From d4b892baddeeda4bdc09cad71c336ec29cecca76 Mon Sep 17 00:00:00 2001 From: Will King Date: Thu, 4 Mar 2021 14:42:43 -0800 Subject: [PATCH] Wrote the section on laws of motion --- CurrentWriting/sections/00_Introduction.tex | 7 ++ CurrentWriting/sections/01_LawsOfMotion.tex | 74 +++++++++++++++++++ .../sections/02_KesslerSyndrome.tex | 8 ++ .../sections/03_SurvivalAnalysis.tex | 8 ++ .../sections/04_ConstellationOperator.tex | 8 ++ CurrentWriting/sections/05_SocialPlanner.tex | 8 ++ CurrentWriting/sections/SubfileTemplate.tex | 8 ++ 7 files changed, 121 insertions(+) create mode 100644 CurrentWriting/sections/00_Introduction.tex create mode 100644 CurrentWriting/sections/01_LawsOfMotion.tex create mode 100644 CurrentWriting/sections/02_KesslerSyndrome.tex create mode 100644 CurrentWriting/sections/03_SurvivalAnalysis.tex create mode 100644 CurrentWriting/sections/04_ConstellationOperator.tex create mode 100644 CurrentWriting/sections/05_SocialPlanner.tex create mode 100644 CurrentWriting/sections/SubfileTemplate.tex diff --git a/CurrentWriting/sections/00_Introduction.tex b/CurrentWriting/sections/00_Introduction.tex new file mode 100644 index 0000000..bf92334 --- /dev/null +++ b/CurrentWriting/sections/00_Introduction.tex @@ -0,0 +1,7 @@ +\documentclass[../Main.tex]{subfiles} +\graphicspath{{\subfix{Assets/img/}}} + +\begin{document} +Introduction goes here. +Don't include much yet. +\end{document} diff --git a/CurrentWriting/sections/01_LawsOfMotion.tex b/CurrentWriting/sections/01_LawsOfMotion.tex new file mode 100644 index 0000000..26cbb05 --- /dev/null +++ b/CurrentWriting/sections/01_LawsOfMotion.tex @@ -0,0 +1,74 @@ +\documentclass[../Main.tex]{subfiles} +\graphicspath{{\subfix{Assets/img/}}} + +\begin{document} +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}. + +\subsection{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 live indefinitely, these facts give us the following 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 +\end{align} +Where $l^i(\cdot)$ represents the rate at which satellites are destroyed by collisions. +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{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 many of the constellations that +will add to the overall risk have not been concieved by their designers yet. + + +Consequent to these reasons, I believe the loss function $l^i$ should have the following properties related +to satellite stocks. +\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} + + +\subsection{Debris} +Debris is generated by various processes, including: +\begin{itemize} + \item Naturally occuring debris + \item Satellite launches, operations, failures, or intentional destruction. + \item Collisions between satellites + \item Collisions between satellites and debris + \item Collisions between debris +\end{itemize} +Debris leaves orbit when atmospheric drag slows it down enough to reenter the atmosphere. + +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} +I formulate this more specifically as: +\begin{align} + D_{t+1} = (1-\delta)D_t + g(D_t) + \sum^N_{i=1} \gamma l^i(\{s^j_t\},D_t) + \Gamma x^i_t +\end{align} +where $\Gamma,\gamma$ 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. + +\end{document} diff --git a/CurrentWriting/sections/02_KesslerSyndrome.tex b/CurrentWriting/sections/02_KesslerSyndrome.tex new file mode 100644 index 0000000..75008c1 --- /dev/null +++ b/CurrentWriting/sections/02_KesslerSyndrome.tex @@ -0,0 +1,8 @@ +\documentclass[../Main.tex]{subfiles} +\graphicspath{{\subfix{Assets/img/}}} + +\begin{document} +Introduction goes here +\subsection{testing} +This is a subsection of the introduction. +\end{document} diff --git a/CurrentWriting/sections/03_SurvivalAnalysis.tex b/CurrentWriting/sections/03_SurvivalAnalysis.tex new file mode 100644 index 0000000..75008c1 --- /dev/null +++ b/CurrentWriting/sections/03_SurvivalAnalysis.tex @@ -0,0 +1,8 @@ +\documentclass[../Main.tex]{subfiles} +\graphicspath{{\subfix{Assets/img/}}} + +\begin{document} +Introduction goes here +\subsection{testing} +This is a subsection of the introduction. +\end{document} diff --git a/CurrentWriting/sections/04_ConstellationOperator.tex b/CurrentWriting/sections/04_ConstellationOperator.tex new file mode 100644 index 0000000..75008c1 --- /dev/null +++ b/CurrentWriting/sections/04_ConstellationOperator.tex @@ -0,0 +1,8 @@ +\documentclass[../Main.tex]{subfiles} +\graphicspath{{\subfix{Assets/img/}}} + +\begin{document} +Introduction goes here +\subsection{testing} +This is a subsection of the introduction. +\end{document} diff --git a/CurrentWriting/sections/05_SocialPlanner.tex b/CurrentWriting/sections/05_SocialPlanner.tex new file mode 100644 index 0000000..75008c1 --- /dev/null +++ b/CurrentWriting/sections/05_SocialPlanner.tex @@ -0,0 +1,8 @@ +\documentclass[../Main.tex]{subfiles} +\graphicspath{{\subfix{Assets/img/}}} + +\begin{document} +Introduction goes here +\subsection{testing} +This is a subsection of the introduction. +\end{document} diff --git a/CurrentWriting/sections/SubfileTemplate.tex b/CurrentWriting/sections/SubfileTemplate.tex new file mode 100644 index 0000000..75008c1 --- /dev/null +++ b/CurrentWriting/sections/SubfileTemplate.tex @@ -0,0 +1,8 @@ +\documentclass[../Main.tex]{subfiles} +\graphicspath{{\subfix{Assets/img/}}} + +\begin{document} +Introduction goes here +\subsection{testing} +This is a subsection of the introduction. +\end{document}