Orbital resonance

A orbital resonance , in astronomy, takes place when two objects orbiting around a third have periods of revolution whose report/ratio is a simple whole fraction. It is a mechanical particular case of Résonance.

Stability of the orbits

Since the publication of laws of Newton, the problem of stability of the orbits worried many mathematicians, while starting with Laplace. As the solution of the problem of two bodies does not take into account mutual interactions between planets, from the small interactions surely will accumulate and end up changing the orbits. Or then, it remains to discover new mechanisms who maintain the stability of the unit. It was also Laplace who found the first answers to explain the remarkable dance of the Jupiter moons. One can say that this field of investigation remained very active since and there remain always mysteries to be elucidated (for example interactions of the small moons with the particles of the rings of giant planets).

In general, resonance can:

to relate to either only one parameter, or any combination of the parameters of orbit;
to act on scales of time very different, comparable with the periods of the secular orbits, or , going until the 10⁴ - 10⁶ of years;
it can just as easily be the cause of the stability of the orbits that the source of theirs destabilization.

Types of resonance

The periodic gravitational influence of the planets (the moons) can destabilize their orbits. It is what makes it possible to explain the existence of bands in the Ceinture of asteroids where the number of bodies is considerably weaker. These bands, called gaps of Kirkwood, would have been created by a resonance with the orbit of Jupiter which would have caused the ejection of the body being there.

Resonance can have the opposite effect: it can allow the stabilization of orbits and to protect certain bodies from gravitational disturbances. Thus Pluto and the others Plutino S are protected from the ejection of their orbit by a resonance 3:2 with giant planet Neptune. Other objects of the Ceinture of Kuiper are also in other resonances with this planet: 1:2, 4:5… The Trojan asteroids can even be regarded as being in resonance 1:1 with their planet.

When several objects have their orbital period in a relationship with simple entireties, one speaks about resonance of Laplace . It is the case, for example, of the the Moon S of Jupiter, Ganymède, Europe and Io which is in a resonance 1: - 3: 2.

Commensurability of the periods of revolution

There exist only five resonances of this type concerning the major planets or moons in the Solar system (much larger number relates to the asteroids, the rings and the small satellites):

2:3 Neptune - Pluto;
4:2 Mimas - Téthys (the Saturn moons);
2:1 Encelade - Dioné (the Saturn moons);
4:3 Titan - Hypérion (the Saturn moons);
1: - 3: 2 Io - Europe - Ganymède (the Jupiter moons), the single resonance of Laplace.

The simple whole relations between the periods of revolution hide more complex relations:

the points of conjunction can oscillate around the values of balance defined by resonance;
taking into account the eccentricity S of the orbits, the Node S or the pericenters can change.

Like an illustration, for very known resonance 1:2 Io-Europe, if the periods of revolution were really in this exact report/ratio, the average movements (opposite of the period) would satisfy the following equation: : $n_ {Io} - 2 \ cdot n_ {Have} = 0$

However, while checking with the data one obtains $-0.7395^o jour^ {- 1}$ , a value well too large to be neglected.

In fact, resonance is exact but it must also include the Précession pericenter $\ dowry \ omega$ The corrected equation (which belongs to the relations of Laplace) is

$n_ {Io} - 2 \ cdot n_ {Have} + \ dowry {\ Omega} _ {Io} = 0$

In other words, the average movement of Io is well the double of that of Europe by taking account of the precession of the pericenter. An observer located on the pericenter would have seen the moons arriving at the conjunction at the same place. Other resonances satisfy the similar equations except for the Mimas-Téthys pair. In this last case, resonance satisfies the following expression

$4 \ cdot n_ {Th} - 3 \ cdot n_ {Semi} - \ Omega_ {Th} - \ Omega_ {Semi} = 0$

The point of conjunction oscillates around a point halfway between the nodes of the two moons.

The resonance of Laplace

The most remarkable resonance, that of the three Galiléennes moons, includes the relation which constrained the position of the moons on their orbits:

$\ Phi_L = \ lambda_ {Io} - 3 \ cdot \ lambda_ {Have} + 2 \ cdot \ lambda_ {Ga} = 180^o$

where $\ lambda$ is average longitudes of the moons. This constraint makes impossible triple conjunction of the moons. The graph illustrates the positions of the moons after 1,2 and 3 periods of Io.

See too

synchronous Rotation