Effects of Rotational Motion on the Ballistic Coefficient of Space Debris. Thomas Benjamin Walsh. Auburn, Alabama August 4, 2012.

Transcription

1 Effects of Rotational Motion on the Ballistic Coefficient of Space Debis by Thomas Benjamin Walsh A thesis submitted to the Gaduate Faculty of Aubun Univesity in patial fulfillment of the euiements fo the Degee of Maste of Science Aubun, Alabama August 4, 0 Appoved by D. John Cochan, J., Chai, Pofesso of Aeospace Engineeing D. David Cicci, Pofesso of Aeospace Engineeing D. Andew Sinclai, Associate Pofesso of Aeospace Engineeing

2 Abstact Accuate pedication of the futue states of opeational spacecaft and space debis ae necessay fo conjunctional analysis. Pediction of the states of debis fo a few obital peiods is possible with impoved models of the uppe atmosphee if the objects have spheical geometies. Howeve, spacecaft and othe objects with complex geometies that tumble thoughout thei obit pesent a difficult poblem because the otational motion affects thei obital motion and vice vesa. In this thesis, the coupled tanslational and otational motions of objects, specifically space debis, ae studied using a digital simulation based on a six-degee-of-feedom, igid body model. In paticula, focus is on vaiations due to otation of an object's ballistic coefficient, which is the poduct of a coefficient of dag and a efeence aea divided by the object's mass. If the density is well modeled, the ballistic coefficient is the pincipal unknown in the dag foce. The esults of numeous simulations show a pedictable elationship between an object's ballistic coefficient and its nodal egession. The simulation is also used to poduce data fo an obit detemination pocess in which the ballistic coefficient is estimated. Results ae pesented that show continuous estimation of the ballistic coefficient, which is fundamental to accuately pedicting futue states of debis. ii

3 Acknowledgements The autho wishes to expess his most sincee appeciation fo his adviso, D. John Cochan, J., fo his valuable guidance thoughout the development of this thesis. D. Cochan's unpaalleled expetise in education and the subject matte coveed in this thesis have given the autho a ewading expeience. He would also like to thank the membes of his committee, D. David Cicci and D. Andew Sinclai, fo thei helpful insight thoughout the eseach and eview pocesses. The autho would like to thank his family and fiends in both Akansas and Alabama fo thei patience and continuous suppot. iii

4 Table of Contents Abstact...ii Acknowledgments...iii List of Symbols... vi List of Figues...vii Chapte : Intoduction.... Backgound.... Focus of this Reseach... Chapte : Review of Liteatue and Low Eath Obit Space Debis The Space Envionment Foces Toues... 6 Chapte 3: Development of the Analysis Pogam Intoduction Euations of Motion Tanslational Euations Rotational Euations Effective Aea Numeical Integation... 7 iv

5 3.4. Integation Pefomance... 7 Chapte 4: Analysis Theoy and Results Analysis Appoach Aveage Effective Aea Existing Ballistic Coefficient Data Relationship of Effective Aea Vaiations with Inclination Impact Analysis... 5 Chapte 5: Obit Detemination Methodology Data Geneation Pogam Powe-Density Matix and Pocess Noise Obit Detemination Results Chapte 6: Conclusions Bibliogaphy Appendix A: ECI Coodinate System and Altenate Relative Motion Euations Deivation Appendix B: Attitude Dynamics... 7 v

6 List of Symbols X Y Z î ĵ kˆ ˆ V V el v X, Y, Z a a X, Y, Z a dag ECI X-position ECI Y-position ECI Z-position x-diection unit vecto y-diection unit vecto z-diection unit vecto Geocentic ange Geocentic position vecto Geocentic position unit vecto ECI Velocity Relative velocity between object and atmosphee Velocity components ECI Acceleation Acceleation components Acceleation due to dag a J Acceleation due to zonal hamonic J µ Standad gavitational paamete G Univesal gavitational constant m Mass of object m Mass of pimay souce of gavitational attaction m Mass of object undegoing gavitational attaction R e Radius of Eath U Gavitational potential U Gavitational potential due to zonal hamonic J P nm Legende polynomial J n Zonal hamonic coefficient F Foce on object Foce due to dag C D A Dag coefficient Dag aea, Effective Aea Atmospheic density Toue on object Toue due to gavity-gadient vi

7 List of Figues Figue : Eath-centeed inetial coodinate system with obital elements... 0 Figue : Obital plane with semi-majo axis, a, occupied focus, F, and peigee, P... 0 Figue 3: ECI coodinate system with two bodies... Figue 4: ECI Coodinate System with Spheical Coodinates... 5 Figue 5: Body centeed coodinate system... 8 Figue 6: Rotation of body centeed coodinate system though Eule angles... 9 Figue 7: Coodinate System fo Gavity-Gadient Toue... 3 Figue 8: Pefomance Indicatos vs. Integation Time... 8 Figue 9: Spacecaft Enegies vs. Time... 9 Figue 0: Effective Aea vs. Time Ove Obit fo Cylinde... 3 Figue : Aveage Effective Aea Ovelaid on Instantaneous Effective Aea fo Cylinde Figue : Ballistic Coefficient vs. Time fo Satellite Figue 3: Ballistic Coefficient vs. Time fo Satellite Figue 4: Ballistic Coefficient vs. Time fo Satellite Figue 5: Ballistic Coefficient vs. Time fo Satellite Figue 6: Aveage Aea & Sine of R.A.A.N. vs. Time fo 0 Inclination fo Cylinde Figue 7: Aveage Aea & Sine of R.A.A.N. vs. Time fo 30 Inclination fo Cylinde vii

8 Figue 8: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Cylinde... 4 Figue 9: Aveage Aea & Sine of R.A.A.N. vs. Time fo 60 Inclination fo Cylinde... 4 Figue 0: Aveage Aea & Sine of R.A.A.N. vs. Time fo 80 Inclination fo Cylinde... 4 Figue : Aveage Aea & Sine of R.A.A.N. vs. Time fo 0 Inclination fo Cone Figue : Aveage Aea & Sine of R.A.A.N. vs. Time fo 0 Inclination fo Cone Figue 3: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Cone Figue 4: Aveage Aea & Sine of R.A.A.N. vs. Time fo 60 Inclination fo Cone Figue 5: Aveage Aea & Sine of R.A.A.N. vs. Time fo 80 Inclination fo Cone Figue 6: Aveage Aea & Sine of R.A.A.N. vs. Time fo 0 Inclination fo Flat Plate Figue 7: Aveage Aea & Sine of R.A.A.N. vs. Time fo 30 Inclination fo Flat Plate Figue 8: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Flat Plate Figue 9: Aveage Aea & Sine of R.A.A.N. vs. Time fo 60 Inclination fo Flat Plate Figue 30: Aveage Aea & Sine of R.A.A.N. vs. Time fo 80 Inclination fo Flat Plate Figue 3: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Cylinde with Random Initial Attitude Conditions fo 50 Cases Figue 3: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Cylinde with Random Tue Anomaly fo 50 Cases Figue 33: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Cone with Random Tue Anomaly and Initial Attitude Conditions fo 50 Cases... 5 Figue 34: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Flat Plate with Random Tue Anomaly and Initial Attitude Conditions fo 50 Cases... 5 viii

9 Figue 35: Impacted Cylinde Case vs. Time Figue 36: Impacted Cylinde Case vs. Time Figue 37: Impacted Cylinde Case 3 vs. Time Figue 38: Impacted Cylinde Case 4 vs. Time Figue 39: Impacted Cylinde Case 5 vs. Time Figue 40: MEKF Results fo 0 Days with Synthetic Ballistic Coefficient... 6 Figue 4: MEKF Results fo One Half Yea with Convegent Ballistic Coefficient Figue 4: MEKF Results fo One Half Yea with Dynamic Ballistic Coefficient Figue B.: Eule Angles vs. Time... 7 Figue B.: Rotational Rates vs. Time... 7 Figue B.3: Angula Momentum vs. Time Figue B.4: Instantaneous Aea vs. Time ix

10 Chapte Intoduction. Backgound Since mankind stated launching spacecaft into geocentic obits in the 950's, the theat of obital collisions has been of inceasing pevalence and impotance. Inceased undestanding in the fields of geopotential and atmospheic density have led to highly accuate obital popagation schemes. Although these schemes take into account a geat many petubations, they typically do not account fo the vaiability of the aea used in the calculation of atmospheic dag. This aea is the pojection of the suface aea of the object onto a plane nomal to the elative velocity of the object with espect to the atmosphee. Heein, it is called the "pojected aea" o "effective aea." Unless a satellite is spheical o tidally locked, this aea can play a majo ole in changing the magnitude of the dag foce. An example of such change occued in late Septembe, 0, when NASA's Uppe Atmospheic Reseach Satellite plummeted back to Eath ove the pacific ocean. Scientists and enginees tacking the satellite had difficulty pedicting e-enty locations because the satellite began tumbling, so that the aea used in detemining the magnitude of dag was highly vaiable. This poblem extends beyond e-enty events to objects in the uppe eaches of Eath's atmosphee. Anytime the pojected aea is not known, accuate dag calculations cannot be pefomed. Histoically, the calculation of the vaiable pojected aea has been left out of popagation schemes due to the complex time vaying natue of the vaiables necessay to compute it. With moden computational

11 powe, it is feasible to use high fidelity dynamic models to compute vaiables that wee once deemed constant.. Focus of this Reseach To ceate a bette undestanding of the pojected aea, a digital simulation was developed that includes coupling of the otational o "tumbling" motion of a satellite with the obital motion of its cente of mass. The focus of this wok is the vaiability of the pojected aea and its effects on the ballistic coefficient. Since the othe paametes in the calculation of the ballistic coefficient ae constant, the diffeence between the two paametes is simply a atio. Chaacteistic tends of the ballistic coefficient ae shown that demonstate a elationship between the ballistic coefficient's time evolving behavio and the oientation of the obit it follows. Analyses ae also conducted that povide insight into the emegence of cetain chaacteistics, such as feuency and amplitude changes, that ae obseved in actual ballistic coefficient data. An obit detemination pogam is employed to show that with a known atmospheic density, estimates of the aveage ballistic coefficient with an aveaged pojected aea can be obtained. The analysis and esults in this thesis should pove useful in not only detemining the ballistic coefficient of satellites with unknown attitude conditions, but also in pedicting what the ballistic coefficient should be in the nea futue.

12 Chapte Review of Liteatue and Low Eath Obit. Space Debis The tem space debis is esticted to manmade objects that ae typically divided into thee size categoies: "lage" objects geate than 0 cm diamete, "isk" objects between and 0 cm diamete, and "small" objects less than cm diamete. While all thee sizes of debis can cause catastophic damage to active obital payloads, it is only feasible to catalog lage and some isk sized objects. Objects in the isk categoy ae typically the most dangeous, as they ae too small to tack but lage enough to cause substantial damage. It is estimated that ove 9,000 lage objects, 500,000 isk objects, and upwads of tens of millions of small objects ae cuently in obit about the Eath []. Souces fo debis can include deelict spacecaft, depleted uppe-stages of launch vehicles, and ejecta fom obital collisions. It is clea that without mitigation, the numbe of debis objects, and the theat to opeating spacecaft will continue to incease. Smalle objects may be hamless, but the elative speed in a collision of a small object and a spacecaft in obit can be nealy 5 km/s and lead to enomous eleases of enegy. The lagest concentations of obital debis lie in low Eath obit, aound 00 to,000 km altitude, and in geostationay obit, aound 35,000 km altitude. Nealy all debis can be attibuted to Russia, China, and the United States []. Due to the theat of debis colliding with opeating spacecaft and the manned Intenational Space Station, it is necessay that debis objects ae tacked accuately and thei positions catalogued. Seveal U.S. govenmental and militay agencies tack moe than 0,000 of the lage objects. They also pefom analyses and pedict close misses 3

13 and collisions between objects. Of couse, active payloads must be monitoed closely to avoid collisions with objects in the debis field, but it is also helpful to know of collisions between two debis objects as well. Simulations can be used to pedict whee the new debis fom these collisions will eside as well as the numbe of objects ceated.. The Space Envionment The motion of any object in geocentic obit may be descibed using well known physical laws. If the object's physical chaacteistics and those of its envionment ae known, the study of its motion can be boken down into linea and angula displacements caused by foces and moments, espectively. It is necessay to undestand the phenomena that causes these actions in ode to model them mathematically with an acceptable degee of accuacy... Foces Geneally, the lagest magnitude envionmental foce exeted on an object in obit is gavity. Gavity is an attactive foce between any two objects in the univese that have mass. As such, all objects in the univese that have mass attact the object unde inspection at any given time. Fo obital velocities, Newton's law of gavitation fo two point masses can be used to model spacecaft motion. This is an acceptable way to achieve accuate esults if the masses, velocities, and enegies of the objects in the system ae sufficiently small. 4

14 The pincipal aeodynamic foce expeienced by objects in low Eath obit is dag. Stictly dissipative, dag is esponsible fo ciculaizing such an object's obit and eventually causing it to eente the Eath's atmosphee. Fo decades, pehaps the geatest challenge to be ovecome in detemining an accuate model fo obital motion has been that of accuately modeling the uppe atmosphee. Atmospheic densities vay not only with altitude, but with the complex sola cycle. Sola output, the intensity of the adiation emitted by the sun, vaies in an appoximate yea cycle of the numbe and locations of sunspots. Many atmospheic density models, such as those in the Jacchia family (Jacchia 70, Jacchia 77) include models of the sola cycle. Howeve, a substantial amount of eo in density models fo peiods of time seveal days in length can still be expeienced. The dag coefficient multiplied by the pojected aea and divided by the mass of the object fom the ballistic coefficient. Along with density models, the ballistic coefficient has been a value long sought afte by investigatos to incease the accuacy of atmospheic dag models. Histoically, accuate estimates of the dag coefficient have been obtained in laboatoy tests fo atmospheic vehicles. Howeve, accuate eceation of the envionment of space in a laboatoy is much moe difficult and expensive, leaving the dag coefficient to be estimated in othe ways. Values of C D ae typically between and 3 when the pojected aea of an object is used as the efeence aea. The most common value fo the dag coefficient used in obital analyses is.. This value was ecommended by Cook as acceptable fo satellites of unknown geometies [4]. Assuming that the mass of the object is known, the emaining paamete in the calculation of dag is the pojected aea, A. Fo most satellites, space stations, and payloads the geometies ae well known. 5

15 Hence, fo any given oientation of such an object with espect to the velocity, an A can be calculated. Fo an opeational spacecaft, the challenge in detemining the pojected aea in these cases is detemining the oientation. Howeve, due to the uncetain histoies of many objects in the debis field, accuate geomety and obital oientation ae geneally unknown, making the estimation of A vey difficult. The last foce that is consideed of impotance in this investigation is that due to the oblateness of the Eath. Due to the Eath not being entiely spheical and having a ough suface with lage amounts of mass displacements, i.e. mountains, oceans, continents; its gavitational field is not unifom. The conventional appoach to modeling the Eath's gavitational field is to use Legende polynomials. The J zonal hamonic, which is symmetic and invaiant about the Eath's axis of otation, is used to model the Eath's oblateness, o bulging about the euato. The foce due to this bulging esults in peiodic changes in the eccenticities and semi-majo axes of the obits spacecaft and debis, as well as secula changes in the ight ascension of the ascending node, known as nodal pecession o egession. Depending on an obit's inclination and shape, the ascending node can pecess o egess at ates geate than 6 degees pe day... Toues Thee ae seveal envionmental factos that can cause changes in the oientation of an object. The two that have the geatest effect on an object in low Eath obit ae the gavity-gadient and aeodynamic toues. Gavity-gadient toue, like gavitational petubations, is due to the non-unifomity of the Eath's gavitational field and the finite size of an obiting object. The gadient is due pimaily to the deceasing magnitude of 6

16 Eath's gavitational field with distance, which esults in a geate foce on potions of an obiting object that ae close to the cente of the Eath. The geneation of a toue due to this effect of couse euies that the object be non-symmetical about at least one axis. The diection in which the toue acts on an object depends entiely on its cuent oientation. In Chapte 3 it will be shown that the magnitude of the gavity-gadient toue is invesely popotional to the cube of the magnitude of the object's geometic position vecto [3]. Fo obits below about 400 km altitude, aeodynamic toue becomes the pimay petubation affecting otational motion [5]. Atmospheic densities at these altitudes allow fo inteactions to be handled by using a fee-molecule flow model. Feemolecula flow, deived fom the kinetic theoy of gases, is based on the fact that in nea- Eath space, the molecula mean fee path is much lage than any paticula object's physical dimensions. This teatment allows fo incoming and outgoing paticles to be handled sepaately. As such, two limiting cases aise in the study of the molecula eflections: specula and diffuse. Specula eflections dictate that the molecule bounces off a suface without any enegy loss, and that it leaves the suface in the same plane it aived in with eual speed. The momentum exchanged is in the suface-nomal diection, othogonal to the suface plane. In the case of diffuse eflections, molecules adhee to the suface upon collision. Then, they depat the suface at a late time with an enegy that is detemined by the sufaces tempeatue. Few molecules exhibit eithe model exactly, but fall somewhee in between [6]. The calculation of aeodynamic toues on obiting objects is difficult fo complex object geometies, but fomulas ae available fo simple geometies like cylindes, cones, and flat plates. 7

17 Chapte 3 Development of the Analysis Pogam 3. Intoduction The numeical analysis of low Eath obit debis pesented in this thesis was pefomed by a pogam witten in Fotan 77. This pogam is a six degee-of-feedom (6DoF) digital simulation that numeically integates a set of odinay diffeential euations of motion. At use specified time incements, infomation such as position, attitude, and angula momentum can be extacted fom the simulation and ecoded. 3. Euations of Motion Heein, obiting objects ae modeled as igid bodies. Hence, the euations of motion that detemine the linea and angula motion expeienced by an obiting object as functions of time ae well known. The euations descibe the tanslational motion of the object's cente of mass and the otation of its pincipal axis system. The integation scheme is designed to integate fist-ode, odinay diffeential euations. Thus, the euations ae witten to descibe changes in velocity (acceleation) and changes in position (velocity) fo both tanslational and otational motion. The phase "six-degeesof-feedom" stems fom thee possible linea movements and thee possible otational movements. The euations that descibe these movements ae deived about two sets of othogonal, odeed tiplet axes. Six fist-ode diffeential euations ae euied to descibe the point mass motion of the object in one set of axes. Typically, an additional six euations ae euied to descibe the otational motion of the object in the othe set 8

18 of axes. Conventionally Eule angles ae used to define oientation of the igid bodies. To ovecome singulaities in the otational motion euations, the thee euations that descibe the otational velocity ae eplaced by fou euations fo Eule paametes. All tolled, thee ae thiteen euations of motion: six fo linea motion and seven fo otational motion. 3.. Tanslational Euations The application of Newton's laws of motion euies the use of an inetial coodinate system. Although an Eath-centeed coodinate system is not tuly inetial, acceleations due to foces neglected in consideing the Eath and an obiting object as a system ae small in magnitude compaed to those due to the inteaction between the Eath and the object. Thus, to study the elative motion of a spacecaft o debis object, an Eath-centeed coodinate system is appopiate. The Eath-centeed system can also be fixed in oientation. So, the six tanslational euations ae deived using an Eathcenteed inetial (ECI) coodinate system. An explanation of why the ECI system is appopiate to use is given in Appendix A. 9

19 Figue : Eath-centeed inetial coodinate system with obital elements Figue : Obital plane with semi-majo axis, a, occupied focus, F, and peigee, P Figues and show five obital elements and the Catesian coodinate system OXYZ used as the efeence system fo the euations of motion. Elements shown ae the inclination, i, the ight-ascension of the ascending node, Ω o R.A.A.N., agument of the peigee, ω, the tue anomaly, υ, and the semi-majo axis, a. The sixth obital element, 0

20 eccenticity, o simply e, falls between 0 and and is a measue of how much the obit deviates fom a cicle. Togethe they fom the shape of an obit and the position of the object along the obit. The position is also given by the coodinates X, Y, and Z o the position vecto (X,Y,Z). The fist thee euations fo linea motion ae athe tivial. The time ate of change of a position component is eual to the velocity coesponding to that component, v x, y, z. X& v Y& vy Z& v Z X The thee euations which goven the acceleation of the cente of mass of the object ae boken down into fou pats: two-body gavitational foce due to the Eath, gavitational petubations due to the Eath, aeodynamic dag, and foces due to othe souces, such as the Sun and moon. ma F + F + F + F 3.4 gav oblate dag othe The foces acting on the object ae labeled accoding to thei souce. The foces ae caused by gavity, gav, the oblateness of the Eath, oblate, atmospheic dag, dag, and thid-body objects, othe. The sum of the foces is eual to the mass, m, times the acceleation of the object, a, accoding to Newton's second law. The euations of motion fo the acceleation caused by two-body gavitational foce ae deived fom Newton's law of univesal gavitation. 3.5

21 The foce F gav is a function of the univesal gavitational constant, G, the pimay and seconday masses, m and m, espectively, and the elative position vecto of the two masses,. Conside an inetial coodinate system that contains two point masses. Figue 3: Inetial coodinate system with two bodies The oigin, I, is a fixed point. R and R epesent the inetial position vectos of the two point masses, while R cm is the inetial position vecto of the system's cente of mass. Taking into account mutual gavitational attaction only, the inetial euations of motion fo the position vectos of the masses ae given by: m R && mm G ˆ

22 Hee, the subscipts and efe to the two point masses, m and m. The vecto pointing fom m to m is v, and ˆ is the magnitude of that vecto. Though vecto addition, it is clea that: R + v R v R R R and R ae the inetial position vectos of the two point masses. Taking the second deivative of euation 3.9 yields the acceleation of the elative position vecto of m with espect to m. & R && R && 3.0 & m m ( m m ) + G ˆ G ˆ G ˆ 3. We now assume that m is the mass of the Eath and m is the mass of the satellite. Theefoe the mass of m is negligible in compaison to m and the following appoximation is made: ( m m ) G µ G 3. + m This poduct will be efeed to as the standad gavitational paamete, o simply µ. The standad gavitational paamete used in all simulations has a value of km 3 /s. We can now state the euation fo the elative motion of the satellite when it and the Eath ae both consideed to attact like point masses: 3.3 To obtain a moe geneal esult, it is convenient to assume that the cente of mass of the Eath-object system is O in the ECI system and is "inetial." It is also assumed that the 3

23 4 Eath is a igid body with a gavitational potential U. Fo two-body gavity, pat of U is U B so that: B U & & whee [ ] ˆ ˆ ˆ Z Y X Z U Y U X U U B B B B K J I and U B µ Fo the case of a moe geneal gavitational field, we conside U to be a moe geneal function of the position vecto. Hee, the del symbol epesents the mathematical gadient opeato. In Catesian coodinates, the acceleation components ae the components of B U. [ ] ( ) Z v Z Z U Y v Y Y U X X Z Y X v X X U Z B Y B X B µ µ µ µ + + & && & && & & & Since the Eath is non-spheical, tems must be added to the ight-hand side of the above potential to accuately model the gavitational field. It is convenient to expess these tems in spheical coodinates (, θ, λ), due to the natue of planets being nealy spheical. U fo the Eath can now be expessed in the fom [7]:

24 µ U + B(, θ, λ) whee B is: 3.0 n R e J ( ) n Pn 0 sinθ µ B(, θ, λ) 3. n n n Re ( + ) ( ) Cnm cos mλ S nm sin mλ Pnm sinθ m Figue 4: ECI coodinate system with spheical coodinates This full fom vesion of the appoximation of Eath's gavitational field consists of Legende polynomials, P nm, and the adius of Eath, R e. Zonal hamonic coefficients ae J n, C nm, and S nm ae tesseal hamonic coefficients fo n m and sectoal hamonic coefficients fo n m. It should be noted that the cases fo n 0 is simply the two-body gavitation and n is not pesent because the oigin coincides with the cente of Eath's mass. Tesseal and sectoal hamonics ae geneally smalle than zonal hamonics and not nealy as impotant fo accuate simulations of low Eath obits. As such, the potential function can be appoximated as: 3. 5

25 6 Fom Fig. 4, it can be seen that Z θ sin. The second zonal hamonic is of concen fo this pogam, so fo n the Legende polynomial is: ( ) 3 sin 3 sin,0 Z P θ θ 3.3 It follows that the second tem in the above appoximation fo the gavitational potential, U, is: Z R J Z J R U e e µ µ 3.4 Taking the gadient gives the component acceleations of the J zonal hamonic to be added to the two-body gavitational acceleations , 5 7, 5 7, Z Z R J a Y Y Z R J a X X Z R J a e z J e y J e x J µ µ µ This completes the second phase of summing the foces that affect low Eath obits in the analysis pogam. The next acceleations to be added to the object's euations of motion ae those caused by aeodynamic dag. Using a simila appoach to two-body gavitation, the acceleation expeienced by objects in the uppe atmosphee is given by: 3.8 The atmospheic density is found using an exponential model that eads in an altitude and outputs a density. This model is fomed by a combination of the U.S.

26 Standad Atmosphee (976), CIRA-7, and CIRA-7 with exospheic tempeatue with the tempeatue set at 000 Kelvin [7]. The velocity tem, V el, epesents the elative velocity between the object and the atmosphee. This is calculated using the Eath's otational velocity, ωe. Vel V ωe ω kˆ e ωe Anothe modification to the dag euations is the use of the peviously defined ballistic coefficient, β. Taking the patial deivatives of the total acceleations gives the component acceleations to be added into the linea euations of motion. a a a dag, X dag, Y dag, Z [( v + ω Y ) + ( v ω X ) + v ] ( v + ω Y ) βρ 3.3 X e Y e Z X e βρ [( v X + ωey ) + ( vy ωe X ) + vz ] ( vy ωe X ) 3.3 βρ [( v X + ωey ) + ( vy ωe X ) + vz ] ( vz ) 3.33 Adding these acceleations accoding to thei components esults in the final thee euations fo linea motion. These fist-ode odinay diffeential euations ae to be numeically integated to popagate the location of the cente of mass of objects in low Eath obit. µ X v& X + a J, X + a dag, X µ Y v& + a + a 3.35 Y J, Y dag, Y

27 3.. Rotational Euations As mentioned peviously, thee ae seven euations of motion that ae used to descibe the otational motion of an object in this investigation. Fo convenience, these euations ae deived using the cente of mass of the object and assuming that the bodyfixed axis system Cx B y B z B is a pincipal coodinate system of the object. Figue 5: Body centeed coodinate system This coodinate system coincides with the Eath-centeed inetial system when the attitude angles ae all zeo. In a manne simila to the deivation of the tanslational euations of motion, the fist euations consideed ae those that define the angula position vaiables. Fist, Eule angles ae consideed. These angles ae functions of themselves and the angula velocity of the object. Eule angles ae used to define the initial angula oientation of the object. The set of thee angles (ψ, θ, ϕ) that ae used define a 3-- tansfomation fom the ECI system to the pincipal Cx B y B z B system. 8

28 Figue 6: Rotation of body centeed coodinate system though Eule angles That is, the z-axis otation takes place fist though an angle of ψ, followed by a otation about the new y-axis though θ, and finally a otation about the new x-axis though ϕ. These angles ae conveted into a uatenion, the elements of which ae Eule paametes, fo integation puposes. The advantage of using uatenion algeba in attitude dynamics is that thee ae no singulaities in the diffeential euation fo the Eule paametes. Quatenion algeba was developed in 843 by Si William Rowan Hamilton [8]. A uatenion consists of fou tems, i.e. a vecto and scala pat. The vecto pat can be diectly coelated to the axis that a given otation is about, while the scala pat is the cosine of one-half the angle swept though in the otation. The uatenion used in this analysis is a unit uatenion whose values ange fom - to. The initial uatenion fo integation is ceated though the use of a diection cosine matix, o DCM. A DCM is a matix that houses the cosines of the angles between a vecto and the coodinate system. Fo example, it can tansfom the components of a vecto in the inetial fame to a bodyfixed fame. The tanspose of a DCM tansfoms that vecto back to the oiginal 9

29 0 coodinate fame. The following DCM is ceated using the thee Eule angles that descibe the attitude of the pinciple axis system. As denoted by the subscipt, it tansfoms vectos in the inetial fame to the body fame. Fo bevity, cosine has been abbeviated as "c" and sine as "s" in the following expession. + + ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( φ θ φ ψ θ ψ φ θ ψ φ ψ φ φ θ θ ψ φ ψ φ ψ φ θ φ ψ θ θ ψ θ ψ c c s c s s c s c c s s s c s s s c c s c s s c s c s c c BI C 3.37 The components of this DCM ae used to ceate the components of the unit uatenion that is to be integated. The components of the unit uatenion ae Eule paametes as defined in euations 3.38 though 3.4. The subscipts of C ij denote the location of the element in the DCM, whee i is the ow and j is the column C C C C C C C C C The diffeential euation fo a unit uatenion is a function of angula velocity and the cuent uatenion. This is whee most of the advantages can be ealized. Using the tilde notation, a matix of angula velocities is fomed as such: ῶ 3.4 The diffeential euation is given by [8]:

30 & Ω 3.43 whee Ω 0 ~ T ω ω ω v 3.44 Euation 3.43 povides fou scala diffeential euations that popagate the components of a uatenion,. ( ) ( ) ( ) ( ) ω ω ω ω ω ω ω ω ω ω ω ω & & & & Afte integation, the Eule paametes ae tansfomed into Eule angles and ecoded as an output. Tansfomation fomulas may be obtained by euating components of the DCM in tems of uatenion components with that of the DCM in tems of the Eule angles. The uatenion fom of the DCM is: ( ) ( ) ( ) ( ) ( ) ( ) BI C 3.49 The Eule angles ae found by elating selected elements of each DCM. ( ) ( ) tan ) tan( C C ψ ψ

31 The last thee euations that descibe otational motion ae fo the popagation of the otational angula momentum vecto. This angula momentum is the poduct of the object's inetia dyadic, I, with its angula velocity. H I ω 3.53 It is a vecto uantity that is conseved if thee ae no extenal toues on the object. Since thee ae toues acting the objects in this simulation, the angula momentum can vay geatly in magnitude and diection depending on the case. Since the pincipal axis system is otating with espect to the inetial fame, the time ate of change of angula momentum is eual to the toue about the cente of mass [8]. & H ~ ω H + T 3.54 In component fom, E gives the final thee diffeential euations fo the otational motion of the object, H& H& H& x y z ω H z ω H x ω H y y z x ω H y ω H z ω H x z x y + T + T x y + T z whee T x, T y, and T z ae body-fixed components of T. Afte popagating the angula momentum, the new angula velocities ae fomed by multiplying the invese of the inetia matix, I, times the new angula momentum vecto. ω x H x 3.58 The final step in completing the euations of motion is to detemine expessions fo the toues acting on an object. The gavity-gadient toue stems fom the fact that the foce due to gavity is non-unifom ove a body. The deivation begins by using

32 Newton's law of univesal gavitation in mass element fom to wite the element foces, df v i, as a function of mass elements dm i. Fo the case of an abitay oigin of the body axes, Fig. 7 is used in the deivation. v µ dm 3.59 i dfi 3 i i The toue on an object is due to this foce acting on an element of mass at a distance d i fom the geometic cente. The following euation takes advantage of ρ, the vecto fom the geometic cente to the cente of mass, and d i, the vecto fom the cente of mass to the mass element. d T i d d F ( ρ + d ) d F 3.60 i i i i \ Figue 7: Coodinate System fo Gavity-Gadient Toue The diffeential toue in E may be integated ove the entie object to obtain: 3.6 The position of the mass element in the inetial fame can be expessed in tems of the position of the cente of mass. This is impotant because toues act about the cente of mass. 3

33 + d + ρ + d i i ' i 3.6 Fo cases involving space debis, the dimensions of the object ae much smalle than the distances between the gavitational souces, i.e. >> ρ + ' i d i. This allows the use of an appoximation of the distance between the mass element and the inetial oigin. 3 ( ) ( ) ' ' ' ρ d d 3 ( + d ) ( ) i ρ i ρ i i i i 3.63 Substituting the above euation into the mass element integal yields the gavity-gadient toue fo the coodinate system in Fig. 7. ( bˆ 3 ) ( i bˆ ρ + )( i bˆ ) dmi µ m µ TGG To futhe simplify the expession, the fist tem can be eliminated by choosing the geometic cente of objects to coincide with thei cente of mass. The unit vecto bˆ is the oientation of the body axes with espect to the obiting fame. This vecto can be detemined by taking the fist column of the poduct of two diection cosine matices. C C C 3.65 BO BI T OI The integal in E may be evaluated and expessed in tems of the moment of inetia tenso to yield the gavity-gadient toue [7]. T GG [ bˆ ( bˆ )] 3µ I The Catesian components of the above toue ae the expessions fo gavity-gadient toue that ae pat of the components T x, T y, and T z fo Es though

34 T GG 3µ 3 ( I ZZ I YY ) bb3 ( I XX I ZZ ) bb3 ( ) I YY I XX bb 3.67 The aeodynamic toue model adopted fo this effot takes advantage of the aeodynamic foce pe unit mass calculated in the euations of motion. Multiplying the acceleations by the mass gives the aeodynamic foces in the inetial coodinate system. The foces must then be tansfeed into the body centeed coodinate system by use of a diection cosine matix. If we let CP be the vecto pointing fom the cente of mass to the use-defined cente of pessue, the coss-poduct of it and the aeodynamic foces esults in the aeodynamic toues. F CP T dag aeo [ CPx CPy CPz ] m [ a a a ] CP F dag, x dag dag, y m CPy CPz CPx dag, z a a a dag, z dag, x dag, y CPz CPx CPy a a a dag, y dag, z dag, x 3.70 The esultant toue added to the ate of change of the angula momentum is now the summation of the gavity gadient and aeodynamic toues. T T T x y z T GG, x + aeo, x 3.7 T T GG, y GG, z T + T + T aeo, y aeo, z Effective Aea To detemine the effective aea of a tumbling object in obit, it is convenient to take advantage of the simple shapes that make up the geate object. In the case of a 5

35 ight-cicula cylinde, it becomes a cicle when viewed fom the x-diection, and a ectangle when viewed fom the y o z-diections. Fo a ight-cicula cone, these become a cicle and tiangle, espectively. The cosine of the angle between the nomal diection of these shapes and the elative wind detemine the magnitude of the contibution they make to the total aea. Since the nomal diection fom the cicula base of the cylinde o cone ae in the x-diection, the dot poduct between an x-diection unit vecto and the elative wind give the cosine of the angle between the two vectos. V C cosα T BI [ 0 0] V whee V [ V V V ] x y z T 3.74 VxC cosα + VyC V + V C z The cosine of the angle, cos α, is simply a function of the inetial velocity and thee diection cosines. It can now be applied to the individual aeas to detemine the entie total effective aea. α A A base cos + Aside cos 3.76 α If the angle between the nomal diection of the base of the object eaches 90 degees, the cosine of that angle will be zeo and the esultant total aea will be the aea of the side of the object. Likewise, any deviation of the angle fom zeo degees will add to the total aea via the side without egad to the oientation. This allows the use of the Pythagoean identity as a means of finding the multiplie fo the contibution of the side, A side. 6

36 3.4 Numeical Integation Popagation of the euations of motion though time is achieved by numeical integation. The algoithm used is a modified vesion of the Runge-Kutta-Fehlbeg fifth ode solution method. It is a six-step, vaiable time-step algoithm that poducts a fouth and fifth ode solution. The diffeence of these solutions is compaed to a toleance which detemines if the time-step will incease o decease. When the compaison is below the desied accuacy toleance, the fifth-ode solution is epoted and the algoithm continues, with an incease in the time step. This algoithm poves to be time efficient while maintaining accuacy [8]. Depending on the initial conditions, mainly the otational velocities, one low Eath obit simulation takes a faction of a second to un. Modifications include logic to set a maximum numbe of iteations and a maximum time step Integation Pefomance To detemine the ate at which integation eo contaminates the esults, stability tests must be pefomed on the analysis pogam. To have constant obital elements, angula momentum, and enegy, a numbe of suboutines ae withheld fom the pogam's nomal opeation. In effect, it educes the pogam to a two-body popagation scheme with an object tumbling at a constant ate. An abitay integation time of two yeas was chosen because it is much longe than any poposed ballistic coefficient o position foecasting window. The obit popagated is a nea-cicula, 700 km altitude obit with the initial ight ascension of the ascending node at 0 degees and inclination of 45 degees. Thee attibutes, which would be constant in an ideal scenaio, wee chosen 7

37 as indicatos of integation accuacy decay: angula momentum magnitude, H, specific obital enegy, ε, and otational kinetic enegy, T ot. All thee indicatos inceased seculaly thoughout the integation, but at vey slow ates. Tanslational kinetic enegy, T tans, and potential enegy, U tans, vaied peiodically as expected fo an elliptic obit. The specific obital enegy is a combination of the tanslational and otational enegies. ε T + U + T 3.77 tans tans ot The angula momentum magnitude inceased by 4.3E-3 J-s, an incease of pecent. Rotational kinetic enegy inceased by.5e-4 J, an incease of pecent. The specific obital enegy inceased in accodance with the incease in otational kinetic enegy. The following plot show these pefomance paametes plotted against time. Figue 8: Pefomance Indicatos vs. Integation Time The next plot shows the enegy components vesus time. It is woth noting that the tanslational kinetic and potential enegies counte-balance each othe and account fo a constant total tanslational enegy thoughout the integation. The obital elements, except fo the tue anomaly, emained nealy constant as in an ideal scenaio. 8

38 Figue 9: Spacecaft Enegies vs. Time 9

39 Chapte 4 Analysis Theoy and Results 4. Analysis Appoach The pincipal objective of the this thesis is to obtain a geate undestanding of the factos that cause time vaiations of the ballistic coefficients of debis objects. Of paticula inteest ae the causes of anothe pojected aea vaiations. Anothe objective is to investigate the feasibility of using obit detemination techniues to detect apid changes in the ballistic coefficient. A possible eason fo sudden changes in a ballistic coefficient a collision of the tacked object of debis with anothe that esults in a change of the angula momentum of the tacked object. Anothe eason is that thee ae unmodeled density vaiations that change the dag on the object, but the changes ae attibuted to the ballistic coefficient. The pojected aea of a tacked object is an impotant vaiable in the dag model, but often it is ovelooked and assumed to be constant. A bette undestanding of vaiables such as the pojected aea will lead to moe accuate foce and pediction models that can be used to pedict the state of an object fo use in evasive maneuves. Sibet, et al. concluded that accuate position calculations as well as an incease in state vecto update feuency can esult in a nealy complete eduction of nea collision events [9]. Due to the numbe of pieces of debis and the limited infomation available about each individual piece, cetain assumptions must be made to obtain typical esults. Hee, it is assumed that all tacked objects in the simulation pogam have a mass of 00 30

40 kilogams. Thee geometies ae used to epesent the objects: a cylinde, a ight cicula cone, and a flat plate. The cylinde is epesentative of a satellite bus o spent launch vehicle stage. Right cicula cones ae also common in ockety, especially fo e-enty vehicles. A flat plat geomety is epesentative of a sepaated sola panel o a potion of a spacecaft. In all simulations, the thee geometies have lengths extending in the body-x diection of 0 metes. The cylinde and cone have base diametes of mete, while the flat-plate has a width of mete. It is assumed that the flat-plate is infinitely thin. The thee geometies ae meant to give divesity to the simulations, that is, of couse still less than the divese set of geometies of actual space debis. 4.. Aveage Effective Aea Since the simulation pogam integates non-linea euations of motion with high accuacy, a vey small time step is geneally euied. It is not uncommon fo this timestep to be less than one second, meaning thee ae thousands of oppotunities to ecod the dynamical states and effective aea pe obit. When collecting data on the dynamical states, this can be vey advantageous and esult in high fidelity aea data as shown in Fig. 0. The obit popagated to geneate data fo Fig. 0 is nealy cicula with the object at an altitude of 700 km. The obit is inclined at 45 degees with initial ight ascension of ascending node, tue anomaly, and agument of peigee of zeo degees. 3

41 Figue 0: Effective Aea vs. Time Ove Obit fo Cylinde Howeve, the effective aea may vay so much ove an extended peiod of time that without manipulation, finding tends can be difficult. This poblem is alleviated by calculating the aveage aea ove a time peiod, and epoting that value less feuently. The aveage effective aea, A, is calculated by summing the aea at each time, A(t), times the time-step, time as indicated in E. 4.. t, and dividing the sum by the diffeence of the total obsevational A t final final ( ) t t 4. t initial A t t t initial 3

42 Figue : Aveage Effective Aea Ovelaid on Instantaneous Effective Aea fo Cylinde The obsevational time used to obtain the aveage effective aea in Fig. is 8 hous. Clealy, an aveage effective aea is a much moe epesentative facto that scales the amount of dag on an object is obseved. Afte popagating the motion fo ten days using both the actual and aveage effectives aeas, the final positions ae compaed. Table : Diffeence in Positions Detemined Using Instantaneous Aea vs. Aveage Aea x y z elative Act. Aea (km) Ave. Aea (km) Diffeence (m)

43 Linea intepolation was used between aveage aea data points to obtain the instantaneous aea fo continuous dag calculations. Since the magnitude of the position eo is only 8.87 metes, using and studying the aveage effective aea can be consideed easonable fo some puposes. Fo collision avoidance the actual aea is pobably bette if chaacteized well. 4.. Existing Ballistic Coefficient Data The Ai Foce Space Command Space Analysis Cente is one of the govenmental agencies tasked with tacking debis and pefoming conjunctional analyses on the catalog of objects including active payloads. Reseaches at the cente have developed a method to extact the ballistic coefficient estimates fo the objects they tack. Although the Ai Foce's softwae and methods ae not public domain, they have povided a sample of ballistic coefficient data sets fo 7 pieces of debis. The samples ae fo the peiod Januay, 005 to Januay, 007 and ae pesented as pecent deviations fom the mean fo the two yeas. The following figues ae pesented without edit. The title contains the NORAD catalog numbe and name of the object. Also stated in the title is the tem "DCA Values," whee DCA stands fo dynamic calibation atmosphee. This efes to the Ai Foce's method of calibating the tempeatue values fo thei synthetic atmosphee by obseving the foces on satellites whose chaacteistics ae well established. 34

44 0046 THOR DEBRIS DB/Bave DCA Values DB/BAve % YEAR Figue : Ballistic Coefficient vs. Time fo Satellite DELTA DEBRIS DB/Bave DCA Values DB/BAve % Figue 3: Ballistic Coefficient vs. Time fo Satellite 84 35

45 030 DELTA DEBRIS DB/Bave DCA Values DB/BAve % YEAR Figue 4: Ballistic Coefficient vs. Time fo Satellite CZ-4 DEBRIS DB/Bave DCA Values DB/BAve % Figue 5: Ballistic Coefficient vs. Time fo Satellite

46 These fou examples wee chosen because they easily exhibit the exteme vaiability of the ballistic coefficient as well as some insight into the natue. The data in Fig. shows well-behaved peiodicity fo the fist yea, followed by a lage incease in the mean value of the ballistic coefficient in Novembe of the fist yea. That incease is followed by high feuency vaiation at what appeas to be a diffeent mean value. The data in Fig. 3 exhibits an almost peiodic elationship that looks like it should be pedictable. The data in Fig. 4 shows a much moe complex time vaiation that looks to be somewhat pedictable. Howeve, the data in Fig. 5 shows exteme jumps in value with noisy data thoughout. Some of the ballistic coefficient data exhibits almost peiod vaiations with peiods on the ode of months. Inteest was found in these vaiations because the peiod of the nodal egession of an object obiting the Eath vaies fom to 8 months depending on the obit's inclination. Thus, an investigation into the vaiation of the effective aea of an object and its elation to inclination and the egession of the ascending node was conducted. Of inteest in the following section ae the inclinations of the obits of the objects that wee used to calculate ballistic coefficient plots. Table : Inclinations of Example Satellites [0] Satellite 0046 Satellite 84 Satellite 030 Satellite 0857 Inclination Relationship of Effective Aea Vaiations with Inclination The ight ascension of the ascending node of a pogade obit about the Eath vaies peiodically and seculaly. The secula change is a otation of the obital plane 37

47 about the Z-axis of the ECI coodinate system and is known as nodal egession (o pecession fo etogade obits). The cause of the egession is the non-spheicity of the Eath and in tun its gavitational field. The ate at which an obit egesses depends pincipally on the semi-majo axis, eccenticity of the obit, and its inclination. The aveage pecession ate of the ight ascension of the ascending node is given by E. 4. [8]. 3 J Ω & µ e ( cos i) 7 4. ( e ) a Since the obits simulated in this thesis ae nealy cicula, inclination is the main facto detemining the peiod of the motion of the ight ascension of the ascending node in the esults pesented hee. The "nodal ate" is geneally between.5 and 8 degees pe day, esulting in peiods anging fom 40 to 45 days, espectively. Figues 6 though 30 contain plots of time histoies of the aveage effective aea and the sine of the ight ascension of the ascending node. In each case, the initial obit of the object is nealy cicula with the object initially placed at 700 kilometes altitude and at the ascending node. Fo calculating the semi-majo axis in all simulations, the adius of the Eath used is km. The initial tue anomaly, ight ascension of ascending node, and agument of peigee ae all zeo. The initial attitude with espect to the inetial fame and angula ates ae zeo. Afte the popagation begins, envionmental toue cause the object to tumble in a "andom" manne. All simulations wee popagated fo 365 days with vaious initial inclinations. The total popagation time was chosen to exhibit the peiodicity of the effective aea and its elationship to the nodal egession. 38

48 Duing a yea the ight ascension of the ascending completes at least one peiod fo all inclinations. The aveage of the effective aea was taken evey 8 hous. The sine of the ight ascension of the ascending node is pesented with a negative phase shift of 45 degees. This phase shift aligns the effective aea and the sine of the ascending node to show thei simila peiodicities. Figues 6 though 34 contain the phase shifted sine of R.A.A.N. 39

49 Figue 6: Aveage Aea & Sine of R.A.A.N. vs. Time fo 0 Inclination fo Cylinde Figue 7: Aveage Aea & Sine of R.A.A.N. vs. Time fo 30 Inclination fo Cylinde 40

50 Figue 8: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Cylinde Figue 9: Aveage Aea & Sine of R.A.A.N. vs. Time fo 60 Inclination fo Cylinde 4

51 Figue 0: Aveage Aea & Sine of R.A.A.N. vs. Time fo 80 Inclination fo Cylinde As can be seen, the peiod of the aveage effective aea is clealy elated to the peiod of the ight ascension of the ascending node. Also, the amplitude gows with the inceases in inclination. These factos should pove useful in detemining a bandwidth in which the ballistic coefficient should eside. Of couse, the pattens may change with vaying initial conditions. The following figues contain aea data fo a ight cicula cone with the same initial conditions. Since the cente of pessue of the cone is not at the same location as its cente of mass, aeodynamic toues play a ole in causing otational motion of the cone. 4

52 Figue : Aveage Aea & Sine of R.A.A.N. vs. Time fo 0 Inclination fo Cone Figue : Aveage Aea & Sine of R.A.A.N. vs. Time fo 30 Inclination fo Cone 43

53 Figue 3: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Cone Figue 4: Aveage Aea & Sine of R.A.A.N. vs. Time fo 60 Inclination fo Cone 44

54 Figue 5: Aveage Aea & Sine of R.A.A.N. vs. Time fo 80 Inclination fo Cone The behavio of the vaiable effective aea of the ight cicula cone is makedly simila to that of the cylinde. The same symmeties in moments of inetia ae pesent in both the cone and the cylinde, but only half of the side aea is pesent in the cone. Again, the amplitudes of the effective aeas inceased with the inclination. The aeodynamic toue seems to have played a mino ole in the otational motion at this altitude. The next set of plots is geneated fom the same simulations using a flat plate as the object. A flat plate has the same moment of inetia symmeties as the othe geometies, so motion is expected to be simila. The main diffeence occus in the aea calculation, which is now a function of only one suface. 45

55 Figue 6: Aveage Aea & Sine of R.A.A.N. vs. Time fo 0 Inclination fo Flat Plate Figue 7: Aveage Aea & Sine of R.A.A.N. vs. Time fo 30 Inclination fo Flat Plate 46

56 Figue 8: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Flat Plate Figue 9: Aveage Aea & Sine of R.A.A.N. vs. Time fo 60 Inclination fo Flat Plate 47

57 Figue 30: Aveage Aea & Sine of R.A.A.N. vs. Time fo 80 Inclination fo Flat Plate The plots also exhibit peiodic tends that follow the phase shifted sine of the ascending node. Although the tends ae simila to those of the cylinde and ight cicula cone, the peaks coesponding to the minimum of the sine of R.A.A.N. ae fa less amplified. Since the geomety is so diffeent fom a cylinde o cone, it is expected to see diffeent chaacteistics. Howeve, seeing the same ualitative tends coesponding to the ascending node is supising and encouaging to anyone seeking the ballistic coefficient. This also bodes well fo the application of this wok to the LEO debis field since the geometies ae unknown and almost cetainly uite vaied. Tends emeged between the aveage effective aea and the ascending node fo a numbe of diffeent inclinations with the initial attitude conditions set to zeo fo all cases. To detemine if these tends holds fo othe conditions, futhe testing is euied. In the following simulation, a unifom pseudo-andom numbe geneato was employed 48

58 to andomize the initial conditions of the attitude of the object befoe integation began. As befoe, the object was placed at the ascending node. The yea-long simulation was epeated 50 times, each with uniue initial attitude conditions. Figue 3: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Cylinde with Random Initial Attitude Conditions fo 50 Cases Obviously, the same tend of following the phase shifted sine of the ascending node occus. The aea falls within a bandwidth that is lagest when the phase shifted sine of R.A.A.N. eaches its lowest value, coesponding to a -45 degee ight ascension of the ascending node. A simila simulation was conducted to detemine if the initial position of the satellite plays a ole in the behavio of the aveage effective aea. This time, the tue anomaly was andomized. 49

59 Figue 3: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Cylinde with Random Tue Anomaly fo 50 Cases As can be seen in Fig. 3, the peviously stated tend holds. The chaacteistics of the plot ae vey simila to Fig. 3, which stengthens the agument that the effective aveage aea, and in tun the ballistic coefficient, has the same feuency as the sine of R.A.A.N. In anothe benchmak of validity, ageements with the actual ballistic coefficient data can be seen. Fo a yea's woth of data fom satellite 84 in Fig. 3, the ballistic coefficient peaked thee times. Satellite 84's obit is inclined 98.77, o nealy 0 fom pola. This is the same case as an 80 obit, which fo all thee geometies the aveage aea peaked thee times as well. The same conclusion can be dawn between satellite 0046 in Fig. and the chaacteistics of the 60 effective aveage aea plots. 50

60 Fo Figs. 33 and 34, simulations wee un fo the ight cicula cone and the flat plate. The popagation time was one half yea, epeated 50 times. The tue anomaly and initial attitude conditions wee selected fom a pseudo-andom unifom distibution anging fom -80 to 80. Figue 33: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Cone with Random Tue Anomaly and Initial Attitude Conditions fo 50 Cases 5

61 Figue 34: Aveage Aea & Sine of R.A.A.N. vs. Time fo 45 Inclination fo Flat Plate with Random Tue Anomaly and Initial Attitude Conditions fo 50 Cases It appeas that no matte the geomety, thee is a substantial dependence on the sine of the phase shifted ight ascension of the ascending node. Like the cylinde, the aveage aea falls within a bandwidth fo alteing initial conditions of the cone and flat plate. 4.3 Impact Analysis One of the possible easons fo a sudden change in the ballistic coefficient seen in Figs. and 5 is an impact with anothe object. Relative speeds fo objects in low Eath obit can appoach 5 km/s, esulting in lage amounts of kinetic enegy in the smallest of objects. Of inteest to this investigation is the change in angula momentum due to the impact. Anothe way in which a change in the ballistic coefficient could occu is the 5

62 addition, o eduction, of mass by an inelastic collision. Not only would this change the ballistic coefficient inheently, but could uite possibly affect the otational behavio by alteing the moments of inetia. The impact as simulated heein is based on the assumption that a small obiting mass is absobed by the object unde inspection. The total mass change of the system is negligible, and the impact takes place a cetain distance fom the cente of mass of the object, causing a change in angula momentum. The object used fo this analysis is the cylinde. The angula momentum added to the system is fomed by taking the coss poduct of the impact vecto and the linea momentum exchanged between the impacting paticle and the cylinde. H impact m V 4.3 impact p el In the above euation, V el is the elative velocity between the cylinde and the impacting paticle, impact is the vecto pointing fom the cente of mass to the position of impact, m p is the mass of the impacting paticle, and H impact is the angula momentum added to the object. The mass of the impacting paticle was chosen to be gam, and the impact location was 4 metes fom the cente of mass. These paametes wee held constant while the impacting paticle's obit was chosen using the pseudo-andom numbe geneato. Impacting paticle velocities anged fom 0 to 00 pecent of the object's velocity since they have the same eccenticity and altitude. The popagation fo each impacted cylinde case lasted one half yea. The esulting aveage aea is plotted ove the aveage aea fo the same case without an impact, the contol case. The cylinde's obit was inclined 45 fo each case with a nealy cicula shape and an altitude of 700 km. 53

63 The angula obital elements ae initialized at 0 degees, and the impact occued afte one day of integation. The cases shown ae fo diffeent completions of the algoithm in which the elative velocity is uniue in each case. These cases demonstate the similaities of the effective aea's time-vaying behavio between uniue impacts, as well as the diffeences that occu. Figue 35: Impacted Cylinde Case vs. Time 54

64 Figue 36: Impacted Cylinde Case vs. Time Figue 37: Impacted Cylinde Case 3 vs. Time 55

65 Figue 38: Impacted Cylinde Case 4 vs. Time Figue 39: Impacted Cylinde Case 5 vs. Time 56

66 It is clea that a substantial eduction in the aveage effective aea occus afte impact fo each case. No cases aose whee the obits of the object and impacting paticle wee the same, esulting in zeo elative velocity. The peiodic tend occus at the same feuency, but the amplitude ange is much lage. The tend in Fig. 37 is of utmost impotance, as it shows that in impact is a plausible cause fo the eatic behavio of the ballistic coefficient exhibited by satellite 0046 in the second yea of Fig.. Also, the behavio of the aea in Fig. 36 is vey simila to that of satellite 030 in Fig. 4. This indicates that satellite 030 was tumbling with substantial angula momentum duing the two-yea peiod since thee ae no ualitative changes in its behavio. 57

67 Chapte 5 Obit Detemination 5. Methodology In this chapte, obit detemination techniues ae shown to pove useful in detemining the aveage effective aea, which futhe validates the use of an aveage aea fo tacking puposes. Also, simulations ae poduced to test the obit detemination pogam's ability to detect lage chaacteistic changes in the ballistic coefficient. The pocess outlined hee makes use of a modified extended Kalman filte (MEKF) witten in Fotan 77. The pogam tacks and updates seven states: the inetial position vecto, the inetial velocity vecto, and the ballistic coefficient. The MEKF is a well-known tool used fo obit and tajectoy detemination. The paticula algoithm used was taken fom Tapley []. Since the MEKF estimates and update the position to the most accuate location, it does not ely on a vaiable ballistic coefficient to calculate the coect dag acceleations to popagate the obital position accuately. In othe wods, the MEKF is a tool used in this wok to estimate values of the ballistic coefficient, not estimate accuate positions of the object. The ballistic coefficient that the MEKF detemines is a best fit to the data it is given. It will be shown late that this "best fit" is vey close to the ballistic coefficient calculated using the aveage effective aea. 5.. Data Geneation Pogam The MEKF essentially opeates by popagating the state vecto, eading an obsevation, and applying updates to the state afte evey obsevation. This event diven 58

68 pocess continues, fo simulation puposes, until thee ae no moe obsevations to ead. Thus, obsevations must be geneated fo the pogam to ead. The data geneation pogam uses the 6DoF simulation to povide the motion of the cylinde fo a given time peiod. Evey 5 seconds, a ange measuement is taken fom the neaest ada station on Eath. This only occus if the cylinde is in the line of sight to the ada station, which is esticted to 60 downwad fom zenith in any diection. The location of these ada stations ae geneated andomly and stoed befoe the pocess begins. The 0 ada stations modeled ae compaable in numbe to the stations in the space suveillance netwok used by the U.S. govenment []. The aveage effective aea is also ecoded duing this pocess fo late compaison with the estimated ballistic coefficient. This aea is simply multiplied by the atio of the dag coefficient to the cylinde mass to yield the aveage ballistic coefficient. The obit used fo data geneation was nealy cicula with an initial altitude of 700 km. The obit was inclined 45 degees with all othe angula obital elements initialized at zeo degees. 5.. Powe-Density Matix and Pocess Noise One dawback that plagues the MEKF is a tendency to divege, o poduce incoect state updates, afte many obsevations. This occus because the pogam is based on a vaiable covaiance matix. Afte time has passed and many obsevations have been pocessed, this covaiance matix becomes small. The esult is the MEKF having moe confidence in the cuent state than the obsevations. To avoid this poblem, a diagonal matix the same size as the covaiance matix is added to the diffeential euation govening the covaiance matix's ate of change. 59

69 P & T AP + PA + Q 5. In E. 5., P is the covaiance matix, A is the state elation matix, and Q is the powedensity matix. This diagonal matix is known as the powe-density matix, Q, descibed by Vallado as the second moment (o covaiance) of the pocess noise [3].The pocess noise epesents un-modeled acceleations acting on the object. It is the fact that pocess noise is pesent in the model that allows the MEKF to update the ballistic coefficient. The ballistic coefficient does not change though popagation, but its value is estimated using the obseved ange to the ange calculated by the model used to popagate the six tanslational states. The addition of the powe-density matix to the covaiance's diffeential euation has a lage effect when the covaiance becomes small, which is when it is most necessay to take action. The effect is an incease in the elements of the covaiance matix. This pocess is epeated evey time the covaiance matix becomes small, esulting in non-divegence of the MEKF. The values that occupy the powe-density matix wee detemined by a binayencoded genetic algoithm made available at Aubun Univesity. The optimization pogam's goal is to minimize the oot-mean-suae eo of the diffeence between the ange measuements and the anges calculated in the filte. The design vaiables that the genetic algoithm vaied to meet this goal ae the 7 diagonal elements of the powedensity matix. The efeence obit, which was nealy cicula with an initial object altitude at 700 km, was popagated fo 50 days. The algoithm evaluated 00 membes fo 0 geneations. Theefoe, the values fo the powe-density matix found by the genetic algoithm ae the best found fo the 4000 cases evaluated. The last diagonal position, which is coupled to the ballistic coefficient, can be left at zeo. This allows the 60

70 estimated ballistic coefficient to convege to a value, and not vay evey time the covaiance matix was affected by the powe-density matix. These two options will be efeed to as a convegent ballistic coefficient option and a dynamic ballistic coefficient option. The diagonals of the powe-density matix detemined by the genetic algoithm ae shown in Table 5.. Table 5.: Diagonal Values of Powe-Density Matix (Q) Q Q Q 33 Q E E E E-09 Q 55 Q 66 Q E E E Obit Detemination Results The fist simulation used a synthetic ballistic coefficient that was not calculated by the 6DoF. This was done to show that the MEKF does an exemplay job of finding the mean of the ballistic coefficient. The obit used to geneate the obsevations was inclined at 45 degees. The powe-density option of a convegent ballistic coefficient was used. The output of the obit detemination pogam consists of two plots: the aveage and estimated ballistic coefficients vesus time and the measuement esiduals (O-C) vesus time. The O-C plot shows the diffeence between the ange measuement calculated by the MEKF and the ange measuement given by the obsevation at that time. If the filte is woking coectly, these values ae typically less than 0 metes and only influenced by the andom noise intoduced in the data geneation pogam. 6

71 Figue 40: MEKF Results fo 0 Days with Synthetic Ballistic Coefficient The synthetic ballistic coefficient consisted of a mean value of 0. m /kg plus a sine function with a peiod of 0 days and amplitude of 0.04 m /kg. Figue 40 shows that the MEKF conveged on the mean value of 0. m /kg and that the filte did not divege as shown by the O-C plot. The MEKF conveged to a nea pefect answe within days. A simila simulation was epeated twice with actual ballistic coefficient data. The simulations lasted one-half yea, with the only diffeence being the powe-density matix option. 6

72 Figue 4: MEKF Results fo One Half Yea with Convegent Ballistic Coefficient Figue 4: MEKF Results fo One Half Yea with Dynamic Ballistic Coefficient 63