@ -10,7 +10,7 @@ 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 $ .
In my notation, this is that $ R^ i ( S _ t, D _ t ) = 0 ~ \forall i $ .
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.
@ -22,11 +22,12 @@ They define it in terms of a ``kessler region'', the set of satellite stocks and
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 \}
\kappa = \left \{ S_ t, D_ t :
\lim _ { k\rightarrow \infty } D_ { t+k} \left (S_ { t+k-1} , D_ { t+k-1} , X_ t\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 .
better than the definition proposed by \cite { Adilov2018}
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
@ -38,7 +39,7 @@ The former is a global emergency, while the latter is effectively non-existant.
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 .
constitutes is in the kessler region .
@ -49,13 +50,15 @@ fashions than \cite{RaoRondina2020}, for which I term the regions
First, define the $ \epsilon $ -kessler region as:
\begin { align}
\kappa = \left \{ \{ s^ j_ t\} , D_ t :
\kappa _ \epsilon = \left \{ S_ t_ 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} .
an upper bound on the derivative of debris generation\footnote {
Note that the non-proto-kessler region is defined by a lipshitz-like condition
} .
It is easily shown that this criteria is sufficient to guarantee Rao and Rondina's criteria.
@ -84,45 +87,45 @@ of the kessler region would capture this behavior, but the $\epsilon$-kessler de
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)
Also, 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 :
\kappa _ \text { proto} = \left \{ S_ t_ 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 $ .
of various proto-kesslerian regions.
% 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
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-kessler 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
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?
% I believe, but have not verified, that some choices of $ \varepsilon $ , although permitting cycles,
% would relegate them to levels with minimal economic impact.
% 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\% .
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 between periods $ t $ 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.
youainti
4 years ago