A deep dive into the 2 g+ h resonance: separatrices, manifolds and phase space structure of navigation satellites

Jérôme Daquin; Edoardo Legnaro; Ioannis Gkolias; Christos Efthymiopoulos

doi:10.1007/s10569-021-10060-6

A deep dive into the 2 g+ h resonance: separatrices, manifolds and phase space structure of navigation satellites

Jérôme Daquin, Edoardo Legnaro, Ioannis Gkolias, Christos Efthymiopoulos

Résultats de recherche: Contribution à un journal/une revue › Article › Revue par des pairs

Résumé

Despite extended past studies, several questions regarding the resonant structure of the medium-Earth orbit (MEO) region remain hitherto unanswered. This work describes in depth the effects of the 2 g+ h lunisolar resonance. In particular, (i) we compute the correct forms of the separatrices of the resonance in the inclination-eccentricity (i, e) space for fixed semi-major axis a. This allows to compute the change in the width of the 2 g+ h resonance as the altitude increases. (ii) We discuss the crucial role played by the value of the inclination of the Laplace plane, i_L. Since i_L is comparable to the resonance’s separatrix width, the parametrization of all resonance bifurcations has to be done in terms of the proper inclination i_p, instead of the mean one. (iii) The subset of circular orbits constitutes an invariant subspace embedded in the full phase space, the center manifold C, where actual navigation satellites lie. Using i_p as a label, we compute its range of values for which C becomes a normally hyperbolic invariant manifold (NHIM). The structure of invariant tori in C allows to explain the role of the initial phase h noticed in several works. (iv) Through Fast Lyapunov Indicator (FLI) cartography, we portray the stable and unstable manifolds of the NHIM as the altitude increases. Manifold oscillations dominate in phase space between a= 24,000 km and a= 30,000 km as a result of the sweeping of the 2 g+ h resonance by the [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext. resonances. The noticeable effects of the latter are explained as a consequence of the relative inclination of the Moon’s orbit with respect to the ecliptic. The role of the phases [InlineEquation not available: see fulltext.] in the structures observed in the FLI maps is also clarified. Finally, (v) we discuss how the understanding of the manifold dynamics could inspire end-of-life disposal strategies.

langue originale	Anglais
Numéro d'article	6
journal	Celestial Mechanics & Dynamical Astronomy
Volume	134
Numéro de publication	1
Les DOIs	https://doi.org/10.1007/s10569-021-10060-6
Etat de la publication	Publié - févr. 2022
Modification externe	Oui

Financement

J.D. is a postdoctoral researcher of the ?Fonds de la Recherche Scientifique??FNRS. J.D. acknowledges the support of the ERC project 677793 ?Stable and Chaotic Motions in the Planetary Problem? leaded by Prof.?Gabriella Pinzari. I.G. acknowledges the support of the ERC project 679086 COMPASS ?Control for Orbit Manoeuvring through Perturbations for Application to Space Systems.? E.?L. has been supported by the Marie Curie Initial Training Network Stardust-R, Grant Agreement Number 813644 under the H2020 research and innovation program. C.?E. was partially supported by EU-ITN Stardust-R and MIUR-PRIN 20178CJA2B ?New Frontiers of Celestial Mechanics: theory and Applications.? The authors acknowledge feedback and discussions with Elisa Maria Alessi. J.D. is a postdoctoral researcher of the \u2018Fonds de la Recherche Scientifique\u2019\u2014FNRS. J.D. acknowledges the support of the ERC project 677793 \u2018Stable and Chaotic Motions in the Planetary Problem\u2019 leaded by Prof. Gabriella Pinzari. I.G. acknowledges the support of the ERC project 679086 COMPASS \u2018Control for Orbit Manoeuvring through Perturbations for Application to Space Systems.\u2019 E. L. has been supported by the Marie Curie Initial Training Network Stardust-R, Grant Agreement Number 813644 under the H2020 research and innovation program. C. E. was partially supported by EU-ITN Stardust-R and MIUR-PRIN 20178CJA2B \u2018New Frontiers of Celestial Mechanics: theory and Applications.\u2019 The authors acknowledge feedback and discussions with Elisa Maria Alessi.

Bailleurs de fonds	Numéro du bailleur de fonds
Stable and Chaotic Motions in the Planetary Problem?
MIUR-PRIN
European Resuscitation Council
Fonds De La Recherche Scientifique - FNRS
Horizon 2020 Framework Programme	813644, 677793
H2020 research and innovation program	MIUR-PRIN 20178CJA2B
European Research Council	679086

Accès au document

10.1007/s10569-021-10060-6

Autres fichiers et liens

Lien vers la publication sur Scopus

Contient cette citation

@article{3c80c7b482204400b1a9dd36f5283029,

title = "A deep dive into the 2 g+ h resonance: separatrices, manifolds and phase space structure of navigation satellites",

abstract = "Despite extended past studies, several questions regarding the resonant structure of the medium-Earth orbit (MEO) region remain hitherto unanswered. This work describes in depth the effects of the 2 g+ h lunisolar resonance. In particular, (i) we compute the correct forms of the separatrices of the resonance in the inclination-eccentricity (i, e) space for fixed semi-major axis a. This allows to compute the change in the width of the 2 g+ h resonance as the altitude increases. (ii) We discuss the crucial role played by the value of the inclination of the Laplace plane, iL. Since iL is comparable to the resonance{\textquoteright}s separatrix width, the parametrization of all resonance bifurcations has to be done in terms of the proper inclination ip, instead of the mean one. (iii) The subset of circular orbits constitutes an invariant subspace embedded in the full phase space, the center manifold C, where actual navigation satellites lie. Using ip as a label, we compute its range of values for which C becomes a normally hyperbolic invariant manifold (NHIM). The structure of invariant tori in C allows to explain the role of the initial phase h noticed in several works. (iv) Through Fast Lyapunov Indicator (FLI) cartography, we portray the stable and unstable manifolds of the NHIM as the altitude increases. Manifold oscillations dominate in phase space between a= 24,000 km and a= 30,000 km as a result of the sweeping of the 2 g+ h resonance by the [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext. resonances. The noticeable effects of the latter are explained as a consequence of the relative inclination of the Moon{\textquoteright}s orbit with respect to the ecliptic. The role of the phases [InlineEquation not available: see fulltext.] in the structures observed in the FLI maps is also clarified. Finally, (v) we discuss how the understanding of the manifold dynamics could inspire end-of-life disposal strategies.",

keywords = "Chaotic dynamics, GNSS, Manifold dynamics, Medium- Earth orbit, Orbital resonances, Satellite orbits",

author = "J{\'e}r{\^o}me Daquin and Edoardo Legnaro and Ioannis Gkolias and Christos Efthymiopoulos",

note = "Funding Information: J.D. is a postdoctoral researcher of the {\textquoteleft}Fonds de la Recherche Scientifique{\textquoteright}—FNRS. J.D. acknowledges the support of the ERC project 677793 {\textquoteleft}Stable and Chaotic Motions in the Planetary Problem{\textquoteright} leaded by Prof. Gabriella Pinzari. I.G. acknowledges the support of the ERC project 679086 COMPASS {\textquoteleft}Control for Orbit Manoeuvring through Perturbations for Application to Space Systems.{\textquoteright} E. L. has been supported by the Marie Curie Initial Training Network Stardust-R, Grant Agreement Number 813644 under the H2020 research and innovation program. C. E. was partially supported by EU-ITN Stardust-R and MIUR-PRIN 20178CJA2B {\textquoteleft}New Frontiers of Celestial Mechanics: theory and Applications.{\textquoteright} The authors acknowledge feedback and discussions with Elisa Maria Alessi. Funding Information: J.D. is a postdoctoral researcher of the ?Fonds de la Recherche Scientifique??FNRS. J.D. acknowledges the support of the ERC project 677793 ?Stable and Chaotic Motions in the Planetary Problem? leaded by Prof.?Gabriella Pinzari. I.G. acknowledges the support of the ERC project 679086 COMPASS ?Control for Orbit Manoeuvring through Perturbations for Application to Space Systems.? E.?L. has been supported by the Marie Curie Initial Training Network Stardust-R, Grant Agreement Number 813644 under the H2020 research and innovation program. C.?E. was partially supported by EU-ITN Stardust-R and MIUR-PRIN 20178CJA2B ?New Frontiers of Celestial Mechanics: theory and Applications.? The authors acknowledge feedback and discussions with Elisa Maria Alessi. Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer Nature B.V.",

year = "2022",

month = feb,

doi = "10.1007/s10569-021-10060-6",

language = "English",

volume = "134",

journal = "Celestial Mechanics \& Dynamical Astronomy",

issn = "0923-2958",

publisher = "Springer Netherlands",

number = "1",

}

TY - JOUR

T1 - A deep dive into the 2 g+ h resonance

T2 - separatrices, manifolds and phase space structure of navigation satellites

AU - Daquin, Jérôme

AU - Legnaro, Edoardo

AU - Gkolias, Ioannis

AU - Efthymiopoulos, Christos

N1 - Funding Information: J.D. is a postdoctoral researcher of the ‘Fonds de la Recherche Scientifique’—FNRS. J.D. acknowledges the support of the ERC project 677793 ‘Stable and Chaotic Motions in the Planetary Problem’ leaded by Prof. Gabriella Pinzari. I.G. acknowledges the support of the ERC project 679086 COMPASS ‘Control for Orbit Manoeuvring through Perturbations for Application to Space Systems.’ E. L. has been supported by the Marie Curie Initial Training Network Stardust-R, Grant Agreement Number 813644 under the H2020 research and innovation program. C. E. was partially supported by EU-ITN Stardust-R and MIUR-PRIN 20178CJA2B ‘New Frontiers of Celestial Mechanics: theory and Applications.’ The authors acknowledge feedback and discussions with Elisa Maria Alessi. Funding Information: J.D. is a postdoctoral researcher of the ?Fonds de la Recherche Scientifique??FNRS. J.D. acknowledges the support of the ERC project 677793 ?Stable and Chaotic Motions in the Planetary Problem? leaded by Prof.?Gabriella Pinzari. I.G. acknowledges the support of the ERC project 679086 COMPASS ?Control for Orbit Manoeuvring through Perturbations for Application to Space Systems.? E.?L. has been supported by the Marie Curie Initial Training Network Stardust-R, Grant Agreement Number 813644 under the H2020 research and innovation program. C.?E. was partially supported by EU-ITN Stardust-R and MIUR-PRIN 20178CJA2B ?New Frontiers of Celestial Mechanics: theory and Applications.? The authors acknowledge feedback and discussions with Elisa Maria Alessi. Publisher Copyright: © 2022, The Author(s), under exclusive licence to Springer Nature B.V.

PY - 2022/2

Y1 - 2022/2

N2 - Despite extended past studies, several questions regarding the resonant structure of the medium-Earth orbit (MEO) region remain hitherto unanswered. This work describes in depth the effects of the 2 g+ h lunisolar resonance. In particular, (i) we compute the correct forms of the separatrices of the resonance in the inclination-eccentricity (i, e) space for fixed semi-major axis a. This allows to compute the change in the width of the 2 g+ h resonance as the altitude increases. (ii) We discuss the crucial role played by the value of the inclination of the Laplace plane, iL. Since iL is comparable to the resonance’s separatrix width, the parametrization of all resonance bifurcations has to be done in terms of the proper inclination ip, instead of the mean one. (iii) The subset of circular orbits constitutes an invariant subspace embedded in the full phase space, the center manifold C, where actual navigation satellites lie. Using ip as a label, we compute its range of values for which C becomes a normally hyperbolic invariant manifold (NHIM). The structure of invariant tori in C allows to explain the role of the initial phase h noticed in several works. (iv) Through Fast Lyapunov Indicator (FLI) cartography, we portray the stable and unstable manifolds of the NHIM as the altitude increases. Manifold oscillations dominate in phase space between a= 24,000 km and a= 30,000 km as a result of the sweeping of the 2 g+ h resonance by the [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext. resonances. The noticeable effects of the latter are explained as a consequence of the relative inclination of the Moon’s orbit with respect to the ecliptic. The role of the phases [InlineEquation not available: see fulltext.] in the structures observed in the FLI maps is also clarified. Finally, (v) we discuss how the understanding of the manifold dynamics could inspire end-of-life disposal strategies.

AB - Despite extended past studies, several questions regarding the resonant structure of the medium-Earth orbit (MEO) region remain hitherto unanswered. This work describes in depth the effects of the 2 g+ h lunisolar resonance. In particular, (i) we compute the correct forms of the separatrices of the resonance in the inclination-eccentricity (i, e) space for fixed semi-major axis a. This allows to compute the change in the width of the 2 g+ h resonance as the altitude increases. (ii) We discuss the crucial role played by the value of the inclination of the Laplace plane, iL. Since iL is comparable to the resonance’s separatrix width, the parametrization of all resonance bifurcations has to be done in terms of the proper inclination ip, instead of the mean one. (iii) The subset of circular orbits constitutes an invariant subspace embedded in the full phase space, the center manifold C, where actual navigation satellites lie. Using ip as a label, we compute its range of values for which C becomes a normally hyperbolic invariant manifold (NHIM). The structure of invariant tori in C allows to explain the role of the initial phase h noticed in several works. (iv) Through Fast Lyapunov Indicator (FLI) cartography, we portray the stable and unstable manifolds of the NHIM as the altitude increases. Manifold oscillations dominate in phase space between a= 24,000 km and a= 30,000 km as a result of the sweeping of the 2 g+ h resonance by the [InlineEquation not available: see fulltext.] and [InlineEquation not available: see fulltext. resonances. The noticeable effects of the latter are explained as a consequence of the relative inclination of the Moon’s orbit with respect to the ecliptic. The role of the phases [InlineEquation not available: see fulltext.] in the structures observed in the FLI maps is also clarified. Finally, (v) we discuss how the understanding of the manifold dynamics could inspire end-of-life disposal strategies.

KW - Chaotic dynamics

KW - GNSS

KW - Manifold dynamics

KW - Medium- Earth orbit

KW - Orbital resonances

KW - Satellite orbits

UR - http://www.scopus.com/inward/record.url?scp=85123069776&partnerID=8YFLogxK

U2 - 10.1007/s10569-021-10060-6

DO - 10.1007/s10569-021-10060-6

M3 - Article

AN - SCOPUS:85123069776

SN - 0923-2958

VL - 134

JO - Celestial Mechanics & Dynamical Astronomy

JF - Celestial Mechanics & Dynamical Astronomy

IS - 1

M1 - 6

ER -

A deep dive into the 2 g+ h resonance: separatrices, manifolds and phase space structure of navigation satellites

Résumé

Financement

Accès au document

Autres fichiers et liens

Empreinte digitale

Contient cette citation