Orbital resonance

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Orbital Resonance is also the title of a science fiction novel by John Barnes.

In celestial mechanics, an orbital resonance occurs when two orbiting bodies exert a regular, periodic gravitational influence on each other.

Contents

1 History
2 Types of resonance
3 Mean motion resonances in the Solar System

3.1 The Laplace resonance

4 'Near' mean motion resonances
5 See also
6 External links
7 References

History

Ever since the discovery of Newton's laws of motion in the 17th century, the stability of planetary orbits has preoccupied many mathematicians, starting with Laplace. The stable orbits that arise in a two-body approximation ignore the influence of other bodies. These added interactions, even when very small, might add up over longer periods to significantly change the orbital parameters and leading to a completely different configuration of the Solar System. Or, it was thought, some other stabilising mechanisms might be there. It was Laplace who found the first answers explaining the remarkable dance of the Galilean moons (see below). It is fair to say that this general field of study has remained very active since then, with plenty more yet to be understood (e.g. how interactions of moonlets with particles of the rings of giant planets result in maintaining the rings).

Types of resonance

In general, an orbital resonance may

involve one or any combination of the orbit parameters (e.g. eccentricity versus semimajor axis, or eccentricity versus orbit inclination).
act on any time scale from short term, commensurable with the orbit periods to secular (measured in 10^{4^{to 10⁶ years).}}
lead to either long term stabilisation of the orbits or be the cause of their destabilization.

A mean motion orbital resonance occurs when two bodies have periods of revolution that are a simple integer ratio of each other. Depending on the details, this can either stabilize or destabilize the orbit. Stabilization occurs when the two bodies move in such a synchronised fashion that they never closely approach. For instance:

Pluto and the Plutinos are in stable orbits, despite crossing the orbit of the much larger Neptune. This is because a 3:2 resonance keeps them always at a large distance from it. Other (much more numerous) Neptune-crossing bodies that were not in resonance were ejected from that region by strong perturbations due to Neptune.
The Trojan asteroids may be regarded as being protected by a 1:1 resonance with Jupiter.
The extrasolar planets Gliese 876b and Gliese 876c are in a 2:1 orbital resonance

Orbital resonances can also destabilize one of the orbits. For small bodies, destabilization is actually far more likely. For instance:

There is a series of almost empty lanes in the asteroid belt called Kirkwood gaps corresponding to mean-motion resonances with Jupiter. Almost all asteroids in those regions have been ejected by the repeated perturbations.

A Laplace resonance occurs when three or more orbiting bodies have a simple integer ratio between their orbital periods. For example, Jupiter's moons Ganymede, Europa, and Io are in a 1:2:4 orbital resonance.

A Secular resonance occurs when the precession of two orbits is synchronised (usually a precession of the perihelion or ascending node). A small body in secular resonance with a much larger one (e.g. a planet) will precess at the same rate as the large body. Over long times (a million years, or so) a secular resonance will change the eccentricity and inclination of the small body. A prominent example is the

ν₆ secular resonance between asteroids and Saturn. Asteroids which approach it have their eccentricity slowly increased until they become mars-crossers, at which point they are usually ejected from the asteroid belt due to a close pass to Mars. This resonance forms the inner and "side" boundaries of the main asteroid belt around 2 AU, and at inclinations of about 20°.

Secular resonances can be linear

Mean motion resonances in the Solar System

There are only six known mean motion resonances in the Solar system involving planets or satellites (a much greater number involve asteroids, rings and moonlets).

2:3 Neptune-Pluto
4:2 Mimas-Tethys (Saturn’s moons)
2:1 Enceladus-Dione (Saturn’s moons)
4:3 Titan-Hyperion (Saturn's moons)
1:2:4 Io-Europa-Ganymede (Jupiter’s moons); the only Laplace resonance
1:4:6 Charon-Nix-Hydra (Pluto's moons)

The simple integer ratios between periods are a convenient simplification hiding more complex relations:

the point of conjunction can oscillate (librate) around an equilibrium point defined by the resonance.
given non-zero eccentricities, the nodes or periapsides can drift (a resonance related, short period, not secular precession).

As illustration of the latter, consider the well known 1:2 resonance of Io-Europa. If the orbiting periods were in this relation, the mean motions [n\,\!] (inverse of periods, often expressed in degrees per day) would satisfy the following

[n_ - 2\cdot n_ = 0 ]

Substituting the data (from the wikipedia) one will get −0.7395° day⁻¹, a value substantially different from zero!

Actually, the resonance is perfect but it involves also the precession of perijove (the point closest to Jupiter) [\dot\omega] The correct equation (part of the Laplace equations) is:

[n_ - 2\cdot n_ + \dot\omega_ = 0 ]

In other words, the mean motion of Io is indeed double of that of Europa taking into account the precession of the perijove. An observer sitting on the (drifting) perijove will see the moons coming into conjunction in the same place (elongation). The other pairs listed above satisfy the same type of equation with the exception of Mimas-Tethys resonance. In this case, the resonance satisfies the equation

[4\cdot n_ - 2\cdot n_ - \Omega_- \Omega_= 0]

The point of conjunctions librates around the midpoint between the nodes of the two moons.

The Laplace resonance

Illustration of Io-Europa-Ganymede resonance. The dark moon is Ganymede, the gray moon is Europa, and the yellow moon is Io

The most remarkable resonance involving Io-Europa-Ganymede includes the following relation locking the orbital phase of the moons:

[\Phi_L ][= \lambda_ - 3\cdot\lambda_ + 2\cdot\lambda_ ][= 180^\circ]

where [\lambda] are mean longitudes of the moons. This relation makes a triple conjunction impossible. The graphic illustrates the positions of the moons after 1, 2 and 3 Io periods.

'Near' mean motion resonances

Other near resonances exist among the moons including:

Saturn system

(5:3) Rhea-Dione

Uranus system

The absence of (precise) resonances in the Uranus system, given their abundance in the Saturn and Jupiter systems is actually a bit of enigma.

External links

Malhotra Orbital Resonances and Chaos in the Solar System, preprint [link]

References

Murray, Dermot Solar System Dynamics, Cambridge University Press, ISBN 0-521-57597-4

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