Icarus 231 (2014) 273–286 Contents lists available at ScienceDirect Icarus journal homepage: www.elsevier.com/locate/icarus Atlas of three body mean motion resonances in the Solar System Tabaré Gallardo Departamento de Astronomía, Instituto de Física, Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay article info abstract Article history: We present a numerical method to estimate the strengths of arbitrary three body mean motion reso- Received 22 July 2013 nances between two planets in circular coplanar orbits and a massless particle in an arbitrary orbit. This Revised 14 December 2013 method allows us to obtain an atlas of the three body resonances in the Solar System showing where are Accepted 17 December 2013 located and how strong are thousands of resonances involving all the planets from 0 to 1000 au. This atlas Available online 27 December 2013 confirms the dynamical relevance of the three body resonances involving Jupiter and Saturn in the aster- oid belt but also shows the existence of a family of relatively strong three body resonances involving Ura- Keywords: nus and Neptune in the far Trans-Neptunian region and relatively strong resonances involving terrestrial Asteroids, dynamics and jovian planets in the inner planetary system. We calculate the density of relevant resonances along Celestial mechanics Resonances, orbital the Solar System resulting that the main asteroid belt is located in a region of the planetary system with Kuiper belt the lowest density of three body resonances. The method also allows the location of the equilibrium points showing the existence of asymmetric librations (r – 0 or 180). We obtain the functional depen- dence of the resonance’s strength with the order of the resonance and the eccentricity and inclination of the particle’s orbit. We identify some objects evolving in or very close to three body resonances with Earth–Jupiter, Saturn–Neptune and Uranus–Neptune apart from Jupiter–Saturn, in particular the NEA 2009 SJ18 is evolving in the resonance 1 À 1E À 1J and the centaur 10199 Chariklo is evolving under the influence of the resonance 5 À 2S À 2N. Ó 2013 Elsevier Inc. All rights reserved. 1. Introduction critical angles, identified thousands of asteroids in TBRs with Jupi- ter and Saturn, concluding there are more asteroids in TBRs than in Fifteen years ago, concentrated over a period of about a year, a two body resonances. succession of papers were published (Murray et al., 1998; Nesv- The approximate nominal position in semimajor axis of the orny´ and Morbidelli, 1998, 1999; Morbidelli and Nesvorny´ , 1999) TBRs taking arbitrary pairs of planets is very simple if we ignore showing an intense theoretical and numerical work on three body the secular motion of the perihelia and nodes of the three bodies. mean motion resonances (TBRs) involving an asteroid and two When these slow secular motions are taken into account each of massive planets. These papers, that have their roots on earlier the nominal TBRs split in a multiplet of resonances all them very works devoted to the study of an asteroid in zero order TBRs near the nominal one (Morbidelli, 2002). The challenge is to obtain (Wilkens, 1933; Okyay, 1935; Aksnes, 1988), stated the relevance the strength, width or libration timescale that give us the dynam- that the TBRs involving Jupiter–Saturn and also Mars–Jupiter have ical relevance of these resonances. Analytical planar theories in the long term stability in the asteroid’s region. In spite of being developed by Murray et al. (1998) and Nesvorny´ and Morbidelli weaker than the two body resonances they are much more numer- (1999) allowed to describe and understand the dynamics of the ous generating several dynamical features in the asteroidal popu- TBRs involving Jupiter and Saturn in the asteroidal region. These lation, like concentrations for some values of semimajor axes, theories are appropriated to study in detail specific resonances anomalous amplitude librations and chaotic evolutions (Nesvorny´ with Jupiter and Saturn but its application to any arbitrary reso- and Morbidelli, 1998). In particular, both borders of the main aster- nance involving any planet is not trivial, which possibly explains oid belt exhibit chaotic diffusion due to the superposition of the absence of papers published on this topic since these years several weak two-body and TBRs (Murray and Holman, 1997; with the exceptions of a few ones devoted to specific scenarios Morbidelli and Nesvorny´ , 1999). More recently, Smirnov and (Guzzo, 2005; de La Fuente Marcos and de La Fuente Marcos, Shevchenko (2013) in a massive numerical integration of 249,567 2008; Cachucho et al., 2010; Quillen, 2011). asteroids by 105 years and looking at the time evolution of the In this paper, from a different approach, we will obtain a global view of all dynamically relevant TBRs involving all the planets ta- ken by pairs along all the Solar System. Our method is not E-mail address: gallardo@fisica.edu.uy analytical but numerical and it is based on an estimation of the 0019-1035/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.icarus.2013.12.020 274 T. Gallardo / Icarus 231 (2014) 273–286 1-2J+1S, e=0.0 1-2J+1S, e=0.1 ) ) σ σ ( ( ρ ρ 0 90 180 270 360 0 90 180 270 360 σ σ 1-2J+1S, e=0.2 1-2J+1S, e=0.3 ) ) σ σ ( ( ρ ρ 0 90 180 270 360 0 90 180 270 360 σ σ 1-2J+1S, e=0.4 1-2J+1S, e=0.5 ) ) σ σ ( ( ρ ρ 0 90 180 270 360 0 90 180 270 360 σ σ Fig. 1. Evolution of the shape of qðrÞ for growing eccentricities for the zero order resonance 1-2J + 1S at a ¼ 3:8045 au. Numerical integrations of test particles show that the asymmetric librations around r 90 and r 270 are stable and also exist large amplitude librations around r ¼ 180 that wrap both asymmetric libration centers. The librations around r ¼ 0 are unstable. strength of the resonances which is obtained evaluating the effects is oscillating over time, being ki the mean longitudes and ki integers. of the mutual perturbations in all possible spatial configurations of More precisely, for massless particles with inclined orbits there are the three bodies satisfying the resonant condition. It allows us to other possible definitions for the critical angle involving combina- appreciate the effect of arbitrary eccentricity and inclination of tions of its X0 and -0, but it is possible to show that after an appro- the massless particle’s orbit on the resonance’s strength but, in or- priate averaging procedure the leading term in the expansion of the der to reduce the number of parameters involved in the problem, resulting disturbing function will be the one whose argument is gi- we impose coplanar and circular orbits for the two perturbing ven by Eq. (1) (Nesvorny´ and Morbidelli, 1999). In this work we will planets. Nevertheless, the method can be extended to arbitrary use the following notation: the order of the resonance is planetary orbits. In the next section we start describing our numer- q ¼jk0 þ k1 þ k2j, we call p ¼jk0jþjk1jþjk2j and we note ical method and we explore how the calculated strengths depend k0 þ k1P1 þ k2P2 the resonance involving planets P1 and P2. The on the parameters of the problem. In Section 3 we analyze the approximate nominal location of the resonance assuming unper- distribution of TBRs along the Solar System and discuss its turbed Keplerian motions is dynamical effects providing some examples. In Section 4 we qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi present the conclusions. À3=2 k1 3 k2 3 a0 ’ ð1 þ m1Þ=a1 À ð1 þ m2Þ=a2 ð2Þ k0 k0 2. Numerical approximation to the disturbing function for three body resonances where ai and mi are the mean semimajor axes of the planets and its masses expressed in solar masses respectively. In the real Solar Sys- Three body resonances between a massless particle with an tem the actual location depends on the precession of the perihelia arbitrary orbit given by ða0; e; i; X0; -0Þ and two planets P1 and P2 and the gravitational effects of the other planets not taken into in circular coplanar orbits occur when the critical angle account. In the very far Trans-Neptunian Region (TNR), depending on the resonance, the actual location could be shifted something r ¼ k k þ k k þ k k ðk þ k þ k Þ- ð1Þ 0 0 1 1 2 2 0 1 2 0 between 0.1 au and 1 au. T. Gallardo / Icarus 231 (2014) 273–286 275 1-3J+1S, e=0.01 1-3J+1S, e=0.1 ) ) σ σ ( ( ρ ρ 0 90 180 270 360 0 90 180 270 360 σ σ 1-3J+1S, e=0.2 1-3J+1S, e=0.3 ) ) σ σ ( ( ρ ρ 0 90 180 270 360 0 90 180 270 360 σ σ 1-3J+1S, e=0.4 1-3J+1S, e=0.5 ) ) σ σ ( ( ρ ρ 0 90 180 270 360 0 90 180 270 360 σ σ Fig. 2. Evolution of the shape of qðrÞ for growing eccentricities for the first order resonance 1-3J + 1S at a ¼ 2:7518 au, the resonance where is evolving 485 Genua. For large eccentricities asymmetric equilibrium points appear, which are stable according to numerical integrations of test particles. There are also large amplitude librations around r ¼ 180 that wrap both asymmetric libration centers. 2.758 2.756 2.754 2.752 2.75 mean a (au) 2.748 360 270 180 (1-3J+1S) σ 90 0 0 5000 10000 15000 20000 25000 30000 35000 40000 time (yrs) Fig.