Öpik-type collision probability for high-inclination orbits

The classical Öpik theory provides an estimate of the collision probability between two bodies on bound, heliocentric or planetocentric orbits under restrictive assumptions of: (i) constant eccentricity and inclination, and (ii) uniform circulation of the longitude of node and argument of pericenter. These assumptions are violated whenever either of the orbits has a large inclination with respect to the local Laplace plane or large eccentricity, and their motion is perturbed by an exterior (tidal) gravitational field of a planet or the Sun. In this situation, known as the Lidov–Kozai regime, the eccentricity and inclination values exhibit large and correlated oscillations. At the same time, the longitude of node and the argument of pericenter may have strongly nonlinear time evolution, with the latter being even bound to a small interval of values. Here we develop a new Öpik-type collision probability theory which is valid even for highly inclined and/or eccentric orbits of the projectile. We assume that the orbit of the target is circular and in the local Laplace plane. Such a generalized setting is necessary, as an example, to correctly estimate the terrestrial impact fluxes of sporadic micrometeoroids on high-inclination orbits (notably those from the toroidal source and the associated helion and anti-helion arcs).

► We generalize the classical Öpik collision theory for orbits with high inclination and eccentricity values.
► We test the theory by providing comparison with the Öpik theory and results of numerical experiments.
► The new theory is needed for specific applications in planetary science such as Earth impacts of micrometeoroids from toroidal complex.