Two Bodies per Galaxy: a Ring and a Sphere
Reducing each galaxy to a single ring (disk+gas) and a single sphere (bulge), and looking at the differential
Each galaxy is collapsed to two objects: a ring carrying the disk+gas mass at a single characteristic radius, and a sphere of fixed density carrying the bulge. Comparing each real component velocity to this reduced model, two regularities appear across the calibration galaxies: the disk+gas differential rises to a maximum near R/R_ring ≈ 1 — exactly where the ring is placed — and the bulge differentials all peak at the same place, near R/R_ring ≈ 0.2–0.3. The components organize themselves around the model, not randomly.
1. The reduction
We strip each galaxy down to its essentials — two bodies, each with one mass, one radius, one velocity:
SPHERE (bulge): one universal density ρ = 4.08×10⁸ M☉/kpc³ → R_bulge = (3M_bulge/4πρ)^(1/3), V_bulge = √(G·M_bulge/R_bulge)
The bulge density is fixed at the median of the SPARC bulges, so the sphere has no free parameter per galaxy: its size follows from its mass alone. The disk becomes a single ring at the radius where, on average, its mass sits.
2. The differential
For each component we plot the real rotation velocity minus the velocity of its reduced model, against normalized radius. A flat zero line would mean the reduction is exact; departures show where and how the real, extended matter differs from its idealized one-radius stand-in.
3. What the two patterns say
The differential is offset below zero — a known consequence of the reduction: a ring that concentrates all the disk mass at one radius produces a higher circular velocity than the same mass spread out, so the real, extended matter sits below it. But the shape of the offset is the signal:
Disk+gas: the curves are not flat — they rise toward a clear maximum around R/R_ring ≈ 1. The reduced ring is closest to the real disk precisely at the radius where the ring is placed, exactly as it should be. The single-radius approximation is at its best where the mass actually concentrates.
Bulge: the bulge differentials all peak at the same inner location (R/R_ring ≈ 0.2–0.3), then fall together along a common curve. Despite very different bulge masses, the reduced sphere captures the same characteristic inner scale for all of them — the fixed-density assumption produces a consistent, repeatable behaviour rather than scatter.
We are not far. Two objects per galaxy — one ring, one sphere of universal density — already reproduce where each component matters: the disk near one ring-radius, the bulge at a fixed inner scale. The remaining systematic offset is the expected price of concentrating extended mass at a single radius, not a failure of the picture. The next step is to compare each component to the velocity of its mass enclosed at each radius rather than its total mass at one radius — which should turn the offset curves into a differential centred on zero, with departures that carry physical meaning.
All velocities are real SPARC data (Lelli, McGaugh & Schombert 2016), Υ = 0.5; 19 calibration galaxies present in SPARC. The negative offset is a geometric artefact of the single-radius reduction, stated plainly — the result here is the organization of the differentials (the disk maximum near R_ring, the common bulge peak), not their absolute level.
BeeTheory.com — Two bodies per galaxy: ring and sphere · Data: Lelli, McGaugh & Schombert 2016 · Initial generation: 21 May 2026 with Claude.ai · © Technoplane S.A.S. 2026