1、For office use onlyT1 _T2 _T3 _T4 _Team Control Number52214Problem ChosenBFor office use onlyF1 _F2 _F3 _F4 _2016aMCM/ICMSummary Sheet(Your teams 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 s
2、chool, advisor, or team members on this page. summarySpace junk has reached into a unbelievable number these days, as a result of high speed development in aviation industry. Thus, the space debris bring mankind serious obstacles in space exploration. Therefore, clearing the space debris has become
3、a top-task to be dealt with.To begin with, we have studied the time-space distribution pattern of debris, we have predicted the variation of fragments with time. After knowing the distribution feature of fragments density in altitude and latitude, we have learned roughly that space junks distributio
4、n mainly ranges from 800km to 1200km of orbit altitude, on which we are able to calculate flux to illustrate the collision of space fragments. Next step, we analysed the successful launching rate, and by combining two aspects, as a sort of risk parameter, we can assess the risk of different projects
5、. We adopt three normal methods for clearing space debris before making a cost analysis and a profit assessment respectively. .After assessing profit in each method, we draw a conclusion that Large satellites is the most profitable one. These two assessment can be later used as an essential indicato
6、r for all projects. Following step is that we use analytic hierarchy approach to assess three projects profit, safety, efficiency, applicability, by which we learn what weight coefficient can be when consider four indicators comprehensively and the advantages and disadvantages of each projects indic
7、ator. Using these weight coefficient as a indicator for fuzzy comprehensive evaluation of weight coefficient, we reach three projects assessment with weight average principle. The results follows, Space-based water jets is fair, High energy lasers is better and Large satellites between two.Since fra
8、gments will change with time, we last step apply Markov model to revise.Then comes our conclusion, that is clearing space junk is worthwhile for private company.content1.Introduction11.1 Overview 11.2 Assumptions21.3 Nomenclature22 Model Theory32.1 Evolutionary modeling Thought.32.1.1The basic sub-m
9、odels modeling thought32.1.2 Debris at a future time in the evolution42.2 Space debris density distribution model52.2.1 The basic concept of the track52.2.2 Spatial density distribution52.3 Risk Assessment Model82.3.1 Mathematical model of the collision probability82.3.2 Launch success rate analysis
10、103. Based on the economic evaluation model.113.1 cost analysis 113.2 Benefit Analysis163.3 Analytic Hierarchy Model Select weight173.4 Fuzzy Comprehensive Evaluation Model203.5 Fixed time-based Markov model224 Final Remarks234.1 Strengths and Weaknesses234.2 Future Model Development234.3 Conclusion
11、s24The Executive Summary25Economic-optimized model for space junk1.Introduction 1.1 OverviewSpace junk is brought by human beings in aerospace activity, which generally refers to the fragments made from any disabled payload, rock body that distribute in spacecraft orbit, and from explosion and colli
12、sion. Normally, they appear in different size and mass, show various distribution patterns in different altitude. Thereby, the running velocity can be different too.Figure 1Status of space debrisSpace junk has reached into a unbelievable number these days, as a result of high speed development in av
13、iation industry. Besides, it will have collision with spacecraft in a remarkable speed, bringing safety problem to spacecraft itself. For example, The issue itself became more widely discussed in the news media when the Russian satellite Kosmos-2251 and the USA satellite Iridium-33 collided on 10 Fe
14、bruary, 2009.Clearing the space junk, therefore, has become a top-task to be dealt with, but at the same time, is also worthwhile for private company. Being a private high-tech company, working out clearing space debris project should make sure the technique could be fully carried out, consider whic
15、h theoretical project would be adopted and improved. Given understanding how to eliminate the fragments, such as space-based Water jets and high energy lasers and large satellites designed to sweep up the debris, among others, the difference of application scope, operation efficiency and cost, high-
16、tech company is able to combine several projects according to its own demand. In addition, it is also obligated to asses if there is a appealing attraction in terms of economic profit through including cost and profit in economic point of qualitative or quantitative evaluation scheme, combing the ri
17、sk of unsuccessful launching, the collision and explosion in orbit. 1.2 AssumptionsThe accuracy of our models rely on certain key, simplifying assumptions. These assumptionsare listed below:The density of space debris is distributed over the longitudes uniform.High track density area similar variati
18、on with altitude and time with low orbit.Risk mainly the risk of collision risk and launch time, and does not occur other unforeseen circumstances.The private company has sufficient fundraising ability, not because of funding problems led to the stagnation of the project.Launch success rate can be u
19、sed instead of the average of recent years.Probability of collision is small in space.Ignores the Space Debris Mitigation of natural factors.In the evaluation system, our weight setting is feasible and science.The costs and benefits of private companies chosen are reasonable.1.3 Nomenclature calcula
20、ted period the growth rate of parameter J in the X year.other factors weight coefficient the X years model latitude latitude the number of space fragments at concerned altitude() Density with respect to the different heights of space debris distribution range of inclination i() Density with respect
21、to the distribution of near-Earth heights()Density with respect to the distribution of eccentricity e parametric latitude=probability of collision in the N time. mean anomaly of H the average differential flux of particle projectile cross sectional area fluence risky probability of launching probabi
22、lity of unsuccessful launching probability of collision2 Model Theory2.1 Evolutionary modeling Thought2.1.1 The basic sub-models modeling thoughtAccording to access to information, divided into the following types of space debrisFigure 2 Classification of space debrisIf there is no time to clean up
23、the debris will continue to grow over time, so we need to change with time rate of debris, building debris evolutionary models in the coming time, to determine its quantity.The so-called space debris is produced by the fragmentation of the events and non-events fragmentation, fragmentation of the ev
24、ent is defined by the explosions, collisions and other causes of debris. Non-fragmentation events in the debris into large diameter produced fragments (diameter 2 cm) and small fragments (diameter 2 cm) in both cases. Thereby dividing the basic model of fragmentation, large and small debris debris m
25、odels models were considered as of the distribution of 31 December 1998 fragments in each model 1. Taking into account the effect of atmospheric drag and other debris orbit perturbation, establish orbital debris transition model. Changes in the track model to 400 km orbit altitude for the sector, co
26、nsidered separately atmospheric drag and solar radiation pressure dominated the two cases.2.1.2 Debris at a future time in the evolutionConsider fragmentation events and non-fragmentation events and related factors, the growth rate of the introduction of events:=1=55(1)Wherein, X is calculated year
27、and X 1900; as weights; parameters for the desired growth rate in the first X years.Taking into account other confounding factors, with represent these factors, such as the contribution of the collision between the fragments, taken at a future time in power and other debris environment.Using mathema
28、tical language to describe the evolution of the debris as follows: =1(,)In the formula, represents the first model year. Figure 3 Space debris density versus time2.2 Space debris density distribution model 2.2.1 The basic concept of the trackIn reality, we generally only study of near-Earth orbit, b
29、ecause of near-earth orbit human space activity is concentrated, it is carried out including Earth observation, communications, meteorological research, broadcasting and other space activities in a very important natural resource. We can take its height in the range of approximately 200km 2000km. An
30、other reason is that we can calculate based on the model of the near-Earth orbit apogee height, so more research in LEO orbit.The following figure shows the relationship between the geometric parameters of the orbit, will be used here only gives the basic relationship, assuming known data are: semi-
31、major axis a, eccentricity e.(2)Figure 4 Geometrical relation on of the ellipse orbitsCan be obtained: half normal chord=(12)Perigee height=(1)Perigee height=(1+)From Keplers third law, cycle operation of spacecraft in orbit T satisfies the relationship=23Where is the Earths gravitational constant ,
32、 =3986003/22.2.2 Spatial density distributionSpace refers to the probability density of space debris that appears on each spatial unit, can be used to describe the spatial distribution of debris in space, we remove space debris research method premise is further important factor affecting our econom
33、ic model. Suppose the spatial density of space debris in the longitude is evenly distributed, so the space density distribution of space debris is only a function of latitude and altitude h. Suppose is less than the height dimension h smaller than (,) the total amount of space debris is defined:(,)=
34、2Thus the combination of the spatial density and type definitions:(h, ) = lim0(5)(4)(6)(7)(8)Wherein is the volume unit by uniform =2 ,hypothesis, longitude integrated, there=22Where r = h + R is the distance from the center of the earth, R is the radius of the Earth. and =(,)thereby=(,)= (,)22 (,)2
35、(+)2It can be seen that the construction space density distribution, the key lies in the constructor (,)When constructing spatial density, takes the following parameters () as an initial condition:：Total number of space debris at the height of interest；P( )：Density with respect to the near-Earth hei
36、ght of the distribution;P( )：Eccentricity e relative density distribution;P( )：Density with respect to the different heights of space debris distribution range of inclination i.Here，p( ), P( )and P( ) are independent of each other, they satisfy the following conditions:=1()=1()=1() Function p(h, ) t
37、hrough a two-stage configuration, the first consideration in the orbital altitude (h，h+ )within the scope of a thin layer of spherical space debris Average number N(h,h+ ); Second, consider the space debris in the first step on the basis of the given the distribution of the Earths latitude, is N(h,h
38、+ , ),the following relationship exists between these parameters:(9)(10)(12)(13)(14)(11)(,+,)=(,+)Clearly, the following relation holds：(,)=,()()()among them =(,+) (,+)Is the probability of debris falling into the area h，h+，Where and are falling debris ,+ (,+) (,+)probability latitude layer , FL and
39、 h，h+ of. ( ,+ ) can be determined by the following method. Located at +latitude debris layer , within the residence time , debris + and orbital period is T, then( , )=+Let latitude parameters, namely, =+Where f and are true anomaly and orbit perigee angular distance. and latitude there is a relatio
40、nship:(sin )=or =(/)Can be obtained = 22By symmetry orbits, available=2where = 2(,)Among them(,)=(1)212(+)2Can be obtained ( , )：+( , )= +1(,) 22(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)The linear relationship between the flight time and the mean anomaly between, is determined by the followin
41、g method. (,+)The perigee height and eccentricity e, according to the track dynamics can be obtained apogee height , according and , ( , ) determination can be divided into the following six +cases:1. ，at this time ；+ =02. ，at this time ； +=3. ，at this time ；+=+4. ，at this time ； + =15. ，at this tim
42、e ；+=+6. ，at this time 。+ =0Wherein: and respectively h and h+ corresponding + mean anomaly.Based on the above analysis, it can be drawn from the function p (h, ) expression:（27）(,)=() (,+)(,)()()among them：(sin ) ()= ()22 (28)Figure 5Density variation with height latitude2.3 Risk Assessment Model2.
43、3.1 Mathematical model of the collision probabilitySpatial density of space debris environment usually fragments (Density) and flux (Flux) will be described. Description density of space debris in the distribution of space, the size of the density value represents the size of the local region within
44、 the spacecraft and space debris collision risk. Flux space debris refers to the unit time by unit. Number of cross-sectional area of space debris, debris flux and the spacecraft suffered the impact of space debris is proportional to the risk. Research on space debris flux density and can help us to
45、 determine the probability that we launch a spacecraft to collide with debris occurred.According to kinetic theory of gases, known as cross-sectional area of the space object at a constant rate v run on space debris density D of the track, the average differential impacting particle flux , units of ; the corresponding fluence = 21 =In the limited time the average number of collisions c can be given by the following relationship:=According to the Poisson probability distribution theory, n times impact event occurred i
