Atlas of the mean motion resonances in the Solar System
Introduction
For many years the dynamical studies of MMRs were restricted to low order resonances because these are the most evident in the asteroid belt. High order resonances, however, started to appear in studies of highly eccentric orbits like those of comets (Chambers, 1997), NEAs (Morbidelli and Nesvorný, 1999), trans-neptunians (Robutel and Laskar, 2001) and specially meteors streams (Emel'yanenko, 1992), becoming the capture in high order MMR a not so uncommon phenomena. It is laborious to identify which one of the hundreds of MMRs that theoretically exist near the semimajor axis of the orbit we are studying is the one affecting the body's motion. This difficulty is due to the absence of a simple method that adequately weighs the strength of each resonance. Neither we have a global view of the strength of the resonances with all the planets over all Solar System. Authors have opted to plot the resonance's strength as a function that decreases as resonance's order increases (Nesvorný and Morbidelli, 1998), but this criteria gives equal strength for all resonances of the same order which is unrealistic. For zero inclination orbits it is possible to compute the widths in semimajor axis of the MMRs with the planets as a function of the eccentricity (Dermott and Murray, 1983, Morbidelli et al., 1995, Nesvorný et al., 2002) but no simple method exists to compute the widths in the case of non-zero inclination orbits. We present here a method to estimate the strength of the mean motion resonant orbits with arbitrary orbital elements. The method is a modification and an extension of the author's method recently proposed (Gallardo, 2006). Based on this principle we compute the strength of the resonances with all the planets form Mercury to Neptune for all ranges of semimajor axis, from the Sun up to 300 AU assuming typical orbital eccentricities and inclinations of populations of small bodies like near-Earth asteroids (NEAs), centaurs, trans-neptunian objects (TNOs) and scattered disk objects (SDOs).
This work is organized as follows. In Section 2 we show how to evaluate numerically the disturbing function for a resonant orbit. In Section 3 we analyze the possible shapes of and the location of the equilibrium points defined by its minima. In Section 4 we define the strength function for a given resonance and we analyze its shape. In Section 5 we calculate SR for thousands of resonances with all the planets except Pluto, we analyze the generated atlas and look for real objects showing the predicted behavior. In Section 6 we present the conclusions.
Section snippets
Numerical evaluation of the resonant disturbing function
Given a planet of mass and radius vector in an heliocentric frame and a small body at r the disturbing function is: Since Laplace's times astronomers looked for an analytical expression for as an explicit function of the orbital elements. We will refer the reader to Gallardo (2006), for example, for a detailed explanation of the general form of the disturbing function. In this paper we will assume circular and zero inclination orbits for the planets designed here
The shape of and the libration centers
The equations of the resonant motion show that the time evolution of the semimajor axis, , is proportional to then the shape of is crucial because it defines the location of stable and unstable equilibrium points. For specific values of the minima of define the stable equilibrium points also known as libration centers around which there exist the librations. The unstable equilibrium points are defined by the maxima.
We found that at low the function
Numerical estimation of the resonance's strength
For a given resonant orbit defined by parameters the disturbing function is determined. We define the strength function SR as: being the mean value of with respect to σ and the minimum value of . This definition is in agreement with the coefficients of the resonant terms of the expansion of the disturbing function for low orbits because for this case is a sinusoid with an amplitude given by . This definition of strength
Atlas of MMRs and examples
For a given resonance with a given planet its strength SR is a function of as we have explained in the previous section. Then, a specific small body will experience the effects of the resonances according to its orbital elements. We take typical orbital elements for different populations and calculate the strength of all resonances verifying and order with all the planets from Mercury to Neptune. In the region between 0 and 6 AU we used typical orbital elements of
Conclusions
According to the shape of the resonant disturbing function all resonances can be classified in three different groups: (a) type , (b) odd order interior resonances and (c) even order interior resonances and all exterior resonances excluding the resonances. Horseshoe trajectories wrapping two libration centers are only possible for the first class. For high inclination orbits the shape of is generally very different from the low inclination case and it becomes strongly dependent
Acknowledgements
The author acknowledges the criticism given by D. Nesvorný and another anonymous referee. This work was developed in the framework of the “Proyecto CSIC I + D, Dinamica Secular de Sistemas Planetarios y Cuerpos Menores.”
References (23)
- et al.
Transient co-orbital asteroids
Icarus
(2004) - et al.
Evolution of near-Earth asteroids close to mean motion resonances
Planet. Space Sci.
(2001) Why Halley-types resonate but long-period comets don't: A dynamical distinction between short and long-period comets
Icarus
(1997)A numerical survey of transient co-orbitals of the terrestrial planets
Icarus
(2000)- et al.
The scattered disk population as a source of Oort cloud comets: Evaluation of its current and past role in populating the Oort cloud
Icarus
(2004) The occurrence of high order mean motion resonances and Kozai mechanism in the scattered disk
Icarus
(2006)- et al.
Evidence for an extended scattered disk
Icarus
(2002) - et al.
Numerous weak resonances drive asteroids toward terrestrial planets orbits
Icarus
(1999) - et al.
The resonant structure of the Kuiper belt and the dynamics of the first five trans-neptunian objects
Icarus
(1995) - et al.
Mean motion resonances in the trans-neptunian region
Icarus
(2001)
Frequency map and global dynamics in the Solar System I
Icarus
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