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For office use only

________________

Team Control Number

42745

Problem Chosen

For office use only

________________

2016

MCM/ICM

Summary Sheet

(Your team's summary should be included as the first page of your electronic submission.)

Type a summary of your results on this page. Do not include the name of your school, advisor, or

team members on this page.

- ‘Kowalski, visual of debris at 9’o clock;’

- ‘Half of North America just lost their Facebook.’

Well, the lines above from the movie Gravity might be slightly exaggerate, but it would more or less

be real if we do nothing about the threatening space debris.

In this paper, we focus on evaluating potential commercial opportunities in removing space debris that

is currently revolving around the Earth in low Earth orbit since the growing number of aerospace

debris has been a major concern around the world community. Three models are presented to analyze

this problem from distinguishing perspectives, namely, the potential economical profits, the

probability of collision and the role of the policy maker. We presume that a private firm discussed

here can both provide insurance services, covering collision for launched satellites, and remove

hazardous space debris. We reduce the spatial debris challenge to a two dimensional problem. Gamma

distribution is used to fit the distribution of space debris.

The first of the three models proposed presents a time-dependent evaluation method that projects

potential economic profits for 15 consecutive years. The revenue function is determined by the

insurance commissions collected from the satellites owner; the cost function can be split into two

parts, the debris removing cost, composed by land-based facilities cost and space-based facilities cost,

and the potential collision related compensation. The Cobb-Douglas utility function is employed here

to estimate the dollar value of probable compensation.

The second of the three models helps to better assess the compensation level by estimating the

probability of a collision between the existing debris and lately launched satellites. The dots, or the

Matlab generated debris, are distributed in a 1000*1000 matrix, following the gamma p.d.f and some

other prerequisites. Then the probability, according to the law of large numbers, is reasonably

approximated by multiple run-times.

The third of the three models jumps out of the two parties previously discussed and shifts to a third

body, the policy maker. A game theory approach is introduced in this model to help the policy maker

better estimate the potential risks and benefits. The payoff matrix is developed and compared under

different circumstances to present the effectiveness under pivotal ‘what if’ scenarios.

Admittedly, the potential profits projected in the first model appear to be quite sensitive to the

insurance commission determinant. One plausible explanation is that the insurance commission

collected each year is the only source of revenue. Also, we did not consider the time value of money.

Based on the data and figures returned from the models above, we conclude with confidence that the

commercial opportunities do exist if the private firm chooses the land-based and space-based

combined alternative. Additionally, the insurance commission determinant shall be set to no less than

4%; and 5% or more is recommended. Policy makers shall implement new industrial standards

towards orbit objects’ design and procedures to attain a Pareto improvement.

数学中国

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本内容由645617861l同学与工作人员共同整理，恭喜同学成功完成数学中国第一期威客项目

数学中国：www.madio.net，最专业的数学建模平台

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Executive Summary

We shall first imagine the following scenario, ‘Half of North America just lost their

Facebook.’ The famous line from Gravity may not be exaggerate. Satellites, generally deemed

as the main actor in space, are facing tremendous threats caused by space debris currently. If

not controlled, the space debris is forecasted to be the main obstacle for further exploration of

the universe.

Considering the significance and urgency of this issue, a chorus of furious space specialists

pilloried the inaction of some politicians and developed multiple ways to eliminate space

debris. Meanwhile, numerous projects to remove space debris have been proposed. To

consider the business opportunities and risks in this field, we assume that a private firm

involved can both provide insurance services covering collision for launched satellites and

remove hazardous space debris; then we develop several models to simulate circumstances

and evaluate the feasibility and possible executive blueprint of space debris eliminating

projects.

We raised three models from distinguishing perspective. To be precise, the possible

commercial profits, the probability of collision between existing debris and lately launched

satellites, and the efficiency of public policies. Those models employ knowledge from several

subjects, such as mathematics, astrophysics, economy etc.. The simulated debris distribution

is as follows:

We classified the prevailing methods of removing space debris into two categories, namely,

the land-based alternative and the space-based alternative. From the data and figures returned

from the models, the projected profits of a private firm are emphatically analyzed for each

alternative.

The land-based alternative requires a huge amount of initial investment and, thus, needs a

relatively long period of time to gain a net profit. It also should be noticed that this alternative

数学中国

www.madio.net

本内容由645617861l同学与工作人员共同整理，恭喜同学成功完成数学中国第一期威客项目

数学中国：www.madio.net，最专业的数学建模平台

备战美赛，美赛资料下载：http://www.madio.net/forum-108-1.html

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can control the quantity of debris ideally, with a total raise of less than 1000 in the 15 years

consecutive simulation.

On the other side, the space-based alternative requires relatively less initial investment. Yet,

the rocket deployment cost incurred when sending a debris remover into low earth orbit is

fairly costly. One fatal weakness of this alternative is that the material used to construct, say,

a Solar Sail, a kind of space debris capturer, put a lower altitude limit of the debris that can be

removed. Hence, this alternative may fail to control the quantity of debris located at low

altitude. In the 15 years consecutive simulation, the results prove our theoretical analysis and

the debris in inner layer seems to be out of control in the end.

Due to the obvious advantages and weaknesses of each alternative, it makes sense to run this

model for a third time with these two alternatives combined, which means we deploy land-

based alternative and space-based alternative simultaneously. The result shows that the time

needed to earn a profit is noticeably reduced while the total quantity of space debris in the low

earth orbit can be controlled with satisfaction.

A game theory analysis is also carried out in the third model. We compare the Nash

Equilibrium solution, or the optimal solution, with and without implementing new industrial

policies. And the outcome is that by implementing new industrial policies and standards, the

policy maker can help the two players reach a better state and promote the social welfare.

Further conducted sensitivity analysis also reveals several interesting points. First, the

implementation of new industrial standards toward the design and procedure of orbiting

objects can greatly reduce the increasing speed of space debris. If the increasing rate

decreases from 5% to 4%, the ending space debris is reduced from 12302 to 9932, net of

2370, which is quite significant. A pattern of diminishing return for each 1% decrease is also

shown from data gathered.

The application of recyclable rockets that could be used in space-based alternative does not

make a huge difference to the firm’s net profit. We compare the projected profits for the firm

by letting the rockets recycle times be 3, 6, 9 and 12. The disparity in accumulated profits

caused by different rockets’ lifetime is not significant since the difference between the

maximum and the minimum is only 15.4% of the total profits. Further, total profits increased

by only 0.38 billion for each additional lifetime increase.

The sensitivity test of the insurance commission parameter shows that this variable can be

essential for the firm to make a profit in the early stage of simulations. By setting the variable

equal to 4%, the time needed to make a net profit is 5 years while the profits are all positive if

we increase that variable by only 1%.

Therefore, we shall formally conclude that the combined alternative is an optimal method to

tackle the space debris issue for the private firm. Furthermore, the insurance commission

parameter shall be set to equal or greater 4% and is recommended to be equal or above 5% to

promote the robustness of the projected net profits. The policy makers are also suggested to

implement new industrial standards to reduce space debris, in order to reach a better outcome,

as the game theory approach proposed.

数学中国

www.madio.net

本内容由645617861l同学与工作人员共同整理，恭喜同学成功完成数学中国第一期威客项目

数学中国：www.madio.net，最专业的数学建模平台

备战美赛，美赛资料下载：http://www.madio.net/forum-108-1.html

美赛讨论加入群：2017美赛官方群287336869

Contents

1 Introduction 3

1.1 Background……………………………………............................................... 3

1.2 Restatement of the Problem…………………………………………………... 3

2 Existing Methods 3

2.1 Land-Based Methods…………………………………………………………. 3

2.2 Space-Based Methods………………………………………………………… 4

3 Terminologies and Notations 4

3.1 Terminologies………………………………………………………………..... 4

3.2 Notation Table……………………………………………………………........ 5

4 Model One – A Time-Dependent Evaluation Model 6

4.1 Assumptions………………………………………………………………….. 6

4.2 Establishing the Model……………………………………………………….. 6

4.2.1 Revenue Function …………………………………………………….... 7

4.2.2 Cost Function …………………………………...................................... 7

5. Model Two – A Model to Determine Pr(t) 9

5.1 Further Assumptions …………………………………………………………. 9

5.2 Some Thoughts ………………………………………………………………. 9

5.3 Establishing the Debris Distribution Model …………..................................... 10

5.4 Space Debris Removal Process ……………………………………………….. 11

5.5 Space Debris Growth Process ………………………………………………… 12

5.6 Satellites Launching and Crashing ……………………................................... 12

6. Results 13

6.1 Land-Based Removal Alternative alone………………..................................... 13

6.2 Space-Based Removal Alternative alone……………….................................... 14

6.3 The Combination of the Two Alternatives ………………………………....... 15

7. Some ‘What If’ Scenarios and Sensitivity Analysis 16

7.1 Scenario 1: New Industrial Standards toward Design and Procedures ……….. 16

7.2 Scenario 2: Recyclable Rocket ……………………………………………….. 17

7.3 A Sensitivity Test of  …...……………………………........………………… 18

8. A Game Theory Approach 19

9. Conclusions 20

10. Strengths and Further Improvements 21

References 22

数学中国

www.madio.net

本内容由645617861l同学与工作人员共同整理，恭喜同学成功完成数学中国第一期威客项目

数学中国：www.madio.net，最专业的数学建模平台

备战美赛，美赛资料下载：http://www.madio.net/forum-108-1.html

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