Orbits Will King

Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Dynamic Launch Decision for Satellite Constellation Operators

Will King

Washington State University william.f.king@wsu.edu

December 1, 2020

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Introduction

ESA – Sep. 2019

For the first time ever, ESA has performed a ’collision avoidance manoeuvre’ to protect one of its satellites from colliding with a ’mega constellation’ #SpaceTraffic (Agency, 2019)

In 1978,Donald Kessler and Burton Cour-Palais identified a potential threat to the new frontier of Earth Orbit. They suggested that if there are enough objects in orbit, debris colliding with other debris and artificial satellites could create debris at an increasing rate, leading to an uncontrollable cascade of collisions, now termed kessler syndrome (Kessler & Cour-Palais, 1978). My goal is to evaluate how the organization of satellite operations into “constellations” affects pollution dynamics and the incentives of operators to deviate from socially optimal behaviors.

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Overview

1 Introduction
2 Literature
3 Model

Constellation Operator’s Problem Social Planner’s Problem

4 Analysis

Welfare Analysis Survival Rates Kessler Syndrome

5 Conclusion

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Literature

(Macauley, 1998) : Estimates the welfare loss due to inefficient allocation of geostationary orbit slots. (Adilov et al., 2015) : Two period model evaluating launch decisions. (Adilov et al., 2018) : Develop an economic Kessler syndrome where pollution is sufficient to halt launches. (Rao & Rondina, 2020) : A widely cited working paper developing the first dynamic model of orbit allocations. Originates in Rao’s dissertation from 2015. (Adilov et al., 2019) : Develops a dynamic model evaluating competitive interactions between firms. (Rao et al., 2020) : Estimates the impact of implementing satellite taxes on future profitability of the satellite industry.

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

High level description

This model is the first dynamic model to incorporate effects from organization as constellations. These effects enter in two forms:

1 Economies of scale in value production.

2 Collision avoidance efficiencies from constellation planning. Key features of this model are:

The assumption that each constellation creates utility without competitive interactions (i.e. monopolistically). Each satellite within a constellation is considered identical. Only the number of satellites contributes to the value produced.

These features simplify computation significantly.

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Mathematical Terms

Symbol  Details  Description

N N > 0

Number of constellations

s_{t}^{i} i ∈ {1, …, N}

Satellite stock of i in t

x_{t}^{i} Ditto

Launches of satellites in t by i

S_{t}

Total number of satellites in t

D_{t} D_{t} ≥ 0

Level of debris in t

m, M m > 0, M > 0 Debris generated from launches and collisions respectively

g(D_{t})

Debris generated from collisions with debris

δ δ ∈ (0, 1)

Decay rate of debris

l^{i}(s_{t}^{i}, S_{t}, D_{t}) l^{i}() ∈ (0, 1)

Rate of satellite loss in i due to collisions

u^{i}(s_{t}^{i}, S_{t}, D_{t}) Utility generated by satellite stock s_{t}^{i} given

S_{t}, D_{t}.

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Constellation Operator’s Problem

V^{i}(s_{t}^{i}, S_{t}, D_{t}) = max_{x_{t}^{i}≥0} u^{i}(s_{t}^{i}, S_{t}, D_{t}) − Fx_{t}^{i} + βV^{i}(s_{t+1}^{i}, S_{t+1}, D_{t+1})  (1)

Subject To:

D_{t+1} =(1 − δ)D_{t} + mX_{t} + M · (∑_{i=1}^{N}s_{t}^{i}l^{i}(s_{t}^{i}, S_{t}, D_{t})) + g(D_{t}) (2)
s_{t+1}^{i} =[1 − l^{i}(s_{t}^{i}, S_{t}, D_{t})]s_{t}^{i} + x_{t}^{i}  (3)
S_{t} = ∑_{i=1}^{N}s_{t}^{i} X_{t} = ∑_{i=1}^{N}x_{t}^{i}  (4)

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Solving Constellation’s Problem I

The general envelope conditions are:

\frac{∂V^{i}}{∂s_{t}^{i}} − \frac{d u^{i}}{ds_{t}^{i}} =β [\frac{∂V^{i}}{∂s_{t+1}^{i}} \frac{∂s_{t+1}^{i}}{∂s_{t}^{i}} + \frac{∂V^{i}}{∂S_{t+1}} \frac{∂S_{t+1}}{∂s_{t}^{i}} + \frac{∂V^{i}}{∂D_{t+1}} \frac{∂D_{t+1}}{∂s_{t}^{i}}] (5)

\frac{∂V^{i}}{∂S_{t}} − \frac{d u^{i}}{dS_{t}} =β [\frac{∂V^{i}}{∂s_{t+1}^{i}} \frac{∂s_{t+1}^{i}}{∂S_{t}} + \frac{∂V^{i}}{∂S_{t+1}} \frac{∂S_{t+1}}{∂S_{t}} + \frac{∂V^{i}}{∂D_{t+1}} \frac{∂D_{t+1}}{∂S_{t}}] (6)

\frac{∂V^{i}}{∂D_{t}} − \frac{d u^{i}}{dD_{t}} =β [\frac{∂V^{i}}{∂s_{t+1}^{i}} \frac{∂s_{t+1}^{i}}{∂D_{t}} + \frac{∂V^{i}}{∂S_{t+1}} \frac{∂S_{t+1}}{∂D_{t}} + \frac{∂V^{i}}{∂D_{t+1}} \frac{∂D_{t+1}}{∂D_{t}}] (7)
∇V_{t}^{i} − ∇u_{t}^{i} =βA∇V_{t+1}^{i}  (8)

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Solving Constellation’s Problem II

The optimality conditions is:

\frac{F}{β} =\frac{∂V^{i}}{∂s_{t+1}^{i}} + \frac{∂V^{i}}{∂S_{t+1}} + m\frac{∂V^{i}}{∂D_{t+1}}  (9)

Iterating both forward and backward one period gives the system

\frac{F}{β} =\frac{∂V^{i}}{∂s_{t}^{i}} + \frac{∂V^{i}}{∂S_{t}} + m\frac{∂V^{i}}{∂D_{t}}  (10)
\frac{F}{β} =\frac{∂V^{i}}{∂s_{t+1}^{i}} + \frac{∂V^{i}}{∂S_{t+1}} + m\frac{∂V^{i}}{∂D_{t+1}}  (11)

\frac{F}{β} =\frac{∂V^{i}}{∂s_{t+2}^{i}} + \frac{∂V^{i}}{∂S_{t+2}} + m\frac{∂V^{i}}{∂D_{t+2}}  (12)

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Solving Constellation’s Problem III

Thus by iterating eqs. (5) to (7) to match eqs. (10) to (12), we can simplify from 9 equations with 9 unknowns to 3 equations with 3 unknowns allowing us to solve for ∇_{[s_{t}^{i},S_{t},D_{t}]}V_{t} in terms of derivatives of the utility function and derivatives of the laws of motion. Substituting ∇V_{t} into the equation below (eq. (10)) provides the euler equation that characterizes the policy function x_{t}^{i}(s_{T}^{i}, S_{t}, D_{t}).

\frac{F}{β} =\frac{∂V^{i}}{∂s_{t}^{i}} + \frac{∂V^{i}}{∂S_{t}} + m\frac{∂V^{i}}{∂D_{t}}

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Social Planner’s Problem

We can address the social planner’s problem in much the same way.

W ({s_{t}^{i}}, D_{t}) = max_{{x_{t}^{i}}_{i=1}^{N}≥0} (∑_{i=1}^{N}u^{i}(s_{t}^{i}, S_{t}, D_{t})) − FX_{t} + βW ({s_{t+1}^{i}}, D_{t+1}) (13)

Subject To:

D_{t+1} =(1 − δ)D_{t} + mX_{t} + M · (∑_{i=1}^{N}s_{t}^{i}l^{i}(s_{t}^{i}, S_{t}, D_{t})) + g(D_{t})  (14)
s_{t+1}^{i} =[1 − l^{i}(s_{t}^{i}, S_{t}, D_{t})]s_{t}^{i} + x_{t}^{i}  (15)
S_{t} = ∑_{i=1}^{N}s_{t}^{i} X_{t} = ∑_{i=1}^{N}x_{t}^{i}  (16)

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Solving Planner’s Problem I

The N + 1 Envelope Conditions are:

\frac{∂W}{∂s_{t}^{i}} = ∑_{j=1}^{N} \frac{d u^{j}}{ds_{t}^{i}} + β [∑_{j=1}^{N} \frac{∂W}{∂s_{t+1}^{j}} \frac{∂s_{t+1}^{j}}{∂s_{t}^{i}} + \frac{∂W}{∂D_{t+1}} \frac{∂D_{t+1}}{∂s_{t}^{i}} ] ∀i ∈ \{1, …, N\} (17)

\frac{∂W}{∂D_{t}} = ∑_{j=1}^{N} \frac{d u^{j}}{dD_{t}} + β [∑_{j=1}^{N} \frac{∂W}{∂s_{t+1}^{j}} \frac{∂s_{t+1}^{j}}{∂D_{t}} + \frac{∂W}{∂D_{t+1}} \frac{∂D_{t+1}}{∂D_{t}} ]  (18)

(19)

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Solving Planner’s Problem II

The N Optimality Conditions are:

0 = − F + β [∑_{j=1}^{N} \frac{∂W}{∂s_{t+1}^{j}} \frac{∂s_{t+1}^{j}}{∂x_{t}^{i}} + \frac{∂W}{∂D_{t+1}} \frac{∂D_{t+1}}{∂x_{t}^{i}} ] ∀i ∈ \{1, …, N\} (20)

Iterating eq. (20) one period forward (from t + 1 to t + 2) for i = 1 and and substituting in the correctly iterated envelope conditions provides the final equation for a system of N + 1 optimality conditions for ∇W_{t}. Once again, iterating eq. (20) backwards from t + 1 to t and substituting in ∇W_{t} will allow you to find the N euler equations characterizing the policy functions {x_{t}^{i}}.

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Welfare analysis

A standard result in the models mentioned in the slide on previous work is that of how free entry or competitive use results in launching more than the socially optimal number of satellites. I suspect that result will hold true in this model. Unfortunately I have not been able to do more than these derivations. The welfare analysis will involve some numerical methods at some point as it gets very messy.

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Survival Rates

One key analysis in Rao and Rondina, 2020 is about the survival rates of satellites. Define the survival rate for a constellation and the society to be:

R_{i} =\frac{s_{t+1}^{i} − x_{t}^{i}}{s_{t}^{i}} = 1 − l^{i}(s_{t}^{i}, S_{t}, D_{t})  (21)

R =\frac{S_{t+1} − X_{t}}{S_{t}} = \frac{∑_{i=1}^{N} s_{t}^{i}[1 − l^{i}(s_{t}^{i}, S_{t}, D_{t})]}{S_{t}}  (22)

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

The marginal survival rates when a given constellation i changes size are:

\frac{∂R_{i}}{∂s_{t}^{i}} = − (\frac{∂l^{i}}{∂s_{t}^{i}} + \frac{∂l^{i}}{∂S_{t}} \frac{∂S_{t}}{∂s_{t}^{i}} ) ≤ 0  (23)

\frac{∂R}{∂s_{t}^{i}} = ∑_{i=1}^{N}[\frac{R_{i}}{S_{t}} ] − \frac{R}{S_{t}} + ∑_{i=1}^{N} \frac{s_{t}^{i}}{S_{t}} \frac{∂R_{i}}{∂s_{t}^{i}}  (24)

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Thus society’s marginal survival rate is less than the weighted arithemetic mean of survival rates for individually growing constellations when:

∑_{i=1}^{N}[\frac{R_{i}}{S_{t}} ] − \frac{R}{S_{t}} + ∑_{i=1}^{N} \frac{s_{t}^{i}}{S_{t}} \frac{∂R_{i}}{∂s_{t}^{i}} ≤ ∑_{i=1}^{N} \frac{s_{t}^{i}}{S_{t}} \frac{∂R_{i}}{∂s_{t}^{i}}  (25)

∑_{i=1}^{N}R_{i} − R ≤0  (26)

∑_{i=1}^{N}[1 − l^{i}(s_{t}^{i}, S_{t}, D_{t})] − ∑_{i=1}^{N}s_{t}^{i}[1 − l^{i}(s_{t}^{i}, S_{t}, D_{t})] ≤0  (27)

∑_{i=1}^{N}(1 − s_{t}^{i})[1 − l^{i}(s_{t}^{i}, S_{t}, D_{t})] ≤0  (28)

This condition is met as every constellation consists of at least one satellite.

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Economic Kessler Syndrome

Adilov et al., 2018 develop a description of economic kessler syndrom as when the debris and satellite stocks are such that it is not profitable to launch. Mathematically this is:

ϑ_{3} = {({s_{t}^{i}}, D_{t}) : X_{t}({s_{t}^{i}}, D_{t}) = 0}  (29)

This definition has the benefit that it can be found through a numerical search directly on the euler equations developed previously. Again, I have not been able to implement this analysis.

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

Conclusion

Summary: In this paper I have described a model and general set of euler equations describing the decisions facing satellite constellation operators. Additionally I have established that negative pollution externalities exist, consitent with other models. This model provides a basis for analyses of competitive and non-competitive interaction between constellation operators, and for the analysis of policy interventions. Future Work: There remains significant work to finalize the model, including exploring a numerical model, clarifying existence criteria, and verifying if constellation operators are likely to overuse orbits.

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

References I

Adilov, N., Alexander, P. J., & Cunningham, B. M. (2015). An economic analysis of earth orbit pollution. Environmental and Resource Economics, 60(1), 81–98. https://doi.org/10.1007/s10640-013-9758-4

Adilov, N., Alexander, P. J., & Cunningham, B. M. (2018). An economic “kessler syndrome”: A dynamic model of earth orbit debris. Economics Letters, 166, 79–82. https://doi.org/10.1016/j.econlet.2018.02.025

Adilov, N., Cunningham, B., Alexander, P., Duvall, J., & Shiman, D. (2019). Left for dead: Anti-competitive behavior in orbital space. Economic Inquiry, 57. https://doi.org/10.1111/ecin.12790

Agency, E. S. (ESAOperations). (2019, September 2). 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. https://twitter.com/esaoperations.

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

References II

Kessler, D. J., & Cour-Palais, B. G. (1978). Collision frequency of artificial satellites: The creation of a debris belt. Journal of Geophysical Research: Space Physics, 83(A6), https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JA083iA06p02637, 2637–2646. https://doi.org/10.1029/JA083iA06p02637

Macauley, M. K. (1998). Allocation of orbit and spectrum resources for regional communications: What’s at stake? The Journal of Law and Economics, 41(S2), 737–764. https://doi.org/10.1086/467411

Rao, A., Burgess, M. G., & Kaffine, D. (2020). Orbital-use fees could more than quadruple the value of the space industry. Proceedings of the National Academy of Sciences, 117(23), https://www.pnas.org/content/117/23/12756.full.pdf, 12756–12762. https://doi.org/10.1073/pnas.1921260117

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Introduction Literature Model

Constellation Operator’s Problem Social Planner’s Problem

Analysis

Welfare Analysis Survival Rates Kessler Syndrome

Conclusion References

References III

Rao, A., & Rondina, G. (2020). Cost in space:debris and collision risk in the orbital commons (Working Paper) [Middlebury College — UC San Diego]. NA. Middlebury College — UC San Diego.

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