Computational and Applied Mathematics

, Volume 37, Supplement 1, pp 338–364 | Cite as

A study of Earth–Moon trajectories based on analytical expressions for the velocity increments

  • Luiz Arthur Gagg FilhoEmail author
  • Sandro da Silva Fernandes


A study of Earth–Moon bi-impulsive trajectories is presented in this paper. The motion of the space vehicle is described by the classic planar circular restricted three-body problem. The velocity increments are computed through analytical expressions, which are derived from the development of the Jacobi Integral expression. To determine the trajectories, a new two-point boundary value problem (TPBVP) with prescribed value of Jacobi Integral is formulated. Internal and external trajectories are determined through the solution of this new TPBVP for several times of flight. A relation between the Jacobi Integral and the Kepler’s energy at arrival is derived and several kinds of study are performed. Critical values of the Jacobi Integral, for which the Kepler’s energy of the space vehicle on the arrival trajectory becomes negative, are calculated for several configurations of arrival at the low Moon orbit in both directions: clockwise and counterclockwise. Results show that the proposed method allows the estimation of the fuel consumption before solving the TPBVP, and it facilitates the determination of trajectories with large time of flight. However, increasing values of the time of flight are not necessarily related with the increase of the Jacobi Integral value, which means that the obtaining of new trajectories becomes more difficult as the Jacobi Integral increases. Moreover, the proposed method provides results to be used as initial guess for more complex models and for optimization algorithms in order to minimize the total fuel consumption. For this case, this paper presents an example where an internal trajectory with large time of flight is optimized considering the Sun’s attraction.


Earth–Moon trajectories Jacobi Integral Ballistic capture Three-body problem Multiple revolution trajectories 

Mathematics Subject Classification

70F07 Three-body problems 70M20 Orbital Mechanics 



This research is supported by Grant 2012/25308-5, So Paulo Research Foundation (FAPESP), by Grant 2012/21023-6, So Paulo Research Foundation (FAPESP), and by CNPq under contract 304913/2013-8.


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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  1. 1.Departamento de MatemáticaInstituto Tecnológico de AeronáuticaSão José dos CamposBrazil

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