todays work

CurrentWriting/sections/01_LawsOfMotion.tex

@@ -7,24 +7,40 @@ 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 live indefinitely, these facts give us the following law of motion for each 
-constellation $i$.
+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
+    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.
-Note that it is reasonable to assume that the loss of satellites to collisions should be
-increasing in the level of debris: $\parder{l^i}{D_t}{} >0$.
+Where $l^i(\cdot)$ represents the rate at which satellites are destroyed by collisions, i.e.
+the satellite loss function.
+%Assumption: 
 
 \subsubsection{Collision Efficiencies}
 %TODO: Explain bit about constellation collision efficiencies.
@@ -34,33 +50,34 @@ I assume that when designing a constellation, the operator chooses to minimize c
 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?
+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.
-Thus although an operator may choos to minimize their total risk when launching 
-a constellation, later constellation formation will often add to the total risk.
+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 abandon at the time.\cref{ListOfOrbitalIncidents}
+    and one of them was abandoned at the time.\cref{ListOfOrbitalIncidents}
 }
-but this is due to the fact that satellites that evasive maneuvers are usually taken
+but this may be due to the fact that evasive maneuvers are usually taken
 when collisions appear reasonably possible.
 
 
-Consequent to these reasons, I believe the loss function $l^i$ should have the
-following properties related to satellite stocks.
+These collision efficiencies can be represented in the satellite loss function $l^i$ 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}
-Notably, an additional satellite in any constellation increases the probability of loosing
+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 of the additional satellite.
+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: $\parder{l^i}{D_t}{} >0$.
 
 \subsection{Debris}
 Debris is generated by various processes, including:
@@ -76,7 +93,7 @@ Debris is generated by various processes, including:
         all generate more debris.
 \end{itemize}
 It leaves orbit when atmospheric drag slows it down enough to reenter the atmosphere.
-Because the atmosphere is so negligible for many orbits, reentry can easily take decades
+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.
@@ -89,6 +106,7 @@ For simplicity, I formulate this more specifically as:
                 + \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