ATLAS OF THREE BODY MEAN MOTION RESONANCES IN THE SOLAR SYSTEM

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

The present plots, tables and programs give the location and estimated strength in relative units of the three-body resonances between a massless particle and two planets. The estimated strength Delta rho was calculated following Gallardo (2014, Icarus 231, 273-286). It depends on the eccentricity, inclination and the argument of perihelion (w) of the particle's orbit. For more details see preprint here.

Plots

From 0 to 1000 au. Code colors: resonances involving Mercury as the most inner planet = red, Venus=green, Earth=blue, Mars=pink, Jupiter=black, Saturn=red, Uranus = green.
From 2 au to 3.5 au. Blue: two-body resonances, red: three-body resonances. Two-body and three-body resonances are not to the same scale.

Tables

Atlas from 0 to 1000 au calculated assuming e=0.15, i=6, w=60 (long file 4MB).
Atlas from 0 to 100 au for ZERO ORDER three-body resonances.

Programs

Program RHOSIGMA to compute rho(sigma) and strength Delta rho of a resonant orbit. Source code f77 and executable for windows and linux.
Program ATLAS3BR to compute all resonances and its strength Delta rho in an interval of semimajor axis for a specific set of values (e,i,w). Source code f77 and executable for windows and linux.
Three Body Resonance locator for Android.
Three Body Resonance locator for web browser.

Use

Suppose you are studying an asteroid or comet and you want to know if it is in a resonant motion. You have two options:

Look at the tables for the strongest resonances near the semimayor axis of the body's orbit. Then compute the corresponding critical angle
```
sigma = k0*lambda_ast + k1*lambda_pla1 + k2*lambda_pla2 - (k0+k1+k2)*longper_ast 
```
and follow its time evolution.
Or run the program ATLAS3BR and compute the strengths of all resonances near the semimajor axis of the particle according to the values of (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 corresponding to the strongest resonances in the interval.

Limitations

The semimajor axis of the actual resonances can be something different from this theory because we have not taken into account the shift in "a" generated by the time evolution of the perihelion and node of the particle's orbit. For larger semimajor axes we have larger errors in "a".