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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} 
