**THE DYNAMICS OF CHAOS AND **

DISPOSAL OF SPACE DEBRIS

DISPOSAL OF SPACE DEBRIS

### State of the Art

Recent studies on space debris show the occurrence of a plethora of dynamical phenomena: the overlapping of resonances and the onset of chaos, the existence of a web-like structure of luni-solar resonances, the chaotic variation of the orbital elements, the occurrence of bifurcations of equilibria, the existence of libration regions which lead to excursions in the eccentricity, the chaotic transport in the phase space. These intricate dynamics exists, for examples, in the neighbourhood of GNSS constellations. Despite the fact that the MEO region will be populated by four complete constellations, namely GPS, GLONASS, Galileo and BeiDou, there are no internationally agreed mitigation guidelines, as for LEO and GEO. Thus, it is mandatory to understand the dynamics of these objects. Past approaches were based mainly on deriving single or double-averaged (over short period terms) forms of the equations of motion. Such derivations, dating back to the 60’s, have been refined over the years. These approaches are able to identify critical solutions depending on the particle’s major semi-axis, eccentricity and inclination, as well as the main secular frequencies of motion, as induced by the Earth’s oblateness, as well as luni-solar secular perturbations. However, an important drawback of these analytical approaches is their failure to characterise the chaotic nature of the orbital dynamics close to particular resonances. The study of the chaotic dynamics, as well as its consequences for the long term diffusion of populations of space debris, becomes particularly important for highly inclined orbits, where, luni-solar resonances lead, in particular, to domains of extended resonance overlap. A complete investigation for every Earth orbital region is still incomplete. Furthermore, the use of perturbation theory will allow the study of the separation of nearby orbits in the fully non-linear regime.

### Objectives

WP3 covers all activities related to dynamic modelling, resonances and long term evolution of space objects in different orbital regimes and how this can be exploited to dispose of space debris. It includes the coupling of natural dynamics with passive and/or active disposal strategies.

### Major Results

We have developed a mathematical model to study the orbital and rotational dynamics of a wide variety of objects: from satellites and space debris around the Earth to asteroids. Using numerical and analytical techniques, we have obtained several results on the resonant dynamics of the model as we vary eccentricity and semi-major axis (different orbital element regions). We have shown interesting properties for large eccentricities and included dissipative effects that tend to stabilize the long-term evolution of the system.

On the other hand, we have produced an integrator that allows to compute long-term evolution of orbits around the Earth that are affected by lunisolar perturbations. This has been done averaging the mean anomaly of the orbit (single average approach) so we obtain an expansion of the equations up to any order in the eccentricity. We are using this integrator to analyse the effect of lunisolar perturbations on highly eccentric orbits (HEO) and highly inclined orbits.

We also developed a program to simulate various break-up events, with the aim to study the long term evolution of different objects with the same precursor. Besides, by using a (computer assisted) normalization procedure, we have successfully studied the evolution of space debris’ families in order to identify the initial parent body. We have analysed the different behaviour of space objects with close initial conditions, due to the lunisolar resonances.

The Academy of Athens (research group by the ESR 5 Edoardo Legnaro, and profs. M. Harsoula and Christos Efthymiopoulos, as well as 3-month secondment of ESR 4 Tudor Vartolomei) has worked in the characterization of the regular and chaotic dynamics, as well as the theoretical modelling and exploration of disposal strategies near lunisolar resonances for space debris at MEO. The work included:

- Production of a complete cartography, in terms of the chaotic indicator FLI (Fast Lyapunov Indicator), of the MEO region (semi-major axis from 20000 to 32000 km), identification of all major lunisolar resonances as well as their domains of resonance overlap.
- Computation of theoretical models (secular Hamiltonian and secular equations of motions) including the effects of the J2-terms, lunar and solar tide.
- Quantitative characterization of the timescales and initial conditions for the fast disposal of space debris through the stable and unstable invariant manifolds of the normally hyperbolic central manifolds at the inclination-dependent resonances.
- Construction of a computer-algebraic symbolic code for the computation of high order resonant normal forms around each inclination-dependent resonance.
- Preparation of two publications in peer-reviewed journals.

The work of ESR6 focused on the analysis of the dynamics of a space debris of the Earth with moderate to high area-to-mass ratio. The model problem takes into account the two-body classical Keplerian problem, the perturbation due to the non-sphericity of the Earth, attraction of the Sun and Moon and the Solar Radiation Pressure (SRP) effect. The geopotential is expressed as a truncated spherical harmonics expansion, as proposed in [1]. Series expansions for the lunisolar perturbations are considered as in [2]. In a similar way, a series expansion for the SRP is obtained and the effect of particular resonances which occur whenever there is a commensurability between the rates of change of some angular quantities is analyzed. When the debris is located close to a resonance, the effect of a particular term is *enhanced* and it accumulates over time, resulting in some peculiar behaviour.

The resulting problem is expressed in both Cartesian and Hamiltonian formulations. Expressing the model in classical Cartesian coordinates is implemented more easily on a calculator, but propagating an orbit is more computationally expensive. On the other hand the Hamiltonian formulation is way more efficient, but requires some additional theoretical studies. We provide the location of the equlibria by drawing the phase portrait of a *toy model *consisting of an Hamiltonian including the secular effect of the geopotential and a dominant SRP *resonant term. *To this end, a dimensional reduction is performed through a change of variables, reducing the starting 3-degree-of-freedom Hamiltonian to a 1-degree-of-freedom one, albeit depending parametrically on some constant values.

Finally, a series of tests was performed using the *toy model. *Such tests show that the main effect of SRP is a periodical variation of the eccentricity, which could led the debris to a collision with the Earth. Such tests are validated numerically by propagating an orbit and its tangent map in the Cartesian framework, in order to compute Fast Lyapunov Indicators (FLIs), a powerful chaos indicator which allows to discern between chaotic and regular motion.

Our study confirms the results obtained by authors such as [3] and [4] using different techniques, and presents a rather complex behaviour, with phase spaces which cannot be reduced to ones similar to a simple pendulum.

The results open new interesting scenarios in the study of resonances in the space debris problem. The techniques used by ESR6 can be applied to gain further understanding of phenomena such as the overlapping of resonances, both secular and semi-secular, and bifurcations due the variation of certain constants of motions. The output of this proposed research will be of fundamental importance for the contribution of ESR6 to WP9 and for designing disposal scenarios.

__References__

*Theory of Satellite Geodesy: Applications of Satellites to Geodesy*. Wal-tam, Massachusetts: Blaisdell Publishing Company, 1966[2] A. Celletti, C. Efthymiopoulos, F. Gachet, C. Galeş, and G. Pucacco,

*Dynamical models and the onset of chaos in space debris,*International Journal of Non-Linear Mechanics, vol. 90, pp. 147–163, 2017. [Online]. Available: https://www.sciencedirect.com/science/article/abs/pii/S0020746216301172[3] S. Valk, A. Lemaitre, and L. Anselmo,

*Analytical and semi-analytical investigations of geosynchronous space debris with high area-to-mass ratios,*Advances in Space Research, vol. 41, no. 7, pp. 1077–1090, 2008. [Online]. Available: https://www.sciencedirect.com/science/article/abs/pii/S0273117707010435[4] E. Alessi, C. Colombo, and A. Rossi,

*Phase space description of the dynamics due to the coupled effect of the planetary oblateness and the solar radiation pressure perturbations*, Celestial Mechanics and Dynamical Astronomy, vol. 131, no. 43, 2019. [Online]. Available: https://link.springer.com/article/10.1007/s10569-019-9919-z

We have created a wikipage that contains theoretical aspects and examples of our work on WP3.

### Research Output

- Margheri, A. and Misquero, M. (2020),
*A dissipative Kepler problem with a family of singular drags.*Mech. Dyn. Astr. 132, 17, 29 March 2020, https://doi.org/10.1007/s10569-020-9956-7 - Ortega, R. and Misquero, M. (2020),
*Some Rigorous Results on the 1:1 resonance of the spin-orbit Problem*, SIAM J. Appl. Dyn. Syst., In press, Accepted on 2 July 2020, Preprint available at https://www.ugr.es/~ecuadif/files/MisqueroOrtega.pdf - Misquero, M. (2020
*), The spin-spin model and the capture into the double synchronous resonance*, Nonlinearity, In press, Accepted on 26 October 2020, Preprint available at https://arxiv.org/abs/2010.09354 - Celletti, A., Pucacco, G., & Vartolomei, T. (2020).
*Proper elements for space debris,*Submitted at Celestial Mechanics and Dynamical Astronomy. - Efthymiopoulos, C., Legnaro E., Daquin, J., and Gkolias, I.: A deep-dive into the 2g+h resonance (preprint).
- Legnaro, E., and Efthymiopoulos, C.: Analytical theory for Inclination dependent Lunisolar Resonances (preprint).
- Paoli, R. (2020)
*The dynamics around extended bodies: tools and techniques.*Presented at the Stardust-R Global Virtual Workshop I (GVW-I), Online. 7-10 Sep. 2020, https://arnold.dm.unipi.it/wordpress/index.php/timetable/event/roberto-paoli/. - Peñarroya, P., Vyas, S., Paoli, R., & Kajak, K. M. (2020)
*Survey of Landing Methods on Small Bodies : Benefits of Robotics Manipulators to the Field.*Poster presented at the International Symposium on Artificial Intelligence, Robotics and Automation in Space (i-SAIRAS), Online. 19-23 Oct. 2020, DOI: 5281/zenodo.4275978. - Paoli, R. (2020)
*The dynamics around the Earth and other extended bodies*. Presented at The Conferences of the Doctoral Schools of the Romanian University Consortium, Online. 22-23 Oct. 2020, https://profs.info.uaic.ro/~CSDCU_MIF2020/index.php/matematica/. - Paoli, R. & Gales, C. (2021)
*Semi-analytic theory for Solar Radiation Pressure semi-secular resonances.*(In preparation). - Paoli, R. & Gales, C. (2021)
*Spherical Harmonic coefficients for some constant density polyhedra and their gravitational influence.*(In preparation).