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In this model there are two types of entities subject to laws of motion; 
i.e. constellation-level satellite stocks and debris.
These laws are the foundations to the results found in \cref{SEC:Kessler,SEC:Survival}, and 
are crucial elements of the models presented in sections \cref{SEC:Operator,SEC:Planner}.

\subsubsection{Mathematical Preliminaries}
Throughout the remainder of the paper, the following notation will be used.
Superscripts $s^i$ denote satellite constellations while
subscripts $s_t$ denote time periods.
\begin{itemize}
    \item $s^i_t$ represents the number of satellites in a constellation $i$ in period $t$.
        This is often referred to as the satellite ``stock'' of a constellation.
    \item $x^i_t$ represents the number of satellites launched as part of constellation $i$ 
        in period $t$
    \item $D_t$ represents the level of debris at period $t$.
\end{itemize}
In the case of satellite stocks, often the set of stocks for each constellation needs to 
be discussed. 
I've used curly braces around to denote this set, i.e. $\{ s^j_t \}$ represents the set
of constellations stocks, ordered by index $j$.

\subsubsection{Satellite Stocks}
Each constellation consists of a number of satellites in orbit, controlled by the same operator and
operated for the same purpose.
Satellites can be destroyed by collisions with other satellites or debris.
Of course, satellite stocks can be increased by launching more satellites.
Assuming satellites are not actively deorbited, we get the 
following general law of motion for each constellation $i$.
\begin{align}
    s^i_{t+1} = \left( 1 - l^i(\{s^j_t\}, D_t)\right)s^i_t + x^i_t
    %Couple of Notes:
    %  This does not allow for natural decay of satellites.
    %  Nor does it include a deorbit decision.
    %  Representing those might be:
    %  - \eta s^i_t - y^i_t
\end{align}
Where $l^i(\cdot)$ represents the rate at which satellites are destroyed by collisions.
%Assumption: 

\subsubsection{Collision Efficiencies}
%TODO: Explain bit about constellation collision efficiencies.
As demonstrated by \cite{reiland2020}, there are constellation designs by which an operator can
minimize the risk of intra-constellation collisions.
I assume that when designing a constellation, the operator chooses to minimize collision risks,
and as a result, there is a greater relative risk of inter-constellation collision.

It is reasonable to ask why operators would not use the same techniques to reduce 
inter-constellation collision risks.
While some of the steps could be taken, a fundamental issue arises in that constellations
are operated for different purposes and require different orbital properties.
%Maybe 2 operators can place themselves in low risk orbits, but adding a 3rd increases the risk to all of them.
%This could be explained as Coordination across time (time travel doesn't exist yet)
This coordination is also complicated by the fact that constellations are not
designed nor launched at the same time.
Consequently an operator may choos to minimize their total risk when launching 
a constellation, the later launch of constellations may lead to a suboptimal orbit design.
It is important to note that satellite-on-satellite collisions are rare\footnote{
    I am only aware of one collision between satellites, 
    and one of them was abandoned at the time.\cref{ListOfOrbitalIncidents}
}
but this may be due to the fact that evasive maneuvers are usually taken
when collisions appear reasonably possible.


These collision efficiencies can be represented in the satellite destruction rate $l^i(\cdot)$ when:
\begin{align}
    \parder{l^i}{s^k_t}{} > 0 ~~\forall k \in \{1,\dots,N)\\
    \parder{l^i}{s^j_t}{} > \parder{l^i}{s^i_t}{}  ~~\forall j\neq i
\end{align}
Note that an additional satellite in any constellation increases the probability of loosing
a satellite from a given constellation, and this risk is lower 
for the home constellation of the additional satellite.

Note that it is reasonable to assume that the loss of satellites to collisions should be
increasing in the level of debris: $\parder{l^i}{D_t}{} >0$.

\subsubsection{Debris}
Debris is generated by various processes, including:
\begin{itemize}
    \item Naturally occuring debris is captured from interplanetary space.
    \item Satellite launches, operations, failures, or intentional destruction.
    \item Collisions between 
        \begin{itemize}
            \item Two satellites
            \item A satellite and debris
            \item Two pieces of debris
        \end{itemize}
        all generate more debris.
\end{itemize}
It leaves orbit when atmospheric drag slows it down enough to reenter the atmosphere.
Because the atmosphere is negligible for many orbits, reentry can easily take decades
or centuries.

These effects can be represented by the following general law of motion.
\begin{align}
    D_{t+1} = (1-\delta)D_t + g(D_t) + \gamma(\{s^j_t\},D_t) + \Gamma(\{x^j_t\})
\end{align}
For simplicity, I formulate this more specifically as:
\begin{align}
    D_{t+1} = (1-\delta)D_t + g(D_t) 
                + \sum^N_{i=1} \vec \gamma l^i(\{s^j_t\},D_t)
                + \vec \Gamma \sum^n_{j=1} \{x^j_t\}
\end{align}
%WORKING HERE
where $\vec \Gamma,\vec \gamma$ represent the debris generated by each 
launch and collision respectively,
while $\delta,g(\cdot)$ represent the decay rate of debris and the 
autocatalysis\footnote{
    Using terminology from \cite(RaoRondina2020).
} of debris generation.

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