Analytical limit on the maximum semi-major axis of a planet, if it is to be able to scatter particles as far inwards as the position of the exozodi at r_zodi, as a function of the particle’s initial eccentricity. This is calculated using Eq. (6), and the particle’s initial eccentricity is a proxy for the Tisserand parameter value, as described in Sect. 3.1. This limit only strictly exists for planets on circular orbits and vanishes for e_i > 0.46 (or T_i < 2), or for very low eccentricity particles that do not cross the planet’s orbit (e_i > 0.04 or T_i < 3).