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@Misc{EsaTweet,
author = {ESA},
title = {For the first time ever, ESA has performed a 'collision avoidance manoeuvre' to protect one of its satellites from colliding with a 'mega constellation' \#SpaceTraffic},
addendum = {https://twitter.com/esaoperations},
date = {2019-09-02},
nameaddon = {\@ESAOperations},
}
@Electronic{ArsTechnicaStatement,
author = {Brodkin, Jon},
howpublished = {Online Article},
language = {English},
note = {Statement from SpaceX to ARS Technica},
organization = {Ars Technica},
title = {SpaceX satellite was on “collision course” until ESA satellite was re-routed},
url = {https://arstechnica.com/information-technology/2019/09/spacex-satellite-was-on-collision-course-until-esa-satellite-was-re-routed/},
date = {2019-09-03},
}
@Electronic{EsaBlog,
author = {ESA},
howpublished = {Online},
language = {English},
organization = {European Space Agency},
title = {ESA spacecraft dodges large constellation},
url = {http://www.esa.int/Safety_Security/ESA_spacecraft_dodges_large_constellation},
date = {2019-09-03},
}
@PhdThesis{Rao:Dissertation,
author = {Rao, Akhil},
school = {University of Colorado},
title = {The Economics of Orbit Use: Theory, Policy, and Practice},
year = {2019},
}
@Article{adilov_alexander_cunningham_2015,
author = {Adilov and Alexander and Cunningham},
journal = {Environmental and Resource Economics},
title = {An Economic Analysis of Earth Orbit Pollution},
year = {2015},
issn = {0924-6460},
number = {1},
pages = {81–98},
volume = {60},
doi = {10.1007/s10640-013-9758-4},
publisher = {Environmental and Resource Economics},
}
@Article{Macauley_1998,
author = {Macauley, Molly K},
journal = {The Journal of Law and Economics},
title = {Allocation of Orbit and Spectrum Resources for Regional Communications: What's At Stake?},
year = {1998},
issn = {0022-2186},
number = {S2},
pages = {737–764},
volume = {41},
abstract = {Contentious debate surrounds allocation of the geostationary orbit and electromagneticspectrum, two resources used by communications satellites. An extensive economicsliterature alleges that the nonmarket administrative allocative procedures now in place arehighly inefficient, but no research has empirically estimated the welfare loss. This paperdevelops a conceptual framework and a computerized model to estimate the economic valueof the resources, the size and distribution of welfare costs associated with the presentregulatory regime, and the potential gains from more market-like allocation.
Key Words: outer space, communications satellites, pricing natural resources
JEL Classification Nos.: H4, Q2},
doi = {10.1086/467411},
publisher = {The Journal of Law and Economics},
}
@InBook{brillinger_2001,
author = {Brillinger, David R.},
pages = {105–116},
title = {Space Debris: Flux in a Two Dimensional Orbit},
year = {2001},
doi = {10.1007/978-3-0348-8326-9_8},
}
@Article{Adilov2018a,
author = {Adilov and Alexander and Cunningham},
title = {Corrigendum to “An economic “Kessler Syndrome”: A dynamic model of earth orbit debris” [Econom. Lett. 166 (2018) 79–82]},
year = {2018},
issn = {0165-1765},
pages = {185},
volume = {170},
doi = {10.1016/j.econlet.2018.04.012},
}
@Misc{Kessler1990,
author = {Donald Kessler},
title = {Orbital debris environment for spacecraft in low earth orbit},
year = {1990},
doi = {10.2514/6.1990-1353},
}
@Article{Adilov,
author = {Adilov and Alexander and Cunningham},
title = {Earth Orbit Debris: An Economic Model},
year = {2015},
issn = {1556-5068},
doi = {10.2139/ssrn.2264915},
}
@Article{Kessler1978,
author = {Kessler, Donald J. and Cour-Palais, Burton G.},
journal = {Journal of Geophysical Research: Space Physics},
title = {Collision frequency of artificial satellites: The creation of a debris belt},
year = {1978},
number = {A6},
pages = {2637-2646},
volume = {83},
abstract = {As the number of artificial satellites in earth orbit increases, the probability of collisions between satellites also increases. Satellite collisions would produce orbiting fragments, each of which would increase the probability of further collisions, leading to the growth of a belt of debris around the earth. This process parallels certain theories concerning the growth of the asteroid belt. The debris flux in such an earth-orbiting belt could exceed the natural meteoroid flux, affecting future spacecraft designs. A mathematical model was used to predict the rate at which such a belt might form. Under certain conditions the belt could begin to form within this century and could be a significant problem during the next century. The possibility that numerous unobserved fragments already exist from spacecraft explosions would decrease this time interval. However, early implementation of specialized launch constraints and operational procedures could significantly delay the formation of the belt.},
doi = {10.1029/JA083iA06p02637},
eprint = {https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JA083iA06p02637},
url = {https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/JA083iA06p02637},
}
@Electronic{FAA2020,
author = {NA},
howpublished = {Online},
month = oct,
note = {Describes altitude of LEO and GEO},
organization = {Federal Aviation Administration},
url = {https://www.faa.gov/space/additional_information/faq/#s1},
year = {2020},
}
@TechReport{RaoRondina2020,
author = {Rao and Rondina},
institution = {NA},
title = {Cost in Space:Debris and Collision Risk in the Orbital Commons},
year = {2020},
month = feb,
note = {Middlebury College | UC San Diego},
type = {Working Paper},
}
@Article{Adilov2019,
author = {Adilov and Cunningham and Alexander and Duvall and Shiman},
journal = {Economic Inquiry},
title = {LEFT FOR DEAD: ANTI‐COMPETITIVE BEHAVIOR IN ORBITAL SPACE},
year = {2019},
month = {04},
volume = {57},
doi = {10.1111/ecin.12790},
}
@Article{Rao2020,
author = {Rao and Burgess and Kaffine},
journal = {Proceedings of the National Academy of Sciences},
title = {Orbital-use fees could more than quadruple the value of the space industry},
year = {2020},
issn = {0027-8424},
number = {23},
pages = {12756--12762},
volume = {117},
abstract = {The commercial satellite industry is rapidly expanding. A side effect of this expansion is a growing buildup of space debris that imposes costly collision risk on satellite operators. Proposed solutions to this debris have been primarily technological, but the core of the problem is incentives{\textemdash}satellites are being launched without consideration of the collision risks they impose on other operators. We show that this incentive problem can be solved with an internationally harmonized {\textquotedblleft}orbital-use fee{\textquotedblright} (OUF){\textemdash}a tax on orbiting satellites. Using a coupled physical{\textendash}economic model, we project that an optimally designed OUF could more than quadruple the long-run value of the satellite industry by 2040.The space industry{\textquoteright}s rapid recent growth represents the latest tragedy of the commons. Satellites launched into orbit contribute to{\textemdash}and risk damage from{\textemdash}a growing buildup of space debris and other satellites. Collision risk from this orbital congestion is costly to satellite operators. Technological and managerial solutions{\textemdash}such as active debris removal or end-of-life satellite deorbit guidelines{\textemdash}are currently being explored by regulatory authorities. However, none of these approaches address the underlying incentive problem: satellite operators do not account for costs they impose on each other via collision risk. Here, we show that an internationally harmonized orbital-use fee can correct these incentives and substantially increase the value of the space industry. We construct and analyze a coupled physical{\textendash}economic model of commercial launches and debris accumulation in low-Earth orbit. Similar to carbon taxes, our model projects an optimal fee that rises at a rate of 14\% per year, equal to roughly $235,000 per satellite-year in 2040. The long-run value of the satellite industry would more than quadruple by 2040{\textemdash}increasing from around $600 billion under business as usual to around $3 trillion. In contrast, we project that purely technological solutions are unlikely to fully address the problem of orbital congestion. Indeed, we find debris removal sometimes worsens economic damages from congestion by increasing launch incentives. In other sectors, addressing the tragedy of the commons has often been a game of catch-up with substantial social costs. The infant space industry can avert these costs before they escalate.},
doi = {10.1073/pnas.1921260117},
eprint = {https://www.pnas.org/content/117/23/12756.full.pdf},
publisher = {National Academy of Sciences},
url = {https://www.pnas.org/content/117/23/12756},
}
@Article{GrzelkaWagner2019,
author = {Grzelka, Zachary and Wagner, Jeffrey},
journal = {Environmental and Resource Economics},
title = {Managing Satellite Debris in Low-Earth Orbit: Incentivizing Ex Ante Satellite Quality and Ex Post Take-Back Programs},
year = {2019},
issn = {0924-6460},
number = {1},
pages = {319–336},
volume = {74},
doi = {10.1007/s10640-019-00320-3},
publisher = {Environmental and Resource Economics},
}
@Misc{Kennedy1962,
author = {John F. Kennedy},
month = sep,
title = {Address at Rice University on the Nation's Space Effort},
year = {1962},
url = {https://er.jsc.nasa.gov/seh/ricetalk.htm},
}
@Article{adilov_alexander_cunningham_2018,
author = {Adilov and Alexander and Cunningham},
journal = {Economics Letters},
title = {An economic “Kessler Syndrome”: A dynamic model of earth orbit debris},
year = {2018},
issn = {0165-1765},
pages = {79–82},
volume = {166},
doi = {10.1016/j.econlet.2018.02.025},
publisher = {Economics Letters},
}
@Comment{jabref-meta: databaseType:bibtex;}

542
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\documentclass{article}
%%%%%%%%%Packages%%%%%%%%%%%%%%%
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\title{Dynamic Launch Decisions for Satellite Constellation Operators}
\author{William King}
\institute{Washington State University}
\begin{document}
\maketitle
\begin{abstract}
Over the last 10 years new technology has make low earth orbits (LEOs) more accessible, and
the resulting increase in LEO satellites has increased the risk of collision.
Because debris in orbit generates more debris through collisions with objects in orbit
and the debris created during launch and operation imposes a negative externality
on other operators,
optimal use of orbits is believed to not occur under free entry.
This paper develops a dynamic model of satellite operation incorporating two effects not considered
in previous models.
The first effect is complementarity between same-purpose, single operator fleets (called constellations).
The second effect is collision avoidance efficiencies that exist within constellations.
The primary result is a theoretical model and the resulting analysis of the difference in survival ratios between
constellation operators and society.
\end{abstract}
\keywords{Orbits, Pollution, Economies of Scale, Externality }
\jel{Q29, Q58, L25}
\newpage
% ---------------------------------------------------------------------------------------
\section{Introduction}
% Motivating Example (ESA - SpaceX)
In September of 2019, the European Space Agency (ESA) released a tweet explaining that they had performed an
adjustment maneuver to avoid a collision with a SpaceX Starlink Satellite in Low Earth Orbit (LEO)\autocite{EsaTweet}.
While later reports\autocite{ArsTechnicaStatement} described it as the result of miscommunications,
ESA used the opportunity to highlight the difficulties arising from coordinating avoidance maneuvers and how
such coordination will become more difficult as the size and number of
single purpose, single operator satellite fleets (satellite constellations) increase in low earth orbit\autocite{EsaBlog}.
% Background on issues of congestion and pollution
% Kessler Syndrome
In spite of the fact that there is a lot of maneuvering room in outer space,
%\footnote{``Space is big. Really big. You just won’t believe how vastly hugely mind bogglingly big it is.
%I mean, you may think it’s a long way down the road to the chemist,
%but that’s just peanuts to space.''\cite{DouglasAdams}}
the repeated interactions of periodic orbits make collisions probable.
Consequently, objects in orbit are subject to both a congestion effect and a pollution effect.
Congestion effects are primarily derived from avoiding collisions between artificial satellites.
Pollution in orbit consists of debris, both natural and man-made, which increases
the probability of an unforseen collision.
The defining dynamic of pollution in orbit is that it self-propogates as debris collides with itself
and orbiting satellites to generate more debris.
This dynamic underlies a key concern, originally explored by Kessler and Cour-Palais \autocite{Kessler1978}
that with sufficient mass in orbit (through satellite launches), the debris generating process
could undergo a runaway effect rendering various orbital regions unusable.
This cascade of collisions is often known as Kessler syndrome and theoretically
may take place over various timescales.
% ---------------
Orbits may be divided into three primary groups,
Low Earth Orbit (LEO, less than 2,400km in altitude\autocite{FAA2020}),
Medium Earth Orbit (MEO), and High Earth Orbit (HEO) with Geostationary Earth Orbit (GEO)
considered a particular classification of orbit.
While the topic of LEO allocation has historically remained somewhat unexplored, the last 6 years has seen
a variety of new empircal studies and theoretical models published.
In general, three primary, related topics appear in the literature:
Allocative Efficiency, Externality Mitigation, and Economic vs Physical Kessler Syndromes.
% ---------------
Although Kessler and Cour-Palais determined that a runaway pollution effect could make a set of orbits
physically unusable, Adilov et al \autocite{adilov_alexander_cunningham_2018} %Kessler Syndrome
have shown that economic benefits provided by orbits will drop sufficiently to make the net marginal
benefit of new launches negative before the physical kessler syndrome occurs.
% ---------------
%Allocative efficiency
The primary concern is to establish wether or not orbits will be overused
due to their common-pool nature, and if allocation procedures are efficient.
The earliest theoretical model I have found, due to Adilov, Alexander, and
Cunningham \autocite{adilov_alexander_cunningham_2015}, examines pollution
using a two-period salop model, incorporating the effects of launch debris on
survival into the second period.
They find that the social planner generates debris and launches at lower rates
than a free entry market.
This same result was found by Rao and Rondina \autocite{RaoRondina2020} in
the context of an infinite period dynamic model.
They approach the problem in the case where numerous operators in a free entry environment
can each launch a single, identical satellite.
% ---------------
In addition to analyzing the allocative results, a significant area of interest is
what impact various policy interventions can have.
The policies analyzed and methods used have been widely varied.
Macauley \autocite{Macauley_1998} provided the first evidence of suboptimal behavior in orbit
by estimating the welfare lose due to the current method of assigning GEO slots to operators.
The potential losses due to anti-competitive behavior was highlighted by Adilov et al \autocite{Adilov2019},
who have analyzed the opportunities for strategic
``warehousing'' of non-functional satellites as a means of increasing competitive advantage by
denying operating locations to competitors in GEO.
Grzelka and Wagner \autocite{GrzelkaWagner2019} explore methods of encouraging satellite quality (in terms of debris)
and cleanup.
Finally, Rao and Rondina \autocite{RaoRondina2020b} estimate that achieving socially optimal
behavior through orbital use fees could increase the value generated by the space industry by a factor of 4.
% ---------------
This paper's objective is to devlop a dynamic model which incoporates
complementary effects of constellations as well as collision avoidance efficiencies of constellations,
thus addressing a gap in the current literature.
In addition, I examine if there exists a negative externality related to changes in stock size, and
establish a condition related to average behavior that describes this externality.
Finally, I lay foundations for the derivation of profit maximizing launch rules.
The paper is organized as follows.
Section \ref{Model} describes the mathematical organization of the model
for the cases of independent constellation operators and a social planner
operating the same constellations.
%It also includes a brief digression into the free entry conditions.
Section \ref{Comparisons} evaluates the differences between the
constellation operators and social planner models, particularly
the difference between marginal survival rates .
%Of particular interest is the difference in launch rates and marginal survival rates.
%Section \ref{Kessler} ...
Section \ref{Conclusion} concludes with a discussion of potential extensions and
topics which have not yet been addressed.
% ---------------------------------------------------------------------------------------
\section{Model}\label{Model}
%Intuitive description
The dynamic model is an extension of Rao and Rondina's working paper \autocite{RaoRondina2020},
specifically their non-stochastic model.
For a given orbital shell (a set of orbits that interact regularly), I assume there are $N$ operators,
each of which has the potential to launch and operate a satellite
constellation consisting of some endogenosly chosen number of identical satellites.
These satellites are not only identical within a constellation, but across constellations.
% -------------------
Each constellation operator has a personal satellite stock $s^i_t$ in each period, and chooses the
number of launches in that time period $x^i_t$.
For simplicity, each launch is assumed to have a fixed cost $F$.
In the aggregate, the satellite stock and launches for each period are represented by:
\begin{align}
S_t =&\sum_{i=1}^N s^i_t \\
X_t =&\sum_{i=1}^N x^i_t
\end{align}
% -------------------
Satellites in a constellation are damaged or destroyed at the rate $l^i(s^i_t,S_t,D_t)$,
which is assumed to be increasing in $s^i_t$, $S_t$, and $D_t$ (debris, see below).
One key difference from the previous models of Rao and Redina \autocite{RaoRondina2020} and
Adilov et al \autocite{adilov_alexander_cunningham_2018} is that this model allows the rate of
collision within constellations and between constellations to be different.
This reflects the assumption that an operator can and will put more effort into protecting the satellites within
the constellation from each other.
One example of how this can be acomplished is that while choosing the orbits for a constellation,
it is possible for an operator to chose a set of trajectories that best meet their needs and
minimizes the risk of collision within the constellation.
Mathematically this is represented by the inclusion of $s^i_t$ in $l^i$.
Together with the launch rate, we obtain a law of motion for both constellation-level
and society-level satellite stocks.
\begin{align}
s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
S_{t+1} =& X_t + \sum^N_{i=1} \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t
\end{align}
% -------------------
The level of debris in each period is represented by $D_t$, and is assumed to pose a latent risk.
In particular, it is assumed that once debris is created, the risk it provides is only avoidable
through not launching future satellites.
In addition to natually occuring debris, debris is generated through the following three mechanisms.
\begin{itemize}
\item At launch, various processes can shed debris.
Examples include leftover rocket stages, explosions during launch and deployment,
and slag from solid rocket boosters.
\item When destroyed, satellites will fragment and produce debris.
\item Debris can collide with other debris, forming more but smaller debris.
\end{itemize}
This provides the following law of debris dynamics.
\begin{align}
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)
\end{align}
where $\delta$ represents the decay of debris -- through reentering the atmosphere -- for a given shell,
$M$ represents the debris generated from each collision,
$m$ represents the debris generated from each launch,
and $g(D_t)$ represents the new fragments from debris colliding with other debris.
% -------------------
Each constellation $i \in {1,\dots,N}$ produces value for their operator at each period according to the function:
\begin{align}
u^i(s^i_t, S_t, D_t) = u^i(s^i_t)
\end{align}
For computational simplicity, it is assumed that benefits provided are wholely dependent on the number
of satellites in operation.
The approach presented in the appendix is generalizable to the case where benefits are conditional on
the total satellite and debris stocks.
Complementarity within a constellation appears when $\parder{u^i}{s^i_t}{2} > 0$ for some values of $s^i_t,S_t, D_t$.
% ---------------------------------------------
\subsection{Constellation Operator's Program}
The aformentioned aspects of the model form the following bellman equation for each constellation operator.
\begin{align}
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}) \\
\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 % Is this also a state variable?
\end{align}
The system of envelope conditions is linear and can be written as a matrix equation.
In Appendix \ref{APX:Derivations:Constellation} I begin development of the euler equation
in a generalizable way.
Unfortunately repeated errors in the mathematics has prevented me from achieving more in the
launch rate analysis to date.
%The resulting euler equation is:
%\begin{align}
% 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} % TODO: This could also be changed to a matrix form with a row-vector.
%Where $A$ is the matrix of partial derivatives of the laws of motion corresponding to
%the envelope conditions.
%The matrix $A$ and the derivation of the euler equation
%is described in Appendix \ref{APX:Derivations:Constellation}
% ---------------------------------------------------------------------------------------
%\subsubsection{Free Entry}
%Operators are assumed to enter the market as long as the value of entering is above 0.
% ---------------------------------------------
\subsection{Social Planner's Program}
The social planner (or fleet planner to use Rao and Rondina's terminology), is tasked with
maximizing the sum of the operators' benefits $W(\{s^i_t\},S_t,D_t) = \sum^N_{i=1} V^i(s^i_t,S_t,D_t)$.
%Crucial assumption. $\beta^i =\beta \forall i$
Often, in polluting environments, there is an ambient population that is harmed by pollution.
Very rarely does satellite debris pose a hazard to those on earth, thus in this model
the only population whom's welfare is addressed are the satellite operators themselves.
\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}
%Goal: Add the euler equation.
Due to the aformentioned errors, I have not begun a derivation of the optimal launch rate
for the social planner at this point.
I expect it to be solvable using the same approach as for the constellation operators
outlined in Appendix \ref{APX:Derivations:Constellation}.
% ---------------------------------------------------------------------------------------
%\section{Convergence Properties}\label{Convergence}
% ---------------------------------------------------------------------------------------
\section{Comparisons}\label{Comparisons}
% Marginal survival.
In line with theory on common-pool resources, we expect there to be a negative externality
incurred by increasing the satellite stock.
The details of this externality can be observed in the marginal suvival rate.
Define the survival rate for a constellation and the society to be:
\begin{align}
R_i =& \frac{s^i_{t+1}- x^i_t}{s^i_t} = 1- l^i(s^i_t,S_t,D_t) \\
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}
\end{align}
The marginal survival rates when a given constellation $i$ changes size are:
\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}{} \label{EQ:iii} \\
\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{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}{} \label{EQ:i}
\end{align}
Note that $ \sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}$ is the average marginal survival rate
across constellation operators.
The derivation of equation \ref{EQ:i} is in Appendix \ref{APX:Derivations:Survival}.
Direct comparison between the marginal survival rates of an individual operator and the social planner's fleet
cannot proceede further without specifying the functional loss forms $l^i(\cdot)$
and specifying which firm the comparison is with.
In spite of this, conditions on the average effects can be specified as follows.
Society's marginal survival rate is greater than the average marginal survival rate when:
% NOTE: Should I do this using absolute value arguments? I don't think so.
\begin{align}
\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}{}
\geq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\
\sum_{i=1}^N R_i - R \geq& 0\\
\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\\
\sum_{i=1}^N (S_t - s^i_t) [1- l^i(s^i_t,S_t,D_t)] \geq& 0 \label{EQ:ii}
\end{align}
Which is always true as $S_t > s^i_t$ and $l^i(\cdot) \in [0,1]$ for all $i$.
%As we are discussing arithemetic means of rates, it makes sense.
%A geometric mean might behave differently.
If a single constellation makes up the whole stock of satellites, then \eref{EQ:ii} reduces
to a tautology.
As $\parder{R_i}{s^i_t}{} < 0$ from \eref{EQ:iii} and the assumptions on collision mechanics, we see
that the average marginal survival rate acts as a lower bound on the marginal societal survival rate.
Assuming that survival rates are not increased by adding another satellite i.e. $\parder{R}{s^i_t}{}<0$ then gives
us the following bounds on societal rates $\sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{}<\parder{R}{s^i_t}{}<0$
%By way of interpretation, this means adding a satellite to a constellation has a larger impact
%on the society's survival rate than on the average survival rate across constellations.
%This result is consistent with previous results establishing a negative externality
This suggests that some operators experience marginal changes to their own satellite
stocks much more intensely than society as a whole does.
%Do it again using a geometric mean.
%Welfare
% TODO: The efforts to establish optimal launch rates is holding back this section.
Once optimal launch rates have been determined, an evaluation of the welfare effects of open access
policy can be evaluated, in line with much of the current literature.
% ---------------------------------------------------------------------------------------
%\section{Kessler Syndrome}\label{Kessler}
%Discuss the impacts of Kessler Syndrome (an numerical example where this will occur?)
%Raou and Rondina
%Adilov
% TODO: Not enough material to discuss this yet. I think I'll need relative launch rate information.
% ---------------------------------------------------------------------------------------
%\section{Numerical Model}\label{Numerical}
% ---------------------------------------------------------------------------------------
\section{Concluding Remarks}\label{Conclusion}
The dynamic model developed in this paper provides insight into the incentives faced by
constellation operators in comparison with a social planner and, when completed, should provide
insight on how self-perpetuating externalities drive sub-optimal behavior.
At this point, major work remains in developing optimal launch rates and verifying if
the expected difference in optimal launch rates between individual operators and a social planner exist,
as occurs in other models.
In addition to the remaining work on fleshing out the model, the following extensions and applications of the
model will fill gaps in the literature or complement current work:
\begin{itemize}
\item Asymmetric constellation sizes: What are the impacts on social welfare when a variety of
constellation sizes exist
\item Policy interventions: Various policy proposals to reduce negative externalities have been proposed,
including launch quotas, launch taxes, and orbit use fees \autocite{RaoRondina2020b}.
\item Introduction of stochastics: There are various ways that stochastics can enter the model, from the scales
determining debris generation to the per-period satellite collision rate.
\item Differentiation of satellites and launch methods: Different launch methods and satellite features can
affect the accumulation of debris.
\item Richer satellite lifetimes: the current satellite lifetime of [launch, operate] could be extended
to include stages such as development and disposal.
In particular, a multiperiod develoment cycle with sunk costs incurred along the way may
exacerbate problems where stable equilibria are overshot.
This will allow for more policy interventions to be analyzed.
\item Strategic behavior: Concerns include whether constellation network effects can be used to prevent new entrants
in the case of competition for a satellite services market.
\end{itemize}
While computationally complicated, the results so far imply that there is a defined difference between
the risks faced at the constellation operator's level and the level of society as a whole.
%While I expect there to be a socially suboptimal launch rate under open access, as
Although not a common topic in economics, orbit use has properties that requires
current study in order to determine and drive optimal behavior, before there are no more viable orbits to use.
\newpage
\printbibliography
\newpage
\appendix
\section{Derivations} \label{APX:Derivations}
%\subsection{Useful Mathematical Notes}\label{APX:Derivations:Useful}
%To fill in with a set of useful mathematical notes for use throughout.
%\subsubsection{Useful Derivatives}
\subsection{Constellation Operator}\label{APX:Derivations:Constellation}
Given the following bellman equation
\begin{align}
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}) \\
\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}
Giving the optimality condition:
\begin{align}
\frac{F}{\beta} =& 2\parder{V^i}{S_{t+1}}{}
+ m\parder{V^i}{D_{t+1}}{}
+ \parder{V^i}{s^i_{t+1}}{}
\end{align}
Assuming $\parder{u^i}{S_t}{} = 0$ and $\parder{u^i}{D_t}{} = 0$, then the envelope conditions are:
\begin{align}
\parder{V^i}{s^i_{t}}{} - \parder{u^i}{s^i_t}{}=& \beta\left[
\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{s^i_t}{}
+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{s^i_t}{}
+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{s^i_t}{}
\right] \\
\parder{V^i}{S_{t}}{} =& \beta\left[
\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{S_t}{}
+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{S_t}{}
+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{S_t}{}
\right] \\
\parder{V^i}{D_{t}}{} =& \beta\left[
\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{D_t}{}
+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{D_t}{}
+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{D_t}{}
\right]
\end{align}
Note the linearity of the equations.
This allows us to rewrite the system as the following matrix expression.
\begin{align}
\beta
\left[
\begin{matrix}
\parder{s^i_{t+1}}{s^i_t}{} & \parder{S_{t+1}}{s^i_t}{} & \parder{D_{t+1}}{s^i_t}{} \\
\parder{s^i_{t+1}}{S_t}{} & \parder{S_{t+1}}{S_t}{} & \parder{D_{t+1}}{S_t}{} \\
\parder{s^i_{t+1}}{D_t}{} & \parder{S_{t+1}}{D_t}{} & \parder{D_{t+1}}{D_t}{}
\end{matrix}
\right]
\left[
\begin{matrix}
\parder{V^i}{s^i_{t+1}}{} \\
\parder{V^i}{S_{t+1}}{} \\
\parder{V^i}{D_{t+1}}{}
\end{matrix}
\right]
=&
\left[
\begin{matrix}
\parder{V^i}{s^i_{t}}{} - \parder{u^i}{s^i_t}{} \\
\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}
1- l^i(\cdot) - s^i_t \parder{l^i}{s^i_t}{}
& 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}{}
& M\left[\parder{l^i}{s^i_t}{} + \sum^N_{j=1} \parder{l^i}{S_t}{} \right] \\
- s^i_t \parder{l^i}{S_t}{}
& - \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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The goal of this paper is to examine the impact of network externalities on the optimal decision making for satellite operators
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@Misc{EsaTweet,
author = {European Space Agency},
title = {For the first time ever, ESA has performed a 'collision avoidance manoeuvre' to protect one of its satellites from colliding with a 'mega constellation' \#SpaceTraffic},
addendum = {https://twitter.com/esaoperations},
date = {2019-09-02},
nameaddon = {\@ESAOperations},
}
@Electronic{ArsTechnicaStatement,
author = {Brodkin, Jon},
howpublished = {Online Article},
language = {English},
note = {Statement from SpaceX to ARS Technica},
organization = {Ars Technica},
title = {SpaceX satellite was on “collision course” until ESA satellite was re-routed},
url = {https://arstechnica.com/information-technology/2019/09/spacex-satellite-was-on-collision-course-until-esa-satellite-was-re-routed/},
date = {2019-09-03},
}
@Electronic{EsaBlog,
author = {ESA},
howpublished = {Online},
language = {English},
organization = {European Space Agency},
title = {ESA spacecraft dodges large constellation},
url = {http://www.esa.int/Safety_Security/ESA_spacecraft_dodges_large_constellation},
date = {2019-09-03},
}
@PhdThesis{Rao:Orbit,
author = {Rao, Akhil},
school = {University of Colorado},
title = {The Economics of Orbit Use: Theory, Policy, and Practice},
year = {2019},
}
@Article{adilov_alexander_cunningham_2015,
author = {Adilov, Nodir and Alexander, Peter J. and Cunningham, Brendan M.},
journal = {Environmental and Resource Economics},
title = {An Economic Analysis of Earth Orbit Pollution},
year = {2015},
issn = {0924-6460},
number = {1},
pages = {81–98},
volume = {60},
doi = {10.1007/s10640-013-9758-4},
publisher = {Environmental and Resource Economics},
}
@Article{Macauley_1998,
author = {Macauley, Molly K},
journal = {The Journal of Law and Economics},
title = {Allocation of Orbit and Spectrum Resources for Regional Communications: What's At Stake?},
year = {1998},
issn = {0022-2186},
number = {S2},
pages = {737–764},
volume = {41},
abstract = {Contentious debate surrounds allocation of the geostationary orbit and electromagneticspectrum, two resources used by communications satellites. An extensive economicsliterature alleges that the nonmarket administrative allocative procedures now in place arehighly inefficient, but no research has empirically estimated the welfare loss. This paperdevelops a conceptual framework and a computerized model to estimate the economic valueof the resources, the size and distribution of welfare costs associated with the presentregulatory regime, and the potential gains from more market-like allocation.
Key Words: outer space, communications satellites, pricing natural resources
JEL Classification Nos.: H4, Q2},
doi = {10.1086/467411},
publisher = {The Journal of Law and Economics},
}
@InBook{brillinger_2001,
author = {Brillinger, David R.},
pages = {105–116},
title = {Space Debris: Flux in a Two Dimensional Orbit},
year = {2001},
doi = {10.1007/978-3-0348-8326-9_8},
}
@Article{Grzelka2019,
author = {Zachary Grzelka and Jeffrey Wagner},
journal = {Environmental and Resource Economics},
title = {Managing Satellite Debris in Low-Earth Orbit: Incentivizing Ex Ante Satellite Quality and Ex Post Take-Back Programs},
year = {2019},
month = {feb},
number = {1},
pages = {319--336},
volume = {74},
doi = {10.1007/s10640-019-00320-3},
publisher = {Springer Science and Business Media {LLC}},
}
@Article{Adilov2018,
author = {Nodir Adilov and Peter J. Alexander and Brendan M. Cunningham},
title = {An economic “Kessler Syndrome”: A dynamic model of earth orbit debris},
year = {2018},
issn = {0165-1765},
pages = {79-82},
volume = {166},
doi = {10.1016/j.econlet.2018.02.025},
}
@Article{Adilov2018a,
author = {Nodir Adilov and Peter J. Alexander and Brendan M. Cunningham},
title = {Corrigendum to “An economic “Kessler Syndrome”: A dynamic model of earth orbit debris” [Econom. Lett. 166 (2018) 79–82]},
year = {2018},
issn = {0165-1765},
pages = {185},
volume = {170},
doi = {10.1016/j.econlet.2018.04.012},
}
@Misc{Kessler1990,
author = {Donald Kessler},
title = {Orbital debris environment for spacecraft in low earth orbit},
year = {1990},
doi = {10.2514/6.1990-1353},
}
@Article{Adilov,
author = {Nodir Adilov and Peter J. Alexander and Brendan Michael Cunningham},
title = {Earth Orbit Debris: An Economic Model},
year = {2015},
issn = {1556-5068},
doi = {10.2139/ssrn.2264915},
}
@Article{Kessler1978,
author = {Kessler, Donald J. and Cour-Palais, Burton G.},
journal = {Journal of Geophysical Research: Space Physics},
title = {Collision frequency of artificial satellites: The creation of a debris belt},
year = {1978},
number = {A6},
pages = {2637-2646},
volume = {83},
abstract = {As the number of artificial satellites in earth orbit increases, the probability of collisions between satellites also increases. Satellite collisions would produce orbiting fragments, each of which would increase the probability of further collisions, leading to the growth of a belt of debris around the earth. This process parallels certain theories concerning the growth of the asteroid belt. The debris flux in such an earth-orbiting belt could exceed the natural meteoroid flux, affecting future spacecraft designs. A mathematical model was used to predict the rate at which such a belt might form. Under certain conditions the belt could begin to form within this century and could be a significant problem during the next century. The possibility that numerous unobserved fragments already exist from spacecraft explosions would decrease this time interval. However, early implementation of specialized launch constraints and operational procedures could significantly delay the formation of the belt.},
doi = {10.1029/JA083iA06p02637},
eprint = {https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JA083iA06p02637},
url = {https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/JA083iA06p02637},
}
@Electronic{FAA2020,
author = {NA},
howpublished = {Online},
month = oct,
note = {Describes altitude of LEO and GEO},
organization = {Federal Aviation Administration},
url = {https://www.faa.gov/space/additional_information/faq/#s1},
year = {2020},
}
@Article{adilov_alexander_cunningham_2018,
author = {Adilov, Nodir and Alexander, Peter J. and Cunningham, Brendan M.},
journal = {Economics Letters},
title = {An economic “Kessler Syndrome”: A dynamic model of earth orbit debris},
year = {2018},
issn = {0165-1765},
pages = {79–82},
volume = {166},
doi = {10.1016/j.econlet.2018.02.025},
publisher = {Economics Letters},
}
@TechReport{RaoRondina2020,
author = {Akhil Rao; Giacomo Rondina},
institution = {NA},
title = {Cost in Space:Debris and Collision Risk in the Orbital Commons},
year = {2020},
month = feb,
note = {Middlebury College | UC San Diego},
type = {Working Paper},
}
@Article{Adilov2019,
author = {Adilov, Nodir and Cunningham, Brendan and Alexander, Peter and Duvall, Jerry and Shiman, Daniel},
journal = {Economic Inquiry},
title = {LEFT FOR DEAD: ANTI‐COMPETITIVE BEHAVIOR IN ORBITAL SPACE},
year = {2019},
month = {04},
volume = {57},
doi = {10.1111/ecin.12790},
}
@Article{RaoRondina2020b,
author = {Rao, Akhil and Burgess, Matthew G. and Kaffine, Daniel},
journal = {Proceedings of the National Academy of Sciences},
title = {Orbital-use fees could more than quadruple the value of the space industry},
year = {2020},
issn = {0027-8424},
number = {23},
pages = {12756--12762},
volume = {117},
abstract = {The commercial satellite industry is rapidly expanding. A side effect of this expansion is a growing buildup of space debris that imposes costly collision risk on satellite operators. Proposed solutions to this debris have been primarily technological, but the core of the problem is incentives{\textemdash}satellites are being launched without consideration of the collision risks they impose on other operators. We show that this incentive problem can be solved with an internationally harmonized {\textquotedblleft}orbital-use fee{\textquotedblright} (OUF){\textemdash}a tax on orbiting satellites. Using a coupled physical{\textendash}economic model, we project that an optimally designed OUF could more than quadruple the long-run value of the satellite industry by 2040.The space industry{\textquoteright}s rapid recent growth represents the latest tragedy of the commons. Satellites launched into orbit contribute to{\textemdash}and risk damage from{\textemdash}a growing buildup of space debris and other satellites. Collision risk from this orbital congestion is costly to satellite operators. Technological and managerial solutions{\textemdash}such as active debris removal or end-of-life satellite deorbit guidelines{\textemdash}are currently being explored by regulatory authorities. However, none of these approaches address the underlying incentive problem: satellite operators do not account for costs they impose on each other via collision risk. Here, we show that an internationally harmonized orbital-use fee can correct these incentives and substantially increase the value of the space industry. We construct and analyze a coupled physical{\textendash}economic model of commercial launches and debris accumulation in low-Earth orbit. Similar to carbon taxes, our model projects an optimal fee that rises at a rate of 14\% per year, equal to roughly $235,000 per satellite-year in 2040. The long-run value of the satellite industry would more than quadruple by 2040{\textemdash}increasing from around $600 billion under business as usual to around $3 trillion. In contrast, we project that purely technological solutions are unlikely to fully address the problem of orbital congestion. Indeed, we find debris removal sometimes worsens economic damages from congestion by increasing launch incentives. In other sectors, addressing the tragedy of the commons has often been a game of catch-up with substantial social costs. The infant space industry can avert these costs before they escalate.},
doi = {10.1073/pnas.1921260117},
eprint = {https://www.pnas.org/content/117/23/12756.full.pdf},
publisher = {National Academy of Sciences},
url = {https://www.pnas.org/content/117/23/12756},
}
@Comment{jabref-meta: databaseType:bibtex;}

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\title{Dynamic Launch Decisions for Satellite Constellation Operators}
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Over the last 10 years new technology has make low earth orbits (LEOs) more accessible, and
the resulting increase in LEO satellites has increased the risk of collision.
Because debris in orbit generates more debris through collisions with objects in orbit
and the debris created during launch and operation imposes a negative externality
on other operators,
optimal use of orbits is believed to not occur under free entry.
This paper develops a dynamic model of satellite operation incorporating two effects not considered
in previous models.
The first effect is complementarity between satellites within the same operator's fleet (called a constellation).
The second effect is collision avoidance efficiencies that exist within constellations.
The primary result is a theoretical model and the resulting analysis of the difference in survival ratios between
constellation operators and society.
\end{abstract}
\keywords{Orbits, Pollution, Economies of Scale, Externality }
\jel{Q29, Q58, L25}
\textbf{Acknowledgments:} I am the sole author and have recieved no contributions from others as of yet.
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\documentclass[leqno]{article}
%% Created with wxMaxima 19.07.0
\setlength{\parskip}{\medskipamount}
\setlength{\parindent}{0pt}
\usepackage{iftex}
\ifPDFTeX
% PDFLaTeX or LaTeX
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\DeclareUnicodeCharacter{00B5}{\ensuremath{\mu}}
\else
% XeLaTeX or LuaLaTeX
\usepackage{fontspec}
\fi
\usepackage{graphicx}
\usepackage{color}
\usepackage{amsmath}
\usepackage{grffile}
\usepackage{ifthen}
\newsavebox{\picturebox}
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\newcommand{\includeimage}[1]{
\savebox{\picturebox}{\includegraphics{#1}}
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}
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\usepackage{animate} % This package is required because the wxMaxima configuration option
% "Export animations to TeX" was enabled when this file was generated.
\definecolor{labelcolor}{RGB}{100,0,0}
\begin{document}
\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{4em}\color{red}\bfseries
(\% i1)
\end{minipage}
\begin{minipage}[t]{\textwidth}\color{blue}
A: matrix([1 -l -s*l\_s,1-l -s*l\_s - Nsl\_S,M*(l\_s + Nl\_S)],[-s*l\_S,-Nsl\_S,M*Nl\_S],[-s*l\_D,-NSl\_D,1-delta + M*Nl\_D + g\_D]);
\end{minipage}
%%% OUTPUT:
\[\displaystyle \tag{A}
\begin{pmatrix}-{l_s} s-l+1 & -{l_s} s-l-{{\mathit{Nsl}}_S}+1 & M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \\
-{l_S} s & -{{\mathit{Nsl}}_S} & M\, {{\mathit{Nl}}_S}\\
-{l_D} s & -{{\mathit{NSl}}_D} & {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\end{pmatrix}\mbox{}
\]
%%%%%%%%%%%%%%%
\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{4em}\color{red}\bfseries
(\% i2)
\end{minipage}
\begin{minipage}[t]{\textwidth}\color{blue}
invert(A);
\end{minipage}
%%% OUTPUT:
\[\displaystyle \tag{\% o2}
\begin{pmatrix}\frac{M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) } & \frac{-\left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) -M\, {{\mathit{NSl}}_D} \left( {l_s}+{{\mathit{Nl}}_S}\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) } & \frac{M\, {{\mathit{Nl}}_S} \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +M\, {{\mathit{Nsl}}_S} \left( {l_s}+{{\mathit{Nl}}_S}\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }\\
\frac{\left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s}{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) } & \frac{\left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \left( -{l_s} s-l+1\right) +M\, {l_D} \left( {l_s}+{{\mathit{Nl}}_S}\right) s}{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) } & \frac{-M\, {{\mathit{Nl}}_S} \left( -{l_s} s-l+1\right) -M\, {l_S} \left( {l_s}+{{\mathit{Nl}}_S}\right) s}{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }\\
\frac{{{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s}{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) } & \frac{{{\mathit{NSl}}_D} \left( -{l_s} s-l+1\right) -{l_D} s\, \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) } & \frac{{l_S} s\, \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) -{{\mathit{Nsl}}_S} \left( -{l_s} s-l+1\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }\end{pmatrix}\mbox{}
\]
%%%%%%%%%%%%%%%
\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{4em}\color{red}\bfseries
(\% i3)
\end{minipage}
\begin{minipage}[t]{\textwidth}\color{blue}
b: matrix([b1],[b2],[b3]);
\end{minipage}
%%% OUTPUT:
\[\displaystyle \tag{b}
\begin{pmatrix}\mathit{b1}\\
\mathit{b2}\\
\mathit{b3}\end{pmatrix}\mbox{}
\]
%%%%%%%%%%%%%%%
\noindent
%%%%%%%%%%%%%%%
%%% INPUT:
\begin{minipage}[t]{4em}\color{red}\bfseries
(\% i8)
\end{minipage}
\begin{minipage}[t]{\textwidth}\color{blue}
x : invert(A).b;
\end{minipage}
%%% OUTPUT:
\[\displaystyle \tag{x}
\begin{pmatrix}\frac{\mathit{b2}\, \left( -\left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) -M\, {{\mathit{NSl}}_D} \left( {l_s}+{{\mathit{Nl}}_S}\right) \right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }+\frac{\mathit{b3}\, \left( M\, {{\mathit{Nl}}_S} \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +M\, {{\mathit{Nsl}}_S} \left( {l_s}+{{\mathit{Nl}}_S}\right) \right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }+\frac{\mathit{b1}\, \left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }\\
\frac{\mathit{b2}\, \left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \left( -{l_s} s-l+1\right) +M\, {l_D} \left( {l_s}+{{\mathit{Nl}}_S}\right) s\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }+\frac{\mathit{b3}\, \left( -M\, {{\mathit{Nl}}_S} \left( -{l_s} s-l+1\right) -M\, {l_S} \left( {l_s}+{{\mathit{Nl}}_S}\right) s\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }+\frac{\mathit{b1}\, \left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }\\
\frac{\mathit{b3}\, \left( {l_S} s\, \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) -{{\mathit{Nsl}}_S} \left( -{l_s} s-l+1\right) \right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }+\frac{\mathit{b2}\, \left( {{\mathit{NSl}}_D} \left( -{l_s} s-l+1\right) -{l_D} s\, \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) \right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }+\frac{\mathit{b1}\, \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }{\left( \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) {l_S} s-M\, {{\mathit{Nl}}_S} {l_D} s\right) \left( -{l_s} s-l-{{\mathit{Nsl}}_S}+1\right) +\left( M\, {{\mathit{NSl}}_D} {{\mathit{Nl}}_S}-{{\mathit{Nsl}}_S} \left( {g_D}-delta+M\, {{\mathit{Nl}}_D}+1\right) \right) \left( -{l_s} s-l+1\right) +M\, \left( {l_s}+{{\mathit{Nl}}_S}\right) \left( {{\mathit{NSl}}_D} {l_S} s-{{\mathit{Nsl}}_S} {l_D} s\right) }\end{pmatrix}\mbox{}
\]
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\title{Dynamic Launch Decisions for Satellite Constellation Operators}
%Alternate title? Constellations in Orbit
%\author{William King}
\institute{Washington State University}
\begin{document}
\maketitle
\begin{abstract}
Over the last decades, new technology has make low earth orbits (LEOs) more accessible, and
the resulting increase in LEO satellites has increased the risk of collision.
Because debris in orbit generates more debris through collisions with objects in orbit
and the debris created during launch and operation imposes a negative externality
on other operators,
optimal use of orbits is believed to not occur under free entry.
This paper develops a dynamic model of satellite operation incorporating two effects not considered
in previous models.
The first effect is complementarity between satellites within the same operator's fleet (called a constellation).
The second effect is collision avoidance efficiencies that exist within constellations.
The primary result is a theoretical model and the resulting analysis of the difference in survival rates between
constellation operators and society.
\end{abstract}
\keywords{Orbits, Pollution, Economies of Scale, Externality }
\jel{Q29, Q58, L25}
\textbf{Acknowledgments:} I am the sole author and have recieved no contributions from others as of yet.
This paper has been approved for dual submission in Econs 529 and Econs 594 by the instructors.
\newpage
\tableofcontents
\newpage
% ---------------------------------------------------------------------------------------
\section{Introduction}
% Motivating Example (ESA - SpaceX)
In September of 2019, the European Space Agency (ESA) released a tweet explaining that they had performed an
maneuver to avoid a collision with a SpaceX Starlink Satellite in Low Earth Orbit (LEO)\autocite{EsaTweet}.
While later reports\autocite{ArsTechnicaStatement} described it as the result of miscommunications,
ESA used the opportunity to highlight the difficulties arising from coordinating avoidance maneuvers and how
such coordination will become more difficult as the size and number of
single purpose, single operator satellite fleets (satellite constellations) increase in low earth orbit\autocite{EsaBlog}.
% Background on issues of congestion and pollution
% Kessler Syndrome
In spite of the fact that there is a lot of maneuvering room in outer space,
%\footnote{``Space is big. Really big. You just won’t believe how vastly hugely mind bogglingly big it is.
%I mean, you may think it’s a long way down the road to the chemist,
%but that’s just peanuts to space.''\cite{DouglasAdams}}
the repeated interactions of periodic orbits make collisions probable.
Consequently, objects in orbit are subject to both a congestion effect and a pollution effect.
Congestion effects are primarily derived from avoiding collisions between artificial satellites.
Pollution in orbit consists of debris, both natural and man-made, which increases
the probability of an unforeseen collision.
The defining dynamic of pollution in orbit is that it self-propagates as debris collides with itself
and orbiting satellites to generate more debris.
This dynamic underlies a key concern, originally explored by Kessler and Cour-Palais \autocite{Kessler1978}
that with sufficient mass in orbit (through satellite launches), the debris generating process
could undergo a runaway effect rendering various orbital regions unusable.
This cascade of collisions is often known as Kessler syndrome and
may take place over various timescales.
% ---------------
Orbits may be divided into three primary groups,
Low Earth Orbit (LEO, less than 2,400km in altitude\autocite{FAA2020}),
Medium Earth Orbit (MEO), and High Earth Orbit (HEO) with Geostationary Earth Orbit (GEO)
considered a particular classification of orbit.
While the topic of LEO allocation has historically remained somewhat unexplored, the last 6 years has seen
a variety of new empirical studies and theoretical models published.
In general, three primary, related topics appear in the literature:
Allocative Efficiency, Policy Intervention, and the occurrence of Kessler Syndrome.
% ---------------
%Allocative efficiency
The primary concern is to establish whether or not orbits will be overused
due to their common-pool nature, and if allocation procedures are efficient.
The earliest theoretical model I have found, due to Adilov, Alexander, and
Cunningham \autocite{adilov_alexander_cunningham_2015}, examines pollution
using a two-period salop model, incorporating the effects of launch debris on
survival into the second period.
They find that the social planner generates debris and launches at lower rates
than a free entry market.
This same result was found by Rao and Rondina \autocite{RaoRondina2020} in
the context of an infinite period dynamic model.
They approach the problem in the case where numerous operators in a free entry environment
can each launch a single, identical satellite.
% ---------------
In addition to analyzing the allocative results, a significant area of interest is
what impact various policy interventions can have.
The policies analyzed and methods used have been widely varied.
Macauley \autocite{Macauley_1998} provided the first evidence of sub-optimal behavior in orbit
by estimating the welfare lose due to the current method of assigning GEO slots to operators.
The potential losses due to anti-competitive behavior was highlighted by Adilov et al \autocite{Adilov2019},
who have analyzed the opportunities for strategic
``warehousing'' of non-functional satellites as a means of increasing competitive advantage by
denying operating locations to competitors in GEO.
Grzelka and Wagner \autocite{GrzelkaWagner2019} explore methods of encouraging satellite quality (in terms of debris)
and cleanup.
Finally, Rao and Rondina \autocite{Rao2020} estimate that achieving socially optimal
behavior through orbital use fees could increase the value generated by the space industry by a factor of four.
% ---------------
Although Kessler and Cour-Palais determined that a runaway pollution effect could make a set of orbits
physically unusable, Adilov et al \autocite{adilov_alexander_cunningham_2018} %Kessler Syndrome
have shown that economic benefits provided by orbits will drop sufficiently to make the net marginal
benefit of new launches negative before the physical kessler syndrome occurs.
%TODO: Discuss how various definitions have been proposed in the economics literature to match the models.
% ---------------
This paper's objective is to %develop a dynamic model which incorporates
lay the foundations necessary to explore the effects of organizing satellites as constellations ,
particularly through collision avoidance efficiencies and economies of scale in utility production.
No model as of yet has examined these aspects of orbit use.
The primary analytical result aside from developing the preliminary model and characterizing general solutions
is to examine if there exists a negative externality related to survival rates.
% ---------------
The paper is organized as follows.
Section \ref{Model} describes the mathematical organization of the model
for the cases of independent constellation operators and a social planner
operating the same constellations.
%It also includes a brief digression into the free entry conditions.
Section \ref{Analysis} evaluates the differences between the
constellation operators and social planner models, particularly
the difference between marginal survival rates .
%Of particular interest is the difference in launch rates and marginal survival rates.
%Section \ref{Kessler} ...
Section \ref{Conclusion} concludes with a discussion of potential extensions and
topics which have not yet been addressed.
% ---------------------------------------------------------------------------------------
\section{Model}\label{Model}
%Intuitive description
The dynamic model is an extension of Rao and Rondina's working paper \autocite{RaoRondina2020}
(specifically their non-stochastic model)
to include how operators deal with constellations.
\subsection{Model Outline}
For a given orbital shell (a set of orbits that interact regularly), I assume there are $N$ operators,
each of which has the potential to launch and operate a satellite
constellation consisting of some endogenously chosen number of identical satellites.
% -------------------
Each constellation operator has a personal satellite stock $s^i_t$ in each period, and chooses the
number of launches in that time period $x^i_t$.
For simplicity, each launch is assumed to have a fixed cost $F$.
In the aggregate, the satellite stock and launches for each period are represented by:
\begin{align}
S_t =&\sum_{i=1}^N s^i_t \\
X_t =&\sum_{i=1}^N x^i_t
\end{align}
% -------------------
Satellites in a constellation are damaged or destroyed at the rate $l^i(s^i_t,S_t,D_t)$,
which is assumed to be increasing in $s^i_t$, $S_t$, and $D_t$ (debris, see below).
One key difference from the previous models of Rao and Rondina \autocite{RaoRondina2020} and
Adilov et al \autocite{adilov_alexander_cunningham_2018} is that this model allows the rate of
collision within constellations and between constellations to be different.
This reflects the assumption that an operator can and will put more effort into protecting the satellites within
the constellation from each other.
One example of how this can be accomplished is that while choosing the orbits for a constellation,
it is possible for an operator to chose a set of trajectories that best meet their needs and
minimizes the risk of collision within the constellation.
Mathematically this is represented by the inclusion of $s^i_t$ in $l^i$.
Together with the launch rate, we obtain a law of motion for both constellation-level
and society-level satellite stocks.
\begin{align}
s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
S_{t+1} =& X_t + \sum^N_{i=1} \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t
\end{align}
%Discuss first derivatives
%The case where there
% -------------------
The level of debris in each period is represented by $D_t$, and is assumed to pose a latent risk.
In particular, it is assumed that once debris is created, the risk it provides is only avoidable
through not launching future satellites.
%\footnote{This is one important extension as avoiding debris reduces the operational lifetime
% of satellites and may affect optimal taxation.
In addition to naturally occurring debris, debris is generated through the following three mechanisms.
\begin{itemize}
\item At launch, various processes can shed debris.
Examples include leftover rocket stages, explosions during launch and deployment,
and slag from solid rocket boosters.
\item When destroyed, satellites will fragment and produce debris.
\item Debris can collide with other debris, forming more but smaller debris.
\end{itemize}
This provides the following law of debris dynamics.
\begin{align}
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)
\end{align}
where $\delta$ represents the proportional decay of debris
-- through reentering the atmosphere -- for a given shell,
$M$ represents the debris generated from each collision,
$m$ represents the debris generated from each launch,
and $g(D_t)$ represents the new fragments from debris colliding with other debris.
% -------------------
Each constellation $i \in {1,\dots,N}$ produces value for their operator at each period according to the function:
\begin{align}
u^i(s^i_t, S_t, D_t)
\end{align}
Productive economies of scale within a constellation appear when
$\parder{u^i}{s^i_t}{2} > 0$ for some values of $s^i_t,S_t, D_t$.
Of note is that firms are assumed to produce value monopolistically, i.e. there are no substitution nor
complementary effects between constellations.
This allows us to examine the effects of economies of scale and collision avoidance efficiencies
without incorporating the effects of competition.
The period value function may incorporate the effects of orbit and congestion debris, accounting
for their effect in producing value to the operator.
Adilov et al analyzed the effects of competition between operators in launch decisions \autocite{Adilov2019}.
A similar approach could be used, but would add significant complexity to the model.
One key note is the choice of the word ``value'' as opposed to ``profit''.
Historically, space operations have been motivated by objectives other than profit,
such as national security, scientific inquisitiveness, to enhance hobbies such as amature radio,
or to quote President John F. Kennedy,
``\dots because [it] is hard.''\autocite{Kennedy1962}.
This choice of terminology acknowledges that orbit use is not exclusively commercial
and there may be interference between commercial and non-commercial operations.
% ---------------------------------------------
\subsection{Constellation Operator's Program}
%The aforementioned aspects of the model form the following bellman equation for each constellation operator.
%\begin{align}
% 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}) \\
% \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 % Is this also a state variable?
%\end{align}
%The system of envelope conditions is linear and can be written as a matrix equation.
%In Appendix \ref{APX:Derivations:Constellation} I develop the euler equation
%in a generalizable way.
Often, in polluting environments, there is an ambient population that is harmed by pollution.
Very rarely does satellite debris pose a hazard to those on earth, thus in this model
the only population who's welfare is addressed are the satellite operators themselves.
Each operator faces the following problem:
\input{./includes/Appendix_constellation_program}
% ---------------------------------------------
\subsection{Social Planner's Program}
The social planner (or fleet planner to use Rao and Rondina's terminology), is tasked with
maximizing the sum of the operators' benefits $W(\{s^i_t\},S_t,D_t) = \sum^N_{i=1} V^i(s^i_t,S_t,D_t)$.
%\begin{align}
% W(\{s^i_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}
%
%%Goal: Add the euler equation.
%The derivation of the euler equation, and conditions on it's existence are
%outlined in Appendix \ref{APX:Derivations:Fleet}.
\input{./includes/Appendix_planner_program}
% ---------------------------------------------------------------------------------------
\section{Analysis}\label{Analysis}
%Describe analysis types
%Survival ratios
%two firm model
\subsection{Survival Ratios}\label{Survival}
% Marginal survival.
In line with theory on common-pool resources, we expect there to be a negative externality
incurred by increasing the satellite stock.
Some details of this externality can be observed in the marginal survival rate.
Define the survival rate for a constellation and the society to be:
\begin{align}
R_i =& \frac{s^i_{t+1}- x^i_t}{s^i_t} = 1- l^i(s^i_t,S_t,D_t) \\
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}
\end{align}
The marginal survival rates when a given constellation $i$ changes size are:
\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}{} \label{EQ:iii} \\
\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{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}{} \label{EQ:i}
\end{align}
Note that $ \sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}$ is the average marginal survival rate
across constellation operators.
The derivation of equation \ref{EQ:i} is in Appendix \ref{APX:Derivations:Survival_Direct}.
Direct comparison between the marginal survival rates of an individual operator and the social planner's fleet
cannot proceed further without specifying the functional loss forms $l^i(\cdot)$
and specifying which firm will be compared to society.
In spite of this, conditions on the average effects can be specified as follows.
Society's marginal survival rate is greater than the weighted, arithmetic mean of marginal survival rates
of the constellation when:
\begin{align}
\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}{}
\leq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\
\sum_{i=1}^N R_i - R \leq& 0\\
\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\\
\sum_{i=1}^N (1 - s^i_t) [1- l^i(s^i_t,S_t,D_t)] \leq& 0 \label{EQ:ii}
\end{align}
which is true if every constellation has at least one satellite.
Based on the definition of constellation survival rate, $s^i_t =0 \Rightarrow R_i = \frac{0}{0}$
i.e. the survival rate is undefined.
In combination with the physical reality that there cannot be a negative number
of satellites in a constellation, we are left to conclude that a meaningful constellation
has at least one satellite.
As $\parder{R_i}{s^i_t}{} < 0$ from the assumptions on collision mechanics, we can interpret
this result as that the marginal survival rate of the entire satellite fleet is lower
than the weighted arithmetic mean of marginal survival rates across constellations.
This demonstrates the negative externality of satellite operation, and is a very general condition,
consistent with other orbital pollution models.
Note that it does allow for some constellations to have a lower marginal survival rate than the fleet,
but it can be true as a general condition.
%TODO: Some more analysis can be done by comparing the case of avoidance efficiencies vs non-efficiencies.
%\subsubsection{Average Effects}
%TODO: Review and rewrite this section, including discussing the implications
%As we are analyzing survival rates, a geometric mean is better used to describe average effects.
%By weighting the geometric mean with constellation sizes, we get:
%\begin{align}
% 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]
%\end{align}
%The marginal effect is assumed to be negative, thus
%\begin{align}
% 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]
% \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] \\
% 0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t
% \left( \ln(1-l^i)
% - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
% \right)
% - \sum^N_{j=1} s_t^j \ln(1-l^j) \right] \\
% 0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t
% \left( \ln(R_i)
% - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
% \right)
% - \sum^N_{j=1} s_t^j \ln(R_j) \right] \\
% 0 > & \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
% - \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) \\
% 0 > & \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{} - \ln R_G \\
% \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}{}
% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
%\end{align}
%Welfare
% TODO: Develop overarching results.
% ---------------------------------------------------------------------------------------
\subsection{Kessler Syndrome}\label{Kessler}
%Current plan: Explain the kessler region in this model
%Rao's physical approach
%Adilov's economic approach
Rao and Rondina \autocite{RaoRondina2020} interpret their model in terms of a physical
kessler syndrome, while Adilove et al \autocite{adilov_alexander_cunningham_2018}
develop the concept of an economic kessler syndrome.
Generalizing Rao's approach, we define the kessler region as the set of states such that
the debris stock will tend to infinity, and kessler syndrome as when the state is in
the kessler region.
Formally, the kessler region is:
\begin{align}
\vartheta_1 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
\lim_{t \rightarrow \infty} D_{t+1} = \infty \right\}
\end{align}
I suspect, but have not been able to prove, that an equivalent condition is:
\begin{align}
\vartheta_2 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
\parder{(D_{t+1}-D_t)}{D_t}{} > 0 \right\}
\end{align}
If the assumption holds, then a condition for a physical kessler region in this model is:
\begin{align}
\vartheta_2 =
\left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
-\delta
+ m\parder{X_t(\{s^i_t\},D_t)}{D_t}{}
+ M\cdot \left( \sum^N_{i=1} \parder{l^i}{D_t}{} \right)
+ g(D_t) > 0 \right\}
\end{align}
Adilov defines an economic kessler syndrome (and thus kessler region) along the lines of
\begin{align}
\vartheta_3 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) = 0 \right\}
\end{align}
This represents the conditions under which adding satellites to the orbit becomes unprofitable.
He establishes general conditions under which an economic kessler syndrom precedes a
physical kessler syndrome.
The benefit of this definition is that the euler equation defining $X_t(\cdot)$
can be searched for the states that imply $X_t = 0, \forall t$
\footnote{I have yet to conduct such a search, but plan on doing so as part of a numerical simulation.}.
% ---------------------------------------------------------------------------------------
%\subsection{Numerical Model}\label{Numerical}
% 2-firm model: Symmetric
% 2-firm model: asymetric sizes or payoffs.
% ---------------------------------------------------------------------------------------
\section{Concluding Remarks}\label{Conclusion}
%TODO: rewrite and update.
The dynamic model developed in this paper provides insight into the incentives faced by
constellation operators in comparison with a social planner and, when completed, should provide
insight on how self-perpetuating externalities drive sub-optimal behavior.
At this point, major work remains in identifying optimal launch rates and verifying if
the expected difference in optimal launch rates between individual operators and a social planner exist,
as occurs in other models.
In addition to the remaining work on fleshing out the model, work on the following extensions and applications of the
model can fill gaps in the literature or complement current work.
Notable areas of interest for future research include:
\begin{itemize}
\item Asymmetric constellation sizes: What are the impacts on social welfare when a variety of
constellation sizes exist?
\item Policy interventions: Various policy proposals to reduce negative externalities have been proposed,
including launch quotas, launch taxes, and orbit use fees \autocite{RaoRondina2020b}.
% \item Introduction of stochastics: There are various ways that stochastics can enter the model, from the scales
% determining debris generation to the per-period satellite collision rate.
% \item Differentiation of satellites and launch methods: Different launch methods and satellite features can
% affect the accumulation of debris.
% \item Richer satellite lifetimes: the current satellite lifetime of [launch, operate] could be extended
% to include stages such as development and disposal.
% In particular, a multi-period development cycle with sunk costs incurred along the way may
% exacerbate problems where stable equilibria are overshot.
% This will allow for more policy interventions to be analyzed.
\item Strategic behavior: Concerns include whether constellation network effects can be used to prevent new entrants
in the case of competition for a satellite services market.
\end{itemize}
While computationally complicated, the results so far imply that there is a defined difference between
the risks faced at the constellation operator's level and the level of society as a whole.
Although not a common topic in economics, orbit use has properties that requires
current study in order to identify optimal behavior, inform policies, and prevent kessler syndrome
before there are no more viable orbits to use.
\newpage
\printbibliography
\newpage
\appendix
\section{Derivations} \label{APX:Derivations}
%\subsection{Useful Mathematical Notes}\label{APX:Derivations:Useful}
%To fill in with a set of useful mathematical notes for use throughout.
%\subsubsection{Useful Derivatives}
%\subsection{Constellation Operator}\label{APX:Derivations:Constellation}
%\input{./includes/Appendix_constellation_program}
%\subsection{Fleet Planner}\label{APX:Derivations:Fleet}
%\input{./includes/Appendix_planner_program}
\subsection{Survival Rates}\label{APX:Derivations:Survival_Direct}
\input{./includes/Appendix_Survival_direct}
%\subsection{Survival Rates: Geometric Mean Analysis}\label{APX:Derivations:Survival_Geometric}
%\input{./includes/Appendix_Survival_geometric}
%TODO
\end{document}

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%----------------------------------------------------------------------------------------
% TITLE PAGE
%----------------------------------------------------------------------------------------
\title[Orbits]{Dynamic Launch Decision for Satellite Constellation Operators}
%Constellations in orbit
\author{Will King} % Your name
\institute[WSU] % Your institution as it will appear on the bottom of every slide, may be shorthand to save space
{
Washington State University \\ % Your institution for the title page
\medskip
\textit{william.f.king@wsu.edu} % Your email address
}
\date{\today} % Date, can be changed to a custom date
\begin{document}
\begin{frame}
\titlepage % Print the title page as the first slide
\end{frame}
\section{Introduction}
\begin{frame}
\frametitle{Introduction}
\begin{block}{ESA -- Sep. 2019}
For the first time ever, ESA has performed a 'collision avoidance manoeuvre' to protect one of its satellites from colliding with a 'mega constellation' \#SpaceTraffic \autocite{EsaTweet}
\end{block}
In 1978,Donald Kessler and Burton Cour-Palais identified a potential threat to the new frontier of Earth Orbit.
They suggested that if there are enough objects in orbit, debris colliding with other debris and artificial
satellites could create debris at an increasing rate, leading to an uncontrollable cascade of collisions,
now termed kessler syndrome \autocite{Kessler1978}.
My goal is to evaluate how the organization of satellite operations into ``constellations'' affects
pollution dynamics and the incentives of operators to deviate from socially optimal behaviors.
\end{frame}
\begin{frame}
\frametitle{Overview}
\tableofcontents
\end{frame}
%----------------------------------------------------------------------------------------
% PRESENTATION SLIDES
%----------------------------------------------------------------------------------------
\section{Literature}
\begin{frame}
\frametitle{Literature}
\begin{itemize}
\item \autocite{Macauley_1998} : Estimates the welfare loss due to inefficient allocation
of geostationary orbit slots.
\item \autocite{adilov_alexander_cunningham_2015} : Two period model evaluating launch decisions.
\item \autocite{adilov_alexander_cunningham_2018} : Develop an economic Kessler syndrome
where pollution is sufficient to halt launches.
\item \autocite{RaoRondina2020} : A widely cited working paper developing the
first dynamic model of orbit allocations. Originates in Rao's dissertation from 2015.
\item \autocite{Adilov2019} : Develops a dynamic model evaluating competitive interactions
between firms.
\item \autocite{Rao2020} : Estimates the impact of implementing satellite taxes on future
profitability of the satellite industry.
\end{itemize}
\end{frame}
%----------------------------------------------------------------------------------------
\section{Model}
\begin{frame}
\frametitle{High level description}
This model is the first dynamic model to incorporate effects from organization as constellations.
These effects enter in two forms:
\begin{enumerate}
\item Economies of scale in value production.
\item Collision avoidance efficiencies from constellation planning.
\end{enumerate}
Key features of this model are:
\begin{itemize}
\item The assumption that each constellation creates utility without
competitive interactions (i.e. monopolistically).
\item Each satellite within a constellation is considered identical.
Only the number of satellites contributes to the value produced.
\end{itemize}
These features simplify computation significantly.
\end{frame}
%------------------------------------------------
\begin{frame}
\frametitle{Mathematical Terms}
\begin{tabular}{| p{0.17\linewidth} | p{0.2\linewidth} p{0.5\linewidth} | }
\hline
Symbol & Details & Description \\
\hline
$N$ & $N>0$ & Number of constellations \\
\hline
$s^i_t$ & $i \in \{1,\dots,N\}$ & Satellite stock of $i$ in $t$ \\
\hline
$x^i_t$ & Ditto & Launches of satellites in $t$ by $i$ \\
\hline
$S_t$ & & Total number of satellites in $t$ \\
\hline
$D_t$ & $D_t \geq 0$ & Level of debris in $t$ \\
\hline
$m,M$ & $m>0,M>0$ & Debris generated from launches and collisions respectively \\
\hline
$g(D_t)$ & & Debris generated from collisions with debris \\
\hline
$\delta$ & $\delta \in (0,1)$ & Decay rate of debris \\
\hline
$l^i(s^i_t,S_t,D_t)$ & $l^i() \in (0,1)$ & Rate of satellite loss in $i$ due to collisions \\
\hline
$u^i(s^i_t,S_t,D_t)$ & & Utility generated by satellite stock $s^i_t$ given $S_t,D_t$.\\
\hline
\end{tabular}
\end{frame}
%------------------------------------------------
\subsection{Constellation Operator's Problem}
\begin{frame}
\frametitle{Constellation Operator's Problem}
\begin{align}
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}) \\
\text{Subject To:}& \notag\\
D_{t+1} =& (1-\delta) D_t + m X_t + M\cdot \left( \sum^N_{i=1} s^i_t l^i(s^i_t,S_t,D_t) \right) + g(D_t) \label{law_motion:debris}\\
s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \label{law_motion:private_stock}\\
& S_t =\sum_{i=1}^N s^i_t ~~~ X_t =\sum_{i=1}^N x^i_t
\end{align}
\end{frame}
%------------------------------------------------
\begin{frame}[allowframebreaks]
\frametitle{Solving Constellation's Problem}
The general envelope conditions are:
\begin{align}
\parder{V^i}{s^i_{t}}{} - \der{u^i}{s^i_t}{}=& \beta\left[
\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{s^i_t}{}
+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{s^i_t}{}
+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{s^i_t}{}
\right] \label{EQ:env1}\\
\parder{V^i}{S_{t}}{} - \der{u^i}{S_t}{} =& \beta\left[
\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{S_t}{}
+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{S_t}{}
+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{S_t}{}
\right] \label{EQ:env2}\\
\parder{V^i}{D_{t}}{} - \der{u^i}{D_t}{} =& \beta\left[
\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{D_t}{}
+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{D_t}{}
+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{D_t}{}
\right] \label{EQ:env3} \\
\nabla V^i_t - \nabla u_t^i =& \beta A \nabla V^i_{t+1}
\end{align}
The optimality conditions is:
\begin{align}
\frac{F}{\beta} =& \parder{V^i}{s^i_{t+1}}{}
+ \parder{V^i}{S_{t+1}}{}
+ m\parder{V^i}{D_{t+1}}{}
\end{align}
Iterating both forward and backward one period gives the system
\begin{align}
\frac{F}{\beta} =& \parder{V^i}{s^i_{t}}{}
+ \parder{V^i}{S_{t}}{}
+ m\parder{V^i}{D_{t}}{} \label{EQ:opt1}\\
\frac{F}{\beta} =& \parder{V^i}{s^i_{t+1}}{}
+ \parder{V^i}{S_{t+1}}{}
+ m\parder{V^i}{D_{t+1}}{} \label{EQ:opt2}\\
\frac{F}{\beta} =& \parder{V^i}{s^i_{t+2}}{}
+ \parder{V^i}{S_{t+2}}{}
+ m\parder{V^i}{D_{t+2}}{} \label{EQ:opt3}
\end{align}
Thus by iterating \cref{EQ:env1,EQ:env2,EQ:env3} to match \cref{EQ:opt1,EQ:opt2,EQ:opt3},
we can simplify from 9 equations with 9 unknowns to 3 equations with 3 unknowns
allowing us to solve for $\nabla_{[s^i_t,S_t,D_t]} V_t$
in terms of derivatives of the utility function and derivatives of the laws of motion.
Substituting $\nabla V_t$ into the equation below (\cref{EQ:opt1}) provides the euler equation
that characterizes the policy function $x^i_t(s^i_T,S_t,D_t)$.
\begin{align}
\frac{F}{\beta} =& \parder{V^i}{s^i_{t}}{}
+ \parder{V^i}{S_{t}}{}
+ m\parder{V^i}{D_{t}}{} \notag
\end{align}
\end{frame}
%------------------------------------------------
\subsection{Social Planner's Problem}
\begin{frame}
\frametitle{Social Planner's Problem}
We can address the social planner's problem in much the same way.
\begin{align}
W(\{s^i_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}\}, D_{t+1}) \\
\text{Subject To:}& \notag\\
D_{t+1} =& (1-\delta) D_t + m X_t + M\cdot \left( \sum^N_{i=1} s^i_t 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}
\end{frame}
%------------------------------------------------
\begin{frame}[allowframebreaks]
\frametitle{Solving Planner's Problem}
The $N+1$ Envelope Conditions are:
\begin{align}
\parder{W}{s_t^i}{} =& \sum^N_{j=1} \der{u^j}{s_t^i}{}
+ \beta \left[ \sum^N_{j=1} \parder{W}{s_{t+1}^j}{} \parder{s_{t+1}^j}{s_t^i}{}
+ \parder{W}{D_{t+1}}{} \parder{D_{t+1}}{s_t^i}{} \right]
~~~ \forall i \in \{1,\dots,N\} \label{EQ:S:env1}\\
\parder{W}{D_t}{} =& \sum^N_{j=1} \der{u^j}{D_t}{}
+ \beta \left[ \sum^N_{j=1} \parder{W}{s_{t+1}^j}{} \parder{s_{t+1}^j}{D_t}{}
+ \parder{W}{D_{t+1}}{} \parder{D_{t+1}}{D_t}{} \right] \\
\end{align}
The $N$ Optimality Conditions are:
\begin{align}
0 =& -F + \beta \left[ \sum^N_{j=1} \parder{W}{s^j_{t+1}}{} \parder{s^j_{t+1}}{x^i_t}{}
+ \parder{W}{D_{t+1}}{} \parder{D_{t+1}}{x^i_t}{}\right]
~~~ \forall i \in \{1,\dots,N\} \label{EQ:S:opt1}
\end{align}
Iterating \cref{EQ:S:opt1} one period forward (from $t+1$ to $t+2$) for $i=1$ and and substituting in the
correctly iterated envelope conditions provides
the final equation for a system of $N+1$ optimality conditions for $\nabla W_t$.
Once again, iterating \cref{EQ:S:opt1} backwards from $t+1$ to $t$ and substituting in $\nabla W_t$ will allow
you to find the $N$ euler equations characterizing the policy functions $\{x^i_t\}$.
%----------------------------------------------------------------------------------------
\section{Analysis}
\subsection{Welfare Analysis}
\end{frame}
\begin{frame}
\frametitle{Welfare analysis}
A standard result in the models mentioned in the slide on previous work is that of how free
entry or competitive use results in launching more than the socially optimal number of satellites.
I suspect that result will hold true in this model.
\textit{Unfortunately I have not been able to do more than these derivations.
The welfare analysis will involve some numerical methods at some point as it gets very messy.}
\end{frame}
%------------------------------------------------
\subsection{Survival Rates}
\begin{frame}
\frametitle{Survival Rates}
One key analysis in \cite{RaoRondina2020} is about the survival rates of satellites.
Define the survival rate for a constellation and the society to be:
\begin{align}
R_i =& \frac{s^i_{t+1}- x^i_t}{s^i_t} = 1- l^i(s^i_t,S_t,D_t) \\
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}
\end{align}
\end{frame}
\begin{frame}
\frametitle{Survival Rates}
The marginal survival rates when a given constellation $i$ changes size are:
\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)
\leq 0 \\
\parder{R}{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}{} \label{EQ:i}
\end{align}
\end{frame}
\begin{frame}
\frametitle{Survival Rates}
Thus society's marginal survival rate is less than the weighted arithemetic mean of
survival rates for individually growing constellations when:
\begin{align}
\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}{}
\leq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\
\sum_{i=1}^N R_i - R \leq& 0\\
\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\\
\sum_{i=1}^N (1 - s^i_t) [1- l^i(s^i_t,S_t,D_t)] \leq& 0
\end{align}
This condition is met as every constellation consists of at least one satellite.
\end{frame}
%------------------------------------------------
\begin{frame}
\subsection{Kessler Syndrome}
\frametitle{Economic Kessler Syndrome}
\cite{adilov_alexander_cunningham_2018} develop a description of economic kessler syndrom
as when the debris and satellite stocks are such that it is not profitable to launch.
Mathematically this is:
\begin{align}
\vartheta_3 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) = 0 \right\}
\end{align}
This definition has the benefit that it can be found through a numerical search directly on
the euler equations developed previously.
\textit{Again, I have not been able to implement this analysis.}
\end{frame}
%------------------------------------------------
\section{Conclusion}
\begin{frame}
\frametitle{Conclusion}
\textbf{Summary:}
In this paper I have described a model and general set of euler equations describing
the decisions facing satellite constellation operators.
Additionally I have established that negative pollution externalities exist, consitent
with other models.
This model provides a basis for analyses of competitive and non-competitive
interaction between constellation operators, and for the analysis of policy interventions.
\textbf{Future Work:}
There remains significant work to finalize the model, including exploring a numerical model,
clarifying existence criteria, and verifying if constellation operators are likely to overuse orbits.
\end{frame}
%----------------------------------------------------------------------------------------
\begin{frame}[allowframebreaks]
\frametitle{References}
\printbibliography
\end{frame}
\end{document}

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\defcounter {refsection}{0}\relax
\beamer@sectionintoc {3}{Model}{5}{0}{3}
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\beamer@subsectionintoc {3}{1}{Constellation Operator's Problem}{7}{0}{3}
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220
Writing/2020-125-02_presentation/References.bib
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@ -0,0 +1,220 @@
% Encoding: UTF-8
@Misc{EsaTweet,
author = {European_Space_Agency},
title = {For the first time ever, ESA has performed a `collision avoidance manoeuvre' to protect one of its satellites from colliding with a 'mega constellation' \#SpaceTraffic},
addendum = {https://twitter.com/esaoperations},
date = {2019-09-02},
nameaddon = {\@ESAOperations},
}
@Electronic{ArsTechnicaStatement,
author = {Brodkin, Jon},
howpublished = {Online Article},
language = {English},
note = {Statement from SpaceX to ARS Technica},
organization = {Ars Technica},
title = {SpaceX satellite was on “collision course” until ESA satellite was re-routed},
url = {https://arstechnica.com/information-technology/2019/09/spacex-satellite-was-on-collision-course-until-esa-satellite-was-re-routed/},
date = {2019-09-03},
}
@Electronic{EsaBlog,
author = {European_Space_Agency},
howpublished = {Online},
language = {English},
organization = {European Space Agency},
title = {ESA spacecraft dodges large constellation},
url = {http://www.esa.int/Safety_Security/ESA_spacecraft_dodges_large_constellation},
date = {2019-09-03},
}
@PhdThesis{Rao:dissertation,
author = {Rao, Akhil},
school = {University of Colorado},
title = {The Economics of Orbit Use: Theory, Policy, and Practice},
year = {2019},
}
@Article{adilov_alexander_cunningham_2015,
author = {Adilov, Nodir and Alexander, Peter J. and Cunningham, Brendan M.},
journal = {Environmental and Resource Economics},
title = {An Economic Analysis of Earth Orbit Pollution},
year = {2015},
issn = {0924-6460},
number = {1},
pages = {81–98},
volume = {60},
doi = {10.1007/s10640-013-9758-4},
publisher = {Environmental and Resource Economics},
}
@Article{Macauley_1998,
author = {Macauley, Molly K},
journal = {The Journal of Law and Economics},
title = {Allocation of Orbit and Spectrum Resources for Regional Communications: What's At Stake?},
year = {1998},
issn = {0022-2186},
number = {S2},
pages = {737–764},
volume = {41},
abstract = {Contentious debate surrounds allocation of the geostationary orbit and electromagneticspectrum, two resources used by communications satellites. An extensive economicsliterature alleges that the nonmarket administrative allocative procedures now in place arehighly inefficient, but no research has empirically estimated the welfare loss. This paperdevelops a conceptual framework and a computerized model to estimate the economic valueof the resources, the size and distribution of welfare costs associated with the presentregulatory regime, and the potential gains from more market-like allocation.
Key Words: outer space, communications satellites, pricing natural resources
JEL Classification Nos.: H4, Q2},
doi = {10.1086/467411},
publisher = {The Journal of Law and Economics},
}
@InBook{brillinger_2001,
author = {Brillinger, David R.},
pages = {105–116},
title = {Space Debris: Flux in a Two Dimensional Orbit},
year = {2001},
doi = {10.1007/978-3-0348-8326-9_8},
}
@Article{Grzelka2019,
author = {Zachary Grzelka and Jeffrey Wagner},
journal = {Environmental and Resource Economics},
title = {Managing Satellite Debris in Low-Earth Orbit: Incentivizing Ex Ante Satellite Quality and Ex Post Take-Back Programs},
year = {2019},
month = {feb},
number = {1},
pages = {319--336},
volume = {74},
doi = {10.1007/s10640-019-00320-3},
publisher = {Springer Science and Business Media {LLC}},
}
@Article{Adilov2018,
author = {Nodir Adilov and Peter J. Alexander and Brendan M. Cunningham},
title = {An economic “Kessler Syndrome”: A dynamic model of earth orbit debris},
year = {2018},
issn = {0165-1765},
pages = {79-82},
volume = {166},
comment = {Might be a working paper?},
doi = {10.1016/j.econlet.2018.02.025},
}
@Article{Adilov2018a,
author = {Nodir Adilov and Peter J. Alexander and Brendan M. Cunningham},
title = {Corrigendum to “An economic “Kessler Syndrome”: A dynamic model of earth orbit debris” [Econom. Lett. 166 (2018) 79–82]},
year = {2018},
issn = {0165-1765},
pages = {185},
volume = {170},
doi = {10.1016/j.econlet.2018.04.012},
}
@Misc{Kessler1990,
author = {Donald Kessler},
title = {Orbital debris environment for spacecraft in low earth orbit},
year = {1990},
doi = {10.2514/6.1990-1353},
}
@Article{Adilov2015,
author = {Nodir Adilov and Peter J. Alexander and Brendan Michael Cunningham},
title = {Earth Orbit Debris: An Economic Model},
year = {2015},
issn = {1556-5068},
doi = {10.2139/ssrn.2264915},
}
@Article{Kessler1978,
author = {Kessler, Donald J. and Cour-Palais, Burton G.},
journal = {Journal of Geophysical Research: Space Physics},
title = {Collision frequency of artificial satellites: The creation of a debris belt},
year = {1978},
number = {A6},
pages = {2637-2646},
volume = {83},
abstract = {As the number of artificial satellites in earth orbit increases, the probability of collisions between satellites also increases. Satellite collisions would produce orbiting fragments, each of which would increase the probability of further collisions, leading to the growth of a belt of debris around the earth. This process parallels certain theories concerning the growth of the asteroid belt. The debris flux in such an earth-orbiting belt could exceed the natural meteoroid flux, affecting future spacecraft designs. A mathematical model was used to predict the rate at which such a belt might form. Under certain conditions the belt could begin to form within this century and could be a significant problem during the next century. The possibility that numerous unobserved fragments already exist from spacecraft explosions would decrease this time interval. However, early implementation of specialized launch constraints and operational procedures could significantly delay the formation of the belt.},
doi = {10.1029/JA083iA06p02637},
eprint = {https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JA083iA06p02637},
url = {https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/JA083iA06p02637},
}
@Electronic{FAA2020,
author = {NA},
howpublished = {Online},
month = oct,
note = {Describes altitude of LEO and GEO},
organization = {Federal Aviation Administration},
url = {https://www.faa.gov/space/additional_information/faq/#s1},
year = {2020},
}
@Article{adilov_alexander_cunningham_2018,
author = {Adilov, Nodir and Alexander, Peter J. and Cunningham, Brendan M.},
journal = {Economics Letters},
title = {An economic “Kessler Syndrome”: A dynamic model of earth orbit debris},
year = {2018},
issn = {0165-1765},
pages = {79–82},
volume = {166},
doi = {10.1016/j.econlet.2018.02.025},
publisher = {Economics Letters},
}
@TechReport{RaoRondina2020,
author = {Rao, Ahkil and Rondina, Giacomo},
institution = {NA},
title = {Cost in Space:Debris and Collision Risk in the Orbital Commons},
year = {2020},
month = feb,
note = {Middlebury College | UC San Diego},
type = {Working Paper},
}
@Article{Adilov2019,
author = {Adilov, Nodir and Cunningham, Brendan and Alexander, Peter and Duvall, Jerry and Shiman, Daniel},
journal = {Econ Inq},
title = {LEFT FOR DEAD: ANTI‐COMPETITIVE BEHAVIOR IN ORBITAL SPACE},
year = {2019},
month = {04},
volume = {57},
doi = {10.1111/ecin.12790},
}
@Article{Rao2020,
author = {Rao and Burgess and Kaffine},
journal = {Proceedings of the National Academy of Sciences},
title = {Orbital-use fees could more than quadruple the value of the space industry},
year = {2020},
issn = {0027-8424},
number = {23},
pages = {12756--12762},
volume = {117},
abstract = {The commercial satellite industry is rapidly expanding. A side effect of this expansion is a growing buildup of space debris that imposes costly collision risk on satellite operators. Proposed solutions to this debris have been primarily technological, but the core of the problem is incentives{\textemdash}satellites are being launched without consideration of the collision risks they impose on other operators. We show that this incentive problem can be solved with an internationally harmonized {\textquotedblleft}orbital-use fee{\textquotedblright} (OUF){\textemdash}a tax on orbiting satellites. Using a coupled physical{\textendash}economic model, we project that an optimally designed OUF could more than quadruple the long-run value of the satellite industry by 2040.The space industry{\textquoteright}s rapid recent growth represents the latest tragedy of the commons. Satellites launched into orbit contribute to{\textemdash}and risk damage from{\textemdash}a growing buildup of space debris and other satellites. Collision risk from this orbital congestion is costly to satellite operators. Technological and managerial solutions{\textemdash}such as active debris removal or end-of-life satellite deorbit guidelines{\textemdash}are currently being explored by regulatory authorities. However, none of these approaches address the underlying incentive problem: satellite operators do not account for costs they impose on each other via collision risk. Here, we show that an internationally harmonized orbital-use fee can correct these incentives and substantially increase the value of the space industry. We construct and analyze a coupled physical{\textendash}economic model of commercial launches and debris accumulation in low-Earth orbit. Similar to carbon taxes, our model projects an optimal fee that rises at a rate of 14\% per year, equal to roughly $235,000 per satellite-year in 2040. The long-run value of the satellite industry would more than quadruple by 2040{\textemdash}increasing from around $600 billion under business as usual to around $3 trillion. In contrast, we project that purely technological solutions are unlikely to fully address the problem of orbital congestion. Indeed, we find debris removal sometimes worsens economic damages from congestion by increasing launch incentives. In other sectors, addressing the tragedy of the commons has often been a game of catch-up with substantial social costs. The infant space industry can avert these costs before they escalate.},
doi = {10.1073/pnas.1921260117},
eprint = {https://www.pnas.org/content/117/23/12756.full.pdf},
publisher = {National Academy of Sciences},
url = {https://www.pnas.org/content/117/23/12756},
}
@Article{GrzelkaWagner2019,
author = {Grzelka, Zachary and Wagner, Jeffrey},
journal = {Environmental and Resource Economics},
title = {Managing Satellite Debris in Low-Earth Orbit: Incentivizing Ex Ante Satellite Quality and Ex Post Take-Back Programs},
year = {2019},
issn = {0924-6460},
number = {1},
pages = {319–336},
volume = {74},
doi = {10.1007/s10640-019-00320-3},
publisher = {Environmental and Resource Economics},
}
@Misc{Kennedy1962,
author = {John F. Kennedy},
month = sep,
title = {Address at Rice University on the Nation's Space Effort},
year = {1962},
url = {https://er.jsc.nasa.gov/seh/ricetalk.htm},
}
@Comment{jabref-meta: databaseType:bibtex;}

220
Writing/2020-125-02_presentation/References.bib.bak
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@ -0,0 +1,220 @@
% Encoding: UTF-8
@Misc{EsaTweet,
author = {European_Space_Agency},
title = {For the first time ever, ESA has performed a `collision avoidance manoeuvre' to protect one of its satellites from colliding with a 'mega constellation' \#SpaceTraffic},
addendum = {https://twitter.com/esaoperations},
date = {2019-09-02},
nameaddon = {\@ESAOperations},
}
@Electronic{ArsTechnicaStatement,
author = {Brodkin, Jon},
howpublished = {Online Article},
language = {English},
note = {Statement from SpaceX to ARS Technica},
organization = {Ars Technica},
title = {SpaceX satellite was on “collision course” until ESA satellite was re-routed},
url = {https://arstechnica.com/information-technology/2019/09/spacex-satellite-was-on-collision-course-until-esa-satellite-was-re-routed/},
date = {2019-09-03},
}
@Electronic{EsaBlog,
author = {European_Space_Agency},
howpublished = {Online},
language = {English},
organization = {European Space Agency},
title = {ESA spacecraft dodges large constellation},
url = {http://www.esa.int/Safety_Security/ESA_spacecraft_dodges_large_constellation},
date = {2019-09-03},
}
@PhdThesis{Rao:dissertation,
author = {Rao, Akhil},
school = {University of Colorado},
title = {The Economics of Orbit Use: Theory, Policy, and Practice},
year = {2019},
}
@Article{adilov_alexander_cunningham_2015,
author = {Adilov, Nodir and Alexander, Peter J. and Cunningham, Brendan M.},
journal = {Environmental and Resource Economics},
title = {An Economic Analysis of Earth Orbit Pollution},
year = {2015},
issn = {0924-6460},
number = {1},
pages = {81–98},
volume = {60},
doi = {10.1007/s10640-013-9758-4},
publisher = {Environmental and Resource Economics},
}
@Article{Macauley_1998,
author = {Macauley, Molly K},
journal = {The Journal of Law and Economics},
title = {Allocation of Orbit and Spectrum Resources for Regional Communications: What's At Stake?},
year = {1998},
issn = {0022-2186},
number = {S2},
pages = {737–764},
volume = {41},
abstract = {Contentious debate surrounds allocation of the geostationary orbit and electromagneticspectrum, two resources used by communications satellites. An extensive economicsliterature alleges that the nonmarket administrative allocative procedures now in place arehighly inefficient, but no research has empirically estimated the welfare loss. This paperdevelops a conceptual framework and a computerized model to estimate the economic valueof the resources, the size and distribution of welfare costs associated with the presentregulatory regime, and the potential gains from more market-like allocation.
Key Words: outer space, communications satellites, pricing natural resources
JEL Classification Nos.: H4, Q2},
doi = {10.1086/467411},
publisher = {The Journal of Law and Economics},
}
@InBook{brillinger_2001,
author = {Brillinger, David R.},
pages = {105–116},
title = {Space Debris: Flux in a Two Dimensional Orbit},
year = {2001},
doi = {10.1007/978-3-0348-8326-9_8},
}
@Article{Grzelka2019,
author = {Zachary Grzelka and Jeffrey Wagner},
journal = {Environmental and Resource Economics},
title = {Managing Satellite Debris in Low-Earth Orbit: Incentivizing Ex Ante Satellite Quality and Ex Post Take-Back Programs},
year = {2019},
month = {feb},
number = {1},
pages = {319--336},
volume = {74},
doi = {10.1007/s10640-019-00320-3},
publisher = {Springer Science and Business Media {LLC}},
}
@Article{Adilov2018,
author = {Nodir Adilov and Peter J. Alexander and Brendan M. Cunningham},
title = {An economic “Kessler Syndrome”: A dynamic model of earth orbit debris},
year = {2018},
issn = {0165-1765},
pages = {79-82},
volume = {166},
comment = {Might be a working paper?},
doi = {10.1016/j.econlet.2018.02.025},
}
@Article{Adilov2018a,
author = {Nodir Adilov and Peter J. Alexander and Brendan M. Cunningham},
title = {Corrigendum to “An economic “Kessler Syndrome”: A dynamic model of earth orbit debris” [Econom. Lett. 166 (2018) 79–82]},
year = {2018},
issn = {0165-1765},
pages = {185},
volume = {170},
doi = {10.1016/j.econlet.2018.04.012},
}
@Misc{Kessler1990,
author = {Donald Kessler},
title = {Orbital debris environment for spacecraft in low earth orbit},
year = {1990},
doi = {10.2514/6.1990-1353},
}
@Article{Adilov2015,
author = {Nodir Adilov and Peter J. Alexander and Brendan Michael Cunningham},
title = {Earth Orbit Debris: An Economic Model},
year = {2015},
issn = {1556-5068},
doi = {10.2139/ssrn.2264915},
}
@Article{Kessler1978,
author = {Kessler, Donald J. and Cour-Palais, Burton G.},
journal = {Journal of Geophysical Research: Space Physics},
title = {Collision frequency of artificial satellites: The creation of a debris belt},
year = {1978},
number = {A6},
pages = {2637-2646},
volume = {83},
abstract = {As the number of artificial satellites in earth orbit increases, the probability of collisions between satellites also increases. Satellite collisions would produce orbiting fragments, each of which would increase the probability of further collisions, leading to the growth of a belt of debris around the earth. This process parallels certain theories concerning the growth of the asteroid belt. The debris flux in such an earth-orbiting belt could exceed the natural meteoroid flux, affecting future spacecraft designs. A mathematical model was used to predict the rate at which such a belt might form. Under certain conditions the belt could begin to form within this century and could be a significant problem during the next century. The possibility that numerous unobserved fragments already exist from spacecraft explosions would decrease this time interval. However, early implementation of specialized launch constraints and operational procedures could significantly delay the formation of the belt.},
doi = {10.1029/JA083iA06p02637},
eprint = {https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JA083iA06p02637},
url = {https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/JA083iA06p02637},
}
@Electronic{FAA2020,
author = {NA},
howpublished = {Online},
month = oct,
note = {Describes altitude of LEO and GEO},
organization = {Federal Aviation Administration},
url = {https://www.faa.gov/space/additional_information/faq/#s1},
year = {2020},
}
@Article{adilov_alexander_cunningham_2018,
author = {Adilov, Nodir and Alexander, Peter J. and Cunningham, Brendan M.},
journal = {Economics Letters},
title = {An economic “Kessler Syndrome”: A dynamic model of earth orbit debris},
year = {2018},
issn = {0165-1765},
pages = {79–82},
volume = {166},
doi = {10.1016/j.econlet.2018.02.025},
publisher = {Economics Letters},
}
@TechReport{RaoRondina2020,
author = {Rao, Ahkil and Rondina, Giacomo},
institution = {NA},
title = {Cost in Space:Debris and Collision Risk in the Orbital Commons},
year = {2020},
month = feb,
note = {Middlebury College | UC San Diego},
type = {Working Paper},
}
@Article{Adilov2019,
author = {Adilov, Nodir and Cunningham, Brendan and Alexander, Peter and Duvall, Jerry and Shiman, Daniel},
journal = {Econ Inq},
title = {LEFT FOR DEAD: ANTI‐COMPETITIVE BEHAVIOR IN ORBITAL SPACE},
year = {2019},
month = {04},
volume = {57},
doi = {10.1111/ecin.12790},
}
@Article{Rao2020,
author = {Rao and Burgess and Kaffine},
journal = {Proceedings of the National Academy of Sciences},
title = {Orbital-use fees could more than quadruple the value of the space industry},
year = {2020},
issn = {0027-8424},
number = {23},
pages = {12756--12762},
volume = {117},
abstract = {The commercial satellite industry is rapidly expanding. A side effect of this expansion is a growing buildup of space debris that imposes costly collision risk on satellite operators. Proposed solutions to this debris have been primarily technological, but the core of the problem is incentives{\textemdash}satellites are being launched without consideration of the collision risks they impose on other operators. We show that this incentive problem can be solved with an internationally harmonized {\textquotedblleft}orbital-use fee{\textquotedblright} (OUF){\textemdash}a tax on orbiting satellites. Using a coupled physical{\textendash}economic model, we project that an optimally designed OUF could more than quadruple the long-run value of the satellite industry by 2040.The space industry{\textquoteright}s rapid recent growth represents the latest tragedy of the commons. Satellites launched into orbit contribute to{\textemdash}and risk damage from{\textemdash}a growing buildup of space debris and other satellites. Collision risk from this orbital congestion is costly to satellite operators. Technological and managerial solutions{\textemdash}such as active debris removal or end-of-life satellite deorbit guidelines{\textemdash}are currently being explored by regulatory authorities. However, none of these approaches address the underlying incentive problem: satellite operators do not account for costs they impose on each other via collision risk. Here, we show that an internationally harmonized orbital-use fee can correct these incentives and substantially increase the value of the space industry. We construct and analyze a coupled physical{\textendash}economic model of commercial launches and debris accumulation in low-Earth orbit. Similar to carbon taxes, our model projects an optimal fee that rises at a rate of 14\% per year, equal to roughly $235,000 per satellite-year in 2040. The long-run value of the satellite industry would more than quadruple by 2040{\textemdash}increasing from around $600 billion under business as usual to around $3 trillion. In contrast, we project that purely technological solutions are unlikely to fully address the problem of orbital congestion. Indeed, we find debris removal sometimes worsens economic damages from congestion by increasing launch incentives. In other sectors, addressing the tragedy of the commons has often been a game of catch-up with substantial social costs. The infant space industry can avert these costs before they escalate.},
doi = {10.1073/pnas.1921260117},
eprint = {https://www.pnas.org/content/117/23/12756.full.pdf},
publisher = {National Academy of Sciences},
url = {https://www.pnas.org/content/117/23/12756},
}
@Article{GrzelkaWagner2019,
author = {Grzelka, Zachary and Wagner, Jeffrey},
journal = {Environmental and Resource Economics},
title = {Managing Satellite Debris in Low-Earth Orbit: Incentivizing Ex Ante Satellite Quality and Ex Post Take-Back Programs},
year = {2019},
issn = {0924-6460},
number = {1},
pages = {319–336},
volume = {74},
doi = {10.1007/s10640-019-00320-3},
publisher = {Environmental and Resource Economics},
}
@Misc{Kennedy1962,
author = {John F. Kennedy},
month = sep,
title = {Address at Rice University on the Nation's Space Effort},
year = {1962},
url = {https://er.jsc.nasa.gov/seh/ricetalk.htm},
}
@Comment{jabref-meta: databaseType:bibtex;}

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\title{Dynamic Launch Decisions for Satellite Constellation Operators}
%Alternate title? Constellations in Orbit
%\author{William King}
\institute{Washington State University}
\begin{document}
\maketitle
\begin{abstract}
%Justification.
Over the last decades, new technology has made low earth orbits (LEOs) more accessible, and
the resulting increase in LEO satellites has increased the risk of collision.
%Discuss pollution externality
Orbital operations produce an externality through the creation of debris during launch,
operation, and collisions which contributes to the risk of destruction.
%Discuss debris propagation
This effect is compounded as debris in orbit generates more debris through collisions with objects in orbit,
possibly leading to a runaway effect called kessler syndrome.
%Describe contribution
This paper develops a dynamic model of satellite operation incorporating two effects not considered
in previous models: complementary network-like effects between satellites within
the same operator's fleet (called a constellation) and collision avoidance efficiencies realized within constellations.
%Describe the state of the results
The primary result is a preliminary model and the resulting analysis of the difference in satellite
survival rates between constellations and and the societal fleet.
\end{abstract}
\keywords{Orbits, Pollution, Economies of Scale, Externality }
\jel{Q29, Q58, L25}
\textbf{Acknowledgments:} I am the sole author and have received no contributions from others as of yet.
This paper has been approved for dual submission in Econs 529 and Econs 594 by the instructors.
\newpage
\tableofcontents
\newpage
% ---------------------------------------------------------------------------------------
\section{Introduction}
% Motivating Example (ESA - SpaceX)
In September of 2019, the European Space Agency (ESA) released a tweet explaining that they had performed an
maneuver to avoid a collision with a SpaceX Starlink Satellite in Low Earth Orbit (LEO)\autocite{EsaTweet}.
While later reports\autocite{ArsTechnicaStatement} described it as the result of miscommunications,
ESA used the opportunity to highlight the difficulties arising from coordinating avoidance maneuvers and how
such coordination will become more difficult as the size and number of
single purpose, single operator satellite fleets (satellite constellations) increase in low earth orbit\autocite{EsaBlog}.
% Background on issues of congestion and pollution
% Kessler Syndrome
In spite of the fact that there is a lot of maneuvering room in outer space,
%\footnote{``Space is big. Really big. You just won’t believe how vastly hugely mind bogglingly big it is.
%I mean, you may think it’s a long way down the road to the chemist,
%but that’s just peanuts to space.''\cite{DouglasAdams}}
the repeated interactions of periodic orbits make collisions probable.
Consequently, objects in orbit are subject to both a congestion effect and a pollution effect.
Congestion effects are primarily derived from avoiding collisions between artificial satellites.
Pollution in orbit consists of debris, both natural and man-made, which increases
the probability of an unforeseen collision.
The defining feature of pollution in orbit is that it self-propagates as debris collides with itself
and orbiting satellites to generate more debris.
This dynamic underlies a key concern, originally explored by Kessler and Cour-Palais \autocite{Kessler1978}
that with sufficient mass in orbit (through satellite launches), the debris generating process
could undergo a runaway effect rendering various orbital regions unusable.
This cascade of collisions is often known as Kessler syndrome and
may take place over various timescales.
% ---------------
%Discuss how various definitions of kessler syndrome
% have been proposed in the economics literature to match the models.
%Not sure if the following contributes much given the previous paragraph.
%Although Kessler and Cour-Palais determined that a runaway pollution effect could make a set of orbits
%physically unusable, Adilov et al \autocite{adilov_alexander_cunningham_2018} %Kessler Syndrome
%have shown that economic benefits provided by orbits will drop sufficiently to make the net marginal
%benefit of new launches negative before the physical kessler syndrome occurs.
% ---------------
Orbits may be divided into three primary groups,
Low Earth Orbit (LEO),
Medium Earth Orbit (MEO), and High Earth Orbit (HEO) where Geostationary Earth Orbit (GEO)
considered a particular classification of HEO.
While the topic of LEO allocation has historically remained somewhat unexplored, the last 6 years has seen
a variety of new empirical studies and theoretical models published.
% ---------------
%Allocative efficiency
Macauley provided the first evidence of sub-optimal behavior in orbit
by estimating the welfare loss due to the current method of assigning GEO slots to operators\autocite{Macauley_1998}.
The potential losses due to anti-competitive behavior were highlighted by Adilov et al ,
who have analyzed the opportunities for strategic
``warehousing'' of non-functional satellites as a means of increasing competitive advantage by
denying operating locations to competitors in GEO\autocite{Adilov2019}.
The primary concern expressed in many of the published papers is whether or not orbits will be overused
due to their common-pool nature, and which policies may prevent kessler syndrome.
On this topic, Adilov, Alexander, and Cunningham examine pollution
using a two-period salop model, incorporating the effects of launch debris on
survival into the second period\autocite{adilov_alexander_cunningham_2015}.
They find that the social planner generates debris and launches at lower rates
than a free entry market.
This same result was found by Rao and Rondina in
the context of an infinite period dynamic model.
%Potential Edit
Their approach is defined by the assumption that there are
numerous operators in a free entry environment who
can each launch a single, identical constellation\autocite{RaoRondina2020}.
Rao, Burgess, and Kaffine use this model to estimate that achieving socially optimal
behavior through orbital use fees could increase the value generated by the
space industry by a factor of four\autocite{Rao2020}.
% ---------------
%In addition to analyzing the allocative results, a significant area of interest is
%what impact various policy interventions can have.
%The policies and methods used to analyze their impact have been widely varied.
%Other topics of interest include
%Grzelka and Wagner \autocite{GrzelkaWagner2019} explore methods of encouraging satellite quality (in terms of debris)
%and cleanup.
% ---------------
My %FP
objective is to explore the effects from organizing satellites into constellations
on satellite launch decisions and operation.
%I %FP
%do this by extending Rao and Rondina's dynamic satellite operators model\autocite{RaoRondina2020}
%to account for non-symmetric constellation sizes and
%incorporate the effects of both economies of scale as satellites in constellations complement each other and
%collision avoidance efficiencies where satellites are less likely to collide with constellation members.
Although not explored in this paper, I %FP
hope to lay the groundwork for an
analysis regarding pigouvian taxation as a solution to the externality of orbital debris.
%Explain what the article does.
The primary results of this paper are:
preliminary development of the extended dynamic model,
characterization of the general solutions to both the constellation operators' problems and
the fleet planner's problem,
and an analysis of survival rates within constellations and the entire fleet.
%Contribution statement
%Adds to raoRondina2020 and adilov2018 in extedning to more diverse situations.
This work is most closely related to Rao and Rondina's model\autocite{RaoRondina2020} and the
dynamic model developed by Adilov et all \autocite{adilov_alexander_cunningham_2018}.
%Similarities
% - Rao
% - Law of debris:
% - law of motion for stocks
% - Adilov
% - law of Debris
% - constellations
%Differences
% - Rao
% - constellation
% - avoicance efficiencies
% - Adilov
% - Allows for non-firm participants
% - avoidance efficiencies
It is distinguished from the two models mentioned previously by accounting for
collision avoidance efficiencies where satellites are less likely to collide with constellation members,
as neither of the mentioned models accounts for this behavior.
Additionally, it differs from Rao et al's model in that it allows constellations to be of different sizes.
Adilov et al permit constellations, but assume that all constellation operators are profit maximizing firms.
I explicitly provide a way to account for non-commercial space activities, such as military satellites.
One key similarity of all three models is the form of the intertemporal laws of motion of both constellation
sizes and debris.
For debris, this involves accounting for existing debris, debris from launches, and debris from collisions.
In the case of the fleet or constellation sizes, they all account for loss due to collisions
and additions through launches.
% ---------------
%TODO: Needs rewritten after everything else.
The paper is organized as follows.
In section \ref{Model} %describes the mathematical organization of the model
the underlying mathematical model is given for both constellation operators and a societal fleet planner.
Section \ref{Analysis} %Examines marginal survival rate.
examines how externalities generated by operating satellite constellations differ between
constellation operators and fleet planners.
It also examines various definitions of kessler syndrome and how that might be examined in this model.
The paper concludes in section \ref{Conclusion}, %concludes with a discussion of potential extensions and
%topics which have not yet been addressed.
with a discussion of outstanding issues, limitations to the model, and some areas of future interest.
The appendix \ref{APX:Derivations} contains mathematical derivations.
% ---------------------------------------------------------------------------------------
\section{Model}\label{Model}
%Intuitive description
This infinite period, dynamic model is an extension of Rao and Rondina's working paper\autocite{RaoRondina2020}
to include how operators deal with constellations.
In summary, each constellation operator has a utility function and a loss function that depend
on the number of satellites in the constellation, the total number of satellites in the societal fleet,
and the amount of debris in orbit.
The loss function describes the degradation and destruction of satellites within the constellation,
and plays a critical role in the laws of motion of the satellite.
The utility function is used to describe how increases in constellation size affect utility production, given
the fleet size and debris levels.
\subsection{Model Description}
For a given set of orbits that interact regularly (an orbital ``shell''), I %FP
assume there are $N$ operators,
each of which has the potential to launch and operate a satellite
constellation consisting of some endogenously chosen number of identical satellites.
% -------------------
Each constellation $i$ is described by the number of satellites
in period $t$, where this satellite stock is denoted by $s^i_t$.
Each operator of the constellation $i$ chooses the number of launches $x^i_t$ in each time period $t$.
For simplicity, each launch is assumed to have a fixed cost $F$.
In the aggregate, the satellite stock and launches for each period are represented by:
\begin{align}
S_t =&\sum_{i=1}^N s^i_t \\
X_t =&\sum_{i=1}^N x^i_t
\end{align}
% -------------------
Satellites in a constellation are damaged or destroyed by collisions at the rate $l^i(s^i_t,S_t,D_t) \in (0,1)$.
This includes collisions both within and without constellations.
I %FP
assume that:
\begin{align}
\parder{l^i}{D_t}{} >& 0 \\
\parder{l^i}{S_t}{} >&
\der{l^i}{s^i_t}{} = \parder{l^i}{s^i_t}{} + \parder{l^i}{S_t}{} > 0 \label{EQ:xx}
\end{align}
Equation \ref{EQ:xx} represents one of the key distinctions from previous dynamic models, in that
the marginal risk of collision from adding a satellite to one's own constellation is
lower than the marginal risk of collision from other operators adding satellites.
The effects due to collision avoidance efficiencies within constellations will be examined in section \ref{Analysis}.
For any numerical examination, this assumption requires that:
\begin{align}
0 > \parder{l^i}{s^i_t}{} > -\parder{l^i}{S_t}{}
\end{align}
This functional assumption, as described in \cref{EQ:xx}, is justified by the fact that when adding
satellites to a constellation, an operator can choose to place the satellites in orbits that will
have nearly zero probability of colliding with another satellite in the constellation.
Operators who experience a collision between two of their own satellites experience
a higher cost than if one satellite collides with the satellite of another operator,
thus we would expect more care to be given to the internal organization of constellations.
Consequent to this ex-ante optimal organization within constellations,
the majority of collisions observed should occur between satellites of different constellations
and not within the same constellation.
Between the launch rate and destruction rate, I %FP
obtain a law of motion for both constellation-level
and society-level satellite stocks:
\begin{align}
s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
S_{t+1} =& X_t + \sum^N_{i=1} \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t
\end{align}
where next period satellite stock equals the surviving satellite stock plus the total number of launches.
% -------------------
The level of debris in each period is represented by $D_t$, and is assumed to pose a latent risk.
In particular, I %FP we can
assume that once debris is created, the risk it provides is only avoidable
through not launching future satellites.
%\footnote{This is one important extension as avoiding debris reduces the operational lifetime
% of satellites and may affect optimal taxation.
In addition to naturally occurring debris, new debris is generated through the following three mechanisms.
\begin{itemize}
\item At launch, various processes can shed debris.
Examples include leftover rocket stages, explosions during launch and deployment,
and slag from solid rocket boosters.
\item When destroyed, satellites will fragment and produce debris.
\item Debris can collide with other debris, forming more but smaller debris.
\end{itemize}
This provides the following law of debris dynamics.
\begin{align}
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)
\end{align}
where $\delta$ represents the proportional decay of debris
-- through reentering the atmosphere -- for a given shell,
$M$ represents the debris generated from each collision,
$m$ represents the debris generated from each launch,
and $g(D_t)$ represents the new fragments from debris colliding with other debris.
The parameters $\delta, M,$ and $m$ are assumed to be exogenously determined and non-stochastic.
% -------------------
%Describe the situation in which operators operate
Satellite operators -- whether commercial, governmental, research, or hobbyist\footnote{
Notable examples of hobby satellites are the amateur (HAM) radio OSCAR satellites} --
expect to receive some utility from satellite operation.
Because there are both firm and non-firm operators, we cannot denote this utility as
exclusively profit utility nor consumption utility.
Firms, such as television or internet providers experience this utility as profit, while
government, research institutions, or hobbyists operating satellites will experience this utility as
consumption of the service provided.
The choice of terminology acknowledges that the utility derived from orbit use is neither exclusively
productive nor consumptive,
and there may be interference between productive commercial and consumptive non-commercial operations.
Mathematically, this is represented by time-separable utility function:
\begin{align}
u^i(s^i_t, S_t, D_t)
\end{align}
For simplicity, each constellation produces utility such that it is not affected by
the size of any other given constellation.
In the case that the constellation operator is a profit maximizing firm, this implies that
they are a monopolist in their market.
The period utility function may incorporate the effects of orbital congestion ($S_t$) or debris ($D_t$),
accounting for their effect in producing value to the operator.
Productive economies of scale within a constellation appear when
$\parder{u^i}{s^i_t}{2} > 0$ for some values of $s^i_t,S_t, D_t$,
and represents situations such as those of satellite-based internet providers
that require a minimum number of satellites in the constellation to provide a given level of service.
%Adilov et al analyzed the effects of competition between operators in launch decisions \autocite{Adilov2019}.
%A similar approach could be used, but would add significant complexity to the model.
% ---------------------------------------------
\subsection{Constellation Operator's Program}
%The aforementioned aspects of the model form the following bellman equation for each constellation operator.
%\begin{align}
% 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}) \\
% \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 % Is this also a state variable?
%\end{align}
%The system of envelope conditions is linear and can be written as a matrix equation.
%In Appendix \ref{APX:Derivations:Constellation} I develop the euler equation
%in a generalizable way.
Often, in polluting environments, there is an ambient population that is harmed by pollution.
Very rarely does satellite debris pose a hazard to those on earth, thus in this model
the only population who's welfare is addressed are the satellite operators themselves.
Each operator faces the following problem:
\input{./includes/Appendix_constellation_program}
% ---------------------------------------------
\subsection{Social Planner's Program}
The social planner (or fleet planner to use Rao and Rondina's terminology), is tasked with
maximizing the sum of the operators' benefits $W(\{s^i_t\},S_t,D_t) = \sum^N_{i=1} V^i(s^i_t,S_t,D_t)$
as satellite debris rarely poses a threat to the welfare of those on earth.
%\begin{align}
% W(\{s^i_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}
%
%%Goal: Add the euler equation.
%The derivation of the euler equation, and conditions on it's existence are
%outlined in Appendix \ref{APX:Derivations:Fleet}.
\input{./includes/Appendix_planner_program}
% ---------------------------------------------------------------------------------------
\section{Analysis}\label{Analysis}
\subsection{Survival Ratios}\label{Survival}
In line with basic theories of common-pool resources,
we expect there to be a negative externality incurred on other constellations
when a constellation increases their own satellite stock (resource usage).
This externality comes from two effects, congestion and pollution.
Congestion, due to size of the societal fleet, may affect the utility achieved by other satellite operators
and it increases the probability of a satellite on satellite collision.
Pollution, the debris in all future periods, increase the rate of degradation and destruction
of satellites.
When a constellation operator increases their satellite stock, the other operators
experience a loss of welfare through both congestion and pollution.
One way to measure the effects of satellite operations is through survival rates.
% Marginal survival.
The survival rate for a constellation $i$ is defined as $R_i = 1-l^i(\cdot)$, the proportion of satellites
that were not lost (degraded nor destroyed) between period $t$ and $t+1$.
Thus the marginal survival rate represents the additional loss of
satellites due to a slightly larger constellation or fleet stock.
Mathematically the survival rates for a constellation and for society's fleet are defined as:
\begin{align}
R_i =& \frac{s^i_{t+1}- x^i_t}{s^i_t} = 1- l^i(s^i_t,S_t,D_t) \\
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} \label{EQ:socsurv}
\end{align}
In this case, the fleet survival rate \cref{EQ:socsurv}, represents the proportion of satellites
in period $t+1$ that survived from period $t$.
The marginal survival rates when a given constellation $i$ changes size are:
\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}{} \label{EQ:iii} \\
\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{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}{} \label{EQ:i}
\end{align}
Note that $ \sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}$ is the weighted, average marginal survival rate
across constellation operators.
The derivation of \cref{EQ:i} is in Appendix \ref{APX:Derivations:Survival_Direct}.
Direct comparison between the marginal survival rates of an individual operator and the social planner's fleet
cannot proceed further without specifying the functional loss forms $l^i(\cdot)$
and specifying which firm will be compared to society.
In spite of this, conditions on the average effects can be developted as follows.
The marginal survival rate of the fleet is greater than the weighted, arithmetic mean of marginal survival rates
of the constellations when:
\begin{align}
\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}{}
\leq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\
\sum_{i=1}^N R_i - R \leq& 0\\
\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\\
\sum_{i=1}^N (1 - s^i_t) [1- l^i(s^i_t,S_t,D_t)] \leq& 0 \label{EQ:ii}
\end{align}
which is true if every constellation has at least one satellite.
As any constellation of interest has at least one satellite
and $\parder{R_i}{s^i_t}{} < 0$ from the assumption on collision mechanics that $\der{l^i}{s_t^i}{}>0$,
we conclude that the marginal survival rate of the entire satellite fleet is lower
than the weighted arithmetic mean of marginal survival rates across constellations.
Note that it is possible for some constellations to have a lower marginal survival rate than the fleet,
but the survival rate for many operators must be higher than the societal rate.
Consequently, we would expect many operators to underestimate the impact of their behaviors on others
if they use their own observed or expected risk factors to estimate the risk they impose on others.
%%%Note on this section:
%%% So there is probably more insight into how to define survival rates in regards to geometric or harmonic
%%% means.
%%% The societal survival rate I chose is a simple and straightforward way of analyzing the issue,
%%% but there are probably other ways to define a fleet survival rate.
%%% I am interested in looking at weighted geometric or harmonic means as well.
%TODO2: Some more analysis can be done by comparing the case of avoidance efficiencies vs non-efficiencies.
%\subsubsection{Average Effects}
%TODO2: Review and rewrite this section, including discussing the implications
%As we are analyzing survival rates, a geometric mean is better used to describe average effects.
%By weighting the geometric mean with constellation sizes, we get:
%\begin{align}
% 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]
%\end{align}
%The marginal effect is assumed to be negative, thus
%\begin{align}
% 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]
% \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] \\
% 0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t
% \left( \ln(1-l^i)
% - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
% \right)
% - \sum^N_{j=1} s_t^j \ln(1-l^j) \right] \\
% 0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t
% \left( \ln(R_i)
% - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
% \right)
% - \sum^N_{j=1} s_t^j \ln(R_j) \right] \\
% 0 > & \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
% - \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) \\
% 0 > & \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{} - \ln R_G \\
% \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}{}
% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
%\end{align}
%Welfare
% TODO3: Develop overarching results.
% ---------------------------------------------------------------------------------------
\subsection{Kessler Syndrome}\label{Kessler}
%Current plan: Explain the kessler region in this model
%Rao's physical approach
%Adilov's economic approach
Rao and Rondina\autocite{RaoRondina2020} interpret their model in terms of a physical
kessler syndrome, while Adilov et al\autocite{adilov_alexander_cunningham_2018}
develop the concept of an economic kessler syndrome.
Generalizing Rao's approach, I %FP
define the kessler region as the set of states such that
the debris stock will tend to infinity, and kessler syndrome as when the state is in
the kessler region.
Formally, the kessler region is:
\begin{align}
\vartheta_1 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
\lim_{t \rightarrow \infty} D_{t+1} = \infty \right\}
\end{align}
I suspect, but have not been able to prove, that an equivalent condition is:
\begin{align}
\vartheta_2 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
\parder{(D_{t+1}-D_t)}{D_t}{} > 0 \right\}
\end{align}
If the assumption holds, then a condition for a physical kessler region in this model is:
\begin{align}
\vartheta_2 =
\left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
m\parder{X_t(\{s^i_t\},D_t)}{D_t}{}
+ M\cdot \left( \sum^N_{i=1} \parder{l^i}{D_t}{} \right)
+ g(D_t) > \delta \right\}
\end{align}
Adilov et al\autocite{adilov_alexander_cunningham_2018} define an economic kessler syndrome
(and thus kessler region) along the lines of
\begin{align}
\vartheta_3 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) = 0 \right\}
\end{align}
This represents the conditions under which adding satellites to the orbit becomes unprofitable.
They are able to establish conditions under which an economic kessler syndrome precedes a
physical kessler syndrome.
Some modification of the conditions are required to get them to match the terminology in this
model, but I have not yet completed that work.
The benefit of this definition is that the euler equation defining $X_t(\cdot)$
can be searched for the states that imply $X_t = 0, \forall t$
\footnote{I have yet to conduct such a search, but plan on doing so as part of a numerical simulation.}.
% ---------------------------------------------------------------------------------------
%\subsection{Numerical Model}\label{Numerical}
% 2-firm model: Symmetric
% 2-firm model: asymetric sizes or payoffs.
% ---------------------------------------------------------------------------------------
\section{Summary and Concluding Remarks}\label{Conclusion}
%Summary
%Restate topic and objective
Although significant work remains to describe the impacts of organizing satellites as constellations,
I have been able to achieve
%model not complete
many of preliminary milestones.
%conditions for the existence of an euler equation
% - kessler region analysis
Foremost among these is the section which characterizes the general euler equation and provides
a simple set of conditions for existence.
This has opened a possible numerical approach to determining the economic kessler region.
%survival rates R analysis
Additionally, we have identified some preliminary results constraining the fleet's marginal survival rate
to be less than the weighted arithmetic average of the constellations' marginal survival rate.
This result -- consistent with the assumptions on avoidance efficiencies -- highlights the nature
of the externality imposed by operating and launching satellites.
%In spite of this
%Limitations
%Change the state space to include the quantities in each satellite constellation.
There are three primary limitations within the model.
The first is the implicit assumption on $u(\cdot)$ that firms operating constellations
act monopolistically, i.e. they do not compete in the same market.
This is an unreasonable assumption as there are already firms attempting to compete in LEO
as satellite internet providers, most notably SpaceX's Starlink and OneWeb.
%Computational difficulty - I believe that algebraic solutions require either a very
%simple model with strict assumptions or significante algebraic work.
%Computational solutions depend on the accuracy of the chosen functional form.
The second primary limitation is that of computational difficulty, due to the large state space
of the model.
Even the simple constellation operator's problem presented here requires intensive
algebra to define the euler equation.
The typical response to this issue is to use computational methods to estimate
the value and policy functions for both the operators and the fleet planner, but this has the disadvantage
of reducing generalizability.
%The model doesn't track individual satellite lifetimes.
% - Agent-based modeling?
The third limitation is that the model doesn't track individual satellites through their lifetime, particularly
the decision to deorbit or park the satellite.
Thus I ignore satellite both ex-ante and ex-post heterogeneity, preventing the analysis of
how policies affect satellite disposal decisions.
%Policy Implications
%Discuss application to pigouvian taxation.
% - Does optimal taxation depend on
% - Avoidance efficiencies? This affects the externalities of congestion, and maybe pollution?
% - Relation between constellation size and fleet size? A larger firm may internalize more of the externality.
% - In-Network economies of scale? If the tax is targeted to affect marginal utility, this may become more difficult
% with economies of scale in value production.
The ultimate goal of developing this model is to facilitate policy analyses geared towards optimizing
the productive use of orbits.
As previous work has suggested that taxation may be an appropriate policy response to encourage
optimal use, I hope to be able to address the following questions with this model,
at least in specific (computational) cases:
\begin{enumerate}
\item Do avoidance efficiencies affect the optimal tax schedule for a given constellation operator?
E.g. one constellation may be able to almost completely eliminate the chance of a within constellation
collision, while another may not. Should they be taxed at different rates?
% \item Does the optimal tax rate depend on the relative size of a constellation to the fleet?
%As the case of the fleet planner is similar to having a single constellation
%in orbit, but having many constellations in orbit leads to pollution issues
%Would a quota on operators give similar enough results to be an effective policy step?
\item Do productive economies of scale require a non-linear tax schedule to optimize orbit use?
\item How does the decay rate $\delta$ (which depends on constellation altitude)
affect the optimal tax schedule?
\end{enumerate}
%Future Research Implications
%Areas of interest
% - Strategic behavior of firms: Preemptive entry
One concern, tangential to work by Adilov, et al\autocite{Adilov2019} is that there may be ways for firms
to increase barriers to entry for competitors by holding more satellites in orbit.
If this is the case, it begs the question of whether this will move the satellite stock closer
to kessler syndrome through an increase in the fleet stock of satellites, or if
the avoidance efficiencies are sufficient to move it farther from kessler syndrome.
This is a crucial question to answer as it could inform policies regarding launch quotas and
taxation.
%Add stochastics
% - incorporate risk adversion
Finally, a glaring issue is that the model is deterministic, and thus doesn't include
risk aversion.
The variety of satellite operators that currently exist include militaries operating
intelligence and communications satellites.
One would expect that the critical nature of these constellations would imply a high level
of risk aversion in these operators, making this an important area of study.
%TODO: Concluding paragraph?
%The dynamic model developed in this paper provides insight into the incentives faced by
%constellation operators in comparison with a social planner and, when completed,
%should provide insight on how self-perpetuating externalities drive sub-optimal behavior.
%At this point, major work remains in identifying optimal launch rates and verifying if
%the expected difference in optimal launch rates between individual operators and a social planner exist,
%as occurs in other models.
%In addition to the remaining work on fleshing out the model, work on the following extensions and applications of the
%model can fill gaps in the literature or complement current work.
%Notable areas of interest for future research include:
%\begin{itemize}
% \item Asymmetric constellation sizes: What are the impacts on social welfare when a variety of
% constellation sizes exist?
% \item Policy interventions: Various policy proposals to reduce negative externalities have been proposed,
% including launch quotas, launch taxes, and orbit use fees \autocite{RaoRondina2020b}.
%% \item Introduction of stochastics: There are various ways that stochastics can enter the model, from the scales
%% determining debris generation to the per-period satellite collision rate.
%% \item Differentiation of satellites and launch methods: Different launch methods and satellite features can
%% affect the accumulation of debris.
%% \item Richer satellite lifetimes: the current satellite lifetime of [launch, operate] could be extended
%% to include stages such as development and disposal.
%% In particular, a multi-period development cycle with sunk costs incurred along the way may
%% exacerbate problems where stable equilibria are overshot.
%% This will allow for more policy interventions to be analyzed.
% \item Strategic behavior: Concerns include whether constellation network effects can be used to prevent new entrants
% in the case of competition for a satellite services market.
%\end{itemize}
%
%While computationally complicated, the results so far imply that there is a defined difference between
%the risks faced at the constellation operator's level and the level of society as a whole.
%Although not a common topic in economics, orbit use has properties that requires
%current study in order to identify optimal behavior, inform policies, and prevent kessler syndrome
%before there are no more viable orbits to use.
\newpage
\printbibliography
\newpage
\appendix
\section{Derivations} \label{APX:Derivations}
%\subsection{Useful Mathematical Notes}\label{APX:Derivations:Useful}
%To fill in with a set of useful mathematical notes for use throughout.
%\subsubsection{Useful Derivatives}
%\subsection{Constellation Operator}\label{APX:Derivations:Constellation}
%\input{./includes/Appendix_constellation_program}
%\subsection{Fleet Planner}\label{APX:Derivations:Fleet}
%\input{./includes/Appendix_planner_program}
\subsection{Survival Rates}\label{APX:Derivations:Survival_Direct}
\input{./includes/Appendix_Survival_direct}
%\subsection{Survival Rates: Geometric Mean Analysis}\label{APX:Derivations:Survival_Geometric}
%\input{./includes/Appendix_Survival_geometric}
\end{document}

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%%%%%%%%%%%%CUSTOMIZATION%%%%%%%%%%%%%%%%%%%%%%
\title{Dynamic Launch Decisions for Satellite Constellation Operators}
%Alternate title? Constellations in Orbit
%\author{William King}
\institute{Washington State University}
\begin{document}
\maketitle
\begin{abstract}
%Justification.
Over the last decades, new technology has made low earth orbits (LEOs) more accessible, and
the resulting increase in LEO satellites has increased the risk of collision.
%Discuss pollution externality
Orbital operations produce an externality through the creation of debris during launch,
operation, and collisions which contributes to the risk of destruction.
%Discuss debris propagation
This effect is compounded as debris in orbit generates more debris through collisions with objects in orbit,
possibly leading to a runaway effect called kessler syndrome.
%Describe contribution
This paper develops a dynamic model of satellite operation incorporating two effects not considered
in previous models: complementary network-like effects between satellites within
the same operator's fleet (called a constellation) and collision avoidance efficiencies realized within constellations.
%Describe the state of the results
The primary result is a preliminary model and the resulting analysis of the difference in satellite
survival rates between constellations and and the societal fleet.
\end{abstract}
\keywords{Orbits, Pollution, Economies of Scale, Externality }
\jel{Q29, Q58, L25}
\textbf{Acknowledgments:} I am the sole author and have recieved no contributions from others as of yet.
This paper has been approved for dual submission in Econs 529 and Econs 594 by the instructors.
\newpage
\tableofcontents
\newpage
% ---------------------------------------------------------------------------------------
\section{Introduction}
% Motivating Example (ESA - SpaceX)
In September of 2019, the European Space Agency (ESA) released a tweet explaining that they had performed an
maneuver to avoid a collision with a SpaceX Starlink Satellite in Low Earth Orbit (LEO)\autocite{EsaTweet}.
While later reports\autocite{ArsTechnicaStatement} described it as the result of miscommunications,
ESA used the opportunity to highlight the difficulties arising from coordinating avoidance maneuvers and how
such coordination will become more difficult as the size and number of
single purpose, single operator satellite fleets (satellite constellations) increase in low earth orbit\autocite{EsaBlog}.
% Background on issues of congestion and pollution
% Kessler Syndrome
In spite of the fact that there is a lot of maneuvering room in outer space,
%\footnote{``Space is big. Really big. You just won’t believe how vastly hugely mind bogglingly big it is.
%I mean, you may think it’s a long way down the road to the chemist,
%but that’s just peanuts to space.''\cite{DouglasAdams}}
the repeated interactions of periodic orbits make collisions probable.
Consequently, objects in orbit are subject to both a congestion effect and a pollution effect.
Congestion effects are primarily derived from avoiding collisions between artificial satellites.
Pollution in orbit consists of debris, both natural and man-made, which increases
the probability of an unforeseen collision.
The defining feature of pollution in orbit is that it self-propagates as debris collides with itself
and orbiting satellites to generate more debris.
This dynamic underlies a key concern, originally explored by Kessler and Cour-Palais \autocite{Kessler1978}
that with sufficient mass in orbit (through satellite launches), the debris generating process
could undergo a runaway effect rendering various orbital regions unusable.
This cascade of collisions is often known as Kessler syndrome and
may take place over various timescales.
% ---------------
%Discuss how various definitions of kessler syndrome
% have been proposed in the economics literature to match the models.
%Not sure if the following contributes much given the previous paragraph.
%Although Kessler and Cour-Palais determined that a runaway pollution effect could make a set of orbits
%physically unusable, Adilov et al \autocite{adilov_alexander_cunningham_2018} %Kessler Syndrome
%have shown that economic benefits provided by orbits will drop sufficiently to make the net marginal
%benefit of new launches negative before the physical kessler syndrome occurs.
% ---------------
Orbits may be divided into three primary groups,
Low Earth Orbit (LEO),
Medium Earth Orbit (MEO), and High Earth Orbit (HEO) where Geostationary Earth Orbit (GEO)
considered a particular classification of HEO.
While the topic of LEO allocation has historically remained somewhat unexplored, the last 6 years has seen
a variety of new empirical studies and theoretical models published.
% ---------------
%Allocative efficiency
Macauley provided the first evidence of sub-optimal behavior in orbit
by estimating the welfare loss due to the current method of assigning GEO slots to operators\autocite{Macauley_1998}.
The potential losses due to anti-competitive behavior were highlighted by Adilov et al ,
who have analyzed the opportunities for strategic
``warehousing'' of non-functional satellites as a means of increasing competitive advantage by
denying operating locations to competitors in GEO\autocite{Adilov2019}.
The primary concern expressed in many of the published papers is whether or not orbits will be overused
due to their common-pool nature, and which policies may prevent kessler syndrome.
On this topic, Adilov, Alexander, and Cunningham examine pollution
using a two-period salop model, incorporating the effects of launch debris on
survival into the second period\autocite{adilov_alexander_cunningham_2015}.
They find that the social planner generates debris and launches at lower rates
than a free entry market.
This same result was found by Rao and Rondina in
the context of an infinite period dynamic model.
%Potential Edit
Their approach is defined by the assumption that there are
numerous operators in a free entry environment who
can each launch a single, identical constellation\autocite{RaoRondina2020}.
Rao, Burgess, and Kaffine use this model to estimate that achieving socially optimal
behavior through orbital use fees could increase the value generated by the
space industry by a factor of four\autocite{Rao2020}.
% ---------------
%In addition to analyzing the allocative results, a significant area of interest is
%what impact various policy interventions can have.
%The policies and methods used to analyze their impact have been widely varied.
%Other topics of interest include
%Grzelka and Wagner \autocite{GrzelkaWagner2019} explore methods of encouraging satellite quality (in terms of debris)
%and cleanup.
% ---------------
My %FP
objective is to explore the effects from organizing satellites into constellations
on satellite launch decisions and operation.
%I %FP
%do this by extending Rao and Rondina's dynamic satellite operators model\autocite{RaoRondina2020}
%to account for non-symmetric constellation sizes and
%incorporate the effects of both economies of scale as satellites in constellations complement each other and
%collision avoidance efficiencies where satellites are less likely to collide with constellation members.
Although not explored in this paper, I %FP
hope to lay the groundwork for an
analysis regarding pigouvian taxation as a solution to the externality of orbital debris.
%Explain what the article does.
The primary results of this paper are:
preliminary development of the extended dynamic model,
characterization of the general solutions to both the constellation operators' problems and
the fleet planner's problem,
and an analysis of survival rates within constellations and the entire fleet.
%Contribution statement
%Adds to raoRondina2020 and adilov2018 in extedning to more diverse situations.
This work is most closely related to Rao and Rondina's model\autocite{RaoRondina2020} and the
dynamic model developed by Adilov et all \autocite{adilov_alexander_cunningham_2018}.
%Similarities
% - Rao
% - Law of debris:
% - law of motion for stocks
% - Adilov
% - law of Debris
% - constellations
%Differences
% - Rao
% - constellation
% - avoicance efficiencies
% - Adilov
% - Allows for non-firm participants
% - avoidance efficiencies
It is distinguished from the two models mentioned previously by accounting for
collision avoidance efficiencies where satellites are less likely to collide with constellation members,
as neither of the mentioned models accounts for this behavior.
Additionally, it differs from Rao et al's model in that it allows constellations to be of different sizes.
Adilov et al permit constellations, but assume that all constellation operators are profit maximizing firms.
I explicitly provide a way to account for non-commercial space activities, such as military satellites.
One key similarity of all three models is the form of the intertemporal laws of motion of both constellation
sizes and debris.
For debris, this involves accounting for existing debris, debris from launches, and debris from collisions.
In the case of the fleet or constellation sizes, they all account for loss due to collisions
and additions through launches.
% ---------------
%TODO: Needs rewritten after everything else.
The paper is organized as follows.
In section \ref{Model} %describes the mathematical organization of the model
the underlying mathematical model is given for both constellation operators and a societal fleet planner.
Section \ref{Analysis} %Examines marginal survival rate.
examines how externalities generated by operating satellite constellations differ between
constellation operators and fleet planners.
It also examines various definitions of kessler syndrome and how that might be examined in this model.
The paper concludes in section \ref{Conclusion}, %concludes with a discussion of potential extensions and
%topics which have not yet been addressed.
with a discussion of outstanding issues, limitations to the model, and some areas of future interest.
The appendix \ref{APX:Derivations} contains mathematical derivations.
% ---------------------------------------------------------------------------------------
\section{Model}\label{Model}
%Intuitive description
This infinite period, dynamic model is an extension of Rao and Rondina's working paper\autocite{RaoRondina2020}
to include how operators deal with constellations.
In summary, each constellation operator has a utility function and a loss function that depend
on the number of satellites in the constellation, the total number of satellites in the societal fleet,
and the amount of debris in orbit.
The loss function describes the degredation and destruction of satellites within the constellation,
and plays a critical role in the laws of motion of the satellite.
The utility function is used to describe how increases in constellation size affect utility production, given
the fleet size and debris levels.
\subsection{Model Description}
For a given set of orbits that interact regularly (an orbital ``shell''), I %FP
assume there are $N$ operators,
each of which has the potential to launch and operate a satellite
constellation consisting of some endogenously chosen number of identical satellites.
% -------------------
Each constellation $i$ is described by the number of satellites
in period $t$, where this satellite stock is denoted by $s^i_t$.
Each operator of the constellation $i$ chooses the number of launches $x^i_t$ in each time period $t$.
For simplicity, each launch is assumed to have a fixed cost $F$.
In the aggregate, the satellite stock and launches for each period are represented by:
\begin{align}
S_t =&\sum_{i=1}^N s^i_t \\
X_t =&\sum_{i=1}^N x^i_t
\end{align}
% -------------------
Satellites in a constellation are damaged or destroyed by collisions at the rate $l^i(s^i_t,S_t,D_t) \in (0,1)$.
This includes collisions both within and without constellations.
I %FP
assume that:
\begin{align}
\parder{l^i}{D_t}{} >& 0 \\
\parder{l^i}{S_t}{} >&
\der{l^i}{s^i_t}{} = \parder{l^i}{s^i_t}{} + \parder{l^i}{S_t}{} > 0 \label{EQ:xx}
\end{align}
Equation \ref{EQ:xx} represents one of the key distinctions from previous dynamic models, in that
the marginal risk of collision from adding a satellite to one's own constellation is
lower than the marginal risk of collision from other operators adding satellites.
The effects due to collision avoidance efficiencies within constellations will be examined in section \ref{Analysis}.
For any numerical examination, this assumption requires that:
\begin{align}
0 > \parder{l^i}{s^i_t}{} > -\parder{l^i}{S_t}{}
\end{align}
This functional assumption, as described in \cref{EQ:xx}, is justified by the fact that when adding
satellites to a constellation, an operator can choose to place the satellites in orbits that will
have nearly zero probability of colliding with another satellite in the constellation.
Operators who experience a collision between two of their own satellites experience
a higher cost than if one satellite collides with the satellite of another operator,
thus we would expect more care to be given to the internal organization of constellations.
Consequent to this ex-ante optimal organization within constellations,
the majority of collisions observed should occur between satellites of different constellations
and not within the same constellation.
Between the launch rate and destruction rate, I%FP
obtain a law of motion for both constellation-level
and society-level satellite stocks:
\begin{align}
s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \\
S_{t+1} =& X_t + \sum^N_{i=1} \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t
\end{align}
Where next period satellite stock equals the surviving satellite stock plus the total number of launches.
% -------------------
The level of debris in each period is represented by $D_t$, and is assumed to pose a latent risk.
In particular, I%FP we can
assume that once debris is created, the risk it provides is only avoidable
through not launching future satellites.
%\footnote{This is one important extension as avoiding debris reduces the operational lifetime
% of satellites and may affect optimal taxation.
In addition to naturally occurring debris, new debris is generated through the following three mechanisms.
\begin{itemize}
\item At launch, various processes can shed debris.
Examples include leftover rocket stages, explosions during launch and deployment,
and slag from solid rocket boosters.
\item When destroyed, satellites will fragment and produce debris.
\item Debris can collide with other debris, forming more but smaller debris.
\end{itemize}
This provides the following law of debris dynamics.
\begin{align}
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)
\end{align}
where $\delta$ represents the proportional decay of debris
-- through reentering the atmosphere -- for a given shell,
$M$ represents the debris generated from each collision,
$m$ represents the debris generated from each launch,
and $g(D_t)$ represents the new fragments from debris colliding with other debris.
The parameters $\delta, M,$ and $m$ are assumed to be exogenously determined and non-stochastic.
% -------------------
%Describe the situation in which operators operate
Satellite operators -- whether commercial, governmental, research, or hobbyist\footnote{
Notable examples of hobby satellites are the amateur (HAM) radio OSCAR satellites} --
expect to recieve some utility from satellite operation.
Because there are both firm and non-firm operators, we cannot denote this utility as
exclusively profit utility nor consuption utility.
Firms, such as televison or internet providers experience this utility as profit, while
government, research institutions, or hobbyists operating satellites will experience this utility as
consumption of the service provided.
The choice of terminology acknowledges that the utility derived from orbit use is neither exclusively
productive nor consumptive,
and there may be interference between productive commercial and consumptive non-commercial operations.
Mathematically, this is represented by time-seperable utility function:
\begin{align}
u^i(s^i_t, S_t, D_t)
\end{align}
For simplicity, each constellation produces utility such that it is not affected by
the size of any other given constellation.
In the case that the constellation operator is a profit maximizing firm, this implies that
they are a monopolist in their market.
The period utility function may incorporate the effects of orbital congestion ($S_t$) or debris ($D_t$),
accounting for their effect in producing value to the operator.
Productive economies of scale within a constellation appear when
$\parder{u^i}{s^i_t}{2} > 0$ for some values of $s^i_t,S_t, D_t$,
and represents situations such as those of satellite-based internet providers
that require a minimum number of satellites in the constellation to provide a given level of service.
%Adilov et al analyzed the effects of competition between operators in launch decisions \autocite{Adilov2019}.
%A similar approach could be used, but would add significant complexity to the model.
% ---------------------------------------------
\subsection{Constellation Operator's Program}
%The aforementioned aspects of the model form the following bellman equation for each constellation operator.
%\begin{align}
% 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}) \\
% \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 % Is this also a state variable?
%\end{align}
%The system of envelope conditions is linear and can be written as a matrix equation.
%In Appendix \ref{APX:Derivations:Constellation} I develop the euler equation
%in a generalizable way.
Often, in polluting environments, there is an ambient population that is harmed by pollution.
Very rarely does satellite debris pose a hazard to those on earth, thus in this model
the only population who's welfare is addressed are the satellite operators themselves.
Each operator faces the following problem:
\input{./includes/Appendix_constellation_program}
% ---------------------------------------------
\subsection{Social Planner's Program}
The social planner (or fleet planner to use Rao and Rondina's terminology), is tasked with
maximizing the sum of the operators' benefits $W(\{s^i_t\},S_t,D_t) = \sum^N_{i=1} V^i(s^i_t,S_t,D_t)$
as satellite debris rarely poses a threat to the welfare of those on earth.
%\begin{align}
% W(\{s^i_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}
%
%%Goal: Add the euler equation.
%The derivation of the euler equation, and conditions on it's existence are
%outlined in Appendix \ref{APX:Derivations:Fleet}.
\input{./includes/Appendix_planner_program}
% ---------------------------------------------------------------------------------------
\section{Analysis}\label{Analysis}
\subsection{Survival Ratios}\label{Survival}
In line with basic theories of common-pool resources,
we expect there to be a negative externality incurred on other constellations
when a constellation increases their own satellite stock.
This externality comes from two effects, congestion and pollution.
Congestion, due to size of the societal fleet, may affect the utility achieved by other satellite operators
and it increases the probability of a satellite on satellite collision.
Pollution, the debris in all future periods, increase the rate of degradation and destruction
of satellites.
When a constellation operator increases their satellite stock and the other operators
experience a loss of welfare through both congestion and pollution.
One way to measure the effects of satellite launches is through survival rates.
% Marginal survival.
The survival rate for a constellation $i$ is defined as $R_i = 1-l^i(\cdot)$, the proportion of satellites
that were not lost (degraded nor destroyed) between period $t$ and $t+1$.
Thus the marginal survival rate represents the additional loss of
satellites due to a slightly larger constellation or fleet stock.
Mathematically the survival rates for a constellation and for society's fleet are defined as:
\begin{align}
R_i =& \frac{s^i_{t+1}- x^i_t}{s^i_t} = 1- l^i(s^i_t,S_t,D_t) \\
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} \label{EQ:socsurv}
\end{align}
In this case, the fleet survival rate \cref{EQ:socsurv}, represents the proportion of satellites
in period $t+1$ that survived from period $t$.
The marginal survival rates when a given constellation $i$ changes size are:
\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}{} \label{EQ:iii} \\
\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{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}{} \label{EQ:i}
\end{align}
Note that $ \sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}$ is the average marginal survival rate
across constellation operators.
The derivation of \cref{EQ:i} is in Appendix \ref{APX:Derivations:Survival_Direct}.
Direct comparison between the marginal survival rates of an individual operator and the social planner's fleet
cannot proceed further without specifying the functional loss forms $l^i(\cdot)$
and specifying which firm will be compared to society.
In spite of this, conditions on the average effects can be specified as follows.
The marginal survival rate of the fleet is greater than the weighted, arithmetic mean of marginal survival rates
of the constellations when:
\begin{align}
\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}{}
\leq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\
\sum_{i=1}^N R_i - R \leq& 0\\
\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\\
\sum_{i=1}^N (1 - s^i_t) [1- l^i(s^i_t,S_t,D_t)] \leq& 0 \label{EQ:ii}
\end{align}
which is true if every constellation has at least one satellite.
As any constellation of interest has at least one satellite
and $\parder{R_i}{s^i_t}{} < 0$ from the assumption on collision mechanics that $\der{l^i}{s_t^i}{}>0$,
we conclude that the marginal survival rate of the entire satellite fleet is lower
than the weighted arithmetic mean of marginal survival rates across constellations.
Note that it is possible for some constellations to have a lower marginal survival rate than the fleet,
but it is true as a general condition.
Consequently, we would expect many operators to underestimate the impact of their behaviors on others
if they use their own observed or expected risk factors to estimate the risk they impose on others.
%%%Note on this section:
%%% So there is probably more insight into how to define survival rates in regards to geometric or harmonic
%%% means.
%%% The societal survival rate I chose is a simple and straightforward way of analyzing the issue,
%%% but there are probably other ways to define a fleet survival rate.
%%% I am interested in looking at weighted geometric or harmonic means as well.
%TODO2: Some more analysis can be done by comparing the case of avoidance efficiencies vs non-efficiencies.
%\subsubsection{Average Effects}
%TODO2: Review and rewrite this section, including discussing the implications
%As we are analyzing survival rates, a geometric mean is better used to describe average effects.
%By weighting the geometric mean with constellation sizes, we get:
%\begin{align}
% 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]
%\end{align}
%The marginal effect is assumed to be negative, thus
%\begin{align}
% 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]
% \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] \\
% 0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t
% \left( \ln(1-l^i)
% - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
% \right)
% - \sum^N_{j=1} s_t^j \ln(1-l^j) \right] \\
% 0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t
% \left( \ln(R_i)
% - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
% \right)
% - \sum^N_{j=1} s_t^j \ln(R_j) \right] \\
% 0 > & \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
% - \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) \\
% 0 > & \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{} - \ln R_G \\
% \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}{}
% - \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
%\end{align}
%Welfare
% TODO3: Develop overarching results.
% ---------------------------------------------------------------------------------------
\subsection{Kessler Syndrome}\label{Kessler}
%Current plan: Explain the kessler region in this model
%Rao's physical approach
%Adilov's economic approach
Rao and Rondina\autocite{RaoRondina2020} interpret their model in terms of a physical
kessler syndrome, while Adilove et al\autocite{adilov_alexander_cunningham_2018}
develop the concept of an economic kessler syndrome.
Generalizing Rao's approach, I%FP
define the kessler region as the set of states such that
the debris stock will tend to infinity, and kessler syndrome as when the state is in
the kessler region.
Formally, the kessler region is:
\begin{align}
\vartheta_1 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
\lim_{t \rightarrow \infty} D_{t+1} = \infty \right\}
\end{align}
I suspect, but have not been able to prove, that an equivalent condition is:
\begin{align}
\vartheta_2 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
\parder{(D_{t+1}-D_t)}{D_t}{} > 0 \right\}
\end{align}
If the assumption holds, then a condition for a physical kessler region in this model is:
\begin{align}
\vartheta_2 =
\left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) \wedge (\{s^i_t\},D_t) \Rightarrow
-\delta
+ m\parder{X_t(\{s^i_t\},D_t)}{D_t}{}
+ M\cdot \left( \sum^N_{i=1} \parder{l^i}{D_t}{} \right)
+ g(D_t) > 0 \right\}
\end{align}
Adilov et al\autocite{adilov_alexander_cunningham_2018} define an economic kessler syndrome
(and thus kessler region) along the lines of
\begin{align}
\vartheta_3 = \left\{ (\{s^i_t\},D_t) : X_t(\{s^i_t\},D_t) = 0 \right\}
\end{align}
This represents the conditions under which adding satellites to the orbit becomes unprofitable.
They are able to establish conditions under which an economic kessler syndrom precedes a
physical kessler syndrome.
Some modification of the conditions are required to get them to match the terminology in this
model, but I have not yet completed that work.
The benefit of this definition is that the euler equation defining $X_t(\cdot)$
can be searched for the states that imply $X_t = 0, \forall t$
\footnote{I have yet to conduct such a search, but plan on doing so as part of a numerical simulation.}.
% ---------------------------------------------------------------------------------------
%\subsection{Numerical Model}\label{Numerical}
% 2-firm model: Symmetric
% 2-firm model: asymetric sizes or payoffs.
% ---------------------------------------------------------------------------------------
\section{Summary and Concluding Remarks}\label{Conclusion}
%Summary
%Restate topic and objective
Although significant work remains to describe the impacts of organizing satellites as constellations,
I have been able to achieve
%model not complete
many of the preliminary milestones.
%conditions for the existence of an euler equation
% - kessler region analysis
Foremost among these is the section which characterizes the general euler equation and provides
a simple set of conditions for existence.
This has opened a possible numerical approach to determining the economic kessler region.
%survival rates R analysis
Additionally, we have identified some preliminary results constraining the fleet's marginal survival rate
to be less than the weighted arithmetic average of the constellations' marginal survival rate.
This result -- consistent with the assumptions on avoidance efficiencies -- highlights the nature
of the externality imposed by operating and launching satellites.
In spite of this
%Limitations
%Change the state space to include the quantities in each satellite constellation.
There are three primary limitations within the model.
The first is the implicit assumption on $u(\cdot)$ that firms operating constellations
act monopolistically, i.e. they do not compete in the same market.
This is an unreasonable assumption as there are already firms attempting to compete in LEO
as satellite internet providers, most notably SpaceX's Starlink and OneWeb.
%Computational difficulty - I believe that algebraic solutions require either a very
%simple model with strict assumptions or significante algebraic work.
%Computational solutions depend on the accuracy of the chosen functional form.
The second primary limitation is that of computational difficulty, due to the large state space
of the model.
Even the simple constellation operator's problem presented here requires intensive
algrebra to define the euler equation.
The typical response to this issue is to use computational methods to estimate
the value and policy functions for both the operators and the fleet planner, but this has the disadvantage
of reducing generalizability.
%The model doesn't track individual satellite lifetimes.
% - Agent-based modeling?
The third limitation is that the model doesn't track individual satellites through their lifetime, particularly
the decision to deorbit or park the satellite.
Thus I ignore satellite both ex-ante and ex-post heterogeneity, preventing the analysis of
how policies affect satellite disposal decisions.
%Policy Implications
%Discuss application to pigouvian taxation.
% - Does optimal taxation depend on
% - Avoidance efficiencies? This affects the externalities of congestion, and maybe pollution?
% - Relation between constellation size and fleet size? A larger firm may internalize more of the externality.
% - In-Network economies of scale? If the tax is targeted to affect marginal utility, this may become more difficult
% with economies of scale in value production.
The ultimate goal of developing this model is to facilitate policy analyses geared towards optimizing
the productive use of orbits.
As previous work has suggested that taxiation may be an appropriate policy response to encourage
optimal use, I hope to be able to address the following questions with this model,
at least in specific (computational) cases:
\begin{enumerate}
\item Do avoidance efficiencies affect the optimal tax schedule for a given constellation operator?
E.g. one constellation may be able to almost completely eliminate the chance of a within constellation
collision, while another may not. Should they be taxed at different rates?
% \item Does the optimal tax rate depend on the relative size of a constellation to the fleet?
%As the case of the fleet planner is similar to having a single constellation
%in orbit, but having many constellations in orbit leads to pollution issues
%Would a quota on operators give similar enough results to be an effective policy step?
\item Do productive economies of scale require a non-linear tax schedule to optimize orbit use?
\item How does the decay rate $\delta$ (dependent on constellation altitude) affect the optimal
tax schedule?
\end{enumerate}
%Future Research Implications
%Areas of interest
% - Strategic behavior of firms: Preemptive entry
One concern, tangental to work by Adilove, Duval et al\autocite{Adilov2019} is that there may be ways for firms
to increase barriers to entry for competitors by holding more satellites in orbit.
If this is the case, it begs the question of whether this will move the satellite stock closer
to kessler syndrome through an increase in the fleet stock of satellites, or if
the avoidance efficiencies are sufficient to move it farther from kessler syndrome.
This is a crucial question to answer as it could inform policies regarding launch quotas and
taxation.
%Add stochastics
% - incorporate risk adversion
Finally, a glaring issue is that the model is deterministic, and thus doesn't include
risk adversion.
The variety of satellite operators that currently exist include militaries operating
intellegence and communications satellites.
One would expect that the critical nature of these constellations would imply a high level
of risk adversion to these operators, making this an important area of study.
%The dynamic model developed in this paper provides insight into the incentives faced by
%constellation operators in comparison with a social planner and, when completed,
%should provide insight on how self-perpetuating externalities drive sub-optimal behavior.
%At this point, major work remains in identifying optimal launch rates and verifying if
%the expected difference in optimal launch rates between individual operators and a social planner exist,
%as occurs in other models.
%In addition to the remaining work on fleshing out the model, work on the following extensions and applications of the
%model can fill gaps in the literature or complement current work.
%Notable areas of interest for future research include:
%\begin{itemize}
% \item Asymmetric constellation sizes: What are the impacts on social welfare when a variety of
% constellation sizes exist?
% \item Policy interventions: Various policy proposals to reduce negative externalities have been proposed,
% including launch quotas, launch taxes, and orbit use fees \autocite{RaoRondina2020b}.
%% \item Introduction of stochastics: There are various ways that stochastics can enter the model, from the scales
%% determining debris generation to the per-period satellite collision rate.
%% \item Differentiation of satellites and launch methods: Different launch methods and satellite features can
%% affect the accumulation of debris.
%% \item Richer satellite lifetimes: the current satellite lifetime of [launch, operate] could be extended
%% to include stages such as development and disposal.
%% In particular, a multi-period development cycle with sunk costs incurred along the way may
%% exacerbate problems where stable equilibria are overshot.
%% This will allow for more policy interventions to be analyzed.
% \item Strategic behavior: Concerns include whether constellation network effects can be used to prevent new entrants
% in the case of competition for a satellite services market.
%\end{itemize}
%
%While computationally complicated, the results so far imply that there is a defined difference between
%the risks faced at the constellation operator's level and the level of society as a whole.
%Although not a common topic in economics, orbit use has properties that requires
%current study in order to identify optimal behavior, inform policies, and prevent kessler syndrome
%before there are no more viable orbits to use.
\newpage
\printbibliography
\newpage
\appendix
\section{Derivations} \label{APX:Derivations}
%\subsection{Useful Mathematical Notes}\label{APX:Derivations:Useful}
%To fill in with a set of useful mathematical notes for use throughout.
%\subsubsection{Useful Derivatives}
%\subsection{Constellation Operator}\label{APX:Derivations:Constellation}
%\input{./includes/Appendix_constellation_program}
%\subsection{Fleet Planner}\label{APX:Derivations:Fleet}
%\input{./includes/Appendix_planner_program}
\subsection{Survival Rates}\label{APX:Derivations:Survival_Direct}
\input{./includes/Appendix_Survival_direct}
%\subsection{Survival Rates: Geometric Mean Analysis}\label{APX:Derivations:Survival_Geometric}
%\input{./includes/Appendix_Survival_geometric}
\end{document}

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% Encoding: UTF-8
@Misc{EsaTweet,
author = {ESA},
title = {For the first time ever, ESA has performed a 'collision avoidance manoeuvre' to protect one of its satellites from colliding with a 'mega constellation' \#SpaceTraffic},
addendum = {https://twitter.com/esaoperations},
date = {2019-09-02},
nameaddon = {\@ESAOperations},
}
@Electronic{ArsTechnicaStatement,
author = {Brodkin, Jon},
howpublished = {Online Article},
language = {English},
note = {Statement from SpaceX to ARS Technica},
organization = {Ars Technica},
title = {SpaceX satellite was on “collision course” until ESA satellite was re-routed},
url = {https://arstechnica.com/information-technology/2019/09/spacex-satellite-was-on-collision-course-until-esa-satellite-was-re-routed/},
date = {2019-09-03},
}
@Electronic{EsaBlog,
author = {ESA},
howpublished = {Online},
language = {English},
organization = {European Space Agency},
title = {ESA spacecraft dodges large constellation},
url = {http://www.esa.int/Safety_Security/ESA_spacecraft_dodges_large_constellation},
date = {2019-09-03},
}
@PhdThesis{Rao:Dissertation,
author = {Rao, Akhil},
school = {University of Colorado},
title = {The Economics of Orbit Use: Theory, Policy, and Practice},
year = {2019},
}
@Article{adilov_alexander_cunningham_2015,
author = {Adilov and Alexander and Cunningham},
journal = {Environmental and Resource Economics},
title = {An Economic Analysis of Earth Orbit Pollution},
year = {2015},
issn = {0924-6460},
number = {1},
pages = {81–98},
volume = {60},
doi = {10.1007/s10640-013-9758-4},
publisher = {Environmental and Resource Economics},
}
@Article{Macauley_1998,
author = {Macauley, Molly K},
journal = {The Journal of Law and Economics},
title = {Allocation of Orbit and Spectrum Resources for Regional Communications: What's At Stake?},
year = {1998},
issn = {0022-2186},
number = {S2},
pages = {737–764},
volume = {41},
abstract = {Contentious debate surrounds allocation of the geostationary orbit and electromagneticspectrum, two resources used by communications satellites. An extensive economicsliterature alleges that the nonmarket administrative allocative procedures now in place arehighly inefficient, but no research has empirically estimated the welfare loss. This paperdevelops a conceptual framework and a computerized model to estimate the economic valueof the resources, the size and distribution of welfare costs associated with the presentregulatory regime, and the potential gains from more market-like allocation.
Key Words: outer space, communications satellites, pricing natural resources
JEL Classification Nos.: H4, Q2},
doi = {10.1086/467411},
publisher = {The Journal of Law and Economics},
}
@InBook{brillinger_2001,
author = {Brillinger, David R.},
pages = {105–116},
title = {Space Debris: Flux in a Two Dimensional Orbit},
year = {2001},
doi = {10.1007/978-3-0348-8326-9_8},
}
@Article{Adilov2018a,
author = {Adilov and Alexander and Cunningham},
title = {Corrigendum to “An economic “Kessler Syndrome”: A dynamic model of earth orbit debris” [Econom. Lett. 166 (2018) 79–82]},
year = {2018},
issn = {0165-1765},
pages = {185},
volume = {170},
doi = {10.1016/j.econlet.2018.04.012},
}
@Misc{Kessler1990,
author = {Donald Kessler},
title = {Orbital debris environment for spacecraft in low earth orbit},
year = {1990},
doi = {10.2514/6.1990-1353},
}
@Article{Adilov,
author = {Adilov and Alexander and Cunningham},
title = {Earth Orbit Debris: An Economic Model},
year = {2015},
issn = {1556-5068},
doi = {10.2139/ssrn.2264915},
}
@Article{Kessler1978,
author = {Kessler, Donald J. and Cour-Palais, Burton G.},
journal = {Journal of Geophysical Research: Space Physics},
title = {Collision frequency of artificial satellites: The creation of a debris belt},
year = {1978},
number = {A6},
pages = {2637-2646},
volume = {83},
abstract = {As the number of artificial satellites in earth orbit increases, the probability of collisions between satellites also increases. Satellite collisions would produce orbiting fragments, each of which would increase the probability of further collisions, leading to the growth of a belt of debris around the earth. This process parallels certain theories concerning the growth of the asteroid belt. The debris flux in such an earth-orbiting belt could exceed the natural meteoroid flux, affecting future spacecraft designs. A mathematical model was used to predict the rate at which such a belt might form. Under certain conditions the belt could begin to form within this century and could be a significant problem during the next century. The possibility that numerous unobserved fragments already exist from spacecraft explosions would decrease this time interval. However, early implementation of specialized launch constraints and operational procedures could significantly delay the formation of the belt.},
doi = {10.1029/JA083iA06p02637},
eprint = {https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JA083iA06p02637},
url = {https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/JA083iA06p02637},
}
@Electronic{FAA2020,
author = {NA},
howpublished = {Online},
month = oct,
note = {Describes altitude of LEO and GEO},
organization = {Federal Aviation Administration},
url = {https://www.faa.gov/space/additional_information/faq/#s1},
year = {2020},
}
@TechReport{RaoRondina2020,
author = {Rao and Rondina},
institution = {NA},
title = {Cost in Space:Debris and Collision Risk in the Orbital Commons},
year = {2020},
month = feb,
note = {Middlebury College | UC San Diego},
type = {Working Paper},
}
@Article{Adilov2019,
author = {Adilov and Cunningham and Alexander and Duvall and Shiman},
journal = {Economic Inquiry},
title = {LEFT FOR DEAD: ANTI‐COMPETITIVE BEHAVIOR IN ORBITAL SPACE},
year = {2019},
month = {04},
volume = {57},
doi = {10.1111/ecin.12790},
}
@Article{Rao2020,
author = {Rao and Burgess and Kaffine},
journal = {Proceedings of the National Academy of Sciences},
title = {Orbital-use fees could more than quadruple the value of the space industry},
year = {2020},
issn = {0027-8424},
number = {23},
pages = {12756--12762},
volume = {117},
abstract = {The commercial satellite industry is rapidly expanding. A side effect of this expansion is a growing buildup of space debris that imposes costly collision risk on satellite operators. Proposed solutions to this debris have been primarily technological, but the core of the problem is incentives{\textemdash}satellites are being launched without consideration of the collision risks they impose on other operators. We show that this incentive problem can be solved with an internationally harmonized {\textquotedblleft}orbital-use fee{\textquotedblright} (OUF){\textemdash}a tax on orbiting satellites. Using a coupled physical{\textendash}economic model, we project that an optimally designed OUF could more than quadruple the long-run value of the satellite industry by 2040.The space industry{\textquoteright}s rapid recent growth represents the latest tragedy of the commons. Satellites launched into orbit contribute to{\textemdash}and risk damage from{\textemdash}a growing buildup of space debris and other satellites. Collision risk from this orbital congestion is costly to satellite operators. Technological and managerial solutions{\textemdash}such as active debris removal or end-of-life satellite deorbit guidelines{\textemdash}are currently being explored by regulatory authorities. However, none of these approaches address the underlying incentive problem: satellite operators do not account for costs they impose on each other via collision risk. Here, we show that an internationally harmonized orbital-use fee can correct these incentives and substantially increase the value of the space industry. We construct and analyze a coupled physical{\textendash}economic model of commercial launches and debris accumulation in low-Earth orbit. Similar to carbon taxes, our model projects an optimal fee that rises at a rate of 14\% per year, equal to roughly $235,000 per satellite-year in 2040. The long-run value of the satellite industry would more than quadruple by 2040{\textemdash}increasing from around $600 billion under business as usual to around $3 trillion. In contrast, we project that purely technological solutions are unlikely to fully address the problem of orbital congestion. Indeed, we find debris removal sometimes worsens economic damages from congestion by increasing launch incentives. In other sectors, addressing the tragedy of the commons has often been a game of catch-up with substantial social costs. The infant space industry can avert these costs before they escalate.},
doi = {10.1073/pnas.1921260117},
eprint = {https://www.pnas.org/content/117/23/12756.full.pdf},
publisher = {National Academy of Sciences},
url = {https://www.pnas.org/content/117/23/12756},
}
@Article{GrzelkaWagner2019,
author = {Grzelka, Zachary and Wagner, Jeffrey},
journal = {Environmental and Resource Economics},
title = {Managing Satellite Debris in Low-Earth Orbit: Incentivizing Ex Ante Satellite Quality and Ex Post Take-Back Programs},
year = {2019},
issn = {0924-6460},
number = {1},
pages = {319–336},
volume = {74},
doi = {10.1007/s10640-019-00320-3},
publisher = {Environmental and Resource Economics},
}
@Misc{Kennedy1962,
author = {John F. Kennedy},
month = sep,
title = {Address at Rice University on the Nation's Space Effort},
year = {1962},
url = {https://er.jsc.nasa.gov/seh/ricetalk.htm},
}
@Article{adilov_alexander_cunningham_2018,
author = {Adilov and Alexander and Cunningham},
journal = {Economics Letters},
title = {An economic “Kessler Syndrome”: A dynamic model of earth orbit debris},
year = {2018},
issn = {0165-1765},
pages = {79–82},
volume = {166},
doi = {10.1016/j.econlet.2018.02.025},
publisher = {Economics Letters},
}
@Comment{jabref-meta: databaseType:bibtex;}

196
Writing/CurrentWriting/References.bib.bak
Unescape Escape View File

@ -0,0 +1,196 @@
% Encoding: UTF-8
@Misc{EsaTweet,
author = {European Space Agency},
title = {For the first time ever, ESA has performed a 'collision avoidance manoeuvre' to protect one of its satellites from colliding with a 'mega constellation' \#SpaceTraffic},
addendum = {https://twitter.com/esaoperations},
date = {2019-09-02},
nameaddon = {\@ESAOperations},
}
@Electronic{ArsTechnicaStatement,
author = {Brodkin, Jon},
howpublished = {Online Article},
language = {English},
note = {Statement from SpaceX to ARS Technica},
organization = {Ars Technica},
title = {SpaceX satellite was on “collision course” until ESA satellite was re-routed},
url = {https://arstechnica.com/information-technology/2019/09/spacex-satellite-was-on-collision-course-until-esa-satellite-was-re-routed/},
date = {2019-09-03},
}
@Electronic{EsaBlog,
author = {European Space Agency},
howpublished = {Online},
language = {English},
organization = {European Space Agency},
title = {ESA spacecraft dodges large constellation},
url = {http://www.esa.int/Safety_Security/ESA_spacecraft_dodges_large_constellation},
date = {2019-09-03},
}
@PhdThesis{Rao:Dissertation,
author = {Rao, Akhil},
school = {University of Colorado},
title = {The Economics of Orbit Use: Theory, Policy, and Practice},
year = {2019},
}
@Article{adilov_alexander_cunningham_2015,
author = {Adilov and Alexander and Cunningham},
journal = {Environmental and Resource Economics},
title = {An Economic Analysis of Earth Orbit Pollution},
year = {2015},
issn = {0924-6460},
number = {1},
pages = {81–98},
volume = {60},
doi = {10.1007/s10640-013-9758-4},
publisher = {Environmental and Resource Economics},
}
@Article{Macauley_1998,
author = {Macauley, Molly K},
journal = {The Journal of Law and Economics},
title = {Allocation of Orbit and Spectrum Resources for Regional Communications: What's At Stake?},
year = {1998},
issn = {0022-2186},
number = {S2},
pages = {737–764},
volume = {41},
abstract = {Contentious debate surrounds allocation of the geostationary orbit and electromagneticspectrum, two resources used by communications satellites. An extensive economicsliterature alleges that the nonmarket administrative allocative procedures now in place arehighly inefficient, but no research has empirically estimated the welfare loss. This paperdevelops a conceptual framework and a computerized model to estimate the economic valueof the resources, the size and distribution of welfare costs associated with the presentregulatory regime, and the potential gains from more market-like allocation.
Key Words: outer space, communications satellites, pricing natural resources
JEL Classification Nos.: H4, Q2},
doi = {10.1086/467411},
publisher = {The Journal of Law and Economics},
}
@InBook{brillinger_2001,
author = {Brillinger, David R.},
pages = {105–116},
title = {Space Debris: Flux in a Two Dimensional Orbit},
year = {2001},
doi = {10.1007/978-3-0348-8326-9_8},
}
@Article{Adilov2018a,
author = {Adilov and Alexander and Cunningham},
title = {Corrigendum to “An economic “Kessler Syndrome”: A dynamic model of earth orbit debris” [Econom. Lett. 166 (2018) 79–82]},
year = {2018},
issn = {0165-1765},
pages = {185},
volume = {170},
doi = {10.1016/j.econlet.2018.04.012},
}
@Misc{Kessler1990,
author = {Donald Kessler},
title = {Orbital debris environment for spacecraft in low earth orbit},
year = {1990},
doi = {10.2514/6.1990-1353},
}
@Article{Adilov,
author = {Adilov and Alexander and Cunningham},
title = {Earth Orbit Debris: An Economic Model},
year = {2015},
issn = {1556-5068},
doi = {10.2139/ssrn.2264915},
}
@Article{Kessler1978,
author = {Kessler, Donald J. and Cour-Palais, Burton G.},
journal = {Journal of Geophysical Research: Space Physics},
title = {Collision frequency of artificial satellites: The creation of a debris belt},
year = {1978},
number = {A6},
pages = {2637-2646},
volume = {83},
abstract = {As the number of artificial satellites in earth orbit increases, the probability of collisions between satellites also increases. Satellite collisions would produce orbiting fragments, each of which would increase the probability of further collisions, leading to the growth of a belt of debris around the earth. This process parallels certain theories concerning the growth of the asteroid belt. The debris flux in such an earth-orbiting belt could exceed the natural meteoroid flux, affecting future spacecraft designs. A mathematical model was used to predict the rate at which such a belt might form. Under certain conditions the belt could begin to form within this century and could be a significant problem during the next century. The possibility that numerous unobserved fragments already exist from spacecraft explosions would decrease this time interval. However, early implementation of specialized launch constraints and operational procedures could significantly delay the formation of the belt.},
doi = {10.1029/JA083iA06p02637},
eprint = {https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JA083iA06p02637},
url = {https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/JA083iA06p02637},
}
@Electronic{FAA2020,
author = {NA},
howpublished = {Online},
month = oct,
note = {Describes altitude of LEO and GEO},
organization = {Federal Aviation Administration},
url = {https://www.faa.gov/space/additional_information/faq/#s1},
year = {2020},
}
@TechReport{RaoRondina2020,
author = {Rao and Rondina},
institution = {NA},
title = {Cost in Space:Debris and Collision Risk in the Orbital Commons},
year = {2020},
month = feb,
note = {Middlebury College | UC San Diego},
type = {Working Paper},
}
@Article{Adilov2019,
author = {Adilov and Cunningham and Alexander and Duvall and Shiman},
journal = {Economic Inquiry},
title = {LEFT FOR DEAD: ANTI‐COMPETITIVE BEHAVIOR IN ORBITAL SPACE},
year = {2019},
month = {04},
volume = {57},
doi = {10.1111/ecin.12790},
}
@Article{Rao2020,
author = {Rao and Burgess and Kaffine},
journal = {Proceedings of the National Academy of Sciences},
title = {Orbital-use fees could more than quadruple the value of the space industry},
year = {2020},
issn = {0027-8424},
number = {23},
pages = {12756--12762},
volume = {117},
abstract = {The commercial satellite industry is rapidly expanding. A side effect of this expansion is a growing buildup of space debris that imposes costly collision risk on satellite operators. Proposed solutions to this debris have been primarily technological, but the core of the problem is incentives{\textemdash}satellites are being launched without consideration of the collision risks they impose on other operators. We show that this incentive problem can be solved with an internationally harmonized {\textquotedblleft}orbital-use fee{\textquotedblright} (OUF){\textemdash}a tax on orbiting satellites. Using a coupled physical{\textendash}economic model, we project that an optimally designed OUF could more than quadruple the long-run value of the satellite industry by 2040.The space industry{\textquoteright}s rapid recent growth represents the latest tragedy of the commons. Satellites launched into orbit contribute to{\textemdash}and risk damage from{\textemdash}a growing buildup of space debris and other satellites. Collision risk from this orbital congestion is costly to satellite operators. Technological and managerial solutions{\textemdash}such as active debris removal or end-of-life satellite deorbit guidelines{\textemdash}are currently being explored by regulatory authorities. However, none of these approaches address the underlying incentive problem: satellite operators do not account for costs they impose on each other via collision risk. Here, we show that an internationally harmonized orbital-use fee can correct these incentives and substantially increase the value of the space industry. We construct and analyze a coupled physical{\textendash}economic model of commercial launches and debris accumulation in low-Earth orbit. Similar to carbon taxes, our model projects an optimal fee that rises at a rate of 14\% per year, equal to roughly $235,000 per satellite-year in 2040. The long-run value of the satellite industry would more than quadruple by 2040{\textemdash}increasing from around $600 billion under business as usual to around $3 trillion. In contrast, we project that purely technological solutions are unlikely to fully address the problem of orbital congestion. Indeed, we find debris removal sometimes worsens economic damages from congestion by increasing launch incentives. In other sectors, addressing the tragedy of the commons has often been a game of catch-up with substantial social costs. The infant space industry can avert these costs before they escalate.},
doi = {10.1073/pnas.1921260117},
eprint = {https://www.pnas.org/content/117/23/12756.full.pdf},
publisher = {National Academy of Sciences},
url = {https://www.pnas.org/content/117/23/12756},
}
@Article{GrzelkaWagner2019,
author = {Grzelka, Zachary and Wagner, Jeffrey},
journal = {Environmental and Resource Economics},
title = {Managing Satellite Debris in Low-Earth Orbit: Incentivizing Ex Ante Satellite Quality and Ex Post Take-Back Programs},
year = {2019},
issn = {0924-6460},
number = {1},
pages = {319–336},
volume = {74},
doi = {10.1007/s10640-019-00320-3},
publisher = {Environmental and Resource Economics},
}
@Misc{Kennedy1962,
author = {John F. Kennedy},
month = sep,
title = {Address at Rice University on the Nation's Space Effort},
year = {1962},
url = {https://er.jsc.nasa.gov/seh/ricetalk.htm},
}
@Article{adilov_alexander_cunningham_2018,
author = {Adilov and Alexander and Cunningham},
journal = {Economics Letters},
title = {An economic “Kessler Syndrome”: A dynamic model of earth orbit debris},
year = {2018},
issn = {0165-1765},
pages = {79–82},
volume = {166},
doi = {10.1016/j.econlet.2018.02.025},
publisher = {Economics Letters},
}
@Comment{jabref-meta: databaseType:bibtex;}

16
Writing/CurrentWriting/includes/Appendix_Survival_direct.tex
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@ -0,0 +1,16 @@
\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}

28
Writing/CurrentWriting/includes/Appendix_Survival_geometric.tex
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@ -0,0 +1,28 @@
As we are analyzing survival rates, a geometric mean is better used to describe average effects.
By weighting the geometric mean with constellation sizes, we get:
\begin{align}
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]
\end{align}
The marginal effect is assumed to be negative, thus-
\begin{align}
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]
\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] \\
0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t-
\left( \ln(1-l^i)-
- \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
- \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
\right)-
- \sum^N_{j=1} s_t^j \ln(1-l^j) \right] \\
0 > \parder{R_G}{s^i_t}{} =& \frac{R_G}{S_t^2} \left[ S^t-
\left( \ln(R_i)-
- \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
- \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}
\right)-
- \sum^N_{j=1} s_t^j \ln(R_j) \right] \\
0 > & \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
- \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) \\
0 > & \ln R_i - \frac{s^i_t}{1-l^i} \parder{l^i}{s^i_t}{}
- \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{} - \ln R_G \\
\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}{}
- \sum^N_{j=1} \frac{s^j_t}{1-l^j} \parder{l^j}{S_t}{}--
\end{align}

16
Writing/CurrentWriting/includes/Appendix_Useful.tex
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@ -0,0 +1,16 @@
\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}

118
Writing/CurrentWriting/includes/Appendix_constellation_program.tex
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%Given the following bellman equation
\begin{align}
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}) \\
\text{Subject To:}& \notag\\
D_{t+1} =& (1-\delta) D_t + m X_t + M\cdot \left( \sum^N_{i=1} s^i_t l^i(s^i_t,S_t,D_t) \right) + g(D_t) \label{law_motion:debris}\\
s^i_{t+1} =& \left[ 1-l^i(s^i_t,S_t,D_t) \right] s^i_t + x^i_t \label{law_motion:private_stock}\\
S_t =&\sum_{i=1}^N s^i_t \\
X_t =&\sum_{i=1}^N x^i_t
\end{align}
Where $V^i(\cdot)$ represents the value function for the constellation $i$ and $\beta$ represents a common
discount factor across operators.
\subsubsection{Characterizing solutions}
These give the optimality condition:
\begin{align}
\frac{F}{\beta} =& \parder{V^i}{s^i_{t+1}}{}
+ \parder{V^i}{S_{t+1}}{}
+ m\parder{V^i}{D_{t+1}}{}
\end{align}
Iterating both forward and backward one condition gives the system
\begin{align}
\frac{F}{\beta} =& \parder{V^i}{s^i_{t}}{}
+ \parder{V^i}{S_{t}}{}
+ m\parder{V^i}{D_{t}}{} \label{EQ:vi}\\
\frac{F}{\beta} =& \parder{V^i}{s^i_{t+1}}{}
+ \parder{V^i}{S_{t+1}}{}
+ m\parder{V^i}{D_{t+1}}{} \label{EQ:vii}\\
\frac{F}{\beta} =& \parder{V^i}{s^i_{t+2}}{}
+ \parder{V^i}{S_{t+2}}{}
+ m\parder{V^i}{D_{t+2}}{} \label{EQ:viii}
\end{align}
The general envelope conditions are:
\begin{align}
\parder{V^i}{s^i_{t}}{} - \parder{u^i}{s^i_t}{}=& \beta\left[
\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{s^i_t}{}
+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{s^i_t}{}
+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{s^i_t}{}
\right] \\
\parder{V^i}{S_{t}}{} - \parder{u^i}{S_t}{} =& \beta\left[
\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{S_t}{}
+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{S_t}{}
+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{S_t}{}
\right] \\
\parder{V^i}{D_{t}}{} - \parder{u^i}{D_t}{} =& \beta\left[
\parder{V^i}{s^i_{t+1}}{} \parder{s^i_{t+1}}{D_t}{}
+ \parder{V^i}{S_{t+1}}{} \parder{S_{t+1}}{D_t}{}
+ \parder{V^i}{D_{t+1}}{} \parder{D_{t+1}}{D_t}{}
\right]
\end{align}
Note the linearity of the equations.
This allows us to rewrite the system of envelope conditions as the following matrix expression.
\begin{align}
\beta
\left[
\begin{matrix}
\parder{s^i_{t+1}}{s^i_t}{} & \parder{S_{t+1}}{s^i_t}{} & \parder{D_{t+1}}{s^i_t}{} \\
\parder{s^i_{t+1}}{S_t}{} & \parder{S_{t+1}}{S_t}{} & \parder{D_{t+1}}{S_t}{} \\
\parder{s^i_{t+1}}{D_t}{} & \parder{S_{t+1}}{D_t}{} & \parder{D_{t+1}}{D_t}{}
\end{matrix}
\right]
\left[
\begin{matrix}
\parder{V^i}{s^i_{t+1}}{} \\
\parder{V^i}{S_{t+1}}{} \\
\parder{V^i}{D_{t+1}}{}
\end{matrix}
\right]
=&
\left[
\begin{matrix}
\parder{V^i}{s^i_{t}}{} - \parder{u^i}{s^i_t}{} \\
\parder{V^i}{S_{t}}{} - \parder{u^i}{S_t}{}\\
\parder{V^i}{D_{t}}{} - \parder{u^i}{D_t}{}
\end{matrix}
\right] \\
\beta A \nabla_{[s^i_{t+1},S_{t+1},D_{t+1}]} V^i =& \nabla_{[s^i_t,S_t,D_t]} V^i - \nabla_{[s^i_t,S_t,D_t]} u^i \\
\beta A \nabla V^i_{t+1} =& \nabla V^i_t - \nabla u_t^i ~~\text{for conciseness}
\end{align}
Solving for $\nabla V^i_{t+1}$, we get
\begin{align}
\nabla V^i_{t+1} =& (\beta A)^{-1} (\nabla V^i_t - \nabla u_t^i) \label{EQ:iv}
\end{align}
By iterating \eref{EQ:iv} one period, we get:
\begin{align}
\beta A \nabla V^i_{t+2} =& \nabla V^i_{t+1} - \nabla u_{t+1}^i \\
\nabla V^i_{t+2} =& (\beta A)^{-1}
\left( (\beta A)^{-1} (\nabla V^i_t - \nabla u_t^i)- \nabla u_{t+1}^i \right) \label{EQ:v}
\end{align}
With \cref{EQ:iv,EQ:v} substituted into the system of \cref{EQ:vi,EQ:vii,EQ:viii}, we can
now solve for the optimal, functional form of $\nabla_{[s^i_t,S_t,D_t]} V^i$.
Substituting this back into \cref{EQ:vi} gives the euler equation for the optimal launch function
$x^i_t(s^i_t,S_t,D_t)$.
\subsubsection{Conditions for existence of a solution}
For any given set of functional forms $l^i(\cdot),g(\cdot)$ and coefficients $m,M$,
one must verify that $A$ is invertible for all
values of the state and choice variables $s^i_t,S_t,D_t$, and $x^i_t$.
For the laws of motion \cref{law_motion:debris,law_motion:private_stock}, the matrix $A$ above is:
\begin{align}
\left[
\begin{matrix}
1- l^i(\cdot) - s^i_t \parder{l^i}{s^i_t}{}
& 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}{}
& M\left[\parder{l^i}{s^i_t}{} + \sum^N_{j=1} \parder{l^i}{S_t}{} \right] \\
- s^i_t \parder{l^i}{S_t}{}
& - \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}

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\begin{align}
W(\{s^i_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}\}, D_{t+1}) \\
\text{Subject To:}& \notag\\
D_{t+1} =& (1-\delta) D_t + m X_t + M\cdot \left( \sum^N_{i=1} s^i_t 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}
Solving for the euler equation follows the steps laid out in
the section
% appendix section \ref{APX:Derivations:Constellation}
for constellation operators.
\subsubsection{Characterizing solutions}
The $N+1$ Envelope Conditions are:
\begin{align}
\parder{W}{s_t^i}{} =& \sum^N_{j=1} \der{u^j}{s_t^i}{}
+ \beta \left[ \sum^N_{j=1} \parder{W}{s_{t+1}^j}{} \parder{s_{t+1}^j}{s_t^i}{}
+ \parder{W}{D_{t+1}}{} \parder{D_{t+1}}{s_t^i}{} \right]
~~~ \forall i \in \{1,\dots,N\} \\
\parder{W}{D_t}{} =& \sum^N_{j=1} \der{u^j}{D_t}{}
+ \beta \left[ \sum^N_{j=1} \parder{W}{s_{t+1}^j}{} \parder{s_{t+1}^j}{D_t}{}
+ \parder{W}{D_{t+1}}{} \parder{D_{t+1}}{D_t}{} \right] \\
\nabla W_t - \sum^N_{j=1} \nabla u^j_t =& \beta B \cdot \nabla W_{t+1}
\end{align}
Assuming $B$ is non-singular, we again find that:
\begin{align}
\nabla W_{t+1} =& (\beta B)^{-1} (\nabla W_t - \sum^N_{j=1} \nabla u^j_t) \label{EQ:viii}
\end{align}
The $N$ Optimality Conditions are:
\begin{align}
0 =& -F + \beta \left[ \sum^N_{j=1} \parder{W}{s^j_{t+1}}{} \parder{s^j_{t+1}}{x^i_t}{}
+ \parder{W}{D_{t+1}}{} \parder{D_{t+1}}{x^i_t}{}\right]
~~~ \forall i \in \{1,\dots,N\} \label{EQ:ix}\\
\frac{F}{\beta} \vect{1} =& C \nabla W_{t+1}
% = C(\beta B)^{-1} (\nabla W_t - \sum^N_{j=1} \nabla u^j_t) \label{EQ:viii}
\end{align}
Where $C$ is a $N \times N+1$ matrix.
Iterating \cref{EQ:ix} one period forward (from $t+1$ to $t+2$) for $i=1$ and and substituting in \cref{EQ:viii}
twice provides the final equation for a system of $N+1$ equations for $\nabla W_t$.
Finally, iterating \cref{EQ:ix} one period backward (from $t+1$ to $t$) for all $i$, and substituting the
previously found values for $\nabla W_t$ into these optimality conditions defines the system of euler equations
that characterize $\{x^i_t\}$.

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{"sorting":"name-up","project":{"title":"zettlr","format":"pdf","cslStyle":"","pdf":{"author":"Generated by Zettlr","keywords":"","papertype":"a4paper","pagenumbering":"arabic","tmargin":3,"rmargin":3,"bmargin":3,"lmargin":3,"margin_unit":"cm","lineheight":"1.2","mainfont":"Times New Roman","sansfont":"Arial","fontsize":12,"toc":true,"tocDepth":2,"titlepage":true,"textpl":""}},"icon":null}

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zettlr_orbits/Background Knowledge/Initial Thoughts.md
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# How I think orbit decisions are made.
There is a company (operator) with some $n\in N_0$ satellites in orbit.
They feel there would be benefit of adding another satellite.
They can estimate with relative accuracy the revenue improvements from adding and operating the satellite.
They have knowledge about the cost of operating the satellite.
They have a prior belief about the risks the satellite will face post launch.
This belief updates as situation changes.
The satellite can be in one of the following stages:
- Development
- Launch
- Operation
- Defunct
- Destroyed
- Deorbited
Satellite state also includes
- current velocity/trajectory (orbit)
- current location
- current consummables status (fuel, film, some forms of inertia)
- current renewables status (battery, digitial storage space with up/download link)
- current operational status
Sattelite control includes
- consumable use
- renewables use
- renuwables renewal
and can influence
- operational status (except current location)
- stage
Risk takes the form of:
- Development Risk (to slow of development, miss out on opportunities)
- Launch Risk (doesn't deploy properly, fire end points to space)
- Operational risk (lose control, move out of optimal path, interference by other satellites)
- Latent existence risk (unknown space debris) (possibly ignorable?)
- Observed existence risk (known space debris, other satellites)
- Kessler risk (kessler syndrome)
- Liability
The operator is then faced with the following decision during operation:
How do I maximize operational revenue while minimizing the avoidable and unavoidable risk, over time.
Also, how do I make an optimal stopping decision to choose when to deorbit?
Other interesting insights
The uninformed probability that you survive X time is exponential in debris, cet.paribus., but is not exponential conditional on increasing or decreasing satellite stock
- gamma? where no launches get an exponential form, but increased/decreased launches gives a gamma?
The chance of transferring to a defunct state is not going to have the conditional indepencece behavior of exposure to debris.
Modeling ideas
Don't need to model orbit period by period. Usually movement etc takes place over multiple orbits.
Debris risk could probably be handled as density bands at various latitudes, and integrated across for a given time (related to how much time is spend in each band)
Other Satellite risk - Not sure how to model it.
Assume constelations attempt to place at a uniform density (this might not be true).
How I think orbit decisions are made.
There is a company (operator) with some $n\in N_0$ satellites in orbit.
They feel there would be benefit of adding another satellite.
They can estimate with relative accuracy the revenue improvements from adding and operating the satellite.
They have knowledge about the cost of operating the satellite.
They have a prior belief about the risks the satellite will face post launch.
This belief updates as situation changes.
The satellite can be in one of the following stages:
- Development
- Launch
- Operation
- Defunct
- Destroyed
- Deorbited
Satellite state also includes
- current velocity/trajectory (orbit)
- current location
- current consummables status (fuel, film, some forms of inertia)
- current renewables status (battery, digitial storage space with up/download link)
- current operational status
Sattelite control includes
- consumable use
- renewables use
- renuwables renewal
and can influence
- operational status (except current location)
- stage
Risk takes the form of:
- Development Risk (to slow of development, miss out on opportunities)
- Launch Risk (doesn't deploy properly, fire end points to space)
- Operational risk (lose control, move out of optimal path, interference by other satellites)
- Latent existence risk (unknown space debris) (possibly ignorable?)
- Observed existence risk (known space debris, other satellites)
- Kessler risk (kessler syndrome)
- Liability
The operator is then faced with the following decision during operation:
How do I maximize operational revenue while minimizing the avoidable and unavoidable risk, over time.
Also, how do I make an optimal stopping decision to choose when to deorbit?
Other interesting insights
The uninformed probability that you survive X time is exponential in debris, cet.paribus., but is not exponential conditional on increasing or decreasing satellite stock
- gamma? where no launches get an exponential form, but increased/decreased launches gives a gamma?
The chance of transferring to a defunct state is not going to have the conditional indepencece behavior of exposure to debris.
Modeling ideas
Don't need to model orbit period by period. Usually movement etc takes place over multiple orbits.
Debris risk could probably be handled as density bands at various latitudes, and integrated across for a given time (related to how much time is spend in each band)
Other Satellite risk - Not sure how to model it.
Assume constelations attempt to place at a uniform density (this might not be true).

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zettlr_orbits/Background Knowledge/Questions.md
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I have a couple of questions about the following
How are orbits legally allocated to satellite operators?
How would one represent a socially optimal allocation?
Is there a mechanism that will encourage operators and countries to self allocate appropriately?
What infrastructure is required for that?
What form would enforcement take?
How would potential collisions/avoidance work?
Is there a set of legal standards that would encourage optimal behavior?

15
zettlr_orbits/Other/Organization.md
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So here is how I have things organized.
Literature is in my Calibre Library, under a virtual library.
This zettlr_orbits is supposed to control my
The Folder structure is as follows.
./ ## Contains Everything
zettlr/ ## Contains my knowledge base on the topic
Writing/ ## Contains my past written work on the subject.
CurrentWriting ## Includes any current presentations or papers I'm writing
YYYY-MM-DD_type ## The generic style of my presentations etc. Once in a written state
Code/ ## Contains any code I've developed spedifically for the project.
This is all going to have version control using Git

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zettlr_orbits/Other/StartPoints.md
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The purpose of this section is to provide a set of starting points for my orbit allocation project.
Here are

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{"sorting":"name-up","project":null,"icon":null}

0
zettlr_orbits/Working Sections/20210303164215.md
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