diff --git a/CurrentWriting/Main.tex b/CurrentWriting/Main.tex index d3767a0..c07fbdc 100644 --- a/CurrentWriting/Main.tex +++ b/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. -% -% -% -% -% -% -% diff --git a/CurrentWriting/sections/00_Introduction.tex b/CurrentWriting/sections/00_Introduction.tex index bf92334..0f0531c 100644 --- a/CurrentWriting/sections/00_Introduction.tex +++ b/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} diff --git a/CurrentWriting/sections/02_KesslerSyndrome.tex b/CurrentWriting/sections/02_KesslerSyndrome.tex index fb1467c..2ec52ca 100644 --- a/CurrentWriting/sections/02_KesslerSyndrome.tex +++ b/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. diff --git a/CurrentWriting/sections/06_KesslerRegion.tex b/CurrentWriting/sections/06_KesslerRegion.tex index 635b331..880c9d8 100644 --- a/CurrentWriting/sections/06_KesslerRegion.tex +++ b/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} diff --git a/CurrentWriting/sections/08_CurrentLiterature.tex b/CurrentWriting/sections/08_CurrentLiterature.tex deleted file mode 100644 index f3ed18f..0000000 --- a/CurrentWriting/sections/08_CurrentLiterature.tex +++ /dev/null @@ -1,17 +0,0 @@ -\documentclass[../Main.tex]{subfiles} -\graphicspath{{\subfix{Assets/img/}}} - -\begin{document} -%% Summary of literature -% List of major research -% -% -% -% -% -% -% - - - -\end{document}