scholarly journals Resonance libration and width at arbitrary inclination

2020 ◽  
Vol 493 (2) ◽  
pp. 2854-2871
Author(s):  
F Namouni ◽  
M H M Morais
Keyword(s):  
Analytical Tool ◽  
Resonance Width ◽  
Disturbing Function ◽  
Three Body Problem ◽  
Mean Motion Resonances ◽  
Resonance Modes ◽  
Pendulum Model ◽  
Powerful Analytical Tool ◽  
The Mean ◽  
Three Body

ABSTRACT We apply the analytical disturbing function for arbitrary inclination derived in our previous work to characterize resonant width and libration of mean motion resonances at arbitrary inclination obtained from direct numerical simulations of the three-body problem. We examine the 2:1 and 3:1 inner Jupiter and 1:2 and 1:3 outer Neptune resonances and their possible asymmetric librations using a new analytical pendulum model of resonance that includes the simultaneous libration of multiple arguments and their second harmonics. The numerically derived resonance separatrices are obtained using the mean exponential growth factor of nearby orbits (megno chaos indicator). We find that the analytical and numerical estimates are in agreement and that resonance width is determined by the first few fundamental resonance modes that librate simultaneously on the resonant time-scale. Our results demonstrate that the new pendulum model may be used to ascertain resonance width analytically, and more generally, that the disturbing function for arbitrary inclination is a powerful analytical tool that describes resonance dynamics of low as well as high inclination asteroids in the Solar system.

scholarly journals On the Stability of Collinear Points of the RTBP with Triaxial and Oblate Primaries and Relativistic Effects

2021 ◽  
pp. 56-73
Author(s):  
S. E. Abd El-Bar
Keyword(s):  
Characteristic Equation ◽  
Body Problem ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Relativistic Effects ◽  
Three Body Problem ◽  
Total Potential ◽  
The Mean ◽  
The Stability ◽  
Three Body

Under the influence of some different perturbations, we study the stability of collinear equilibrium points of the Restricted Three Body Problem. More precisely, the perturbations due to the triaxiality of the bigger primary and the oblateness of the smaller primary, in addition to the relativistic effects, are considered. Moreover, the total potential and the mean motion of the problem are obtained. The equations of motion are derived and linearized around the collinear points. For studying the stability of these points, the characteristic equation and its partial derivatives are derived. Two real and two imaginary roots of the characteristic equation are deduced from the plotted figures throughout the manuscript. In addition, the instability of the collinear points is stressed. Finally, we compute some selected roots corresponding to the eigenvalues which are based on some selected values of the perturbing parameters in the Tables 1, 2.

scholarly journals 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

2019 ◽  
Vol 7 (2) ◽  
pp. 57
Author(s):  
A. Arantza Jency ◽  
Ram Krishan Sharma
Keyword(s):  
Body Problem ◽  
Critical Mass ◽  
Oblate Spheroid ◽  
Motion Equation ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Lagrangian Points ◽  
Mean Motion ◽  
The Mean ◽  
Three Body

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.   

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

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.

Special cases of the restricted problem of three bodies

1966 ◽  
Vol 25 ◽  
pp. 187-193 ◽  
Author(s):  
J. Schubart
Keyword(s):  
Restricted Problem ◽  
Body Problem ◽  
Disturbing Function ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Long Period ◽  
Period Effects ◽  
Special Cases ◽  
Three Body

The long-period effects in nearly commensurable cases of the restricted three-body problem were studied according to the ideas of Poincaré. The secular and critical terms of the disturbing function were isolated by a numerical averaging process, by use of an IBM 7094 computer.

scholarly journals Explicit Semianalytical Theory of Asteroid Motion

1999 ◽  
Vol 172 ◽  
pp. 87-96
Author(s):  
I.V. Tupikova ◽  
A.A. Vakhidov ◽  
M. Soffel
Keyword(s):  
Closed Form ◽  
Explicit Form ◽  
Minor Planets ◽  
Mean Motion Resonances ◽  
Resonant Case ◽  
Proper Elements ◽  
The Mean ◽  
The Masses ◽  
Three Body ◽  
Asteroid Orbit

AbstractA new semianalytical theory of asteroid motion is presented. The theory is developed on the basis of Kaula’s expansion of the disturbing function including terms up to the second order with respect to the masses of disturbing bodies. The theory is constructed in explicit form that gives the possibility to study separately the influence of different perturbations in the dynamics of minor planets. The mean-motion resonances with major planets as well as mixed three-body resonances can also be taken into account. For the non-resonant case the formulas obtained can be used for deriving the second transformation to calculate the proper elements of an asteroid orbit in closed form with respect to inclinations and eccentricities.

scholarly journals Mean Motion Resonances in The Asteroid Belt

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.

scholarly journals A new expansion of planetary disturbing function and applications to interior, co-orbital and exterior resonances with planets

2022 ◽  
Vol 21 (12) ◽  
pp. 311
Author(s):  
Han-Lun Lei
Keyword(s):  
Semimajor Axis ◽  
Disturbing Function ◽  
Circular Orbits ◽  
Mean Motion Resonances ◽  
Mean Motion ◽  
Planetary Disturbing Function ◽  
Resonant Dynamics ◽  
The Mean ◽  
Analytical Expansion ◽  
Orbital Resonances

Abstract In this study, a new expansion of planetary disturbing function is developed for describing the resonant dynamics of minor bodies with arbitrary inclinations and semimajor axis ratios. In practice, the disturbing function is expanded around circular orbits in the first step and then, in the second step, the resulting mutual interaction between circular orbits is expanded around a reference point. As usual, the resulting expansion is presented in the Fourier series form, where the force amplitudes are dependent on the semimajor axis, eccentricity and inclination, and the harmonic arguments are linear combinations of the mean longitude, longitude of pericenter and longitude of ascending node of each mass. The resulting new expansion is valid for arbitrary inclinations and semimajor axis ratios. In the case of mean motion resonant configuration, the disturbing function can be easily averaged to produce the analytical expansion of resonant disturbing function. Based on the analytical expansion, the Hamiltonian model of mean motion resonances is formulated, and the resulting analytical developments are applied to Jupiter’s inner and co-orbital resonances and Neptune’s exterior resonances. Analytical expansion is validated by comparing the analytical results with the associated numerical outcomes.

scholarly journals Long-term evolution of the Galilean satellites: the capture of Callisto into resonance

2020 ◽  
Vol 639 ◽  
pp. A40 ◽  
Author(s):  
Giacomo Lari ◽  
Melaine Saillenfest ◽  
Marco Fenucci
Keyword(s):  
Galilean Satellites ◽  
Tidal Dissipation ◽  
Mean Motion Resonances ◽  
Tidal Forces ◽  
Recent Estimate ◽  
The Mean ◽  
A Chain ◽  
The Stability ◽  
Three Body

Context. The Galilean satellites have very complex orbital dynamics due to the mean-motion resonances and the tidal forces acting in the system. The strong dissipation in the couple Jupiter–Io is spread to all the moons involved in the so-called Laplace resonance (Io, Europa, and Ganymede), leading to a migration of their orbits. Aims. We aim to characterize the future behavior of the Galilean satellites over the Solar System lifetime and to quantify the stability of the Laplace resonance. Tidal dissipation permits the satellites to exit from the current resonances or be captured into new ones, causing large variation in the moons’ orbital elements. In particular, we want to investigate the possible capture of Callisto into resonance. Methods. We performed hundreds of propagations using an improved version of a recent semi-analytical model. As Ganymede moves outwards, it approaches the 2:1 resonance with Callisto, inducing a temporary chaotic motion in the system. For this reason, we draw a statistical picture of the outcome of the resonant encounter. Results. The system can settle into two distinct outcomes: (A) a chain of three 2:1 two-body resonances (Io–Europa, Europa–Ganymede, and Ganymede–Callisto), or (B) a resonant chain involving the 2:1 two-body resonance Io–Europa plus at least one pure 4:2:1 three-body resonance, most frequently between Europa, Ganymede, and Callisto. In case A (56% of the simulations), the Laplace resonance is always preserved and the eccentricities remain confined to small values below 0.01. In case B (44% of the simulations), the Laplace resonance is generally disrupted and the eccentricities of Ganymede and Callisto can increase up to about 0.1, making this configuration unstable and driving the system into new resonances. In all cases, Callisto starts to migrate outward, pushed by the resonant action of the other moons. Conclusions. From our results, the capture of Callisto into resonance appears to be extremely likely (100% of our simulations). The exact timing of its entrance into resonance depends on the precise rate of energy dissipation in the system. Assuming the most recent estimate of the dissipation between Io and Jupiter, the resonant encounter happens at about 1.5 Gyr from now. Therefore, the stability of the Laplace resonance as we know it today is guaranteed at least up to about 1.5 Gyr.

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

2016 ◽  
Vol 18 (10) ◽  
pp. 2315-2403 ◽  
Author(s):  
Jacques Féjoz ◽  
Marcel Guàrdia ◽  
Vadim Kaloshin ◽  
Pablo Roldán
Keyword(s):  
Body Problem ◽  
Three Body Problem ◽  
Mean Motion Resonances ◽  
Mean Motion ◽  
Kirkwood Gaps ◽  
Three Body ◽  
And Diffusion

scholarly journals The Satellite Case of the Three-Body Problem

1979 ◽  
Vol 81 ◽  
pp. 77-83
Author(s):  
Gen'ichiro Hori
Keyword(s):  
Body Problem ◽  
Disturbing Function ◽  
The Sun ◽  
Satellite Theory ◽  
Three Body Problem ◽  
Two Factors ◽  
Three Body

In the lunar or satellite theory, the disturbing function is developed in the form in which each term consists of two factors. The first factor depends on the position of the sun, and the second one that of the satellite.

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