BeeTheory · Residual Analysis · 2025

Reading the Residuals:
Why Some Galaxies Are Above and Others Below the Prediction

In the 20-galaxy SPARC fit, 15 galaxies are underestimated and 3 overestimated, excluding the two large outliers. This asymmetry is not random. Two physical parameters drive the pattern — and they point directly to the missing physics in the current BeeTheory model.

0. The Two Groups — Stated First

Above the line — model overestimates V_BT > V_f

BeeTheory predicts more dark mass than the galaxy actually has. The 3 galaxies within the core sample are:

NGC 4051 — +15.8% — low gas fraction, Seyfert AGN
NGC 0100 — +6.7% — low surface brightness edge-on
NGC 0300 — +0.9% — flocculent spiral, isolated

Common trait: low gas fraction, f_gas ≈ 0.37, no bulge, pure stellar disk. The model attributes too much dark mass because it only sees the stellar disk — not the fact that the disk alone cannot generate as much field as assumed.

Below the line — model underestimates V_BT < V_f

BeeTheory predicts less dark mass than the galaxy actually has. This concerns 15 galaxies, including all 7 with bulges:

All 7 bulge galaxies — underestimated without exception
NGC 3621 — −16.2% — very gas-rich, f_gas = 0.82
NGC 3521 — −17.1% — large extended gas disk
NGC 0925, NGC 3198 — gas-rich late spirals

Common traits: either has a bulge, not fully modelled, or high gas fraction, f_gas ≈ 0.50–0.82. The extended HI disk is not included as a BeeTheory source.

1. The Two Physical Drivers

−0.68

Pearson r: error vs gas fraction, non-bulge galaxies

100%

Bulge galaxies underestimated, 7 of 7

+0.04

Pearson r: error vs Σ_d, no signal

Two independent physical effects drive the residuals — one for galaxies with bulges, one for gas-rich systems. Surface density Σ_d has essentially no predictive power for the residuals once bulge presence is controlled for.

Driver 1 — Bulge Presence

Every galaxy with a detected bulge is underestimated. The current model assigns 15% of stellar mass to the bulge — a rough guess. The actual bulge mass fraction varies from about 5% in late spirals to about 40% in Sa galaxies.

More importantly, the bulge generates a BeeTheory dark field with a short coherence length, ℓ_b ≪ ℓ_d, intense at small r and contributing significantly to V_f at the flat-rotation radius. This is not fully captured in the current model.

In the Milky Way, the two-regime analysis showed that the bulge contributes about 35% of total dark mass at r = 8 kpc, despite having only 18% of the baryonic mass. The same amplification occurs in these galaxies — but the model uses a generic K_b and bulge fraction rather than galaxy-specific values.

Driver 2 — Gas Fraction

In non-bulge galaxies, the residual correlates with gas fraction at r = −0.68. The direction is clear: more gas means the model underestimates the velocity.

The current BeeTheory model uses the stellar disk as source, with scale R_d. But in gas-rich galaxies, the HI disk extends to R_HI ≈ 1.7–3 × R_d. This extended gas disk is a BeeTheory source that was not included.

Its larger scale radius means a larger ℓ_gas and a different dark field profile. When gas dominates the baryonic mass, ignoring the gas disk significantly underestimates the total dark field.

The apparent paradox — why does more gas mean less predicted dark mass?

The model uses K_d = K₀/R_d, where R_d is the stellar disk scale radius. When f_gas is high, the gas disk extends beyond the stellar disk, but the model only sees the stellar disk.

The stellar disk alone generates a dark field calibrated to V_f. In reality, the gas disk also contributes, and since gas extends further, its effective ℓ is larger, creating a different and more extended dark field profile. The model attributes all dark mass to a single stellar source and underestimates the total.

2. Quantitative Analysis — The Correlations

Residual Error vs Gas Fraction f_gas — Non-Bulge Galaxies Only

No bulge — used in correlation Has bulge — excluded from gas correlation Best-fit trend

Sorted Residuals — The Bulge Signal

Has bulge, always underestimated No bulge Zero line

Comparison: Above vs Below — Mean Properties

Property	Overestimated ↑	Underestimated ↓	Interpretation
N galaxies	3	15	Systematic underestimate dominates
Mean \|error\|	+7.8%	−7.1%	Symmetric magnitude, asymmetric count
Mean f_gas	0.37	0.50	Gas-rich means underestimated
Mean Σ_d, L_⊙/pc²	146	247	Denser disks mean underestimated, mostly a bulge effect
Has bulge	0 / 3	7 / 15	All bulge galaxies underestimated
Mean Hubble type T	5.7, Sc	5.1, Sc	No signal. T is not a driver.

3. The Mechanism — What the BeeTheory Formula Misses

3.1 The Gas Disk as a Missing BeeTheory Source

The current formula uses the stellar disk as the only source of dark field:

Current formula — stellar disk source only \(\rho_{\mathrm{dark}}(r)=\frac{K_0}{R_d^\star}\int \Sigma_0^\star e^{-R’/R_d^\star}\,\mathrm{kernel}\,dR’\)

But every baryonic mass element is a BeeTheory source. In gas-rich galaxies, the HI disk contains as much mass as the stellar disk and extends to R_HI ≈ 1.7R_d^★. The correct formula should be:

Corrected formula — stellar + gas disk sources \(\rho_{\mathrm{dark}}(r)=\frac{K_0}{R_d^\star}\int\Sigma_\star e^{-R’/R_d^\star}\mathrm{kernel}_d\,dR’+\frac{K_0}{R_{\mathrm{HI}}}\int\Sigma_{\mathrm{HI}}e^{-R’/R_{\mathrm{HI}}}\mathrm{kernel}_{\mathrm{gas}}\,dR’\) \(R_{\mathrm{HI}}\approx1.7R_d^\star,\qquad \mathrm{kernel}_{\mathrm{gas}}=\frac{(1+\alpha_{\mathrm{gas}}D)e^{-\alpha_{\mathrm{gas}}D}}{D^2},\qquad \alpha_{\mathrm{gas}}=\frac{1}{c_{\mathrm{gas}}R_{\mathrm{HI}}}\)

The gas disk source has a longer coherence length than the stellar disk. Its dark field is more extended and contributes differently at large r.

It should increase the predicted dark mass for gas-rich galaxies, reducing the underestimate.
It should reduce the overestimate for low-gas pure-stellar-disk galaxies.
It explains why NGC 0925, NGC 3198, and NGC 3621 are underestimated.

3.2 The Bulge Dark Field — A Compact, Intense Source

In BeeTheory, a compact source generates a more intense dark field per unit mass, because the Yukawa kernel gives more weight to small distances. The bulge, concentrated within about 1–2 kpc, generates a dark field with a short coherence length ℓ_b ≪ ℓ_d.

Bulge dark field — high-intensity compact source \(\rho_{\mathrm{dark,bulge}}(r)=K_b\int_0^{R_b}\rho_{0,b}e^{-r’/r_b}\frac{(1+\alpha_bD)e^{-\alpha_bD}}{D^2}4\pi r’^2\,dr’\) \(\alpha_b=\frac{1}{\ell_b}\gg \alpha_d=\frac{1}{\ell_d}\)

The current model uses Kb = 1.055 kpc⁻¹, calibrated on the Milky Way, and assigns 15% of stellar mass to the bulge — both rough estimates.

Why the bulge is hard to model in SPARC galaxies

SPARC provides stellar disk scale radius R_d and total luminosity, but not a reliable bulge-to-disk decomposition for all galaxies. Separating the bulge from the disk requires 2D photometric fitting, available for some SPARC galaxies but not uniformly.

4. What This Predicts — The Corrected Model

If the two identified physical effects are included — gas disk as a separate BeeTheory source, and bulge with galaxy-specific mass fraction — the residual pattern should disappear.

Correction for overestimated galaxies

For NGC 4051 and NGC 0100, the stellar-disk-only formula is nearly correct because gas is low. The correction is small.
The overestimate may come from slightly overestimated stellar mass-to-light ratio Υ★.
NGC 0300 is already essentially correct at +0.9%.

Correction for underestimated galaxies

Gas-rich galaxies such as NGC 3621, NGC 0925, and NGC 3198 should improve when an HI disk source is included.
Using R_HI = 1.7R_d and K_HI = K₀/R_HI can add about 10–15% dark field.
Bulge galaxies such as NGC 3521 and NGC 0891 require galaxy-specific bulge mass fractions.

Complete corrected BeeTheory formula — three sources \(\rho_{\mathrm{dark}}(r)=\underbrace{\frac{K_0}{R_d^\star}\int\Sigma_\star e^{-R’/R_d^\star}K_d\,dR’}_{\mathrm{stellar\ disk}}+\underbrace{\frac{K_0}{R_{\mathrm{HI}}}\int\Sigma_{\mathrm{HI}}e^{-R’/R_{\mathrm{HI}}}K_{\mathrm{gas}}\,dR’}_{\mathrm{HI\ gas\ disk}}+\underbrace{K_b\int\rho_{0,b}e^{-r’/r_b}K_{\mathrm{bulge}}\,dr’}_{\mathrm{bulge}}\)

This three-source formula still uses only two universal constants — K₀ = 0.3759 and c = 6.40 — because R_HI and r_b are measured baryonic properties of each galaxy, not free parameters.

Positive scientific conclusion

The residual pattern is not random noise. It is structured, explainable, and points to well-defined missing physics.

A model that fails in a structured way is more valuable than one that fails randomly: it tells you exactly what to improve. In this case, the BeeTheory framework is correct in structure; what is missing is the inclusion of all baryonic sources, not just the dominant stellar disk.

The prediction is clear: add the HI gas disk and proper bulge decomposition to the formula, and the 18/20 success rate should improve, including the two outliers CamB and NGC 3741.

Data: Lelli, McGaugh, Schombert, AJ 152, 157, 2016.

BeeTheory model: Dutertre, 2023, extended 2025.

HI disk scaling: R_HI/R_d ≈ 1.7, Broeils & Rhee 1997; Swaters et al. 2009; Lelli et al. 2014.

Reading the Residuals:Why Some Galaxies Are Above and Others Below the Prediction