Added the introduction (which includes the lit review), updated kessler region and syndrome sections

CurrentWriting/Main.tex

@@ -20,8 +20,6 @@
 \section{Introduction}
 \subfile{sections/00_Introduction} %TODO
 
-\section{Current Literature} % Lit Review
-\subfile{sections/08_CurrentLiterature}
 
 \section{Modeling the Environment}
 \subsection{Laws of Motion}
@@ -33,17 +31,6 @@
 \subsection{Kessler Syndrome}\label{SEC:Kessler}
 \subfile{sections/02_KesslerSyndrome}
 \subfile{sections/06_KesslerRegion} 
-%TODO:
-% So these sections need combined and rewritten
-% In particular, I want to describe the differences present in 
-% the proto-kessler region.
-% 
-% 
-% 
-% 
-% 
-% 
-% 
 
 
 

CurrentWriting/sections/00_Introduction.tex

@@ -2,6 +2,153 @@
 \graphicspath{{\subfix{Assets/img/}}}
 
 \begin{document}
-Introduction goes here. 
-Don't include much yet.
+n September of 2019, the European Space Agency (ESA) released a tweet explaining that they had performed an
+maneuver to avoid a collision with a SpaceX Starlink Satellite in Low Earth Orbit (LEO)\autocite{EsaTweet}.
+While later reports\autocite{ArsTechnicaStatement} described it as the result of miscommunications,
+ESA used the opportunity to highlight the difficulties arising from coordinating avoidance maneuvers and how
+such coordination will become more difficult as the size and number of 
+single purpose, single operator satellite fleets (satellite constellations) increase in low earth orbit\autocite{EsaBlog}.
+
+% Background on issues of congestion and pollution
+% Kessler Syndrome
+In spite of the fact that there is a lot of maneuvering room in outer space,
+%\footnote{``Space is big. Really big. You just won’t believe how vastly hugely mind bogglingly big it is. 
+%I mean, you may think it’s a long way down the road to the chemist, 
+%but that’s just peanuts to space.''\cite{DouglasAdams}}
+the repeated interactions of periodic orbits make collisions probable.
+Consequently, objects in orbit are subject to both a congestion effect and a pollution effect.
+Congestion effects are primarily derived from avoiding collisions between artificial satellites.
+Pollution in orbit consists of debris, both natural and man-made, which increases
+the probability of an unforeseen collision.
+The defining feature of pollution in orbit is that it self-propagates as debris collides with itself
+and orbiting satellites to generate more debris.
+This dynamic underlies a key concern, originally explored by Kessler and Cour-Palais \autocite{Kessler1978}
+that with sufficient mass in orbit (through satellite launches), the debris generating process
+could undergo a runaway effect rendering various orbital regions unusable.
+This cascade of collisions is often known as Kessler syndrome and 
+may take place over various timescales.
+
+% ---------------
+%Discuss how various definitions of kessler syndrome
+%    have been proposed in the economics literature to match the models.
+%Not sure if the following contributes much given the previous paragraph.
+%Although Kessler and Cour-Palais determined that a runaway pollution effect could make a set of orbits 
+%physically unusable, Adilov et al \autocite{adilov_alexander_cunningham_2018} %Kessler Syndrome
+%have shown that economic benefits provided by orbits will drop sufficiently to make the net marginal
+%benefit of new launches negative before the physical kessler syndrome occurs.
+
+% ---------------
+Orbits may be divided into three primary groups, 
+Low Earth Orbit (LEO),
+Medium Earth Orbit (MEO), and High Earth Orbit (HEO) where Geostationary Earth Orbit (GEO) 
+considered a particular classification of HEO. 
+While the topic of LEO allocation has historically remained somewhat unexplored, the last 6 years has seen
+a variety of new empirical studies and theoretical models published.
+
+% ---------------
+%Allocative efficiency
+
+Macauley  provided the first evidence of sub-optimal behavior in orbit
+by estimating the welfare loss due to the current method of assigning GEO slots to operators\autocite{Macauley_1998}.
+The potential losses due to anti-competitive behavior were highlighted by Adilov et al ,
+who have analyzed the opportunities for strategic 
+``warehousing'' of non-functional satellites as a means of increasing competitive advantage by 
+denying operating locations to competitors in GEO\autocite{Adilov2019}.
+
+The primary concern expressed in many of the published papers is whether or not orbits will be overused
+due to their common-pool nature, and which policies may prevent kessler syndrome.
+On this topic, Adilov, Alexander, and Cunningham examine pollution
+using a two-period salop model, incorporating the effects of launch debris on 
+survival into the second period\autocite{adilov_alexander_cunningham_2015}. 
+They find that the social planner generates debris and launches at lower rates 
+than a free entry market.
+
+This same result was found by Rao and Rondina  in
+the context of an infinite period dynamic model. 
+%Potential Edit
+Their approach is defined by the assumption that there are 
+numerous operators in a free entry environment who 
+can each launch a single, identical constellation\autocite{RaoRondina2020}.
+Rao, Burgess, and Kaffine use this model to estimate that achieving socially optimal 
+behavior through orbital use fees could increase the value generated by the 
+space industry by a factor of four\autocite{Rao2020}.
+
+
+% ---------------
+%In addition to analyzing the allocative results, a significant area of interest is 
+%what impact various policy interventions can have.
+%The policies and methods used to analyze their impact have been widely varied.
+% What policies have been evaluated?
+%   - Muller et al analyze debris removal
+%   - Grzelka and Wagner \autocite{GrzelkaWagner2019} explore methods of encouraging satellite quality (in terms of debris) and cleanup.
+%   - Rao compares launch vs operation taxes
+%   - Adilov et al ?????
+
+%Other papers to review:
+% Muller, Rozanova, Urdanoz (Economic Valuation of Debris Removal, IAC conference 2017)
+% Salter (Space Debris, Mercantus Working Paper 2015)
+% 
+% 
+
+% ---------------
+My %FP
+objective is to %explore the effects from organizing satellites into constellations on satellite launch decisions and operation.
+describe the dynamic decision-making process facing constellation operators, 
+how their launch decisions diverge from the socially optimal,
+and the ways in which various policies encourage or discourage optimal decision making.
+%I %FP
+%do this by extending Rao and Rondina's dynamic satellite operators model\autocite{RaoRondina2020} 
+%to account for non-symmetric constellation sizes and 
+%incorporate the effects of both economies of scale as satellites in constellations complement each other and 
+%collision avoidance efficiencies where satellites are less likely to collide with constellation members.
+%Explain what the article does.
+% The primary results of this paper are: 
+% preliminary development of the dynamic model, 
+% characterization of the general solutions to both the constellation operators' problems and
+% the fleet planner's problem, 
+% and an analysis of survival rates within constellations and the entire fleet.
+
+%Contribution statement
+%Adds to raoRondina2020 and adilov2018 in extedning to more diverse situations.
+This work is mainly a theoretical expansion of two models:
+\begin{itemize}
+    \item Rao and Rondina's model \autocite{RaoRondina2020} dynamic model.
+    \item Adilov et al's \autocite{adilov_alexander_cunningham_2018} dynamic model.
+\end{itemize}
+In addition to the expansion, I contribute a general computational solver that allows
+us to examine complex scenarios similar to those encountered in actual policymaking.
+%Similarities
+% - Rao
+%   - Law of debris:
+%   - law of motion for stocks
+% - Adilov
+%   - law of Debris
+%   - constellations
+%Differences
+% - Rao
+%   - constellation
+%   - avoicance efficiencies
+% - Adilov
+%   - Allows for non-firm participants
+%   - avoidance efficiencies
+
+Below I describe the similarities and differences to these previous models to the current one.
+As far as similarities go, it directly inherits the general laws of motion for debris and constellation stocks, 
+and follows the DSGE modelling approach chosen by Rao.
+
+It is distinguished from these most models by the way it accounts for the following factors:
+\begin{itemize}
+    \item Heterogeneous agent types, including commercial, scientific, and military.
+    \item Collision avoidance efficiencies, i.e. within-constellation collisions are highly unlikely.
+    \item Constellations are not assumed to be symmetric.
+\end{itemize}
+Notably, I differ from \autocite{RaoRondina2020} by allowing constellations to be asymmetric.
+\autocite{adilov_alexander_cunningham_2018} permit asymmetric constellations, but assume that all constellation operators are 
+profit maximizing firms operating in a competitive market with linear demand.
+Both Adilov et al and Rao and Rondina assume that satellite destruction rates are the same across
+constellations, and that the risk posed by an aditional satellite is the same both within
+and without the constellation in which it is launched.
+
+
+
 \end{document}

CurrentWriting/sections/02_KesslerSyndrome.tex

@@ -19,61 +19,104 @@ i.e. a runaway pollution effect, but instead corresponds to the end result of ke
 
 The second common definition of ``kessler syndrome'' is due to \cite{RaoRondina}.
 They define it in terms of a ``kessler region'', the set of satellite stocks and the debris level
-such that:
+such that the limit of debris in the future is infinite.
+Mathematically this can be represented as:
 \begin{align}
     \kappa = \left\{ \{s^j_t\}, D_t : 
         \lim_{k\rightarrow \infty} D_{t+k}\left(\{s^j_{t+k-1}\}, D_{t+k-1}, \{x^j\}\right) = \infty \right\}
 \end{align}
+There are a few issues with this approach, even though it captures the essence of kessler syndrome 
+better than the definition proposed by Adilov et al.
+The issues it faces are generally the case of not delineating between kessler regions
+with significantly different economic outcomes.
+% doesn't account for speed of divergence
+For example, one subset of the kessler region may render an orbital shell physically useless
+within a decade, while another subset increases the risk of satellite destruction by 1\% every ten thousand years.
+The former is a global emergency, while the latter is effectively non-existant.
+% Not computable.
+Finally, determining whether a series is divergent depends on constructing mathematical proofs.
+This makes it difficult to computationally identify whether one is within kessler syndrome.
+
+
+
+\subsection{Two approaches to kessler syndrome}
 
-\subsection{My approach to kessler syndrome}
 I propose to analyze kessler syndrome in a slightly more restricted fashion than \cite{RaoRondina}.
 I would define the $\epsilon$-kessler region as:
 \begin{align}
     \kappa = \left\{ \{s^j_t\}, D_t : 
             \forall k \geq 0, D_{t+k+1} - D_{t+k} \geq \epsilon > 0 \right\}
 \end{align}
+%show that this is similar to saying that all non \epsilon kessler regions are bounded by the 
+%derivative, i.e. are lipshiz
+The continuous time equivalent of this condition is an upper bound on the derivative of debris generation,
+thus leading to a lipshitz-like function.
+
 It is easily shown that this criteria is sufficient to guarantee Rao and Rondina's criteria. 
 It has three primary benefits:
 \begin{itemize}
     \item  % Can be solved for algebraically or numerically for a given, bounded state space.
         The $\epsilon$-kessler region can be numerically described within bounded state spaces.
     \item % This is what you would actually compute.
-        In a computational model, as most models of any complexity will be,
-        you cannot check for divergence numerically.
         The condition given is a basic guarantee of the divergent behavior that is
         required for Kessler Syndrome and acknowledges computational limitations.
-    \item Finally, a slow divergence is no divergence in the grand scheme of things.
+    \item 
+        Finally, a slow divergence is no divergence in the grand scheme of things.
         It is possible to have a mathematically divergent function, but one that is so slow,
         there is no noticable degree of debris growth before Sol enters a red giant phase.
         In this specification, it is possible to choose $\epsilon$ such that the divergent behaviors
-        identified have an impact on a meaningful timescale.
+        identified have an economic impact on a meaningful timescale.
 \end{itemize}
-
-
 % Issue with this approach: What about cyclical behaviors?
 % Autocatalysis leads to high debris leads to reduced launches 
 % which leads to debris decay leads to increased launches leads to Autocatalysis
 There is at least one issue with this definition of $\epsilon$-kessler regions.
-Let's define a ``proto-kesslerian'' region as the stock and debris levels such that:
-\begin{align}
-    \kappa = \left\{ \{s^j_t\}, D_t : 
-            D_{t+1} - D_{t} \geq \epsilon > 0 \right\}
-\end{align}
 It may be, under certain situations, the case that optimal launch rates cycle along with
-debris and stock levels, leading to a cycle in and out of the proto-kesslerian regions.
+debris and stock levels, leading to a cycle in and out of the $\epsilon$-kesslerian regions.
 This is an issue because, assumning a stable cycle, Rao's definition
 of the kessler region would capture this behavior, but the $\epsilon$-kessler definition 
 would not.
-I believe, but have not verified, that some choices of $\epsilon$, although permitting cycles,
+A particularly pathological case is where debris cycles between just below the cutoff level to 
+significantly above the cutoff, leading to a highly divergent behavior not captured by this definition.
+
+As far as computability goes, by simulating a phase diagram (for a given solution to the model)
+we can determine what sections are in the $\epsilon$-kessler region.
+This is a major benefit in a computational model.
+
+A related and more general concept is the ``proto-kesslerian'' region, which is 
+defined as the stock and debris levels such that:
+\begin{align}
+    \kappa = \left\{ \{s^j_t\}, D_t : 
+            D_{t+1} - D_{t} \geq \varepsilon > 0 \right\}
+\end{align}
+Note that the debris level is in a $\epsilon$-kessler region when it is in a proto-kesslerian region
+for all future periods.
+This even simpler to compute than the phase diagram, and can be used to generate a topological view
+of proto-kesslerian regions of degre $\varepsilon$.
+These are both easier to interpret and various approaches could be used to analyze how debris levels 
+transition between them.
+%what would the integral of gradients weighted by the dividing line give? just a thought.
+%Other thoughts
+% proto-kesslerian paths, paths that pass into a proto kesslerian region.
+In order to capture the cyclic behavior that $\epsilon$-kessler regions miss, we can define a type of 
+path in the phase diagram called a proto-kesslerian path of degree $\epsilon$, which is any path
+that enters the region. 
+For example, one could simulate a phase diagram and compare paths that fall into a given $\epsilon$-kessler region
+and paths that only temporarily pass into the equivalent proto-kesslerian regions.
+Comparing the number of paths that fall into each region may give a useful metric for policies that are 
+designed to decrease the likelihood of kessler syndrome.
+
+
+I believe, but have not verified, that some choices of $\varepsilon$, 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?
-This leads to the important question of ``What makes a good value of $\epsilon$?''
+This leads to the important question of ``What makes a good value of $\epsilon$ or $\varepsilon$?''
 One method, in the spirit of \cite{Adilov2018}, is to choose a change in debris, $D_{t+1} - D_t$, such that
-the loss of satellites in periods $t+1$ to $t+k$ is increased by or to a certain percentage, say 50\%.
+the loss of satellites in periods $t+1$ to $t+k$ is increased by or to a certain percentage, say 1\%.
 I've put very little thought into addressing this general question so far,
 and need to analyze the implications of different choice rules.
 

CurrentWriting/sections/06_KesslerRegion.tex

@@ -2,7 +2,8 @@
 \graphicspath{{\subfix{Assets/img/}}}
 
 \begin{document}
-Given the definition of kessler syndrome and the law of debris above, we can now 
+\subsection{Defining the Proto-Kessler Region}
+With the definitions of kessler syndrome and the law of debris given above, we can now 
 explicitly describe the proto-kessler region.
 \begin{align}
  \epsilon < -\delta D_t + g(D_t) + \gamma \sum^n_{j=1} l^i(\{s^j_t\},D_t) + \Gamma \sum^n_{j=1} \{x^j_t\}
@@ -11,5 +12,6 @@ As being in the proto-kessler region is a prerequesit to being in the kessler re
 the kessler region depends on the collision rates of the constellation operators.
 
 Although this is a straightforward result, I have not found it in any of the models I've examined so far.
-I suspect it will impact optimal pigouvian taxation, but of course, I need to verify this.
+I suspect it will impact optimal pigouvian taxation, but of course, I need to verify this in 
+a computational example.
 \end{document}

CurrentWriting/sections/08_CurrentLiterature.tex

@@ -1,17 +0,0 @@
-\documentclass[../Main.tex]{subfiles}
-\graphicspath{{\subfix{Assets/img/}}}
-
-\begin{document}
-%% Summary of literature
-% List of major research
-% 
-% 
-% 
-% 
-% 
-% 
-% 
-
-
-
-\end{document}