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81 lines
3.6 KiB
TeX
81 lines
3.6 KiB
TeX
\documentclass[../Main.tex]{subfiles}
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\graphicspath{{\subfix{Assets/img/}}}
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\begin{document}
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In this model there are two types of entities subject to laws of motion;
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i.e. constellation-level satellite stocks and debris.
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These laws are the foundations to the results found in \cref{SEC:Kessler,SEC:Survival}, and
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are crucial elements of the models presented in sections \cref{SEC:Operator,SEC:Planner}.
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\subsubsection{Satellite Stocks}
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Each constellation consists of a number of satellites in orbit, controlled by the same operator and
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operated for the same purpose.
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Satellites can be destroyed by collisions with other satellites or debris.
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Of course, satellite stocks can be increased by launching more satellites.
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Assuming satellites live indefinitely, these facts give us the following law of motion for each
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constellation $i$.
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\begin{align}
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S^i_{t+1} = \left( 1 - l^i(\{s^j_t\}, D_t)\right)s^i_t + x^i_t
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%Couple of Notes:
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% This does not allow for natural decay of satellites.
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% Nor does it include a deorbit decision.
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%
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%
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\end{align}
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Where $l^i(\cdot)$ represents the rate at which satellites are destroyed by collisions.
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Note that it is reasonable to assume that the loss of satellites to collisions should be
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increasing in the level of debris: $\parder{l^i}{D_t}{} >0$.
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\paragraph{Collision Efficiencies}
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%TODO: Explain bit about constellation collision efficiencies.
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As demonstrated by \cite{reiland2020}, there are constellation designs by which an operator can
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minimize the risk of intra-constellation collisions.
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I assume that when designing a constellation, the operator chooses to minimize collision risks,
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and as a result, there is a greater relative risk of inter-constellation collision.
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It is reasonable to ask why operators would not use the same techniques to reduce
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inter-constellation collision risks?
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While some of the steps could be taken, a fundamental issue arises in that constellations
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are operated for different purposes and require different orbital properties.
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%Maybe 2 operators can place themselves in low risk orbits, but adding a 3rd increases the risk to all of them.
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%This could be explained as Coordination across time (time travel doesn't exist yet)
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This coordination is also complicated by the fact that constellations are not
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designed nor launched at the same time.
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Consequent to these reasons, I believe the loss function $l^i$ should have the following properties related
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to satellite stocks.
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\begin{align}
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\parder{l^i}{s^k_t}{} > 0 ~~\forall k \in \{1,\dots,N)\\
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\parder{l^i}{s^j_t}{} > \parder{l^i}{s^i_t}{} ~~\forall j\neq i
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\end{align}
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\subsection{Debris}
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Debris is generated by various processes, including:
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\begin{itemize}
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\item Naturally occuring debris
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\item Satellite launches, operations, failures, or intentional destruction.
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\item Collisions between satellites
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\item Collisions between satellites and debris
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\item Collisions between debris
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\end{itemize}
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Debris leaves orbit when atmospheric drag slows it down enough to reenter the atmosphere.
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These effects can be represented by the following general law of motion.
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\begin{align}
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D_{t+1} = (1-\delta)D_t + g(D_t) + \gamma(\{s^j_t\},D_t) + \Gamma(\{x^j_t\})
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\end{align}
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I formulate this more specifically as:
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\begin{align}
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D_{t+1} = (1-\delta)D_t + g(D_t)
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+ \sum^N_{i=1} \gamma l^i(\{s^j_t\},D_t)
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+ \Gamma \sum^n_{j=1} \{x^j_t\}
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\end{align}
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where $\Gamma,\gamma$ represent the debris generated by each launch and collision respectively,
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while $\delta,g(\cdot)$ represent the decay rate of debris and the
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autocatalysis\footnote{
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Using terminology from \cite(RaoRondina2020).
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} of debris generation.
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\end{document}
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