A Universal Reference Curve for Disk+Gas Galaxies

One saturating shape, anchored at the reference ring, flat in the limit

BeeTheory.com · Real data · 21 May 2026

Result first

Normalized by a single visible reference ring — radius R_ring, velocity V_ring — the rotation curves of 143 bulgeless galaxies share one shape. It is captured by a single saturating law that passes through (1, 1) by construction and flattens to a constant in the limit: V/V_ring rises from the centre, equals 1 at the ring radius, and levels off near 1.60 far out. No per-galaxy tuning — one universal curve.

1. The reference ring

Each disk+gas galaxy is reduced to one ring carrying its visible mass at the mass-weighted radius. This fixes two natural reference quantities: R_ring (where the mass sits) and V_ring = √(G·M_ring/R_ring) (the ring’s own circular velocity). Dividing every measured rotation point by these two values puts all galaxies on the same axes — and anchors them at the common point (1, 1).

2. The reference curve

Across four decades of mass, the normalized points follow one shape. Two requirements pin it down: it must pass through (1, 1), and it must become horizontal far out — real rotation curves plateau, they do not climb without end. A logarithm passes through (1,1) but rises forever; a saturating rational form does both:

V / V_ring = 1 + 0.60 · (x − 1) / (x + 0.74), x = R / R_ring

At x = 1 the numerator vanishes, so V/V_ring = 1 exactly — the curve is anchored at the ring. As x → ∞ the ratio tends to 1, so V/V_ring approaches the flat asymptote 1.60. The single shape rises, crosses the ring at unity, and levels off.

Saturating reference curve over normalized disk+gas curves — The 143 normalized curves (green) with the saturating reference law (red), anchored at (1, 1) and flattening toward 1.60. The dotted blue line is a plain logarithm — it fits as well near the ring but rises without limit, missing the plateau.

3. The shape in numbers

R / R_ring	V / V_ring
0.3	0.59
0.5	0.76
1.0	1.00 (anchor)
2.0	1.22
3.0	1.32
5.0	1.42
→ ∞	1.60 (flat)

What it captures — and what it does not

The law captures the shared shape: a rise from the centre, unity at the ring, a flat plateau beyond. It does so with no free parameter per galaxy and with the right asymptotic behaviour (horizontal, not divergent). What it does not capture is the spread: real plateaus range from about 0.8 to 2.0 in these units, while the single curve traces only the central trend (typical deviation ≈ 0.30 in V/V_ring). The reference curve is the backbone of the family; individual galaxies scatter around it.

Honesty note

All points are real SPARC data (Lelli, McGaugh & Schombert 2016); R_ring and V_ring come from the fixed per-galaxy reference table. The saturating form was fitted to the pooled normalized points of all 143 disk+gas galaxies; it is an empirical description of the shared curve shape, not a derivation from first principles. The asymptote 1.60 is the median plateau level, not a universal constant each galaxy reaches.

BeeTheory.com — A universal reference curve for disk+gas galaxies · Data: Lelli, McGaugh & Schombert 2016 · Initial generation: 21 May 2026 with Claude.ai · © Technoplane S.A.S. 2026