Elsevier

Icarus

Volume 184, Issue 1, September 2006, Pages 29-38
Icarus

Atlas of the mean motion resonances in the Solar System

https://doi.org/10.1016/j.icarus.2006.04.001Get rights and content

Abstract

The aim of this work is to present a systematic survey of the strength of the mean motion resonances (MMRs) in the Solar System. We know by applying simple formulas where the resonances with the planets are located but there is no indication of the strength that these resonances have. We propose a numerical method for the calculation of this strength and we present an atlas of the MMRs constructed with this method. We found there exist several resonances unexpectedly strong and we look and find in the small bodies population several bodies captured in these resonances. In particular in the inner Solar System we find one asteroid in the resonance 6:5 with Venus, five asteroids in resonance 1:2 with Venus, three asteroids in resonance 1:2 with Earth and six asteroids in resonance 2:5 with Earth. We find some new possible co-orbitals of Earth, Mars, Saturn, Uranus and Neptune. We also present a discussion about the behavior of the resonant disturbing function and where the stable equilibrium points can be found at low and high inclination resonant orbits.

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 R(σ) for a resonant orbit. In Section 3 we analyze the possible shapes of R(σ) and the location of the equilibrium points defined by its minima. In Section 4 we define the strength function SR(e,i,ω) 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 mP and radius vector rP in an heliocentric frame and a small body at r the disturbing function is:R=k2mP(1|rPr|rrPrP3). Since Laplace's times astronomers looked for an analytical expression for R 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 R(σ) and the libration centers

The equations of the resonant motion show that the time evolution of the semimajor axis, da/dt, is proportional to R/σ then the shape of R(σ) is crucial because it defines the location of stable and unstable equilibrium points. For specific values of (e,i,ω) the minima of R(σ) 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 (e,i) the function R(σ)

Numerical estimation of the resonance's strength SR(e,i,ω)

For a given resonant orbit defined by parameters (a,e,i,ϖ,Ω) the disturbing function R(σ) is determined. We define the strength function SR as:SR(a,e,i,ω)=RRmin being R the mean value of R(σ) with respect to σ and Rmin the minimum value of R(σ). This definition is in agreement with the coefficients of the resonant terms of the expansion of the disturbing function for low (e,i) orbits because for this case R(σ) is a sinusoid with an amplitude given by RRmin. This definition of strength

Atlas of MMRs and examples

For a given resonance with a given planet its strength SR is a function of (e,i,ω) 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 (e,i) for different populations and calculate the strength of all resonances verifying |p+q|<100 and order q<100 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 R(σ) all resonances can be classified in three different groups: (a) type 1:n, (b) odd order interior resonances and (c) even order interior resonances and all exterior resonances excluding the 1:n resonances. Horseshoe trajectories wrapping two libration centers are only possible for the first class. For high inclination orbits the shape of R(σ) 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.”

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