In the latest issue of SciAm there’s an article on the whole Pluto/Planet thing, and in that article in a couple of places, it mentions that some planets have “locked” each other into “harmonic” (I think that was the word) orbits and so can never collide even though their orbits cross. The specific examples were Neptune/Pluto and a pair of planets orbiting some star other than our own.
So somehow, these planets influence each other so that their orbits obtain periods with ratios such that the two never occupy their orbits’ crossing points at the same time.
My question is, how does this work itself out? How do planets (through gravity I assume) “lock” other planets into particular orbital periods?
Well if you think about it, the planets that are locked in harmonic orbit are the rare few that have had their orbits perfected… And those who were unfortunate to be off by a tad bit were long ago destroyed. I guess the simplest answer is because they were in the right place at the right time during the big bang, and anything less fortunate is long gone…
That is what I would have thought–that if they weren’t in harmonic orbits, they wouldn’t be there. But the wording of the article seemed to imply that somehow gravity works things out such that orbits that cross actually affect each other so as to develop into harmonic periods.
I interpret “cross” to mean that Pluto’s orbit is sometimes inside Neptune’s with respect to the Sun. If so, they do cross. I don’t know if they could ever collide.
Celestial mechanics is one of those things that’s pretty easy in principle (mostly just Kepler’s Laws, Newton’s Laws of Motion, and a basic understanding of gravitation) but fiendishly complex in application, owing to the difficulty of dealing with gravitational influences in a multi-body system (see the n-body problem) and resultant complex mathematical modeling of the behavior of such a system.
Harmonics are systems in which the periodic behavior of one part of the system naturally drives behavior of another. A system of gears, for instance, can be considered by their very nature harmonic, owing to the ratio of radii of the elements of the gear train. However, aside from backlash or slop there are no variable elements in the train. We would call this an explicitly deterministic system; by knowing the qualities of each element in the gear train, you can calculate the speed of any member from that of any other member as a combination of the ratios. However, if you introduced a variable or compliant member into the train–say, a rubber shaft–you’d have an additional time-varying element which would add considerable complexity to the model of the train. Add a bunch of these, and figuring out the resultant behavior becomes very difficult and erratic.
Since planets are tied together only by the gravitational attractions between them–which vary proportional to the square of the distance between them–then their behavior can be extremely complex, like a bunch of marbles tied together with rubber bands. Here and here are some animations of the behavior of an exaggerated n-body system.
When planets are widely separated and there is a single dominant body (which is a fair representation of the Solar System) then the influences (called perturbations) that cause planets to deviate from their orbits are measureable but small; even massy Jupiter isn’t going to throw the much smaller Earth or Mars out of joint. However, in a much tighter system, such as the Galilean moons, orbital resonances can develop which drives (and maintains) the bodies in specific ratios; this happens because when they tend to deviate, the get “out of joint”, like a rubber band stretched too tight, and tend to return to their preferred orbit. (The Galilean system falls into a Laplace resonance.) Anything that falls out of these resonance patterns will almost certainly be ejected from the system (or at least, thrown into a very eccentric and likely erratic orbit.) This obviously occurs over fairly long time scales, depending on the size of the bodies and tightness of orbits; anywhere from tens of thousands of years to millions, but in terms of astronomical time scales systems tend to fall into order rapidly.
On a larger scale, the Solar System falls into an apparent (and possibly coincidental) resonance. Whether this is meaningful and what impact this would have on the development of the present system is a topic of hot debate for planetologists and celestical mechanics geeks, but no doubt resonances played a significant impact in clearing out the system of extraneous junk that would otherwise pose a hazard to planets. (The Solar System is surrounded by a large cloud of small-scale debris called the Oort Cloud from which most long-duration comments originate.)
Planets and moons can also fall into a sort of resonance called tidal locking or rotational resonance, where they always face the same side toward the central body (the Moon toward the Earth) or rotate with a specified ratio (Mercury about the Sun). And other effects, such as relativistic frame dragging can cause or influece resonance patterns in ways that are very complex to predict.
So…not a simple answer, and when you get into the actual math, it can get downright ugly (or beautiful, depending on how well you like nonlinear differential equations and Bode plots).
Pluto and Neptune are locked into a 2:3 orbit, meaning that for every three orbits of the Sun that Neptune completes, Pluto completes two. Further, the plane of Pluto’s orbit is highly inclined with respect to the ecliptic (approximately the plane of the rest of the planets in the solar system), which means that whilst the planes of the orbits of Neptune and Pluto do cross, the planets themselves never do actually get very close to each other; the minimum seperation between the two planets is 17 times the Earth-Sun distance.
As to why this should happen, the simple answer is that this is the situation that minimised the influence of Neptune on Pluto. Essentially, you can imagine the situation where Neptune and Pluto start off in a random configuration, where they have the opportunity to interact with each other at some non-zero distance. Pluto then gets a ‘kick’ from Neptune, which pushes it into a slightly different orbit. This continues until the ‘kick’ from Neptune is minimised, and the orbit of both Neptune and Pluto are stable, which in this case is in a 2:3 resonance.
there are some interesting moons of Jupiter or Saturn that have some interesting dynamics, which include IIRC switching orbits with each other, and same time to revolve around the planet as each other.
The earth-moon are also sort of in that category around the sun
You’re thinking of Janus-Epimetheus, which can best be described as co-orbiting. Here is a page describing their orbits, although it simplifies things a bit. Because Janus has about 4 times the mass of Epimetheus, it only changes orbit by 1/4 the amount.
Not really. But Earth is in a co-orbiting relationship with an asteroid named Cruithne