The Cost of Future Collisions in LEO - Star Technology and Research

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where F ki (m) is the cumulative distribution (8), H n is the average altitude of the
asset A n , and
k n =
ω n
2π 2 .
R n
Fig. 6. Post-collision increase of the debris flux on sun-sync
assets: a) m > 0.5 g, b) m > 1 g, c) m > 2 g.
The flux of fragments with masses between m and m + dm is calculated as
Ψ n dm, where
Ψ n ≈ k ∑
n
β nk P ki f ki (m) g ki (H n ). (22)
P c
k,i
Formulas (17) and (21) characterize the current state of the LEO debris field
in terms of the average statistically expected production of small fragments in a
catastrophic collision between large objects. Fig. 6 shows how the fluxes on sunsync
assets will increase on average in three different fragment mass ranges after
a catastrophic collision. The virtual fluxes peak around 900 km, because this is
where the primary sources of the future fragments are concentrated.
It is interesting to note that a fairly accurate approximation for the overall
flux can be obtained by replacing the weighted distributions (14) in formulas (21)–
(22) with the base distributions (12) centered around the middle of the altitude
overlap. It works because the altitude overlaps are typically small compared to the
scale heights of the altitude distributions, and because the number of objects is
large.
5. The cost of a catastrophic collision
The cost of a catastrophic collision includes an immediate loss, if an asset was
destroyed in the collision, and a delayed loss, if other assets were damaged later