Tisserand periapsis bound ($q\ge q_{\min }(T)$)

For bound planet-crossing Kepler orbits, $T=2/(q+Q)+2\sqrt{2qQ/(q+Q)}\cos I$, with the planet radius taken as one. At fixed $q<1$, $T$ is maximized by $I=0$ and $Q=1$. For $2<T<3$ this gives $q\ge (1-\sqrt{3-T}{)}^2/[1+2\sqrt{3-T}-(3-T)]$. This form remains regular at $T=\sqrt{8}$. It is an invariant-based accessibility bound, not a guarantee that a scattering history attains it; for $0<T\le 2$ it gives no positive universal pericentre floor.