Evolution of k ≡ w_eff/R_BHL over ∼3 orbits for cut angles: θ = −μ (red), θ = −ψ (purple), and θ = −90° (blue). Bottom x-axis: time; top x-axis: orbital radius r. Left panel: rays in the orbital plane. Right panel: rays perpendicular to the plane (z-direction). In each panel, the apastron passage is indicated with a grey dashed vertical line. Curves are computed from discrete hydro snapshots (only every few time steps retained), so the small-scale jaggedness reflects the sampling cadence rather than intrinsic variability.