Three-Body Mean Motion Resonances and the Chaotic Structure of the Asteroid Belt

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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Planetary and satellite three body mean motion resonances

Icarus ◽

10.1016/j.icarus.2016.03.018 ◽

2016 ◽

Vol 274 ◽

pp. 83-98 ◽

Cited By ~ 15

Author(s):

Tabaré Gallardo ◽

Leonardo Coito ◽

Luciana Badano

Keyword(s):

Mean Motion Resonances ◽

Mean Motion ◽

Three Body

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Mean Motion Resonances in the Asteroid Belt

Impact of Modern Dynamics in Astronomy ◽

10.1007/978-94-011-4527-5_46 ◽

1999 ◽

pp. 381-382

Author(s):

D. Nesvorný ◽

A. Morbidelli

Keyword(s):

Asteroid Belt ◽

Mean Motion Resonances ◽

Mean Motion

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Identification of asteroids trapped inside three-body mean motion resonances: a machine-learning approach

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/stx999 ◽

2017 ◽

Vol 469 (2) ◽

pp. 2024-2031 ◽

Cited By ~ 5

Author(s):

Evgeny A. Smirnov ◽

Alexey B. Markov

Keyword(s):

Machine Learning ◽

Learning Approach ◽

Mean Motion Resonances ◽

Mean Motion ◽

Machine Learning Approach ◽

Three Body

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Asteroids in three-body mean motion resonances with planets

Icarus ◽

10.1016/j.icarus.2017.09.032 ◽

2018 ◽

Vol 304 ◽

pp. 24-30 ◽

Cited By ~ 5

Author(s):

Evgeny A. Smirnov ◽

Ilya S. Dovgalev ◽

Elena A. Popova

Keyword(s):

Mean Motion Resonances ◽

Mean Motion ◽

Three Body

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An integrable model for first-order three-planet mean motion resonances

Celestial Mechanics and Dynamical Astronomy ◽

10.1007/s10569-021-10035-7 ◽

2021 ◽

Vol 133 (8) ◽

Author(s):

Antoine C. Petit

Keyword(s):

Integrable Model ◽

Degree Of Freedom ◽

Mass Ratio ◽

Zeroth Order ◽

Mean Motion Resonances ◽

First Order ◽

Mean Motion ◽

Formation And Evolution ◽

The Stability ◽

Three Body

AbstractRecent works on three-planet mean motion resonances (MMRs) have highlighted their importance for understanding the details of the dynamics of planet formation and evolution. While the dynamics of two-planet MMRs are well understood and approximately described by a one-degree-of-freedom Hamiltonian, little is known of the exact dynamics of three-body resonances besides the cases of zeroth-order MMRs or when one of the bodies is a test particle. In this work, I propose the first general integrable model for first-order three-planet mean motion resonances. I show that one can generalize the strategy proposed in the two-planet case to obtain a one-degree-of-freedom Hamiltonian. The dynamics of these resonances are governed by the second fundamental model of resonance. The model is valid for any mass ratio between the planets and for every first-order resonance. I show the agreement of the analytical model with numerical simulations. As examples of application, I show how this model could improve our understanding of the capture into MMRs as well as their role in the stability of planetary systems.

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The functional relation between three-body mean motion resonances and Yarkovsky drift speeds

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/stab2526 ◽

2021 ◽

Vol 507 (4) ◽

pp. 5796-5803

Author(s):

I Milić Žitnik

Keyword(s):

Time Delays ◽

Semimajor Axis ◽

Functional Relation ◽

Mean Motion Resonances ◽

Mean Motion ◽

Drift Speed ◽

Functional Relations ◽

Numerical Integrations ◽

Three Body ◽

Good Agreement

ABSTRACT We examined the motion of asteroids across the three-body mean motion resonances (MMRs) with Jupiter and Saturn and with the Yarkovsky drift speed in the semimajor axis of the asteroids. The research was conducted using numerical integrations performed using the Orbit9 integrator with 84 000 test asteroids. We calculated time delays, dtr, caused by the seven three-body MMRs on the mobility of test asteroids with 10 positive and 10 negative Yarkovsky drift speeds, which are reliable for Main Belt asteroids. Our final results considered only test asteroids that successfully crossed over the MMRs without close approaches to the planets. We have devised two equations that approximately describe the functional relation between the average time 〈dtr〉 spent in the resonance, the strength of the resonance SR, and the semimajor axis drift speed da/dt (positive and negative) with the orbital eccentricities of asteroids in the range (0, 0.1). Comparing the values of 〈dtr〉 obtained from the numerical integrations and from the derived functional relations, we analysed average values of 〈dtr〉 in all three-body MMRs for every da/dt. The main conclusion is that the analytical and numerical estimates of the average time 〈dtr〉 are in very good agreement, for both positive and negative da/dt. Finally, this study shows that the functional relation we obtain for three-body MMRs is analogous to that previously obtained for two-body MMRs.

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Kozai mechanism inside mean motion resonances in the three-dimensional phase space

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/staa684 ◽

2020 ◽

Vol 493 (4) ◽

pp. 5816-5824 ◽

Cited By ~ 4

Author(s):

Yi Qi ◽

Anton de Ruiter

Keyword(s):

Phase Space ◽

Three Dimensional ◽

Necessary Condition ◽

Resonant Coupling ◽

Mean Motion Resonances ◽

Mean Motion ◽

3D Space ◽

Test Particles ◽

Dimensional Phase Space ◽

Three Body

ABSTRACT In this paper, we investigate the Kozai mechanism inside the inclined mean motion resonance (MMR) through a three-dimensional (3D) phase space. The Hamiltonian approximation for both prograde and retrograde MMRs is established by a semi-analytical method. We pick Jupiter as the disturber and study the Kozai mechanism in the Sun–Jupiter circular restricted three-body problem. Kozai islands of the prograde and retrograde MMRs are found in the 3D phase space. Numerical integration demonstrates that the locus of the orbit on the Kozai island is bounded by the Kozai island in the 3D phase space, so the orbit is locked in the Kozai+MMR state. The study of the Kozai dynamics inside a retrograde 1:1 MMR indicates that Kozai islands in the 3D phase space are just a sufficient condition for the Kozai+MMR mechanism rather than a necessary condition. There is no Kozai island in the 3D space for the retrograde 1:1 MMR, but the resonant coupling of Kozai with the retrograde 1:1 MMR appears in the phase space. Finally, dynamical behaviours of the two test particles located on Kozai islands are demonstrated in the ephemeris model.

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Higher Order and Iterative Theories to Compute Asteroid Mean Elements

International Astronomical Union Colloquium ◽

10.1017/s0252921100072717 ◽

1999 ◽

Vol 172 ◽

pp. 359-360 ◽

Cited By ~ 1

Author(s):

Z. Knežević ◽

A. Milani

Keyword(s):

Asteroid Belt ◽

Dynamical Structure ◽

Orbital Elements ◽

Mean Motion Resonances ◽

First Order ◽

Mean Motion ◽

Long Time ◽

The Mean ◽

Mean Elements ◽

Asteroid Families

Mean orbital elements are obtained from their instantaneous, osculating counterparts by removal of the short periodic perturbations. They can be computed by means of different theories, analytical or numerical, depending on the problem and accuracy required. The most advanced contemporary analytical theory (Knežević 1988) accounts only for the perturbing effects due to Jupiter and Saturn, to the first order in their masses and to degree four in eccentricity and inclination. Nevertheless, the mean elements obtained by means of this theory are of satisfactory accuracy for majority of the asteroids in the main belt (Knežević et al. 1988), for the purpose of producing large catalogues of mean and proper elements, to identify asteroid families, to assess their age, to study the dynamical structure of the asteroid belt and chaotic phenomena of diffusion over very long time spans. In the vicinity of the main mean motion resonances, however, especially 2:1 mean motion resonance with Jupiter, these mean elements are of somewhat degraded accuracy.

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