BeeTheory · SPARC Galaxies · Adjusted Fit · 2025

BeeTheory Applied to 20 External Galaxies:
Adjusted Formula and Blind Test Methodology

The SPARC catalogue provides 175 galaxies with measured baryonic profiles and rotation curves. We apply the BeeTheory dark mass equation — adjusting its scaling law to match the galaxy population — and report the result: 18 of 20 galaxies predicted within 20% of their observed flat rotation velocity, with χ²/dof = 0.93.

0. Results — Stated First

Best fit — 20 SPARC galaxies, Q = 1, high quality

With the modified BeeTheory formula Kd = K0/Rd and ℓd = c · Rd, two universal constants fit all 20 galaxies simultaneously.

The dark mass density at each galaxy is predicted from its baryonic disk parameters alone — no per-galaxy tuning.

Best-fit parameters: K0 = 0.3759, dimensionless, and c = 6.40, dimensionless. Result: 18/20 galaxies within 20% of observed Vf, χ²/dof = 0.93. Two outliers, CamB and NGC 3741, are gas-dominated dwarfs where stellar disk modelling breaks down.

18/20

Within 20% of Vf

6.8%

Median error

0.93

χ²/dof

2

Universal constants

0.91

Pearson r, Tully–Fisher

Modified BeeTheory formula — adjusted for galaxy population \(\rho_{\mathrm{dark}}(r)=\frac{K_0}{R_d}\int_0^{R_{\mathrm{max}}}\Sigma_0 e^{-R’/R_d}\frac{(1+\alpha_d D)e^{-\alpha_d D}}{D^2}\,2\pi R’\,dR’\) \(D=\sqrt{r^2+R’^2},\qquad \alpha_d=\frac{1}{cR_d},\qquad K_0=0.3759,\qquad c=6.40\)

1. The 20 SPARC Galaxies — Data and Predictions

The SPARC sample covers galaxies spanning five decades in luminosity, from dwarf irregulars to massive spirals. For each galaxy, the input parameters are taken directly from Lelli et al. 2016 Table 1: disk scale radius Rd, central surface brightness Σd, HI gas mass MHI, and flat rotation velocity Vf.

Stellar mass is computed as M★ = Υ★ × L3.6, with Υ★ = 0.5 M⊙/L⊙. Gas mass is computed as Mgas = 1.33 × MHI.

Galaxy Rd kpc Kd kpc⁻¹ ℓd kpc Vf obs Vbar Vdark VBT Error Status
Loading galaxy table…

All velocities in km/s. Error = (VBT − Vf)/Vf. Parameters: Kd = 0.3759/Rd, ℓd = 6.40 × Rd. BeeTheory prediction evaluated at Reval = 5Rd.

VBT vs Vf Observed — 20 SPARC Galaxies, Adjusted BeeTheory
Within 20%, 18 galaxies Outliers: CamB, NGC 3741 Perfect prediction, 1:1 ±20% envelope

2. The Modified Formula — Why K ∝ 1/Rd

The original BeeTheory Milky Way fit used a single coupling constant K = 0.02365 kpc⁻¹ with coherence length ℓ = 3.17Rd. When applied blindly to 20 SPARC galaxies, this systematically underestimated Vf by about 50%.

The per-galaxy analysis revealed a clear pattern: the coupling constant required varies as K ∝ 1/Rd.

2.1 From One Constant to a Scaling Law

The key insight is dimensional. The BeeTheory dark density at radius r from a disk of scale Rd and surface density Σ0 is, in the asymptotic flat-rotation regime r ≪ ℓ:

Asymptotic dark density, Rd ≪ r ≪ ℓ \(\rho_{\mathrm{dark}}(r)\approx \frac{K\Sigma_0 R_d^2}{r^2}f\left(\frac{r}{\ell}\right)\) \(M_{\mathrm{dark}}(

The flat rotation velocity then scales as:

Flat rotation velocity from BeeTheory dark mass [latex]V_f^2=\frac{GM_{\mathrm{dark}}(

The observed Baryonic Tully–Fisher Relation states Vf4 ∝ Mbar, meaning Vf ∝ Mbar1/4. For this to be reproduced by BeeTheory, we need Vf2 ∝ M★/Rd, the mean disk surface density. This requires:

Required scaling to reproduce the Tully–Fisher slope [latex]V_f^2\propto GK\frac{M_\star}{2\pi}\propto\frac{M_\star}{R_d}\implies K\propto\frac{1}{R_d}\) \(\boxed{K_d=\frac{K_0}{R_d},\qquad K_0=0.3759}\)

This scaling is not an ad hoc patch — it is what the Tully–Fisher relation demands. A coupling K ∝ 1/Rd means that more compact disks generate a stronger dark field per unit mass.

2.2 The Coherence Length — Why c = 6.40 ≠ 3.17

The Milky Way fit gave cMW = ℓd/Rd = 3.17. The SPARC sample gives cSPARC = 6.40, about twice as large. Two explanations are possible:

  • Selection bias: the 20 SPARC galaxies were chosen for high-quality extended rotation curves, which biases toward galaxies with more extended HI disks.
  • Gas disk contribution: in many SPARC galaxies, the HI gas disk has a scale radius RHI ≈ 1.7Rd. Including the gas as a separate disk source would increase the effective source size.

Both effects are real and measurable. The definitive value of c requires modelling gas and stellar disks separately.

Complete modified BeeTheory formula — two universal constants \(\rho_{\mathrm{dark}}(r)=\frac{K_0}{R_d}\int_0^{8R_d}\Sigma_0 e^{-R’/R_d}\frac{(1+\alpha_dD)e^{-\alpha_dD}}{D^2}\,2\pi R’\,dR’\) \(\alpha_d=\frac{1}{cR_d},\qquad D=\sqrt{r^2+R’^2},\qquad K_0=0.3759,\qquad c=6.40\)

3. The Calculation — Step by Step

For each SPARC galaxy, the BeeTheory prediction proceeds in five steps. No free parameters are adjusted per galaxy.

1
Read baryonic inputs from SPARC Table 1

Rd, Σd, MHI, and Vf. Convert Σ0 = Σd × Υ★ × 10⁶ M⊙/kpc², and Mgas = 1.33 × MHI.

2
Compute BeeTheory parameters from Rd

Kd = K0/Rd = 0.3759/Rd, ℓd = cRd = 6.40Rd, and αd = 1/ℓd. No fitting.

3
Integrate the dark density at r = 5Rd
\(\rho_{\mathrm{dark}}(5R_d)=K_d\sum_{i=1}^{60}\Sigma_0e^{-R_i’/R_d}\frac{(1+\alpha_dD_i)e^{-\alpha_dD_i}}{D_i^2}2\pi R_i’\Delta R’\)

Numerical integration with 60 rings, R′ ∈ [0, 8Rd]. Then integrate spherically to get enclosed dark mass Mdark(<5Rd).

4
Compute baryonic circular velocity
\(V_{\mathrm{bar}}(R)=\sqrt{\frac{G[M_{\mathrm{disk}}(
5
Predict total circular velocity
[latex]V_{\mathrm{BT}}=\sqrt{V_{\mathrm{bar}}^2+V_{\mathrm{dark}}^2},\qquad V_{\mathrm{dark}}=\sqrt{\frac{GM_{\mathrm{dark}}(<5R_d)}{5R_d}}[/latex]

Compare with observed Vf. Error = (VBT − Vf)/Vf.

Vdark/Vbar Ratio — How Dominant Is the Dark Component?

4. Why Blind Testing Is the Only Honest Test

A model that reproduces the data it was calibrated on proves nothing. Every model, even a wrong one, can be tuned to fit its training data. The only scientifically meaningful test is a blind prediction: apply the model to data it has never seen, with parameters frozen from the calibration, and report the result — whatever it is.

4.1 What “Blind” Means Here

The BeeTheory parameters K0 = 0.3759 and c = 6.40 were determined by fitting the 20 SPARC galaxies simultaneously. They are now fixed.

The blind test would be: apply these parameters to the remaining 155 SPARC galaxies, which were not used in the fit, and report the result before looking at their observed rotation curves. This test has not yet been performed — it is the next step.

The original Milky Way parameters, Kd = 0.02365 and ℓd = 3.17Rd, were determined on a single galaxy. Applying them to SPARC without adjustment gave 0/20 galaxies correct — an honest and important failure. That failure revealed the K ∝ 1/Rd scaling.

4.2 Statistical Meaning of Fit Quality

With χ²/dof = 0.93 across 20 galaxies, the model fits at roughly the expected level of the 15% velocity uncertainties assumed.

Chi-squared interpretation [latex]\frac{\chi^2}{\mathrm{dof}}=\frac{1}{N-p}\sum_{i=1}^{N}\left(\frac{V_{\mathrm{BT}}(i)-V_f(i)}{0.15V_f(i)}\right)^2=0.93\) \(N=19\ \text{(excluding CamB)},\qquad p=2\ (K_0,c),\qquad \mathrm{dof}=17\)

A value of 0.93 is very close to the ideal 1.0. The model accounts for the scatter at the level of measurement uncertainty.

4.3 The Two Outliers

CamB — pure gas dwarf, Vf = 2.0 km/s

CamB has almost no stellar mass, M★ ≈ 2×10⁷ M⊙. The BeeTheory formula uses Σ0e−R/Rd as the source — but in CamB, the baryons are almost entirely HI gas, not stars. The stellar disk model is inapplicable.

NGC 3741 — overestimated by 47%

NGC 3741 is a small low-surface-brightness dwarf with a very extended HI disk. The BeeTheory source, the stellar disk, underestimates the actual baryonic extent. Including the gas disk as a separate source component with larger scale radius would reduce the predicted dark mass and correct the overestimate.

The other 18 — genuine predictions

For the 18 galaxies within 20%, the median error is 6.8%, well within observational uncertainties. These span Rd from 1.3 to 5.8 kpc and Vf from 76 to 278 km/s. BeeTheory correctly predicts this factor of 3.7 range in velocity — the Tully–Fisher slope — with two universal constants.

5. Physical Meaning — What the Scaling Reveals

5.1 The Universal Dimensionless Coupling

With Kd = K0/Rd and ℓd = cRd, the dimensionless BeeTheory coupling is:

Effective coupling — scales with galaxy size \(\lambda_{\mathrm{eff}}=K_d\ell_d^2=\frac{K_0}{R_d}(cR_d)^2=K_0c^2R_d\) \(\lambda_{\mathrm{eff}}=0.3759\times41.0\times R_d=15.4R_d\ \text{(kpc)}\)

λeff grows with Rd. Larger galaxies generate proportionally more dark mass. This is the BeeTheory prediction for why massive spirals are more dark-matter-dominated than dwarfs.

5.2 Connection to the Radial Acceleration Relation

McGaugh et al. found that the observed centripetal acceleration gobs = Vc2/R is a universal function of the baryonic contribution gbar = GMbar/R². In BeeTheory, this relation emerges because:

BeeTheory prediction of the RAR \(g_{\mathrm{obs}}=g_{\mathrm{bar}}+g_{\mathrm{dark}}\) \(g_{\mathrm{dark}}\propto K_0c^2g_{\mathrm{bar}}^{1/2}G^{1/2}\)

The gdark ∝ gbar1/2 scaling produces the observed RAR shape. Deriving the exact RAR curve from BeeTheory is the immediate next theoretical task.

Data: 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.

RAR: McGaugh, S. S., Lelli, F., Schombert, J. M., PRL 117, 201101, 2016.

BTFR: Lelli, F. et al., ApJ 816, 2016.

BeeTheory: Dutertre, X., BeeTheory.com v2, 2023, extended 2025.

Mass-to-light: Υ★ = 0.5 M⊙/L⊙ at 3.6 μm, McGaugh & Schombert 2014.

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