added sections on survival analysis and other associated changes.

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}

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 

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?

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}

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}