On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited

Abstract

In the framework of the planar and circular restricted three-body problem, we consider an asteroid that orbits the Sun in quasi-satellite motion with a planet. A quasi-satellite trajectory is a heliocentric orbit in co-orbital resonance with the planet, characterized by a nonzero eccentricity and a resonant angle that librates around zero. Likewise, in the rotating frame with the planet, it describes the same trajectory as the one of a retrograde satellite even though the planet acts as a perturbator. In the last few years, the discoveries of asteroids in this type of motion made the term “quasi-satellite” more and more present in the literature. However, some authors rather use the term “retrograde satellite” when referring to this kind of motion in the studies of the restricted problem in the rotating frame. In this paper, we intend to clarify the terminology to use, in order to bridge the gap between the perturbative co-orbital point of view and the more general approach in the rotating frame. Through a numerical exploration of the co-orbital phase space, we describe the quasi-satellite domain and highlight that it is not reachable by low eccentricities by averaging process. We will show that the quasi-satellite domain is effectively included in the domain of the retrograde satellites and neatly defined in terms of frequencies. Eventually, we highlight a remarkable high eccentric quasi-satellite orbit corresponding to a frozen ellipse in the heliocentric frame. We extend this result to the eccentric case (planet on an eccentric motion) and show that two families of frozen ellipses originate from this remarkable orbit.

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Notes

  1. 1.

    The two firsts are works of the Copenhagen group that extensively explored periodic orbit solutions in the planar restricted three-body problem with two equal masses. The two lasts are the first numerical explorations of all the solutions of the restricted three-body problem that recovered and completed the precedent works.

  2. 2.

    Let us still mention that the “quasi-satellite” terminology has already been used in the paper of Danielsson and Ip (1972) but this was to describe the resonant behaviour of the near-Earth Object 1685 Toro and therefore was completely disconnected to retrograde satellite motion.

  3. 3.

    As we have to take into account the degree of freedom \((\lambda ', {{\widetilde{\varLambda }}}')\), we have \({{\mathscr {M}}}\subset {{\mathbb {T}}}\times {{\mathbb {R}}}\times {{\mathbb {C}}}^2\times {{\mathbb {T}}}\times {{\mathbb {R}}}\).

  4. 4.

    See the Sect. 3.2.

  5. 5.

    See Sect. 4 for further details on the family f.

  6. 6.

    In practice, the matrix \({{\mathcal {M}}}(\theta _0,u_0)\) is provided by a numerical differentiation of the equations of motion at the fixed point \((\theta _0,u_0)\).

  7. 7.

    Floquet theory; for further details, see Meyer and Hall (1992).

  8. 8.

    In practice, the numerical algorithm of the Poincaré map provides g as in the Eq. (20), while the frequency \(\nu \) is obtained via the monodromy matrix \(d\varPi _T\) [see Eq. (21)] that is calculated with a numerical differentiation algorithm on the Poincaré map.

  9. 9.

    See Fig. 5.

Abbreviations

RF:

Rotating frame with the planet

AP:

Averaged problem

RAP:

Reduced averaged problem

RS:

Retrograde satellite

TP:

Tadpole

HS:

Horseshoe

QS:

Quasi-satellite

sRS:

“Satellized” retrograde satellite

QS\(_{b}~\) :

Binary quasi-satellite

QS\(_{h}~\) :

Heliocentric quasi-satellite

\(L_1\), \(L_2\), \(L_3\) :

Circular Eulerian aligned configurations

\(L_4\), \(L_5\) :

Circular Lagranian equilateral configurations

\({{{\mathscr {L}}}_{4}^{l}}\), \({{{\mathscr {L}}}_{5}^{l}}\) :

In the RF, long period families that originate from \(L_4\) and \(L_5\).

\({{{\mathscr {L}}}_{3}}\), \({{{\mathscr {L}}}_{4}^{s}}\), \({{{\mathscr {L}}}_{5}^{s}}\) :

In the RF, short period families that originate from \(L_3\), \(L_4\) and \(L_5\).

Family f :

In the RF, one-parameter family of simple-periodic symmetrical retrograde satellite orbits that extends from an infinitesimal neighbourhood of the planet to the collision with the Sun. For \({\varepsilon }<0.0477\), it is stable but contains two particular orbits where the frequencies \(\nu \) and \(1-g\) are in 1 : 3 resonance. These two orbits decomposed the neighbourhood of the family f in three domains: sRS, QS\(_{b}~\)and QS\(_{h}~\).

1, \(\nu \), g :

Frequencies, respectively, associated with the fast variations (the mean longitudes \(\lambda \) and \(\lambda '\)), the semi-fast component of the dynamics (oscillation of the resonant angle \(\theta \)) and the secular evolution of a trajectory (precession of the periaster argument \(\omega \)).

\({{{\mathcal {N}}}_{L_4}^{u}}\), \({{{\mathcal {N}}}_{L_5}^{u}}\) :

In the RAP, the AP and the RF, families of \(2\pi /\nu \)-periodic orbits parametrized by \(|u|\le 0\) and that originate from \(L_4\) and \(L_5\). Moreover, they correspond to \({{{\mathscr {L}}}_{4}^{l}}\) and \({{{\mathscr {L}}}_{5}^{l}}\) in the RF.

\({{{\mathcal {G}}}_{L_3}^{e_0}}\), \({{{\mathcal {G}}}_{L_4}^{e_0}}\), \({{{\mathcal {G}}}_{L_5}^{e_0}}\) :

In the RAP, families of fixed points parametrized by \(e_0\) and that originate from \(L_3\), \(L_4\) and \(L_5\). In the AP and the RF, these fixed points correspond to periodic orbits of frequency, respectively, g and \(1-g\). Moreover, they correspond to \({{{\mathscr {L}}}_{3}}\), \({{{\mathscr {L}}}_{4}^{s}}\) and \({{{\mathscr {L}}}_{5}^{s}}\) in the RF.

\({{{\mathcal {G}}}_{QS}^{e_0}}\) :

In the RAP, family of fixed points parametrized by \(e_0\). In the AP and the RF, these fixed points correspond to periodic orbits of frequency, respectively, g and \(1-g\). Moreover, this family corresponds to a part of the family f that belongs to the QS\(_{h}~\)domain.

\({G_{L_3}}\), \({G_{L_4}}\), \({G_{L_5}}\), \({G_{QS}}\) :

In the RAP, fixed points that belong to \({{{\mathcal {G}}}_{L_3}^{e_0}}\), \({{{\mathcal {G}}}_{L_4}^{e_0}}\), \({{{\mathcal {G}}}_{L_5}^{e_0}}\) and \({{{\mathcal {G}}}_{QS}^{e_0}}\) and characterized by \(g=0\). In the AP, sets of fixed points (also denoted as “circles of fixed points”) parametrized by \(\omega (t=0)\). In the RF, sets of \(2\pi \)-periodic orbits parametrized by \(\big (\lambda '-\omega \big )_{t=0}\).

\({G_{L_3,1}^{e'}}\), \({G_{L_3,2}^{e'}}\), \({G_{L_4,1}^{e'}}\), \({G_{L_4,2}^{e'}}\), \({G_{L_5,1}^{e'}}\), \({G_{L_5,2}^{e'}}\), \({G_{QS,1}^{e'}}\), \({G_{QS,2}^{e'}}\) :

In the AP with \(e'\ge 0\), families of fixed points that originate from the circles of fixed points \({G_{L_3}}\), \({G_{L_4}}\), \({G_{L_5}}\), \({G_{QS}}\) when \(e'=0\).

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Acknowledgements

This work has been developed during the Ph.D thesis of Alexandre Pousse at the “Astronomie et Systèmes Dynamiques”, IMCCE, Observatoire de Paris.

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Correspondence to Alexandre Pousse.

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A. Pousse: supported by the H2020-ERC project 677793 StableChaoticPlanetM.

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Pousse, A., Robutel, P. & Vienne, A. On the co-orbital motion in the planar restricted three-body problem: the quasi-satellite motion revisited. Celest Mech Dyn Astr 128, 383–407 (2017). https://doi.org/10.1007/s10569-016-9749-1

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Keywords

  • Restricted three-body problem
  • Co-orbital motion
  • Quasi-satellite
  • Averaged Hamiltonian