ATLAS OF MEAN MOTION RESONANCES IN THE SOLAR SYSTEM

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

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). 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

Tables

Atlas from 0 to 6 AU calculated for typical NEAs' orbits: e=0.46, i=15, w=60.
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=0.5, i=20, w=60.

Programs

Program RSIGMA to compute R(sigma) and strength SR of a resonant orbit. Source code f77 and executable for windows.
Program ATLAS to compute all resonances and its strength SR in an interval of semimajor axis for a specific set of values (e,i,w). Source code f77 and executable for windows.

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 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.


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

lambda = longper + mean anomaly


Minima of R(sigma) give the libration centers.

Please, send an email to gallardo@fisica.edu.uy saying if you agree or disagree with this convention.

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

More on the resonant diisturbing function here.
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

Please, report comments and errors to gallardo@fisica.edu.uy.