Stability map as a function of P₁∕P₂ and μ = (m₁ + m₂)∕m₀, for circular coplanar orbits, taking ζ = 60° as initial condition (vertical dashed line in Fig. 1). Coloured pixels are long-term stable with respect to the integration time of 10⁵ orbits. Grey pixels are long-term unstable, and white pixels are short-term unstable. The black curve represents the mutual Hill radius of the planets, , while the vertical line represents P₁∕P₂ ≈ 1.04 which is close to the estimated value for the candidates of the TOI-178 system discussed in Sect. 3. The colour code is the amplitude of libration: black for zero (lagrangian equilibria), orange for horseshoe orbits. The bottom-right corner shows stable orbits outside of the co-orbital area, in yellow. The purple line is the stability criterion from the overlapping of first-order MMRs (Deck et al. 2013). Black dots represent orbits that are stable over 10¹⁰ orbital periods; see the text for more details.