The Two-Body Reduction Across the Full Sample
Ring and sphere applied to all 175 SPARC galaxies, split by composition and coloured by Hubble type
Reducing every galaxy to one ring (disk+gas) and, where present, one sphere (bulge) and taking the differential against each component’s real velocity, the full sample splits cleanly: 143 galaxies with disk+gas only and 32 with a bulge. In the disk-only family the differential is ordered by Hubble type — dwarfs hug the zero line, concentrated spirals fall furthest. In the bulged family the bulge differential crosses above zero at the centre for several galaxies and peaks at a common inner radius — the first sign of points on both sides of the line.
1. Disk+gas galaxies (143)
The differential is not random — it is sorted by morphology. Late-type dwarfs (blue) sit in a tight band just below zero: their diffuse disk+gas is well captured by a single ring. Earlier, more concentrated spirals (gold) dive much lower, because their mass is centrally piled up and a single-radius ring departs more from the real, extended distribution. The vertical position in this plot is essentially a measure of how concentrated the visible matter is.
2. Bulged galaxies (32)
Disk panel: as in the bulgeless family, the differential climbs toward a maximum around R/R_ring ≈ 1 — the ring best matches the disk where the disk mass actually concentrates. These galaxies being massive, the spread is wider.
Bulge panel: this is the new feature. Every bulge differential peaks at the same inner location (R/R_ring ≈ 0.1–0.3), and several rise above zero there — meaning the real bulge turns faster than the fixed-density sphere predicts at the centre, then falls below it outward. For the first time the differential lands on both sides of the line, organized around a shared inner scale rather than scattered.
One ring and one sphere of universal density already capture the structure of every galaxy’s components across four decades of mass. The disk differential measures concentration (ordered by type); the bulge differential, centred on a common inner radius and crossing zero, behaves like a genuine sub-system with its own characteristic scale. The disk’s overall negative offset remains the geometric cost of concentrating extended mass at one radius — but the morphological ordering and the bulge’s zero-crossing are physical, repeatable signals, not noise.
All velocities are real SPARC data (Lelli, McGaugh & Schombert 2016), Υ = 0.5. Sample: 175 galaxies (143 disk+gas, 32 bulged). Component masses use the SPARC velocity decomposition; the single bulge density is the sample median. The negative offset of the disk panels is the stated artefact of the one-radius reduction — the results read here are the ordering and the zero-crossings, not absolute levels.
BeeTheory.com — Two-body reduction across the full sample · Data: Lelli, McGaugh & Schombert 2016 · Initial generation: 21 May 2026 with Claude.ai · © Technoplane S.A.S. 2026