A Reference Curve for the Bulge
The same saturating shape, now anchored on the bulge — and it nearly matches the disk’s
Normalized by the bulge reference — radius R_bulge, velocity V_bulge — the rotation curves of the 32 bulged galaxies follow a single saturating law that passes through (1, 1) and flattens to 1.51 far out. Strikingly, this is almost the same curve found for disk+gas (asymptote 1.60): both components, each scaled to its own reference, rise about 50% above it and level off.
1. The bulge reference
Each bulge is reduced to a sphere of universal density carrying the bulge mass, fixing two reference quantities: R_bulge (its equivalent radius) and V_bulge = √(G·M_bulge/R_bulge). Dividing every measured rotation point by these puts all 32 bulged galaxies on common axes, anchored at (1, 1). Because the bulge is the innermost component, most points sit at x = R/R_bulge > 1 — the curve is sampled mostly outward of the bulge.
2. The reference curve
The same two requirements apply — pass through (1, 1), flatten in the limit — and the same saturating form satisfies them:
At x = 1, V/V_bulge = 1 exactly. As x → ∞, V/V_bulge → 1.51, a flat asymptote. The fit (typical deviation 0.27) is slightly tighter than for the disk.
3. Disk and bulge — almost the same law
| Component | amplitude | transition | asymptote |
|---|---|---|---|
| Disk + gas (ring) | 0.60 | 0.74 | 1.60 |
| Bulge (sphere) | 0.51 | 1.46 | 1.51 |
The two asymptotes — 1.60 for the disk, 1.51 for the bulge — are close. Each component, reduced to its own reference object and scaled by its own radius and velocity, climbs to roughly 1.5 times its reference and then plateaus. The shape is shared across two physically very different components.
A single normalized shape describes both a thin extended disk and a compact central bulge, once each is referred to its own scale. This is the kind of regularity a unified mechanism should produce: the excess velocity beyond the visible reference is not arbitrary but follows one saturating form, the same for both components up to a common factor near 1.5. It does not prove the mechanism — but it shows the two pieces of every galaxy obey the same simple rule.
All points are real SPARC data (Lelli, McGaugh & Schombert 2016); R_bulge and V_bulge come from the fixed per-galaxy table, where the bulge is assigned a single universal density (the sample median). The saturating form was fitted to the pooled normalized points of all 32 bulged galaxies — an empirical description of the shared shape, not a first-principles derivation. As with the disk, the curve traces the central trend; individual galaxies scatter around it, and high-bulge-fraction galaxies climb well above the asymptote.
BeeTheory.com — A reference curve for the bulge · Data: Lelli, McGaugh & Schombert 2016 · Initial generation: 21 May 2026 with Claude.ai · © Technoplane S.A.S. 2026