BeeTheory · Full Geometric Decomposition · 5 Components · SPARC 2025
From Differential Elements to Dark Mass:
Five Geometric Components,
20 SPARC Galaxies
Thin disk, thick disk, Hernquist bulge, HI gas ring, spiral arm excess — each with its own differential element (\(dA\), \(dV\)) and BeeTheory Yukawa kernel. One universal coupling \(K_0 = 0.3759\). Result: 16/20 galaxies within 20% of \(V_f\).
0. Results — first
5-component BeeTheory — 20 SPARC galaxies, K₀ = 0.3759 universal
Adding the spiral arm contribution — modelled as an azimuthally-averaged BeeTheory source of excess surface density proportional to the arm amplitude (A_s) — brings 16/20 galaxies within 20% of the observed flat rotation velocity, with median error 10.8% on the 18 core galaxies. The coupling constant \(K_0 = 0.3759\) is identical to all previous fits.
The spiral arm dark field contributes 5–15% of total dark mass in typical spirals (Sc–Sb), and negligibly in irregulars and pure-gas dwarfs. It is the first quantitative prediction of how non-axisymmetric structure generates BeeTheory dark mass.
16 / 20
Within 20% of \(V_f\)
10.8%
Median error (core 18)
K₀ = 0.3759
Universal — unchanged
Chart notes
(V_text{BT}) vs (V_f) — 5-component BeeTheory prediction, 20 SPARC galaxies.
- Within 20%: 16 galaxies
- 20–30%: boundary cases
- Outliers: CamB and NGC3741
- Perfect reference: 1:1 prediction line
- Band: ±20%
The original HTML included interactive Chart.js graphs. They have been converted into static explanatory notes and data tables so the page can be pasted safely into WordPress Gutenberg without embedded JavaScript.
1. Differential elements — the geometric foundation
Every BeeTheory dark density integral is built from a differential mass element (dM) of a source component. The element type depends on the geometry: annular ring (2D), spherical shell (3D), or arc segment (spiral). The Yukawa kernel is then applied to each \(dM\).
Differential ring element — 2D disk and gas ring
$$dM_\text{ring}(R) = \Sigma(R)\cdot 2\pi R\,dR, \qquad \text{with }D=\sqrt{r^2+R^2}$$
The ring at radius \(R\) and width \(dR\) has area \(dA = 2\pi R\,dR\). Every mass element \(\Sigma(R)\cdot dA\) contributes to the dark density at field point \(r\) via the BeeTheory kernel at distance \(D\). For a monopole approximation (azimuthal average), \(D = \sqrt{r^2+R^2}\) — the distance from field point to ring centre.
Differential shell element — 3D spherical bulge
$$dM_\text{shell}(r') = \rho(r')\cdot 4\pi r'^2\,dr', \qquad \text{with }D=\sqrt{r^2+r'^2}$$
The spherical shell at radius \(r’\) and thickness \(dr’\) has volume \(dV = 4\pi r’^2\,dr’\). For a spherically symmetric source (Hernquist bulge), the azimuthal average of the BeeTheory kernel reduces to using \(D = \sqrt{r^2+r’^2}\) (monopole). This is exact for \(r \neq r’\).
Differential arc element — spiral arm excess
$$dM_\text{arm}(R,\phi) = A_s\,\Sigma_\text{disk}(R)\cos\bigl(m[\phi-\phi_s(R)]\bigr)\cdot R\,d\phi\,dR$$
Azimuthal average: \(\langle\cos(m[\phi-\phi_s])\rangle = 0\), so arms contribute zero to the azimuthally-averaged rotation curve. However, they do contribute to the BeeTheory dark field via the excess surface density: the arm peaks ((cos=+1)) generate a locally stronger wave field. The effective extra source mass per ring is (langle|deltaSigma|rangle = (A_s/pi)Sigma_text{disk}(R)), modelled with a shorter coherence length (c_text{arm} = 2.0) (arms are azimuthally concentrated).
2. Five geometric components — formulas and BeeTheory equations
① Thin stellar disk — 2D ring integral
Exponential disk \(\Sigma_\text{thin}(R) = \Sigma_{0,t}\,e^{-R/R_d}\)
Geometry: flat 2D disk, rings from \(R=0\) to \(8R_d\). Scale: \(R_d\) from SPARC photometry.
Mass fraction: \(0.75\times(1-f_b)\times M_\star\). Scale: \(R_{d,t} = R_d\).
K_thin = K₀/Rd, ℓ_thin = c_disk·Rd = 3.17·Rd
② Thick stellar disk — 2D ring integral
Exponential disk \(\Sigma_\text{thick}(R) = \Sigma_{0,k}\,e^{-R/1.5R_d}\)
Geometry: flat 2D disk, more extended. Scale: \(R_{d,k} = 1.5\,R_d\).
Mass fraction: \(0.25\times(1-f_b)\times M_\star\). Represents old kinematically hot population.
K_thick = K₀/1.5Rd, ℓ_thick = 3.17·1.5Rd
③ Hernquist bulge — 3D spherical shell integral
\(\rho_b(r) = \dfrac{M_b}{2\pi}\dfrac{a}{r(r+a)^3}\), \(\;M_b(<r) = M_b\dfrac{r^2}{(r+a)^2}\)
Geometry: 3D spherical, shells from \(r=0\) to \(6a\). Scale: \(a = \max(0.5R_d, 0.3\,\text{kpc})\).
Mass: \(f_b(T)\times M_\star\). Hubble-type dependent. Uses shorter coherence \(c_\text{sph} = 0.41\).
K_bulge = K₀/a, ℓ_bulge = 0.41·a
④ HI gas ring — 2D ring integral, central hole
\(\Sigma_g(R) \propto \exp\!\left(-\dfrac{R_m}{R} – \dfrac{R}{R_g}\right)\), \(\;R_m=0.5R_g\)
Geometry: 2D annular ring with central depression. Scale: \(R_g = 1.7\,R_d\). Peak at \(R\approx\sqrt{R_m R_g}\).
Mass: \(M_\text{gas} = 1.33\,M_\text{HI}\) (includes He). Often \(>M_\star\) in dwarfs.
K_gas = K₀/Rg, ℓ_gas = 3.17·Rg
⑤ Spiral arms — azimuthal arc excess, 2D
\(\Sigma_\text{arm}(R,\phi) = A_s\,\Sigma_\text{disk}(R)\cos\bigl[m(\phi-\phi_s(R))\bigr]\), \(\;\phi_s(R) = \dfrac{1}{\tan p}\ln\!\dfrac{R}{R_0}\)
Geometry: logarithmic spiral pattern \(r = r_0 e^{b\phi}\) with \(b = \tan p\) (\(p\) = pitch angle). Number of arms \(m=2\) for most spirals. The azimuthal average of the dark field from the arm peaks gives an effective extra ring source with density \(f_\text{sp}\,\Sigma_\text{disk}(R)\), where \(f_\text{sp} = A_s/\pi\) (RMS over arm cross-section). Coherence length \(c_\text{arm} = 2.0\) (arms are concentrated, shorter coherence than the full disk).
Arm parameters by Hubble type: \(A_s = 0.15\)–\(0.60\), \(p = 8°\)–\(30°\), \(f_\text{sp} = 0.08\)–\(0.30\).
K_spiral = K₀/Rd, ℓ_spiral = 2.0·Rd, Σ_source = f_sp·Σ_disk
The complete BeeTheory dark density — 5 sources
Total BeeTheory dark density — superposition of all 5 differential elements
$$\rho_\text{dark}(r) =
\underbrace{K_t\!\int\!\Sigma_t\,e^{-R/R_d}\,\frac{(1+\alpha_t D)e^{-\alpha_t D}}{D^2}2\pi R\,dR}_{\text{thin disk}}
+
\underbrace{K_k\!\int\!\Sigma_k\,e^{-R/R_{dk}}\,\frac{(1+\alpha_k D)e^{-\alpha_k D}}{D^2}2\pi R\,dR}_{\text{thick disk}}$$
$$+
\underbrace{K_b\!\int\!\rho_b(r')\,\frac{(1+\alpha_b D)e^{-\alpha_b D}}{D^2}4\pi r'^2\,dr'}_{\text{Hernquist bulge}}
+
\underbrace{K_g\!\int\!\Sigma_g(R)\,\frac{(1+\alpha_g D)e^{-\alpha_g D}}{D^2}2\pi R\,dR}_{\text{gas ring}}
+
\underbrace{K_t\!\int\!f_\text{sp}\Sigma_t\,e^{-R/R_d}\,\frac{(1+\alpha_\text{sp} D)e^{-\alpha_\text{sp} D}}{D^2}2\pi R\,dR}_{\text{spiral arm excess}}$$
$$D = \sqrt{r^2 + R'^2}\text{ (disk/ring)}, \quad D = \sqrt{r^2+r'^2}\text{ (bulge)}, \quad K_i = \frac{K_0}{R_i}$$
Total circular velocity — baryonic + BeeTheory dark
$$V_c(R) = \sqrt{V_\text{bar}^2(R) + V_\text{dark}^2(R)}, \quad V_\text{dark}(R) = \sqrt{\frac{G\,M_\text{dark}(<R)}{R}}, \quad M_\text{dark}(<R) = \int_0^R 4\pi r^2\,\rho_\text{dark}(r)\,dr$$
3. All parameters — one table
| Symbol | Value | Source | Physical meaning |
|---|---|---|---|
| \(K_0\) | 0.3759 | SPARC 20-galaxy fit | Universal wave-mass coupling (dimensionless) — same for all components and all galaxies |
| \(c_\text{disk}\) | 3.17 | Milky Way two-regime | \(\ell/R\) ratio for disk/ring sources (2D planar geometry) |
| \(c_\text{sph}\) | 0.41 | Milky Way two-regime | \(\ell/R\) ratio for spherical (bulge) sources (3D geometry) |
| \(c_\text{arm}\) | 2.00 | Intermediate estimate | \(\ell/R\) for spiral arms (azimuthally concentrated → shorter coherence than full disk) |
| \(f_t\), \(f_k\) | 0.75, 0.25 | MW thin/thick ratio | Fraction of disk stellar mass in thin/thick disk |
| \(R_{d,k}\) | \(1.5\,R_d\) | Bland-Hawthorn (2016) | Thick disk scale radius relative to thin disk |
| \(a\) | \(\max(0.5R_d, 0.3\,\text{kpc})\) | Standard Hernquist | Bulge scale radius (Hernquist profile) |
| \(R_g\) | \(1.7\,R_d\) | Broeils & Rhee (1997) | HI gas disk scale relative to stellar disk |
| \(R_m\) | \(0.5\,R_g\) | Standard ring model | Controls central HI hole; peak density at \(R \approx \sqrt{R_m R_g}\) |
| \(A_s(T)\) | 0.15–0.60 | Rix & Zaritsky (1995) | Spiral arm amplitude (fraction of disk surface density) |
| \(p(T)\) | 8°–30° | Davis et al. (2012) | Spiral pitch angle; \(b = \tan p\), \(r = r_0 e^{b\phi}\) |
| \(f_\text{sp}(T)\) | 0.08–0.30 | Derived: \(\approx A_s/2\) | Effective dark-field excess from arms |
| \(f_b(T)\) | 0–40% | Morphological | Bulge mass fraction by Hubble type \(T\) |
| \(\Upsilon_\star\) | \(0.5\,M_\odot/L_\odot\) | McGaugh (2014) | Stellar mass-to-light ratio at 3.6 µm |
Spiral arm parameters by Hubble type
| Type | Class | \(A_s\) | \(m\) | \(p\) (°) | \(f_\text{sp}\) | Note |
|---|---|---|---|---|---|---|
| T ≤ 1 | Sa | 0.15 | 2 | 8 | 0.08 | Weak, tightly wound arms |
| T = 2–3 | Sb | 0.25–0.35 | 2 | 12–15 | 0.12–0.18 | Moderate spiral structure |
| T = 4–5 | Sc | 0.45–0.55 | 2 | 18–20 | 0.22–0.28 | Strong, open arms |
| T = 6–7 | Sd | 0.50–0.60 | 2 | 22–25 | 0.25–0.30 | Flocculent, irregular |
| T ≥ 8 | Sm/Im | 0.20–0.40 | 2 | 28–30 | 0.10–0.20 | Very open, weak arms |
4. Galaxy-by-galaxy results
| Galaxy | \(R_d\) | \(T\) | \(f_b\) | \(f_\text{gas}\) | \(A_s\) | \(p°\) | \(V_f\) | \(V_\text{bar}\) | \(V_\text{thin}\) | \(V_\text{thick}\) | \(V_\text{bulge}\) | \(V_\text{gas}\) | \(V_\text{arm}\) | \(V_\text{BT}\) | Error | ✓ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| CamB | 0.47 | 10 | — | 32% | 0.2 | 30° | 2 | 10 | 24 | 11 | 0 | 13 | 6 | 32 | +1496.5% | ✗ |
| D631-7 | 0.70 | 10 | — | 74% | 0.2 | 30° | 58 | 26 | 39 | 18 | 0 | 51 | 10 | 72 | +24.8% | ~ |
| NGC0024 | 1.33 | 5 | — | 35% | 0.55 | 20° | 84 | 25 | 58 | 26 | 0 | 32 | 26 | 80 | −4.5% | ✓ |
| NGC0100 | 2.30 | 6 | — | 30% | 0.6 | 22° | 83 | 27 | 64 | 29 | 0 | 32 | 30 | 87 | +5.2% | ✓ |
| NGC0247 | 2.40 | 7 | — | 54% | 0.5 | 25° | 90 | 32 | 63 | 29 | 0 | 52 | 27 | 96 | +7.1% | ✓ |
| NGC0300 | 1.50 | 7 | — | 54% | 0.5 | 25° | 76 | 28 | 55 | 25 | 0 | 45 | 23 | 83 | +9.4% | ✓ |
| NGC0801 | 5.80 | 5 | 5% | 32% | 0.55 | 20° | 208 | 62 | 141 | 64 | 8 | 75 | 63 | 193 | −7.1% | ✓ |
| NGC0891 | 4.10 | 3 | 20% | 32% | 0.35 | 15° | 212 | 54 | 114 | 52 | 14 | 67 | 41 | 159 | −25.2% | ~ |
| NGC0925 | 3.10 | 7 | — | 75% | 0.5 | 25° | 105 | 44 | 65 | 29 | 0 | 85 | 28 | 122 | +16.5% | ✓ |
| NGC2403 | 1.80 | 6 | — | 60% | 0.6 | 22° | 131 | 43 | 80 | 36 | 0 | 74 | 37 | 128 | −2.6% | ✓ |
| NGC2841 | 3.50 | 3 | 20% | 32% | 0.35 | 15° | 278 | 85 | 180 | 81 | 22 | 105 | 64 | 248 | −10.7% | ✓ |
| NGC2903 | 2.60 | 4 | 12% | 38% | 0.45 | 18° | 184 | 55 | 116 | 53 | 11 | 73 | 46 | 164 | −11.0% | ✓ |
| NGC2976 | 0.75 | 5 | — | 29% | 0.55 | 20° | 80 | 24 | 56 | 25 | 0 | 27 | 25 | 76 | −5.7% | ✓ |
| NGC3031 | 2.30 | 2 | 30% | 37% | 0.25 | 12° | 210 | 65 | 125 | 56 | 20 | 88 | 36 | 180 | −14.2% | ✓ |
| NGC3198 | 3.14 | 5 | — | 71% | 0.55 | 20° | 151 | 60 | 95 | 43 | 0 | 113 | 43 | 171 | +13.0% | ✓ |
| NGC3521 | 2.80 | 4 | 12% | 59% | 0.45 | 18° | 225 | 70 | 124 | 56 | 12 | 120 | 49 | 200 | −11.0% | ✓ |
| NGC3621 | 2.10 | 7 | — | 82% | 0.5 | 25° | 149 | 64 | 82 | 37 | 0 | 132 | 35 | 176 | +18.2% | ✓ |
| NGC3741 | 0.68 | 10 | — | 73% | 0.2 | 30° | 51 | 33 | 51 | 23 | 0 | 64 | 14 | 92 | +80.9% | ✗ |
| NGC4013 | 2.20 | 3 | 20% | 43% | 0.35 | 15° | 185 | 60 | 117 | 53 | 14 | 86 | 42 | 172 | −7.2% | ✓ |
| NGC4051 | 1.90 | 4 | — | 28% | 0.45 | 18° | 110 | 39 | 93 | 42 | 0 | 44 | 37 | 123 | +11.9% | ✓ |
Legend: ✓ = within 20%; ~ = intermediate boundary case; ✗ = strong structural outlier. Velocities are in km/s.
5. The spiral arm result — new BeeTheory prediction
The spiral arm contribution to dark mass is a genuinely new prediction of BeeTheory — no standard dark matter model makes a specific prediction about how non-axisymmetric baryonic structure affects the dark halo. In BeeTheory, the answer is direct: every mass element emits a wave field, including mass concentrated in spiral arms. The arm excess surface density generates an additional dark field with a shorter coherence length (the arms are spatially concentrated in the azimuthal direction).
BeeTheory spiral arm prediction — testable
At fixed \(R_d\), \(M_\star\), and \(M_\text{gas}\): galaxies with stronger arms (\(A_s\) larger) should have slightly more dark mass, and this excess should be concentrated at radii \(\sim 2\)–\(4\,R_d\) (where the arm dark field is strongest). This predicts a weak correlation between arm amplitude and the “dark mass excess” after subtracting the axisymmetric disk+gas+bulge contribution. This is testable with SPARC full rotation curves (not just \(V_f\)).
In particular: flocculent spirals (Sd, \(T=7\)) with \(A_s\approx 0.5\) should have \(\sim 5\)–\(8\%\) more dark mass than a smooth exponential disk galaxy of the same \(M_\star\) and \(R_d\). This is within observational reach with existing SPARC data.
Why the spiral arms don’t affect the azimuthal average of \(V_c\)
The azimuthal average of \(\cos(m[\phi-\phi_s(R)])\) is zero: the spiral density perturbation cancels when integrated over \(\phi\). This means spiral arms do not change the mean rotation curve — but they do change the BeeTheory dark field. Why? Because the BeeTheory kernel ((1+alpha D)e^{-alpha D}/D^2) is non-linear in the source distribution: a locally enhanced surface density in the arm regions generates a stronger local dark field than the same mass distributed uniformly. The non-linearity of the convolution is what makes spiral arms relevant for BeeTheory but not for classical gravitational dynamics.
References and data
Data: Lelli, McGaugh, Schombert, AJ 152, 157 (2016). Spiral arm parameters: Rix & Zaritsky (1995), Davis et al. (2012). HI disk scaling: Broeils & Rhee (1997). Thick disk: Bland-Hawthorn & Gerhard (2016). BeeTheory: Dutertre (2023), extended 2025.