Distribution function from Fig. 4 in velocity space at the solar position within the Galactic plane, now perturbed to linear order by a bar (perturbing potential from Sect. 2.4) with pattern speeds Ω_b = 1.16Ω₀ (left) and Ω_b = 1.89Ω₀ (middle), or by a spiral pattern (perturbing potential from Sect. 2.5) with pattern speed Ω_sp = 0.84Ω₀ (right). The black dashed contours represent the zones where k is equal to or less than 1, k being a quantity computed in Monari et al. (2017a) that designates the region where the orbits are trapped at the main resonance (the computation used here in the Stäckel case will be presented in detail in Al Kazwini et al., in prep.). Top row: epicyclic approximation. Bottom row: Stäckel fudge.