The Hubble sequence from cone occupation of the PG flow

Interactive model after E.P.J. de Haas: every tracer stays on its cone θ₀; morphology is set by how the flow occupies the cones

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Tuning fork

Flow on the cones

v_r = √(2GM/r) − Hr v_φ² = ½(2 + sin θ₀) v_r²(R) − v_r²(r)

Bulge mass M

Bulge radius R

Redshift z

Cone-occupation width σ

Isotropic occupation

Azimuthal strength κ

Bar (two-Lagrangian nesting)

Critical radius r_c
Pitch α at 10 kpc
Bar zone / ring R₂

Cone occupation (latitude −90° … +90°)

Equatorial pitch α(r), 0 – 15 kpc (0 – 90°)

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Flow speed

Arms Bulge Auto-rotate

About this model

PG flow lines satisfy dθ/dt = 0, so every tracer stays on its own cone θ₀. Ellipticals: isotropic or broad occupation with weak azimuthal flow (|v_φ| ≲ |v_r|); S0: several adjacent cones without an arm pattern; spirals: a thin equatorial cone. Arms are streamlines of the flow and open as M and R grow, because r_c = (2GM/H²)^1/3 moves out and v_r/v_φ scales with r/R. Bars are the two-Lagrangian nested spiral (R₁ = 0.22 R₂, M₁ = 0.07 M₂, NGC 3741 ratios) ending in a ring at R₂.

Display choices: stage presets, occupation widths, κ for ellipticals, and an adaptive tracer clock (3–45 Myr per frame). In this model the bulge sphere grows with R, so Sc carries the largest bulge — opposite to the classical visual criterion.

de Haas: JHEPGC 11 (2025), doi:10.4236/jhepgc.2025.114097 · JHEPGC 12 (2026), doi:10.4236/jhepgc.2026.121022 · PG Hubble sequence (2025) · Maurer–Cartan connection (2026).