Level curves of the Hamiltonians (left panel) and (right panel) on the (ϖ,log a_per) plane for a test mass at a_res = 1 AU in 4:1 mean motion resonance with an outer perturber with μ = 3 × 10^-6, ϖ′ = 0 and e′ = 0.1. The mass of the parent star is set at M_∗ = 1 M_⊙. The black solid line experiencing a significant change in a_peri indicates the trajectory with the initial conditions ϖ = 0 and e = 0.05. The lower edge of the plot is at the location of the radius of the star, here taken to be the radius of the Sun (R_⊙). The white dotted line indicates the location of the Roche limit, calculated using a density of the test particle of ρ_tp = 2 g / cm³. The addition of the General Relativity potential reduces drastically the efficiency of the planetary perturbation in driving the test particle to collide with the star.