On the Advantages of Using a Strict Hierarchy to Model Astrodynamical Problems

In this paper an algorithm is developed that combines the capabilities and advantages of several different astrodynamical models of increasing complexity. Splitting these models in a strict hierarchical order yields a clearer grasp on what is available. With the effort of developing a comprehensive model overhead, the equations for the spacecraft motion in simpler models can be readily obtained as particular cases. The proposed algorithm embeds the circular and elliptic restricted three-body problems, the four-body bicircular and concentric models, an averaged n-body model, and, at the top hierarchic ladder, the full ephemeris spice-based restricted n-body problem. The equations of motion are reduced to the assignment of 13 time-varying coefficients, which multiply the states and the gravitational potential to reproduce the proper vector field. This approach is powerful because it allows, for instance, an efficient and quick way to check solutions for different dynamics and parameters. It is shown how a gradual increase of the dynamics complexity greatly improves accuracy, the chances of success and the convergence rate of a continuation algorithm, applied to low-energy transfers.

[1]  Gerard Gómez,et al.  Solar system models with a selected set of frequencies , 2002 .

[2]  F. Topputo.  On optimal two-impulse Earth–Moon transfers in a four-body model , 2013 .

[3]  G. Muntané,et al.  A note on the dynamics around the L1,2 Lagrange points of the Earth-Moon system in a complete solar system model , 2012 .

[4]  V. Szebehely.  Theory of Orbits: The Restricted Problem of Three Bodies , 1968 .

[5]  J. Masdemont,et al.  A note on the dynamics around the Lagrange collinear points of the Earth–Moon system in a complete Solar System model , 2013 .

[6]  Stefano Campagnola,et al.  Low-Thrust Approach and Gravitational Capture at Mercury , 2004 .

[7]  Francesco Topputo,et al.  Trajectory refinement of three-body orbits in the real solar system model , 2017 .

[8]  Josep J. Masdemont,et al.  Dynamics in the center manifold of the collinear points of the restricted three body problem , 1999 .

[9]  F. Topputo,et al.  Method to Design Ballistic Capture in the Elliptic Restricted Three-Body Problem , 2010 .

[10]  J. Masdemont,et al.  Libration Point Orbits and Applications Libration Point Orbits: a Survey from the Dynamical Point of View I. Dynamics and Phase Space around the Libration Points , 2022 .

[11]  F. Topputo,et al.  Analysis of ballistic capture in Sun–planet models , 2015 .

[12]  D. Richardson,et al.  Analytic construction of periodic orbits about the collinear points , 1980 .

[13]  John T. Betts,et al.  Practical Methods for Optimal Control and Estimation Using Nonlinear Programming , 2009 .

[14]  J. K. Miller,et al.  Sun-Perturbed Earth-to-Moon Transfers with Ballistic Capture , 1993 .

[15]  Bruce Conway.  Spacecraft Trajectory Optimization: Preface , 2010 .

[16]  G. Gómez,et al.  The dynamics around the collinear equilibrium points of the RTBP , 2001 .

[17]  P. Kuchynka,et al.  The Planetary and Lunar Ephemerides DE430 and DE431 , 2014 .

[18]  X. Hou,et al.  On quasi-periodic motions around the collinear libration points in the real Earth–Moon system , 2011 .

[19]  Gerard Gómez,et al.  DYNAMICAL SUBSTITUTES OF THE LIBRATION POINTS FOR SIMPLIFIED SOLAR SYSTEM MODELS , 2003 .

[20]  Maria Aymerich Andreu.  The Quasi-Bicircular Problem , 1998 .