ATLAS OF TWO BODY MEAN MOTION RESONANCES IN THE SOLAR SYSTEM

Tabare Gallardo
Departamento Astronomia, Facultad de Ciencias, Uruguay
www.fisica.edu.uy/~gallardo

BREAKING NEWS: There is an updated new website with new codes and plots HERE

The present plots, tables and programs give the location and strength of the MMRs between a massless particle and a planet. They were calculated considering fixed orbits for both two bodies. No variations in perihelion nor longitude of the node was considered. Mean semimajor axes were taken for the planets which were assumed with e=i=0. The strength SR(e,i,w) was calculated following Gallardo (2006, Atlas of MMRs in the solar system, Icarus 184, 29-38, preprint here). An improved and expanded version of the theory can be found in Gallardo 2019 (preprint arxiv here). It depends on the eccentricity, inclination and the argument of perihelion (w) of the body's orbit.

Plots

(Mercury=red, Venus=green, Earth=blue, Mars=pink, Jupiter=black, Saturn=red, Uranus=green, Neptune=blue)

0 to 2 AU, 2 to 4 AU, 4 to 6 AU calculated for typical NEAs' orbits: e=0.46, i=15, w=60.
6 to 25 AU calculated for typical centaurs' orbits: e=0.46, i=32, w=60.
25 to 48 AU calculated for typical TNOs' orbits: e=0.20, i=10, w=60.
40 to 100 AU calculated for typical SDOs' orbits: q=32 AU, i=20, w=60.
100 to 300 AU (linear scale) calculated for typical SDOs' orbits: q=32 AU, i=20, w=60.
0 to 300 AU composite view of the previous figures.
more plots
EXTREME resonances: circular, polar, retrograde, very eccentric ...

Tables

Atlas from 0 to 6 AU calculated for typical NEAs' orbits: e=0.46, i=15, w=60. Html version.
Atlas from 6 to 25 AU calculated for typical centaurs' orbits: e=0.46, i=32, w=60.
Atlas from 25 to 48 AU calculated for typical TNOs' orbits: e=0.20, i=10, w=60.
Atlas from 40 to 100 AU calculated for typical SDOs' orbits: q=32 AU, i=20, w=60.
Atlas from 100 to 300 AU calculated for typical SDOs' orbits: q=32 AU, i=20, w=60.
Atlas from 0 to 410 AU calculated for e=0.2, i=10.

Programs

NEW Program SUPERATLAS (2019) this code replaces the old ATLAS.
Program ATLAS (2018) to compute all resonances and their strength SR in an interval of semimajor axis for a specific set of values (e,i,w). Source code f77 and executable for windows and linux. SEE ABOVE.
NEW Program RESONALYZER (2019) this code replaces the old RSIGMA.
VERSION 2018: Program RSIGMAV3 to compute R(sigma) and strength SR of a resonant orbit. Source code f77 and executable for windows and linux.
VERSION 2018: Program SReiw to compute curves SR(e) for a given orbital inclination and argument of perihelion. Source code f77 and executable for windows and linux.

How to ...

How to use all these numerology?
Suppose you are studying an asteroid or comet and you want to know if it is in a resonant motion. You have two possibilities:

Look at the tables for the strongest resonances near the semimayor axis of the body's orbit. Then compute the corresponding critical angle
sigma = (p+q)*lambda_planet - p*lambda - q*longper
and follow its time evolution.
Or run the program ATLAS (or SUPERATLAS) and compute the strength according to the (e,i,w) of the particle's orbit. This is the best choice. Then compute and look at the time evolution of the critical angles.

Conventions


degree p<0: exterior resonances (|p+q|<|p|)
degree p>0: interior resonances (|p+q|>|p|)
order q>=0


Examples:

resonance 2:3 is an exterior resonance given by p=-3, q=1, then |-3+1|:|-3| = 2:3
resonance 3:2 is an interior resonance given by p=2, q=1, then |2+1|:|2| = 3:2
trojans 1:1 are given by p=-1, q=0
resonance 1:2N means Neptune makes 2 revolutions and the particle 1 revolution.
resonance 2:1N means Neptune makes 1 revolution and the particle 2 revolutions.

critical angle:    sigma = (p+q)*lambda_planet - p*lambda - q*longper
lambda = longper + mean anomaly

Minima of R(sigma) give the libration centers.

Limitations

Planetary eccentricities were not taken into account so real motions should depart from this theory for very small eccentricities (e less than e_planet). Only one planet is taken into account then secular or resonant effects due to other planets can modify the theoretical libration centers given by the minima of R(sigma).

Links

to sites related to MMRs:

News

See also Evaluating the signatures of the mean motion resonances in the Solar System. Gallardo 2007.
This atlas allowed us to find a new population of asteroids captured in the resonance 1:2 with Mars. See also this website.
Lykawka and Mukai (2007) have found there is a correlation between Strength and Stickiness
Soja et at. (2011) have found a relationship between strength and width.
de la Fuente Marcos and de la Fuente Marcos (2013) confirmed Crantor as horseshoe of Uranus.
Resonance capture at direct and retrograde orbits studied by Namouni and Morais (2015) are in good agreement with the predictions given by this method.
Milic Žitnik and Novakovic (2016) have found a correlation between strength SR and semimajor axis mobility of asteroids.
Resonances in the asteroid and trans–Neptunian belts: A brief review. Gallardo 2018.
Strength, stability and three dimensional structure of mean motion resonances in the Solar System. Gallardo 2019, Icarus 317. Version arxiv.
Three-dimensional phase structures of mean motion resonances. Lei 2019.
Three dimensional structure of mean motion resonances beyond Neptune. Gallardo 2019.

Back to the web of the atlas of resonances