Length l of the robotic swarm (before inverted aerobraking) is:
All values are counted into the direction of the robotic swarm, i.e. all are positive (with the possible, but not very useful exception of initial velocity).
To compute l, take the point of view of the robotic swarm, i.e. subtract the speed of the robotic swarm from all given velocities. The speed of the robotic swarm is therefore zero in our new coordinate system. The space ship has an initial velocity of the speed of the kinetic fuel (in earth based coordinates) coming towards the swarm, minus any forward velocity the space craft got from going to orbital height. The space craft is then decelerated (a becomes negative) to its final velocity. During this time, the space ship travels a distance of l in relation to the swarm. Hence l is the necessary length of the swarm.
Remember that a final velocity of more than 80% the speed of the kinetic fuel becomes more and more inefficient (see ImpulseTransfer).
Estimating the necessary number of robots in the swarm is more
difficult, because it depends on how long before impact on the
aerobrake kinetic fuel can be put in space, and at what speed (speed in
relation to the swarm). Which in turn depends on the capabilities of
the aerobrake and its needs to create and maintain the plasma shield
and the strength of the plasma shield.
How to turn kinetic propellant into gas / plasma before impact?
Project Orion told me not to worry too much about creation of the
protective plasma layer. So wether to use pulsed or continuous impact
is not as important as I feared. But it also confirmed my opinion that
a pusher plate / heat shield can not withstand impacts of solid
particles (see cite of shrapnels from duds).
So far I have basically two ideas I consider to be realistic:
(Still a lot of work to do here ...)