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

32
Fig. C2. Velocity distributions for fragments of a) Fengyun-1C, b) Cosmos-2251, c) Iridium-33.
The altitude profiles result from the ejection of the fragments from the collision
point. Fig. C2 shows the distributions of the ejection velocities of the fragments
of the three satellites using 2.5-m/s bins. These distributions can be roughly approximated
by the following formula
n(v) = dN v
dv
= N f
v s
·
ka
1 + a 2 , a = ( v
v s
) b
, (C3)
where n(v)dv is the number of fragments ejected at velocities between v and v +dv,
N v is the number of fragments ejected at velocities below v, N f is the total number
of fragments, v s is a characteristic velocity, b ≈ 1.6, and k ≈ 0.57 is a normalization
parameter, such that
∫ ∞
0
n(v) dv = N f .
(C4)
The characteristic velocity v s is approximately 50 m/s for Fengyun-1C, 35 m/s for
Cosmos-2251, and 20 m/s for Iridium-33. Again, the values of b and v s are not
precise and can be floated within certain ranges.
We note that the scale hight h s of the altitude density profile and the characteristic
ejection velocity v s are related approximately as
h s ≈ 2v s /ω,
(C5)
where ω is an average angular rate of the orbital motion. This is not surprising:
faster ejections should result in wider altitude distributions.
The ejection patterns in the observed events were not symmetric. However,
when we consider all possible collision conditions, the average ejection pattern
should be more symmetric. In a model case of a spherically symmetric ejection
from a circular orbit with the velocity distribution (C3), the altitude residence
density of the cloud of fragments can be calculated as
n(h) = 1 ∫ ∞ ∫ π ∫ π/2
n(v)dv dφ ρ(h) cos ψ dψ
4π 0
−π −π/2
(C6)