  This paper presents an analytical model for collision probability assessments
between de-orbiting or injecting space objects and satellite constellations.
Considering the first to be subjected to a continuous tangential acceleration,
its spiraling motion would result in a series of close approaches in the
proximity of a constellation. The proposed methodology involves the integration
of the collision probability density function on the encounter plane, from
which two analytical formulas, one for the number of close approaches and one
for their respective average collision probability, are obtained. The
mathematical description of the crossing dynamics relies on the assumption of
circular orbits and independent collision probabilities, but does not require
to propagate the satellites' orbit. A comparison with a conventional
propagation method has been performed for validation purposes, proving its
accuracy also in case of elliptical crossing orbits. The model developed has
been used to assess the risk connected to constellation's satellites
replacement, once they have reached their programmed End-of-Life. The
environmental impact of the full replacement of 12 approved constellations is
analysed by means of average collision probability. In particular, it is shown
that the key features for space exploitation sustainability are the maximum
propulsion available from the thruster, the selection of an optimal crossing
orbit and the true anomaly phases between constellations' and crossing
satellites. The consequences of an in-orbit collision are also investigated by
assessing the collision risk generated by the formation of a debris cloud. The
results corroborate the need for international standards for space traffic
management as an exponentially increasing satellites population could trigger a
chain reaction of collisions, making LEO inaccessible for decades.
