### Abstract

triangle configuration, eight natural (i.e. with zero thrust) equilibrium points are identified, four of which are close to the asteroid, two stable and two unstable, when considering as primaries the Sun and any other two bodies of the Solar System. Artificial equilibria, which can be seen as low thrust perturbations of the natural ones, are then taken into account with the aim of identifying their linearly stable subset. The Lyapunov stability of these marginally stable points is then analysed using basic KAM (Kolmogorov Arnold Moser) theory and Arnold’s stability theorem. In order to apply such theorem an iterative procedure to reduce the Hamiltonian into Birkhoff’s Normal Form is applied up to fourth order, explicitly defining, at each step, the generating function of a symplectic transformation. Despite the complexity of this process, Normal Forms are a

fundamental, necessary step for any application of KAM theory; such theory, transforming a non-integrable system into a sum of perturbed integrable ones, enables the computation of a high order analytical approximation of the system dynamics, plus an estimation of the discrepancy from the initial model. As an application of KAM theory, a proof of the nonlinear stability for the low thrust generated equilibrium points under non resonant conditions is found using Arnold’s stability theorem. Results show that Lyapunov stability is guaranteed along the linearly stable domain with the exception of a set of points with zero measure where the conditions to apply Arnold‘s theorem are not satisfied.

Original language | English |
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Number of pages | 8 |

Publication status | Published - 3 Oct 2011 |

Event | 62nd International Astronautical Congress 2011 - Cape Town, South Africa Duration: 3 Oct 2011 → 7 Oct 2011 |

### Conference

Conference | 62nd International Astronautical Congress 2011 |
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Country | South Africa |

City | Cape Town |

Period | 3/10/11 → 7/10/11 |

### Fingerprint

### Keywords

- four body problem
- low-thrust propulsion
- L4
- nonlinear stability
- Arnold's Theorem

### Cite this

*Nonlinearly stable equilibria in the Sun-Jupiter-Trojan-Spacecraft four body problem*. Paper presented at 62nd International Astronautical Congress 2011, Cape Town, South Africa.

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**Nonlinearly stable equilibria in the Sun-Jupiter-Trojan-Spacecraft four body problem.** / Ceccaroni, Marta; Biggs, James.

Research output: Contribution to conference › Paper

TY - CONF

T1 - Nonlinearly stable equilibria in the Sun-Jupiter-Trojan-Spacecraft four body problem

AU - Ceccaroni, Marta

AU - Biggs, James

PY - 2011/10/3

Y1 - 2011/10/3

N2 - The Trojan asteroids have been highlighted as a main target for future discovery missions, which will enable key questions about the formation of our Solar system to be answered. Programs like the Japanese Jupiter and Trojan Asteroids Exploration Programme are already testing technology demonstrators like the IKAROS spacecraft to enable future interplanetary missions to Jupiter and the Trojans. In this paper an analytic analysis of the stability of the Low thrust Sun Jupiter Asteroid Spacecraft system, is presented, from a Hamiltonian point of view. Setting the three primaries in the stable Lagrangian equilateraltriangle configuration, eight natural (i.e. with zero thrust) equilibrium points are identified, four of which are close to the asteroid, two stable and two unstable, when considering as primaries the Sun and any other two bodies of the Solar System. Artificial equilibria, which can be seen as low thrust perturbations of the natural ones, are then taken into account with the aim of identifying their linearly stable subset. The Lyapunov stability of these marginally stable points is then analysed using basic KAM (Kolmogorov Arnold Moser) theory and Arnold’s stability theorem. In order to apply such theorem an iterative procedure to reduce the Hamiltonian into Birkhoff’s Normal Form is applied up to fourth order, explicitly defining, at each step, the generating function of a symplectic transformation. Despite the complexity of this process, Normal Forms are afundamental, necessary step for any application of KAM theory; such theory, transforming a non-integrable system into a sum of perturbed integrable ones, enables the computation of a high order analytical approximation of the system dynamics, plus an estimation of the discrepancy from the initial model. As an application of KAM theory, a proof of the nonlinear stability for the low thrust generated equilibrium points under non resonant conditions is found using Arnold’s stability theorem. Results show that Lyapunov stability is guaranteed along the linearly stable domain with the exception of a set of points with zero measure where the conditions to apply Arnold‘s theorem are not satisfied.

AB - The Trojan asteroids have been highlighted as a main target for future discovery missions, which will enable key questions about the formation of our Solar system to be answered. Programs like the Japanese Jupiter and Trojan Asteroids Exploration Programme are already testing technology demonstrators like the IKAROS spacecraft to enable future interplanetary missions to Jupiter and the Trojans. In this paper an analytic analysis of the stability of the Low thrust Sun Jupiter Asteroid Spacecraft system, is presented, from a Hamiltonian point of view. Setting the three primaries in the stable Lagrangian equilateraltriangle configuration, eight natural (i.e. with zero thrust) equilibrium points are identified, four of which are close to the asteroid, two stable and two unstable, when considering as primaries the Sun and any other two bodies of the Solar System. Artificial equilibria, which can be seen as low thrust perturbations of the natural ones, are then taken into account with the aim of identifying their linearly stable subset. The Lyapunov stability of these marginally stable points is then analysed using basic KAM (Kolmogorov Arnold Moser) theory and Arnold’s stability theorem. In order to apply such theorem an iterative procedure to reduce the Hamiltonian into Birkhoff’s Normal Form is applied up to fourth order, explicitly defining, at each step, the generating function of a symplectic transformation. Despite the complexity of this process, Normal Forms are afundamental, necessary step for any application of KAM theory; such theory, transforming a non-integrable system into a sum of perturbed integrable ones, enables the computation of a high order analytical approximation of the system dynamics, plus an estimation of the discrepancy from the initial model. As an application of KAM theory, a proof of the nonlinear stability for the low thrust generated equilibrium points under non resonant conditions is found using Arnold’s stability theorem. Results show that Lyapunov stability is guaranteed along the linearly stable domain with the exception of a set of points with zero measure where the conditions to apply Arnold‘s theorem are not satisfied.

KW - four body problem

KW - low-thrust propulsion

KW - L4

KW - nonlinear stability

KW - Arnold's Theorem

UR - http://iac2011.com/

M3 - Paper

ER -