Computational and Applied Mathematics

, Volume 37, Supplement 1, pp 1–6 | Cite as

Celestial mechanics, spacecrafts, and 50th years of the first humans on the Moon

  • Vivian Martins Gomes
  • Cristiano Fiorilo de Mello
  • Elbert E. N. MacauEmail author
  • Antonio Fernando Bertachini de Almeida Prado
  • Othon Cabo Winter


Aerospace engineering is a relatively new topic in engineering. It deals with several aspects of activities related to space. It includes Astrodynamics, which is a field that studies the motion of spacecraft, like guidance and control, which studies different forms to guide the motion of a spacecraft, etc. All those fields started from the Celestial Mechanics, one of the first topics studied in Astronomy. The first studies had the goal to record and explain the motions of the stars, with special attention given to the motion of some “irregular” stars, which showed later to be planets. Considering the advances in the technology, those earlier studies generated the space activities that are well known nowadays. Different topics, like orbital and attitude maneuvers and determination of spacecrafts, mission design, etc. are covered. The present Focus Issue publishes several papers related to aerospace engineering in general and can be useful for further studies and planning of space missions.


Celestial mechanics Astrodynamics Orbital dynamics 

Mathematics Subject Classification



  1. Belbruno EA, Miller JK (1993) Sun-perturbed Earth-to-Moon transfers with ballistic capture. J Guid Control Dyn 16(4):770–775CrossRefGoogle Scholar
  2. Brophy JR, Noca M (1998) Eletric propulsion for solar system exploration. J Propuls Power 14:700–707CrossRefGoogle Scholar
  3. Camargo BCB, Winter OC, Foryta DW (2018) Comp Appl Math. 10:10. CrossRefGoogle Scholar
  4. Carvalho JPS, Vilhena de Moraes R, Prado AFBA (2010) Some orbital characteristics of lunar artificial satellites. Celest Mech Dyn Astron 108(4):371–388MathSciNetCrossRefGoogle Scholar
  5. CasalinoL ColasurdoG (2002) Missions to asteroids using solar eletric propusion. Acta Astronaut 50(11):705–711CrossRefGoogle Scholar
  6. D’Amario LA, Byrnes DV, Stanford RH (1982) Interplanetary trajectory optimization with application to Galileo. J Guid Control Dyn 5(5):465–471CrossRefGoogle Scholar
  7. de Almeida AK, de Almeida Prado AFB, de Moraes RV et al (2017) Comp Appl Math. CrossRefGoogle Scholar
  8. DeMelo CF, Macau EEN, Winter OC (2009) Strategy for plane change of Earth orbits using lunar gravity and trajectories derived of family G. Celest Mech Dyn Astron 103:281–299CrossRefGoogle Scholar
  9. Domingos RC, Winter OC, Yokoyama T (2006) Stable satellites around extrasolar giant planets. Mon Not R Astron Soc (Print) 373:1227–1234CrossRefGoogle Scholar
  10. Dunham D, Davis S (1985) Optimization of a multiple Lunar–Swingby trajectory sequence. J Astronaut Sci 33(3):275–288Google Scholar
  11. Eckel KG (1963) Optimum transfer in a central force field with n impulses. Astronaut Acta 9(5/6):302–324Google Scholar
  12. Farquhar R, Muhonen D, Church LC (1985) Trajectories and orbital maneuvers for the ISEE-3/ICE comet mission. J Astronaut Sci 33(3):235–254Google Scholar
  13. Farquhard RW, Dunham DW (1981) A new trajectory concept for exploring the earth’s geomagnetic tail. J Guid Control Dyn 4(2):192–196CrossRefGoogle Scholar
  14. Flandro G (1966) Fast reconnaissance missions to the outer solar system utilizing energy derived from the gravitational field of Jupiter. Astronaut Acta 12:4Google Scholar
  15. Gagg Filho LA, Fernandes SS (2017) Comp Appl Math. CrossRefGoogle Scholar
  16. Garcia RV, Silva WR, Pardal PCPM et al (2017) Comp Appl Math. CrossRefGoogle Scholar
  17. Garcia RV, Kuga HK, Silva WR et al (2018) Comp Appl Math. CrossRefGoogle Scholar
  18. Gomes VM, Domingos RC (2015) Studying the lifetime of orbits around Moons in elliptic motion. Mat Aplicada e Comput 2015:1–9zbMATHGoogle Scholar
  19. Gomes VM, Prado AFBA (2008) Swing-by maneuvers for a cloud of particles with planets of the solar system. WSEAS Trans Appl Theor Mech 03:869–878Google Scholar
  20. Gomes VM, Formiga JKS, Moraes RV (2013) Studying close approaches for a cloud of particles considering atmospheric drag. Math Probl Eng (Print) 2013:1–10Google Scholar
  21. Gross LR, Prussing JE (1974) Optimal multiple-impulse direct ascent fixed-time rendezvous. AIAA J 12(7):885–889CrossRefGoogle Scholar
  22. Hoelker RF, Silber R (1959) The bi-elliptic transfer between circular co-planar orbits. Tech Memo, Army Ballistic Missile Agency. Redstone Arsenal, Alabama, pp 2–59Google Scholar
  23. Hohmann W (1925) Die Erreichbarkeit der Himmelskörper. Verlag Oldenbourg in München. ISBN: 3-486- 23106-5Google Scholar
  24. Kluever CA, Pierson BL (1994) Optimal low-thrust three-dimensional earth-moon trajectories. J Guid Control Dyn 18(4):830–837CrossRefGoogle Scholar
  25. Macau EEN (2000) Using chaos to guide a spacecraft to the moon. Acta Astronaut Inglaterra 47(12):871–878CrossRefGoogle Scholar
  26. Macau EEN, Grebogi C (2006) Control of chaos and its relevancy to spacecraft steering. Philos Trans R Soc Math Phys Eng Sci Grã Bretanha 364:2463–2481MathSciNetCrossRefGoogle Scholar
  27. Machuy AL, Prado A, Stuchi TJ (2007) Numerical study of the time required for the gravitational capture in the bi-circular four-body problem. Adv Space Res 40(1):118–124. CrossRefGoogle Scholar
  28. Namouni F, Morais H (2017) Comp Appl Math. CrossRefGoogle Scholar
  29. Neto EV, Prado AFBA (1998) Time-of-flight analyses for the gravitational capture maneuver. J Guid Control Dyn 21(1):122–126. CrossRefzbMATHGoogle Scholar
  30. Prado AFBAA (2007) comparison of the “patched-conics” approach and the restricted problem for swing-bys. Adv Space Res 40:113–117CrossRefGoogle Scholar
  31. Prussing JE (1969) Optimal four-impulse fixed-time rendezvous in the vicinity of a circular orbit. AIAA J 7(5):928–935CrossRefGoogle Scholar
  32. Prussing JE, Chiu JH (1986) Optimal multiple-impulse time-fixed rendezvous between circular orbits. J Guid Control Dyn 9(1):17–22CrossRefGoogle Scholar
  33. Quang AD, Quang Long D (2018) Comp Appl Math. CrossRefGoogle Scholar
  34. Rodler F, Lopez-Morales M, Ribas I (2012) Weighing the non-transiting hot Jupiter Tau BOOb. Astrophys J Lett. CrossRefGoogle Scholar
  35. Roth HL (1967) Minimization of the velocity increment for a bi-elliptic transfer with plane change. Astronaut Acta 13(2):119–130zbMATHGoogle Scholar
  36. Salazar FJT, De Melo CF, Macau EEN, Winter OC (2012) Three-body problem, its Lagrangian points and how to exploit them using an alternative transfer to L4 and L5. Celest Mech Dyn Astron 114:201–213MathSciNetCrossRefGoogle Scholar
  37. Salazar FJT, Masdemont JJ, Gómez G, Macau EE, Winter OC (2014) Zero, minimum and maximum relative radial acceleration for planar formation flight dynamics near triangular libration points in the Earth-Moon system. Adv Space Res 54:1838–1857CrossRefGoogle Scholar
  38. Salazar FJT, Macau EEN, Winter OC (2015a) Chaotic dynamics in a low-energy transfer strategy to the equilateral equilibrium points in the Earth-Moon system. Int J Bifurcat Chaos Appl Sci Eng 25:1550077MathSciNetCrossRefGoogle Scholar
  39. Salazar FJT, Winter OC, Macau EE, Masdemont JJ, Gómez G (2015b) Natural formations at the Earth–Moon triangular point in perturbed restricted problems. Adv Space Res 56:144–162CrossRefGoogle Scholar
  40. Salazar FJT, Winter OC, McInnes CR (2017) Comp Appl Math. CrossRefGoogle Scholar
  41. Sanchez DM, Prado AFBA, Yokoyama T (2014) On the effects of each term of the geopotential perturbation along the time I: quasi-circular orbits. Adv Space Res 54:1008–1018CrossRefGoogle Scholar
  42. Shternfeld A (1959) Soviet space science. Basic Books Inc, New York, pp 109–111Google Scholar
  43. Sukhanov AA, Prado AFBA (2001) Constant tangential low-thrust trajectories near an oblate planet. J Guid Control Dyn 24(4):723–731. CrossRefGoogle Scholar
  44. Ting L (1960) Optimum orbital transfer by several impulses. Astronaut Acta 6(5):256–265zbMATHGoogle Scholar
  45. Wolszczan A, Frail DA (1992) A planetary system around the millisecond pulsar PSR1257 + 12. Nature 355(6356):145–147CrossRefGoogle Scholar
  46. Zanardi MC, Celestino CC, Borderes Motta G et al (2018) Comp Appl Math.

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Authors and Affiliations

  • Vivian Martins Gomes
    • 1
  • Cristiano Fiorilo de Mello
    • 2
  • Elbert E. N. Macau
    • 3
    • 4
    Email author
  • Antonio Fernando Bertachini de Almeida Prado
    • 3
  • Othon Cabo Winter
    • 1
  1. 1.Unesp, Univ. Estadual PaulistaGuaratinguetáBrazil
  2. 2.Universidade Federal de Minas Gerais, UFMGBelo HorizonteBrazil
  3. 3.Instituto Nacional de Pesquisas Espaciais, INPESão José dos CamposBrazil
  4. 4.Federal University of Sao Paulo, UNIFESPSão José dos CamposBrazil

Personalised recommendations