ATLAS OF TWO BODY MEAN MOTION RESONANCES IN THE SOLAR SYSTEM
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)
- 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 ...
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.
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.
- Program ATLAS (updated in 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.
- NEW VERSION 2018: Program RSIGMAV3 to compute R(sigma) and strength SR of a resonant orbit. Source code f77 and executable for windows and linux.
- NEW 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.
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:
- 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
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.
degree p<0: exterior resonances (|p+q|<|p|)
degree p>0: interior resonances (|p+q|>|p|)
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
Links to sites related to MMRs:
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web of the atlas of resonances