Öpik-type collision frequency for Kozai-driven projectiles: Target bodies on inclined circular orbits

Existing \"Opik-type collision-frequency methods already incorporate the Kozai-driven secular evolution of the orbital elements of high-inclination projectiles. However, these methods generally assume that the target body's orbit lies in the reference plane defined by the orbital plane of the perturbing body. The present paper extends the semi-analytical framework developed by Vokrouhlick\'y et al. (2012) for a target body on a circular orbit in the reference plane to the case of a target body on a circular orbit with a non-zero inclination relative to that plane. The target body's nodal precession rate is prescribed to be constant and may be zero. In this geometry, whether the two orbits intersect depends not only on the secular state of the projectile's orbit but also on the relative nodal longitude between the two orbits. To account for this dependence, the relative nodal longitude is introduced as an additional geometrical variable, and the framework is extended accordingly. When the target body's circular orbit lies in the reference plane, the present formulation analytically reduces to the zero-inclination case described by Vokrouhlick\'y et al. (2012). The numerical results also confirm this reduction. For the two cases with inclined target-body orbits, the collision frequencies computed with the present framework are used to predict semi-analytical decay curves for the fraction of projectiles remaining. These predicted curves closely match those obtained from direct dynamical simulations.