# How do higher numbers affect orbital resonance?

When talking about orbital resonance, the most common resonances in our Solar System are 1:2 (think Jupiter's moons) and 2:3 (think Pluto and Neptune). However, the lack of larger numbers within these resonances confuses me. I accumulated some data about resonances and I figured out that the difference between the numbers gives more stabilized resonances. For example, a 1:29 resonance would not be stable due to the massive separation of the objects from each other and the gravitational meddling of other bodies. However, that also brings up a different question, and the one I will be asking. A 28:29 resonance seems somewhat stable; after all the objects are close together and will have high gravitational influence on each other. Space is big, so it is unlikely the objects would collide, so why do we not see any 7:8 resonances or other resonances involving high numbers in our Solar system?

• What is it about a "28:29 resonance" that "seems somewhat stable"? They spend a long time in close proximity pulling on each other in the radial direction, could that also result in a rapidly growing eccentricity for both? And radial forces are like increasing or decreasing the central body's attractive force, which will lead to dephasing. On the other hand a 1:2 or 2:3 might be a "happy medium", both weak enough and short-enough impulse that the stabilizing effects are stronger than the destabilizing effects.
– uhoh
Commented Feb 24, 2021 at 14:21
• Technically a 28:29 or a 234261:234262 resonance could be called a 1:1 resonance due to its close proximity to a true 1:1. And 62:125 would be 1:2 and so on. Commented Feb 24, 2021 at 14:59
• I guess, but that doesn't explain the lack of 3:4 resonances in our Solar System.. Could you explain that to me? Commented Feb 24, 2021 at 17:35
• The Thule Asteroids are in a 3:4 resonance: en.wikipedia.org/wiki/279_Thule. Commented Feb 24, 2021 at 22:32
• Oh, I didn't know that. Thanks for telling me. Commented Mar 1, 2021 at 2:46

## General Explanation

In the history of our Solar System, the orbits of Solar System bodies have gradually evolved, passing through many "higher number" orbital resonance ratios. Only the stabilizing effects of the most stable "lower number" resonance ratios have been enough to overcome the forces causing orbital migration.

When we look at the orbital resonances in the current Solar System, we get a snapshot in time that just shows us the resonances that have been powerful enough to overcome other forces that cause orbital migration.

A rigorous mathematical explanation for why lower-number resonances have a greater effect than higher number resonances may be resonance-type specific and most have enough math to choke a cat. Intuitively, if you push a child on a swing 28 times for every 29 times they swing, you aren't going to have much of an effect. Orbital resonances are similar.

## Specific Example (Mercury Spin-Orbit Resonance):

Common wisdom holds that orbiting bodies will eventually achieve a 1:1 orbital/spin resonance with a main body, like the Earth's Moon. Though the Moon was spinning faster in the past, tidal torque caused it's spin to slow until the spin rate was equal to the orbital period. As the Moon spun down, it passed through many other ratios of orbital periods.

The case for Mercury's orbital/spin resonance around the Sun is a lot more complicated. Scientists and astronomers are still working out how and why Mercury was caught in a 3:2 spin/orbit resonance as it spun down, rather than passing through the resonance. From this excellent news article:

Tidal dissipation in the planet and the ensuing deceleration of its spin inevitably carry the planet through a sequence of the spin-orbit resonances. The question then becomes in which of these resonances the planet should eventually get trapped...

the international team revisited the problem of tidal evolution of Mercury's spin and found that the 3:2 resonance is indeed the most probable end-state. The frequency-dependent tidal torque acts as an efficient trap for the planet trying to traverse a resonance. The efficiency of the trap strongly depends on the value of orbital eccentricity, as well as on the temperature and viscosity of Mercury's mantle.

Note:

Forces causing migration of planetary orbits include (but are not limited to):

1. Loss of planetary angular momentum due to gaseous drag in the early proto-planetary circumstellar accretion disk.

2. Tidal forces resulting in exchanges of angular momentum.

3. Collisions.

4. Various n-body gravitational interactions.

5. The Yarkovsky effect.

6. Resonances with other bodies.