BeeTheory · Galactic Application · Technical Note XXXIII

Census of the 23 Galaxies:
Visible vs Dynamical Mass

For each of the 23 galaxies of the calibration sample (Milky Way + 22 SPARC), we compute the total visible mass from observational data ($M_\star$ from Spitzer photometry, $M_\text{gas}$ from HI surveys, $M_\text{bulge}$ for early types) and compare it with the dynamical mass inferred from the observed flat rotation velocity. The difference — the “missing mass” — is what any gravity theory must explain. We sort by missing mass, identify the galaxies with no deficit, and group the most extreme cases by category.

1. The result first

Mass deficit across the 23 calibration galaxies

Galaxies with $M_\text{visible} \geq M_\text{dynamical}$2 / 23 (CamB, DDO064)
Galaxies with mass deficit ($M_\text{dyn} > M_\text{vis}$)21 / 23
Median ratio $M_\text{dynamical}/M_\text{visible}$7.7
Range of mass deficit ratiosFrom $0.03$ to $13.6$
Worst-deficit categoryLSB Sd galaxies — median ratio $times 13.4$
Best-fit categoryCompact dwarfs Im — some have $M_\text{vis} \approx M_\text{dyn}$

2. Methodology

Visible mass is computed from observational inputs available in SPARC (Lelli et al. 2016):

$$M_\text{visible} \;=\; \underbrace{\Upsilon \cdot 2\pi\,\Sigma_d\,R_d^2}_{M_\star} \;+\; \underbrace{1.33\,M_{\text{HI}}}_{M_\text{gas}} \;+\; \underbrace{0.25\,M_\star \text{ if } T \leq 3}_{M_\text{bulge}}$$

with $\Upsilon = 0.5\,M_\odot/L_\odot$ at $3.6\,\mu$m (standard mass-to-light, McGaugh & Schombert 2014), and a $1.33$ factor for helium correction on the HI gas mass. The bulge is included only for early-type galaxies (Hubble T ≤ 3).

Dynamical mass is computed from the observed flat rotation velocity $V_f$ at a characteristic radius:

$$M_\text{dynamical} \;=\; \frac{V_f^2 \cdot R_\text{eff}}{G}, \qquad R_\text{eff} = 5\,R_d$$

$R_\text{eff} = 5\,R_d$ is the characteristic radius of the flat plateau for an exponential disk — far enough that the rotation curve has settled to $V_f$. This is a uniform convention applied identically to all 23 galaxies.

3. The full table — sorted by missing mass

#GalaxyType$R_d$ (kpc) $V_f$ (km/s)$M_\text{visible}$$M_\text{dyn}$ Missing massRatio
1CamBIm0.472.0$6.7 \times 10^7$$2.2 \times 10^6$$-6.5 \times 10^7$$\times 0.03$
2DDO064Im0.3326.0$2.7 \times 10^8$$2.6 \times 10^8$$-7.9 \times 10^6$$\times 0.97$
3ESO444-G084Im0.5527.0$2.2 \times 10^8$$4.7 \times 10^8$$+2.5 \times 10^8$$\times 2.2$
4DDO154Im0.6047.0$6.8 \times 10^8$$1.5 \times 10^9$$+8.6 \times 10^8$$\times 2.3$
5DDO170Im1.1038.0$6.0 \times 10^8$$1.9 \times 10^9$$+1.3 \times 10^9$$\times 3.1$
6DDO168Im0.6952.0$4.3 \times 10^8$$2.2 \times 10^9$$+1.7 \times 10^9$$\times 5.1$
7D631-7Im0.7057.7$6.9 \times 10^8$$2.7 \times 10^9$$+2.0 \times 10^9$$\times 3.9$
8DDO161Im1.1055.0$1.2 \times 10^9$$3.9 \times 10^9$$+2.6 \times 10^9$$\times 3.2$
9F565-V2Im1.0053.0$3.2 \times 10^8$$3.3 \times 10^9$$+2.9 \times 10^9$$\times 10.1$
10F563-V2Im1.1059.0$5.8 \times 10^8$$4.4 \times 10^9$$+3.9 \times 10^9$$\times 7.7$
11F563-V1Im1.2064.0$5.1 \times 10^8$$5.7 \times 10^9$$+5.2 \times 10^9$$\times 11.2$
12F567-2Im1.8067.0$9.5 \times 10^8$$9.4 \times 10^9$$+8.4 \times 10^9$$\times 9.9$
13F568-V1Im2.1082.0$1.3 \times 10^9$$1.6 \times 10^{10}$$+1.5 \times 10^{10}$$\times 12.2$
14ESO116-G012Sd2.1093.0$3.2 \times 10^9$$2.1 \times 10^{10}$$+1.8 \times 10^{10}$$\times 6.6$
15F561-1Im2.5087.0$1.8 \times 10^9$$2.2 \times 10^{10}$$+2.0 \times 10^{10}$$\times 12.3$
16F563-1Im2.7092.0$2.1 \times 10^9$$2.7 \times 10^{10}$$+2.4 \times 10^{10}$$\times 12.9$
17F568-3Sd3.00108.0$3.0 \times 10^9$$4.1 \times 10^{10}$$+3.8 \times 10^{10}$$\times 13.6$
18F574-1Sd3.60107.0$3.8 \times 10^9$$4.8 \times 10^{10}$$+4.4 \times 10^{10}$$\times 12.8$
19F568-1Sd3.20115.0$3.7 \times 10^9$$4.9 \times 10^{10}$$+4.6 \times 10^{10}$$\times 13.4$
20NGC3198Sc3.14151.0$1.6 \times 10^{10}$$8.3 \times 10^{10}$$+6.7 \times 10^{10}$$\times 5.1$
21F571-8Sd4.50125.0$6.1 \times 10^9$$8.2 \times 10^{10}$$+7.6 \times 10^{10}$$\times 13.4$
22Milky WaySbc2.60229.0$6.6 \times 10^{10}$$1.6 \times 10^{11}$$+9.3 \times 10^{10}$$\times 2.4$
23NGC2841Sb3.50278.0$4.0 \times 10^{10}$$3.1 \times 10^{11}$$+2.7 \times 10^{11}$$\times 7.8$
Galaxies sorted by missing mass in ascending order. Green rows: no deficit. Red rows: worst deficits (LSB Sd galaxies). Blue row: Milky Way. Mass deficit ratios span from $0.03$ (CamB anomaly) to $13.6$ (F568-3).

4. Visualization

Visible mass vs dynamical mass — 23 galaxies Sorted by missing mass (increasing). Red bars = mass to be explained by BeeTheory wave field. CamB×0.0DDO064×1.0ESO444-G084×2.2DDO154×2.3DDO170×3.1DDO168×5.1D631-7×3.9DDO161×3.2F565-V2×10.1F563-V2×7.7F563-V1×11.2F567-2×9.9F568-V1×12.2ESO116-G012×6.6F561-1×12.3F563-1×12.9F568-3×13.6F574-1×12.8F568-1×13.4NGC3198×5.1F571-8×13.4Milky Way×2.4NGC2841×7.8 10^610^710^810^910^1010^1110^12 Mass (M_⊙, log scale) M_visibleMissing mass (M_dyn − M_visible)×N: ratio M_dyn / M_visible
Visible mass (filled bars, colored by type) and missing mass (red overlay) for each galaxy, sorted ascending. The two galaxies at the top (CamB, DDO064) show no mass deficit — their visible mass alone matches or exceeds the dynamical requirement. The bottom of the chart is dominated by NGC2841 (Sb), Milky Way, and several LSB Sd galaxies (F571-8, F568-1, F574-1, F568-3) where the missing mass dominates by an order of magnitude.
Dynamical mass vs visible mass — 23 galaxies grouped by Hubble type The 1:1 line shows where visible mass would suffice. Most galaxies sit at ×3 to ×30 above. 10^710^810^910^1010^1110^710^810^910^1010^11 ×3×10×30 CamBDDO154F568-1NGC3198MWNGC2841 M_visible (M_⊙) M_dynamical from V_f (M_⊙) Im (dwarfs)Sd (LSB)ScSbc/Sb
Dynamical vs visible mass for all 23 galaxies. The dashed line is the 1:1 relation — where pure baryonic matter would suffice. Most galaxies sit between the $\times 3$ and $\times 30$ deficit lines. The dwarf Im galaxies are scattered near 1:1 (some even below it), while LSB Sd galaxies cluster around $times 13$, and massive types (Milky Way, NGC2841) range from $times 2$ to $times 8$.

5. Galaxies with no mass deficit

Only two galaxies of the 23 have $M_\text{visible} \geq M_\text{dynamical}$:

Galaxy$M_\text{vis}$$M_\text{dyn}$RatioComment
CamB$6.7 \times 10^7\,M_\odot$$2.2 \times 10^6\,M_\odot$$\times 0.03$Anomaly: $V_f = 2$ km/s is extremely low. SPARC literature notes CamB as outlier — likely $V_f$ measurement systematics from extreme face-on inclination or HI mapping limit.
DDO064$2.7 \times 10^8\,M_\odot$$2.6 \times 10^8\,M_\odot$$\times 0.97$Compact gas-rich dwarf irregular. Visible mass alone explains the rotation curve to within $3%$. No need for BeeTheory wave field at this scale.

DDO064 is the cleanest test

DDO064’s $M_\text{dyn}/M_\text{vis} \approx 1$ ratio shows that purely baryonic gravity can suffice in some regimes. The challenge for BeeTheory is to explain why it doesn’t suffice for the other 21 galaxies — without over-predicting the rotation curve of compact dwarfs like DDO064.

6. Categorization by deficit severity

The 23 galaxies fall into four natural categories based on the magnitude of their visible mass deficit:

CategoryMass deficit rangeMembersMedian ratio
Group A — No deficit$M_\text{dyn} \leq M_\text{vis}$CamB, DDO064$\times 0.5$
Group B — Mild deficit$\times 1$ to $\times 5$ESO444-G084, DDO154, DDO170, D631-7, DDO161, NGC3198, Milky Way$\times 3.1$
Group C — Severe deficit$\times 5$ to $\times 10$DDO168, ESO116-G012, F563-V2, F567-2, NGC2841$\times 7.7$
Group D — Extreme deficit$\times 10$ to $\times 14$F565-V2, F563-V1, F568-V1, F561-1, F563-1, F568-3, F574-1, F568-1, F571-8$\times 12.8$
Group D is overwhelmingly composed of LSB galaxies of the Fornax / Schombert (“F” prefix) sample — low surface brightness disks where visible matter is diluted over large radii while rotation velocity remains high.

7. The worst-deficit cases — what they have in common

The nine galaxies of Group D (extreme deficit, $\times 10$ to $\times 14$) share specific physical properties:

  • Low surface brightness. Central surface density $Sigma_d$ between $15$ and $40,L_odot/text{pc}^2$ — about a factor 10–30 lower than the Milky Way ($sim 400,L_odot/text{pc}^2$).
  • Late Hubble type. Almost all are Sd (T = 8) or Im (T = 10) — no bulge, very extended disks.
  • Substantial gas fraction. $M_text{gas} gtrsim M_star$ in most cases — these are gas-rich systems.
  • Modest rotation velocities. $V_f$ between $50$ and $125$ km/s — neither dwarf nor massive, but middle-range. Yet their visible mass would predict $V_f$ closer to $20$–$35$ km/s under pure Newtonian gravity.

These are the systems where the discrepancy between visible mass and dynamical mass is most extreme. They are also the systems where the BeeTheory wave field, in its current universal-parameter form, fails most severely (Note XXXII on F568-1 documented this in detail).

Pattern: deficit correlates inversely with surface density

The lower the surface density, the higher the deficit ratio. LSB galaxies — with diluted visible matter spread over large radii — require the largest wave-field response per unit visible mass. This is the empirical signature that the BeeTheory coupling must scale with surface density, not be a universal constant. The next note will derive and test this scaling.

8. Summary

1. Of the 23 calibration galaxies, $21$ exhibit a clear visible mass deficit; $2$ are explained by visible matter alone.

2. The deficit ratio $M_\text{dyn}/M_\text{vis}$ has a median of $7.7$ and spans from $0.03$ (CamB anomaly) to $13.6$ (F568-3).

3. The deficit is not random: it correlates with galaxy type and surface density. Compact gas-rich dwarfs (Im, T = 10) have small deficits; LSB disks (Sd, T = 8) have the largest.

4. The four-group categorization sorts the sample into manageable physical classes. Group D (LSB Sd) is the most demanding for any gravity theory; Group A (compact dwarfs) is the simplest.

5. The systematic correlation between surface density and deficit ratio is a strong constraint for BeeTheory. It suggests that the wave coupling $lambda$ and/or the coherence length $ell_text{wave}$ depend on local surface density rather than being universal constants.

References. Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023). · Notes XXX–XXXII — BeeTheory.com (2026). · Lelli, F., McGaugh, S. S., Schombert, J. M. — SPARC: 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves, AJ 152, 157 (2016). · McGaugh, S. S., Schombert, J. M. — Color-Mass-to-Light-Ratio Relations for Disk Galaxies, AJ 148, 77 (2014). · de Blok, W. J. G., McGaugh, S. S. — The dark and visible matter content of low surface brightness disc galaxies, MNRAS 290, 533 (1997). · Schombert, J. M., Bothun, G. D., Schneider, S. E., McGaugh, S. S. — A catalog of low surface brightness galaxies, AJ 103, 1107 (1992). [F-galaxies catalog].

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