Tabare Gallardo
Departamento Astronomia, Facultad de Ciencias, Uruguay

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 2018 (preprint arxiv here). It depends on the eccentricity, inclination and the argument of perihelion (w) of the body's orbit.


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



We have tested these programs for retrogtrade orbits and they work fine. For example here is a plot of the strongest resonances near the semimajor axis of comet Halley. They were calculated for i=162 degrees.

This is the resonant disturbing function R(sigma) for the retrograde object 2005 NP82 in resonance 5:6 with Jupiter (left) and the time evolution of the critical angle (rigth). According to R(sigma) the equilibrium point is at sigma = 320 approximately. The numerical integration confirms it.

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:


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


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.


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).


to sites related to MMRs:


Back to the web of the atlas of resonances