This lecture discusses mathematical models of space debris. Models can be used for prediction, such as forecasting debris amounts, or for understanding, by examining factors in isolation. While models provide an idealized representation, they have limitations and cannot perfectly predict outcomes. A good model suits its intended purpose, balances accuracy, transparency and flexibility, and is parameterized based on available data. The lecture focuses on defining the appropriate uses and standards for effective debris modeling.

1. Space Environment
Lecture 35 – Space Debris (Vol. 3)
Introduction to modelling (part 2)
Professor Hugh Lewis
SESA3038 Space Environment

2. Overview of lecture 35
• In the previous lecture, we began our look at space debris modelling and
models, and we answered the question:
– What are mathematical models?
• In lecture 35 we will look at what models can do and what models cannot
do, and we will try to answer the question:
– What makes a good model?
• As mentioned in the previous lecture, following this introduction to
mathematical modelling, we will use the remainder of this week’s lectures
to develop a simple debris model and, subsequently, to add complexity
• Note that, again, many of the slides in this lecture are marked with red
borders, indicating their importance and relevance to our topic.
Space Environment – Space Debris (Vol. 3)

3. Mathematical modelling
• What are mathematical models?
• What mathematical models can do
• What mathematical models cannot do
• What is a good model?
The content for this lecture is based on my experience and understanding, but much is also based on:
M.J. Keeling and P. Rohani. “Modeling Infectious Diseases in Humans and Animals” Princeton University Press, New Jersey, USA, 2008
Space Environment – Space Debris (Vol. 3)

4. What models can do
Models have two distinct roles:
• PREDICTION
• UNDERSTANDING
• These roles are related to the properties of accuracy and transparency, so are
often in conflict
– We require a high degree of accuracy from any predictive model
– Transparency is a more important quality of models used to improve our
understanding
Space Environment – Space Debris (Vol. 3)

5. What models can do
Prediction:
• Perhaps the most obvious use of models
– We need the model to be as accurate as possible
– The model includes all known and relevant complexities and
heterogeneities in the population
Space Environment – Space Debris (Vol. 3)

6. What models can do
Prediction:
• BUT predictions of space debris can vary widely from model to model, from
scenario to scenario, and even within the same scenario.
• Indicates a lack of repeatability (accuracy) and the presence of bias:
– Not enough data or poor parameterisation
– Measurement error
– Large amounts of uncertainty
– Using too many variables (over complicated)
• “Experts and their models are no better than a layperson at making quality
decisions within these systems.”
SO:
• WE NEED TO ASK THE RIGHT QUESTIONS AND BUILD THE RIGHT
MODELS TO OBTAIN THE ANSWERS
Space Environment – Space Debris (Vol. 3)

7. What models can do
Prediction:
• The wrong questions:
• “How many objects will be in orbit in the year 2050?”
• “Which objects will be involved in collisions over the next five years?”
• The right questions:
• “What can we do to reduce the amount of debris?”
• “What mitigation measures are the most cost-effective?”
• “Is debris growth under control?”
Space Environment – Space Debris (Vol. 3)

8. What models can do
Prediction:
• Guide decisions where a trade-off exists between two or more alternative
control strategies (be conscious that different models can provide conflicting
advice)
• Accurate models can also be used as a diagnostic tool:
• The failure to predict behaviour in a particular location (e.g. in a particular
orbit) can help to identify underlying parameters and behaviour that is
different from the norm
Space Environment – Space Debris (Vol. 3)

9. What models can do
Understanding:
• Models provide an “ideal world” in which individual factors can be examined in
isolation and where the behaviour of every element is recorded in perfect detail
• Insights gained are often robust and generic
• Can be applied to a wide variety of problems
• Can help us to design more sophisticated predictive models
• Can help us to gather more relevant data
• Can help us to identify elements that are important and elements that can be
neglected
• Building from simple models to more complex ones can enable an
understanding of the rich complexities and dynamics that are observed in the
real world
Space Environment – Space Debris (Vol. 3)

10. What models cannot do
• MODELS HAVE LIMITATIONS
• IT IS IMPOSSIBLE TO BUILD A FULLY ACCURATE MODEL
• There will always be some element of behaviour or quirk of the environment
that is UNKNOWN or UNKNOWABLE
• We will never be able to predict the precise evolution of the debris
population, or which objects will be involved in a collision
• The best we can achieve:
– Models that provide CONFIDENCE INTERVALS
– Models that can determine the risks for different GROUPS or TYPES of
object
Space Environment – Space Debris (Vol. 3)

11. What is a good model?
• Two points define a good model:
1. The model should be SUITED TO ITS PURPOSE (i.e. it should be as simple
as possible, but no simpler)
2. The model should have an appropriate BALANCE of accuracy, transparency
and flexibility
• A model designed to help us understand should concentrate on the
characteristics of interest while simplifying all others
• A model built for accurate prediction should represent the full system dynamics
and include all relevant features of the environment and the objects in it
• Determining what is relevant is difficult!
Space Environment – Space Debris (Vol. 3)

12. What is a good model?
• Other factors needed for a good model:
• All relevant and important elements of the model should be
PARAMETERIZABLE (where needed) from available data
– Early in the history of a system, it might be difficult or impossible to
produce a good predictive model because of the lack of data (e.g. a novel
epidemic)
• If we are only interested in understanding the system, there is less need for
the model to accurately represent a particular scenario and availability of
data is less important.
• What makes a model “good” or “right” is subjective and context dependent
Space Environment – Space Debris (Vol. 3)

13. Recap of lecture 35
• In this lecture we looked at what models can do. In particular we focused
on the two primary purposes of models:
– Prediction, e.g. for guiding policy decisions or for diagnosing problems
– Understanding, e.g. to help identify the important elements and
parameters
• We also looked at what models cannot do, and highlighted the fact that it
is impossible to build a perfect model
• Finally, we tried to answer the question, what makes a good model?
– The model should be suited to its purpose, have the right balance of
accuracy, transparency and flexibility, and is parameterizable
• Note that, again, many of the slides in this lecture were marked with red
borders, indicating their importance and relevance to our topic.
Space Environment – Space Debris (Vol. 3)