Characterization of Low-Energy Quasiperiodic Orbits in the Elliptic Restricted 4-Body Problem with Orbital Resonance
Abstract
:1. Introduction
2. The Elliptic Restricted 4-Body Problem
2.1. Dynamical Model
- The center of mass of the system is indicated as O;
- The relative motion of with respect to describes a Keplerian orbit onto the plane with semimajor axis a and eccentricity e;
- The relative motion of with respect to describes a Keplerian orbit onto the plane with semimajor axis and eccentricity ;
- The orbital plane is tilted of an angle with respect to ;
2.2. Hamiltonian Formalism
3. Classification of Low-Energy Trajectories in the Elliptic Restricted 4-Body Problem
3.1. Persistence of the Topological Properties
3.2. Normal Forms for the Elliptic Restricted 4-Body Problem
3.3. Topological Characterization of Low-Energy Trajectories
- Lissajous quasiperiodic orbits, characterized by , which evolve inside the equilibrium region;
- Transit trajectories, corresponding to the hyperbolic segments , which cross the equilibrium region twice, once towards and once towards , in a finite interval of time;
- Bouncing trajectories, corresponding to the hyperbolic segments , which never cross the equilibrium region;
- Long-term ballistic captures, characterized by either or , which cross the equilibrium region twice, over an indefinitely long interval of time.
4. Resonant Terms and Quasiperiodic Solutions
4.1. Identification of Resonant Terms
4.2. Determination of Stationary Points
5. Numerical Analysis
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A. Terms of the Power Series Expansion
Appendix B. Canonical Transformations
Appendix B.1. Second Order
Appendix B.2. Third Order
References
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Variable | Symbol | Value |
---|---|---|
Mass of Jupiter | 1.899 × 10 kg | |
Mass of Europa | 4.799 × 10 kg | |
Mass of Io | 8.932 × 10 kg | |
Jupiter–Europa semimajor axis | a | 6.711 × 10 km |
Jupiter–Io semimajor axis | 4.218 × 10 km | |
Jupiter–Europa eccentricity | e | 0.0094 |
Jupiter–Io eccentricity | 0.0041 | |
Inclination between the orbital planes | 0.430 deg | |
Coordinate of the libration point | 6.081 × 10 km | |
Resonant Hamiltonian terms | 10 |
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Carletta, S.; Pontani, M.; Teofilatto, P. Characterization of Low-Energy Quasiperiodic Orbits in the Elliptic Restricted 4-Body Problem with Orbital Resonance. Aerospace 2022, 9, 175. https://doi.org/10.3390/aerospace9040175
Carletta S, Pontani M, Teofilatto P. Characterization of Low-Energy Quasiperiodic Orbits in the Elliptic Restricted 4-Body Problem with Orbital Resonance. Aerospace. 2022; 9(4):175. https://doi.org/10.3390/aerospace9040175
Chicago/Turabian StyleCarletta, Stefano, Mauro Pontani, and Paolo Teofilatto. 2022. "Characterization of Low-Energy Quasiperiodic Orbits in the Elliptic Restricted 4-Body Problem with Orbital Resonance" Aerospace 9, no. 4: 175. https://doi.org/10.3390/aerospace9040175
APA StyleCarletta, S., Pontani, M., & Teofilatto, P. (2022). Characterization of Low-Energy Quasiperiodic Orbits in the Elliptic Restricted 4-Body Problem with Orbital Resonance. Aerospace, 9(4), 175. https://doi.org/10.3390/aerospace9040175