@ -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} .
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
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.
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.
The benefit of this approach is that it is algebraically simple.
It was used in to show that firms will stop launching before
It was used in to show that firms will stop launching before
orbits are rendered physically useless.
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.
such that the limit of debris in the future is infinite.
Mathematically this can be represented as:
Mathematically this can be represented as:
\begin { align} \begin { align}
\kappa = \left \{ \{ s^ j_ t\} , D_ t :
\kappa = \left \{ S_ 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 \}
\lim _ { k\rightarrow \infty } D_ { t+k} \left (S_ { t+k-1} , D_ { t+k-1} , X_ t\right )
= \infty \right \}
\end { align} \end { align}
There are a few issues with this approach, even though it captures the essence of kessler syndrome
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
The issues it faces are generally the case of not delineating between kessler regions
with significantly different economic outcomes.
with significantly different economic outcomes.
% doesn't account for speed of divergence% 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
The last disadvantage I'd like to mention is that determining whether a
series is divergent depends on constructing mathematical proofs.
series is divergent depends on constructing mathematical proofs.
This makes it difficult to computationally identify whether a given state
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:
First, define the $ \epsilon $ -kessler region as:
\begin { align} \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 \}
\forall k \geq 0, D_ { t+k+1} - D_ { t+k} \geq \epsilon > 0 \right \}
\end { align} \end { align}
% show that this is similar to saying that all non \epsilon kessler regions are bounded by the % show that this is similar to saying that all non \epsilon kessler regions are bounded by the
% derivative, i.e. are lipshiz% derivative, i.e. are lipshiz
The continuous time equivalent of this condition is defining the non-kessler regions by
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.
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.
would not.
A particularly pathological case is where debris cycles between just below the cutoff level to
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.
significantly above the cutoff, leading to a highly divergent behavior not captured by this definition.
Also, by simulating a phase diagram (for a given solution to the model)
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.
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
A related and more general concept is the ``proto-kesslerian'' region, which is
defined as the stock and debris levels such that:
defined as the stock and debris levels such that:
\begin { align} \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 \}
D_ { t+1} - D_ { t} \geq \varepsilon > 0 \right \}
\end { align} \end { align}
% Note that the debris level is in a $ \epsilon $ -kessler region when it is in a proto-kesslerian region% Note that the debris level is in a $ \epsilon $ -kessler region when it is in a proto-kesslerian region
% for all future periods.% for all future periods.
This even simpler to compute than the phase diagram, and can be used to generate a topological view
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 % These are both easier to interpret and various approaches could be used to analyze how debris levels
% transition between them.% transition between them.
% % % what would the integral of gradients weighted by the dividing line measure? just a thought.% % % what would the integral of gradients weighted by the dividing line measure? just a thought.
% Other thoughts% Other thoughts
% proto-kesslerian paths, paths that pass into a proto kesslerian region.% 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
In order to capture the cyclic behavior that $ \epsilon $ -kessler regions miss,
path in the phase diagram called a proto-kesslerian path of degree $ \epsilon $ , which is any path
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.
that enters the region.
For example, one could simulate a phase diagram and compare paths that fall into a given $ \epsilon $ -kessler 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.
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.
designed to decrease the likelihood of kessler syndrome.
I believe, but have not verified, that some choices of $ \varepsilon $ , although permitting cycles,
% I believe, but have not verified, that some choices of $ \varepsilon $ , although permitting cycles,
would relegate them to levels with minimal economic impact.
% 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 ?% Area of research: What makes a good \epsilon ?
This leads to the important question of ``What makes a good value of $ \epsilon $ or $ \varepsilon $ ?''
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
One method, in the spirit of \cite { Adilov2018} ,
the loss of satellites in periods $ t + 1 $ to $ t + k $ is increased by or to a certain percentage, say 1\% .
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,
I've put very little thought into addressing this general question so far,
and need to analyze the implications of different choice rules.
and need to analyze the implications of different choice rules.
youainti
4 years ago