diff --git a/CurrentWriting/Main.pdf b/CurrentWriting/Main.pdf index cb0888f..69641e2 100644 Binary files a/CurrentWriting/Main.pdf and b/CurrentWriting/Main.pdf differ diff --git a/CurrentWriting/Main.tex b/CurrentWriting/Main.tex index 8f7ba02..a6d5673 100644 --- a/CurrentWriting/Main.tex +++ b/CurrentWriting/Main.tex @@ -19,23 +19,46 @@ \newpage \section{Introduction} -\subfile{sections/00_Introduction} +\subfile{sections/00_Introduction} %TODO \section{Laws of Motion} -\subfile{sections/01_LawsOfMotion} +\subfile{sections/01_LawsOfMotion} %DONE \section{Kessler Syndrome}\label{SEC:Kessler} \subfile{sections/02_KesslerSyndrome} -\section{Survival Analysis}\label{SEC:Survival} -\subfile{sections/03_SurvivalAnalysis} +\section{Motion Results}\label{SEC:MotionResults} +The following are two results due to the laws of motion presented. -\section{Constellation Operator's Program}\label{SEC:Operator} -\subfile{sections/04_ConstellationOperator} +\subsection{Kessler Regions} +\subfile{sections/06_KesslerRegion} %TODO -\section{Social Planner's Program}\label{SEC:Planner} -\subfile{sections/05_SocialPlanner} +\subsection{Survival Analysis}\label{SEC:Survival} +\subfile{sections/03_SurvivalAnalysis} %TODO +\section{Model} +\subsection{Constellation Operator's Program}\label{SEC:Operator} +\subfile{sections/04_ConstellationOperator} %TODO + +\subsection{Social Planner's Program}\label{SEC:Planner} +\subfile{sections/05_SocialPlanner} %TODO + +\section{Computation} +No work has been done here so far. + +\section{Assumptions and Caveats} +I hope to write a section clearly explaining assumptions, caveats, and shortcomings here. +%time periods are long enough for debris to disperse after collisions. +%Only a single type of debris +%With my current computational idea; each constellation provides the same risk to each other constellation +% That can be easily adjusted in the computational models. + +\section{Appedicies} +\subsection{Mathematical Notation} +Needs completed. +\subsection{Derivations} +\subsubsection{Marginal Survival Rates}\label{APX:Derivations:SurvivalRates} +\subfile{sections/appedicies/apx_01_MarginalSurvivalRates} \end{document} diff --git a/CurrentWriting/sections/01_LawsOfMotion.tex b/CurrentWriting/sections/01_LawsOfMotion.tex index 26cbb05..9138694 100644 --- a/CurrentWriting/sections/01_LawsOfMotion.tex +++ b/CurrentWriting/sections/01_LawsOfMotion.tex @@ -63,7 +63,9 @@ These effects can be represented by the following general law of motion. \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 + D_{t+1} = (1-\delta)D_t + g(D_t) + + \sum^N_{i=1} \gamma l^i(\{s^j_t\},D_t) + + \Gamma \sum^n_{j=1} \{x^j_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 diff --git a/CurrentWriting/sections/02_KesslerSyndrome.pdf b/CurrentWriting/sections/02_KesslerSyndrome.pdf deleted file mode 100644 index 8bfe422..0000000 Binary files a/CurrentWriting/sections/02_KesslerSyndrome.pdf and /dev/null differ diff --git a/CurrentWriting/sections/02_KesslerSyndrome.tex b/CurrentWriting/sections/02_KesslerSyndrome.tex index d82e252..fb1467c 100644 --- a/CurrentWriting/sections/02_KesslerSyndrome.tex +++ b/CurrentWriting/sections/02_KesslerSyndrome.tex @@ -67,6 +67,7 @@ would not. I believe, but have not verified, that some choices of $\epsilon$, although permitting cycles, would relegate them to levels with minimal economic impact. %Maybe can be studies by phase or flow diagrams? +%Consider where it cycles between just below epsilon and then to a large increase in debris? %Area of research: What makes a good \epsilon? diff --git a/CurrentWriting/sections/03_SurvivalAnalysis.tex b/CurrentWriting/sections/03_SurvivalAnalysis.tex index 75008c1..c98b806 100644 --- a/CurrentWriting/sections/03_SurvivalAnalysis.tex +++ b/CurrentWriting/sections/03_SurvivalAnalysis.tex @@ -2,7 +2,60 @@ \graphicspath{{\subfix{Assets/img/}}} \begin{document} -Introduction goes here -\subsection{testing} -This is a subsection of the introduction. +In his dissertation \cite{RaoDissertation} briefly examines the "survival rates" of a satellite constellation. +I've applied this to my model and extended the results. + +% Marginal survival. +The survival rate for a constellation $i$ is defined as $R_i = 1-l^i(\cdot)$, the proportion of satellites- +that were not lost (degraded nor destroyed) between period $t$ and $t+1$. +Thus the marginal survival rate represents the additional loss of +satellites due to a slightly larger constellation or fleet stock. + +Let $S_t = \sum^n_{j=1} s^j_t$. +Then the survival rates for a constellation and for society's fleet are respectively defined as: +\begin{align} + R_i =& \frac{s^i_{t+1}- x^i_t}{s^i_t} = 1- l^i(s^i_t,S_t,D_t) \\ + R =& \frac{S_{t+1}- X_t}{S_t} = \frac{\sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] }{S_t} \label{EQ:socsurv} +\end{align} +In this case, the fleet survival rate \cref{EQ:socsurv}, represents the proportion of satellites- +in period $t+1$ that survived from period $t$. + +The marginal survival rates when a given constellation $i$ changes size are: +\begin{align} + \parder{R_i}{s^i_t}{} =& -\left(\parder{l^i}{s^i_t}{} + \parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{} \right) + = - \parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{} \label{EQ:iii} \\ + \parder{R}{s^i_t}{} =& \frac{S_t \sum_{i=1}^N \left( [1-l^i(s^i_t,S_t,D_t)] + + s^i_t [ -\parder{l^i}{s^i_t}{} -\parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{}] \right) + - \left( \sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] \right)} + {(S_t)^2} \notag{}\\ + =& \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t} + +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \label{EQ:i} +\end{align} +Note that $ \sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}$ is the weighted, average marginal survival rate- +across constellation operators. +The derivation of \cref{EQ:i} is in Appendix \ref{APX:Derivations:SurvivalRates}. +Direct comparison between the marginal survival rates of an individual operator and the social planner's fleet +cannot proceed further without specifying the functional loss forms $l^i(\cdot)$ +and specifying which firm will be compared to society. +In spite of this, conditions on the average effects can be developted as follows. + +The marginal survival rate of the fleet is less than the weighted, arithmetic mean of marginal survival rates- +of the constellations when: +\begin{align} + \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t} + +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} + \leq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\ + \sum_{i=1}^N R_i - R \leq& 0\\ + \sum_{i=1}^N [1- l^i(s^i_t,S_t,D_t)] - \sum_{i=1}^N s^i_t [1- l^i(s^i_t,S_t,D_t)] \leq& 0\\ + \sum_{i=1}^N (1 - s^i_t) [1- l^i(s^i_t,S_t,D_t)] \leq& 0 \label{EQ:ii} +\end{align} +which is true if every constellation has at least one satellite. +As any constellation of interest has at least one satellite +and $\parder{R_i}{s^i_t}{} < 0$ from the assumption on collision mechanics that $\der{l^i}{s_t^i}{}>0$, +we conclude that the marginal survival rate of the entire satellite fleet is lower +than the weighted arithmetic mean of marginal survival rates across constellations. +Note that it is possible for some constellations to have a lower marginal survival rate than the fleet, +but the survival rate for many operators must be higher than the societal rate. +Consequently, we would expect many operators to underestimate the impact of their behaviors on others +if they use their own observed or expected risk factors to estimate the risk they impose on others. \end{document} diff --git a/CurrentWriting/sections/appedicies/apx_01_MarginalSurvivalRates.tex b/CurrentWriting/sections/appedicies/apx_01_MarginalSurvivalRates.tex new file mode 100644 index 0000000..5f39028 --- /dev/null +++ b/CurrentWriting/sections/appedicies/apx_01_MarginalSurvivalRates.tex @@ -0,0 +1,24 @@ +\documentclass[../Main.tex]{subfiles} +\graphicspath{{\subfix{Assets/img/}}} + +\begin{document} +Derivations related to the marginal survival rate analysis. + +\begin{align*} + \parder{R_i}{s^i_t}{} =& -\left(\parder{l^i}{s^i_t}{} + \parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{} \right) + = - \parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{} \\ + \parder{R}{s^i_t}{} =& \frac{S_t \sum_{i=1}^N + \left( [1-l^i(s^i_t,S_t,D_t)] + s^i_t [ -\parder{l^i}{s^i_t}{} -\parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{}] \right) + - \left( \sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] \right)}{(S_t)^2} \\ + =& \sum_{i=1}^N \left[ \frac{S_t [1-l^i(s^i_t,S_t,D_t)]}{(S_t)^2} + - \frac{ s^i_t[1-l^i(s^i_t,S_t,D_t)] }{(S_t)^2} \right] + +\sum_{i=1}^N \frac{ s^i_t S_t [ -\parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{}] }{(S_t)^2} \\ + =& \sum_{i=1}^N \left[ \frac{S_t - s^i_t}{(S_t)^2}[1-l^i(s^i_t,S_t,D_t)] \right] + +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \\ + =& \sum_{i=1}^N \left[ \frac{1}{S_t}[1-l^i(s^i_t,S_t,D_t)] \right] - \frac{R}{S_t} + +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \\ + =& \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t} + +\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} +\end{align*} + +\end{document}