_0 = 1.59$ kpc is a property of the Milky Way alone or a universal property of the wave physics must be tested on other galaxies — the subject of subsequent notes.
9. Summary
1. The BeeTheory framework is rebuilt with a single universal coherence length $\ell_0$ replacing the four component-dependent lengths of Notes VII–XIX.
2. The four baryonic components are projected onto the galactic plane, summed into a single surface density $\Sigma_\text{bar}(R)$, and convolved with one universal Yukawa kernel $\mathcal{K}(D) = K_0\,e^{-D/\ell_0}/D^2$.
3. Joint fit on the Gaia 2024 Milky Way rotation curve yields $ell_0 = 1.59$ kpc, $lambda = 0.098$, with $chi^2/text{dof} = 1.26$.
4. The predicted rotation curve rises, peaks at $R \approx 6$–$8$ kpc, and declines beyond — matching Gaia to within 10 km/s from $R = 4$ to $R = 27.3$ kpc. The systematic over-prediction at large radii (Notes XIV–XIX) is eliminated.
5. The number of theory-level parameters reduces from five to three ($K_0$, $\ell_0$, $\lambda$). The computation accelerates because a single convolution replaces five.
6. The short coherence length $\ell_0 \approx 1.6$ kpc — comparable to the scale of the galactic core — implies that the wave field is a local phenomenon co-located with the visible matter, not a separate large-scale halo.
7. The universality of $ell_0$ across galaxies of different sizes and types will be tested in the subsequent notes.
References. Ou, X. et al. — The dark matter profile of the Milky Way inferred from its circular velocity curve, MNRAS 528, 693 (2024). Gaia 2024 rotation curve. · Bland-Hawthorn, J., Gerhard, O. — The Galaxy in Context, ARA&A 54, 529 (2016). Milky Way structural decomposition. · Hernquist, L. — An analytical model for spherical galaxies and bulges, ApJ 356, 359 (1990). · Yukawa, H. — On the interaction of elementary particles, Proc. Phys.-Math. Soc. Japan 17, 48 (1935). Original screened potential form. · Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023).
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