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

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are synthesized from the probability-weighted and time-averaged streams from all
possible catastrophic collisions. However, we can treat them as physical streams
for the purpose of statistical damage calculations.
We will now focus on a very narrow range of fragment masses around 1 g
that are believed to be at the “threshold of lethality” in terms of the impacts on
the operational satellites. For short-term projections in this range, we will separate
mass and altitude distributions by setting
We will also use cumulative distributions
and
ρ ki (m, H) = f ki (m) g ki (H). (7)
F ki (m) =
G ki (H) =
∫ ∞
m
∫ ∞
H
f ki (µ) dµ (8)
g ki (h) dh. (9)
Appendix B suggests a simple power law distribution for the number of fragments
heavier than m,
F ki (m) = κ ki M k (m c /m) γ , (10)
where m c = 1 g is a characteristic mass, κ ki is the average statistically expected
yield of fragments heavier than m c per unit mass of the object B k produced in
a collision with the object B i , M k is the mass of the object B k , and γ ≈ 0.8.
Based on the data from the Fengyun-1C and Cosmos-Iridium events, an estimate
of κ ki ≈ 24/kg is derived in Appendix B. For each object, the average yield will
depend on its composition and design. The corresponding distribution density is
f ki (m) = κ ki M k γ m γ c /m γ+1 . (11)
Analysis in Appendix C suggests the following approximation for the altitude
distribution of collision fragments
n(h, h 0 ) = k 0
h s
(
1 + |h − h 0|
h s
) −b
, (12)
where h 0 is the collision altitude, h s is the scale height of the distribution, b ≈ 2.37,
and k 0 = (b − 1)/2 ≈ 0.69 is a normalization coefficient, such that
∫ ∞
0
n(h, h 0 ) dh = 1.
Characteristic values of h s are estimated in Appendix C for tracked collision fragments,
however, there is no data on small untracked fragments. It is anticipated