(a) Spiral density waves formed by the precessing ellipses of the elliptical closed orbits of the x₁ family, from the model of Eq. (8) for a pattern speed Ω_sp = 15 km s⁻¹ kpc⁻¹ and density of the spiral potential ρ₀ = 5 × 10⁷ M_⊙ kpc⁻³, between the second ILR and the 4:1 resonance. (b) Same as in (a), but with the ellipses of the x₁ family inside the second ILR. Superposed are circles corresponding to the second ILR (green circle), 4:1 resonance (blue circle), and corotation (black circle). The imposed spiral arms (in red) are derived from the minima of the spiral potential of Eq. (8). The coincidence of the imposed spirals with the spiral density wave created by the precessing ellipses is nearly complete.