  Neglecting small fragments in space debris evolutionary models can lead to a
significant underestimation of the collision risk for operational satellites.
However, when scaling down to the millimeter range, the debris population grows
to over one million objects, making deterministic approaches too
computationally expensive. On the contrary, probabilistic models provide a more
efficient alternative, which however typically work under some simplifying
assumptions on the dynamics, limiting their field of applicability. This work
proposes an extension of the density-based collision risk models to any orbital
dynamics and impact geometry. The impact rate with a target satellite is
derived from a multi-dimensional phase space density function in Keplerian
elements, which discretely varies over both phase space and time. The
assumption of a bin-wise constant cloud density allows for the analytical
transformation of the six-dimensional distribution in orbital elements into the
three-dimensional spatial density function, guaranteeing an efficient and
accurate evaluation of the fragments flux. The proposed method is applied to
the assessment of the collision risk posed by real fragmentation clouds in
different orbital regions on a high-risk target object. The effect on the
impact rate of the additional model features, compared to previous
probabilistic formulations, is discussed.
