|
|
\documentclass[12pt]{article}
|
|
|
|
|
|
%%%%%%%%%Packages%%%%%%%%%%%%%%%
|
|
|
\usepackage{geometry}
|
|
|
\geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
|
|
|
\setlength{\parskip}{7pt}
|
|
|
\setlength{\parindent}{6pt}
|
|
|
%\setlength{\parindent}{0pt}
|
|
|
|
|
|
\usepackage{subfiles}
|
|
|
|
|
|
\usepackage{setspace}
|
|
|
\doublespacing
|
|
|
|
|
|
\usepackage{amsmath}
|
|
|
\usepackage{mathtools}
|
|
|
\usepackage{amsthm}
|
|
|
\usepackage{amssymb}
|
|
|
\usepackage{thmtools, thm-restate}
|
|
|
\usepackage{cleveref}
|
|
|
\usepackage{harpoon}
|
|
|
\newcommand*{\vect}[1]{\overrightharp{\ensuremath{#1}}}
|
|
|
|
|
|
%Add institute to title
|
|
|
\usepackage{etoolbox}
|
|
|
\makeatletter
|
|
|
\providecommand{\institute}[1]{% add institute to \maketitle
|
|
|
\apptocmd{\@author}{\end{tabular}
|
|
|
\par
|
|
|
\begin{tabular}[t]{c}
|
|
|
#1}{}{}
|
|
|
}
|
|
|
\makeatother
|
|
|
|
|
|
|
|
|
|
|
|
%%%%%%%%%%%%SETUP THEOREMS%%%%%%%%%%%%%%%%%%%%%%
|
|
|
\declaretheorem[within=subsection]{theorem}
|
|
|
|
|
|
|
|
|
%%%%%%%%%%%%FORMATTING%%%%%%%%%%%%%%%%%%%%%
|
|
|
%Math formatting
|
|
|
\newcommand{\bb}[1]{\mathbb{#1}}
|
|
|
\newcommand{\parder}[3]{\ensuremath{ \frac{\partial^{#3} #1}{\partial #2~^{#3}}}}
|
|
|
\newcommand{\der}[3]{\ensuremath{ \frac{d^{#3} #1}{d #2~^{#3}}}}
|
|
|
|
|
|
%Stats Related
|
|
|
%\newcommand{\likeli}[2]{\text{L}\left( #1 | #2 \right)}
|
|
|
\newcommand{\pr}[1]{\text{Pr}\left( #1 \right)}
|
|
|
|
|
|
%These are some formatting/reminder commands
|
|
|
\newcommand{\todo}[1]{
|
|
|
\textbf{\#TODO: \underline{#1}}
|
|
|
}
|
|
|
|
|
|
|
|
|
%%%%%%%%%%%%%%%Math Operators%%%%%%%%%%%%%%%%%%
|
|
|
\DeclareMathOperator{\argmax}{argmax}
|
|
|
\DeclareMathOperator*{\plim}{plim}
|
|
|
\DeclareMathOperator*{\adj}{adj}
|
|
|
|
|
|
|
|
|
%%%%%%%%%%%%%%INTERNAL REFERENCES%%%%%%%%%%%%%%
|
|
|
\newcommand{\eref}[1]{Eq. \ref{#1}}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
%%%%%%%%%%%%%%Bibliography%%%%%%%%%%%%%%%
|
|
|
\usepackage[backend=biber,
|
|
|
,sorting=none
|
|
|
,style=authoryear-ibid
|
|
|
,autocite=footnote
|
|
|
]{biblatex}
|
|
|
\addbibresource{References.bib}
|
|
|
%%%%% Adjust this at some point.
|
|
|
%This is how to perform citations.
|
|
|
% Use \cite{ref} to get a numerical reference to the bibliography
|
|
|
% Use \citetitle{ref} to get the title as a citation
|
|
|
% Use \fullcite{ref} to insert the full reference.
|
|
|
% Use \autcite{ref} to get formatted references
|
|
|
|
|
|
%%%%%%%%%%%%%%Other commands%%%%%%%%%%%%%%
|
|
|
\providecommand{\keywords}[1]{\textbf{\textit{Keywords:}} #1}
|
|
|
\providecommand{\jel}[1]{\textbf{\textit{JEL Codes:}} #1}
|
|
|
|
|
|
|
|
|
%%%%%%%%%%%%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 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}
|
|
|
|
|
|
|