The Plateau Height Scales with V_ring
Letting the asymptote depend on the galaxy’s own rotation velocity
The single flat asymptote of the reference curve (1.60 for every galaxy) was a compromise. In fact the plateau height — how far V/V_ring rises above the visible ring — correlates with V_ring: smaller, slower galaxies rise higher, larger ones stay near 1. Making the asymptote a function of V_ring captures this, with the right physical meaning: dwarfs carry proportionally more missing velocity than massive spirals.
1. The correlation
Measuring each galaxy’s plateau (median V/V_ring in the outer third) against its V_ring shows a clear trend: the strongest single correlation among all geometric parameters tested (r = −0.60, stronger than mass, radius, or extent). The sign is negative — high V_ring means a low normalized plateau.
The fitted asymptote is:
| V_ring (km/s) | asymptote | type |
|---|---|---|
| 20 | 1.83 | dwarf |
| 50 | 1.53 | — |
| 100 | 1.30 | — |
| 200 | 1.07 | spiral |
| 300 | 0.93 | massive (declining) |
2. The curve with a variable asymptote
Inserting this into the saturating form gives a family of curves, all anchored at (1, 1) but flattening at different heights according to V_ring:
The fit improves modestly — median deviation 0.287 versus 0.299 for the fixed asymptote — but the gain is mainly conceptual: the plateau is no longer an arbitrary constant. It is set by the galaxy’s own rotation velocity, and the trend reproduces a known fact, that smaller galaxies are more dark-matter-dominated.
A massive spiral (high V_ring) plateaus near 1 — its visible ring nearly accounts for its rotation. A dwarf (low V_ring) plateaus near 1.8 — most of its rotation needs something beyond the visible ring. The single law now encodes this gradient with one extra coefficient, turning the fixed asymptote into a physical scaling.
All velocities are real SPARC data (Lelli, McGaugh & Schombert 2016); V_ring and R_ring from the fixed per-galaxy table. The correlation (r = −0.60, R² ≈ 0.36) is real but partly mechanical: V_ring sits in the denominator of V/V_ring, so some anticorrelation is built in. The improvement in fit is small; the value here is the physical reading, not a large numerical gain. A cleaner test — correlating the absolute excess (V_plateau − V_ring, in km/s) with V_ring — would separate the physical effect from the mechanical one.
BeeTheory.com — The plateau height scales with V_ring · Data: Lelli, McGaugh & Schombert 2016 · Initial generation: 21 May 2026 with Claude.ai · © Technoplane S.A.S. 2026