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%%%%%%%%%%%%CUSTOMIZATION%%%%%%%%%%%%%%%%%%%%%%
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\title{Dynamic Launch Decisions for Satellite Constellation Operators}
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%Alternate title? Constellations in Orbit
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%\author{William King}
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\institute{Washington State University}
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\begin{document}
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\maketitle
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\begin{abstract}
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Over the last decades, new technology has make low earth orbits (LEOs) more accessible, and
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the resulting increase in LEO satellites has increased the risk of collision.
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Because debris in orbit generates more debris through collisions with objects in orbit
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and the debris created during launch and operation imposes a negative externality
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on other operators,
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optimal use of orbits is believed to not occur under free entry.
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This paper develops a dynamic model of satellite operation incorporating two effects not considered
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in previous models.
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The first effect is complementarity between satellites within the same operator's fleet (called a constellation).
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The second effect is collision avoidance efficiencies that exist within constellations.
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The primary result is a theoretical model and the resulting analysis of the difference in survival rates between
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constellation operators and society.
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\end{abstract}
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\keywords{Orbits, Pollution, Economies of Scale, Externality }
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\jel{Q29, Q58, L25}
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\textbf{Acknowledgments:} I am the sole author and have recieved no contributions from others as of yet.
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This paper has been approved for dual submission in Econs 529 and Econs 594 by the instructors.
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\newpage
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\tableofcontents
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\newpage
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% ---------------------------------------------------------------------------------------
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\section{Introduction}
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% Motivating Example (ESA - SpaceX)
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In September of 2019, the European Space Agency (ESA) released a tweet explaining that they had performed an
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maneuver to avoid a collision with a SpaceX Starlink Satellite in Low Earth Orbit (LEO)\autocite{EsaTweet}.
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While later reports\autocite{ArsTechnicaStatement} described it as the result of miscommunications,
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ESA used the opportunity to highlight the difficulties arising from coordinating avoidance maneuvers and how
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such coordination will become more difficult as the size and number of
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single purpose, single operator satellite fleets (satellite constellations) increase in low earth orbit\autocite{EsaBlog}.
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% Background on issues of congestion and pollution
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% Kessler Syndrome
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In spite of the fact that there is a lot of maneuvering room in outer space,
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%\footnote{``Space is big. Really big. You just won’t believe how vastly hugely mind bogglingly big it is.
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%I mean, you may think it’s a long way down the road to the chemist,
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%but that’s just peanuts to space.''\cite{DouglasAdams}}
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the repeated interactions of periodic orbits make collisions probable.
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Consequently, objects in orbit are subject to both a congestion effect and a pollution effect.
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Congestion effects are primarily derived from avoiding collisions between artificial satellites.
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Pollution in orbit consists of debris, both natural and man-made, which increases
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the probability of an unforeseen collision.
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The defining dynamic of pollution in orbit is that it self-propagates as debris collides with itself
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and orbiting satellites to generate more debris.
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This dynamic underlies a key concern, originally explored by Kessler and Cour-Palais \autocite{Kessler1978}
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that with sufficient mass in orbit (through satellite launches), the debris generating process
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could undergo a runaway effect rendering various orbital regions unusable.
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This cascade of collisions is often known as Kessler syndrome and
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may take place over various timescales.
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% ---------------
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Orbits may be divided into three primary groups,
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Low Earth Orbit (LEO, less than 2,400km in altitude\autocite{FAA2020}),
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Medium Earth Orbit (MEO), and High Earth Orbit (HEO) with Geostationary Earth Orbit (GEO)
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considered a particular classification of orbit.
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While the topic of LEO allocation has historically remained somewhat unexplored, the last 6 years has seen
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a variety of new empirical studies and theoretical models published.
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In general, three primary, related topics appear in the literature:
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Allocative Efficiency, Policy Intervention, and the occurrence of Kessler Syndrome.
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% ---------------
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%Allocative efficiency
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The primary concern is to establish whether or not orbits will be overused
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due to their common-pool nature, and if allocation procedures are efficient.
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The earliest theoretical model I have found, due to Adilov, Alexander, and
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Cunningham \autocite{adilov_alexander_cunningham_2015}, examines pollution
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using a two-period salop model, incorporating the effects of launch debris on
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survival into the second period.
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They find that the social planner generates debris and launches at lower rates
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than a free entry market.
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This same result was found by Rao and Rondina \autocite{RaoRondina2020} in
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the context of an infinite period dynamic model.
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They approach the problem in the case where numerous operators in a free entry environment
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can each launch a single, identical satellite.
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% ---------------
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In addition to analyzing the allocative results, a significant area of interest is
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what impact various policy interventions can have.
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The policies analyzed and methods used have been widely varied.
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Macauley \autocite{Macauley_1998} provided the first evidence of sub-optimal behavior in orbit
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by estimating the welfare lose due to the current method of assigning GEO slots to operators.
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The potential losses due to anti-competitive behavior was highlighted by Adilov et al \autocite{Adilov2019},
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who have analyzed the opportunities for strategic
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``warehousing'' of non-functional satellites as a means of increasing competitive advantage by
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denying operating locations to competitors in GEO.
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Grzelka and Wagner \autocite{GrzelkaWagner2019} explore methods of encouraging satellite quality (in terms of debris)
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and cleanup.
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Finally, Rao and Rondina \autocite{Rao2020} estimate that achieving socially optimal
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behavior through orbital use fees could increase the value generated by the space industry by a factor of four.
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% ---------------
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Although Kessler and Cour-Palais determined that a runaway pollution effect could make a set of orbits
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physically unusable, Adilov et al \autocite{adilov_alexander_cunningham_2018} %Kessler Syndrome
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have shown that economic benefits provided by orbits will drop sufficiently to make the net marginal
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benefit of new launches negative before the physical kessler syndrome occurs.
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%TODO: Discuss how various definitions have been proposed in the economics literature to match the models.
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% ---------------
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This paper's objective is to %develop a dynamic model which incorporates
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lay the foundations necessary to explore the effects of organizing satellites as constellations ,
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particularly through collision avoidance efficiencies and economies of scale in utility production.
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No model as of yet has examined these aspects of orbit use.
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The primary analytical result aside from developing the preliminary model and characterizing general solutions
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is to examine if there exists a negative externality related to survival rates.
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% ---------------
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The paper is organized as follows.
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Section \ref{Model} describes the mathematical organization of the model
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for the cases of independent constellation operators and a social planner
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operating the same constellations.
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%It also includes a brief digression into the free entry conditions.
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Section \ref{Analysis} evaluates the differences between the
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constellation operators and social planner models, particularly
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the difference between marginal survival rates .
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%Of particular interest is the difference in launch rates and marginal survival rates.
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%Section \ref{Kessler} ...
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Section \ref{Conclusion} concludes with a discussion of potential extensions and
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topics which have not yet been addressed.
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% ---------------------------------------------------------------------------------------
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\section{Model}\label{Model}
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%Intuitive description
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The dynamic model is an extension of Rao and Rondina's working paper \autocite{RaoRondina2020}
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(specifically their non-stochastic model)
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to include how operators deal with constellations.
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\subsection{Model Outline}
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For a given orbital shell (a set of orbits that interact regularly), I assume there are $N$ operators,
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each of which has the potential to launch and operate a satellite
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constellation consisting of some endogenously chosen number of identical satellites.
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% -------------------
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Each constellation operator has a personal satellite stock $s^i_t$ in each period, and chooses the
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number of launches in that time period $x^i_t$.
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For simplicity, each launch is assumed to have a fixed cost $F$.
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In the aggregate, the satellite stock and launches for each period are represented by:
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\begin{align}
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S_t =&\sum_{i=1}^N s^i_t \\
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X_t =&\sum_{i=1}^N x^i_t
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\end{align}
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% -------------------
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Satellites in a constellation are damaged or destroyed at the rate $l^i(s^i_t,S_t,D_t)$,
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which is assumed to be increasing in $s^i_t$, $S_t$, and $D_t$ (debris, see below).
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One key difference from the previous models of Rao and Rondina \autocite{RaoRondina2020} and
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Adilov et al \autocite{adilov_alexander_cunningham_2018} is that this model allows the rate of
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collision within constellations and between constellations to be different.
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This reflects the assumption that an operator can and will put more effort into protecting the satellites within
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the constellation from each other.
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One example of how this can be accomplished is that while choosing the orbits for a constellation,
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it is possible for an operator to chose a set of trajectories that best meet their needs and
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minimizes the risk of collision within the constellation.
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Mathematically this is represented by the inclusion of $s^i_t$ in $l^i$.
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Together with the launch rate, we obtain a law of motion for both constellation-level
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and society-level satellite stocks.
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\begin{align}
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s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
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S_{t+1} =& X_t + \sum^N_{i=1} \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t
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\end{align}
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%Discuss first derivatives
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%The case where there
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% -------------------
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The level of debris in each period is represented by $D_t$, and is assumed to pose a latent risk.
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In particular, it is assumed that once debris is created, the risk it provides is only avoidable
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through not launching future satellites.
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%\footnote{This is one important extension as avoiding debris reduces the operational lifetime
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% of satellites and may affect optimal taxation.
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In addition to naturally occurring debris, debris is generated through the following three mechanisms.
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\begin{itemize}
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\item At launch, various processes can shed debris.
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Examples include leftover rocket stages, explosions during launch and deployment,
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and slag from solid rocket boosters.
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\item When destroyed, satellites will fragment and produce debris.
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\item Debris can collide with other debris, forming more but smaller debris.
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\end{itemize}
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This provides the following law of debris dynamics.
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\begin{align}
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D_{t+1} =& (1-\delta) D_t + m X_t + M\cdot \left( \sum^N_{i=1} l^i(s^i_t,S_t,D_t) \right) + g(D_t)
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\end{align}
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where $\delta$ represents the proportional decay of debris
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-- through reentering the atmosphere -- for a given shell,
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$M$ represents the debris generated from each collision,
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$m$ represents the debris generated from each launch,
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and $g(D_t)$ represents the new fragments from debris colliding with other debris.
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% -------------------
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Each constellation $i \in {1,\dots,N}$ produces value for their operator at each period according to the function:
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\begin{align}
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u^i(s^i_t, S_t, D_t)
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\end{align}
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Productive economies of scale within a constellation appear when
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$\parder{u^i}{s^i_t}{2} > 0$ for some values of $s^i_t,S_t, D_t$.
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Of note is that firms are assumed to produce value monopolistically, i.e. there are no substitution nor
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complementary effects between constellations.
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This allows us to examine the effects of economies of scale and collision avoidance efficiencies
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without incorporating the effects of competition.
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The period value function may incorporate the effects of orbit and congestion debris, accounting
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for their effect in producing value to the operator.
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Adilov et al analyzed the effects of competition between operators in launch decisions \autocite{Adilov2019}.
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A similar approach could be used, but would add significant complexity to the model.
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One key note is the choice of the word ``value'' as opposed to ``profit''.
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Historically, space operations have been motivated by objectives other than profit,
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such as national security, scientific inquisitiveness, to enhance hobbies such as amature radio,
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or to quote President John F. Kennedy,
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``\dots because [it] is hard.''\autocite{Kennedy1962}.
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This choice of terminology acknowledges that orbit use is not exclusively commercial
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and there may be interference between commercial and non-commercial operations.
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% ---------------------------------------------
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\subsection{Constellation Operator's Program}
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%The aforementioned aspects of the model form the following bellman equation for each constellation operator.
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%\begin{align}
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% V^i(s^i_t,S_t,D_t) =& \max_{x^i_t \geq 0} ~~ u^i(s^i_t) - Fx^i_t + \beta V^i(s^i_{t+1}, S_{t+1}, D_{t+1}) \\
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% \text{Subject To:}& \notag\\
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% D_{t+1} =& (1-\delta) D_t + m X_t + M\cdot \left( \sum^N_{i=1} l^i(s^i_t,S_t,D_t) \right) + g(D_t) \\
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% s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
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% S_t =&\sum_{i=1}^N s^i_t \\
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% X_t =&\sum_{i=1}^N x^i_t % Is this also a state variable?
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%\end{align}
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%The system of envelope conditions is linear and can be written as a matrix equation.
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%In Appendix \ref{APX:Derivations:Constellation} I develop the euler equation
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%in a generalizable way.
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Often, in polluting environments, there is an ambient population that is harmed by pollution.
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Very rarely does satellite debris pose a hazard to those on earth, thus in this model
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the only population who's welfare is addressed are the satellite operators themselves.
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Each operator faces the following problem:
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\input{./includes/Appendix_constellation_program}
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% ---------------------------------------------
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\subsection{Social Planner's Program}
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The social planner (or fleet planner to use Rao and Rondina's terminology), is tasked with
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maximizing the sum of the operators' benefits $W(\{s^i_t\},S_t,D_t) = \sum^N_{i=1} V^i(s^i_t,S_t,D_t)$.
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%\begin{align}
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% W(\{s^i_t\},D_t) =& \max_{\{x^i_t\}^N_{i=1} \geq 0}
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% ~~ \left(\sum^N_{i=1} u^i(s^i_t,S_t,D_t)\right) - FX_t
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% + \beta W(\{s^i_{t+1}\}, S_{t+1}, D_{t+1}) \\
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% \text{Subject To:}& \notag\\
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% D_{t+1} =& (1-\delta) D_t + m X_t + M\cdot \left( \sum^N_{i=1} l^i(s^i_t,S_t,D_t) \right) + g(D_t) \\
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% s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
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% S_t =&\sum_{i=1}^N s^i_t \\
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% X_t =&\sum_{i=1}^N x^i_t
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%\end{align}
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%
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%%Goal: Add the euler equation.
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%The derivation of the euler equation, and conditions on it's existence are
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%outlined in Appendix \ref{APX:Derivations:Fleet}.
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\input{./includes/Appendix_planner_program}
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% ---------------------------------------------------------------------------------------
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\section{Analysis}\label{Analysis}
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%Describe analysis types
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%Survival ratios
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%two firm model
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\subsection{Survival Ratios}\label{Survival}
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% Marginal survival.
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In line with theory on common-pool resources, we expect there to be a negative externality
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incurred by increasing the satellite stock.
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Some details of this externality can be observed in the marginal survival rate.
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Define the survival rate for a constellation and the society to be:
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\begin{align}
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R_i =& \frac{s^i_{t+1}- x^i_t}{s^i_t} = 1- l^i(s^i_t,S_t,D_t) \\
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R =& \frac{S_{t+1}- X_t}{S_t} = \frac{\sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] }{S_t}
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\end{align}
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The marginal survival rates when a given constellation $i$ changes size are:
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\begin{align}
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\parder{R_i}{s^i_t}{} =& -\left(\parder{l^i}{s^i_t}{} + \parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{} \right)
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= - \parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{} \label{EQ:iii} \\
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\parder{R}{s^i_t}{} =& \frac{S_t \sum_{i=1}^N
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\left( [1-l^i(s^i_t,S_t,D_t)] + s^i_t [ -\parder{l^i}{s^i_t}{} -\parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{}] \right)
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- \left( \sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] \right)}{(S_t)^2} \\
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=& \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
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+\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \label{EQ:i}
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\end{align}
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Note that $ \sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}$ is the average marginal survival rate
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across constellation operators.
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The derivation of equation \ref{EQ:i} is in Appendix \ref{APX:Derivations:Survival_Direct}.
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Direct comparison between the marginal survival rates of an individual operator and the social planner's fleet
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cannot proceed further without specifying the functional loss forms $l^i(\cdot)$
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and specifying which firm will be compared to society.
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In spite of this, conditions on the average effects can be specified as follows.
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Society's marginal survival rate is greater than the weighted, arithmetic mean of marginal survival rates
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of the constellation when:
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\begin{align}
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\sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
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+\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}
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\leq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\
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\sum_{i=1}^N R_i - R \leq& 0\\
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\sum_{i=1}^N [1- l^i(s^i_t,S_t,D_t)] - \sum_{i=1}^N s^i_t [1- l^i(s^i_t,S_t,D_t)] \leq& 0\\
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\sum_{i=1}^N (1 - s^i_t) [1- l^i(s^i_t,S_t,D_t)] \leq& 0 \label{EQ:ii}
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\end{align}
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which is true if every constellation has at least one satellite.
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Based on the definition of constellation survival rate, $s^i_t =0 \Rightarrow R_i = \frac{0}{0}$
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i.e. the survival rate is undefined.
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In combination with the physical reality that there cannot be a negative number
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of satellites in a constellation, we are left to conclude that a meaningful constellation
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has at least one satellite.
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As $\parder{R_i}{s^i_t}{} < 0$ from the assumptions on collision mechanics, we can interpret
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this result as that the marginal survival rate of the entire satellite fleet is lower
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than the weighted arithmetic mean of marginal survival rates across constellations.
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This demonstrates the negative externality of satellite operation, and is a very general condition,
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consistent with other orbital pollution models.
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Note that it does allow for some constellations to have a lower marginal survival rate than the fleet,
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but it can be true as a general condition.
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%TODO: Some more analysis can be done by comparing the case of avoidance efficiencies vs non-efficiencies.
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%\subsubsection{Average Effects}
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%TODO: Review and rewrite this section, including discussing the implications
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%As we are analyzing survival rates, a geometric mean is better used to describe average effects.
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%By weighting the geometric mean with constellation sizes, we get:
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%\begin{align}
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% R_G = \exp \left[ \frac{1}{S_t} \sum^N_{j=1} s_t^j \ln(1-l^j(s^j_t,S_t,D_t)) \right]
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%\end{align}
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%The marginal effect is assumed to be negative, thus
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%\begin{align}
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% 0 > \parder{R_G}{s^i_t}{} =& \exp \left[ \frac{1}{S_t} \sum^N_{j=1} s_t^j \ln(1-l^j(s^j_t,S_t,D_t)) \right]
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% \left[ \parder{}{s^i_t}{} \frac{1}{S_t} \sum^N_{j=1} s_t^j \ln(1-l^j(s^j_t,S_t,D_t)) \right] \\
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% 0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t
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% \left( \ln(1-l^i)
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% - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
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% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
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% \right)
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% - \sum^N_{j=1} s_t^j \ln(1-l^j) \right] \\
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% 0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t
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% \left( \ln(R_i)
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% - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
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% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
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% \right)
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% - \sum^N_{j=1} s_t^j \ln(R_j) \right] \\
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% 0 > & \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
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% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{} - \sum^N_{j=1} \frac{s_t^j}{S_t} \ln(R_j) \\
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% 0 > & \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
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% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{} - \ln R_G \\
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% \ln \frac{R_G}{R_i} =& \ln R_G - \ln R_i > - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
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% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
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%\end{align}
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%Welfare
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% TODO: Develop overarching results.
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% ---------------------------------------------------------------------------------------
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\subsection{Kessler Syndrome}\label{Kessler}
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%Current plan: Explain the kessler region in this model
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%Rao's physical approach
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%Adilov's economic approach
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Rao and Rondina \autocite{RaoRondina2020} interpret their model in terms of a physical
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kessler syndrome, while Adilove et al \autocite{adilov_alexander_cunningham_2018}
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develop the concept of an economic kessler syndrome.
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Generalizing Rao's approach, we define the kessler region as the set of states such that
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the debris stock will tend to infinity, and kessler syndrome as when the state is in
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the kessler region.
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Formally, the kessler region is:
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\begin{align}
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\vartheta_1 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
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\lim_{t \rightarrow \infty} D_{t+1} = \infty \right\}
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\end{align}
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I suspect, but have not been able to prove, that an equivalent condition is:
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\begin{align}
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\vartheta_2 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
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\parder{(D_{t+1}-D_t)}{D_t}{} > 0 \right\}
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\end{align}
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If the assumption holds, then a condition for a physical kessler region in this model is:
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\begin{align}
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\vartheta_2 =
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\left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
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-\delta
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+ m\parder{X_t(\{s^i_t\},D_t)}{D_t}{}
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+ M\cdot \left( \sum^N_{i=1} \parder{l^i}{D_t}{} \right)
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+ g(D_t) > 0 \right\}
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\end{align}
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Adilov defines an economic kessler syndrome (and thus kessler region) along the lines of
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\begin{align}
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\vartheta_3 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) = 0 \right\}
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\end{align}
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This represents the conditions under which adding satellites to the orbit becomes unprofitable.
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He establishes general conditions under which an economic kessler syndrom precedes a
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physical kessler syndrome.
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The benefit of this definition is that the euler equation defining $X_t(\cdot)$
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can be searched for the states that imply $X_t = 0, \forall t$
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\footnote{I have yet to conduct such a search, but plan on doing so as part of a numerical simulation.}.
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% ---------------------------------------------------------------------------------------
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%\subsection{Numerical Model}\label{Numerical}
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% 2-firm model: Symmetric
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% 2-firm model: asymetric sizes or payoffs.
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% ---------------------------------------------------------------------------------------
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\section{Concluding Remarks}\label{Conclusion}
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%TODO: rewrite and update.
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The dynamic model developed in this paper provides insight into the incentives faced by
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constellation operators in comparison with a social planner and, when completed, should provide
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insight on how self-perpetuating externalities drive sub-optimal behavior.
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At this point, major work remains in identifying optimal launch rates and verifying if
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the expected difference in optimal launch rates between individual operators and a social planner exist,
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as occurs in other models.
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In addition to the remaining work on fleshing out the model, work on the following extensions and applications of the
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model can fill gaps in the literature or complement current work.
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Notable areas of interest for future research include:
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\begin{itemize}
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\item Asymmetric constellation sizes: What are the impacts on social welfare when a variety of
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constellation sizes exist?
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\item Policy interventions: Various policy proposals to reduce negative externalities have been proposed,
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including launch quotas, launch taxes, and orbit use fees \autocite{RaoRondina2020b}.
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% \item Introduction of stochastics: There are various ways that stochastics can enter the model, from the scales
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% determining debris generation to the per-period satellite collision rate.
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% \item Differentiation of satellites and launch methods: Different launch methods and satellite features can
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% affect the accumulation of debris.
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% \item Richer satellite lifetimes: the current satellite lifetime of [launch, operate] could be extended
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% to include stages such as development and disposal.
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% In particular, a multi-period development cycle with sunk costs incurred along the way may
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% exacerbate problems where stable equilibria are overshot.
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% This will allow for more policy interventions to be analyzed.
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\item Strategic behavior: Concerns include whether constellation network effects can be used to prevent new entrants
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in the case of competition for a satellite services market.
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\end{itemize}
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While computationally complicated, the results so far imply that there is a defined difference between
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the risks faced at the constellation operator's level and the level of society as a whole.
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Although not a common topic in economics, orbit use has properties that requires
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current study in order to identify optimal behavior, inform policies, and prevent kessler syndrome
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before there are no more viable orbits to use.
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\newpage
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\printbibliography
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\newpage
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\appendix
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\section{Derivations} \label{APX:Derivations}
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%\subsection{Useful Mathematical Notes}\label{APX:Derivations:Useful}
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%To fill in with a set of useful mathematical notes for use throughout.
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%\subsubsection{Useful Derivatives}
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%\subsection{Constellation Operator}\label{APX:Derivations:Constellation}
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%\input{./includes/Appendix_constellation_program}
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%\subsection{Fleet Planner}\label{APX:Derivations:Fleet}
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%\input{./includes/Appendix_planner_program}
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\subsection{Survival Rates}\label{APX:Derivations:Survival_Direct}
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\input{./includes/Appendix_Survival_direct}
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%\subsection{Survival Rates: Geometric Mean Analysis}\label{APX:Derivations:Survival_Geometric}
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%\input{./includes/Appendix_Survival_geometric}
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%TODO
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\end{document}
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