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 Survey of asteroids in retrograde mean motion resonances with planets

2019 ◽  
Vol 630 ◽  
pp. A60 ◽  
Author(s):  
Miao Li ◽  
Yukun Huang ◽  
Shengping Gong
Keyword(s):  
Solar System ◽  
Integration Time ◽  
Potential Candidate ◽  
Resonant Condition ◽  
Mean Motion Resonances ◽  
Orbital Resonance ◽  
Mean Motion ◽  
Resonant Dynamics ◽  
Orbital Resonances

Aims. Asteroids in mean motion resonances (MMRs) with planets are common in the solar system. In recent years, increasingly more retrograde asteroids are discovered, several of which are identified to be in resonances with planets. We here systematically present the retrograde resonant configurations where all the asteroids are trapped with any of the eight planets and evaluate their resonant condition. We also discuss a possible production mechanism of retrograde centaurs and dynamical lifetimes of all the retrograde asteroids. Methods. We numerically integrated a swarm of clones (ten clones for each object) of all the retrograde asteroids (condition code U < 7) from −10 000 to 100 000 yr, using the MERCURY package in the model of solar system. We considered all of the p/−q resonances with eight planets where the positive integers p and q were both smaller than 16. In total, 143 retrograde resonant configurations were taken into consideration. The integration time was further extended to analyze their dynamical lifetimes and evolutions. Results. We present all the meaningful retrograde resonant configurations where p and q are both smaller than 16 are presented. Thirty-eight asteroids are found to be trapped in 50 retrograde mean motion resonances (RMMRs) with planets. Our results confirm that RMMRs with giant planets are common in retrograde asteroids. Of these, 15 asteroids are currently in retrograde resonances with planets, and 30 asteroids will be captured in 35 retrograde resonant configurations. Some particular resonant configurations such as polar resonances and co-orbital resonances are also identified. For example, Centaur 2005 TJ50 may be the first potential candidate to be currently in polar retrograde co-orbital resonance with Saturn. Moreover, 2016 FH13 is likely the first identified asteroid that will be captured in polar retrograde resonance with Uranus. Our results provide many candidates for the research of retrograde resonant dynamics and resonance capture. Dynamical lifetimes of retrograde asteroids are investigated by long-term integrations, and only ten objects survived longer than 10 Myr. We confirmed that the near-polar trans-Neptunian objects 2011 KT19 and 2008 KV42 have the longest dynamical lifetimes of the discovered retrograde asteroids. In our long-term simulations, the orbits of 12 centaurs can flip from retrograde to prograde state and back again. This flipping mechanism might be a possible explanation of the origins of retrograde centaurs. Generally, our results are also helpful for understanding the dynamical evolutions of small bodies in the solar system.

scholarly journals On the divergence of first-order resonance widths at low eccentricities

2020 ◽  
Vol 496 (3) ◽  
pp. 3152-3160 ◽  
Author(s):  
Renu Malhotra ◽  
Nan Zhang
Keyword(s):  
Planetary Systems ◽  
Circular Orbits ◽  
Mean Motion Resonances ◽  
Order Resonance ◽  
First Order ◽  
Resonance Zone ◽  
Mean Motion ◽  
Orbital Resonances ◽  
Theoretical Analyses

ABSTRACT Orbital resonances play an important role in the dynamics of planetary systems. Classical theoretical analyses found in textbooks report that libration widths of first-order mean motion resonances diverge for nearly circular orbits. Here, we examine the nature of this divergence with a non-perturbative analysis of a few first-order resonances interior to a Jupiter-mass planet. We show that a first-order resonance has two branches, the pericentric and the apocentric resonance zone. As the eccentricity approaches zero, the centres of these zones diverge away from the nominal resonance location but their widths shrink. We also report a novel finding of ‘bridges’ between adjacent first-order resonances: at low eccentricities, the apocentric libration zone of a first-order resonance smoothly connects with the pericentric libration zone of the neighbouring first-order resonance. These bridges may facilitate resonant migration across large radial distances in planetary systems, entirely in the low-eccentricity regime.

scholarly journals Four-billion year stability of the Earth–Mars belt

2020 ◽  
Vol 500 (1) ◽  
pp. 1151-1157
Author(s):  
Yukun Huang (黄宇坤) ◽  
Brett Gladman
Keyword(s):  
Semimajor Axis ◽  
Orbital Stability ◽  
Terrestrial Planet ◽  
Small Bodies ◽  
Mean Motion Resonances ◽  
Mean Motion ◽  
The Earth ◽  
Steady State Model ◽  
Near Earth Objects

ABSTRACT Previous work has demonstrated orbital stability for 100 Myr of initially near-circular and coplanar small bodies in a region termed the ‘Earth–Mars belt’ from 1.08 &lt; a &lt; 1.28 au. Via numerical integration of 3000 particles, we studied orbits from 1.04–1.30 au for the age of the Solar system. We show that on this time-scale, except for a few locations where mean-motion resonances with Earth affect stability, only a narrower ‘Earth–Mars belt’ covering a ∼ (1.09, 1.17) au, e &lt; 0.04, and I &lt; 1° has over half of the initial orbits survive for 4.5 Gyr. In addition to mean-motion resonances, we are able to see how the ν3, ν4, and ν6 secular resonances contribute to long-term instability in the outer (1.17–1.30 au) region on Gyr time-scales. We show that all of the (rather small) near-Earth objects (NEOs) in or close to the Earth–Mars belt appear to be consistent with recently arrived transient objects by comparing to a NEO steady-state model. Given the &lt;200 m scale of these NEOs, we estimated the Yarkovsky drift rates in semimajor axis and use these to estimate that a diameter of ∼100 km or larger would allow primordial asteroids in the Earth–Mars belt to likely survive. We conclude that only a few 100-km sized asteroids could have been present in the belt’s region at the end of the terrestrial planet formation.

Regular and Chaotic Dynamics in the Mean-Motion Resonances:

Asteroids III ◽  
2002 ◽  
pp. 379-394
Author(s):  
D. Nesvorný ◽  
S. Ferraz-Mello ◽  
M. Holman ◽  
A. Morbidelli
Keyword(s):  
Chaotic Dynamics ◽  
Mean Motion Resonances ◽  
Mean Motion ◽  
The Mean

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.

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

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.

scholarly journals Higher Order and Iterative Theories to Compute Asteroid Mean Elements

1999 ◽  
Vol 172 ◽  
pp. 359-360 ◽  
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.

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 The onset of instability in resonant chains

2020 ◽  
Vol 494 (4) ◽  
pp. 4950-4968 ◽  
Author(s):  
Gabriele Pichierri ◽  
Alessandro Morbidelli
Keyword(s):  
Dynamical Mechanism ◽  
Mean Motion Resonances ◽  
Secondary Resonances ◽  
Mean Motion ◽  
Large N ◽  
Excited System ◽  
Resonant Systems ◽  
The Mean ◽  
The Stability ◽  
Mean Motion Resonance

ABSTRACT There is evidence that most chains of mean motion resonances of type k:k − 1 among exoplanets become unstable once the dissipative action from the gas is removed from the system, particularly for large N (the number of planets) and k (indicating how compact the chain is). We present a novel dynamical mechanism that can explain the origin of these instabilities and thus the dearth of resonant systems in the exoplanet sample. It relies on the emergence of secondary resonances between a fraction of the synodic frequency 2π(1/P1 − 1/P2) and the libration frequencies in the mean motion resonance. These secondary resonances excite the amplitudes of libration of the mean motion resonances, thus leading to an instability. We detail the emergence of these secondary resonances by carrying out an explicit perturbative scheme to second order in the planetary masses and isolating the harmonic terms that are associated with them. Focusing on the case of three planets in the 3:2–3:2 mean motion resonance as an example, a simple but general analytical model of one of these resonances is obtained, which describes the initial phase of the activation of one such secondary resonance. The dynamics of the excited system is also briefly described. Finally, a generalization of this dynamical mechanism is obtained for arbitrary N and k. This leads to an explanation of previous numerical experiments on the stability of resonant chains, showing why the critical planetary mass allowed for stability decreases with increasing N and k.

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