Orbital resonance - basic aspects

Question

In the following lines, I will ask you some questions regarding the notion of orbital resonance. I know that the orbital resonance of two celestial body represents the driving of a dynamical system by a periodic force at a frequency which is a rational multiple of the natural frequency, and also, it is the simple report between their periods of revolution around the Sun, for exemple. Even so, I would like to better understand the phenomenon of orbital resonance, and for this purpose, I hope that someone can help me with a few answers:

1. In Orbital resonances and chaos in the Solar System¹ (also here), Renu Malhotra explains:

In fact, there are many mathematical similarities with the playground analogy, including the fact of nonlinearity of the oscillations, which plays a fundamental role in the long term evolution of orbits in the planetary system. But there is also an important difference: in the playground, the child adjusts her driving frequency to remain in tune - hence in resonance - with the natural frequency which changes with the amplitude of the swing. Such self-tuning is sometimes realized in the Solar system; but it is more often and more generally the case that resonances come-and-go. And, as we shall see, resonances can be the source of both instability and long term stability".

a) How can a celestial body (an asteroid) be capture into orbital resonance and how can it escape from one?

b) How can self-tuning be achieved in a orbital resonance?

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¹Solar system Formation and Evolution. ASP Conference Series, Vol. 149, 1998, D. Lazzaro et al., eds.

Answer 1

There are two main contexts for orbital resonance, and these can be thought of as "Jupiter scattering asteroids" and "Jupiter's moons in a 1:2:4 resonance"

In the case of scattering, orbital perturbations from Earth, Mars, Saturn, Jupiter can cause asteroids to change their orbital characteristics over time. Each planet is constantly perturbing all the asteroids and the effect can be to alter the inclination, eccentricity or the period of the asteroid. Moreover random interactions between asteroids can pass momentum from one asteroid to another. The effect of these small interactions is like a random walk (over a sufficiently long period of time) and can cause an asteroid to approach a big resonance with Jupiter.

At this point, the asteroid will get a significant perturbation, This tends to push it out of resonance. Of course the asteroid is too small to have an effect on Jupiter's orbit, so the asteroid is pushed away from the resonance. This is the cause of the Kirkwood gaps in the asteroid belt.

The moons of Jupiter are a different matter. The moons are of comparative size so changes in orbit of one moon can have a effect on another moon, which can, in turn feedback to alter the orbit of the first moon.

As the moons of Jupiter are tidally affected by Jupiter, their orbits can be altered by Jovian tides. These can cause the orbits of a moon to increase. But when the moon reaches a resonance with another moon, further increases in the moon's orbit will feed forward to the other moon, raising its orbit too in such a way as to maintain the resonance. And the other moon will feed back to the first to resist the tidal effects, also maintaining the resonance.

The moons of Jupiter are not large enough to have a scattering effect on each other, but they can have stabilising effect on each other's orbits. It is through this feedback mechanism that a self-tuning is created.

A body can leave resonance either by scattering (in the case of the Asteroid belt, the resonance is the cause of the scattering) or if the forces and perturbations (tidal forces, perterbations from the sun or other planets) become significant enough to overcome the feedback effect that maintains the resonance.

Most asteroids are not in a stable resonance with Jupiter, the big exceptions are the trojan asteroids. These are in a 1:1 resonance. Some seem to have formed there. They were part of the dusty disk that was near the young Jupiter. Others have probably become captured, through multi-body interactions. That is an asteroid has a close encounter with Jupiter, its moons or its Trojans and by chance, this gives it the right orbital energy to enter into a horseshoe orbit and later into a Trojan orbit. A single body can't capture in this way, but interactions between Jupiter, moons and other bodies might.