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Writing/CurrentWriting/includes/Appendix_constellation_program.tex create mode 100644 Writing/CurrentWriting/includes/Appendix_planner_program.tex create mode 100644 zettlr_orbits/.ztr-directory create mode 100644 zettlr_orbits/Background Knowledge/Initial Thoughts.md create mode 100644 zettlr_orbits/Background Knowledge/Questions.md create mode 100644 zettlr_orbits/Other/Organization.md create mode 100644 zettlr_orbits/Other/StartPoints.md create mode 100644 zettlr_orbits/Working Sections/.ztr-directory create mode 100644 zettlr_orbits/Working Sections/20210303164215.md diff --git a/Assets/References.bib b/Assets/References.bib new file mode 100644 index 0000000..c275edc --- /dev/null +++ b/Assets/References.bib @@ -0,0 +1,196 @@ +% 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;} diff --git a/Inciteful.xyz_results.bib b/Inciteful.xyz_results.bib new file mode 100644 index 0000000..d24096c --- /dev/null +++ b/Inciteful.xyz_results.bib @@ -0,0 +1,542 @@ +% Encoding: UTF-8 +@article{16916285, + incitefulid = {16916285}, + title = {Orbital-use fees could more than quadruple the value of the space industry.}, + magid = {3028737585}, + url = {http://doi.org/10.1073/pnas.1921260117}, + doi = {10.1073/pnas.1921260117}, + author = {Akhil Rao and Matthew G. Burgess and Daniel T. Kaffine}, + year = {2020}, + journal = {Proceedings of the National Academy of Sciences of the United States of America}, + volume = {117}, + issue = {23} +} + +@article{46960957, + incitefulid = {46960957}, + title = {Large constellations of small satellites: A survey of near future challenges and missions}, + magid = {3083163740}, + url = {http://doi.org/10.3390/aerospace7090133}, + doi = {10.3390/aerospace7090133}, + author = {Giacomo Curzi and Dario Modenini and Paolo Tortora}, + year = {2020}, + volume = {7}, + issue = {9} +} + +@article{7397687, + incitefulid = {7397687}, + title = {LEFT FOR DEAD: ANTI‐COMPETITIVE BEHAVIOR IN ORBITAL SPACE}, + magid = {2939271905}, + url = {http://doi.org/10.1111/ecin.12790}, + doi = {10.1111/ecin.12790}, + author = {Nodir Adilov and Brendan Cunningham and Peter J. Alexander and Jerry B. Duvall and Daniel R. Shiman}, + year = {2019}, + journal = {Economic Inquiry}, + volume = {57}, + issue = {3} +} + +@article{86503421, + incitefulid = {86503421}, + title = {Cooperative space object tracking via universal Kalman consensus filter}, + magid = {2935663069}, + url = {http://doi.org/10.1016/j.actaastro.2019.03.080}, + doi = {10.1016/j.actaastro.2019.03.080}, + author = {Yun-peng Hu and Wenshan Su and Lei Chen and Yan-Gang Liang}, + year = {2019}, + journal = {Acta Astronautica}, + volume = {160} +} + +@article{47677809, + incitefulid = {47677809}, + title = {The quest to conquer Earth's space junk problem.}, + magid = {2891788317}, + url = {http://doi.org/10.1038/d41586-018-06170-1}, + doi = {10.1038/d41586-018-06170-1}, + author = {Alexandra Witze}, + year = {2018}, + journal = {Nature}, + volume = {561}, + issue = {7721} +} + +@article{90020925, + incitefulid = {90020925}, + title = {Can catch shares prevent fisheries collapse}, + magid = {2047928207}, + url = {http://doi.org/10.1126/science.1159478}, + doi = {10.1126/science.1159478}, + author = {Christopher Costello and Steven D. Gaines and John Lynham}, + year = {2008}, + journal = {Science}, + volume = {321}, + issue = {5896} +} + +@article{67541791, + incitefulid = {67541791}, + title = {The Space Debris Environment and Satellite Manufacturing}, + magid = {2229692713}, + author = {Walter Tam}, + year = {2015} +} + +@article{122417116, + incitefulid = {122417116}, + title = {LEGEND - a three-dimensional LEO-to-GEO debris evolutionary model}, + magid = {2050608102}, + url = {http://doi.org/10.1016/j.asr.2003.02.027}, + doi = {10.1016/j.asr.2003.02.027}, + author = {J.-C. Liou and D.T Hall and P.H. Krisko and J.N Opiela}, + year = {2004}, + journal = {Advances in Space Research}, + volume = {34}, + issue = {5} +} + +@article{41781267, + incitefulid = {41781267}, + title = {Taxing congestion of the space commons}, + magid = {3046381038}, + url = {http://doi.org/10.1016/j.actaastro.2020.07.036}, + doi = {10.1016/j.actaastro.2020.07.036}, + author = {Sylvain Béal and Marc Deschamps and Hervé Moulin}, + year = {2020}, + journal = {Acta Astronautica}, + volume = {177} +} + +@article{68211731, + incitefulid = {68211731}, + title = {Space Junk: Behavioural Economics and the Prioritisation of Solutions}, + magid = {3033048442}, + url = {http://doi.org/10.1080/10242694.2020.1772552}, + doi = {10.1080/10242694.2020.1772552}, + author = {Peter J. Phillips and Gabriela Pohl}, + year = {2020}, + journal = {Defence and Peace Economics} +} + +@article{125437261, + incitefulid = {125437261}, + title = {Instability of the Present LEO Satellite Populations}, + magid = {1986744954}, + url = {http://doi.org/10.1016/j.asr.2007.04.081}, + doi = {10.1016/j.asr.2007.04.081}, + author = {J.-C. Liou and Nicholas L. Johnson}, + year = {2008}, + journal = {Advances in Space Research}, + volume = {41}, + issue = {7} +} + +@article{166315978, + incitefulid = {166315978}, + title = {An economic “Kessler Syndrome”: A dynamic model of earth orbit debris}, + magid = {2789696176}, + url = {http://doi.org/10.1016/j.econlet.2018.02.025}, + doi = {10.1016/j.econlet.2018.02.025}, + author = {Nodir Adilov and Peter J. Alexander and Brendan Cunningham}, + year = {2018}, + journal = {Economics Letters}, + volume = {166} +} + +@article{28500679, + incitefulid = {28500679}, + title = {Reduction Over Time to Facilitate Peer-to-Peer Barter Relationships}, + magid = {2051749409}, + url = {http://doi.org/10.1093/ietisy/e89-d.1.181}, + doi = {10.1093/ietisy/e89-d.1.181}, + author = {Kenji Saito and E. Morino and Jun Murai}, + year = {2006}, + journal = {IEICE Transactions on Information and Systems}, + volume = {89}, + issue = {1} +} + +@article{201467079, + incitefulid = {201467079}, + title = {The Problem of Social Cost}, + magid = {2131740028}, + author = {Ronald H. Coase}, + year = {2016} +} + +@article{56077441, + incitefulid = {56077441}, + title = {The Economics of Welfare}, + magid = {1579344322}, + author = {Arthur Cecil Pigou}, + year = {1920} +} + +@article{50032059, + incitefulid = {50032059}, + title = {A multi-spacecraft formation approach to space debris surveillance}, + magid = {2466447974}, + url = {http://doi.org/10.1016/j.actaastro.2016.05.040}, + doi = {10.1016/j.actaastro.2016.05.040}, + author = {Leonard Felicetti and M. Reza Emami}, + year = {2016}, + journal = {Acta Astronautica}, + volume = {127} +} + +@article{102686645, + incitefulid = {102686645}, + title = {Orbital debris: Threat analysis and satellite security compliances}, + magid = {2781208112}, + url = {http://doi.org/10.1109/i2ct.2017.8226277}, + doi = {10.1109/i2ct.2017.8226277}, + author = {Uttam Banik and Shaikh Muhammad Rizwan Ali and Mahbubur Rahman}, + year = {2017} +} + +@article{66562802, + incitefulid = {66562802}, + title = {Multiple Debris Orbital Collision Avoidance}, + magid = {2949670998}, + url = {http://doi.org/10.1109/aero.2019.8742114}, + doi = {10.1109/aero.2019.8742114}, + author = {Ahmed Refaat Hamed and Ahmed Badawy and Adel A. Omer and Mahmoud M. Ashry and Wessam M. Hussein}, + year = {2019} +} + +@article{118162223, + incitefulid = {118162223}, + title = {Risks in Space from Orbiting Debris}, + magid = {1622616609}, + url = {http://doi.org/10.1126/science.1121337}, + doi = {10.1126/science.1121337}, + author = {J.-C. Liou and Nicholas L. Johnson}, + year = {2006}, + journal = {Science}, + volume = {311}, + issue = {5759} +} + +@article{46739820, + incitefulid = {46739820}, + title = {Orbital Debris: the Growing Threat to Space Operations}, + magid = {768267261}, + author = {Nicholas L. 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b/Writing/2020-11-12_Paper/DynamicConstellationOrbits.tex.bak new file mode 100644 index 0000000..1ffaca9 --- /dev/null +++ b/Writing/2020-11-12_Paper/DynamicConstellationOrbits.tex.bak @@ -0,0 +1,607 @@ +\documentclass{article} + +%%%%%%%%%Packages%%%%%%%%%%%%%%% +\usepackage{geometry} +%\geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1.2in,rmargin=1.2in} +\setlength{\parskip}{7pt} +\setlength{\parindent}{6pt} +%\setlength{\parindent}{0pt} + + +\usepackage{amsmath} +\usepackage{mathtools} +\usepackage{amsthm} +\usepackage{amssymb} +\usepackage{thmtools, thm-restate} +\usepackage{cleveref} + +%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{\likeli}[2]{\text{L}\left( #1 | #2 \right)} + +\newcommand{\induction}[3]{ + \textbf{#1}\\ + \underline{Basis Step:}\\ + #2 + + \underline{Induction Step:}\\ + #3 +} +\newcommand{\mle}[1]{\hat{#1}_{\text{MLE}}} +\newcommand{\pr}[1]{\text{Pr}\left( #1 \right)} +\newcommand{\thm}[3]{ + \begin{restatable}[#1]{theorem}{#2} + \label{thm:#2} + #3 + \end{restatable} +} + +%These are some formatting/reminder commands +\newcommand{\todo}[1]{ + \textbf{\#TODO: \underline{#1}} +} +\newcommand{\define}[1]{%for now just highlight, in future, link. + \underline{\textbf{#1}} +} + + +%%%%%%%%%%%%%%%Math Operators%%%%%%%%%%%%%%%%%% +\DeclareMathOperator{\argmax}{argmax} +\DeclareMathOperator*{\plim}{plim} +\DeclareMathOperator*{\adj}{adj} + + +%%%%%%%%%%%%%%INTERNAL REFERENCES%%%%%%%%%%%%%% +\newcommand{\eref}[1]{Eq. \ref{#1}} %Already defined + + + + +%%%%%%%%%%%%%%Bibliography%%%%%%%%%%%%%%% +\usepackage[backend=biber,sorting=none,style=numeric]{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 + +\providecommand{\keywords}[1]{\textbf{\textit{Keywords:}} #1} +\providecommand{\jel}[1]{\textbf{\textit{JEL Codes:}} #1} + + +%%%%%%%%%%%%CUSTOMIZATION%%%%%%%%%%%%%%%%%%%%%% +\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} + + diff --git a/Writing/2020-11-12_Paper/Linalg.html b/Writing/2020-11-12_Paper/Linalg.html new file mode 100644 index 0000000..8fdfe86 --- /dev/null +++ b/Writing/2020-11-12_Paper/Linalg.html @@ -0,0 +1,4753 @@ + + + + Linalg + + + + + + + + +

+

Please enable JavaScript in order to get a 2d display of the equations embedded in this web page.

+
+ + + + + + + + +
+ (%i1) + + + + 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] +
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+ Created with + wxMaxima. +

+ + diff --git a/Writing/2020-11-12_Paper/ObjectiveState b/Writing/2020-11-12_Paper/ObjectiveState new file mode 100644 index 0000000..c05d3bd --- /dev/null +++ b/Writing/2020-11-12_Paper/ObjectiveState @@ -0,0 +1,4 @@ +[first] +The goal of this paper is to examine the impact of network externalities on the optimal decision making for satellite operators + +[second] diff --git a/Writing/2020-11-12_Paper/References.bib.bak b/Writing/2020-11-12_Paper/References.bib.bak new file mode 100644 index 0000000..ce82639 --- /dev/null +++ b/Writing/2020-11-12_Paper/References.bib.bak @@ -0,0 +1,198 @@ +% 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 = {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;} diff --git a/Writing/2020-11-12_Paper/Title/Title_DynamicConstellationOrbits.aux b/Writing/2020-11-12_Paper/Title/Title_DynamicConstellationOrbits.aux new file mode 100644 index 0000000..b0f82be --- /dev/null +++ b/Writing/2020-11-12_Paper/Title/Title_DynamicConstellationOrbits.aux @@ -0,0 +1,5 @@ +\relax +\abx@aux@refcontext{none/global//global/global} +\@writefile{toc}{\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax } +\@writefile{lof}{\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse {pagetracker}\boolfalse {backtracker}\relax } +\@writefile{lot}{\boolfalse {citerequest}\boolfalse {citetracker}\boolfalse 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b/Writing/2020-11-12_Paper/Title/Title_DynamicConstellationOrbits.tex @@ -0,0 +1,130 @@ +\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{setspace} +\doublespacing + +\usepackage{amsmath} +\usepackage{mathtools} +\usepackage{amsthm} +\usepackage{amssymb} +\usepackage{thmtools, thm-restate} +\usepackage{cleveref} + +%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{\likeli}[2]{\text{L}\left( #1 | #2 \right)} + +\newcommand{\induction}[3]{ + \textbf{#1}\\ + \underline{Basis Step:}\\ + #2 + + \underline{Induction Step:}\\ + #3 +} +\newcommand{\mle}[1]{\hat{#1}_{\text{MLE}}} +\newcommand{\pr}[1]{\text{Pr}\left( #1 \right)} +\newcommand{\thm}[3]{ + \begin{restatable}[#1]{theorem}{#2} + \label{thm:#2} + #3 + \end{restatable} +} + +%These are some formatting/reminder commands +\newcommand{\todo}[1]{ + \textbf{\#TODO: \underline{#1}} +} +\newcommand{\define}[1]{%for now just highlight, in future, link. + \underline{\textbf{#1}} +} + + +%%%%%%%%%%%%%%%Math Operators%%%%%%%%%%%%%%%%%% +\DeclareMathOperator{\argmax}{argmax} +\DeclareMathOperator*{\plim}{plim} +\DeclareMathOperator*{\adj}{adj} + + +%%%%%%%%%%%%%%INTERNAL REFERENCES%%%%%%%%%%%%%% +\newcommand{\eref}[1]{Eq. \ref{#1}} %Already defined + + + + +%%%%%%%%%%%%%%Bibliography%%%%%%%%%%%%%%% +\usepackage[backend=biber,sorting=none,style=numeric]{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} +\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 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. +This paper has been approved for dual submission in Econs 529 and Econs 594 by the instructors. + +\end{document} + + diff --git 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index 0000000..e9ac106 --- /dev/null +++ b/Writing/2020-11-12_Paper/linalg/Linalg.tex @@ -0,0 +1,130 @@ +\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} +\newlength{\pictureboxwidth} +\newlength{\pictureboxheight} +\newcommand{\includeimage}[1]{ + \savebox{\picturebox}{\includegraphics{#1}} + \settoheight{\pictureboxheight}{\usebox{\picturebox}} + \settowidth{\pictureboxwidth}{\usebox{\picturebox}} + \ifthenelse{\lengthtest{\pictureboxwidth > .95\linewidth}} + { + \includegraphics[width=.95\linewidth,height=.80\textheight,keepaspectratio]{#1} + } + { + \ifthenelse{\lengthtest{\pictureboxheight>.80\textheight}} + { + \includegraphics[width=.95\linewidth,height=.80\textheight,keepaspectratio]{#1} + + } + { + \includegraphics{#1} + } + } +} +\newlength{\thislabelwidth} +\DeclareMathOperator{\abs}{abs} +\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{} +\] +%%%%%%%%%%%%%%% +\end{document} diff --git 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+++ b/Writing/2020-12-19_Paper/DynamicConstellationOrbits.tex @@ -0,0 +1,578 @@ +\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{\mle}[1]{\hat{#1}_{\text{MLE}}} +\newcommand{\pr}[1]{\text{Pr}\left( #1 \right)} +\newcommand{\thm}[3]{ + \begin{restatable}[#1]{theorem}{#2} + \label{thm:#2} + #3 + \end{restatable} +} + +%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=numeric]{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} + 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} + + diff --git a/Writing/2020-125-02_presentation/OrbitAllocation.aux b/Writing/2020-125-02_presentation/OrbitAllocation.aux new file mode 100644 index 0000000..e38da04 --- /dev/null +++ b/Writing/2020-125-02_presentation/OrbitAllocation.aux @@ -0,0 +1,189 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline 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+[0] Config.pm:304> INFO - This is Biber 2.14 (beta) +[0] Config.pm:307> INFO - Logfile is 'OrbitAllocation.blg' +[105] biber:322> INFO - === Tue Dec 1, 2020, 02:52:55 +[122] Biber.pm:415> INFO - Reading 'OrbitAllocation.bcf' +[239] Biber.pm:943> INFO - Found 8 citekeys in bib section 0 +[260] Biber.pm:4256> INFO - Processing section 0 +[261] Utils.pm:75> INFO - Globbing data source 'References.bib' +[262] Utils.pm:91> INFO - Globbed data source 'References.bib' to References.bib +[281] Biber.pm:4455> INFO - Looking for bibtex format file 'References.bib' for section 0 +[284] bibtex.pm:1653> INFO - LaTeX decoding ... +[351] bibtex.pm:1471> INFO - Found BibTeX data source 'References.bib' +[493] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized' +[493] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable' +[493] Biber.pm:4084> INFO - Sorting list 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