Vertical height H and vertical velocity υ_z as a function of ϕ Top panel: H/H₀ as a function of ϕ, where H₀ = c_s/Ω₀, obtained solving Eq. (29). Bottom panel: υ_z/c_s as a function of ϕ, obtained from Eq. (27) after solving Eq. (29), calculated at an elevation z = H above the midplane − c_s is the most natural normalisation² for υ_z. Solution to Eq. (29) was obtained solving for periodic solutions across one orbit, i.e. H(a, 0) = H(a, 2π), using a shooting method for solving the boundary value problem.