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131 lines
7.2 KiB
TeX
131 lines
7.2 KiB
TeX
\documentclass[../Main.tex]{subfiles}
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\graphicspath{{\subfix{Assets/img/}}}
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\begin{document}
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In \cite{Kessler1978} the authors described and forecasted what has come to be
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known as ``kessler syndrome'', where debris collides with itself in such a way that
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the overall debris level grows exponentially.
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A few methods have been used to model this behavior in the economics literature.
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The first one I want to explain was developed by \cite{Adilov2018}.
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They characterize kessler syndrome as the point in time at which an orbit is
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unusable as each satellite in orbit will be destroyed within a single time period.
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In my notation, this is that $l^i(\{s^j_t\}, D_t) = 1$.
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The benefit of this approach is that it is algebraically simple.
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It was used in to show that firms will stop launching before
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orbits are rendered physically useless.
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Unfortunately, it does not convey the original intent of ``kessler syndrome'',
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i.e. a runaway pollution effect, but instead corresponds to the end result of kessler syndrome.
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The second definition of ``kessler syndrome'' is due to \cite{RaoRondina2020}.
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They define it in terms of a ``kessler region'', the set of satellite stocks and the debris level
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such that the limit of debris in the future is infinite.
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Mathematically this can be represented as:
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\begin{align}
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\kappa = \left\{ \{s^j_t\}, D_t :
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\lim_{k\rightarrow \infty} D_{t+k}\left(\{s^j_{t+k-1}\}, D_{t+k-1}, \{x^j\}\right) = \infty \right\}
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\end{align}
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There are a few issues with this approach, even though it captures the essence of kessler syndrome
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better than the definition proposed by Adilov et al.
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The issues it faces are generally the case of not delineating between kessler regions
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with significantly different economic outcomes.
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% doesn't account for speed of divergence
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For example, one subset of the kessler region may render an orbital shell physically useless
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within a decade, while another subset increases the risk of satellite
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destruction by 1\% every ten thousand years.
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The former is a global emergency, while the latter is effectively non-existant.
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% Not computable.
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The last disadvantage I'd like to mention is that determining whether a
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series is divergent depends on constructing mathematical proofs.
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This makes it difficult to computationally identify whether a given state
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constitutes as kessler syndrome.
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\subsection{Two approaches to kessler syndrome}
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I propose to analyze kessler syndrome in two slightly more restricted
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fashions than \cite{RaoRondina2020}, for which I term the regions
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the $\epsilon$-kessler region and the proto-kesslerian region.
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First, define the $\epsilon$-kessler region as:
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\begin{align}
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\kappa = \left\{ \{s^j_t\}, D_t :
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\forall k \geq 0, D_{t+k+1} - D_{t+k} \geq \epsilon > 0 \right\}
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\end{align}
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%show that this is similar to saying that all non \epsilon kessler regions are bounded by the
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%derivative, i.e. are lipshiz
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The continuous time equivalent of this condition is defining the non-kessler regions by
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an upper bound on the derivative of debris generation\footnote{A lipshitz-like condition}.
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It is easily shown that this criteria is sufficient to guarantee Rao and Rondina's criteria.
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It has three primary benefits:
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\begin{itemize}
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\item % Can be solved for algebraically or numerically for a given, bounded state space.
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The $\epsilon$-kessler region can be numerically described within bounded state spaces.
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\item % This is what you would actually compute.
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The condition given is a basic guarantee of the divergent behavior that is
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required for Kessler Syndrome and acknowledges computational limitations.
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\item
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Finally, a slow divergence is no divergence in the grand scheme of things.
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It is possible to have a mathematically divergent function, but one that is so slow,
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there is no noticable degree of debris growth before Sol enters a red giant phase.
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In this specification, it is possible to choose $\epsilon$ such that the divergent behaviors
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identified have an economic impact on a meaningful timescale.
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\end{itemize}
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% Issue with this approach: What about cyclical behaviors?
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% Autocatalysis leads to high debris leads to reduced launches
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% which leads to debris decay leads to increased launches leads to Autocatalysis
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There is at least one issue with this definition of $\epsilon$-kessler regions.
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It may be, under certain situations, the case that optimal launch rates cycle along with
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debris and stock levels, leading to a cycle in and out of the $\epsilon$-kesslerian regions.
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This is an issue because, assumning a stable cycle, Rao's definition
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of the kessler region would capture this behavior, but the $\epsilon$-kessler definition
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would not.
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A particularly pathological case is where debris cycles between just below the cutoff level to
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significantly above the cutoff, leading to a highly divergent behavior not captured by this definition.
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As far as computability goes, by simulating a phase diagram (for a given solution to the model)
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we can determine what sections are in the $\epsilon$-kessler region.
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This is a major benefit in a computational model.
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A related and more general concept is the ``proto-kesslerian'' region, which is
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defined as the stock and debris levels such that:
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\begin{align}
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\kappa = \left\{ \{s^j_t\}, D_t :
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D_{t+1} - D_{t} \geq \varepsilon > 0 \right\}
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\end{align}
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%Note that the debris level is in a $\epsilon$-kessler region when it is in a proto-kesslerian region
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%for all future periods.
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This even simpler to compute than the phase diagram, and can be used to generate a topological view
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of proto-kesslerian regions of degre $\varepsilon$.
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%These are both easier to interpret and various approaches could be used to analyze how debris levels
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%transition between them.
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%%%what would the integral of gradients weighted by the dividing line measure? just a thought.
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%Other thoughts
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% proto-kesslerian paths, paths that pass into a proto kesslerian region.
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In order to capture the cyclic behavior that $\epsilon$-kessler regions miss, we can define a type of
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path in the phase diagram called a proto-kesslerian path of degree $\epsilon$, which is any path
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that enters the region.
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For example, one could simulate a phase diagram and compare paths that fall into a given $\epsilon$-kessler region
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and paths that only temporarily pass into the equivalent proto-kesslerian regions.
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Comparing the number of paths that fall into each region may give a useful metric for policies that are
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designed to decrease the likelihood of kessler syndrome.
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I believe, but have not verified, that some choices of $\varepsilon$, although permitting cycles,
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would relegate them to levels with minimal economic impact.
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%Maybe can be studies by phase or flow diagrams?
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%Consider where it cycles between just below epsilon and then to a large increase in debris?
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%Area of research: What makes a good \epsilon?
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This leads to the important question of ``What makes a good value of $\epsilon$ or $\varepsilon$?''
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One method, in the spirit of \cite{Adilov2018}, is to choose a change in debris, $D_{t+1} - D_t$, such that
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the loss of satellites in periods $t+1$ to $t+k$ is increased by or to a certain percentage, say 1\%.
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I've put very little thought into addressing this general question so far,
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and need to analyze the implications of different choice rules.
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\end{document}
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