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1. Space Environment
Lecture 34 – Space Debris (Vol. 3)
Introduction to modelling (part 1)
Professor Hugh Lewis
SESA3038 Space Environment

2. Overview of lecture 34
• We begin our look at space debris modelling and models in this lecture.
• This lecture is a brief introduction to mathematical modelling, in general,
and together with the next lecture, will answer questions such as:
– What are mathematical models?
– What makes a good model?
• Following this introduction we will use the remainder of this week’s
lectures to develop a simple debris model and, subsequently, to add
complexity
• Note that all 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 are mathematical models?
A model…
• Is a conceptual tool that explains how an object or system of objects will behave
• Is a refined and precise mathematical description of the system
• Allows scientists to translate between behaviour at various scales, or extrapolate
from a known set of conditions to another
• PREDICTION: e.g. predict the population-level dynamics from an object-level
knowledge of space environment and orbital factors, or predict the long-term
behaviour from early behaviour, to understand the outcome of a collision or
the benefits of debris mitigation perhaps
Space Environment – Space Debris (Vol. 3)

5. What are mathematical models?
Models come in a variety of forms…
• Highly complex models that need a variety of experts to create them (e.g. a
space debris model might need expertise in hypervelocity impacts, orbital
mechanics, solar dynamics, etc.)
• Simple models that can be easily understood, modified and adapted.
What sort of model is the most appropriate?
• Depends on:
• The question being asked
• The level of precision or generality required
• The available data
• The time frame by which results are needed
Space Environment – Space Debris (Vol. 3)

6. What are mathematical models?
“All models are wrong, some are useful”
• Even the most complex model will make some simplifying assumptions
• We want models that capture the ESSENTIAL features of a system
• It is difficult to say definitively which model is “right”
• The measures of usefulness of any model are subjective
Formulating a model is a trade-off between
• Three important and often conflicting elements:
• ACCURACY
• TRANSPARENCY
• FLEXIBILITY
Space Environment – Space Debris (Vol. 3)

7. What are mathematical models?
Accuracy
• The ability to reproduce the observed data and reliably predict future dynamics
• Can be qualitative (e.g. needed to advise on future mitigation policies)
• Can be quantitative (e.g. needed to provide insights into problem dynamics)
• Generally, accuracy improves with model complexity
• Generally, accuracy improves with the inclusion of more
HETEROGENEITIES and relevant debris detail
• Be wary: accuracy is not the same as precision (e.g. darts on a dart board)
Space Environment – Space Debris (Vol. 3)

8. What are mathematical models?
Accuracy
• Be wary: accuracy is not the same as precision (e.g. darts on a dart board)
• Adding complexity is difficult because of…
– Increased computational power
– The need to have the mechanistic understanding of detailed processes
– The sometimes limited availability of necessary parameters and data
– Hence: ACCURACY IS ALWAYS LIMITED
Machine Learning seems to provide a way around some of these limitations but don’t be fooled…
Space Environment – Space Debris (Vol. 3)

9. What are mathematical models?
Transparency
• Comes from being able to understand (either analytically or numerically) how
the various model components influence the dynamics and interact
• It is achieved by adding or removing components and building upon general
intuitions from simpler models
– As the number of model components increases, it becomes more difficult to
assess the role of each component and its interactions with the whole.
– TRANSPARENCY IS OFTEN IN DIRECT OPPOSITION TO ACCURACY
Space Environment – Space Debris (Vol. 3)

10. What are mathematical models?
Flexibility
• Measures the ease with which the model can be adapted to new situations
• This is a vital quality if the model is used to evaluate mitigation policies or
predict future debris levels in an ever-changing environment.
• Most mechanistic models are based on well-understood debris evolution
principles and are therefore highly flexible.
• “Black box” time-series tools (e.g regression models, Machine Learning
models) may be able to accurately reproduce a given time-series of reported
debris, but are not often flexible.
The first Microsoft Excel activity looks at this issue in more depth
Space Environment – Space Debris (Vol. 3)

11. Recap of lecture 34
• We began our look at space debris modelling and models in this lecture.
• We briefly introduced mathematical modelling and answered the question:
– What are mathematical models?
• In the next lecture we will look for answers to additional questions:
– What models can do
– What models cannot do
– What makes a good model?
• As mentioned at the beginning, all 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)