Computational and Applied Mathematics

, Volume 37, Supplement 1, pp 133–143 | Cite as

Orbital effects in a cloud of space debris making a close approach with the earth

  • Jorge Kennety S. FormigaEmail author
  • Vivian Martins Gomes
  • Rodolpho Vilhena de Moraes


The present paper has the goal of studying the changes of the orbital parameters of each individual element of a cloud of particles that makes a close approach with the Earth. Clouds of particles are formed when natural or man-made bodies explode for some reason. After an explosion like that, the center of mass of the cloud follows the same orbit of the body that generated the explosion, but the individual particles have different trajectories. The cloud is specified by a distribution of semi-major axis and eccentricity of their particles. This cloud is assumed to pass close to the Earth, making a close approach that modifies the trajectory of every particle that belongs to the cloud. The present paper makes simulations based in the “Patched-Conics” model to obtain the new trajectories of each particle. Then, it is possible to map the new distribution of the Keplerian elements of the particles that constituted the cloud, using the previous distribution as initial conditions. These information are important when planning satellite missions having a spacecraft passing close to a cloud of this type, because it is possible to obtain values for the density and amplitude of the cloud, so finding the risks of collision and the possible maneuvers that need to be made in the spacecraft to avoid the collisions.


Orbital Maneuver Close approach Restricted three-body problem Space trajectories 

Mathematics Subject Classification

70F05 70F07 70F15 



The authors wish to express their appreciation for the support provided by Grants #2016/15675-1, 2011/13101-4, 2011/08171-3 from São Paulo Research Foundation (FAPESP); Grants # 473164/2013-2 and 309106/2015-0, 308823/2015-0 from the National Council for Scientific and Technological Development (CNPq) and Institute of Science and Technology, UNESP-São Paulo State University.


  1. Broucke RA (1988) The Celestial mechanics of gravity assist. In: AIAA/AAS Astrodynamics Conf. Minneapolis, vol 88Google Scholar
  2. Broucke RA, Prado AFBA (1993) Jupiter Swing-By Trajectories Passing Near the Earth. Space Flight Mech. 82:1159–1176Google Scholar
  3. Colombo C, McInnes CR (2011) Evolution of swarms of smart dust spacecraft. New Trends in Astrod and Appl VI. Courant Institute of Mathematical Sciences, New YorkGoogle Scholar
  4. D’Amario LA, Byrnes DV, Stanford RH (1982) Interplanetary trajectory optimization with application to Galileo. J Guid Control Dyn 5–5:465–471CrossRefGoogle Scholar
  5. Felipe G, Prado AFBA (2004) Trajectory selection for a spacecraft performing a two-dimensional swing-by. Adv Space Res 34(11):2256–2261CrossRefGoogle Scholar
  6. Formiga JKS, Prado AFBA (2011) Orbital characteristics due to the three dimensional swing-by in the Sun–Jupiter system. In: 10th International conference on computational intelligence man–machine systems and cybernetics. Jakarta, pp 61–69Google Scholar
  7. Formiga JKS, Santos DPS (2015) Orbital maneuvers to reach and explore a triple asteroid. Comp. Math. Appl. doi: 10.1007/s40314-016-0307-y
  8. Formiga JKS, Prado AFBA (2014) For Studying sequences of resonant orbits to perform successive close approaches with the Moon. J Braz Mech Sci Eng Soc. doi: 10.1007/s40430-014-0254-8
  9. Flandro G (1996) Fast reconnaissance missions to the outer solar system utilizing energy derived from the gravitational field of Jupiter. Astronaut Acta 12–4:329–337Google Scholar
  10. Hoots FR, Hansen BW (2014) Satellite breakup debris cloud characterization. In: 24th AAS/AIAA Space Flight Mechanics Meeting. AAS, Santa Fe, pp 14–329Google Scholar
  11. Gor’kavyi N, Ozernoy L, Mather J, Taidakova T (1997) Quasi-stationary states of dust flows under Poynting-Robertson drag: new analytical and numerical solutions. Astrophys J 1(488):268–276CrossRefGoogle Scholar
  12. Gomes VM, Prado AFBA (2008) Swing-By Maneuvers for a Cloud of Particles with Planets of the Solar System. WSEAS Trans Appl Theor Mech 3(11):869–878Google Scholar
  13. Gomes VM, Prado AFBA (2010) A study of the impact of the initial energy in a close approach of a cloud of particles. WSEAS Trans Math 9:811–820Google Scholar
  14. Gomes VM, Prado AFBA, Justyna G (2013) Dynamics of space particles and spacecrafts passing by theatmosphere of the Earth. Sci World J 2013:1–6CrossRefGoogle Scholar
  15. Kresák L (1954) On a criterion concerning the perturbing action of the Earth on meteor streams. Bull Astron Inst Czechoslov 5:45–49Google Scholar
  16. Krisko PH (2007) The predicted growth of the low-Earth orbit space debris environment—an assessment of future risk for spacecraft. Proc Inst Mech Eng Part G J Aer Eng 221(6):975–985. doi: 10.1243/09544100JAERO192 CrossRefGoogle Scholar
  17. Letizia F, Colombo C, Lewis HG, McInnes CR (2013) Debris cloud evolution in Low Earth Orbit. In: 64th International Astronautical Congress. International Astronautical Federation. IAC-13.A6.P.12Google Scholar
  18. Lewis H, Swinerd G, Williams N, Gittins G (2001) DAMAGE: a dedicated GEO debris model framework. Third European Conference on Space Debris. Darmstadt, pp 373–378Google Scholar
  19. Longman RW, Schneider AM (1970) Use of Jupiter’s moons for gravity assist. J Spacecr Rockets 7:570–576CrossRefGoogle Scholar
  20. Longuski JM, Williams SN (1991) The last grand tour opportunity to Pluto. J Astronaut Sci 39:359–365Google Scholar
  21. Lynam AE, Kloster KW, Longuski JM (2011) Multiple-satellite-aided capture trajectories at Jupiter using the Laplace resonance. Celest Mech Dyn Astron 109:59–84MathSciNetCrossRefGoogle Scholar
  22. Martin C, Walker R, Klinkrad H (2004) The sensitivity of the ESA DELTA model. Adv Space Res 34(5):969–974. doi: 10.1016/j.asr.2003.02.028 CrossRefGoogle Scholar
  23. McInnes CR (2000) Simple analytic model of the long term evolution of nanosatellite constellations. J Guid Cont Dyn 23(2):332–338CrossRefGoogle Scholar
  24. Liou JC (2011) An active debris removal parametric study for LEO environment remediation. Adv Space Res 47(11):1865–1876. doi: 10.1016/j.asr.2011.02.003 CrossRefGoogle Scholar
  25. Pardini C, Anselmo L (2013) Review of past on-orbit collisions among cataloged objects and examination of the catastrophic fragmentation concept. In: International Astronautical Congress. IAC-13.A6.2.9, BeijingGoogle Scholar
  26. Prado AFBA, Broucke R (1996) Transfer orbits in the Earth-moon system using a regularized model. J Guid Control Dyn 19(4):929–933CrossRefGoogle Scholar
  27. Prado AFBA, Broucke RA (1995) Classification of swing-by trajectories using the moon. Appl Mech Rev 48(11):S138–S142CrossRefGoogle Scholar
  28. Rossi A, Cordelli A, Pardini C (1995) Modelling the space debris evolution: two new computer codes. Adv. Astr. Sci. Space Flight Mech. AlbuqGoogle Scholar
  29. Sampaio JC, Wnuk E, Vilhena de Moraes R, Fernandes SS (2014) Resonant Orbital Dynamics in LEO Region: Space Debris in Focus. Math. Prob. Eng. 2014:1–12. doi: 10.1155/2014/929810 CrossRefGoogle Scholar
  30. Swenson BL (1991) Neptune atmospheric probe mission. In: AIAA7AAS astrodynamics conference, Hilton Head, pp 92–4371Google Scholar
  31. White AE, Lewis HG (2013) The many futures of active debris removal. Acta Astr 95:189–197. doi: 10.1016/j.actaastro.2013.11.009 CrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  • Jorge Kennety S. Formiga
    • 1
    Email author
  • Vivian Martins Gomes
    • 2
  • Rodolpho Vilhena de Moraes
    • 3
  1. 1.Instituto de Ciência e TecnologiaUNESP, Universidade Estadual PaulistaSão PauloBrazil
  2. 2.Grupo de Dinâmica Orbital e PlanetologiaUNESP, Universidade Estadual PaulistaSão PauloBrazil
  3. 3.Universidade Federal de São Paulo/ICTSão PauloBrazil

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