Estimations of ϵ_⋆p,1 as a function of a₁ for typical compact multi-planet systems. This figure displays estimations of ϵ_⋆p,1 for three-planet systems with uniform planetary masses of m₁-m₃ = 5m_⊕ and varying orbital separations δa = {16, 20, 25, 30} R_H. Two sets of stellar parameters, resulting in J₂ = 7.97 × 10⁻⁴ and J₂ = 1.46 × 10⁻⁴, respectively, are adopted. The selection of δa= 16R_H represents the minimum spacing required for these systems to maintain stability over 1 Gyr (Zhu & Dong 2021). The grade-shaded histogram (presented using the right y-axis) illustrates the distribution of a₁ for the Kepler sample of multi-planet systems, which are retrieved from the NASA Exoplanet Archive.