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

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and Ḣ1 and Ḣ2 are the decay rates at the altitudes H 1 and H 2 according to (32). As
all fragments from the layer dH 1 move to the layer dH 2 , the average spatial density
of fragments changes from n 1 to n 2 , conserving the total number of fragments,
n 1 dH 1 4πR 2 1
= n 2 dH 2 4πR 2 2. (34)
Taking into account (33), relation (34) translates into a curious “time-delayed”
form of the continuity equation for the vertical flow of fragments,
n 1 Ḣ 1 R 2 1
= n 2 Ḣ 2 R 2 2. (35)
If we disregard the change in the orbit radius compared to the change in the
air density, we will arrive at a remarkably simple relation describing the long-term
evolution of the fragment density,
n 1 ρ a (H 1 ) ≈ n 2 ρ a (H 2 ). (36)
For practical calculations, it is easier to integrate (32) backward in time, or
use a “reverse” equation
Ḣ = λ ρ a (H). (37)
Now, we can calculate the density n at any altitude H and time t as
n(H, t) ≈ n(H t , 0) ρ a (H t )/ρ a (H), (38)
where H t is the altitude reached at time t along the solution of equation (37) with
the initial altitude H, and n(H t , 0) is the initial density at the altitude H t at time
t = 0. A typical solution of equation (37) for small fragments with b c ∼ 0.5 m 2 /kg
is plotted in Fig. 8.
Fig. 8. A typical altitude function for small fragments.