Resonances

The swing:

To set a swing in motion, it must be pushed at each passage, every second time, or every third time. If the movement is to be maintained, the swing must be pushed in the same position each time: the pushing must be in resonance with the swing. The period of the pushes (time between two pushes) must be equal to the period of the swing (duration of a swing motion), or equal to twice (or triple…) the period of the swing.

Planets:

When planets orbit around a star (or satellites ? around a planet), there are resonances when there is a commensurable ratio between the orbital parameters.

The simplest are resonances between the orbital periods ? .

2:1 resonance
The two bodies are in 2:1 resonance. The inner body completes an orbit in one year and the outer body takes two years. They always encounter each other at the same location and the disturbances will accumulate.
(LAOG/H. Beust)

There is resonance of order n:m, where n and m are integers, if one planet completes n orbits while the other completes m. There can also be resonance between the rotation of a body on itself and its orbital movement.

When the orbits are eccentric and/or inclined, other “rates” can become involved. The mutual disturbances between the planets change the orbital planes, this is precession ? (the movement of the axis of a spinning top). The precession ? rates of two bodies can resonate with each other. The precession ? rate of a body may also resonate with the rate of revolution of another body...

Gravitation is an extremely simple law giving rise to highly complex phenomena.

Resonances are everywhere in the Solar System:

The Moon’s rotation movement is in 1:1 resonance with its orbital motion. It takes the same time to complete a rotation around its axis as one orbit around the Earth. This is why, from Earth, we always see the same side of the Moon.

Mercury’s rotation around its own axis is in 3:2 resonance with its orbit around the Sun. The planet completes 3 rotations on its axis while performing 2 orbits of the Sun.

Neptune and Pluto are in 3:2 resonance. Neptune completes three orbits of the Sun, while Pluto completes two. The angular frequency of Neptune, $n_N$, is greater than the angular frequency of Pluto, $n_P$. It can be written $2 \times n_N = 3 \times n_P$.

In the rings of Saturn, there are countless resonances between satellites ? or between satellites ? and rings.

What are the roles of resonance?

The results of resonance vary widely:

  • We have known for a long time that, in the asteroid ? belt, there are voids, in other words zones containing no objects. these locations are resonances with Jupiter. The role of resonances with Jupiter has been to expel the objects that enter this zone.
  • The location of the 3:2 resonance with Neptune has the opposite role. At the location of this resonance, a large number of small bodies have accumulated, including Pluto. This resonance is thus an area of stability .
  • The resonance between the orbital period ? and the period of rotation of the Moon stabilises  its synchronous rotation with Earth. This configuration prevents the internal friction in the Moon caused by the tidal forces of the Earth when it was rotating faster. This friction slowed the Moon down to this equilibrium position. This is a highly effective mechanism: in the Solar System, all the satellites ? close to their planet have synchronous rotation.
  • In the rings of Saturn, a large number of structures: sharp edges, ripples, density waves , are created by resonance with satellites ? .
Waves in the rings of Saturn
Resonance between the orbits of the particles and small satellites ? create density waves or vertical ripples (bottom right).
(NASA/Cassini)