You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
81 lines
4.3 KiB
TeX
81 lines
4.3 KiB
TeX
\documentclass[../Main.tex]{subfiles}
|
|
\graphicspath{{\subfix{Assets/img/}}}
|
|
|
|
\begin{document}
|
|
In \cite{Kessler1978} the authors described and forecasted what has come to be
|
|
known as ``kessler syndrome'', where debris collides with itself in such a way that
|
|
the overall debris level grows exponentially.
|
|
A few methods have been used to model this behavior in the economics literature.
|
|
|
|
The first one I want to explain was developed by \cite{Adilov2018}.
|
|
They characterize kessler syndrome as the point in time at which an orbit is
|
|
unusable as each satellite launched will be destroyed within a single time period.
|
|
In my notation, this is that $l^i(\{s^j_t\}, D_t) = 1$.
|
|
The benefit of this approach is that it is algebraically simple.
|
|
It was used in this role to show that firms will stop launching before
|
|
orbits are rendered physically useless.
|
|
Unfortunately, it does not convey the original intent of ``kessler syndrome'',
|
|
i.e. a runaway pollution effect, but instead corresponds to the end result of kessler syndrome.
|
|
|
|
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:
|
|
\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}
|
|
|
|
\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}
|
|
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.
|
|
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.
|
|
\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.
|
|
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,
|
|
would relegate them to levels with minimal economic impact.
|
|
%Maybe can be studies by phase or flow diagrams?
|
|
|
|
|
|
%Area of research: What makes a good \epsilon?
|
|
This leads to the important question of ``What makes a good value of $\epsilon$?''
|
|
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\%.
|
|
I've put very little thought into addressing this general question so far,
|
|
and need to analyze the implications of different choice rules.
|
|
|
|
|
|
\end{document}
|