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1. Space Environment
Lecture 37 – Space Debris (Vol. 3)
Talent’s analytic model
Professor Hugh Lewis
SESA3038 Space Environment

2. Overview of lecture 37
• In the previous lecture we introduced a simple systems model of space
debris that should allow us to make predictions of, and understand, the
growth of the space debris population
• In this lecture we introduce a model published by David Talent in 1992
that mirrors the systems model of space debris (but features some
additional complexity that leads to different forms of behaviour)
• We show how we can use this model to understand the fundamental
behaviour of the space debris population, especially the equilibria that
arise
• We show results from the 1992 paper to illustrate the predictive
capabilities
Space Environment – Space Debris (Vol. 3)

3. Systems thinking Space Environment – Space Debris (Vol. 3)
4. Space debris system
Number of
orbital
objects (N)
Collisions Re-entries
Launches
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
= 𝐴𝐴
The inflow due to launches
does not depend on the
number of objects already
in the system
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
= 𝐶𝐶𝑁𝑁2
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
= −𝐵𝐵𝐵𝐵
The collision rate is proportional to
the square of the number of objects
(c.f. the number of football matches
in a round-robin tournament)
A, B and C are parameters
that affect the flows in this
system.
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
= 𝐴𝐴 − 𝐵𝐵𝐵𝐵 + 𝐶𝐶𝑁𝑁2

4. Talent’s model Space Environment – Space Debris (Vol. 3)
• Same differential equation approach as the
systems model
• Also known as “Particles-in-a-box” model:
• 𝑁𝑁: number of objects in the environment
• 𝐴𝐴: deposition coefficient
• Launches
• Fragmentations
• Retrievals/removals
• 𝐵𝐵: atmospheric decay coefficient
• 𝐶𝐶: collision coefficient
𝑑𝑑𝑁𝑁
𝑑𝑑𝑑𝑑
= 𝐴𝐴 + 𝐵𝐵𝐵𝐵 + 𝐶𝐶𝑁𝑁2

5. Talent’s model Space Environment – Space Debris (Vol. 3)
• Same differential equation approach as the
systems model
• Also known as “Particles-in-a-box” model:
• 𝑁𝑁: number of objects in the environment
• 𝐴𝐴: deposition coefficient
• Launches
• Fragmentations
• Retrievals/removals
• 𝐵𝐵: atmospheric decay coefficient
• 𝐶𝐶: collision coefficient
𝑑𝑑𝑁𝑁
𝑑𝑑𝑑𝑑
= 𝐴𝐴 + 𝐵𝐵𝐵𝐵 + 𝐶𝐶𝑁𝑁2

6. Talent’s model Space Environment – Space Debris (Vol. 3)
• Equilibrium populations:
• Roots of the equation:
• are:
• where:
• If 𝑞𝑞 > 0: sinks > sources  conditionally stable
• If 𝑞𝑞 = 0: sinks ≈ sources  instability threshold
• If 𝑞𝑞 < 0: sinks < sources  unconditionally unstable
𝑑𝑑𝑁𝑁
𝑑𝑑𝑑𝑑
= 𝐴𝐴 + 𝐵𝐵𝐵𝐵 + 𝐶𝐶𝑁𝑁2
𝑁𝑁1,2 =
−𝐵𝐵 ± 𝐵𝐵2 − 4𝐴𝐴𝐴𝐴
2𝐶𝐶
𝑞𝑞 = 𝐵𝐵2 − 4𝐴𝐴𝐴𝐴 = sink terms – source terms

7. Talent’s model Space Environment – Space Debris (Vol. 3)
• Equilibrium populations:

8. Talent’s model results Space Environment – Space Debris (Vol. 3)
• Short-term v long-term

9. Overview of lecture 37
• In this lecture we introduced a model published by David Talent in 1992
that mirrors the systems model of space debris (but features some
additional complexity that leads to different forms of behaviour)
• We showed how we can use this model to understand the fundamental
behaviour of the space debris population, especially the equilibria that
arise
• We showed results from the 1992 paper to illustrate the predictive
capabilities
Space Environment – Space Debris (Vol. 3)

10. Activities
• You can read the paper by David Talent
from 1992 (available on Blackboard):
– Especially section 3: “Preliminary
space debris considerations for the
development of large constellations
of satellites in LEO”
• Further reading (not on Blackboard):
– Book: “Thinking in Systems” by
Donella H. Meadows
• Remember the Microsoft Excel
activities that build on systems thinking
and Talent’s model
Space Environment – Space Debris (Vol. 3)