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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 in orbit 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 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 definition of ``kessler syndrome'' is due to \cite{RaoRondina2020}.
They define it in terms of a ``kessler region'', the set of satellite stocks and the debris level
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.
The last disadvantage I'd like to mention is that  determining whether a 
series is divergent depends on constructing mathematical proofs.
This makes it difficult to computationally identify whether a given state
constitutes as kessler syndrome.



\subsection{Two approaches to kessler syndrome}
I propose to analyze kessler syndrome in two slightly more restricted 
fashions than \cite{RaoRondina2020}, for which I term the regions
 the $\epsilon$-kessler region and the proto-kesslerian region.

First, 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 defining the non-kessler regions by
an upper bound on the derivative of debris generation\footnote{A lipshitz-like condition}.


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.
        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 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.
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 $\epsilon$-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.
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 measure? 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 $\epsilon$ 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 1\%.
I've put very little thought into addressing this general question so far,
and need to analyze the implications of different choice rules.


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