About this model
Imposed: the jet cone θ₀ and vφ/vesc = tan θ₀ (0.105 for 6°). Calibrated: r₀ and ε∞; the opening angle bounds |ε∞| ≲ 0.17 and measured helix wavelengths would fix it through λ ≈ 2πr ε cos θ₀/(1+ε). Derived: helical streamlines on fixed cones, winding that diverges at r₀, bipolar symmetry, nested spine (0.3 θ₀, w > 0 everywhere) and sheath. Interpretive: the hand-over from broad inflow to the jet cone (Bernoulli redirection; PG streamlines keep θ fixed), the collimation break and HST-1 as recollimation, and the jet’s relativistic speed.
Display conventions: log-radius zoom, tracer speeds not physical, jet cones widened for visibility, markers schematic. M = 6.5×10⁹ M⊙, rg ≈ 3.1×10⁻⁴ pc; photon ring √27 rg; HST-1 ≈ 200 pc deprojected.
de Haas (2025), “A Single Painlevé–Gullstrand Geometry Explains the Launch, Collimation, and Helical Structure of the M87 Jet”, with revised Sec. 2.2 · Hada et al., Astron. Astrophys. Rev. 32, 5 (2024).