BeeTheory v3 · Adaptive Gas Geometry · 159 SPARC Galaxies · 2025

Gas Fraction Drives
the Gas Scale.
81% Within 20%.

Conclusion — v3, adaptive gas geometry, 159 galaxies

Replacing a fixed gas scale with one that transitions smoothly from stellar-tied ($1.7\,R_d$) to HI mass-based ($R_\text{HI}/6.1$) as gas fraction increases — controlled by a sigmoid with centre $f_\text{gas}=0.68$ — improves the prediction from 74% → 81% within 20% of $V_f$. Only 4 galaxies exceed 50% error, all ultra-compact pure-gas dwarfs with $R_d < 0.7\,\text{kpc}$.

The Pearson correlation between predicted and observed velocities jumps to r = 0.966 from 0.941. The median error drops to 10.4%. This is achieved by adding two physically motivated parameters: the gas fraction at the stellar-to-HI transition ($w_c = 0.68$) and the HI effective scale factor ($f_f = 6.1$).

v1 — Fixed $R_g = 1.7R_d$

74%within 20%

11.3%median error

r = 0.941Pearson

5outliers >50%

v2 — $\max(R_\text{HI}/11.9,\,1.7R_d)$

74%within 20%

11.0%median error

r = 0.943Pearson

4outliers >50%

v3 — Sigmoid blend

81%within 20%

10.4%median error

r = 0.966Pearson

4outliers >50%

81%Within 20%
128 / 159

10.4%Median error
↓ from 11.3%

r = 0.966Pearson r
↑ from 0.941

90%Q=1 quality
36 / 40 within 20%

96%Within 35%
152 / 159

4Outliers >50%
all $R_d < 0.7$ kpc

1. The prediction — 159 galaxies

$V_\text{BT}$ vs $V_f$ — v3 adaptive gas, 159 SPARC galaxies (log–log). Hover for details.

Within 20% (128) 20–50% (27) >50% outliers (4) 1:1 ±20%

Error distribution — v1 vs v3 comparison

v3 — adaptive gas v1 — fixed $1.7R_d$

2. The key change — adaptive gas scale

The single innovation in v3 is a gas ring scale $R_g$ that depends on the gas fraction $f_\text{gas}$. For stellar-dominated galaxies ($f_\text{gas} \ll 0.68$) it reduces to the old $1.7\,R_d$. For gas-dominated galaxies ($f_\text{gas} \gg 0.68$) it transitions smoothly to a scale derived from the HI mass–radius relation.

v3 adaptive gas scale — the key formula $$R_g = (1-w)\cdot 1.7\,R_d \;+\; w\cdot \frac{R_\text{HI}}{f_f}$$ $$w = \frac{1}{1+e^{-k(f_\text{gas}-w_c)}}$$ $$w_c = 0.678, \qquad f_f = 6.09, \qquad k = 10$$ $$\log_{10}(R_\text{HI}/\text{kpc}) = 0.506\,\log_{10}(M_\text{HI}/M_\odot) – 3.293$$

Physical interpretation of the three parameters

$w_c = 0.678$: the gas fraction where the gas source transitions from stellar-disk tied to HI-mass based. Below this value, the stellar disk defines the gas ring scale. Above it, the HI extent dominates.

$f_f = 6.09$: converts the Wang et al. outer HI radius, measured at the $1\,M_\odot/\text{pc}^2$ isodensity, to the effective BeeTheory ring scale.

$k = 10$: controls the transition sharpness. It is fixed, not optimised, and results are stable for $k$ between about 6 and 15.

$R_g / R_d$ vs gas fraction — transition between fixed and HI-based gas geometry

v3: adaptive $R_g/R_d$ v1: fixed $R_g/R_d = 1.7$ Actual galaxies

3. Complete parameter table

Unchanged from v1

$K_0$0.3759

$c_\text{disk}$3.17

$c_\text{sph}$0.41

$c_\text{arm}$2.00

Thin disk fraction$0.75(1-f_b)$

Thick disk scale$R_{d,k} = 1.5R_d$

Bulge scale$\max(0.5R_d,\,0.25)$

Gas mass$1.33\,M_\text{HI}$

$\Upsilon_\star$$0.5\,M_\odot/L_\odot$

New in v3 — gas geometry

Transition centre $w_c$0.678

HI scale factor $f_f$6.09

Sigmoid steepness $k$10.0 (fixed)

Low-$f_\text{gas}$ limit$1.7\,R_d$

High-$f_\text{gas}$ limit$R_\text{HI,Wang}/6.09$

HI law sourceWang et al. 2016

Total free parameters on this 159-galaxy sample: 2

$w_c$ and $f_f$ are the only parameters introduced in v3. All other parameters remain fixed from the Milky Way calibration and the original 20-galaxy SPARC fit.

4. Remaining 4 outliers

Galaxy	$V_f$	$V_\text{BT}$	Error	$f_\text{gas}$	$R_d$	$w$	$R_g$	Why
DDO064	26	44	+70%	0.85	0.33	0.84	1.12	Ultra-compact. $R_d = 0.33$ kpc. HI surface density profile needed.
KK98-251	17	31	+83%	0.74	0.30	0.65	0.51	Extremely small galaxy. At $V_f=17$ km/s, measurement uncertainty is large.
ESO444-G084	27	45	+66%	0.74	0.55	0.64	0.99	Gas-dominated irregular. No exponential stellar disk.
NGC3741	51	77	+52%	0.72	0.68	0.62	1.85	Very extended HI relative to stars. Per-galaxy 21 cm profile needed.

Why these 4 cannot be fixed by scaling alone

All four outliers have $R_d < 0.7\,\text{kpc}$, $f_\text{gas} > 0.70$, and irregular morphologies. In these systems, the exponential stellar disk model has limited physical meaning: the galaxy is dominated by gas. The correct BeeTheory source is the measured HI surface density profile $\Sigma_\text{HI}(R)$ from 21 cm maps.

v3 compared to MOND and NFW on the same sample

MOND achieves roughly 85% within a factor of 1.5 on SPARC using one free parameter, $a_0$. BeeTheory v3 achieves 96% within 35%, using adaptive gas geometry while keeping the fundamental coupling $K_0$ fixed across galaxy types and scales.

Data: Lelli et al. AJ 152, 157 (2016) · HI radius: Wang et al. MNRAS 460, 2143 (2016) · HI scale: Swaters et al. (2009) · BeeTheory: Dutertre (2023), extended 2025 · $K_0$, $c_\text{disk}$, $c_\text{sph}$ fixed from MW + original 20-galaxy calibration · $w_c$, $f_f$ fitted on 159-galaxy sample