The motion of a celestial body orbiting a body of much larger mass (such as a planet around a star, or a satellite around a planet) is a simple periodic motion (uniform circular motion).
It can happen that two planets orbit the star with revolution times (the period of their motion) whose ratio is that of two small integers (such as 2:1, 2:3, etc.)
In this case their mutual attraction also becomes periodic: it increases when the two planets come closer, and decreases when they move apart.
If these variations of the gravitational force felt by each planet are significant for its orbit, the orbit can change. At this point two scenarios are possible:
The motion of the planets becomes unstable: the orbits change slowly until the two bodies no longer influence each other appreciably. In the case of a larger and a smaller body, this effect amounts to "driving" the smaller body out of certain regions of space (see the examples below)
The two planets never come close enough to influence each other violently, and enter a stable resonance. The changes in the orbits are compensated by opposite changes over each revolution.
NB: in the figure the orbits are to scale, as are the diameters of the moons. The diameter of Jupiter, instead, is deliberately exaggerated. Distances are in thousands of km.
Examples
In 2006 a discussion among astronomers took away Pluto's status as a planet. Indeed, according to the definition of the International Astronomical Union, a planet must have "cleared" the zone of its orbit of other bodies: it is the repeated interactions (in a sense, a 1:1 resonance with the bodies travelling along the same orbit) that allow it to do so.
main body
orbiting body
period of revolution (days)
Jupiter
Ganymede
7.15
Jupiter
Europa
3.55
Jupiter
Io
1.77
Saturn
Tethys
1.89
Saturn
Mimas
0.94
Saturn
Hyperion
21.28
Saturn
Titan
15.95
Sun
Pluto
90613
Sun
Neptune
60223
Many celestial bodies of the solar system are in an approximate resonance relation. Among the best known (see the table alongside):
Neptune and Pluto are in 2:3 resonance. Their orbits show many resonant features.
Jupiter's moons Ganymede, Europa and Io are in orbital resonance with each other, with ratios 4:2:1.
Saturn's moons Hyperion and Titan are in 4:3 resonance.
Saturn's moons Tethys and Mimas are in 4:2 resonance.
Jupiter and the Greek and Trojan asteroids are in 1:1 resonance. This means that these bodies all lie on the same orbit. The asteroids survive stably under the influence of Jupiter because they occupy particular positions of stable equilibrium (called the Lagrangian points L4 and L5).
Saturn's rings show empty bands. In satellite images they appear as darker rings. These gaps are caused by the unstable resonance between the ring particles and the nearest moons. The particles initially present in these regions are moved into the adjacent zones by the periodic attraction exerted by the moons.
Similar empty bands are observed in the asteroid belt between Mars and Jupiter, mainly because of the influence of Jupiter.
Photograph of Saturn taken by the Voyager 1 probe on 16 November 1980, from a distance of 5.3 million km from the planet. The rings and the Cassini division are clearly visible (the Cassini division is the dark band between the rings, which contains much less material than the bright parts, as can be seen by noticing that the planet is visible through it) Source: NASA
Photograph of Saturn taken by the Voyager 1 probe on 13 November 1980, from a distance of 1.5 million km from the planet. The detail of the structure of the rings is more clearly visible. There are several empty zones (gaps) due to orbital resonances. Source: NASA
