Revisiting The Averaged Problem in The Case of Mean-Motion Resonances of The Restricted Three-Body Problem. Global Rigorous Treatment and Application To The Co-Orbital Motion.

Three Body Problem ◽

Mean Motion ◽

Rigorous Treatment ◽

Three Body ◽

Suitable Approximation ◽

Mean Motion Resonance

Abstract A classical approach to the restricted three-body problem is to analyze the dynamics of the massless body in the synodic reference frame. A different approach is represented by the perturbative treatment: in particular the averaged problem of a mean-motion resonance allows to investigate the long-term behavior of the solutions through a suitable approximation that focuses on a particular region of the phase space. In this paper, we intend to bridge a gap between the two approaches in the specific case of mean-motion resonant dynamics, establish the limit of validity of the averaged problem, and take advantage of its results in order to compute trajectories in the synodic reference frame. After the description of each approach, we develop a rigorous treatment of the averaging process, estimate the size of the transformation and prove that the averaged problem is a suitable approximation of the restricted three-body problem as long as the solutions are located outside the Hill's sphere of the secondary. In such a case, a rigorous theorem of stability over finite but large timescales can be proven. We establish that a solution of the averaged problem provides an accurate approximation of the trajectories on the synodic reference frame within a finite time that depend on the minimal distance to the Hill's sphere of the secondary. The last part of this work is devoted to the co-orbital motion (i.e., the dynamics in 1:1 mean-motion resonance) in the circular-planar case. In this case, an interpretation of the solutions of the averaged problem in the synodic reference frame is detailed and a method that allows to compute co-orbital trajectories is displayed.

Location and stability of the triangular Lagrange points in photo-gravitational elliptic restricted three body problem with the more massive primary as an oblate spheroid

International Journal of Advanced Astronomy ◽

10.14419/ijaa.v7i2.29814 ◽

2019 ◽

Vol 7 (2) ◽

pp. 57

Author(s):

A. Arantza Jency ◽

Ram Krishan Sharma

Keyword(s):

Body Problem ◽

Critical Mass ◽

Oblate Spheroid ◽

Motion Equation ◽

Three Body Problem ◽

Lagrangian Points ◽

Mean Motion ◽

The Mean ◽

The triangular Lagrangian points of the elliptic restricted three-body problem (ERTBP) with oblate and radiating more massive primary are studied. The mean motion equation used here is different from the ones employed in many studies on the perturbed ERTBP. The effect of oblateness on the mean motion equation varies. This change influences the location and stability of the triangular Lagrangian points. The points tend to shift in the y-direction. The influence of the oblateness on the critical mass ratio is also altered. But the eccentricity limit for stability remains the same.

Stationary solutions, critical mass, Tadpole orbits in the circular restricted three-body problem with the more massive primary as an oblate spheroid

Indian Journal of Science and Technology ◽

10.17485/ijst/v13i39.1396 ◽

2020 ◽

Vol 13 (39) ◽

pp. 4168-4188

Author(s):

A Arantza Jency

Keyword(s):

Body Problem ◽

Equilibrium Points ◽

Critical Mass ◽

Equations Of Motion ◽

Oblate Spheroid ◽

Three Body Problem ◽

Mean Motion ◽

The Mean ◽

Background: The location and stability of the equilibrium points are studied for the Planar Circular Restricted Three-Body Problem where the more massive primary is an oblate spheroid. Methods: The mean motion of the equations of motion is formulated from the secular perturbations as derived by(1) and used in(2–4). The singularities of the equations of motion are found for locating the equilibrium points. Their stability is analysed using the linearized variational equations of motion at the equilibrium points. Findings: As the effect of oblateness in the mean motion expression increases, the location and stability of the equilibrium points are affected by the oblateness of the more massive primary. It is interesting to note that all the three collinear points move towards the more massive primary with oblateness. It is a new result. Among the shifts in the locations of the five equilibrium points, the y–location of the triangular equilibrium points relocate the most. It is very interesting to note that the eccentricities (e) of the orbits around L1 and L3 increase, while it decreases around L2 with the addition of oblateness with the new mean motion. The decrease in e is significant in Saturn-Mimas system from 0.95036 to 0.87558. Similarly, the value of the critical mass ratio mc, which sets the limit for the linear stability of the triangular points, further reduces significantly from 0:285: : :A1 to 0:365: : :A1 with the new mean motion. The mean motion sz in the z-direction increases significantly with the new mean motion from 9A1/4 to 9A1/2.

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

Celestial Mechanics and Dynamical Astronomy ◽

10.1007/s10569-016-9749-1 ◽

2017 ◽

Vol 128 (4) ◽

pp. 383-407 ◽

Cited By ~ 6

Author(s):

Alexandre Pousse ◽

Philippe Robutel ◽

Alain Vienne

Keyword(s):

Body Problem ◽

Orbital Motion ◽

Satellite Motion ◽

Three Body Problem ◽

Orbital Motion Around the Collinear Libration Points of the Restricted Three-Body Problem

Journal of Advances in Mathematics and Computer Science ◽

10.9734/jamcs/2018/43370 ◽

2018 ◽

Vol 29 (1) ◽

pp. 1-16

Author(s):

A. H. Ibrahim ◽

M. N. Ismail ◽

A. S. Zaghrout ◽

S. H. Younis ◽

M. O. El Shikh

Keyword(s):

Body Problem ◽

Orbital Motion ◽

Libration Points ◽

Three Body Problem ◽

A general solution to non-collinear equilibria in terms of largest root (κ) of confocal oblate spheroid

International Journal of Advanced Astronomy ◽

10.14419/ijaa.v4i1.5587 ◽

2016 ◽

Vol 4 (1) ◽

pp. 1

Author(s):

M Javed Idrisi

Keyword(s):

General Solution ◽

Body Problem ◽

Oblate Spheroid ◽

Three Body Problem ◽

Mean Motion ◽

Three Body ◽

Infinitesimal Mass

This paper deals with the existence of non-collinear equilibria in restricted three-body problem when less massive primary is an oblate spheroid and the potential of oblate spheroid is in terms of largest root of confocal oblate spheroid. This is found that the non-collinear equilibria are the solution of the equations r1 = n-2/3 and κ = 1 – a2, where r1 is the distance of the infinitesimal mass from more massive primary, n is mean-motion of primaries, a is semi axis of oblate spheroid and κ is the largest root of the equation of confocal oblate spheroid passes through the infinitesimal mass.

Phase structure of co-orbital motion with Jupiter

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/staa1069 ◽

2020 ◽

Vol 494 (4) ◽

pp. 4695-4705 ◽

Cited By ~ 2

Author(s):

Yi Qi ◽

Anton de Ruiter

Keyword(s):

Analytical Method ◽

Phase Structure ◽

Body Problem ◽

Orbital Motion ◽

Orbital Dynamics ◽

The Sun ◽

Three Body Problem ◽

Phase Structures ◽

ABSTRACT In this paper, we investigate the dynamics of the inclined co-orbital motion with Jupiter through a torus phase structure in the Sun–Jupiter circular restricted three-body problem. A semi-analytical method to establish the Hamiltonian approximation for the inclined co-orbital motion is proposed. Phase structures of different kinds of co-orbital behaviours are shown in the torus space clearly. Based on numerical computation, we analyse the evolution and the connection of different co-orbital dynamics. Summarizing results and conclusions in this paper, we find two main principles throughout the investigation of the co-orbital motion: (i) the libration amplitude of the resonant angle for the co-orbital motion is bounded by the corresponding Hamiltonian isosurface in the torus space and (ii) the co-orbital behaviour is influenced by collision curves, and with the decrease of the Hamiltonian value, the influence is more significant.