\"Opik-type collision frequency for Kozai-driven projectiles: Target bodies on eccentric and inclined orbits

Abstract

For high-inclination projectiles undergoing Lidov--Kozai secular evolution, coupled variations in eccentricity and inclination complicate estimates of long-term collision frequencies with a target body. Previous collision-frequency studies for Kozai-driven projectiles have considered target-body orbits that are either circular or confined to the reference plane. The present study considers the general case in which the target body's orbit has both non-zero eccentricity and non-zero inclination. For each complete Lidov--Kozai cycle, the initial relative nodal longitude and the target body's initial argument of periapsis serve as two independent orientation variables. The single-cycle collision frequency is evaluated for each pair of initial values. Under the incommensurability condition considered here, the long-term mean is then obtained by explicitly averaging these values over both orientation variables. For the special cases in which the target body's orbit is circular or lies in the reference plane, the method is shown analytically to reduce to the corresponding formulations of previous studies. Numerical comparisons confirm these reductions. For the general eccentric and inclined case, the resulting collision frequency predicts a projectile survival curve that agrees well with the fraction of projectiles remaining in independent direct dynamical integrations. Finally, it is shown that, for a fixed dynamical setup and under the incommensurability condition, the long-term mean collision frequency associated with a given closed Hamiltonian level curve is independent of both the projectile's initial secular state along that level curve and its initial longitude of ascending node.