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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}


