Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem

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.

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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 ◽

Restricted Three Body Problem ◽

Three Body Problem ◽

Mean Motion ◽

The Mean ◽

Three Body

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.

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Mean Motion Resonances in The Asteroid Belt

International Astronomical Union Colloquium ◽

10.1017/s025292110007281x ◽

1999 ◽

Vol 172 ◽

pp. 381-382

Author(s):

D. Nesvorný ◽

A. Morbidelli

Keyword(s):

Asteroid Belt ◽

Mean Motion Resonances ◽

Mean Motion ◽

Lyapunov Time ◽

Asteroidal Belt ◽

Kirkwood Gaps ◽

The Mean ◽

Three Body

The Kirkwood gaps in the main asteroidal belt (2 – 3.5 AU) coincide with the mean motion resonances with Jupiter (4/1, 3/1, 5/2, 7/3, 2/1). Similarly, several narrower gaps are observed in the outer asteroid belt (3.5 – 4 AU) at places of 11/6, 9/5, 7/4 and 5/3 Jovian resonances (Holman and Murray 1996). As it is now generally accepted, the formation and preservation of these gaps is due to the chaos of the resonant space and efficient ejection of the primordial and collisionaly injected bodies towards high eccentricities and planet-crossing orbits.The Jovian mean motion resonances are not the most important in what concerns the chaos of the observed (i.e. remaining) asteroid population. It was estimated by Šidlichovský and Nesvorný (1998) that about 40% of known objects have the Lyapunov time less than 105 years. It was later found (Nesvorný and Morbidelli 1998, 1999; Morbidelli and Nesvorný 1999) that the resonances responsible for this chaos are, in decreasing order of importance: 1) three-body resonances with Jupiter and Saturn, 2) exterior resonances with Mars, 3) moderate order Jovian resonances, and 4) three-body resonances with Mars and Jupiter.

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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 ◽

Restricted Three Body Problem ◽

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.

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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.

10.21203/rs.3.rs-614015/v1 ◽

2021 ◽

Author(s):

Alexandre Pousse ◽

Elisa Maria Alessi

Keyword(s):

Reference Frame ◽

Body Problem ◽

Orbital Motion ◽

Restricted Three Body Problem ◽

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.

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