BeeTheory · Foundations · Technical Note XIV
Step 1 — Milky Way:
Applying the BeeTheory Yukawa Kernel
The methodology of Note XII is applied to the Milky Way using the explicit Yukawa-form wave kernel $\mathcal{K}(D) = K_0\,(1+\alpha D)\,e^{-\alpha D}/D^2$, with integration carried out separately for each baryonic component according to its geometry. The result is compared point-by-point to the Gaia 2024 rotation curve and to the standard model’s “missing mass”. This note establishes the baseline of the framework as applied with the new, fully geometric formulation.
1. The result first
Baseline result — Milky Way with explicit Yukawa kernel
Rotation curve. The model reproduces the Gaia 2024 velocity to within 2 km/s at $R = 4$ kpc, but over-predicts by $+33$ km/s at the solar radius ($R = 8$ kpc) and by $+64$ km/s at $R = 27.3$ kpc. $\chi^2/\text{dof} = 1.27$.
Missing mass. The BeeTheory wave-field mass matches the standard model’s “missing mass” to within 5% at $R = 4$ kpc, but exceeds it by a factor of 2.2 at $R = 27.3$ kpc. The model produces too much dark-field mass at large radii.
Local density. $\rho_\text{wave}(R_\odot) = 0.72$ GeV/cm³, compared to the observed range $0.39$–$0.45$ GeV/cm³. Over-predicted by roughly a factor of 1.7.
What this means
The explicit Yukawa formulation produces a rotation curve that is too flat at large radii. The wave-field decay length $\ell$ is too long, allowing the wave field to keep contributing mass beyond the visible disk. This is the structural baseline before the surface-density refinement identified in Note XI is incorporated.
2. What we set out to compute
The Milky Way is the natural test case because it is the galaxy on which the global coupling $lambda$ was originally calibrated, and because two independent observations exist to compare against:
(a) The rotation curve $V_c(R)$ from Gaia 2024 (Ou et al., MNRAS 528), which measures the circular velocity at ten radii from $R = 2$ kpc to $R = 27.3$ kpc with statistical uncertainties of $7$–$17$ km/s. This is the velocity that BeeTheory must reproduce, by combining the baryonic contribution $V_\text{bar}(R)$ with the wave-field contribution $V_\text{wave}(R)$.
(b) The “missing mass” $M_text{missing}(. The standard interpretation invokes particle dark matter to provide this mass. BeeTheory predicts instead that this mass is the wave-field $M_\text{wave}( (c) The local dark-matter density at the Sun, measured at $\rho \approx 0.39$–$0.45$ GeV/cm³ from kinematic studies of the solar neighbourhood. BeeTheory predicts a value for $\rho_\text{wave}(R_\odot)$ from the same wave-field calculation. The agreement (or disagreement) on these three observables tests three different aspects of the model: its rotation-curve shape, its enclosed-mass profile, and its local density normalisation.
3. The wave kernel — explicit form
Each baryonic mass element generates a BeeTheory wave field, with intensity at a field point separated by distance $D$ given by:
Yukawa-form wave kernel
$$\mathcal{K}(D) \;=\; K_0 \cdot \frac{(1 + \alpha D)\,e^{-\alpha D}}{D^2}, \qquad \alpha = \frac{1}{\ell}$$
The exponential damping $e^{-\alpha D}$ ensures the wave field has finite total mass — without it, the wave-field integral would diverge at infinity. The prefactor $(1 + alpha D)$ comes from the regularised wave function of Note I; together with the exponential, it makes the spatial structure of the kernel quasi-Newtonian at $D ll ell$ and exponentially suppressed at $D gg ell$.
The characteristic length $\ell$ depends on the component generating the field:
| Component | Coherence length $\ell$ (kpc) | Geometric scale |
|---|---|---|
| Bulge (3D Hernquist) | $\ell_b = c_\text{sph}\,r_b = 0.41 \times 0.61$ | $0.25$ |
| Thin disk | $\ell_\text{thin} = c_\text{disk}\,R_d = 3.17 \times 2.6$ | $8.24$ |
| Thick disk | $\ell_\text{thick} = c_\text{disk}\,(1.5\,R_d)$ | $12.36$ |
| Gas ring | $\ell_\text{gas} = c_\text{disk}\,(1.7\,R_d)$ | $14.01$ |
| Spiral arms | $\ell_\text{arm} = c_\text{arm}\,R_d = 2.0 \times 2.6$ | $5.20$ |
4. Integration geometry, component by component
For each component, we integrate the source density convolved with the kernel, using the appropriate volume element for the geometry. The result is the wave-field density $\rho_\text{wave}^{(i)}(r)$ at the field point.
4.1 Bulge — spherical shell integration
The bulge is a three-dimensional spherical distribution. Each thin shell of radius $r’$ contains mass $\rho_b(r’)\,4\pi r’^2\,dr’$ and contributes to the field at radial distance $r$ from the centre. In the monopolar approximation, the effective separation between the field point and a generic point on the shell is $D = \sqrt{r^2 + r’^2}$:
$$\rho_\text{wave}^{(b)}(r) \;=\; \int_0^{6r_b} \rho_b(r’) \cdot K_0\,\frac{(1 + \alpha_b D)\,e^{-\alpha_b D}}{D^2} \cdot 4\pi r’^2 \, dr’, \quad D = \sqrt{r^2 + r’^2}$$
With $\rho_b(r’) = M_b r_b / [2\pi r'(r’+r_b)^3]$ (Hernquist) and $\alpha_b = 1/0.25 = 4.0$ kpc$^{-1}$. The integration is cut off at $6\,r_b$, beyond which the density is numerically negligible.
4.2 Thin and thick disks — concentric ring integration
Each disk is a thin axisymmetric distribution. The disk is decomposed into concentric rings at galactocentric radius $R’$ and width $dR’$, each carrying mass $\Sigma(R’)\,2\pi R’\,dR’$. The contribution to the field point at radius $r$ (in the disk plane) requires the effective separation $D = \sqrt{r^2 + R’^2}$ in the same monopolar approximation:
$$\rho_\text{wave}^{(\text{thin})}(r) \;=\; \int_0^{8R_d} \Sigma_\text{thin}(R’) \cdot K_0\,\frac{(1 + \alpha_\text{thin} D)\,e^{-\alpha_\text{thin} D}}{D^2} \cdot 2\pi R’ \, dR’, \quad D = \sqrt{r^2 + R’^2}$$
with $\Sigma_\text{thin}(R’) = (M_\text{thin}/2\pi R_d^2)\,e^{-R’/R_d}$ and $\alpha_\text{thin} = 1/8.24 = 0.121$ kpc$^{-1}$. The thick disk is identical with its own scale: $\Sigma_\text{thick}$, $R_\text{thick} = 3.9$ kpc, $\alpha_\text{thick} = 0.081$ kpc$^{-1}$.
4.3 Gas ring — ring integration with central depletion
The gas distribution has a central hole (negligible HI at $R \lesssim 2$ kpc) and extends further than the stellar disk. The profile $\Sigma_\text{gas}(R’) = \Sigma_0\,\exp(-R_\text{hole}/R’ – R’/R_g)$ captures both features:
$$\rho_\text{wave}^{(\text{gas})}(r) \;=\; \int_{R_\text{hole}}^{8R_g} \Sigma_\text{gas}(R’) \cdot K_0\,\frac{(1 + \alpha_\text{gas} D)\,e^{-\alpha_\text{gas} D}}{D^2} \cdot 2\pi R’ \, dR’$$
with $R_g = 4.42$ kpc, $R_\text{hole} = 2.21$ kpc, $\alpha_\text{gas} = 0.071$ kpc$^{-1}$. The longer coherence length reflects the more extended gas distribution.
4.4 Spiral arms — ring integration with reduced amplitude and narrower kernel
The spiral arms carry $10\%$ of the thin-disk surface density and have their own coherence length $\ell_\text{arm} = 5.2$ kpc, narrower than that of the disk to reflect the azimuthal concentration of the arms:
$$\rho_\text{wave}^{(\text{arm})}(r) \;=\; \int_0^{8R_d} 0.10\,\Sigma_\text{thin}(R’) \cdot K_0\,\frac{(1 + \alpha_\text{arm} D)\,e^{-\alpha_\text{arm} D}}{D^2} \cdot 2\pi R’ \, dR’$$
4.5 Total wave-field density
$$\rho_\text{wave}(r) \;=\; \lambda \,\sum_{i} \rho_\text{wave}^{(i)}(r), \quad \lambda = 0.189$$
5. From the wave density to the rotation curve
Once $\rho_\text{wave}(r)$ is known at every radius, the total enclosed wave-field mass is obtained by radial integration:
$$M_\text{wave}(R) \;=\; \int_0^{R} 4\pi r^2 \rho_\text{wave}(r) \, dr$$
The predicted circular velocity then combines the baryonic and wave-field contributions in quadrature, by the Newtonian relation:
$$V_c^2(R) \;=\; V_\text{bar}^2(R) \;+\; \frac{G\,M_\text{wave}(R)}{R}$$
6. Rotation curve — point-by-point results
| $R$ (kpc) | $V_\text{bar}$ (km/s) | $M_\text{wave}/10^{10}$ | $V_\text{wave}$ (km/s) | $V_\text{tot}$ (km/s) | $V_\text{obs}$ (Gaia 2024) | $\Delta$ |
|---|---|---|---|---|---|---|
| 2.0 | 157.8 | 0.67 | 120.0 | 198.3 | 250 ± 12 | -51.7 |
| 4.0 | 164.1 | 2.54 | 165.4 | 233.0 | 235 ± 10 | -2.0 |
| 6.0 | 163.5 | 5.17 | 192.5 | 252.6 | 230 ± 8 | +22.6 |
| 8.0 | 157.3 | 8.13 | 209.1 | 261.7 | 229 ± 7 | +32.7 |
| 10.0 | 148.7 | 11.18 | 219.3 | 265.0 | 224 ± 8 | +41.0 |
| 12.0 | 139.6 | 14.15 | 225.2 | 265.0 | 217 ± 9 | +48.0 |
| 15.0 | 126.7 | 18.29 | 229.0 | 261.8 | 208 ± 10 | +53.8 |
| 20.0 | 109.3 | 24.10 | 227.7 | 252.6 | 195 ± 12 | +57.6 |
| 25.0 | 96.7 | 28.54 | 221.6 | 241.8 | 180 ± 15 | +61.8 |
| 27.3 | 92.0 | 30.18 | 218.1 | 236.7 | 173 ± 17 | +63.7 |
The model matches the observation excellently at $R = 4$ kpc (Δ = −2 km/s) but over-predicts increasingly with radius. At the solar radius, the over-prediction is +33 km/s (4.7σ above the Gaia uncertainty). At the outer boundary $R = 27.3$ kpc, the over-prediction reaches +64 km/s (3.8σ). The predicted curve is too flat — the wave-field continues to contribute mass beyond the visible disk, because the exponential cutoff at $D \sim \ell$ allows it.
7. Wave mass versus the standard model’s “missing mass”
For each radius, we compare three quantities: the baryonic mass enclosed (visible matter only), the dynamical mass required by the observed velocity (Newton’s law applied to $V_\text{obs}$), and the BeeTheory wave-field mass. The difference between the second and the first is what the standard model calls “missing mass”:
$$M_\text{missing}(
The ratio $M_text{wave}/M_text{missing}$ tells us how well the BeeTheory wave field replaces particle dark matter on a radius-by-radius basis:
| $R$ (kpc) | $M_\text{bar}(| $M_\text{dyn}( | $M_\text{missing}$ | $M_\text{wave}$ (BT) | Ratio $M_\text{wave}/M_\text{miss}$ | |
|---|---|---|---|---|---|
| 2.0 | 1.16e+10 | 2.91e+10 | 1.75e+10 | 6.69e+09 | 0.38 |
| 4.0 | 2.51e+10 | 5.13e+10 | 2.63e+10 | 2.54e+10 | 0.97 |
| 6.0 | 3.73e+10 | 7.38e+10 | 3.65e+10 | 5.17e+10 | 1.42 |
| 8.0 | 4.60e+10 | 9.75e+10 | 5.15e+10 | 8.13e+10 | 1.58 |
| 10.0 | 5.14e+10 | 1.17e+11 | 6.52e+10 | 1.12e+11 | 1.71 |
| 12.0 | 5.44e+10 | 1.31e+11 | 7.70e+10 | 1.41e+11 | 1.84 |
| 15.0 | 5.60e+10 | 1.51e+11 | 9.49e+10 | 1.83e+11 | 1.93 |
| 20.0 | 5.56e+10 | 1.77e+11 | 1.21e+11 | 2.41e+11 | 1.99 |
| 25.0 | 5.43e+10 | 1.88e+11 | 1.34e+11 | 2.85e+11 | 2.13 |
| 27.3 | 5.37e+10 | 1.90e+11 | 1.36e+11 | 3.02e+11 | 2.22 |
Quantitative reading
At $R = 4$ kpc the wave field essentially matches the missing mass (ratio $0.97$). Between $R = 6$ and $R = 8$ kpc the model already exceeds the missing mass by 40–60%. Beyond $R = 15$ kpc, the wave-field mass is roughly double what the standard model invokes as dark matter. The model produces extra mass at large radii — exactly the symptom of a coherence length $\ell$ that is too long for the visible stellar disk.
8. Component contributions to the wave field at the solar radius
Evaluating each component’s contribution to $\rho_\text{wave}(R_\odot = 8\,\text{kpc})$ shows which baryonic source dominates the wave field there:
| Component | $\rho_\text{wave}(R_\odot)$ ($M_\odot$/kpc³) | Fraction of total |
|---|---|---|
| Thin stellar disk | $6.05 \times 10^7$ | 60.6% |
| Thick stellar disk | $1.91 \times 10^7$ | 19.1% |
| Gas ring | $1.62 \times 10^7$ | 16.2% |
| Spiral arms | $4.15 \times 10^6$ | 4.1% |
| Bulge | $1.55 \times 10^{-5}$ | $\sim$0% |
The thin stellar disk dominates the wave field at the Sun’s position (60%), with the thick disk and gas ring contributing roughly equally (16–19%). The bulge contributes negligibly because $\ell_b = 0.25$ kpc is much smaller than $R_\odot = 8$ kpc — the exponential suppression in the kernel kills the bulge contribution at this distance.
Converting the total density to particle-physics units gives $\rho_\text{wave}(R_\odot) = 0.717$ GeV/cm³, to be compared with the kinematic measurement of $0.39$–$0.45$ GeV/cm³ from Read 2014 and subsequent analyses. The prediction over-shoots the observed local density by a factor of $1.6$–$1.8$ — consistent with the rotation-curve over-prediction at the same radius.
9. What this baseline establishes
The mechanism works in principle
Around $R = 4$ kpc — the central body of the disk — the integrated wave field equals the standard model’s missing mass to within 5%, and the rotation curve is reproduced to within 2 km/s. The wave kernel, applied to the visible baryons, produces a gravitational mass quantitatively comparable to particle dark matter at this radius. No new particle is needed; the wave field of the visible matter accounts for the missing gravity.
But the curve is too flat at large radii
Beyond the central disk, the model over-predicts the rotation velocity by an amount that grows monotonically with radius. The wave field continues to accumulate mass beyond the visible disk because the kernel’s coherence length $\ell_\text{thin} = 8.24$ kpc is comparable to the disk size itself, allowing significant contributions at $D = 15$–$25$ kpc. The Gaia rotation curve, in contrast, declines slightly beyond $R \sim 10$ kpc — a feature the current formulation does not reproduce.
A baseline, not a final answer
This calculation establishes the baseline of the model with $\ell_i$ depending linearly on $R_d$ alone. The diagnostic of Note XI identified that $\Sigma_d$ — the central surface density — must enter the determination of $\ell_i$ to correct the curve at large radii. The denser the disk, the more localised the wave response should be. Incorporating this refinement is the subject of subsequent notes. The Milky Way baseline reported here is what those notes must improve upon.
10. Summary
1. The Milky Way rotation curve is computed by integrating each baryonic component against the Yukawa wave kernel $mathcal{K}(D) = K_0(1 + alpha D),e^{-alpha D}/D^2$, with appropriate geometry: spherical shells for the bulge, concentric rings for the disks, gas, and spiral arms.
2. At $R = 4$ kpc, the BeeTheory wave-field mass agrees with the standard model’s “missing mass” to within 5% (ratio 0.97), and the predicted velocity matches Gaia 2024 to within 2 km/s.
3. At the solar radius ($R = 8$ kpc), the model over-predicts the rotation velocity by $+33$ km/s and the local dark-matter density by a factor of 1.6 — both consistent with each other.
4. Beyond $R = 15$ kpc, the predicted wave-field mass exceeds the standard model’s missing mass by a factor of 2 or more. The predicted rotation curve does not decline as the Gaia data require.
5. The thin stellar disk dominates the wave field at the Sun’s position (60% of $\rho_\text{wave}$). The bulge contributes negligibly. The decomposition is consistent with the integration geometry described.
6. The over-prediction at large $R$ is the structural signature of $\ell_i$ being too long. Note XI identified that $\Sigma_d$ must enter the coherence-length formula. The refinement of $\ell_i$ via $\Sigma_d$ is the next step.
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). MW structural decomposition. · Read, J. I. — The Local Dark Matter Density, J. Phys. G 41, 063101 (2014). Local DM density measurements. · Freeman, K. C. — On the disks of spiral and S0 galaxies, ApJ 160, 811 (1970). · Hernquist, L. — An analytical model for spherical galaxies and bulges, ApJ 356, 359 (1990). · Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023).
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