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%%%%%%%%%%%%CUSTOMIZATION%%%%%%%%%%%%%%%%%%%%%%
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\title{Dynamic Launch Decisions for Satellite Constellation Operators}
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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 10 years 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 same-purpose, single operator fleets (called constellations).
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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 ratios 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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\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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adjustment 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 unforseen collision.
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The defining dynamic of pollution in orbit is that it self-propogates 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 theoretically
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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 empircal 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, Externality Mitigation, and Economic vs Physical Kessler Syndromes.
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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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% ---------------
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%Allocative efficiency
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The primary concern is to establish wether 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 suboptimal 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{RaoRondina2020b} 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 4.
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% ---------------
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This paper's objective is to devlop a dynamic model which incoporates
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complementary effects of constellations as well as collision avoidance efficiencies of constellations,
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thus addressing a gap in the current literature.
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In addition, I examine if there exists a negative externality related to changes in stock size, and
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establish a condition related to average behavior that describes this externality.
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Finally, I lay foundations for the derivation of profit maximizing launch rules.
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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{Comparisons} 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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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 endogenosly chosen number of identical satellites.
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These satellites are not only identical within a constellation, but across constellations.
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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 Redina \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 acomplished 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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% -------------------
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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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In addition to natually occuring 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 decay of debris -- 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) = u^i(s^i_t)
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\end{align}
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For computational simplicity, it is assumed that benefits provided are wholely dependent on the number
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of satellites in operation.
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The approach presented in the appendix is generalizable to the case where benefits are conditional on
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the total satellite and debris stocks.
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Complementarity within a constellation appears when $\parder{u^i}{s^i_t}{2} > 0$ for some values of $s^i_t,S_t, D_t$.
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% ---------------------------------------------
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\subsection{Constellation Operator's Program}
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The aformentioned 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 begin development of the euler equation
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in a generalizable way.
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Unfortunately repeated errors in the mathematics has prevented me from achieving more in the
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launch rate analysis to date.
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%The resulting euler equation is:
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%\begin{align}
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% F \det(A) =&
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% [\adj(A) (D_{[s^i_t,S_t,D_t]} V^i - b)]\big|_{1} \\
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% &+ 2 [\adj(A) (D_{[s^i_t,S_t,D_t]} V^i - b)]\big|_{2} \notag\\
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% &+ m [\adj(A) (D_{[s^i_t,S_t,D_t]} V^i - b)]\big|_{3} \notag
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%\end{align} % TODO: This could also be changed to a matrix form with a row-vector.
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%Where $A$ is the matrix of partial derivatives of the laws of motion corresponding to
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%the envelope conditions.
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%The matrix $A$ and the derivation of the euler equation
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%is described in Appendix \ref{APX:Derivations:Constellation}
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% ---------------------------------------------------------------------------------------
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%\subsubsection{Free Entry}
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%Operators are assumed to enter the market as long as the value of entering is above 0.
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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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%Crucial assumption. $\beta^i =\beta \forall i$
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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 whom's welfare is addressed are the satellite operators themselves.
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\begin{align}
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W(\{s^i_t\},S_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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%Goal: Add the euler equation.
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Due to the aformentioned errors, I have not begun a derivation of the optimal launch rate
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for the social planner at this point.
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I expect it to be solvable using the same approach as for the constellation operators
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outlined in Appendix \ref{APX:Derivations:Constellation}.
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% ---------------------------------------------------------------------------------------
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%\section{Convergence Properties}\label{Convergence}
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% ---------------------------------------------------------------------------------------
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\section{Comparisons}\label{Comparisons}
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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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The details of this externality can be observed in the marginal suvival 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}.
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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 proceede further without specifying the functional loss forms $l^i(\cdot)$
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and specifying which firm the comparison is with.
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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 average marginal survival rate when:
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% NOTE: Should I do this using absolute value arguments? I don't think so.
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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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\geq& \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 \geq& 0\\
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\sum_{i=1}^N S_t [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)] \geq& 0\\
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\sum_{i=1}^N (S_t - s^i_t) [1- l^i(s^i_t,S_t,D_t)] \geq& 0 \label{EQ:ii}
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\end{align}
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Which is always true as $S_t > s^i_t$ and $l^i(\cdot) \in [0,1]$ for all $i$.
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%As we are discussing arithemetic means of rates, it makes sense.
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%A geometric mean might behave differently.
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If a single constellation makes up the whole stock of satellites, then \eref{EQ:ii} reduces
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to a tautology.
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As $\parder{R_i}{s^i_t}{} < 0$ from \eref{EQ:iii} and the assumptions on collision mechanics, we see
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that the average marginal survival rate acts as a lower bound on the marginal societal survival rate.
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Assuming that survival rates are not increased by adding another satellite i.e. $\parder{R}{s^i_t}{}<0$ then gives
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us the following bounds on societal rates $\sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{}<\parder{R}{s^i_t}{}<0$
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%By way of interpretation, this means adding a satellite to a constellation has a larger impact
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%on the society's survival rate than on the average survival rate across constellations.
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%This result is consistent with previous results establishing a negative externality
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This suggests that some operators experience marginal changes to their own satellite
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stocks much more intensely than society as a whole does.
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%Do it again using a geometric mean.
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%Welfare
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% TODO: The efforts to establish optimal launch rates is holding back this section.
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Once optimal launch rates have been determined, an evaluation of the welfare effects of open access
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policy can be evaluated, in line with much of the current literature.
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% ---------------------------------------------------------------------------------------
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%\section{Kessler Syndrome}\label{Kessler}
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%Discuss the impacts of Kessler Syndrome (an numerical example where this will occur?)
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%Raou and Rondina
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%Adilov
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% TODO: Not enough material to discuss this yet. I think I'll need relative launch rate information.
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% ---------------------------------------------------------------------------------------
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%\section{Numerical Model}\label{Numerical}
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% ---------------------------------------------------------------------------------------
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\section{Concluding Remarks}\label{Conclusion}
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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 developing 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, the following extensions and applications of the
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model will fill gaps in the literature or complement current work:
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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 multiperiod develoment 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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%While I expect there to be a socially suboptimal launch rate under open access, as
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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 determine and drive optimal behavior, 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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Given the following bellman equation
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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,S_t,D_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
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|
\end{align}
|
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|
Giving the optimality condition:
|
|
|
\begin{align}
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|
\frac{F}{\beta} =& 2\parder{V^i}{S_{t+1}}{}
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|
+ m\parder{V^i}{D_{t+1}}{}
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|
+ \parder{V^i}{s^i_{t+1}}{}
|
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|
\end{align}
|
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|
Assuming $\parder{u^i}{S_t}{} = 0$ and $\parder{u^i}{D_t}{} = 0$, then the envelope conditions are:
|
|
|
\begin{align}
|
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|
\parder{V^i}{s^i_{t}}{} - \parder{u^i}{s^i_t}{}=& \beta\left[
|
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|
\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{s^i_t}{}
|
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|
+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{s^i_t}{}
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|
+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{s^i_t}{}
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|
\right] \\
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|
\parder{V^i}{S_{t}}{} =& \beta\left[
|
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|
\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{S_t}{}
|
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|
+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{S_t}{}
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|
+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{S_t}{}
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|
\right] \\
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|
\parder{V^i}{D_{t}}{} =& \beta\left[
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|
\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{D_t}{}
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|
+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{D_t}{}
|
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|
+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{D_t}{}
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|
\right]
|
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|
\end{align}
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|
Note the linearity of the equations.
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|
This allows us to rewrite the system as the following matrix expression.
|
|
|
\begin{align}
|
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|
\beta
|
|
|
\left[
|
|
|
\begin{matrix}
|
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|
\parder{s^i_{t+1}}{s^i_t}{} & \parder{S_{t+1}}{s^i_t}{} & \parder{D_{t+1}}{s^i_t}{} \\
|
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|
\parder{s^i_{t+1}}{S_t}{} & \parder{S_{t+1}}{S_t}{} & \parder{D_{t+1}}{S_t}{} \\
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|
\parder{s^i_{t+1}}{D_t}{} & \parder{S_{t+1}}{D_t}{} & \parder{D_{t+1}}{D_t}{}
|
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|
\end{matrix}
|
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|
\right]
|
|
|
\left[
|
|
|
\begin{matrix}
|
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|
\parder{V^i}{s^i_{t+1}}{} \\
|
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|
\parder{V^i}{S_{t+1}}{} \\
|
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|
\parder{V^i}{D_{t+1}}{}
|
|
|
\end{matrix}
|
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|
\right]
|
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|
=&
|
|
|
\left[
|
|
|
\begin{matrix}
|
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|
\parder{V^i}{s^i_{t}}{} - \parder{u^i}{s^i_t}{} \\
|
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|
\parder{V^i}{S_{t}}{} \\
|
|
|
\parder{V^i}{D_{t}}{}
|
|
|
\end{matrix}
|
|
|
\right] \\
|
|
|
A D_{[s^i_{t+1},S_{t+1},D_{t+1}]} V^i =& D_{[s^i_t,S_t,D_t]} V^i - b
|
|
|
\end{align}
|
|
|
The matrix $A$ above is equivalent to
|
|
|
\begin{align}
|
|
|
\left[
|
|
|
\begin{matrix}
|
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|
1- l^i(\cdot) - s^i_t \parder{l^i}{s^i_t}{}
|
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|
& 1-l^i(\cdot) - s^i_t \parder{l^i}{s^i_t}{} - \sum_{j=1}^N s^j_t \parder{l^j}{S_t}{}
|
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|
& M\left[\parder{l^i}{s^i_t}{} + \sum^N_{j=1} \parder{l^i}{S_t}{} \right] \\
|
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|
- s^i_t \parder{l^i}{S_t}{}
|
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|
& - \sum_{j=1}^N s^j_t \parder{l^j}{S_t}{}
|
|
|
& M \sum^N_{j=1} \parder{l^i}{S_t}{} \\
|
|
|
- s^i_t \parder{l^i}{D_t}{}
|
|
|
& - \sum_{j=1}^N s^j_t \parder{l^j}{D_t}{}
|
|
|
& (1-\delta) + M \sum^N_{j=1} \parder{l^i}{D_t}{} + \parder{g}{D_t}{} \\
|
|
|
\end{matrix}
|
|
|
\right]
|
|
|
\end{align}
|
|
|
Solving this directly is difficult.
|
|
|
We can use the fact that $A^{-1} = \frac{\adj(A)}{\det{A}}$, assuming $A$ is invertible.
|
|
|
\begin{align}
|
|
|
D_{[s^i_{t+1},S_{t+1},D_{t+1}]} V^i =& \frac{\adj(A)}{\beta \det(A)} (D_{[s^i_t,S_t,D_t]} V^i - b)
|
|
|
\end{align}
|
|
|
Using each entry from $D_{[s^i_{t+1},S_{t+1},D_{t+1}]} V^i$ in the optimality condition and the notation
|
|
|
$B|_{i,j}$ to represent the element $b_{i,j}$ from the matrix $B$, we get the condition:
|
|
|
\begin{align}
|
|
|
\frac{F}{\beta} \beta \det(A) = F \det(A) =&
|
|
|
[\adj(A) (D_{[s^i_t,S_t,D_t]} V^i - b)]\big|_{1} \\
|
|
|
&+ 2 [\adj(A) (D_{[s^i_t,S_t,D_t]} V^i - b)]\big|_{2} \notag\\
|
|
|
&+ m [\adj(A) (D_{[s^i_t,S_t,D_t]} V^i - b)]\big|_{3} \notag
|
|
|
\end{align}
|
|
|
A little work remains to develop the euler equation that characterizes the optimal launch decision.
|
|
|
|
|
|
Of course, for any given set of functional forms $l^i,g$, one must verify if $A$ is invertible.
|
|
|
|
|
|
|
|
|
\subsection{Fleet Planner}
|
|
|
\begin{align}
|
|
|
W(\{s^i_t\},S_t,D_t) =& \max_{\{x^i_t\}^N_{i=1} \geq 0}
|
|
|
~~\left( \sum^N_{i=1} u^i(s^i_t,S_t,D_t)\right) - FX_t
|
|
|
+ \beta W(\{s^i_{t+1}\}, S_{t+1}, D_{t+1}) \\
|
|
|
\text{Subject To:}& \notag\\
|
|
|
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) \\
|
|
|
s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
|
|
|
S_t =&\sum_{i=1}^N s^i_t \\
|
|
|
X_t =&\sum_{i=1}^N x^i_t
|
|
|
\end{align}
|
|
|
This is expected to follow the constellation operator's results closely.
|
|
|
|
|
|
|
|
|
|
|
|
\subsection{Survival Rates}\label{APX:Derivations:Survival}
|
|
|
\begin{align}
|
|
|
\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)
|
|
|
= - \parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{} \\
|
|
|
\parder{R}{s^i_t}{} =& \frac{S_t \sum_{i=1}^N
|
|
|
\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)
|
|
|
- \left( \sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] \right)}{(S_t)^2} \\
|
|
|
=& \sum_{i=1}^N \left[ \frac{S_t [1-l^i(s^i_t,S_t,D_t)]}{(S_t)^2}
|
|
|
- \frac{ s^i_t[1-l^i(s^i_t,S_t,D_t)] }{(S_t)^2} \right]
|
|
|
+\sum_{i=1}^N \frac{ s^i_t S_t [ -\parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{}] }{(S_t)^2} \\
|
|
|
=& \sum_{i=1}^N \left[ \frac{S_t - s^i_t}{(S_t)^2}[1-l^i(s^i_t,S_t,D_t)] \right]
|
|
|
+\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \\
|
|
|
=& \sum_{i=1}^N \left[ \frac{1}{S_t}[1-l^i(s^i_t,S_t,D_t)] \right] - \frac{R}{S_t}
|
|
|
+\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \\
|
|
|
=& \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
|
|
|
+\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}
|
|
|
\end{align}
|
|
|
|
|
|
|
|
|
\end{document}
|
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|