BeeTheory · Foundations · Technical Note XII
Formalisation:
The Galactic-Scale BeeTheory Computation
This note formalises the BeeTheory framework as it is applied to a disk galaxy. It specifies the observational inputs, the geometric decomposition of the baryonic distribution, the integral equations defining the wave field for each component, and the chain of operations that yields the predicted rotation curve. The procedure is strictly unidirectional: the observed baryonic structure determines the wave field, which determines the rotation curve — never the reverse.
1. The computation in one diagram
A unidirectional chain
Observed photometry $\;\longrightarrow\;$ Baryonic decomposition $(\rho_\text{bar})$
$\big\downarrow$
Wave-field convolution $\;\longrightarrow\;$ Wave density $(\rho_\text{wave})$
$\big\downarrow$
Mass integration $\;\longrightarrow\;$ Enclosed wave mass $(M_\text{wave})$
$\big\downarrow$
Newtonian relation $\;\longrightarrow\;$ Predicted rotation curve $(V_c)$
No step is inverted. The rotation curve $V_c(R)$ is never used as an input.
2. Observational inputs
For each galaxy, the computation requires five published observables. These are the only galaxy-specific quantities; everything else is computed from them. No fitting against the rotation curve is performed at this stage.
| Symbol | Quantity | Source |
|---|---|---|
| $T$ | Hubble morphological type | Catalogue (de Vaucouleurs et al. 1991, SPARC) |
| $R_d$ | Stellar disk scale length (kpc) | Spitzer 3.6 µm photometry (SPARC) |
| $\Sigma_d$ | Central disk surface brightness ($L_\odot/\text{pc}^2$) | Spitzer 3.6 µm photometry (SPARC) |
| $M_\text{HI}$ | Total atomic hydrogen mass ($M_odot$) | 21-cm radio observations (SPARC) |
| $\Upsilon_\star$ | Stellar mass-to-light ratio at 3.6 µm | Fixed universal: $0.5\,M_\odot/L_\odot$ (McGaugh 2014) |
For the Milky Way, $R_d$, $Sigma_d$, and $M_text{HI}$ are replaced by the analogous values determined from internal stellar surveys (Bovy & Rix 2013) and 21-cm maps. The same five-quantity input vector is used.
3. Baryonic decomposition — five geometric components
From the five observational inputs, the baryonic mass is partitioned into five distinct geometric components. Each component carries its own density profile and characteristic scale.
3.1 Total stellar and gas masses
$$M_\star \;=\; 2\pi\,R_d^2\,\Sigma_d\,\Upsilon_\star$$
$$M_\text{gas} \;=\; 1.33\,M_\text{HI} \qquad \text{(He correction; Arnett 1996)}$$
3.2 Component masses and scales
| Component | Mass | Scale | Activation |
|---|---|---|---|
| Bulge | $M_b = 0.20\,M_\star$ | $r_b = \max(0.5\,R_d,\,0.3\text{ kpc})$ | If $T \leq 4$ |
| Thin disk | $M_\text{thin} = 0.75\,(M_\star – M_b)$ | $R_d$ | Always |
| Thick disk | $M_\text{thick} = 0.25\,(M_\star – M_b)$ | $1.5\,R_d$ | Always |
| Gas ring | $M_\text{gas} = 1.33\,M_\text{HI}$ | $R_g = 1.7\,R_d$ (Broeils & Rhee 1997) | Always |
| Spiral arms | $M_\text{arm} = 0.10\,M_\text{thin}$ (effective) | $R_d$ (follows thin disk) | Always |
3.3 Density profiles
Bulge (3D Hernquist)
$$\rho_b(r) \;=\; \frac{M_b\,r_b}{2\pi\,r\,(r + r_b)^3}$$
Thin and thick stellar disks (2D exponential)
$$\Sigma_\text{thin}(R) \;=\; \frac{M_\text{thin}}{2\pi\,R_d^2}\,e^{-R/R_d}$$
$$\Sigma_\text{thick}(R) \;=\; \frac{M_\text{thick}}{2\pi\,(1.5\,R_d)^2}\,e^{-R/(1.5R_d)}$$
Gas ring (2D exponential with central hole)
$$\Sigma_\text{gas}(R) \;=\; \frac{M_\text{gas}}{2\pi\,R_g^2}\,\exp\!\left(-\frac{R_\text{hole}}{R} – \frac{R}{R_g}\right), \quad R_\text{hole} = 0.5\,R_g$$
Spiral arm excess (2D, follows thin disk)
$$\Sigma_\text{arm}(R) \;=\; 0.10\;\Sigma_\text{thin}(R)$$
4. The wave kernel
Each baryonic mass element generates a BeeTheory wave field. The field at a point $vec{r}$ produced by a source element at $vec{r},’$ separated by $D = |vec{r} – vec{r},’|$ is governed by the Yukawa-type kernel derived from the regularized wave function of Note I:
BeeTheory wave kernel
$$\mathcal{K}_i(D) \;=\; K_0\,\frac{(1 + \alpha_i\,D)\,e^{-\alpha_i\,D}}{D^2}, \qquad \alpha_i \;=\; \frac{1}{\ell_i}$$
Here $K_0$ is the universal wave-mass amplitude (a single dimensionless number) and $\ell_i$ is the coherence length of component $i$. The kernel encodes a quasi-Newtonian $1/D^2$ behaviour at short separations, modulated by an exponential cutoff at scales beyond $\ell_i$. The form $(1 + \alpha D)\,e^{-\alpha D}$ ensures continuity and finite total enclosed mass at infinity.
4.1 Component coherence lengths
The coherence length of each component is set by its natural geometric scale, multiplied by a dimensionless constant specific to its dimensionality:
| Component | Coherence length | Geometric constant |
|---|---|---|
| Bulge (3D sphere) | $\ell_b = c_\text{sph}\,r_b$ | $c_\text{sph}$ |
| Thin disk (2D) | $\ell_\text{thin} = c_\text{disk}\,R_d$ | $c_\text{disk}$ |
| Thick disk (2D) | $\ell_\text{thick} = c_\text{disk}\,(1.5\,R_d)$ | $c_\text{disk}$ |
| Gas ring (2D) | $\ell_\text{gas} = c_\text{disk}\,R_g$ | $c_\text{disk}$ |
| Spiral arms (2D, azimuthally concentrated) | $\ell_\text{arm} = c_\text{arm}\,R_d$ | $c_\text{arm}$ |
The three geometric constants $(c_\text{sph},\,c_\text{disk},\,c_\text{arm})$ are universal — they do not vary from galaxy to galaxy. Together with the global wave-mass amplitude $K_0$ and the wave-field coupling $\lambda$, they constitute the full set of theory-level parameters.
5. Wave-field convolution — integral equations per component
The wave-field density at a position $\vec{r}$ is the convolution of the baryonic source distribution with the wave kernel. For a galactically symmetric system (axially symmetric, monopolar approximation), each baryonic component contributes additively:
Total wave-field density at radius $r$
$$\rho_\text{wave}(r) \;=\; \lambda \;\sum_{i \in \{\text{thin, thick, gas, arm, bulge}\}} \rho_\text{wave}^{(i)}(r)$$
The five integrals are written below, one per component. Each integral converts a baryonic mass distribution into a wave-field mass distribution at the same spatial point.
5.1 Bulge — 3D shell integration
$$\rho_\text{wave}^{(b)}(r) \;=\; \int_0^{r_\text{max}} \rho_b(r’)\;\mathcal{K}_b\!\left(\sqrt{r^2 + r’^2}\right)\;4\pi r’^2\,dr’$$
The integration is over concentric spherical shells of radius $r’$. The field point at radius $r$ from the centre sees each shell at an effective separation $D = \sqrt{r^2 + r’^2}$ in the monopolar approximation. The integration extends out to $r_\text{max} = 6\,r_b$, beyond which the bulge density is numerically negligible.
5.2 Thin disk — 2D ring integration
$$\rho_\text{wave}^{(\text{thin})}(r) \;=\; \int_0^{R_\text{max}} \Sigma_\text{thin}(R’)\;\mathcal{K}_\text{thin}\!\left(\sqrt{r^2 + R’^2}\right)\;2\pi R’\,dR’$$
The disk is decomposed into concentric rings of radius $R’$ and infinitesimal width $dR’$, each carrying surface mass $\Sigma_\text{thin}(R’)\,2\pi R’\,dR’$. The same monopolar approximation applies: the wave field at radius $r$ from the centre receives contributions from each ring at effective separation $D = \sqrt{r^2 + R’^2}$. The integration range is $R_\text{max} = 8\,R_d$.
5.3 Thick disk — 2D ring integration
$$\rho_\text{wave}^{(\text{thick})}(r) \;=\; \int_0^{R_\text{max}} \Sigma_\text{thick}(R’)\;\mathcal{K}_\text{thick}\!\left(\sqrt{r^2 + R’^2}\right)\;2\pi R’\,dR’$$
Identical to the thin disk integration, with $\Sigma_\text{thick}(R’)$ as the source density and a kernel parameter $\alpha_\text{thick} = 1/(c_\text{disk}\,\cdot 1.5\,R_d)$. The wider radial extent of the thick disk results in a slightly broader wave-coherence range.
5.4 Gas ring — 2D ring integration with central depletion
$$\rho_\text{wave}^{(\text{gas})}(r) \;=\; \int_{R_\text{hole}}^{R_\text{max}} \Sigma_\text{gas}(R’)\;\mathcal{K}_\text{gas}\!\left(\sqrt{r^2 + R’^2}\right)\;2\pi R’\,dR’$$
The gas distribution has a central hole, captured by the radial cutoff at $R_\text{hole} = 0.5\,R_g$ in the lower bound of integration. Outside this cutoff, the gas extends further than the stellar disk; this is reflected in the larger characteristic scale $R_g = 1.7\,R_d$, which feeds into the coherence length $\ell_\text{gas} = c_\text{disk}\,R_g$.
5.5 Spiral arm excess — 2D ring integration with reduced amplitude
$$\rho_\text{wave}^{(\text{arm})}(r) \;=\; \int_0^{R_\text{max}} \Sigma_\text{arm}(R’)\;\mathcal{K}_\text{arm}\!\left(\sqrt{r^2 + R’^2}\right)\;2\pi R’\,dR’$$
The spiral arms are treated as an axially averaged enhancement of the thin disk surface density at the level of $10\%$, with their own coherence length $\ell_\text{arm} = c_\text{arm}\,R_d$. The kernel is therefore narrower than the thin disk kernel, reflecting the azimuthal concentration of the spiral structure.
6. Enclosed wave mass and predicted rotation curve
Once the total wave-field density $\rho_\text{wave}(r)$ is known, the enclosed wave-field mass within a sphere of radius $R$ is obtained by radial integration:
Enclosed wave-field mass
$$M_\text{wave}(R) \;=\; \int_0^{R} 4\pi\,r^2\,\rho_\text{wave}(r)\,dr$$
The predicted circular velocity at radius $R$ then follows from the Newtonian relation, combining the baryonic and wave-field contributions in quadrature:
Predicted circular velocity
$$V_c^2(R) \;=\; V_\text{bar}^2(R) \;+\; \frac{G\,M_\text{wave}(R)}{R}$$
The baryonic velocity $V_\text{bar}(R)$ is itself the quadratic sum of contributions from the four disk-like components (Freeman 1970 formula for each exponential profile) and the bulge (Hernquist enclosed mass formula):
$$V_\text{bar}^2(R) \;=\; V_\text{bulge}^2 + V_\text{thin}^2 + V_\text{thick}^2 + V_\text{gas}^2$$
where each $V_i(R)$ is the standard Newtonian circular velocity of the corresponding mass distribution.
7. Theory-level parameters
The complete BeeTheory framework, as applied to galaxies, contains five parameters at the theory level. These are universal: they do not vary from galaxy to galaxy.
| Symbol | Meaning | Role |
|---|---|---|
| $K_0$ | Wave-mass amplitude | Sets the dimensionless scale of the wave kernel |
| $c_\text{sph}$ | 3D geometric constant | Ratio $\ell/r_\text{scale}$ for spherical sources (bulge) |
| $c_\text{disk}$ | 2D geometric constant | Ratio $\ell/R_\text{scale}$ for disk and ring sources |
| $c_\text{arm}$ | Spiral geometric constant | Ratio $\ell/R_d$ for the azimuthally concentrated arm excess |
| $\lambda$ | Global wave-field coupling | Scales the total wave-field density |
Universality of the parameters
All five parameters are global. The same numerical values apply to the Milky Way, to dwarf irregulars, to massive spirals. The galaxy-specific information enters only through the five observational inputs $(T,\,R_d,\,\Sigma_d,\,M_\text{HI},\,\Upsilon_\star)$. The model does not contain any per-galaxy tunable parameter.
8. The unidirectional nature of the computation
An open chain — no feedback
The entire computation flows from inputs to outputs, in one direction. The photometric and 21-cm observations determine the baryonic decomposition. The baryonic decomposition determines the wave-field density. The wave-field density determines the enclosed wave mass. The enclosed wave mass determines the predicted rotation curve. At no point does the rotation curve influence any earlier step of the computation.
This unidirectionality has three important consequences.
(a) Once the five theory-level parameters are fixed, the rotation curve is a strict prediction, not a fit. Comparison with the observed rotation curve is a test, not a calibration.
(b) The model has no mechanism for galaxy-by-galaxy adjustment. Every modification of the rotation-curve prediction must come from a modification of the input vector $(T,\,R_d,\,\Sigma_d,\,M_\text{HI},\,\Upsilon_\star)$ or from a change to the universal theory-level parameters $(K_0,\,c_\text{sph},\,c_\text{disk},\,c_\text{arm},\,\lambda)$.
(c) Calibrating $lambda$ on a reference galaxy is not the same as fitting it to that galaxy’s rotation curve. The calibration determines a single global number; the rotation curve at all other radii of the reference galaxy, and the rotation curves of all other galaxies, are then strict predictions of the calibrated framework.
9. The role of central surface density (Note XI revision)
The diagnostic of Note XI identified that the residual prediction error correlates strongly with the central baryonic surface density $\Sigma_d$, independently of the disk scale length $R_d$. The formalisation presented above is the version of the model before this finding is incorporated — it uses only $R_d$ in the coherence length expressions $\ell_i = c_i\,R_d$.
Where the refinement will enter
In the refined model, the coherence lengths $\ell_i$ will depend on both $R_d$ and $\Sigma_d$, replacing the strict linear relation $\ell_i = c_i\,R_d$ with a function $\ell_i = c_i\,R_d\,\phi(\Sigma_d/\Sigma_\text{ref})$ that absorbs the residual identified in Note XI. The functional form of $phi$ and its parameters will be determined in subsequent notes, first on the 22-galaxy calibration set, then validated by blind prediction on the remaining SPARC sample.
The unidirectional structure of the computation is preserved by this refinement: $\Sigma_d$ is an observational input, the modified coherence lengths feed into the same convolution integrals, and the rotation curve emerges as before. Only one operational link is added — the dependence of $\ell_i$ on a second observable.
10. Summary of the methodology
1. Inputs. Five observables per galaxy: Hubble type $T$, disk scale $R_d$, surface brightness $\Sigma_d$, HI mass $M_\text{HI}$, and the universal stellar mass-to-light ratio $\Upsilon_\star$.
2. Baryonic decomposition. Five components: bulge (if $T \leq 4$), thin disk, thick disk, gas ring, spiral arm excess. Each carries an analytical density profile.
3. Wave kernel. Universal Yukawa-type form $\mathcal{K}_i(D) = K_0\,(1 + \alpha_i D)\,e^{-\alpha_i D}/D^2$ with coherence length $\ell_i = c_i\,R_\text{scale}$ determined by the geometric extent of each component.
4. Convolution. Each component generates a wave-field density via a one-dimensional integral over rings (2D components) or shells (3D bulge). The total wave-field density is the sum of the five components, scaled by the global coupling $\lambda$.
5. Output. The enclosed wave mass $M_\text{wave}(R)$ is integrated and combined with the baryonic velocity $V_\text{bar}(R)$ to yield the predicted rotation curve $V_c(R)$.
6. Theory-level parameters. $(K_0,\,c_\text{sph},\,c_\text{disk},\,c_\text{arm},\,\lambda)$ — universal, no per-galaxy tuning. A refinement under study will add a dependence on $\Sigma_d$.
7. Direction. Inputs → baryons → wave field → rotation curve. No feedback. The rotation curve is a prediction, not a fit.
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). · 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). · Broeils, A. H., Rhee, M.-H. — Short 21-cm WSRT observations of spiral and irregular galaxies, A&A 324, 877 (1997). · McGaugh, S. S. — The third law of galactic rotation, Galaxies 2, 601 (2014). · Bovy, J., Rix, H.-W. — A direct dynamical measurement of the Milky Way’s disk surface density profile, ApJ 779, 115 (2013). · Arnett, D. — Supernovae and Nucleosynthesis, Princeton (1996). · Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023).
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