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In September of 2019, the European Space Agency (ESA) released a tweet explaining that they had performed an
maneuver to avoid a collision with a SpaceX Starlink Satellite in Low Earth Orbit (LEO)\autocite{EsaTweet}.
While later reports\autocite{ArsTechnicaStatement} described it as the result of miscommunications,
ESA used the opportunity to highlight the difficulties arising from coordinating avoidance maneuvers and how
such coordination will become more difficult as the size and number of 
single purpose, single operator satellite fleets (satellite constellations) increase in low earth orbit\autocite{EsaBlog}.


% Background on issues of congestion and pollution
% Kessler Syndrome
In spite of the fact that there is a lot of maneuvering room in outer space,
%\footnote{``Space is big. Really big. You just won’t believe how vastly hugely mind bogglingly big it is. 
%I mean, you may think it’s a long way down the road to the chemist, 
%but that’s just peanuts to space.''\cite{DouglasAdams}}
the repeated interactions of periodic orbits make collisions probable.
Consequently, objects in orbit are subject to both a congestion effect and a pollution effect.
Congestion effects are primarily derived from avoiding collisions between artificial satellites.
Pollution in orbit consists of debris, both natural and man-made, which increases
the probability of an unforeseen collision.
The defining feature of pollution in orbit is that it self-propagates as debris collides with itself
and orbiting satellites to generate more debris.
This dynamic underlies a key concern, originally explored by Kessler and Cour-Palais \autocite{Kessler1978}
that with sufficient satellites in orbit, the debris generating process
could undergo a runaway effect rendering some orbital regions unusable.
This cascade of collisions is often known as Kessler syndrome and 
may take place over various timescales, from weeks to decades.

% ---------------
%Discuss how various definitions of kessler syndrome
%    have been proposed in the economics literature to match the models.
%Not sure if the following contributes much given the previous paragraph.
%Although Kessler and Cour-Palais determined that a runaway pollution effect could make a set of orbits 
%physically unusable, Adilov et al \autocite{adilov_alexander_cunningham_2018} %Kessler Syndrome
%have shown that economic benefits provided by orbits will drop sufficiently to make the net marginal
%benefit of new launches negative before the physical kessler syndrome occurs.

% ---------------

%Orbits may be divided into three primary groups, 
%Low Earth Orbit (LEO),
%Medium Earth Orbit (MEO), and High Earth Orbit (HEO) where Geostationary Earth Orbit (GEO) 
%is considered a particular subset of HEOs. 
%While the topic of LEO allocation has historically remained somewhat unexplored, the last 6 years has seen
%a variety of new empirical studies and theoretical models published.

% ---------------
%Allocative efficiency

Macauley  provided the first evidence of sub-optimal behavior in orbit
by estimating the welfare loss due to the current method of assigning GEO slots to operators\autocite{Macauley_1998}.
The potential losses due to anti-competitive behavior were highlighted by Adilov et al,
who have analyzed the opportunities for strategic 
``warehousing'' of non-functional satellites as a means of increasing competitive advantage by 
denying operating locations to competitors in GEO\autocite{Adilov2019}.

The primary concern expressed in many recently published papers is whether or not orbits will be overused
due to their common-pool nature, and which policies may prevent kessler syndrome.
On this topic, Adilov, Alexander, and Cunningham examine pollution
using a two-period salop model, incorporating the effects of launch debris on 
survival into the second period\autocite{Adilov2015}. 
They find that the social planner generates debris and launches at lower rates 
than a free entry market.

This same result was found by Rao and Rondina  in
the context of an infinite period dynamic model. 
%Potential Edit
Their approach is defined by the assumption that there are 
numerous operators in a free entry environment who 
can each launch a single, identical constellation\autocite{RaoRondina2020}.
Rao, Burgess, and Kaffine use this model to estimate that achieving socially optimal 
behavior through orbital use fees could increase the value generated by the 
space industry by a factor of four\autocite{Rao2020} in the long run.


% ---------------
%In addition to analyzing the allocative results, a significant area of interest is 
%what impact various policy interventions can have.
%The policies and methods used to analyze their impact have been widely varied.
% What policies have been evaluated?
%   - Muller et al analyze debris removal
%   - Grzelka and Wagner \autocite{GrzelkaWagner2019} explore methods of encouraging satellite quality (in terms of debris) and cleanup.
%   - Rao compares launch vs operation taxes
%   - Adilov et al ?????

%Other papers to review:
% Muller, Rozanova, Urdanoz (Economic Valuation of Debris Removal, IAC conference 2017)
% Salter (Space Debris, Mercantus Working Paper 2015)
% 
% 

% ---------------
My %FP
objective is to %explore the effects from organizing satellites into constellations on satellite launch decisions and operation.
describe the dynamic decision-making process facing constellation operators, 
how their launch decisions diverge from the socially optimal,
and the ways in which various policies encourage or discourage optimal decision making.
%I %FP
%do this by extending Rao and Rondina's dynamic satellite operators model\autocite{RaoRondina2020} 
%to account for non-symmetric constellation sizes and 
%incorporate the effects of both economies of scale as satellites in constellations complement each other and 
%collision avoidance efficiencies where satellites are less likely to collide with constellation members.
%Explain what the article does.
% The primary results of this paper are: 
% preliminary development of the dynamic model, 
% characterization of the general solutions to both the constellation operators' problems and
% the fleet planner's problem, 
% and an analysis of survival rates within constellations and the entire fleet.
%Contribution statement
%Adds to raoRondina2020 and adilov2018 in extedning to more diverse situations.
The heterogeneity that I permit is the distinguishing feature of the model and the major
justification for this work, as orbits are used by many different types of operators.
Specifically, I permit:
\begin{itemize}
    \item Heterogeneous agent types including commercial, scientific, and military.
    \item Asymetric constellations.
    \item Inter- and intra- constellation risk to differ.
\end{itemize}
each of which are important qualities of the current orbital environment.
None of these aspects are considered in the papers that I have reviewed so far.

This work is mainly a theoretical expansion of two dynamic models by
\cite{RaoRondina2020} as well as \cite{Adilov2018}
%This model inherits the laws of motion for debris and constellation stocks from 
%the aformentioned models and follows the DSGE modelling approach chosen by Rao.
In addition to the models' expansion, I contribute a general computational solver to analyse the
complex situations that arise in practice.

%Similarities
% - Rao
%   - Law of debris:
%   - law of motion for stocks
% - Adilov
%   - law of Debris
%   - constellations
%Differences
% - Rao
%   - constellation
%   - avoicance efficiencies
% - Adilov
%   - Allows for non-firm participants
%   - avoidance efficiencies



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