Numerical analysis of the conditions for jet-like outflows for A_γ = 2, 4, 8 (blue, green, and red). The bottom panel shows condition 1 (Eq. (B.1)), which is satisfied beyond the radii marked by the vertical dashed lines for A_γ = 4 and 8. The top panel shows condition 2 (Eq. (B.5)); the horizontal dashed line indicates the escape velocity threshold v_w(r′, r_H/2)/v_e(r_H/2) = 1. The shaded red region highlights the radial range where both conditions are simultaneously satisfied for A_γ = 8, identifying the region from which material is ejected from the planet’s vicinity.