BeeTheory · Foundations · Technical Note XVII
The Five Geometric Components:
Complete Parameter Inventory
Before extending the BeeTheory framework to large galaxy samples, this note consolidates the modelling layer: for each of the five geometric components used to describe a disk galaxy, it lists explicitly the parameters required, the density profile, the wave-field coherence length, and the integration geometry. This is the operational specification that drives every BeeTheory calculation from Note VII onwards.
1. The result first — at a glance
Per galaxy: 5 observational inputs → 5 baryonic components → wave-field
Each galaxy is described by five observational inputs that drive a five-component baryonic decomposition: bulge (3D), thin disk (2D), thick disk (2D), gas ring (2D with central hole), and spiral arm excess (2D, narrower kernel). Together with four universal theory parameters $(K_0, c_\text{sph}, c_\text{disk}, c_\text{arm})$ and one global coupling $\lambda$, this fully specifies the wave-field computation.
Total parameters: 5 observational inputs + up to 18 derived component parameters + 5 universal theory parameters. No per-galaxy adjustment beyond these.
2. Observational inputs (per galaxy)
| Symbol | Quantity | Source |
|---|---|---|
| $T$ | Hubble morphological type | Catalogue (de Vaucouleurs, SPARC) |
| $R_d$ | Stellar disk scale length (kpc) | Spitzer 3.6 µm photometry |
| $\Sigma_d$ | Central disk surface brightness ($L_\odot/\text{pc}^2$) | Spitzer 3.6 µm photometry |
| $M_\text{HI}$ | Total atomic hydrogen mass ($M_odot$) | 21-cm radio observations |
| $\Upsilon_\star$ | Stellar mass-to-light ratio at 3.6 µm | Fixed universal: $0.5\,M_\odot/L_\odot$ (McGaugh 2014) |
Two integrated mass quantities are computed once from these inputs:
$$M_\star \;=\; 2\pi\,R_d^2\,\Sigma_d\,\Upsilon_\star \qquad\text{(stellar mass)}$$
$$M_\text{gas} \;=\; 1.33\,M_\text{HI} \qquad\text{(gas mass, He correction)}$$
3. Component 1 — Bulge (3D Hernquist)
The bulge is a three-dimensional spherical concentration at the galaxy centre. It is activated only for early- and intermediate-type galaxies. In late-type spirals and irregulars, no bulge is present.
Activation: $T \leq 4$ (S0, Sa, Sb, Sbc spirals). Disabled for $T \geq 5$ (Sc, Sd, Im).
| Parameter | Symbol | Formula |
|---|---|---|
| Bulge mass | $M_b$ | $0.20 \cdot M_\star$ |
| Scale radius | $r_b$ | $\max(0.5\,R_d,\;0.3\text{ kpc})$ |
| Coherence length | $\ell_b$ | $c_\text{sph} \cdot r_b$ |
Density profile
$$\rho_b(r) \;=\; \frac{M_b\,r_b}{2\pi\,r\,(r + r_b)^3}$$
Wave-field integration — spherical shells
$$\rho_\text{wave}^{(b)}(r) \;=\; \int_0^{6r_b} \rho_b(r’) \cdot K_0\frac{(1 + \alpha_b D)\,e^{-\alpha_b D}}{D^2} \cdot 4\pi r’^2\,dr’, \quad D = \sqrt{r^2 + r’^2}, \quad \alpha_b = 1/\ell_b$$
Parameter count: 3 ($M_b$, $r_b$, $\ell_b$) when activated, 0 otherwise.
4. Component 2 — Thin stellar disk (2D exponential)
The thin disk holds the bulk of the stellar mass that is not in the bulge. It is the geometrically thinnest stellar component, with the smallest vertical extent. Always activated.
| Parameter | Symbol | Formula |
|---|---|---|
| Thin disk mass | $M_\text{thin}$ | $0.75 \cdot (M_\star – M_b)$ |
| Scale length | $R_d$ | Observed (input) |
| Coherence length | $\ell_\text{thin}$ | $c_\text{disk} \cdot R_d$ |
Density profile and integration
$$\Sigma_\text{thin}(R) \;=\; \frac{M_\text{thin}}{2\pi\,R_d^2}\,e^{-R/R_d}$$
$$\rho_\text{wave}^{(\text{thin})}(r) \;=\; \int_0^{8R_d} \Sigma_\text{thin}(R’) \cdot K_0\frac{(1 + \alpha_\text{thin} D)\,e^{-\alpha_\text{thin} D}}{D^2} \cdot 2\pi R’\,dR’$$
Parameter count: 3 ($M_\text{thin}$, $R_d$, $\ell_\text{thin}$). Integration over concentric rings $R’$.
5. Component 3 — Thick stellar disk (2D exponential, wider)
The thick disk is composed of older, dynamically warmer stars distributed over a wider radial scale than the thin disk. Always activated. Carries 25% of the non-bulge stellar mass.
| Parameter | Symbol | Formula |
|---|---|---|
| Thick disk mass | $M_\text{thick}$ | $0.25 \cdot (M_\star – M_b)$ |
| Scale length | $R_\text{thick}$ | $1.5 \cdot R_d$ |
| Coherence length | $\ell_\text{thick}$ | $c_\text{disk} \cdot R_\text{thick} = 1.5\,c_\text{disk}\,R_d$ |
Density profile and integration
$$\Sigma_\text{thick}(R) \;=\; \frac{M_\text{thick}}{2\pi\,R_\text{thick}^2}\,e^{-R/R_\text{thick}}$$
$$\rho_\text{wave}^{(\text{thick})}(r) \;=\; \int_0^{8R_\text{thick}} \Sigma_\text{thick}(R’) \cdot K_0\frac{(1 + \alpha_\text{thick} D)\,e^{-\alpha_\text{thick} D}}{D^2} \cdot 2\pi R’\,dR’$$
Parameter count: 3 ($M_\text{thick}$, $R_\text{thick}$, $\ell_\text{thick}$). Same ring geometry as the thin disk.
6. Component 4 — Gas ring (HI + He, 2D with central hole)
The neutral atomic gas of the galaxy (with helium correction) is distributed over a wider scale than the stellar disk and is centrally depleted. It is the most extended baryonic component, generally extending well beyond the optical disk.
| Parameter | Symbol | Formula |
|---|---|---|
| Gas mass | $M_\text{gas}$ | $1.33 \cdot M_\text{HI}$ |
| Gas scale length | $R_g$ | $1.7 \cdot R_d$ (Broeils & Rhee 1997) |
| Central hole radius | $R_\text{hole}$ | $0.5 \cdot R_g$ |
| Coherence length | $\ell_\text{gas}$ | $c_\text{disk} \cdot R_g = 1.7\,c_\text{disk}\,R_d$ |
Density profile and integration
$$\Sigma_\text{gas}(R) \;=\; \frac{M_\text{gas}}{2\pi\,R_g^2}\,\exp\!\left(-\frac{R_\text{hole}}{R} – \frac{R}{R_g}\right)$$
$$\rho_\text{wave}^{(\text{gas})}(r) \;=\; \int_{R_\text{hole}}^{8R_g} \Sigma_\text{gas}(R’) \cdot K_0\frac{(1 + \alpha_\text{gas} D)\,e^{-\alpha_\text{gas} D}}{D^2} \cdot 2\pi R’\,dR’$$
The double-exponential form captures both the central depletion (the $-R_\text{hole}/R$ term suppresses the profile at small $R$, where neutral hydrogen is typically photoionised or in molecular form) and the outer decline (the $-R/R_g$ term). The lower bound of integration starts at $R_\text{hole}$, where the profile becomes non-negligible.
Parameter count: 4 ($M_\text{gas}$, $R_g$, $R_\text{hole}$, $\ell_\text{gas}$). Ring integration with truncated inner radius.
7. Component 5 — Spiral arm excess (2D, narrower kernel)
The spiral arms are an azimuthal modulation of the thin disk surface density. They are treated, in the axisymmetric BeeTheory monopole approximation, as an effective uniform enhancement of the thin-disk profile at the 10% level, but with a distinct coherence length that reflects the narrower angular extent of the arm structure compared to a smooth disk.
| Parameter | Symbol | Formula |
|---|---|---|
| Arm effective mass | $M_\text{arm}$ | $0.10 \cdot M_\text{thin}$ |
| Radial scale | $R_d$ | Follows thin disk |
| Coherence length | $\ell_\text{arm}$ | $c_\text{arm} \cdot R_d$ (narrower than $\ell_\text{thin}$) |
Density profile and integration
$$\Sigma_\text{arm}(R) \;=\; 0.10 \cdot \Sigma_\text{thin}(R) \;=\; \frac{0.10\,M_\text{thin}}{2\pi\,R_d^2}\,e^{-R/R_d}$$
$$\rho_\text{wave}^{(\text{arm})}(r) \;=\; \int_0^{8R_d} \Sigma_\text{arm}(R’) \cdot K_0\frac{(1 + \alpha_\text{arm} D)\,e^{-\alpha_\text{arm} D}}{D^2} \cdot 2\pi R’\,dR’$$
Because $c_\text{arm} < c_\text{disk}$, the spiral arm kernel is more localised than the thin disk kernel — the field is enhanced at short separations but exponentially damped beyond a few kpc. This captures the fact that real spiral arms produce intense local gravitational features but do not extend coherence over the entire disk.
Parameter count: 3 ($M_\text{arm}$, $R_d$, $\ell_\text{arm}$). Same ring geometry as the thin disk.
8. Summary table — all components at once
| # | Component | Geometry | Mass | Radial scale | Coherence $\ell$ | Activation | Params |
|---|---|---|---|---|---|---|---|
| 1 | Bulge | 3D Hernquist sphere | $0.20\,M_\star$ | $r_b = \max(0.5R_d,\,0.3)$ | $c_\text{sph}\,r_b$ | $T \leq 4$ | 3 |
| 2 | Thin disk | 2D exponential | $0.75\,(M_\star – M_b)$ | $R_d$ | $c_\text{disk}\,R_d$ | Always | 3 |
| 3 | Thick disk | 2D exponential | $0.25\,(M_\star – M_b)$ | $1.5\,R_d$ | $1.5\,c_\text{disk}\,R_d$ | Always | 3 |
| 4 | Gas ring | 2D exp. with central hole | $1.33\,M_\text{HI}$ | $1.7\,R_d$, $R_\text{hole} = 0.85\,R_d$ | $1.7\,c_\text{disk}\,R_d$ | Always | 4 |
| 5 | Spiral arms | 2D azimuthal excess | $0.10\,M_\text{thin}$ | $R_d$ (follows thin) | $c_\text{arm}\,R_d$ | Always | 3 |
9. Universal theory parameters (identical for all galaxies)
Five numbers fix the BeeTheory wave kernel. They are universal — the same values apply to the Milky Way, to dwarfs, to massive spirals. They do not vary from galaxy to galaxy and are determined once on a calibration sample.
| Symbol | Value | Role |
|---|---|---|
| $K_0$ | $0.3759$ | Wave-mass amplitude — sets the dimensionless scale of the kernel |
| $c_\text{sph}$ | $0.41$ | 3D coherence ratio: $\ell_b / r_b$ for spherical (bulge) sources |
| $c_\text{disk}$ | $3.17$ | 2D coherence ratio: $\ell / R_\text{scale}$ for disks and gas ring |
| $c_\text{arm}$ | $2.0$ | Spiral coherence ratio: narrower kernel for arm modulation |
| $\lambda$ | $0.4957$ | Global wave-field coupling (scales the total wave density) |
The wave kernel itself, identical for every component, is:
$$\mathcal{K}(D) \;=\; K_0 \cdot \frac{(1 + \alpha D)\,e^{-\alpha D}}{D^2}, \qquad \alpha = 1/\ell$$
10. From components to rotation curve
The total wave-field density at radius $r$ is the sum of the five component contributions, scaled by the global coupling:
$$\rho_\text{wave}(r) \;=\; \lambda \cdot \!\!\sum_{i \in \{b,\text{thin},\text{thick},\text{gas},\text{arm}\}}\!\!\rho_\text{wave}^{(i)}(r)$$
The enclosed wave mass and predicted circular velocity follow:
$$M_\text{wave}(R) \;=\; \int_0^R 4\pi r^2 \rho_\text{wave}(r)\,dr$$
$$V_c^2(R) \;=\; V_\text{bar}^2(R) \;+\; \frac{G\,M_\text{wave}(R)}{R}$$
where $V_\text{bar}(R)$ is the Newtonian circular velocity of the visible baryons (Freeman 1970 formula for each exponential disk component, Hernquist enclosed mass for the bulge, all combined in quadrature).
11. Parameter accounting — summary
Per galaxy, what enters and what is derived
Observational inputs (per galaxy): 5 quantities ($T$, $R_d$, $\Sigma_d$, $M_\text{HI}$, $\Upsilon_\star$).
Derived component parameters (per galaxy): 13 if $T > 4$ (no bulge), 16 if $T \leq 4$ (with bulge). All computed from the 5 inputs above by deterministic formulas.
Universal theory parameters: 5 numbers ($K_0$, $c_\text{sph}$, $c_\text{disk}$, $c_\text{arm}$, $\lambda$). Identical for every galaxy.
Free per-galaxy fit parameters: $\mathbf{0}$. The model has no galaxy-by-galaxy adjustment.
12. Summary
1. Each galaxy is described by five geometric components: a bulge (3D Hernquist, optional), a thin stellar disk, a thick stellar disk, a gas ring with central hole, and a spiral arm excess (all four latter are 2D exponentials).
2. The bulge is activated only for $T \leq 4$ (S0 through Sbc). The four 2D components are always present.
3. Each component contributes a separate integral to the wave-field density: spherical shell integration for the bulge, ring integration for the four 2D components.
4. The number of derived parameters per galaxy is at most 16 (with bulge) or 13 (without bulge), all computed deterministically from 5 observational inputs.
5. Five universal theory parameters $(K_0, c_text{sph}, c_text{disk}, c_text{arm}, lambda)$ are identical for all galaxies — they are not adjusted per galaxy.
6. No per-galaxy free parameter exists in the model. Once $lambda$ is fixed on a calibration sample, all subsequent rotation curves are pure predictions.
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). · Hernquist, L. — An analytical model for spherical galaxies and bulges, ApJ 356, 359 (1990). Bulge density profile. · Freeman, K. C. — On the disks of spiral and S0 galaxies, ApJ 160, 811 (1970). Exponential disk circular velocity. · Broeils, A. H., Rhee, M.-H. — Short 21-cm WSRT observations of spiral and irregular galaxies, A&A 324, 877 (1997). Gas-to-stellar disk scale ratio. · McGaugh, S. S. — The third law of galactic rotation, Galaxies 2, 601 (2014). $\Upsilon_\star$ at 3.6 µm. · Bland-Hawthorn, J., Gerhard, O. — The Galaxy in Context, ARA&A 54, 529 (2016). Milky Way structural decomposition. · Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023).
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