Orbital resonance

Contents

Venus/Earth

The most precise orbital resonance is between Venus and Earth. These planets meet 5 times during 8 Earth years and during 13 Venus years. Then their resonance ratio is 13:8 ... After those 8 ea-years, the planets meet on almost a same place. It differs by 2.586 Earth-days, in counter-orbital (retrograde) direction, which is only 2.55°. It takes 239.8 years to rotate the resonance shape by 1/5 of the circle (which is 30x the 8-year cycle), then 1199 years to rotate one full round (which is 150x the 8-year cycle minus 1 year).
This difference makes the higher-order wave of the resonance. Only in the case of Earth/Venus, this higher-order wave among direct neighbours is in retrograde (counter-orbital) direction. All outer planets have got the higher-order wave in prograde direction among neighbours.
It takes 243 years (30x the 8-year cycle + 2 meetings) between the planets meet on an almost-same place.

The resonance shape gets little modified occasionally, mostly by perturbations by Jupiter, but is always restores briefly. Hence, the resonance keeps the orbits stable despite of all perturbations.

Note the start of Fibonacci sequence of 5,8,13, that approaches the Golden section ratio.
The actual difference of the resonance rate from a Golden section is only 0.46%...


The shape, traced by the bary-center of the two planets (by this sub-system), look like these:

From 1902 to 1910 Compared 2 versions
(5 interleaved) 		After 240 years (1/5 round) Meet-point distance chart
The distances during meet-points vary due to the distance of meet-points from Earth perihelion and probably also due to the different inclinations of the orbits.
The longer trend (growing distance) is probably caused by the change of excentricities of the orbits?
In progress compared with trajectories


The plots are 1 dot per 1 earthday. From the dot density, you can see the speed of the barycenter at various parts of the trajectory. The point moves fast when far from the Sun, and almost stops when it is nearest. This event is a lot dynamical: the barycenter starts approaching Sun in orbital direction, then it stops almost on one place for 2-3 weeks, then it starts receding from Sun and moving in the orbital direction...


When compared with 1/5 round turn after 240 years, the shape almost closes onto its previous version, but not completelly, probably due to disturbances by larger planets, mostly by Jupiter... Yet despite these disturbances, the resonance keeps the orbits more stable (so that the resonance shape restores briefly (within few years) after the perturbations), and on a larger scale, there is no chaos.
 

Tidal locking

The Venus spin/rotation seems to be tidaly locked by Earth:
The Venus sidereal day (when compared to fixed star background) is -243.0185 ea-days in counter-orbital (retrograde) direction. After the 13 orbits of Venus, when it meets Earth for 5th time during the resonance cycle, it makes almost exactly 12 day spins relative to the star background. Since this, the ratio between Earth year and Venus day is almost 8/12, which is 2/3, more exactly 0.66533.
The apparent Solar day on Venus is 116.75 days, and it takes almost exactly 5 apparent solar days between consecutive meetings of Earth and Venus (25 during the whole resonance cycle), so that Venus shows always almost same face to Earth during each meeting, and shows that same face to Earth and Sun during heliocentric oposition of Earth and Venus.
One reason for the counter-orbital (retrograde) spin direction seems, that the tidal force is highest, when the planets meet (and are nearest to each other), and in these times, the Earth moves retrograde from Venus point of view, same as if Venus meets Mercury or is nearest to Jupiter, they also move retrograde from Venus point of view. The other reason may be the absence of any moon on Venus...

Jupiter/Saturn

Jupiter and Saturn resonate in 5:2 rate, they meet 3 times during 5 Jupiter years and 2 Saturn years, with a difference of 242 ea-days in orbital (prograde) direction (which is 9.7309°, or 1.125% of the cycle).
The difference makes the shape rotate, and close almost after 854 ea-years, which is almost 29 saturn-years and 72 jupiter-years.

Bary-trace Compared with 2nd After 854 years Compared with next step
Compared with inner
planet trajectories 		Meet-point distance chart
Note, that Mars just does not pass through the bary-center trajectory of its outer neighbours. The Asteroid belt is in the place, where a planet could have been, but either could not form or was destroyed, since it passed through the bary-center of Jupiter and Saturn...

Saturn/Uranus

Saturn and Uranus resonate in 20:7 rate (which is close to 3:1), with a difference of 7.75° (after 588.8 years of the cycle) in prograde direction.
Bary-trace Meet-point distance chart
Note, that the Jupiter just does not pass through the bary-center trace of its outer neighbours...

Uranus/Neptune

Uranus and Neptune resonate probably in 51:26 rate (which is close to 2:1), with a difference of 4.278° (after 4286.8 years of the cycle) in prograde direction.
 
Bary-trace Saturn meets their
barycenter 		again after 30 years
Note, that Saturn passes through the barycenter trajectory of its outer neighbours, and seems to meet that bary-center regularly...

Neptune/Pluto

Neptune/Pluto resonate in 3:2 rate, with a difference of 2.7° - 3.1° in retrograde direction during most meetings in present era. Anyhow (according to DE406), before some 4 millenia the drift has occured sometimes in prograde direction also.
No bary-trace is available, since their weight is incomparable and the barycenter between Neptune and Pluto is almost inside the Neptune, same as barycenter between Jupiter and Mars is almost inside the Jupiter...

The mid-point trace (which has got no physical meaning) looks like this:

 

Other planets

Mercury/Venus resonate in mean 23:9 rate.

Barycenter trace Mid-point trace 10 succesive Mid-point traces
 
Their weight is little comparable, so that trajectory trace little differs from Venus orbital trajectory.
When drawing middle-point between Mercury and Venus (which has got no physical meaning), it reveals the excentric shape of 14 petals, which rotates in prograde direction, by one petal after 10 these shapes.

Mercury orbit is highly excentrical...
Mercury is also one of few planets, that pass through the trajectory of barycenter of its outer neighbours (other one is Saturn), and seems to meet it regularly (only) at some times.

The resonance of Earth/Mars is little close to 15:8, but it seems the least perfect of the planetary resonances...
The barycenter trajectory between Earth and Mars seems erratic, probably due to disturbations to Mars by asteroids and Jupiter...


It takes 14.96 earth-years to complete this figure, here compared to trajectories...
Note the Venus trajectory, as it just does not touch the Earth/Mars barycenter trace...
Note, that the figure is little irregular...

The resonance of Venus/Mars is 3:1. Venus meets Mars (heliocentric conjunction) two times during this cycle, with an asymetric lengths (353.9 days and 314.8 days at some times), which probably depends on Mars excentricity. Total time between 2 meetings (3 Venus years, 1 Mars year) is 668.7825 ea-days, with a difference of -8.544° (in retrograde direction).
The whole shape rotates after 32.9 years by 1/3 of circle, and after 98.7 years by 1 whole circle...

Meet-point distance chart
Trace between 2 meetings Trace figure closes later Figure after rotating 1/3 round
 
Mercury sometimes meets the barycenter of Venus and Mars, but even at closest approaches it is separated vertically (Z ecliptic coordinate) by at least 3,318,844 km (at 12.9.414 AD)

The resonance of Mercury/Earth is 29:7, with 22 nodes drifting in retrograde direction after 46 years by 1/7 of the circle...

Their weight is not much comparable, so the bary-center trajectory runs almost in Earth center. The mid-point trajectory (which has got no physical meaning), looks like this:
Midpoint trajectory
between Emb and Mercury 		6.593 figures (after 46 years)
 
 
Resonance can be found also between non-neighbouring planets in the outer group:

Ratio between Jupiter/Uranus is 7:1, with difference of -4.48475° in retrograde direction.
Ratio between Jupiter/Neptune is 14:1, with difference of +3.0176° in prograde direction.
Ratio between Saturn/Neptune is 28:5, with difference of +2.16647° in prograde direction.

Jupiter/Uranus 7:1 Jupiter/Neptune 14:1 Saturn/Neptune 28:5
 

Resonance groups

There is no apparent resonance between Mars and Jupiter (ratio of orbital periods 6.30792 which is close to 82/13), so it is the place, that separates two resonating groups of planets:
Within the group, the middle planets resonate on both sides and even with non-neighbour peers.

Conclusion

There is no chaos in planetary motions, but instead the cycles are more complex then was previously thought.
The perturbations (see Angular momentum article) average in longer-scale cycles, and the planetary motions are stabilized by the resonance.
 
Written 2006 June 27 - July 11 by Semi.