Nonlinear Dynamics

, Volume 90, Issue 4, pp 2545–2556 | Cite as

Exact solutions and adiabatic invariants for equations of satellite attitude motion under Coulomb torque

  • Vladimir S. Aslanov
Original Paper


This paper focuses on the attitude dynamics of a defunct axisymmetric satellite under the action of Coulomb forces to enable active space debris removal. Touchless Coulomb interaction occurs between an active spacecraft and the passive satellite at a fixed separation distance. The recently developed multi-sphere method of Schaub and Stevenson allows providing a simplified electrostatic force and torque model between non-spherical space objects. The existence of torques between charged bodies makes it necessary to study the attitude motion of the passive satellite for ensuring the safety of the space debris removal. The goal is first to deduce the equations of motion in the canonical form which is suitable for analytical analysis and then to construct a phase portrait, and to obtain exact solutions using Jacobi elliptic functions. Finally, for the disturbed motion of the system of two bodies, when the distance between the active spacecraft and the defunct satellite (or) and charge voltage changes slowly over time, adiabatic invariants are found in terms of the complete elliptic integrals. In this case, the adiabatic invariants are approximately first integrals of the disturbed system and they remain approximately constant for long time intervals during which the parameters change considerably. For a plane motion, the adiabatic invariants used to obtain an analytical solution for envelope of a deflection angle of the defunct satellite. This work extends the theory to the three-dimensional tumbling motion of a satellite on an orbit. The obtained results can be applied to study an opportunity of the space debris removal by the Coulomb interaction with the active spacecraft as a pusher.


Electrostatic toque Satellite Adiabatic invariants Exact solutions Space debris 



Multi-Sphere Method



This study was supported by the Russian Foundation for Basic Research (RFBR15-01-01456-A).


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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Samara National Research UniversitySamaraRussia

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